text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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{"url":"https:\/\/eccc.weizmann.ac.il\/keyword\/19430\/","text":"Under the auspices of the Computational Complexity Foundation (CCF)\n\nREPORTS > KEYWORD > INVARIANT SUBSPACES:\nReports tagged with invariant subspaces:\nTR17-021 | 11th February 2017\nNeeraj Kayal, Vineet Nair, Chandan Saha, S\u00e9bastien Tavenas\n\n#### Reconstruction of full rank Algebraic Branching Programs\n\nAn algebraic branching program (ABP) A can be modelled as a product expression $X_1\\cdot X_2\\cdot \\dots \\cdot X_d$, where $X_1$ and $X_d$ are $1 \\times w$ and $w \\times 1$ matrices respectively, and every other $X_k$ is a $w \\times w$ matrix; the entries of these matrices are linear forms ... more >>>\n\nTR18-191 | 10th November 2018\nNeeraj Kayal, Chandan Saha\n\n#### Reconstruction of non-degenerate homogeneous depth three circuits\n\nA homogeneous depth three circuit $C$ computes a polynomial\n$$f = T_1 + T_2 + ... + T_s ,$$ where each $T_i$ is a product of $d$ linear forms in $n$ variables over some underlying field $\\mathbb{F}$. Given black-box access to $f$, can we efficiently reconstruct (i.e. proper learn) a ... more >>>\n\nISSN 1433-8092 | Imprint","date":"2020-11-30 02:19:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7813860177993774, \"perplexity\": 2183.8965483724073}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141204453.65\/warc\/CC-MAIN-20201130004748-20201130034748-00359.warc.gz\"}"} | null | null |
namespace Ui {
class ConnectomeGenerateDialog;
}
class ConnectomeGenerateDialog : public QDialog, private Ui::ConnectomeGenerateDialog
{
Q_OBJECT
public:
ConnectomeGenerateDialog(QWidget *parent = 0);
~ConnectomeGenerateDialog() {qDeleteAll(this->children());}
bool useFreesurfer() { return UseFreesurferBox->isChecked(); }
QString getFileName() { return FileNameEdit->text(); }
private slots:
void onLineEditTextChange();
void onUseFSCheck(bool status);
void onBrowseButtonPress();
};
#endif // CONNECTOMEGENERATEDIALOG_H
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,771 |
Show all 9 Topics
Article Quality Management, Change Management, Time Management, Complexity, Cost Control, Resource Management 1 October 2017
Counting on Precision
By Blumerman, Lisa The United States census is a complex, large-scale, multiyear exercise in program management. The scope of accurately and cost effectively counting more than 330 million people across more than 3.7…
Article Quality Management, Cost Control 1 November 2016
Building Better
By Parsi, Novid Metrics about the global construction sector poised for growth. But some organizations are pursuing new approaches to boost efficiency and productivity.
Article Quality Management, Cost Control 1 May 2015
Reformat and reboot
By Hendershot, Steve The annual cost of failed U.S. government IT projects is estimated to be as high as US$20 billion. Why haven't well-funded government IT projects experienced success? Find out what measures are…
Article Quality Management, Cost Control June 2009
By Kopp, Quentin Executive support is vital to securing the resources needed to implement projects. But the executive's role does not end after the resources are obtained and allocated. This article features the…
Article Quality Management, Cost Control 1 August 2002
Lost costs
By Zafar, Zartab | Rasmussen, Dean Lost efficiency on construction projects can affect productivity.Project managers should periodically examine the productivity rate and learn the causes of inefficiency.Two methods for gauging…
Article Quality Management, Risk Management, Cost Control May 2000
A balance sheet for projects
By Goodpasture, John C. | Hulett, David A project delivers value to its investors within a prescribed limit of risk. This article describes a way of obtaining this prescribed level of risk at the outset of the project. From management's…
Article Program Management, Quality Management, Cost Control September 1998
Aussie project management
By Eager, David This article reports on the project of staging the 2000 Olympic Games in Sydney, Australia. Infrastructure preparation is to include the construction of an Olympic Village, new sporting facilities,…
Article Quality Management, Time Management, Cost Control February 1996
Project struggle #1
With this column, PM Network welcomes a new contributor, Dr. Will B. Struggles, PMP, to its pages. In a series of articles, Dr. Struggles will be sharing his thoughts on the obstacles, challenges,…
Article Quality Management, Cost Control, Scope Management December 1994
The WBS
By Webster, Francis Marion This tutorial covers the basic principles of developing a work breakdown structure (WBS). A WBS starts with a dynamic vision of the project, perhaps in the form of a drawing, diagram, or computer…
Article Quality Management, Cost Control, Scope Management May 1994
The earned value concept
By Fleming, Quentin W. | Koppelman, Joel M. A project manager truly understands his job once he has a tangible metric called 'Done' for the project, a process for managing changes, and a way to know how much of the job has been accomplished… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,964 |
Immanuel Lutheran teaches and responds to the love of the Triune God the Father, creator of all that exists; Jesus Christ, the Son, who became human to suffer and die for the sins of all mankind and to rise to life again in victory over death and the devil; and the Holy Spirit, who creates faith through God's Word and Sacraments. These three persons of the Trinity are coequal and coeternal, one God.
Immanuel accepts and preaches the Biblical teaching of Martin Luther that inspired the reformation of the Christian Church in the 16th century. These teachings can be summarized in three phrases: Grace Alone, Faith Alone, Scripture Alone.
The Bible is God's inerrant and infallible Word, in which He reveals His Law and His Gospel of salvation in Jesus Christ. It is the sole rule and norm for Christian teaching.
The word "Synod" in the Lutheran Church--Missouri Synod (LCMS) comes from Greek words that mean "walking together." The term has a rich meaning in our church body, because congregations voluntarily choose to belong to the synod. Though diverse in our service, we all hold to a shared confession of Jesus Christ as taught in the Holy Scripture and the Lutheran Confessions.
The Lutheran Confessions are statements of belief that were transcribed and shared by the church leaders in during the 16th century. These confessions are contained in The Book of Concord. Immanuel and all members of the LCMS accept and believe these confessions to be a correct and accurate interpretation and presentation of biblical doctrine.
The Lord's Supper is celebrated at this congregation in the confession and glad confidence that, as He says, our Lord gives into our mouths not only bread and wine but His very body and blood to eat and to drink for the forgiveness of sins and to strengthen our union with Him and with one another. Our Lord invites to His table those who trust His words, repent of all sin, and set aside any refusal to forgive and love as He forgives and loves us, that they may show forth His death until He comes. Because those who eat and drink our Lord's body and blood unworthily do so to their great harm and because Holy Communion is a confession of the faith which is confessed at this alter, any who are not yet instructed, in doubt, or who hold a confession differing from that of this congregation and The Lutheran Chuch-Missouri Synod, and yet desire to receive the sacrament, are asked first to speak with the pastor or an usher. For further study, see Matthew 5:23f.; 10:32f.; 18:15-35; 26:26-29; 1Cor. 11:17-34. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,518 |
The Business Rate Supplements Act 2009 (c 7) is an Act of the Parliament of the United Kingdom. It creates a power to impose business rate supplements. It gives effect to the proposals contained in the command paper "Business rate supplements: a White Paper" (Cm 7230).
Sections 28 to 32 came into force on 2 July 2009. The rest of the Act, except for section 16(5) and Schedule 2, came into force, in England, on 19 August 2009.
References
Halsbury's Statutes,
External links
The Business Rate Supplements Act 2009, as amended from the National Archives.
The Business Rate Supplements Act 2009, as originally enacted from the National Archives.
Explanatory notes to the Business Rate Supplements Act 2009.
United Kingdom Acts of Parliament 2009
Business in the United Kingdom
Taxation in the United Kingdom
2009 in economics | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 108 |
namespace Specify.Stories
{
/// <summary>
/// A User Story template base class for a BDDfy Story allowing representation as a class rather than an attribute.
/// </summary>
public abstract class UserStory : Story
{
private const string As_a_prefix = "As a";
private const string I_want_prefix = "I want";
private const string So_that_prefix = "So that";
/// <summary>
/// Gets or sets the 'As a' clause.
/// </summary>
/// <value>As a.</value>
public string AsA
{
get { return Narrative1; }
set { Narrative1 = CleanseProperty(value, As_a_prefix); }
}
/// <summary>
/// Gets or sets the 'I want' clause.
/// </summary>
/// <value>The i want.</value>
public string IWant
{
get { return Narrative2; }
set { Narrative2 = CleanseProperty(value, I_want_prefix); }
}
/// <summary>
/// Gets or sets the 'So that' clause.
/// </summary>
/// <value>The so that.</value>
public string SoThat
{
get { return Narrative3; }
set { Narrative3 = CleanseProperty(value, So_that_prefix); }
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,689 |
SO WHAT'S WITH THE NAME?
Is it time to re-paint your stucco home here in the Phoenix area? Crash of Rhinos is one of the most trusted names in painting here in the Valley. When you need local painters you can turn to for helpful advice and great service, we're here for you.
We handle every step of the home exterior painting process, from removing old paint and prep work to applying coats and repairing damaged surfaces. The only thing you need to do is decide what color you want.
This may seem like the easiest part of the job, but it's actually a lot harder than you might think! In this blog post, we'll walk you through what you need to think about when choosing an exterior paint color—and how Crash of Rhinos can help.
Before you can even start to think about colors, you really need to think about the paint itself. Too many homeowners we talk to think that all paint is the same. Trust us, your local painters: it's not, and picking the wrong type or brand of paint could lead to future headaches.
For example, some brands of paint are perfect for places like Boston or Seattle, where summer temperatures are mild and days are overcast. Those paints don't do so well, however, here in Phoenix, where the scorching sun bakes the paint and it's exposed to sunshine almost every day of the year.
Some paints provide full cover after just one coat, while others require several coats in order to be fully applied. The difference is in the quality of the pigments they contain. Choosing a paint containing the proper pigments will save you on the amount of paint used.
The next concern is how long the paint is going to last. The solids (the remains on the wall after the paint has dried) play a major role in the durability of the paint. The higher percentage of solids there is (over 45% is better), the more dense and durable the coating.
The change of seasons, such as from summer to winter, may cause some serious problems to the exterior of your home if the paint is not of the right quality.
The paints which have a capability to endure these kinds of weather changes are the ones with an all-acrylic binder. The binder is what holds the pigments together and provides the right protection from the outside environment.
At Crash of Rhinos, we use Dunn-Edwards paints because it meets or exceeds the three standards listed above. Unlike some paints, Dunn-Edwards is specially formulated to stand up to lots of sunshine and heat. It's perfect for the exterior of stucco homes here in Phoenix.
Pay attention to elements like the roof, the chimney, the driveway ,and any stone and brick features on your home that won't be painted. What color are they? Their color should play a determining role in your choice of colors.
For example, if a part of the façade is made of stone, use a similar color, so that you can achieve visual harmony.
If the roof has an intense color (like terra-cotta), choose a more neutral color for the exterior. Be careful not to accent any unappealing elements like the downspouts or the air conditioners.
The landscape and the surroundings of your house can have great influence on what your exterior color looks like. Do you have trees with colorful blossoms, flowering shrubs or green plantings? Their color may have a beautiful impact on your exterior.
What does your HOA allow?
If your neighborhood has an HOA, you should start by finding out what colors they allow you to paint your house. The last thing you want to do is have your local painters finish up the work, only to get slapped with a non-compliance fine from your association.
Ok, but where to start? You don't even know who runs the HOA! With Crash of Rhinos' professional paint advisor services, we make this easy. We handle communicating with your HOA for you. We'll find out the allowed colors and present them to you as options.
Here in Phoenix, historic homes don't have restrictions on the color they can be painted, unless otherwise regulated by an HOA. Still, the city's Historic Preservation Office has some tips for selecting a color and treatment.
Look for an already established color pattern in the neighborhood. You don't need to imitate the exact colors used by your neighbors. You just need to be in the same ballpark. Crash of Rhinos can help you find similar colors that give your home its individual style.
By doing this, your local painters will help your house blend in with the neighborhood while still preserving your right to paint your house in a color or shade that you like.
Ok, so you've now ruled out many colors, and you're down to your top 3. Great! Now, how do you know which color will look right on the exterior of your home?
Testing a swatch on your home's exterior.
We'll only present you with colors that are in compliance with the rules of your HOA and match what you're interested in, as well as the colors of the neighborhood. This will help you narrow down which color you want to try out.
As part of every free estimate we provide, we take pictures of your home that can be later used to create real-to-life digital renderings of your home and what it will look like painted.
As part of your consultation, Crash of Rhinos can paint a test swatch on your home, somewhere inconspicuous. This swatch of color will help you see what the color looks like in "real life" and in realistic lighting conditions. After all, one color can look different in the bright light of morning than it does in the shadows of the late afternoon.
Step #4: Make your final selection and let your local painters get to work!
Once you've picked your favorite paint color, it's time for the Crash of Rhinos team to take things from there! With our thorough painting process, your new favorite color will be on your home's exterior in no time.
Everything starts with our FREE estimate. To schedule yours, call us at (602) 540-7471, or request a visit from us.
What should you expect from the exterior paint used on your home? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,360 |
\section{Introduction} \label{sec:intro}
Precision cosmology with the Lyman-$\alpha$ forest is an area of increasing interest, both for its
ability to constrain the evolution of the Universe \citep{McDonald2007,Busca2013,Slosar2013,
Font-Ribera2014a,Bautista2017,duMasdesBourboux2017}
and for its unique insight into the
small-scale power spectrum that is a potential diagnostic of warm dark matter, neutrinos, and other particle physics phenomena
\citep{Seljak2005,Seljak2006b,Seljak2006a,Viel2006b,Viel2013,Palanque-Delabrouille2015,Irsic2017}.
For the latter studies it is essential to explore the effect of both particle physics and
astrophysical parameters on the measured power spectrum, since there are important degeneracies with the
thermal and ionization history of the intergalactic medium (IGM) that need to be considered
\citep{McDonald2005b,McQuinn2016,Walther2018b}.
Ideally one runs a series of cosmological simulations to forward model the parameters into mock observables that can be compared with data \citep{McDonald2005a,Viel2006a,Borde2014}.
However, the mock observations suffer from intrinsic sample variance due to the finite simulation box size; sparse sampling of
random phases and amplitudes of large-scale modes in the initial conditions propagate to an uncertainty in the final mock power
spectrum. This uncertainty can affect even small-scale modes through non-linear gravitational coupling. In practice, model uncertainty is the limiting factor on large scales where it exceeds the observational uncertainty \citep{Palanque-Delabrouille2015}
Suppressing the uncertainties can be achieved either by expanding the box size, or by averaging many simulated realizations of the same volume size; either way
a more precise mean power spectrum (or other observable) can be achieved. But each simulation is expensive, especially as the
box size is increased. To extract the best possible constraints, one has to tension this need for large sample sizes or volumes for
a single point in parameter space against the requirement to adequately sample the parameter space itself.
This paper examines an approach to lessen the tension, by initializing simulations with carefully chosen amplitudes and
phases for each mode, thus reducing the total volume or number of simulations required to achieve a given statistical accuracy on the
mean mock power spectrum. This approach will free up computing time per parameter instantiation to increase the density of samples in parameter space. Methods, such as emulators (Bird et al. in prep.; Rogers et al. in prep.), which interpolate between hydrodynamic simulations predictions, can then be more accurate. Interpolation is necessary because only a limited number of simulations can feasibly be run.
We minimize the number of simulation realizations by running {\it paired, fixed} simulations \citep{Pontzen2016, Angulo2016}.
The difference between the standard simulations and the paired-fixed simulations is in the input density field.
For a Gaussian random field, the amplitude of each mode is drawn from a Rayleigh distribution with zero mean and variance equal to the power spectrum at that scale, and the phase drawn with uniform probability between 0 and $2 \pi$.
Instead, for paired-fixed simulations, we fix the amplitudes of the Fourier modes to the square root of the power spectrum, and invert the phase of one simulation in a pair relative to the other.
The fixing reduces the variance of the power spectrum on the largest scales, which remain in or near the linear regime, while the pairing cancels leading-order errors due to non-linear evolution of chance correlations between large-scale modes.
On its own pairing is a benign manipulation in the sense that both realisations in the pair are legitimate, equally-likely draws from a Gaussian random field \citep{Pontzen2016}. Fixing, on the other hand, generates an ensemble that is quantitatively different from Gaussian; \cite{Angulo2016} argued that, to any order in perturbation theory, the differences affect only the variance on the power spectrum and cannot propagate to other observables such as the power spectrum or observable measures of non-Gaussianity. Combined, pairing and fixing thus formally removes the leading order and next-to-leading order uncertainties that generate scatter in predicted mock observations from simulations. In \cite{Villaescusa2018}, we thoroughly explored the statistics of paired-fixed simulations for both dark matter evolution and galaxy formation, finding that there are no statistically significant biases in the approach, and in many settings it does yield significantly enhanced statistical accuracy.
In this paper, we apply these techniques for the first time to the study of the power spectrum of fluctuations in the Lyman-$\alpha$ (Ly$\alpha$) forest, as measured from hydrodynamical simulations. The Ly$\alpha$\ forest is the collection of absorption features present in distant quasar spectra due to intervening neutral hydrogen absorbing Ly$\alpha$\ photons as the quasar light redshifts. Physically, the forest comprises a relatively smooth, low density environment of hydrogen which traces the underlying matter density of the Universe well in the redshift range $2<z<5$ \citep{Croft1998, 1996ApJ...457L..51H}. At higher redshifts $z>5$ and especially beyond reionization the neutral hydrogen is too optically thick \citep{1965ApJ...142.1633G}; at lower redshift $z<2$ the forest cannot be observed with optical instruments. Although use of the Ly$\alpha$\ forest as a tracer of the matter density is limited to a finite redshift interval, it is a unique probe of a wide range of otherwise inaccessible redshifts and scales. It has the unique advantage of tracing the matter density at small scales, down to 10s of kpc, in regions where the evolution is still close to linear.
This paper is structured as follows.
In Section \ref{sec:sims} we describe the simulations used in this analysis,
before presenting results in Section \ref{sec:results}.
We conclude in Section \ref{sec:conc} with a brief summary and discussion.
\section{Simulations} \label{sec:sims}
In this section we describe the simulations used in this study.
We start by describing the different sets of initial conditions
(Gaussian, fixed and paired-fixed); we will then discuss the hydrodynamical
simulations and the method to simulate Ly$\alpha$\ forest sightlines from the simulations.
\subsection{Gaussian, Fixed and Paired-Fixed Random Fields}
In this paper we will follow the naming convention in \cite{Villaescusa2018},
which contains more detailed definitions including
power spectrum conventions.
Here we present a brief summary of the required definitions to interpret our results.
Given a density field $\rho(\vec{x})$, the density contrast is defined as
\begin{equation}
\delta(\vec{x})=\frac{\rho(\vec{x})-\bar{\rho}}{\bar{\rho}}~,
\end{equation}
where $\bar{\rho}=\langle \rho(\vec{x})\rangle$.
In a simulation we take discretized values of these quantities at
the center of a cell $\vec{x_i}$, $\delta_i = \delta(\vec{x_i})$, and
define discrete Fourier modes $\tilde\delta_n$ as
\begin{equation}
\tilde \delta_n = \sum_i \delta_i ~ e^{i \vec{x}_i \cdot \vec{k}_n}
= A_n ~ e^{i \theta_n} ~,
\end{equation}
where $\vec{k}_n$ identifies the wavenumber, $A_n$ is the amplitude of the
discrete Fourier mode and $\theta_n$ is the phase.
For a Gaussian density field, each $\theta_n$ is a random variable distributed
uniformly between 0 and $2\pi$ and each $A_n$ follows a Rayleigh distribution
\begin{equation}
p(A_n)~{\rm d}A_n=\frac{A_n}{\sigma_n^2}~e^{-A_n^2/2\sigma_n^2}{\rm d}A_n ~,
\label{Eq:Rayleigh}
\end{equation}
with $\langle A_n \rangle = \sqrt{\frac{\pi}{2}} ~\sigma_n$, and $\sigma_n^2$
is proportional to the power spectrum $P(|\vec{k}_n|)$
(with the constant of proportionality
depending on the particular conventions adopted; see \cite{Villaescusa2018}).
For generating initial conditions (ICs) representing a discretized Gaussian density field for a simulation of the Universe, the mode amplitudes ($A_n$) and phases ($\theta_n$) are chosen randomly from their distributions above.
This randomness generates a source of variance in the final mock power spectrum from the simulation; the amplitudes $A_n$ propagate
directly, while the phases $\theta_n$ become significant during non-linear evolution because their values determine how
modes of different scales interact with each other.
In this paper, we explore alternative approaches to generating ICs such that these variance effects are minimised.
Similar to \citeauthor{Villaescusa2018}, we define Gaussian, paired Gaussian, fixed
and paired-fixed fields as follows:
\begin{itemize}
\item {\bf Gaussian field}: A field with $\tilde\delta_n=A_n~e^{i\theta_n}$,
where $A_n$ follows the Rayleigh distribution of Eq. \ref{Eq:Rayleigh} and $\theta_n$ is a random variable distributed uniformly between 0 and $2\pi$.
\item {\bf Paired Gaussian field}: A pair of Gaussian fields $\delta$ and $\delta'$,
where $\tilde\delta_n=A_n~e^{i\theta_n}$ and
$\tilde\delta_n'=A_n~e^{i(\theta_n+\pi)}=-\tilde\delta_n$;
the values of $A_n$ and $\theta_n$ are the same for the two fields
and $A_n$ is drawn from the Rayleigh distribution of Eq. \ref{Eq:Rayleigh}.
Because the leading-order uncertainties are typically from the amplitudes rather than
the phases, we do not use paired fields without fixing the amplitudes in this work.
\item {\bf Fixed field}: A field with $\tilde\delta_n=A_n~e^{i\theta_n}$,
where we fix $A_n = \sqrt{2}~\sigma_n$. The phase $\theta_n$ remains a uniform random
variable between $0$ and $2\pi$.
\item {\bf Paired-fixed field}: A pair of fields,
$\tilde\delta_n=A_n~e^{i\theta_n}$ and
$\tilde\delta_n'=A_n~e^{i(\theta_n+\pi)}=-\tilde\delta_n$,
where the values of $A_n$ are the same for the two fields
and fixed to $A_n = \sqrt{2}~\sigma_n$, and the values of $\theta_n$ are also the same for the two fields, as defined, rendering $\tilde\delta_n$ and $\tilde\delta_n'$ exactly out of phase with respect to each other. Here $\theta_n$ remains a uniform random variable between $0$ and $2\pi$.
\end{itemize}
The purpose of this paper is to compare the statistical accuracy of paired-fixed simulations with standard simulations (based on Gaussian fields), in the context of Ly$\alpha$\ forest power spectrum analyses.
\subsection{Hydrodynamical simulations} \label{sec:hydro}
We perform hydrodynamical simulations with the
\textsc{Gadget-3} code, a modified version of the publicly available code \textsc{Gadget-2} \citep{Springel2005}. The code incorporates radiative cooling by primordial hydrogen and helium, alongside heating by the UV background following the model of \cite{Viel_2013}. The \texttt{quick-lya} flag is enabled, so gas particles with overdensities larger than 1000 and temperatures below $10^5$ K are converted into collisionless particles \citep{Viel_2004}. This approach focuses the computational effort on low-density gas associated with the Ly$\alpha$\ forest, and is much more efficient than either allowing dense gas to accumulate or adopting a realistic galaxy formation and feedback model, to which the forest is largely insensitive. Thus, this computational efficiency has no significant effect on the accuracy of estimated Ly$\alpha$\ forest observables.
The initial conditions are generated at $z=99$ by displacing cold dark matter (CDM) and gas particles from their initial positions in a regular grid, according to the Zel'dovich approximation. We account for the different power spectra and growth factors of CDM and baryons by rescaling the $z=0$ linear results according to the procedure described in \cite{Zennaro_2016}.
Our simulations evolve $512^3$ CDM and $512^3$ gas particles from $z=99$ down to $z=2$ within a periodic box of $40 \ h^{-1}\text{Mpc}$, storing snapshots at redshifts 4, 3 and 2. We run a total of 100 simulations (25 fixed pairs, and 50 standard simulations). In order to study the dependence of our results on volume we also run an identical set of simulations as those described above but with $256^3$ CDM and $256^3$ gas particles in a box of size $20\ h^{-1}\text{Mpc}$.
The values of the cosmological parameters, the same for all simulations, are in good agreement with results from \cite{Planck}: $\Omega_{\rm m}=0.3175$, $\Omega_{\rm b}=0.049$, $\Omega_\nu=0$, $\Omega_\Lambda=0.6825$, $h=0.67$, $n_s=0.9624$, $\sigma_8=0.834$.
\subsection{Modeling Ly$\alpha$\ absorption} \label{sec:skewers}
In order to study {Ly$\alpha$} forest flux statistics we generate a set of
mock spectra containing only the {Ly$\alpha$} absorption line from neutral
hydrogen for each snapshot in our simulation suite. The spectra are
calculated on a square grid of 160 000 spectra for the 40 Mpc
box. Each spectrum extends the full length of the box, and maintains the periodic
boundary conditions of the underlying simulation, with sizes in velocity space of \([3\,793, 4\,267,
4\,698]\,\mathrm{km}\,\mathrm{s}^{-1}\) respectively at \(z = [2, 3,
4]\). In each spectrum, the optical depth \(\tau\) is calculated by
taking the product of the column density of neutral hydrogen and the
atomic absorption coefficient for the Ly$\alpha$\ line \citep{humlicek79}
appropriately redshifted for both its cosmological and peculiar
velocities. Measurements are made in bins of velocity width \(10\,\mathrm{km}\,\mathrm{s}^{-1}\) using the code \texttt{fake\_spectra} \citep{bird17}.
Once we have computed the Ly$\alpha$\ optical depth, we derive the transmitted flux fraction (or flux for short):
\begin{equation}
F = e^{-\tau} ~.
\end{equation}
We then calculate $\bar{F}$, the mean flux over all pixels in our box. Next, we adopt the common
practice of rescaling the optical depth of each simulation, $\tau \to \alpha \tau$ where $\alpha$
is chosen to fix $\bar{F}$ to a target value. This is normally adopted when making comparisons to data because it
reduces the impact of uncertainties in the UV background.
In our case, we set $\alpha$ in each simulation
such that $\bar{F}$ is fixed to the mean over all simulations. We verified that, if we do not
renormalize $\tau$, the benefits of paired and fixed simulations remain (in fact they even
increase further on small scales).
After renormalization, we calculate the fluctuations around the mean, $\delta_F$:
\begin{equation}
\delta_F = \frac{F}{\bar F} - 1~.
\end{equation}
Finally, we calculate flux power spectra using the code \texttt{lyman\_alpha} \citep{Rogers2018a, Rogers2018b}. Studies of the Ly$\alpha$\ forest either adopt 1D or 3D power spectra depending on the density of available observational skewers; the former approach is computationally simpler and captures small-scale line-of-sight information, while the latter approach is required to capture larger-scale correlations that do not fit within a single quasar spectrum. In this work we will study the effects of paired-fixed simulations in both settings.
To create a mock power spectrum, the first step is to generate the Fourier transform (\(\tilde{\delta}_F\)) of the fluctuation field \(\delta_F\). For 1D power spectra, the transformation is taken separately for each skewer along the line-of-sight direction; otherwise we apply the 3D transformation to the entire box. In both cases we are able to exploit fast Fourier transforms since all velocity bins are of equal width and spectra are evenly sampled in the transverse direction. We then estimate power spectra by averaging $|\tilde\delta_F|^2$ over all skewers for the 1D power spectrum, or in $|\vec{k}|$ shells for the 3D power spectrum.
When working with paired simulations, we average the power spectra to form a single estimate from each pair. This leaves us with $25$ independent estimates from our ensemble of $50$ paired-fixed simulations, a point that will be discussed further below.
\section{Results} \label{sec:results}
In this Section we will present the statistical performance of our different simulations in predicting Ly$\alpha$\ power spectra.
To quantify the performance of any given ensemble, we calculate the mean of the measured power across all realizations, $\bar{P}(k)$
\begin{equation}
\bar{P}(k) = \frac{1}{N_{\mathrm{est}}} \sum_{i=1}^{N_{\mathrm{est}}} P_i(k) \,,
\end{equation}
where $N_{\mathrm{est}}$ is the number of independent estimates of $\bar{P}(k)$ and $P_i(k)$ are the individual
estimates. We also calculate the single-estimate variance in $P(k)$:
\begin{equation}
\sigma^2(P(k)) = \frac{1}{N_{\mathrm{est}}-1} \sum_{i=1}^{N_{\mathrm{est}}} \left( P_i(k) - \bar{P}(k) \right)^2\,,
\end{equation}
As noted above, in the case of paired (or paired-fixed) simulations $N_{\mathrm{est}} = N_{\mathrm{sim}}/2$, while in the case of
a standard ensemble, $N_{\mathrm{est}} = N_{\mathrm{sim}}$ where in both cases $N_{\mathrm{sim}} = 50$, the number of simulations. This difference arises because paired power spectrum estimates are not independent and instead the {\it average} of each pair is regarded as providing a single sample $P_i(k)$.
We are actually interested in the expected uncertainty on the
mean power spectrum, $(\Delta \bar{P})^2$, estimated as
\begin{equation}
(\Delta \bar{P})^2 \equiv \frac{\sigma^2(P(k))}{N_{\mathrm{est}}}\textrm{.}\label{eq:Pbar-errors}
\end{equation}
The ratio of the standard to the paired-fixed uncertainties gives the most useful performance metric; in our case
where $N_{\mathrm{sim}}$ is the same between the two ensembles, this reduces to
\begin{equation}
\frac{(\Delta \bar{P}_{\mathrm{S}})^2}{(\Delta \bar{P}_{\mathrm{PF}})^2} = \frac{\sigma^2_\mathrm{S}(P(k))}{2\sigma^2_{\mathrm{PF}}(P(k))}\,,
\label{eq:varratio}
\end{equation}
where S and PF subscripts refer to the standard (Gaussian) and paired-fixed ensembles respectively. This ratio can also be thought of as the computational speed-up for achieving a fixed target accuracy (assuming the uncertainty on the mean power spectrum scales proportional to $\sqrt{N_{\mathrm{sim}}}$). It limits to unity when the paired-fixed approach is not providing any improvement.
From our paired-fixed simulations we can also form a fixed ensemble simply by
discarding one of each pair. Thus we will also show results for a fixed ensemble
with $N_{\mathrm{est}}=N_{\mathrm{sim}}=25$. Once again the estimated accuracy of
the mean power is given by Equation~(\ref{eq:varratio}), including the factor $2$
renormalisation for the relative number of estimates.
To ensure that the manipulations of the initial conditions do not introduce any bias,
we also calculate $\bar{P}_\mathrm{S}
- \bar{P}_\mathrm{PF}$. For finite $N_{\mathrm{sim}}$ this difference
contains a residual statistical uncertainty; one should expect
\begin{equation}
\langle (\bar{P}_\mathrm{S} - \bar{P}_\mathrm{PF})^2 \rangle = (\Delta
\bar P_{\mathrm{S}})^2 + (\Delta \bar P_{\mathrm{PF}})^2\label{eq:pbar-difference}\,,
\end{equation}
given the independence of the two ensembles. Therefore we check that $\bar{P}_\mathrm{S} - \bar{P}_\mathrm{PF}$ remains comparable in magnitude to this expected residual.
\subsection{Matter Power Spectrum}\label{subsec:Pm}
We start by discussing the matter power spectrum (i.e. the 3D power spectrum of the overdensity field without any transformation to Ly$\alpha$\ flux); \cite{Villaescusa2018} also discussed this quantity, but not at the redshifts in which we are interested in this work. Our simulations also have different baryonic physics implementations.
\begin{figure}
\centering
\includegraphics[width=0.8 \textwidth]{Pk_m_40Mpc_hydro.pdf}
\caption{Matter power spectrum measured at $z=2, 3$ and $4$.
The top panel shows the mean power spectrum in the two sets of simulations, the standard simulations, $\bar{P}_{\rm S}$, with the $1\sigma$ uncertainty on the mean, and the paired-fixed simulations, $\bar{P}_{\rm PF}$; the means are indistinguishable.
The middle panel shows the fractional difference between the mean power measured in the two sets of simulations, with a shaded region showing the expected $1\sigma$ uncertainty on that difference which is given by Eq. 10.
Across all scales, there is no evidence for bias in the estimate of the paired-fixed simulations relative to the traditional simulations.
The bottom panel shows the ratio between the uncertainty on the mean of the standard and paired-fixed simulations,
$(\Delta \bar P_\mathrm{S})^2 / (\Delta \bar P_\mathrm{PF})^2$, with dashed lines showing instead the effect of
fixing without pairs $(\Delta \bar P_\mathrm{S})^2 / (\Delta \bar P_\mathrm{F})^2$. These ratios summarise how fast the paired-fixed (or fixed) simulations converge on the mean power spectrum relative to the standard simulations.
As a typical example of improvement, for wavenumbers $k=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$, we find estimates of the matter power spectra converge $390$ and $2.0$ times faster in the paired-fixed simulations compared with the standard simulations.
}
\label{fig:pkm3d40}
\end{figure}
In Figure \ref{fig:pkm3d40} we show the matter power spectrum measured
at $z=2, 3, \mathrm{and} \;4$. In the top panel we show the average power spectrum
in the two ensembles: the set of 50 standard simulations
$\bar{P}_{\rm S}$ and the set of 25 paired-fixed simulations
$\bar{P}_{\rm PF}$. Error bars show the uncertainties
$\Delta \bar P_{\rm S}$ in the mean of the standard simulations,
computed using Equation (\ref{eq:Pbar-errors}). The uncertainties are
small due to the large number of simulations. The uncertainties in the paired-fixed simulation are too small to be plotted in the top panel, as we will shortly discuss.
The middle panel shows the difference between the mean power
measured in the two sets of simulations, as a fraction of the total
power in the standard simulations. Additionally we show, as a grey band, the $\pm 1\sigma$
uncertainty at $z=3$\footnote{We find that our calculations of this expected
scatter as a fraction of the mean power is almost independent of
redshift, as the fractional uncertainties depend primarily just on the number
of modes in each power spectrum bin.} defined by
Equation~(\ref{eq:pbar-difference}). The two approaches to estimating
the non-linear power spectrum thus agree well, consistent with
\cite{Angulo2016} and \cite{Villaescusa2018}. In the bottom panel we quantify the statistical improvement achieved
by the paired-fixed simulations with respect to the standard
simulations. We show the ratios of the uncertainty on the means, comparing the standard
and paired-fixed simulations, $(\Delta \bar{P}_{\rm S})^2/(\Delta \bar
P_{\rm PF})^2$,
Equation (\ref{eq:varratio})\footnote{Note, this is the square of the quantity shown in the bottom panels in \cite{Villaescusa2018}.}. This ratio represents how quickly the
paired-fixed simulations converge on the true mean relative
to the standard simulations. We also show the ratio of the uncertainty on the means of the standard simulations and fixed (but not paired)
simulations, shown as the dashed lines. The dotted horizontal line
shows a value of 1, indicating the level where fixed and paired-fixed
simulations do not bring any statistical improvement over standard
simulations.
Comparing the solid and dashed lines shows that both fixing and
pairing the simulations contributes to reducing uncertainty on the mean power spectrum at low $k$.
As a typical example of improvement, for wavenumbers
$k=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$, we find
estimates of the matter power spectra converge $390$ and $2$ times
faster in paired-fixed simulations compared with standard
simulations.
Broadly this agrees with results from
\cite{Villaescusa2018} but, unlike in the earlier work, we find that
the factor $\simeq 2$ improvement in a paired-fixed simulation
continues to $k = 10\ h\text{Mpc}^{-1}$. To isolate the cause of
this difference, we first considered the effect of box size. Our
simulations are of boxes with side length $40\ h\text{Mpc}^{-1}$,
which is intermediate between boxes considered by
\cite{Villaescusa2018}. We therefore ran a suite of smaller
$20\ h\text{Mpc}^{-1}$ simulations for direct comparison to the smallest boxes of
that earlier work. However we found similar results at large $k$ in this additional
suite, and conclude that box size is not the driver for the different behavior.
The only remaining difference between our present simulations and the hydrodynamic simulations of \cite{Villaescusa2018}
lies in the drastically different baryonic physics implementation.
As discussed in Section
\ref{sec:hydro}, we do not implement feedback but instead use an
efficient prescription to convert high density gas into collisionless
particles. By contrast, \cite{Villaescusa2018} adopts a
state-of-the-art star formation and feedback prescription
which results in significant mass transport due to galactic
outflows. We verified that the $20\ h^{-1}\text{Mpc}$ dark-matter-only runs of \cite{Villaescusa2018}, like our Ly$\alpha$\ runs,
also generate substantial improvements from pairing and fixing at high $k$.
It therefore seems that potential efficiency gains at $k \gg 1 \ h\text{Mpc}^{-1}$
can be erased by stochastic noise from sufficiently
energetic feedback and winds.
Our practical conclusions will be insensitive to this regime.
Uncertainties
at high $k$ are dominated by observational rather than theoretical
uncertainty, due to the rapidly shrinking absolute magnitude of
simulation sample variance. Because the sample variance is intrinsically small, it is only important that
the power spectrum remains unbiased at these scales.
\subsection{3D Ly$\alpha$\ Power Spectrum}\label{subsec:P3D}
We next turn our attention to the 3D Ly$\alpha$\ power spectrum. On large,
quasi-linear scales, the 3D Ly$\alpha$\ power spectrum is proportional to the matter power
spectrum, with an amplitude set by the scale-independent bias
parameter, and an angular dependence described by the \cite{Kaiser1987}
model of linear redshift-space distortions. In this work,
we include these distortions but consider the angle-averaged (i.e. monopole)
power spectrum rather than divide the power spectrum into angular bins.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{Pk_lya3d_40Mpc_hydro.pdf}
\caption{Same as Figure \ref{fig:pkm3d40}, but for the 3D Ly$\alpha$\ flux power spectrum. As a typical example of improvement, for wavenumbers $k=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$, we find estimates of the 3D Ly$\alpha$\ flux power spectra converge $12$ and $2.0$ times faster in a paired-fixed ensemble compared with a standard ensemble.
}
\label{fig:varratio3d40}
\end{figure}
We present the measured 3D Ly$\alpha$\ forest power spectrum in Figure
\ref{fig:varratio3d40}. The panels are computed and arranged as in
Figure \ref{fig:pkm3d40}, but starting from the flux $\delta_F$ instead of
the overdensity $\delta$.
In the top panel the
uncertainties on the mean power spectrum of the standard simulations
are again small due to the large number of simulations. The middle panel shows the fractional difference between the
paired-fixed and standard simulation power spectra, with the grey
shaded region again indicating the expected $1\sigma$ scatter.
Several measured points scatter outside this envelope (most notably at
$k<0.3\ h\text{Mpc}^{-1}$) but this is consistent with a statistical
fluctuation driven by scatter in the standard ensemble. At high $k$
the difference in the mean is sub-percent. Comparing the central panels of Figures~\ref{fig:pkm3d40} and~\ref{fig:varratio3d40} reveals that the shape of the $1\sigma$ contour is different between the Ly$\alpha$\ and matter power cases. At high $k$, the matter power enters a highly non-linear regime where modes are strongly coupled, whereas
the flux arises from low density clouds that are still quasi-linear. In the case of flux, modes are near-decoupled and the
uncertainty decays proportionally to the increasing number of modes per $k$ shell.
The bottom panel of Figure~\ref{fig:varratio3d40} quantifies the
statistical improvement achieved by the paired-fixed
simulations with respect to the standard simulations, shown as the solid line. The
dashed line represents simulations that are just fixed. The dotted
horizontal line shows a value of 1, indicating the level where fixed
and paired-fixed simulations do not bring any statistical improvement
over standard simulations. As a typical example of improvement, for
wavenumbers $k=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$,
we find estimates of the 3D Ly$\alpha$\ flux power spectra converge $12$
and $2$ times faster in a paired-fixed ensemble compared with a
standard ensemble.
There is a minimum level of improvement at
$k \simeq 1\ h\text{Mpc}^{-1}$ where the paired-fixed approach does not outperform the standard ensemble at $z=3$. The relative performance
improvements then {\it increases} with increasing $k$ beyond this
point, which is surprising. Previously it has been argued that the
improvements generated by the paired-fixed approach can be understood
within the framework of standard perturbation theory
\citep{Angulo2016,Pontzen2016} which more obviously applies to
intermediate quasi-linear scales than the $k>1\ h\text{Mpc}^{-1}$
regime.
At present we cannot fully explain why pairing and fixing improves the
Ly$\alpha$\ power spectrum accuracy in this regime. It might be that
super-sample covariance terms couple high $k$ to low
$k$ in a unique way given that the forest is dominated by underdense
regions (unlike the matter power spectrum; \citealp{Takada2013}).
However, we did not
investigate further for this work because the improvements are not
particularly relevant for observational studies; the absolute
magnitude of the model uncertainties in the $k>1\ h\text{Mpc}^{-1}$ rapidly become
smaller than observational uncertainties (see top panel of Figure
\ref{fig:varratio3d40} for a visualization of the magnitude of the model uncertainties). Thus the improvements in accuracy at low $k$
are of most practical benefit to future work.
\subsection{1D Ly$\alpha$\ Power Spectrum}\label{subsec:P1D}
Most hydrodynamical simulations of the Ly$\alpha$\ forest have been used to
predict the 1D power spectrum
\citep{McDonald2005a,Viel2006a,Borde2014,Lukic2015,Walther2018b}.
As described in Section \ref{sec:skewers}, the
1D spectrum captures small-scale information along the line of sight
without the computational complexity of cross-correlating between
different skewers.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Pk_lya1d_40Mpc_hydro.pdf}
\caption{Same as Figure \ref{fig:pkm3d40}, but for the 1D Ly$\alpha$\ flux power spectrum.
The line-of-sight wavenumber $k_\parallel$ corresponds to an integral over many
3D wavenumbers $k$, so there is not a direct correspondence between the scales. As a typical example of improvement, for wavenumbers $k_{\parallel}=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$, we find estimates of the 1D Ly$\alpha$\ flux power spectra converge $34$ and $1.7$ times faster in a paired-fixed ensemble compared with a standard ensemble.
}
\label{fig:varratio1d40}
\end{figure}
We present the measured 1D Ly$\alpha$\ power spectrum in Figure
\ref{fig:varratio1d40}, using again the same panel layout as Figure
\ref{fig:pkm3d40}. In the top panel the uncertainties on the mean
power spectrum of the standard simulations are, as usual, small due to the
large number of simulations. The middle panel shows the fractional
uncertainty in the paired-fixed simulations relative to the
standard simulations, with the grey shaded region indicating the $1\sigma$
uncertainty on this difference.
Once again there is overall good agreement, with the strongest
deviation arising at $k=0.9\ h\text{Mpc}^{-1}$ at a significance of
almost $2\sigma$. However the difference between our standard and
paired-fixed ensemble remains less than $2\%$, and (given the large number
of independent $k$ bins) we believe the
difference to be consistent with the expected level of statistical
fluctuations.
The bottom panel quantifies the statistical improvement achieved by
the paired-fixed simulations (solid line) with respect to the standard
simulations. The dashed line represents simulations that are fixed but
not paired. As a typical example of improvement, for wavenumbers
$k_{\parallel}=0.25\ h\text{Mpc}^{-1}$ and $2\ h\text{Mpc}^{-1}$ at $z=3$,
we find estimates of the 1D Ly$\alpha$\ flux power spectra converge $34$ and
$1.7$ times faster in a paired-fixed ensemble compared with a
standard ensemble.
As with the 3D Ly$\alpha$\ flux power (Section
\ref{subsec:P3D}) there is a steadily increasing improvement at high
$k$, forming a local minimum at $k_{\parallel} \simeq 1\ h\text{Mpc}^{-1}$;
since the 1D power spectrum mixes multiple modes from the 3D power
spectrum, this is consistent with projecting the improvement discussed
in Section \ref{subsec:P3D}.
\section{Discussion and Conclusions} \label{sec:conc}
We have shown that using the method of paired-fixed simulations reduces the uncertainty of the mean mock Ly$\alpha$\ forest power spectrum measured in hydrodynamical simulations, and therefore requires less computing time to estimate the expected value of the considered quantity, which is needed to evaluate the likelihood of the data.
As a typical example of improvement, for wavenumbers $k=0.25\ h\text{Mpc}^{-1}$ at $z=3$, we find estimates of the 1D and 3D power spectra converge $34$ and $12$ times faster in a paired-fixed ensemble compared with a standard ensemble.
The largest improvements are at small k, where model uncertainties are larger than observational uncertainties. The improvements are minimal at $k=1\ h\text{Mpc}^{-1}$, but at these scales the observational uncertainties are larger than the model uncertainties, so reduced uncertainty on the mean mock power spectrum is unnecessary. This suggests that running a large simulation box size with paired-fixed ICs will provide the most accurate mock Ly$\alpha$\ forest power spectrum over the largest range of scales.
By reducing the computational time required to achieve a target accuracy for mock power spectra, the method frees up resources for a more thorough exploration of astrophysical and cosmological parameters. It is essential to be able to sample efficiently over this space, with at least three parameters describing the shape and amplitude of the input linear power spectrum and at least two for astrophysical effects to span the reionisation redshift and level of heat injection.
In future, likelihoods for datasets such as eBOSS and DESI will likely take advantage of emulators which
interpolate predicted power spectra within these high-dimensional parameter spaces (\cite{Heitmann16,Walther2018b}, Rogers et al in prep, Bird et al in prep). In this context, freeing up CPU time by using the paired-fixed approach will allow for a denser sampling of training points for the emulator. That in turn will beat down interpolation errors and thus lead to more accurate inferences from forthcoming data.
\section*{Acknowledgements} \label{sec:ack}
It is a pleasure to thank An\v{z}e Slosar for useful discussions. The work of LA, FVN and SG is supported by the Simons Foundation. AP was supported by the Royal Society. AFR was supported by an STFC Ernest Rutherford Fellowship, grant reference ST/N003853/1. AP and AFR were further supported by STFC Consolidated Grant no ST/R000476/1.
This work was partially enabled by funding from the University College London (UCL) Cosmoparticle Initiative.
This work was supported by collaborative
visits funded by the Cosmology and Astroparticle Student and Postdoc Exchange
Network (CASPEN).
The simulations were run on the Gordon cluster at the San Diego Supercomputer Center supported by the Simons Foundation.
\software{
The code used in this project is available from
\url{https://github.com/andersdot/LyA-InvertPhase}.
This research utilized the following open-source \textsl{Python} packages:
\textsl{Astropy} (\citealt{Astropy-Collaboration:2013}),
\textsl{scipy}(\citealt{scipy:2001}),
\textsl{matplotlib} (\citealt{Hunter:2007}),
\textsl{fake\_spectra} (\citealt{bird17}) and
\textsl{numpy} (\citealt{Van-der-Walt:2011}).
It also utilized the python pylians libraries, publicly available at \url{https://github.com/ franciscovillaescusa/Pylians}, and \textsl{lyman\_alpha} \citep{Rogers2018a, Rogers2018b}}
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Galepsus or Galepsos () was a Greek city located in the region of Edonis in ancient Thrace and later in Macedon. It was located east of Phagres and about 17 km from Amphipolis. It belonged to the Delian League and it was founded as a colony of Thasos. After the conquest of Amphipolis it was occupied by Brasidas in 424 BCE, but recovered by Cleon in the ensuing year. Perseus of Macedon, fleeing the Romans who had defeated him at Pydna, sailed the mouth of the Strymon, and towards Galepsus, staying there before moving on to Samothrace.
It was named after Galepsos who was a descendant of Thasos and of Telephe.
The site of Galepsus is near the modern Kariani.
See also
Galepsus (Chalcidice)
References
Populated places in ancient Macedonia
Populated places in ancient Thrace
Former populated places in Greece
Greek colonies in Thrace
Thasian colonies
Members of the Delian League | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,118 |
\chapter{Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions}\label{chap:LNCIME}
\chaptermark{Stable solutions to some elliptic problems}
\centerline{Xavier Cabr\'e\footnote{Universitat Polit\`ecnica de Catalunya, Departament de Matem\`{a}tiques, Diagonal 647,
08028 Barcelona, Spain
({\tt xavier.cabre@upc.edu})
}\footnote{ ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain}
and Giorgio Poggesi\footnote{
Dipartimento di Matematica ed Informatica ``U.~Dini'',
Universit\` a di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
({\tt giorgio.poggesi@unifi.it})
}
}
\vspace{1cm}
\begin{quotation}
{\bf Abstract.}
{\small These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017.
The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar.
The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J.~Simons on the flatness of stable minimal cones in low dimensions.
Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E.~De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone.
The third problem was raised by H.~Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic.
Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.}
\end{quotation}
\bigskip
The abstract
above gives
an account of the topics treated in these lecture notes.
\section{Minimal cones}
\label{sec:mincones}
In this section we discuss two classical results on the theory of minimal surfaces: Simons flatness result on stable minimal cones in low dimensions and the Bombieri-De Giorgi-Giusti counterexample in high dimensions.
The main purpose of these lecture notes is to present the main ideas and computations leading to these deep results -- and to related ones in subsequent sections. Therefore, to save time for this purpose, we do not consider the most general classes of sets or functions (defined through weak notions), but instead we assume them to be regular enough.
Throughout the notes, for certain results we will refer to three other expositions: the books of Giusti \cite{G}
and of Colding and Minicozzi \cite{ColMinLN}, and the CIME lecture notes of Cozzi and Figalli \cite{CF}.
The notes \cite{CabCapThree} by the first author and Capella have a similar spirit to the current ones and may complement them.
\begin{definition}[Perimeter]
Let $E \subset \mathbb{R}^n$ be an open set, regular enough. For a given open ball $B_R$ we define the {\it perimeter of} $E$ {\it in} $B_R$ as
\begin{equation*}
P(E, B_R) := H_{n-1} (\partial E \cap B_R ),
\end{equation*}
where $H_{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure (see Figure~\ref{fig:1}).
\end{definition}
The interested reader can learn from \cite{G, CF} a more general notion of perimeter (defined by duality or in a weak sense) and the concept of set of finite perimeter.
\begin{definition}[Minimal set]\label{def:minimal set}
We say that an open set (regular enough) $E \subset \mathbb{R}^n$ is a {\it minimal set} (or a {\it set of minimal perimeter}) if and only if, for every given open ball $B_R$, it holds that
$$P(E, B_R)\leq P(F, B_R)$$
for every open set $F\subset\mathbb{R}^n$ (regular enough) such that $E \setminus B_R = F\setminus B_R$.
\end{definition}
In other words, $E$ has least perimeter in $B_R$ among all (regular) sets which agree with $E$ outside $B_R$.
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE1}
\caption{The perimeter of $E$ in $B_R$}
\label{fig:1}
\end{figure}
To proceed, one considers small perturbations of a given set $E$
and computes the {\it first and second variations of the perimeter functional}.
To this end,
let $\{\phi_t\}$ be a one-parameter family of maps $\phi_t:\mathbb{R}^n \to \mathbb{R}^n$
such that $\phi_0$ is the identity $I$ and all the maps $\phi_t-I$ have compact support (uniformly) contained in $B_R$.
Consider the sets $E_t=\phi_t(E)$. We are interested in the perimeter functional $P(E_t, B_R)$.
One proceeds by choosing $\phi_t=I+t\xi\nu$, which shifts the original set $E$ in the normal direction $\nu$ to its boundary. Here $\nu$ is the outer normal to $E$ and is extended in a neighborhood of $\partial E$ to agree with the gradient of the signed distance function to $\partial E$, as in \cite{G} or in our Subsection \ref{subsec:SimLem} below. On the other hand, $\xi$ is a scalar function with compact support in $B_R$ (see Figure \ref{fig:2}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE2}
\caption{A normal deformation $E_t$ of $E$}
\label{fig:2}
\end{figure}
It can be proved (see chapter 10 of \cite{G}) that the first and second variations of perimeter are given by
\begin{eqnarray}
\left.\frac{d}{dt} P(E_t, B_R) \right|_{t=0}
&=&\int_{\partial E}{\mathcal H}\xi dH_{n-1},
\label{eq:1-1v}\\
\left.\frac{d^2}{dt^2} P(E_t, B_R) \right|_{t=0}
&=&\int_{\partial E}\left\{|\delta\xi|^2-(c^2-{\mathcal H}^2)\xi^2
\right\}dH_{n-1},
\label{eq:1-1vBIS}
\end{eqnarray}
where ${\mathcal H}={\mathcal H}(x)$ is the {\it mean curvature} of $\partial E$ at $x$ and $c^2=c^2(x)$ is the sum of the squares
of the $n-1$ principal
curvatures $k_1, \dots, k_{n-1}$ of $\partial E$ at $x$. More precisely,
$$
{\mathcal H}(x)= k_1 + \dots + k_{n-1} \quad \mbox{ and } \quad c^2= k_1^2 + \dots + k_{n-1}^2 .
$$
In \eqref{eq:1-1vBIS}, $\delta$ (sometimes denoted by $\nabla_T$) is the tangential gradient to the surface
$\partial E$, given by
\begin{equation}\label{def:tangentialgradient}
\delta \xi = \nabla_T \xi = \nabla \xi - (\nabla \xi \cdot \nu) \nu
\end{equation}
for any function $\xi$ defined in a neighborhood of $\partial E$. Here $\nabla$ is the usual Euclidean gradient and $\nu$ is always the normal vector to $\partial E$.
Being $\de$ the tangential gradient, one can check that $\de \xi_{| \partial E}$ depends only on $\xi_{| \partial E}$. It can be therefore computed for functions $\xi: \partial E \to \mathbb{R}$ defined only on $\partial E$ (and not necessarily in a neighborhood of $\partial E$).
\begin{definition}\label{def:servepertesiLNminimizingminimal}
\begin{enumerate}[(i)]
\item We say that $\partial E$ is a {\it minimal surface} (or a {\it stationary surface}) if the first variation of perimeter vanishes for all balls $B_R$. Equivalently, by \eqref{eq:1-1v}, ${\mathcal H} = 0$ on $\partial E$.
\item We say that $\partial E$ is a {\it stable minimal surface} if ${\mathcal H} =0$ and the second variation of perimeter is nonnegative for all balls $B_R$.
\item We say that $\partial E$ is a {\it minimizing minimal surface} if $E$ is a minimal set as in Definition~\ref{def:minimal set}.
\end{enumerate}
\end{definition}
We warn the reader that in some books or articles ``minimal surface'' may mean ``minimizing minimal surface''.
\begin{remark}
\begin{enumerate}[(i)]
\item If $\partial E$ is a minimal surface (i.e., ${\mathcal H} = 0$), the second variation
of perimeter \eqref{eq:1-1vBIS} becomes
\begin{equation}
\left.\frac{d^2}{dt^2} P(E_t, B_R) \right|_{t=0} = \int_{\partial E}\left\{|\delta\xi|^2-c^2\xi^2\right\}dH_{n-1} .
\label{eq:minimal-1-2v2}
\end{equation}
\item If $\partial E$ is a minimizing minimal surface, then $\partial E$ is a stable minimal surface. In fact, in this case the function $P(E_t, B_R)$
has a global minimum at $t=0$.
\end{enumerate}
\end{remark}
\subsection{The Simons cone. Minimality}\label{subsec 1.1:MinimalitySimcones}
\begin{definition}[The Simons cone]
The Simons cone ${\mathcal C}_S \subset \mathbb{R}^{2m}$ is the set
\begin{equation}\label{def:simonscone}
{\mathcal C}_S =\{ x\in\mathbb{R}^{2m}\, : \, x_1^2+\ldots +x_m^2 = x_{m+1}^2
+\ldots +x_{2m}^2 \} .
\end{equation}
In what follows we will also use the following notation:
\begin{equation*}
{\mathcal C}_S =\{ x=(x',x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : |x'|^2=|x''|^2 \}.
\end{equation*}
Let us consider the open set
\begin{equation*}
E_S= \left\{x \in \mathbb{R}^{2m} : u(x):= |x'|^2 - |x''|^2 < 0 \right\},
\end{equation*}
and notice that $\partial E_S = {\mathcal C}_S$ (see Figure \ref{fig:3}).
\end{definition}
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE3}
\caption{The set $E_S$ and the Simons cone ${\mathcal C}_S$}
\label{fig:3}
\end{figure}
\begin{exercise}
Prove that the Simons cone has zero mean curvature for every integer $m \ge 1$. For this, use the following fact (that you may also try to prove): if
$$
E=\left\{x \in \mathbb{R}^n : u(x) < 0 \right\}
$$
for some function $u: \mathbb{R}^n \to \mathbb{R}$, then the mean curvature of $\partial E$ is given by
\begin{equation}
\label{mean curvature formula}
{\mathcal H} = \left. \mathop{\mathrm{div}} \left( \frac{\nabla u}{| \nabla u|} \right) \right|_{\partial E}.
\end{equation}
\end{exercise}
\begin{remark}\label{remark:nominimindim2}
It is easy to check that, in $\mathbb{R}^2$, ${\mathcal C}_S$ is not a minimizing minimal surface. In fact, referring to Figure \ref{fig:4}, the shortest way to go from $P_1$ to $P_2$ is through the straight line. Thus, if we consider as a competitor in $B_R$ the interior of the set
$$F:= \overline{E}_S \cup \overline{T}_1 \cup \overline{T}_2,$$
where $T_1$ is the triangle with vertices $O$, $P_1$, $P_2$, and $T_2$ is the symmetric of $T_1$ with respect to $O$, we have that $F$ has less perimeter in $B_R$ than $E_S$.
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE4}
\caption{The Simons cone ${\mathcal C}_S$ is not a minimizer in $\mathbb{R}^2$}
\label{fig:4}
\end{figure}
\end{remark}
In 1969 Bombieri, De Giorgi, and Giusti proved the following result.
\begin{theorem}[Bombieri-De Giorgi-Giusti~\cite{BdGG}]
\label{thm:SimCone}
If $2m\geq 8$, then $E_S$ is a minimal set in $\mathbb{R}^{2m}$. That is, if $2m\geq 8$, the Simons cone ${\mathcal C}_S$ is a minimizing minimal surface.
\end{theorem}
The following is a clever proof of Theorem \ref{thm:SimCone} found in 2009 by G.~De Philippis and E.~Paolini (\cite{DePP}). It is based on a {\it calibration argument}.
Let us first define
\begin{equation}\label{def:utilde1}
\tilde{u} = |x'|^4 - |x''|^4 ;
\end{equation}
clearly we have that
\begin{equation*}
E_S= \left\{x \in \mathbb{R}^{2m} : \tilde{u}(x) < 0 \right\} \, \mbox{ and } \, \partial E_S = {\mathcal C}_S.
\end{equation*}
Let us also consider the vector field
\begin{equation}\label{def:Xvectorfield}
X= \frac{\nabla \tilde{u}}{ |\nabla \tilde{u}| } .
\end{equation}
\begin{exercise}\label{ex:utilde}
Check that if $m \ge 4$, $\mathop{\mathrm{div}} X$ has the same sign as $\tilde{u}$ in $\mathbb{R}^{2m}$.
\end{exercise}
\begin{proof}[Proof of Theorem \ref{thm:SimCone}]
By Exercise \ref{ex:utilde} we know that if $m \ge 4$, $\mathop{\mathrm{div}} X$ has the same sign as $\tilde{u}$, where $\tilde{u}$ and $X$ are defined in \eqref{def:utilde1} and \eqref{def:Xvectorfield}.
Let $F$ be a competitor for $E_S$ in a ball $B_R$, with $F$ regular enough. We have that
$F \setminus B_R = E_S \setminus B_R$.
Set $\Omega := F \setminus E_S $ (see Figure \ref{fig:5}). By using the fact that $\mathop{\mathrm{div}} X \geq 0$ in $\Omega$ and the divergence theorem, we deduce that
\begin{equation}
\label{calideph}
0 \leq \int_{\Omega} \mathop{\mathrm{div}} X \, dx= \int_{\partial E_S \cap \overline{\Omega} } X \cdot \nu_\Omega \, dH_{n-1} + \int_{\partial F \cap \overline{\Omega}} X \cdot \nu_\Omega \, dH_{n-1}.
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE5}
\caption{A calibration proving that the Simons cone ${\mathcal C}_S$ is minimizing}
\label{fig:5}
\end{figure}
Since $X= \nu_{E_S}= - \nu_\Omega$ on $\partial E_S \cap \overline{\Omega}$, and $|X| \leq 1$ (since in fact $|X|= 1$) everywhere (and hence in particular on $\partial F \cap \overline{\Omega}$), from \eqref{calideph} we conclude
\begin{equation}\label{eq:dimcalibration1}
H_{n-1} (\partial E_s \cap \overline{\Omega}) \le H_{n-1} (\partial F \cap \overline{\Omega}) .
\end{equation}
With the same reasoning it is easy to prove that \eqref{eq:dimcalibration1} holds also for
$\Omega := E_S \setminus F$.
Putting both inequalities together, we conclude that $P(E_S,B_R) \leq P(F, B_R)$.
Notice that the proof works for competitors $F$ which are regular enough (since we applied the divergence theorem). However, it can be generalized to very general competitors by using the generalized definition of perimeter, as in \cite[Theorem~1.5]{DePP}.
\end{proof}
Theorem \ref{thm:SimCone} can also be proved with another argument -- but still very much related to the previous one and that also uses the function $\tilde{u} = |x'|^4 - |x''|^4 $. It consists of going to one more dimension $\mathbb{R}^{2m+1}$ and working with the minimal surface equation for graphs, \eqref{equa:Hgraph} below. This is done in Theorem 16.4 of \cite{G} (see also the proof of Theorem 2.2 in \cite{CabCapThree}).
In the proof above we used a vector field $X$ satisfying the following three properties (with $E= E_S$):
\begin{enumerate}[(i)]
\item $\mathop{\mathrm{div}} X \geq 0$ in $B_R \setminus E$ and $\mathop{\mathrm{div}} X \leq 0$ in $E \cap B_R$;
\item $X= \nu_{E}$ on $\partial E \cap B_R$;
\item $|X| \leq 1$ in $B_R$.
\end{enumerate}
\begin{definition}[Calibration]\label{def:calibrationprima}
If $X$ satisfies the three properties above we say that $X$ is a \textit{calibration} for $E$ in $B_R$.
\end{definition}
\begin{exercise}
Use a similar argument to that of our last proof and build a calibration to show that a hyperplane in $\mathbb{R}^n$ is a minimizing minimal surface.
\end{exercise}
In an appendix, and with the purpose that the reader gets acquainted with another calibration, we present one which solves the isoperimetric problem: balls minimize perimeter among sets of given volume in $\mathbb{R}^n$.
Note that the first variation (or Euler-Lagrange equation) for this problem is, by Lagrange multipliers,
${\mathcal H} = c$, where $c \in \mathbb{R}$ is a constant.
The following is an alternative proof of Theorem \ref{thm:SimCone}. It uses a {\it foliation argument}, as explained below.
This second proof is probably more transparent (or intuitive) than the previous one and it is used often in minimal surfaces theory, but requires to know the existence of a (regular enough) minimizer (something that was not necessary in the previous proof). This existence result is available and can be proved with tools of the Calculus of Variations (see \cite{CF,G}).
The proof also requires the use of the following important fact. If $\Sigma_1$, $\Sigma_2 \subset B_R$ are two connected hypersurfaces (regular enough), both satisfying ${\mathcal H}=0$, and such that $\Sigma_1 \cap \Sigma_2 \neq \varnothing$ and $\Sigma_1$ lies on one side of $\Sigma_2$, then $\Sigma_1 \equiv \Sigma_2$ in $B_R$. Lying on one side can be defined as $\Sigma_1 = \partial F_1$, $\Sigma_2 = \partial F_2$, and $F_1 \subset F_2$. The same result holds if $F_1$ satisfies ${\mathcal H}=0$ and $F_2$ satisfies ${\mathcal H} \geq 0$.
This result can be proved writing both surfaces as graphs in a neighborhood of a common point $P \in \Sigma_1 \cap \Sigma_2$. The minimal surface equation ${\mathcal H}=0$ then becomes
\begin{equation}\label{equa:Hgraph}
\mathop{\mathrm{div}} \left( \frac{\nabla \fhi_1}{ \sqrt{1+| \nabla \fhi_1|^2 } } \right) =0
\end{equation}
for $\fhi_1: \Omega \subset \mathbb{R}^{n-1} \to \mathbb{R}$ such that $\left( y', \fhi_1 (y') \right) \subset \Omega \times \mathbb{R} $ is a piece of $\Sigma_1$ (after a rotation and translation). Then, assuming that $\fhi_2$ also satisfies \eqref{equa:Hgraph} -- or the appropriate inequality --, one can see that $\fhi_1 - \fhi_2$ is a (super)solution of a second order linear elliptic equation. Since $\fhi_1 - \fhi_2 \geq 0$ (due to the ordering of $\Sigma_1$ and $\Sigma_2$), the strong maximum principle leads to $\fhi_1 - \fhi_2 \equiv 0$ (since $(\fhi_1 -\fhi_2)(0)=0$ at the touching point). See Section 7 of Chapter 1 of \cite{ColMinLN} for more details.
\begin{alternative}[of Theorem \ref{thm:SimCone}]
Note that the hypersurfaces
$$\left\{ x \in \mathbb{R}^{2m} : \tilde{u} (x) = \lambda \right\} , $$
with $\lambda \in \mathbb{R}$, form a foliation of $\mathbb{R}^{2m}$, where $\tilde{u}$ is the function defined in \eqref{def:utilde1}.
Let $F$ be a minimizer of the perimeter in $B_R$ among sets that coincide with $E_S$ on $\partial B_R$, and assume that it is regular enough. Since $F$ is a minimizer, in particular $\partial F$ is a solution of the minimal surface equation ${\mathcal H}=0$.
Since $2m \geq 8$, by \eqref{mean curvature formula} and Exercise \ref{ex:utilde}, the leaves of our foliation $\left\{ x \in \mathbb{R}^{2m} : \tilde{u} (x) = \lambda \right\}$ are subsolutions of the same equation for $\lambda>0$, and supersolutions for $\lambda<0$ .
If $F \not\equiv E_S$, there will be a first leaf (starting either from $\lambda= + \infty$ or from $\lambda = - \infty$) $\left\{ x \in \mathbb{R}^{2m} : \tilde{u} (x) = \lambda_* \right\}$, with $\lambda_* \neq 0$, that touches $\partial F$ at a point in $\overline{B}_R$ that we call $P$ (see Figure \ref{fig:6}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE6}
\caption{The foliation argument to prove that the Simons cone ${\mathcal C}_S$ is minimizing}
\label{fig:6}
\end{figure}
The point $P$ cannot belong to $\partial B_R$, since it holds that
$$\partial F \cap \partial B_R = {\mathcal C}_S \cap \partial B_R = \left\{ x: \tilde{u} (x) = 0 \right\} \cap \partial B_R ,$$
and the level sets of $\tilde{u}$ do not intersect each other.
Thus, $P$ must be an interior point of $B_R$. But then we arrive at a contradiction, by the ``strong maximum principle'' argument commented right before this proof, applied with
$\Sigma_1= \partial F$ and $\Sigma_2= \left\{ x \in \mathbb{R}^{2m} : \tilde{u} (x) = \lambda_* \right\} $.
As an exercise, write the details to prove the existence of a first leaf touching $\partial F$ at an interior point.
This same foliation argument will be used, in a simpler setting for graphs and the Allen-Cahn equation, in the proof of Theorem \ref{alba} in the next section.
\qed
\end{alternative}
\begin{remark}
The previous foliation argument gives more than the minimality of ${\mathcal C}_S$. It gives {\it uniqueness for the Dirichlet (or Plateau) problem} associated to the minimal surface equation with ${\mathcal C}_S$ as boundary value on $\partial B_R$.
\end{remark}
\begin{remark}\label{rem:Foliation B-DG-G}
In our alternative proof of Theorem \ref{thm:SimCone} we used a clever foliation made of subsolutions and supersolutions. This sufficed to prove in a simple way Theorem \ref{thm:SimCone}, but required to (luckily) find the auxiliary function $\tilde{u} = |x'|^4 - |x''|^4 $.
Instead, in \cite{BdGG}, Bombieri, De Giorgi, and Giusti considered the foliation made of exact solutions to the minimal surface equation ${\mathcal H}=0$, when $2 m \ge 8$. To this end, they proceeded as in the following exercise and wrote the minimal surface equation, for surfaces with rotational symmetry in $x'$ and in $x''$, as an ODE in $\mathbb{R}^2$, finding Equation \eqref{equa:odest} below. They then showed that the solutions of such ODE in the $(s,t)$-plane do not intersect each other (and neither the Simons cone), and thus form a foliation (see Remark \ref{rem:dopohardy} for more information on this).
\end{remark}
\begin{exercise}
Let us set $s= |x'|$ and $t=|x''|$ for $x=(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m$.
Check that the following two ODEs are equivalent to the minimal surface equation ${\mathcal H}=0$ written in the $(s,t)$-variables for surfaces with rotational symmetry in $x'$ and in $x''$.
\begin{enumerate}[(i)]
\item As done in \cite{BdGG}, if we set a parametric representation $ s=s(\tau)$, $t = t(\tau)$, we find
\begin{equation}\label{equa:odest}
s''t'-s't''+(m-1) \left( (s')^2 + (t')^2 \right) \left( \frac{s'}{t} - \frac{t'}{s} \right) = 0 ;
\end{equation}
\item as done in \cite{Da}, if we set $s=e^{z(\theta)} \cos (\theta)$, $t = e^{z(\theta)} \sin (\theta)$ we get
$$
z''= \left( 1+(z')^2 \right) \left( (2m -1) - \frac{2(m-1) \cos (2 \theta)}{ \sin (2 \theta)} \, z' \right) .
$$
\end{enumerate}
The previous ODEs can be found starting from \eqref{mean curvature formula} when $u=u(s,t)$ depends only on $s$ and $t$. Alternatively, they can also be found computing the first variation of the perimeter functional in $\mathbb{R}^{2m}$ written in the $(s,t)$-variables:
\begin{equation}\label{equa:stenergy}
c \int s^{m-1} t^{m-1} \, dH_1 (s,t),
\end{equation}
for some positive constant $c$, that becomes
$$
c \int e^{(2m-1)z(\theta)} \, \cos^{m-1} (\theta) \, \, \sin^{m-1} (\theta) \, \, \sqrt{1+ \left( z'(\theta) \right)^2 } \, d \theta
$$
with the parametrization in point (ii).
\end{exercise}
\begin{remark}
For $n \ge 8$, there exist other minimizing cones, such as some of the {\it Lawson's cones}, defined by
$$
{\mathcal C}_L= \left\{ y= (y', y'') \in \mathbb{R}^k \times \mathbb{R}^{n-k} : |y'|^2= c_{n,k} \, |y''|^2 \right\}
$$
for $k \ge 2$ and $n-k \ge 2$.
For details, see \cite{Da}.
\end{remark}
Notice that if $\partial E$ is a {\it cone} (i.e., $\lambda \partial E = \partial E$ for every $\lambda>0$), in the expressions \eqref{eq:1-1v}, \eqref{eq:1-1vBIS}, and \eqref{eq:minimal-1-2v2} we will always consider $\xi$ with compact support outside the origin (thus, not changing the possible singularity of the cone at the origin).
The next theorem was proved by Simons in 1968\footnote{Theorem \ref{thmcone} was proved in 1965 by De Giorgi for $n=3$, in 1966 by Almgren for $n=4$, and finally in 1968 by Simons in any dimension $n \le 7$.} (it is Theorem 10.10 in \cite{G}).
It is a crucial result towards the regularity theory of minimizing minimal surfaces.
\begin{theorem}[Simons~\cite{S}]\label{thmcone}
Let $E \subset \mathbb{R}^n$ be an open set such that $\partial E$ is a stable minimal cone and
$\partial E\setminus\{0\}$ is regular.
Thus, we are assuming ${\mathcal H}=0$ and
\begin{equation}
\int_{\partial E}\left\{|\delta\xi|^2-c^2\xi^2\right\}dH_{n-1}\geq 0
\label{eq:1-2v2}
\end{equation}
for every $\xi\in C^1(\partial E)$ with
compact support outside the origin.
If $3 \leq n\leq 7$, then $\partial E$ is a hyperplane.
\end{theorem}
\begin{remark}\label{remarkchemiserve:1.17}
Simons result (Theorem~\ref{thmcone}), together with a blow-up argument and a monotonicity formula (as main tools), lead to the flatness of every minimizing minimal surface in all of $\mathbb{R}^n$ if $n \le 7$ (see \cite[Theorem 17.3]{G} for a proof). The same tools also give the analyticity of every minimal surface that is minimizing in a given ball of $\mathbb{R}^n$ if $n \le 7$ (see \cite[Theorem 10.11]{G} for a detailed proof). See also \cite{CF} for a great shorter exposition of these results.
\end{remark}
The dimension $7$ in Theorem \ref{thmcone} is optimal, since by Theorem \ref{thm:SimCone} the Simons cone provides a counterexample in dimension $8$.
The following is a very rough explanation of why the minimizer of the Dirichlet (or Plateau) problem is the Simons cone (and thus passes through the origin) in high dimensions -- in opposition with low dimensions, as in Figure \ref{fig:4}, where the minimizer stays away from the origin. In the perimeter functional written in the $(s,t)$-variables \eqref{equa:stenergy}, the Jacobian $s^{m-1} t^{m-1}$ becomes smaller and smaller near the origin as $m$ gets larger. Thus, lengths near $(s,t)=(0,0)$ become smaller as the dimension $m$ increases.
In order to prove Theorem \ref{thmcone}, we start with some important preliminaries.
Recalling \eqref{def:tangentialgradient}, for $i=1 \dots ,n$, we define the tangential derivative
$$
\de_i \xi := \partial_i \xi - \nu^i \, \nu^k \xi_k ,
$$
where $\nu= \nu_E = (\nu^1, \dots, \nu^n): \partial E \to \mathbb{R}^n$ is the exterior normal to $E$ on $\partial E$,
$\partial_i \xi = \partial_{x_i} \xi = \xi_i$ are Euclidean partial derivatives, and we used the standard convention of sum $\sum_{k=1}^{n}$ over repeated indices.
As mentioned right after definition \eqref{def:tangentialgradient}, even if to compute $\partial_i \xi$ requires to extend $\xi$ to a neighborhood of $\partial E$, $\de_i \xi$ is well defined knowing $\xi$ only on
$\partial E$ -- since it is a tangential derivative. Note also that we have $n$ tangential derivatives $\de_1, \dots, \de_n$ and, thus, they are linearly dependent, since $\partial E$ is $(n-1)$-dimensional. However, it is easy to check (as an exercise) that
\begin{equation*}
|\de \xi|^2= \sum_{i=1}^n |\de_i \xi|^2 .
\end{equation*}
We next define {\it the Laplace-Beltrami operator} on $\partial E$ by
\begin{equation}\label{def:Laplace-Beltrami}
\De_{LB} \, \xi := \sum_{i=1}^n \delta_i\delta_i \xi ,
\end{equation}
acting on functions $\xi: \partial E \to \mathbb{R}$.
For the reader knowing Riemannian calculus, one can check that
$$
\De_{LB} \, \xi = {\mathop{\mathrm{div}}}_T (\nabla_T \xi) = {\mathop{\mathrm{div}}}_T (\de \xi) ,
$$
where $\nabla_T=\de$ is the tangential gradient introduced in \eqref{def:tangentialgradient} and $\mathop{\mathrm{div}}_T$ denotes the (tangential) divergence on the manifold $\partial E$.
According to \eqref{mean curvature formula}, we have that
$$
{\mathcal H} = {\mathop{\mathrm{div}}}_T \nu = \sum_{i=1}^n \de_i \nu^i.
$$
We will also use the following {\it formula of integration by parts}:
\begin{equation}\label{eq:Giustitypo}
\int_{\partial E} \de_i \phi \, dH_{n-1} = \int_{\partial E} {\mathcal H} \phi \nu^i dH_{n-1}
\end{equation}
for every (smooth) hypersurface $\partial E$ and $\phi \in C^1 (\partial E)$ with compact support.
Equation \eqref{eq:Giustitypo} is proved in Giusti's book \cite[Lemma 10.8]{G}. However, there are
two typos in \cite[Lemma 10.8]{G}: ${\mathcal H}$ is missed in the identity above, and there is an error of a sign in the proof of \cite[Lemma 10.8]{G}.
Replacing $\phi$ by $\phi \fhi$ in \eqref{eq:Giustitypo}, we deduce that
\begin{equation}\label{equa:intparts2}
\int_{\partial E} \phi \, \de_i \fhi \, dH_{n-1} = - \int_{\partial E} (\de_i \phi) \fhi \, dH_{n-1} + \int_{\partial E} {\mathcal H} \phi \fhi \nu^i dH_{n-1} .
\end{equation}
From this, replacing $\phi$ by $\de_i \phi$ in \eqref{equa:intparts2} and using that $\sum_{i=1}^n \nu^i \de_i \phi = \nu \cdot \de \phi=0$, we also have
\begin{equation}\label{equa:intLB}
\int_{\partial E} \de \phi \cdot \de \fhi \, dH_{n-1} = \sum_{i=1}^n \int_{\partial E} \de_i \phi \, \de_i \fhi \, dH_{n-1} = - \int_{\partial E} (\De_{LB} \, \phi) \fhi \, dH_{n-1} .
\end{equation}
\begin{remark}\label{rem:Jacobi operator}
For a minimal surface $\partial E$, the second variation of perimeter given by \eqref{eq:minimal-1-2v2} can also be rewritten, after \eqref{equa:intLB}, as
$$
\int_{\partial E} \left\{ - \De_{LB} \, \xi -c^2 \xi \right\} \xi \, dH_{n-1}.
$$
The operator $- \De_{LB} \, -c^2$ appearing in this expression is called {\it the Jacobi operator}. It is the linearization at the minimal surface $\partial E$ of the minimal surface equation ${\mathcal H} =0$.
\end{remark}
Towards the proof of Simons theorem, let us now take $\xi=\tilde{c}\eta$ in
\eqref{eq:1-2v2}, where $\tilde{c}$ and $\eta$ are still arbitrary ($\eta$ with
compact support outside the origin) and will be chosen later.
We obtain
\begin{eqnarray*}
0 &\le& \int_{\partial E} \left\{ |\delta\xi|^2-c^2 \xi^2 \right\} dH_{n-1} \\
&=& \int_{\partial E} \left\{ \tilde{c}^2 |\delta \eta|^2 + \eta^2 | \delta \tilde{c} |^2
+ \tilde{c} \de \tilde{c} \cdot \de \eta^2 - c^2 \tilde{c}^2 \eta^2 \right\} dH_{n-1} \\
&=& \int_{\partial E} \left\{ \tilde{c}^2|\delta\eta|^2 - ( \De_{LB} \, \tilde{c} +c^2 \tilde{c} ) \tilde{c} \eta^2 \right\} dH_{n-1} ,
\end{eqnarray*}
where at the last step we used integration by parts \eqref{equa:intparts2}.
This leads to the inequality
\begin{equation*}
\int_{\partial E} \left\{ \De_{LB} \, \tilde{c} +c^2 \tilde{c} \right\} \tilde{c} \eta^2 dH_{n-1} \le \int_{\partial E} \tilde{c}^2|\delta\eta|^2 dH_{n-1} ,
\end{equation*}
where the term $ \De_{LB} \, \tilde{c} +c^2 \tilde{c}$ appearing in the first integral is the linearized or Jacobi operator at $\partial E$ acting on $\tilde{c}$.
Now we make the choice $\tilde{c} = c$ and we arrive, as a consequence of stability, to
\begin{equation}\label{eq:Disuguaglianza prima di Teo Simons}
\int_{\partial E} \left\{ \frac{1}{2} \De_{LB} \, c^2 - | \delta c|^2 +c^4 \right\} \eta^2 dH_{n-1} \le \int_{\partial E} c^2|\delta\eta|^2 dH_{n-1} .
\end{equation}
At this point, Simons proof of Theorem \ref{thmcone} uses the following inequality for the Laplace-Beltrami operator $\Delta_{LB}$ of $c^2$
(recall that $c^2$ is the sum of the squares of the principal curvatures
of $\partial E$), in the case
when $\partial E$ is a stationary cone.
\begin{lemma}[Simons lemma~\cite{S}]
\label{lmsimons}
Let $E\subset\mathbb{R}^n$ be an open set such that $\partial E$
is a cone with zero mean curvature and $\partial E\setminus\{0\}$ is regular.
Then, $c^2$ is homogeneous of degree $-2$ and, in $\partial E\setminus\{0\}$, we have
\[
\frac{1}{2} \De_{LB} \, c^2 - |\delta c|^2 + c^4 \geq \frac{2c^2}{|x|^2}.
\]
\end{lemma}
In Subsection \ref{subsec:SimLem} we will give an outline of the proof of this result.
We now use Lemma \ref{lmsimons} to complete the proof of Theorem \ref{thmcone}.
\begin{proof}[Proof of Theorem \ref{thmcone}]
By using \eqref{eq:Disuguaglianza prima di Teo Simons} together with Lemma~\ref{lmsimons} we obtain
\begin{equation}
0\leq\int_{\partial
E}c^2\left\{|\delta\eta|^2-\frac{2\eta^2}{|x|^2}\right\}dH_{n-1}
\label{eq:1-mcstb}
\end{equation}
for every $\eta\in C^1(\partial E)$ with compact
support outside the origin.
By approximation, the same holds for $\eta$ Lipschitz instead of $C^1$.
If $r=|x|$, we now choose $\eta$ to be the Lipschitz function
$$
\eta=
\bigg \{
\begin{array}{rl}
r^{- \alpha} \quad \text{if } r \le 1 \ \\
r^{- \beta} \quad \text{if } r \ge 1 . \\
\end{array}
$$
By directly computing
\begin{equation}\label{equaz:provaetadelta}
|\de \eta|^2=
\bigg \{
\begin{array}{rl}
\alpha^2 r^{- 2 \alpha -2} \quad \text{if } r \le 1 \ \\
\beta^2 r^{- 2 \beta -2} \quad \text{if } r \ge 1 , \\
\end{array}
\end{equation}
we realize that if
\begin{equation}\label{equaz:provaalfabeta}
\alpha^2<2 \, \mbox{ and } \, \beta^2<2 ,
\end{equation}
then in \eqref{eq:1-mcstb} we have $|\de \eta|^2 - 2 \eta^2 / r^2 <0$.
If $\eta$ were an admissible function in \eqref{eq:1-mcstb}, we would then conclude that $c^2 \equiv 0$ on
$\partial E$. This is equivalent to $\partial E$ being an hyperplane.
Now, for $\eta$ to have compact support and hence be admissible, we need to cut-off $\eta$ near $0$ and infinity. As an exercise, one can check that the cut-offs work (i.e., the tails in the integrals tend to zero) if (and only if)
\begin{equation}\label{equazion:cutoffex}
\int_{\partial E}c^2 | \de \eta|^2 dH_{n-1}<\infty,
\end{equation}
or equivalently, since they have the same homogeneity,
$$
\int_{\partial E}c^2\frac{\eta^2}{|x|^2} \, dH_{n-1}<\infty.
$$
By recalling that the Jacobian on $\partial E$ (in spherical coordinates) is $r^{(n-1)-1}$, \eqref{equaz:provaetadelta}, and that, by Lemma~\ref{lmsimons}, $c^2$ is homogeneous of degree $-2$, we deduce that
\eqref{equazion:cutoffex} is satisfied if $n-6- 2\alpha >-1$ and $n-6 - 2 \beta <-1$. That is, if
\begin{equation}
\alpha < \frac{n -5 }{2} \quad\textrm{ and }\quad \frac{n-5}{2} < \beta .
\label{eq:1-ineqs}
\end{equation}
If $3\le n\le 7$ then $(n -5 )^2/4<2$, i.e., $ - \sqrt{2} < (n -5) / 2 < \sqrt{2}$, and thus we can choose $\alpha$ and $\beta$ satisfying
\eqref{eq:1-ineqs} and \eqref{equaz:provaalfabeta}. It then follows that $c^2\equiv 0$, and
hence $\partial E$ is a hyperplane.
\end{proof}
The argument in the previous proof (leading to the dimension $n\le 7$) is very much related to a well known result: Hardy's inequality in $\mathbb{R}^n$ -- which is presented next.
\subsection{Hardy's inequality}
As already noticed in Remark \ref{rem:Jacobi operator}, for a minimal surface $\partial E$ the second variation of perimeter \eqref{eq:minimal-1-2v2} can also be rewritten, by integrating by parts, as
$$
\int_{\partial E} \left\{ - \De_{LB} \, \xi -c^2 \xi \right\} \xi dH_{n-1} .
$$
This involves the linearized or Jacobi operator $- \De_{LB} \, -c^2$. If $x= |x| \sigma= r \sigma$, with
$\sigma \in S^{n-1}$, then $c^2= d(\sigma) / |x|^2$ (if $\partial E$ is a cone and thus $c^2$ is homogeneous of degree $-2$), where $d(\sigma)$ depends only on the angles $\sigma$. Thus, we are in the presence of the ``Hardy-type operator''
$$- \De_{LB} \, - \frac{d(\sigma)}{|x|^2} ;$$
notice that $\De_{LB} \, $ and $d(\sigma) / |x|^2 $ scale in the same way.
Thus, for all admissible functions $\xi$,
$$0 \le \int_{\partial E} \left\{ |\de \xi|^2 - \frac{d(\sigma)}{|x|^2} \, \xi^2 \right\} dH_{n-1} , \quad \mbox{if $\partial E$ is a stable minimal cone.}$$
Let us analyze the simplest case when $\partial E = \mathbb{R}^n$ and $d \equiv constant$. Then, the validity or not of the previous inequality is given by Hardy's inequality, stated and proved next.
\begin{proposition}[Hardy's inequality]\label{Prop:Hardy inequality}
If $n \ge 3$ and $\xi \in C_{c}^{1}(\mathbb{R}^n \setminus \lbrace 0 \rbrace)$, then
\begin{equation}\label{Hardyin}
\frac{(n-2)^{2}}{4}\int _{\mathbb{R}^n}\frac{\xi ^{2}}{|x|^{2}} \,dx
\leq \int _{\mathbb{R}^n}\left|\nabla \xi \right|^{2}dx .
\end{equation}
In addition, $(n-2)^{2} / 4$ is the best constant in this inequality and it is not achieved by any $0 \not\equiv \xi \in H^1 (\mathbb{R}^n)$.
Moreover, if $a> (n-2)^{2} / 4$, then the Dirichlet spectrum of $- \De_{LB} \, - a / |x|^2$ in the unit ball $B_1$ goes all the way to $- \infty$. That is,
\begin{equation}\label{equa:quotient}
\inf \frac{ \int_{B_1} \lbrace |\nabla \xi|^2 - a \, \frac{\xi^2}{|x|^2} \rbrace dx }{\int_{B_1} |\xi|^2 dx} = - \infty ,
\end{equation}
where the infimum is taken over $0 \not\equiv \xi \in H_0^1 (B_1)$.
\end{proposition}
\begin{proof}
Using spherical coordinates, for a given $\sigma \in S^{n-1}$ we can write
\begin{equation}\label{har1}
\int_0^{+\infty} r^{n-1} r^{-2} \xi^2(r \sigma) \, dr = - \frac{1}{n-2} \int_0^{+\infty} r^{n-2} 2 \xi (r \sigma) \xi_r (r \sigma) \, dr .
\end{equation}
Here we integrated by parts, using that $r^{n-3}= \left( r^{n-2} / (n-2) \right)'$.
Now we apply the Cauchy-Schwarz inequality in the right-hand side to obtain
\begin{multline}\label{har2}
- \int_0^{+\infty} r^{n-2} \xi \xi_r \, r^{\frac{n-3}{2} } \, r^{- \frac{n-3}{2} } \, dr \le
\\
\left( \int_0^{+\infty} r^{n-3} \xi^2 \, dr\right)^{\frac{1}{2}} \, \left(\int_0^{+\infty} r^{n-1} \xi^2_r \, dr \right)^{\frac{1}{2}}.
\end{multline}
Putting together \eqref{har1} and \eqref{har2} we get
$$\int_0^{+\infty} r^{n-3} \xi^2 \, dr \le \frac{2}{n-2} \left( \int_0^{+\infty} r^{n-3} \xi^2 \, dr\right)^{\frac{1}{2}} \, \left(\int_0^{+\infty} r^{n-1} \xi^2_r \, dr \right)^{\frac{1}{2}},$$
that is,
$$\frac{(n-2)^2}{4} \int_0^{+\infty} r^{n-1} \, \frac{\xi^2 }{r^2} \, dr \le \int_0^{+\infty} r^{n-1} \xi^2_r \, dr . $$
By integrating in $\sigma$ we conclude \eqref{Hardyin}.
An inspection of the equality cases in the previous proof shows that the best constant is not achieved.
Let us now consider $(n-2)^2 / 4 < \alpha^2 < a$ with $\alpha \searrow (n-2) / 2$. Take
$$
\xi = r^{- \alpha} -1
$$
and cut it off near the origin to be admissible. If we consider the main terms in the quotient \eqref{equa:quotient}, we get
$$
\frac{ \int (\alpha^2 - a) r^{-2 \alpha -2} dx }{ \int r^{-2 \alpha } dx } .
$$
Thus it is clear that, as $\alpha \searrow (n-2) / 2$, the denominator remains finite independently of the cut-off, while the numerator is as negative as we want after the cut-off. Hence, the quotient tends to $- \infty$.
\end{proof}
\begin{remark}\label{rem:dopohardy}
As we explained in Remark \ref{rem:Foliation B-DG-G}, in \cite{BdGG}, Bombieri, De Giorgi, and Giusti used a foliation made of exact solutions to the minimal surface equation ${\mathcal H}=0$ when $2 m \ge 8$.
These are the solutions of the ODE \eqref{equa:odest} starting from points $\left( s(0),t(0) \right)= \left( s_0,0 \right) $ in the $s$-axis and with vertical derivative $\left( s'(0), t'(0) \right) = \left( 0,1 \right) $.
They showed that, for $2m \ge 8$, they do not intersect each other, neither intersect the Simons cone ${\mathcal C}_S$. Instead, in dimensions $4$ and $6$ they do not produce a foliation and, in fact, each of them crosses infinitely many times ${\mathcal C}_S$, as showed in Figure \ref{fig:add2}. This reflects the fact that the linearized operator $- \De_{LB} \, -c^2$ on ${\mathcal C}_S$ has infinitely many negative eigenvalues, as in the simpler situation of Hardy's inequality in the last statement of Proposition \ref{Prop:Hardy inequality}.
\end{remark}
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{ADDITIONALFIGURE2}
\caption{Behaviour of the solutions to ${\mathcal H} = 0$ in dimensions $2$, $4$, and $6$}
\label{fig:add2}
\end{figure}
\subsection{Proof of the Simons lemma}\label{subsec:SimLem}
As promised, in this section we present the proof of Lemma \ref{lmsimons} with almost all details.
We follow the proof contained in Giusti's book \cite{G}, where more details can be found (Simons lemma is Lemma 10.9 in \cite{G}). We point out that in the proof of \cite{G} there are the following two typos:
\begin{itemize}
\item as already noticed before, the identity in the statement of \cite[Lemma 10.8]{G} is missing ${\mathcal H}$ in the second integrand. We wrote the corrected identity in equation \eqref{eq:Giustitypo} of these notes;
\item the label (10.18) is missing in line -8, page 122 of \cite{G}.
\end{itemize}
Alternative proofs of Lemma \ref{lmsimons} using intrinsic Riemaniann tensors can be found in the original paper of Simons \cite{S} from 1968 and also in the book of Colding and Minicozzi \cite{ColMinLN}.
\begin{notations}
We denote by $d(x)$ the signed distance function to $\partial E$, defined by
$$d(x) :=
\bigg \{
\begin{array}{rl}
\operatorname{dist} (x, \partial E) , & \ \ x \in \mathbb{R}^n \setminus E , \\
- \operatorname{dist} (x, \partial E) , & \ \ x \in E . \\
\end{array}
$$
As we are assuming $E \setminus \{ 0 \}$ to be regular, we have that $d(x)$ is $C^2$ in a neighborhood of $\partial E \setminus \{ 0 \} $.
The normal vector to $\partial E$ is given by
$$\nu= \nabla d= \frac{ \nabla d}{| \nabla d |};$$
we write
$$\nu= (\nu^1, \dots, \nu^n) = (d_1, \dots, d_n) ,$$
where we adopt the abbreviated notation
$$w_i=w_{x_i} = \partial_i w \, \mbox{ and } \, w_{ij}=w_{x_i x_j} = \partial_{ij} w $$
for partial derivatives in $\mathbb{R}^n$.
As introduced after Theorem \ref{thmcone}, we will use the tangential derivatives
\begin{equation*}
\de_i := \partial_i - \nu^i \nu^k \partial_k
\end{equation*}
for $i=1, \dots, n$, and thus
$$
\de_i w = w_i - \nu^i \nu^k w_k,
$$
where we adopted the summation convention over repeated indices.
Finally, recall the Laplace-Beltrami operator defined in \eqref{def:Laplace-Beltrami}:
\begin{equation*}
\De_{LB} \, := \de_i \de_i.
\end{equation*}
\end{notations}
\begin{remark}
Since
\begin{equation}\label{equazione:nu2}
1= | \nu |^2 = \sum_{k=1}^n d_k^2 ,
\end{equation}
it holds that
$$
d_{jk} d_k =0 \, \mbox{ for } j=1, \dots,n.
$$
Thus, we have
$$
\de_i \nu^j= \de_i d_j = d_{ij} - d_i d_k d_{kj}= d_{ij}=d_{ji},
$$
which leads to
\begin{equation*}
\de_i \nu^j= \de_j \nu^i .
\end{equation*}
\end{remark}
\begin{exercise}
Using $\de_i \nu^j = d_{ij}$, verify that
\begin{equation*}
{\mathcal H}= \de_i \nu^i ,
\end{equation*}
\begin{equation}
\label{eq:def c indici}
c^2= \de_i \nu^j \de_j \nu^i = \sum_{i,j=1}^n (\de_i \nu^j)^2 .
\end{equation}
\end{exercise}
The identities
\begin{equation*}
\nu^i \de_i = 0 ,
\end{equation*}
\begin{equation}
\label{eq:utileperSimonslemmaproof}
\nu^i \de_j \nu^i = 0 , \, \mbox{ for } j=1,\dots,n ,
\end{equation}
will be used often in the following computations. The first one follows from the definition of $\de_i$, while the second is immediate from \eqref{equazione:nu2}.
The next lemma will be useful in what follows.
\begin{lemma}
The following equations hold for every smooth hypersurface $\partial E$:
\begin{equation}
\label{eq:Giusti 1}
\de_i \de_j = \de_j \de_i + (\nu^i \de_j \nu^k - \nu^j \de_i \nu^k)\de_k ,
\end{equation}
\begin{equation}
\label{eq:Giusti 2}
\De_{LB} \, \nu^j + c^2 \nu^j = \de_j {\mathcal H} \, (=0 \, \mbox{ if $\partial E$ is stationary}) ,
\end{equation}
for all indices $i$ and $j$.
\end{lemma}
For a proof of this lemma, see \cite[Lemma 10.7]{G}.
Equation \eqref{eq:Giusti 2} is an important one. It says that the normal vector $\nu$ to a minimal surface solves the Jacobi equation $\left( \De_{LB} \, + c^2 \right) \nu \equiv 0$ on $\partial E$. This reflects the invariance of the equation ${\mathcal H} = 0$ by translations (to see this, write a perturbation made by a small translation as a normal deformation, as in Figure \ref{fig:2}).
If $\partial E$ is stationary, from \eqref{eq:Giusti 1} and by means of simple calculations, one obtains that
\begin{equation}
\label{eq:Giusti 3}
\De_{LB} \, \de_k = \de_k \De_{LB} \, - 2 \nu^k (\de_i \nu^j) \de_i \de_j - 2 (\de_k \nu^j)(\de_j \nu^i) \de_i .
\end{equation}
Equation \eqref{eq:Giusti 3} is the formula with the missed label (10.18) in \cite{G}.
We are ready now to give the
\begin{sketch}[of Lemma \ref{lmsimons}]
By \eqref{eq:def c indici} we can write that
$$
\frac{1}{2} \De_{LB} \, c^2 = (\de_i \nu^j) \De_{LB} \, \de_i \nu^j + \sum_{i,j,k} (\de_k \de_i \nu^j)^2.
$$
Then, using \eqref{eq:Giusti 2}, \eqref{eq:Giusti 3}, and the fact ${\mathcal H}=0$, we have
$$
\frac{1}{2} \De_{LB} \, c^2 = - (\de_i \nu^j) \de_i (c^2 \nu^j) - 2 (\de_i \nu^j) (\de_k \nu^l) (\de_l \nu^j) (\de_i \nu^k) + \sum_{i,j,k} (\de_k \de_i \nu^j)^2 ,
$$
and by \eqref{eq:Giusti 1}
$$
\frac{1}{2} \De_{LB} \, c^2 = -c^4 - 2 \nu^i \nu^l (\de_j \de_l \nu^k) (\de_k \de_i \nu^j) + \sum_{i,j,k} (\de_k \de_i \nu^j)^2.
$$
Now, if $x_0 \in \partial E \setminus \{ 0 \}$, we can choose the $x_n$-axis to be the same direction as $\nu(x_0)$. Thus, $\nu(x_0)=(0,\dots,0,1)$ and at $x_0$ we have
$$
\nu^n=1 , \, \de_n=0 ,
$$
$$
\nu^{\alpha}=0 , \, \de_\alpha= \partial_\alpha \, \mbox{ for } \alpha=1,\dots,n-1 .
$$
Hence, computing from now on always at the point $x_0$, and by using \eqref{eq:Giusti 1} and \eqref{eq:utileperSimonslemmaproof}, we get
\begin{eqnarray*}
\frac{1}{2} \De_{LB} \, c^2 &=& -c^4 + \sum_{\alpha,\beta,\ga} (\de_\ga \de_\alpha \nu^\beta)^2 + 2 \sum_{\alpha, \ga} (\de_\ga \de_\alpha \nu^n)^2 - 2 \sum_{\alpha, \beta} (\de_\alpha \de_\beta \nu^n)^2
\\
&=& -c^4 + \sum_{\alpha,\beta,\ga} (\de_\ga \de_\alpha \nu^\beta)^2 ,
\end{eqnarray*}
where all the greek indices indicate summation from $1$ to $n-1$.
On the other hand, we have
$$
| \de c |^2 = \frac{1}{c^2} (\de_\alpha \nu^\beta) (\de_\ga \de_\alpha \nu^\beta) (\de_\sigma \nu^\tau) (\de_\ga \de_\sigma \nu^\tau) ,
$$
and hence
\begin{equation*}
\frac{1}{2} \De_{LB} \, c^2 + c^4 - | \de c |^2 = \frac{1}{2 c^2} \sum_{\alpha, \beta, \ga, \sigma, \tau} \left[ (\de_\sigma \nu^{\tau}) (\de_\ga \de_\alpha \nu^\beta) - (\de_\alpha \nu^\beta) (\de_\ga \de_\sigma \nu^\tau) \right]^2 .
\end{equation*}
Now remember that $\partial E$ is a cone with vertex at the origin, and thus
$<x, \nu> = 0$ on $\partial E$.
Since we took $\nu= (0, \dots , 0 , 1)$ at $x_0$, we may choose coordinates in such a way that $x_0$ lies on the $(n-1)$-axis. In particular, $\nu^{n-1} = 0$ at $x_0$ and
$$
0 = \de_i <x, \nu> = <\de_i x, \nu>+<x, \de_i \nu> = <x, \de_i \nu>,
$$
which leads to
$$
\de_i \nu^{n-1} = 0 \, \mbox { at } \, x_0 .
$$
If now the letters $A,B,S,T$ run from $1$ to $n-2$, he have
\begin{equation*}
\begin{split}
\frac{1}{2} \De_{LB} \, c^2 + c^4 - | \de c |^2 = & \frac{1}{2 c^2} \sum_{A,B,S,T, \ga} \left[ (\de_S \nu^T) (\de_\ga \de_A \nu^B) - (\de_A \nu^B) (\de_\ga \de_S \nu^T) \right]^2
\\
& + \frac{2}{c^2} \sum_{S,T, \ga, \alpha} (\de_S \nu^T)^2 (\de_\ga \de_{n-1} \nu^{\alpha})^2 \ge 2 \sum_{\alpha, \ga} ( \de_\ga \de_{n-1} \nu^{\alpha} )^2.
\end{split}
\end{equation*}
From \eqref{eq:Giusti 1}, $\de_i \de_{n-1} = \de_{n-1} \de_i $ and $\de_{n-1} = \partial_{n-1} = \pm \left( x^j /|x| \right) \partial_j$ at $x_0$.
Since $\partial E$ is a cone, $\nu$ is homogeneous of degree 0 and hence $\de_i \nu^\alpha$ is homogeneous of degree $-1$. Thus, by Euler's theorem on homogeneous functions, we have
$$
\de_{n-1} \de_i \nu^{\alpha} = \pm \frac{x^j \partial_j }{|x|} \de_i \nu^{\alpha} = \mp \frac{1}{|x|} \de_i \nu^{\alpha} ,
$$
and hence
$$
2 \sum_{i, \alpha } ( \de_i \de_{n-1} \nu^{\alpha} )^2 = \frac{2}{ | x |^2} \sum _{i, \alpha} (\de_i \nu^\alpha)^2 = \frac{2 c^2}{| x |^2} .
$$
The proof is now completed.
\qed
\end{sketch}
\subsection{Comments on: harmonic maps, free boundary problems, and nonlocal minimal surfaces}
Here we briefly sketch arguments and results similar to the previous ones on minimal surfaces, now for three other elliptic problems.
\subsubsection{Harmonic maps}
Consider the energy
\begin{equation}
\label{eq:harm_enrg}
E(u)=\frac{1}{2}\int_{\Omega}|Du|^2dx
\end{equation}
for $H^1$ maps $u:\Omega\subset\mathbb{R}^n \to \overline{S^N_+} $ from a domain
$\Omega$ of $\mathbb{R}^n$ into the closed upper hemisphere
$$\overline{S^N_+}=\{y\in \mathbb{R}^{N+1}: |y|=1, y_{N+1}\ge 0\}.$$
A critical point of $E$ is called a (weakly) {\it harmonic map}.
When a map minimizes $E$ among all maps with values into $\overline{S^N_+}$ and with same boundary
values on $\partial \Omega$, then it is called a
minimizing harmonic map.
From the energy \eqref{eq:harm_enrg} and the restriction $|u| \equiv 1$, one finds that the equation for harmonic maps is given by
\begin{equation*}
-\Delta u=|Du|^2u \qquad\text{in }\Omega.
\end{equation*}
In 1983, J\"ager and Kaul proved the following theorem, that we state here without proving it (see the original paper \cite{JK} for the proof).
\begin{theorem}[J\"ager-Kaul~\cite{JK}]\label{thm:equator}
The equator map
\begin{equation*}
u_*:B_1\subset\mathbb{R}^n \rightarrow \overline{S^n_+},
\quad x\mapsto \left(x/|x|,0\right)
\end{equation*}
is a minimizing harmonic map on the class
$$
{\mathcal C}=\{u\in H^1(B_1\subset\mathbb{R}^n,S^n) : u=u_* \text{ on }\partial B_1\}
$$
if and only if $n\ge 7$.
\end{theorem}
We just mention that the proof of the ``if" in Theorem \ref{thm:equator} uses a calibration argument.
Later, Giaquinta and Sou\v cek \cite{GiaSouLN}, and independently Schoen and Uhlenbeck \cite{SU2}, proved the following result.
\begin{theorem}[Giaquinta-Sou\v cek \cite{GiaSouLN}; Schoen-Uhlenbeck \cite{SU2}]\label{thm:reg_harm}
\hfill\\
Let $u:B_1\subset\mathbb{R}^n\to \overline{S^N_+}$ be a minimizing harmonic map,
homogeneous of degree zero,
into the closed upper hemisphere $\overline{S^N_+}$. If $3\le n \le 6$,
then $u$ is constant.
\end{theorem}
Now we will show an outline of the proof of Theorem~\ref{thm:reg_harm} following~\cite{GiaSouLN}. More details can also be found in Section 3 of \cite{CabCapThree}. This theorem gives an alternative proof of one part of the statement of Theorem~\ref{thm:equator}. Namely, that the equator map $u_*$
is not minimizing for $3\le n\le 6$.
\begin{sketch}[of Theorem~\ref{thm:reg_harm}]
After stereographic projection (with respect to the south pole)
$P$ from $S^{N}\subset\mathbb{R}^{N+1}$ to $\mathbb{R}^N$, for the new
function $v=P\circ u: B_1\subset\mathbb{R}^n\to\mathbb{R}^N$,
the energy \eqref{eq:harm_enrg} (up to a constant factor)
is given by
\begin{equation*}
E(v):=\int_{B_1}\frac{|Dv|^2}{(1+|v|^2)^2}dx.
\end{equation*}
In addition, we have $|v|\le 1$ since the image of $u$ is contained
in the closed upper hemisphere.
By testing the function
$$
\xi(x)=v(x)\eta(|x|),
$$
where
$\eta$ is a smooth radial function with compact support in $B_1$, in the equation of the first variation of the energy, that is
\begin{equation*}
\delta E(v) \xi=0 ,
\end{equation*}
one can deduce that either $v$ is constant (and
then the proof is finished) or
$$
|v|\equiv 1 ,
$$
that we assume from now on.
Since $v$ is a minimizer, we have that the second variation of the energy satisfies
$$\delta^2 E(v) (\xi,\xi) \ge 0 .$$
By choosing here the function
$$
\xi(x)=v(x)|Dv(x)|\eta(|x|),
$$
where
$\eta$ is a smooth radial function with compact support in $B_1$
(to be chosen later), and setting
$$
c(x):=|Dv(x)|,
$$
one can conclude the proof by similar arguments as in the previous section and by using Lemma \ref{lem:harm}, stated next.
\qed
\end{sketch}
\begin{lemma}\label{lem:harm}
If $v$ is a harmonic map, homogeneous of degree
zero, and with $|v|\equiv 1$, we have
\begin{equation*}
\frac{1}{2}\Delta c^2-|Dc|^2 + c^4 \geq \frac{c^2}{|x|^2}+\frac{c^4}{n-1},
\end{equation*}
where $c:=|Dv|$.
\end{lemma}
This lemma is the analogue result of Lemma~\ref{lmsimons} for
minimal cones. See \cite{GiaSouLN} for a proof of the lemma,
which also follows from Bochner identity (see \cite{SU2}).
\subsubsection{Free boundary problems}
Consider the one-phase free boundary problem:
\begin{equation}\label{000:eq:freboundpb}
\left\{
\begin{array}{rcll}
\Delta u &=& 0 & \quad\mbox{in } E \\
u &=& 0 & \quad\mbox{on } \partial E \\
| \nabla u | &=& 1 & \quad\mbox{on } \partial E \setminus \lbrace 0 \rbrace ,\\
\end{array}\right.
\end{equation}
where $u$ is homogeneous of degree one and positive in the domain $E \subset \mathbb{R}^n$ and $\partial E$ is a cone.
We are interested in solutions $u$ that are stable for the Alt-Caffarelli energy functional
\begin{equation*}
E_{B_1}(u) = \int_{B_1} \left\lbrace | \nabla u |^2 + \mathbbm{1}_{ \left\lbrace u> 0 \right\rbrace } \right\rbrace dx
\end{equation*}
with respect to compact domain deformations that do not contain the origin. More precisely, we say that $u$ is {\it stable} if for any smooth vector field $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with $0 \notin {\rm supp} \Psi \subset B_1$ we have
\begin{equation*}
\left.\frac{d^2}{dt^2} E_{B_1} \left(u \left(x + t \Psi(x) \right) \right) \right|_{t=0} \ge 0.
\end{equation*}
The following result due to Jerison and Savin is contained in \cite{JS}, where a detailed proof can be found.
\begin{theorem}[Jerison-Savin \cite{JS}]\label{JSfreebp}
The only stable, homogeneous of degree one, solutions of \eqref{000:eq:freboundpb} in dimension $n\le 4$ are the one-dimensional solutions $u=(x \cdot \nu )^+$, $\nu \in S^{n-1}$.
\end{theorem}
In dimension $n=3$ this result had been established by Caffarelli, Jerison, and Kenig \cite{CJK}, where they conjectured that it remains true up to dimension $n \le 6$.
On the other hand, in dimension $n=7$, De Silva and Jerison \cite{DJ} provided an example of a nontrivial minimizer.
The proof of Jerison and Savin of Theorem \ref{JSfreebp} is similar to Simons proof of the rigidity of stable minimal cones in low dimensions: they find functions $c$ (now involving the second derivatives of $u$) which satisfy appropriate differential inequalities for the linearized equation.
Here, the linearized problem is the following:
$$
\left\{
\begin{array}{rcll}
\Delta v &=& 0 & \quad\mbox{in } E \\
v_{\nu} + {\mathcal H} v &=& 0 & \quad\mbox{on } \partial E \setminus \left\lbrace 0 \right\rbrace . \\
\end{array}\right.
$$
For the function
$$
c^2= \Vert D^2 u \Vert^2 = \sum_{i,j =1}^n u_{ij}^2 ,
$$
they found the following interior inequality which is similar to the one of the Simons lemma:
$$\frac{1}{2} \Delta c^2 - |\nabla c|^2 \geq 2 \, \frac{n-2}{n-1} \, \frac{c^2}{|x|^2}+ \frac{2}{n-1} \, |\nabla c|^2.$$
In addition, they also need to prove a boundary inequality involving $c_\nu$.
Furthermore, to establish Theorem \ref{JSfreebp} in dimension $n=4$, a more involved function $c$ of the second derivatives of $u$ is needed.
\subsubsection{Nonlocal minimal surfaces}
Nonlocal minimal surfaces, or $\alpha$-minimal surfaces (where $\alpha \in (0,1)$), have been introduced in 2010 in the seminal paper of Caffarelli, Roquejoffre, and Savin \cite{CRS}. These surfaces are connected to fractional perimeters and diffusions and, as $\alpha \nearrow 1$, they converge to classical minimal surfaces.
We refer to the lecture notes \cite{CF} and the survey \cite{DipVal}, where more references can be found.
For $\alpha$-minimal surfaces and all $\alpha \in (0,1)$, the analogue of Simons flatness result is only known in dimension $2$ by a result for minimizers of Savin and Valdinoci \cite{SV}.
\section{The Allen-Cahn equation}\label{ALLENCA}
This section concerns the {\it Allen-Cahn equation}
\begin{equation}\label{AlCa}
- \Delta u= u - u^3 \quad \mbox{in $\mathbb{R}^{n}$}.
\end{equation}
By using equation \eqref{AlCa} and the maximum principle it can be proved that any solution satisfies
$|u| \le 1$. Then, by the strong maximum principle we have that either $|u|<1$ or $u \equiv \pm 1$.
Since $u \equiv \pm 1$ are trivial solutions, from now on we consider $u:\mathbb{R}^{n}\to (-1,1)$.
We introduce the class of {\it 1-d solutions}:
$$
u(x)= u_* (x \cdot e) \quad \mbox{ for a vector } \, e \in \mathbb{R}^n, |e|=1,
$$
where
$$
u_* (y)= \tanh\left(\frac{y}{\sqrt{2}} \right).
$$
The solution $u_*$ is sometimes referred to as the {\it layer solution} to \eqref{AlCa};
see Figure \ref{fig:8}.
The fact that $u$ depends only on one variable can be rephrased also by saying that all the level sets
$\{u=s\}$ of $u$ are
hyperplanes.
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE8}
\caption{The increasing, or layer, solution to the Allen-Cahn equation}
\label{fig:8}
\end{figure}
\begin{exercise}
Check that the $1$-d functions introduced above are solutions of the Allen-Cahn equation.
\end{exercise}
\begin{remark}\label{rem:1dthenconditions}
Let us take $e= e_n = (0, \dots, 0,1)$ and consider the 1-d solution
$u(x)=u_* (x_n) = \tanh ( x_n / \sqrt{2} )$. It is clear that the following two relations hold:
\begin{equation}\label{monoton}
u_{x_n} >0 \quad \mbox{in }\mathbb{R}^{n} ,
\end{equation}
\begin{equation}\label{limiting}
\lim_{x_n\rightarrow\pm\infty}u(x', x_n)=\pm 1
\quad \text{ for all }x'\in \mathbb{R}^{n-1}.
\end{equation}
\end{remark}
The energy functional associated to equation \eqref{AlCa} is
\begin{equation*}
E_\Omega (u):=\int_\Omega \left\{ \frac{1}{2} |\nabla u|^2
+ G(u)\right\} dx ,
\end{equation*}
where $G$ is the double-well potential in Figure \ref{fig:7}:
\begin{equation*}
G(u)= \frac{1}{4} \left( 1- u^2 \right)^2.
\end{equation*}
\begin{definition}[Minimizer]\label{definit:MinimizAC}
A function $u:\mathbb{R}^{n}\to (-1,1)$ is said to be a {\it minimizer} of \eqref{AlCa}
when
$$E_{B_R} (u) \leq E_{B_R} (v)$$
for every open ball $B_R$ and functions $v: \overline{B_R}\to \mathbb{R}$ such that $v\equiv u$ on
$\partial B_R$.
\end{definition}
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE7}
\caption{The double-well potential in the Allen-Cahn energy}
\label{fig:7}
\end{figure}
\smallskip
\noindent
{\bf Connection with the theory of minimal surfaces.}
The Allen-Cahn equation has its origin in the theory of phase transitions and it is used as a model for some nonlinear reaction-diffusion processes.
To better understand this, let $\Omega\subset \mathbb{R}^n$ be a bounded domain, and consider the Allen-Cahn equation with parameter $\varepsilon > 0$,
\begin{equation}\label{eq:Allen-Cahnwithparameter}
- \varepsilon^2 \Delta u= u - u^3 \, \mbox{ in } \, \Omega ,
\end{equation}
with associated energy functional given by
\begin{equation}\label{eq:energyACwithparameter}
E_{\varepsilon}(u)= \int_{\Omega} \left\lbrace \frac{\varepsilon}{2} \, | \nabla u|^2 + \frac{1}{\varepsilon} \, G(u) \right\rbrace dx .
\end{equation}
Assume now that there are two populations (or chemical states) A and B and that $u$ is a density measuring the percentage of the two populations at every point: if $u(x) = 1$ (respectively, $u(x)= -1$) at a point $x$, we have only population A at $x$ (respectively, population B); $u(x)=0$ means that at $x$ we have $50\%$ of population A and $50\%$ of population $B$.
By \eqref{eq:energyACwithparameter}, it is clear that in order to minimize $E_{\varepsilon}$ as $\varepsilon$ tends to $0$, $G(u)$ must be very small. From Figure \ref{fig:7} we see that this happens when $u$ is close to $\pm 1$. These heuristics are indeed formally confirmed by a celebrated theorem of Modica and Mortola. It states that, if $u_\varepsilon$ is a family of minimizers of $E_\varepsilon$,
then, up to a subsequence, $u_\varepsilon$ converges in $L^1_{\mathrm{loc}} (\Omega)$, as $\varepsilon$ tends to $0$,
to
$$
u_0 = \mathbbm{1}_{ \Omega_+ } - \mathbbm{1}_{ \Omega_- }
$$
for some disjoint sets $\Omega_{\pm}$ having as common boundary a surface $\Gamma$. In addition, $\Gamma$ is a minimizing minimal surface.
Therefore, the result of Modica-Mortola establishes that the two populations tend to their total separation, and in such a (clever) way that the interface surface $\Gamma$ of separation has least possible area.
Finally, notice that the $1$-d solution of \eqref{eq:Allen-Cahnwithparameter},
$$
x \mapsto u_* (\frac{x \cdot e}{\varepsilon}) ,
$$
makes a very fast transition from $-1$ to $1$ in a scale of order $\varepsilon$. Accordingly, in Figure \ref{fig:add1}, $u_\varepsilon$ will make this type of fast transition across the limiting minimizing minimal surface $\Gamma$.
The interested reader can see \cite{AAC} for more details.
\smallskip
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{ADDITIONALFIGURE1potato}
\caption{The zero level set of $u_\varepsilon$, the limiting function $u_0$, and the minimal surface $\Gamma$}
\label{fig:add1}
\end{figure}
\subsection{Minimality of monotone solutions with limits \texorpdfstring{$\pm 1$}{+-1}}
The following fundamental result shows that monotone solutions with limits $\pm 1$ are minimizers (as in Definition \ref{definit:MinimizAC}).
\begin{theorem}[Alberti-Ambrosio-Cabr\'e \cite{AAC}]\label{alba}
Suppose that $u$ is a solution of \eqref{AlCa} satisfying the monotonicity hypothesis \eqref{monoton} and the condition \eqref{limiting} on limits.
Then, $u$ is a minimizer of \eqref{AlCa} in $\mathbb{R}^n$.
\end{theorem}
See \cite{AAC} for the original proof of the Theorem~\ref{alba}. It uses a calibration built from a foliation and avoids the use of the strong maximum principle, but it is slightly involved. Instead, the simple proof that we give here was suggested to the first author (after one of his lectures on \cite{AAC}) by L.~Caffarelli.
It uses a simple foliation argument together with the strong maximum principle, as in the alternative proof of Theorem \ref{thm:SimCone} given in Subsection \ref{subsec 1.1:MinimalitySimcones}.
\begin{proof}[Proof of Theorem \ref{alba}]
Denoting $x= (x' ,x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, let us consider the functions
$$
u^t (x):= u(x', x_n +t), \, \mbox{ for } \, t\in\mathbb{R}.
$$
By the monotonicity assumption \eqref{monoton} we have that
\begin{equation}\label{eq:monfoliation}
u^t < u^{t'} \, \mbox{ in } \mathbb{R}^n , \, \mbox{ if } t<t'.
\end{equation}
Thus, by \eqref{limiting} we have that the graphs of $u^t= u^t (x)$, $t \in \mathbb{R}$,
form a foliation filling all of $\mathbb{R}^n \times (-1,1)$.
Moreover, we have that for every $t \in \mathbb{R}$, $u^t$ are solutions of $-\Delta u^t =u^t - (u^t)^3$ in $\mathbb{R}^n$.
By simple arguments of the Calculus of Variations, given a ball $B_R$ it can be proved that there exists
a minimizer $v: \overline{B}_R \to \mathbb{R}$ of $E_{B_R}$ such that $v=u$ on $\partial B_R$. In particular, $v$ satisfies
\begin{equation*}
\left\{
\begin{array}{rcll}
- \Delta v &=& v - v^3 & \quad\mbox{in } B_R \\
|v| &<& 1 & \quad\mbox{in } \overline{B}_R \\
v &=& u & \quad\mbox{on } \partial B_R .\\
\end{array}\right.
\end{equation*}
By \eqref{limiting}, we have that the graph of $u^t$ in the compact set $\overline{B}_R$ is above the graph of $v$ for $t$ large enough, and it is below the graph of $v$ for $t$ negative enough
(see Figure~\ref{fig:9}).
If $v \not\equiv u$, assume that $v<u$ at some point in $B_R$ (the situation $v>u$ somewhere in $B_R$ is done similarly). It follows that, starting from $t= - \infty$, there will exist a first $t_* <0$
such that $u^{t_*}$ touches $v$ at a point $P \in \overline{B}_R$. This means that
$u^{t_*} \le v$ in $\overline{B}_R$ and $u^{t_*} (P) = v(P)$.
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE9}
\caption{The foliation $\lbrace u^t \rbrace$ and the minimizer $v$}
\label{fig:9}
\end{figure}
By \eqref{eq:monfoliation}, $t_* <0$, and the fact that $v=u = u^0$ on $\partial B_R$, the point $P$ cannot belong to
$\partial B_R$. Thus, $P$ will be an interior point of $B_R$.
But then we have that $u^{t_*}$ and $v$ are two solutions of the same semilinear equation (the Allen-Cahn equation), the graph of $u^{t_*}$ stays below that of $v$, and they touch each other at the interior point $\left( P, v(P) \right) $. This is a contradiction with the strong maximum principle.
Here we leave as an exercise (stated next) to verify that the difference of two solutions of $- \Delta u = f(u)$ satisfies a linear elliptic equation to which we can apply the strong maximum principle.
This leads to $u^{t_*} \equiv v$, which contradicts $u^{t_*} < v = u^0$ on $\partial B_R$.
\end{proof}
\begin{exercise}
Prove that the difference $w:= v_1 - v_2$ of two solutions of a semilinear equation $- \Delta v= f(v)$, where $f$ is a Lipschitz function, satisfies a linear equation of the form $\Delta w + c(x) \, w = 0$, for some function $c \in L^{\infty}$. Verify that, as a consequence, this leads to $u^{t_*} \equiv v$ in the previous proof.
\end{exercise}
By recalling Remark \ref{rem:1dthenconditions}, we immediately get the following corollary.
\begin{corollary}
The 1-d solution $u(x)=u_* (x \cdot e)$ is a minimizer of \eqref{AlCa} in $\mathbb{R}^n$, for every unit vector $e \in \mathbb{R}^n$.
\end{corollary}
As a corollary of Theorem \ref{alba}, we easily deduce the following important energy estimates.
\begin{corollary}[Energy upper bounds; Ambrosio-Cabr\'e \cite{ambcab}]\label{tupper}
Let $u$ be a solution of \eqref{AlCa} satisfying
\eqref{monoton} and \eqref{limiting} (or more generally, let $u$ be a minimizer in $\mathbb{R}^n$).
Then, for all $R \ge 1$ we have
\begin{equation}\label{upperbou}
E_{B_R} (u) \leq C R^{n-1}
\end{equation}
for some constant $C$ independent of $R$.
In particular, since $G \ge 0$, we have that
$$
\int_{B_R} |\nabla u|^2 \,dx
\leq C R^{n-1}
$$
for all $R \ge 1$.
\end{corollary}
\begin{remark}
The proof of Corollary \ref{tupper} is trivial for $1$-d solutions. Indeed, it is easy to check that $\int_{- \infty}^{+ \infty} \left\lbrace \frac{1}{2} (u_* ')^2 + \frac{1}{4}(1- u_*^2)^2 \right\rbrace dy < \infty$ and, as a consequence, by applying Fubini's theorem on a cube larger than $B_R$, that \eqref{upperbou} holds.
This argument also shows that the exponent $n-1$ in \eqref{upperbou} is optimal (since it cannot be improved for $1$-d solutions).
\end{remark}
The estimates in Corollary \ref{tupper} are fundamental in the proofs of a conjecture of De Giorgi that we treat in the next subsection.
The estimate \eqref{upperbou} was first proved by Ambrosio and the first author in \cite{ambcab}. Later on, in \cite{AAC} Alberti, Ambrosio, and the first author discovered that monotone solutions with limits are minimizers (Theorem \ref{alba} above). This allowed to simplify the original proof of the energy estimates found in \cite{ambcab}, as follows.
\begin{proof}[Proof of Corollary \ref{tupper}]
Since $u$ is a minimizer by Theorem \ref{alba} (or by hypothesis), we can
perform a simple energy comparison argument. Indeed, let
$\phi_R\in C^\infty(\mathbb{R}^{n})$ satisfy $0\le\phi_R\le 1$ in $\mathbb{R}^{n}$,
$\phi_R\equiv 1$ in $B_{R-1}$,
$\phi_R\equiv 0$ in $\mathbb{R}^{n}\setminus B_R$, and
$\Vert\nabla\phi_R\Vert_\infty\leq 2$. Consider
$$
v_R:=(1-\phi_R) u+\phi_R.
$$
Since $v_R \equiv u$ on $\partial B_R$, we can compare
the energy of $u$ in $B_R$ with that of $v_R$.
We obtain
\begin{multline*}
\int_{B_R}\left\{\frac{1}{2} |\nabla u|^2+ G(u) \right\}dx \leq
\int_{B_R}\left\{\frac{1}{2}|\nabla v_R|^2+ G(v_R) \right\}dx
\\ =\int_{B_R\setminus B_{R-1}}
\left\{\frac{1}{2}|\nabla v_R|^2+ G(v_R) \right\}dx \leq
C|B_R\setminus B_{R-1}| \le CR^{n-1}
\end{multline*}
for every $R \ge 1$, with $C$ independent of $R$. In the second inequality of the chain above we used that $\frac{1}{2}|\nabla v_R|^2+ G(v_R) \le C$ in $B_R\setminus B_{R-1}$ for some constant $C$ independent
of~$R$. This is a consequence of the following exercise.
\end{proof}
\begin{exercise}
Prove that if $u$ is a solution of a semilinear equation $- \Delta u = f(u)$ in $\mathbb{R}^n$ and $|u| \le 1$ in $\mathbb{R}^n$, where $f$ is a continuous nonlinearity, then $|u| + | \nabla u| \le C$ in $\mathbb{R}^n$ for some constant $C$ depending only on $n$ and $f$. See \cite{ambcab}, if necessary, for a proof.
\end{exercise}
\subsection{A conjecture of De Giorgi}
In 1978, E. De Giorgi \cite{DG} stated the following conjecture:
\smallskip\noindent
{\em
{\rm {\bf Conjecture} (DG).}
Let $u:\mathbb{R}^{n}\to (-1,1)$ be a solution of the Allen-Cahn equation \eqref{AlCa}
satisfying the monotonicity condition \eqref{monoton}.
Then, $u$ is a $1$-d solution (or equivalently, all level sets $\{u=s\}$ of $u$ are
hyperplanes), at least if $n\leq 8$.
}
\smallskip
This conjecture was proved in 1997 for $n=2$ by Ghoussoub and Gui \cite{GG}, and in 2000 for $n=3$ by
Ambrosio and Cabr\'e \cite{ambcab}.
Next we state a deep result of Savin \cite{Sa} under the only assumption of minimality. This is the semilinear analogue of Simons Theorem \ref{thmcone} and Remark \ref{remarkchemiserve:1.17} on minimal surfaces. As we will see, Savin's result leads to a proof of Conjecture (DG) for $n \le 8$ if the additional condition \eqref{limiting} on limits is assumed.
\begin{theorem}[Savin \cite{Sa}]\label{teosavin}
Assume that $n\leq 7$ and that $u$ is a minimizer of \eqref{AlCa} in~$\mathbb{R}^n$.
Then, $u$ is a $1$-d solution.
\end{theorem}
The hypothesis $n\leq 7$ on its statement is sharp.
Indeed, in 2017 Liu, Wang, and Wei \cite{LWW} have shown the existence of a minimizer in $\mathbb{R}^8$ whose level sets are not hyperplanes. Its zero level set is asymptotic at infinity to the Simons cone. However, a canonical solution described in Subsection \ref{subsec:saddlesha} (and whose zero level set is exactly the Simons cone) is still not known to be a minimizer in $\mathbb{R}^8$.
Note that Theorem \ref{teosavin} makes no assumptions on the monotonicity or
the limits at infinity of the solution. To prove Conjecture (DG) using Savin's result
(Theorem~ \ref{teosavin}), one needs to make the further assumption \eqref{limiting} on the limits only to guarantee, by Theorem~\ref{alba}, that the solution is actually a minimizer.
Then, Theorem~\ref{teosavin} (and the gain of one more dimension, $n=8$, thanks to the monotonicity of the solution)
leads to the proof of Conjecture (DG) for monotone solutions with limits $\pm 1$.
However, for $4\leq n\leq 8$ the conjecture in its original statement (i.e., without the limits $\pm 1$ as hypothesis) is still open.
To our knowledge no clear evidence is known about its validity
or not.
The proof of Theorem \ref{teosavin} uses an improvement of flatness result for the Allen-Cahn equation developed by Savin, as well as Theorem \ref{thmcone} on the non-existence of stable minimal cones in dimension $n \le 7$.
Instead, the proofs of Conjecture (DG) in dimensions $2$ and $3$ are much simpler. They use the energy estimates of Corollary \ref{tupper} and a Liouville-type theorem developed in \cite{ambcab} (see also \cite{AAC}). As explained next, the idea of the proof originates in the paper \cite{BCN} by Berestycki, Caffarelli, and Nirenberg.
\smallskip
\noindent
{\bf Motivation for the proof of Conjecture (DG) for $n \le 3$.}
In \cite{BCN} the authors made the following heuristic observation. From the equation
$- \Delta u = f(u)$ and the monotonicity assumption \eqref{monoton}, by differentiating we find that
\begin{equation}
\label{eq:remforseno}
u_{x_n}>0 \, \mbox{ and } \, L u_{x_n} := \left( - \Delta - f'(u) \right) u_{x_n}=0 \, \mbox{ in } \mathbb{R}^n .
\end{equation}
If we were in a bounded domain $\Omega\subset \mathbb{R}^n$ instead of $\mathbb{R}^n$ (and we forgot about boundary conditions), from \eqref{eq:remforseno}, we would deduce that $u_{x_n}$ is the first eigenfunction of $L$ and that its first eigenvalue is $0$. As a consequence, such eigenvalue is simple. But then, since we also have that
$$
L u_{x_i} = \left( - \Delta - f'(u) \right) u_{x_i}=0 \quad \mbox{for } \, i=1, \dots, n-1,
$$
the simplicity of the eigenvalue would lead to
\begin{equation}\label{eq:remforseno2}
u_{x_i}= c_i u_{x_n} \quad \mbox{for } \, i=1, \dots, n-1 ,
\end{equation}
where $c_i$ are constants. Now, we would conclude that $u$ is a 1-d solution, by the following exercise.
\begin{exercise}
Check that \eqref{eq:remforseno2}, with $c_i$ being constants, is equivalent to the fact that $u$
is a 1-d solution.
\end{exercise}
To make this argument work in the whole $\mathbb{R}^n$, one needs a Liouville-type theorem. For $n=2$ it was proved in \cite{BCN} and \cite{GG}. Later, \cite{ambcab} used it to prove Conjecture (DG) in $\mathbb{R}^3$ after proving the crucial energy estimate \eqref{upperbou}. The Liouville theorem requires the right hand side of \eqref{upperbou} to be bounded by $C R^2= C R^{3-1}$.
\smallskip
In 2011, del Pino, Kowalczyk, and Wei \cite{dP} established
that Conjecture~(DG) does not hold for $n\geq 9$
-- as suggested in De Giorgi's original statement.
\begin{theorem}[del Pino-Kowalczyk-Wei \cite{dP}]\label{thm:delP K W}
If $n \ge 9$, there exists a solution of \eqref{AlCa}, satisfying \eqref{monoton} and \eqref{limiting}, and which is not a $1$-d solution.
\end{theorem}
The proof in \cite{dP} uses crucially the minimal graph in $\mathbb{R}^9$ built by Bombieri, De Giorgi, and Giusti in \cite{BdGG}. This is a minimal surface in $\mathbb{R}^9$ given by the graph of a function $\phi: \mathbb{R}^8 \to \mathbb{R}$ which is antisymmetric with respect to the Simons cone.
The solution of Theorem \ref{thm:delP K W} is built in such a way that its zero level set stays at finite distance from the Bombieri-De Giorgi-Giusti graph.
We consider next a similar object to the previous minimal graph, but in the context of the Allen-Cahn equation: a solution $u: (\mathbb{R}^8 =) \mathbb{R}^{2m} \to \mathbb{R}$ which is antisymmetric with respect to the Simons cone.
\subsection{The saddle-shaped solution vanishing on the Simons cone}\label{subsec:saddlesha}
As in Section \ref{sec:mincones}, let $m \ge 1$ and denote by ${\mathcal C}_S$
the Simons cone \eqref{def:simonscone}.
For $x=(x_1,\dots, x_{2m})\in\mathbb{R}^{2m}$, $s$ and $t$ denote the two
radial variables
\begin{equation}\label{coor}
s = \sqrt{x_1^2+...+x_m^2} \quad \mbox{ and } \quad t = \sqrt{x_{m+1}^2+...+x_{2m}^2}.
\end{equation}
The Simons cone is given by
$$
{\mathcal C}_S=\{s=t\}=\partial E, \quad\text{ where }
E =\{s>t\}.
$$
\begin{definition}[Saddle-shaped solution]\label{def:saddlesolution}
We say that $u:\mathbb{R}^{2m}\rightarrow\mathbb{R}$ is a {\it saddle-shaped
solution} (or simply a saddle solution) of the Allen-Cahn equation
\begin{equation}\label{eq2m}
-\Delta u= u- u^3 \quad {\rm in }\;\mathbb{R}^{2m}
\end{equation}
whenever $u$ is a
solution of
\eqref{eq2m} and, with $s$ and $t$ defined by \eqref{coor},
\renewcommand{\labelenumi}{$($\alph{enumi}$)$}
\begin{enumerate}
\item $u$ depends only on the variables $s$ and $t$. We write
$u=u(s,t)$;
\item $u>0$ in $E:=\{s>t\}$;
\item $u(s,t)=-u(t,s)$ in $\mathbb{R}^{2m}$.
\end{enumerate}
\end{definition}
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE10}
\caption{The saddle-shaped solution $u$ and the Simons cone ${\mathcal C}_S$}
\label{fig:10}
\end{figure}
\begin{remark}
Notice that if $u$ is a saddle-shaped solution, then we have $u=0$ on ${\mathcal C}_S$ (see Figure~\ref{fig:10}).
\end{remark}
While the existence of a saddle-shaped solution is easily established, its uniqueness is more delicate. This was accomplished in 2012 by the first author in \cite{Ca}.
\begin{theorem}[Cabr\'e \cite{Ca}] \label{unique}
For every even dimension $2m\geq 2$,
there exists a unique saddle-shaped solution $u$ of \eqref{eq2m}.
\end{theorem}
Due to the minimality of the Simons cone when $2m \ge 8$ (and also because of the minimizer from \cite{LWW} referred to after Theorem \ref{teosavin}), the saddle-shaped solution is expected to be a minimizer when $2m \ge 8$:
\begin{open problem}
Is the saddle-shaped solution a minimizer of \eqref{eq2m} in $\mathbb{R}^8$, or at least in higher even dimensions?
\end{open problem}
Nothing is known on this open problem except for the following result.
It establishes stability (a weaker property than minimality) for $2m \ge 14$. Below, we sketch its proof.
\begin{theorem}[Cabr\'e \cite{Ca}]\label{stab}
If $2m\geq 14$, the saddle-shaped solution $u$ of \eqref{eq2m} is stable in $\mathbb{R}^{2m}$, in the sense of the following definition.
\end{theorem}
\begin{definition}[Stability]
We say that a
solution $u$ of $- \Delta u = f(u)$ in $\mathbb{R}^n$ is {\it stable} if the second variation of the energy with respect to compactly
supported perturbations $\xi$ is nonnegative. That is, if
\begin{equation*}
\int_{{\mathbb{R}}^{n}}\left\{ |\nabla\xi|^2-f'(u)\xi^2\right\}dx\geq 0 \quad \mbox{ for all } \;\xi\in C^1_c(\mathbb{R}^{n}) .
\end{equation*}
\end{definition}
In the rest of this section, we will take $n=2m$ and $f$ to be the Allen-Cahn nonlinearity, i.e.,
$f (u) = u- u^3 .$
\begin{sketch}[of Theorem \ref{stab}]
Notice that
\begin{equation}\label{eqst}
u_{ss}+u_{tt}+(m-1){\Big (}\frac{u_s}{s}+\frac{u_t}{t}{\Big
)}+ f(u) =0 ,
\end{equation}
for $s>0$ and $t>0$,
is equation \eqref{eq2m} expressed in the
$(s,t)$ variables.
Let us introduce the function
\begin{equation}\label{0001:eq:deffi}
\varphi := t^{-b}u_s -s^{-b}u_t .
\end{equation}
Differentiating \eqref{eqst} with respect to $s$ (and to $t$), one finds equations satisfied by $u_s$ (and by $u_t$) -- and which involve a zero order term with coefficient $f'(u)$. These equations, together with some more delicate monotonicity properties of the saddle-shaped solution established in \cite{Ca}, can be used to prove the following fact.
For $2m\geq 14$, one can choose $b>0$ in \eqref{0001:eq:deffi} (see \cite{Ca} for more details) such that $\varphi$ is a positive supersolution of the linearized problem, i.e.:
\begin{equation}\label{posivar}
\varphi >0 \quad \text{in } \{st> 0\},
\end{equation}
\begin{equation}\label{superpf}
\{\Delta + f'(u)\} \varphi \leq 0 \quad \text{in } \mathbb{R}^{2m}\setminus\{st=0\}=\{st>0\} .
\end{equation}
Next, using \eqref{posivar} and \eqref{superpf}, we can verify the stability
condition of $u$ for any $C^1$ test function
$\xi = \xi(x)$ with compact support in $\{st>0\}$. Indeed, multiply \eqref{superpf}
by $\xi^2/\varphi$ and integrate by parts to get
\begin{eqnarray*}
\int_{\{st>0\}} f'(u)\,\xi^2 \, dx
& = &\int_{\{st>0\}} f'(u) \varphi\, \frac{\xi^2}{\varphi} \, dx \\
& \leq & \int_{\{st>0\}} -\Delta \varphi \, \frac{\xi^2}{\varphi} \, dx \\
& = &\int_{\{st>0\}}
\nabla \varphi \, \nabla\xi\, \frac{2\xi}{\varphi} \, dx
-\int_{\{st>0\}} \frac{|\nabla \varphi|^2}{\varphi^2}\, \xi^2 \, dx .
\end{eqnarray*}
Now, using the Cauchy-Schwarz inequality, we are led to
\begin{equation*}
\int_{\{st>0\}} f'(u)\,\xi^2 \, dx \leq \int_{\{st>0\}} |\nabla\xi|^2 \, dx .
\end{equation*}
Finally, by a cut-off argument we can prove that this same inequality holds also for every function $\xi \in C^1_c(\mathbb{R}^{2m})$.
\qed
\end{sketch}
\begin{remark}
Alternatively to the variational proof seen above, another way to establish stability from the existence of a positive supersolution to the linearized problem is by using the maximum principle (see \cite{BNV} for more details).
\end{remark}
\section{Blow-up problems}
In this final section, we consider positive solutions of the semilinear problem
\begin{equation}\label{pb}
\left\{
\begin{array}{rcll}
-\Delta u &=& f(u) & \quad\mbox{in } \Omega \\
u &>& 0 & \quad\mbox{in } \Omega \\
u &=& 0 & \quad\mbox{on } \partial\Omega ,\\
\end{array}\right.
\end{equation}
where $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain, $n\geq 1$, and $f: \mathbb{R}^+ \to \mathbb{R}$ is $C^1$.
The associated energy functional is
\begin{equation}\label{eq:energiaultimasec}
E_\Omega (u):=\int_\Omega \left\{ \frac{1}{2} |\nabla u|^2 - F(u) \right\} dx ,
\end{equation}
where $F$ is such that $F' = f$.
\subsection{Stable and extremal solutions. A singular stable solution for \texorpdfstring{$n \ge 10$}{n>=10}}
We define next the class of stable solutions to \eqref{pb}. It includes any local minimizer, i.e., any minimizer of \eqref{eq:energiaultimasec} under small perturbations vanishing on $\partial \Omega$.
\begin{definition}[Stability]
A solution $u$ of \eqref{pb} is said to be \emph{stable} if
the second variation of the energy with respect to $C^1$ perturbations $\xi$ vanishing on $\partial \Omega$ is nonnegative. That is, if
\begin{equation} \label{LN:stability}
\int_{\Omega} f'(u)\xi^2\,dx
\leq
\int_{\Omega} |\nabla\xi|^2\,dx \quad\textrm{for all } \xi \in C^1 (\overline{\Omega} ) \textrm{ with } \xi_{|\partial \Omega} \equiv 0.
\end{equation}
\end{definition}
There are many nonlinearities for which \eqref{pb} admits a (positive) stable solution.
Indeed, replace $f(u)$ by $\lambda f(u)$
in \eqref{pb}, with $\lambda\geq 0$:
\begin{equation}\label{pbla}
\left\{
\begin{array}{rcll}
-\Delta u &=& \lambda f(u) & \quad\mbox{in $\Omega$}\\
u &=& 0 & \quad\mbox{on $\partial\Omega$.}\\
\end{array}\right.
\end{equation}
Assume that $f$ is positive, nondecreasing, and superlinear at $+\infty$, that is,
\begin{equation} \label{hypf}
f(0) > 0, \quad f'\geq 0 \quad\textrm{and}\quad\lim_{t\to +\infty}\frac{f(t)}t=+\infty.
\end{equation}
Note that also in this case we look for positive solutions (when $\lambda>0$), since $f>0$.
We point out that, for $\lambda>0$, $u\equiv 0$ is not a solution.
\begin{proposition}\label{prop:extremal}
Assuming \eqref{hypf}, there exists an extremal parameter $\lambda^*\in (0,+\infty)$ such that if $0\le\lambda<\lambda^*$ then
\eqref{pbla} admits a minimal stable classical solution $u_\lambda$. Here ``minimal'' means the smallest among all the solutions, while ``classical'' means of class $C^2$. Being classical is a consequence of $u_\lambda \in L^{\infty} ( \Omega)$ if $\lambda < \lambda^*$.
On the other hand, if $\lambda>\lambda^*$ then \eqref{pbla} has no classical solution.
The family of classical solutions $\left\{ u_{\lambda }:0\le \lambda <\lambda ^{*}\right\}$
is increasing in $\lambda$, and its limit as $\lambda\uparrow\lambda^{*}$ is a weak solution $u^*=u_{\lambda^*}$ of \eqref{pbla} for $\lambda=\lambda^*$.
\end{proposition}
\begin{definition}[Extremal solution]
The function $u^*$ given by Proposition \ref{prop:extremal} is called \emph{the extremal solution} of \eqref{pbla}.
\end{definition}
For a proof of Proposition \ref{prop:extremal} see the book \cite{Dupaigne} by L. Dupaigne.
The definition of weak solution (the sense in which $u^*$ is a solution) requires $u^* \in L^1 (\Omega)$, $f(u^* ) \mathrm{dist}(\cdot, \partial \Omega) \in L^1 (\Omega)$, and the equation to be satisfied in the distributional sense after multiplying it by test functions vanishing on $\partial \Omega$ and integrating by parts twice (see \cite{Dupaigne}). Other useful references regarding extremal and stable solutions are \cite{Brezis}, \cite{Cextremal}, and \cite{CabBound}.
Since 1996, Brezis has raised several questions regarding stable and extremal solutions; see for instance \cite{Brezis}. They have led to interesting works, some of them described next. One of his questions is the following.
\medskip
\noindent
\textbf{Question} (Brezis). Depending on the dimension $n$ or on the domain $\Omega$, is the extremal solution $u^*$ of \eqref{pbla} bounded (and therefore classical) or is it unbounded?
More generally, one may ask the same question for the larger class of stable solutions to \eqref{pb}.
\medskip
\begin{figure}[htbp]
\centering
\includegraphics[scale=.25]{FIGURE11}
\caption{The family of stable solutions $u_\lambda$ and the extremal solution $u^*$}
\label{fig:11}
\end{figure}
The following is an explicit example of stable unbounded (or singular) solution.
It is easy to check that, for $n\geq 3$, the function $\tilde u=-2\log\vert{x}\vert$ is a solution of \eqref{pb} in $\Omega =B_1$,
the unit ball, for $f(u)= 2(n-2) e^u$.
Let us now consider the linearized operator at $\tilde u$, which is given by
$$
-\Delta -2(n-2)e^{\tilde u} =
-\Delta -\frac{2(n-2)}{\vert{x}\vert^2}.
$$
If $n\ge 10$, then its first Dirichlet eigenvalue in $B_1$ is nonnegative.
This is a consequence of {\it Hardy's inequality} \eqref{Hardyin}:
$$
\frac{(n-2)^2}{4}\int_{B_1}\frac{\xi^2}{\vert{x}\vert^2} \, dx \;\; \leq\;\;
\int_{B_1} \vert\nabla \xi\vert^2 dx
\qquad\quad
\mbox{ for every } \xi\in H^1_0(B_1),
$$
and the fact that $2(n-2)\leq (n-2)^2/4$ if $n\geq 10$.
Thus we proved the following result.
\begin{proposition}\label{prop:dimensopt}
For $n\geq 10$, $\tilde u = -2\log\vert{x}\vert$ is an $H^1_0(B_1)$ stable weak solution of $-\Delta u = 2 (n-2) e^u $ in $B_1$, $u>0$ in $B_1$, $u=0$ on $\partial B_1$.
\end{proposition}
Thus, in dimensions $n\geq 10$ there exist unbounded $H^1_0$ stable weak solutions of \eqref{pb},
even in the unit ball and for the exponential nonlinearity.
It is believed that $n \ge 10$ could be the optimal dimension for this fact, as we describe next.
\subsection{Regularity of stable solutions. The Allard and Michael-Simon Sobolev inequality}
The following results give $L^{\infty}$ bounds for stable solutions. To avoid technicalities we state the bounds for the extremal solution but, more generally, they also apply to every stable weak solution of \eqref{pb} which is the pointwise limit of a sequence of bounded stable solutions to similar equations (see \cite{Dupaigne}).
\begin{theorem}[Crandall-Rabinowitz \cite{CR}]
\label{thm:CranRab}
Let $u^*$ be the extremal solution of \eqref{pbla} with $f(u)=e^u$ or $f(u)=(1+u)^p$, $p>1$. If $n\le 9$, then $u^* \in L^\infty (\Omega)$.
\end{theorem}
\begin{sketch}[in the case $f(u)= e^u$]
Use the equation in \eqref{pbla} for the classical solutions $u=u_\lambda$ ($\lambda < \lambda^*$), together with the stability condition \eqref{LN:stability} for the test function $\xi=e^{\alpha u} -1$ (for a positive exponent $\alpha$ to be chosen later).
More precisely, start from \eqref{LN:stability} -- with $f'$ replaced by $\lambda f'$ -- and to proceed with $\int_\Omega \alpha^2 e^{2 \alpha u} |\nabla u|^2$, write $ \alpha^2 e^{2 \alpha u} | \nabla u|^2 = ( \alpha / 2) \nabla \left( e^{2 \alpha u} -1 \right) \nabla u$, and integrate by parts to use \eqref{pbla}.
For every $\alpha < 2$, verify that this leads, after letting $\lambda \uparrow \lambda^*$,
to $e^{u^*} \in L^{2 \alpha +1} (\Omega)$. As a consequence, by Calder\'{o}n-Zygmund theory and Sobolev embeddings, $u^* \in W^{2, 2 \alpha +1} (\Omega) \subset L^\infty (\Omega)$ if $2 ( 2 \alpha +1) > n$. This requires that $n \le 9$.
\qed
\end{sketch}
Notice that the nonlinearities $f(u)=e^u$ or $f(u)=(1+u)^p$ with $p>1$ satisfy~\eqref{hypf}.
In the radial case $\Omega=B_1$ we have the following result.
\begin{theorem}[Cabr\'e-Capella \cite{CabrCappLN}]
\label{corolrad}
Let $u^*$ be the extremal solution of \eqref{pbla}. Assume that $f$ satisfies \eqref{hypf} and that $\Omega=B_1$.
If $1\leq n\leq 9$, then $u^* \in L^\infty (B_1)$.
\end{theorem}
As mentioned before, this theorem also holds for every $H_0^1 (B_1)$ stable weak solution of \eqref{pb}, for any $f \in C^1$.
Thus, in view of Proposition \ref{prop:dimensopt}, the dimension $n \le 9$ is optimal in this result.
We turn now to the nonradial case and we present the currently known results. First, in 2000 Nedev solved the case $n \le 3$.
\begin{theorem}[Nedev \cite{ND}]\label{thm3pre}
Let $f$ be convex and satisfy \eqref{hypf}, and $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain.
If $n\leq 3$, then $u^* \in L^\infty(\Omega)$.
\end{theorem}
In 2010, Nedev's result was improved to dimension four:
\begin{theorem}[Cabr\'e \cite{Cabre}; Villegas \cite{V}]\label{thm3}
Let $f$ satisfy \eqref{hypf}, $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain, and $1 \le n\leq 4$.
If $n\in \{3,4\}$ assume either that $f$ is a convex nonlinearity or that $\Omega$ is a convex domain.
Then, $u^* \in L^\infty(\Omega)$.
\end{theorem}
For $3\leq n \leq 4$, \cite{Cabre} requires $\Omega$ to be convex, while $f$ needs not be
convex. Some years later, S. Villegas~\cite{V} succeeded to use both \cite{Cabre} and \cite{ND} when $n=4$
to remove the requirement that $\Omega$ is convex by further assuming that $f$ is convex.
\begin{open problem}
For every $\Omega$ and for every $f$ satisfying \eqref{hypf}, is the extremal solution $u^*$ -- or, in general, $H_0^1$ stable weak solutions of \eqref{pb} -- always bounded in dimensions $5,6,7,8,9$?
\end{open problem}
We recall that the answer to this question is affirmative when $\Omega = B_1$, by Theorem \ref{corolrad}. We next sketch the proof of this radial result, as well as the regularity theorem in the nonradial case up to $n \le 4$.
In the case $n=4$, we will need the following remarkable result.
\begin{theorem}[Allard; Michael and Simon]
\label{Sobolev}
Let $M\subset \mathbb{R}^{m+1}$ be an immersed smooth $m$-dimensional
compact hypersurface without boundary.
Then, for every $p\in [1, m)$,
there exists a constant $C = C(m,p)$ depending only on the dimension $m$
and the exponent $p$ such that, for every $C^\infty$ function $v : M \to \mathbb{R}$,
\begin{equation}
\label{MSsob}
\left( \int_M |v|^{p^*} \,dV \right)^{1/p^*} \leq C(m,p)
\left( \int_ M (|\nabla v|^p + | {\mathcal H} v|^p) \,dV \right)^{1/p},
\end{equation}
where ${\mathcal H}$ is the mean curvature of $M$ and $p^* = mp/(m - p)$.
\end{theorem}
This theorem dates from 1972 and has its origin in an important result of Miranda from 1967. It stated that \eqref{MSsob} holds with ${\mathcal H}=0$ if $M$ is a minimal surface in $\mathbb{R}^{m+1}$.
See the book \cite{Dupaigne} for a proof of Theorem \ref{Sobolev}.
\begin{remark}
Note that this Sobolev inequality contains a term involving the mean curvature of $M$ on its right-hand side. This fact makes, in a remarkable way, that the constant $C(m,p)$ in the inequality does not depend on the geometry of the manifold~$M$.
\end{remark}
\begin{sketch}[of Theorems \ref{corolrad}, \ref{thm3pre}, and \ref{thm3}]
For Theorem \ref{thm3pre} the test function to be used is $\xi=h(u)$, for some $h$ depending on $f$ (as in the proof of Theorem~\ref{thm:CranRab}).
Instead, for Theorems \ref{corolrad} and \ref{thm3}, the proofs start by writing
the stability condition \eqref{LN:stability} for the test
function $\xi= \tilde{c} \eta$, where $\eta|_{\partial\Omega}\equiv0$.
This was motivated by the analogous computation that we have presented for minimal surfaces right after Remark \ref{rem:Jacobi operator}.
Integrating by parts, one easily deduces that
\begin{equation}\label{stabc}
\int _{\Omega} \left( \Delta \tilde{c} +f'(u) \tilde{c} \right) \tilde{c} \eta ^{2}\,dx \le
\int _{\Omega} \tilde{c}^{2}\left|\nabla \eta\right|^{2}\,dx.
\end{equation}
Next, a key point is to choose a function $\tilde{c}$ satisfying an appropriate equation for
the linearized operator $\Delta + f'(u)$. In the radial case (Theorem \ref{corolrad}) the choice of $\tilde{c}$ and the final choice of $\xi$ are
\begin{equation*}
\tilde{c} =u_{r} \quad\text{ and }\quad\xi=u_r r(r^{-\alpha}-(1/2)^{-\alpha})_{+} ,
\end{equation*}
where $r=|x|$, $\alpha>0$, and $\xi$ is later truncated near the origin to make it Lipschitz.
The proof in the radial case is quite simple after computing the equation satisfied by $u_r$.
For the estimate up to dimension 4 in the nonradial case (Theorem~\ref{thm3}), \cite{Cabre} takes
\begin{equation}\label{choice4}
\tilde{c} =\left|\nabla u\right| \quad\text{ and }\quad\xi=\left|\nabla u\right|\varphi(u) ,
\end{equation}
where, in dimension $n=4$, $\varphi$ is chosen depending on the solution $u$ itself.
We make the choice \eqref{choice4} and, in particular, we take
$\tilde{c} =\left|\nabla u\right|$ in \eqref{stabc}.
It is easy to check that, in the set
$\left\{\left|\nabla u\right|>0\right\}$, we have
\begin{equation}\label{dipassaggio}
\left(\Delta + f'(u)\right)|\nabla u|=\frac{1}{|\nabla u|}
\left(\sum_{i,j}u_{ij}^2-\sum_i\left(\sum_{j}u_{ij}\frac{u_j}
{|\nabla u|}\right)^2\right).
\end{equation}
Taking an orthonormal basis in which the last vector is the normal $\nabla u/|\nabla u|$ to the level set of $u$ (through a given point $x \in \Omega$), and the other vectors are the principal directions of the level set at $x$, one easily sees that \eqref{dipassaggio} can be written as
\begin{equation}\label{eq:grad}
\left( \Delta+f'(u)\right)\left|\nabla u\right|=
\frac{1}{\left|\nabla u\right|} \left(\left|\nabla_T\left|\nabla u\right|\right|^2+
\left|A\right|^2\left|\nabla u\right|^2\right)\quad \text{in}\
\Omega\cap\left\{\left|\nabla u\right|>0\right\},
\end{equation}
where $\left|A\right|^2=\left|A\left(x\right)\right|^2$ is the squared norm of
the second fundamental form of the level set of $u$ passing through a
given point $x\in\Omega\cap\left\{\left|\nabla u\right|>0\right\}$, i.e., the
sum of the squares of the principal curvatures of the level set. In the notation of the first section on minimal surfaces, $|A|^2 = c^2$.
On the other hand, as in that section $\nabla_T = \de$ denotes the tangential gradient to the level set.
Thus, \eqref{eq:grad} involves geometric information of the level sets
of $u$.
Therefore, using the stability condition \eqref{stabc},
we conclude that
\begin{equation}\label{semi1}
\int_{\left\{\left|\nabla u\right|>0\right\}}
\left( |\nabla_T |\nabla u||^2 +|A|^2|\nabla u|^2\right)\eta^2\,dx
\leq \int_\Omega |\nabla u|^2 |\nabla \eta|^2 \,dx .
\end{equation}
Let us define
$$
T:=\max_{\overline{\Omega} }u=\left\|u\right\|_{L^\infty (\Omega)}
\quad\text{ and }\quad \Gamma_s:=\left\{x\in\Omega:u(x)=s\right\}
$$
for $s\in(0,T)$.
We now use \eqref{semi1} with
$\eta=\varphi(u)$,
where $\varphi$ is a Lipschitz function in $\left[0,T\right]$
with $\varphi(0)=0$.
The right hand side of \eqref{semi1} becomes
\begin{eqnarray*}
\int_{\Omega}\left|\nabla u\right|^2\left|\nabla\eta\right|^2dx&=&
\int_{\Omega}\left|\nabla u\right|^4\varphi'(u)^2dx\\
&=&\int_{0}^{T}\left(\int_{\Gamma_s}\left|\nabla u\right|^3\,dV_s
\right)\varphi'(s)^2\,ds,
\end{eqnarray*}
by the {\it coarea formula}. Thus, \eqref{semi1} can be written as
\begin{eqnarray*}
&&\hspace{-1cm} \int_{0}^{T}\left(\int_{\Gamma_s}\left|\nabla u\right|^3\,dV_s
\right)\varphi'(s)^2\,ds\\
&&\geq\int_{\left\{\left|\nabla
u\right|>0\right\}}\left(\left|\nabla_T\left|
\nabla u\right|\right|^2+\left|A\right|^2\left|\nabla
u\right|^2\right)\varphi(u)^2dx\\
&&=\int_{0}^{T}\left(\int_{\Gamma_s\cap\left\{\left|\nabla
u\right|>0\right\}}\frac{1}{\left|\nabla u\right|}
\left(\left|\nabla_T\left|
\nabla u\right|\right|^2+\left|A\right|^2\left|\nabla
u\right|^2\right)\,dV_s\right)\varphi(s)^2\,ds\\
&&=\int_{0}^{T}\left(\int_{\Gamma_s\cap\left\{\left|\nabla
u\right|>0\right\}}
\left( 4\left|\nabla_T\left|
\nabla u\right|^{1/2}\right|^2+\left(\left|A\right|\left|\nabla
u\right|^{1/2}\right)^{2}\right) \,dV_s\right)\varphi(s)^2\,ds .
\end{eqnarray*}
We conclude that
\begin{equation}
\int_0^T h_1(s) \varphi(s)^2 \,ds
\leq
\int_0^T h_2(s) \varphi'(s)^2 \,ds,
\label{semi3}
\end{equation}
for all Lipschitz functions
$\varphi:\left[0,T\right]\rightarrow\mathbb{R}$ with
$\varphi(0)=0$, where
\begin{equation*}
h_1(s):=\int_{\Gamma_s}
\left( 4|\nabla_T |\nabla u|^{1/2}|^2 +\left( |A||\nabla u|^{1/2} \right)^2\right) \,dV_s\, ,\
h_2(s):=\int_{\Gamma_s} |\nabla u|^3\,dV_s
\end{equation*}
for every regular value $s$ of $u$.
We recall that, by Sard's theorem, almost every $s\in(0,T)$ is a regular value of $u$.
Inequality \eqref{semi3}, with $h_1$ and $h_2$ as defined above, leads to a bound for $T$
(that is, to an $L^{\infty}$ estimate and hence to Theorem \ref{thm3}) after choosing an
appropriate test function $\varphi$ in \eqref{semi3}.
In dimensions 2 and 3 we can choose a simple function $\varphi$ in \eqref{semi3} and use well known geometric inequalities
about the curvature of manifolds (note that $h_1$ involves the curvature
of the level sets of $u$). Instead, in dimension 4 we need to use the geometric Sobolev inequality of Theorem \ref{Sobolev} on each level set of $u$.
Note that ${\mathcal H}^2 \le (n-1) |A|^2$.
This gives the following lower bound for $h_1 (s)$:
$$
c(n) \left( \int_{\Gamma_s} | \nabla u |^{\frac{n-1}{n-3} } \right)^{\frac{n-3}{n-1}} \le h_1 (s).
$$
Comparing this with $h_2 (s)$, which appears in the right hand side of \eqref{semi3}, we only know how to derive an $L^{\infty}$-estimate for $u$ (i.e., a bound on $T= \max u$) when the exponent $(n-1) / (n-3)$ in the above inequality is larger than or equal to the exponent $3$ in $h_2 (s)$.
This requires $n \le 4$.
See \cite{Cabre} for details on how the proof is finished.
\qed
\end{sketch}
\section{Appendix: a calibration giving the optimal isoperimetric inequality}
Our first proof of Theorem \ref{thm:SimCone} used a calibration. To understand better the concept and use of ``calibrations'', we present here another one. It leads to a proof of the isoperimetric problem.
The isoperimetric problems asks which sets in $\mathbb{R}^n$ minimize perimeter for a given volume. Making the first variation of perimeter (as in Section \ref{sec:mincones}), but now with a volume constraint, one discovers that a minimizer $\Omega$ should satisfy ${\mathcal H}=c$ (with $c$ a constant), at least in a weak sense, where ${\mathcal H}$ is the mean curvature of $\partial \Omega$. Obviously, balls satisfy this equation -- they have constant mean curvature.
The isoperimetric inequality states that the unique minimizers are, indeed, balls. In other words, we have:
\begin{theorem}[The isoperimetric inequality]\label{thm:servepertesiLNisoperimetric}
We have
\begin{equation}
\label{eq:isoperimetric}
\frac{|\partial \Omega|}{| \Omega|^{\frac{n-1}{n}}} \ge \frac{|\partial B_1|}{| B_1|^{\frac{n-1}{n}}}
\end{equation}
for every bounded smooth domain $\Omega \subset \mathbb{R}^n$.
In addition, if equality holds in \eqref{eq:isoperimetric}, then $\Omega$ must be a ball.
\end{theorem}
In 1996 the first author found the following proof of the isoperimetric problem. It uses a calibration (for more details see \cite{Caisoperimetric}).
\begin{sketch}[of the isoperimetric inequality]
The initial idea was to characterize the perimeter $|\partial \Omega|$ as in \eqref{calideph}-\eqref{eq:dimcalibration1}, that is, as
$$
| \partial \Omega | = \sup_{ \Vert X \Vert_{L^\infty} \le 1} \int_{ \partial \Omega} X \cdot \nu \, dH_{n-1} .
$$
Taking $X$ to be a gradient, we have that
$$
| \partial \Omega | = \int_{ \partial \Omega} \nabla u \cdot \nu \, dH_{n-1} = \int_{\partial \Omega} u_\nu \, dH_{n-1} ,
$$
for every function $u$ such that $u_\nu=1$ on $\partial \Omega$.
Let us take $u$ to be the solution of
\begin{equation}\label{eqappisop}
\left\{
\begin{array}{rcll}
\Delta u &=& c & \quad\mbox{in } \Omega \\
u_\nu &=& 1 & \quad\mbox{on } \partial\Omega ,\\
\end{array}\right.
\end{equation}
where $c$ is a constant that, by the divergence theorem, is given by
$$
c= \frac{|\partial \Omega|}{| \Omega| }.
$$
It is known that there exists a unique solution $u$ to \eqref{eqappisop} (up to an additive constant).
Now let us see that $X=\nabla u$ (where $X$ was the notation that we used in the proof of Theorem \ref{thm:SimCone}) can play the role of a calibration.
In fact, in analogy with Definition~\ref{def:calibrationprima} we have:
\begin{enumerate}[]
\item(i-bis) \,$\mathop{\mathrm{div}} \nabla u = \frac{|\partial \Omega|}{| \Omega| } \, \mbox{ in } \, \Omega $;
\item(ii-bis) \,$\nabla u \cdot \nu = 1 \, \mbox{ on } \, \partial \Omega$;
\item(iii-bis) \,$B_1 (0) \subset \nabla u (\Gamma_u)$, where
$$\Gamma_u = \left\{ x \in \Omega : u(y) \ge u(x) + \nabla u(x) \cdot (y-x) \, \mbox{ for every } \, y \in \overline{\Omega} \right\} $$
is the {\it lower contact set of $u$}, that is, the set of the points of $\Omega$ at which the tangent plane to $u$ stays below $u$ in $\Omega$.
\end{enumerate}
The relations (i-bis) and (ii-bis) follow immediately from \eqref{eqappisop}.
In the following exercise, we ask to establish (iii-bis) and finish the proof of \eqref{eq:isoperimetric}.
We point out that this proof also gives that $\Omega$ must be a ball if equality holds in \eqref{eq:isoperimetric}.
\qed
\end{sketch}
\begin{exercise}
Establish (iii-bis) above. For this, use a foliation-contact argument (as in the alternative proof of Theorem \ref{thm:SimCone} and in the proof of Theorem \ref{alba}), foliating now $\mathbb{R}^n \times \mathbb{R}$ by parallel hyperplanes.
Next, finish the proof of \eqref{eq:isoperimetric}. For this, consider the measures of the two sets in (iii-bis), compute $| \nabla u (\Gamma_u)|$ using the {\it area formula}, and control $\det D^2 u$ using the geometric-arithmetic means inequality.
\end{exercise}
\begin{quote}
{\bf Acknowledgments.}
{\small The authors wish to thank Lorenzo Cavallina for producing the figures of this work.
The first author is member of the Barcelona Graduate School of Mathematics and is supported by MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P. He is also part of the Catalan research group 2017 SGR 1392.
The second author was partially supported by PhD funds of the Universit\`{a} di Firenze and he is a member of the Gruppo Nazionale Analisi Matematica Probabilit\'{a} e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was partially written while the second author was visiting the Departament de Matem\`{a}tiques of the Universitat Polit\`ecnica de Catalunya, that he wishes to thank for hospitality and support.}
\end{quote}
\chapter{Littlewood's fourth principle}
\label{chap:Littlewoodfourthprinciple}
\centerline{Rolando Magnanini\footnote{Dipartimento di Matematica e Informatica ``U.~Dini'',
Universit\` a di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
({\tt magnanin@math.unifi.it}).}
and Giorgio Poggesi\footnote{
Dipartimento di Matematica ed Informatica ``U.~Dini'',
Universit\` a di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
({\tt giorgio.poggesi@unifi.it}).
}
}
\vspace{1cm}
\begin{quotation}
{\bf Abstract.}
{\small In Real Analysis, Littlewood's three principles are known as heuristics that help teach the essentials of measure theory and reveal the analogies between the concepts of topological space and continuous function on one side and those of measurable space and measurable function on the other one. They are based on important and rigorous statements, such as Lusin's and Egoroff-Severini's theorems, and have ingenious and elegant proofs. We shall comment on those theorems and show how their proofs
can possibly be made simpler by introducing a \textit{fourth principle}. These alternative
proofs make even more manifest those analogies and show that Egoroff-Severini's theorem can be considered as the natural generalization of the classical Dini's monotone convergence theorem.
}
\end{quotation}
\section{Introduction.}
John Edenson Littlewood (9 June 1885 - 6 September 1977) was a British mathematician.
In 1944, he wrote an influential textbook, \textit{Lectures on the Theory of Functions} (\cite{Littlewood}), in which he proposed three principles as guides for working in real analysis; these are heuristics to help teach the essentials of measure theory, as Littlewood himself wrote in \cite{Littlewood}:
\begin{quotation}
The extent of knowledge [of real analysis] required is nothing like so great as is sometimes supposed. There are three principles, roughly expressible in the following terms:
every (measurable) set is nearly a finite sum of intervals;
every function (of class $L^{\lambda}$) is nearly continuous;
every convergent sequence is nearly uniformly convergent.
Most of the results of the present section are fairly intuitive applications of these ideas, and the student armed with them should be equal to most occasions when real variable theory is called for. If one of the principles would be the obvious means to settle a problem if it were ``quite'' true, it is natural to ask if the ``nearly'' is near enough, and for a problem that is actually soluble it generally is.
\end{quotation}
To benefit our further discussion, we shall express Littlewood's principles
and their rigorous statements in forms
that are slightly different from those originally stated.
The first principle descends directly from the very definition of Lebesgue measurability of a set.
\begin{principle1}
Every measurable set is nearly closed.
\end{principle1}
The second principle relates the measurability of a function to the more familiar property of continuity.
\begin{principle2}
Every measurable function is nearly continuous.
\end{principle2}
The third principle connects the pointwise convergence of a sequence of functions to
the standard concept of uniform convergence.
\begin{principle3}
Every sequence of measurable functions that converges pointwise almost everywhere is nearly
uniformly convergent.
\end{principle3}
These principles are based on important theorems that give a rigorous meaning to the term ``nearly''.
We shall recall these in the next section along with their ingenious proofs that give a taste of the
standard arguments used in Real Analysis.
\par
In Section 3, we will discuss a {\it fourth principle} that associates the concept of finiteness
of a function to that of its boundedness.
\begin{principle4}
Every measurable function that is finite almost everywhere is nearly
bounded.
\end{principle4}
In the mathematical literature (see \cite{DiB}, \cite{CD}, \cite{Littlewood}, \cite{Ma}, \cite{Royden}, \cite{Ru}, \cite{Ta}), the proof of the second principle is based on the third; it can be easily seen that the fourth principle can be derived from the second.
\par
However, we shall see that the fourth principle can also be proved independently;
this fact makes possible a proof of the second principle \textit{without} appealing for the third,
that itself can be derived from the second, by a totally new proof based on \textit{Dini's monotone convergence theorem}.
\par
As in \cite{Littlewood}, to make our discussion as simple as possible, we shall consider the Lebesgue measure $m$ for the real line $\mathbb {R}$.
\section{The three principles}
We recall the definitions of {\it inner} and {\it outer measure} of
a set $E\subseteq\mathbb{R}$: they are
\begin{eqnarray*}
&&m_i(E)=\sup\{ |K|: K \mbox{ is compact and } K\subseteq E\},\\
&&m_e(E)=\inf\{|A| : A\mbox{ is open and } A\supseteq E\},
\end{eqnarray*}
where the number $|K|$ is the infimum of the total lengths of all the finite unions of open intervals that contain $K$; accordingly, $|A|$ is the supremum of the total lengths of all the finite unions of closed intervals contained in $A$.
It always holds that $m_i(E)\le m_e(E)$. The set $E$ is (Lebesgue) measurable if and only if
$m_i(E)=m_e(E)$;
when this is the case, the {\it measure} of $E$ is $m(E)=m_i(E)=m_e(E)$; thus $m(E)\in[0,\infty]$
and it can be proved that $m$ is a measure on the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb{R}$.
\par
By the properties of the supremum, it is easily seen that, for any pair of subsets $E$ and $F$ of $\mathbb{R}$, $m_e(E\cup F)\le m_e(E)+m_e(F)$ and $m_e(E)\le m_e(F)$ if $E\subseteq F$.
\vskip.1cm
The first principle is a condition for the measurability of subsets of $\mathbb{R}$.
\begin{theorem}[First Principle]
\label{th:first}
Let $E\subset\mathbb{R}$ be a set of finite outer measure.
\par
Then, $E$ is measurable if and only if for every $\varepsilon > 0$ there exist two sets $K$ and $F$, with
$K$ closed (compact), $K \cup F = E$ and $m_e(F) < \varepsilon$.
\end{theorem}
This is what is meant by \textit{nearly closed}.
\begin{proof}
If $E$ is measurable, for any $\varepsilon>0$ we can find a compact set $K\subseteq E$ and an open
set $A\supseteq E$ such that
$$
m(K)>m(E)-\varepsilon/2 \ \mbox{ and } \ m(A)<m(E)+\varepsilon/2.
$$
The set $A\setminus K$ is open and contains $E\setminus K$. Thus, by setting $F=E\setminus K$, we have $E=K\cup F$ and
$$
m_e(F)\le m(A)-m(K)<\varepsilon.
$$
\par
Viceversa, for every $\varepsilon>0$ we have:
$$
m_e(E)=m_e(K\cup F)\le m_e(K)+m_e(F)<m(K)+\varepsilon\le m_i(E)+\varepsilon.
$$
Since $\varepsilon$ is arbitrary, then $m_e(E)\le m_i(E)$.
\end{proof}
The second and third principles concern measurable functions from (measurable) subsets of $\mathbb{R}$
to the {\it extended real line} $\overline{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}\cup\{-\infty\}$,
that is functions are allowed to have values $+\infty$ and $-\infty$.
\par
Let $f:E\to\overline{\mathbb{R}}$ be a function defined on a measurable subset $E$ of $\mathbb{R}$.
We say that $f$ is \textit{measurable} if the \textit{level sets} defined by
$$
L(f,t)=\{ x\in E: f(x)>t\}
$$
are measurable subsets of $\mathbb{R}$ for every $t\in\mathbb{R}$. It is easy to verify that if we replace $L(f,t)$ with $L^*(f,t)=\{ x\in E: f(x) \geq t\}$ we have an equivalent definition.
\par
Since the countable union and intersection of measurable sets are measurable, it is not hard to show that
the pointwise infimum and supremum of a sequence of measurable functions $f_n:E\to\overline{\mathbb{R}}$
are measurable functions as well as the function defined for any $x\in E$ by
$$
\limsup_{n\to\infty} f_n(x)=\inf_{k\ge 1}\sup_{n\ge k} f_n(x).
$$
\par
Since the countable union of sets of measure zero has measure zero and the difference between $E$ and any set of measure zero is measurable, the same definitions and conclusions hold even
if the functions $f$ and $f_n$ are defined \textit{almost everywhere} (denoted for short by a.e.), that is
if the subsets of $E$ in which they are not defined have measure zero. In the same spirit,
we say that a function or a sequence of functions satisfies a given property \textit{a.e.} in $E$, if that property holds with the exception of a subset of measure zero.
\par
As already mentioned, the third principle is needed to prove the second and is
known as \textit{Egoroff's theorem} or \textit{Egoroff-Severini's theorem}.\footnote{Dmitri Egoroff, a Russian physicist and geometer and Carlo Severini, an Italian mathematician, published independent proofs of this theorem in 1910 and 1911
(see \cite{Eg} and \cite{Severini}); Severini's assumptions are more restrictive. Severini's result is not very well-known, since it is hidden in a paper on orthogonal polynomials, published in Italian.}
\begin{theorem}[Third Principle; Egoroff-Severini]
\label{th:Egoroff}
Let $E\subset \mathbb{R}$ be a measurable set with finite measure and
let $f:E\to\overline{\mathbb{R}}$ be measurable and finite a.e. in $E$.
The sequence of measurable functions $f_n:E\to\overline{\mathbb{R}}$ converges a.e. to $f$ in $E$ for $n\to\infty$ if and only if, for every $\varepsilon> 0$, there exists a closed set $K \subseteq E$ such that $m(E \setminus K) < \varepsilon$ and $f_n$ converges uniformly to $f$ on $K$.
\end{theorem}
This is what we mean by \textit{nearly uniformly convergent}.
\begin{proof}
If $f_n \to f$ a.e. in $E$ as $n\to\infty$, the subset of $E$ in which $f_n \to f$ pointwise
has the same measure as $E$; hence, without loss of generality, we can assume that $f_n(x)$
converges to $f(x)$ for every $x\in E$.
Consider the functions defined by
\begin{equation}
\label{defgn}
g_n(x)=\sup_{k\ge n} |f_k(x)-f(x)|, \ \ x\in E
\end{equation}
and the sets
\begin{equation}
\label{defEnm}
E_{n,m} =\left\{x \in E : g_n(x) <\frac1{m} \right\}
\ \mbox{ for } \ n, m\in\mathbb{N}.
\end{equation}
Observe that, if $x\in E$, then $g_n(x)\to 0$ as $n\to\infty$ and hence for any $m\in\mathbb{N}$
$$
E=\bigcup_{n=1}^\infty E_{n,m}.
$$
As $E_{n,m}$ is increasing with $n$, the monotone convergence theorem implies that $m(E_{n,m})$ converges to $m(E)$ for $n\to\infty$ and for any $m \in\mathbb{N}$.
Thus, for every $\varepsilon>0$ and $m \in\mathbb{N}$, there exists an index $\nu=\nu(\varepsilon, m)$ such that
$
m(E \setminus E_{\nu,m}) < \varepsilon/ {2^{m+1}}.
$
\par
The measure of the set $F = \bigcup\limits_{m = 1}^{\infty} (E \setminus E_{\nu,m})$ is arbitrary small, in fact
$$
m(F) \leq \sum_{m=1}^{\infty} m(E \setminus E_{\nu,m}) < \varepsilon/ 2.
$$
Also, since $E\setminus F$ is measurable, by Thorem \ref{th:first} there exists a compact set $K \subseteq E \setminus F$ such that $m(E \setminus F) - m(K) < \varepsilon/ 2$, and hence
$$
m(E \setminus K) = m(E\setminus F)+m(F) - m(K) < \varepsilon.
$$
\par
Since $K \subseteq E \setminus F = \bigcap\limits_{m=1}^{\infty} E_{\nu(\varepsilon, m), m}$ we have that
$$
|f_n(x) - f(x)| < \frac{1}{m} \ \mbox{ for any } \ x\in K \ \mbox{ and } \ n \ge \nu(\varepsilon, m),
$$
by the definitions of $E_{\nu,m}$ and $g_{n}$; this means that $f_n$ converges uniformly to $f$ on $K$ as $n\to\infty$.
\vskip.1cm
Viceversa, if for every $\varepsilon> 0$ there is a closed set $K \subseteq E$ with
$m(E \setminus K) < \varepsilon$ and $f_n \rightarrow f$ uniformly on $K$, then by
choosing $\varepsilon=1/m$ we can say that there is a closed set $K_m \subseteq E$ such that $f_n \to f$ uniformly on $K_m$ and $m(E \setminus K_m) < 1 / m$.
\par
Therefore, $f_n(x) \rightarrow f(x)$ for any $x$ in the set $F = \bigcup\limits_{m=1}^\infty K_m$ and
$$
m(E \setminus F) = m\Bigl(\bigcap_{m=1}^\infty (E \setminus K_m)\Bigr) \leq m(E \setminus K_m) <\frac1{m} \ \mbox{ for any } \ m\in\mathbb{N},
$$
which implies that $m(E \setminus F) = 0$. Thus, $f_n\to f$ a.e. in $E$ as $n\to\infty$.
\end{proof}
The second principle corresponds to \textit{Lusin's theorem} (see \cite{Lu}),\footnote{N. N. Lusin or Luzin was a student of Egoroff. For biographical notes on Egoroff and Lusin see \cite{GK}.} that we state
here in a form similar to Theorems \ref{th:first} and \ref{th:Egoroff}.
\begin{theorem}[Second Principle; Lusin]
\label{th:Lusin}
Let $E\subset \mathbb{R}$ be a measurable set with finite measure and
let $f:E\to\overline{\mathbb{R}}$ be finite a.e. in $E$.
\par
Then, $f$ is measurable in $E$ if and only if, for every $\varepsilon>0$, there exists a closed set $K \subseteq E$ such that $m(E \setminus K) < \varepsilon$ and the restriction of $f$ to $K$ is continuous.
\end{theorem}
This is what we mean by \textit{nearly continuos}.
\vskip.1cm
The proof of Lusin's theorem is done by approximation by simple functions. A \textit{simple function}
is a measurable function that has a finite number of real values.
If $c_1, \ldots, c_n$ are the \textit{distinct} values of a simple function $s$,
then $s$ can be conveniently represented as
$$
s = \sum\limits_{j = 1}^{n} c_{j}\mathcal{X}_{E_j},
$$
where $\mathcal{X}_{E_j}$ is the characteristic function of the set $E_j = \left\{ x \in E \textrm{ : } s(x) = c_j \right\}$. Notice that the $E_j$'s form a
covering of $E$ of pairwise disjoint measurable sets.
\par
Simple functions play
a crucial role in Real Analysis; this is mainly due to the following result of which we shall
omit the proof.
\begin{theorem}[Approximation by Simple Functions, \cite{Royden}, \cite{Ru}]
\label{th:approximation}
Let $E\subseteq\mathbb{R}$ be a measurable set and let $f: E\to [0,+\infty]$ be a measurable function.
\par
Then, there exists an increasing sequence of non-negative simple functions $s_n$ that
converges pointwise to $f$ in $E$ for $n\to\infty$.
\par
Moreover, if $f$ is bounded, then $s_n$ converges to $f$ uniformly in $E$.
\end{theorem}
We can now give the proof of Lusin's theorem.
\begin{proof}
Any measurable function $f$ can be decomposed as $f=f^+-f^-$, where $f^+=\max(f,0)$ and
$f^-=\max(-f,0)$ are measurable and non-negative functions. Thus, we can always suppose that $f$
is non-negative and hence, by Theorem \ref{th:approximation}, it
can be approximated pointwise by a sequence of simple functions.
\par
We first prove that a simple function $s$ is nearly continuos.
Since the sets $E_j$ defining $s$ are measurable, if we fix $\varepsilon> 0$ we can find closed subsets $K_j$ of $E_j$ such that $m(E_j \setminus K_j) < \varepsilon/n$ for $j=1,\dots, n$.
The union $K$ of the sets $K_j$ is also a closed set and, since the $E_j$'s cover $E$, we have that $m(E \setminus K) < \varepsilon$. Since the closed sets $K_j$ are pairwise disjoint (as the $E_j$'s are pairwise disjoint) and $s$ is constant on $K_j$ for all $j=1,\dots,n$, we conclude that $s$ is continuous in $K$.
\par
Now, if $f$ is measurable and non-negative,
let $s_n$ be a sequence of simple functions that converges pointwise to $f$ and fix an $\varepsilon>0$.
\par
As the $s_n$'s are nearly continuous, for any natural number $n$, there exists a closed set $K_n \subseteq E$ such that $m(E \setminus K_n) < \varepsilon/{2^{n+1}}$ and $s_n$ is continuous in $K_n$. By Theorem \ref{th:Egoroff}, there exists a closed set $K_0 \subseteq E$ such that $m(E \setminus K_0) < \varepsilon/2$ and $s_n$ converges uniformly to $f$ in $K_0$ as $n\to\infty$. Thus, in the set
$$
K = \bigcap\limits_{n=0}^\infty K_n
$$
the functions $s_n$ are all continuous and converge uniformly to $f$. Therefore $f$ is continuous in $K$
and
$$
m(E \setminus K) = m\Bigl(\bigcup\limits_{n=0}^\infty (E \setminus K_n)\Bigr) \leq \sum_{n=0}^{\infty} m(E \setminus K_n) < \varepsilon.
$$
\par
Viceversa, if $f$ is nearly continuous, fix an $\varepsilon>0$ and let $K$ be a closed subset of $E$ such that $m(E \setminus K) < \varepsilon$ and $f$ is continuous in $K$. For any $t \in \mathbb{R}$, we have:
$$
L^*(f,t)= \left\{x \in K : f(x) \geq t\right\} \cup \left\{x \in E \setminus K : f(x) \geq t\right\}.
$$
The former set in this decomposition is closed, as the restriction of $f$ to $K$ is continuous,
while the latter is clearly a subset of $E \setminus K$ and hence its outer measure must be less than $\varepsilon$. By Theorem \ref{th:first}, $L^*(f,t)$ is measurable (for any $t\in\mathbb{R}$), which means that $f$
is measurable.
\end{proof}
\section{The fourth principle}
We shall now present alternative proofs of Theorems \ref{th:Egoroff} and \ref{th:Lusin}.
They are based on a fourth principle, that corresponds to the following theorem.
\begin{theorem}[Fourth Principle]
\label{th:fourth}
Let $E\subset \mathbb{R}$ be a measurable set with finite measure and let $f:E\to\overline{\mathbb{R}}$ be a measurable function.
\par
Then, $f$ is finite a.e. in $E$ if and only if, for every $\varepsilon > 0$, there exists a closed set $K \subseteq E$ such that $m(E \setminus K) < \varepsilon$ and $f$ is bounded on $K$.
\end{theorem}
This is what we mean by \textit{nearly bounded}.
\begin{proof}
If $f$ is finite a.e., we have that
$$
m(\left\{x \in E : |f(x)| = \infty\right\})=0.
$$
As $f$ is measurable, $|f|$ is also measurable and so are the sets
$$
L(|f|,n) = \left\{x \in E :|f(x)| > n\right\}, \ n\in\mathbb{N}.
$$
Observe that the sequence of sets $L(|f|,n)$ is decreasing and
$$
\bigcap\limits_{n=1}^\infty L(|f|,n) = \left\{x \in E :|f(x)| = \infty\right\}.
$$
As $m(L(|f|,1)) \leq m(E) < \infty$, we can
apply the (downward) monotone convergence theorem and infer that
$$
\lim_{n \to \infty} m(L(|f|,n)) = m(\left\{x \in E :|f(x)| = \infty\right\}) = 0.
$$
\par
Thus, if we fix $\varepsilon>0$, there is an $n_{\varepsilon} \in \mathbb N$ such that $m(L(|f|,n_{\varepsilon})) < \frac{\varepsilon}{2}$. Also, we can find a closed subset $K$ of the measurable set
$E \setminus L(|f|,n_{\varepsilon})$ such that $m(E \setminus L(|f|,n_{\varepsilon}))-m(K)< \frac{\varepsilon}{2}$. Finally, since $K \subseteq E \setminus L(|f|,n_{\varepsilon})$, $|f|$ is obviously bounded by $n_\varepsilon$ on $K$
and
$$
m(E\setminus K)=m(E \setminus L(|f|,n_{\varepsilon}))+m(L(|f|,n_{\varepsilon})\setminus K)<\varepsilon.
$$
\par
Viceversa, if $f$ is nearly bounded, then for any $n \in\mathbb{N}$ there exists a closed set $K_n \subseteq E$ such that $m(E \setminus K_n) < 1/n$ and $f$ is bounded (and hence finite) in $K_n$.
Thus, $\left\{x \in E :|f(x)| =\infty\right\} \subseteq E \setminus K_n$ for any $n \in\mathbb{N}$, and hence $$
m(\left\{x \in E:|f(x)| = \infty\right\}) \leq \lim_{n\to\infty} m(E \setminus K_n) =0,
$$
that is $f$ is finite a.e..
\end{proof}
\begin{remark}
Notice that this theorem can also be derived from Theorem \ref{th:Lusin}.
In fact,
without loss of generality, the closed set $K$ provided by Theorem \ref{th:Lusin} can be
taken to be compact and hence, $f$ is surely bounded on $K$, being continuous on a compact set.
\end{remark}
More importantly for our aims, Theorem \ref{th:fourth} enables us to prove Theorem \ref{th:Lusin} \textit{without} using Theorem \ref{th:Egoroff}.
\begin{proof}[Alternative proof of Theorem \ref{th:Lusin}]
The proof runs similarly to that presented in Section 2. If $f$ is measurable, without loss of generality, we can assume that $f$ is non-negative and hence $f$ can be approximated pointwise by a sequence of simple functions $s_n$, which we know are nearly continuous. Thus, for any $\varepsilon>0$, we can still construct the sequence of closed subsets $K_n$ of $E$ such that $m(E \setminus K_n) < \varepsilon/{2^{n+1}}$ and $s_n$ is continuous in $K_n$.
\par
Now, as $f$ is finite a.e., Theorem \ref{th:fourth} implies that it is nearly bounded, that is we can find a closed subset $K_0$ of $E$ in which $f$ is bounded and $m(E \setminus K_0) < \varepsilon/ 2$.
We apply the second part of the Theorem \ref{th:approximation} and infer that $s_n$ converges uniformly to $f$ in $K_0$. As seen before, we conclude that $f$ is continuous in the intersection $K$ of all the $K_n$'s,
because in $K$ it is the uniform limit of the sequence of continuous functions $s_n$. As before
$m(E\setminus K)<\varepsilon$.
\par
The reverse implication remains unchanged.
\end{proof}
In order to give our alternative proof of Theorem \ref{th:Egoroff}, we need to recall
a classical result for sequences of continuous functions.
\begin{theorem}[Dini]
\label{th:Dini}
Let $K$ be a compact subset of $\mathbb{R}$ and let be given a sequence of continuous functions $f_n:K\to \mathbb{R}$ that converges pointwise and monotonically in $K$ to a function $f:K\to\mathbb{R}$.
\par
If $f$ is also continuous, then $f_n$ converges uniformly to $f$.
\end{theorem}
\begin{proof}
We shall prove the theorem when $f_n$ is monotonically increasing.
For each $n \in \mathbb{N}$, set $h_n = f - f_n$; as $n\to\infty$ the continuos functions $h_n$ decrease pointwise to $0$ on $K$.
Fix $\varepsilon>0$. The sets $A_n = \left\{x \in K:h_n(x) < \varepsilon\right\}$ are relatively open in $K$, since the $h_n$'s are continuous; also, $A_n \subseteq A_{n+1}$ for every $n\in\mathbb{N}$, since the ${h_n}$'s decrease; finally,
the $A_n$'s cover $K$, since the $h_n$ converge pointwise to $0$.
\par
By compactness, $K$ is then covered by a finite number $m$ of the $A_n$'s, which means that $A_m = K$ for some $m\in\mathbb{N}$. This implies that $|f(x) - f_n(x)| < \varepsilon$ for all $n \geq m$ and $x \in K$, as desired.
\end{proof}
\begin{remark}
\label{semicont}
The conclusion of Theorem \ref{th:Dini} still holds true if we assume that the sequence of $f_n$'s is increasing (respectively decreasing) and $f$ and all the $f_n$'s are lower (respectively upper) semicontinuous. (We say that $f$ is lower semicontinuous if the level sets
$L_+(f,t)$ are open for every $t \in \mathbb{R}$; $f$ is upper semicontinuous if $-f$ is lower semicontinuous.)
\end{remark}
Now, Theorem \ref{th:Egoroff} can be proved by appealing to Theorems \ref{th:Lusin} and \ref{th:Dini}.
\begin{proof}[Alternative proof of Theorem \ref{th:Egoroff}]
As in the classical proof of this theorem, we can always assume that $f_n(x) \to f(x)$ for every $x\in E$.
\par
Consider the functions and sets defined in \eqref{defgn} and \eqref{defEnm}, respectively.
We shall first show that there exists an $\nu\in\mathbb{N}$ such that $g_n$ is nearly bounded for every
$n\ge \nu$. In fact, as already observed, since $g_n\to 0$ pointwise in $E$ as $n\to\infty$, we have that
$$
E=\bigcup_{n=1}^\infty E_{n,1},
$$
and the $E_{n,1}$'s increase with $n$. Hence, if we fix $\varepsilon>0$, there is a $\nu\in\mathbb{N}$ such that $m(E\setminus E_\nu)<\varepsilon/2$. Since
$E_\nu$ is measurable, by Theorem \ref{th:first} we can find a closed subset $K$ of $E_\nu$ such that
$m(E_\nu\setminus K)<\varepsilon/2$.
\par
Therefore, $m(E\setminus K)<\varepsilon$ and for every $n\ge \nu$
$$
0\le g_n(x)\le g_\nu(x)<1, \ \mbox{ for any } \ x\in K.
$$
\par
Now, being $g_n$ nearly bounded in $E$ for every $n\ge \nu$, the alternative proof of Theorem \ref{th:Lusin} implies that $g_n$ is nearly continuous in $E$, that is for every $n\ge \nu$ there exists a closed
subset $K_n$ of $E$ such that $m(E\setminus K_n)<\varepsilon/2^{n-\nu+1}$ and $g_n$ is continuous on $K_n$.
The set
$$
K=\bigcap_{n=\nu}^\infty K_n
$$
is closed, $m(E\setminus K)<\varepsilon$ and on $K$ the functions $g_n$ are continuos for any $n\ge \nu$
and monotonically descrease to $0$ as $n\to\infty$.
\par
By Theorem \ref{th:Dini}, the $g_n$'s converge to $0$ uniformly on $K$.
This means that the $f_n$'s converge to $f$ uniformly on $K$ as $n\to\infty$.
\par
The reverse implication remains unchanged.
\end{proof}
\begin{remark}
Egoroff's theorem can be considered, in a sense, as the \textit{natural} substitute of Dini's theorem,
in case the monotonicity assumption is removed.
In fact, notice that the sequence of the $g_n$'s defined in \eqref{defgn} is decreasing; however, the $g_n$'s are in general no longer upper semicontinuous (they are only lower semicontinuous) and Dini's theorem (even in the form described in Remark \ref{semicont}) cannot be applied. In spite of that, the $g_n$'s \textit{remain} measurable if the $f_n$'s are so.
\end{remark}
Of course, all the proofs presented in Sections 2 and 3 still work if we replace the real line $\mathbb{R}$
by an Euclidean space of any dimension.
\vskip.1cm
Theorems \ref{th:Egoroff}, \ref{th:Lusin} and \ref{th:fourth} can also be extended to general measure spaces not necessarily endowed with a topology: the intersted reader can refer to \cite{CD}, \cite{Ox}, \cite{Royden} and \cite{Ta}.
\chapter*{Acknowledgements}
First and foremost I thank my advisor Rolando Magnanini for introducing me to mathematical research. I really appreciated his constant guidance and encouraging criticism, as well as the many remarkable opportunities that he gave me during the PhD period.
I am also indebted with all the professors who, in these three years, have invited me at their institutions in order to collaborate in mathematical research:
Xavier Cabr\'e at the Universitat Polit\`{e}cnica de Catalunya (UPC) in Barcelona, Daniel Peralta-Salas at the Instituto de Ciencias Matem\'aticas (ICMAT) in Madrid, Shigeru Sakaguchi and Kazuhiro Ishige at the Tohoku University in Sendai, Lorenzo Brasco at the Universit\`{a} di Ferrara.
It has been a honour for me to interact and work with them.
I would like to thank Paolo Salani, Andrea Colesanti, Chiara Bianchini, and Elisa Francini for their kind and constructive availability.
Also, I thank my friend and colleague Diego Berti, with whom I shared Magnanini as advisor, for the long and good time spent together.
Special thanks to my partner Matilde, my parents, and the rest of my family. Their unfailing and unconditional support has been extremely important to me.
\tableofcontents
\mainmatter
\chapter*{Introduction}
\markboth{Introduction}{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
\begin{quotation}
The thesis is divided in two parts. The first
is based on the results obtained in \cite{MP, MP2, Pog} and further improvements that give the title to the thesis. The following introduction is dedicated {\it only} to this part.
The second part, entitled ``Other works'', contains a couple of papers, as they were published.
The first are the lecture notes \cite{CP} to which the author of this thesis collaborated. They
have several points of contact with the first part.
The second (\cite{MPLittlewood}) is an article on a different topic, published during my studies as a PhD student.
\end{quotation}
The pioneering symmetry results obtained by A. D. Alexandrov \cite{Al1}, \cite{Al2} and J. Serrin \cite{Se} are now classical but still influential. The former -- the well-known Soap Bubble Theorem -- states that a compact hypersurface, embedded in $\mathbb{R}^N$, that has constant mean curvature must be a sphere. The latter -- Serrin's symmetry result -- has to do with certain overdetermined problems for partial differential equations.
In its simplest formulation, it states that the overdetermined boundary value problem
\begin{eqnarray}
\label{serrin1}
&\Delta u=N \ \mbox{ in } \ \Omega, \quad u=0 \ \mbox{ on } \ \Gamma, \\
\label{serrin2}
&u_\nu=R \ \mbox{ on } \ \Gamma,
\end{eqnarray}
admits a solution
for some positive constant $R$ if and only if $\Omega$ is a ball of radius $R$ and, up to translations, $u(x)=(|x|^2-R^2)/2$. Here, $\Omega$ denotes a bounded domain in $\mathbb{R}^N$, $N\ge 2$, with sufficiently smooth boundary $\Gamma$, say $C^2$, and $u_\nu$ is the outward normal derivative of $u$ on $\Gamma$.
This result inaugurated a new and fruitful field in mathematical research at the confluence of Analysis and Geometry.
Both problems have many applications to other areas of mathematics and natural sciences. In fact (see the next section), Alexandrov's theorem is related with soap bubbles and the classical isoperimetric problem, while Serrin's result -- as Serrin himself explains in \cite{Se} -- was actually motivated by two concrete problems in mathematical physics regarding the torsion of a straight solid bar and the tangential stress of a fluid on the walls of a rectilinear pipe.
\bigskip
\noindent{\bf Some motivations.}
Let us consider a free soap bubble in the space, that is a closed surface $\Gamma$ in $\mathbb{R}^3$ made of a soap film enclosing a domain $\Omega$ (made of air) of fixed volume.
It is known that (see \cite{Roslibro}) if the surface is in equilibrium then it must be a minimizing minimal surface (in the terminology of Chapter \ref{chap:LNCIME}).
This means that its area within any compact region increases when the surface is perturbed within that region.
Thus, soap bubbles solve the classical isoperimetric problem, that asks which sets $\Omega$ in $\mathbb{R}^N$ (if they exist) minimize the surface area for a given volume.
Put in other terms, soap bubbles attain the sign of equality in the classical isoperimetric inequality. Hence, they must be spheres (see also Theorem \ref{thm:servepertesiLNisoperimetric}).
A standard proof of the isoperimetric inequality that hinges on rearrangement techniques can be found in \cite[Section 1.12]{PZ}.
Here, we just want to underline the relation between the isoperimetric problem and Alexandrov's theorem.
In this introduction, we adopt the terminology and techniques pertaining to \textit{shape derivatives}, which will later be useful also to give a motivation to Serrin's problem.
See Chapter \ref{chap:LNCIME}, for a more general treatment pertaining the theory of minimal surfaces.
Thus, we assume the existence of a set $\Omega$ whose boundary $\Gamma$ (of class $C^2$) minimizes its surface area among sets of given volume.
Then, we consider the family of domains $\Omega_t$ such that $\Omega_0= \Omega $, where
\begin{equation}\label{intro:eq:evolutiondomains1}
\Omega_t = \mathcal{M}_t (\Omega)
\end{equation}
and $\mathcal{M}_t: \mathbb{R}^N \to \mathbb{R}^N$ is a mapping such that
\begin{equation}\label{intro:eq:evolutiondomains2}
\mathcal{M}_0 (x) =x, \; \mathcal{M}'_0 (x) = \phi(x) \nu(x) .
\end{equation}
Here, the symbol $'$ means differentiation with respect to $t$, $\phi$ is any compactly supported continuous function, and $\nu$ is a proper extension of the unit normal vector field to a tubular neighborhood of $\Gamma$.
Thus, we consider the area and volume functionals (in the variable $t$)
\begin{equation*}
A(t) = | \Gamma_t | = \int_{\Gamma_t} \, dS_x \quad \text{ and } \quad V(t) = | \Omega_t| = \int_{\Omega_t} \, dx ,
\end{equation*}
where $\Gamma_t$ is the boundary of $\Omega_t$, and $|\Gamma_t|$, $|\Omega_t|$ denote indifferently the $N-1$-dimensional measure of $\Gamma_t$ and the $N$-dimensional measure of $\Omega_t$.
Since $\Omega_0 = \Omega$ is the domain that minimizes $A(t)$ among all the domains in the one-parameter family $\left\lbrace \Omega_t \right\rbrace_{t \in \mathbb{R}}$ that have prescribed volume $|\Omega_t|=V$, the method of {\it Lagrange multipliers} informs us that there exists a number $\lambda$ such that
$$
A'(0) - \lambda V'(0) = 0.
$$
Now by applying Hadamard's variational formula (see \cite[Chapter 5]{HP}) we can directly compute that the derivatives of $V$ and $A$ are given by:
\begin{equation*}
V'(0)= \int_{\Gamma} \phi (x) \, dS_x
\end{equation*}
and
\begin{equation*}
A'(0) = \int_{\Gamma} H(x) \, \phi(x) \, dS_x ,
\end{equation*}
where $H$ is the mean curvature of $\Gamma$.
Therefore, we obtain that
$$
\int_{\Gamma} \left( H(x) - \lambda \right) \, \phi(x) \, dS_x = 0.
$$
Since $\phi$ is arbitrary, we conclude that $H \equiv \lambda$ on $\Gamma$.
The value $\lambda$ can be computed by using {\it Minkowski's identity},
\begin{equation*}
\int_\Gamma H \, <x, \nu> \,dS_x = |\Gamma| ,
\end{equation*}
and equals the number $H_0$ given by
$$
H_0= \frac{| \Gamma|}{N | \Omega|} .
$$
This argument proves that soap bubbles must have constant mean curvature. In turn, Alexandrov's theorem informs us that they must be spheres.
Similar arguments can be applied to optimization of other physical quantities, which also satisfy inequalities of isoperimetric type.
A celebrated example, which is connected with the overdetermined problem \eqref{serrin1}, \eqref{serrin2}, is given by {\it Saint Venant's Principle}, explained below.
To this aim we introduce the {\it torsional rigidity} $\tau(\Omega)$ of a bar of cross-section $\Omega$, that after some normalizations, can be defined as
$$
\max \left\lbrace Q(v) : 0 \not\equiv v \in W_0^{1,2} (\Omega) \right\rbrace , \, \text{ where } \, Q(v)= \frac{ \left( N \, \int_\Omega v \, dx \right)^2}{\int_\Omega |\nabla v|^2 \, dx } .
$$
As mentioned in \cite{Se}, this quantity is related to the solution $u$ of \eqref{serrin1}. In fact, it turns out that $u$ realizes the maximum above, and we have that
\begin{equation*}
\tau(\Omega) = Q(u) = \int_\Omega | \nabla u|^2 \, dx = -N \int_\Omega u \, dx.
\end{equation*}
Saint Venant's Principle (or the isoperimetric inequality for the torsional rigidity) states that the ball maximizes $\tau(\Omega)$ among sets of given volume.
The standard proof of this result also uses rearrangement techniques and can be found in \cite[Section 1.12]{PZ}.
Here, in analogy to what done just before,
we simply show the relation between Saint Venant's Principle and problem \eqref{serrin1},\eqref{serrin2}.
Thus, we suppose that a set $\Omega$ (of class $C^{1, \alpha}$) maximizing $\tau(\Omega)$ among sets of given volume exists and consider the one-parameter family $\Omega_t$ such that $\Omega_0= \Omega$ already defined.
This time, as the parameter $t$ changes, we must describe how the solution $u(t,x)$ of \eqref{serrin1} in $\Omega_t$ changes, since the torsional rigidity of $\Omega_t$ reads as
\begin{equation*}
T(t)= \tau (\Omega_t)= - N \int_{\Omega_t} u (t,x) \, dx .
\end{equation*}
By using the method of Lagrange multipliers as before, we deduce that there exists a number $\lambda$ such that
\begin{equation*}
T'(0) + \lambda V'(0) = 0.
\end{equation*}
By Hadamard's variational formula (see \cite[Chapter 5]{HP}), we can compute
\begin{equation*}
T'(0)= -N \int_{\Omega} u'(x) \, dx - N \int_{\Gamma} u(x) \phi(x) \, dS_x,
\end{equation*}
where $u(x)= u(0,x)$, and $u'(x)$ is the derivative of $u(t,x)$ with respect to $t$, evaluated at $t=0$.
Moreover, it turns out that $u'$ is the solution of the problem
\begin{equation*}
\Delta u' = 0 \, \mbox{ in } \Omega , \, u'= u_\nu \phi \, \mbox{ on } \Gamma.
\end{equation*}
Thus,
we immediately find that
$$
T'(0) = -N \int_{\Omega} u' \, dx = - \int_{\Omega} u' \Delta u \, dx = - \int_{\Gamma} u' u_\nu \, dS_x ,
$$
after an integration by parts in the last equality.
By using this last formula together with the fact that $ u'= u_\nu \phi$ on $\Gamma$,
and recalling the already computed expression for $V'(0)$, we conclude that
$$
0= T'(0) + \lambda V'(0) = \int_{\Gamma} \left( - u_\nu^2 + \lambda \right) \phi \, dS_x .
$$
Since $\phi$ is arbitrary, we deduce that $u_\nu^2 \equiv \lambda$ on $\Gamma$. By the identity
\begin{equation*}
\int_\Gamma u_\nu\,dS_x=N\,|\Omega|,
\end{equation*}
we then compute $\lambda=R^2$,
where
$$
R=\frac{N |\Omega|}{| \Gamma|}.
$$
The core of this thesis concerns symmetry and stability results for the Soap Bubble Theorem, Serrin's problem, and some other related problems. Most of the material and techniques presented in this part are contained in the papers \cite{MP, MP2, Pog}, that are already published or in print. However, in the process of writing the thesis, we realized that some of the stability results described in those papers could be sensibly improved.
Thus, we present our results in this new (technically equivalent) form.
\bigskip
\noindent{\bf Integral identities and symmetry.}
The Soap Bubble Theorem and Serrin's result share several common features. To prove his result, Alexandrov introduced his reflection principle, an elegant geometric technique that also works for other symmetry results concerning curvatures.
Serrin's proof hinges on his method of moving planes, an adaptation and refinement of the reflection principle. That method proves to be a very flexible tool, since it allows to prove
radial symmetry for positive solutions of a far more general class of non-linear equations, that includes the semi-linear equation
\begin{equation}
\label{semilinear}
\Delta u=f(u),
\end{equation}
where $f$ is a locally Lipschitz continuous non-linearity.
\par
Also, alternative proofs of both symmetry results can be given, based on certain integral identities and inequalities.
In fact, in the same issue of the journal in which \cite{Se} is published, H.~F.~Weinberger \cite{We} gave a different proof of Serrin's symmetry result for problem \eqref{serrin1}-\eqref{serrin2} based on integration by parts, a maximum principle for an appropriate $P$-function, and the Cauchy-Schwarz inequality.
These ideas can be extended to the Soap Bubble Theorem,
using R. C. Reilly's argument (\cite{Re}), in which the relevant hypersurface is
regarded as the zero level surface of the solution of \eqref{serrin1}.
The connection between \eqref{serrin1}-\eqref{serrin2} and the Soap Bubble problem is then hinted by the simple differential identity
$$
\Delta u=|\nabla u|\,\mathop{\mathrm{div}}\frac{\nabla u}{| \nabla u|}+\frac{\langle \nabla^2u\,\nabla u, \nabla u\rangle}{|\nabla u|^2};
$$
here, $\nabla u$ and $\nabla^2u$ are the gradient and the hessian matrix of $u$, as standard. If we agree
to still denote by $\nu$ the vector field $\nabla u/|\nabla u|$ (that on $\Gamma$ coincides with the outward unit normal), the above identity and \eqref{serrin1} inform us that
\begin{equation*}
u_{\nu\nu}+(N-1)\,H\,u_\nu=N,
\end{equation*}
on every {\it non-critical} level surface of $u$, and hence on $\Gamma$. In fact, a well known formula states that
the mean curvature $H$ (with respect to the inner normal) of a regular level surface of $u$ equals
$$
\frac1{N-1}\,\mathop{\mathrm{div}}\frac{\nabla u}{|\nabla u|}.
$$
\par
In both problems, the radial symmetry of $\Gamma$ will follow from that of the solution $u$ of \eqref{serrin1}. In fact, we will show that in Newton's inequality,
\begin{equation*}
(\Delta u)^2\le N\,|\nabla^2 u|^2,
\end{equation*}
that holds pointwise in $\Omega$ by the Cauchy-Schwarz inequality, the equality sign is identically attained in $\Omega$. The point is that such equality holds if and only if
$u$ is a quadratic polynomial $q$ of the form
\begin{equation*}
q(x)=\frac12\, (|x-z|^2-a),
\end{equation*}
for some choice of $z\in\mathbb{R}^N$ and $a\in\mathbb{R}$ (see Lemma \ref{lem:sphericaldetector}). The boundary condition in \eqref{serrin1} will then
tell us that $\Gamma$ must be a sphere centered at $z$.
\par
The starting points of our analysis are the following two integral identities (see Theorems \ref{thm:serrinidentity} and \ref{thm:identitySBT}):
\begin{equation}
\label{intro:idwps}
\int_{\Omega} (-u) \left\{ |\nabla ^2 u|^2- \frac{ (\Delta u)^2}{N} \right\} dx=
\frac{1}{2}\,\int_\Gamma \left( u_\nu^2- R^2\right) (u_\nu-q_\nu)\,dS_x
\end{equation}
and
\begin{multline}
\label{intro:H-fundamental}
\frac1{N-1}\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}dx+
\frac1{R}\,\int_\Gamma (u_\nu-R)^2 dS_x = \\
\int_{\Gamma}(H_0-H)\, (u_\nu)^2 dS_x.
\end{multline}
Here, $R$ and $H_0$ are the already mentioned reference constants:
\begin{equation*}
R=\frac{N\,|\Omega|}{|\Gamma|}, \quad H_0=\frac1{R}=\frac{|\Gamma|}{N\,|\Omega|}.
\end{equation*}
Identities \eqref{intro:idwps} and \eqref{intro:H-fundamental} hold regardless of how the point $z$ or the constant $a$ are chosen for $q$.
Identity \eqref{intro:idwps}, claimed in \cite[Remark 2.5]{MP} and proved in \cite{MP2}, puts together and refine Weinberger's identities and some remarks of L.~E.~Payne and P.~W.~Schaefer \cite{PS}. Identity \eqref{intro:H-fundamental} was proved in \cite[Theorem 2.2]{MP} by polishing the arguments contained in \cite{Re}.
It is thus evident that each of the two identities gives spherical symmetry if respectively $u_\nu=R$ or $H=H_0$ on $\Gamma$, since Newton's inequality holds with the equality sign (notice that in \eqref{intro:idwps} $-u>0$ by the strong maximum principle). The same conclusion is also achieved if we only assume that $u_\nu$ or $H$ are constant on $\Gamma$, since those constants must equal $R$ and $H_0$, as already observed.
Thus, \eqref{intro:idwps} and \eqref{intro:H-fundamental} give new elegant proofs of Alexandrov's and Serrin's results.
Moreover,
they have several advantages that we shall describe in this and the next section.
One advantage is that we can obtain symmetry under weaker assumptions.
In fact, observe that to get spherical symmetry it is enough to prove that the right-hand sides in the two identities are non-positive. For instance, in the case of the Soap Bubble Theorem (see Theorem \ref{th:SBT}) one gets the spherical symmetry if
$$\int_{\Gamma}(H_0-H)\, (u_\nu)^2 dS_x \le 0 ,$$
which is certainly true if $H\ge H_0$ on $\Gamma$.
Such slight generalization has been exploited in \cite{Pog} to improve a symmetry result of Garofalo and Sartori (\cite{GS}) for the p-capacitary potential with constant normal derivative. In fact, in \cite{Pog} the star-shaped assumption used in \cite{GS} has been removed.
Another advantage is that \eqref{intro:idwps} and \eqref{intro:H-fundamental} tell us something more about Saint Venant's principle and the classical isoperimetric problem described in the previous section.
In fact, it has been noticed in \cite[Theorem 7]{Mag} that, to infer that a domain $\Omega$ is a ball, it is sufficient to require that, under the
flow \eqref{intro:eq:evolutiondomains1}-\eqref{intro:eq:evolutiondomains2} with the choice $\phi= (u_\nu - q_\nu)/2$, the function
$$
t \to T(t) + R^2 (V(t) - V)
$$
has non-positive derivative at $t=0$. In particular, the same conclusion holds if $t=0$ is a critical point.
In fact, with that choice of $\phi$, \eqref{intro:eq:evolutiondomains2} gives that
$$
T'(0)+ R^2 V'(0)= \frac{1}{2} \int_\Gamma (u_\nu - R^2) \, (u_\nu - q_\nu) \, dS_x ,
$$
and the conclusion follows from \eqref{intro:idwps}.
For the case of the classical isoperimetric problem we can deduce a similar statement.
In fact we can infer that a domain $\Omega$ is a ball if, under the
flow \eqref{intro:eq:evolutiondomains1}-\eqref{intro:eq:evolutiondomains2} with the choice $\phi=- u_\nu^2$, the function
$$
t \to A(t) - H_0 (V(t) - V)
$$
has non-positive derivative at $t=0$. In particular, the same conclusion holds if $t=0$ is a critical point.
In fact, with that choice of $\phi$, \eqref{intro:eq:evolutiondomains2} gives that
\begin{equation*}
A'(0) - H_0 V'(0)= \int_\Gamma (H_0 - H) \, ( u_\nu^2) \, dS_x ,
\end{equation*}
and the conclusion follows from \eqref{intro:H-fundamental}.
Thus, \eqref{intro:idwps} and \eqref{intro:H-fundamental} inform us that the flows generated respectively by $\phi= (u_\nu - q_\nu)/2$ and $\phi= - u_\nu^2$ are quite privileged.
\bigskip
\noindent{\bf Stability results.}
The greatest benefit
produced by our identities are undoubtedly the stability results for the Soap Bubble Theorem, Serrin's problem, and other related overdetermined problems.
Technically speaking, there are several ways to describe the closeness of a domain to a ball. In this thesis, we mainly privilege
the following: find two concentric balls $B_{\rho_i} (z)$ and $B_{\rho_e} (z)$, centered at $z \in \Omega$ with radii $\rho_i$ and $\rho_e$, such that
\begin{equation}
\label{balls}
B_{\rho_i} (z) \subseteq \Omega \subseteq B_{\rho_e} (z)
\end{equation}
and
\begin{equation}
\label{stability}
\rho_e-\rho_i\le \psi(\eta),
\end{equation}
where
$\psi:[0,\infty)\to[0,\infty)$ is a continuous function vanishing at $0$ and $\eta$ is a suitable measure of the deviation of $u_\nu$ or $H$ from being a constant.
The landmark results of this thesis are the following stability estimates:
\begin{equation}\label{intro:eq:Improved-Serrin-stability}
\rho_e-\rho_i\le C \, \Vert u_\nu - R \Vert_{2,\Gamma}^{\tau_N}
\end{equation}
and
\begin{equation}
\label{intro:eq:SBT-improved-stability}
\rho_e-\rho_i\le C\,\Vert H_0-H\Vert_{2,\Gamma}^{\tau_N} .
\end{equation}
In \eqref{intro:eq:Improved-Serrin-stability} (see Theorem \ref{thm:Improved-Serrin-stability} for details), $\tau_2 = 1$, $\tau_3$ is arbitrarily close to one,
and $\tau_N = 2/(N-1)$ for $N \ge 4$.
In \eqref{intro:eq:SBT-improved-stability} (see Theorem \ref{thm:SBT-improved-stability} for details), $\tau_N=1$ for $N=2, 3$, $\tau_4$ is arbitrarily close to one, and
$\tau_N = 2/(N-2) $ for $N\ge 5$.
The constants $C$ depend on the dimension $N$, the diameter $d_\Omega$, the radii $r_i$, $r_e$ of the uniform interior and exterior sphere conditions, and the distance $\de_\Gamma (z)$ of $z$ to $\Gamma$.
As pointed out in Remarks \ref{rem:removing-distance-convex} and \ref{remarkprova:nuova scelte z}, the dependence on $\de_\Gamma (z)$ of the constants in \eqref{intro:eq:Improved-Serrin-stability} and \eqref{intro:eq:SBT-improved-stability} can (so far) be removed when $\Omega$ is convex or by assuming some additional requirements.
The stability results for Serrin's problem so far obtained in the literature can be divided into two groups, depending on the method employed. One is based on a quantitative study of the method of moving planes and hinges on the use of Harnack's inequality. In \cite{ABR}, where the stability issue was considered for the first time, \eqref{balls} and \eqref{stability} are obtained with
$\psi(\eta)=C\,|\log \eta|^{-1/N}$ and $\eta=\Vert u_\nu-c\Vert_{C^1(\Gamma)}$.
The estimate was later improved in \cite{CMV} to get
\begin{equation*}
\psi(\eta)=C\,\eta^{\tau_N} \quad \mbox{and} \quad \eta=\sup_{\substack{x,y \in \Gamma\\ \ x \neq y}} \frac{|u_\nu(x) - u_\nu(y)|}{|x-y|}.
\end{equation*}
The exponent $\tau_N \in (0,1)$ can be computed for a general setting and, if $\Omega$ is convex,
is arbitrarily close to $1/(N+1)$.
It should be also stressed that the method works in the more general case of the semilinear equation \eqref{semilinear}.
The approach of using integral identities and inequalities, in the wake of Weinberger's proof of symmetry, was inaugurated in \cite{BNST}, and later improved in \cite{Fe} and \cite{MP2}. This method has worked so far only for problem \eqref{serrin1}, though.
The main result in \cite{BNST} states that (if a uniform bound on $u_\nu$ is assumed) $\Omega$ can be approximated in measure by a finite number of mutually disjoint balls $B_i$. The error in the approximation is
$\psi(\eta)=C\,\eta^{1/(4N+9)}$ where $\eta=\Vert u_\nu-c\Vert_{1,\Gamma}$. There, it is also proved that
\eqref{balls} and \eqref{stability} hold with
$\psi(\eta)=C\,\eta^{1/2(4N+9)(N-1)}$ and $\eta=\Vert u_\nu-c\Vert_{\infty, \Gamma}$.
In \cite[Theorem 1.1]{MP2} we have obtained
$\psi(\eta)= C\,\eta^{\frac{2}{N+2} }$ where $\eta = \Vert u_\nu - R \Vert_{2,\Gamma}$, which already improves on \cite{ABR, CMV, BNST}.
While writing this thesis, we obtained \eqref{intro:eq:Improved-Serrin-stability}, that improves on \cite{MP2} (for every $N \ge 2$) to the extent that it gains the (optimal) Lipschitz stability in the case $N=2$.
In this thesis we will also consider a weaker $L^1$-deviation (as in \cite{BNST}) and prove (see Theorem \ref{thm:Serrin-stability}) the inequality
\begin{equation}
\label{stability-Serrin}
\rho_e-\rho_i\le C\,\Vert u_\nu-R\Vert_{1,\Gamma}^{\tau_N/2},
\end{equation}
where $\tau_N$ is the same appearing in \eqref{intro:eq:Improved-Serrin-stability}.
Also this inequality is new and refines one stated in \cite[Theorem 3.6]{MP2}
in which $\tau_N /2$ was replaced by $1/(N+2)$.
Of course, that inequality was already better than that obtained in \cite{BNST}.
In \cite{Fe} instead, a different measure of closeness to spherical symmetry is adopted, by considering a slight modification of the so-called Fraenkel asymmetry:
\begin{equation}
\label{intro:asymmetry}
{\mathcal A}(\Omega)=\inf\left\{\frac{|\Omega\Delta B^x|}{|B^x|}: x \mbox{ center of a ball $B^x$ with radius $R$} \right\}.
\end{equation}
Here, $\Omega\Delta B^x$ denotes the symmetric difference of $\Omega$ and $B^x$.
It is then obtained a Lipschitz-type estimate: ${\mathcal A}(\Omega)\le C\,\Vert u_\nu-R\Vert_{2,\Gamma}$.
The control given by $\rho_e-\rho_i$ is stronger than that given by ${\mathcal A}(\Omega)$ (see Section \ref{sec:stabbyasymmetrySBT}).
\medskip
Let us now turn our attention to the Soap Bubble Theorem and comment on our estimate \eqref{intro:eq:SBT-improved-stability}.
The only stability result based on Alexandrov's reflection principle has been obtained in \cite{CV} and states that there exist two positive constants $C$, $\varepsilon$ such that \eqref{balls} and \eqref{stability} hold for $\psi(\eta)=C\eta$ if $\eta= \Vert H_0-H\Vert_{\infty,\Gamma} < \varepsilon .$
In \cite{CM} and \cite{KM}, similar results (again with the uniform deviation) are obtained, based on the proof of the Soap Bubble Theorem via Heintze-Karcher's inequality (that holds if $\Gamma$ is mean convex) given in \cite{Ro}. Moreover, in \cite{KM} \eqref{balls} and \eqref{stability} are obtained also with $\psi(\eta)= C \, \eta$ and $\eta= \Vert H_0-H\Vert_{2,\Gamma}$, for surfaces that are small normal deformations of spheres.
Further advances have been obtained in \cite[Theorem 1.2]{MP2} and \cite[Theorem 4.1]{MP}. In fact, based on Reilly's proof (\cite{Re}) of the Soap Bubble Theorem, \eqref{balls} and \eqref{stability} are shown to hold for general surfaces respectively with
$$
\psi(\eta)= C\, \eta^{\tau_N} \quad \mbox{and} \quad \eta=\Vert H_0-H\Vert_{2,\Gamma}.
$$
$$
\psi(\eta) = C\, \eta^{\tau_N /2} \quad \mbox{and} \quad \eta = \int_{\Gamma}(H_0-H)^+\,dS_x;
$$
Here, $\tau_N=1$ for $N=2,3$, and $\tau_N=2/(N+2)$ for $N \ge 4$.
Both these estimates are improved
by \eqref{intro:eq:SBT-improved-stability}
and
\begin{equation}
\label{the-estimate}
\rho_e-\rho_i\le C\,\left\{\int_{\Gamma}(H_0-H)^+\,dS_x\right\}^{\tau_N /2}
\end{equation}
(see Theorem \ref{thm:SBT-stability}), which are original material of this thesis. In \eqref{the-estimate}, $\tau_N$ is the same appearing in \eqref{intro:eq:SBT-improved-stability}.
If we compare the exponents in \eqref{intro:eq:SBT-improved-stability} and \eqref{the-estimate} to those obtained in \cite[Theorem 1.2]{MP2} and \cite[Theorem 4.1]{MP}, we notice that the dependence of $\tau_N$ on $N$ has become virtually continuous, in the sense that $\tau_N\to 1$, if $N$ ``approaches'' $4$ from below or from above.
\par
As a final important achievement, by arguments similar to those of \cite{Fe}, we report in Theorem \ref{th:asymmetry} the (optimal) inequality for the asymmetry \eqref{intro:asymmetry} obtained in \cite[Theorem 4.6]{MP2}:
\begin{equation}\label{eq:introasymmetrystab}
{\mathcal A}(\Omega)\le C\,\Vert H_0-H\Vert_{2,\Gamma}.
\end{equation}
\medskip
In the remaining part of this introduction, we shall pinpoint the key remarks in the proof of \eqref{intro:eq:Improved-Serrin-stability} and \eqref{intro:eq:SBT-improved-stability}.
\par
First, notice that the function $h=q-u $ is harmonic and we have that
\begin{equation*}
|\nabla ^2 u|^2-\frac{(\Delta u)^2}{N} = |\nabla^2 h|^2 .
\end{equation*}
Thus, \eqref{intro:idwps} reads as:
\begin{equation}
\label{intro:idwps-h}
\int_{\Omega} (-u)\, |\nabla ^2 h|^2\,dx=
\frac{1}{2}\,\int_\Gamma ( R^2-u_\nu^2)\, h_\nu\,dS_x.
\end{equation}
Also, since $h=q$ on $\Gamma$, if we choose $z$ in $\Omega$, it holds that
\begin{equation*}
\max_{\Gamma} h-\min_{\Gamma} h=\frac12\,(\rho_e^2-\rho_i^2)\ge \frac{r_i}{2} \, (\rho_e-\rho_i).
\end{equation*}
Now, observe that \eqref{intro:idwps-h} holds regardless of the choice of the parameters $z$ and $a$ defining $q$. We will thus complete the first step of our proof by choosing $z\in\Omega$ in a way that the oscillation of $h$ on $\Gamma$ can be bounded in terms of the volume integral in \eqref{intro:idwps-h}.
\par
To carry out this plan, we use three ingredients. First, as done in Lemmas \ref{lem:Lp-estimate-oscillation-generic-v} and \ref{lem:L2-estimate-oscillation}, we show that the oscillation of $h$ on $\Gamma$, and hence $\rho_e - \rho_i$ can be bounded from above in the following way:
\begin{equation}\label{intro:eq:lemmanuovooscillationLpnorm}
\rho_e - \rho_i \le C (N, p, d_\Omega, r_i, r_e) \, \Vert h - h_\Omega \Vert_{p, \Omega}^{p/(N+p)} ,
\end{equation}
where $h_\Omega$ is the mean value of $h$ on $\Omega$ and $p \in \left[1, \infty \right)$.
We emphasize that this inequality is new and generalizes to any $p$ the estimate obtained in \cite[Lemma 3.3]{MP} for $p=2$.
Secondly (see Lemma \ref{lem:relationdist}), we easily obtain the bound
\begin{equation*}
\frac{r_i}{2} \,\de_\Gamma(x) \le -u \ \mbox { on } \ \overline{\Omega}.
\end{equation*}
Thirdly, we choose $z \in \Omega$ as a minimum (or any critical) point of $u$ and we apply two integral inequalities to $h$ and its first (harmonic) derivatives. One is the Hardy-Poincar\'e-type inequality
\begin{equation}
\label{boas-straube}
\Vert v \Vert_{r, \Omega} \le C(N, r, p, \alpha, d_\Omega, r_i, \de_\Gamma (z)) \, \Vert \de_\Gamma(x)^\alpha \, \nabla v(x) \Vert_{p, \Omega},
\end{equation}
that is applied to the first (harmonic) derivatives of $h$. It holds for any harmonic function $v$
in $\Omega$ that is zero at some given point in $\Omega$ (in our case that point will be $z$, since $\nabla h(z)=0$), when $r, p, \alpha$ are three numbers such that either $1 \le p \le r \le \frac{Np}{N-p(1 - \alpha )}$, $p(1 - \alpha)<N$, $0 \le \alpha \le 1$ (see Lemma \ref{lem:John-two-inequalities}),
or $1 \le r=p <\infty$ and $0 \le \alpha \le 1$ (see Lemma \ref{lem:two-inequalities}).
The other one is applied to
$h- h_\Omega$ and is the Poincar\'e-type inequality
\begin{equation}
\label{classicalpoincare}
\Vert v \Vert_{r, \Omega} \le C(N, r, p, d_\Omega, r_i) \, \Vert \nabla v(x) \Vert_{p, \Omega}
\end{equation}
that holds for any function $v\in W^{1,p}(\Omega)$ with zero mean value on $\Omega$, where $r$ and $p$ must satisfy the same inequalities mentioned above for \eqref{boas-straube} when $\alpha=0$ (see
Lemmas \ref{lem:John-two-inequalities} and \ref{lem:two-inequalities}).
This third step is accomplished in Theorem \ref{thm:serrin-W22-stability}, where all the details can be found.
\par
Thus, putting together the above arguments gives that (see Theorem \ref{thm:serrin-W22-stability}) there exists a constant $C$ depending on $N$, $d_\Omega$, $r_i$, $r_e$, $\de_\Gamma (z)$
such that
\begin{equation}
\label{the-estimate-for-h}
\rho_e - \rho_i \le C \left(\int_{\Omega} (-u)\, |\nabla ^2 h|^2\,dx\right)^{\tau_N /2},
\end{equation}
where $\tau_N$ is that appearing in \eqref{intro:eq:Improved-Serrin-stability}.
We mention that in dimension $N=2$ there is no need to use \eqref{intro:eq:lemmanuovooscillationLpnorm}, thanks to Sobolev imbedding theorem (see item (i) of Theorem \ref{thm:serrin-W22-stability}).
Next, we work on the right-hand side of \eqref{intro:idwps-h}. The important observation is that, if $u_\nu-R$ tends to $0$, also $h_\nu$ does. Quantitatively, this fact can be expressed by the inequality
\begin{equation}\label{intro:eq:tracehnu}
\Vert h_\nu\Vert_{2,\Gamma}\le C(N, d_\Omega, r_i, r_e, \de_\Gamma (z) ) \,\Vert u_\nu-R\Vert_{2,\Gamma},
\end{equation}
that can be derived (see the proof of Theorem \ref{thm:Improved-Serrin-stability}) by exploiting arguments contained in \cite{Fe}.
Thus, after an application of H\"older's inequality to the right-hand side of \eqref{intro:idwps-h}, by \eqref{intro:eq:tracehnu} we deduce that
$$
\int_\Omega (-u) |\nabla^2 h|^2 dx \le C(N, d_\Omega, r_i, r_e, \de_\Gamma (z) ) \,\Vert u_\nu-R\Vert_{2,\Gamma}^2 ,
$$
and \eqref{intro:eq:Improved-Serrin-stability} will follow by \eqref{the-estimate-for-h}.
\smallskip
Let us now sketch the proof of \eqref{intro:eq:SBT-improved-stability}.
We fix again $z$ at a local minimum point of $u$ in $\Omega$.
In this way, $z \in \Omega$ and $h=q-u$ is such that $\nabla h(z)=0.$
Thus, as before we can apply to $h-h_\Omega$ and to its first (harmonic) derivatives the Poincar\'e-type inequalities \eqref{classicalpoincare} and \eqref{boas-straube} (with $\alpha=0$).
In this way, by exploiting again \eqref{intro:eq:lemmanuovooscillationLpnorm}, we get that (see Theorem \ref{thm:SBT-W22-stability}) there exists a constant $C$ depending on $N$, $d_\Omega$, $r_i$, $r_e$, $\de_\Gamma (z)$
such that
\begin{equation}
\label{eq:step2}
\rho_e - \rho_i \le C \| \nabla^2 h \|^{\tau_N}_{2, \Omega},
\end{equation}
where $\tau_N$ is that appearing in \eqref{intro:eq:SBT-improved-stability}.
We mention that when $N=2,3$, there is no need to use \eqref{intro:eq:lemmanuovooscillationLpnorm}, thanks to Sobolev imbedding theorem (see item (i) of Theorem \ref{thm:SBT-W22-stability}).
Next, we work on the right-hand side of \eqref{intro:H-fundamental} and we notice that
it can be rewritten as (see Theorem \ref{thm:identitySBT})
$$
\int_{\Gamma}(H_0-H)\,(u_\nu-q_\nu)\,u_\nu\,dS_x+
\int_{\Gamma}(H_0-H)\, (u_\nu-R)\,q_\nu\, dS_x.
$$
Hence, \eqref{intro:H-fundamental} can be written in terms of $h$ as
\begin{multline}
\label{intro:identity-SBT-h}
\frac1{N-1}\int_{\Omega} |\nabla ^2 h|^2 dx+
\frac1{R}\,\int_\Gamma (u_\nu-R)^2 dS_x = \\
-\int_{\Gamma}(H_0-H)\,h_\nu\,u_\nu\,dS_x+
\int_{\Gamma}(H_0-H)\, (u_\nu-R)\,q_\nu\, dS_x.
\end{multline}
Discarding the first summand at the left-hand side of \eqref{intro:identity-SBT-h} and applying H\"older's inequality and \eqref{intro:eq:tracehnu} to its right-hand side yield that
$$
\Vert u_\nu-R\Vert_{2,\Gamma}\le C (N, d_\Omega, r_i, r_e, \de_\Gamma (z) ) \,\Vert H_0-H\Vert_{2,\Gamma}.
$$
Thus, by discarding the second summand at the left-hand side of \eqref{intro:identity-SBT-h} we get
\begin{equation*}
\| \nabla ^2 h \|_{2,\Omega} \le C (N, d_\Omega, r_i, r_e, \de_\Gamma (z) ) \| H_0 - H \|_{2, \Gamma}.
\end{equation*}
Hence, \eqref{intro:eq:SBT-improved-stability} easily follows by putting together this last inequality and \eqref{eq:step2}.
\bigskip
\noindent{\bf Plan of the work.}
The core of the thesis comprises four chapters.
Chapter \ref{chapter:integral identities and symmetry results} contains integral identities and related symmetry results.
In Section \ref{sec:integral identities} we prove \eqref{intro:idwps} and \eqref{intro:H-fundamental}, as well as other related identities.
Section \ref{sec:symmetry results via integral identities} contains the corresponding relevant symmetry results.
Chapter \ref{chap:PogAA} describes symmetry theorems that concern overdetermined problems for $p$-harmonic functions in exterior and punctured domains. They are essentially the results proved in \cite{Pog}.
In Chapter \ref{chapter:various estimates} we collect all the estimates for the torsional rigidity density $u$ and the harmonic function $h$ that will be useful to derive our stability results.
Section \ref{sec:gradient estimate for torsion} is dedicated to the function $u$ and contains pointwise estimates for $u$ and its gradient.
Section \ref{sec:harmonic functions in weighted spaces} collects Hardy-Poincar\'e inequalities as well as a proof of a trace inequality for harmonic functions which is the key ingredient to get \eqref{intro:eq:tracehnu}.
In Section \ref{sec:estimates for the oscillation of harmonic functions} we present the crucial lemma which allows to prove \eqref{intro:eq:lemmanuovooscillationLpnorm}.
All the inequalities for the particular harmonic function $h= q- u$ that are necessary to get our stability estimates are presented in Section \ref{sec:estimates for h}, as a consequence of Sections \ref{sec:gradient estimate for torsion}, \ref{sec:harmonic functions in weighted spaces}, \ref{sec:estimates for the oscillation of harmonic functions}.
Chapter \ref{chapter:Stability results} contains all the stability results for the spherical configuration.
Section \ref{sec:stability Serrin} is devoted to the case of Serrin's problem \eqref{serrin1}, \eqref{serrin2} and contains the proof of inequalities \eqref{intro:eq:Improved-Serrin-stability} and \eqref{stability-Serrin}.
Section \ref{sec:Alexandrov's SBT} is devoted to the case of Alexandrov's Soap Bubble Theorem and contains the proof of \eqref{intro:eq:SBT-improved-stability} and \eqref{the-estimate}.
In Section \ref{sec:stabbyasymmetrySBT} we consider the asymmetry ${\mathcal A}(\Omega)$
defined in \eqref{intro:asymmetry}, we prove \eqref{eq:introasymmetrystab}, and we compare ${\mathcal A}(\Omega)$ with $\rho_e -\rho_i$.
Finally, Section \ref{sec:Other stability results mean curvature} collects other related stability results.
As already mentioned, the second part of the thesis, entitled ``Other works'', contains a couple of papers as they were published. For the sake of coherence with the topic of this dissertation, we decided not to present them in the main part.
In chapter \ref{chap:LNCIME} are reported the lecture notes \cite{CP} of a CIME summer course taught by Prof. Xavier Cabr\'e in Cetraro during the week of June 19-23, 2017. The author of the present thesis attended that course and then collaborated in writing those notes.
The second paper, reported in Chapter \ref{chap:Littlewoodfourthprinciple}, is \cite{MPLittlewood}. That is a short article stimulated by some remarks made while the author of the present thesis was attending (as undergraduate student) the class ``Analisi Matematica III'' taught by Prof. Rolando Magnanini at the Universit\`a di Firenze.
\chapter{Integral identities and symmetry}\label{chapter:integral identities and symmetry results}
In this chapter, we shall present a number of integral identities which our symmetry and stability results are based on. In them, the torsional rigidity density $u$, i.e. the solution of \eqref{serrin1}, plays the main role. Their proofs are combinations of the divergence theorem, differential identities, and some ad hoc manipulations.
Before we start, we set some relevant notations.
By $\Omega\subset\mathbb{R}^N$, $N\ge 2$, we shall denote a bounded domain, that is a connected bounded open set, and call $\Gamma$ its boundary.
By $|\Omega|$ and $|\Gamma|$, we will denote indifferently the $N$-dimensional Lebesgue measure of $\Omega$
and the surface measure of $\Gamma$. When $\Gamma$ is of class $C^1$, $\nu$ will denote the (exterior) unit normal vector field to $\Gamma$ and, when $\Gamma$ is
of class $C^2$, $H(x)$ will denote its mean curvature (with respect to $-\nu(x)$) at $x\in\Gamma$.
\par
As already done in the introduction, we set $R$ and $H_0$ to be the two reference constants given by
\begin{equation}
\label{def-R-H0}
R=\frac{N\,|\Omega|}{|\Gamma|}, \quad H_0=\frac1{R}=\frac{|\Gamma|}{N\,|\Omega|},
\end{equation}
and we use the letter $q$ to denote the quadratic polynomial defined by
\begin{equation}
\label{quadratic}
q(x)=\frac12\, (|x-z|^2-a),
\end{equation}
where $z$ is any point in $\mathbb{R}^N$ and $a$ is any real number.
\section{Integral identities for the torsional rigidity density}\label{sec:integral identities}
We start by recalling the following classical Rellich-Pohozaev identity (\cite{Poh}).
\begin{lem}[Rellich-Pohozaev identity]
Let $\Omega$ be a domain with boundary $\Gamma$ of class $C^1$.
For every function $u \in C^2(\Omega) \cap C^1(\overline{\Omega})$, it holds that
\begin{multline}
\label{eq:idPohozaevgeneral}
\frac{N-2}{2} \int_{\Omega} |\nabla u|^2 \, dx - \int_{\Omega} <x , \nabla u > \Delta u \, dx =
\\
\frac{1}{2} \int_\Gamma |\nabla u|^2 <x, \nu> \, dS_x - \int_\Gamma <x,\nabla u> u_\nu \, dS_x.
\end{multline}
\end{lem}
\begin{proof}
By direct computation it is easy to verify the following differential identity:
$$
\mathop{\mathrm{div}} \left\lbrace \frac{| \nabla u|^2}{2} \, x - <x, \nabla u> \nabla u \right\rbrace = \frac{N-2}{2} \, |\nabla u|^2 - <x,\nabla u> \Delta u.
$$
Thus, \eqref{eq:idPohozaevgeneral} easily follows by integrating over $\Omega$ and applying the divergence theorem.
\end{proof}
We now focus our attention on the solution $u$ of \eqref{serrin1}.
In this case, it is easy to check that \eqref{eq:idPohozaevgeneral} becomes the identity stated in the following corollary.
\begin{cor}[Rellich-Pohozaev identity for the torsional rigidity density]
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^{1, \alpha}$, $0< \alpha \le 1$.
Then the solution $u$ of \eqref{serrin1} satisfies the identity
\begin{equation}
\label{Pohozaev}
(N+2) \int_{\Omega} |\nabla u|^2 \, dx =\int_\Gamma (u_\nu)^2 \, q_\nu\,dS_x ,
\end{equation}
where $q$ denotes the quadratic polynomial defined in \eqref{quadratic}.
\end{cor}
\begin{proof}
If $\Gamma$ is of class $C^{1,\alpha}$, then $u\in C^{1,\alpha}(\overline{\Omega})\cap C^2(\Omega)$, and hence \eqref{eq:idPohozaevgeneral} holds.
Since $u$ satisfies \eqref{serrin1}, the following differential identity holds
$$
\mathop{\mathrm{div}} \left( u \, x - u \, \nabla u \right) = < \nabla u, x> - | \nabla u |^2.
$$
By integrating over $\Omega$, applying the divergence theorem, and recalling the boundary
condition in \eqref{serrin1}, we obtain that
\begin{equation*}
\int_\Omega | \nabla u|^2 \, dx = \int_{\Omega} <\nabla u , x> \, dx.
\end{equation*}
Thus, the left-hand side of \eqref{eq:idPohozaevgeneral} becomes
$$
- \left( \frac{N+2}{2} \right) \int_\Omega | \nabla u|^2.
$$
On the other hand, by using the fact that $\nabla u = u_\nu \, \nu$ on $\Gamma$, the right-hand side of \eqref{eq:idPohozaevgeneral} becomes
$$
- \frac{1}{2} \int_\Gamma (u_\nu)^2 <x, \nu> ,
$$
and the conclusion follows.
\end{proof}
Following the tracks of Weinberger \cite{We} we introduce the P-function,
\begin{equation}
\label{P-function}
P = \frac{1}{2}\,|\nabla u|^2 - u,
\end{equation}
and we easily compute that
\begin{equation}
\label{differential-identity}
\Delta P = |\nabla^2 u|^2-\frac{(\Delta u)^2}{N}.
\end{equation}
We are now ready to provide the proof of the fundamental identity for Serrin's problem.
\begin{thm}[Fundamental identity for Serrin's problem, \cite{MP,MP2}]
\label{thm:serrinidentity}
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^{1,\alpha}$, $0<\alpha\le 1$, and $R$ be the positive constant defined in \eqref{def-R-H0}.
Then the solution $u$ of \eqref{serrin1} satisfies
\begin{equation}
\label{idwps}
\int_{\Omega} (-u) \left\{ |\nabla ^2 u|^2- \frac{ (\Delta u)^2}{N} \right\} dx=
\frac{1}{2}\,\int_\Gamma \left( u_\nu^2- R^2\right) (u_\nu-q_\nu)\,dS_x .
\end{equation}
\end{thm}
\begin{proof}
First, suppose that $\Gamma$ is of class $C^{2,\alpha}$, so that $u\in C^{2,\alpha}(\overline{\Omega})$. Integration by parts then gives:
$$
\int_\Omega (u\,\Delta P-P\,\Delta u)\,dx=\int_\Gamma(u\,P_\nu-u_\nu\,P)\,dS_x.
$$
Thus, since $u$ satisfies \eqref{serrin1}, we have that
\begin{equation}
\label{parts}
\int_\Omega (-u)\,\Delta P\,dx=-N\,\int_\Omega P\,dx+\frac12\,\int_\Gamma u_\nu^3\,dS_x,
\end{equation}
being $P=|\nabla u|^2/2=u_\nu^2/2$ on $\Gamma$.
\par
Next, notice that we can write
$
u_\nu^3 = (u_\nu^2 - R^2)(u_\nu - q_\nu) + R^2 (u_\nu - q_\nu) + u_\nu^2 q_\nu.
$
By the divergence theorem and \eqref{Pohozaev} we then compute:
\begin{multline*}
N \int_{\Omega} P\,dx=\frac{N}{2} \int_\Omega |\nabla u|^2 dx-\int_\Omega u\,\Delta u\,dx =
\left( \frac{N}{2} + 1 \right) \int_{\Omega} |\nabla u|^2 \, dx
=
\\
\frac{1}{2} \int_\Gamma u_\nu^2 q_\nu \, dS_x .
\end{multline*}
Thus, this identity, \eqref{parts}, and \eqref{differential-identity} give \eqref{idwps}, since
$$
\int_\Gamma (u_\nu-q_\nu)\,dS_x=0,
$$
being $u-q$ harmonic in $\Omega$.
\par
If $\Gamma$ is of class $C^{1,\alpha}$, then $u\in C^{1,\alpha}(\overline{\Omega})\cap C^2(\Omega)$. Thus, by a standard approximation argument, we conclude that \eqref{idwps} holds also in this case.
\end{proof}
\begin{rem}
{\rm
The assumptions on the regularity of $\Gamma$ can further be weakened. For instance, if $\Omega$ is a
(bounded) convex domain, then inequality \eqref{bound-M-convex} below gives that $u_\nu$ is essentially bounded on $\Gamma$ with respect to the $(N-1)$-dimensional Hausdorff measure on $\Gamma$. Thus, an approximation argument again gives that \eqref{idwps} holds true.
}
\end{rem}
We now present a couple of integral identities, involving the torsional rigidity density $u$ and a harmonic function $v$, which are necessary to establish a useful trace inequality (see Lemma \ref{lem:genericv-trace inequality} below).
\begin{lem}\label{lem:identityfortraceineq}
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^{1,\alpha}$, $0<\alpha\le 1$, and let $u$ be the solution of \eqref{serrin1}. Then, for every harmonic function $v$ in $\Omega$ the following two identities hold:
\begin{equation}\label{eq:nuovaidentityforidentityfortraceinequality}
\int_{\Gamma} v^2 u_{\nu} \, dS_x = N \int_{\Omega} v^2 dx + 2 \int_{\Omega} (-u) |\nabla v|^2 dx ;
\end{equation}
\begin{equation}\label{eq:identityfortraceinequality}
\int_{\Gamma} |\nabla v|^2 u_{\nu} dS_x = N \int_{\Omega} |\nabla v|^2 dx + 2 \int_{\Omega} (-u) |\nabla^2 v|^2 dx.
\end{equation}
\end{lem}
\begin{proof}
We begin with the following differential identity:
\begin{equation*}
\mathop{\mathrm{div}}\,\{v^2 \nabla u - u \, \nabla(v^2)\}= v^2 \Delta u - u \, \Delta (v^2)= N \, v^2 - 2 u \, |\nabla v|^2,
\end{equation*}
that holds for any $v$ harmonic function in $\Omega$, if $u$ satisfies \eqref{serrin1}.
Next, we integrate on $\Omega$ and, by the divergence theorem,
we get \eqref{eq:nuovaidentityforidentityfortraceinequality}.
If we use \eqref{eq:nuovaidentityforidentityfortraceinequality} with $v=v_{i}$, and hence we sum up over $i=1,\dots, N$, we obtain \eqref{eq:identityfortraceinequality}.
\end{proof}
We now prove several integral identities which involve the mean curvature function $H$.
We start with the following one, that can be obtained by polishing the arguments contained in \cite{Re}.
\begin{thm}
\label{th:fundamental-identity}
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^2$ and
denote by $H$ the mean curvature of $\Gamma$.
\par
If $u$ is the solution of \eqref{serrin1},
then the following identity holds:
\begin{equation}\label{fundamental}
\frac1{N-1}\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}\,dx = N |\Omega| -\int_{\Gamma}H\, (u_\nu)^2\,dS_x.
\end{equation}
\end{thm}
\begin{proof}
Let $P$ be given by \eqref{P-function}. By the divergence theorem we can write:
\begin{equation}
\label{div-p-function}
\int_{\Omega} \Delta P \,dx = \int_{\Gamma} P_\nu \, dS_x.
\end{equation}
To compute $P_\nu$, we observe that $\nabla u$ is
parallel to $\nu$ on $\Gamma$, that is $\nabla u= u_\nu \,\nu$ on $\Gamma$. Thus,
$$
P_\nu=\langle D^2 u\, \nabla u, \nu\rangle-u_\nu=u_\nu \langle (D^2 u)\,\nu,\nu\rangle-u_\nu=u_{\nu\nu}\, u_\nu-u_\nu.
$$
We recall Reilly's identity from the introduction,
\begin{equation*}
u_{\nu\nu}+(N-1)\,H\,u_\nu=N ,
\end{equation*}
from which we obtain that
$$
u_{\nu\nu}\, u_\nu+(N-1)\,H\, (u_\nu)^2=N\,u_\nu,
$$
and hence
$$
P_\nu=(N-1)\,u_\nu-(N-1)\,H\, (u_\nu)^2
$$
on $\Gamma$.
\par
Therefore, \eqref{fundamental} follows from this identity, \eqref{differential-identity}, \eqref{div-p-function} and the formula
\begin{equation}
\label{volume}
\int_\Gamma u_\nu\,dS_x=N\,|\Omega|,
\end{equation}
that is an easy consequence of the divergence theorem.
\end{proof}
\begin{thm}[Identities for the Soap Bubble Theorem, \cite{MP,MP2}]
\label{thm:identitySBT}
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^2$ and
denote by $H$ the mean curvature of $\Gamma$.
Let $R$ and $H_0$ be the two positive constants defined in \eqref{def-R-H0}.
If $u$ is the solution of \eqref{serrin1}, then the following two identities hold true:
\begin{multline}
\label{H-fundamental}
\frac1{N-1}\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}dx+
\frac1{R}\,\int_\Gamma (u_\nu-R)^2 dS_x = \\
\int_{\Gamma}(H_0-H)\, (u_\nu)^2 dS_x ,
\end{multline}
\begin{multline}
\label{identity-SBT2}
\frac1{N-1}\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}dx+
\frac1{R}\,\int_\Gamma (u_\nu-R)^2 dS_x = \\
\int_{\Gamma}(H_0-H)\,(u_\nu-q_\nu)\,u_\nu\,dS_x+
\int_{\Gamma}(H_0-H)\, (u_\nu-R)\,q_\nu\, dS_x.
\end{multline}
\end{thm}
\begin{proof}
Since, by \eqref{volume}, we have that
$$
\frac1{R}\,\int_\Gamma u_\nu^2\,dS_x=
\frac1{R}\,\int_\Gamma (u_\nu-R)^2\,dS_x+N |\Omega|,
$$
then
\begin{multline*}
\int_\Gamma H\,(u_\nu)^2\,dS_x=H_0\,\int_\Gamma (u_\nu)^2\,dS_x+
\int_\Gamma \left( H-H_0\right)\,(u_\nu)^2\,dS_x= \\
\frac1{R}\,\int_\Gamma (u_\nu-R)^2\,dS_x+N |\Omega|+
\int_\Gamma ( H-H_0)\,(u_\nu)^2\,dS_x.
\end{multline*}
Thus, \eqref{H-fundamental} follows from this identity and \eqref{fundamental} at once.
Identity \eqref{identity-SBT2} can be proved just by rearranging the right-hand side of \eqref{H-fundamental}.
In fact, straightforward calculations that use \eqref{def-R-H0} and Minkowski's identity
\begin{equation}
\label{minkowski}
\int_\Gamma H\,q_\nu\,dS_x=|\Gamma|
\end{equation}
tell us that
\begin{multline*}
\int_{\Gamma}(H_0-H)\,(u_\nu-q_\nu)\,u_\nu\,dS_x+\\
\int_{\Gamma}(H_0-H)\, (u_\nu-R)\,q_\nu\, dS_x=
\int_\Gamma \left( H_0-H \right) u_\nu^2 \, dS_x ,
\end{multline*}
and hence \eqref{identity-SBT2} follows.
\end{proof}
We finally show that, if $\Gamma$ is mean-convex, that is $H \ge 0$, \eqref{fundamental-identity2} can also be rearranged into an identity leading to Heintze-Karcher's inequality \eqref{heintze-karcher} below (see \cite{HK}).
\begin{thm}[\cite{MP}]
\label{thm:identityheintze-karcher}
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^2$ and
denote by $H$ the mean curvature of $\Gamma$.
Let $u$ be the solution of \eqref{serrin1}.
\par
If $\Gamma$ is mean-convex, then we have the following identity:
\begin{multline}
\label{heintze-karcher-identity}
\frac1{N-1}\,\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}\,dx +\int_\Gamma\frac{(1-H\,u_\nu)^2}{H}\,dS_x =
\\
\int_\Gamma\frac{dS_x}{H}-N |\Omega|.
\end{multline}
\end{thm}
\begin{proof}
Since \eqref{volume} holds, from \eqref{fundamental} we obtain that
\begin{equation}
\label{fundamental-identity2}
\frac1{N-1}\int_{\Omega} \left\{ |\nabla ^2 u|^2-\frac{(\Delta u)^2}{N}\right\}\,dx = \int_{\Gamma}(1-H u_\nu)\, u_\nu\,dS_x.
\end{equation}
Notice that \eqref{fundamental-identity2} still holds without the assumption $H\ge0$.
Since $u_\nu$ and $H$ are continuous, the identity
\begin{equation}\label{eq:HKsenzastrictly}
\frac{(1-H\,u_\nu)^2}{H}=-(1-H\,u_\nu) u_\nu+\frac1{H}-u_\nu,
\end{equation}
holds pointwise for $H\ge0$.
If the integral $\int_\Gamma \frac{dS_x}{H}$ is infinite, then \eqref{heintze-karcher-identity} trivially holds. Otherwise $1/H$ is finite (and hence $H > 0$) almost everywhere
on $\Gamma$ and hence, \eqref{eq:HKsenzastrictly} holds almost everywhere.
Thus, by integrating \eqref{eq:HKsenzastrictly} on $\Gamma$, summing the result up to \eqref{fundamental-identity2} and taking into account \eqref{volume},
we get \eqref{heintze-karcher-identity}.
\end{proof}
\section{Symmetry results}\label{sec:symmetry results via integral identities}
The quantity at the right-hand side of \eqref{differential-identity}, that we call Cauchy-Schwarz deficit for the hessian matrix $\nabla^2 u$, will play the role of detector of spherical symmetry, as is clear from the following lemma. In what follows, $I$ denotes the identity matrix.
\begin{lem}[Spherical detector]
\label{lem:sphericaldetector}
Let $\Omega \subseteq \mathbb{R}^N$ be a domain and $u \in C^2 (\Omega)$.
Then it holds that
\begin{equation}
\label{newton}
|\nabla^2 u|^2 - \frac{(\Delta u)^2}{ N} \ge 0 \quad \mbox{in } \Omega,
\end{equation}
and the equality sign holds if and only if $u$ is a quadratic polynomial.
Moreover, if $u$ is the solution of \eqref{serrin1}, the equality sign holds if and only if $\Gamma$ is a sphere (and hence $\Omega$ is a ball) of radius $R$ given by \eqref{def-R-H0} and
$$
u(x)= \frac{1}{2} \, (|x|^2 - R^2),
$$
up to a translation.
\end{lem}
\begin{proof}
We regard the matrices $\nabla^2 u$ and $I$ as vectors in $\mathbb{R}^{N^2}$.
Inequality \eqref{newton} is then the classical Cauchy-Schwarz inequality.
For the characterization of the equality case, we first consider $u$ solution of \eqref{serrin1} and we set $w(x)= u(x) - |x|^2 /2$. Since $w$ is harmonic, direct computations show that
$$
0=|\nabla^2 u|^2 - \frac{(\Delta u)^2}{ N} = | \nabla w|^2.
$$
Thus, $w$ is affine and $u$ is quadratic. Therefore, $u$ can be written in the form
\eqref{quadratic},
for some $z\in\mathbb{R}^N$ and $a\in\mathbb{R}$.
\par
Since $u=0$ on $\Gamma$, then $|x-z|^2=a$ for $x\in\Gamma$, that is $\Gamma$ must be a sphere centered at $z$.
Moreover, $a$ must be positive and
$$
\sqrt{a}\,|\Gamma|=\int_\Gamma |x-z|\,dS_x=\int_\Gamma (x-z)\cdot\nu(x)\,dS_x=N\,|\Omega|.
$$
In conclusion, $\Gamma$ is a sphere centered at $z$ with radius $R$.
If $u$ is any $C^2$ function for which the equality holds in \eqref{newton}, we can still
conclude that $u$ is a quadratic polynomial.
In fact, being \eqref{newton} the classical Cauchy-Schwarz inequality in $\mathbb{R}^{N^2}$, if the equality sign holds we have that $\nabla^2 u (x) = \lambda(x) \, I$.
Since
$$u_{iij}= u_{iji} = 0 \quad \mbox{and} \quad u_{jjj}= \lambda_j = u_{iij} , $$
we immediately realize that the function
$\lambda (x)$ must be a constant $\lambda$.
Notice that, this conclusion surely holds if $u$ is of class $C^3$. However, even if $u$ is of class $C^2$, the last two differential identities still hold in the sense of distributions and the same conclusion follows.
Thus, by integrating directly the system $\nabla^2 u = \lambda \, I$, we obtain that $u$ must be a quadratic polynomial.
\end{proof}
As an immediate corollary of Theorem \ref{thm:serrinidentity} we have the following more general version of Serrin's symmetry result.
\begin{thm}[Symmetry for the torsional rigidity density, \cite{MP2}]
Let $\Omega \subset \mathbb R^N$ be a bounded domain with boundary $\Gamma$ of class $C^{1,\alpha}$, $0<\alpha\le 1$. Let $R$ be the positive constant defined in \eqref{def-R-H0}, and $u$ be the solution of \eqref{serrin1}.
If the right-hand side of \eqref{idwps} is non-positive,
$\Gamma$ must be a sphere (and hence $\Omega$ a ball) of radius $R$.
The same conclusion clearly holds if either $u_\nu$ is constant on $\Gamma$ or $u_\nu - q_\nu = 0$ on $\Gamma$.
\end{thm}
\begin{proof}
If the right-hand side of \eqref{idwps} is non-positive, then the integrand at the left-hand side must be zero,
being non-negative by \eqref{newton} and the maximum principle for $u$. Then \eqref{newton} must hold with the equality sign, since $u<0$ on $\Omega$, by the strong maximum principle. The conclusion follows from Lemma \ref{lem:sphericaldetector}.
\par
Finally, if $u_\nu\equiv c$ on $\Gamma$ for some constant $c$, then
$$
c\,|\Gamma|=\int_\Gamma u_\nu\,dS_x=N\,|\Omega|,
$$
that is $c=R$, and hence we can apply the previous argument.
The same conclusion clearly holds if $u_\nu-q_\nu=0$ on $\Gamma$. Notice that in this case a simpler alternative proof is available. In fact, since $u-q$ is harmonic in $\Omega$ and $(u-q)_\nu=0$ on $\Gamma$, then $u-q$ must be constant on $\overline{\Omega}$.
\end{proof}
\begin{rem}
{\rm
Based on the work of Vogel \cite{Vo}, the regularity assumption on $\Omega$ can be dropped, if \eqref{serrin1} and the conditions $u=0$, $u_\nu=c$ on $\Gamma$ (for some constant $c$) are stated in a weak sense.
In fact, if $u \in W_{loc}^{1,p} \left( \Omega \right)$ is a function such that
$$
\int_{\Omega} \nabla u \cdot \nabla \phi \, dx + N \int_\Omega \phi \, dx =0
$$
for every $\phi \in C_0^\infty \left( \Omega \right)$,
and for all $\varepsilon >0$ there exists a neighborhood $U_\varepsilon \supset \Gamma$ such that $|u(x)| < \varepsilon$ and $\left| |\nabla u| - c \right| < \varepsilon$ for a.e. $x \in U_\varepsilon \cap \Omega $, \cite[Theorem 1]{Vo} ensures that $\Gamma$ must be of class $C^2$.
}
\end{rem}
As an immediate corollary of
Theorem \ref{thm:identitySBT} we have the following
more general version of the Soap Bubble Theorem.
\begin{thm}[Soap Bubble-type Theorem, \cite{MP}]
\label{th:SBT}
Let $\Gamma\subset\mathbb{R}^N$ be a surface of class $C^2$, which is the boundary of a bounded domain $\Omega\subset\mathbb{R}^N$, and let $u$ be the solution of \eqref{serrin1}.
Let $R$ and $H_0$ be the two positive constants defined in \eqref{def-R-H0}.
If the right-hand side of \eqref{H-fundamental} is non-positive, then $\Gamma$ must be a sphere
of radius $R$.
In particular, the same conclusion holds if either the mean curvature $H$ of $\Gamma$ satisfies the inequality $H\ge H_0$ or $H$ is constant on $\Gamma$.
\end{thm}
\begin{proof}
If the right-hand side in \eqref{H-fundamental} is non-positive (and this certainly happens if $H\ge H_0$ on $\Gamma$), then both summands at the left-hand side must
be zero, being non-negative.
\par
The fact that also the first summand is zero gives that the Cauchy-Schwarz deficit for the hessian matrix $\nabla^2 u$ must be identically zero and the conclusion follows from Lemma \ref{lem:sphericaldetector}.
\par
If $H$ equals some constant, instead, then \eqref{minkowski} tells us that the constant must equal $H_0$, and hence we can apply the previous argument.
\end{proof}
\begin{rem}
{\rm
(i) As pointed out in the previous proof, the assumption of the theorem also gives that the second summand at the left-hand side of \eqref{H-fundamental} must be zero, and hence $u_\nu = R$ on $\Gamma$. Thus, Serrin's overdetermining condition \eqref{serrin2} holds {\it before showing that $\Omega$ is a ball}.
\par
(ii) We observe that the assumption that $H\ge H_0$ on $\Gamma$ gives that $H\equiv H_0$, anyway, if $\Omega$ is strictly star-shaped with respect to some origin $p$. In fact, by Minkowski's identity \eqref{minkowski}, we obtain that
$$
0\le\int_\Gamma [H(x)-H_0] \langle(x-p), \nu(x)\rangle\,dS_x=|\Gamma|-H_0 \int_\Gamma \langle(x-p), \nu(x)\rangle\,dS_x=0,
$$
and we know that $\langle(x-p), \nu(x)\rangle>0$ for $x\in\Gamma$.
}
\end{rem}
We now show that \eqref{heintze-karcher-identity} implies Heintze-Karcher's inequality (see \cite{HK}).
We mention that our proof is slightly different from that of A. Ros in \cite{Ro} and relates the equality case for Heintze-Karcher's inequality to a new overdetermined problem, which is described in item (ii) of the following theorem. As it will be clear, \eqref{heintze-karcher-identity} also gives an alternative proof of Alexandrov's theorem via the characterization of the equality sign in
Heintze-Karcher's inequality \eqref{heintze-karcher}.
\begin{thm}[Heintze-Karcher's inequality and a related overdetermined problem, \cite{MP}]
\label{thm:heintze-karcher}
Let $\Gamma\subset\mathbb{R}^N$ be a mean-convex surface of class $C^2$, which is the boundary of a bounded domain $\Omega\subset\mathbb{R}^N$, and denote by $H$ its mean curvature. Then,
\begin{enumerate}[(i)]
\item Heintze-Karcher's inequality
\begin{equation}
\label{heintze-karcher}
\int_\Gamma \frac{dS_x}{H}\ge N |\Omega|
\end{equation}
holds and the equality sign is attained if and only if $\Omega$ is a ball;
\item $\Omega$ is a ball if and only if the solution $u$ of \eqref{serrin1} satisfies
\begin{equation}\label{equa:overdeteitemi}
u_\nu(x)=1/H(x) \quad \mbox{for every} \quad x\in\Gamma .
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
(i) Both summands at the left-hand side of \eqref{heintze-karcher-identity} are non-negative and hence \eqref{heintze-karcher} follows.
If the right-hand side is zero, those summands must be zero. The vanishing of the first summand implies that $\Omega$ is a ball, as already noticed. Note in passing that the vanishing of the second summand gives that $u_\nu=1/H$ on $\Gamma$, which also implies radial symmetry, by item (ii).
(ii) It is clear that, if $\Omega$ is a ball, then \eqref{equa:overdeteitemi} holds. Conversely, it is easy to check that the right-hand side and the second summand of the left-hand side of \eqref{heintze-karcher-identity} are zero when \eqref{equa:overdeteitemi} occurs.
Thus, the conclusion follows from Lemma \ref{lem:sphericaldetector}.
\par
Notice that assumption \eqref{equa:overdeteitemi} implies that $H$ must be positive, since $u_{\nu}$ is positive and finite.
\end{proof}
\begin{rem}
{\rm
(i) As already noticed in \cite{Ro}, the characterization of the equality sign of Heintze-Karcher's inequality given in item (i) of Theorem \ref{thm:heintze-karcher}, leads to a different proof of the Soap Bubble Theorem. In fact, if $H$ equals some constant on $\Gamma$, by using Minkowski's identity \eqref{minkowski} we know that such a constant must have the value $H_0$ in \eqref{def-R-H0}.
Thus, \eqref{heintze-karcher} holds with the equality sign, and hence $\Gamma$ is a sphere by item (i) of Theorem \ref{thm:heintze-karcher}.
(ii) Item (ii) of Theorem \ref{thm:heintze-karcher} could be proved also by using \eqref{fundamental-identity2} instead of \eqref{heintze-karcher-identity}. In fact, it is immediate to check that the right-hand side of \eqref{fundamental-identity2} is zero when \eqref{equa:overdeteitemi} occurs.
}
\end{rem}
We conclude this section by giving an alternative proof of Serrin's theorem that can have its own interest. Unfortunately, it only works for star-shaped domains.
\begin{thm}[Theorem 2.4, \cite{MP}]
Let $\Omega\subset\mathbb{R}^N$, $N\ge2$, be a bounded domain with boundary $\Gamma$ of class $C^2$.
Assume that $\langle(x-p), \nu(x)\rangle>0$ for every $x\in\Gamma$ and some $p\in\Omega$, and let $u$ be the solution of \eqref{serrin1}.
\par
Then, $\Omega$ is a ball if and only if $u$ satisfies \eqref{serrin2}.
\end{thm}
\begin{proof}
It is clear that, if $\Omega$ is a ball, then \eqref{serrin2} holds. Conversely, we shall check that the right-hand side of \eqref{fundamental} is zero when \eqref{serrin2} occurs.
\par
Let $u_\nu$ be constant on $\Gamma$; by \eqref{volume} we know that that constant equals the value $R$ given in \eqref{def-R-H0}. Also, notice that $1-H u_\nu\ge 0$ on $\Gamma$. In fact, the function $P$ in \eqref{P-function}
is subharmonic in $\Omega$, since $\Delta P\ge 0$ by \eqref{differential-identity}. Thus, it attains its maximum on $\Gamma$, where it is constant.
We thus have that
$$
0\le P_\nu=u_\nu u_{\nu \nu}-u_\nu=(N-1)\,(1-H u_\nu)\,u_\nu \ \mbox{ on } \ \Gamma.
$$
Now,
\begin{multline*}
0\le\int_\Gamma [1-H(x) u_\nu(x)]\,\langle (x-p),\nu(x)\rangle\,dS_x=\\
\int_\Gamma [1-H(x) R]\,\langle (x-p),\nu(x)\rangle\,dS_x=0,
\end{multline*}
by \eqref{minkowski}. Thus, $1-H u_\nu\equiv 0$ on $\Gamma$ and hence item (ii) of Theorem \ref{thm:heintze-karcher} applies.
\end{proof}
\chapter{Radial symmetry for \texorpdfstring{$p$}{p}-harmonic functions in exterior and punctured domains}\label{chap:PogAA}
\chaptermark{Radial symmetry for $p$-harmonic functions}
In this chapter, we collect symmetry results obtained in \cite{Pog}, involving $p$-harmonic functions in exterior and punctured domains.
\section{Motivations and statement of results}
The \textit{electrostatic $p$-capacity} of a bounded open set $\Omega \subset \mathbb{R}^N$, $1<p<N$, is defined by
\begin{equation}\label{def:Capacity}
\mathop{\mathrm{Cap_p}}_p(\Omega)= \inf \left\lbrace \int_{\mathbb{R}^N} | \nabla w|^p : \; w \in C^{\infty}_{0} (\mathbb{R}^N), \; w \ge 1 \; \text{in} \; \Omega \right\rbrace
\end{equation}
Under appropriate sufficient conditions, there exists a unique minimizing function $u$ of \eqref{def:Capacity}; such function is called the {\it $p$-capacitary potential} of $\Omega$, and satisfies
\begin{equation}
\label{Problemcapacity}
\Delta_p u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overline{\Omega} , \; u=1 \, \mbox{ on } \Gamma, \; \lim_{|x|\rightarrow \infty} u(x)=0 ,
\end{equation}
where $\Gamma$ is the boundary of $\Omega$ and $\Delta_p$ denotes the $p$-Laplace operator defined by
\begin{equation*}
\Delta_p u = \mathop{\mathrm{div}} \left( |\nabla u|^{p-2} \nabla u \right).
\end{equation*}
It is well known that the $p$-capacity could be equivalently defined by means of the $p$-capacitary potential $u$ as
\begin{equation}\label{eq:caponboundary}
\mathop{\mathrm{Cap_p}}_p(\Omega) = \int_{\mathbb{R}^N \setminus \overline{\Omega} } |\nabla u|^p \, dx = \int_\Gamma | \nabla u|^{p-1} \, dS_x,
\end{equation}
where the second equality follows by integration by parts (see the proof of Lemma~\ref{lem:asymptoticcap}).
In Section \ref{sec:exterior} we consider Problem \eqref{Problemcapacity} under Serrin's overdetermined condition given by
\begin{equation}\label{eq:overdetermination}
| \nabla u| = c \mbox{ on } \Gamma,
\end{equation}
and we prove the following result.
\begin{thm}[\cite{Pog}]\label{thm:Serrinesterno}
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain (i.e., a connected bounded open set).
For $1<p<N$, the problem \eqref{Problemcapacity}, \eqref{eq:overdetermination} admits a weak solution if and only if $\Omega$ is a ball.
\end{thm}
By a weak solution in the statement of Theorem \ref{thm:Serrinesterno} we mean a function $u \in W_{loc}^{1,p} \left( \mathbb{R}^N \setminus \overline{\Omega} \right)$ such that
$$
\int_{\mathbb{R}^N \setminus \overline{\Omega}} |\nabla u|^{p-2} \, \nabla u \cdot \nabla \phi \, dx=0
$$
for every $\phi \in C_0^\infty \left( \mathbb{R}^N \setminus \overline{\Omega} \right)$
and satisfying the boundary conditions in the weak sense, i.e.: for all $\varepsilon >0$ there exists a neighborhood $U_\varepsilon \supset \Gamma$ such that $|u(x)- 1| < \varepsilon$ and $\left| |\nabla u| - c \right| < \varepsilon$ for a.e. $x \in U_\varepsilon \cap \left( \mathbb{R}^N \setminus \overline{\Omega} \right)$.
Notice the complete absence in Theorem \ref{thm:Serrinesterno} of any smoothness assumption as well as of any other assumption on the domain $\Omega$.
Under the additional assumption that $\Omega$ is star-shaped, Theorem \ref{thm:Serrinesterno} has been proved by Garofalo and Sartori in \cite{GS}, by extending to the case $1<p<N$ the tools developed in the case $p=2$ by Payne and Philippin (\cite{PP2}, \cite{Ph}).
The proof in \cite{GS}, which combines integral identities and a maximum principle for an appropriate $P$-function, bears a resemblance to Weinberger's proof (\cite{We}) of symmetry for the archetype torsion problem \eqref{serrin1}
under Serrin's overdetermined condition
\eqref{serrin2}.
To prove Theorem \ref{thm:Serrinesterno}, we improve on the arguments used in \cite{GS} and we exploit, as a new crucial ingredient, Theorem \ref{th:SBT}, that is the Soap Bubble-type Theorem proved via integral identities in Section \ref{sec:symmetry results via integral identities}.
Symmetry for Problem \eqref{Problemcapacity}, \eqref{eq:overdetermination} was first obtained by Reichel (\cite{Re1}, \cite{Re2}) by adapting the method of moving planes introduced by Serrin (\cite{Se}) to prove symmetry for the overdetermined torsion problem \eqref{serrin1}, \eqref{serrin2}.
In Reichel's works (\cite{Re1}, \cite{Re2}) the star-shapedness assumption is not requested, but the domain $\Omega$ is a priori assumed to be $C^{2,\alpha}$ and the solution $u$ is assumed to be of class $C^{1, \alpha}(\mathbb{R}^N \setminus \Omega)$.
For completeness let us mention that many alternative proofs and improvements in various directions of symmetry results for Serrin's problems relative to the equations \eqref{serrin1} and \eqref{Problemcapacity} have been obtained in the years and can be found in the literature: for the overdetermined torsion problem \eqref{serrin1}, \eqref{serrin2} see for example \cite{PS}, \cite{GL}, \cite{DP}, \cite{BH}, \cite{BNST}, \cite{FK}, \cite{CiS}, \cite{WX}, \cite{MP2}, \cite{BC}, and the surveys \cite{Mag}, \cite{NT}, \cite{Ka}; for the exterior overdetermined problem \eqref{Problemcapacity}, \eqref{eq:overdetermination} where the domain $\Omega$ is assumed to be convex see \cite{MR}, \cite{BCS}, \cite{BC}, and \cite{FMP}.
In Section \ref{sec:casoconforme}, we establish the result corresponding to Theorem \ref{thm:Serrinesterno} in the special case $1<p=N$.
In this case, the problem corresponding to \eqref{Problemcapacity} is (see e.g. \cite{CC}):
\begin{equation}
\label{ConformalProblemcapacity}
\Delta_N u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overline{\Omega} , \; u=1 \, \mbox{ on } \Gamma, \; u(x) \sim - \ln{|x|} \text{ as } |x|\to \infty ,
\end{equation}
where $\sim$ means that
\begin{equation}\label{eq:confasymppre}
c_1 \le \frac{u(x)}{ ( - \ln{|x|} ) } \le c_2, \text{ as } |x| \to \infty
\end{equation}
for some positive constants $c_1$,$c_2$.
What we prove is the following.
\begin{thm}[\cite{Pog}]\label{thm:conforme}
Let $\Omega \subset \mathbb{R}^N$, $1<N$, be a bounded domain.
The problem \eqref{ConformalProblemcapacity} with the overdetermined condition \eqref{eq:overdetermination}
admits a weak solution if and only if $\Omega$ is a ball.
\end{thm}
A proof of Theorem \ref{thm:conforme} that uses the method of moving planes is contained in \cite{Re2}, under the additional a priori smoothness assumptions $\Gamma \in C^{2, \alpha}$, $u \in C^{1,\alpha}(\mathbb{R}^N \setminus \Omega)$.
Our proof via integral identities seems to be new and cannot be found in the literature unless for the classical case $p=N=2$, which has been treated with similar arguments in \cite{Mar} (for piecewise smooth domains) and \cite{MR} (for Lipschitz domains). Moreover, in Theorem \ref{thm:conforme} no assumptions on the domain $\Omega$ are made.
We mention that a related symmetry result for the $N$-capacitary potential in a bounded (smooth) star-shaped ring domain has been established in \cite{PP3}.
In Section \ref{sec:Interior}, we show how the same ideas used in our proof of Theorem \ref{thm:Serrinesterno} can be adapted to give a symmetry result for a similar problem in a bounded punctured domain. More precisely, we prove the following theorem concerning the problem
\begin{equation}\label{Pdelta}
- \Delta_p u = K \, \de_0 \, \mbox{ in } \Omega , \, u=c \, \mbox{ on } \Gamma,
\end{equation}
under Serrin's overdetermined condition
\begin{equation}\label{eq:interno-overdetermination}
| \nabla u| = 1 \mbox{ on } \Gamma ,
\end{equation}
where with $\de_0$ we denote the Dirac delta centered at the origin $0 \in \Omega$ and $K$ is some positive normalization constant.
\begin{thm}[\cite{Pog}]
\label{thm:Serrininterno}
Let $\Omega \subset \mathbb{R}^N$ be a bounded star-shaped domain.
For $1<p<N$, the problem \eqref{Pdelta}, \eqref{eq:interno-overdetermination}
admits a weak solution if and only if $\Omega$ is a ball centered at the origin.
\end{thm}
It should be noticed, however, that in Theorem \ref{thm:Serrininterno} we need to assume $\Omega$ to be star-shaped, restriction that is not present in the proofs of Payne and Schaefer (\cite{PS})(for the case $p=2$), Alessandrini and Rosset (\cite{AR}), and Enciso and Peralta-Salas (\cite{EP}).
We mention that the proof in \cite{AR} uses an adaptation of the method of moving planes, the proof in \cite{EP} is in the wake of Weinberger, and both of them also cover the special case $p=N$.
For $p=2$, Problem \eqref{Problemcapacity} arises naturally in electrostatics. In this context $u$ is the (normalized) potential of the electric field $\nabla u$ generated by a conductor $\Omega$.
We recall that when $\Omega$ is in the electric equilibrium, the electric field in the interior of $\Omega$ is null, and hence the electric potential $u$ is constant in $\Omega$ (i.e. $u \equiv 1$ in $\Omega$); moreover, the electric charges present in the conductor are distributed on the boundary $\Gamma$ of $\Omega$.
It is also known that the electric field $\nabla u$ on $\Gamma$ is orthogonal to $\Gamma$ -- i.e. $\nabla u = u_\nu \, \nu$, where
$\nu$ denotes the outer unit normal with respect to $\Omega$ and $u_\nu$ denotes the derivative of $u$ in the direction $\nu$
-- and its intensity is given\footnote{More precisely, in free space the intensity of the electric field on $\Gamma$ is given by $|\nabla u|_{| \Gamma} = - u_\nu = \frac{\rho}{\varepsilon_0}$, where $\rho$ is the surface charge density over $\Gamma$ and $\varepsilon_0$ is the vacuum permittivity. We ignored the constant $\varepsilon_0$ to be coherent with the mathematical definition of capacity given in \eqref{def:Capacity}.} by the surface charge density over $\Gamma$. In this context the capacity is defined as the total electric charge needed to induce the potential $u$, that is
$$\mathop{\mathrm{Cap_p}}(\Omega)= \int_\Gamma | \nabla u | \, dS_x, $$
in accordance with \eqref{eq:caponboundary} for $p=2$.
Following this physical interpretation Theorem \ref{thm:Serrinesterno} simply states that the electric field on the boundary $\Gamma$ of the conductor $\Omega$ is constant -- or equivalently that the charges present in the conductor are uniformly distributed on $\Gamma$ (i.e. the surface charge density is constant over $\Gamma$) -- if and only if $\Omega$ is a round ball.
Another result of interest in the same context that is related to Problem \eqref{Problemcapacity}, \eqref{eq:overdetermination} is a Poincar\'e's theorem known as the isoperimetric inequality for the capacity, stating that, among sets having given volume, the ball minimizes $\mathop{\mathrm{Cap_p}}(\Omega)$.
We mention that a proof of this inequality that hinges on rearrangement techniques can be found in \cite[Section 1.12]{PZ} (see also \cite{Ja} for a useful review of that proof).
Here, -- as done in the introduction for the isoperimetric problem and Saint Venant's principle -- we just want to underline the relation present between this result and Problem \eqref{Problemcapacity}, \eqref{eq:overdetermination} (for $p=2$).
In fact, once that the existence of a minimizing set $\Omega$ is established, we can show through the technique of shape derivatives that the solution of \eqref{Problemcapacity} in $\Omega$ also satisfies the overdetermined condition \eqref{eq:overdetermination} on $\Gamma$; the reasoning is the following.
We consider the evolution of the domains $\Omega_t$ given by
\begin{equation*}
\Omega_t = \mathcal{M}_t (\Omega),
\end{equation*}
where $\Omega= \Omega_0$ is fixed, and $\mathcal{M}_t: \mathbb{R}^N \to \mathbb{R}^N$ is a mapping such that
$$
\mathcal{M}_0 (x) =x, \; \mathcal{M}'_0 (x) = \phi(x) \nu(x),
$$
where the symbol $'$ means differentiation with respect to $t$, $\phi$ is any compactly supported continuous function, and $\nu$ is a proper extension of the unit normal vector field to a tubular neighborhood of $\Gamma_0$.
Thus, we consider $u(t,x)$, solution of Problem \eqref{Problemcapacity} in $\Omega= \Omega_t$, and the two functions (in the variable $t$) $\mathop{\mathrm{Cap_p}} (\Omega_t)$ and $| \Omega_t|$.
Since $\Omega=\Omega_0$ is the domain that minimizes $\mathop{\mathrm{Cap_p}}(\Omega_t)$ among all the domains in the one-parameter family $\left\lbrace \Omega_t \right\rbrace_{t \in \mathbb{R}}$ that have prescribed volume $|\Omega_t|=V$, by using the method of Lagrange multipliers and Hadamard's variational formula (see \cite[Chapter 5]{HP}), standard computations lead to prove that there exists a number $\lambda$ such that
\begin{equation*}
\int_{\Gamma} \phi(x) \left[ u_\nu^2 (x) - \lambda \right] \, dS_x = 0,
\end{equation*}
where we have set $u(x)=u(0,x)$.
Since $\phi$ is arbitrary, we deduce that $u_\nu^2 \equiv \lambda$ on $\Gamma$, that is, $u$ satisfies the overdetermined condition \eqref{eq:overdetermination} on $\Gamma$.
Other applications of Problem \eqref{Problemcapacity} are related to quantum theory, acoustic, theory of musical instruments, and the study of heat, electrical and fluid flow (see for example \cite{CFG}, \cite{DZH}, \cite{BC} and references therein).
Finally, we just mention that also Problem \eqref{Pdelta} arises in electrostatics for $p=2$: in this case, Theorem \ref{thm:Serrininterno} states that the electric field on a conducting hypersurface enclosing a charge is constant if and only if the conductor is a sphere centered at the charge.
\section{The exterior problem: proof of Theorem \ref{thm:Serrinesterno}}\label{sec:exterior}
\sectionmark{The exterior problem: proof of Theorem \ref{thm:Serrinesterno}}
In order to prove Theorem \ref{thm:Serrinesterno}, we start by collecting all the necessary ingredients.
In this section $u$ denotes a weak solution to \eqref{Problemcapacity},\eqref{eq:overdetermination}, in the sense explained in the Introduction, and it holds that $1<p<N$.
\begin{rem}[On the regularity]\label{rem:regularity}
{\rm
Due to the degeneracy or singularity of the $p$-Laplacian (when $p \neq 2$) at the critical points of $u$, $u$ is in general only $C^{1, \alpha}_{loc}(\mathbb{R}^N \setminus \overline{\Omega})$ (see \cite{Di}, \cite{Le}, \cite{To}), whereas it is $C^{\infty}$ in a neighborhood of any of its regular points thanks to standard elliptic regularity theory (see \cite{GT}). However, as already noticed in \cite{GS}, the additional assumption given by the weak boundary condition \eqref{eq:overdetermination} ensures that $u$ can be extended to a $C^2$-function in a neighborhood of $\Gamma$, so that by using the work of Vogel \cite{Vo} we get that $\Gamma$ is of class $C^2$. Thus, by \cite[Theorem 1]{Li} it turns out that $u$ is $C^{1, \alpha}_{loc} (\mathbb{R}^N \setminus \Omega)$.
As a consequence we can now interpret the boundary condition in \eqref{Problemcapacity} and the one in \eqref{eq:overdetermination} in the classical strong sense.
}
\end{rem}
\begin{rem}
{\rm
More precisely, by using Vogel's work \cite{Vo}, which is based on the deep results on free boundaries contained in \cite{AC} and \cite{ACF}, one can prove that $\Gamma$ is of class $C^{2,\alpha}$ from each side. Even if
here we do not need this refinement, it should be noticed that in light of this remark the arguments contained in \cite{Re2} give an alternative and complete proof of Theorem \ref{thm:Serrinesterno} (and also of Theorem \ref{thm:conforme}). In fact, the smoothness assumptions of \cite[Theorem 1]{Re2} are satisfied.
}
\end{rem}
By using the ideas contained in \cite{GS} together with a result of Kichenassamy and V\'eron (\cite{KV}), we can recover the following useful asymptotic expansion for $u$ as $|x|$ tends to infinity.
\begin{lem}[Asymptotic expansion, \cite{Pog}]\label{lem:asymptoticcap}
As $|x|$ tends to infinity it holds that
\begin{equation}\label{eq:cap-asymptotic u}
u(x) = \frac{p-1}{N-p} \left( \frac{\mathop{\mathrm{Cap_p}}_p (\Omega)}{\omega_N} \right)^{\frac{1}{p-1}} |x|^{- \frac{N-p}{p-1}} + o (|x|^{- \frac{N-p}{p-1}}).
\end{equation}
\end{lem}
The computations leading to determine the constant of proportionality in \eqref{eq:cap-asymptotic u} originate from \cite{GS}, where they have been used to give a complete proof -- which works without invoking \cite{KV} -- of a different but related result
(\cite[Theorem~3.1]{GS}), which is described in more details in Remark \ref{rem:Garofaoloasymptotic} below.
We mention that, later, in \cite{CoS} the same computations have been treated with tools of convex analysis and used together with the result contained in \cite[Remark 1.5]{KV} to prove Lemma \ref{lem:asymptoticcap} for convex sets.
\begin{proof}[Proof of Lemma \ref{lem:asymptoticcap}]
As noticed in \cite{GS}, if $u$ is a solution of \eqref{Problemcapacity}, then the weak comparison principle for the $p$-Laplacian (see \cite{HKM}) implies the existence of positive constants $c_1$,$c_2$, $R_0$ such that
$$
c_1 \mu(x) \le u(x) \le c_2 \mu(x), \quad \text{ if } |x| \ge R_0,
$$
where $\mu(x)$ denotes the radial fundamental solution of the $p$-Laplace operator given by
\begin{equation*}
\mu(x) = \frac{(p-1)}{(N-p)} \frac{1}{\omega_N^{\frac{1}{p-1}}} |x|^{\frac{p-N}{p-1}}.
\end{equation*}
Thus we can apply the result of Kichenassamy and V\'eron (\cite[Remark 1.5]{KV}) and state that there exists a constant $\ga$ such that
\begin{equation}\label{eq:passKV}
\lim_{|x| \to \infty} \frac{u(x)}{\mu(x)} = \ga, \quad \lim_{|x| \to \infty} |x|^{\frac{N-p}{p-1} + | \alpha|} D^\alpha \left( u - \ga \mu \right)=0,
\end{equation}
for all multi-indices $\alpha=(\alpha_1, \dots, \alpha_N)$ with $|\alpha|= \alpha_1+ \dots + \alpha_N \ge 1$.
To establish \eqref{eq:cap-asymptotic u} now it is enough to prove that
\begin{equation}\label{eq:0gamma}
\ga = \mathop{\mathrm{Cap_p}}_p (\Omega)^{\frac{1}{p-1}};
\end{equation}
this can be easily done, as already noticed in \cite{GS}, through the following integration by parts that holds true by the $p$-harmonicity of $u$:
\begin{equation}\label{eq:capintegrperasymptotic}
- \int_{\Gamma} |\nabla u|^{p-2} u_{\nu} \, dS_x = - \lim_{R \to \infty} \int_{\partial B_R} |\nabla u|^{p-2} u_{\nu_{B_R}} \, dS_x .
\end{equation}
Here as in the rest of the chapter $\nu_{B_R}$ denotes the outer unit normal with respect to the ball $B_R$ of radius $R$, $\nu$ is the outer unit normal with respect to $\Omega$, and $u_\nu$ (resp.
$u_{\nu_{B_R}}$) is the derivative of $u$ in the direction $\nu$ (resp. $\nu_{B_R}$).
The left-hand side of \eqref{eq:capintegrperasymptotic} is exactly $\mathop{\mathrm{Cap_p}}_p (\Omega)$ as it is clear by \eqref{eq:caponboundary} and the fact that
\begin{equation}\label{eq:capnormagradmenodernormale}
| \nabla u|= - u_\nu \text{ on } \Gamma ;
\end{equation}
moreover, the limit in the right-hand side of \eqref{eq:capintegrperasymptotic} can be explicitly computed by using the second equation in \eqref{eq:passKV} (with $|\alpha|=1$) and it turns out to be $\ga^{p-1}$. Thus, \eqref{eq:0gamma} is proved and \eqref{eq:cap-asymptotic u} follows.
For completeness, we explain here how to prove the second identity in \eqref{eq:caponboundary}: we take the limit for $R \to \infty$ of the following integration by parts made on $B_R \setminus \overline{\Omega} $ and we note that the integral on $\partial B_R$ converge to zero due to
\eqref{eq:passKV}:
\begin{equation*}
\int_{B_R \setminus \overline{\Omega} } |\nabla u|^p \, dx = \int_{\partial B_R} u | \nabla u|^{p-2} u_{\nu_{B_R}} \, dS_x - \int_{\Gamma} |\nabla u|^{p-2} u_\nu \, dS_x .
\end{equation*}
The desired identity is thus proved just by recalling \eqref{eq:capnormagradmenodernormale}.
\end{proof}
It is well known that the value $c$ of $| \nabla u|$ on $ \Gamma$ appearing in the overdetermined condition \eqref{eq:overdetermination} can be explicitly computed.
\begin{lem}[Explicit value of $c$ in \eqref{eq:overdetermination}]
The constant $c$ appearing in \eqref{eq:overdetermination} equals
\begin{equation}\label{eq:valoreunu serrinesterno}
c = \frac{N-p}{p-1} \, H_0,
\end{equation}
where $H_0$ is the constant defined in \eqref{def-R-H0}.
Moreover, the following explicit expression of the $p$-capacity of $\Omega$ holds:
\begin{equation}\label{eq:cap in terms isop}
\mathop{\mathrm{Cap_p}}_p (\Omega) = \left( \frac{N-p}{p-1} \right)^{p-1} \frac{| \Gamma|^p}{(N | \Omega|)^{p-1}}.
\end{equation}
\end{lem}
\proof
It is enough to use \eqref{eq:caponboundary} together with the following Rellich-Pohozaev-type identity:
\begin{equation}\label{eq:Poho}
(N-p)\int_{\mathbb{R}^N \setminus \overline{\Omega} } |\nabla u|^p \, dx = (p-1)\int_\Gamma |\nabla u|^p <x, \nu> \, dS_x .
\end{equation}
In fact, by using \eqref{eq:overdetermination} in \eqref{eq:caponboundary} and \eqref{eq:Poho} we deduce respectively that
\begin{equation*}
\mathop{\mathrm{Cap_p}}_p (\Omega) = c^{p-1} | \Gamma| \quad \text{and} \quad \mathop{\mathrm{Cap_p}}_p (\Omega)=\frac{p-1}{N-p} c^p N | \Omega| ,
\end{equation*}
from which we get \eqref{eq:valoreunu serrinesterno} and \eqref{eq:cap in terms isop}.
Equation \eqref{eq:Poho} comes directly by taking the limit for $R \to \infty$ of the following integration by parts made on $B_R \setminus \overline{\Omega} $ and noting that the integrals on $\partial B_R$ converge to zero due to the asymptotic going of $u$ at infinity given by \eqref{eq:cap-asymptotic u}:
\begin{multline}\label{eq:provaconforme}
(N-p)\int_{B_R \setminus \overline{\Omega} } |\nabla u|^p \, dx =
\\
p \int_\Gamma |\nabla u|^{p-2} <x, \nabla u> u_\nu \, dS_x - \int_\Gamma |\nabla u|^p <x, \nu> \, dS_x -
\\
p \int_{\partial B_R } |\nabla u|^{p-2} <x, \nabla u> u_{\nu_{B_R}} \, dS_x + \int_{\partial B_R } |\nabla u|^p <x, \nu_{B_R}> \, dS_x .
\end{multline}
Equation \eqref{eq:Poho} is in fact proved just by recalling that
\begin{equation}\label{eq:capgraddernormnormale}
\nabla u = u_\nu \, \nu \text{ on }\Gamma.
\end{equation}
\endproof
\vspace{1pt}
\noindent
\textbf{The P-function.}
As last ingredient, we introduce the $P$-function
\begin{equation}
\label{def:P}
P= \frac{|\nabla u|^p}{u^{\frac{p(N-1)}{(N-p)}}}.
\end{equation}
Notice that in the radial case, i.e. if $\Omega= B_R (x_0)$ is a ball of radius $R$ centered at the point $x_0$, we have that
\begin{equation}
\label{eq:esplicitaradiale1}
u(x) = \left( \frac{R}{|x - x_0 |} \right)^{\frac{N-p}{p-1}}
\end{equation}
and thus
$$
P \equiv \left( \frac{N-p}{p-1} \right)^{p} R^{-p}.
$$
In \cite{GS} the authors have studied extensively the properties of the function $P$.
In particular, in \cite[Theorem 2.2]{GS} it is proved that $P$ satisfies the strong maximum principle, i.e. the function $P$ cannot attain a local maximum at an interior point of $\mathbb{R}^N \setminus \overline{ \Omega}$, unless $P$ is constant. We mention that this property for the case $p=2$ was first established in \cite{PP2}.
\vspace{1pt}
Now that we collected all the ingredients, we are in position to give the proof of Theorem \ref{thm:Serrinesterno}.
\begin{proof}[Proof of Theorem \ref{thm:Serrinesterno}]
By \eqref{eq:cap-asymptotic u} it is easy to check that
\begin{equation*}
\lim_{|x| \rightarrow \infty} P(x)= \left( \frac{N-p}{p-1} \right)^{\frac{p(N-1)}{N-p}} \left( \frac{ \omega_N }{ \mathop{\mathrm{Cap_p}}_p(\Omega)} \right)^{\frac{p}{N-p}} ,
\end{equation*}
from which, by using \eqref{eq:cap in terms isop}, we get
\begin{equation}\label{eq:prova2}
\lim_{|x| \rightarrow \infty} P(x)= \left( \frac{N-p}{p-1} \right)^{p} \left( \frac{ \omega_N \left( N | \Omega| \right)^{p-1} }{ | \Gamma|^p } \right)^{\frac{p}{N-p}}.
\end{equation}
Moreover, by recalling the boundary condition in
\eqref{Problemcapacity}, \eqref{eq:overdetermination}, and \eqref{eq:valoreunu serrinesterno}
we can compute that
\begin{equation}\label{eq:prova}
P_{| \Gamma} = \left( \frac{N-p}{p-1} \right)^p \left( \frac{| \Gamma| }{ N | \Omega| } \right)^p.
\end{equation}
By using the classical isoperimetric inequality (see, e.g., \cite{BZ})
\begin{equation}\label{eq:ISOPerimetrica}
\frac{| \Gamma|^{\frac{N}{N-1}}}{N \omega_N^{\frac{1}{N-1}}} \ge |\Omega|,
\end{equation}
by \eqref{eq:prova2} and \eqref{eq:prova} it is easy to check that
$$
\lim_{|x| \rightarrow \infty} P(x) \le P_{| \Gamma}.
$$
Hence, by the strong maximum principle proved in \cite[Theorem 2.2]{GS}, $P$ attains its maximum on $\Gamma$ and thus we can affirm that
\begin{equation}\label{eq:00}
P_{\nu} \le 0 ,
\end{equation}
where, $\nu$ is still the outer unit normal with respect to $\Omega$.
If we directly compute $P_{\nu}$ and we use \eqref{eq:capgraddernormnormale}, we find
\begin{equation}\label{eq:01}
P_\nu = p \, u^{- \frac{p(N-1)}{N-p}} |\nabla u|^{p-2} \left\lbrace u_{\nu \nu} u_\nu - \frac{N-1}{N-p} \, |\nabla u|^2 \, \frac{u_\nu}{u} \right\rbrace.
\end{equation}
By the well known differential identity
\begin{equation*}
\Delta_p u = | \nabla u|^{p-2} \left\lbrace (p-1) u_{\nu \nu} + (N-1) H u_\nu \right\rbrace \quad \mbox{ on $\Gamma$ }
\end{equation*}
and the $p$-harmonicity of $u$ we deduce that
\begin{equation}\label{eq:02}
u_{\nu \nu}= - \frac{N-1}{p-1} H u_\nu.
\end{equation}
By combining \eqref{eq:00}, \eqref{eq:01}, and \eqref{eq:02} we get
\begin{equation*}
p (N-1) u^{- \frac{p(N-1)}{N-p}} |\nabla u|^p \left\lbrace \frac{- u_{\nu }}{(N-p) u} - \frac{H}{p-1} \right\rbrace \le 0,
\end{equation*}
from which, by using the fact that $u=1$ on $\Gamma$, \eqref{eq:overdetermination}, \eqref{eq:capnormagradmenodernormale}, and \eqref{eq:valoreunu serrinesterno} we get
$$H_0 -H \le 0.$$
We can now conclude by using Theorem \ref{th:SBT}.
\end{proof}
\begin{rem}
{\rm
Since the solution of \eqref{Problemcapacity} in a ball is explicitly known, as a corollary of Theorem \ref{thm:Serrinesterno} we get that $u$ is spherically symmetric about the center $x_0$ of (the ball) $\Omega$ and it is given by \eqref{eq:esplicitaradiale1} with $R= \frac{N-p}{(p-1)c}$.
}
\end{rem}
\begin{rem}\label{rem:Garofaoloasymptotic}
{\rm
As already mentioned before, in \cite{GS} a result slightly different from Lemma~\ref{lem:asymptoticcap} is used. In fact, in \cite[Theorem 3.1]{GS} it is proved, independently from the work \cite{KV}, that if $P$ takes its supremum at infinity, then the asymptotic expansion \eqref{eq:cap-asymptotic u} holds true and hence
$\lim_{|x| \to \infty} P(x)$ exists and it is given by \eqref{eq:prova2}; clearly, that result would be sufficient to complete the proof of Theorem \ref{thm:Serrinesterno} without invoking Lemma \ref{lem:asymptoticcap}.
}
\end{rem}
\begin{rem}
{\rm
In \cite{GS}, instead of the classical isoperimetric inequality, the authors use the nonlinear version of the isoperimetric inequality for the capacity mentioned in the Introduction stating that, for any bounded open set $A$, if $B_\rho$ denotes a ball such that $|B_\rho|= \frac{ \omega_N }{N} \rho^N=|A|$, it holds that
\begin{equation}\label{eq:Poincare}
\mathop{\mathrm{Cap_p}}_p (A) \ge \mathop{\mathrm{Cap_p}}_p (B_\rho) ,
\end{equation}
with equality if and only if $A$ is a ball (for a proof see, e.g., \cite{Ge}).
Since the $p$-capacity of a ball is known explicitly (see, e.g., \cite[Equation (4.7)]{GS}):
\begin{equation*}
\mathop{\mathrm{Cap_p}}_p(B_\rho)= \omega_N \left( \frac{N-p}{p-1} \right)^{p-1} \rho^{N-p} ,
\end{equation*}
if we take $B_\rho$ such that $|B_\rho|= |\Omega|$, for $\Omega$ \eqref{eq:Poincare} becomes
\begin{equation}\label{eq:isop cap}
\mathop{\mathrm{Cap_p}}_p (\Omega) \ge \omega_N \left( \frac{N-p}{p-1} \right)^{p-1} \left( \frac{N |\Omega|}{\omega_N} \right)^{\frac{N-p}{N}}.
\end{equation}
Now we notice that, since \eqref{eq:cap in terms isop} holds, \eqref{eq:isop cap} is equivalent to the classical isoperimetric inequality \eqref{eq:ISOPerimetrica}.
In fact, if we put \eqref{eq:cap in terms isop} in \eqref{eq:isop cap}, it is easy to check that \eqref{eq:isop cap} becomes exactly \eqref{eq:ISOPerimetrica}.
}
\end{rem}
\section{The case \texorpdfstring{$1<p=N$}{1<p=N}: proof of Theorem \ref{thm:conforme}}\label{sec:casoconforme}
\sectionmark{The case \texorpdfstring{$1<p=N$}{1<p=N}: proof of Theorem \ref{thm:conforme}}
In the present section, we consider $u$ solution to the exterior problem \eqref{ConformalProblemcapacity}, \eqref{eq:overdetermination}, and $1<N$.
In order to give the proof of Theorem \ref{thm:conforme}, we collect all the necessary ingredients in the following remark.
\begin{rem}
{\rm
(i)(On the regularity). We notice that the regularity results invoked in Remark \ref{rem:regularity} hold when $p=N$, too.
(ii)(Asymptotic expansion). By \eqref{eq:confasymppre}, the result of Kichenassamy and V\'eron (\cite[Remark 1.5]{KV}) applies also in this case and hence we have that \eqref{eq:passKV} holds with
$$
\mu(x)= - \frac{ \ln{| x |} }{\omega_N^{\frac{1}{N-1}}}.
$$
It is easy to check that also Identity \eqref{eq:capintegrperasymptotic} still holds (with $p$ replaced by $N$); by computing the limit in the right-hand side, and by using \eqref{eq:overdetermination} and the fact that $|\nabla u| = - u_\nu$ in the left-hand side, \eqref{eq:capintegrperasymptotic} leads to
$$
c^{N-1} | \Gamma|= \ga^{N-1},
$$
from which we deduce the following asymptotic expansion for $u$ at infinity:
\begin{equation}\label{eq:conformasymptoticexp}
u(x)= - c \left( \frac{|\Gamma|}{\omega_N} \right)^{\frac{1}{N-1}} \, \ln{|x|} + O(1).
\end{equation}
}
\end{rem}
We are ready now to prove Theorem \ref{thm:conforme}.
\begin{proof}[Proof of Theorem \ref{thm:conforme}]
We just find the analogous of the Rellich-Pohozaev-type identity \eqref{eq:provaconforme} when $p=N$; since $u$ is $N$-harmonic, now the vector field
$$X= N <x, \nabla u> | \nabla u|^{N-2} \nabla u - | \nabla u|^N x$$
is divergence-free and thus integration by parts leads to
\begin{equation}\label{eq:conformeintegrbyparts}
\int_{\Gamma} <X, \nu> \, dS_x= \lim_{R \to \infty} \int_{\partial B_R} <X, \nu_{B_R}> \, dS_x.
\end{equation}
For the left-hand side in \eqref{eq:conformeintegrbyparts}, by using that $\nabla u = u_\nu \, \nu$ on $\Gamma$ and \eqref{eq:overdetermination} we immediately find that
$$
\int_{\Gamma} <X, \nu> \, dS_x= (N-1) c^N N|\Omega|.
$$
For the right-hand side in \eqref{eq:conformeintegrbyparts}, by the asymptotic expansion \eqref{eq:conformasymptoticexp} we easily compute that
$$
\lim_{R \to \infty} \int_{\partial B_R} <X, \nu_{B_R}> \, dS_x = (N-1) c^N \frac{|\Gamma|^{\frac{N}{N-1}}}{\omega_N^{\frac{1}{N-1}}}.
$$
Thus, \eqref{eq:conformeintegrbyparts} becomes
$$
N |\Omega| = \frac{|\Gamma|^\frac{N}{N-1}}{\omega_N^{\frac{1}{N-1}}},
$$
that is the equality case of the classical isoperimetric inequality. Hence $\Omega$ must be a ball.
\end{proof}
\begin{rem}
{\rm
As a corollary of Theorem \ref{thm:conforme} we get that $u$ is spherically symmetric about the center $x_0$ of (the ball) $\Omega$ and it is given by
$$
u(x)= - c \left( \frac{|\Gamma|}{\omega_N} \right)^{\frac{1}{N-1}} \, \ln{|x - x_0|} ,
$$
up to an additive constant.
}
\end{rem}
\section{The interior problem: proof of Theorem \ref{thm:Serrininterno}}\label{sec:Interior}
\sectionmark{The interior problem: proof of Theorem \ref{thm:Serrininterno}}
In the present section $u$ is a weak solution to \eqref{Pdelta}, \eqref{eq:interno-overdetermination}, and $1<p<N$.
By a weak solution of \eqref{Pdelta}, \eqref{eq:interno-overdetermination} we mean a function $u \in W^{1,p}_{loc} \left( \Omega \setminus \left\lbrace 0 \right\rbrace \right)$ such that
$$
\int_{\Omega} |\nabla u|^{p-2} \, \nabla u \cdot \nabla \phi \, dx=0
$$
for every $\phi \in C_0^\infty \left( \Omega \setminus \left\lbrace 0 \right\rbrace \right)$ and satisfying the boundary condition in \eqref{Pdelta} and the one in \eqref{eq:interno-overdetermination} in the weak sense explained after Theorem \ref{thm:Serrinesterno}.
In order to give our proof of Theorem \ref{thm:Serrininterno}, we collect all the necessary ingredients in the following remark.
\begin{rem}
{\rm
(i)(On the regularity). The regularity results presented for the exterior problem in
Remark \ref{rem:regularity} hold in the same way also for the interior problem, so that, reasoning as explained there, we can affirm that $u$ can be extended to a $C^2$-function in a neighborhood of $\Gamma$, $\Gamma$ is of class $C^2$, and $u \in C^{1, \alpha}_{loc} (\overline{\Omega})$.
(ii)(Explicit value of $K$ in \eqref{Pdelta}). It is easy to show that the normalization constant $K$ that appears in \eqref{Pdelta} must take the value
$$
K= |\Gamma |,
$$
to be compatible with the overdetermined condition \eqref{eq:interno-overdetermination} (see for example \cite{EP}).
(iii)(Asymptotic expansion). As a direct application of \cite[Theorem 1.1]{KV}, we deduce that the asymptotic behaviour of the solution $u$ of \eqref{Pdelta} near the origin is given by
\begin{equation}
\label{asymptoticu}
u(x) = \frac{p-1}{N-p} \left( \frac{|\Gamma|}{ \omega_N} \right)^{\frac{1}{p-1}} \, |x|^{- \frac{N-p}{p-1}} + o(|x|^{- \frac{N-p}{p-1}}).
\end{equation}
We mention that this expansion has been used also in \cite{EP}.
}
\end{rem}
\vspace{1pt}
\noindent
\textbf{The P-function.}
We consider again the P-function defined in \eqref{def:P}; by virtue of the weak $p$-harmonicity of $u$ in $\Omega \setminus \lbrace 0 \rbrace$, \cite[Theorem 2.2]{GS} ensures that the strong maximum principle holds for $P$ in $\Omega \setminus \lbrace 0 \rbrace$.
\vspace{1pt}
We are ready now to prove Theorem \ref{thm:Serrininterno}; as announced in the Introduction, the proof uses arguments similar to those used in the proof of Theorem \ref{thm:Serrinesterno}.
\begin{proof}[Proof of Theorem \ref{thm:Serrininterno}]
As already noticed in \cite{EP}, concerning the existence of a weak solution to the overdetermined problem \eqref{Pdelta}, \eqref{eq:interno-overdetermination}, the actual value of the function $u$ on $\Gamma$ is irrelevant, since any function differing from $u$ by a constant is $p$-harmonic whenever $u$ is.
Thus, let us now fix the constant $c$ that appears in \eqref{Pdelta} as
\begin{equation}\label{def:cSBT}
c= \frac{p-1}{N-p} \left( \frac{N |\Omega|}{|\Gamma|} \right);
\end{equation}
with this choice and by recalling \eqref{eq:interno-overdetermination} we have that
$$
P_{| \Gamma} = \left( \frac{p-1}{N-p} \right)^{- \frac{p(N-1)}{N-p}} \left( \frac{N| \Omega|}{| \Gamma|} \right)^{- \frac{p(N-1)}{N-p}}.
$$
Moreover, by using \eqref{asymptoticu} we find also that
$$
\lim_{|x| \rightarrow 0} P(x)= \left( \frac{p-1}{N-p} \right)^{- \frac{p(N-1)}{N-p}}\left( \frac{|\Gamma|}{\omega_N} \right)^{- \frac{p}{N-p}}.
$$
By the isoperimetric inequality \eqref{eq:ISOPerimetrica} it is easy to check that
$$
\lim_{|x| \rightarrow 0} P(x) \le P_{| \Gamma} ,
$$
and hence, by the maximum principle proved in \cite[Theorem 2.2]{GS}, we realize that $P$ attains its maximum on $\Gamma$. We thus have that
\begin{equation}\label{eq:MPtoP}
0\le P_{\nu}.
\end{equation}
By a direct computation, with exactly the same manipulations used for the exterior problem in the proof of Theorem \ref{thm:Serrinesterno}, we find that
\begin{equation}\label{eq:PnuSBT}
P_{\nu} = p (N-1) u^{- \frac{p(N-1)}{N-p}} |\nabla u|^p \left\lbrace \frac{- u_{\nu}}{(N-p) u} - \frac{H}{p-1} \right\rbrace.
\end{equation}
By coupling \eqref{eq:MPtoP} with \eqref{eq:PnuSBT}, we can deduce that
\begin{equation*}
H \le \frac{p-1}{N-p} \left(\frac{- u_\nu}{u}\right),
\end{equation*}
that by using \eqref{def:cSBT}, \eqref{Pdelta}, \eqref{eq:interno-overdetermination}, and the fact that $|\nabla u|= - u_\nu$ on $\Gamma$, leads to
\begin{equation}\label{eq:penultima SBT}
H \le H_0,
\end{equation}
where $H_0$ is the constant defined in \eqref{def-R-H0}.
Since $\Omega$ is star-shaped with respect to a point $z \in \Omega$ (possibly distinct from $0$), we have that $<(x - z) ,\nu >$ is non-negative on $\Gamma$.
Thus, multiplying \eqref{eq:penultima SBT} by $<(x - z) ,\nu>$,
and integrating over $\Gamma$, we get
\begin{equation*}
\int_\Gamma H <(x - z ) ,\nu> \, dS_x \le |\Gamma|.
\end{equation*}
By recalling the regularity of $\Gamma$ and {\it Minkowski's identity}
\begin{equation*}
\int_\Gamma H <(x - z ) ,\nu> \, dS_x = |\Gamma| ,
\end{equation*}
we deduce -- as already noticed in \cite[Proof of Theorem 1.1]{GS} -- that the equality sign must hold in \eqref{eq:penultima SBT}, that is
$$
H \equiv H_0.
$$
Thus, the conclusion follows by the classical Alexandrov's Soap Bubble Theorem (\cite{Al2}) or, if we want, again by Theorem \ref{th:SBT}.
\end{proof}
\begin{rem}
{\rm
As a corollary of Theorem \ref{thm:Serrininterno} we get that $u$ is spherically symmetric about the center $0$ of (the ball) $\Omega$ and it is given by
$$
u(x) = \frac{p-1}{N-p} \left( \frac{|\Gamma|}{ \omega_N} \right)^{\frac{1}{p-1}} \, |x|^{- \frac{N-p}{p-1}} ,
$$
up to an additive constant.
}
\end{rem}
\chapter{Some estimates for harmonic functions}\label{chapter:various estimates}
As already sketched in the introduction, the desired stability estimates for the spherical symmetry of $\Omega$ will be obtained by linking the oscillation on $\Gamma$ of the harmonic function $h= q - u$, where $u$ is the solution of \eqref{serrin1} and $q$ is the quadratic polynomial defined in \eqref{quadratic},
to the integrals
$$
\int_\Omega (-u)|\nabla^2 h|^2 dx \quad \mbox{and} \quad \int_\Omega |\nabla^2 h|^2 dx.
$$
\par
In order to fulfill this agenda, in Sections \ref{sec:harmonic functions in weighted spaces} and \ref{sec:estimates for the oscillation of harmonic functions} we collect some useful estimates for harmonic functions that have their own interest.
Then, in Section \ref{sec:estimates for h} we deduce some inequalities for the particular harmonic function $h= q- u$ that will be useful for the study of the stability issue addressed in Chapter \ref{chapter:Stability results}.
In Sections \ref{sec:estimates for the oscillation of harmonic functions} and \ref{sec:estimates for h}, some pointwise estimates for the torsional rigidity density will be useful. We collect them in the following section.
Before we start, we set some relevant notations.
For a point $z\in\Omega$, $\rho_i$ and $\rho_e$ denote the radius of the largest ball centered at $z$ and contained in $\Omega$ and that of the smallest ball that contains $\Omega$ with the same center, that is
\begin{equation}
\label{def-rhos}
\rho_i=\min_{x\in\Gamma}|x-z| \ \mbox{ and } \ \rho_e=\max_{x\in\Gamma}|x-z|.
\end{equation}
The diameter of $\Omega$ is indicated by $d_\Omega$, while $\de_\Gamma (x)$ denotes the distance of a point $x$ to the boundary $\Gamma$.
We recall that if $\Gamma$ is of class $C^2$, $\Omega$ has the properties of the uniform interior and exterior sphere condition, whose respective radii we have designated by $r_i$ and $r_e$. In other words,
there exists $r_e > 0$ (resp. $r_i>0$) such that for each $p \in \Gamma$ there exists a ball contained in $\mathbb{R}^N \setminus \overline{\Omega}$ (resp. contained in $\Omega$) of radius $r_e$ (resp. $r_i$) such that
its closure intersects $\Gamma$ only at $p$.
Finally, if $\Gamma$ is of class $C^{1,\alpha}$, the unique solution of \eqref{serrin1} is of class at least
$C^{1,\alpha}(\overline{\Omega})$.
Thus, we can define
\begin{equation}
\label{bound-gradient}
M = \max_{\overline{\Omega}} |\nabla u| = \max_\Gamma u_\nu .
\end{equation}
\section{Pointwise estimates for the torsional rigidity density}\label{sec:gradient estimate for torsion}
We start with the following lemma in which we relate $u(x)$ with the distance function $\de_\Gamma (x)$.
\begin{lem}
\label{lem:relationdist}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain such that $\Gamma$ is made of regular points for the Dirichlet problem, and let $u$ be the solution of \eqref{serrin1}. Then
$$
-u(x)\ge\frac12\,\de_\Gamma(x)^2 \ \mbox{ for every } \ x\in\overline{\Omega}.
$$
Moreover, if $\Gamma$ is of class $C^{2}$, then it holds that
\begin{equation}
\label{eq:relationdist}
-u(x) \ge \frac{r_i}{2}\,\de_\Gamma(x)\ \mbox{ for every } \ x\in\overline{\Omega}.
\end{equation}
\end{lem}
\begin{proof}
If every point of $\Gamma$ is regular, then a unique solution $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ exists for \eqref{serrin1}. Now, for $x\in\Omega$, let $r=\de_\Gamma(x)$ and consider the ball $B=B_r(x)$.
Let $w$ be the solution of \eqref{serrin1} in $B$, that is $w(y)=(|y-x|^2-r^2)/2$. By comparison we have that $w\ge u$ on $\overline{B}$ and hence, in particular, $w(x) \ge u(x)$. Thus, we infer the first inequality in the lemma.
\par
If $\Gamma$ is of class $C^{2}$, \eqref{eq:relationdist} certainly holds if $\de_\Gamma(x)\ge r_i$. If $\de_\Gamma (x) < r_i$, instead, let $z$ be the closest point in $\Gamma$ to $x$ and call $B$ the ball of radius $r_i$ touching $\Gamma$ at $z$ and containing $x$. Up to a translation, we can always suppose that the center of the ball $B$ is the origin $0$. If $w$ is the solution of \eqref{serrin1} in $B$, that is $w(y)=\left(|y|^2- r_i^2 \right)/2$, by comparison we have that $w \ge u$ in $B$, and hence
$$
-u(x) \ge \frac12\,(|x|^2-r_i^2)=
\frac12\,( r_i + |x| )(r_i-|x|)\ge\frac12\,r_i\,(r_i-|x|).
$$
This implies \eqref{eq:relationdist}, since $r_i-|x|=\de_\Gamma(x)$.
\end{proof}
In the remaining part of this section we present a simple method to estimate the number $M$ (defined in \eqref{bound-gradient})
in a quite general domain. The following lemma results from a simple inspection and by the uniqueness for the Dirichlet problem.
\begin{lem}[Torsional rigidity density in an annulus]\label{soluzioneincoronacircolare}
\label{lem:torsion-annulus}
Let $A=A_{r,R}\subset \mathbb{R}^N$ be the annulus centered at the origin and radii $0<r<R$,
and set $\ka=r/R$.
\par
Then, the solution $w$ of the Dirichlet problem
\begin{equation*}
\label{torsion-annulus}
\Delta w = N \ \textrm{ in } \ A, \quad w = 0 \ \textrm{ on } \ \partial A,
\end{equation*}
is defined for $r\le |x|\le R$ by
\begin{equation*}
w(x) =
\begin{cases}
\displaystyle\frac12\, |x|^2 +\frac{R^2}{2}\,(1-\ka^2)\,\frac{\log(|x|/r)}{\log\ka} -\frac{r^2}{2}
\ &\mbox{ for } \ N=2,
\vspace{5pt} \\
\displaystyle\frac12\,|x|^2 +\frac12\,\frac{R^2}{1-\ka^{N-2}}\,\left\{ (1-\ka^2)\,(|x|/r)^{2-N}+\ka^N-1\right\} \ &\mbox{ for } \ N \ge 3.
\end{cases}
\end{equation*}
\end{lem}
\medskip
\begin{thm}[A bound for the gradient on $\Gamma$, \cite{MP}]
\label{thm:boundary-gradient}
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain that satisfies the uniform interior and exterior conditions with radii $r_i$ and $r_e$ and let $u\in C^1(\overline{\Omega})\cap C^2(\Omega)$ be a solution
of \eqref{serrin1} in $\Omega$.
\par
Then, we have that
\begin{equation}
\label{gradient-estimate}
r_i\le |\nabla u| \le
c_N\,\frac{d_\Omega(d_\Omega+r_e)}{r_e} \ \mbox{ on } \ \Gamma,
\end{equation}
where $d_\Omega$ is the diameter of $\Omega$ and
$c_N=3/2$ for $N=2$ and $c_N=N/2$ for $N\ge 3$.
\end{thm}
\begin{proof}
We first prove the first inequality in \eqref{gradient-estimate}. Fix any $p\in\Gamma$.
Let $B=B_{r_i}$ be the interior ball touching $\Gamma$ at $p$ and place the origin of cartesian axes at the center of $B$.
\par
If $w$ is the solution of \eqref{serrin1} in $B$, that is $w(x)= (|x|^2-r_i^2)/2$, by comparison we have that $w\ge u$ on $\overline{\Omega}$ and hence, since $u(p)=w(p)=0$, we obtain:
$$
u_\nu(p)\ge w_\nu(p)=r_i.
$$
\par
To prove the second inequality, we place the origin of axes at the center of the exterior ball $B=B_{r_e}$ touching $\Gamma$ at $p$. Denote by $A$ the smallest annulus containing $\Omega$, concentric with $B$ and having $\partial B$ as internal boundary and let $R$ be the radius of its external boundary.
\par
If $w$ is the solution of \eqref{serrin1} in $A$, by comparison we have that $w\le u$ on $\overline{\Omega}$.
Moreover, since $u(p)=w(p)=0$, we have that
$$
u_\nu(p)\le w_\nu(p).
$$
By Lemma \ref{lem:torsion-annulus} we then compute that
$$
w_\nu(p) =\frac{R (R-r_e)}{r_e}\,f(\ka)
$$
where, for $0<\ka<1$,
\begin{equation*}
f(\ka)=
\begin{cases}
\displaystyle \frac{2\ka^2\log(1/\ka)+\ka^2-1}{2(1-\ka) \log(1/\ka)}
\ &\mbox{ for } \ N=2,
\vspace{5pt} \\
\displaystyle\frac{2 \ka^N-N \ka^2+N-2}{2(1-\ka)(1-\ka^{N-2})} \ &\mbox{ for } \ N \ge 3.
\end{cases}
\end{equation*}
Notice that $f$ is bounded since it can be extended to a continuous function on $[0,1]$.
Tedious calculations yield that
$$
\sup_{0<\ka<1} f(\ka)=
\begin{cases}
\frac32 \ &\mbox{ for } \ N=2, \\
\frac{N}2 \ &\mbox{ for } \ N\ge 3.
\end{cases}
$$
Finally, observe that $R\le d_\Omega+r_e$.
\end{proof}
\begin{rem}\label{rem:boundgradienttorsionconvexr_eremoving}
{\rm
(i) Notice that, when $\Omega$ is convex, we can choose $r_e=+\infty$ in \eqref{gradient-estimate} and obtain
\begin{equation}
\label{bound-M-convex}
M\le c_N\,d_\Omega.
\end{equation}
(ii) To the best of our knowledge, inequality \eqref{gradient-estimate}, established in \cite{MP}, was not
present in the literature for general smooth domains.
Other estimates are given in \cite{PP} for planar strictly convex domains (but the same argument can be generalized to general dimension for strictly mean convex domains) and in \cite{CM} for strictly mean convex domains in general dimension.
In particular, in \cite[Lemma 2.2]{CM} the authors prove that there exists a universal constant $c_0$ such that
\begin{equation*}
| \nabla u | \leq c_0 | \Omega |^{1/N} \mbox{ in } \overline{\Omega}.
\end{equation*}
}
\end{rem}
\section{Harmonic functions in weighted spaces}\label{sec:harmonic functions in weighted spaces}
To start, we briefly report on some Hardy-Poincar\'e-type inequalities for harmonic functions that are present in the literature.
The following lemma can be deduced from the work of Boas and Straube \cite{BS}, which improves a result of Ziemer \cite{Zi}.
In what follows, for a set $A$ and a function $v: A \to \mathbb{R}$, $v_A$ denotes the {\it mean value of $v$ in $A$} that is
$$
v_A= \frac{1}{|A|} \, \int_A v \, dx.
$$
Also, for a function $v:\Omega \to \mathbb{R}$ we define
$$
\Vert \de_\Gamma^\alpha \, \nabla v \Vert_{p,\Omega} = \left( \sum_{i=1}^N \Vert \de_\Gamma^\alpha \, v_i \Vert_{p,\Omega}^p \right)^\frac{1}{p} \quad \mbox{and} \quad
\Vert \de_\Gamma^\alpha \, \nabla^2 v \Vert_{p,\Omega} = \left( \sum_{i,j=1}^N \Vert \de_\Gamma^\alpha \, v_{ij} \Vert_{p,\Omega}^p \right)^\frac{1}{p},
$$
for $0 \le \alpha \le 1$ and $p \in [1, \infty)$.
\begin{lem}[\cite{BS}]
\label{lem:two-inequalities}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^{0,\alpha}$,
$0 \le \alpha \le 1$, let $z$ be a point in $\Omega$, and consider $p \in \left[ 1, \infty \right)$. Then,
\par
(i) there exists a positive constant $ \mu_{p, \alpha} ( \Omega, z ) $, such that
\begin{equation}
\label{harmonic-quasi-poincare}
\Vert v \Vert_{p,\Omega} \le \mu_{p, \alpha} ( \Omega, z)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega},
\end{equation}
for every harmonic function $v$ in $\Omega$ such that $v(z)=0$;
\par
(ii)
there exists a positive constant, $\overline{\mu}_{p, \alpha} (\Omega)$ such that
\begin{equation}
\label{harmonic-poincare}
\Vert v - v_\Omega \Vert_{p,\Omega} \le \overline{\mu}_{p, \alpha} (\Omega)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega},
\end{equation}
for every function $v$ which is harmonic in $\Omega$.
\par
In particular, if $\Gamma$ has a Lipschitz boundary, the number $\alpha$ can be replaced by any exponent in $[0,1]$.
\end{lem}
\begin{proof}
The assertions (i) and (ii) are easy consequences of a general result of Boas and Straube (see \cite{BS}).
In case (i), we apply \cite[Example 2.5]{BS}). In case (ii), \cite[Example 2.1]{BS} is appropriate. In fact, \cite[Example 2.1]{BS} proves \eqref{harmonic-poincare} for every function $v \in L^p (\Omega) \cap W^{1,p}_{loc} (\Omega)$. The addition of the harmonicity of $v$ in \eqref{harmonic-poincare} clearly gives a better constant.
\par
The (solvable) variational problems
\begin{equation*}
\mu_{p, \alpha} (\Omega, z) =
\min \left\{ \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega} : \Vert v \Vert_{p, \Omega} = 1, \,\Delta v =0 \text{ in } \Omega, \, v(z) = 0 \right\}
\end{equation*}
and
\begin{equation*}
\overline{\mu}_{p, \alpha} (\Omega) =
\min \left\{ \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega} : \Vert v \Vert_{p, \Omega} = 1, \,\Delta v =0 \text{ in } \Omega, \, v_\Omega = 0 \right\}
\end{equation*}
then characterize the two constants.
\end{proof}
\begin{rem}
{\rm
When $\alpha=0$ we understand the boundary of $\Omega$ to be locally the graph of a continuous function. In this case, \eqref{harmonic-quasi-poincare} is exactly the result contained in \cite{Zi}.
Also, in the case $p=2$ and $\alpha=0$ from \eqref{harmonic-quasi-poincare} and \eqref{harmonic-poincare} we recover the Poincar\'e-type inequalities proved and used in \cite{MP}.
}
\end{rem}
\medskip
The following corollary describes a couple of applications of the Hardy-Poincar\'e inequalities that we have just presented.
\begin{cor}\label{cor:Poincareaigradienti}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^{0,\alpha}$, $0 \le \alpha \le 1$, and let $v$ be a harmonic function in $\Omega$.
(i) If $z$ is a critical point of $v$ in $\Omega$, then it holds that
\begin{equation*}
\Vert \nabla v \Vert_{p, \Omega} \le \mu_{p, \alpha} ( \Omega, z)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla^2 v \Vert_{p, \Omega}.
\end{equation*}
(ii) If
$$\int_\Omega \nabla v \, dx = 0,$$
then it holds that
\begin{equation*}
\Vert \nabla v \Vert_{p, \Omega} \le \overline{\mu}_{p, \alpha} (\Omega)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla^2 v \Vert_{p, \Omega}.
\end{equation*}
\end{cor}
\begin{proof}
Since $\nabla v (z)=0$ (respectively $\int_\Omega \nabla v \, dx =0$), we can apply \eqref{harmonic-quasi-poincare} (respectively \eqref{harmonic-poincare}) to each first partial derivative $v_i$ of $v$, $i=1, \dots, N$. If we raise to the power of $p$ those inequalities and sum over $i=1, \dots, N$, the conclusion easily follows.
\end{proof}
We now present a simple lemma which will be useful in the sequel to manipulate the left-hand side of Hardy-Poincar\'e-type inequalities.
\begin{lem}\label{lem:mediaor-mediauguale}
Let $\Omega$ be a domain with finite measure and let $A \subseteq \Omega$ be a set of positive measure. If $v \in L^p(\Omega)$, then for every $\lambda \in \mathbb{R}$
\begin{equation}\label{eq:mediaor-mediauguale}
\Vert v - v_A \Vert_{p,\Omega} \le \left[ 1+ \left( \frac{|\Omega|}{|A|} \right)^{\frac{1}{p}} \right] \, \Vert v - \lambda \Vert_{p,\Omega}.
\end{equation}
\end{lem}
\begin{proof}
By H\"older's inequality, we have that
\begin{equation*}
|v_A - \lambda| \le \frac1{|A|} \int_A|v-\lambda| \, dx \le
|A|^{-1/p} \Vert v- \lambda \Vert_{p,A} \le
|A|^{-1/p} \Vert v - \lambda \Vert_{p,\Omega}.
\end{equation*}
Since $|v_A - \lambda|$ is constant, we then infer that
$$
\Vert v_A-\lambda\Vert_{p,\Omega}=|\Omega|^{1/p} |v_A - \lambda| \le \left( \frac{|\Omega|}{|A|} \right)^{\frac{1}{p}} \, \Vert v- \lambda \Vert_{p,\Omega}.
$$
Thus, \eqref{eq:mediaor-mediauguale} follows by an application of the triangular inequality.
\end{proof}
Other versions of Hardy-Poincar\'e inequalities different from those presented in Lemma \ref{lem:two-inequalities} can be deduced from the work of Hurri-Syrj\"anen \cite{HS}, which was stimulated by \cite{BS}.
In order to state these results, we introduce the notions of $b_0$-John domain and $L_0$-John domain with base point $z \in \Omega$. Roughly speaking, a domain is a $b_0$-John domain (resp. a $L_0$-John domain with base point $z$) if it is possible to travel from one point of the domain to another (resp. from $z$ to another point of the domain) without going too close to the boundary.
A domain $\Omega$ in $\mathbb{R}^N$ is a {\it $b_0$-John domain}, $b_0 \ge 1$, if each pair of distinct points $a$ and $b$ in $\Omega$ can be joined by a curve $\ga: \left[0,1 \right] \rightarrow \Omega$ such that
\begin{equation*}
\de_\Gamma (\ga(t)) \ge b_0^{-1} \min{ \left\lbrace |\ga(t) - a|, |\ga(t) - b| \right\rbrace }.
\end{equation*}
A domain $\Omega$ in $\mathbb{R}^N$ is a {\it $L_0$-John domain with base point $z \in \Omega$}, $L_0 \ge 1$, if each point $x \in \Omega$ can be joined to $z$ by a curve $\ga: \left[0,1 \right] \rightarrow \Omega$ such that
\begin{equation*}
\de_\Gamma (\ga(t)) \ge L_0^{-1} |\ga(t) - x|.
\end{equation*}
It is known that, for bounded domains, the two definitions are quantitatively equivalent (see \cite[Theorem 3.6]{Va}).
The two notions could be also defined respectively through the so-called {\it $b_0$-cigar} and {\it $L_0$-carrot} properties (see \cite{Va}).
\begin{lem}\label{lem:John-two-inequalities}
Let $\Omega \subset \mathbb{R}^N$ be a bounded $b_0$-John domain, and consider three numbers $r, p, \alpha$ such that $1 \le p \le r \le \frac{Np}{N-p(1 - \alpha )}$, $p(1 - \alpha)<N$, $0 \le \alpha \le 1$.
Then,
(i) there exists a positive constant $ \mu_{r,p, \alpha} ( \Omega, z ) $, such that
\begin{equation}
\label{John-harmonic-quasi-poincare}
\Vert v \Vert_{r,\Omega} \le \mu_{r, p, \alpha} ( \Omega, z)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega},
\end{equation}
for every function $v$ which is harmonic in $\Omega$ and such that $v(z)=0$;
\par
(ii) there exists a positive constant, $\overline{\mu}_{r, p, \alpha} (\Omega)$ such that
\begin{equation}
\label{John-harmonic-poincare}
\Vert v - v_\Omega \Vert_{r,\Omega} \le \overline{\mu}_{r, p, \alpha} (\Omega)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega},
\end{equation}
for every function $v$ which is harmonic in $\Omega$.
\end{lem}
\begin{proof}
In \cite[Theorem 1.3]{HS} it is proved that there exists a constant $c=c(N,\, r, \, p,\, \alpha, \, \Omega)$ such that
\begin{equation}\label{eq:risultatodiHSconr-media}
\Vert v - v_{r, \Omega} \Vert_{r,\Omega} \le c \, \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega} ,
\end{equation}
for every $v \in L^1_{loc}(\Omega)$ such that $\de_\Gamma^\alpha \, \nabla v \in L^p(\Omega)$.
Here, $v_{r,\Omega}$ denotes the {\it $r$-mean of $v$ in $\Omega$} which is defined -- following \cite{IMW} -- as the unique minimizer of the problem
$$\inf_{\lambda \in \mathbb{R}} \Vert v - \lambda \Vert_{r,\Omega}.$$
Notice that, in the case $r=2$, $v_{2, \Omega}$ is the classical mean value of $v$ in $\Omega$, i.e. $v_{2,\Omega} = v_\Omega$,
as can be easily verified.
By using Lemma \ref{lem:mediaor-mediauguale} with $A= \Omega$ and $\lambda = v_{r, \Omega}$ we thus prove \eqref{John-harmonic-poincare} for every $v \in L^1_{loc}(\Omega)$ such that $\de_\Gamma^\alpha \, \nabla v \in L^p(\Omega)$.
The assumption of the harmonicity of $v$ in \eqref{John-harmonic-poincare} clearly gives a better constant.
Inequality \eqref{John-harmonic-quasi-poincare} can be deduced from \eqref{John-harmonic-poincare} by applying Lemma \ref{lem:mediaor-mediauguale} with $A= B_{\de_\Gamma (z)} (z)$ and $\lambda = v_\Omega$ and recalling that, since $v$ is harmonic, by the mean value property it holds that
$v(z)= v_{B_{\de_\Gamma (z)} (z)}.$
The (solvable) variational problems
\begin{equation*}
\mu_{r, p, \alpha} (\Omega, z) =
\min \left\{ \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega} : \Vert v \Vert_{r, \Omega} = 1, \,\Delta v =0 \text{ in } \Omega, \, v(z) = 0 \right\}
\end{equation*}
and
\begin{equation*}
\overline{\mu}_{r, p, \alpha} (\Omega) =
\min \left\{ \Vert \de_\Gamma^{\alpha} \, \nabla v \Vert_{p, \Omega} : \Vert v \Vert_{r, \Omega} = 1, \,\Delta v =0 \text{ in } \Omega, v_{\Omega} = 0 \right\}
\end{equation*}
then characterize the two constants.
\end{proof}
\begin{rem}
{\rm
When $r=p$, \eqref{John-harmonic-quasi-poincare} and \eqref{John-harmonic-poincare} become exactly \eqref{harmonic-quasi-poincare} and \eqref{harmonic-poincare}.
However, it should be noticed that \eqref{harmonic-quasi-poincare} and \eqref{harmonic-poincare} proved in \cite{BS} also hold true without the restriction $p (1-\alpha) < N$ assumed in \cite{HS}.
}
\end{rem}
From Lemma \ref{lem:John-two-inequalities} we can derive estimates for the derivatives of harmonic functions, as already done in Corollary \ref{cor:Poincareaigradienti} for Lemma \ref{lem:two-inequalities}.
\begin{cor}\label{cor:JohnPoincareaigradienti}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded $b_0$-John domain and let $v$ be a harmonic function in $\Omega$. Consider three numbers $r, p, \alpha$ such that $1 \le p \le r \le \frac{Np}{N-p(1 - \alpha )}$, $p(1 - \alpha)<N$, $0 \le \alpha \le 1$.
(i) If $z$ is a critical point of $v$ in $\Omega$, then it holds that
\begin{equation*}
\Vert \nabla v \Vert_{r, \Omega} \le \mu_{r, p, \alpha} ( \Omega, z)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla^2 v \Vert_{p, \Omega}.
\end{equation*}
(ii) If
$$\int_\Omega \nabla v \, dx = 0,$$
then it holds that
\begin{equation*}
\Vert \nabla v \Vert_{r, \Omega} \le \overline{\mu}_{r, p, \alpha} (\Omega)^{-1} \Vert \de_\Gamma^{\alpha} \, \nabla^2 v \Vert_{p, \Omega}.
\end{equation*}
\end{cor}
\begin{proof}
Since $\nabla v (z)=0$ (respectively $\int_\Omega \nabla v \, dx = 0,$), we can apply \eqref{harmonic-quasi-poincare} (respectively \eqref{harmonic-poincare}) to each first partial derivative $v_i$ of $v$, $i=1, \dots, N$. If we raise to the power of $r$ those inequalities and sum over $i=1, \dots, N$, the conclusion easily follows in view of the inequality
$$
\sum_{i=1}^N x_i^{\frac{r}{p}} \le \left( \sum_{i=1}^N x_i \right)^{\frac{r}{p}}
$$
that holds for every $(x_1, \dots, x_N) \in \mathbb{R}^N$ with $x_i \ge 0$ for $i=1, \dots, N$, since $r/p \ge 1$.
\end{proof}
\begin{rem}[Tracing the geometric dependence of the constants]\label{rem:stime mu HS in item ii}
{\rm
In this remark, we explain how to trace the dependence on a few geometrical parameters of the constants in the relevant inequalities.
(i) The proof of \cite{HS} has the benefit of giving an explicit upper bound for the constant $c$ appearing in \eqref{eq:risultatodiHSconr-media}, from which, by following the steps of our proof, we can deduce explicit estimates for $\mu_{r,p,\alpha}(\Omega,z)^{-1}$ and $\overline{\mu}_{r, p, \alpha} (\Omega)^{-1}$.
In fact, we easily show that
\begin{equation*}
\overline{\mu}_{r, p, \alpha} (\Omega)^{-1} \le k_{N,\, r, \, p,\, \alpha} \, b_0^N |\Omega|^{\frac{1-\alpha}{N} +\frac{1}{r} +\frac{1}{p} } ,
\end{equation*}
\begin{equation*}
\mu_{r,p,\alpha}(\Omega,z)^{-1} \le k_{N,\, r, \, p, \,\alpha} \, \left(\frac{b_0}{\de_\Gamma (z)^{\frac{1}{r} }}\right)^N |\Omega|^{\frac{1-\alpha}{N} +\frac{2}{r} +\frac{1}{p} } .
\end{equation*}
A better estimate for $\mu_{r,p,\alpha}(\Omega,z)$ can be obtained for $L_0$-John domains with base point $z$. Since the computations are tedious and technical we present them in Appendix \ref{appendix:estimates-mu-for-L0John} and here we just report the final estimate, that is,
\begin{equation}\label{eq:estimatemu-r-p-al-generalizingFeappendix}
\mu_{r,p,\alpha}(\Omega,z)^{-1} \le k_{N,r,p,\alpha} \, L_0^N |\Omega|^{\frac{1-\alpha}{N} +\frac{1}{r} +\frac{1}{p} } .
\end{equation}
(ii) In the sequel we will also need to trace the dependence on relevant geometrical quantities of the constants $\overline{\mu}_{p, 0} (\Omega)$ and $\mu_{p,0}(\Omega,z)$ appearing in \eqref{harmonic-poincare} and \eqref{harmonic-quasi-poincare} in the case $\alpha=0$.
In \cite[Theorem 8.5]{HS1},
where the author proves \eqref{harmonic-poincare} in the case $\alpha=0$ for $b_0$-John domains,
an explicit upper bound for $\overline{\mu}_{p,0} (\Omega)^{-1}$, in terms of $b_0$ and $d_{\Omega}$ only, can be found. We warn the reader that the definition of John domain used there is different from the definitions that we gave in this thesis, but it is equivalent in view of \cite[Theorem 8.5]{MarS}. Explicitly, by putting together \cite[Theorem 8.5]{HS1} and \cite[Theorem 8.5]{MarS} one finds that
\begin{equation*}
\overline{\mu}_{p,0} (\Omega)^{-1} \le k_{N, \, p} \, b_0^{3N(1 + \frac{N}{p})} \, d_\Omega.
\end{equation*}
Reasoning as in the proof of \eqref{John-harmonic-quasi-poincare}, from this estimate one can also deduce a bound for $\mu_{p,0}(\Omega,z)$. In fact, by applying Lemma \ref{lem:mediaor-mediauguale} with $A= B_{\de_\Gamma (z)} (z)$ and $\lambda = v_\Omega$ and recalling the mean value property of $v$, from \eqref{harmonic-poincare} and the bound for $\overline{\mu}_{p,0} (\Omega)$, we easily compute that
\begin{equation*}
\mu_{p,0} (\Omega, z)^{-1} \le k_{N, \, p} \, \left( \frac{|\Omega|}{\de_\Gamma (z)^N} \right)^\frac{1}{r} \,b_0^{3N(1 + \frac{N}{p})} \, d_\Omega.
\end{equation*}
A better estimate for $\mu_{p,0}(\Omega,z)$ can be obtained for $L_0$-John domains with base point $z$, that is,
\begin{equation}\label{eq:estimatemu-p-0-generalizingFeappendix}
\mu_{p, 0}(\Omega,z)^{-1} \le k_{N, \, p} \, L_0^{3N(1 + \frac{N}{p})} \, d_\Omega .
\end{equation}
Complete computations to obtain \eqref{eq:estimatemu-p-0-generalizingFeappendix} can be found in Appendix \ref{appendix:estimates-mu-for-L0John}.
(iii) A domain of class $C^{2}$ is obviously a $b_0$-John domain and a $L_0$-John domain with base point $z$ for every $z \in \Omega$. In fact, by the definitions, it is not difficult to prove the following bounds
\begin{equation*}
b_0 \le \frac{d_\Omega}{r_i} ,
\end{equation*}
\begin{equation*}
L_0 \le \frac{d_\Omega}{\min[r_i, \de_\Gamma (z)] } .
\end{equation*}
Thus, for $C^2$-domains items (i) and (ii) inform us that the following estimates hold
\begin{equation*}
\overline{\mu}_{r, p, \alpha} (\Omega)^{-1} \le k_{N,\, r, \, p,\, \alpha} \, \left( \frac{d_\Omega}{r_i} \right)^N |\Omega|^{\frac{1-\alpha}{N} +\frac{1}{r} +\frac{1}{p} } ,
\end{equation*}
\begin{equation*}
\mu_{r,p,\alpha}(\Omega,z)^{-1} \le k_{N,r,p,\alpha} \, \left( \frac{d_\Omega}{\min[r_i, \de_\Gamma (z)] } \right)^N |\Omega|^{\frac{1-\alpha}{N} +\frac{1}{r} +\frac{1}{p} } ,
\end{equation*}
\begin{equation*}
\overline{\mu}_{p,0} (\Omega)^{-1} \le k_{N, \, p} \, \frac{d_\Omega^{3N(1 + \frac{N}{p}) + 1 } }{r_i^{3N(1 + \frac{N}{p})} } ,
\end{equation*}
\begin{equation*}
\mu_{p, 0}(\Omega,z)^{-1} \le k_{N, \, p} \, \frac{d_\Omega^{3N(1 + \frac{N}{p}) + 1 } }{\min[r_i, \de_\Gamma (z)]^{3N(1 + \frac{N}{p})} } .
\end{equation*}
}
\end{rem}
\medskip
To conclude this section, as a consequence of Lemma \ref{lem:identityfortraceineq}, we present a weighted trace inequality.
We mention that the following proof modifies an idea of W. Feldman \cite{Fe} for our purposes.
\begin{lem}[A trace inequality for harmonic functions]
\label{lem:genericv-trace inequality}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^2$ and let $v$ be a harmonic function in $\Omega$.
(i) If $z$ is a critical point of $v$ in $\Omega$, then it holds that
\begin{equation*}
\int_{\Gamma} |\nabla v|^2 dS_x \le \frac{2}{r_i} \left(1+\frac{N}{r_i\, \mu_{2,2, \frac{1}{2} }(\Omega,z)^2 } \right) \int_{\Omega} (-u) |\nabla^2 v|^2 dx.
\end{equation*}
(ii) If
$$\int_\Omega \nabla v \, dx = 0,$$
then it holds that
\begin{equation*}
\int_{\Gamma} |\nabla v|^2 dS_x \le \frac{2}{r_i} \left(1+\frac{N}{r_i\, \overline{\mu}_{2,2, \frac{1}{2} }(\Omega)^2 } \right) \int_{\Omega} (-u) |\nabla^2 v|^2 dx.
\end{equation*}
\end{lem}
\begin{proof}
Since the term $u_\nu$ at the left-hand side of \eqref{eq:identityfortraceinequality} can be bounded from below by $r_i$, by an adaptation of Hopf's lemma (see Theorem \ref{thm:boundary-gradient}), it holds that
\begin{equation*}
r_i \, \int_{\Gamma} |\nabla v|^2 dS_x \le N \int_{\Omega} |\nabla v|^2 dx + 2 \int_{\Omega} (-u) |\nabla^2 v|^2 dx.
\end{equation*}
Thus, the conclusion follows from this last formula, Corollary \ref{cor:JohnPoincareaigradienti} with $r=p=2$ and $\alpha=1/2$, and \eqref{eq:relationdist}.
\end{proof}
\section{An estimate for the oscillation of harmonic functions}\label{sec:estimates for the oscillation of harmonic functions}
In this section, we single out the key lemma that will produce most of the stability estimates of Chapter \ref{chapter:Stability results} below. It contains an inequality for the oscillation of a harmonic function $v$ in terms of its $L^p$-norm and of a bound for its gradient.
We point out that the following lemma is new and generalizes the estimates proved and used in \cite{MP, MP2} for $p=2$.
To this aim, we define the {\it parallel set} as
$$
\Omega_\sigma=\{ y\in\Omega: \de_\Gamma (y) >\sigma\} \quad \mbox{ for } \quad 0<\sigma \le r_i.
$$
\begin{lem}
\label{lem:Lp-estimate-oscillation-generic-v}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^2$ and let $v$ be a harmonic function in $\Omega$ of class $C^1 (\overline{\Omega})$. Let $G$ be an upper bound for the gradient of $v$ on $\Gamma$.
\par
Then, there exist two constants $a_{N,p}$ and $\alpha_{N,p}$ depending only on $N$ and $p$ such that if
\begin{equation}
\label{smallness-generic-v}
\Vert v - v_{\Omega} \Vert_{p, \Omega} \le \alpha_{N,p} \, r_{i}^{\frac{N+p}{p}} G
\end{equation}
holds, we have that
\begin{equation}
\label{Lp-stability-generic-v}
\max_{\Gamma} v - \min_{\Gamma} v \le a_{N,p} \, G^{ \frac{N}{N+p} } \, \Vert v - v_{\Omega} \Vert_{p, \Omega}^{ p/(N+p) }.
\end{equation}
\end{lem}
\begin{proof}
Since $v$ is harmonic it attains its extrema on the boundary $\Gamma$.
Let $x_i$ and $x_e$ be points in $\Gamma$ that respectively minimize and maximize $v$ on $\Gamma$ and, for
$$
0<\sigma \le r_i,
$$
define the two points in $y_i, y_e\in\partial\Omega_\sigma$ by
$y_j=x_j-\sigma\nu(x_j)$, $j=i, e$.
\par
By the fundamental theorem of calculus we have that
\begin{equation}\label{eq:prova-TFCI-generic v}
v(x_j)= v(y_j) + \int_0^\sigma \langle\nabla v(x_j-t\nu(x_j)),\nu(x_j)\rangle\,dt.
\end{equation}
\par
Since $v$ is harmonic and $y_j\in \overline{ \Omega }_\sigma $, $j=i, e$, we can use the mean value property for the balls with radius $\sigma$ centered at $y_j$ and obtain:
\begin{multline*}
|v(y_j) - v_{\Omega}| \le \frac1{|B|\, \sigma^N}\,\int_{B_\sigma(y_j)}|v - v_{\Omega} |\,dy\le \\
\frac{1}{ \left[ |B|\, \sigma^N \right]^{1/p} } \, \left[\int_{B_\sigma(y_j)}|v - v_{\Omega} |^p\,dy\right]^{1/p}\le
\frac1{ \left[ |B|\, \sigma^N \right]^{1/p} } \, \left[\int_{\Omega}|v- v_{\Omega}|^p\,dy\right]^{1/p}
\end{multline*}
after an application of H\"older's inequality and by the fact that $B_\sigma(y_j) \subseteq \Omega$. This and \eqref{eq:prova-TFCI-generic v} then yield that
\begin{equation*}
\max_{\Gamma} v - \min_{\Gamma} v \le 2 \, \left[ \frac{\Vert v - v_{\Omega} \Vert_{p, \Omega} }{ |B|^{1/p} \, \sigma^{N/p}}+ \sigma \, G \right] ,
\end{equation*}
for every $0<\sigma \le r_i$.
Here we used that
$|\nabla v|$ attains its maximum on $\Gamma$, being $v$ harmonic.
\par
Therefore, by minimizing the right-hand side of the last inequality, we can conveniently choose
$$
\sigma=\left(\frac{N\,\Vert v - v_{\Omega} \Vert_{p, \Omega} }{ p \, |B|^{1/p}\, G }\right)^{p/(N+p)}
$$
and obtain \eqref{Lp-stability-generic-v}, if $\sigma \le r_i$; \eqref{smallness-generic-v} will then follow. The explicit computation immediately shows that
\begin{equation}
\label{eq:costantia_Nal_Nlemmagenericv}
a_{N,p}= \frac{ 2 (N+p) }{N^{\frac{N}{N+2}} p^{\frac{p}{N+p}} } \,|B|^{\frac{1}{N+p}}
\quad \mbox{and} \quad \alpha_{N,p}= \frac{ p }{N} \, |B|^{\frac{1}{p} } .
\end{equation}
Notice that, the fact that \eqref{Lp-stability-generic-v} holds if \eqref{smallness-generic-v} is verified, remains true even if we replace in \eqref{smallness-generic-v} and \eqref{Lp-stability-generic-v} $v_\Omega$ by any $\lambda \in \mathbb{R}$.
\end{proof}
In the following corollary, we present a geometric sufficient condition on the domain that makes \eqref{smallness-generic-v} verified.
To this aim, for $x,$ $y \in \Omega$ we denote by $d_\Omega (x, y)$ the {\it intrinsic distance of $x$ to $y$ in $\Omega$} induced by the euclidean metric,
that is
\begin{multline*}
d_\Omega (x, y) =
\\
\inf \left\lbrace \int_0^1 |\ga' (t)| \, dt : \, \ga: \left[ 0, 1 \right] \to \Omega \text{ piecewise $C^1$, $\ga(0)=x$, $\ga(1)=y$} \right\rbrace .
\end{multline*}
\begin{cor}
Under the assumptions of Lemma \ref{lem:Lp-estimate-oscillation-generic-v}, \eqref{smallness-generic-v} holds true -- and hence also \eqref{Lp-stability-generic-v} does -- if the following inequality is verified
\begin{equation}\label{eq:nuovacondizionegeometricainlemmaosc}
\int_\Omega d_\Omega (x_M,x)^p \, dx \le \alpha_{N,p}^p \, r_{i}^{N+p},
\end{equation}
for a point $x_M$ such that $v(x_M)= v_{\Omega}$.
\end{cor}
\begin{proof}
The fact that \eqref{smallness-generic-v} holds true if \eqref{eq:nuovacondizionegeometricainlemmaosc} is verified follows from the inequality
\begin{equation}\label{eq:disuguaglianzanuovacondistanzaintrinseca}
\Vert v - v_{\Omega} \Vert_{p, \Omega} \le G \, \Vert d_\Omega (x_M,x) \Vert_{p,\Omega},
\end{equation}
that can be proved by using the fundamental theorem of calculus.
In fact, if $\ga: \left[ 0, 1 \right] \to \Omega$ is any piecewise $C^1$ curve from $x_M$ to $x$, the fundamental theorem of calculus informs us that
$$
v(x)- v(x_M)= \int_0^1 < \nabla v( \ga (t)), \ga'(t)> \, dt .
$$
Thus, we deduce that
$$
|v(x)- v(x_M)| \le G \, \int_0^1 | \ga'(t) | \, dt ,
$$
and \eqref{eq:disuguaglianzanuovacondistanzaintrinseca} easily follows from the definition
of $d_\Omega(x,y)$, since $v(x_M) = v_\Omega$.
\end{proof}
\section{Estimates for \texorpdfstring{$h=q-u$}{h=q-u}}\label{sec:estimates for h}
We now turn back our attention to the harmonic function $h = q-u$.
The following lemma is immediate.
\begin{lem}
Let $u$ be the solution of \eqref{serrin1} and set $h=q-u$, where $q$ is defined in \eqref{quadratic}.
Then, $h$ is harmonic in $\Omega$ and it holds that:
\begin{equation}\label{L2-norm-hessian}
|\nabla^2 h|^2 = | \nabla ^2 u|^2 - \frac{(\Delta u)^2}{N}.
\end{equation}
Moreover, if the center $z$ of the polynomial $q$
is chosen in $\Omega$, then the oscillation of $h$ on $\Gamma$ can be bounded from below as follows:
\begin{equation}
\label{oscillationvolume}
\max_{\Gamma} h-\min_{\Gamma} h \ge \frac12\,(|\Omega|/|B|)^{1/N}(\rho_e-\rho_i) ,
\end{equation}
or also:
\begin{equation}
\label{oscillation}
\max_{\Gamma} h-\min_{\Gamma} h \ge \frac{r_i}{2}(\rho_e-\rho_i) ,
\end{equation}
if $\Gamma$ is of class $C^2$.
\end{lem}
\begin{proof}
Since by a direct computation it is immediate to check that $\Delta q = N$, the harmonicity of $h$ follows.
Simple and direct computations also give \eqref{L2-norm-hessian}.
Notice that $h=q$ on $\Gamma$. Thus, by choosing $z$ in $\Omega$, from \eqref{def-rhos} we get
$$\max_{\Gamma} h-\min_{\Gamma} h=\frac12\,(\rho_e^2-\rho_i^2) .$$
Hence, \eqref{oscillationvolume} follows from the inequality $\rho_e+\rho_i\ge\rho_e\ge (|\Omega|/|B|)^{1/N}$, that holds true since $B_{\rho_e}\supseteq\Omega$.
If $\Gamma$ is of class $C^2$, \eqref{oscillation} follows by noting that
$\rho_e+\rho_i \ge \rho_e \ge r_i$, or, if we want, also from \eqref{oscillationvolume} and the trivial inequality $|\Omega| \ge |B| \, r_i^N$.
\end{proof}
By exploiting the additional information that we have about $h$, we now modify Lemma \ref{lem:Lp-estimate-oscillation-generic-v} to
directly link $\rho_e - \rho_i$ to the $L^p$-norm of $h$.
We do it in the following lemma which generalizes to the case of any $L^p$-norm \cite[Lemma 3.3]{MP}, that holds for $p=2$.
\begin{lem}
\label{lem:L2-estimate-oscillation}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary of class $C^2$.
Set $h=q-u$, where $u$ is the solution of \eqref{serrin1} and $q$ is any quadratic polynomial as in \eqref{quadratic} with $z\in\Omega$.
\par
Then, there exists a positive constant $C$ such that
\begin{equation}
\label{L2-stability}
\rho_e-\rho_i\le C \, \Vert h - h_{\Omega} \Vert_{p, \Omega}^{ p/(N+p) }.
\end{equation}
The constant $C$ depends on $N$, $p$, $d_\Omega$, $r_i$, $r_e$. If $\Omega$ is convex the dependence on $r_e$ can be removed.
\end{lem}
\begin{proof}
By direct computations it is easy to check that
\begin{equation*}
| \nabla h | \le M + d_\Omega \quad \mbox{on } \overline{\Omega},
\end{equation*}
where $M$ is the maximum of $|\nabla u|$ on $\overline{\Omega}$, as defined in \eqref{bound-gradient}.
Thus, we can apply Lemma \ref{lem:Lp-estimate-oscillation-generic-v} with $v=h$ and $G= M + d_\Omega$.
By means of \eqref{oscillation} we deduce that
\eqref{L2-stability} holds with
\begin{equation}\label{eq:constantCmaxnuovolemmaoscillation}
C= 2 \, a_{N,p} \, \frac{ (M + d_\Omega )^{ \frac{N}{N+p} } }{ r_i } ,
\end{equation}
if
\begin{equation*}
\Vert h - h_{\Omega} \Vert_{p, \Omega} \le \alpha_{N,p} \, (M + d_\Omega) \, r_{i}^{\frac{N+p}{p}} .
\end{equation*}
Here,
$a_{N,p}$ and $\alpha_{N,p}$ are the constants defined in \eqref{eq:costantia_Nal_Nlemmagenericv}.
On the other hand, if
\begin{equation*}
\Vert h - h_{\Omega} \Vert_{p, \Omega} > \alpha_{N,p} \, (M + d_\Omega) \, r_{i}^{\frac{N+p}{p}} ,
\end{equation*}
it is trivial to check that \eqref{L2-stability} is verified with
$$C= \frac{d_\Omega}{ \left[ \alpha_{N,p} \, (M + d_\Omega) \right]^{\frac{p}{N+p}} \, r_{i} }.$$
Thus, \eqref{L2-stability} always holds true if we choose
the maximum between this constant and that in \eqref{eq:constantCmaxnuovolemmaoscillation}. We then can easily see that the following constant will do:
\begin{equation*}
C= \max{ \left\lbrace 2 \, a_{N,p} , \alpha_{N,p}^{- \frac{p}{N+p}} \right\rbrace } \, \frac{d_\Omega^\frac{N}{N+p} }{r_i} \, \left( 1 + \frac{M}{d_\Omega} \right)^{ \frac{N}{N+p} } .
\end{equation*}
Now, by means of \eqref{gradient-estimate}, we obtain the constant
\begin{equation*}
C= \max{ \left\lbrace 2 \, a_{N,p} , \alpha_{N,p}^{- \frac{p}{N+p}} \right\rbrace } \, \frac{d_\Omega^\frac{N}{N+p} }{r_i} \, \left( 1 + c_N \, \frac{d_\Omega + r_e}{r_e} \right)^{ \frac{N}{N+p} }.
\end{equation*}
If $\Omega$ is convex, the dependence on $r_e$ can be avoided and we can choose
\begin{equation*}
C= \left( 1 + c_N \right)^{ \frac{N}{N+p} } \, \max{ \left\lbrace 2 \, a_{N,p} , \alpha_{N,p}^{- \frac{p}{N+p}} \right\rbrace } \, \frac{d_\Omega^\frac{N}{N+p} }{r_i},
\end{equation*}
in light of \eqref{bound-M-convex}.
\end{proof}
For Serrin's overdetermined problem, Theorem \ref{thm:serrin-W22-stability} below will be crucial.
There, we associate the oscillation of $h$, and hence $\rho_e - \rho_i$, with the weighted $L^2$-norm of its Hessian matrix.
To this aim, we now choose the center $z$ of the quadratic polynomial $q$ in \eqref{quadratic} to be any critical point of $u$ in $\Omega$. Notice that the (global) minimum point of $u$ is always attained in $\Omega$.
With this choice we have that $\nabla h (z) =0$.
We emphasize that the result that we present here improves (for every $N \ge 2$) the exponents of estimates obtained in \cite{MP2}.
\begin{thm}
\label{thm:serrin-W22-stability}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^{2}$
and $z$ be any critical point in $\Omega$ of the solution $u$ of \eqref{serrin1}.
Consider the function $h=q-u$, with $q$ given by \eqref{quadratic}.
There exists a positive constant $C$ such that
\begin{equation}\label{eq:C-provastab-serrin-W22}
\rho_e-\rho_i\le C\, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega}^{\tau_N} ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_2 = 1$;
\item $\tau_3$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{eq:C-provastab-serrin-W22} holds with $\tau_3 = 1- \theta$;
\item $\tau_N = 2/(N-1)$ for $N \ge 4$.
\end{enumerate}
The constant $C$
depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$,
and $\theta$ (only in the case $N=3$).
\end{thm}
\begin{proof}
For the sake of clarity, we will always use the letter $c$ to denote the constants in all the inequalities appearing in the proof.
Their explicit computation will be clear by following the steps of the proof.
(i)
Let $N=2$.
By the Sobolev immersion theorem (see for instance \cite[Theorem 3.12]{Gi} or \cite[Chapter 5]{Ad}), we have that there is a constant $c$ such that, for any $v\in W^{1,4 }(\Omega)$, we have that
\begin{equation}
\label{eq:immersionSerrinN2}
\frac{|v(x)-v(y)|}{|x-y|^{\frac{1}{2} } }\le c\,\Vert v\Vert_{W^{1,4}(\Omega)} \ \mbox{ for any $x, y\in\overline{\Omega}$ with $x\not=y$}.
\end{equation}
Applying \eqref{harmonic-poincare} with $v=h$, $p=4$, and $\alpha=0$ leads to
$$
\Vert h - h_{\Omega} \Vert_{W^{1,4}(\Omega)}\le c \, \Vert \nabla h\Vert_{4,\Omega}.
$$
Since $\nabla h(z)=0$, we can apply item (i) of Corollary \ref{cor:JohnPoincareaigradienti} with $r=4$, $p=2$, and $\alpha=1/2$ to $h$ and obtain that
$$
\Vert \nabla h \Vert_{4,\Omega} \le c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega} .
$$
Thus, we have that
$$
\Vert h - h_{\Omega} \Vert_{W^{1,4}(\Omega)}\le c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega} .
$$
By using the last inequality together with \eqref{eq:immersionSerrinN2}, by choosing $v= h- h_{\Omega}$ and noting that $|x-y| \le d_\Omega$ for any $x, y\in\overline{\Omega}$, we have that
\begin{equation*}
\max_\Gamma h-\min_\Gamma h \le
c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega}.
\end{equation*}
Thus, by recalling \eqref{oscillation} we get that \eqref{eq:C-provastab-serrin-W22} holds with $\tau_2=1$.
(ii) Let $N=3$.
By applying to the function $h$ \eqref{harmonic-poincare} with $r= \frac{3(1- \theta)}{\theta}$,
$p=3(1 -\theta )$, $\alpha=0$, and item (i) of Corollary \ref{cor:JohnPoincareaigradienti} with
$r=3 ( 1 - \theta )$, $p=2$, $\alpha=1/2$, we get
$$
\Vert h - h_{\Omega} \Vert_{\frac{3( 1 - \theta)}{\theta}, \Omega} \le c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega}.
$$
Thus, by recalling Lemma \ref{lem:L2-estimate-oscillation} we have that \eqref{eq:C-provastab-serrin-W22} holds true with $\tau_3 = 1- \theta $.
(iii) Let $N \ge 4$. Since $\nabla h(z)=0$, we can apply to $h$ item (i) of Corollary \ref{cor:JohnPoincareaigradienti} with $r=\frac{2N}{N-1}$, $p=2$, $\alpha=1/2$, and obtain that
\begin{equation*}
\Vert \nabla h \Vert_{\frac{2N}{N-1},\Omega} \le c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega} .
\end{equation*}
Being $N \ge 4$, we can apply \eqref{John-harmonic-poincare} with $v=h$, $r=\frac{2N}{N-3}$, $p=\frac{2N}{N-1}$, $\alpha=0$, and get
\begin{equation*}
\Vert h- h_{ \Omega} \Vert_{\frac{2N}{N-3} } \le c \, \Vert \nabla h \Vert_{\frac{2N}{N-1},\Omega}.
\end{equation*}
Thus,
$$
\Vert h- h_{ \Omega} \Vert_{\frac{2N}{N-3} } \le c \, \Vert \de_\Gamma^{\frac{1}{2} } \, \nabla^2 h \Vert_{2,\Omega},
$$
and by Lemma \ref{lem:L2-estimate-oscillation} we get that \eqref{eq:C-provastab-serrin-W22} holds with $\tau_N = 2/(N-1)$.
\end{proof}
\begin{rem}[On the constant $C$]\label{rem:dipendenzecostanti}
{\rm
The constant $C$ can be shown to depend only on the parameters mentioned in the statement of Theorem \ref{thm:serrin-W22-stability}.
In fact, the parameters $\mu_{p, 0} (\Omega,z)$, $\overline{\mu}_{p, 0} (\Omega)$, $\mu_{r,p, \alpha} (\Omega,z)$, $\overline{\mu}_{r,p,\alpha} (\Omega)$, can be estimated by using item (iii) of Remark \ref{rem:stime mu HS in item ii}.
To remove the dependence on the volume, then one can use the trivial bound
$$
|\Omega|^{1/N} \le |B|^{1/N} d_\Omega /2 .
$$
We recall that if $\Omega$ has the strong local Lipschitz property (for the definition see \cite[Section 4.5]{Ad}), the immersion constant (that we used in the proof of item (i) of Theorem \ref{thm:serrin-W22-stability}) depends only on $N$ and the two Lipschitz parameters of the definition (see \cite[Chapter 5]{Ad}).
In our case $\Omega$ is of class $C^2$, hence obviously it has the strong local Lipschitz property and the two Lipschitz parameters can be easily estimated in terms of $\min \lbrace r_i,r_e \rbrace $.
}
\end{rem}
In the case of Alexandrov's Soap Bubble Theorem, we have to deal with \eqref{H-fundamental} or \eqref{identity-SBT2}, where just the unweighted $L^2$-norm of the Hessian of $h$ appears.
Thus, the appropriate result in this case is Theorem \ref{thm:SBT-W22-stability} below, in which we
improve (for every $N\ge 4$) the exponents of estimates obtained in \cite{MP}.
\begin{thm}
\label{thm:SBT-W22-stability}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain with boundary $\Gamma$ of class $C^{2}$
and $z$ be any critical point in $\Omega$ of the solution $u$ of \eqref{serrin1}.
Consider the function $h=q-u$, with $q$ given by \eqref{quadratic}.
There exists a positive constant $C$ such that
\begin{equation}
\label{W22-stability}
\rho_e-\rho_i\le C\, \Vert \nabla^2 h\Vert_{2,\Omega}^{\tau_N} ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_N=1$ for $N=2$ or $3$;
\item $\tau_4$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{W22-stability} holds with $\tau_4= 1- \theta $;
\item $\tau_N = 2/(N-2) $ for $N\ge 5$.
\end{enumerate}
The constant $C$ depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$, and $\theta$ (only in the case $N=4$).
\end{thm}
\begin{proof}
As done in the proof of Theorem \ref{thm:serrin-W22-stability}, for the sake of clarity, we will always use the letter $c$ to denote the constants in all the inequalities appearing in the proof.
Their explicit computation will be clear by following the steps of the proof. By reasoning as described in Remark \ref{rem:dipendenzecostanti}, one can easily check that those constants depend only on the geometric parameters of $\Omega$ mentioned in the statement of the theorem.
(i) Let $N=2$ or $3$.
By the Sobolev immersion theorem (see for instance \cite[Theorem 3.12]{Gi} or \cite[Chapter 5]{Ad}), we have that there is a constant $c$ such that, for any $v\in W^{2,2}(\Omega)$, we have that
\begin{equation}
\label{eq:immersionSBTN23}
\frac{|v(x)-v(y)|}{|x-y|^\ga}\le c\, \Vert v\Vert_{W^{2,2}(\Omega)} \ \mbox{ for any $x, y\in\overline{\Omega}$ with $x\not=y$},
\end{equation}
where $\ga$ is any number in $(0,1)$ for $N=2$ and $\ga=1/2$ for $N=3$.
Since $\nabla h(z)=0$, we can apply item (i) of Corollary \ref{cor:Poincareaigradienti} with $p=2$ and $\alpha= 0$ to $h$ and obtain that
\begin{equation*}
\int_{\Omega} |\nabla h|^2 \, dx \le c \, \int_{\Omega} | \nabla^2 h|^2 \, dx.
\end{equation*}
Using this last inequality together with \eqref{harmonic-poincare} with $v= h$, $p=2$, $\alpha=0$, leads to
$$
\Vert h - h_{\Omega} \Vert_{W^{2,2}(\Omega)}\le c \,\Vert \nabla^2 h\Vert_{2,\Omega}.
$$
Hence, by using \eqref{eq:immersionSBTN23} with $v= h-h_{ \Omega}$ and noting that $|x-y| \le d_\Omega$ for any $x, y\in\overline{\Omega}$, we get that
\begin{equation*}
\max_\Gamma h-\min_\Gamma h \le c \, \Vert \nabla^2 h\Vert_{2,\Omega}.
\end{equation*}
Thus, by recalling \eqref{oscillation} we get that \eqref{W22-stability} holds with $\tau_N=1$.
(ii) Let $N=4$.
By applying \eqref{harmonic-poincare} with $r= \frac{4( 1 - \theta)}{\theta}$, $p= 4 ( 1 -\theta )$, $\alpha=0$, and item (i) of Corollary \ref{cor:JohnPoincareaigradienti} with $r=4 ( 1 - \theta )$, $p=2$, $\alpha= 0$, for $v=h$ we get
$$
\Vert h - h_{ \Omega} \Vert_{\frac{4( 1 - \theta)}{\theta}, \Omega} \le c \, \Vert \nabla^2 h \Vert_{2,\Omega}.
$$
Thus, by Lemma \ref{lem:L2-estimate-oscillation} we conclude that \eqref{W22-stability} holds with $\tau_4= 1- \theta$.
(iii) Let $N\ge 5$. Applying \eqref{John-harmonic-poincare} with $r= \frac{2N}{N-4}$, $p= \frac{2N}{N-2}$, $\alpha=0$, and item (i) of Corollary \ref{cor:JohnPoincareaigradienti} with $r={\frac{2N}{N-2}}$, $p=2$, $\alpha=0$, by choosing $v=h$ we get that
$$
\Vert h - h_{ \Omega} \Vert_{\frac{2N}{N-4},\Omega} \le c \, \Vert\nabla^2 h\Vert_{2,\Omega}.
$$
By Lemma \ref{lem:L2-estimate-oscillation}, we have that \eqref{W22-stability} holds
true with $\tau_N = 2/(N-2)$.
\end{proof}
\begin{rem}\label{rem:removing-distance-convex}
{\rm
The parameter $\de_\Gamma (z)$ is used only to give an estimate of the parameters $\mu_{p, 0} (\Omega,z)$ and $\mu_{r,p,\alpha} (\Omega,z)$ in terms of an explicit geometrical quantity.
Moreover, in case $\Omega$ is convex, by using $\de_\Gamma (z)$ we
are able to completely remove the dependence of the constants on $z$.
Indeed, in this case $u$ has a unique minimum point $z$ in $\Omega$, since $u$ is analytic and the level sets of $u$ are convex by a result in \cite{Ko} (see \cite{MS}, for a similar argument), and hence we have only one choice for the point $z$.
Thus, an estimate of $\de_\Gamma(z)$ from below can be obtained, first, by putting together arguments in \cite[Lemma 2.6]{BMS} and \cite[Remark 2.5]{BMS}, to obtain that
$$
\de_\Gamma(z)\ge \frac{k_N}{|\Omega|\, d_\Omega^{N-1}}\,\left[ \max_{\overline{\Omega}}(-u) \right]^N.
$$
Secondly, if we set $x$ to be a point in $\Omega$ such that $\de_\Gamma (x)= r_i$, by recalling Lemma \ref{lem:relationdist} we easily find that
\begin{equation*}
\max_{\overline{\Omega}}(-u) \ge - u(x) \ge \frac{r_i^2}{2}.
\end{equation*}
All in all, we have that
$$
\de_\Gamma(z)\ge k_N\,\frac{r_i^{2N}}{|\Omega|\, d_\Omega^{N-1}}.
$$
Thus, if $\Omega$ is convex, we can affirm that the constants $C$, appearing in Theorems \ref{thm:serrin-W22-stability} and \ref{thm:SBT-W22-stability}, depend on $N$, $r_i$, and $d_\Omega$, only. In fact, as already noticed in item (i) of Remark \ref{rem:boundgradienttorsionconvexr_eremoving}, in this case also the parameter $r_e$ can be removed, being $r_e=+\infty$.
}
\end{rem}
Even if $\Omega$ is not convex, the dependence on $\de_\Gamma (z)$ can still be removed in some other cases. In fact, we can do it by choosing the point $z$ appearing in \eqref{quadratic} differently, as explained in the following remark.
\par
\begin{rem}
\label{remarkprova:nuova scelte z}
{\rm
(i) We can choose $z$ as the center of mass of $\Omega$.
In fact, if $z$ is the center of mass of $\Omega$, we have that
\begin{multline*}
\int_\Omega \nabla h(x)\,dx=\int_\Omega [x-z-\nabla u(x)]\,dx=\\
\int_\Omega x\,dx-|\Omega|\,z-\int_\Gamma u(x)\,\nu(x)\,dS_x=0.
\end{multline*}
Thus, we can use item (ii) of Corollaries \ref{cor:Poincareaigradienti}, \ref{cor:JohnPoincareaigradienti} instead of item (i). In this way, in the estimates of Theorems \ref{thm:serrin-W22-stability} and \ref{thm:SBT-W22-stability} we simply obtain the same constants with $\mu_{p, 0}(\Omega,z)$ and $\mu_{r,p, \alpha}(\Omega,z)$ replaced by $\overline{\mu}_{p, 0}(\Omega)$ and $\overline{\mu}_{r, p, \alpha}(\Omega)$. Thus, we removed the presence of $\de_\Gamma (z)$ and hence the dependence on $z$.
It should be noticed that, in this case the extra assumption that $z \in \Omega$ is needed, since we want that the ball $B_{\rho_i}(z)$ be contained in $\Omega$.
\par
(ii) As done in \cite{Fe}, another possible way to choose $z$ is $z= x_0 - \nabla u (x_0)$,
where $x_0 \in \Omega$ is any point such that $\de_\Gamma (x_0) \ge r_i$. In fact, we obtain that $\nabla h (x_0)=0$ and we can thus use \eqref{harmonic-quasi-poincare} and \eqref{John-harmonic-quasi-poincare}, with $\mu_{p, 0}(\Omega,z)$ and $\mu_{r,p,\alpha}(\Omega,z)$ replaced by $\mu_{p, 0}(\Omega,x_0)$ and $\mu_{r,p,\alpha}(\Omega,x_0)$.
Thus, by recalling item (iii) of Remark \ref{rem:stime mu HS in item ii} it is clear that we removed the dependence on $\de_\Gamma (x_0)$ in our constants that, as already noticed, comes from the estimation of the parameters $\mu_{p, 0} (\Omega, x_0)$ and $\mu_{r,p,\alpha} (\Omega, x_0)$.
\par
As in item (i), we should additionally require that $z \in \Omega$, to be sure that the ball $B_{\rho_i}(z)$ be contained in $\Omega$.
}
\end{rem}
\chapter{Stability results}\label{chapter:Stability results}
In this chapter, we collect our results on the stability of the spherical configuration by putting together the identities derived in Chapter \ref{chapter:integral identities and symmetry results} and the estimates obtained in Chapter \ref{chapter:various estimates}.
\section{Stability for Serrin's overdetermined problem}\label{sec:stability Serrin}
In light of \eqref{L2-norm-hessian}, \eqref{idwps} can be rewritten in terms of the harmonic function $h$, as stated in the following.
\begin{lem}
Under the same assumptions of Theorem \ref{thm:serrinidentity}, if we set $h=q-u$, then
it holds that
\begin{equation}\label{idwps-h}
\int_{\Omega} (-u)\, |\nabla ^2 h|^2\,dx=
\frac{1}{2}\,\int_\Gamma ( R^2-u_\nu^2)\, h_\nu\,dS_x.
\end{equation}
\end{lem}
\begin{proof}
Simple computations give that $h_\nu=q_\nu-u_\nu$.
By using this identity and \eqref{L2-norm-hessian}, \eqref{idwps-h} easily follows from \eqref{idwps}.
\end{proof}
\par
In light of \eqref{eq:relationdist}, Theorem \ref{thm:serrin-W22-stability} gives an estimate from below of the left-hand side of \eqref{idwps-h}. Now, we will take care of its right-hand side and prove our main result for Serrin's problem. The result that we present here, improves (for every $N \ge 2$) the exponents in the estimate obtained in \cite[Theorem 1.1]{MP2}.
\begin{thm}[Stability for Serrin's problem]
\label{thm:Improved-Serrin-stability}
Let $\Omega\subset\mathbb{R}^N$, $N\ge2$, be a bounded domain with boundary $\Gamma$ of class $C^2$ and $R$ be the constant defined in \eqref{def-R-H0}.
Let $u$ be the solution of problem \eqref{serrin1} and
$z\in\Omega$ be any of its critical points.
\par
There exists a positive constant $C$ such that
\begin{equation}
\label{general improved stability serrin C}
\rho_e-\rho_i\le C\,\Vert u_\nu - R \Vert_{2,\Gamma}^{\tau_N} ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_2 = 1$;
\item $\tau_3$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{general improved stability serrin C} holds with $\tau_3 = 1- \theta$;
\item $\tau_N = 2/(N-1)$ for $N \ge 4$.
\end{enumerate}
The constant $C$ depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$,
and $\theta$ (only in the case $N=3$).
\end{thm}
\begin{proof}
We have that
$$
\int_\Gamma ( R^2-u_\nu^2)\, h_\nu\,dS_x\le (M+R)\,\Vert u_\nu-R\Vert_{2,\Gamma} \Vert h_\nu\Vert_{2,\Gamma},
$$
after an an application of H\"older's inequality.
Thus, by item (i) of Lemma \ref{lem:genericv-trace inequality} with $v=h$, \eqref{idwps-h}, and this inequality, we infer that
\begin{multline*}
\Vert h_\nu\Vert_{2,\Gamma}^2\le \frac{2}{r_i} \left(1+\frac{N}{r_i\, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2 } \right) \int_{\Omega} (-u) |\nabla^2h|^2 dx \le \\
\frac{M+R}{r_i} \left(1+\frac{N}{r_i\, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2} \right)\Vert u_\nu-R\Vert_{2,\Gamma} \Vert h_\nu\Vert_{2,\Gamma},
\end{multline*}
and hence
\begin{equation}
\label{ineq-feldman}
\Vert h_\nu\Vert_{2,\Gamma}\le \frac{M+R}{r_i} \left(1+\frac{N}{r_i\, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2} \right)\Vert u_\nu-R\Vert_{2,\Gamma}.
\end{equation}
Therefore,
\begin{multline*}
\int_\Omega |\nabla ^2 h|^2 \de_\Gamma (x)\, dx \le \frac{2}{r_i}\int_\Omega (-u) |\nabla^2 h|^2 dx \le \\
\left(\frac{M+R}{r_i}\right)^2 \left(1+\frac{N}{r_i \, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2} \right)\Vert u_\nu-R\Vert_{2,\Gamma}^2,
\end{multline*}
by Lemma \ref{lem:relationdist}.
These inequalities and Theorem \ref{thm:serrin-W22-stability} then give the desired conclusion.
We recall that $\mu_{2,2,\frac{1}{2} }(\Omega,z)$ appearing in the constant in the last inequality can be estimated in terms of $d_\Omega$ and $\min \left[ r_i, \de_\Gamma (z) \right]$, by proceeding as described in Remark \ref{rem:dipendenzecostanti}.
The ratio $R$ can be estimated (from above) in terms of $|\Omega|^{1/N}$ -- just by using the isoperimetric inequality; in turn, $|\Omega|^{1/N}$ can be bounded in terms of $d_\Omega$ by proceeding as described in Remark \ref{rem:dipendenzecostanti}.
Finally, as usual, $M$ can be estimated by means of \eqref{gradient-estimate}.
\end{proof}
If we want to measure the deviation of $u_\nu$ from $R$ in $L^1$-norm, we get a smaller (reduced by one half) stability exponent. The following result improves \cite[Theorem 3.6]{MP2}.
\begin{thm}[Stability with $L^1$-deviation]
\label{thm:Serrin-stability}
Theorem \ref{thm:Improved-Serrin-stability} still holds
with \eqref{general improved stability serrin C} replaced by
\begin{equation*}
\rho_e-\rho_i\le C\,\Vert u_\nu - R \Vert_{1,\Gamma}^{\tau_N/2} .
\end{equation*}
\end{thm}
\begin{proof}
Instead of applying H\"older's inequality to the right-hand side of \eqref{idwps-h}, we just use the rough bound:
\begin{equation*}
\int_{\Omega} (-u)\, |\nabla ^2 h|^2\,dx \le
\frac{1}{2}\, \left( M + R \right)\,(M + d_{\Omega}) \, \int_\Gamma \left| u_\nu - R \right| \, dS_x,
\end{equation*}
since $(u_\nu+R)\,|h_\nu|\le ( M + R)\,(M + d_{\Omega})$ on $\Gamma$. The conclusion then follows from similar arguments.
\end{proof}
\begin{rem}
{\rm
If $\Omega$ is convex, in view of Remark \ref{rem:removing-distance-convex} we can claim that
the constants $C$ of Theorems \ref{thm:Improved-Serrin-stability} and \ref{thm:Serrin-stability} depend only on $N$, $r_i$, $d_\Omega$ (and $\theta$ only in the case $N=3$).
The dependence of $C$ on $\de_\Gamma (z)$ can be removed also in the cases described in Remark \ref{remarkprova:nuova scelte z}.
Regarding the case described in item (i) of that remark, we notice that, since in that situation $z$ is chosen as the center of mass of $\Omega$, in the proof of Theorem \ref{thm:Improved-Serrin-stability} we will use item (ii) of Lemma \ref{lem:genericv-trace inequality} instead of item (i), so that $\mu_{2,2,\frac{1}{2}} (\Omega, z)$ will be replaced by $\overline{\mu}_{2,2,\frac{1}{2}} (\Omega)$.
}
\end{rem}
\bigskip
Since the estimates in Theorems \ref{thm:Improved-Serrin-stability} and \ref{thm:Serrin-stability} do not depend on the particular critical point chosen, as a corollary, we obtain results of closeness to a union of balls:
here, we just illustrate the instance of Theorem \ref{thm:Improved-Serrin-stability}.
\begin{cor}[Closeness to an aggregate of balls]
\label{cor:Serrin-stability-aggregate}
Let $\Gamma$, $R$ and $u$ be as in Theorem \ref{thm:Improved-Serrin-stability}.
Then, there exist points $z_1, \dots, z_n$ in $\Omega$, $n\ge 1$, and corresponding numbers
\begin{equation*}
\rho_i^j=\min_{x\in\Gamma}|x-z_j| \ \mbox{ and } \ \rho_e^j=\min_{x\in\Gamma}|x-z_j|,
\quad j=1, \dots, n,
\end{equation*}
such that
\begin{equation*}
\bigcup_{j=1}^n B_{\rho_i^j}(z_j)\subset\Omega\subset \bigcap_{j=1}^n B_{\rho_e^j}(z_j)
\end{equation*}
and
$$
\max_{1\le j\le n}(\rho_e^j-\rho_i^j)\le C\,\Vert u_\nu - R \Vert_{2,\Gamma}^{\tau_N} .
$$
Here, the exponent $\tau_N$ and the constant $C$ are those of Theorem \ref{thm:Improved-Serrin-stability}.
\par
The number $n$ can be chosen as the number of connected components of the set $\mathcal{M}$ of all the local minimum points of the solution $u$ of \eqref{serrin1}.
\end{cor}
\begin{proof}
We pick one point $z_j$ from
each connected component of the set of local minimum points of $u$. By applying Theorem \ref{thm:Improved-Serrin-stability} to each $z_j$, the conclusion is then evident.
\end{proof}
\begin{rem}\label{rem:serrin-stimequantitativefissandoparametri}
{\rm
The estimates presented in Theorems \ref{thm:Improved-Serrin-stability}, \ref{thm:Serrin-stability} may be interpreted as stability estimates, once that we fixed some a priori bounds on the relevant parameters: here, we just illustrate the case of Theorem \ref{thm:Improved-Serrin-stability}.
Given three positive constants $\overline{d}$, $\underline{r}$, and $\underline{\de}$, let ${\mathcal D}={\mathcal D} (\overline{d}, \underline{r}, \underline{\de})$ be the class of bounded domains $\Omega\subset\mathbb{R}^N$ with boundary of class $C^{2}$, such that
$$
d_{ \Omega} \le \overline{d}, \quad r_i(\Omega), r_e(\Omega)\ge \underline{r}, \quad \de_\Gamma (z) \ge \underline{\de}.
$$
Then, for every $\Omega\in {\mathcal D} $, we have that
$$
\rho_e-\rho_i\le C\, \Vert u_\nu - R \Vert_{2,\Gamma}^{\tau_N},
$$
where $\tau_N$ is that appearing in \eqref{general improved stability serrin C} and $C$ is a constant depending on $N$, $\overline{d}$, $\underline{r}$, $\underline{\de}$ (and $\theta$ only in the case $N=3$).
\par
If we relax the a priori assumption that $\Omega\in {\mathcal D}$ (in particular if we remove the lower bound $\underline{r}$), it may happen that,
as the deviation $\Vert u_\nu - R \Vert_{2,\Gamma}$ tends to $0$, $\Omega$ tends to the ideal configuration of two or more disjoint balls, while $C$ diverges since $\underline{r}$ tends to $0$. The configuration of more balls connected with tiny (but arbitrarily long) tentacles has been quantitatively studied in \cite{BNST}.
}
\end{rem}
\section{Stability for Alexandrov's Soap Bubble Theorem}\label{sec:Alexandrov's SBT}
This section is devoted to the stability issue for the Soap Bubble Theorem.
As already noticed, in light of \eqref{L2-norm-hessian}, \eqref{identity-SBT2} can be rewritten in terms of $h$, as stated in the following.
\begin{lem}
Under the same assumptions of Theorem \ref{thm:identitySBT}, if we set $h=q-u$, then
it holds that
\begin{multline}\label{identity-SBT-h}
\frac1{N-1}\int_{\Omega} |\nabla ^2 h|^2 dx+
\frac1{R}\,\int_\Gamma (u_\nu-R)^2 dS_x = \\
-\int_{\Gamma}(H_0-H)\,h_\nu\,u_\nu\,dS_x+
\int_{\Gamma}(H_0-H)\, (u_\nu-R)\,q_\nu\, dS_x.
\end{multline}
\end{lem}
Next, we derive the following lemma, that parallels and is a useful consequence of Lemma \ref{lem:genericv-trace inequality}.
\begin{lem}
\label{lem:improving-SBT}
Let $\Omega\subset\mathbb{R}^N$, $N\ge2$, be a bounded domain with boundary $\Gamma$ of class $C^2$.
Denote by $H$ the mean curvature of $\Gamma$ and let $H_0$ be the constant defined in \eqref{def-R-H0}.
\par
Then, the following inequality holds:
\begin{equation}
\label{improving-SBT}
\Vert u_\nu-R\Vert_{2,\Gamma} \le
R\left\{ d_\Omega +\frac{M (M+R)}{r_i}\left(1+\frac{N}{r_i \, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2 }\right)\right\} \Vert H_0-H\Vert_{2,\Gamma}.
\end{equation}
\end{lem}
\begin{proof}
Discarding the first summand on the left-hand side of \eqref{identity-SBT-h} and applying H\"older's inequality on its right-hand side gives that
$$
\frac1{R} \Vert u_\nu-R\Vert_{2,\Gamma}^2 \le \Vert H_0-H\Vert_{2,\Gamma}\left( M \Vert h_\nu\Vert_{2,\Gamma} + d_\Omega \,\Vert u_\nu-R\Vert_{2,\Gamma} \right),
$$
since $u_\nu\le M$ and $|q_\nu|\le d_\Omega $ on $\Gamma$. Thus, inequality \eqref{ineq-feldman}
implies that
\begin{multline*}
\Vert u_\nu-R\Vert_{2,\Gamma}^2 \le
\\
R\left\{ d_\Omega +\frac{M (M+R)}{r_i}\left(1+\frac{N}{r_i \, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2}\right)\right\} \Vert H_0-H\Vert_{2,\Gamma} \Vert u_\nu-R\Vert_{2,\Gamma},
\end{multline*}
from which \eqref{improving-SBT} follows at once.
\end{proof}
We are now ready to prove our main result for Alexandrov's Soap Bubble Theorem. The result that we present here, improves (for every $N \ge 4$) the exponents of estimates obtained in \cite[Theorem 1.2]{MP2}.
\begin{thm}[Stability for the Soap Bubble Theorem]
\label{thm:SBT-improved-stability}
Let $N\ge 2$ and let $\Gamma$ be a surface of class $C^{2}$, which is the boundary of a bounded domain $\Omega\subset\mathbb{R}^N$. Denote by $H$ the mean curvature of $\Gamma$ and let $H_0$ be the constant defined in \eqref{def-R-H0}.
\par
Then,
for some point $z\in\Omega$
there exists a positive constant $C$ such that
\begin{equation}
\label{SBT-improved-stability-C}
\rho_e-\rho_i \le C\,\Vert H_0-H\Vert_{2,\Gamma}^{\tau_N} ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_N=1$ for $N=2$ or $3$;
\item $\tau_4$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{SBT-improved-stability-C} holds with $\tau_4= 1- \theta $;
\item $\tau_N = 2/(N-2) $ for $N\ge 5$.
\end{enumerate}
The constant $C$ depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$, and $\theta$ (only in the case $N=4$).
\end{thm}
\begin{proof}
As before, we choose $z\in\Omega$ to be any local minimum point of $u$ in $\Omega$.
Discarding the second summand on the left-hand side of \eqref{identity-SBT-h} and applying H\"older's inequality on its right-hand side, as in the previous proof, gives that
\begin{multline*}
\frac1{N-1}\,\int_\Omega |\nabla^2 h|^2 dx\le \\
R\left\{ d_\Omega +\frac{M (M+R)}{r_i}\left(1+\frac{N}{r_i \, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2}\right)\right\} \Vert H_0-H\Vert_{2,\Gamma} \Vert u_\nu-R\Vert_{2,\Gamma}, \le \\
R^2\left\{ d_\Omega +\frac{M (M+R)}{r_i}\left(1+\frac{N}{r_i \, \mu_{2,2,\frac{1}{2} }(\Omega,z)^2 }\right)\right\}^2 \Vert H_0-H\Vert_{2,\Gamma}^2,
\end{multline*}
where the second inequality follows from Lemma \ref{lem:improving-SBT}.
\par
The conclusion then follows from
Theorem \ref{thm:SBT-W22-stability}.
\end{proof}
\begin{rem}
{\rm
Estimates similar to those of Theorem \ref{thm:SBT-improved-stability} can also be obtained as a direct corollary of Theorem \ref{thm:Improved-Serrin-stability}, by means of \eqref{improving-SBT}. As it is clear, in this way the exponents $\tau_N$ would be worse than those obtained in Theorem \ref{thm:SBT-improved-stability}.
}
\end{rem}
We now present a stability result with a weaker deviation (at the cost of getting a smaller stability exponent), analogous to Theorem \ref{thm:Serrin-stability}.
To this aim, we notice that from \eqref{H-fundamental} and \eqref{L2-norm-hessian} we can easily deduce the following inequality
\begin{equation}
\label{fundamental-stability2}
\frac1{N-1}\int_{\Omega} |\nabla^2 h |^2 \,dx + \int_{\Gamma}(H_0-H)^-\, (u_\nu)^2\,dS_x \le
\int_{\Gamma}(H_0-H)^+\, (u_\nu)^2\,dS_x
\end{equation}
(here, we use the positive and negative part functions $(t)^+=\max(t,0)$ and $(t)^-=\max(-t,0)$). That inequality tells us that, if we have an {\it a priori} bound $M$ for
$u_\nu$ on $\Gamma$, then its left-hand side is small if the integral
\begin{equation*}
\int_{\Gamma}(H_0-H)^+\,dS_x
\end{equation*}
is also small. In particular, if $H$ is not too much smaller than $H_0$, then it cannot be too much larger than $H_0$ and the Cauchy-Schwarz deficit cannot be too large.
It is clear that,
$$
\int_\Gamma (H_0-H)^+\,dS_x,
$$
is a deviation weaker than $\Vert H_0-H\Vert_{1,\Gamma}$.
The result that we present here, improves (for every $N \ge 4$) the estimates obtained in \cite[Theorem 4.1]{MP}.
\begin{thm}[Stability with $L^1$-type deviation]
\label{thm:SBT-stability}
Theorem \ref{thm:SBT-improved-stability} still holds with \eqref{SBT-improved-stability-C} replaced by
\begin{equation*}
\rho_e-\rho_i \le C\, \left\lbrace \int_\Gamma (H_0-H)^+\,dS_x \right\rbrace^{\tau_N /2} .
\end{equation*}
\end{thm}
\begin{proof}
From \eqref{fundamental-stability2}, we infer that
\begin{equation*}
\Vert \nabla^2 h\Vert_{2,\Omega}\le M\,\sqrt{N-1}\, \left\lbrace \int_\Gamma (H_0-H)^+\,dS_x \right\rbrace^{1/2}
\end{equation*}
The conclusion then follows from Theorem \ref{thm:SBT-W22-stability}.
\end{proof}
\begin{rem}\label{rem:ontheconstants-SBT-removingdelta-seconvexocasidiremark}
{\rm
(i) If $\Omega$ is convex, the dependence of the constants $C$ in Theorems \ref{thm:SBT-improved-stability} and \ref{thm:SBT-stability} can be reduced to the parameters $N$, $r_i$, $d_\Omega$ (and $\theta$ only in the case $N=4$), as described in Remark \ref{rem:removing-distance-convex}. The dependence on the parameter $\de_\Gamma (z)$ can be removed also in the cases described in Remark \ref{remarkprova:nuova scelte z}.
(ii) Results of closeness to a union of balls analogous to Corollary \ref{cor:Serrin-stability-aggregate} can be easily derived also for Theorems \ref{thm:SBT-improved-stability} and \ref{thm:SBT-stability}.
}
\end{rem}
\begin{rem}\label{rem:stimequantitativefissandoparametri}
{\rm
The estimates presented in Theorems \ref{thm:SBT-improved-stability}, \ref{thm:SBT-stability} may be interpreted as stability estimates, once some a priori information is available: here, we just illustrate the case of Theorem \ref{thm:SBT-improved-stability}.
Given three positive constants $\overline{d}$, $\underline{r}$, and $\underline{\de}$, let ${\mathcal S}={\mathcal S}(\overline{d}, \underline{r}, \underline{\de})$ be the class of connected surfaces $\Gamma\subset\mathbb{R}^N$ of class $C^{2}$, where $\Gamma$ is the boundary of a bounded domain $\Omega$, such that
$$
d_{ \Omega} \le \overline{d}, \quad r_i(\Omega), r_e(\Omega)\ge \underline{r}, \quad \de_\Gamma (z) \ge \underline{\de}.
$$
Then, for every $\Gamma\in{\mathcal S}$ we have that
$$
\rho_e-\rho_i\le C\, \Vert H_0-H\Vert_{2,\Gamma}^{\tau_N},
$$
where $\tau_N$ is that appearing in \eqref{SBT-improved-stability-C} and the constant $C$ depends on $N$, $\overline{d}$, $\underline{r}$, $\underline{\de}$ (and $\theta$ only in the case $N=4$).
\par
If we relax the a priori assumption that $\Gamma\in{\mathcal S}$ (in particular if we remove the lower bound
$\underline{r}$), it may happen that,
as the deviation $\Vert H_0 - H\Vert_{2,\Gamma}$ tends to $0$, $\Omega$ tends to the ideal configuration of two or more mutually tangent balls, while $C$ diverges since $\underline{r}$ tends to $0$. Such a configuration can be observed, for example, as limit of sets created by truncating (and then smoothly completing) unduloids with very thin necks. This phenomenon (called bubbling) has been quantitatively studied in \cite{CM} by considering strictly mean convex surfaces and by using the uniform deviation $\Vert H_0 - H\Vert_{\infty,\Gamma}$.
}
\end{rem}
\section{Quantitative bounds for an asymmetry}\label{sec:stabbyasymmetrySBT}
Another consequence of Lemma \ref{lem:improving-SBT} is the following inequality that shows an optimal stability exponent for any $N\ge 2$. The number ${\mathcal A}(\Omega)$, defined as
\begin{equation}
\label{asymmetry}
{\mathcal A}(\Omega)=\inf\left\{\frac{|\Omega\Delta B^x|}{|B^x|}: x \mbox{ center of a ball $B^x$ with radius $R$} \right\} ,
\end{equation}
is some sort of asymmetry similar to the so-called Fraenkel asymmetry (see \cite{Fr}). Here, $\Omega\Delta B^x$ denotes the symmetric difference of $\Omega$ and $B^x$, and $R$ is the constant defined in \eqref{def-R-H0}.
\begin{thm}[Stability by asymmetry, \cite{MP2}]
\label{th:asymmetry}
Let $N\ge 2$ and let $\Gamma$ be a surface of class $C^{2}$, which is the boundary of a bounded domain $\Omega\subset\mathbb{R}^N$. Denote by $H$ the mean curvature of $\Gamma$ and let $H_0$ be the constant defined in \eqref{def-R-H0}.
\par
Then it holds that
\begin{equation}
\label{eq:SBTstabinmisura}
{\mathcal A}(\Omega)\le C \, \Vert H_0 - H \Vert_{2,\Gamma},
\end{equation}
for some positive constant $C$.
\end{thm}
\begin{proof}
We use \cite[inequality (2.14)]{Fe}: we have that
\begin{equation*}
\frac{|\Omega \Delta B_R^z|}{|B_R^z|} \le C \, \Vert u_\nu - R \Vert_{2,\Gamma},
\end{equation*}
where $B_R^z$ is a ball of radius $R$ (as defined in \eqref{def-R-H0}) and centered at the point $z$ described in item (ii) of Remark \ref{remarkprova:nuova scelte z}. Hence, we obtain that
$$
{\mathcal A}(\Omega)\le C \, \Vert u_\nu - R \Vert_{2,\Gamma},
$$
by the definition \eqref{asymmetry}.
Thus, thanks to \eqref{improving-SBT}, we obtain \eqref{eq:SBTstabinmisura}.
\par
Here, if we estimate $L_0$ as described in item (iii) of Remark \ref{rem:stime mu HS in item ii}, we can see that $C$ depends on $N, d_\Omega, r_i, |\Gamma|/|\Omega|$.
\end{proof}
\begin{rem}[On the asymmetry ${\mathcal A}(\Omega)$]{\rm
Notice that, for any $x\in \Omega$, we have that
$$
\frac{|\Omega \Delta B_R^x|}{|B_R^x|} \le \frac{|B_{\rho_e}^x \setminus B_{\rho_i}^x|}{|B_R^x|}=
\frac{\rho_e^N-\rho_i^N}{R^N} \le \frac{N \, \rho_e^{N-1}}{R^N} \, (\rho_e - \rho_i),
$$
and $\rho_e\le d_\Omega$.
Thus, if $d_\Omega/R$ remains bounded and $(\rho_e - \rho_i)/R$ tends to $0$, then the ratio $|\Omega \Delta B_R^x|/|B_R^x|$ does it too.
\par
The converse is not true in general. For example, consider a lollipop made by a ball and a stick with fixed length $L$ and vanishing width; as that width vanishes, the ratio $d_\Omega/R$ remains bounded, while $|\Omega \Delta B_R^x|/|B_R^x|$ tends to zero and $\rho_e - \rho_i \ge L >0$.
\bigskip
If we fix $r_i$, $r_e$, and $d_\Omega$, we have the following result.
}
\end{rem}
\begin{thm}[\cite{MP2}]
\label{thm:comparing asymmetry}
Let $\Omega\subset\mathbb{R}^N$, $N\ge 2$, be a bounded domain satisfying the uniform interior and exterior sphere conditions with radii $r_i$ and $r_e$, and set $\underline{r}=\min{ \left[ r_i, r_e \right] }$.
\par
Then, we have that
$$
\rho_e - \rho_i \le \max{ \left[ 2 \, d_\Omega , \frac{d_\Omega^2 }{2 \, \underline{r} } \right] } \, {\mathcal A}(\Omega)^{\frac{1}{N}} .
$$
\end{thm}
\begin{proof}
Let $x$ be any point in $\Omega$.
It is clear that
\begin{equation}
\label{maxcasiincomparing asymmetry}
\max(\rho_e - R, R- \rho_i) \ge \frac{\rho_e - \rho_i}{2}.
\end{equation}
If that maximum is $\rho_e - R$, at a point $y$ where the ball centered at $x$ with radius $\rho_e$ touches $\Gamma$, we consider the interior touching ball $B_{r_i}$ .
\par
If $2 r_i < (\rho_e - \rho_i)/2$ then $B_{r_i} \subset \Omega \setminus B_R^x$ and hence \begin{equation*}
\frac{|\Omega \Delta B_R^x|}{|B_R^x|} \ge \left( \frac{r_i}{R} \right)^N.
\end{equation*}
If, else, $2 r_i \ge (\rho_e - \rho_i)/2$, $B_{r_i}$ contains a ball of radius $(\rho_e - \rho_i)/4$ still touching $\Gamma$ at $y$. Such a ball is contained in $\Omega \setminus B_R^x$, and hence
\begin{equation*}
\frac{|\Omega \Delta B_R^x|}{|B_R^x|} \ge \left( \frac{\rho_e - \rho_i}{4 \, R} \right)^N.
\end{equation*}
\par
Thus, we proved that
\begin{equation}\label{eq:asymmetrynuovatogliendoepsilon}
\frac{|\Omega \Delta B_R^x|}{|B_R^x|} \ge \min \left\lbrace \left( \frac{r_i}{R} \right)^N , \left( \frac{\rho_e - \rho_i}{4 \, R} \right)^N \right\rbrace .
\end{equation}
\par
If, else, the maximum in \eqref{maxcasiincomparing asymmetry} is $R- \rho_i$, we proceed similarly, by reasoning on the exterior ball $B_{r_e}$ and $\mathbb{R}^N\setminus\overline{\Omega}$, and we obtain \eqref{eq:asymmetrynuovatogliendoepsilon} with $r_i$ replaced by $r_e$.
Thus, by recalling that $\underline{r}=\min{ \left[ r_i, r_e \right] }$, it always holds that
\begin{equation*}
\frac{|\Omega \Delta B_R^x|}{|B_R^x|} \ge \min \left\lbrace \left( \frac{ \underline{r} }{R} \right)^N , \left( \frac{\rho_e - \rho_i}{4 \, R} \right)^N \right\rbrace ,
\end{equation*}
from which, since $x$ was arbitrarily chosen in $\Omega$, we deduce that
$$
\rho_e - \rho_i \le 4 \, R \, {\mathcal A}(\Omega)^{\frac{1}{N}} \quad \mbox{ if } \quad {\mathcal A}(\Omega)\le \left(
\frac{ \underline{r} }{R} \right)^N .
$$
On the other hand, if
$$
{\mathcal A}(\Omega) > \left( \frac{ \underline{r} }{R} \right)^N
$$
it is trivial to check that
$$
\rho_e - \rho_i \le \frac{d_\Omega \, R}{ \underline{r} } \, {\mathcal A}(\Omega)^{\frac{1}{N}}
$$
holds.
Thus, we deduce that
$$
\rho_e - \rho_i \le \max{ \left[ 4 \, R , \frac{d_\Omega \, R}{ \underline{r} } \right] } \, {\mathcal A}(\Omega)^{\frac{1}{N}} .
$$
By proceeding as described at the end of the proof of Theorem \ref{thm:Improved-Serrin-stability}, we can easily obtain the estimate $R \leq d_\Omega/2$, from which the conclusion easily follows.
\end{proof}
\begin{rem}
{\rm
Theorem \ref{thm:comparing asymmetry} and \eqref{eq:SBTstabinmisura} give the inequality
$$
\rho_e - \rho_i \le C \, \Vert H_0 - H \Vert_{2,\Gamma}^{1/N}
$$
that, for any $N\ge 2$, is poorer than that obtained in Theorem \ref{thm:SBT-improved-stability}.
}
\end{rem}
\section{Other related stability results}\label{sec:Other stability results mean curvature}
If we suppose that $\Gamma$ is mean convex, then we can use Theorem \ref{thm:identityheintze-karcher} to obtain a stability result for Heintze-Karcher inequality.
The following theorem improves (for every $N\ge 4$) the exponents obtained in \cite[Theorem 4.5]{MP}.
\begin{thm}[Stability for Heintze-Karcher's inequality]
\label{thm:stability-hk}
Let $\Gamma$ be a surface of class $C^2$, which is the boundary of a bounded domain $\Omega\subset\mathbb{R}^N$, $N\ge 2$. Denote by $H$ its mean curvature and suppose that $H\ge0$ on $\Gamma$.
Then, there exist a point $z\in\Omega$ and a positive constant $C$ such that
\begin{equation}
\label{stability-hk-C}
\rho_e-\rho_i\le C\,\left(\int_\Gamma\frac{dS_x}{H}-N\,|\Omega|\right)^{\tau_N / 2} ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_N=1$ for $N=2$ or $3$;
\item $\tau_4$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{stability-hk-C} holds with $\tau_4= 1- \theta $;
\item $\tau_N = 2/(N-2) $ for $N\ge 5$.
\end{enumerate}
The constant $C$ depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$, and $\theta$ (only in the case $N=4$).
\end{thm}
\begin{proof}
As before, we choose $z\in\Omega$ be any local minimum point of $u$ in $\Omega$.
Moreover, by \eqref{heintze-karcher-identity} and \eqref{heintze-karcher}, we have that
$$
\frac1{N-1}\,\int_\Omega |\nabla^2 h|^2\,dx\le \int_\Gamma\frac{dS_x}{H}-N\,|\Omega|.
$$
Thus, the conclusion follows from Theorem \ref{thm:SBT-W22-stability}.
\end{proof}
Since the deficit of Heintze-Karcher's inequality can be written as
$$
\int_\Gamma\frac{dS_x}{H}-N\,|\Omega| = \int_\Gamma \left( \frac{1}{H} - u_\nu \right) \, dS_x ,
$$
where $u$ is the solution of \eqref{serrin1},
Theorem \ref{thm:stability-hk} also gives a stability estimate for the overdetermined boundary value problem mentioned in item (ii) of Theorem \ref{thm:heintze-karcher}, as stated next. It is clear that the deviation $\int_\Gamma \left( 1/H - u_\nu \right) \, dS_x$ is weaker than $\Vert 1/H - u_\nu \Vert_{1, \Gamma}$. The following theorem improves \cite[Theorem 4.8]{MP}.
\begin{thm}[Stability for a related overdetermined problem]
\label{thm:OBVP-stability}
Let $\Gamma$, $\Omega$, and $H$ be as in Theorem \ref{thm:stability-hk}.
Let $u$ be the solution of problem \eqref{serrin1} and $z \in \Omega$ be any of its critical points.
\par
There exists a positive constant $C$ such that
\begin{equation}
\label{OBVT-stability-gener-C}
\rho_e-\rho_i\le C\, \left\lbrace \int_\Gamma \left( \frac{1}{H} - u_\nu \right) \, dS_x \right\rbrace^{ \tau_N / 2 } ,
\end{equation}
with the following specifications:
\begin{enumerate}[(i)]
\item $\tau_N=1$ for $N=2$ or $3$;
\item $\tau_4$ is arbitrarily close to one, in the sense that for any $\theta>0$, there exists a positive constant $C$ such that \eqref{OBVT-stability-gener-C} holds with $\tau_4= 1- \theta $;
\item $\tau_N = 2/(N-2) $ for $N\ge 5$.
\end{enumerate}
The constant $C$ depends on $N$, $r_i$, $r_e$, $d_\Omega$, $\de_\Gamma (z)$, and $\theta$ (only in the case $N=4$).
\end{thm}
\begin{rem}
{\rm
It is clear that analogs of items (i) and (ii) of Remark \ref{rem:ontheconstants-SBT-removingdelta-seconvexocasidiremark} hold also for
Theorems \ref{thm:stability-hk} and \ref{thm:OBVP-stability}.
}
\end{rem}
\begin{appendices}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,000 |
{"url":"http:\/\/www.talkstats.com\/showthread.php\/10978-Ratio-of-uniforms-is-Cauchy","text":"Thread: Ratio of uniforms is Cauchy\n\n1. Ratio of uniforms is Cauchy\n\nHi all!\n\nI have U is uniform(0,1) and V is uniform(-1,1) such that U^2 + V^2 <= 1. Furthermore, let X=V\/U. Show that X is Cauchy.\n\nI tried to solve this problem using transformation:\nX = V\/U => V = XY\nY = U => U = Y\nThen the Jacobian is |J|=y and the joint pdf of X and Y is just f(x,y)=.5*y. This seems to be wrong since integrated out y won't give me a Cauchy.\n\nThen I tried using the CDF:\nP(X <= x) = P(V\/U <= x) = P(V <= Ux) = x -.5 but this does not work either.\n\nObviously I have to incorporate the condition that U^2 + V^2 <= 1 but I have no idea how and I would really appreciate some help.\n\nThanks,\nJenny\n\n2. Dont we need to know what is joint pdf of <X,Y> ?\n\nAre they independant?\n\n3. I mean joint pdf of <U, V>. Without this problem is not solvable.\n\n4. Originally Posted by JennySton\nHi all!\n\nI have U is uniform(0,1) and V is uniform(-1,1) such that U^2 + V^2 <= 1. Furthermore, let X=V\/U. Show that X is Cauchy.\n\nI tried to solve this problem using transformation:\nX = V\/U => V = XY\nY = U => U = Y\nThen the Jacobian is |J|=y and the joint pdf of X and Y is just f(x,y)=.5*y. This seems to be wrong since integrated out y won't give me a Cauchy.\n\nThen I tried using the CDF:\nP(X <= x) = P(V\/U <= x) = P(V <= Ux) = x -.5 but this does not work either.\n\nObviously I have to incorporate the condition that U^2 + V^2 <= 1 but I have no idea how and I would really appreciate some help.\n\nThanks,\nJenny\n1) I'm betting the joint pdf is $\\frac{2}{\\pi}$; one over the area of the circle that U,V are defined on.\n\n2) you then want\n\n$U^2+V^2\\leq 1\\Rightarrow Y^2+(XY)^2\\leq 1\\Rightarrow Y^2(1+X^2)\\leq 1\\Rightarrow Y\\leq\\sqrt{ \\frac{1}{1+X^2}}$\n\n3) then compute\n\n$\\int\\limits_{0}^{ \\sqrt{\\frac{1}{1+X^2}}} \\frac{2}{\\pi}ydy$\n\n5. Thanks for your help! I see what I did wrong!\n\nPosting Permissions\n\n\u2022 You may not post new threads\n\u2022 You may not post replies\n\u2022 You may not post attachments\n\u2022 You may not edit your posts","date":"2013-05-24 08:03:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 3, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7943415641784668, \"perplexity\": 1333.768772049887}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368704368465\/warc\/CC-MAIN-20130516113928-00019-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Q: Thread needs to use Invoke method to access UI controls but BackGroundworker doesn't. Why? BackgroundWorker can access UI controls directly, but Thread's cannot, Why?
Isn't BackgroundWorker a thread? If it isn't, what is it? Also, why is direct accessing restricted? Is restricting direct access Microsoft's choosen way for doing something or must it be done this way?
A: According to MSDN, "The BackgroundWorker class allows you to run an operation on a separate, dedicated thread."
BackgroundWorker acutally uses Form.Invoke() to switch control to the main UI thread to report progress. In the fact BackgroundWorker is just a helper class that you could easily write yourself, it uses a second thread to do the work but then it switches to the main UI thread to access GUI.
Also from MSDN: "You can listen for events that report the progress of your operation and signal when your operation is completed." This means that the background operation runs on a separate thread and you can't access GUI from that thread. But you can access GUI controls from progress event because it is raised on the main UI thread. Whenever you call BackgroundWorker.ReportProgress() on the worker thread, it raises BackgroundWorker.ProgressChanged event on the main thread.
Also, you ask why GUI cannot be accessed from other threads. It is because whole Windows GUI subsystem is not thread safe. As far as I know this is pretty normal in other platforms (outside Windows) as well, so it definitely isn't Microsoft specific. It is probably from historical reasons and probably has something to do with code efficiency. (A thread safe GUI would be slower.) But I can't provide any strong information source as a reference.
A: A background worker will throw a cross-thread exception just as a new thread would. A backgroundworker is just a fancy way to start a new thread.
If you are using the ProgressChanged event then you do not need to invoke it because there is some .NET magic behind the scenes doing it for you.
More information on background workers:
http://msdn.microsoft.com/en-us/library/system.componentmodel.backgroundworker(v=vs.100).aspx
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,088 |
package org.javaplus.jmp.taglib;
import javax.mail.MessagingException;
import javax.mail.Part;
import javax.mail.internet.MimeUtility;
import javax.servlet.jsp.JspException;
import javax.servlet.jsp.JspTagException;
import org.javaplus.jmp.util.MessageUtil;
/**
* <code>HeaderTag</code> implements tags that create and add a header
* field to the {@link javax.mail.Part} instance associated with our most
* immediate enclosing instance of a tag whose implementaion class is
* a subclass of {@link PartTag}.
*
* @see org.javaplus.jmp.taglib.PartTag
* @see javax.mail.Part
*
* @author Stephen Suen
*/
public class HeaderTag extends MessagingTagSupport {
private String name = null;
private String value = null;
/**
* Creates new instance of tag handler
*/
public HeaderTag() {
super();
}
/**
* Returns the header field name.
* @return the name
*/
public String getName() {
return name;
}
/**
* Sets the header field name to the speicified value.
* @param name the new header name
*/
public void setName(String name) {
this.name = normalizeAttribute(name);
}
/**
* Returns the header field value.
* @return the value
* @throws JspException if error occurred.
*/
public String getValue() throws JspException {
return value;
}
/**
* Sets the header field value to the specified value.
* @param value the new value
*/
public void setValue(String value) {
this.value = value;
}
/**
* <p>
* Encodes the specified value if necessary. Subclasses should override
* this method to provide necessary encoding required for the particular
* header fields, and then call this method before returning. </p>
* <p>
* The algorithm used here is:</p>
* <ul>
* <li>Remove any leading and trailing whitespaces.</li>
* <li>Replace every line break character , including <code>'\r'</code>
* and <code>'\n'</code>, with a space character.
* </li>
* <li>Fold the result string by calling
* {@link javax.mail.internet.MimeUtility#fold(int,String)}</li>
* </ul>
* <p>
* This method will be called by {@link #doEndTag()} method to encode
* the specified <code>value</code> attribute before adding the header
* field to the enclosing <code>Part</code> instance.</p>
*
* @param value
* @return the encoded value
* @throws JspException if the value cannot be encoded
* @see #doEndTag()
*/
protected String encodeValue(String value) throws JspException {
// remove leading and trailing whitespaces
value = normalizeAttribute(value);
if (value == null) {
return null;
}
// replace any line break with space
StringBuffer sb = new StringBuffer(value.length());
for (int i = 0; i < value.length(); i++) {
char c = value.charAt(i);
if (c == '\r' || c == '\n') {
sb.append(' ');
} else {
sb.append(c);
}
}
value = sb.toString();
int used = this.name.length();
return MimeUtility.fold(used, value);
}
//
// Overrided BodyTag method ----------------------------------------------//
//
/**
* Performs any necessary validation on the <code>name</code> and
* <code>value</code> attribute. An empty value or a value containing
* only whitespaces is not allowed for both <code>name</code> and
* <code>value</code> attribute. An <code>JspException</code>
* will be thrown to indicate such an error.
*
* @return the value returned from the superclass
* @throws JspException if an error occurs
*/
@Override
public int doStartTag() throws JspException {
int result = super.doStartTag();
if (name == null) {
throw new JspTagException(MessageUtil.getMessage(
"headertag_null_name", name, value));
}
return result;
}
@Override
public int doEndTag() throws JspException {
// if value attribute not set, use body content as value
if (value == null) {
value = getBodyContentString();
}
value = encodeValue(value);
if (value == null) {
throw new JspTagException(MessageUtil.getMessage(
"headertag_null_value", name, value));
}
// add this header field to the Part instance associated with our
// most immediate enclosing PartTag instance if any,
// otherwise, throw exception indicating the nesting error.
MessagingTagSupport parentTag = getParentMessagingTag();
if (parentTag == null || !(parentTag instanceof PartTag)) {
throw new JspException(MessageUtil.getMessage(
"headertag_nesting_error", name, value));
} else {
Part part = ((PartTag) parentTag).getPart();
try {
part.addHeader(name, value);
} catch (MessagingException ex) {
throw new JspException(MessageUtil.getMessage(
"headertag_addheader_failed", name, value),
ex);
}
}
return super.doEndTag();
}
@Override
public void release() {
this.name = null;
this.value = null;
super.release();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 513 |
Denmark's OUH upgrades orthopaedic clinical care technology with EOS system
Denmark's Odense University Hospital (OUH) has become the first hospital in Scandinavia to deploy EOS imaging's EOS low-dose 2D / 3D system for orthopaedic clinical care.
By admin-demo
Denmark's Odense University Hospital (OUH) has become the first hospital in Scandinavia to deploy EOS imaging's EOS low-dose 2D / 3D system for orthopaedic clinical care.
"EOS low-dose 2D / 3D imaging system calculates a range of clinical parameters required for diagnosis and surgical planning."
EOS's imaging system delivers lower radiation doses, when compared to digital X-ray and CT scans, and also provides full-body 2D and 3D images of the patient's skeleton in a natural standing or seated position.
OUH radiology department head Jens Karstoft said the hospital is incorporating the system into its clinical practice, as the advanced orthopaedic imaging technology it uses reduces radiation exposure levels for patients.
"With the EOS system, we not only require [a few] scans to precisely diagnose even complex conditions, but also expose patients to a substantially lower radiation dose," Karstoft added.
EOS low-dose 2D / 3D imaging system calculates a range of clinical parameters required for diagnosis and surgical planning.
The parameters enable more informed diagnosis and treatment of orthopaedic conditions, such as scoliosis, degenerative spine and lower limb joint conditions.
According to various study reports, doses delivered by EOS's system are up to nine times less, compared to computed radiography X-ray, and 20 times lower than low dose CT scans.
EOS imaging CEO Marie Meynadier said: "As such, we are pleased to have the EOS system, based upon Nobel Prize-winning technology, now available to clinicians and patients in Scandinavia to enable high-quality imaging with optimised safety." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,503 |
Ḥā petit ṭāʾ et deux points souscrits est une lettre additionnelle de l'alphabet arabe utilisée dans l'écriture du khowar. Elle est composée d'un ḥā diacrité d'un petit ṭāʾ et deux points souscrits ou d'un jīm diacrité d'un petit ṭāʾ et deux points souscrits au lieu d'un point souscrit.
Utilisation
En khowar, représente une consonne affriquée rétroflexe sourde , transcrite c point souscrit avec l'alphabet latin.
Notes et références
Bibliographie
Ha petit ta et deux points souscrits | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,470 |
Труккаццано () — коммуна в Италии, в провинции Милан области Ломбардия.
Население составляет 4353 человека, плотность населения составляет 198 чел./км². Занимает площадь 22 км². Почтовый индекс — 20060. Телефонный код — 02.
Покровителями коммуны почитаются Пресвятая Богородица (Madonna di Rezzano), празднование во второе воскресение мая, и святой архангел Божий Михаил, празднование 29 сентября.
Ссылки
Официальный сайт населённого пункта | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,657 |
Q: using svn diff command in code Hi i have a problem using svn command in code behind :
public void SvnDiff(int rev1, int rev2)
{
try
{
var p = new Process();
p.StartInfo.UseShellExecute = false;
p.StartInfo.RedirectStandardOutput = true;
p.StartInfo.FileName = "svn";
string arg = string.Format("diff -r {0}:{1} --summarize --xml > SvnDiff.xml", rev1, rev2);
Console.WriteLine(arg);
p.StartInfo.Arguments = arg;
p.Start();
p.WaitForExit();
}
catch (Exception e)
{
Console.WriteLine(e);
}
}
When i use this command in cmd, it's working fine.
svn diff -r 2882:2888 --summarize --xml > SvnDiff.xml
but when i run my method i got this message:
svn: E020024: Error resolving case of '>'
What can i do right now to solve this ?
Thanks for any advice
A: You can read all text and then write to the file as @Scott suggested, but it can be problematic if the output is too large.
You can instead write as the output is generated. Create a local StreamWriter for the file, and a method to write whenever new output data is available:
StreamWriter redirectStream = new StreamWriter("SvnDiff.xml")
void Redirect(object Sender, DataReceivedEventArgs e)
{
if ((e.Data != null)&&(redirectStream != null))
redirectStream.WriteLine(e.Data);
}
and when you start the process:
p.OutputDataReceived += new DataReceivedEventHandler(Redirect); // handler here to redirect
p.Start();
p.BeginOutputReadLine();
p.WaitForExit();
redirectStream.Flush();
redirectStream.Close();
A: The > SvnDiff.xml part of your command line aren't arguments being passed to SVN. They're redirecting standard output. To do that, take a look at the documentation:
Since you already have RedirectStandardOutput set properly, you just need to take advantage of it. Simply put
string output = p.StandardOutput.ReadToEnd();
in front of your p.WaitForExit(); line.
Then, just use File.WriteAllText("SvnDiff.xml", output);
A: Try with this....
var p = new Process();
p.StartInfo.UseShellExecute = false;
p.StartInfo.RedirectStandardOutput = true;
p.StartInfo.FileName = "cmd.exe";
string arg = string.Format("/K svn diff -r {0}:{1} --summarize --xml > SvnDiff.xml", rev1, rev2);
Console.WriteLine(arg);
p.StartInfo.Arguments = arg;
p.Start();
p.WaitForExit();
if it works then change /K with /C
A: Thanks for answers, I use StandardOutput and it work :)
var p = new Process();
p.StartInfo.UseShellExecute = false;
p.StartInfo.RedirectStandardOutput = true;
p.StartInfo.FileName = "svn";
p.StartInfo.Arguments = string.Format("diff -r {0}:{1} --summarize --xml", rev1, rev2);
p.Start();
string output = p.StandardOutput.ReadToEnd();
p.WaitForExit();
File.WriteAllText("SvnDiff.xml", output);
Thanks for all of you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 35 |
Buchanan Island ist eine unbewohnte Insel im australischen Northern Territory. Sie ist 52,7 Kilometer vom australischen Festland entfernt. Die Insel wird zu den Tiwi-Inseln gezählt.
Die Insel ist sehr flach. Die höchste Erhebung liegt nur einen Meter über dem Meeresspiegel.
Die Insel umrundet ein Sandstrand.
Einzelnachweise
Insel (Australien und Ozeanien)
Insel (Northern Territory)
Insel (Timorsee)
Tiwi Islands Region | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,876 |
\section{Introduction} \label{sec:intro}
Hammond and Lewis \cite{Ha-Le} found some elementary results for
$2$-colored partitions mod $5$.
Let $E(q) = \prod_{n=1}^\infty(1-q^n)$, and
$$
\sum_{n=0}^\infty p_{-2}(n) q^n = \frac{1}{E(q)^2},
$$
which is the generating function for pairs of partitions $(\pi_1,\pi_2)$
(or $2$-colored partitions). Throughout this paper we refer to
such pairs of partitions as {\it bipartitions}. It is not hard to show that
\begin{equation}
p_{-2}(5n+2) \equiv
p_{-2}(5n+3) \equiv
p_{-2}(5n+4) \equiv 0 \pmod{5}.
\mylabel{eq:2colcongs}
\end{equation}
Hammond and Lewis \cite{Ha-Le} found a crank for these congruences. By crank
we mean a statistic that divides the relevant partitions into equinumerous
classes.
They define
\begin{equation}
\mbox{birank}(\pi_1,\pi_2) = \#(\pi_1) - \#(\pi_2),
\mylabel{eq:birankdef}
\end{equation}
where $\#(\pi)$ denotes the number of parts in the partition $\pi$.
They show that the residue of the birank mod $5$
divides the bipartitions of $n$ into $5$ equal classes
provided $n\equiv2$, $3$ or $4\pmod{5}$.
The proof is elementary. It relies on Jacobi's triple product identity and the
method of \cite{Ga88}, which uses roots of unity.
We have found two other analogs.
\bigskip
\noindent
\textbf{First Analog - The Dyson-birank}
\noindent
Dyson \cite{Dy44} defined the \textit{rank} of a partition as
the largest part minus the number of parts. We define
\begin{equation}
\mbox{Dyson-birank}(\pi_1,\pi_2) = \mbox{rank}(\pi_1)
+ 2\,\mbox{rank}(\pi_2).
\mylabel{eq:Dysonbirankdef}
\end{equation}
In Section \sect{Dbirank}, we show that the residue of the Dyson-birank mod $5$
divides the bipartitions of $n$ into $5$ equal classes
provided $n\equiv2$, or $4\pmod{5}$. Unfortunately the Dyson-birank
does not work if $n\equiv3\pmod{5}$. Nonetheless, for the other
residue classes this is a surprising and deep result because
of the nature of the rank generating function. The proof
depends on known results for the rank mod $5$ due to
Atkin and Swinnerton-Dyer \cite{At-Sw}.
\bigskip
\noindent
\textbf{Second Analog - The $5$-core-birank}
\noindent
In \cite{Ga-Ki-St} new statistics were defined in terms of $t$-cores which
gave new combinatorial interpretations of Ramanujan's partition
congruences mod $5$, $7$ and $11$. For example, for a partition $\pi$
the \textit{$5$-core-crank} is defined as
\begin{equation}
\mbox{$5$-core-crank}(\pi) = r_1 + 2 r_2 - 2 r_3 - r_4,
\mylabel{eq:GKScrankdef}
\end{equation}
where $r_j$ is the number of cells labelled $j$ in the $5$-residue
diagram of $\pi$. Then in \cite{Ga-Ki-St} we proved combinatorially that
the residue of the $5$-core-crank divides the partitions of $5n+4$
into $5$ equal classes.
We define
\begin{equation}
\mbox{$5$-core-birank}(\pi_1,\pi_2) = \mbox{$5$-core-crank}(\pi_1)
+ 2\,(\mbox{$5$-core-crank}(\pi_2)).
\mylabel{eq:GKSbirankdef}
\end{equation}
In Section \sect{5cbirank}, we show that the $5$-core-birank divides the
bipartitions of $n$ into
$5$ equal classes
for $n\equiv2$, $3$ or $4\pmod{5}$. This is quite a surprising result.
The proof relies on the $5$-dissection
of the $5$-core-crank generating function for
$5$-cores.
The crank of a partition is defined to be the largest part if
it contains no ones and otherwise it is the difference between number of parts larger
than the number of ones, and the number of ones. The crank gives a combinatorial
of Ramanujan's partition congruences mod $5$, $7$ and $11$ and solves a
problem of Dyson \cite{Dy44}, \cite[p.52]{Dy96}. See \cite{An-Ga88}. This crank
is different to the $5$-core crank given in \cite{Ga-Ki-St}. It is natural
to ask whether there is a crank analog of the birank. This question has been
answered in part by Andrews \cite{An08}. In Section \sect{bicrank},
we consider Andrews result and how it may be extended. In \cite{An08},
Andrews also considered congruences for more general multipartitions. In
Section \sect{ghlbirank}, we give multipartition analogs of the Hammond-Lewis birank which
explain these more general congruences. In Section \sect{mcranks}, we extend
Andrews bicrank to multicranks of what we call extended multipartitions, and
give alternative explanations of our multipartition congruences. In Section \sect{end},
we close with some further problems.
\subsection*{Notation}
For a partition $\pi$ we denote the sum of parts by $\abs{\pi}$.
We will use the standard $q$-notation.
$$
(z;q)_n=(z)_n=
\begin{cases}
\prod_{j=0}^{n-1}(1-zq^j), & n>0 \\
1, & n=0,
\end{cases}
$$
and
$$
(z;q)_\infty=(z)_\infty = \lim_{n\to\infty} (z;q)_n
=\prod_{n=1}^\infty (1-z q^{(n-1)}),
$$
where $\abs{q}<1$. We will also you the following notation
for Jacobi-type theta-products.
$$
J_{a,m}(q) := (q^a;q^m)_\infty (q^{m-a};q^m)_\infty (q^m;q^m)_\infty.
$$
\section{The Hammond-Lewis Birank} \label{sec:HLbirank}
For completeness we include some details of the Hammond-Lewis birank.
For a bipartition $\bpi=\tpi$ we denote the sum of parts by
\begin{equation}
\abs{\bpi} = \abs{\pi_1} + \abs{\pi_2}.
\mylabel{eq:abspi}
\end{equation}
We denote the Hammond-Lewis birank by
\begin{equation}
\mbox{HL-birank}(\bpi) = \#(\pi_1) - \#(\pi_2),
\mylabel{eq:HLbirankdef}
\end{equation}
where $\#(\pi)$ denotes the number of parts in the partition $\pi$.
The HL-birank generating function is
\begin{equation}
\sum_{\bpi=(\pi_1,\pi_2)} z^{\HLbr} q^{\abs{\bpi}} = \frac{1}{(zq;q)_\infty (z^{-1}q;q)_\infty}.
\mylabel{eq:HLgenfunc}
\end{equation}
We let $N_{\HL}(m,t,n)$ denote the number of bipartitions $\bpi=\tpi$
with HL-birank congruent to $m\pmod{t}$.
Suppose $\zeta$ is primitive $5$th root of unity.
By letting $z=\zeta$ in \eqn{HLgenfunc}
and
using Jacobi's triple product identity, Hammond and Lewis found that
\begin{align}
\sum_{n=0}^\infty
\sum_{k=0}^4 \zeta^k N_{\HL}(k,5,n) q^n
&= \frac{1}{(\zeta q;q)_\infty (\zeta^{-1}q;q)_\infty}
= \frac{(\zeta^2q;q)_\infty (\zeta^{-2}q;q)_\infty(q;q)_\infty}
{(q^5;q^5)_\infty},\mylabel{eq:birankmod5}\\
&=(q^{25};q^{25})_\infty\left(
\frac{1}{J_{1,5}(q^5)} + (\zeta + \zeta^{-1})
\frac{q}{J_{2,5}(q^5)}
\right).
\nonumber
\end{align}
Since the coefficient of $q^n$ on the right side of \eqn{birankmod5}
is zero when $n\equiv2$, $3$ or $4\pmod{5}$, Hammond and Lewis's main result
follows.
\begin{theorem}
\cite{Ha-Le}
\label{thm:thm1}
The residue of the HL-birank mod $5$
divides the bipartitions of $n$ into $5$ equal classes
provided $n\equiv2$, $3$ or $4\pmod{5}$.
\end{theorem}
We illustrate Theorem \thm{thm1} for the case $n=3$.
\begin{equation*}
\begin{array}{rr}
\mbox{Bipartitions of $3$} & \mbox{HL-birank (mod $5$)}\\
(3,-) & 1-0 \equiv 1\\
(2+1,-) & 2-0 \equiv 2\\
(1+1+1,-) & 3-0 \equiv 3\\
(2,1) & 1-1 \equiv 0\\
(1+1,1) & 2-1 \equiv 1\\
(1,2) & 1-1 \equiv 0\\
(1,1+1) & 1-2 \equiv4\\
(-,3) & 0-1 \equiv 4\\
(-,2+1) & 0-2 \equiv 3\\
(-,1+1+1) & 0-3 \equiv 2\\
\end{array}
\end{equation*}
Thus
$$
N_{\HL}(0,5,3) = N_{\HL}(1,5,3) = N_{\HL}(2,5,3) = N_{\HL}(3,5,3) =
N_{\HL}(4,5,3) = 2,
$$
and we see that the residue of the HL-birank mod $5$ divides the $10$
bipartitions of $3$ into $5$ equal classes.
\section{The Dyson-Birank} \label{sec:Dbirank}
Dyson \cite{Dy44}, \cite[p.52]{Dy96} defined the rank of a partition as the largest
part minus the number of parts.
We define the Dyson-analog of the
birank for bipartitions $\bpi=\tpi$ by
\begin{equation}
\mbox{Dyson-birank}(\bpi) = \mbox{rank}(\pi_1)
+ 2\,\mbox{rank}(\pi_2).
\mylabel{eq:Dysonbirankdef2}
\end{equation}
In this section we prove
\begin{theorem}
\label{thm:thm2}
The residue of the Dyson-birank mod $5$
divides the bipartitions of $n$ into $5$ equal classes
provided $n\equiv2$, or $4\pmod{5}$.
\end{theorem}
We let $N_{\DY}(m,t,n)$ denote the number of bipartitions $\bpi=\tpi$
with Dyson-birank congruent to $m\pmod{t}$.
We illustrate Theorem \thm{thm2} for the case $n=2$.
\begin{equation*}
\begin{array}{rr}
\mbox{Bipartitions of $2$} & \mbox{Dyson-birank (mod $5$)}\\
(2,-) & 1+0 \equiv 1\\
(1+1,-) & -1+0 \equiv 4\\
(1,1) & 0+0 \equiv 0\\
(-,2) & 0+2 \equiv 2\\
(-,1+1) & 0-2 \equiv 3\\
\end{array}
\end{equation*}
Thus
$$
N_{\DY}(0,5,2) = N_{\DY}(1,5,2) = N_{\DY}(2,5,2) = N_{\DY}(3,5,2) =
N_{\DY}(4,5,2) = 1,
$$
and we see that the residue of the Dyson-birank mod $5$ divides the $5$
bipartitions of $2$ into $5$ equal classes. We note that Theorem
\thm{thm2} does not hold for $n\equiv 3\pmod{5}$. The first counterexample
occurs when $n=13$. The Dyson-birank mod $5$ fails to divide the
1770 bipartitions of $13$ into $5$ equal classes. We have
$$
N_{\DY}(0,5,13) = 358,\quad\mbox{but}\quad
N_{\DY}(1,5,13) = N_{\DY}(2,5,13) = N_{\DY}(3,5,13) =
N_{\DY}(4,5,13) = 353.
$$
To prove Theorem \thm{thm2} we need the $5$-dissection of the rank generating
function when $z=\zeta$.
The Dyson-rank generating function is
\begin{equation}
f(z,q) = \sum_{\pi} z^{\srank(\pi)} q^{\abs{\pi}} = 1 +
\sum_{n=1}^\infty \frac{q^{n^2}}{(zq;q)_n (z^{-1}q;q)_n}.
\mylabel{eq:rankgenfunc}
\end{equation}
We let $N(m,t,n)$ denote the number of ordinary partitions of $n$
with rank congruent to $m$ mod $t$. Then
\begin{align}
f(\zeta,q)=&\sum_{n=0}^\infty
\sum_{k=0}^4 \zeta^k N(k,5,n) q^n
=1+\sum_{n=1}^\infty \frac{q^{n^2}}{(\zeta q;q)_n ({\zeta}^{-1}q;q)_n}
\mylabel{eq:fdissect}\\
&= (A(q^5) - (3 + \zeta^2 + \zeta^3)\,\phi(q^5))
+ q\,B(q^5)
+ q^2 (\zeta + \zeta^4)\, C(q^5) \nonumber\\
&\qquad
+q^3 ( (1+\zeta^2+\zeta^3)\, D(q^5) + (1 + 2\zeta^2 + 2\zeta^3)\,\psi(q^5)),
\nonumber
\end{align}
where
\begin{align}
A(q) &= \frac{E^2(q)\,J_{2,5}(q)}{J_{1,5}^2(q)},\mylabel{eq:Adef}\\
B(q) &= \frac{E^2(q)}{J_{1,5}(q)},\mylabel{eq:Bdef}\\
C(q) &= \frac{E^2(q)}{J_{2,5}(q)},\mylabel{eq:Cdef}\\
D(q) &= \frac{E^2(q)\,J_{1,5}(q)}{J_{2,5}^2(q)},\mylabel{eq:Ddef}\\
\phi(q) &=-1+\sum_{n=0}^\infty\frac{q^{5n^2}}{(q;q^5)_{n+1}(q^4;q^5)_n}
\mylabel{eq:phidef}\\
&= \frac{q}{E(q^5)}\sum_{m=-\infty}^\infty
(-1)^m\frac{q^{\frac{15}2m(m+1)}}{1-q^{5m+1}},\nonumber \\
\noalign{\mbox{and}}
\psi(q)
&=\dfrac1{q}\biggl\{-1+\sum_{n=0}^\infty\frac{q^{5n^2}}{(q^2;q^5)_{n+1}(q^3;q^5)_n}
\biggr\} \mylabel{eq:psidef}\\
&=\frac{q}{E(q^5)} \sum_{m=-\infty}^\infty
(-1)^m\frac{q^{\frac{15}2m(m+1)}}{1-q^{5m+2}}.
\nonumber
\end{align}
Equation \eqn{fdissect} has an unusual history. It is originally due
to Ramanujan since appears in the Lost Notebook. It is closely related
to Dyson's conjectures on the rank \cite{Dy44}, which were proved
by Atkin and Swinnerton-Dyer \cite{At-Sw}. As pointed out in \cite{Ga88} and
\cite{Ga88b},
equation \eqn{fdissect} is actually equivalent to one of
Atkin and Swinnerton-Dyer's main results. Dyson, Atkin and Swinnerton-Dyer
were unaware of Ramanujan's result.
The Dyson-birank generating function is
\begin{equation}
\sum_{\bpi=(\pi_1,\pi_2)} z^{\DYbr} q^{\abs{\bpi}} = f(z,q) \, f(z^2,q),
\mylabel{eq:DYgenfunc}
\end{equation}
where $f(z,q)$ is the generating function for the Dyson rank of ordinary partitions
given in \eqn{rankgenfunc}.
Thus we have
\begin{equation}
\sum_{n=0}^\infty \sum_{k=0}^4 \zeta^k N_{\DY}(k,5,n) q^n
= f(\zeta,q)\, f(\zeta^2,q).
\mylabel{eq:Dbrmod5}
\end{equation}
Using only \eqn{fdissect} and the fact that
\begin{equation}
B^2(q) = A(q)\,C(q),\qquad C^2(q) = B(q)\,D(q),
\mylabel{eq:ABCDfact}
\end{equation}
we find that the coefficient of $q^n$ in the $q$-expansion of
$f(\zeta,q)\, f(\zeta^2,q)$ is zero if
$n\equiv2$, or $4\pmod{5}$. Theorem \thm{thm2} then follows from \eqn{Dbrmod5}.
Although Theorem \thm{thm2} does not hold when $n\equiv 3\pmod{5}$,
there is some simplification in the product $f(\zeta,q)\, f(\zeta^2,q)$.
We find that
\begin{equation}
\sum_{n=0}^\infty \sum_{k=0}^4 \zeta^k N_{\DY}(k,5,5n+3) q^n
= 5 \, \phi(q)\, \psi(q),
\mylabel{eq:phipsiid}
\end{equation}
using the fact that
\begin{equation}
A(q) \, D(q) = B(q) \, C(q).
\mylabel{eq:ABCDfact2}
\end{equation}
\section{The $5$-Core-Birank} \label{sec:5cbirank}
For ordinary partitions $\pi$ the {\it $5$-core-crank} is defined
by
\begin{equation}
\mbox{$5$-core-crank}(\pi) = r_1 + 2 r_2 - 2 r_3 - r_4,
\mylabel{eq:GKScrankdef2}
\end{equation}
where $r_j$ is the number of cells labelled $j$ in the $5$-residue
diagram of $\pi$. See \cite[Prop.1,p.7]{Ga-Ki-St}. We define the
$5$-core-crank
analog for bipartitions $\bpi=\tpi$ by
\begin{equation}
\mbox{$5$-core-birank}(\bpi) = \mbox{$5$-core-crank}(\pi_1)
+ 2\,(\mbox{$5$-core-crank}(\pi_2)).
\mylabel{eq:GKSbirankdef2}
\end{equation}
In this section we prove
\begin{theorem}
\label{thm:thm3}
The residue of the $5$-core-birank mod $5$
divides the bipartitions of $n$ into $5$ equal classes
provided $n\equiv2$, $3$ or $4\pmod{5}$.
\end{theorem}
We let $N_{\FC}(m,t,n)$ denote the number of bipartitions $\bpi=\tpi$
with $5$-core-birank congruent to $m\pmod{t}$.
We illustrate Theorem \thm{thm3} for the case $n=3$.
\begin{equation*}
\begin{array}{rr}
\mbox{Bipartitions of $3$} & \mbox{$5$-core-birank (mod $5$)}\\
(3,-) & 3+0 \equiv 3\\
(2+1,-) & 0+0 \equiv 0\\
(1+1+1,-) & -3+0 \equiv 2\\
(2,1) & 1+0 \equiv 1\\
(1+1,1) & -1+0 \equiv 4\\
(1,2) & 0+2 \equiv 2\\
(1,1+1) & 0-2 \equiv3\\
(-,3) & 0+6 \equiv 1\\
(-,2+1) & 0+0 \equiv 0\\
(-,-6) & 0-3 \equiv 4\\
\end{array}
\end{equation*}
Thus
$$
N_{\FC}(0,5,3) = N_{\FC}(1,5,3) = N_{\FC}(2,5,3) = N_{\FC}(3,5,3) =
N_{\FC}(4,5,3) = 2,
$$
and we see that the residue of the $5$-core-birank mod $5$ divides the $10$
bipartitions of $3$ into $5$ equal classes. We note that
although the Dyson-birank does not in general divide the
bipartitions of $5n+3$ into $5$ equal classes the $5$-core-birank does.
To prove Theorem \thm{thm3} we need the $5$-dissection of the
$5$-core-crank generating
function when $z=\zeta$.
The $5$-core-crank generating function is
\begin{equation}
\Phi(z,q) = \sum_{\pi} z^{\fccrank(\pi)} q^{\abs{\pi}} = \frac{1}{E^5(q^5)}
T(z,q),
\mylabel{eq:fccrankgenfunc}
\end{equation}
where
\begin{equation}
T(z,q) := \sum_{\mbox{$\pi$ a $5$-core}}
z^{\mbox{$5$-core-crank($\pi$)}} q^{\abs{\pi}}
= \sum_{\substack{\vec{n}\in\Z^5\\\vec{n}\cdot\vec{1}=0}}
z^{n_1 + 3n_2 + n_3} q^{\tfrac{5}{2}||\vec{n}||^2 + \vec{b}\cdot\vec{n}},
\mylabel{eq:Tdef}
\end{equation}
where $\vec{1}=(1,1,1,1,1)$ and $\vec{b}=(0,1,2,3,4)$.
Equation \eqn{fccrankgenfunc} can be proved combinatorially and
in a straightforward manner using Bijections 1 and 2 from
\cite[pp.2-3]{Ga-Ki-St} and \cite[(4.2), p.6]{Ga-Ki-St}.
We need the $5$-dissection of $T(\zeta,q)$:
\begin{equation}
T(\zeta,q) = W(q^5)(1 + q R(q^5) + q^2 (\zeta^2+\zeta^3) R(q^5)^2
- q^3 (\zeta^2+\zeta^3) R(q^5)^3),
\mylabel{eq:T5dissect}
\end{equation}
where
\begin{align}
W(q) &:=
J_{2,5}(q)^3( J_{10,25}(q) - q (1 + \zeta^2 + \zeta^3) J_{5,25}(q) ),
\mylabel{eq:Wdef}\\
\noalign{\mbox{and}}
R(q) &:= \frac{ J_{1,5}(q)}{J_{2,5}(q)}.\mylabel{eq:Rdef}
\end{align}
We will prove \eqn{T5dissect} in the next section. Theorem \thm{thm3}
follows easily from \eqn{T5dissect}.
The $5$-core-birank generating function is
\begin{equation}
\sum_{\bpi=(\pi_1,\pi_2)} z^{\fcbr} q^{\abs{\bpi}} =
\frac{1}{E^{10}(q^5)}\, T(z,q) \, T(z^2,q),
\mylabel{eq:fcbrgenfunc}
\end{equation}
where $T(z,q)$ is the generating function for the
$5$-core-crank of partitions that are $5$-cores given in \eqn{Tdef}.
Thus we have
\begin{equation}
\sum_{n=0}^\infty \sum_{k=0}^4 \zeta^k N_{\FC}(k,5,n) q^n
= \frac{1}{E^{10}(q^5)}\,T(\zeta,q)\, T(\zeta^2,q).
\mylabel{eq:FCbrmod5}
\end{equation}
From \eqn{T5dissect} we find that
\begin{equation}
T(\zeta,q) \, T(\zeta^2,q) = W^2(q^5)( 1 + 2\,q^5\,R^5(q^5)
+ q R(q^5)( 2 - q^5\,R^5(q^5)).
\mylabel{eq:TTid}
\end{equation}
Since coefficient of $q^n$ in the $q$-expansion of
$T(\zeta,q) \, T(\zeta^2,q)$ is zero when
$n\equiv2$, $3$ or $4\pmod{5}$, Theorem \thm{thm3} then follows
from \eqn{FCbrmod5}.
\section{A Theta-Function Identity} \label{sec:thetaid}
In this section we will prove the following theta-function identity.
\begin{equation}
U(z,q) = F_0(q)\, S_0(z,q) + F_1(q)\, S_1(z,q) + F_2(q)\, S_2(z,q)
+ F_3(q)\, S_3(z,q) + F_4(q)\, S_4(z,q),
\mylabel{eq:Uid}
\end{equation}
where
\begin{align}
F_0(q) &= W(q^{10})\,(1 + q^{2}\,R(q^{10}) + (\zeta^2+\zeta^3)\,q^{4}\,R(q^{10})^2 -
(\zeta^2+\zeta^3)\,q^{6}\,R(q^{10})^3),\mylabel{eq:f0def}\\
F_1(q) &= W(q^{10})\,(\zeta^4 + \zeta\,q^{2}\,R(q^{10}) + (1+\zeta)\,q^{4},R(q^{10})^2 -
(\zeta^2+\zeta^3)\,q^{6}\,R(q^{10})^3),\mylabel{eq:f1def}\\
F_2(q) &= W(q^{10})\,(1 + \zeta^4\,q^{2}\,R(q^{10}) + (1+\zeta)\,q^{4}\,R(q^{10})^2 -
(1+\zeta^4)\,q^{6}\,R(q^{10})^3),\mylabel{eq:f2def}\\
F_3(q) &= W(q^{10})\,(1 + \zeta\,q^{2}\,R(q^{10}) + (1+\zeta^4)\,q^{4}\,R(q^{10})^2 -
(1+\zeta)\,q^{6}\,R(q^{10})^3),\mylabel{eq:f3def}\\
F_4(q) &= W(q^{10})\,(\zeta + \zeta^4\,q^{2}\,R(q^{10}) + (1+\zeta^4)\,q^{4}\,R(q^{10})^2 -
(\zeta^2+\zeta^3)\,q^{6}\,R(q^{10})^3),\mylabel{eq:f4def}
\end{align}
\begin{align}
S_0(z,q) &= \sum_{n=-\infty}^{\infty} z^{5n}q^{25n^2+20n},
\mylabel{eq:S0def} \\
S_1(z,q) &= \sum_{n=-\infty}^{\infty} z^{5n+1}q^{25n^2+30n+5},
\mylabel{eq:S1def} \\
S_2(z,q) &= \sum_{n=-\infty}^{\infty} z^{5n+2}q^{25n^2+40n+12},
\mylabel{eq:S2def} \\
S_3(z,q) &= \sum_{n=-\infty}^{\infty} z^{5n+3}q^{25n^2+50n+21},
\mylabel{eq:S3def} \\
S_4(z,q) &= \sum_{n=-\infty}^{\infty} z^{5n+4}q^{25n^2+60n+32},
\mylabel{eq:S4def}
\end{align}
and
\begin{equation}
U(z,q)
= \sum_{\substack{\vec{n}\in\Z^5}}
z^{\vec{n}\cdot\vec{1}}
\zeta^{n_1 + 3n_2 + n_3}
q^{5||\vec{n}||^2 + 2\vec{b}\cdot\vec{n}},
\mylabel{eq:Udef}
\end{equation}
where $W(q)$ and $R(q)$ defined in \eqn{Wdef} and \eqn{Rdef} respectively,
and the vectors
$\vec{n}=(n_0,n_1,n_2,n_3,n_4$, $\vec{1}=(1,1,1,1,1)$
and $\vec{b}=(0,1,2,3,4)$ as before.
We note that \eqn{T5dissect} follows from \eqn{Uid} by taking the
coefficient of $z^0$ and replacing $q$ by $q^{1/2}$. Equation
\eqn{T5dissect} was the crucial identity needed in the proof of
Theorem \thm{thm3}.
We prove the identity \eqn{Uid} using standard techniques.
We show that both sides satisfy the same functional equation and
both sides agree for enough values of the parameter $z$.
Most of these evaluations can be proved by elementary means using
Jacobi's triple product. For one evaluation we will need the theory
of modular functions.
We define the following Jacobi theta function
\begin{equation}
\Theta(z,q) = \sum_{n=-\infty}^\infty z^n q^{n^2},
\mylabel{eq:Thetadef}
\end{equation}
for $z\ne0$ and $\abs{q}<1$. We will need Jacobi's triple product
identity
\begin{equation}
\sum_{n=-\infty}^\infty z^n q^{n^2}
=(-zq;q^2)_\infty (-z^{-1}q;q^2)_\infty (q^2;q^2)_\infty,
\mylabel{eq:jtprod}
\end{equation}
and the well-known functional equation
\begin{equation}
\Theta(zq^2,q) = z^{-1} q^{-1} \Theta(z,q),
\mylabel{eq:Thetafid}
\end{equation}
for $z\ne0$ and $0<\abs{q}<1$.
From the definition \eqn{Udef} we have
\begin{equation}
U(z,q) = \Theta(z\zeta^4,q^5) \, \Theta(zq^2\zeta,q^5) \, \Theta(zq^4,q^5)
\, \Theta(z\zeta q^6,q^5) \, \Theta(z\zeta^4 q^8,q^5).
\mylabel{eq:Uprodid}
\end{equation}
From \eqn{jtprod} and \eqn{Thetafid} we have
\begin{equation}
U(z q^{10},q) = z^{-5} q^{-45} U(z,q),
\mylabel{eq:Ufid}
\end{equation}
and
\begin{equation}
U(z,q) = 0\quad\mbox{for}\quad
z = -q^5\,\zeta,\quad -q^3\,\zeta^4, -q, -\zeta^4\,q^9, -\zeta\,q^7.
\mylabel{eq:Uzeros}
\end{equation}
Let $V(z,q)$ denote the function of the right side of \eqn{Uid}.
Each $S_j(z,q)$ can be written in terms of the theta function
$\Theta(z,q)$ and we find that
$S_j(z q^{10},q) = z^{-5} q^{-45} S_j(z,q)$ for each $j$ so that
\begin{equation}
V(z q^{10},q) = z^{-5} q^{-45} V(z,q).
\mylabel{eq:Vfid}
\end{equation}
Hence the left and right sides of \eqn{Uid} satisfy the same
functional equation (i.e.\ \eqn{Ufid}, \eqn{Vfid}).
In view of \cite[Lemma 2]{At-Sw} or \cite[Lemma 1]{Hi-Ga-Bo},
it suffices to show that \eqn{Uid} holds for $6$ distinct
values of $z$ with $\abs{q}^{10} < \abs{z} \le 1$.
We claim that
\begin{equation}
V(z,q) = 0\quad\mbox{for}\quad
z = -q^5\,\zeta,\quad -q^3\,\zeta^4, -q, -\zeta^4\,q^9, -\zeta\,q^7.
\mylabel{eq:Vzeros}
\end{equation}
Using \eqn{jtprod} we can easily evaluate each $S_j(z,q)$ for
these values of $z$.
\begin{align}
S_0(-\zeta q^5,q) &= -q^{-20}J_{2,5}(q^{10}),
\mylabel{eq:Sv1}\\
S_1(-\zeta q^5,q) &= \zeta q^{-20} J_{2,5}(q^{10}),
\mylabel{eq:Sv2}\\
S_2(-\zeta q^5,q) &= -\zeta^2 q^{-18} J_{1,5}(q^{10}),
\mylabel{eq:Sv3}\\
S_3(-\zeta q^5,q) &= 0,
\mylabel{eq:Sv4}\\
S_4(-\zeta q^5,q) &= \zeta^4 q^{-18} J_{1,5}(q^{10}),
\mylabel{eq:Sv5}\\
\noalign{\medskip}
S_0(-q^3\zeta^4,q) &= -q^{-10}J_{1,5}(q^{10}),
\mylabel{eq:Sv6}\\
S_1(-q^3\zeta^4,q) &= \zeta^4 q^{-12} J_{2,5}(q^{10}),
\mylabel{eq:Sv7}\\
S_2(-q^3\zeta^4,q) &= -\zeta^3 q^{-12} J_{2,5}(q^{10}),
\mylabel{eq:Sv8}\\
S_3(-q^3\zeta^4,q) &= \zeta^2 q^{-10} J_{1,5}(q^{10}),
\mylabel{eq:Sv9}\\
S_4(-q^3\zeta^4,q) &= 0 ,
\mylabel{eq:Sv10}\\
\noalign{\medskip}
S_0(-q,q) &= 0,
\mylabel{eq:Sv11}\\
S_1(-q,q) &= q^{-4}J_{1,5}(q^{10}),
\mylabel{eq:Sv12}\\
S_2(-q,q) &= -q^{-6}J_{2,5}(q^{10}),
\mylabel{eq:Sv13}\\
S_3(-q,q) &= q^{-6}J_{2,5}(q^{10}),
\mylabel{eq:Sv14}\\
S_4(-q,q) &= -q^{-4}J_{1,5}(q^{10}),
\mylabel{eq:Sv15}\\
\noalign{\medskip}
S_0(-\zeta^4q^9,q) &= -q^{-40}J_{1,5}(q^{10}),
\mylabel{eq:Sv16}\\
S_1(-\zeta^4q^9,q) &= 0
\mylabel{eq:Sv17}\\
S_2(-\zeta^4q^9,q) &= \zeta^3q^{-40}J_{1,5}(q^{10}),
\mylabel{eq:Sv18}\\
S_3(-\zeta^4q^9,q) &= -\zeta^2q^{-42}J_{2,5}(q^{10}),
\mylabel{eq:Sv19}\\
S_4(-\zeta^4q^9,q) &= \zeta q^{-42}J_{2,5}(q^{10}),
\mylabel{eq:Sv20}\\
\noalign{\medskip}
S_0(-\zeta q^7,q) &= -q^{-30}J_{2,5}(q^{10}),
\mylabel{eq:Sv21}\\
S_1(-\zeta q^7,q) &= \zeta q^{-28}J_{1,5}(q^{10}),
\mylabel{eq:Sv22}\\
S_2(-\zeta q^7,q) &= 0,
\mylabel{eq:Sv23}\\
S_3(-\zeta q^7,q) &= -\zeta^3q^{-28}J_{1,5}(q^{10}),
\mylabel{eq:Sv24}\\
S_4(-\zeta q^7,q) &= \zeta^4q^{-30}J_{2,5}(q^{10}).
\mylabel{eq:Sv25}
\end{align}
The verification of \eqn{Vzeros} is just a routine calculation.
Thus both sides of \eqn{Uid} agree for $5$ distinct values of $z$
in the region $\abs{q}^{10} < \abs{z} \le 1$.
We show that both sides agree for $z=-1$, and then our
identity \eqn{Uid} will follow. To achieve this we use the theory
of modular functions. Since this is a standard technique we just
sketch some of the details.
First, we calculate the $5$-dissection of each theta function on
the right side of \eqn{Uprodid} when $z=-1$. By \eqn{jtprod} we find
that
\begin{align}
\Theta(-\zeta^4,q^5) &= J_{1,2}(q^{50})+(1+\zeta^2+\zeta^3)q^{5}J_{3,10}(q^{25})
+(\zeta^2+\zeta^3)q^{20}J_{1,10}(q^{25}),
\mylabel{eq:Peval1}\\
\Theta(-\zeta q^2,q^5)&= J_{27,50}(q^{5})+\zeta^3q^{16}J_{7,50}(q^{5})
-\zeta q^{7}J_{13,50}(q^{5})-\zeta^4q^{3}J_{17,50}(q^{5})
+q^{24}\zeta^2J_{7,50}(q^{5}),
\mylabel{eq:Peval2}\\
\Theta(-q^4,q^5) &= J_{43,50}(q^{5})-q\,J_{19,50}(q^{5})+q^{12}J_{9,50}(q^{5})
+q^{28}J_{1,50}(q^{5}) -q^{9}J_{11,50}(q^{5}),
\mylabel{eq:Peval3}\\
q\,\Theta(-\zeta q^6,q^5)&= -\zeta^4J_{21,50}(q^{5})+q\,J_{19,50}(q^{5})
-\zeta q^{12}J_{9,50}(q^{5})
-\zeta^2q^{28}J_{1,50}(q^{5}) +\zeta^3q^{9}J_{23,50}(q^{5}),
\mylabel{eq:Peval4}\\
q^{3} \Theta(-\zeta^4 q^8,q^5)
&=-\zeta J_{23,50}(q^{5})-\zeta^4q^{16}J_{7,50}(q^{5})
+\zeta^2q^{7}J_{27,50}(q^{5}) +q^{3}J_{17,50}(q^{5})
-\zeta^3q^{24}J_{3,50}(q^{5}).
\mylabel{eq:Peval5}
\end{align}
Next, we evaluate each $S_j(-1,q)$ using \eqn{jtprod}
\begin{align}
S_0(-1,q) &= S_1(-1,q) = J_{1,10}(q^{5}),\mylabel{eq:Seval1}\\
S_2(-1,q) &= S_4(-1,q) = -q^{-3}J_{3,10}(q^{5}),\mylabel{eq:Seval2}\\
S_3(-1,q) &= q^{-4}J_{1,2}(q^{25}).\mylabel{eq:Seval3}
\end{align}
For $0\le r \le 4$, we define the operator $\mathcal{U}_{r,5}$ by
\begin{equation}
\mathcal{U}_{r,5}\left(\sum_{n} a(n) q^n \right)
= \sum_{n} a(5n+r) q^n.
\mylabel{eq:Uop}
\end{equation}
To show that \eqn{Uid} holds for $z=-1$ we need to prove $5$ identities
\begin{equation}
\mathcal{U}_{r,5}\left(q^4\,U(-1,q)\right)
=
\mathcal{U}_{r,5}\left(q^4\,V(-1,q)\right),
\mylabel{eq:UVrid}
\end{equation}
for $0\le r \le 4$. It turns out that each of these identities is equivalent
to a modular function identity for the group $\Gamma_1(50)$. We provide
some detail for the case $r=0$. Using \eqn{Seval1}--\eqn{Seval3} we
find that
\begin{align}
\mathcal{U}_{0,5}\left(q^4\,V(-1,q)\right)
&=
\left(J_{2,5}(q^{10}) - q^2 (1 + \zeta^2 + \zeta^3) J_{1,5}(q^{10})\right)
\mylabel{eq:U0Vid}\\
&\times \left( J_{1,2}(q^5) J_{4,10}^3(q)
+ q (-1 + \zeta^2 + \zeta^3) J_{2,10}^2(q) J_{3,10}(q) J_{4,10}(q)\right. \nonumber\\
&\hphantom{XXXXXXXXXXXXXX}\left. - 2 q^2 (\zeta^2+\zeta^3) J_{2,10}^3 J_{1,10}(q) \right).
\nonumber
\end{align}
We can utilize \eqn{Peval1}--\eqn{Peval5} to write the left side
of the $r=0$ case of \eqn{UVrid} as a sum of $135$ explicit theta products
\begin{align}
&\mathcal{U}_{0,5}\left(q^4\,U(-1,q)\right) \mylabel{eq:U0Uid}\\
&\quad =
q^{10} J_{3,50}^2(q) J_{19,50}^2(q) J_{25,50}(q) + \cdots
+ 2 q^{10} (\zeta^2 + \zeta^3) J_{1,50}(q) J_{9,50}(q) J_{13,50}(q) J_{17,50}(q)
J_{25,50}(q).
\nonumber
\end{align}
We have to prove that the right side of \eqn{U0Vid} equals the right side of \eqn{U0Vid}.
After dividing both sides by $J_{2,5}(q^{10}) J_{1,2}(q^5) J_{4,10}^3(q)$ we find that this
is equivalent to showing that a certain linear combination of $140$ generalized
eta-quotients
simplifies to the constant $1$
\begin{equation}
(1+\zeta^2+\zeta^3)\,\eta_{50,10} \, \eta_{50,20}^{-1}
+ \cdots
\eta_{50,4}^{-3} \, \eta_{50,5}^{-2} \, \eta_{50,6}^{-3} \, \eta_{50,10}^{-4} \,
\eta_{50,14}^{-3} \, \eta_{50,15}^{-2} \, \eta_{50,16}^{-3} \, \eta_{50,20}^{-5} \,
\eta_{50,24}^{-3} \, \eta_{50,23}^{2} \, \eta_{50,21}^{2} = 1.
\mylabel{eq:bigcombo}
\end{equation}
Here
\begin{equation}
\eta_{n,m} = \eta_{n,m}(\tau)
= \exp(\pi i P_2(m/n) n\tau)\,
\prod_{k\equiv\pm m\pmod{n}}(1 - \exp(2\pi ik\tau)
= q^{n P_2(m/n)/2} J_{m,n}(q),
\mylabel{eq:getadef}
\end{equation}
where $P_2(t) = \{t\}^2 - \{t\} + \tfrac{1}{6}$, and $q=\exp(2\pi i\tau)$.
Using \cite[Theorem 2.9, p.7]{Choi06}, \cite[Theorem 3, p.126]{Ro94} that
each generalised eta-quotient in \eqn{bigcombo} is indeed a modular
function on $\Gamma_1(50)$. As usual we need the valence formula
\begin{equation}
\sum_{z\in\mathcal{F}} \ORD(f;z,\Gamma) = 0,
\mylabel{eq:valform}
\end{equation}
provided $f$ is a nontrivial modular function on $\Gamma$, and
$\mathcal{F}$ is a fundamental set for $\Gamma$. Using MAGMA, the following
is a complete set of inequivalent cusps for $\Gamma_1(50)$
\begin{align}
\mathcal{C}
&=\{\infty, \tfrac{0}{1},\, \tfrac{1}{10},\, \tfrac{1}{9},\, \tfrac{2}{17},\, \tfrac{3}{25},\,
\tfrac{1}{8},\, \tfrac{2}{15},\, \tfrac{3}{22},\, \tfrac{4}{29},\, \tfrac{5}{36},\,
\tfrac{6}{43},\, \tfrac{7}{50},\, \tfrac{3}{20},\, \tfrac{2}{13},\, \tfrac{3}{19},\,
\tfrac{4}{25},\, \tfrac{1}{6},\, \tfrac{6}{35},\, \tfrac{9}{52},\, \tfrac{13}{75},\,
\tfrac{4}{23},\, \tfrac{7}{39},\, \tfrac{9}{50},\, \tfrac{7}{38},\, \tfrac{3}{16},\,
\mylabel{eq:CG}\\
&\qquad\tfrac{5}{26},\, \tfrac{1}{5},\, \tfrac{27}{125},\, \tfrac{7}{32},\, \tfrac{11}{50},\,
\tfrac{12}{53},\, \tfrac{17}{75},\, \tfrac{6}{25},\, \tfrac{1}{4},\, \tfrac{13}{50},\,
\tfrac{53}{200},\, \tfrac{67}{250},\, \tfrac{27}{100},\, \tfrac{11}{40},\, \tfrac{18}{65},\,
\tfrac{7}{25},\, \tfrac{3}{10},\, \tfrac{7}{20},\, \tfrac{9}{25},\, \tfrac{11}{30},\,
\tfrac{19}{50},\, \tfrac{49}{125},\, \tfrac{2}{5},\, \tfrac{21}{50},\, \nonumber\\
&\qquad \tfrac{11}{25},\,
\tfrac{11}{20},\, \tfrac{26}{45},\, \tfrac{3}{5},\, \tfrac{59}{85},\, \tfrac{7}{10}\},
\nonumber
\end{align}
with corresponding widths
\begin{align}
&\{
1,\, 50,\, 5,\, 50,\, 50,\, 2,\, 25,\, 10,\, 25,\, 50,\, 25,\, 50,\, 1,\, 5,\, 50,\, 50,\, 2,\, 25,\, 10,\, 25,\, 2,\, 50,\, 50,\, 1,\, 25,\,
\mylabel{eq:WG}\\
&\qquad 25,\, 25,\, 10,\, 2,\, 25,\, 1,\, 50,\, 2,\, 2,\, 25,\, 1,\, 1,\, 1,\, 1,\, 5,\, 10,\, 2,\, 5,\, 5,\, 2,\, 5,\, 1,\, 2,\, 10,\, 1,\, 2,\, 5,\,
\nonumber\\
&\qquad 10,\, 10,\, 10,\, 5\}.
\nonumber
\end{align}
Using known results for the invariant order of generalized eta-quotients
at cusps \cite[(2.3), p.7]{Choi06}, \cite[pp.127-128]{Ro94} we have calculated the
order at each cusp of every function in \eqn{bigcombo}. As check we verified
that the total Order of each function is zero. With $\mathcal{J}$ being
the set generalized eta-quotients ocurring in \eqn{bigcombo} we calculated
\begin{equation}
\sum_{c\in\mathcal{C}\setminus\{\infty\}}
\min_{ f\in \mathcal{J}}(\ORD(f;c;\Gamma_1(50)),0) = -145.
\mylabel{eq:minords}
\end{equation}
Hence, by the valence formula \eqn{valform} it suffices to verify
\eqn{bigcombo} (or equivalently \eqn{UVrid} with $r=0$)
up to $q^{145}$, since generalized eta-quotients have no poles or zeros in the upper-half
plane. We have actually verified
the result up to $q^{200}$. All calculations, except for \eqn{CG},and \eqn{WG},
were done using MAPLE. The calculations needed to verify
\eqn{UVrid} for $r=1$, $2$, $3$, $4$ are similiar and have been carried out.
This conpletes our proof of \eqn{Uid}.
\section{The Andrews Bicrank and Extensions} \label{sec:bicrank}
For a partition $\pi$, let $\ell(\pi)$ denote the largest part of $\pi$,
$\varpi(\pi)$ denote the number of ones in $\pi$,
and $\mu(\pi)$ denote the number of parts of $\pi$ larger than $\varpi(\pi)$.
The crank of $\pi$ is given by
\begin{equation}
\mbox{crank}(\pi) =
\begin{cases}
\ell(\pi), &\mbox{if $\varpi(\pi)=0$}, \\
\mu(\pi) - \varpi(\pi), &\mbox{if $\varpi(\pi)>0$}.
\end{cases}
\mylabel{eq:crankdef}
\end{equation}
The crank gives a combinatorial interpretation
of Ramanujan's partition congruences mod $5$, $7$ and $11$ and solves a
problem of Dyson \cite{Dy44}, \cite[p.52]{Dy96}. See \cite{An-Ga88}.
In \cite{An08}, Andrews gave a combinatorial interpretation of the
congruence
\begin{equation}
p_{-2}(5n+3) \equiv 0 \pmod{5},
\mylabel{eq:p25n3}
\end{equation}
in terms of the crank. This result is a crank analog of the Dyson-birank
but is more complicated since it on involves positive and negative weights.
This complication is because of the nature of the generating function
for the crank. Let $M(m,n)$ denote the number of partitions of $n$ with
crank $m$. Then
\begin{equation}
\sum_{n\ge0} \sum_m M(m,n) z^m q^n = (1-z)q +
\frac{(q;q)_\infty}{(zq;q)_\infty (z^{-1}q;q)_\infty}.
\mylabel{eq:Mgf}
\end{equation}
Define $M'(m,n)$ by
\begin{equation}
\sum_{n\ge0} \sum_m M'(m,n) z^m q^n =
\frac{(q;q)_\infty}{(zq;q)_\infty (z^{-1}q;q)_\infty}
= 1 + (z -1 + z^{-1})q + (z^2 + z^{-2})q^2 + \cdots.
\mylabel{eq:Mvgf}
\end{equation}
We need to interpret $M'(m,n)$ combinatorially. To this we need to
definition of partition.
To the set of partitions we need to add two additional partitions of $1$ which we denote
by $1_a$ and $1_b$. We call this new set $\mathcal{E}$, the set of extended partitions.
\begin{equation}
\mathcal{E} = \{(-), 1_a, 1_b, 1, 2, 1+1, 3, 2+1, 1+1+1, \cdots\}.
\mylabel{eq:eptns}
\end{equation}
We have $\abs{1_a}=\abs{1_b}=1$.
Here as usual $(-)$ is the empty partition of $0$. For these extended partitions
define a weight function $w(\pi)$ defined by
\begin{equation}
w(\pi) =
\begin{cases}
-1, &\mbox{if $\pi=1_b$}.\\
\abs{\pi}, &\mbox{otherwise}.
\end{cases}
\mylabel{eq:wdef}
\end{equation}
Thus for the three extended partitions of $1$ we have
$w(1)=w(1_a)=1$, and $w(1_b)=-1$, and the total weight is still $p(1)=1$.
Therefore
\begin{equation}
\sum_{\substack{\pi\in\mathcal{E}\\ \abs{\pi}=n}} w(\pi) = p(n).
\mylabel{eq:totw}
\end{equation}
We also extend the definition of crank by $\mbox{crank}(1_a)=1$,
and $\mbox{crank}(1_b)=0$. Recall that for ordinary partition of $1$ we have
$\mbox{crank}(1_b)=-1$.
We now have our desired combinatorial interpretation of $M'(m,n)$.
\begin{equation}
F(z,q) = \sum_{\pi\in\mathcal{E}} w(\pi) z^{\mbox{crank}(\pi)} q^{\abs{\pi}}
= \sum_{n\ge0} \sum_m M'(m,n) z^m q^n =
\frac{(q;q)_\infty}{(zq;q)_\infty (z^{-1}q;q)_\infty}.
\mylabel{eq:Mvid}
\end{equation}
In other words,
\begin{equation}
M'(m,n) =
\sum_{\substack{\pi\in\mathcal{E}\\ \abs{\pi}=n,\, \mbox{crank}(\pi)=m}} w(\pi).
\mylabel{eq:Mvid2}
\end{equation}
We note that the function $F(z,q)$ (at least as an infinite product)
occured in Ramanujan's Lost Notebook.
We define the set of extended bipartitions by $\mathcal{E} \times \mathcal{E}$,i.e.\ an
extended bipartition is simply a pair of extended partitions. For
an extended bipartition $\pi=(\pi_1,\pi_2)$ we define
a sum of parts function and a weight function in the natural way
\begin{equation}
\abs{\pi} = \abs{\pi_1} + \abs{\pi_2},\quad\mbox{and}\qquad
w(\pi) = w(\pi_1) \, w(\pi_2).
\mylabel{eq:ewdef}
\end{equation}
We denote Andrews's bicrank function by $\mbox{bicrank}_1$.
We give a variant which we call $\mbox{bicrank}_2$.
For an extended bipartition $\pi=(\pi_1,\pi_2)$ we define
\begin{align}
\mbox{bicrank}_1(\pi) &= \mbox{crank}(\pi_1) + \mbox{crank}(\pi_2),
\mylabel{eq:bic1}\\
\mbox{bicrank}_2(\pi) &= \mbox{crank}(\pi_1) + 2\,\mbox{crank}(\pi_2),
\mylabel{eq:bic2}
\end{align}
Amazingly together these two bicrank functions give a
new interpretation for all three congruences in \eqn{2colcongs}.
For $j=1$, $2$ we define $M_j(m,t,n)$ by
\begin{equation}
M_j(m,t,n) =
\sum_{\substack{\pi\in\mathcal{E}\times\mathcal{E}\\
\abs{\pi}=n,\, \mbox{bicrank}_j(\pi)\equiv m\pmod{t}}} w(\pi).
\mylabel{eq:Mjdef}
\end{equation}
In other words, $M_j(m,t,n)$ is the number of extended bipartitions of $n$
with $\mbox{bicrank}_j$ congruent to $m$ mod $t$ counted by the weight $w$.
In this section we prove
\begin{theorem}
\label{thm:thm4}
$$
\hphantom{x}
$$
\begin{enumerate}
\item[(i)]
The residue of the $\mbox{bicrank}_1(\pi)$ mod $5$
divides the extended bipartitions of $n$ into $5$ classes of equal weight
provided $n\equiv3\pmod{5}$.
\item[(ii)]
The residue of the $\mbox{bicrank}_2(\pi)$ mod $5$
divides the extended bipartitions of $n$ into $5$ classes of equal weight
provided $n\equiv2$ or $4\pmod{5}$.
\end{enumerate}
\end{theorem}
We illustrate the first case of Theorem \thm{thm4} (i).
There are $18$ extended bipartitions of $3$ giving a total weight of $p_{-2}(3)=10$.
\begin{equation*}
\begin{array}{rrr}
\mbox{Extended bipartitions of $3$} & \mbox{bicrank}_1 \pmod{5} & \mbox{weight}=w \\
(2+1,-) & 0 \equiv 0 & 1\\
(-,2+1) & 0+0 \equiv 0 & 1\\
(2,1) & 2-1 \equiv 1 & 1\\
(1,2) &-1+2 \equiv 1 & 1\\
(1+1+1,-)& -3 \equiv 2 & 1\\
(2,1_b) & 2+0 \equiv 2 & -1\\
(1+1,1) & -2-1 \equiv 2 & 1\\
(1_b,2) & 0+2 \equiv 2 & -1\\
(1,1+1) &-1-2 \equiv 2 & 1\\
(-,1+1+1) &0-3 \equiv 2 & 1\\
(3,-) & 3 \equiv 3 & 1\\
(2,1_a) & 2+1 \equiv 3 & 1\\
(1+1,1_b)&-2+0 \equiv 3 & -1\\
(1_a,2) & 1+2 \equiv 3 & 1\\
(1_b,1+1)& 0-2 \equiv 3 & -1\\
(-,3) & 0+3 \equiv 3 & 1\\
(1+1,1_a)& -2+1 \equiv 4 & 1\\
(1_a,1+1)& 1-2 \equiv 4 & 1\\
\end{array}
\end{equation*}
Thus
$$
M_1(0,5,3) = M_1(1,5,3) = M_1(2,5,3) = M_1(3,5,3) =
M_1(4,5,3) = 2,
$$
and we see that the residue of the $\mbox{bicrank}_1$ mod $5$ divides the $18$
bipartitions of $3$ into $5$ classes of equal total weight $2$.
We illustrate the first case of Theorem \thm{thm4} (ii).
There are $13$ extended bipartitions of $2$ giving a total weight of $p_{-2}(2)=5$.
\begin{equation*}
\begin{array}{rrr}
\mbox{Extended bipartitions of $2$} & \mbox{bicrank}_2 \pmod{5} & \mbox{weight}=w \\
(2,-) & 2+0 \equiv 2 & 1\\
(1+1,-) &-2+0 \equiv 3 & 1\\
(1,1) & -1-2 \equiv 2 & 1\\
(1,1_a) & -1+2 \equiv 1 & 1\\
(1,1_b) & -1+0 \equiv 4 &-1\\
(1_a,1) & 1-2 \equiv 4 & 1\\
(1_a,1_a)& 1+2 \equiv 3 & 1\\
(1_a,1_b) & 1+0 \equiv 1 &-1\\
(1_b,1) & 0-2 \equiv 3 &-1\\
(1_b,1_a) & 0+2 \equiv 2 &-1\\
(1_b,1_b) & 0+0 \equiv 0 & 1\\
(-,2) & 0+4 \equiv 4 & 1\\
(-,1+1) & 0-4 \equiv 1 & 1\\
\end{array}
\end{equation*}
Thus
$$
M_2(0,5,2) = M_2(1,5,2) = M_2(2,5,2) = M_2(3,5,2) =
M_2(4,5,2) = 1,
$$
and we see that the residue of the $\mbox{bicrank}_2$ mod $5$ divides the $13$
bipartitions of $2$ into $5$ classes of equal total weight $1$.
Theorem \thm{thm4} (i) is due to Andrews \cite{An08}.
Theorem \thm{thm4} (ii) is a natural extension, and its proof is analogous.
For completeness we include a sketch of the proof. We define
\begin{equation}
M'(r,t,n) = \sum_{m\equiv r\pmod{t}} M'(m,n),
\end{equation}
which is the number of ordinary partitions of $n$ with crank congruent
to $r$ mod $t$ when $n\ne1$. When $n=1$ it is counting extended partitions.
Then
\begin{align}
F(\zeta,q)=&\sum_{n=0}^\infty
\sum_{k=0}^4 \zeta^k M'(k,5,n) q^n
= \frac{(q;q)_\infty}{(\zeta q;q)_\infty (\zeta^{-1}q;q)_\infty}
\mylabel{eq:Fdissect}\\
&= A(q^5)
- q (\zeta + \zeta^4)^2 \,B(q^5)
+ q^2 (\zeta^2 + \zeta^3)\, C(q^5) -q^3 (\zeta+\zeta^4)\, D(q^5),
\nonumber
\end{align}
where
$F(z,q)$ is given in \eqn{Mvid}, $A(q)$, $B(q)$, $C(q)$, and $D(q)$ are
given in \eqn{Adef}--\eqn{Ddef}. Equation \eqn{Fdissect} appears
in Ramanujan's Lost Notebook \cite[p.20]{Ra88} and is
proved in \cite[(1.30)]{Ga88}.
The two bicrank generating functions are given by
\begin{align}
\sum_{\bpi=(\pi_1,\pi_2)} z^{\bcra} q^{\abs{\bpi}} &= F(z,q)^2,
\mylabel{eq:bcr1genfunc}\\
\sum_{\bpi=(\pi_1,\pi_2)} z^{\bcrb} q^{\abs{\bpi}} &= F(z,q) \, F(z^2,q),
\mylabel{eq:bcr2genfunc}
\end{align}
where $F(z,q)$ is the generating function for the crank of
extended partitions \eqn{Mvid}.
Thus we have
\begin{align}
\sum_{n=0}^\infty \sum_{k=0}^4 \zeta^k M_{1}(k,5,n) q^n
&= F(\zeta,q)^2,
\mylabel{eq:bcr1mod5}\\
\sum_{n=0}^\infty \sum_{k=0}^4 \zeta^k M_{1}(k,5,n) q^n
&= F(\zeta,q)\, F(\zeta^2,q).
\mylabel{eq:bcr2mod5}
\end{align}
Using only \eqn{Fdissect} and equations
\eqn{ABCDfact} and \eqn{ABCDfact2}
we easily find that
find that the coefficient of $q^n$ in the $q$-expansion of
$F(\zeta,q)^2$ is zero if
$n\equiv3\pmod{5}$, ant that the
the coefficient of $q^n$ in the $q$-expansion of
$F(\zeta,q)\, F(\zeta^2,q)$ is zero if
$n\equiv2$, or $4\pmod{5}$. Both parts of Theorem \thm{thm4} then follow
from equations \eqn{bcr1mod5} and \eqn{bcr2mod5}.
\section{A Multirank Analog of the Hammond-Lewis Birank} \label{sec:ghlbirank}
Let $\mathcal{P}$ denote the set of partitions. A multipartition
with $r$ components or an $r$-colored partition of $n$ is simply an
$r$-tuple
\begin{equation}
\vec{\pi}=(\pi_1,\pi_2,\dots,\pi_r) \in
\mathcal{P} \times \mathcal{P} \times \cdots \times \mathcal{P}
=\mathcal{P}^r,
\end{equation}
where
\begin{equation}
\sum_{k=1}^r \abs{\pi_k}=n.
\end{equation}
It is clear that the number of $r$-colored partitions of $n$ is
$p_{-r}(n)$ where
\begin{equation}
\sum_{n\ge0} p_{-r}(n) q^n = \frac{1}{E(q)^r}.
\end{equation}
There are two elementary and well-known congruences.
\begin{theorem}
\label{thm:thm5}
Let $t>3$ be prime.
\begin{enumerate}
\item[(i)] If
$24n+1$ is a quadratic nonresidue mod $t$, then
then
\begin{equation}
p_{1-t}(n) \equiv 0 \pmod{t}.
\end{equation}
\item[(ii)] If
$8n+1$ is not a quadratic residue mod $t$,
then
\begin{equation}
p_{3-t}(n) \equiv 0 \pmod{t}.
\end{equation}
\end{enumerate}
\end{theorem}
These results follow easily from identities of Euler and Jacobi.
Theorem \thm{thm5} (i) follows from
\begin{equation}
\sum_{n\ge0} p_{1-t}(n) q^{24n+1} = \frac{qE(q^{24})}{E(q^{24})^t}
\equiv \frac{1}{E(q^{24t})} \sum_{n=-\infty}^\infty (-1)^n q^{(6n+1)^2}
\pmod{t}.
\end{equation}
Here we have used Euler's Pentagonal Number Theorem \cite[Thm 353]{Ha-Wr-BOOK}
\begin{equation}
E(q) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n+1)/2}.
\mylabel{eq:Epent}
\end{equation}
Theorem \thm{thm5} (ii) follows from
\begin{equation}
\sum_{n\ge0} p_{3-t}(n) q^{8n+1} = \frac{qE^3(q^{8})}{E(q^{8})^t}
\equiv \frac{1}{E(q^{8t})} \sum_{n\ge0}(-1)^n (2n+1) q^{(2n+1)^2}
\pmod{t},
\end{equation}
where we have used Jacobi's Identity \cite[Thm 237]{Ha-Wr-BOOK}
\begin{equation}
E(q)^3 = \sum_{n\ge0} (-1)^n (2n+1) q^{n(n+1)/2}.
\mylabel{eq:jacid}
\end{equation}
Theorem \thm{thm6} (ii) is Theorem 1 in \cite{An08}.
In this section we construct analogs of the Hammond-Lewis birank to
combinatorially explain the two congruences in Theorem \thm{thm5}.
Andrews's bicrank \cite{An08} (see also equation \eqn{bic1}) gave
a combinatorial interpretation of Theorem \thm{thm5} (ii) for the
case $t=5$, and $n\equiv 3\pmod{5}$. The Hammond-Lewis birank gave
a combinatorial interpretation of Theorem \thm{thm5} (ii) for
the case $t=5$, and all relelvant $n$.
For even $r$, we define
the generalized Hammond-Lewis multirank by
\begin{equation}
\mbox{gHL-multirank}(\vec{\pi})
= \sum_{k=1}^{r/2} k\,\left(\#(\pi_k) - \#(\pi_{r+1-k})\right),
\mylabel{eq:gHLmrankdef}
\end{equation}
for $\vec{\pi}=(\pi_1,\pi_2,\dots,\pi_{r})$ a multipartition with
$r$ components. The $r=2$ case corresponds to the Hammond-Lewis birank.
In this section we prove
\begin{theorem}
\label{thm:thm6}
Let $t>3$ be prime.
\begin{enumerate}
\item[(i)]
The residue of the generalized-Hammond-Lewis-multirank mod $t$
divides the multipartitions of $n$ with $r=t-1$ components into $t$ equal classes
provided $24n+1$ is a quadratic nonresidue mod $t$.
\item[(ii)]
The residue of the generalized-Hammond-Lewis-multirank mod $t$
divides the multipartitions of $n$ with $r=t-3$ components into $t$ equal classes
provided $8n+1$ is not a quadratic residue mod $t$.
\end{enumerate}
\end{theorem}
We illustrate Theorem \thm{thm6} (ii) for $t=7$ and $n=2$.
\begin{equation*}
\begin{array}{rr}
\mbox{Multipartitions of $2$} & \mbox{generalized-HL-multirank}\\
\mbox{with $4$ components} & \mbox{(mod $7$)}\\
(-, -, -, 1+1) & -2\equiv5\\
( -, -, -, 2) & -1\equiv6\\
( -, -, 1, 1) & -3\equiv4\\
(-, -, 1+1,-) & -4\equiv3\\
( -, -, 2,-) & -2\equiv5\\
( -, 1, -, 1) & 1\equiv1\\
( -, 1, 1,-) & 0\equiv0\\
(-, 1+1, -,-) & 4\equiv4\\
( -, 2, -,-) & 2\equiv2\\
( 1, -, -, 1) & 0\equiv0\\
( 1, -, 1,-) & -1\equiv6\\
( 1, 1, -,-) & 3\equiv3\\
(1+1, -, -,-) & 2\equiv2\\
( 2, -, -,-) & 1\equiv1\\
\end{array}
\end{equation*}
We see that the residue of generalized-Hammond-Lewis-multirank mod $7$
divides the $14$ $4$-colored partitions of $2$ into $7$ equal classes.
Both parts of Theorem \thm{thm6} are easy to prove. For (i),
we need only Euler's pentagonal number theorem \eqn{Epent}.
We let $\zeta_t$ be a primitive $t$-th root
of unity. We have
\begin{align}
\sum_{\vec{\pi}\in\mathcal{P}^{t-1}} \zeta_t^{\gHLmr} q^{\abs{\vec{\pi}}}
&= \prod_{k=1}^{(t-1)/2} \frac{1}{(\zeta_t^k q;q)_\infty (\zeta_t^{-k}q;q)_\infty}
\mylabel{eq:gHLgenfunc}\\
&= \frac{(q;q)_\infty}
{(q^t;q^t)_\infty}.
\end{align}
From \eqn{Epent} we have
\begin{equation}
\sum_{\vec{\pi}\in\mathcal{P}^{t-1}} \zeta_t^{\gHLmr} q^{24\abs{\vec{\pi}}+1}
=
\frac{1}{(q^{24t};q^{24t})_\infty}
\sum_{n=-\infty}^\infty (-1)^n q^{(6n+1)^2}.
\mylabel{eq:gHLgenfunc2}
\end{equation}
We see that in the $q$-expansion on the right side of \eqn{gHLgenfunc2}
the coefficient of $q^n$ is zero when $n$ is a quadratic nonresidue mod $t$.
Theorem \thm{thm6} (i) follows.
For part (ii) of Theorem \thm{thm6} we only need Jacobi's triple
product identity \eqn{jtprod}.
We have
\begin{align}
\sum_{\vec{\pi}\in\mathcal{P}^{t-3}} \zeta_t^{\gHLmr} q^{\abs{\vec{\pi}}}
&= \prod_{k=1}^{(t-3)/2} \frac{1}{(\zeta_t^k q;q)_\infty (\zeta_t^{-k}q;q)_\infty}
\mylabel{eq:gHLgenfunc3}\\
&= \frac{(\zeta_t^{(t-1)/2}q;q)_\infty (\zeta_t^{-(t-1)/2}q;q)_\infty (q;q)_\infty}
{(q^t;q^t)_\infty}\nonumber\\
&= \frac{\displaystyle \sum_{m=-\infty}^\infty (-1)^{m+1}
(\zeta_t^{(m+1)(t-1)/2} - \zeta_t^{-m(t-1)/2}) q^{m(m+1)/2}}
{(1 - \zeta_t^{(t-1)/2}) (q^t;q^t)_\infty},
\nonumber
\end{align}
and
\begin{align}
&\sum_{\vec{\pi}\in\mathcal{P}^{t-3}} \zeta_t^{\gHLmr} q^{8\abs{\vec{\pi}}+1}
\mylabel{eq:gHLgenfunc4}\\
&\quad = \frac{1}{(1 - \zeta_t^{(t-1)/2}) (q^{8t};q^{8t})_\infty}
\sum_{m=-\infty}^\infty (-1)^{m+1}
\zeta_t^{-m(t-1)/2}
(\zeta_t^{(2m+1)(t-1)/2} - 1) q^{(2m+1)^2}.
\nonumber
\end{align}
We see that in the $q$-expansion on the right side of \eqn{gHLgenfunc4}
the coefficient of $q^n$ is zero when $n$ is not a quadratic residue mod $t$,
i.e.\ when $n$ is either a quadratic nonresidue or $n\equiv0\pmod{t}$.
Theorem \thm{thm6} (ii) follows.
\section{Multicranks} \label{sec:mcranks}
In this section we give some extensions of the bicrank to multipartitions
and provide alternative interpretations for some of the congruences given
in Theorem \thm{thm5}. We define two multicranks. These multicranks are defined
in terms of cartesian products of extended partitions and ordinary partitions.
In Section \sect{bicrank}, we defined the set of extended partitions $\mathcal{E}$
and its associated crank and weight function. Recall from Section \sect{ghlbirank}
that $\mathcal{P}$ denotes the set of ordinary partitions, and $\mathcal{P}\subset\mathcal{E}$.
Let $r$ be a positive even integer.
For an extended multipartition
\begin{equation}
\vec{\pi}=(\pi_1,\pi_2,\dots,\pi_{r}) \in
\mathcal{E} \times \cdots \times
\mathcal{E} \times \mathcal{P} \times \cdots \times \mathcal{P}
=\mathcal{E}^{r/2} \times \mathcal{P}^{r/2},
\end{equation}
we define multicrank-I by
\begin{equation}
\mbox{multicrank-I}(\vec{\pi}) =
\sum_{k=1}^{r/2} k\cdot\mbox{crank}(\pi_k).
\mylabel{eq:mcrankIdef}
\end{equation}
For an extended multipartition
\begin{equation}
\vec{\pi}=(\pi_1,\pi_2,\dots,\pi_{r}) \in
\mathcal{E} \times \mathcal{E}
\times \mathcal{P} \times \cdots \times \mathcal{P}
= \mathcal{E} \times \mathcal{E} \times \mathcal{P}^{r-2},
\end{equation}
we define multicrank-II by
\begin{equation}
\mbox{multicrank-II}(\vec{\pi}) =
\sum_{k=1}^{2} k\cdot\mbox{crank}(\pi_k) +
\sum_{k=3}^{r} k\,\left(\#(\pi_k) - \#(\pi_{r-k+3})\right).
\mylabel{eq:mcrankIIdef}
\end{equation}
We note that the $\mbox{bicrank}_2$ corresponds to the multicrank-II when
$r=2$.
For both types of extended multipartitions we define
a sum of parts function and a weight function in the natural way
\begin{equation}
\abs{\vec{\pi}} = \sum_{k=1}^{r} \abs{\pi_k},\quad\mbox{and}\qquad
w(\vec{\pi}) = \prod_{k=1}^{r} w(\pi_k).
\mylabel{eq:ewmdef}
\end{equation}
We have
\begin{equation}
\sum_{\substack{\vec{\pi}\in \mathcal{E}^{r/2}\mathcal{P}^{r/2}\\
\abs{\vec{\pi}}=n}} w(\vec{\pi})
=
\sum_{\substack{\vec{\pi}\in \mathcal{E}^{2}\mathcal{P}^{r-2}\\
\abs{\vec{\pi}}=n}} w(\vec{\pi})
= p_{-r}(n).
\mylabel{eq:totw1}
\end{equation}
\begin{theorem}
\label{thm:thm7}
Let $t>3$ be prime.
\begin{enumerate}
\item[(i)]
The residue of the multicrank-I mod $t$
divides the extended multipartitions of $n$ from
$\mathcal{E}^{(t-1)/2}\times\mathcal{P}^{(t-1)/2}$
into $t$ equal classes of equal weight
provided $24n+1$ is a quadratic nonresidue mod $t$.
\item[(ii)]
The residue of the multicrank-I mod $t$
divides the extended multipartitions of $n$ from
$\mathcal{E}^{(t-3)/2}\times\mathcal{P}^{(t-3)/2}$
into $t$ equal classes of equal weight
provided $8n+1$ is not a quadratic residue mod $t$.
\item[(iii)]
The residue of the multicrank-II mod $t$
divides the extended multipartitions of $n$ from
$\mathcal{E}^{2}\times\mathcal{P}^{t-5}$
into $t$ equal classes of equal weight
provided $8n+1$ is a quadratic nonresidue mod $t$.
\end{enumerate}
\end{theorem}
In view of \eqn{totw1}, Theorem \thm{thm7} (i), (ii) provides
alternative combinatorial intepretations of our congruences
for multipartitions given in Theorem \thm{thm5} (i), (ii).
The result in part (iii) is weaker than (ii). We include it
since it is a generalization of the $\mbox{bicrank}_2$.
The proof of Theorem \thm{thm7} is very similar to Theorem \thm{thm6}.
We have
\begin{align}
\sum_{\vec{\pi}\in
\mathcal{E}^{(t-1)/2}\times\mathcal{P}^{(t-1)/2}}
\zeta_t^{\mcrI} w(\vec{\pi}) q^{\abs{\vec{\pi}}}
&= \left(\prod_{k=1}^{(t-1)/2} F(\zeta_t^k,q) \right) \frac{1}{E^{(t-1)/2}(q)}
\mylabel{eq:mcIgenfunc}\\
&= \prod_{k=1}^{(t-1)/2} \frac{1}{(\zeta_t^k q;q)_\infty (\zeta_t^{-k}q;q)_\infty}
\nonumber\\
&= \frac{(q;q)_\infty}
{(q^t;q^t)_\infty},
\nonumber
\end{align}
where $F(z,q)$ is the crank generating function given in \eqn{Mvid}.
Theorem \thm{thm7} (i) then follows from \eqn{gHLgenfunc2}.
Similarly,
\begin{align}
\sum_{\vec{\pi}\in
\mathcal{E}^{(t-3)/2}\times\mathcal{P}^{(t-3)/2}}
\zeta_t^{\mcrI} w(\vec{\pi}) q^{\abs{\vec{\pi}}}
&= \left(\prod_{k=1}^{(t-3)/2} F(\zeta_t^k,q) \right) \frac{1}{E^{(t-3)/2}(q)}
\mylabel{eq:mcIgenfunc2}\\
&= \prod_{k=1}^{(t-3)/2} \frac{1}{(\zeta_t^k q;q)_\infty (\zeta_t^{-k}q;q)_\infty}.
\nonumber
\end{align}
Theorem \thm{thm7} (ii) then follows from \eqn{gHLgenfunc3}, \eqn{gHLgenfunc4}.
We have
\begin{align}
\sum_{\vec{\pi}\in
\mathcal{E}^{2}\times\mathcal{P}^{t-5}}
\zeta_t^{\mcrII} w(\vec{\pi}) q^{\abs{\vec{\pi}}}
&= F(\zeta_t,q)\,F(\zeta_t^2,q)\,
\prod_{k=3}^{(t-1)/2} \frac{1}{(\zeta_t^k q;q)_\infty (\zeta_t^{-k}q;q)_\infty}.
\mylabel{eq:mcIIgenfunc}\\
& = \frac{(q)_\infty^3}{(q^t;q^t)_\infty}.
\nonumber
\end{align}
Then
\begin{equation}
\sum_{\vec{\pi}\in
\mathcal{E}^{2}\times\mathcal{P}^{t-5}}
\zeta_t^{\mcrII} w(\vec{\pi}) q^{8\abs{\vec{\pi}}+1}
= \frac{1}{E(q^{8t})} \sum_{n\ge0}(-1)^n (2n+1) q^{(2n+1)^2},
\mylabel{eq:mcIIgenfunc2}
\end{equation}
and
Theorem \thm{thm7} (iii) follows.
\section{Concluding Remarks} \label{sec:end}
The two main results of this paper are the combinatorial inpretations
of the 2-colored partition congruences \eqn{2colcongs} in terms
of the Dyson-birank and the $5$-core-birank. The author has been unable to
extended these two results to higher dimensional multipartitions. The extensions
of the Hammond-Lewis birank and Andrews bicrank are much easier because the
generating functions involved are simple infinite products.
It seems unlikely that a combinatorial proof of \eqn{T5dissect}
is possible. This identity gives the $5$-dissection of the $5$-core-crank
generating function when $z=\zeta_5$. The proof given in the paper relies
on a heavy use of the theory of modular functions. A more elementary
proof is desirable. In \cite{Ga-Ki-St}, a combinatorial proof is given that the
residue of $5$-core-crank mod $5$ divides the $5$-cores of $5n+4$ into $5$
equal classes. It would interesting to see if the methods of \cite{Ga-Ki-St}
could be extended to give a combinatorial proof of Theorem \thm{thm3},
which is our result for the $5$-core-birank.
It is clear that the generalized-Hammond-Lewis multiranks and our multicranks
are related. For instance, from equations \eqn{gHLgenfunc}, \eqn{mcIgenfunc},
\eqn{gHLgenfunc3}, and \eqn{mcIgenfunc2} we have
\begin{align}
\sum_{\vec{\pi}\in\mathcal{P}^{t-1}} \zeta_t^{\gHLmr} q^{\abs{\vec{\pi}}}
&=
\sum_{\vec{\pi}\in
\mathcal{E}^{(t-1)/2}\times\mathcal{P}^{(t-1)/2}}
\zeta_t^{\mcrI} w(\vec{\pi}) q^{\abs{\vec{\pi}}},
\mylabel{eq:gHLmcrid1}\\
\sum_{\vec{\pi}\in\mathcal{P}^{t-3}} \zeta_t^{\gHLmr} q^{\abs{\vec{\pi}}}
&=
\sum_{\vec{\pi}\in
\mathcal{E}^{(t-3)/2}\times\mathcal{P}^{(t-3)/2}}
\zeta_t^{\mcrI} w(\vec{\pi}) q^{\abs{\vec{\pi}}}.
\mylabel{eq:gHLmcrid2}
\end{align}
It would interesting to find a combinatorial proof these identities.
However what would be more interesting is to find bijective proofs of
Theorems \thm{thm4} and \thm{thm6}. This is a reasonable problem since
the generating functions involved are simple infinite products.
\noindent
\textbf{Acknowledgement}
\noindent
I would like to thank \dots.
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,688 |
Looking for information about ongoing M+ exhibitions and programmes? Go to the West Kowloon Cultural District website. There, you'll find opening hours and venue information, as well as further details about the museum.
For now, we are a museum with a temporary venue—the M+ Pavilion. Our building, designed by Herzog & de Meuron, is slated to be completed in 2019, when M+ joins Hong Kong's skyline as part of the West Kowloon Cultural District.
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Q: How to extract patterns from a file and fill an bash array with them? My intent is to write a shell script to extract a pattern ,using regular expressions, from a file and fill an array with all the ocurrences of the pattern in order to foreach it.
What is the best way to achieve this?
I am trying to do it using sed. And a problem I am facing is that the patterns can have newlines and these newlines must be considered, eg:
File content:
"My name
is XXX"
"My name is YYY"
"Today
is
the "
When I extract all patterns between double quotes, including the double quotes, the output of the first ocurrence must be:
"My name
is XXX"
A:
fill an array with all the ocurrences of the pattern
First convert your file to have meaningful delimiter, ex. null byte, with ex. GNU sed with -z switch:
sed -z 's/"\([^"]*\)"[^"]*/\1\00/g'
I've added the [^"]* on the end, so that characters not between " are removed.
After it it becomes more trivial to parse it.
You can get the first element with:
head -z -n1
Or sort and count the occurrences:
sort -z | uniq -z -c
Or load to an array with bash's maparray:
maparray -d '' -t arr < <(<input sed -z 's/"\([^"]*\)"[^"]*/\1\00/'g))
Alternatively you can use ex. $'\01' as the separator, as long as it's unique, it becomes simple to parse such data in bash.
Handling such streams is a bit hard in bash. You can't set variable value in shell with embedded null byte. Also expect sometimes warnings on command substitutions. Usually when handling data with arbitrary bytes, I convert it with xxd -p to plain ascii and back with xxd -r -p. With that, it becomes easier.
The following script:
cat <<'EOF' >input
"My name
is XXX"
"My name is YYY"
"Today
is
the "
EOF
sed -z 's/"\([^"]*\)"[^"]*/\1\x00/g' input > input_parsed
echo "##First element is:"
printf '"'
<input_parsed head -z -n1
printf '"\n'
echo "##Elemets count are:"
<input_parsed sort -z | uniq -z -c
echo
echo "##The array is:"
mapfile -d '' -t arr <input_parsed
declare -p arr
will output (the formatting is a bit off, because of the non-newline delimetered output from uniq):
##First element is:
"My name
is XXX"
##Elemets count are:
1 My name
is XXX 1 My name is YYY 1 Today
is
the
##The array is:
declare -a arr=([0]=$'My name\nis XXX' [1]="My name is YYY" [2]=$'Today\nis\nthe ')
Tested on repl.it.
A: This may be what you're looking for, depending on the answers to the questions I posted in a comment:
$ readarray -d '' -t arr < <(grep -zo '"[^"]*"' file)
$ printf '%s\n' "${arr[0]}"
"My name
is XXX"
$ declare -p arr
declare -a arr=([0]=$'"My name \nis XXX"' [1]="\"My name is YYY\"" [2]=$'"Today\nis\nthe "')
It uses GNU grep for -z.
A: Sed can extract your desired pattern with or without newlines.
But if you want to store the multiple results into a bash array,
it may be easier to make use of bash regex.
Then please try the following:
lines=$(< "file") # slurp all lines
re='"[^"]+"' # regex to match substring between double quotes
while [[ $lines =~ ($re)(.*) ]]; do
array+=("${BASH_REMATCH[1]}") # push the matched pattern to the array
lines=${BASH_REMATCH[2]} # update $lines with the remaining part
done
# report the result
for (( i=0; i<${#array[@]}; i++ )); do
echo "$i: ${array[$i]}"
done
Output:
0: "My name
is XXX"
1: "My name is YYY"
2: "Today
is
the "
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,853 |
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A moped was stolen between Oct. 27-28 while parked in the South Quad moped parking. There are no leads and the case is closed. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,502 |
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<h3>Uses of <a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a> in <a href="../../../../../squidpony/squidgrid/gui/gdx/package-summary.html">squidpony.squidgrid.gui.gdx</a></h3>
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<td class="colFirst"><code><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a></code></td>
<td class="colLast"><span class="typeNameLabel">TextPanel.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/TextPanel.html#borderStyle">borderStyle</a></span></code> </td>
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<td class="colFirst"><code>static <a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a></code></td>
<td class="colLast"><span class="typeNameLabel">UIUtil.CornerStyle.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html#valueOf-java.lang.String-">valueOf</a></span>(<a href="http://docs.oracle.com/javase/8/docs/api/java/lang/String.html?is-external=true" title="class or interface in java.lang">String</a> name)</code>
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<td class="colFirst"><code>static <a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a>[]</code></td>
<td class="colLast"><span class="typeNameLabel">UIUtil.CornerStyle.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html#values--">values</a></span>()</code>
<div class="block">Returns an array containing the constants of this enum type, in
the order they are declared.</div>
</td>
</tr>
</tbody>
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<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing methods, and an explanation">
<caption><span>Methods in <a href="../../../../../squidpony/squidgrid/gui/gdx/package-summary.html">squidpony.squidgrid.gui.gdx</a> with parameters of type <a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a></span><span class="tabEnd"> </span></caption>
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<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
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<td class="colFirst"><code>static void</code></td>
<td class="colLast"><span class="typeNameLabel">UIUtil.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.html#drawMarginsAround-com.badlogic.gdx.graphics.glutils.ShapeRenderer-com.badlogic.gdx.scenes.scene2d.Actor-float-com.badlogic.gdx.graphics.Color-squidpony.squidgrid.gui.gdx.UIUtil.CornerStyle-">drawMarginsAround</a></span>(com.badlogic.gdx.graphics.glutils.ShapeRenderer renderer_,
com.badlogic.gdx.scenes.scene2d.Actor a,
float margin,
com.badlogic.gdx.graphics.Color color,
<a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a> cornerStyle)</code>
<div class="block">Draws margins around an actor.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static void</code></td>
<td class="colLast"><span class="typeNameLabel">UIUtil.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.html#drawMarginsAround-com.badlogic.gdx.graphics.glutils.ShapeRenderer-float-float-float-float-float-com.badlogic.gdx.graphics.Color-squidpony.squidgrid.gui.gdx.UIUtil.CornerStyle-">drawMarginsAround</a></span>(com.badlogic.gdx.graphics.glutils.ShapeRenderer renderer_,
float botLeftX,
float botLeftY,
float width,
float height,
float margin,
com.badlogic.gdx.graphics.Color color,
<a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a> cornerStyle)</code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static void</code></td>
<td class="colLast"><span class="typeNameLabel">UIUtil.</span><code><span class="memberNameLink"><a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.html#drawMarginsAround-com.badlogic.gdx.graphics.glutils.ShapeRenderer-float-float-float-float-float-com.badlogic.gdx.graphics.Color-squidpony.squidgrid.gui.gdx.UIUtil.CornerStyle-float-float-">drawMarginsAround</a></span>(com.badlogic.gdx.graphics.glutils.ShapeRenderer renderer_,
float botLeftX,
float botLeftY,
float width,
float height,
float margin,
com.badlogic.gdx.graphics.Color color,
<a href="../../../../../squidpony/squidgrid/gui/gdx/UIUtil.CornerStyle.html" title="enum in squidpony.squidgrid.gui.gdx">UIUtil.CornerStyle</a> cornerStyle,
float zoomX,
float zoomY)</code> </td>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,701 |
Q: UITableView crashing with reloadRowsAtIndexPaths I have an iPad iOS app where a tableview is displayed on the left side of the view and a data-entry form is displayed on the right side. The right-hand data entry form is comprised of another tableview with UITextFields within the cells.
When a particular textfield is changed then I need to change the corresponding row within the left-hand table. I am managing this using notifications. Most of the time it all works fine, but sometimes the app crashes when I attempt to update the particular row within the left-hand table.
The left-side tableview is being managed by a different view controller to the right-hand table.
The right-hand table textfield has the following method within its view controller:
-(void)textFieldDidChange:(id)sender
{
switch (((UIView*)sender).tag) {
case TAG_QE_PAGECAPTIONTEXTFIELD:
NSLog(@" = = = = = Caption did change %@",((UITextField*)sender).text);
[getModelForm() updateCurrentPageCaption:((UITextField*)sender).text forLanguage:@"English"];
[[NSNotificationCenter defaultCenter] postNotificationName: @"pageCaptionWasChanged" object: @"PageEdit"];
break;
default:
break;
}
}
The viewcontroller for the left-hand table establishes a listener for the notification with the following code:
[[NSNotificationCenter defaultCenter]
addObserver:self
selector:@selector(eventPageCaptionWasChanged:)
name: @"pageCaptionWasChanged"
object: nil];
The view controller for the left-hand table then has the following method:
-(void)eventPageCaptionWasChanged:(id)alteredText
{
UITableView* tableView = (UITableView*)[ self.view viewWithTag:TAG_QE_PAGELIST_TABLEVIEW ];
int row = (int)getModelForm().currentPageNo;
NSLog(@"HPSQuestionnaireEditorController got eventPageCaptionWasChanged with %@ for row %i",alteredText,row);
NSIndexPath* indexPath = [NSIndexPath indexPathForRow:row inSection:0];
NSArray* array = [NSArray arrayWithObject:indexPath];
int modelPageCount = getModelForm().pages.count;
NSLog(@"eventPageCaptionWasChanged modelPageCount=%i",modelPageCount);
[tableView reloadRowsAtIndexPaths:array withRowAnimation:UITableViewRowAnimationFade];
}
The app intermittently crashes on the [tableView reloadRowsAtIndexPaths line. The exception is:
* Assertion failure in -[UITableView _endCellAnimationsWithContext:], /SourceCache/UIKit_Sim/UIKit-1914.84/UITableView.m:1037
Can anyone help?
Thanks very much.
A: It seems like the page numbers are starting at 1, but the table indexes start at zero, so when you look for a row numbered 2 the table is actually searching for a 3rd item i.e. 0,1,2 and is therefore is getting an out of bounds exception.
Easiest way to fix it would be to change this line.
int row = (int)getModelForm().currentPageNo;
to this
int row = (int)getModelForm().currentPageNo -1;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,635 |
var crypto = require('crypto');
module.exports = function (pass, salt) {
var hash = crypto.createHash('sha512');
hash.update(pass, 'utf8');
hash.update(salt, 'utf8');
return hash.digest('base64');
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,696 |
"""
.. topic:: Design Notes
1. Commands are constrained to this ABNF grammar::
command = command-name *[wsp option] *[wsp [dquote] field-name [dquote]]
command-name = alpha *( alpha / digit )
option = option-name [wsp] "=" [wsp] option-value
option-name = alpha *( alpha / digit / "_" )
option-value = word / quoted-string
word = 1*( %01-%08 / %0B / %0C / %0E-1F / %21 / %23-%FF ) ; Any character but DQUOTE and WSP
quoted-string = dquote *( word / wsp / "\" dquote / dquote dquote ) dquote
field-name = ( "_" / alpha ) *( alpha / digit / "_" / "." / "-" )
It does not show that :code:`field-name` values may be comma-separated. This is because Splunk strips commas from
the command line. A search command will never see them.
2. Search commands targeting versions of Splunk prior to 6.3 must be statically configured as follows:
.. code-block:: text
:linenos:
[command_name]
filename = command_name.py
supports_getinfo = true
supports_rawargs = true
No other static configuration is required or expected and may interfere with command execution.
3. Commands support dynamic probing for settings.
Splunk probes for settings dynamically when :code:`supports_getinfo=true`.
You must add this line to the commands.conf stanza for each of your search
commands.
4. Commands do not support parsed arguments on the command line.
Splunk parses arguments when :code:`supports_rawargs=false`. The
:code:`SearchCommand` class sets this value unconditionally. You cannot
override it.
**Rationale**
Splunk parses arguments by stripping quotes, nothing more. This may be useful
in some cases, but doesn't work well with our chosen grammar.
5. Commands consume input headers.
An input header is provided by Splunk when :code:`enableheader=true`. The
:class:`SearchCommand` class sets this value unconditionally. You cannot
override it.
6. Commands produce an output messages header.
Splunk expects a command to produce an output messages header when
:code:`outputheader=true`. The :class:`SearchCommand` class sets this value
unconditionally. You cannot override it.
7. Commands support multi-value fields.
Multi-value fields are provided and consumed by Splunk when
:code:`supports_multivalue=true`. This value is fixed. You cannot override
it.
8. This module represents all fields on the output stream in multi-value
format.
Splunk recognizes two kinds of data: :code:`value` and :code:`list(value)`.
The multi-value format represents these data in field pairs. Given field
:code:`name` the multi-value format calls for the creation of this pair of
fields.
================= =========================================================
Field name Field data
================= =========================================================
:code:`name` Value or text from which a list of values was derived.
:code:`__mv_name` Empty, if :code:`field` represents a :code:`value`;
otherwise, an encoded :code:`list(value)`. Values in the
list are wrapped in dollar signs ($) and separated by
semi-colons (;). Dollar signs ($) within a value are
represented by a pair of dollar signs ($$).
================= =========================================================
Serializing data in this format enables streaming and reduces a command's
memory footprint at the cost of one extra byte of data per field per record
and a small amount of extra processing time by the next command in the
pipeline.
9. A :class:`ReportingCommand` must override :meth:`~ReportingCommand.reduce`
and may override :meth:`~ReportingCommand.map`. Map/reduce commands on the
Splunk processing pipeline are distinguished as this example illustrates.
**Splunk command**
.. code-block:: text
sum total=total_date_hour date_hour
**Map command line**
.. code-block:: text
sum __GETINFO__ __map__ total=total_date_hour date_hour
sum __EXECUTE__ __map__ total=total_date_hour date_hour
**Reduce command line**
.. code-block:: text
sum __GETINFO__ total=total_date_hour date_hour
sum __EXECUTE__ total=total_date_hour date_hour
The :code:`__map__` argument is introduced by
:meth:`ReportingCommand._execute`. Search command authors cannot influence
the contents of the command line in this release.
.. topic:: References
1. `Search command style guide <http://docs.splunk.com/Documentation/Splunk/6.0/Search/Searchcommandstyleguide>`__
2. `Commands.conf.spec <http://docs.splunk.com/Documentation/Splunk/5.0.5/Admin/Commandsconf>`_
"""
from __future__ import absolute_import, division, print_function, unicode_literals
from .environment import *
from .decorators import *
from .validators import *
from .generating_command import GeneratingCommand
from .streaming_command import StreamingCommand
from .eventing_command import EventingCommand
from .reporting_command import ReportingCommand
from .external_search_command import execute, ExternalSearchCommand
from .search_command import dispatch, SearchMetric
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,689 |
\section{Introduction}
Recently, topology in non-Hermitian systems attracts much attention because of its significant effects on electronic structures \cite{shen2018topological,gong2018topological,kawabata2018symmetry,lee2016anomalous,kawabata2019non,hatano1996localization}, and also non-equilibrium dynamics, particularly focusing on $\mathcal{PT}$-symmetric systems\cite{kawabata2018parity}.
Non-Hermiticity of Hamiltonian implies the quantum system has energy gain or loss.
Therefore, open quantum systems or non-equilibrium systems are the platform of the non-Hermitian quantum physics and have been intensively studied \cite{rotter2009non,daley2014quantum,hatano2019exceptional}.
Recently, it is found that many-body or disordered systems are also the platform of non-Hermitian physics due to its finite lifetime of excitations \cite{kozii2017non,shen2018topological,papaj2019nodal,zyuzin2018flat,moors2019disorder,yoshida2018non,michishita2019relation,yoshida2019symmetry,kimura2019chiral}.
For many-body or disordered systems, non-Hermiticity of the Hamiltonian appears via the dulled spectrum function and the novel quasiparticle dispersion.
Especially, the realization of the exceptional points are widely discussed \cite{kozii2017non,kawabata2019non,yoshida2019symmetry,okugawa2019topological,budich2019symmetry}.
The band structure of the non-Hermitian Hamiltonian possesses exceptional points,
where the eigenvectors as well as the eigenenergies are degenerate, and thus, they cannot span the full Hilbert space \cite{shen2018topological}.
This remarkable feature of the non-Hermite energy band is experimentally observable via angular-resolved photoemiession spectroscopy (ARPES) or quasiparticle interference experiment (QPI) measurements.
A theoretical study of a Weyl fermion system with dissipation, which can be realized in cold atomic systems or photonic systems, showed that spin-dependent gain or loss in this system leads to the formation of a Weyl exceptional ring (WER) and the generation of a flat band inside the WER \cite{xu2017weyl,cerjan2019experimental,mcclarty2019non}.
For solid state systems, a WER and the flat band inside the WER due to impurity potentials in Weyl semimetals (WSMs) were considered\cite{zyuzin2018flat}. In this previous study, it was shown that the tilt of the Weyl cones in the disordered WSMs leads to the formation of WERs. This result is applicable to disordered WSMs with tilted Weyl cones.
However, Weyl semimetal materials with large tilts of Weyl cones are rather limited.
Therefore, for the experimental verification of WERs in Weyl semimetals, it is desirable to explore for more general mechanisms of WERs.
In this paper, we propose the general prescription for the generation of the disorder-induced WERs.
In our scheme, WERs appear regardless of the detail of scatterers as long as the impurity scatterings are not in the unitarity limit. For example, magnetic impurities or the tilt of Weyl cones are not necessary for its realization.
The condition for the generation of WERs holds ubiquitously, and they are realizable for almost all of WSMs except for very special cases.
Because of the generality and simplicity of our proposal, its experimental detection via ARPES or QPI measurements for the well-established WSM materials, such as TaAs or Co$_3$Sn$_2$S$_2$, is feasible \cite{lv2015experimental,wang2018large}.
We, furthermore, investigate a fate of WERs and the flat bands inside WERs at Weyl semimetal-insulator transition induced by tuning symmetry-breaking fields, and propose that the two WERs merge into a hybrid exceptional ring, at the transition point where a topological Dirac semimetal (TDSM) realizes, leading to a flat band without vorticity of the phase of complex-energy eigenvalues inside the hybrid point ring.
This paper is organized as follow. In Sec.~\ref{NH hamiltonian}, we introduce a model for a disordered Weyl semimetal. Here, we define the energy-dependent quasiparticle Hamiltonian and derive the energy-independent effective quasiparticle Hamiltonian, which is generally non-Hermitian. In Sec.~\ref{disorder induced WER}, we discuss disorder-induced WERs by using the self-consistent $T$-matrix approximation. In this section, we verify that as long as the system is deviated from the unitarity limit, WERs generally appear. In addition, we clarify that {\it intra-valley multiple scatterings} play a crucial role for the realization of WERs. In Sec~\ref{Topo-phase transition}, we discuss a fate of WER at topological phase transition point. Here, we show that a hybrid point ring and a flat band without vorticity of the phase of complex-energy eigenvalues are realized just at the transition point.
In Sec~\ref{hybrid point ring TDSM}, we discuss the realization of the hybrid point ring in generic TDSM materials.
\section{Non-Hermitian Quasiparticle Hamiltonian}
\label{NH hamiltonian}
In this section, we define the energy-dependent quasiparticle Hamiltonian, which can be non-Hermitian. By extracting low energy parts in the vicinity of the poles of the retarded Green's function, we can obtain a general procedure for constructing the effective energy-independent quasiparticle Hamiltonian, which can also be non-Hermitian.
We first introduce the energy-dependent quasiparticle Hamiltonian $\tilde{H}(\epsilon,{\bm p})$:
\begin{align}
\tilde{H}(\epsilon,{\bm p}) &= H_0({\bm p}) + \Sigma(\epsilon), \label{eq:nonh}
\end{align}
where $H_0({\bm p}$) is the single-particle Hamiltonian and $\Sigma(\epsilon)$ is the self-energy\cite{shen2018quantum}. With the use of this quasiparticle Hamiltonian, the retarded Green's function can be defined as $G^{\rm R}({\bm p},\epsilon) \equiv [\epsilon - \tilde{H}(\epsilon,{\bm p})]^{-1}$.
Because of the imaginary part of the retarded self-energy associated with the finite lifetime of quasiparticles, the energy-dependent Hamiltonian is generally non-Hermitian.
The complex poles of the Green's function is determined by ${\rm det} [\epsilon - \tilde{H}(\epsilon,{\bm p})]$.
In low energy regions, the self-energy can be written as $ \Sigma(\epsilon) = \Sigma_0+ \Sigma_1\epsilon +\mathcal{O}(\epsilon^2)$.
In the low energy region, we can introduce the energy-independent quasiparticle Hamiltonian ${\cal H}_{\rm eff}({\bm p})$:
\begin{align}
{\cal H}_{\rm eff}({\bm p}) &= [1 - \Sigma_1]^{-1} (H_0({\bm p})+ \Sigma_0), \label{eq:heff}
\end{align}
whose eigenvalues ${\cal E}_n({\bm p})$ determine the complex poles of the Green's function because of ${\rm det} [\epsilon - \tilde{H}({\bm p},\epsilon)] \simeq {\rm det} [\epsilon -{\cal H}_{\rm eff}({\bm p})] =0$.
Note that the quasiparticle Hamiltonian $\tilde{H}(\epsilon,{\bm p})$ and ${\cal H}_{\rm eff}({\bm p})$ can be {\it non-Hermitian} and that its spectrum can be complex.
The non-Hermitian matrix can be non-diagonalizable at certain momenta.
These points are called exceptional points (EPs) in the mathematical physics literature, which are topologically stable.
The topological exceptional points can appear in the quasiparticle spectrum.
In the following sections, we show that these EPs naturally appears in disordered WSMs.
Finally, we comment on the origin of the non-Hermiticity of disordered systems. First of all, we note that the eigenvalue of the quasiparticle Hamiltonian is not the eigenvalues of Hamiltonian, but the poles of the dressed Green's function. Thus, its non-Hermiticity does not mean the existence of the total energy gain and loss in quantum systems. The non-Hermiticity of many body or disordered systems reflects
finite life-time of quasiparticles, and in the case of multi-band systems such as WSMs, leads to drastic change of the spectral functions of
quasiparticles,
which are directly observable via ARPES or QPI measurements.
For deeper understanding of non-Hermiticity of disordered system, we consider
more rigorous approaches,
e.g. the exact-diagonalization for sufficiently large systems. In a large real-space system, the eigenvalues of the Hamiltonian are real.
However, we focus on the quasiparticle spectrum obtained from the poles of the Green's function.
More precisely, we can calculate the real-space Green's function $G({\bm x},{\bm x}')$ with impurities and $G_0({\bm x},{\bm x}')=G({\bm x}-{\bm x}')\delta({\bm x}-{\bm x}')$ without impurities. Then, with the use of the Fourier transformation, we obtain the Fourior transformed Green's function $G({\bm p},{\bm p}') = {\cal FT}[G({\bm x},{\bm x}')]$ and $G_0({\bm p},{\bm p}') = {\cal FT}[G_0({\bm x},{\bm x}')]$ where ${\cal FT}$ is the Fourier transformation. The self-energy in momentum space is defined by $\Sigma ({\bm p},{\bm p}') = [G_0^{-1}({\bm p},{\bm p}') ] -[G^{-1}({\bm p},{\bm p}') ]$. If we consider the self-energy with ${\bm p}={\bm p}'$, we can obtain the poles of the Green's function $G({\bm p}) \equiv G({\bm p},{\bm p}) $, which is measured by ARPES experiments.
Since the inverse of the Fourier transformed Green's function $ {\cal FT}[G(({\bm x},{\bm x}') )]$ can have the imaginary part, the self-energy can have the imaginary part. In terms of physical understanding, this means that the one-particle Green's function is smeared with the impurities even in a sufficiently large system. This smearing effect arises from the self-energy. Therefore, even without gain/loss in the entire system, there is a complex spectrum defined by the poles of the dressed Green's function, since the energy gain/loss for quasiparticles exists because of scattering processes.
We also would like to point out that Ref. [12] considered large systems in real space and the authors compared the results between the numerical and analytical calculations. This also means that we can use the complex spectrum of the non-Hermitian quasiparticle Hamiltonian in both numerical and analytical calculations.
\section{Weyl exceptional ring and topological flat band in disordered WSM}
\label{disorder induced WER}
\subsection{Weyl exceptional ring : effective quasiparticle Hamiltonian }
Here, we discuss WERs and disorder-induced tilted Weyl cones by constructing energy-independent effective Hamiltonian. We consider a minimal two band lattice model of WSM. The minimal model of WSM is given by
\begin{align}
\label{hamWSM}
H_0({\bm p})= {\bm R}({\bm p}) \cdot {\bm \sigma},
\end{align}
where ${\bm \sigma} = (\sigma_x,\sigma_y,\sigma_z)^T$ is a vector of the Pauli matrices and ${\bm R}({\bm p}) = (v \sin p_x, v \sin p_y, \gamma (\cos p_z -m))^T$. This simple model of a WSM describes low energy physics of two adjacent WPs. For example, if we regard that the Pauli matrices is in the orbital space, this two-band model can describe the two WPs near $L$-point in Weyl ferromagnet, Co$_3$Sn$_2$S$_2$ \cite{ozawa2019two}.
The model Eq.~(\ref{hamWSM}) exhibits the transition between the WSM phase and the band insulator phase, depending on the parameter $m$.
In the case that $|m| < 1$, WPs appear in ${\bm p}=(0,0,\pm \cos^{-1} m)$. At $|m|=1$, the positive and negative chiral fermions overlap in the Brillouin zone (BZ).
For $|m|>1$, the band gap opens, resulting in the band insulator.
We, now, consider effects of disorder potentials in this system, which are incorporated into the self-energy.
As will be shown later, the self-energy is expressed as $\Sigma(\epsilon) = \Sigma_0(\epsilon) \sigma_0 + \Sigma_z(\epsilon) \sigma_z$. To obtain the energy-independent quasiparticle Hamiltonian, we expand it in the energy $\epsilon$ as
\begin{eqnarray}
\Sigma_0(\epsilon) &\equiv& \Sigma_0^0 + \Sigma_0^1\epsilon+\mathcal{O}(\epsilon^2),\\
\Sigma_z(\epsilon) &\equiv& \Sigma_z^0 + \Sigma_z^1 \epsilon+\mathcal{O}(\epsilon^2).
\end{eqnarray}
As shown in Eq.~(\ref{eq:heff}), the quasiparticle Hamiltonian in low-energy region ${\cal H}_{\rm eff}({\bm p})$ is written as
\begin{align}
{\cal H}_{\rm eff}({\bm p})= {\bm R}'({\bm p}) \cdot {\bm \sigma} + ab(p_z) \sigma_0,
\end{align}
where
\begin{eqnarray}
{\bm R}'({\bm p}) &=& (aR_x'({\bm p}),aR_y'({\bm p}),aR_z'({\bm p})),
\end{eqnarray}
\begin{eqnarray}
R_x'(p_x,p_y)&=&(1-\Sigma_0^1)R_x(p_x)+i\Sigma_z^1R_y(p_y),\\
R_y'(p_x,p_y)&=&(1-\Sigma_0^1)R_y(p_y)-i\Sigma_z^1R_x(p_x),\\
R_z'(p_z)&=&(1-\Sigma_0^1)(R_z(p_z)+\Sigma_z^0)+\Sigma_z^1\Sigma_0^0,
\end{eqnarray}
with the renormalizatiuon coefficient $a \equiv 1/ \left( (1-\Sigma_0^1)^2-(\Sigma_z^1)^2) \right)$ and $b(p_z) \equiv (1-\Sigma_0^1)\Sigma_0^0+\Sigma_z^1(R_z(p_z)+\Sigma_z^0)$. The $b(p_z)$-term leads to tilted of Weyl cone \cite{papaj2019nodal}.
The eigenvalue of ${\cal H}_{\rm eff}$ is expressed as
\begin{align}
{\cal E}_\pm({\bm p}) &= a\left(\pm \sqrt{ R_x'(p_x,p_y)^2 + R_y'(p_x,p_y)^2 + R_z'(p_z)^2} +b(p_z)\right). \label{eq:ep}
\end{align}
The non-Hermitian Hamiltonian is not diagonalizable when the first term in Eq.~(\ref{eq:ep}) is zero since there is only one eigenvector.
The exceptional points form WER on ${\bm p}_d$, where the equations
${\rm Re} \: [R_z'(p_{dz})] = 0$ and
$R_x'(p_{xd},p_{yd})^2+R_y'(p_{xd},p_{yd})^2 = {\rm Im} \: [ R_z'(p_{dz})]^2$ are satisfied.
The above discussion indicates that the imaginary part of $\Sigma_z(\epsilon)$ has an important role for the appearance of the WER.
\subsection{Disorder-induced Weyl exceptional ring}
Here, we clarify that WERs appear regardless of the detail of the scattering potentials, as long as there are {\it intra-valley multiple scattering processes}.
As shown below, neither the orbital dependence of scattering potentials nor magnetic impurities is necessary, in contrast to two-dimensional Dirac systems, where these specific features of scattering potentials are necessary for the realization of exceptional rings\cite{kozii2017non}.
We consider randomly distributed impurities and assume the scattering potential $V({\bm x})=V_0(\sigma_0+\beta \sigma_z)\sum_a \delta({\bm x}-{\bm x}_a) $. Here, ${\bm x}_a$ expresses a site of scatterers and $\beta$ is difference of scattering potential between two degrees of freedom (orbital or spin).
Especially, if Pauli matrices is in spin space, the finite $\beta$ means magnetic impurities.
The self-energy $\Sigma(\epsilon)$ derived from the self-consistent $T$-matrix approximation is given as
\begin{eqnarray}
\label{selfenergy}
\Sigma(\epsilon)&=&n_{\rm imp}T(\epsilon)=\Sigma_0(\epsilon)\sigma_0+\Sigma_z(\epsilon)\sigma_z,\\
\label{T}
T(\epsilon)&=&T_0(\epsilon)\sigma_0+T_z(\epsilon)\sigma_z\nonumber\\
&=&V_0\left(\sigma_0+ \beta \sigma_z \right) \nonumber \\
&+&V_0\left(\sigma_0+ \beta \sigma_z \right)\int_{\rm BZ} \frac{d^3 p}{(2\pi )^3}G(\epsilon,{\bm p})T(\epsilon),
\end{eqnarray}
where $T(\epsilon)$ and $n_{\rm imp}$ are the $T$-matrix and the concentration of scatterers, respectively. Here, we can decompose the integrated Green's function as $\int_{\rm BZ} \frac{d^3 p}{(2\pi )^3}G(\epsilon,{\bm p})\equiv \overline{G}_0(\epsilon)+\overline{G}_z(\epsilon)\sigma_z$. The off-diagonal components of the integrated Green's function are canceled since the off-diagonal components of Hamiltonian is odd functions of momentum. The $\Sigma_0$ term broadens the spectrum and does not qualitatively change the dispersion of quasiparticles. On the other hand, $\Sigma_z$ term drastically changes the dispersion relation of quasiparticles as discussed in the previous section.
The expression of integrated Green function is
\begin{eqnarray}
\label{integrated G0}
\overline{G}_0&=&\int_{\rm BZ} \frac{d^3 p}{(2\pi )^3}\frac{\epsilon-\Sigma_0}{\tilde{E}_+(\epsilon,{\bm p})-\tilde{E}_-(\epsilon,{\bm p})}\nonumber\\&\times&\left(\frac{1}{\epsilon-\tilde{E}_+(\epsilon,{\bm p})}-\frac{1}{\epsilon-\tilde{E}_-(\epsilon,{\bm p})}\right),\\
\label{integrated Gz}
\overline{G}_z&=&\int_{\rm BZ} \frac{d^3 p}{(2\pi )^3}\frac{ \gamma \cos p_z-m+\Sigma_z}{\tilde{E}_+(\epsilon,{\bm p})-\tilde{E}_-(\epsilon,{\bm p})}\nonumber\\&\times&\left(\frac{1}{\epsilon-\tilde{E}_+(\epsilon,{\bm p})}-\frac{1}{\epsilon-\tilde{E}_-(\epsilon,{\bm p})}\right).
\end{eqnarray}
where $\tilde{E}_\pm(\epsilon,{\bm p})$ is the complex eigen-energy of the energy-dependent quasiparticle Hamiltonian $\tilde{H}(\epsilon,{\bm p})$ defined in Eq.~(\ref{eq:nonh}), which is given by
\begin{eqnarray}
&&\tilde{E}_\pm(\epsilon,{\bm p}) =\nonumber\\ &\pm& \sqrt{(\gamma(\cos p_z-m)+\Sigma_z(\epsilon))^2+v^2(\sin^2 p_x+\sin^2 p_y)}+\Sigma_0.\nonumber\\
\end{eqnarray}
For $\epsilon={\rm Re}\Sigma_0(\epsilon),\; p_z=\pm \cos^{-1} (m-{\rm Re}\Sigma_z(\epsilon))/\gamma,\; \sin^2 p_x+\sin^2 p_y =|{\rm Im} \Sigma_z(\epsilon)/v|$, the energy-dependent quasiparticle Hamiltonian is not diagonalizable. These exceptional points form a WER. In the surface area enclosed by this WER
$\sin^2 p_x+\sin^2 p_y <|{\rm Im} \Sigma_z(\epsilon)/v|$, the real part of complex-energy eigenvalue vanishes and flat band characterized by vorticity of the phase of the complex-energy eigenvalues is realized.
\begin{figure}[t]
\includegraphics[width=8cm]{FIG1.eps}
\vspace{-0.6cm}
\caption{The relation between the generation of the $\Sigma_z$ and the parameter $m$.}
\label{fig01}
\end{figure}
\begin{figure}[b]
\includegraphics[width=7cm]{FIG2.eps}
\caption{The $m$-dependence of the radius of a WER. The parameters are set as, $\gamma=1,\; v=0.6,\; n_{\rm imp}=0.05,\; V_0=3,\; \beta=0$. In this paper, we set the $p$-mesh size $35 \times 35 \times 35$. }
\label{fig0}
\end{figure}
The self-consistent equations (\ref{T}) can be transformed as
\begin{eqnarray}
\label{Sigma_0}
\Sigma_0(\epsilon)=n_{\rm imp}V_0\frac{1-(1-\beta^2)V_0\overline{G}_0}{\left( 1-V_0\left(\overline{G}_0+\beta \overline{G}_z\right) \right)^2-V_0^2(\overline{G}_z+\beta \overline{G}_0)^2},\nonumber\\\\
\label{Sigma_z}
\Sigma_z(\epsilon)=n_{\rm imp}V_0\frac{\beta+(1-\beta^2)V_0\overline{G}_z}{\left( 1-V_0\left(\overline{G}_0+\beta \overline{G}_z\right) \right)^2-V_0^2(\overline{G}_z+\beta \overline{G}_0)^2}.\nonumber\\
\end{eqnarray}
In the similar manner to the two dimensional Dirac system, the orbital dependence of scattering potential or magnetic impurities leads to the generation of exceptional points \cite{kozii2017non,papaj2019nodal}. Interestingly, even when scattering potential does not have orbital dependence or magnetic character, i.e. $\beta=0$,
$\Sigma_z(\epsilon)$ naturally appears in the case that $m\neq 0$. To see this more clearly, we consider isotropic scattering potentials $\beta=0$. In this case, the self-consistent equation (\ref{Sigma_z}) becomes
\begin{eqnarray}
\label{Sigma_z_nonb}
\Sigma_z(\epsilon)&=&n_{\rm imp}V_0\frac{V_0\overline{G}_z}{\left( 1-V_0\overline{G}_0 \right)^2-V_0^2\overline{G}_z^2}.
\end{eqnarray}
Eq. (\ref{integrated Gz}) shows $\overline{G}_z$ becomes finite in the case that the parameter $m$ is non-zero.
Therefore, we can conclude that WERs can be realized in the case of $m\neq0$ even when $\beta=0$.
Here, we clarify the physical meaning of the parameter $m$, which is related to one of the conditions for the generation of WERs.
The parameter $m$ lifts two cosine bands on $p_x=p_y=0$ and determines the position of WPs.
The finite $m$ means that the position of WPs deviates from the symmetric points of BZ : ${\bm p}_{\rm sym}=(0,0,\pm \pi/2)$, which is defined by
\begin{eqnarray}
H_0(-{\bm p}+{\bm p}_{\rm sym})=-H_0({\bm p}+{\bm p}_{\rm sym}).
\end{eqnarray}
As shown in FIG.~\ref{fig01}, when WPs are located on the symmetric points of the BZ, the contributions to the integrated Green's function $\overline{G}_z$ from the region $p_z>|p_{z{\rm sym}}|$ and the region $p_z<|p_{z{\rm sym}}|$ are exactly canceled.
FIG.~\ref{fig0} shows the $m$-dependence of the radius of WER , which indicates that
the radius increases as the function of the distance between the WPs and the symmetric points.
Therefore, the deviation of the position of WPs from the symmetric points of the BZ is a key ingredient for the realization of large WERs.
We stress that this condition for large WERs are satisfied for many of candidate materials of WSMs, and $m=0$ is rarely realized in real systems.
\subsection{Origin of WER and disappearance of WER at unitarity limit}
In this subsection, we verify that the physical origins of WERs found in the previous subsection are {\it intra-valley multiple scattering processes}.
We also discuss disorder-induced tilted Weyl cones and their asymmetric density of states (DOS).
Moreover, we show that WERs disappear in the unitarity limit.
\begin{figure}[t]
\includegraphics[width=9cm]{FIG3.eps}
\caption{Figure (a) and (b) are the $\epsilon$ dependence of ${\rm Im}\Sigma_z(\epsilon)$ and the density of states $\rho(\epsilon)$ respectively. The red (solid line), green (dashed line) and blue (dotted line) curves correspond to $(V_0/\gamma,n_{\rm imp})=(300,0.01),\;(3,0.05),\;(-3,0.05)$. The other parameters are set as, $\gamma=1,\; v=0.6,\; m=0.3,\beta=0$.} \label{fig1}
\end{figure}
\begin{figure}[b]
\includegraphics[width=9cm]{FIG4.eps}
\caption{(a) Radius of a WER, $|{\rm Im}\Sigma_z(\epsilon)/v|$, at $\epsilon={\rm Re}\Sigma_0(\epsilon)$. The parameters are the same as those used in FIG.1. (b) DOS at the three points indicated by colors, red (solid line), green (dashed line), and blue (dotted line), in FIG.3(a) : $(V_0/\gamma,\beta)=(3,0),\;(3,0.4),\;(3,0.8)$. }
\label{fig2}
\end{figure}
\begin{figure}[b]
\includegraphics[width=9cm]{FIG5.eps}
\caption{The left figure is the spectral function at $p_x=p_y=0$. The right one is the enlarged figure around the Weyl points. The red line indicates the dispersion relation of quasiparticles. It shows that the disorder-induced tilted of Weyl cone appears.}
\label{fig3}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=16cm]{FIG6.eps}
\caption{Zero energy points ${\rm Re}\tilde{E}({\bm p})=0$ and the phase of the energy eigenvalues : (a) WSM phase ($|m|<1$), (b) phase boundary ($|m|=1$), (c) insulator phase ($|m|>1$).}
\label{fig5}
\end{center}
\end{figure*}
FIG.\ref{fig1} (a) and (b) show that the $\epsilon$-dependence of ${\rm Im}\Sigma_z(\epsilon)$ and the density of states (DOS) $\rho(\epsilon)$, respectively. It is shown that disorder makes the DOS asymmetric. This asymmetric behavior depends on the sign of scattering potentials, which implies that the asymmetry is due to the formation of impurity bands. The red curves (solid line) in FIG.\ref{fig1} (a) is ${\rm Im}\Sigma_z(\epsilon)$ in the unitarity limit : $V_0\to \infty ,\; n_{\rm imp}\to 0$. In this limit, the symmetry of DOS $\rho(-\epsilon)=\rho(\epsilon)$ is recovered. In the unitarity limit, the self-consistent equations for (\ref{Sigma_0}) and (\ref{Sigma_z}) become
\begin{eqnarray}
\label{Sigma_0unitarity}
\Sigma_0(\epsilon)&=&-n_{\rm imp}V_0\frac{\overline{G}_0(\epsilon)}{\overline{G}_0(\epsilon)^2-\overline{G}_z(\epsilon)^2},\\
\label{Sigma_zunitarity}
\Sigma_z(\epsilon)&=&n_{\rm imp}V_0\frac{\overline{G}_z(\epsilon)}{\overline{G}_0(\epsilon)^2-\overline{G}_z(\epsilon)^2}.
\end{eqnarray}
In the unitarity limit, the self-consistent equations no longer depend on $\beta$. Here, we assume the $\epsilon$-dependence of the integrated Green's function as : $\overline{G}_0(-\epsilon)=-\overline{G}_0^*(\epsilon),\;\overline{G}_z(-\epsilon)=\overline{G}_z^*(\epsilon)$. From Eqs.~(\ref{Sigma_0unitarity}) and (\ref{Sigma_zunitarity}), we can easily find $\Sigma_0(-\epsilon)=-\Sigma_0^*(\epsilon),\;\Sigma_z(-\epsilon)=\Sigma_z^*(\epsilon)$. These conditions and Eqs. (\ref{Sigma_0unitarity}) and (\ref{Sigma_zunitarity}) are self-consistent because $\tilde{E}_-(-\epsilon,\bm{p})=-\tilde{E}_+^*(\epsilon,\bm{p})$ is satisfied. Therefore, in the unitarity limit, ${\rm Im}\Sigma_z(\epsilon)=0$ at $\epsilon={\rm Re}\Sigma(\epsilon)=0$, which implies that WERs do not appear\cite{com1}.
As discussed in Appendix, within the self-consistent Born approximation, WERs do not appear for $\beta=0$\cite{suppliment}. The disappearance of WER in the unitarity limit or in the Born approximation implies that the origins of the generation of WERs are multiple scattering processes with moderate strength of impurity potentials.
In addition, as discussed in Appendix, we found that intra-valley scatterings lead to the generation of WER. Therefore, we conclude {\it intra-valley multiple scatterings} in WSM generally leads to the realization of WERs\cite{suppliment}. Combining the results obtained in the previous section, we can conclude that
the deviation from the unitarity limit and the deviation of WPs from the symmetric points of the BZ are crucial conditions for realization of WERs.
This is our main result.
These conditions are general and do not depend on the details of scattering potentials.
Neither magnetic impurities nor the orbital dependence of scattering potentials are necessary. For most of real candidate materials of WSMs, WPs are not located on symmetric points of the BZ. Therefore, for almost all Weyl material, WERs can be realized by generic disorder potentials, and the experimental detection of them are feasible.
In FIG.~\ref{fig2} (a), the radius of WER ($|{\rm Im}\Sigma_z(\epsilon)/v|$ at $\epsilon={\rm Re}\Sigma_0(\epsilon)$) is shown.
The results in FIG.\ref{fig2} (a) show that the radius of the WER is increased by $\beta$ in the manner similar to the two-dimensional Dirac system \cite{papaj2019nodal}.
Usually, the generation of a flat band is accompanied with the singular behavior of the DOS\cite{marchenko2018extremely}.
In FIG.~\ref{fig2} (b), the DOS for three different values of $\beta $ are shown.
In FIG.\ref{fig2} (b), we can not see the signature of the realization of flat bands. As discussed in Appendix, the growth of a flat band inside the WER leads to the increase of the DOS. However, it is relatively small \cite{marchenko2018extremely}.
This is because that the formation of the flat band inside the WER can be regarded as split of WPs. Therefore, the increment of DOS due to the generation of flat band inside WER is negligibly small in as shown in FIG.\ref{fig2} (b).
The asymmetry of the DOS near $\epsilon \simeq 0$ is originated from energy dependence of the self energy and disorder-induced tilted Weyl cones. Actually, as shown in FIG.~\ref{fig3}, disorder potentials induce the tilts of Weyl cones, which breaks the symmetry of the DOS: $\rho(-\epsilon)=\rho(\epsilon)$.
Here, we discuss the difference between our study and previous ones. Ref. [13] discusses WSMs with tilted Weyl cones and shows that the tilt of Weyl cones leads to the realization of WERs. This scenario is applicable to multilayer system or Type II WSM. On the other hand, our proposal is applicable to more general disordered WSM materials.
We comment on the relation between our two band model and real candidate materials. If we regard that the Pauli-matrices is in orbital space, our two band model can be seen as the effective Hamiltonian of Weyl ferromagnet Co$_3$Sn$_2$S$_2$\cite{xu2017weyl,ozawa2019two}. Actually, our two band model can describe the low energy physics of this material governed by WPs near $L$-points. The simple band structure derived from half-metalic character and low carrier density of this material can be considered as that Co$_3$Sn$_2$S$_2$ is good platform for demonstration of Weyl physics in condensed matter\cite{liu2018giant}. Therefore, the experimental verification of WERs via ARPES or QPI measurements is feasible.
To close this section, we discuss the relation between the generation of WERs and a disorder-driven semimetal to diffusive metal transition. In WSMs, the semimetalic featrure of the DOS ($\rho(\epsilon)\propto \epsilon^2$) is robust against weak disorder~\cite{fradkin1986critical}. Within the perturbative approach, a certain strength of disorder potentials leads to the phase transition to a diffusive metal phase~\cite{kobayashi2014density,bera2016dirty,shapourian2016phase}. In the semimetalic region, WERs can not be realized because the complex selfenergy can not be introduced due to the causality. As the asymmetric DOS shows (FIG.~\ref{fig3}), the state obtained in our calculation is in a diffusive metallic phase. Thus WERs are realized. Recently, it is verified that there is a non-perturbative contribution to the DOS which renders the semimetal state unstable~\cite{pixley2016rare,pixley2017single}. We can expect that WERs can be realizable even in this region~\cite{com2}.
\section{fate of WER and flat band at topological phase transition point}
\label{Topo-phase transition}
In this section, we discussed a fate of WERs at a topological phase transition point. We introduce the self-energy $\Sigma(\epsilon) \simeq \Sigma_0 \sigma_0+\Sigma_z \sigma_z$ by hand in the two band model which is discussed in the previous section.
In FIG. \ref{fig5}, we show disorder-induced flat bands inside WER and the phase of complex-energy eigenvalues for the WSM phase $|m|<1$, the phase boundary $|m|=1$ and the band insulator phase $|m|>1$. We can see that flat bands without vorticity of the phase of complex energy eigenvalues is realized at the topological phase transition point. At the phase transition point, a vortex and an anti-vortex of the phase of the complex-energy eigenvalues are overlapped and the vorticities are canceled. However, at the topological phase transition point, non-Hermitian Hamiltonian is still non-diagonalizable at the edge of the flat band.
The set of $\bm{p}$-points at the edge of the flat band constitutes a hybrid point ring.
In contrast to EPs which are accompanied by a quantized vortex of the phase of complex-energy eigenvalues, hybrid points are not accompanied by a vortex. Hybrid points are also topological defects of the energy spectrum of non-Hermitian systems.
As shown in FIG.~\ref{fig5}, the hybrid point ring is realized at the topological phase transition point.
\section{hybrid point ring in topological Dirac semimetal}
\label{hybrid point ring TDSM}
In previous section, we demonstrate that merging of two WERs leads to the realization of a hybrid point ring.
In this section, we verifies its realization in TDSM systems.
We consider Cd$_3$As$_2$ which is a typical TDSM system with a simple band structure and topologically protected Dirac points (DPs)\cite{borisenko2014experimental,kobayashi2015topological}. Since these topologically protected DPs can not be gapped out as long as the four-fold rotational symmetry is retained, this material can be considered as an ideal platform for the realization of hybrid point ring and flat band without vorticity of the phase of the complex-energy eigenvalues. The effective Hamiltonian of this material is the same as the sum of two systems of the lattice model considered in Sec.~\ref{disorder induced WER}.
\begin{figure}[t]
\includegraphics[width=8cm]{FIG7.eps}
\caption{Spectral functions of a TDSM. The left and right figures show the spectral functions at $V_0/|t_0|=0.5,\;3$ respectively. The upper panel shows the spectral functions at $\epsilon={\rm Re}\Sigma_0(\epsilon),\;p_z=\cos^{-1}\left( -(t_0+2t_2+{\rm Re}\Sigma_z(\epsilon))/t_1 \right)$. The lower panel shows the spectral functions at $\epsilon={\rm Re}\Sigma_0(\epsilon),\; p_y=0$. The parameters are set to $n_{\rm imp}=0.05,\; \beta=0 ,\;V_0=3,\;V_0=1,\; t_0=-1,\;t_1=1,\;t_2=0.3,\;\Lambda=0.7$.}
\label{fig6}
\end{figure}
The effective Hamiltonian is given by
\begin{eqnarray}
\label{hamiltonian TDSM}
H_0^{{\rm TDSM}}({\bm p})=
\begin{pmatrix}
h({\bm p})&&0\\
0&&-h({\bm p})
\end{pmatrix}
=h({\bm p})s_{z},
\end{eqnarray}
where $h({\bm p})=m_p\sigma_z+\Lambda(\sigma_x \sin p_xa+\sigma_y \sin p_ya),\;m_k=t_0+t_1 \cos p_za +t_2 (\cos p_xa+\cos p_ya)$ \cite{wang2013three}. $\sigma_i\;(i=x,y,z)$ and $s_i\;(i=x,y,z)$ are Pauli matrices in orbital and spin space, respectively. $t_1$ and $t_2$ are hopping integrals.
$\Lambda$ describes hybridization of $s$ and $p$ orbital due to anti-symmetric spin-orbit interaction. The effective Hamiltonian can be regarded as the sum of two WSM models with opposite chirality. The sum of positive and negative chiral Fermions leads to DPs, which appear at ${\bm p} = (0,0,\pm \cos^{-1} (-(t_0+2t_2)/t_1))$.
To verify the realization of the hybrid point ring in the TDSM, we introduce disorder potentials in this model by using the self-consistent $T$-matrix approximation. We consider the scattering potential $V({\bm x})=V_0(\sigma_0+\beta \sigma_z)\sum_a \delta({\bm x}-{\bm x}_a) $. Here, ${\bm x}_a$ expresses a site of scatterers and $\beta$ is the difference between scattering potentials of $s$ and $p$ orbitals. In FIG.~\ref{fig6}, the spectral function of the TDSM is shown. The flat band appears even when $\beta=0$. Asymmetry of scattering potentials in orbital space increases the radius of the hybrid point ring in a manner similar to the case of two-band Weyl model discussed in the previous sections.
\section{Conclusion}
In this paper, we have investigated disorder-induced WERs, clarifying the general scheme for their realization.
In our scheme, WERs are realized regardless of the details of scattering potentials, and also the tilt of the Weyl cone is not necessary.
The position of the WPs and multiple scattering processes play important roles for the generation of the WERs.
The conditions for the realization WERs are summarized as follows: (i) the positions of WPs deviate from the symmetric points of the BZ, (ii) The scattering potentials deviate from the unitarity limit.
We note that the condition (i) is satisfied for almost all candidate materials of WSMs.
Therefore, we can conclude that WERs are realizable almost all WSMs and experimentally detectable.
In particular, our two-band model is relevant to the Weyl ferromagnet, Co$_3$Sn$_2$S$_2$.
In addition, we have verified that the generation of the flat band inside the WER leads to the small increase of the DOS.
Moreover, the energy dependence of the self-energy leads to the disorder-induced tilted Weyl cone and asymmetry of the DOS. These features can be experimentally verified by STM measurements.
Finally, a fate of WER at a topological phase transition point was discussed.
We have found that, at the transition point where the TDSM state is realized, merging of two WERs with a vortex and an anti-vortex of complex energy eigen-values leads to the realization of the hybrid point ring and topological flat band without vorticity.
We have demonstrated the realization of the hybrid point ring and the flat band in the case of Cd$_3$As$_2$.
Our theoretical proposal can directly be detected via ARPES or QPI measurement.
These measurement are useful for detection of the anisotropic broadening of quasiparticle spectrum which is related evolving exceptional or hybrid rings. We believe that it become a large step for non-Hermitian physics in disordered systems.
\section{Acknowledgments}
TM thanks T. Mizushima, K. Nomura, A. Ozawa, Y. Ominato and K. Kobayashi for faithful discussion. TM acknowledges illuminating discussions with T. Yoshida and J. H. Pixley,.
The calculations were partially performed by the supercomputing system SGI ICE X at the Japan Atomic Energy Agency. Especially, J. H. Pixley inform us of Refs. [36] and [37], which are useful for the deeper understanding our results.
TM was supported by a JSPS Fellowship for Young Scientists.
YN was partially supported by the ''Topological Materials Science'' (No. 18H04228) KAKENHI on Innovative Areas from JSPS of Japan.
SF was supported by the Grant-in-Aids for Scientific
Research from JSPS of Japan (Grants No. 17K05517), and KAKENHI on Innovative Areas ``Topological Materials Science'' (No.~JP15H05852) and "J-Physics" (No.~JP18H04318).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,528 |
First Look: Toy Story Land Attractions at Disney's Hollywood Studios
Posted by Mickey News | Apr 21, 2016 | Walt Disney World Resort News | 0 |
In Toy Story Land, you'll find yourself shrunk to the size of a toy to explore the world of Andy's backyard with your favorite Toy Story characters, including Woody and Buzz. There are two new themed attractions being developed for Toy Story Land.
The first – Slinky Dog Dash – will be a family coaster attraction you'll want to ride again and again. The attraction features a coaster track that Andy has built all over his backyard using his Mega Coaster Play Kit, but as you know, he has a pretty amazing imagination, so he's combined it with some of his other toys, according to Imagineer Kathy Mangum. Take a look at a concept video of Slinky Dog Dash that Kathy shared earlier.
On Slinky Dog Dash, you will zip, dodge and dash around many turns and drops that Andy has created to really make Slinky and his coils stretch to his limits.
The second all-new attraction in Toy Story Land will be Alien Swirling Saucers. This attraction is designed as a toy play set that Andy got from Pizza Planet, inspired by the first Toy Story film . Aliens are flying around in their toy flying saucers and trying to capture your rocket toy vehicle with "The Claw."
As you rotate around the toy planets and satellites as part of the game, you'll swirl to the beat of fun "Space Jazz" music developed just for this experience, Mangum said. The music, the lighting, and the sound effects will add to the flurry of your adventure while "The Claw" looms ominously over you.
Also today, Mangum shared an update that beginning Memorial Day weekend, even more guests will be able to experience Toy Story Mania with the addition of a third track.
Stay tuned to Mickey News for more details on Toy Story Land.
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Life Twice Lived Review
Posted by Mike Aiken
Death is pretty final. But if your child died in an accident would you take up the chance of creating a near perfect match using their DNA? That's the conceit explored by Gail Louw's play 'A Life Twice Given' which provides a skilful adaption for the stage of David Daniel's book of the same name.
We know there's a flatworm that you can cut in half and both parts will regenerate, carry identical traits and live happily ever after. It's the ultimate 'two for the price of one' in the natural world. But you wouldn't do that with a human would you? Of course, you might not think it's possible. On the other hand, advances in DNA engineering may move that a step close. So would you, if you could, opt for human cloning?
Gail Louw's staging is taut and unfussy. This throws maximum attention on the moral dilemma played out in dialogues between the couple throughout the play. It is reminiscent of a Bernard Shaw drama where the points of view can appear more important than the characterisation. That wasn't quite true here.
We see the son making shapes with his hands, wheeling his scarf like a kite in blue light. There's klezmer violin in the background. His parents, Lisa and David, have those cute lives we recognise from US soaps. They occupy successful professional lives and have a holiday home in the woods by a river. She wears a summer dress with rippling cool lines. He's kitted out in blue shirt and smart suit. Time for a trip downstream.
Suddenly it's all red lighting and roaring air ambulances from every direction: their son is killed in a river accident. Nothing can bring him back. But David, in a wild moment, cuts a piece from his dead son's ear and preserves it at the university lab in liquid nitrogen. When his wife wakes from a coma he starts to raise the idea of 'bringing their son back' by cloning the DNA onto a new human egg.
The play then takes us through a series of debates between Lisa and David about the scientific possibility, the moral and religious dilemmas, and the ethical probity of the plan. Will it work and, it it works, will all hell be let lose in wider society?
An eccentric, unorthodox but scientific rabbi meets them in Prague five hours before the new millennium to discuss the issue: "A good man can be a partner in God's creation" he argues. She says: "You're trying to twist yourself into a pretzel!" Would the 'new' child be identical to their dead son? Who would be the 'new' boy's parents?
Time periods are hinted at through short passages from rock songs dating from the 1980s to the present day. We start with 'Billie Jean' who is "not my lover", of course, but more importantly "the kid is not my son" – or is he? Who will be the true parents of a newly cloned boy? Would he be a 'new' child or a replica of the old? How would he grapple with the dilemmas of being a clone? We end in the dystopian future of the 2020s.
This play is an example of theatre providing insights into the complexity of emotional and moral issues like human cloning. We should not underestimate the difficulties of performing this type of play. The ferocity of ideas can overshadow a cast. Instead we had deft but controlled acting which let the story breath. Of course, works like this lend themselves to debating styles that are common in forum theatre which involves direct audience participation in unravelling and shaping contentious issues. Indeed, plenty of us hung around in the auditorium at the end, trying to figure out our own views.
Gail Louw is unafraid to dig into complex social issues emerging from scientific advances and she avoids simple prescriptive answers. In this adaption of 'A Life Twice Given' she reinforces her reputation as a writer of distinction with the work realised by tight directing from John Burrows. Do see it again. It's a play worth twice seen!
Rialto, Sunday 27th October 2019
Photos by Sheila Burnett
Mike Aiken
Mike lives in Brighton. This is a full time occupation. He's also a researcher, writer and activist. Any time left over he spends hanging around cafes and pubs listening to people on their phones. He loves theatre that pokes into difficult places. You won't find him on Facebook.
Metz, Fri 6th Dec
Klaus Johann Grobe, Thurs 5th Dec
Life Twice Lived Review - Brighton Source | {
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} | 5,384 |
\section{Introduction} \label{sec:introduction}
Hybrid video codecs like H.264/AVC \cite{Richardson2003} are able to effectively compress video sequences and provide a good viewing experience at manageable data rates. This is achieved by irrelevance reduction on the one hand side and redundancy reduction on the other hand side. Whilst irrelevance reduction results from quantization, the redundancy in a video sequence is removed by prediction, transformations and entropy coding. In this context, the area actually being encoded is always predicted from already transmitted and decoded areas. Afterwards, the prediction residual is transformed and transmitted to the decoder. Since prediction takes place at a very early stage of the encoding, it has a very high influence on the coding efficiency.
Current video codecs use two strategies for taking advantage of the correlations within a video sequence for prediction. To cope with temporal similarities, motion compensated prediction \cite{Dufaux1995} is used and spatial similarities are exploited by skillfully continuing the signal from already transmitted areas into the area currently encoded. Although the codecs can switch between spatial and temporal prediction for selecting a rate-distortion optimal mode, the fact that a video sequence possesses temporal and spatial correlations at the same time is ignored. Up to now, only few prediction algorithms exist that can make use of spatial as well as temporal redundancies. The algorithms from Jiang \cite{Jiang2009} and Matsuda \cite{Matsuda2010} are two of them to mention.
In \cite{Seiler2008c} we introduced a different spatio-temporal prediction algorithm, the spatially refined motion compensation. This algorithm operates in two stages and uses motion compensated prediction for exploiting temporal correlations and a spatial refinement step for including spatial correlations into the prediction. Even though we already proposed modifications of this algorithms with a reduced complexity \cite{Seiler2010}, the computational load produced by spatial refinement is still far too large for real-time implementations. In order to cope with this, we now propose a method for reusing the H.264/AVC deblocking filter \cite{List2003} for spatio-temporal prediction.
\section{Spatially Refined Motion Compensated Prediction Principles} \label{sec:prediction}
\begin{figure}
\psfrag{m}[t][t][0.9]{$m$}%
\psfrag{n}[t][t][0.9]{$n$}%
\psfrag{x}[t][t][0.9]{$x$}%
\psfrag{y}[t][t][0.9]{$y$}%
\psfrag{t}[t][t][0.9]{$t$}%
\psfrag{B}[l][l][0.9]{$\mathcal{B}$}%
\psfrag{R}[l][l][0.9]{$\mathcal{R}$}%
\psfrag{x0}[t][t][0.9]{$x_0$}%
\psfrag{y0}[t][t][0.9]{$y_0$}%
\psfrag{tau}[t][t][0.9]{$\tau$}%
\psfrag{tau1}[t][t][0.9]{$\tau-1$}%
\centering
\includegraphics[width=0.35\textwidth]{graphics/refinement_area.eps}
\caption{\emph{Relationship between currently predicted macroblock at position $\left(x_0,y_0\right)$ and already decoded neighboring blocks. The predicted block is divided into sub-blocks for motion compensation.}}
\label{fig:refinement_area}
\end{figure}
Before spatio-temporal prediction with the utilization of the H.264/AVC deblocking filter is introduced, we will briefly review the basic ideas of spatially refined motion compensated prediction. In order to exploit spatial as well as temporal redundancies for prediction, spatially refined motion compensation operates in two stages. During the first stage, pure motion compensated prediction is carried out. Here, all features of motion compensated prediction that have been developed in the recent years like fractional-pel accuracy, multiple reference frames or sub-block partitioning can be used. Using this, the temporal similarities between the individual frames can be exploited well. But, motion compensated prediction takes place without considering spatial redundancies. In order to account for this, the motion compensated signal is spatially refined in the second step. For this, the spatially adjacent, already decoded macroblocks are regarded. If the macroblocks are processed in line-scan order, the macroblocks to the left and above are available. Fig.\ \ref{fig:refinement_area} shows this relationship between the motion compensated block and the already decoded neighboring blocks for the example that the macroblock at position $\left(x_0,y_0\right)$ in frame $t=\tau$ is predicted. In this example, the feature of H.264/AVC to divide a macroblock into sub-blocks for motion compensation is included.
The originally proposed algorithm for spatial refinement \cite{Seiler2008c} obtains the spatio-temporal prediction by iteratively generating a joint model for the motion compensated block and the already decoded neighboring blocks by Frequency Selective Approximation (FSA). The model results from a weighted superposition of two-dimensional basis functions. Although this algorithm yields a significantly increased coding efficiency, it possesses the drawback of a very high computational complexity. In \cite{Seiler2010} an alternative model generation is proposed for an accelerated refinement, but the complexity is still too high for real-time implementations.
In order to cope with this, in the following we introduce an algorithm that reuses a slightly modified version of the H.264/AVC deblocking filter \cite{List2003} for spatial refinement. Due to the reuse of this highly optimized filter, spatial refinement can be carried out at a very high speed.
\section{Using the H.264/AVC Deblocking Filter for Spatial Refinement} \label{sec:deblocking}
\begin{figure}
\vspace{-0.5cm}
\begin{center}
\includegraphics[width=0.35\textwidth]{graphics/block_diagram.eps}
\end{center}\vspace{-0.5cm}
\caption{\emph{Simplified block diagram of a hybrid video encoder with spatial refinement.}}
\label{fig:coder}
\end{figure}
The original intention of the H.264/AVC deblocking filter \cite{List2003} is to reduce the artifacts which result from quantization. \mbox{Fig.\ \ref{fig:coder}} shows the original position of the deblocking filter in a simplified block diagram of a hybrid video encoder with spatial refinement of the motion compensated predicted signal. The deblocking filter aims at reducing blocking artifacts before the decoded image is copied to the reference buffer from which the motion compensated prediction is carried out. Due to this, using the deblocking filter has three advantages. First, the decoded output image contains less artifacts that might be visible to the viewer. Second, the motion compensated predicted signal fits the input macroblock more, since the motion compensation is performed on the deblocked data. Hence, the coding efficiency is increased. Third, compared to a post-processing deblocking filter, one frame buffer less is required.
Blocking artifacts arise as the individual macroblocks, or even the sub-blocks are quantized independently of each other. Due to this, the pixel values in different blocks differ unnaturally strong and the block boundaries become visible. The deblocking filter aims at detecting unnatural edges at block boundaries and at softening them, if required.
But, by taking a close look at motion compensated prediction, one can discover that the predicted signal contains similar artifacts. In the process of motion compensated prediction, in the reference frame the best fitting block, or respectively sub-block if macroblock partitioning is used, is determined for the currently regarded block. However, for selecting a block in the reference frame, only the current block is considered and the neighboring blocks are ignored. Hence, the block that is selected for motion compensated prediction may not fit the neighboring, already decoded macroblocks. In this case, the transition between the prediction signal and the neighboring blocks is similar to a blocking artifact. Obviously, such a motion compensated block cannot be a good predictor since the spatial correlations to the neighboring blocks are not considered. For resolving this problem, we propose to use a deblocking filter for spatial refinement of the motion compensated block and for incorporating spatial redundancies into the prediction process.
\begin{figure}
\vspace{-0.5cm}
\psfrag{p3}[t][t][0.85]{$p_3$}%
\psfrag{p2}[t][t][0.85]{$p_2$}%
\psfrag{p1}[t][t][0.85]{$p_1$}%
\psfrag{p0}[t][t][0.85]{$p_0$}%
\psfrag{q0}[t][t][0.85]{$q_0$}%
\psfrag{q1}[t][t][0.85]{$q_1$}%
\psfrag{q2}[t][t][0.85]{$q_2$}%
\psfrag{q3}[t][t][0.85]{$q_3$}%
\psfrag{Sub-block}[t][t][0.8]{Sub-block}%
\psfrag{boundary}[t][t][0.8]{boundary}%
\centering
\includegraphics[width=0.3\textwidth]{graphics/deblocking.eps}
\caption{\emph{Samples covered by the deblocking filter.}}
\label{fig:deblocking}
\end{figure}
The filter used for spatial refinement is similar to the strong deblocking filter from H.264/AVC \cite{List2003} and covers always eight samples, four on the one side of the sub-block boundary and four on the other. As illustrated in Fig.\ \ref{fig:deblocking}, these samples are denoted by $p_0,\ldots, p_3$ and $q_0,\ldots, q_3$. Independently of the actual sub-block partitioning, the filtering is carried out for all sub-block boundaries, first in vertical direction, then in horizontal direction. In this process, the filtering starts with the boundary to the already reconstructed neighboring macroblocks. Using this, spatial similarities can be drawn from the already decoded blocks into the predictor.
According to \cite{List2003}, two thresholds are required for controlling the amount of filtering. Originally, the thresholds
\begin{eqnarray}
\alpha &=& 0.8 \left(2^{h/6}-1\right) \\
\beta &=& 0.5h-7
\end{eqnarray}
depend on the quantization parameter. Since the amount of deblocking that is required for spatial refinement does not depend on the quantization, the parameter $h$ can be chosen freely. The choice of $h$ will be discussed in Section \ref{sec:results}. Alg.\ \ref{algo:deblocking} lists the steps that are necessary for deriving the filtered values $p^\prime_0,p^\prime_1,p^\prime_2,p^\prime_3,q^\prime_0,q^\prime_1,q^\prime_2,q^\prime_3$ from the original values $p_0,p_1,p_2,p_3,q_0,q_1,q_2,q_3$. Comparing Alg.\ \ref{algo:deblocking} with the original deblocking filter from \cite{List2003} one can discover that the calculations for deriving the filtered values are identical with the only exception that the clipping of the filtered values is not used. As motion compensated samples can differ strongly from the original samples, limiting the range of the filtered values would not be advisable. Due to the similarities, the proposed refinement can run as efficiently as the H.264/AVC deblocking filter and the optimized implementations can be reused. After filtering, the original values of the motion compensated block are replaced by the filtered values.
\begin{algorithm}[t]
\caption{Deblocking filter for spatial refinement}
\small
\label{algo:deblocking}
\begin{algorithmic}
\item[\algorithmicinput] $p_0,p_1,p_2,p_3,q_0,q_1,q_2,q_3$
\STATE $p^\prime_i=p_i, \ \ q^\prime_i=q_i, \ \ \forall i$
\IF {$\left|p_0-q_0\right|< \alpha \ \& \ \left|p_0-p_1\right|<\beta \ \& \ \left|q_0-q_1\right|<\beta$}
\STATE $p_0^\prime = p_0+\left(4\left(q_0-p_0\right)+\left(p_1-q_1\right)+4\right)>\!\!>3$
\STATE $q_0^\prime = q_0-\left(4\left(q_0-p_0\right)+\left(p_1-q_1\right)+4\right)>\!\!>3$
\IF {$\left|p_0-p_2\right|<\beta$}
\STATE $p_1^\prime = p_1+\left(p_2+\left(\left(p_0+q_0+1\right)>\!\!>1\right)-2p_1\right)>\!\!>1$
\ENDIF
\IF {$\left|q_0-q_2\right|<\beta$}
\STATE $q_1^\prime = q_1+\left(q_2+\left(\left(q_0+p_0+1\right)>\!\!>1\right)-2q_1\right)>\!\!>1$
\ENDIF
\ENDIF
\IF {$\left|p_0-q_0\right| < \left(\alpha>\!\!>2\right)+2$}
\IF {$\left|p_0-p_2\right|<\beta$}
\STATE $p_0^\prime = \left(p_2+2p_1+2p_0+2q_0+q_1+4\right)>\!\!>3$
\STATE $p_1^\prime = \left(p_2+p_1+p_0+q_0+2\right)>\!\!>2$
\STATE $p_2^\prime = \left(2p_3+3p_2+p_1+p_0+q_0+4\right)>\!\!>3$
\ELSE
\STATE $p_0^\prime = \left(2p_1+p_0+q_1+2\right)>\!\!>2$
\ENDIF
\IF {$\left|q_0-q_2\right|<\beta$}
\STATE $q_0^\prime = \left(q_2+2q_1+2q_0+2p_0+p_1+4\right)>\!\!>3$
\STATE $q_1^\prime = \left(q_2+q_1+q_0+p_0+2\right)>\!\!>2$
\STATE $q_2^\prime = \left(2q_3+3q_2+q_1+q_0+p_0+4\right)>\!\!>3$
\ELSE
\STATE $q_0^\prime = \left(2q_1+q_0+p_1+2\right)>\!\!>2$
\ENDIF
\ENDIF
\item[\algorithmicoutput] $p^\prime_0,p^\prime_1,p^\prime_2,p^\prime_3,q^\prime_0,q^\prime_1,q^\prime_2,q^\prime_3$
\end{algorithmic}\vspace{2mm}
\end{algorithm}
These steps are repeated for all vertical and horizontal sub-block boundaries until all motion compensated samples are replaced by the filtered ones. As the already filtered samples are included in the filtering of subsequent samples, the spatial information propagates into the motion compensated block. Finally, the refined block is used for prediction. Since the refined block exploits temporal as well as spatial correlations, it is a better predictor for the block to be encoded and thus increases the coding efficiency. It has to be noted that the proposed usage of the deblocking filter is not intended for replacing the original H.264/AVC deblocking filter. The aim of the original filter is to allay artifacts resulting from quantization whereas the objective of spatial refinement is to exploit spatial as well as temporal redundancies for prediction. As we will show in the next section, the highest coding efficiency can be achieved if spatial refinement and the original deblocking filter are used together.
\section{Simulations and Results}\label{sec:results}
\begin{figure}
\vspace{-3mm}
\centering
\input{graphics/crew_720p.tex}\vspace{-2mm}
\caption{\emph{Rate-distortion curves for first $399$ P-frames of the 720p-sequence ``Crew''.}}
\label{fig:rd_curve}
\end{figure}
For evaluating the coding efficiency and the acceleration of the spatial refinement step that can be achieved by the proposed algorithm, we implemented it into H.264/AVC reference encoder JM10.2 running in Main Profile. In addition to the novel refinement, refinement by FSA \cite{Seiler2008c} is considered for comparison. The tests are carried out on four test sequences: two CIF-sequences (``Discovery City'', ``Vimto'') of $100$ frames length and two 720p-sequences (``Crew'', ``Jets'') of $400$ frames length. All sequences are coded in IPPP order and with different quantization parameters (QP). In this process, a wide quality range from QP $16$ to QP $43$ with spacing $3$ is regarded for the CIF-sequences whereas only high qualities with QPs in the range between $16$ and $28$, also with spacing $3$, are regarded for the high resolution sequences. Since not all macroblocks benefit from spatial refinement, one bit per macroblock has to be added as side information to signalize the decoder if spatial refinement should be used or not. Thus, as shown in Fig. \ref{fig:coder}, the encoder now can switch between three modes: spatial prediction, temporal prediction, and spatio-temporal prediction. We have tested two different scenarios: first, the original H.264/AVC deblocking filter is switched off and the artifacts resulting from quantization are not removed within the prediction loop. Second, the original H.264/AVC deblocking filter is turned on. In this case the deblocking filter is used twice, on the one hand side for reducing quantization artifacts, on the other hand side for spatial refinement of the motion compensated block.
The parameters for spatial refinement by FSA are chosen according to \cite{Seiler2008c}. For the proposed deblocked refinement, only the parameter $h$ can be varied. This parameter controls the amount of filtering, i.\ e.\ at which point a transition is regarded as blocking artifact or as natural edge. For large values of $h$ the thresholds $\alpha$ and $\beta$ also become large and the filtering of the pixels takes place more often than for small values. We have tested different values of $h$ from the range between $10$ and $43$ and discovered that the overall improvement only varies marginally if values from the range between $19$ and $40$ are selected. Hence, we decided to select $h=28$.
Fig.\ \ref{fig:rd_curve} shows the rate-distortion curves of the 720p-se\-quence ``Crew'' for the cases that pure motion compensated prediction is used and that spatial refinement by FSA and the proposed algorithm are included, each time with the original H.264/AVC deblocking filter turned on and off. Taking a look at the pure motion compensated prediction, one can recognize the effect of the original deblocking filter since the coding efficiency is higher for lower qualities. But, with increasing quality the quantization artifacts reduce and the influence of the original deblocking filter becomes void. By regarding the curves for the cases that spatial refinement is used, it becomes apparent that the coding efficiency can be significantly increased by spatial refinement, independently of the actual refinement algorithm.
In order to condense the results, Table \ref{tab:psnr_gain} lists the average rate reduction or respectively the average $\mathrm{ PSNR}$ gain that can be achieved by spatial refinement, compared to pure motion compensated prediction. For this, the Bj{\o}ntegaard metric \cite{Bjontegaard2001} is used. For the case that the original H.264/AVC deblocking filter is turned off, spatial refinement yields very large gains over pure motion compensated prediction. In this case, the blocking artifacts resulting from quantization propagate to the motion compensated prediction, causing a lower coding efficiency. Although spatial refinement can also deal with this problem, it is advisable to switch the original deblocking filter on. In this case, an average rate reduction of up to $15.7\%$ can be achieved by spatial refinement using FSA and up to $7.2\%$ by the proposed deblocked refinement.
\begin{table}
\fontsize{8}{7}\selectfont
\centering
\begin{tabular}{|l|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{FSA \cite{Seiler2008c}} & \multicolumn{2}{c|}{Proposed} \\ \hline
H.264/AVC&&&&\\ deblocking & off & on & off & on \\ \hline
\multicolumn{5}{|l|}{Avg.\ Rate Reduction}\\ \hline
``D.\ City''& $-10.34\%$ & $-5.37\%$ & $-13.65\%$ & $-5.51\%$ \\ \hline
``Vimto'' & $-13.40\%$ & $-5.77\%$ & $-12.15\%$ & $-2.72\%$ \\ \hline
``Crew'' & $-16.89\%$ & $-10.95\%$ & $-15.69\%$ & $-7.22\%$ \\ \hline
``Jets'' & $-22.20\%$ & $-9.97\%$ & $-19.03\%$ & $-6.00\%$ \\ \hline
\multicolumn{5}{|l|}{ Avg.\ $\mathrm{ PSNR}$ gain}\\ \hline
``D.\ City'' & $0.74 \, \mathrm{dB}$ & $0.35 \, \mathrm{dB}$ & $0.99 \, \mathrm{dB}$ & $0.36 \, \mathrm{dB}$ \\ \hline
``Vimto'' & $0.66 \, \mathrm{dB}$ & $0.26 \, \mathrm{dB}$ & $0.59 \, \mathrm{dB}$ & $0.12 \, \mathrm{dB}$ \\ \hline
``Crew'' & $0.81 \, \mathrm{dB}$ & $0.45 \, \mathrm{dB}$ & $0.75 \, \mathrm{dB}$ & $0.29 \, \mathrm{dB}$ \\ \hline
``Jets'' & $0.48 \, \mathrm{dB}$ & $0.18 \, \mathrm{dB}$ & $0.41 \, \mathrm{dB}$ & $0.11 \, \mathrm{dB}$ \\ \hline
\end{tabular}
\caption{\emph{Average relative rate reduction and average $\, \mathrm{PSNR}$ gain, calculated according to \cite{Bjontegaard2001} for spatial refinement by FSA \cite{Seiler2008c} and the proposed algorithm.}}
\label{tab:psnr_gain}
\end{table}
Comparing the results for spatial refinement by FSA and the proposed deblocked refinement, it becomes obvious that FSA always can achieve a considerably higher coding efficiency. But, as mentioned before, FSA is computationally very expensive. Due to all the optimization that has been carried out for developing the H.264/AVC deblocking filter, the proposed refinement algorithm operates very fast. To quantify the speed-up, Table \ref{tab:refinement_time} shows the time per macroblock that is required for spatial refinement. The simulations have been carried out on one core of an Intel Core2 Quad, running at $2.4 \, \mathrm{GHz}$ and equipped with $8 \, \mathrm{GB}$ RAM. Since the proposed refinement algorithm only requires $0.075 \, \mathrm{msec}$ per macroblock it is approximately $900$ times faster than the original FSA refinement and a real-time implementation becomes realistic. This huge acceleration also can justify the lower quality of the spatial refinement
\begin{table}
\fontsize{8}{7}\selectfont
\centering
\begin{tabular}{|l|c|}
\hline
& Processing time\\ \hline
FSA \cite{Seiler2008c} & $67.1 \, \mathrm{msec}$\\ \hline
Proposed & $0.075 \, \mathrm{msec}$ \\ \hline\hline
Acceleration factor & $895$\\ \hline
\end{tabular}
\caption{\emph{Processing time per macroblock for spatial refinement.}}
\label{tab:refinement_time}
\end{table}
\section{Conclusion} \label{sec:conclusion}
Within the scope of this paper we proposed a method for reusing the H.264/AVC deblocking filter for spatio-temporal prediction in video coding. For this, the motion compensated prediction is spatially refined. By exploiting spatial as well as temporal correlations for prediction, a mean rate reduction of up to $7.2\%$ can be achieved by the proposed algorithm. Since the novel algorithm reuses the highly optimized H.264/AVC deblocking filter, the processing time for the refinement is very small and a real-time implementation is possible.
Although the proposed algorithm already yields an increased coding efficiency, future research has to focus on adapting the deblocking filter to the needs of spatial refinement and to take the actural partitioning of the macroblocks into into account for refinement.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,805 |
Q: Running c++ in Windows as .exe I wrote a c++ program and converted it to a .exe file with the hopes to let it run on windows without needing to install a compiler. I ran well on my main computer (which has a compiler so it ran as expected) but did not run on another device. The error message shown was "libgcc_s_dw2-1.dll was not found"
How can I run a c++ program (or .exe file) from a flashdrive or other medium on any windows machine without extra installation.
A: C++ programs require additional c++ standard and runtime libraries to run. Most linker links to these libraries dynamically. That is some dll are required for running the program. You can run the program from a portable flash drive by copying those dll to the same folder as the application.
Alternatively, some linkers also have option to statically link, that is incorporate the code in dll ( kind of ) to your binary exe.
So you can either find out what all dll is required and copy them to your local folder ( maybe use something like dependancy walker or this newer one, i haven't actually used the newer alternative). Or use static linking like shown in this answer.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 315 |
\section{Preliminaries }
The aim of this paper is to characterize the subsets $C$ of a locally
convex space $(X,\tau)$ whose normal cone $N_{C}$ is a maximal monotone
operator.
Here $(X,\tau)$ is a non-trivial (that is, $X\neq\{0\}$) real Hausdorff
separated locally convex space (LCS for short), $X^{\ast}$ is its
topological dual usually endowed with the weak-star topology denoted
by $w^{*}$, $(X^{*},w^{*})^{*}$ is identified with $X$, $\left\langle x,x^{\ast}\right\rangle :=x^{\ast}(x)=:c(x,x^{\ast})$,
for $x\in X$, $x^{\ast}\in X^{\ast}$ denotes the \emph{duality product}
or \emph{coupling }of $X\times X^{\ast}$, and $\operatorname*{Graph}T=\{(x,x^{*})\in X\times X^{*}\mid x^{*}\in T(x)\}$
stands for the \emph{graph} of $T:X\rightrightarrows X^{*}$.
Rockafellar showed in \cite[Theorem\ A]{MR0262827} that when $X$
is a Banach space and $f:X\to\overline{\mathbb{R}}$ is proper convex
lower semicontinuous then its \emph{convex subdifferential} $\partial f:X\rightrightarrows X^{*}$,
defined by $x^{*}\in\partial f(x)$ if $f(x)$ is finite and for every
$y\in X$, $f(y)\ge f(x)+\langle y-x,x^{*}\rangle$, is maximal monotone
($\partial f\in\mathfrak{M}(X)$ for short). In particular the \emph{normal
cone} to $C$ which is given by $N_{C}=\partial\iota_{C}\in\mathfrak{M}(X)$,
whenever $C\subset X$ is closed convex. Here $\iota_{C}(x)=0$, for
$x\in C$; $\iota_{C}(x)=+\infty$, for $x\in X\setminus C$ denotes
the \emph{indicator function} of $C$.
Therefore it is interesting to find when a normal cone is maximal
monotone outside the Banach space context.
Our main argument stems from the explicit form of the normal cone
Fitzpatrick function (see Theorem \ref{fnc} below) and our characterization
of maximal monotone operators as representable and of type NI (see
\cite[Theorem\ 2.3]{MR2207807} or \cite[Theorem\ 3.4]{MR2453098}).
Recall that the \emph{Fitzpatrick function} $\varphi_{T}:X\times X^{*}\rightarrow\overline{\mathbb{R}}$
of a multi-valued operator $T:X\rightrightarrows X^{*}$ is given
by (see \cite{MR1009594})
\begin{equation}
\varphi_{T}(x,x^{*}):=\sup\{\langle x-a,a^{*}\rangle+\langle a,x^{*}\rangle\mid(a,a^{*})\in\operatorname*{Graph}T\},\ (x,x^{*})\in X\times X^{*}.\label{ff}
\end{equation}
As usual, given a LCS $(E,\mu)$ and $A\subset E$ we denote by ``$\operatorname*{conv}A$''
the \emph{convex hull} of $A$, ``$\operatorname*{span}A$'' the
linear hull of $A$, ``$\operatorname*{cl}_{\mu}(A)=\overline{A}^{\mu}$''
the $\mu-$\emph{closure} of $A$, ``$\operatorname*{int}_{\mu}A$''
the $\mu-$\emph{topological interior }of $A$, ``$\operatorname*{core}A$''
the \emph{algebraic interior} of $A$. The use of the $\mu-$notation
is not enforced when the topology $\mu$ is clearly understood.
For $f,g:E\rightarrow\overline{\mathbb{R}}$ we set $[f\leq g]:=\{x\in E\mid f(x)\leq g(x)\}$;
the sets $[f=g]$, $[f<g]$, and $[f>g]$ being defined in a similar
manner. We write $f\ge g$ shorter for $f(z)\ge g(z)$, for every
$z\in E$.
For a multi-function $T:X\rightrightarrows X^{*}$, $D(T)=\operatorname*{Pr}_{X}(\operatorname*{Graph}T)$,
$R(T)=\operatorname*{Pr}_{X^{*}}(\operatorname*{Graph}T)$ stand for
the domain and the range of $T$ respectively, where $\operatorname*{Pr}_{X}$,
$\operatorname*{Pr}_{X^{*}}$ denote the projections of $X\times X^{*}$
onto $X$, $X^{*}$ respectively. When no confusion can occur, $T:X\rightrightarrows X^{\ast}$
will be identified with $\operatorname*{Graph}T\subset X\times X^{*}$.
The restriction of an operator $T:X\rightrightarrows X^{*}$ to $U\subset X$
is the operator $T|_{U}:X\rightrightarrows X^{*}$ defined by $\operatorname*{Graph}(T|_{U})=\operatorname*{Graph}T\cap(U\times X^{*})$.
The operator $T^{+}:X\rightrightarrows X^{\ast}$ whose graph is $\operatorname*{Graph}(T^{+}):=[\varphi_{T}\le c]$
describes all $(x,x^{*})\in X\times X^{*}$ that are monotonically
related (m.r. for short) to $T$, that is $(x,x^{*})\in[\varphi_{T}\le c]$
iff, for every $(a,a^{*})\in\operatorname*{Graph}T$, $\langle x-a,x^{*}-a^{*}\rangle\ge0$.
We consider the following classes of functions and operators on $(X,\tau)$
\begin{description}
\item [{$\Lambda(X)$}] the class formed by proper convex functions $f:X\rightarrow\overline{\mathbb{R}}$.
Recall that $f$ is \emph{proper} if $\operatorname*{dom}f:=\{x\in X\mid f(x)<\infty\}$
is nonempty and $f$ does not take the value $-\infty$,
\item [{$\Gamma_{\tau}(X)$}] the class of functions $f\in\Lambda(X)$
that are $\tau$\textendash lower semi-continuous (\emph{$\tau$\textendash }lsc
for short),
\item [{$\mathcal{M}(X)$}] the class of non-empty monotone operators $T:X\rightrightarrows X^{\ast}$
($\operatorname*{Graph}T\neq\emptyset$). Recall that $T:X\rightrightarrows X^{\ast}$
is \emph{monotone} if, for all $(x_{1},x_{1}^{*}),(x_{2},x_{2}^{*})\in\operatorname*{Graph}T$,
$$\left\langle x_{1}-x_{2},x_{1}^{\ast}-x_{2}^{\ast}\right\rangle \geq0$$
or, equivalently, $\operatorname*{Graph}T\subset\operatorname*{Graph}(T^{+})$.
\item [{$\mathfrak{M}(X)$}] the class of maximal monotone operators $T:X\rightrightarrows X^{*}$.
The maximality is understood in the sense of graph inclusion as subsets
of $X\times X^{*}$. It is easily seen that $T\in\mathfrak{M}(X)$
iff $T=T^{+}$.
\end{description}
To a proper function $f:(X,\tau)\rightarrow\overline{\mathbb{R}}$
we associate the following notions:
\begin{description}
\item [{$\operatorname*{Epi}f:=\{(x,t)\in X\times\mathbb{R}\mid f(x)\leq t\}$}] is
the \emph{epigraph} of $f$,
\item [{$\operatorname*{conv}f:X\rightarrow\overline{\mathbb{R}}$,}] the
\emph{convex hull} of $f$, which is the greatest convex function
majorized by $f$, $(\operatorname*{conv}f)(x):=\inf\{t\in\mathbb{R}\mid(x,t)\in\operatorname*{conv}(\operatorname*{Epi}f)\}$
for $x\in X$,
\item [{$\operatorname*{cl}_{\tau}\operatorname*{conv}f:X\rightarrow\overline{\mathbb{R}}$,}] the
\emph{$\tau-$lsc convex hull} of $f$, which is the greatest \emph{$\tau$\textendash }lsc
convex function majorized by $f$, $(\operatorname*{cl}_{\tau}\operatorname*{conv}f)(x):=\inf\{t\in\mathbb{R}\mid(x,t)\in\operatorname*{cl}_{\tau}(\operatorname*{conv}\operatorname*{Epi}f)\}$
for $x\in X$,
\item [{$f^{\ast}:X^{\ast}\rightarrow\overline{\mathbb{R}}$}] is the \emph{convex
conjugate} of $f:X\rightarrow\overline{\mathbb{R}}$ with respect
to the dual system $(X,X^{\ast})$, $f^{\ast}(x^{\ast}):=\sup\{\left\langle x,x^{\ast}\right\rangle -f(x)\mid x\in X\}$
for $x^{\ast}\in X^{\ast}$.
\end{description}
Accordingly, $\sigma_{C}(x^{*}):=\sup\{\langle x,x^{*}\rangle\mid x\in C\}=\iota_{C}^{*}(x^{*})$,
for $x^{*}\in X^{*}$. Recall that $f^{**}:=(f^{*})^{*}=\operatorname*{cl}\operatorname*{conv}f$
whenever $\operatorname*{cl}\operatorname*{conv}f$ (or equivalently
$f^{*}$) is proper, where for functions defined in $X^{*}$, the
conjugates are taken with respect to the dual system $(X^{*},X)$.
Throughout this article the conventions $\infty-\infty=\infty$, $\sup\emptyset=-\infty$,
and $\inf\emptyset=\infty$ are enforced while the use of the topology
notation is avoided when the topology is clearly understood.
All the considerations and results of this paper can be done with
respect to a separated dual system of vector spaces $(X,Y)$.
\section{Support points and the maximality of the normal cone}
Given $(X,\tau)$ a LCS and $C\subset X$, we denote by
\[
\operatorname*{Supp}C=\{x\in C\mid N_{C}(x)\neq\{0\}\}
\]
the set of \emph{support points of }$C$ and by
\[
C^{\#}:=\{x\in X\mid\forall(a,a^{*})\in\operatorname*{Graph}N_{C},\ \langle x-a,a^{*}\rangle\le0\}
\]
the \emph{portable hull of }$C$ which is the intersection of all
the supporting half-spaces that contain $C$ and are supported at
points in $C$.
\noindent \begin{center}
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\rput(8.444531,0.29){\Large $C^{\#}$}
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\end{pspicture}}
\par\end{center}
From their definitions, $\operatorname*{Supp}\emptyset=\operatorname*{Supp}X=\emptyset$,
$\emptyset^{\#}=X$, and
\[
x\in C^{\#}\Leftrightarrow\forall a\in\operatorname*{Supp}C,\forall a^{*}\in N_{C}(a),\ \langle x-a,a^{*}\rangle\le0,
\]
with the remarks that, for every $C\subset X$, one has $C^{\#}\neq\emptyset$
and $\operatorname*{cl}_{\tau}\operatorname*{conv}C\subset(\operatorname*{cl}_{\tau}\operatorname*{conv}C)^{\#}\subset C^{\#}$;
while $C^{\#}=X$ when $\operatorname*{Supp}C=\emptyset$, e.g., $X^{\#}=X$.
\begin{theorem} \label{fnc} Let $X$ be a LCS. For every $C\subset X$,
$(x,x^{*})\in X\times X^{*}$
\begin{equation}
\varphi_{N_{C}}(x,x^{*})=\iota_{C^{\#}}(x)+\sigma_{C}(x^{*}).\label{eq:fnc}
\end{equation}
In particular, for every $\emptyset\neq C\subset X$, $(x,x^{*})\in X\times X^{*}$,
$\varphi_{N_{C}}(x,0)=\iota_{C^{\#}}(x)\ge0$, $\varphi_{N_{C}}(x,x^{*})\le\iota_{C}(x)+\sigma_{C}(x^{*})$
and
\begin{equation}
C^{\#}:=\operatorname*{Pr}\,\!\!_{X}(\operatorname*{dom}\varphi_{N_{C}})=\{x\in X\mid\varphi_{N_{C}}(x,0)=(\le)0\}=D(N_{C}^{+}).\label{eq:}
\end{equation}
\end{theorem}
\strut
The last part of Theorem \ref{fnc} says that any monotone extension
$T$ of $N_{C}$ has $D(T)\subset C^{\#}$ or that we can extend $N_{C}$
monotonically only inside $C^{\#}\times X^{*}$.
\begin{theorem} \label{ncmm} Let $X$ be a LCS and let $C\subset X$.
The following are equivalent
\medskip
\emph{(i)} $N_{C}\in\mathfrak{M}(X)$,
\medskip
\emph{(ii) }$\varphi_{N_{C}}(x,x^{*})=\iota_{C}(x)+\sigma_{C}(x^{*})$,
$(x,x^{*})\in X\times X^{*}$,
\medskip
\emph{(iii)} $C^{\#}\subset(=)C$
\medskip
\emph{(iv)} $C=\{x\in X\mid\varphi_{N_{C}}(x,0)\le0\}$. \end{theorem}
Concerning the previous result note that $C$ is non-empty closed
convex whenever $N_{C}\in\mathfrak{M}(X)$. Also, subpoint (iv) can
be restated equivalently as
\medskip
(iv)$'$ $C$ is non-empty and $C=(\supset)\{x\in X\mid\varphi_{N_{C}}(x,0)\le(=)0\}$.
\strut
Theorem \ref{ncmm} has strong ties with the separation theorem. Assume
that $C\subset X$ is closed and convex. Then for every $x\in X\setminus C$
there is $n^{*}\in X^{*}\setminus\{0\}$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})=\sup\{\langle u,n^{*}\rangle\mid u\in C\}$.
In the next theorem we see that the maximality of $N_{C}$ is equivalent
to the possibility of picking, in the previous separation argument,
of a non-zero $n^{*}$ that attains its global maximum on $C$, that
is, $n^{*}\in R(N_{C})$ which is also called a \emph{support functional}
of $C$ (see e.g. \cite{MR0154092}).
\begin{theorem} \label{ncsep} Let $X$ be a LCS and let $C\subset X$.
Then $N_{C}\in\mathfrak{M}(X)$ (or $C=C^{\#}$) iff for every $x\in X\setminus C$
there is $n^{*}\in D(\partial\sigma_{C})=R(N_{C})$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})$.
\end{theorem}
\begin{corollary} \label{Bic} Let $X$ be a LCS and let $C\subset X$
be non-empty closed and convex. If $X$ is a Banach space or $\operatorname*{int}C\neq\emptyset$,
or $C$ is weakly compact then $N_{C}\in\mathfrak{M}(X)$. \end{corollary}
Note that $C^{\#}$, the portable hull of $C$, is the smallest set
formed by intersecting half-spaces that contain $C$ and are supported
at points in $C$, and, at the same time, the largest set on which
the normal cone $N_{C}$ can be extended monotonically.
\begin{proposition} \label{eNC} Let $X$ be a LCS. For every $C\subset X$,
$N_{C^{\#}}|_{C}=N_{C}$, $\operatorname*{Graph}N_{C}\subset\operatorname*{Graph}N_{C^{\#}}$,
$N_{C^{\#}}\in\mathfrak{M}(X)$, and $C^{\#\#}:=(C^{\#})^{\#}=C^{\#}$.
\end{proposition}
\begin{remark}\emph{ Every (maximal) monotone extension of $N_{C}$
has the domain contained in $C^{\#}$, that is, $N_{C}\subset T\in\mathcal{M}(X)\Rightarrow D(T)\subset D(N_{C}^{+})=C^{\#}$.
Therefore $N_{C^{\#}}$ is a maximal monotone extension of $N_{C}$
with the largest possible domain. In general, $N_{C^{\#}}$ is not
the only maximal monotone extension of $N_{C}$ (even with the largest
possible domain or with a normal cone structure). For example, for
$C=(0,1]\subset\mathbb{R}$, $N_{C}$ admits $N_{\overline{C}}$ and
$N_{C^{\#}}$ as two different maximal monotone extensions; moreover
$N_{C}$ has an infinity of maximal monotone extensions with the largest
possible domain $C^{\#}=(-\infty,1]$.} \end{remark}
The maximality of the subdifferential allows us to reprove and extend
some of the Bishop-Phelps results (see \cite{MR0154092}).
\begin{theorem} Let $X$ be a LCS. If, for every closed convex $C\subset X$,
$N_{C}\in\mathfrak{M}(X)$, then, for every closed convex $C\subset X$,
$\operatorname*{Supp}C$ is dense in $\operatorname*{bd}C$. \end{theorem}
\begin{remark} The previous results still holds for a fixed closed
convex $C\subset X$ if, for every $U\subset X$ closed convex such
that $C\cap\operatorname*{int}U\neq\emptyset$, $N_{C\cap U}\in\mathfrak{M}(X)$.
\end{remark}
\begin{theorem} Let $X$ be a LCS and let $C\subset X$ be closed
convex, free of lines, and finite-dimensional, that is, $\operatorname*{dim}(\operatorname*{span}C)<\infty$.
Then $N_{C}\in\mathfrak{M}(X)$ and $\operatorname*{cl}_{w^{*}}R(N_{C})=\operatorname*{cl}_{w^{*}}(\operatorname*{dom}\sigma_{C})$.
If, in addition, $C$ is bounded, then $\operatorname*{cl}_{w^{*}}R(N_{C})=X^{*}$.
\end{theorem}
\begin{proposition} under review \end{proposition}
\begin{corollary} Let $X$ be a LCS such that for every $f\in\Gamma(X)$,
$\operatorname*{Graph}\partial f\neq\emptyset$. Then for every closed
convex bounded $C\subset X$ the support functionals of $C$ are weak-star
dense in $X^{*}$. \end{corollary}
\begin{corollary} Let $(X,\|\cdot\|)$ be a Banach space. Then for
every closed convex bounded $C\subset X$ the support functionals
of $C$ are strongly dense in $X^{*}$.
If $C\subset X$ is merely closed and convex then the support functionals
of $C$ are strongly dense in the domain of $\sigma_{C}$. In other
words for every $\epsilon>0$, $f\in X^{*}$ such that $\sup_{C}f<+\infty$
there is $g\in X^{*}$, $x_{0}\in C$ such that $g(x_{0})=\sup_{C}g$
(that is, $g\in N_{C}(x_{0})$) and $\|f-g\|\le\epsilon$. \end{corollary}
\section{Partial portable hulls and sum representability}
Let $(X,\tau)$ be a LCS and $C,S\subset X$. We denote by $C_{S}^{\#}$
the \emph{partial portable hull of $C$ on $S$} which is the intersection
of all the (supporting) half-spaces that contain $C$ and are supported
at points in $S$,
\[
C_{S}^{\#}:=\{x\in X\mid\forall a\in C\cap S,\ a^{*}\in N_{C}(a),\ \langle x-a,a^{*}\rangle\le0\}.
\]
Equivalently
\[
x\in C_{S}^{\#}\Leftrightarrow\forall a\in S\cap\operatorname*{Supp}C,\ a^{*}\in N_{C}(a),\ \langle x-a,a^{*}\rangle\le0,
\]
with the remarks that $C_{S}^{\#}=X$ when $S\cap\operatorname*{Supp}C=\emptyset$
and that, for every $C,S\subset X$, $C_{S}^{\#}\neq\emptyset$, $C\subset\operatorname*{cl}_{\tau}\operatorname*{conv}C\subset C^{\#}\subset C_{S}^{\#}$,
$C_{S}^{\#}=C_{S\cap C}^{\#}=C_{S\cap\operatorname*{Supp}C}^{\#}$.
Note also that $C_{S}^{\#}=C^{\#}$ whenever $S\supset\operatorname*{Supp}C$.
\noindent \begin{center}
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\par\end{center}
\begin{proposition} \label{ncs} Let $X$ be a LCS and let $C,S\subset X$.
Then
\emph{(i)} $\operatorname*{Graph}(N_{C}|_{S})\subset\operatorname*{Graph}N_{C_{S}^{\#}}$
and
\[
N_{C}|_{S}=N_{C_{S}^{\#}}|_{S}\ {\rm iff}\ S\cap C=S\cap C_{S}^{\#}.
\]
In particular $N_{C^{\#}}|_{C}=N_{C}$.
\emph{(ii)} $(C_{S}^{\#})_{S}^{\#}=(C_{S}^{\#})^{\#}=C_{S}^{\#}$.
In particular $N_{C_{S}^{\#}}\in\mathfrak{M}(X)$. \end{proposition}
\begin{theorem} under review \end{theorem}
Recall the following notion ( see Definition 14 in \cite{liro-arxiv})
\begin{definition} \emph{Let $(X,\tau)$ be a LCS. An operator }$T:X\rightrightarrows X^{*}$\emph{
is }representable\emph{ }in\emph{ $C\subset X$ or $C-$}representable\emph{
if $C\cap D(T)\ne\emptyset$ and there is $h\in\Gamma_{\tau\times w^{*}}(X\times X^{*})$
such that $h\ge c$ and $[h=c]\cap C\times X^{*}=\operatorname*{Graph}(T|_{C})$.}
\end{definition}
Using the partial portable hull we can recover the representability
of the sum between a representable operator and the normal cone (see
Theorem 35 in \cite{liro-arxiv})
\begin{theorem} \label{psi-Tnc} Let $X$ be a LCS, let $T:X\rightrightarrows X^{*}$,
and let $C\subset X$ be closed convex such that $D(T)\cap\operatorname*{int}C\neq\emptyset$.
Then $\psi_{T+N_{C}}=\psi_{T|_{C}}\square_{2}\sigma_{C_{D(T)}^{\#}\times\{0\}}$
or
\begin{equation}
\psi_{T+N_{C}}(x,x^{*})=\min\{\psi_{T|_{C}}(x,x^{*}-u^{*})+\sigma_{C_{D(T)}^{\#}}(u^{*})\mid u^{*}\in X^{*}\},\ (x,x^{*})\in X\times X^{*},\label{psiTNC}
\end{equation}
\begin{equation}
[\psi_{T+N_{C}}=c]=[\psi_{T|_{C}}=c]+N_{C}.\label{TNCrepres}
\end{equation}
In particular, if, in addition, $T$ is $C-$representable then $T+N_{C}$
is representable. \end{theorem}
\section{Concluding remarks}
Let $(X,\tau)$ be a LCS and let us call a set $C\subset X$ \emph{portable}
if $C=C^{\#}$ (or $N_{C}\in\mathfrak{M}(X)$). Some of the results
of this paper can be summarized as follows:
\begin{itemize}
\item $C\subset X$ is portable iff for every $x\in X\setminus C$ there
is $n^{*}\in D(\partial\sigma_{C})=R(N_{C})$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})$
or, equivalently, $[\sigma_{C-x}<0]\cap R(N_{C})\neq\emptyset$.
\item If $C\subset X$ is closed convex and $X$ is a Banach space or $\operatorname*{int}C\neq\emptyset$,
or $C$ is weakly compact then $C$ is portable.
\item For every $C,S\subset X$, $C_{S}^{\#}$ is portable.
\item If every closed convex $C\subset X$ is portable then for every closed
convex $C\subset X$, $\operatorname*{Supp}C$ is dense in $\operatorname*{bd}C$.
\item Every closed convex, free of lines, and finite-dimensional set is
portable.
\end{itemize}
The results in this article are an incentive for studying the following
problems:
\begin{description}
\item [{(P1)}] Find cha racterizations of all LCS $X$ with the property
that, for every closed convex $C\subset X$, $N_{C}\in\mathfrak{M}(X)$.
\item [{(P2)}] Find characterizations of all LCS $X$ with the property
that, for every $f\in\Gamma(X)$, $\operatorname*{Graph}\partial f\neq\emptyset$.
\item [{(P3)}] Find characterizations of all LCS $X$ with the property
that, for every $f\in\Gamma(X)$, $\partial f\in\mathfrak{M}(X)$.
\end{description}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,505 |
\section{Introduction}
\label{sec:intro}
A unified view of human diseases and cellular functions across a broad range of human tissues is essential, not only for understanding basic biology but also for interpreting genetic variation and developing therapeutic strategies~\citep{Yeger2015,Okabe2014,Greene2015,Gtex2015}.
In particular, the precise functions of proteins frequently depend on the tissue, and different proteins can have different cellular functions in different tissues~\citep{Lois2002,Rakyan2008,Magger2012,Guan2012,Fagerberg2014,Yeger2015,Hu2016}.
While our view of the human protein-protein interaction network as a key source for studying protein function, is constantly expanding, much less is known about networks that form in biologically important environments such as within distinct tissues or in specific diseases~\citep{Yeger2015}. Although incredibly influential, current computational methods for extracting functional information from protein interaction networks lack tissue specificity as they assume that cellular function is constant across organs and tissues~\citep{Barutcuoglu2006,Mostafavi2008,Radivojac2013,Stojanova2013,Kramer2014,Zitnik2015}. In other words, cellular functions in heart are assumed to be the same as functions in skin. The methods are, hence, less successful in constructing {\em accurate maps of both where and how proteins act.} In particular, existing network-based methods are probably not the ultimate representation of human tissues for three reasons. (1) First, current methods for cellular function prediction on networks~\citep{Mostafavi2009,Radivojac2013,Zitnik2015,Vidulin2016} do not model networks with regards to patterns that span tissues, organs, and cellular systems. This means that a complex tissue involving a multiscale hierarchy of cellular subsystems is not readily captured by current models~\citep{Dutkowski2012,Carvunis2014}. (2) Second, many genome-scale functional maps~\citep{Lopes2011,Rolland2014,Kotlyar2015,Kitsak2016,Costanzo2016,Wang2016} are descriptive maps of physical or functional protein connectivity that do not, by themselves, predict cellular function. (3) Third, only few computational approaches~\citep{Magger2012,Guan2012,Ganegoda2014,Antanaviciute2015} used tissue-specific information to identify novel genes and relationships between genes. However, their focus was to leverage tissue specificity to improve prediction of global cellular functions and global gene-disease associations. As such, these approaches account for tissue specificity, but they do not resolve the challenge of predicting gene-function relationships that might be specific to a particular tissue. To be able to predict a range of tissue-specific functions one needs to design scalable multiscale models that can relate tissues to each other, extract rich feature representations for proteins in each tissue-specific network, and then use the extracted features for tissue-specific cellular function prediction.
\xhdr{Present work}
We present {\em{OhmNet}}\xspace, an algorithm for hierarchy-aware unsupervised feature learning in multi-layer networks. Our focus is on learning features of proteins in different tissues. We represent each tissue as a network, where nodes represent proteins. Tissue networks act as layers in a multi-layer network, where we use a hierarchy to model dependencies between the layers (\emph{i.e.}\xspace, tissues) (Figure~\ref{fig:ohmnet}). We then develop a computational framework that learns features of each node (\emph{i.e.}\xspace, protein) by taking into consideration connections between the nodes within each layer, together with inter-layer relationships between proteins active on different layers. More precisely, our approach embeds each protein in each tissue in a $d$-dimensional feature space such that proteins with similar network neighborhoods in similar tissues are embedded closely together.
In {\em{OhmNet}}\xspace, we define an objective function that is independent of the downstream prediction task, meaning that the feature representations are learned in a purely unsupervised way. This results in task-independent features, that, as we show, outperform task-specific approaches in predictive accuracy. Furthermore, since our features are not designed for a specific downstream prediction task, they generalize across a wide variety of tasks and tissues. For example, we use the learned features to study protein functions across different cellular systems (\eg, cell types, tissues, organs, and organ systems).
\begin{figure}
\begin{center}
\includegraphics[width=0.47\textwidth]{{zitnik-fig1}.pdf}
\caption{\textbf{A multi-layer network with four layers, where each layer represents a tissue-specific protein-protein interaction network.} The hierarchy $M$ encodes biological similarities between the tissues at multiple scales. {\em{OhmNet}}\xspace embeds each node in a $d$-dimensional feature space, which we use for tissue-specific protein function prediction. For example, layers $G_i$, $G_j$, $G_k$ and $G_l$, might represent brain tissue-specific interaction networks in cerebrum, hypothalamus, tegmentum, and medulla.
}
\label{fig:ohmnet}
\end{center}
\end{figure}
{\em{OhmNet}}\xspace builds on recent success of unsupervised representation learning methods based on neural architectures~\citep{Mikolov2013,Grover2016}. In particular, we develop a new form of structured regularization, which makes {\em{OhmNet}}\xspace especially suitable for multi-layer interdependent networks. Our key contribution lies in modeling the tissue taxonomy constraints by encoding relationships between the tissues in a tissue hierarchy and then using the structured regularization with the tissue hierarchy (Figure~\ref{fig:ohmnet}). This way {\em{OhmNet}}\xspace effectively learns multiscale feature representations for proteins that are consistent with the tissue hierarchy.
Our experiments focus on three tasks defined on a multi-layer tissue network: (i) a multi-label node classification task, where every protein is assigned zero, one or more tissue-specific cellular functions; (ii) a transfer learning task, where we predict cellular functions for a protein in one tissue based on classifiers trained on features from other tissues; and (iii) a network embedding visualization task, where we create meaningful tissue-specific visualizations that lay out proteins on a two-dimensional space. Since the multiscale protein feature vectors returned by {\em{OhmNet}}\xspace are task-independent, we use {\em{OhmNet}}\xspace one time only to learn the features for proteins in every tissue and at every scale of the tissue hierarchy. We can then solve the cellular function prediction task for any tissue using the appropriate tissue-specific protein features.
We contrast {\em{OhmNet}}\xspace's performance with that of state-of-the-art approaches for feature learning~\citep{Nickel2011,Tang2015,Cannistraci2013,Grover2016}, approaches for tissue-independent cellular function prediction~\citep{Mostafavi2008,Zuberi2013}, and approaches for prioritization of disease-causing genes in tissue-specific protein interaction networks~\citep{Magger2012,Guan2012}, which we adapted for the cellular function prediction task. We experiment with a multi-layer network having 107 genome-wide tissue-specific protein interaction layers, and we consider a tissue hierarchy describing 219 cellular systems in the human body. Experiments demonstrate that tissue-specific protein interaction layers provide the necessary protein and tissue context for predicting cellular function. {\em{OhmNet}}\xspace outperforms alternative approaches by up to 14.9\% on multi-label classification and up to 20.3\% on transfer learning. Another notable finding is that {\em{OhmNet}}\xspace outperforms alternative approaches, which are based on non-hierarchical versions of the same dataset, alluding to the benefits of modeling hierarchical tissue organization. We observe that neglecting the existence of tissues or aggregating tissue-specific interaction networks into a single network discards important biological information and affects performance on multi-label classification and transfer learning tasks. Finally, we exemplify the utility of {\em{OhmNet}}\xspace for exploring the multiscale structure of tissues. In a case study on nine brain tissue networks, we show that {\em{OhmNet}}\xspace's features inherently encode a multiscale brain organization.
The rest of the paper is organized as follows. In Section~\ref{sec:related}, we briefly survey related work in feature learning for networks. We present the technical details of {\em{OhmNet}}\xspace in Section~\ref{sec:ohmnet}. In Section~\ref{sec:data}, we describe the multi-layer tissue network and the tissue hierarchy. We empirically evaluate {\em{OhmNet}}\xspace in Section~\ref{sec:results} and conclude with directions for future work in Section~\ref{sec:conclusion}.
\section{Related work}
\label{sec:related}
We have seen in Section~\ref{sec:intro} that despite the abundance of methods for cellular function prediction, only a few, if any, take into account biologically important contexts given by human tissues. We now turn our focus to the problem of feature learning in networks.
Most approaches for automatic (\emph{i.e.}\xspace, non-hand-engineered) feature learning in networks can be categorized into matrix factorization and neural network embedding based approaches. In matrix factorization, a network is expressed as a data matrix where the entries represent relationships. The data matrix is projected to a low dimensional space using linear techniques based on SVD~\citep{Tang2012}, or non-linear techniques based on multi-dimensional scaling~\citep{Tenenbaum2000,Belkin2001,Hou2014}. These methods have two important drawbacks. First, they do not account for important structures typically exhibited in networks such as high sparsity and skewed degree distribution. Second, matrix factorization methods perform a global factorization of the data matrix while a local-centric method might often yield more useful feature representations~\citep{Kramer2014}.
Limitations of matrix factorization are overcome by neural network embeddings. Recent studies focused on embedding nodes into low-dimensional vector spaces by first using random walks to construct the network neighborhood of every node in the graph, and then optimizing an objective function with network neighborhoods as input~\citep{Perozzi2014,Tang2015,Grover2016}. The objective function is carefully designed to preserve both the local and global network structures. A state-of-the-art neural network embedding algorithm is the Node2vec algorithm~\citep{Grover2016}, which learns feature representations as follows: it scans over the nodes in a network, and for every node it aims to embed it such that the node's features can predict nearby nodes, that is, node's feature predict which other nodes are part of its network neighborhood. Node2vec can explore different network neighborhoods to embed nodes based on the principles of homophily (\emph{i.e.}\xspace, network communities) as well as structural equivalence (\emph{i.e.}\xspace, structural roles of nodes).
However, a challenging problem for neural network embedding-based methods is to learn features in multi-layer networks. Existing methods can learn features in multi-layer networks either by treating each layer independently of other layers, or by aggregating the layers into a single (weighted) network. However, neglecting the existence of multiple layers or aggregating the layers into a single network, alters topological properties of the system as well as the importance of individual nodes with respect to the entire network structure~\citep{De2016}. This is a major shortcoming of prior work that can lead to a wrong identification of the most versatile nodes~\citep{De2015} and overestimation of the importance of more marginal nodes~\citep{De2014}. As we shall show, this shortcoming also affects predictive accuracy of the learned features. Our approach {\em{OhmNet}}\xspace overcomes this limitation since it learns features in a multi-layer network in the context of the entire system structure, bridging together different layers and generalizing methods developed for learning features in single-layer networks.
In biological domains, measures based on similarities of nodes' extended network neighborhoods are well established for predicting protein functions. Several approaches use graphlets~\citep{Przulj2007} to systematically describe network structure around each node. This is done by counting how many instances of small subgraph patterns occur in the network neighborhood of a given node. Graphlet-based methods, such as graphlet degree vectors~\citep{Hayes2013}, can thus be seen as an alternative approach for extracting feature representations for nodes. In contrast to neural embedding-based methods, such as {\em{OhmNet}}\xspace, which learn continuous feature representations, graphlet-based methods return discrete counts of motif occurrences. Further, graphlet-based methods in their current form cannot be applied to multi-layer networks without collapsing the network layers into one network.
Finally, there exists recent work for task-dependent feature learning based on graph-specific deep network architectures~\citep{Zhai2015,Li2015,Xiaoyi2014,Wang2016b}. Our approach differs from those approaches in two important ways. First, those architectures are task-dependent, meaning they directly optimize the objective function for a downstream prediction task, such as cellular function prediction in a particular tissue, using several layers of non-linear transformations. Second, those architectures do not model rich graph structures, such as multi-layer networks with hierarchies.
\begin{methods}
\section{Feature learning in multi-layer networks}
\label{sec:ohmnet}
We formulate feature learning in multi-layer networks as a maximum likelihood optimization problem. Let $V$ be a given set of $N$ nodes (\eg, proteins) $\{u_1, u_2, \dots, u_N\},$ and let there be $K$ types of edges (\eg, protein interactions in different tissues) between pairs of nodes $u_1, u_2, \dots, u_N$. A multi-layer network is a general system in which each biological context is represented by a distinct {\em layer} $i$ (where $i = 1, 2, \dots, K$) of a system (Figure~\ref{fig:ohmnet}). We use the term {\em single-layer network (layer)} for the network $G_i = (V_i, E_i)$ that indicates the edges $E_i$ between nodes $V_i \subseteq V$ within the same layer $i$. Our analysis is general and applies to any (un)directed, (un)weighted multi-layer network.
We take into account the possibility that a node $u_k$ from layer $i$ can be related to any other node $u_h$ in any other layer $j$. We encode information about the dependencies between layers in a hierarchical manner that we use in the learning process. Let the hierarchy be a directed tree $\mathcal{M}$ defined over a set $M$ of elements by the parent-child relationships given by $\pi : M \rightarrow M,$ where $\pi(i)$ is the parent of element $i$ in the hierarchy (Figure~\ref{fig:ohmnet}). Let $T \subset M$ be the set of all leaves in the hierarchy. Let $T_i$ be the set of all leaves in the sub-hierarchy rooted at $i$. We assume that each layer $G_i$ is attached to one leaf in the hierarchy. As a result, the hierarchy $\mathcal{M}$ has exactly $K$ leaves. For convenience, let $C_i$ denote the set of all children of element $i$ in the hierarchy.
The problem of feature learning in a multi-layer network is to learn functions $f_1, f_2, \dots, f_K$, such that each function $f_i : V_i \rightarrow \mathbb{R}^d$ maps nodes in $V_i$ to feature representations in $\mathbb{R}^d$. Here, $d$ is a parameter specifying the number of dimensions in the feature representation of one node. Equivalently, $f_i$ is a matrix of $|V_i| \times d$ parameters.
We proceed by describing {\em{OhmNet}}\xspace, our approach for feature learning in multi-layer networks. {\em{OhmNet}}\xspace has two components:
\begin{itemize}
\item {\bf single-layer network objectives}, in which nodes with similar network neighborhoods in each layer are embedded close together,
\item {\bf hierarchical dependency objectives}, in which nodes in nearby layers in the hierarchy are encouraged to share similar features.
\end{itemize}
We start by describing the model that considers the layers independently of each other. We then extend the model to encourage nodes which are nearby in the hierarchy to have similar features.
\subsection{Single-layer network objectives}
We start by formalizing the intuition that nodes with similar network neighborhoods in each layer should share similar features. For that, we specify one objective for each layer in a given multi-layer network. We shall later discuss how {\em{OhmNet}}\xspace incorporates the dependencies between different layers.
Our goal is to take layer $G_i$ and learn $f_i$ which embeds nodes from similar network regions, or nodes with similar structural roles, closely together. In {\em{OhmNet}}\xspace, we aim to achieve this goal by specifying the following objective function for each layer $G_i$. Given a node $u \in V_i$, the objective function $\omega_i$ seeks to predict, which nodes are members of $u$'s network neighborhood $N_{i}(u)$ based on the learned node features $f_i$:
\begin{equation}
\omega_i(u) = \log Pr(N_{i}(u) | f_i(u)),
\end{equation}
where the conditional likelihood of every node-neighborhood node pair is modeled independently as:
\begin{equation}
Pr(N_{i}(u) | f_i(u)) = \prod_{v \in N_i(u)} Pr(v | f_i(u)).
\end{equation}
The conditional likelihood is a softmax unit parameterized by a dot product of nodes' features, which corresponds to a single-layer feed-forward neural network:
\begin{equation}
Pr(v | f_i(u)) = \frac{\exp(f_i(v) f_i(u))}{\sum_{z \in V_i} \exp(f_i(z) f_i(u))}.
\end{equation}
Given a node $u$, maximization of $\omega_i(u)$ tries to maximize classification of nodes in $u$'s network neighborhood based on $u$'s learned representation.
The objective $\Omega_i$ is defined for each layer $i$:
\begin{equation}
\Omega_i = \sum_{u \in V_i} \omega_i(u), \;\;\; \textrm{for } i=1, 2, \dots, K.
\label{eq:omega-def}
\end{equation}
The objective is inspired by the intuition that nodes with similar network neighborhoods tend to have similar meanings, or roles, in a network. It formalizes this intuition by encouraging nodes in similar network neighborhoods to share similar features.
We found that a flexible notion of a network neighborhood $N_i$ is crucial to achieve excellent predictive accuracy on a downstream cellular function prediction task~\citep{Grover2016}. For that reason, we use a randomized procedure to sample many different neighborhoods of a given node $u$. Technically, the network neighborhood $N_{i}(u)$ is a set of nodes that appear in an appropriately biased random walk defined on layer $G_i$ and started at node $u$~\citep{Grover2016}. The neighborhoods $N_i(u)$ are not restricted to just immediate neighbors but can have vastly different structures depending on the sampling strategy.
Next, we expand {\em{OhmNet}}\xspace's single-layer network objectives to leverage information provided by the tissue taxonomy and this way inform embeddings across different layers.
\subsection{Hierarchical dependency objectives}
So far, we specified $K$ layer-by-layer objectives each of which estimates node features in its layer independently of node features in other layers. This means that nodes in different layers representing the same entity have features that are learned independently of each other.
To harness the dependencies between the layers, we expand {\em{OhmNet}}\xspace with terms that encourage sharing of protein features between the layers. Our approach is based on the assumption that nearby layers in the hierarchy are semantically close to each other and hence proteins/nodes in them should share similar features. For example, in the tissue multi-layer network, we model the fact that the ``medulla'' layer is part of the ``brainstem'' layer, which is, in turn, part of the ``brain'' layer. We use the dependencies among the layers to define a joint objective for regularization of the learned features of proteins.
We propose to use the hierarchy in the learning process by incorporating a recursive structure into the regularization term for every element in the hierarchy $\mathcal{M}$. Specifically, we propose the following form of regularization for node $u$ that resides in element $i$ of the hierarchy $\mathcal{M}$:
\begin{equation}
c_i(u) = \frac{1}{2} \| f_i(u) - f_{\pi(i)}(u) \|_2^2.
\end{equation}
This recursive form of regularization enforces the features of node $u$ in the hierarchy $i$ to be similar to the features of node $u$ in $i$'s parent $\pi(i)$ under the Euclidean norm. When regularizing features of all nodes across all elements of the hierarchy, we obtain:
\begin{equation}
C_i = \sum_{u \in L_i} c_i(u), \;\;\; \textrm{where } L_i = \cup_{j \in T_i} V_j
\end{equation}
In words, we specify the features for both leaf as well as internal, \emph{i.e.}\xspace, non-leaf, elements in the hierarchy, and we regularize the features of sibling (\emph{i.e.}\xspace, sharing the same parent) hierarchy elements towards features in the common parent element in the hierarchy.
\xhdr{Node features at multiple scales}
It is important to notice that {\em{OhmNet}}\xspace's structured regularization allows us to learn feature representations at multiple scales. For example, consider a multi-layer network in Figure~\ref{fig:hie-toy}, consisting of four layers that are interrelated by a two-level hierarchy. {\em{OhmNet}}\xspace learns the mappings $f_i$, $f_j$, $f_k$, and $f_l$ that map nodes in each layer into a $d$-dimensional feature space. Additionally, {\em{OhmNet}}\xspace also learns the mapping $f_2$ representing features for nodes appearing in the hierarchy leaves $T_2$, \emph{i.e.}\xspace, $V_i \cup V_j$, at an intermediate scale, and the mapping $f_1$ representing features for nodes appearing in the hierarchy leaves $T_1$, \emph{i.e.}\xspace, $V_i \cup V_j \cup V_k \cup V_l$, at the highest scale.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{{zitnik-fig2}.pdf}
\caption{\textbf{A multi-layer network with four layers.} Relationships between the layers are encoded by a two-level hierarchy $\mathcal{M}$. Leaves of the hierarchy correspond to the network layers. Given networks $G_i$ and hierarchy $\mathcal{M}$, {\em{OhmNet}}\xspace learns node embeddings captured by functions $f_i$.
}
\label{fig:hie-toy}
\end{center}
\end{figure}
The modeling of relationships between layers in a multi-layer network has several implications:
\begin{itemize}
\item First, the model encourages nodes which are in nearby layers in the hierarchy to share similar features.
\item Second, the model shares statistical strength across the hierarchy as nodes in different layers representing the same protein share features through ancestors in the hierarchy.
\item Third, this model is more efficient than the fully pairwise model. In the fully pairwise model, the dependencies between layers are modeled by pairwise comparisons of nodes across all pairs of layers, which takes $O(K^2 N)$ time, where $K$ is the number of layers and $N$ is the number of nodes. In contrast, {\em{OhmNet}}\xspace models inter-layer dependencies according to the parent-child relationships specified by the hierarchy, which takes only $O(|M|N)$ time. Since {\em{OhmNet}}\xspace's hierarchy is a tree, it holds that $|M| \ll K^2$, meaning that the proposed model scales more easily to large multi-layer networks than the fully pairwise model.
\item Finally, the hierarchy is a natural way to represent and model biological systems spanning many different biological scales~\citep{Carvunis2014,Greene2015,Yu2016}.
\end{itemize}
\subsection{Full {\em{OhmNet}}\xspace model}
Given a multi-layer network consisting of layers $G_1, G_2, \dots, G_K$, and a hierarchy encoding relationships between the layers, the {\em{OhmNet}}\xspace's goal is to learn the functions $f_1, f_2, \dots, f_K$ that map from nodes in each layer to feature representations. {\em{OhmNet}}\xspace achieves this goal by fitting its feature learning model to a given multi-layer network and a given hierarchy, \emph{i.e.}\xspace, by finding the mapping functions $f_1, f_2, \dots, f_K$ that maximize the data likelihood.
Given the data, {\em{OhmNet}}\xspace aims to solve the following maximum likelihood optimization problem:
\begin{equation}
\max_{f_1, f_2, \dots, f_{|M|}} \sum_{i \in T} \Omega_i - \lambda \sum_{j \in M} C_j,
\label{eq:opt-problem}
\end{equation}
which includes the single-layer network objectives for all network layers, and the hierarchical dependency objectives for all hierarchy elements. In Eq.~(\ref{eq:opt-problem}), parameter $\lambda$ is a user-specified parameter representing the regularization strength. While the optimization problem in Eq.~(\ref{eq:opt-problem}) is non-convex due to the non-convexity of the single-layer objective~\citep{Grover2016}, stochastic gradient with negative sampling can be used to efficiently solve the problem.
One appealing property of {\em{OhmNet}}\xspace is that by solving the problem in Eq.~(\ref{eq:opt-problem}) we obtain estimates for functions $f_1, f_2, \dots, f_K$ located in the leaf elements of the hierarchy (\emph{i.e.}\xspace, layers of a given multi-layer network), as well as estimates for functions $f_{K+1}, f_{K+2}, \dots, f_{|M|}$ located in the internal elements of the hierarchy.
\subsection{The {\em{OhmNet}}\xspace algorithm}
The pseudocode for {\em{OhmNet}}\xspace is given in Algorithm~\ref{alg:ohmnet}.
In the first phase, {\em{OhmNet}}\xspace applies the Node2vec's algorithm~\citep{Grover2016} to construct network neighborhoods for each node in every layer. Given a layer $G_i$ and a node $u \in V_i$, the algorithm simulates a user-defined number of fixed length random walks started at node $u$ (step 4 in Algorithm~\ref{alg:ohmnet}).
In the second phase, {\em{OhmNet}}\xspace uses an iterative approach in which features associated with each object in the hierarchy are iteratively updated by fixing the rest of the features. The iterative approach has the advantage that it can easily incorporate the closed-form updates developed for the internal objects of the hierarchy (step 11 in Algorithm~\ref{alg:ohmnet}), thereby accelerating the convergence of {\em{OhmNet}}\xspace algorithm. For each leaf object $i$, {\em{OhmNet}}\xspace isolates the terms in the optimization problem in Eq.~(\ref{eq:opt-problem}) that depend on the model parameters defining function $f_i$. {\em{OhmNet}}\xspace then optimizes Eq.~(\ref{eq:opt-problem}) by performing one epoch of stochastic gradient descent (SGD1) over $f_i$'s model parameters (step 15 in Algorithm~\ref{alg:ohmnet}).
The two phases of {\em{OhmNet}}\xspace are executed sequentially. The {\em{OhmNet}}\xspace algorithm scales to large multi-layer networks because each phase is parallelizable and executed asynchronously. The choice to model the dependencies between network layers using the hierarchical model requires $O(|M| N)$ time instead of the fully pairwise model, which requires $O(K^2 N)$ time.
\begin{algorithm}
\caption{\textbf{The {\em{OhmNet}}\xspace algorithm.}}
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE Multi-layer network, $(G_1, G_2, \dots, G_K)$ with $G_i = (V_i, E_i)$, Hierarchy, $\mathcal{M}$, Feature representation size, $d$, Network neighborhood strategy, $S$, Regularization strength, $\lambda$\\[1mm]
\FOR {$i \in T$}
\FOR {$u \in V_i$}
\STATE $N_{i}(u) = \textrm{Node2vecWalk}(G_i, u, S)$ \citep{Grover2016}
\ENDFOR
\ENDFOR
\WHILE {$f_1, f_2, \dots, f_{|M|}$ not converged}
\FOR {$i \in M$}
\IF {$i \in T$}
\FOR {$u \in V_i$}
\STATE $f_i(u) \!= \!\textrm{SGD1}(N_{i}(u), d, \lambda)$ by Eq.~(\ref{eq:opt-problem})
\ENDFOR
\ELSE
\FOR {$u \in \cup_{j \in T_i} V_j$}
\STATE $f_i(u) \!=\! \frac{1}{|C_{i}| + 1} ( f_{\pi(i)}(u) \!+\! \sum_{c \in C_{i}} \! f_c(u))$
\ENDFOR
\ENDIF
\ENDFOR
\ENDWHILE
\RETURN $f_1, f_2, \dots, f_{|M|}$
\end{algorithmic}
\label{alg:ohmnet}
\end{algorithm}
\end{methods}
\section{Tissue-specific interactome data}
\label{sec:data}
To construct the human protein-protein interaction (PPI) network, tissue-specific network layers, tissue hierarchy, and tissue-specific gene-function relationships, we downloaded and used standard protein, tissue, and function information from various reputable data sources.
\subsection{Tissue hierarchy}\label{sec:tissue-hierarchy}
We retrieved the mapping of tissues in the Human Protein Reference Database (HPRD)~\citep{Prasad2009} to tissues in the BRENDA Tissue Ontology~\citep{Chang2014} from \cite{Greene2015}. The data is provided as a supplementary dataset in \cite{Greene2015}. The hierarchical relationships between tissues were then determined by the directed acyclic graph structure of the BRENDA Tissue Ontology. Examples of tissues included: muscle, adrenal cortex, bone marrow, and spleen (Figure~\ref{fig:tissue-hierarchy}).
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{{zitnik-fig3}.pdf}
\caption{\textbf{The tissue hierarchy considered in this work.} The tissue hierarchy is a directed tree defined over $|M| = 219$ tissue terms from the BRENDA Tissue Ontology. Edges in the tree point from children to parents based on ontological relationships: ``develops\_from'', ``is\_a'', ``part\_of'', and ``related\_to''. The $K = 107$ tissues with tissue-specific protein interaction networks are the blue leaves in the tree.}
\label{fig:tissue-hierarchy}
\end{center}
\end{figure}
\subsection{Tissue-specific interaction networks}\label{sec:interactome-data}
We took the gene-to-tissue mapping compiled by \cite{Greene2015}. \citeauthor{Greene2015} mapped genes to HPRD tissues based on low-throughput tissue-specific gene expression data. The gene-to-tissue mapping was then combined with the human PPI network. The resulting multi-layer tissue network had 107 layers, each layer corresponded to a PPI network specific to a particular tissue. Details are provided next.
The human PPI network was collected from \cite{Orchard2013,Rolland2014,Chatr2015,Prasad2009,Ruepp2010,Menche2015}. Considered were physical protein-protein interactions with supported by experimental evidence. It should be noted that interactions based on gene expression and evolutionary data were not considered. The global (unweighted) human PPI network has 21,557 proteins interconnected by 342,353 interactions. The reader is referred to \cite{Menche2015} for a detailed description of the data.
For each of 107 tissues, a tissue-specific human PPI network was constructed based on the global PPI network. For a given tissue, every edge in the global PPI network was labeled as specifically co-expressed in that tissue using the criterion developed by \cite{Greene2015}. \citeauthor{Greene2015} labeled each edge as specifically co-expressed if either both proteins are specific to that tissue or one protein is tissue-specific and the other is ubiquitous. Lists of specifically co-expressed proteins were retrieved from \cite{Greene2015}. Finally, the PPI network specific to a particular tissue is a subnetwork of the global PPI network, induced by the set of specifically co-expressed edges in that tissue.
\subsection{Tissue-specific cellular functions and gene annotations}
\label{sec:go-data}
Associations between tissues and cellular functions were retrieved from \cite{Greene2015}. \citeauthor{Greene2015} manually curated biological processes in the Gene Ontology\citep{Ashburner2000} (GO) and mapped them to tissues in the BRENDA Tissue Ontology~\citep{Chang2014} based on whether a given biological process is specifically active in a given tissue. The data is provided as a supplementar dataset in \cite{Greene2015}. An example of a cellular function-tissue pair is ``low-density lipoprotein particle remodeling'' in the blood plasma tissue.
All gene annotations were propagated along the ontology hierarchy. Considered are functions with at least 15 annotated proteins~\citep{Guan2012}. In total, there are 584 tissue-specific cellular functions covering 48 distinct tissues. Each tissue-specific function is assigned to one or more leaves in the tissue hierarchy (Section~\ref{sec:tissue-hierarchy}).
\section{Results}
\label{sec:results}
The {\em{OhmNet}}\xspace's objective in Eq.~(\ref{eq:opt-problem}) is independent of any downstream task. This flexibility offered by {\em{OhmNet}}\xspace makes the learned feature representations suitable for a variety of analytics tasks discussed below.
\subsection{Prediction of tissue-specific cellular functions}\label{sec:classify}
\xhdr{Experimental setup}
We view the problem of predicting cellular functions as solving a multi-label node classification task. Here, every node (\emph{i.e.}\xspace, protein) is assigned one or more labels (\emph{i.e.}\xspace, cellular functions from the GO) from a finite set of labels (\emph{i.e.}\xspace, all cellular functions in the GO, see Section~\ref{sec:go-data}).
We apply {\em{OhmNet}}\xspace, which for every node in every layer learns a separate feature vector in an unsupervised way. Thus, for every layer and every function we then train a separate one-vs-all linear classifier using the modified Huber loss with elastic net regularization. Using cross validation, we observe 90\% of proteins and all their cellular functions across the layers during the training phase. The task is then to predict the tissue-specific functions for the remaining 10\% of proteins.
We evaluate the performance of {\em{OhmNet}}\xspace against the following feature learning approaches:
\begin{itemize}
\item RESCAL tensor decomposition~\citep{Nickel2011}: This is a tensor factorization approach that takes the multi-layer network structure into account. Given $X_i$, a normalized Laplacian matrix of layer $G_i$, matrix $X_i$ is factorized as: $X_i = A R_i A^T$, for $i = 1, 2, \dots, K.$ Here, matrix $A$ contains $d$-dimensional feature representation for nodes.
\item Minimum curvilinear embedding~\citep{Cannistraci2013}: This is a non-linear unsupervised framework that embeds nodes in a low-dimensional space. The approach was originally developed for protein interaction prediction, aiming to embed protein pairs representing good candidate interactions closer to each other. It utilizes a network denoising method as well as structural information provided by the PPI network topology.
\item LINE~\citep{Tang2015}: This approach first learns $d/2$ dimensions based on immediate network neighbors of nodes, and then the next $d/2$ dimensions based on network neighbors at a 2-hop distance.
\item Node2vec~\citep{Grover2016}: This approach learns $d$-dimensional features for nodes based on a biased random walk procedure that flexibly explores network neighborhoods of nodes.
\end{itemize}
In addition, we evaluate the performance of {\em{OhmNet}}\xspace against the following tissue-specific/agnostic function prediction approaches:
\begin{itemize}
\item GeneMania~\citep{Zuberi2013}: This is a supervised approach that takes a multi-layer network as input and directly predicts cellular functions in two separate phases. In the first phase, it aggregates the layers into one weighted network by weighting the layers according to their utility for predicting a given function. It then uses a label propagation algorithm on the weighted network to predict the function.
\item Tissue-specific network propagation~\citep{Magger2012}: This approach assigns a prior score to proteins associated with known functions that are phenotypically similar to the query function. This score is then propagated through a network in an iterative process. The approach was developed for tissue-specific disease gene prioritization.
\item Network-based tissue-specific SVM~\citep{Guan2012}: This approach adopts the network-based candidate gene prediction scheme. Essentially, the connection weights in a network to all positive examples (\emph{i.e.}\xspace, genes already known to be related to a phenotype) are utilized as features for linear support vector machine (SVM) classification. The approach was developed for tissue-specific phenotype and disease gene prioritization.
\end{itemize}
The parameter settings for every approach are determined using internal cross validation procedure with a grid search over candidate parameter values. Specifically, $d=128$ is used in all experiments.
Last, we aim to evaluate the benefit of our proposed multi-layer representation of the tissue networks. To this end we also consider two additional network representations:
\begin{itemize}
\item Independent layers: This approach learns features for nodes in each layer by running LINE or Node2vec algorithm on one layer at a time and independently of other layers in the network.
\item Collapsed layers: This approach first aggregates the layers into a single network by connecting nodes representing the same entity in different layers to each other. It then learns feature for nodes in the aggregated network.
\end{itemize}
\xhdr{Experimental results}
Table~\ref{tab:other-approaches} and Figure~\ref{fig:tissue-auroc} give the area under the curve (AUC) scores of tissue-specific protein function prediction.
From the results, we see how modeling the tissues and their hierarchy spanning multiple biological scales allows {\em{OhmNet}}\xspace to outperform other benchmark approaches. {\em{OhmNet}}\xspace outperforms GeneMania~\citep{Mostafavi2008,Zuberi2013} by 10.7\%, which can be explained by GeneMania's inability to weight layers in the tissue network according to a multiscale tissue organization that is consistent with the tissue taxonomy constraints. We also compared {\em{OhmNet}}\xspace to two other methods~\citep{Guan2012,Magger2012} that were so far demonstrated as useful for mining tissue-specific protein relationships. {\em{OhmNet}}\xspace has produced more accurate predictions, surpassing other methods by up to 12.0\% (AUROC) and up to 26.8\% (AUPRC).
Independent modeling of the layers showed worse performance than collapsing the layers into one network. We observed that Collapsed LINE achieved a gain of 3.3\% over Independent LINE, and Collapsed Node2vec achieved a gain of 7.4\% over Independent Node2vec. However, approaches that neglect the existence of tissues or collapse tissue-specific protein interaction networks into a single network discard important information about the rich hierarchy of biological systems, giving {\em{OhmNet}}\xspace a 14.0\% gain over Collapsed LINE, and a 8.5\% gain over Collapsed Node2vec in AUC scores. This result is a good illustration of how tissue specificity is related to specialization of protein function~\citep{Greene2015}, and approaches able to directly profile proteins' distinct interaction neighborhoods in different tissues can leverage this specificity to generate more accurate hypotheses about tissue-specific protein actions.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{{zitnik-fig4}.pdf}
\caption{\textbf{Area under ROC curve (AUROC) scores for tissue-specific cellular function prediction by {\em{OhmNet}}\xspace.} Numbers in the brackets are counts of tissue-specific cellular functions per tissue.}
\label{fig:tissue-auroc}
\end{center}
\end{figure}
\begin{table}
\processtable{Area under ROC curve (AUROC) and area under precision-recall curve (AUPRC) scores for tissue-specific cellular function prediction. Values in the brackets are halves of the interquartile distance. {\em{OhmNet}}\xspace's results are statistically significant with a p-value of less than 0.05.
\label{tab:other-approaches}}{
\setlength{\tabcolsep}{4pt}
\begin{tabular}{lcc}\toprule
Approach & AUROC & AUPRC \\\midrule
Tensor decomposition & \multirow{2}{*}{0.674 ($\pm$ 0.124)} & \multirow{2}{*}{0.235 ($\pm$ 0.052)} \\
\citep{Nickel2011} & & \\
Minimum curvilinear embedding & \multirow{2}{*}{0.674 ($\pm$ 0.064)} & \multirow{2}{*}{0.248 ($\pm$ 0.071)} \\
\citep{Cannistraci2013} & & \\
Independent LINE & \multirow{2}{*}{0.642 ($\pm$ 0.053)} & \multirow{2}{*}{0.261 ($\pm$ 0.068)} \\
\citep{Tang2015} & & \\
Collapsed LINE & \multirow{2}{*}{0.663 ($\pm$ 0.047)} & \multirow{2}{*}{0.271 ($\pm$ 0.053)} \\
\citep{Tang2015} & & \\
Independent Node2vec & \multirow{2}{*}{0.649 ($\pm$ 0.063)} & \multirow{2}{*}{0.283 ($\pm$ 0.052)} \\
\citep{Grover2016} & & \\
Collapsed Node2vec & \multirow{2}{*}{0.697 ($\pm$ 0.085)} & \multirow{2}{*}{0.298 ($\pm$ 0.061)} \\
\citep{Grover2016} & & \\\midrule
GeneMania & \multirow{2}{*}{0.683 ($\pm$ 0.077)} & \multirow{2}{*}{0.274 ($\pm$ 0.094)} \\
\citep{Zuberi2013} & & \\
Network-based tissue-specific SVM & \multirow{2}{*}{0.701 ($\pm$ 0.091)} & \multirow{2}{*}{0.281 ($\pm$ 0.059)} \\
\citep{Guan2012} & & \\
Tissue-specific network propagation & \multirow{2}{*}{0.675 ($\pm$ 0.051)} & \multirow{2}{*}{0.265 ($\pm$ 0.083)} \\
\citep{Magger2012} & & \\\midrule
{\em{OhmNet}}\xspace & \multirow{2}{*}{0.756 ($\pm$ 0.067)} & \multirow{2}{*}{0.336 ($\pm$ 0.045)} \\
(Section~\ref{sec:ohmnet}) \\\botrule
\end{tabular}}{}
\end{table}
\subsection{Transfer of cellular functions to a new tissue}\label{sec:transfer}
\xhdr{Experimental setup}
In the transfer learning setting, we attempt to transfer knowledge learned in one or more {\em source layers} and use it for prediction in a {\em target layer}.
As before, we apply {\em{OhmNet}}\xspace to obtain a separate feature vector for every node and every layer in an unsupervised way. We then consider, in turn, every tissue as a target layer and {\em all other tissues} as source layers. For every function and every source layer, we train a separate classifier using the same classification model as in Section~\ref{sec:classify}. We then predict functions for the target layer using only classifiers trained on the source layers. That is, we aim to predict cellular functions taking place in the target tissue without having access to any cellular function gene annotation in that tissue, \emph{i.e.}\xspace, we pretend the target tissue has no annotations. Prediction for one node in the target layer is the weighted average of predictions of the classifiers trained on source layers. Weights reflect hierarchy-based distances of source tissues from the target tissue. They are determined by the closed-form expressions mathematically equivalent to {\em{OhmNet}}\xspace's regularization (details omitted due to space constraints).
\begin{table}
\processtable{Area under ROC curve (AUROC) scores for transfer learning. Shown are the scores for ten tissues with best performance on cellular function prediction task. ``Non-transfer'': a classifier is trained on a target tissue and then used to predict cellular functions in the same tissue (Section~\ref{sec:classify}). ``Transfer'': classifers are trained on all non-target tissues and then used to predict cellular functions in the target tissue (Section~\ref{sec:transfer}). \label{tab:transfer}}{
\setlength{\tabcolsep}{5pt}
\begin{tabular}{l|c|c}\toprule
Target tissue & AUROC (Non-transfer) & AUROC (Transfer) \\\midrule
Natural killer cell & 0.834 ($\pm$ 0.076) & 0.776 ($\pm$ 0.063) \\
Placenta & 0.830 ($\pm$ 0.082) & 0.758 ($\pm$ 0.068) \\
Spleen & 0.803 ($\pm$ 0.030) & 0.779 ($\pm$ 0.043) \\
Liver & 0.803 ($\pm$ 0.047) & 0.741 ($\pm$ 0.025) \\
Forebrain & 0.796 ($\pm$ 0.036) & 0.755 ($\pm$ 0.037) \\
Macrophage & 0.789 ($\pm$ 0.037) & 0.724 ($\pm$ 0.024) \\
Epidermis & 0.785 ($\pm$ 0.030) & 0.749 ($\pm$ 0.032) \\
Hematopoietic stem cell & 0.784 ($\pm$ 0.035) & 0.744 ($\pm$ 0.036) \\
Blood plasma & 0.784 ($\pm$ 0.027) & 0.703 ($\pm$ 0.039) \\
Smooth muscle & 0.778 ($\pm$ 0.031) & 0.729 ($\pm$ 0.041) \\\midrule
Average & 0.799 & 0.746 \\\botrule
\end{tabular}}{}
\end{table}
\xhdr{Experimental results}
Table~\ref{tab:transfer} shows the classification accuracy results for transfer learning based on {\em{OhmNet}}\xspace. Since transfer tasks are more difficult than non-transfer tasks (Section~\ref{sec:classify}), it is expected that the AUC scores will decrease on transfer tasks. Results in Table~\ref{tab:transfer} confirm these expectations; however, we observe a very graceful degradation in performance leading to an only 7\% average decrease in the AUC scores. We get the smallest performance differences for target tissues with many biologically similar source tissues (\emph{i.e.}\xspace, source layers) in the tissue network. For example, performance difference for the forebrain is only 5.2\%, which is due to the fact that there are nine other layers in the tissue network closely related to the forebrain, such as the cerebellum and the midbrain. Considering all 48 tissues with tissue-specific cellular functions, {\em{OhmNet}}\xspace outperforms all comparison methods on most transfer tasks, achieving a gain of up to 20.3\% over the closest benchmark in AUC scores (scores not shown). Notice that we exclude GeneMania in the comparison because it is not amenable to transfer learning. This result suggests that considering the relationships between tissues when learning features for proteins has a significant impact on transfer performance.
Generally speaking, we observed that the transferability of classifiers decreased when the tree-based distance between the source and the target tissue in the tissue hierarchy increased, which is consistent with the empirical evidence in transfer learning~\citep{Yosinski2014}. This also matches our intuition that a source tissue should be most informative for predicting cellular functions in an anatomically close target tissue (\eg, source and target tissues are both part of the same organ).
\subsection{The multiscale model of brain tissues}\label{sec:case-study}
We have seen in Section~\ref{sec:tissue-hierarchy} that human tissues have a multi-level hierarchical organization. The tissue hierarchy categorizes tissues into: cell types, groups of cells with similar structure and function; organs, groups of tissues that work together to perform a specific activity; and organ systems, groups of two or more tissues that work together for the good of the entire body. We now aim to empirically demonstrate this fact and show that {\em{OhmNet}}\xspace in fact can discover embeddings that obey this organization.
We first construct a multi-layer brain network by integrating nine brain-specific protein interaction networks (\eg, the cerebellum, frontal lobe, brainstem, and other brain tissues). Each of nine brain-specific networks is one layer in the multi-layer network. The layers are organized according to a two-level hierarchy (Figure~\ref{fig:brain-case}{\color{red}a}). We run {\em{OhmNet}}\xspace on this multi-layer network to find node features in a purely unsupervised way. We then map the nodes to the 2-D space based on the learned features. This way we assign every node in every layer to a point in the two-dimensional space based solely on the node's learned features. We then visualize the points and color them based on the layer they belong to.
Figure~\ref{fig:brain-case}{\color{red}b} shows the example for the brainstem tissues: substantia nigra, pons, midbrain, and medulla oblongata. Laying out these tissue-specific networks is very challenging as the four brainstem tissues are very closely related to each other in the human body. However, the visualization using {\em{OhmNet}}\xspace performs quite well. Notice how points of the same color are closely distributed, and how well regions of the same color are separated from each other. In the brainstem example, this means that {\em{OhmNet}}\xspace generates a meaningful layout of the brainstem tissue-specific networks, in which proteins belonging to the same tissues are clustered together.
Figure~\ref{fig:brain-case}{\color{red}c} shows the example for the brain, which is located one level up from the brainstem in the tissue hierarchy. Again, {\em{OhmNet}}\xspace produces a meaningful layout of the nine brain tissue-specific networks.
Additionally, we repeated this analysis by visualizing protein features learned by running principal component analysis (PCA) or non-negative matrix factorization (NMF) algorithm on the brain-specific PPI networks. Acknowledging the subjective nature of this analysis, we observed that visualizations using PCA or NMF were not very meaningful, as proteins belonging to the same tissue were not clustered together (data not shown).
{\em{OhmNet}}\xspace's result in Figure~\ref{fig:brain-case} is especially appealing because of two reasons. First, it shows that {\em{OhmNet}}\xspace can learn node features that adhere to a given hierarchy of layers. In the brain example, {\em{OhmNet}}\xspace learns the protein features that expose the multiscale tissue hierarchy. Second, it shows that {\em{OhmNet}}\xspace can generate meaningful visualizations of network embeddings despite the fact that {\em{OhmNet}}\xspace's objective is independent of the visualization task.
\begin{figure*}
\begin{center}
\includegraphics[width=\textwidth]{{zitnik-fig5}.pdf}
\caption{\textbf{Visualization of the brain tissue-specific protein interaction networks.} {\bf A.} The two-level brain tissue hierarchy as specified by the BRENDA Tissue Ontology~\citep{Chang2014} and used in the case study in Section~\ref{sec:case-study}. Leaves of the hierarchy (in blue) represent nine brain tissues each of which is associated with a tissue-specific protein interaction network. {\bf B.} Visualization of the brainstem-specific networks. The proteins are mapped to the 2-D space using the t-SNE package with learned features as input. Color of a node indicates the tissue of the protein. {\bf C.} Visualization of the brain-specific networks. The proteins are mapped and colored using the same procedure as in B.}
\label{fig:brain-case}
\end{center}
\end{figure*}
\section{Conclusion}
\label{sec:conclusion}
We presented {\em{OhmNet}}\xspace, an approach for unsupervised feature learning in multi-layer networks. We use {\em{OhmNet}}\xspace to learn state-of-the-art task-independent protein features on a multi-layer network with 107 tissues. {\em{OhmNet}}\xspace models tissue interdependence up and down a tissue hierarchy spanning dozens of biological scales. The learned features achieve excellent accuracy on the cellular function prediction task, allow us to transfer functions to unannotated tissues, and provide insights into tissues.
There are several directions for future work. Our approach assumes the dependencies between layers are given in the form of a hierarchy. In several biological scenarios, the dependencies are given in the form of a graph, and we hope to extend the approach to handle graph-based dependencies. As the learned protein features are independent of any downstream task, it would be interesting to see whether our approach performs equally well for gene-disease association prediction and disease pathway detection.
\section*{Funding}
This research has been supported in part by NSF IIS-1149837, NIH BD2K U54EB020405, DARPA SIMPLEX N66001 and Chan Zuckerberg Biohub.
\vspace{3mm}
{\em \noindent Conflict of Interest:} none declared.
\bibliographystyle{natbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,580 |
\section{INTRODUCTION}
4U 1820-30 is a low mass X-ray binary (LMXB) located near the
center of the globular cluster NGC 6624. The binary orbital period is
$P_{1}\simeq685\s$, revealed in X-ray
observations as a modulation with $\sim 2-3\%$ peak to peak amplitude \citep{1987ApJ...312L..17S}. Subsequently,
\citet{1997ApJ...482L..69A} discovered a $\sim16\% $ peak to
peak modulation (period $ 687.6\pm2.4$ s) in the UV band from HST.
This short period, low amplitude variation is very stable, with
$\dot{P}/P=(-3.47\pm1.48)\times10^{-8}yr^{-1}$
\citep{2001ApJ...563..934C}, which is consistent with the earlier
measurement of $\dot{P}/P=(-5.3\pm1.1)\times10^{-8}yr^{-1}$ from
\citet{1993A&A...279L..21V}; this
stability led \citet{2001ApJ...563..934C} to suggestion that this
modulation reflects the orbital period of the binary.
Both the short binary period and the type I X-ray bursts observed in
this system imply that the secondary star is a helium white dwarf, of mass
$m_2=(0.05-0.08) M_{\bigodot}$, accreting mass onto a primary neutron
star \citep{1987ApJ...322..842R}. The distance to the source is
estimated to be $7.6\pm0.4 \kpc$ \citep{2003A&A...399..663K}.
It is striking that neither the magnitude nor the sign of the period
derivative is consistent with the prediction
$\dot{P}/P>+8.8\times10^{-8}yr^{-1}$ of the standard evolution
scenario for compact binaries overflowing their Roche lobe
\citep{1987ApJ...322..842R}. It has been suggested that the negative
period derivative is only apparent, i.e., that it is not intrinsic to
the binary, but instead reflects the acceleration of the binary in the
gravitational potential of the globular cluster which houses the binary
\citep{1993MNRAS.260..686V}. However, quantitative estimates
show that the acceleration, while of roughly the right magnitude, is
unlikely to be large enough, by itself, to explain the large
discrepancy between the evolution scenario and the observations
\citep{1993MNRAS.260..686V,1993ApJ...413L.117K, 2001ApJ...563..934C}.
A second striking property of 4U 1820-303 is the much larger
luminosity variation, by factor of $\gtrsim2$, seen at a period of $P_{3}\simeq 171$ days. Analysis of the RXTE ASM data shows
that this long period modulation does not exhibit a significant period
derivative, $\dot P_3/P_3<2.2\times10^{-4}yr^{-1}$\citep{2001ApJ...563..934C}. The ratio between this long
period and the binary orbital period is $\simeq 2\times10^4$, which
appears to be too high to be due to disk precession at the mass ratio
of the system \citep{1998MNRAS.299L..32L, 1999MNRAS.308..207W}.
In this paper we adopt the assumption of
\citet{1988IAUS..126..347G}, that the $171$ day period is due to the
presence of a third body in the system. The third (outer) star
modulates the eccentricity of the binary at long term period
$P_{3}\simeq P^{2}_{2}/(eP_{1})$, where $P_{2}$ is the orbital period of
the third star and $e$ is the eccentricity of the inner binary. Taking into account only perturbations from the third star, the binary orbital period of $685~\s$ and $\sim171$ day
long-term modulation imply that the orbital period of the third star
must be $\sim 1$ day. The presence of additional sources of precession, such as that due to tidal distortion of the white dwarf secondary, requires a stronger perturbation from the third body and hence a smaller orbit in order to modulate eccentricity of the inner binary at the $171$ day period. We show that the luminosity modulation arises
from variations in the eccentricity of the inner binary associated with
libration around a stable fixed point in the Kozai resonance.
Tidal dissipation in the white dwarf, driven by the eccentricity
of the binary orbit, tends to decrease both the eccentricity and the
semimajor axis (hence period) of the binary, which we suggest is
responsible, in part, for the anomalous observed period
derivative---note that \citet{1987ApJ...322..842R} did not treat the
effects of tidal dissipation. The combination of tidal dissipation and
mass transfer will result in a lower value of $\dot{P}/P$ than that
produced by conservative mass transfer alone.
For rapid enough dissipation, or, expressed another way, for low
enough values of the tidal dissipation parameter $Q$, $\dot P<0$ could
result. We do not favor this as the explanation for the observed
negative period derivative; we show that such rapid dissipation damps
eccentricity within $10^{-3}$ of the system's lifetime. Subsequently
the mass transfer takes over the evolution of the semimajor axis. In
other words, we would be incredibly lucky to observe the system in the
short time that $e$ is significant, in the absence of another
perturbing influence. We also show that, given the most recent estimates for
the acceleration of millisecond pulsars in the
gravitational field of the globular cluster, the cluster gravity does not appear to
contribute significantly to the observed period derivative of 1820-30.
Thus it appears that, while both tidal dissipation and acceleration in the gravitational field of the cluster contribute negatively to the period derivative, they can not fully explain it. Since we favor the hierarchical triple model as an explanation for the origin of $171$day period of luminosity variations, we suggest that the apparent negative period derivative, which is a $2-\sigma$ result, may either be an observational artifact or due to the some yet not understood physical processes.
The relation between the luminosity variations and the period
derivative is deeper; we argue that the (intrinsic) increase in the
semimajor axis of the binary (driven by Roche lobe overflow) leads to
trapping of the system deep in the Kozai resonance. The resonance
transfers angular momentum from the inner binary to the third star,
and back, periodically, without affecting the semimajor axis of either
orbit. However, the dissipation associated with the strong tides when
the forced eccentricity is largest does remove energy from the orbit
of the inner binary. This energy loss peaks when the mutual
inclination is small. It is well known that this coupled Kozai-tidal
evolution tends to leave the system with a mutual inclination between
the two orbits near the Kozai critical value ($\sim40^\circ$); see,
for example, Figure 4 in \citet{2007ApJ...670..820W} or Figure 7 in
\citet{2007ApJ...669.1298F}. We show that the period of small
oscillations is naturally $\sim170$ days when the mutual inclination is close to the Kozai critical value. Whether the evolution of the inclination in systems like 1820-30, which, unlike the planetary systems, is know to undergo Roche lobe overflow, is a question we are currently investigating.
This paper is organized as follows. In \S \ref{sec:dynamics} we
develop an analytic understanding of the system, describing the
resonance dynamics, calculating the location of the fixed point as a
function of the system parameters (stellar masses, orbital radii, and
the mutual inclination of the two orbits), and the frequency (or
period) of small oscillations. In \S \ref{sec:capture} we describe a
possible dynamical path by which the system arrived at its present
configuration. The dynamical history relies crucially on both the
Roche lobe overflow (which drives the system into resonance) and the
tidal dissipation, which tends to drive the mutual inclination toward
the Kozai critical value. In section \ref{sec:numerics} we describe
the results of numerical integrations of the equations of motion,
presenting a fiducial model that reproduces the observed properties of
4U 1820-30. We also demonstrate trapping in the case of an expanding
inner binary orbit, and detrapping in the case of a shrinking binary
orbit. In \S \ref{sec:constraints} we use the model to put constraints
on the ratio of the tidal dissipation parameter
$Q$ and the tidal Love number ($k_2$) of the Helium white dwarf for our fiducial eccentricity. We discuss our results, and those of previous workers,
in \S \ref{sec:discussion}. We present our conclusion in the final
section. We give the details of the numerical model in the appendix A. In appendix B we discuss in details adiabatic invariance of the action and how it governs the evolution of the system by comparing analytic and numerical analysis.
\section{UNDERSTANDING THE DYNAMICS OF THE $4U 1820-30$ SYSTEM} \label{sec:dynamics}
The presence of a third body orbiting the center of mass of a tight
binary will induce changes in the orbital elements of the binary,
changes that take place over a variety of time scales. The changes are
particularly dramatic if the mutual inclination of the two orbits is
large. \citet{1962AJ.....67..591K} showed that when the
initial inclination between inner and outer orbits has values between
some critical inclination $\ic$ and $180^{o}-\ic$, both the
eccentricity of the inner binary and the mutual inclination undergo
periodic oscillations known as Kozai cycles.
The period of the Kozai cycles is much longer than either the binary's
orbital period, or the period of the outer orbit. This justifies the
use of the secular approximation, which involves averaging the
equations of motion over the orbital periods of inner and outer
binaries; as a result, the averaged equations of motion predict that
the semimajor axes of both binaries are unchanged.
If the luminosity variations in 4U 1820-30 are due to Kozai cycles,
the semimajor-axis ratio $a_{out}/a \approx 8$, so in our analytic work we use the
quadrupole approximation for the potential experienced by the inner
binary due to the third body. In our numerical work we keep terms to octupole order, but we show that the higher order terms change the quantitative results only slightly.
The angular momentum of the outer binary is much greater than that of
the inner, so that the orientation of the outer binary is, to a good
approximation, also a constant of the motion. In that case, after the
averaging procedure, the final Hamiltonian has one degree of freedom.
Kozai cycles are the consequence of a $1 : 1$ resonance between the
precession rates of the longitude of the ascending node $\Omega$ and
the longitude of the periastron $\varpi$ of the inner binary. The
condition for Kozai resonance, $\dot{\varpi}-\dot{\Omega}=0$, is
satisfied only for high inclination orbits; for low inclinations, the
line of nodes precesses in a retrograde sense ($\dot\Omega<0$), while
the apsidal line precesses in a prograde sense.
We employ Delaunay variables to describe the motion of the inner
binary. The angular variables are the mean anomaly $l$, the argument
of periastron $\omega$, and the longitude of the ascending node
$\Omega$; of these, only $\omega$ appears in the averaged
Hamiltonian. Their respective conjugate momenta are:
\bea \La &=&m_1
m_2 \sqrt{\frac{Ga}{m_1+m_2}}\\ \G &=&\La\sqrt{1-e^2}\\ \Ha
&=&\G\cos{\imath}. \eea
The longitude of periastron is $\varpi\equiv\Omega+\omega$. Recall
that we are assuming that the semimajor axis of the outer binary is
large enough that the total angular momentum is dominated by that of
the outer binary, so that $\imath$ is effectively the mutual
inclination between the two binary orbits. We occasionally refer to
the elements of the third star, using a subscript 'out' to distinguish
them from those of the inner binary.
After averaging over $l$ and $l_{out}$, the Hamiltonian describing the
motion of a tight binary orbited by a third body, allowing for the
effects of both tidal and rotational bulges on the secondary, and for
the apsidal precession induced by general relativistic effects, is
\citep{1997AJ....113.1915I,2000ApJ...535..385F,2007ApJ...669.1298F}
\bea \label{eq:hamiltonian}
H&=&\frac{-3A}{2}\Bigg[-{5\over3}-3\frac{\Ha^{2}}{\La^{2}}+\frac{\G^{2}}{\La^{2}}+5\frac{\Ha^{2}}{\G^{2}}+5\cos2\omega\left(1-\frac{\G^{2}}{\La^{2}}-\frac{\Ha^{2}}{\G^{2}}+\frac{\Ha^{2}}{\La^{2}}\right)\Bigg]\nonumber\\ &&-B\frac{\La}{\G}-k_2C\left(35\frac{\La^{9}}{\G^{9}}-30\frac{\La^{7}}{\G^{7}}+3\frac{\La^{5}}{\G^{5}}\right)
-k_2D\frac{\La^{3}}{\G^{3}},
\eea
where the term proportional to $A$ is the Kozai term, the term
proportional to $B$ enforces the average apsidal precession due to
general relativity, and the terms proportional to $C$ and $D$
represent the tidal and rotational bulges, respectively; the explicit
appearance of the tidal Love number $k_2$ in the latter two terms
highlights the fact that these terms represent the effects of the
white dwarf's tidal and rotational bulges. The expressions for the
constants are
\bea
A&=&{1\over8}\Phi\,
{m_2m_3\over(m_1+m_2)^2}
\left({a\over a_{out}}\right)^3
{1\over(1-e^2_{out})^{3/2}}\\
B&=&{3\over 2}\Phi
{m_2\over m_1} {r_s\over a}\label{eq:GRa}\\
C&=&{1\over 16}\Phi
{m_1\over m_1+m_2}\left({R_2\over a}\right)^{5}\\
D&=&{1\over12}\Phi
\left({R_2\over a}\right)^5f(\tilde\Omega_{spin})\label{eq:spin},
\eea
where
\be
\Phi\equiv{G(m_1+m_2)m_1\over a}.
\ee
Recall that the semimajor axis and eccentricity of the outer body's
orbit are denoted by $a_{out}$ and $e_{out}$. The quantity
$r_s\equiv2Gm_1/c^2$ in equation (\ref{eq:GRa}) is the Schwarzschild
radius of the neutron star.
As just noted, the term proportional to $D$ accounts for the
rotational bulge produced by the spin of the white dwarf. The spin is
projected onto the triad defined by the Laplace-Runge-Lenz vector,
pointing along the apsidal line from the white dwarf at apoapse toward
the neutron star, and denoted by a subscript $e$, the total angular
momentum vector, subscript $h$, and their cross product, denoted by
$q$. We have scaled the spin to the orbital frequency (or mean motion)
$n$, so that, e.g., $\tilde\Omega_e\equiv\Omega_e/n$. We do so because
we anticipate that for small eccentricity the white dwarf will be
tidally locked. Then
$f(\tilde\Omega_{spin})\equiv2\tilde\Omega^2_h-\tilde\Omega_e^2-\tilde\Omega_q^2$
is a dimensionless quantity of order unity.
For the fiducial values of the system parameters listed in table 1, $A\approx1.73\times10^{44}$,
the ratios $B/A\approx 0.53$, $C/A\approx1.82$, and $D/A\approx2.54$.
\input{table1}
\subsection{The Kozai mechanism}
We start our discussion of the dynamics of the system by focusing on
understanding the Kozai mechanism, neglecting forces due to the tidal
and rotational bulges of the Helium white dwarf in the inner binary,
and the effects of general relativity.
We locate the resonance by looking for a fixed point of the
Hamiltonian; since we are neglecting the tidal and rotational bulges,
and the general relativistic precession, we set $B=C=D=0$ and
differentiate the Hamiltonian with respect to $\omega$, to find
$\omega_f=0,90^\circ,180^\circ,270^\circ$. The fixed points at $\omega_f=90^\circ$ and
$\omega_f=270^\circ$ are stable. Differentiating the Hamiltonian with
respect to $\G$, substituting $\omega=90^\circ$ (or $270^\circ$) and setting the
result equal to zero, we find $\G_f^4=(5/3)\Ha^2\La^2$. In terms of
the eccentricity,
\be \label{eq:fixed_e
e_f = \sqrt{1-{5\over3}\cos^2\imath_f},
\ee
where the subscript $f$ indicates that this is the eccentricity of the
stable fixed point. The frequency of small oscillations around the
fixed point (small librations) is
\be
\omega_0\equiv \left[\left({\partial^2 H\over \partial
\omega^2}\right)_{\omega_f,\G_f}\left({\partial^2H\over\partial\G^2}\right)_{\omega_f,\G_f}\right]^{1/2}.
\ee Performing the derivatives, \be \label{eq:omega_0}
\omega_0=\omega_A\left(18+90{\Ha^2\La^2\over\G_f^4}\right)^{1/2}
\left(1-{\G^2_f\over \La^2}-{\Ha^2\over\G_f^2}+{\Ha^2\over
\La^2}\right)^{1/2}, \ee
where we have defined
\be
\omega_A \equiv \sqrt{30A^2\over \La^2}=\sqrt{15\over32}n{m_3\over m_1+m_2}
\left({a\over a_{out}}\right)^3{1\over(1-e^2_{out})^{3/2}}.
\ee
The last factor in equation (\ref{eq:omega_0}) is $e_f\sin\imath_f$.
In terms of the eccentricity,
\be \label{eq:omega_0e
\omega_0 = {3\over2}\sqrt{15}\,n{m_3\over
m_1+m_2}\left({a \over a_{out}}\right)^3{e_f\sin\imath_f\over
(1-e_{out}^2)^{3/2}} . \ee
From equation (\ref{eq:fixed_e}) we see that the critical inclination for a
Kozai resonance to occur, in the absence of other dynamical effects,
is $\ic = \cos^{-1} \sqrt{3/5}\approx 39.2^\circ$. If $\imath>\ic$,
orbits started at $\omega=90^\circ$ with $e<e_f$ will librate around
the fixed point, so that $\omega$ remains between $0^\circ$ and
$180^\circ$ (or an even more restricted range). From
equation (\ref{eq:omega_0}) or (\ref{eq:omega_0e}), the period of small
oscillations $P_0\sim 1/e_f$, a point that will be important later.
In contrast, orbits started at $\omega=0$ and $e>0$ will circulate
($\omega$ will range from $0$ to $360^\circ$). Librating and
circulating orbits are separated by the separatrix, an orbit that
neither librates nor circulates. The width of the separatrix (as
measured by the excursion in $e$) depends only on the initial
inclination: $e_{sep}=[1-(5/3)\cos^{2} \imath]^{1/2}$.
Examples of librating and circulating orbits (for a system including
the effects of GR and tidal bulges) are shown in section
\ref{sec:numerics}.
Note that, even for systems with $\imath<\ic$, where no stable Kozai
fixed point exists, both the mutual inclination and the eccentricity
of the inner binary can undergo oscillations with significant
amplitude (although reduced compared to the case with $\imath>\ic$).
Kozai cycles will be substantial only as long as the perturbation from
the outer body dominates over the other sources of apsidal precession
in the inner binary orbit, a point we now address.
\subsection{Kozai cycles in the presence of additional forces}
The physical effects represented by the terms proportional to $B$,
$C$, and $D$ are capable of suppressing Kozai oscillations. We
investigate their effects in this section.
As an aside, there is a small apsidal precession introduced by
dissipative effects in the He white dwarf, but this precession rate is
negligible compared to the other three. We mention it here because
tidal dissipation has a major role to play in the capture (or
otherwise) of the system into the Kozai resonance.
The equations for the precession rates due to the Kozai mechanism,
that due to general relativity, and the tidal and rotational bulges of
the white dwarf, are:
\pagebreak
\bea \label{eq: eqn for omega_dots}
\dot{\omega}_{Kozai}&=&{3\over4}n
\left({m_3\over m_1+m_2}\right)
\left({a\over a_{out}}\right)^3
{1\over(1-e_{out}^2)^{3/2}}\times
{1\over\sqrt{1-e^2}}\nonumber\\
&&\qquad\left[ 2(1-e^2)+5\sin^2\omega(e^2-\sin^{2}\imath)\right] \label{eq:kozai}\\
\dot{\omega}_{GR}&=&{3\over2}n\left({m_1+m_2\over m_1}\right)
\left({r_s\over a}\right)
{1\over(1-e^2)}\label{eq:GR}\\
\dot{\omega}_{TB}&=&{15\over16}n\,k_2{m_1\over m_2}
\left({R_2\over a}\right)^5{8+12e^2+e^4\over(1-e^2)^5}\label{eq:tb}\\
\dot{\omega}_{RB}&=&{n\,k_2\over4}{m_1+m_2\over m_2}
\left({R_2\over a}\right)^5{1\over(1-e^2)^2}
\Bigg[
\left(2\tilde\Omega^{2}_{h}-\tilde\Omega^{2}_{e}-\tilde\Omega^{2}_{q}\right)+2\tilde\Omega_{h}\cot\imath\left(\tilde\Omega_{e}\sin\omega +\tilde\Omega_{q}\cos\omega\right)
\Bigg].\label{eq:rb}
\eea
The Kozai term (equation \ref{eq:kozai}) can be either positive or
negative, depending on the value of $\sin \imath$. Both the white
dwarf tidal bulge and the GR terms are positive, so both tend to
suppress Kozai oscillations. The term induced by the white dwarf
rotational bulge, on the other hand, can be of either sign, depending
on the orientation of the white dwarf spin. If the white dwarf is
tidally locked and if its spin is aligned (which we assume in our
analytic model, but not in our numerical models), this term
contributes positive $\dot{\omega}$. In case of non-aligned spins the
precession rate may be negative (as we will see).
\subsubsection{The tidal bulge and the tidal Love number $k_2$}
The tidal bulge of the white dwarf in 4U 1820-30 dominates the non-Kozai
apsidal precession rate, for physically plausible values of $k_2$. We
remind the reader that in Newtonian gravitational theory the tidal
Love number $k_2$ is a dimensionless constant that relates the mass
multipole moment created by tidal forces on a spherical celestial body
to the gravitational tidal field in which it is immersed; in other
words, $k_2$ encodes information about body's internal
structure.\footnote{In a confusing usage, the apsidal precession
constant, which is a factor of two smaller than the tidal Love
number, but which we do not employ, is also denoted by $k_2$.}
We use $k_2=0.01$, which is computed by Arras (private communication) as the ratio of the potential due to the perturbed mass distribution, to the external potential causing the perturbed mass, under the assumption that our He white dwarf is a fluid object.
Soft X-ray observations of the source indicate a rather small absorption, consistent with that expected to be produced by the interstellar medium of the Galaxy; this rules out any significant outflows from the accretion disk or the surface of the white dwarf. This implies an absence of mass loss through the $L_2$ Lagrangian point of the white dwarf, which puts an upper limit on the eccentricity of the inner binary; according to \citet{2005MNRAS.358..544R}, for our system parameters, the upper limit on the eccentricity of inner binary is $e_{max}\simeq 0.07$.
If 4U 1820-30 has a non-zero but small eccentricity, as indicated by the
observed luminosity variations, then in the absence of a third body,
the precession rate of the binary orbit is dominated by the tidal
bulge induced in the white dwarf by the gravity of the neutron star;
from equations (\ref{eq:GR}) and (\ref{eq:tb}), the tidal bulge induces a
precession rate at least a few times that induced by GR:
\be
{\dot\omega_{TB}\over\dot\omega_{GR}} \approx 4
\left({k_2\over0.01}\right)
\left({a\over 1.32\times10^{10}\cm}\right)^{-4}.
\ee
In order for the Kozai mechanism to produce significant variations in
$e$, the Kozai-induced precession rate must be comparable to or larger
than the sum of the precession rates produced by the other terms. For
physically realistic values of $k_2$, as we have just seen, the
precession rate induced by the tidal bulge of the white dwarf is by
far the largest, so if the Kozai effect is to be important, it must
produce a precession rate larger than $\dot \omega_{TB}$.
\subsection{Libration around the fixed point and the frequency of small oscillations}
\subsubsection{Why libration?}
For the values of the tidal Love number $k_2$ and eccentricity listed in table 1, the period of
the precession rate induced by the tidal bulge,
$P_{TB}=2\pi/\dot\omega_{\rm TB}$, is a factor of ten shorter that the
period of the observed luminosity variations. If this term set the
rate of precession, and the eccentricity varied as a result of this
precession, then the variations in X-ray luminosity would occur with a
period substantially shorter than the observed $170$ days.
In order to produce a much longer period, some other term must tend to
produce a negative precession rate. When this negative precession rate
is added to that produced by the tidal bulge, the resulting period can
be much longer than that produced by the tidal bulge alone.
Under the assumption that the white dwarf is tidally locked (we show later it is not), the only
term capable of producing a negative precession rate is the Kozai
term. Hence we are led to look for a cancellation between the Kozai
precession rate and the precession rate induced by the tidal bulge.
However, it is not enough to ask for a rough cancellation. To get the
observed precession rate, the sum of all the terms must cancel to
better than $10\%$. This requires some fine tuning of the mutual
inclination, a rather unsatisfactory situation.
On the other hand, if the system is captured into libration, then the
sum of all the precession terms is exactly zero. If the system is deep
in the resonance, then the period of libration is simply the period
associated with small oscillations around the fixed point. We show
here that the period of small oscillations is naturally around $170$
days, if the mutual inclination is near the critical value for Kozai
oscillations.
\subsubsection{The frequency of small oscillations}
Setting the first derivative of the Hamiltonian
(\ref{eq:hamiltonian}) with respect to $\omega$ and $\G$ to zero, we
find the following expression for the location of the stable fixed
points in the limit of small eccentricity: \bea \omega_f
&=&90^\circ\ ,\ 270^\circ\\ e_f
&=&\sqrt{\frac{18-30\frac{\Ha^2}{\La^2}-{B\over A}-120k_2{C\over
A}-3k_2{D\over A}}{60\frac{\Ha^2}{\La^2}+{3\over2}{B\over
A}+840k_2{C\over A}+\frac{15}{2}k_2{D\over A}}}. \label{eq:efix}
\eea
We can write the second of these as
\be
e_f =\sqrt{30\left[\cos^2\ic-\cos^2\imath\right]\over{60\frac{\Ha^2}{\La^2}+{3\over2}{B\over A}+840k_2{C\over A}+\frac{15}{2}k_2{D\over A}}},
\ee
where
\be
\cos^2\ic \equiv {3\over 5}-{1\over 30}{B\over A} - 4k_2 {C\over A} -{1\over 10} k_2 {D\over A}.
\ee
Evaluating the second derivative of the Hamiltonian at the fixed point
we obtain the expression for the frequency of small oscillation around
the fixed point:
\bea \label{eq:period}
\omega_0=&\omega_A&\Bigg[\left(18+90{\Ha^2\La^2\over G_f^4}\right)+2{B\over A}\frac{\La^3}{G_f^3}+k_2{C\over
A}\left(3150\frac{\La^{11}}{G_f^{11}}-1680\frac{\La^9}{G_f^9}+90\frac{\La^7}{G_f^7}\right)+12k_2{D\over
A}\frac{\La^5}{G_f^{5}}\Bigg]^{1/2}\nonumber\\
&\times& e_f\sin\imath_f,
\eea
which should be compared to equation (\ref{eq:omega_0}). As in the pure
Kozai case, the period of small oscillations $P_0\sim1/e_f$.
Figure \ref{Fig:Pi} shows $P_0$ as a function of the initial
inclination. As the initial inclination increases above the critical
value, the period of small oscillations decreases rapidly. Increasing
the initial inclination increases the magnitude of the Kozai torque;
in the absence of other torques, and for inclinations above the
critical inclination, increasing the magnitude of the Kozai torque is
analogous to increasing the restoring force in a harmonic oscillator,
thereby increasing the frequency of oscillation. When there are other
torques in the problem, the critical inclination will change; for
example, the presence of a tidal bulge on the secondary increases the
critical inclination.
Very near the critical inclination, the effective restoring force is
small, $\sim e_f\sin \imath_f$, so the frequency of small oscillations
is small, and the period of oscillations is large---hence the rapid
increase in $P_0$ as the inclination decreases toward the critical
inclination ($\ic\approx44^\circ$ in Figure \ref{Fig:Pi}).
Figure \ref{Fig:Pa} shows $P_0$ as a function of $a_{out}$. As
expected from the $n_{out}\sim a_{out}^3$ dependance of $\omega_A$,
the period of eccentricity oscillations increases rather rapidly with
$a_{out}$.
\begin{figure}
\epsscale{0.7}
\plotone{f1.eps}
\caption{
The period of small oscillations vs. the initial
inclination for a system similar to 4U 1820-30, with $k_2=0.01$. The
critical inclination is $\ic\approx 44.7^\circ$. At large
inclinations, well above $\ic$, the period of libration is of order
days. Only if $\imath\approx\ic$ is the period of order $170$
days. The solid line is the prediction of equation (\ref{eq:period}); the solid circles come from numerical integration of the equations of motion to quadrupole order, while the open squares come from integration accurate to octupole order.
\label{Fig:Pi}}
\end{figure}
\begin{figure}
\epsscale{0.7}
\plotone{f2.eps}
\caption{
The period of small oscillations vs. $a_{out}$. As the semimajor axis of the outer binary, $a_{out}$
increases, the period of small oscillations is increasing too, which
is expected from the $n_{out}\sim a_{out}^3$ dependance of
$\dot\omega_{Kozai}$. The solid line is the prediction of
equation (\ref{eq:period}), while the solid circles and open squares are from numerical
integrations accurate to quadrupole and octupole order respectively.
\label{Fig:Pa}}
\end{figure}
\section{MASS TRANSFER, TIDAL DISSIPATION, AND CAPTURE INTO LIBRATION} \label{sec:capture}
We have shown that physically plausible values of $k_2$ lead to a
precession frequency $\dot\omega_{TB}$ that is much larger than the
observed frequency of luminosity variations in 4U 1820-30. We then
appealed to an equally large precession, of the opposite sign,
produced by the Kozai interaction, to cancel the prograde precession
caused by the tidal bulge. In order to avoid fine tuning, we argued
that the system has to be in libration, so that the observed low
frequency actually arises from libration, rather than precession of
the apsidal line of the binary orbit.
Whether the tidal bulge or GR effects produce a larger precession
rate, we argue that it is no coincidence that the magnitude of the
Kozai precession rate is equal to the sum of the other precession
rates: the system will evolve so as to capture the orbit into
resonance, in which the sum of all the precession rates is zero.
Capture into libration in the Kozai resonance is a natural consequence
of semimajor axis expansion, the latter driven by mass loss from the
white dwarf as a result of its overflowing its Roche lobe. The action
$\int \G d\omega$ is an adiabatic invariant (for detailed discussion see appendix B), since the semimajor axis
of the binary orbit is expanding on the accretion time scale $m_2/\dot
m_2\approx 10^7\yr$, much greater than either the orbital or
precession time scale. In contrast to mass transfer, tidal dissipation
tends to shrink the semi-major axis; if this effect dominates,
trapping into the Kozai resonance is not possible.
How does expansion of the inner orbit lead to capture into libration?
As $a$ increases, the mutual torque between the two orbits will
increase as well---the inner orbit is expanding, effectively moving
closer to the outer orbit. This increasing torque corresponds to a
deepening of the Kozai potential, and an expansion in the size of the
separatrix of the Kozai resonance. Orbits other than the separatrix
have a fixed action, while the action of the separatrix is
increasing. If the increase in the action of the separatrix grows to
exceed the action of an initially circulating orbit, that circulating
orbit will be captured into resonance, and begin to librate. As $a$
continues to expand, the captured orbit will move closer and closer to
the fixed point of the resonance, librating with the frequency of
small oscillations.
More quantitatively, mass transfer tends to increase $a$ \citep{1982ApJ...254..616R}:
\be
\dot a_{MT}={1\over2/3-1/q}{3\times2^{23/6}\over5c^5}
\Big[{K\over0.4242}\Big]^{3/2}
{m_1(G(m_1+m_2))^{3/2}\over a^{9/2}},
\ee
where $q=m_1/m_2$ and $K=k\theta_{\gamma'}/(\mu m_p)$; $k$ is the
Boltzmann constant, $\mu$ is the mean molecular weight, $m_p$ is the
mass of the proton and $\theta_{\gamma'}$ is the polytropic
temperature. The parameter $K$ is given by the following mass-radius
relation: \be K=N_nGm_2^{1-(1/n)}R^{(3/n)-1}, \ee where $N_n$ is a
tabulated numerical coefficient (for $n=1.5$ it is 0.4242;
\citet{1939isss.book.....C}).
Tidal dissipation in the white dwarf will tend to reduce the semimajor
axis of the binary. In the limit of small eccentricity,
\be \label{eq:tidal dissipation
\left({da\over dt}\right)_{TD}\approx-{38\over3}na{k_2\over Q}{m_1\over m_2}
\left({R_2\over a}\right)^5 e^2 .
\ee
We argue that the orbit must be expanding. $(e/ \dot e )_{TD}$ is $~100$ times shorter than $(a/ \dot a )_{TD}$, so unless something excites $e$ (such as third body or thermal tides) we are unlikely to catch the system in a phase where periastron, $r_p=a(1-e)$, is increasing while $a$ is decreasing.
\section{NUMERICAL RESULTS}\label{sec:numerics}
\subsection{Numerical model using the quadrupole approximation}
Our numerical model treats the gravitational effects of the
third body in the quadrupole approximation. We average over the orbital
periods of both the inner binary and the outer companion. We demonstrate in section \ref{sec:octupole} and in figures \ref{Fig:Pi} and \ref{Fig:Pa} that treating the effects of the third body in octupole approximation does not qualitatively change our findings. We include
the following dynamical effects:
\begin{itemize}
\item periastron advance due to general relativity;
\item periastron advance arising from quadrupole distortions of the
helium white dwarf due to both tides and rotation;
\item orbital decay due to tidal dissipation in the white dwarf;
\item loss of binary orbital angular momentum due to gravitational radiation;
\item conservative mass transfer from the helium white dwarf to the neutron
star primary driven by the emission of gravitational radiation.
\end{itemize}
\noindent Note that the Kozai mechanism described in the previous
section is included in the three body gravitational dynamics. The
equations used in our model are listed in the appendix.
\subsection{Results}
We use as fiducial parameters $m_1=1.4M_\odot$, $m_2=0.067M_\odot$,
and $m_3=0.55M_\odot$. The semimajor axis of the inner binary is
$a=1.32\times10^{10}\cm$, chosen to match the observed orbital period
of $685\s$. The radius of the Helium white dwarf is
$R_2=2.2\times10^9\cm$, while the fiducial Love number is $k_2=0.01$.
To reproduce the $171$ day eccentricity oscillations (Figure
\ref{Fig:ecc}), we use the following initial parameters:
$a_{out}=8.0a$ (yielding $P_{out}=0.15$ day). We start with
$e_0=0.009$, $\omega_0=90^\circ$, $\imath_{init}=44.715^\circ$, and
$e_{out,0}=10^{-4}$.
Figure \ref{Fig:ecc} shows the eccentricity oscillations of the inner
binary, with a period of $~171$ days, over a decade. The amplitude of
the eccentricity oscillations is of order of $7\times10^{-3}$, which
is sufficient to enhance mass transfer enough to produce the observed
luminosity oscillations of a factor of $\gtrsim2$ \citep{2007MNRAS.377.1006Z};
see their Figure 3.
The amplitude of the eccentricity oscillations depends on the initial
eccentricity, as illustrated in Figure \ref{Fig:ps_all}; a lower
initial eccentricity produces eccentricity oscillations with higher
amplitude. If the system circulates, the amplitude of the eccentricity
oscillations is larger still.
Having the system trapped in libration about the fixed point explains
both the origin of the $171$ day period luminosity variations, as well
as the small amplitude of the eccentricity oscillations; the
observations require that magnitude of the eccentricity oscillations
be small so as to avoid overly large luminosity variations---a point
we return to below.
\begin{figure}
\epsscale{0.6}
\plotone{f3.eps}
\plotone{f3_b.eps}
\caption{
The eccentricity as a function of time (upper
panel) and the phase space ($e$ versus $\omega$) for our fiducial
model. The period of the eccentricity oscillations is $171$ days,
and the amplitude of the eccentricity oscillation is sufficient to
produce the observed factor of $2-3$ variation in luminosity.
\label{Fig:ecc}}
\end{figure}
\begin{figure}
\epsscale{0.7}
\plotone{f4.eps}
\caption{
Phase portrait for four different initial
eccentricities at initial inclination $\imath=44.715^o$ and initial
$\omega=90^\circ$. Our fiducial orbit, with
a libration period of $171$ days and the amplitude of the eccentricity oscillations sufficient to produce observed variations in the luminosity, is labeled as the ``current orbit in libration''; it is the same orbit presented on
Figure \ref{Fig:ecc}. The unlabelled librating orbit is very near the fixed point; it has a period of the eccentricity oscillations that is close to but shorter than the observed period, while the variations induced in the luminosity are too small compared to the observations. The
circulating orbit produces luminosity variations that are too large, as well as having an oscillation period that is too long.
\label{Fig:ps_all}}
\end{figure}
\subsection{Resonant trapping and detrapping of 4U 1820-30}
\noindent
The mass transfer rate is determined by the inner binary mass and
semimajor axis. These parameters are reasonably well constrained from
observations \citep{1987ApJ...312L..17S, 1997ApJ...482L..69A,
1987ApJ...322..842R}. The amount of tidal dissipation is
parameterized by the tidal dissipation factor $Q$, which for white
dwarfs is not well constrained at all. If we know the value of
period derivative, $\dot P$, we can constrain $Q$ (or more precisely,
$(e/0.009)^2Q/k_2$, see equation \ref{eq:tidal dissipation}) for the white dwarf
in the system.
We argued at the end of \S \ref{sec:capture} that the intrinsic $\dot
P$ must be positive, since a shrinking binary orbit and a decaying eccentricity quickly lead to mass transfer driving expansion of the binary orbit. There is a second argument
against an intrinsic negative $\dot P$: if the orbit of the inner
binary is shrinking, an initially librating orbit will quickly become
circulating, and the period of luminosity variations will change
dramatically. If we tune $Q$ to the value that reproduces the observed
negative period derivative ($Q=2.5\times10^7$, assuming $k_2=0.01$) and let the
system evolve, the system is driven out of libration after about
$~1500\yr$, as shown in Figure \ref{Fig:kick_out}. As the figure shows, the eccentricity of the inner binary decreases significantly due to tidal dissipation, which in turn reduces the strength of tidal dissipation. With tidal dissipation weakened, mass transfer will dominate the evolution of the semimajor axis and, as expected from the standard evolutionary scenario, the semimajor axis starts to expand (not shown in the figure). As long as there is some small eccentricity in the inner binary there is some tidal dissipation present that tends to slow down the expansion rate of the semimajor axis.
The reason for the detrapping is rather subtle. First, we note that the decrease in $e$ is \underline{not} due to direct tidal damping; Equation (\ref{eq:e_dot_TD}) predicts $(e/\dot e)_{TD} \sim 10^5 \yr$, while $e$ changes by factor of $2$ in $~2000 \yr$. To verify this, we have set $\dot e_{TD}=0$, and verified that integration yields the same result. The reason for such a short time scale for decrease in $e$ lies in the fact that the spins do not remain tidally locked throughout the evolution of the system and the evolution of the eccentricity is rather strongly influenced by the their lack of pseudo-synchronism. Detailed discussion and figures are given in appendix B.
\begin{figure}
\epsscale{0.49}
\plotone{f5.eps}
\plotone{f5_b.eps}
\plotone{f5_c.eps}
\caption{
a) $\omega$ vs $t$. We start the evolution of the system by placing
the system in libration. Here we use $Q=2.5\times10^{7}$, which is the value
required to reproduce the observed negative period derivative. The
action of the separatrix is decreasing, and the system is ejected from
the resonance after about $1500\yr$. b) $e$ vs $\omega$, phase space
evolution plot, showing that the orbit evolves from libration to
rotation, with the transition occurring between the $1500$ and $1700\yr$ snapshots. c) $\dot{P}/P$ vs $t$. The period derivative remains negative only for about $1700\yr$, which is $10^{-3}$ of the lifetime of the system; the eccentricity damps sufficiently that the $\dot m$ term takes over.
\label{Fig:kick_out}}
\end{figure}
On the other hand, if the observed negative period is not an intrinsic
property of the system, in other words, if the effect of mass transfer
wins over the effect of tidal dissipation, the action of the
separatrix increases with time, and trapping will occur.
Figure \ref{Fig:trap_omega} shows a system initially put on a
circulating orbit. As the integration proceeds, the separatrix
expands, eventually capturing the orbit, which then librates for the
duration of the integration.
\begin{figure}
\epsscale{0.55}
\plotone{f6.eps}
\plotone{f6_b.eps}
\caption{
a) $\omega$ vs $t$. The system is placed in
circulation; after about $27000\yr$ it gets trapped in libration.
Here we have used $Q=5\times10^7$, while the initial eccentricity is $e_0=0.032$; all
other parameters are the same as used in Figure
\ref{Fig:ecc}. b) $e$ vs $\omega$ for the same integration.
\label{Fig:trap_omega}}
\end{figure}
\subsection{Numerical model using octupole approximation}\label{sec:octupole}
In this subsection we treat gravitational effects of the third body in the octupole approximation. As in the case of the quadrupole approximation, we derive our equations of motion from the double averaged Hamiltonian and we include all of the previously listed dynamical effects. As Figure \ref{Fig:ecc_oct} demonstrates, the octupole approximation does not change qualitatively our previous findings. All parameters, except the initial inclination, used in the octupole approximation are listed in table 1. In order to produce the $171$ days period of the eccentricity oscillations,
and the amplitude of the eccentricity oscillation that produces the observed factor of $2-3$ variation in luminosity, a slightly higher inclination is required ($\imath = 45.1^o$).
\begin{figure}
\epsscale{0.6}
\plotone{f7.eps}
\plotone{f7_b.eps}
\caption{
The eccentricity as a function of time (upper
panel) and the phase space ($e$ versus $\omega$) in the octupole approximation. To produce the $171$ days period of the eccentricity oscillations,
and the amplitude of the eccentricity oscillation that is sufficient to
produce the observed factor of $2-3$ variation in luminosity, slightly higher inclination is required ($\imath = 45.1^o$).
\label{Fig:ecc_oct}}
\end{figure}
\section{ON THE VALUE OF $Q$ AND THE ORIGIN OF THE SMALL (OR NEGATIVE) $\dot P$} \label{sec:constraints}
The standard theory of Roche lobe overflow predicts $\dot
P/P\ge+8.8\times10^{-8}\yr^{-1}$. The measured $\dot
P/P=(-3.47\pm1.48)\times10^{-8}\yr^{-1}$ is eight standard deviations
away from this value. We have argued in previous section that $\dot
P/P$ should be positive, but even if it is two or three standard
deviation from the measured value, it is still five below the
predicted value. The origin of this discrepancy has been a puzzle
since it was discovered.
The suggestion that the binary has a finite eccentricity immediately
suggests a reason for the low value of $\dot P$: tidal dissipation in
the white dwarf will tend to reduce the semimajor axis of the orbit,
contributing a substantial negative term to $\dot P$.
The tidal dissipation could in fact dominate the orbital evolution,
overcoming the effects of mass transfer as seen in Figure \ref{Fig:kick_out}. We do not argue for this point of view, however, because it would be unlikely that the system could be observed in a stage of the evolution that last only $10^{-3}$ of its lifetime. In addition, we believe that the system is trapped in libration.
The observed $\dot P/P$ consists of at least three parts:
\be \label{eq: pdot
\left({\dot P\over P}\right)_{obs} =
\left({\dot P\over P}\right)_{Roche}+
\left({\dot P\over P }\right)_{accel}+
\left({\dot P\over P}\right)_{TD}.
\ee
The values of the observed and Roche terms were given above, and, as noted there, they are not consistent with each other. The second term on the right hand side of equation (\ref{eq: pdot}) represents the acceleration experienced by 1820-30 in the gravitational field of its host globular cluster, while the third term on the right represents the effects of tidal dissipation in the white dwarf secondary.
A natural explanation for the observed negative $\dot P$ might be provided by a combination of the last two effects, but still allow for the system to be trapped in resonance. First, tidal dissipation reduces the intrinsic $\dot P/P$ substantially from
that expected due to Roche lobe overflow alone, but leaves $\dot
P/P>0$. We then we appeal to the argument of \citet{1993MNRAS.260..686V},
that the $(\dot P/P)_{accel}$ term produces an {\em apparent} negative total $\dot P$. Indeed, given the most recent published estimate of $a_{max}/c=7.9\times10^{-8}yr^{-1}$ from \citet{1993MNRAS.260..686V}, it is plausible that we would observe a negative period derivative, while the intrinsic (or physical) period derivative is in fact postive.
However, recent estimates for the cluster acceleration from millisecond pulsar timing suggest a maximum of $a_{max}/c=1.3\times10^{-9}yr^{-1}$ (Lynch and Ransom, private communication), an order of magnitude smaller than the estimate from \citet{1993MNRAS.260..686V}; if the smaller value holds up, the observed negative period derivative is difficult to understand in the context of current models.
Given that the measured negative period dervative is significant only at the two-sigma level, and that there is no clear physical explanation for such an orbital decay, it is worth considering the possibility that the observed value is in error. If we ignore the observed negative period derivative, and simply assume that the intrinsic $\dot P$ is postive, we find a lower limit on $Q$ given by $(e/0.009)^2Q/k_2>3.15\times10^9$. We can get a firmer lower limit on $Q$ by requiring the system to remain trapped in a resonance for a considerable fraction of its lifetime. Given that $m_2=0.067M_\odot$ and $\dot m_2\approx 10^{-8}\rm \ M_\odot \ {\rm yr^{-1}}$, the lifetime during which this system can sustain its high X-ray luminosity is estimated to be $~7$ million years, so a reasonable fraction of its lifetime to remain trapped in a resonance is at least $10^5\yr$. According to our model for $(e/0.009)^2Q/k_2> 4.0\times10^9$ the system remains trapped in the resonance for more than $10^5\yr$ (see Figure \ref{Fig:detrap}), and as Figure \ref{Fig:mt_rate} demonstrates, the mass transfer rate remains within roughly $10\%$ of its nominal value $\dot{m}_2\thickapprox10^{-8}\rm \ M_\odot \ {\rm yr^{-1}}$. In this case, the eccentricity of the inner binary will never damp down to a fixed point because it is indirectly driven up by semimajor axis expansion due to mass loss on a timescale of order $~10^4\yr$, which is at least an order of magnitude shorter then the timescale for eccentricity damping due to tidal dissipation. As expected from the evolutionary scenario the intrinsic period derivative is positive, but because of the effect of tidal dissipation it is smaller than that due to Roche lobe overflow alone (see Figure \ref{Fig:mt_rate}); the nominal value for the period derivative due to Roche lobe overflow alone for our system parameters is $\dot{P}/P\thickapprox 1.3\times10^{-7}\yr^{-1}$ \citep{1987ApJ...322..842R}.
Finally, we note that if the inner binary is in fact expanding, the eccentricity will tend to increase as well. If the eccentricity is large enough, then Roche lobe overflow will occur through both the inner and outer Lagrange points, in contradiction with the low observed x-ray absorption. Figure \ref{Fig:detrap} shows that the eccentricity, while increasing with time, remains smaller than $0.07$, consistent with the lack of mass loss through the outer ($L_2$) Lagrange point \citep{2005MNRAS.358..544R}.
\begin{figure}
\epsscale{0.55}
\plotone{f8.eps}
\plotone{f8_b.eps}
\caption{
The eccentricity as a function of time (upper
panel) and the argument of periastron as a function of time ( $\omega$ versus $t$, lower panel) in the quadrupole approximation using $(e/0.009)^2Q/k_2=4.5\times10^9$. The system remains trapped in the resonance for more than $10^5\yr$ which is a considerable fraction of the system lifetime. The eccentricity stays under the limit of $0.07$, a constraint imposed by the absence of $L_2$ mass loss.
\label{Fig:detrap}}
\end{figure}
\begin{figure}
\epsscale{0.55}
\plotone{f9.eps}
\plotone{f9_b.eps}
\caption{
The mass transfer rate as a function of time (upper panel) and $\dot{P}/P$ (lower panel, solid line) as a function of time in the quadrupole approximation using $(e/0.009)^2Q/k_2=4.5\times10^9$. The system remains trapped in the resonance for more than $10^5\yr$, which is a reasonable fraction of the system lifetime. The mass transfer rate is within $~10\%$ of its nominal value $\dot{m}_2 \thickapprox10^{-8}\rm \ M_\odot \ {\rm yr^{-1}}$. $\dot{P}/P$ is lower than that due to Roche lobe overflow alone (dashed line), but still $>0$.
\label{Fig:mt_rate}}
\end{figure}
\subsection{The nature of the third body}
If the outer star is a white dwarf or a main sequence star, its mass
is constrained to be $\lesssim0.5M_{\bigodot}$ by the lack of an
optical detection \citep{2001ApJ...563..934C}. The lack of absorbing
material along the line of sight to the X-ray source indicates that
the third star is not overflowing {\em its} Roche lobe. From the Roche
lobe fitting formula of \citep{1983ApJ...268..368E},
\be
R_3<R_{L}\approx{0.49q^{2/3}\over0.6q^{2/3}+\ln(1+q^{1/3})}a_{out},
\ee
where $q$ is the mass ratio of the third star to the total mass of the
inner binary. This translates to $R_3\lesssim0.36R_\odot$. From table
9 in \citet{2007ApJ...663..573B}, this implies
$m_3\lesssim0.39M_\odot$. We conclude that the only stars with mass
$\gtrsim0.4M_{\bigodot}$ that will fit into the outer orbit is a white
dwarf or neutron star (or black hole). This leaves open the
possibility that the third star is a main sequence star with
$m\lesssim0.4M_\odot$.
According to \citet{2008msah.conf..101I}, her Table 1, the fraction of
hierarchical triples consisting of a neutron star primary, a white
dwarf secondary, and a white dwarf tertiary formed via
binary\textemdash binary encounters is about $1.4\times10^{-3}$. The
fraction of triples consisting of a neutron star-white dwarf binary
orbited by a $0.4M_\odot$ (or lower) main sequence star is
similar. The fraction of neutron star-white dwarf-neutron star systems
is $2.1\times10^{-5}$. If this is the primary channel for formation of
triple star systems, the third star is likely to be either a white
dwarf or a low mass ($m<0.4M_\odot$) main sequence star.
\section{DISCUSSION} \label{sec:discussion}
The origin of the $170$ day luminosity variations in 4U 1820-30 was
first attributed to the presence of a third body in the system by
\citet{1988IAUS..126..347G}; this possibility was expanded upon by
\citet{2001ApJ...563..934C} and more recently by
\citet{2007MNRAS.377.1006Z}. \citet{2007MNRAS.377.1006Z} used a
numerical model that calculates the time evolution of an isolated
hierarchical triple of point masses, using secular perturbation theory
up to octupole terms. Their model neglects the effects of tidal and
rotational distortion of the white dwarf, tidal friction, mass
transfer and gravitational radiation from the inner binary. Their
calculations do include the GR periastron precession of the inner
binary.
\citet{2007MNRAS.377.1006Z} find a configuration that reproduces the
171 day oscillations (assuming they are due to variations in
$e$). They note that the GR precession rate is near $170$ days, and
then choose a rather low neutron star plus white dwarf mass of
$1.29+0.07M_\odot$. With this choice, the period of the GR precession
is $\sim168$ days. This period is very near, but slightly shorter
than, the observed 171 day period. To arrive at the longer period,
they choose the location and inclination of the third body so that the
Kozai torque results in a retrograde precession. When added to the GR
precession, this retrograde Kozai precession ensures the period of
eccentricity oscillations will be longer than 168 days. They are
driven to a much lower magnitude for the Kozai torque than employed in
this paper; they use $a_{out}=8.66a$ and $i_0=40.96^\circ$. They start
with $\omega=0^\circ$ and $e=10^{-4}$, ensuring that their solution
circulates rather than librating.
They note that the apparent near equality between the Kozai and GR
precession rates is ``a very remarkable coincidence'', but go on to
say that they do not have any explanation for this coincidence.
We have argued that the origin of the $171$ day period of the
luminosity variation of LMBX 4U 1820-30 arises from libration in the
Kozai resonance. This trapping explains why the Kozai precession rate
is comparable to the sum of the other precession rates in the
problem. If $k_2$ is small enough, then the largest precession
frequency in the absence of a third body is that given by general
relativity. In that case, the Kozai and GR precession rates will sum
to zero, i.e., the magnitude of the two precession rates will be
equal. Hence if $k_2$ is small, then the expansion of the orbit of the
inner binary naturally explains the ``remarkable coincidence'' noted
by \citet{2007MNRAS.377.1006Z}. We stress that, independent of the value
of $k_2$, the natural state of the system is likely to be libration
rather than circulation.
Trapping into libration is a consequence of mass-transfer driven
orbital expansion in the inner binary. We have pointed out that the
apparent negative period derivative, if it were intrinsic to the
system, would not last for reasonable fraction of the system's lifetime. We find this to be an
untenable situation.
The observed negative period derivative of the inner binary allows us
to constrain the tidal dissipation factor $Q$ yielding a very firm
lower limit of $(e/0.009)^2Q/k_2>3.15\times10^9$. We argue, however, that $(e/0.009)^2Q/k_2$ has to be yet higher, to trap and maintain the system in libration
around the stable Kozai fixed point. Our finding indicates that if 4U
1820-30 is indeed a triple system, the negative period derivative is
not an intrinsic property of the system. However, as we showed in section \ref{sec:constraints} it does not arise from the
acceleration of the gravitation field of the globular cluster in which
4U 1820-30 resides, as suggested by \citet{1993MNRAS.260..686V}.
In general, the eccentric orbit of a close binary system similar to 4U
1820-30 could lead to a time-dependent irradiation of the secondary
which could, in turn, give rise to a thermal tide
\citep{2010ApJ...714....1A}. A thermal tidal torque opposes the
gravitational tidal torque, tending to force the secondary away from
synchronous rotation and to enhance the orbital eccentricity. An
asynchronous spin may cause large tidal heating rates, depositing heat
in the interior of the secondary. In addition, the irradiation of the
stellar surface by the neutron star (or by the accretion disk) will
reduce the heat flux from the center of the white dwarf outward, so
these irradiated white dwarfs will be hotter than passively cooling
white dwarfs. Since they are hotter, they will have larger radii. The
interplay between the two tidal torques would eventually set the
equilibrium spin state. As long as this equilibrium state is not
reached, the resulting bulge may oscillate, causing a periodic
exchange of angular momentum between the orbit and the spin of the
white dwarf. This might provide an alternate mechanism for producing
the luminosity variations in 4U 1820-30. Since this period is very stable, $\dot
P_3/P_3<2.2\times10^{-4}$ according to \citet{2001ApJ...563..934C}, we are currently looking into possibility of such an interplay between gravitational and thermal tidal
torque as an explanation for $171$ day period in 4U 1820-30.
We anticipate that the resonance trapping mechanism we have described
in this paper is generic in Roche lobe overflow binaries in triple
systems. The exact nature of the librating orbit will vary with the
properties of the particular system. For example, for a binary with a
larger semimajor axis, such that $\dot\omega_{TB}>\dot\omega_{GR}$,
resonant trapping will lead to $|\dot\omega_{kozai}|=\dot\omega_{GR}$.
\section{CONCLUSIONS\label{sec:conclusions}}
This paper provides an estimate for a lower limit of the tidal dissipation
parameter $Q$ for a Helium white dwarf. It also elucidates the
possible evolutionary history of 4U 1820-30, i.e., how the system arrived
at a state where the secular dynamics are not dominated by the effects
of the white dwarf's tidal bulge, despite the fact that the white
dwarf is overflowing its Roche lobe in an orbit with a period of
$685\s$.
We suggest that the system is trapped in the Kozai resonance. This
resonance trapping is responsible for the observed $171$ day period,
which we interpret as the period of small oscillations around a stable
fixed point in the Kozai resonance. If the system is not librating,
one requires very fine tuning to get the 171 day period.
We provide lower limit on the tidal dissipation rate, as measured by
the factor $Q$; $(e/0.009)^2Q/k_2>4\times10^9$.
Further exploration of the long term (tidal and mass overflow-driven)
evolution of this and similar short period ultra compact X-ray
binaries is clearly warranted. Inclusion of the thermal tides into dynamics of these systems may introduce an alternative explanation for origin of long period modulation of the light curve.
For the particular case of 4U 1820-30, better modelling of the
gravitational potential in the host globular cluster, NGC 6624, would
allow for an upper limit on $Q$. We are pursuing both lines
of investigation.
\acknowledgments The authors are grateful to Natasha Ivanova, Cole
Miller, Doug Hamilton, Andrew Cumming and Phil Arras for helpful discussions. This
research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France, and of NASA's Astrophysics Data System.
The authors are supported in part by the Canada Research Chair program and
by NSERC of Canada.
\vspace{15mm}
\section*{APPENDIX A\\
EQUATIONS OF MOTION}
\noindent The equations of motion we employ model the Kozai interaction, the dynamical effects of the tidal bulge of the He white dwarf, GR periastron precession, the rotational bulge of the He white dwarf, conservative mass transfer driven by the emission of gravitational radiation, and tidal dissipation. We do not consider tides raised on the neutron star primary. Detailed derivation of the equations representing Kozai cycles with tidal friction and GR periastron precession can be found in \citet{1998ApJ...499..853E} and \citet{2001ApJ...562.1012E}. Stellar masses are denoted by $m_1$ (the mass of the neutron star primary), $m_2$ (the mass of the white dwarf secondary) $M\equiv m_1+m_2$ (the inner binary mass), $m_3$ (the mass of the outer companion), and the reduced mass of the inner binary $\mu=m_{1}m_2/(m_{1}+m_2)$. The mean motion of the inner binary is $n=2\pi/P=[GM/a^{3}]^{1/2}$. The inner binary orbital elements are: semimajor axis $a$, eccentricity $e$, mutual inclination between the inner binary and the outer binary orbit $\imath$, the argument of periastron $\omega$, the longitude of ascending node $\Omega$. $k_{2}$ is the tidal Love number, $Q$ is the tidal dissipation factor, and $R_2$ is the radius of the white dwarf. The orbital parameters of the outer binary are denoted $a_{out}$ and $e_{out}$. $G$ is Newtons constant and $c$ is the speed of light.
Changes in the semimajor axis of the inner binary $a$ are caused by tidal dissipation and mass transfer:
\be \label{eq:a_dot}
\dot{a}=\dot{a}_{TD}+\dot{a}_{MT},
\ee
where
\bea \label{eq:a_dots}
\dot{a}_{TD}&=&-2a\frac{1}{t_{F}}\left[\frac{1+\frac{15}{2}e^{2}+\frac{45}{8}e^{4}+\frac{5}{16}e^{6}}{(1-e^{2})^{\frac{13}{2}}}-\frac{\Omega_{h}}{n}\frac{1+3e^{2}+\frac{3}{8}e^{4}}{(1-e^{2})^{5}}\right]\label{eq:a_dot_TD}\\
&-&2a\frac{e^{2}}{1-e^{2}}\frac{9}{t_{F}}\left[\frac{1+\frac{15}{4}e^{2}+\frac{15}{8}e^{4}+\frac{5}{64}e^{6}}{(1-e^{2})^{\frac{13}{2}}}-\frac{11\Omega_{h}}{18n}\frac{1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}}{(1-e^{2})^{5}}\right]\\
\dot{a}_{MT}&=&-\frac{2}{3} a\frac{\dot{m}_{2}}{m_2}
\eea
with $\dot{m}_{2}$ given by:
\be \label{eq:m_dot}
\dot{m}_{2}=-6.21\times 10^{-4}(\frac{m_{1}}{M_{\odot}})^{\frac{2}{3}}\left(\frac{P_{periastron}}{minutes}\right)^{\frac{-14}{3}} \frac{M_{\odot}}{\yr}.
\ee
For the zero eccentricity case, equation \ref{eq:m_dot} is derived in detail in \citet{1987ApJ...322..842R}, where instead of the dependancy on periastron period $P_{periastron}$ they consider dependancy on binary period.
The tidal friction time scale is
\be \label{eq:tF}
t_{F}=\frac{1}{6}\left(\frac{a}{R_2}\right)^{5}\frac{1}{n}\frac{m_{2}}{m_{1}}\frac{Q}{k_{2}}.
\ee
The eccentricity of the inner binary $e$ is affected by the Kozai torque and by tidal dissipation:
\be \label{eq:e_dot}
\dot{e}_{in}=\dot{e}_{Kozai}+\dot{e}_{TD},
\ee
where
\bea \label{eq:e_dots}
\dot{e}_{Kozai}&=&\frac{15}{8}\frac{Gm_3}{a_{out}^3(1-e^{2}_{out})^{\frac{3}{2}}n}e\sqrt{1-e^{2}}\sin2\omega\sin^{2}\imath \\
\dot{e}_{TD}&=& -\frac{9e}{t_{F}}\left[\frac{1+\frac{15}{4}e^{2}+\frac{15}{8}e^{4}+\frac{5}{64}e^{6}}{(1-e^{2})^{\frac{13}{2}}}-\frac{11\Omega_{h}}{18n}\frac{1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}}{(1-e^{2})^{5}}\right]\label{eq:e_dot_TD}.\\
\eea
The mutual inclination between the inner and the outer binary orbit, $\imath$ is affected by Kozai torques, by the rotational bulge, and by tidal dissipation:
\be \label{eq:i_dot}
\dot{\imath}=\dot{\imath}_{Kozai}+\dot{\imath}_{RB}+\dot{\imath}_{TD},
\ee
where
\bea \label{eq:i_dots}
\dot{\imath}_{Kozai}&=&-\frac{15}{8}\frac{Gm_3}{a_{out}^3(1-e^{2}_{out})^{\frac{3}{2}}n}\frac{e^{2}}{\sqrt{1-e^{2}}}\sin2\omega\sin\imath\cos\imath \\
\dot{\imath}_{RB}&=&\frac{m_{1}k_{2}R_2^{5}}{2\mu na^{5}}\frac{\Omega_{h}(\Omega_{q}\sin\omega-\Omega_{e}\cos\omega)}{(1-e^2)^5}\\
\dot{\imath}_{TD}&=&-\frac{\Omega_{e}\sin\omega}{2nt_{F}}\frac{1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}}{(1-e^{2})^{5}}-\frac{\Omega_{q}\cos\omega}{2nt_{F}}\frac{1+\frac{9}{2}e^{2}+\frac{5}{8}e^{4}}{(1-e^{2})^{5}}.
\eea
Besides the negative precession rate of the argument of periastron due to Kozai cycles, the total precession rate of the argument of periastron has additional positive contributions from the tidal bulge, GR, the rotational bulge, and the tidal dissipation:
\be \label{eq: omega_dot}
\dot{\omega}_{in}=\dot{\omega}_{Kozai}+\dot{\omega}_{TB}+\dot{\omega}_{GR}+\dot{\omega}_{RB}+\dot{\omega}_{TD},
\ee
\noindent where
\bea \label{eq:omega_dots}
\dot{\omega}_{Kozai}&=&\frac{3}{4}\frac{Gm_3}{a_{out}^3(1-e^{2}_{out})^{\frac{3}{2}}n}\frac{1}{\sqrt{1-e^{2}}}\Bigg[ 2(1-e^{2})+5\sin^{2}\omega(e^{2}-\sin^{2}\imath)\Bigg] \\
\dot{\omega}_{TB}&=&\frac{15(GM)^{\frac{1}{2}}}{16a^{\frac{13}{2}}}\frac{8+12e^{2}+e^{4}}{(1-e^{2})^{5}}\frac{m_{1}}{m_{2}}k_{2}R_2^{5}\\
\dot{\omega}_{GR}&=&\frac{3(GM)^{\frac{3}{2}}}{a^{\frac{5}{2}}c^{2}(1-e^{2})}\\
\dot{\omega}_{RB}&=&\frac{M^{\frac{1}{2}}}{4G^{\frac{1}{2}}a^{\frac{7}{2}}(1-e^{2})^{2}}\frac{k_{2}R_2^{5}}{m_{2}}\\
&\times&\Bigg[\left(2\Omega^{2}_{h}-\Omega^{2}_{e}-\Omega^{2}_{q}\right)+2\Omega_{h}\cot\imath\left(\Omega_{e}\sin\omega +\Omega_{q}\cos\omega\right)\Bigg]\\
\dot{\omega}_{TD}&=&-\frac{\Omega_e\cos(\omega)\cot\imath}{2nt_F}\frac{1+\frac{3}{2}e^2+\frac{1}{8}e^4}{(1-e^2)^5}+\frac{\Omega_q\sin(\omega)\cot\imath}{2nt_F}\frac{1+\frac{9}{2}e^2+\frac{5}{8}e^4}{(1-e^2)^5}.
\eea
The precession of the longitude of ascending node is caused by Kozai cycles, rotational bulge and tidal dissipation:
\be \label{eq:Omega_dot}
\dot{\Omega}_{in}=\dot{\Omega}_{Kozai}+\dot{\Omega}_{RB}+\dot{\Omega}_{TD},
\ee
where
\bea \label{eq:Omega_dots}
\dot{\Omega}_{Kozai}&=&-\frac{Gm_3}{a_{out}^3(1-e^{2}_{out})^{\frac{3}{2}}n}\frac{\cos\iota}{4\sqrt{1-e^{2}}}\Bigg(3+12e^{2}-15e^{2}\cos^{2}\omega\Bigg)\\
\dot{\Omega}_{RB}&=&\frac{m_{1}k_{2}R_2^{5}}{2\mu na^{5}\sin\imath}\frac{\Omega_{h}}{(1-e^{2})^{2}}\Bigg(-\Omega_{q}\cos\omega-\Omega_{e}\sin\omega\Bigg)\\
\dot{\Omega}_{TD}&=&\frac{\Omega_{e}\cos\omega}{2n\sin\imath t_{F}}\frac{1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}}{(1-e^{2})^{5}}-\frac{\Omega_{q}\sin\omega}{2n\sin\imath t_{F}}\frac{1+\frac{9}{2}e^{2}+\frac{5}{8}e^{4}}{(1-e^{2})^{5}}.
\eea
\vspace{20mm}
\section*{APPENDIX B\\
ADIABATIC INVARIANCE OF THE ACTION}
\noindent Time-dependent Hamiltonians, even those with just one degree of freedom, can be difficult to solve. However, for Hamiltonians where the time dependance is sufficiently slow, the problem is easier to tackle due to the existence of variables that are almost constant. The approximate constants are the action variables of the Hamiltonian, when the slow time dependance is neglected. Suppose that the time dependance enters through a time dependent parameter $\kappa(t)$. If the parameter $\kappa$ varies very slowly with time, treating $\kappa$ as time-independent parameter allows us to find action-angle variables following the standard prescription. These action-angle variables are function of time through $\kappa(t)$, which leads to the action no longer being a constant of motion. However, when $\kappa$ varies slowly with time, the action is nearly constant. Such an action is known as an adiabatic invariant.
As described in section \ref{sec:capture}, capture in the resonance is a natural consequence of semimajor axis expansion driven by mass transfer from the white dwarf. The Hamiltonian of our system (see equation \ref{eq:hamiltonian}) is a function of the semimajor axis, which is a parameter of $H$, playing the role of $\kappa(t)$. In our case the semimajor axis is not the only parameter varying with time; the masses of the inner binary vary with time as well. Here we show, both analytically and via numerical integration, that the change in the eccentricity is coupled to the change in the semimajor axis. When the semimajor axis expands (respectively, contracts) the eccentricity of the stable fixed point increases (decreases). We also demonstrate that the timescale for the change in the eccentricity is a factor of $\gtrsim 150$ shorter than the timescale for the semimajor axis.
To find the action we expand our Hamiltonian (equation \ref{eq:hamiltonian}) around the fixed point:
\bea
\G=\G_f + \Delta \G\\
\omega=\omega_f +\Delta \omega
\eea
\noindent Since we are expanding around the resonance, all terms $\propto \Delta \G$ vanish. After some algebra we find:
\bea \label{eq:exp_hamil}
H&=&-A\Big[-10+12\frac{\G_f^2} {\La^2} \cos^2{\imath_f}+9\frac{\G_f^2}{\La^2}+15\cos^2{\imath_f}+\frac{B}{A} \frac{\La}{\G_f}+k_2\frac{C}{A}\left( 35\frac{\La^9}{\G_f^9}-30\frac{\La^7}{\G_f^7}+3\frac{\La^5}{\G_f^5}\right)+k_2\frac{D}{A} \frac{\La^3}{\G_f^3}\Big]\nonumber\\&&-\frac{A}{2\La^2}\Big[18-24\cos^2{\imath_f}+2\frac{B}{A}\frac{\La^3}{\G_f^3}+30k_2\frac{C}{A}\left(105\frac{\La^11}{\G_f^11}-56\frac{\La^9}{G_f^9}+3\frac{\La^7}{G_f^7}\right)+12k_2\frac{D}{A}\frac{\La^5}{G_f^5}\Big]\Delta\G^2\nonumber\\&&-15A\left(1-\frac{\G_f^2}{\La^2}\right)\sin^2{\imath_f}\Delta\omega^2
\eea
\noindent which is similar to the Hamiltonian of the harmonic oscillator. Written more compactly (and implicitly defining $\alpha(t)$, $\beta(t)$ and $\C(t)$):
\be\label{eq:hamil_for_ho}
H= \C(t)+\frac{\alpha(t)}{2}\Delta\G^2+\frac{\beta(t)}{2}\Delta\omega^2=H_0.
\ee
\noindent We solve for $\Delta\G$ and evaluate the integral
\be
J=\frac{2}{\pi}\int_0^{\Delta\omega_{max}}\left(\frac{2(H_0-\C(t))}{\alpha}-\frac{\beta}{\alpha}\right)^{\frac{1}{2}}\,d \Delta\omega
\ee
\noindent where $\Delta\omega_{max}=\left(2(H_0-\C(t))/\beta\right)^{\frac{1}{2}}$. We find:
\be\label{eq:action_ho}
J=\frac{H_0-\C(t)}{(\alpha\beta)^{\frac{1}{2}}}.
\ee
\noindent Plugging in the corresponding terms from equation (\ref{eq:exp_hamil}) yields:
\be
J=\frac{\La(a,m_1,m_2)}{e_f\sin{\imath_f}}\! \, \frac{P_1(e_f, a, m_1, m_2, \imath_f)}{P_2^\frac{1}{2}(e_f, a, m_1, m_2, \imath_f)},
\ee
\noindent where
\bea
P_1&=&\frac{H_0}{A}-10-12(1-e_f^2)\cos^2{\imath_f}+9(1-e_f^2)+15\cos^2{\imath_f}+\frac{B}{A}(1+\frac{1}{2}e_f^2)+4k_2\frac{C}{A}(2+15e_f^2)\nonumber\\&&+k_2\frac{D}{A}(1+\frac{3}{2}e_f^2)\\
\eea
and
\be
P_2=30\Big[18-24\cos^2{\imath_f}+2\frac{B}{A}(1+\frac{3}{2}e_f^2)+120k_2\frac{C}{A}(13+84e_f^2)+12k_2\frac{D}{A}(1+\frac{5}{2}e_f^2).
\ee
Since the action $J$ is an adiabatic invariant, we have:
\be\label{eq:dJdt}
\frac{dJ}{dt}=\frac{\partial J}{\partial e}\dot{e}+\frac{\partial J}{\partial a}\dot{a}+\left(\frac{\partial J}{\partial m_2}-\frac{\partial J}{\partial m_1}\right)\dot{m}_2=0.
\ee
The partial derivatives are:
\bea
\frac{\partial J}{\partial e}&=&-\frac{J}{e_f}\left(1-\frac{e_f}{P_1}\frac{\partial P_1}{\partial e}+\frac{1}{2}\frac{e_f}{P_2}\frac{\partial P_2}{\partial e}\right)=\C_e\frac{J}{e_f}\\
\frac{\partial J}{\partial a}&=&\frac{J}{a}\left(1+\frac{a}{P_1}\frac{\partial P_1}{\partial a}-\frac{1}{2}\frac{a}{P_2}\frac{\partial P_2}{\partial a}\right)=\C_a\frac{J}{a}\\
\frac{\partial J}{\partial m_2}&=&\frac{J}{m_2}\left(1+\frac{m_2}{P_1}\frac{\partial P_1}{\partial m_2}-\frac{1}{2}\frac{m_2}{P_2}\frac{\partial P_2}{\partial m_2}\right)=\C_{m_2}\frac{J}{m_2}\\
\frac{\partial J}{\partial m_1}&=&\frac{J}{m_1}\left(1+\frac{m_1}{P_1}\frac{\partial P_1}{\partial m_1}-\frac{1}{2}\frac{m_1}{P_2}\frac{\partial P_2}{\partial m_1}\right)=\C_{m_1}\frac{J}{m_1}.
\eea
Plugging these partial derivatives back into equation (\ref{eq:dJdt}) yields:
\be\label{eq:all_dots}
0=\C_e\frac{\dot{e}}{e_f}+\C_a\frac{\dot{a}}{a}+\left(\C_{m_2}-\frac{m_2}{m_1}\C_{m_1}\right)\frac{\dot{m}_2}{m_2}.
\ee
The inner binary orbit is eccentric, which makes the mass transfer rate proportional to periastron distance $r_p=a(1-e)$. Hence the $\dot{m_2}$ term can be decoupled into two terms, one proportional to $\dot e$ and the other proportional to $\dot a$:
\be\label{eq:m2decouple}
\frac{\dot{m}_2}{m_2}=\frac{\dot{r_p}}{r_p}=-\frac{3}{2}\frac{\dot a}{a}+\frac{3}{2}\frac{\dot e}{1-e}.
\ee
Combining equations (\ref{eq:all_dots}) and (\ref{eq:m2decouple}) and solving for $\dot e$ leads to:
\be
\frac{\dot e}{e_f}=\frac{\frac{3}{2}\left(\C_{m_2}-\frac{m_2}{m_1}\C_{m_1}\right)-\C_a}{\C_e+\frac{3}{2}\frac{e_f}{1-e_f}\left(\C_{m_2}-\frac{m_2}{m_1}\C_{m_1}\right)}\frac{\dot a}{a}.
\ee
Plugging in the numerical values:
\be\label{eq:edot_vs_adot}
\frac{\dot e}{e_f}\approx150\frac{\dot a}{a}.
\ee
Defining the time scales for the eccentricity and the semimajor axis to decay or increase (depending on the value of $Q$) as $\tau_e=e_f/\dot e$ and $\tau_a=a/\dot a$, the timescales in equation (\ref{eq:edot_vs_adot}) are related by:
\be
\tau_e\sim6.7\times10^{-3}\tau_a.
\ee
\begin{figure}
\epsscale{0.55}
\plotone{f10.eps}
\caption{
$\dot e$ as a function of time for the case where the semimajor axis is expanding, $Q=8\times10^7$. We start integration exactly at the fixed point, where initial eccentricity is $e_{f,0}=0.01555$. All other parameters are as listed in table 1. The solid line comes from the analytic estimate where the action $J$ is considered to be an adiabatic invariant. The dashed line is a result of numerical integration. As expected from the action being adiabatic invariant, $\dot e$ is positive. The difference in the magnitude of $\dot e$ within first $2\times10^5\yr$ is a result of our simplified analytic calculation that does not include spin dynamics. Since our analytic estimate is valid for small eccentricities, here we stop the integration when $e>0.1$ \label{Fig:e_dot8Q7}}
\end{figure}
To demonstrate that the eccentricity evolution is indeed a consequence of the action being an adiabatic invariant, we follow the evolution of the orbit around the fixed point $e_f=0.0155$ and $\omega_f=90^\circ$. Figure \ref{Fig:e_dot8Q7} shows $\dot e$ as a function of time in a case where the semimajor axis is increasing, meaning that tidal dissipation is sufficiently weak so that the evolution of the semimajor axis is dominated by mass transfer ($Q=8\times10^7$). The solid line presents $\dot e$ predicted by equation (\ref{eq:edot_vs_adot}). For $t\gtrsim10^{5}\yr$ the numerical integration gives $\dot e \approx 10^{-7}\yr^{-1}$ corresponding to a timescale $150$ times shorter than the timescale for the semimajor axis.
Despite the fact that the semimajor axis is expanding, the numerical integration shows a transient phase (roughly the first 2000 years) where $de/dt<0$, and a longer phase ($\sim 10^5\yr$) where $\dot e$ is larger than predicted by equation (\ref{eq:edot_vs_adot}). There are contributions to the eccentricity evolution which we have ignored in our analytic treatment; for example the spin of the white dwarf is not locked during the evolution of the system. These un-modelled contributions are the source of the transient behaviour.
To support this statement, we illustrate the eccentricity evolution in various cases where we turn off different dynamical effects in Figure \ref{Fig:e_all8Q7}. The solid line presents a result from the numerical integration that includes all dynamical effects in our model, while the dotted line is the same integration with the $\dot e_{TD}$ term set to $0$; the result shows that direct tidal dissipation on the eccentricity (equation \ref{eq:e_dot_TD}) is not dynamically significant. The dashed line presents the case where $\dot a_{TD}$ is set to $0$ (see equation \ref{eq:a_dot_TD}); The result shows that $\dot a_{TD}$ has a significant influence on the eccentricity evolution. The dash-dotted line shows the eccentricity when tidal dissipation factor $Q$ is set to infinity, but only in the differential equations that govern spin evolution. The long dash-dotted line shows the eccentricity evolution in the case where $Q$ is set to infinity in the equations that govern the evolution of the spins together with $\dot a_{TD}$ being set to $0$. The latter three cases demonstrate that the eccentricity starts increasing immediately with the semimajor axis expansion, which is exactly the behaviour predicted by the analytic analysis. After $5.5\times10^5\yr$ the eccentricity becomes $\gtrsim0.1$ and since our analytic estimate is valid only for small eccentricities we stop the integration here.
The cause of the transient behaviour is the fact that the spin of the white dwarf is not, contrary to our choice of initial conditions, tidally locked during the evolution of the system. Whether the spin settles down in some Cassini state or other stable configuration later during the evolution of the system is a possibility open to further investigation.
\begin{figure}
\epsscale{0.55}
\plotone{f11.eps}
\caption{
The eccentricity of the fixed point of the inner binary as a function of time. Initial conditions are the same as in Figure \ref{Fig:e_dot8Q7}. The solid line is a result of the numerical integration including all dynamical effects, while the dotted line is a result of the same integration but with $\dot e_{TD}=0$; these two results demonstrate that direct tidal dissipation on the eccentricity does not significantly affect the evolution of eccentricity. The dashed line is the result of integration where $\dot a_{TD}=0$; this term has a more significant effect on the evolution of the eccentricity. The dash-dotted line presents the case where we set $Q=\infty$, but only in the equations that govern the spin evolution. The long dash-dotted line shows the eccentricity evolution when $Q=\infty$ for the spins and $\dot a_{TD}=0$; in this case we have the fastest increase in the eccentricity. \label{Fig:e_all8Q7}}
\end{figure}
\clearpage
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{"url":"http:\/\/www.atorahlife.com\/vcsco\/category\/probability-and-statistics\/","text":"## Statistical Significance of a Voter Survey\n\nIn a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let p\u0302 be the fraction of survey respondents who preferred the incumbent.\n\nGive an estimate of p.\n\np\u0302 = the sample mean = 215 \/ 400 = 0.5375\n\nThis is also the estimate of the population mean, p.\n\nCalculate the Standard Error (SE) of the estimate, p\u0302.\n\nThis is a Bernoulli sample. So, the sample variance for a Bernoulli sample is:\n\nsx2 = p\u0302 * (1- ?p\u0302 )\n\nsx2 = 0.5375*(1-0.5375)\n\nsx2 = 0.2486\n\nSo, the SE of the estimated mean is:\n\nSE = sqrt(sx2 \/ n)\n\nSE = sqrt(0.2486 \/ 400)\n\nSE = 0.0249\n\nWhat is the p-value for the test H0 : p = 0.5 vs. H1 : p != 0.5?\n\nThe first thing to note is that, because the alternate hypothesis is !=, this is a two-sided hypothesis.\n\nNext, compute the t-statistic assuming the null hypothesis:\n\nt = ( (Sample Mean) \u2013 (Null Hypothesis) \/ SE )\n\nt = (0.5375 \u2013 0.5) \/ 0.0249\n\nt = 1.506\n\nThe p-value for the two sided hypothesis is, therefore,\n\np = 2 * NORMDIST(-1.506, 0, 1, 1)\n\np = .1321\n\nNote that we use the negative t-statistic in this calculation of area under the curve because we are interested in the area that is inconsistent with the null hypothesis.\n\nWhat is the p-value for the test H0 : p = 0.5 vs. H1 : p > 0.5?\n\nThis time, we have a one-sided hypothesis test. So, the answer is half of what it was for the two sided test:\n\np = NORMDIST(-1.506, 0, 1, 1)\n\np = 0.066\n\nThe results of the one-sided hypothesis test differs from the two sided test because in the one-sided test, we are only interested in the case where the average is > the null hypothesis. In this case, we\u2019re interested in the case where the incumbent may actually have less than 50% of the population support. In the two-sided case, we were interested in the case where the incumbent had support from less than 50% of the population as well as the case where the incumbent had more than 57.5% of the population\u2019s support.\n\nDo the survey results contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey?\n\nIt depends on the chosen significance level. For significance levels > 0.066, there is statistical evidence that the incumbent is ahead.","date":"2018-02-22 06:49:40","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.863047182559967, \"perplexity\": 618.1835554995864}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891814036.49\/warc\/CC-MAIN-20180222061730-20180222081730-00677.warc.gz\"}"} | null | null |
Love South of Heaven
By: Robert C. Koehler Source: iViews Dec 22, 2015 No Comments
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Category: Americas, Faith & Spirituality, Featured, Life & Society Topics: Violence, War Values: Love Views: 651
Write about love, as in love thy enemy, and the social recoil sounds like this:
"There is no nexus at which we can speak with ISIS. Singing Kumbaya while being led to a beheading can't work."
"Any thug who threatens a cop gets what he deserves. One bullet or ten — I could care less. If a thug will threaten a cop or a prison guard, he will kill or maim me or mine without hesitation for very little reason. You want to give these thugs 'civil rights' — I want to give them a funeral. My way insures me and mine do not get killed or maimed. Your way insures I probably will."
These are responses to recent columns, in which I have tried to address the American and global hell created by the belief that violence, rather than endlessly begetting itself and spewing consequences far beyond conventional perception, actually solves problems in something other than the shortest of short terms. This is tricky. "Love thy enemy," or words to that effect, may be the foundation of Christianity and every other major religion, but they're utterly misunderstood and belittled in the realm of popular culture and I doubt they've ever been taken seriously at the level of government.
It's what they do in heaven. Sing Kumbaya, play the harp, love the other dead people (who, of course, went through a vetting process to get in similar to what we impose on refugees from Syria or Iraq). Here on Earth . . . come on, get real. The cynics cry "Trump! Trump!" because he tells it like it is, the way a junior high bully would. It's simple. It's linear. A bullet for a thug and the thug is dead. Problem solved.
Of course, a bigger problem is also created, but to relate this problem — ISIS, for instance — to one's own actions, or the actions of one's country, is way too complicated, so the cynics choose to stay simple.
How do we counter this simplistic-mindedness?
"The usual way to generate force is to create anger, desire and fear," writes Thich Nhat Hanh, the Vietnamese Buddhist monk, teacher, peace activist. "But these are dangerous sources of energy because they are blind . . ."
Let's pause mid-quote and summon the memory of our own impulsive emotions, our own anger and fear and blindness. Now let's arm those emotions. Whether or not we're "justified" in what we do next, the person on the receiving end is certain to have lost his or her humanity, at least for a terrible instant.
But what could happen next is so much worse: When these emotions become collective, the result is mob mentality. And when they become institutionalized — buried deep in the nation's soul — the inevitable result is war . . . and war . . . and war. And it's self-perpetuating. The dehumanized enemy strikes back, perhaps with horrendous actions, which of course justify what we do next. Eventually one side or the other "wins" and "peace" prevails for a moment or two, but it's always a broken and temporary thing, requiring armed guards at the perimeter. This is peace with fear.
And it's a way of life, humanity's normal: being perpetually armed, perpetually terrified, perpetually blind.
But Hanh's quote continues: ". . . whereas the force of love springs from awareness, and does not destroy its own aims. Out of love and the willingness to act, strategies and tactics will be created naturally from the circumstances of the struggle."
The force of love springs from awareness. What, oh God, does this mean? What, especially, does it mean beyond personal acts of big-hearted decency? Is love always distorted, often beyond recognition, when it is institutionalized?
Consider, for instance, the idea of the "penitentiary." With roots in the word "penitent," it was conceived by early 19th century prison reformers to be a place of resurrection — spiritual rebirth — for wayward souls. Maybe there was always a moralistic lunacy attached to the concept. In any case, it's no accident that the concept degraded over the decades to the word "pen" and the incarcerated have pretty much lost all their humanity.
"Prison must be something they fear, not just a momentary . . . way station on the road to the next crime," my correspondent, quoted above, a former prison guard (I think), wrote in his reply to my column from last week, in which I discuss an inmate's beating death by guards. "Today's prisons are a joke. The guards live in fear of the inmates — not the other way around. . . . Beatings are all that will keep some inmates in line. Who ever said there is no such thing as a bad boy was a lunatic. There are bad boys — more than you want to contemplate, and all they understand is superior violence."
I quote him in order to let his words percolate next to those of Thich Nhat Hanh. "The force of love springs from awareness." Again I ask, what does this mean? What does it mean in a world where violence is the answer to so many of our problems and a large percentage of the population is angry, fearful — and armed? What does it mean in a war- and prison-dependent economy, stoked by a too-often clueless media with a financial stake in more of the same? What does it mean in a world where cynicism rules?
I reach out to the planet's peacemakers. I know there are millions of you, enduring hardship and risking your lives to free us, to free the planet, from our self-inflicted hell.
"The careless habits of mind and heart that allow us to pollute and waste also allow us to treat other human beings as disposable," the editors of Commonweal wrote last June, commenting on the papal encyclical "Laudato Si." "'A true ecological approach,' (Pope) Francis writes, 'always becomes a social approach; it must integrate questions of justice in debates on the environment, so as to hear both the cry of the earth and the cry of the poor.'"
I would add: the cry of the refugees, the cry of the warriors, the cry of the inmates, the cry of the police, the cry of the prison guards . . . the cry of all humanity. Let us listen, let us reach out, let us look one another in the eyes no matter how difficult this proves to be.
© 2015 TRIBUNE CONTENT AGENCY, INC.
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The Cheapening of Human Life
War and Poverty: a Compromise with Hell
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{"url":"https:\/\/cracku.in\/blog\/cat-previous-year-data-interpretation-questions\/","text":"# CAT Previous Year Data Interpretation Questions\n\n1\n5328\n\nCAT Previous Year Data Interpretation Questions:\n\nData Interpretation questions and answers from previous year question papers. Important Data interpretation (DI) asked problems in CAT exam.\n\nCAT Data Interpretation Questions PDF\n\nInstructions (1 \u2013 5):\u00a0Read the following information carefully and answer the questions which follow.\n\nThe following bar graph represents the production, exports and per capita consumption of wheat in Sri Lanka for five years 2011 to 2015.\n\nConsumption = Production \u2013 Exports\nPer capita consumption = $$\\frac{consumption}{population}$$\n\nQuestion 1: The ratio of exports to consumption is highest for which year?\n\na) 2011\nb) 2012\nc) 2013\nd) 2014\n\nQuestion 2: Which year saw the highest percentage change in production?\n\na) 2012\nb) 2013\nc) 2014\nd) 2015\n\nQuestion 3: The population of Sri Lanka was highest in which year?\n\na) 2014\nb) 2013\nc) 2015\nd) 2012\n\nQuestion 4: What was the population of Sri Lanka in 2013\n\na) 2.5 million\nb) 2.5 lacs\nc) 3 million\nd) 3 lacs\n\nQuestion 5: The percentage change in consumption is highest for which year?\n\na) 2012\nb) 2014\nc) 2015\nd) It was same for two years\n\nInstructions (6 \u2013 10):\u00a0In a film industry every movie belongs to one of the five genres Action, Comedy, Drama, Horror and Science Fiction. Also, every movie falls in one category Hit or Flop.\n\nFollowing table shows the number of movies from each genre released in this movie industry in the year 2015.\n\nFollowing table gives the hit and flop status of movies released in the year 2015 including those of a superstar called KRK.\n\nQuestion 6: What is the total number of hit movies of KRK across the genres Comedy, Drama and Horror in the year 2015?\n\na) 36\nb) 39\nc) 42\nd) 46\n\nQuestion 7: What percentage of the total movies that released in the year 2015 were flops?\n\na) 49.25%\nb) 55.65%\nc) 62.33%\nd) 66.48%\n\nQuestion 8: What percentage of the total movies that released in the year 2015 were flops?\n\na) 49.25%\nb) 55.65%\nc) 62.33%\nd) 66.48%\n\nQuestion 9: Across how many genres were the number of hits more than the number of flops for the actor KRK?\n\na) 1\nb) 2\nc) 3\nd) 4\n\nQuestion 10: How many movies of KRK were flops across the genres Action, Horror and Science fiction?\na) 16\nb) 23\nc) 29\nd) None of these\n\nSolutions for\u00a0CAT Previous Year Data Interpretation Questions:\n\nSolutions (1 \u2013 5):\n\nIn 2011\nProduction = 200, Exports = 110 so consumption = 90\nIn 2012\nProduction = 240, Exports = 130 so consumption = 120\nIn 2013\nProduction = 220, Exports = 120 so consumption = 100\nIn 2014,\nProduction = 300, Exports = 170 so consumption = 130\nIn 2015\nProduction = 250, Exports = 140 so consumption = 110\nClearly the required ratio is highest for 2014.\n\nIn 2011\nProduction = 200\nIn 2012\nProduction = 240\nIn 2013\nProduction = 220\nIn 2014,\nProduction = 300\nIn 2015\nProduction = 250\nSo the percentage change in production is highest for the year 2014.\n\nPopulation = Consumption\/per capita consumption\nIn 2011\nConsumption = 90, Per capita consumption = 40\nIn 2012\nConsumption = 120, Per capita consumption = 50\nIn 2013\nConsumption = 100, Per capita consumption = 40\nIn 2014,\nConsumption = 130, Per capita consumption = 30\nIn 2015\nConsumption = 110, Per capita consumption = 50\nClearly, this ratio is highest for the year 2014.\n\nIn 2013\nConsumption = 100, Per capita consumption = 40\nSo required population = 100\/40 = 2.5 million.\n\nIn 2011\nProduction = 200, Exports = 110 so consumption = 90\nIn 2012\nProduction = 240, Exports = 130 so consumption = 110\nIn 2013\nProduction = 220, Exports = 120 so consumption = 100\nIn 2014,\nProduction = 300, Exports = 170 so consumption = 130\nIn 2015\nProduction = 250, Exports = 140 so consumption = 110\nSo the percentage change in consumption is highest for the year 2014.\n\nSolutions (6 \u2013 10):\n\nWe fill the following table to answer this data interpretation set.\nAs we know the total number of movies in each category, we can find the hit movies in each category as the % given in the first row second table * number of movies.\nFlop movies = Total Movies \u2013 Hit Movies\nFlop movies by KRK = % given in 3rd row * Flop Movies\nTotal movies by KRK = Flop Movies \/ (1 \u2013 % given in second row)\nHit movies by KRK = Total movies by KRK \u2013 Flop movies by KRK","date":"2022-06-27 01:49:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4314296245574951, \"perplexity\": 5706.75156550519}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103324665.17\/warc\/CC-MAIN-20220627012807-20220627042807-00443.warc.gz\"}"} | null | null |
Haga är den stadsdel i Umeå som är belägen närmast norr om centrum. Det är en av Umeås äldsta stadsdelar, vars södra delar (närmast centrum) domineras av villabebyggelse från 1940-talet, medan det längre norrut finns ett varierat utbud av två- och trevånings hyres- och bostadsrättshus från sent 1940-tal och framåt. Haga och stadsdelen Sandbacka räknas ofta ihop, under benämningen Haga/Sandbacka, eller ibland lite felaktigt bara Haga.
Geografi
Haga avgränsas i norr mot stadsdelen Sandbacka av gatan Sandaparken och i väst av länsväg 503 (tidigare E4) och det före detta regementsområdet för I20 (Västerbottens regemente), som numera hyser Umestans företagspark. I söder gränsar Haga mot järnvägen (Botniabanan), där gång- och cykeltunneln mot centrum domineras av konstverket Lev!. I öster finns friluftsområdet Gammlia och naturparken Stadsliden: vid dess kant ligger Umeå Sporthall (Gammliahallen) och Umeå Energi Arena (före detta Gammliavallen) – hemmaplan för fotbollslagen Umeå IK och Umeå FC.
Stadsdelen korsas i nord-sydlig riktning av Östra Kyrkogatan och i öst-västlig riktning av Hissjövägen (Länsväg 363). Dessutom sträcker sig ett genomgående parkstråk från Hagaskolan i nordväst längs Djupbäcken, först i sydostlig riktning och sedan rakt söderut fram till järnvägen och Järnvägsallén/Holmsundsvägen.
Historia
När Umeå efter stadsbranden 1888 fick en ny stadsplan (1998) – där den tidigare så brokiga stadskärnan indelades i färre och större kvarter och bebyggdes med mer "stadsmässiga" hus – rymdes inte längre hela stadsbefolkningen i centrala Umeå. Tomtpriserna ökade och människor med lägre inkomst fick söka sig till stadens utkanter. Ett av de områden där "oreglerad bebyggelse" växte fram var Haga, som kom att betraktas som Umeås första förstad.
Hagas äldsta, västra del ligger direkt norr om Umeå centrum. Före stadsbranden fanns där bara någon enstaka jordbruksfastighet och ett antal lador, men redan 1893 redovisas ett 40-tal fastighetsägare på Haga, huvudsakligen längs befintliga kommunikationsleder som den gamla landsvägen mot Ersmark (nuvarande Generalsgatan). Vid slutet av 1800-talet fanns också ett antal byggnader på Fredrikshög – ett område öster om Östra Kyrkogatan som idag motsvaras av bland annat kvarteret Focken vid Fredrikshögsgatan – som från 1930-talet kom att ingå i Haga.
Arbetare, hantverkare och fattiga
Befolkningen utgjordes vid denna tid främst av arbetare, enklare hantverkare och gårdsägare (jordbrukare). Husen var vanligtvis en- eller tvåfamiljshus av trä med omgivande trädgård. Ända fram till 1920-talet låg Haga utanför stadsplanelagt område, vilket gjorde stadsdelen till något av stadsbornas "avstjälpningsplats – där förlades stadens sopförbränningsanläggning, förorenande industriverksamhet som garveri och fattigstugan, Hagagården. Den kommunala servicen var också länge eftersatt. Centrala Umeå fick elektricitet åren 1892–1899, men Haga fick vänta till 1903, och även utbyggnaden av vatten- och avloppsnät släpade efter.
Järnvägsarbetare och militärer
När järnvägen nådde Umeå 1896 slog sig många järnvägsarbetare ned på det närbelägna Haga. Nya inflyttare följde på etableringen av Dragonregementet (1998) och Västerbottens regemente (1908) – vilket ännu idag kan spåras i kvartersnamn som Banvakten och Konduktören och på gatunamn som Bangatan, Kaptensgatan och Majorsgatan. En ny landsväg byggdes norrut, Mickelsträskvägen (nuvarande Östra Kyrkogatan).
År 1914 tillkallades arkitekt Per Olof Hallman från Överintendentsämbetet för att planlägga förstäderna Haga, Öbacka och Öppen plats (ett område Väst på stan), och hans planförslag för Haga godkändes 1922. Hallman bröt mot det vanliga rutnätssystemet och införde en mer kontinental stil, med ett friare kvarters- och gatunät som utgick från den naturliga terrängen, befintliga gator (som Korsgatan och Generalsgatan) och befintliga byggnader som de egnahemstomter som 1919 upplåtits vid det så kallade Sandahemmanet vid nuvarande Armbågakroken. Värt att notera är att Haga-området så sent som 1924 användes för renkapplöpningar och som vinterbete för renar från Rans sameby i Ammarnästrakten.
Ett centrum anlades vid Hagaplan, och kring Djupbäcken anlades ett grönområde med "brandavskiljande" syfte, Hagaparken. Ursprungsplanen föreskrev öppen trähusbebyggelse med högst två våningar, men när Östra Haga byggdes under 1930-talet tilläts även inredd vind. Vid det laget hade antalet fastigheter ökat till ett hundratal. Med förtätningen blev jordbrukarna färre, men järnvägsanställda och militärer allt fler. Hagabor skaffade extrainkomster via "Rum för resande", det byggdes kaféer och småskola – och 1925 invigdes Umeås nya arena för fotboll och friidrott, Gammliavallen.
Bostadsbrist och barnrikehus
Umeå befolkning ökade nu snabbt, från knappt 4 000 år 1900 till drygt 13 000 i slutet av 1930-talet, och med ökande bostadsbrist började staden planera för nybyggande på Haga. 1939 beslöt fullmäktige att upplåta tre kvarter vid Hagaparken för att bygga egnahem för barnrika familjer, och 28 små enplanshus byggdes. År 1942 antogs stadsarkitekt Kjell Wretlings nya stadsplan – som nu omfattade även området öster och norr om Hagaparken, samt Sandbacka upp till kvarteren ovanför Norra kyrkogården.
År 1946 stod HSB:s första fyra trevåningshus, i betong med slätputsade fasader, klara på Stenmarksvägen i kvarteret Huggaren på Haga. Nästa stora projekt var Gustav Garvares gata, som sträcker sig mellan Hagaparken och den nya Sandaparken. På Gustav Garvares uppförde HSB och Riksbyggen åren 1947–49 åtta trevåningshus i betong med 133 moderna lägenheter – med badrum och balkong, parkettgolv i vardagsrummen, elspis och kylskåp – främst avsedda för barnfamiljer och äldre. Nu gjorde också samhällsservicen entré, med mjölkaffär, charkuteriaffär, färghandel och fiskaffär (synliga på bilden till höger).
I Hagaparken mot Hissjövägen byggdes 1952 Umeå första friliggande förskola, ritad av arkitekt Bruno Mathsson. 1954 stod folkskolan Hagaskolan färdig, med 12 klassrum, 4 specialrum och plats för 338 elever.
Det beslöts att folkparken skulle flyttas från Erikslund (östra delen av vad som nu blivit Sandaparken) till det nya idrotts- och industriområdet vid Rothoffsvägen nedanför Gammlia, intill Gammliavallen. Hösten 1958 utökades området med Umeås första fullstora sporthall, Gammliahallen, som inte bara blev en arena för sport utan också för danser och musikarrangemang – inte minst för den jazzfestival som fick sin premiär 1968. Två år senare, 1970, invigdes där också en ny simhall).
Stadsdelen växer mot norr
Haga utvidgades också österut mot Stadsliden. Längs gatan Skogsbrynet i västra kanten av Gammliaskogen byggdes 1948–1950 1,5-plansvillor i trä, puts eller tegel, med utsikt över staden. När stenkrossanläggningen vid gatans sydligaste del avvecklades i slutet av 1950-talet byggdes fem punkthus i åtta våningar, inflyttningsklara 1962.
Sedan E4 fått ny sträckning genom Umeå i Västra Esplanadens fortsättning mellan militärområdet och Haga, utvidgades området i början av 1970-talet ytterligare. På Sandabrånet, norr om Hagaskolan, byggdes flerbostadshus, radhus och kedjehus, vanligtvis i gult tegel. Just radhusbyggande präglade årtiondet, och många sådana byggdes för att ersätta äldre småhus och småindustri. Inför bostadsmässan Bo 87 byggdes området Sandahöjd vid Stadslidens högt placerade utlöpare mot norr – som tidigare hyst Umeå camping – där två punkthus nu utgör ett landmärke i staden.
Andra associationer
Haga blev under några år riksbekant på grund av den serie överfallsvåldtäkter som inleddes där 1998 av den så kallade Hagamannen.
Källor
Externa länkar
Umea.se:s webbplats om Haga
Stadsdelar i Umeå | {
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Online ticket sales for the 2016 Beer & Cider Festival are now officially open.
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Lay the patient supine and notify the patient's physician.
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Initiate core strengthening exercises to maintain intraabdominal pressure. | {
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\section{Introduction}
Mathematical descriptions of the models described through differential
equations with derivatives of non-integer orders have proved to be a very
useful instrument for modeling of various viscoelasticity cases, stability
theory, controllability theory, and other related fields. Time-delays are
often related with physico-chemical processes, electric networks, hydraulic
networks, heredity in population growth, the economy and other related
industries. In general, a peculiarity of the adequate mathematical models is
that the rate of change of these processes depends on past history.
Differential systems describing these models are called time-delay
differential equations. The qualitative theory of linear time-delay equations
is well investigated. Recently, the time-delay differential equations has been
considered. in \cite{diblik3}-\cite{diblik8}. In \cite{diblik6}-\cite{mah2}
authors derived the exact expressions of solutions of linear continuous and
discrete delay equations by proposing the concepts of delayed matrix
functions. On the other hand, stability concepts and relative controllability
problems of linear time-delay differential equations were investigated in
\cite{shuklin}-\cite{med1}.
The unification of differential equations with delay and differential
equations with fractional derivative is provided by differential equations,
including both delay and non-integer derivatives, so called time-delay
fractional differenial equations. In applications, this unification is useful
for creating highly adequate models of some systems with memory. One can
notice that works on this field involve Riemann-Liouville and Caputo type
fractional derivatives. Besides these derivatives, there is an other
fractional derivative, involving the logarithmic function, so called Hadamard
fractional derivative. For the literature on the related field of fractional
time-delay equations of Caputo type and Riemann--Liouville type, we refer the
researcher to \cite{wang2}-\cite{wang5}.
It is known that (\cite{kst}, page 235, \cite{wang5}) a solution of a Hadamard
fractional linear syste
\begin{align*}
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) & =\lambda
y\left( t\right) +f\left( t\right) ,\ \ t\in\left( 1,T\right] ,h>0,\\
\left( ^{H}I_{1^{+}}^{1-\alpha}y\right) \left( 1^{+}\right) & =a\in
R,\lambda\in R,0<\alpha<1,
\end{align*}
has the for
\[
y\left( t\right) =a\left( \ln t\right) ^{\alpha-1}E_{\alpha,\alpha}\left[
\lambda\left( \ln t\right) ^{\alpha}\right] +\int_{1}^{t}\left( \ln
\frac{t}{s}\right) ^{\alpha-1}E_{\alpha,\alpha}\left[ \lambda\left(
\ln\frac{t}{s}\right) ^{\alpha}\right] f\left( s\right) \frac{ds}{s}.
\]
However, we find that there exists only one \cite{yang} work on the
representation of explicit solutions of Hadamard type fractional order delay
linear differential equations. In \cite{yang} authors studied the Hadamard
type fractional linear time-delay syste
\begin{align}
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) & =\mathcal{A
_{1}y\left( t-h\right) ,\ \ t\in\left( 1,T\right] ,h>0,\nonumber\\
y\left( t\right) & =\varphi\left( t\right) ,\ \ 1\leq t\leq
h,\label{ld1}\\
\left( ^{H}I_{1^{+}}^{1-\alpha}y\right) \left( 1^{+}\right) & =a\in
R^{n},\nonumber
\end{align}
where $\mathcal{A}_{01}$ is constant $n\times n$ square matrix.
Motivated by the above researches, we investigate a new class of Hadamard-type
fractional delay differential equations. We extend to consider an explicit
representation of solutions of a Hadamard type fractional time-delay
differential equation of the following form by introducing a new delayed M-L
type function with logarith
\begin{equation}
\left\{
\begin{array}
[c]{c
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) =\mathcal{A
_{0}y\left( t\right) +\mathcal{A}_{1}y\left( \frac{t}{h}\right) +f\left(
t\right) ,\ \ t\in\left( 1,T\right] ,h>0,\\
y\left( t\right) =\varphi\left( t\right) ,\ \ \frac{1}{h}<t\leq1,\\
\left( ^{H}I_{1^{+}}^{1-\alpha}y\right) \left( h^{+}\right) =a\in
\mathbb{R}^{n},
\end{array}
\right. \label{de11
\end{equation}
where $\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( \cdot\right) $ is the
Hadamard derivative of order $\alpha\in\left( 0,1\right) $, $\mathcal{A
_{0},\mathcal{A}_{01}\in\mathbb{R}^{n\times n}$ denote constant matricies, and
$\varphi:\left[ \frac{1}{h},1\right] \rightarrow\mathbb{R}^{n}$ is an
arbitrary Hadamard differentiable vector function, $f\in C\left( \left[
1,T\right] ,\mathbb{R}^{n}\right) $, $T=h^{l}$ for a fixed natural number
$l$.
The second purpose of this paper is to study the existence and stability of
solutions for a Hadamard type fractional delay differential equatio
\begin{equation}
\left\{
\begin{array}
[c]{c
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) =\mathcal{A
_{0}y\left( t\right) +\mathcal{A}_{1}y\left( \frac{t}{h}\right) +f\left(
t,y\left( t\right) \right) ,\ \ t\in\left( 1,T\right] ,h>0,\\
y\left( t\right) =\varphi\left( t\right) ,\ \ \frac{1}{h}<t\leq1,\\
\left( ^{H}I_{1^{+}}^{1-\alpha}y\right) \left( 1^{+}\right) =a\in
\mathbb{R}^{n},
\end{array}
\right. \label{de2
\end{equation}
At the end of this section, we state the main contribution of the paper as follows:
(i) We propose delayed M-L type functions $Y_{h,\alpha,\beta}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) $ with logarithms, by means of the
matrix equations (\ref{exp1}). We show that for $\mathcal{A}_{1}=\Theta$ the
function $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) $ coincide with M-L type function with two parameters $\left( \ln
t-\ln s\right) ^{\beta-1}E_{\alpha,\beta}\left( \mathcal{A}_{0}\left( \ln
t-\ln s\right) ^{\alpha}\right) $. For $\mathcal{A}_{0}=\Theta$ delayed M-L
type function $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) $ coincide with delayed M-L type matrix function with two
parameters $E_{h,\alpha,\beta}^{\mathcal{A}_{1}}\left( \ln t-\ln h\right)
,.$introduced in (\ref{ml2}).
(ii) We explicitly write the solution of Hadamard type fractional delay linear
system (\ref{de11}) via delayed perturbation of M-L type function with
logarithm. Using this representation we study existence uniqueness and
Ulam-Hyers stability of nonlinear equation (\ref{de2}).
\section{Preliminaries}
Let $0<a<b<\infty$ and $C\left[ a,b\right] $ be the Banach space of all
continuous functions $y:[a,b]\rightarrow R^{n}$ with the norm $\left\Vert
y\right\Vert _{C}:=\max\left\{ \left\Vert y\left( t\right) \right\Vert
:t\in\left[ a,b\right] \right\} $. For $0\leq\gamma<1$, we denote the space
$C_{\gamma,\ln}\left( a,b\right] $ by the weighted Banach space of the
continuous function $y:[a,b]\rightarrow R$, which is given b
\[
C_{\gamma,\ln}\left( a,b\right] :=\left\{ y\left( t\right) :\left(
\ln\frac{t}{a}\right) ^{\gamma}y\left( t\right) \in C\left[ a,b\right]
\right\} ,
\]
endowed with the norm $\left\Vert y\right\Vert _{\gamma}:=\sup\left\{ \left(
\ln\frac{t}{a}\right) ^{\gamma}\left\Vert y\left( t\right) \right\Vert
:t\in\left( a,b\right] \right\} $.
The following definitions and lemmas will be used in this paper.
\begin{definition}
Hadamard fractional integral of order $\alpha\in R^{+}$ of function $y\left(
t\right) $ is defined by
\[
\left( ^{H}I_{a^{+}}^{\alpha}y\right) \left( t\right) =\frac{1
{\Gamma\left( \alpha\right) }\int_{a}^{t}\left( \ln\frac{t}{s}\right)
^{\alpha-1}y\left( s\right) \frac{ds}{s},\ 0<a<t\leq b,
\]
where $\Gamma$ is the Gamma function.
\end{definition}
\begin{definition}
Hadamard fractional derivative of order $\alpha\in\left[ n-1,n\right) $,
$n\in Z^{+}$ of function $y\left( t\right) $ is defined by
\[
\left( ^{H}D_{a^{+}}^{\alpha}y\right) \left( t\right) =\frac{1
{\Gamma\left( n-\alpha\right) }\left( t\frac{d}{dt}\right) ^{n}\int
_{a}^{t}\left( \ln\frac{t}{s}\right) ^{n-\alpha+1}y\left( s\right)
\frac{ds}{s},\ 0<a<t\leq b.
\]
\end{definition}
\begin{lemma}
If $a,\gamma,\beta>0$ then
\begin{itemize}
\item $\left( ^{H}I_{a^{+}}^{\gamma}\left( \ln\frac{t}{a}\right) ^{\beta
-1}\right) \left( t\right) =\dfrac{\Gamma\left( \beta\right) }
{\Gamma\left( \beta+\gamma\right) }\left( \ln\frac{t}{a}\right)
^{\beta+\gamma-1}.$
\item $\left( ^{H}D_{a^{+}}^{\gamma}\left( \ln\frac{t}{a}\right) ^{\beta
-1}\right) \left( t\right) =\dfrac{\Gamma\left( \beta\right) }
{\Gamma\left( \beta-\gamma\right) }\left( \ln\frac{t}{a}\right)
^{\beta-\gamma-1}.$
\item For $0<\beta<1$, $\left( ^{H}D_{a^{+}}^{\beta}\left( \ln\frac{t
{a}\right) ^{\beta-1}\right) \left( t\right) =0.$
\end{itemize}
\end{lemma}
\begin{definition}
\label{def:01} M-L type matrix function with two parameters $e_{\alpha,\beta
}\left( \mathcal{A}_{0};t\right) :\mathbb{R}\rightarrow\mathbb{R}^{n\times
n}$ is defined by
\[
e_{\alpha,\beta}\left( \mathcal{A}_{0};t\right) :=t^{\beta-1}E_{\alpha
,\beta}\left( \mathcal{A}_{0};t\right) :=t^{\beta-1
{\displaystyle\sum\limits_{k=0}^{\infty}}
\frac{\mathcal{A}_{0}^{k}t^{\alpha k}}{\Gamma\left( k\alpha+\beta\right)
},\ \ \ \alpha,\beta>0,t\in\mathbb{R}.
\]
\end{definition}
\begin{definition}
\label{def:11} Two parameters delayed M-L type matrix function $E_{h,\alpha
,\beta}^{\mathcal{A}_{01}}\left( \ln t\right) :\mathbb{R}^{+}\rightarrow
\mathbb{R}^{n\times n}$ with logarithm is defined by
\begin{equation}
E_{h,\alpha,\beta}^{\mathcal{A}_{1}}\left( \ln t\right) :=\left\{
\begin{tabular}
[c]{ll
$\Theta,$ & $-\infty<t\leq\frac{1}{h},$\\
$I\frac{\left( \ln t+\ln h\right) ^{\beta-1}}{\Gamma\left( \beta\right)
},$ & $\frac{1}{h}<t\leq1,$\\
$I\frac{\left( \ln t+\ln h\right) ^{\beta-1}}{\Gamma\left( \beta\right)
}+\mathcal{A}_{1}\frac{\left( \ln t\right) ^{\alpha+\beta-1}}{\Gamma\left(
\alpha+\beta\right) }+...+\mathcal{A}_{1}^{p}\frac{\left( \ln t-\left(
p-1\right) \ln h\right) ^{p\alpha+\beta-1}}{\Gamma\left( p\alpha
+\beta\right) },$ & $h^{p-1}<t\leq h^{p}.
\end{tabular}
\ \ \right. \label{ml2
\end{equation}
\end{definition}
Our definition of the two parameters delayed M-L type matrix function with
logarithm differs substantially from the definition given in \cite{yang}.
In order to give a definition of delayed M-L type matrix functions with
logarithm, we introduce the following matrices $Y_{\alpha,\beta,k},$
$k=0,1,2,...
\begin{align}
Y_{\alpha,\beta,0}\left( t,s\right) & =\left( \ln\frac{t}{s}\right)
^{\beta-1}E_{\alpha,\beta}\left( \mathcal{A}_{0};\ln\frac{t}{s}\right)
,\ \ \nonumber\\
Y_{\alpha,\beta,1}\left( t,sh\right) & =\int_{sh}^{t}e_{\alpha,\alpha
}\left( \mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A}_{1}\frac{1
{\Gamma\left( \beta\right) }\left( \ln\frac{r}{sh}\right) ^{\beta-1
\frac{dr}{r},\nonumber\\
Y_{\alpha,\beta,k}\left( t,sh^{k}\right) & =\int_{sh^{k}}^{t
e_{\alpha,\alpha}\left( \mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A
_{1}Y_{\alpha,\beta,k-1}\left( \frac{r}{h},sh^{k-1}\right) \frac{dr}{r}.
\label{for1
\end{align}
\begin{definition}
\label{def:22} Let $\mathcal{A}_{0},\mathcal{A}_{1}\in\mathbb{R}^{n\times n}$
be fixed matrices and $k\in\mathbb{N}\cup\left\{ 0\right\} $. Delayed M-L
type function $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
\cdot,\cdot\right) :\mathbb{R\times R\rightarrow R}^{n}$ with logarithm
generated by $\mathcal{A}_{0},\mathcal{A}_{1}$\ is defined by
\begin{equation}
Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) :
{\displaystyle\sum\limits_{j=0}^{\infty}}
Y_{\alpha,\beta,j}\left( t,sh^{j}\right) H\left( t-sh^{j}\right) =\left\{
\begin{tabular}
[c]{ll
$\Theta,$ & $-\infty<t<s,$\\
$I,$ & $t=s,$\\
$Y_{\alpha,\beta,0}\left( t,s\right) +Y_{\alpha,\beta,1}\left( t,sh\right)
+...+Y_{\alpha,\beta,k}\left( t,sh^{k}\right) ,$ & $sh^{k}<t\leq sh^{k+1},
\end{tabular}
\ \ \ \ \right. \label{exp1
\end{equation}
where $H\left( t\right) $ is a Heaviside function: $H\left( t\right)
=\left\{
\begin{array}
[c]{c
1,\ \ t>0,\\
0,\ \ \ t\leq0.
\end{array}
\right. $.
\end{definition}
\begin{lemma}
\label{lem:2}Let $a,b>-1$. For $sh^{k}<t\leq sh^{k+1},$ $k\in\mathbb{N
\cup\{0\}$, one has
\begin{align}
\int_{r}^{t}\left( \ln\frac{t}{s}\right) ^{a}\left( \ln\frac{s}{r}\right)
^{b}\frac{ds}{s} & =\left( \ln\frac{t}{r}\right) ^{a+b+1}\mathcal{A
_{1}[a+1,b+1],\label{inq1}\\
\frac{1}{\Gamma\left( 1-\alpha\right) }\int_{sh^{k}}^{t}\left( \ln\frac
{t}{r}\right) ^{-\alpha}Y_{\alpha,\beta,k}\left( r,sh^{k}\right) \frac
{dr}{r} & =\int_{sh^{k}}^{t}E_{\alpha,1}\left( \mathcal{A}_{0};\ln\frac
{t}{r}\right) \mathcal{A}_{1}Y_{\alpha,\beta,k-1}\left( \frac{r}{h
,sh^{k-1}\right) \frac{dr}{r}. \label{inq2
\end{align}
\end{lemma}
\begin{proof}
Let $\ln\dfrac{s}{r}=\tau\ln\dfrac{t}{r}.$ Then $s=r\left( \dfrac{t
{r}\right) ^{\tau}$, $ds=r\left( \dfrac{t}{r}\right) ^{\tau}\ln\dfrac{t
{r}d\tau$. So we have
\[
\ln\frac{t}{s}=\ln\left( \dfrac{t}{r}\right) ^{1-\tau}=\left(
1-\tau\right) \ln\left( \dfrac{t}{r}\right) ,
\]
and
\begin{align*}
\int_{r}^{t}\left( \ln\frac{t}{s}\right) ^{a}\left( \ln\frac{s}{r}\right)
^{b}\frac{ds}{s} & =\int_{0}^{1}\left( \ln\frac{t}{r}\right) ^{a}\left(
1-\tau\right) ^{a}\left( \ln\frac{t}{r}\right) ^{b}\tau^{b}\ln\dfrac{t
{r}d\tau\\
& =\left( \ln\frac{t}{r}\right) ^{a+b+1}\mathcal{B}\left[ a+1,b+1\right]
.
\end{align*}
To prove (\ref{inq2}), firstly using (\ref{inq1}) we calculate it for $k=0:$
\begin{align}
\frac{1}{\Gamma\left( 1-\alpha\right) }\int_{s}^{t}\left( \ln\frac{t
{r}\right) ^{-\alpha}Y_{\alpha,\beta,0}\left( r,s\right) \frac{dr}{r} &
=\frac{1}{\Gamma\left( 1-\alpha\right) }\frac{1}{\Gamma\left( \beta\right)
}\int_{s}^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha}\left( \ln\frac{r
{s}\right) ^{\beta-1}\frac{dr}{r}\nonumber\\
& =\frac{\left( \ln t-\ln s\right) ^{-\alpha+\beta}}{\Gamma\left(
-\alpha+\beta+1\right) }. \label{form1
\end{align}
Similarly, for any $k\in\mathbb{N}$we have
\begin{align*}
& \frac{1}{\Gamma\left( 1-\alpha\right) }\int_{sh^{k}}^{t}\left( \ln
\frac{t}{r}\right) ^{-\alpha}Y_{\alpha,\beta,k}\left( r,sh^{k}\right)
\frac{dr}{r}\\
& =\frac{1}{\Gamma\left( 1-\alpha\right) }\int_{sh^{k}}^{t}\left( \ln
\frac{t}{r}\right) ^{-\alpha}\int_{sh^{k}}^{r}e_{\alpha,\alpha}\left(
\mathcal{A}_{0};\ln\frac{r}{r_{1}}\right) \mathcal{A}_{1}Y_{\alpha,\beta
,k-1}\left( \frac{r_{1}}{h},sh^{k-1}\right) H\left( r-sh^{k}\right)
\frac{dr_{1}}{r_{1}}\frac{dr}{r}\\
& =\frac{1}{\Gamma\left( 1-\alpha\right) }\int_{sh^{k}}^{t}\int_{r_{1}
^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha}e_{\alpha,\alpha}\left(
\mathcal{A}_{0};\ln\frac{r}{r_{1}}\right) \frac{dr}{r}\mathcal{A
_{1}Y_{\alpha,\beta,k-1}\left( \frac{r_{1}}{h},sh^{k-1}\right) H\left(
r-sh^{k}\right) \frac{dr_{1}}{r_{1}}\\
& =\int_{sh^{k}}^{t}E_{\alpha,1}\left( \mathcal{A}_{0};\ln\frac{t}{r_{1
}\right) \mathcal{A}_{1}Y_{\alpha,\beta,k-1}\left( \frac{r_{1}}{h
,sh^{k-1}\right) \frac{dr_{1}}{r_{1}}.
\end{align*}
\end{proof}
\begin{lemma}
\label{lem:1}If $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ are permutable
matrices, then
\begin{equation}
Y_{\alpha,\beta,k}\left( t,sh^{k}\right) =\mathcal{A}_{1}^{k}\sum
_{n=0}^{\infty}\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \mathcal{A}_{0}^{n}\frac{\left( \ln t-\ln sh^{k}\right)
^{n\alpha+k\alpha+\beta-1}}{\Gamma\left( n\alpha+k\alpha+\beta\right) }.
\label{id1
\end{equation}
\end{lemma}
\begin{proof}
The proof is based on the inequality (\ref{inq1}). For $k=1,$we have
\begin{align*}
Y_{\alpha,\beta,1}\left( t,sh\right) & =\int_{sh}^{t}e_{\alpha,\alpha
}\left( \mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A}_{1}Y_{\alpha
,\beta,0}\left( \frac{r}{h},s\right) \frac{dr}{r}\\
& =\mathcal{A}_{1}\int_{sh}^{t}\sum_{n=0}^{\infty}\mathcal{A}_{0}^{n}\left(
\ln\frac{t}{r}\right) ^{\alpha n+\alpha-1}\frac{1}{\Gamma\left( \alpha
n+\alpha\right) }\sum_{k=0}^{\infty}\mathcal{A}_{0}^{k}\left( \ln\frac
{r}{sh}\right) ^{\alpha n+\beta-1}\frac{1}{\Gamma\left( \alpha
k+\beta\right) }\frac{dr}{r}\\
& =\mathcal{A}_{1}\int_{sh}^{t}\sum_{n=0}^{\infty}\sum_{k=0}^{k
\mathcal{A}_{0}^{k}\left( \ln\frac{t}{r}\right) ^{\alpha k+\alpha-1}\frac
{1}{\Gamma\left( \alpha k+\alpha\right) }\mathcal{A}_{0}^{n-k}\left(
\ln\frac{r}{sh}\right) ^{\alpha\left( n-k\right) +\beta-1}\frac{1
{\Gamma\left( \alpha\left( n-k\right) +\beta\right) }\frac{dr}{r}\\
& =\mathcal{A}_{1}\sum_{n=0}^{\infty}\sum_{k=0}^{k}\mathcal{A}_{0}^{n
\frac{1}{\Gamma\left( \alpha k+\alpha\right) \Gamma\left( \alpha\left(
n-k\right) +\beta\right) }\int_{sh}^{t}\left( \ln\frac{t}{r}\right)
^{\alpha k+\alpha-1}\left( \ln\frac{r}{sh}\right) ^{\alpha\left(
n-k\right) +\beta-1}\frac{dr}{r}\\
& =\mathcal{A}_{1}\sum_{n=0}^{\infty}\sum_{k=0}^{k}\mathcal{A}_{0}^{n
\frac{\left( \ln t-\ln sh\right) ^{\alpha n+\alpha+\beta-1}}{\Gamma\left(
\alpha k+\alpha\right) \Gamma\left( \alpha\left( n-k\right) +\beta\right)
}\mathcal{A}_{01}\left[ \alpha k+\alpha,\alpha\left( n-k\right)
+\beta\right] \\
& =\mathcal{A}_{1}\sum_{n=0}^{\infty}\left(
\begin{array}
[c]{c
n+1\\
1
\end{array}
\right) \mathcal{A}_{0}^{n}\frac{\left( \ln t-\ln sh\right) ^{\alpha
n+\alpha+\beta-1}}{\Gamma\left( \alpha n+\alpha+\beta\right) }.
\end{align*}
For $k=2,$ we get
\begin{align*}
Y_{\alpha,\beta,2}\left( t,sh^{2}\right) & =\int_{sh^{2}}^{t
e_{\alpha,\alpha}\left( \mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A
_{1}Y_{\alpha,\beta,1}\left( \frac{r}{h},sh\right) dr\\
& =\mathcal{A}_{1}\int_{sh^{2}}^{t}\sum_{n=0}^{\infty}\sum_{k=0
^{n}\mathcal{A}_{0}^{k}\left( \ln\frac{t}{r}\right) ^{k\alpha+\alpha-1
\frac{1}{\Gamma\left( k\alpha+\alpha\right) }\mathcal{A}_{1}\left(
\begin{array}
[c]{c
n+1-k\\
1
\end{array}
\right) \\
& \ \ \ \ \times\mathcal{A}_{0}^{n-k}\left( \ln\frac{r}{sh^{2}}\right)
^{n\alpha-k\alpha+\alpha+\beta-1}\frac{1}{\Gamma\left( n\alpha-k\alpha
+\alpha+\beta\right) }\\
& =\mathcal{A}_{1}^{2}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\left(
\begin{array}
[c]{c
n+1-k\\
1
\end{array}
\right) \mathcal{A}_{0}^{n}\frac{1}{\Gamma\left( k\alpha+\beta\right)
\Gamma\left( n\alpha-k\alpha+\alpha+\beta\right) }\int_{sh^{2}}^{t}\left(
\ln\frac{t}{r}\right) ^{k\alpha+\alpha-1}\left( \ln\frac{r}{sh^{2}}\right)
^{n\alpha-k\alpha+\alpha+\beta-1}dr\\
& =\mathcal{A}_{1}^{2}\sum_{n=0}^{\infty}\left( \sum_{k=0}^{n}\left(
\begin{array}
[c]{c
n+1-k\\
1
\end{array}
\right) \right) \mathcal{A}_{0}^{n}\left( \ln\frac{r}{sh^{2}}\right)
^{n\alpha+2\alpha+\beta-1}\frac{1}{\Gamma\left( n\alpha+2\alpha+\beta\right)
}\\
& =\mathcal{A}_{1}^{2}\sum_{n=0}^{\infty}\left(
\begin{array}
[c]{c
n+2\\
2
\end{array}
\right) \mathcal{A}_{0}^{n}\frac{\left( \ln t-\ln sh^{2}\right)
^{n\alpha+2\alpha+\beta-1}}{\Gamma\left( n\alpha+2\alpha+\beta\right) }.
\end{align*}
Using the Mathematical Induction in a similar manner we can get (\ref{id1}).
\end{proof}
According to Lemma \ref{lem:1} in the case $\mathcal{A}_{0}\mathcal{A
_{1}=\mathcal{A}_{1}\mathcal{A}_{0}$ delayed M-L type function $Y_{h,\alpha
,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) $ has a simple
form:
\begin{equation}
Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right)
:=\left\{
\begin{tabular}
[c]{ll
$\Theta,$ & $-\infty\leq t<s,$\\
$I,$ & $t=s,$\\
{\displaystyle\sum\limits_{i=0}^{\infty}}
\mathcal{A}_{0}^{i}\dfrac{\left( \ln t-\ln s\right) ^{i\alpha+\beta-1
}{\Gamma\left( i\alpha+\beta\right) }
{\displaystyle\sum\limits_{i=1}^{\infty}}
\left(
\begin{array}
[c]{c
i\\
1
\end{array}
\right) \mathcal{A}_{0}^{i-1}\mathcal{A}_{1}\dfrac{\left( \ln t-\ln
sh\right) ^{i\alpha+\beta-1}}{\Gamma\left( i\alpha+\beta\right) }$ & \\
$+...
{\displaystyle\sum\limits_{i=p}^{\infty}}
\left(
\begin{array}
[c]{c
i\\
p
\end{array}
\right) \mathcal{A}_{0}^{i-p}\mathcal{A}_{1}^{p}\dfrac{\left( \ln t-\ln
sh^{p}\right) ^{i\alpha+\beta-1}}{\Gamma\left( i\alpha+\beta\right) },$ &
$sh^{p}<t\leq sh^{p+1}.
\end{tabular}
\ \ \ \ \ \ \ \ \ \ \ \ \right. \label{perf1
\end{equation}
Next lemma shows some special cases of the delayed M-L type function.
\begin{lemma}
\label{lem:3}Let $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) $ be defined by (\ref{exp1}). Then the following holds true:
\begin{description}
\item[(i)] if $\mathcal{A}_{0}=\Theta$ then $Y_{h,\alpha,\beta}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,1\right) =E_{h,\alpha,\beta}^{\mathcal{A}_{1
}\left( \ln\frac{t}{h}\right) ,\ \ h^{k-1}<\frac{t}{h}\leq h^{k}$,
\item[(ii)] if $\mathcal{A}_{1}=\Theta$ then $Y_{h,\alpha,\beta
^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) =\left( \ln\frac{t
{s}\right) ^{\beta-1}E_{\alpha,\beta}\left( \mathcal{A}_{0}\left( \ln
\frac{t}{s}\right) ^{\alpha}\right) =e_{\alpha,\beta}\left( \mathcal{A
_{0};\ln\frac{t}{s}\right) ,$
\item[(iii)] if $\alpha=\beta=1$ and $\mathcal{A}_{0}\mathcal{A
_{1}=\mathcal{A}_{1}\mathcal{A}_{0}$ then $Y_{h,1,1}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) =e^{\mathcal{A}_{0}\left( \ln t-\ln
s\right) }e_{h}^{\mathcal{A}_{11}\left( \ln t-\ln h\right) }$,
$\mathcal{A}_{11}=\mathcal{A}_{1}e^{-\mathcal{A}_{0}\ln h},$ $sh^{k}<t\leq
sh^{k+1}$.
\end{description}
\end{lemma}
\begin{proof}
(i) If $\mathcal{A}_{0}=\Theta,$ then the formula (\ref{for1})
\begin{align*}
Y_{\alpha,\beta,0}\left( t,s\right) & =e_{\alpha,\beta}\left( \Theta
,\ln\frac{t}{s}\right) =I\frac{\left( \ln t-\ln s\right) ^{\beta-1}
{\Gamma\left( \beta\right) },\\
Y_{\alpha,\beta,1}\left( t,sh\right) & =\int_{sh}^{t}e_{\alpha,\alpha
}\left( \Theta,\ln\frac{t}{r}\right) \mathcal{A}_{1}Y_{0}\left( \frac{r
{h},s\right) \frac{dr}{r}=\frac{1}{\Gamma\left( \alpha\right) \Gamma\left(
\beta\right) }\mathcal{A}_{1}\int_{sh}^{t}\left( \ln\frac{t}{r}\right)
^{\alpha-1}\left( \ln\frac{r}{sh}\right) ^{\beta-1}\frac{dr}{r}\\
& =\frac{1}{\Gamma\left( \alpha\right) \Gamma\left( \beta\right)
}\mathcal{A}_{1}\left( \ln\frac{t}{sh}\right) ^{\alpha+\beta-1
\mathcal{B}\left[ \alpha,\beta\right] =\frac{1}{\Gamma\left( \alpha
+\beta\right) }\mathcal{A}_{1}\left( \ln\frac{t}{sh}\right) ^{\alpha
+\beta-1},\\
Y_{\alpha,\beta,2}\left( t,sh^{2}\right) & =\int_{sh^{2}}^{t
e_{\alpha,\alpha}\left( \Theta,\ln\frac{t}{r}\right) \mathcal{A}_{1
Y_{1}\left( \frac{r}{h},sh\right) dr=\frac{1}{\Gamma\left( \alpha\right)
\Gamma\left( \alpha+\beta\right) }\mathcal{A}_{1}^{2}\int_{sh^{2}
^{t}\left( \ln\frac{t}{r}\right) ^{\alpha-1}\left( \ln\frac{r}{sh^{2
}\right) ^{\alpha+\beta-1}\frac{dr}{r}\\
& =\frac{1}{\Gamma\left( \alpha\right) \Gamma\left( \alpha+\beta\right)
}\mathcal{A}_{1}^{2}\left( \ln\frac{t}{sh^{2}}\right) ^{2\alpha+\beta
-1}\mathcal{B}\left[ \alpha,\alpha+\beta\right] =\frac{1}{\Gamma\left(
2\alpha+\beta\right) }\mathcal{A}_{1}^{2}\left( \ln\frac{t}{sh^{2}}\right)
^{2\alpha+\beta-1},\\
Y_{\alpha,\beta,k}\left( t,sh^{k}\right) & =\mathcal{A}_{1}^{k}\left(
\ln\frac{t}{sh^{k}}\right) ^{k\alpha+\beta-1}\frac{1}{\Gamma\left(
k\alpha+\beta\right) },\ \ k\geq0.
\end{align*}
So $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,1\right) $
coincides with $E_{h,\alpha,\beta}^{\mathcal{A}_{1}}\left( \ln t-\ln
h\right) :$
\begin{align*}
Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,1\right) &
{\displaystyle\sum\limits_{i=0}^{k}}
\mathcal{A}_{1}^{i}\left( \ln\frac{t}{h^{i}}\right) ^{i\alpha+\beta-1
\dfrac{1}{\Gamma\left( i\alpha+\beta\right) }=I\frac{\left( \ln t\right)
^{\beta-1}}{\Gamma\left( \beta\right) }+\mathcal{A}_{1}\frac{\left( \ln
t-\ln h\right) ^{\alpha+\beta-1}}{\Gamma\left( \alpha+\beta\right)
}+...+\mathcal{A}_{1}^{k}\frac{\left( \ln t-k\ln h\right) ^{k\alpha+\beta
-1}}{\Gamma\left( k\alpha+\beta\right) }\\
& =E_{h,\alpha,\beta}^{\mathcal{A}_{1}}\left( \ln t-\ln h\right)
,\ \ h^{k-1}<\frac{t}{h}\leq h^{k}.
\end{align*}
(ii) Trivially, from definition of $Y_{h,\alpha,\beta}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) $ we have: if $\mathcal{A}_{1
=\Theta$, then
\[
Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right)
=\left( \ln\frac{t}{s}\right) ^{\beta-1}E_{\alpha,\beta}\left(
\mathcal{A}_{0}\left( \ln\frac{t}{s}\right) ^{\alpha}\right) .
\]
(iii) By (\ref{perf1}) for the case $\alpha=\beta=1$ and $\mathcal{A
_{0}\mathcal{A}_{1}=\mathcal{A}_{1}\mathcal{A}_{0}$, we have
\begin{align*}
Y_{h,1,1}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) &
{\displaystyle\sum\limits_{i=0}^{\infty}}
\mathcal{A}_{0}^{i}\dfrac{\left( \ln t-\ln s\right) ^{i}}{\Gamma\left(
i+1\right) }
{\displaystyle\sum\limits_{i=0}^{\infty}}
\left(
\begin{array}
[c]{c
i+1\\
1
\end{array}
\right) \mathcal{A}_{0}^{i}\mathcal{A}_{1}\dfrac{\left( \ln t-\ln sh\right)
^{i+1}}{\Gamma\left( i+1\right) }\\
& +...
{\displaystyle\sum\limits_{i=0}^{\infty}}
\left(
\begin{array}
[c]{c
i+k\\
k
\end{array}
\right) \mathcal{A}_{0}^{i}\mathcal{A}_{1}^{k}\dfrac{\left( \ln t-\ln
sh^{k}\right) ^{i+k}}{\Gamma\left( i+k+1\right) }\\
& =e^{\mathcal{A}_{0}\left( \ln t-\ln s\right) }+e^{\mathcal{A}_{0}\left(
\ln t-\ln sh\right) }\mathcal{A}_{1}\left( \ln t-\ln sh\right)
+...+e^{\mathcal{A}_{0}\left( \ln t-\ln sh^{k}\right) }\mathcal{A}_{1
^{k}\frac{1}{k!}\left( \ln t-k\ln h\right) ^{k}\\
& =e^{\mathcal{A}_{0}\left( \ln t-\ln s\right) }e_{h}^{\mathcal{A
_{11}\left( \ln t-\ln h\right) }.
\end{align*}
\end{proof}
It turns out that $Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) $ is a delayed perturbation of the Cauchy matrix with logarithm of
the homogeneous equation (\ref{de11}) with $f=0.$
\begin{lemma}
\label{lem:4}$Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1
}:\mathbb{R\times R}\rightarrow\mathbb{R}^{n\times n}$ is a solution of
\begin{equation}
^{H}D_{h^{+}}^{\alpha}Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,s\right) =\left( \ln\frac{t}{s}\right) ^{-\alpha+\beta-1
\frac{1}{\Gamma\left( -\alpha+\beta\right) }+\mathcal{A}_{0}Y_{h,\alpha
,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) +\mathcal{A
_{1}Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( \frac{t
{h},s\right) . \label{form4
\end{equation}
\end{lemma}
\begin{proof}
According to (\ref{form1}) we have
\begin{align}
\left( ^{H}D_{1^{+}}^{\alpha}Y_{\alpha,\beta,0}\left( t,s\right) \right)
\left( t\right) & =\frac{1}{\Gamma\left( 1-\alpha\right) }\left(
t\frac{d}{dt}\right) \int_{s}^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha
}Y_{\alpha,\beta,0}\left( r,s\right) \frac{dr}{r}\nonumber\\
& =\frac{1}{\Gamma\left( 1-\alpha\right) }\left( t\frac{d}{dt}\right)
\int_{s}^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha}e_{\alpha,\beta}\left(
r,s\right) \frac{dr}{r}\nonumber\\
& =\left( t\frac{d}{dt}\right) E_{\alpha,1-\alpha+\beta}\left(
\mathcal{A}_{0};\ln\frac{t}{s}\right) \nonumber\\
& =\left( \ln\frac{t}{s}\right) ^{-\alpha+\beta-1}\frac{1}{\Gamma\left(
-\alpha+\beta\right) }+\mathcal{A}_{0}e_{\alpha,\beta}\left( \mathcal{A
_{0};\ln\frac{t}{s}\right) \nonumber\\
& =\left( \ln\frac{t}{s}\right) ^{-\alpha+\beta-1}\frac{1}{\Gamma\left(
-\alpha+\beta\right) }+\mathcal{A}_{0}Y_{\alpha,\beta,0}\left( t,s\right)
.\label{form2
\end{align}
On the other hand for any $k\in\mathbb{N}$
\begin{align}
\left( ^{H}D_{1^{+}}^{\alpha}Y_{\alpha,\beta,k}\left( t,sh^{k}\right)
\right) \left( t\right) & =\frac{1}{\Gamma\left( 1-\alpha\right)
}\left( t\frac{d}{dt}\right) \int_{1}^{t}\left( \ln\frac{t}{r}\right)
^{-\alpha}Y_{\alpha,\beta,k}\left( r,s\right) \frac{dr}{r}\nonumber\\
& =\left( t\frac{d}{dt}\right) \int_{sh^{k}}^{t}E_{\alpha,1}\left(
\mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A}_{1}Y_{\alpha,\beta
,k-1}\left( \frac{r}{h},sh^{k-1}\right) \frac{dr}{r}\nonumber\\
& =\int_{sh^{k}}^{t}\left( t\frac{d}{dt}\right) E_{\alpha,1}\left(
\mathcal{A}_{0};\ln\frac{t}{r}\right) \mathcal{A}_{1}Y_{\alpha,\beta
,k-1}\left( \frac{r}{h},sh^{k-1}\right) \frac{dr}{r}\nonumber\\
& +\mathcal{A}_{1}Y_{\alpha,\beta,k-1}\left( \frac{t}{h},sh^{k-1}\right)
\nonumber\\
& =\mathcal{A}_{0}Y_{\alpha,\beta,k}\left( t,sh^{k}\right) +\mathcal{A
_{1}Y_{\alpha,\beta,k-1}\left( \frac{t}{h},sh^{k-1}\right) .\label{form3
\end{align}
From (\ref{form2}) and (\ref{form3}) it follows that for $sh^{k}<t\leq
sh^{k+1}$
\begin{align*}
^{H}D_{1^{+}}^{\alpha}Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{01
}\left( t,s\right) & =\ ^{H}D_{1^{+}}^{\alpha}Y_{\alpha,\beta,0}\left(
t,s\right) +\ ^{H}D_{1^{+}}^{\alpha}Y_{\alpha,\beta,1}\left( t,sh\right)
+...+\ ^{H}D_{1^{+}}^{\alpha}Y_{\alpha,\beta,k}\left( t,sh^{k}\right) \\
& =\left( \ln\frac{t}{s}\right) ^{-\alpha+\beta-1}\frac{1}{\Gamma\left(
-\alpha+\beta\right) }+\mathcal{A}_{0}Y_{\alpha,\beta,0}\left( t,s\right)
+\mathcal{A}_{0}Y_{\alpha,\beta,1}\left( t,sh\right) +\mathcal{A
_{1}Y_{\alpha,\beta,0}\left( \frac{t}{h},s\right) \\
& +...+\mathcal{A}_{0}Y_{\alpha,\beta,k}\left( t,sh^{k}\right)
+\mathcal{A}_{1}Y_{\alpha,\beta,k-1}\left( \frac{t}{h},sh^{k-1}\right) \\
& =\left( \ln\frac{t}{s}\right) ^{-\alpha+\beta-1}\frac{1}{\Gamma\left(
-\alpha+\beta\right) }++\mathcal{A}_{0}Y_{h,\alpha,\beta}^{\mathcal{A
_{0},\mathcal{A}_{01}}\left( t,s\right) +\mathcal{A}_{1}Y_{h,\alpha,\beta
}^{\mathcal{A}_{0},\mathcal{A}_{01}}\left( \frac{t}{h},s\right) .
\end{align*}
The proof is completed.
\end{proof}
\begin{theorem}
\label{thm:1}The solution $y(t)$ of (\ref{de11}) with zero initial condition
has a form
\[
y\left( t\right) =\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) f\left( s\right) \frac{ds
{s},\ \ t\geq0.
\]
\end{theorem}
\begin{proof}
Assume that any solution of nonhomogeneous system $y\left( t\right) $ has
the form
\begin{equation}
y\left( t\right) =\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) h\left( s\right) \frac{ds
{s},\ \ t\geq0,\label{rp1
\end{equation}
where $h\left( s\right) ,$ $1\leq s\leq t\leq T$ is an unknown continuous
vector function and $y(1)=0$. Having Hadamard fractional differentiation on
both sides of (\ref{rp1}) , for $1<t\leq h$ we have
\begin{align*}
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) & =\mathcal{A
_{0}y\left( t\right) +\mathcal{A}_{1}y\left( \frac{t}{h}\right) +f\left(
t\right) \\
& =\mathcal{A}_{0}\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0
,\mathcal{A}_{1}}\left( t,s\right) h\left( s\right) \frac{ds
{s}+\mathcal{A}_{1}\int_{1}^{t/h}Y_{h,\alpha,\alpha}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( \frac{t}{h},s\right) h\left( s\right)
\frac{ds}{s}+f\left( t\right) \\
& =\mathcal{A}_{0}\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0
,\mathcal{A}_{1}}\left( t,s\right) h\left( s\right) \frac{ds}{s}+f\left(
t\right) .
\end{align*}
On the other hand, according to Lemma \ref{lem:2}, we have
\begin{align*}
\left( ^{H}D_{1^{+}}^{\alpha}y\right) \left( t\right) & =\frac{1
{\Gamma\left( 1-\alpha\right) }\left( t\frac{d}{dt}\right) \int_{1
^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha}\left( \int_{1}^{r
Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( r,s\right)
h\left( s\right) \frac{ds}{s}\right) \frac{dr}{r}\\
& =\frac{1}{\Gamma\left( 1-\alpha\right) }\left( t\frac{d}{dt}\right)
\int_{1}^{t}\int_{s}^{t}\left( \ln\frac{t}{r}\right) ^{-\alpha
Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( r,s\right)
h\left( s\right) \frac{dr}{r}\frac{ds}{s}\\
& =c\left( t\right) +\frac{1}{\Gamma\left( 1-\alpha\right) }\int_{1
^{t}\left( t\frac{d}{dt}\right) \int_{s}^{t}\left( \ln\frac{t}{r}\right)
^{-\alpha}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
r,s\right) h\left( s\right) \frac{dr}{r}\frac{ds}{s}\\
& =h\left( t\right) +\mathcal{A}_{0}\int_{1}^{t}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}}\left( t,s\right) h\left( s\right)
\frac{dr}{r}.
\end{align*}
Therefore, $h(t)\equiv f(t)$. The proof is completed.
\end{proof}
\begin{theorem}
\label{thm:2}Let $p=0,1,...,l$. A solution $y\in C\left( \left( \left(
p-1\right) h,ph\right] ,\mathbb{R}^{n}\right) $ of (\ref{de11}) with $f=0$
has a form
\[
y\left( t\right) =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,\frac{1}{h}\right) a+\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) \left( \left(
^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \left( s\right)
-\mathcal{A}_{0}\varphi\left( s\right) \right) \frac{ds}{s}.
\]
\end{theorem}
\begin{proof}
We are looking for a solution, which depends on an unknown constant $c,$ and
an Hadamard differentiable vector function $g\left( t\right) ,$ of the form
\[
y\left( t\right) =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,\frac{1}{h}\right) c+\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) g\left( s\right)
\frac{ds}{s},
\]
Moreover, $y\left( t\right) $ satisfies initial conditions
\begin{align*}
y\left( t\right) & =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,\frac{1}{h}\right) c+\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) g\left( s\right)
\frac{ds}{s}:=\varphi\left( t\right) ,\ \ \frac{1}{h}<t\leq1,\\
\left( ^{H}I_{\frac{1}{h}^{+}}^{1-\alpha}y\right) \left( \frac{1}{h
^{+}\right) & =a.
\end{align*}
We have
\begin{align*}
a & =\left( ^{H}I_{\frac{1}{h}^{+}}^{1-\alpha}y\right) \left( \frac{1
{h}^{+}\right) =\lim_{t\rightarrow\frac{1}{h}^{+}}\left( ^{H}I_{\frac{1
{h}^{+}}^{1-\alpha}y\right) \left( t\right) \\
& =\lim_{t\rightarrow\frac{1}{h}^{+}}\left( \frac{1}{\Gamma\left(
1-\alpha\right) }\int_{\frac{1}{h}}^{t}\left( \ln t-\ln s\right) ^{-\alpha
}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,\frac{1
{h}\right) c\frac{dt}{t}\right) \\
& =\lim_{t\rightarrow\frac{1}{h}^{+}}\left( \frac{1}{\Gamma\left(
1-\alpha\right) }\int_{\frac{1}{h}}^{t}\left( \ln t-\ln s\right) ^{-\alpha
}e_{\alpha,\alpha}\left( \mathcal{A}_{0},\ln t\right) c\frac{dt}{t}\right)
=c.
\end{align*}
Thus $c=a$. Since $\frac{1}{h}<t\leq1$, we obtain that
\[
Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right)
=\left\{
\begin{tabular}
[c]{ll
$\left( \ln\frac{t}{s}\right) ^{\alpha-1}E_{\alpha,\alpha}\left(
\mathcal{A}_{0}\left( \ln\frac{t}{s}\right) ^{\alpha}\right) ,$ & $\frac
{1}{h}\leq s<t\leq1,\ $\\
$\Theta,$ & $t<s\leq h.
\end{tabular}
\ \ \ \ \ \right.
\]
Consequently, on interval $\frac{1}{h}<t\leq1$, we can easily derive
\begin{align}
\varphi\left( t\right) & =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0
,\mathcal{A}_{1}}\left( t,\frac{1}{h}\right) a+\int_{\frac{1}{h}
^{1}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right)
g\left( s\right) \frac{ds}{s}\label{q1}\\
& =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,\frac{1
{h}\right) a+\int_{\frac{1}{h}}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) g\left( s\right) \frac{ds}{s
+\int_{t}^{1}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) g\left( s\right) \frac{ds}{s}\nonumber\\
& =\left( \ln th\right) ^{\alpha-1}E_{\alpha,\alpha}\left( \mathcal{A
_{0}\left( \ln th\right) ^{\alpha}\right) a+\int_{\frac{1}{h}}^{t}\left(
\ln\frac{t}{s}\right) ^{\alpha-1}E_{\alpha,\alpha}\left( \mathcal{A
_{0}\left( \ln\frac{t}{s}\right) ^{\alpha}\right) g\left( s\right)
\frac{ds}{s}.\nonumber
\end{align}
Having differentiated (\ref{q1}) in Hadamard sense, we obtain
\begin{align*}
\left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \left( t\right) &
=\mathcal{A}_{0}\left( \ln th\right) ^{\alpha-1}E_{\alpha,\alpha}\left(
\mathcal{A}_{0}\left( \ln th\right) ^{\alpha}\right) a+\mathcal{A}_{0
\int_{\frac{1}{h}}^{t}\left( \ln\frac{t}{s}\right) ^{\alpha-1
E_{\alpha,\alpha}\left( \mathcal{A}_{0}\left( \ln\frac{t}{s}\right)
^{\alpha}\right) g\left( s\right) \frac{ds}{s}+g\left( t\right) \\
& =\mathcal{A}_{0}\varphi\left( t\right) +g\left( t\right) .
\end{align*}
Therefore, $g\left( t\right) =\left( ^{H}D_{1^{+}}^{\alpha}\varphi\right)
\left( t\right) -\mathcal{A}_{0}\varphi\left( t\right) $ and the desired
formula holds.
\end{proof}
Combining Theorems \ref{thm:1} and \ref{thm:2} together we get the following result.
\begin{corollary}
A solution $y\in C\left( \left[ 1,T\right] \cap\left( h^{p-1
,h^{p}\right] ,\mathbb{R}^{n}\right) $ of (\ref{de11}) has a form
\begin{align*}
y\left( t\right) & =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,\frac{1}{h}\right) a+\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) \left[ \left(
^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \left( s\right)
-\mathcal{A}_{0}\varphi\left( s\right) \right] \frac{ds}{s}\\
& +\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) f\left( s\right) \frac{ds}{s}.
\end{align*}
\end{corollary}
\section{Existence Uniqueness and Stability}
In this section, we consider the following equivalent integral form of the
nonlinear Cauchy problem for fractional time-delay differential equations with
Hadamard derivative (\ref{de2})
\begin{align}
y\left( t\right) & =Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1
}\left( t,\frac{1}{h}\right) a+\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,s\right) \left[ \left(
^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \left( s\right)
-\mathcal{A}_{0}\varphi\left( s\right) \right] \frac{ds}{s}\nonumber\\
& +\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) f\left( s,y\left( s\right) \right) \frac{ds}{s}. \label{ine1
\end{align}
Let us introduce the conditions under which existence and uniqueness of the
integral equation $\left( \ref{ine1}\right) $ will be investigated.
\begin{enumerate}
\item[(A1)] $f:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}$ be a
function such that $f\left( t,y\right) \in C_{\gamma,\ln}\left[ 1,T\right]
$ with $\gamma<\alpha$ for any $y\in\mathbb{R}^{n};$
\item[(A2)] There exists a positive constant $L_{f}>0$ such that
\[
\left\Vert f\left( t,y_{1}\right) -f\left( t,y_{2}\right) \right\Vert \leq
L_{f}\left\Vert y_{1}-y_{2}\right\Vert ,
\]
for each $\left( t,y_{1}\right) ,\left( t,y_{2}\right) \in\left[
1,T\right] \times\mathbb{R}^{n}$.
\end{enumerate}
From (A1) and (A2), it follows that
\[
\left\Vert f\left( t,y\right) \right\Vert \leq L_{f}\left\Vert y\right\Vert
+L_{2}\ \ \ \ \text{for some }L_{2}>0.
\]
To prove existence uniqueness and stability of (\ref{ine1}) we use the
following estimation of $Y_{\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{01
}\left( t,s\right) .$
\begin{lemma}
\label{lem:5}We have for $sh^{p}<t\leq sh^{p+1},$ $p=0,1,...,$
\[
\left\Vert Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) \left( t,s\right) \right\Vert \leq Y_{h,\alpha,\beta
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \leq Y_{1,\alpha,\beta}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
t,1\right) .
\]
\end{lemma}
\begin{proof}
Indeeed,
\begin{align*}
\left\Vert Y_{h,\alpha,\beta}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) \right\Vert & \leq Y_{h,\alpha,\beta}^{\left\Vert \mathcal{A
_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,s\right)
=\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t-\ln sh^{k}\right) ^{n\alpha
+k\alpha+\beta-1}}{\Gamma\left( n\alpha+k\alpha+\beta\right) }\\
& \leq\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t\right) ^{n\alpha+k\alpha+\beta-1
}{\Gamma\left( n\alpha+k\alpha+\beta\right) }\\
& =\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t\right) ^{n\alpha+k\alpha+\beta-1
}{\Gamma\left( n\alpha+k\alpha+\beta\right) }\\
& =Y_{1,\alpha,\beta}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}_{1}\right\Vert }\left( t,1\right) .
\end{align*}
\end{proof}
Our first result on existence and uniqueness of (\ref{ine1}) is based on the
Banach contraction principle.
\begin{theorem}
\label{thm:e}Assume that (A1), (A2) hold. If
\[
L_{f}\Gamma\left( 1-\gamma\right) \left( \ln T\right) ^{\gamma
Y_{\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}_{1}\right\Vert }\left( T,1\right) <1,
\]
then the Cauchy problem (\ref{de2}) has a unique solution on $[1,T]$.
\end{theorem}
\begin{proof}
We define an operator $\Theta$ on $\mathcal{B}_{r}:=\left\{ y\in
C_{\gamma,\ln}\left[ 1,T\right] :\left\Vert y\right\Vert _{\gamma}\leq
r\right\} $ as follows
\begin{align*}
\left( \Theta y\right) \left( t\right) & =Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}}\left( t,\frac{1}{h}\right) a+\int_{\frac
{1}{h}}^{1}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) \left[ \left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right)
\left( s\right) -\mathcal{A}_{0}\varphi\left( s\right) \right] \frac
{ds}{s}\\
& +\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) f\left( s,y\left( s\right) \right) \frac{ds}{s},
\end{align*}
where $r\geq\dfrac{M_{2}}{1-M_{1}}$,
\begin{align*}
M_{2} & :=\left( \ln T\right) ^{\gamma}Y_{1,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
T,1\right) \left\Vert a\right\Vert +\Gamma\left( 1-\gamma\right) \left(
\ln T\right) ^{\gamma}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A
_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left( T,1\right)
\left\Vert \left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right)
-\mathcal{A}_{0}\varphi\right\Vert _{\gamma,\ln}\\
& \ \ \ +L_{2}\left( \ln T\right) ^{\gamma+1}Y_{1,\alpha,\alpha
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( T,1\right) ,\\
M_{1} & :=L_{f}\Gamma\left( 1-\gamma\right) \left( \ln T\right) ^{\gamma
}Y_{\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}\right\Vert }\left( T,1\right) .
\end{align*}
It is obvious that $\Theta$ is well defined due to (A1). Therefore, the
existence of a solution of the Cauchy problem (\ref{de2}) is equivalent to
that of the operator $\Theta$ has a fixed point on $\mathcal{B}_{r}$. We will
use the Banach contraction principle to prove that $\Theta$ has a fixed point.
The proof is divided into two steps.
\textit{Step 1}. $\Theta y\in\mathcal{B}_{r}$ for any $y\in\mathcal{B}_{r}$.
Indeed, for any $y\in\mathcal{B}_{r}$ and any $\delta>0,$ by (A3)
\begin{align}
\left\Vert \left( \ln t\right) ^{\gamma}\left( \Theta y\right) \left(
t\right) \right\Vert & \leq\left( \ln t\right) ^{\gamma}Y_{1,\alpha
,\alpha}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,\frac{1}{h}\right) \left\Vert a\right\Vert
+\left( \ln t\right) ^{\gamma}\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \left\Vert \left( \left( ^{H
D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \right) \left( s\right)
-\mathcal{A}_{0}\varphi\left( s\right) \right\Vert \frac{ds}{s}\nonumber\\
& +\left( \ln t\right) ^{\gamma}\int_{1}^{t}Y_{h,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
t,s\right) \left\Vert f\left( s,y\left( s\right) \right) \right\Vert
\frac{ds}{s} \label{est1
\end{align}
Firstly, we estimate the first integral:
\begin{align}
& \left( \ln t\right) ^{\gamma}\int_{\frac{1}{h}}^{1}Y_{h,\alpha,\alpha
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \left\Vert \left( \left( ^{H
D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) \right) \left( s\right)
-\mathcal{A}_{0}\varphi\left( s\right) \right\Vert \frac{ds}{s}\nonumber\\
& \leq\left( \ln t\right) ^{\gamma}\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\int_{\frac{1}{h}}^{1}\frac{\left( \ln t-\ln
sh^{k}\right) ^{n\alpha+k\alpha+\alpha-1}}{\Gamma\left( n\alpha
+k\alpha+\alpha\right) }\left( \ln s\right) ^{-\gamma}\frac{ds
{s}\left\Vert \left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right)
-\mathcal{A}_{0}\varphi\right\Vert _{\gamma,\ln}\nonumber\\
& \leq\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t\right) ^{n\alpha+k\alpha
+\alpha-\gamma}}{\Gamma\left( n\alpha+k\alpha+\alpha-\gamma+1\right)
}\left\Vert \left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right)
-\mathcal{A}_{0}\varphi\right\Vert _{\gamma,\ln}\nonumber\\
& =\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}\right\Vert }\left( t,1\right) \left\Vert \left(
^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right) -\mathcal{A}_{0
\varphi\right\Vert _{\gamma,\ln}. \label{est2
\end{align}
Similarly,
\begin{align}
& \left( \ln t\right) ^{\gamma}\int_{\frac{1}{h}}^{t}Y_{h,\alpha,\alpha
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \left\Vert f\left( s,y\left( s\right)
\right) \right\Vert \frac{ds}{s}\nonumber\\
& \leq\left( \ln t\right) ^{\gamma}\int_{\frac{1}{h}}^{t}Y_{h,\alpha
,\alpha}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \left( L_{1}\left\Vert y\left(
s\right) \right\Vert +L_{2}\right) \frac{ds}{s}\nonumber\\
& \leq L_{f}\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
}Y_{h,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,\frac{1}{h}\right)
\left\Vert y\right\Vert _{\gamma,\ln}+L_{2}\left( \ln t\right) ^{\gamma
+1}Y_{1,\alpha,\alpha}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}_{1}\right\Vert }\left( t,\frac{1}{h}\right) . \label{est3
\end{align}
Inserting (\ref{est2}) and (\ref{est3}) into (\ref{est1}) we get
\begin{align*}
& \left\Vert \left( \ln t\right) ^{\gamma}\left( \Theta y\right) \left(
t\right) \right\Vert \\
& \leq\left( \ln t\right) ^{\gamma}Y_{1,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
t,1\right) \left\Vert a\right\Vert +\Gamma\left( 1-\gamma\right) \left(
\ln t\right) ^{\gamma}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A
_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,1\right)
\left\Vert ^{H}D_{1^{+}}^{\alpha}\varphi-\mathcal{A}_{0}\varphi\right\Vert
_{\gamma,\ln}\\
& +L_{2}\left( \ln t\right) ^{\gamma+1}Y_{1,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
t,1\right) +L_{f}\Gamma\left( 1-\gamma\right) \left( \ln t\right)
^{\gamma}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,1\right) \left\Vert
y\right\Vert _{\gamma,\ln}\\
& \leq M_{2}+M_{1}\left\Vert y\right\Vert _{\gamma,\ln}\leq M_{2}+M_{1}r\leq
r
\end{align*}
\textit{Step 2.} Let $y,z\in C_{\gamma,\ln}\left[ 1,T\right] $. Then similar
to the estimation (\ref{est3}) we get
\begin{align*}
& \left\Vert \left( \ln t\right) ^{\gamma}\left( \left( \Theta y\right)
\left( t\right) -\left( \Theta z\right) \left( t\right) \right)
\right\Vert \leq\left( \ln t\right) ^{\gamma}\int_{1}^{t}Y_{h,\alpha,\alpha
}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( t,s\right) \left\Vert f\left( s,y\left( s\right)
\right) -f\left( s,z\left( s\right) \right) \right\Vert \frac{ds}{s}\\
& \leq\left( \ln t\right) ^{\gamma}\int_{1}^{t}\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t-\ln sh^{k}\right) ^{n\alpha
+k\alpha+\alpha-1}}{\Gamma\left( n\alpha+k\alpha+\alpha\right) }\left\Vert
f\left( s,y\left( s\right) \right) -f\left( s,z\left( s\right) \right)
\right\Vert \frac{ds}{s}\\
& \leq L_{f}\left( \ln t\right) ^{\gamma}\int_{1}^{t}\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t-\ln s\right) ^{n\alpha+k\alpha
+\alpha-1}}{\Gamma\left( n\alpha+k\alpha+\alpha\right) }\left( \ln
s\right) ^{-\gamma}\left( \ln s\right) ^{\gamma}\left\Vert y\left(
s\right) -z\left( s\right) \right\Vert \frac{ds}{s}\\
& \leq L_{f}\left( \ln t\right) ^{\gamma}\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\int_{1}^{t}\frac{\left( \ln t-\ln s\right)
^{n\alpha+k\alpha+\alpha-1}}{\Gamma\left( n\alpha+k\alpha+\alpha\right)
}\left( \ln s\right) ^{-\gamma}\frac{ds}{s}\left\Vert y-z\right\Vert
_{\gamma,\ln}\\
& \leq L_{f}\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
}\sum_{k=0}^{p
{\displaystyle\sum\limits_{n=0}^{\infty}}
\left(
\begin{array}
[c]{c
n+k\\
k
\end{array}
\right) \left\Vert \mathcal{A}_{1}\right\Vert ^{k}\left\Vert \mathcal{A
_{0}\right\Vert ^{n}\frac{\left( \ln t\right) ^{n\alpha+k\alpha
+\alpha-\gamma}}{\Gamma\left( n\alpha+k\alpha+\alpha-\gamma+1\right)
}\left\Vert y-z\right\Vert _{\gamma,\ln}\\
& =L_{f}\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,1\right) \left\Vert
y-z\right\Vert _{\gamma,\ln},
\end{align*}
which implies that
\begin{equation}
\left\Vert \Theta y-\Theta z\right\Vert _{\gamma,\ln}\leq L_{f}\Gamma\left(
1-\gamma\right) \left( \ln T\right) ^{\gamma}Y_{1,\alpha,\alpha-\gamma
+1}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A
_{1}\right\Vert }\left( T,1\right) \left\Vert y-z\right\Vert _{\gamma,\ln
}.\label{fp
\end{equation}
Hence, the operator $\Theta$ is contraction on $\mathcal{B}_{r}$ and the proof
is competed by using the Banach fixed point theorem.
\end{proof}
Secondly, we discuss the Ulam-Hyers stability for the problems (\ref{de2}) by
means of integral operator given b
\[
y\left( t\right) =\left( \Theta y\right) \left( t\right) ,\ \
\]
where $\Theta$ is defined by (\ref{ine1}).
Define the following nonlinear operator $Q:C_{\gamma,\ln}([1,T],\mathbb{R
^{n})\rightarrow C_{\gamma,\ln}([1,T],\mathbb{R}^{n})$
\[
Q\left( y\right) \left( t\right) :=\left( ^{H}D_{1^{+}}^{\alpha}y\right)
\left( t\right) -\mathcal{A}_{0}y\left( t\right) -\mathcal{A}_{1}y\left(
\frac{t}{h}\right) -f\left( t,y\left( t\right) \right) .
\]
For some $\varepsilon>0$, we look at the following inequality:
\begin{equation}
\left\Vert Q\left( y\right) \right\Vert _{\gamma,\ln}\leq\varepsilon.
\label{ul1
\end{equation}
\begin{definition}
We say that the equation (\ref{ine1}) is Ulam-Hyers stable, if there exist
$V>0$ such that for every solution $y^{\ast}\in C_{\gamma,\ln}([\frac{1
{h},T],\mathbb{R}^{n})$ of the inequality (\ref{ul1}), there exists a unique
solution $y\in C_{\gamma,\ln}([\frac{1}{h},T],\mathbb{R}^{n})$ of problem
(\ref{ine1}) with
\begin{equation}
\left\Vert y-y^{\ast}\right\Vert _{\gamma,\ln}\leq V\varepsilon. \label{ul2
\end{equation}
\end{definition}
\begin{theorem}
\label{Thm:3}Under the assumptions of Theorem \ref{thm:e}, the problem
(\ref{ine1}) is stable inUlam-Hyers sense.
\end{theorem}
\begin{proof}
Let $y\in C_{\gamma,\ln}([\frac{1}{h},T],\mathbb{R}^{n})$ be the solution of
\ the problem (\ref{ine1}). Let $y^{\ast}$ be any solution satisfying
(\ref{ul1}):
\[
\left( ^{H}D_{1^{+}}^{\alpha}y^{\ast}\right) \left( t\right)
=\mathcal{A}_{0}y^{\ast}\left( t\right) +\mathcal{A}_{1}y^{\ast}\left(
\frac{t}{h}\right) +f\left( t,y^{\ast}\left( t\right) \right) +Q\left(
y^{\ast}\right) \left( t\right) .
\]
So
\[
y^{\ast}\left( t\right) =\Theta\left( y^{\ast}\right) \left( t\right)
+\int_{1}^{t}Y_{h,\alpha,\alpha}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left(
t,s\right) Q\left( y^{\ast}\right) \left( s\right) \frac{ds}{s}.
\]
It follows that
\begin{align*}
& \left( \ln t\right) ^{\gamma}\left\Vert \Theta\left( y^{\ast}\right)
\left( t\right) -y^{\ast}\left( t\right) \right\Vert \leq\left( \ln
t\right) ^{\gamma}\int_{1}^{t}\left\Vert Y_{h,\alpha,\alpha}^{\mathcal{A
_{0},\mathcal{A}_{1}}\left( t,s\right) \right\Vert \left\Vert Q\left(
y^{\ast}\right) \left( s\right) \right\Vert \frac{ds}{s}\\
& \leq\left( \ln T\right) ^{\gamma+1}Y_{1,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
T,1\right) \varepsilon.
\end{align*}
Therefore, we deduce by the fixed-point property (\ref{fp}) of the operator
$\Theta,$ that
\begin{align}
\left( \ln t\right) ^{\gamma}\left\Vert y\left( t\right) -y^{\ast}\left(
t\right) \right\Vert & \leq\left( \ln t\right) ^{\gamma}\left\Vert
\Theta\left( y\right) \left( t\right) -\Theta\left( y^{\ast}\right)
\left( t\right) \right\Vert +\left( \ln t\right) ^{\gamma}\left\Vert
\Theta\left( y^{\ast}\right) \left( t\right) -y^{\ast}\left( t\right)
\right\Vert \nonumber\\
& \leq L_{f}\Gamma\left( 1-\gamma\right) \left( \ln t\right) ^{\gamma
}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( t,1\right) \left\Vert
y-y^{\ast}\right\Vert _{\gamma,\ln}+\left( \ln T\right) ^{\gamma
+1}Y_{1,\alpha,\alpha}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}_{1}\right\Vert }\left( T,1\right) \varepsilon, \label{ul7
\end{align}
and
\[
\left\Vert y-y^{\ast}\right\Vert _{\gamma,\ln}\leq\frac{\left( \ln T\right)
^{\gamma+1}Y_{1,\alpha,\alpha}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( T,1\right) }{1-L_{f
\Gamma\left( 1-\gamma\right) \left( \ln T\right) ^{\gamma}Y_{1,\alpha
,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert ,\left\Vert
\mathcal{A}_{1}\right\Vert }\left( T,1\right) }\varepsilon.
\]
Thus, the problem (\ref{de2}) is Ulam-Hyers stable with.
\[
V=\frac{\left( \ln T\right) ^{\gamma+1}Y_{1,\alpha,\alpha}^{\left\Vert
\mathcal{A}_{0}\right\Vert ,\left\Vert \mathcal{A}_{1}\right\Vert }\left(
T,1\right) }{1-L_{f}\Gamma\left( 1-\gamma\right) \left( \ln T\right)
^{\gamma}Y_{1,\alpha,\alpha-\gamma+1}^{\left\Vert \mathcal{A}_{0}\right\Vert
,\left\Vert \mathcal{A}_{1}\right\Vert }\left( T,1\right) }.
\]
\end{proof}
\section{Example}
In this section, we give an example to illustrate the obtained theoretical
result. Let $\alpha=0.3$, $h=1.2,$ $k=4.$ Conside
\begin{align}
\left( ^{H}D_{1^{+}}^{0.3}y\right) \left( t\right) & =\mathcal{A
_{1}y\left( \frac{t}{1.2}\right) ,\ \ t\in\left( 1,2.0736\right]
,\nonumber\\
y\left( t\right) & =\varphi\left( t\right) ,\ \ \frac{1}{1.2}\leq
t\leq1,\nonumber\\
\left( ^{H}I_{\frac{1}{1.2}^{+}}^{0.7}y\right) \left( \frac{1}{1.2
^{+}\right) & =a\in R^{n}, \label{ex2
\end{align}
wher
\[
\mathcal{A}_{1}=\left(
\begin{tabular}
[c]{ll
$2$ & $1$\\
$3$ & $5
\end{tabular}
\ \ \ \right) ,\ \ \ a=\left(
\begin{array}
[c]{c
1\\
2
\end{array}
\right)
\]
The solution of (\ref{ex2}) can be represented by $Y_{h,\alpha,\alpha
}^{\mathcal{A}_{0},\mathcal{A}_{1}}\left( t,\frac{1}{h}\right)
=E_{h,\alpha,\alpha}^{\mathcal{A}_{1}}\left( \ln t\right)
\[
y\left( t\right) =E_{h,\alpha,\alpha}^{\mathcal{A}_{1}}\left( \ln t\right)
a+\int_{\frac{1}{h}}^{1}E_{h,\alpha,\alpha}^{\mathcal{A}_{1}}\left( \ln
\frac{t}{s}\right) \left( ^{H}D_{\frac{1}{h}^{+}}^{\alpha}\varphi\right)
\left( s\right) \frac{ds}{s},
\]
wher
\[
E_{h,\alpha,\alpha}^{\mathcal{A}_{1}}\left( \ln t\right) =\left\{
\begin{tabular}
[c]{ll
$0,$ & $-\infty<x\leq\frac{1}{1.2},$\\
$I\dfrac{\left( \ln t+\ln1.2\right) ^{-0.7}}{\Gamma\left( 0.3\right) },$ &
$\frac{1}{1.2}<x\leq1,$\\
$I\dfrac{\left( \ln t+\ln1.2\right) ^{-0.7}}{\Gamma\left( 0.3\right)
}+\mathcal{A}_{1}\dfrac{\left( \ln t\right) ^{-0.4}}{\Gamma\left(
0.6\right) },$ & $1<x\leq1.2,$\\
$I\dfrac{\left( \ln t+\ln1.2\right) ^{-0.7}}{\Gamma\left( 0.3\right)
}+\mathcal{A}_{1}\dfrac{\left( \ln t\right) ^{-0.4}}{\Gamma\left(
0.6\right) }+\mathcal{A}_{1}^{2}\dfrac{\left( \ln t-\ln1.2\right) ^{-0.1
}{\Gamma\left( 0.9\right) },$ & $1.2<x\leq\left( 1.2\right) ^{2},$\\
$I\dfrac{\left( \ln t+\ln1.2\right) ^{-0.7}}{\Gamma\left( 0.3\right)
}+\mathcal{A}_{1}\dfrac{\left( \ln t\right) ^{-0.4}}{\Gamma\left(
0.6\right) }+\mathcal{A}_{1}^{2}\dfrac{\left( \ln t-\ln1.2\right) ^{-0.1
}{\Gamma\left( 0.9\right) }$ & \\
$+\mathcal{A}_{1}^{3}\dfrac{\left( \ln t-2\ln1.2\right) ^{0.2}
{\Gamma\left( 1.2\right) }$ & $\left( 1.2\right) ^{2}<x\leq\left(
1.2\right) ^{3},$\\
$I\dfrac{\left( \ln t+\ln1.2\right) ^{-0.7}}{\Gamma\left( 0.3\right)
}+\mathcal{A}_{1}\dfrac{\left( \ln t\right) ^{-0.4}}{\Gamma\left(
0.6\right) }+\mathcal{A}_{1}^{2}\dfrac{\left( \ln t-\ln1.2\right) ^{-0.1
}{\Gamma\left( 0.9\right) }$ & \\
$+\mathcal{A}_{1}^{3}\dfrac{\left( \ln t-2\ln1.2\right) ^{0.2}
{\Gamma\left( 1.2\right) }+\mathcal{A}_{1}^{4}\dfrac{\left( \ln
t-3\ln1.2\right) ^{0.5}}{\Gamma\left( 1.5\right) }$ & $\left( 1.2\right)
^{3}<x\leq\left( 1.2\right) ^{4}.
\end{tabular}
\ \ \ \right.
\]
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,517 |
\section{Introduction}
\section{Introduction}
\subsection{Motivation and set-up} We recall the classical definition of a
{\it Diophantine $m$-tuple\/} as a vector $\(a_1, \ldots, a_m\) \in {\mathbb N}^m$ such
that all shifted products $a_ia_j+1$, $1 \leqslant i < j \leqslant m$, are perfect squares.
The long-standing conjecture on the finiteness of the set
of Diophantine quintuples, after a series of intermediate results by various
authors, has been established in a striking work of Dujella~\cite{Duj}, who has also shown the
the non-existence of Diophantine sextuples. More recently, He, Togb{\'e} and
Ziegler~\cite{HTZ} have show
the non-existence of Diophantine quintuples is shown, see also~\cite{BCM}.
Quite naturally, these results
have suggested to study the generalisation of this notion to other algebraic domains
such as, for example, the set of rational numbers or points on curves, as well as in
many other directions, see,
for example,~\cite{DKRM, DKMS, Gup, KN} and references therein. The notion also readily extends to
the setting of finite fields, see~\cite{DK,HKMNOSS,MR-S}.
Let $q$ be an odd prime power and let ${\mathbb F}_q$ be the finite field of $q$ elements.
For $r \in {\mathbb F}_q^*$, we say that an $m$-tuple $\(a_1, \ldots, a_m\) \in {\mathbb F}_q^m$
form a {\it Diophantine $m$-tuple in ${\mathbb F}_q$ with a shift $r$}
if all $m(m-1)/2$ shifted products $a_ia_j + r$ are perfect squares in ${\mathbb F}_q$.
\begin{remark}
We note that it is customary to exclude zero values from the domain from which
$a_1, \ldots, a_m$ are drawn. However in the counting results below this makes
no difference, while this simplifies the notation. In particular,
the total number of such $m$-tuples over ${\mathbb F}_q$ with a zero entry (which is at most $mq^{m-1}$)
can be absorbed in the error term of our asymptotic formula.
\end{remark}
Let $N_r(m,q)$ be the number of distinct Diophantine $m$-tuple in ${\mathbb F}_q$ with a shift $r$.
It has been shown by Dujella and Kazalicki~\cite{DK} that for $r=1$ and a prime $p$
we have
\begin{equation}
\label{eq:D-K Bound}
N_1(m,p) = 2^{-m(m-1)/2} p^m + o(p^m).
\end{equation}
Using some ideas of Dujella and Kazalicki~\cite{DK},
Mani and Rubinstein-Salzedo~\cite[Theorem~5.1]{MR-S} have given explicit formulas
for $N_r(2,q) $ and $N_r(3,q)$ and for $m \geqslant 4$ presented
a more precise than~\eqref{eq:D-K Bound} asymptotic formula
\begin{equation}
\label{eq:MR-S Bound}
N_r(m,q) = 2^{-m(m-1)/2} q^m + O\(q^{m-1/2}\),
\end{equation}
where the implied constant may depend on $m$,
which also holds for any $r \in{\mathbb F}_q^*$ (and it is also easy to see
that for any odd prime power $q$ rather than just for a prime $q = p$ as in~\cite{MR-S}).
We also observe that the bound~\eqref{eq:MR-S Bound} can be derived within
the initial approach of Dujella and Kazalicki~\cite{DK} if one appeals to
a version of the Lang-Weil bound~\cite{LaWe}.
\subsection{New bound}
Here we show that using some simple arguments the bound on the error term
in~\eqref{eq:MR-S Bound} can be improved.
\begin{theorem}
\label{thm:Nmq}
For a fixed $m\geqslant 4$, uniformly over $r \in{\mathbb F}_q^*$, we have
$$
N_(m,q) = 2^{-m(m-1)/2} q^m + O\(q^{m-1}\),
$$
where the implied constant may depend on $m$.
\end{theorem}
As in~\cite{MR-S} our proof is based on an application of the {\it Weil bound\/} for multiplicative
character sums with polynomials, see, for example,~\cite[Theorem~11.23]{IwKow}.
\section{Proof of Theorem~\ref{thm:Nmq}}
\subsection{Preliminary transformations}
Since there are $O\(q^{m-1}\)$ choices of $m$-tuples $\(a_1, \ldots, a_m\) \subseteq {\mathbb F}_q^m$ for which $a_ia_j + r = 0$ for some
$1 \leqslant i < j \leqslant m$, or with $a_i = 0$ for some
$1 \leqslant i \leqslant m$, following the argument of~\cite{MR-S},
we write
$$
N_r(m,q) = 2^{-m(m-1)/2} \sum_{a_1, \ldots, a_m\in {\mathbb F}_q^*}
\prod_{1 \leqslant i < j \leqslant m}\(1 + \chi\(a_i a_j + r\)\) + O\(q^{m-1}\),
$$
where $\chi$ is the quadratic character of ${\mathbb F}_q$, we refer to~\cite[Chapter~3]{IwKow} for a
background on characters.
Therefore
\begin{equation}
\label{eq:N and R}
N_r(m,q) = 2^{-m(m-1)/2} q^m + \sum_{\substack{\bm{\varepsilon} \in \{0,1\}^{m(m-1)/2}\\ \bm{\varepsilon} \ne \mathbf 0}}
R\(\bm{\varepsilon}\)
+ O\(q^{m-1}\),
\end{equation}
where for $\bm{\varepsilon} = \(\varepsilon_{i,j}\)_{1 \leqslant i < j \leqslant m} \in \{0,1\}^m$
\begin{equation}
\label{eq:Reps}
R\(\bm{\varepsilon}\) = \sum_{a_1, \ldots, a_m\in {\mathbb F}_q^*}
\prod_{1 \leqslant i < j \leqslant m}\chi\(a_i a_j + r\)^{\varepsilon_{i,j}} .
\end{equation}
We now fix $\bm{\varepsilon} \in \{0,1\}^{m(m-1)/2}$ with $\bm{\varepsilon} \ne \mathbf 0$ and
estimate $R\(\bm{\varepsilon}\)$. Renumbering the variables $a_1, \ldots, a_m$,
we see that without loss of generality, we can assume that
\begin{equation}
\label{eq:eps12}
\varepsilon_{1,2}=1.
\end{equation}
We now consider the following two cases depending on vanishing and
non-vanishing of the exponents $\varepsilon_{i,j}$ with $2 \leqslant i < j \leqslant m$.
\subsection{Vanishing exponents $\varepsilon_{i,j}$ with $2 \leqslant i < j \leqslant m$} Assume that
\begin{equation}
\label{eq:vanish}
\varepsilon_{i,j} = 0, \qquad \text{for all $2 \leqslant i < j \leqslant m$.}
\end{equation}Then we see that under the conditions~\eqref{eq:vanish}
the expression for $R\(\bm{\varepsilon}\)$ in~\eqref{eq:Reps} simplifies as
\begin{align*}
R\(\bm{\varepsilon}\) & = \sum_{a_1, \ldots, a_m\in {\mathbb F}_q^*}
\prod_{2 \leqslant j \leqslant m}\chi\(a_1 a_j + r\)^{\varepsilon_{1,j}} \\
& = \sum_{a_1 \in {\mathbb F}_q^*} \prod_{2 \leqslant j \leqslant m} \sum_{a_j \in {\mathbb F}_q^*} \chi\(a_1 a_j + r\)^{\varepsilon_{1,j}}.
\end{align*}
Hence, estimating the sums over $a_3, \ldots, a_m$ trivially as $q-1$ and recalling
our assumption~\eqref{eq:eps12}, we obtain
$$
|R\(\bm{\varepsilon}\)| \leqslant (q-1)^{m-2}
\sum_{a_1 \in {\mathbb F}_q^*} \left| \sum_{a_2 \in {\mathbb F}_q^*} \chi\(a_1 a_2 + r\)\right| .
$$
Clearly, for every $a_1 \in {\mathbb F}_q^*$ we have
\begin{align*}
\sum_{a_2 \in {\mathbb F}_q^*} \chi\(a_1 a_2 + r\) & = \sum_{a \in {\mathbb F}_q^*} \chi\(a + r\)\\
& = \sum_{a \in {\mathbb F}_q} \chi\(a \) - \chi(1) = \sum_{a \in {\mathbb F}_q} \chi\(a\) - 1=-1.
\end{align*}
Hence we obtain
\begin{equation}
\label{eq:Bound I}
|R\(\bm{\varepsilon}\)| \leqslant (q-1)^{m-1}
\end{equation}
in this case.
\subsection{Non-vanishing exponents $\varepsilon_{i,j}$ with $2 \leqslant i < j \leqslant m$} We now assume that
\begin{equation}
\label{eq:non-vanish}
\varepsilon_{i,j} \ne 0, \qquad \text{for some $2 \leqslant i < j \leqslant m$.}
\end{equation}
We write $R\(\bm{\varepsilon}\)$ as
$$
R\(\bm{\varepsilon}\)
= \sum_{a_1, \ldots, a_m\in {\mathbb F}_q^*} \prod_{2 \leqslant j \leqslant m}\chi\(a_1 a_j + r\)^{\varepsilon_{1,j}} \prod_{2\leqslant i < j \leqslant m} \chi\(a_i a_j + r\)^{\varepsilon_{i,j}} .
$$
Observe that for any $b \in {\mathbb F}_q^*$ the map
$$(a_1, a_2, … , a_m) \mapsto (a_1/b, a_2b, … , a_mb)$$
is a permutation on ${\mathbb F}_p^m$. Hence
\begin{align*}
R\(\bm{\varepsilon}\)
& = (p-1)^{-1} \sum_{b \in {\mathbb F}_q^*} \sum_{a_1, \ldots, a_m\in {\mathbb F}_q^*}
\prod_{2 \leqslant j \leqslant m}\chi\(a_1 a_j + r\)^{\varepsilon_{1,j}} \\
&\qquad \qquad \qquad \qquad \qquad \qquad
\prod_{2\leqslant i < j \leqslant m} \chi\(a_i a_j b^2 + r\)^{\varepsilon_{i,j}},
\end{align*}
which we now rearrange as
$$
R\(\bm{\varepsilon}\)
= (q-1)^{-1} \sum_{a_2, \ldots, a_m\in {\mathbb F}_q^*}
S(a_2,…, a_m) T(a_2,…, a_m),
$$
where
\begin{align*}
& S(a_2,…, a_m) = \sum_{a_1 \in {\mathbb F}_q^*}
\chi\(\prod_{2 \leqslant j \leqslant m}\(a_1 a_j + r\)^{\varepsilon_{1,j}}\),\\
& T(a_2,…, a_m) = \sum_{b\in {\mathbb F}_q^*}
\chi\( \prod_{2\leqslant i < j \leqslant m}\(a_i a_j b^2 + r\)^{\varepsilon_{i,j}}\).
\end{align*}
We now examine the polynomials
\begin{align*}
& F_{a_2,…, a_m}(X) = \prod_{2 \leqslant j \leqslant m}\(a_j X + r\)^{\varepsilon_{1,j}}, \\
& G_{a_2,…, a_m}(X) = \prod_{2\leqslant i < j \leqslant m}\(a_i a_j X^2 + r\)^{\varepsilon_{i,j}}.
\end{align*}
Because of our assumptions~\eqref{eq:eps12} and~\eqref{eq:non-vanish}
both these polynomials are of positive degree.
Furthermore, it is clear that there are at most $O\(q^{m-2}\)$ choices for $(m-1)$-tuples $(a_2,…, a_m)\in {\mathbb F}_q^{m-1}$
for which at least one of the polynomials $ F_{a_2,…, a_m}(X)$ and $G_{a_2,…, a_m}(X)$
is a perfect square in the algebraic closure of ${\mathbb F}_q$. In this case we estimate
both sums $S(a_2,…, a_m) $ and $T(a_2,…, a_m)$ trivially as $q-1$.
Hence, the contribution to $R\(\bm{\varepsilon}\)$ from such sums is
\begin{equation}
\label{eq:bad a_i}
A = O\((q-1)^{-1} q^{m-2} (q-1)^2\) = O(q^{m-1}).
\end{equation}
For other choices of $(a_2,…, a_m)\in {\mathbb F}_q^{m-1}$, by the Weil bound, see, for example,~\cite[Theorem~11.23]{IwKow}, we have
$$
S(a_2,…, a_m), T(a_2,…, a_m) = O(q^{1/2}).
$$
Hence, the contribution to $R\(\bm{\varepsilon}\)$ from such sums is
\begin{equation}
\label{eq:good a_i}
B = O\((q-1)^{-1} q^{m-1} \(q^{1/2}\)^2\) = O(q^{m-1}).
\end{equation}
Combining~\eqref{eq:bad a_i} and~\eqref{eq:good a_i}
we arrive to
\begin{equation}
\label{eq:Bound II}
R\(\bm{\varepsilon}\) = A + B = O(q^{m-1})
\end{equation}
in this case.
\subsection{Concluding the proof} Substituting the bounds~\eqref{eq:Bound I}
and~\eqref{eq:Bound II} in~\eqref{eq:N and R} we immediately obtain the desired result.
\section{Comments}
It is easy to see that all implied constants can be evaluated explicitly.
Hence one can use our argument to estimate the smallest $q$ (in terms of $m$)
for which $N_r(m,q) > 0$ for all $r \in {\mathbb F}_q^*$. However the inductive approach of
Dujella and Kazalicki~\cite[Theorem~17]{DK} seems to be more effective for this question.
Since we have multivariate character sums, it is also natural to try improve Theorem~\ref{thm:Nmq}
via the use of some version of the {\it Deligne bound\/}, see, for example~\cite{Katz}.
Unfortunately, our polynomials have a high dimensional
singularity locus, which seems to prevent this approach.
\section*{Acknowledgement}
The authors would like to thank Andrej Dujella for several very useful comments and suggestions.
This work was supported by he Australian Research Council (Discovery Project DP200100355).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,152 |
* Write a program that extracts from a given text all palindromes, e.g. "ABBA", "lamal", "exe".
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,113 |
House and Lot Mt. Eden Road(#153061) 04/06/2019 10:00 AM EDT CLOSED!
This is a live on site auction with pre-bidding and simulcast bidding available. Please carefully read all the terrms and conditions and agree to them before placing your bids.
9816 MT. EDEN ROAD SELLS AT 10:00 AM. One story frame home located in the community of Travisville between Southville and Mt. Eden. There are two bedrooms, one bath, kitchen, living room, and a basement on a nice lot with a large rear deck. The home has an updated electric furnace, hardwood in bedrooms, living room and hallway, and has a carport with storage area. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,101 |
Are you kidding me? Are you now beginning to see why my hindquarters are chapped over church signs? I actually saw this one on a church sign in my neighborhood. In fact, it was the very church where I was pastoring at the time. When I drove by and saw it, I was horrified and immediately changed it, which went over like a you-know-what in church. At the time, the deacons took turns each week changing the sign. The fellow who's turn it was to change the sign on this occasion obviously possessed a "works" mentality and not one of grace. In his arrogant opinion, if you were not working in the church, you were "useless" (a direct quote). I blatantly disagreed, and continually proclaimed a message of grace and acceptance. People do not come to church looking for a job, they are looking for Jesus, and did NOT see it in this man's life. Sadly, the author of this sign became one of the largest thorns in my side in this church. He finally left after three very unsuccessful (and one very ugly) attempts to run me off. Unfortunately, he had planted many seeds of discord before his departure from the flock that exist even today, nearly 2 years after my own departure.
For the record, I took over the "Sign Ministry" at the church after this incident with very few objections.
Churches turned me off forever when they started in on their current anti-gay shtick. They now focus on hate and money instead of love, and I am sure Jesus would tear them all down for that if he returned today.
naah - too much generalization on the "anti-gay schtick" thing. a few churches and church people have messed up, but as a whole it's still the best way to live life. "love God, love others" is rich with meaning, and might be one of the few church signs that would get it right somehow. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,822 |
Sewadjenre Nebiryraw I fou un faraó egipci de la dinastia XVI, que va governar a Tebes. El seu nom d'Horus fou Sewadjtawy, el seu Nebti fou Netjerkheperu; el seu nom d'Horus d'or Neferkhau, el seu nom de tron o Nesut biti fou Sewadjenre; i el seu nom Sa Ra fou Nebiryraw. Se suposa que era fill de Sobekhotep VIII i fou pare de Semenre i de Sewoserenre Bebiankh.
Va succeir a Sankhenre (Mentuhotep VI) segons l'ordre en què són esmentats al papir de Torí. Segons aquest, va regnar 26 anys i mesos (vers 1625-1600 aC) i el va succeir Nebiryraw II. Es desconeix on és enterrat.
Hi ha constància d'aquest faraó en diversos escarabats (algun d'aquests al Museu Petrie: UC 11608, UC 11609, UC 11610, UC 11611 i UC 11612). També en una estela a Tebes (Karnak) descoberta el 1927, que conté el tractat en el qual es dona el càrrec d'alcalde d'Elkab (Nekheb) a Sebeknakht, per a ell i els seus descendents, i en una daga trobada a Diòspolis Parva.
Faraons de la dinastia XVI | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,955 |
\section{Introduction and main results}
\subsection {The model}
\label{S1.1}
{\bf 1.\ Polymer configuration.} The polymer is modeled by a directed path drawn from
the set
\begin{equation}
\label{pidef}
\Pi=\Big\{\pi=(k,\pi_{k})_{k\in{\mathbb N}_0}\colon\,\pi_0=0,\,\mathrm{sign}(\pi_{k-1})
+\mathrm{sign}(\pi_k) \neq 0,\,\pi_k\in{\mathbb Z}\,\,\forall\,k\in{\mathbb N}\Big\}
\end{equation}
of directed paths in ${\mathbb N}_0\times{\mathbb Z}$ that start at the origin and visit the interface
${\mathbb N}_0\times\{0\}$ when switching from the lower halfplane to the upper halfplane, and
vice versa. Let $P^*$ be the path measure on $\Pi$ under which the excursions away from
the interface are i.i.d., lie above or below the interface with equal probability, and
have a length distribution $\rho$ on ${\mathbb N}$ with a \emph{polynomial tail}:
\begin{equation}
\label{rhocond}
\lim_{ {n\to\infty} \atop{\rho(n)>0} }
\frac{\log\rho(n)}{\log n} = -\alpha \mbox{ for some } \alpha\in [1,\infty).
\end{equation}
The support of $\rho$ is assumed to satisfy the following non-sparsity condition
\begin{equation}
\lim_{m\rightarrow\infty}\frac1m \log \sum_{n>m} \rho(n)=0.
\end{equation}
Denote by $\Pi_n,P^*_n$ the restriction of $\Pi,P^*$ to $n$-step paths that end at the
interface.
\medskip\noindent
{\bf 2.\ Disorder.} Let $\hat E$ and $\bar E$ be subsets of ${\mathbb R}$. The edges of the paths in $\Pi$
are labeled by an i.i.d.\ sequence of $\hat E$-valued random variables $\hat \omega=(\hat \omega_i)_{i\in{\mathbb N}}$
with common law $\hat \mu$, modeling the random monomer types. The sites at the interface are
labeled by an i.i.d.\ sequence of $\bar E$-valued random variables $\bar \omega=(\bar \omega_i)_{i\in{\mathbb N}}$
with common law $\bar \mu$, modeling the random charges. In the sequel we abbreviate $\omega
=(\omega_{i})_{i\in{\mathbb N}}$ with $\omega_i=(\hat \omega_i,\bar \omega_i)$ and assume that $\hat \omega$ and $\bar \omega$ are
independent. We further assume, without loss of generality, that both $\hat \omega_1$ and $\bar \omega_1$
have zero mean, unit variance, and satisfy
\begin{equation}
\label{mgffin}
\hat M(t) = \log \int_{\hat E} e^{-t\hat \omega_1} \hat \mu(d\hat \omega_1) < \infty \quad \forall\,t\in{\mathbb R},
\qquad
\bar M(t)= \log \int_{\bar E} e^{-t\bar \omega_1} \bar \mu(d\bar \omega_1) < \infty \quad \forall\,t\in{\mathbb R}.
\end{equation}
We write ${\mathbb P}$ for the law of $\omega$, and ${\mathbb P}_{\hat \omega}$ and ${\mathbb P}_{\bar \omega}$ for the laws of $\hat \omega$
and $\bar \omega$.
\medskip\noindent
{\bf 3.\ Path measure.} Given $n\in{\mathbb N}$ and $\omega$, the \emph{quenched copolymer with pinning}
is the path measure given by
\begin{equation}
\label{copwad}
\widetilde P^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n(\pi)
= \frac{1}{\tilde Z^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n}
\,\exp\left[\widetilde H_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}(\pi)\right]\,P^*_n(\pi),
\qquad \pi\in\Pi_n,
\end{equation}
where $\hat\beta,\hat h,\bar\beta\geq0$ and $\bar h\in{\mathbb R}$ are parameters, $\widetilde Z^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n$
is the normalizing partition sum, and
\begin{equation}
\label{copadhamil}
\widetilde H_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}(\pi)
= \hat\beta\sum_{i=1}^n (\hat \omega_i+\hat h)\,\Delta_i+\sum_{i=1}^{n} (\bar\beta\,\bar \omega_i-\bar h)\delta_i
\end{equation}
is the interaction Hamiltonian, where $\delta_i=1_{\{\pi_i=0\}} \in \{0,1\}$ and
$\Delta_i=\mathrm{sign}(\pi_{i-1},\pi_i)\in \{-1,1\}$ (the $i$-th edge is below
or above the interface).
\medskip\noindent
{\bf Key example:}
The choice $\hat E=\bar E=\{-1,1\}$ corresponds to the situation where the upper halfplane
consists of oil, the lower halfplane consists of water, the monomer types are either
hydrophobic ($\hat \omega_i=1$) or hydrophilic ($\hat \omega_i=-1$), and the charges are either positive
($\bar \omega_i=1$) or negative ($\bar \omega_i=-1$); see Fig.~\ref{fig-copolex}. In \eqref{copadhamil},
$\hat\beta$ and $\bar\beta$ are the \emph{strengths} of the monomer-solvent and monomer-interface
interactions, while $\hat h$ and $\bar h$ are the \emph{biases} of these interactions. If $P^*$
is the law of the directed simple random walk on ${\mathbb Z}$, then \eqref{rhocond} holds
with $\alpha=\tfrac32$.
\begin{figure}[htbp]
\vspace{-.50cm}
\begin{center}
\includegraphics[scale = .5]{copadpict3.pdf}
\end{center}
\vspace{0cm}
\vspace{-10.5cm}
\caption{\small A directed polymer near a linear interface, separating oil in the upper
halfplane and water in the lower halfplane. Hydrophobic monomers in the polymer are
light shaded, hydrophilic monomers are dark shaded. Positive charges at the interface
are light shaded, negative charges are dark shaded.}
\label{fig-copolex}
\vspace{-.1cm}
\end{figure}
In the literature, the model without the monomer-interface interaction ($\bar\beta=\bar h=0$) is
called the \emph{copolymer model}, while the model without the monomer-solvent interaction
($\hat h=\hat\beta=0$) is called the \emph{pinning model} (see Giacomin~\cite{Gi07} and den
Hollander~\cite{dHo09} for an overview). The model with both interactions is referred to
as the \emph{copolymer with pinning model}. In the sequel, if $k$ is a quantity associated
with the combined model, then $\hat k$ and $\bar k$ denote the analogous quantities in
the copolymer model, respectively, the pinning model.
\subsection{Quenched excess free energy and critical curve}
\label{S1.2}
The \emph{quenched free energy} per monomer
\begin{equation}
\label{copadfe}
f^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h) = \lim_{n\to\infty} \frac1n\,\log\widetilde
Z^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n
\end{equation}
exists $\omega$-a.s.\ and in ${\mathbb P}$-mean (see e.g.\ Giacomin~\cite{Gi04}). By restricting
the partition sum $\widetilde Z_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}$ to paths that stay above
the interface up to time $n$, we obtain, using the law of large numbers for $\hat \omega$, that
$f^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)\geq \hat\beta\,\hat h$. The {\em quenched excess free energy} per
monomer
\begin{equation}
\label{copadefe}
g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h) = f^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)-\hat\beta\,\hat h
\end{equation}
corresponds to the Hamiltonian
\begin{equation}
\label{copadmodhamil}
H_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}(\pi)=\hat\beta\sum_{i=1}^n (\hat \omega_i+\hat h)\,\left[\Delta_i-1\right]
+\sum_{i=1}^n (\bar\beta\,\bar \omega_i-\bar h)\delta_i
\end{equation}
and has two phases
\begin{equation}
\label{copadquephases}
\begin{aligned}
{\mathcal L}^{\rm que} &= \left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in[0,\infty)^3\times{\mathbb R}\colon\,
g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)>0\right\},\\
{\mathcal D}^{\rm que} &= \left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in[0,\infty)^3\times{\mathbb R}\colon\,
g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\},
\end{aligned}
\end{equation}
called the {\em quenched localized phase} (where the strategy of staying close to the
interface is optimal) and the {\em quenched delocalized phase} (where the strategy of
wandering away from the interface is optimal). The map $\hat h\mapsto g^{\rm que}(\hat\beta,\hat h,
\bar\beta,\bar h)$ is non-increasing and convex for every $\hat\beta,\bar\beta\geq0$ and $\bar h\in{\mathbb R}$. Hence,
${\mathcal L}^{\rm que}$ and ${\mathcal D}^{\rm que}$ are separated by a single curve
\begin{equation}
\label{copadcc}
h_c^{\rm que}(\hat\beta,\bar\beta,\bar h)
= \inf\left\{\hat h\geq 0\colon\,g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\},
\end{equation}
called the {\em quenched critical curve}.
In the sequel we write $\hat g^{\rm que}(\hat\beta,\hat h)$, $\hat h_c^{\rm que}(\hat\beta)$, $\hat
{\mathcal L}^{\rm que}$, $\hat{\mathcal D}^{\rm que}$ for the analogous quantities in the copolymer model
($\bar\beta=\bar h=0$), and $\bar g^{\rm que}(\bar\beta,\bar h)$, $\bar h_c^{\rm que}(\bar\beta)$, $\bar{\mathcal L}^{\rm que}$,
$\bar{\mathcal D}^{\rm que}$ for the analogous quantities in the pinning model ($\hat\beta= \hat h=0$).
\subsection{Annealed excess free energy and critical curve}
\label{S1.3}
The {\em annealed excess free energy} per monomer is given by
\begin{equation}
\label{copadaefe}
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h) = \lim_{n\to\infty} \frac1n \log Z_n^{\hat\beta,\hat h,\bar\beta,\bar h}
= \lim_{n\to\infty} \frac1n \log {\mathbb E}\left(Z_n^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}\right),
\end{equation}
where ${\mathbb E}$ is the expectation w.r.t.\ the joint disorder distribution ${\mathbb P}$. This also
has two phases,
\begin{equation}
\label{copadannphases}
\begin{aligned}
{\mathcal L}^{\rm ann} &= \left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in [0,\infty)^3\times{\mathbb R}\colon\,
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)>0\right\},\\
{\mathcal D}^{\rm ann} &= \left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in [0,\infty)^3\times{\mathbb R}\colon\,
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\},
\end{aligned}
\end{equation}
called the {\em annealed localized phase} and the {\em annealed delocalized phase},
respectively. The two phases are separated by the {\em annealed critical curve}
\begin{equation}
\label{copadacc}
h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h) = \inf\left\{\hat h\geq 0\colon\,
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\}.
\end{equation}
Let ${\mathcal N}(g) = \sum_{n\in{\mathbb N}} e^{-ng}\,\rho(n),\, g\in{\mathbb R}$. We will show in Section~\ref{S3.2}
that
\begin{equation}
\label{ganncomb}
\begin{aligned}
&g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h) \mbox{ is the unique $g$-value at which}\\
&\log \left[\tfrac12 {\mathcal N}(g)+\tfrac12 {\mathcal N}\big(g-[\hat M(2\hat\beta)-2\hat\beta \hat h]\big)\right]
+\bar M(-\bar\beta)-\bar h \mbox{ changes sign}.
\end{aligned}
\end{equation}
\noindent
{\bf Remark:} The annealed model is exactly solvable. In fact, sharp asymptotics estimates on the
annealed partition function that go beyond the free energy can be derived by using the techniques
in Giacomin~\cite{Gi07}, Section 2.2. We will derive variational formulas for the annealed and
the quenched free energies. The annealed variational problem turns out to be easy, but we need
it as an object of comparison in our study of the quenched variational formula.
It follows from \eqref{ganncomb} that for the copolymer model ($\bar\beta=\bar h=0$)
\begin{equation}
\label{copannfeg}
\begin{aligned}
\hat{g}^{\rm\, ann}(\hat\beta,\hat h) &= 0 \vee [\hat M(2\hat\beta)-2\hat\beta\hat h],\\
\hat h_c^{\rm ann}(\hat\beta) &= (2\hat\beta)^{-1}\hat M(2\hat\beta),
\end{aligned}
\end{equation}
and for the pinning model ($\hat\beta=\hat h=0$)
\begin{equation}
\label{pinannfeg}
\begin{aligned}
&\bar g^{\rm ann}(\bar\beta,\bar h) \mbox{ is the unique $g$-value for which }
{\mathcal N}(g) = e^{-(0\vee[\bar M(-\bar\beta)-\bar h])},\\
&\bar h_c^{\rm ann}(\bar\beta) = \bar M(-\bar\beta).
\end{aligned}
\end{equation}
For more details on these special cases, see Giacomin~\cite{Gi07} and den
Hollander~\cite{dHo09}, and references therein.
\subsection{Main results}
\label{S1.4}
Our variational characterization of the excess free energies and the critical curves
is contained in the following theorem. For technical reasons, in the sequel we
\emph{exclude} the case $\hat\beta>0$, $\hat h=0$ for the quenched version.
\begin{theorem}
\label{freeenegvar}
Assume {\rm (\ref{rhocond})} and {\rm (\ref{mgffin})}.\\
(i) For every $\alpha \geq 1$ and $\hat\beta,\hat h,\bar\beta\geq 0$, there are lower semi-continuous,
convex and non-increasing functions
\begin{equation}
\label{Sannounce}
\begin{aligned}
&g\mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g),\\
&g\mapsto S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g),
\end{aligned}
\end{equation}
given by explicit variational formulas such that, for every $\bar h\in{\mathbb R}$,
\begin{equation}
\label{gquegannvarfor}
\begin{aligned}
g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h) &= \inf\{g\in{\mathbb R}\colon\,S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0\},\\
g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h) &= \inf\{g\in{\mathbb R}\colon\,S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0\}.
\end{aligned}
\end{equation}
(ii) For every $\alpha \geq 1$, $\hat\beta>0$, $\bar\beta\geq 0$ and $\bar h\in{\mathbb R}$,
\begin{equation}
\label{ccquccannvarfor}
\begin{aligned}
h_c^\mathrm{que}(\hat\beta,\bar\beta,\bar h)
&= \inf\left\{\hat h>0\colon\,S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq 0\right\},\\
h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h)
&= \inf\left\{\hat h\geq 0\colon\,S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq0\right\}.
\end{aligned}
\end{equation}
\end{theorem}
\noindent
The variational formulas for $S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ and $S^\mathrm{ann}(\hat\beta,\hat h,
\bar\beta;g)$ are given in Theorems~\ref{varfloc}--\ref{varflocann} in Section~\ref{S3}.
Figs.~\ref{fig-varfe}--\ref{fig-copadvarhs} in Sections~\ref{S3} and \ref{Sec5} show
how these functions depend on $\hat\beta$, $\hat h$, $\bar\beta$ and $g$, which is crucial for our analysis.
Next, we state seven corollaries that are consequences of the variational formulas.
The content of these corollaries will be discussed in Section~\ref{S1.6}. The
first corollary looks at the excess free energies. Put
\begin{equation}
\label{lann12}
\begin{split}
\bar h_\ast(\hat\beta,\hat h,\bar\beta)&=\bar M(-\bar\beta)+\log\left(\tfrac12\left[1+{\mathcal N}\big(|\hat M(2\hat\beta)-2\hat\beta\hat h|\big)
\right]\right),\cr
{\mathcal L}^{\rm ann}_1
&=\left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in[0,\infty)^3\times{\mathbb R}\colon\,(\hat\beta,\hat h)\in\hat
{\mathcal L}^{\rm ann}\right\},\cr
{\mathcal L}^{\rm ann}_2
&=\left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in[0,\infty)^3\times{\mathbb R}\colon\,(\hat\beta,\hat h)\in\hat
{\mathcal D}^{\rm ann},\bar h<\bar h_\ast(\hat\beta,\hat h,\bar\beta)\right\}.
\end{split}
\end{equation}
\begin{corollary}
\label{freeeneggap}
(i) For every $\alpha\geq 1$, $\hat\beta>0$ and $\bar\beta\geq0$, $g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h)$
and $g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h)$ are the unique $g$-values that solve the equations
\begin{equation}
\begin{split}
S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)
&= \bar h, \quad \text{if } \hat h>0 , \,\bar h\leq
S^{\rm que}(\hat\beta,\hat h,\bar\beta;0),
\cr
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)
&=\bar h, \quad \text{if } \hat h\geq 0,\,
\bar h\leq \bar h_{\ast}(\hat\beta,\hat h,\bar\beta).
\end{split}
\end{equation}
(ii) The annealed localized phase ${\mathcal L}^{\rm ann}$ admits the decomposition ${\mathcal L}^{\rm ann}
={\mathcal L}^{\rm ann}_1\cup {\mathcal L}^{\rm ann}_2$.\\
\noindent
(iii) On ${\mathcal L}^{\rm ann}$,
\begin{equation}
g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h) < g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h),
\end{equation}
with the possible exception of the case where $m_\rho=\sum_{n\in{\mathbb N}} n\rho(n)=\infty$ and
$\bar h=\bar h_\ast(\hat\beta,\hat h,\bar\beta)$.\\
(iv) For every $\alpha\geq 1$ and $\hat\beta,\hat h,\bar\beta\geq0$,
\begin{equation}
g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h)
\left\{\begin{array}{ll}
=\hat g^{\rm ann}(\hat\beta,\hat h), &\mbox{if } \bar h\geq \bar h_\ast(\hat\beta,\hat h,\bar\beta),\\
>\hat g^{\rm ann}(\hat\beta,\hat h), &\mbox{otherwise.}
\end{array}
\right.
\end{equation}
\end{corollary}
The next four corollaries look at the critical curves.
\begin{corollary}
\label{hquehannvarfor}
For every $\alpha\geq 1$, $\hat\beta>0$ and $\bar\beta\geq0~$, the maps
\begin{equation}
\label{Sannounce*}
\begin{aligned}
&\hat h\mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0),\\
&\hat h \mapsto S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0),
\end{aligned}
\end{equation}
are convex and non-increasing on $(0,\infty)$. Both critical curves are continuous and
non-increasing in $\bar h$. Moreover (see Figs.~{\rm \ref{fig-copadvarhc}--\ref{fig-copadvarhannc}}),
\begin{equation}
\label{ccqueflatp}
h_c^\mathrm{que}(\hat\beta,\bar\beta,\bar h)
=\left\{\begin{array}{ll}
\infty, &\mbox{if } \bar h\leq\bar h_c^{\rm que}(\bar\beta)-\log 2,\\
\hat h_c^{\rm ann}(\hat\beta/\alpha), &\mbox{if } \bar h> s^{\ast}(\hat\beta,\bar\beta,\alpha),\\
h_\ast^{\rm que}(\hat\beta,\bar\beta,\bar h), &\mbox{otherwise},
\end{array}
\right.
\end{equation}
and
\begin{equation}
\label{ccannflatp}
h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h)
=\left\{\begin{array}{ll}
\infty, &\mbox{if } \bar h\leq\bar h_c^{\rm ann}(\bar\beta)-\log 2,\\
\hat h_c^{\rm ann}(\hat\beta), &\mbox{if } \bar h> \bar h_c^{\rm ann}(\bar\beta),\\
h_\ast^{\rm ann}(\hat\beta,\bar\beta,\bar h), &\mbox{otherwise}.
\end{array}
\right.
\end{equation}
where
\begin{equation}
\label{sast}
s^{\ast}(\hat\beta,\bar\beta,\alpha) = S^\mathrm{que}\left(\hat\beta,\hat h_c^{\rm ann}
(\hat\beta/\alpha),\bar\beta;0\right) \in \left([\bar h_c^{\rm que}(\bar\beta)-\log 2]\vee0,\infty\right],
\end{equation}
and $h_\ast^{\rm que}(\hat\beta,\bar\beta,\bar h)$ and $h_\ast^{\rm ann}
(\hat\beta,\bar\beta,\bar h)$ are the unique $\hat h$-values that solve the equations
\begin{equation}
\label{qccth}
\begin{split}
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0) &= \bar h,\cr
S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0) &= \bar h.
\end{split}
\end{equation}
In particular, both $h_\ast^{\rm que}(\hat\beta,\bar\beta,\bar h)$ and $h_\ast^{\rm ann}
(\hat\beta,\bar\beta,\bar h)$ are convex and strictly decreasing functions of $\bar h$.
\end{corollary}
\begin{figure}[htbp]
\vspace{.8cm}
\begin{minipage}[hbt]{7.6cm}
\centering
\setlength{\unitlength}{0.50cm}
\begin{picture}(12,6)(0,0)
\put(0,0){\line(12,0){12}}
\put(4,0){\line(0,6){6}}
{\thicklines
\qbezier(2,5)(5,1.6)(9,1)
\qbezier(9,1)(11,1)(12,1)
\qbezier(2,5)(1,5)(0,5)
}
\qbezier[30](2,5)(2,3)(2,0)
\qbezier[30](2,1)(5,1)(9,1)
\qbezier[10](8.9,1)(8.9,.5)(8.9,0)
\put(1,5.2){$\infty$}
\put(7.6,-1){$s^{\ast}(\hat\beta,\bar\beta,\alpha)$}
\put(2,6.3){$h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$}
\put(12.2,-.3){$\bar h$}
\put(-1,.9){$\hat h^{\rm ann}_c(\frac{\hat\beta}{\alpha})$}
\put(-.5,-1){$\bar h_c^{\rm que}(\bar\beta)-\log 2$}
\end{picture}
\vspace{.6cm}
\begin{center}
(a)
\end{center}
\end{minipage}
\begin{minipage}[hbt]{8.6cm}
\centering
\setlength{\unitlength}{0.50cm}
\begin{picture}(12,6)(0,0)
\put(0,0){\line(12,0){12}}
\put(4,0){\line(0,6){6}}
{\thicklines
\qbezier(2,5)(5,1.3)(12,1.2)
\qbezier(2,5)(1,5)(0,5)
}
\qbezier[30](2,5)(2,3)(2,0)
\qbezier[30](2,1)(5,1)(12,1)
\put(1,5.2){$\infty$}
\put(2,6.3){$h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$}
\put(12.2,-.3){$\bar h$}
\put(-1,.9){$\hat h^{\rm ann}_c(\frac{\hat\beta}{\alpha})$}
\put(-.5,-1){$\bar h_c^{\rm que}(\bar\beta)-\log 2$}
\end{picture}
\vspace{.6cm}
\begin{center}
(b)
\end{center}
\end{minipage}
\vspace{-0cm}
\caption{\small Qualitative picture of the map $\bar h \mapsto h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$
for $\hat\beta>0$ and $\bar\beta\geq0$ when: (a) $s^{\ast}(\hat\beta,\bar\beta,\alpha)<\infty$; (b)
$s^{\ast}(\hat\beta,\bar\beta,\alpha)=\infty$.}
\label{fig-copadvarhc}
\end{figure}
\begin{figure}[htbp]
\vspace{1.5cm}
\begin{center}
\setlength{\unitlength}{0.50cm}
\begin{picture}(22,9)(0,0)
\put(-1,0){\line(22,0){22}}
\put(8,0){\line(0,9){9}}
{\thicklines
\qbezier(3,8)(10,2.6)(17,2)
\qbezier(17,2)(20,2)(21,2)
\qbezier(3,8)(1,8)(-1,8)
}
\qbezier[40](3,8)(3,3)(3,0)
\qbezier[20](16.8,0)(16.8,1)(16.8,2)
\qbezier[60](3,2)(12,2)(17,2)
\put(1.3,8.2){$\infty$}
\put(15.9,-1){$\bar h^{\rm ann}_c(\bar\beta)$}
\put(5.6,9.5){$h^\mathrm{ann}_c(\hat\beta,\bar\beta,\bar h)$}
\put(0.2,1.9){$\hat h^\mathrm{ann}_c(\hat\beta)$}
\put(21.5,-.3){$\bar h$}
\put(.81,-1){$\bar h^{\rm ann}_c(\bar\beta)-\log 2$}
\end{picture}
\end{center}
\vspace{0.2cm}
\caption{\small Qualitative picture of the map $\bar h \mapsto h^\mathrm{ann}_c
(\hat\beta,\bar\beta,\bar h)$ for $\hat\beta,\bar\beta\geq0$.}
\label{fig-copadvarhannc}
\end{figure}
\begin{corollary}
\label{hcubstrict}
For every $\alpha>1$, $\hat\beta>0$ and $\bar\beta\geq0$,
\begin{equation}
h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)
\left\{\begin{array}{ll}
< h^\mathrm{ann}_c(\hat\beta,\bar\beta,\bar h)\leq\infty, &\mbox{if } \bar h>\bar h^{\rm que}_c(\bar\beta)-\log 2,\\
= h^\mathrm{ann}_c(\hat\beta,\bar\beta,\bar h)=\infty, &\mbox{otherwise.}
\end{array}
\right.
\end{equation}
\end{corollary}
\begin{corollary}
\label{hclbstrict}
For every $\alpha>1$, $\hat\beta>0$ and $\bar\beta\geq0$,
\begin{equation}
h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)
\left\{\begin{array}{ll}
> \hat h^\mathrm{ann}_c(\hat\beta/\alpha), &\mbox{ if } \bar h< s^{\ast}(\hat\beta,\bar\beta,\alpha),\\
= \hat h^\mathrm{ann}_c(\hat\beta/\alpha), &\mbox{ otherwise.}
\end{array}
\right.
\end{equation}
\end{corollary}
\begin{corollary}
\label{critqueann}
(i) For every $\alpha\geq 1$ and $\hat\beta,\bar\beta\geq 0$,
\begin{equation}
\begin{aligned}
\inf\left\{\bar h\in{\mathbb R}\colon\,g^\mathrm{ann}(\hat\beta,\hat h_c^{\rm ann}(\hat\beta),\bar\beta,\bar h)=0 \right\}
&= \bar h^{\rm ann}_c(\bar\beta),\\
\inf\left\{\hat h\geq 0\colon\,g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h_c^{\rm ann}(\bar\beta))=0 \right\}
&= \hat h^{\rm ann}_c(\hat\beta).
\end{aligned}
\end{equation}
(ii) For every $\alpha\geq 1$, $\hat\beta>0$ and $\bar\beta=0$,
\begin{equation}
\inf\left\{\bar h\in{\mathbb R}\colon g^\mathrm{que}(\hat\beta,\hat h_c^{\rm que}(\hat\beta),\bar\beta,\bar h)=0 \right\}
=0.
\end{equation}
\end{corollary}
The last two corollaries concern the typical path behavior. Let $P^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n$
denote the path measure associated with the Hamiltonian $H_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}$ defined
in \eqref{copadmodhamil}. Write $\mathcal M_n = \mathcal M_n(\pi) = |\{1\leq i\leq n\colon\,\pi_i=0\}|$
to denote the number of times the polymer returns to the interface up to time $n$. Define
\begin{equation}
\label{dquedecomp}
\begin{split}
{\mathcal D}^{\rm que}_1 = \left\{(\hat\beta,\hat h,\bar\beta,\bar h)\in{\mathcal D}^{\rm que}\colon\,
\bar h\leq s^\ast(\hat\beta,\bar\beta,\alpha)\right\}.
\end{split}
\end{equation}
\begin{corollary}
\label{delocpathprop}
For every $(\hat\beta,\hat h,\bar\beta,\bar h)\in \mathrm{int}({\mathcal D}^\mathrm{que}_1) \cup ({\mathcal D}^{\rm que}
\setminus{\mathcal D}^{\rm que}_1)$ and $c>\alpha/[-(S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)-\bar h)] \in
(0,\infty)$,
\begin{equation}
\label{ubdint}
\lim_{n\to\infty} P^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n\left(\mathcal M_n \geq c\log n\right)
= 0 \qquad \omega-a.s.
\end{equation}
\end{corollary}
\begin{corollary}
\label{locpathprop}
For every $(\hat\beta,\hat h,\bar\beta,\bar h) \in {\mathcal L}^\mathrm{que}$,
\begin{equation}
\label{LLNpath}
\lim_{n\to\infty}
P^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}_n\left(|\tfrac1n\mathcal M_n-C| \leq \varepsilon \right) = 1
\qquad \omega-a.s.\quad \forall\,\varepsilon>0,
\end{equation}
where
\begin{equation}
\label{densid}
-\frac{1}{C} = \frac{\partial}{\partial g}\,
S^\mathrm{que}\big(\hat\beta,\hat h,\bar\beta;g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h)\big)
\in (-\infty,0),
\end{equation}
provided this derivative exists. (By convexity, at least the left-derivative and the
right-derivative exist.)
\end{corollary}
\subsection{Discussion}
\label{S1.6}
{\bf 1.}
The copolymer and pinning versions of Theorem \ref{freeenegvar} are obtained by putting
$\bar\beta=\bar h=0$ and $\hat\beta=\hat h=0$, respectively. The copolymer version of Theorem \ref{freeenegvar}
was proved in Bolthausen, den Hollander and Opoku \cite{BodHoOp11}.
\medskip\noindent
{\bf 2.}
Corollary~\ref{freeeneggap}(i) identifies the range of parameters for which the free energies
given by \eqref{gquegannvarfor} are the $g$-values where the variational formulas equal $\bar h$.
Corollary~\ref{freeeneggap}(ii) shows that the annealed combined model is localized when
the annealed copolymer model is localized. On the other hand, if the annealed copolymer
model is delocalized, then a sufficiently attractive pinning interaction is needed for
the annealed combined model to become localized, namely, $\bar h<\bar h_\ast(\hat\beta,\hat h,\bar\beta)$. It
is an open problem to identify a similar threshold for the quenched combined model.
\medskip\noindent
{\bf 3.}
In Bolthausen, den Hollander and Opoku~\cite{BodHoOp11} it was shown with the help of
the variational approach that for the copolymer model there is a gap between the quenched
and the annealed excess free energy in the localized phase of the annealed copolymer model.
It was argued that this gap can also be deduced with the help of a result in Giacomin
and Toninelli~\cite{GiTo06a,GiTo06b}, namely, the fact that the map $\hat h\mapsto\hat
g^\mathrm{que}(\hat\beta,\hat h)$ drops below a quadratic as $\hat h \uparrow \hat h_c^\mathrm{que}(\hat\beta)$
(i.e., the phase transition is ``at least of second order''). Indeed, $g^\mathrm{que}
\leq g^\mathrm{ann}$, $\hat h\mapsto \hat g^\mathrm{que}(\hat\beta,\hat h)$ is convex and strictly decreasing
on $(0,\hat h_c^\mathrm{que}(\hat\beta)]$, and $\hat h\mapsto\hat g^\mathrm{ann}(\hat\beta,\hat h)$
is linear and strictly decreasing on $(0,\hat h_c^\mathrm{ann}(\hat\beta)]$. The quadratic bound
implies that the gap is present for $\hat h$ slightly below $\hat h_c^\mathrm{ann}(\hat\beta)$, and
therefore it must be present for all $\hat h$ below $\hat h_c^\mathrm{ann}(\hat\beta)$. Now, the
same arguments as in \cite{GiTo06a,GiTo06b} show that also $\hat h\mapsto g^\mathrm{que}
(\hat\beta,\hat h,\bar\beta,\bar h)$ drops below a quadratic as $\hat h \uparrow h_c^\mathrm{que}(\hat\beta,\bar\beta,\bar h)$. However, $\hat h\mapsto g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h)$ is {\em not} linear on
$(0,h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h)]$ (see \eqref{ganncomb}), and so there is no similar
proof of Corollary~\ref{freeeneggap}(iii). Our proof underscores the \emph{robustness}
of the variational approach. We expect the gap to be present also when $m_\rho=\infty$
and $\bar h=\bar h_*(\hat\beta,\hat h,\bar\beta)$, but this remains open.
\medskip\noindent
{\bf 4.} Corollary~\ref{freeeneggap}(iv) gives a natural interpretation for $\bar h_\ast
(\hat\beta,\hat h,\bar\beta)$, namely, this is the critical value below which the pinning interaction
has an effect in the annealed model and above which it has not.
\medskip\noindent
{\bf 5.}
The precise shape of the quenched critical curve for the combined model was not well
understood (see e.g.\ Giacomin~\cite{Gi07}, Section 6.3.2, and Caravenna, Giacomin and
Toninelli~\cite{CaGiTo10}, last paragraph of Section 1.5). In particular, in \cite{Gi07}
two possible shapes were suggested for $\bar\beta=0$, as shown in Fig.~\ref{fig-giaco}.
Corollary~\ref{hquehannvarfor} rules out line 2, while it proves line 1 in the following
sense: (1) this line holds for all $\bar\beta\geq0$; (2) for $\bar h<\bar h^{\rm que}_c(\bar\beta)-\log 2$,
the combined model is \emph{fully localized}; (3) conditionally on $s^\ast(\hat\beta,\bar\beta,
\alpha)<\infty$, for $\bar h \geq s^\ast(\hat\beta,\bar\beta,\alpha)$ the quenched critical curve concides
with $\hat h_c^{\rm ann}(\hat\beta/\alpha)$ (see Fig.~\ref{fig-copadvarhc}). In the literature
$\hat h\mapsto \hat h_c^{\rm ann}(\hat\beta/\alpha)$ is called the \emph{Monthus-line}. Thus, when we sit at
the far ends of the $\bar h$-axis, the critical behavior of the quenched combined model is
determined either by the copolymer interaction (on the far right) or by the pinning
interaction (on the far left). Only in between is there a non-trivial competition between
the two interactions.
\begin{figure}[htbp]
\vspace{.5cm}
\begin{center}
\setlength{\unitlength}{.68cm}
\begin{picture}(14,7)(0,0)
\put(0,0){\line(14,0){14}}
\put(7.2,0){\line(0,7){7}}
{\thicklines
\qbezier(-.61,6.8)(1.8,2.6)(14,1.7)
\qbezier(-.61,5.2)(3,2.9)(5,2.6)
\qbezier(5,2.6)(12,2.6)(14,2.6)
}
\qbezier[25](7.2,1.5)(12,1.5)(14,1.5)
\put(6,7.5){$h^\mathrm{que}_c(\hat\beta,0,\bar h)$}
\put(7.3,2.8){$\hat h^\mathrm{que}_c(\hat\beta)$}
\put(4.5,1.3){$\hat h^\mathrm{ann}_c(\hat\beta/\alpha)$}
\put(14.5,-.3){$\bar h$}
\put(-1,6.6){$1$}
\put(-1,5){$2$}
\end{picture}
\end{center}
\vspace{-0.4cm}
\caption{\small Possible qualitative pictures of the map $\bar h \mapsto h^\mathrm{que}_c
(\hat\beta,0,\bar h)$ for $\hat\beta>0$.}
\label{fig-giaco}
\end{figure}
\medskip\noindent
{\bf 6.}
The threshold values $\bar h=\bar h^{\rm que}(\bar\beta)-\log 2$ and $\bar h=\bar h^{\rm ann}(\bar\beta)-\log 2$
(see Figs.~\ref{fig-copadvarhc}--\ref{fig-copadvarhannc}) are the critical points for
the quenched and the annealed pinning model when the polymer is allowed to stay in the
upper halfplane only. In the literature this restricted pinning model is called the
\emph{wetting model} (see Giacomin~\cite{Gi07}, den Hollander \cite{dHo09}).
These values of $\bar h$ are the transition points at which the quenched and the annealed
critical curves of the combined model change from being finite to being infinite. Thus,
we recover the critical curves for the wetting model from those of the combined model
by putting $\hat h=\infty$.
\medskip\noindent
{\bf 7.}
It is known from the literature that the pinning model undergoes a transition between
\emph{disorder relevance} and \emph{disorder irrelevance}. In the former regime, there
is a gap between the quenched and the annealed critical curve, in the latter there is
not. (For some authors disorder relevance also incorperates a difference in behaviour
of the annealed and the quenched free energies as the critical curves are approached.)
The transition depends on $\alpha$, $\bar\beta$ and $\bar \mu$ (the pinning disorder law).
In particular, if $\alpha>
\tfrac32$, then the disorder is relevant for all $\bar\beta>0$, while if $\alpha\in (1,\tfrac32)$,
then there is a critical threshold $\bar\beta_c\in (0,\infty]$ such that the disorder is
irrelevant for $\bar\beta\leq \bar\beta_c$ and relevant for $\bar\beta>\bar\beta_c$. The transition is absent
in the copolymer model (at least when the copolymer disorder law $\hat\mu$ has finite exponential
moments): the disorder is relevant for all $\alpha>1$. However,
Corollary~\ref{hcubstrict} shows that in the combined model the transition occurs for
all $\alpha>1$, $\hat\beta>0$ and $\bar\beta\geq 0$. Indeed, the disorder is relevant for $\bar h>
\bar h^{\rm que}(\bar\beta)-\log 2$ and is irrelevant for $\bar h\leq \bar h^{\rm que}(\bar\beta)-\log 2$.
\medskip\noindent
{\bf 8.}
The quenched critical curve is bounded from below by the Monthus-line (as the critical
curve moves closer to the Monthus-line, the copolymer interaction more and more dominates
the pinning interaction). Corollary~\ref{hclbstrict} and Fig.~\ref{fig-copadvarhc} show
that the critical curve stays above the Monthus-line as long as $\bar h<s^\ast(\hat\beta,\bar\beta,
\alpha)$. If $s^\ast(\hat\beta,\bar\beta,\alpha)=\infty$, then the quenched critical curve is
everywhere above the Monthus-line (see Fig.~\ref{fig-copadvarhc}(b)). A \emph{sufficient}
condition for $s^\ast(\hat\beta,\bar\beta,\alpha)<\infty$ is
\begin{equation}
\label{neccond}
\sum_{n\in{\mathbb N}}\rho(n)^{\frac1\alpha}<\infty.
\end{equation}
We do not know whether $s^\ast(\hat\beta,\bar\beta,\alpha)<\infty$ always. For $\bar\beta=0$,
Toninelli~\cite{To07} proved that, under condition \eqref{neccond}, the quenched
critical curve coincides with the Monthus-line for $\bar h$ large enough.
\medskip\noindent
{\bf 9.}
As an anonymous referee pointed out, line 2 of Fig. \ref{fig-giaco} can be disproved
by combining results for the copolymer model proved in Bolthausen, den Hollander and Opoku
\cite{BodHoOp11} and Giacomin \cite{Gi07}, Section 6.3.2, with a fractional moment estimate.
Let us present the argument for the case $\bar\beta=0$. The key is the observation that
\begin{enumerate}
\item[(1)] $h_c^{\rm que}(\hat \beta,0,0)=\hat h_c^{\rm que}(\hat\beta)>\hat h_c^{\rm ann}
(\hat\beta/\alpha)$ (proved in \cite{BodHoOp11}).
\item[(2)] $\lim_{\bar h\rightarrow\infty} h_c^{\rm que}(\hat \beta,0,\bar h)
=\hat h_c^{\rm ann}(\hat\beta/\alpha).$
\end{enumerate}
The proof for (2) goes as follows:
Note from Giacomin \cite{Gi07}, Section 6.3.2, that $h_c^{\rm que}(\hat \beta,0,\bar h)
\geq\hat h_c^{\rm ann}(\hat\beta/\alpha)$. The reverse of this inequality as $\bar h\rightarrow\infty$
follows from the fractional moment estimate
\begin{equation}\label{simproof}
\begin{aligned}
{\mathbb E}\left[\left(Z_n^{\hat\beta,\hat h,0,\bar h,\omega}\right)^\gamma\right]
& \leq \sum_{N=1}^n\sum_{0=k_0<k_1<\ldots<k_N=n}\prod_{i=1}^N\rho(k_i-k_{i-1})^\gamma e^{-\bar h\gamma}
2^{1-\gamma},
\end{aligned}
\end{equation}
valid for $\hat h=\hat h^{\rm ann}_c(\gamma\hat\beta)$ and $\gamma\in(0,1)$, for the combined
partition sum where the path starts and ends at the interface.
Indeed, for any $\gamma>\frac1\alpha$, if $\bar h>0$ is large enough so that
$\sum_{n\in{\mathbb N}}\rho(n)^\gamma e^{-h\gamma}
2^{1-\gamma}\leq 1$, then the right-hand side is the partition function for a homogeneous
pinning model with a defective excursion length distribution and therefore has zero free
energy (see e.g. \cite{Gi07}, Section 2.2). Hence $h_c^{\rm que}(\hat \beta,0,\bar h)
\leq\hat h_c^{\rm ann}(\gamma\hat\beta)$. Let $\bar h\rightarrow \infty$ followed by
$\gamma\downarrow\frac1\alpha$ to get (2). But (1) and (2) together with convexity of
$\bar h\mapsto h_c^{\rm que}(\hat \beta,0,\bar h)$ disprove line 2 of Fig. \ref{fig-giaco}.
Although the above fractional moment estimate extends to the case $\bar\beta>0$, it is not
clear to us how the rare stretch strategy used in \cite{Gi07}, Section 6.3.2, can be
used to arrive at the lower bound $h^{\rm que}_c(\hat\beta,\bar\beta,\bar h)\geq \hat
h^{\rm ann}_c(\hat\beta/\alpha)$, for the case $\bar\beta>0$, since the polymer may hit pinning
disorder with very large absolute value upon visiting or exiting a rare stretch in the copolymer
disorder making the pinning contribution to the energy of order greater than $O(1)$.
Moreover, it is not clear to us how to arrive at the lower bound $\,h^{\rm que}_c(\hat\beta,
\bar\beta,\bar h)\geq \hat h^{\rm que}_c(\hat\beta),$ for the case $\bar\beta>0$,
based on the argument in \cite{BodHoOp11} that gave rise to the inequality in (1).
\medskip\noindent
{\bf 10.}
Corollary~\ref{critqueann}(i) shows that the critical curve for the annealed
combined model taken at the $\bar h$-value where the annealed copolymer model is
critical coincides with the annealed critical curve of the pinning model, and
vice versa. For the quenched combined model a similar result is expected, but
this remains open. One of the questions that was posed in Giacomin \cite{Gi07},
Section 6.3.2, for the quenched combined model is whether an arbitrary small
pinning bias $-\bar h>0$ can lead to localization for $\bar\beta=0$, $\hat\beta>0$ and $\hat h
=\hat h^{\rm que}_c(\hat\beta)$. This question is answered in the affirmative by
Corollary~\ref{critqueann}(ii).
\medskip\noindent
{\bf 11.}
Giacomin and Toninelli~\cite{GiTo06} showed that in ${\mathcal L}^{\rm que}$ the longest
excursion under the quenched path measure $P_n^{\hat\beta,\hat h,\bar\beta,\bar h,\omega}$ is of order
$\log n$. No information was obtained about the path behavior in ${\mathcal D}^{\rm que}$.
Corollary~\ref{delocpathprop} says that in ${\mathcal D}^{\rm que}$ (which is the region
on or above the critial curve in Fig.~\ref{fig-copadvarhc}), with the exception
of the piece of the critical curve over the interval $(-\infty,s_*(\hat\beta,\bar\beta,\alpha))$,
the total number of visits to the interface up to time $n$ is at most of order $\log n$.
On this piece, the number may very well be of larger order. Corollary~\ref{locpathprop}
says that in ${\mathcal L}^{\rm que}$ this number is proportional to $n$, with a variational
formula for the proportionality constant. Since on the piece of the critical curve
over the interval $[s_*(\hat\beta,\bar\beta,\alpha),\infty)$ the number is of order $\log n$,
the phase transition is expected to be first order on this piece.
\medskip\noindent
{\bf 12.}
Smoothness of the free energy in the localized phase, finite-size corrections, and
a central limit theorem for the free energy can be found in \cite{GiTo06}.
P\'etr\'elis~\cite{Pert09} studies the weak interaction limit of the combined model.
\subsection{Outline}
\label{S1.7}
The present paper uses ideas from Cheliotis and den Hollander~\cite{ChdHo10} and Bolthausen,
den Hollander and Opoku~\cite{BodHoOp11}. The proof of Theorem \ref{freeenegvar} uses
large deviation principles derived in Birkner~\cite{Bi08} and Birkner, Greven and den
Hollander~\cite{BiGrdHo10}. The quenched variational formula and its proof are given in
Section \ref{S3.1}, the annealed variational formula and its proof in Section~\ref{S3.2}.
Section~\ref{S3.3} contains the proof of Theorem~\ref{freeenegvar}. The proofs of Corollaries
\ref{freeeneggap}--\ref{locpathprop} are given in Sections~\ref{Sec4}--\ref{Sec6}.
The latter require certain technical results, which are proved in
Appendices~\ref{appA}--\ref{appc}.
\section{Large Deviation Principle (LDP)}
\label{S2}
Let $E$ be a Polish space, playing the role of an alphabet, i.e., a set of \emph{letters}.
Let $\widetilde{E} = \cup_{k\in{\mathbb N}} E^k$ be the set of \emph{finite words} drawn from
$E$, which can be metrized to become a Polish space.
Fix $\nu \in {\mathcal P}(E)$, and $\rho\in{\mathcal P}({\mathbb N})$ satisfying (\ref{rhocond}). Let $X=(X_k)_{k\in{\mathbb N}}$
be i.i.d.\ $E$-valued random variables with marginal law $\nu$, and $\tau=(\tau_i)_{i\in{\mathbb N}}$
i.i.d.\ ${\mathbb N}$-valued random variables with marginal law $\rho$. Assume that $X$ and
$\tau$ are independent, and write ${\mathbb P}\otimes P^\ast$ to denote their joint law. Cut words
out of the letter sequence $X$ according to $\tau$ (see Fig.~\ref{fig-cutting}), i.e., put
\begin{equation}
\label{Tdefs}
T_0=0 \quad \mbox{ and } \quad T_i=T_{i-1}+\tau_i,\quad i\in{\mathbb N},
\end{equation}
and let
\begin{equation}
\label{eqndefYi}
Y^{(i)} = \bigl( X_{T_{i-1}+1}, X_{T_{i-1}+2},\dots, X_{T_{i}}\bigr),
\quad i \in {\mathbb N}.
\end{equation}
Under the law ${\mathbb P}\otimes P^\ast$, $Y = (Y^{(i)})_{i\in{\mathbb N}}$ is an i.i.d.\ sequence of words
with marginal distribution $q_{\rho,\nu}$ on $\widetilde{E}$ given by
\begin{equation}
\label{q0def}
\begin{aligned}
{\mathbb P}\otimes P^\ast\big(Y^{(1)}\in(dx_1,\dots,dx_n)\big)
&=q_{\rho,\bar \mu}\big((dx_1,\dots,dx_n)\big) \\[0.2cm]
& = \rho(n) \,\nu(dx_1) \times \dots \times \nu(dx_n), \qquad
n\in{\mathbb N},\,x_1,\dots,x_n\in E.
\end{aligned}
\end{equation}
\begin{figure}[htbp]
\vspace{-.7cm}
\begin{center}
\includegraphics[scale = .7]{words_with_symbols.pdf}
\vspace{-17.1cm}
\caption{\small Cutting words out of a sequence of letters according to renewal times.}
\label{fig-cutting}
\end{center}
\vspace{-.2cm}
\end{figure}
The reverse operation of \emph{cutting} words out of a sequence of letters is
\emph{glueing} words together into a sequence of letters. Formally, this is done
by defining a \emph{concatenation} map $\kappa$ from $\widetilde{E}^{\mathbb N}$ to $E^{{\mathbb N}}$.
This map induces in a natural way a map from ${\mathcal P}(\widetilde{E}^{\mathbb N})$ to ${\mathcal P}(E^{{\mathbb N}})$,
the sets of probability measures on $\widetilde{E}^{\mathbb N}$ and $E^{{\mathbb N}}$ (endowed with
the topology of weak convergence). The concatenation $q_{\rho,\nu}^{\otimes{\mathbb N}}\circ
\kappa^{-1}$ of $q_{\rho,\nu}^{\otimes{\mathbb N}}$ equals $\nu^{\otimes{\mathbb N}}$, as is evident from
(\ref{q0def}).
\subsection{Annealed LDP}
\label{S2.1}
Let ${\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$ be the set of probability measures
on $\widetilde{E}^{\mathbb N}$ that are invariant under the left-shift $\widetilde{\theta}$
acting on $\widetilde{E}^{\mathbb N}$. For $N\in{\mathbb N}$, let $(Y^{(1)},\dots,Y^{(N)})^\mathrm{per}$
be the periodic extension of the $N$-tuple $(Y^{(1)},\dots,Y^{(N)})\in\widetilde{E}^N$
to an element of $\widetilde{E}^{\mathbb N}$. The \emph{empirical process of $N$-tuples of words} is
defined as
\begin{equation}
\label{eqndefRN}
R_N^X = \frac{1}{N} \sum_{i=0}^{N-1}
\delta_{\widetilde{\theta}^i (Y^{(1)},\dots,Y^{(N)})^\mathrm{per}}
\in \mathcal{P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N}),
\end{equation}
where the supercript $X$ indicates that the words $Y^{(1)},\dots,Y^{(N)}$ are cut from
the latter sequence $X$. For $Q\in{\mathcal P}^\mathrm{inv}(\widetilde{E}^{\mathbb N})$, let $H(Q \mid
q_{\rho,\nu}^{\otimes{\mathbb N}})$ be the \emph{specific relative entropy of $Q$ w.r.t.\
$q_{\rho,\nu}^{\otimes{\mathbb N}}$} defined by
\begin{equation}
\label{spentrdef}
H(Q \mid q_{\rho,\nu}^{\otimes{\mathbb N}}) = \lim_{N\rightarrow\infty} \frac{1}{N}\,
h(\pi_N Q \mid q_{\rho,\nu}^N),
\end{equation}
where $\pi_N Q \in {\mathcal P}(\widetilde{E}^N)$ denotes the projection of $Q$ onto the first
$N$ words, $h(\,\cdot\mid\cdot\,)$ denotes relative entropy, and the limit is
non-decreasing.
For the applications below we will need the following tilted version of $\rho$:
\begin{equation}
\label{rhoz}
\rho_g(n)=\frac{e^{-gn}~\rho(n)}{{\mathcal N}(g)} \qquad \text{with}
\qquad {\mathcal N}(g)=\sum_{n\in{\mathbb N}} e^{-gn}\,\rho(n), \quad g\geq 0.
\end{equation}
Note that, for $g>0$, $\rho_g$ has a tail that is exponentially bounded. The following
result relates the relative entropies with $q_{\rho_g,\nu}^{\otimes{\mathbb N}}$ and
$q_{\rho,\nu}^{\otimes{\mathbb N}}$ as reference measures.
\begin{lemma} {\rm \cite{BodHoOp11}}
\label{rhozre}
For $Q\in{\mathcal P}^\mathrm{inv}(\widetilde{E}^{\mathbb N})$ and $g\geq0$,
\begin{equation}
\label{rhozrho}
H(Q\mid q_{\rho_g,\nu}^{\otimes{\mathbb N}})
= H(Q\mid q_{\rho,\nu}^{\otimes{\mathbb N}})+\log {\mathcal N}(g)+g~{\mathbb E}_Q(\tau_1).
\end{equation}
\end{lemma}
\noindent
This result shows that, for $g\geq0$, $m_Q={\mathbb E}_Q(\tau_1)<\infty$ whenever $H(Q\mid
q_{\rho_g,\nu}^{\otimes{\mathbb N}})<\infty$, which is a special case of \cite{Bi08}, Lemma 7.
The following \emph{annealed LDP} is standard (see e.g.\ Dembo and Zeitouni~\cite{DeZe98},
Section 6.5).
\begin{theorem}
\label{aLDP}
For every $g \geq 0$, the family $({\mathbb P}\otimes P^\ast_g)(R_N^\cdot \in \cdot\,)$,
$N\in{\mathbb N}$, satisfies the LDP on ${\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$ with rate $N$ and
with rate function $I_g^{\mathrm{ann}}$ given by
\begin{equation}
\label{Ianndef}
I_g^{\mathrm{ann}}(Q)= H\big(Q \mid q_{\rho_g,\nu}^{\otimes{\mathbb N}}\big),
\qquad Q \in {\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N}).
\end{equation}
This rate function is lower semi-continuous, has compact level sets, has a unique
zero at $q_{\rho_g,\nu}^{\otimes{\mathbb N}}$, and is affine.
\end{theorem}
It follows from Lemma~\ref{rhozre} that
\begin{equation}\label{Iannz}
I_g^{\rm ann}(Q)=I^{\rm ann}(Q)+\log{\mathcal N}(g)+gm_Q,
\end{equation}
where $I^{\rm ann}(Q)=H(Q\mid q_{\rho,\nu}^{\otimes {\mathbb N}})$, the annealed rate function
for $g=0$.
\subsection{Quenched LDP}
\label{S2.2}
To formulate the quenched analogue of Theorem~\ref{aLDP}, we need some more notation.
Let ${\mathcal P}^{\mathrm{inv}}(E^{{\mathbb N}})$ be the set of probability measures on $E^{{\mathbb N}}$
that are invariant under the left-shift $\theta$ acting on $E^{{\mathbb N}}$. For $Q\in
{\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$ such that $m_Q < \infty$, define
\begin{equation}
\label{PsiQdef}
\Psi_Q = \frac{1}{m_Q} E_Q\left(\sum_{k=0}^{\tau_1-1}
\delta_{\theta^k\kappa(Y)}\right) \in {\mathcal P}^{\mathrm{inv}}(E^{{\mathbb N}}).
\end{equation}
Think of $\Psi_Q$ as the shift-invariant version of $Q\circ\kappa^{-1}$ obtained
after \emph{randomizing} the location of the origin. This randomization is necessary
because a shift-invariant $Q$ in general does not give rise to a shift-invariant
$Q\circ\kappa^{-1}$.
For ${\rm tr} \in {\mathbb N}$, let $[\cdot]_{\rm tr} \colon\,\widetilde{E} \to [\widetilde{E}]_{\rm tr}
= \cup_{n=1}^{\rm tr} E^n$ denote the \emph{truncation map} on words defined by
\begin{equation}
\label{trunword}
y = (x_1,\dots,x_n) \mapsto [y]_{\rm tr} = (x_1,\dots,x_{n \wedge {\rm tr}}),
\qquad n\in{\mathbb N},\,x_1,\dots,x_n\in E,
\end{equation}
i.e., $[y]_{\rm tr}$ is the word of length $\leq{\rm tr}$ obtained from the word $y$ by dropping
all the letters with label $>{\rm tr}$. This map induces in a natural way a map from
$\widetilde E^{\mathbb N}$ to $[\widetilde{E}]_{\rm tr}^{\mathbb N}$, and from ${\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$
to ${\mathcal P}^{\mathrm{inv}}([\widetilde{E}]_{\rm tr}^{\mathbb N})$. Note that if $Q\in{\mathcal P}^{\mathrm{inv}}
(\widetilde{E}^{\mathbb N})$, then $[Q]_{\rm tr}$ is an element of the set
\begin{equation}
\label{Pfin}
{\mathcal P}^{\mathrm{inv,fin}}(\widetilde{E}^{\mathbb N}) = \{Q\in{\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})
\colon\,m_Q<\infty\}.
\end{equation}
Define (w-lim means weak limit)
\begin{equation}
\label{Rdef}
{\mathcal R} = \left\{Q\in{\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})\colon\,
{\rm w}-\lim_{N\to\infty} \frac{1}{N}
\sum_{k=0}^{N-1} \delta_{\theta^k\kappa(Y)}=\nu^{\otimes{\mathbb N}} \quad Q-a.s.\right\},
\end{equation}
i.e., the set of probability measures in ${\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$ under
which the concatenation of words almost surely has the same asymptotic statistics as
a typical realization of $X$.
\begin{theorem}
\label{qLDP}
{\rm (Birkner~\cite{Bi08}; Birkner, Greven and den Hollander~\cite{BiGrdHo10})}
Assume {\rm (\ref{rhocond}--\ref{mgffin})}. Then, for $\nu^{\otimes {\mathbb N}}$--a.s.\
all $X$ and all $g \in [0,\infty)$, the family of (regular) conditional probability
distributions $P^\ast_g(R_N^X \in \cdot \mid X)$, $N\in{\mathbb N}$, satisfies the LDP on
${\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$ with rate $N$ and with deterministic rate
function $I^{\mathrm{que}}_g$ given by
\begin{equation}
\label{eqgndefinitionIalgz}
I^\mathrm{que}_g(Q) = \left\{\begin{array}{ll}
I^{\rm ann}_g(Q),
&\mbox{if } Q\in{\mathcal R},\\
\infty,
&\mbox{otherwise},
\end{array}
\right.
\text{ when } g>0,
\end{equation}
and
\begin{equation}
\label{eqgndefinitionIalg}
I^\mathrm{que}(Q) = \left\{\begin{array}{ll}
I^\mathrm{fin}(Q),
&\mbox{if } Q\in{\mathcal P}^{\mathrm{inv,fin}}(\widetilde{E}^{\mathbb N}),\\
\lim_{{\rm tr} \to \infty} I^\mathrm{fin}\big([Q]_{\rm tr}\big),
&\mbox{otherwise},
\end{array}
\right.
\text{ when } g=0,
\end{equation}
where
\begin{equation}
\label{eqnratefctexplicitalg}
I^\mathrm{fin}(Q) = H(Q \mid q_{\rho,\nu}^{\otimes{\mathbb N}})
+ (\alpha-1) \, m_Q \,H\big(\Psi_{Q} \mid \nu^{\otimes{\mathbb N}}\big).
\end{equation}
This rate function is lower semi-continuous, has compact level sets, has a unique zero
at $q_{\rho_g,\nu}^{\otimes{\mathbb N}}$, and is affine.
\end{theorem}
It was shown in \cite{Bi08}, Lemma 2, that
\begin{equation}
\label{Requiv}
\Psi_{Q} =\nu^{\otimes{\mathbb N}} \quad \Longleftrightarrow \quad
Q\in{\mathcal R} \qquad \mbox{ on } {\mathcal P}^{\mathrm{inv,fin}}(\widetilde{E}^{\mathbb N}),
\end{equation}
which explains why the restriction $Q\in{\mathcal R}$ appears in
\eqref{eqgndefinitionIalgz}. For more background, see \cite{BiGrdHo10}.
Note that $I^\mathrm{que}(Q)$ requires a truncation approximation when
$m_Q=\infty$, for which case there is no closed form expression like in
\eqref{eqnratefctexplicitalg}. As we will see later on, the cases
$m_Q<\infty$ and $m_Q=\infty$ need to be separated. For later reference
we remark that, for all $Q\in{\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})$,
\begin{equation}
\label{truncapproxcont}
\begin{aligned}
I^\mathrm{ann}(Q)
&= \lim_{{\rm tr} \to \infty} I^\mathrm{ann}([Q]_{\rm tr}),\\
I^\mathrm{que}(Q)
&= \lim_{{\rm tr} \to \infty} I^\mathrm{que}([Q]_{\rm tr}),
\end{aligned}
\end{equation}
as shown in \cite{BiGrdHo10}, Lemma A.1.
\section{Variational formulas for excess free energies}
\label{S3}
This section uses the LDP of Section~\ref{S2} to derive variational formulas for the
excess free energy of the quenched and the annealed version of the combined model.
The quenched version is treated in Section~\ref{S3.1}, the annealed version
in Section~\ref{S3.2}. The results in Sections~\ref{S3.1}--\ref{S3.2} are used
in Section~\ref{S3.3} to prove Theorem~\ref{freeenegvar}.
In the combined model words are made up of letters from the alphabet $E=\hat E\times\bar E$,
where $\hat E$ and $\bar E$ are subsets of ${\mathbb R}$, and are cut from the letter sequence $\omega
=((\hat \omega_i,\bar \omega_i))_{i\in{\mathbb N}}$, where $\hat \omega=(\hat \omega_i)_{i\in{\mathbb N}}$ and $\bar \omega=(\bar \omega_i)_{i\in{\mathbb N}}$
are i.i.d.\ sequences of $\hat E$-valued and $\bar E$-valued random variables with joint
common law $\nu=\hat \mu\otimes\bar \mu$. Let $\hat\pi$ and $\bar\pi$ be the projection maps
from $E$ onto $\hat E$ and $\bar E$, respectively, i.e $\hat\pi((\hat \omega_1,\bar \omega_1))=\hat \omega_1$ and
$\bar\pi((\hat \omega_1,\bar \omega_1))=\bar \omega_1$ for $(\hat \omega_1,\bar \omega_1)\in E$. These maps extend naturally
to $E^{\mathbb N}$, $\widetilde E$, $\widetilde E^{\mathbb N}$, ${\mathcal P}\big(\widetilde E\big)$ and ${\mathcal P}\big(\widetilde E^{\mathbb N}\big)$. For instance,
if $\xi\in E^{\mathbb N}$, i.e., $\xi=((\hat \omega_i,\bar \omega_i))_{i\in{\mathbb N}}$, then $\hat\pi\xi=\hat \omega=(\hat \omega_i)_{i
\in{\mathbb N}}$ and $\bar\pi\xi=\bar \omega=(\bar \omega_i)_{i\in{\mathbb N}}$.
As before, we will write $k$, $\hat k$ and $\bar k$ for a quantity $k$ associated with
the copolymer with pinning model, the copolymer model, respectively, the pinning model.
For instance, if $Q\in{\mathcal P}^{\rm inv}\big(\widetilde E^{\mathbb N}\big)$, $\hat{Q}\in{\mathcal P}^{\rm inv}\big(\widetilde{\hat E}^{\mathbb N}
\big)$ and $\bar Q\in{\mathcal P}^{\rm inv}\big(\widetilde{\bar E}^{\mathbb N}\big)$, then the rate functions $I^{\rm ann}
(Q)=H(Q|q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}})$, $\hat I^{\rm ann}(\hat{Q})=H(\hat{Q}|q_{\rho,
\hat \mu}^{\otimes{\mathbb N}})$, $\bar I^{\rm ann}(\bar Q)=H(\bar Q|q_{\rho,\bar \mu}^{\otimes{\mathbb N}})$ and the
sets ${\mathcal R}$, $\hat{\mathcal R}$, $\bar{\mathcal R}$ are defined as in \eqref{Rdef}.
The LDPs of the laws of the empirical processes $R^{\hat \omega}_N=\hat\pi R^{\omega}_N$ and
$R^{\bar \omega}_N=\bar\pi R^{\omega}_N$ can be derived from those of $R^{\omega}_N$ via the
contraction principle (see e.g.\ Dembo and Zeutouni~\cite{DeZe98}, Theorem 4.2.1),
because the projection maps $\hat\pi$ and $\bar\pi$ are continuous. In particular,
for any $\hat{Q}\in{\mathcal P}^{\rm inv}\big(\widetilde{\hat E}^{\mathbb N}\big)$ and $\bar Q\in{\mathcal P}^{\rm inv}\big(\widetilde{\bar E}^{\mathbb N}
\big)$
\begin{equation}
\label{conprin}
\hat I^{\rm que}(\hat{Q})
= \inf_{Q\in{\mathcal P}^{\rm inv}\big(\widetilde E^{\mathbb N}\big)\colon\,\atop\hat\pi Q=\hat{Q} } I^{\rm que}(Q),
\qquad
\bar I^{\rm que}(\bar Q)
= \inf_{Q\in{\mathcal P}^{\rm inv}\big(\widetilde E^{\mathbb N}\big)\colon\,\atop\bar\pi Q=\bar Q }
I^{\rm que}(Q),
\end{equation}
where $\hat\pi Q=Q\circ(\hat\pi)^{-1}$ and $\bar\pi Q=Q\circ(\bar\pi)^{-1}$. Similarly,
we may express $\hat I^{\rm ann}$ and $\bar I^{\rm ann}$ in terms of $I^{\rm ann}$.
\subsection{Quenched excess free energy}
\label{S3.1}
Abbreviate
\begin{equation}
\label{Csetdef}
{\mathcal C}^{\rm fin} = \left\{Q\in{\mathcal P}^\mathrm{inv}(\widetilde{E}^{\mathbb N})\colon\,
I^\mathrm{ann}(Q)<\infty, \, m_Q<\infty\right\}.
\end{equation}
\begin{theorem}
\label{varfloc}
Assume {\rm (\ref{rhocond})} and {\rm (\ref{mgffin})}. Fix $\hat\beta$, $\hat h>0$, $\bar\beta\geq0$
and $\bar h\in{\mathbb R}$.\\
(i) The quenched excess free energy is given by
\begin{equation}
\label{gvarexp}
g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h)
= \inf\left\{g\in{\mathbb R}\colon\,S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0\right\},
\end{equation}
where
\begin{equation}
\label{Sdef}
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g) = \sup_{Q\in{\mathcal C}^\mathrm{fin}\cap{\mathcal R}}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-g m_Q-I^\mathrm{ann}(Q)\right]
\end{equation}
with
\begin{eqnarray}
\label{pinPhidef}
\Phi(Q) &=& \int_{\bar E} \bar \omega_1\, (\bar\pi_{1,1}Q)(d\bar \omega_1)\\
\label{coPhidef}
\Phi_{\hat\beta,\hat h}(Q)
&=& \int_{\widetilde{\hat E}} (\hat\pi_1 Q)(d\hat \omega)\,\log \phi_{\hat\beta, \hat h}(\hat \omega),\\
\label{phidef}
\phi_{\hat\beta, \hat h}(\hat \omega)
&=& \tfrac12\left(1+\exp\left[-2\hat\beta \hat h\,\tau_1-2\hat\beta\,\sum_{k=1}^{\tau_1}
\hat \omega_k\right]\right).
\end{eqnarray}
Here, the map $\bar\pi_{1,1}\colon\,\widetilde E^{\mathbb N}\to \bar E$ is the projection onto the first
letter of the first word in the sentence consisting of words cut out from $\bar \omega$, i.e.,
$\bar\pi_{1,1}Q=Q\circ(\bar\pi_{1,1})^{-1}$, while the map $\hat\pi_1\colon\,\widetilde E^{\mathbb N}\to\widetilde{\hat E}$
is the projection onto the first word in the sentence consisting of words cut out
from $\hat \omega$, i.e., $\hat\pi_1 Q=Q\circ(\hat\pi_1)^{-1}$, and $\tau_1$ is the length
of the first word.\\
(ii) An alternative variational formula at $g=0$ is $S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)
=S_{*}^\mathrm{que}(\hat\beta,\hat h,\bar\beta)$ with
\begin{equation}
\label{Sdefalt}
S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta) = \sup_{Q\in{\mathcal C}^\mathrm{fin}}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-I^\mathrm{que}(Q)\right].
\end{equation}
(iii) The map $g \mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is lower semi-continuous,
convex and non-increasing on ${\mathbb R}$, is infinite on $(-\infty,0)$, and is finite, continuous
and strictly decreasing on $(0,\infty)$.
\end{theorem}
\begin{proof}
The proof is an adaptation of the proof of Theorem 3.1 in \cite{BodHoOp11} and comes
in 3 steps.
\medskip\noindent
{\bf 1.}
Suppose that $\pi\in\Pi_n$ has $t_n=t_n(\pi)$ excursions away from the interface. If
$k_i$ denote the times at which $\pi$ visits the interface, then the Hamiltonian reads
\begin{equation}
\label{copadHamexcsplit}
\begin{split}
H_n^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(\pi)
&=\hat\beta\sum_{k=1}^n(\hat \omega_k+\hat h)\left[{\rm sign}(\pi_{k-1},\pi_k)-1
\right]+\sum_{k=1}^{n} (\bar\beta \bar \omega_k-\bar h)1_{\{\pi_k=0\}}\cr
&= \sum_{i=1}^{t_n}\left[\bar\beta \bar \omega_{k_{i}}-\bar h-2\,\hat\beta\,1_{A^-_i}
\sum_{k\in I_i}(\hat \omega_k+\hat h) \right],
\end{split}
\end{equation}
where $A^-_i$ is the event that the $i$-th excursion is below the interface and $I_i
=(k_{i-1},k_i]\cap{\mathbb N}$. Since each excursion has equal probability to lie below or
above the interface, the $i$-th excursion contributes
\begin{equation}
\label{ithexcontri}
\phi_{\hat\beta, \hat h}(\hat \omega_{I_i})\,e^{\bar\beta \bar \omega_{k_{i}}-\bar h}
=\tfrac12\left(1+\exp\left[-2\,\hat\beta\sum_{k\in I_i}
(\hat \omega_k+\hat h)\right]\right)\,e^{\bar\beta \bar \omega_{k_{i}}-\bar h}
\end{equation}
to the partition sum $Z_{n}^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}$, where $\hat \omega_{I_i}$ is the word in
$\widetilde{\hat E}$ cut out from $\hat \omega$ by the $i$-th excursion interval $I_i$. Consequently, we have
\begin{equation}
Z_{n}^{\hat\beta, \hat h,\bar\beta,\bar h,\omega} = \sum_{N\in{\mathbb N}}\,\,\sum_{0=k_0<k_1<\cdots<k_N=n}\,\,
\prod_{i=1}^N \rho(k_i-k_{i-1})\,e^{(\bar\beta \bar \omega_{k_{i}}-\bar h)}
e^{\log \phi_{\hat\beta, \hat h}(\hat \omega_{I_i})}.
\end{equation}
Therefore, summing over $n$, we get
\begin{equation}
\label{copadZFNrel}
\sum_{n\in{\mathbb N}} Z_{n}^{\hat\beta, \hat h,\bar\beta,\bar h,\omega} e^{-gn}
= \sum_{N\in{\mathbb N}} F_N^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(g),
\qquad g\geq0,
\end{equation}
with
\begin{equation}
\label{FNdef}
\begin{aligned}
F_N^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(g)&=
e^{-N\bar h}
\sum_{0=k_0<k_1<\cdots<k_N<\infty}\,\,
\left(\prod_{i=1}^N \rho(k_i-k_{i-1})\right)\,e^{-g(k_i-k_{i-1})}\\
&\qquad\qquad\times
\exp\left[\sum_{i=1}^N \left( \log \phi_{\hat\beta, \hat h}(\hat \omega_{I_i})
+\bar\beta~\bar \omega_{k_{i}}\right)\right]\\
&= \left(\mathcal{N}(g)\,e^{-\bar h}\right)^N
\sum_{0=k_0<k_1<\cdots<k_N<\infty}\,\,
\left(\prod_{i=1}^N \rho_g(k_i-k_{i-1})\right)\\
&\qquad\qquad\times
\exp\left[\sum_{i=1}^N \left( \log \phi_{\hat\beta, \hat h}(\hat \omega_{I_i})
+\bar\beta~\bar \omega_{k_{i}}\right)\right]\\
&= \left(\mathcal{N}(g)\,e^{-\bar h}\right)^N
\,E^*_g\left(\exp\left [N\left(\Phi_{\hat\beta, \hat h}(R_N^{\omega})+\bar\beta\,
\Phi(R_N^{\omega})\right)\right]\right),
\end{aligned}
\end{equation}
where
\begin{equation}
\label{copadempprocomega}
R_N^{\omega}\big((k_i)_{i=0}^N\big) = \frac{1}{N} \sum_{i=1}^N
\delta_{\widetilde{\theta}^i(\omega_{I_1},\ldots,\omega_{I_N})^{\mathrm{per}}}
\end{equation}
denotes the \emph{empirical process of $N$-tuples of words} cut out from $\omega$ by
the $N$ successive excursions, and $\Phi_{\hat\beta,\hat h},\Phi$ are defined in
(\ref{pinPhidef}--\ref{phidef}).
\medskip\noindent
{\bf 2.}
The left-hand side of (\ref{copadZFNrel}) is a power series with radius of convergence
$g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)$ (recall \eqref{copadefe}). Define
\begin{equation}
\label{Slimdef}
\begin{aligned}
s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)
&= \log \mathcal{N}(g) + \limsup_{N\to\infty} \frac{1}{N} \log
E^*_g\left(\exp\left [N\left(\Phi_{\hat\beta, \hat h}(R_N^{\omega})+\bar\beta\,
\Phi(R_N^{\omega})\right)\right]\right)
\end{aligned}
\end{equation}
and note that the limsup exists and is constant (possibly infinity) $\omega$-a.s.\
because it is measurable w.r.t.\ the tail sigma-algebra of $\omega$ (which is trivial).
Note from \eqref{FNdef} and \eqref{Slimdef} that
\begin{equation}
\label{copadSlim}
\begin{aligned}
s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)-\bar h
&= \limsup_{N\to\infty} \frac{1}{N} \log F_N^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(g).
\end{aligned}
\end{equation}
By \eqref{copadefe}, the left-hand side of (\ref{copadZFNrel}) is a power series that
converges for $g>g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h)$ and diverges for $g<g^\mathrm{que}
(\hat\beta,\hat h,\bar\beta,\bar h)$. Further, it follows from the first equality of \eqref{FNdef} and
\eqref{copadSlim}, that the map $g\mapsto s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is non-increasing.
In particular, it is strictly decreasing when finite. This we show in Step 3 below.
Hence we have
\begin{equation}
\label{roc}
g^\mathrm{que}(\hat\beta,\hat h,\bar\beta,\bar h)
= \inf\left\{g\in{\mathbb R}\colon\,s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0\right\}.
\end{equation}
\medskip\noindent
{\bf 3.}
We claim that, for any $\hat\beta, \hat h>0$ and $\bar\beta\geq0$, the map $g\mapsto \bar
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is finite on $(0,\infty)$ and infinite on
$(-\infty,0)$ (see Fig.~\ref{fig-varfe}), and
\begin{equation}
\label{seqsbar}
s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)= S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)\quad \forall\, g\in{\mathbb R}.
\end{equation}
Note from the contraction principle in \eqref{conprin} that $\hat I^{\rm ann}(\hat\pi Q)$
and $\bar I^{\rm ann}(\bar\pi Q)$ are finite whenever $I^{\rm ann}(Q)<\infty$. Therefore,
for any $\hat\beta>0$, $\bar\beta\geq0$ and $\hat h>0$, it follows from Lemmas~\ref{pinphifin} and
\ref{finitephicop} in Appendix~\ref{appA} that $\bar\beta\Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)<\infty$
whenever $I^{\rm ann}(Q)<\infty$. This implies that the map $g\mapsto S^\mathrm{que}
(\hat\beta,\hat h,\bar\beta;g)$ is convex and lower semi-continuous, since, by \eqref{Sdef},
$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is the supremum of a family of functions that are
finite and linear (and hence continuous) in $g$. Now the fact that $g\mapsto
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is strictly
decreasing when finite follows as follows: Suppose that $g_1<g_2$, and
$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g_1)<\infty$. Then it follows from the fact $m_Q\geq 1$
and \eqref{Sdef} that
\begin{equation}
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g_2)- S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g_1)\leq-(g_2-g_1)<0.
\end{equation}
Further, we will show in Section \ref{Sec6.1} that $s^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)<\infty$,
for all $g>0$. This and convexity imply continuity on $(0,\infty).$ These prove
(iii) of the theorem.
The rest of the proof follows from the claim in \eqref{seqsbar}, whose
proof we defer to Appendix~\ref{appB}.
\end{proof}
\begin{figure}[htbp]
\vspace{1.5cm}
\begin{minipage}[hbt]{6cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(-2,0){\line(10,0){10}}
\put(0,-6){\line(0,10){14}}
{\thicklines
\qbezier(2,-2.5)(0,1.5)(0,4)
\qbezier(-3,7)(-1.5,7)(-.2,7)
}
\put(8.5,-0.2){$g$}
\put(-2,9.5){$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$}
\put(-2,7.5){$\infty$}
\put(0,4){\circle*{.5}}
\put(0,7){\circle{.5}}
\put(-3,-.2){$\bar h$}
\end{picture}
\vspace{1.2cm}
\begin{center}
\qquad (1) $\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})<\hat h < h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$
\end{center}
\end{minipage}
\begin{minipage}[hbtb]{4.3cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(-2,0){\line(8,0){10.5}}
\put(0,-6){\line(0,12){14}}
{\thicklines
\qbezier(2.5,-4)(1,-2.5)(0,0)
\qbezier(-3,7)(-1.5,7)(-.2,7)
}
\put(9,-0.2){$g$}
\put(-2,9.5){$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$}
\put(-2,7.5){$\infty$}
\put(0,0){\circle*{.5}}
\put(0,7){\circle{.5}}
\put(-3,-.2){$\bar h$}
\end{picture}
\vspace{1.2cm}
\begin{flushright}
\qquad (2) $\hat h = h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$
\end{flushright}
\end{minipage}
\begin{minipage}[hbt]{4.3cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(-2,0){\line(8,0){10}}
\put(0,-6){\line(0,14){14}}
{\thicklines
\qbezier(2.5,-6.5)(1,-5.5)(0,-2)
\qbezier(-3,7)(-1.5,7)(-.2,7)
}
\put(8.5,-0.2){$g$}
\put(-2,9.5){$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$}
\put(-2,7.5){$\infty$}
\put(0,-2){\circle*{.5}}
\put(0,7){\circle{.5}}
\put(-3,-.2){$\bar h$}
\end{picture}
\vspace{1.2cm}
\begin{flushright}
\qquad (3) $\hat h > h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$
\end{flushright}
\end{minipage}
\vspace{-.4cm}
\begin{center}
\caption{\small Qualitative picture of the map $g \mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$
for $\hat\beta,\hat h>0$, $\bar\beta\geq0$. The horizontal axis is located at the point $\bar h\leq
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$ on
the vertical axis. The map $g\mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is strictly decreasing when finite. For
$\hat h = h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$ and $\bar h> S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0),$
$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ jumps at $g=0$ from a value below $\bar h$ to
infinity. Since the map $g\mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ is lower semi-continuous
and decreasing, it follows that $\lim_{g\downarrow 0}S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)=S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$,
which is finite if $\hat h>\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})$ and infinite if
$\hat h<\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})$ (see Lemma 5.2).
For $\hat h<\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})$, $g \mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)$ has the shape
as in (1) but tends to infinity as $g\downarrow0$ . For $\hat h=\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})$ the picture
is the same as for $\hat h<\hat h_c^{\rm ann}(\frac{\hat\beta}{\alpha})$ if $s^\ast(\hat\beta,\hat h,\alpha)=\infty$, and takes the
shape in (1) if $s^\ast(\hat\beta,\hat h,\alpha)<\infty$.}
\label{fig-varfe}
\end{center}
\vspace{-.5cm}
\end{figure}
Analogues of Theorem~\ref{varfloc} also hold for the copolymer model and the pinning
model. The copolymer analogue is obtained by putting $\bar\beta=\bar h=0$, which leads to
analogous variational formulas for $\hat S^{\rm que}(\hat\beta,\hat h;g)$ and $\hat g^{\rm que}
(\hat\beta,\hat h)$. In the variational formula for $\hat S^{\rm que}(\hat\beta,\hat h;g)$ we replace
${\mathcal C}^{\rm fin}\cap{\mathcal R}$ by $\hat{\mathcal C}^{\rm fin}\cap\hat{\mathcal R}$ in \eqref{Sdef}. This
replacement is a consequence of the contraction principle in \eqref{conprin}. Although
the contraction principle holds on ${\mathcal P}^{\rm inv}(\widetilde E^{\mathbb N})$, it turns out that the $Q\notin
{\mathcal C}^{\rm fin}\cap{\mathcal R}$ play no role in \eqref{Sdef}. Similarly, Theorem~\ref{varfloc}
reduces to the pinning model upon putting $\hat\beta=\hat h=0$. The variational formula for
$\bar S^{\rm que}(\bar\beta;g)$ is the same as that in \eqref{Sdef}, with ${\mathcal C}^{\rm fin}
\cap{\mathcal R}$ replaced by $\bar{\mathcal C}^{\rm fin}\cap\bar{\mathcal R}$.
\subsection{Annealed excess free energy}
\label{S3.2}
We next present the variational formula for the annealed excess free energy. This will
serve as an \emph{object of comparison} in our study of the quenched model. Define
\begin{equation}
\label{cndef}
{\mathcal N}(\hat\beta, \hat h,\bar\beta;g) = \tfrac12e^{ \bar M(-\bar\beta)}\left
(\sum_{n\in{\mathbb N}} \rho(n)\,e^{-ng}+\sum_{n\in{\mathbb N}}\rho(n)\,
e^{-n\left(g-[\hat M(2\,\hat\beta)-2\,\hat\beta\hat h]\right)}\right)
\end{equation}
(recall \eqref{mgffin}).
\begin{theorem}
\label{varflocann}
Assume {\rm (\ref{rhocond})} and {\rm (\ref{mgffin})}. Fix $\hat\beta,\hat h,\bar\beta\geq0$
and $\bar h\in{\mathbb R}$.\\
(i) The annealed excess free energy is given by
\begin{equation}
\label{ganvarexp}
g^\mathrm{ann}(\hat\beta,\hat h,\bar\beta,\bar h)
= \inf\left\{g\in{\mathbb R}\colon\,S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0\right\},
\end{equation}
where
\begin{equation}
\label{Sandef}
S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)
= \sup_{Q\in{\mathcal C}^\mathrm{fin}}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-g m_Q-I^\mathrm{ann}(Q)\right].
\end{equation}
(ii) The map $g \mapsto S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)$ is lower semi-continuous,
convex and non-increasing on ${\mathbb R}$. Furthermore, it is infinite on $(-\infty,
\hat{g}^{\rm ann}(\hat\beta,\hat h))$, and finite, continuous and strictly decreasing
on $[\hat{g}^{\rm ann}(\hat\beta,\hat h),\infty)$ (recall \eqref{copannfeg}).
\end{theorem}
\begin{proof}
The proof comes in 3 steps.
\medskip\noindent
Replacing $Z^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}_n$ by $ Z^{\hat\beta, \hat h,\bar\beta,\bar h}_n={\mathbb E}(Z^{\hat\beta, \hat h,\bar\beta,
\bar h,\omega}_n)$ in \eqref{copadZFNrel}, we obtain from \eqref{FNdef} that
\begin{equation}
\label{FNdefann}
F_N^{\hat\beta, \hat h,\bar\beta,\bar h}(g) = {\mathbb E}\left(F_N^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(g)\right)
= {\mathcal N}(\hat\beta, \hat h,\bar\beta;g)^N e^{-\bar h N}.
\end{equation}
It therefore follows from \eqref{copadSlim} and \eqref{FNdefann} that
\begin{equation}
\begin{split}
\label{copadsa1}
s^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h
&=\limsup_{N\rightarrow\infty}\frac1N\log F_N^{\hat\beta, \hat h,\bar\beta,\bar h}(g)
=\log {\mathcal N}(\hat\beta, \hat h,\bar\beta;g)-\bar h,
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}\label{copadsdef}
s^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)
=\limsup_{N\rightarrow\infty}\frac1N\log \left(e^{N\bar h}F_N^{\hat\beta, \hat h,\bar\beta,\bar h}(g)\right)
=\log {\mathcal N}(\hat\beta, \hat h,\bar\beta;g).
\end{split}
\end{equation}
Note from \eqref{cndef} and \eqref{copadsdef} that the map $g\mapsto s^{\rm ann}(\hat\beta,\hat h,
\bar\beta;g)$ is non-increasing. Moreover, for any $\hat\beta,\hat h,\bar\beta\geq 0$ and $\bar h\in{\mathbb R}$, we see
from \eqref{copadZFNrel} after replacing $ Z^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}_n$ by $ Z^{\hat\beta,\hat h,\bar\beta,
\bar h}_n$ that $g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)$ is the smallest $g$-value at which $s^{\rm ann}
(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign, i.e.,
\begin{equation}
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)
=\inf\left\{g\in{\mathbb R}\colon s^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h<0 \right\}.
\end{equation}
The proof of (i) and (ii) will follow once we show that
\begin{equation}
\label{anneqvfm}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)=s^{\rm ann}(\hat\beta,\hat h,\bar\beta;g) \quad \forall\,g\in{\mathbb R},
\end{equation}
since \eqref{cndef}, \eqref{copadsdef} and \eqref{anneqvfm} show that the map $g\mapsto
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)$ is infinite whenever $g<\hat{g}^{\rm\, ann}(\hat\beta,\hat h) =
0\vee[M(2\hat\beta)-2\hat\beta \hat h]$, and is finite otherwise. Lower semi-continuity and convexity
of this map follow from \eqref{Sandef}, because the function under the supremum is
linear and finite in $g$, while convexity and finiteness imply continuity. The proof
of \eqref{anneqvfm} follows from the arguments in \cite{BodHoOp11}, Theorem 3.2, as we
show in steps 2--3.
\medskip\noindent
{\bf 2.}
For the case $g<\hat{g}^{\rm ann}(\hat\beta,\hat h)$, note from \eqref{cndef} that ${\mathcal N}(\hat\beta,
\hat h,\bar\beta;g)=\infty$ for all $\hat\beta,\hat h,\bar\beta\geq0$ and $\bar h\in{\mathbb R}$. To show that $S^{\rm ann}
(\hat\beta,\hat h,\bar\beta;g)=\infty$ for this case, we proceed as in steps (II) and (III) of the proof
of \cite{BodHoOp11}, Theorem 3.2, by evaluating the functional under the supremum
in \eqref{Sandef} at $Q^L_{\hat\beta}=(q_{\hat\beta}^L)^{\otimes{\mathbb N}}$ with
\begin{equation}
q_{\hat\beta}^L(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n))
=\delta_{Ln}\left[\hat \mu_{\hat\beta}(d\hat \omega_1)\times\cdots\times\hat \mu_{\hat\beta}(d\hat \omega_n)\right]
\times\left[\bar \mu(d\bar \omega_1)\times\cdots\times\bar \mu(d\bar \omega_n)\right],
\end{equation}
where $L,n\in{\mathbb N}$, $\hat \omega_1,\ldots,\hat \omega_n\in\hat E$, $\bar \omega_1,\ldots,\bar \omega_n\in\bar E$, and (recall
\eqref{mgffin})
\begin{equation}
\hat \mu_{\hat\beta}(d\hat \omega_1)=e^{-2\hat\beta\hat \omega_1- \hat M(2\hat\beta)}\hat \mu(d\hat \omega_1).
\end{equation}
Note from \eqref{pinPhidef} that $\Phi(Q^L_{\hat\beta})=0$ because $\bar \mu$ has zero mean. This
leads to a lower bound on $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)$ that tends to infinity as $L\to
\infty$. To get the desired lower bound, we have to distinguish between the cases
$\hat g^{\rm ann}(\hat\beta,\hat h)=0$ and $\hat g^{\rm ann}(\hat\beta,\hat h)>0$. For $\hat g^{\rm ann}
(\hat\beta,\hat h)=0$ use $Q_0^L$, for $\hat g^{\rm ann}(\hat\beta,\hat h)>0$ with $\hat\beta>0$ use $Q_{\hat\beta}^L$.
\medskip\noindent
{\bf 3.}
For the case $g\geq\hat{g}^{\rm ann}(\hat\beta,\hat h)$, we proceed as in step 1 and 2 of the
proof of Theorem 3.2 of \cite{BodHoOp11}. Note that $\Phi_{\hat\beta,\hat h}(Q)$ and $\Phi(Q)$
defined in (\ref{pinPhidef}--\ref{phidef}) are functionals of $\pi_1 Q$, where $\pi_1 Q$
is the first-word marginal of $Q$. Moreover, by (\ref{spentrdef}),
\begin{equation}
\label{Hiidred}
\inf_{ {Q\in{\mathcal P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb N})} \atop {\pi_1 Q=q} }
H(Q \mid q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}})
= h(q \mid q_{\rho,\hat \mu\otimes\bar \mu}) \qquad \forall\,q \in {\mathcal P}(\widetilde{E})
\end{equation}
with the infimum \emph{uniquely} attained at $Q = q^{\otimes{\mathbb N}}$, where the right-hand
side denotes the relative entropy of $q$ w.r.t.\ $q_{\rho,\hat \mu\otimes\bar \mu}$. (The uniqueness
of the minimum is easily deduced from the strict convexity of relative entropy on
finite cylinders.) Consequently, the variational formula in \eqref{Sandef} becomes
\begin{equation}
\label{varredann}
\begin{split}
S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)
&= \sup_{ {q \in {\mathcal P}(\widetilde{E})} \atop
{m_q<\infty,\,h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty} }
\Big\{\int_{\widetilde{E}} q(d\omega)\,[\bar\beta \bar \omega_1+\log\phi_{\hat\beta, \hat h}(\hat \omega)]
-gm_Q -h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})\Big\}\cr
&=\sup_{ {q \in {\mathcal P}(\widetilde{E})} \atop
{m_q<\infty,\,h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty} }
\Big\{ \int_{\widetilde{E}} q(d\omega)\,[\bar\beta \bar \omega_1+\log\phi_{\hat\beta, \hat h}(\hat \omega)-g\tau(\omega)]\\
&\qquad\qquad\qquad\qquad\qquad - \int_{\widetilde{E}} q(d\omega)
\log \left(\frac{q(d\omega)}{q_{\rho,\hat \mu\otimes\bar \mu}(d\omega)}\right)\Big\} \cr
&=\bar M(-\bar\beta)+ \log\hat{\mathcal N}(\hat\beta, \hat h;g)-\inf_{ {q \in {\mathcal P}(\widetilde{E})}
\atop {m_q<\infty,\,h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty} }
h(q \mid q_{\hat\beta, \hat h,\bar\beta;g}),
\end{split}
\end{equation}
where (by an abuse of notation) $\omega=((\hat \omega_i,\bar \omega_i))_{i=1}^{\tau(\omega)}$ is the disorder in
the first word, $\phi_{\hat\beta, \hat h}(\hat \omega)$ is defined in (\ref{phidef}), $m_q=\int_{\widetilde
{E}} q(d\omega)\tau(\omega)$, $\tau(\omega)$ is the length of the word $\omega$, and
\begin{equation}
\label{uniqmax}
\begin{split}
q_{\hat\beta, \hat h,\bar\beta;g}(d(\hat \omega_1,\bar \omega_1),\cdots,d(\hat \omega_n,\bar \omega_n))
&=\frac{\rho(n)\phi_{\hat\beta, \hat h}(\hat \omega)e^{\bar\beta \bar \omega_1-ng}}
{\hat{\mathcal N}(\hat\beta, \hat h;g)e^{\bar M(-\bar\beta)}}(\hat \mu\otimes\bar \mu)^n
(d(\hat \omega_1,\bar \omega_1),\cdots,d(\hat \omega_n,\bar \omega_n)),\cr
\hat{\mathcal N}(\hat\beta, \hat h;g)
&=\tfrac12\left[\sum_{n\in{\mathbb N}}\rho(n)e^{-ng}
+\sum_{n\in{\mathbb N}}\rho(n)e^{-n(g-[M(2\hat\beta)-2\hat\beta \hat h])}\right].
\end{split}
\end{equation}
Note from \eqref{cndef} that ${\mathcal N}(\hat\beta, \hat h,\bar\beta;g)=\hat{\mathcal N}(\hat\beta, \hat h;g)e^{\bar M(-\bar\beta)}$.
The infimum in the last equality of \eqref{varredann} is uniquely attained at $q=q_{\hat\beta,
\hat h,\bar\beta;g}$. Therefore the variational problem in \eqref{Sandef} for $g\geq\hat{g}^{\rm
ann}(\hat\beta,\hat h)$ takes the form
\begin{equation}
\label{scopadf}
\begin{split}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)
&=\log\left(\tfrac12\left(\sum_{n\in{\mathbb N}}\rho(n)
e^{-gn}+\sum_{n\in{\mathbb N}}\rho(n)e^{-n\left(g-[\hat M(2\,\hat\beta)-2\,\hat\beta\hat h]\right)}\right)
e^{\bar M(-\bar\beta)}\right)\cr
&=\log {\mathcal N}(\hat\beta, \hat h,\bar\beta;g) = s^{\rm ann}(\hat\beta,\hat h,\bar\beta;g).
\end{split}
\end{equation}
The last formula proves \eqref{ganncomb}.
\end{proof}
As in the quenched model, there are analogous versions of Theorem~\ref{varflocann}
for the annealed copolymer model and the annealed pinning model. These are obtained
by putting either $\bar\beta=\bar h=0$ or $\hat\beta=\hat h=0$, replacing ${\mathcal C}^\mathrm{fin}$
by $\hat{\mathcal C}^\mathrm{fin}$ and $\bar{\mathcal C}^\mathrm{fin}$, respectively. The copolymer
version of Theorem~\ref{varflocann} was derived in \cite{BodHoOp11}, Theorem 3.2,
and the pinning version (for $g=0$ only) in \cite{ChdHo10}, Theorem 1.3.
Putting $\bar\beta=\bar h=0$, we get the copolymer analogue of \eqref{scopadf}:
\begin{equation}
\label{Scopann}
\hat S^{\rm ann}(\hat\beta,\hat h;g)
=\log\left(\tfrac12\left[\sum_{n\in{\mathbb N}}\rho(n)\,e^{-n g}
+ \sum_{n\in{\mathbb N}}\rho(n)\,e^{-n(g-[\hat M(2\,\hat\beta)-2\,\hat\beta\,\hat h])}\right]\right).
\end{equation}
This expression, which was obtained in \cite{BodHoOp11}, is plotted in
Fig.~\ref{fig-varfe1}. Putting $\hat\beta=\hat h=0$, we get the pinning analogue:
\begin{equation}
\label{Sanpin}
\begin{split}
\bar S^{\rm ann}(\bar\beta;g)
&= \bar M(-\bar\beta)+\log\left(\sum_{n\in{\mathbb N}} \rho(n)\,e^{-ng}\right).
\end{split}
\end{equation}
\begin{figure}[htbp]
\vspace{2cm}
\begin{minipage}[hbt]{5cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(0,0){\line(8,0){8}}
\put(0,-4){\line(0,12){12}}
{\thicklines
\qbezier(3,-5)(2,-3.5)(1,-1)
\qbezier(-3,7)(-1.5,7)(.8,7)
}
\qbezier[40](1,-1)(1,-3)(1,7)
\put(8.3,-0.2){$g$}
\put(-3,9){$\hat S^\mathrm{ann}(\hat\beta, \hat h;g)$}
\put(-2,7.5){$\infty$}
\put(1,-1){\circle*{.5}}
\put(-1,-0.2){$0$}
\put(1,7){\circle{.5}}
\end{picture}
\vspace{1.2cm}
\begin{center}
{\qquad (1) $\hat h <\hat h^\mathrm{ann}_c(\hat\beta)$}
\end{center}
\end{minipage}
\begin{minipage}[hbtb]{5cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(0,0){\line(8,0){8.5}}
\put(0,-4){\line(0,12){12}}
{\thicklines
\qbezier(3,-4.5)(1,-2.5)(0,0)
\qbezier(-3,7)(-1.5,7)(-.2,7)
}
\qbezier[40](4,0)(4,0)(0,0)
\qbezier[40](4,0)(4,0)(0,0)
\put(8.7,-0.2){$g$}
\put(-3,9){$ \hat S^\mathrm{ann}(\hat\beta, \hat h;g)$}
\put(-2,7.5){$\infty$}
\put(0,0){\circle*{.5}}
\put(-1,-0.2){$0$}
\put(0,7){\circle{.5}}
\end{picture}
\vspace{1.3cm}
\begin{center}
\qquad (2) $\hat h = \hat h^\mathrm{ann}_c(\hat\beta)$
\end{center}
\end{minipage}
\begin{minipage}[hbt]{5cm}
\centering
\setlength{\unitlength}{0.3cm}
\begin{picture}(8,8)(-7,-2.2)
\put(0,0){\line(8,0){8}}
\put(0,-4){\line(0,12){12}}
{\thicklines
\qbezier(4,-5)(1,-3.5)(0,-1)
\qbezier(-3,7)(-1.5,7)(-.2,7)
}
\put(8.3,-0.2){$g$}
\put(-3,9){$\hat S^\mathrm{ann}(\hat\beta, \hat h;g)$}
\put(-2,7.5){$\infty$}
\put(0,-1){\circle*{.5}}
\put(-1,-0.2){$0$}
\put(0,7){\circle{.5}}
\end{picture}
\vspace{1.2cm}
\begin{center}
\qquad (3) $\hat h > \hat h^\mathrm{ann}_c(\hat\beta)$
\end{center}
\end{minipage}
\vspace{-.4cm}
\begin{center}
\caption{\small Qualitative picture of the map $g \mapsto \hat S^\mathrm{ann}(\hat\beta,\hat h;g)$
for $\hat\beta,\hat h\geq0$.
Compare with Fig.~\ref{fig-varfe}.}
\label{fig-varfe1}
\end{center}
\vspace{-.6cm}
\end{figure}
The map $g \mapsto S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)$ has the same qualitative picture
as in Fig.~\ref{fig-varfe1}, with the following changes: the horizontal axis is
located at $\bar h$ instead of zero, and $\hat h_c^{\rm ann}(\hat\beta)$ is replaced by
$h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$.
Subtracting $\bar h$ from \eqref{Scopann} and \eqref{Sanpin}, we get from \eqref{ganvarexp}
that the excess free energies $\hat g^{\rm ann}(\hat\beta,\hat h)$ and $\bar g^{\rm ann}(\bar\beta,\bar h)$
take the form given in \eqref{copannfeg} and \eqref{pinannfeg}, respectively. The
following lemma summarizes their relationship.
\begin{lemma}
\label{grela}
For every $\bar\beta,\hat h,\hat\beta\geq 0$ and $\bar h\in{\mathbb R}$ (recall \eqref{lann12})
\begin{equation}
\label{anneferel}
\begin{split}
g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)\left\{\begin{array}{ll}
=\hat g^{\rm ann}(\hat\beta,\hat h), &\mbox{ if }\quad \bar h\geq \bar h_\ast(\hat\beta, \hat h,\bar\beta),\\
> \hat g^{\rm ann}(\hat\beta,\hat h), &\mbox{ if } \quad \bar h< \bar h_\ast(\hat\beta, \hat h,\bar\beta),\\
\leq\bar g^{\rm ann}(\bar\beta,\bar h), &\mbox{ if } \quad \hat h>\hat h_c^{\rm ann}(\hat\beta),\\
=\bar g^{\rm ann}(\bar\beta,\bar h), &\mbox{ if } \quad \hat h=\hat h_c^{\rm ann}(\hat\beta),\\
\geq\bar g^{\rm ann}(\bar\beta,\bar h), &\mbox{ if } \quad \hat h<\hat h_c^{\rm ann}(\hat\beta).
\end{array}
\right.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Note from \eqref{Sandef} and (\ref{scopadf}--\ref{Scopann}) that $S^{\rm ann}(\hat\beta,\hat h,
\bar\beta;g)-\bar h$ is $\hat S^{\rm ann}(\hat\beta,\hat h;g)$ shifted by $\bar M(-\bar\beta)-\bar h$. We see
from Fig.~\ref{fig-varfe1} that if $\bar h\geq \bar h_\ast(\hat\beta,\hat h,\bar\beta)$, then the map
$g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at the same value of $g$ as
the map $g\mapsto\hat S^{\rm ann}(\hat\beta,\hat h;g)$ does. Hence $g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)
=\hat g^{\rm ann}(\hat\beta,\hat h)$ whenever $\bar h\geq\bar h_\ast(\hat\beta,\hat h,\bar\beta)$. On the other
hand, if $\bar h<\bar h_\ast(\hat\beta, \hat h,\bar\beta)$, then the map $g\mapsto\hat S^{\rm ann}(\hat\beta,\hat h;g)$
changes sign before the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ does, i.e.,
$S^{\rm ann}(\hat\beta,\hat h,\bar\beta;\hat g^{\rm ann}(\hat\beta,\hat h))-\bar h>0$, and hence $g^{\rm ann}
(\hat\beta,\hat h,\bar\beta,\bar h)>\hat g^{\rm ann}(\hat\beta,\hat h)$.
The rest of the proof follows from a comparison of \eqref{scopadf} and \eqref{Sanpin}.
Note that, for $\hat h>\hat h_c^{\rm ann}(\hat\beta)$, we have $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h
<\bar S^{\rm ann}(\bar\beta;g)-\bar h$, which implies that $g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)\leq
\bar g^{\rm ann}(\bar\beta,\bar h)$. For $\hat h=\hat h_c^{\rm ann}(\hat\beta)$, we have $S^{\rm ann}
(\hat\beta,\hat h,\bar\beta;g)-\bar h = \bar S^{\rm ann}(\bar\beta;g)-\bar h$, which implies that $g^{\rm ann}
(\hat\beta,\hat h,\bar\beta,\bar h)=\bar g^{\rm ann}(\bar\beta,\bar h)$. Finally, for $\hat h<\hat h_c^{\rm ann}(\hat\beta)$
we have $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h > \bar S^{\rm ann}(\bar\beta;g)-\bar h$, which implies
that $g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)\geq \bar g^{\rm ann}(\bar\beta,\bar h)$.
\end{proof}
\subsection{Proof of Theorem~\ref{freeenegvar}}
\label{S3.3}
\begin{proof}
Throughout the proof $\hat\beta>0$, $\bar\beta\geq0$ and $\bar h\in{\mathbb R}$ are fixed.\\
(i) Use Theorems~\ref{varfloc}(i,iii).\\
(ii) Recall from \eqref{copadcc} and \eqref{gvarexp} that
\begin{equation}
\label{scc}
\begin{split}
h_c^{\rm que}(\hat\beta,\bar\beta,\bar h)
&=\inf\left\{\hat h>0\colon\, g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\}\cr
&=\inf\left\{\hat h>0\colon\, S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq0\right\}.
\end{split}
\end{equation}
Indeed, it follows from \eqref{gvarexp} that $g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)=0$ is equivalent
to saying that the map $g\mapsto S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at zero.
This sign change can happen while $S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)-\bar h$ is either zero or
negative (see Fig.~\ref{fig-varfe}(2--3)). The corresponding expression for $h_c^{\rm ann}
(\hat\beta,\bar\beta,\bar h)$ is obtained in a similar way.
\end{proof}
\section{Key lemma and proof of Corollary~\ref{freeeneggap}}
\label{Sec4}
The following lemma will be used in the proof of Corollary~\ref{freeeneggap}.
\begin{lemma}
\label{auxlemgap}
Fix $\alpha\geq 1$, $\hat\beta,\hat h>0$ and $\bar\beta\geq0$. Then, for $g>0$,
\begin{equation}
\label{squesanninq}
S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)<S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)
\left\{\begin{array}{ll}
\text{if } (\hat\beta,\hat h)\in \hat{\mathcal D}^{\rm ann},\\
\text{if } (\hat\beta,\hat h)\in \hat{\mathcal L}^{\rm ann} \text{ and } m_\rho<\infty,\\
\text{if } g\neq \hat g^{\rm ann}(\hat\beta,\hat h),(\hat\beta,\hat h)\in \hat{\mathcal L}^{\rm ann}
\text{ and } m_\rho=\infty.
\end{array}
\right.
\end{equation}
\end{lemma}
Lemma~\ref{auxlemgap} is proved in Section~\ref{Sec4.2.2}. In Section~\ref{Sec4.2.1} we
use Lemma~\ref{auxlemgap} to prove Corollary~\ref{freeeneggap}.
\subsection{Proof of Corollary~\ref{freeeneggap}}
\label{Sec4.2.1}
\begin{proof}
(ii) Throughout the proof, $\alpha\geq 1$, $\hat\beta,\hat h>0$ and $\bar\beta\geq0$.
It follows from \eqref{anneferel} that ${\mathcal L}^{\rm ann}_1\subset{\mathcal L}^{\rm ann}$.
Note that,
for $(\hat\beta,\hat h)\in\hat{\mathcal L}^{\rm ann}$, the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$
changes sign at some $g\geq\hat g^{\rm ann}(\hat\beta,\hat h)>0$, i.e., $g^{\rm ann}
(\hat\beta,\hat h,\bar\beta,\bar h)\geq\hat g^{\rm ann}(\hat\beta,\hat h)>0$ for all $\bar\beta\geq0$ and $\bar h\in{\mathbb R}$ (see \eqref{anneferel}). Hence
${\mathcal L}^{\rm ann}_1\subset{\mathcal L}^{\rm ann}$.
Note from \eqref{scopadf} and \eqref{Scopann} that
\begin{equation}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h = \hat S^{\rm ann}(\hat\beta, \hat h;g)+\bar M(-\bar\beta)-\bar h.
\end{equation}
Furthermore, note from Fig.~\ref{fig-varfe1}(2--3) that, for $(\hat\beta,\hat h)\in\hat{\mathcal D}^{\rm ann}$,
the map $g\mapsto \hat S^{\rm ann}(\hat\beta,\hat h;g)$ changes sign at $g=0$ while $\hat
S^{\rm ann}(\hat\beta,\hat h;0)$ is either negative or zero. In either case, we need
\begin{equation}
\bar h<\bar M(-\bar\beta)+\log \left(\tfrac12\left[1+\sum_{n\in{\mathbb N}}\rho(n)
e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right]\right)=\bar h_\ast(\hat\beta, \hat h,\bar\beta)
\end{equation}
to ensure that the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at a positive
$g$-value. This concludes the proof that ${\mathcal L}^{\rm ann}={\mathcal L}^{\rm ann}_1\cup {\mathcal L}^{\rm ann}_2$.
\medskip\noindent
(i) As we saw in the proof of (ii), for the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$
to reach zero we need that $\bar h\leq \bar h_\ast(\hat\beta, \hat h,\bar\beta)$. Thus, for this range of
$\bar h$-values, we know that the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h~$ changes
sign when it is zero. The proof for $g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)$ follows from
Fig.~\ref{fig-varfe}.
\medskip\noindent
(iii) We first consider the cases: (a) $(\hat\beta,\hat h,\bar\beta,\bar h)\in{\mathcal L}^{\rm ann}_2$; (b)
$(\hat\beta,\hat h,\bar\beta,\bar h)\in{\mathcal L}^{\rm ann}_1$ and $m_\rho<\infty$. In these cases we have that
$\hat h<h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h)$ by (ii). It follows from (\ref{scopadf}--\ref{Scopann})
and Fig.~\ref{fig-varfe1} that the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes
sign at some $g>0$ while it is either zero or negative. In either case the finiteness of
the map $g\mapsto S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ on $(0,\infty)$ and \eqref{squesanninq}
imply that $g\mapsto S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at a smaller value of
$g$ than $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ does. This concludes the proof for
cases (a--b).
We next consider the case: (c) $(\hat\beta,\hat h,\bar\beta,\bar h)\in{\mathcal L}^{\rm ann}$ with $(\hat\beta,\hat h)\in\hat
{\mathcal L}^{\rm ann}$, $\bar h\neq\bar h_\ast(\hat\beta,\hat h,\bar\beta)$ and $m_\rho=\infty$. We know from
\eqref{squesanninq} that $S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)<S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)$ for
$g>0$ and $g\neq\hat g^{\rm ann}(\hat\beta,\hat h)$. If $\bar h>\bar h_\ast(\hat\beta, \hat h,\bar\beta)$, then the
map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at $\hat g^{\rm ann}(\hat\beta,\hat h)$
while jumping from $<0$ to $\infty$. By the continuity of the map $g\mapsto S^{\rm que}
(\hat\beta,\hat h,\bar\beta;g)$ on $(0,\infty)$, this implies that the map $g\mapsto S^{\rm que}(\hat\beta,\hat h,
\bar\beta;g)-\bar h$ changes sign at a $g$-value smaller than $\hat g^{\rm ann}(\hat\beta,\hat h)$. Furthermore, if $\bar h<\bar h_\ast(\hat\beta,\hat h,\bar\beta)$, then the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$
changes sign at a $g$-value larger than $\hat g^{\rm ann}(\hat\beta,\hat h)$, while it is zero.
Since $S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)<S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)$ for $g>\hat g^{\rm ann}
(\hat\beta,\hat h)$, we have that $g^{\rm que}(\hat\beta,\hat h,\bar\beta,\bar h)<g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)$.
\medskip\noindent
(iv) The proof follows from Lemma \ref{grela}.
\end{proof}
\subsection{Proof of Lemma~\ref{auxlemgap}}
\label{Sec4.2.2}
\begin{proof}
The proof comes in five steps. Step 1 proves the strict inequality in \eqref{squesanninq},
using a claim about the finiteness of $I^{\rm ann}$ at some specific $Q$ in combination
with arguments from Birker~\cite{Bi08}. Steps 2-5 are used to prove the claim about the finiteness of $I^{\rm ann}$. Note that for $0<g<\hat g^\mathrm{ann}(\hat\beta,\hat h)$ the claim trivially follows from Theorems~\ref{varfloc}(iii) and \ref{varflocann}(ii), since
$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)<\infty$ and $S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)=\infty$
for this range of $g$-values. Thus, what remains to be considered is the case
$g\geq \hat g^\mathrm{ann}(\hat\beta,\hat h)$.
\medskip\noindent
{\bf 1.}
For $g\geq \hat g^\mathrm{ann}(\hat\beta,\hat h)$, note from \eqref{uniqmax} and the remark below
it that there is a unique maximizer $Q_{\hat\beta, \hat h,\bar\beta;g}=(q_{\hat\beta,\hat h,\bar\beta;g})^{\otimes{\mathbb N}}$
for the variational formula for $S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;g)$ in \eqref{Sandef}, where
\begin{equation}
\label{qbhbb}
\begin{split}
&q_{\hat\beta, \hat h,\bar\beta;g}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n))\cr
&=\dfrac{\frac12\;\rho(n)e^{-gn}(1+e^{-2\hat\beta\big[n\hat h
+\sum_{i=1}^n\hat \omega_i\big]})e^{\bar\beta\bar \omega_1}}{ {\mathcal N}(\hat\beta, \hat h,\bar\beta;g)}
(\hat \mu\otimes\bar \mu)^{\otimes n}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n)),
\end{split}
\end{equation}
where
\begin{equation}
\label{cnhat}
{\mathcal N}(\hat\beta, \hat h,\bar\beta;g)
=\tfrac12e^{\bar M(-\bar\beta)}\left[\sum_{n\in{\mathbb N}}\rho(n)e^{-gn}
+\sum_{n\in{\mathbb N}}\rho(n) e^{-n(g-[\hat M(2\hat\beta)-2\hat\beta \hat h])}\right]
=e^{\bar M(-\bar\beta)} \hat{\mathcal N}(\hat\beta, \hat h;g).
\end{equation}
Note further that $Q_{\hat\beta, \hat h,\bar\beta;g}\notin{\mathcal R}$. We claim that, for $g\geq\hat g^{\rm ann}
(\hat\beta,\hat h)$ and under the conditions in \eqref{squesanninq},
\begin{equation}
\label{iannfclaim}
H(Q_{\hat\beta, \hat h,\bar\beta;g} \mid q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}})
=h(q_{\hat\beta, \hat h,\bar\beta;g} \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty.
\end{equation}
This will be proved in Step 2. Let $M<\infty$ be such that $h(q_{\hat\beta, \hat h,\bar\beta;g} \mid
q_{\rho,\hat \mu\otimes\bar \mu})<M$. Then the set
\begin{equation}
\mathcal{A}_M = \left\{Q\in{\mathcal P}^{\rm inv}(\widetilde{E}^{\rm {\mathbb N}})
\colon\, H(Q \mid q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}})\leq M \right\}
\end{equation}
is compact in the weak topology, and contains $Q_{\hat\beta,\hat h,\bar\beta;g}$ in its interior. It
follows from Birkner \cite{Bi08}, Remark 8, that $\mathcal{A}_M\cap{\mathcal R}$ is a closed
subset of ${\mathcal P}^{\rm inv}(\widetilde{E}^{\rm {\mathbb N}})$. This in turn implies that there exists
a $\delta>0$ such that $B_\delta(Q_{\hat\beta, \hat h,\bar\beta;g})$ (the $\delta$-ball around $Q_{\hat\beta,
\hat h,\bar\beta;g}$) satisfies $B_\delta(Q_{\hat\beta, \hat h,\bar\beta;g})\cap \mathcal{A}_M\subset{\mathcal R}^c$.
Let
\begin{equation}
\bar\delta = \sup\left\{0\leq \delta'\leq \delta\colon\,
B_{\delta'}(Q_{\hat\beta, \hat h,\bar\beta;g})\cap \mathcal{A}_M
= B_{\delta'}(Q_{\hat\beta, \hat h,\bar\beta;g})\right\}.
\end{equation}
Then ${\mathcal R}\subset B_{\bar\delta}(Q_{\hat\beta, \hat h,\bar\beta;g})^c$. Therefore, for $g\geq\hat
g^{\rm ann}(\hat\beta,\hat h)$ and under the conditions in \eqref{squesanninq}, we get that
\begin{equation}
\begin{split}
S^{\rm que}(\hat\beta,\hat h,\bar\beta;g)
&=\sup_{Q\in{\mathcal C}^{\rm fin}\cap{\mathcal R}}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-g m_Q-I^{\rm ann}(Q)\right]\cr
&\leq \sup_{Q\in{\mathcal C}^{\rm fin}\cap B_{\bar\delta}(Q_{\hat\beta, \hat h,\bar\beta;g})^c}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-g m_Q-I^{\rm ann}(Q)\right]\cr
&< \sup_{Q\in{\mathcal C}^{\rm fin}}
\left[\bar\beta \Phi(Q)+\Phi_{\hat\beta, \hat h}(Q)-g m_Q-I^{\rm ann}(Q)\right]\cr
&= S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g) = \log {\mathcal N}(\hat\beta,\hat h,\bar\beta;g).
\end{split}
\end{equation}
The strict inequality follows because no maximizing sequence in ${\mathcal C}^{\rm fin}\cap
B_{\bar\delta}(Q_{\hat\beta,\hat h,\bar\beta;g})^c$ can have $Q_{\hat\beta, \hat h,\bar\beta;g}$ as its limit
($Q_{\hat\beta, \hat h,\bar\beta;g}$ being the unique maximizer of the variational problem in
the second inequality).
\medskip\noindent
{\bf 2.} Let us now turn to the proof of the claim in \eqref{iannfclaim}. For $g\geq\hat
g^{\rm ann}(\hat\beta,\hat h)$, it follows from \eqref{qbhbb} and \eqref{cnhat} that
\begin{equation}
\label{Hfinatqmax}
\begin{split}
h(q_{\hat\beta, \hat h\bar\beta;g} \mid q_{\rho,\hat \mu\otimes\bar \mu})\leq I+II,
\end{split}
\end{equation}
where
\begin{equation}
\label{anexp}
\begin{split}
I &= \bar\beta\int_{\bar E} \bar \omega_1 e^{\bar\beta \bar \omega_1-\bar M(-\bar\beta)}\bar \mu(d\bar \omega_1)
- \log{\mathcal N}(\hat\beta, \hat h,\bar\beta;g),\cr
II &= \frac{1}{\hat{\mathcal N}(\hat\beta, \hat h;g)} \sum_{n\in{\mathbb N}} \rho(n)e^{-ng} A(n),\cr
A(n) &= \tfrac12\int_{\hat E^n}\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]
\log\left(\tfrac12 \left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\right)
\hat \mu^{\otimes n}(d\hat \omega).
\end{split}
\end{equation}
The inequality in \eqref{Hfinatqmax} follows from \eqref{qbhbb} after replacing $e^{-gn}$
by 1. It is easy to see that $I<\infty$, because for $g\geq \hat g^{\rm ann}(\hat\beta,\hat h)$
we have that ${\mathcal N}(\hat\beta, \hat h,\bar\beta;g)<\infty$. Furthermore, since $\bar \mu$ has a finite moment
generating function, it follows from the H\"older inequality that $\int_{{\mathbb R}} \bar \omega_1
e^{\bar\beta \bar \omega_1-\bar M(-\bar\beta)}\bar \mu(d\bar \omega_1)<\infty$. We proceed to show that $II<\infty$.
\medskip\noindent
{\bf 3.}
We first estimate $A(n)$. Note that
\begin{equation}
\begin{split}
A(n) &= \tfrac12\int_{\hat E^n}\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]
\log\left(\tfrac12 \left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\right)
\hat \mu^{\otimes n}(d\hat \omega)\cr
&=\tfrac12\int_{\hat E^n}\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]
\log\left(\frac{e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}}{2}
\left[1+e^{2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\right)
\hat \mu^{\otimes n}(d\hat \omega)\cr
&= A_1(n)+A_2(n),
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
A_1(n) &= -\hat\beta\sum_{k=1}^n\int_{\hat E^n}(\hat \omega_k+\hat h)
\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\hat \mu^{\otimes n}(d\hat \omega),\cr
A_2(n) &= \tfrac12\int_{\hat E^n}\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]
\log\left( \tfrac12 \left[1+e^{2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\right)
\hat \mu^{\otimes n}(d\hat \omega).
\end{split}
\end{equation}
The finiteness of $II$ will follow once we show that
\begin{equation}
\sum_{n\in{\mathbb N}}\rho(n)e^{-gn}[A_1(n)+A_2(n)]<\infty.
\end{equation}
\medskip\noindent
{\bf 4.}
We start with the estimation of $A_2(n)$. Put $u_n(\hat \omega)=-2\hat\beta\sum_{i=1}^n(\hat \omega_i+h)$ and,
for $n\in{\mathbb N}$ and $m\in {\mathbb N}_0$, define
\begin{equation}
B_{m,n} = \left\{\hat \omega\in\hat E^n\colon -(m+1)< u_n(\hat \omega)\leq -m\right\},
\qquad m_n=m_n(\hat\beta,\hat h)=\lceil 4\hat\beta \hat h n \rceil.
\end{equation}
Then note that
\begin{equation}\label{A2n}
\begin{split}
A_2(n) &=\tfrac12\int_{\hat E^n}\left[1+e^{u_n(\hat \omega)}\right]
\log\left( \tfrac12 \left[1+e^{-u_n(\hat \omega)}\right]\right)\hat \mu^{\otimes n}(d\hat \omega)\cr
&\leq \int_{\hat E^n}\left[1\vee e^{u_n(\hat \omega)}\right]
\log\left(1\vee e^{-u_n(\hat \omega)}\right)\hat \mu^{\otimes n}(d\hat \omega)\cr
&=-\int_{u_n\leq0} u_n(\hat \omega)\hat \mu^{\otimes n}(d\hat \omega)\cr
&=-\sum_{m\in{\mathbb N}_0} \int_{B_{m,n}} u_n(\hat \omega)\hat \mu^{\otimes n}(d\hat \omega)\cr
&\leq \sum_{m=0}^{m_n} (m+1) + \sum_{m>m_n} (m+1){\mathbb P}_{\hat \omega}(B_{m,n})\cr
&\leq 4m_n^2 + \sum_{v\in{\mathbb N}} (m_n+v+1){\mathbb P}_{\hat \omega}(B_{m_n+v,n}).
\end{split}
\end{equation}
The second inequality uses that $-(m+1)< u_n\leq -m$ on $B_{m,n}$ and ${\mathbb P}_{\hat \omega}(B_{m_n+v,n})
\leq1$. Estimate
\begin{equation}
\label{A2npre}
\begin{split}
{\mathbb P}_{\hat \omega}(B_{m_n+v,n})
&={\mathbb P}_{\hat \omega}\left(\frac{m_n+v}{2\hat\beta} \leq \sum_{k=1}^n(\hat \omega_k+\hat h)
< \frac{m_n+v+1}{2\hat\beta}\right)\cr
&\leq {\mathbb P}_{\hat \omega}\left(\sum_{k=1}^n\hat \omega_k\geq \frac{m_n+v}{2\hat\beta}-n\hat h\right)\cr
&\leq {\mathbb P}_{\hat \omega}\left(\sum_{k=1}^n\hat \omega_k\geq \frac{4\hat\beta n \hat h+v}{2\hat\beta}-n\hat h\right)\cr
& ={\mathbb P}_{\hat \omega}\left(\sum_{k=1}^n\hat \omega_k\geq \frac{v}{2\hat\beta}+n\hat h\right)\cr
&\leq e^{-C\left(\frac{v}{2\hat\beta}+n\right)}.
\end{split}
\end{equation}
The last inequality uses \cite{BodHoOp11}, Lemma D.1, where $C$ is a positive constant
depending on $\hat h$ only. Inserting \eqref{A2npre} into \eqref{A2n},
we get
\begin{equation}
A_2(n)\leq 4 m_n^2+(m_n+1)e^{-Cn}\frac{e^{-1/2\hat\beta}}{1-e^{-1/2\hat\beta}}
+e^{-Cn}\sum_{v\in{\mathbb N}}v\, e^{-v/2\hat\beta}.
\end{equation}
Furthermore, using that $g>0$, we get
\begin{equation}
\begin{split}
\sum_{n\in{\mathbb N}}\rho(n) e^{-ng} A_2(n)
&\leq 4\sum_{n\in{\mathbb N}}\rho(n) e^{-ng} m_n^2
+ \frac{e^{-1/2\hat\beta}}{1-e^{-1/2\hat\beta}}\sum_{n\in{\mathbb N}}\rho(n) (m_n+1)e^{-n[g+C]}\cr
&\qquad + \sum_{n\in{\mathbb N}}\rho(n) e^{-n[g+C]} \sum_{v\in{\mathbb N}}v\, e^{-v/2\hat\beta}<\infty.
\end{split}
\end{equation}
\medskip\noindent
{\bf 5.} We proceed with the estimation of $A_1(n)$:
\begin{equation}
\begin{split}
A_1(n) &= -\hat\beta\sum_{k=1}^n\int_{\hat E^n} (\hat \omega_k+\hat h)
\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\hat \mu^{\otimes n}(d\hat \omega)\cr
&\leq -\hat\beta\sum_{k=1}^n\int_{\hat E^n}\hat \omega_k
\left[1+e^{-2\hat\beta\sum_{i=1}^n(\hat \omega_i+\hat h)}\right]\hat \mu^{\otimes n}(d\hat \omega)\cr
&= -n\hat\beta\; e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}\;{\mathbb E}_{\hat \mu_{\hat\beta}}(\hat \omega_1),
\end{split}
\end{equation}
where $\hat \mu_{\hat\beta}(d\hat \omega_1)=e^{-2\hat\beta\hat \omega_1-M(2\hat\beta)}\hat \mu(d\hat \omega_1)$. The right-hand side is
non-negative because ${\mathbb E}_{\hat \mu_{\hat\beta}}(\hat \omega_1)\leq0$, and so
\begin{equation}
\sum_{n\in{\mathbb N}}\rho(n) e^{-ng} A_1(n) \leq -\hat\beta {\mathbb E}_{\hat \mu_{\hat\beta}}(\hat \omega_1)
\sum_{n\in{\mathbb N}} n\rho(n)\,e^{-n(g-[M(2\hat\beta)-2\hat\beta \hat h])}.
\end{equation}
This bound is finite if
\begin{enumerate}
\item $g>\hat g^{\rm ann}(\hat\beta,\hat h)=\hat M(2\hat\beta)-2\hat\beta \hat h$;
\item $g=\hat g^{\rm ann}(\hat\beta,\hat h)$ and $m_\rho<\infty$.
\end{enumerate}
This concludes the proof since, if $(\hat\beta,\hat h)\in\hat{\mathcal D}^{\rm ann}$, then $\hat g^{\rm ann}
(\hat\beta,\hat h)=0$ and we only want the finiteness for $g>0$.
\end{proof}
For the pinning model, the associated unique maximizer $\bar Q_{\bar\beta;g}$ for the variational
formula for $\bar S^{\rm ann}(\bar\beta;g)$ satisfies $ H(\bar Q_{\bar\beta;g}|q^{\otimes{\mathbb N}}_{\rho,\bar \mu})
<\infty$ for $g \geq 0$. However, this does not imply separation between $\bar S^{\rm que}
(\bar\beta;0)$ and $\bar S^{\rm ann}(\bar\beta;0)$, since we may have $\bar Q_{\bar\beta;0}\in\bar{\mathcal R}$ for
$m_\rho=\infty$. The separation occurs at $g=0$ as soon as $m_\rho<\infty$, since this will
imply that $\bar Q_{\bar\beta;0}\notin\bar{\mathcal R}$.
\section{Proof of Corollary \ref{hquehannvarfor}}
\label{Sec5}
To prove Corollary~\ref{hquehannvarfor} we need some further preparation, formulated as
Lemmas~\ref{asyptann}--\ref{hizlimts} below. These lemmas, together with the proof of
Corollary~\ref{hquehannvarfor}, are given in Section~\ref{Sec5.1}. Section~\ref{Sec5.2}
contains the proof of the first two lemmas, and Appendix~\ref{appc} the proof of the third
lemma.
\subsection{Key lemmas and proof of Corollary~\ref{hquehannvarfor}}
\label{Sec5.1}
\begin{lemma}
\label{asyptann}
For $\hat\beta,\bar\beta\geq0$,
\begin{equation}
\label{Stildeann}
\begin{split}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)=\left\{\begin{array}{ll}
\bar M(-\bar\beta), &\mbox{ if }\quad \hat h=\hat h_c^{\rm ann}(\hat\beta),\\
\infty, &\mbox{ if }\quad \hat h<\hat h_c^{\rm ann}(\hat\beta),\\
\bar M(-\bar\beta)-\log 2, &\mbox{ if }\quad \hat h=\infty.
\end{array}
\right.
\end{split}
\end{equation}
Furthermore, the map $\hat h\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)$ is strictly convex and
strictly decreasing on $[\hat h_c^{\rm ann}(\hat\beta),\infty)$.
\end{lemma}
\begin{figure}[htbp]
\vspace{3.5cm}
\begin{center}
\setlength{\unitlength}{0.48cm}
\begin{picture}(10,4)(0,-4)
\put(0,0){\line(14,0){14}}
\put(0,-4.5){\line(0,5){10}}
{\thicklines
\qbezier(4,3.5)(7.5,-2.8)(14,-3.5)
\qbezier(3.8,5)(3,5)(0,5)
}
\qbezier[30](0,-4)(4,-4)(14,-4)
\qbezier[30](4,0)(4,2)(4,3.5)
\qbezier[30](0,3.5)(2,3.5)(4,3.5)
\put(14.5,-0.2){$\hat h$}
\put(2.8,-1){$\hat h_c^{\rm ann}(\hat\beta)$}
\put(-3,6.3){$S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0)$}
\put(-3,3.2){$\bar h^{\rm ann}_c(\bar\beta)$}
\put(7,.5){$h^\mathrm{ann}_c(\hat\beta,\bar\beta,\bar h)$}
\put(-1,-.4){$\bar h$}
\put(2,5.3){$\infty$}
\put(4,5){\circle{.3}}
\put(4,3.5){\circle*{.3}}
\put(-6,-4.2){$\bar h^{\rm ann}_c(\bar\beta)-\log 2$}
\end{picture}
\end{center}
\vspace{-0.2cm}
\caption{\small Qualitative picture of $\hat h \mapsto S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0)$
for $\hat\beta>0$ and $\bar\beta\geq0$. }
\label{fig-copadannvarhs}
\vspace{0cm}
\end{figure}
\begin{lemma}
\label{queasympt}
For every $\hat\beta>0$ and $\bar\beta\geq0$ (see Fig.~{\rm \ref{fig-copadannvarhs}}),
\begin{equation}
S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)=S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta)
\left\{\begin{array}{ll}
=\infty, &\mbox{ for }\quad \hat h<\hat h_c^{\rm ann}(\hat\beta/\alpha)\\
>0, &\mbox{ for }\quad \hat h=\hat h_c^{\rm ann}(\hat\beta/\alpha)\\
<\infty, &\mbox{ for }\quad \hat h>\hat h_c^{\rm ann}(\hat\beta/\alpha).
\end{array}
\right.
\end{equation}
\end{lemma}
\begin{figure}[htbp]
\vspace{3.5cm}
\begin{minipage}[hbt]{8.1cm}
\centering
\setlength{\unitlength}{0.48cm}
\begin{picture}(10,4)(0,-4)
\put(2,0){\line(10,0){10}}
\put(2,-4.5){\line(0,9){9}}
{\thicklines
\qbezier(6,2.5)(7.5,-2.8)(11.5,-3.5)
\qbezier(2,3.5)(4.5,3.5)(5.8,3.5)
}
\qbezier[30](2,-4)(6,-4)(11.5,-4)
\qbezier[30](2,2.5)(4,2.5)(6,2.5)
\qbezier[30](6,0)(6,1.5)(6,2.5)
\put(12.5,-0.2){$\hat h$}
\put(-.9,5.5){$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$}
\put(-1.8,2.4){\small $s_*(\hat\beta,\bar\beta,\alpha)$}
\put(7.5,.3){$h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$}
\put(4,-1){$\hat h_c^\mathrm{ann}(\frac{\hat\beta}{\alpha})$}
\put(1,-.4){$\bar h$}
\put(4,3.64){$\infty$}
\put(6,2.5){\circle*{.4}}
\put(6,3.5){\circle{.4}}
\put(-3.8,-4){$\bar h^{\rm que}_c(\bar\beta)-\log 2$}
\end{picture}
\begin{center}
(a)
\end{center}
\end{minipage}
\begin{minipage}[hbt]{8.1cm}
\centering
\setlength{\unitlength}{0.48cm}
\begin{picture}(10,4)(0,-4)
\put(2,0){\line(10,0){10}}
\put(2,-4.5){\line(0,9){9}}
{\thicklines
\qbezier(6,3.5)(7.5,-2.8)(11.5,-3.5)
\qbezier(2,3.5)(4.5,3.5)(6,3.5)
}
\qbezier[30](2,-4)(6,-4)(11.5,-4)
\qbezier[30](6,0)(6,1.5)(6,3.5)
\put(12.5,-0.2){$\hat h$}
\put(-.9,5.5){$S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$}
\put(-1.8,3.4){\small $s_*(\hat\beta,\bar\beta,\alpha)$}
\put(7.5,.3){$h^\mathrm{que}_c(\hat\beta,\bar\beta,\bar h)$}
\put(4,-1){$\hat h_c^\mathrm{ann}(\frac{\hat\beta}{\alpha})$}
\put(1,-.4){$\bar h$}
\put(4,3.64){$\infty$}
\put(-3.8,-4){$\bar h^{\rm que}_c(\bar\beta)-\log 2$}
\end{picture}
\begin{center}
(b)
\end{center}
\end{minipage}
\vspace{-0.2cm}
\caption{\small Qualitative picture of $\hat h \mapsto S^\mathrm{que}(\hat\beta,\bar\beta,\hat h;0)$ for
$\hat\beta>0$ and $\bar\beta\geq0$: (a) $s_*(\hat\beta,\bar\beta,\alpha)<\infty$; (b) $s_*(\hat\beta,\bar\beta,\alpha)=\infty$.}
\label{fig-copadvarhs}
\vspace{-.20cm}
\end{figure}
\begin{lemma}
\label{hizlimts}
For every $\hat\beta>0$ and $\bar\beta\geq0$ (see Fig.~{\rm \ref{fig-copadvarhs}}),
\begin{equation}\label{hzilim1}
\begin{split}
S^{\rm que}(\hat\beta,\infty-,\bar\beta;0)
& = \lim_{\hat h\to\infty} S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)
=S^{\rm que}(\hat\beta,\infty,\bar\beta;0) = \bar h_c^{\rm que}(\bar\beta)-\log 2.
\end{split}
\end{equation}
\end{lemma}
We now give the proof of Corollary~\ref{hquehannvarfor}.
\begin{proof}
Throughout the proof $\hat\beta>0$, $\bar\beta\geq0$ and $\bar h\in{\mathbb R}$ are fixed. Note from \eqref{phidef}
that the map $\hat h\mapsto\log\phi_{\hat\beta,\hat h}(\hat \omega)$ is strictly decreasing and convex for
all $\hat \omega\in\widetilde{\hat E}$. It therefore follows from \eqref{Sdef} and \eqref{Sandef}
that the maps $\hat h \mapsto S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$ and $\hat h \mapsto S^\mathrm{ann}
(\hat\beta,\hat h,\bar\beta;0)$ are strictly decreasing when finite (because $\tau(\omega)\geq 1$) and
convex (because sums and suprema of convex functions are convex).
Recall from \eqref{copadacc} and \eqref{gvarexp} that
\begin{equation}
\label{sancc}
\begin{split}
h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)
&=\inf\left\{\hat h\geq0\colon\,g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)=0\right\}\cr
&=\inf\left\{\hat h\geq0\colon\,S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq0\right\}.
\end{split}
\end{equation}
Indeed, it follows from \eqref{gvarexp} that $g^{\rm ann}(\hat\beta,\hat h,\bar\beta,\bar h)=0$ is equivalent
to saying that the map $g\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ changes sign at zero.
This change of sign can happen while $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;g)-\bar h$ is either zero or
negative (see e.g.\ Fig.~\ref{fig-varfe}(2--3)).
For $\bar h\geq\bar h^{\rm ann}_c(\bar\beta)$, it follows from Lemma~\ref{asyptann} and
Fig.~\ref{fig-copadannvarhs} that $\hat h=\hat h^{\rm ann}_c(\hat\beta)$ is the smallest
value of $\hat h$ at which $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq 0$ and hence $h^{\rm ann}_c(\hat\beta,
\bar\beta,\bar h)=\hat h^{\rm ann}_c(\hat\beta)$. Furthermore, note from Fig.~\ref{fig-copadannvarhs}
that the map $\hat h\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)$ is strictly decreasing and convex
on $[\hat h^{\rm ann}_c(\hat\beta),\infty)$ and has the interval $\left(\bar h^{\rm ann}_c(\bar\beta)
-\log 2,\bar h^{\rm ann}_c(\bar\beta)\right]$ as its range. In particular, $S^{\rm ann}(\hat\beta,\hat
h^{\rm ann}_c(\hat\beta),\bar\beta;0)=\bar h^{\rm ann}_c(\bar\beta)$ and $S^{\rm ann}(\hat\beta,\infty,\bar\beta;0)
=\bar h^{\rm ann}_c(\bar\beta)-\log 2$. Therefore, for $\bar h\in \left(\bar h^{\rm ann}_c(\bar\beta)-\log 2,
\bar h^{\rm ann}_c(\bar\beta)\right]$, the map $\hat h\mapsto S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h$ changes
sign at the unique value of $\hat h$ at which $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)=\bar h$.
For $\bar h\leq \bar h^{\rm ann}_c(\bar\beta)-\log 2$, it follows from Fig.~\ref{fig-copadannvarhs}
that $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h>0$ for all $\hat h\in[0,\infty)$. It therefore follows
from \eqref{sancc} that
\begin{equation}
\begin{split}
h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)
&=\inf\left\{\hat h\geq0\colon\,S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h\leq0\right\}=\infty.
\end{split}
\end{equation}
The proof for $h_c^{\rm que}(\hat\beta,\bar\beta,\bar h)$ follows from that of $h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$
after replacing $S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0)$, $\bar h_c^{\rm ann}(\bar\beta)-\log 2$ and
$\bar h_c^{\rm ann}(\bar\beta)$ by $S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;0)$, $\bar h_c^{\rm que}(\bar\beta)-\log 2$
and $s^{\ast}(\hat\beta,\bar\beta,\alpha)$, respectively.
Since both the quenched and the annealed free energies are convex functions of $\bar h$
(by H\"older inequality), $\bar h \mapsto h_\ast^{\rm que}(\hat\beta,\bar\beta,\bar h)$ and $\bar h \mapsto
h_\ast^{\rm ann}(\hat\beta,\bar\beta,\bar h)$ are also convex and strictly decreasing.
\end{proof}
\subsection{Proof of Lemmas~\ref{asyptann}--\ref{queasympt}}
\label{Sec5.2}
{\bf Proof of Lemma~\ref{asyptann}:}\\
\begin{proof}
Note from \eqref{scopadf} that
\begin{equation}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)
=\bar M(-\bar\beta)+\log\left(\tfrac12\left[1+\sum_{n\in{\mathbb N}}\rho(n)
e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right]\right),
\end{equation}
which implies the claim.
\end{proof}
\noindent
{\bf Proof of Lemma~\ref{queasympt}:}\\
\begin{proof}
Throughout the proof $\hat\beta,\hat h>0$ and $\bar\beta\geq0$ are fixed. The proof uses arguments
from \cite{BodHoOp11}, Theorem 3.3 and Section 6. Note from \eqref{copadSlim},
\eqref{seqsbar} and Lemma~\ref{varlem} that
\begin{equation}
\label{copadSlimref}
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)
= \bar h + \limsup_{N\to\infty} \frac{1}{N} \log F_N^{\hat\beta, \hat h,\bar\beta,\bar h,\omega}(g)
= \log \mathcal{N}(g) + \limsup_{N\to\infty} \frac{1}{N}\log S^{\omega}_N(g),
\end{equation}
where
\begin{equation}
S_N^\omega (g) =E^*_g\left(\exp\left [N\left(\Phi_{\hat\beta, \hat h}(R_N^{\omega})+\bar\beta\,
\Phi(R_N^{\omega})\right)\right]\right).
\end{equation}
It follows from the fractional-moment argument in \cite{BodHoOp11}, Eq.\ (6.4), that
\begin{equation}
\label{braceterm}
{\mathbb E}([S_N^\omega (g)]^t)\leq \left(2^{1-t}\sum_{n\in{\mathbb N}}\rho_g(n)^{t}
e^{ \bar M(-\bar\beta t)}\right)^{N}<\infty, \quad g\geq 0,
\end{equation}
where $t\in[0,1]$ is chosen such that $\hat M(2\hat\beta t)-2\hat\beta \hat h t\leq 0$. Abbreviate the
term inside the brackets of \eqref{braceterm} by $K_t$ and note that
\begin{equation}
\begin{split}
{\mathbb P}\left(\frac{t}{N}\log S_N^\omega(g)\geq\log K_t+\epsilon\right)
&={\mathbb P}\left( [S_N^\omega(g)]^t\geq K_t^N e^{N\epsilon}\right)\cr
&\leq {\mathbb E}([S_N^\omega(g)]^t)K_t^{-N} e^{-N\epsilon}\cr
&\leq e^{-N\epsilon}, \qquad \epsilon>0.
\end{split}
\end{equation}
Therefore, for $g>0$, this estimate together with the Borel-Cantelli lemma shows that
$\omega$-a.s.\ (recall \eqref{rhoz})
\begin{equation}
\begin{split}
S^\mathrm{que}(\hat\beta,\hat h,\bar\beta;g)
&= \log{\mathcal N}(g)+\limsup_{N\rightarrow\infty}\frac1N\log S_N^\omega(g)
\leq \frac1t \log K_t+\log{\mathcal N}(g)\cr
&= \frac{1-t}{t}\log 2 + \frac1t \log\left(\sum_{n\in{\mathbb N}}
\rho_g(n)^{t}\right)+\frac1t \bar M(-\bar\beta t)+\log{\mathcal N}(g)<\infty.
\end{split}
\end{equation}
This estimate also holds for $g=0$ when $\sum_{n\in{\mathbb N}}\rho(n)^{t}<\infty$. This is the
case for any pair $t\in(1/\alpha,1]$ and $\hat h>\hat h_c^{\rm ann}(\hat\beta/\alpha)$ satisfying
$\hat M(2\hat\beta t)-2\hat\beta \hat h t\leq 0$ (recall \eqref{rhocond}). Therefore we conclude that
$S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)<\infty$ whenever $\hat h>\hat h_c^{\rm ann}(\hat\beta/\alpha)$.
To prove that $S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)=\infty$ for $\hat h<\hat h_c^{\rm ann}(\hat\beta/\alpha)$,
we replace $q_{\bar\beta}^L$ in \cite{BodHoOp11}, Eq.\ (6.8), by
\begin{equation}
q_{\hat\beta}^L(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n))
= \delta_{n,L} \left[\hat \mu_{\hat\beta/\alpha}(\hat \omega_1)\times\cdots\times\hat \mu_{\hat\beta/\alpha}(d\hat \omega_n)\right]
\times\left[\bar \mu(d\bar \omega_1)\times\cdots\times\bar \mu(d\bar \omega_n)\right],
\end{equation}
where
\begin{equation}
\hat \mu_{\hat\beta/\alpha}(d\hat \omega_1)=e^{-(2\hat\beta/\alpha)\hat \omega_1-\hat M(2\hat\beta/\alpha)}\hat \mu(d\hat \omega_1).
\end{equation}
With this choice the rest of the argument in \cite{BodHoOp11}, Section 6.2, goes through
easily.
Finally, to prove that $S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)>0$ at $\hat h=\hat h_c^{\rm ann}(\hat\beta/\alpha)$
we proceed as follows. Adding, respectively, $\bar\beta \Phi(Q)$ and $\bar\beta\sum_{n\in{\mathbb N}}\int_{\bar E^n}
\bar \omega_1 q(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n))$ to the functionals being optimized in
\cite{BodHoOp11}, Eqs.\ (6.19--6.20), we get the following analogue of \cite{BodHoOp11},
Eq.\ (6.21),
\begin{equation}
\begin{split}
&q_{\hat\beta, \hat h,\bar\beta}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n))\cr
&=\frac{1}{\hat{\mathcal N}(\hat\beta, \hat h)\,e^{ \bar M(-\bar\beta/\alpha)}}
\left[\phi_{\hat\beta, \hat h}((\hat \omega_1,\ldots\hat \omega_n))\,e^{\bar\beta \bar \omega_1}\right]^{1/\alpha}
q_{\rho, \hat \mu\otimes\bar \mu}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_n,\bar \omega_n)),
\end{split}
\end{equation}
where
\begin{equation}
\hat{\mathcal N}(\hat\beta, \hat h)=\sum_{n\in{\mathbb N}}\rho(n)\int_{\hat E^n}\hat \mu(d\hat \omega_1)\times\cdots\times\hat \mu(\hat \omega_n)
\left\{\tfrac12\left(1+e^{-2\hat\beta\sum_{k=1}^n(\hat \omega_k+\hat h)}\right)\right\}^{1/\alpha}.
\end{equation}
Note from \cite{BodHoOp11}, Eqs.\ (6.23--6.29), that $1<\hat{\mathcal N}(\hat\beta,\hat h_c^{\rm ann}(\hat\beta/\alpha))
\leq 2^{1-(1/\alpha)}$. Therefore it follows from \cite{BodHoOp11}, Steps 1 and 2 in Section 6.3, that
\begin{equation}
S^{\rm que}(\hat\beta,\hat h_c^{\rm ann}(\hat\beta/\alpha),\bar\beta;0)= S_*^{\rm que}(\hat\beta,\hat h_c^{\rm ann}
(\hat\beta/\alpha),\bar\beta)\geq \alpha\log \hat{\mathcal N}(\hat\beta,\hat h_c^{\rm ann}(\hat\beta/\alpha))+\alpha \bar M(-\bar\beta/\alpha)>0.
\end{equation}
\end{proof}
\section{Proofs of Corollaries~\ref{hcubstrict}--\ref{locpathprop}}
\label{Sec6}
\subsection{Proof of Corollary \ref{hcubstrict}}
\label{Sec6.1}
\begin{proof}
Throughout the proof, $\alpha>1$, $\hat\beta>0$, $\bar\beta\geq0$ and $\bar h\in{\mathbb R}$ are fixed.
The proof for $\bar h^{\rm que}_c(\bar\beta)-\log 2<\bar h\leq\bar h^{\rm ann}_c(\bar\beta)-
\log 2$ is trivial, since $h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)=\infty$ and $h_c^{\rm que}
(\hat\beta,\bar\beta,\bar h)<\infty$. The rest of the proof will follow once we show that
$h_c^{\rm que}(\hat\beta,\bar\beta,\bar h)<h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$ for $\bar h^{\rm ann }_c(\bar\beta)
-\log 2<\bar h\leq\bar h^{\rm ann}_c(\bar\beta)$. This is because, for $\bar h\geq\bar h^{\rm ann}_c(\bar\beta)$,
$h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)=\hat h^{\rm ann}_c(\hat\beta)$ and $\bar h\mapsto h^{\rm que}
(\hat\beta,\bar\beta,\bar h)$ is non-increasing. Furthermore, the map $\bar h\mapsto h^{\rm ann}
(\hat\beta,\bar\beta,\bar h)$ is also non-increasing, and so $h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)\geq
\hat h^{\rm ann}_c(\hat\beta)$.
For $\bar h^{\rm ann }_c(\bar\beta)-\log 2<\bar h\leq\bar h^{\rm ann}_c(\bar\beta)$, it follows
from Corollary~\ref{hquehannvarfor} that $h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$ is the unique
$\hat h$-value that solves the equation $S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)=\bar h$. Note that
for $\hat h\geq \hat h_c^{\rm ann}(\hat\beta)=\hat M(2\hat\beta)/2\hat\beta$, which is the range of
$\hat h$-values attainable by $h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$, the measure
$q_{\hat\beta,\hat h,\bar\beta;0}$ (recall \eqref{uniqmax}) is well-defined and is the unique
minimizer of the last variational formula in \eqref{varredann}, for $g=0$.
Hence, for $\bar h^{\rm ann}_c(\bar\beta)-\log 2<\bar h\leq\bar h^{\rm ann}_c(\bar\beta)$,
$h_c^{\rm ann}(\hat\beta,\bar\beta,\bar h)$ is the unique $\hat h$-value that solves the equation
\begin{equation}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h=\bar M(-\bar\beta)+\log\hat{\mathcal N}(\hat\beta, \hat h;0)-\bar h=0.
\end{equation}
Again, it follows from \eqref{truncapproxcont} that, for any $Q\in{\mathcal P}^{\rm inv}
(\widetilde{E}^{\mathbb N})$,
\begin{equation}
\begin{split}
I^{\rm que}(Q)
&=\sup_{{\rm tr}\in{\mathbb N}}I^{\rm que}([Q]_{{\rm tr}})=\sup_{{\rm tr}\in{\mathbb N}}
\left[H\left([Q]_{{\rm tr}}|q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}}\right)
+(\alpha-1)m_{[Q]_{{\rm tr}}}
H\left(\Psi_{[Q]_{{\rm tr}}}|(\hat \mu\otimes\bar \mu)^{\otimes{\mathbb N}}\right)\right]\cr
&\geq H\left(Q \mid q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}}\right)
+(\alpha-1)m_{[Q]_{{\rm tr}}}
H\left(\Psi_{[Q]_{{\rm tr}}}|(\hat \mu\otimes\bar \mu)^{\otimes{\mathbb N}}\right),
\quad {\rm tr}\in{\mathbb N}.
\end{split}
\end{equation}
Furthermore, it follows from \eqref{spentrdef} and the remark below it that
\begin{equation}
\label{red1}
H(Q \mid q_{\rho,\hat \mu\otimes\bar \mu}^{\otimes{\mathbb N}})
\geq h(\pi_1Q \mid q_{\rho,\hat \mu\otimes\bar \mu}),
\qquad H(\Psi_{[Q]_{{\rm tr}}} \mid (\hat \mu\otimes\bar \mu)^{\otimes{\mathbb N}})
\geq h(\widetilde\pi_{1}
\Psi_{[Q]_{{\rm tr}}} \mid \hat \mu\otimes\bar \mu),
\end{equation}
where $\widetilde\pi_1$ is the projection onto the first \emph{letter} and ${\rm tr}\in{\mathbb N}$.
Moreover, it follows from \eqref{PsiQdef} that
\begin{equation}
\label{red2}
\widetilde\pi_1\Psi_Q = \widetilde\pi_1\Psi_{(\pi_1 Q)^{\otimes{\mathbb N}}}.
\end{equation}
Since $m_Q = m_{(\pi_1 Q)^{\otimes{\mathbb N}}}= m_{\pi_1 Q}$, (\ref{red1}--\ref{red2}) combine
with \eqref{Sdefalt} to give
\begin{equation}
\label{upbound}
\begin{split}
&S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta)\cr
&\leq \sup_{ {q\in{\mathcal P}(\widetilde{E})} \atop h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty;
\,{m_q<\infty} }
\left[\int_{\widetilde{E}} q(d\omega)[\bar\beta \bar \omega_1+\log\phi_{\hat\beta,\hat h}(\hat \omega)]
- h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})\right.\\
&\qquad\qquad\qquad \left. - (\alpha-1) m_{[q]_{{\rm tr}}}
h(\widetilde\pi_1\Psi_{[q]_{{\rm tr}}} \mid \hat \mu\otimes\bar \mu) \right],
\end{split}
\end{equation}
where
\begin{equation}
\phi_{\hat\beta, \hat h}(\hat \omega)
= \tfrac12 \left(1 + e^{-2\hat\beta \hat h m-2\hat\beta [\hat \omega_1+\dots+\hat \omega_m]}\right)
\end{equation}
and
\begin{equation}
(\widetilde\pi_1\Psi_{[q]_{{\rm tr}}})(d(\hat \omega_1,\bar \omega_1))
= \frac{1}{m_{[q]_{{\rm tr}}}} \sum_{m\in{\mathbb N}} [r]_{{\rm tr}}(m)
\sum_{k=1}^m q_m(E^{k-1},d(\hat \omega_1,\bar \omega_1),E^{m-k})
\end{equation}
with the notation
\begin{equation}
q(d\omega) = r(m)q_m(d(\hat \omega_1,\bar \omega_1),\dots,d(\hat \omega_m,\bar \omega_m)),
\qquad \omega=((\hat \omega_1,\bar \omega_1),\dots,(\hat \omega_m,\bar \omega_m)),
\end{equation}
and
\begin{equation}
[r]_{{\rm tr}}(m)=\left\{\begin{array}{ll}
r(m) &\mbox{ if } 1\leq m<{\rm tr}\\
\sum_{n={\rm tr}}^\infty r(n) & \mbox{ if } m={\rm tr}\\
0 & \mbox{otherwise.}
\end{array}
\right.
\end{equation}
Therefore, for $\hat h\geq\hat h^{\rm ann}_c(\hat\beta)$, after combining the first two terms in the
supremum in \eqref{upbound}, as in \eqref{varredann}, we obtain
\begin{equation}
\begin{split}
S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta)
&\leq \bar M(-\bar\beta)+ \log\hat{\mathcal N}(\hat\beta, \hat h;0) \cr
&\qquad - \inf_{ {q\in{\mathcal P}(\widetilde{E})} \atop h(q\mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty;
\,{m_q<\infty} } \left[h(q \mid q_{\hat\beta, \hat h,\bar\beta})
+ (\alpha-1) m_{[q]_{{\rm tr}}}
h(\widetilde\pi_1\Psi_{[q]_{{\rm tr}}} \mid \hat \mu\otimes \bar \mu) \right].
\end{split}
\end{equation}
Hence, for $\hat h\geq\hat h^{\rm ann}_c(\hat\beta)$, it follows from \eqref{varredann} that
\begin{equation}
\begin{split}
&S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta)
\leq S^\mathrm{ann}(\hat\beta,\hat h,\bar\beta;0)\cr
&\qquad - \inf_{ {q\in{\mathcal P}(\widetilde{E})} \atop
h(q \mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty;~ {m_q<\infty} }
\left[h(q \mid q_{\hat\beta, \hat h,\bar\beta})
+ (\alpha-1) m_{[q]_{{\rm tr}}}
h(\widetilde\pi_1\Psi_{[q]_{{\rm tr}}} \mid\hat \mu\otimes \bar \mu) \right].
\end{split}
\end{equation}
The first term in the variational formula achieves its minimal value zero at
$q=q_{\hat\beta,\hat h,\bar\beta}$ (or along a minimizing sequence converging to $q_{\hat\beta, \hat h,\bar\beta}$).
However, via some simple computations we obtain
\begin{equation}
\begin{split}
\widetilde\pi_1\Psi_{[q_{\hat\beta, \hat h, \bar\beta}]_{{\rm tr}}}(d\hat \omega_1,d\bar \omega_1)
&= \dfrac{C_{{\rm tr}}(\hat \omega_1,\hat\beta, \hat h)}
{A_{{\rm tr}}(\hat\beta,\hat h)}\hat \mu(d\hat \omega_1)\bar \mu_{\bar\beta}(d\bar \omega_1)
+ \dfrac{B_{{\rm tr}}(\hat \omega_1,\hat\beta, \hat h)}
{A_{{\rm tr}}(\hat\beta,\hat h)}\hat \mu(d\hat \omega_1)\bar \mu(d\bar \omega_1),
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
A_{{\rm tr}}(\hat\beta,\hat h)
&= \tfrac12\left(\sum_{n=1}^{{\rm tr-1}}n[1+e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}]\rho(n)
+ {\rm tr}\sum_{n={\rm tr}}^{\infty}[1+e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}]\rho(n)\right),\cr
B_{{\rm tr}}(\hat \omega_1,\hat\beta, \hat h)&=\sum_{m=1}^{{\rm tr}-1}(m-1)\rho(m)
\left[1+e^{(m-1)[\hat M(2\hat\beta)-2\hat\beta \hat h]-2\hat\beta(\hat \omega_1+\hat h)}\right]\cr
&\qquad +({\rm tr}-1)\dfrac{1+e^{({\rm tr}-1)[\hat M(2\hat\beta)-2\hat\beta \hat h]
-2\hat\beta(\hat \omega_1+\hat h)}}{1+e^{{\rm tr}[\hat M(2\hat\beta)-2\hat\beta \hat h]}}
\sum_{m={\rm tr}}^{\infty}\rho(m)
\left[1+e^{m[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right],\cr
C_{{\rm tr}}(\hat \omega_1,\hat\beta, \hat h)
&=\sum_{m=1}^{{\rm tr}-1}\rho(m)\left[1+e^{(m-1)[\hat M(2\hat\beta)-2\hat\beta \hat h]
-2\hat\beta(\hat \omega_1+\hat h)}\right]\cr
&\qquad +\dfrac{1+e^{({\rm tr}-1)[\hat M(2\hat\beta)-2\hat\beta \hat h]
-2\hat\beta(\hat \omega_1+\hat h)}}{1+e^{{\rm tr}[\hat M(2\hat\beta)-2\hat\beta \hat h]}}
\sum_{m={\rm tr}}^{\infty}\rho(m)\left[1+e^{m[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right],
\end{split}
\end{equation}
and $\bar \mu_{\bar\beta}(d\bar \omega_1)=e^{\bar\beta \bar \omega_1-\bar M(-\bar\beta)}\bar \mu(d\bar \omega_1)$. Here we use that
\begin{equation}
q_{\hat\beta, \hat h, \bar\beta}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_m,\bar \omega_m))
=r(m)q_{m}(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_m,\bar \omega_m))
\end{equation}
with
\begin{equation}
\begin{split}
r(m) &=\dfrac{1+e^{m[\hat M(2\hat\beta)-2\hat\beta \hat h]}}{2\hat{\mathcal N}(\hat\beta,\hat h;0)}\rho(m),\cr
q_m(d(\hat \omega_1,\bar \omega_1),\ldots,d(\hat \omega_m,\bar \omega_m))
&=\dfrac{\left(1+e^{-2\hat\beta\sum_{i=1}^m(\hat \omega_i+\hat h)}\right)e^{\bar\beta \bar \omega_1}}
{e^{\bar M(-\bar\beta)}\left(1+e^{m[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right)}
\prod_{i=1}^m\hat \mu(d\hat \omega_i)\bar \mu(d\bar \omega_i).
\end{split}
\end{equation}
Note that $\widetilde\pi_1\Psi_{[q_{\hat\beta,\hat h_c^{\rm ann}(\hat\beta),0}]_{{\rm tr}}}
=(\tfrac12[\hat \mu+\hat \mu_{\hat\beta}])\otimes\bar \mu$, where $\hat \mu_{\hat\beta}(d\hat \omega_1)=e^{-2\hat\beta\hat \omega_1-\hat M(2\hat\beta)}
\hat \mu(d\hat \omega_1)$. Thus $\widetilde\pi_1\Psi_{[q_{\hat\beta,\hat h,\bar\beta}]_{{\rm tr}}}\neq\hat \mu\otimes \bar \mu$
for $\hat h\geq\hat h_c^{\rm ann}(\hat\beta)$, and so we have
\begin{equation}
\begin{split}
&S_*^\mathrm{que}(\hat\beta,h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h),\bar\beta)-\bar h
\leq S^\mathrm{ann}(\hat\beta,h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h),\bar\beta;0)-\bar h\cr
&-\inf_{ {q\in{\mathcal P}(\widetilde{E})} \atop h(q\mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty;
\,{m_q<\infty}}\left[h(q \mid q_{\hat\beta,h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h),\bar\beta})
+ (\alpha-1) m_{[q]_{{\rm tr}}} h(\widetilde\pi_1
\Psi_{[q]_{{\rm tr}} }\mid\hat \mu\otimes \bar \mu) \right]\cr
&=\qquad - \inf_{ {q\in{\mathcal P}(\widetilde{E})} \atop h(q\mid q_{\rho,\hat \mu\otimes\bar \mu})<\infty;
\,{m_q<\infty}}\left[h(q \mid q_{\hat\beta,h_c^\mathrm{ann}(\hat\beta,\bar\beta,\bar h),\bar\beta})
+ (\alpha-1) m_{[q]_{{\rm tr}}} h(\widetilde\pi_1
\Psi_{[q]_{{\rm tr}}} \mid\hat \mu\otimes \bar \mu) \right]<0.
\end{split}
\end{equation}
Since $S_*^\mathrm{que}(\hat\beta,h_c^\mathrm{que}(\hat\beta,\bar\beta,\bar h),\bar\beta)-\bar h = 0$ and since
$\hat h\mapsto S_*^\mathrm{que}(\hat\beta,\hat h,\bar\beta)$ is strictly decreasing on $(\hat h_c^\mathrm{ann}
(\hat\beta/\alpha),\infty)$, it follows that $h_c^\mathrm{que}(\hat\beta,\bar\beta,\bar h)<h_c^\mathrm{ann}
(\hat\beta,\bar\beta,\bar h)$ for $\bar h^{\rm ann}_c(\bar\beta)-\log 2<\bar h\leq\bar h^{\rm ann}_c(\bar\beta)$.
\end{proof}
\subsection{Proof of Corollary~\ref{hclbstrict}}
\begin{proof}
The map $\hat h\mapsto S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)$ is strictly decreasing and convex on
$(\hat h^{\rm ann}_c(\hat\beta/\alpha),\infty)$ (recall Fig.~\ref{fig-copadvarhs}). Therefore,
for $\bar h<s^\ast(\hat\beta,\bar\beta,\alpha)$, the $\hat h$-value that solves the equation $S^{\rm que}
(\hat\beta,\hat h,\bar\beta;0)=\bar h$ is strictly greater than $\hat h^{\rm ann}_c(\hat\beta/\alpha)$, which
proves that $h^{\rm que}_c(\hat\beta,\bar\beta,\bar h)>\hat h^{\rm ann}_c(\hat\beta/\alpha)$. The proof for
$\bar h\geq s^\ast(\hat\beta,\bar\beta,\alpha)$ follows from Corollary~\ref{hquehannvarfor} and
\eqref{ccqueflatp}.
\end{proof}
\subsection{Proof of Corollary~\ref{critqueann}}
\begin{proof}
(i) Note from \eqref{scopadf} that
\begin{equation}
\label{s6.37}
S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)-\bar h=\bar M(-\hat\beta)-\bar h+\log\left(\tfrac12
\left[1+\sum_{n\in{\mathbb N}}\rho(n)e^{n[\hat M(2\hat\beta)-2\hat\beta \hat h]}\right]\right).
\end{equation}
Note from \eqref{s6.37} and (\ref{Scopann}--\ref{Sanpin}) that
$S^{\rm ann}(\hat\beta,\hat h^{\rm ann}_c(\hat\beta),\bar\beta;0)-\bar h=\bar S^{\rm ann}(\bar\beta;0)-\bar h$ and
$S^{\rm ann}(\hat\beta,\hat h,\bar\beta;0)$ $-\bar h^{\rm ann}_c(\bar\beta)=\hat S^{\rm ann}(\hat\beta,\hat h;0)$.
These observations, together with the remark below Theorem~\ref{varflocann}, conclude
the proof for (i).\\
(ii) Recall from \eqref{Sdefalt} that
\begin{equation}
\begin{split}
S_*^\mathrm{que}(\hat\beta,\hat h,0) &= \sup_{Q\in{\mathcal C}^\mathrm{fin}}
\left[\Phi_{\hat\beta, \hat h}(Q)-I^\mathrm{que}(Q)\right]\cr
&=\sup_{\hat{Q}\in\hat{\mathcal C}^\mathrm{fin}}
\left[\Phi_{\hat\beta, \hat h}(\hat{Q})-\hat I^\mathrm{que}(\hat{Q})\right]
=\hat S^{\rm que}(\hat\beta,\hat h;0).
\end{split}
\end{equation}
The second equality uses the remark below Theorem~\ref{varfloc}. Hence $\tilde
h_c^{\rm que}(\hat\beta,0)=S_\ast^\mathrm{que}(\hat\beta,\hat h^{\rm que}_c(\hat\beta),0)=\hat
S^\mathrm{que}(\hat\beta,\hat h^{\rm que}_c(\hat\beta);0)=0$ by \cite{BodHoOp11}, Theorem 1.1(ii).
\end{proof}
\subsection{Proofs of Corollaries \ref{delocpathprop} and \ref{locpathprop}}
\begin{proof}
The proofs are similar to those of Corollaries 1.6--1.7 in \cite{BodHoOp11}, Section 8.
For the former, all that is needed is $S^{\rm que}(\hat\beta,\hat h,\bar\beta;0)-\bar h<0$, which holds for
$(\hat\beta,\hat h,\bar\beta,\bar h)\in {\rm int}({\mathcal D}^{\rm que}_1)\cup({\mathcal D}^{\rm que}\setminus{\mathcal D}^{\rm que}_1)$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,459 |
#ifndef BlobResourceHandle_h
#define BlobResourceHandle_h
#include "net/FileStreamClient.h"
#include "net/StreamReader.h"
#include "third_party/WebKit/public/platform/WebBlobData.h"
#include "third_party/WebKit/public/platform/WebURLRequest.h"
#include <wtf/PassRefPtr.h>
#include <wtf/Vector.h>
#include <wtf/text/WTFString.h>
#include <memory>
namespace blink {
class WebURLRequest;
class WebURLResponse;
class WebURLLoaderClient;
class WebURLLoader;
struct WebURLError;
}
namespace net {
class BlobDataWrap {
public:
BlobDataWrap();
~BlobDataWrap();
const Vector<blink::WebBlobData::Item*>& items() const { return m_items; }
void appendItem(blink::WebBlobData::Item* item) { m_items.append(item); }
int m_ref;
String m_contentType;
private:
Vector<blink::WebBlobData::Item*> m_items;
};
struct BlobTempFileInfo {
String tempUrl;
Vector<char> data;
int refCount;
};
class StreamWrap;
class BlobResourceLoader : public FileStreamClient {
public:
static BlobResourceLoader* createAsync(BlobDataWrap*, const blink::WebURLRequest&, blink::WebURLLoaderClient*, blink::WebURLLoader*);
static void loadResourceSynchronously(BlobDataWrap*, const blink::WebURLRequest&, blink::WebURLLoader*, blink::WebURLError&, blink::WebURLResponse&, Vector<char>& data);
void start();
int readSync(char*, int);
bool aborted() const { return m_aborted; }
// ResourceHandle methods.
/*virtual*/ void cancel() /*override*/;
/*virtual*/ void continueDidReceiveResponse() /*override*/;
~BlobResourceLoader();
private:
BlobResourceLoader(BlobDataWrap*, const blink::WebURLRequest&, blink::WebURLLoaderClient*, blink::WebURLLoader* loader, bool async);
// FileStreamClient methods.
virtual void didGetSize(long long) override;
virtual void didOpen(bool) override;
virtual void didRead(int) override;
void doNotifyFinish();
void doStart();
void getSizeForNext();
void seek();
void consumeData(const char* data, int bytesRead);
void failed(int errorCode);
void readAsync();
void readDataAsync(const blink::WebBlobData::Item&);
void readFileAsync(const blink::WebBlobData::Item&);
int readDataSync(const blink::WebBlobData::Item&, char*, int);
int readFileSync(const blink::WebBlobData::Item&, char*, int);
void notifyResponse();
void notifyResponseOnSuccess();
void notifyResponseOnError();
void notifyReceiveData(const char*, int);
void notifyFail(int errorCode);
void notifyFinish();
blink::WebURLRequest m_request;
blink::WebURLLoader* m_loader;
blink::WebURLLoaderClient* m_client;
BlobDataWrap* m_blobData;
bool m_async;
std::unique_ptr<StreamWrap> m_streamWrap;
Vector<char> m_buffer;
Vector<long long> m_itemLengthList;
int m_errorCode;
bool m_aborted;
long long m_rangeOffset;
long long m_rangeEnd;
long long m_rangeSuffixLength;
long long m_totalRemainingSize;
long long m_currentItemReadSize;
unsigned m_sizeItemCount;
unsigned m_readItemCount;
bool m_fileOpened;
friend class LoaderWrap;
enum RunType {
kStart,
kNotifyFinish
};
LoaderWrap* m_doNotifyFinishAsynTaskWeakPtr;
LoaderWrap* m_doStartAsynTaskWeakPtr;
void asynTaskFinish(RunType runType);
bool* m_isDestroied;
};
} // namespace WebCore
#endif // BlobResourceHandle_h
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,973 |
Civil Registration of Births, Marriages and Deaths begin
The civil registration of Birth, Marriage, or Death (BMD) events began in Ireland on 1 January 1864. This was considerably later than in England. (Civil records for members of the Church of Ireland began in 1845).
These civil records were collected according to the registration district and not the civil parish. An ancestor's Birth, Marriage, or Death (BMD) record would appear under a district name that could bear no resemblance to their home parish. (Deaths were registered in the district where an individual died, e.g. in a hospital located across a county boundary).
John Grenham's Placename Search gives the corresponding Registrar's District.
Irish civil registration records are now FREE to SEARCH online (for genealogical purposes; respecting the 100-year rule).
READ MORE: Guide to Irish Civil Records
LEARN MORE: How to locate a parish
Joannem (John) Dyer
Ireland XO DOB:1st Jun 1814
John McCullough Graham
Ireland XO DOB:22nd Sep 1832
John O'Dwyer
Ireland XO DOB:1741
Portland Row, 27-28 Portland Row. Dublin D01 YF59
Montpelier House, Montpelier hill | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,886 |
Lars Bukdahl (* 14. srpna 1968 Risskov) je dánský básník a literární kritik. Je synem socioložky Lisy Togebyové a filozofa Jørgena K. Bukdahla.
Životopis
Studoval na Aarhus Katedralskole. V roce 1994 získal titul z filozofie a srovnávací literatury na Kodaňské univerzitě a grant od Statens Kunstfond. Od roku 1996 pracoval jako recenzent pro Weekendavisen. Také vyučoval na kodaňské univerzitě a na Forfatterskolen. Publikoval v Hvedekorn, Den Blå Port, Passage a Øverste Kirurgiske. Od roku 2008 je literárním redaktorem ve Hvedekorn.
Dílo
Readymade! : stævnemøder og moderne digte, 1987 (básně)
Guldhornene, Borgen, 1988 (román)
Mestertyvenes tid, Borgen, 1989 (básně)
Kys mig, Borgen, 1990 (básně)
Brat Amerika, Borgen, 1990 (román)
Skyer på græs, Borgen, 1991 (básně)
Spiller boccia med kongen, Borgen, 1994 (básně)
Næseblod i Sofus City, Borgen, 1996 (básně)
Rimses den ene og Remses den anden, Borgen, 1997 (básně)
116 chok for sheiken, Borgen, 2001 (básně)
Generationsmaskinen. Dansk litteratur som yngst 1990-2004, Borgen, 2004 (kritika)
GO-GO! Readymade 2-3, Borgen, 2006 (básně)
Reference
Dánští spisovatelé
Dánští básníci
Narození v roce 1968
Muži
Absolventi Kodaňské univerzity
Žijící lidé
Básníci tvořící dánsky | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,213 |
\section{Introduction}\label{sec:intro}
\input{./journal/intro.tex}
\section{Signatures of curves}\label{sec:sig}
\subsection{Invariants of planar curves}\label{sec:diffex}
\input{./journal/curvature.tex}
\subsection{Differential signatures}\label{sec:diffsig}
\input{./journal/diffsig.tex}
\subsection{Joint signatures}\label{sec:jointsig}
\input{./journal/jointsig.tex}
\section{Witness sets for signatures}\label{sec:nag}
\subsection{Background}\label{sec:nagbackground}
\input{./journal/nagbackground.tex}
\subsection{A general equality test}\label{sec:equality}
\input{./journal/equality.tex}
\section{Implementation, examples, and experiments}\label{sec:implementation}
\input{./journal/exp.tex}
\section*{Acknowledgments}
{\small Research of T.~Duff is supported in part by NSF DMS-1719968, a fellowship from the Algorithms and Randomness Center at Georgia Tech, and by the Max Planck Institute for Mathematics in the Sciences in Leipzig.}
{\small Research of M.~Ruddy was supported in part by the Max Planck Institute for Mathematics in the Sciences in Leipzig.}
\bibliographystyle{acm}
\subsection{Witness sets for signatures}
\label{subsec:witness}
Our implementation of Algorithm~\ref{alg:equality-test} treats only the special case where the domain of each rational map is some Cartesian product of irreducible plane curves, say $X_i = C_i^k$ for some integer $k.$ For the purpose of our implementation, the various ingredients for the input to Algorithm~\ref{alg:equality-test} are easily provided. Suppose $\mathcal{I}_{C_i}=\langle f_i\rangle$ for $i=0,1.$ Then the reduced regular sequence we need is given by $\left( f_0 (x_1,y_1), \ldots , f_0 (x_k,y_k) \right).$ Sampling from $X_0$ amounts to sampling $k$ times from $C_0$; we sample the curve $C_0$ using homotopy continuation from a linear-product start system~\cite[Sec 8.4.3]{SW05}.\\\\
It remains to discuss computation of the witness set for the image of the signature map.
We now summarize the relevant techniques from numerical algebraic geometry in this setting.
For a \emph{generic plane curve} of degree $d,$
\[
f(x,y;p) = p_{0,0} + p_{1,0} x + p_{0,1} y + p_{1,1} x\, y + \cdots + p_{0,d} y^d,
\]
it is natural to consider the signature map which is a rational function in the parameters $p=(p_{0,0}, \ldots , p_{0,d}).$
We may write the parametric signature map as $\Phi (x,y;p) = (\Phi_1 (x,y;p), \ldots, \Phi_m (x,y;p)).$
There is an associated incidence correspondence
\[
V_\Phi = \{ (x_1,y_1,\ldots , x_k, y_k, p, L) \in (\mathbb{C}^2)^k \times \mathbb{C}^{\binom{d+2}{2}} \times \mathbb{G}_{k,m} \mid f(x_i, y_i, p) = 0 , \Phi (x,y; p) \in L \},
\]
where $\mathbb{G}_{k,m}$ denotes the Grassmannian of codimension-$k$ affine subspaces of $\mathbb{C}^m.$ For generic $L,$ the fiber over $(p, L)$ of the projection $\pi: V_\Phi \rightarrow \mathbb{C}^{\binom{d+2}{2}} \times \mathbb{G}_{k,m}$ is naturally identified with a pseudowitness set for the signature map of the curve corresponding to $p$, denoted $\Phi (\cdots ; p).$
\begin{proposition}\label{prop:VPHI}
The incidence variety $V_\Phi$ is irreducible.
\end{proposition}
\begin{proof}
The fibers of the coordinate projection $\pi_1: V_{\Phi} \to (\mathbb{C}^2)^k \times \mathbb{C}^{\binom{d+2}{2}}$ given by $(x,y,p, L) \mapsto (x,y,p)$ are affine-linear spaces.
Thus, a Zariski-open subset of $V_\Phi $ is an affine bundle over some base $B_1\subset (\mathbb{C}^2)^k\times \mathbb{C}^{\binom{d+2}{2}}.$
Let $X$ denote the Zariski closure of the image of $\pi_1.$
We may then consider the coordinate projection $\pi_2: X \to \mathbb{C}^{\binom{d+2}{2}},$ whose image is Zariski-dense in $\mathbb{C}^{\binom{d+2}{2}}.$
Once again, $\pi_2$ restricts to an affine bundle over a base $B_2 \subset \mathbb{C}^{\binom{d+2}{2}},$ which is now easily seen to be connected in the complex topology.
Therefore, considering both $\pi_2$ and $\pi_1,$ there exists a dense Zariski-open subset $U \subset V_\Phi$ which is connected in the complex topology.
Therefore $U$ is irreducible in the Zariski topology, giving that $\overline{U} = V_\Phi $ is also irreducible.
\end{proof}
The fiber $\pi^{-1}(p,L)$ of the projection defined above is a zero-dimensional subset of $V_\Phi$ with cardinality $N$ given by the product of degree of the signature variety and the size of the symmetry group for a generic curve of degree $d$. Consider the subset $B\subset \pi(V_\Phi)$ where each fiber has the same cardinality, i.e. \[
B = \{(p,L)\, :\, |\pi^{-1}(p,L)| = N\}.
\]
Note that a generic $(p,L)\in\mathbb{C}^{\binom{d+2}{2}} \times \mathbb{G}_{k,m}$ lies in $B$.
For a fixed $(p,L)\in B$, the \emph{monodromy group} ${\mathcal M} (\pi ; p,L)$ is a permutation group which acts on the fiber of $\pi^{-1}(p,L)$ by lifting loops in $B$ based at $(p,L)$ to $V_\Phi$.
Proposition \ref{prop:VPHI} acts transitively on the points in the pseudowitness set.
It now follows that, for the curve of interest given by $p_1\in \mathbb{C}^{\binom{d+2}{2}},$ we may compute a pseudo-witness set for the signature $\Phi_1 = \Phi (\cdots ; p_1)$ using the following steps, which are standard in numerical algebraic geometry:
\begin{itemize}
\item[1)] Fix generic $(x_0,y_0)\in \mathbb{C}^{2k}$, and find $(p_0, L_0)$ so that $(x,y,p_0, L_0) \in V_\Phi$ by solving linear systems of equations.
\item[2)] Using the transitivity of the monodromy group, complete $(x_0,y_0)$ to a pseudowitness set for the curve given by $p_0.$
\item[3)] The pseudowitness set for $p_1$ will consist of finite endpoints as $t\to 1$ of the homotopy (see.~\cite{MS89})
\begin{equation}
\label{eq:homotopy2}
H_p(x;t) = \left(
\begin{array}{c}
f(x,y; t p_1 + (1-t) p_0 )\\
L \circ \Phi (x, y; t p_1 + (1-t) p_0)
\end{array}\right) =0.
\end{equation}
\end{itemize}
Viewed as a subgroup of the symmetric group $S_N,$ it natural to ask how the monodromy group ${\mathcal M} (\Phi ; p, L)$ depends on the type of signature map $\Phi $ and the generic curve degree $d.$
As soon as $d$ is large enough, a generic curve specified by $p$ will have a trivial symmetry group.
In our experiments, our computations show that each monodromy group ${\mathcal M} (\Phi ; p, L)$ is the entire symmetric group $S_N$ in such cases.
In general, we have that ${\mathcal M} (\Phi ; p, L)$ is a subgroup of the \emph{wreath product} $S_{N_1} \wr S_{N_2},$ where $N_1$ is the size of the generic symmetry group and $N_2 = \deg \imclo{\Phi_{p}}.$
Thus, for families of curves with a nontrivial symmetry group, ${\mathcal M} (\Phi ; p, L)$ is \emph{imprimitive}, in which case the \emph{decomposable monodromy} technique from~\cite{AR16} may be used to speed up witness set computation.
Finally, we note that Proposition~\ref{prop:VPHI} and the homotopy from Equation~\eqref{eq:homotopy2} can be considered in more structured settings---for instance, when the curves $f(x,y;p)$ are drawn from a linear subspace of $p\in \mathbb{C}^{\binom{d+2}{2}},$ or when the image slices $L$ are multiprojective as in the sense of Definition~\ref{def:multi-witness-set}.
We leave the study of monodromy groups in these settings as an interesting direction for further research.
\section{Constructing a Fundamental Set of Invariants}
Suppose we are in the setting of a smooth Lie group action $G$ (where $\dim G=r$) on a manifold $M$ of dimension $n$. For a point $z\in M$ denote $\mathcal{O}_z$ as its orbit. Our goal will be construct a \textit{fundamental} set of a invariants.
\begin{definition}
A collection of invariants $\mathcal{I}=\{I_1,\hdots , I_k\}$ for the action of $G$ on $M$ forms a \textbf{fundamental set} if they are functionally independent and any invariants can be expressed as a smooth function of $I_1,\hdots I_k$.
\end{definition}
We will assume that the action of $G$ on $M$ is
\begin{itemize}
\item \textit{free}, meaning $\{g\in G\, |\, g\cdot z=z\}=\{e\}$ for all $z\in M$, and
\item \textit{regular}: $\mathcal{O}_z$ all have the same dimension and each point in $M$ has arbitrarily small neighborhoods having pathwise-connected intersection with each group orbit.
\end{itemize}
These conditions can be relaxed by either considering the \textit{local} action of $G$ on a neighborhood of $M$, in which case the following discussion still follows but only locally, or one can consider actions which are simply \textit{semiregular} (all orbits have same dimension), and construct a fundamental sets of invariants using cross-sections (see \cite{HK07}).
\begin{definition}
A \textbf{moving frame} is a smooth, $G$-equivariant map $\rho : M\rightarrow G$. In other words $\rho(g\cdot z)= \rho(z) \cdot g^{-1}$.\footnote{Alternatively one can call this a \textit{right} moving frame and similarly define a \textit{left} moving frame.}
\end{definition}
A moving frame exists if and only if the action of $G$ is free and regular. We can construct a moving frame by choosing a \textit{cross-section} to the orbits of $G$ on $M$.
\begin{definition}
A \textbf{cross-section} is an $(n-r)$ dimensional submanifold $K\subset M$ that intersects each orbit transversally in exactly one point.
\end{definition}
Then the map $\rho : M\rightarrow G$ where $\rho(z)=g$ is the unique group element such that $g\cdot z\in (\mathcal{O}_z\cap K)$.\footnote{Any moving frame $\rho$ has an associated cross-section given by $K=\rho^{-1}(e)$.} Suppose we have a cross-section given by the $r$ equations
$$
Z_1(z)=c_1, \quad \hdots\quad Z_r(z)=c_r,
$$
where $c_1, \hdots, c_r$ are constants. A moving frame can then be defined by solving
$$
Z_1(g\cdot z)=c_1, \quad \hdots\quad Z_r(g\cdot z)=c_r,
$$
for the group parameters $g=(g_1, \hdots g_r)$ in terms of the coordinates $z=(z_1,\hdots, z_n)$. Transversality and the Implicit Function Theorem imply the existence of a solution in a neighborhood of each $z\in M$.
\begin{example}
Consider the action $G$ of rotation on $\mathbb{R}^2\backslash\{(0,0)\}$. This action is given by the map
$$
(x,y) \mapsto (cx-sy,sx+cy),
$$
where $c^2+s^2=1$. Then the set
$$
K=\{(x,y)\in \mathbb{R}^2\,|\, x=0, y\geq 0\}
$$
is a cross-section for this action. The moving frame is obtained by solving the equations
$$
cx-sy=0, \quad c^2+s^2=1.
$$
Thus the map is given by $c=\frac{y}{\sqrt{x^2+y^2}}, s=\frac{x}{\sqrt{x^2+y^2}}$.
\end{example}
The \textit{invariantization} of any function $F(z_1,\hdots,z_n)$ is the function $\iota F$ which agrees with $F$ on the cross-section $K$. This can be obtained via the moving frame map as $\iota F(z)= F(\rho(z)\cdot z)$. In particular one can invariantize the coordinate functions to obtain the set $\mathcal{I}=\{\iota z_1, \hdots, \iota z_n\}$, and then
$$
\iota F(z_1,\hdots z_n)=F(\iota z_1, \hdots, \iota z_n).
$$
Since for any invariants function $\iota I= I$, we can rewrite any invariant in terms of the invariantized coordinate function, meaning the set $\mathcal{I}$ is a fundamental set of invariants.
\begin{example}
Consider the moving frame map in the previous example for rotations of $\mathbb{R}^2$. The invariantized coordinate functions are given by
$$
\iota x= \rho(x,y)\cdot x=0 \quad \text{and} \quad \iota y = \rho(x,y) \cdot y=\left(\frac{x}{\sqrt{x^2+y^2}}\right)x+\left(\frac{y}{\sqrt{x^2+y^2}}\right)y=\sqrt{x^2+y^2}.
$$
The invariant $\iota x=0$ is known as a \textit{phantom invariant}, and the invariant $\iota y$ is a fundamental invariant for this action. Thus any invariant can be written as a smooth function of $\sqrt{x^2+y^2}$.
\end{example}
\section{Prolonged Action on Derivatives}
In this section we consider the action of $G$ on the class of $p$-dimensional submanifolds of $M$. The action of $G$ on $M$ induces an action on the $n$th order submanifold jet bundle $J^n=J^n(M,p)$ (for more details see \cite{olver:purple}). This action is called the $n$th order \textit{prolongation} of $G$. The action can be defined recursively through implicit differentiation.
Most `interesting' actions are not free, but we can still construct meaningful invariants by prolonging the action to the higher-dimensional jet space. Provided $G$ acts effectively, the prolonged action of $G$ on $J^n$ is locally free. Thus we can use a cross-section to construct a fundamental set of (differential) invariants for this action.
\begin{example}
Consider the action of rotation on curves in $\mathbb{R}^2$. Locally we can consider a curve as the graph of a smooth function $y=f(x)$. For any curve $C\subset \mathbb{R}^2$, a lift of the curve to its jet space is given by
$$
j_C(x,y)=\left(x,y,y^{(1)},y^{(2)},\hdots y^{(n)}\right)
$$
where $y^{(i)}$ denotes the $i$th derivative of $y$ with respect to $x$ at $(x,y)$ and $j_C:\mathbb{R}^2\rightarrow J^n$. Here we can consider $J^n=\mathbb{R}^{n+2}$ as Euclidean space with coordinates $\left(x,y,y^{(1)},\hdots y^{(n)}\right)$.\footnote{Technically this is `patch' of the full jet space $J^n$. Intuitively it functions similarly to an affine chart of projective space.} The action of rotation is given by the coordinate functions $\overline{x}=cx-sy,\, \overline{y}=sx+cy$. Applying the operator
$$
\frac{d}{d\overline{x}}=\frac{dx}{d\overline{x}}\frac{d}{dx}=\left(\frac{1}{c-s\frac{dy}{dx}}\right)\frac{d}{dx}
$$
yields
$$
\overline{y}^{(1)}=\frac{d\overline{y}}{d\overline{x}}=\frac{d}{d\overline{x}}(\overline{y})=\left(\frac{1}{c-s\frac{dy}{dx}}\right)\frac{d}{dx}(\overline{y})=\frac{s+c\frac{dy}{dx}}{c-s\frac{dy}{dx}}.
$$
Thus the prolonged action on $J^1$ is given by
$$
g\cdot \left(x,y,y^{(1)}\right)=\left(cx-sy,sx+cy,\frac{s+cy^{(1)}}{c-sy^{(1)}}\right).
$$
One could recursively define the action on $J^n$ by repeatedly applying $\frac{d}{d\overline{x}}$ to $\overline{y}^{(1)}$. The moving frame defined in previous examples yields the invariantized coordinate functions:
$$
\iota x=0, \quad \iota y = \sqrt{x^2+y^2}, \quad \iota y^{(1)}=-\frac{yy^{(1)}+x}{xy^{(1)}-y}.
$$
The set $\{\iota y, \iota y^{(1)}\}$ is a fundamental set of invariants for the prolonged action of $G$ on $J^1$.
\end{example}
Under certain regularity conditions around a point, we can locally assume that a $p$-dimensional submanifold of $M$ is given by the graph of $n-p$ coordinate functions
$$
z_{n-p}(z_1,\hdots, z_p),\hdots, z_{n}(z_1,\hdots, z_p).
$$
Denote the dependent variables as $x^1,\hdots, x^p$ and the independent variables as $u^1,\hdots, u^{n-p}$. We can consider a point $z\in M$ as $(x^1,\hdots , x^p, u^1,\hdots u^{n-p})$. Then $u^{\alpha}_J$ is the $k$th order partial derivative of $u^\alpha$ with respect to $J=(j_1,j_2,\hdots, j_k)$ where $\alpha \in \{1,\hdots, n-p\}$ and $j_i\in \{1,\hdots, p\}$ for all $i$. Thus there are $p$ differential operators $D_i=\frac{d}{dx^i}$ associated with $p$-dimensional submanifolds of $M$. A moving frame also invariantizes these differential operators.
\begin{definition}
An \textbf{invariant differential operator} for the action of $G$ is a differential operator $D=\sum_{i=1}^p f_i(z)\frac{d}{dx^i}$ such that
$$
\overline{D}=\sum_{i=1}^p f_i(\overline{x}^1,\hdots , \overline{x}^p, \overline{u}^1,\hdots \overline{u}^{n-p})\frac{d}{d\overline{x}^i}=D.
$$
\end{definition}
Note that applying an invariant differential operator to an invariant function yields an invariant function. Let $H$ be any invariant function. Then $DH(z)=\sum_{i=1}^p f_i(z)H_i,$ and
\begin{align*}
DH(g\cdot z)&=\sum_{i=1}^p f_i(\overline{x}^1,\hdots , \overline{x}^p, \overline{u}^1,\hdots \overline{u}^{n-p})H_i(\overline{x}^1,\hdots , \overline{x}^p, \overline{u}^1,\hdots \overline{u}^{n-p})\\
&=\sum_{i=1}^p f_i(\overline{x}^1,\hdots , \overline{x}^p, \overline{u}^1,\hdots \overline{u}^{n-p})\frac{d}{d\overline{x}^i}\left(H(\overline{x}^1,\hdots , \overline{x}^p, \overline{u}^1,\hdots \overline{u}^{n-p})\right)\\
&=\overline{D}H(g\cdot z)=\overline{D}H(z)=DH(z).
\end{align*}
A moving frame can be used to construct a set of differential operators from the operators $D_1,\hdots, D_p$. First note that the operator $\overline{D}_i=\frac{d}{d\overline{x}^i}$ is given by
$$
\overline{D}_i=\sum_{k=1}^p((\overline{D}x)^{-1})_{(i,k)}D_k
$$
where $\overline{D}x$ is the Jacobian matrix of the transformation $x\mapsto \overline{x}$, i.e. $\overline{D}x=\left(\frac{d\overline{x}^i}{dx^k}\right)$. Then the invariantization of $D_i$ along the moving frame is given by $\mathcal{D}_i= \overline{D}_i|_{g=\rho(z)}.$
\begin{example}
For on curves in $\mathbb{R}^2$ there is a single differential operator $\frac{d}{dx}$. The action of rotation on $\mathbb{R}^2$ transforms this operator into
$$
\frac{d}{d\overline{x}}=\frac{dx}{d\overline{x}}\frac{d}{dx}=\left(\frac{1}{c-sy^{(1)}}\right)\frac{d}{dx}.
$$
The moving frame given by $c=\frac{y}{\sqrt{x^2+y^2}}, s=\frac{x}{\sqrt{x^2+y^2}}$ yields the differential operator
$$
\mathcal{D}=\left(\frac{\sqrt{x^2+y^2}}{y-xy^{(1)}}\right)\frac{d}{dx}.
$$
Note that this is an invariant differential operator:
\begin{align*}
\overline{\mathcal{D}}&=\left(\frac{\sqrt{x^2+y^2}}{(sx+cy)-(cx-sy)\frac{s+cy^{(1)}}{c-sy^{(1)}}} \right)\frac{d}{d\overline{x}}\\
&=\left(\frac{\sqrt{x^2+y^2}}{(sx+cy)-(cx-sy)\frac{s+cy^{(1)}}{c-sy^{(1)}}} \right)\left(\frac{1}{c-sy^{(1)}}\right)\frac{d}{dx}\\
&=\left(\frac{\sqrt{x^2+y^2}}{(c-sy^{(1)})(sx+cy)-(cx-sy)(s+cy^{(1)})} \right)\frac{d}{dx}\\
&=\left(\frac{\sqrt{x^2+y^2}}{y-xy^{(1)}} \right)\frac{d}{dx}\\.
\end{align*}
\end{example}
\section{Differential Invariant Algebras}
For a free action of $G$ on $J^n$, denote a moving frame $\rho:J^n \rightarrow G$ as an $n$th order moving frame. Then the differential invariants constructed by the moving frame are of order $\leq n$. Note that the differential invariant operators take differential invariants of order $n$ to differential invariants of order $n+1$. Alternatively one can prolong the action to $J^{n+1}$ to a construct a fundamental set of differential invariants using the same moving frame. The invariants constructed via the associated moving frame of order $n+1$ and invariant differentiation are usually not the same. They do however have interesting relationships as explored in [[[DifferentialInvariantAlgebras]]].
The key fact here is that an $n$-th order moving frame produces an $n+1$-th order differential invariant for each coordinate $u^\alpha_J$, where $|J|=n+1$, and hence the number of functionally independent differential invariants of order $n+1$ is equal to the size of $\{u^\alpha_J\, |\, |J|=n+1\}$. Invariant differentiation of this set provides a functionally independent set of $n+2$-th order differential invariants equal to the size of $\{u^\alpha_J\, |\, |J|=n+2\}$. In this case the invariant differential operators allow one to construct a fundamental set of invariants for the action of $G$ on $J^{n+2}$ from the fundamental set of invariants for the action of $G$ on $J^{n+1}$.
\begin{definition}
A set of differential invariants $\mathcal{I}$ is \textbf{generating} if any other differential invariant can be written as a smooth function of the invariants in $\mathcal{I}$ or their invariant derivatives.
\end{definition}
\begin{theorem}
For the free action of $G$ on $J^n$, an $n$-th order moving frame on $J^{n+1}$ produces a generating set of differential invariants.
\end{theorem}
In practice, the syzygies between the differential invariants often imply a smaller subset will be generating.
\begin{example}
For the action of rotation on curves the moving frame was of order $0$, but also produced a first order differential invariant by considering the action on $J^1$:
$$
\iota y^{(1)}=-\frac{yy^{(1)}+x}{xy^{(1)}-y}.
$$
We could also apply the invariant differential operator to $\iota y =\sqrt{x^2+y^2}$ to produce another differential invariant of first order:
$$
\mathcal{D}(\iota y)=\left(\frac{\sqrt{x^2+y^2}}{y-xy^{(1)}}\right)\frac{d}{dx}(\sqrt{x^2+y^2})=-\frac{yy^{(1)}+x}{xy^{(1)}-y}.
$$
In this case the two differential invariants coincide. The above theorem guarantees that $\iota y$ and $\iota y^{(1)}$ are a generating set of differential invariants, but we see that $\iota y^{(1)}$ can be obtained from $\iota y$ by applying $\mathcal{D}$. Therefore $\sqrt{x^2+y^2}$ generates the `algebra' of differential invariants for the action of rotation on $\mathbb{R}^2$.
\end{example}
\begin{example}
See Example 3.1 in http://www-users.math.umn.edu/~olver/mf_/mfm.pdf for a derivation of Euclidean curvature using a moving frame for the action of rotations and translations on curves in $\mathbb{R}^2$.
\end{example}
\section{Equivalence and Symmetry}
In this section we see how the above can be used to construct signatures of $p$-dimensional submanifolds, which determine their equivalence classes under $G$ (locally). The key idea is to use the fact that the functional relatioships between the differential invariants for a particular submanifold completely determine its equivalence class (See \cite{FO99}). Because a finite set of differential invariants can generate the entire differential algebra through invariant differentiation, a finite set of functional relationships will completely determine all functional relationships between the differential invariants.
Consider a generic $p$-dimensional submanifold $S$, and suppose that we have an $n$th order moving frame $\rho:J^n \rightarrow G$ for the free action of $G$ on $J^n$. Further suppose that $\dim J^n - \dim G \geq p$. Then there are at least $p$ independent differential invariants of order $n$. This implies that, when restricted to $S$, the $(n+1)$-th order differential invariants can (locally) be written as smooth functions of $n$th order differential invariants. Thus, by invariant differentiation and the chain rule, every higher order differential invariant on $S$ is determined by these functions.
For example, we can think of the $n$-th and $n+1$-th order differential invariants as defining a map on $S$
$$
\Theta(x^1,x^2,\hdots ,x^p)=(I_n^1,I_n^2,\hdots, I_n^p, I_n^{p+1}, \hdots, I_{n+1}^1, I^2_{n+1} \hdots),
$$
where the subscript denotes the order of the differential invariants and $\Theta: S\rightarrow \mathbb{R}^m$ where $m=\dim J^{n+1}-\dim G$. Denote the image of $\Theta$ as $\mathcal{S}$, i.e. $\mathcal{S}=\Theta(S)$, and call it the \textit{differential signature} of $S$. Then $\mathcal{S}$ is a submanifold of $\mathbb{R}^m$ with coordinates $(I_n^1,I_n^2,\hdots, I_n^p, I_n^{p+1}, \hdots, I_{n+1}^1, I^2_{n+1} \hdots)$ of dimension $p$. Thus we can locally write it as the graph of functions in $p$ variables:
\begin{align*}
I_n^{p+1}&=F_n^{p+1}(I_n^1,I_n^2,\hdots, I_n^p)\\
&\vdots\\
I_{n+1}^1&=F_{n+1}^1(I_n^1,I_n^2,\hdots, I_n^p)\\
I_{n+1}^2&=F_{n+1}^2(I_n^1,I_n^2,\hdots, I_n^p)\\
&\vdots\\
\end{align*}
Thus the differential signature encodes the fundamental functional relationships between the differential invariants. We can see that the $n+2$-th order differential invariants are completely determined by invariant differentiating the above functions. For instance
\begin{align*}
\mathcal{D}(I_{n+1}^1)&=\sum_{i=1}^p \mathcal{D}(I_n^i)\frac{\partial F_{n+1}^1}{\partial I^i_n}(I_n^1,I_n^2,\hdots, I_n^p)\\
&=\sum_{i=1}^p F_{n+1}^i\frac{\partial F_{n+1}^1}{\partial I^i_n}.\\
\end{align*}
Therefore the differential signature encodes all functional relationships between the differential invariants. In \cite{FO99} they show that this is enough to determine (local) equivalence of submanifolds.
\begin{theorem}
For sufficiently regular $p$-dimensional submanifolds $S, \overline{S}$ and an $n$th order moving frame $\rho$ where $\dim J^n - \dim G \geq p$, the differential signatures $\mathcal{S}$ and $\overline{\mathcal{S}}$ are parameterized by the $n$th and $n+1$th order differential invariants restricted to $S$ and $\overline{S}$ respectively. In other words $S$ and $\overline{S}$ and (locally) equivalence under $G$ if and only if $\mathcal{S}=\overline{\mathcal{S}}$.
\end{theorem}
When $\dim J^n - \dim G < p$, one can take the $n$th, $n+1$th and $n+2$th order differential invariants. Because of syzygies between the invariants, it is unlikely one needs the entire collection to define a signature.
\begin{example}
For the action of rotation on curves, the moving frame of order $0$ produced one $0$th order differential invariant:
$$
\iota y =\sqrt{x^2+y^2}.
$$
Since the dimension of curves is equal to the number of invariants produced, we can take as our signature the map
$$
\Theta: \mathbb{R}^2 \rightarrow \mathbb{R}^2
$$
defined by the generating invariant and its invariant derivative $(x,y)\mapsto \left(\sqrt{x^2+y^2},-\frac{yy^{(1)}+x}{xy^{(1)}-y}\right).$ For a particular curve $C$, the image of $C$ under $\Theta$ is its differential signature $\mathcal{S}=\Theta(C)$. Locally we can write the signature of $C$ as
$$
\iota y^{(1)} = F(\iota y)
$$
for some smooth function $F$. Since any differential invariant can be obtained by differentiating $\iota y$ with respect to $\mathcal{D}$, this functional relationship completely determines all other differential invariants on $C$. For example, $\{\iota y,\iota y^{(1)}, \mathcal{D}\left(\iota y^{(1)}\right)\}$ is a fundamental set of differential invariants for the action of rotations on $J^2$, and, on $C$, we can write
$$
\mathcal{D}\left(\iota y^{(1)}\right)=\iota y^{(1)} F_{\iota y}(\iota y)=F(\iota y)F_{\iota y}(\iota y).
$$
Any two smooth curves will be locally equivalent under a rotation if and only if they have the same signature.
\end{example}
The size of the pre-image of a generic point of the differential signature $\mathcal{S}$ also encodes information about the symmetry group of $S$ under $G$. If $S$ has a discrete symmetry group $G_S$ where $|G_S|=k$, then the map $\Theta$ is generically a $k:1$ mapping. The group action of $G$ on $J^n$ is free, and thus for most $p\in S$,
$$
|\{ q\in S\, |\, g\cdot q=p\}|=|\{\Theta^{-1}(\Theta(p))\}|=k.
$$
When $G_S$ is infinite, the dimension of $G_S$ also influences the dimension of $\mathcal{S}$. In many cases, if $\dim G_S=k$, then $\dim \mathcal{S}$ drops by $k$.
\section{Joint Invariants}
Another approach to constructing invariants and determining equivalence of $p$-dimensional submanifolds of $M$ under the action of $G$ is by considering the induced action on the product space $M^m=M\times M\times \hdots \times M$ given by
$$
g\cdot (q_1,q_2,\hdots, q_m)=(g\cdot q_1, g\cdot q_2, \hdots, g\cdot q_m)
$$
for $g\in G$ and $(q_1,\hdots, q_m)\in M$. The same treatment of moving frames (to construct fundamental sets of invariants) and signatures (to use the differential invariant algebra to determine equivalence and symmetry) can be applied to this case as well: $p$-dimensional submanifolds of $M^m$. The jet space of In particular if we consider $p$-dimensional submanifolds $S$ of $M$, then these also correspond to $mp$-dimensional submanifolds of $M^m$ of the form
$$
S\times S\times \hdots \times S.
$$
The action of $G$ preserves the set of such submanifolds of $M^m$. Additionally the $n$-th order jet space of this special case is given by $(J^n)^{\times m}$, a subset (preserved by the induced action of $G$) of the larger jet space of all submanifolds of $M^m$.
One reason for doing this is to attempt to define a differential signature map with only $0$th order, or simply lower order, differential invariants. We call $0$th order invariants of the product space $M^m$, \textit{joint invariants} and differential invariants on the product space $(J^n)^{\times m}$ \textit{joint differential invariants}.
This is particularly useful when the action of $G$ is transitive on $M$, but becomes free when taking a sufficiently high product space $M^m$. In this case, if $\dim M=n$ we must have that
$$
r< mn,
$$
which implies that there exists a fundamental set of invariants of size $mn-r$. However a signature map is defined by the functional relationships between invariants. Thus the number of invariants needed must be greater than the dimension of the submanifold considered, and for $p$-dimensional submanifolds of $M$ we must have
$$
mp < mn-r.
$$
\begin{example}
For example consider curves in $\mathbb{R}^2$ under the action of $G=\mathcal{S}\mathcal{E}(2)$, the special Euclidean group of rotations and translations. Here $p=1$ and $r=\dim \mathcal{S}\mathcal{E}(2)=3.$ The action of $G$ on $\mathbb{R}^2\times \mathbb{R}^2$ is free and there is one fundamental invariant, $(x_1-x_2)^2+(y_1-y_2)^2$ (since $2(2)-3=1$). However a curve $C\subset \mathbb{R}^2$ defines a $2$-dimensional submanifold $C\times C\subset \mathbb{R}^2\times \mathbb{R}^2$.
To have any hope of defining a signature map using joint invariants, we must consider the action of $G$ on $(\mathbb{R}^2)^4$ since
$$
4(1) \leq 4(2)-3.
$$
The six distances between points in each copy of $\mathbb{R}^2$ generate the joint differential algebra (for details see \cite{Olv00}) and define a signature map for submanifolds $(C)^4\subset (\mathbb{R}^2)^4$. Thus this is a signature map for curves $C\subset \mathbb{R}^2$ since $C_1 \cong_G C_2$ if and only if $(C_1)^4 \cong_G (C_2)^4$.
Note that since $4(2)-3=5$, these six distances are not functionally independent. There exists a functional relationship between them for all curves (called a \textit{universal syzygy}) along with a functional relationship specific to curve one is restricting to. One reason for using all six, however, is that the corresponding signature manifold $\mathcal{S}$ will have more symmetry.
\end{example}
For a detailed exposition on the example of $\mathcal{S}\mathcal{E}(2)$ see Example 3.6 and Theorem 8.5 in \cite{Olv00} (http://www-users.math.umn.edu/~olver/mf_/ji.pdf).
\section{Nested Formulas for Euclidean Signature Maps}
The rational \textbf{Euclidean} differential signature map (not special Euclidean!) is given in jet notation by
$$
(\kappa^2,\kappa_s^2)=\left(\frac{\left(y^{(2)}\right)^2}{\left(1+\left(y^{(1)}\right)^2\right)^3},\frac{\left(y^{(3)}\left(1+\left(y^{(1)}\right)^2\right)-3y^{(1)}\left(y^{(2)}\right)^2\right)^2}{\left(1+\left(y^{(1)}\right)^2\right)^6}\right).
$$
Alternatively we can write this in terms of the partial derivatives of $F(x,y)$ and try to simplify the expression:
\footnotesize{
\begin{align*}
\kappa^2&=\frac{\left(F_x^2F_{yy}-2F_{xy}F_xF_y+F_{xx}F_y^2\right)^2}{\left(F_x^2+F_y^2\right)^3}\\
\kappa_s^2&=\left(1/\left(F_x^2+F_y^2\right)^6\right)\left((-F_x^2F_y^3-F_y^5)F_{xxx}+(3F_x^3F_y^2+3F_y^4F_x)F_{xxy}+\right.\\
&\left.(-3F_x^4F_y-3F_x^2F_y^3)F_{xyy}+(F_x^5+F_x^3F_y^2)F_{yyy}-3((F_x^2-F_y^2)F_{xy}-F_xF_y(F_{xx}-F_{yy}))(F_x^2F_{yy}-2F_{xy}F_xF_y+F_{xx}F_y^2)\right)^2
\end{align*}
}
The two-point Euclidean joint differential signature map is given in jet coordinates by
$$
\left(d_{12}, \iota y^{(1)}_1, \iota y^{(1)}_2 \right) = \left((x_1-x_2)^2+(y_1-y_2)^2, \frac{(y_2-y_1)y^{(1)}_1-(x_1-x_2)}{(x_1-x_2)y^{(1)}_1-(y_1-y_2)},\frac{(y_2-y_1)y^{(1)}_2-(x_1-x_2)}{(x_1-x_2)y^{(1)}_2-(y_1-y_2)}\right)
$$
Alternatively we can write this in terms of the partial derivatives of $F(x,y)$ and try to simplify the expression:
\begin{align*}
\left(d_{12}, \iota y^{(1)}_1, \iota y^{(1)}_2 \right) = &\left((x_1-x_2)^2+(y_1-y_2)^2,\right. \\
&\left.\frac{(y_2-y_1)F_x(x_1,y_1)+(x_1-x_2)F_y(x_1,y_1)}{(x_1-x_2)F_x(x_1,y_1)+(y_1-y_2)F_y(x_1,y_1)},\frac{(y_2-y_1)F_x(x_2,y_2)+(x_1-x_2)F_y(x_2,y_2)}{(x_1-x_2)F_x(x_2,y_2)+(y_1-y_2)F_y(x_2,y_2)}\right)
\end{align*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,635 |
\section{Introduction}\label{intro}
The notion of relatively hyperbolic groups was introduced in \cite{Gro87}
and has been studied by many authors
(see for example \cite{Bow12}, \cite{D-S05b}, \cite{Far98} and \cite{Osi06a}).
When a countable group is relatively hyperbolic,
relative quasiconvexity for subgroups can be defined.
There are several equivalent definitions of relative hyperbolicity for countable groups
and those of relative quasiconvexity for subgroups (see \cite[Section 3 and Section 6]{Hru10}).
In this paper we adopt definitions in terms of geometrically finite convergence actions.
We regard a conjugacy invariant collection of subgroups of a countable group
relative to which the group is hyperbolic as a structure on the group
and call such a collection a relatively hyperbolic structure on the group.
In this paper we discuss the notion of
the universal relatively hyperbolic structure on a group
which is used in order to characterize relatively hyperbolic structures on the group.
In particular we give a characterization of
the universal relatively hyperbolic structure on a finitely generated group
(Corollary \ref{char-universal}).
We also study relations between relatively hyperbolic structures on a group
and relative quasiconvexity for subgroups of the group.
Indeed we give two theorems:
\begin{itemize}
\item (Theorem \ref{inf-rhs}) when a countable group is not virtually cyclic
and admits a proper relatively hyperbolic structure,
it has two families of infinitely many relatively hyperbolic structures
such that structures in one family have pairwise distinct collections of
relatively quasiconvex subgroups while
structures in the other family have the same collection of
relatively quasiconvex subgroups;
\item (Theorem \ref{qc-updown2}) two relatively hyperbolic structures on
a finitely generated group have the same collection of relatively quasiconvex subgroups
if and only if these structures are equal when we ignore virtually infinite cyclic subgroups.
\end{itemize}
The following are contents of this paper.
Section 2 gives preliminaries. We recall several facts about convergence actions,
and give definitions of relatively hyperbolic structures on a countable group
and relatively quasiconvex subgroups.
In Section 3, blow-ups and blow-downs of relatively hyperbolic structures are defined.
By using these notions,
we define a partial order on the set of all relatively hyperbolic structures on a countable group.
In Section 4, the universal relatively hyperbolic structure is defined in order to characterize relatively hyperbolic structure on a countable group.
In Section 5, we consider how relative quasiconvexity for subgroups varies when we blow up and down a relatively hyperbolic structure.
In Section 6, cardinality of the set of relatively hyperbolic structures is studied and
Theorem \ref{inf-rhs} is proved.
In section 7, when we consider a finitely generated group,
it is shown that the partially ordered set of all relatively hyperbolic structures
on the group is a directed set (Proposition \ref{fiberprod}).
Also Corollary \ref{char-universal} is proved.
In section 8, we show Theorem \ref{qc-updown2}.
In Appendix A, examples of torsion-free countable groups without the universal relatively hyperbolic structure are given.
In Appendix B, we discuss a condition for a characterization of relatively hyperbolic structures by using mapping class groups.
Here we introduce several notations.
In this paper
$G$ denote a countable group with the discrete topology.
Let $L$ be a subgroup of $G$.
Let $\H$ and $\K$ be two conjugacy invariant collections of infinite subgroups of $G$.
We define $L$-conjugacy invariant collections
$\H_L$ and $L \wedge \H$ of infinite subgroups of $L$,
and $G$-conjugacy invariant collections
$\H_{\ne}$ and $\H\wedge \K$ of infinite subgroups of $G$ as
\begin{align*}
\H_L=&\{ H\in \H ~|~ H \subset L\};\\
L \wedge \H=&\left\{ P ~\left|~
P\text{ is an infinite subgroup of }L,
P=L \cap H\text{ for some }H \in \H
\right.
\right\};\\
\H_{\ne}=&\H \setminus \{ H \in \H ~|~ H\text{ is virtually infinite cyclic}\};\\
\H\wedge \K=&\left\{ P ~\left|~
P\text{ is an infinite subgroup of }G,
P=H \cap K\text{ for some }H\in \H\text{ and }K\in \K \right.
\right\}.
\end{align*}
\section{Relatively hyperbolic structures on a group
and relative quasiconvexity for subgroups}\label{Relatively}
First we recall some definitions and properties related to convergence actions (refer to \cite{G-M87}, \cite{Tuk94}, \cite{Tuk98}, \cite{Bow12} and \cite{Bow99b}).
Let $G$ have a continuous action on a compact metrizable space $X$.
The action is called a convergence action if $X$ has infinitely many points and for each infinite sequence $\{g_i\}$ of mutually different elements of $G$, there exist a subsequence $\{g_{i_j}\}$ of $\{g_i\}$ and two points $r,a\in X$ such that $g_{i_j}|_{X\setminus \{r\}}$ converges to $a$ uniformly on each compact subset of $X\setminus \{r\}$
and also $g_{i_j}^{-1}|_{X\setminus \{a\}}$ converges to $r$ uniformly on each compact subset of $X\setminus \{a\}$.
The sequence $\{g_{i_j}\}$ is called a convergence sequence and also the points $r$ and $a$ are called the repelling point of $\{g_{i_j}\}$ and the attracting point of $\{g_{i_j}\}$, respectively.
We fix a convergence action of $G$ on a compact metrizable space $X$.
The set of all repelling points and attracting points is equal to the limit set $\Lambda(G,X)$ (\cite[Lemma 2M]{Tuk94}).
The cardinality of $\Lambda(G,X)$ is $0$, $1$, $2$ or $\infty$ (\cite[Theorem 2S, Theorem 2T]{Tuk94}).
If $\#\Lambda(G,X)=\infty$, then the action of $G$ on $X$ is called a
non-elementary convergence action and also the induced action of $G$ on $\Lambda(G,X)$ is a minimal non-elementary convergence action.
We remark that $\#\Lambda(G,X)=0$ if $G$ is finite by definition.
Also $G$ is virtually infinite cyclic if $\#\Lambda(G,X)=2$ (see \cite[Lemma 2Q,Lemma 2N and Theorem 2I]{Tuk94}).
An element $l$ of $G$ is said to be loxodromic
if it is of infinite order and has exactly two fixed points.
For a loxodromic element $l\in G$, the sequence $\{l^i\}_{i\in \N}$ is a convergence sequence with the repelling point $r$ and the attracting point $a$,
which are distinct and fixed by $l$.
We call a subgroup $H$ of $G$ a parabolic subgroup if
it is infinite, fixes exactly one point and has no loxodromic elements.
Such a point is called a parabolic point.
A parabolic point is said to be bounded if its maximal parabolic subgroup
acts cocompactly on its complement.
We call a point $r$ of $X$ a conical limit point if there exists a convergence sequence $\{g_i\}$ with the attracting point $a\in X$ such that the sequence $\{g_i(r)\}$ converges to a different point from $a$.
The convergence action is said to be geometrically finite
if every point of $X$ is either a conical limit point or a bounded parabolic point.
Since $X$ has infinitely many points, every geometrically finite convergence action is non-elementary.
Also since all conical limit points and all bounded parabolic points belong to the limit set, every geometrically finite convergence action is minimal.
Based on \cite[Definition 1]{Bow12} and \cite[Theorem 0.1]{Yam04}
(see also \cite[Definition 3.1]{Hru10}),
we define relatively hyperbolic structures on a countable group from a dynamical viewpoint.
\begin{dfn}\label{def-rhs}
Let $\H$ be a conjugacy invariant collection of infinite subgroups of $G$.
The group $G$ is said to be hyperbolic relative to $\H$
if there exists a convergence action of $G$ on a compact metrizable space
satisfying the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] the set of all maximal parabolic subgroups of the action is equal to $\H$;
\item[(2)] one of the following holds:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] the limit set of the action is a finite set;
\item[(ii)] the limit set of the action is an infinite set and the induced convergence action of $G$ on the limit set is geometrically finite.
\end{enumerate}
\end{enumerate}
Such a collection $\H$ is called a relatively hyperbolic structure on $G$.
We remark that any relatively hyperbolic structure on $G$ has only
finitely many conjugacy classes (see \cite[Theorem 1B]{Tuk98}).
A relatively hyperbolic structure is said to be trivial (resp. proper) if it is (resp. is not) equal to $\{G\}$.
We denote the set of all relatively hyperbolic structures on $G$ by $\RHS(G)$.
\end{dfn}
Let $G$ be endowed with a relatively hyperbolic structure $\H$ and
consider a convergence action of $G$ satisfying the conditions in Definition \ref{def-rhs}.
It essentially follows from \cite[Theorem 9.4]{Bow12}
and \cite[Theorem 0.1]{Yam04} that the limit set of such a convergence action
is uniquely determined as a $G$-space independently of the choice of a convergence action.
We denote such a $G$-space by $\partial (G,\H)$, which is called the Bowditch boundary.
Indeed we have the following:
\begin{prp}\label{boundary}
Let $\H$ be a relatively hyperbolic structure on $G$.
Consider a convergence action of $G$ on a compact metrizable space $X$ satisfying the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] the set of all maximal parabolic subgroups of the action is equal to $\H$;
\item[(2)] one of the following holds:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] the limit set of the action is a finite set;
\item[(ii)] the limit set of the action is an infinite set and the induced convergence action of $G$ on the limit set is geometrically finite.
\end{enumerate}
\end{enumerate}
Then the limit set $\Lambda(G,X)$ of the action is uniquely determined
as a $G$-space independently of the choice of a convergence action and a compact metrizable space $X$.
\end{prp}
\begin{proof}
If $\Lambda(G,X)$ is an infinite set, then the assertion follows from \cite[Theorem 9.4]{Bow12} and \cite[Theorem 0.1]{Yam04}.
Now we suppose that the limit set $\Lambda(G,X)$ is a finite set.
Under this assumption, the proof is divided into the following three cases.
In the case where $\Lambda(G,X)$ is empty, the group $G$ is finite and $\H$ is empty.
Hence the limit set of every convergence action of $G$ satisfying the conditions above is empty.
In the case where $\Lambda(G,X)$ consists of one point, $\H$ is equal to $\{G\}$.
The limit set of every convergence action of $G$ satisfying the conditions above consists of one point.
In the case where $\Lambda(G,X)$ consists of two points, it follows from \cite[Theorem 2Q and Theorem 2I]{Tuk94} that there exists a loxodromic element $l \in G$ such that the subgroup $\langle l \rangle$ generated by $l$ is of finite index.
Replacing $l$ by its power if necessary, we may assume that $\langle l \rangle$ is a finite index normal subgroup of $G$.
For every convergence action of $G$
such that the limit set consists of two points,
$l$ is loxodromic and the limit set consists of the two fixed point of $l$.
For each element $g \in G$, either $glg^{-1}=l$ or $glg^{-1}=l^{-1}$ holds.
If the former holds, then $g$ acts trivially on the limit set.
If the latter holds, then $g$ interchanges the two points.
Thus the action of $G$ on the limit set is independent of a choice of the convergence action and a compact metrizable space $X$.
\end{proof}
The following are special examples of relatively hyperbolic structures.
\begin{itemize}
\setlength{\itemsep}{0mm}
\item $G$ is infinite if and only if it has the trivial relatively hyperbolic structure $\{G\}$.
Moreover, $\partial(G, \{G\})$ consists of a single point.
\item $G$ is hyperbolic if and only if
the empty collection $\emptyset$ is a relatively hyperbolic structure on $G$.
Moreover, $\partial(G, \emptyset)$ is the Gromov boundary of $G$
(see \cite[8.2]{Gro87} and also \cite[Theorem 3.4 and Theorem 3.7]{Fre95}).
\end{itemize}
We can refine Proposition \ref{boundary} for the case of
virtually infinite cyclic groups.
We note that a virtually infinite cyclic group $G$ has the maximal finite normal subgroup $K$ and $G/K$ is isomorphic to either $\Z$ or $\Z/2\Z*\Z/2\Z$ (see for example \cite[Lemma 11.4]{Hem76}).
\begin{crl}
Let $G$ be a virtually infinite cyclic group and $K$ be the maximal finite normal subgroup of $G$.
Then $G/K$ is isomorphic to $\Z$ if and only if the action of $G$ on $\partial(G,\emptyset)$ is trivial.
\end{crl}
\begin{proof}
We put $G_0=G/K$ and define a generating set $S_0$ of $G_0$ as follows.
\begin{align*}
S_0=\left\{
\begin{array}{ll}
\{t\} & ~{\text{if}}~ G_0 ~{\text{is isomorphic to}}~ \Z \\
\{a, b\} & ~{\text{if}}~ G_0 ~{\text{is isomorphic to}}~ \Z/2\Z*\Z/2\Z,
\end{array}
\right.
\end{align*}
where $a,b \in G_0$ satisfy $a^2=b^2=1$.
The Cayley graph $\Gamma(G_0, S_0)$ is hyperbolic and the Gromov boundary $\partial\Gamma(G_0, S_0)$ consists of two points.
We consider a compact metrizable space $\overline{\Gamma(G_0, S_0)}=\Gamma(G_0, S_0) \cup \partial\Gamma(G_0, S_0)$ and the action of $G$ on $\overline{\Gamma(G_0, S_0)}$ which is induced from the action of $G_0$ on $\Gamma(G_0, S_0)$.
This action is a convergence action and the limit set $\Lambda(G, \overline{\Gamma(G_0, S_0)})$ is equal to $\partial\Gamma(G_0, S_0)$.
If $G_0$ is isomorphic to $\Z$, then the action of $G$ on $\partial\Gamma(G_0, S_0)$ is trivial.
If $G_0$ is isomorphic to $\Z/2\Z*\Z/2\Z$, then the element $a \in G_0$ acts on $\Gamma(G_0, S_0)$ as an inversion and hence the action of $G$ on $\partial\Gamma(G_0, S_0)$ is nontrivial.
Since $\partial\Gamma(G_0, S_0)$ is $\partial(G,\emptyset)$ by Proposition \ref{boundary}, the assertion follows.
\end{proof}
We recall the notion of relative quasiconvexity for subgroups of a countable group with a relatively hyperbolic structure in accordance with \cite[Definition 1.6]{Dah03}.
\begin{dfn}\label{def-relqc}
Let $G$ be endowed with a relatively hyperbolic structure $\H$.
A subgroup $L$ is said to be quasiconvex relative to $\H$ in $G$ if one of the following holds:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] $\partial(G, \H)$ is a finite set;
\item[(2)] $\partial(G, \H)$ is an infinite set and $\Lambda (L, \partial(G,\H))$ is a finite set;
\item[(3)] Both $\partial(G, \H)$ and $\Lambda (L, \partial(G,\H))$ are infinite sets and the induced convergence action of $L$ on $\Lambda (L, \partial(G,\H))$ is geometrically finite.
\end{enumerate}
We denote the set of all subgroups of $G$
that are quasiconvex relative to $\H$ in $G$ by $\RQC(G, \H)$.
\end{dfn}
\begin{rem}\label{ele-relqc}
Let $G$ be endowed with a relatively hyperbolic structure $\H$.
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(I)]
It immediately follows from the definition that for every $H \in \H$, every subgroup of $H$ is quasiconvex relative to $\H$ in $G$.
Also since the limit set of a convergence action of a virtually abelian group is a finite set (see \cite[Theorem 2U]{Tuk94}), every virtually abelian subgroup of $G$ is quasiconvex relative to $\H$ in $G$.
In particular every finite subgroup of $G$ is quasiconvex relative to $\H$ in $G$.
\item[(II)] When $\H=\emptyset$,
a subgroup $L$ of $G$ is quasiconvex relative to $\emptyset$ in $G$
if and only if it is quasiconvex in the ordinary sense (see \cite[Proposition 4.3]{Bow99b}).
\item[(III)] If a subgroup $L$ of $G$ is quasiconvex relative to $\H$, then $L \wedge \H$ is a relatively hyperbolic structure on $L$ by \cite[Theorem 9.1]{Hru10}.
\end{enumerate}
\end{rem}
\section{A partial order on the set of relatively hyperbolic structures}
We define a relation $\to$ on
the set of conjugacy invariant collections of infinite subgroups of $G$ as follows.
For two conjugacy invariant collections $\H$ and $\K$ of infinite subgroups of $G$,
$\K \to \H$ holds if for every $K \in \K$, there exists $H \in \H$ such that $K \subset H$.
When $\K \to \H$ holds, $\K$ is called a blow-up of $\H$ and $\H$ is called a blow-down of $\K$
(compare with \cite{M-O-Y5}).
We show the following.
\begin{prp}\label{order}
The relation $\to$ defines a partial order on $\RHS(G)$.
\end{prp}
Note that the outer automorphism group $\Out(G)$ of $G$ acts naturally on $\RHS(G)$ and that this action preserves the order $\to$.
\begin{dfn}
Let $G$ be a group and let $\H$ be a conjugacy invariant collection of infinite subgroups of $G$.
The collection $\H$ is said to be almost malnormal in $G$ if the following hold:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] the intersection of every pair of two elements of $\H$ is finite.
\item[(2)] every element of $\H$ is equal to its normalizer in $G$.
\end{enumerate}
\end{dfn}
A subgroup $H$ of $G$ is said to be almost malnormal in $G$
if $H \cap gHg^{-1}$ is finite for every $g \in G \setminus H$.
Hence the condition (2) above can be replaced by the condition that every element of $\H$ is almost malnormal in $G$.
\begin{lmm}\label{almal}
Let $G$ have a convergence action on a compact metrizable space $X$.
Then the collection of maximal parabolic subgroups is almost malnormal in $G$.
\end{lmm}
\begin{proof}
Let $H$ be a maximal parabolic subgroup fixing a parabolic point $p$.
We have $\Fix(H)=\{p\}$ and $\Stab_G(p)=H$.
Suppose that $g \in G$ normalizes $H$.
Then we have $\Fix(H)=\{g(p)\}$ and hence $g$ belongs to $H$.
Take two different maximal parabolic subgroups $H_1$ and $H_2$ fixing parabolic points $p_1$ and $p_2$, respectively.
Then the intersection $H_1 \cap H_2$ contains no loxodromic elements and
fixes two different points $p_1$ and $p_2$.
Hence $H_1 \cap H_2$ is finite by \cite[Lemma 3B]{Tuk98}.
\end{proof}
\begin{lmm}\label{order'}
The relation $\to$ defines a partial order
on the set of almost malnormal and conjugacy invariant collections
of infinite subgroups of $G$.
\end{lmm}
\begin{proof}
The relation $\to$ is obviously reflexive and transitive.
We show that it is antisymmetric.
Let $\H$ and $\K$ be almost malnormal and
conjugacy invariant collections of infinite subgroups of $G$ such that $\H \to \K$
and $\K \to \H$. We prove that $\H=\K$.
It suffices to show that $\H \subset \K$.
Let $H$ be an element of $\H$.
Then there exists an element $K$ of $\K$ containing $H$.
Since $\K \to \H$ by the assumption, there exists an element $H'$ of $\H$ containing $K$.
We have $H \subset K \subset H'$.
Since $\H$ is almost malnormal in $G$ and $H$ is infinite, this implies that $H=K=H'$. Therefore $H$ belongs to $\K$.
\end{proof}
Proposition \ref{order} immediately follows from Lemma \ref{almal} and Lemma \ref{order'}.
When we consider sets of representatives of conjugacy classes
of almost malnormal and conjugacy invariant collections
of infinite subgroups of $G$,
the following is convenient:
\begin{lmm}\label{rep}
Let $\K$ and $\H$ be almost malnormal and conjugacy invariant collections
of infinite subgroups of $G$ such that $\K \to \H$.
If $\{ H_\lambda ~|~ \lambda \in \Lambda\}$ is a set of representatives of conjugacy classes of $\H$, then the following hold:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] if we take a set $\{ K_{\lambda, \mu} ~|~ \mu \in M_{\lambda}\}$ of representatives of $H_\lambda$-conjugacy classes of $\K_{H_\lambda}$ for each $\lambda \in \Lambda$, then the set $\{ K_{\lambda, \mu} ~|~ \lambda \in \Lambda, \mu \in M_{\lambda}\}$ is a set of representatives of conjugacy classes of $\K$.
\item[(2)] if the set of conjugacy classes of $\K$ is finite, then the set of $H_\lambda$-conjugacy classes of $\K_{H_\lambda}$ is finite for each $\lambda \in \Lambda$.
\end{enumerate}
\end{lmm}
\begin{proof}
(1) Suppose that there exist $\lambda, \lambda' \in \Lambda$, $\mu \in M_\lambda$ and $\mu' \in M_{\lambda'}$ such that $gK_{\lambda,\mu}g^{-1}=K_{\lambda',\mu'}$ for some $g\in G$.
Then the intersection $gH_\lambda g^{-1} \cap H_{\lambda'}$ is infinite.
Since $\H$ is almost malnormal in $G$, this implies that $\lambda=\lambda'$ and $g \in H_\lambda$.
Since $\K_{H_\lambda}$ is also almost malnormal in $H_\lambda$ for every $\lambda \in \Lambda$, we have $\mu=\mu'$ and $g \in K_{\lambda,\mu}$.
On the other hand, since $\K \to \H$ and $\{ H_\lambda ~|~ \lambda \in \Lambda\}$ is a set of representatives of conjugacy classes of $\H$, for every $K\in \K$, there exist $g\in G$ and $\lambda \in \Lambda$ such that $gKg^{-1} \subset H_\lambda$.
Hence there exist $h \in H_\lambda$ and $\mu \in M_\lambda$ such that $hgK(hg)^{-1}=K_{\lambda, \mu}$.
\noindent
(2) It follows from (1) that if the set of $H_\lambda$-conjugacy classes of $\K_{H_\lambda}$ is infinite for some $\lambda \in \Lambda$, then the set of conjugacy classes of $\K$ is infinite.
\end{proof}
\section{The universal relatively hyperbolic structure on a group}
B. Bowditch \cite[Theorem 7.11]{Bow12} characterized relatively hyperbolic structures on a hyperbolic group among conjugacy invariant collections of infinite subgroups.
If $G$ is a hyperbolic group, then every relatively hyperbolic structure is a blow-down of the empty collection $\emptyset$.
Taking this into account, we introduce the following notion.
\begin{dfn}\label{def-universal}
A relatively hyperbolic structure $\K$ on $G$ is called a universal relatively hyperbolic structure on $G$ if every relatively hyperbolic structure on $G$ is a blow-down of $\K$.
\end{dfn}
Since a universal relatively hyperbolic structure on $G$ is
the greatest structure with respect to the order $\to$, it is unique if it exists.
The universal relatively hyperbolic structure is characterized in Corollary \ref{char-universal} for the case of finitely generated groups and we have several finitely generated groups with the universal relatively hyperbolic structure (see Remark (II) in Section \ref{A common}).
Now we state a characterization of relatively hyperbolic structures on a countable group with the universal relatively hyperbolic structure as follows. This is a consequence of \cite[Theorem 1.1]{Yan11} (refer to Lemma \ref{rep}).
\begin{prp}\label{rhs}
Let $G$ have the universal relatively hyperbolic structure $\K$ and
let $\H$ be a conjugacy invariant collection of infinite subgroups of $G$.
Then $\H$ is a relatively hyperbolic structure on $G$ if and only if the following are satisfied:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(0)] $\K \to \H$;
\item[(1)] $\H$ is almost malnormal in $G$;
\item[(2)] $\H$ has only finitely many conjugacy classes;
\item[(3)] every element of $\H$ is quasiconvex relative to $\K$ in $G$.
\end{enumerate}
\end{prp}
\begin{rem}\label{rem-rhs}
We make remarks about the conditions (0) and (3) in Proposition \ref{rhs}.
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(I)] Proposition \ref{rhs} extends Bowditch's characterization of relatively hyperbolic structures on a hyperbolic group \cite[Theorem 7.11]{Bow12}. Indeed, when $G$ is a hyperbolic group, the universal relatively hyperbolic structure of $G$ is the empty collection $\emptyset$ and thus the condition (0) in Proposition \ref{rhs} can be omitted.
However, we cannot omit the condition (0) in Proposition \ref{rhs} in general
(see Proposition \ref{pA}).
\item[(II)] If $G$ is a finitely generated group, then the condition (3) in Proposition \ref{rhs} can be replaced the following condition:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(3)'] every element of $\H$ is finitely generated and undistorted in $G$.
\end{enumerate}
Indeed if $\H$ is a relatively hyperbolic structure on $G$, then every element of $\H$ is finitely generated by \cite[Proposition 2.29]{Osi06a} and it is undistorted in $G$ by \cite[Lemma 5.4]{Osi06a}.
On the other hand, if we suppose that the condition (3)' holds, then it follows from \cite[Theorem 1.5]{Hru10} that every element of $\H$ is quasiconvex relative to $\K$ in $G$.
\item[(III)] There exist finitely generated groups $G$ with the universal relatively hyperbolic structure $\K$ such that every finitely generated subgroup of $G$ is quasiconvex relative to $\K$ in $G$ (see Remark (III) in Section \ref{A common}).
For such groups, we can replace the condition (3) in Proposition \ref{rhs} by the following condition:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(3)''] every element of $\H$ is finitely generated.
\end{enumerate}
We do not know whether (3) in Proposition \ref{rhs} can be replaced by (3)'' whenever $G$ is a finitely generated group with the universal relatively hyperbolic structure.
\end{enumerate}
\end{rem}
\section{Relative quasiconvexity under blowing up and down a relatively hyperbolic structure}
We characterize how relative quasiconvexity for subgroups varies when we blow up and down a relatively hyperbolic structure as follows:
\begin{prp}\label{qc-updown}
Let $\K$ and $\H$ be two relatively hyperbolic structures on $G$ such that $\K\to \H$.
Then for every subgroup $L$ of $G$, the following conditions (i), (ii), (iii) are equivalent:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] $L$ is quasiconvex relative to $\K$ in $G$;
\item[(ii)] $L$ is quasiconvex relative to $\H$ in $G$
and $L\cap H$ is quasiconvex relative to $\K$ in $G$
for every $H \in \H$;
\item[(iii)] $L$ is quasiconvex relative to $\H$ in $G$
and $L\cap H$ is quasiconvex relative to $\K_H$ in $H$
for every $H \in \H$.
\end{enumerate}
\end{prp}
The equivalence of the conditions (i) and (ii) follows from \cite[Theorem 1.3]{Yan11}
and Lemma \ref{rep}.
The equivalence of the conditions (ii) and (iii) is implied by the following:
\begin{lmm}\label{relqc-rest}
Let $\K$ and $\H$ be relatively hyperbolic structures on $G$ such that $\K \to \H$.
Fix an element $H$ of $\H$.
Then for every subgroup $L$ of $H$, the following conditions (i), (ii) are equivalent:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] $L$ is quasiconvex relative to $\K$ in $G$;
\item[(ii)] $L$ is quasiconvex relative to $\K_H$ in $H$.
\end{enumerate}
\end{lmm}
\begin{proof}
First we remark that $\K_H$ is a relatively hyperbolic structure on $H$ by
\cite[Corollary 3.4]{Yan11}.
It follows from the definition that $\K_H \subset H\wedge \K$.
Let $K$ be an element of $\K$ such that $H \cap K$ is infinite.
Since $\K \to \H$, there exists $H' \in \H$ containing $K$.
Then $H \cap H'$ is infinite.
Since $\H$ is a relatively hyperbolic structure on $G$, this implies that $H=H'$.
Hence $K$ is contained in $H$ and it belongs to $\K_H$.
Thus we showed that $\K_H=H\wedge \K$.
Hence the assertion follows from \cite[Corollary 9.3]{Hru10}.
\end{proof}
In order to give a criterion for
two relatively hyperbolic structures on a countable group
to give the same set of relatively quasiconvex subgroups,
we show the following:
\begin{prp}\label{nqc}
Let $G$ be endowed with a relatively hyperbolic structure $\K$.
Then every subgroup of $G$ is quasiconvex relative to $\K$ in $G$ if and only if either $G$ is virtually cyclic or $\K$ is trivial.
\end{prp}
\begin{proof}
The `if' part follows from Remark (I) in Section \ref{Relatively}.
We prove the `only if' part.
Suppose that $G$ is not virtually cyclic and that $\K$ is proper.
Then it follows from \cite[Corollary 3.2]{M-O-Y4}
that there exists a subgroup $L$ of $G$ which is a free group of rank two
and strongly quasiconvex relative to $\K$ in $G$.
Then $L$ has a subgroup $L'$ which is not finitely generated.
Assume that $L'$ is quasiconvex relative to $\K$ in $G$.
Then $L' \wedge \K$ is a relatively hyperbolic structure on $L'$
by \cite[Theorem 1.2 (1)]{Hru10}.
Since $L$ is strongly quasiconvex relative to $\K$ in $G$,
we have $L \wedge \K=\emptyset$ and hence $L' \wedge \K=\emptyset$.
This implies that $L'$ is a hyperbolic group, which contradicts the fact
that hyperbolic groups are finitely generated.
Therefore $L'$ is not quasiconvex relative to $\K$ in $G$.
\end{proof}
For a relatively hyperbolic structure $\H$ on $G$,
note that $\H_{\ne}$ is a relatively hyperbolic structure on $G$
(see \cite[Theorem 2.40]{Osi06a}).
If two relatively hyperbolic structures $\K$ and $\H$ on $G$
satisfy $\K_{\ne}=\H_{\ne}$, then we have $\RQC(G,\K)=\RQC(G,\H)$
(\cite[Corollary 1.3]{MP08}). On the other hand we have the following:
\begin{crl}\label{qc-updown1}
Let $\K$ and $\H$ be two relatively hyperbolic structures on $G$.
If $\K \to \H$ and $\RQC(G,\K)=\RQC(G,\H)$, then we have $\K_{\ne}=\H_{\ne}$.
\end{crl}
\noindent
For the case of finitely generated groups,
this is refined as in Theorem \ref{qc-updown2}.
\begin{proof}[Proof of Corollary \ref{qc-updown1}]
We suppose that $\K \to \H$ and $\RQC(G,\K)=\RQC(G,\H)$.
It follows from the equivalence between the conditions (i) and (iii) in Proposition \ref{qc-updown}
that for every $H \in \H_{\ne}$, every subgroup of $H$ is quasiconvex relative to $\K_H$ in $H$.
Since $H$ is not virtually cyclic, we have $\K_{H}=\{H\}$ by Proposition \ref{nqc}.
This implies that $\H_{\ne} \to \K_{\ne}$ and hence $\K_{\ne}=\H_{\ne}$.
\end{proof}
\section{Cardinality of the set of relatively hyperbolic structures}
First we consider virtually infinite cyclic groups.
\begin{prp}\label{virtinfcyclic}
The group $G$ is virtually infinite cyclic if and only if $\RHS(G)=\{\emptyset, \{ G \}\}$.
\end{prp}
In order to prove this,
we need the following:
\begin{lmm}\label{inf-ind}
Let $\H$ be an almost malnormal and conjugacy invariant collection of infinite subgroups of $G$.
If $\H$ is not equal to $\{G\}$, then $\H$ consists of infinite index subgroups of $G$.
\end{lmm}
\begin{proof}
Assume that an element $H\in \H$ is a finite index subgroup of $G$.
Then $H$ contains a finite index normal subgroup $H'$ of $G$.
Since $G$ is infinite, $H'$ is also infinite.
For every $g \in G$, we have $H \cap gHg^{-1} \supset H'$.
In particular for every $g \in G \setminus H$,
$H \cap gHg^{-1}$ is infinite.
This contradicts the assumption that $\H$ is almost malnormal in $G$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{virtinfcyclic}]
Since $G$ is infinite and hyperbolic if and only if
$\{ G \}$ and $\emptyset$ are elements of $\RHS(G)$,
we suppose that $G$ is infinite and hyperbolic.
If $G$ is virtually infinite cyclic with a relatively hyperbolic structure $\H \neq \emptyset$, then every element of $\H$ is a finite index subgroup of $G$.
Hence $\H$ is trivial by Lemma \ref{inf-ind}.
On the other hand, if $G$ is not virtually infinite cyclic then it follows from \cite[Corollary 1.7]{Osi06b} that $G$ has a virtually infinite cyclic subgroup $H$ which is hyperbolically embedded into $G$ relative to the empty collection $\emptyset$.
\end{proof}
Second we consider countable groups which are not virtually cyclic.
\begin{thm}\label{inf-rhs}
Suppose that $G$ is not virtually cyclic and admits a proper relatively hyperbolic structure $\H$.
Then there exists a sequence $(\H_n)_{n \in \N \cup \{0\}}$ of proper relatively hyperbolic structures on $G$ satisfying the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] $\H_0=\H$ and $\H_{n} \subsetneqq \H_{n+1}$ for every $n \in \N \cup \{0\}$;
\item[(2)] if $i < j$, then $\RQC(G, \H_i)$ is a proper subset of $\RQC(G, \H_j)$.
\end{enumerate}
On the other hand there exists a sequence $(\K_n)_{n \in \N \cup \{0\}}$ of proper relatively hyperbolic structures on $G$ satisfying the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] $\K_0=\H$ and $\K_{n} \subsetneqq \K_{n+1}$ for every $n \in \N \cup \{0\}$;
\item[(2)] $\RQC(G, \K_n)=\RQC(G, \H)$ for every $n\in \N \cup \{0\}$.
\end{enumerate}
In particular $\RHS(G)/\Out(G)$ has infinitely many elements.
\end{thm}
\begin{proof}
We construct $\H_n$ inductively.
We put $\H_0=\H$.
Suppose that $\H_n$ is constructed.
Since $\H_n$ is proper, it follows from \cite[Theorem 1.2]{M-O-Y4} that there exists a finitely generated and virtually non-abelian free subgroup $V$ of $G$ which is hyperbolically embedded into $G$ relative to $\H_n$.
We put $\H_{n+1}=\H_n \cup \{ P ~|~ P=gVg^{-1}$ for some $g \in G\}$.
If $i < j$, then it follows from \cite[Theorem 1.1(1) and Corollary 1.3]{MP08}
that $\RQC(G, \H_i)$ is a proper subset of $\RQC(G, \H_j)$.
Also we have a sequence $(\K_n)_{n\in \N \cup \{0\}}$ with the desired properties by using \cite[Corollary 1.7]{Osi06b} instead of \cite[Theorem 1.2]{M-O-Y4} on the above argument.
\end{proof}
For finitely generated groups, we have the following.
\begin{prp}\label{fingen}
If $G$ is a finitely generated group,
then $\RHS(G)$ is countable.
\end{prp}
\begin{proof}
For a set $A$, we denote by $F(A)$ the set of all finite subsets of $A$.
We suppose that $G$ is a finitely generated group and
construct an injective map $i:\RHS(G) \to F(F(G))$ as follows.
Let $\H$ be a relatively hyperbolic structure on $G$.
We take a finite set $\{ H_i ~|~ i \in \{1, \ldots, n\} \}$ of representatives of conjugacy classes of $\H$.
Since $G$ is finitely generated,
$\H$ consists of finitely generated subgroups of $G$ by
\cite[Proposition 2.29]{Osi06a}.
Hence we can choose a finite generating set $S_i$ of $H_i$
for each $i\in\{1, \ldots, n\}$.
We define $i(\H)=\{ S_i ~|~ i \in \{1, \ldots, n\} \}$.
Since $G$ is finitely generated, it is countable and hence $F(F(G))$ is also countable.
Therefore $\RHS(G)$ is also countable.
\end{proof}
On the other hand we have the following:
\begin{prp}
Let $G_k$ be an infinite countable group for each positive integer $k$ and let $G=*_{k\in \N}G_k$.
Then $\RHS(G)$ is uncountable.
Moreover if we suppose that for every $k \in \N$, $G_k$ is freely indecomposable
and not isomorphic to $G_j$ for every $j \in \N \setminus \{k\}$,
then $\RHS(G) \slash \Out(G)$ is also uncountable.
\end{prp}
\begin{proof}
We denote by $\{0,1\}^\N$ the set of all maps from $\N$ to $\{0,1\}$.
For each $\sigma\in\{0,1\}^\N$, we consider a relatively hyperbolic structure $\H_\sigma$ represented by $\{*_{\sigma(k)=0}G_k, *_{\sigma(k)=1}G_k\}$.
Since we have $\H_{\sigma_1}\neq\H_{\sigma_2}$ for $\sigma_1, \sigma_2 \in \{0,1\}^\N$ such that $\sigma_1\neq \sigma_2$, the former assertion follows from the fact that $\{0,1\}^\N$ is uncountable.
Moreover if we suppose that for every $k \in \N$, $G_k$ is freely indecomposable
and not isomorphic to $G_j$ for every $j \in \N \setminus \{k\}$, then it follows from Kurosh subgroup theorem (see for example \cite[Chapter IV, Theorem 1.10]{L-S77}) that no automorphisms of $G$ transform $\H_{\sigma_1}$ to $\H_{\sigma_2}$.
\end{proof}
\section{A common blow-up of two relatively hyperbolic structures}\label{A common}
In the case of finitely generated groups, we have the following:
\begin{prp}\label{fiberprod}
Let $G$ be a finitely generated group.
If $\H$ and $\K$ are relatively hyperbolic structures on $G$,
then $\H\wedge \K$ is also a relatively hyperbolic structure on $G$.
\end{prp}
In the above, $\H\wedge \K$ is a common blow-up of two relatively hyperbolic structures $\H$ and $\K$.
In particular, the partially ordered set $(\RHS(G), \to)$ is a directed set.
In order to prove Proposition \ref{fiberprod},
we need the following:
\begin{lmm}\label{rest}
Let $G$ be a finitely generated group with two relatively hyperbolic structures $\H$ and $\K$.
Then for every $H\in \H$, $H\wedge \K$ is a relatively hyperbolic structure on $H$.
\end{lmm}
\begin{proof}
Since $H$ is finitely generated by \cite[Proposition 2.29]{Osi06a}, it follows from \cite[Lemma 5.4]{Osi06a} that $H$ is undistorted in $G$.
Therefore $H\wedge \K$ is a relatively hyperbolic structure on $H$ by \cite[Theorem 1.8]{D-S05b} (see also \cite[Theorem 1.5 and Theorem 9.1]{Hru10}).
\end{proof}
\begin{proof}[Proof of Proposition \ref{fiberprod}]
We denote the conjugacy classes of $\H$ by $\H_1, \ldots, \H_n$.
Then we have $\H=\bigsqcup_{i=1}^n \H_i$ and $\bigsqcup_{i=1}^{n} (\H_i \wedge \K)=\H \wedge \K$.
We show that for each $l \in \{1, \ldots, n+1\}$, $(\bigsqcup_{i=1}^{l-1} (\H_i \wedge \K)) \cup (\bigsqcup_{i=l}^{n} \H_i)$ is a relatively hyperbolic structure on $G$.
The proof is done by induction on $l$.
When $l=1$, the assertion obviously holds.
Suppose that $m \in \{1, \ldots, n\}$ and that $(\bigsqcup_{i=1}^{m-1} (\H_i \wedge \K)) \cup (\bigsqcup_{i=m}^{n} \H_i)$ is a relatively hyperbolic structure on $G$.
Each element $H$ of $\H_m$ is hyperbolic relative to $H \wedge \K$ by Lemma \ref{rest}.
Since we have $\bigsqcup_{H \in \H_m} (H \wedge \K)=\H_m \wedge \K$,
it follows from \cite[Corollary1.14]{D-S05b} that $G$ is hyperbolic relative to
$(\bigsqcup_{i=1}^{m} (\H_i \wedge \K)) \cup (\bigsqcup_{i=m+1}^{n} \H_i)$.
\end{proof}
Now we can characterize the universal relatively hyperbolic structure for the case of finitely generated groups.
\begin{crl}\label{char-universal}
Let $G$ be a finitely generated group and
let $\K$ be a relatively hyperbolic structure on $G$.
Then the following are equivalent:
\begin{enumerate}
\item[(i)] $\K$ is the universal relatively hyperbolic structure on $G$;
\item[(ii)] every element of $\K$ has no proper relatively hyperbolic structures;
\item[(iii)] no relatively hyperbolic structure on $G$ other than $\K$ is a blow-up of $\K$.
\end{enumerate}
\end{crl}
\begin{proof}
The implication $(i)\Rightarrow (iii)$ follows from Proposition \ref{order}.
We prove that $(iii)\Rightarrow (i)$.
Suppose that no relatively hyperbolic structure on $G$ other than $\K$ is a blow-up of $\K$.
Let $\H$ be an arbitrary relatively hyperbolic structure on $G$.
Then it follows from Proposition \ref{fiberprod} that
$\H\wedge \K$ is a relatively hyperbolic structure on $G$
such that $\H\wedge \K \to \H$ and $\H\wedge \K \to \K$.
We have $\H \wedge \K=\K$ by the assumption and hence $\H$ is a blow-down of $\K$.
Next we prove that $(ii)\Rightarrow (iii)$.
Suppose that every element of $\K$ has no proper relatively hyperbolic structures.
Let $\H$ be an arbitrary relatively hyperbolic structure on $G$ with $\H \to \K$.
We prove that $\H=\K$.
Let $K$ be an arbitrary element of $\K$.
It follows from \cite[Corollary 3.4]{Yan11} that $\H_K$
is a relatively hyperbolic structure on $K$.
The assumption on $\K$ implies that $\H_K=\{K\}$.
Hence $K$ is an element of $\H$ and this implies that $\K \to \H$.
It follows from the assumption on $\K$ that $\H \to \K$ and hence we have $\H=\K$
by Proposition \ref{order}.
Finally we prove that $(iii)\Rightarrow (ii)$.
Suppose that there exists an element $K$ of $\K$
which has a proper relatively hyperbolic structure $\K'$.
We set $\H=(\K \setminus \{ L ~|~ L=gKg^{-1}$ for some $g \in G\})
\cup \{ P ~|~ P=gQg^{-1}$ for some $Q \in \K'$ and $g \in G\}$.
It follows from \cite[Corollary 1.14]{D-S05b} that $\H$
is a relatively hyperbolic structure on $G$.
It follows from the construction of $\H$ that $\H \neq \K$ and $\H$ is a blow-up of $\K$.
\end{proof}
\begin{rem}\label{rem-universal}
We make remarks on the universal relatively hyperbolic structure.
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(I)] Since the action of the outer automorphism group $\Out(G)$ on $\RHS(G)$ preserves the order $\to$, the universal relatively hyperbolic structure on $G$ is invariant under the action of $\Out(G)$ (see also \cite[Lemma 4.23(4)]{D-S08}).
For finitely generated groups,
the property of having no proper relatively hyperbolic structures
is a quasi-isometric invariant by \cite[Theorem 1.2]{Dru09}.
In view of \cite[Theorem 4.8]{B-D-M09}, Corollary \ref{char-universal} implies that for finitely generated groups the existence of the universal relatively hyperbolic structure is invariant under quasi-isometry.
For a finitely generated group $G$ with the universal relatively hyperbolic structure, relationship between splitting of $G$ and $\Out(G)$ is described in \cite[Theorem 1.12]{D-S08}.
\item[(II)] If $G$ is infinite and has no proper relatively hyperbolic structures,
then the trivial relatively hyperbolic structure $\K=\{G\}$
is a unique relatively hyperbolic structure on $G$ and hence it is universal.
The examples of such groups are $\Z^n\ (n \ge 2)$, $\SL(n,\Z)\ (n \ge 3)$
and the mapping class group of an orientable surface of genus $g$ with $p$ punctures,
where $3g+p \ge 5$ (see \cite[Theorem 1.2 and p.557]{B-D-M09}
and \cite[Section 8]{K-N04} for details and other examples).
There exists a criterion for countable groups to have no proper relatively hyperbolic structures
(see \cite[Theorem 1]{K-N04} together with \cite[Definition 1]{Bow12},
and also \cite[Theorem 2]{A-A-S07}).
\indent
On the other hand, Corollary \ref{char-universal} enables us to recognize that each of the following finitely generated groups has the universal relatively hyperbolic structure $\K$, which is proper:
\begin{itemize}
\setlength{\itemsep}{0mm}
\item each hyperbolic group with $\K=\emptyset$;
\item each geometrically finite Kleinian group with the collection $\K$ of all maximal parabolic subgroups that are not virtually infinite cyclic (note that every maximal parabolic subgroup of a Kleinian group is virtually abelian (see for example \cite[Proposition 2.2]{M-T98}));
\item each free product $A*B$ with the collection $\K$ of all conjugates of $A$ and $B$, where $A$ and $B$ are finitely generated groups having no proper relatively hyperbolic structures;
\item each one-relator product $A*B \slash \langle\langle r^m \rangle\rangle$ with the collection $\K$ of all conjugates of $A$ and $B$, where $A$ and $B$ are finitely generated groups having no proper relatively hyperbolic structures, $r$ is a cyclically reduced word of length at least 2 and $m \ge 6$ (see \cite[Theorem 4.1]{MP-W11b});
\item each limit group with the collection $\K$ of all maximal abelian non-cyclic subgroups (\cite[Theorem 0.3]{Dah03}).
\end{itemize}
\item [(III)] The following are examples of finitely generated groups $G$
with the universal relatively hyperbolic structure $\K$ such that
every finitely generated subgroup of $G$ is quasiconvex relative to $\K$ in $G$
(refer to the above (II)):
\begin{itemize}
\setlength{\itemsep}{0mm}
\item each finitely generated free group (\cite[Section 2]{Sho91}) and the fundamental group of each closed hyperbolic surface (\cite[Proposition 2]{Pit93});
\item geometrically finite Kleinian groups of the second kind acting on $\mathbb{H}^3$ (\cite[Proposition 7.1]{Mor84});
\item each free product $A*B$ where $A$ and $B$ are finitely generated groups having no proper relatively hyperbolic structures;
\item each one-relator product $A*B \slash \langle\langle r^m \rangle\rangle$ where $A$ and $B$ are finitely generated groups having no proper relatively hyperbolic structures, $r$ is a cyclically reduced word of length $|r| \ge 2$ and $m > 3|r|$ (\cite[Theorem 1.7]{MP-W11b});
\item each limit group (\cite[Proposition 4.6]{Dah03}).
\end{itemize}
\item [(IV)] There exist finitely generated groups which do not have the universal relatively hyperbolic structure.
An example of such groups is the so-called Dunwoody's inaccessible group (see \cite[Section 6]{B-D-M09}).
Note that Dunwoody's inaccessible group is finitely generated, not finitely presentable and has torsions.
We can also obtain torsion-free countable non-finitely generated groups without the universal relatively hyperbolic structure (see Proposition \ref{freeprod}).
However, it is unknown whether every finitely presented (resp. torsion-free finitely generated) group has the universal relatively hyperbolic structure (see \cite[Question 1.5]{B-D-M09}).
\end{enumerate}
\end{rem}
\section{Relatively hyperbolic structures with the same set of relatively quasiconvex subgroups}
We determine when two relatively hyperbolic structures
have the same collection of relatively quasiconvex subgroups.
\begin{thm}\label{qc-updown2}
Let $G$ be a finitely generated group and let
$\K$ and $\H$ be two relatively hyperbolic structures on $G$.
Then we have the following:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(1)] $\RQC(G,\K)\cap\RQC(G,\H)=\RQC(G,\K\wedge\H)$.
\item[(2)] The following conditions are equivalent:
\begin{enumerate}
\setlength{\itemsep}{0mm}
\item[(i)] $\RQC(G,\K)=\RQC(G,\H)$;
\item[(ii)] $\RQC(G,\K)=\RQC(G,\H)=\RQC(G,\K\wedge\H)$;
\item[(iii)] $\K_{\ne}= \H_{\ne}$.
\end{enumerate}
\end{enumerate}
\end{thm}
\noindent
We remark that
the equivalence between (i) and (iii) in (2) follows from \cite[Corollary 1.3]{MP08}
when $\K$ is a subcollection of $\H$.
\begin{proof}
(1) Let $L$ be a subgroup of $G$.
First we suppose that $L$ is quasiconvex relative to $\K\wedge\H$ in $G$.
Since both $\K$ and $\H$ are blow-downs of $\K\wedge\H$, it follows from Proposition \ref{qc-updown} (i) $\Rightarrow$ (ii) that $L$ is quasiconvex relative to $\K$ in $G$ and quasiconvex relative to $\H$ in $G$.
Next we suppose that $L$ is quasiconvex relative to $\K$ in $G$ and quasiconvex relative to $\H$ in $G$.
Let $H$ be an element of $\H$.
Since $G$ is finitely generated, $H$ is undistorted in $G$ by \cite[Lemma 5.4]{Osi06a}.
It follows from \cite[Theorem 1.5]{Hru10} that $H$ is quasiconvex relative to $\K$ in $G$.
Therefore we have $L \cap H$ is also quasiconvex relative to $\K$ in $G$ by \cite[Theorem 1.2 (2)]{Hru10}.
Since $H$ and $L\cap H$ are quasiconvex relative to $\K$ in $G$ and $L\cap H$ is a subgroup of $H$,
$L \cap H$ is quasiconvex relative to $H \wedge \K$ in $H$ by \cite[Corollary 9.3]{Hru10}
(see also Lemma \ref{relqc-rest}).
Since we have $H \wedge \K=(\K \wedge \H)_H$, $L \cap H$ is quasiconvex relative to $(\K \wedge \H)_H$ in $H$.
Hence it follows from Proposition \ref{qc-updown} (iii) $\Rightarrow$ (i) that $L$ is quasiconvex relative to $\K\wedge\H$ in $G$.
(2) Since we have $\RQC(G,\K) \cap \RQC(G,\H)=\RQC(G, \K\wedge \H)$, the
conditions (i) and (ii) are equivalent.
The condition (iii) implies the condition (i)
by \cite[Corollary 1.3]{MP08}.
Finally we prove that the condition (ii) implies the condition (iii).
Since we have $\K \wedge \H \to \H$ and $\K \wedge \H \to \K$,
it follows from Corollary \ref{qc-updown1} that $\H_{\ne}=(\K \wedge \H)_{\ne}=\K_{\ne}$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,747 |
\section{Introduction}
\label{sec:intro}
Filaments are widely existing structures in cold dense interstellar medium \citep[e.g.][]{ragan12, dirienzo15, wang16}. They are observed to have multiple substructures and velocity components, which should considerably facilitate the mass aggregation and star-forming processes therein \citep[e.g.][]{hacar13,henshaw14,shimajiri19}. The intersected and converging filaments could have more significant association with the formation of clustered and massive young stars \citep{peretto14, lopez14, williams18, chen19a, tmorales19, hu21, liu21}. In several representative cases, sharp velocity gradients have been observed along the filaments towards the central star-forming cores \citep{peretto13,kirk13,hacar17}, which could represent converging flows feeding young stellar objects (YSOs). Besides the converging motion, the filaments could also have longitudinal collapse to generate dense clumps and cores at two ends, as shown in both theoretical \citep{cw15} and observational \citep{dewangan19,yuan20} works.
Although the mass transfer flows are identified from velocity gradients in a number of regions, their connection to individual star-forming cores are still not widely confirmed. It is uncertain whether a filament can directly support the accretion to an YSO or merely influence the core evolution in less direct ways, such as increasing the perturbation and compressing the gas. An external mass supply instead of isolated core collapse is also speculated in the competitive accretion model \citep{bonnell06,clark08}. But the detailed process of the gas transfer into the individual star-forming cores is still largely unknown, and should be explored in subsequent observations.
Orion A molecular cloud is an ideal target to study the mass transfer and accretion process. Its main structure is a well-known dense and compact integral-shaped filament (ISF) elongated from north to south \citep[][etc.]{johnstone99,salji15,hacar18}. It has an average distance of 390 pc \citep{grobschedl18}, and a length scale of 6 pc. There are dozens of dense star-forming cores on the filament structure \citep{li13,takahashi13,lopez13,kirk17}. The cloud contains complex substructures which are mostly in a cold and quiescent state \citep{hacar18, zhang20}. Over a spatial scale of 0.5 pc around the cloud, there are nearly 400 protostars and 3000 more evolved (Class-II to III) young stars \citep{megeath12}. The young stars have an overall coherent spatial distribution with the molecular cloud \citep{stutz16, grobschedl18}, and would be influencing the gas components in both radiative and dynamical ways \citep{megeath12,dario16,qiu18}.
MMS-7 is a dense molecular core in Orion A OMC-3 region \citep{takahashi06}. We investigated this region with a joint analysis of different molecular tracers, including the ${\rm N_2H^+}$ (1-0) observed with ALMA and ${\rm C^{18}O}$ (2-1) observed with the Submillimeter Array (SMA). From these data we can more completely inspect the dense gas around the center as well as more extended structures. In addition, the ALMA spectra can exhibit the gas velocity field with a much better sensitivity and spatial resolution than the previous observations, thereby reveal the interplay between the filament-based gas motion and the star-forming cores in more details.
In Sec.\,\ref{sec:obs}, we described the observation. In Sec.\,\ref{sec:result}, we presented the filament and dense core structures and the velocity distribution. The dynamical origin of the velocity distribution is discussed in detail in Sec.\,\ref{sec:discussion}, including the overall dynamical condition in Sec.\,\ref{sec:dv_origin}, energy scales in Sec.\,\ref{sec:energy_scale}, the velocity field of each filament in Sec.\,\ref{sec:vel_profile}, gravitational instability in Sec.\,\ref{sec:m_vir}, and evolutionary state in Sec.\,\ref{sec:evolve}. A summary is given in Sec.\,\ref{sec:summary}.
\section{Observation and Data Reduction}\label{sec:obs}
The ${\rm N_2H^+}$ (1-0) data around MMS-7 is from a large scale mapping over the Orion A OMC-2,3 region \citep[][also see Figure 1a]{zhang20}. The ALMA observation of OMC-2/3 (proposal ID 2013.1.00662.S, PI: Diego Mardones) was carried out during November 2014 and August 2015 in Band 3 using the 12-m main array and the Atacama Compact Array (ACA). The baselines of the 12-m main array and the ACA observations range from 13.6 to 340.0\,m and 6.8 to 87.4\,m, respectively. We combined the UV datasets of the 12-m array and ACA and CLEANed it in CASA \citep{mcmullin07} using the interactive model. The restored images was corrected by Primary-beam collection. The synthesized full-width half-maximum beam size is $3\times3''$, the maximum spatial coverage is $94''$. The standard deviation noise level per channel is $\sigma=7$ mJy beam$^{-1}$ (corresponding to $T_{\rm b}=0.14$ K).
The SMA data is publicly available in the SMA data archive. The observation is performed in March 2014 in the compact array. The receiver 0 is used with the local oscillation frequency tuned at 225 GHz. The three isotopologue CO $(2-1)$ lines are included in the sidebands. The velocity resolution is 0.7 km s$^{-1}$. The sensitivity in the line observation is 0.5 K. The synthesized images have a beam size of $6''.1\times4''.5$ with a position angle of $39^\circ$ to the northeast. The spatial structures smaller than $30''$ can be restored by the SMA baselines.
We also referred to the JCMT/SCUBA $450\,{\rm \mu m}$ continuum image \citep{johnstone99} to inspect the global gas distribution. The image has a beam size of $9''$ and an rms level of 0.13 Jy beam$^{-1}$.
\section{Result}\label{sec:result}
\subsection{The Gas Distribution around MMS-7}
Figure 1a shows the integrated emission of the ${\rm N_2H^+}~(1_{0}-0_{1})$ transition overlaid on the IRAC $8\,{\rm \mu m}$ emission for the entire OMC-2,3 region. The overall gas structures are presented in a relevant study \citep{zhang20}.
Figure 1b shows the zoom-in ${\rm N_2H^+}$ emission region (white contours) around MMS-7 overlaid on the Spitzer/IRAC $8\,{\rm \mu m}$ image. The $8\,{\rm \mu m}$ image shows two point sources, IRAC-2407 and IRAC-2405, which were presented in \citet{takahashi06} and are included in the Orion YSO catalogue \citep{megeath12}. The ${\rm N_2H^+}$ emission exhibits three major filament structures extending from MMS-7 to the Northwest (NW), South, and North, respectively. The three filaments are all closely overlapped with the infrared-dark region in the $8\,{\rm \mu m}$ image. The SMA 1.3 mm continuum emission (green contours) exhibits three dense cores. The central one (MMS-7) coincides with the ${\rm N_2H^+}$ emission peak. The second core (denoted as Core-2) overlaps with IRAC-2405. The other three cores, Core-3, 4, and 5 are located on the NW filament. Core-5 is not fully covered by our mapping field, but can still be seen to have a higher intensity than Core-3 and Core-4, thus could be more massive. The NW filament thus has a similarity with the end clumps observed in other filaments \citep[e.g.][]{dewangan19,yuan20}.
The bright young star IRAC-2407 was found to drive a bi-polar outflow \citep{takahashi06}, which is also labeled in Figure 1b. The outflow is likely causing a dissipation of the North filament, sweeping it away from the outflow direction. The central core MMS-7 is displaced from IRAC-2407 and its outflow, and has no other overlapped IR sources. MMS-7 thus should be an independent prestellar core. The three ${\rm N_2H^+}$ filaments could be originally intersected at MMS-7, but later become separated due to the presence of two IR sources (YSOs). This overall morphology is comparable to the frequently observed filament-hub systems despite the different spatial scales \citep[e.g.][]{williams18, chen19a, tmorales19}.
Figure 1c shows the SMA ${\rm C^{18}O}$~(2-1) emission overlapped on the ${\rm NH_3}~(1,1)$ emission (gray contours) and kinetic temperature map obtained from our previous Very Large Array (VLA) observation \citep{li13}. The ${\rm C^{18}O}~(2-1)$ emission shows a similar spatial distribution with the 1.3 mm continuum. The emission peak overlaps with Core-2, the other two emission features are near MMS-7 and Core-3, respectively. Although the ${\rm C^{18}O}$ and ${\rm NH_3}$ emission regions do not fully overlap with the ${\rm N_2H^+}$ filaments, they all have a bulk spatial extension from Southeast to Northwest that is in parallel with the NW filament. The kinetic temperature estimated from the ${\rm NH_3}$ inversion shows a quite uniform distribution around $13\pm2$ K over the MMS-7 region \citep{li13}, suggesting that the dense filaments and cores are not significantly affected by the stellar heating.
Figure 1d shows the ${\rm N_2H^+}$ emission overlaid on the dust continuum emission of the JCMT/SCUBA $450\,{\rm \mu m}$ band. The $450\,{\rm \mu m}$ emission has a similar morphology with the ${\rm N_2H^+}$ filaments and meanwhile has a broader and more continuous spatial distribution. The $450\,{\rm \mu m}$ emission region should represent the parental clump of the filaments and cores. We estimated the effective radius of the clump using $A_{\rm clump}=\pi r_{\rm eff}^2$, where $A_{\rm clump}$ is the clump area within the 30\%-contour level.
We calculated the gas column density and core mass from the dust continuum emission using the gray-body emission model \cite{hildebrand83},
\begin{equation}\label{equ:m_dust}
\begin{aligned}
S_{\nu} & = \kappa_\nu B_{\nu}(T_{\rm d}) \Omega \mu m_{\rm H} N_{\rm tot} \\
& = \frac{\kappa_{\nu} B_{\nu}(T_{\rm d}) M_{\Omega} }{D^2},
\end{aligned}
\end{equation}
wherein $S_{\nu}$ is the flux density at the frequency $\nu$ within solid angle $\Omega$, $N_{\rm tot}=N({\rm H_2}+{\rm HI})$ is the total column density, $B_{\nu}(T_{\rm d})$ is the Planck function of the dust temperature $T_{\rm d}$. We assumed that $T_{\rm d}$ is comparable to the ${\rm NH_3}$ kinetic temperature of $T=13$ K. $\mu$=$2.33$ is the mean molecular weight \citep{myers83a}, $m_{\rm H}$ is the atomic hydrogen mass, and $\kappa_\nu$ is the dust opacity, which is related to the frequency as $\kappa_\nu=\kappa_{\rm 230 GHz}(\nu/{\rm 230 GHz})^{\beta}$. The reference value of $\kappa_{\rm 230 GHz}=0.009$ cm$^2$ g$^{-1}$ is provided by the dust model for the grains with coagulation for $10^5$ years with accreted ice mantles at a density of $10^6\,{\rm cm^{-3}}$ \citep{ossenkopf94}, and the average index of $\beta=1.9$ in Orion A \citep{sadavoy16} is adopted for the calculation. We calculated $N_{\rm tot}$ and $M_{\rm core}$ for the three dense cores, MMS-7, Core-1, and Core-2 from the SMA 1.3 mm continuum emission. We measured the core area $A_{\rm core}$ above the detection limit, and derived the effective core radius from the from the expression of $A_{\rm core}=\pi r_{\rm eff}^2$. The finally adopted core radius is also deconvolved with the beam size, i.e. $r_{\rm core}^2$=$r_{\rm eff}^2-r_{\rm beam}^2$. The parental clump mass is estimated from the $450\,{\rm \mu m}$ emission also using Equation \ref{equ:m_dust}. We also derived the mass of each filament assuming that the its mass ratio to MMS-7 is equal to ${\rm N_2H^+}$ intensity ratio, that is $M_{\rm fil}/M_{\rm MMS-7} = S_{\rm fil}({\rm N_2H^+})/S_{\rm MMS-7}({\rm N_2H^+})$. The physical parameters are shown in Table 1.
Figure 1e shows the ${\rm N_2H^+}$ (1-0) and ${\rm C^{18}O}$ spectra at the MMS-7 center. We fitted the ${\rm N_2H^+}$ hyperfine components (HFCs) using Pyspeckit\footnote{Toolkit for fitting and manipulating spectroscopic data in python.} program, wherein the physical parameters of the HFCs are adopted from \citet{caselli95}. The molecular column density can be estimated as \citep{caselli02, henshaw14}:
\begin{equation}\label{equ:n_mol}
\begin{aligned}
N_{\rm mol} & = \frac{8\pi}{\lambda^3 A_{\rm ul}} \frac{g_l}{g_u} \times \frac{1}{J(T_{\rm ex})-J(T_{\rm bg})} \times \frac{1}{1-\exp(-h\nu /k_{\rm B} T_{\rm ex})} \\
& \times\frac{Q_{\rm rot}(T_{\rm ex})}{g_l \exp(-El/k_{\rm B} T_{\rm ex})} \int T_{\rm b} {\rm d}v,
\end{aligned}
\end{equation}
wherein $A_{\rm ul}$ is the Einstein coefficient \citep{schoier05}, $g_u$ and $g_l$ are the statistical weights of the upper and lower states, respectively; $J_\nu(T)$ is the equivalent Rayleigh-Jeans temperature. $Q_{\rm rot}(T)$ is the partition function. In calculation, we also assume that $T_{\rm ex}$ is similar with the ${\rm NH_3}$ kinetic temperature.
Figure 1f shows the column density profile along the NW filament. The sampling direction is indicated by the solid line in Figure 1d. The $N_{\rm tot}$ profile is estimated from the $450\,{\rm \mu m}$ continuum. It has a peak value of $N_{\rm tot}=4.1\times 10^{23}$ cm$^{-2}$ at MMS-7 and a FWHM width of $\sim 10''$. The surrounding gas has average $N_{\rm tot}=1.4 \times 10^{23}$ cm$^{-2}$ on the NW filament (offset$>10''$) and $1.0\times 10^{23}$ cm$^{-2}$ out of the filaments. The $N({\rm N_2H^+})$ and $N_{\rm tot}$ exhibit similar spatial profiles, suggesting that ${\rm N_2H^+}$ and $450\,{\rm \mu m}$ continuum emissions could trace the nearly same gas-and-dust components.
Figure 1g shows the molecular abundance ($X_{\rm mol}$) profile along the NW filament estimated from $X_{\rm mol}=N_{\rm mol}/N_{\rm tot}$. The ${\rm N_2H^+}$ abundance varies between 0.4 to $1.2\times10^{-9}$, and the three cores have quite similar values of $X({\rm N_2H^+})=(0.8-1.0)\times10^{-9}$. In comparison, the ${\rm C^{18}O}$ abundance is peaked at Core-2 and rapidly declines towards MMS-7. The abundances of individual cores are presented in Table 1. Core-2 has high $X({\rm C^{18}O})$ probably due to its association with the protostar IRAC-2405. MMS-7 and Core-3 have no detectable IR counterparts thus could be younger and have a certain CO depletions. MMS-7 in particular has the highest mass but the lowest ${\rm C^{18}O}$ abundance, suggesting the absence of stellar heating.
\subsection{The Velocity Distribution}\label{sec:vel_profile}
Figure 2a shows the velocity field overlaid on the integrated ${\rm N_2H^+}$ emission region. The velocity is measured by fitting the ${\rm N_2H^+}~(1_0-0_1)$ transition with a gaussian profile pixel by pixel. The velocity field shows a noticeable increasing trend along the NW filament towards the center, with a variation from $v_{\rm lsr}=11$ to 9.2 km s$^{-1}$. The Southern and Northern filaments also exhibit less prominent but similar velocity gradient, with $v_{\rm lsr}$ variation from 10.75 to 9.75 km s$^{-1}$\ towards the edges near the center.
Figure 2b shows the ${\rm N_2H^+}~(1_0-0_1)$ line width ($\Delta v$) distribution. It exhibits peak values at MMS-7 center and several other positions. There are four $\Delta v$ peaks on the NW filament. They are coincident with intensity peaks. The $\Delta v$ peak on the North filament also overlaps with the emission peak. The $\Delta v$ peaks have an average increasing scale of 0.2 km s$^{-1}$, which is much smaller than the $v_{\rm lsr}$ increasing scale towards the center (Figure 2a). The $v_{\rm lsr}$ and $\Delta v$ distributions suggest that although the filaments have a bulk motion, they still remain a quite low internal velocity dispersion.
Figure 2c and 2d show the position-velocity (PV) diagrams along the NW filament and the direction from North to South, respectively. As shown in Figure 2c, the velocity profile along the NW filament varies from $v-v_{\rm sys}$=$-1$ to $-2$ km s$^{-1}$\ in spatial range from offset$=45''$ to $0''$. The velocity gradient is not linear but has a steeper increase as approaching the center. The velocity profile of ${\rm C^{18}O}$ (contours) also shows an increasing trend from offset$=20''$ to $0''$, which is roughly in parallel with the ${\rm N_2H^+}$ emission. Since the ${\rm C^{18}O}$ is closely associated with the 1.3 mm dense cores, it suggests that the cores should have a coherent motion with the filaments.
The velocity gradient from South to North is presented in Figure 2d. This direction partly covers the South and North filaments. The South one extends from offset=0 to $30''$ and the North one from offset=$-40''$ to $-20''$. The two filaments also have a similar velocity gradient of $v-v_{\rm sys}=0$ to $-1.2$ km s$^{-1}$\ towards the inner region. Since IRAC-2407 and 2405 (Core-3) are located at positions where the velocity gradient becomes steeper, the velocity features should therefore suggest the gas motion to have an interaction with these two YSOs, respectively. Meanwhile the two filaments are not terminated at the YSOs, but extend over them to the center. They could therefore have a convergent motion together with the NW filament towards MMS-7.
\section{Discussion}\label{sec:discussion}
\subsection{Dynamical Origin of the Velocity Profiles}\label{sec:dv_origin}
As shown in Figure 2c and 2d, the three ${\rm N_2H^+}$ filaments exhibit comparable velocity increase towards the center. The ${\rm C^{18}O}$ emission also partly follows the ${\rm N_2H^+}$ velocity profile around MMS-7. Moreover, for each filament, the velocity gradient becomes steeper within an area of offset=$\pm 20''$, as indicated by the vertical dashed lines. Since the ${\rm C^{18}O}$ and ${\rm N_2H^+}$ are both dense-gas tracers, they should reveal the bulk motion of the dense filaments and cores instead of activities driven by star formation, such as outflow and envelope expansion. The star-forming activities could also have an inconsistency with the observed line widths. The ${\rm N_2H^+}\,(1_0-0_1)$ transition only has a limited increase from $\Delta v=0.4$ to 0.65 km s$^{-1}$\ towards the center (Figure 2b), which is much lower than the scale of radial velocity increase. Since the velocity gradient is clearly seen in all three filaments, it suggests that the dense gas should have an overall contraction towards the center due to the self gravity.
Moreover, in order to generate the observed gas motion, there are two requirements for the gas structure. First, the clump should have a non-spherical distribution so that a fraction of mass can obtain a bulk motion towards the direction where the ambient density is lower. Second, the low-density region should be on the front side of the clump so that the compressed gas can have a blueshift. This spatial layout is not difficult to be satisfied in real condition. A schematic view of the gas distribution and motion is shown in Figure 3.
Although the density profile along the line of sight is largely unknown, its spatial distribution shown in Figure 3 is still supported by two factors. First, as seen in Figure 2c and 2d, a fraction of ${\rm C^{18}O}$ emission around the central position is not associated with the blueshifted ${\rm N_2H^+}$ velocity gradient, but has a different velocity of $v_{\rm lsr}\simeq-1~{\rm km\,s^{-1}}$. This component could trace the more quiescent dense gas behind the blueshift gas. Second, the ${\rm N_2H^+}$ emission is weakened towards IRAC-2407, and the velocity profile of the Northern filament also appears to be disturbed by this source (Figure 2d). These features suggests that this YSO is causing a gas dissipation on the front side.
\subsection{Contraction Velocities and Energy Scales}\label{sec:energy_scale}
The filament contraction could be essentially driven by the self gravity and it requires two major physical conditions in this process: (1) the self gravity should provide sufficient kinetic energy to account for the current gas motion; (2) the clump should have a gravitational instability to initiate the contraction.
To make a semi-quantitative calculation, we assume that the gas is contracting from an initial radius of $r_0$ to a smaller radius $r$. The gravitational energy release in this process can be approximated to be
\begin{equation}\label{equ:gmr}
\Delta\mathcal{W} = GM_c^2\left(\frac{1}{r} - \frac{1}{r_0}\right),
\end{equation}
where $M_c$ is the contracting gas mass. It should include MMS-7 and its surrounding gas. As shown in Figure 3b and 3c, the blueshifted gas is mainly distributed within $20''$ around the center. The gas mass in this area is $5.5\,M_\odot$ as calculated from the $450\,{\rm \mu m}$ emission using Equation \ref{equ:m_dust}. The released kinetic energy can be estimated as
\begin{equation}\label{equ:mv2}
\epsilon \Delta \mathcal{W} = \frac{1}{2}M_c [v(r)^2 - v(r_0)^2],
\end{equation}
wherein $\epsilon$ represents the efficiency of energy conversion. Assuming that the contracting gas has a final velocity of $v_{\rm real}$ and an inclination of $\theta$ from the line of sight, the observed radial velocity would be $v_{\rm obs}=v_{\rm real} \cos \theta$. From Equation \ref{equ:gmr} and \ref{equ:mv2}, the radius and velocity can have a relation of
\begin{equation}\label{equ:vr}
v_{\rm obs}(r)^2 - v_{\rm obs}(r_0)^2 = 2 G M_c \epsilon \cos^2 \theta (r^{-1} - r_0^{-1}).
\end{equation}
Since the three parameters $\epsilon$, $M_c$, and $\theta$ are fully degenerated, their production on the right side is considered as one coefficient, $c_0 = 2 G M_c \epsilon \cos^2 \theta$. The other two parameters, $r_0$ and $v_{\rm obs}(r_0)$ are the initial radius and velocity, respectively. We expect $r_0$ to be much larger than the spatial extent of the contracting gas ($r<20''$), and the initial velocity nearly zero. In this case, the central part of the $v_{\rm obs}(r)$ curve would approach $v(r)^2 \propto 1/r$.
We compared the $v(r)$ function and velocity profile of the filaments as shown in Figure 4. For each filament, one can let $v(r)$ curve well coincide with the velocity profile in the range of $r<20''$ only by adjusting $c_0$. The three $v(r)$ curves have comparable $c_0$ values of $c_0/2G=(0.6\pm0.3)\,M_\odot$. The contracting mass is then constrained to be
\begin{equation}\label{equ:m_c}
\begin{aligned}
M_c & = \frac{c_0}{2G}\frac{1}{\epsilon \cos^2 \theta} \\
\quad & = (5.0\pm2.5) \left(\frac{\epsilon}{0.5}\right)^{-1} \left(\frac{\cos \theta}{\cos 60^\circ}\right)^{-2} M_\odot.
\end{aligned}
\end{equation}
From this equation we can satisfy the observed mass scale $5.5\,M_\odot$ using reasonable $\theta$ and $\epsilon$ values. Although the individual parameters are still uncertain, the calculation shows that the gravitational contraction could release sufficient kinetic energy to account for the observed gas motion.
\subsection{Velocity Variations}\label{sec:vel_profile}
Despite the agreement in energy scales, the velocity distribution still has two deviation features from the modeled $v(r)$ curves that should be further examined. First, the theoretical $v(r)$ function will approach infinity towards the center, while the observed velocity profiles are terminated at final velocities around the center. For the NW filament (Figure 4b), the velocity profile reaches the maximum value in the area of $r<4$ arcsec around the center. This area is comparable to the size of MMS-7 core ($r=3.7$ arcsec). As an reasonable explanation this energy release of $\epsilon\Delta\mathcal{W}$ could be largely transferred to MMS-7, and accelerating this core as an entire body to the final velocity.
Second, although the three filaments have similar velocity increasing features towards the center, their final velocities still have a difference. This could be also attributed to IRAC-2405 and 2407. As shown in Figure 4d, the North filament becomes weaker at IRAC-2407, but still connected to MMS-7, and the gas structure well follows the $v(r)$ curve. This suggests that the gas motion is still towards the center and would reach the same velocity with MMS-7. The South filament, according to its velocity gradient, could have a similar connection to MMS-7 except for a more significant interruption by IRAC-2406 (Core-2). The two YSOs could have dissipated the surrounding ${\rm N_2H^+}$ component but not largely affect the global inward motion towards MMS-7. For the NW filament, its outer part ($r>20''$) could also be affected by the local dense cores. As seen in Figure 4b, the emission region is largely deviated from the $v(r)$ curve in the spatial range between Core-3 and Core-4 with a scale of $\Delta v\simeq 1.0$ km s$^{-1}$, suggesting that the gas motion could be affected by these two cores.
Although the filaments have these systemic velocity variations, the ${\rm N_2H^+}$ $(1_0-0_1)$ line profile is usually single-peaked and has relatively small line width of $\Delta v<0.65$ km s$^{-1}$\ all over the region. There are no evident multiple velocity components in MMS-7 region. The three filaments thus should all be one-dimensional compact filaments. Multiple velocity components in a filament would indicate intertwined substructures therein, and their formation could depend on initial conditions of turbulence and density fluctuations in a molecular cloud as well as magnetic field. In Orion A, the incidence of multiple velocity components vary from region to region, and may have an overall increasing trend towards the OMC-2 and Orion KL regions \citep[][Figure B.3 therein]{hacar18}, wherein the major fraction of dense gas over the ISF is being assembled.
\subsection{Clump and Filaments: Instabilities}\label{sec:m_vir}
In order to initiate the contraction, the clump should have a gravitational instability. To estimate this, we adopted the modified virial mass $M_{\rm vir}^B$ \citep{bertoldi92,pillai11,sanhueza17,liu18}, which takes the density profile, thermal, turbulent, and magnetic pressure into account, estimated as
\begin{equation} \label{equ:m_vir}
M_{\rm vir}^B = \frac{5 R_{\rm eff}}{G} \left(\sigma_{\rm tot}^2+\frac{\sigma_A^2}{6}\right),
\end{equation}
where $R_{\rm eff}$ is the effective radius, $\sigma_{\rm tot}$ is the total velocity dispersion, $\sigma_A$ is the Alfv\'{e}nic velocity. It is determined by the magnetic field strength $B$ and average gas density $\overline{\rho}$ as $\sigma_A=B/\sqrt{4\pi \overline{\rho}}$. Here we adopt the average value of $B=0.19$ mG in OMC-3 \citep{poidevin10} and the average density of the $450\,{\rm \mu m}$ clump. The velocity dispersion consists of the thermal and non-thermal parts:
\begin{equation}
\begin{aligned}
\sigma_{\rm tot} & = \sqrt{\sigma_{\rm th}^2 + \sigma_{\rm nt}^2} \\
\quad & = \sqrt{\frac{k_{\rm B} T_{\rm kin}}{\mu m_{\rm H}} + \frac{\Delta v^2}{8 \ln 2} - \frac{k_{\rm B} T_{\rm kin}}{m_{\rm mol}}},
\end{aligned}
\end{equation}
wherein the thermal part is
\begin{equation}
\sigma_{\rm th}=\sqrt{\frac{k_{\rm B} T_{\rm kin}}{\mu m_{\rm H}}},
\end{equation}
and the non-thermal part is
\begin{equation}
\sigma_{\rm nt} = \sqrt{\frac{\Delta v^2}{8 \ln 2} - \frac{k_{\rm B} T_{\rm kin}}{m_{\rm mol}}},
\end{equation}
and $m_{\rm mol}$ is the molecular mass. In calculation we adopted a kinetic temperature of $T_{\rm kin}=13$ K as measured from the ${\rm NH_3}$ inversion lines \citep[][also see Fig.\,1c]{li13}.
The virial mass is sensitive to $\sigma_{\rm tot}$ and could have large variation if using different molecular lines. Besides the ALMA ${\rm N_2H^+}$ (1-0) line, we also considered the ${\rm N_2H^+}$ line observed by the NRO 45-meter telescope \citep{tatematsu08}, which has a resolution of $\sim18$ arcsec (beam size) and could better recover the extended gas. For the ${\rm N_2H^+}$, we measured the line width to be $\Delta v=0.9$ km s$^{-1}$\ at MMS-7 and 1.3 km s$^{-1}$\ over the clump surface within $r=33''$. We also inspected the NRO ${\rm C^{18}O}$ (1-0) lines \citep{feddersen20} and measured corresponding values to be $\Delta v=2.2$ and 2.5 km s$^{-1}$, respectively. The derived $M_{\rm vir}^B$ values are listed in Table 1. The single-dish ${\rm N_2H^+}$ emission has a similar spatial morphology with the dust continuum in OMC-3 \citep[][Figure 3 and 4 therein]{tatematsu08}, and its line width is close to the value observed with ALMA herein. This suggests that the cold dense gas component maintains a low turbulent state at both large and small scales, i.e., from the entire clump to its inner structures. The large virial ratio of $M_{\rm clump}/M_{\rm vir}^B=2.6$ suggests a considerable instability due to the self gravity.
On the other hand, since the dense gas in the clump is concentrated on the filaments, they should be the major ingredient to determine the clump stability. We estimated the critical linear mass density for the filament \citep{stodolkiewicz63, ostriker64} using
\begin{equation}\label{equ:eta_crit}
\eta_{\rm crit} = \frac{2 \sigma_{\rm tot}^2}{G}.
\end{equation}
The observed gas mass per unit length can be estimated from
\begin{equation} \label{equ:m_obs}
\eta_{\rm obs} = M_{\rm fil} / (Z_{\rm fil}/\sin \theta),
\end{equation}
where $Z_{\rm fil}$ is the filament length. In calculation we assume $\theta=60^\circ$. The derived $\eta$ of each filament is shown in Table 1. For all three filaments we have $\eta_{\rm obs}/\eta_{\rm crit}$ slightly exceeding 1.0. With the uncertainty in $\theta$ taken into account, $\eta_{\rm obs}$ could reach the maximum value at $\theta=90^\circ$. Even in this extreme case, the ratio of $\eta_{\rm obs}/\eta_{\rm obs}$ is still much lower than the virial ratio ($M_{\rm clump}/M_{\rm vir}^B=2.6$). We therefore expect the clump to have a less instability than in the spherical case.
\subsection{Evolutionary State and Timescales}\label{sec:evolve}
We estimated the current infall rate of the clump contraction from its average radius $r_{\rm in}$, velocity $v_{\rm in}$, and density $n_c$ using
\begin{equation}
\begin{aligned}
\dot{M}_{\rm in} & \simeq 4 \pi r_{\rm in}^2 \mu m_0 n_{\rm c} v_{\rm in} \\
\quad & = 8.0 \times 10^{-5} \left(\frac{r_{\rm in}}{4000\,{\rm AU}} \right)^2 \left(\frac{n_c}{5\times10^5\,{\rm cm^{-3}}} \right) \left(\frac{v_{\rm in}}{0.6\,{\rm km\,s^{-1}}} \right)\, M_\odot \,{\rm year}^{-1}.
\end{aligned}
\end{equation}
In calculation we adopted the average radius of $r=10''$ where the velocity gradient becomes significant. At this infall rate, the contraction can maintain a timescale up to $t_{\rm in}=M_{\rm clump}/\dot{M}_{\rm in}=3.6\times10^5$ years.
We considered two typical theoretical collapse models to compare the observation. First, the free-fall collapse represents the most rapid scenario for the clump evolution. It has a time scale of
\begin{equation}
\begin{aligned}
t_{\rm ff} & =\left(\frac{3\pi}{32 G \mu m_{\rm H}n_c}\right)^{1/2} \\
\quad & = 5\times10^4 \left(\frac{n_c}{5.0\times10^5\,{\rm cm^{-3}}}\right)^{-1/2} {\rm years},
\end{aligned}
\end{equation}
and consequently an infall rate of
\begin{equation}
\begin{aligned}
\langle\dot{M}_{\rm ff}\rangle & = M_{\rm clump}/t_{\rm ff} \\
\quad & = 6 \times 10^{-4} \left(\frac{M_{\rm clump}}{30\,M_\odot}\right)\left(\frac{n_c}{5.0\times10^5\,{\rm cm^{-3}}}\right)^{1/2}\,M_\odot \,{\rm year}^{-1}.
\end{aligned}
\end{equation}
The second case is the longitudinal collapse of a linear filament. The timescale can be estimated as \citep{pon12, cw15}
\begin{equation}\label{equ:t_fil}
\begin{aligned}
t_{\rm fil} & =(0.49+0.26A_{\rm O})(G\rho_{\rm O})^{-1/2} \\
\quad & = 3.5\times10^5 \left( \frac{0.49+0.26 A_{\rm O}}{5.7} \right)\left(\frac{\overline{n}}{1.0\times10^6\,{\rm cm^{-3}}}\right)^{-1/2} {\rm years},
\end{aligned}
\end{equation}
wherein $A_{\rm O}\equiv Z_{\rm O}/R_{\rm O}$ is the filament aspect ratio between its half-length $Z_{\rm O}=Z_{\rm fil}/2$ and average radius. In Equation \ref{equ:t_fil} the default $t_{\rm fil}$ is estimated using $A_{\rm O}=20$, which is the value of the NW filament. Comparing the three timescales, we have $t_{\rm fil}\simeq t_{\rm in} \gg t_{\rm ff}$. It suggests that the current infall rate is more inclined to the filamentary case. This result shows an agreement with the estimation of instabilities (Sec. \ref{sec:m_vir}).
During the filament contraction, the average velocity is estimated to be $\overline{v}_{\rm fil}=Z_{\rm O}/t_{\rm fil}=0.4$ km s$^{-1}$, which is comparable to the observed velocity more distant from the center ($r>10''$). For the inner region ($r<10''$), the one-dimensional kinematics would be less applicable since the three filaments are intersected, and the end point of each filament is fixed at MMS-7. The three filaments are contracting towards a common center and would therefore have the observed steeper velocity increase.
During the filament evolution, the local jeans instability may cause the filament to have a fragmentation. For the NW filament, the fragmentation could be taking place and leading to the formation of the four dense cores. Following \citet{wiseman98} and \citet{takahashi13}, the jeans length of a cylindrical gas body can be estimated as
\begin{equation}
\lambda_{\rm frag}\simeq\frac{20 \sigma_{\rm tot}}{\sqrt{4\pi G \rho}}.
\end{equation}
Adopting the number density and $\Delta v$ values in Table 1, we can derive $\lambda_{\rm frag}=24''$. With the projection effect taking into account, this distance is in a favorable agreement with the average spacing of the cores on the NW filament ($\sim20''$).
Third, the filament can also be oscillating along its transverse direction and this can occur simultaneously during the fragmentation \citep{clarke17,stutz16,liu19a}. This may also be ongoing in MMS-7 region, since the filaments indeed exhibit a feature of velocity fluctuation. As seen in Figure 4, the NW and north filaments have a velocity fluctuation with a scale of 0.2 to 0.3 km s$^{-1}$\ and a spatial scale of $\sim10$ arcsec (0.02 pc) as labelled in red lines throughout the emission feature. The fluctuation has a velocity scale. The fragmentation model \citep{clarke17} derived a relation between the dominant perturbation wavelength $\lambda_{\rm D}$ and the time scale $\tau_{\rm crit}$ for the filament to become supercritical that is $\lambda_{\rm D}\sim 2\tau_{\rm crit} a_0$, wherein $a_0$ is the isothermal sound speed. For the NW filament, if adopting $\lambda_{\rm D}=10''$, we can derive $\tau_{\rm crit}=10^4$ years, which is lower than $t_{\rm in}$ for an order of magnitude. The contrast between the two time scales suggests that the filament should be actually stable against oscillation, otherwise it would be already dissipated. Although the oscillation could be taking place, it should be lower than a certain threshold level to significantly interrupt the global structure or the contraction towards the center. If there is no other perturbations, the inflow would be able to transport a considerable fraction of the clump mass into the center.
\section{Summary}\label{sec:summary}
We investigated the gas structures in Orion OMC-3 MMS-7 region from dust continuum and molecular lines, in particular the ALMA ${\rm N_2H^+}~(1-0)$. These tracers revealed key properties of the multi-filament structures, as specified in four major aspects:
(1) Three dense and compact filaments are observed in ${\rm N_2H^+}~(1-0)$ emission. They compose the spine structure of the parental clump and tend to have an intersection at the clump center. There are at least seven dense cores and stellar objects being formed in this 0.2-pc region, with the central core MMS-7 being the most massive one ($3.6\,M_\odot$). Besides the two infrared sources (IRAC-2405 and 2407), the other five objects could all be starless or prestellar cores.
(2) The three filaments all exhibit a noticeable velocity gradient towards the center as measured from the ${\rm N_2H^+}~(1_0-0_1)$ transition. The radial velocity reaches the maximum blueshift of $v-v_{\rm sys}=-2.0$ km s$^{-1}$\ at the center. In the mean time the filaments have relatively small velocity dispersion ($\Delta v<0.65$ km s$^{-1}$). Since the dense gas has no evident star-forming activities (e.g. shell expansion and outflow), the velocity gradient should mainly trace the inward gas motion, namely the filament contraction towards MMS-7.
(3) The infall timescale ($3.6\times 10^5$ years) and mass transfer rate ($8\times10^{-5}\,M_\odot\,{\rm year}^{-1}$) are both much less prominent than those in a spherical free-fall collapse. The infall property and critical-mass analysis both suggest that the clump instability should mainly depend on the internal filaments. Although the filaments in particular the NW one has a likely fragmentation, it does not seem to disturb the global filament structures and contraction towards MMS-7.
In general, the MMS-7 region shows a similarity with frequently observed filament-hub systems. And in this case the filamentary contraction towards an individual prestellar core is more evidently identified. Although there are multiple dense cores and young stars, the major fraction of the mass would still be transferred to the central object. This scenario is comparable to the competitive accretion, and it further implies that the intersection of the convergent filaments could be the major site to assemble the most massive core.
\section{Acknowledgement}
This work is supported by the China Ministry of Science and Technology under State Key Development Program for Basic Research (973 program) No. 2012CB821802, the National Natural Science Foundation of China No. 11403041, No. 11373038, No. 11373045, Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB09010302, and the Young Researcher Grant of National Astronomical Observatories, Chinese Academy of Sciences.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,260 |
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Body-Worn Cameras and Internal Accountability at a Police Agency
Marthinus Koen, Brooke Mathna
AM J QUALITATIVE RES, Volume 3, Issue 2, pp. 1-22
DOI: https://doi.org/10.29333/ajqr/6363
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Existing research on body-worn cameras have primarily focused on certain policing outcomes (e.g., citizen complaints and use-of-force), however, only a handful of research to date has considered how the implementation of body-worn cameras have impacted internal organizational processes at police departments. Using semi-structured interviews, a survey, and ride-along observations, we examined how body-worn cameras impacted the way police officers were held or felt accountable for their behavior. The study was conducted at the Sunnyvale Police Department (pseudonym), a small city agency in the United States that had been using cameras for two and a half years. Particularly, we describe how body-worn cameras impacted accountability at Sunnyvale within different organizational contexts that included reporting, citizen interactions, training, and supervision. Consistent with the hopes of reformers, body-worn cameras did seem to raise the general sense of accountability as they became a part of training, citizen encounters, reporting, and supervision. However, these changes were not like reformers would have imagined, as the department did not intently use cameras in a way to hold officers any more accountable for their conduct and performance on the street.
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The Perceptions of Private and Suburban Public Schools among Middle Class Urban Parents: Evidence from Albany, New York
Paul T. Knudson
AM J QUALITATIVE RES, Volume 3, Issue 2, pp. 23-41
Substantial research has examined urban middle class parents and their motivations for enrolling their children in city public schools. Less research, however, has explored how these types of parents view private schools and suburban public schools. Using in-depth interviews with parents and guided by the frameworks of rational choice and social identity theories, this paper explores this topic. Findings suggest that both instrumental and philosophical factors affect parents' perceptions of private schools and suburban public schools. Parents tended to be skeptical of the purported, superior quality of education in both private and suburban public schools. They also felt these schools offered neither advantages surrounding the social/peer climate, nor in college preparation. In addition, parents believed that enrolling their children in either private schools or suburban public schools would undermine their egalitarian and inclusive social and political values.
Keywords: Urban Education, Schools, School Choice, Private and Suburban Public Schools.
Cognitive Considerations In Interface Design For Children With Asperger's Syndrome: A Proposed Budgeting Tool
Fatih Demir
The characteristics of a particular user need to be evaluated to inform the design in the field of Human-Computer Interaction (HCI) design (Demir et al., 2017). This research consists of an evaluation of the characteristics of Asperger Syndrome (AS) and a budgeting tool has been proposed for children with the AS. The research results show that simplicity, consistency, color-coding and efficient labeling in design helps to reduce the cognitive load of the children with AS, thus, make the design more usable and useful for them.
Keywords: HCI, Asperger's Syndrome, Budgeting, Cognitive Considerations, User Experience.
Refugee Perspectives on Integration in Germany
Cuneyt Gurer
This study will examine the integration process from the perspectives of refugees by focusing on individual and social level experiences. Data for this study comes from individual-level semi-structured and focus group discussions collected over a period of seven months in Germany and covers refugees who applied for asylum after 2015. Keeping in mind the fact that the meaning of integration differs for different audiences, this study identifies five distinct phases of refugee integration in Germany. In each phase, refugees face certain challenges that hinder their integration into local communities. These phases, with some minor differences, may also be applicable to other countries, and therefore this study offers a general framework to analyze the integration process from refugee perspectives. Individual and social dimensions of integration analyzed and individual adjustment and coping mechanisms demonstrated throughout the study. This study suggests that refugees should be more involved in the definition of integration and more importantly, social and professional interaction through "entry points" should be encouraged for successful integration of refugees.
Keywords: Refugees, Integration, Adaptation, Integration Policies.
"Relational Transformation": A Grounded Theory of the Processes Clinical Nurse-Educators (CNEs) Use to Assist Students Bridge the Theory-Practice Gap
Joseph Osuji, Jane Onyiapat, Mohamed El-Hussein, Peace Iheanacho, Chinenye Ogbogu, Nneka Ubochi, Ada Obiekwu
The theory-practice gap debate has permeated nursing literature for many decades. While some scholars insist on the existence of a gap and argue that the real value of nursing can only be represented by a broad theoretical framework that explains what nurses do; others have categorically rejected the need for nursing theories. The gap between theory and practice in nursing education has been highlighted as a pressing problem in developing countries where nursing education is in transition from hospital-based training to universities. Clinical Nurse Educators (CNEs) have important roles to play in assisting students to deconstruct their practices in relation to nursing theories. The goal of this study was to construct a theory that explains the processes clinical nurse educators employ with their students to bridge the theory-practice gap in nursing education. We sought to answer two major questions: How do nurse educators assist their students to bridge the gap between theory and practice? How can we theoretically explain the process of bridging the theory-practice gap? The overall design for this research study was qualitative, rooted in the tradition of classical grounded theory (GT). This method of research has a unique feature in its ability to make possible the generation of theory that explains a process. Participants in this study were clinical nurse educators in Departments of Nursing Sciences, at universities in South Eastern Nigeria. Participants interview transcripts constituted data for analysis. The core category that emerged from participants' data was "relational transformation" where clinical instructors facilitate a transformative learning process, mediated within the context of the instructor-student relationship and aimed at assisting students to transform into professionals. This relational process involves assisting students to connect classroom theory to what they experience in the clinical areas, through questioning, mentoring, role modeling, and reflection.
Keywords: Theory-Practice Gap, Nursing Theory, Transformation, Professional Socialization, Grounded Theory.
Stained Glass Windows and Rainbows: My Journey to Socially Just Perceptions of and Actions toward LGBTQ Students
Kristina Andrews
As a classroom teacher I have an ethical and moral obligation to provide a socially just and equitable learning environment and educational experience for all my students. Auto-ethnography is a research method grounded in modern philosophy. Auto-ethnography enables me to link the personal and cultural through inquiry and voice. In this article I explore the state of understanding my personal role and experiences as a conservative Christian educator in learning about and interacting with LGBTQ students. By reflecting and evaluating my personal experiences and attitudes I am able to identify my former biases, discriminatory tendencies, and stereotypes about LGBTQ community members. Therefore, throughout my analysis, I sought a means that would enable me to maintain my personal religious values while eliminating biases, stereotypes, and unjust perceptions. As a result, I recognize and embrace my religious identify as a conservative Christian educator who accepts my LGBTQ students without judgement or bias in order to be respectful, socially just, equitable, and nondiscriminatory.
Keywords: Auto-Ethnography, Christian Educator, LGBTQ.
Terri Ratini
(no abstract) | {
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A β-oxidação é um processo catabólico de ácidos graxos que consiste na sua oxidação mitocondrial. Eles sofrem remoção, por oxidação, de sucessivas unidades de dois átomos de carbono na forma de acetil-CoA. Como exemplo pode ser citado o ácido palmítico, um ácido graxos de 16 carbonos, que vai sofrer sete reações oxidativas, perdendo em cada uma delas dois átomos de carbono na forma de acetil-CoA. Ao final desse processo os dois carbonos restantes estarão na forma de acetil-CoA.
A β-oxidação é dividida em quatro reações sequenciais:
Oxidação, na qual o acil-CoA é oxidado a enoil-CoA, com redução de FAD a FADH2
Hidratação, na qual uma dupla ligação é hidratada e ocorre a formação de 3-hidroxiacil-CoA
Oxidação de um grupo hidroxila a carbonila, tendo como resultado uma beta-cetoacil-CoA e NADH
Cisão, em que o β-cetoacil-CoA reage com uma molécula de CoA formando um acetil-CoA e um acil-CoA que continua no ciclo até ser convertido a acetil-CoA
Mas quando a cadeia de ácidos graxos for ímpar, o produto final da β-oxidação será o propionil-CoA, esse composto, através da incorporação de CO2 e gasto energético através de quebras de ligações do ATP, se transforma em succinil-CoA, que é um composto do Ciclo de Krebs.
Após a β-oxidação, os resíduos acetila do acetil-CoA são oxidados até chegarem a CO2, o que ocorre no ciclo do ácido cítrico. Os acetil-coa vindos da oxidação vão entrar nessa via junto com os acetil-coA provenientes da desidrogenação e descarboxilação do piruvato pelo complexo enzimático da piruvato desidrogenase. Nessa etapa haverá produção de NADH e FADH2 para suprir de elétrons a cadeia respiratória da mitocôndria, que os levará ao oxigênio. Junto a esse fluxo de está a fosforilação do ADP em ATP. Com isso a energia gerada na oxidação de ácidos graxos vai ser conservada na forma de ATP.
A ativação do ácido graxo
A oxidação de ácidos graxos começa com a formação de uma ligação de tioéster entre o grupo carboxilo do grupo de ácido graxos e o tiol do CoA. Esta reação é catalisada pela acetil-CoA sintase. A reação pode ter lugar na mitocôndria. Este é o caso para os ácidos graxos de cadeia curta, que podem difundir-se através da membrana daquela organela. Para moléculas de cadeia longa, a reação tem lugar no folheto citoplasmático da membrana mitocondrial. A reação é acompanhada por hidrólise de uma molécula de ATP em AMP e pirofosfato. Esta reação é prontamente reversível: o pirofosfato é hidrolisado para que sua concentração citosólica seja baixa. Isto ajuda a dirigir a reação de ativação no sentido da formação do acetil-CoA.
Saldo energético
A oxidação de ácidos graxos produz muito mais energia que a oxidação de carboidratos. Uma molécula de palmitato, por exemplo, produz um saldo líquido de 108 ATPs, enquanto uma molécula de glicose produz apenas 32.
Similaridades entre beta-oxidação e ciclo do ácido cítrico mostradas como projeções de Fischer e como modelo poligonal
As reações da beta oxidação e parte do ciclo dos ácido cítrico apresentam similaridades estruturais em três das quatro reações da beta oxidação: a oxidação por FAD, a hidratação e a oxidação por NAD+. Tal comparação pode ser feita observando as reações utilizando projeções de Fischer, e também de maneira bastante clara usando o modelo poligonal. Cada enzima dessas etapas apresenta similaridades estruturais.
Formação de corpos cetônicos no humano
Em situações de baixa concentração de glicose no sangue (como jejum prolongado) a β-oxidação é uma alternativa para a produção de energia (pois libera FADH2 e NADH).Consequentemente, há muita produção de acetil-CoA. O Ciclo de Krebs não consegue absorver todo esse substrato, estando prejudicado, uma vez que seus intermediários estão envolvidos na gliconeogênese. Essas moléculas de acetil-CoA se condensam , formando Corpos cetônicos, essa condensação acaba liberando Coenzima A, o que é essencial para que haja continuidade no Ciclo de Krebs. Essa produção ocorre principalmente no fígado, que por sua vez não possui a capacidade de degradar corpos cetônicos (evita ciclo fútil, pois nesse caso o fígado realizaria a síntese e a degradação desses corpos, e os outros órgãos do corpo não poderiam obter a energia da quebra dessas moléculas). Os corpos cetônicos podem ser usados como fonte de energia no cérebro em casos de desnutrição, nos quais a disponibilidade de glicose é mínima.
Estrutura dos intermediários da cetogênese em projeções de Fischer e modelo poligonal
Os intermediários da reação de condensação de acetil CoA em projeções de Fischer mostram as mudanças químicas passo a passo. Tal imagem pode ser representada utilizando-se o modelo poligonal.
Os tecidos extra-hepáticos usam os corpos cetônicos como combustíveis
O D-β-hidroxibutirato é oxidado até acetoacetato pela D-β-hidroxibutirato desidrogenase nos tecidos extra-hepáticos. O acetoacetato é ativado para formar o éster da coenzima A por transferência do CoA do succinil-CoA, um intermediário do ciclo do ácido cítrico, numa reação catalisada pela β-cetoacil-CoA transferase. O acetoacetil-CoA é então clivado pela tiolase para liberar duas moléculas de acetil-CoA que entram no ciclo do ácido cítrico.
Estrutura dos intermediários da degradação de corpos cetônicos em projeção de Fischer e modelo poligonal
Os intermediários de reação em projeção de Fischer mostram as mudanças químicas passo a passo. Tal imagem pode ser comparada à representação utilizando o modelo poligonal.
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Proving Fault In An Accident Claim In Massachusetts
Even though Massachusetts is a no-fault state for insurance purposes, this does not mean you cannot file an accident claim against another driver. There are circumstances that allow an accident victim to seek compensation from another driver instead of looking to the victim's insurance provider as the sole source of compensation. For example, if your medical bills exceed $2,000 or you suffered a permanent injury, you may be able to file an accident claim.
However, you must prove the other driver caused the accident to recover compensation for your medical bills, lost income, and other damages. Proving fault in an accident can be challenging, especially in certain car accident cases. Accidents involving commercial vehicles or multiple vehicles are often more difficult to investigate. An experienced Massachusetts car accident attorney can help you gather the evidence you need to prove fault in an accident.
What Evidence Can Help in Proving Fault in an Accident?
Attorneys use a variety of evidence when they are focusing on proving fault in an accident case. Some examples of the evidence we may use in your case include:
Police Report — When the police officer investigates the accident, the officer may cite the other driver with a traffic offense that contributed to the cause of the crash. Even if the officer does not issue a citation, he will typically indicate whether each driver contributed to the cause of the crash. Therefore, a police report stating the other driver was at fault can be valuable when we are negotiating a settlement with the insurance company for the other driver.
Accident Scene Evidence — Evidence from the accident scene can be extremely useful, especially if we must recreate the accident to determine fault. Examples of evidence from the accident scene include photographs, videos, measurements of skid marks, notation of the position of each vehicle after impact, and information about the surroundings that could have been a factor in the cause of the crash. If possible, you should always take photographs and videos of the accident scene immediately after the crash.
Eyewitness Statements — Statements from eyewitnesses can be very compelling, especially at trial. Jurors often give more weight to an eyewitness account because unlike the litigants the witness does not have a financial stake in the outcome of the trial.
Operator's Report — In some cases, a driver may admit fault in his or her operator's report. It is very important to obtain copies of the reports and review the reports to determine if they can be used as evidence of the other driver claiming responsibility for the collision.
Medical Records — Medical records are also important evidence in proving fault in an accident case. Your injuries can help substantiate the position of the vehicles and the speed of the vehicles at the time of the crash.
Expert Testimony — In complex accident cases, we may retain experts who can recreate the accident to demonstrate cause. The expert's testimony can provide convincing evidence that convinces a jury that the other driver was at fault for the crash.
Depending on the type of accident and the vehicles involved, there could be other evidence we use to prove fault. For example, in a commercial vehicle accident, we would work quickly to obtain the information on the truck's black box and the records from the trucking company. It is important to work with an experienced car accident attorney who understands the type of evidence and how to obtain the evidence to prove that you were not at fault for the collision.
Call for a Free Consultation with a Massachusetts Accident Attorney
If a negligent driver injured you in an accident, we can help. Contact Jim Glaser Law by calling 888-743-2079 for a free case evaluation. Our attorneys can review your case and provide you with the options for filing an injury claim to recover compensation for your injuries.
Jim Glaser Law2018-08-08T12:11:28-04:00May 30th, 2018|Accident Lawyer|Comments Off on Proving Fault In An Accident Claim In Massachusetts
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\section{Introduction}
Optimal transport provides a convenient framework for density interpolation as a convex optimization problem. Its most remarkable feature is its sensitivity to horizontal displacement, which generally allows one to retrieve translations when interpolating between two densities. This property has motivated the application of optimal transport to many imaging problems, especially in the context of physical sciences and fluid dynamics. A typical example comes from satellite image interpolation in oceanography. In this case, one is interested in reconstructing the evolution of a quantity of interest such as Sea Surface Temperature (SST) or Sea Surface Height (SSH) between two given observations.
As highlighted in \cite{hug2015multi}, for this type of applications one needs to include appropriate regularization terms to avoid the appearance of unphysical phenomena such as mass concentration in the reconstructed density evolution.
In this paper we propose a finite element approach to solve the dynamical formulation of optimal transport with quadratic cost on unstructured meshes (and therefore can be easily implemented on complex domains) and that can be easily modified to include different type of regularizations which are relevant for the dynamic reconstruction and interpolation of physical quantities. For some choices of finite element spaces, using the framework introduced in \cite{lavenant2019unconditional}, we can prove convergence of our discrete solutions to the ones of the continuous problem.
The dynamical formulation of optimal transport inspired some of the first numerical methods for this problem. This reads as follows: given two probability measures $\rho_0, \rho_1 \in \mc{P}(D)$ on a compact domain $D\subset \mathbb{R}^d$, find the curve $t\in[0,1] \mapsto \rho(t,\cdot) \in \mc{P}(D)$ which solves
\begin{equation}\label{eq:dynamic}
\inf_{\rho,v} \left \{ \int_0^1\int_D \frac{|v(t,\cdot)|^2}{2}\mathrm{d} \rho(t,\cdot) \mathrm{d} t \,;\, \partial_t{\rho}+ \mathrm{div}_x(\rho v) =0 ,\rho(0) = \rho_0,\,\rho(1) = \rho_1 \right\}
\end{equation}
where $v:[0,1]\times D \rightarrow \mathbb{R}^d$ is a time-dependent velocity field on $D$ tangent to the boundary $\partial D$, and $|\cdot|$ denotes the Euclidean norm. In other words, problem \eqref{eq:dynamic} selects the curve of minimal kinetic energy with fixed endpoints $\rho_0$ and $\rho_1$.
Benamou and Brenier \cite{benamou2000computational} realized that introducing the momentum $m \coloneqq \rho v$, problem \eqref{eq:dynamic} can be recast into a convex optimization problem in the variables $(\rho, m)$, with a linear constraint, since the continuity equation becomes
\begin{equation}\label{eq:continuity}
\partial_t{\rho} + \mathrm{div}_x m = 0 \,.
\end{equation}
If we define $\sigma \coloneqq (\rho,m)$, regarded as a measure on $[0,1]\times D$, this constraint is equivalent to $\mathrm{div}\,\sigma = 0$, where now $\mathrm{div}$ denotes the divergence operator on the space-time domain $[0,1] \times D$. Introducing the dual variable $q = (a,b)$ where $a \in C([0,1]\times D)$ and $b \in C([0,1]\times D;\mathbb{R}^{d})$, the kinetic energy minimized in \eqref{eq:dynamic} can be written in the form
\[
\sup_q \left \{ \int_0^1 \int_D q \cdot \mathrm{d} \sigma \,;\, a+\frac{|b|^2}{2} \leq 0 \right\}.
\]
Combining this expression with \eqref{eq:dynamic} we obtain a saddle point problem in the variables $(q,\sigma)$ with a nonlinear constraint on $q$ and a linear one on $\sigma$.
The numerical method proposed in \cite{benamou2000computational} involves discretizing $q$ and $\sigma$ by their values on a regular grid, and expressing the constraint on $\sigma$ via a Lagrange multiplier; then the dual problem can be solved by an Augmented Lagrangian ADMM approach, optimizing separately in $q$ and the Lagrange multiplier and then performing a gradient descent step on $\sigma$. Disregarding the discretization in space-time, the convergence of the method has been studied in \cite{guittet2003time,hug2017convergence}. The same approach was used to discretize different problems related to optimal transport (e.g., gradient flows \cite{benamou2016augmented}, mean field games \cite{benamou2015augmented}, unbalanced optimal transport \cite{gallouet2019unbalanced}) using a finite element discretization in space-time. Importantly, in these cases the numerical method is obtained by discretizing the several steps of the augmented Lagrangian approach rather than as a discrete optimization algorithm. This implies that in general it is difficult to establish the convergence of the discrete algorithms. Moreover, for these type of methods, convergence results towards the continuous solutions with mesh refinement are only available for specific settings (e.g., the $L^1$-type optimal transport problems studied in \cite{igbida2017augmented}), but they are not available for the optimal transport problem \eqref{eq:dynamic}.
Papadakis, Peyré, and Oudet proposed in \cite{papadakis2014optimal} a staggered finite difference discretization on regular grids of the optimal transport problem \eqref{eq:dynamic}, and they considered different proximal splitting algorithms to solve it. The computational bottleneck for these methods as well as for the original augmented Lagrangian approach is the \modif{projection onto the space of divergence-free vector fields $\sigma$, which amounts to solving a Poisson equation at each iteration.} This however can be avoided by exploiting the Helmholtz decomposition of vector fields, as recently showed in \cite{henry2019primal}, or adding regularization terms as in \cite{li2018computations}. Recently, Carrillo and collaborators \cite{carrillo2019primal} proposed a finite difference scheme similar to that in \cite{papadakis2014optimal} (in the context of the discretization of Wasserstein gradient flows), for which they could also prove its convergence with mesh refinement, but only upon strong regularity assumptions on the solutions of the continuous problem.
In \cite{lavenant2018dynamical} a numerical scheme was proposed using tools from finite element and finite volume methods, where one explicitly constructs a duality structure for the discrete variables. Later Lavenant \cite{lavenant2019unconditional} proved convergence of this scheme, unconditionally with respect to the time/space step size, to the solutions of the optimal transport problem, proposing a general framework for convergence of discretizations of problem \eqref{eq:dynamic} between two arbitrary probability measures. This filled a critical gap for the analysis of discrete dynamical transport models, since previously convergence results were only known in case of sufficiently smooth solutions (as in \cite{carrillo2019primal}) or conditional to the relative time/space step sizes (e.g., in the context of finite volume methods, combining the results in \cite{erbar2020computation} and \cite{gladbach2018scaling}).
\subsection{Contributions and structure of the paper}
In this paper we propose a mixed finite element discretization of \eqref{eq:dynamic} which generalizes to the finite element setting the finite difference scheme proposed by Papadakis et al.\ \cite{papadakis2014optimal}. We derive our method by discretizing a saddle point formulation of the dynamic optimal transport problem on Hilbert spaces, where one looks for a solution $(q, \sigma) \in L^2([0,1]\times D;\mathbb{R}^{d+1})^2$. Nonetheless, we stress that the method we obtain is still well-defined when the initial and final data are arbitrary probability measures. By using $H(\mathrm{div})$-conforming spaces for the variable $\sigma$, we are able to construct discrete solutions that satisfy exactly the weak form of the continuity equation \eqref{eq:continuity}.
\modif{Using the framework of \cite{lavenant2019unconditional}, we also show that our discrete solutions, for specific choices of finite element spaces, converge towards the solutions of the optimal transport problem between two arbitrary measures, and therefore even when the solution $\sigma$ is only a measure (see Theorem \ref{th:convergence}). Such a result carries over also to a slight modification of the finite difference scheme proposed in \cite{papadakis2014optimal}, which can be viewed as a particular instance of our discretization on a uniform quadrilateral grid (see Remark \ref{rm:convpeyre}).}
\modif{
Finally, as in \cite{papadakis2014optimal}, we solve the discrete problem using a proximal splitting algorithm \cite{pock2009algorithm}. Importantly, this is not only a discretization of the same algorithm applied to the continuous saddle point formulation as in previous works, but also a genuine optimization scheme applied to the finite dimensional problem. Furthermore, we observe numerically that the proposed modification of the finite difference scheme in \cite{papadakis2014optimal} (which we derived to prove convergence with mesh refinement) also yields a remarkable speedup for the convergence of the proximal splitting algorithm itself, keeping approximately the same computational cost per iteration.}
The paper is structured as follows. We establish the notation in Section \ref{sec:notation}. In Section \ref{sec:dynamical} we give the precise formulation of problem \eqref{eq:dynamic} and describe the proximal splitting algorithm applied to the continuous problem in the Hilbert space setting. In Section \ref{sec:fe} we introduce and discuss the main finite element tools we use for our method. In Section \ref{sec:discrete} we define our finite element discretization of problem \eqref{eq:dynamic} and state the convergence result. In Section \ref{sec:proximal} we detail the steps required for solving our discrete optimal transport problem with a proximal splitting algorithm. In Section \ref{sec:regularization} we describe how to introduce regularization terms in the formulation. Finally in Section \ref{sec:num} we present some numerical results.
\section{Notation}\label{sec:notation}
Throughout the paper we will denote by $D\subset \mathbb{R}^d$ a convex polytope, with $d\in\{2,3\}$, and by $\Omega \coloneqq [0,1] \times D$ the space-time domain. For differential operators such as $\nabla$ or $\mathrm{div}$, we use the subscript $x$ to emphasize that these are defined on $D$ rather than $\Omega$, but we will drop this subscript when this is clear from the context.
We use the standard notation for Sobolev spaces on $D$ or $\Omega$. In particular, $L^p(D;\mathbb{R}^d)$ denotes the space of functions $f:D\rightarrow \mathbb{R}^d$ whose Euclidean norm $|f|$ is in $L^p(D)$. We use a similar notation for functions taking values on a subset $K \subset \mathbb{R}^d$, or defined on $\Omega$. We denote by $H(\mathrm{div};D)$ the space of vector fields $f:D \rightarrow \mathbb{R}^d$ in $L^2(D;\mathbb{R}^d)$ whose divergence is in $L^2(D)$. Similarly, $H(\mathrm{div}; \Omega)$ the space of vector fields $f: \Omega \rightarrow \mathbb{R}^{d+1}$ in $L^2(\Omega;\mathbb{R}^{d+1})$ whose divergence is in $L^2(\Omega)$.
Finally, we denote by $\mc{M}(D)$ the set of finite signed measures on $D$, by $\mc{M}_+(D) \subset \mc{M}(D)$ the convex subset of positive measures; by $\mc{P}(D) \subset \mc{M}_+(D)$ the set of positive measures of total mass equal to one; and by $C(D)$ the space of continuous functions on $D$. We use a similar notation for the spaces of measures and continuous functions on $\Omega$. We use $\langle \cdot,\cdot \rangle$ to denote either the duality pairing between measures and continuous functions or the $L^2$ inner product, on either $D$ or $\Omega$, according to the context.
\section{Dynamical formulation of optimal transport}\label{sec:dynamical}
The dynamical optimal transport problem \eqref{eq:dynamic} can be formulated as a saddle point problem on the space of measures $\sigma \coloneqq (\rho,m)\in \mc{M}(\Omega) \times \mc{M}(\Omega)^{d}$. This can be written as follows
\begin{equation}\label{eq:BB}
\inf_{\sigma \in \mc{C}} \mc{A}(\sigma) ,\quad \mc{A}(\sigma) \coloneqq \sup_{q \in C(\Omega; K)} \langle q, \sigma\rangle,
\end{equation}
where $\mc{C}$ is the set of measures $\sigma \in \mc{M}(\Omega)^{d+1}$ satisfying $\mr{div}\, \sigma = 0$ in distributional sense with boundary conditions
\begin{equation}\label{eq:boundary}
\sigma \cdot n_{\partial \Omega} = \mc{X} \,, \quad \mc{X} \coloneqq
\left \{
\begin{array}{ll}
\rho_0 & \text{ on } \{0\} \times D,\\
\rho_1 & \text{ on } \{1\} \times D,\\
0 & \text{ otherwise, }
\end{array}
\right.
\end{equation}
with $\rho_0,\rho_1 \in \mc{P}(D)$, and where $C(\Omega; K)$ is the space of continuous functions on $\Omega$ taking value in the convex set
\begin{equation}\label{eq:K}
K \coloneqq \left \{ (a,b) \in \mathbb{R}\times \mathbb{R}^{d} \,; \, a +\frac{|b|^2}{2} \leq 0 \right \}.
\end{equation}
It will be convenient to treat time and space as separate variables. In particular we will also use the action defined by
\[
A(\rho,m) \coloneqq \sup_{(a,b) \in C(D; K)} \langle \rho,a\rangle + \langle m,b\rangle\,,
\]
for any $(\rho,m) \in \mathcal{M}(D)^{d+1}$. Then, $A(\rho,m)$ is finite if and only if $m$ has a density with respect to $\rho$ and in that case $A(\rho,m) = \int_D B(\rho,m) $, where $B:\mathbb{R} \times \mathbb{R}^d \rightarrow [0, +\infty]$ is the function given by
\[
B(a,b) \coloneqq \left\{
\begin{array}{ll}
\frac{|b|^2}{2a} & \text{if } a>0, \\
0 & \text{if } a=0, b=0, \\
+\infty & \text{if } a =0, b\neq 0 \text{ or } a<0\,.
\end{array}
\right.
\]
Due to the definition of the function $B$, any saddle point of problem (\ref{eq:BB}) must satisfy $\rho\ge0$.
The value of the infimum of problem \eqref{eq:BB} coincides with $W^2_2(\rho_0,\rho_1)/2$, where $W_2(\cdot,\cdot)$ denotes the Wasserstein distance associated with the $L^2$ cost (see Theorem 5.28 in \cite{santambrogio2015optimal}). Moreover the infimum itself is attained by a measure $\sigma=(\rho,m)$, where $\rho$ is known as the Wasserstein geodesic between $\rho_0$ and $\rho_1$ (see proposition 5.32 in \cite{santambrogio2015optimal}).
We refer the reader to \cite{santambrogio2015optimal} for more details on the links between the dynamical formulation and the Wasserstein distance.
\subsection{Hilbert space setting and proximal splitting}\label{sec:ham}
Before discussing the discretization of problem \eqref{eq:BB}, we review its reformulation on Hilbert spaces, and discuss the convergence of the proximal splitting algorithm.
\begin{proposition}[Guittet \cite{guittet2003time}; Hug et al. \cite{hug2017convergence}] \label{prop:hilbert}
Suppose $\rho_0,\rho_1 \in L^2(D)$. Then problem \eqref{eq:BB} is equivalent to
\begin{equation}\label{eq:BB1}
\inf_{\sigma \in \mc{C}} \sup_{q \in L^2(\Omega; K)} \langle q, \sigma\rangle\,,
\end{equation}
where $\mc{C}$ is the set of functions $\sigma \in H(\mr{div};\Omega)$ satisfying $\mr{div}\, \sigma = 0$ in weak sense with boundary conditions given by \eqref{eq:boundary}. Moreover, assuming that $\mathrm{supp}(\rho_0) \cup \mathrm{supp}(\rho_1) \subset \accentset{\circ}{D}$, there exists a saddle point $(\sigma^*,q^*) \in \mc{C} \times L^2(\Omega;K)$ solving problem \eqref{eq:BB1}.
\end{proposition}
The equivalence of problem \eqref{eq:BB1} to \eqref{eq:BB} can be easily deduced by a regularization argument on $\sigma$ and then applying Lusin's theorem as in Proposition 5.18 in \cite{santambrogio2015optimal}. The proof for the existence of a saddle point problem is more delicate and can be found in \cite{hug2017convergence}.
In order to apply a proximal splitting algorithm to solve problem \eqref{eq:BB1}, we first write it in the form
\begin{equation}\label{eq:BB2}
\inf_{\sigma \in L^2(\Omega;\mathbb{R}^{d+1}) } \sup_{q \in L^2(\Omega; \mathbb{R}^{d+1})}
\langle q, \sigma\rangle + \iota_{\mc{C }} (\sigma) - \iota_{\mc{K}} (q)\,,
\end{equation}
where $\iota$ denotes the convex indicator function and
\[
\mc{K} \coloneqq L^2(\Omega;K) = \{ q \in L^2(\Omega;\mathbb{R}^{d+1})\,; \, q \in K ~ \text{a.e.} \}.
\]
Note in particular that $\mc{C}$ and $\mc{K}$ are closed convex sets of $L^2$.
We apply to \eqref{eq:BB2} the primal-dual projection algorithm proposed in \cite{pock2009algorithm}. In particular, given $\tau_1, \tau_2>0$ and an admissible $(\sigma^0, q^0)\in \mc{C}\times \mc{K}$, we define the sequence $\{(\sigma^k,q^k)\}_k$ by the two-step algorithm:
\begin{subequations}\label{eq:PS}
\begin{align}
\text{\textbf{Step 1}}: \qquad & \sigma^{k+1} = P_{\mc{C}} (\sigma^k- \tau_1 q^k)\,.\\
\text{\textbf{Step 2}}: \qquad &q^{k+1} = P_{\mc{K}}(q^k +\tau_2(2 \sigma^{k+1}-\sigma^k))\,.
\end{align}
\end{subequations}
where $P_{\mc{C}}$ and $P_{\mc{K}}$ are the $L^2$ projections on the closed convex sets $\mc{C}$ and $\mc{K}$, respectively. The projection onto $\mc{C}$ amounts to computing the Helmholtz decomposition of $\sigma^k- \tau_1 q^k$ and selecting the divergence-free part, whereas the projection onto $\mc{K}$ is a pointwise projection applied to a representative of $q^k +\tau_2(2 \sigma^{k+1}-\sigma^k)$.
The proof of convergence in \cite{pock2009algorithm} holds also in our setting. More precisely, the following convergence theorem holds.
\begin{theorem}[Pock et al.\ \cite{pock2009algorithm}]\label{th:cp}
If $\tau_1 \tau_2 <1$ then $(\sigma^k, q^k)\rightarrow (\sigma^*, q^*)\in \mc{C}\times \mc{K}$ which solves \eqref{eq:BB1}.
\end{theorem}
Discretizing problem \eqref{eq:BB2}, and consequently the proximal splitting algorithm \eqref{eq:PS}, with finite elements requires choosing finite-dimensional spaces for $\sigma$ and $q$ so that the steps in \eqref{eq:PS} are well-posed and \modif{computationally feasible}. However, satisfying these requirements is not enough to guarantee convergence of the discrete solutions to the ones of the infinite dimensional problem. Hereafter we will identify a class of finite element spaces for which the theory developed in \cite{lavenant2019unconditional} applies, which allows us to deduce convergence to the solutions of problem \eqref{eq:BB}, i.e.\ even when $\rho_0$ and $\rho_1$ are arbitrary probability measures and the Hilbert space setting presented in this section is not well-defined.
\section{Mixed finite element setting}\label{sec:fe}
\subsection{Finite element spaces on $D$}\label{sec:femd}
We recall that $D$ is a convex polytope in $\mathbb{R}^d$, with $d\in\{2,3\}$. We consider a triangulation of $D$ which we denote $\mc{T}_h$, i.e. a decomposition of $D$ in either simplicial or quadrilateral \modif{(disjoint)} elements, where $h$ is the maximum diameter of the elements in $\mc{T}_h$. We assume that there exists a constant $C_{mesh}$ such that
\begin{equation}\label{eq:Cmesh}
|h|^d \leq C_{mesh} |T|\,, \quad \forall\, T\in \mc{T}_h\,.
\end{equation}
This implies that the mesh is quasiuniform, meaning that the ratio of any two element diameters is uniformly bounded by a constant depending only on $C_{mesh}$, and shape-regular,
that is, for each element $T\in \mc{T}_h$, the ratio of its diameter and the diameter of the largest inscribed ball is uniformly bounded by a constant depending only on $C_{mesh}$ (see, e.g., \cite{arnold2006finite}).
For any $T\in \mc{T}_h$, we denote by $\mc{P}_k(T)$ the space of polynomials of degree up to $k$ on $T$. If $T$ is a quadrilateral element, i.e., in general, if $T$ is obtained by an affine transformation $\phi:I^d \rightarrow T $ where $I$ is the unit interval, then we define $\mc{P}_{k_1,\ldots k_d}(I^d)\coloneqq \mc{P}_{k_1}(I) \otimes \ldots \otimes \mc{P}_{k_d}(I)$ and $\mc{P}_{k_1,\ldots k_d}(T) \coloneqq \mc{P}_{k_1,\ldots k_d}(I^d) \circ \phi^{-1}$.
We now define the finite element spaces $Q_h$ and $V_h$ which will serve to construct approximations of the density $\rho$ and the momentum $m$, respectively. We set
\[
Q_{h} \coloneqq \{ \varphi \in L^{2}(D)\,;\, \varphi|_T \in \mc{P}_0(T), ~ \forall\, T\in \mc{T}_h\},
\]
\[
V_{h} \coloneqq \{ v \in H(\mr{div};D) \,;\, v|_T \in V_h(T), ~ \forall\, T\in \mc{T}_h\}.
\]
where $V_h(T)$ is the so-called shape function space. We distinguish two cases:
\begin{enumerate}
\item for simplicial elements (triangles or tetrahedrons), we take $V_h(T)$ to be either
\[
\mc{RT}_0(T) \coloneqq \{ v = v_0 + v_1 \hat{x} \,;\, v_0 \in (\mc{P}_0(T))^d\,, v_1\in\mc{P}_0(T) \} \subset (\mc{P}_1(T))^d,
\]
where $\hat{x} = (x_1,\ldots,x_d)\in (\mc{P}_1(T))^d$,
which generates the lowest order Raviart-Thomas space; or $\mc{BDM}_1(T) = (\mc{P}_1(T))^d$, which generates the lowest order Brezzi-Douglas-Marini $H(\mr{div})$-conforming space;
\item for quadrilateral elements, we set $T = \phi( I^d)$, where $I$ is an interval and $\phi$ an affine transformation, and we take $V_h(T)$ to be the tensor product space which generates the lowest order Raviart-Thomas space on quadrilateral elements. This is defined as follows:
\[
{\mc{RT}}_{[0]}(T) \coloneqq \left \{
\begin{array}{ll}
\mc{P}_{1,0}(T) e_1 + \mc{P}_{0,1}(T) e_2 & \text{if } d =2\,,\\
\mc{P}_{1,0,0}(T) e_1 + \mc{P}_{0,1,0}(T) e_2 + \mc{P}_{0,0,1}(T) e_3& \text{if } d =3\,,
\end{array}
\right.
\]
where $\{e_i\}_i$ is the basis for $\mathbb{R}^d$ aligned with the edges of $T$.
\end{enumerate}
In other words, the space $V_h$ is chosen as one of the standard lowest order $H(\mr{div})$-conforming spaces. In fact, the property of being piece-wise linear will be crucial in the following, namely to prove the convergence result in Theorem \ref{th:convergence} (see, in particular, Proposition \ref{prop:properties} in the appendix). A graphical representation of the degrees of freedom associated with these spaces is shown in figure \ref{fig:dofs}.
Importantly, with the choices mentioned above, one can define projection operators $\Pi_{Q_h}: L^2(D) \rightarrow Q_h$ and \modif{ $\Pi_{V_h}: \mathcal{V}_D \subset H(\mr{div};D)\rightarrow V_h$ that commute with the divergence operator \cite{arnold2006finite,boffi2013mixed}, where $\mathcal{V}_D$ is a dense subset of sufficiently smooth vector fields.
By an appropriate regularization procedure of such operators (see, e.g., Section 5.4 in \cite{arnold2006finite}), one can construct bounded projections $\tilde{\Pi}_{Q_h}: L^2(D) \rightarrow Q_h$ and $\tilde{\Pi}_{V_h}: H(\mr{div};D)\rightarrow V_h$ satisfying a similar property.}
In other words, the following diagram commutes
\[
\centering
\begin{tikzcd}
H(\mr{div};D) \arrow[d,"\tilde{\Pi}_{V_h}"]\arrow[r, "\mr{div}"] & L^2(D)\arrow[d,"\tilde{\Pi}_{Q_h}"] \\V_h\arrow[r, "\mr{div}"] & Q_h
\end{tikzcd}
\]
As a consequence, the divergence operator is surjective onto $Q_h$ when restricted on $V_h$, i.e.\ $\mr{div}\, V_h = Q_h$.
Finally, we let $Q_h^+\subset Q_h$ the convex subset of non-negative piecewise constant functions.
\begin{remark}\label{rem:interpolators}
\modif{For the proof of Theorem \ref{th:convergence} in the appendix, we will consider as commuting projections $\Pi_{V_h}$ and $\Pi_{Q_h}$ the canonical projections defined in Section 5.2 of \cite{arnold2006finite}. Here, we will only need the explicit definition of $\Pi_{Q_h}$,} which is given by
\begin{equation}\label{eq:PiQh}
\Pi_{Q_h} \rho |_T = \frac{1}{|T|} \int_T \rho \,,
\end{equation}
for any $T\in \mc{T}_h$. Note, in particular, that $\Pi_{Q_h}$ is well-defined on $\mc{M}(D)$ and its restriction on $\mc{M}_+(D)$ is surjective onto $Q_h^+$.
\end{remark}
\begin{figure}
\centering
\subfloat[{$\mc{RT}_0$}]{\includegraphics[width=.22\textwidth]{RT}} \qquad
\subfloat[{$\mc{BDM}_1$}]{\includegraphics[width=.22\textwidth]{BDM}} \qquad
\subfloat[{${\mc{RT}}_{[0]}$}]{\includegraphics[width=.22\textwidth]{Q}}
\caption{Degrees of freedom for different choices of shape function space $V_h(T)$}
\label{fig:dofs}
\end{figure}
\subsection{Finite element spaces on $\Omega$}
We now introduce finite element spaces on the space-time domain $[0,1]\times D$. We first define a decomposition ${\mc{T}}_{h,\tau}$, obtained by a tensor product construction. In other words, we assume that ${\mc{T}}_{h,\tau}$ is obtained by tensor product of a triangulation $\mc{T}_h$ of $D$ and a decocomposition of $[0,1]$ of maximum size $\tau$, so that any element $S\in {\mc{T}}_{h,\tau}$ is of the form $S = [t_0,t_1] \otimes T$ where $T\in \mc{T}_h$.
We now define the finite element spaces $F_{h,\tau}$ and $Z_{h,\tau}$ on the space-time domain. The space $Z_{h,\tau}$ will be constructed using the standard tensor product construction based on the spaces $Q_h$ and $V_h$ defined on $D$, and continuous $\mc{P}_1$ and discontinuous $\mc{P}_0$ spaces on $[0,1]$. In our discretization, the space-time vector field $(\rho,m)$ will be an element of $Z_{h,\tau}$ whereas $F_{h,\tau}$ will be the space of discrete Lagrange multipliers associated with the continuity equation, which is equivalent to the constraint that the space-time divergence of $(\rho,m)$ is zero.
More precisely, we define
\[
F_{h,\tau} \coloneqq \{ \phi\in L^{2}(\Omega)\,;\, \phi|_S \in \mc{P}_0({S}), ~ \forall\, {S}\in {\mc{T}}_{h,\tau}\},
\]
\[
Z_{h,\tau} \coloneqq \{ v \in H(\mr{div};\Omega) \,;\, v|_S \in Z_{h,\tau}({S}), ~ \forall\, {S}\in {\mc{T}}_{h,\tau}\}.
\]
For ${S} = [t_0,t_1] \otimes T$, the shape function space $Z_{h,\tau}({S})$ is built by defining a shape function space for the density, in the space-time domain, which is given by
\[
Q_{h,\tau}(S) \coloneqq \mc{P}_1([t_0,t_1])\otimes Q_h(T)
\]
(i.e.\ the density is piecewise linear in time), and a shape function space for the momentum, in the space-time domain, which is given by
\[
V_{h,\tau}(S) \coloneqq \mc{P}_0([t_0,t_1])\otimes V_{h}(T) \,
\]
(i.e.\ the momentum is piecewise constant in time). Then, we set
\[
Z_{h,\tau}({S}) \coloneqq ( Q_{h,\tau}(S) \,\hat{t} )\oplus V_{h,\tau}(S)\,,
\]
where $\hat{t}$ is the unit vector oriented in the time direction. The spaces $F_{h,\tau}$ and $Z_{h,\tau}$ inherit from $Q_h$ and $V_h$ the commuting diagram property mentioned above. In particular, there exist bounded projections $\tilde{\Pi}_{F_{h,\tau}}: L^2(\Omega) \rightarrow F_{h,\tau}$ and $\tilde{\Pi}_{Z_{h,\tau}}: H(\mr{div};\Omega)\rightarrow Z_{h,\tau}$ for which the following diagram commutes
\begin{equation}\label{eq:commdiag}
\centering
\begin{tikzcd}
H(\mr{div};\Omega) \arrow[d,"\tilde{\Pi}_{Z_{h,\tau}}"]\arrow[r, "\mr{div}"] & L^2(\Omega)\arrow[d,"\tilde{\Pi}_{F_{h,\tau}}"] \\ Z_{h,\tau}\arrow[r, "\mr{div}"] & F_{h,\tau}
\end{tikzcd}
\end{equation}
where the divergence is the one associated with the space-time domain $\Omega$. Then, as before, the divergence operator is surjective onto $F_{h,\tau}$ when restricted on $Z_{h,\tau}$, i.e.\ $\mr{div}\, Z_{h,\tau} = F_{h,\tau}$. Note that the precise definition for the projection operators on tensor product meshes can be found in \cite{Arnold14}.
\subsection{Discrete projection on the divergence-free subspace}\label{sec:proj}
Denote by $\mc{B}$ the kernel of the divergence operator on $H(\mr{div};\Omega)$. Given $\xi \in L^2(\Omega)$ we define the projection $P_{\mc{B}}( \xi)$ to be the divergence-free vector field $\sigma$ minimizing the $L^2$ distance from $\xi$. This can be obtained solving the following problem for $(\sigma,\phi) \in H(\mr{div};\Omega) \times L^2(\Omega)$
\begin{equation}\label{eq:mixed1}
\left \{
\begin{array}{ll}
\langle \sigma,v \rangle + \langle \phi,\mr{div}\, v \rangle = \langle \xi,v\rangle & \forall v \in H(\mr{div};\Omega)\,,\\
\langle \mr{div}\, \sigma, \psi \rangle = 0 & \forall \psi \in L^2(\Omega)\,.
\end{array}\right.
\end{equation}
Let $\mc{B}_{h,\tau}$ be the kernel of the divergence operator restricted on $Z_{h,\tau}$. We define the projection $P_{\mc{B}_{h,\tau}}( \xi)$ to be the divergence-free vector field $\sigma_{h,\tau} \in Z_{h,\tau}$ minimizing the $L^2$ distance from $\xi$. This can be obtained solving the following problem for $(\sigma_{h,\tau} ,{\phi}_{h,\tau} ) \in Z_{h,\tau} \times F_{h,\tau}$
\begin{equation}\label{eq:mixed2}
\left \{
\begin{array}{ll}
\langle {\sigma}_{h,\tau} ,v \rangle + \langle {\phi}_{h,\tau} ,\mr{div}\, v \rangle = \langle \xi,v\rangle & \forall v \in Z_{h,\tau}\,,\\
\langle \mr{div}\, {\sigma}_{h,\tau} , \psi \rangle = 0 & \forall \psi \in F_{h,\tau}\,.
\end{array}\right.
\end{equation}
The commuting diagram \eqref{eq:commdiag} implies well-posedness of the discrete system. In particular, it implies the following inf-sup condition: there exists a constant $\beta>0$ independent of $h$ and $\tau$ such that
\[
\inf_{\phi \in F_{h,\tau}} \sup_{\sigma \in Z_{h,\tau}} \frac{\langle \phi,\mathrm{div}\, \sigma \rangle}{\| \sigma \|_{H(\mathrm{div})} \| \phi \|_{L^2} } \geq \beta\,,
\]
see for example proposition 5.4.2 in \cite{boffi2013mixed}. Then, problem \eqref{eq:mixed2} is well-posed, i.e.\ it has a unique solution $(\sigma_{h,\tau},\phi_{h,\tau})$ which verifies $\sigma_{h,\tau}\in \mathcal{B}$ and
\begin{subequations}\label{eq:stabest}
\[ \| \sigma_{h,\tau} \|_{L^2} \leq C_1 \| \xi\|_{L^2} \,, \]%
\[ \| \phi_{h,\tau} \|_{L^2} \leq C_2 \| \xi\|_{L^2} \,, \]%
\[ \| \sigma_{h,\tau} -\sigma \|_{L^2} +\| \phi_{h,\tau} -\phi \|_{L^2} \leq C_3 \| \xi_{h,\tau} - \xi \|_{L^2} \,, \]
\end{subequations}
where $C_1, C_2, C_3>0$ are constants independent of $h$ and $\tau$, $\xi_{h,\tau}$ is the $L^2$ projection of $\xi$ onto $Z_{h,\tau}$ and $(\sigma,\phi)$ is the unique solution of problem \eqref{eq:mixed1} (e.g., these results can be derived as particular cases of Theorems 4.3.2, 5.2.1 and 5.2.5 in \cite{boffi2013mixed}).
In the following we will need to compute the discrete version of the $L^2$ projection onto $\mc{C}$. In particular we define
\begin{equation}\label{eq:Cht}
\mc{C}_{h,\tau} \coloneqq \{ \sigma \in \mc{B}_{h,\tau}\,, \,\sigma\cdot n_{\partial \Omega} = {\mc{X}}_{h,\tau} \}\,,
\end{equation}
where, since $\Pi_{Q_h}$ can be defined on $\mc{M}(D)$ (see equation \eqref{eq:PiQh}), we set
\begin{equation}\label{eq:boundaryd}
{\mc{X}}_{h,\tau} \coloneqq
\left \{
\begin{array}{ll}
\Pi_{Q_h} \rho_0 & \text{ on } \{0\} \times D,\\
\Pi_{Q_h} \rho_1 & \text{ on } \{1\} \times D,\\
0 & \text{ otherwise. }
\end{array}
\right.
\end{equation}
The well-posedness results described above for the $L^2$ projections onto $\mc{B}$ and $\mc{B}_{h,\tau}$ hold also for the $L^2$ projections onto $\mc{C}$ and $\mc{C}_{h,\tau}$ up to adding \modif{Neumann} boundary conditions to the spaces $H(\mr{div};\Omega)$ and $Z_{h,\tau}$, and replacing $L^2(\Omega)$ and $F_{h,\tau}$ by $L^2(\Omega)/ \mathbb{R}$ and $F_{h,\tau}/\mathbb{R}$, respectively.
\section{Discrete dynamical formulation and convergence}\label{sec:discrete}
In this section we formulate the discrete problem and state a convergence result obtained by applying the theory developed in \cite{lavenant2019unconditional}. For this, we need to introduce a space for the discrete dual variable $q$. We adopt the same notation as for the spaces defined in Section \ref{sec:fe}. In particular, we set for $r\in\{0,1\}$,
\[
X_{h}^r \coloneqq \{\phi \in L^2(D)\,;\, \phi|_T \in X_h^r(T), ~ \forall\, T\in \mc{T}_h\}.
\]
The superscript $r$ denotes the polynomial order of the shape function space $X_h^r(T)$. We distinguish two cases:
\begin{enumerate}
\item for simplicial elements (triangles or tetrahedrons), we take $X_h^r(T)\coloneqq \mc{P}_r(T)$.
\item for quadrilateral elements, we set $T = \phi( I^d)$, where $I$ is an interval and $\phi$ an affine transformation, and we take $X^r_h(T)\coloneqq \mc{P}_r(I)^d\circ \phi^{-1}$.
\end{enumerate}
The associated space-time space is defined by
\[
{X}_{h,\tau}^r \coloneqq \{\phi \in L^2(\Omega)\,;\, \phi|_S \in {X}^r_{h,\tau}({S}), ~ \forall\, {S}\in {\mc{T}}_{h,\tau}\},
\]
with ${X}_{h,\tau}^r({S}) =\mc{P}_0([t_0,t_1]) \otimes X_h^r(T)$. In order to simplify the notation, we will omit the superscript $r$ when not relevant to the discussion.
\begin{remark} The choice $r\in\{0,1\}$ is dictated by computational feasibility of the algorithm. In fact, for these cases, we can compute explicitly the projection on $\mc{K}\cap {X}^r_{h,\tau}$ \modif{(with respect to appropriate inner products)} as it will be explained in the next section. On the other hand, we restrict ourselves to piecewise constant functions in time since this is crucial for the convergence of the algorithm, as shown in \cite{lavenant2019unconditional}.
\end{remark}
The discrete action (at fixed time) is defined as follows:
\[
A_h(\rho,m) \coloneqq \sup_{(a,b) \in (X_{h})^{d+1}} \{ \langle \rho,a\rangle + \langle m,b\rangle\,;\, (a,b) \in K \, a.e. \}
\]
for any $(\rho,m) \in Q_h \times V_h$. By construction, $A_h:Q_h\times V_h \rightarrow [0, +\infty]$ is a proper convex function $-1$-positively homogeneous in its first variable and $2$-positively homogeneous in its second variable. Moreover, it is non-increasing in its first argument, i.e.\ $A_h(\rho_1+\rho_2,m) \leq A_h(\rho_1,m)$ for any $\rho_1,\rho_2 \in Q_h^+$ and $m\in V_h$. In fact, suppose that $A_h(\rho_1 + \rho_2,m)<+\infty$. Then there exists $(a^*,b^*) \in \modif{(X_h)^{d+1}\cap \mc{K}}$ such that $\langle \rho_1+\rho_2,a^*\rangle + \langle m,b^*\rangle = A_h(\rho_1+\rho_2,m)$; in particular $a^*\leq 0$. Then
\[
A_h(\rho_1,m) \geq A_h(\rho_1+\rho_2,m)-\langle \rho_2,a^* \rangle \geq A_h(\rho_1+\rho_2,m)\,,
\]
and by a similar reasoning we obtain that if $A(\rho_1+\rho_2,m) = +\infty$ then we also have $A(\rho_1,m) = +\infty$.
The space-time discretization of problem \eqref{eq:BB} is given by
\begin{equation}\label{eq:BBd}
\inf_{\substack{\sigma \in \mc{C}_{h,\tau},\\\modif{\rho\geq 0}}} \mc{A}_{h,\tau}(\sigma) ,\quad \mc{A}_{h,\tau} (\sigma) \coloneqq \sup_{q \in (X^r_{h,\tau})^{d+1}\cap \mc{K}} \langle q, \sigma\rangle\,.
\end{equation}
Note that, by definition, $\mc{A}_{h,\tau}$ is convex and non-negative. Therefore, problem \eqref{eq:BBd} always admits minimizers.
Suppose that the time discretization is given by a decomposition of the interval $[0,1]$ in $N$ elements, i.e.\ fixing the points $0=t_0 <t_1 <\ldots < t_{N+1} =1$. Given $\sigma = (\rho,m) \in Z_{h,\tau}$, we can identify the density $\rho$ with the collection $\{ \rho_i\}_{i=0}^{N+1}$ with $\rho_i \in Q_h$, and the momentum $m$ with the collection $\{ m_i\}_{i=1}^{N+1}$ with $m_i \in V_h$. Since $q$ is piecewise constant in time, we have the following equivalent formulation
\begin{equation}\label{eq:spacetimeaction}
\mc{A}_{h,\tau}(\sigma) = \sum_{i=1}^{N+1}A_h \left(\frac{\rho_i+\rho_{i-1}}{2},m_i \right) |t_{i}-t_{i-1}| \,.
\end{equation}
Note that in order to obtain \eqref{eq:spacetimeaction} from \eqref{eq:BBd}, we relied on the particular choice of finite element spaces for density (piecewise linear in time), momentum (piecewise constant in time) and the corresponding dual variables (piecewise constant in time).
\begin{remark}[Continuity constraint] The choice of a $H(\mathrm{div})$-conforming finite element space for $\sigma$ implies that the weak form of the continuity equation $\partial_t{\rho} + \mathrm{div}_x\, m =0$ is satisfied exactly by any solution of the discrete saddle point problem \eqref{eq:BBd} (this is also directly implied by the definition of the constraint set $\mathcal{C}_{h,\tau}$ in \eqref{eq:Cht}) .
\end{remark}
\begin{remark}[Positivity constraint] \label{rem:pos} \modif{Note that removing the positivity constraint in the formulation \eqref{eq:BBd}, we obtain a different scheme. In that case, since the action is evaluated on the mean density (in time), the positivity constraint $\rho\ge0$ is then only enforced on $\frac{\rho_i+\rho_{i-1}}{2}$, rather than on each $\rho_i$ separately.}
\end{remark}
The objects introduced until now define a finite dimensional model of optimal transport in the sense of Definition 2.5 in \cite{lavenant2019unconditional}. The framework developed therein can be used to deduce a convergence result for our scheme.
\begin{theorem}\label{th:convergence}
Let $\rho_0,\rho_1 \in \mc{P}(D)$ be given and $\{{\mc{T}}_{h,\tau}\}_{h,\tau > 0}$ a family of tensor-product decomposition of $\Omega$ such that the time discretization is uniform, i.e.\ $t_{i} -t_{i-1} = \tau$ for all $i=1, \ldots, N+1$, and the space discretization $\mc{T}_h$ satisfies equation \eqref{eq:Cmesh}. Let $\sigma_{h,\tau}$ be a minimizer of problem \eqref{eq:BBd} associated with ${\mc{T}}_{h,\tau}$ and for $r=1$. Then, as $h,\tau \rightarrow 0$, up to extraction of a subsequence, $\sigma_{h,\tau}$ converges weakly to $\sigma \in \mc{M}(\Omega)^{d+1}$ a minimizer of problem \eqref{eq:BB}.
\end{theorem}
The proof is essentially an extension of the one presented in \cite{lavenant2019unconditional} and is postponed to the appendix.
\begin{remark}[Stability]\label{rem:stability} The existence of bounded projections sastifying the commuting diagram \eqref{eq:commdiag} ensures stability of the projection onto $\mathcal{C}_{h,\tau}$ (see \eqref{eq:stabest}). Such commuting projections are also crucial to estabilish the convergence result in Theorem \ref{th:convergence}: in \cite{lavenant2019unconditional}, they are used to sample the continuous solution into a discrete one satisfying the continuity equation, therefore providing an admissible candidate for the discrete problem. Nonetheless, due to the nonlinear constraint $q\in \mathcal{K}$, one cannot apply the standard linear theory in \cite{boffi2013mixed}, for example, so the commuting diagram condition does not imply directly a stability result analogous to \eqref{eq:stabest} for the saddle point problem \eqref{eq:BBd} (even if we see it as a discretization of the Hilbert space formulation in Proposition \ref{prop:hilbert}). \modif{Numerically (see Section \ref{sec:num}) the finite element pairs considered here $(Z_{h,\tau},X_{h,\tau}^r)$ appear to be stable when $r=1$, but strong oscillations may occur for $r=0$ and $V_h = \mathcal{BDM}_1$, providing empirical evidence of} the instablity of the discretization for this case.
\end{remark}
\begin{remark}\label{rm:convpeyre}
Suppose that $D = [0,1]^d$ and that ${\mc{T}}_{h,\tau}$ is a uniform quadrilateral discretization of $\Omega = [0,1]^{d+1}$. Then for $r=0$ \modif{and removing the constraint $\rho \geq 0$ (see Remark \ref{rem:pos})}, the discrete problem \eqref{eq:BBd} coincides with the discretization proposed in \cite{papadakis2014optimal}. Theorem \ref{th:convergence} shows that modifying this method with $r=1$ \modif{and adding the positivity constraint at all times}, one can prove convergence to the solution of the continuous problem \eqref{eq:BB}.
\end{remark}
\section{The proximal splitting algorithm}\label{sec:proximal}
\modif{We now describe in detail the discrete version of the proximal splitting algorithm introduced in Section \ref{sec:ham}, in the simplest setting where we remove the additional positivity constraint on the density, i.e.\ we solve
\[
\inf_{\substack{\sigma \in \mc{C}_{h,\tau}}} \mc{A}_{h,\tau}(\sigma) ,\quad \mc{A}_{h,\tau} (\sigma) \coloneqq \sup_{q \in (X_{h,\tau})^{d+1}\cap \mc{K}} \langle q, \sigma\rangle\,.
\]
As mentioned in Remark \ref{rem:pos}, this amounts to enforcing positivity only on the mean density in time between consecutive time-steps. Using this formulation rather than \eqref{eq:BBd} we can reproduce the structure of the continuous version of the scheme, described in Section \ref{sec:ham}. Note, however, that one can actually solve problem \eqref{eq:BBd} with a similar strategy, e.g., by first reformulating the problem intrudicing a Lagrange multiplier to enforce the continuity equation, and then applying the same proximal splitting algorithm considered here but with the new variables and with an appropriate choice of norms.}
We start by defining
\[
\mc{K}_{h,\tau}^r \coloneqq \mc{K} \cap ({X}_{h,\tau}^r)^{d+1} \coloneqq \{ q \in ({X}_{h,\tau}^r)^{d+1}\,; \, q \in K \, a.e.\}\,.
\]
We write the discrete problem as follows:
\begin{equation}\label{eq:BBdprox}
\inf_{\sigma \in L^2(\Omega;\mathbb{R}^{d+1}) } \sup_{q \in L^2(\Omega; \mathbb{R}^{d+1})}
\langle q, \sigma\rangle + \iota_{\mc{C}_{h,\tau}} (\sigma) - \iota_{\mc{K}_{h,\tau}} (q)\,,
\end{equation}
where $\mc{C}_{h,\tau}$ is defined in \eqref{eq:Cht}. Then, the proximal splitting algorithm of section \ref{sec:ham} applied to problem \eqref{eq:BBdprox} can be formulated as follows: given $\tau_1, \tau_2>0$ and an admissible $(\sigma^0, q^0)\in \mc{C}_{h,\tau} \times \mc{K}_{h,\tau}$, we define the sequence $\{(\sigma^k,q^k)\}_k$ by performing iteratively the following two steps:
\begin{subequations}\label{eq:PSd}
\begin{align}
\text{\textbf{Step 1}}: \qquad & \sigma^{k+1} = P_{\mc{C}_{h,\tau}} (\sigma^k- \tau_1 q^k)\,.\\
\text{\textbf{Step 2}}: \qquad &q^{k+1} = P_{\mc{K}_{h,\tau}}(q^k +\tau_2 (2 \sigma^{k+1}-\sigma^k))\,.
\end{align}
\end{subequations}
The convergence result in Theorem \ref{th:cp} clearly holds also in the discrete setting and gives convergence of the algorithm to a discrete saddle point $(\sigma_{h,\tau}, q_{h,\tau})$, if the condition $\tau_1 \tau_2 <1$ is satisfied.
The two steps in the algorithm can be computed as follows.
\\
\paragraph{\textbf{Step 1}} As discussed in Section \ref{sec:proj}, the projection $P_{\mc{C}_{h,\tau}}$ can be computed modifying the system given by \eqref{eq:mixed2} by adding the Neumann boundary conditions associated with the function \eqref{eq:boundaryd}.
\\
\paragraph{\textbf{Step 2}} Since $P_{\mc{K}_{h,\tau}}$ is an $L^2$ projection, we have that $P_{\mc{K}_{h,\tau}} = P_{\mc{K}_{h,\tau}} \circ P_{({X}_{h,\tau}^r)^{d+1}}$, where $P_{({X}_{h,\tau}^r)^{d+1}}$ denotes the $L^2$ projection onto $({X}_{h,\tau}^r)^{d+1}$. This means that we only need to be able to compute $P_{\mc{K}_{h,\tau}}$ when applied to an element of ${X}_{h,\tau}^r$. In addition, since ${X}_{h,\tau}^r$ is discontinuous across elements, we can compute the projection element by element, and since functions in ${X}^r_{h,\tau}(S)$ are constant along the time direction, we can also eliminate the time variable in the projection. \modif{In other words, we only need to solve for each element $[t_0,t_1]\times S$ a problem in the form
\begin{equation}\label{eq:projectionS}
\xi_{\mc{K}} \coloneqq \mr{argmin} \{ \| \xi - q \|_{L^2(T)}^2 \,;\, q \in ({X}_{h}^r(T))^{d+1}\,, ~ q(x) \in K ~\forall x\in T\}
\end{equation}
for a given $\xi \in (X_h^r(T))^{d+1}$.}
We distinguish two cases:
\begin{enumerate}
\item if $r=0$, the projection \eqref{eq:projectionS} is just the projection of a vector $\xi \in \mathbb{R}^{d+1}$ onto the convex set $K$;
\item \modif{if $r=1$, any $\xi\in (X_h^1)^{d+1}$ is fully determined by its value on the vertices $\{v_i\}_i$ of $T$, and the condition $\xi\in \mathcal{K}$, is equivalent to $\xi(v_i) \in K$, by convexity of the set $K$ (see equation \eqref{eq:K}). However the problem is coupled in these variables when computing the projection in the $L^2$ norm. Here, we use instead a different projection and we simply set
\[
\xi_{\mc{K}}(v_i) = \mr{argmin} \{ | \xi(v_i) - q |^2 \,; q \in K\}\,.
\]
Note that this is a variational crime, but it can be avoided by reformulating the algorithm using as inner product on $X_h^1$ a weighted $\ell^2$ inner product on the degrees of freedom.}
\end{enumerate}
In both cases we only need to compute for each degree of freedom the projection of a given vector $(\bar{a},\bar{b}) \in \mathbb{R} \times \mathbb{R}^{d}$ onto $K$. If $(\bar{a},\bar{b})\notin K$, such a projection is given explicitly by the vector
\[
\left (-\frac{\mu^2}{2},\mu \,\frac{\bar{b}}{|\bar{b}|}\right)
\]
where $\mu \geq 0$ is the largest real root of the third order polynomial
\[
x \mapsto \frac{x^3}{2} + x (\bar{a}+1) - |\bar{b}|\,.
\]
\begin{remark}
As for the finite difference discretization studied in \cite{papadakis2014optimal}, different optimization techniques could be applied to solve problem \eqref{eq:BBd}. In particular, it should be noted that the ADMM approach orginally proposed by Benamou and Brenier \cite{benamou2000computational} could also be applied. This would lead to a very similar algorithm to \eqref{eq:PSd}, but it would require the introduction of an additional variable which avoids coupling of the degrees of freedom in the optimization step with respect to $q$. In other words, this is needed in order to be able to perform the projection on $K$ for each degree of freedom separately. More details on this issue can be found in \cite{papadakis2014optimal} for the discretization studied therein, and they hold also in the finite element setting.
\end{remark}
\section{Regularization} \label{sec:regularization}
The optimal transport problem does not involve any regularizing effect on the interpolation between two measures.
In fact, one can even expect a loss of regularity in some cases, namely if one is interpolating between two smooth densities on a smooth but non-convex domain. Such a loss of regularity (which is often unphysical when the density represents a physical quantity) can be avoided introducing additional regularization terms in the formulation.
In this section we describe how to do so, and how these modifications translate at the algorithmic level.
We consider the Hilbert space setting discribed in Section \ref{sec:ham} and we study problems in the form
\begin{equation}\label{eq:BBr}
\inf_{\sigma \in \mc{C}} \mc{A}(\sigma) + \alpha \mc{R}(\sigma)
\end{equation}
where $\mc{R}:L^2(\Omega) \rightarrow \mathbb{R}$ is a convex, proper and l.s.c. functional, and $\alpha>0$. For this type of problem, we can still apply the proximal splitting algorithm \eqref{eq:PS} replacing the projection onto $\mc{C}$ by $\mr{prox}_{\tau_1 \mc F}$, the proximal operator of $\mc{F} \coloneqq \iota_{\mc C} + \alpha \mc{R}$, defined by
\[
\mr{prox}_{\tau_1 \mc{F}}(\xi) = \underset{\eta\in L^2(\Omega; \mathbb{R}^{d+1})}{\mr{argmin}} \frac{\|\xi- \eta\|^2}{2\tau_1} + \mc{F}(\eta)\,.
\]
This leads to the so-called PDGH algorithm, which for $\tau_1 \tau_2<1$ can be seen just as a proximal point method applied to a monotone operator \cite{chambolle2016introduction}, and therefore we still have convergence in the Hilbert space setting. As mentioned in \cite{lavenant2019unconditional} convergence of the discrete problem with mesh refinement is more delicate and will not be discussed here.
\subsection{Mixed $L^2$-Wasserstein distance}
Define for any $\sigma = (\rho,m) \in L^2(\Omega) \times L^2(\Omega;\mathbb{R}^d)$
\[
\mc{R}(\sigma) \coloneqq \left \{
\begin{array}{ll}
\frac{1}{2}\| \partial_t \rho \|^2_{L^2(\Omega)} & \text{if } \partial_t\rho \in L^2(\Omega) \,,\\
+\infty & \text{otherwise}\,.
\end{array}
\right.
\]
With this functional, problem \eqref{eq:BBr} yields an interpolation between the Wasserstein distance and the $L^2$ distance.
It was originally considered in \cite{benamou2001mixed}, where a conjugate gradient method was proposed to compute the minimizers.
Let $V \coloneqq H^1([0,1]; L^2(D)) \times L^2([0,1];H(\mr{div};D))$ and let $\accentset{\circ}{V}$ be the same space with homogenous boundary conditions on the fluxes. For any $\xi \in L^2(\Omega)^{d+1}$, $\sigma = \mr{prox}_{\tau_1 \mc{F}}(\xi)$ is obtained by solving the following system for $(\sigma,\phi) \in V \times L^2(\Omega)/\mathbb{R}$
\[
\left \{
\begin{array}{ll}
\langle \sigma,v \rangle + \alpha \tau_1 \langle \partial_t \rho, \partial_t v_t \rangle + \langle \phi,\mr{div}\, v \rangle = \langle \xi,v\rangle \,, & \forall v \in \accentset{\circ}{V}\,,\\
\langle \mr{div}\, \sigma, \psi \rangle = 0 \, , & \forall \psi \in L^2(\Omega)/\mathbb{R}\,, \\
\sigma \cdot n_{\partial \Omega} = \mc{X}\,,
\end{array}\right.
\]
where $v_t = v\cdot \hat{t}$ is the component of $v$ in the time direction. Well-posedness can be obtained by standard methods for saddle point problems \cite{boffi2013mixed} and it translates directly into well-posedness of the discrete system obtained by replacing $V$ with $Z_{h,\tau}$, $L^2(\Omega)$ with $F_{h,\tau}$, and $\mc{X}$ with $ \mc{X}_{h,\tau}$.
\subsection{$H^1$ regularization}
Define for any $\sigma = (\rho,m) \in L^2(\Omega) \times L^2(\Omega;\mathbb{R}^d)$
\begin{equation}\label{eq:regh1}
\mc{R}(\sigma) \coloneqq \left \{
\begin{array}{ll}
\frac{1}{2}\| \nabla_x \rho \|^2_{L^2(\Omega)}& \text{if } \rho \in L^2([0,1];H^1(D))\,, \\
+\infty & \text{otherwise}\,.
\end{array}
\right.
\end{equation}
In this case we set $V \coloneqq H(\mr{div};\Omega)$, $W \coloneqq L^2([0,1];H(\mr{div}_x;D))$ and let $\accentset{\circ}{V}$ and $\accentset{\circ}{W}$ be the same spaces with homogenous boundary conditions on the fluxes. Then, for any $\xi \in L^2(\Omega)^{d+1}$, $\sigma = \mr{prox}_{\tau_1 \mc{F}}(\xi)$ is obtained by solving the following system for $(\sigma,\eta,\phi) \in V \times \accentset{\circ}{W} \times L^2(\Omega)/\mathbb{R}$
\[
\left \{
\begin{array}{ll}
\langle \sigma,v \rangle - \alpha \tau_1 \langle \mr{div}_x \eta, v_t \rangle + \langle \phi,\mr{div}\, v \rangle = \langle \xi,v\rangle \,, & \forall v \in \accentset{\circ}{V}\,,\\
\langle \rho, \mr{div}_x w \rangle + \langle \eta, w \rangle = 0 \,, & \forall w\in \accentset{\circ}{W}\,,\\
\langle \mr{div}\, \sigma, \psi \rangle = 0 \, , & \forall \psi \in L^2(\Omega)/\mathbb{R}\,, \\
\sigma \cdot n_{\partial \Omega} = \mc{X}\,,
\end{array}\right.
\]
where $v_t = v\cdot \hat{t}$ is the component of $v$ in the time direction.
As before, well-posedness can be obtained by standard methods for saddle point problems \cite{boffi2013mixed}.
We introduce the space ${W}_{h,\tau} \subset L^2([0,1];H(\mr{div}_x;D))$ whose shape functions on $S = [t_0,t_1] \otimes T$ are given by
\[{W}_{h,\tau}(S) \coloneqq \mc{P}_1([t_0,t_1])\otimes V_h(T).
\]
We denote by $\accentset{\circ}{{W}}_{h,\tau}$ the same space with the boundary conditions $\eta \cdot n_{\partial \Omega} = 0$ on $[0,1] \times \partial D$.
Denote by $\nabla_x^h : L^2(\Omega) \rightarrow \accentset{\circ}{{W}}_{h,\tau}$ the adjoint of $-\mr{div}_x$ defined by
\[
\langle \nabla_x^h \phi, \eta \rangle = -\langle \phi, \mr{div}_x \eta \rangle \,, \quad \forall \, (\phi, \eta) \in L^2(\Omega)\times \accentset{\circ}{{W}}_{h,\tau}\,.
\]
We define a discrete version of \eqref{eq:regh1} as follows:
\[
\mc{R}_{h,\tau} (\sigma) \coloneqq \frac{1}{2} \| \nabla_x^h \rho\|^2_{L^2(\Omega)}.
\]
Let $\mc{F}_{h,\tau} \coloneqq \iota_{\mc{C}_{h,\tau}} + \alpha \mc{R}_{h,\tau}$.
Then for any $\xi \in L^2(\Omega)^{d+1}$, $\sigma = \mr{prox}_{\tau_1 \mc{F}_{h,\tau}}(\xi)$ is obtained by solving the following system for $(\sigma,\eta,\phi) \in \accentset{\circ}{V}_{h,\tau} \times \accentset{\circ}{W}_{h,\tau} \times F_{h,\tau}/\mathbb{R}$:
\[
\left \{
\begin{array}{ll}
\langle \sigma,v \rangle - \alpha \tau_1\langle \mr{div}_x \eta, v_t\rangle + \langle \phi,\mr{div}\, v \rangle = \langle \xi,v\rangle \,, & \forall v \in \accentset{\circ}{V}_{h,\tau}\,,\\
\langle \rho, \mr{div}_x w \rangle + \langle \eta, w \rangle =0 \,, & \forall w \in \accentset{\circ}{W}_{h,\tau}\,,\\
\langle \mr{div}\, \sigma, \psi \rangle = 0 \, , & \forall \psi \in F_{h,\tau}/\mathbb{R}\,, \\
\sigma \cdot n_{\partial \Omega} = \mc{X}_{h,\tau}\,.
\end{array}\right.
\]
\section{Numerical results}\label{sec:num}
In this section we describe two numerical tests that demonstrate the behaviour of the proposed discretization both qualitatively and in terms of convergence of the algorithm. For both tests the time discretization is uniform, but we will use different meshes and finite element spaces for the discretization in space. \modif{For all tests, we set $\tau_1 = \tau_2 = 1$ as parameters of the proximal splitting algorithm \eqref{eq:PSd}}. The results shown hereafter have been obtained using the finite element software Firedrake \cite{Rathgeber2016} (see \cite{McRae2016,Bercea2016}, for the tensor product constructions) and the linear solver for the mixed Poisson equation is based on PETSc \cite{petsc-user-ref,petsc-efficient}. The code to perform the tests in this section can be found in the repository
\url{https://github.com/andnatale/dynamic-ot.git}.
\subsection{Qualitative behaviour and convergence of the proximal-splitting algorithm}
\modif{We set $D=[0,1]^2$, and consider either a structured triangular mesh, an unstructured one, or a uniform Cartesian mesh (shown in figures \ref{fig:tristrcomp}, \ref{fig:triunstrcomp} and \ref{fig:quadcomp}), and $\tau \coloneqq |t_{i+1} - t_i| = 1/20$. The initial and final densities are given by
\begin{equation}\label{eq:bcscos}
\rho_0(x) \propto \frac{3}{2} + \cos(2\pi |x-x_0|)
, \quad \rho_1(x) \propto \frac{3}{2} - \cos(2\pi |x-x_0|) \,,
\end{equation}
where $x_0 = (0.5,0.5)$, and they are normalized so that the total mass is equal to one. In figures \ref{fig:tristrcomp}, \ref{fig:triunstrcomp} and \ref{fig:quadcomp}, the interpolation at time $t=0.5$ is shown for different choices of spaces $V_h$ and $X_h$ and different meshes. The discretization corresponding to the couple $V_h = \mc{BDM}_1$ and $X_h^0$ yields a very oscillatory solution both on the structured and unstructured mesh. Oscillations appear also for $V_h = \mc{RT}_0$, although the qualitative features of the solution are well captured. For this latter case, the oscillations seem to be very sensitive to the structure of the mesh and are attenuated when choosing $X_h^1$ instead of $X_h^0$. On the Cartesian mesh the scheme does not generate any oscillations, with the choice of the space $X_h^1$ leading to slightly more diffusive results. Note that the appearance of oscillations is not related to the positivity constraint since in the case considered here the interpolation is strictly positive. On the other hand, we remark that for tests leading to pure translation of compactly supported densities (not shown) the oscillations disappear almost entirely even for the couple $V_h = \mc{BDM}_1$, $X_h^0$.
}
\modif{
In figure \ref{fig:Gaussianconv}, the different schemes are compared in terms of convergence of the proximal splitting algorithm. For each mesh and combination of spaces, we compute a reference solution $\sigma^*$ corresponding to $10^4$ iterations of the algorithm and we estimate the error at the $n$th iteration by $\|\sigma^n-\sigma^*\|_{L^2(\Omega)}$. The cases corresponding to $X_h^1$ appear to converge significantly faster than those corresponding to $X_h^0$. Note that in the case $V_h=\mc{RT}_{[0]}$, $X_h^0$, the resulting discretization as well as the optimization algorithm coincide with the ones proposed in \cite{papadakis2014optimal}, since here we consider a uniform Cartesian grid. Also in this case, replacing $X_h^0$ with $X_h^1$ (besides providing a convergence guarantee, see Remark \ref{rm:convpeyre}) yields a considerable speedup of the algorithm.
}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\begin{minipage}{0.29\linewidth}
\subfloat{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{mesh}} \hfill
\end{minipage} %
\begin{minipage}{0.58\linewidth}
\subfloat[{$\mc{RT}_0, X_h^0$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_RT0X0}}
\subfloat[{$\mc{RT}_{0}, X_h^1$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_RT0X1}}\\
\subfloat[{$\mc{BDM}_1, X_h^0$}]{\includegraphics[scale=.2,trim = 250 50 250 50,clip]{Finalr_BDM0X0}}
\subfloat[{$\mc{BDM}_1, X_h^1$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_BDM0X1}}
\end{minipage}%
\begin{minipage}{0.1\linewidth}\hspace{-1em}
\subfloat{\includegraphics[scale=.27, trim = 760 90 160 120,clip]{Final_scale}}
\end{minipage}
\caption{Comparison between optimal transport interpolations of the densities in \eqref{eq:bcscos} for different spaces on a structured triangular mesh. Note that in the case $V_h = \mc{BDM}_1$, $X_h^0$, the data exceeds the color map range.}
\label{fig:tristrcomp}
\end{figure}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\begin{minipage}{0.29\linewidth}
\subfloat{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{meshuns}} \hfill
\end{minipage} %
\begin{minipage}{0.58\linewidth}
\subfloat[{$\mc{RT}_0, X_h^0$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_uns_RT0X0}}
\subfloat[{$\mc{RT}_{0}, X_h^1$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_uns_RT0X1}}\\
\subfloat[{$\mc{BDM}_1, X_h^0$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_uns_BDM0X0}}
\subfloat[{$\mc{BDM}_1, X_h^1$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_uns_BDM0X1}}
\end{minipage}%
\begin{minipage}{0.1\linewidth}\hspace{-1em}
\subfloat{\includegraphics[scale=.27, trim = 760 90 160 120,clip]{Final_scale}}
\end{minipage}
\caption{Comparison between optimal transport interpolations of the densities in \eqref{eq:bcscos} for different spaces on an unstructured triangular mesh. Note that in the case $V_h = \mc{BDM}_1$, $X_h^0$, the data exceeds the color map range.}
\label{fig:triunstrcomp}
\end{figure}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\begin{minipage}{0.28\linewidth}
\subfloat{\includegraphics[scale=.2, trim = 250 30 250 50,clip]{meshcart}} \hfill
\end{minipage} %
\begin{minipage}{0.59\linewidth}
\subfloat[\scriptsize{$\mc{RT}_{[0]}, X_h^0$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_RTCF0X0}}
\subfloat[\scriptsize{$\mc{RT}_{[0]}, X_h^1$}]{\includegraphics[scale=.2, trim = 250 50 250 50,clip]{Finalr_RTCF0X1}}
\end{minipage}%
\begin{minipage}{0.1\linewidth}\hspace{-1em}
\subfloat{\includegraphics[scale=.27, trim = 760 90 160 120,clip]{Final_scale}}
\end{minipage}
\caption{Comparison between optimal transport interpolations of the densities in \eqref{eq:bcscos} for different spaces on an uniform Cartesian mesh. }
\label{fig:quadcomp}
\end{figure}
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[$\mc{RT}_0$(str.)]{\includegraphics[scale=.65,trim = 0 0 5 0,clip]{rConvergence_RT.png}}
\subfloat[$ \mc{BDM}_1$(str.)]{\includegraphics[scale=.65,trim = 32 0 5 0 ,clip]{rConvergence_BDM.png}}
\subfloat[$\mc{RT}_{[0]}$]{\includegraphics[scale=.65,trim = 32 0 5 0,clip]{rConvergence_RTCF.png}}\\
\subfloat[$\mc{RT}_0$(unstr.)]{\includegraphics[scale=.65,trim = 0 0 5 0,clip]{rConvergenceuns_RT.png}}
\subfloat[$\mc{BDM}_1$(unstr.)]{\includegraphics[scale=.65,trim = 32 0 5 0 ,clip]{rConvergenceuns_BDM.png}}
\caption{Convergence of the proximal splitting algorithm measured by $\|\sigma_{n}-\sigma^*\|_{L^2(\Omega)}$ for different spaces $V_h$ and $X_h^r$ on the structured (str.) and unstructured (unstr.) triangular mesh, and on the Cartesian mesh.}
\label{fig:Gaussianconv}
\end{figure}
\subsection{Non-convex domain}
We now consider a non-convex polygonal domain $D$, with the spatial mesh $\mc{T}_h$ represented in figure \ref{fig:Zmesh} and $\tau \coloneqq |t_{i+1} - t_i| = 1/30$. Note that even if the case of a non-convex domain is beyond the domain of applicability of the convergence results presented in this paper, our scheme is still well-defined for this case. The boundary conditions are given by
\[
\rho_0(x) = \operatorname{exp} \left(-\frac{|x-x_0|^2}{2 s^2}\right)
, \quad \rho_1(x) = \operatorname{exp} \left(-\frac{|x-x_1|^2}{2 s^2} \right) \,,
\]
with $s=0.1$, $x_0 = (0.5,0.1)$ and $x_1 = (0.5, 0.9)$. Such boundary conditions are illustrated in figure \ref{fig:Zmesh}. In this case, the exact density interpolation is not absolutely continuous, since mass concentrates on the segment connecting the two non-convex corners of the domain. Note that we have therefore refined the mesh along the diagonal where we expect the mass to concentrate.
In figure \ref{fig:Zevol}, \ref{fig:Zevolreg} and \ref{fig:ZevolregL} we show the density evolution up to time $t=0.5$ (the other half of the time evolution being symmetric in space given the boundary conditions and the domain shape) for the non-regularized case, the $H^1$ regularization and the $L^2$ regularization, respectively. For both regularizations the density profile appears to be smoothened, but only the $H^1$ regularization avoids concentration at the corners.
The proximal operator of the projection on the continuity equation is more expensive computationally for the $H^1$ regularization than for the other two cases, since we have to solve a larger mixed system at each iteration. However, for both regularizations, the proximal splitting algorithm itself converges much faster than the non-regularized case, as it can be seen in figure \ref{fig:Zconv}.
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[]{\includegraphics[scale=.29,trim = 350 110 312 120,clip]{Final_zmesh.png}}
\subfloat[$t = 0$]{\includegraphics[scale=.245,trim = 345 110 303 120,clip]{Z00reg.jpg}}
\subfloat[$t = 1$]{\includegraphics[scale=.26,trim = 353 123 353 120,clip]{Z10reg.jpg}}
\subfloat[]{\includegraphics[scale=.3,trim = 760 130 200 120,clip]{Z10reg.jpg}}
\caption{Mesh, and initial and final density for the non-convex domain test.}
\label{fig:Zmesh}
\end{figure}
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[$t = 0$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z00.jpg}}
\subfloat[$t = 0.1$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z01.jpg}}
\subfloat[$t = 0.2$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z02.jpg}}\\
\subfloat[$t = 0.3$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z03.jpg}}
\subfloat[$t = 0.4$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z04.jpg}}
\subfloat[$t = 0.5$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z05.jpg}}
\caption{Density evolution on the non-convex domain without regularization, $V_h = \mc{RT}_0$, $X_h^1$ (color scale is rescaled to fit data range).}
\label{fig:Zevol}
\end{figure}
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[$t = 0$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z00Lreg.jpg}}
\subfloat[$t = 0.1$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z01Lreg.jpg}}
\subfloat[$t = 0.2$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z02Lreg.jpg}}\\
\subfloat[$t = 0.3$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z03Lreg.jpg}}
\subfloat[$t = 0.4$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z04Lreg.jpg}}
\subfloat[$t = 0.5$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z05Lreg.jpg}}
\caption{Density evolution on the non-convex domain with $L^2$ regularization, $\alpha=0.002$, $V_h = \mc{RT}_0$, $X_h^1$ (color scale is rescaled to fit data range).}
\label{fig:ZevolregL}
\end{figure}
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[$t = 0$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z00reg.jpg}}
\subfloat[$t = 0.1$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z01reg.jpg}}
\subfloat[$t = 0.2$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z02reg.jpg}}\\
\subfloat[$t = 0.3$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z03reg.jpg}}
\subfloat[$t = 0.4$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z04reg.jpg}}
\subfloat[$t = 0.5$]{\includegraphics[scale=.25,trim = 350 110 300 120,clip]{Z05reg.jpg}}
\caption{Density evolution on the non-convex domain with $H^1$ regularization, $\alpha=0.002$, $V_h = \mc{RT}_0$, $X_h^1$ (color scale is rescaled to fit data range).}
\label{fig:Zevolreg}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=.65,trim = 0 0 5 0,clip]{RT_Z.png}
\caption{Convergence of the proximal splitting algorithm measured by $\|\sigma_{n+1}-\sigma_n\|_{L^2(
\Omega)}$ for non-convex domain test without regularization (a); with the $H^1$ regularization and $\alpha = 0.002$ (b); with the $L^2$ regularization and $\alpha = 0.002$ (c).}
\label{fig:Zconv}
\end{figure}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 50 |
using System;
using WebFormsMvp;
using ZobShop.ModelViewPresenter.Administration.ProductsList;
namespace ZobShop.ModelViewPresenter.ShoppingCart.Summary
{
public interface ICartSummaryView : IView<CartSummaryVIewModel>
{
event EventHandler MyInit;
event EventHandler<DeleteProductEventArgs> RemoveFromCart;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,686 |
Q: What kind of issue may request a "cascade persist operation" for a Doctrine entity? I have a Like entity with a getDislike() function in my Symfony2-Doctrine2 project. Once I retrieve a Like entity and I call the getDislike() function in an echo, nothing is printed.
echo '<br />'. $like->getDislike(); //Nothing is printed (1, 0, true, false)
echo '<br />'. $like->getId(); // the entity id is printed
If I call the same function with the same entity in an if condition,
if($like->getDislike() == FALSE){ //throw an exception
....}
then the following error is returned:
A new entity was found through the relationship 'Acme\ArticleBundle\Entity\Article#likes' that
was not configured to cascade persist operations for entity: Acme\ArticleBundle\Entity
\Like@00000000653f908100000000749fcbfe. Explicitly persist the new entity or configure cascading
persist operations on the relationship. If you cannot find out which entity causes the problem
implement 'Acme\ArticleBundle\Entity\Like#__toString()' to get a clue.
Why the preceding error happens! The Article entity is not mentioned in my code (of course there is a relationship many Like to one Article and vice versa in the doctrine scheme).
Any idea?
Upon request, I add the Like entity file:
namespace Acme\ArticleBundle\Entity;
use Doctrine\ORM\Mapping as ORM;
use Doctrine\Common\Collections\ArrayCollection;
/**
* Acme\ArticleBundle\Entity\Like
*
* @ORM\Table()
* @ORM\Entity
*/
class Like
{
/**
* @ORM\ManyToOne(targetEntity="Article")
* @ORM\JoinColumn(name="article_id", referencedColumnName="id")
*/
protected $article;
/**
* @ORM\Column(type="boolean")
*/
protected $dislike;
//.... other fields
/**
* Set dislike
*
* @param boolean $dislike
*/
public function setDislike($dislike)
{
$this->dislike = $dislike;
}
/**
* Get dislike
*
* @return boolean
*/
public function getDislike()
{
return $this->dislike;
}
//.... other methods
}
Here is the article Entity:
namespace Acme\ArticleBundle\Entity;
use Symfony\Component\Validator\Constraints as Assert;
use Doctrine\ORM\Mapping as ORM;
use Doctrine\Common\Collections\ArrayCollection;
/**
* Acme\ArticleBundle\Entity\Article
*
*
* @ORM\Table()
* @ORM\Entity(repositoryClass="Acme\ArticleBundle\Repository\ArticleRepository")
*/
class Article
{
/**
*
* @ORM\OneToMany(targetEntity="Like", mappedBy="article")
*/
protected $likes;
//.... other fields
public function __construct() {
...
$this->likes = new \Doctrine\Common\Collections\ArrayCollection();
...
}
}
And here is the controller code:
public function likeDislikeItemAction(){
if ($this->get('security.context')->isGranted('IS_AUTHENTICATED_FULLY')){
$user = $this->get('security.context')->getToken()->getUser();
$entityType = "AcmeArticleBundle:Article"; // This static for debugging
$entityId = 1; // This static for debugging
$doctrine = $this->getDoctrine();
$item = $doctrine
->getRepository($entityType)
->findOneById($entityId);
if(is_object($item)){
$itemlikes = $item->getLikes();
$totvotes = sizeof($itemlikes);
$numlikes = $this->getNumLikes($itemlikes);
$numdislikes = $totvotes - $numlikes;
$dislike_value = 'like'; // This static for debugging
if($dislike_value == 'like') $isDislike = FALSE;
else $isDislike = TRUE;
$like = $this->getLikeDislikeEntity($doctrine, $user, $entityType, $entityId);
if(is_object($like)) {
$test = $like->getDislike();
echo 'test val ' . $test;
}
}
}
return NULL;
}
private function getLikeDislikeEntity($doctrine, $user, $entityType, $entityId){
if ($entityType == "AcmeArticleBundle:Article") {
return $doctrine->getRepository("AcmeArticleBundle:Like")
->findOneBy(array('author' => $user->getId(), 'article' =>$entityId));
}
return null;
}
A: Finally, I've found the error. I interpreted the error message ("A new entity was found through the relationship 'Acme\ArticleBundle\Entity\Article#likes'") and so I put the code:
$itemlikes = $item->getLikes();
after the code that retrieves the Like entity, i.e.:
$like = $this->getLikeDislikeEntity($doctrine, $user, $entityType, $entityId);
Then, the error disappeared!
Maybe, the problem is that I'm opening two copy of the same row in the DB (one from the list obtained with getLikes() and another with the function to get the specific Like instance, i.e. getLikeDislikeEntity). I have to try getReference() from EntityManager to see if it doesn't provide the same error!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,443 |
Le monument à Lénine à Kiev est une statue dédiée à Vladimir Lénine, le fondateur de l'Union soviétique à Kiev, la capitale de l'Ukraine. Le monument à Lénine, plus grand que nature (3,45 mètres), est construit par le sculpteur soviétique Sergueï Merkourov à partir de la même pierre rouge carélienne que le mausolée de Lénine. Il est exposé à l'Exposition universelle de New York de 1939 et érigé dans la rue principale Krechtchatyk de Kiev (à l'intersection du boulevard Shevchenko, en face du marché Bessarabsky) le 5 décembre 1946.
La statue a été violemment renversée de son piédestal et écrasée par une foule en colère le 8 décembre 2013, dans le cadre des événements d'Euromaïdan, lorsque de nombreuses autres statues soviétiques ont été renversées. Le socle reste en place et est parfois devenu un site d'œuvres d'art politiques et d'échanges. Depuis 2016, diverses sculptures ou installations sont exposées devant le socle.
Depuis 2015, (tous) les monuments liés à des thèmes et/ou des personnes communistes sont illégaux en Ukraine.
Période post-soviétique
Selon le décret de l'ancien président de l'Ukraine, Viktor Iouchtchenko, ce monument de l'Union soviétique et de la période communiste associée aurait dû être enlevé après l'indépendance de l'Ukraine. Néanmoins, en raison de la résistance du Parti communiste d'Ukraine, dont les membres ont été élus à la Verhovna Rada, le dernier monument de Kiev à Lénine est resté debout.
Démolition
Le décembre 2013, un groupe d'hommes masqués tente de renverser la statue lors de la vague de protestations d'Euromaïdan. La police réagit immédiatement en déployant une petite unité de police antiémeute Berkout qui est attaquée, submergée et forcée de fuir.
Par la suite, le 8 décembre 2013, plusieurs Ukrainiens prétendant être affiliés au parti politique Svoboda ont renversé la statue, sous le regard silencieux de la police de Kiev. La statue s'est ensuite cassée à la suite de sa chute.
Après la chute de la statue de Lénine, la foule a commencé à chanter l'hymne national de l'Ukraine . Plus tard, des morceaux du monument ont été ramassés par les manifestants comme souvenirs.
Réponse
L'enlèvement ou la destruction des monuments et statues de Lénine a pris une ampleur particulière après la destruction de la statue à Lénine de Kiev. Sous la devise "Ленінопад" (Léninopad), des militants ont abattu une douzaine de monuments communistes dans la région de Kiev, Jytomyr, Khmelnytskyi et ailleurs, ou les ont endommagés au cours des manifestations d'EuroMaidan au printemps 2014. Dans d'autres villes et villages, les monuments ont été enlevés par des équipements lourds organisés et transportés vers des casses ou des dépotoirs.
Réponse politique
Un député ukrainien du parti UDAR, Valeriy Karpuntsov, a annoncé que la police ukrainienne avait commencé les arrestations de personnes présentes dans la zone lors de la chute du dernier monument à Lénine à Kiev.
Le gouverneur de l'oblast de Kharkiv, Mikhaïl Dobkine, a tweeté le 8 décembre 2013 à propos du lancement d'une campagne de financement participatif pour restaurer le monument : . Il a déclaré qu'il allouerait la somme de hryvnias pour la restauration du monument.
Opinion publique
La plupart des habitants de Kiev (69%) avaient une attitude négative à l'égard du retrait du monument de Lénine lors des actions de protestation de masse, tandis que 13% avaient une attitude positive et 15% restaient indifférents.
Site de l'ancien monument depuis son renversement
Le 15 mai 2015, le président ukrainien Petro Porochenko a signé un projet de loi pour le retrait de tous les monuments à thème communiste en Ukraine.
Bien que la statue ait été retirée, le socle du monument est resté en place. Ce socle est devenu parfois un site d'exposition d'œuvres d'art et d'échanges politiques.
Références
Articles connexes
Léninopad
Liste de monuments dédiés à Lénine
Décommunisation en Ukraine
Liens externes
BBC | L'Ukraine commet un statue-cide
Carte des statues de Lénine décapitées ou totalement tombées lors du Leninopad en Ukraine
Sculpture en plein air en Ukraine
Œuvre vandalisée
Histoire de Kiev
2013 en Ukraine
Euromaïdan
Œuvre d'art à Kiev
Monument ou mémorial en l'honneur de Vladimir Ilitch Lénine
Bâtiment détruit en Ukraine
Patrimoine classé d'Ukraine | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,785 |
/*=================================\
* ArduinoFileBrowser\Program.cs
*
* The Coestaris licenses this file to you under the MIT license.
* See the LICENSE file in the project root for more information.
*
* Created: 06.08.2017 20:09
* Last Edited: 09.09.2017 21:10:45
*=================================*/
using System;
using System.Windows.Forms;
namespace FileBrowser
{
static class Program
{
[STAThread]
static void Main()
{
Application.EnableVisualStyles();
Application.SetCompatibleTextRenderingDefault(false);
Application.Run(new SelectPortForm());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,384 |
\section{Introduction}
Evaluating or determining bounds on the various communication capacities of a
quantum channel is one of the main concerns of quantum information theory
\cite{H13book,W15book}. One can consider supplementing a channel with an
additional resource such as free entanglement
\cite{PhysRevLett.83.3081,ieee2002bennett}\ or classical communication
\cite{BBPSSW96EPP,BDSW96}, and such a consideration leads to different kinds
of capacities. Supplementing a channel with free classical communication, with
the goal being to communicate quantum information or private classical
information reliably, is of particular relevance due to its connection with
the operational setting of quantum key distribution \cite{bb84,E91}. The
former is called the local operations and classical communication (LOCC)
assisted quantum capacity, while the latter is called the secret-key-agreement capacity.
The relevance of these latter capacities is that an upper bound on them can
serve as a benchmark to determine whether one has experimentally implemented a
working quantum repeater \cite{L15}, which is a device needed for the
practical implementation of quantum key distribution. A first result in this
direction, building on earlier developments in \cite{CW04,C06,HHHO05,HHHO09},
is due to \cite{TGW14IEEE,TGW14Nat} (see also \cite{Wilde2016}), in which it
was shown that the squashed entanglement of a quantum channel is an upper
bound on both its LOCC-assisted quantum capacity and its secret-key-agreement
capacity. Some follow-up works \cite{PLOB15,WTB16}\ then considered other
entanglement measures such as relative entropy of entanglement and established
their relevance as bounds on these capacities in certain cases. There has been
an increasing interest in this topic in recent years, with a series of papers
developing it further
\cite{TGW14IEEE,TGW14Nat,STW16,PLOB15,Goodenough2015,TSW16,AML16,WTB16,Christandl2017,Wilde2016,BA17,RGRKVRHWE17,KW17,TSW17,RKBKMA17}%
.
In this paper, we develop this topic even further, in the following ways:
\begin{enumerate}
\item First, we define the amortized entanglement of a quantum channel as the
largest difference in the entanglement between the output and the input of the
channel, with entanglement quantified by some entanglement measure
\cite{H42007}. We note that amortized entanglement is closely related to ideas
put forth in \cite{BHLS03,LHL03,Christandl2017,BGMW17}, which were used
therein to give bounds on the performance of adaptive protocols (see also the
very recent paper \cite{RKBKMA17}\ for related ideas).
\item We then prove several properties of the amortized entanglement, while
considering special cases in which the entanglement measure is set to the
relative entropy of entanglement \cite{VP98}\ or the Rains relative entropy
\cite{R99,R01,AdMVW02}.
These latter quantities are shown to be single-letter upper bounds on the
secret-key-agreement and PPT-assisted quantum capacities of
a quantum channel, respectively.
Another important property that we establish in these
special cases is that the amortized entanglement obeys a uniform continuity
bound of the flavor in \cite{Winter15,S16cont}, with a dependence on the
output dimension of the two channels under consideration and the diamond norm
of their difference \cite{K97}.
\item These latter results lead to upper bounds on the secret-key-agreement
capacity of approximately teleportation-simulable channels (channels that are
close in diamond norm to a teleportation-simulable channel \cite{BDSW96,HHH99,CDP09}%
). Similarly, we find upper bounds on the positive-partial-transpose
(PPT)\ assisted quantum capacity for approximately PPT-simulable channels
(defined later). The main idea behind obtaining these bounds is broadly
similar to the approach of approximately degradable channels put forth in
\cite{SSWR14}.
\item We next showcase the aforementioned bounds for a simple qubit channel,
which is a convex combination of an amplitude damping channel and a
depolarizing channel (note that this channel is considered in the concurrent
work \cite{LKDW17} as well). The main finding here is that the upper bounds
from approximate simulation are reasonably close to lower bounds on the
capacities whenever the noise in the channel is low, and this result is
consistent with that which was found in earlier work \cite{SSWR14,LLS17}.
\item Finally, we discuss how many of the concepts developed in our paper can
be extended to general resource theories \cite{BG15,fritz_2015,RKR15,KR16}. In
particular, we discuss the amortized resourcefulness of a quantum channel and
prove how it leads to an upper bound on the amount of resourcefulness that can
be extracted from multiple calls to a quantum channel by interleaving calls to
it with free channels. We also introduce the notion of a $\nu$%
-freely-simulable channel as a generalization of the concept of a
teleportation-simulable channel.
\end{enumerate}
At the end of the paper, we conclude with a summary and open questions. The
rest of our paper proceeds in the order given above,
Appendix~\ref{sec:supp-lemmas} provides some supplementary lemmas that are
needed to establish the uniform continuity bound mentioned above, and
Appendix~\ref{sec:cov-TP-sim} discusses the relation between approximate
covariance \cite{LKDW17} and approximately teleportation-simulable channels, as well as
showing how to simulate the twirling of a channel \cite{BDSW96} via a generalized
teleportation protocol. Throughout our paper, we use notation and concepts
that are by now standard in quantum information theory, and we point the
reader to \cite{W15book}\ for background.
\section{Amortized entanglement of a quantum channel}
\begin{figure}[ptb]
\begin{center}
\includegraphics[
width=3.5405in
]{amortized-ent.pdf}
\end{center}
\caption{The amortized entanglement of a quantum channel $\mathcal{N}_{A\to
B}$ is the largest difference in the entanglement between the output state on
systems $A^{\prime}:BB^{\prime}$ and the input state on systems $A^{\prime
}A:B^{\prime}$, the former of which is generated by the quantum channel
$\mathcal{N}_{A\rightarrow B}$.}%
\label{fig:amortized}%
\end{figure}
We begin by defining the amortized entanglement of a quantum channel as the
largest difference that can be achieved between the entanglement of an output
and input state of a quantum channel (see Figure~\ref{fig:amortized}\ for a
visual illustration of the scenario to which amortized entanglement
corresponds). Definition~\ref{def:amortized-ent} below applies to any
entanglement measure, which, as in \cite{H42007}, we define to be any function
of a bipartite quantum state that is monotone with respect to an LOCC channel,
i.e., a quantum channel that can be implemented by local operations and
classical communication (LOCC). As a minimal requirement, we also take an
entanglement measure to be equal to zero when evaluated on a product state and non-negative in general.
\begin{definition}
[Amortized entanglement of a quantum channel]\label{def:amortized-ent}For a
quantum channel $\mathcal{N}_{A\rightarrow B}$ and an entanglement measure
$E$, we define the channel's amortized entanglement as follows:%
\begin{equation}
E_{A}(\mathcal{N})\equiv\sup_{\rho_{A^{\prime}AB^{\prime}}}E(A^{\prime
};BB^{\prime})_{\theta}-E(A^{\prime}A;B^{\prime})_{\rho},
\end{equation}
where $\theta_{A^{\prime}BB^{\prime}}\equiv\mathcal{N}_{A\rightarrow B}%
(\rho_{A^{\prime}AB^{\prime}})$.
\end{definition}
As we stressed in the introduction, the quantity $E_{A}(\mathcal{N})$ is
closely related to ideas from prior work
\cite{BHLS03,LHL03,Christandl2017,BGMW17}, as well as the very recent
\cite{RKBKMA17}. Intuitively, the amortized entanglement of a channel captures
the largest difference in entanglement that can be generated between the
output and input of the channel.
Recall that the quantum relative entropy $D(\varsigma\Vert\xi)$ for a state
$\varsigma$ and a positive semi-definite operator $\xi$ is defined as
\cite{U62,Lindblad1973}%
\begin{equation}
D(\varsigma\Vert\xi)\equiv\operatorname{Tr}\{\varsigma\left[ \log
_{2}\varsigma-\log_{2}\xi\right] \},
\end{equation}
whenever $\operatorname{supp}(\varsigma)\subseteq\operatorname{supp}(\xi)$ and
it is equal to $+\infty$ otherwise. We obtain two special cases of amortized
entanglement by considering the relative entropy of entanglement
\cite{VP98}\ and the Rains relative entropy \cite{R99,R01}\ as the underlying
entanglement measures. The former entanglement measure is relevant in the
context of secret-key distillation \cite{HHHO05,HHHO09}\ and the latter in the
context of entanglement distillation \cite{R99,R01}, both tasks performed with
respect to a bipartite state. Sections~\ref{sec:amortized-rel-ent-SKA} and
\ref{sec:amortized-Rains-q-comm} show how the amortized measures given below
are relevant in the context of secret-key-agreement and quantum communication
assisted by classical communication, respectively, both tasks performed with
respect to a quantum channel.
\begin{definition}
[Amortized relative entropy of entanglement]For a quantum channel
$\mathcal{N}_{A\rightarrow B}$, its amortized relative entropy of entanglement
is defined as follows:%
\begin{equation}
E_{AR}(\mathcal{N})\equiv\sup_{\rho_{A^{\prime}AB^{\prime}}}E_{R}(A^{\prime
};BB^{\prime})_{\theta}-E_{R}(A^{\prime}A;B^{\prime})_{\rho},
\end{equation}
where $\theta_{A^{\prime}BB^{\prime}}\equiv\mathcal{N}_{A\rightarrow B}%
(\rho_{A^{\prime}AB^{\prime}})$ and the relative entropy of entanglement
$E_{R}(C;D)_{\tau}$\ of a bipartite state $\tau_{CD}$ is defined as
\cite{VP98}%
\begin{equation}
E_{R}(C;D)_{\tau}\equiv\inf_{\sigma_{CD}\in\operatorname{SEP}(C:D)}D(\tau
_{CD}\Vert\sigma_{CD}),
\end{equation}
with $\operatorname{SEP}$ denoting the set of separable states \cite{W89}.
\end{definition}
\begin{definition}
[Amortized Rains relative entropy]\label{def:amortized-Rains}For a quantum
channel $\mathcal{N}_{A\rightarrow B}$, its amortized Rains relative entropy
is defined as follows:%
\begin{equation}
R_{A}(\mathcal{N})\equiv\sup_{\rho_{A^{\prime}AB^{\prime}}}R(A^{\prime
};BB^{\prime})_{\theta}-R(A^{\prime}A;B^{\prime})_{\rho},
\end{equation}
where $\theta_{A^{\prime}BB^{\prime}}\equiv\mathcal{N}_{A\rightarrow B}%
(\rho_{A^{\prime}AB^{\prime}})$ and the Rains relative entropy $R(C;D)_{\tau}%
$\ of a bipartite state $\tau_{CD}$ is defined as \cite{R99,R01}%
\begin{equation}
R(C;D)_{\tau}\equiv\inf_{\sigma_{CD}\in\operatorname{PPT}^{\prime}(C:D)}%
D(\tau_{CD}\Vert\sigma_{CD}),
\end{equation}
with $\operatorname{PPT}^{\prime}(C\!:\!D)$ denoting the Rains set
\cite{AdMVW02}:%
\begin{equation}
\operatorname{PPT}^{\prime}(C\!:\!D)=\{\sigma_{CD}:\sigma_{CD}\geq
0\wedge\left\Vert T_{D}(\sigma_{CD})\right\Vert _{1}\leq1\},
\end{equation}
and $T_{D}$ denotes the partial transpose of system $D$.
\end{definition}
Observe that $\operatorname{SEP}\subset\operatorname{PPT}^{\prime}$. Also,
note that the quantities $E_{AR}(\mathcal{N})$ and $R_{A}(\mathcal{N})$
involve an optimization over mixed states on systems $A^{\prime}AB^{\prime}$,
and we do not have an upper bound on the dimension of the $A^{\prime}$ or
$B^{\prime}$ systems. So these quantities could be difficult to calculate in
general. One of the main contributions of our paper (see
Section~\ref{sec:approx-TP-bound}) is to show how this quantity can be
approximated well in certain cases.
\subsection{Amortized entanglement versus the entanglement of a channel}
For any entanglement measure $E$, the entanglement of the channel is defined
as \cite{TGW14IEEE,TGW14Nat,TWW14,PLOB15,RKBKMA17}%
\begin{equation}
E(\mathcal{N})\equiv\sup_{\psi_{A^{\prime}A}}E(A^{\prime};B)_{\theta},
\end{equation}
where%
\begin{equation}
\theta_{A^{\prime}B}\equiv\mathcal{N}_{A\rightarrow B}(\psi_{A^{\prime}A}),
\end{equation}
and $\psi_{A^{\prime}A}$ is an arbitrary pure bipartite state with system $A'$ isomorphic to the channel input system~$A$. It suffices to optimize over pure states of the above form instead of general mixed states, due to purification, Schmidt decomposition, and monotonicity of the entanglement measure $E$ with respect to local operations (one of which is partial trace). Particular measures of interest are a channel's
relative entropy of entanglement and the Rains relative entropy:
\begin{align}
E_{R}(\mathcal{N}) & =\sup_{\psi_{A^{\prime}A}}E_{R}(A^{\prime}%
;B)_{\theta},\\
R(\mathcal{N}) & =\sup_{\psi_{A^{\prime}A}}R(A^{\prime};B)_{\theta%
}.
\end{align}
The amortized entanglement of a channel is never smaller than that channel's
entanglement. That is, we always have the following inequality:%
\begin{equation}
E_{A}(\mathcal{N})\geq E(\mathcal{N}), \label{eq:amortized->=usual}%
\end{equation}
by taking $B^{\prime}$ to be a trivial system in
Definition~\ref{def:amortized-ent}.
The squashed entanglement $E_{\operatorname{sq}}$\ is a special entanglement
measure that obeys many desirable properties \cite{CW04,KW04,C06,BCY11,LW14}
(see also the various discussions in \cite{T99,T02}). One can also define the
dynamic version of this entanglement measure as the squashed entanglement of a
channel \cite{TGW14IEEE}, denoted as $E_{\operatorname{sq}}(\mathcal{N})$. A
particular property of squashed entanglement was established as
\cite[Theorem~7]{TGW14IEEE}. We remark here (briefly) that \cite[Theorem~7]%
{TGW14IEEE} implies the following inequality for the amortized version of
squashed entanglement%
\begin{equation}
E_{A,\operatorname{sq}}(\mathcal{N})\leq E_{\operatorname{sq}}(\mathcal{N}),
\end{equation}
which by \eqref{eq:amortized->=usual}, implies the following equality for
squashed entanglement:%
\begin{equation}
E_{A,\operatorname{sq}}(\mathcal{N})=E_{\operatorname{sq}}(\mathcal{N}).
\end{equation}
Thus, the squashed entanglement is rather special, in the sense that
amortization does not enhance its value.
\subsection{Convexity of a channel's amortized entaglement}
An entanglement measure $E$ is convex with respect to states \cite{H42007} if for all bipartite states $\rho^{0}_{CD}$ and $\rho^{1}_{CD}$ and $\lambda \in [0,1]$, the following equality holds:
\begin{equation}
E(C;D)_{\rho^\lambda} \leq
\lambda E(C;D)_{\rho^0} +
(1-\lambda) E(C;D)_{\rho^1},
\end{equation}
where $\rho^\lambda_{CD} =
\lambda \rho^0_{CD} +
(1-\lambda) \rho^1_{CD}$.
As the following proposition states, this property extends to amortized entanglement:
\begin{proposition}
[Convexity] Let $E$ be an entanglement measure that is convex with respect to states. Then the amortized entanglement $E_{A}$\ of a channel is convex with respect to channels, in the sense that the following inequality holds for all quantum channels $\mathcal{N}^0$ and $\mathcal{N}^1$ and
$\lambda \in [0,1]$:
\begin{equation}
E_{A}(\mathcal{N}^\lambda)\leq \lambda E_A(\mathcal{N}^0)+(1-\lambda)E_A(\mathcal{N}^1),
\end{equation}
where $\mathcal{N}^\lambda = \lambda \mathcal{N}^0+(1-\lambda)\mathcal{N}^1$.
\end{proposition}
\begin{proof}
Let $\rho_{A'AB'}$ be a state and set $\tau_{A'BB'}=\lambda \mathcal{N}^0_{A\rightarrow B}(\rho_{A'AB'})+(1-\lambda)\mathcal{N}^1(\rho_{A'AB'})$. Then consider that
\begin{align}
&E(A';BB')_{\tau}-E(A'A;B')_{\rho}
\nonumber
\\&= E(A';BB')_{\tau}-\lambda E(A'A;B')_{\rho}-(1-\lambda)E(A'A;B')_{\rho} \\
&\leq \lambda E(A';BB')_{\mathcal{N}^0(\rho)}+(1-\lambda)E(A';BB')_{\mathcal{N}^1(\rho)}-\lambda E(A'A;B')_{\rho}-(1-\lambda)E(A'A;B')_{\rho}\\
&= \lambda (E(A';BB')_{\mathcal{N}^0(\rho)}-E(A'A;B')_{\rho})+(1-\lambda) (E(A';BB')_{\mathcal{N}^1(\rho)}-E(A'A;B')_{\rho})\\
&\leq \lambda E_A(\mathcal{N}^0)+(1-\lambda)E_A(\mathcal{N}^1)\label{eq:convex_ineq}
\end{align}
The first equality follows from expanding the second term. The first inequality follows from the convexity of the entanglement measure $E$. The second equality from rearrangement of the terms. The second inequality follows from taking the supremum over all states $\rho_{A'AB'}$. Since the inequality in $\eqref{eq:convex_ineq}$ holds for all states $\rho_{A'AB'}$, the proof is complete.
\end{proof}
\subsection{Faithfulness of a channel's amortized entanglement}
An entanglement measure $E$ is faithful if it is equal to zero if and only if the state on which it is evaluated is a separable state. A quantum channel $\mathcal{N}$ is entanglement-breaking
\cite{HSR03}
if for all input states $\rho_{RA}$, the output state $(\operatorname{id}_R \otimes \mathcal{N}_{A\to B})(\rho_{RA})$ is a separable state. The following proposition extends the faithfulness property of entanglement measures to amortized entanglement and entanglement-breaking channels:
\begin{proposition}
[Faithfulness] Let $E$ be an entanglement measure that is equal to zero for all separable states. If a channel $\mathcal{N}$ is entanglement-breaking, then its amortized entanglement $E_A(\mathcal{N})$ is equal to zero. If the entanglement measure $E$ is faithful and the amortized entanglement $E_{A}(\mathcal{N})$ of a channel $\mathcal{N}$ is equal to zero, then the channel $\mathcal{N}$ is entanglement-breaking.
\end{proposition}
\begin{proof}
We begin by proving the first statement above: for an arbitrary entanglement measure $E$, if a channel $\mathcal{N}$ is entanglement breaking, then its amortized entanglement is equal to zero.
Let $\rho_{A'AB'}$ be an arbitrary input state to the channel, and let $\sigma_{A'BB'}=\mathcal{N}(\rho_{A'AB'})$ denote the output of the channel.
Recall that any entanglement-breaking channel can be represented as a measurement of the input system, followed by the preparation of a state on the output system, conditioned on the outcome of the measurement \cite{HSR03}. As such, the channel itself can be implemented by LOCC from the sender to the receiver. Then consider that
\begin{equation}
E(A';BB')_{\sigma}-E(A'A;B')_{\rho}
\leq E(A'A;B')_{\rho}-E(A'A;B')_{\rho}=0.
\end{equation}
The inequality follows from the fact that $E$ is an entanglement measure and is thus monotone with respect to LOCC. Given that we always have $E_{A}(\mathcal{N}) \geq 0$, we conclude that $E_{A}(\mathcal{N}) = 0$.
Now we prove the second statement above: if the entanglement measure $E$ is faithful and the amortized entanglement $E_{A}(\mathcal{N})$ of a channel $\mathcal{N}$ is equal to zero, then the channel $\mathcal{N}$ is entanglement-breaking. From \eqref{eq:amortized->=usual}, we have that $E_A(\mathcal{N})\geq E(\mathcal{N})$, which in turn implies that $E(\mathcal{N})=0$.
Since the underlying entanglement measure $E$ is faithful and
$E(A';B)_{\mathcal{N}(\rho)} \leq E(\mathcal{N})$ for all mixed input states $\rho_{A'A}$, we conclude that for all mixed input states $ \rho_{A'A}$, the output state $\mathcal{N}_{A \to B}(\rho_{A'A})$ is separable. Thus, $\mathcal{N}$ is entanglement breaking.
\end{proof}
\subsection{(Sub)additivity of a channel's amortized entanglement}
\begin{proposition}
[Subadditivity]\label{prop:amortized-subadditive}For any entanglement
measurement $E$, the amortized entanglement $E_{A}$\ of a channel is a
subadditive function of quantum channels, in the sense that the following
inequality holds for quantum channels $\mathcal{N}$ and $\mathcal{M}$:%
\begin{equation}
E_{A}(\mathcal{N}\otimes\mathcal{M})\leq E_{A}(\mathcal{N})+E_{A}%
(\mathcal{M}). \label{eq:amortized-subadd-gen}%
\end{equation}
\end{proposition}
\begin{proof}
Let $A_{1}$ and $B_{1}$ denote the respective input and output systems for
quantum channel$~\mathcal{N}$, and let $A_{2}$ and $B_{2}$ denote the
respective input and output quantum systems for quantum channel$~\mathcal{M}%
$.\ Let $\rho_{A^{\prime}A_{1}A_{2}B^{\prime}}$ denote a state to consider at
the input of $\mathcal{N}\otimes\mathcal{M}$, when optimizing the amortized
entanglement. Let $\theta_{A^{\prime}B_{1}B_{2}B^{\prime}}=(\mathcal{N}%
_{A_{1}\rightarrow B_{1}}\otimes\mathcal{M}_{A_{2}\rightarrow B_{2}}%
)(\rho_{A^{\prime}A_{1}A_{2}B^{\prime}})$, which is the state at the output of
the channel $\mathcal{N}\otimes\mathcal{M}$ when inputting $\rho_{A^{\prime
}A_{1}A_{2}B^{\prime}}$. Define the intermediary state $\tau_{A^{\prime}%
A_{1}B_{2}B^{\prime}}=\mathcal{M}_{A_{2}\rightarrow B_{2}}(\rho_{A^{\prime
}A_{1}A_{2}B^{\prime}})$. Then consider that%
\begin{align}
& E(A^{\prime};B_{1}B_{2}B^{\prime})_{\theta}-E(A^{\prime}A_{1}%
A_{2};B^{\prime})_{\rho}\nonumber\\
& =E(A^{\prime};B_{1}B_{2}B^{\prime})_{\theta}-E(A^{\prime}A_{1}%
;B_{2}B^{\prime})_{\tau}+E(A^{\prime}A_{1};B_{2}B^{\prime})_{\tau}%
-E(A^{\prime}A_{1}A_{2};B^{\prime})_{\rho}\\
& \leq E_{A}(\mathcal{N})+E_{A}(\mathcal{M}).
\label{eq:amortized-subadd-critical-ineq}%
\end{align}
The first equality follows by adding and subtracting $E(A^{\prime}A_{1}%
;B_{2}B^{\prime})_{\tau}$. The second inequality follows because the states
$\tau_{A^{\prime}A_{1}B_{2}B^{\prime}}$ and $\theta_{A^{\prime}B_{1}%
B_{2}B^{\prime}}$ are particular states to consider at the respective input
and output for the amortized entanglement of the channel $\mathcal{N}$, by
making the identifications $A^{\prime}\leftrightarrow A^{\prime}$, $B^{\prime
}\leftrightarrow B^{\prime}B_{2}$, $B\leftrightarrow B_{1}$, and
$A\leftrightarrow A_{1}$, while the states $\rho_{A^{\prime}A_{1}%
A_{2}B^{\prime}}$\ and$\ \tau_{A^{\prime}A_{1}B_{2}B^{\prime}}$ are particular
states to consider at the respective input and output for the amortized
entanglement of the channel $\mathcal{M}$, by making the identifications
$A^{\prime}\leftrightarrow A^{\prime}A_{1}$, $B^{\prime}\leftrightarrow
B^{\prime}$, $B\leftrightarrow B_{2}$, and $A\leftrightarrow A_{2}$. Since the
inequality in \eqref{eq:amortized-subadd-critical-ineq} holds for all states
$\rho_{A^{\prime}A_{1}A_{2}B^{\prime}}$, we can conclude the inequality in \eqref{eq:amortized-subadd-gen}.
\end{proof}
\bigskip
An immediate consequence of Proposition~\ref{prop:amortized-subadditive}\ is
the following inequality:%
\begin{equation}
\sup_{\mathcal{M}}\left[ E_{A}(\mathcal{N}\otimes\mathcal{M})-E_{A}%
(\mathcal{M})\right] \leq E_{A}(\mathcal{N}),
\end{equation}
where the supremum is with respect to a quantum channel $\mathcal{M}$. This
inequality demonstrates that no other channel can help to enhance the
amortized entanglement of a quantum channel. See \cite{SSW08,WY16}\ for
related notions, i.e., potential capacity.
An entanglement measure $E$ is additive with respect to states \cite{H42007}%
\ if the following equality holds%
\begin{equation}
E(C_{1}C_{2};D_{1}D_{2})_{\tau}=E(C_{1};D_{1})_{\xi}+E(C_{2};D_{2})_{\zeta},
\end{equation}
where $\tau_{C_{1}C_{2}D_{1}D_{2}}=\xi_{C_{1}D_{1}}\otimes\zeta_{C_{2}D_{2}}$
and $\xi_{C_{1}D_{1}}$ and $\zeta_{C_{2}D_{2}}$ are bipartite states. It is
subadditive if%
\begin{equation}
E(C_{1}C_{2};D_{1}D_{2})_{\tau}\leq E(C_{1};D_{1})_{\xi}+E(C_{2};D_{2}%
)_{\zeta},
\end{equation}
and this latter property holds for both the relative entropy of entanglement
and the Rains relative entropy \cite{H42007}. The following proposition states
that amortized entanglement is additive if the underlying entanglement measure
is additive:
\begin{proposition}
[Additivity]\label{prop:amortized-additive}For any entanglement measurement
$E$ that is additive with respect to states, the amortized entanglement
$E_{A}$\ of a channel is an additive function of quantum channels, in the
sense that the following equality holds for quantum channels $\mathcal{N}$ and
$\mathcal{M}$:%
\begin{equation}
E_{A}(\mathcal{N}\otimes\mathcal{M})=E_{A}(\mathcal{N})+E_{A}(\mathcal{M}).
\label{eq:additivity-amortized}%
\end{equation}
\end{proposition}
\begin{proof}
The inequality $\leq$ holds for all channels as shown in
Proposition~\ref{prop:amortized-subadditive}. To see the other inequality, let
$\rho_{A_{1}^{\prime}A_{1}B_{1}^{\prime}}\otimes\kappa_{A_{2}^{\prime}%
A_{2}B_{2}^{\prime}}$ be an arbitrary state to consider for $E_{A}%
(\mathcal{N}\otimes\mathcal{M})$, and let%
\begin{equation}
\theta_{A_{1}^{\prime}B_{1}B_{1}^{\prime}A_{2}^{\prime}B_{2}B_{2}^{\prime}%
}=\mathcal{N}_{A_{1}\rightarrow B_{1}}(\rho_{A_{1}^{\prime}A_{1}B_{1}^{\prime
}})\otimes\mathcal{M}_{A_{2}\rightarrow B_{2}}(\kappa_{A_{2}^{\prime}%
A_{2}B_{2}^{\prime}}).
\end{equation}
Then%
\begin{align}
E_{A}(\mathcal{N}\otimes\mathcal{M}) & \geq E(A_{1}^{\prime}A_{2}^{\prime
};B_{1}B_{2}B_{1}^{\prime}B_{2}^{\prime})_{\theta}-E(A_{1}^{\prime}%
A_{2}^{\prime}A_{1}A_{2};B_{1}^{\prime}B_{2}^{\prime})_{\rho\otimes\kappa}\\
& =E(A_{1}^{\prime};B_{1}B_{1}^{\prime})_{\theta}-E(A_{1}^{\prime}A_{1}%
;B_{1}^{\prime})_{\rho}+E(A_{2}^{\prime};B_{2}B_{2}^{\prime})_{\theta}%
-E(A_{2}^{\prime}A_{2};B_{2}^{\prime})_{\kappa}%
\end{align}
The equality follows from the assumption that the underlying entanglement
measure is additive with respect to states. Since the above inequality holds
for all input states $\rho_{A_{1}^{\prime}A_{1}B_{1}^{\prime}}$ and
$\kappa_{A_{2}^{\prime}A_{2}B_{2}^{\prime}}$, we can conclude
\eqref{eq:additivity-amortized} after applying
Definition~\ref{def:amortized-ent}.
\end{proof}
\subsection{Amortized entanglement and teleportation simulation}
\label{sec:TP-simulation-def}Teleportation simulation of a quantum channel is
one of the earliest and most central insights in quantum information theory
\cite{BDSW96}, and it is a key tool used to establish upper bounds on
capacities of quantum channels assisted by local operations and classical
communication (LOCC) \cite{BDSW96,WPG07,NFC09,Mul12}. The basic idea behind
this tool is that a quantum channel can be simulated by the action of a
teleportation protocol \cite{PhysRevLett.70.1895,prl1998braunstein,Werner01}%
\ on a resource state $\omega_{RB}$\ shared between the sender $A$\ and
receiver $B$. More generally, a channel $\mathcal{N}_{A\rightarrow B}$ with
input system $A$ and output system $B$ is defined to be
teleportation-simulable with associated resource state $\omega_{RB}$ if the
following equality holds for all input states $\rho_{A}$ \cite[Eq.(11)]{HHH99}:%
\begin{equation}
\mathcal{N}_{A\rightarrow B}(\rho_{A})=\mathcal{L}_{ARB\rightarrow B}(\rho
_{A}\otimes\omega_{RB}),
\end{equation}
where $\mathcal{L}_{ARB\rightarrow B}$ is a quantum channel consisting of
LOCC\ between the sender, who has systems $A$ and $R$, and the receiver, who
has system $B$ ($\mathcal{L}_{ARB\rightarrow B}$ can also be considered a
generalized teleportation protocol, as in \cite{Werner01}).
Whenever the underlying entanglement measure is subadditive with respect to
quantum states, then one can easily bound the amortized entanglement
$E_{A}(\mathcal{N})$\ from above for channels that are teleportation-simulable:
\begin{proposition}
\label{prop:tp-upper-bound}Let $E_{S}$ be an entanglement measure that is
subadditive with respect to states, and let $E_{AS}$ denote its amortized
version. If a channel $\mathcal{N}_{A\rightarrow B}$\ is
teleportation-simulable with associated state $\omega_{RB}$, then the
following bound holds%
\begin{equation}
E_{AS}(\mathcal{N})\leq E_{S}(R;B)_{\omega},
\end{equation}
where $E_{AS}(\mathcal{N})$ denotes the amortized entanglement defined through
$E_{S}$ and Definition~\ref{def:amortized-ent}.
\end{proposition}
\begin{proof}
By the definition of a teleportation-simulable channel, we have that%
\begin{equation}
\mathcal{N}_{A\rightarrow B}(\rho_{A})=\mathcal{L}_{ARB\rightarrow B}(\rho
_{A}\otimes\omega_{RB}),
\end{equation}
where $\mathcal{L}_{ARB\rightarrow B}$ is an LOCC\ channel. Then for any input
state $\rho_{A^{\prime}B^{\prime}A}$, we have that%
\begin{align}
E_{S}(A^{\prime};BB^{\prime})_{\mathcal{L}(\rho\otimes\omega)}-E_{S}%
(A^{\prime}A;B^{\prime})_{\rho} & \leq E_{S}(A^{\prime}AR;B^{\prime}%
B)_{\rho\otimes\omega}-E_{S}(A^{\prime}A;B^{\prime})_{\rho}\\
& \leq E_{S}(A^{\prime}A;B^{\prime})_{\rho}+E_{S}(R;B)_{\omega}%
-E_{S}(A^{\prime}A;B^{\prime})_{\rho}\\
& =E_{S}(R;B)_{\omega}.
\end{align}
The first inequality follows from monotonicity of $E_{S}$ with respect to LOCC
channels (the fact that $E_{S}$ is an entanglement measure). The second
inequality follows from the assumption that $E_{S}$ is subadditive.
\end{proof}
Proposition~\ref{prop:tp-upper-bound}\ implies that the amortized entanglement
of a channel never exceeds the entanglement of the maximally entangled state,
whenever the underlying entanglement measure is subadditive. This follows
because any channel can be simulated by teleportation using the maximally
entangled state as the resource state, along with local processing. In
particular, Alice could apply the channel locally to her system and then
teleport it to Bob; also, she could first teleport to Bob and then he could
perform the local processing. So this leads to the following upper bound on
amortized entanglement in this case:
\begin{proposition}[Dimension bound]
\label{prop:tp-dim-upper-bound}Let $E_{S}$ be an entanglement measure that is
subadditive with respect to states, and let $E_{AS}$ denote its amortized
version. Let $\mathcal{N}_{A\rightarrow B}$ be a quantum channel. The
following bound holds%
\begin{equation}
E_{AS}(\mathcal{N})\leq\min\{E_{S}(A;\bar{A})_{\Phi},E_{S}(B;\bar{B})_{\Phi
}\},
\end{equation}
where $E_{AS}(\mathcal{N})$ denotes the amortized entanglement defined through
$E_{S}$ and Definition~\ref{def:amortized-ent}, $\bar{A}$ is a system
isomorphic to the channel input system $A$, $\bar{B}$ is a system isomorphic
to the channel output system $B$, and $\Phi$ denotes the maximally entangled
state. For the amortized relative entropy of entanglement and the amortized
Rains relative entropy, the above implies that%
\begin{equation}
E_{AR}(\mathcal{N}),\ R_{A}(\mathcal{N})\leq\log_{2}\min\{|A|,|B|\},
\end{equation}
because these underlying entanglement measures are equal to $\log_{2}d$ when
evaluated on a maximally entangled state of Schmidt rank $d$ \cite{H42007}.
\end{proposition}
In certain cases, the inequality in Proposition~\ref{prop:tp-upper-bound}\ is
actually an equality:
\begin{proposition}
\label{prop:TP-sim-amortized-equal}Let $E_{S}$ be an entanglement measure that
is subadditive with respect to states, and let $E_{AS}$ denote its amortized
version. If a channel $\mathcal{N}_{A\rightarrow B}$\ is
teleportation-simulable with associated state $\omega_{RB}=\mathcal{N}%
_{A\rightarrow B}(\rho_{RA})$ for some input state $\rho_{RA}$, then the
following equality holds%
\begin{equation}
E_{AS}(\mathcal{N})=E_{S}(R;B)_{\omega}.
\end{equation}
\end{proposition}
\begin{proof}
From Proposition~\ref{prop:tp-upper-bound}, we have that $E_{AS}%
(\mathcal{N})\leq E_{S}(R;B)_{\omega}$. The other inequality follows by
picking $\rho_{A^{\prime}B^{\prime}A}=\rho_{RA}$, with the identification
$A^{\prime}\leftrightarrow R$ and $B^{\prime}\leftrightarrow\emptyset$ (i.e.,
$B^{\prime}$ is a trivial system), and then we find that%
\begin{equation}
E_{AS}(\mathcal{N})=\sup_{\rho_{A^{\prime}AB^{\prime}}}E_{S}(A^{\prime
};BB^{\prime})_{\theta}-E_{S}(A^{\prime}A;B^{\prime})_{\rho}\geq
E_{S}(R;B)_{\omega}.
\end{equation}
This concludes the proof.
\end{proof}
\begin{remark}
For several channels with sufficient symmetry, such as covariant channels, one
can pick the input state $\rho_{RA}$ in
Proposition~\ref{prop:TP-sim-amortized-equal}\ to be the maximally entangled
state~$\Phi_{RA}$ \cite[Section~7]{CDP09}.
\end{remark}
\subsection{Uniform continuity of amortized relative entropy of entanglement
and amortized Rains relative entropy}
The following theorem establishes that both the amortized relative entropy of
entanglement and the amortized Rains relative entropy obey a uniform
continuity bound. This bound will play a central role in bounding the
respective secret-key-agreement and LOCC-assisted quantum capacities of
approximately teleportation-simulable channels (see
Sections~\ref{sec:amortized-rel-ent-SKA} and \ref{sec:amortized-Rains-q-comm}).
Before we state the theorem, we recall that the diamond norm of the difference
of two quantum channels $\mathcal{N}_{A\rightarrow B}$ and $\mathcal{M}%
_{A\rightarrow B}$ is defined as \cite{K97}%
\begin{align}
\left\Vert \mathcal{N}_{A\rightarrow B}-\mathcal{M}_{A\rightarrow
B}\right\Vert _{\Diamond} & =\sup_{\rho_{RA}}\left\Vert \left[
\operatorname{id}_{R}\otimes(\mathcal{N}_{A\rightarrow B}-\mathcal{M}%
_{A\rightarrow B})\right] (\rho_{RA})\right\Vert _{1}\\
& =\max_{\psi_{RA}}\left\Vert \left[ \operatorname{id}_{R}\otimes
(\mathcal{N}_{A\rightarrow B}-\mathcal{M}_{A\rightarrow B})\right] (\psi
_{RA})\right\Vert _{1},
\end{align}
with $\left\Vert X\right\Vert _{1}=\operatorname{Tr}\{\sqrt{X^{\dag}X}\}$ and
the second equality, with an optimization restricted to pure states $\psi
_{RA}$ with $\left\vert R\right\vert =\left\vert A\right\vert $, follows from
the convexity of the trace norm and the Schmidt decomposition. The diamond
norm is a well established and operationally meaningful measure of the
distinguishability of two quantum channels.
\begin{theorem}
\label{thm:continuity}Let $\varepsilon\in\lbrack0,1]$. Let $E$ refer to either
the relative entropy of entanglement or the Rains relative entropy, and let
$E_{A}$ refer to their amortized versions. For channels $\mathcal{N}%
_{A\rightarrow B}$ and $\mathcal{M}_{A\rightarrow B}$ such that%
\begin{equation}
\frac{1}{2}\left\Vert \mathcal{N}_{A\rightarrow B}-\mathcal{M}_{A\rightarrow
B}\right\Vert _{\Diamond}\leq\varepsilon, \label{eq:diamond-norm-bound}%
\end{equation}
the following bound holds%
\begin{equation}
\left\vert E_{A}(\mathcal{N})-E_{A}(\mathcal{M})\right\vert \leq
2\varepsilon\log_{2}\left\vert B\right\vert +g(\varepsilon),
\end{equation}
where $\left\vert B\right\vert $ is the dimension of the channel output system
$B$ and $g(\varepsilon)\equiv\left( \varepsilon+1\right) \log_{2}%
(\varepsilon+1)-\varepsilon\log_{2}\varepsilon$.
\end{theorem}
\begin{proof}
Our proof follows the general approach from \cite{Winter15}, but it has some
additional observations needed for our context. For a state $\rho_{A^{\prime
}AB^{\prime}}$, let us define%
\begin{equation}
E_{A}(\rho,\mathcal{N})\equiv E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{N}}%
}-E(A^{\prime}A;B^{\prime})_{\rho},
\end{equation}
where $\theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}}\equiv\mathcal{N}%
_{A\rightarrow B}(\rho_{A^{\prime}AB^{\prime}})$. Then consider that%
\begin{equation}
\left\vert E_{A}(\rho,\mathcal{N})-E_{A}(\rho,\mathcal{M})\right\vert
=\left\vert E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{N}}}-E(A^{\prime
};BB^{\prime})_{\theta^{\mathcal{M}}}\right\vert ,
\end{equation}
where $\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\equiv\mathcal{M}%
_{A\rightarrow B}(\rho_{A^{\prime}AB^{\prime}})$. Our intent now is to prove
that the following bound holds for all states $\rho_{A^{\prime}AB^{\prime}}$%
\begin{equation}
\left\vert E_{A}(\rho,\mathcal{N})-E_{A}(\rho,\mathcal{M})\right\vert
\leq2\varepsilon\log_{2}\left\vert B\right\vert +g(\varepsilon).
\label{eq:bound-for-each-state}%
\end{equation}
Since the bound in \eqref{eq:diamond-norm-bound} holds, we can conclude that%
\begin{equation}
\frac{1}{2}\left\Vert \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}}%
-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right\Vert _{1}\equiv
\varepsilon_{0}\leq\varepsilon.
\end{equation}
Let us suppose that $\varepsilon_{0}>0$. Otherwise, the bound in
\eqref{eq:bound-for-each-state}\ trivially holds. Let us define the states
$\Omega_{A^{\prime}BB^{\prime}}$ as%
\begin{equation}
\Omega_{A^{\prime}BB^{\prime}}=\frac{\left[ \theta_{A^{\prime}BB^{\prime}%
}^{\mathcal{N}}-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right] _{+}%
}{\operatorname{Tr}\{\left[ \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}%
}-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right] _{+}\}}=\frac
{1}{\varepsilon_{0}}\left[ \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}%
}-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right] _{+},
\end{equation}
where $\left[ \cdot\right] _{+}$ denotes the positive part of an operator. Note that the equality
$\operatorname{Tr}\{\left[ \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}%
}-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right] _{+}\} = \varepsilon_{0}$ follows from the fact that
$\theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}%
}$ and $\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}$ are states (see \cite{W15book} for more details).
Since%
\begin{align}
\theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}} & =\theta_{A^{\prime}%
BB^{\prime}}^{\mathcal{N}}-\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}%
+\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\\
& \leq\left[ \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}}-\theta_{A^{\prime
}BB^{\prime}}^{\mathcal{M}}\right] _{+}+\theta_{A^{\prime}BB^{\prime}%
}^{\mathcal{M}}\\
& =\left( 1+\varepsilon_{0}\right) \left( \frac{1}{1+\varepsilon_{0}%
}\left[ \theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}}-\theta_{A^{\prime
}BB^{\prime}}^{\mathcal{M}}\right] _{+}+\frac{1}{1+\varepsilon_{0}}%
\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right) \\
& =\left( 1+\varepsilon_{0}\right) \left( \frac{\varepsilon_{0}%
}{1+\varepsilon_{0}}\Omega_{A^{\prime}BB^{\prime}}+\frac{1}{1+\varepsilon_{0}%
}\theta_{A^{\prime}BB^{\prime}}^{\mathcal{M}}\right) ,
\end{align}
we can define%
\begin{equation}
\xi_{A^{\prime}BB^{\prime}}\equiv\frac{\varepsilon_{0}}{1+\varepsilon_{0}%
}\Omega_{A^{\prime}BB^{\prime}}+\frac{1}{1+\varepsilon_{0}}\theta_{A^{\prime
}BB^{\prime}}^{\mathcal{M}},
\end{equation}
and it follows that%
\begin{equation}
\xi_{A^{\prime}BB^{\prime}}=\frac{\varepsilon_{0}}{1+\varepsilon_{0}}%
\Omega_{A^{\prime}BB^{\prime}}^{\prime}+\frac{1}{1+\varepsilon_{0}}%
\theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}},
\end{equation}
where the state $\Omega_{A^{\prime}BB^{\prime}}^{\prime}$ is defined as%
\begin{equation}
\Omega_{A^{\prime}BB^{\prime}}^{\prime}\equiv\frac{1}{\varepsilon_{0}}\left[
\left( 1+\varepsilon_{0}\right) \xi_{A^{\prime}BB^{\prime}}-\theta
_{A^{\prime}BB^{\prime}}^{\mathcal{N}}\right] .
\end{equation}
Consider that%
\begin{equation}
\Omega_{A^{\prime}B^{\prime}}=\Omega_{A^{\prime}B^{\prime}}^{\prime}
\label{eq:critical-omega}%
\end{equation}
because%
\begin{align}
\operatorname{Tr}_{B}\left\{ \frac{\varepsilon_{0}}{1+\varepsilon_{0}}%
\Omega_{A^{\prime}BB^{\prime}}+\frac{1}{1+\varepsilon_{0}}\theta_{A^{\prime
}BB^{\prime}}^{\mathcal{M}}\right\} & =\frac{\varepsilon_{0}}%
{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}}+\frac{1}{1+\varepsilon_{0}%
}\theta_{A^{\prime}B^{\prime}}^{\mathcal{M}}\\
& =\frac{\varepsilon_{0}}{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}%
}+\frac{1}{1+\varepsilon_{0}}\rho_{A^{\prime}B^{\prime}},\\
\operatorname{Tr}_{B}\left\{ \frac{\varepsilon_{0}}{1+\varepsilon_{0}}%
\Omega_{A^{\prime}BB^{\prime}}^{\prime}+\frac{1}{1+\varepsilon_{0}}%
\theta_{A^{\prime}BB^{\prime}}^{\mathcal{N}}\right\} & =\frac
{\varepsilon_{0}}{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}}^{\prime
}+\frac{1}{1+\varepsilon_{0}}\theta_{A^{\prime}B^{\prime}}^{\mathcal{N}}\\
& =\frac{\varepsilon_{0}}{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}%
}^{\prime}+\frac{1}{1+\varepsilon_{0}}\rho_{A^{\prime}B^{\prime}},
\end{align}
and so%
\begin{equation}
\frac{\varepsilon_{0}}{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}}%
+\frac{1}{1+\varepsilon_{0}}\rho_{A^{\prime}B^{\prime}}=\frac{\varepsilon_{0}%
}{1+\varepsilon_{0}}\Omega_{A^{\prime}B^{\prime}}^{\prime}+\frac
{1}{1+\varepsilon_{0}}\rho_{A^{\prime}B^{\prime}},
\end{equation}
from which we can conclude \eqref{eq:critical-omega} since $\varepsilon_{0}%
>0$. By convexity of relative entropy of entanglement and the Rains relative
entropy \cite{H42007}, we have that%
\begin{equation}
E(A^{\prime};BB^{\prime})_{\xi}\leq\frac{1}{1+\varepsilon_{0}}E(A^{\prime
};BB^{\prime})_{\theta^{\mathcal{M}}}+\frac{\varepsilon_{0}}{1+\varepsilon
_{0}}E(A^{\prime};BB^{\prime})_{\Omega},
\end{equation}
and from Lemma~\ref{lem:rel-ent-mixture-ent-bound}, we have that%
\begin{equation}
\frac{1}{1+\varepsilon_{0}}E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{N}}%
}+\frac{\varepsilon_{0}}{1+\varepsilon_{0}}E(A^{\prime};BB^{\prime}%
)_{\Omega^{\prime}}\leq E(A^{\prime};BB^{\prime})_{\xi}+h_{2}\!\left(
\frac{\varepsilon_{0}}{1+\varepsilon_{0}}\right) ,
\end{equation}
So this means that%
\begin{align}
E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{N}}} & \leq\left(
1+\varepsilon_{0}\right) E(A^{\prime};BB^{\prime})_{\xi}+\left(
1+\varepsilon_{0}\right) h_{2}\!\left( \frac{\varepsilon_{0}}{1+\varepsilon
_{0}}\right) -\varepsilon_{0}E(A^{\prime};BB^{\prime})_{\Omega^{\prime}}\\
& \leq E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{M}}}+\varepsilon
_{0}E(A^{\prime};BB^{\prime})_{\Omega}+g(\varepsilon_{0})-\varepsilon
_{0}E(A^{\prime};BB^{\prime})_{\Omega^{\prime}}\\
& =E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{M}}}+g(\varepsilon
_{0})+\varepsilon_{0}\left[ E(A^{\prime};BB^{\prime})_{\Omega}-E(A^{\prime
};BB^{\prime})_{\Omega^{\prime}}\right] ,
\end{align}
where we used that $\left( 1+\varepsilon_{0}\right) h_{2}\!\left(
\frac{\varepsilon_{0}}{1+\varepsilon_{0}}\right) =g(\varepsilon_{0})$.
Finally, consider that%
\begin{align}
E(A^{\prime};BB^{\prime})_{\Omega}-E(A^{\prime};BB^{\prime})_{\Omega^{\prime
}} & \leq2\log_{2}\left\vert B\right\vert +E(A^{\prime};B^{\prime})_{\Omega
}-E(A^{\prime};B^{\prime})_{\Omega^{\prime}}\\
& =2\log_{2}\left\vert B\right\vert .
\end{align}
The first inequality follows from Lemma~\ref{lem:dim-bound}, the
LOCC\ monotonicity of relative entropy of entanglement and the Rains relative
entropy, and \eqref{eq:critical-omega}. So this implies that%
\begin{align}
E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{N}}}-E(A^{\prime};BB^{\prime
})_{\theta^{\mathcal{M}}} & \leq g(\varepsilon_{0})+2\varepsilon_{0}\log
_{2}\left\vert B\right\vert \\
& \leq g(\varepsilon)+2\varepsilon\log_{2}\left\vert B\right\vert ,
\end{align}
the latter inequality holding because $g(\cdot)$ is a monotone increasing
function. The other inequality%
\begin{equation}
E(A^{\prime};BB^{\prime})_{\theta^{\mathcal{M}}}-E(A^{\prime};BB^{\prime
})_{\theta^{\mathcal{N}}}\leq g(\varepsilon)+2\varepsilon\log_{2}\left\vert
B\right\vert
\end{equation}
follows immediately using similar steps, and this now establishes
\eqref{eq:bound-for-each-state}. Then we have that the following bound holds
for all input states $\rho_{A^{\prime}B^{\prime}A}$:%
\begin{align}
E_{A}(\rho,\mathcal{N}) & \leq\sup_{\rho}E_{A}(\rho,\mathcal{M}%
)+2\varepsilon\log_{2}\left\vert B\right\vert +g(\varepsilon)\\
& =E_{A}(\mathcal{M})+2\varepsilon\log_{2}\left\vert B\right\vert
+g(\varepsilon).
\end{align}
Since the bound holds for all input states $\rho_{A^{\prime}B^{\prime}A}$, we
can conclude that%
\begin{equation}
E_{A}(\mathcal{N})\leq E_{A}(\mathcal{M})+2\varepsilon\log_{2}\left\vert
B\right\vert +g(\varepsilon).
\end{equation}
In a similar way, we obtain the opposite inequality%
\begin{equation}
E_{A}(\mathcal{M})\leq E_{A}(\mathcal{N})+2\varepsilon\log_{2}\left\vert
B\right\vert +g(\varepsilon),
\end{equation}
and this completes the proof.
\end{proof}
\section{Amortized relative entropy of entanglement and secret key agreement}
\label{sec:amortized-rel-ent-SKA}In this section, we prove that the amortized
relative entropy of entanglement is an upper bound on the secret-key-agreement
capacity of a quantum channel. We begin by reviewing the structure of a
secret-key-agreement protocol \cite{TGW14IEEE,TGW14Nat}, how such a protocol
can be purified along the lines observed in \cite{HHHO05,HHHO09}, the critical
performance parameters for such a protocol, and then we finally give a proof
for the aforementioned claim. Note that the proof bears some similarities with
proofs in prior works \cite{WTB16,Christandl2017,BGMW17}, as well as an argument that
appeared recently in \cite{DW17} in a different context.
\subsection{Protocol for secret key agreement}
\label{sec:SKA-protocol}Here we review the structure of a secret-key-agreement
protocol, along the lines discussed in \cite{TGW14IEEE,TGW14Nat}:
A sender Alice and a receiver Bob are spatially separated and are connected by
a quantum channel $\mathcal{N}_{A\rightarrow B}$. They begin by performing an
LOCC channel $\mathcal{L}_{\emptyset\rightarrow A_{1}^{\prime}A_{1}%
B_{1}^{\prime}}^{(1)}$, which leads to a separable state $\rho_{A_{1}^{\prime
}A_{1}B_{1}^{\prime}}^{(1)}$, where $A_{1}^{\prime}$ and $B_{1}^{\prime}$ are
systems that are finite-dimensional but arbitrarily large and $A_{1}$ is a
system that can be fed into the first channel use. Alice transmits system
$A_{1}$ into the first channel, leading to a state $\sigma_{A_{1}^{\prime
}B_{1}B_{1}^{\prime}}^{(1)}\equiv\mathcal{N}_{A_{1}\rightarrow B_{1}}%
(\rho_{A_{1}^{\prime}A_{1}B_{1}^{\prime}}^{(1)})$. They then perform the LOCC
channel $\mathcal{L}_{A_{1}^{\prime}B_{1}B_{1}^{\prime}\rightarrow
A_{2}^{\prime}A_{2}B_{2}^{\prime}}^{(2)}$, which leads to the state%
\begin{equation}
\rho_{A_{2}^{\prime}A_{2}B_{2}^{\prime}}^{(2)}\equiv\mathcal{L}_{A_{1}%
^{\prime}B_{1}B_{1}^{\prime}\rightarrow A_{2}^{\prime}A_{2}B_{2}^{\prime}%
}^{(2)}(\sigma_{A_{1}^{\prime}B_{1}B_{1}^{\prime}}^{(1)}).
\end{equation}
Alice feeds in the system $A_{2}$ to the second channel use $\mathcal{N}%
_{A_{2}\rightarrow B_{2}}$, leading to the state $\sigma_{A_{2}^{\prime}%
B_{2}B_{2}^{\prime}}^{(2)}\equiv\mathcal{N}_{A_{2}\rightarrow B_{2}}%
(\rho_{A_{2}^{\prime}A_{2}B_{2}^{\prime}}^{(1)})$. This process
continues:\ the protocol uses the channel $n$ times. In general, we have the
following states for all $i\in\{2,\ldots,n\}$:%
\begin{align}
\rho_{A_{i}^{\prime}A_{i}B_{i}^{\prime}}^{(i)} & \equiv\mathcal{L}%
_{A_{i-1}^{\prime}B_{i-1}B_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}%
B_{i}^{\prime}}^{(i)}(\sigma_{A_{i-1}^{\prime}B_{i-1}B_{i-1}^{\prime}}%
^{(i-1)}),\\
\sigma_{A_{i}^{\prime}B_{i}B_{i}^{\prime}}^{(i)} & \equiv\mathcal{N}%
_{A_{i}\rightarrow B_{i}}(\rho_{A_{i}^{\prime}A_{i}B_{i}^{\prime}}^{(i)}),
\end{align}
where $\mathcal{L}_{A_{i-1}^{\prime}B_{i-1}B_{i-1}^{\prime}\rightarrow
A_{i}^{\prime}A_{i}B_{i}^{\prime}}^{(i)}$ is an LOCC\ channel. The final step
of the protocol consists of an LOCC channel $\mathcal{L}_{A_{n}^{\prime}%
B_{n}B_{n}^{\prime}\rightarrow K_{A}K_{B}}^{(n+1)}$, which produces the key
systems $K_{A}$ and $K_{B}$ for Alice and Bob, respectively. The final state
of the protocol is then as follows:%
\begin{equation}
\omega_{K_{A}K_{B}}\equiv\mathcal{L}_{A_{n}^{\prime}B_{n}B_{n}^{\prime
}\rightarrow K_{A}K_{B}}^{(n+1)}(\sigma_{A_{n}^{\prime}B_{n}B_{n}^{\prime}%
}^{(n)}).
\end{equation}
The goal of the protocol is that the final state $\omega_{K_{A}K_{B}}$ is
close to a secret-key state. Figure~\ref{fig:private-code}\ depicts such a
protocol. It may not yet be clear exactly what we mean by \textquotedblleft
close to a secret-key state,\textquotedblright\ but our approach is standard
and we clarify this point in the following two sections.
\begin{figure}[ptb]
\begin{center}
\includegraphics[
width=6.5638in
]{LOCC-assisted-private-comm.pdf}
\end{center}
\caption{A protocol for secret key agreement over a quantum channel.}%
\label{fig:private-code}%
\end{figure}
\subsection{Purifying a secret-key-agreement protocol}
Related to the observations in \cite{HHHO05,HHHO09}, any protocol of the above
form can be purified in the following sense. The initial state $\rho
_{A_{1}^{\prime}A_{1}B_{1}^{\prime}}^{(1)}$ is a separable state of the
following form:%
\begin{equation}
\rho_{A_{1}^{\prime}A_{1}B_{1}^{\prime}}^{(1)}\equiv\sum_{y_{1}}p_{Y_{1}%
}(y_{1})\tau_{A_{1}^{\prime}A_{1}}^{y_{1}}\otimes\zeta_{B_{1}}^{y_{1}}.
\end{equation}
The classical random variable $Y_{1}$ corresponds to a message exchanged
between Alice and Bob to establish this state. It can be purified in the
following way:%
\begin{equation}
|\psi^{(1)}\rangle_{A_{1}^{\prime}A_{1}S_{A_{1}}B_{1}^{\prime}S_{B_{1}}Y_{1}%
}\equiv\sum_{y_{1}}\sqrt{p_{Y_{1}}(y_{1})}|\tau^{y_{1}}\rangle_{A_{1}^{\prime
}A_{1}S_{A_{1}}}\otimes|\zeta^{y_{1}}\rangle_{B_{1}S_{B_{1}}}\otimes
|y_{1}\rangle_{Y_{1}},
\end{equation}
where $S_{A_{1}}$ and $S_{B_{1}}$ are local \textquotedblleft
shield\textquotedblright\ systems that in principle could be held by Alice and
Bob, respectively, $|\tau^{y_{1}}\rangle_{A_{1}^{\prime}A_{1}S_{A_{1}}}$ and
$|\zeta^{y_{1}}\rangle_{B_{1}S_{B_{1}}}$ purify $\tau_{A_{1}^{\prime}A_{1}%
}^{y_{1}}$ and $\zeta_{B_{1}}^{y_{1}}$, respectively, and Eve possesses system
$Y_{1}$, which contains a coherent classical copy of the classical data
exchanged. Each LOCC\ channel $\mathcal{L}_{A_{i-1}^{\prime}B_{i-1}%
B_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}B_{i}^{\prime}}^{(i)}$\ can be
written in the following form \cite{Wat15}, for all $i\in\{2,\ldots,n\}$:%
\begin{equation}
\mathcal{L}_{A_{i-1}^{\prime}B_{i-1}B_{i-1}^{\prime}\rightarrow A_{i}^{\prime
}A_{i}B_{i}^{\prime}}^{(i)}\equiv\sum_{y_{i}}\mathcal{E}_{A_{i-1}^{\prime
}\rightarrow A_{i}^{\prime}A_{i}}^{y_{i}}\otimes\mathcal{F}_{B_{i-1}%
B_{i-1}^{\prime}\rightarrow B_{i}^{\prime}}^{y_{i}},
\label{eq:LOCC-as-separable}%
\end{equation}
where $\{\mathcal{E}_{A_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}}^{y_{i}%
}\}_{y_{i}}$ and $\{\mathcal{F}_{B_{i-1}B_{i-1}^{\prime}\rightarrow
B_{i}^{\prime}}^{y_{i}}\}_{y_{i}}$ are collections of completely positive,
trace non-increasing maps such that the map in \eqref{eq:LOCC-as-separable} is
trace preserving. Such an LOCC\ channel can be purified to an isometry in the
following way:%
\begin{equation}
U_{A_{i-1}^{\prime}B_{i-1}B_{i-1}^{\prime}\rightarrow A_{i}^{\prime}%
A_{i}S_{A_{i}}B_{i}^{\prime}S_{B_{i}}Y_{i}}^{\mathcal{L}^{(i)}}\equiv
\sum_{y_{i}}U_{A_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}S_{A_{i}}%
}^{\mathcal{E}^{y_{i}}}\otimes U_{B_{i-1}B_{i-1}^{\prime}\rightarrow
B_{i}^{\prime}S_{B_{i}}}^{\mathcal{F}^{y_{i}}}\otimes|y_{i}\rangle_{Y_{i}},
\label{eq:LOCC-as-separable-iso-ext}%
\end{equation}
where $\{U_{A_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}S_{A_{i}}%
}^{\mathcal{E}^{y_{i}}}\}_{y_{i}}$ and $\{U_{B_{i-1}B_{i-1}^{\prime
}\rightarrow B_{i}^{\prime}S_{B_{i}}}^{\mathcal{F}^{y_{i}}}\}_{y_{i}}$ are
collections of linear operators (each of which is a contraction, i.e.,
$\left\Vert U_{A_{i-1}^{\prime}\rightarrow A_{i}^{\prime}A_{i}S_{A_{i}}%
}^{\mathcal{E}^{y_{i}}}\right\Vert _{\infty},\left\Vert U_{B_{i-1}%
B_{i-1}^{\prime}\rightarrow B_{i}^{\prime}S_{B_{i}}}^{\mathcal{F}^{y_{i}}%
}\right\Vert _{\infty}\leq1$) such that the linear operator in
\eqref{eq:LOCC-as-separable-iso-ext} is an isometry, and $Y_{i}$ is a system
containing a coherent classical copy of the classical data exchanged in this
round, the system $Y_{i}$\ being held by Eve. The final LOCC\ channel can be
written similarly as%
\begin{equation}
\mathcal{L}_{A_{n}^{\prime}B_{n}B_{n}^{\prime}\rightarrow K_{A}K_{B}}%
^{(n+1)}\equiv\sum_{y_{n+1}}\mathcal{E}_{A_{n}^{\prime}\rightarrow K_{A}%
}^{y_{n+1}}\otimes\mathcal{F}_{B_{n}B_{n}^{\prime}\rightarrow K_{B}}^{y_{n+1}%
},
\end{equation}
and it can be purified to an isometry similarly as%
\begin{equation}
U_{A_{n}^{\prime}B_{n}B_{n}^{\prime}\rightarrow K_{A}S_{A_{n+1}}%
K_{B}S_{B_{n+1}}}^{\mathcal{L}^{(n+1)}}\equiv\sum_{y_{n+1}}U_{A_{n}^{\prime
}\rightarrow K_{A}S_{A_{n+1}}}^{\mathcal{E}^{y_{n+1}}}\otimes U_{B_{n}%
B_{n}^{\prime}\rightarrow K_{B}S_{B_{n+1}}}^{\mathcal{F}^{y_{n+1}}}%
\otimes|y_{n+1}\rangle_{Y_{n+1}}.
\end{equation}
Furthermore, each channel use $\mathcal{N}_{A_{i}\rightarrow B_{i}}$, for all
$i\in\{1,\ldots,n\}$ is purified by an isometry $U_{A_{i}\rightarrow
B_{i}E_{i}}^{\mathcal{N}}$, such that Eve possesses the environment system
$E_{i}$.
\subsection{Performance of a secret-key-agreement protocol}
\label{sec:secret-key-performance}At the end of the purified protocol, Alice
possesses the key system $K_{A}$ and the shield systems $S_{A}\equiv S_{A_{1}%
}\cdots S_{A_{n+1}}$, Bob possesses the key system $K_{B}$ and the shield
systems $S_{B}\equiv S_{B_{1}}\cdots S_{B_{n+1}}$, and Eve possesses the
environment systems $E^{n}\equiv E_{1}\cdots E_{n}$ as well as the coherent
copies $Y^{n+1}\equiv Y_{1}\cdots Y_{n+1}$ of the classical data exchanged.
The state at the end of the purified protocol is a pure state $|\omega
\rangle_{K_{A}S_{A}K_{B}S_{B}E^{n}Y^{n+1}}$. Fix $n,K\in\mathbb{N}$ and
$\varepsilon\in\lbrack0,1]$. The original protocol is an $(n,K,\varepsilon)$
protocol if%
\begin{equation}
F(\omega_{K_{A}K_{B}E^{n}Y^{n+1}},\overline{\Phi}_{K_{A}K_{B}}\otimes
\xi_{E^{n}Y^{n+1}})\geq1-\varepsilon, \label{eq:fidelity-tripartite-setting}%
\end{equation}
where the fidelity $F(\tau,\kappa)\equiv\left\Vert \sqrt{\tau}\sqrt{\kappa
}\right\Vert _{1}^{2}$ \cite{U76}, the maximally correlated state
$\overline{\Phi}_{K_{A}K_{B}}$ is defined as%
\begin{equation}
\overline{\Phi}_{K_{A}K_{B}}\equiv\frac{1}{K}\sum_{k=1}^{K}|k\rangle\langle
k|_{K_{A}}\otimes|k\rangle\langle k|_{K_{B}},
\end{equation}
and $\xi_{E^{n}Y^{n+1}}$ is an arbitrary state.
By the observations of \cite{HHHO05,HHHO09}\ (understood as a clever
application of Uhlmann's theorem for fidelity \cite{U76}), rather than
focusing on the tripartite scenario, one can focus on the bipartite scenario,
in which the goal is to produce an approximate private state of Alice and
Bob's systems. The criterion in \eqref{eq:fidelity-tripartite-setting} is
fully equivalent to%
\begin{equation}
F(\omega_{K_{A}S_{A}K_{B}S_{B}},\gamma_{K_{A}S_{A}K_{B}S_{B}})\geq
1-\varepsilon,
\end{equation}
where $\gamma_{K_{A}S_{A}K_{B}S_{B}}$ is a private state \cite{HHHO05,HHHO09}
of the following form:%
\begin{equation}
U_{K_{A}S_{A}K_{B}S_{B}}(\Phi_{K_{A}K_{B}}\otimes\theta_{S_{A}S_{B}}%
)U_{K_{A}S_{A}K_{B}S_{B}}^{\dag},
\end{equation}
with $U_{K_{A}S_{A}K_{B}S_{B}}$ a twisting unitary of the form $U_{K_{A}%
S_{A}K_{B}S_{B}}=\sum_{i,j=1}^{K}|i\rangle\langle i|_{K_{A}}\otimes
|j\rangle\langle j|_{K_{B}}\otimes U_{S_{A}S_{B}}^{ij}$, $\Phi_{K_{A}K_{B}}$
is a maximally entangled state of the form%
\begin{equation}
\Phi_{K_{A}K_{B}}\equiv\frac{1}{K}\sum_{i,j=1}^{K}|i\rangle\langle j|_{K_{A}%
}\otimes|i\rangle\langle j|_{K_{B}}, \label{eq:MES}%
\end{equation}
and $\theta_{S_{A}S_{B}}$ is an arbitrary state of the shield systems. The
main idea in this latter picture is that we can use the techniques of
entanglement theory to understand private communication protocols
\cite{HHHO05,HHHO09}.
A rate $R$ is achievable for secret key agreement if for all $\varepsilon
\in(0,1)$, $\delta>0$, and sufficiently large$~n$, there exists an
$(n,2^{n\left( R-\delta\right) },\varepsilon)$ protocol. The
secret-key-agreement capacity of $\mathcal{N}_{A\rightarrow B}$, denoted as
$P^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$, is equal to the supremum
of all achievable rates.
\subsection{Teleportation-simulable channels and reduction by teleportation}
\label{sec:TP-sim-reduct}An implication of channel simulation via
teleportation, as discussed in Section~\ref{sec:TP-simulation-def}, is that
the performance of a general protocol that uses the channel $n$ times, with
each use interleaved by local operations and classical communication (LOCC),
can be bounded from above by the performance of a protocol with a much simpler
form:\ the simplified protocol consists of a single round of LOCC acting on
$n$ copies of the resource state $\omega_{RB}$ \cite{BDSW96,NFC09,Mul12}. This
is called reduction by teleportation. Note that reduction by teleportation is
a very general procedure and clearly can be used more generally in any
LOCC-assisted protocol trying to accomplish an arbitrary
information-processing task. Of course, a secret-key-agreement protocol is one
particular kind of protocol of the above form, as considered in the follow-up
works \cite{PLOB15,WTB16}, and so the general reduction method of
\cite{BDSW96,NFC09,Mul12}\ applies to this particular case.
\subsection{Amortized relative entropy of entanglement as a bound for
secret-key-agreement protocols}
The main goal of this section is to show that the amortized relative entropy
of entanglement is an upper bound on the rate of secret key that can be
extracted by a secret-key-agreement protocol. We begin by establishing the
following theorem:
\begin{proposition}
\label{prop:weak-converse-bound}The following weak-converse bound holds for an
$(n,K,\varepsilon)$ secret-key-agreement protocol conducted over a quantum
channel $\mathcal{N}$:%
\begin{equation}
(1-\varepsilon)\frac{\log_{2}K}{n}\leq E_{AR}(\mathcal{N})+\frac{1}{n}%
h_{2}(\varepsilon), \label{eq:amortized-weak-converse-bound}%
\end{equation}
where $E_{AR}(\mathcal{N})$ is the amortized relative entropy of entanglement and $h_{2}(\varepsilon)\equiv-\varepsilon\log_{2}\varepsilon-\left(1-\varepsilon
\right)\log_{2}\!\left(1-\varepsilon\right)$ denotes the binary entropy.
\end{proposition}
\begin{proof}
To see how the amortized relative entropy of entanglement gives an upper bound
on the performance of a secret-key-agreement protocol, consider the following
steps. We suppose that we are dealing with an $(n,K,\varepsilon)$
secret-key-agreement protocol as described previously. First, at the end of
the protocol one can perform a privacy test \cite{WTB16}\ (see also
\cite{HHHO09,HHHLO08,PhysRevLett.100.110502}), which untwists the twisting
unitary of the ideal target private state and projects onto the maximally
entangled state of the key systems (cf.,
Section~\ref{sec:secret-key-performance}). Let $F^{\ast}$ denote the
probability of the actual state $\omega_{K_{A}S_{A}K_{B}S_{B}}$\ at the end of
the protocol passing this test. By \cite[Lemma~9]{WTB16}, this probability is
larger than $1-\varepsilon$. Let $p_{\operatorname{SEP}}$ denote the
probability that a given separable state $\sigma_{K_{A}S_{A}K_{B}S_{B}}$\ of
systems $K_{A}S_{A}K_{B}S_{B}$ passes the test. This probability is no larger
than $1/K$, following from results of \cite{HHHO09} (reviewed as
\cite[Lemma~10]{WTB16}). Then we find that%
\begin{align}
-h_{2}(\varepsilon)+(1-\varepsilon)\log_{2}K & \leq D(\{1-\varepsilon
,\varepsilon\}\Vert\{1/K,1-1/K\})\\
& \leq D(\{F^{\ast},1-F^{\ast}\}\Vert\{p_{\operatorname{SEP}}%
,1-p_{\operatorname{SEP}}\})\\
& \leq D(\omega_{K_{A}S_{A}K_{B}S_{B}}\Vert\sigma_{K_{A}S_{A}K_{B}S_{B}}).
\end{align}
The first inequality follows because%
\begin{align}
D(\{1-\varepsilon,\varepsilon\}\Vert\{1/K,1-1/K\}) & =(1-\varepsilon
)\log_{2}\left( \frac{1-\varepsilon}{1/K}\right) +\varepsilon\log_{2}\left(
\frac{\varepsilon}{1-1/K}\right) \\
& =-h_{2}(\varepsilon)+\left( 1-\varepsilon\right) \log_{2}K+\varepsilon
\log_{2}\left( \frac{K}{K-1}\right) \\
& \geq-h_{2}(\varepsilon)+\left( 1-\varepsilon\right) \log_{2}K.
\end{align}
with the last inequality above following because $K\geq1$ (note that the
singular case of $K=1$ is not particularly interesting because the rate is
zero and the protocol is thus trivial). The second inequality follows because
the distributions $\{F^{\ast},1-F^{\ast}\}$ and $\{p_{\operatorname{SEP}%
},1-p_{\operatorname{SEP}}\}$ are more distinguishable than the distributions
$\{1-\varepsilon,\varepsilon\}$ and $\{1/K,1-1/K\}$, due to the conditions
$F^{\ast}\geq1-\varepsilon$ and $p_{\operatorname{SEP}}\leq1/K$ (we are
assuming without loss of generality that $1-\varepsilon\geq1/K$ for this
statement; if it is not the case, then the code is in a rather sad state of
affairs with poor performance and there is no need to give a converse bound in
this case---the bound would simply be $\log_{2}K<-\log_{2}(1-\varepsilon)$).
The final inequality follows from monotonicity of quantum relative entropy
with respect to the privacy test (understood as a measurement channel). Since
the above chain of inequalities holds for all separable states $\sigma
_{K_{A}S_{A}K_{B}S_{B}}$, we find that%
\begin{equation}
-h_{2}(\varepsilon)+(1-\varepsilon)\log_{2}K\leq E_{R}(K_{A}S_{A};K_{B}%
S_{B})_{\omega},
\end{equation}
which can be rewritten as\footnote{Alternatively, the bound in
\eqref{eq:first-bound-final-state} can be established for all $\varepsilon
\in(0,1)$ and $K\geq1$ by using several methods from
\cite{WR12,MW12,WTB16,KW17}. In more detail, we could make use of the
$\varepsilon$-relative entropy of entanglement \cite{BD11}\ as an intermediary
step to get that $\log_{2}K\leq E_{R}^{\varepsilon}(K_{A}S_{A};K_{B}%
S_{B})_{\omega}\leq\frac{1}{1-\varepsilon}\left[ E_{R}(K_{A}S_{A};K_{B}%
S_{B})_{\omega}+h_{2}(\varepsilon)\right] $, with the first bound following
from \cite[Theorem~11]{WTB16} and the second from \cite{WR12,MW12,KW17}.}%
\begin{equation}
(1-\varepsilon)\log_{2}K\leq E_{R}(K_{A}S_{A};K_{B}S_{B})_{\omega}%
+h_{2}(\varepsilon). \label{eq:first-bound-final-state}%
\end{equation}
From the monotonicity of the relative entropy of entanglement with respect to
LOCC \cite{VP98}, we find that%
\begin{align}
E_{R}(K_{A}S_{A};K_{B}S_{B})_{\omega} & \leq E_{R}(A_{n}^{\prime};B_{n}%
B_{n}^{\prime})_{\sigma^{(n)}}\\
& =E_{R}(A_{n}^{\prime};B_{n}B_{n}^{\prime})_{\sigma^{(n)}}-E_{R}%
(A_{1}^{\prime}A_{1};B_{1}^{\prime})_{\rho^{(1)}}\\
& =E_{R}(A_{n}^{\prime};B_{n}B_{n}^{\prime})_{\sigma^{(n)}}+\left[
\sum_{i=2}^{n}E_{R}(A_{i}^{\prime}A_{i};B_{i}^{\prime})_{\rho^{(i)}}%
-E_{R}(A_{i}^{\prime}A_{i};B_{i}^{\prime})_{\rho^{(i)}}\right] \nonumber\\
& \qquad-E_{R}(A_{1}^{\prime}A_{1};B_{1}^{\prime})_{\rho^{(1)}}\\
& \leq\sum_{i=1}^{n}E_{R}(A_{i}^{\prime};B_{i}B_{i}^{\prime})_{\sigma^{(i)}%
}-E_{R}(A_{i}^{\prime}A_{i};B_{i}^{\prime})_{\rho^{(i)}}\\
& \leq nE_{AR}(\mathcal{N}). \label{eq:last-bound-amortized-proof}%
\end{align}
The first equality follows because the state $\rho_{A_{1}^{\prime}A_{1}%
B_{1}^{\prime}}^{(1)}$ is a separable state with vanishing relative entropy of
entanglement. The second equality follows by adding and subtracting terms. The
second inequality follows because $E_{R}(A_{i}^{\prime}A_{i};B_{i}^{\prime
})_{\rho^{(i)}}\leq E_{R}(A_{i-1}^{\prime};B_{i-1}B_{i-1}^{\prime}%
)_{\sigma^{(i-1)}}$ for all $i\in\{2,\ldots,n\}$, due to monotonicity of the
relative entropy of entanglement with respect to LOCC. The final inequality
follows because each term $E_{R}(A_{n}^{\prime};B_{n}B_{n}^{\prime}%
)_{\sigma^{(i)}}-E_{R}(A_{i}^{\prime}A_{i};B_{i}^{\prime})_{\rho^{(i)}}$ is of
the form in the amortized relative entropy of entanglement, so that optimizing
over all inputs of the form $\rho^{(i)}$\ cannot exceed $E_{AR}(\mathcal{N})$.
Combining \eqref{eq:first-bound-final-state}\ and
\eqref{eq:last-bound-amortized-proof}, we arrive at the inequality in \eqref{eq:amortized-weak-converse-bound}.
\end{proof}
\bigskip
Taking the limit in Proposition~\ref{prop:weak-converse-bound}\ as
$n\rightarrow\infty$ and then as $\varepsilon\rightarrow0$ leads to the
following asymptotic statement:
\begin{theorem}
\label{thm:amortized-rel-ent-bound}The secret-key-agreement capacity
$P^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$ of a quantum channel
$\mathcal{N}_{A\rightarrow B}$ cannot exceed its amortized relative entropy of
entanglement:%
\begin{equation}
P^{\leftrightarrow}(\mathcal{N})\leq E_{AR}(\mathcal{N}).
\end{equation}
\end{theorem}
\begin{remark}
Interestingly, a similar approach using the sandwiched R\'{e}nyi relative
entropy \cite{MDSFT13,WWY13}\ gives the following upper bound for all
$\alpha>1$%
\begin{equation}
\frac{\alpha}{\alpha-1}\log_{2}(1-\varepsilon)\leq n\widetilde{E}_{AR}%
^{\alpha}(\mathcal{N})-\log_{2}K, \label{eq:alpha-rel-ent-bound-1}%
\end{equation}
where%
\begin{equation}
\widetilde{E}_{AR}^{\alpha}(\mathcal{N})=\sup_{\rho_{A^{\prime}AB^{\prime}}%
}\widetilde{E}_{R}^{\alpha}(A^{\prime};BB^{\prime})_{\omega}-\widetilde{E}%
_{R}^{\alpha}(A^{\prime}A;B^{\prime})_{\rho}, \label{eq:alpha-rel-ent-bound-2}%
\end{equation}
and $\widetilde{E}_{R}^{\alpha}$ denotes the sandwiched R\'{e}nyi relative
entropy of entanglement \cite{WTB16},\ defined from the sandwiched R\'{e}nyi
relative entropy \cite{MDSFT13,WWY13}. One of the main results of
\cite{Christandl2017} is the bound $\widetilde{E}_{AR}^{\alpha}(\mathcal{N}%
)\leq E_{\max}(\mathcal{N})$, where $E_{\max}(\mathcal{N})$ denotes the
channel's max-relative entropy of entanglement (cf., \cite{D09}).\ The authors
of \cite{Christandl2017} observed that this latter bound in turn implies the
following bound for all $\alpha>1$:%
\begin{equation}
\frac{\alpha}{\alpha-1}\log_{2}(1-\varepsilon)\leq nE_{\max}(\mathcal{N}%
)-\log_{2}K.
\end{equation}
\end{remark}
\section{Amortized Rains relative entropy and PPT-assisted quantum
communication}
\label{sec:amortized-Rains-q-comm}The main goal of this section is to prove
that the amortized Rains relative entropy from
Definition~\ref{def:amortized-Rains} is an upper bound on a channel's
PPT-assisted quantum capacity. By this, we mean that a sender and receiver are
allowed to use a channel many times, and between every channel use, they are
allowed free usage of channels that are positive-partial-transpose
(PPT)\ preserving. In what follows, we detail these concepts, and then we
state the main theorem (Theorem~\ref{thm:amortized-rel-ent-bound-PPT}).
\subsection{Positive-partial-transpose preserving quantum channels}
A quantum channel $\mathcal{P}$ is a positive-partial-transpose (PPT)
preserving channel from systems $A\!:\!B$ to systems $A^{\prime}%
\!:\!B^{\prime}$ if the map $T_{B^{\prime}}\circ\mathcal{N}_{AB\rightarrow
A^{\prime}B^{\prime}}\circ T_{B}$ is completely positive and trace
preserving~\cite{R01}, where $T_{B}$ and $T_{B^{\prime}}$ denote the partial
transpose map. For a given basis $\{|i\rangle\}_{i}$, the transpose map is a
positive map, specified by $\rho\rightarrow\sum_{i,j}|i\rangle\langle
j|\rho|i\rangle\langle j|$. In what follows, we call PPT-preserving channels
\textquotedblleft PPT\ channels\textquotedblright\ as an abbreviation. It has
been known for a long time that PPT\ channels contain the set of LOCC channels
\cite{R01}, and so an immediate operational consequence of this containment is
that any general upper bound on the performance of a PPT-assisted protocol
serves as an upper bound on the performance of an LOCC-assisted protocol
\cite{R01}. This fact and the fact that PPT\ channels are simpler to analyze
mathematically than LOCC were some of the main motivations for introducing
this class of channels \cite{R01}.
PPT\ channels preserve the set $\operatorname{PPT}^{\prime}$ discussed in
Definition~\ref{def:amortized-Rains} \cite{R01,AdMVW02}. For this reason and
since the relative entropy is monotone with respect to quantum channels
\cite{Lindblad1975}, it follows that the Rains relative entropy is monotone
with respect to PPT\ channels \cite{R01,AdMVW02}, in the sense that%
\begin{equation}
R(A;B)_{\rho}\geq R(A^{\prime};B^{\prime})_{\mathcal{P}(\rho)},
\end{equation}
where $\rho_{AB}$ is a bipartite state and $\mathcal{P}_{AB\rightarrow
A^{\prime}B^{\prime}}$ is a PPT\ channel.
The notion of PPT\ channels then leads to a more general notion of the
teleportation simulation of a quantum channel:
\begin{definition}
[$\omega$-PPT-simulable channel]\label{def:PPT-sim-chan}A channel
$\mathcal{N}_{A\rightarrow B}$ with input system $A$ and output system $B$ is
defined to be PPT-simulable with associated resource state $\omega_{RB}$
($\omega$-PPT-simulable for short) if the following equality holds for all
input states $\rho_{A}$:%
\begin{equation}
\mathcal{N}_{A\rightarrow B}(\rho_{A})=\mathcal{P}_{ARB\rightarrow B}(\rho
_{A}\otimes\omega_{RB}),
\end{equation}
where $\mathcal{P}_{ARB\rightarrow B}$ is a PPT quantum channel with respect
to the bipartite cut $AR|B$ at the input. Note that every
teleportation-simulable channel with associated resource state $\omega_{RB}$
is PPT-simulable with associated resource state $\omega_{RB}$.
\end{definition}
\subsection{Protocols for PPT-assisted quantum communication and their
performance}
\label{sec:PPT-protocols}The structure of an $(n,M,\varepsilon)$ protocol for
PPT-assisted quantum communication is quite similar to that for an
$(n,K,\varepsilon)$ protocol for secret key agreement, which we discussed
previously in Section~\ref{sec:SKA-protocol}. In fact, such a PPT-assisted
protocol is exactly as outlined in Section~\ref{sec:SKA-protocol} and
Figure~\ref{fig:private-code}, but each LOCC\ channel is replaced with a
PPT\ channel. Let us denote the final state of the protocol by $\omega
_{M_{A}M_{B}}$ instead of $\omega_{K_{A}K_{B}}$. Fixing $n,M\in\mathbb{N}$ and
$\varepsilon\in\lbrack0,1]$, the protocol is an $(n,M,\varepsilon)$
PPT-assisted quantum communication protocol if%
\begin{equation}
F(\omega_{M_{A}M_{B}},\Phi_{M_{A}M_{B}})\geq1-\varepsilon,
\end{equation}
where the maximally entangled state $\Phi_{M_{A}M_{B}}$ is defined in \eqref{eq:MES}.
A rate $R$ is achievable for PPT-assisted quantum communication if for all
$\varepsilon\in(0,1)$, $\delta>0$, and sufficiently large $n$, there exists an
$(n,2^{n\left( R-\delta\right) },\varepsilon)$ protocol. The PPT-assisted
quantum capacity of $\mathcal{N}_{A\rightarrow B}$, denoted as
$Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$, is equal to
the supremum of all achievable rates.
We can also consider the whole development above when we only allow the
assistance of LOCC\ channels instead of PPT\ channels. In this case, we have
similar notions as above, and then we arrive at the LOCC-assisted quantum
capacity $Q^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$. It then
immediately follows that%
\begin{equation}
Q^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})\leq Q^{\mathbf{PPT}%
,\leftrightarrow}(\mathcal{N}_{A\rightarrow B})
\end{equation}
because every LOCC\ channel is a PPT\ channel. We also have the following
bound%
\begin{equation}
Q^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})\leq P^{\leftrightarrow
}(\mathcal{N}_{A\rightarrow B}),
\end{equation}
as observed in \cite{TWW14}, because a maximally entangled state, the target
state of an LOCC-assisted quantum communication protocol, is a particular kind
of private state.
For channels that are $\omega$-PPT-simulable, as in
Definition~\ref{def:PPT-sim-chan}, PPT-assisted protocols simplify immensely,
just as was the case for teleportation-simulable channels. Indeed, in this
case, PPT-assisted protocols can be reduced to the action of a single
PPT\ channel on $n$ copies of the resource state $\omega_{RB}$, and this
reduction is helpful in bounding the performance of PPT-assisted protocols
conducted over such channels.
\subsection{Amortized Rains relative entropy as a bound for PPT-assisted
quantum communication protocols}
We can employ an argument nearly identical to that given in the proof of
Proposition~\ref{prop:weak-converse-bound}\ in order to establish that the
amortized Rains relative entropy is an upper bound on the rate at which
maximal entanglement can be extracted by a PPT-assisted quantum communication
protocol. Indeed, we simply replace the privacy test therein by a maximal
entanglement test (i.e., a measurement specified by a projection onto the
maximally entangled state or its complement). By the definition of an
$(n,M,\varepsilon)$ protocol and the fidelity, the probability for the final
state $\omega_{M_{A}M_{B}}$ of the protocol to pass this test is larger than
$1-\varepsilon$. Furthermore, due to \cite[Lemma~2]{R99}, the bound
$\operatorname{Tr}\{\Phi_{M_{A}M_{B}}\sigma_{M_{A}M_{B}}\}\leq1/M$ holds for
all states $\sigma_{M_{A}M_{B}}\in\operatorname{PPT}^{\prime}(M_{A}%
\!:\!M_{B})$. These bounds and the same reasoning employed in the proof of
Proposition~\ref{prop:weak-converse-bound}\ allow us to conclude the following
weak-converse bound for any PPT-assisted quantum communication protocol:
\begin{proposition}
\label{prop:weak-converse-bound-PPT}The following weak-converse bound holds
for an $(n,M,\varepsilon)$ PPT-assisted quantum communication protocol
conducted over a quantum channel $\mathcal{N}$:%
\begin{equation}
(1-\varepsilon)\frac{\log_{2}M}{n}\leq R_{A}(\mathcal{N})+\frac{1}{n}%
h_{2}(\varepsilon),
\end{equation}
where $R_{A}(\mathcal{N})$ denotes the amortized Rains relative entropy from
Definition~\ref{def:amortized-Rains}.
\end{proposition}
Taking the limit in Proposition~\ref{prop:weak-converse-bound-PPT}\ as
$n\rightarrow\infty$ and then as $\varepsilon\rightarrow0$ leads to the
following asymptotic statement:
\begin{theorem}
\label{thm:amortized-rel-ent-bound-PPT}The PPT-assisted quantum capacity
$Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$ of a quantum
channel $\mathcal{N}_{A\rightarrow B}$ cannot exceed its amortized Rains
relative entropy:%
\begin{equation}
Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N})\leq R_{A}(\mathcal{N}).
\end{equation}
\end{theorem}
Furthermore, we obtain a bound for PPT-assisted quantum communication similar
to that stated in
\eqref{eq:alpha-rel-ent-bound-1}--\eqref{eq:alpha-rel-ent-bound-2}\ by
employing similar reasoning.
\section{Approximately teleportation- and PPT-simulable channels}
\label{sec:approx-TP-bound}
We now define approximately teleportation- and PPT-simulable channels:
\begin{definition}
[Approximately teleportation- and PPT-simulable channels]%
\label{def:approx-TP-sim}A quantum channel $\mathcal{N}_{A\rightarrow B}$ is
$\varepsilon$-approximately teleportation-simulable with associated resource
state $\omega_{RB}$ if there exists a channel $\mathcal{M}_{A\rightarrow B}$
that is exactly teleportation-simulable with associated resource state
$\omega_{RB}$ such that%
\begin{equation}
\frac{1}{2}\left\Vert \mathcal{N}_{A\rightarrow B}-\mathcal{M}_{A\rightarrow
B}\right\Vert _{\Diamond}\leq\varepsilon.
\end{equation}
For short, we say that $\mathcal{N}_{A\rightarrow B}$ is $\left(
\varepsilon,\omega_{RB}\right) $-approximately teleportation-simulable. The
same definition applies for an $\left( \varepsilon,\omega_{RB}\right)
$-approximately PPT-simulable channel, but the difference is that
$\mathcal{M}_{A\rightarrow B}$ is exactly PPT-simulable with associated
resource state $\omega_{RB}$. Also, if a channel is $\left( \varepsilon
,\omega_{RB}\right) $-approximately teleportation-simulable, then it is
$\left( \varepsilon,\omega_{RB}\right) $-approximately PPT-simulable.
\end{definition}
In Appendix~\ref{sec:cov-TP-sim}, we discuss the relation between the notion of
approximately teleportation-simulable channels and the recently introduced
notion of approximately covariant channels \cite{LKDW17}. Therein, we also
discuss channel twirling and how to simulate this procedure via a generalized
teleportation protocol.
The following theorem is an immediate consequence of
Proposition~\ref{prop:tp-upper-bound}, Theorem~\ref{thm:continuity}, and
Theorem~\ref{thm:amortized-rel-ent-bound}, and it constitutes one of the main
results of our paper:
\begin{theorem}
\label{thm:approx-TP-sim-bound}If a channel $\mathcal{N}_{A\rightarrow B}$ is
$\left( \varepsilon,\omega_{RB}\right) $-approximately
teleportation-simulable, then its secret-key-agreement capacity
$P^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})$ is bounded from above as%
\begin{equation}
P^{\leftrightarrow}(\mathcal{N}_{A\rightarrow B})\leq E_{R}(R;B)_{\omega
}+2\varepsilon\log_{2}\left\vert B\right\vert +g(\varepsilon).
\end{equation}
\end{theorem}
Similarly, the following theorem is an immediate consequence of
Proposition~\ref{prop:tp-upper-bound}, Theorem~\ref{thm:continuity}, and
Theorem~\ref{thm:amortized-rel-ent-bound-PPT}:
\begin{theorem}
\label{thm:approx-TP-sim-bound-PPT}If a channel $\mathcal{N}_{A\rightarrow B}$
is $\left( \varepsilon,\omega_{RB}\right) $-approximately PPT-simulable,
then its PPT-assisted quantum capacity $Q^{\mathbf{PPT},\leftrightarrow
}(\mathcal{N}_{A\rightarrow B})$ is bounded from above as%
\begin{equation}
Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}_{A\rightarrow B})\leq
R(R;B)_{\omega}+2\varepsilon\log_{2}\left\vert B\right\vert +g(\varepsilon).
\end{equation}
\end{theorem}
In the next section, we apply the bounds from
Theorems~\ref{thm:approx-TP-sim-bound}\ and \ref{thm:approx-TP-sim-bound-PPT}%
\ to an example qubit channel.
\section{Bounds on the assisted capacities of a particular qubit channel}
\label{sec:examples}
In this section, we apply the bounds from
Theorems~\ref{thm:approx-TP-sim-bound}\ and \ref{thm:approx-TP-sim-bound-PPT}%
\ to a particular qubit channel $\mathcal{N}_{p}$, which we define to be a
convex mixture of an amplitude damping channel and a depolarizing channel:%
\begin{equation}
\mathcal{N}_{p}(\rho)=p\mathcal{A}_{p}(\rho)+(1-p)\mathcal{D}_{p}(\rho),
\label{eqn:channel}%
\end{equation}
where $p\in\lbrack0,1]$ and $\rho$ is an input qubit density operator (this is
the same channel considered in concurrent work \cite{LKDW17}). The amplitude
damping channel $\mathcal{A}_{p}$ is defined as
\begin{align}
\mathcal{A}_{p}(\rho) & =K_{1}\rho K_{1}^{\dagger}+K_{2}\rho K_{2}^{\dagger
},\\
K_{1} & =|0\rangle\langle0|+\sqrt{1-p}|1\rangle\langle1|,\\
K_{2} & =\sqrt{p}|0\rangle\langle1|.
\end{align}
Also, $\mathcal{D}_{p}$ denotes the qubit depolarizing channel:%
\begin{equation}
\mathcal{D}_{p}(\rho)=(1-p)\rho+\frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z),
\end{equation}
where $X$, $Y$, and $Z$ are the Pauli operators.
Let $\Phi(\mathcal{M})$ denote the Choi state associated with a channel
$\mathcal{M}$, i.e., the state that results from sending one share of a
maximally entangled state through the channel. It has been known for many
years now \cite{BDSW96,HHH99,CDP09}\ that the depolarizing channel is
teleportation-simulable with associated resource state $\Phi(\mathcal{D}_{p}%
)$. Thus, for small values of $p$, we should expect for a convex mixture of
the depolarizing channel and the amplitude channel to be approximately
teleportation-simulable with associated resource state $\Phi(\mathcal{D}_{p}%
)$, given that $\mathcal{N}_{p}$ is intuitively close to $\mathcal{D}_{p}$ for
small values of $p$. Indeed, it follows from the results of \cite[Section~4.2]{LKDW17} and
the discussion in Appendix~\ref{sec:cov-TP-sim} that the channel
$\mathcal{N}_{p}$ is $(p^{2}/2,\Phi(\overline{\mathcal{N}}_{p}))$%
-approximately teleportation-simulable, where $\overline{\mathcal{N}}_{p}$
denotes the following teleportation-simulable channel:%
\begin{equation}
\overline{\mathcal{N}}_{p}(\rho)=\frac{1}{2}\left[ \mathcal{N}_{p}%
(\rho)+X\mathcal{N}_{p}(X\rho X)X\right] .
\end{equation}
We can thus apply Theorems~\ref{thm:approx-TP-sim-bound} and
\ref{thm:approx-TP-sim-bound-PPT} to arrive at the following bounds on the
secret-key-agreement capacity $P^{\leftrightarrow}(\mathcal{N}_{p})$ and the
PPT-assisted quantum capacity $Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}%
_{p})$:%
\begin{align}
P^{\leftrightarrow}(\mathcal{N}_{p}) & \leq E_{R}(A;B)_{\Phi(\overline
{\mathcal{N}}_{p})}+p^{2}+g(p^{2}/2),\\
Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}_{p}) & \leq R(A;B)_{\Phi
(\overline{\mathcal{N}}_{p})}+p^{2}+g(p^{2}/2).
\end{align}
By noting that the set $\operatorname{PPT}^{\prime}$ contains the set of PPT
states and applying the well known result that PPT\ states are equal to
separable states for $2\times2$ systems \cite{P96,Horodecki19961}, we can
conclude that%
\begin{equation}
P^{\leftrightarrow}(\mathcal{N}_{p}),\ Q^{\mathbf{PPT},\leftrightarrow
}(\mathcal{N}_{p})\leq E_{\mathbf{PPT}}(A;B)_{\Phi(\overline{\mathcal{N}}%
_{p})}+p^{2}+g(p^{2}/2),\label{eq:qubit-chan-upper-bound}%
\end{equation}
where $E_{\mathbf{PPT}}$ denotes the relative entropy to PPT\ states.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=10cm]{convx_amp_dep.pdf}
\end{center}
\caption{Upper and lower bounds on the secret-key-agreement capacity
$P^{\leftrightarrow}(\mathcal{N}_{p})$ and the PPT-assisted quantum capacity
$Q^{\mathbf{PPT},\leftrightarrow}(\mathcal{N}_{p})$\ of the channel defined in
(\ref{eqn:channel}). The vertical axis represents rate in either private bits
or qubits per channel use. The horizontal axis represents the value of the
channel parameter $p$.}%
\label{fig:plots}%
\end{figure}
The upper bound in \eqref{eq:qubit-chan-upper-bound} is plotted in
Figure~\ref{fig:plots}. To calculate $E_{\mathbf{PPT}}(A;B)_{\Phi
(\overline{\mathcal{N}}_{p})}$, we have made use of the relative entropy
optimization techniques put forward recently in \cite{FF17}.
We also consider lower bounds on the assisted capacities. Note that both
$P^{\leftrightarrow}(\mathcal{N}_{p})$ and $Q^{\mathbf{PPT},\leftrightarrow
}(\mathcal{N}_{p})$ are bounded from below by the coherent information
\cite{PhysRevA.54.2629}\ and the negative CB-entropy \cite{DJKR06}\ of the
channel (note that the latter is sometimes called \textquotedblleft reverse
coherent information\textquotedblright). These lower bounds are a direct
consequence of the developments in \cite{DW05}. In Figure~\ref{fig:plots}, we
also plot the coherent information of the channel, with the input state being
the maximally entangled state. For the negative CB-entropy, we optimize over
the input states of the channel and can exploit symmetry to simplify this optimization.
Here we elaborate on how to simplify the calculation of the negative
CB-entropy for our example. As discussed in \cite{DJKR06}, it is possible to
write the negative CB-entropy of a channel $\mathcal{N}_{A\rightarrow B}$ as
the following optimization:%
\begin{equation}
-H_{\operatorname{CB}}(\mathcal{N})=\sup_{\rho}H(B|E)_{\mathcal{U}%
^{\mathcal{N}}(\rho)},
\end{equation}
where $\mathcal{U}_{A\rightarrow BE}^{\mathcal{N}}$ denotes an isometric
channel that extends $\mathcal{N}_{A\rightarrow B}$. Due to the concavity of
conditional entropy \cite{LR73,PhysRevLett.30.434}, it immediately follows
that $H(B|E)_{\mathcal{U}^{\mathcal{N}}(\rho)}$ is concave with respect to the
input density operator $\rho$, and thus the calculation of
$-H_{\operatorname{CB}}(\mathcal{N})$ is a concave optimization problem. For
our example, we can further simplify the calculation of $-H_{\operatorname{CB}%
}(\mathcal{N}_{p})$ by exploiting the symmetry of $\mathcal{N}_{p}$. To see
this symmetry, consider that both the amplitude damping channel and the
depolarizing channel are covariant with respect to $I$ and $Z$, in the sense
that%
\begin{equation}
\mathcal{A}_{p}(U\rho U^{\dag})=U\mathcal{A}_{p}(\rho)U^{\dag},\qquad
\mathcal{D}_{p}(U\rho U^{\dag})=U\mathcal{D}_{p}(\rho)U^{\dag},
\end{equation}
where $U$ can be $I$ or $Z$. This observation then implies that $\mathcal{N}%
_{p}$ is covariant with respect to $I$ and $Z$. By invoking an observation
stated in \cite{Hol06}, it follows that%
\begin{equation}
\mathcal{U}^{\mathcal{N}_{p}}(Z\rho Z)=(Z\otimes\bar{Z})\mathcal{U}%
^{\mathcal{N}_{p}}(\rho)(Z\otimes\bar{Z}),
\end{equation}
where $\mathcal{U}^{\mathcal{N}_{p}}$ is an isometric channel that extends
$\mathcal{N}_{p}$ and $\bar{Z}$ is a unitary representation of $Z$. We can
then exploit the invariance of conditional entropy with respect to local
unitaries and its concavity to find that%
\begin{align}
H(B|E)_{\mathcal{U}^{\mathcal{N}_{p}}(\rho)} & =\frac{1}{2}\left[
H(B|E)_{\mathcal{U}^{\mathcal{N}_{p}}(\rho)}+H(B|E)_{(Z\otimes\bar
{Z})\mathcal{U}^{\mathcal{N}_{p}}(\rho)(Z\otimes\bar{Z})}\right] \\
& =\frac{1}{2}\left[ H(B|E)_{\mathcal{U}^{\mathcal{N}_{p}}(\rho
)}+H(B|E)_{\mathcal{U}^{\mathcal{N}_{p}}(Z\rho Z)}\right] \\
& \leq H(B|E)_{\mathcal{U}^{\mathcal{N}_{p}}(\frac{1}{2}(\rho+Z\rho Z))}.
\end{align}
Since $\frac{1}{2}(\rho+Z\rho Z)$ has no off-diagonal elements with respect to
the standard basis, the above calculation reduces the optimization of the
negative CB-entropy for $\mathcal{N}_{p}$\ to an optimization over a single parameter.
The optimized negative CB-entropy is plotted in Figure \ref{fig:plots}. Our
findings are consistent with those in earlier works \cite{SSWR14,LLS17}: the
upper bound from \eqref{eq:qubit-chan-upper-bound} is closer to the lower
bounds in the low-noise regime (small values of $p$) than it is in the
high-noise regime.
\section{Generalizations to other resource theories}
In this section, we discuss how to generalize several of the concepts in our
paper to general resource theories \cite{BG15,fritz_2015,RKR15,KR16}. This
generalization has already been considered in the context of the resource
theory of coherence \cite{BGMW17}, and in fact, we note here that the recent
developments in \cite{BGMW17} were what served as the inspiration for our
present paper. In short, a resource theory consists of a few basic
elements.\ There is a set $F$\ of free quantum states, i.e., those that the
players involved are allowed to access without any cost. Related to these,
there is a set of free channels, and they should have the property that a free
state remains free after a free channel acts on it. Once these are defined, it
follows that any state that is not free is considered resourceful, i.e.,
useful in the context of the resource theory. We can also then define a
measure $V$ of the resourcefulness of a quantum state, and some fundamental
properties that it should satisfy are that
\begin{enumerate}
\item it should be monotone non-increasing under the action of a free channel and
\item it should be equal to zero when evaluated on a free state.
\end{enumerate}
\noindent A typical choice of a resourcefulness measure of a state $\rho$
satisfying these requirements is the relative entropy of
resourcefulness:\ $\inf_{\sigma\in F}D(\rho\Vert\sigma)$.
With these basic aspects established and given a measure $V$ of the
resourcefulness of a quantum state, we might be interested in quantifying how
resourceful a channel $\mathcal{N}$\ is. One way of doing so is to define the
amortized resourcefulness of a quantum channel as follows, generalizing the
amortized entanglement from Definition~\ref{def:amortized-ent}:%
\begin{equation}
V_{A}(\mathcal{N})=\sup_{\rho_{RA}}V((\operatorname{id}_{R} \otimes
\mathcal{N}_{A\to B})(\rho_{RA}))-V(\rho_{RA}).
\end{equation}
\begin{figure}[ptb]
\begin{center}
\includegraphics[
width=6.5638in
]{resource-theory-protocol.pdf}
\end{center}
\caption{A protocol for extracting resourcefulness from a channel
$\mathcal{N}$ by accessing it with free operations between each channel use.}%
\label{fig:resource-theory-protocol}%
\end{figure}
Suppose now that we have a protocol that accesses the channel $\mathcal{N}$ a
total of $n$\ times and between each channel use, we allow for a free channel
to be applied. Such protocols generalize those that we considered in
Sections~\ref{sec:SKA-protocol}\ and \ref{sec:PPT-protocols}. Let $\omega$
denote the final state generated by the protocol, let $\rho_{R_{i}A_{i}}$
denote the state before the $i$th channel use, and let $\sigma_{R_{i}B_{i}}$
denote the state after the $i$th channel use. See
Figure~\ref{fig:resource-theory-protocol} for a depiction of such a protocol.
Then by applying the same reasoning in the proofs of
Propositions~\ref{prop:weak-converse-bound} and
\ref{prop:weak-converse-bound-PPT} (but now using properties 1 and 2 above),
we find the following bound:%
\begin{align}
V(\omega) & \leq V(\sigma_{R_{n}B_{n}})\\
& =V(\sigma_{R_{n}B_{n}})-V(\rho_{R_{1}A_{1}})\\
& =V((\operatorname{id}_{R}\otimes\mathcal{N}_{A_{n}\rightarrow B_{n}}%
)(\rho_{R_{n}A_{n}}))-V(\rho_{R_{1}A_{1}})+\sum_{i=2}^{n}V(\rho_{R_{i}A_{i}%
})-V(\rho_{R_{i}A_{i}})\\
& \leq V((\operatorname{id}_{R}\otimes\mathcal{N}_{A_{n}\rightarrow B_{n}%
})(\rho_{R_{n}A_{n}}))-V(\rho_{R_{1}A_{1}})\nonumber\\
& \qquad+\sum_{i=2}^{n}V((\operatorname{id}_{R}\otimes\mathcal{N}%
_{A_{i-1}\rightarrow B_{i-1}})(\rho_{R_{i-1}A_{i-1}}))-V(\rho_{R_{i}A_{i}})\\
& =\sum_{i=1}^{n}V((\operatorname{id}_{R}\otimes\mathcal{N}_{A_{i}\rightarrow
B_{i}})(\rho_{R_{i}A_{i}}))-V(\rho_{R_{i}A_{i}})\\
& \leq nV_{A}(\mathcal{N}), \label{eq:amortized-bound-resource-theory}%
\end{align}
which serves as a limitation on how much of the resource we can extract by
invoking the channel $n$ times in such a way. If $V(\omega)$ can be connected
to meaningful operational parameters such as the closeness of the final state
$\omega$ to a desired target state and the number of basic units of a
resource, as was the case in Propositions~\ref{prop:weak-converse-bound} and
\ref{prop:weak-converse-bound-PPT}, then the above bound would be even more
interesting in the context of a given resource theory.
We can also define $\nu$-freely-simulable channels as a generalization of the
teleportation-simulable channels of \cite{BDSW96,HHH99} and the $\omega
$-PPT-simulable channels introduced in Definition~\ref{def:PPT-sim-chan}:
\begin{definition}
[$\nu$-freely-simulable channel]A quantum channel $\mathcal{N}$ is $\nu
$-freely-simulable if there exists a resourceful state $\nu$ and a free
channel $\mathcal{F}$ such that the following equality holds for all input
states $\rho$:%
\begin{equation}
\mathcal{N}(\rho)=\mathcal{F}(\rho\otimes\nu).
\end{equation}
\end{definition}
\begin{figure}[ptb]
\begin{center}
\includegraphics[
width=6.5638in
]{resource-theory-protocol-freely-sim.pdf}
\end{center}
\caption{A protocol for extracting resourcefulness from a $\nu$-freely
simulable channel $\mathcal{N}$ by accessing it with free operations between
each channel use. Any such protocol can be understood as the action of a
single free channel (everything within the dotted box) acting on the resource
state $\nu^{\otimes n}$.}%
\label{fig:resource-theory-protocol-freely-sim}%
\end{figure}
For $\nu$-freely simulable channels, protocols of the form discussed
previously simplify significantly, as depicted in
Figure~\ref{fig:resource-theory-protocol-freely-sim}. The reduction depicted
in Figure~\ref{fig:resource-theory-protocol-freely-sim} generalizes reduction
by teleportation \cite{BDSW96,Mul12}\ reviewed in
Section~\ref{sec:TP-sim-reduct}\ as well as the more general approach of
\textquotedblleft quantum simulation\textquotedblright\ put forward in
\cite{DM14}, as the reduction applies in the context of any resource theory.
By employing property~1 above and inspecting
Figure~\ref{fig:resource-theory-protocol-freely-sim}, it is immediate that the
following bound holds%
\begin{equation}
V(\omega)\leq V(\nu^{\otimes n}),
\end{equation}
which is just the statement that the amount of resourcefulness that can be
extracted from the channel is limited by the resourcefulness of the underlying
state $\nu$. If the resourcefulness measure is subadditive with respect to
quantum states, then we arrive at the following bound for any protocol of the
above form:%
\begin{equation}
\frac{1}{n}V(\omega)\leq V(\nu).
\end{equation}
We think that it would be very interesting to work out some applications or
consequences of the above observations in the context of several resource
theories, such as thermodynamics \cite{BHORS13}, asymmetry \cite{MS14}, or
non-Gaussianity. We could certainly also consider approximately $\nu
$-freely-simulable channels in order to find bounds on the extraction rates
that are possible from protocols that use resourceful channels that are close
to $\nu$-freely-simulable ones.
\section{Conclusion}
In this paper, we introduced the amortized entanglement of a channel as the
largest difference in entanglement between the output and input of a quantum
channel. We proved several properties of amortized entanglement and considered
special cases of the measures such as amortized relative entropy of
entanglement and amortized Rains relative entropy. One property of especial
interest is the uniform continuity of the latter two special cases, in which
an upper bound on the deviation of the amortized entanglement of two channels
is given in terms of the output dimension of the channels and the diamond norm
of their difference. This uniform continuity bound and the notion of
approximately teleportation- and PPT-simulable channels then immediately leads
to an upper bound on the secret-key-agreement and LOCC-assisted quantum
capacities of such channels. We applied these notions to an example channel,
which consists of a convex mixture of an amplitude damping channel and a
depolarizing channel, and we found that the upper bound is reasonably close to
lower bounds on the capacities whenever the noise in the channel is
sufficiently low. Finally, we discussed how to generalize many of the notions
in the paper to more general resource theories, introducing concepts such as
amortized resourcefulness of a channel and $\nu$-freely-simulable channels.
For future work, we think it would be interesting to explore the
aforementioned generalization further, in the context of other resource
theories such as thermodynamics, asymmetry, or non-Gaussianity.
\bigskip
\textbf{Acknowledgements.} We thank Stefan B\"auml, Siddhartha Das, Nilanjana
Datta, Felix Leditzky, Iman Marvian, and Andreas Winter for discussions
related to this paper. We also acknowledge support from the Office of Naval Research.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,786 |
Chairman Smith Congratulates Jacksonville Mayor-elect Alvin Brown
NEWS FROM THE FLORIDA DEMOCRATIC PARTY
For Immediate Release: May 18, 2011
Contact: Eric Jotkoff, (850) 222-3411
Florida Democratic Party Chairman Rod Smith released the following statement congratulating Mayor-elect Alvin Brown on his election as Mayor of Jacksonville, Florida's largest city. One of Chairman Smith's first acts upon being elected to lead the Florida Democratic Party was to endorse Alvin Brown's campaign:
"I would like to congratulate Alvin Brown on his election as the next Mayor of Jacksonville.
"Alvin Brown has shown himself to be the leader Jacksonville needs to move the city forward. He understands that quality public schools attract quality jobs, which is why Alvin received strong support from Democrats, Republicans and Independents.
"As Mayor, Alvin will continue working to move Jacksonville in a new direction, uniting the city behind his vision to improve the quality of life for all the residents of Jacksonville." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,708 |
Produced by Ted Garvin, Thomas Berger and the Online
Distributed Proofreading Team.
AMONG MY BOOKS
First Series
by JAMES RUSSELL LOWELL
* * * * *
To F.D.L.
Love comes and goes with music in his feet,
And tunes young pulses to his roundelays;
Love brings thee this: will it persuade thee, Sweet,
That he turns proser when he comes and stays?
* * * * *
CONTENTS.
DRYDEN
WITCHCRAFT
SHAKESPEARE ONCE MORE
NEW ENGLAND TWO CENTURIES AGO
LESSING
ROUSSEAU AND THE SENTIMENTALISTS
* * * * *
DRYDEN.[1]
Benvenuto Cellini tells us that when, in his boyhood, he saw a salamander
come out of the fire, his grandfather forthwith gave him a sound beating,
that he might the better remember so unique a prodigy. Though perhaps in
this case the rod had another application than the autobiographer chooses
to disclose, and was intended to fix in the pupil's mind a lesson of
veracity rather than of science, the testimony to its mnemonic virtue
remains. Nay, so universally was it once believed that the senses, and
through them the faculties of observation and retention, were quickened
by an irritation of the cuticle, that in France it was customary to whip
the children annually at the boundaries of the parish, lest the true
place of them might ever be lost through neglect of so inexpensive a
mordant for the memory. From this practice the older school of critics
would seem to have taken a hint for keeping fixed the limits of good
taste, and what was somewhat vaguely called _classical_ English. To mark
these limits in poetry, they set up as Hermae the images they had made to
them of Dryden, of Pope, and later of Goldsmith. Here they solemnly
castigated every new aspirant in verse, who in turn performed the same
function for the next generation, thus helping to keep always sacred and
immovable the _ne plus ultra_ alike of inspiration and of the vocabulary.
Though no two natures were ever much more unlike than those of Dryden and
Pope, and again of Pope and Goldsmith, and no two styles, except in such
externals as could be easily caught and copied, yet it was the fashion,
down even to the last generation, to advise young writers to form
themselves, as it was called, on these excellent models. Wordsworth
himself began in this school; and though there were glimpses, here and
there, of a direct study of nature, yet most of the epithets in his
earlier pieces were of the traditional kind so fatal to poetry during
great part of the last century; and he indulged in that alphabetic
personification which enlivens all such words as Hunger, Solitude,
Freedom, by the easy magic of an initial capital.
"Where the green apple shrivels on the spray,
And pines the unripened pear in summer's kindliest ray,
Even here Content has fixed her smiling reign
With Independence, child of high Disdain.
Exulting 'mid the winter of the skies,
Shy as the jealous chamois, Freedom flies,
And often grasps her sword, and often eyes."
Here we have every characteristic of the artificial method, even to the
triplet, which Swift hated so heartily as "a vicious way of rhyming
wherewith Mr. Dryden abounded, imitated by all the bad versifiers of
Charles the Second's reign." Wordsworth became, indeed, very early the
leader of reform; but, like Wesley, he endeavored a reform within the
Establishment. Purifying the substance, he retained the outward forms
with a feeling rather than conviction that, in poetry, substance and form
are but manifestations of the same inward life, the one fused into the
other in the vivid heat of their common expression. Wordsworth could
never wholly shake off the influence of the century into which he was
born. He began by proposing a reform of the ritual, but it went no
further than an attempt to get rid of the words of Latin original where
the meaning was as well or better given in derivatives of the Saxon. He
would have stricken out the "assemble" and left the "meet together." Like
Wesley, he might be compelled by necessity to a breach of the canon; but,
like him, he was never a willing schismatic, and his singing robes were
the full and flowing canonicals of the church by law established.
Inspiration makes short work with the usage of the best authors and
ready-made elegances of diction; but where Wordsworth is not possessed by
his demon, as Moliere said of Corneille, he equals Thomson in verbiage,
out-Miltons Milton in artifice of style, and Latinizes his diction beyond
Dryden. The fact was, that he took up his early opinions on instinct, and
insensibly modified them as he studied the masters of what may be called
the Middle Period of English verse.[2] As a young man, he disparaged
Virgil ("We talked a great deal of nonsense in those days," he said when
taken to task for it later in life); at fifty-nine he translated three
books of the Aeneid, in emulation of Dryden, though falling far short of
him in everything but closeness, as he seems, after a few years, to have
been convinced. Keats was the first resolute and wilful heretic, the true
founder of the modern school, which admits no cis-Elizabethan authority
save Milton, whose own English was formed upon those earlier models.
Keats denounced the authors of that style which came in toward the close
of the seventeenth century, and reigned absolute through the whole of the
eighteenth, as
"A schism,
Nurtured by foppery and barbarism,
... who went about
Holding a poor decrepit standard out,
Marked with most flimsy mottoes, and in large
The name of one Boileau!"
But Keats had never then[3] studied the writers of whom he speaks so
contemptuously, though he might have profited by so doing. Boileau would
at least have taught him that _flimsy_ would have been an apter epithet
for the _standard_ than for the mottoes upon it. Dryden was the author of
that schism against which Keats so vehemently asserts the claim of the
orthodox teaching it had displaced. He was far more just to Boileau, of
whom Keats had probably never read a word. "If I would only cross the
seas," he says, "I might find in France a living Horace and a Juvenal in
the person of the admirable Boileau, whose numbers are excellent, whose
expressions are noble, whose thoughts are just, whose language is pure,
whose satire is pointed, and whose sense is just. What he borrows from
the ancients he repays with usury of his own, in coin as good and almost
as universally valuable."[4]
Dryden has now been in his grave nearly a hundred and seventy years; in
the second class of English poets perhaps no one stands, on the whole, so
high as he; during his lifetime, in spite of jealousy, detraction,
unpopular politics, and a suspicious change of faith, his pre-eminence
was conceded; he was the earliest complete type of the purely literary
man, in the modern sense; there is a singular unanimity in allowing him a
certain claim to _greatness_ which would be denied to men as famous and
more read,--to Pope or Swift, for example; he is supposed, in some way or
other, to have reformed English poetry. It is now about half a century
since the only uniform edition of his works was edited by Scott. No
library is complete without him, no name is more familiar than his, and
yet it may be suspected that few writers are more thoroughly buried in
that great cemetery of the "British Poets." If contemporary reputation be
often deceitful, posthumous fame may be generally trusted, for it is a
verdict made up of the suffrages of the select men in succeeding
generations. This verdict has been as good as unanimous in favor of
Dryden. It is, perhaps, worth while to take a fresh observation of him,
to consider him neither as warning nor example, but to endeavor to make
out what it is that has given so lofty and firm a position to one of the
most unequal, inconsistent, and faulty writers that ever lived. He is a
curious example of what we often remark of the living, but rarely of the
dead,--that they get credit for what they might be quite as much as for
what they are,--and posterity has applied to him one of his own rules of
criticism, judging him by the best rather than the average of his
achievement, a thing posterity is seldom wont to do. On the losing side
in politics, it is true of his polemical writings as of Burke's,--whom in
many respects he resembles, and especially in that supreme quality of a
reasoner, that his mind gathers not only heat, but clearness and
expansion, by its own motion,--that they have won his battle for him in
the judgment of after times.
To us, looking back at him, he gradually becomes a singularly interesting
and even picturesque figure. He is, in more senses than one, in language,
in turn of thought, in style of mind, in the direction of his activity,
the first of the moderns. He is the first literary man who was also a man
of the world, as we understand the term. He succeeded Ben Jonson as the
acknowledged dictator of wit and criticism, as Dr. Johnson, after nearly
the same interval, succeeded him. All ages are, in some sense, ages of
transition; but there are times when the transition is more marked, more
rapid; and it is, perhaps, an ill fortune for a man of letters to arrive
at maturity during such a period, still more to represent in himself the
change that is going on, and to be an efficient cause in bringing it
about. Unless, like Goethe, he is of a singularly uncontemporaneous
nature, capable of being _tutta in se romita_, and of running parallel
with his time rather than being sucked into its current, he will be
thwarted in that harmonious development of native force which has so much
to do with its steady and successful application. Dryden suffered, no
doubt, in this way. Though in creed he seems to have drifted backward in
an eddy of the general current; yet of the intellectual movement of the
time, so far certainly as literature shared in it, he could say, with
Aeneas, not only that he saw, but that himself was a great part of it.
That movement was, on the whole, a downward one, from faith to
scepticism, from enthusiasm to cynicism, from the imagination to the
understanding. It was in a direction altogether away from those springs
of imagination and faith at which they of the last age had slaked the
thirst or renewed the vigor of their souls. Dryden himself recognized
that indefinable and gregarious influence which we call nowadays the
Spirit of the Age, when he said that "every Age has a kind of universal
Genius."[5] He had also a just notion of that in which he lived; for he
remarks, incidentally, that "all knowing ages are naturally sceptic and
not at all bigoted, which, if I am not much deceived, is the proper
character of our own."[6] It may be conceived that he was even painfully
half-aware of having fallen upon a time incapable, not merely of a great
poet, but perhaps of any poet at all; for nothing is so sensitive to the
chill of a sceptical atmosphere as that enthusiasm which, if it be not
genius, is at least the beautiful illusion that saves it from the
baffling quibbles of self-consciousness. Thrice unhappy he who, horn to
see things as they might be, is schooled by circumstances to see them as
people say they are,--to read God in a prose translation. Such was
Dryden's lot, and such, for a good part of his days, it was by his own
choice. He who was of a stature to snatch the torch of life that flashes
from lifted hand to hand along the generations, over the heads of
inferior men, chose rather to be a link-boy to the stews.
As a writer for the stage, he deliberately adopted and repeatedly
reaffirmed the maxim that
"He who lives to please, must please to live."
Without earnest convictions, no great or sound literature is conceivable.
But if Dryden mostly wanted that inspiration which comes of belief in and
devotion to something nobler and more abiding than the present moment and
its petulant need, he had, at least, the next best thing to that,--a
thorough faith in himself. He was, moreover, a man of singularly open
soul, and of a temper self-confident enough to be candid even with
himself. His mind was growing to the last, his judgment widening and
deepening, his artistic sense refining itself more and more. He confessed
his errors, and was not ashamed to retrace his steps in search of that
better knowledge which the omniscience of superficial study had
disparaged. Surely an intellect that is still pliable at seventy is a
phenomenon as interesting as it is rare. But at whatever period of his
life we look at Dryden, and whatever, for the moment, may have been his
poetic creed, there was something in the nature of the man that would not
be wholly subdued to what it worked in. There are continual glimpses of
something in him greater than he, hints of possibilities finer than
anything he has done. You feel that the whole of him was better than any
random specimens, though of his best, seem to prove. _Incessu patet_, he
has by times the large stride of the elder race, though it sinks too
often into the slouch of a man who has seen better days. His grand air
may, in part, spring from a habit of easy superiority to his competitors;
but must also, in part, be ascribed to an innate dignity of character.
That this pre-eminence should have been so generally admitted, during his
life, can only be explained by a bottom of good sense, kindliness, and
sound judgment, whose solid worth could afford that many a flurry of
vanity, petulance, and even error should flit across the surface and be
forgotten. Whatever else Dryden may have been, the last and abiding
impression of him is, that he was thoroughly manly; and while it may be
disputed whether he was a great poet, it may be said of him, as
Wordsworth said of Burke, that "he was by far the greatest man of his
age, not only abounding in knowledge himself, but feeding, in various
directions, his most able contemporaries."[7]
Dryden was born in 1631. He was accordingly six years old when Jonson
died, was nearly a quarter of a century younger than Milton, and may have
personally known Bishop Hall, the first English satirist, who was living
till 1656. On the other side, he was older than Swift by thirty-six, than
Addison by forty-one, and than Pope by fifty-seven years. Dennis says
that "Dryden, for the last ten years of his life, was much acquainted
with Addison, and drank with him more than he ever used to do, probably
so far as to hasten his end," being commonly "an extreme sober man." Pope
tell us that, in his twelfth year, he "saw Dryden," perhaps at Will's,
perhaps in the street, as Scott did Burns. Dryden himself visited Milton
now and then, and was intimate with Davenant, who could tell him of
Fletcher and Jonson from personal recollection. Thus he stands between
the age before and that which followed him, giving a hand to each. His
father was a country clergyman, of Puritan leanings, a younger son of an
ancient county family. The Puritanism is thought to have come in with the
poet's great-grandfather, who made in his will the somewhat singular
statement that he was "assured by the Holy Ghost that he was elect of
God." It would appear from this that Dryden's self-confidence was an
inheritance. The solid quality of his mind showed itself early. He
himself tells us that he had read Polybius "in English, with the pleasure
of a boy, before he was ten years of age, and yet even then _had some
dark notions of the prudence with which he conducted his design_."[8] The
concluding words are very characteristic, even if Dryden, as men commonly
do, interpreted his boyish turn of mind by later self-knowledge. We thus
get a glimpse of him browsing--for, like Johnson, Burke, and the full as
distinguished from the learned men, he was always a random reader[9]--in
his father's library, and painfully culling here and there a spray of his
own proper nutriment from among the stubs and thorns of Puritan divinity.
After such schooling as could be had in the country, he was sent up to
Westminster School, then under the headship of the celebrated Dr. Busby.
Here he made his first essays in verse, translating, among other school
exercises of the same kind, the third satire of Persius. In 1650 he was
entered at Trinity College, Cambridge, and remained there for seven
years. The only record of his college life is a discipline imposed, in
1652, for "disobedience to the Vice-Master, and contumacy in taking his
punishment, inflicted by him." Whether this punishment was corporeal, as
Johnson insinuates in the similar case of Milton, we are ignorant. He
certainly retained no very fond recollection of his Alma Mater, for in
his "Prologue to the University of Oxford," he says:--
"Oxford to him a dearer name shall be
Than his own mother university;
Thebes did his green, unknowing youth engage,
He chooses Athens in his riper age."
By the death of his father, in 1654, he came into possession of a small
estate of sixty pounds a year, from which, however, a third must be
deducted, for his mother's dower, till 1676. After leaving Cambridge, he
became secretary to his near relative, Sir Gilbert Pickering, at that
time Cromwell's chamberlain, and a member of his Upper House. In 1670 he
succeeded Davenant as Poet Laureate,[10] and Howell as Historiographer,
with a yearly salary of two hundred pounds. This place he lost at the
Revolution, and had the mortification to see his old enemy and butt,
Shadwell, promoted to it, as the best poet the Whig party could muster.
If William was obliged to read the verses of his official minstrel,
Dryden was more than avenged. From 1688 to his death, twelve years later,
he earned his bread manfully by his pen, without any mean complaining,
and with no allusion to his fallen fortunes that is not dignified and
touching. These latter years, during which he was his own man again, were
probably the happiest of his life. In 1664 or 1665 he married Lady
Elizabeth Howard, daughter of the Earl of Berkshire. About a hundred
pounds a year were thus added to his income. The marriage is said not to
have been a happy one, and perhaps it was not, for his wife was
apparently a weak-minded woman; but the inference from the internal
evidence of Dryden's plays, as of Shakespeare's, is very untrustworthy,
ridicule of marriage having always been a common stock in trade of the
comic writers.
The earliest of his verses that have come down to us were written upon
the death of Lord Hastings, and are as bad as they can be,--a kind of
parody on the worst of Donne. They have every fault of his manner,
without a hint of the subtile and often profound thought that more than
redeems it. As the Doctor himself would have said, here is Donne outdone.
The young nobleman died of the small-pox, and Dryden exclaims
pathetically,--
"Was there no milder way than the small-pox,
The very filthiness of Pandora's box?"
He compares the pustules to "rosebuds stuck i' the lily skin about," and
says that
"Each little pimple had a tear in it
To wail the fault its rising did commit."
But he has not done his worst yet, by a great deal. What follows is even
finer:--
"No comet need foretell his change drew on,
Whose corpse might seem a constellation.
O, had he died of old, how great a strife
Had been who from his death should draw their life!
Who should, by one rich draught, become whate'er
Seneca, Cato, Numa, Caesar, were,
Learned, virtuous, pious, great, and have by this
An universal metempsychosis!
Must all these aged sires in one funeral
Expire? all die in one so young, so small?"
It is said that one of Allston's early pictures was brought to him, after
he had long forgotten it, and his opinion asked as to the wisdom of the
young artist's persevering in the career he had chosen. Allston advised
his quitting it forthwith as hopeless. Could the same experiment have
been tried with these verses upon Dryden, can any one doubt that his
counsel would have been the same? It should be remembered, however, that
he was barely turned eighteen when they were written, and the tendency of
his style is noticeable in so early an abandonment of the participial
_ed_ in _learned_ and _aged_. In the next year he appears again in some
commendatory verses prefixed to the sacred epigrams of his friend, John
Hoddesdon. In these he speaks of the author as a
"Young eaglet, who, thy nest thus soon forsook,
So lofty and divine a course hast took
As all admire, before the down begin
To peep, as yet, upon thy smoother chin."
Here is almost every fault which Dryden's later nicety would have
condemned. But perhaps there is no schooling so good for an author as his
own youthful indiscretions. After this effort Dryden seems to have lain
fallow for ten years, and then he at length reappears in thirty-seven
"heroic stanzas" on the death of Cromwell. The versification is smoother,
but the conceits are there again, though in a milder form. The verse is
modelled after "Gondibert." A single image from nature (he was almost
always happy in these) gives some hint of the maturer Dryden:--
"And wars, like mists that rise against the sun,
Made him but greater seem, not greater grow."
Two other verses,
"And the isle, when her protecting genius went,
Upon his obsequies loud sighs conferred,"
are interesting, because they show that he had been studying the early
poems of Milton. He has contrived to bury under a rubbish of verbiage one
of the most purely imaginative passages ever written by the great Puritan
poet.
"From haunted spring and dale,
Edged with poplar pale,
The parting genius is with sighing sent."
This is the more curious because, twenty-four years afterwards, he says,
in defending rhyme: "Whatever causes he [Milton] alleges for the
abolishment of rhyme, his own particular reason is plainly this, that
rhyme was not his talent; he had neither the ease of doing it nor the
graces of it: which is manifest in his _Juvenilia_, ... where his rhyme
is always constrained and forced, and comes hardly from him, at an age
when the soul is most pliant, and the passion of love makes almost every
man a rhymer, though not a poet."[11] It was this, no doubt, that
heartened Dr. Johnson to say of "Lycidas" that "the diction was harsh,
the rhymes uncertain, and the numbers unpleasing." It is Dryden's excuse
that his characteristic excellence is to argue persuasively and
powerfully, whether in verse or prose, and that he was amply endowed with
the most needful quality of an advocate,--to be always strongly and
wholly of his present way of thinking, whatever it might be. Next we
have, in 1660, "Astraea Redux" on the "happy restoration" of Charles II.
In this also we can forebode little of the full-grown Dryden but his
defects. We see his tendency to exaggeration, and to confound physical
with metaphysical, as where he says of the ships that brought home the
royal brothers, that
"The joyful London meets
The princely York, himself alone a freight,
The Swiftsure groans beneath great Gloster's weight"
and speaks of the
"Repeated prayer
Which stormed the skies and ravished Charles from thence."
There is also a certain everydayness, not to say vulgarity, of phrase,
which Dryden never wholly refined away, and which continually tempts us
to sum up at once against him as the greatest poet that ever was or could
be made wholly out of prose.
"Heaven would no bargain for its blessings drive"
is an example. On the other hand, there are a few verses almost worthy of
his best days, as these:--
"Some lazy ages lost in sleep and ease,
No action leave to busy chronicles;
Such whose _supine felicity_ but makes
In story chasms, in epochas mistakes,
O'er whom Time gently shakes his wings of down,
Till with his silent sickle they are mown,"
These are all the more noteworthy, that Dryden, unless in argument, is
seldom equal for six lines together. In the poem to Lord Clarendon (1662)
there are four verses that have something of the "energy divine" for
which Pope praised his master.
"Let envy, then, those crimes within you see
From which the happy never must be free;
Envy that does with misery reside,
The joy and the revenge of ruined pride."
In his "Aurengzebe" (1675) there is a passage, of which, as it is a good
example of Dryden, I shall quote the whole, though my purpose aims mainly
at the latter verses:--
"When I consider life, 't is all a cheat;
Yet, fooled with Hope, men favor the deceit,
Trust on, and think to-morrow will repay;
To-morrow's falser than the former day,
Lies worse, and, while it says we shall be blest
With some new joys, cuts off what we possest.
Strange cozenage! none would live past years again,
Yet all hope pleasure in what yet remain,
And from the dregs of life think to receive
What the first sprightly running could not give.
I'm tired of waiting for this chymic gold
Which fools us young and beggars us when old."
The "first sprightly running" of Dryden's vintage was, it must be
confessed, a little muddy, if not beery; but if his own soil did not
produce grapes of the choicest flavor, he knew where they were to be had;
and his product, like sound wine, grew better the longer it stood upon
the lees. He tells us, evidently thinking of himself, that in a poet,
"from fifty to threescore, the balance generally holds even in our colder
climates, for he loses not much in fancy, and judgment, which is the
effect of observation, still increases. His succeeding years afford him
little more than the stubble of his own harvest, yet, if his constitution
be healthful, his mind may still retain a decent vigor, and the gleanings
of that of Ephraim, in comparison with others, will surpass the vintage
of Abiezer."[12] Since Chaucer, none of our poets has had a constitution
more healthful, and it was his old age that yielded the best of him. In
him the understanding was, perhaps, in overplus for his entire good
fortune as a poet, and that is a faculty among the earliest to mature. We
have seen him, at only ten years, divining the power of reason in
Polybius.[13] The same turn of mind led him later to imitate the French
school of tragedy, and to admire in Ben Jonson the most correct of
English poets. It was his imagination that needed quickening, and it is
very curious to trace through his different prefaces the gradual opening
of his eyes to the causes of the solitary pre-eminence of Shakespeare. At
first he is sensible of an attraction towards him which he cannot
explain, and for which he apologizes, as if it were wrong. But he feels
himself drawn more and more strongly, till at last he ceases to resist
altogether, and is forced to acknowledge that there is something in this
one man that is not and never was anywhere else, something not to be
reasoned about, ineffable, divine; if contrary to the rules, so much the
worse for _them_. It may be conjectured that Dryden's Puritan
associations may have stood in the way of his more properly poetic
culture, and that his early knowledge of Shakespeare was slight. He tells
us that Davenant, whom he could not have known before he himself was
twenty-seven, first taught him to admire the great poet. But even after
his imagination had become conscious of its prerogative, and his
expression had been ennobled by frequenting this higher society, we find
him continually dropping back into that _sermo pedestris_ which seems, on
the whole, to have been his more natural element. We always feel his
epoch in him, that he was the lock which let our language down from its
point of highest poetry to its level of easiest and most gently flowing
prose. His enthusiasm needs the contagion of other minds to arouse it;
but his strong sense, his command of the happy word, his wit, which is
distinguished by a certain breadth and, as it were, power of
generalization, as Pope's by keenness of edge and point, were his,
whether he would or no. Accordingly, his poetry is often best and his
verse more flowing where (as in parts of his version of the twenty-ninth
ode of the third book of Horace) he is amplifying the suggestions of
another mind.[14] Viewed from one side, he justifies Milton's remark of
him, that "he was a good rhymist, but no poet." To look at all sides, and
to distrust the verdict of a single mood, is, no doubt, the duty of a
critic. But how if a certain side be so often presented as to thrust
forward in the memory and disturb it in the effort to recall that total
impression (for the office of a critic is not, though often so
misunderstood, to say _guilty_ or _not guilty_ of some particular fact)
which is the only safe ground of judgment? It is the weight of the whole
man, not of one or the other limb of him, that we want. _Expende
Hannibalem_. Very good, but not in a scale capacious only of a single
quality at a time, for it is their union, and not their addition, that
assures the value of each separately. It was not this or that which gave
him his weight in council, his swiftness of decision in battle that
outran the forethought of other men,--it was Hannibal. But this prosaic
element in Dryden will force itself upon me. As I read him, I cannot help
thinking of an ostrich, to be classed with flying things, and capable,
what with leap and flap together, of leaving the earth for a longer or
shorter space, but loving the open plain, where wing and foot help each
other to something that is both flight and run at once. What with his
haste and a certain dash, which, according to our mood, we may call
florid or splendid, he seems to stand among poets where Rubens does among
painters,--greater, perhaps, as a colorist than an artist, yet great here
also, if we compare him with any but the first.
We have arrived at Dryden's thirty-second year, and thus far have found
little in him to warrant an augury that he was ever to be one of the
_great_ names in English literature, the most perfect type, that is, of
his class, and that class a high one, though not the highest. If Joseph
de Maistre's axiom, _Qui n'a pas vaincu a trente ans, ne vaincra jamais_,
were true, there would be little hope of him, for he has won no battle
yet. But there is something solid and doughty in the man, that can rise
from defeat, the stuff of which victories are made in due time, when we
are able to choose our position better, and the sun is at our back.
Hitherto his performances have been mainly of the _obbligato_ sort, at
which few men of original force are good, least of all Dryden, who had
always something of stiffness in his strength. Waller had praised the
living Cromwell in perhaps the manliest verses he ever wrote,--not _very_
manly, to be sure, but really elegant, and, on the whole, better than
those in which Dryden squeezed out melodious tears. Waller, who had also
made himself conspicuous as a volunteer Antony to the country squire
turned Caesar,
("With ermine clad and purple, let him hold
A royal sceptre made of Spanish gold,")
was more servile than Dryden in hailing the return of _ex officio_
Majesty. He bewails to Charles, in snuffling heroics,
"Our sorrow and our crime
To have accepted life so long a time,
Without you here."
A weak man, put to the test by rough and angry times, as Waller was, may
be pitied, but meanness is nothing but contemptible under any
circumstances. If it be true that "every conqueror creates a Muse,"
Cromwell was unfortunate. Even Milton's sonnet, though dignified, is
reserved if not distrustful. Marvell's "Horatian Ode," the most truly
classic in our language, is worthy of its theme. The same poet's Elegy,
in parts noble, and everywhere humanly tender, is worth more than all
Carlyle's biography as a witness to the gentler qualities of the hero,
and of the deep affection that stalwart nature could inspire in hearts of
truly masculine temper. As it is little known, a few verses of it may be
quoted to show the difference between grief that thinks of its object and
grief that thinks of its rhymes:--
"Valor, religion, friendship, prudence died
At once with him, and all that's good beside,
And we, death's refuse, nature's dregs, confined
To loathsome life, alas! are left behind.
Where we (so once we used) shall now no more,
To fetch day, press about his chamber-door,
No more shall hear that powerful language charm,
Whose force oft spared the labor of his arm,
No more shall follow where he spent the days
In war or counsel, or in prayer and praise.
* * * * *
I saw him dead; a leaden slumber lies,
And mortal sleep, over those wakeful eyes;
Those gentle rays under the lids were fled,
Which through his looks that piercing sweetness shed;
That port, which so majestic was and strong,
Loose and deprived of vigor stretched along,
All withered, all discolored, pale, and wan,
How much another thing! no more That Man!
O human glory! vain! O death! O wings!
O worthless world! O transitory things!
Yet dwelt that greatness in his shape decayed
That still, though dead, greater than Death he laid,
And, in his altered face, you something feign
That threatens Death he yet will live again."
Such verses might not satisfy Lindley Murray, but they are of that higher
mood which satisfies the heart. These couplets, too, have an energy
worthy of Milton's friend:--
"When up the armed mountains of Dunbar
He marched, and through deep Severn, ending war."
"Thee, many ages hence, in martial verse
Shall the English soldier, ere he charge, rehearse."
On the whole, one is glad that Dryden's panegyric on the Protector was so
poor. It was purely official verse-making. Had there been any feeling in
it, there had been baseness in his address to Charles. As it is, we may
fairly assume that he was so far sincere in both cases as to be thankful
for a chance to exercise himself in rhyme, without much caring whether
upon a funeral or a restoration. He might naturally enough expect that
poetry would have a better chance under Charles than under Cromwell, or
any successor with Commonwealth principles. Cromwell had more serious
matters to think about than verses, while Charles might at least care as
much about them as it was in his base good-nature to care about anything
but loose women and spaniels. Dryden's sound sense, afterwards so
conspicuous, shows itself even in these pieces, when we can get at it
through the tangled thicket of tropical phrase. But the authentic and
unmistakable Dryden first manifests himself in some verses addressed to
his friend Dr. Charlton in 1663. We have first his common sense which has
almost the point of wit, yet with a tang of prose:--
"The longest tyranny that ever swayed
Was that wherein our ancestors betrayed
Their freeborn reason to the Stagyrite,
And made his torch their universal light.
_So truth, while only one supplied the state,
Grew scarce and dear and yet sophisticate.
Still it was bought, like emp'ric wares or charms,
Hard words sealed up with Aristotle's arms_."
Then we have his graceful sweetness of fancy, where he speaks of the
inhabitants of the New World:--
"Guiltless men who danced away their time,
Fresh as their groves and happy as their clime."
And, finally, there is a hint of imagination where "mighty visions of the
Danish race" watch round Charles sheltered in Stonehenge after the battle
of Worcester. These passages might have been written by the Dryden whom
we learn to know fifteen years later. They have the advantage that he
wrote them to please himself. His contemporary, Dr. Heylin, said of
French cooks, that "their trade was not to feed the belly, but the
palate." Dryden was a great while in learning this secret, as available
in good writing as in cookery. He strove after it, but his thoroughly
English nature, to the last, would too easily content itself with serving
up the honest beef of his thought, without regard to daintiness of flavor
in the dressing of it.[15] Of the best English poetry, it might be said
that it is understanding aerated by imagination. In Dryden the solid part
too often refused to mix kindly with the leaven, either remaining lumpish
or rising to a hasty puffiness. Grace and lightness were with him much
more a laborious achievement than a natural gift, and it is all the more
remarkable that he should so often have attained to what seems such an
easy perfection in both. Always a hasty writer,[16] he was long in
forming his style, and to the last was apt to snatch the readiest word
rather than wait for the fittest. He was not wholly and unconsciously
poet, but a thinker who sometimes lost himself on enchanted ground and
was transfigured by its touch. This preponderance in him of the reasoning
over the intuitive faculties, the one always there, the other flashing in
when you least expect it, accounts for that inequality and even
incongruousness in his writing which makes one revise his judgment at
every tenth page. In his prose you come upon passages that persuade you
he is a poet, in spite of his verses so often turning state's evidence
against him as to convince you he is none. He is a prose-writer, with a
kind of Aeolian attachment. For example, take this bit of prose from the
dedication of his version of Virgil's Pastorals, 1694: "He found the
strength of his genius betimes, and was even in his youth preluding to
his Georgicks and his Aeneis. He could not forbear to try his wings,
though his pinions were not hardened to maintain a long, laborious
flight; yet sometimes they bore him to a pitch as lofty as ever he was
able to reach afterwards. But when he was admonished by his subject to
descend, he came down gently circling in the air and singing to the
ground, like a lark melodious in her mounting and continuing her song
till she alights, still preparing for a higher flight at her next sally,
and tuning her voice to better music." This is charming, and yet even
this wants the ethereal tincture that pervades the style of Jeremy
Taylor, making it, as Burke said of Sheridan's eloquence, "neither prose
nor poetry, but something better than either." Let us compare Taylor's
treatment of the same image: "For so have I seen a lark rising from his
bed of grass and soaring upwards, singing as he rises, and hopes to get
to heaven and climb above the clouds; but the poor bird was beaten back
by the loud sighings of an eastern wind, and his motion made irregular
and inconstant, descending more at every breath of the tempest than it
could recover by the libration and frequent weighing of his wings, till
the little creature was forced to sit down and pant, and stay till the
storm was over, and then it made a prosperous flight, and did rise and
sing as if it had learned music and motion of an angel as he passed
sometimes through the air about his ministries here below." Taylor's
fault is that his sentences too often smell of the library, but what an
open air is here! How unpremeditated it all seems! How carelessly he
knots each new thought, as it comes, to the one before it with an _and_,
like a girl making lace! And what a slidingly musical use he makes of the
sibilants with which our language is unjustly taxed by those who can only
make them hiss, not sing! There are twelve of them in the first twenty
words, fifteen of which are monsyllables. We notice the structure of
Dryden's periods, but this grows up as we read. It gushes, like the song
of the bird itself,--
"In profuse strains of unpremeditated art."
Let us now take a specimen of Dryden's bad prose from one of his poems. I
open the "Annus Mirabilis" at random, and hit upon this:--
'Our little fleet was now engaged so far,
That, like the swordfish in the whale, they fought.
The combat only seemed a civil war,
Till through their bowels we our passage wrought.'
Is this Dryden, or Sternhold, or Shadwell, those Toms who made him say
that "dulness was fatal to the name of Tom"? The natural history of
Goldsmith in the verse of Pye! His thoughts did not "voluntary move
harmonious numbers." He had his choice between prose and verse, and seems
to be poetical on second thought. I do not speak without book. He was
more than half conscious of it himself. In the same letter to Mrs.
Steward, just cited, he says, "I am still drudging on, always a poet and
never a good one"; and this from no mock-modesty, for he is always
handsomely frank in telling us whatever of his own doing pleased him.
This was written in the last year of his life, and at about the same time
he says elsewhere: "What judgment I had increases rather than diminishes,
and thoughts, such as they are, come crowding in so fast upon me that my
only difficulty is to choose or to reject, to run them into verse or to
give them the other harmony of prose; I have so long studied and
practised both, that they are grown into a habit and become familiar to
me."[17] I think that a man who was primarily a poet would hardly have
felt this equanimity of choice.
I find a confirmation of this feeling about Dryden in his early literary
loves. His taste was not an instinct, but the slow result of reflection
and of the manfulness with which he always acknowledged to himself his
own mistakes. In this latter respect few men deal so magnanimously with
themselves as he, and accordingly few have been so happily inconsistent.
_Ancora imparo_ might have served him for a motto as well as Michael
Angelo. His prefaces are a complete log of his life, and the habit of
writing them was a useful one to him, for it forced him to think with a
pen in his hand, which, according to Goethe, "if it do no other good,
keeps the mind from staggering about." In these prefaces we see his taste
gradually rising from Du Bartas to Spenser, from Cowley to Milton, from
Corneille to Shakespeare. "I remember when I was a boy," he says in his
dedication of the "Spanish Friar," 1681, "I thought inimitable Spenser a
mean poet in comparison of Sylvester's _Du Bartas_, and was rapt into an
ecstasy when I read these lines:--
'Now when the winter's keener breath began
To crystallize the Baltic ocean,
To glaze the lakes, to bridle up the floods,
And periwig with snow[18] the baldpate woods.'
I am much deceived if this be not abominable fustian." Swift, in his
"Tale of a Tub," has a ludicrous passage in this style: "Look on this
globe of earth, you will find it to be a very complete and fashionable
dress. What is that which some call _land_, but a fine coat faced with
green? or the _sea_, but a waistcoat of water-tabby? Proceed to the
particular works of creation, you will find how curious journeyman Nature
has been to trim up the vegetable _beaux_; observe how _sparkish a
periwig adorns the head of a beech_, and what a fine doublet of white
satin is worn by the birch." The fault is not in any inaptness of the
images, nor in the mere vulgarity of the things themselves, but in that
of the associations they awaken. The "prithee, undo this button" of Lear,
coming where it does and expressing what it does, is one of those touches
of the pathetically sublime, of which only Shakespeare ever knew the
secret. Herrick, too, has a charming poem on "Julia's petticoat," the
charm being that he lifts the familiar and the low to the region of
sentiment. In the passage from Sylvester, it is precisely the reverse,
and the wig takes as much from the sentiment as it adds to a Lord
Chancellor. So Pope's proverbial verse,
"True wit is Nature to advantage drest,"
unpleasantly suggests Nature under the hands of a lady's-maid.[19] We
have no word in English that will exactly define this want of propriety
in diction. _Vulgar_ is too strong, and _commonplace_ too weak. Perhaps
_bourgeois_ comes as near as any. It is to be noticed that Dryden does
not unequivocally condemn the passage he quotes, but qualifies it with an
"if I am not much mistaken." Indeed, though his judgment in substantials,
like that of Johnson, is always worth having, his taste, the negative
half of genius, never altogether refined itself from a colloquial
familiarity, which is one of the charms of his prose, and gives that air
of easy strength in which his satire is unmatched. In his "Royal Martyr"
(1669), the tyrant Maximin says to the gods:--
"Keep you your rain and sunshine in the skies,
And I'll keep back my flame and sacrifice;
_Your trade of Heaven shall soon be at a stand,
And all your goods lie dead upon your hand,_"--
a passage which has as many faults as only Dryden was capable of
committing, even to a false idiom forced by the last rhyme. The same
tyrant in dying exclaims:--
"And after thee I'll go,
Revenging still, and following e'en to th' other world my blow,
And, _shoving back this earth on which I sit,
I'll mount and scatter all the gods I hit._"
In the "Conquest of Grenada" (1670), we have:--
"This little loss in our vast body shews
So small, that half _have never heard the news;
Fame's out of breath e'er she can fly so far
To tell 'em all that you have e'er made war_."[20]
And in the same play,
"That busy thing,
_The soul, is packing up_, and just on wing
Like parting swallows when they seek the spring,"
where the last sweet verse curiously illustrates that inequality (poetry
on a prose background) which so often puzzles us in Dryden. Infinitely
worse is the speech of Almanzor to his mother's ghost:--
"I'll rush into the covert of the night
And pull thee backward by the shroud to light,
Or else I'll squeeze thee like a bladder there,
And make thee groan thyself away to air."
What wonder that Dryden should have been substituted for Davenant as the
butt of the "Rehearsal," and that the parody should have had such a run?
And yet it was Dryden who, in speaking of Persius, hit upon the happy
phrase of "boisterous metaphors";[21] it was Dryden who said of Cowley,
whom he elsewhere calls "the darling of my youth,"[22] that he was "sunk
in reputation because he could never forgive any conceit which came in
his way, but swept, like a drag-net, great and small."[23] But the
passages I have thus far cited as specimens of our poet's coarseness (for
poet he surely was _intus_, though not always _in cute_) were written
before he was forty, and he had an odd notion, suitable to his healthy
complexion, that poets on the whole improve after that date. Man at
forty, he says, "seems to be fully in his summer tropic, ... and I
believe that it will hold in all great poets that, though they wrote
before with a certain heat of genius which inspired them, yet that heat
was not perfectly digested."[24] But artificial heat is never to be
digested at all, as is plain in Dryden's case. He was a man who warmed
slowly, and, in his hurry to supply the market, forced his mind. The
result was the same after forty as before. In "Oedipus" (1679) we find,
"Not one bolt
Shall err from Thebes, but more be called for, more,
_New-moulded thunder of a larger size!_"
This play was written in conjunction with Lee, of whom Dryden relates[25]
that, when some one said to him, "It is easy enough to write like a
madman," he replied, "No, it is hard to write like a madman, but easy
enough to write like a fool,"--perhaps the most compendious lecture on
poetry ever delivered. The splendid bit of eloquence, which has so much
the sheet-iron clang of impeachment thunder (I hope that Dryden is not in
the Library of Congress!) is perhaps Lee's. The following passage almost
certainly is his:--
"Sure 'tis the end of all things! Fate has torn
The lock of Time off, and his head is now
The ghastly ball of round Eternity!"
But the next, in which the soul is likened to the pocket of an indignant
housemaid charged with theft, is wholly in Dryden's manner:--
"No; I dare challenge heaven to turn me outward,
And shake my soul quite empty in your sight."
In the same style, he makes his Don Sebastian (1690) say that he is as
much astonished as "drowsy mortals" at the last trump,
"When, called in haste, _they fumble for their limbs_,"
and propose to take upon himself the whole of a crime shared with another
by asking Heaven _to charge the bill_ on him. And in "King Arthur,"
written ten years after the Preface from which I have quoted his
confession about Dubartas, we have a passage precisely of the kind he
condemned:--
"Ah for the many souls as but this morn
Were clothed with flesh and warmed with vital blood,
But naked now, or _shirted_ but with air."
Dryden too often violated his own admirable rule, that "an author is not
to write all he can, but only all he ought."[26] In his worst images,
however, there is often a vividness that half excuses them. But it is a
grotesque vividness, as from the flare of a bonfire. They do not flash
into sudden lustre, as in the great poets, where the imaginations of poet
and reader leap toward each other and meet half-way.
English prose is indebted to Dryden for having freed it from the cloister
of pedantry. He, more than any other single writer, contributed, as well
by precept as example, to give it suppleness of movement and the easier
air of the modern world. His own style, juicy with proverbial phrases,
has that familiar dignity, so hard to attain, perhaps unattainable except
by one who, like Dryden, feels that his position is assured. Charles
Cotton is as easy, but not so elegant; Walton as familiar, but not so
flowing; Swift as idiomatic, but not so elevated; Burke more splendid,
but not so equally luminous. That his style was no easy acquisition
(though, of course, the aptitude was innate) he himself tells us. In his
dedication of "Troilus and Cressida" (1679), where he seems to hint at
the erection of an Academy, he says that "the perfect knowledge of a
tongue was never attained by any single person. The Court, the College,
and the Town must all be joined in it. And as our English is a
composition of the dead and living tongues, there is required a perfect
knowledge, not only of the Greek and Latin, but of the Old German,
French, and Italian, and to help all these, a conversation with those
authors of our own who have written with the fewest faults in prose and
verse. But how barbarously we yet write and speak your Lordship knows,
and I am sufficiently sensible in my own English.[27] For I am often put
to a stand in considering whether what I write be the idiom of the
tongue, or false grammar and nonsense couched beneath that specious name
of _Anglicism_, and have no other way to clear my doubts but by
translating my English into Latin, and thereby trying what sense the
words will bear in a more stable language." _Tantae molis erat_. Five
years later: "The proprieties and delicacies of the English are known to
few; it is impossible even for a good wit to understand and practise them
without the help of a liberal education, long reading and digesting of
those few good authors we have amongst us, the knowledge of men and
manners, _the freedom of habitudes and conversation with the best company
of both sexes_, and, in short, without wearing off the rust which he
contracted while he was laying in a stock of learning." In the passage I
have italicized, it will be seen that Dryden lays some stress upon the
influence of women in refining language. Swift, also, in his plan for an
Academy, says: "Now, though I would by no means give the ladies the
trouble of advising us in the reformation of our language, yet I cannot
help thinking that, since they have been left out of all meetings except
parties at play, or where worse designs are carried on, our conversation
has very much degenerated."[28] Swift affirms that the language had grown
corrupt since the Restoration, and that "the Court, which used to be the
standard of propriety and correctness of speech, was then, and, I think,
has ever since continued, the worst school in England."[29] He lays the
blame partly on the general licentiousness, partly upon the French
education of many of Charles's courtiers, and partly on the poets. Dryden
undoubtedly formed his diction by the usage of the Court. The age was a
very free-and-easy, not to say a very coarse one. Its coarseness was not
external, like that of Elizabeth's day, but the outward mark of an inward
depravity. What Swift's notion of the refinement of women was may be
judged by his anecdotes of Stella. I will not say that Dryden's prose did
not gain by the conversational elasticity which his frequenting men and
women of the world enabled him to give it. It is the best specimen of
every-day style that we have. But the habitual dwelling of his mind in a
commonplace atmosphere, and among those easy levels of sentiment which
befitted Will's Coffee-house and the Bird-cage Walk, was a damage to his
poetry. Solitude is as needful to the imagination as society is wholesome
for the character. He cannot always distinguish between enthusiasm and
extravagance when he sees them. But apart from these influences which I
have adduced in exculpation, there was certainly a vein of coarseness in
him, a want of that exquisite sensitiveness which is the conscience of
the artist. An old gentleman, writing to the Gentleman's Magazine in
1745, professes to remember "plain John Dryden (before he paid his court
with success to the great) in one uniform clothing of Norwich drugget. I
have eat tarts at the Mulberry Garden with him and Madam Reeve, when our
author advanced to a sword and Chadreux wig."[30] I always fancy Dryden
in the drugget, with wig, lace ruffles, and sword superimposed. It is the
type of this curiously incongruous man.
The first poem by which Dryden won a general acknowledgment of his power
was the "Annus Mirabilis," written in his thirty-seventh year. Pepys,
himself not altogether a bad judge, doubtless expresses the common
opinion when he says: "I am very well pleased this night with reading a
poem I brought home with me last night from Westminster Hall, of
Dryden's, upon the present war; a very good poem."[31] And a very good
poem, in some sort, it continues to be, in spite of its amazing
blemishes. We must always bear in mind that Dryden lived in an age that
supplied him with no ready-made inspiration, and that big phrases and
images are apt to be pressed into the service when great ones do not
volunteer. With this poem begins the long series of Dryden's prefaces, of
which Swift made such excellent, though malicious, fun that I cannot
forbear to quote it. "I do utterly disapprove and declare against that
pernicious custom of making the _preface_ a bill of fare to the book. For
I have always looked upon it as a high point of indiscretion in
monster-mongers and other retailers of strange sights to hang out a fair
picture over the door, drawn after the life, with a most eloquent
description underneath; this has saved me many a threepence.... Such is
exactly the fate at this time of _prefaces_.... This expedient was
admirable at first; our great Dryden has long carried it as far as it
would go, and with incredible success. He has often said to me in
confidence, 'that the world would never have suspected him to be so great
a poet, if he had not assured them so frequently, in his prefaces, that
it was impossible they could either doubt or forget it.' Perhaps it may
be so; however, I much fear his instructions have edified out of their
place, and taught men to grow wiser in certain points where he never
intended they should."[32] The _monster-mongers_ is a terrible thrust,
when we remember some of the comedies and heroic plays which Dryden
ushered in this fashion. In the dedication of the "Annus" to the city of
London is one of those pithy sentences of which Dryden is ever afterwards
so full, and which he lets fall with a carelessness that seems always to
deepen the meaning: "I have heard, indeed, of some virtuous persons who
have ended unfortunately, but never of any virtuous nation; Providence is
engaged too deeply when the cause becomes so general." In his "account"
of the poem in a letter to Sir Robert Howard he says: "I have chosen to
write my poem in quatrains or stanzas of four in alternate rhyme, because
I have ever judged them more noble and of greater dignity, both for the
sound and number, than any other verse in use amongst us.... The learned
languages have certainly a great advantage of us in not being tied to the
slavery of any rhyme.... But in this necessity of our rhymes, I have
always found the couplet verse most easy, though not so proper for this
occasion; for there the work is sooner at an end, every two lines
concluding the labor of the poet." A little further on: "They [the
French] write in alexandrines, or verses of six feet, such as amongst us
is the old translation of Homer by Chapman: all which, by lengthening
their chain,[33] makes the sphere of their activity the greater." I have
quoted these passages because, in a small compass, they include several
things characteristic of Dryden. "I have ever judged," and "I have always
found," are particularly so. If he took up an opinion in the morning, he
would have found so many arguments for it before night that it would seem
already old and familiar. So with his reproach of rhyme; a year or two
before he was eagerly defending it;[34] again a few years, and he will
utterly condemn and drop it in his plays, while retaining it in his
translations; afterwards his study of Milton leads him to think that
blank verse would suit the epic style better, and he proposes to try it
with Homer, but at last translates one book as a specimen, and behold, it
is in rhyme! But the charm of this great advocate is, that, whatever side
he was on, he could always find excellent reasons for it, and state them
with great force, and abundance of happy illustration. He is an exception
to the proverb, and is none the worse pleader than he is always pleading
his own cause. The blunder about Chapman is of a kind into which his
hasty temperament often betrayed him. He remembered that Chapman's
"Iliad" was in a long measure, concluded without looking that it was
alexandrine, and then attributes it generally to his "Homer." Chapman's
"Iliad" is done in fourteen-syllable verse, and his "Odyssee" in the very
metre that Dryden himself used in his own version,[35] I remark also what
he says of the couplet, that it was easy because the second verse
concludes the labor of the poet. And yet it was Dryden who found it hard
for that very reason. His vehement abundance refused those narrow banks,
first running over into a triplet, and, even then uncontainable, rising
to an alexandrine in the concluding verse. And I have little doubt that
it was the roominess, rather than the dignity, of the quatrain which led
him to choose it. As apposite to this, I may quote what he elsewhere says
of octosyllabic verse: "The thought can turn itself with greater ease in
a larger compass. When the rhyme comes too thick upon us, it straightens
the expression: we are thinking of the close, when we should be employed
in adorning the thought. It makes a poet giddy with turning in a space
too narrow for his imagination."[36]
Dryden himself, as was not always the case with him, was well satisfied
with his work. He calls it his best hitherto, and attributes his success
to the excellence of his subject, "incomparably the best he had ever had,
_excepting only the Royal Family_." The first part is devoted to the
Dutch war; the last to the fire of London. The martial half is infinitely
the better of the two. He altogether surpasses his model, Davenant. If
his poem lack the gravity of thought attained by a few stanzas of
"Gondibert," it is vastly superior in life, in picturesqueness, in the
energy of single lines, and, above all, in imagination. Few men have read
"Gondibert," and almost every one speaks of it, as commonly of the dead,
with a certain subdued respect. And it deserves respect as an honest
effort to bring poetry back to its highest office in the ideal treatment
of life. Davenant emulated Spenser, and if his poem had been as good as
his preface, it could still be read in another spirit than that of
investigation. As it is, it always reminds me of Goldsmith's famous
verse. It is remote, unfriendly, solitary, and, above all, slow. Its
shining passages, for there are such, remind one of distress-rockets sent
up at intervals from a ship just about to founder, and sadden rather than
cheer.[37]
The first part of the "Annus Mirabilis" is by no means clear of the false
taste of the time,[38] though it has some of Dryden's manliest verses and
happiest comparisons, always his two distinguishing merits. Here, as
almost everywhere else in Dryden, measuring him merely as poet, we recall
what he, with pathetic pride, says of himself in the prologue to
"Aurengzebe":--
"Let him retire, betwixt two ages cast,
The first of this, the hindmost of the last."
What can be worse than what he says of comets?--
"Whether they unctuous exhalations are
Fired by the sun, or seeming so alone,
Or each some more remote and slippery star
Which loses footing when to mortals shown."
Or than this, of the destruction of the Dutch India ships?--
"Amidst whole heaps of spices lights a ball,
And now their odors armed against them fly;
Some preciously by shattered porcelain fall,
And some by aromatic splinters die."
Dear Dr. Johnson had his doubts about Shakespeare, but here at least was
poetry! This is one of the quatrains which he pronounces "worthy of our
author."[39]
But Dryden himself has said that "a man who is resolved to praise an
author with any appearance of justice must be sure to take him on the
strongest side, and where he is least liable to exceptions." This is true
also of one who wishes to measure an author fairly, for the higher wisdom
of criticism lies in the capacity to admire.
Leser, wie gefall ich dir?
Leser, wie gefaellst du mir?
are both fair questions, the answer to the first being more often
involved in that to the second than is sometimes thought. The poet in
Dryden was never more fully revealed than in such verses as these:--
"And threatening France, placed like a painted Jove,[40]
Kept idle thunder in his lifted hand";
"Silent in smoke of cannon they come on";
"And his loud guns speak thick, like angry men";
"The vigorous seaman every port-hole plies,
And adds his heart to every gun he fires";
"And, though to me unknown, they sure fought well,
Whom Rupert led, and who were British born."
This is masculine writing, and yet it must be said that there is scarcely
a quatrain in which the rhyme does not trip him into a platitude, and
there are too many swaggering with that _expression forte d'un sentiment
faible_ which Voltaire condemns in Corneille,--a temptation to which
Dryden always lay too invitingly open. But there are passages higher in
kind than any I have cited, because they show imagination. Such are the
verses in which he describes the dreams of the disheartened enemy:--
"In dreams they fearful precipices tread,
Or, shipwrecked, labor to some distant shore,
Or in dark churches walk among the dead";
and those in which he recalls glorious memories, and sees where
"The mighty ghosts of our great Harries rose,
And armed Edwards looked with anxious eyes."
A few verses, like the pleasantly alliterative one in which he makes the
spider, "from the silent ambush of his den," "feel far off the trembling
of his thread," show that he was beginning to study the niceties of
verse, instead of trusting wholly to what he would have called his
natural _fougue_. On the whole, this part of the poem is very good war
poetry, as war poetry goes (for there is but one first-rate poem of the
kind in English,--short, national, eager as if the writer were personally
engaged, with the rapid metre of a drum beating the charge,--and that is
Drayton's "Battle of Agincourt"),[41] but it shows more study of Lucan
than of Virgil, and for a long time yet we shall find Dryden bewildered
by bad models. He is always imitating--no, that is not the word, always
emulating--somebody in his more strictly poetical attempts, for in that
direction he always needed some external impulse to set his mind in
motion. This is more or less true of all authors; nor does it detract
from their originality, which depends wholly on their being able so far
to forget themselves as to let something of themselves slip into what
they write.[42] Of absolute originality we will not speak till authors
are raised by some Deucalion-and-Pyrrha process; and even then our faith
would be small, for writers who have no past are pretty sure of having no
future. Dryden, at any rate, always had to have his copy set him at the
top of the page, and wrote ill or well accordingly. His mind (somewhat
solid for a poet) warmed slowly, but, once fairly heated through, he had
more of that good-luck of self-oblivion than most men. He certainly gave
even a liberal interpretation to Moliere's rule of taking his own
property wherever he found it, though he sometimes blundered awkwardly
about what was properly _his_; but in literature, it should be
remembered, a thing always becomes his at last who says it best, and thus
makes it his own.[43]
Mr. Savage Landor once told me that he said to Wordsworth: "Mr.
Wordsworth, a man may mix poetry with prose as much as he pleases, and it
will only elevate and enliven; but the moment he mixes a particle of
prose with his poetry, it precipitates the whole." Wordsworth, he added,
never forgave him. The always hasty Dryden, as I think I have already
said, was liable, like a careless apothecary's 'prentice, to make the
same confusion of ingredients, especially in the more mischievous way. I
cannot leave the "Annus Mirabilis" without giving an example of this.
Describing the Dutch prizes, rather like an auctioneer than a poet, he
says that
"Some English wool, vexed in a Belgian loom,
And into cloth of spongy softness made,
Did into France or colder Denmark doom,
To ruin with worse ware our staple trade."
One might fancy this written by the secretary of a board of trade in an
unguarded moment; but we should remember that the poem is dedicated to
the city of London. The depreciation of the rival fabrics is exquisite;
and Dryden, the most English of our poets, would not be so thoroughly
English if he had not in him some fibre of _la nation boutiquiere_. Let
us now see how he succeeds in attempting to infuse science (the most
obstinately prosy material) with poetry. Speaking of "a more exact
knowledge of the longitudes," as he explains in a note, he tells us that,
"Then we upon our globe's last verge shall go,
And view the ocean leaning on the sky;
From thence our rolling neighbors we shall know,
And on the lunar world securely pry."
Dr. Johnson confesses that he does not understand this. Why should he,
when it is plain that Dryden was wholly in the dark himself! To
understand it is none of my business, but I confess that it interests me
as an Americanism. We have hitherto been credited as the inventors of the
"jumping-off place" at the extreme western verge of the world. But Dryden
was beforehand with us. Though he doubtless knew that the earth was a
sphere (and perhaps that it was flattened at the poles), it was always a
flat surface in his fancy. In his "Amphitryon," he makes Alcmena say:--
"No, I would fly thee to the ridge of earth,
And leap the precipice to 'scape thy sight."
And in his "Spanish Friar," Lorenzo says to Elvira that they "will travel
together to the ridge of the world, and then drop together into the
next." It is idle for us poor Yankees to hope that we can invent
anything. To say sooth, if Dryden had left nothing behind him but the
"Annus Mirabilis," he might have served as a type of the kind of poet
America would have produced by the biggest-river-and-tallest-mountain
recipe,--longitude and latitude in plenty, with marks of culture
scattered here and there like the _carets_ on a proof-sheet.
It is now time to say something of Dryden as a dramatist. In the
thirty-two years between 1662 and 1694 he produced twenty-five plays, and
assisted Lee in two. I have hinted that it took Dryden longer than most
men to find the true bent of his genius. On a superficial view, he might
almost seem to confirm that theory, maintained by Johnson, among others,
that genius was nothing more than great intellectual power exercised
persistently in some particular direction which chance decided, so that
it lay in circumstance merely whether a man should turn out a Shakespeare
or a Newton. But when we come to compare what he wrote, regardless of
Minerva's averted face, with the spontaneous production of his happier
muse, we shall be inclined to think his example one of the strongest
cases against the theory in question. He began his dramatic career, as
usual, by rowing against the strong current of his nature, and pulled
only the more doggedly the more he felt himself swept down the stream.
His first attempt was at comedy, and, though his earliest piece of that
kind (the "Wild Gallant," 1663) utterly failed, he wrote eight others
afterwards. On the 23d February, 1663, Pepys writes in his diary: "To
Court, and there saw the 'Wild Gallant' performed by the king's house;
but it was ill acted, and the play so poor a thing as I never saw in my
life almost, and so little answering the name, that, from the beginning
to the end, I could not, nor can at this time, tell certainly which was
the Wild Gallant. The king did not seem pleased at all the whole play,
nor anybody else." After some alteration, it was revived with more
success. On its publication in 1669 Dryden honestly admitted its former
failure, though with a kind of salvo for his self-love. "I made the town
my judges, and the greater part condemned it. After which I do not think
it my concernment to defend it with the ordinary zeal of a poet for his
decried poem, though Corneille is more resolute in his preface before
'Pertharite,'[44] which was condemned more universally than this.... Yet
it was received at Court, and was more than once the divertisement of his
Majesty, by his own command." Pepys lets us amusingly behind the scenes
in the matter of his Majesty's divertisement. Dryden does not seem to see
that in the condemnation of something meant to amuse the public there can
be no question of degree. To fail at all is to fail utterly.
"_Tous les genres sont permis, hors le genre ennuyeux._"
In the reading, at least, all Dryden's comic writing for the stage must
be ranked with the latter class. He himself would fain make an exception
of the "Spanish Friar," but I confess that I rather wonder at than envy
those who can be amused by it. His comedies lack everything that a comedy
should have,--lightness, quickness of transition, unexpectedness of
incident, easy cleverness of dialogue, and humorous contrast of character
brought out by identity of situation. The comic parts of the "Maiden
Queen" seem to me Dryden's best, but the merit even of these is
Shakespeare's, and there is little choice where even the best is only
tolerable. The common quality, however, of all Dryden's comedies is their
nastiness, the more remarkable because we have ample evidence that he was
a man of modest conversation. Pepys, who was by no means squeamish (for
he found "Sir Martin Marall" "the most entire piece of mirth ... that
certainly ever was writ ... very good wit therein, not fooling"), writes
in his diary of the 19th June, 1668: "My wife and Deb to the king's
play-house to-day, thinking to spy me there, and saw the new play
'Evening Love,' of Dryden's, which, though the world commends, she likes
not." The next day he saw it himself, "and do not like it, it being very
smutty, and nothing so good as the 'Maiden Queen' or the 'Indian Emperor'
of Dryden's making. _I was troubled at it_." On the 22d he adds: "Calling
this day at Herringman's,[45] he tells me Dryden do himself call it but a
fifth-rate play." This was no doubt true, and yet, though Dryden in his
preface to the play says, "I confess I have given [yielded] too much to
the people in it, and am ashamed for them as well as for myself, that I
have pleased them at so cheap a rate," he takes care to add, "not that
there is anything here that I would not defend to an ill-natured judge."
The plot was from Calderon, and the author, rebutting the charge of
plagiarism, tells us that the king ("without whose command they should no
longer be troubled with anything of mine") had already answered for him
by saying, "that he only desired that they who accused me of theft would
always steal him plays like mine." Of the morals of the play he has not a
word, nor do I believe that he was conscious of any harm in them till he
was attacked by Collier, and then, (with some protest against what he
considers the undue severity of his censor) he had the manliness to
confess that he had done wrong. "It becomes me not to draw my pen in the
defence of a bad cause, when I have so often drawn it for a good
one."[46] And in a letter to his correspondent, Mrs. Thomas, written only
a few weeks before his death, warning her against the example of Mrs.
Behn, he says, with remorseful sincerity: "I confess I am the last man in
the world who ought in justice to arraign her, who have been myself too
much a libertine in most of my poems, which I should be well contented I
had time either to purge or to see them fairly burned." Congreve was less
patient, and even Dryden, in the last epilogue he ever wrote, attempts an
excuse:--
"Perhaps the Parson stretched a point too far,
When with our Theatres he waged a war;
He tells you that this very moral age
Received the first infection from the Stage,
But sure a banished Court, with lewdness fraught,
The seeds of open vice returning brought.
* * * * *
Whitehall the naked Venus first revealed,
Who, standing, as at Cyprus, in her shrine,
The strumpet was adored with rites divine.
* * * * *
The poets, who must live by courts or starve,
Were proud so good a Government to serve,
And, mixing with buffoons and pimps profane,
Tainted the Stage for some small snip of gain."
Dryden least of all men should have stooped to this palliation, for he
had, not without justice, said of himself "The same parts and application
which have made me a poet might have raised me to any honors of the
gown." Milton and Marvell neither lived by the Court, nor starved.
Charles Lamb most ingeniously defends the Comedy of the Restoration as
"the sanctuary and quiet Alsatia of hunted casuistry," where there was no
pretence of representing a real world.[47] But this was certainly not so.
Dryden again and again boasts of the superior advantage which his age had
over that of the elder dramatists, in painting polite life, and
attributes it to a greater freedom of intercourse between the poets and
the frequenters of the Court.[48] We shall be less surprised at the
_kind_ of refinement upon which Dryden congratulated himself, when we
learn (from the dedication of "Marriage a la Mode") that the Earl of
Rochester was its exemplar: "The best comic writers of our age will join
with me to acknowledge that they have copied the gallantries of courts,
the delicacy of expression, and the decencies of behavior from your
Lordship." In judging Dryden, it should be borne in mind that for some
years he was under contract to deliver three plays a year, a kind of bond
to which no man should subject his brain who has a decent respect for the
quality of its products. We should remember, too, that in his day
_manners_ meant what we call _morals_, that custom always makes a larger
part of virtue among average men than they are quite aware, and that the
reaction from an outward conformity which had no root in inward faith may
for a time have given to the frank expression of laxity an air of honesty
that made it seem almost refreshing. There is no such hotbed for excess
of license as excess of restraint, and the arrogant fanaticism of a
single virtue is apt to make men suspicious of tyranny in all the rest.
But the riot of emancipation could not last long, for the more tolerant
society is of private vice, the more exacting will it be of public
decorum, that excellent thing, so often the plausible substitute for
things more excellent. By 1678 the public mind had so far recovered its
tone that Dryden's comedy of "Limberham" was barely tolerated for three
nights. I will let the man who looked at human nature from more sides,
and therefore judged it more gently than any other, give the only excuse
possible for Dryden:--
"Men's judgments are
A parcel of their fortunes, and things outward
Do draw the inward quality after them
To suffer all alike."
Dryden's own apology only makes matters worse for him by showing that he
committed his offences with his eyes wide open, and that he wrote
comedies so wholly in despite of nature as never to deviate into the
comic. Failing as clown, he did not scruple to take on himself the office
of Chiffinch to the palled appetite of the public. "For I confess my
chief endeavours are to delight the age in which I live. If the humour
of this be for low comedy, small accidents, and raillery, I will force my
genius to obey it, though with more reputation I could write in verse. I
know I am not so fitted by nature to write comedy; I want that gayety of
humour which is requisite to it. My conversation is slow and dull, my
humour saturnine and reserved: In short, I am none of those who endeavour
to break jests in company or make repartees. So that those who decry
my comedies do me no injury, except it be in point of profit: Reputation
in them is the last thing to which I shall pretend."[49] For my own part,
though I have been forced to hold my nose in picking my way through these
ordures of Dryden, I am free to say that I think them far less morally
mischievous than that _corps-de-ballet_ literature in which the most
animal of the passions is made more temptingly naked by a veil of French
gauze. Nor does Dryden's lewdness leave such a reek in the mind as the
filthy cynicism of Swift, who delighted to uncover the nakedness of our
common mother.
It is pleasant to follow Dryden into the more congenial region of heroic
plays, though here also we find him making a false start. Anxious to
please the king,[50] and so able a reasoner as to convince even himself
of the justice of whatever cause he argued, he not only wrote tragedies
in the French style, but defended his practice in an essay which is by
far the most delightful reproduction of the classic dialogue ever written
in English. Eugenius (Lord Buckhurst), Lisideius (Sir Charles Sidley),
Crites (Sir E. Howard), and Neander (Dryden) are the four partakers in
the debate. The comparative merits of ancients and moderns, of the
Shakespearian and contemporary drama, of rhyme and blank verse, the value
of the three (supposed) Aristotelian unities, are the main topics
discussed. The tone of the discussion is admirable, midway between
bookishness and talk, and the fairness with which each side of the
argument is treated shows the breadth of Dryden's mind perhaps better
than any other one piece of his writing. There are no men of straw set up
to be knocked down again, as there commonly are in debates conducted upon
this plan. The "Defence" of the Essay is to be taken as a supplement to
Neander's share in it, as well as many scattered passages in subsequent
prefaces and dedications. All the interlocutors agree that "the sweetness
of English verse was never understood or practised by our fathers," and
that "our poesy is much improved by the happiness of some writers yet
living, who first taught us to mould our thoughts into easy and
significant words, to retrench the superfluities of expression, and to
make our rhyme so properly a part of the verse that it should never
mislead the sense, but itself be led and governed by it." In another
place he shows that by "living writers" he meant Waller and Denham.
"Rhyme has all the advantages of prose besides its own. But the
excellence and dignity of it were never fully known till Mr. Waller
taught it: he first made writing easily an art; first showed us to
conclude the sense, most commonly in distiches, which in the verse before
him runs on for so many lines together that the reader is out of breath
to overtake it."[51] Dryden afterwards changed his mind, and one of the
excellences of his own rhymed verse is, that his sense is too ample to be
concluded by the distich. Rhyme had been censured as unnatural in
dialogue; but Dryden replies that it is no more so than blank verse,
since no man talks any kind of verse in real life. But the argument for
rhyme is of another kind. "I am satisfied if it cause delight, for
delight is the chief if not the only end of poesy [he should have said
_means_]; instruction can be admitted but in the second place, for poesy
only instructs as it delights.... The converse, therefore, which a poet
is to imitate must be heightened with all the arts and ornaments of
poesy, and must be such as, strictly considered, could never be supposed
spoken by any without premeditation.... Thus prose, though the rightful
prince, yet is by common consent deposed as too weak for the government
of serious plays, and, he failing, there now start up two competitors;
one the nearer in blood, which is blank verse; the other more fit for the
ends of government, which is rhyme. Blank verse is, indeed, the nearer
prose, but he is blemished with the weakness of his predecessor. Rhyme
(for I will deal clearly) has somewhat of the usurper in him; but he is
brave and generous, and his dominion pleasing."[52] To the objection that
the difficulties of rhyme will lead to circumlocution, he answers in
substance, that a good poet will know how to avoid them.
It is curious how long the superstition that Waller was the refiner of
English verse has prevailed since Dryden first gave it vogue. He was a
very poor poet and a purely mechanical versifier. He has lived mainly on
the credit of a single couplet,
"The soul's dark cottage, battered and decayed.
Lets in new light through chinks that Time hath made,"
in which the melody alone belongs to him, and the conceit, such as it is,
to Samuel Daniel, who said, long before, that the body's
"Walls, grown thin, permit the mind
To look out thorough and his frailty find."
Waller has made worse nonsense of it in the transfusion. It might seem
that Ben Jonson had a prophetic foreboding of him when he wrote: "Others
there are that have no composition at all, but a kind of tuning and
rhyming fall, in what they write. It runs and slides and only makes a
sound. Women's poets they are called, as you have women's tailors.
They write a verse as smooth, as soft, as cream
In which there is no torrent, nor scarce stream.
You may sound these wits and find the depth of them with your
middle-finger."[53] It seems to have been taken for granted by Waller, as
afterwards by Dryden, that our elder poets bestowed no thought upon their
verse. "Waller was smooth," but unhappily he was also flat, and his
importation of the French theory of the couplet as a kind of thought-coop
did nothing but mischief.[54] He never compassed even a smoothness
approaching this description of a nightingale's song by a third-rate poet
of the earlier school,--
"Trails her plain ditty in one long-spun note
Through the sleek passage of her open throat,
A clear, unwrinkled song,"--
one of whose beauties is its running over into the third verse. Those
poets indeed
"Felt music's pulse in all her arteries ";
and Dryden himself found out, when he came to try it, that blank verse
was not so easy a thing as he at first conceived it, nay, that it is the
most difficult of all verse, and that it must make up in harmony, by
variety of pause and modulation, for what it loses in the melody of
rhyme. In what makes the chief merit of his later versification, he but
rediscovered the secret of his predecessors in giving to rhymed
pentameters something of the freedom of blank verse, and not mistaking
metre for rhythm.
Voltaire, in his Commentary on Corneille, has sufficiently lamented the
awkwardness of movement imposed upon the French dramatists by the gyves
of rhyme. But he considers the necessity of overcoming this obstacle, on
the whole, an advantage. Difficulty is his tenth and superior muse. How
did Dryden, who says nearly the same thing, succeed in his attempt at the
French manner? He fell into every one of its vices, without attaining
much of what constitutes its excellence. From the nature of the language,
all French poetry is purely artificial, and its high polish is all that
keeps out decay. The length of their dramatic verse forces the French
into much tautology, into bombast in its original meaning, the stuffing
out a thought with words till it fills the line. The rigid system of
their rhyme, which makes it much harder to manage than in English, has
accustomed them to inaccuracies of thought which would shock them in
prose. For example, in the "Cinna" of Corneille, as originally written,
Emilie says to Augustus,--
"Ces flammes dans nos coeurs des longtemps etoient nees,
Et ce sont des secrets de plus de quatre annees."
I say nothing of the second verse, which is purely prosaic surplusage
exacted by the rhyme, nor of the jingling together of _ces, des, etoient,
nees, des,_ and _secrets_, but I confess that _nees_ does not seem to be
the epithet that Corneille would have chosen for _flammes_, if he could
have had his own way, and that flames would seem of all things the
hardest to keep secret. But in revising, Corneille changed the first
verse thus,--
"Ces flammes dans nos coeurs _sans votre ordre_ etoient nees."
Can anything be more absurd than flames born to order? Yet Voltaire, on
his guard against these rhyming pitfalls for the sense, does not notice
this in his minute comments on this play. Of extravagant metaphor, the
result of this same making sound the file-leader of sense, a single
example from "Heraclius" shall suffice:--
"La vapeur de mon sang ira grossir la foudre
Que Dieu tient deja prete a le reduire en poudre."
One cannot think of a Louis Quatorze Apollo except in a full-bottomed
periwig, and the tragic style of their poets is always showing the
disastrous influence of that portentous comet. It is the _style perruque_
in another than the French meaning of the phrase, and the skill lay in
dressing it majestically, so that, as Cibber says, "upon the head of a
man of sense, _if it became him_, it could never fail of drawing to him a
more partial regard and benevolence than could possibly be hoped for in
an ill-made one." It did not become Dryden, and he left it off.[55]
Like his own Zimri, Dryden was "all for" this or that fancy, till he took
up with another. But even while he was writing on French models, his
judgment could not be blinded to their defects. "Look upon the 'Cinna'
and the 'Pompey,' they are not so properly to be called plays as long
discourses of reason of State, and 'Polieucte' in matters of religion is
as solemn as the long stops upon our organs; ... their actors speak by
the hour-glass like our parsons.... I deny not but this may suit well
enough with the French, for as we, who are a more sullen people, come to
be diverted at our plays, so they, who are of an airy and gay temper,
come thither to make themselves more serious."[56] With what an air of
innocent unconsciousness the sarcasm is driven home! Again, while he was
still slaving at these bricks without straw, he says: "The present French
poets are generally accused that, wheresoever they lay the scene, or in
whatever age, the manners of their heroes are wholly French. Racine's
Bajazet is bred at Constantinople, but his civilities are conveyed to him
by some secret passage from Versailles into the Seraglio." It is curious
that Voltaire, speaking of the _Berenice_ of Racine, praises a passage in
it for precisely what Dryden condemns: "Il semble qu'on entende
_Henriette_ d'Angleterre elle-meme parlant au marquis de _Vardes_. La
politesse de la cour de _Louis XIV_., l'agrement de la langue Francaise,
la douceur de la versification la plus naturelle, le sentiment le plus
tendre, tout se trouve dans ce peu de vers." After Dryden had broken
away from the heroic style, he speaks out more plainly. In the Preface
to his "All for Love," in reply to some cavils upon "little, and not
essential decencies," the decision about which he refers to a master of
ceremonies, he goes on to say: "The French poets, I confess, are strict
observers of these punctilios; ... in this nicety of manners does the
excellency of French poetry consist. Their heroes are the most civil
people breathing, but their good breeding seldom extends to a word of
sense. All their wit is in their ceremony; they want the genius which
animates our stage, and therefore 't is but necessary, when they cannot
please, that they should take care not to offend.... They are so careful
not to exasperate a critic that they never leave him any work, ... for no
part of a poem is worth our discommending where the whole is insipid, as
when we have once tasted palled wine we stay not to examine it glass by
glass. But while they affect to shine in trifles, they are often careless
in essentials.... For my part, I desire to be tried by the laws of my own
country." This is said in heat, but it is plain enough that his mind was
wholly changed. In his discourse on epic poetry he is as decided, but
more temperate. He says that the French heroic verse "runs with more
activity than strength.[57] Their language is not strung with sinews like
our English; it has the nimbleness of a greyhound, but not the bulk and
body of a mastiff. Our men and our verses overbear them by their weight,
and _pondere, non numero_, is the British motto. The French have set up
purity for the standard of their language, and a masculine vigor is that
of ours. Like their tongue is the genius of their poets,--light and
trifling in comparison of the English."[58]
Dryden might have profited by an admirable saying of his own, that "they
who would combat general authority with particular opinion must first
establish themselves a reputation of understanding better than other
men." He understood the defects much better than the beauties of the
French theatre. Lessing was even more one-sided in his judgment upon
it.[59] Goethe, with his usual wisdom, studied it carefully without
losing his temper, and tried to profit by its structural merits. Dryden,
with his eyes wide open, copied its worst faults, especially its
declamatory sentiment. He should have known that certain things can never
be transplanted, and that among these is a style of poetry whose great
excellence was that it was in perfect sympathy with the genius of the
people among whom it came into being. But the truth is, that Dryden had
no aptitude whatever for the stage, and in writing for it he was
attempting to make a trade of his genius,--an arrangement from which the
genius always withdraws in disgust. It was easier to make loose thinking
and the bad writing which betrays it pass unobserved while the ear was
occupied with the sonorous music of the rhyme to which they marched.
Except in "All for Love," "the only play," he tells us, "which he wrote
to please himself,"[60] there is no trace of real passion in any of his
tragedies. This, indeed, is inevitable, for there are no characters, but
only personages, in any except that. That is, in many respects, a noble
play, and there are few finer scenes, whether in the conception or the
carrying out, than that between Antony and Ventidius in the first
act.[61]
As usual, Dryden's good sense was not blind to the extravagances of his
dramatic style. In "Mac Flecknoe" he makes his own Maximin the type of
childish rant,
"And little Maximins the gods defy";
but, as usual also, he could give a plausible reason for his own mistakes
by means of that most fallacious of all fallacies which is true so far as
it goes. In his Prologue to the "Royal Martyr" he says:--
"And he who servilely creeps after sense
Is safe, but ne'er will reach an excellence.
* * * * *
But, when a tyrant for his theme he had,
He loosed the reins and let his muse run mad,
And, though he stumbles in a full career,
Yet rashness is a better fault than fear;
* * * * *
They then, who of each trip advantage take,
Find out those faults which they want wit to make."
And in the Preface to the same play he tells us: "I have not everywhere
observed the equality of numbers in my verse, partly by reason of my
haste, but more especially because I _would not have my sense a slave to
syllables_." Dryden, when he had not a bad case to argue, would have had
small respect for the wit whose skill lay in the making of faults, and
has himself, where his self-love was not engaged, admirably defined the
boundary which divides boldness from rashness. What Quintilian says of
Seneca applies very aptly to Dryden: "Velles eum suo ingenio dixisse,
alieno judicio."[62] He was thinking of himself, I fancy, when he makes
Ventidius say of Antony,--
"He starts out wide
And bounds into a vice that bears him far
From his first course, and plunges him in ills;
But, when his danger makes him find his fault,
Quick to observe, and full of sharp remorse,
He censures eagerly his own misdeeds,
Judging himself with malice to himself,
And not forgiving what as man he did
Because his other parts are more than man."
But bad though they nearly all are as wholes, his plays contain passages
which only the great masters have surpassed, and to the level of which no
subsequent writer for the stage has ever risen. The necessity of rhyme
often forced him to a platitude, as where he says,--
"My love was blind to your deluding art,
But blind men feel when stabbed so near the heart."[63]
But even in rhyme he not seldom justifies his claim to the title of
"glorious John." In the very play from which I have just quoted are these
verses in his best manner:--
"No, like his better Fortune I'll appear,
With open arms, loose veil, and flowing hair,
Just flying forward from her rolling sphere."
His comparisons, as I have said, are almost always happy. This, from the
"Indian Emperor," is tenderly pathetic:--
"As callow birds,
Whose mother's killed in seeking of the prey,
Cry in their nest and think her long away,
And, at each leaf that stirs, each blast of wind,
Gape for the food which they must never find."
And this, of the anger with which the Maiden Queen, striving to hide her
jealousy, betrays her love, is vigorous:--
"Her rage was love, and its tempestuous flame,
Like lightning, showed the heaven from whence it came."
The following simile from the "Conquest of Grenada" is as well expressed
as it is apt in conception:--
"I scarcely understand my own intent;
But, silk-worm like, so long within have wrought,
That I am lost in my own web of thought."
In the "Rival Ladies," Angelina, walking in the dark, describes her
sensations naturally and strikingly:--
"No noise but what my footsteps make, and they
Sound dreadfully and louder than by day:
They double too, and every step I take
Sounds thick, methinks, and more than one could make."
In all the rhymed plays[64] there are many passages which one is rather
inclined to like than sure he would be right in liking them. The
following verses from "Aurengzebe" are of this sort:--
"My love was such it needed no return,
Rich in itself, like elemental fire,
Whose pureness does no aliment require."
This is Cowleyish, and _pureness_ is surely the wrong word; and yet it is
better than mere commonplace. Perhaps what oftenest turns the balance in
Dryden's favor, when we are weighing his claims as a poet, is his
persistent capability of enthusiasm. To the last he kindles, and
sometimes _almost_ flashes out that supernatural light which is the
supreme test of poetic genius. As he himself so finely and
characteristically says in "Aurengzebe," there was no period in his life
when it was not true of him that
"He felt the inspiring heat, the absent god return."
The verses which follow are full of him, and, with the exception of the
single word _underwent_, are in his luckiest manner:--
"One loose, one sally of a hero's soul,
Does all the military art control.
While timorous wit goes round, or fords the shore,
He shoots the gulf, and is already o'er,
And, when the enthusiastic fit is spent,
Looks back amazed at what he underwent."[65]
Pithy sentences and phrases always drop from Dryden's pen as if unawares,
whether in prose or verse. I string together a few at random:--
"The greatest argument for love is love."
"Few know the use of life before 't is past."
"Time gives himself and is not valued."
"Death in itself is nothing; but we fear
To be we know not what, we know not where."
"Love either finds equality or makes it;
Like death, he knows no difference in degrees."
"That's empire, that which I can give away."
"Yours is a soul irregularly great,
Which, wanting temper, yet abounds in heat."
"Forgiveness to the injured does belong,
But they ne'er pardon who have done the wrong."
"Poor women's thoughts are all extempore."
"The cause of love can never be assigned,
'T is in no face, but in the lover's mind."[66]
"Heaven can forgive a crime to penitence,
For Heaven can judge if penitence be true;
But man, who knows not hearts, should make examples."
"Kings' titles commonly begin by force,
Which time wears off and mellows into right."
"Fear's a large promiser; who subject live
To that base passion, know not what they give."
"The secret pleasure of the generous act
Is the great mind's great bribe."
"That bad thing, gold, buys all good things."
"Why, love does all that's noble here below."
"To prove religion true,
If either wit or sufferings could suffice,
All faiths afford the constant and the wise."
But Dryden, as he tells us himself,
"Grew weary of his long-loved mistress, Rhyme;
Passion's too fierce to be in fetters bound,
And Nature flies him like enchanted ground."
The finest things in his plays were written in blank verse, as vernacular
to him as the alexandrine to the French. In this he vindicates his claim
as a poet. His diction gets wings, and both his verse and his thought
become capable of a reach which was denied them when set in the stocks of
the couplet. The solid man becomes even airy in this new-found freedom:
Anthony says,
"How I loved,
Witness ye days and nights, and all ye hours
That _danced away with down upon your feet_."
And what image was ever more delicately exquisite, what movement more
fadingly accordant with the sense, than in the last two verses of the
following passage?
"I feel death rising higher still and higher,
Within my bosom; every breath I fetch
Shuts up my life within a shorter compass,
_And, like the vanishing sound of bells, grows less
And less each pulse, till it be lost in air_."[67]
Nor was he altogether without pathos, though it is rare with him. The
following passage seems to me tenderly full of it:--
"Something like
That voice, methinks, I should have somewhere heard;
But floods of woe have hurried it far off
Beyond my ken of soul."[68]
And this single verse from "Aurengzebe":--
"Live still! oh live! live even to be unkind!"
with its passionate eagerness and sobbing repetition, is worth a
ship-load of the long-drawn treacle of modern self-compassion.
Now and then, to be sure, we come upon something that makes us hesitate
again whether, after all, Dryden was not grandiose rather than great, as
in the two passages that next follow:--
"He looks secure of death, superior greatness,
Like Jove when he made Fate and said, Thou art
The slave of my creation."[69]
"I'm pleased with my own work; Jove was not more
With infant nature, when his spacious hand
Had rounded this huge ball of earth and seas,
To give it the first push and see it roll
Along the vast abyss."[70]
I should say that Dryden is more apt to dilate our fancy than our
thought, as great poets have the gift of doing. But if he have not the
potent alchemy that transmutes the lead of our commonplace associations
into gold, as Shakespeare knows how to do so easily, yet his sense is
always up to the sterling standard; and though he has not added so much
as some have done to the stock of bullion which others afterwards coin
and put in circulation, there are few who have minted so many phrases
that are still a part of our daily currency. The first line of the
following passage has been worn pretty smooth, but the succeeding ones
are less familiar:--
"Men are but children of a larger growth,
Our appetites as apt to change as theirs,
And full as craving too and full as vain;
And yet the soul, shut up in her dark room,
Viewing so clear abroad, at home sees nothing;
But, like a mole in earth, busy and blind,
Works all her folly up and casts it outward
In the world's open view."[71]
The image is mixed and even contradictory, but the thought obtains grace
for it. I feel as if Shakespeare would have written _seeing_ for
_viewing_, thus gaining the strength of repetition in one verse and
avoiding the sameness of it in the other. Dryden, I suspect, was not much
given to correction, and indeed one of the great charms of his best
writing is that everything seems struck off at a heat, as by a superior
man in the best mood of his talk. Where he rises, he generally becomes
fervent rather than imaginative; his thought does not incorporate itself
in metaphor, as in purely poetic minds, but repeats and reinforces itself
in simile. Where he _is_ imaginative, it is in that lower sense which the
poverty of our language, for want of a better word, compels us to call
_picturesque_, and even then he shows little of that finer instinct which
suggests so much more than it tells, and works the more powerfully as it
taxes more the imagination of the reader. In Donne's "Relic" there is an
example of what I mean. He fancies some one breaking up his grave and
spying
"A bracelet of bright hair about the bone,"--
a verse that still shines there in the darkness of the tomb, after two
centuries, like one of those inextinguishable lamps whose secret is
lost.[72] Yet Dryden sometimes showed a sense of this magic of a
mysterious hint, as in the "Spanish Friar":--
"No, I confess, you bade me not in words;
The dial spoke not, but it made shrewd signs,
And pointed full upon the stroke of murder."
This is perhaps a solitary example. Nor is he always so possessed by the
image in his mind as unconsciously to choose even the picturesquely
imaginative word. He has done so, however, in this passage from "Marriage
a la Mode":--
"You ne'er mast hope again to see your princess,
Except as prisoners view fair walks and streets,
And careless passengers going by their grates."
But after all, he is best upon a level, table-land, it is true, and a
very high level, but still somewhere between the loftier peaks of
inspiration and the plain of every-day life. In those passages where he
moralizes he is always good, setting some obvious truth in a new light by
vigorous phrase and happy illustration. Take this (from "Oedipus") as a
proof of it:--
"The gods are just,
But how can finite measure infinite?
Reason! alas, it does not know itself!
Yet man, vain man, would with his short-lined plummet
Fathom the vast abyss of heavenly justice.
Whatever is, is in its causes just,
Since all things are by fate. But purblind man
Sees but a part o' th' chain, the nearest links,
His eyes not carrying to that equal beam
That poises all above."
From the same play I pick an illustration of that ripened sweetness of
thought and language which marks the natural vein of Dryden. One cannot
help applying the passage to the late Mr. Quincy:--
"Of no distemper, of no blast he died,
But fell like autumn fruit that mellowed long,
E'en wondered at because he dropt no sooner;
Fate seemed to wind him up for fourscore years;
Yet freshly ran he on ten winters more,
Till, like a clock worn out with eating Time,
The wheels of weary life at last stood still."[73]
Here is another of the same kind from "All for Love":--
"Gone so soon!
Is Death no more? He used him carelessly,
With a familiar kindness; ere he knocked,
Ran to the door and took him in his arms,
As who should say, You're welcome at all hours,
A friend need give no warning."
With one more extract from the same play, which is in every way his best,
for he had, when he wrote it, been feeding on the bee-bread of
Shakespeare, I shall conclude. Antony says,
"For I am now so sunk from what I was,
Thou find'st me at my lowest water-mark.
The rivers that ran in and raised my fortunes
Are all dried up, or take another course:
What I have left is from my native spring;
I've a heart still that swells in scorn of Fate,
And lifts me to my banks."
This is certainly, from beginning to end, in what used to be called the
_grand_ style, at once noble and natural. I have not undertaken to
analyze any one of the plays, for (except in "All for Love") it would
have been only to expose their weakness. Dryden had _no_ constructive
faculty; and in every one of his longer poems that required a plot, the
plot is bad, always more or less inconsistent with itself, and rather
hitched-on to the subject than combining with it. It is fair to say,
however, before leaving this part of Dryden's literary work, that Horne
Tooke thought "Don Sebastian" "the best play extant."[74]
Gray admired the plays of Dryden, "not as dramatic compositions, but as
poetry."[75] "There are as many things finely said in his plays as almost
by anybody," said Pope to Spence. Of their rant, their fustian, their
bombast, their bad English, of their innumerable sins against Dryden's
own better conscience both as poet and critic, I shall excuse myself from
giving any instances.[76] I like what is good in Dryden so much, and it
_is_ so good, that I think Gray was justified in always losing his temper
when he heard "his faults criticised."[77]
It is as a satirist and pleader in verse that Dryden is best known, and
as both he is in some respects unrivalled. His satire is not so sly as
Chaucer's, but it is distinguished by the same good-nature. There is no
malice in it. I shall not enter into his literary quarrels further than
to say that he seems to me, on the whole, to have been forbearing, which
is the more striking as he tells us repeatedly that he was naturally
vindictive. It was he who called revenge "the darling attribute of
heaven." "I complain not of their lampoons and libels, though I have been
the public mark for many years. I am vindictive enough to have repelled
force by force, if I could imagine that any of them had ever reached me."
It was this feeling of easy superiority, I suspect, that made him the
mark for so much jealous vituperation. Scott is wrong in attributing his
onslaught upon Settle to jealousy because one of the latter's plays had
been performed at Court,--an honor never paid to any of Dryden's.[78] I
have found nothing like a trace of jealousy in that large and benignant
nature. In his vindication of the "Duke of Guise," he says, with honest
confidence in himself: "Nay, I durst almost refer myself to some of the
angry poets on the other side, whether I have not rather countenanced and
assisted their beginnings than hindered them from rising." He seems to
have been really as indifferent to the attacks on himself as Pope
pretended to be. In the same vindication he says of the "Rehearsal," the
only one of them that had any wit in it, and it has a great deal: "Much
less am I concerned at the noble name of Bayes; that's a brat so like his
own father that he cannot be mistaken for any other body. They might as
reasonably have called Tom Sternhold Virgil, and the resemblance would
have held as well." In his Essay on Satire he says: "And yet we know that
in Christian charity all offences are to be forgiven as we expect the
like pardon for those we daily commit against Almighty God. And this
consideration has often made me tremble when I was saying our Lord's
Prayer; for the plain condition of the forgiveness which we beg is the
pardoning of others the offences which they have done to us; for which
reason I have many times avoided the commission of that fault, even when
I have been notoriously provoked."[79] And in another passage he says,
with his usual wisdom: "Good sense and good-nature are never separated,
though the ignorant world has thought otherwise. Good-nature, by which I
mean beneficence and candor, is the product of right reason, which of
necessity will give allowance to the failings of others, by considering
that there is nothing perfect in mankind." In the same Essay he gives his
own receipt for satire: "How easy it is to call rogue and villain, and
that wittily! but how hard to make a man appear a fool, a blockhead, or a
knave, without using any of those opprobrious terms!... This is the
mystery of that noble trade.... Neither is it true that this fineness of
raillery is offensive: a witty man is tickled while he is hurt in this
manner, and a fool feels it not.... There is a vast difference between
the slovenly butchering of a man and the fineness of a stroke that
separates the head from the body, and leaves it standing in its place. A
man may be capable, as Jack Ketch's wife said of his servant, of a plain
piece of work, of a bare hanging; but to make a malefactor die sweetly
was only belonging to her husband. I wish I could apply it to myself, if
the reader would be kind enough to think it belongs to me. The character
of Zimri in my 'Absalom' is, in my opinion, worth the whole poem. It is
not bloody, but it is ridiculous enough, and he for whom it was intended
was too witty to resent it as an injury.... I avoided the mention of
great crimes, and applied myself to the representing of blind sides and
little extravagances, to which, the wittier a man is, he is genrally the
more obnoxious."
Dryden thought his genius led him that way. In his elegy on the satirist
Oldham, whom Hallam, without reading him, I suspect, ranks next to
Dryden,[80] he says:--
"For sure our souls were near allied, and thine
Cast in the same poetic mould with mine;
One common note in either lyre did strike,
And knaves and fools we both abhorred alike."
His practice is not always so delicate as his theory; but if he was
sometimes rough, he never took a base advantage. He knocks his antagonist
down, and there an end. Pope seems to have nursed his grudge, and then,
watching his chance, to have squirted vitriol from behind a corner,
rather glad than otherwise if it fell on the women of those he hated or
envied. And if Dryden is never dastardly, as Pope often was, so also he
never wrote anything so maliciously depreciatory as Pope's unprovoked
attack on Addison. Dryden's satire is often coarse, but where it is
coarsest, it is commonly in defence of himself against attacks that were
themselves brutal. Then, to be sure, he snatches the first ready cudgel,
as in Shadwell's case, though even then there is something of the
good-humor of conscious strength. Pope's provocation was too often the
mere opportunity to say a biting thing, where he could do it safely. If
his victim showed fight, he tried to smooth things over, as with Dennis.
Dryden could forget that he had ever had a quarrel, but he never slunk
away from any, least of all from one provoked by himself.[81] Pope's
satire is too much occupied with the externals of manners, habits,
personal defects, and peculiarities. Dryden goes right to the rooted
character of the man, to the weaknesses of his nature, as where he says
of Burnet:--
"Prompt to assail, and careless of defence,
Invulnerable in his impudence,
He dares the world, and, eager of a name,
He thrusts about and _justles into fame_.
So fond of loud report that, not to miss
Of being known (his last and utmost bliss),
_He rather would be known for what he is_."
It would be hard to find in Pope such compression of meaning as in the
first, or such penetrative sarcasm as in the second of the passages I
have underscored. Dryden's satire is still quoted for its
comprehensiveness of application, Pope's rather for the elegance of its
finish and the point of its phrase than for any deeper qualities.[82] I
do not remember that Dryden ever makes poverty a reproach.[83] He was
above it, alike by generosity of birth and mind. Pope is always the
_parvenu_, always giving himself the airs of a fine gentleman, and, like
Horace Walpole and Byron, affecting superiority to professional
literature. Dryden, like Lessing, was a hack-writer, and was proud, as an
honest man has a right to be, of being able to get his bread by his
brains. He lived in Grub Street all his life, and never dreamed that
where a man of genius lived was not the best quarter of the town. "Tell
his Majesty," said sturdy old Jonson, "that his soul lives in an alley."
Dryden's prefaces are a mine of good writing and judicious criticism. His
_obiter dicta_ have often the penetration, and always more than the
equity, of Voltaire's, for Dryden never loses temper, and never
altogether qualifies his judgment by his self-love. "He was a more
universal writer than Voltaire," said Horne Tooke, and perhaps it is true
that he had a broader view, though his learning was neither so extensive
nor so accurate. My space will not afford many extracts, but I cannot
forbear one or two. He says of Chaucer, that "he is a perpetual fountain
of good sense,"[84] and likes him better than Ovid,--a bold confession in
that day. He prefers the pastorals of Theocritus to those of Virgil.
"Virgil's shepherds are too well read in the philosophy of Epicurus and
of Plato"; "there is a kind of rusticity in all those pompous verses,
somewhat of a holiday shepherd strutting in his country buskins";[85]
"Theocritus is softer than Ovid, he touches the passions more delicately,
and performs all this out of his own fund, without diving into the arts
and sciences for a supply. Even his Doric dialect has an incomparable
sweetness in his clownishness, like a fair shepherdess, in her country
russet, talking in a Yorkshire tone."[86] Comparing Virgil's verse with
that of some other poets, he says, that his "numbers are perpetually
varied to increase the delight of the reader, so that the same sounds are
never repeated twice together. On the contrary, Ovid and Claudian, though
they write in styles different from each other, yet have each of them but
one sort of music in their verses. All the versification and little
variety of Claudian is included within the compass of four or five lines,
and then he begins again in the same tenor, perpetually closing his sense
at the end of a verse, and that verse commonly which they call golden, or
two substantives and two adjectives with a verb betwixt them to keep the
peace. Ovid, with all his sweetness, has as little variety of numbers and
sound as he; he is always, as it were, upon the hand-gallop, and his
verse runs upon carpet-ground."[87] What a dreary half-century would have
been saved to English poetry, could Pope have laid these sentences to
heart! Upon translation, no one has written so much and so well as Dryden
in his various prefaces. Whatever has been said since is either expansion
or variation of what he had said before. His general theory may be stated
as an aim at something between the literalness of metaphrase and the
looseness of paraphase. "Where I have enlarged," he says, "I desire the
false critics would not always think that those thoughts are wholly mine,
but either _they are secretly in the poet_, or may be fairly deduced from
him." Coleridge, with his usual cleverness of _assimilation_, has
condensed him in a letter to Wordsworth: "There is no medium between a
prose version and one on the avowed principle of _compensation_ in the
widest sense, i.e. manner, genius, total effect."[88]
I have selected these passages, not because they are the best, but
because they have a near application to Dryden himself. His own
characterization of Chaucer (though too narrow for the greatest but one
of English poets) is the best that could be given of himself: "He is a
perpetual fountain of good sense." And the other passages show him a
close and open-minded student of the art he professed. Has his influence
on our literature, but especially on our poetry, been on the whole for
good or evil? If he could have been read with the liberal understanding
which he brought to the works of others, I should answer at once that it
had been beneficial. But his translations and paraphrases, in some ways
the best things he did, were done, like his plays, under contract to
deliver a certain number of verses for a specified sum. The
versification, of which he had learned the art by long practice, is
excellent, but his haste has led him to fill out the measure of lines
with phrases that add only to dilute, and thus the clearest, the most
direct, the most manly versifier of his time became, without meaning it,
the source (_fons et origo malorum_) of that poetic diction from which
our poetry has not even yet recovered. I do not like to say it, but he
has sometimes smothered the childlike simplicity of Chaucer under
feather-beds of verbiage. What this kind of thing came to in the next
century, when everybody ceremoniously took a bushel-basket to bring a
wren's egg to market in, is only too sadly familiar. It is clear that his
natural taste led Dryden to prefer directness and simplicity of style. If
he was too often tempted astray by Artifice, his love of Nature betrays
itself in many an almost passionate outbreak of angry remorse. Addison
tells us that he took particular delight in the reading of our old
English ballads. What he valued above all things was Force, though in his
haste he is willing to make a shift with its counterfeit, Effect. As
usual, he had a good reason to urge for what he did: "I will not excuse,
but justify myself for one pretended crime for which I am liable to be
charged by false critics, not only in this translation, but in many of my
original poems,--that I Latinize too much. It is true that when I find an
English word significant and sounding, I neither borrow from the Latin or
any other language; but when I want at home I must seek abroad. If
sounding words are not of our growth and manufacture, who shall hinder me
to import them from a foreign country? I carry not out the treasure of
the nation which is never to return; but what I bring from Italy I spend
in England: here it remains, and here it circulates; for if the coin be
good, it will pass from one hand to another. I trade both with the living
and the dead for the enrichment of our native language. We have enough in
England to supply our necessity; but if we will have things of
magnificence and splendor, we must get them by commerce.... Therefore, if
I find a word in a classic author, I propose it to be naturalized by
using it myself, and if the public approve of it the bill passes. But
every man cannot distinguish betwixt pedantry and poetry; every man,
therefore, is not fit to innovate."[89] This is admirably said, and with
Dryden's accustomed penetration to the root of the matter. The Latin has
given us most of our canorous words, only they must not be confounded
with merely sonorous ones, still less with phrases that, instead of
supplementing the sense, encumber it. It was of Latinizing in this sense
that Dryden was guilty. Instead of stabbing, he "with steel invades the
life." The consequence was that by and by we have Dr. Johnson's poet,
Savage, telling us,--
"In front, a parlor meets my entering view,
Opposed a room to sweet refection due";
Dr. Blacklock making a forlorn maiden say of her "dear," who is out
late,--
"Or by some apoplectic fit deprest
Perhaps, alas! he seeks eternal rest";
and Mr. Bruce, in a Danish war-song, calling on the vikings to "assume
their oars." But it must be admitted of Dryden that he seldom makes the
second verse of a couplet the mere trainbearer to the first, as Pope was
continually doing. In Dryden the rhyme waits upon the thought; in Pope
and his school the thought courtesies to the tune for which it is
written.
Dryden has also been blamed for his gallicisms.[90] He tried some, it is
true, but they have not been accepted.
I do not think he added a single word to the language; unless, as I
suspect, he first used _magnetism_ in its present sense of moral
attraction. What he did in his best writing was to use the English as if
it were a spoken, and not merely an inkhorn language; as if it were his
own to do what he pleased with it, as if it need not be ashamed of
itself.[91]
In this respect, his service to our prose was greater than any other man
has ever rendered. He says he formed his style upon Tillotson's (Bossuet,
on the other hand, formed _his_ upon Corneille's); but I rather think he
got it at Will's, for its great charm is that it has the various freedom
of talk.[92] In verse, he had a pomp which, excellent in itself, became
pompousness in his imitators. But he had nothing of Milton's ear for
various rhythm and interwoven harmony. He knew how to give new
modulation, sweetness, and force to the pentameter; but in what used to
be called pindarics, I am heretic enough to think he generally failed.
His so much praised "Alexander's Feast" (in parts of it, at least) has no
excuse for its slovenly metre and awkward expression, but that it was
written for music. He himself tells us, in the epistle dedicatory to
"King Arthur," "that the numbers of poetry and vocal music are sometimes
so contrary, that in many places I have been obliged to cramp my verses
and make them ragged to the reader that they may be harmonious to the
hearer." His renowned ode suffered from this constraint, but this is no
apology for the vulgarity of conception in too many passages.[93]
Dryden's conversion to Romanism has been commonly taken for granted as
insincere, and has therefore left an abiding stain on his character,
though the other mud thrown at him by angry opponents or rivals brushed
off so soon as it was dry. But I think his change of faith susceptible of
several explanations, none of them in any way discreditable to him. Where
Church and State are habitually associated, it is natural that minds even
of a high order should unconsciously come to regard religion as only a
subtler mode of police.[94] Dryden, conservative by nature, had
discovered before Joseph de Maistre, that Protestantism, so long as it
justified its name by continuing to be an active principle, was the
abettor of Republicanism. I think this is hinted in more than one passage
in his preface to "The Hind and Panther." He may very well have preferred
Romanism because of its elder claim to authority in all matters of
doctrine, but I think he had a deeper reason in the constitution of his
own mind. That he was "naturally inclined to scepticism in philosophy,"
he tells us of himself in the preface to the "Religio Laici"; but he was
a sceptic with an imaginative side, and in such characters scepticism and
superstition play into each other's hands. This finds a curious
illustration in a letter to his sons, written four years before his
death: "Towards the latter end of this month, September, Charles will
begin to recover his perfect health, according to his Nativity, which,
casting it myself, I am sure is true, and all things hitherto have
happened accordingly to the very time that I predicted them." Have we
forgotten Montaigne's votive offerings at the shrine of Loreto?
Dryden was short of body, inclined to stoutness, and florid of
complexion. He is said to have had "a sleepy eye," but was handsome and
of a manly carriage. He "was not a very genteel man, he was intimate with
none but poetical men.[95] He was said to be a very good man by all that
knew him: he was as plump as Mr. Pitt, of a fresh color and a down look,
and not very conversible." So Pope described him to Spence. He still
reigns in literary tradition, as when at Will's his elbow-chair had the
best place by the fire in winter, or on the balcony in summer, and when a
pinch from his snuff-box made a young author blush with pleasure as would
now-a-days a favorable notice in the "Saturday Review." What gave and
secures for him this singular eminence? To put it in a single word, I
think that his qualities and faculties were in that rare combination
which makes character. This gave _flavor_ to whatever he wrote,--a very
rare quality.
Was he, then, a great poet? Hardly, in the narrowest definition. But he
was a strong thinker who sometimes carried common sense to a height where
it catches the light of a diviner air, and warmed reason till it had
wellnigh the illuminating property of intuition. Certainly he is not,
like Spenser, the poets' poet, but other men have also their rights. Even
the Philistine is a man and a brother, and is entirely right so far as he
sees. To demand more of him is to be unreasonable. And he sees, among
other things, that a man who undertakes to write should first have a
meaning perfectly defined to himself, and then should be able to set it
forth clearly in the best words. This is precisely Dryden's praise,[96]
and amid the rickety sentiment looming big through misty phrase which
marks so much of modern literature, to read him is as bracing as a
northwest wind. He blows the mind clear. In ripeness of mind and bluff
heartiness of expression, he takes rank with the best. His phrase is
always a short-cut to his sense, for his estate was too spacious for him
to need that trick of winding the path of his thought about, and planting
it out with clumps of epithet, by which the landscape-gardeners of
literature give to a paltry half-acre the air of a park. In poetry, to be
next-best is, in one sense, to be nothing; and yet to be among the first
in any kind of writing, as Dryden certainly was, is to be one of a very
small company. He had, beyond most, the gift of the right word. And if he
does not, like one or two of the greater masters of song, stir our
sympathies by that indefinable aroma so magical in arousing the subtile
associations of the soul, he has this in common with the few great
writers, that the winged seeds of his thought embed themselves in the
memory and germinate there. If I could be guilty of the absurdity of
recommending to a young man any author on whom to form his style, I
should tell him that, next to having something that will not stay unsaid,
he could find no safer guide than Dryden.
Cowper, in a letter to Mr. Unwin (5th January, 1782), expresses what I
think is the common feeling about Dryden, that, with all his defects, he
had that indefinable something we call Genius. "But I admire Dryden most
[he had been speaking of Pope], who has succeeded by mere dint of genius,
and in spite of a laziness and a carelessness almost peculiar to himself.
His faults are numberless, and so are his beauties. His faults are those
of a great man, and his beauties are such (at least sometimes) as Pope
with all his touching and retouching could never equal." But, after all,
perhaps no man has summed him up so well as John Dennis, one of Pope's
typical dunces, a dull man outside of his own sphere, as men are apt to
be, but who had some sound notions as a critic, and thus became the
object of Pope's fear and therefore of his resentment. Dennis speaks of
him as his "departed friend, whom I infinitely esteemed when living for
the solidity of his thought, for the spring and the warmth and the
beautiful turn of it; for the power and variety and fulness of his
harmony; for the purity, the perspicuity, the energy of his expression;
and, whenever these great qualities are required, for the pomp and
solemnity and majesty of his style."[97]
Footnotes:
[1] The Dramatick Works of John Dryden, Esq. In six volumes. London:
Printed for Jacob Tonson, in the Strand. MDCCXXXV. 18mo.
The Critical and Miscellaneous Prose-Works of John Dryden, now first
collected. With Notes and Illustrations. An Account of the Life and
Writings of the Author, grounded on Original and Authentick
Documents; and a Collection of his Letters, the greatest Part of
which has never before been published. By Edmund Malone, Esq. London:
T. Cadell and W. Davies, in the Strand. 4 vols. 8vo.
The Poetical Works of John Dryden. (Edited by Mitford.) London: W.
Pickering. 1832. 6 vols. 18mo.
[2] His "Character of a Happy Warrior" (1806), one of his noblest
poems, has a dash of Dryden in it,--still more his "Epistle to Sir
George Beaumont (1811)."
[3] He studied Dryden's versification before writing his "Lamia."
[4] On the Origin and Progress of Satire. See Johnson's
counter-opinion in his life of Dryden.
[5] Essay on Dramatick Posey.
[6] Life of Lucian.
[7] "The great man must have that intellect which puts in motion the
intellect of others."--Landor, _Im. Con._, Diogenes and Plato.
[8] Character of Polybius (1692).
[9] "For my own part, who must confess it to my shame that I never
read anything but for pleasure." Life of Plutarch (1683).
[10] Gray says petulantly enough that "Dryden was as disgraceful to
the office, from his character, as the poorest scribbler could have
been from his verses."--Gray to Mason, 19th December, 1757.
[11] Essay on the Origin and Progress of Satire.
[12] Dedication of the Georgics.
[13] Dryden's penetration is always remarkable. His general judgment
of Polybius coincides remarkably with that of Mommsen. (Roem. Gesch.
II. 448, _seq_.)
[14] "I have taken some pains to make it my masterpiece in English."
Preface to Second Miscellany. Fox said that it "was better than the
original." J.C. Scaliger said of Erasmus: "Ex alieno ingenio poeta,
ex suo versificator."
[15] In one of the last letters he ever wrote, thanking his cousin
Mrs. Steward for a gift of marrow-puddings, he says: "A chine of
honest bacon would please my appetite more than all the
marrow-puddings; for I like them better plain, having a very vulgar
stomach." So of Cowley he says: "There was plenty enough, but ill
sorted, whole pyramids of sweetmeats for boys and women, but little
of solid meat for men." The physical is a truer antitype of the
spiritual man than we are willing to admit, and the brain is often
forced to acknowledge the inconvenient country-cousinship of the
stomach.
[16] In his preface to "All for Love," he says, evidently alluding to
himself: "If he have a friend whose hastiness in writing is his
greatest fault, Horace would have taught him to have minced the
matter, and to have called it readiness of thought and a flowing
fancy." And in the Preface to the Fables he says of Homer: "This
vehemence of his, I confess, is more suitable to my temper." He makes
other allusions to it.
[17] Preface to the Fables.
[18] _Wool_ is Sylvester's word. Dryden reminds us of Burke in this
also, that he always quotes from memory and seldom exactly. His
memory was better for things than for words. This helps to explain
the length of time it took him to master that vocabulary at last so
various, full, and seemingly extemporaneous. He is a large quoter,
though, with his usual inconsistency, he says, "I am no admirer of
quotations." (Essay on Heroic Plays.)
[19] In the _Epimetheus_ of a poet usually as elegant as Gray
himself, one's finer sense is a little jarred by the
"Spectral gleam their snow-white _dresses_."
[20] This probably suggested to Young the grandiose image in his
"Last Day" (B. ii.):--
"Those overwhelming armies....
Whose rear lay wrapt in night, while breaking dawn
Roused the broad front and called the battle on."
This, to be sure, is no plagiarism; but it should be carried to
Dryden's credit that we catch the poets of the next half-century
oftener with their hands in his pockets than in those of any one
else.
[21] Essay on Satire.
[22] Ibid.
[23] Preface to Fables. Men are always inclined to revenge themselves
on their old idols in the first enthusiasm of conversion to a purer
faith. Cowley had all the faults that Dryden loads him with, and yet
his popularity was to some extent deserved. He at least had a theory
that poetry should soar, not creep, and longed for some expedient, in
the failure of natural wings, by which he could lift himself away
from the conventional and commonplace. By beating out the substance
of Pindar very thin, he contrived a kind of balloon which, tumid with
gas, did certainly mount a little, _into_ the clouds, if not above
them, though sure to come suddenly down with a bump. His odes,
indeed, are an alternation of upward jerks and concussions, and smack
more of Chapelain than of the Theban, but his prose is very
agreeable,--Montaigne and water, perhaps, but with some flavor of the
Gascon wine left. The strophe of his ode to Dr. Scarborough, in which
he compares his surgical friend, operating for the stone, to Moses
striking the rock, more than justifies all the ill that Dryden could
lay at his door. It was into precisely such mud-holes that Cowley's
Will-o'-the-Wisp had misguided him. Men may never wholly shake off a
vice but they are always conscious of it, and hate the tempter.
[24] Dedication of Georgics.
[25] In a letter to Dennis, 1693.
[26] Preface to Fables.
[27] More than half a century later, Orrery, in his "Remarks" on
Swift, says: "We speak and we write at random; and if a man's common
conversation were committed to paper, he would be startled _for_ _to_
find himself guilty in _so few_ sentences of so many solecisms and
such false English." I do not remember _for to_ anywhere in Dryden's
prose. _So few_ has long been denizened; no wonder, since it is
nothing more than _si peu_ Anglicized.
[28] Letter to the Lord High Treasurer.
[29] Ibid. He complains of "manglings and abbreviations." "What does
your Lordship think of the words drudg'd, disturb'd, rebuk'd,
fledg'd, and a thousand others?" In a contribution to the "Tatler"
(No. 230) he ridicules the use of _'um_ for _them_, and a number of
slang Footnote: phrases, among which is _mob_. "The war," he says,
"has introduced abundance of polysyllables, which will never be able
to live many more campaigns." _Speculations, operations,
preliminaries, ambassadors, pallisadoes, communication,
circumvallation, battalions_, are the instances he gives, and all are
now familiar. No man, or body of men, can dam the stream of language.
Dryden is rather fond of _'em_ for _them_, but uses it rarely in his
prose. Swift himself prefers _'tis_ to _it is_, as does Emerson
still. In what Swift says of the poets, he may be fairly suspected of
glancing at Dryden, who was his kinsman, and whose prefaces and
translation of Virgil he ridicules in the "Tale of a Tub." Dryden is
reported to have said of him, "Cousin Swift is no poet." The Dean
began his literary career by Pindaric odes to Athenian Societies and
the like,--perhaps the greatest mistake as to his own powers of which
an author was ever guilty. It was very likely that he would send
these to his relative, already distinguished, for his opinion upon
them. If this was so, the justice of Dryden's judgment must have
added to the smart. Swift never forgot or forgave: Dryden was
careless enough to do the one, and large enough to do the other.
[30] Both Malone and Scott accept this gentleman's evidence without
question, but I confess suspicion of a memory that runs back more
than eighty-one years, and recollects a man before he had any claim
to remembrance. Dryden was never poor, and there is at Oxford a
portrait of him painted in 1664, which represents him in a superb
periwig and laced band. This was "before he had paid his court with
success to the great." But the story is at least _ben trovato_, and
morally true enough to serve as an illustration. Who the "old
gentleman" was has never been discovered. Of Crowne (who has some
interest for us as a sometime student at Harvard) he says: "Many a
cup of metheglm have I drank with little starch'd Johnny Crown; we
called him so, from the stiff, unalterable primness of his long
cravat." Crowne reflects no more credit on his Alma Mater than
Downing. Both were sneaks, and of such a kind as, I think, can only
be produced by a debauched Puritanism. Crowne, as a rival of Dryden,
is contemptuously alluded to by Cibber in his "Apology."
[31] Diary, III. 390. Almost the only notices of Dryden that make him
alive to me I have found in the delicious book of this
Polonius-Montaigne, the only man who ever had the courage to keep a
sincere journal, even under the shelter of cipher.
[32] Tale of a Tub, Sect. V. Pepys also speaks of buying the "Maiden
Queen" of Mr. Dryden's, which he himself, in his preface, seems to
brag of, and indeed is a good play.--18th January, 1668.
[33] He is fond of this image. In the "Maiden Queen" Celadon tells
Sabina that, when he is with her rival Florimel, his heart is still
her prisoner, "it only draws a longer chain after it." Goldsmith's
fancy was taken by it; and everybody admires in the "Traveller" the
extraordinary conceit of a heart dragging a lengthening chain. The
smoothness of too many rhymed pentameters is that of thin ice over
shallow water; so long as we glide along rapidly, all is well; but if
we dwell a moment on any one spot, we may find ourselves knee-deep in
mud. A later poet, in trying to improve on Goldsmith, shows the
ludicrousness of the image:--
"And round my heart's leg ties its galling chain."
To write imaginatively a man should have--imagination!
[34] See his epistle dedicatory to the "Rival Ladies" (1664). For the
other side, see particularly a passage in his "Discourse on Epic
Poetry" (1697).
[35] In the same way he had two years before assumed that Shakespeare
"was the first who, to shun the pains of continued rhyming, invented
that kind of writing which we call blank verse!" Dryden was never, I
suspect, a very careful student of English literature. He seems never
to have known that Surrey translated a part of the "Aeneid" (and with
great spirit) into blank verse. Indeed, he was not a scholar, in the
proper sense of the word, but he had that faculty of rapid
assimilation without study, so remarkable in Coleridge and other rich
minds, whose office is rather to impregnate than to invent. These
brokers of thought perform a great office in literature, second only
to that of originators.
[36] Essay on Satire. What he has said just before this about Butler
is worth noting. Butler had had a chief hand in the "Rehearsal," but
Dryden had no grudges where the question was of giving its just
praise to merit.
[37] The conclusion of the second canto of Book Third is the best
continuously fine passage. Dryden's poem has nowhere so much meaning
in so small space as Davenant, when he says of the sense of honor
that,
"Like Power, it grows to nothing, growing less."
Davenant took the hint of the stanza from Sir John Davies. Wyatt
first used it, so far as I know, in English.
[38] Perhaps there is no better lecture on the prevailing vices of
style and thought (if thought this frothy ferment of the mind may be
called) than in Cotton Mather's "Magnalia." For Mather, like a true
provincial, appropriates only the mannerism, and, as is usual in such
cases, betrays all its weakness by the unconscious parody of
exaggeration.
[39] The Doctor was a capital judge of the substantial value of the
goods he handled, but his judgment always seems that of the thumb and
forefinger. For the shades, the disposition of colors, the beauty of
the figures, he has as good as no sense whatever. The critical parts
of his Life of Dryden seem to me the best of his writing in this
kind. There is little to be gleaned after him. He had studied his
author, which he seldom did, and his criticism is sympathetic, a
thing still rarer with him. As illustrative of his own habits, his
remarks on Dryden's reading are curious.
[40] Perhaps the hint was given by a phrase of Corneille, _monarque
en peinture_. Dryden seldom borrows, unless from Shakespeare, without
improving, and he borrowed a great deal. Thus in "Don Sebastian" of
suicide:--
"Brutus and Cato might discharge their souls,
And give them furloughs for the other world;
But we, like sentries, are obliged to stand
In starless nights and wait the appointed hour."
The thought is Cicero's, but how it is intensified by the "starless
nights"! Dryden, I suspect, got it from his favorite, Montaigne, who
says, "Que nous ne pouvons abandonner cette garnison du monde, sans
le commandement exprez de celuy qui nous y a mis." (L. ii. chap. 3.)
In the same play, by a very Drydenish verse, he gives new force to an
old comparison:--
"And I should break through laws divine and human.
And think 'em cobwebs spread for little man,
_Which all the bulky herd of Nature breaks_."
[41] Not his solemn historical droning under that title, but
addressed "To the Cambrio-Britons on their harp."
[42] "Les poetes euxmemes s'animent et s'echauffent par la lecture
des autres poetes. Messieurs de Malherbe, Corneille, &c., se
disposoient au travail par la lecture des poetes qui etoient de leur
gout."--Vigneul, Marvilliana, I. 64, 65.
[43] For example, Waller had said,
"Others may use the ocean as their road,
Only the English _make it their abode_;
* * * * *
We _tread on billows with a steady foot_"--
long before Campbell. Campbell helps himself to both thoughts,
enlivens them into
"Her march is o'er the mountain wave,
Her home is on the deep,"
and they are his forevermore. His "leviathans afloat" he _lifted_
from the "Annus Mirabilis"; but in what court could Dryden sue?
Again, Waller in another poem calls the Duke of York's flag
"His dreadful streamer, like a comet's hair";
and this, I believe, is the first application of the celestial
portent to this particular comparison. Yet Milton's "imperial ensign"
waves defiant behind his impregnable lines, and even Campbell flaunts
his "meteor flag" in Waller's face. Gray's bard might be sent to the
lock-up, but even he would find bail.
"C'est imiter quelqu'un que de planter des choux."
[44] Corneille's tragedy of "Pertharite" was acted unsuccessfully in
1659. Racine made free use of it in his more fortunate "Andromaque."
[45] Dryden's publisher.
[46] Preface to the Fables.
[47] I interpret some otherwise ambiguous passages in this charming
and acute essay by its title: "On the _artificial_ comedy of the last
century."
[48] See especially his defence of the epilogue to the Second Part of
the "Conquest of Granada" (1672).
[49] Defence of an Essay on Dramatick Poesy.
[50] "The favor which heroick plays have lately found upon our
theatres has been wholly derived to them from the countenance and
approbation they have received at Court." (Dedication of "Indian
Emperor" to Duchess of Monmouth.)
[51] Dedication of "Rival Ladies."
[52] Defence of the Essay. Dryden, in the happiness of his
illustrative comparisons, is almost unmatched. Like himself, they
occupy a middle ground between poetry and prose,--they are a cross
between metaphor and simile.
[53] Discoveries.
[54] What a wretched rhymer he could be we may see in his
_alteration_ of the "Maid's Tragedy" of Beaumont and Fletcher:--
"Not long since walking in the field,
My nurse and I, we there beheld
A goodly fruit; which, tempting me,
I would have plucked: but, trembling, she,
Whoever eat those berries, cried,
In less than half an hour died!"
What intolerable seesaw! Not much of Byron's "fatal facility" in
_these_ octosyllabics!
[55] In more senses than one. His last and best portrait shows him
in his own gray hair.
[56] Essay on Dramatick Poesy.
[57] A French hendecasyllable verse runs exactly like our ballad
measure:--
A cobbler there was and he lived in a stall, ...
_La raison, pour marcher, n'a souvent qu'une voye._
(Dryden's note.)
The verse is not a hendecasyllable. "Attended watchfully to her
recitative (Mile. Duchesnois), and find that, in nine lines out of
ten, 'A cobbler there was,' &c, is the tune of the French
heroics."--_Moore's Diary_, 24th April, 1821.
[58] "The language of the age is never the language of poetry, except
among the French, whose verse, where the thought or image does not
support it, differs in nothing from prose."--Gray to West.
[59] Diderot and Rousseau, however, thought their language unfit for
poetry, and Voltaire seems to have half agreed with them. No one has
expressed this feeling more neatly than Fauriel: "Nul doute que l'on
ne puisse dire en prose des choses eminemment poetiques, tout comme
il n'est que trop certain que l'on peut en dire de fort prosaiques en
vers, et meme en excellents vers, en vers elegamment tournes, et en
beau langage. C'est un fait dont je n'ai pas besoin d'indiquer
d'exemples: aucune litterature n'en fournirait autant que le
notre."--Hist. de la Poesie Provencale, II. 237.
[60] Parallel of Poetry and Painting.
[61] "Il y a seulement la scene de _Ventidius_ et d'_Antoine_ qui est
digne de Corneille. C'est la le sentiment de milord Bolingbroke et
de tous les bons auteurs; c'est ainsi que pensait
Addisson."--Voltaire to M. De Fromont, 15th November, 1735.
[62] Inst. X., i. 129.
[63] Conquest of Grenada, Second Part.
[64] In most he mingles blank verse.
[65] Conquest of Grenada.
[66] This recalls a striking verse of Alfred de Musset:--
"La muse est toujours belle.
Meme pour l'insense, meme pour l'impuissant,
_Car sa beaute pour nous, c'est notre amour pour elle._"
[67] Rival Ladies.
[68] Don Sebastian.
[69] Don Sebastian.
[70] Cleomenes.
[71] All for Love.
[72] Dryden, with his wonted perspicacity, follows Ben Jonson in
calling Donne "the greatest wit, though not the best poet, of our
nation." (Dedication of Eleonora.) Even as a poet Donne
"Had in him those brave translunary things
That our first poets had."
To open vistas for the imagination through the blind wall of the
senses as he could sometimes do, is the supreme function of poetry.
[73] My own judgment is my sole warrant for attributing these
extracts from Oedipus to Dryden rather than Lee.
[74] Recollections of Rogers, p. 165.
[75] Nicholls's Reminiscences of Gray. Pickering's edition of Gray's
Works, Vol. V. p. 35.
[76] Let one suffice for all. In the "Royal Martyr," Porphyrius.
awaiting his execution, says to Maximin, who had wished him for a
son-in-law:--
"Where'er thou stand'st, I'll level at that place
My gushing blood, and spout it at thy face;
Thus not by marriage we our blood will join;
Nay, more, my arms shall throw my head at thine."
"It is no shame," says Dryden himself, "to be a poet, though it is to
be a bad one."
[77] Gray, _ubi supra_, p. 38.
[78] Scott had never seen Pepys's Diary when he wrote this, or he
would have left it unwritten: "Fell to discourse of the last night's
work at Court, where the ladies and Duke of Monmouth acted the
'Indian Emperor,' wherein they told me these things most remarkable
that not any woman but the Duchess of Monmouth and Mrs. Cornwallis
did anything but like fools and stocks, but that these two did do
most extraordinary well: that not any man did anything well but
Captain O'Bryan, who spoke and did well, but above all things did
dance most incomparably."--14th January, 1668.
[79] See also that noble passage in the "Hind and Panther"
(1572-1591), where this is put into verse. Dryden always thought in
prose.
[80] Probably on the authority of this very epitaph, as if epitaphs
were to be believed even under oath! A great many authors live
because we read nothing but their tombstones. Oldham was, to borrow
one of Dryden's phrases, "a bad or, which is worse, an indifferent
poet."
[81] "He was of a nature exceedingly humane and compassionate easily
forgiving injuries, and capable of a prompt and sincere
reconciliation with them that had offended him."--Congress.
[82] Coleridge says excellently: "You will find this a good gauge or
criterion of genius,--whether it progresses and evolves, or only
spins upon itself. Take Dryden's Achitophel and Zimri; every line
adds to or modifies the character, which is, as it were, a-building
up to the very last verse; whereas in Pope's Timon, &c. the first two
or three couplets contain all the pith of the character, and the
twenty or thirty lines that follow are so much evidence or proof of
overt acts of jealousy, or pride, or whatever it may be that is
satirized." (Table-Talk, 192.) Some of Dryden's best satirical hits
are let fall by seeming accident in his prose, as where he says of
his Protestant assailants, "Most of them love all whores but her of
Babylon." They had first attacked him on the score of his private
morals.
[83] That he taxes Shadwell with it is only a seeming exception, as
any careful reader will see.
[84] Preface to Fables.
[85] Dedication of the Georgics.
[86] Preface to Second Miscellany.
[87] Ibid.
[88] Memoirs of Wordsworth, Vol. II. p. 74 (American edition).
[89] A Discourse of Epick Poetry "If the _public_ approve." "On ne
peut pas admettre dans le developpement des langues aucune revolution
artificielle et sciemment executee; il n'y a pour elles ni conciles,
ni assemblees deliberantes; on ne les reforme pas comme une
constitution vicieuse."--Renan, De l'Origine du Langage, p 95.
[90] This is an old complaint. Puttenham sighs over such innovation
in Elizabeth's time, and Carew in James's. A language grows, and is
not made. Almost all the new-fangled words with which Jonson taxes
Marston in his "Poetaster" are now current.
[91] Like most idiomatic, as distinguished from correct writers, he
knew very little about the language historically or critically. His
prose and poetry swarm with locutions that would have made Lindley
Murray's hair stand on end. _How_ little he knew is plain from his
criticising in Ben Jonson the use of _ones_ in the plural, of "Though
Heaven should speak with all _his_ wrath," and be "as false English
for _are_, though the rhyme hides it." Yet all are good English, and
I have found them all in Dryden's own writing! Of his sins against
idiom I have a longer list than I have room for. And yet he is one of
our highest authorities for _real_ English.
[92] To see what he rescued us from in pedantry on the one hand, and
vulgarism on the other, read Feltham and Tom Brown--if you can.
[93] "Cette ode mise en musique par Purcell (si je ne me trompe),
passe en Angleterre pour le chef-d'oeuvre de la poesie la plus
sublime et la plus variee; et je vous avoue que, comme je sais mieux
l'anglais que le grec, j'aime cent fois mieux cette ode que tout
Pindare."--Voltaire to M. De Chabanon, 9 mars, 1772.
Dryden would have agreed with Voltaire. When Chief-Justice Marlay,
then a young Templar, "congratulated him on having produced the
finest and noblest Ode that had ever been written in any language,
You are right, young gentleman' (replied Dryden), 'a nobler Ode never
_was_ produced, nor ever _will_.'"--Malone.
[94] This was true of Coleridge, Wordsworth, and still more of
Southey who in some respects was not unlike Dryden.
[95] Pope's notion of gentility was perhaps expressed in a letter
from Lord Cobham to him: "I congratulate you upon the fine weather.
'T is a strange thing that people of condition and men of parts must
enjoy it in common with the rest of the world." (Ruffhead's Pope, p
276, _note_.) His Lordship's naive distinction between people of
condition and men of parts is as good as Pope's between genteel and
poetical men. I fancy the poet grinning savagely as he read it.
[96] "Nothing is truly sublime," he himself said, "that is not just
and proper."
[97] Dennis in a letter to Tonson, 1715.
WITCHCRAFT.[98]
Credulity, as a mental and moral phenomenon, manifests itself in widely
different ways, according as it chances to be the daughter of fancy or
terror. The one lies warm about the heart as Folk-lore, fills moonlit
dells with dancing fairies, sets out a meal for the Brownie, hears the
tinkle of airy bridle-bells as Tamlane rides away with the Queen of
Dreams, changes Pluto and Proserpine into Oberon and Titania, and makes
friends with unseen powers as Good Folk; the other is a bird of night,
whose shadow sends a chill among the roots of the hair: it sucks with the
vampire, gorges with the ghoule, is choked by the night-hag, pines away
under the witch's charm, and commits uncleanness with the embodied
Principle of Evil, giving up the fair realm of innocent belief to a murky
throng from the slums and stews of the debauched brain. Both have
vanished from among educated men, and such superstition as comes to the
surface now-a-days is the harmless Jacobitism of sentiment, pleasing
itself with the fiction all the more because there is no exacting reality
behind it to impose a duty or demand a sacrifice. And as Jacobitism
survived the Stuarts, so this has outlived the dynasty to which it
professes an after-dinner allegiance. It nails a horseshoe over the door,
but keeps a rattle by its bedside to summon a more substantial watchman;
it hangs a crape on the beehives to get a taste of ideal sweetness, but
obeys the teaching of the latest bee-book for material and marketable
honey. This is the aesthetic variety of the malady, or rather, perhaps,
it is only the old complaint robbed of all its pain, and lapped in waking
dreams by the narcotism of an age of science. To the world at large it is
not undelightful to see the poetical instincts of friends and neighbors
finding some other vent than that of verse. But there has been a
superstition of very different fibre, of more intense and practical
validity, the deformed child of faith, peopling the midnight of the mind
with fearful shapes and phrenetic suggestions, a monstrous brood of its
own begetting, and making even good men ferocious in imagined
self-defence.
Imagination, has always been, and still is, in a narrower sense, the
great mythologizer; but both its mode of manifestation and the force with
which it reacts on the mind are one thing in its crude form of childlike
wonder, and another thing after it has been more or less consciously
manipulated by the poetic faculty. A mythology that broods over us in our
cradles, that mingles with the lullaby of the nurse and the
winter-evening legends of the chimney-corner, that brightens day with the
possibility of divine encounters, and darkens night with intimations of
demonic ambushes, is of other substance than one which we take down from
our bookcase, sapless as the shelf it stood on, and remote from all
present sympathy with man or nature as a town history. It is something
like the difference between live metaphor and dead personification.
Primarily, the action of the imagination is the same in the mythologizer
and the poet, that is, it forces its own consciousness on the objects of
the senses, and compels them to sympathize with its own momentary
impressions. When Shakespeare in his "Lucrece" makes
"The threshold grate the door to have him heard,"
his mind is acting under the same impulse that first endowed with human
feeling and then with human shape all the invisible forces of nature, and
called into being those
"Fair humanities of old religion,"
whose loss the poets mourn. So also Shakespeare no doubt projected
himself in his own creations; but those creations never became so
perfectly disengaged from him, so objective, or, as they used to say,
extrinsical, to him, as to react upon him like real and even alien
existences. I mean permanently, for momentarily they may and must have
done so. But before man's consciousness had wholly disentangled itself
from outward objects, all nature was but a many-sided mirror which gave
back to him a thousand images more or less beautified or distorted,
magnified or diminished, of himself, till his imagination grew to look
upon its own incorporations as having an independent being. Thus, by
degrees, it became at last passive to its own creations. You may see
imaginative children every day anthropomorphizing in this way, and the
dupes of that super-abundant vitality in themselves, which bestows
qualities proper to itself on everything about them. There is a period of
development in which grown men are childlike. In such a period the fables
which endow beasts with human attributes first grew up; and we luckily
read them so early as never to become suspicious of any absurdity in
them. The Finnic epos of "Kalewala" is a curious illustration of the same
fact. In that every thing has the affections, passions, and consciousness
of men. When the mother of Lemminkaeinen is seeking her lost son,--
"Sought she many days the lost one,
Sought him ever without finding;
Then the roadways come to meet her,
And she asks them with beseeching:
'Roadways, ye whom God hath shapen,
Have ye not my son beholden,
Nowhere seen the golden apple,
Him, my darling staff of silver?'
Prudently they gave her answer,
Thus to her replied the roadways:
'For thy son we cannot plague us,
We have sorrows too, a many,
Since our own lot is a hard one
And our fortune is but evil,
By dog's feet to be run over,
By the wheel-tire to be wounded,
And by heavy heels down-trampled.'"
It is in this tendency of the mind under certain conditions to confound
the objective with subjective, or rather to mistake the one for the
other, that Mr. Tylor, in his "Early History of Mankind," is fain to seek
the origin of the supernatural, as we somewhat vaguely call whatever
transcends our ordinary experience. And this, no doubt, will in many
cases account for the particular shapes assumed by certain phantasmal
appearances, though I am inclined to doubt whether it be a sufficient
explanation of the abstract phenomenon. It is easy for the arithmetician
to make a key to the problems that he has devised to suit himself. An
immediate and habitual confusion of the kind spoken of is insanity; and
the hypochondriac is tracked by the black dog of his own mind. Disease
itself is, of course, in one sense natural, as being the result of
natural causes; but if we assume health as the mean representing the
normal poise of all the mental facilities, we must be content to call
hypochondria subternatural, because the tone of the instrument is
lowered, and to designate as supernatural only those ecstasies in which
the mind, under intense but not unhealthy excitement, is snatched
sometimes above itself, as in poets and other persons of imaginative
temperament. In poets this liability to be possessed by the creations of
their own brains is limited and proportioned by the artistic sense, and
the imagination thus truly becomes the shaping faculty, while in less
regulated or coarser organizations it dwells forever in the _Nifelheim_
of phantasmagoria and dream, a thaumaturge half cheat, half dupe. What
Mr. Tylor has to say on this matter is ingenious and full of valuable
suggestion, and to a certain extent solves our difficulties. Nightmare,
for example, will explain the testimony of witnesses in trials for
witchcraft, that they had been hag-ridden by the accused. But to prove
the possibility, nay, the probability, of this confusion of objective
with subjective is not enough. It accounts very well for such apparitions
as those which appeared to Dion, to Brutus, and to Curtius Rufus. In such
cases the imagination is undoubtedly its own _doppel-gaenger_, and sees
nothing more than the projection of its own deceit. But I am puzzled, I
confess, to explain the appearance of the _first_ ghost, especially among
men who thought death to be the end-all here below. The thing once
conceived of, it is easy, on Mr. Tylor's theory, to account for all after
the first. If it was originally believed that only the spirits of those
who had died violent deaths were permitted to wander,[99] the conscience
of a remorseful murderer may have been haunted by the memory of his
victim, till the imagination, infected in its turn, gave outward reality
to the image on the inward eye. After putting to death Boetius and
Symmachus, it is said that Theodoric saw in the head of a fish served at
his dinner the face of Symmachus, grinning horribly and with flaming
eyes, whereupon he took to his bed and died soon after in great agony of
mind. It is not safe, perhaps, to believe all that is reported of an
Arian; but supposing the story to be true, there is only a short step
from such a delusion of the senses to the complete ghost of popular
legend. But, in some of the most trustworthy stories of apparitions, they
have shown themselves not only to persons who had done them no wrong in
the flesh, but also to such as had never even known them. The _eidolon_
of James Haddock appeared to a man named Taverner, that he might interest
himself in recovering a piece of land unjustly kept from the dead man's
infant son. If we may trust Defoe, Bishop Jeremy Taylor twice examined
Taverner, and was convinced of the truth of his story. In this case,
Taverner had formerly known Haddock. But the apparition of an old
gentleman which entered the learned Dr. Scott's study, and directed him
where to find a missing deed needful in settling what had lately been its
estate in the West of England, chose for its attorney in the business an
entire stranger, who had never even seen its original in the flesh.
Whatever its origin, a belief in spirits seems to have been common to all
the nations of the ancient world who have left us any record of
themselves. Ghosts began to walk early, and are walking still, in spite
of the shrill cock-crow of _wir haben ja aufgeklaert._ Even the ghost in
chains, which one would naturally take to be a fashion peculiar to
convicts escaped from purgatory, is older than the belief in that
reforming penitentiary. The younger Pliny tells a very good story to this
effect: "There was at Athens a large and spacious house which lay under
the disrepute of being haunted. In the dead of the night a noise
resembling the clashing of iron was frequently heared, which, if you
listened more attentively, sounded like the rattling of chains; at first
it seemed at a distance, but approached nearer by degrees; immediately
afterward a spectre appeared, in the form of an old man, extremely meagre
and ghastly, with a long beard and dishevelled hair, rattling the chains
on his feet and hands.... By this means the house was at last deserted,
being judged by everybody to be absolutely uninhabitable; so that it was
now entirely abandoned to the ghost. However, in hopes that some tenant
might be found who was ignorant of this great calamity which attended it,
a bill was put up giving notice that it was either to be let or sold. It
happened that the philosopher Athenodorus came to Athens at this time,
and, reading the bill, inquired the price. The extraordinary cheapness
raised his suspicion; nevertheless, when he heared the whole story, he
was so far from being discouraged that he was more strongly inclined to
hire it, and, in short, actually did so. When it grew towards evening, he
ordered a couch to be prepared for him in the fore part of the house,
and, after calling for a light, together with his pen and tablets, he
directed all his people to retire. But that his mind might not, for want
of employment, be open to the vain terrors of imaginary noises and
spirits, he applied himself to writing with the utmost attention. The
first part of the night passed with usual silence, when at length the
chains began to rattle; however, he neither lifted up his eyes nor laid
down his pen, but diverted his observation by pursuing his studies with
greater earnestness. The noise increased, and advanced nearer, till it
seemed at the door, and at last in the chamber. He looked up and saw the
ghost exactly in the manner it had been described to him; it stood before
him, beckoning with the finger. Athenodorus made a sign with his hand
that it should wait a little, and threw his eyes again upon his papers;
but the ghost still rattling his chains in his ears, he looked up and saw
him beckoning as before. Upon this he immediately arose, and with the
light in his hand followed it. The ghost slowly stalked along, as if
encumbered with his chains, and, turning into the area of the house,
suddenly vanished. Athenodorus, being thus deserted, made a mark with
some grass and leaves where the spirit left him. The next day he gave
information of this to the magistrates, and advised them to order that
spot to be dug up. This was accordingly done, and the skeleton of a man
in chains was there found; for the body, having lain a considerable time
in the ground, was putrefied and mouldered away from the fetters. The
bones, being collected together, were publicly buried, and thus, after
the ghost was appeased by the proper ceremonies, the house was haunted no
more."[100] This story has such a modern air as to be absolutely
disheartening. Are ghosts, then, as incapable of invention as dramatic
authors? But the demeanor of Athenodorus has the grand air of the
classical period, of one _qui connait son monde_, and feels the
superiority of a living philosopher to a dead Philistine. How far above
all modern armament is his prophylactic against his insubstantial
fellow-lodger! Now-a-days men take pistols into haunted houses. Sterne,
and after him Novalis, discovered that gunpowder made all men equally
tall, but Athenodorus had found out that pen and ink establish a
superiority in spiritual stature. As men of this world, we feel our
dignity exalted by his keeping an ambassador from the other waiting till
he had finished his paragraph. Never surely did authorship appear to
greater advantage. Athenodorus seems to have been of Hamlet's mind:
"I do not set my life at a pin's fee,
And, for my soul, what can it do to that,
Being a thing immortal, as itself?"[101]
A superstition, as its name imports, is something that has been left to
stand over, like unfinished business, from one session of the world's
_witenagemot_ to the next. The vulgar receive it implicitly on the
principle of _omne ignotum pro possibili_, a theory acted on by a much
larger number than is commonly supposed, and even the enlightened are too
apt to consider it, if not proved, at least rendered probable by the
hearsay evidence of popular experience. Particular superstitions are
sometimes the embodiment by popular imagination of ideas that were at
first mere poetic figments, but more commonly the degraded and distorted
relics of religious beliefs. Dethroned gods, outlawed by the new dynasty,
haunted the borders of their old dominions, lurking in forests and
mountains, and venturing to show themselves only after nightfall. Grimm
and others have detected old divinities skulking about in strange
disguises, and living from hand to mouth on the charity of Gammer Grethel
and Mere l'Oie. Cast out from Olympus and Asgard, they were thankful for
the hospitality of the chimney-corner, and kept soul and body together by
an illicit traffic between this world and the other. While Schiller was
lamenting the Gods of Greece, some of them were nearer neighbors to him
than he dreamed; and Heine had the wit to turn them to delightful
account, showing himself, perhaps, the wiser of the two in saving what he
could from the shipwreck of the past for present use on this prosaic Juan
Fernandez of a scientific age, instead of sitting down to bewail it. To
make the pagan divinities hateful, they were stigmatized as cacodaemons;
and as the human mind finds a pleasure in analogy and system, an infernal
hierarchy gradually shaped itself as the convenient antipodes and
counterpoise of the celestial one. Perhaps at the bottom of it all there
was a kind of unconscious manicheism, and Satan, as Prince of Darkness,
or of the Powers of the Air, became at last a sovereign, with his great
feudatories and countless vassals, capable of maintaining a not unequal
contest with the King of Heaven. He was supposed to have a certain power
of bestowing earthly prosperity, but he was really, after all, nothing
better than a James II. at St. Germains, who could make Dukes of Perth
and confer titular fiefs and garters as much as he liked, without the
unpleasant necessity of providing any substance behind the shadow. That
there should have been so much loyalty to him, under these disheartening
circumstances, seems to me, on the whole, creditable to poor human
nature. In this case it is due, at least in part, to that instinct of the
poor among the races of the North, where there was a long winter, and too
often a scanty harvest,--and the poor have been always and everywhere a
majority,--which made a deity of Wish. The _Acheronta-movebo_ impulse
must have been pardonably strong in old women starving with cold and
hunger, and fathers with large families and a small winter stock of
provision. Especially in the transition period from the old religion to
the new, the temptation must have been great to try one's luck with the
discrowned dynasty, when the intruder was deaf and blind to claims that
seemed just enough, so long as it was still believed that God personally
interfered in the affairs of men. On his death-bed, says Piers Plowman,
"The poore dare plede and prove by reson
To have allowance of his lord; by the law he it claimeth;
* * * * *
Thanne may beggaris as beestes after boote waiten
That al hir lif han lyved in langour and in defaute
But God sente hem som tyme som manere joye,
Outher here or ellis where, kynde wolde it nevere."
He utters the common feeling when he says that it were against nature.
But when a man has his choice between here and elsewhere, it may be
feared that the other world will seem too desperately far away to be
waited for when hungry ruin has him in the wind, and the chance on earth
is so temptingly near. Hence the notion of a transfer of allegiance from
God to Satan, sometimes by a written compact, sometimes with the ceremony
by which homage is done to a feudal superior.
Most of the practices of witchcraft--such as the power to raise storms,
to destroy cattle, to assume the shape of beasts by the use of certain
ointments, to induce deadly maladies in men by waxen images, or love by
means of charms and philtres--were inheritances from ancient paganism.
But the theory of a compact was the product of later times, the result,
no doubt, of the efforts of the clergy to inspire a horror of any lapse
into heathenish rites by making devils of all the old gods. Christianity
may be said to have invented the soul as an individual entity to be saved
or lost; and thus grosser wits were led to conceive of it as a piece of
property that could be transferred by deed of gift or sale, duly signed,
sealed, and witnessed. The earliest legend of the kind is that of
Theophilus, chancellor of the church of Adana in Cilicia some time during
the sixth century. It is said to have been first written by Eutychianus,
who had been a pupil of Theophilus, and who tells the story partly as an
eyewitness, partly from the narration of his master. The nun Hroswitha
first treated it dramatically in the latter half of the tenth century.
Some four hundred years later Rutebeuf made it the theme of a French
miracle-play. His treatment of it is not without a certain poetic merit.
Theophilus has been deprived by his bishop of a lucrative office. In his
despair he meets with Saladin, _qui parloit au deable quant il voloit_.
Saladin tempts him to deny God and devote himself to the Devil, who, in
return, will give him back all his old prosperity and more. He at last
consents, signs and seals the contract required, and is restored to his
old place by the bishop. But now remorse and terror come upon him; he
calls on the Virgin, who, after some demur, compels Satan to bring back
his deed from the infernal muniment-chest (which must have been
fire-proof beyond any skill of our modern safe-makers), and the bishop
having read it aloud to the awe-stricken congregation, Theophilus becomes
his own man again. In this play, the theory of devilish compact is
already complete in all its particulars. The paper must be signed with
the blood of the grantor, who does feudal homage (_or joing tes mains, et
si devien mes hom_), and engages to eschew good and do evil all the days
of his life. The Devil, however, does not imprint any stigma upon his new
vassal, as in the later stories of witch-compacts. The following passage
from the opening speech of Theophilus will illustrate the conception to
which I have alluded of God as a liege lord against whom one might seek
revenge on sufficient provocation,--and the only revenge possible was to
rob him of a subject by going over to the great Suzerain, his deadly
foe:--
"N'est riens que por avoir ne face;
Ne pris riens Dieu et sa manace.
Irai me je noier ou pendre?
Ie ne m'en puis pas a Dieu prendre,
C'on ne puet a lui avenir.
* * * * *
Mes il s'est en si haut lieu mis,
Por eschiver ses anemis
C'on n'i puet trere ni lancier.
Se or pooie a lui tancier,
Et combattre et escrimir,
La char li feroie fremir.
Or est la sus en son solaz,
Laz! chetis! et je sui es laz
De Povrete et de Soufrete."[102]
During the Middle Ages the story became a favorite topic with preachers,
while carvings and painted windows tended still further to popularize it,
and to render men's minds familiar with the idea which makes the nexus of
its plot. The plastic hands of Calderon shaped it into a dramatic poem
not surpassed, perhaps hardly equalled, in subtile imaginative quality by
any other of modern times.
In proportion as a belief in the possibility of this damnable
merchandising with hell became general, accusations of it grew more
numerous. Among others, the memory of Pope Sylvester II, was blackened
with the charge of having thus bargained away his soul. All learning fell
under suspicion, till at length the very grammar itself (the last volume
in the world, one would say, to conjure with) gave to English the word
_gramary_ (enchantment), and in French became a book of magic, under the
alias of _Grimoire_. It is not at all unlikely that, in an age when the
boundary between actual and possible was not very well defined, there
were scholars who made experiments in this direction, and signed
contracts, though they never had a chance to complete their bargain by an
actual delivery. I do not recall any case of witchcraft in which such a
document was produced in court as evidence against the accused. Such a
one, it is true, was ascribed to Grandier, but was not brought forward at
his trial. It should seem that Grandier had been shrewd enough to take a
bond to secure the fulfilment of the contract on the other side; for we
have the document in fac-simile, signed and sealed by Lucifer, Beelzebub,
Satan, Elimi, Leviathan, and Astaroth, duly witnessed by Baalberith,
Secretary of the Grand Council of Demons. Fancy the competition such a
state paper as this would arouse at a sale of autographs! Commonly no
security appears to have been given by the other party to these
arrangements but the bare word of the Devil, which was considered, no
doubt, every whit as good as his bond. In most cases, indeed, he was the
loser, and showed a want of capacity for affairs equal to that of an
average giant of romance. Never was comedy acted over and over with such
sameness of repetition as "The Devil is an Ass." How often must he have
exclaimed (laughing in his sleeve):--
"_I_ to such blockheads set my wit,
_I_ damn such fools!--go, go, you're bit!"
In popular legend he is made the victim of some equivocation so gross
that any court of equity would have ruled in his favor. On the other
hand, if the story had been dressed up by some mediaeval Tract Society,
the Virgin appears in person at the right moment _ex machina_, and
compels him to give up the property he had honestly paid for. One is
tempted to ask, Were there no attorneys, then, in the place he came from,
of whom he might have taken advice beforehand? On the whole, he had
rather hard measure, and it is a wonder he did not throw up the business
in disgust. Sometimes, however, he was more lucky, as with the unhappy
Dr. Faust; and even so lately as 1695, he came in the shape of a "tall
fellow with black beard and periwig, respectable looking and well
dressed," about two o'clock in the afternoon, to fly away with the
Marechal de Luxembourg, which, on the stroke of five, he punctually did
as per contract, taking with him the window and its stone framing into
the bargain. The clothes and wig of the involuntary aeronaut were, in the
handsomest manner, left upon the bed, as not included in the bill of
sale. In this case also we have a copy of the articles of agreement,
twenty-eight in number, by the last of which the Marechal renounces God
and devotes himself to the enemy. This clause, sometimes the only one,
always the most important in such compacts, seems to show that they first
took shape in the imagination, while the struggle between Paganism and
Christianity was still going on. As the converted heathen was made to
renounce his false gods, none the less real for being false, so the
renegade Christian must forswear the true Deity. It is very likely,
however, that the whole thing may be more modern than the assumed date of
Theophilus would imply, and if so, the idea of feudal allegiance gave the
first hint, as it certainly modified the particulars, of the ceremonial.
This notion of a personal and private treaty with the Evil One has
something of dignity about it that has made it perennially attractive to
the most imaginative minds. It rather flatters than mocks our feeling of
the dignity of man. As we come down to the vulgar parody of it in the
confessions of wretched old women on the rack, our pity and indignation
are mingled with disgust. One of the most particular of these confessions
is that of Abel de la Rue, convicted in 1584. The accused was a novice in
the Franciscan Convent at Meaux. Having been punished by the master of
the novices for stealing some apples and nuts in the convent garden, the
Devil appeared to him in the shape of a black dog, promising him his
protection, and advising him to leave the convent. Not long after going
into the sacristy, he saw a large volume fastened by a chain, and further
secured by bars of iron. The name of this book was _Grimoire_. Thrusting
his hands through the bars, he contrived to open it, and having read a
sentence (which Bodin carefully suppresses), there suddenly appeared to
him a man of middle stature, with a pale and very frightful countenance,
clad in a long black robe of the Italian fashion, and with faces of men
like his own on his breast and knees. As for his feet they were like
those of cows. He could not have been the most agreeable of companions,
_ayant le corps et haleine puante_. This man told him not to be afraid,
to take off his habit, to put faith in him, and he would give him
whatever he asked. Then laying hold of him below the arms, the unknown
transported him under the gallows of Meaux, and then said to him with a
trembling and broken voice, and having a visage as pale as that of a man
who has been hanged, and a very stinking breath, that he should fear
nothing, but have entire confidence in him, that he should never want for
anything, that his own name was Maitre Rigoux, and that he would like to
be his master; to which De la Rue made answer that he would do whatever
he commanded, and that he wished to be gone from the Franciscans.
Thereupon Rigoux disappeared, but returning between seven and eight in
the evening, took him round the waist and carried him back to the
sacristy, promising to come again for him the next day. This he
accordingly did, and told De la Rue to take off his habit, get him gone
from the convent, and meet him near a great tree on the high-road from
Meaux to Vaulx-Courtois. Rigoux met him there and took him to a certain
Maitre Pierre, who, after a few words exchanged in an undertone with
Rigoux, sent De la Rue to the stable, after his return whence he saw no
more of Rigoux. Thereupon Pierre and his wife made him good cheer,
telling him that for the love of Maitre Rigoux they would treat him well,
and that he must obey the said Rigoux, which he promised to do. About two
months after, Maitre Pierre, who commonly took him to the fields to watch
cattle, said to him there that they must go to the Assembly, because he
(Pierre) was out of powders, to which he made answer that he was willing.
Three days later, about Christmas eve, 1575, Pierre having sent his wife
to sleep out of the house, set a long branch of broom in the
chimney-corner, and bade De la Rue go to bed, but not to sleep. About
eleven they heard a great noise as of an impetuous wind and thunder in
the chimney: which hearing, Maitre Pierre told him to dress himself, for
it was time to be gone. Then Pierre took some grease from a little box
and anointed himself under the arm-pits, and De la Rue on the palms of
his hands, which incontinently felt as if on fire, and the said grease
stank like a cat three weeks or a month dead. Then, Pierre and he
bestriding the branch, Maitre Rigoux took it by the butt and drew it up
chimney as if the wind had lifted them. And, the night being dark, he saw
suddenly a torch before them lighting them, and Maitre Rigoux was gone
unless he had changed himself into the said torch. Arrived at a grassy
place some five leagues from Vaulx-Courtois, they found a company of some
sixty people of all ages, none of whom he knew, except a certain Pierre
of Dampmartin and an old woman who was executed, as he had heard, about
five years ago for sorcery at Lagny. Then suddenly he noticed that all
(except Rigoux, who was clad as before) were dressed in linen, though
they had not changed their clothes. Then, at command of the eldest among
them, who seemed about eighty years old, with a white beard and almost
wholly bald, each swept the place in front of himself with his broom.
Thereupon Rigoux changed into a great he-goat, black and stinking, around
whom they all danced backward with their faces outward and their backs
towards the goat. They danced about half an hour, and then his master
told him they must adore the goat who was the Devil _et ce fait et dict,
veit que ledict Bouc courba ses deux pieds de deuant et leua son cul en
haut, et lors que certaines menues graines grosses comme testes
d'espingles, qui se conuertissoient en poudres fort puantes, sentant le
soulphre et poudre a canon et chair puant meslees ensemble seroient
tombees sur plusieurs drappeaux en sept doubles._ Then the oldest, and so
the rest in order, went forward on their knees and gathered up their
cloths with the powders, but first each _se seroit incline vers le Diable
et iceluy baise en la partie honteuse de son corps._ They went home on
their broom, lighted as before. De la Rue confessed also that he was at
another assembly on the eve of St. John Baptist. With the powders they
could cause the death of men against whom they had a spite, or their
cattle. Rigoux before long began to tempt him to drown himself, and,
though he lay down, yet rolled him some distance towards the river. It is
plain that the poor fellow was mad or half-witted or both. And yet Bodin,
the author of the _De Republica,_ reckoned one of the ablest books of
that age, believed all this filthy nonsense, and prefixes it to his
_Demonomanie,_ as proof conclusive of the existence of sorcerers.
This was in 1587. Just a century later, Glanvil, one of the most eminent
men of his day, and Henry More, the Platonist, whose memory is still dear
to the lovers of an imaginative mysticism, were perfectly satisfied with
evidence like that which follows. Elizabeth Styles confessed, in 1664,
"that the Devil about ten years since appeared to her in the shape of a
handsome Man, and after of a black Dog. That he promised her Money, and
that she should live gallantly, and have the pleasure of the World for
twelve years, if she would with her Blood sign his Paper, which was to
give her soul to him and observe his Laws and that he might suck her
Blood. This after Four Solicitations, the Examinant promised him to do.
Upon which he pricked the fourth Finger of her right hand, between the
middle and upper Joynt (where the Sign at the Examination remained) and
with a Drop or two of her Blood, she signed the Paper with an O. Upon
this the Devil gave her sixpence and vanished with the Paper. That since
he hath appeared to her in the Shape of a _Man_, and did so on
_Wednesday_ seven-night past, but more usually he appears in the Likeness
of a _Dog_, and _Cat_, and a _Fly_ like a Millar, in which last he
usually sucks in the Poll about four of the Clock in the Morning, and did
so _Jan_. 27, and that it is pain to her to be so suckt. That when she
hath a desire to do harm she calls the Spirit by the name of _Robin_, to
whom, when he appeareth, she useth these words, _O Sathan, give me my
purpose_. She then tells him what she would have done. And that he should
so appear to her was part of her Contract with him." The Devil in this
case appeared as a black (dark-complexioned) man "in black clothes, with
a little band,"--a very clerical-looking personage. "Before they are
carried to their meetings they anoint their Foreheads and Hand-Wrists
with an Oyl the Spirit brings them (which smells raw) and then they are
carried in a very short time, using these words as they pass, _Thout,
tout a tout, throughout and about_. And when they go off from their
Meetings they say, _Rentum, Tormentum_. That at every meeting before the
Spirit vanisheth away, he appoints the next meeting place and time, and
at his departure there is a foul smell. At their meeting they have
usually Wine or good Beer, Cakes, Meat or the like. They eat and drink
really when they meet, in their Bodies, dance also and have some Musick.
The Man in black sits at the higher end, and _Anne Bishop_ usually next
him. He useth some words before meat, and none after; his Voice is
audible but very low. The Man in black sometimes plays on a Pipe or
Cittern, and the Company dance. At last the Devil vanisheth, and all are
carried to their several homes in a short space. At their parting they
say, _A Boy! merry meet, merry part!_" Alice Duke confessed "that Anne
Bishop persuaded her to go with her into the Churchyard in the
Night-time, and being come thither, to go backward round the Church,
which they did three times. In their first round they met a Man in black
Cloths who went round the second time with them; and then they met a
thing in the Shape of a great black Toad which leapt up against the
Examinant's Apron. In their third round they met somewhat in the shape of
a Rat, which vanished away." She also received sixpence from the Devil,
and "her Familiar did commonly suck her right Breast about seven at night
in the shape of a little Cat of a dunnish Colour, which is as smooth as a
Want [mole], and when she is suckt, she is in a kind of Trance." Poor
Christian Green got only fourpence half-penny for her soul, but her
bargain was made some years later than that of the others, and
quotations, as the stock-brokers would say, ranged lower. Her familiar
took the shape of a hedgehog. Julian Cox confessed that "she had been
often tempted by the Devil to be a Witch, but never consented. That one
Evening she walkt about a Mile from her own House and there came riding
towards her three Persons upon three Broomstaves, born up about a yard
and a half from the ground. Two of them she formerly knew, which was a
Witch and a Wizzard that were hanged for Witchcraft several years before.
The third person she knew not. He came in the shape of a black Man, and
tempted her to give him her Soul, or to that effect, and to express it by
pricking her Finger and giving her name in her Blood in token of it." On
her trial Judge Archer told the jury, "he had heard that a Witch could
not repeat that Petition in the Lord's Prayer, viz. _And lead us not into
temptation_, and having this occasion, he would try the Experiment." The
jury "were not in the least measure to guide their Verdict according to
it, because it was not legal Evidence." Accordingly it was found that the
poor old trot could say only, _Lead us into temptation, or Lead us not
into no temptation_. Probably she used the latter form first, and,
finding she had blundered, corrected herself by leaving out both the
negatives. The old English double negation seems never to have been heard
of by the court. Janet Douglass, a pretended dumb girl, by whose
contrivance five persons had been burned at Paisley, in 1677, for having
caused the sickness of Sir George Maxwell by means of waxen and other
images, having recovered her speech shortly after, declared that she "had
some smattering knowledge of the Lord's prayer, which she had heard the
witches repeat, it seems, by her vision, in the presence of the Devil;
and at his desire, which they observed, they added to the word _art_ the
letter _w_, which made it run, 'Our Father which wart in heaven,' by
which means the Devil made the application of the prayer to himself." She
also showed on the arm of a woman named Campbell "an _invisible_ mark
which she had gotten from the Devil." The wife of one Barton confessed
that she had engaged "in the Devil's service. She renounced her baptism,
and did prostrate her body to the foul spirit, and received his mark, and
got a new name from him, and was called _Margaratus_. She was asked if
she ever had any pleasure in his company? 'Never much,' says she, 'but
one night going to a dancing upon Pentland Hills, in the likeness of a
rough tanny [tawny] dog, playing on a pair of pipes; the spring he
played,' says she, 'was _The silly bit chicken, gar cast it a pickle, and
it will grow meikle._'"[103] In 1670, near seventy of both sexes, among
them fifteen children, were executed for witchcraft at the village of
Mohra in Sweden. Thirty-six children, between the ages of nine and
sixteen, were sentenced to be scourged with rods on the palms of their
hands, once a week for a year. The evidence in this case against the
accused seems to have been mostly that of children. "Being asked whether
they were sure that they were at any time carried away by the Devil, they
all declared they were, begging of the Commissioners that they might be
freed from that intolerable slavery." They "used to go to a Gravel pit
which lay hardby a Cross-way and there they put on a vest over their
heads, and then danced round, and after ran to the Cross-way and called
the Devil thrice, first with a still Voice, the second time somewhat
louder, and the third time very loud, with these words, _Antecessour,
come and carry us to Blockula_. Whereupon immediately he used to appear,
but in different Habits; but for the most part they saw him in a gray
Coat and red and blue Stockings. He had a red Beard, a highcrowned Hat,
with linnen of divers Colours wrapt about it, and long Garters upon his
Stockings." "They must procure some Scrapings of Altars and Filings of
Church-Clocks [bells], and he gives them a Horn with some Salve in it
wherewith they do anoint themselves." "Being asked whether they were sure
of a real personal Transportation, and whether they were awake when it
was done, they all answered in the Affirmative, and that the Devil
sometimes laid something down in the Place that was very like them. But
one of them confessed that he did only take away her Strength, and her
Body lay still upon the Ground. Yet sometimes he took even her Body with
him." "Till of late they never had that power to carry away Children, but
only this year and the last, and the Devil did at this time force them to
it. That heretofore it was sufficient to carry but one of their Children
or a Stranger's Child, which yet happened seldom, but now he did plague
them and whip them if they did not procure him Children, insomuch that
they had no peace or quiet for him; and whereas formerly one Journey a
Week would serve their turn from their own town to the place aforesaid,
now they were forced to run to other Towns and Places for Children, and
that they brought with them some fifteen, some sixteen Children every
night. For their journey they made use of all sorts of Instruments, of
Beasts, of Men, of Spits, and Posts, according as they had opportunity.
If they do ride upon Goats and have many Children with them," they have a
way of lengthening the goat with a spit, "and then are anointed with the
aforesaid Ointment. A little Girl of Elfdale confessed, That, naming the
name of JESUS, as she was carried away, she fell suddenly upon the Ground
and got a great hole in her Side, which the Devil presently healed up
again. The first thing they must do at Blockula was that they must deny
all and devote themselves Body and Soul to the Devil, and promise to
serve him faithfully, and confirm all this with an Oath. Hereupon they
cut their Fingers, and with their Bloud writ their Name in his Book. He
caused them to be baptized by such Priests as he had there and made them
confirm their Baptism with dreadful Oaths and Imprecations. Here-upon the
Devil gave them a Purse, wherein their filings of Clocks [bells], with a
Stone tied to it, which they threw into the Water, and then they were
forced to speak these words: _As these filings of the Clock do never
return to the Clock from which they are taken, so may my soul never
return to Heaven_. The diet they did use to have there was Broth with
Colworts and Bacon in it, Oatmeal-Bread spread with Butter, Milk, and
Cheese. Sometimes it tasted very well, sometimes very ill. After Meals,
they went to Dancing, and in the mean while Swore and Cursed most
dreadfully, and afterward went to fighting one with another. The Devil
had Sons and Daughters by them, which he did marry together, and they did
couple and brought forth Toads and Serpents. If he hath a mind to be
merry with them, he lets them all ride upon Spits before him, takes
afterwards the Spits and beats them black and blue, and then laughs at
them. They had seen sometimes a very great Devil like a Dragon, with fire
about him and bound with an Iron Chain, and the Devil that converses with
them tells them that, if they confess anything, he will let that great
Devil loose upon them, whereby all _Sweedland_ shall come into great
danger. The Devil taught them to milk, which was in this wise: they used
to stick a knife in the Wall and hang a kind of Label on it, which they
drew and stroaked, and as long as this lasted the Persons that they had
Power over were miserably plagued, and the Beasts were milked that way
till sometimes they died of it. The minister of Elfdale declared that one
Night these Witches were to his thinking upon the crown of his Head and
that from thence he had had a long-continued Pain of the Head. One of the
Witches confessed, too, that the Devil had sent her to torment the
Minister, and that she was ordered to use a Nail and strike it into his
Head, but it would not enter very deep. They confessed also that the
Devil gives them a Beast about the bigness and shape of a young Cat,
which they call a _Carrier_, and that he gives them a Bird too as big as
a Raven, but white. And these two Creatures they can send anywhere, and
wherever they come they take away all sorts of Victuals they can get.
What the Bird brings they may keep for themselves; but what the Carrier
brings they must reserve for the Devil. The Lords Commissioners were
indeed very earnest and took great Pains to persuade them to show some of
their Tricks, but to no Purpose; for they did all unanimously confess,
that, since they had confessed all, they found that all their Witchcraft
was gone, and that the Devil at this time appeared to them very terrible
with Claws on his Hands and Feet, and with Horns on his Head and a long
Tail behind." At Blockula "the Devil had a Church, such another as in the
town of Mohra. When the Commissioners were coming, he told the Witches
they should not fear them, for he would certainly kill them all. And they
confessed that some of them had attempted to murther the Commissioners,
but had not been able to effect it."
In these confessions we find included nearly all the particulars of the
popular belief concerning witchcraft, and see the gradual degradation of
the once superb Lucifer to the vulgar scarecrow with horns and tail. "The
Prince of Darkness _was_ a gentleman." From him who had not lost all his
original brightness, to this dirty fellow who leaves a stench, sometimes
of brimstone, behind him, the descent is a long one. For the dispersion
of this foul odor Dr. Henry More gives an odd reason. "The Devil also, as
in other stories, leaving an ill smell behind him, seems to imply the
reality of the business, those adscititious particles he held together in
his visible vehicle being loosened at his vanishing and so offending the
nostrils by their floating and diffusing themselves in the open Air." In
all the stories vestiges of Paganism are not indistinct. The three
principal witch gatherings of the year were held on the days of great
pagan festivals, which were afterwards adopted by the Church. Maury
supposes the witches' Sabbath to be derived from the rites of Bacchus
Sabazius, and accounts in this way for the Devil's taking the shape of a
he-goat. But the name was more likely to be given from hatred of the
Jews, and the goat may have a much less remote origin. Bodin assumes the
identity of the Devil with Pan, and in the popular mythology both of
Kelts and Teutons there were certain hairy wood-demons called by the
former _Dus_ and by the latter _Scrat_. Our common names of _Deuse_ and
_Old Scratch_ are plainly derived from these, and possibly _Old Harry_ is
a corruption of _Old Hairy_. By Latinization they became Satyrs. Here, at
any rate, is the source of the cloven hoof. The belief in the Devil's
appearing to his worshippers as a goat is very old. Possibly the fact
that this animal was sacred to Thor, the god of thunder, may explain it.
Certain it is that the traditions of Vulcan, Thor, and Wayland[104]
converged at last in Satan. Like Vulcan, he was hurled from heaven, and
like him he still limps across the stage in Mephistopheles, though
without knowing why. In Germany, he has a horse's and not a cloven
foot,[105] because the horse was a frequent pagan sacrifice, and
therefore associated with devil-worship under the new dispensation. Hence
the horror of hippophagism which some French gastronomes are striving to
overcome. Everybody who has read "Tom Brown," or Wordsworth's Sonnet on a
German stove, remembers the Saxon horse sacred to Woden. The raven was
also his peculiar bird, and Grimm is inclined to think this the reason
why the witch's familiar appears so often in that shape. It is true that
our _Old Nick_ is derived from _Nikkar_, one of the titles of that
divinity, but the association of the Evil One with the raven is older,
and most probably owing to the ill-omened character of the bird itself.
Already in the apocryphal gospel of the "Infancy," the demoniac Son of
the Chief Priest puts on his head one of the swaddling-clothes of Christ
which Mary has hung out to dry, and forthwith "the devils began to come
out of his mouth and to fly away as _crows_ and serpents."
It will be noticed that the witches underwent a form of baptism. As the
system gradually perfected itself among the least imaginative of men, as
the superstitious are apt to be, they could do nothing better than
describe Satan's world as in all respects the reverse of that which had
been conceived by the orthodox intellect as Divine. Have you an
illustrated Bible of the last century? Very good. Turn it upside down,
and you find the prints on the whole about as near nature as ever, and
yet pretending to be something new by a simple device that saves the
fancy a good deal of trouble. For, while it is true that the poetic fancy
plays, yet the faculty which goes by that pseudonyme in prosaic minds
(and it was by such that the details of this Satanic commerce were pieced
together) is hard put to it for invention, and only too thankful for any
labor-saving contrivance whatsoever. Accordingly, all it need take the
trouble to do was to reverse the ideas of sacred things already engraved
on its surface, and behold, a kingdom of hell with all the merit and none
of the difficulty of originality! "Uti olim Deus populo suo Hierosolymis
Synagogas erexit ut in iis ignarus legis divinae populus erudiretur,
voluntatemque Dei placitam ex verbo in iis praedicato hauriret; ita et
Diabolus in omnibus omnino suis actionibus simiam Dei agens, gregi suo
acherontico conventus et synagogas, quas satanica sabbata vocant,
indicit.... Atque de hisce Conventibus et Synagogis Lamiarum nullus
Antorum quos quidem evolvi, imo nec ipse Lamiarum Patronus [here he
glances at Wierus] scilicet ne dubiolum quidem movit. Adeo ut tuto
affirmari liceat conventus a diabolo certo institui. Quos vel ipse,
tanquam praeses collegii, vel per daemonem, qui ad cujuslibet sagae
custodiam constitutus est, ... vel per alios Magos aut sagas per unum aut
duos dies antequam fiat congregatio denunciat.... Loci in quibus solent a
daemone coetus et conventicula malefica institui plerumque sunt
sylvestres, occulti, subterranei, et ab hominum conversatione remoti....
Evocatae hoc modo et tempore Lamiae, ... daemon illis persuadet eas non
posse conventiculis interesse nisi nudum corpus unguento ex corpusculis
infantum ante baptismum necatorum praeparato illinant, idque propterea
solum illis persuadet ut ad quam plurimas infantum insontium caedes eas
alliciat.... Unctionis ritu peracto, abiturientes, ne forte a maritis in
lectis desiderantur, vel per incantationem somnum, aurem nimirum
vellicando dextra manu prius praedicto unguine illita, conciliant maritis
ex quo non facile possunt excitari; vel daemones personas quasdam
dormientibus adumbrant, quas, si contigeret expergisci, suas uxores esse
putarent; vel interea alius daemon in forma succubi ad latus maritorum
adjungitur qui loco uxoris est.... Et ita sine omni remora insidentes
baculo, furcae, scopis, aut arundini vel tauro, equo, sui, hirco, aut
cani, _quorum omnium exempla prodidit Remig_. L.I.c. 14, devehuntur a
daemone ad loca destinata.... Ibi daemon praeses conventus in solio sedet
magnifico, forma terrifica, ut plurimum hirci vel canis. Ad quem
advenientes viri juxta ac mulieres accedunt reverentiae exhibendae et
adorandi gratia, non tamen uno eodemque modo. Interdum complicatis
genubus supplices; interdum obverso incedentes tergo et modo retrogrado,
in oppositum directo illi reverentiae quam nos praestare solemus. In
signum homagii (sit honor castis auribus) Principem suum hircum in
[obscaenissimo quodam corporis loco] summa cum reverentia sacrilego ore
osculantur. Quo facto, sacrificia daemoni faciunt multis modis. Saepe
liberos suos ipsi offerunt. Saepe communione sumpta benedictam hostiam in
ore asservatam et extractam (horreo dicere) daemoni oblatam coram eo pede
conculcant. His et similibus flagitiis et abominationibus execrandis
commissis, incipiunt mensis assidere et convivari de cibis insipidis,
insulsis,[106] furtivis, quos daemon suppeditat, vel quos singulae
attulere, inderdum tripudiant ante convivium, interdum post illud.... Nec
mensae sua deest benedictio coetu hoc digna, verbis constans plane
blasphemis quibus ipsum Beelzebub et creatorem et datorem et
conservatorem omnium profitentur. Eadem sententia est gratiarum actionis.
Post convivium, dorsis invicem obversis ... choreas ducere et cantare
fescenninos in honorem daemonis obscaenissimos, vel ad tympanum
fistulamve sedentis alicujus in bifida arbore saltare ... tum suis
amasiis daemonibus foedissime commisceri. Ultimo pulveribus (quos aliqui
scribunt esse cineres hirci illis quem daemon assumpserat et quem adorant
subito coram illius flamma absumpti) vel venenis aliis acceptis, saepe
etiam cuique indicto nocendi penso, et pronunciato Pseudothei daemonis
decreto, ULCISCAMINI VOS, ALIOQUI MORIEMINI. Duabus aut tribus horis in
hisce ludis exactis circa Gallicinium daemon convivas suas
dimittit."[107] Sometimes they were baptized anew. Sometimes they
renounced the Virgin, whom they called in their rites _extensam
mulierem_. If the Ave Mary bell should ring while the demon is conveying
home his witch, he lets her drop. In the confession of Agnes Simpson the
meeting place was North Berwick Kirk. "The Devil started up himself in
the pulpit, like a meikle black man, and calling the row [roll] every one
answered, _Here_. At his command they opened up three graves and cutted
off from the dead corpses the joints of their fingers, toes, and nose,
and parted them amongst them, and the said Agnes Simpson got for her part
a winding-sheet and two joints. The Devil commanded them to keep the
joints upon them while [till] they were dry, and then to make a powder of
them to do evil withal." This confession is sadly memorable, for it was
made before James I., then king of Scots, and is said to have convinced
him of the reality of witchcraft. Hence the act passed in the first year
of his reign in England, and not repealed till 1736, under which, perhaps
in consequence of which, so many suffered.
The notion of these witch-gatherings was first suggested, there can be
little doubt, by secret conventicles of persisting or relapsed pagans, or
of heretics. Both, perhaps, contributed their share. Sometimes a
mountain, as in Germany the Blocksberg,[108] sometimes a conspicuous oak
or linden, and there were many such among both Gauls and Germans sacred
of old to pagan rites, and later a lonely heath, a place where two roads
crossed each other, a cavern, gravel-pit, or quarry, the gallows, or the
churchyard, was the place appointed for their diabolic orgies. That the
witch could be conveyed bodily to these meetings was at first admitted
without any question. But as the husbands of accused persons sometimes
testified that their wives had not left their beds on the alleged night
of meeting, the witchmongers were put to strange shifts by way of
accounting for it. Sometimes the Devil imposed on the husband by a
_deceptio visus_; sometimes a demon took the place of the wife; sometimes
the body was left and the spirit only transported. But the more orthodox
opinion was in favor of corporeal deportation. Bodin appeals triumphantly
to the cases of Habbakuk (now in the Apocrypha, but once making a part of
the Book of Daniel), and of Philip in the Acts of the Apostles. "I find,"
he says, "this ecstatic ravishment they talk of much more wonderful than
bodily transport. And if the Devil has this power, as they confess, of
ravishing the spirit out of the body, is it not more easy to carry body
and soul without separation or division of the reasonable part, than to
withdraw and divide the one from the other without death?" The author of
_De Lamiis_ argues for the corporeal theory. "The evil Angels have the
same superiority of natural power as the good, since by the Fall they
lost none of the gifts of nature, but only those of grace." Now, as we
know that good angels can thus transport men in the twinkling of an eye,
it follows that evil ones may do the same. He fortifies his position by a
recent example from secular history. "No one doubts about John Faust, who
dwelt at Wittenberg, in the time of the sainted Luther, and who, seating
himself on his cloak with his companions, was conveyed away and borne by
the Devil through the air to distant kingdoms."[109] Glanvin inclines
rather to the spiritual than the material hypothesis, and suggests "that
the Witch's anointing herself before she takes her flight may perhaps
serve to keep the body tenantable and in fit disposition to receive the
spirit at its return." Aubrey, whose "Miscellanies" were published in
1696, had no doubts whatever as to the physical asportation of the witch.
He says that a gentleman of his acquaintance "was in Portugal _anno_
1655, when one was burnt by the inquisition for being brought thither
from Goa, in East India, in the air, in an incredible short time." As to
the conveyance of witches through crevices, keyholes, chimneys, and the
like, Herr Walburger discusses the question with such comical gravity
that we must give his argument in the undiminished splendor of its
jurisconsult latinity. The first sentence is worthy of Magister
Bartholomaeus Kuckuk. "Haec realis delatio trahit me quoque ad illam
vulgo agitatam quaestionem: _An diabolus Lamias corpore per angusta
foramina parietum, fenestrarum, portarum aut per cavernas ignifluas ferre
queant?_" (Surely if _tace_ be good Latin for a candle, _caverna
igniflua_ should be flattering to a chimney.) "Resp. Lamiae praedicto
modo saepius fatentur sese a diabolo per caminum aut alia loca angustiora
scopis insidentes per aerem ad montem Bructerorum deferri. Verum
deluduntur a Satana istaec mulieres hoc casu egregie nec revera rimulas
istas penetrant, sed solummodo daemon praecedens latenter aperit et
claudit januas vel fenestras corporis earum capaces, per quas eas
intromittit quae putant se formam animalculi parvi, mustelae, catti,
locustae, et aliorum induisse. At si forte contingat ut per parietem se
delatam confiteatur Saga, tunc, si non totum hoc praestigiosum est,
daemonem tamen maxima celeritate tot quot sufficiunt lapides eximere et
sustinere aliosne ruant, et postea eadem celeritate iterum eos in suum
locum reponere, existimo: cum hominum adspectus hanc tartarei latomi
fraudem nequeat deprendere. Idem quoque judicium esse potest de
translatione per caminum. Siquidem si caverna igniflua justae
amplitudinis est ut nullo impedimento et haesitatione corpus humanum eam
perrepere possit, diabolo impossibile non esse per eam eas educere. Si
vero per inproportionatum (ut ita loquar) corporibus spatium eas educit
tunc meras illusiones praestigiosas esse censeo, nec a diabolo hoc unquam
effici posse. Ratio est, quoniam diabolus essentiam creaturae seu lamiae
immutare non potest, multo minus efficere ut majus corpus penetret per
spatium inproportionatum, alioquin corporum penetratio esset admittenda
quod contra naturam et omne Physicorum principium est." This is fine
reasoning, and the _ut ita loquar_ thrown in so carelessly, as if with a
deprecatory wave of the hand for using a less classical locution than
usual, strikes me as a very delicate touch indeed.
Grimm tells us that he does not know when broomsticks, spits, and similar
utensils were first assumed to be the canonical instruments of this
nocturnal equitation. He thinks it comparatively modern, but I suspect it
is as old as the first child that ever bestrode his father's staff, and
fancied it into a courser shod with wind, like those of Pindar. Alas for
the poverty of human invention! It cannot afford a hippogriff for an
everyday occasion. The poor old crones, badgered by inquisitors into
confessing they had been where they never were, were involved in the
further necessity of explaining how the devil they got there. The only
steed their parents had ever been rich enough to keep had been of this
domestic sort, and they no doubt had ridden in this inexpensive fashion,
imagining themselves the grand dames they saw sometimes flash by, in the
happy days of childhood, now so far away. Forced to give a _how_, and
unable to conceive of mounting in the air without something to sustain
them, their bewildered wits naturally took refuge in some such simple
subterfuge, and the broomstave, which might make part of the poorest
house's furniture, was the nearest at hand. If youth and good spirits
could put such life into a dead stick once, why not age and evil spirits
now? Moreover, what so likely as an _emeritus_ implement of this sort to
become the staff of a withered beldame, and thus to be naturally
associated with her image? I remember very well a poor half-crazed
creature, who always wore a scarlet cloak and leaned on such a stay,
cursing and banning after a fashion that would infallibly have burned her
two hundred years ago. But apart from any adventitious associations of
later growth, it is certain that a very ancient belief gave to magic the
power of imparting life, or the semblance of it, to inanimate things, and
thus sometimes making servants of them. The wands of the Egyptian
magicians were turned to serpents. Still nearer to the purpose is the
capital story of Lucian, out of which Goethe made his _Zauberlehrling_,
of the stick turned water-carrier. The classical theory of the witch's
flight was driven to no such vulgar expedients, the ointment turning her
into a bird for the nonce, as in Lucian and Apuleius. In those days, too,
there was nothing known of any camp-meeting of witches and wizards, but
each sorceress transformed herself that she might fly to her paramour.
According to some of the Scotch stories, the witch, after bestriding her
broomsticks must repeat the magic formula, _Horse and Hattork!_ The
flitting of these ill-omened night-birds, like nearly all the general
superstitions relating to witchcraft, mingles itself and is lost in a
throng of figures more august.[110] Diana, Bertha, Holda, Abundia,
Befana, once beautiful and divine, the bringers of blessing while men
slept, became demons haunting the drear of darkness with terror and
ominous suggestion. The process of disenchantment must have been a long
one, and none can say how soon it became complete. Perhaps we may take
Heine's word for it, that
"Genau bei Weibern
Weiss man niemals wo der Engel
Aufhoert und der Teufel anfaengt."
Once goblinized, Herodias joins them, doomed still to bear about the
Baptist's head; and Woden, who, first losing his identity in the Wild
Huntsman, sinks by degrees into the mere _spook_ of a Suabian baron,
sinfully fond of field-sports, and therefore punished with an eternal
phantasm of them, "the hunter and the deer a shade." More and more
vulgarized, the infernal train snatches up and sweeps along with it every
lawless shape and wild conjecture of distempered fancy, streaming away at
last into a comet's tail of wild-haired hags, eager with unnatural hate
and more unnatural lust, the nightmare breed of some exorcist's or
inquisitor's surfeit, whose own lie has turned upon him in sleep.
As it is painfully interesting to trace the gradual degeneration of a
poetic faith into the ritual of unimaginative Tupperism, so it is amusing
to see pedantry clinging faithfully to the traditions of its prosaic
nature, and holding sacred the dead shells that once housed a moral
symbol. What a divine thing the _out_side always has been and continues
to be! And how the cast clothes of the mind continue always to be in
fashion! We turn our coats without changing the cut of them. But was it
possible for a man to change not only his skin but his nature? Were there
such things as _versipelles, lycanthropi, werwolfs,_ and _loupgarous?_ In
the earliest ages science was poetry, as in the later poetry has become
science. The phenomena of nature, imaginatively represented, were not
long in becoming myths. These the primal poets reproduced again as
symbols, no longer of physical, but of moral truths. By and by the
professional poets, in search of a subject, are struck by the fund of
picturesque material lying unused in them, and work them up once more as
narratives, with appropriate personages and decorations. Thence they take
the further downward step into legend, and from that to superstition. How
many metamorphoses between the elder Edda and the Nibelungen, between
Arcturus and the "Idyls of the King"! Let a good, thorough-paced proser
get hold of one of these stories, and he carefully desiccates them of
whatever fancy may be left, till he has reduced them to the proper
dryness of fact. King Lycaon, grandson by the spindleside of Oceanus,
after passing through all the stages I have mentioned, becomes the
ancestor of the werwolf. Ovid is put upon the stand as a witness, and
testifies to the undoubted fact of the poor monarch's own
metamorphosis:--
"Territus ipse fugit, nactusque silentia ruris
Exululat, frustraque loqui conatur."
Does any one still doubt that men may be changed into beasts? Call
Lucian, call Apuleius, call Homer, whose story of the companions of
Ulysses made swine of by Circe, says Bodin, _n'est pas fable_. If that
arch-patron of sorcerers, Wierus, is still unconvinced, and pronounces
the whole thing a delusion of diseased imagination, what does he say to
Nebuchadnezzar? Nay, let St. Austin be subpoenaed, who declares that "in
his time among the Alps sorceresses were common, who, by making
travellers eat of a certain cheese, changed them into beasts of burden
and then back again into men." Too confiding tourist, beware of
_Gruyere_, especially at supper! Then, there was the Philosopher
Ammonius, whose lectures were constantly attended by an ass,--a
phenomenon not without parallel in more recent times, and all the more
credible to Bodin, who had been professor of civil law.
In one case we have fortunately the evidence of the ass himself. In
Germany, two witches who kept an inn made an ass of a young actor,--not
always a very prodigious transformation it will be thought by those
familiar with the stage. In his new shape he drew customers by his
amusing tricks,--_voluptates mille viatoribus exhibebat_. But one day
making his escape (having overheard the secret from his mistresses), he
plunged into the water and was disasinized to the extent of recovering
his original shape. "Id Petrus Damianus, vir sua aetate inter primos
numerandus, cum rem sciscitatus est diligentissime ex hero, _ex asino_,
ex mulieribus sagis confessis factum, Leoni VII. Papae narravit, et
postquam diu in utramque partem coram Papa fuit disputatum, hoc tandem
posse fieri fuit constitum." Bodin must have been delighted with this
story, though perhaps as a Protestant he might have vilipended the
infallible decision of the Pope in its favor. As for lycanthropy, that
was too common in his own time to need any confirmation. It was notorious
to all men. "In Livonia, during the latter part of December, a villain
goes about summoning the sorcerers to meet at a certain place, and if
they fail, the Devil scourges them thither with an iron rod, and that so
sharply that the marks of it remain upon them. Their captain goes before;
and they, to the number of several thousands, follow him across a river,
which passed, they change into wolves, and, casting themselves upon men
and flocks, do all manner of damage." This we have on the authority of
Melancthon's son-in-law, Gaspar Peucerus. Moreover, many books published
in Germany affirm "that one of the greatest kings in Christendom, not
long since dead, was often changed into a wolf." But what need of words?
The conclusive proof remains, that many in our own day, being put to the
torture, have confessed the fact, and been burned alive accordingly. The
maintainers of the reality of witchcraft in the next century seem to have
dropped the _werwolf_ by common consent, though supported by the same
kind of evidence they relied on in other matters, namely, that of ocular
witnesses, the confession of the accused, and general notoriety. So
lately as 1765 the French peasants believed the "wild beast of the
Gevaudan" to be a _loupgarou_, and that, I think, is his last appearance.
The particulars of the concubinage of witches with their familiars were
discussed with a relish and a filthy minuteness worthy of Sanchez. Could
children be born of these devilish amours? Of course they could, said one
party; are there not plenty of cases in authentic history? Who was the
father of Romulus and Remus? nay, not so very long ago, of Merlin?
Another party denied the possibility of the thing altogether. Among these
was Luther, who declared the children either to be supposititious, or
else mere imps, disguised as innocent sucklings, and known as
_Wechselkinder_, or changelings, who were common enough, as everybody
must be aware. Of the intercourse itself Luther had no doubts.[111] A
third party took a middle ground, and believed that vermin and toads
might be the offspring of such amours. And how did the Demon, a mere
spiritual essence, contrive himself a body? Some would have it that he
entered into dead bodies, by preference, of course, those of sorcerers.
It is plain, from the confession of De la Rue, that this was the theory
of his examiners. This also had historical evidence in its favor. There
was the well-known leading case of the Bride of Corinth, for example. And
but yesterday, as it were, at Crossen in Silesia, did not Christopher
Monig, an apothecary's servant, come back after being buried, and do
duty, as if nothing particular had happened, putting up prescriptions as
usual, and "pounding drugs in the mortar with a mighty noise"?
Apothecaries seem to have been special victims of these Satanic pranks,
for another appeared at Reichenbach not long before, affirming that, "he
had poisoned several men with his drugs," which certainly gives an air of
truth to the story. Accordingly the Devil is represented as being
unpleasantly cold to the touch. "Caietan escrit qu'une sorciere demanda
un iour au diable pourquoy il ne se rechauffoit, qui fist response qu'il
faisoit ce qu'il pouuoit." Poor Devil! But there are cases in which the
demon is represented as so hot that his grasp left a seared spot as black
as charcoal. Perhaps some of them came from the torrid zone of their
broad empire, and others from the thrilling regions of thick-ribbed ice.
Those who were not satisfied with the dead-body theory contented
themselves, like Dr. More, with that of "adscititious particles," which
has, to be sure, a more metaphysical and scholastic flavor about it. That
the demons really came, either corporeally or through some diabolic
illusion that amounted to the same thing, and that the witch devoted
herself to him body and soul, scarce anybody was bold enough to doubt. To
these familiars their venerable paramours gave endearing nicknames, such
as My little Master, or My dear Martin,--the latter, probably, after the
heresy of Luther, and when the rack was popish. The famous witch-finder
Hopkins enables us to lengthen the list considerably. One witch whom he
convicted, after being "kept from sleep two or three nights," called in
five of her devilish servitors. The first was "_Holt_, who came in like a
white kitling"; the second "_Jarmara_, like a fat spaniel without any
legs at all"; the third, "_Vinegar Tom_, who was like a long-tailed
greyhound with an head like an oxe, with a long tail and broad eyes, who,
when this discoverer spoke to and bade him to the place provided for him
and his angells, immediately transformed himself into the shape of a
child of foure yeares old, without a head, and gave half a dozen turnes
about the house and vanished at the doore"; the fourth, "_Sack and
Sugar_, like a black rabbet"; the fifth, "_News_, like a polcat." Other
names of his finding were Elemauzer, Pywacket, Peck-in-the-Crown,
Grizzel, and Greedygut, "which," he adds, "no mortal could invent." The
name of _Robin_, which we met with in the confession of Alice Duke, has,
perhaps, wider associations than the woman herself dreamed of; for,
through Robin des Bois and Robin Hood, it may be another of those
scattered traces that lead us back to Woden. Probably, however, it is
only our old friend Robin Goodfellow, whose namesake Knecht Ruprecht
makes such a figure in the German fairy mythology. Possessed persons
called in higher agencies,--Thrones, Dominations, Princedoms, Powers; and
among the witnesses against Urbain Grandier we find the names of
Leviathan, Behemoth, Isaacarum, Belaam, Asmodeus, and Beherit, who spoke
French very well, but were remarkably poor Latinists, knowing, indeed,
almost as little of the language as if their youth had been spent in
writing Latin verses.[112] A shrewd Scotch physician tried them with
Gaelic, but they could make nothing of it.
It was only when scepticism had begun to make itself uncomfortably
inquisitive, that the Devil had any difficulty in making himself visible
and even palpable. In simpler times, demons would almost seem to have
made no inconsiderable part of the population. Trithemius tells of one
who served as cook to the Bishop of Hildesheim (one shudders to think of
the school where he had graduated as _Cordon bleu_), and who delectebatur
esse cum hominibus, loquens, interrogans, respondens familiariter
omnibus, aliquando visibiliter, aliquando invisibiliter apparens. This
last feat of "appearing invisibly" would have been worth seeing. In 1554,
the Devil came of a Christmas eve to Lawrence Doner, a parish priest in
Saxony, and asked to be confessed. "Admissus, horrendas adversus Christum
filium Dei blasphemias evomuit. Verum cum virtute verbi Dei a parocho
victus esset, intolerabili post se relicto foetore abiit." Splendidly
dressed, with two companions, he frequented an honest man's house at
Rothenberg. He brought with him a piper or fiddler, and contrived feasts
and dances under pretext of wooing the goodman's daughter. He boasted
that he was a foreign nobleman of immense wealth, and, for a time, was as
successful as an Italian courier has been known to be at one of our
fashionable watering-places. But the importunity of the guest and his
friends at length displicuit patrifamilias, who accordingly one evening
invited a minister of the Word to meet them at supper, and entered upon
pious discourse with him from the word of God. Wherefore, seeking other
matter of conversation, they said that there were many facetious things
more suitable to exhilarate the supper-table than the interpretation of
Holy Writ, and begged that they might be no longer bored with Scripture.
Thoroughly satisfied by their singular way of thinking that his guests
were diabolical, paterfamilias cries out in Latin worthy of Father Tom,
"Apagite, vos scelerati nebulones!" This said, the tartarean impostor and
his companions at once vanished with a great tumult, leaving behind them
a most unpleasant foetor and the bodies of three men who had been hanged.
Perhaps if the clergyman-cure were faithfully tried upon the next
fortune-hunting count with a large real estate in whiskers and an
imaginary one in Barataria, he also might vanish, leaving a strong smell
of barber's-shop, and taking with him a body that will come to the
gallows in due time. It were worth trying. Luther tells of a demon who
served as _famulus_ in a monastery, fetching beer for the monks, and
always insisting on honest measure for his money. There is one case on
record where the Devil appealed to the courts for protection in his
rights. A monk, going to visit his mistress, fell dead as he was passing
a bridge. The good and bad angel came to litigation about his soul. The
case was referred by agreement to Eichard, Duke of Normandy, who decided
that the monk's body should be carried back to the bridge, and his soul
restored to it by the claimants. If he persevered in keeping his
assignation, the Devil was to have him, if not, then the Angel. The monk,
thus put upon his guard, turns back and saves his soul, such as it
was.[113] Perhaps the most impudent thing the Devil ever did was to open
a school of magic in Toledo. The ceremony of graduation in this
institution was peculiar. The senior class had all to run through a
narrow cavern, and the venerable president was entitled to the hindmost,
if he could catch him. Sometimes it happened that he caught only his
shadow, and in that case the man who had been nimble enough to do what
Goethe pronounces impossible, became the most profound magician of his
year. Hence our proverb of _the Devil take the hindmost_, and Chamisso's
story of Peter Schlemihl.
There is no end of such stories. They were repeated and believed by the
gravest and wisest men down to the end of the sixteenth century; they
were received undoubtingly by the great majority down to the end of the
seventeenth. The Devil was an easy way of accounting for what was beyond
men's comprehension. He was the simple and satisfactory answer to all the
conundrums of Nature. And what the Devil had not time to bestow his
personal attention upon, the witch was always ready to do for him. Was a
doctor at a loss about a case? How could he save his credit more cheaply
than by pronouncing it witchcraft, and turning it over to the parson to
be exorcised? Did a man's cow die suddenly, or his horse fall lame?
Witchcraft! Did one of those writers of controversial quartos, heavy as
the stone of Diomed, feel a pain in the small of his back? Witchcraft!
Unhappily there were always ugly old women; and if you crossed them in
any way, or did them a wrong, they were given to scolding and banning.
If, within a year or two after, anything should happen to you or yours,
why, of course, old Mother Bombie or Goody Blake must be at the bottom of
it. For it was perfectly well known that there were witches, (does not
God's law say expressly, "Suffer not a _witch_ to live?") and that they
could cast a spell by the mere glance of their eyes, could cause you to
pine away by melting a waxen image, could give you a pain wherever they
liked by sticking pins into the same, could bring sickness into your
house or into your barn by hiding a Devil's powder under the threshold;
and who knows what else? Worst of all, they could send a demon into your
body, who would cause you to vomit pins, hair, pebbles, knives,-indeed,
almost anything short of a cathedral,-without any fault of yours, utter
through you the most impertinent things _verbi ministro_, and, in short,
make you the most important personage in the parish for the time being.
Meanwhile, you were an object of condolence and contribution to the whole
neighborhood. What wonder if a lazy apprentice or servant-maid (Bekker
gives several instances of the kind detected by him) should prefer being
possessed, with its attendant perquisites, to drudging from morning till
night? And to any one who has observed how common a thing in certain
states of mind self-connivance is, and how near it is to self-deception,
it will not be surprising that some were, to all intents and purposes,
really possessed. Who has never felt an almost irresistible temptation,
and seemingly not self-originated, to let himself go? to let his mind
gallop and kick and curvet and roll like a horse turned loose? in short,
as we Yankees say, "to speak out in meeting"? Who never had it suggested
to him by the fiend to break in at a funeral with a real character of the
deceased, instead of that Mrs. Grundyfied view of him which the clergyman
is so painfully elaborating in his prayer? Remove the pendulum of
conventional routine, and the mental machinery runs on with a whir that
gives a delightful excitement to sluggish temperaments, and is, perhaps,
the natural relief of highly nervous organizations. The tyrant Will is
dethroned, and the sceptre snatched by his frolic sister Whim. This state
of things, if continued, must become either insanity or imposture. But
who can say precisely where consciousness ceases and a kind of automatic
movement begins, the result of over-excitement? The subjects of these
strange disturbances have been almost always young women or girls at a
critical period of their development. Many of the most remarkable cases
have occurred in convents, and both there and elsewhere, as in other
kinds of temporary nervous derangement, have proved contagious.
Sometimes, as in the affair of the nuns of Loudon, there seems every
reason to suspect a conspiracy; but I am not quite ready to say that
Grandier was the only victim, and that some of the energumens were not
unconscious tools in the hands of priestcraft and revenge. One thing is
certain: that in the dioceses of humanely sceptical prelates the cases of
possession were sporadic only, and either cured, or at least hindered
from becoming epidemic, by episcopal mandate. Cardinal Mazarin, when
Papal vice-legate at Avignon, made an end of the trade of exorcism within
his government.
But scepticism, down to the beginning of the eighteenth century, was the
exception. Undoubting and often fanatical belief was the rule. It is easy
enough to be astonished at it, still easier to misapprehend it. How could
sane men have been deceived by such nursery-tales? Still more, how could
they have suffered themselves, on what seems to us such puerile evidence,
to consent to such atrocious cruelties, nay, to urge them on? As to the
belief, we should remember that the human mind, when it sails by _dead
reckoning_, without the possibility of a fresh observation, perhaps
without the instruments necessary to take one, will sometimes bring up in
very strange latitudes. Do we of the nineteenth century, then, always
strike out boldly into the unlandmarked deep of speculation and shape our
courses by the stars, or do we not sometimes con our voyage by what seem
to us the firm and familiar headlands of truth, planted by God himself,
but which may, after all, be no more than an insubstantial mockery of
cloud or airy juggle of mirage? The refraction of our own atmosphere has
by no means made an end of its tricks with the appearances of things in
our little world of thought. The men of that day believed what they saw,
or, as our generation would put it, what they _thought_ they saw. Very
good. The vast majority of men believe, and always will believe, on the
same terms. When one comes along who can partly distinguish the thing
seen from that travesty or distortion of it which the thousand disturbing
influences within him and without him would _make_ him see, we call him a
great philosopher. All our intellectual charts are engraved according to
his observations, and we steer contentedly by them till some man whose
brain rests on a still more unmovable basis corrects them still further
by eliminating what his predecessor thought _he_ saw. We must account for
many former aberrations in the moral world by the presence of more or
less nebulous bodies of a certain gravity which modified the actual
position of truth in its relation to the mind, and which, if they have
now vanished, have made way, perhaps, for others whose influence will in
like manner be allowed for by posterity in their estimate of us. In
matters of faith, astrology has by no means yet given place to astronomy,
nor alchemy become chemistry, which knows what to seek for and how to
find it. In the days of witchcraft all science was still in the condition
of _May-be;_ it is only just bringing itself to find a higher
satisfaction in the imperturbable _Must-be_ of law. We should remember
that what we call _natural_ may have a very different meaning for one
generation from that which it has for another. The boundary between the
"other" world and this ran till very lately, and at some points runs
still, through a vast tract of unexplored border-land of very uncertain
tenure. Even now the territory which Reason holds firmly as Lord Warden
of the marches during daylight, is subject to sudden raids of Imagination
by night. But physical darkness is not the only one that lends
opportunity to such incursions; and in midsummer 1692, when Ebenezer
Bapson, looking out of the fort at Gloucester in broad day, saw shapes of
men, sometimes in blue coats like Indians, sometimes in white waistcoats
like Frenchmen, it seemed _more_ natural to most men that they should be
spectres than men of flesh and blood. Granting the assumed premises, as
nearly every one did, the syllogism was perfect.
So much for the apparent reasonableness of the belief, since every man's
logic is satisfied with a legitimate deduction from his own postulates.
Causes for the cruelty to which the belief led are not further to seek.
Toward no crime have men shown themselves so cold-bloodedly cruel as in
punishing difference of belief, and the first systematic persecutions for
witchcraft began with the inquisitors in the South of France in the
thirteenth century. It was then and there that the charge of sexual
uncleanness with demons was first devised. Persecuted heretics would
naturally meet in darkness and secret, and it was easy to blacken such
meetings with the accusation of deeds so foul as to shun the light of day
and the eyes of men. They met to renounce God and worship the Devil. But
this was not enough. To excite popular hatred and keep it fiercely alive,
fear must be mingled with it; and this end was reached by making the
heretic also a sorcerer, who, by the Devil's help, could and would work
all manner of fiendish mischief. When by this means the belief in a
league between witch and demon had become firmly established, witchcraft
grew into a well-defined crime, hateful enough in itself to furnish
pastime for the torturer and food for the fagot. In the fifteenth
century, witches were burned by thousands, and it may well be doubted if
all paganism together was ever guilty of so many human sacrifices in the
same space of time. In the sixteenth, these holocausts were appealed to
as conclusive evidence of the reality of the crime, terror was again
aroused, the more vindictive that its sources were so vague and
intangible, and cruelty was the natural consequence. Nothing but an
abject panic, in which the whole use of reason, except as a mill to grind
out syllogisms, was altogether lost, will account for some chapters in
Bodin's _Demonomanie_. Men were surrounded by a forever-renewed
conspiracy whose ramifications they could not trace, though they might
now and then lay hold on one of its associates. Protestant and Catholic
might agree in nothing else, but they were unanimous in their dread of
this invisible enemy. If fright could turn civilized Englishmen into
savage Iroquois during the imagined <DW64> plots of New York in 1741 and
of Jamaica in 1865, if the same invisible omnipresence of Fenianism shall
be able to work the same miracle, as it perhaps will, next year in
England itself, why need we be astonished that the blows should have
fallen upon many an innocent head when men were striking wildly in
self-defence, as they supposed, against the unindictable Powers of
Darkness, against a plot which could be carried on by human agents, but
with invisible accessories and by supernatural means? In the seventeenth
century an element was added which pretty well supplied the place of
heresy as a sharpener of hatred and an awakener of indefinable suspicion.
Scepticism had been born into the world, almost more hateful than heresy,
because it had the manners of good society and contented itself with a
smile, a shrug, an almost imperceptible lift of the eyebrow,--a kind of
reasoning especially exasperating to disputants of the old school, who
still cared about victory, even when they did not about the principles
involved in the debate.
The Puritan emigration to New England took place at a time when the
belief in diabolic agency had been hardly called in question, much less
shaken. The early adventurers brought it with them to a country in every
way fitted, not only to keep it alive, but to feed it into greater vigor.
The solitude of the wilderness (and solitude alone, by dis-furnishing the
brain of its commonplace associations, makes it an apt theatre for the
delusions of imagination), the nightly forest noises, the glimpse,
perhaps, through the leaves, of a painted savage face, uncertain whether
of redman or Devil, but more likely of the latter, above all, that
measureless mystery of the unknown and conjectural stretching away
illimitable on all sides and vexing the mind, somewhat as physical
darkness does, with intimation and misgiving,--under all these
influences, whatever seeds of superstition had in any way got over from
the Old World would find an only too congenial soil in the New. The
leaders of that emigration believed and taught that demons loved to dwell
in waste and wooded places, that the Indians did homage to the bodily
presence of the Devil, and that he was especially enraged against those
who had planted an outpost of the true faith upon this continent hitherto
all his own. In the third generation of the settlement, in proportion as
living faith decayed, the clergy insisted all the more strongly on the
traditions of the elders, and as they all placed the sources of goodness
and religion in some inaccessible Other World rather than in the soul of
man himself, they clung to every shred of the supernatural as proof of
the existence of that Other World, and of its interest in the affairs of
this. They had the countenance of all the great theologians, Catholic as
well as Protestant, of the leaders of the Reformation, and in their own
day of such men as More and Glanvil and Baxter.[114] If to all these
causes, more or less operative in 1692, we add the harassing excitement
of an Indian war (urged on by Satan in his hatred of the churches), with
its daily and nightly apprehensions and alarms, we shall be less
astonished that the delusion in Salem Village rose so high than that it
subsided so soon.
I have already said that it was religious antipathy or clerical interest
that first made heresy and witchcraft identical and cast them into the
same expiatory fire. The invention was a Catholic one, but it is plain
that Protestants soon learned its value and were not slow in making it a
plague to the inventor. It was not till after the Reformation that there
was any systematic hunting out of witches in England. Then, no doubt, the
innocent charms and rhyming prayers of the old religion were regarded as
incantations, and twisted into evidence against miserable beldames who
mumbled over in their dotage what they had learned at their mother's
knee. It is plain, at least, that this was one of Agnes Simpson's crimes.
But as respects the frivolity of the proof adduced, there was nothing to
choose between Catholic and Protestant. Out of civil and canon law a net
was woven through whose meshes there was no escape, and into it the
victims were driven by popular clamor. Suspicion of witchcraft was
justified by general report, by the ill-looks of the suspected, by being
silent when accused, by her mother's having been a witch, by flight, by
exclaiming when arrested, _I am lost!_ by a habit of using imprecations,
by the evidence of two witnesses, by the accusation of a man on his
death-bed, by a habit of being away from home at night, by fifty other
things equally grave. Anybody might be an accuser,--a personal enemy, an
infamous person, a child, parent, brother, or sister. Once accused, the
culprit was not to be allowed to touch the ground on the way to prison,
was not to be left alone there lest she have interviews with the Devil
and get from him the means of being insensible under torture, was to be
stripped and shaved in order to prevent her concealing some charm, or to
facilitate the finding of witch-marks. Her right thumb tied to her left
great-toe, and _vice versa_, she was thrown into the water. If she
floated, she was a witch; if she sank and was drowned, she was lucky.
This trial, as old as the days of Pliny the Elder, was gone out of
fashion, the author of _De Lamiis_ assures us, in his day, everywhere but
in Westphalia. "On halfproof or strong presumption," says Bodin, the
judge may proceed to torture. If the witch did not shed tears under the
rack, it was almost conclusive of guilt. On this topic of torture he
grows eloquent. The rack does very well, but to thrust splinters between
the nails and flesh of hands and feet "is the most excellent gehenna of
all, and practised in Turkey." That of Florence, where they seat the
criminal in a hanging chair so contrived that if he drop asleep it
overturns and leaves him hanging by a rope which wrenches his arms
backwards, is perhaps even better, "for the limbs are not broken, and
without trouble or labor one gets out the truth." It is well in carrying
the accused to the chamber of torture to cause some in the next room to
shriek fearfully as if on the rack, that they may be terrified into
confession. It is proper to tell them that their accomplices have
confessed and accused them ("though they have done no such thing") that
they may do the same out of revenge. The judge may also with a good
conscience lie to the prisoner and tell her that if she admit her guilt,
she may be pardoned. This is Bodin's opinion, but Walburger, writing a
century later, concludes that the judge may go to any extent _citra
mendacium_, this side of lying. He may tell the witch that he will be
favorable, meaning to the Commonwealth; that he will see that she has a
new house built for her, that is, a wooden one to burn her in; that her
confession will be most useful in saving her life, to wit, her life
eternal. There seems little difference between the German's white lies
and the Frenchman's black ones. As to punishment, Bodin is fierce for
burning. Though a Protestant, he quotes with evident satisfaction a
decision of the magistrates that one "who had eaten flesh on a Friday
should be burned alive unless he repented, and if he repented, yet he was
hanged out of compassion." A child under twelve who will not confess
meeting with the Devil should be put to death if convicted of the fact,
though Bodin allows that Satan made no express compact with those who had
not arrived at puberty. This he learned from the examination of Jeanne
Harvillier, who deposed, "that, though her mother dedicated her to Satan
so soon as she was born, yet she was not married to him, nor did he
demand that, or her renunciation of God, till she had attained the age of
twelve."
There is no more painful reading than this, except the trials of the
witches themselves. These awaken, by turns, pity, indignation, disgust,
and dread,--dread at the thought of what the human mind may be brought to
believe not only probable, but proven. But it is well to be put upon our
guard by lessons of this kind, for the wisest man is in some respects
little better than a madman in a strait-waistcoat of habit, public
opinion, prudence, or the like. Scepticism began at length to make itself
felt, but it spread slowly and was shy of proclaiming itself. The
orthodox party was not backward to charge with sorcery whoever doubted
their facts or pitied their victims. Bodin says that it is good cause of
suspicion against a judge if he turn the matter into ridicule, or incline
toward mercy. The mob, as it always is, was orthodox. It was dangerous to
doubt, it might be fatal to deny. In 1453 Guillaume de Lure was burned at
Poitiers on his own confession of a compact with Satan, by which he
agreed "to preach and did preach that everything told of sorcerers was
mere fable, and that it was cruelly done to condemn them to death." This
contract was found among his papers signed "with the Devil's own claw,"
as Howell says speaking of a similar case. It is not to be wondered at
that the earlier doubters were cautious. There was literally a reign of
terror, and during such _regimes_ men are commonly found more eager to be
informers and accusers than of counsel for the defence. Peter of Abano is
reckoned among the earliest unbelievers who declared himself openly.[115]
Chaucer was certainly a sceptic, as appears by the opening of the Wife of
Bath's Tale. Wierus, a German physician, was the first to undertake
(1563) a refutation of the facts and assumptions on which the
prosecutions for witchcraft were based. His explanation of the phenomena
is mainly physiological. Mr. Leckie hardly states his position correctly,
in saying, "that he never dreamed of restricting the sphere of the
supernatural." Wierus went as far as he dared. No one can read his book
without feeling that he insinuates much more than he positively affirms
or denies. He would have weakened his cause if he had seemed to
disbelieve in demoniacal possession, since that had the supposed warrant
of Scripture; but it may be questioned whether he uses the words _Satan_
and _Demon_ in any other way than that in which many people still use the
word _Nature_. He was forced to accept certain premises of his opponents
by the line of his argument. When he recites incredible stories without
comment, it is not that he believes them, but that he thinks their
absurdity obvious. That he wrote under a certain restraint is plain from
the Colophon of his book, where he says: "Nihil autem hic ita assertum
volo, quod aequiori judicio Catholicae Christi Ecclesiae non omnino
submittam, palinodia mox spontanea emendaturus, si erroris alicubi
convincar." A great deal of latent and timid scepticism seems to have
been brought to the surface by his work. Many eminent persons wrote to
him in gratitude and commendation. In the Preface to his shorter treatise
_De Lamiis_ (which is a mere abridgment), he thanks God that his labors
had "in many places caused the cruelty against innocent blood to
slacken," and that "some more distinguished judges treat more mildly and
even absolve from capital punishment the wretched old women branded with
the odious name of witches by the populace." In the _Pseudomonarchia
Daemonum_, he gives a kind of census of the diabolic kingdom,[116] but
evidently with secret intention of making the whole thing ridiculous, or
it would not have so stirred the bile of Bodin. Wierus was saluted by
many contemporaries as a Hercules who destroyed monsters, and himself not
immodestly claimed the civic wreath for having saved the lives of
fellow-citizens. Posterity should not forget a man who really did an
honest life's work for humanity and the liberation of thought. From one
of the letters appended to his book we learn that Jacobus Savagius, a
physician of Antwerp, had twenty years before written a treatise with the
same design, but confining himself to the medical argument exclusively.
He was, however, prevented from publishing it by death. It is pleasant to
learn from Bodin that Alciato, the famous lawyer and emblematist, was one
of those who "laughed and made others laugh at the evidence relied on at
the trials, insisting that witchcraft was a thing impossible and
fabulous, and so softened the hearts of judges (in spite of the fact that
an inquisitor had caused to burn more than a hundred sorcerers in
Piedmont), that all the accused escaped." In England, Reginald Scot was
the first to enter the lists in behalf of those who had no champion. His
book, published in 1584, is full of manly sense and spirit, above all, of
a tender humanity that gives it a warmth which we miss in every other
written on the same side. In the dedication to Sir Roger Manwood he says:
"I renounce all protection and despise all friendship that might serve
towards the suppressing or supplanting of truth." To his kinsman, Sir
Thomas Scot, he writes: "My greatest adversaries are _young ignorance_
and _old custom_; for what folly soever tract of time hath fostered, it
is so superstitiously pursued of some, as though no error could be
acquainted with custom." And in his Preface he thus states his motives:
"God that knoweth my heart is witness, and you that read my book shall
see, that my drift and purpose in this enterprise tendeth only to these
respects. First, that the glory and power of God be not so abridged and
abased as to be thrust into the hand or lip of a lewd old woman, whereby
the work of the Creator should be attributed to the power of a creature.
Secondly, that the religion of the Gospel may be seen to stand without
such peevish trumpery. Thirdly, that lawful favor and Christian
compassion be rather used towards these poor souls than rigor and
extremity. Because they which are commonly accused of witchcraft are the
least sufficient of all other persons to speak for themselves, as having
the most base and simple education of all others, the extremity of their
age giving them leave to dote, their poverty to beg, their wrongs to
chide and threaten (as being void of any other way of revenge), their
humor melancholical to be full of imaginations, from whence chiefly
proceedeth the vanity of their confessions.... And for so much as the
mighty help themselves together, and the poor widow's cry, though it
reach to Heaven, is scarce heard here upon earth, I thought good
(according to my poor ability) to make intercession that some part of
common rigor and some points of hasty judgment may be advised upon."....
The case is nowhere put with more point, or urged with more sense and
eloquence, than by Scot, whose book contains also more curious matter, in
the way of charms, incantations, exorcisms, and feats of legerdemain,
than any other of the kind.
Other books followed on the same side, of which Bekker's, published about
a century later, was the most important. It is well reasoned, learned,
and tedious to a masterly degree. But though the belief in witchcraft
might be shaken, it still had the advantage of being on the whole
orthodox and respectable. Wise men, as usual, insisted on regarding
superstition as of one substance with faith, and objected to any scouring
of the shield of religion, lest, like that of Cornelius Scriblerus, it
should suddenly turn out to be nothing more than "a paltry old sconce
with the nozzle broke off." The Devil continued to be the only recognized
Minister Resident of God upon earth. When we remember that one man's
accusation on his death-bed was enough to constitute grave presumption of
witchcraft, it might seem singular that dying testimonies were so long of
no avail against the common credulity. But it should be remembered that
men are mentally no less than corporeally gregarious, and that public
opinion, the fetish even of the nineteenth century, makes men, whether
for good or ill, into a mob, which either hurries the individual judgment
along with it, or runs over and tramples it into insensibility. Those who
are so fortunate as to occupy the philosophical position of spectators
_ab extra_ are very few in any generation.
There were exceptions, it is true, but the old cruelties went on. In 1610
a case came before the tribunal of the _Tourelle_, and when the counsel
for the accused argued at some length that sorcery was ineffectual, and
that the Devil could not destroy life, President Seguier told him that he
might spare his breath, since the court had long been convinced on those
points. And yet two years later the grand-vicars of the Bishop of
Beauvais solemnly summoned Beelzebuth, Satan, Motelu, and Briffaut, with
the four legions under their charge, to appear and sign an agreement
never again to enter the bodies of reasonable or other creatures, under
pain of excommunication! If they refused, they were to be given over to
"the power of hell to be tormented and tortured more than was customary,
three thousand years after the judgment." Under this proclamation they
all came in, like reconstructed rebels, and signed whatever document was
put before them. Toward the middle of the seventeenth century, the safe
thing was still to believe, or at any rate to profess belief. Sir Thomas
Browne, though he had written an exposure of "Vulgar Errors," testified
in court to his faith in the possibility of witchcraft. Sir Kenelm Digby,
in his "Observations on the Religio Medici," takes, perhaps, as advanced
ground as any, when he says: "Neither do I deny there are witches; I only
reserve my assent till I meet with stronger motives to carry it." The
position of even enlightened men of the world in that age might be called
semi-sceptical. La Bruyere, no doubt, expresses the average of opinion:
"Que penser de la magie et du sortilege? La theorie en est obscurcie, les
principes vagues, incertains, et qui approchent du visionnaire; mais il y
a des faits embarrassants, affirmes par des hommes graves qui les ont
vus; les admettre tous, ou les nier tous, parait un egal inconvenient, et
j'ose dire qu'en cela comme en toutes les choses extraordinaires et qui
sorteut des communes regles, il y a un parti a trouver entre les ames
credules et les esprits forts."[117] Montaigne, to be sure, had long
before declared his entire disbelief, and yet the Parliament of
Bourdeaux, his own city, condemned a man to be burned as a _noueeur
d'aiguillettes_ so lately as 1718. Indeed, it was not, says Maury, till
the first quarter of the eighteenth century that one might safely publish
his incredulity in France. In Scotland, witches were burned for the last
time in 1722. Garinet cites the case of a girl near Amiens possessed by
three demons,--Mimi, Zozo, and Crapoulet,--in 1816.
The two beautiful volumes of Mr. Upham are, so far as I know, unique in
their kind. It is, in some respects, a clinical lecture on human nature,
as well as on the special epidemical disease under which the patient is
laboring. He has written not merely a history of the so-called Salem
Witchcraft, but has made it intelligible by a minute account of the place
where the delusion took its rise, the persons concerned in it, whether as
actors or sufferers, and the circumstances which led to it. By deeds,
wills, and the records of courts and churches, by plans, maps, and
drawings, he has recreated Salem Village as it was two hundred years ago,
so that we seem wellnigh to talk with its people and walk over its
fields, or through its cart-tracks and bridle-roads. We are made partners
in parish and village feuds, we share in the chimney-corner gossip, and
learn for the first time how many mean and merely human motives, whether
consciously or unconsciously, gave impulse and intensity to the passions
of the actors in that memorable tragedy which dealt the death-blow in
this country to the belief in Satanic compacts. Mr. Upham's minute
details, which give us something like a photographic picture of the
in-door and out-door scenery that surrounded the events he narrates, help
us materially to understand their origin and the course they inevitably
took. In this respect his book is original and full of new interest. To
know the kind of life these people led, the kind of place they dwelt in,
and the tenor of their thought, makes much real to us that was
conjectural before. The influences of outward nature, of remoteness from
the main highways of the world's thought, of seclusion, as the
foster-mother of traditionary beliefs, of a hard life and unwholesome
diet in exciting or obscuring the brain through the nerves and stomach,
have been hitherto commonly overlooked in accounting for the phenomena of
witchcraft. The great persecutions for this imaginary crime have always
taken place in lonely places, among the poor, the ignorant, and, above
all, the ill-fed.
One of the best things in Mr. Upham's book is the portrait of Parris, the
minister of Salem Village, in whose household the children who, under the
assumed possession of evil spirits, became accusers and witnesses, began
their tricks. He is shown to us pedantic and something of a martinet in
church discipline and ceremony, somewhat inclined to magnify his office,
fond of controversy as he was skilful and rather unscrupulous in the
conduct of it, and glad of any occasion to make himself prominent. Was he
the unconscious agent of his own superstition, or did he take advantage
of the superstition of others for purposes of his own? The question is
not an easy one to answer. Men will sacrifice everything, sometimes even
themselves, to their pride of logic and their love of victory. Bodin
loses sight of humanity altogether in his eagerness to make out his case,
and display his learning in the canon and civil law. He does not scruple
to exaggerate, to misquote, to charge his antagonists with atheism,
sorcery, and insidious designs against religion and society, that he may
persuade the jury of Europe to bring in a verdict of guilty.[118] Yet
there is no reason to doubt the sincerity of his belief. Was Parris
equally sincere? On the whole, I think it likely that he was. But if we
acquit Parris, what shall we say of the demoniacal girls? The probability
seems to be that those who began in harmless deceit found themselves at
length involved so deeply, that dread of shame and punishment drove them
to an extremity where their only choice was between sacrificing
themselves, or others to save themselves. It is not unlikely that some of
the younger girls were so far carried along by imitation or imaginative
sympathy as in some degree to "credit their own lie." Any one who has
watched or made experiments in animal magnetism knows how easy it is to
persuade young women of nervous temperaments that they are doing that by
the will of another which they really do by an obscure volition of their
own, under the influence of an imagination adroitly guided by the
magnetizer. The marvellous is so fascinating, that nine persons in ten,
if once persuaded that a thing is possible, are eager to believe it
probable, and at last cunning in convincing themselves that it is proven.
But it is impossible to believe that the possessed girls in this case did
not know how the pins they vomited got into their mouths. Mr. Upham has
shown, in the case of Anne Putnam, Jr., an hereditary tendency to
hallucination, if not insanity. One of her uncles had seen the Devil by
broad daylight in the novel disguise of a blue boar, in which shape, as a
tavern sign, he had doubtless proved more seductive than in his more
ordinary transfigurations. A great deal of light is let in upon the
question of whether there was deliberate imposture or no, by the
narrative of Rev. Mr. Turell of Medford, written in 1728, which gives us
all the particulars of a case of pretended possession in Littleton, eight
years before. The eldest of three sisters began the game, and found
herself before long obliged to take the next in age into her confidence.
By and by the youngest, finding her sisters pitied and caressed on
account of their supposed sufferings while she was neglected, began to
play off the same tricks. The usual phenomena followed. They were
convulsed, they fell into swoons, they were pinched and bruised, they
were found in the water, on the top of a tree or of the barn. To these
places they said they were conveyed through the air, and there were those
who had seen them flying, which shows how strong is the impulse which
prompts men to conspire with their own delusion, where the marvellous is
concerned. The girls did whatever they had heard or read that was common
in such cases. They even accused a respectable neighbor as the cause of
their torments. There were some doubters, but "so far as I can learn,"
says Turell, "the greater number believed and said they were under the
evil hand, or possessed by Satan." But the most interesting fact of all
is supplied by the confession of the elder sister, made eight years later
under stress of remorse. Having once begun, they found returning more
tedious than going o'er. To keep up their cheat made life a burden to
them, but they could not stop. Thirty years earlier, their juggling might
have proved as disastrous as that at Salem Village. There, parish and
boundary feuds had set enmity between neighbors, and the girls, called on
to say who troubled them, cried out upon those whom they had been wont to
hear called by hard names at home. They probably had no notion what a
frightful ending their comedy was to have; but at any rate they were
powerless, for the reins had passed out of their hands into the sterner
grasp of minister and magistrate. They were dragged deeper and deeper, as
men always are by their own lie.
The proceedings at the Salem trials are sometimes spoken of as if they
were exceptionally cruel. But, in fact, if compared with others of the
same kind, they were exceptionally humane. At a time when Baxter could
tell with satisfaction of a "_reading_ parson" eighty years old, who,
after being kept awake five days and nights, confessed his dealings with
the Devil, it is rather wonderful that no mode of torture other than
mental was tried at Salem. Nor were the magistrates more besotted or
unfair than usual in dealing with the evidence. Now and then, it is true,
a man more sceptical or intelligent than common had exposed some
pretended demoniac. The Bishop of Orleans, in 1598, read aloud to Martha
Brossier the story of the Ephesian Widow, and the girl, hearing Latin,
and taking it for Scripture, went forthwith into convulsions. He found
also that the Devil who possessed her could not distinguish holy from
profane water. But that there were deceptions did not shake the general
belief in the reality of possession. The proof in such cases could not
and ought not to be subjected to the ordinary tests. "If many natural
things," says Bodin, "are incredible and some of them incomprehensible,
_a fortiori_ the power of supernatural intelligences and the doings of
spirits are incomprehensible. But error has risen to its height in this,
that those who have denied the power of spirits and the doings of
sorcerers have wished to dispute physically concerning supernatural or
metaphysical things, which is a notable incongruity." That the girls were
really possessed, seemed to Stoughton and his colleagues the most
rational theory,--a theory in harmony with the rest of their creed, and
sustained by the unanimous consent of pious men as well as the evidence
of that most cunning and least suspected of all sorcerers, the Past,--and
how confront or cross-examine invisible witnesses, especially witnesses
whom it was a kind of impiety to doubt? Evidence that would have been
convincing in ordinary cases was of no weight against the general
prepossession. In 1659 the house of a man in Brightling, Sussex, was
troubled by a demon, who set it on fire at various times, and was
continually throwing things about. The clergy of the neighborhood held a
day of fasting and prayer in consequence. A maid-servant was afterwards
detected as the cause of the missiles. But this did not in the least
stagger Mr. Bennet, minister of the parish, who merely says: "There was a
_seeming blur_ cast, though not on the whole, yet upon some part of it,
for their servant-girl was at last found throwing some things," and goes
off into a eulogium on the "efficacy of prayer."
In one respect, to which Mr. Upham first gives the importance it
deserves, the Salem trials were distinguished from all others. Though
some of the accused had been terrified into confession, yet not one
persevered in it, but all died protesting their innocence, and with
unshaken constancy, though an acknowledgment of guilt would have saved
the lives of all. This martyr proof of the efficacy of Puritanism in the
character and conscience may be allowed to outweigh a great many sneers
at Puritan fanaticism. It is at least a testimony to the courage and
constancy which a profound religious sentiment had made common among the
people of whom these sufferers were average representatives. The accused
also were not, as was commonly the case, abandoned by their friends. In
all the trials of this kind there is nothing so pathetic as the picture
of Jonathan Cary holding up the weary arms of his wife during her trial,
and wiping away the sweat from her brow and the tears from her face.
Another remarkable fact is this, that while in other countries the
delusion was extinguished by the incredulity of the upper classes and the
interference of authority, here the reaction took place among the people
themselves, and here only was an attempt made at some legislative
restitution, however inadequate. Mr. Upham's sincere and honest
narrative, while it never condescends to a formal plea, is the best
vindication possible of a community which was itself the greatest
sufferer by the persecution which its credulity engendered.
If any lesson may be drawn from the tragical and too often disgustful
history of witchcraft, it is not one of exultation at our superior
enlightenment or shame at the shortcomings of the human intellect. It is
rather one of charity and self-distrust. When we see what inhuman
absurdities men in other respects wise and good have clung to as the
corner-stone of their faith in immortality and a divine ordering of the
world, may we not suspect that those who now maintain political or other
doctrines which seem to us barbarous and unenlightened, may be, for all
that, in the main as virtuous and clear-sighted as ourselves? While we
maintain our own side with an honest ardor of conviction, let us not
forget to allow for mortal incompetence in the other. And if there are
men who regret the Good Old Times, without too clear a notion of what
they were, they should at least be thankful that we are rid of that
misguided energy of faith which justified conscience in making men
unrelentingly cruel. Even Mr. Leckie softens a little at the thought of
the many innocent and beautiful beliefs of which a growing scepticism has
robbed us in the decay of supernaturalism. But we need not despair; for,
after all, scepticism is first cousin of credulity, and we are not
surprised to see the tough doubter Montaigne hanging up his offerings in
the shrine of our Lady of Loreto. Scepticism commonly takes up the room
left by defect of imagination, and is the very quality of mind most
likely to seek for sensual proof of supersensual things. If one came from
the dead, it could not believe; and yet it longs for such a witness, and
will put up with a very dubious one. So long as night is left and the
helplessness of dream, the wonderful will not cease from among men. While
we are the solitary prisoners of darkness, the witch seats herself at the
loom of thought, and weaves strange figures into the web that looks so
familiar and ordinary in the dry light of every-day. Just as we are
flattering ourselves that the old spirit of sorcery is laid, behold the
tables are tipping and the floors drumming all over Christendom. The
faculty of wonder is not defunct, but is only getting more and more
emancipated from the unnatural service of terror, and restored to its
proper function as a minister of delight. A higher mode of belief is the
best exorciser, because it makes the spiritual at one with the actual
world instead of hostile, or at best alien. It has been the grossly
material interpretations of spiritual doctrine that have given occasion
to the two extremes of superstition and unbelief. While the resurrection
of the body has been insisted on, that resurrection from the body which
is the privilege of all has been forgotten. Superstition in its baneful
form was largely due to the enforcement by the Church of arguments that
involved a _petitio principii_, for it is the miserable necessity of all
false logic to accept of very ignoble allies. Fear became at length its
chief expedient for the maintenance of its power; and as there is a
beneficent necessity laid upon a majority of mankind to sustain and
perpetuate the order of things they are born into, and to make all new
ideas manfully prove their right, first, to be at all, and then to be
heard, many even superior minds dreaded the tearing away of vicious
accretions as dangerous to the whole edifice of religion and society. But
if this old ghost be fading away in what we regard as the dawn of a
better day, we may console ourselves by thinking that perhaps, after all,
we are not so _much_ wiser than our ancestors. The rappings, the trance
mediums, the visions of hands without bodies, the sounding of musical
instruments without visible fingers, the miraculous inscriptions on the
naked flesh, the enlivenment of furniture,--we have invented none of
them, they are all heirlooms. There is surely room for yet another
schoolmaster, when a score of seers advertise themselves in Boston
newspapers. And if the metaphysicians can never rest till they have taken
their watch to pieces and have arrived at a happy positivism as to its
structure, though at the risk of bringing it to a no-go, we may be sure
that the majority will always take more satisfaction in seeing its hands
mysteriously move on, even if they should err a little as to the precise
time of day established by the astronomical observatories.
Footnotes:
[98] Salem Witchcraft, with an Account of Salem Village, and a
History of Opinions on Witchcraft and Kindred Subjects. By Charles W.
Upham. Boston: Wiggin and Lunt. 1867. 2 vols.
Ioannis Wieri de praestigiis daemonum, et incantationibus ac
veneficiis libri sex, postrema editione sexta aucti et recogniti.
Accessit liber apologeticus et pseudomonarchia daemonum. Cum rerum et
verborum copioso indice. Cum Caes. Maiest. Regisq: Galliarum gratia
et privelegio. Basiliae ex officina Oporiniani, 1583.
Scot's Discovery of Witchcraft: proving the common opinions of
Witches contracting with Divels, Spirits, or Familiars; and their
power to kill, torment, and consume the bodies of men, women, and
children, or other creatures by diseases or otherwise; their flying
in the Air, &c.; To be but imaginary Erronious conceptions and
novelties; Wherein also the lewde, unchristian practises of
Witchmongers, upon aged, melancholy, ignorant and superstitious
people in extorting confessions by inhumane terrors and Tortures, is
notably detected. Also The knavery and confederacy of Conjurors. The
impious blasphemy of Inchanters. The imposture of Soothsayers, and
infidelity of Atheists. The delusion of Pythonists, Figure-casters,
Astrologers, and vanity of Dreamers. The fruitlesse beggarly art of
Alchimistry. The horrible art of Poisoning and all the tricks and
conveyances of juggling and lieger-demain are fully deciphered. With
many other things opened that have long lain hidden: though very
necessary to be known for the undeceiving of Judges, Justices, and
Juries, and for the preservation of poor, aged, deformed, ignorant
people; frequently taken, arraigned, condemned and executed for
Witches, when according to a right understanding, and a good
conscience, Physick, Food, and necessaries should be administered to
him. Whereunto is added a treatise upon the nature and substance of
Spirits and Divels &c., all written and published in Anno 1584. By
Reginald Scot, Esquire. Printed by R.C. and are to be sold by Giles
Calvert dwelling at the Black Spread-Eagle, at the West-End of Pauls,
1651.
De la Demonomanie des Sorciers. A Monseigneur M. Chrestofe De Thou,
Chevalier, Seigneur de Coeli, premier President en la Cour de
Parlement et Conseiller du Roy en son prive Conseil. Reveu, Corrige,
et augmente d'une grande partie. Par I. Bodin Angevin. A Paris: Chez
Iacques Du Puys, Libraire Iure, a la Samaritaine. M.D.LXXXVII. Avec
privilege du Roy.
Magica, seu mirabilium historiarum de Spectris et Apparitionibus
spirituum: Item, de magicis et diabolicis incantationibus. De
Miraculis, Oraculis, Vaticiniis, Divinationibus, Praedictionibus,
Revelationibus et aliis eiusmodi multis ac varijs praestigijs,
ludibrijs et imposturis malorum Daemonum. Libri II. Ex probatis et
fide dignis historiarum scriptoribus diligenter collecti. Islebiae,
cura, Typis et sumptibus Henningi Grossij Bibl. Lipo. 1597. Cum
privilegio.
The displaying of supposed Witchcraft wherein is affirmed that there
are many sorts of Deceivers and Impostors, and divers persons under a
passive delusion of Melancholy and Fancy. But that there is a
corporeal league made betwixt the Devil and the Witch, or that he
sucks on the Witch's body, has carnal copulation, or that Witches are
turned into Cats, Dogs, raise Tempests or the like is utterly denied
and disproved. Wherein is also handled, The existence of Angels and
Spirits, the truth of Apparitions, the Nature of Astral and Sydereal
Spirits, the force of Charms and Philters; with other abstruse
matters. By John Webster, Practitioner in Physick. Falsa etenim
opiniones Hominum non solum surdos sed et coecos faciunt, ita ut
videre nequeant quae aliis perspicua apparent. Galen. lib. 8, de
Comp. Med. London: Printed by I.M. and are to be sold by the
booksellers in London. 1677.
Sadducismus Triumphatus: or Full and Plain Evidence concerning
Witches and Apparitions. In two Parts. The First treating of their
Possibility; the Second of their Real Existence. By Joseph Glanvil,
late Chaplain in Ordinary to His Majesty, and Fellow of the Royal
Society. The third edition. The advantages whereof above the former,
the Reader may understand out of Dr H. More's Account prefixed
therunto. With two Authentick, but wonderful Stories of certain
Swedish Witches. Done into English by A. Horneck DD. London, Printed
for S.L. and are to be sold by Anth. Baskerville at the Bible, the
corner of Essex-street, without Temple-Bar. M.DCLXXXIX.
Demonologie ou Traitte des Demons et Sorciers: De leur puissance et
impuissance: Par Fr. Perraud. Ensemble L'Antidemon de Mascon, ou
Histoire Veritable de ce qu'un Demon a fait et dit, il y a quelques
annees en la maison dudit Sr. Perreaud a Mascon. I. Jacques iv. 7, 8.
"Resistez au Diable, et il s'enfuira de vous. Approchez vous de Dieu,
et il s'approchera de vous." A Geneve, chez Pierre Aubert. M,DC,LIII.
The Wonders of the Invisible World. Being an account of the tryals of
several witches lately executed in New-England. By Cotton Mather,
D.D. To which is added a farther account of the tryals of the New
England Witches. By Increase Mather, D.D., President of Harvard
College. London: John Russell Smith, Soho Square. 1862. (First
printed in Boston, 1692.)
I.N.D.N.J.C. Dissertatio Juridica de Lamiis earumque processu
criminali, _Von Hexen und dem Peinl. Process wider dieselben_, Quam,
auxiliante Divina Gratia, Consensu et Authoritate Magnifici JCtorum
Ordinis in illustribus Athenis Salanis sub praesidio Magnifici,
Nobilissimi, Amplissimi, Consultissimi, atque Excellentissimi Dn.
Ernesti Frider. _Schroeter_ hereditarii in _Wickerstaedt_, JCti et
Antecessoris hujus Salanae Famigeratissimi, Consiliarii Saxonici,
Curiae Provincialis, Facultatis Juridicae, et Scabinatus Assessoris
longe Gravissimi, Domini Patroni Praeceptoris et Promotoris sui nullo
non honoris et observantiae cultu sancte devenerandi, colendi,
publicae Eruditorum censurae subjicit Michael Paris _Walburger_,
Groebziga Anhaltinus, in Acroaterio JCtorum ad diem 1. Maj. A. 1670.
Editio Tertia. Jenae, Typis Pauli Ehrichii, 1707.
Histoire de Diables de Loudun, ou de la Possession des Religieuses
Ursulines, et de la condemnation et du suplice d'Urbain Grandier,
Cure de la meme ville. Cruels effets de la Vengeance du Cardinal de
Richelieu. A Amsterdam Aux depens de la Compagnie. M.DCC.LII.
A view of the Invisible World, or General History of Apparitions.
Collected from the best Authorities, both Antient and Modern, and
attested by Authors of the highest Reputation and Credit. Illustrated
with a Variety of Notes and parallel Cases; in which some Account of
the Nature and Cause of Departed Spirits visiting their former
Stations by returning again into the present World, is treated in a
Manner different to the prevailing Opinions of Mankind. And an
Attempt is made from Rational Principles to account for the Species
of such supernatural Appearances, when they may be suppos'd
consistent with the Divine Appointment in the Government of the
World. With the sentiments of Monsieur Le Clerc, Mr. Locke, Mr.
Addison, and Others on this important Subject. In which some humorous
and diverting instances are remark'd, in order to divert that Gloom
of Melancholy that naturally arises in the Human Mind, from reading
or meditating on such Subjects Illustrated with suitable Cuts.
London: Printed in the year M,DCC,LII. [Mainly from DeFoe's "History
of Apparitions."]
Satan's Invisible World discovered; or, a choice Collection of Modern
Relations, proving evidently, against the Atheists of this present
Age, that there are Devils, Spirits, Witches and Apparitions, from
Authentic Records, Attestations of Witnesses, and undoubted Verity.
To which is added that marvellous History of Major Weir and his
Sister, the Witches of Balgarran, Pittenweem and Calder, &c. By
George Sinclair, late Professor of Philosophy in Glasgow. No man
should be vain that he can injure the merit of a Book; for the
meanest rogue may burn a City or kill a Hero; whereas he could never
build the one, or equal the other. Sir George M'Kenzie, Edinburgh:
Sold by P. Anderson, Parliament Square. M.DCC.LXXX.
La Magie et l'Astrologie dans I'Antiquite et au Moyen Age, ou Etude
sur les superstitions paiennes qui se sont perpetuees jusqu'a nos
jours. Par L.F. Alfred Maury. Troisieme Edition revue et corrigee.
Paris: Didier. 1864.
[99] Lucian, in his "Liars," puts this opinion into the mouth of
Arignotus. The theory by which Lucretius seeks to explain
apparitions, though materialistic, seems to allow some influence also
to the working of imagination. It is hard otherwise to explain how
his _simulacra_, (which are not unlike the _astral spirits_ of later
times) should appear in dreams.
Quae simulacra....
.... nobis vigilantibus obvia mentes
terrificant atque in somnis, cum saepe figuras
contuimur miras simulacraque luce carentum
quae nos horrifice languentis saepe sopore
excierunt.
_De Rer. Nat._ IV. 33-37, ed. Munro.
[100] Pliny's Letters, VII. 27. Melmoth's translation.
[101] Something like this is the speech of Don Juan, after the statue
of Don Gonzales has gone out:
"Pero todas son ideas
Que da a la imaginacion
El temor; y temer muertos
Es muy villano temor.
Que si un cuerpo noble, vivo,
Con potencias y razon
Y con alma no se tema,
?Quien cuerpos muertos temio?"
_El Burlador de Sevilla_, A. iii. s. 15.
[102] Theatre Francais au Moyen Age (Monmerque et Michel), pp. 139,
140.
[103]
"There sat Auld Nick in shape o' beast,
A towzy tyke, black, grim, an' large,
To gie them music was his charge."
[104] Hence, perhaps, the name Valant applied to the Devil, about the
origin of which Grimm is in doubt.
[105] One foot of the Greek Empusa was an ass's hoof.
[106] Salt was forbidden at these witch-feasts.
[107] De Lamiis, p. 59 _et seq_.
[108] If the _Blokula_ of the Swedish witches be a reminiscence of
this, it would seem to point back to remote times and heathen
ceremonies. But it is so impossible to distinguish what was put into
the mind of those who confessed by their examining torturers from
what may have been there before, the result of a common superstition,
that perhaps, after all, the meeting on mountains may have been
suggested by what Pliny says of the dances of Satyrs on Mount Atlas.
[109] Wierus, whose book was published not long after Faust's death,
apparently doubted the whole story, for he alludes to it with an _ut
fertur,_ and plainly looked on him as a mountebank.
[110] See Grimm's D.M., under _Hexenfart, Wutendes Heer_, &c.
[111] Some Catholics, indeed, affirmed that he himself was the son of
a demon who lodged in his father's house under the semblance of a
merchant. Wierus says that a bishop preached to that effect in 1565,
and gravely refutes the story.
[112] Footnote: Melancthon, however, used to tell of a possessed girl
in Italy who knew no Latin, but the Devil in her, being asked by
Bonaroico, a Bolognese professor, what was the best verse in Virgil,
answered at once:--
"Discite justitiam moniti, et non temnere divos,"--
a somewhat remarkable concession on the part of a fallen angel.
[113] This story seems mediaeval and Gothic enough, but is hardly
more so than bringing the case of the Furies _v._ Orestes before the
Areopagus, and putting Apollo in the witness-box, as Aeschylus has
done. The classics, to be sure, are always so classic! In the
_Eumenides_, Apollo takes the place of the good angel. And why not?
For though a demon, and a lying one, he has crept in to the calendar
under his other nnme of Helios as St. Hellas. Could any of his
oracles have foretold this?
[114] Mr. Leckie, in his admirable chapter on Witchcraft, gives a
little more credit to the enlightenment of the Church of England in
this matter than it would seem fairly to deserve. More and Glanvil
were faithful sons of the Church; and if the persecution of witches
was especially rife during the ascendency of the Puritans, it was
because they happened to be in power while there was a reaction
against Sadducism. All the convictions were under the statute of
James I., who was no Puritan. After the restoration, the reaction was
the other way, and Hobbism became the fashion. It is more
philosophical to say that the age believes this and that, than that
the particular men who live in it do so.
[115] I have no means of ascertaining whether he did or not. He was
more probably charged with it by the inquisitors. Mr. Leckie seems to
write of him only upon hearsay, for he calls him Peter "of Apono,"
apparently translating a French translation of the Latin "Aponus."
The only book attributed to him that I have ever seen is itself a
kind of manual of magic.
[116] "With the names and surnames," says Bodin, indignantly, "of
seventy-two princes, and of seven million four hundred and five
thousand nine hundred and twenty-six devils, _errors excepted_."
[117] Cited by Maury, p. 221, note 4.
[118] There is a kind of compensation in the fact that he himself
lived to be accused of sorcery and Judaism.
SHAKESPEARE ONCE MORE.
It may be doubted whether any language be rich enough to maintain more
than one truly great poet,--and whether there be more than one period,
and that very short, in the life of a language, when such a phenomenon as
a great poet is possible. It may be reckoned one of the rarest pieces of
good-luck that ever fell to the share of a race, that (as was true of
Shakespeare) its most rhythmic genius, its acutest intellect, its
profoundest imagination, and its healthiest understanding should have
been combined in one man, and that he should have arrived at the full
development of his powers at the moment when the material in which he was
to work--that wonderful composite called English, the best result of the
confusion of tongues--was in its freshest perfection. The
English-speaking nations should build a monument to the misguided
enthusiasts of the Plain of Shinar; for, as the mixture of many bloods
seems to have made them the most vigorous of modern races, so has the
mingling of divers speeches given them a language which is perhaps the
noblest vehicle of poetic thought that ever existed.
Had Shakespeare been born fifty years earlier, he would have been cramped
by a book-language not yet flexible enough for the demands of rhythmic
emotion, not yet sufficiently popularized for the natural and familiar
expression of supreme thought, not yet so rich in metaphysical phrase as
to render possible that ideal representation of the great passions which
is the aim and end of Art, not yet subdued by practice and general
consent to a definiteness of accentuation essential to ease and congruity
of metrical arrangement. Had he been born fifty years later, his ripened
manhood would have found itself in an England absorbed and angry with the
solution of political and religious problems, from which his whole nature
was averse, instead of in that Elizabethan social system, ordered and
planetary in functions and degrees as the angelic hierarchy of the
Areopagite, where his contemplative eye could crowd itself with various
and brilliant picture, and whence his impartial brain--one lobe of which
seems to have been Normanly refined and the other Saxonly
sagacious--could draw its morals of courtly and worldly wisdom, its
lessons of prudence and magnanimity. In estimating Shakespeare, it should
never be forgotten, that, like Goethe, he was essentially observer and
artist, and incapable of partisanship. The passions, actions, sentiments,
whose character and results he delighted to watch and to reproduce, are
those of man in society as it existed; and it no more occurred to him to
question the right of that society to exist than to criticise the divine
ordination of the seasons. His business was with men as they were, not
with man as he ought to be,--with the human soul as it is shaped or
twisted into character by the complex experience of life, not in its
abstract essence, as something to be saved or lost. During the first half
of the seventeenth century, the centre of intellectual interest was
rather in the other world than in this, rather in the region of thought
and principle and conscience than in actual life. It was a generation in
which the poet was, and felt himself, out of place. Sir Thomas Browne,
out most imaginative mind since Shakespeare, found breathing-room, for a
time, among the "_O altitudines!_" of religious speculation, but soon
descended to occupy himself with the exactitudes of science. Jeremy
Taylor, who half a century earlier would have been Fletcher's rival,
compels his clipped fancy to the conventual discipline of prose, (Maid
Marian turned nun,) and waters his poetic wine with doctrinal eloquence.
Milton is saved from making total shipwreck of his large-utteranced
genius on the desolate Noman's Land of a religious epic only by the lucky
help of Satan and his colleagues, with whom, as foiled rebels and
republicans, he cannot conceal his sympathy. As purely poet, Shakespeare
would have come too late, had his lot fallen in that generation. In mind
and temperament too exoteric for a mystic, his imagination could not have
at once illustrated the influence of his epoch and escaped from it, like
that of Browne; the equilibrium of his judgment, essential to him as an
artist, but equally removed from propagandism, whether as enthusiast or
logician, would have unfitted him for the pulpit; and his intellectual
being was too sensitive to the wonder and beauty of outward life and
Nature to have found satisfaction, as Milton's could, (and perhaps only
by reason of his blindness,) in a world peopled by purely imaginary
figures. We might fancy him becoming a great statesman, but he lacked the
social position which could have opened that career to him. What we mean
when we say _Shakespeare_, is something inconceivable either during the
reign of Henry the Eighth, or the Commonwealth, and which would have been
impossible after the Restoration.
All favorable stars seem to have been in conjunction at his nativity.
The Reformation had passed the period of its vinous fermentation, and its
clarified results remained as an element of intellectual impulse and
exhilaration; there were small signs yet of the acetous and putrefactive
stages which were to follow in the victory and decline of Puritanism. Old
forms of belief and worship still lingered, all the more touching to
Fancy, perhaps, that they were homeless and attainted; the light of
sceptic day was baffled by depths of forest where superstitious shapes
still cowered, creatures of immemorial wonder, the raw material of
Imagination. The invention of printing, without yet vulgarizing letters,
had made the thought and history of the entire past contemporaneous;
while a crowd of translators put every man who could read in inspiring
contact with the select souls of all the centuries. A new world was thus
opened to intellectual adventure at the very time when the keel of
Columbus had turned the first daring furrow of discovery in that
unmeasured ocean which still girt the known earth with a beckoning
horizon of hope and conjecture, which was still fed by rivers that flowed
down out of primeval silences, and which still washed the shores of
Dreamland. Under a wise, cultivated, and firm-handed monarch also, the
national feeling of England grew rapidly more homogeneous and intense,
the rather as the womanhood of the sovereign stimulated a more chivalric
loyalty,--while the new religion, of which she was the defender, helped
to make England morally, as it was geographically, insular to the
continent of Europe.
If circumstances could ever make a great national poet, here were all the
elements mingled at melting-heat in the alembic, and the lucky moment of
projection was clearly come. If a great national poet could ever avail
himself of circumstances, this was the occasion,--and, fortunately,
Shakespeare was equal to it. Above all, we may esteem it lucky that he
found words ready to his use, original and untarnished,--types of thought
whose sharp edges were unworn by repeated impressions. In reading
Hakluyt's Voyages, we are almost startled now and then to find that even
common sailors could not tell the story of their wanderings without
rising to an almost Odyssean strain, and habitually used a diction that
we should be glad to buy back from desuetude at any cost. Those who look
upon language only as anatomists of its structure, or who regard it as
only a means of conveying abstract truth from mind to mind, as if it were
so many algebraic formulae, are apt to overlook the fact that its being
alive is all that gives it poetic value. We do not mean what is
technically called a living language,--the contrivance, hollow as a
speaking-trumpet, by which breathing and moving bipeds, even now, sailing
o'er life's solemn main, are enabled to hail each other and make known
their mutual shortness of mental stores,--but one that is still hot from
the hearts and brains of a people, not hardened yet, but moltenly ductile
to new shapes of sharp and clear relief in the moulds of new thought. So
soon as a language has become literary, so soon as there is a gap between
the speech of books and that of life, the language becomes, so far as
poetry is concerned, almost as dead as Latin, and (as in writing Latin
verses) a mind in itself essentially original becomes in the use of such
a medium of utterance unconsciously reminiscential and reflective, lunar
and not solar, in expression and even in thought. For words and thoughts
have a much more intimate and genetic relation, one with the other, than
most men have any notion of; and it is one thing to use our mother-tongue
as if it belonged to us, and another to be the puppets of an
overmastering vocabulary. "Ye know not," says Ascham, "what hurt ye do to
Learning, that care not for Words, but for Matter, and so make a Divorce
betwixt the Tongue and the Heart." _Lingua Toscana in bocca Romana_ is
the Italian proverb; and that of poets should be, _The tongue of the
people in the mouth of the scholar_. I imply here no assent to the early
theory, _or,_ at any rate, practice, of Wordsworth, who confounded
plebeian modes of thought with rustic forms of phrase, and then atoned
for his blunder by absconding into a diction more Latinized than that of
any poet of his century.
Shakespeare was doubly fortunate. Saxon by the father and Norman by the
mother, he was a representative Englishman. A country boy, he learned
first the rough and ready English of his rustic mates, who knew how to
make nice verbs and adjectives courtesy to their needs. Going up to
London, he acquired the _lingua aulica_ precisely at the happiest moment,
just as it was becoming, in the strictest sense of the word,
_modern,_--just as it had recruited itself, by fresh impressments from
the Latin and Latinized languages, with new words to express the new
ideas of an enlarging intelligence which printing and translation were
fast making cosmopolitan,--words which, in proportion to their novelty,
and to the fact that the mother-tongue and the foreign had not yet wholly
mingled, must have been used with a more exact appreciation of their
meaning.[119] It was in London, and chiefly by means of the stage, that a
thorough amalgamation of the Saxon, Norman, and scholarly elements of
English was brought about. Already, Puttenham, in his "Arte of English
Poesy," declares that the practice of the capital and the country within
sixty miles of it was the standard of correct diction, the _jus et norma
loquendi._ Already Spenser had almost re-created English poetry,--and it
is interesting to observe, that, scholar as he was, the archaic words
which he was at first overfond of introducing are often provincialisms of
purely English original. Already Marlowe had brought the English unrhymed
pentameter (which had hitherto justified but half its name, by being
always blank and never verse) to a perfection of melody, harmony, and
variety which has never been surpassed. Shakespeare, then, found a
language already to a certain extent _established_, but not yet fetlocked
by dictionary and grammar mongers,--a versification harmonized, but which
had not yet exhausted all its modulations, nor been set in the stocks by
critics who deal judgment on refractory feet, that will dance to Orphean
measures of which their judges are insensible. That the language was
established is proved by its comparative uniformity as used by the
dramatists, who wrote for mixed audiences, as well as by Ben Jonson's
satire upon Marston's neologisms; that it at the same time admitted
foreign words to the rights of citizenship on easier terms than now is in
good measure equally true. What was of greater import, no arbitrary line
had been drawn between high words and low; vulgar then meant simply what
was common; poetry had not been aliened from the people by the
establishment of an Upper House of vocables, alone entitled to move in
the stately ceremonials of verse, and privileged from arrest while they
forever keep the promise of meaning to the ear and break it to the sense.
The hot conception of the poet had no time to cool while he was debating
the comparative respectability of this phrase or that; but he snatched
what word his instinct prompted, and saw no indiscretion in making a king
speak as his country nurse might have taught him.[120] It was Waller who
first learned in France that to talk in rhyme alone comported with the
state of royalty. In the time of Shakespeare, the living tongue resembled
that tree which Father Huc saw in Tartary, whose leaves were
languaged,--and every hidden root of thought, every subtilest fibre of
feeling, was mated by new shoots and leafage of expression, fed from
those unseen sources in the common earth of human nature.
The Cabalists had a notion, that whoever found out the mystic word for
anything attained to absolute mastery over that thing. The reverse of
this is certainly true of poetic expression; for he who is thoroughly
possessed of his thought, who imaginatively conceives an idea or image,
becomes master of the word that shall most amply and fitly utter it.
Heminge and Condell tell us, accordingly, that there was scarce a blot in
the manuscripts they received from Shakespeare; and this is the natural
corollary from the fact that such an imagination as his is as
unparalleled as the force, variety, and beauty of the phrase in which it
embodied itself.[121] We believe that Shakespeare, like all other great
poets, instinctively used the dialect which he found current, and that
his words are not more wrested from their ordinary meaning than followed
necessarily from the unwonted weight of thought or stress of passion they
were called on to support. He needed not to mask familiar thoughts in the
weeds of unfamiliar phraseology; for the life that was in his mind could
transfuse the language of every day with an intelligent vivacity, that
makes it seem lambent with fiery purpose, and at each new reading a new
creation. He could say with Dante, that "no word had ever forced him to
say what he would not, though he had forced many a word to say what _it_
would not,"--but only in the sense that the mighty magic of his
imagination had conjured out of it its uttermost secret of power or
pathos. When I say that Shakespeare used the current language of his day,
I mean only that he habitually employed such language as was universally
comprehensible,--that he was not run away with by the hobby of any theory
as to the fitness of this or that component of English for expressing
certain thoughts or feelings. That the artistic value of a choice and
noble diction was quite as well understood in his day as in ours is
evident from the praises bestowed by his contemporaries on Drayton, and
by the epithet "well-languaged" applied to Daniel, whose poetic style is
as modern as that of Tennyson; but the endless absurdities about the
comparative merits of Saxon and Norman-French, vented by persons
incapable of distinguishing one tongue from the other, were as yet
unheard of. Hasty generalizers are apt to overlook the fact, that the
Saxon was never, to any great extent, a literary language. Accordingly,
it held its own very well in the names of common things, but failed to
answer the demands of complex ideas, derived from them. The author of
"Piers Ploughman" wrote for the people,--Chaucer for the court. We open
at random and count the Latin[122] words in ten verses of the "Vision"
and ten of the "Romaunt of the Rose," (a translation from the French,)
and find the proportion to be seven in the former and five in the latter.
The organs of the Saxon have always been unwilling and stiff in learning
languages. He acquired only about as many British words as we have Indian
ones, and I believe that more French and Latin was introduced through the
pen and the eye than through the tongue and the ear. For obvious reasons,
the question is one that must be decided by reference to prose-writers,
and not poets; and it is, we think, pretty well settled that more words
of Latin original were brought into the language in the century between
1550 and 1650 than in the whole period before or since,--and for the
simple reason, that they were absolutely needful to express new modes and
combinations of thought.[123] The language has gained immensely, by the
infusion, in richness of synonyme and in the power of expressing nice
shades of thought and feeling, but more than all in light-footed
polysyllables that trip singing to the music of verse. There are certain
cases, it is true, where the vulgar Saxon word is refined, and the
refined Latin vulgar, in poetry,--as in _sweat_ and _perspiration_; but
there are vastly more in which the Latin bears the bell. Perhaps there
might be a question between the old English _again-rising_ and
_resurrection;_ but there can be no doubt that _conscience_ is better
than _inwit_, and _remorse_ than _again-bite_. Should we translate the
title of Wordsworth's famous ode, "Intimations of Immortality," into
"Hints of Deathlessness," it would hiss like an angry gander. If, instead
of Shakespeare's
"Age cannot wither her,
Nor custom stale her infinite variety,"
we should say, "her boundless manifoldness," the sentiment would suffer
in exact proportion with the music. What homebred English could ape the
high Roman fashion of such togated words as
"The multitudinous sea incarnadine,"--
where the huddling epithet implies the tempest-tossed soul of the
speaker, and at the same time pictures the wallowing waste of ocean more
vividly than the famous phrase of Aeschylus does its rippling sunshine?
Again, _sailor_ is less poetical than _mariner_, as Campbell felt, when
he wrote,
"Ye mariners of England,"
and Coleridge, when he chose
"It was an ancient mariner,"
rather than
"It was an elderly seaman";
for it is as much the charm of poetry that it suggest a certain
remoteness and strangeness as familiarity; and it is essential not only
that we feel at once the meaning of the words in themselves, but also
their melodic meaning in relation to each other, and to the sympathetic
variety of the verse. A word once vulgarized can never be rehabilitated.
We might say now a _buxom_ lass, or that a chambermaid was _buxom_, but
we could not use the term, as Milton did, in its original sense of
_bowsome_,--that is, _lithe, gracefully bending_.[124]
But the secret of force in writing lies not so much in the pedigree of
nouns and adjectives and verbs, as in having something that you believe
in to say, and making the parts of speech vividly conscious of it. It is
when expression becomes an act of memory, instead of an unconscious
necessity, that diction takes the place of warm and hearty speech. It is
not safe to attribute special virtues (as Bosworth, for example, does to
the Saxon) to words of whatever derivation, at least in poetry. Because
Lear's "oak-cleaving thunderbolts," and "the all-dreaded thunder-stone"
in "Cymbeline" are so fine, we would not give up Milton's Virgilian
"fulmined over Greece," where the verb in English conveys at once the
idea of flash and reverberation, but avoids that of riving and
shattering. In the experiments made for casting the great bell for the
Westminster Tower, it was found that the superstition which attributed
the remarkable sweetness and purity of tone in certain old bells to the
larger mixture of silver in their composition had no foundation in fact
It was the cunning proportion in which the ordinary metals were balanced
against each other, the perfection of form, and the nice gradations of
thickness, that wrought the miracle. And it is precisely so with the
language of poetry. The genius of the poet will tell him what word to use
(else what use in his being poet at all?); and even then, unless the
proportion and form, whether of parts or whole, be all that Art requires
and the most sensitive taste finds satisfaction in, he will have failed
to make what shall vibrate through all its parts with a silvery
unison,--in other words, a poem.
I think the component parts of English were in the latter years of
Elizabeth thus exquisitely proportioned one to the other. Yet Bacon had
no faith in his mother-tongue, translating the works on which his fame
was to rest into what he called "the universal language," and affirming
that "English would bankrupt all our books." He was deemed a master of
it, nevertheless; and it is curious that Ben Jonson applies to him in
prose the same commendation which he gave Shakespeare in verse, saying,
that he "performed that in our tongue which may be compared or preferred
either to _insolent Greece or haughty Rome_"; and he adds this pregnant
sentence: "In short, within his view and about his time were all the wits
born that could honor a language or help study. Now things daily fall:
wits grow downwards, eloquence grows backwards." Ben had good reason for
what he said of the wits. Not to speak of science, of Galileo and Kepler,
the sixteenth century was a spendthrift of literary genius. An attack of
immortality in a family might have been looked for then as scarlet-fever
would be now. Montaigne, Tasso, and Cervantes were born within fourteen
years of each other; and in England, while Spenser was still delving over
the _propria quae maribus_, and Raleigh launching paper navies,
Shakespeare was stretching his baby hands for the moon, and the little
Bacon, chewing on his coral, had discovered that impenetrability was one
quality of matter. It almost takes one's breath away to think that
"Hamlet" and the "Novum Organon" were at the risk of teething and measles
at the same time. But Ben was right also in thinking that eloquence had
grown backwards. He lived long enough to see the language of verse become
in a measure traditionary and conventional. It was becoming so, partly
from the necessary order of events, partly because the most natural and
intense expression of feeling had been in so many ways satisfied and
exhausted,--but chiefly because there was no man left to whom, as to
Shakespeare, perfect conception gave perfection of phrase. Dante, among
modern poets, his only rival in condensed force, says: "Optimis
conceptionibus optima loquela conveniet; sed optimae conceptiones non
possunt esse nisi ubi scientia et ingenium est; ... et sic non omnibus
versificantibus optima loquela convenit, cum plerique sine scientia et
ingenio versificantur."[125]
Shakespeare must have been quite as well aware of the provincialism of
English as Bacon was; but he knew that great poetry, being universal in
its appeal to human nature, can make any language classic, and that the
men whose appreciation is immortality will mine through any dialect to
get at an original soul. He had as much confidence in his home-bred
speech as Bacon had want of it, and exclaims:--
"Not marble nor the gilded monuments
Of princes shall outlive this powerful rhyme."
He must have been perfectly conscious of his genius, and of the great
trust which he imposed upon his native tongue as the embodier and
perpetuator of it. As he has avoided obscurities in his sonnets, he would
do so _a fortiori_ in his plays, both for the purpose of immediate effect
on the stage and of future appreciation. Clear thinking makes clear
writing, and he who has shown himself so eminently capable of it in one
case is not to be supposed to abdicate intentionally in others. The
difficult passages in the plays, then, are to be regarded either as
corruptions, or else as phenomena in the natural history of Imagination,
whose study will enable us to arrive at a clearer theory and better
understanding of it.
While I believe that our language had two periods of culmination in
poetic beauty,--one of nature, simplicity, and truth, in the ballads,
which deal only with narrative and feeling,--another of Art, (or Nature
as it is ideally reproduced through the imagination,) of stately
amplitude, of passionate intensity and elevation, in Spenser and the
greater dramatists,--and that Shakespeare made use of the latter as he
found it, I by no means intend to say that he did not enrich it, or that
any inferior man could have dipped the same words out of the great poet's
inkstand. But he enriched it only by the natural expansion and
exhilaration of which it was conscious, in yielding to the mastery of a
genius that could turn and wind it like a fiery Pegasus, making it feel
its life in every limb. He enriched it through that exquisite sense of
music, (never approached but by Marlowe,) to which it seemed eagerly
obedient, as if every word said to him,
"_Bid me_ discourse, I will enchant thine ear,"--
as if every latent harmony revealed itself to him as the gold to Brahma,
when he walked over the earth where it was hidden, crying, "Here am I,
Lord! do with me what thou wilt!" That he used language with that
intimate possession of its meaning possible only to the most vivid
thought is doubtless true; but that he wantonly strained it from its
ordinary sense, that he found it too poor for his necessities, and
accordingly coined new phrases, or that, from haste or carelessness, he
violated any of its received proprieties, I do not believe. I have said
that it was fortunate for him that he came upon an age when our language
was at its best; but it was fortunate also for us, because our costliest
poetic phrase is put beyond reach of decay in the gleaming precipitate in
which it united itself with his thought.
That the propositions I have endeavored to establish have a direct
bearing in various ways upon the qualifications of whoever undertakes to
edit the works of Shakespeare will, I think, be apparent to those who
consider the matter. The hold which Shakespeare has acquired and
maintained upon minds so many and so various, in so many vital respects
utterly unsympathetic and even incapable of sympathy with his own, is one
of the most noteworthy phenomena in the history of literature. That he
has had the most inadequate of editors, that, as his own Falstaff was the
cause of the wit, so he has been the cause of the foolishness that was in
other men, (as where Malone ventured to discourse upon his metres, and
Dr. Johnson on his imagination,) must be apparent to every one,--and also
that his genius and its manifestations are so various, that there is no
commentator but has been able to illustrate him from his own peculiar
point of view or from the results of his own favorite studies. But to
show that he was a good common lawyer, that he understood the theory of
colors, that he was an accurate botanist, a master of the science of
medicine, especially in its relation to mental disease, a profound
metaphysician, and of great experience and insight in politics,--all
these, while they may very well form the staple of separate treatises,
and prove, that, whatever the extent of his learning, the range and
accuracy of his knowledge were beyond precedent or later parallel, are
really outside the province of an editor.
We doubt if posterity owe a greater debt to any two men living in 1623
than to the two obscure actors who in that year published the first folio
edition of Shakespeare's plays. But for them, it is more than likely that
such of his works as had remained to that time unprinted would have been
irrecoverably lost, and among them were "Julius Caesar," "The Tempest,"
and "Macbeth." But are we to believe them when they assert that they
present to us the plays which they reprinted from stolen and
surreptitious copies "cured and perfect of their limbs," and those which
are original in their edition "absolute in their numbers as he
[Shakespeare] conceived them"? Alas, we have read too many theatrical
announcements, have been taught too often that the value of the promise
was in an inverse ratio to the generosity of the exclamation-marks, too
easily to believe that! Nay, we have seen numberless processions of
healthy kine enter our native village unheralded save by the lusty shouts
of drovers, while a wretched calf, cursed by stepdame Nature with two
heads, was brought to us in a triumphal car, avant-couriered by a band of
music as abnormal as itself, and announced as the greatest wonder of the
age. If a double allowance of vituline brains deserve such honor, there
are few commentators on Shakespeare that would have gone afoot, and the
trumpets of Messieurs Heminge and Condell call up in our minds too many
monstrous and deformed associations.
What, then, is the value of the first folio as an authority? For eighteen
of the plays it is the only authority we have, and the only one also for
four others in their complete form. It is admitted that in several
instances Heminge and Condell reprinted the earlier quarto impressions
with a few changes, sometimes for the better and sometimes for the worse;
and it is most probable that copies of those editions (whether
surreptitious or not) had taken the place of the original prompter's
books, as being more convenient and legible. Even in these cases it is
not safe to conclude that all or even any of the variations were made by
the hand of Shakespeare himself. And where the players printed from
manuscript, is it likely to have been that of the author? The probability
is small that a writer so busy as Shakespeare must have been during his
productive period should have copied out their parts for the actors
himself, or that one so indifferent as he seems to have been to the
immediate literary fortunes of his works should have given much care to
the correction of copies, if made by others. The copies exclusively in
the hands of Heminge and Condell were, it is manifest, in some cases,
very imperfect, whether we account for the fact by the burning of the
Globe Theatre or by the necessary wear and tear of years, and (what is
worthy of notice) they are plainly more defective in some parts than in
others. "Measure for Measure" is an example of this, and we are not
satisfied with being told that its ruggedness of verse is intentional, or
that its obscurity is due to the fact that Shakespeare grew more
elliptical in his style as he grew older. Profounder in thought he
doubtless became; though in a mind like his, we believe that this would
imply only a more absolute supremacy in expression. But, from whatever
original we suppose either the quartos or the first folio to have been
printed, it is more than questionable whether the proof-sheets had the
advantage of any revision other than that of the printing-office.
Steevens was of opinion that authors in the time of Shakespeare never
read their own proof-sheets; and Mr. Spedding, in his recent edition of
Bacon, comes independently to the same conclusion.[126] We may be very
sure that Heminge and Condell did not, as vicars, take upon themselves a
disagreeable task which the author would have been too careless to
assume.
Nevertheless, however strong a case may be made out against the Folio of
1623, whatever sins of omission we may lay to the charge of Heminge and
Condell, or of commission to that of the printers, it remains the only
text we have with any claims whatever to authenticity. It should be
deferred to as authority in all cases where it does not make Shakespeare
write bad sense, uncouth metre, or false grammar, of all which we believe
him to have been more supremely incapable than any other man who ever
wrote English. Yet we would not speak unkindly even of the blunders of
the Folio. They have put bread into the mouth of many an honest editor,
publisher, and printer for the last century and a half; and he who loves
the comic side of human nature will find the serious notes of a
_variorum_ edition of Shakespeare as funny reading as the funny ones are
serious. Scarce a commentator of them all, for more than a hundred years,
but thought, as Alphonso of Castile did of Creation, that, if he had only
been at Shakespeare's elbow, he could have given valuable advice; scarce
one who did not know off-hand that there was never a seaport in
Bohemia,--as if Shakespeare's world were one which Mercator could have
projected; scarce one but was satisfied that his ten finger-tips were a
sufficient key to those astronomic wonders of poise and counterpoise, of
planetary law and cometary seeming-exception, in his metres; scarce one
but thought he could gauge like an ale-firkin that intuition whose edging
shallows may have been sounded, but whose abysses, stretching down amid
the sunless roots of Being and Consciousness, mock the plummet; scarce
one but could speak with condescending approval of that prodigious
intelligence so utterly without congener that our baffled language must
coin an adjective to qualify it, and none is so audacious as to say
Shakesperian of any other. And yet, in the midst of our impatience, we
cannot help thinking also of how much healthy mental activity this one
man has been the occasion, how much good he has indirectly done to
society by withdrawing men to investigations and habits of thought that
secluded them from baser attractions, for how many he has enlarged the
circle of study and reflection; since there is nothing in history or
politics, nothing in art or science, nothing in physics or metaphysics,
that is not sooner or later taxed for his illustration. This is partially
true of all great minds, open and sensitive to truth and beauty through
any large arc of their circumference; but it is true in an unexampled
sense of Shakespeare, the vast round of whose balanced nature seems to
have been equatorial, and to have had a southward exposure and a summer
sympathy at every point, so that life, society, statecraft, serve us at
last but as commentaries on him, and whatever we have gathered of
thought, of knowledge, and of experience, confronted with his marvellous
page, shrinks to a mere foot-note, the stepping-stone to some hitherto
inaccessible verse. We admire in Homer the blind placid mirror of the
world's young manhood, the bard who escapes from his misfortune in poems
all memory, all life and bustle, adventure and picture; we revere in
Dante that compressed force of lifelong passion which could make a
private experience cosmopolitan in its reach and everlasting in its
significance; we respect in Goethe the Aristotelian poet, wise by
weariless observation, witty with intention, the stately _Geheimerrath_
of a provincial court in the empire of Nature. As we study these, we seem
in our limited way to penetrate into their consciousness and to measure
and master their methods; but with Shakespeare it is just the other way;
the more we have familiarized ourselves with the operations of our own
consciousness, the more do we find, in reading him, that he has been
beforehand with us, and that, while we have been vainly endeavoring to
find the door of his being, he has searched every nook and cranny of our
own. While other poets and dramatists embody isolated phases of character
and work inward from the phenomenon to the special law which it
illustrates, he seems in some strange way unitary with human nature
itself, and his own soul to have been the law and life-giving power of
which his creations are only the phenomena. We justify or criticise the
characters of other writers by our memory and experience, and pronounce
them natural or unnatural; but he seems to have worked in the very stuff
of which memory and experience are made, and we recognize his truth to
Nature by an innate and unacquired sympathy, as if he alone possessed the
secret of the "ideal form and universal mould," and embodied generic
types rather than individuals. In this Cervantes alone has approached
him; and Don Quixote and Sancho, like the men and women of Shakespeare,
are the contemporaries of every generation, because they are not products
of an artificial and transitory society, but because they are animated by
the primeval and unchanging forces of that humanity which underlies and
survives the forever-fickle creeds and ceremonials of the parochial
corners which we who dwell in them sublimely call The World.
That Shakespeare did not edit his own works must be attributed, we
suspect, to his premature death. That he should not have intended it is
inconceivable. Is there not something of self-consciousness in the
breaking of Prospero's wand and burying his book,--a sort of sad
prophecy, based on self-knowledge of the nature of that man who, after
such thaumaturgy, could go down to Stratford and live there for years,
only collecting his dividends from the Globe Theatre, lending money on
mortgage, and leaning over his gate to chat and bandy quips with
neighbors? His mind had entered into every phase of human life and
thought, had embodied all of them in living creations;--had he found all
empty, and come at last to the belief that genius and its works were as
phantasmagoric as the rest, and that fame was as idle as the rumor of the
pit? However this may be, his works have come down to us in a condition
of manifest and admitted corruption in some portions, while in others
there is an obscurity which may be attributed either to an idiosyncratic
use of words and condensation of phrase, to a depth of intuition for a
proper coalescence with which ordinary language is inadequate, to a
concentration of passion in a focus that consumes the lighter links which
bind together the clauses of a sentence or of a process of reasoning in
common parlance, or to a sense of music which mingles music and meaning
without essentially confounding them. We should demand for a perfect
editor, then, first, a thorough glossological knowledge of the English
contemporary with Shakespeare; second, enough logical acuteness of mind
and metaphysical training to enable him to follow recondite processes of
thought; third, such a conviction of the supremacy of his author as
always to prefer his thought to any theory of his own; fourth, a feeling
for music, and so much knowledge of the practice of other poets as to
understand that Shakespeare's versification differs from theirs as often
in kind as in degree; fifth, an acquaintance with the world as well as
with books; and last, what is, perhaps, of more importance than all, so
great a familiarity with the working of the imaginative faculty in
general, and of its peculiar operation in the mind of Shakespeare, as
will prevent his thinking a passage dark with excess of light, and enable
him to understand fully that the Gothic Shakespeare often superimposed
upon the slender column of a single word, that seems to twist under it,
but does not,--like the quaint shafts in cloisters,--a weight of meaning
which the modern architects of sentences would consider wholly
unjustifiable by correct principle.
Many years ago, while yet Fancy claimed that right in me which Fact has
since, to my no small loss, so successfully disputed, I pleased myself
with imagining the play of Hamlet published under some _alias_, and as
the work of a new candidate in literature. Then I _played_, as the
children say, that it came in regular course before some well-meaning
doer of criticisms, who had never read the original, (no very wild
assumption, as things go,) and endeavored to conceive the kind of way in
which he would be likely to take it. I put myself in his place, and tried
to write such a perfunctory notice as I thought would be likely, in
filling his column, to satisfy his conscience. But it was a _tour de
force_ quite beyond my power to execute without grimace. I could not
arrive at that artistic absorption in my own conception which would
enable me to be natural, and found myself, like a bad actor, continually
betraying my self-consciousness by my very endeavor to hide it under
caricature. The path of Nature is indeed a narrow one, and it is only the
immortals that seek it, and, when they find it, do not find themselves
cramped therein. My result was a dead failure,--satire instead of comedy.
I could not shake off that strange accumulation which we call self, and
report honestly what I saw and felt even to myself, much less to others.
Yet I have often thought, that, unless we can so far free ourselves from
our own prepossessions as to be capable of bringing to a work of art some
freshness of sensation, and receiving from it in turn some new surprise
of sympathy and admiration,--some shock even, it may be, of instinctive
distaste and repulsion,--though we may praise or blame, weighing our
_pros_ and _cons_ in the nicest balances, sealed by proper authority, yet
we shall not criticise in the highest sense. On the other hand, unless we
admit certain principles as fixed beyond question, we shall be able to
render no adequate judgment, but only to record our impressions, which
may be valuable or not, according to the greater or less ductility of the
senses on which they are made. Charles Lamb, for example, came to the old
English dramatists with the feeling of a discoverer. He brought with him
an alert curiosity, and everything was delightful simply because it was
strange. Like other early adventurers, he sometimes mistook shining sand
for gold; but he had the great advantage of not feeling himself
responsible for the manners of the inhabitants he found there, and not
thinking it needful to make them square with any Westminster Catechism of
aesthetics. Best of all, he did not feel compelled to compare them with
the Greeks, about whom he knew little, and cared less. He took them as he
found them, described them in a few pregnant sentences, and displayed his
specimens of their growth, and manufacture. When he arrived at the
dramatists of the Restoration, so far from being shocked, he was charmed
with their pretty and unmoral ways; and what he says of them reminds us
of blunt Captain Dampier, who, in his account of the island of Timor,
remarks, as a matter of no consequence, that the natives "take as many
wives as they can maintain, and as for religion, they have none."
Lamb had the great advantage of seeing the elder dramatists as they were;
it did not lie within his province to point out what they were not.
Himself a fragmentary writer, he had more sympathy with imagination where
it gathers into the intense focus of passionate phrase than with that
higher form of it, where it is the faculty that shapes, gives unity of
design and balanced gravitation of parts. And yet it is only this higher
form of it which can unimpeachably assure to any work the dignity and
permanence of a classic; for it results in that exquisite something
called Style, which, like the grace of perfect breeding, everywhere
pervasive and nowhere emphatic, makes itself felt by the skill with which
it effaces itself, and masters us at last with a sense of indefinable
completeness. On a lower plane we may detect it in the structure of a
sentence, in the limpid expression that implies sincerity of thought; but
it is only where it combines and organizes, where it eludes observation
in particulars to give the rarer delight of perfection as a whole, that
it belongs to art. Then it is truly ideal, the _forma mentis aeterna,_
not as a passive mould into which the thought is poured, but as the
conceptive energy which finds all material plastic to its preconceived
design. Mere vividness of expression, such as makes quotable passages,
comes of the complete surrender of self to the impression, whether
spiritual or sensual, of the moment. It is a quality, perhaps, in which
the young poet is richer than the mature, his very inexperience making
him more venturesome in those leaps of language that startle us with
their rashness only to bewitch us the more with the happy ease of their
accomplishment. For this there are no existing laws of rhetoric, for it
is from such felicities that the rhetoricians deduce and codify their
statutes. It is something which cannot be improved upon or cultivated,
for it is immediate and intuitive. But this power of expression is
subsidiary, and goes only a little way toward the making of a great poet.
Imagination, where it is truly creative, is a faculty, and not a quality;
it looks before and after, it gives the form that makes all the parts
work together harmoniously toward a given end, its seat is in the higher
reason, and it is efficient only as a servant of the will. Imagination,
as it is too often misunderstood, is mere fantasy, the image-making
power, common to all who have the gift of dreams, or who can afford to
buy it in a vulgar drug as De Quincey bought it.
The true poetic imagination is of one quality, whether it be ancient or
modern, and equally subject to those laws of grace, of proportion, of
design, in whose free service, and in that alone, it can become art.
Those laws are something which do not
"Alter when they alteration find,
And bend with the remover to remove."
And they are more clearly to be deduced from the eminent examples of
Greek literature than from any other source. It is the advantage of this
select company of ancients that their works are defecated of all turbid
mixture of contemporaneousness, and have become to us pure _literature_,
our judgment and enjoyment of which cannot be vulgarized by any
prejudices of time or place. This is why the study of them is fitly
called a liberal education, because it emancipates the mind from every
narrow provincialism whether of egoism or tradition, and is the
apprenticeship that every one must serve before becoming a free brother
of the guild which passes the torch of life from age to age. There would
be no dispute about the advantages of that Greek culture which Schiller
advocated with such generous eloquence, if the great authors of antiquity
had not been degraded from teachers of thinking to drillers in grammar,
and made the ruthless pedagogues of root and inflection, instead of
companions for whose society the mind must put on her highest mood. The
discouraged youth too naturally transfers the epithet of _dead_ from the
languages to the authors that wrote in them. What concern have we with
the shades of dialect in Homer or Theocritus, provided they speak the
spiritual _lingua franca_ that abolishes all alienage of race, and makes
whatever shore of time we land on hospitable and homelike? There is much
that is deciduous in books, but all that gives them a title to rank as
literature in the highest sense is perennial. Their vitality is the
vitality not of one or another blood or tongue, but of human nature;
their truth is not topical and transitory, but of universal acceptation;
and thus all great authors seem the coevals not only of each other, but
of whoever reads them, growing wiser with him as he grows wise, and
unlocking to him one secret after another as his own life and experience
give him the key, but on no other condition. Their meaning is absolute,
not conditional; it is a property of _theirs_, quite irrespective of
manners or creed; for the highest culture, the development of the
individual by observation, reflection, and study, leads to one result,
whether in Athens or in London. The more we know of ancient literature,
the more we are struck with its modernness, just as the more we study the
maturer dramas of Shakespeare, the more we feel his nearness in certain
primary qualities to the antique and classical. Yet even in saying this,
I tacitly make the admission that it is the Greeks who must furnish us
with our standard of comparison. Their stamp is upon all the allowed
measures and weights of aesthetic criticism. Nor does a consciousness of
this, nor a constant reference to it, in any sense reduce us to the mere
copying of a bygone excellence; for it is the test of excellence in any
department of art, that it can never be bygone, and it is not mere
difference from antique models, but the _way_ in which that difference is
shown, the direction it takes, that we are to consider in our judgment of
a modern work. The model is not there to be copied merely, but that the
study of it may lead us insensibly to the same processes of thought by
which its purity of outline and harmony of parts were attained, and
enable us to feel that strength is consistent with repose, that
multiplicity is not abundance, that grace is but a more refined form of
power, and that a thought is none the less profound that the limpidity of
its expression allows us to measure it at a glance. To be possessed with
this conviction gives us at least a determinate point of view, and
enables us to appeal a case of taste to a court of final judicature,
whose decisions are guided by immutable principles. When we hear of
certain productions, that they are feeble in design, but masterly in
parts, that they are incoherent, to be sure, but have great merits of
style, we know that it cannot be true; for in the highest examples we
have, the master is revealed by his plan, by his power of making all
accessories, each in its due relation, subordinate to it, and that to
limit style to the rounding of a period or a distich is wholly to
misapprehend its truest and highest function. Donne is full of salient
verses that would take the rudest March winds of criticism with their
beauty, of thoughts that first tease us like charades and then delight us
with the felicity of their solution; but these have not saved him. He is
exiled to the limbo of the formless and the fragmentary. To take a more
recent instance,--Wordsworth had, in some respects, a deeper insight, and
a more adequate utterance of it, than any man of his generation. But it
was a piece-meal insight and utterance; his imagination was feminine, not
masculine, receptive, and not creative. His longer poems are Egyptian
sand-wastes, with here and there an oasis of exquisite greenery, a grand
image, Sphinx-like, half buried in drifting commonplaces, or the solitary
Pompey's Pillar of some towering thought. But what is the fate of a poet
who owns the quarry, but cannot build the poem? Ere the century is out he
will be nine parts dead, and immortal only in that tenth part of him
which is included in a thin volume of "beauties." Already Moxon has felt
the need of extracting this essential oil of him; and his memory will be
kept alive, if at all, by the precious material rather than the
workmanship of the vase that contains his heart. And what shall we
forebode of so many modern poems, full of splendid passages, beginning
everywhere and leading nowhere, reminding us of nothing so much as the
amateur architect who planned his own house, and forgot the staircase
that should connect one floor with another, putting it as an afterthought
on the outside?
Lichtenberg says somewhere, that it was the advantage of the ancients to
write before the great art of writing ill had been invented; and
Shakespeare may be said to have had the good luck of coming after Spenser
(to whom the debt of English poetry is incalculable) had reinvented the
art of writing well. But Shakespeare arrived at a mastery in this respect
which sets him above all other poets. He is not only superior in degree,
but he is also different in kind. In that less purely artistic sphere of
style which concerns the matter rather than the form his charm is often
unspeakable. How perfect his style is may be judged from the fact that it
never curdles into mannerism, and thus absolutely eludes imitation.
Though here, if anywhere, the style is the man, yet it is noticeable
only, like the images of Brutus, by its absence, so thoroughly is he
absorbed in his work, while he fuses thought and word indissolubly
together, till all the particles cohere by the best virtue of each. With
perfect truth he has said of himself that he writes
"All one, ever the same,
Putting invention in a noted weed,
That every word doth almost tell his name."
And yet who has so succeeded in imitating him as to remind us of him by
even so much as the gait of a single verse?[127] Those magnificent
crystallizations of feeling and phrase, basaltic masses, molten and
interfused by the primal fires of passion, are not to be reproduced by
the slow experiments of the laboratory striving to parody creation with
artifice. Mr. Matthew Arnold seems to think that Shakespeare has damaged
English poetry. I wish he had! It is true he lifted Dryden above himself
in "All for Love"; but it was Dryden who said of him, by instinctive
conviction rather than judgment, that within his magic circle none dared
tread but he. Is he to blame for the extravagances of modern diction,
which are but the reaction of the brazen age against the degeneracy of
art into artifice, that has characterized the silver period in every
literature? We see in them only the futile effort of misguided persons to
torture out of language the secret of that inspiration which should be in
themselves. We do not find the extravagances in Shakespeare himself. We
never saw a line in any modern poet that reminded us of him, and will
venture to assert that it is only poets of the second class that find
successful imitators. And the reason seems to us a very plain one. The
genius of the great poet seeks repose in the expression of itself, and
finds it at last in style, which is the establishment of a perfect mutual
understanding between the worker and his material.[128] The secondary
intellect, on the other hand, seeks for excitement in expression, and
stimulates itself into mannerism, which is the wilful obtrusion of self,
as style is its unconscious abnegation. No poet of the first class has
ever left a school, because his imagination is incommunicable; while,
just as surely as the thermometer tells of the neighborhood of an
iceberg, you may detect the presence of a genius of the second class in
any generation by the influence of his mannerism, for that, being an
artificial thing, is capable of reproduction. Dante, Shakespeare, Goethe,
left no heirs either to the form or mode of their expression; while
Milton, Sterne, and Wordsworth left behind them whole regiments uniformed
with all their external characteristics. We do not mean that great poetic
geniuses may not have influenced thought, (though we think it would be
difficult to show how Shakespeare had done so, directly and wilfully,)
but that they have not infected contemporaries or followers with
mannerism. The quality in him which makes him at once so thoroughly
English and so thoroughly cosmopolitan is that aeration of the
understanding by the imagination which he has in common with all the
greater poets, and which is the privilege of genius. The modern school,
which mistakes violence for intensity, seems to catch its breath when it
finds itself on the verge of natural expression, and to say to itself,
"Good heavens! I had almost forgotten I was inspired!" But of Shakespeare
we do not even suspect that he ever remembered it. He does not always
speak in that intense way that flames up in Lear and Macbeth through the
rifts of a soil volcanic with passion. He allows us here and there the
repose of a commonplace character, the consoling distraction of a
humorous one. He knows how to be equable and grand without effort, so
that we forget the altitude of thought to which he has led us, because
the slowly receding <DW72> of a mountain stretching downward by ample
gradations gives a less startling impression of height than to look over
the edge of a ravine that makes but a wrinkle in its flank.
Shakespeare has been sometimes taxed with the barbarism of profuseness
and exaggeration. But this is to measure him by a Sophoclean scale. The
simplicity of the antique tragedy is by no means that of expression, but
is of form merely. In the utterance of great passions, something must be
indulged to the extravagance of Nature; the subdued tones to which pathos
and sentiment are limited cannot express a tempest of the soul The range
between the piteous "no more but so," in which Ophelia compresses the
heart-break whose compression was to make her mad, and that sublime
appeal of Lear to the elements of Nature, only to be matched, if matched
at all, in the "Prometheus," is a wide one, and Shakespeare is as truly
simple in the one as in the other. The simplicity of poetry is not that
of prose, nor its clearness that of ready apprehension merely. To a
subtile sense, a sense heightened by sympathy, those sudden fervors of
phrase, gone ere one can say it lightens, that show us Macbeth groping
among the complexities of thought in his conscience-clouded mind, and
reveal the intricacy rather than enlighten it, while they leave the eye
darkened to the literal meaning of the words, yet make their logical
sequence, the grandeur of the conception, and its truth to Nature clearer
than sober daylight could. There is an obscurity of mist rising from the
undrained shallows of the mind, and there is the darkness of
thunder-cloud gathering its electric masses with passionate intensity
from the clear element of the imagination, not at random or wilfully, but
by the natural processes of the creative faculty, to brood those flashes
of expression that transcend rhetoric, and are only to be apprehended by
the poetic instinct.
In that secondary office of imagination, where it serves the artist, not
as the reason that shapes, but as the interpreter of his conceptions into
words, there is a distinction to be noticed between the higher and lower
mode in which it performs its function. It may be either creative or
pictorial, may body forth the thought or merely image it forth. With
Shakespeare, for example, imagination seems immanent in his very
consciousness; with Milton, in his memory. In the one it sends, as if
without knowing it, a fiery life into the verse,
"Sei die Braut das Wort,
Braeutigam der Geist";
in the other it elaborates a certain pomp and elevation. Accordingly, the
bias of the former is toward over-intensity, of the latter toward
over-diffuseness. Shakespeare's temptation is to push a willing metaphor
beyond its strength, to make a passion over-inform its tenement of words;
Milton cannot resist running a simile on into a fugue. One always fancies
Shakespeare _in_ his best verses, and Milton at the key-board of his
organ. Shakespeare's language is no longer the mere vehicle of thought,
it has become part of it, its very flesh and blood. The pleasure it gives
us is unmixed, direct, like that from the smell of a flower or the flavor
of a fruit. Milton sets everywhere his little pitfalls of bookish
association for the memory. I know that Milton's manner is very grand. It
is slow, it is stately, moving as in triumphal procession, with music,
with historic banners, with spoils from every time and every region, and
captive epithets, like huge Sicambrians, thrust their broad shoulders
between us and the thought whose pomp they decorate. But it is manner,
nevertheless, as is proved by the ease with which it is parodied, by the
danger it is in of degenerating into mannerism whenever it forgets
itself. Fancy a parody of Shakespeare,--I do not mean of his words, but
of his _tone_, for that is what distinguishes the master. You might as
well try it with the Venus of Melos. In Shakespeare it is always the
higher thing, the thought, the fancy, that is pre-eminent; it is Caesar
that draws all eyes, and not the chariot in which he rides, or the throng
which is but the reverberation of his supremacy. If not, how explain the
charm with which he dominates in all tongues, even under the
disenchantment of translation? Among the most alien races he is as
solidly at home as a mountain seen from different sides by many lands,
itself superbly solitary, yet the companion of all thoughts and
domesticated in all imaginations.
In description Shakespeare is especially great, and in that instinct
which gives the peculiar quality of any object of contemplation in a
single happy word that colors the impression on the sense with the mood
of the mind. Most descriptive poets seem to think that a hogshead of
water caught at the spout will give us a livelier notion of a
thunder-shower than the sullen muttering of the first big drops upon the
roof. They forget that it is by suggestion, not cumulation, that profound
impressions are made upon the imagination. Milton's parsimony (so rare in
him) makes the success of his
"Sky lowered, and, muttering thunder, some sad drops
Wept at completion of the mortal sin."
Shakespeare understood perfectly the charm of indirectness, of making his
readers seem to discover for themselves what he means to show them. If he
wishes to tell that the leaves of the willow are gray on the under side,
he does not make it a mere fact of observation by bluntly saying so, but
makes it picturesquely reveal itself to us as it might in Nature:--
"There is a willow grows athwart the flood,
That shows his _hoar_ leaves in the glassy stream."
Where he goes to the landscape for a comparison, he does not ransack wood
and field for specialties, as if he were gathering simples, but takes one
image, obvious, familiar, and makes it new to us either by sympathy or
contrast with his own immediate feeling. He always looked upon Nature
with the eyes of the mind. Thus he can make the melancholy of autumn or
the gladness of spring alike pathetic:--
"That time of year thou mayst in me behold,
When yellow leaves, or few, or none, do hang
Upon those boughs that shake against the cold,
Bare ruined choirs where late the sweet birds sang."
Or again:--
"From thee have I been absent in the spring,
When proud-pied April, dressed in all his trim,
Hath put a spirit of youth in everything,
That heavy Saturn leaped and laughed with him."
But as dramatic poet, Shakespeare goes even beyond this, entering so
perfectly into the consciousness of the characters he himself has
created, that he sees everything through their peculiar mood, and makes
every epithet, as if unconsciously, echo and re-echo it. Theseus asks
Hermia,--
"Can you endure the livery of a nun,
For aye to be in shady cloister mewed,
To live a _barren_ sister all your life,
Chanting faint hymns to the _cold fruitless_ moon?"
When Romeo must leave Juliet, the private pang of the lovers becomes a
property of Nature herself, and
"_Envious_ streaks
Do lace the _severing_ clouds in yonder east."
But even more striking is the following instance from Macbeth:--
"The raven himself is hoarse
That croaks the fatal enterance of Duncan
Under your battlements."
Here Shakespeare, with his wonted tact, makes use of a vulgar
superstition, of a type in which mortal presentiment is already embodied,
to make a common ground on which the hearer and Lady Macbeth may meet.
After this prelude we are prepared to be possessed by her emotion more
fully, to feel in her ears the dull tramp of the blood that seems to make
the raven's croak yet hoarser than it is, and to betray the stealthy
advance of the mind to its fell purpose. For Lady Macbeth hears not so
much the voice of the bodeful bird as of her own premeditated murder, and
we are thus made her shuddering accomplices before the fact. Every image
receives the color of the mind, every word throbs with the pulse of one
controlling passion. The epithet _fatal_ makes us feel the implacable
resolve of the speaker, and shows us that she is tampering with her
conscience by putting off the crime upon the prophecy of the Weird
Sisters to which she alludes. In the word _battlements_, too, not only is
the fancy led up to the perch of the raven, but a hostile image takes the
place of a hospitable; for men commonly speak of receiving a guest under
their roof or within their doors. That this is not over-ingenuity, seeing
what is not to be seen, nor meant to be seen, is clear to me from what
follows. When Duncan and Banquo arrive at the castle, their fancies, free
from all suggestion of evil, call up only gracious and amiable images.
The raven was but the fantastical creation of Lady Macbeth's over-wrought
brain.
"This castle hath a pleasant seat, the air
Nimbly and sweetly doth commend itself
Unto our gentle senses.
This _guest_ of summer,
The _temple-haunting_ martlet, doth approve
By his _loved mansionry_ that the heaven's breath
Smells _wooingly_ here; no jutty, frieze,
Buttress, or coigne of vantage, but this bird
Hath made his pendent bed and procreant cradle."
The contrast here cannot but be as intentional as it is marked. Every
image is one of welcome, security, and confidence. The summer, one may
well fancy, would be a very different hostess from her whom we have just
seen expecting _them_. And why _temple-haunting_, unless because it
suggests sanctuary? _O immaginativa, che si ne rubi delle cose di fuor_,
how infinitely more precious are the inward ones thou givest in return!
If all this be accident, it is at least one of those accidents of which
only this man was ever capable. I divine something like it now and then
in Aeschylus, through the mists of a language which will not let me be
sure of what I see, but nowhere else. Shakespeare, it is true, had, as I
have said, as respects English, the privilege which only first-comers
enjoy. The language was still fresh from those sources at too great a
distance from which it becomes fit only for the service of prose.
Wherever he dipped, it came up clear and sparkling, undefiled as yet by
the drainage of literary factories, or of those dye-houses where the
machine-woven fabrics of sham culture are up to the last
desperate style of sham sentiment. Those who criticise his diction as
sometimes extravagant should remember that in poetry language is
something more than merely the vehicle of thought, that it is meant to
convey the sentiment as much as the sense, and that, if there is a beauty
of use, there is often a higher use of beauty.
What kind of culture Shakespeare had is uncertain; how much he had is
disputed; that he had as much as he wanted, and of whatever kind he
wanted, must be clear to whoever considers the question. Dr. Farmer has
proved, in his entertaining essay, that he got everything at second-hand
from translations, and that, where his translator blundered, he loyally
blundered too. But Goethe, the man of widest acquirement in modern times,
did precisely the same thing. In his character of poet he set as little
store by useless learning as Shakespeare did. He learned to write
hexameters, not from Homer, but from Voss, and Voss found them faulty;
yet somehow _Hermann und Dorothea_ is more readable than _Luise_. So far
as all the classicism then attainable was concerned, Shakespeare got it
as cheap as Goethe did, who always bought it ready-made. For such
purposes of mere aesthetic nourishment Goethe always milked other
minds,--if minds those ruminators and digesters of antiquity into asses'
milk may be called. There were plenty of professors who were forever
assiduously browsing in vales of Enna and on Pentelican <DW72>s among the
vestiges of antiquity, slowly secreting lacteous facts, and not one of
them would have raised his head from that exquisite pasturage, though Pan
had made music through his pipe of reeds. Did Goethe wish to work up a
Greek theme? He drove out Herr Boettiger, for example, among that fodder
delicious to him for its very dryness, that sapless Arcadia of
scholiasts, let him graze, ruminate, and go through all other needful
processes of the antiquarian organism, then got him quietly into a corner
and milked him. The product, after standing long enough, mantled over
with the rich Goethean cream, from which a butter could be churned, if
not precisely classic, quite as good as the ancients could have made out
of the same material. But who has ever read the _Achilleis_, correct in
all _un_essential particulars as it probably is?
It is impossible to conceive that a man, who, in other respects, made
such booty of the world around him, whose observation of manners was so
minute, and whose insight into character and motives, as if he had been
one of God's spies, was so unerring that we accept it without question,
as we do Nature herself, and find it more consoling to explain his
confessedly immense superiority by attributing it to a happy instinct
rather than to the conscientious perfecting of exceptional powers till
practice made them seem to work independently of the will which still
directed them,--it is impossible that such a man should not also have
profited by the converse of the cultivated and quick-witted men in whose
familiar society he lived, that he should not have over and over again
discussed points of criticism and art with them, that he should not have
had his curiosity, so alive to everything else, excited about those
ancients whom university men then, no doubt, as now, extolled without too
much knowledge of what they really were, that he should not have heard
too much rather than too little of Aristotle's _Poetics_, Quinctilian's
_Rhetoric_, Horace's _Art of Poetry_, and the _Unities_, especially from
Ben Jonson,--in short, that he who speaks of himself as
"Desiring this man's art and that man's scope,
With what he most enjoyed contented least,"
and who meditated so profoundly on every other topic of human concern,
should never have turned his thought to the principles of that art which
was both the delight and business of his life, the bread-winner alike for
soul and body. Was there no harvest of the ear for him whose eye had
stocked its garners so full as wellnigh to forestall all after-comers?
Did he who could so counsel the practisers of an art in which he never
arrived at eminence, as in Hamlet's advice to the players, never take
counsel with himself about that other art in which the instinct of the
crowd, no less than the judgment of his rivals, awarded him an easy
pre-eminence? If he had little Latin and less Greek, might he not have
had enough of both for every practical purpose on this side pedantry? The
most extraordinary, one might almost say contradictory, attainments have
been ascribed to him, and yet he has been supposed incapable of what was
within easy reach of every boy at Westminster School. There is a
knowledge that comes of sympathy as living and genetic as that which
comes of mere learning is sapless and unprocreant, and for this no
profound study of the languages is needed.
If Shakespeare did not know the ancients, I think they were at least as
unlucky in not knowing him. But is it incredible that he may have laid
hold of an edition of the Greek tragedians, _Graece et Latine_, and then,
with such poor wits as he was master of, contrived to worry some
considerable meaning out of them? There are at least one or two
coincidences which, whether accidental or not, are curious, and which I
do not remember to have seen noticed. In the _Electra_ of Sophocles,
which is almost identical in its leading motive with _Hamlet_, the Chorus
consoles Electra for the supposed death of Orestes in the same
commonplace way which Hamlet's uncle tries with him.
[Greek: Thnaetou pephukas patros, Aelektra phronei;
Thnaetos d' Orestaes; oste mae lian stene,
Pasin gar aemin tout' opheiletai pathein.]
"Your father lost a father;
That father lost, lost his....
But to persever
In obstinate condolement is a course
Of impious stubbornness....
'T is common; all that live must die."
Shakespeare expatiates somewhat more largely, but the sentiment in both
cases is almost verbally identical. The resemblance is probably a chance
one, for commonplace and consolation were always twin sisters, whom
always to escape is given to no man; but it is nevertheless curious. Here
is another, from the _Oedipus Coloneus_:--
[Greek: Tois toi dikaiois cho brachus nika megan.]
"Thrice is he armed that hath his quarrel just."
Hamlet's "prophetic soul" may be matched with the [Greek: promantis
thumos] of Peleus, (Eurip. Androm. 1075,) and his "sea of troubles," with
the [Greek: kakon pelagos] of Theseus in the _Hippolytus_, or of the
Chorus in the _Hercules Furens_. And, for manner and tone, compare the
speeches of Pheres in the _Alcestis_, and Jocasta in the _Phoenissae_,
with those of Claudio in _Measure for Measure_, and Ulysses in _Troilus
and Cressida_.
The Greek dramatists were somewhat fond of a trick of words in which
there is a reduplication of sense as well as of assonance, as in the
_Electra_:--
[Greek: Alektra gaeraskousan anumenaia te].
So Shakespeare:--
"Unhouseled, disappointed, unaneled";
and Milton after him, or, more likely, after the Greek:--
"Unrespited, unpitied, unreprieved."[129]
I mention these trifles, in passing, because they have interested me, and
therefore may interest others. I lay no stress upon them, for, if once
the conductors of Shakespeare's intelligence had been put in connection
with those Attic brains, he would have reproduced their message in a form
of his own. They would have inspired, and not enslaved him. His
resemblance to them is that of consanguinity, more striking in expression
than in mere resemblance of feature. The likeness between the
Clytemnestra--[Greek: gunaikos androboulon elpizon kear]--of Aeschylus
and the Lady Macbeth of Shakespeare was too remarkable to have escaped
notice. That between the two poets in their choice of epithets is as
great, though more difficult of proof. Yet I think an attentive student
of Shakespeare cannot fail to be reminded of something familiar to him in
such phrases as "flame-eyed fire," "flax-winged ships," "star-neighboring
peaks," the rock Salmydessus,
"Rude jaw of the sea,
Harsh hostess of the seaman, step-mother
Of ships,"
and the beacon with its "_speaking eye_ of fire." Surely there is more
than a verbal, there is a genuine, similarity between the [Greek:
anaerithmon gelasma] and "the unnumbered beach" and "multitudinous sea."
Aeschylus, it seems to me, is willing, just as Shakespeare is, to risk
the prosperity of a verse upon a lucky throw of words, which may come up
the sices of hardy metaphor or the ambsace of conceit. There is such a
difference between far-reaching and far-fetching! Poetry, to be sure, is
always that daring one step beyond, which brings the right man to
fortune, but leaves the wrong one in the ditch, and its law is, Be bold
once and again, yet be not over-bold. It is true, also, that masters of
language are a little apt to play with it. But whatever fault may be
found with Shakespeare in this respect will touch a tender spot in
Aeschylus also. Does he sometimes overload a word, so that the language
not merely, as Dryden says, bends under him, but fairly gives way, and
lets the reader's mind down with the shock as of a false step in taste?
He has nothing worse than [Greek: pelagos anthoun nekrois]. A criticism,
shallow in human nature, however deep in Campbell's Rhetoric, has blamed
him for making persons, under great excitement of sorrow, or whatever
other emotion, parenthesize some trifling play upon words in the very
height of their passion. Those who make such criticisms have either never
felt a passion or seen one in action, or else they forget the exaltation
of sensibility during such crises, so that the attention, whether of the
senses or the mind, is arrested for the moment by what would be
overlooked in ordinary moods. The more forceful the current, the more
sharp the ripple from any alien substance interposed. A passion that
looks forward, like revenge or lust or greed, goes right to its end, and
is straightforward in its expression; but a tragic passion, which is in
its nature unavailing, like disappointment, regret of the inevitable, or
remorse, is reflective, and liable to be continually diverted by the
suggestions of fancy. The one is a concentration of the will, which
intensifies the character and the phrase that expresses it; in the other,
the will is helpless, and, as in insanity, while the flow of the mind
sets imperatively in one direction, it is liable to almost ludicrous
interruptions and diversions upon the most trivial hint of involuntary
association. I am ready to grant that Shakespeare sometimes allows his
characters to spend time, that might be better employed, in carving some
cherry-stone of a quibble;[130] that he is sometimes tempted away from
the natural by the quaint; that he sometimes forces a partial, even a
verbal, analogy between the abstract thought and the sensual image into
an absolute identity, giving us a kind of serious pun. In a pun our
pleasure arises from a gap in the logical nexus too wide for the reason,
but which the ear can bridge in an instant. "Is that your own hare, or a
wig?" The fancy is yet more tickled where logic is treated with a mock
ceremonial of respect.
"His head was turned, _and so_ he chewed
His pigtail till he died."
Now when this kind of thing is done in earnest, the result is one of
those ill-distributed syllogisms which in rhetoric are called conceits.
"Hard was the hand that struck the blow,
Soft was the heart that bled."
I have seen this passage from Warner cited for its beauty, though I
should have thought nothing could be worse, had I not seen General
Morris's
"Her heart and morning broke together
In tears."
Of course, I would not rank with these Gloucester's
"What! will the aspiring blood of Lancaster
Sink in the ground? I thought it would have mounted";
though as mere rhetoric it belongs to the same class.[131]
It might be defended as a bit of ghastly humor characteristic of the
speaker. But at any rate it is not without precedent in the two greater
Greek tragedians. In a chorus of the _Seven against Thebes_ we have:--
[Greek: en de gaia.
Zoa phonoruto
Memiktai, _karta d' eis' omaimoi_.]
And does not Sophocles make Ajax in his despair quibble upon his own name
quite in the Shakespearian fashion, under similar circumstances? Nor does
the coarseness with which our great poet is reproached lack an Aeschylean
parallel. Even the Nurse in _Romeo and Juliet_ would have found a true
gossip in her of the _Agamemnon_, who is so indiscreet in her confidences
concerning the nursery life of Orestes. Whether Raleigh is right or not
in warning historians against following truth too close upon the heels,
the caution is a good one for poets as respects truth to Nature. But it
is a mischievous fallacy in historian or critic to treat as a blemish of
the man what is but the common tincture of his age. It is to confound a
spatter of mud with a moral stain.
But I have been led away from my immediate purpose. I did not intend to
compare Shakespeare with the ancients, much less to justify his defects
by theirs. Shakespeare himself has left us a pregnant satire on
dogmatical and categorical aesthetics (which commonly in discussion soon
lose their ceremonious tails and are reduced to the internecine dog and
cat of their bald first syllables) in the cloud-scene between Hamlet and
Polonius, suggesting exquisitely how futile is any attempt at a cast-iron
definition of those perpetually metamorphic impressions of the beautiful
whose source is as much in the man who looks as in the thing he sees. In
the fine arts a thing is either good in itself or it is nothing. It
neither gains nor loses by having it shown that another good thing was
also good in itself, any more than a bad thing profits by comparison with
another that is worse. The final judgment of the world is intuitive, and
is based, not on proof that a work possesses some of the qualities of
another whose greatness is acknowledged, but on the immediate feeling
that it carries to a high point of perfection certain qualities proper to
itself. One does not flatter a fine pear by comparing it to a fine peach,
nor learn what a fine peach is by tasting ever so many poor ones. The boy
who makes his first bite into one does not need to ask his father if or
how or why it is good. Because continuity is a merit in some kinds of
writing, shall we refuse ourselves to the authentic charm of Montaigne's
want of it? I have heard people complain of French tragedies because they
were so very French. This, though it may not be to some particular
tastes, and may from one point of view be a defect, is from another and
far higher a distinguished merit. It is their flavor, as direct a
telltale of the soil whence they drew it as that of French wines is.
Suppose we should tax the Elgin marbles with being too Greek? When will
people, nay, when will even critics, get over this self-defrauding trick
of cheapening the excellence of one thing by that of another, this
conclusive style of judgment which consists simply in belonging to the
other parish? As one grows older, one loses many idols, perhaps comes at
last to have none at all, though he may honestly enough uncover in
deference to the worshippers before any shrine. But for the seeming loss
the compensation is ample. These saints of literature descend from their
canopied remoteness to be even more precious as men like ourselves, our
companions in field and street, speaking the same tongue, though in many
dialects, and owning one creed under the most diverse masks of form.
Much of that merit of structure which is claimed for the ancient tragedy
is due, if I am not mistaken, to circumstances external to the drama
itself,--to custom, to convention, to the exigencies of the theatre. It
is formal rather than organic. The _Prometheus_ seems to me one of the
few Greek tragedies in which the whole creation has developed itself in
perfect proportion from one central germ of living conception. The motive
of the ancient drama is generally outside of it, while in the modern (at
least in the English) it is necessarily within. Goethe, in a thoughtful
essay,[132] written many years later than his famous criticism of Hamlet
in _Wilhelm Meister_, says that the distinction between the two is the
difference between _sollen_ and _wollen_, that is, between _must_ and
_would_. He means that in the Greek drama the catastrophe is foreordained
by an inexorable Destiny, while the element of Freewill, and consequently
of choice, is the very axis of the modern. The definition is conveniently
portable, but it has its limitations. Goethe's attention was too
exclusively fixed on the Fate tragedies of the Greeks, and upon
Shakespeare among the moderns. In the Spanish drama, for example, custom,
loyalty, honor, and religion are as imperative and as inevitable as doom.
In the _Antigone_, on the other hand, the crisis lies in the character of
the protagonist. In this sense it is modern, and is the first example of
true character-painting in tragedy. But, from whatever cause, that
exquisite analysis of complex motives, and the display of them in action
and speech, which constitute for us the abiding charm of fiction, were
quite unknown to the ancients. They reached their height in Cervantes and
Shakespeare, and, though on a lower plane, still belong to the upper
region of art in Le Sage, Moliere, and Fielding. The personages of the
Greek tragedy seem to be commonly rather types than individuals. In the
modern tragedy, certainly in the four greatest of Shakespeare's
tragedies, there is still something very like Destiny, only the place of
it is changed. It is no longer above man, but in him; yet the catastrophe
is as sternly foredoomed in the characters of Lear, Othello, Macbeth, and
Hamlet as it could be by an infallible oracle. In Macbeth, indeed, the
Weird Sisters introduce an element very like Fate; but generally it may
be said that with the Greeks the character is involved in the action,
while with Shakespeare the action is evolved from the character. In the
one case, the motive of the play controls the personages; in the other,
the chief personages are in themselves the motive to which all else is
subsidiary. In any comparison, therefore, of Shakespeare with the
ancients, we are not to contrast him with them as unapproachable models,
but to consider whether he, like them, did not consciously endeavor,
under the circumstances and limitations in which he found himself, to
produce the most excellent thing possible, a model also in its own
kind,--whether higher or lower in degree is another question. The only
fair comparison would be between him and that one of his contemporaries
who endeavored to anachronize himself, so to speak, and to subject his
art, so far as might be, to the laws of classical composition. Ben Jonson
was a great man, and has sufficiently proved that he had an eye for the
external marks of character; but when he would make a whole of them, he
gives us instead either a bundle of humors or an incorporated idea. With
Shakespeare the plot is an interior organism, in Jonson an external
contrivance. It is the difference between man and tortoise. In the one
the osseous structure is out of sight, indeed, but sustains the flesh and
blood that envelop it, while the other is boxed up and imprisoned in his
bones.
I have been careful to confine myself to what may be called Shakespeare's
ideal tragedies. In the purely historical or chronicle plays, the
conditions are different, and his imagination submits itself to the
necessary restrictions on its freedom of movement. Outside the tragedies
also, the _Tempest_ makes an exception worthy of notice. If I read it
rightly, it is an example of how a great poet should write allegory,--not
embodying metaphysical abstractions, but giving us ideals abstracted from
life itself, suggesting an under-meaning everywhere, forcing it upon us
nowhere, tantalizing the mind with hints that imply so much and tell so
little, and yet keep the attention all eye and ear with eager, if
fruitless, expectation. Here the leading characters are not merely
typical, but symbolical,--that is, they do not illustrate a class of
persons, they belong to universal Nature. Consider the scene of the play.
Shakespeare is wont to take some familiar story, to lay his scene in some
place the name of which, at least, is familiar,--well knowing the reserve
of power that lies in the familiar as a background, when things are set
in front of it under a new and unexpected light. But in the _Tempest_ the
scene is laid nowhere, or certainly in no country laid down on any map.
Nowhere, then? At once nowhere and anywhere,--for it is in the soul of
man, that still vexed island hung between the upper and the nether world,
and liable to incursions from both. There is scarce a play of
Shakespeare's in which there is such variety of character, none in which
character has so little to do in the carrying on and development of the
story. But consider for a moment if ever the Imagination has been so
embodied as in Prospero, the Fancy as in Ariel, the brute Understanding
as in Caliban, who, the moment his poor wits are warmed with the glorious
liquor of Stephano, plots rebellion against his natural lord, the higher
Reason. Miranda is mere abstract Womanhood, as truly so before she sees
Ferdinand as Eve before she was wakened to consciousness by the echo of
her own nature coming back to her, the same, and yet not the same, from
that of Adam. Ferdinand, again, is nothing more than Youth, compelled to
drudge at something he despises, till the sacrifice of will and
abnegation of self win him his ideal in Miranda. The subordinate
personages are simply types; Sebastian and Antonio, of weak character and
evil ambition; Gonzalo, of average sense and honesty; Adrian and
Francisco, of the walking gentlemen who serve to fill up a world. They
are not characters in the same sense with Iago, Falstaff, Shallow, or
Leontius; and it is curious how every one of them loses his way in this
enchanted island of life, all the victims of one illusion after another,
except Prospero, whose ministers are purely ideal. The whole play,
indeed, is a succession of illusions, winding up with those solemn words
of the great enchanter who had summoned to his service every shape of
merriment or passion, every figure in the great tragi-comedy of life, and
who was now bidding farewell to the scene of his triumphs. For in
Prospero shall we not recognize the Artist himself,--
"That did not better for his life provide
Than public means which public manners breeds,
Whence comes it that his name receives a brand,"--
who has forfeited a shining place in the world's eye by devotion to his
art, and who, turned adrift on the ocean of life in the leaky carcass of
a boat, has shipwrecked on that Fortunate Island (as men always do who
find their true vocation) where he is absolute lord, making all the
powers of Nature serve him, but with Ariel and Caliban as special
ministers? Of whom else could he have been thinking, when he says,--
"Graves, at my command,
Have waked their sleepers, oped, and let them forth,
By my so potent art"?
Was this man, so extraordinary from whatever side we look at him, who ran
so easily through the whole scale of human sentiment, from the homely
commonsense of, "When two men ride of one horse, one _must_ ride behind,"
to the transcendental subtilty of,
"No, Time, thou shalt not boast that I do change;
Thy pyramids, built up with newer might,
To me are nothing novel, nothing strange;
They are but dressings of a former sight,"--
was he alone so unconscious of powers, some part of whose magic is
recognized by all mankind, from the school-boy to the philosopher, that
he merely sat by and saw them go without the least notion what they were
about? Was he an inspired idiot, _votre bizarre Shakespeare_? a vast,
irregular genius? a simple rustic, warbling his _native_ wood-notes wild,
in other words, insensible to the benefits of culture? When attempts have
been made at various times to prove that this singular and seemingly
contradictory creature, not one, but all mankind's epitome, was a
musician, a lawyer, a doctor, a Catholic, a Protestant, an atheist, an
Irishman, a discoverer of the circulation of the blood, and finally, that
he was not himself, but somebody else, is it not a little odd that the
last thing anybody should have thought of proving him was an artist?
Nobody believes any longer that immediate inspiration is possible in
modern times (as if God had grown old),--at least, nobody believes it of
the prophets of those days, of John of Leyden, or Reeves, or
Muggleton,--and yet everybody seems to take it for granted of this one
man Shakespeare. He, somehow or other, without knowing it, was able to do
what none of the rest of them, though knowing it all too perfectly well,
could begin to do. Everybody seems to get afraid of him in turn. Voltaire
plays gentleman usher for him to his countrymen, and then, perceiving
that his countrymen find a flavor in him beyond that of _Zaire_ or
_Mahomet_, discovers him to be a _Sauvage ivre, sans le moindre etincelle
de bon gout, et sans le moindre connoissance des regles_. Goethe, who
tells us that _Goetz von Berlichingen_ was written in the Shakespearian
manner,--and we certainly should not have guessed it, if he had not
blabbed,--comes to the final conclusion, that Shakespeare was a poet, but
not a dramatist. Chateaubriand thinks that he has corrupted art. "If, to
attain," he says, "the height of tragic art, it be enough to heap
together disparate scenes without order and without connection, to
dovetail the burlesque with the pathetic, to set the water-carrier beside
the monarch and the huckster-wench beside the queen, who may not
reasonably flatter himself with being the rival of the greatest masters?
Whoever should give himself the trouble to retrace a single one of his
days, ... to keep a journal from hour to hour, would have made a drama in
the fashion of the English poet." But there are journals and journals, as
the French say, and what goes into them depends on the eye that gathers
for them. It is a long step from St. Simon to Dangeau, from Pepys to
Thoresby, from Shakespeare even to the Marquis de Chateaubriand. M. Hugo
alone, convinced that, as founder of the French Romantic School, there is
a kind of family likeness between himself and Shakespeare, stands boldly
forth to prove the father as extravagant as the son. Calm yourself, M.
Hugo, you are no more a child of his than Will Davenant was! But, after
all, is it such a great crime to produce something absolutely new in a
world so tedious as ours, and so apt to tell its old stories over again?
I do not mean new in substance, but in the manner of presentation. Surely
the highest office of a great poet is to show us how much variety,
freshness, and opportunity abides in the obvious and familiar. He invents
nothing, but seems rather to _re_-discover the world about him, and his
penetrating vision gives to things of daily encounter something of the
strangeness of new creation. Meanwhile the changed conditions of modern
life demand a change in the method of treatment. The ideal is not a
strait-waistcoat. Because _Alexis and Dora_ is so charming, shall we have
no _Paul and Virginia?_ It was the idle endeavor to reproduce the old
enchantment in the old way that gave us the pastoral, sent to the garret
now with our grandmothers' achievements of the same sort in worsted.
Every age says to its poets, like a mistress to her lover, "Tell me what
I am like"; and he who succeeds in catching the evanescent expression
that reveals character--which is as much as to say, what is intrinsically
human--will be found to have caught something as imperishable as human
nature itself. Aristophanes, by the vital and essential qualities of his
humorous satire, is already more nearly our contemporary than Moliere;
and even the _Trouveres_, careless and trivial as they mostly are, could
fecundate a great poet like Chaucer, and are still delightful reading.
The Attic tragedy still keeps its hold upon the loyalty of scholars
through their imagination, or their pedantry, or their feeling of an
exclusive property, as may happen, and, however alloyed with baser
matter, this loyalty is legitimate and well bestowed. But the dominion of
the Shakespearian is even wider. It pushes forward its boundaries from
year to year, and moves no landmark backward. Here Alfieri and Leasing
own a common allegiance; and the loyalty to him is one not of guild or
tradition, but of conviction and enthusiasm. Can this be said of any
other modern? of robust Corneille? of tender Racine? of Calderon even,
with his tropical warmth and vigor of production? The Greeks and he are
alike and alone in this, and for the same reason, that both are
unapproachably the highest in their kind. Call him Gothic, if you like,
but the inspiring mind that presided over the growth of these clustered
masses of arch and spire and pinnacle and buttress is neither Greek nor
Gothic,--it is simply genius lending itself to embody the new desire of
man's mind, as it had embodied the old. After all, to be delightful is to
be classic, and the chaotic never pleases long. But manifoldness is not
confusion, any more than formalism is simplicity. If Shakespeare rejected
the unities, as I think he who complains of "Art made tongue-tied by
Authority" might very well deliberately do, it was for the sake of an
imaginative unity more intimate than any of time and place. The antique
in itself is not the ideal, though its remoteness from the vulgarity of
everyday associations helps to make it seem so. The true ideal is not
opposed to the real, nor is it any artificial heightening thereof, but
lies _in_ it, and blessed are the eyes that find it! It is the _mens
divinior_ which hides within the actual, transfiguring matter-of-fact
into matter-of-meaning for him who has the gift of second-sight. In this
sense Hogarth is often more truly ideal than Raphael, Shakespeare often
more truly so than the Greeks. I think it is a more or less conscious
perception of this ideality, as it is a more or less well-grounded
persuasion of it as respects the Greeks, that assures to him, as to them,
and with equal justice, a permanent supremacy over the minds of men. This
gives to his characters their universality, to his thought its
irradiating property, while the artistic purpose running through and
combining the endless variety of scene and character will alone account
for his power of dramatic effect. Goethe affirmed, that, without
Schroeder's prunings and adaptations, Shakespeare was too undramatic for
the German theatre,--that, if the theory that his plays should be
represented textually should prevail, he would be driven from the boards.
The theory has prevailed, and he not only holds his own, but is acted
oftener than ever. It is not irregular genius that can do this, for
surely Germany need not go abroad for what her own Werners could more
than amply supply her with.
But I would much rather quote a fine saying than a bad prophecy of a man
to whom I owe so much. Goethe, in one of the most perfect of his shorter
poems, tells us that a poem is like a painted window. Seen from without,
(and he accordingly justifies the Philistine, who never looks at them
otherwise,) they seem dingy and confused enough; but enter, and then
"Da ist's auf einmal farbig helle,
Geschicht' und Zierath glaenzt in Schnelle."
With the same feeling he says elsewhere in prose, that "there is a
destructive criticism and a productive. The former is very easy; for one
has only to set up in his mind any standard, any model, however narrow"
(let us say the Greeks), "and then boldly assert that the work under
review does not match with it, and therefore is good for nothing,--the
matter is settled, and one must at once deny its claim. Productive
criticism is a great deal more difficult; it asks, What did the author
propose to himself? Is what he proposes reasonable and comprehensible?
and how far has he succeeded in carrying it out?" It is in applying this
latter kind of criticism to Shakespeare that the Germans have set us an
example worthy of all commendation. If they have been sometimes
over-subtile, they at least had the merit of first looking at his works
as wholes, as something that very likely contained an idea, perhaps
conveyed a moral, if we could get at it. The illumination lent us by most
of the English commentators reminds us of the candles which guides hold
up to show us a picture in a dark place, the smoke of which gradually
makes the work of the artist invisible under its repeated layers.
Lessing, as might have been expected, opened the first glimpse in the new
direction; Goethe followed with his famous exposition of Hamlet; A.W.
Schlegel took a more comprehensive view in his Lectures, which Coleridge
worked over into English, adding many fine criticisms of his own on
single passages; and finally, Gervinus has devoted four volumes to a
comment on the plays, full of excellent matter, though pushing the moral
exegesis beyond all reasonable bounds.[133] With the help of all these,
and especially of the last, I shall apply this theory of criticism to
Hamlet, not in the hope of saying anything new, but of bringing something
to the support of the thesis, that, if Shakespeare was skilful as a
playwright, he was even greater as a dramatist,--that, if his immediate
business was to fill the theatre, his higher object was to create
something which, by fulfilling the conditions and answering the
requirements of modern life, should as truly deserve to be called a
work of art as others had deserved it by doing the same thing in
former times and under other circumstances. Supposing him to have
accepted--consciously or not is of little importance--the new terms of
the problem which makes character the pivot of dramatic action, and
consequently the key of dramatic unity, how far did he succeed?
Before attempting my analysis, I must clear away a little rubbish. Are
such anachronisms as those of which Voltaire accuses Shakespeare in
Hamlet, such as the introduction of cannon before the invention of
gunpowder, and making Christians of the Danes three centuries too soon,
of the least bearing aesthetically? I think not; but as they are of a
piece with a great many other criticisms upon the great poet, it is worth
while to dwell upon them a moment.
The first demand we make upon whatever claims to be a work of art (and we
have a right to make it) is that it shall be _in keeping_. Now this
propriety is of two kinds, either extrinsic or intrinsic. In the first I
should class whatever relates rather to the body than the soul of the
work, such as fidelity to the facts of history, (wherever that is
important,) congruity of costume, and the like,--in short, whatever might
come under the head of _picturesque_ truth, a departure from which would
shock too rudely our preconceived associations. I have seen an Indian
chief in French boots, and he seemed to me almost tragic; but, put upon
the stage in tragedy, he would have been ludicrous. Lichtenberg, writing
from London in 1775, tells us that Garrick played Hamlet in a suit of the
French fashion, then commonly worn, and that he was blamed for it by some
of the critics; but, he says, one hears no such criticism during the
play, nor on the way home, nor at supper afterwards, nor indeed till the
emotion roused by the great actor has had time to subside. He justifies
Garrick, though we should not be able to endure it now. Yet nothing would
be gained by trying to make Hamlet's costume true to the assumed period
of the play, for the scene of it is laid in a Denmark that has no dates.
In the second and more important category, I should put, first,
co-ordination of character, that is, a certain variety in harmony of the
personages of a drama, as in the attitudes and coloring of the figures in
a pictorial composition, so that, while mutually relieving and setting
off each other, they shall combine in the total impression; second, that
subordinate truth to Nature which makes each character coherent in
itself; and, third, such propriety of costume and the like as shall
satisfy the superhistoric sense, to which, and to which alone, the higher
drama appeals. All these come within the scope of _imaginative_ truth. To
illustrate my third head by an example. Tieck criticises John Kemble's
dressing for Macbeth in a modern Highland costume, as being ungraceful
without any countervailing merit of historical exactness. I think a
deeper reason for his dissatisfaction might be found in the fact, that
this garb, with its purely modern and British army associations, is out
of place on Fores Heath, and drags the Weird Sisters down with it from
their proper imaginative remoteness in the gloom of the past to the
disenchanting glare of the foot-lights. It is not the antiquarian, but
the poetic conscience, that is wounded. To this, exactness, so far as
concerns ideal representation, may not only not be truth, but may even be
opposed to it. Anachronisms and the like are in themselves of no account,
and become important only when they make a gap too wide for our illusion
to cross unconsciously, that is, when they are anacoluthons to the
imagination. The aim of the artist is psychologic, not historic truth. It
is comparatively easy for an author to _get up_ any period with tolerable
minuteness in externals, but readers and audiences find more difficulty
in getting them down, though oblivion swallows scores of them at a gulp.
The saving truth in such matters is a truth to essential and permanent
characteristics. The Ulysses of Shakespeare, like the Ulysses of Dante
and Tennyson, more or less harmonizes with our ideal conception of the
wary, long-considering, though adventurous son of Laertes, yet Simon Lord
Lovat is doubtless nearer the original type. In Hamlet, though there is
no Denmark of the ninth century, Shakespeare has suggested the prevailing
rudeness of manners quite enough for his purpose. We see it in the single
combat of Hamlet's father with the elder Fortinbras, in the vulgar
wassail of the king, in the English monarch being expected to hang
Rosencrantz and Guildenstern out of hand merely to oblige his cousin of
Denmark, in Laertes, sent to Paris to be made a gentleman of, becoming
instantly capable of any the most barbarous treachery to glut his
vengeance. We cannot fancy Ragnar Lodbrog or Eric the Red matriculating
at Wittenberg, but it was essential that Hamlet should be a scholar, and
Shakespeare sends him thither without more ado. All through the play we
get the notion of a state of society in which a savage nature has
disguised itself in the externals of civilization, like a Maori deacon,
who has only to strip and he becomes once more a tattooed pagan with his
mouth watering for a spare-rib of his pastor. Historically, at the date
of Hamlet, the Danes were in the habit of burning their enemies alive in
their houses, with as much of their family about them as might be to make
it comfortable. Shakespeare seems purposely to have dissociated his play
from history by changing nearly every name in the original legend. The
motive of the play--revenge as a religious duty--belongs only to a social
state in which the traditions of barbarism are still operative, but, with
infallible artistic judgment, Shakespeare has chosen, not untamed Nature,
as he found it in history, but the period of transition, a period in
which the times are always out of joint, and thus the irresolution which
has its root in Hamlet's own character is stimulated by the very
incompatibility of that legacy of vengeance he has inherited from the
past with the new culture and refinement of which he is the
representative. One of the few books which Shakespeare is known to have
possessed was Florio's Montaigne, and he might well have transferred the
Frenchman's motto, _Que scais je_? to the front of his tragedy; nor can I
help fancying something more than accident in the fact that Hamlet has
been a student at Wittenberg, whence those new ideas went forth, of whose
results in unsettling men's faith, and consequently disqualifying them
for promptness in action, Shakespeare had been not only an eye-witness,
but which he must actually have experienced in himself.
One other objection let me touch upon here, especially as it has been
urged against Hamlet, and that is the introduction of low characters and
comic scenes in tragedy. Even Garrick, who had just assisted at the
Stratford Jubilee, where Shakespeare had been pronounced divine, was
induced by this absurd outcry for the proprieties of the tragic stage to
omit the grave-diggers' scene from Hamlet. Leaving apart the fact that
Shakespeare would not have been the representative poet he is, if he had
not given expression to this striking tendency of the Northern races,
which shows itself constantly, not only in their literature, but even in
their mythology and their architecture, the grave-diggers' scene always
impresses me as one of the most pathetic in the whole tragedy. That
Shakespeare introduced such scenes and characters with deliberate
intention, and with a view to artistic relief and contrast, there can
hardly be a doubt. We must take it for granted that a man whose works
show everywhere the results of judgment sometimes acted with forethought.
I find the springs of the profoundest sorrow and pity in this hardened
indifference of the grave-diggers, in their careless discussion as to
whether Ophelia's death was by suicide or no, in their singing and
jesting at their dreary work.
"A pickaxe and a spade, a spade,
For--and a shrouding-sheet:
O, a pit of clay for to be made
For such a guest is meet!"
_We_ know who is to be the guest of this earthen hospitality,--how much
beauty, love, and heartbreak are to be covered in that pit of clay. All
we remember of Ophelia reacts upon us with tenfold force, and we recoil
from our amusement at the ghastly drollery of the two delvers with a
shock of horror. That the unconscious Hamlet should stumble on _this_
grave of all others, that it should be _here_ that he should pause to
muse humorously on death and decay,--all this prepares us for the
revulsion of passion in the next scene, and for the frantic confession,--
"I loved Ophelia; forty thousand brothers
Could not with all _their_ quantity of love
Make up my sum!"
And it is only here that such an asseveration would be true even to the
feeling of the moment; for it is plain from all we know of Hamlet that he
could not so have loved Ophelia, that he was incapable of the
self-abandonment of a true passion, that he would have analyzed this
emotion as he does all others, would have peeped and botanized upon it
till it became to him a mere matter of scientific interest. All this
force of contrast, and this horror of surprise, were necessary so to
intensify his remorseful regret that he should believe himself for once
in earnest. The speech of the King, "O, he is mad, Laertes," recalls him
to himself, and he at once begins to rave:--
"Zounds! show me what thou'lt do!
Woul't weep? woul't fight? woul't fast? woul't tear thyself?
Woul't drink up eysil? eat a crocodile?"
It is easy to see that the whole plot hinges upon the character of
Hamlet, that Shakespeare's conception of this was the ovum out of which
the whole organism was hatched. And here let me remark, that there is a
kind of genealogical necessity in the character,--a thing not altogether
strange to the attentive reader of Shakespeare. Hamlet seems the natural
result of the mixture of father and mother in his temperament, the
resolution and persistence of the one, like sound timber wormholed and
made shaky, as it were, by the other's infirmity of will and
discontinuity of purpose. In natures so imperfectly mixed it is not
uncommon to find vehemence of intention the prelude and counterpoise of
weak performance, the conscious nature striving to keep up its
self-respect by a triumph in words all the more resolute that it feels
assured beforehand of inevitable defeat in action. As in such slipshod
housekeeping men are their own largest creditors, they find it easy to
stave off utter bankruptcy of conscience by taking up one unpaid promise
with another larger, and at heavier interest, till such self-swindling
becomes habitual and by degrees almost painless. How did Coleridge
discount his own notes of this kind with less and less specie as the
figures lengthened on the paper! As with Hamlet, so it is with Ophelia
and Laertes. The father's feebleness comes up again in the wasting
heartbreak and gentle lunacy of the daughter, while the son shows it in a
rashness of impulse and act, a kind of crankiness, of whose essential
feebleness we are all the more sensible as contrasted with a nature so
steady on its keel, and drawing so much water, as that of Horatio,--the
foil at once, in different ways, to both him and Hamlet. It was natural,
also, that the daughter of self-conceited old Polonius should have her
softness stiffened with a fibre of obstinacy; for there are two kinds of
weakness, that which breaks, and that which bends. Ophelia's is of the
former kind; Hero is her counterpart, giving way before calamity, and
rising again so soon as the pressure is removed.
I find two passages in Dante that contain the exactest possible
definition of that habit or quality of Hamlet's mind which justifies the
tragic turn of the play, and renders it natural and unavoidable from the
beginning. The first is from the second canto of the _Inferno_:--
"E quale e quei che disvuol cio che volle,
E per nuovi pensier sangia proposta,
Si che del cominciar tutto si tolle;
Tal mi fec' io in quella oscura costa;
Perche pensando consumai la impresa
Che fu nel cominciar cotanto tosta."
"And like the man who unwills what he willed,
And for new thoughts doth change his first intent,
So that he cannot anywhere begin,
Such became I upon that <DW72> obscure,
Because with thinking I consumed resolve,
That was so ready at the setting out."
Again, in the fifth of the _Purgatorio_:--
"Che sempre l' uomo in cui pensier rampoglia
Sovra pensier, da se dilunga il segno,
Perche la foga l' un dell' altro insolla."
"For always he in whom one thought buds forth
Out of another farther puts the goal.
For each has only force to mar the other."
Dante was a profound metaphysician, and as in the first passage he
describes and defines a certain quality of mind, so in the other he tells
us its result in the character and life, namely, indecision and
failure,--the goal _farther_ off at the end than at the beginning. It is
remarkable how close a resemblance of thought, and even of expression,
there is between the former of these quotations and a part of Hamlet's
famous soliloquy:--
"Thus conscience [i.e. consciousness] doth make cowards of us all;
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought,
And enterprises of great pitch and moment
With this regard their currents turn awry,
And lose the name of action!"
It is an inherent peculiarity of a mind like Hamlet's that it should be
conscious of its own defect. Men of his type are forever analyzing their
own emotions and motives. They cannot do anything, because they always
see two ways of doing it. They cannot determine on any course of action,
because they are always, as it were, standing at the cross-roads, and see
too well the disadvantages of every one of them. It is not that they are
incapable of resolve, but somehow the band between the motive power and
the operative faculties is relaxed and loose. The engine works, but the
machinery it should drive stands still. The imagination is so much in
overplus, that thinking a thing becomes better than doing it, and thought
with its easy perfection, capable of everything because it can accomplish
everything with ideal means, is vastly more attractive and satisfactory
than deed, which must be wrought at best with imperfect instruments, and
always falls short of the conception that went before it. "If to do,"
says Portia in the _Merchant of Venice_,--"if to do were as easy as to
know what 't were good to do, chapels had been churches, and poor men's
cottages princes' palaces." Hamlet knows only too well what 't were good
to do, but he palters with everything in a double sense: he sees the
grain of good there is in evil, and the grain of evil there is in good,
as they exist in the world, and, finding that he can make those
feather-weighted accidents balance each other, infers that there is
little to choose between the essences themselves. He is of Montaigne's
mind, and says expressly that "there is nothing good or ill, but thinking
makes it so." He dwells so exclusively in the world of ideas that the
world of facts seems trifling, nothing is worth the while; and he has
been so long objectless and purposeless, so far as actual life is
concerned, that, when at last an object and an aim are forced upon him,
he cannot deal with them, and gropes about vainly for a motive outside of
himself that shall marshal his thoughts for him and guide his faculties
into the path of action. He is the victim not so much of feebleness of
will as of an intellectual indifference that hinders the will from
working long in any one direction. He wishes to will, but never wills.
His continual iteration of resolve shows that he has no resolution. He is
capable of passionate energy where the occasion presents itself suddenly
from without, because nothing is so irritable as conscious irresolution
with a duty to perform. But of deliberate energy he is not capable; for
there the impulse must come from within, and the blade of his analysis is
so subtile that it can divide the finest hair of motive 'twixt north and
northwest side, leaving him desperate to choose between them. The very
consciousness of his defect is an insuperable bar to his repairing it;
for the unity of purpose, which infuses every fibre of the character with
will available whenever wanted, is impossible where the mind can never
rest till it has resolved that unity into its component elements, and
satisfied itself which on the whole is of greater value. A critical
instinct so insatiable that it must turn upon itself, for lack of
something else to hew and hack, becomes incapable at last of originating
anything except indecision. It becomes infallible in what _not_ to do.
How easily he might have accomplished his task is shown by the conduct of
Laertes. When _he_ has a death to avenge, he raises a mob, breaks into
the palace, bullies the king, and proves how weak the usurper really was.
The world is the victim of splendid parts, and is slow to accept a
rounded whole, because that is something which is long in completing,
still longer in demonstrating its completion. We like to be surprised
into admiration, and not logically convinced that we ought to admire. We
are willing to be delighted with success, though we are somewhat
indifferent to the homely qualities which insure it. Our thought is so
filled with the rocket's burst of momentary splendor so far above us,
that we forget the poor stick, useful and unseen, that made its climbing
possible. One of these homely qualities is continuity of character, and
it escapes present applause because it tells chiefly, in the long run, in
results. With his usual tact, Shakespeare has brought in such a character
as a contrast and foil to Hamlet. Horatio is the only complete _man_ in
the play,--solid, well-knit, and true; a noble, quiet nature, with that
highest of all qualities, judgment, always sane and prompt; who never
drags his anchors for any wind of opinion or fortune, but grips all the
closer to the reality of things. He seems one of those calm,
undemonstrative men whom we love and admire without asking to know why,
crediting them with the capacity of great things, without any test of
actual achievement, because we feel that their manhood is a constant
quality, and no mere accident of circumstance and opportunity. Such men
are always sure of the presence of their highest self on demand. Hamlet
is continually drawing bills on the future, secured by his promise of
himself to himself, which he can never redeem. His own somewhat feminine
nature recognizes its complement in Horatio, and clings to it
instinctively, as naturally as Horatio is attracted by that fatal gift of
imagination, the absence of which makes the strength of his own
character, as its overplus does the weakness of Hamlet's. It is a happy
marriage of two minds drawn together by the charm of unlikeness. Hamlet
feels in Horatio the solid steadiness which he misses in himself; Horatio
in Hamlet that need of service and sustainment to render which gives him
a consciousness of his own value. Hamlet fills the place of a woman to
Horatio, revealing him to himself not only in what he says, but by a
constant claim upon his strength of nature; and there is great
psychological truth in making suicide the first impulse of this quiet,
undemonstrative man, after Hamlet's death, as if the very reason for his
being were taken away with his friend's need of him. In his grief, he for
the first and only time speaks of himself, is first made conscious of
himself by his loss. If this manly reserve of Horatio be true to Nature,
not less so are the communicativeness of Hamlet, and his tendency to
soliloquize. If self-consciousness be alien to the one, it is just as
truly the happiness of the other. Like a musician distrustful of himself,
he is forever tuning his instrument, first overstraining this cord a
little, and then that, but unable to bring them into unison, or to profit
by it if he could.
We do not believe that Horatio ever thought he "was not a pipe for
Fortune's finger to play what stop she please," till Hamlet told him so.
That was Fortune's affair, not his; let her try it, if she liked. He is
unconscious of his own peculiar qualities, as men of decision commonly
are, or they would not be men of decision. When there is a thing to be
done, they go straight at it, and for the time there is nothing for them
in the whole universe but themselves and their object. Hamlet, on the
other hand, is always studying himself. This world and the other, too,
are always present to his mind, and there in the corner is the little
black kobold of a doubt making mouths at him. He breaks down the bridges
before him, not behind him, as a man of action would do; but there is
something more than this. He is an ingrained sceptic; though his is the
scepticism, not of reason, but of feeling, whose root is want of faith in
himself. In him it is passive, a malady rather than a function of the
mind. We might call him insincere: not that he was in any sense a
hypocrite, but only that he never was and never could be in earnest.
Never could be, because no man without intense faith in something ever
can. Even if he only believed in himself, that were better than nothing;
for it will carry a man a great way in the outward successes of life,
nay, will even sometimes give him the Archimedean fulcrum for moving the
world. But Hamlet doubts everything. He doubts the immortality of the
soul, just after seeing his father's spirit, and hearing from its mouth
the secrets of the other world. He doubts Horatio even, and swears him to
secrecy on the cross of his sword, though probably he himself has no
assured belief in the sacredness of the symbol. He doubts Ophelia, and
asks her, "Are you honest?" He doubts the ghost, after he has had a
little time to think about it, and so gets up the play to test the guilt
of the king. And how coherent the whole character is! With what perfect
tact and judgment Shakespeare, in the advice to the players, makes him an
exquisite critic! For just here that part of his character which would be
weak in dealing with affairs is strong. A wise scepticism is the first
attribute of a good critic. He must not believe that the fire-insurance
offices will raise their rates of premium on Charles River, because the
new volume of poems is printing at Riverside or the University Press. He
must not believe so profoundly in the ancients as to think it wholly out
of the question that the world has still vigor enough in its loins to
beget some one who will one of these days be as good an ancient as any of
them.
Another striking quality in Hamlet's nature is his perpetual inclination
to irony. I think this has been generally passed over too lightly, as if
it were something external and accidental, rather assumed as a mask than
part of the real nature of the man. It seems to me to go deeper, to be
something innate, and not merely factitious. It is nothing like the grave
irony of Socrates, which was the weapon of a man thoroughly in
earnest,--the _boomerang_ of argument, which one throws in the opposite
direction of what he means to hit, and which seems to be flying away from
the adversary, who will presently find himself knocked down by it. It is
not like the irony of Timon, which is but the wilful refraction of a
clear mind twisting awry whatever enters it,--or of Iago, which is the
slime that a nature essentially evil loves to trail over all beauty and
goodness to taint them with distrust: it is the half-jest, half-earnest
of an inactive temperament that has not quite made up its mind whether
life is a reality or no, whether men were not made in jest, and which
amuses itself equally with finding a deep meaning in trivial things and a
trifling one in the profoundest mysteries of being, because the want of
earnestness in its own essence infects everything else with its own
indifference. If there be now and then an unmannerly rudeness and
bitterness in it, as in the scenes with Polonius and Osrick, we must
remember that Hamlet was just in the condition which spurs men to sallies
of this kind: dissatisfied, at one neither with the world nor with
himself, and accordingly casting about for something out of himself to
vent his spleen upon. But even in these passages there is no hint of
earnestness, of any purpose beyond the moment; they are mere cat's-paws
of vexation, and not the deep-raking ground-swell of passion, as we see
it in the sarcasm of Lear.
The question of Hamlet's madness has been much discussed and variously
decided. High medical authority has pronounced, as usual, on both sides
of the question. But the induction has been drawn from too narrow
premises, being based on a mere diagnosis of the _case_, and not on an
appreciation of the character in its completeness. We have a case of
pretended madness in the Edgar of _King Lear_; and it is certainly true
that that is a charcoal sketch, coarsely outlined, compared with the
delicate drawing, the lights, shades, and half-tints of the portraiture
in Hamlet. But does this tend to prove that the madness of the latter,
because truer to the recorded observation of experts, is real, and meant
to be real, as the other to be fictitious? Not in the least, as it
appears to me. Hamlet, among all the characters of Shakespeare, is the
most eminently a metaphysician and psychologist. He is a close observer,
continually analyzing his own nature and that of others, letting fall his
little drops of acid irony on all who come near him, to make them show
what they are made of. Even Ophelia is not too sacred, Osrick not too
contemptible for experiment. If such a man assumed madness, he would play
his part perfectly. If Shakespeare himself, without going mad, could so
observe and remember all the abnormal symptoms as to be able to reproduce
them in Hamlet, why should it be beyond the power of Hamlet to reproduce
them in himself? If you deprive Hamlet of reason, there is no truly
tragic motive left. He would be a fit subject for Bedlam, but not for the
stage. We might have pathology enough, but no pathos. Ajax first becomes
tragic when he recovers his wits. If Hamlet is irresponsible, the whole
play is a chaos. That he is not so might be proved by evidence enough,
were it not labor thrown away.
This feigned madness of Hamlet's is one of the few points in which
Shakespeare has kept close to the old story on which he founded his play;
and as he never decided without deliberation, so he never acted without
unerring judgment, Hamlet _drifts_ through the whole tragedy. He never
keeps on one tack long enough to get steerage-way, even if, in a nature
like his, with those electric streamers of whim and fancy forever
wavering across the vault of his brain, the needle of judgment would
point in one direction long enough to strike a course by. The scheme of
simulated insanity is precisely the one he would have been likely to hit
upon, because it enabled him to follow his own bent, and to drift with an
apparent purpose, postponing decisive action by the very means he adopts
to arrive at its accomplishment, and satisfying himself with the show of
doing something that he may escape so much the longer the dreaded
necessity of really doing anything at all. It enables him to _play_ with
life and duty, instead of taking them by the rougher side, where alone
any firm grip is possible,--to feel that he is on the way toward
accomplishing somewhat, when he is really paltering with his own
irresolution. Nothing, I think, could be more finely imagined than this.
Voltaire complains that he goes mad without any sufficient object or
result. Perfectly true, and precisely what was most natural for him to
do, and, accordingly, precisely what Shakespeare meant that he should do.
It was delightful to him to indulge his imagination and humor, to prove
his capacity for something by playing a part: the one thing he could not
do was to bring himself to _act_, unless when surprised by a sudden
impulse of suspicion,--as where he kills Polonius, and there he could not
see his victim. He discourses admirably of suicide, but does not kill
himself; he talks daggers, but uses none. He puts by the chance to kill
the king with the excuse that he will not do it while he is praying, lest
his soul be saved thereby, though it is more than doubtful whether he
believed it himself. He allows himself to be packed off to England,
without any motive except that it would for the time take him farther
from a present duty: the more disagreeable to a nature like his because
it _was_ present, and not a mere matter for speculative consideration.
When Goethe made his famous comparison of the acorn planted in a vase
which it bursts with its growth, and says that in like manner Hamlet is a
nature which breaks down under the weight of a duty too great for it to
bear, he seems to have considered the character too much from one side.
Had Hamlet actually killed himself to escape his too onerous commission,
Goethe's conception of him would have been satisfactory enough. But
Hamlet was hardly a sentimentalist, like Werther; on the contrary, he saw
things only too clearly in the dry north-light of the intellect. It is
chance that at last brings him to his end. It would appear rather that
Shakespeare intended to show us an imaginative temperament brought face
to face with actualities, into any clear relation of sympathy with which
it cannot bring itself. The very means that Shakespeare makes use of to
lay upon him the obligation of acting--the ghost--really seems to make it
all the harder for him to act; for the spectre but gives an additional
excitement to his imagination and a fresh topic for his scepticism.
I shall not attempt to evolve any high moral significance from the play,
even if I thought it possible; for that would be aside from the present
purpose. The scope of the higher drama is to represent life, not everyday
life, it is true, but life lifted above the plane of bread-and-butter
associations, by nobler reaches of language, by the influence at once
inspiring and modulating of verse, by an intenser play of passion
condensing that misty mixture of feeling and reflection which makes the
ordinary atmosphere of existence into flashes of thought and phrase whose
brief, but terrible, illumination prints the outworn landscape of
every-day upon our brains, with its little motives and mean results, in
lines of tell-tale fire. The moral office of tragedy is to show us our
own weaknesses idealized in grander figures and more awful results,--to
teach us that what we pardon in our selves as venial faults, if they seem
to have but slight influence on our immediate fortunes, have arms as long
as those of kings, and reach forward to the catastrophe of our lives,
that they are dry-rotting the very fibre of will and conscience, so that,
if we should be brought to the test of a great temptation or a stringent
emergency, we must be involved in a ruin as sudden and complete as that
we shudder at in the unreal scene of the theatre. But the primary
_object_ of a tragedy is not to inculcate a formal moral. Representing
life, it teaches, like life, by indirection, by those nods and winks that
are thrown away on us blind horses in such profusion. We may learn, to be
sure, plenty of lessons from Shakespeare. We are not likely to have
kingdoms to divide, crowns foretold us by weird sisters, a father's death
to avenge, or to kill our wives from jealousy; but Lear may teach us to
draw the line more clearly between a wise generosity and a loose-handed
weakness of giving; Macbeth, how one sin involves another, and forever
another, by a fatal parthenogenesis, and that the key which unlocks
forbidden doors to our will or passion leaves a stain on the hand, that
may not be so dark as blood, but that will not out; Hamlet, that all the
noblest gifts of person, temperament, and mind slip like sand through the
grasp of an infirm purpose; Othello, that the perpetual silt of some one
weakness, the eddies of a suspicious temper depositing their one
impalpable layer after another, may build up a shoal on which an heroic
life and an otherwise magnanimous nature may bilge and go to pieces. All
this we may learn, and much more, and Shakespeare was no doubt well aware
of all this and more; but I do not believe that he wrote his plays with
any such didactic purpose. He knew human nature too well not to know that
one thorn of experience is worth a whole wilderness of warning,--that,
where one man shapes his life by precept and example, there are a
thousand who have it shaped for them by impulse and by circumstances. He
did not mean his great tragedies for scarecrows, as if the nailing of one
hawk to the barn-door would prevent the next from coming down souse into
the hen-yard. No, it is not the poor bleaching victim hung up to moult
its draggled feathers in the rain that he wishes to show us. He loves the
hawk-nature as well as the hen-nature; and if he is unequalled in
anything, it is in that sunny breadth of view, that impregnability of
reason, that looks down all ranks and conditions of men, all fortune and
misfortune, with the equal eye of the pure artist.
Whether I have fancied anything into Hamlet which the author never
dreamed of putting there I do not greatly concern myself to inquire.
Poets are always entitled to a royalty on whatever we find in their
works; for these fine creations as truly build themselves up in the brain
as they are built up with deliberate forethought. Praise art as we will,
that which the artist did not mean to put into his work, but which found
itself there by some generous process of Nature of which he was as
unaware as the blue river is of its rhyme with the blue sky, has somewhat
in it that snatches us into sympathy with higher things than those which
come by plot and observation. Goethe wrote his _Faust_ in its earliest
form without a thought of the deeper meaning which the exposition of an
age of criticism was to find in it: without foremeaning it, he had
impersonated in Mephistopheles the genius of his century. Shall this
subtract from the debt we owe him? Not at all. If originality were
conscious of itself, it would have lost its right to be original. I
believe that Shakespeare intended to impersonate in Hamlet not a mere
metaphysical entity, but a man of flesh and blood: yet it is certainly
curious how prophetically typical the character is of that introversion
of mind which is so constant a phenomenon of these latter days, of that
over-consciousness which wastes itself in analyzing the motives of action
instead of acting.
The old painters had a rule, that all compositions should be pyramidal in
form,--a central figure, from which the others <DW72> gradually away on
the two sides. Shakespeare probably had never heard of this rule, and, if
he had, would not have been likely to respect it more than he has the
so-called classical unities of time and place. But he understood
perfectly the artistic advantages of gradation, contrast, and relief.
Taking Hamlet as the key-note, we find in him weakness of character,
which, on the one hand, is contrasted with the feebleness that springs
from overweening conceit in Polonius and with frailty of temperament in
Ophelia, while, on the other hand, it is brought into fuller relief by
the steady force of Horatio and the impulsive violence of Laertes, who is
resolute from thoughtlessness, just as Hamlet is irresolute from overplus
of thought.
If we must draw a moral from Hamlet, it would seem to be, that Will is
Fate, and that, Will once abdicating, the inevitable successor in the
regency is Chance. Had Hamlet acted, instead of musing how good it would
be to act, the king might have been the only victim. As it is, all the
main actors in the story are the fortuitous sacrifice of his
irresolution. We see how a single great vice of character at last draws
to itself as allies and confederates all other weaknesses of the man, as
in civil wars the timid and the selfish wait to throw themselves upon the
stronger side.
"In Life's small things be resolute and great
To keep thy muscles trained: know'st thou when Fate
Thy measure takes? or when she'll say to thee,
'I find thee worthy, do this thing for me'?"
I have said that it was doubtful if Shakespeare had any conscious moral
intention in his writings. I meant only that he was purely and primarily
poet. And while he was an English poet in a sense that is true of no
other, his method was thoroughly Greek, yet with this remarkable
difference,--that, while the Greek dramatists took purely national themes
and gave them a universal interest by their mode of treatment, he took
what may be called cosmopolitan traditions, legends of human nature, and
nationalized them by the infusion of his perfectly Anglican breadth of
character and solidity of understanding. Wonderful as his imagination and
fancy are, his perspicacity and artistic discretion are more so. This
country tradesman's son, coming up to London, could set high-bred wits,
like Beaumont, uncopiable lessons in drawing gentlemen such as are seen
nowhere else but on the canvas of Titian; he could take Ulysses away from
Homer and expand the shrewd and crafty islander into a statesman whose
words are the pith of history. But what makes him yet more exceptional
was his utterly unimpeachable judgment, and that poise of character which
enabled him to be at once the greatest of poets and so unnoticeable a
good citizen as to leave no incidents for biography. His material was
never far-sought; (it is still disputed whether the fullest head of which
we have record were cultivated beyond the range of grammar-school
precedent!) but he used it with a poetic instinct which we cannot
parallel, identified himself with it, yet remained always its born and
questionless master. He finds the Clown and Fool upon the stage,--he
makes them the tools of his pleasantry, his satire, and even his pathos;
he finds a fading rustic superstition, and shapes out of it ideal Pucks,
Titanias, and Ariels, in whose existence statesmen and scholars believe
forever. Always poet, he subjects all to the ends of his art, and gives
in Hamlet the churchyard ghost, but with the cothurnus on,--the messenger
of God's revenge against murder; always philosopher, he traces in Macbeth
the metaphysics of apparitions, painting the shadowy Banquo only on the
o'erwrought brain of the murderer, and staining the hand of his
wife-accomplice (because she was the more refined and higher nature) with
the disgustful blood-spot that is not there. We say he had no moral
intention, for the reason, that, as artist, it was not his to deal with
the realities, but only with the shows of things; yet, with a temperament
so just, an insight so inevitable as his, it was impossible that the
moral reality, which underlies the _mirage_ of the poet's vision, should
not always be suggested. His humor and satire are never of the
destructive kind; what he does in that way is suggestive only,--not
breaking bubbles with Thor's hammer, but puffing them away with the
breath of a Clown, or shivering them with the light laugh of a genial
cynic. Men go about to prove the existence of a God! Was it a bit of
phosphorus, that brain whose creations are so real, that, mixing with
them, we feel as if we ourselves were but fleeting magic-lantern shadows?
But higher even than the genius we rate the character of this unique man,
and the grand impersonality of what he wrote. What has he told us of
himself? In our self-exploiting nineteenth century, with its melancholy
liver-complaint, how serene and high he seems! If he had sorrows, he has
made them the woof of everlasting consolation to his kind; and if, as
poets are wont to whine, the outward world was cold to him, its biting
air did but trace itself in loveliest frost-work of fancy on the many
windows of that self-centred and cheerful soul.
Footnotes:
[119] As where Ben Jonson is able to say,--
"Man may securely sin, but safely never."
[120] "Vulgarem locutionem anpellamus cam qua infantes adsuefiunt ab
adsistentibus cum primitus distinguere voces incipiunt: vel, quod
brevius dici potest, vulgarem locutionem asserimus _quam sine omni
regula, nutricem imitantes accepimus_." Dantes, _de Vulg. Eloquio_,
Lib I. cap. i.
[121] Gray, himself a painful corrector, told Nicholls that "nothing
was done so well as at the first concoction,"--adding, as a reason,
"We think in words." Ben Jonson said, it was a pity Shakespeare had
not blotted more, for that he sometimes wrote nonsense,--and cited in
proof of it the verse,
"Caesar did never wrong but with just cause."
The last four words do not appear in the passage as it now stands,
and Professor Craik suggests that they were stricken out in
consequence of Jonson's criticism. This is very probable; but we
suspect that the pen that blotted them was in the hand of Master
Heminge or his colleague. The moral confusion in the idea was surely
admirably characteristic of the general who had just accomplished a
successful _coup d'etat_, the condemnation of which he would fancy
that he read in the face of every honest man he met, and which he
would therefore be forever indirectly palliating.
[122] We use the word _Latin_ here to express words derived either
mediately or immediately from that language.
[123] The prose of Chaucer (1390) and of Sir Thomas Malory
(translating from the French, 1470) is less Latinized than that of
Bacon, Browne, Taylor, or Milton. The glossary to Spenser's
_Shepherd's Calendar_ (1679) explains words of Teutonic and Romanic
root in about equal proportions. The parallel but independent
development of Scotch is not to be forgotten.
[124] I believe that for the last two centuries the Latin radicals of
English have been more familiar and homelike to those who use them
than the Teutonic. Even so accomplished a person as Professor Crail,
in his _English of Shakespeare_, derives _head_, through the German
_haupt_, from the Latin _caput_! I trust that its genealogy is
nobler, and that it is of kin with _coelum, tueri_, rather than with
the Greek [kephalae], if Suidas be right in tracing the origin of
that to a word meaning _vacuity_. Mr. Craik suggests, also, that
_quick_ and _wicked_ may be etymologically identical, _because_ he
fancies a relationship between _busy_ and the German _boese_, though
_wicked_ is evidently the participial form of A. S. _wacan_, (German
_weichen_,) _to bend, to yield_, meaning _one who has given way to
temptation_, while _quick_ seems as clearly related to _wegan_,
meaning _to move_, a different word, even if radically the same. In
the "London Literary Gazette" for November 13,1858, I find an extract
from Miss Millington's "Heraldry in History, Poetry, and Romance," in
which, speaking of the motto of the Prince of Wales,--_De par Houmaut
ich diene_,--she says; "The precise meaning of the former word
[_Houmout_] has not, I think, been ascertained." The word is plainly
the German _Hochmuth_, and the whole would read, _De par (Aus)
Hochmuth ich diene_,--"Out of magnanimity I serve." So entirely lost
is the Saxon meaning of the word _knave_, (A. S. _cnava_, German
_knabe_,) that the name _navvie_, assumed by railway-laborers, has
been transmogrified into _navigator_. I believe that more people
could tell why the month of July was so called than could explain the
origin of the names for our days of the week, and that it is oftener
the Saxon than the French words in Chaucer that puzzle the modern
reader.
[125] _De Vulgari Eloquio_, Lib. II. cap. i. _ad finem_. I quote this
treatise as Dante's, because the thoughts seem manifestly his; though
I believe that in its present form it is an abridgment by some
transcriber, who sometimes copies textually, and sometimes
substitutes his own language for that of the original.
[126] Vol. III. p. 348, _note_. He grounds his belief, not on the
misprinting of words, but on the misplacing of whole paragraphs. We
were struck with the same thing in the original edition of Chapman's
"Biron's Conspiracy and Tragedy." And yet, in comparing two copies of
this edition, I have found corrections which only the author could
have made. One of the misprints which Mr. Spedding notices affords
both a hint and a warning to the conjectural emendator. In the
edition of "The Advancement of Learning" printed in 1605 occurs the
word _dusinesse_. In a later edition this was conjecturally changed
to _business_; but the occurrence of _vertigine_ in the Latin
translation enables Mr. Spedding to print rightly, _dizziness_.
[127] "At first sight, Shakespeare and his contemporary dramatists
seem to write in styles much alike; nothing so easy as to fall into
that of Massinger and the others; whilst no one has ever yet produced
one scene conceived and expressed in the Shakespearian idiom. I
suppose it is because Shakespeare is universal, and, in fact, has no
_manner_."--_Coleridge's Tabletalk_, 214.
[128] Pheidias said of one of his pupils that he had an inspired
thumb, because the modelling-clay yielded to its careless sweep a
grace of curve which it refused to the utmost pains of others.
[129] The best instance I remember is in the _Frogs_, where Bacchus
pleads his inexperience at the oar, and says he is
[Greek: apeiros, athalattotos, asalaminios,]
which might be rendered,
Unskilled, unsea-soned, and un-Salamised.
[130] So Euripides (copied by Theocritus, Id. xxvii.):--
[Greek: Pentheus d' opos mae penthos eisoisei domois] (_Bacchae_,
363.)
[Greek: _Esophronaesen ouk echousa sophronein_]. (_Hippol_., 1037.)
So Calderon: "Y apenas llega, cuando llega a penas."
[131] I have taken the first passage in point that occurred to my
memory. It may not be Shakespeare's, though probably his. The
question of authorship is, I think, settled, so far as criticism can
do it, in Mr. Grant White's admirable essay appended to the Second
Part of Henry VI.
[132] Shakspeare und kein Ende.
[133] I do not mention Ulrici's book, for it seems to me unwieldy and
dull,--zeal without knowledge.
NEW ENGLAND TWO CENTURIES AGO.[134]
The history of New England is written imperishably on the face of a
continent, and in characters as beneficent as they are enduring. In the
Old World national pride feeds itself with the record of battles and
conquests;--battles which proved nothing and settled nothing; conquests
which shifted a boundary on the map, and put one ugly head instead of
another on the coin which the people paid to the tax-gatherer. But
wherever the New-Englander travels among the sturdy commonwealths which
have sprung from the seed of the Mayflower, churches, schools, colleges,
tell him where the men of his race have been, or their influence
penetrated; and an intelligent freedom is the monument of conquests whose
results are not to be measured in square miles. Next to the fugitives
whom Moses led out of Egypt, the little ship-load of outcasts who landed
at Plymouth two centuries and a half ago are destined to influence the
future of the world. The spiritual thirst of mankind has for ages been
quenched at Hebrew fountains; but the embodiment in human institutions of
truths uttered by the Son of man eighteen centuries ago was to be mainly
the work of Puritan thought and Puritan self-devotion. Leave New England
out in the cold! While you are plotting it, she sits by every fireside in
the land where there is piety, culture, and free thought.
Faith in God, faith in man, faith in work,--this is the short formula in
which we may sum up the teaching of the founders of New England, a creed
ample enough for this life and the next. If their municipal regulations
smack somewhat of Judaism, yet there can be no nobler aim or more
practical wisdom than theirs; for it was to make the law of man a living
counterpart of the law of God, in their highest conception of it. Were
they too earnest in the strife to save their souls alive? That is still
the problem which every wise and brave man is lifelong in solving. If the
Devil take a less hateful shape to us than to our fathers, he is as busy
with us as with them; and if we cannot find it in our hearts to break
with a gentleman of so much worldly wisdom, who gives such admirable
dinners, and whose manners are so perfect, so much the worse for us.
Looked at on the outside, New England history is dry and unpicturesque.
There is no rustle of silks, no waving of plumes, no clink of golden
spurs. Our sympathies are not awakened by the changeful destinies, the
rise and fall, of great families, whose doom was in their blood. Instead
of all this, we have the homespun fates of Cephas and Prudence repeated
in an infinite series of peaceable sameness, and finding space enough for
record in the family Bible; we have the noise of axe and hammer and saw,
an apotheosis of dogged work, where, reversing the fairy-tale, nothing is
left to luck, and, if there be any poetry, it is something that cannot be
helped,--the waste of the water over the dam. Extrinsically, it is
prosaic and plebeian; intrinsically, it is poetic and noble; for it is,
perhaps, the most perfect incarnation of an idea the world has ever seen.
That idea was not to found a democracy, nor to charter the city of New
Jerusalem by an act of the General Court, as gentlemen seem to think
whose notions of history and human nature rise like an exhalation from
the good things at a Pilgrim Society dinner. Not in the least. They had
no faith in the Divine institution of a system which gives Teague,
because he can dig, as much influence as Ralph, because he can think, nor
in personal at the expense of general freedom. Their view of human rights
was not so limited that it could not take in human relations and duties
also. They would have been likely to answer the claim, "I am as good as
anybody," by a quiet "Yes, for some things, but not for others; as good,
doubtless, in your place, where all things are good." What the early
settlers of Massachusetts _did_ intend, and what they accomplished, was
the founding here of a _new_ England, and a better one, where the
political superstitions and abuses of the old should never have leave to
take root. So much, we may say, they deliberately intended. No nobles,
either lay or cleric, no great landed estates, and no universal ignorance
as the seed-plot of vice and unreason; but an elective magistracy and
clergy, land for all who would till it, and reading and writing, will ye
nill ye, instead. Here at last, it would seem, simple manhood is to have
a chance to play his stake against Fortune with honest dice, uncogged by
those three hoary sharpers, Prerogative, Patricianism, and Priestcraft.
Whoever has looked into the pamphlets published in England during the
Great Rebellion cannot but have been struck by the fact, that the
principles and practice of the Puritan Colony had begun to react with
considerable force on the mother country; and the policy of the
retrograde party there, after the Restoration, in its dealings with New
England, finds a curious parallel as to its motives (time will show
whether as to its results) in the conduct of the same party towards
America during the last four years.[135] This influence and this fear
alike bear witness to the energy of the principles at work here.
We have said that the details of New England history were essentially dry
and unpoetic. Everything is near, authentic, and petty. There is no mist
of distance to soften outlines, no mirage of tradition to give characters
and events an imaginative loom. So much downright work was perhaps never
wrought on the earth's surface in the same space of time as during the
first forty years after the settlement. But mere work is unpicturesque,
and void of sentiment. Irving instinctively divined and admirably
illustrated in his "Knickerbocker" the humorous element which lies in
this nearness of view, this clear, prosaic daylight of modernness, and
this poverty of stage properties, which makes the actors and the deeds
they were concerned in seem ludicrously small when contrasted with the
semi-mythic grandeur in which we have clothed them, as we look backward
from the crowned result, and fancy a cause as majestic as our conception
of the effect. There was, indeed, one poetic side to the existence
otherwise so narrow and practical; and to have conceived this, however
partially, is the one original and American thing in Cooper. This diviner
glimpse illumines the lives of our Daniel Boones, the man of civilization
and old-world ideas confronted with our forest solitudes,--confronted,
too, for the first time, with his real self, and so led gradually to
disentangle the original substance of his manhood from the artificial
results of culture. Here was our new Adam of the wilderness, forced to
name anew, not the visible creation of God, but the invisible creation of
man, in those forms that lie at the base of social institutions, so
insensibly moulding personal character and controlling individual action.
Here is the protagonist of our New World epic, a figure as poetic as that
of Achilles, as ideally representative as that of Don Quixote, as
romantic in its relation to our homespun and plebeian mythus as Arthur in
his to the mailed and plumed cycle of chivalry. We do not mean, of
course, that Cooper's "Leatherstocking" is all this or anything like it,
but that the character typified in him is ideally and potentially all
this and more.
But whatever was poetical in the lives of the early New-Englanders had
something shy, if not sombre, about it. If their natures flowered, it was
out of sight, like the fern. It was in the practical that they showed
their true quality, as Englishmen are wont. It has been the fashion
lately with a few feeble-minded persons to undervalue the New England
Puritans, as if they were nothing more than gloomy and narrow-minded
fanatics. But all the charges brought against these large-minded and
far-seeing men are precisely those which a really able fanatic, Joseph de
Maistre, lays at the door of Protestantism. Neither a knowledge of human
nature nor of history justifies us in confounding, as is commonly done,
the Puritans of Old and New England, or the English Puritans of the third
with those of the fifth decade of the seventeenth century. Fanaticism,
or, to call it by its milder name, enthusiasm, is only powerful and
active so long as it is aggressive. Establish it firmly in power, and it
becomes conservatism, whether it will or no. A sceptre once put in the
hand, the grip is instinctive; and he who is firmly seated in authority
soon learns to think security, and not progress, the highest lesson of
statecraft. From the summit of power men no longer turn their eyes
upward, but begin to look about them. Aspiration sees only one side of
every question; possession, many. And the English Puritans, after their
revolution was accomplished, stood in even a more precarious position
than most successful assailants of the prerogative of whatever _is_ to
continue in being. They had carried a political end by means of a
religious revival. The fulcrum on which they rested their lever to
overturn the existing order of things (as history always placidly calls
the particular forms of _dis_order for the time being) was in the soul of
man. They could not renew the fiery gush of enthusiasm, when once the
molten metal had begun to stiffen in the mould of policy and precedent.
The religious element of Puritanism became insensibly merged in the
political; and, its one great man taken away, it died, as passions have
done before, of possession. It was one thing to shout with Cromwell
before the battle of Dunbar, "Now, Lord, arise, and let thine enemies be
scattered!" and to snuffle, "Rise, Lord, and keep us safe in our
benefices, our sequestered estates, and our five per cent!" Puritanism
meant something when Captain Hodgson, riding out to battle through the
morning mist, turns over the command of his troop to a lieutenant, and
stays to hear the prayer of a cornet, there was "so much of God in it."
Become traditional, repeating the phrase without the spirit, reading the
present backward as if it were written in Hebrew, translating Jehovah by
"I was" instead of "I am,"--it was no more like its former self than the
hollow drum made of Zisca's skin was like the grim captain whose soul it
had once contained. Yet the change was inevitable, for it is not safe to
confound the things of Caesar with the things of God. Some honest
republicans, like Ludlow, were never able to comprehend the chilling
contrast between the ideal aim and the material fulfilment, and looked
askance on the strenuous reign of Oliver,--that rugged boulder of
primitive manhood lying lonely there on the dead level of the
century,--as if some crooked changeling had been laid in the cradle
instead of that fair babe of the Commonwealth they had dreamed. Truly
there is a tide in the affairs of men, but there is no gulf-stream
setting forever in one direction; and those waves of enthusiasm on whose
crumbling crests we sometimes see nations lifted for a gleaming moment
are wont to have a gloomy trough before and behind.
But the founders of New England, though they must have sympathized
vividly with the struggles and triumphs of their brethren in the mother
country, were never subjected to the same trials and temptations, never
hampered with the same lumber of usages and tradition. They were not
driven to win power by doubtful and desperate ways, nor to maintain it by
any compromises of the ends which make it worth having. From the outset
they were builders, without need of first pulling down, whether to make
room or to provide material. For thirty years after the colonization of
the Bay, they had absolute power to mould as they would the character of
their adolescent commonwealth. During this time a whole generation would
have grown to manhood who knew the Old World only by report, in whose
habitual thought kings, nobles, and bishops would be as far away from all
present and practical concern as the figures in a fairy-tale, and all
whose memories and associations, all their unconscious training by eye
and ear, were New English wholly. Nor were the men whose influence was
greatest in shaping the framework and the policy of the Colony, in any
true sense of the word, fanatics. Enthusiasts, perhaps, they were, but
with then the fermentation had never gone further than the ripeness of
the vinous stage. Disappointment had never made it acetous, nor had it
ever putrefied into the turbid zeal of Fifth Monarchism and sectarian
whimsey. There is no better ballast for keeping the mind steady on its
keel, and saving it from all risk of _crankiness_, than business. And
they were business men, men of facts and figures no less than of
religious earnestness. The sum of two hundred thousand pounds had been
invested in their undertaking,--a sum, for that time, truly enormous as
the result of private combination for a doubtful experiment. That their
enterprise might succeed, they must show a balance on the right side of
the countinghouse ledger, as well as in their private accounts with their
own souls. The liberty of praying when and how they would, must be
balanced with an ability of paying when and as they ought. Nor is the
resulting fact in this case at variance with the _a priori_ theory. They
succeeded in making their thought the life and soul of a body politic,
still powerful, still benignly operative, after two centuries; a thing
which no mere fanatic ever did or ever will accomplish. Sober, earnest,
and thoughtful men, it was no Utopia, no New Atlantis, no realization of
a splendid dream, which they had at heart, but the establishment of the
divine principle of Authority on the common interest and the common
consent; the making, by a contribution from the free-will of all, a power
which should curb and guide the free-will of each for the general good.
If they were stern in their dealings with sectaries, it should be
remembered that the Colony was in fact the private property of the
Massachusetts Company, that unity was essential to its success, and that
John of Leyden had taught them how unendurable by the nostrils of honest
men is the corruption of the right of private judgment in the evil and
selfish hearts of men when no thorough mental training has developed the
understanding and given the judgment its needful means of comparison and
correction. They knew that liberty in the hands of feeble-minded and
unreasoning persons (and all the worse if they are honest) means nothing
more than the supremacy of their particular form of imbecility; means
nothing less, therefore, than downright chaos, a Bedlam-chaos of
monomaniacs and bores. What was to be done with men and women, who bore
conclusive witness to the fall of man by insisting on walking up the
broad-aisle of the meeting-house in a costume which that event had put
forever out of fashion! About their treatment of witches, too, there has
been a great deal of ignorant babble. Puritanism had nothing whatever to
do with it. They acted under a delusion, which, with an exception here
and there (and those mainly medical men, like Wierus and Webster),
darkened the understanding of all Christendom. Dr. Henry More was no
Puritan; and his letter to Glanvil, prefixed to the third edition of the
"Sadducismus Triumphatus," was written in 1678, only fourteen years
before the trials at Salem. Bekker's "Bezauberte Welt" was published in
1693; and in the Preface he speaks of the difficulty of overcoming "the
prejudices in which not only ordinary men, but the learned also, are
obstinate." In Hathaway's case, 1702, Chief-Justice Holt, in charging the
jury, expresses no disbelief in the possibility of witchcraft, and the
indictment implies its existence. Indeed, the natural reaction from the
Salem mania of 1692 put an end to belief in devilish compacts and
demoniac possessions sooner in New England than elsewhere. The last we
hear of it there is in 1720, when Rev. Mr. Turell of Medford detected and
exposed an attempted cheat by two girls. Even in 1692, it was the foolish
breath of Cotton Mather and others of the clergy that blew the dying
embers of this ghastly superstition into a flame; and they were actuated
partly by a desire to bring about a religious revival, which might stay
for a while the hastening lapse of their own authority, and still more by
that credulous scepticism of feeble-minded piety which, dreads the
cutting away of an orthodox tumor of misbelief, as if the life-blood of
faith would follow, and would keep even a stumbling-block in the way of
salvation, if only enough generations had tripped over it to make it
venerable. The witches were condemned on precisely the same grounds that
in our day led to the condemnation of "Essays and Reviews."
But Puritanism was already in the decline when such things were possible.
What had been a wondrous and intimate experience of the soul, a flash
into the very crypt and basis of man's nature from the fire of trial, had
become ritual and tradition. In prosperous times the faith of one
generation becomes the formality of the next. "The necessity of a
reformation," set forth by order of the Synod which met at Cambridge in
1679, though no doubt overstating the case, shows how much even at that
time the ancient strictness had been loosened. The country had grown
rich, its commerce was large, and wealth did its natural work in making
life softer and more worldly, commerce in deprovincializing the minds of
those engaged in it. But Puritanism had already done its duty. As there
are certain creatures whose whole being seems occupied with an egg-laying
errand they are sent upon, incarnate ovipositors, their bodies but bags
to hold this precious deposit, their legs of use only to carry them where
they may safeliest be rid of it, so sometimes a generation seems to have
no other end than the conception and ripening of certain germs. Its blind
stirrings, its apparently aimless seeking hither and thither, are but the
driving of an instinct to be done with its parturient function toward
these principles of future life and power. Puritanism, believing itself
quick with the seed of religious liberty, laid, without knowing it, the
egg of democracy. The English Puritans pulled down church and state to
rebuild Zion on the ruins, and all the while it was not Zion, but
America, they were building. But if their millennium went by, like the
rest, and left men still human; if they, like so many saints and martyrs
before them, listened in vain for the sound of that trumpet which was to
summon all souls to a resurrection from the body of this death which men
call life,--it is not for us, at least, to forget the heavy debt we owe
them. It was the drums of Naseby and Dunbar that gathered the minute-men
on Lexington Common; it was the red dint of the axe on Charles's block
that marked One in our era. The Puritans had their faults. They were
narrow, ungenial; they could not understand the text, "I have piped to
you and ye have not danced," nor conceive that saving one's soul should
be the cheerfullest, and not the dreariest, of businesses. Their
preachers had a way, like the painful Mr. Perkins, of pronouncing the
word _damn_ with such an emphasis as left a doleful echo in their
auditors' ears a good while after. And it was natural that men who
captained or accompanied the exodus from existing forms and associations
into the doubtful wilderness that led to the promised land, should find
more to their purpose in the Old Testament than in the New. As respects
the New England settlers, however visionary some of their religious
tenets may have been, their political ideas savored of the realty, and it
was no Nephelococcygia of which they drew the plan, but of a commonwealth
whose foundation was to rest on solid and familiar earth. If what they
did was done in a corner, the results of it were to be felt to the ends
of the earth; and the figure of Winthrop should be as venerable in
history as that of Romulus is barbarously grand in legend.
I am inclined to think that many of our national characteristics, which
are sometimes attributed to climate and sometimes to institutions, are
traceable to the influences of Puritan descent. We are apt to forget how
very large a proportion of our population is descended from emigrants who
came over before 1660. Those emigrants were in great part representatives
of that element of English character which was most susceptible of
religious impressions; in other words, the most earnest and imaginative.
Our people still differ from their English cousins (as they are fond of
calling themselves when they are afraid we may do them a mischief) in a
certain capacity for enthusiasm, a devotion to abstract principle, an
openness to ideas, a greater aptness for intuitions than for the slow
processes of the syllogism, and, as derivative from this, in minds of
looser texture, a light-armed, skirmishing habit of thought, and a
positive preference of the birds in the bush,--an excellent quality of
character _before_ you have your bird in the hand.
There have been two great distributing centres of the English race on
this continent, Massachusetts and Virginia. Each has impressed the
character of its early legislators on the swarms it has sent forth. Their
ideas are in some fundamental respects the opposites of each other, and
we can only account for it by an antagonism of thought beginning with the
early framers of their respective institutions. New England abolished
caste; in Virginia they still talk of "quality folks." But it was in
making education not only common to all, but in some sense compulsory on
all, that the destiny of the free republics of America was practically
settled. Every man was to be trained, not only to the use of arms, but of
his wits also; and it is these which alone make the others effective
weapons for the maintenance of freedom. You may disarm the hands, but not
the brains, of a people, and to know what should be defended is the first
condition of successful defence. Simple as it seems, it was a great
discovery that the key of knowledge could turn both ways, that it could
open, as well as lock, the door of power to the many. The only things a
New-Englander was ever locked out of were the jails. It is quite true
that our Republic is the heir of the English Commonwealth; but as we
trace events backward to their causes, we shall find it true also, that
what made our Revolution a foregone conclusion was that act of the
General Court, passed in May, 1647, which established the system of
common schools. "To the end that learning may not be buried in the graves
of our forefathers in Church and Commonwealth, the Lord assisting our
endeavors, it is therefore ordered by this Court and authority thereof,
that every township in this jurisdiction, after the Lord hath increased
them to fifty householders, shall then forthwith appoint one within their
towns to teach all such children as shall resort to him to write and
read."
Passing through some Massachusetts village, perhaps at a distance from
any house, it may be in the midst of a piece of woods where four roads
meet, one may sometimes even yet see a small square one-story building,
whose use would not be long doubtful. It is summer, and the flickering
shadows of forest-leaves dapple the roof of the little porch, whose door
stands wide, and shows, hanging on either hand, rows of straw hats and
bonnets, that look as if they had done good service. As you pass the open
windows, you hear whole platoons of high-pitched voices discharging words
of two or three syllables with wonderful precision and unanimity. Then
there is a pause, and the voice of the officer in command is heard
reproving some raw recruit whose vocal musket hung fire. Then the drill
of the small infantry begins anew, but pauses again because some
urchin--who agrees with Voltaire that the superfluous is a very necessary
thing--insists on spelling "subtraction" with an _s_ too much.
If you had the good fortune to be born and bred in the Bay State, your
mind is thronged with half-sad, half-humorous recollections. The a-b abs
of little voices long since hushed in the mould, or ringing now in the
pulpit, at the bar, or in the Senate-chamber, come back to the ear of
memory. You remember the high stool on which culprits used to be elevated
with the tall paper fool's-cap on their heads, blushing to the ears; and
you think with wonder how you have seen them since as men climbing the
world's penance-stools of ambition without a blush, and gladly giving
everything for life's caps and bells. And you have pleasanter memories of
going after pond-lilies, of angling for horn-pouts,--that queer bat among
the fishes,--of nutting, of walking over the creaking snow-crust in
winter, when the warm breath of every household was curling up silently
in the keen blue air. You wonder if life has any rewards more solid and
permanent than the Spanish dollar that was hung around your neck to be
restored again next day, and conclude sadly that it was but too true a
prophecy and emblem of all worldly success. But your moralizing is broken
short off by a rattle of feet and the pouring forth of the whole
swarm,--the boys dancing and shouting,--the mere effervescence of the
fixed air of youth and animal spirits uncorked,--the sedater girls in
confidential twos and threes decanting secrets out of the mouth of one
cape-bonnet into that of another. Times have changed since the jackets
and trousers used to draw up on one side of the road, and the petticoats
on the other, to salute with bow and courtesy the white neckcloth of the
parson or the squire, if it chanced to pass during intermission.
Now this little building, and others like it, were an original kind of
fortification invented by the founders of New England. They are the
martello-towers that protect our coast. This was the great discovery of
our Puritan forefathers. They were the first lawgivers who saw clearly
and enforced practically the simple moral and political truth, that
knowledge was not an alms to be dependent on the chance charity of
private men or the precarious pittance of a trust-fund, but a sacred debt
which the Commonwealth owed to every one of her children. The opening of
the first grammar-school was the opening of the first trench against
monopoly in church and state; the first row of trammels and pothooks
which the little Shearjashubs and Elkanahs blotted and blubbered across
their copy-books, was the preamble to the Declaration of Independence.
The men who gave every man the chance to become a landholder, who made
the transfer of land easy, and put knowledge within the reach of all,
have been called narrow-minded, because they were intolerant. But
intolerant of what? Of what they believed to be dangerous nonsense,
which, if left free, would destroy the last hope of civil and religious
freedom. They had not come here that every man might do that which seemed
good in his own eyes, but in the sight of God. Toleration, moreover, is
something which is won, not granted. It is the equilibrium of neutralized
forces. The Puritans had no notion of tolerating mischief. They looked
upon their little commonwealth as upon their own private estate and
homestead, as they had a right to do, and would no more allow the Devil's
religion of unreason to be preached therein, than we should permit a
prize-fight in our gardens. They were narrow; in other words they had an
edge to them, as men that serve in great emergencies must; for a Gordian
knot is settled sooner with a sword than a beetle.
The founders of New England are commonly represented in the after-dinner
oratory of their descendants as men "before their time," as it is called;
in other words, deliberately prescient of events resulting from new
relations of circumstances, or even from circumstances new in themselves,
and therefore altogether alien from their own experience. Of course, such
a class of men is to be reckoned among those non-existent human varieties
so gravely catalogued by the ancient naturalists. If a man could shape
his action with reference to what should happen a century after his
death, surely it might be asked of him to call in the help of that easier
foreknowledge which reaches from one day to the next,--a power of
prophecy whereof we have no example. I do not object to a wholesome pride
of ancestry, though a little mythical, if it be accompanied with the
feeling that _noblesse oblige_, and do not result merely in a placid
self-satisfaction with our own mediocrity, as if greatness, like
righteousness, could be imputed. We can pardon it even in conquered
races, like the Welsh and Irish, who make up to themselves for present
degradation by imaginary empires in the past whose boundaries they can
extend at will, carrying the bloodless conquests of fancy over regions
laid down upon no map, and concerning which authentic history is
enviously dumb. Those long beadrolls of Keltic kings cannot tyrannize
over us, and we can be patient so long as our own crowns are uncracked by
the shillalah sceptres of their actual representatives. In our own case,
it would not be amiss, perhaps, if we took warning by the example of
Teague and Taffy. At least, I think it would be wise in our orators not
to put forward so prominently the claim of the Yankee to universal
dominion, and his intention to enter upon it forthwith. If we do our
duties as honestly and as much in the fear of God as our forefathers did,
we need not trouble ourselves much about other titles to empire. The
broad foreheads and long heads will win the day at last in spite of all
heraldry, and it will be enough if we feel as keenly as our Puritan
founders did that those organs of empire may be broadened and lengthened
by culture.[136] That our self-complacency should not increase the
complacency of outsiders is not to be wondered at. As _we_ sometimes take
credit to ourselves (since all commendation of our ancestry is indirect
self-flattery) for what the Puritans fathers never were, so there are
others who, to gratify a spite against their descendants, blame them for
not having been what they could not be; namely, before their time in such
matters as slavery, witchcraft, and the like. The view, whether of friend
or foe, is equally unhistorical, nay, without the faintest notion of all
that makes history worth having as a teacher. That our grandfathers
shared in the prejudices of their day is all that makes them human to us;
and that nevertheless they could act bravely and wisely on occasion makes
them only the more venerable. If certain barbarisms and superstitions
disappeared earlier in New England than elsewhere, not by the decision of
exceptionally enlightened or humane judges, but by force of public
opinion, that is the fact that is interesting and instructive for us. I
never thought it an abatement of Hawthorne's genius that he came lineally
from one who sat in judgment on the witches in 1692; it was interesting
rather to trace something hereditary in the sombre character of his
imagination, continually vexing itself to account for the origin of evil,
and baffled for want of that simple solution in a personal Devil.
But I have no desire to discuss the merits or demerits of the Puritans,
having long ago learned the wisdom of saving my sympathy for more modern
objects than Hecuba. My object is to direct the attention of my readers
to a collection of documents where they may see those worthies as they
were in their daily living and thinking. The collections of our various
historical and antiquarian societies can hardly be said to be _published_
in the strict sense of the word, and few consequently are aware how much
they contain of interest for the general reader no less than the special
student. The several volumes of "Winthrop Papers," in especial, are a
mine of entertainment. Here we have the Puritans painted by themselves,
and, while we arrive at a truer notion of the characters of some among
them, and may accordingly sacrifice to that dreadful superstition of
being usefully employed which makes so many bores and bored, we can also
furtively enjoy the oddities of thought and speech, the humors of the
time, which our local historians are too apt to despise as inconsidered
trifles. For myself I confess myself heretic to the established theory of
the gravity of history, and am not displeased with an opportunity to
smile behind my hand at any ludicrous interruption of that sometimes
wearisome ceremonial. I am not sure that I would not sooner give up
Raleigh spreading his cloak to keep the royal Dian's feet from the mud,
than that awful judgment upon the courtier whose Atlantean thighs leaked
away in bran through the rent in his trunk-hose. The painful fact that
Fisher had his head cut off is somewhat mitigated to me by the
circumstance that the Pope should have sent him, of all things in the
world, a cardinal's hat after that incapacitation. Theology herself
becomes less unamiable to me when I find the Supreme Pontiff writing to
the Council of Trent that "they should begin with original sin,
_maintaining yet a due respect for the Emperor_." That infallibility
should thus courtesy to decorum, shall make me think better of it while I
live. I shall accordingly endeavor to give my readers what amusement I
can, leaving it to themselves to extract solid improvement from the
volumes before us, which include a part of the correspondence of three
generations of Winthrops.
Let me premise that there are two men above all others for whom our
respect is heightened by these letters,--the elder John Winthrop and
Roger Williams. Winthrop appears throughout as a truly magnanimous and
noble man in an unobtrusive way,--a kind of greatness that makes less
noise in the world, but is on the whole more solidly satisfying than most
others,--a man who has been dipped in the river of God (a surer baptism
than Styx or dragon's blood) till his character is of perfect proof, and
who appears plainly as the very soul and life of the young Colony. Very
reverend and godly he truly was, and a respect not merely ceremonious,
but personal, a respect that savors of love, shows itself in the letters
addressed to him. Charity and tolerance flow so naturally from the pen of
Williams that it is plain they were in his heart. He does not show
himself a very strong or very wise man, but a thoroughly gentle and good
one. His affection for the two Winthrops is evidently of the warmest. We
suspect that he lived to see that there was more reason in the drum-head
religious discipline which made him, against his will, the founder of a
commonwealth, than he may have thought at first. But for the fanaticism
(as it is the fashion to call the sagacious straitness) of the abler men
who knew how to root the English stock firmly in this new soil on either
side of him, his little plantation could never have existed, and he
himself would have been remembered only, if at all, as one of the jarring
atoms in a chaos of otherwise-mindedness.
Two other men, Emanuel Downing and Hugh Peter, leave a positively
unpleasant savor in the nostrils. Each is selfish in his own
way,--Downing with the shrewdness of an attorney, Peter with that
clerical unction which in a vulgar nature so easily degenerates into
greasiness. Neither of them was the man for a forlorn hope, and both
returned to England when the civil war opened prospect of preferment
there. Both, we suspect, were inclined to value their Puritanism for its
rewards in this world rather than the next. Downing's son, Sir George,
was basely prosperous, making the good cause pay him so long as it was
solvent, and then selling out in season to betray his old commander,
Colonel Okey, to the shambles at Charing Cross. Peter became a colonel in
the Parliament's army, and under the Protectorate one of Cromwell's
chaplains. On his trial, after the Restoration, he made a poor figure, in
striking contrast to some of the brave men who suffered with him. At his
execution a shocking brutality was shown. "When Mr Cook was cut down and
brought to be quartered, one they called Colonel Turner calling to the
Sheriff's men to bring Mr Peters near, that he might see it; and by and
by the Hangman came to him all besmeared in blood, and rubbing his bloody
hands together, he tauntingly asked, _Come, how do you like this, Mr.
Peters? How do you like this work?_"[137] This Colonel Turner can hardly
have been other than the one who four years later came to the hangman's
hands for robbery; and whose behavior, both in the dock and at the
gallows, makes his trial one of the most entertaining as a display of
character. Peter would seem to have been one of those men gifted with
what is sometimes called eloquence; that is, the faculty of stating
things powerfully from momentary feeling, and not from that conviction of
the higher reason which alone can give force and permanence to words. His
letters show him subject, like others of like temperament, to fits of
"hypocondriacal melancholy," and the only witness he called on his trial
was to prove that he was confined to his lodgings by such an attack on
the day of the king's beheading. He seems to have been subject to this
malady at convenience, as some women to hysterics. Honest John Endicott
plainly had small confidence in him, and did not think him the right man
to represent the Colony in England. There is a droll resolve in the
Massachusetts records by which he is "desired to write to Holland for
500_l._ worth of _peter_, & 40_l._ worth of match." It is with a match
that we find him burning his fingers in the present correspondence.
Peter seems to have entangled himself somehow with a Mrs. Deliverance
Sheffield, whether maid or widow nowhere appears, but presumably the
latter. The following statement of his position is amusing enough: "I
have sent Mrs D. Sh. letter, which puts mee to new troubles, for though
shee takes liberty upon my Cossen Downing's speeches, yet (Good Sir) let
mee not be a foole in Israel. I had many good answers to yesterday's
worke [a Fast] and amongst the rest her letter; which (if her owne) doth
argue more wisedome than I thought shee had. You have often sayd I could
not leave her; what to doe is very considerable. Could I with comfort &
credit desist, this seemes best: could I goe on & content myselfe, that
were good.... For though I now seeme free agayne, yet the depth I know
not. Had shee come over with me, I thinke I had bin quieter. This shee
may know, that I have sought God earnestly, that the nexte weeke I shall
bee riper:--I doubt shee gaynes most by such writings: & shee deserves
most where shee is further of. If you shall amongst you advise mee to
write to hir, I shall forthwith; our towne lookes upon mee contracted &
so I have sayd myselfe; what wonder the charge [change?] would make, I
know not." Again: "Still pardon my offensive boldnes: I know not well
whither Mrs Sh. have set mee at liberty or not: my conclusion is, that if
you find I cannot make an honorable retreat, then I shall desire to
advance [Greek: sun Theo]. Of you I now expect your last advise, viz:
whither I must goe on or of, _saluo evangelij honore_: if shee bee in
good earnest to leave all agitations this way, then I stand still & wayt
God's mind concerning mee.... If I had much mony I would part with it to
her free, till wee heare what England doth, supposing I may bee called to
some imployment that will not suit a marryed estate": (here another mode
of escape presents itself, and he goes on:) "for indeed (Sir) some must
looke out & I have very strong thoughts to speake with the Duitch
Governor & lay some way there for a supply &c." At the end of the letter,
an objection to the lady herself occurs to him: "Once more for Mrs Sh: I
had from Mr Hibbins & others, her fellowpassengers, sad discouragements
where they saw her in her trim. I would not come of with dishonor, nor
come on with griefe, or ominous hesitations." On all this shilly-shally
we have a shrewd comment in a letter of Endicott: "I cannot but acquaint
you with my thoughts concerning Mr Peter since hee receaued a letter from
Mrs Sheffield, which was yesterday in the eveninge after the Fast, shee
seeming in her letter to abate of her affeccions towards him & dislikinge
to come to Salem vppon such termes as he had written. I finde now that
hee begins to play her parte, & if I mistake not, you will see him as
greatly in loue with her (if shee will but hold of a little) as euer shee
was with him; but he conceales it what he can as yett. The begininge of
the next weeke you will heare further from him." The widow was evidently
more than a match for poor Peter.
It should appear that a part of his trouble arose from his having
coquetted also with a certain Mrs. Ruth, about whom he was "dealt with by
Mrs Amee, Mr Phillips & 2 more of the Church, our Elder being one. When
Mr Phillips with much violence & sharpnes charged mee home ... that I
should hinder the mayd of a match at London, which was not so, could not
thinke of any kindnes I euer did her, though shee haue had above 300_li._
through my fingers, so as if God uphold me not after an especiall manner,
it will sinke me surely ... hee told me he would not stop my intended
marriage, but assured mee it would not bee good ... all which makes mee
reflect upon my rash proceedings with Mrs Sh." Panurge's doubts and
difficulties about matrimony were not more entertainingly contradictory.
Of course, Peter ends by marrying the widow, and presently we have a
comment on "her trim." In January, 1639, he writes to Winthrop: "My wife
is very thankfull for her apples, & _desires much the new fashioned
shooes_." Eight years later we find him writing from England, where he
had been two years: "I am coming over if I must; my wife comes of
necessity to New England, having run her selfe out of breath here"; and
then in the postscript, "bee sure you never let my wife come away from
thence without my leave, & then you love mee." But life is never pure
comedy, and the end in this case is tragical. Roger Williams, after his
return from England in 1654, writes to John Winthrop, Jr.: "Your brother
flourisheth in good esteeme & is eminent for maintaining the Freedome of
the Conscience as to matters of Beliefe, Religion, & Worship. Your Father
Peters preacheth the same Doctrine though not so zealously as some years
since, yet cries out against New English Rigidities & Persecutions, their
civil injuries & wrongs to himselfe, & their unchristian dealing with him
in excommunicating his distracted wife. All this he tould me in his
lodgings at Whitehall, those lodgings which I was tould were Canterburies
[the Archbishop], but he himselfe tould me that that Library wherein we
were together was Canterburies & given him by the Parliament. His wife
lives from him, not wholy but much distracted. He tells me he had but 200
a yeare & he allowed her 4 score per annum of it. Surely, Sir, the most
holy Lord is most wise in all the trialls he exerciseth his people with.
He tould me that his affliction from his wife stird him up to Action
abroad, & when successe tempted him to Pride, the Bitternes in his
bozome-comforts was a Cooler & a Bridle to him." Truly the whirligig of
time brings about strange revenges. Peter had been driven from England by
the persecutions of Laud; a few years later he "stood armed on the
scaffold" when that prelate was beheaded, and now we find him installed
in the archiepiscopal lodgings. Dr. Palfrey, it appears to me, gives
altogether too favorable an opinion both of Peter's character and
abilities. I conceive him to have been a vain and selfish man. He may
have had the bravery of passionate impulse, but he wanted that steady
courage of character which has such a beautiful constancy in Winthrop. He
always professed a longing to come back to New England, but it was only a
way he had of talking. That he never meant to come is plain from these
letters. Nay, when things looked prosperous in England, he writes to the
younger Winthrop: "My counsell is you should come hither with your family
for certaynly you will bee capable of a comfortable living in this free
Commonwealth. I doo seriously advise it.... G. Downing is worth 500_l_.
per annum but 4_l_. per diem--your brother Stephen worth 2000_l_. & a
maior. I pray come." But when he is snugly ensconced in Whitehall, and
may be presumed to have some influence with the prevailing powers, his
zeal cools. "I wish you & all friends to stay there & rather looke to the
West Indyes if they remoue, for many are here to seeke when they come
ouer." To me Peter's highest promotion seems to have been that he walked
with John Milton at the Protector's funeral. He was, I suspect, one of
those men, to borrow a charitable phrase of Roger Williams, who "feared
God in the main," that is, whenever it was not personally inconvenient.
William Coddington saw him in his glory in 1651: "Soe wee toucke the tyme
to goe to viset Mr Petters at his chamber. I was mery with him & called
him the Arch Bp: of Canterberye, in regard to his adtendance by ministers
& gentlemen, & it passed very well." Considering certain charges brought
against Peter, (though he is said, when under sentence of death, to have
denied the truth of them,) Coddington's statement that he liked to have
"gentlewomen waite of him" in his lodgings has not a pleasant look. One
last report of him we get (September, 1659) in a letter of John
Davenport,--"that Mr Hugh Peters is distracted & under sore horrors of
conscience, crying out of himselfe as damned & confessing haynous
actings."
Occasionally these letters give us interesting glimpses of persons and
things in England. In the letter of Williams just cited, there is a
lesson for all parties raised to power by exceptional causes. "Surely,
Sir, youre Father & all the people of God in England ... are now in the
sadle & at the helme, so high that _non datus descensus nisi cadendo_:
Some cheere up their spirits with the impossibilitie of another fall or
turne, so doth Major G. Harrison ... a very gallant most deserving
heavenly man, but most highflowne for the Kingdom of the Saints & the 5th
Monarchie now risen & their sun never to set againe &c. Others, as, to my
knowledge, the Protector ... are not so full of that faith of miracles,
but still imagine changes & persecutions & the very slaughter of the
witnesses before that glorious morning so much desired of a worldly
Kingdome, if ever such a Kingdome (as literally it is by so many
expounded) be to arise in this present world & dispensation." Poor
General Harrison lived to be one of the witnesses so slaughtered. The
practical good sense of Cromwell is worth noting, the English
understanding struggling against Judaic trammels. Williams gives us
another peep through the keyhole of the past: "It pleased the Lord to
call me for some time & with some persons to practice the Hebrew, the
Greeke, Latine, French & Dutch. The secretarie of the Councell (Mr
Milton) for my Dutch I read him, read me many more languages. Grammar
rules begin to be esteemed a Tyrannie. I taught 2 young Gentlemen, a
Parliament man's sons, as we teach our children English, by words,
phrazes, & constant talke, &c." It is plain that Milton had talked over
with Williams the theory put forth in his tract on Education, and made a
convert of him. We could wish that the good Baptist had gone a little
more into particulars. But which of us knows among the men he meets whom
time will dignify by curtailing him of the "Mr.," and reducing him to a
bare patronymic, as being a kind by himself? We have a glance or two at
Oliver, who is always interesting. "The late renowned Oliver confest to
me in close discourse about the Protestants aifaires &c that he yet feard
great persecutions to the protestants from the Romanists before the
downfall of the Papacie," writes Williams in 1660. This "close discourse"
must have been six years before, when Williams was in England. Within a
year after, Oliver interfered to some purpose in behalf of the
Protestants of Piedmont, and Mr. Milton wrote his famous sonnet. Of the
war with Spain, Williams reports from his letters out of England in 1656:
"This diversion against the Spaniard hath turnd the face & thoughts of
many English, so that the saying now is, Crowne the Protector with
gould,[138] though the sullen yet cry, Crowne him with thornes."
Again in 1654: "I know the Protector had strong thoughts of Hispaniola &
Cuba. Mr Cotton's interpreting of Euphrates to be the West Indies, the
supply of gold (to take off taxes), & the provision of a warmer
_diverticulum & receptaculum_ then N. England is, will make a footing
into those parts very precious, & if it shall please God to vouchsafe
successe to this fleete, I looke to hear of an invitation at least to
these parts for removall from his Highnes who lookes on N. E. only with
an eye of pitie, as poore, cold & useless." The mixture of Euphrates and
taxes, of the transcendental and practical, prophecy taking precedence of
thrift, is characteristic, and recalls Cromwell's famous rule, of fearing
God _and_ keeping your powder dry. In one of the Protector's
speeches,[139] he insists much on his wish to retire to a private life.
There is a curious confirmation of his sincerity in a letter of William
Hooke, then belonging to his household, dated the 13th of April, 1657.
The question of the kingly title was then under debate, and Hooke's
account of the matter helps to a clearer understanding of the reasons for
Cromwell's refusing the title: "The protector is urged _utrinque_ & (I am
ready to think) willing enough to betake himself to a private life, if it
might be. He is a godly man, much in prayer & good discourses, delighting
in good men & good ministers, self-denying & ready to promote any good
work for Christ."[140] On the 5th of February, 1654, Captain John Mason,
of Pequot memory, writes "a word or twoe of newes as it comes from Mr
Eaton, viz: that the Parliament sate in September last; they chose their
old Speaker & Clarke. The Protectour told them they were a free
Parliament, & soe left them that day. They, considering where the
legislative power resided, concluded to vote it on the morrow, & to take
charge of the militia. The Protectour hereing of it, sent for some
numbers of horse, went to the Parliament House, nayld up the doores, sent
for them to the Painted Chamber, told them they should attend the lawes
established, & that he would wallow in his blood before he would part
with what was conferd upon him, tendering them an oath: 140 engaged." Now
it is curious that Mr. Eaton himself, from whom Mason got his news,
wrote, only two days before, an account, differing, in some particulars,
and especially in tone, from Mason's. Of the speech he says, that it
"gave such satisfaction that about 200 have since ingaged to owne the
present Government." Yet Carlyle gives the same number of signers (140)
as Mason, and there is a sentence in Cromwell's speech, as reported by
Carlyle, of precisely the same purport as that quoted by Mason. To me,
that "wallow in my blood" has rather more of the Cromwellian ring in it,
more of the quality of spontaneous speech, than the "rolled into my grave
and buried with infamy" of the official reporter. John Haynes (24th July,
1653) reports "newes from England of astonishing nature," concerning the
dissolution of the Rump. We quote his story both as a contemporaneous
version of the event, and as containing some particulars that explain the
causes that led to it. It differs, in some respects, from Carlyle, and is
hardly less vivid as a picture: "The Parliament of England & Councell of
State are both dissolved, by whom & the manner this: The Lord Cromwell,
Generall, went to the house & asked the Speaker & Bradshaw by what power
they sate ther. They answered by the same power that he woare his sword.
Hee replied they should know they did not, & said they should sitt noe
longer, demanding an account of the vast sommes of money they had
received of the Commons. They said the matter was of great consequence &
they would give him accompt in tenn dayes. He said, Noe, they had sate
too long already (& might now take their ease,) for ther inriching
themselves & impoverishing the Commons, & then seazed uppon all the
Records. Immediatly Lambert, Livetenant Generall, & Hareson Maior
Generall (for they two were with him), tooke the Speaker Lenthall by the
hands, lift him out of the Chaire, & ledd him out of the house, &
commanded the rest to depart, which fortwith was obeied, & the Generall
tooke the keyes & locked the doore." He then goes on to give the reasons
assigned by different persons for the act. Some said that the General
"scented their purpose" to declare themselves perpetual, and to get rid
of him by ordering him to Scotland. "Others say this, that the cries of
the oppressed proveiled much with him.... & hastned the declaracion of
that ould principle, _Salus populi suprema lex_ &c." The General, in the
heat of his wrath, himself snatching the keys and locking the door, has a
look of being drawn from the life. Cromwell, in a letter to General
Fortescue (November,1655), speaks sharply of the disorders and
debauchedness, profaneness and wickedness, commonly practised amongst the
army sent out to the West Indies. Major Mason gives us a specimen: "It is
here reported that some of the soldiers belonging to the ffleet at Boston
ffell upon the watch: after some bickering they comanded them to goe
before the Governour; they retorned that they were Cromwell's boyes."
Have we not, in these days, heard of "Sherman's boys"?
Belonging properly to the "Winthrop Papers," but printed in an earlier
volume (Third Series, Vol. I. pp. 185-198), is a letter of John
Maidstone, which contains the best summary of the Civil War that I ever
read. Indeed, it gives a clearer insight into its causes, and a better
view of the vicissitudes of the Commonwealth and Protectorate, than any
one of the more elaborate histories. There is a singular equity and
absence of party passion in it which gives us faith in the author's
judgment. He was Oliver's Steward of the Household, and his portrait of
him, as that of an eminently fair-minded man who knew him well, is of
great value. Carlyle has not copied it, and, as many of my readers may
never have seen it, I reproduce it here: "Before I pass further, pardon
me in troubling you with the character of his person, which, by reason of
my nearness to him, I had opportunity well to observe. His body was well
compact and strong; his stature under six feet, (I believe about two
inches;) his head so shaped as you might see it a store-house and shop
both, of a vast treasury of natural parts. His temper exceeding fiery, as
I have known, but the flame of it kept down for the most part or soon
allayed with those moral endowments he had. He was naturally
compassionate towards objects in distress, even to an effeminate measure;
though God had made him a heart wherein was left little room for any fear
but what was due to himself, of which there was a large proportion, yet
did he exceed in tenderness toward sufferers. A larger soul, I think,
hath seldom dwelt in a house of clay than his was. I do believe, if his
story were impartially transmitted, and the unprejudiced world well
possessed with it, she would add him to her nine worthies and make that
number a _decemviri_. He lived and died in comfortable communion with
God, as judicious persons near him well observed. He was that Mordecai
that sought the welfare of his people and spake peace to his seed. Yet
were his temptations such, as it appeared frequently that he that hath
grace enough for many men may have too little for himself, the treasure
he had being but in an earthen vessel and that equally defiled with
original sin as any other man's nature is." There are phrases here that
may be matched with the choicest in the life of Agricola; and, indeed,
the whole letter, superior to Tacitus in judicial fairness of tone, goes
abreast of his best writing in condensation, nay, surpasses it in this,
that, while in Tacitus the intensity is of temper, here it is the clear
residuum left by the ferment and settling of thought. Just before,
speaking of the dissolution of Oliver's last Parliament, Maidstone says:
"That was the last which sat during his life, he being compelled to
wrestle with the difficulties of his place so well as he could without
parliamentary assistance, and in it met with so great a burthen as (I
doubt not to say) it drank up his spirits, of which his natural
constitution yielded a vast stock, and brought him to his grave, his
interment being the seed-time of his glory and England's calamity."
Hooke, in a letter of April 16, 1658, has a passage worth quoting: "The
dissolucion of the last Parliament puts the supreme powers upon
difficulties, though the trueth is the Nacion is so ill spirited that
little good is to be expected from these Generall Assemblies. They [the
supreme powers, to wit, Cromwell] have been much in Counsell since this
disappointment, & God hath been sought by them in the effectuall sense of
the need of help from heaven & of the extreme danger impendent on a
miscarriage of their advises. But our expences are so vast that I know
not how they can avoyde a recurrence to another Session & to make a
further tryall.... The land is full of discontents, & the Cavaleerish
party doth still expect a day & nourish hopes of a Revolucion. The
Quakers do still proceed & are not yet come to their period. The
Presbyterians do abound, I thinke, more than ever, & are very bold &
confident because some of their masterpieces lye unanswered, particularly
theire _Jus Divinum Regiminis Ecclesiastici_ which I have sent to Mr.
Davenporte. It hath been extant without answer these many years [only
four, brother Hooke, if we may trust the title-page]. The Anabaptists
abound likewise, & Mr Tombes hath pretended to have answered all the
bookes extant against his opinion. I saw him presenting it to the
Protectour of late. The Episcopall men ply the Common-Prayer booke with
much more boldness then ever since these turnes of things, even in the
open face of the City in severall places. I have spoken of it to the
Protectour but as yet nothing is done in order to their being
suppressed." It should teach us to distrust the apparent size of objects,
which is a mere cheat of their nearness to us, that we are so often
reminded of how small account things seem to one generation for which
another was ready to die. A copy of the _Jus Divinum_ held too close to
the eyes could shut out the universe with its infinite chances and
changes, its splendid indifference to our ephemeral fates. Cromwell, we
should gather, had found out the secret of this historical perspective,
to distinguish between the blaze of a burning tar-barrel and the final
conflagration of all things. He had learned tolerance by the possession
of power,--a proof of his capacity for rule. In 1652 Haynes writes: "Ther
was a Catechise lately in print ther, that denied the divinity of Christ,
yett ther was motions in the house by some, to have it lycenced by
authority. Cromwell mainly oposed, & at last it was voted to bee burnt
which causes much discontent of somme." Six years had made Cromwell
wiser.
One more extract from a letter of Hooke's (30th March, 1659) is worth
giving. After speaking of Oliver's death, he goes on to say: "Many
prayers were put up solemnly for his life, & some, of great & good note,
were too confident that he would not die.... I suppose himselfe had
thoughts that he should have outlived this sickness till near his
dissolution, perhaps a day or two before; which I collect partly by some
words which he was said to speak ... & partly from his delaying, almost
to the last, to nominate his successor, to the wonderment of many who
began sooner to despair of his life.... His eldest son succeedeth him,
being chosen by the Council, the day following his father's death,
whereof he had no expectation. I have heard him say he had thought to
have lived as a country gentleman, & that his father had not employed him
in such a way as to prepare him for such employment; which, he thought,
he did designedly. I suppose his meaning was lest it should have been
apprehended ha had prepared & appointed him for such a place, the burthen
whereof I have several times heard him complaining under since his coming
to the Government, the weighty occasions whereof with continuall
oppressing cares had drunk up his father's spirits, in whose body very
little blood was found when he was opened: the greatest defect visible
was in his heart, which was flaccid & shrunk together. Yet he was one
that could bear much without complaining, as one of a strong constitution
of brain (as appeared when he was dissected) & likewise of body. His son
seemeth to be of another frame, soft & tender, & penetrable with easier
cares by much, yet he is of a sweete countenance, vivacious & candid, as
is the whole frame of his spirit, only naturally inclined to choler. His
reception of multitudes of addresses from towns, cities, & counties doth
declare, among several other indiciums, more of ability in him than
could, ordinarily, have been expected from him. He spake also with
general acceptation & applause when he made his speech before the
Parliament, even far beyond the Lord Fynes....[141] If this Assembly
miss it, we are like to be in an ill condition. The old ways & customs of
England, as to worshipe, are in the hearts of the most, who long to see
the days again which once they saw.... The hearts of very many are for
the house of the Stewarts, & there is a speech as if they would attempt
to call the late King's judges into question.... The city, I hear is full
of Cavaliers." Poor Richard appears to have inherited little of his
father but the inclination to choler. That he could speak far beyond the
Lord Fynes seems to have been not much to the purpose. Rhetoric was not
precisely the medicine for such a case as he had to deal with. Such were
the glimpses which the New England had of the Old. Ishmael must ere-long
learn to shift for himself.
The temperance question agitated the fathers very much as it still does
the children. We have never seen the anti-prohibition argument stated
more cogently than in a letter of Thomas Shepard, minister of Cambridge,
to Winthrop, in 1639: "This also I doe humbly intreat, that there may be
no sin made of _drinking in any case one to another_, for I am confident
he that stands here will fall & be beat from his grounds by his own
arguments; as also that the consequences will be very sad, and the thing
provoking to God & man to make more sins than (as yet is seene) God
himself hath made." A principle as wise now as it was then. Our ancestors
were also harassed as much as we by the difficulties of domestic service.
In a country where land might be had for the asking, it was not easy to
keep hold of servants brought over from England. Emanuel Downing, always
the hard, practical man, would find a remedy in <DW64> slavery. "A warr
with the Narraganset," he writes to Winthrop in 1645, "is verie
considerable to this plantation, ffor I doubt whither it be not synne in
us, having power in our hands, to suffer them to maynteyne the worship of
the devill which their pawwawes often doe; 2lie, If upon a just warre the
Lord should deliver them into our hands, wee might easily have men,
woemen, & children enough to exchange for Moores, which wilbe more
gaynefull pilladge for us than wee conceive, for I doe not see how wee
can thrive untill wee gett into a stock of slaves sufficient to doe all
our buisenes, for our childrens children will hardly see this great
Continent filled with people, soe that our servants will still desire
freedome to plant for them selves, & not stay but for verie great wages.
And I suppose you know verie well how wee shall maynteyne 20 Moores
cheaper than one Englishe servant." The doubt whether it be not sin in us
longer to tolerate their devil-worship, considering how much need we have
of them as merchandise, is delicious. The way in which Hugh Peter grades
the sharp descent from the apostolic to the practical with an _et
cetera_, in the following extract, has the same charm: "Sir, Mr Endecot &
myself salute you in the Lord Jesus &c. Wee have heard of a dividence of
women & children in the bay & would bee glad of a share viz: a young
woman or girle & a boy if you thinke good." Peter seems to have got what
he asked for, and to have been worse off than before; for we find him
writing two years later: "My wife desires my daughter to send to Hanna
that was her mayd, now at Charltowne, to know if shee would dwell with
us, for truly wee are so destitute (having now but an Indian) that wee
know not what to doe." Let any housewife of our day, who does not find
the Keltic element in domestic life so refreshing as to Mr. Arnold in
literature, imagine a household with one wild Pequot woman, communicated
with by signs, for its maid of all work, and take courage. Those were
serious times indeed, when your cook might give warning by taking your
scalp, or _chignon_, as the case might be, and making off with it into
the woods. The fewness and dearness of servants made it necessary to call
in temporary assistance for extraordinary occasions, and hence arose the
common use of the word _help_. As the great majority kept no servants at
all, and yet were liable to need them for work to which the family did
not suffice, as, for instance, in harvest, the use of the word was
naturally extended to all kinds of service. That it did not have its
origin in any false shame at the condition itself, induced by democratic
habits, is plain from the fact that it came into use while the word
_servant_ had a much wider application than now, and certainly implied no
social stigma. Downing and Hooke, each at different times, one of them so
late as 1667, wished to place a son as "servant" with one of the
Winthrops. Roger Williams writes of his daughter, that "she desires to
spend some time in service & liked much Mrs Brenton, who wanted." This
was, no doubt, in order to be well drilled in housekeeping, an example
which might be followed still to advantage. John Tinker, himself the
"servant" or steward of the second Winthrop, makes use of _help_ in both
the senses we have mentioned, and shows the transition of the word from
its restricted to its more general application. "We have fallen a pretty
deal of timber & drawn some by Goodman Rogers's team, but unless your
worship have a good team of your own & a man to go with them, I shall be
much distracted for _help_ ... & when our business is most in haste we
shall be most to seek." Again, writing at harvest, as appears both by the
date and by an elaborate pun,--"I received the _sithes_ you sent but in
that there came not also yourself, it maketh me to _sigth_,"--he says:
"_Help_ is scarce and hard to get, difficult to please, uncertain, &c.
Means runneth out & wages on & I cannot make choice of my _help_."
It may be some consolation to know that the complaint of a decline in the
quality of servants is no modern thing. Shakespeare makes Orlando say to
Adam:
"O, good old man, how well in thee appears
The constant service of the antique world,
When service sweat for duty, not for meed!
Thou art not of the fashion of these times,
When none will sweat but for promotion."
When the faithful old servant is brought upon the stage, we may be sure
he was getting rare. A century later, we have explicit testimony that
things were as bad in this respect as they are now. Don Manuel Gonzales,
who travelled in England in 1730, says of London servants: "As to common
menial servants, they have great wages, are well kept and cloathed, but
are notwithstanding the plague of almost every house in town. They form
themselves into societies or rather confederacies, contributing to the
maintenance of each other when out of place, and if any of them cannot
manage the family where they are entertained, as they please, immediately
they give notice they will be gone. There is no speaking to them, they
are above correction, and if a master should attempt it, he may expect to
be handsomely drubbed by the creature he feeds and harbors, or perhaps an
action brought against him for it. It is become a common saying, _If my
servant ben't a thief, if he be but honest, I can bear with other
things._ And indeed it is very rare in London to meet with an honest
servant."[142] Southey writes to his daughter Edith, in 1824, "All the
maids eloped because I had turned a man out of the kitchen at eleven
o'clock on the preceding night." Nay, Hugh Rhodes, in his _Boke of
Nurture_(1577), speaks of servants "ofte fleeting," i.e. leaving one
master for another.
One of the most curious things revealed to us in these volumes is the
fact that John Winthrop, Jr., was seeking the philosopher's stone, that
universal elixir which could transmute all things to its own substance.
This is plain from the correspondence of Edward Howes. Howes goes to a
certain doctor, professedly to consult him about the method of making a
cement for earthen vessels, no doubt crucibles. His account of him is
amusing, and reminds one of Ben Jonson's Subtle. This was one of the many
quacks who gulled men during that twilight through which alchemy was
passing into chemistry. "This Dr, for a Dr he is, brags that if he have
but the hint or notice of any useful thing not yet invented, he will
undertake to find it out, except some few which he hath vowed not to
meddle with as _vitrum maliabile, perpet. motus, via proxima ad Indos &
lapis philosi_: all, or anything else he will undertake, but for his
private gain, to make a monopoly thereof & to sell the use or knowledge
thereof at too high rates." This breed of pedlers in science is not yet
extinct. The exceptions made by the Doctor show a becoming modesty.
Again: "I have been 2 or 3 times with the Dr & can get but small
satisfaction about your queries.... Yet I must confess he seemed very
free to me, only in the main he was mystical. This he said, that when the
will of God is you shall know what you desire, it will come with such a
light that it will make a harmony among all your authors, causing them
sweetly to agree, & put you forever out of doubt & question." In another
letter: "I cannot discover into _terram incognitam_, but I have had a ken
of it showed unto me. The way to it is, for the most part, horrible &
fearful, the dangers none worse, to them that are _destinati filii_:
sometimes I am travelling that way.... I think I have spoken with some
that have been there."
Howes writes very cautiously: "Dear friend, I desire with all my heart
that I might write plainer to you, but in discovering the mystery, I may
diminish its majesty & give occasion to the profane to abuse it, if it
should fall into unworthy hands." By and by he begins to think his first
doctor a humbug, but he finds a better. Howes was evidently a man of
imaginative temper, fit to be captivated by the alchemistic theory of the
unity of composition in nature, which was so attractive to Goethe.
Perhaps the great poet was himself led to it by his Rosicrucian studies
when writing the first part of Faust. Howes tells his friend that "there
is all good to be found in unity, & all evil in duality & multiplicity.
_Phoenix illa admiranda sola semper existit_, therefore while a man & she
is two, he shall never see her,"--a truth of very wide application, and
too often lost sight of or never seen at all. "The Arabian Philos. I writ
to you of, he was styled among us Dr Lyon, the best of all the
Rosicrucians[143] that ever I met withal, far beyond Dr Ewer: they that
are of his strain are knowing men; they pretend [i.e. claim] to live in
free light, they honor God & do good to the people among whom they live,
& I conceive you are in the right that they had their learning from
Arabia."
Howes is a very interesting person, a mystic of the purest kind, and that
while learning to be an attorney with Emanuel Downing. How little that
perfunctory person dreamed of what was going on under his nose,--as
little as of the spiritual wonders that lay beyond the tip of it! Howes
was a Swedenborgian before Swedenborg. Take this, for example: "But to
our sympathetical business whereby we may communicate our minds one to
another though the diameter of the earth interpose. _Diana non est
centrum omnium_. I would have you so good a geometrician as to know your
own centre. Did you ever yet measure your everlasting self, the length of
your life, the breadth of your love, the depth of your wisdom & the
height of your light? Let Truth be your centre, & you may do it,
otherways not. I could wish you would now begin to leave off being
altogether an outward man; this is but _casa Regentis_; the Ruler can
draw you straight lines from your centre to the confines of an infinite
circumference, by which you may pass from any part of the circumference
to another without obstacle of earth or secation of lines, if you observe
& keep but one & the true & only centre, to pass by it, from it, & to it.
Methinks I now see you _intus et extra_ & talk to you, but you mind me
not because you are from home, you are not within, you look as if you
were careless of yourself; your hand & your voice differ; 'tis my
friend's hand, I know it well; but the voice is your enemy's. O, my
friend, if you love me, get you home, get you in! You have a friend as
well as an enemy. Know them by their voices. The one is still driving or
enticing you out; the other would have you stay within. Be within and
keep within, & all that are within & keep within shall you see know &
communicate with to the full, & shall not need to strain your outward
senses to see & hear that which is like themselves uncertain & too-too
often false, but, abiding forever within, in the centre of Truth, from
thence you may behold & understand the innumerable divers emanations
within the circumference, & still within; for without are falsities,
lies, untruths, dogs &c." Howes was tolerant also, not from want of
faith, but from depth of it. "The relation of your fight with the Indians
I have read in print, but of the fight among yourselves, _bellum
linguarum_ the strife of tongues, I have heard much, but little to the
purpose. I wonder your people, that pretend to know so much, doe not know
that love is the fulfilling of the law, & that against love there is no
law." Howes forgot that what might cause only a ripple in London might
overwhelm the tiny Colony in Boston. Two years later, he writes more
philosophically, and perhaps with a gentle irony, concerning "two
monstrous births & a general earthquake." He hints that the people of the
Bay might perhaps as well take these signs to themselves as lay them at
the door of Mrs. Hutchinson and what not. "Where is there such another
people then [as] in New England, that labors might & main to have Christ
formed in them, yet would give or appoint him his shape & clothe him too?
It cannot be denied that we have conceived many monstrous imaginations of
Christ Jesus: the one imagination says, _Lo, here he is_; the other says,
_Lo, there he is_; multiplicity of conceptions, but is there any one true
shape of Him? And if one of many produce a shape, it is not the shape of
the Son of God, but an ugly horrid metamorphosis. Neither is it a living
shape, but a dead one, yet a crow thinks her own bird the fairest, & most
prefer their own wisdom before God's, Antichrist before Christ." Howes
had certainly arrived at that "centre" of which he speaks and was before
his time, as a man of speculation, never a man of action, may sometimes
be. He was fitter for Plotinus's colony than Winthrop's. He never came to
New England, yet there was always a leaven of his style of thinkers here.
Howes was the true adept, seeking what spiritual ore there might be among
the dross of the hermetic philosophy. What he says sincerely and inwardly
was the cant of those outward professors of the doctrine who were content
to dwell in the material part of it forever. In Jonathan Brewster, we
have a specimen of these Wagners. Is it not curious, that there should
have been a _balneum Mariae_ at New London two hundred years ago? that
_la recherche de l'Absolu_ should have been going on there in a log-hut,
under constant fear that the Indians would put out, not merely the flame
of one little life, but, far worse, the fire of our furnace, and so rob
the world of this divine secret, just on the point of revealing itself?
Alas! poor Brewster's secret was one that many have striven after before
and since, who did not call themselves alchemists,--the secret of getting
gold without earning it,--a chase that brings some men to a four-in-hand
on Shoddy Avenue, and some to the penitentiary, in both cases advertising
its utter vanity. Brewster is a capital specimen of his class, who are
better than the average, because they _do_ mix a little imagination with
their sordidness, and who have also their representatives among us, in
those who expect the Jennings and other ideal estates in England. If
Hawthorne had but known of him! And yet how perfectly did his genius
divine that ideal element in our early New England life, conceiving what
must have been without asking proof of what actually was!
An extract or two will sufficiently exhibit Brewster in his lunes.
Sending back some alchemistic book to Winthrop, he tells him that if his
name be kept secret, "I will write as clear a light, as far as I dare to,
in finding the first ingredience.... The first figure in Flamonell doth
plainly resemble the first ingredience, what it is, & from whence it
comes, & how gotten, as there you may plainly see set forth by 2
resemblances held in a man's hand; for the confections there named is a
delusion, for they are but the operations of the work after some time
set, as the scum of the Red Sea, which is the Virgin's Milk upon the top
of the vessel, white. Red Sea is the sun & moon calcinated & brought &
reduced into water mineral which in some time, & most of the whole time,
is red. 2ndly, the fat of mercurial wind, that is the fat or quintessence
of sun & moon, earth & water, drawn out from them both, & flies aloft &
bore up by the operation of our mercury, that is our fire which is our
air or wind." This is as satisfactory as Lepidus's account of the
generation of the crocodile: "Your serpent of Egypt is bred now of your
mud by the operation of your sun: so is your crocodile." After describing
the three kinds of fire, that of the lamp, that of ashes, and that
against nature, which last "is the fire of fire, that is the secret fire
drawn up, being the quintessence of the sun & moon, with the other
mercurial water joined with & together, which is fire elemental," he
tells us that "these fires are & doth contain the whole mystery of the
work." The reader, perhaps, thinks that he has nothing to do but
forthwith to turn all the lead he can lay his hands on into gold. But no:
"If you had the first ingredience & the proportion of each, yet all were
nothing if you had not the certain times & seasons of the planets &
signs, when to give more or less of this fire, namely a hot & dry, a cold
& moist fire which you must use in the mercurial water before it comes to
black & after into white & then red, which is only done by these fires,
which when you practise you will easily see & perceive, that you shall
stand amazed, & admire at the great & admirable wisdom of God, that can
produce such a wonderful, efficacious, powerful thing as this is to
convert all metallic bodies to its own nature, which may be well called a
first essence. I say by such weak simple means of so little value & so
little & easy labor & skill, that I may say with Artephus, 200 page, it
is of a worke so easy & short, fitter for women & young children than
sage & grave men.... I thank the Lord, I understand the matter perfectly
in the said book, yet I could desire to have it again 12 months hence,
for about that time I shall have occasion to peruse, whenas I come to the
second working which is most difficult, which will be some three or [4]
months before the perfect white, & afterwards, as Artephus saith, I may
burn my books, for he saith it is one regiment as well for the red as for
the white. The Lord in mercy give me life to see the end of it!"--an
exclamation I more than once made in the course of some of Brewster's
periods.
Again, under pledge of profound secrecy, he sends Winthrop a manuscript,
which he may communicate to the owner of the volume formerly lent,
because "it gave me such light in the second work as I should not readily
have found out by study, also & especially how to work the elixir fit for
medicine & healing all maladies which is clean another way of working
than we held formerly. Also a light given how to dissolve any hard
substance into the elixir, which is also another work. And many other
things which in Ribley [Ripley?] I could not find out. More works of the
same I would gladly see ... for, Sir, so it is that any book of this
subject, I can understand it, though never so darkly written, having both
knowledge & experience of the world,[144] that now easily I may
understand their envious carriages to hide it.... You may marvel why I
should give any light to others in this thing before I have perfected my
own. This know, that my work being true thus far by all their writings,
it cannot fail ... for if &c &c you cannot miss if you would, except you
break your glass." He confesses he is mistaken as to the time required,
which he now, as well as I can make out, reckons at about ten years. "I
fear I shall not live to see it finished, in regard partly of the
Indians, who, I fear, will raise wars, as also I have a conceit that God
sees me not worthy of such a blessing, by reason of my manifold
miscarriages." Therefore he "will shortly write all the whole work in few
words plainly which may be done in 20 lines from the first to the last &
seal it up in a little box & subscribe it to yourself ... & will so write
it that neither wife nor children shall know thereof." If Winthrop should
succeed in bringing the work to perfection, Brewster begs him to remember
his wife and children. "I mean if this my work should miscarry by wars of
the Indians, for I may not remove it till it be perfected, otherwise I
should so unsettle the body by removing sun & moon out of their settled
places, that there would then be no other afterworking." Once more he
inculcates secrecy, and for a most comical reason: "For it is such a
secret as is not fit for every one either for secrecy or for parts to use
it, as God's secret for his glory, to do good there with, or else they
may do a great deal of hurt, spending & employing it to satisfy sinful
lusts. Therefore, I intreat you, sir, spare to use my name, & let my
letters I send either be safely kept or burned that I write about it, for
indeed, sir, I am more than before sensible of the evil effects that will
arise by the publishing of it. I should never be at quiet, neither at
home nor abroad, for one or other that would be enquiring & seeking after
knowledge thereof, that I should be tired out & forced to leave the
place: nay, it would be blazed abroad into Europe." How much more comic
is nature than any comedy! _Mutato nomine de te_. Take heart, ambitious
youth, the sun and moon will be no more disconcerted by any effort of
yours than by the pots and pans of Jonathan Brewster. It is a curious
proof of the duality so common (yet so often overlooked) in human
character, that Brewster was all this while manager of the Plymouth
trading-post, near what is now New London. The only professors of the
transmutation of metals who still impose on mankind are to be found in
what is styled the critical department of literature. Their _materia
prima_, or universal solvent, serves equally for the lead of Tupper or
the brass of Swinburne.
In a letter of Sir Kenelm Digby to J. Winthrop, Jr., we find some odd
prescriptions. "For all sorts of agues, I have of late tried the
following magnetical experiment with infallible success. Pare the
patient's nails when the fit is coming on, & put the parings into a
little bag of fine linen or sarsenet, & tie that about a live eel's neck
in a tub of water. The eel will die & the patient will recover. And if a
dog or hog eat that eel, they will also die."
"The man recovered of the bite,
The dog it was that died!"
"I have known one that cured all deliriums & frenzies whatsoever, & at
once taking, with an elixir made of dew, nothing but dew purified &
nipped up in a glass & digested 15 months till all of it was become a
gray powder, not one drop of humidity remaining. This I know to be true,
& that first it was as black as ink, then green then gray, & at 22
months' end it was as white & lustrous as any oriental pearl. But it
cured manias at 15 months' end." Poor Brewster would have been the better
for a dose of it, as well as some in our day, who expect to cure men of
being men by act of Congress. In the same letter Digby boasts of having
made known the properties of _quinquina_, and also of the sympathetic
powder, with which latter he wrought a "famous cure" of pleasant James
Howell, author of the "Letters." I do not recollect that Howell anywhere
alludes to it. In the same letter, Digby speaks of the books he had sent
to Harvard College, and promises to send more. In all Paris he cannot
find a copy of Blaise Viginere _Des Chiffres_. "I had it in my library in
England, but at the plundering of my house I lost it with many other good
books. I have _laid out_ in all places for it." The words we have
underscored would be called a Yankeeism now. The house was Gatehurst, a
fine Elizabethan dwelling, still, or lately, standing. Digby made his
peace with Cromwell, and professes his readiness to spend his blood for
him. He kept well with both sides, and we are not surprised to find Hooke
saying that he hears no good of him from any.
The early colonists found it needful to bring over a few trained
soldiers, both as drillmasters and engineers. Underhill, Patrick, and
Gardner had served in the Low Countries, probably also Mason. As Paris
has been said to be not precisely the place for a deacon, so the camp of
the Prince of Orange could hardly have been the best training-school for
Puritans in practice, however it may have been for masters of casuistic
theology. The position of these rough warriors among a people like those
of the first emigration must have been a droll one. That of Captain
Underhill certainly was. In all our early history, there is no figure so
comic. Full of the pedantry of his profession and fond of noble phrases,
he is a kind of cross between Dugald Dalgetty and Ancient Pistol, with a
slight relish of the _miles gloriosus_. Underhill had taken side with Mr.
Wheelwright in his heretical opinions, and there is every reason why he
should have maintained, with all the ardor of personal interest, the
efficiency of a covenant of grace without reference to the works of the
subject of it. Coming back from a visit to England in 1638, he "was
questioned for some speeches uttered by him in the ship, viz: that they
at Boston were zealous as the scribes and pharisees were and as Paul was
before his conversion, which he denying, they were proved to his face by
a sober woman whom he had seduced in the ship and drawn to his opinion;
but she was afterwards better informed in the truth. Among other
passages, he told her how he came by his assurance, saying that, having
long lain under a spirit of bondage, and continued in a legal way near
five years, he could get no assurance, till at length, as he was taking a
pipe of the good creature tobacco, the spirit fell home upon his heart,
an absolute promise of free grace, with such assurance and joy, as he
never doubted since of his good estate, neither should he, whatsoever sin
he should fall into,--a good preparative for such motions as he
familiarly used to make to some of that sex.... The next day he was
called again and banished. The Lord's day after, he made a speech in the
assembly, showing that as the Lord was pleased to convert Paul as he was
persecuting &c, so he might manifest himself to him as he was making
moderate use of the good creature called tobacco." A week later "he was
privately dealt with upon suspicion of incontinency ... but his excuse
was that the woman was in great trouble of mind, and some temptations,
and that he resorted to her to comfort her." He went to the Eastward,
and, having run himself out there, thought it best to come back to Boston
and reinstate himself by eating his leek. "He came in his worst clothes
(being accustomed to take great pride in his bravery and neatness)
without a band, in a foul linen cap pulled close to his eyes, and,
standing upon a form, he did, with many deep sighs and abundance of
tears, lay open his wicked course, his adultery, his hypocrisy &c. He
spake well, save that his blubbering &c. interrupted him." We hope he was
a sincere penitent, but men of his complexion are apt to be pleased with
such a tragi-comedy of self-abasement, if only they can be chief actors
and conspicuous enough therein. In the correspondence before us Underhill
appears in full turkey-cock proportions. Not having been advanced
according to his own opinion of his merits, he writes to Governor
Winthrop, with an oblique threat that must have amused him somewhat: "I
profess, sir, till I know the cause, I shall not be satisfied, but I hope
God will subdue me to his will; yet this I say that such handling of
officers in foreign parts hath so far subverted some of them as to cause
them turn public rebels against their state & kingdom, which God forbid
should ever be found once so much as to appear in my breast." Why, then
the world's mine oyster, which I with sword will open! Next we hear him
on a point of military discipline at Salem. "It is this: how they have of
their own appointment made them a captain, lieutenant & ensign, & after
such a manner as was never heard of in any school of war, nor in no
kingdom under heaven.... For my part, if there should not be a
reformation in this disordered practise, I would not acknowledge such
officers. If officers should be of no better esteem than for constables
to place them, & martial discipline to proceed disorderly, I would rather
lay down my command than to shame so noble a prince from whom we came."
Again: "Whereas it is somewhat questionable whether the three months I
was absent, as well in the service of the country as of other particular
persons, my request therefore is that this honored Court would be pleased
to decide this controversy, myself alleging it to be the custom of
Nations that, if a Commander be lent to another State, by that State to
whom he is a servant, both his place & means is not detained from him, so
long as he doth not refuse the call of his own State to which he is a
servant, in case they shall call him home." Then bringing up again his
"ancient suit" for a grant of land, he throws in a neat touch of piety:
"& if the honored Court shall vouchsafe to make some addition, that which
hath not been deserved, by the same power of God, may be in due season."
In a postscript, he gives a fine philosophical reason for this desired
addition which will go to the hearts of many in these days of high prices
and wasteful taxation. "The time was when a little went far; then much
was not known nor desired; the reason of the difference lieth only in the
error of judgment, for nature requires no more to uphold it now than when
it was satisfied with less." The valiant Captain interprets the law of
nations, as sovereign powers are wont to do, to suit his advantage in the
special case. We find a parallel case in a letter of Bryan Rosseter to
John Winthrop, Jr., pleading for a remission of taxes. "The lawes of
nations exempt allowed phisitians from personall services, & their
estates from rates & assessments." In the Declaration of the town of
Southampton on Long Island (1673), the dignity of constable is valued at
a juster rate than Underhill was inclined to put upon it. The Dutch, it
seems, demanded of them "to deliver up to them the badge of Civil &
Military power; namely, the Constable's staffe & the Colonel's." Mayor
Munroe of New Orleans did not more effectually magnify his office when he
surrendered the city to General Butler.
Underhill's style is always of the finest. His spelling was under the
purest covenant of grace. I must give a single specimen of it from a
letter whose high moral tone is all the more diverting that it was
written while he was under excommunication for the sin which he
afterwards confessed. It is addressed to Winthrop and Dudley. "Honnored
in the Lord. Youer silenc one more admirse me. I youse chrischan
playnnes. I know you love it. Silenc can not reduce the hart of youer
love'g brother: I would the rightchous would smite me, espeschali youer
slfe & the honnored Depoti to whom I also dereckt this letter together
with youer honnored slfe. Jesos Christ did wayt; & God his Father did dig
and telfe bout the barren figtre before he would cast it of: I would to
God you would tender my soule so as to youse playnnes with me." (As if
anything could be plainer than excommunication and banishment!) "I wrot
to you both, but now [no] answer; & here I am dayli abused by malischous
tongse: John Baker I here hath rot to the honnored depoti how as I was
dronck & like to be cild, & both falc, upon okachon I delt with Wanuerton
for intrushon, & findding them resolutli bent to rout out all gud a <DW41>
us & advanc there superstischous waye, & by boystrous words indeferd to
fritten men to acomplish his end, & he abusing me to my face, dru upon
him with intent to corb his insolent and dasterdli sperrite, but now [no]
danger of my life, although it might hafe bin just with God to hafe
giffen me in the hanse of youer enemise & mine, for they hat the wayse of
the Lord & them that profes them, & therfore layes trapes to cachte the
pore into there deboyst corses, as ister daye on Pickeren their Chorch
Warden caim up to us with intent to mak some of ourse dronc, as is
sospeckted, but the Lord soferd him so to misdemen himslfe as he is likli
to li by the hielse this too month.... My hombel request is that you will
be charitabel of me.... Let justies and merci be goyned.... You may plese
to soggest youer will to this barrer, you will find him tracktabel." The
concluding phrase seems admirably chosen, when we consider the means of
making people "tractable" which the magistrates of the Bay had in their
hands, and were not slow to exercise, as Underhill himself had
experienced.
I cannot deny myself the pleasure of giving one more specimen of the
Captain's "grand-delinquent" style, as I once heard such fine writing
called by a person who little dreamed what a hit he had made. So far as I
have observed, our public defaulters, and others who have nothing to say
for themselves, always rise in style as they sink in self-respect. He is
speaking of one Scott, who had laid claim to certain lands, and had been
called on to show his title. "If he break the comand of the Asembli &
bring not in the counterfit portreture of the King imprest in yello waxe,
anext to his false perpetuiti of 20 mile square, where by he did chet the
Town of Brouckhaven, he is to induer the sentance of the Court of
Asisies." Pistol would have been charmed with that splendid amplification
of the Great Seal. We have seen nothing like it in our day, except in a
speech made to Mr. George Peabody at Danvers, if I recollect, while that
gentleman was so elaborately concealing from his left hand what his right
had been doing. As examples of Captain Underhill's adroitness in phonetic
spelling, I offer _fafarabel_ and _poseschonse_, and reluctantly leave
him.
Another very entertaining fellow for those who are willing to work
through a pretty thick husk of tiresomeness for a genuine kernel of humor
underneath is Coddington. The elder Winthrop endured many trials, but I
doubt if any were sharper than those which his son had to undergo in the
correspondence of this excellently tiresome man. _Tantae molis Romanam
condere gentem!_ The dulness of Coddington, always that of no ordinary
man, became irritable and aggressive after being stung by the gadfly of
Quakerism. Running counter to its proper nature, it made him morbidly
uneasy. Already an Anabaptist, his brain does not seem to have been large
enough to lodge two maggots at once with any comfort to himself. Fancy
John Winthrop, Jr., with all the affairs of the Connecticut Colony on his
back, expected to prescribe alike for the spiritual and bodily ailments
of all the hypochondriacs in his government, and with Philip's war
impending,--fancy him exposed also to perpetual trials like this: "G.F.
[George Fox] hath sent thee a book of his by Jere: Bull, & two more now
which thou mayest communicate to thy Council & officers. Also I remember
before thy last being in England, I sent thee a book written by Francis
Howgall against persecution, by Joseph Nicallson which book thou lovingly
accepted and communicated to the Commissioners of the United Colonies (as
I desired) also J.N. thou entertained with a loving respect which
encouraged me" (fatal hospitality!)--"As a token of that ancient love
that for this 42 years I have had for thee, I have sent thee three
Manuscripts, one of 5 queries, other is of 15, about the love of Jesus
&c. The 3d is why we cannot come to the worship which was not set up by
Christ Jesus, which I desire thee to communicate to the priests to answer
in thy jurisdiction, the Massachusetts, New Plymouth, or elsewhere, &
send their answer in writing to me. Also two printed papers to set up in
thy house. It's reported in Barbadoes that thy brother Sammuell shall be
sent Governour to Antego." What a mere dust of sugar in the last sentence
for such a portentous pill! In his next letter he has other writings of
G. F., "not yet copied, which if thou desireth, when I hear from thee, I
may convey them unto thee. Also sence G. Ffox departure William Edmondson
is arrived at this Island, who having given out a paper to all in
authority, which, my wife having copied, I have here inclosed presented
thee therewith." Books and manuscripts were not all. Coddington was also
glad to bestow on Winthrop any wandering tediousness in the flesh that
came to hand. "I now understand of John Stubbs freedom to visit thee
(with the said Jo: B.) he is a larned man, as witness the battle
door[145] on 35 languages,"--a terrible man this, capable of inflicting
himself on three dozen different kindreds of men. It will be observed
that Coddington, with his "thou desireths," is not quite so well up in
the grammar of his thee-and-thouing as my Lord Coke. Indeed, it is rather
pleasant to see that in his alarm about "the enemy," in 1673, he
backslides into the second person plural. If Winthrop ever looked over
his father's correspondence, he would have read in a letter of Henry
Jacie the following dreadful example of retribution: "The last news we
heard was that the Bores in Bavaria slew about 300 of the Swedish forces
& took about 200 prisoners, of which they put out the eyes of some & cut
out the tonges of others & so sent them to the King of Sweden, which
caused him to lament bytterly for an hour. Then he sent an army &
destroyed those Bores, about 200 or 300 of their towns. Thus we hear."
Think of that, Master Coddington! Could the sinful heart of man always
suppress the wish that a Gustavus might arise to do judgment on the Bores
of Rhode Island? The unkindest part of it was that, on Coddington's own
statement, Winthrop had never persecuted the Quakers, and had even
endeavored to save Robinson and Stevenson in 1659.
Speaking of the execution of these two martyrs to the bee in their
bonnets, John Davenport gives us a capital example of the way in which
Divine "judgments" may be made to work both ways at the pleasure of the
interpreter. As the crowd was going home from the hanging, a drawbridge
gave way, and some lives were lost. The Quakers, of course, made the most
of this lesson to the _pontifices_ in the bearing power of timber,
claiming it as a proof of God's wrath against the persecutors. This was
rather hard, since none of the magistrates perished, and the popular
feeling was strongly in favor of the victims of their severity. But
Davenport gallantly captures these Quaker guns, and turns them against
the enemy himself. "Sir, the hurt that befell so many, by their own
rashness, at the Draw Bridge in Boston, being on the day that the Quakers
were executed, was not without God's special providence in judgment &
wrath, I fear, against the Quakers & their abettors, who will be much
hardened thereby." This is admirable, especially as his parenthesis about
"their own rashness" assumes that the whole thing was owing to natural
causes. The pity for the Quakers, too, implied in the "I fear," is a nice
touch. It is always noticeable how much more liberal those who deal in
God's command without his power are of his wrath than of his mercy. But
we should never understand the Puritans if we did not bear in mind that
they were still prisoners in that religion of Fear which casts out Love.
The nearness of God was oftener a terror than a comfort to them. Yet
perhaps in them was the last apparition of Faith as a wonder-worker in
human affairs. Take away from them what you will, you cannot deny them
_that_, and its constant presence made them great in a way and measure of
which this generation, it is to be feared, can have but a very inadequate
conception. If men now-a-days find their tone antipathetic, it would be
modest at least to consider whether the fault be wholly theirs,--whether
it was they who lacked, or we who have lost. Whether they were right or
wrong in their dealing with the Quakers is not a question to be decided
glibly after two centuries' struggle toward a conception of toleration
very imperfect even yet, perhaps impossible to human nature. If they did
not choose what seems to us the wisest way of keeping the Devil out of
their household, they certainly had a very honest will to keep him out,
which we might emulate with advantage. However it be in other cases,
historic toleration must include intolerance among things to be
tolerated.
The false notion which the first settlers had of the savages by whom the
continent was beflead rather than inhabited, arose in part from what they
had heard of Mexico and Peru, in part from the splendid exaggerations of
the early travellers, who could give their readers an El Dorado at the
cheap cost of a good lie. Hence the kings, dukes, and earls who were so
plenty among the red men. Pride of descent takes many odd shapes, none
odder than when it hugs itself in an ancestry of filthy barbarians, who
daubed themselves for ornament with a mixture of bear's-grease and soot,
or clay, and were called emperors by Captain John Smith and his
compeers. The droll contrast between this imaginary royalty and the
squalid reality is nowhere exposed with more ludicrous unconsciousness
than in the following passage of a letter from Fitz-John Winthrop to his
father, November, 1674: "The bearer hereof, Mr. Danyell, one of the Royal
Indian blood ... does desire me to give an account to yourself of the
late unhappy accident which has happened to him. A little time since, a
careless girl playing with fire at the door, it immediately took hold of
the mats, & in an instant consumed it to ashes, with all the common as
well as his lady's chamber furniture, & his own wardrobe & armory, Indian
plate, & money to the value (as is credibly reported in his estimation)
of more than an hundred pounds Indian.... The Indians have handsomely
already built him a good house & brought him in several necessaries for
his present supply, but that which takes deepest melancholy impression
upon him is the loss of an excellent Masathuset cloth cloak & hat, which
was only seen upon holy days & their general sessions. His journey at
this time is only to intreat your favor & the gentlemen there for a kind
relief in his necessity, having no kind of garment but a short jerkin
which was charitably given him by one of his Common-Councilmen. He
principally aims at a cloak & hat."
"King Stephen was a worthy peer,
His breeches cost him but a crown."
But it will be observed that there is no allusion to any such article of
dress in the costume of this prince of Pequot. Some light is perhaps
thrown on this deficiency by a line or two in one of Williams's letters,
where he says: "I have long had scruples of selling the Natives ought but
what may tend or bring to civilizing: I therefore neither brought nor
shall sell them loose coats nor breeches." Precisely the opposite course
was deemed effectual with the Highland Scotch, between whom and our
Indians there was a very close analogy. They were compelled by law to
adopt the usages of _Gallia Braccata_, and sansculottism made a penal
offence. What impediment to civilization Williams had discovered in the
offending garment it is hard to say. It is a question for Herr
Teufelsdroeck. Royalty, at any rate, in our day, is dependent for much of
its success on the tailor. Williams's opportunities of studying the
Indian character were perhaps greater than those of any other man of his
time. He was always an advocate for justice toward them. But he seems to
have had no better opinion of them than Mr. Parkman,[146] calling them
shortly and sharply, "wolves endowed with men's brains." The same change
of feeling has followed the same causes in their case as in that of the
Highlanders,--they have become romantic in proportion as they ceased to
be dangerous.
As exhibitions of the writer's character, no letters in the collection
have interested us more than those of John Tinker, who for many years was
a kind of steward for John Winthrop and his son. They show him to have
been a thoroughly faithful, grateful, and unselfish servant. He does not
seem to have prospered except in winning respect, for when he died his
funeral charges were paid by the public. We learn from one of his letters
that John Winthrop, Jr., had a <DW64> (presumably a slave) at Paquanet,
for he says that a mad cow there "had almost spoiled the neger & made him
ferfull to tend the rest of the cattell." That such slaves must have been
rare, however, is plain from his constant complaints about the difficulty
of procuring "help," some of which we have already quoted. His spelling
of the word "ferfull" shows that the New England pronunciation of that
word had been brought from the old country. He also uses the word
"creatures" for kine, and the like, precisely as our farmers do now.
There is one very comical passage in a letter of the 2nd of August, 1660,
where he says: "There hath been a motion by some, the chief of the town,
(New London) for my keeping an ordinary, or rather under the notion of a
tavern which, _though it suits not with my genius_, yet am almost
persuaded to accept for some good grounds." Tinker's modesty is most
creditable to him, and we wish it were more common now. No people on the
face of the earth suffer so much as we from impostors who keep
inconveniences, "under the notion of a tavern," without any call of
natural genius thereto; none endure with such unexemplary patience the
superb indifference of inn-keepers, and the condescending inattention of
their gentlemanly deputies. We are the thralls of our railroads and
hotels, and we deserve it.
Richard Saltonstall writes to John Winthrop, Jr., in 1636: "The best
thing that I have to beg your thoughts for at this present is a motto or
two that Mr. Prynne hath writ upon his chamber walls in the Tower." We
copy a few phrases, chiefly for the contrast they make with Lovelace's
famous verses to Althea. Nothing could mark more sharply the different
habits of mind in Puritan and Cavalier. Lovelace is very charming, but he
sings
"The sweetness, mercy, majesty,
And glories of _his_ King,"
to wit, Charles I. To him "stone walls do not a prison make," so long as
he has "freedom in his love, and in his soul is free." Prynne's King was
of another and higher kind: "_Carcer excludit mundum, includit Deum. Deus
est turris etiam in turre: turris libertatis in turre angustiae: Turris
quietis in turre molestice.... Arctari non potest qui in ipsa Dei
infinitate incarceratus spatiatur.... Nil crus sentit in nervo si animus
sit in coelo: nil corpus patitur in ergastulo, si anima sit in Christo_."
If Lovelace has the advantage in fancy, Prynne has it as clearly in depth
of sentiment. There could be little doubt which of the parties
represented by these men would have the better if it came to a
death-grapple.
There is curiously little sentiment in these volumes. Most of the
letters, except where some point of doctrine is concerned, are those of
shrewd, practical men, busy about the affairs of this world, and earnest
to build their New Jerusalem on something more solid than cloud. The
truth is, that men anxious about their souls have not been by any means
the least skilful in providing for the wants of the body. It was far less
the enthusiasm than the common sense of the Puritans which made them what
they were in politics and religion. That a great change should be wrought
in the settlers by the circumstances of their position was inevitable;
that this change should have had some disillusion in it, that it should
have weaned them from the ideal and wonted them to the actual, was
equally so. In 1664, not much more than a generation after the
settlement, Williams prophesies: "When we that have been the eldest are
rotting (to-morrow or next day) a generation will act, I fear, far unlike
the first Winthrops and their models of love. I fear that the common
trinity of the world (profit, preferment, pleasure) will here be the
_tria omnia_ as in all the world beside, that Prelacy and Papacy too will
in this wilderness predominate, that god Land will be (as now it is) as
great a god with us English as god Gold was with the Spaniards. While we
are here, noble sir, let us _viriliter hoc agere, rem agere humanam,
divinam, Christianam_, which, I believe, is all of a most public genius,"
or, as we should now say, true patriotism. If Williams means no play on
the word _humanam_ and _divinam_, the order of precedence in which he
marshals them is noticeable. A generation later, what Williams had
predicted was in a great measure verified. But what made New England
Puritanism narrow was what made Scotch Cameronianism narrow,--its being
secluded from the great movement of the nation. Till 1660 the colony was
ruled and mostly inhabited by Englishmen closely connected with the party
dominant in the mother country, and with their minds broadened by having
to deal with questions of state and European policy. After that time they
sank rapidly into provincials, narrow in thought, in culture, in creed.
Such a pedantic portent as Cotton Mather would have been impossible in
the first generation; he was the natural growth of the third,--the
manifest judgment of God on a generation who thought Words a saving
substitute for Things. Perhaps some injustice has been done to men like
the second Governor Dudley, and it should be counted to them rather as a
merit than a fault, that they wished to bring New England back within
reach of the invigorating influence of national sympathies, and to rescue
it from a tradition which had become empty formalism. Puritanism was
dead, and its profession had become a wearisome cant before the
Revolution of 1688 gave it that vital force in politics which it had lost
in religion.
I have gleaned all I could of what is morally picturesque or
characteristic from these volumes, but New England history has rather a
gregarious than a personal interest. Here, by inherent necessity rather
than design, was made the first experiment in practical democracy, and
accordingly hence began that reaction of the New World upon the Old whose
result can hardly yet be estimated. There is here no temptation to make a
hero, who shall sum up in his own individuality and carry forward by his
own will that purpose of which we seem to catch such bewitching glances
in history, which reveals itself more clearly and constantly, perhaps, in
the annals of New England than elsewhere, and which yet, at best, is but
tentative, doubtful of itself, turned this way and that by chance, made
up of instinct, and modified by circumstance quite as much as it is
directed by deliberate forethought. Such a purpose, or natural craving,
or result of temporary influences, may be misguided by a powerful
character to his own ends, or, if he be strongly in sympathy with it, may
be hastened toward its own fulfilment; but there is no such heroic
element in our drama, and what is remarkable is, that, under whatever
government, democracy grew with the growth of the New England Colonies,
and was at last potent enough to wrench them, and the better part of the
continent with them, from the mother country. It is true that Jefferson
embodied in the Declaration of Independence the speculative theories he
had learned in France, but the impulse to separation came from New
England; and those theories had been long since embodied there in the
practice of the people, if they had never been formulated in distinct
propositions.
I have little sympathy with declaimers about the Pilgrim Fathers, who
look upon them all as men of grand conceptions and superhuman foresight.
An entire ship's company of Columbuses is what the world never saw. It is
not wise to form any theory and fit our facts to it, as a man in a hurry
is apt to cram his travelling-bag, with a total disregard of shape or
texture. But perhaps it may be found that the facts will only fit
comfortably together on a single plan, namely, that the fathers did have
a conception (which those will call grand who regard simplicity as a
necessary element of grandeur) of founding here a commonwealth on those
two eternal bases of Faith and Work; that they had, indeed, no
revolutionary ideas of universal liberty, but yet, what answered the
purpose quite as well, an abiding faith in the brotherhood of man and the
fatherhood of God; and that they did not so much propose to make all
things new, as to develop the latent possibilities of English law and
English character, by clearing away the fences by which the abuse of the
one was gradually discommoning the other from the broad fields of natural
right. They were not in advance of their age, as it is called, for no one
who is so can ever work profitably in it; but they were alive to the
highest and most earnest thinking of their time.
Footnotes:
[135] Written in December, 1864.
[136] It is curious, that, when Cromwell proposed to transfer a
colony from New England to Ireland, one of the conditions insisted on
in Massachusetts was that a college should be established.
[137] State Trials, II. 409. One would not reckon too closely with a
man on trial for his life, but there is something pitiful in Peter's
representing himself as coming back to England "out of the West
Indias," in order to evade any complicity with suspected New England.
[138] Waller put this into verse:--
"Let the rich ore forthwith be melted down
And the state fixed by making him a crown."
[139] The _third_ in Carlyle, 1654.
[140] Collections, Third Series, Vol I. p. 183.
[141] This speech may be found in the Annual Register of 1762.
[142] Collection of Voyages, &c., from the Library of the Earl of
Oxford, Vol. I. p. 151.
[143] Howes writes the word symbolically.
[144] "World" here should clearly be "work."
[145] The title-page of which our learned Marsh has cited for the
etymology of the word.
[146] In his Jesuits in North America.
LESSING[147]
When Burns's humor gave its last pathetic flicker in his "John, don't let
the awkward squad fire over me," was he thinking of actual
brother-volunteers, or of possible biographers? Did his words betray only
the rhythmic sensitiveness of poetic nerves, or were they a foreboding of
that helpless future, when the poet lies at the mercy of the plodder,--of
that bi-voluminous shape in which dulness overtakes and revenges itself
on genius at last? Certainly Burns has suffered as much as most
large-natured creatures from well-meaning efforts to account for him, to
explain him away, to bring him into harmony with those well-regulated
minds which, during a good part of the last century, found out a way,
through rhyme, to snatch a prosiness beyond the reach of prose. Nay, he
has been wronged also by that other want of true appreciation, which
deals in panegyric, and would put asunder those two things which God has
joined,--the poet and the man,--as if it were not the same rash
improvidence that was the happiness of the verse and the misfortune of
the gauger. But his death-bed was at least not haunted by the
unappeasable apprehension of a German for his biographer; and that the
fame of Lessing should have four times survived this cunningest assault
of oblivion is proof enough that its base is broad and deep-set.
There seems to be, in the average German mind, an inability or a
disinclination to see a thing as it really is, unless it be a matter of
science. It finds its keenest pleasure in divining a profound
significance in the most trifling things, and the number of mare's-nests
that have been stared into by the German _Gelehrter_ through his
spectacles passes calculation. They are the one object of contemplation
that makes that singular being perfectly happy, and they seem to be as
common as those of the stork. In the dark forest of aesthetics,
particularly, he finds them at every turn,--"fanno tutto il loco varo."
If the greater part of our English criticism is apt only to skim the
surface, the German, by way of being profound, too often burrows in
delighted darkness quite beneath its subject, till the reader feels the
ground hollow beneath him, and is fearful of caving into unknown depths
of stagnant metaphysic air at every step. The Commentary on Shakespeare
of Gervinus, a really superior man, reminds one of the Roman Campagna,
penetrated underground in all directions by strange winding caverns, the
work of human borers in search of we know not what. Above are the divine
poet's larks and daisies, his incommunicable skies, his broad prospects
of life and nature; and meanwhile our Teutonic _teredo_ worms his way
below, and offers to be our guide into an obscurity of his own
contriving. The reaction of language upon style, and even upon thought,
by its limitations on the one hand, and its suggestions on the other, is
so apparent to any one who has made even a slight study of comparative
literature, that we have sometimes thought the German tongue at least an
accessory before the fact, if nothing more, in the offences of German
literature. The language has such a fatal genius for going
stern-foremost, for yawing, and for not minding the helm without some ten
minutes' notice in advance, that he must be a great sailor indeed who can
safely make it the vehicle for anything but imperishable commodities.
Vischer's _Aesthetik_, the best treatise on the subject, ancient or
modern, is such a book as none but a German could write, and it is
written as none but a German could have written it. The abstracts of its
sections are sometimes nearly as long as the sections themselves, and it
is as hard to make out which head belongs to which tail, as in a knot of
snakes thawing themselves into sluggish individuality under a spring sun.
The average German professor spends his life in making lanterns fit to
guide us through the obscurest passages of all the _ologies_ and _ysics_,
and there are none in the world of such honest workmanship. They are
durable, they have intensifying glasses, reflectors of the most
scientific make, capital sockets in which to set a light, and a handsome
lump of potentially illuminating tallow is thrown in. But, in order to
_see_ by them, the explorer must make his own candle, supply his own
cohesive wick of common-sense, and light it himself. And yet the
admirable thoroughness of the German intellect! We should be ungrateful
indeed if we did not acknowledge that it has supplied the raw material in
almost every branch of science for the defter wits of other nations to
work on; yet we have a suspicion that there are certain lighter
departments of literature in which it may be misapplied, and turn into
something very like clumsiness. Delightful as Jean Paul's humor is, how
much more so would it be if he only knew when to stop! Ethereally deep as
is his sentiment, should we not feel it more if he sometimes gave us a
little less of it,--if he would only not always deal out his wine by
beer-measure? So thorough is the German mind, that might it not seem now
and then to work quite through its subject, and expatiate in cheerful
unconsciousness on the other side thereof?
With all its merits of a higher and deeper kind, it yet seems to us that
German literature has not quite satisfactorily answered that so
long-standing question of the French Abbe about _esprit_. Hard as it is
for a German to be clear, still harder to be light, he is more than ever
awkward in his attempts to produce that quality of style, so peculiarly
French, which is neither wit nor liveliness taken singly, but a mixture
of the two that must be drunk while the effervescence lasts, and will not
bear exportation into any other language. German criticism, excellent in
other respects, and immeasurably superior to that of any other nation in
its constructive faculty, in its instinct for getting at whatever
principle of life lies at the heart of a work of genius, is seldom lucid,
almost never entertaining. It may turn its light, if we have patience,
into every obscurest cranny of its subject, one after another, but it
never flashes light _out_ of the subject itself, as Sainte-Beuve, for
example, so often does, and with such unexpected charm. We should be
inclined to put Julian Schmidt at the head of living critics in all the
more essential elements of his outfit; but with him is not one conscious
at too frequent intervals of the professorial grind,--of that German
tendency to bear on too heavily, where a French critic would touch and go
with such exquisite measure? The Great Nation, as it cheerfully calls
itself, is in nothing greater than its talent for saying little things
agreeably, which is perhaps the very top of mere culture, and in
literature is the next best thing to the power of saying great things as
easily as if they were little German learning, like the elephants of
Pyrrhus, is always in danger of turning upon what it was intended to
adorn and reinforce, and trampling it ponderously to death. And yet what
do we not owe it? Mastering all languages, all records of intellectual
man, it has been able, or has enabled others, to strip away the husks of
nationality and conventionalism from the literatures of many races, and
to disengage that kernel of human truth which is the germinating
principle of them all. Nay, it has taught us to recognize also a certain
value in those very husks, whether as shelter for the unripe or food for
the fallen seed.
That the general want of style in German authors is not wholly the fault
of the language is shown by Heine (a man of mixed blood), who can be
daintily light in German; that it is not altogether a matter of race, is
clear from the graceful airiness of Erasmus and Reuchlin in Latin, and of
Grimm in French. The sense of heaviness which creeps over the reader from
so many German books is mainly due, we suspect to the language, which
seems wellnigh incapable of that aerial perspective so delightful in
first-rate French, and even English, writing. But there must also be in
the national character an insensibility to proportion, a want of that
instinctive discretion which we call tact. Nothing short of this will
account for the perpetual groping of German imaginative literature after
some foreign mould in which to cast its thought or feeling, now trying a
Louis Quatorze pattern, then something supposed to be Shakespearian, and
at last going back to ancient Greece, or even Persia. Goethe himself,
limpidly perfect as are many of his shorter poems, often fails in giving
artistic coherence to his longer works. Leaving deeper qualities wholly
out of the question, Wilhelm Meister seems a mere aggregation of episodes
if compared with such a masterpiece as Paul and Virginia, or even with a
happy improvisation like the Vicar of Wakefield. The second part of
Faust, too, is rather a reflection of Goethe's own changed view of life
and man's relation to it, than an harmonious completion of the original
conception. Full of placid wisdom and exquisite poetry it certainly is;
but if we look at it as a poem, it seems more as if the author had
striven to get in all he could, than to leave out all he might. We cannot
help asking what business have paper money and political economy and
geognosy here? We confess that Thales and the Homunculus weary us not a
little, unless, indeed, a poem be nothing, after all, but a prolonged
conundrum. Many of Schiller's lyrical poems--though the best of them find
no match in modern verse for rapid energy, the very axles of language
kindling with swiftness--seem disproportionately long in parts, and the
thought too often has the life wellnigh squeezed out of it in the
sevenfold coils of diction, dappled though it be with splendid imagery.
In German sentiment, which runs over so easily into sentimentalism, a
foreigner cannot help being struck with a certain incongruousness. What
can be odder, for example, than the mixture of sensibility and sausages
in some of Goethe's earlier notes to Frau von Stein, unless, to be sure,
the publishing them? It would appear that Germans were less sensible to
the ludicrous--and we are far from saying that this may not have its
compensatory advantages--than either the English or the French. And what
is the source of this sensibility, if it be not an instinctive perception
of the incongruous and disproportionate? Among all races, the English has
ever shown itself most keenly alive to the fear of making itself
ridiculous; and among all, none has produced so many humorists, only one
of them, indeed, so profound as Cervantes, yet all masters in their
several ways. What English-speaking man, except Boswell, could have
arrived at Weimar, as Goethe did, in that absurd _Werthermontirung_? And
where, out of Germany, could he have found a reigning Grand Duke to put
his whole court into the same sentimental livery of blue and yellow,
leather breeches, boots, and all, excepting only Herder, and that not on
account of his clerical profession, but of his age? To be sure, it might
be asked also where else in Europe was a prince to be met with capable of
manly friendship with a man whose only decoration was his genius? But the
comicality of the other fact no less remains. Certainly the German
character is in no way so little remarkable as for its humor. If we were
to trust the evidence of Herr Hub's dreary _Deutsche komische und
humoristische Dichtung_, we should believe that no German had even so
much as a suspicion of what humor meant, unless the book itself, as we
are half inclined to suspect, be a joke in three volumes, the _want_ of
fun being the real point thereof. If German patriotism can be induced to
find a grave delight in it, we congratulate Herr Hub's publishers, and
for ourselves advise any sober-minded man who may hereafter "be merry,"
not to "sing psalms," but to read Hub as the more serious amusement of
the two. There are epigrams there that make life more solemn, and, if
taken in sufficient doses, would make it more precarious. Even Jean Paul,
the greatest of German humorous authors, and never surpassed in comic
conception or in the pathetic quality of humor, is not to be named with
his master, Sterne, as a creative humorist. What are Siebenkaes, Fixlein,
Schmelzle, and Fibel, (a single lay-figure to be draped at will with
whimsical sentiment and reflection, and put in various attitudes,)
compared with the living reality of Walter Shandy and his brother Toby,
characters which we do not see merely as puppets in the author's mind,
but poetically projected from it in an independent being of their own?
Heine himself, the most graceful, sometimes the most touching, of modern
poets, and clearly the most easy of German humorists, seems to me wanting
in a refined perception of that inward propriety which is only another
name for poetic proportion, and shocks us sometimes with an
_Unflaethigkeit_, as at the end of his _Deutschland_, which, if it make
Germans laugh, as we should be sorry to believe, makes other people hold
their noses. Such things have not been possible in English since Swift,
and the _persifleur_ Heine cannot offer the same excuse of savage
cynicism that might be pleaded for the Irishman.
I have hinted that Herr Stahr's Life of Lessing is not precisely the kind
of biography that would have been most pleasing to the man who could not
conceive that an author should be satisfied with anything more than truth
in praise, or anything less in criticism. My respect for what Lessing
was, and for what he did, is profound. In the history of literature it
would be hard to find a man so stalwart, so kindly, so sincere,[148] so
capable of great ideas, whether in their influence on the intellect or
the life, so unswervingly true to the truth, so free from the common
weaknesses of his class. Since Luther, Germany has given birth to no such
intellectual athlete,--to no son so German to the core. Greater poets she
has had, but no greater writer; no nature more finely tempered. Nay, may
we not say that great character is as rare a thing as great genius, if it
be not even a nobler form of it? For surely it is easier to embody fine
thinking, or delicate sentiment, or lofty aspiration, in a book than in a
life. The written leaf, if it be, as some few are, a safe-keeper and
conductor of celestial fire, is secure. Poverty cannot pinch, passion
swerve, or trial shake it. But the man Lessing, harassed and striving
life-long, always poor and always hopeful, with no patron but his own
right-hand, the very shuttlecock of fortune, who saw ruin's ploughshare
drive through the hearth on which his first home-fire was hardly kindled,
and who, through all, was faithful to himself, to his friend, to his
duty, and to his ideal, is something more inspiring for us than the most
glorious utterance of merely intellectual power. The figure of Goethe is
grand, it is rightfully pre-eminent, it has something of the calm, and
something of the coldness, of the immortals; but the Valhalla of German
letters can show one form, in its simple manhood, statelier even than
his.
Manliness and simplicity, if they are not necessary coefficients in
producing character of the purest tone, were certainly leading elements
in the Lessing who is still so noteworthy and lovable to us when
eighty-six years have passed since his bodily presence vanished from
among men. He loved clearness, he hated exaggeration in all its forms. He
was the first German who had any conception of style, and who could be
full without spilling over on all sides. Herr Stahr, we think, is not
just the biographer he would have chosen for himself. His book is rather
a panegyric than a biography. There is sometimes an almost comic
disproportion between the matter and the manner, especially in the epic
details of Lessing's onslaughts on the nameless herd of German authors.
It is as if Sophocles should have given a strophe to every bullock slain
by Ajax in his mad foray upon the Grecian commissary stores. He is too
fond of striking an attitude, and his tone rises unpleasantly near a
scream, as he calls the personal attention of heaven and earth to
something which Lessing himself would have thought a very
matter-of-course affair. He who lays it down as an axiom, that "genius
loves simplicity," would hardly have been pleased to hear the "Letters on
Literature" called the "burning thunderbolts of his annihilating
criticism," or the Anti-Goetze pamphlets, "the hurtling arrows that sped
from the bow of the immortal hero." Nor would he with whom accuracy was a
matter of conscience have heard patiently that the Letters "appeared in a
period distinguished for its lofty tone of mind, and in their own
towering boldness they are a true picture of the intrepid character of
the age."[149] If the age was what Herr Stahr represents it to have been,
where is the great merit of Lessing? He would have smiled, we suspect, a
little contemptuously, at Herr Stahr's repeatedly quoting a certificate
from the "historian of the proud Britons," that he was "the first critic
in Europe." Whether we admit or not Lord Macaulay's competence in the
matter, we are sure that Lessing would not have thanked his biographer
for this soup-ticket to a ladleful of fame. If ever a man stood firmly on
his own feet, and asked help of none, that man was Gotthold Ephraim
Lessing.
Herr Stahr's desire to _make_ a hero of his subject, and his love for
sonorous sentences like those we have quoted above, are apt to stand
somewhat in the way of our chance at taking a fair measure of the man,
and seeing in what his heroism really lay. He furnishes little material
for a comparative estimate of Lessing, or for judging of the foreign
influences which helped from time to time in making him what he was.
Nothing is harder than to worry out a date from Herr Stahr's haystacks of
praise and quotation. Yet dates are of special value in tracing the
progress of an intellect like Lessing's, which, little actuated by an
inward creative energy, was commonly stirred to motion by the impulse of
other minds, and struck out its brightest flashes by collision with them.
He himself tells us that a critic should "first seek out some one with
whom he can contend," and quotes in justification from one of Aristotle's
commentators, _Solet Aristoteles quaerere pugnam in suis libris_. This
Lessing was always wont to do. He could only feel his own strength, and
make others feel it,--could only call it into full play in an
intellectual wrestling-bout. He was always anointed and ready for the
ring, but with this distinction, that he was no mere prize-fighter, or
bully for the side that would pay him best, nor even a contender for mere
sentiment, but a self-forgetful champion for the truth as he saw it. Nor
is this true of him only as a critic. His more purely imaginative
works--his Minna, his Emilia, his Nathan--were all written, not to
satisfy the craving of a poetic instinct, nor to rid head and heart of
troublous guests by building them a lodging outside himself, as Goethe
used to do, but to prove some thesis of criticism or morals by which
Truth could be served. His zeal for her was perfectly unselfish. "Does
one write, then, for the sake of being always in the right? I think I
have been as serviceable to Truth," he says, "when I miss her, and my
failure is the occasion of another's discovering her, as if I had
discovered her myself."[150] One would almost be inclined to think, from
Herr Stahr's account of the matter, that Lessing had been an
autochthonous birth of the German soil, without intellectual ancestry or
helpful kindred. That this is the sufficient natural history of no
original mind we need hardly say, since originality consists quite as
much in the power of using to purpose what it finds ready to its hand, as
in that of producing what is absolutely new. Perhaps we might say that it
was nothing more than the faculty of combining the separate, and
therefore ineffectual, conceptions of others, and making them into living
thought by the breath of its own organizing spirit. A great man without a
past, if he be not an impossibility, will certainly have no future. He
would be like those conjectural Miltons and Cromwells of Gray's imaginary
Hamlet. The only privilege of the original man is, that, like other
sovereign princes, he has the right to call in the current coin and
reissue it stamped with his own image, as was the practice of Lessing.
Herr Stahr's over-intensity of phrase is less offensive than amusing when
applied to Lessing's early efforts in criticism. Speaking of poor old
Gottsched, he says: "Lessing assailed him sometimes with cutting
criticism, and again with exquisite humor. In the notice of Gottsched's
poems, he says, among other things, 'The exterior of the volume is so
handsome that it will do great credit to the bookstores, and it is to be
hoped that it will continue to do so for a long time. But to give a
satisfactory idea of the interior surpasses our powers.' And in
conclusion he adds, 'These poems cost two thalers and four groschen. The
two thalers pay for the ridiculous, and the four groschen pretty much for
the useful.'" Again, he tells us that Lessing concludes his notice of
Klopstock's Ode to God "with these inimitably roguish words: 'What
presumption to beg thus earnestly for a woman!' Does not a whole book of
criticism lie in these nine words?" For a young man of twenty-two,
Lessing's criticisms show a great deal of independence and maturity of
thought; but humor he never had, and his wit was always of the
bluntest,--crushing rather than cutting. The mace, and not the scymitar,
was his weapon. Let Herr Stahr put all Lessing's "inimitably roguish
words" together, and compare them with these few intranslatable lines
from Voltaire's letter to Rousseau, thanking him for his _Discours sur
l'Inegalite_: "On n'a jamais employe tant d'esprit a vouloir nous rendre
betes; il prend enviede marcher a quatre pattes quand on lit votre
ouvrage." Lessing from the first was something far better than a wit.
Force was always much more characteristic of him than cleverness.
Sometimes Herr Stahr's hero-worship leads him into positive misstatement.
For example, speaking of Lessing's Preface to the "Contributions to the
History and Reform of the Theatre," he tells us that "his eye was
directed chiefly to the English theatre and Shakespeare." Lessing at that
time (1749) was only twenty, and knew little more than the names of any
foreign dramatists except the French. In this very Preface his English
list skips from Shakespeare to Dryden, and in the Spanish he omits
Calderon, Tirso de Molina, and Alarcon. Accordingly, we suspect that the
date is wrongly assigned to Lessing's translation of _Toda la Vida es
Sueno_. His mind was hardly yet ready to feel the strange charm of this
most imaginative of Calderon's dramas.
Even where Herr Stahr undertakes to give us light on the _sources_ of
Lessing, it is something of the dimmest. He attributes "Miss Sara
Sampson" to the influence of the "Merchant of London," as Mr. Evans
translates it literally from the German, meaning our old friend, "George
Barnwell." But we are strongly inclined to suspect from internal evidence
that Moore's more recent "Gamester" gave the prevailing impulse. And if
Herr Stahr must needs tell us anything of the Tragedy of Middle-Class
Life, he ought to have known that on the English stage it preceded Lillo
by more than a century,--witness the "Yorkshire Tragedy,"--and that
something very like it was even much older in France. We are inclined to
complain, also, that he does not bring out more clearly how much Lessing
owed to Diderot both as dramatist and critic, nor give us so much as a
hint of what already existing English criticism did for him in the way of
suggestion and guidance. But though we feel it to be our duty to say so
much of Herr Stahr's positive faults and negative short-comings, yet we
leave him in very good humor. While he is altogether too full upon
certain points of merely transitory importance,--such as the quarrel with
Klotz,--yet we are bound to thank him both for the abundance of his
extracts from Lessing, and for the judgment he has shown in the choice of
them. Any one not familiar with his writings will be able to get a very
good notion of the quality of his mind, and the amount of his literary
performance, from these volumes; and that, after all, is the chief
matter. As to the absolute merit of his works other than critical, Herr
Stahr's judgment is too much at the mercy of his partiality to be of
great value.
Of Mr. Evans's translation we can speak for the most part with high
commendation. There are great difficulties in translating German prose;
and whatever other good things Herr Stahr may have learned from Lessing,
terseness and clearness are not among them. We have seldom seen a
translation which read more easily, or was generally more faithful. That
Mr. Evans should nod now and then we do not wonder, nor that he should
sometimes choose the wrong word. We have only compared him with the
original where we saw reason for suspecting a slip; but, though we have
not found much to complain of, we have found enough to satisfy us that
his book will gain by a careful revision. We select a few oversights,
mainly from the first volume, as examples. On page 34, comparing Lessing
with Goethe on arriving at the University, Mr. Evans, we think, obscures,
if he does not wholly lose the meaning, when he translates _Leben_ by
"social relations," and is altogether wrong in rendering _Patrizier_ by
"aristocrat." At the top of the next page, too, "suspicious" is not the
word for _bedenklich_. Had he been writing English, he would surely have
said "questionable." On page 47, "overtrodden shoes" is hardly so good as
the idiomatic "down at the heel." On page 104, "A very humorous
representation" is oddly made to "confirm the documentary evidence." The
reverse is meant. On page 115, the sentence beginning "the tendency in
both" needs revising. On page 138, Mr. Evans speaks of the "Poetical
Village-younker of Destouches." This, we think, is hardly the English of
_Le Poete Campagnard_, and almost recalls Lieberkuehn's theory of
translation, toward which Lessing was so unrelenting,--"When I do not
understand a passage, why, I translate it word for word." On page 149,
"Miss Sara Sampson" is called "the first social tragedy of the German
Drama." All tragedies surely are _social_, except the "Prometheus."
_Buergerliche Tragoedie_ means a tragedy in which the protagonist is taken
from common life, and perhaps cannot be translated clearly into English
except by "tragedy of middle-class life." So on page 170 we find Emilia
Galotti called a "Virginia _bourgeoise_," and on page 172 a hospital
becomes a _lazaretto_. On page 190 we have a sentence ending in this
strange fashion: "in an episode of the English original, which Wieland
omitted entirely, one of its characters nevertheless appeared in the
German tragedy." On page 205 we have the Seven Years' War called "a
bloody _process_." This is mere carelessness, for Mr. Evans, in the
second volume, translates it rightly "_lawsuit_." What English reader
would know what "You are intriguing me" means, on page 228? On page 264,
Vol. II., we find a passage inaccurately rendered, which we consider of
more consequence, because it is a quotation from Lessing. "O, out upon
the man who claims, Almighty God, to be a preacher of Thy word, and yet
so impudently asserts that, in order to attain Thy purposes, there was
only one way in which it pleased _Thee_ to make _Thyself_ known to him!"
This is very far from _nur den einzigen Weg gehabt den Du Dir gefallen
lassen ihm kund zu machen!_ The _ihm_ is scornfully emphatic. We hope
Professor Evans will go over his version for a second edition much more
carefully than we have had any occasion to do. He has done an excellent
service to our literature, for which we heartily thank him, in choosing a
book of this kind to translate, and translating it so well. We would not
look such a gift horse too narrowly in the mouth.
Let us now endeavor to sum up the result of Lessing's life and labor with
what success we may.
Gotthold Ephraim Lessing was born (January 22, 1729) at Camenz, in Upper
Lusatia, the second child and eldest son of John Gottfried Lessing, a
Lutheran clergyman. Those who believe in the persistent qualities of
race, or the cumulative property of culture, will find something to their
purpose in his Saxon blood and his clerical and juristic ancestry. It is
worth mentioning, that his grandfather, in the thesis for his doctor's
degree, defended the right to entire freedom of religious belief. The
name first comes to the surface in Parson Clement Lessigk, nearly three
centuries ago, and survives to the present day in a painter of some
distinction. It has almost passed into a proverb, that the mothers of
remarkable children have been something beyond the common. If there be
any truth in the theory, the case of Lessing was an exception, as might
have been inferred, perhaps, from the peculiarly masculine type of his
character and intellect. His mother was in no wise superior, but his
father seems to have been a man somewhat above the pedantic average of
the provincial clergymen of his day, and to have been a scholar in the
ampler meaning of the word. Besides the classics, he had possessed
himself of French and English, and was somewhat versed in the Oriental
languages. The temper of his theology may be guessed from his having
been, as his son tells us with some pride, one of "the earliest
translators of Tillotson." We can only conjecture him from the letters
which Lessing wrote to him, from which we should fancy him as on the
whole a decided and even choleric old gentleman, in whom the wig, though
not a predominant, was yet a notable feature, and who was, like many
other fathers, permanently astonished at the fruit of his loins. He would
have preferred one of the so-called learned professions for his
son,--theology above all,--and would seem to have never quite reconciled
himself to his son's distinction, as being in none of the three careers
which alone were legitimate. Lessing's bearing towards him, always
independent, is really beautiful in its union of respectful tenderness
with unswerving self-assertion. When he wished to evade the maternal eye,
Gotthold used in his letters to set up a screen of Latin between himself
and her; and we conjecture the worthy Pastor Primarius playing over again
in his study at Camenz, with some scruples of conscience, the old trick
of Chaucer's fox:--
"Mulier est hominis confusio;
Madam, the sentence of this Latin is.
Woman is mannes joy and mannes bliss."
He appears to have snatched a fearful and but ill-concealed joy from the
sight of the first collected edition of his son's works, unlike Tillotson
as they certainly were. Ah, had they only been _Opera_! Yet were they not
volumes, after all, and able to stand on their own edges beside the
immortals, if nothing more?
After grinding with private-tutor Mylius the requisite time, Lessing
entered the school of Camenz, and in his thirteenth year was sent to the
higher institution at Meissen. We learn little of his career there,
except that Theophrastus, Plautus, and Terence were already his favorite
authors, that he once characteristically distinguished himself by a
courageous truthfulness, and that he wrote a Latin poem on the valor of
the Saxon soldiers, which his father very sensibly advised him to
shorten. In 1750, four years after leaving the school, he writes to his
father: "I believed even when I was at Meissen that one must learn much
there which he cannot make the least use of in real life (_der Welt_),
and I now [after trying Leipzig and Wittenberg] see it all the more
clearly,"--a melancholy observation which many other young men have made
under similar circumstances. Sent to Leipzig in his seventeenth year, he
finds himself an awkward, ungainly lad, and sets diligently to perfecting
himself in the somewhat unscholastic accomplishments of riding, dancing,
and fencing. He also sedulously frequents the theatre, and wrote a play,
"The Young Scholar," which attained the honor of representation.
Meanwhile his most intimate companion was a younger brother of his
old tutor Mylius, a young man of more than questionable morals,
and who had even written a satire on the elders of Camenz, for
which--over-confidently trusting himself in the outraged city--he had
been fined and imprisoned; so little could the German Muse, celebrated by
Klopstock for her swiftness of foot, protect her son. With this
scandalous person and with play-actors, more than probably of both sexes,
did the young Lessing share a Christmas cake sent him by his mother. Such
news was not long in reaching Camenz, and we can easily fancy how tragic
it seemed in the little parsonage there, to what cabinet councils it gave
rise in the paternal study, to what ominous shaking of the clerical wig
in that domestic Olympus. A pious fraud is practised on the boy, who
hurries home thinly clad through the winter weather, his ill-eaten
Christmas cake wringing him with remorseful indigestion, to receive the
last blessing, if such a prodigal might hope for it, of a broken-hearted
mother. He finds the good dame in excellent health, and softened toward
him by a cold he has taken on his pious journey. He remains at home
several months, now writing Anacreontics of such warmth that his sister
(as volunteer representative of the common hangman) burns them in the
family stove; now composing sermons to convince his mother that "he could
be a preacher any day,"--a theory of that sacred office unhappily not yet
extinct. At Easter, 1747, he gets back to Leipzig again, with some scant
supply of money in his pocket, but is obliged to make his escape thence
between two days somewhere toward the middle of the next year, leaving
behind him some histrionic debts (chiefly, we fear, of a certain
Mademoiselle Lorenz) for which he had confidingly made himself security.
Stranded, by want of floating or other capital, at Wittenberg, he enters
himself, with help from home, as a student there, but soon migrates again
to Berlin, which had been his goal when making his hegira from Leipzig.
In Berlin he remained three years, applying himself to his chosen calling
of author at all work, by doing whatever honest job offered
itself,--verse, criticism, or translation,--and profitably studious in a
very wide range of languages and their literature. Above all, he learned
the great secret, which his stalwart English contemporary, Johnson, also
acquired, of being able to "dine heartily" for threepence.
Meanwhile he continues in a kind of colonial dependence on the parsonage
at Camenz, the bonds gradually slackening, sometimes shaken a little
rudely, and always giving alarming hints of approaching and inevitable
autonomy. From the few home letters of Lessing which remain, (covering
the period before 1753, there are only eight in all,) we are able to
surmise that a pretty constant maternal cluck and shrill paternal warning
were kept up from the home coop. We find Lessing defending the morality
of the stage and his own private morals against charges and suspicions of
his parents, and even making the awful confession that he does not
consider the Christian religion itself as a thing "to be taken on trust,"
nor a Christian by mere tradition so valuable a member of society as "one
who has _prudently_ doubted, and by the way of examination has arrived at
conviction, or at least striven to arrive." Boyish scepticism of the
superficial sort is a common phenomenon enough, but the Lessing variety
of it seems to us sufficiently rare in a youth of twenty. What strikes us
mainly in the letters of these years is not merely the maturity they
show, though that is remarkable, but the tone. We see already in them the
cheerful and never overweening self-confidence which always so pleasantly
distinguished Lessing, and that strength of tackle, so seldom found in
literary men, which brings the mind well home to its anchor, enabling it
to find holding ground and secure riding in any sea. "What care I to live
in plenty," he asks gayly, "if I only live?" Indeed, Lessing learned
early, and never forgot, that whoever would be life's master, and not its
drudge, must make it a means, and never allow it to become an end. He
could say more truly than Goethe, _Mein Acker ist die Zeit_, since he not
only sowed in it the seed of thought for other men and other times, but
cropped it for his daily bread. Above all, we find Lessing even thus
early endowed with the power of keeping his eyes wide open to what he was
after, to what would help or hinder him,--a much more singular gift than
is commonly supposed. Among other jobs of this first Berlin period, he
had undertaken to arrange the library of a certain Herr Ruediger, getting
therefor his meals and "other receipts," whatever they may have been. His
father seems to have heard with anxiety that this arrangement had ceased,
and Lessing writes to him: "I never wished to have anything to do with
this old man longer than _until I had made myself thoroughly acquainted
with his great library_. This is now accomplished, and we have
accordingly parted." This was in his twenty-first year, and we have no
doubt, from the _range_ of scholarship which Lessing had at command so
young, that it was perfectly true. All through his life he was thoroughly
German in this respect also, that he never _quite_ smelted his knowledge
clear from some slag of learning.
In the early part of the first Berlin residence, Pastor Primarius
Lessing, hearing that his son meditated a movement on Vienna, was much
exercised with fears of the temptation to Popery he would be exposed to
in that capital. We suspect that the attraction thitherward had its
source in a perhaps equally catholic, but less theological magnet,--the
Mademoiselle Lorenz above mentioned. Let us remember the perfectly
innocent passion of Mozart for an actress, and be comforted. There is not
the slightest evidence that Lessing's life at this time, or any other,
though careless, was in any way debauched. No scandal was ever coupled
with his name, nor is any biographic chemistry needed to bleach spots out
of his reputation. What cannot be said of Wieland, of Goethe, of
Schiller, of Jean Paul, may be safely affirmed of this busy and
single-minded man. The parental fear of Popery brought him a seasonable
supply of money from home, which enabled him to clothe himself decently
enough to push his literary fortunes, and put on a bold front with
publishers. Poor enough he often was, but never in so shabby a pass that
he was forced to write behind a screen, like Johnson.
It was during this first stay in Berlin that Lessing was brought into
personal relations with Voltaire. Through an acquaintance with the great
man's secretary, Richier, he was employed as translator in the scandalous
Hirschel lawsuit, so dramatically set forth by Carlyle in his Life of
Frederick, though Lessing's share in it seems to have been unknown to
him. The service could hardly have been other than distasteful to him;
but it must have been with some thrill of the _anche io!_ kind that the
poor youth, just fleshing his maiden pen in criticism, stood face to face
with the famous author, with whose name all Europe rang from side to
side. This was in February, 1751. Young as he was, we fancy those cool
eyes of his making some strange discoveries as to the real nature of that
lean nightmare of Jesuits and dunces. Afterwards the same secretary lent
him the manuscript of the _Siecle de Louis XIV._, and Lessing
thoughtlessly taking it into the country with him, it was not forthcoming
when called for by the author. Voltaire naturally enough danced with
rage, screamed all manner of unpleasant things about robbery and the
like, cashiered the secretary, and was, we see no reason to doubt, really
afraid of a pirated edition. _This_ time his cry of wolf must have had a
quaver of sincerity in it. Herr Stahr, who can never keep separate the
Lessing as he then was and the Lessing as he afterwards became, takes
fire at what he chooses to consider an unworthy suspicion of the
Frenchman, and treats himself to some rather cheap indignation on the
subject. For ourselves, we think Voltaire altogether in the right, and we
respect Lessing's honesty too much to suppose, with his biographer, that
it was this which led him, years afterwards, to do such severe justice to
_Merope_, and other tragedies of the same author. The affair happened in
December, 1751, and a year later Lessing calls Voltaire a "great man,"
and says of his _Amalie_, that "it has not only beautiful passages, it is
beautiful throughout, and the tears of a reader of feeling will justify
our judgment." Surely there is no resentment here. Our only wonder would
be at its being written after the Hirschel business. At any rate, we
cannot allow Herr Stahr to shake our faith in the sincerity of Lessing's
motives in criticism,--he could not in the soundness of the criticism
itself,--by tracing it up to a spring at once so petty and so personal.
During a part of 1752,[151] Lessing was at Wittenberg again as student of
medicine, the parental notion of a strictly professional career of some
kind not having yet been abandoned. We must give his father the credit of
having done his best, in a well-meaning paternal fashion, to make his son
over again in his own image, and to thwart the design of nature by
coaxing or driving him into the pinfold of a prosperous obscurity. But
Gotthold, with all his gifts, had no talent whatever for contented
routine. His was a mind always in solution, which the divine order of
things, as it is called, could not precipitate into any of the
traditional forms of crystallization, and in which the time to come was
already fermenting. The principle of growth was in the young literary
hack, and he must obey it or die. His was to the last a _natura
naturans_, never a _naturata_. Lessing seems to have done what he could
to be a dutiful failure. But there was something in him stronger and more
sacred than even filial piety; and the good old pastor is remembered now
only as the father of a son who would have shared the benign oblivion of
his own theological works, if he could only have had his wise way with
him. Even after never so many biographies and review articles, genius
continues to be a marvellous and inspiring thing. At the same time,
considering the then condition of what was pleasantly called literature
in Germany, there was not a little to be said on the paternal side of the
question, though it may not seem now a very heavy mulct to give up one
son out of ten to immortality,--at least the Fates seldom decimate in
_this_ way. Lessing had now, if we accept the common standard in such
matters, "completed his education," and the result may be summed up in
his own words to Michaelis, 16th October, 1754: "I have studied at the
Fuerstenschule at Meissen, and after that at Leipzig and Wittenberg. But I
should be greatly embarrassed if I were asked to tell _what_." As early
as his twentieth year he had arrived at some singular notions as to the
uses of learning. On the 20th of January, 1749, he writes to his mother:
"I found out that books, indeed, would make me learned, _but never make
me a man_." Like most men of great knowledge, as distinguished from mere
scholars, he seems to have been always a rather indiscriminate reader,
and to have been fond, as Johnson was, of "browsing" in libraries.
Johnson neither in amplitude of literature nor exactness of scholarship
could be deemed a match for Lessing; but they were alike in the power of
readily applying whatever they had learned, whether for purposes of
illustration or argument. They resemble each other, also, in a kind of
absolute common-sense, and in the force with which they could plant a
direct blow with the whole weight both of their training and their
temperament behind it. As a critic, Johnson ends where Lessing begins.
The one is happy in the lower region of the understanding: the other can
breathe freely in the ampler air of reason alone. Johnson acquired
learning, and stopped short from indolence at a certain point. Lessing
assimilated it, and accordingly his education ceased only with his life.
Both had something of the intellectual sluggishness that is apt to go
with great strength; and both had to be baited by the antagonism of
circumstances or opinions, not only into the exhibition, but into the
possession of their entire force. Both may be more properly called
original men than, in the highest sense, original writers.
From 1752 to 1760, with an interval of something over two years spent in
Leipzig to be near a good theatre, Lessing was settled in Berlin, and
gave himself wholly and earnestly to the life of a man of letters. A
thoroughly healthy, cheerful nature he most surely had, with something at
first of the careless light-heartedness of youth. Healthy he was not
always to be, not always cheerful, often very far from light-hearted, but
manly from first to last he eminently was. Downcast he could never be,
for his strongest instinct, invaluable to him also as a critic, was to
see things as they really are. And this not in the sense of a cynic, but
of one who measures himself as well as his circumstances,--who loves
truth as the most beautiful of all things and the only permanent
possession, as being of one substance with the soul. In a man like
Lessing, whose character is even more interesting than his works, the
tone and turn of thought are what we like to get glimpses of. And for
this his letters are more helpful than those of most authors, as might be
expected of one who said of himself, that, in his more serious work, "he
must profit by his first heat to accomplish anything." He began, we say,
light-heartedly. He did not believe that "one should thank God only for
good things." "He who is only in good health, and is willing to work, has
nothing to fear in the world." "What another man would call want, I call
comfort." "Must not one often act thoughtlessly, if one would provoke
Fortune to do something for him?" In his first inexperience, the life of
"the sparrow on the house-top" (which we find oddly translated "roof")
was the one he would choose for himself. Later in life, when he wished to
marry, he was of another mind, and perhaps discovered that there was
something in the old father's notion of a fixed position. "The life of
the sparrow on the house-top is only right good if one need not expect
any end to it. If it cannot always last, every day it lasts too
long,"--he writes to Ebert in 1770. Yet even then he takes the manly
view. "Everything in the world has its time, everything may be overlived
and overlooked, if one only have health." Nor let any one suppose that
Lessing, full of courage as he was, found professional authorship a
garden of Alcinoues. From creative literature he continually sought
refuge, and even repose, in the driest drudgery of mere scholarship. On
the 26th of April, 1768, he writes to his brother with something of his
old gayety: "Thank God, the time will soon come when I cannot call a
penny in the world my own but I must first earn it. I am unhappy if it
must be by writing." And again in May, 1771: "Among all the wretched, I
think him the most wretched who must work with his head, even if he is
not conscious of having one. But what is the good of complaining?"
Lessing's life, if it is a noble example, so far as it concerned himself
alone, is also a warning when another is to be asked to share it. He too
would have profited had he earlier learned and more constantly borne in
mind the profound wisdom of that old saying, _Si sit prudentia_. Let the
young poet, however he may believe of his art that "all other pleasures
are not worth its pains," consider well what it is to call down fire from
heaven to keep the pot boiling, before he commit himself to a life of
authorship as something fine and easy. That fire will not condescend to
such office, though it come without asking on ceremonial days to the free
service of the altar.
Lessing, however, never would, even if he could, have so desecrated his
better powers. For a bare livelihood, he always went sturdily to the
market of hack-work, where his learning would fetch him a price. But it
was only in extremest need that he would claim that benefit of clergy. "I
am worried," he writes to his brother Karl, 8th April, 1773, "and work
because working is the only means to cease being so. But you and Vess are
very much mistaken if you think that it could ever be indifferent to me,
under such circumstances, on what I work. Nothing less true, whether as
respects the work itself or the principal object wherefor I work. I have
been in my life before now in very wretched circumstances, yet never in
such that I would have written for bread in the true meaning of the word.
I have begun my 'Contributions' because this work helps me ... to live
from one day to another." It is plain that he does not call this kind of
thing in any high sense writing. Of that he had far other notions; for
though he honestly disclaimed the title, yet his dream was always to be a
poet. But he _was_ willing to work, as he claimed to be, because he had
one ideal higher than that of being a poet, namely, to be thoroughly a
man. To Nicolai he writes in 1758: "All ways of earning his bread are
alike becoming to an honest man, whether to split wood or to sit at the
helm of state. It does not concern his conscience how useful he is, but
how useful he would be." Goethe's poetic sense was the Minotaur to which
he sacrificed everything. To make a study, he would soil the maiden
petals of a woman's soul; to get the delicious sensation of a reflex
sorrow, he would wring a heart. All that saves his egoism from being
hateful is, that, with its immense reaches, it cheats the sense into a
feeling of something like sublimity. A patch of sand is unpleasing; a
desert has all the awe of ocean. Lessing also felt the duty of
self-culture; but it was not so much for the sake of feeding fat this or
that faculty as of strengthening character,--the only soil in which real
mental power can root itself and find sustenance. His advice to his
brother Karl, who was beginning to write for the stage, is two parts
moral to one literary. "Study ethics diligently, learn to express
yourself well and correctly, and cultivate your own character. Without
that I cannot conceive a good dramatic author." Marvellous counsel this
will seem to those who think that wisdom is only to be found in the
fool's paradise of Bohemia!
We said that Lessing's dream was to be a poet. In comparison with success
as a dramatist, he looked on all other achievement as inferior in kind.
In. 1767 he writes to Gleim (speaking of his call to Hamburg): "Such
circumstances were needed to rekindle in me an almost extinguished love
for the theatre. I was just beginning to lose myself in other studies
which would have made me unfit for any work of genius. My _Laocoon_ is
now a secondary labor." And yet he never fell into the mistake of
overvaluing what he valued so highly. His unflinching common-sense would
have saved him from that, as it afterwards enabled him to see that
something was wanting in him which must enter into the making of true
poetry, whose distinction from prose is an inward one of nature, and not
an outward one of form. While yet under thirty, he assures Mendelssohn
that he was quite right in neglecting poetry for philosophy, because
"only a part of our youth should be given up to the arts of the
beautiful. We must practise ourselves in weightier things before we die.
An old man, who lifelong has done nothing but rhyme, and an old man who
lifelong has done nothing but pass his breath through a stick with holes
in it,--I doubt much whether such an old man has arrived at what he was
meant for."
This period of Lessing's life was a productive one, though none of its
printed results can be counted of permanent value, except his share in
the "Letters on German Literature." And even these must be reckoned as
belonging to the years of his apprenticeship and training for the
master-workman he afterwards became. The small fry of authors and
translators were hardly fitted to call out his full strength, but his
vivisection of them taught him the value of certain structural
principles. "To one dissection of the fore quarter of an ass," says
Haydon in his diary, "I owe my information." Yet even in his earliest
criticisms we are struck with the same penetration and steadiness of
judgment, the same firm grasp of the essential and permanent, that were
afterwards to make his opinions law in the courts of taste. For example,
he says of Thomson, that, "as a dramatic poet, he had the fault of never
knowing when to leave off; he lets every character talk so long as
anything can be said; accordingly, during these prolonged conversations,
the action stands still, and the story becomes tedious." Of "Roderick
Random," he says that "its author is neither a Richardson nor a Fielding;
he is one of those writers of whom there are plenty among the Germans and
French." We cite these merely because their firmness of tone seems to us
uncommon in a youth of twenty-four. In the "Letters," the range is much
wider, and the application of principles more consequent. He had already
secured for himself a position among the literary men of that day, and
was beginning to be feared for the inexorable justice of his criticisms.
His "Fables" and his "Miss Sara Sampson" had been translated into French,
and had attracted the attention of Grimm, who says of them (December,
1754): "These Fables commonly contain in a few lines a new and profound
moral meaning. M. Lessing has much wit, genius, and invention; the
dissertations which follow the Fables prove moreover that he is an
excellent critic." In Berlin, Lessing made friendships, especially with
Mendelssohn, Von Kleist, Nicolai, Gleim, and Ramler. For Mendelssohn and
Von Kleist he seems to have felt a real love; for the others at most a
liking, as the best material that could be had. It certainly was not of
the juiciest. He seems to have worked hard and played hard, equally at
home in his study and Baumann's wine-cellar. He was busy, poor, and
happy.
But he was restless. We suspect that the necessity of forever picking up
crumbs, and their occasional scarcity, made the life of the sparrow on
the house-top less agreeable than he had expected. The imagined freedom
was not quite so free after all, for necessity is as short a tether as
dependence, or official duty, or what not, and the regular occupation of
grub-hunting is as tame and wearisome as another. Moreover, Lessing had
probably by this time sucked his friends dry of any intellectual stimulus
they could yield him; and when friendship reaches that pass, it is apt to
be anything but inspiring. Except Mendelssohn and Von Kleist, they were
not men capable of rating him at his true value; and Lessing was one of
those who always burn up the fuel of life at a fearful rate. Admirably
dry as the supplies of Ramler and the rest no doubt were, they had not
substance enough to keep his mind at the high temperature it needed, and
he would soon be driven to the cutting of green stuff from his own
wood-lot, more rich in smoke than fire. Besides this, he could hardly
have been at ease among intimates most of whom could not even conceive of
that intellectual honesty, that total disregard of all personal interests
where truth was concerned, which was an innate quality of Lessing's mind.
Their theory of criticism was, Truth, or even worse if possible, for all
who do not belong to our set; for us, that delicious falsehood which is
no doubt a slow poison, but then so _very_ slow. Their nerves were
unbraced by that fierce democracy of thought, trampling on all
prescription, all tradition, in which Lessing loved to shoulder his way
and advance his insupportable foot. "What is called a heretic," he says
in his Preface to _Berengarius_, "has a very good side. It is a man who
at least _wishes_ to see with his own eyes." And again, "I know not if it
be a duty to offer up fortune and life to the truth; ... but I know it
_is_ a duty, if one undertake to teach the truth, to teach the whole of
it, or none at all." Such men as Gleim and Ramler were mere _dilettanti_,
and could have no notion how sacred his convictions are to a militant
thinker like Lessing. His creed as to the rights of friendship in
criticism might be put in the words of Selden, the firm tread of whose
mind was like his own: "Opinion and affection extremely differ. Opinion
is something wherein I go about to give reason why all the world should
think as I think. Affection is a thing wherein I look after the pleasing
of myself." How little his friends were capable of appreciating this view
of the matter is plain from a letter of Ramler to Gleim, cited by Herr
Stahr. Lessing had shown up the weaknesses of a certain work by the Abbe
Batteux (long ago gathered to his literary fathers as conclusively as
poor old Ramler himself), without regard to the important fact that the
Abbe's book had been translated by a friend. Horrible to think of at
best, thrice horrible when the friend's name was Ramler! The impression
thereby made on the friendly heart may be conceived. A ray of light
penetrated the rather opaque substance of Herr Ramler's mind, and
revealed to him the dangerous character of Lessing. "I know well," he
says, "that Herr Lessing means to speak his own opinion, and"--what is
the dreadful inference?--"and, by suppressing others, to gain air, and
make room for himself. This disposition is not to be overcome."[152]
Fortunately not, for Lessing's opinion always meant something, and was
worth having. Gleim no doubt sympathized deeply with the sufferer by this
treason, for he too had been shocked at some disrespect for La Fontaine,
as a disciple of whom he had announced himself.
Berlin was hardly the place for Lessing, if he could not take a step in
any direction without risk of treading on somebody's gouty foot. This was
not the last time that he was to have experience of the fact that the
critic's pen, the more it has of truth's celestial temper, the more it is
apt to reverse the miracle of the archangel's spear, and to bring out
whatever is toadlike in the nature of him it touches. We can well
understand the sadness with which he said,
"Der Blick des Forscher's fand
Nicht selten mehr als er zu finden wuenschte."
Here, better than anywhere, we may cite something which he wrote of
himself to a friend of Klotz. Lessing, it will be remembered, had
literally "suppressed" Klotz. "What do you apprehend, then, from me? The
more faults and errors you point out to me, so much the more I shall
learn of you; the more I learn of you, the more thankful shall I be....I
wish you knew me more thoroughly. If the opinion you have of my learning
and genius (_Geist_) should perhaps suffer thereby, yet I am sure the
idea I would like you to form of my character would gain. I am not the
insufferable, unmannerly, proud, slanderous man Herr Klotz proclaims me.
It cost me a great deal of trouble and compulsion to be a little bitter
against him."[153] Ramler and the rest had contrived a nice little
society for mutual admiration, much like that described by Goldsmith, if,
indeed, he did not convey it from the French, as was not uncommon with
him. "'What, have you never heard of the admirable Brandellius or the
ingenious Mogusius, one the eye and the other the heart of our
University, known all over the world?' 'Never,' cried the traveller; 'but
pray inform me what Brandellius is particularly remarkable for.' 'You
must be little acquainted with the republic of letters,' said the other,
'to ask such a question. Brandellius has written a most sublime panegyric
on Mogusius.' 'And, prithee, what has Mogusius done to deserve so great a
favor?' 'He has written an excellent poem in praise of Brandellius.'"
Lessing was not the man who could narrow himself to the proportions of a
clique; lifelong he was the terror of the Brandellii and Mogusii, and, at
the signal given by him,
"They, but now who seemed
In bigness to surpass Earth's giant sons,
Now less than smallest dwarfs in narrow room
Throng numberless."
Besides whatever other reasons Leasing may have had for leaving Berlin,
we fancy that his having exhausted whatever means it had of helping his
spiritual growth was the chief. Nine years later, he gave as a reason for
not wishing to stay long in Brunswick, "Not that I do not like Brunswick,
but because nothing comes of being long in a place which one likes."[154]
Whatever the reason, Leasing, in 1760, left Berlin for Breslau, where the
post of secretary had been offered him under Frederick's tough old
General Tauentzien. "I will spin myself in for a while like an ugly worm,
that I may be able to come to light again as a brilliant winged
creature," says his diary. Shortly after his leaving Berlin, he was
chosen a member of the Academy of Sciences there. Herr Stahr, who has no
little fondness for the foot-light style of phrase, says, "It may easily
be imagined that he himself regarded his appointment as an insult rather
than as an honor." Lessing himself merely says that it was a matter of
indifference to him, which is much more in keeping with his character and
with the value of the intended honor.
The Seven Years' War began four years before Lessing took up his abode in
Breslau, and it may be asked how he, as a Saxon, was affected by it. We
might answer, hardly at all. His position was that of armed neutrality.
Long ago at Leipzig he had been accused of Prussian leanings; now in
Berlin he was thought too Saxon. Though he disclaimed any such sentiment
as patriotism, and called himself a cosmopolite, it is plain enough that
his position was simply that of a German. Love of country, except in a
very narrow parochial way, was as impossible in Germany then as in
America during the Colonial period. Lessing himself, in the latter years
of his life, was librarian of one of those petty princelets who sold
their subjects to be shot at in America,--creatures strong enough to
oppress, too weak to protect their people. Whoever would have found a
Germany to love must have pieced it together as painfully as Isis did the
scattered bits of Osiris. Yet he says that "the true patriot is by no
means extinguished" in him. It was the noisy ones that he could not
abide; and, writing to Gleim about his "Grenadier" verses, he advises him
to soften the tone of them a little, he himself being a "declared enemy
of imprecations," which he would leave altogether to the clergy. We think
Herr Stahr makes too much of these anti-patriot flings of Lessing, which,
with a single exception, occur in his letters to Gleim, and with
reference to a kind of verse that could not but be distasteful to him, as
needing no more brains than a drum, nor other inspiration than serves a
trumpet. Lessing undoubtedly had better uses for his breath than to spend
it in shouting for either side in this "bloody lawsuit," as he called it,
in which he was not concerned. He showed himself German enough, and in
the right way, in his persistent warfare against the tyranny of French
taste.
He remained in Breslau the better part of five years, studying life in
new phases, gathering a library, which, as commonly happens, he
afterwards sold at great loss, and writing his _Minna_ and his _Laocooen_.
He accompanied Tauentzien to the siege of Schweidnitz, where Frederick
was present in person. He seems to have lived a rather free-and-easy life
during his term of office, kept shockingly late hours, and learned, among
other things, to gamble,--a fact for which Herr Stahr thinks it needful
to account in a high philosophical fashion. We prefer to think that there
are _some_ motives to which remarkable men are liable in common with the
rest of mankind, and that they may occasionally do a thing merely because
it is pleasant, without forethought of medicinal benefit to the mind.
Lessing's friends (whose names were _not_, as the reader might be tempted
to suppose, Eliphaz, Bildad, and Zophar) expected him to make something
handsome out of his office; but the pitiful result of those five years of
opportunity was nothing more than an immortal book. Unthrifty Lessing, to
have been so nice about your fingers, (and so near the mint, too,) when
your general was wise enough to make his fortune! As if ink-stains were
the only ones that would wash out, and no others had ever been covered
with white kid from the sight of all reasonable men! In July, 1764, he
had a violent fever, which he turned to account in his usual cheerful
way: "The serious epoch of my life is drawing nigh. I am beginning to
become a man, and flatter myself that in this burning fever I have raved
away the last remains of my youthful follies. Fortunate illness!" He had
never intended to bind himself to an official career. To his father he
writes: "I have more than once declared that my present engagement could
not continue long, that I have not given up my old plan of living, and
that I am more than ever resolved to withdraw from any service that is
not wholly to my mind. I have passed the middle of my life, and can think
of nothing that could compel me to make myself a slave for the poor
remainder of it. I write you this, dearest father, and must write you
this, in order that you may not be astonished if, before long, you should
see me once more very far removed from all hopes of, or claims to, a
settled prosperity, as it is called." Before the middle of the next year
he was back in Berlin again.
There he remained for nearly two years, trying the house-top way of life
again, but with indifferent success, as we have reason to think. Indeed,
when the metaphor resolves itself into the plain fact of living just on
the other side of the roof,--in the garret, namely,--and that from hand
to mouth, as was Lessing's case, we need not be surprised to find him
gradually beginning to see something more agreeable in a _fixirtes Glueck_
than he had once been willing to allow. At any rate, he was willing, and
even heartily desirous, that his friends should succeed in getting for
him the place of royal librarian. But Frederick, for some unexplained
reason, would not appoint him. Herr Stahr thinks it had something to do
with the old _Siecle_ manuscript business. But this seems improbable, for
Voltaire's wrath was not directed against Lessing; and even if it had
been, the great king could hardly have carried the name of an obscure
German author in his memory through all those anxious and war-like years.
Whatever the cause, Lessing early in 1767 accepts the position of
Theatrical Manager at Hamburg, as usual not too much vexed with
disappointment, but quoting gayly
"Quod non dant proceres, dabit histrio."
Like Burns, he was always "contented wi' little and canty wi' mair." In
connection with his place as Manager he was to write a series of dramatic
essays and criticisms. It is to this we owe the _Dramaturgie_,--next to
the _Laocooen_ the most valuable of his works. But Lessing--though it is
plain that he made his hand as light as he could, and wrapped his lash in
velvet--soon found that actors had no more taste for truth than authors.
He was obliged to drop his remarks on the special merits or demerits of
players, and to confine himself to those of the pieces represented. By
this his work gained in value; and the latter part of it, written without
reference to a particular stage, and devoted to the discussion of those
general principles of dramatic art on which he had meditated long and
deeply, is far weightier than the rest. There are few men who can put
forth all their muscle in a losing race, and it is characteristic of
Lessing that what he wrote under the dispiritment of failure should be
the most lively and vigorous. Circumstances might be against him, but he
was incapable of believing that a cause could be lost which had once
enlisted his conviction.
The theatrical enterprise did not prosper long; but Lessing had meanwhile
involved himself as partner in a publishing business which harassed him
while it lasted, and when it failed, as was inevitable, left him hampered
with debt. Help came in his appointment (1770) to take charge of the Duke
of Brunswick's library at Wolfenbuettel, with a salary of six hundred
thalers a year. This was the more welcome, as he soon after was betrothed
with Eva Koenig, widow of a rich manufacturer.[155] Her husband's affairs,
however, had been left in confusion, and this, with Lessing's own
embarrassments, prevented their being married till October, 1776. Eva
Koenig was every way worthy of him. Clever, womanly, discreet, with just
enough coyness of the will to be charming when it is joined with
sweetness and good sense, she was the true helpmate of such a man,--the
serious companion of his mind and the playfellow of his affections. There
is something infinitely refreshing to me in the love-letters of these two
persons. Without wanting sentiment, there is such a bracing air about
them as breathes from the higher levels and strong-holds of the soul.
They show that self-possession which can alone reserve to love the power
of new self-surrender,--of never cloying, because never wholly possessed.
Here is no invasion and conquest of the weaker nature by the stronger,
but an equal league of souls, each in its own realm still sovereign. Turn
from such letters as these to those of St. Preux and Julie, and you are
stifled with the heavy perfume of a demirep's boudoir,--to those of
Herder to his Caroline, and you sniff no doubtful odor of professional
unction from the sermon-case. Manly old Dr. Johnson, who could be tender
and true to a plain woman, knew very well what he meant when he wrote
that single poetic sentence of his,--"The shepherd in Virgil grew at last
acquainted with Love, and found him to be a native of the rocks."
In January, 1778, Lessing's wife died from the effects of a difficult
childbirth. The child, a boy, hardly survived its birth. The few words
wrung out of Lessing by this double sorrow are to me as deeply moving as
anything in tragedy. "I wished for once to be as happy (_es so gut
haben_) as other men. But it has gone ill with me!" "And I was so loath
to lose him, this son!" "My wife is dead; and I have had this experience
also. I rejoice that I have not many more such experiences left to make,
and am quite cheerful." "If you had known her! But they say that to
praise one's wife is self-praise. Well, then, I say no more of her! But
if you had known her!" _Quite cheerful!_ On the 10th of August he writes
to Elise Reimarus,--he is writing to a woman now, an old friend of his
and his wife, and will be less restrained: "I am left here all alone. I
have not a single friend to whom I can wholly confide myself.... How
often must I curse my ever wishing to be for once as happy as other men!
How often have I wished myself back again in my old, isolated
condition,--to be nothing, to wish nothing, to do nothing, but what the
present moment brings with it!... Yet I am too proud to think myself
unhappy. I just grind my teeth, and let the boat go as pleases wind and
waves. Enough that I will not overset it myself." It is plain from this
letter that suicide had been in his mind, and, with his antique way of
thinking on many subjects, he would hardly have looked on it as a crime.
But he was too brave a man to throw up the sponge to fate, and had work
to do yet. Within a few days of his wife's death he wrote to Eschenburg:
"I am right heartily ashamed if my letter betrayed the least despair.
Despair is not nearly so much my failing as levity, which often expresses
itself with a little bitterness and misanthropy." A stoic, not from
insensibility or cowardice, as so many are, but from stoutness of heart,
he blushes at a moment's abdication of self-command. And he will not roil
the clear memory of his love with any tinge of the sentimentality so much
the fashion, and to be had so cheap, in that generation. There is a
moderation of sincerity peculiar to Lessing in the epithet of the
following sentence: "How dearly must I pay for the single year I have
lived with a _sensible_ wife!" Werther had then been published four
years. Lessing's grief has that pathos which he praised in sculpture,--he
may writhe, but he must not scream. Nor is this a new thing with him. On
the death of a younger brother, he wrote to his father, fourteen years
before: "Why should those who grieve communicate their grief to each
other purposely to increase it?... Many mourn in death what they loved
not living. I will love in life what nature bids me love, and after death
strive to bewail it as little as I can."
We think Herr Stahr is on his stilts again when he speaks of Lessing's
position at Wolfenbuettel. He calls it an "assuming the chains of feudal
service, being buried in a corner, a martyrdom that consumed the best
powers of his mind and crushed him in body and spirit forever." To crush
_forever_ is rather a strong phrase, Herr Stahr, to apply to the spirit,
if one must ever give heed to the sense as well as the sound of what one
is writing. But eloquence has no bowels for its victims. We have no doubt
the Duke of Brunswick meant well by Lessing, and the salary he paid him
was as large as he would have got from the frugal Frederick. But one
whose trade it was to be a Duke could hardly have had much sympathy with
his librarian after he had once found out what he really was. For even if
he was not, as Herr Stahr affirms, a republican, and we doubt very much
if he was, yet he was not a man who could play with ideas in the light
French fashion. At the ardent touch of his sincerity, they took fire, and
grew dangerous to what is called the social fabric. The logic of wit,
with its momentary flash, is a very different thing from that consequent
logic of thought, pushing forward its deliberate sap day and night with a
fixed object, which belonged to Lessing. The men who attack abuses are
not so much to be dreaded by the reigning house of Superstition as those
who, as Dante says, syllogize hateful truths. As for "the chains of
feudal service," they might serve a Fenian Head-Centre on a pinch, but
are wholly out of place here. The slavery that Lessing had really taken
on him was that of a great library, an Alcina that could always too
easily witch him away from the more serious duty of his genius. That a
mind like his could be buried in a corner is mere twaddle, and of a kind
that has done great wrong to the dignity of letters. Where-ever Lessing
sat, was the head of the table. That he suffered at Wolfenbuettel is true;
but was it nothing to be in love and in debt at the same time, and to
feel that his fruition of the one must be postponed for uncertain years
by his own folly in incurring the other? If the sparrow-life must end,
surely a wee bush is better than nae beild. One cause of Lessing's
occasional restlessness and discontent Herr Stahr has failed to notice.
It is evident from many passages in his letters that he had his share of
the hypochondria which goes with an imaginative temperament. But in him
it only serves to bring out in stronger relief his deep-rooted manliness.
He spent no breath in that melodious whining which, beginning with
Rousseau, has hardly yet gone out of fashion. Work of some kind was his
medicine for the blues,--if not always of the kind he would have chosen,
then the best that was to be had; for the useful, too, had for him a
sweetness of its own. Sometimes he found a congenial labor in rescuing,
as he called it, the memory of some dead scholar or thinker from the
wrongs of ignorance or prejudice or falsehood; sometimes in fishing a
manuscript out of the ooze of oblivion, and giving it, after a critical
cleansing, to the world. Now and then he warmed himself and kept his
muscle in trim with buffeting soundly the champions of that shallow
artificiality and unctuous wordiness, one of which passed for orthodox in
literature, and the other in theology. True religion and creative genius
were both so beautiful to him that he could never abide the mediocre
counterfeit of either, and he who put so much of his own life into all he
wrote could not but hold all scripture sacred in which a divine soul had
recorded itself. It would be doing Lessing great wrong to confound his
controversial writing with the paltry quarrels of authors. His own
personal relations enter into them surprisingly little, for his quarrel
was never with men, but with falsehood, cant, and misleading tradition,
in whomsoever incarnated. Save for this, they were no longer readable,
and might be relegated to that herbarium of Billingsgate gathered by the
elder Disraeli.
So far from being "crushed in spirit" at Wolfenbuettel, the years he spent
there were among the most productive of his life. "Emilia Galotti," begun
in 1758, was finished there and published in 1771. The controversy with
Goetze, by far the most important he was engaged in, and the one in which
he put forth his maturest powers, was carried on thence. His "Nathan the
Wise" (1779), by which almost alone he is known as a poet outside of
Germany, was conceived and composed there. The last few years of his life
were darkened by ill-health and the depression which it brings. His
Nathan had not the success he hoped. It is sad to see the strong,
self-sufficing man casting about for a little sympathy, even for a little
praise. "It is really needful to me that you should have some small good
opinion of it [Nathan], in order to make me once more contented with
myself," he writes to Elise Reimarus in May, 1779. That he was weary of
polemics, and dissatisfied with himself for letting them distract him
from better things, appears from his last pathetic letter to the old
friend he loved and valued most,--Mendelssohn. "And in truth, dear
friend, I sorely need a letter like yours from time to time, if I am not
to become wholly out of humor. I think you do not know me as a man that
has a very hot hunger for praise. But the coldness with which the world
is wont to convince certain people that they do not suit it, if not
deadly, yet stiffens one with chill. I am not astonished that _all_ I
have written lately does not please _you_.... At best, a passage here and
there may have cheated you by recalling our better days. I, too, was then
a sound, slim sapling, and am now such a rotten, gnarled trunk!" This was
written on the 19th of December, 1780; and on the 15th of February, 1781,
Lessing died, not quite fifty-two years old. Goethe was then in his
thirty-second year, and Schiller ten years younger.
* * * * *
Of Lessing's relation to metaphysics the reader will find ample
discussion in Herr Stahr's volumes. We are not particularly concerned
with them, because his interest in such questions was purely speculative,
and because he was more concerned to exercise the powers of his mind than
to analyze them. His chief business, his master impulse always, was to be
a man of letters in the narrower sense of the term. Even into theology he
only made occasional raids across the border, as it were, and that not so
much with a purpose of reform as in defence of principles which applied
equally to the whole domain of thought. He had even less sympathy with
heterodoxy than with orthodoxy, and, so far from joining a party or
wishing to form one, would have left belief a matter of choice to the
individual conscience. "From the bottom of my heart I hate all those
people who wish to found sects. For it is not error, but sectarian error,
yes, even sectarian truth, that makes men unhappy, or would do so if
truth would found a sect."[156] Again he says, that in his theological
controversies he is "much less concerned about theology than about sound
common-sense, and only therefore prefer the old orthodox (at bottom
_tolerant_) theology to the new (at bottom _intolerant_), because the
former openly conflicts with sound common-sense, while the latter would
fain corrupt it. I reconcile myself with my open enemies in order the
better to be on my guard against my secret ones."[157] At another time he
tells his brother that he has a wholly false notion of his (Lessing's)
relation to orthodoxy. "Do you suppose I grudge the world that anybody
should seek to enlighten it?--that I do not heartily wish that every one
should think rationally about religion? I should loathe myself if even in
my scribblings I had any other end than to help forward those great
views. But let me choose my own way, which I think best for this purpose.
And what is simpler than this way? I would not have the impure water,
which has long been unfit to use, preserved; but I would not have it
thrown away before we know whence to get purer.... Orthodoxy, thank God,
we were pretty well done with; a partition-wall had been built between it
and Philosophy, behind which each could go her own way without troubling
the other. But what are they doing now? They are tearing down this wall,
and, under the pretext of making us rational Christians, are making us
very irrational philosophers.... We are agreed that our old religious
system is false; but I cannot say with you that it is a patchwork of
bunglers and half-philosophers. I know nothing in the world in which
human acuteness has been more displayed or exercised than in that."[158]
Lessing was always for freedom, never for looseness, of thought, still
less for laxity of principle. But it must be a real freedom, and not that
vain struggle to become a majority, which, if it succeed, escapes from
heresy only to make heretics of the other side. _Abire ad plures_ would
with him have meant, not bodily but spiritual death. He did not love the
fanaticism of innovation a whit better than that of conservatism. To his
sane understanding, both were equally hateful, as different masks of the
same selfish bully. Coleridge said that toleration was impossible till
indifference made it worthless. Lessing did not wish for toleration,
because that implies authority, nor could his earnest temper have
conceived of indifference. But he thought it as absurd to regulate
opinion as the color of the hair. Here, too, he would have agreed with
Selden, that "it is a vain thing to talk of an heretic, for a man for his
heart cannot think any otherwise than he does think." Herr Stahr's
chapters on this point, bating a little exaltation of tone, are very
satisfactory; though, in his desire to make a leader of Lessing, he
almost represents him as being what he shunned,--the founder of a sect.
The fact is, that Lessing only formulated in his own way a general
movement of thought, and what mainly interests us is that in him we see a
layman, alike indifferent to clerisy and heresy, giving energetic and
pointed utterance to those opinions of his class which the clergy are
content to ignore so long as they remain esoteric. At present the world
has advanced to where Lessing stood, while the Church has done its best
to stand stock-still; and it would be a curious were it not a melancholy
spectacle, to see the indifference with which the laity look on while
theologians thrash their wheatless straw, utterly unconscious that there
is no longer any common term possible that could bring their creeds again
to any point of bearing on the practical life of men. Fielding never made
a profounder stroke of satire than in Squire Western's indignant "Art not
in the pulpit now! When art got up there, I never mind what dost say."
As an author, Lessing began his career at a period when we cannot say
that German literature was at its lowest ebb, only because there had not
yet been any flood-tide. That may be said to have begun with him. When we
say German literature, we mean so much of it as has any interest outside
of Germany. That part of the literary histories which treats of the dead
waste and middle of the eighteenth century reads like a collection of
obituaries, and were better reduced to the conciseness of epitaph, though
the authors of them seem to find a melancholy pleasure, much like that of
undertakers, in the task by which they live. Gottsched reigned supreme on
the legitimate throne of dulness. In Switzerland, Bodmer essayed a more
republican form of the same authority. At that time a traveller reports
eight hundred authors in Zuerich alone! Young aspirant for lettered fame,
in imagination clear away the lichens from their forgotten headstones,
and read humbly the "As I am, so thou must be," on all! Everybody
remembers how Goethe, in the seventh book of his autobiography, tells the
story of his visit to Gottsched. He enters by mistake an inner room at
the moment when a frightened servant brings the discrowned potentate a
periwig large enough to reach to the elbows. That awful emblem of
pretentious sham seems to be the best type of the literature then
predominant. We always fancy it set upon a pole, like Gessler's hat, with
nothing in it that was not wooden, for all men to bow down before. The
periwig style had its natural place in the age of Louis XIV., and there
were certainly brains under it. But it had run out in France, as the
tie-wig style of Pope had in England. In Germany it was the mere
imitation of an imitation. Will it be believed that Gottsched recommends
his Art of Poetry to beginners, in preference to Breitinger's, because it
"_will enable them to produce every species of poem in a correct style_,
while out of that no one can learn to make an ode or a cantata"?
"Whoever," he says, "buys Breitinger's book _in order to learn how to
make poems_, will too late regret his money."[159] Gottsched, perhaps,
did some service even by his advocacy of French models, by calling
attention to the fact that there _was_ such a thing as style, and that it
was of some consequence. But not one of the authors of that time can be
said to survive, nor to be known even by name except to Germans, unless
it be Klopstock, Herder, Wieland, and Gellert. And the latter's
immortality, such as it is, reminds us somewhat of that Lady Gosling's,
whose obituary stated that she was "mentioned by Mrs. Barbauld in her
Life of Richardson 'under the name of Miss M., afterwards Lady G.'"
Klopstock himself is rather remembered for what he was than what he
is,--an immortality of unreadableness; and we much doubt if many Germans
put the "Oberon" in their trunks when they start on a journey. Herder
alone survives, if not as a contributor to literature, strictly so
called, yet as a thinker and as part of the intellectual impulse of the
day. But at the time, though there were two parties, yet within the lines
of each there was a loyal reciprocity of what is called on such occasions
appreciation. Wig ducked to wig, each blockhead had a brother, and there
was a universal apotheosis of the mediocrity of our set. If the greatest
happiness of the greatest number be the true theory, this was all that
could be desired. Even Lessing at one time looked up to Hagedorn as the
German Horace. If Hagedorn were pleased, what mattered it to Horace?
Worse almost than this was the universal pedantry. The solemn bray of one
pedagogue was taken up and prolonged in a thousand echoes. There was not
only no originality, but no desire for it,--perhaps even a dread of it,
as something that would break the _entente cordiale_ of placid mutual
assurance. No great writer had given that tone of good-breeding to the
language which would gain it entrance to the society of European
literature. No man of genius had made it a necessity of polite culture.
It was still as rudely provincial as the Scotch of Allan Ramsay.
Frederick the Great was to be forgiven if, with his practical turn, he
gave himself wholly to French, which had replaced Latin as a cosmopolitan
tongue. It had lightness, ease, fluency, elegance,--in short, all the
good qualities that German lacked. The study of French models was perhaps
the best thing for German literature before it got out of long-clothes.
It was bad only when it became a tradition and a tyranny. Lessing did
more than any other man to overthrow this foreign usurpation when it had
done its work.
The same battle had to be fought on English soil also, and indeed is
hardly over yet. For the renewed outbreak of the old quarrel between
Classical and Romantic grew out of nothing more than an attempt of the
modern spirit to free itself from laws of taste laid down by the _Grand
Siecle_. But we must not forget the debt which all modern prose
literature owes to France. It is true that Machiavelli was the first to
write with classic pith and point in a living language; but he is, for
all that, properly an ancient. Montaigne is really the first modern
writer,--the first who assimilated his Greek and Latin, and showed that
an author might be original and charming, even classical, if he did not
try too hard. He is also the first modern critic, and his judgments of
the writers of antiquity are those of an equal. He made the ancients his
servants, to help him think in Gascon French; and, in spite of his
endless quotations, began the crusade against pedantry. It was not,
however, till a century later, that the reform became complete in France,
and then crossed the Channel. Milton is still a pedant in his prose, and
not seldom even in his great poem. Dryden was the first Englishman who
wrote perfectly easy prose, and he owed his style and turn of thought to
his French reading. His learning sits easily on him, and has a modern
cut. So far, the French influence was one of unmixed good, for it rescued
us from pedantry. It must have done something for Germany in the same
direction. For its effect on poetry we cannot say as much; and its
traditions had themselves become pedantry in another shape when Lessing
made an end of it. He himself certainly learned to write prose of
Diderot; and whatever Herr Stahr may think of it, his share in the
"Letters on German Literature" got its chief inspiration from France.
It is in the _Dramaturgie_ that Lessing first properly enters as an
influence into European literature. He may be said to have begun the
revolt from pseudo-classicism in poetry, and to have been thus
unconsciously the founder of romanticism. Wieland's translation of
Shakespeare had, it is true, appeared in 1762; but Lessing was the first
critic whose profound knowledge of the Greek drama and apprehension of
its principles gave weight to his judgment, who recognized in what the
true greatness of the poet consisted, and found him to be really nearer
the Greeks than any other modern. This was because Lessing looked always
more to the life than the form,--because he knew the classics, and did
not merely cant about them. But if the authority of Lessing, by making
people feel easy in their admiration for Shakespeare, perhaps increased
the influence of his works, and if his discussions of Aristotle have
given a new starting-point to modern criticism, it may be doubted whether
the immediate effect on literature of his own critical essays was so
great as Herr Stahr supposes. Surely "Goetz" and "The Robbers" are nothing
like what he would have called Shakespearian, and the whole _Sturm und
Drang_ tendency would have roused in him nothing but antipathy. Fixed
principles in criticism are useful in helping us to form a judgment of
works already produced, but it is questionable whether they are not
rather a hindrance than a help to living production. Ben Jonson was a
fine critic, intimate with the classics as few men have either the
leisure or the strength of mind to be in this age of many books, and
built regular plays long before they were heard of in France. But he
continually trips and falls flat over his metewand of classical
propriety, his personages are abstractions, and fortunately neither his
precepts nor his practice influenced any one of his greater coevals.[160]
In breadth of understanding, and the gravity of purpose that comes of it,
he was far above Fletcher or Webster, but how far below either in the
subtler, the incalculable, qualities of a dramatic poet! Yet Ben, with
his principles off, could soar and sing with the best of them; and there
are strains in his lyrics which Herrick, the most Catullian of poets
since Catullus, could imitate, but never match. A constant reference to
the statutes which taste has codified would only bewilder the creative
instinct. Criticism can at best teach writers without genius what is to
be avoided or imitated. It cannot communicate life; and its effect, when
reduced to rules, has commonly been to produce that correctness which is
so praiseworthy and so intolerable. It cannot give taste, it can only
demonstrate who has had it. Lessing's essays in this kind were of service
to German literature by their manliness of style, whose example was worth
a hundred treatises, and by the stimulus there is in all original
thinking. Could he have written such a poem as he was capable of
conceiving, his influence would have been far greater. It is the living
soul, and not the metaphysical abstraction of it, that is genetic in
literature. If to do were as easy as to know what were good to be done!
It was out of his own failures to reach the ideal he saw so clearly, that
Lessing drew the wisdom which made him so admirable a critic. Even here,
too, genius can profit by no experience but its own.
For, in spite of Herr Stahr's protest, we must acknowledge the truth of
Lessing's own characteristic confession, that he was no poet. A man of
genius he unquestionably was, if genius may be claimed no less for force
than fineness of mind,--for the intensity of conviction that inspires the
understanding as much as for that apprehension of beauty which gives
energy of will to imagination,--but a poetic genius he was not. His mind
kindled by friction in the process of thinking, not in the flash of
conception, and its delight is in demonstration, not in bodying forth.
His prose can leap and run, his verse is always thinking of its feet. Yet
in his "Minna" and his "Emilia"[161] he shows one faculty of the
dramatist, that of construction, in a higher degree than any other
German.[162] Here his critical deductions served him to some purpose. The
action moves rapidly, there is no speechifying, and the parts are
coherent. Both plays act better than anything of Goethe or Schiller. But
it is the story that interests us, and not the characters. These are not,
it is true, the incorporation of certain ideas, or, still worse, of
certain dogmas, but they certainly seem something like machines by which
the motive of the play is carried on; and there is nothing of that
interplay of plot and character which makes Shakespeare more real in the
closet than other dramatists with all the helps of the theatre. It is a
striking illustration at once of the futility of mere critical insight
and of Lessing's want of imagination, that in the Emilia he should have
thought a Roman motive consistent with modern habits of thought, and that
in Nathan he should have been guilty of anachronisms which violate not
only the accidental truth of fact, but the essential truth of character.
Even if we allowed him imagination, it must be only on the lower plane of
prose; for of verse as anything more than so many metrical feet he had
not the faintest notion. Of that exquisite sympathy with the movement of
the mind, with every swifter or slower pulse of passion, which proves it
another species from prose, the very [Greek: aphroditae kai lura] of
speech, and not merely a higher one, he wanted the fineness of sense to
conceive. If we compare the prose of Dante or Milton, though both were
eloquent, with their verse, we see at once which was the most congenial
to them. Lessing has passages of freer and more harmonious utterance in
some of his most careless prose essays, than can be found in his Nathan
from the first line to the last. In the _numeris lege solutis_ he is
often snatched beyond himself, and becomes truly dithyrambic; in his
pentameters the march of the thought is comparatively hampered and
irresolute. His best things are not poetically delicate, but have the
tougher fibre of proverbs. Is it not enough, then, to be a great
prose-writer? They are as rare as great poets, and if Lessing have the
gift to stir and to dilate that something deeper than the mind which
genius only can reach, what matter if it be not done to music? Of his
minor poems we need say little. Verse was always more or less mechanical
with him, and his epigrams are almost all stiff, as if they were bad
translations from the Latin. Many of them are shockingly coarse, and in
liveliness are on a level with those of our Elizabethan period. Herr
Stahr, of course, cannot bear to give them up, even though Gervinus be
willing. The prettiest of his shorter poems (_Die Namen_)has been
appropriated by Coleridge, who has given it a grace which it wants in the
original. His Nathan, by a poor translation of which he is chiefly known
to English readers, is an Essay on Toleration in the form of a dialogue.
As a play, it has not the interest of Minna or Emilia, though the
Germans, who have a praiseworthy national stoicism where one of their
great writers is concerned, find in seeing it represented a grave
satisfaction, like that of subscribing to a monument. There is a sober
lustre of reflection in it that makes it very good reading; but it wants
the molten interfusion of thought and phrase which only imagination can
achieve.
As Lessing's mind was continually advancing,--always open to new
impressions, and capable, as very few are, of apprehending the
many-sidedness of truth,--as he had the rare quality of being honest with
himself,--his works seem fragmentary, and give at first an impression of
incompleteness. But one learns at length to recognize and value this very
incompleteness as characteristic of the man who was growing lifelong, and
to whom the selfish thought that any share of truth could be exclusively
_his_ was an impossibility. At the end of the ninety-fifth number of the
_Dramaturgie_ he says: "I remind my readers here, that these pages are by
no means intended to contain a dramatic system. I am accordingly not
bound to solve all the difficulties which I raise. I am quite willing
that my thoughts should seem to want connection,--nay, even to contradict
each other,--if only there are thoughts in which they [my readers] find
material for thinking themselves. I wish to do nothing more than scatter
the _fermenta cognitionis_." That is Lessing's great praise, and gives
its chief value to his works,--a value, indeed, imperishable, and of the
noblest kind. No writer can leave a more precious legacy to posterity
than this; and beside this shining merit, all mere literary splendors
look pale and cold. There is that life in Lessing's thought which
engenders life, and not only thinks for us, but makes us think. Not
sceptical, but forever testing and inquiring, it is out of the cloud of
his own doubt that the flash comes at last with sudden and vivid
illumination. Flashes they indeed are, his finest intuitions, and of very
different quality from the equable north-light of the artist. He felt it,
and said it of himself, "Ever so many flashes of lightning do not make
daylight." We speak now of those more rememberable passages where his
highest individuality reveals itself in what may truly be called a
passion of thought. In the "Laocooen" there is daylight of the serenest
temper, and never was there a better example of the discourse of reason,
though even that is also a fragment.
But it is as a nobly original man, even more than as an original thinker,
that Lessing is precious to us, and that he is so considerable in German
literature. In a higher sense, but in the same kind, he is to Germans
what Dr. Johnson is to us,--admirable for what he was. Like Johnson's,
too, but still from a loftier plane, a great deal of his thought has a
direct bearing on the immediate life and interests of men. His genius was
not a St. Elmo's fire, as it so often is with mere poets,--as it was in
Shelley, for example, playing in ineffectual flame about the points of
his thought,--but was interfused with his whole nature and made a part of
his very being. To the Germans, with their weak nerve of sentimentalism,
his brave common-sense is a far wholesomer tonic than the cynicism of
Heine, which is, after all, only sentimentalism soured. His jealousy for
maintaining the just boundaries whether of art or speculation may warn
them to check with timely dikes the tendency of their thought to diffuse
inundation. Their fondness in aesthetic discussion for a nomenclature
subtile enough to split a hair at which even a Thomist would have
despaired, is rebuked by the clear simplicity of his style.[163] But he
is no exclusive property of Germany. As a complete man, constant,
generous, full of honest courage, as a hardy follower of Thought wherever
she might lead him, above all, as a confessor of that Truth which is
forever revealing itself to the seeker, and is the more loved because
never wholly revealable, he is an ennobling possession of mankind. Let
his own striking words characterize him:--
"Not the truth of which any one is, or supposes himself to be, possessed,
but the upright endeavor he has made to arrive at truth, makes the worth
of the man. For not by the possession, but by the investigation, of truth
are his powers expanded, wherein alone his ever-growing perfection
consists. Possession makes us easy, indolent, proud.
"If God held all truth shut in his right hand, and in his left nothing
but the ever-restless instinct for truth, though with the condition of
for ever and ever erring, and should say to me, Choose! I should bow
humbly to his left hand, and say, Father, give! pure truth is for Thee
alone!"
It is not without reason that fame is awarded only after death. The
dust-cloud of notoriety which follows and envelopes the men who drive
with the wind bewilders contemporary judgment. Lessing, while he lived,
had little reward for his labor but the satisfaction inherent in all work
faithfully done; the highest, no doubt, of which human nature is capable,
and yet perhaps not so sweet as that sympathy of which the world's praise
is but an index. But if to perpetuate herself beyond the grave in healthy
and ennobling influences be the noblest aspiration of the mind, and its
fruition the only reward she would have deemed worthy of herself, then is
Lessing to be counted thrice fortunate. Every year since he was laid
prematurely in the earth has seen his power for good increase, and made
him more precious to the hearts and intellects of men. "Lessing," said
Goethe, "would have declined the lofty title of a Genius; but his
enduring influence testifies against himself. On the other hand, we have
in literature other and indeed important names of men who, while they
lived, were esteemed great geniuses, but whose influence ended with their
lives, and who, accordingly, were less than they and others thought. For,
as I have said, there is no genius without a productive power that
continues forever operative."[164]
Footnotes:
[147] G. E. Lessing. _Sein Leben und seine Werke_. Von Adolf Stahr.
Vermehrte und verbesserte Volks-Ausgabe. Dritte Auflage Berlin. 1864.
_The Same_. Translated by E. P. Evans, Ph. D., Professor, &c. in the
University of Michigan. Boston: W. V. Spencer. 1866. 2 vols.
G. E. Lessing's Saemmtliche Schriften, herausgegeben von Karl
Lachmann. 1853-57. 12 Baende.
[148] "If I write at all, it is not possible for me to write
otherwise than just as I think and feel."--Lessing to his father,
21st December, 1767.
[149] "I am sure that Kleist would rather have taken another wound
with him into his grave than have such stuff jabbered over him (_sich
solch Zeug nachschwatzen lassen_)." Lessing to Gleim, 6th September
1759.
[150] Letter to Klotz, 9th June, 1766.
[151] Herr Stahr heads the fifth chapter of his Second Book, "Lessing
at Wittenberg. December, 1751, to November, 1752." But we never feel
quite sure of his dates. The Richier affair puts Lessing in Berlin in
December, 1751, and he took his Master's degree at Wittenberg, 29th
April, 1752. We are told that he finally left Wittenberg "toward the
end" of that year. He himself, writing from Berlin in 1754, says that
he has been absent from that city _nur ein halbes Jahr_ since 1748.
There is only one letter for 1762, dated at Wittenberg, 9th June.
[152] "Ramler," writes Georg Forster, "ist die Ziererei, die
Eigenliebe die Eitelkeit in eigener Person."
[153] Lessing to Von Murr, 25th November, 1768. The whole letter is
well worth reading.
[154] A favorite phrase of his, which Egbert has preserved for us
with its Saxon accent, was, _Es kommt doch nischt dabey heraus_,
implying that one might do something better for a constancy than
shearing twine.
[155] I find surprisingly little about Lessing in such of the
contemporary correspondence of German literary men as I have read. A
letter of Boie to Merck (10 April, 1775) gives us a glimpse of him.
"Do you know that Lessing will probably marry Reiske's widow and come
to Dresden in place of Hagedorn? The restless spirit! How he will get
along with the artists, half of them, too, Italians, is to be
seen.... Liffert and he have met and parted good friends. He has worn
ever since on his finger the ring with the skeleton and butterfly
which Liffert gave him. He is reported to be much dissatisfied with
the theatrical filibustering of Goethe and Lenz, especially with the
remarks on the drama in which so little respect is shown for his
Aristotle, and the Leipzig folks are said to be greatly rejoiced at
getting such an ally."
[156] To his brother Karl, 20th April, 1774.
[157] To the same, 20th March, 1777.
[158] To the same, 2d February, 1774.
[159] Gervinus, IV. 62.
[160] It should be considered, by those sagacious persons who think
that the most marvellous intellect of which we have any record could
not master so much Latin and Greek as would serve a sophomore, that
Shakespeare must through conversation have possessed himself of
whatever principles of art Ben Jonson and the other university men
had been able to deduce from their study of the classics. That they
should not have discussed these matters over their sack at the
Mermaid is incredible; that Shakespeare, who left not a drop in any
orange he squeezed, could not also have got all the juice out of this
one, is even more so.
[161] In "Minna" and "Emilia" Lessing followed the lead of Diderot.
In the Preface to the second edition of Diderot's _Theatre_, he says:
"I am very conscious that my taste, without Diderot's example and
teaching, would have taken quite another direction. Perhaps one more
my own, yet hardly one with which my understanding would in the long
run have been so well content." Diderot's choice of prose was
dictated and justified by the accentual poverty of his mother-tongue,
Lessing certainly revised his judgment on this point (for it was not
equally applicable to German), and wrote his maturer "Nathan" in what
he took for blank verse. There was much kindred between the minds of
the two men. Diderot always seems to us a kind of deboshed Lessing.
Lessing was also indebted to Burke, Hume, the two Wartons, and Hurd,
among other English writers. Not that he borrowed anything of them
but the quickening of his own thought. It should be remembered that
Rousseau was seventeen, Diderot and Sterne sixteen, and Winckelmann
twelve years older than Lessing. Wieland was four years younger.
[162] Goethe's appreciation of Lessing grew with his years. He writes
to Lavater, 18th March, 1781: "Lessing's death has greatly depressed
me. I had much pleasure in him and much hope of him." This is a
little patronizing in tone. But in the last year of his life, talking
with Eckermann, he naturally antedates his admiration, as
reminiscence is wont to do: "You can conceive what an effect this
piece (_Minna_)had upon us young people. It was, in fact, a shining
meteor. It made us aware that something higher existed than anything
whereof that feeble literary epoch had a notion. The first two acts
are truly a masterpiece of exposition, from which one learned much
and can always learn."
[163] Nothing can be droller than the occasional translation by
Vischer of a sentence of Lessing into his own jargon.
[164] Eckermann, Gespraeche mit Goethe, III. 229.
ROUSSEAU AND THE SENTIMENTALISTS.[165]
"We have had the great professor and founder of the philosophy of Vanity
in England. As I had good opportunities of knowing his proceedings almost
from day to day, he left no doubt in my mind that he entertained no
principle either to influence his heart or to guide his understanding but
vanity; with this vice he was possessed to a degree little short of
madness. Benevolence to the whole species, and want of feeling for every
individual with whom the professors come in contact, form the character
of the new philosophy. Setting up for an unsocial independence, this
their hero of vanity refuses the just price of common labor, as well as
the tribute which opulence owes to genius, and which, when paid, honors
the giver and the receiver, and then pleads his beggary as an excuse for
his crimes. He melts with tenderness for those only who touch him by the
remotest relation, and then, without one natural pang, casts away, as a
sort of offal and excrement, the spawn of his disgustful amours, and
sends his children to the hospital of foundlings. The bear loves, licks,
and forms her young, but bears are not philosophers."
This was Burke's opinion of the only contemporary who can be said to
rival him in fervid and sustained eloquence, to surpass him in grace and
persuasiveness of style. Perhaps we should have been more thankful to him
if he had left us instead a record of those "proceedings almost from day
to day" which he had such "good opportunities of knowing," but it
probably never entered his head that posterity might care as much about
the doings of the citizen of Geneva as about the sayings of even a
British Right Honorable. Vanity eludes recognition by its victims in more
shapes, and more pleasing, than any other passion, and perhaps had Mr.
Burke been able imaginatively to translate Swiss Jean Jacques into Irish
Edmund, he would have found no juster equivalent for the obnoxious
trisyllable than "righteous self-esteem." For Burke was himself also, in
the subtler sense of the word, a sentimentalist, that is, a man who took
what would now be called an aesthetic view of morals and politics. No man
who ever wrote English, except perhaps Mr. Ruskin, more habitually
mistook his own personal likes and dislikes, tastes and distastes, for
general principles, and this, it may be suspected, is the secret of all
merely eloquent writing. He hints at madness as an explanation of
Rousseau, and it is curious enough that Mr. Buckle was fain to explain
_him_ in the same way. It is not, we confess, a solution that we find
very satisfactory in this latter case. Burke's fury against the French
Revolution was nothing more than was natural to a desperate man in
self-defence. It was his own life, or, at least, all that made life dear
to him, that was in danger. He had all that abstract political wisdom
which may be naturally secreted by a magnanimous nature and a sensitive
temperament, absolutely none of that rough-and-tumble kind which is so
needful for the conduct of affairs. Fastidiousness is only another form
of egotism; and all men who know not where to look for truth save in the
narrow well of self will find their own image at the bottom, and mistake
it for what they are seeking. Burke's hatred of Rousseau was genuine and
instinctive. It was so genuine and so instinctive as no hatred can be but
that of self, of our own weaknesses as we see them in another man. But
there was also something deeper in it than this. There was mixed with it
the natural dread in the political diviner of the political logician,--in
the empirical, of the theoretic statesman. Burke, confounding the idea of
society with the form of it then existing, would have preserved that as
the only specific against anarchy. Rousseau, assuming that society as it
then existed was but another name for anarchy, would have reconstituted
it on an ideal basis. The one has left behind him some of the profoundest
aphorisms of political wisdom; the other, some of the clearest principles
of political science. The one, clinging to Divine right, found in the
fact that things were, a reason that they ought to be; the other, aiming
to solve the problem of the Divine order, would deduce from that
abstraction alone the claim of anything to be at all. There seems a mere
oppugnancy of nature between the two, and yet both were, in different
ways, the dupes of their own imaginations.
Now let us hear the opinion of a philosopher who _was_ a bear, whether
bears be philosophers or not. Boswell had a genuine relish for what was
superior in any way, from genius to claret, and of course he did not let
Rousseau escape him. "One evening at the Mitre, Johnson said
sarcastically to me, 'It seems, sir, you have kept very good company
abroad,--Rousseau and Wilkes!' I answered with a smile, 'My dear sir, you
don't call Rousseau bad company; do you really think _him_ a bad man?'
Johnson: 'Sir, if you are talking jestingly of this, I don't talk with
you. If you mean to be serious, I think him one of the worst of men, a
rascal who ought to be hunted out of society, as he has been. Three or
four nations have expelled him, and it is a shame that he is protected in
this country. Rousseau, sir, is a very bad man. I would sooner sign a
sentence for his transportation, than that of any felon who has gone from
the Old Bailey these many years. Yes, I should like to have him work in
the plantations.'" _We_ were the plantations then, and Rousseau was
destined to work there in another and much more wonderful fashion than
the gruff old Ursa Major imagined. However, there is always a refreshing
heartiness in his growl, a masculine bass with no snarl in it. The
Doctor's logic is of that fine old crusted Port sort, the native
manufacture of the British conservative mind. Three or four nations
_have_, therefore England ought. A few years later, had the Doctor been
living, if three or four nations had treated their kings as France did
hers, would he have thought the _ergo_ a very stringent one for England?
Mr. Burke, who could speak with studied respect of the Prince of Wales,
and of his vices with that charity which thinketh no evil and can afford
to think no evil of so important a living member of the British
Constitution, surely could have had no unmixed moral repugnance for
Rousseau's "disgustful amours." It was because they were _his_ that they
were so loathsome. Mr. Burke was a snob, though an inspired one. Dr.
Johnson, the friend of that wretchedest of lewd fellows, Richard Savage,
and of that gay man about town, Topham Beauclerk,--himself sprung from an
amour that would have been disgustful had it not been royal,--must also
have felt something more in respect of Rousseau than the mere repugnance
of virtue for vice. We must sometimes allow to personal temperament its
right of peremptory challenge. Johnson had not that fine sensitiveness to
the political atmosphere which made Burke presageful of coming tempest,
but both of them felt that there was something dangerous in this man.
Their dislike has in it somewhat of the energy of fear. Neither of them
had the same feeling toward Voltaire, the man of supreme talent, but both
felt that what Rousseau was possessed by was genius, with its terrible
force either to attract or repel.
"By the pricking of my thumbs,
Something wicked this way comes."
Burke and Johnson were both of them sincere men, both of them men of
character as well as of intellectual force; and we cite their opinions of
Rousseau with the respect which is due to an honest conviction which has
apparent grounds for its adoption, whether we agree with it or no. But it
strikes us as a little singular that one whose life was so full of moral
inconsistency, whose character is so contemptible in many ways, in some
we might almost say so revolting, should yet have exercised so deep and
lasting an influence, and on minds so various, should still be an object
of minute and earnest discussion,--that he should have had such vigor in
his intellectual loins as to have been the father of Chateaubriand,
Byron, Lamartine, George Sand, and many more in literature, in politics
of Jefferson and Thomas Paine,--that the spots he had haunted should draw
pilgrims so unlike as Gibbon and Napoleon, nay, should draw them still,
after the lapse of near a century. Surely there must have been a basis of
sincerity in this man seldom matched, if it can prevail against so many
reasons for repugnance, aversion, and even disgust. He could not have
been the mere sentimentalist and rhetorician for which the
rough-and-ready understanding would at first glance be inclined to
condemn him. In a certain sense he was both of these, but he was
something more. It will bring us a little nearer the point we are aiming
at if we quote one other and more recent English opinion of him.
Mr. Thomas Moore, returning pleasantly in a travelling-carriage from a
trip to Italy, in which he had never forgotten the poetical shop at home,
but had carefully noted down all the pretty images that occurred to him
for future use,--Mr. Thomas Moore, on his way back from a visit to his
noble friend Byron, at Venice, who had there been leading a life so gross
as to be talked about, even amid the crash of Napoleon's fall, and who
was just writing "Don Juan" for the improvement of the world,--Mr. Thomas
Moore, fresh from the reading of Byron's Memoirs, which were so
scandalous that, by some hocus-pocus, three thousand guineas afterward
found their way into his own pocket for consenting to suppress them,--Mr.
Thomas Moore, the _ci-devant_ friend of the Prince Regent, and the author
of Little's Poems, among other objects of pilgrimage visits _Les
Charmettes_, where Rousseau had lived with Madame de Warens. So good an
opportunity for occasional verses was not to be lost, so good a text for
a little virtuous moralizing not to be thrown away; and accordingly Mr.
Moore pours out several pages of octosyllabic disgust at the sensuality
of the dead man of genius. There was no horror for Byron. Toward him all
was suavity and decorous _bienseance_. That lively sense of benefits to
be received made the Irish Anacreon wink with both his little eyes. In
the judgment of a liberal like Mr. Moore, were not the errors of a lord
excusable? But with poor Rousseau the case was very different. The son of
a watchmaker, an outcast from boyhood up, always on the perilous edge of
poverty,--what right had he to indulge himself in any immoralities? So it
is always with the sentimentalists. It is never the thing in itself that
is bad or good, but the thing in its relation to some conventional and
mostly selfish standard. Moore could be a moralist, in this case, without
any trouble, and with the advantage of winning Lord Lansdowne's approval;
he could write some graceful verses which everybody would buy, and for
the rest it is not hard to be a stoic in eight-syllable measure and a
travelling-carriage. The next dinner at Bowood will taste none the worse.
Accordingly he speaks of
"The mire, the strife
And vanities of this man's life,
Who more than all that e'er have glowed
With fancy's flame (and it was his
In fullest warmth and radiance) showed
What an impostor Genius is;
How, with that strong mimetic art
Which forms its life and soul, it takes
All shapes of thought, all hues of heart,
Nor feels itself one throb it wakes;
How, like a gem, its light may shine,
O'er the dark path by mortals trod,
Itself as mean a worm the while
As crawls at midnight o'er the sod;
* * * * *
How, with the pencil hardly dry
From coloring up such scenes of love
And beauty as make young hearts sigh,
And dream and think through heaven they rove," &c., &c.
Very spirited, is it not? One has only to overlook a little
threadbareness in the similes, and it is very good oratorical verse. But
would we believe in it, we must never read Mr. Moore's own journal, and
find out how thin a piece of veneering his own life was,--how he lived in
sham till his very nature had become subdued to it, till he could
persuade himself that a sham could be written into a reality, and
actually made experiment thereof in his Diary.
One verse in this diatribe deserves a special comment,--
"What an impostor Genius is!"
In two respects there is nothing to be objected to in it. It is of eight
syllables, and "is" rhymes unexceptionably with "his." But is there the
least filament of truth in it? We venture to assert, not the least. It
was not Rousseau's genius that was an impostor. It was the one thing in
him that was always true. We grant that, in allowing that a man has
genius. Talent is that which is in a man's power; genius is that in whose
power a man is. That is the very difference between them. We might turn
the tables on Moore, the man of talent, and say truly enough, What an
impostor talent is! Moore talks of the mimetic power with a total
misapprehension of what it really is. The mimetic power had nothing
whatever to do with the affair. Rousseau had none of it; Shakespeare had
it in excess; but what difference would it make in our judgment of Hamlet
or Othello if a manuscript of Shakespeare's memoirs should turn up, and
we should find out that he had been a pitiful fellow? None in the world;
for he is not a professed moralist, and his life does not give the
warrant to his words. But if Demosthenes, after all his Philippies,
throws away his shield and runs, we feel the contemptibleness of the
contradiction. With genius itself we never find any fault. It would be an
over-nicety that would do that. We do not get invited to nectar and
ambrosia so often that we think of grumbling and saying we have better at
home. No; the same genius that mastered him who wrote the poem masters us
in reading it, and we care for nothing outside the poem itself. How the
author lived, what he wore, how he looked,--all that is mere gossip,
about which we need not trouble ourselves. Whatever he was or did,
somehow or other God let him be worthy to write _this_, and that is
enough for us. We forgive everything to the genius; we are inexorable to
the man. Shakespeare, Goethe, Burns,--what have their biographies to do
with us? Genius is not a question of character. It may be sordid, like
the lamp of Aladdin, in its externals; what care we, while the touch of
it builds palaces for us, makes us rich as only men in dream-land are
rich, and lords to the utmost bound of imagination? So, when people talk
of the ungrateful way in which the world treats its geniuses, they speak
unwisely. There is no work of genius which has not been the delight of
mankind, no word of genius to which the human heart and soul have not,
sooner or later, responded. But the man whom the genius takes possession
of for its pen, for its trowel, for its pencil, for its chisel, _him_ the
world treats according to his deserts. Does Burns drink? It sets him to
gauging casks of gin. For, remember, it is not to the practical world
that the genius appeals; it _is_ the practical world which judges of the
man's fitness for its uses, and has a right so to judge. No amount of
patronage could have made distilled liquors less toothsome to Robbie
Burns, as no amount of them could make a Burns of the Ettrick Shepherd.
There is an old story in the _Gesta Romanorum_ of a priest who was found
fault with by one of his parishioners because his life was in painful
discordance with his teaching. So one day he takes his critic out to a
stream, and, giving him to drink of it, asks him if he does not find it
sweet and pure water. The parishioner, having answered that it was, is
taken to the source, and finds that what had so refreshed him flowed from
between the jaws of a dead dog. "Let this teach thee," said the priest,
"that the very best doctrine may take its rise in a very impure and
disgustful spring, and that excellent morals may be taught by a man who
has no morals at all." It is easy enough to see the fallacy here. Had the
man known beforehand from what a carrion fountain-head the stream issued,
he could not have drunk of it without loathing. Had the priest merely
bidden him to _look_ at the stream and see how beautiful it was, instead
of tasting it, it would have been quite another matter. And this is
precisely the difference between what appeals to our aesthetic and to our
moral sense, between what is judged of by the taste and the conscience.
It is when the sentimentalist turns preacher of morals that we
investigate his character, and are justified in so doing. He may express
as many and as delicate shades of feeling as he likes,--for this the
sensibility of his organization perfectly fits him, no other person could
do it so well,--but the moment he undertakes to establish his feeling as
a rule of conduct, we ask at once how far are his own life and deed in
accordance with what he preaches? For every man feels instinctively that
all the beautiful sentiments in the world weigh less than a single lovely
action; and that while tenderness of feeling and susceptibility to
generous emotions are accidents of temperament, goodness is an
achievement of the will and a quality of the life. Fine words, says our
homely old proverb, butter no parsnips; and if the question be how to
render those vegetables palatable, an ounce of butter would be worth more
than all the orations of Cicero. The only conclusive evidence of a man's
sincerity is that he give _himself_ for a principle. Words, money, all
things else, are comparatively easy to give away; but when a man makes a
gift of his daily life and practice, it is plain that the truth, whatever
it may be, has taken possession of him. From that sincerity his words
gain the force and pertinency of deeds, and his money is no longer the
pale drudge 'twixt man and man, but, by a beautiful magic, what erewhile
bore the image and superscription of Caesar seems now to bear the image
and superscription of God. It is thus that there is a genius for
goodness, for magnanimity, for self-sacrifice, as well as for creative
art; and it is thus that by a more refined sort of Platonism the Infinite
Beauty dwells in and shapes to its own likeness the soul which gives it
body and individuality. But when Moore charges genius with being an
impostor, the confusion of his ideas is pitiable. There is nothing so
true, so sincere, so downright and forthright, as genius. It is always
truer than the man himself is, greater than he. If Shakespeare the man
had been as marvellous a creature as the genius that wrote his plays,
that genius so comprehensive in its intelligence, so wise even in its
play, that its clowns are moralists and philosophers, so penetrative that
a single one of its phrases reveals to us the secret of our own
character, would his contemporaries have left us so wholly without record
of him as they have done, distinguishing him in no wise from his
fellow-players?
Rousseau, no doubt, was weak, nay, more than that, was sometimes
despicable, but yet is not fairly to be reckoned among the herd of
sentimentalists. It is shocking that a man whose preaching made it
fashionable for women of rank to nurse their own children should have
sent his own, as soon as born, to the foundling hospital, still more
shocking that, in a note to his _Discours sur l'Inegalite_, he should
speak of this crime as one of the consequences of our social system. But
for all that there was a faith and an ardor of conviction in him that
distinguish him from most of the writers of his time. Nor were his
practice and his preaching always inconsistent. He contrived to pay
regularly, whatever his own circumstances were, a pension of one hundred
_livres_ a year to a maternal aunt who had been kind to him in childhood.
Nor was his asceticism a sham. He might have turned his gift into laced
coats and _chateaux_ as easily as Voltaire, had he not held it too sacred
to be bartered away in any such losing exchange.
But what is worthy of especial remark is this,--that in nearly all that
he wrote his leading object was the good of his kind, and that through
all the vicissitudes of a life which illness, sensibility of temperament,
and the approaches of insanity rendered wretched,--the associate of
infidels, the foundling child, as it were, of an age without belief,
least of all in itself,--he professed and evidently felt deeply a faith
in the goodness both of man and of God. There is no such thing as
scoffing in his writings. On the other hand, there is no stereotyped
morality. He does not ignore the existence of scepticism; he recognizes
its existence in his own nature, meets it frankly face to face, and makes
it confess that there are things in the teaching of Christ that are
deeper than its doubt. The influence of his early education at Geneva is
apparent here. An intellect so acute as his, trained in the school of
Calvin in a republic where theological discussion was as much the
amusement of the people as the opera was at Paris, could not fail to be a
good logician. He had the fortitude to follow his logic wherever it led
him. If the very impressibility of character which quickened his
perception of the beauties of nature, and made him alive to the charm of
music and musical expression, prevented him from being in the highest
sense an original writer, and if his ideas were mostly suggested to him
by books, yet the clearness, consecutiveness, and eloquence with which he
stated and enforced them made them his own. There was at least that
original fire in him which could fuse them and run them in a novel mould.
His power lay in this very ability of manipulating the thoughts of
others. Fond of paradox he doubtless was, but he had a way of putting
things that arrested attention and excited thought. It was, perhaps, this
very sensibility of the surrounding atmosphere of feeling and
speculation, which made Rousseau more directly influential on
contemporary thought (or perhaps we should say sentiment) than any writer
of his time. And this is rarely consistent with enduring greatness in
literature. It forces us to remember, against our will, the oratorical
character of his works. They were all pleas, and he a great advocate,
with Europe in the jury-box. Enthusiasm begets enthusiasm, eloquence
produces conviction for the moment, but it is only by truth to nature and
the everlasting intuitions of mankind that those abiding influences are
won that enlarge from generation to generation. Rousseau was in many
respects--as great pleaders always are--a man of the day, who must needs
become a mere name to posterity, yet he could not but have had in him
some not inconsiderable share of that principle by which man eternizes
himself. For it is only to such that the night cometh not in which no man
shall work, and he is still operative both in politics and literature by
the principles he formulated or the emotions to which he gave a voice so
piercing and so sympathetic.
In judging Rousseau, it would be unfair not to take note of the malarious
atmosphere in which he grew up. The constitution of his mind was thus
early infected with a feverish taint that made him shiveringly sensitive
to a temperature which hardier natures found bracing. To him this rough
world was but too literally a rack. Good-humored Mother Nature commonly
imbeds the nerves of her children in a padding of self-conceit that
serves as a buffer against the ordinary shocks to which even a life of
routine is liable, and it would seem at first sight as if Rousseau had
been better cared for than usual in this regard. But as his self-conceit
was enormous, so was the reaction from it proportionate, and the fretting
suspiciousness of temper, sure mark of an unsound mind, which rendered
him incapable of intimate friendship, while passionately longing for it,
became inevitably, when turned inward, a tormenting self-distrust. To
dwell in unrealities is the doom of the sentimentalist; but it should not
be forgotten that the same fitful intensity of emotion which makes them
real as the means of elation, gives them substance also for torture. Too
irritably jealous to endure the rude society of men, he steeped his
senses in the enervating incense that women are only too ready to burn.
If their friendship be a safeguard to the other sex, their homage is
fatal to all but the strongest, and Rousseau was weak both by inheritance
and early training. His father was one of those feeble creatures for whom
a fine phrase could always satisfactorily fill the void that
non-performance leaves behind it. If he neglected duty, he made up for it
by that cultivation of the finer sentiments of our common nature which
waters flowers of speech with the brineless tears of a flabby remorse,
without one fibre of resolve in it, and which impoverishes the character
in proportion as it enriches the vocabulary. He was a very Apicius in
that digestible kind of woe which makes no man leaner, and had a favorite
receipt for cooking you up a sorrow _a la douleur inassouvie_ that had
just enough delicious sharpness in it to bring tears into the eyes by
tickling the palate. "When he said to me, 'Jean Jacques, let us speak of
thy mother,' I said to him, 'Well, father, we are going to weep, then,'
and this word alone drew tears from him. 'Ah !' said he, groaning, 'give
her back to me, console me for her, fill the void she has left in my
soul!'" Alas! in such cases, the void she leaves is only that she found.
The grief that seeks any other than its own society will erelong want an
object. This admirable parent allowed his son to become an outcast at
sixteen, without any attempt to reclaim him, in order to enjoy unmolested
a petty inheritance to which the boy was entitled in right of his mother.
"This conduct," Rousseau tells us, "of a father whose tenderness and
virtue were so well known to me, caused me to make reflections on myself
which have not a little contributed to make my heart sound. I drew from
it this great maxim of morals, the only one perhaps serviceable in
practice, to avoid situations which put our duties in opposition to our
interest, and which show us our own advantage in the wrong of another,
sure that in such situations, _however sincere may be one's love of
virtue_, it sooner or later grows weak without our perceiving it, _and
that we become unjust and wicked in action without having ceased to be
just and good in soul_."
This maxim may do for that "fugitive and cloistered virtue, unexercised
and unbreathed, that never sallies out and seeks its adversary," which
Milton could not praise,--that is, for a manhood whose distinction it is
not to be manly,--but it is chiefly worth notice as being the
characteristic doctrine of sentimentalism. This disjoining of deed from
will, of practice from theory, is to put asunder what God has joined by
an indissoluble sacrament. The soul must be tainted before the action
become corrupt; and there is no self-delusion more fatal than that which
makes the conscience dreamy with the anodyne of lofty sentiments, while
the life is grovelling and sensual,--witness Coleridge. In his case we
feel something like disgust. But where, as in his son Hartley, there is
hereditary infirmity, where the man sees the principle that might rescue
him slip from the clutch of a nerveless will, like a rope through the
fingers of a drowning man, and the confession of faith is the moan of
despair, there is room for no harsher feeling than pity. Rousseau showed
through life a singular proneness for being convinced by his own
eloquence; he was always his own first convert; and this reconciles his
power as a writer with his weakness as a man. He and all like him mistake
emotion for conviction, velleity for resolve, the brief eddy of sentiment
for the midcurrent of ever-gathering faith in duty that draws to itself
all the affluents of conscience and will, and gives continuity of purpose
to life. They are like men who love the stimulus of being under
conviction, as it is called, who, forever getting religion, never get
capital enough to retire upon and spend for their own need and the common
service.
The sentimentalist is the spiritual hypochondriac, with whom fancies
become facts, while facts are a discomfort because they will not be
evaporated into fancy. In his eyes, Theory is too fine a dame to confess
even a country-cousinship with coarse handed Practice, whose homely ways
would disconcert her artificial world. The very susceptibility that makes
him quick to feel, makes him also incapable of deep and durable feeling.
He loves to think he suffers, and keeps a pet sorrow, a blue-devil
familiar, that goes with him everywhere, like Paracelsus's black dog. He
takes good care, however, that it shall not be the true sulphurous
article that sometimes takes a fancy to fly away with his conjurer. Rene
says: "In my madness I had gone so far as even to wish I might experience
a misfortune, so that my suffering might at least have a real object."
But no; selfishness is only active egotism, and there is nothing and
nobody, with a single exception, which this sort of creature will not
sacrifice, rather than give any other than an imaginary pang to his idol.
Vicarious pain he is not unwilling to endure, nay, will even commit
suicide by proxy, like the German poet who let his wife kill herself to
give him a sensation. Had young Jerusalem been anything like Goethe's
portrait of him in Werther, he would have taken very good care not to
blow out the brains which he would have thought only too precious. Real
sorrows are uncomfortable things, but purely aesthetic ones are by no
means unpleasant, and I have always fancied the handsome young Wolfgang
writing those distracted letters to Auguste Stolberg with a looking-glass
in front of him to give back an image of his desolation, and finding it
rather pleasant than otherwise to shed the tear of sympathy with self
that would seem so bitter to his fair correspondent. The tears that have
real salt in them will keep; they are the difficult, manly tears that are
shed in secret; but the pathos soon evaporates from that fresh-water with
which a man can bedew a dead donkey in public, while his wife is having a
good cry over his neglect of her at home. We do not think the worse of
Goethe for hypothetically desolating himself in the fashion aforesaid,
for with many constitutions it is as purely natural a crisis as
dentition, which the stronger worry through, and turn out very sensible,
agreeable fellows. But where there is an arrest of development, and the
heartbreak of the patient is audibly prolonged through life, we have a
spectacle which the toughest heart would wish to get as far away from as
possible.
We would not be supposed to overlook the distinction, too often lost
sight of, between sentimentalism and sentiment, the latter being a very
excellent thing in its way, as genuine things are apt to be. Sentiment is
intellectualized emotion, emotion precipitated, as it were, in pretty
crystals by the fancy. This is the delightful staple of the poets of
social life like Horace and Beranger, or Thackeray, when he too rarely
played with verse. It puts into words for us that decorous average of
feeling to the expression of which society can consent without danger of
being indiscreetly moved. It is excellent for people who are willing to
save their souls alive to any extent that shall not be discomposing. It
is even satisfying till some deeper experience has given us a hunger
which what we so glibly call "the world" cannot sate, just as a water-ice
is nourishment enough to a man who has had his dinner. It is the
sufficing lyrical interpreter of those lighter hours that should make
part of every healthy man's day, and is noxious only when it palls men's
appetite for the truly profound poetry which is very passion of very soul
sobered by afterthought and embodied in eternal types by imagination.
True sentiment is emotion ripened by a slow ferment of the mind and
qualified to an agreeable temperance by that taste which is the
conscience of polite society. But the sentimentalist always insists on
taking his emotion neat, and, as his sense gradually deadens to the
stimulus, increases his dose till he ends in a kind of moral deliquium.
At first the debaucher, he becomes at last the victim of his sensations.
Among the ancients we find no trace of sentimentalism. Their masculine
mood both of body and mind left no room for it, and hence the bracing
quality of their literature compared with that of recent times, its tonic
property, that seems almost too astringent to palates relaxed by a
daintier diet. The first great example of the degenerate modern tendency
was Petrarch, who may be said to have given it impulse and direction. A
more perfect specimen of the type has not since appeared. An intellectual
voluptuary, a moral _dilettante_, the first instance of that character,
since too common, the gentleman in search of a sensation, seeking a
solitude at Vaucluse because it made him more likely to be in demand at
Avignon, praising philosophic poverty with a sharp eye to the next rich
benefice in the gift of his patron, commending a good life but careful
first of a good living, happy only in seclusion but making a dangerous
journey to enjoy the theatrical show of a coronation in the Capitol,
cherishing a fruitless passion which broke his heart three or four times
a year and yet could not make an end of him till he had reached the ripe
age of seventy and survived his mistress a quarter of a century,--surely
a more exquisite perfection of inconsistency would be hard to find.
When Petrarch returned from his journey into the North of Europe in 1332,
he balanced the books of his unrequited passion, and, finding that he had
now been in love seven years, thought the time had at last come to call
deliberately on Death. Had Death taken him at his word, he would have
protested that he was only in fun. For we find him always taking good
care of an excellent constitution, avoiding the plague with commendable
assiduity, and in the very year when he declares it absolutely essential
to his peace of mind to die for good and all, taking refuge in the
fortress of Capranica, from a wholesome dread of having his throat cut by
robbers. There is such a difference between dying in a sonnet with a
cambric handkerchief at one's eyes, and the prosaic reality of demise
certified in the parish register! Practically it is inconvenient to be
dead. Among other things, it puts an end to the manufacture of sonnets.
But there seems to have been an excellent understanding between Petrarch
and Death, for he was brought to that grisly monarch's door so often,
that, otherwise, nothing short of a miracle or the nine lives of that
animal whom love also makes lyrical could have saved him. "I consent," he
cries, "to live and die in Africa among its serpents, upon Caucasus, or
Atlas, if, while I live, to breathe a pure air, and after my death a
little corner of earth where to bestow my body, may be allowed me. This
is all I ask, but this I cannot obtain. Doomed always to wander, and to
be a stranger everywhere, O Fortune, Fortune, fix me at last to some one
spot! I do not covet thy favors. Let me enjoy a tranquil poverty, let me
pass in this retreat the few days that remain to me!" The pathetic stop
of Petrarch's poetical organ was one he could pull out at pleasure,--and
indeed we soon learn to distrust literary tears, as the cheap subterfuge
for want of real feeling with natures of this quality. Solitude with him
was but the pseudonyme of notoriety. Poverty was the archdeaconry of
Parma, with other ecclesiastical pickings. During his retreat at
Vaucluse, in the very height of that divine sonneteering love of Laura,
of that sensitive purity which called Avignon Babylon, and rebuked the
sinfulness of Clement, he was himself begetting that kind of children
which we spell with a _b_. We believe that, if Messer Francesco had been
present when the woman was taken in adultery, he would have flung the
first stone without the slightest feeling of inconsistency, nay, with a
sublime sense of virtue. The truth is, that it made very little
difference to him what sort of proper sentiment he expressed, provided he
could do it elegantly and with unction.
Would any one feel the difference between his faint abstractions and the
Platonism of a powerful nature fitted alike for the withdrawal of ideal
contemplation and for breasting the storms of life,--would any one know
how wide a depth divides a noble friendship based on sympathy of pursuit
and aspiration, on that mutual help which souls capable of
self-sustainment are the readiest to give or to take, and a simulated
passion, true neither to the spiritual nor the sensual part of man,--let
him compare the sonnets of Petrarch with those which Michel Angelo
addressed to Vittoria Colonna. In them the airiest pinnacles of sentiment
and speculation are buttressed with solid mason-work of thought, and of
an actual, not fancied experience, and the depth of feeling is measured
by the sobriety and reserve of expression, while in Petrarch's all
ingenuousness is frittered away into ingenuity. Both are cold, but the
coldness of the one is self-restraint, while the other chills with
pretence of warmth. In Michel Angelo's, you feel the great architect; in
Petrarch's the artist who can best realize his conception in the limits
of a cherry-stone. And yet this man influenced literature longer and more
widely than almost any other in modern times. So great is the charm of
elegance, so unreal is the larger part of what is written!
Certainly I do not mean to say that a work of art should be looked at by
the light of the artist's biography, or measured by our standard of his
character. Nor do I reckon what was genuine in Petrarch--his love of
letters, his refinement, his skill in the superficial graces of language,
that rhetorical art by which the music of words supplants their meaning,
and the verse moulds the thought instead of being plastic to it--after
any such fashion. I have no ambition for that character of _valet de
chambre_ which is said to disenchant the most heroic figures into mere
every-day personages, for it implies a mean soul no less than a servile
condition. But we have a right to demand a certain amount of reality,
however small, in the emotion of a man who makes it his business to
endeavor at exciting our own. We have a privilege of nature to shiver
before a painted flame, how cunningly soever the colors be laid on. Yet
our love of minute biographical detail, our desire to make ourselves
spies upon the men of the past, seems so much of an instinct in us, that
we must look for the spring of it in human nature, and that somewhat
deeper than mere curiosity or love of gossip. It should seem to arise
from what must be considered on the whole a creditable feeling, namely,
that we value character more than any amount of talent,--the skill to
_be_ something, above that of doing anything but the best of its kind.
The highest creative genius, and that only, is privileged from arrest by
this personality, for there the thing produced is altogether disengaged
from the producer. But in natures incapable of this escape from
themselves, the author is inevitably mixed with his work, and we have a
feeling that the amount of his sterling character is the security for the
notes he issues. Especially we feel so when truth to self, which is
always self-forgetful, and not truth to nature, makes an essential part
of the value of what is offered us; as where a man undertakes to narrate
personal experience or to enforce a dogma. This is particularly true as
respects sentimentalists, because of their intrusive self-consciousness;
for there is no more universal characteristic of human nature than the
instinct of men to apologize to themselves for themselves, and to justify
personal failings by generalizing them into universal laws. A man would
be the keenest devil's advocate against himself, were it not that he has
always taken a retaining fee for the defence; for we think that the
indirect and mostly unconscious pleas in abatement which we read between
the lines in the works of many authors are oftener written to set
themselves right in their own eyes than in those of the world. And in the
real life of the sentimentalist it is the same. He is under the wretched
necessity of keeping up, at least in public, the character he has
assumed, till he at last reaches that last shift of bankrupt
self-respect, to play the hypocrite with himself. Lamartine, after
passing round the hat in Europe and America, takes to his bed from
wounded pride when the French Senate votes him a subsidy, and sheds tears
of humiliation. Ideally, he resents it; in practical coin, he will accept
the shame without a wry face.
George Sand, speaking of Rousseau's "Confessions," says that an
autobiographer always makes himself the hero of his own novel, and cannot
help idealizing, even if he would. But the weak point of all
sentimentalists is that they always have been, and always continue under
every conceivable circumstance to be, their own ideals, whether they are
writing their own lives or no. Rousseau opens his book with the
statement: "I am not made like any of those I have seen; I venture to
believe myself unlike any that exists. If I am not worth more, at least I
am different." O exquisite cunning of self-flattery! It is this very
imagined difference that makes us worth more in our own foolish sight.
For while all men are apt to think, or to persuade themselves that they
think, all other men their accomplices in vice or weakness, they are not
difficult of belief that they are singular in any quality or talent on
which they hug themselves. More than this; people who are truly original
are the last to find it out, for the moment we become conscious of a
virtue it has left us or is getting ready to go. Originality does not
consist in a fidgety assertion of selfhood, but in the faculty of getting
rid of it altogether, that the truer genius of the man, which commerces
with universal nature and with other souls through a common sympathy with
that, may take all his powers wholly to itself,--and the truly original
man could no more be jealous of his peculiar gift, than the grass could
take credit to itself for being green. What is the reason that all
children are geniuses, (though they contrive so soon to outgrow that
dangerous quality,) except that they never cross-examine themselves on
the subject? The moment that process begins, their speech loses its gift
of unexpectedness, and they become as tediously impertinent as the rest
of us.
If there never was any one like him, if he constituted a genus in
himself, to what end write confessions in which no other human being
could ever be in a condition to take the least possible interest? All men
are interested in Montaigne in proportion as all men find more of
themselves in him, and all men see but one image in the glass which the
greatest of poets holds up to nature, an image which at once startles and
charms them with its familiarity. Fabulists always endow their animals
with the passions and desires of men. But if an ox could dictate his
confessions, what glimmer of understanding should we find in those bovine
confidences, unless on some theory of pre existence, some blank misgiving
of a creature moving about in worlds not realized? The truth is, that we
recognize the common humanity of Rousseau in the very weakness that
betrayed him into this conceit of himself; we find he is just like the
rest of us in this very assumption of essential difference, for among all
animals man is the only one who tries to pass for more than he is, and so
involves himself in the condemnation of seeming less.
But it would be sheer waste of time to hunt Rousseau through all his
doublings of inconsistency, and run him to earth in every new paradox.
His first two books attacked, one of them literature, and the other
society. But this did not prevent him from being diligent with his pen,
nor from availing himself of his credit with persons who enjoyed all the
advantages of that inequality whose evils he had so pointedly exposed.
Indeed, it is curious how little practical communism there has been, how
few professors it has had who would not have gained by a general
dividend. It is perhaps no frantic effort of generosity in a philosopher
with ten crowns in his pocket when he offers to make common stock with a
neighbor who has ten thousand of yearly income, nor is it an uncommon
thing to see such theories knocked clean out of a man's head by the
descent of a thumping legacy. But, consistent or not, Rousseau remains
permanently interesting as the highest and most perfect type of the
sentimentalist of genius. His was perhaps the acutest mind that was ever
mated with an organization so diseased, the brain most far-reaching in
speculation that ever kept itself steady and worked out its problems amid
such disordered tumult of the nerves.[166] His letter to the Archbishop
of Paris, admirable for its lucid power and soberness of tone, and his
_Rousseau juge de Jean Jacques_, which no man can read and believe him to
have been sane, show him to us in his strength and weakness, and give us
a more charitable, let us hope therefore a truer, notion of him than his
own apology for himself. That he was a man of genius appears unmistakably
in his impressibility by the deeper meaning of the epoch in which he
lived. Before an eruption, clouds steeped through and through with
electric life gather over the crater, as if in sympathy and expectation.
As the mountain heaves and cracks, these vapory masses are seamed with
fire, as if they felt and answered the dumb agony that is struggling for
utterance below. Just such flashes of eager sympathetic fire break
continually from the cloudy volumes of Rousseau, the result at once and
the warning of that convulsion of which Paris was to be the crater and
all Europe to feel the spasm. There are symptoms enough elsewhere of that
want of faith in the existing order which made the Revolution
inevitable,--even so shallow an observer as Horace Walpole could forebode
it so early as 1765,--but Rousseau more than all others is the
unconscious expression of the groping after something radically new, the
instinct for a change that should be organic and pervade every fibre of
the social and political body. Freedom of thought owes far more to the
jester Voltaire, who also had his solid kernel of earnest, than to the
sombre Genevese, whose earnestness is of the deadly kind. Yet, for good
or evil, the latter was the father of modern democracy, and with out him
our Declaration of Independence would have wanted some of those sentences
in which the immemorial longings of the poor and the dreams of solitary
enthusiasts were at last affirmed as axioms in the manifesto of a nation,
so that all the world might hear.
Though Rousseau, like many other fanatics, had a remarkable vein of
common sense in him, (witness his remarks on duelling, on
landscape-gardening, on French poetry, and much of his thought on
education,) we cannot trace many practical results to his teaching, least
of all in politics. For the great difficulty with his system, if system
it may be called, is, that, while it professes to follow nature, it not
only assumes as a starting-point that the individual man may be made over
again, but proceeds to the conclusion that man himself, that human
nature, must be made over again, and governments remodelled on a purely
theoretic basis. But when something like an experiment in this direction
was made in 1789, not only did it fail as regarded man in general, but
even as regards the particular variety of man that inhabited France. The
Revolution accomplished many changes, and beneficent ones, yet it left
France peopled, not by a new race without traditions, but by Frenchmen.
Still, there could not but be a wonderful force in the words of a man
who, above all others, had the secret of making abstractions glow with
his own fervor; and his ideas--dispersed now in the atmosphere of
thought--have influenced, perhaps still continue to influence, speculative
minds, which prefer swift and sure generalization to hesitating and
doubtful experience.
Rousseau has, in one respect, been utterly misrepresented and
misunderstood. Even Chateaubriand most unfilially classes him and
Voltaire together. It appears to me that the inmost core of his being was
religious. Had he remained in the Catholic Church he might have been a
saint. Had he come earlier, he might have founded an order. His was
precisely the nature on which religious enthusiasm takes the strongest
hold,--a temperament which finds a sensuous delight in spiritual things,
and satisfies its craving for excitement with celestial debauch. He had
not the iron temper of a great reformer and organizer like Knox, who,
true Scotchman that he was, found a way to weld this world and the other
together in a cast-iron creed; but he had as much as any man ever had
that gift of a great preacher to make the oratorical fervor which
persuades himself while it lasts into the abiding conviction of his
hearers. That very persuasion of his that the soul could remain pure
while the life was corrupt, is not unexampled among men who have left
holier names than he. His "Confessions," also, would assign him to that
class with whom the religious sentiment is strong, and the moral nature
weak. They are apt to believe that they may, as special pleaders say,
confess and avoid. Hawthorne has admirably illustrated this in the
penance of Mr. Dimmesdale. With all the soil that is upon Rousseau, I
cannot help looking on him as one capable beyond any in his generation of
being divinely possessed; and if it happened otherwise, when we remember
the much that hindered and the little that helped in a life and time like
his, we shall be much readier to pity than to condemn. It was his very
fitness for being something better that makes him able to shock us so
with what in too many respects he unhappily was. Less gifted, he had been
less hardly judged. More than any other of the sentimentalists, except
possibly Sterne, he had in him a staple of sincerity. Compared with
Chateaubriand, he is honesty, compared with Lamartine, he is manliness
itself. His nearest congener in our own tongue is Cowper.
In the whole school there is a sickly taint. The strongest mark which
Rousseau has left upon literature is a sensibility to the picturesque in
Nature, not with Nature as a strengthener and consoler, a wholesome tonic
for a mind ill at ease with itself, but with Nature as a kind of feminine
echo to the mood, flattering it with sympathy rather than correcting it
with rebuke or lifting it away from its unmanly depression, as in the
wholesomer fellow-feeling of Wordsworth. They seek in her an accessary,
and not a reproof. It is less a sympathy with Nature than a sympathy with
ourselves as we compel her to reflect us. It is solitude, Nature for her
estrangement from man, not for her companionship with him,--it is
desolation and ruin, Nature as she has triumphed over man,--with which
this order of mind seeks communion and in which it finds solace. It is
with the hostile and destructive power of matter, and not with the spirit
of life and renewal that dwells in it, that they ally themselves. And in
human character it is the same. St. Preux, Rene, Werther, Manfred,
Quasimodo, they are all anomalies, distortions, ruins,--so much easier is
it to caricature life from our own sickly conception of it, than to paint
it in its noble simplicity; so much cheaper is unreality than truth.
Every man is conscious that he leads two lives,--the one trivial and
ordinary, the other sacred and recluse; one which he carries to society
and the dinner-table, the other in which his youth and aspiration survive
for him, and which is a confidence between himself and God. Both may be
equally sincere, and there need be no contradiction between them, any
more than in a healthy man between soul and body. If the higher life be
real and earnest, its result, whether in literature or affairs, will be
real and earnest too. But no man can produce great things who is not
thoroughly sincere in dealing with himself, who would not exchange the
finest show for the poorest reality, who does not so love his work that
he is not only glad to give himself for it, but finds rather a gain than
a sacrifice in the surrender. The sentimentalist does not think of what
he does so much as of what the world will think of what he does. He
translates should into would, looks upon the spheres of duty and beauty
as alien to each other, and can never learn how life rounds itself to a
noble completeness between these two opposite but mutually sustaining
poles of what we long for and what we must.
Did Rousseau, then, lead a life of this quality? Perhaps, when we
consider the contrast which every man who looks backward must feel
between the life he planned and the life which circumstance within him
and without him has made for him, we should rather ask, Was this the life
he meant to lead? Perhaps, when we take into account his faculty of
self-deception,--it may be no greater than our own,--we should ask, Was
this the life he believed he led? Have we any right to judge this man
after our blunt English fashion, and condemn him, as we are wont to do,
on the finding of a jury of average householders? Is French reality
precisely our reality? Could we tolerate tragedy in rhymed alexandrines,
instead of blank verse? The whole life of Rousseau is pitched on this
heroic key, and for the most trivial occasion he must be ready with the
sublime sentiments that are supposed to suit him rather than it. It is
one of the most curious features of the sentimental ailment, that, while
it shuns the contact of men, it courts publicity. In proportion as
solitude and communion with self lead the sentimentalist to exaggerate
the importance of his own personality, he comes to think that the least
event connected with it is of consequence to his fellow-men. If he change
his shirt, he would have mankind aware of it. Victor Hugo, the greatest
living representative of the class, considers it necessary to let the
world know by letter from time to time his opinions on every conceivable
subject about which it is not asked nor is of the least value unless we
concede to him an immediate inspiration. We men of colder blood, in whom
self-consciousness takes the form of pride, and who have deified
_mauvaise honte_ as if our defect were our virtue, find it especially
hard to understand that artistic impulse of more southern races to _pose_
themselves properly on every occasion, and not even to die without some
tribute of deference to the taste of the world they are leaving. Was not
even mighty Caesar's last thought of his drapery? Let us not condemn
Rousseau for what seems to us the indecent exposure of himself in his
"Confessions."
Those who allow an oratorical and purely conventional side disconnected
with our private understanding of the facts, and with life, in which
everything has a wholly parliamentary sense where truth is made
subservient to the momentary exigencies of eloquence, should be
charitable to Rousseau. While we encourage a distinction which
establishes two kinds of truth, one for the world, and another for the
conscience, while we take pleasure in a kind of speech that has no
relation to the real thought of speaker or hearer, but to the rostrum
only, we must not be hasty to condemn a sentimentalism which we do our
best to foster. We listen in public with the gravity or augurs to what we
smile at when we meet a brother adept. France is the native land of
eulogy, of truth padded out to the size and shape demanded by
_comme-il-faut_. The French Academy has, perhaps, done more harm by the
vogue it has given to this style, than it has done good by its literary
purism; for the best purity of a language depends on the limpidity of its
source in veracity of thought. Rousseau was in many respects a typical
Frenchman, and it is not to be wondered at if he too often fell in with
the fashion of saying what was expected of him, and what he thought due
to the situation, rather than what would have been true to his inmost
consciousness. Perhaps we should allow something also to the influence of
a Calvinistic training, which certainly helps men who have the least
natural tendency towards it to set faith above works, and to persuade
themselves of the efficacy of an inward grace to offset an outward and
visible defection from it.
As the sentimentalist always takes a fanciful, sometimes an unreal, life
for an ideal one, it would be too much to say that Rousseau was a man of
earnest convictions. But he was a man of fitfully intense ones, as suited
so mobile a temperament, and his writings, more than those of any other
of his tribe, carry with them that persuasion that was in him while he
wrote. In them at least he is as consistent as a man who admits new ideas
can ever be. The children of his brain he never abandoned, but clung to
them with paternal fidelity. Intellectually he was true and fearless;
constitutionally, timid, contradictory, and weak; but never, if we
understand him rightly, false. He was a little too credulous of sonorous
sentiment, but he was never, like Chateaubriand or Lamartine, the lackey
of fine phrases. If, as some fanciful physiologists have assumed, there
be a masculine and feminine lobe of the brain, it would seem that in men
of sentimental turn the masculine half fell in love with and made an idol
of the other, obeying and admiring all the pretty whims of this _folle du
logis_. In Rousseau the mistress had some noble elements of character,
and less taint of the _demi-monde_ than is visible in more recent cases
of the same illicit relation.
Footnotes:
[165] _Histoire des Idees Morales et Politiques en France au XVIIIme
Siecle._ Par M. Jules Barni, Professeur a l'Academie de Geneve, Tome
II. Paris, 1867.
[166] Perhaps we should except Newton.
End of the Project Gutenberg EBook of Among My Books, by James Russell Lowell
*** | {
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} | 7,864 |
HomeNew BikesBike NewsKnow Your Two-wheeler Brand: India Yamaha Motor
Know Your Two-wheeler Brand: India Yamaha Motor
Published On May 5, 2019 By Praveen M. for Yamaha YZF R15 V3
From the nostalgic RX100s to modern R15s and FZs, Yamaha's racing genes have remained unperturbed by changing trends!
Yamaha was originally a musical instruments manufacturer, and the company started off its motorcycle business only in 1955. It gave the famous three tuning forks a whole new meaning where melody, harmony and rhythm were complemented by technology, production and sales. Here's how the Japanese brand's story started in India:
Two-stroke glory:
Three decades after Yamaha came into existence, the Japanese manufacturer entered our shores in 1985 through a joint venture with Escorts. This joint venture gave birth to the legendary RX100, for one. This motorcycle single-handedly etched the brand's name into the hearts of countless enthusiasts. The 100cc two-stroke motor churning out 11.5PS might not be much but that engine in combination with the bike's featherlight dry weight of just 95kg made it truly a force to reckon with. The RX100 dominated numerous motorsport events across the country.
It was then followed by different upgraded models like the RX G and then the RXZ 135 (remember the one with the 5-speed transmission?) but none managed to make as much of a hard-hitting impact as the RX100.
Also Read: Top 5 Reasons Why We Want The Yamaha RX100 To Make A Comeback
Another notable motorcycle that deserves a mention is the RD350. Even though it was not manufactured by Yamaha in India, the bike was so blisteringly fast for its time that it became a prized possession, or at least a dream motorcycle for almost every other two-wheeler enthusiast in India. The one made in India was rebadged as Rajdoot 350, and it was low on power and torque in favour of fuel efficiency. The move did make sense for India (read cost-conscious and mileage-centric market). However, the thrilling performance wasn't everyone's cup of tea, and the high asking price meant that by 1990, the famed RD disappeared from the production line.
A solo journey, facing a brave new world:
The ever-tightening emission norms and competition in the market sounded the death knell for Yamaha's two-stroke motorcycles. So post that, Yamaha tried concentrating on commuters for the Indian market which was ever so hungry for fuel efficient bikes. Remember bikes like the Crux, the 125cc Fazer with that insect-like dual headlamp, and the Gladiator?
While these motorcycles were technologically sound, they never captured audiences' attention as those bikes didn't exactly stand for what people associated Yamaha with - sportiness and performance. In fact, the Japanese brand also tried its luck in the cruiser segment with the Yamaha Enticer, which also failed to make an impression.
Back in the game:
But there was a profound transformation of how the brand was perceived, in 2008. That year marked the arrival of the legendary YZF-R15. The magical combination of R1-inspired styling with fully-faired bodywork and an advanced liquid-cooled gem of an engine really put the Japanese bikemaker back on the centre stage. Also, the R15's agile handling and competitive pricing made it one of the best enthusiast's motorcycles at the time, by a huge margin!
Yamaha didn't just stop with this, as in September (just three months after the R15's launch), it launched the FZ16. While it didn't have an engine that was as powerful as the R15, it still managed to decimate its competition. This was thanks to its air-cooled 153cc motor. It had the show to match the go too, as that fat 140-section rear tyre put it straight into the league of big bikes (or at least it gave that kind of an impression). Needless to say, the FZ sold like hot cakes, and sometime later the brand also launched a slightly sportier-looking variant, the FZ-S. Yamaha also dabbled in the scooter segment with the sporty-looking Ray range and the retro Fascino, which racked up consistently healthy numbers.
A performance-centric future:
After the subsequent iterations of the FZ and R15 range, Yamaha then expanded its performance portfolio with the YZF-R3. Its latest addition is the MT-15 and it's evident that the brand's racing genes have remained unfazed. Going forward, expect Yamaha to start off its launch spree with the facelifted R3, which might make its debut by late 2019 or at the 2020 Auto Expo. If we're lucky we could even witness the arrival of the Tenere 700 under its CBU range as the ADV segment is getting a lot of traction in the Indian two-wheeler market. We wouldn't be surprised if Yamaha commences its CKD operations in India in a bid to make its big bikes even more affordable. Among it Japanese rivals, Yamaha is the only brand that doesn't have CKD operations in India and this has hurt them with respect to sales and presence in the performance bike segment in our country.
The Japanese brand seems to be playing its performance card even in the scooter segment with the scheduled launch of the NMax 155. This scooter will most likely be the first one to sport a liquid-cooled engine in its segment. All these developments are surely bound to rev the heart of enthusiasts!
YZF R15 V3 Images
166 ReviewsRate This Bike
Rs.1.40 Lakh* Get On Road Price
Mileage48.75 kmpl
Engine155 cc
Power19.3 PS
Praveen M.
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World Jawa Day Celebrated With Over 700 Riders | {
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<h2 class="title">C</h2>
<dl>
<dt><span class="memberNameLink"><a href="../game/MainGame.html#camera">camera</a></span> - Variable in class game.<a href="../game/MainGame.html" title="class in game">MainGame</a></dt>
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<dt><span class="memberNameLink"><a href="../game/MenuScreens/GameOverMenu.html#camera">camera</a></span> - Variable in class game.MenuScreens.<a href="../game/MenuScreens/GameOverMenu.html" title="class in game.MenuScreens">GameOverMenu</a></dt>
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<dt><span class="memberNameLink"><a href="../game/MenuScreens/MainMenu.html#camera">camera</a></span> - Variable in class game.MenuScreens.<a href="../game/MenuScreens/MainMenu.html" title="class in game.MenuScreens">MainMenu</a></dt>
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<dt><span class="memberNameLink"><a href="../game/MenuScreens/PauseMenu.html#camera">camera</a></span> - Variable in class game.MenuScreens.<a href="../game/MenuScreens/PauseMenu.html" title="class in game.MenuScreens">PauseMenu</a></dt>
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<dt><span class="memberNameLink"><a href="../game/MainGame.html#checkCollisions--">checkCollisions()</a></span> - Method in class game.<a href="../game/MainGame.html" title="class in game">MainGame</a></dt>
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<div class="block">All game state checks are located here</div>
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<div class="block">Check if <a href="../game/Player.html" title="class in game"><code>Player</code></a> has won by collecting 999990 points.</div>
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<div class="block">Loads all the assets for the menu and sets their position in the game world.</div>
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<dt><span class="memberNameLink"><a href="../game/MenuScreens/MainMenu.html#createUi--">createUi()</a></span> - Method in class game.MenuScreens.<a href="../game/MenuScreens/MainMenu.html" title="class in game.MenuScreens">MainMenu</a></dt>
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<div class="block">Loads all the assets for the menu and sets their position in the game world.</div>
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<dt><span class="memberNameLink"><a href="../game/MenuScreens/PauseMenu.html#createUi--">createUi()</a></span> - Method in class game.MenuScreens.<a href="../game/MenuScreens/PauseMenu.html" title="class in game.MenuScreens">PauseMenu</a></dt>
<dd>
<div class="block">Loads all the assets for the menu and sets their position in the game world.</div>
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Home Cara Sutra Sex Blog Articles Chastity Release – Her Icy Response
Chastity Release – Her Icy Response
By Cara Sutra for Fetish Friday
I watched him washing the dishes from where I stood in the doorway. With his back to me he looked every inch the perfect, devoted partner. He cut a beautiful figure, too. Broad shoulders under his loose fitting shirt tapered to a narrow waist, a testament to his daily workouts, and I knew his pert little butt would be framed perfectly in tight jocks under those light blue jeans.
Stepping into the kitchen I could feel the cool tiles underneath my feet, the only barrier between floor and sole being the sheer stockings I wore under the office suit that day. Softly I padded up behind him and I could tell he now sensed my presence but didn't look around yet – he waited to see what I'd do. I couldn't wait any longer. I reached underneath his shirt which hung loosely outside his jeans, and let my fingertips roam softly over his muscular, chiselled frame. He started at my touch, even though he had known it was coming, but didn't say anything yet. I could feel the muscle definition either side of his spine up to his shoulders, and moving my hands around to his tummy I inwardly delighted at each hard-earned ab over his tummy.
My proximity to his waistline was distracting him, no doubt about that. It had been a while since I'd given him such tactile attention.
"I want to see it. Show it to me."
"Yes Mistress."
He turned around, hands still wet and soapy from the washing up bowl, careful not to get me wet. No getting me wet without my permission. Of course he made me wet without my permission on a regular basis, but he wasn't aware of it at the time – not until his punishment for it was in full flight. He wiped his hands off on his jeans and unbuckled his belt, unbuttoned his fly and pulled down those tight jocks enough to reveal what I couldn't wait to see. Concealed beneath his clothes all day as he kept house for me, his office executive partner, was the shiny metal chastity cage I'd locked him into just a couple of weeks into our now 8-year long relationship.
I smiled up at him in genuine delight but his eyes were turned downward to the floor. I knew he still struggled with the humiliation of being locked into a chastity device for me, despite the fact that it was also the hottest thing he'd ever known. Giving me complete control over his orgasms, his sex life, made him want to wank more furiously than ever – except he couldn't, obviously. One of the keys to the device was permanently on a long chain around my neck and buried between my breasts all day. I only freed him when I felt like it, and even then he'd have to beg pretty damn hard first. The begging for release made the cage feel tighter than ever as his cock strained to get an ironic chastity-related erection. It all amused me so much – and turned me on enormously.
Tilting his chin up so I could look into his embarrassed face properly, I relished the fight between humiliation, hate and the most intense arousal in his eyes. My other hand reached down and felt the weight of his chastity locked cock in my hands. He jumped as my cool hands came into contact with his hot straining balls under the metal device.
"Have you been a good boy today? Given up searching for the other key yet?"
Every padlock comes with two keys, and only one was around my neck. Teasing him about the whereabouts of the other was my favourite game at the moment. Not that he'd ever be allowed to unlock without permission, obviously. Even signs he'd been looking for the key instead of getting on with his chores was reason to give him a lengthy punishment at the end of the cane.
"I… Yes, Mistress."
Despite his predicament with his balls in my manicured grasp, he wanted to say no – I was sure of it. He'd never give up fantasising about unlocking himself, finding the key somewhere while I was at work and being able to take hold of his own surging erect cock and pumping it frenziedly until he ejaculated the thick streams of hot cum which had built in his overfull balls since the last time I'd milked him to empty them safely. It would be the most unsatisfying act he'd ever do, as it would be against his deeply ingrained chastity fetish and it would upset me greatly, but fantasising about being able to freely wank while locked up is all part of the fun.
"Good. Finally. That means it's time to show you where I've hidden it."
His metal bound cock actually leaped that time, his balls yanked upwards as his cock desperately but futilely attempted to reach something resembling an erection. I couldn't help it. I laughed.
My laughter successfully quenching his momentary bid for cock freedom with a heavy dose of humiliation, I continued as I enjoyed every increasing increment of fear building in his eyes.
"You're going to have to work for your temporary freedom though. If you do a good enough job I might even let you cum."
The desperation for release won out over the fear, I could tell by the slightly crazed look on his face brought on by an extra-long term locked into the cage. He'd do anything to get out of it now. He'd probably agree to not even cum, just as long as he could feel cool air against his cock instead of being trapped in the unrelenting metal.
I continued sharply as a swift reminder of his position.
"What do you say?!"
"Thank you, Mistress. Sorry… I… thank you."
Good. He'd learned not to argue, not to challenge, not to question. Obey, agree, be useful. Serve.
"That's better. Remove your jeans and pants and then look in the freezer."
A slightly quizzical look at that but he managed to scramble a semblance of submissive agreement and trust back on to his face before I changed my mind about his release as a punishment. We were going to stay here? In the kitchen? And why the freezer?
I must admit to feeling pretty damn smug about my secondary key hiding place. It was perfect… right at the back of the freezer behind the ice container was a small space. Enough for a couple of very special ice cubes. My lover and I had made them on one of his weekend visits.
Trousers and pants were neatly folded up on the side and I followed as he made his way to the freezer, no doubt bursting with curiosity about what was going to happen next. I motioned to him where to look and let him take out the very special ice cube, in the centre of which the padlock key could only just be made out with a glint here and there.
"You think I made that with water, don't you, my trusting little slut?"
His face paled a little as I shut the freezer door behind him, the ice cube on the palm of his hand. I took it from him.
"Oh, no. That would be far too easy."
I took the ice cube from him and continued.
"This is a very special ice cube indeed. In order to get to the key and earn your release, you're going to suck on a frozen cube of my lover's cum. And you're going to enjoy every last drop as you work for it, because it will make me very, very happy…"
The colour completely drained from his face then. I felt my knickers get incredibly damp. There'd be time enough to punish for that later, but for now I ignored my aching cunt and focused on his beautiful distaste. I pushed him down to his knees in front of me, him just wearing his shirt and the -for now- still locked chastity cage. I was really going to enjoy this.
Read the second part of this story here, where he finally gets release & an orgasm… sort of.
Please note that the above story is a fantasy – before re-enacting in reality please ensure all parties have suitable and thorough sexual health checks
Underwear & padlock images copyright of RuffledSheets.com – used with permission
Beginner's Guide To Penis Chastity & Male Chastity Play The Fifty Shades of Grey Movie: Release Day Review Gratis New Beginnings Erotic Anthology Due For Release 20 March 2014 New erotic book release: Blue Mondays by Emily Dubberley Puppy Play For Mistress – Chastity Tease And Orgasm Denial What I Get Out Of Locking A Man In A Chastity Cage The Conversion: A Male Chastity Story Locked In A Chastity Cage: A Real-Life Journal Keyholding for a New Chastity sub/slave Chastity Keys Tease
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spankfunfor 13 August, 2016 at 4:31 am
Very Enjoyable Start!
Rich 9 April, 2017 at 3:06 am
Cara, what a deliciously, deviously cruel and hot idea! Damn. I'm inventive and kinky and contantly imagining new scenarios, but I wish I'd come up with this one. I love it both as a writer of kink and as a switch who happens to be in the mood for clever femdom. In my mind, the scene continues with the debate raging inside your man's head between eating a man's cum to earn his own ejaculation, and the suffering frustration of having to store his jizz in his balls for days, weeks or months longer after his hopes were raised. Wow, my cock is leaking.
International Male Chastity Day | International Chastity Day Info
List of Shame: The 10 sex toys which need forcing into...
Should I Let A Company Approve My Sex Toy Review Before...
No more page 3? Don't worry, there are plenty more boobs... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,072 |
Scroll down for our newsfeed.
New! -resources for constructors- New!
Niander Wallace.... We are coming for you.
23o18 Ok so here is the link to some of trurl's artworks. enjoy.
22o18 we changed the cloud site to individual passwords. the student password no longer works. if you would like a password please email us. this works better with multiple users.
17o18 if you want to use the cloud site please request a user account from the email address at left.
02o18 a lot has been happening on the business backend. nothing def yet but T&K may have found a home planet. more shortly. and more on the projects.
22m18 here is a link to a paper on a semi formalism for causality modelling in AI I have drafted.
06m18 This is also useful for apprentices.
06m18 considering developing a page or index of resources for constructors. This is a good place to start for beginners.
01m18 cloud is up and running, with a full sql database. also have created a document control system that is totally automated. automation is good.
02a18 SEEKING COLLABORATORS FOR PROJECTS! - -follow the link for more information.
26m18 3D printers are up and running.
this is one of the coolest videos Ive ever seen. enjoy. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,035 |
\section{Introduction}
\label{sec:intro}
In hierarchical galaxy formation models, based on the cold dark matter (CDM) paradigm,
very small dwarf galaxies are excellent candidates for earliest formed galaxies. Numerical
simulations based on these models make definitive predictions about the average density
and shape of the dark matter distribution in such galaxies. These dwarf
systems are typically dark matter dominated, even in the innermost regions,
unlike bright galaxies where the stellar population is dynamically important.
Dwarf galaxies could hence provide an unique opportunity to compare observations
with the CDM simulations.
According to the numerical simulations, mass-density distribution
in the inner parts of the simulated dark matter halos could be well
described by a cusp i.e. r$^{-1}$ power law. This cusp in the density
distribution manifests itself as a steeply rising rotation curve in the
inner regions of galaxies. However, this prediction from CDM simulations
is found to disagree with the observed rotation curves of several
dwarf galaxies (e.g. deBlok \& Bosma 2002); the data indicate a constant
density core dark matter distribution. Apart from the shape of the dark
matter halos, numerical simulations also predict an anti-correlation
between the characteristic density and the virial mass of the dark matter
halos. In hierarchical scenarios, the low mass halos form at the earlier times, when the
background was higher, hence dwarf galaxies should have larger dark
matter densities.
All these predictions of the numerical simulations were however untested
in the faintest dwarf galaxies, as it was widely believed that very faint dwarf
irregular galaxies do not show any systematic rotational motions.
From a systematic study of the kinematics of a sample of dwarfs, C\^{o}t\'{e}, Carignan
\& Freeman (2000) found that normal rotation is seen only in galaxies
brighter than -14 mag, while fainter dwarfs have disturbed kinematics.
This result is consistent with the earlier findings of Lo, Sargent \& Young (1993), who
from a study of a sample of dwarfs (with M$_{B\rm} \sim -9$ to M$_{B\rm} \sim -15$)
found that very faint dwarf irregulars have chaotic velocity fields. However, most of the
previous studies were done with a coarser velocity resolution and modest sensitivities.
Inorder to study the kinematics of very faint dwarf irregular galaxies, we obtained
high velocity resolution and high sensitivity GMRT HI 21 cm-line observations of a sample
of galaxies with M$_B>-$12.5.
The GMRT has a hybrid configuration which simultaneously
provides both high angular resolution ($\sim$ 2$''$ if one uses baselines between the arm antennas)
as well as sensitivity to extended emission (from baselines between the antennas in the central
square). This unique hybrid configuration of the GMRT makes it an excellent facility
to study such galaxies. The velocity resolution used for our observations was $\sim1.6$ kms$^{-1}$
and the typical integration time on each source was $\sim 16~-$18 hrs, which gave a typical rms
of $\sim$ 1.0 mJy/Beam per channel. The galaxies in our sample have typical HI masses $\sim$ 10
$^{6-8}$ M$_\odot$.
We present here the results obtained from our GMRT observations.
\section{Results of The GMRT Observations}
\label{sec:result}
Figs.1[B],3[B] show the velocity fields for some of the galaxies in our sample.
Our GMRT observations clearly show that the earlier conclusions about the kinematics
of faintest galaxies were a consequence of observational bias. Since most of the
earlier studies of faint dwarf galaxies were done with coarser
velocity resolutions ($\sim 6 - 7$ kms$^{-1}$) and modest sensitivities,
hence, such observations could not detect systematic gradients, which
are typically $\sim 10-15$ kms$^{-1}$, in the velocity fields of such faint
dwarfs. On the other hand, our high velocity resolution and high
sensitivity observations found large scale systematic velocity gradients
in even the faintest galaxy in our sample i.e. DDO210. The velocity field for
DDO210, differs significantly from the velocity field derived earlier by
Lo et al.(1993). The pattern seen in the GMRT velocity field of DDO210 (fig.~\ref{fig:camb_mom}[B]) is,
to zeroth order, consistent with that expected from rotation, on the other
hand, the velocity field derived by Lo et al.(1993) based on a coarser velocity
resolution of $\sim$ 6 kms$^{-1}$, is indeed ``chaotic". This difference in the
observed kinematics suggests that high velocity resolution and high sensitivity
is crucial in determining the systematic gradients in the velocity fields of
faint galaxies. For some of the galaxies in our sample this large scale
systematic gradients could be modeled as systematic rotation, hence allowing
us to derive the rotation curve for those galaxies and to determine the structure
of their dark matter halos through mass modelling. The rest of this paper discusses
the results obtained from the detailed mass modelling of two of our
sample galaxies viz. Camelopardalis B (Cam B;M$_B\sim-$10.9) and DDO210 (M$_B\sim-$10.6).
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F1.PS,width=2.5in, height=5.5in}}
\caption{{\bf{[A]}}Integrated HI emission map of Cam B at
$40^{''} \times 38^{''}$ resolution overlayed on the Digitised Sky Survey
Image. The contour levels are $3.7, 8.8, 19.1, 24.3, 29.4, 34.6, 39.8, 44.9, 50.1,
55.2, 60.4, 65.5~\&~ 70.7\times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of Cam~B at $24^{''}\times 22^{''}$
resolution. The contours are in steps of 1~kms$^{-1}$ and
range from 70.0~kms$^{-1}$ (the extreme North East contour)
to 84.0~kms$^{-1}$ (the extreme South West contour).
}
\label{fig:camb_mom}
\end{figure}
\subsection{Camelopardalis B}
\label{ssec:camb}
Fig.~\ref{fig:camb_mom}[A] shows the integrated HI column density image of Cam B at 40$''\times 38''$
resolution overlayed on the optical DSS image. The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 2.2 Mpc) is 5.3$\pm$0.5 $\times10^6$M$_\odot$ and the M$_{HI}$/L$_B$
ratio is found to be 1.4 in solar units.
The velocity field derived from the moment analysis of 24$''\times 22''$ resolution data is shown
in the Fig.~\ref{fig:camb_mom}[B]. The velocity field is regular and the isovelocity contours are approximately
parallel, this is a signature of rigid body rotation. The kinematical major axis of the galaxy has a position
angle $\sim$ 215$^\circ$, i.e. it is well aligned with the major axis of both the HI distribution and the optical
emission.
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F2.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The derived rotation curve for Cam B (dashes)
and the rotation curve after applying
the asymmetric drift correction (dash dots).
{\bf{[B]}}Mass models for Cam B using the corrected rotation curve.
The points are the observed data. The total mass of gaseous disk (dashed line)
is $6.6\times10^6 M_\odot$ (after scaling the total HI mass by a factor of 1.25, to include
the contibution of primordial Helium). The stellar disk (short dash dot line) has
$\Upsilon_V=0.2$, giving a stellar mass of $0.7 \times10^6 M_\odot$. The
best fit total rotation curve for the constant density halo model is shown as
a solid line, while the contribution of the halo itself is shown as a
long dash dot line (the halo density is $\rho_0=13.7\times10^{-3}
M_\odot$ pc$^{-3}$). The best fit total rotation curve for an NFW type halo
(for $c=1.0$ and $\Upsilon_V=0.0$) is shown as a dotted line.
}
\label{fig:camb_vrot}
\end{figure}
The rotation curve for Cam B was derived from the velocity fields at 40$''\times 38''$, 24$''\times 22''$
and 16$''\times 14''$ resolution using the tilted ring method (see Begum, Chengalur and Hopp 2003 for details). The
derived hybrid rotation curve is shown as a dashed curve in Fig.~\ref{fig:camb_vrot}[A]. We find that the peak
inclination corrected rotational velocity for Cam B ($\sim$ 7.0 kms$^{-1}$) is comparable to the observed HI
dispersion i.e. V$_{\rm{max}}/\sigma_{\rm{HI}}\sim1.0$. This implies that the random motions provide significant
dynamical support to the system. In other words, the observed rotational velocities underestimate the total
dynamical mass in the galaxy due to a significant pressure support of the HI gas. Hence,
the observed rotation velocities were corrected for this pressure support, using the usual
``asymmetric drift" correction, before constructing mass models (see Begum et al (2003) for details).
The dot-dashed curve in Fig.~\ref{fig:camb_vrot}[A] shows the ``asymmetric drift" corrected rotation curve.
Using the ``asymmetric drift" corrected curve, mass models for Cam B were derived.
The best fit mass model using a modified isothermal halo is shown
in Fig.~\ref{fig:camb_vrot}[B]. Also shown in the figure is
the best fit total rotation curve for an NFW type halo.
As can be seen, the kinematics of Cam~B is well fit with
a modified isothermal halo while an NFW halo provides
a poor fit to the data.
The ``asymmetric drift" corrected rotation curve for Cam B is
rising till the last measured point (Fig.~\ref{fig:camb_vrot}[B]), hence the core
radius of the isothermal halo could not be constrained
from the data. The best fit model for a constant density halo
gives central halo density ($\rho_0$) of 12.0$\times 10^{-3}~M_
\odot$~pc$^{-3}$.
The derived $\rho_0$ is relatively insensitive to the assumed
mass-to-light ratio of the stellar disk ($\Upsilon_V$). We found that
by changing $\Upsilon_V$ from a value of 0 (minimum disk fit)
to a value of 2.0 (maximum disk fit), $\rho_0$ changes by $<$20\%.
From the last measured point of the observed rotation curve, a total dynamical mass of
1.1$\times 10^8 M_\odot$ is derived, i.e. at the last measured point
more than 90\% of the mass of Cam~B is dark. Futher, the dominance of the
dark matter halo together with the linear shape of the rotation
curve (after correction for ``asymmetric drift'') means that one
cannot obtain a good fit to the rotation curve using an NFW halo
regardless of the assumed $\Upsilon_V$.
\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F3.PS,width=5.5in}}
\caption{{\bf{[A]}} The optical DSS image of DDO210 (greyscales) with
the GMRT 44$^{''}\times37^{''}$ resolution integrated HI
emission map (contours) overlayed.
The contour levels are $0.7, 10.2, 19.8, 29.3, 38.3, 45.5, 57.1,
67.5, 77.1, 86.7, 96.2, 105.7, 121.4~ \& ~124.8 \times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of DDO210 at 29$^{''}\times 23^{''}$
resolution. The contours are in steps of 1~kms$^{-1}$ and
range from $-$145.0~kms$^{-1}$ to $-$133.0~kms$^{-1}$.
}
\label{fig:ddo210_mom}
\end{figure}
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F4.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The hybrid rotation curve (dashes) and the rotation curve
after applying the asymmetric drift correction (dots).
{\bf{[B]}} Mass models for DDO210 using the corrected rotation curve.
The points are the observed data. The total mass of gaseous
disk (dashed line) is $3.6\times10^6 M_\odot$ (including the contibution
of primodial He).The stellar
disk (short dash dot line) has
$\Upsilon_B=3.4$, giving a stellar mass of $9.2 \times10^6 M_\odot$.
The best fit total rotation curve for the constant density
halo model is shown as
a solid line, while the contribution of the halo itself
is shown as a long dash dot line (the halo density is
$\rho_0=29~\times10^{-3} M_\odot$ pc$^{-3}$).
The best fit total rotation curve for an NFW type halo,
using $\Upsilon_B=0.5$, c=5.0 and $v_{200}$=38.0~km s$^{-1}$
is shown as a dotted line. See text for more details.
}
\label{fig:ddo210_vrot}
\end{figure}
\subsection{DDO210}
\label{ssec:camb}
DDO210 is the faintest known gas rich dwarf galaxy in our local group.
Karachentsev et al.(2002), based on the HST observations of the I magnitude of
the tip of the red giant branch, estimated the distance to this galaxy to
be 950$\pm$50 kpc.
Fig.~\ref{fig:ddo210_mom}[A] shows the integrated HI column density image of
DDO210 at 44$''\times 37''$
resolution overlayed on the optical DSS image. The HI isodensity contours are
elongated in eastern and southern half of the galaxy, indicating a density enhancement
in these directions.The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 1.0 Mpc) is 2.8$\pm$0.3 $\times10^6$M$_\odot$
and the M$_{HI}$/L$_B$ ratio is found to be 1.0 in solar units.
The velocity field of DDO210 derived from the moment analysis of 29$''\times 23''$
resolution data cube is shown in Fig.~\ref{fig:ddo210_mom}[B]. The velocity field is
regular and a systematic velocity gradient is seen across the galaxy.
Fig.~\ref{fig:ddo210_vrot}[A] shows the derived rotation curve for DDO210 and the
``asymmetric drift" corrected rotation curve. We find that the ``asymmetric
drift" corrected rotation curve of DDO210 can be well fit with a
modified isothermal halo (with a central density $\rho_0 \sim 29\times10^
{-3}$ $M_\odot$ pc$^{-3}$). In the case of the NFW halo, we find that
there are a range of parameters which provide acceptable fits e.g. (v$_{200}\sim$ 20 kms$^{-1}$,
c $\sim$ 10) to (v$_{200}\sim$ 500 kms$^{-1}$, c$\sim$ 0.001); the halo parameters
could not be uniquely determined from the fit. However, the best fit value of the
concentration parameter c, at any given v$_{200}$ was found to be consistently smaller
than the value predicted by numerical simulations for the $\Lambda$CDM universe (Bullock et al. 2001).
Fig.~\ref{fig:ddo210_vrot}[B] shows the best fit mass models for DDO210 (see Begum \& Chengalur 2004 for details).
\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F5.PS,width=4.0in}}
\caption{Scatter plots of the central halo density against the
circular velocity and the absolute blue magnitude . The data (empty squares)
are from Verheijen~(1997), Broeils~(1992),
C\^{o}t\'{e} et al.~(2000) and Swaters~(1999). The filled
squares are the medians of the binned data, and the straight
lines are the best fits to the data. Cam~B and DDO210 are
shown as crosses.
}
\label{fig:dens}
\end{figure}
\section{Discussion}
\label{sec:result}
Fig.~\ref{fig:dens} shows the core density of isothermal halo against circular
velocity and absolute blue magnitude for a sample of galaxies, spanning a range of
magnitudes from M$_B\sim-10.0$ mag to M$_B\sim-23.0$ mag. Cam B and DDO210 are also
shown in the figure, lying at the low luminosity end of the sample.
As can be seen in the figure,
there is a trend of increasing halo density with a decrease in circular velocity
and absolute magnitude, shown by a best fit line to the data (solid line),
although the correlation is very weak and noisy. Further, as a guide to an
eye, we have also binned the data and plotted the median value (solid points).
Binned data also shows a similar trend. Such a tread is expected in
hierarchical structure formations scenario (e.g. Navarro,
Frenk \& White 1997).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,966 |
With the UK alone now sending over 1 billion text messages per week1 there has never been a better time for business to adopt this technology.
Immediate delivery and cost effective, this is proving to be a very attractive alternative to email.
Just30 introduced this technology very early on and we now pride ourselves on the diversity of applications we have produced.
To find out exactly how we can help you capitalise on this hugely popular media please give us a call and we will happily give you some ideas and discuss how we have already introduced this to existing clients. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
package com.djs.learn.spring_sample.hibernate;
import java.util.List;
import org.apache.log4j.Logger;
import org.springframework.orm.hibernate3.support.HibernateDaoSupport;
import com.djs.learn.spring_sample.db.Item;
import com.djs.learn.spring_sample.db.ItemDao;
/*
This class use spring-modules-cache. But it is discontinued.
<!-- SpringModules framework is dead -->
<dependency>
<groupId>org.springmodules</groupId>
<artifactId>spring-modules-cache</artifactId>
<version>0.8</version>
<exclusions>
<exclusion>
<artifactId>gigaspaces-ce</artifactId>
<groupId>gigaspaces</groupId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>jsk-lib</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>jsk-platform</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>mahalo</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>reggie</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>start</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>boot</artifactId>
</exclusion>
<exclusion>
<groupId>jini</groupId>
<artifactId>webster</artifactId>
</exclusion>
<exclusion>
<groupId>jboss</groupId>
<artifactId>jboss-cache</artifactId>
</exclusion>
<exclusion>
<groupId>jboss</groupId>
<artifactId>jboss-common</artifactId>
</exclusion>
<exclusion>
<groupId>jboss</groupId>
<artifactId>jboss-jmx</artifactId>
</exclusion>
<exclusion>
<groupId>jboss</groupId>
<artifactId>jboss-minimal</artifactId>
</exclusion>
<exclusion>
<groupId>jboss</groupId>
<artifactId>jboss-system</artifactId>
</exclusion>
<exclusion>
<groupId>jcs</groupId>
<artifactId>jcs</artifactId>
</exclusion>
<exclusion>
<groupId>xpp3</groupId>
<artifactId>xpp3_min</artifactId>
</exclusion>
<exclusion>
<groupId>ehcache</groupId>
<artifactId>ehcache</artifactId>
</exclusion>
<exclusion>
<groupId>org.springframework</groupId>
<artifactId>spring</artifactId>
</exclusion>
</exclusions>
</dependency>
<beans xmlns="http://www.springframework.org/schema/beans"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:ehcache="http://www.springmodules.org/schema/ehcache"
xsi:schemaLocation="http://www.springframework.org/schema/beans
http://www.springframework.org/schema/beans/spring-beans-2.5.xsd
http://www.springmodules.org/schema/ehcache
http://www.springmodules.org/schema/cache/springmodules-ehcache.xsd">
<!-- This is to enable Ehcache for itemDao2. -->
<ehcache:config configLocation="classpath:db_ehcache.xml" />
<!-- This method is not recommended, because SpringModules framework is
dead. -->
<ehcache:annotations>
<ehcache:caching id="localCacheModel" cacheName="localCache" />
<ehcache:flushing id="localFlushModel" cacheNames="localCache"
when="before" />
</ehcache:annotations>
<bean id="itemDao2"
class="com.djs.learn.spring_sample.hibernate.ItemDaoSpringHibernateImplA">
<property name="sessionFactory" ref="sessionFactory" />
</bean>
</beans>
*/
public class ItemDaoSpringHibernateImplA extends HibernateDaoSupport implements ItemDao
{
private final Logger log = Logger.getLogger( ItemDaoSpringHibernateImplA.class );
@Override
// @CacheFlush(modelId = "localFlushModel")
public void save( Item item )
{
if (log.isInfoEnabled())
{
log.info( "Save/Update: " + item );
}
// This saveOrUpdate() will do an update action first. If not exit, then do an insert action.
// getHibernateTemplate().saveOrUpdate( item );
getHibernateTemplate().save( item );
}
@Override
// @Cacheable(modelId = "localCacheModel")
public List<Item> query( String itemName )
{
// Here must include Item's full package name.
final String sql = "FROM " + Item.class.getName() + " WHERE ITEM_NAME = ?";
if (log.isInfoEnabled())
{
log.info( "Query: " + itemName );
}
List items = getHibernateTemplate().find( sql, new Object []
{ itemName } );
if (log.isInfoEnabled())
{
log.info( "Result: " + items );
}
return items.size() > 0 ? (List<Item>)items : null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,522 |
Het wapen van De Wold Aa werd op 1 maart 1960 per Koninklijk besluit aan het Drentse waterschap De Wold Aa verleend. In 1995 ging het waterschap met de andere waterschappen De Oude Vaart, Middenveld, Riegmeer en Smilde op in het nieuwe waterschap Meppelerdiep. Hiermee verviel het wapen.
Blazoenering
De blazoenering luidt als volgt:
''In zilver een golvende rechterschuinbalk van azuur, in het hoofdschild vergezeld van een voortschrijdende monnik van natuurlijke kleur, gekleed in een pij van sabel, geschoeid van keel en voor zich uit dragende een paalsgewijze geplaatste schop van azuur, gesteeld van keel, in de schildvoet vergezeld van een roos, eveneens van keel met gouden hart en kelkbladeren van sinopel. Het schild gedekt door een gouden kroon van drie bladeren en twee paarlen.
De heraldische kleuren in het wapen zijn azuur (blauw), goud (geel), keel (rood), sabel (zwart) en zilver. Het schild is gedekt met een gravenkroon.
Geschiedenis
Het waterschap werd in 1932 opgericht met het doel het onderhouden van de Wold Aa. Nadat in de jaren '50 de eerste wapens werden verleend aan waterschappen in Drenthe had ook De Wold Aa interesse in een waterschapswapen. Hierop werd een brief met ontwerp op 26 september 1957 naar de Hoge Raad van Adel verstuurd. De Hoge Raad van Adel had bezwaar tegen de monnik in het ontwerp. Zo beweerde de Hoge Raad van Adel dat de Monniken zich niet met het ontginnen van het land bezighielden. Het tegendeel werd bewezen dat de monniken van Ruinen en Dickninge zich daar wel mee bezig hielden.
Symboliek
De golvende balk in het wapen moet het diepje de Wold Aa voorstellen. Verder verwijst de roos naar de heren van Ruinen die de heerlijkheden Ruinen en Ruinerwold bestuurden. De voormalige heerlijkheden lagen ten slotte in het waterschapgebied. Verder staat er in het wapen ook een monnik met schop die waarschijnlijk het werkgerij aan iemand anders gaat geven, deze herinnert aan de werkzaamheid van de monniken en nonnen van de abdijen Ruinen en Dickninge in het gebied.
Verwant wapen
Wapen van voormalig Nederlands waterschap
Drents symbool | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 263 |
\section{Introduction}
Recently modern information-communication-technology (ICT) has opened us access to large amounts of stored digital data on human communication, which in turn has enabled us to have unprecedented insights into the patterns of human behavior and social interaction. For example we can now study the structure and dynamics of large-scale human communication networks~\cite{Onnela2007a,Onnela2007b,Krings2009,Palla2007} and the laws of mobility~\cite{Gonzalez2008,Song2010a,Song2010b}, as well as the motifs of individual behavior~\cite{Barabasi2005,Malmgren2008,Malmgren2009a,Malmgren2009b,Anteneodo2010,Miritello2011,Karsai2011}. One of the robust findings of these studies is that human activity over a variety of communication channels is inhomogeneous, such that high activity bursts of rapidly occurring events are separated by long periods of inactivity~\cite{Johansen2001,Eckmann2004,Harder2006,Goncalves2008,Zhou2008,Radicchi2009}. This feature is usually characterized by the distribution of inter-event times $\tau$, defined as time intervals between, e.g., consecutive e-mails sent by a single user. This distribution has been found to have a heavy tail and show a power-law decay as $P(\tau)\sim \tau^{-1}$~\cite{Barabasi2005}.
In human behavior obvious causes of inhomogeneity are the circadian and other longer cycles of our lives as results of natural and societal factors. Malmgren~\textit{et al.}~\cite{Malmgren2008,Malmgren2009a} suggested that an approximate power-law scaling found in the inter-event time distribution of human correspondence activity is a consequence of circadian and weekly cycles affecting us all, such that the large inter-event times are attributed to nighttime and weekend inactivity. As an explanation they proposed a cascading inhomogeneous Poisson process, which is a combination of two Poisson processes with different time scales. One of them is characterized by the time-dependent event rate representing the circadian and weekly activity patterns, while the other corresponds to the cascading bursty behavior with a shorter time scale. Their model was able to reproduce an apparent power-law behavior in the inter-event time distribution of email and postal mail correspondence. In addition they calculated the Fano and Allan factors to indicate the existence of some correlations for the email data as well as for their model of inhomogeneous Poisson process, with quite good comparison~\cite{Anteneodo2010}.
However, the question remains whether in addition to the circadian and weekly cycle driven inhomogeneities there are also other correlations due to human task execution that contribute to the inhomogeneities observed in communication patterns, as suggested, e.g., by the queuing models ~\cite{Barabasi2005,Vazquez2006}. There is evidence for this by Goh and Barab\'asi~\cite{Goh2008}, who introduced a measure that indicates the communication patterns to have correlations. Recently, Wu~\textit{et al.} have studied the modified version of the queuing process proposed in~\cite{Barabasi2005} by introducing a Poisson process as the initiator of localized bursty activity~\cite{Wu2010}. This was aimed at explaining the observation that the inter-event time distributions in Short Message (SM) correspondence follow a bimodal combination of power-law and Poisson distributions. The power-law (Poisson) behavior was found dominant for $\tau<\tau_0$ ($\tau>\tau_0$). Since the event rates extracted from the empirical data have the time scales larger than $\tau_0$ (also measured empirically), a bimodal distribution was successfully obtained. However, in their work the effects of circadian and weekly activity patterns were not considered, thus needing to be investigated in detail.
As the circadian and weekly cycles affect human communication patterns in quite obvious ways, taking place mostly during the daytime and differently during the weekends, our aim in this paper is to remove or de-season from the data the temporal inhomogeneities driven by these cycles. Then the study of the remaining de-seasoned data would enable us to get insight to the existence of other human activity driven correlations. This is important for two reasons. First, communication patterns tell about the nature of human behavior. Second, in devising models of human communication behavior the different origins of inhomogeneities should be properly taken into account; is it enough to describe the communication pattern by an inhomogeneous Poissonian process or do we need a model to reflect correlations in other human activities, such as those due to task execution?
In this paper, we provide a systematic method to de-season the circadian and weekly patterns from the mobile phone communication data. Firstly, we extract the circadian and weekly patterns from the time-stamped communication records and secondly, these patterns are removed by rescaling the timings of the communication events, i.e. phone calls and SMs. The rescaling is performed such that the time is dilated or contracted at times of high or low event activity, respectively. Finally, we obtain the inter-event time distributions by using the rescaled timings and comparing them with the original distributions to check how the heavy tail and burstiness behavior are affected. As the main results we find that the de-seasoned data still shows heavy tail inter-event time distributions with power-law scalings thus indicating that human task execution is a possible cause of remaining burstiness in mobile phone communication.
This paper is organized as follows. In Section~\ref{sect:analysis}, we introduce the methods for de-seasoning the circadian and weekly patterns systematically in various ways. By applying these methods the values of burstiness of inter-event time distributions are obtained and subsequently discussed. Finally, we summarize the results in Section~\ref{sect:summary}.
\section{De-seasoning analysis}\label{sect:analysis}
We investigate the effect of circadian and weekly cycles on the heavy-tailed inter-event time distribution and burstiness in human activity by using the mobile phone call (MPC) dataset from a European operator (national market share $\sim 20\%$) with time-stamped records over a period of $119$ days starting from January 2, 2007. The data of January 1, 2007 are not considered due to its rather unusual pattern of human communication. We have only retained links with bidirectional interaction, yielding $N=5.2\times 10^6$ users, $L=10.6\times 10^6$ links, and $C=322\times 10^6$ events (calls). For the analysis of Short Message (SM) dataset, see Appendix.
We perform the de-seasoning analysis by defining first the observable. For an individual service user $i$, $n_i(t)$ denotes the number of events at time $t$, where $t$ ranges from $0$ seconds, i.e. the start of January 2, 2007 at midnight, to $T_f\approx 1.03\times 10^7$ seconds (119 days). The total number of events $s_i\equiv \sum_{t=0}^{T_f}n_i(t)$ is called the strength of user $i$. In general, for a set of users $\Lambda$, the number of events at time $t$ is denoted by $n_\Lambda(t)\equiv \sum_{i\in\Lambda}n_i(t)$. $\Lambda$ can represent one user, a set of users, or the whole population. When the period of cycle $T$ is given, the event rate $\rho_{\Lambda,T}(t)$ with $0\leq t<T$ is defined as
\begin{equation}
\rho_{\Lambda,T}(t)=\frac{T}{s_\Lambda} \sum_{k=0}^{\lfloor T_f/T\rfloor} n_\Lambda(t+kT),\ s_\Lambda=\sum_{t=0}^{T_f} n_\Lambda(t).
\label{eq:eventRate}
\end{equation}
For convenience, we redefine the periodic event rate with period $T$ as $\rho_{\Lambda,T}(t)=\rho_{\Lambda,T}(t+kT)$ with any non-negative integer $k$ for $0\leq t<\infty$. By means of the event rate, we define the rescaled time $t^*(t)$ as following~\cite{Anteneodo2010}
\begin{equation}
t^*(t)=\sum_{0\leq t'<t} \rho_{\Lambda,T}(t').
\label{eq:rescaledTime}
\end{equation}
This rescaling corresponds to the transformation of the time variable by $\rho^*(t^*)dt^*=\rho(t)dt$ with $\rho^*(t^*)=1$. Here $\rho^*(t^*)=1$ means that there exists no cyclic pattern in the frame of rescaled time. The time is dilated (contracted) at the moment of high (low) activity.
In order to check whether the rescaling affects the inter-event time distributions and whether still some burstiness exists we reformulate the inter-event time distributions by using rescaled event times and compare them with the original distributions. The definition of the rescaled inter-event time from the rescaled time is straightforward. Considering two consecutive events of a user $i\in\Lambda$ occurring at times $t_j$ and $t_{j+1}$, the original inter-event time is $\tau \equiv t_{j+1}-t_j$, then the corresponding rescaled inter-event time is defined as follows
\begin{equation}
\tau^* \equiv t^*(t_{j+1})-t^*(t_j)=\sum_{t_j\leq t'<t_{j+1}}\rho_{\Lambda,T}(t').
\label{eq:rescaledTau}
\end{equation}
To find out how much the de-seasoning affects burstiness, we measure the burstiness of events, as proposed in~\cite{Goh2008}, where the burstiness parameter $B$ is defined as
\begin{equation}
B\equiv\frac{\sigma_\tau-m_\tau}{\sigma_\tau+m_\tau}.
\end{equation}
Here $\sigma_\tau$ and $m_\tau$ are the standard deviation and the mean of the inter-event time distribution $P(\tau)$, respectively. The value of $B$ is bounded within the range of $[-1,1]$ such that $B=1$ for the most bursty behavior, $B=0$ for neutral or homogeneous Poisson behavior, and $B=-1$ for completely regular behavior. The burstiness of the original inter-event time distribution, denoted by $B_0$, is to be compared with that of $B_T$ of the rescaled inter-event time distribution for given period $T$. With the de-seasoning the burstiness is expected to decrease and here we are most interested in by what amount the burstiness decreases when using $T=1$ day or $7$ days, i.e. removing the circadian or weekly patterns.
\begin{figure*}[!t]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{fig1a_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig1a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig1b_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig1b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig1c_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig1c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig1d_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig1d_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig1e_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig1e_right.pdf}
\end{tabular}
\caption{De-seasoning MPC patterns of individual users: the original and the rescaled event rates with period of $T=1$ day (left) and the original and the rescaled inter-event time distributions with various periods of $T$ (right). Individual users with the strength $s_i=200$ (a), 400 (b), 800 (c), 1600 (d), and 3197 (e) are analyzed. The original inter-event time distribution of the whole population is also plotted as a dashed curve for comparison.}
\label{fig:call_individual}\end{figure*}
\subsection{Individual de-seasoning}
First, we perform the de-seasoning analysis for individual users with various values of $T$. For some sample individuals in the MPC dataset, we obtain the original and the rescaled event rates. The event rates in case of $T=1$ day are depicted in the left column of Fig.~\ref{fig:call_individual}. The strengths of individual users are 200, 400, 800, 1600, and 3197. In most cases we find the characteristic circadian pattern, i.e. inactive nighttime and active daytime with one peak in the afternoon and another peak in the evening. The rescaled event rate successfully shows the expected de-seasoning effect, i.e. $\rho^*(t^*)=1$, except for weak fluctuations.
The rescaled inter-event time distributions up to $T=28$ days are compared with the original distributions in the right column of Fig.~\ref{fig:call_individual}. Note that the possible minimum value of rescaled inter-event time is $T/s_i$. We find that the rescaled inter-event time distributions still show the heavy tails. For the user with strength $200$, the burstiness decreases from the original value of $B_0\approx 0.202$ to value $B_7\approx 0.174$ (weekly pattern removed), then dropping further to value $B_{28}\approx 0.104$ (i.e. monthly pattern removed). For the most active user with strength $s_i=3197$, the burstiness decays faster as $T$ increases: $B_0\approx 0.469$, $B_7\approx 0.254$, and $B_{28}\approx 0.219$. However, the values of $B$ are overall larger than those of the less active user. The results imply that de-seasoning the circadian and weekly patterns does not considerably affect the temporal burstiness patterns of individuals. Finally, in a limiting case of $T=T_f$, since $n_i(t)$ has the value of either 0 or 1, all $\tau^*$ are the same as $T/s_i$ in Eqs.~(\ref{eq:eventRate}) and (\ref{eq:rescaledTau}), leading to $B_{T_f}=-1$.
\begin{figure*}[!t]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{fig2a_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig2a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig2b_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig2b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig2c_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig2c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig2d_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig2d_right.pdf}
\end{tabular}
\caption{Distributions $P(B_T)$ of the original and rescaled burstiness of invididual users with the same strength (left) and distributions $P(\Delta B_T)$ of the difference in burstiness, defined as $\Delta B_T=B_T-B_0$ (right). The individual users with the strengths $s_i=200$ (a), 400 (b), 800 (c), and 1600 (d) are analyzed. The numbers of users are correspondingly 6397, 1746, 196, and 7.}
\label{fig:call_strength_sep}\end{figure*}
Next, we obtain the distributions $P(B_T)$ of original and rescaled values of burstiness of individual users with the same strength and the distributions $P(\Delta B_T)$ of the difference in burstiness, defined as $\Delta B_T\equiv B_T-B_0$, see Fig.~\ref{fig:call_strength_sep}. The averages and the standard deviations of burstiness distributions for different periods of $T$ are plotted in Fig.~\ref{fig:call_burstiness}(a). The more active users have the larger values of burstiness, while the values of burstiness of the more active users decays faster (slower) than those of the less active users before (after) $T=7$ days. The overall behavior of the distributions shows that de-seasoning the circadian and weekly patterns does not destroy the bursty behavior of most individual users irrespective of their strengths. In addition, we find some exceptional users whose original values of burstiness are negative, indicating for more regular behavior than the Poisson process, and we also find a few individual users whose values of burstiness have grown as a result of de-seasoning, i.e. $\Delta B_T>0$.
\begin{figure*}[!t]
\begin{tabular}{ccc}
\includegraphics[width=.3\columnwidth]{fig3a.pdf}&
\includegraphics[width=.3\columnwidth]{fig3b.pdf}&
\includegraphics[width=.3\columnwidth]{fig3c.pdf}
\end{tabular}
\caption{Burstiness $B_T$ as a function of period of $T$: (a) the average and the standard deviation of $B_T$ obtained from the burstiness distribution in Fig.~\ref{fig:call_strength_sep}, (b) burstiness from groups with the same strength, and (c) burstiness from groups with broad ranges of strength.}
\label{fig:call_burstiness}\end{figure*}
\begin{figure*}[!ht]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{fig4a_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig4a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig4b_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig4b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig4c_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig4c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig4d_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig4d_right.pdf}
\end{tabular}
\caption{De-seasoning of MPC patterns for groups with the same strengths: $s=200$ (a), 400 (b), 800 (c), and 1600 (d). The original and rescaled distributions of burstiness are plotted in Fig.~\ref{fig:call_burstiness}(b).}
\label{fig:call_strength_merge}\end{figure*}
\subsection{De-seasoning the groups of individuals with the same strength}
Here we analyze the group of individual users with the same strength, i.e. $\Lambda_s\equiv \{i|s_i=s\}$. The averaged event rate of a group is measured by merging individual event rates, precisely by obtaining $n_{\Lambda_s}(t)= \sum_{i\in\Lambda_s} n_i(t)$. Figure~\ref{fig:call_strength_merge} shows the original and the rescaled event rates with $T=1$ day (left) and the original and the rescaled inter-event time distributions with various periods of $T$ (right) for groups with strengths $s=200$, 400, 800, and 1600. The values of burstiness decrease only slightly as $T$ increases, but are smaller than those of the original burstiness, as shown in Fig.~\ref{fig:call_burstiness}(b).
The burstiness of groups of individuals with the same strength is larger than the average values of individual burstiness from $P(B)$ of the same strength. For example, $B_0\approx 0.256$ for the group of strength $s=200$ turns out to be larger than $\int P(B_0)dB_0\approx 0.204$. Regarding to this difference, we would like to note that the de-seasoning of individual event times by means of the averaged event rate may cause systematic errors due to the different circadian and weekly patterns between the individual and the group. For resolving this issue, the various data clustering methods, such as self-organizing maps, can be used to classify users' activity patterns beyond their strengths and then perform the de-seasoning separately for the different groups.
\subsection{De-seasoning the groups of individuals with broad ranges of strength}
For the larger scale analysis, we consider the strength dependent grouping of users, i.e. groups of individual users with a broad range of strengths, denoted by $\Lambda_{m_1,m_2}\equiv \{i| m_1\leq s_i<m_2\}$, as similarly done in~\cite{Zhou2008,Radicchi2009}. The values of $m$s are determined in terms of the ratio to the maximum strength $s_{\rm max}=7911$, see Table~\ref{table:call_group} for the details of the groups. We determine the averaged event rates of the groups and some of them are shown in the left column of Fig.~\ref{fig:call_group}. By means of the event rates, we perform the de-seasoning to get the rescaled inter-event time distributions, see the right column of Fig.~\ref{fig:call_group}. It is found that the values of burstiness initially decrease slightly and then stay constant at relatively large values as $T$ increases, shown in Fig.~\ref{fig:call_burstiness}(c). These results again confirm our conclusions that de-seasoning the circadian and weekly patterns does not wipe out the bursty behavior of human communication patterns.
\begin{table}[!h]
\caption{Strength dependent grouping of individuals in the MPC dataset. For each group, the range of strength, also in terms of the ratio to the maximum strength $s_{\rm max}=7911$, the number of users, and its fraction to the whole population except for the user with $s_{\rm max}$ are summarized.}
\label{table:call_group}
\begin{indented}
\item[]\begin{tabular}{rrr}
\hline
group index & strength range ($\%$) & the number of users ($\%$)\\
\hline
0 & 0-79 (0-1) & 2821103 (54.4)\\
1 & 79-158 (1-2) & 1010923 (19.5)\\
2 & 158-316 (2-4) & 843412 (16.3)\\
3 & 316-632 (4-8) & 418460 (8.1)\\
4 & 632-1265 (8-16) & 89718 (1.7)\\
5 & 1265-2531 (16-32) & 5857 (0.1)\\
6 & 2531-5063 (32-64) & 173 (0.003)\\
7 & 5063-7594 (64-96) & 5 (0.0001)\\
whole & 0-7594 (0-96) & 5189651 (100)\\
\hline
\end{tabular}
\end{indented}
\end{table}
\begin{figure*}[!ht]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{fig5a_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig5a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig5b_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig5b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig5c_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig5c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig5d_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig5d_right.pdf}\\
\includegraphics[width=.45\columnwidth]{fig5e_left.pdf}&
\includegraphics[width=.45\columnwidth]{fig5e_right.pdf}
\end{tabular}
\caption{De-seasoning of the MPC patterns for groups of individuals with broad ranges of strength: groups 1 (a), 3 (b), 5 (c), 6 (d), and the whole population (e). For the details of groups, see Table~\ref{table:call_group}. The original and rescaled distributions of burstiness are plotted in Fig.~\ref{fig:call_burstiness}(c).}
\label{fig:call_group}\end{figure*}
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{fig6a.pdf}&
\includegraphics[width=.45\columnwidth]{fig6b.pdf}\\
\includegraphics[width=.45\columnwidth]{fig6c.pdf}&
\includegraphics[width=.45\columnwidth]{fig6d.pdf}\\
\includegraphics[width=.45\columnwidth]{fig6e.pdf}
\end{tabular}
\caption{Power spectra, $P(f)$, of the original and the rescaled event rates with various periods of $T$ for individual users with strengths 200 (a) and 800 (b), for groups with the same strengths of 200 (c) and 800 (d), and for the whole population (e). The circadian and weekly peaks in the original power spectra are successfully removed by the de-seasoning in various ways.}
\label{fig:call_powerSpec}\end{figure*}
\subsection{Power spectra analysis}
In order to see clearly the effect of de-seasoning on the event rates, we compare the power spectra of the rescaled event rates to the original event rates. The power spectrum of the event rate is defined as
\begin{equation}
P(f)=\left|\sum_{t=0}^{T_f}\rho_{\Lambda,T}(t)e^{2\pi ift}\right|^2,
\end{equation}
where $f$ denotes the frequency. In Fig.~\ref{fig:call_powerSpec} we show the results of this comparison, where it is evident that with our de-seasoning methods, the circadian and weekly peaks of the original power spectrum are successfully removed and do not show in the rescaled spectra. In all the cases, ranging from the individual de-seasoning to the whole population de-seasoning, when $T=1$ day, the circadian peak at $1/f=1$ day is removed while the others, i.e. the weekly and monthly peaks remain. For $T=7$ days, the weekly peak at $1/f=7$ days is removed, and so on. This behavior can be understood because the cyclic patterns longer than $T$ will not be affected by de-seasoning with the period $T$.
\section{Summary}\label{sect:summary}
The heavy tails and burstiness of inter-event time distributions in human communication activity are affected by circadian and weekly cycles as well as by correlations rooted in human task execution. To investigate the existence of correlations rooted in human task execution we devised a systematic method to de-season circadian and weekly cycles appearing in human activity and successfully demonstrated their removal. Here the circadian and weekly patterns extracted from the mobile phone call and Short Messages records are used to rescale the timings of events, i.e. the time is dilated or contracted during high or low call or SM activity of individual service users, respectively.
We have found that after de-seasoning circadian and weekly cycles driven inhomogeneities, the heavy tails and burstiness of inter-event time distributions still remain. Hence our results imply that the heavy tails and burstiness are not only the consequence of circadian and weekly cycles but also due to correlations in human task execution. In addition, we calculated the Fano and Allan factors~\cite{Anteneodo2010} for the de-seasoned mobile phone communication data, which yielded further evidence of the remaining burstiness in both mobile phone call and Short Messages activities of human communication. Although beyond the scope of the current study, as the next step one needs to focus on describing the mechanisms of the remaining burstiness in human communication patterns, served well by building models and analyzing their results.
\section*{Appendix}\label{appendix}
We also investigated the effect of circadian and weekly cycles on the heavy-tailed inter-event time distribution by using Short Message (SM) dataset from the same European operator mentioned in the main text. We have only retained links with bidirectional interaction, yielding $N=4.2\times 10^6$ users, $L=8.5\times 10^6$ links, and $C=114\times 10^6$ events (SMs). We have merged some consecutive SMs sent by one user to another within $10$ seconds into one SM event because one longer message can be divided into many short messages due to the length limit of a single SM ($160$ characters)~\cite{Kovanen2009}.
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{figAppend1a_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend1a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend1b_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend1b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend1c_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend1c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend1d_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend1d_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend1e_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend1e_right.pdf}
\end{tabular}
\caption{De-seasoning SM patterns of individual users: the original and the rescaled event rates with period of $T=1$ day (left) and the original and the rescaled inter-event time distributions with various periods of $T$ (right). Individual users with the strength $s_i=200$ (a), 400 (b), 800 (c), 1600 (d), and 3200 (e) are analyzed. The original inter-event time distribution of the whole population is also plotted as a dashed curve for comparison.}
\label{fig:SMS_individual}\end{figure*}
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{figAppend2a_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend2a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend2b_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend2b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend2c_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend2c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend2d_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend2d_right.pdf}
\end{tabular}
\caption{Distributions $P(B_T)$ of the original and rescaled burstiness of invididual users with the same strength (left) and distributions $P(\Delta B_T)$ of the difference in burstiness, defined as $\Delta B_T=B_T-B_0$ (right). The individual users with the strengths $s_i=200$ (a), 400 (b), 800 (c), and 1600 (d) are analyzed. The numbers of users are correspondingly 1434, 344, 62, and 10.}
\label{fig:SMS_strength_sep}\end{figure*}
\begin{figure*}[!h]
\begin{tabular}{ccc}
\includegraphics[width=.3\columnwidth]{figAppend3a.pdf}&
\includegraphics[width=.3\columnwidth]{figAppend3b.pdf}&
\includegraphics[width=.3\columnwidth]{figAppend3c.pdf}
\end{tabular}
\caption{Burstiness $B_T$ as a function of period of $T$: (a) the average and the standard deviation of $B_T$ obtained from the burstiness distribution in Fig.~\ref{fig:SMS_strength_sep}, (b) burstiness from groups with the same strength, and (c) burstiness from groups with broad ranges of strength.}
\label{fig:SMS_burstiness}\end{figure*}
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{figAppend4a_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend4a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend4b_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend4b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend4c_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend4c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend4d_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend4d_right.pdf}
\end{tabular}
\caption{De-seasoning of SM patterns for groups with the same strengths: $s=200$ (a), 400 (b), 800 (c), and 1600 (d). The original and rescaled distributions of burstiness are plotted in Fig.~\ref{fig:SMS_burstiness}(b).}
\label{fig:SMS_strength_merge}\end{figure*}
\begin{table}[!h]
\caption{Strength dependent grouping in the SM dataset. For each group, the range of strength, also in terms of the ratio to the maximum strength $s_{\rm max}=37925$, the number of users, and its fraction to the whole population except for two most active users are summarized.}
\label{table:SMS_group}
\begin{indented}
\item[]\begin{tabular}{rrr}
\hline
group index & strength range ($\%$) & the number of users ($\%$)\\
\hline
0 & 0-94 (0-0.25) & 3685116 (87.2)\\
1 & 94-189 (0.25-0.5) & 294181 (6.9)\\
2 & 189-379 (0.5-1) & 152893 (3.6)\\
3 & 379-758 (1-2) & 63379 (1.5)\\
4 & 758-1517 (2-4) & 22507 (0.53)\\
5 & 1517-3034 (4-8) & 7505 (0.18)\\
6 & 3034-6068 (8-16) & 2393 (0.057)\\
7 & 6068-12136 (16-32) & 316 (0.007)\\
8 & 12136-24272 (32-64) & 30 (0.001)\\
whole & 0-24272 (0-64) & 4228320 (100)\\
\hline
\end{tabular}
\end{indented}
\end{table}
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{figAppend5a_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend5a_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend5b_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend5b_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend5c_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend5c_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend5d_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend5d_right.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend5e_left.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend5e_right.pdf}
\end{tabular}
\caption{De-seasoning of the SM patterns for groups of individuals with broad ranges of strength: groups 1 (a), 3 (b), 5 (c), 7 (d), and the whole population (e). For the details of groups, see Table~\ref{table:SMS_group}. The original and rescaled distributions of burstiness are plotted in Fig.~\ref{fig:SMS_burstiness}(c).}
\label{fig:SMS_group}\end{figure*}
\begin{figure*}[!h]
\begin{tabular}{cc}
\includegraphics[width=.45\columnwidth]{figAppend6a.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend6b.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend6c.pdf}&
\includegraphics[width=.45\columnwidth]{figAppend6d.pdf}\\
\includegraphics[width=.45\columnwidth]{figAppend6e.pdf}
\end{tabular}
\caption{Power spectra, $P(f)$, of the original and the rescaled event rates with various periods of $T$ for individual users with strengths 200 (a) and 800 (b), for groups with the same strengths of 200 (c) and 800 (d), and for the whole population (e). The circadian and weekly peaks in the original power spectra are successfully removed by the de-seasoning in various ways.}
\label{fig:SMS_powerSpec}\end{figure*}
We apply the same de-seasoning analysis described in the main text to the SM dataset. Figure~\ref{fig:SMS_individual} shows the various circadian activity patterns of individual users. The main difference from the MPC dataset is that the activity peak is around 11 PM. This feature becomes evident if the averaged event rates are obtained from the same strength groups or from the groups with broad ranges of strength, shown in the left columns of Figs~\ref{fig:SMS_strength_merge} and~\ref{fig:SMS_group}. For the details of the strength dependent grouping, see Table~\ref{table:SMS_group}.
The inter-event time distributions and their values of burstiness are also compared. At first, the distributions cannot be described by the simple power-law form but by the bimodal combination of power-law and Poisson distributions as suggested by~\cite{Wu2010}. As shown in Fig.~\ref{fig:SMS_burstiness}, the values of burstiness slowly decrease as the period $T$ increases up to 7 days, which implies that de-seasoning the circadian and weekly patterns does not considerably affect the bursty behavior.
Finally, we perform the power spectrum analysis for SM dataset in Fig.~\ref{fig:SMS_powerSpec} and find again that the cyclic patterns longer than $T$ cannot be removed by the de-seasoning with period $T$. All these results confirm our conclusion that the heavy tail and burstiness are not only the consequence of circadian and other longer cycle patterns but also due to other correlations, such as human task execution.
\section*{Acknowledgements}
Financial support by Aalto University postdoctoral program (HJ), from EU's 7th Framework Program's FET-Open to ICTeCollective project no. 238597 and by the Academy of Finland, the Finnish Center of Excellence program 2006-2011, project no. 129670 (MK, KK, JK), as well as by TEKES (FiDiPro) (JK) are gratefully acknowledged. The authors thank A.-L.~Barab\'asi for the mobile phone communication dataset used in this research.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,091 |
Q: Class / Function not returning column data I have a Class to manage the users and everything was working fine until I had to add more columns to the table and modify 2 names. Now one of the modified names, the function that returns the column content is not working, the colum is fill with data but it's not printing anything when I call the function.
I rechecked many times the code, looking for a bad typed name or something, but everything seems to be fine, I can't find the problem...
This is my class:
require_once('aet.php');
class staff {
private $aet;
private $not_working_column;
private $working_column;
public function __construct() {
$this->aet = new aet();
}
private function generate($staff) {
$this->not_working_column = $staff->not_working_column;
$this->working_column = $staff->working_column;
}
private function addInformation($stmt) {
$i = 0;
$stmt->bind_result($not_working_column, $working_column);
while ($stmt->fetch()) {
$arrayStaff[$i] = new staff();
$arrayStaff[$i]->setNotWorkingColum($not_working_column);
$arrayStaff[$i]->setWorkingColumn($working_column);
$i++;
}
}
public function StaffFromEmail($email) {
$mysqli = $this->aet->getAetSql();
if ($stmt = $mysqli->prepare("SELECT * FROM staff WHERE email = ? LIMIT 1")) {
$stmt->bind_param('s', $email);
$stmt->execute();
$stmt->store_result();
if ($stmt->num_rows == 0) {
$exit = false;
}
else {
$arrayStaff = $this->addInformation($stmt);
$this->generate($arrayStaff[0]);
$exit = true;
}
}
return $exit;
}
public function setNotWorkingColum($not_working_column) {
$this->not_working_column = $not_working_column;
}
public function getNotWorkingColum() {
return $this->not_working_column;
}
public function setWorkingColumn($working_column) {
$this->working_column = $working_column;
}
public function getWorkingColumn() {
return $this->working_column;
}
}
And in the form where users can update their info
<div class="item">
<div class="input">
<input type="text" placeholder="" name="staff_info[]" value="<?php echo $staff->getNotWorkingColum(); ?>" />
</div>
</div>
<div class="item">
<div class="input">
<input type="text" placeholder="" name="staff_info[]" value="<?php echo $staff->getWorkingColumn(); ?>" />
</div>
</div>
I've also made a video https://www.youtube.com/watch?v=9S_Uw7IK_xY
A: My fault, while changing the name of a column I missed one:
public function setPersonalPhone($personal_phone) {
$this->phone = $personal_phone;
}
Should be:
public function setPersonalPhone($personal_phone) {
$this->personal_phone = $personal_phone;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 0 |
import mappyfile
# -- General configuration ------------------------------------------------
# If your documentation needs a minimal Sphinx version, state it here.
#
# needs_sphinx = '1.0'
# Add any Sphinx extension module names here, as strings. They can be
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
extensions = [
'sphinx.ext.autodoc',
'sphinx.ext.napoleon'
]
# Add any paths that contain templates here, relative to this directory.
templates_path = ['_templates']
# The suffix(es) of source filenames.
# You can specify multiple suffix as a list of string:
#
# source_suffix = ['.rst', '.md']
source_suffix = '.rst'
# The encoding of source files.
#
# source_encoding = 'utf-8-sig'
# The master toctree document.
master_doc = 'index'
# General information about the project.
project = u'mappyfile'
copyright = u'2019, Seth Girvin'
author = u'Seth Girvin'
# The version info for the project you're documenting, acts as replacement for
# |version| and |release|, also used in various other places throughout the
# built documents.
#
# The short X.Y version.
version = mappyfile.__version__
# The full version, including alpha/beta/rc tags.
release = mappyfile.__version__
# The language for content autogenerated by Sphinx. Refer to documentation
# for a list of supported languages.
#
# This is also used if you do content translation via gettext catalogs.
# Usually you set "language" from the command line for these cases.
language = None
# There are two options for replacing |today|: either, you set today to some
# non-false value, then it is used:
#
# today = ''
#
# Else, today_fmt is used as the format for a strftime call.
#
# today_fmt = '%B %d, %Y'
# List of patterns, relative to source directory, that match files and
# directories to ignore when looking for source files.
# This patterns also effect to html_static_path and html_extra_path
exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store']
# The reST default role (used for this markup: `text`) to use for all
# documents.
#
# default_role = None
# If true, '()' will be appended to :func: etc. cross-reference text.
#
# add_function_parentheses = True
# If true, the current module name will be prepended to all description
# unit titles (such as .. function::).
#
# add_module_names = True
# If true, sectionauthor and moduleauthor directives will be shown in the
# output. They are ignored by default.
#
# show_authors = False
# The name of the Pygments (syntax highlighting) style to use.
pygments_style = 'sphinx'
# A list of ignored prefixes for module index sorting.
# modindex_common_prefix = []
# If true, keep warnings as "system message" paragraphs in the built documents.
# keep_warnings = False
# If true, `todo` and `todoList` produce output, else they produce nothing.
todo_include_todos = False
# -- Options for HTML output ----------------------------------------------
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
#
html_theme = 'alabaster'
# Theme options are theme-specific and customize the look and feel of a theme
# further. For a list of options available for each theme, see the
# documentation.
#
# html_theme_options = {}
# Add any paths that contain custom themes here, relative to this directory.
# html_theme_path = []
# The name for this set of Sphinx documents.
# "<project> v<release> documentation" by default.
#
# html_title = u'mappyfile v0.1'
# A shorter title for the navigation bar. Default is the same as html_title.
#
# html_short_title = None
# The name of an image file (relative to this directory) to place at the top
# of the sidebar.
#
# html_logo = None
# The name of an image file (relative to this directory) to use as a favicon of
# the docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32
# pixels large.
#
# html_favicon = None
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
html_static_path = ['_static']
# Add any extra paths that contain custom files (such as robots.txt or
# .htaccess) here, relative to this directory. These files are copied
# directly to the root of the documentation.
#
# html_extra_path = []
# If not None, a 'Last updated on:' timestamp is inserted at every page
# bottom, using the given strftime format.
# The empty string is equivalent to '%b %d, %Y'.
#
# html_last_updated_fmt = None
# If true, SmartyPants will be used to convert quotes and dashes to
# typographically correct entities.
#
# html_use_smartypants = True
# Custom sidebar templates, maps document names to template names.
#
#html_sidebars = {'**': ['globaltoc.html','searchbox.html','localtoc.html','relations.html']}
#
html_logo = 'images/roll-36697_640.png'
html_theme_options = {
# 'logo': 'images/roll-36697_640.png',
'page_width': '70%',
'github_user': 'geographika',
'github_repo': 'mappyfile',
'github_button': True,
'github_banner': True
}
# Additional templates that should be rendered to pages, maps page names to
# template names.
#
# html_additional_pages = {}
# If false, no module index is generated.
#
# html_domain_indices = True
# If false, no index is generated.
#
# html_use_index = True
# If true, the index is split into individual pages for each letter.
#
# html_split_index = False
# If true, links to the reST sources are added to the pages.
#
# html_show_sourcelink = True
# If true, "Created using Sphinx" is shown in the HTML footer. Default is True.
#
# html_show_sphinx = True
# If true, "(C) Copyright ..." is shown in the HTML footer. Default is True.
#
# html_show_copyright = True
# If true, an OpenSearch description file will be output, and all pages will
# contain a <link> tag referring to it. The value of this option must be the
# base URL from which the finished HTML is served.
#
# html_use_opensearch = ''
# This is the file name suffix for HTML files (e.g. ".xhtml").
# html_file_suffix = None
# Language to be used for generating the HTML full-text search index.
# Sphinx supports the following languages:
# 'da', 'de', 'en', 'es', 'fi', 'fr', 'hu', 'it', 'ja'
# 'nl', 'no', 'pt', 'ro', 'ru', 'sv', 'tr', 'zh'
#
# html_search_language = 'en'
# A dictionary with options for the search language support, empty by default.
# 'ja' uses this config value.
# 'zh' user can custom change `jieba` dictionary path.
#
# html_search_options = {'type': 'default'}
# The name of a javascript file (relative to the configuration directory) that
# implements a search results scorer. If empty, the default will be used.
#
# html_search_scorer = 'scorer.js'
# Output file base name for HTML help builder.
htmlhelp_basename = 'mappyfiledoc'
# -- Options for LaTeX output ---------------------------------------------
latex_elements = {
# The paper size ('letterpaper' or 'a4paper').
#
# 'papersize': 'letterpaper',
# The font size ('10pt', '11pt' or '12pt').
#
# 'pointsize': '10pt',
# Additional stuff for the LaTeX preamble.
#
# 'preamble': '',
# Latex figure (float) alignment
#
# 'figure_align': 'htbp',
}
# Grouping the document tree into LaTeX files. List of tuples
# (source start file, target name, title,
# author, documentclass [howto, manual, or own class]).
latex_documents = [
(master_doc, 'mappyfile.tex', u'mappyfile Documentation',
u'S Girvin', 'manual'),
]
# The name of an image file (relative to this directory) to place at the top of
# the title page.
#
# latex_logo = None
# For "manual" documents, if this is true, then toplevel headings are parts,
# not chapters.
#
# latex_use_parts = False
# If true, show page references after internal links.
#
# latex_show_pagerefs = False
# If true, show URL addresses after external links.
#
# latex_show_urls = False
# Documents to append as an appendix to all manuals.
#
# latex_appendices = []
# It false, will not define \strong, \code, itleref, \crossref ... but only
# \sphinxstrong, ..., \sphinxtitleref, ... To help avoid clash with user added
# packages.
#
# latex_keep_old_macro_names = True
# If false, no module index is generated.
#
# latex_domain_indices = True
# -- Options for manual page output ---------------------------------------
# One entry per manual page. List of tuples
# (source start file, name, description, authors, manual section).
man_pages = [
(master_doc, 'mappyfile', u'mappyfile Documentation',
[author], 1)
]
# If true, show URL addresses after external links.
#
# man_show_urls = False
# -- Options for Texinfo output -------------------------------------------
# Grouping the document tree into Texinfo files. List of tuples
# (source start file, target name, title, author,
# dir menu entry, description, category)
texinfo_documents = [
(master_doc, 'mappyfile', u'mappyfile Documentation',
author, 'mappyfile', 'One line description of project.',
'Miscellaneous'),
]
# Documents to append as an appendix to all manuals.
#
# texinfo_appendices = []
# If false, no module index is generated.
#
# texinfo_domain_indices = True
# How to display URL addresses: 'footnote', 'no', or 'inline'.
#
# texinfo_show_urls = 'footnote'
# If true, do not generate a @detailmenu in the "Top" node's menu.
#
# texinfo_no_detailmenu = False
# -- Options for Epub output ----------------------------------------------
# Bibliographic Dublin Core info.
epub_title = project
epub_author = author
epub_publisher = author
epub_copyright = copyright
# The basename for the epub file. It defaults to the project name.
# epub_basename = project
# The HTML theme for the epub output. Since the default themes are not
# optimized for small screen space, using the same theme for HTML and epub
# output is usually not wise. This defaults to 'epub', a theme designed to save
# visual space.
#
# epub_theme = 'epub'
# The language of the text. It defaults to the language option
# or 'en' if the language is not set.
#
# epub_language = ''
# The scheme of the identifier. Typical schemes are ISBN or URL.
# epub_scheme = ''
# The unique identifier of the text. This can be a ISBN number
# or the project homepage.
#
# epub_identifier = ''
# A unique identification for the text.
#
# epub_uid = ''
# A tuple containing the cover image and cover page html template filenames.
#
# epub_cover = ()
# A sequence of (type, uri, title) tuples for the guide element of content.opf.
#
# epub_guide = ()
# HTML files that should be inserted before the pages created by sphinx.
# The format is a list of tuples containing the path and title.
#
# epub_pre_files = []
# HTML files that should be inserted after the pages created by sphinx.
# The format is a list of tuples containing the path and title.
#
# epub_post_files = []
# A list of files that should not be packed into the epub file.
epub_exclude_files = ['search.html']
# The depth of the table of contents in toc.ncx.
#
# epub_tocdepth = 3
# Allow duplicate toc entries.
#
# epub_tocdup = True
# Choose between 'default' and 'includehidden'.
#
# epub_tocscope = 'default'
# Fix unsupported image types using the Pillow.
#
# epub_fix_images = False
# Scale large images.
#
# epub_max_image_width = 0
# How to display URL addresses: 'footnote', 'no', or 'inline'.
#
# epub_show_urls = 'inline'
# If false, no index is generated.
#
# epub_use_index = True
from pygments.lexer import RegexLexer, bygroups,combined, include
from pygments.token import *
from pygments import highlight
from pygments.formatters import HtmlFormatter
class WKTLexer(RegexLexer):
name = 'wkt'
aliases = ['wkt']
filenames = ['*.wkt']
tokens = {
'root': [
(r'\s+', Text),
(r'[{}\[\]();,-.]+', Punctuation),
(r'(PROJCS)\b', Generic.Heading),
(r'(PARAMETER|PROJECTION|SPHEROID|DATUM|GEOGCS|AXIS)\b', Keyword),
(r'(PRIMEM|UNIT|TOWGS84)\b', Keyword.Constant),
(r'(AUTHORITY)\b', Name.Builtin),
(r'[$a-zA-Z_][a-zA-Z0-9_]*', Name.Other),
(r'[0-9][0-9]*\.[0-9]+([eE][0-9]+)?[fd]?', Number.Float),
(r'0x[0-9a-fA-F]+', Number.Hex),
(r'[0-9]+', Number.Integer),
(r'"(\\\\|\\"|[^"])*"', String.Double),
(r"'(\\\\|\\'|[^'])*'", String.Single),
]
}
import re
builtins = (r'(ANNOTATION|'
r'AUTO|BEVEL|BITMAP|BUTT|CARTOLINE|CC|CENTER|'
r'CHART|CIRCLE|CL|CONTOUR|CR|CSV|'
r'DEFAULT|DD|ELLIPSE|EMBED|FALSE|FEET|FOLLOW|'
r'GIANT|HATCH|HILITE|INCHES|KERNELDENSITY|KILOMETERS|LARGE|LC|'
r'LEFT|LINE|LL|LOCAL|LR|MEDIUM|METERS|MILES|MITER|MULTIPLE|MYGIS|MYSQL|NONE|'
r'NORMAL|OFF|OGR|ON|ONE-TO-ONE|ONE-TO-MANY|ORACLESPATIAL|'
r'PERCENTAGES|PIXMAP|PIXELS|POINT|POLYGON|POSTGIS|POSTGRESQL|'
r'PLUGIN|QUERY|RASTER|RIGHT|ROUND|SDE|SELECTED|SIMPLE|SINGLE|'
r'SMALL|SQUARE|TINY|TRIANGLE|TRUE|TRUETYPE|UC|UL|UNION|UR|UV_ANGLE|UV_MINUS_ANGLE|UV_LENGTH|UV_LENGTH_2|UVRASTER|VECTOR|'
r'WFS|WMS|ALPHA|'
r'GIF|JPEG|JPG|PNG|WBMP|SWF|PDF|GTIFF|PC256|RGB|RGBA|INT16|FLOAT32|GD|'
r'AGG|CAIRO|PNG8|SVG|KML|KMZ|GDAL|UTFGRID'
r')\b')
keywords = (r'(ALIGN|'
r'ALPHACOLOR|ANCHORPOINT|ANGLE|ANTIALIAS|AUTHOR|BACKGROUNDCOLOR|BACKGROUNDSHADOWCOLOR|'
r'BACKGROUNDSHADOWSIZE|BANDSITEM|BROWSEFORMAT|BUFFER|CHARACTER|CLASS|CLASSITEM|'
r'CLASSGROUP|CLUSTER|COLOR|COLORRANGE|COMPOSITE|COMPOP|COMPFILTER|CONFIG|CONNECTION|CONNECTIONOPTIONS|CONNECTIONTYPE|DATA|DATAPATTERN|DATARANGE|DEBUG|'
r'DRIVER|DUMP|EMPTY|ENCODING|END|ERROR|EXPRESSION|EXTENT|EXTENSION|FEATURE|'
r'FILLED|FILTER|FILTERITEM|FOOTER|FONT|FONTSET|FORCE|FORMATOPTION|FROM|GAP|GEOMTRANSFORM|'
r'GRID|GRIDSTEP|GRATICULE|GROUP|HEADER|IMAGE|IMAGECOLOR|IMAGETYPE|IMAGEQUALITY|IMAGEPATH|'
r'IMAGEURL|INCLUDE|INDEX|INITIALGAP|INTERLACE|INTERVALS|JOIN|KEYIMAGE|KEYSIZE|KEYSPACING|LABEL|'
r'LABELCACHE|LABELFORMAT|LABELITEM|LABELMAXSCALE|LABELMAXSCALEDENOM|'
r'LABELMINSCALE|LABELMINSCALEDENOM|LABELREQUIRES|LATLON|LAYER|LEADER|LEGEND|'
r'LEGENDFORMAT|LINECAP|LINEJOIN|LINEJOINMAXSIZE|LOG|MAP|MARKER|MARKERSIZE|'
r'MASK|MAXARCS|MAXBOXSIZE|MAXDISTANCE|MAXFEATURES|MAXINTERVAL|MAXSCALE|MAXSCALEDENOM|MINSCALE|'
r'MINSCALEDENOM|MAXGEOWIDTH|MAXLENGTH|MAXSIZE|MAXSUBDIVIDE|MAXTEMPLATE|'
r'MAXWIDTH|METADATA|MIMETYPE|MINARCS|MINBOXSIZE|MINDISTANCE|'
r'MINFEATURESIZE|MININTERVAL|MINSCALE|MINSCALEDENOM|MINGEOWIDTH'
r'MINLENGTH|MINSIZE|MINSUBDIVIDE|MINTEMPLATE|MINWIDTH|NAME|OFFSET|OFFSITE|'
r'OPACITY|OUTLINECOLOR|OUTLINEWIDTH|OUTPUTFORMAT|OVERLAYBACKGROUNDCOLOR|'
r'OVERLAYCOLOR|OVERLAYMAXSIZE|OVERLAYMINSIZE|OVERLAYOUTLINECOLOR|'
r'OVERLAYSIZE|OVERLAYSYMBOL|PARTIALS|PATTERN|POINTS|POLAROFFSET|POSITION|POSTLABELCACHE|'
r'PRIORITY|PROCESSING|PROJECTION|QUERYFORMAT|QUERYMAP|REFERENCE|REGION|'
r'RELATIVETO|REQUIRES|RESOLUTION|SCALE|SCALEDENOM|SCALETOKEN|SHADOWCOLOR|SHADOWSIZE|'
r'SHAPEPATH|SIZE|SIZEUNITS|STATUS|STYLE|STYLEITEM|SYMBOL|SYMBOLSCALE|'
r'SYMBOLSCALEDENOM|SYMBOLSET|TABLE|TEMPLATE|TEMPLATEPATTERN|TEXT|'
r'TILEINDEX|TILEITEM|TILESRS|TITLE|TO|TOLERANCE|TOLERANCEUNITS|TRANSPARENCY|'
r'TRANSPARENT|TRANSFORM|TYPE|UNITS|UTFDATA|UTFITEM|WEB|WIDTH|WKT|WRAP|IMAGEMODE|VALIDATION|VALUES'
r')\b')
class MapFileLexer(RegexLexer):
name = 'mapfile'
aliases = ['mapfile']
filenames = ['*.map']
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| {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/geometry\/geometry-common-core-15th-edition\/chapter-11-surface-area-and-volume-11-5-volumes-of-pyramids-and-cones-practice-and-problem-solving-exercises-page-729\/5","text":"## Geometry: Common Core (15th Edition)\n\n$200 \\ cm^3$\nThe volume of a square pyramid is: $Volume =\\dfrac{1}{2} \\times b\\times h=\\dfrac{1}{2} \\times S\\times h$ Plug the values of $S$ and $h$ in the formula to obtain: $Volume=\\dfrac{1}{2} \\times (10)^2 \\times (6)\\\\=\\dfrac{1}{2} \\times 600\\\\=200 \\ cm^3$","date":"2020-09-25 03:06:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35187971591949463, \"perplexity\": 457.9436955187429}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400221980.49\/warc\/CC-MAIN-20200925021647-20200925051647-00043.warc.gz\"}"} | null | null |
\section{Introduction}
Metallic glasses (MGs) are principally of great interest as structural materials since they exhibit high strength and hardness. However, most MGs lack ductility, especially under tension where zero ductility prevails. However, in recent years progress has been made in developing bulk metallic glasses (BMGs) exhibiting respectable ductility during cold rolling, bending and compression tests. Upon inhomogeneous deformation, that is at low temperatures and high stresses, the plasticity is manifested by a macroscopic sliding along a localized region called a shear band (SB) having a thickness of about 15 nm or less. It has been found that such SBs contain alternating density changes accompanied by structural changes in the medium range order (MRO) \cite{rosner2014density,schmidt2015quantitative,hieronymus2017shear,Hilke2019275}. The current understanding of how mesoscopic SBs evolve from shear transformation zones (STZs), which are regarded as the main carriers for plasticity in metallic glasses \cite{hassani2019probing}, is that alignments of Eshelby-like quadrupolar stress-field perturbations lead to percolation and thus to SB formation \cite{hieronymus2017shear,dasgupta2012microscopic}. The formation of a SB upon deformation creates an interface between the SB and the matrix. To maintain cohesion at the interface atoms need to be rearranged in the matrix, which consequently should also affect the adjacent matrix regions. Indeed, recent publications report on the existence of so-called shear band affected zones (SBAZs) \cite{cao2009structural,maass2014single,shen2018shear,kuchemann2018shear,liu2018elastic}. To address the relation between the structure of amorphous materials in terms of MRO and their mechanical behavior in more detail, the immediate environment of SBs in BMGs was investigated using fluctuation electron microscopy (FEM) to extract the local information on MRO \cite{treacy1996variable,voyles2002fluctuation,treacy2005fluctuation}. FEM contains information about the four-body correlation of atom pairs (pair-pair correlation function) $g_4(|\vec{r}_1|, |\vec{r}_2|, |\vec{r}|, \theta)$ yielding information about the MRO (cluster size and volume fraction) in amorphous materials \cite{treacy2005fluctuation,Bogle2010}. The obtained MRO profiles measured across SBs from tensile and compressive sides of 3-point bending tests display the impact of the local stress state on the MRO and thus shed more light on the lateral extension of deformation in SB environments.
\section{Methods}
FEM and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) were performed with a Thermo Fisher Scientific FEI Themis 300 G3 transmission electron microscope (TEM) operated at 300\,kV. Nanobeam diffraction patterns (NBDPs) were acquired with parallel illumination using a probe size of 1.3\,nm at full width half maximum (FWHM) in \textmu Probe-STEM mode operated at spotsize 8 with a 50\,\textmu m C2 aperture giving a semi-convergence angle of 0.8\,mrad. The probe size used for the acquisition of the NBDPs was measured directly on the Ceta camera prior to the FEM experiments using Digital Micrograph plugins by D. Mitchell \cite{mitchell2005scripting}. A beam current of 15\,pA and a camera length of 77\,mm were used for the acquisition in combination with an US 2000 CCD camera (Gatan) at binning 4 ($512\times512$ pixels). Data sets consisting of $60\times200$ (Fig.~\ref{fig:FIG2}) or $80\times200$ (Fig.~\ref{fig:FIG3}) individual NBDPs were performed across SBs from tensile and compressive sides of 3-point bending tests. From either 60 or 80 individual NBDPs containing the spatially resolved diffracted intensity $I(|\vec{k}|,\vec{r})$ at a fixed distance $\vec{r}$ from the SB, the normalized variance $V(|\vec{k}|,\vec{r})$ was calculated in the form of the annular mean of variance image ($\Omega_{VImage} (k)$) \cite{daulton2010nanobeam} using
\begin{equation}
V(|\vec{k}|,R = 1.3\,nm) = \frac{\left\langle I^2 (\vec{k},R,\vec{r}) \right\rangle}{\left\langle I (\vec{k},R,\vec{r}) \right\rangle^2}-1
\label{EQ:NVAR}
\end{equation}
where $\left\langle\,\,\right\rangle$ indicates the averaging over different sample positions r or volumes and R denotes the FWHM of the probe and thus the reciprocal space resolution. Since the normalized $V(k,R)$ scales with $1/t$ \cite{yi2011effect,radic2019fluctuation}, a thickness correction was made using the slope of the HAADF signal across the SB \cite{voyles2002fluctuatione}. The individual normalized variances were subsequently plotted against their position. The peak height of the first normalized variance peak was taken as a measure of the MRO volume fraction.
\section{Results}
\begin{figure}
\includegraphics[width=\columnwidth]{FIG1-FINAL.png}%
\caption{Normalized variance profiles of as-cast and deformed states (compressive and tensile side) of Pd$_{40}$Ni$_{40}$P$_{20}$ (a, b) and Vit105 (c, d) showing a reduction in peak height due to deformation.}
\label{fig:FIG1}
\end{figure}
3-point bending test of notched bars were carried out (see supplementary video in Appendix A). A more detailed description is given in reference \cite{geissler2019}. During such deformation tests, the area around the notch is dominated by tensile strain whereas the side opposite to the notch is mainly under compressive strain. In this paper these specific regions are referred to as the tensile and compressive sides, respectively. FIB lamellae containing SBs from each side (tensile and compressive) of the deformed samples were prepared perpendicular to shear steps penetrating through the surfaces. Fig.~\ref{fig:FIG1} shows representative examples of normalized variance profiles from as-cast and deformed matrix states (compressive and tensile side) of Pd$_{40}$Ni$_{40}$P$_{20}$ (Fig.~\ref{fig:FIG1}a,b) and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$ (Vit105) (Fig.~\ref{fig:FIG1}c,d). It is worth noting here that the normalized variance $V(k,R)$ scales with $1/t$, where $t$ is the foil thickness \cite{yi2011effect,radic2019fluctuation,voyles2002fluctuatione}. However, $V(k,R)$ signals of different materials acquired at similar foil thicknesses are directly comparable on an absolute scale. Comparing the peak heights of samples with similar foil thickness $t/\lambda = 0.6$ (Fig.~\ref{fig:FIG1}b,d) as a measure for the MRO volume fraction shows that the MRO in the as-cast state is higher for Vit105 than for Pd$_{40}$Ni$_{40}$P$_{20}$. Moreover, three of the individual variance profiles in Fig.~\ref{fig:FIG1} show a clear reduction in the peak height (by a factor of about 2); however it should be noted that the reduction is less pronounced for the compression side of Pd$_{40}$Ni$_{40}$P$_{20}$ as shown in Fig.~\ref{fig:FIG1}b. This behavior was also observed for severely deformed Pd$_{40}$Ni$_{40}$P$_{20}$ \cite{Zhou2019a,Shahkhali2019}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{FIG2-FINAL.png}%
\caption{HAADF-STEM image showing FIB-prepared lamellae with a shear band in (a) Pd$_{40}$Ni$_{40}$P$_{20}$ and (c) in Vit105 taken from the tensile side of the notched 3-point bending test. Note the shear steps at the surface. The red boxes indicate the regions from which the individual NBDPs were acquired (from top to bottom). The NBDP map has a size of 200 nm x 1000 nm for Pd$_{40}$Ni$_{40}$P$_{20}$ (a) and 300 nm x 2000 nm for Vit105 (b) containing each 60 x 200 individual NBDPs. (b) and (d) display the height of the first normalized variance peaks, calculated from an avergage of 60 NBDPs, versus the distance from the shear band. Note the shallow gradient from left to right. The red profiles display the HAADF intensity indicating the foil thickness along the scanned area.}
\label{fig:FIG2}
\end{figure}
The immediate environments of SBs were inspected by acquiring NBDPs over larger areas. Fig.~\ref{fig:FIG2}a and Fig.~\ref{fig:FIG2}c show HAADF-STEM images of the FIB-prepared lamellae. Since the contrast from the SBs is very faint, their positions are indicated by white dashed lines, ending at the surface steps. The FEM analyses were performed for SBs of the tensile sides in Pd$_{40}$Ni$_{40}$P$_{20}$ and Vit105 corresponding to the red rectangular areas shown in Fig.~\ref{fig:FIG2}a,c. The results; that is, the normalized variances mapped parallel to the dashed grey line are shown in Fig.~\ref{fig:FIG2}b and Fig.~\ref{fig:FIG2}d. A shallow continuous gradient is observed for both BMGs running from one side to the other (Fig.~\ref{fig:FIG2}b and Fig.~\ref{fig:FIG2}d). The peak height, which is taken as a measure for the MRO volume fraction, was reduced by a factor of about 2 after deformation (cf. Fig.~\ref{fig:FIG1}). It is worth noting that thickness effects (see red HAADF profile in Fig.~\ref{fig:FIG2}b and Fig.~\ref{fig:FIG2}d), potentially influencing these gradients in the variance peak heights, were eliminated by a thickness correction \cite{treacy2012examination}. In the case of Fig.~\ref{fig:FIG2}b the HAADF intensity is almost flat showing that there is hardly any influence from the foil thickness. Since there is no plateau level in the variance peak height visible, it is difficult to estimate the lateral extension of such shallow gradients. Fig.~\ref{fig:FIG2}d suggests the extension to be in the range of several microns.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{FIG3-FINAL.png}%
\caption{HAADF-STEM image showing a shear band in (a) Pd$_{40}$Ni$_{40}$P$_{20}$ and (b) Vit 105 taken from the compressive side of the notched bars which were deformed by 3-point bending tests. Note the shear step at the surface in each case. Note the additional surface shear steps (see white arrows in (a)) indicating the existence of more SBs. The red boxes indicate the regions from which the individual NBDPs were acquired (from top to bottom). The NBDP map has a size of 300\,nm x 3000\,nm for Pd$_{40}$Ni$_{40}$P$_{20}$ (a) and 300\,nm x 2000\,nm for Vit105 (b) containing each $80\times200$ individual NBDPs. (b) and (d) display the height of the first normalized variance peaks, calculated from an avergage of 80 NBDPs, versus the distance from the shear band. The red profiles correspond to the HAADF intensity and provide a reference for the foil thickness along the scanned area.}
\label{fig:FIG3}
\end{figure}
In like manner, the regions around the SBs from the compressive side were investigated. Fig.~\ref{fig:FIG3}a and Fig.~\ref{fig:FIG3}c depict HAADF-STEM images of SBs from the compressive side. The results of the FEM analyses are shown in Fig.~\ref{fig:FIG3}b and Fig.~\ref{fig:FIG3}d. In contrast to the tensile side, the measurements of the compressive side display for both BMGs a pronounced dip in the peak height of the normalized variance $V(k,R)$ in the vicinity of the SB. While a symmetric dip with a clear minimum at the SB position is observed for Vit105, the dip minimum for Pd$_{40}$Ni$_{40}$P$_{20}$ is displaced with respect to the SB position. The dip minimum identifies the highest stress concentration in the shear band leading to the greatest reduction of MRO. The lateral extension of the MRO curve is larger than 1\,\textmu m. The influence of the foil thickness is shown by the HAADF signal (red curves in Fig.~\ref{fig:FIG3}b and Fig.~\ref{fig:FIG3}d). Since a thickness correction was carried out on the variance peak heights, the dips in the MRO curves are not thickness artifacts.
\section{Discussion}
In the following we discuss the robustness of this analysis.
\subsection{Error assessment of FEM analysis}
FEM analyses reveal differences in the absolute peak height of the normalized variance $V(k,R)$ in the vicinity of SBs \cite{Hilke2019275} with respect to the undeformed ''as-cast`` matrix by a factor of up to about 2 (Fig.~\ref{fig:FIG1}). Thus, by considering the absolute peak height of the normalized variance as a measure for the MRO volume fraction, changes in the MRO of the SB environment are detectable. Thickness gradients along the SB (parallel to the scan direction) are also an error source since the calculation of the normalized variances results from the averaging of the individual NBDPs along this direction. An error estimation due to such thickness effects (parallel to the SB) can be obtained from the match of very small or high k-values (see 3D plots in the Supplementary Material (Fig. S2)). These errors can be estimated to a maximum of $V = 0.005$ and are thus far below the observed absolute differences in peak height of the normalized variance $V(k,R)$. As the thickness effects perpendicular to the scan direction (across the SB) were already corrected using the HAADF intensities shown in Fig.~\ref{fig:FIG2} and Fig.~\ref{fig:FIG3}, the obtained MRO profiles are free from thickness artifacts.
\subsection{Variance peak height: variations and shapes}
The detected peak height variations of $V(k, R)$ reveal significant differences between the tensile and compressive side of the sample; that is, a pronounced dip of $V(k, R)$ at the SBs of the compressive side. For the tensile side a continuous shallow gradient seems to be characteristic. No clear minimum is visible. While a pronounced $V(k)$-dependence with a clear minimum at the SB position is observed for the compressive side of Vit105, the shape of the $V(k)$-relation for the compressive side of the SBs in Pd$_{40}$Ni$_{40}$P$_{20}$ is asymmetric, showing no overlap between the location of the SB and the minimum of the data. Intuitively, one would expect the minimum to overlap with the position of the SB since the MRO reduction in the SB should be highest. Hidden SBs explain the shift of the dip minimum since two more surface shear steps (see white arrows in Fig.~\ref{fig:FIG3}a) clearly indicate the presence of more SBs within the inspected SB area.
\noindent Different structural changes due to the tension-compression asymmetry in BMGs have also been measured using high-energy synchrotron x-ray scattering \cite{chen2013atomic}. An explanation for the difference in shape of the $V(k)$ curves between compressive and tensile sides may be found in the anharmonicity of the interatomic potential so that atoms or clusters are differently affected at a given stress, i.e. pulling them apart is easier than pushing them together. This fits to the observation that the SBs formed earlier and more abundantly on the tensile side (see Supplementary Video).
Another point that needs to be discussed in connection with the shape of the $V(k)$ curves is the amount of deformation (material flow) carried by each individual SB. This can be estimated by the heights of the shear steps at the surface after deformation. Tab.~\ref{tab:TAB1} lists the heights of the shear steps of each SB investigated in this study. By comparing these data with the FEM results shown in Fig.~\ref{fig:FIG2} and Fig.~\ref{fig:FIG3}, there seems to be no correlation between the shear offset heights and the impact on the SB environment.
Next, we discuss the influence of the testing geometry on the observed deformation behavior. The different geometrical constraint for tension and compression loading defines the complex stress states in the deformation experiments of BMGs (here notched 3-point bending test) and has thus great influence on the deformation behavior (ductile-brittle) \cite{wu2008strength,conner2003shear}. Since similar deformation signatures (MRO gradients in Fig.~\ref{fig:FIG2} and Fig.~\ref{fig:FIG3}) were found for both tested BMGs showing either ductile or brittle behavior, we believe that the observed MRO gradients represent characteristic stress states for the compressive and tensile side of notched 3-point bending tests. In fact, finite element simulations of BMGs under tensile load \cite{chen2013effect,chen2013deformation,chen2014deformation,chen2019effect} showed Von Mieses stress distributions which correspond to the shapes of the $V(K)$ curves observed for the tensile side (Fig.~\ref{fig:FIG2}). This means in conclusion that the MRO memorizes the impact of the stress field.
\begin{table}[htbp]
\caption{Shear step height originating from the investigated SBs.}
\vspace{5pt}
\begin{tabular}{c|c}
Sample / location & Shear offset \\
& height at surface \\
\hline \hline
Pd$_{40}$Ni$_{40}$P$_{20}$ [tensile side] & 130\,nm\\
Vit105 [tensile side] & 40\,nm \\
\hline
Pd$_{40}$Ni$_{40}$P$_{20}$ [compressive side] & 1.7\,\textmu m\\
Vit105 [compressive side] & 200\,nm\\
\end{tabular}
\label{tab:TAB1}
\end{table}
\subsection{Shear-affected zones}
Our results, including previous strain analyses \cite{binkowski2015sub} on individual SBs, show that the SB affected zones are in the range of a few microns as an upper limit. However, shear band environments probed by nanoindentation (hardness), Young's modulus measurements or changes in magnetic domains were reported to extend to $10-160$ microns \cite{cao2009structural,maass2014single,shen2018shear,kuchemann2018shear,liu2018elastic}. Moreover, these experiments also showed an increase in the lateral extension of the affected zones with in-creasing deformation \cite{maass2014single,shen2018shear,pan2011softening}. We see two reasons for this discrepancy: Firstly, the fact that the measurement scales (lateral resolution) are not comparable to each other. Secondly, deformation frequently leads to (hidden) multiple shear banding with SBs branching off the primary one \cite{csopu2020atomic}. Such shear band branches do not always penetrate through the surfaces and thus often remain undetected by more indirect methods (nanoindentation, atomic or magnetic force microscopy). In conclusion our results, having a very good lateral resolution with the ability to visualize the individual SBs, strongly suggest a lateral extension of shear-affected zones in the range of a few microns as an upper limit.
\section{Conclusions}
\noindent Fluctuation electron microscopy revealed a detailed structural picture of the interplay between deformation and MRO structure obtained from two representative BMGs (Pd$_{40}$Ni$_{40}$P$_{20}$ and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$) performed under 3-point bending conditions. (i) Prior to deformation, the amount of MRO was observed to be higher for Vit105 than for Pd$_{40}$Ni$_{40}$P$_{20}$. The degree of MRO was reduced after deformation. Profiling the MRO of shear band environments from compressive and tensile sides revealed characteristic gradients which seem to be material-independent and thus displaying the impact of the local stress state on the MRO. Our results show a lateral extension of shear-affected environments of shear bands in bulk metallic glasses with an upper limit of a few microns.
\section{Acknowledgments}
We gratefully acknowledge financial support by the DFG via SPP 1594 (Topological engineering of ultra-strong glasses, WI 1899/27-2 and GE 1106/11) and WI 1899/29-1 (Coupling of irreversible plastic rearrangements and heterogeneity of the local structure during deformation of metallic glasses, projekt number 325408982). Moreover, DFG funding is acknowledged for TEM equipment via the Major Research Instrumentation Programme under INST 211/719-1 FUGG.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,700 |
Simon Wilderspin
A Look At How Our Players Out On Loan Are Doing
Goalkeeper Rowley has been a consistent performer for the Yeltz whilst Anderson continues to pose a big threat to defences, having scored five goals in just five appearances.
Youngsters Shaun Rowley and Kaiman Anderson continue to impress on loan at Halesowen Town.
Tyrone Barnett moved to Southend just two days after Collins but made an equally impressive impact on his new club, finding the net just five minutes into his debut during his side's 4-2 loss at Swindon Town. The forward has gone on to net three times in as many games for Phil Brown's team.
Collins, who joined the Cobblers on the 5th January, has made a big impression having scored two goals in three games for his new side.
Chris Wilder's side have been in fine form this season and adding the firepower of Collins is sure to boost their chances of promotion to League One.
James Collins has had a productive month down in Northampton as he seeks to add another promotion to his CV.
Caton has signed with National League side Lincoln City and has made an instant impact having made his debut at the weekend in his side's 1-0 win over Guiseley.
Town currently have five players out on loan, with winger James Caton the latest player to temporarily ply his trade away from the Greenhous Meadow.
January is usually a busy month with players coming and going, both on permanent and loan deals. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,247 |
\section{INTRODUCTION}
It is expected that hierarchical gravitational clustering of matter induces
shock waves in baryonic gas in the large-scale structure (LSS) of the Universe \citep{krj96, miniati00}.
Simulations for the LSS formation suggest that
strong shocks ($M_s\ga 10$) form in relatively cooler environments
in voids, filaments, and outside cluster virial radii,
while weak shock ($M_s \la$ several)
are produced by mergers and flow motions in hotter intracluster media (ICMs)
\citep{ryuetal03,psej06,kangetal07,skillman08,hoeft08,vazza09,bruggen12}.
Observationally, the existence of such weak shocks in ICMs has been revealed through
temperature jumps in the X-ray emitting gas and Mpc-scale relics with radio spectra
softening downstream of the shock \citep[see][for reviews]{markevitch07,
feretti12}.
These cosmological shocks are the primary means through which the gravitational
energy released during the LSS formation is dissipated into the gas entropy,
magnetic field, turbulence and nonthermal particles \citep{rkcd08}.
In fact, shocks are ubiquitous in astrophysical environments from the heliosphere to galaxy clusters
and they are thought to be the main `cosmic accelerators' of high energy cosmic-ray (CR) particles \citep{blaeic87}.
In diffusive shock acceleration (DSA) theory,
suprathermal particles are scattered by magnetohydrodynamic (MHD) waves and isotropized in the local wave frames,
and gain energy through multiple crossings of the shock \citep{bell78, dru83, maldru01}.
While most postshock thermal particles are advected downstream,
some suprathermal particles energetic enough to swim against downstream turbulent waves
can cross the shock and be injected into the Fermi first-order process.
Then these streaming CRs generate resonant waves via two-stream instability
and nonresonant waves via CR current-driven instability, which in turn amplify
turbulent magnetic fields in the preshock region \citep{bell78,lucek00, bell04,schure12}.
Thin X-ray synchrotron emitting rims observed in several young supernova
remnants (SNRs) indicate that CR electrons are accelerated to 10-100 TeV and cool radiatively
in the magnetic field of several $100 \muG$ behind
the forward shock \citep[e.g.,][]{parizot06,reynolds12}.
This provides clear evidence for efficient magnetic field amplification during CR acceleration
at strong CR modified shocks.
These plasma physical processes, \ie injection of suprathermal particles into the CR population,
excitation of MHD waves and amplification of turbulent magnetic fields via plasma instabilities,
and further acceleration of CRs via Fermi first-order process are important ingredients of
DSA and should operate
at all types of astrophysical shocks including cosmological shocks in the LSS \citep[e.g.,][]{maldru01, zweibel10,schure12, bruggen12}.
In addition, relativistic particles can be accelerated stochastically by MHD turbulence,
most likely driven in ICMs of merging clusters \citep{petrosian01,cb05,bl07}.
CRs can be also injected into the intergalactic space by radio galaxies \citep{kdlc01}
and through winds from star-forming galaxies \citep{volk99},
and later re-accelerated by turbulence and/or shocks.
Diffuse synchrotron emission from radio halos and relics in galaxy clusters
indicates the presence of GeV electrons gyrating in $\mu$G-level magnetic fields on Mpc scales
\citep[e.g.,][]{ct02,gf04,wrbh10,krj12}.
On the other hand, non-detection of $\gamma$-ray emission from galaxy clusters by Fermi-LAT
and VERITAS observations, combined with radio halo observations, puts rather strong constraints
on the CR proton population and the magnetic field strength in ICMs, if one adopts
the ``hadronic" model, in which inelastic collisions of CR protons with ICM protons produce
the radio emitting electrons and
the $\pi^0$ decay \citep{acke10, ddcb10,jeltema11, arlen12}.
Alternatively, in the ``re-acceleration" model, in which those secondary electrons
produced by p-p collisions are accelerated further by MHD turbulence in ICMs,
the CR proton pressure not exceeding a few \% of the
gas thermal pressure could be consistent with both the Fermi-LAT upper limits from the
GeV $\gamma$-ray flux and the radio properties of cluster halos \citep{brunetti12}.
Recently, amplification of turbulent magnetic fields via plasma instabilities
and injection of CR protons and electrons at non-relativistic
collisonless shocks have been studied, using Particle-in-Cell (PIC) and hybrid plasma
simulations \citep[e.g.][]{riqu09,riqu11,guo10,garat12}.
In PIC simulations, the Maxwell's equations for electric and magnetic fields are solved
along with the equations of motion for ions and electrons, so
the full wave-particle interactions can be followed from first principles.
However, extremely wide ranges of length and time scales need to be resolved mainly
because of the large proton to electron mass ratio.
In hybrid simulations, only the ions are treated kinetically
while the electrons are treated as a neutralizing, massless fluid, alleviating
severe computational requirements.
However, it is still prohibitively expensive to simulate the full extent of DSA
from the thermal energies of background plasma to the relativistic energies of cosmic rays,
following the relevant plasma interactions at the same time.
So we do not yet have full understandings of injection and diffusive scattering of CRs
and magnetic field amplification (MFA) to make precise quantitative predictions for DSA.
Instead, most of kinetic DSA approaches, in which the diffusion-convection equation
for the phase-space distribution of particles is solved, commonly adopt
phenomenological models that may emulate some of those processes
\citep[e.g.,][]{kjg02,bkv09,pzs10,lee12,capri12,kang12}.
Another approximate method is a steady-state Monte Carlo simulation approach, in which
parameterized models for particle diffusion, growth of self-generated MHD turbulence,
wave dissipation and plasma heating are implemented \citep[e.g.,][]{vladimirov08}.
In our previous studies, we performed DSA simulations of CR protons at cosmological shocks,
assuming that the magnetic field strength is uniform in space and constant in time,
and presented the time-asymptotic values
of fractional thermalization, $\delta(M_s)$, and fractional
CR acceleration, $\eta(M_s)$, as a function of the sonic Mach number $M_s$ \citep{kj07, krj09}.
These energy dissipation efficiencies were adopted in a post-processing step
for structure formation simulations in order to estimate the CR generation at cosmological shocks
\citep[e.g.,][]{skillman08, vazza09}.
Recently, \citet{vazza12} have used those efficiencies to include self-consistently
the CR pressure terms in the gasdynamic conservation equations for cosmological simulations.
In this paper, we revisit the problem of DSA efficiency at cosmological shocks,
including phenomenological models for MFA
and drift of scattering centers with Alfv\'en speed in the {\it amplified magnetic field}.
Amplification of turbulent magnetic fields driven by CR streaming instabilities is included through an approximate,
analytic model suggested by \citet{capri12}.
As in our previous works, a thermal leakage injection model and
a Bohm-like diffusion coefficient ($\kappa(p) \propto p$) are adopted as well.
This paper is organized as follows.
The numerical method and phenomenological models for plasma physical processes in DSA theory,
and the model parameters for cosmological shocks are described in Section 2.
We then present the detailed simulation results in Section 3
and summarize the main conclusion in Section 4.
\section{DSA MODEL}
In the diffusion approximation, where the pitch-angle distribution of CRs
is nearly isotropic,
the Fokker-Plank equation of the particle distribution function is reduced to the following
diffusion-convection equation:
\begin{equation}
{\partial f \over \partial t} + (u+u_w) {\partial f \over \partial x}
= {p \over 3} {{\partial (u+u_w)} \over {\partial x}}
{{\partial f} \over {\partial p}}
+ {\partial \over \partial x} \left[\kappa(x,p)
{\partial f \over \partial x} \right],
\label{diffcon}
\end{equation}
where $f(x,p,t)$ is the isotropic part of the pitch-angle averaged CR distribution function,
$\kappa(x,p)$ is the spatial diffusion coefficient
along the direction parallel to the mean magnetic field
and $u_w$ is the drift speed of local Alfv\'enic wave turbulence
with respect to the plasma \citep{skill75}.
Here, we consider quasi-parallel shocks in one-dimensional planar geometry,
in which the mean magnetic field is roughly parallel to the flow direction.
The flow velocity, $u$, is calculated by solving the momentum
conservation equation with dynamical feedback of the CR pressure and self-generated
magnetic fields,
\begin{equation}
{\partial (\rho u) \over \partial t} + {\partial (\rho u^2 + P_g + P_c + P_B) \over \partial x} = 0.
\label{mocon}
\end{equation}
The CR pressure, $P_c$, is calculated self-consistently with the CR distribution function $f$,
while the magnetic pressure, $P_B$, is calculated according to our phenomenological model for
MFA (see Section 2.4)
rather than solving the induction equation \citep{capri09}.
We point that the dynamical effects of magnetic field are not important
with $P_B \la 0.01\rho_0u_s^2$.
The details of our DSA numerical code,
the CRASH (Cosmic-Ray Amr SHock), can be found in \citet{kjg02}.
\subsection{Thermal Leakage Injection}
Injection of protons from the postshock thermal pool into the CR population via wave-particle
interactions is expected to depend on several properties of the shock, including the sonic and Alfv\'enic Mach numbers, the obliquity angle of mean magnetic field, and the strength of pre-existing and
self-excited MHD turbulence.
As in our previous studies, we adopt a simple phenomenological model in which particles
above an
''effective'' injection momentum $p_{\rm inj}$ get
injected to the CR population:
\begin{equation}
p_{\rm inj} \approx 1.17 m_p u_2 \left(1+ {1.07 \over \epsilon_B} \right),
\label{pinj}
\end{equation}
where $\epsilon_B = B_0/B_{\perp}$ is the ratio of
the mean magnetic field along the shock normal, $B_0$, to
the amplitude of the postshock MHD wave turbulence, $B_{\perp}$ \citep{maldru01,kjg02}.
This injection model reflects plasma physical arguments
that the particle speed must be several times larger than
the downstream flow speed, $u_2$, depending on the strength of MHD wave turbulence,
in order for suprathermal particles to leak upstream across the shock transition layer.
Since the physical range of the parameter $\epsilon_B$ is not tightly constrained,
we adopt $\epsilon_B =0.25$ as a canonical value, which results in the injected particle
fraction, $\xi=n_{\rm cr,2}/n_2\sim 10^{-4}-10^{-3}$ for $M_s\ga 3$ (see Figure 3 below).
Previous studies showed that DSA saturates for $\xi \ga 10^{-4}$,
so the acceleration efficiency obtained here may represent an upper limit for
the efficient injection regime \citep[e.g.,][]{kjg02,capri12}.
In fact, this injection fraction is similar to the commonly adopted values for nonlinear
DSA modeling of SNRs \citep[e.g.,][]{bkv09}.
If we adopt a smaller value of $\epsilon_B$ for stronger wave turbulence,
$p_{\rm inj}$ has to be higher, leading to a smaller injection fraction and a lower
acceleration efficiency.
\subsection{Bohm-like Diffusion Model}
In our model, turbulent MHD waves are self-generated efficiently by plasma instabilities driven by
CRs streaming upstream in the shock precursor, so we can assume
that CR particles are resonantly scattered by Alfv\'en waves with fully saturated spectrum.
Then the particle diffusion can be approximated by a Bohm-like diffusion coefficient,
$\kappa_B \sim (1/3) r_g v$,
but with flattened non-relativistic momentum dependence \citep{kj07}:
\begin{equation}
\kappa(x,p) = \kappa^* {B_0 \over B_{\parallel}(x)} \cdot {p \over m_pc},
\label{Bohm}
\end{equation}
where $\kappa^*= m_p c^3/(3eB_0)= (3.13\times 10^{22} {\rm cm^2s^{-1}}) B_0^{-1}$,
and $B_0$ is the magnetic field strength far upstream expressed in units of $\mu$G.
The strength of the parallel component of local magnetic field, $B_{\parallel}(x)$, will be
described in the next section.
Hereafter, we use the subscripts `0', `1', and `2' to denote
conditions far upstream of the shock, immediate upstream and downstream of the subshock, respectively.
\subsection{Magnetic Field Amplification}
It was well known that CRs streaming upstream in the shock precursor excite resonant
Alfv\'en waves with a wavelength ($\lambda$) comparable with the CR gyroradius ($r_g$), and
turbulent magnetic fields can be amplified into the nonlinear regime (\ie $\delta B \gg B_0$) \citep{bell78,lucek00}.
Later, it was discovered that the nonresonant ($\lambda\ll r_g$), fast-growing instability driven
by the CR current ($j_{cr}=e n_{cr} u_s$) can amplify the magnetic field by orders of magnitude,
up to the level consistent with the thin X-ray rims at SNRs \citep{bell04}.
Several plasma simulations have shown that both $B_{\parallel}/B_0$
and $B_{\perp}/B_0$ can increase by a factor of up to $\sim 10-45$ via the Bell's CR current-driven instability \citep{riqu09, riqu10,ohira09}.
Moreover, it was suggested that long-wavelength magnetic fluctuations can grow as well
in the presence of short-scale, circularly-polarized Alfv\'en waves excited by the Bell-type instability \citep{bykov11}.
Recently, \citet{roga12} have also shown that large-scale magnetic fluctuations can grow along
the original field by the $\alpha$ effect driven by the nonresonant instability and
both the parallel and perpendicular components can be further amplified.
There are several other instabilities that may amplify the turbulent magnetic field
on scales greater than the CR gyroradius such as the firehose, filamentation, and acoustic
instabilities \citep[e.g.][]{beresnyak09,dru12,schure12}.
Although Bell's (2004) original study assumed parallel background magnetic field,
it turns out that the non-resonant instability operates for all shocks, regardless of the
inclination angle between the shock normal and the mean background magnetic
field \citep{schure12},
and so the isoptropization of the amplified magnetic field
can be a reasonable approximation \citep{riqu09,roga12}.
Here, we adopt the prescription for MFA due to CR streaming instabilities that was suggested by \citet{capri12},
based on the assumption of isotropization of the amplified magnetic field and the effective Alfv\'en speed in the local, amplified field: $\delta B^2/(8\pi \rho_0 u_s^2)=(2/25)(1-U^{5/4})^2 U^{-1.5}$,
where $\bold{\delta B} = \bold{B} - \bold{B_0}$ and $U= (u_s-u)/u_s$ is the flow speed in the shock rest frame
normalized by the shock speed $u_s$.
In the test-particle regime where the flow structure is not modified,
the upstream magnetic field is not amplified in this model (\ie $U(x)=1$).
In the shock precursor ($x > x_s$, where $x_s$ is the shock position), the MFA factor becomes
\begin{equation}
{\delta B(x)^2 \over B_0^2} = {4\over 25}M_{A,0}^2 { {(1-U(x)^{5/4})^2}\over U(x)^{3/2}},
\label{Bpre}
\end{equation}
where $M_{A,0}= u_s/v_{A,0}$ is the Alfv\'enic Mach number
for the far upstream Alfv\'en speed, and $v_{A,0}= B_0/ \sqrt{4\pi \rho_0}$.
This model predicts that MFA increases with $M_{A,0}$
and the precursor strength (\ie degree of shock modification by CRs) \citep{vladimirov08}.
In the case of a ``moderately modified'' shock, in which the immediate preshock speed is $U_1\approx 0.8$, for example, the amplified magnetic pressure increases to
$\delta B_1^2/8\pi \approx 6.6\times 10^{-3} \rho_0 u_s^2$
and the amplification factor scales as $\delta B_1/B_0 \approx 0.12 M_{A,0}$.
We will show in the next section that the shock structure is modified only moderately
owing to the Alfv\'enic drift, so the magnetic field pressure is less than a few \% of the
shock ram pressure even at strong shocks ($M_s\ga 10$).
For the highest Mach number model considered here, $M_s= 100$, the preshock amplification factor becomes $\delta B_1/B_0 \approx 100$, which is somewhat larger than what was found in the plasma
simulations for the Bell-type current-driven instability \citep{riqu09,riqu10}.
Considering possible MFA beyond the Bell-type instability by other large-scale instabilities
\citep[e.g.][]{bykov11,roga12,schure12}, this level of MFA may not be out of reach.
Note that this recipe is intended to be a heuristic model that may represent qualitatively the
MFA process in the shock precursor.
Assuming that the two perpendicular components of preshock magnetic fields are completely isotropized
and simply compressed across the subshock,
the immediate postshock field strength can be estimated by
\begin{equation}
B_2/B_1=\sqrt{1/3+2/3(\rho_2/\rho_1)^2}.
\label{B2}
\end{equation}
We note that the MFA model described in equations (\ref{Bpre})-(\ref{B2}) is also used
for the diffusion coefficient model given by equation (\ref{Bohm}).
\subsection{Alfv\'enic Drift}
Resonant Alfv\'en waves excited by the cosmic ray streaming are pushed by the
CR pressure gradient ($\partial P_c/\partial x$) and
propagate against the underlying flow in the shock precursor \citep[e.g.][]{skill75,bell78}.
The mean drift speed of scattering centers is commonly approximated as the Alfv\'en speed, \ie
$u_{w,1}(x) \approx + v_A \approx B(x) / \sqrt{4\pi \rho(x)}$, pointing upstream away from the shock,
where $B(x)$ is the local, amplified magnetic field strength estimated by equation (\ref{Bpre}).
For isotropic magnetic fields, the parallel component would be roughly $B_{\parallel}\approx B(x)/\sqrt{3}$.
But we simply use $B(x)$ for the effective Alfv\'en speed,
since the uncertainty in this model is probably greater than the factor of $\sqrt{3}$
(see Section 3 for a further comment on this factor).
In the postshock region the Alfv\'enic turbulence is probably relatively balanced,
so the wave drift can be ignored, that is, $u_{w,2} \approx 0$ \citep{jon93}.
Since the Alfv\'enic drift reduces the velocity difference between upstream and
downstream scattering centers, compared to that of the bulk flow,
the resulting CR spectrum becomes softer than estimated without considering
the wave drift.
Here, we do not consider loss of turbulent magnetic energy and gas heating
due to wave dissipation in order to avoid
introducing additional free parameters to the problem.
\subsection{Set-up for DSA Simulations}
Previous studies have shown that the DSA efficiency depends primarily on the shock sonic Mach number \citep{kangetal07}.
So we considered shocks with a wide range of the sonic Mach number, $M_s=1.5-100$,
propagating into the intergalactic medium (IGM) of different temperature phases, $T_0=10^4-5\times 10^7$ K \citep{krcs05}. Then, the shock speed is given by $u_s=(150\kms)M_s (T_0/10^6{\rm K})^{1/2}$.
We specify the background magnetic field strength by setting the so-called plasma beta, $\beta_P=P_g/P_B$,
the ratio of the gas pressure to the magnetic pressure.
So the upstream magnetic field strength is given as $B_0^2 = 8\pi P_g/\beta_P$,
where $\beta_P \sim 100$ is taken as a canonical value in ICMs \citep[see, e.g.,][]{rkcd08}.
Then, the ratio of the background Alfv\'en speed to the sound speed, $v_{A,0}/c_s=\sqrt{2/(\beta_P \gamma_g)}$
(where $\gamma_g$ is the gas adiabatic index), which determines the significance of Alfv\'enic drift,
depends only on the parameter $\beta_P$.
Moreover, the upstream Alfv\'enic Mach number, $M_{A,0}=u_s/v_{A,0}=M_s\sqrt{\beta_P \gamma_g/2}$, controls the magnetic field
amplification factor as given in equation (\ref{Bpre}).
For $\beta_P=100$ and $\gamma_g=5/3$, the background Aflv\'en speed is about 10 \% of the sound speed,
\ie $v_{A,0}=0.11c_s$ (independent of $M_s$ and $T_0$), and $M_{A,0}=9.1M_s$.
For a higher value $\beta_P$ (\ie weaker magnetic fields), of course,
the Alfv\'enic drift effect will be less significant.
With a fixed value of $\beta_P$, the upstream magnetic field strength
can be specified by the upstream gas pressure, $n_{H,0} T_0$,
as follow:
\begin{equation}
B_0 = 0.28 \muG \left({ {n_{H,0} T_0} \over {10^3\cm3 {\rm K}}}\right)^{1/2}
\left( {100\over \beta_P}\right)^{1/2}.
\label{b0}
\end{equation}
We choose the hydrogen number density, $n_{H,0}=10^{-4} \cm3$,
as the fiducial value to obtain specific
values of magnetic field strength shown in Figures 1 - 2 below.
But this choice does not affects the time asymptotic results shown
in Figures 3 - 4,
since the CR modified shock evolves in a self-similar manner and the time-asymptotic states depend primarily on
$M_s$ and $M_{A,0}$, independent of the specific value of $B_0$.
Since the tension in the magnetic field lines hinders Bell's CR current-driven instability,
MFA occurs if the background field strength satisfies the condition, $B_0 < B_s= (0.87 \muG) (n_{cr}u_s)^{1/2}$ \citep{zweibel10}.
For a typical shock speed of $u_s\sim 10^3 \kms$ formed in the IGM with $n_{H,0}\sim 10^{-6}-10^{-4} \cm3$ with the CR injection fraction, $\xi\sim 10^{-4}-10^{-3}$,
the maximum magnetic field for the growth of nonresonant waves is roughly $B_s\sim 0.1-1 \muG$.
The magnetic field strength estimated by equation (\ref{b0}) is $B_0\approx 0.28\muG$ for $n_{H,0}=10^{-4} \cm3$ and $T_0=10^7$ K (ICMs) and
$B_0\approx 10^{-3} \muG$ for $n_{H,0}=10^{-6} \cm3$ and $T_0=10^4$ K (voids).
Considering the uncertainties in the model and the parameters,
it seems reasonable to assume that MFA via CR streaming instabilities can be effective
at cosmological shocks in the LSS \citep{zweibel10}.
In the simulations, the diffusion coefficient, $\kappa^*$ in equation (\ref{Bohm}), can be normalized with
a specific value of $\kappa_o$.
Then, the related length and time scales are given as $l_o=\kappa_o/u_s$
and $t_o=\kappa_o/u_s^2$, respectively.
Since the flow structure and the CR pressure approach the time-asymptotic self-similar states,
a specific physical value of $\kappa_o$ matters only in the determination
of $p_{\rm max}/m_p c \approx 0.1 u_s^2 t/\kappa^* $ at a given simulation time.
For example, with $\kappa_o=10^6\kappa^*$, the highest momentum reached at time $t$
becomes $p_{\rm max}/m_p c \approx 10^5(t/t_o)$.
It was suggested that non-linear wave damping and dissipation
due to ion-neutral collisions may weaken stochastic scatterings,
leading to slower acceleration and escape of highest energy particles from the shock \citep{pz05}.
Since these processes are not well understood in a quantitative way,
we do not include wave dissipation in the simulations.
Instead we implement a free escape boundary (FEB) at an upstream location
by setting $f(x_{\rm FEB},p)=0$ at $x_{\rm FEB}= 0.5\ l_o$,
which may emulate the escape of the highest energy particles with
the diffusion length, $\kappa(p)/u_s \ga x_{\rm FEB}$.
Under this FEB condition, the CR spectrum and the shock structure including the precursor
approach the time-asymptotic states in the time scale of $t/t_o\sim 1$ \citep{kang12}.
As noted in the introduction, CR protons can be accelerated by
merger and accretion shocks, injected into the intergalactic space by star forming galaxies
and active galaxies, and
accelerated by turbulence. Because of long life time and slow diffusion,
CR protons should be accumulated in the LSS over cosmological times.
So it seems natural to assume that ICMs contains pre-existing populations of CR protons.
But their nature is not well constrained, except that
the pressure of CR protons is less than a few \% of the gas thermal
pressure \citep{arlen12,brunetti12}.
For a model spectrum of pre-existing CR protons,
we adopt a simple power-law form,
$f_0(p)=f_{\rm pre}\cdot (p/p_{\rm inj})^{-s}$ for $p \geq p_{\rm inj}$,
with the slope $s = 4.5$, which corresponds to the slope of the test-particle power-law
momentum spectrum accelerated at $M=3$ shocks.
We note that the slope of the CR proton spectrum
inferred from the radio spectral index (\ie $\alpha_R\approx (s-2)/2 $) of
cluster halos ranges $4.5 \la s \la 5$
\citep[e.g.,][]{jeltema11}.
The amplitude, $f_{\rm pre}$, is set by the ratio of the upstream CR to gas
pressure, $R\equiv P_{c,0}/P_{g,0}$, where $R = 0.05$ is chosen as a canonical value.
Table 1 lists the considered models:
the weak shock models with $T_0\ge10^7$ K,
the strong shock models with $T_0=10^5-10^6$ K,
and the strongest shock models with $T_0=10^4$ K represent shocks formed in hot ICMs,
in the warm-hot intergalactic medium (WHIM) of filaments, and
in voids, respectively.
Simulations start with purely gasdynamic shocks initially at rest at $x_s = 0$.
\section{DSA SIMULATION RESULTS}
Figures 1 - 2 show the spatial profiles of magnetic field strength, $B(x)$, and CR pressure,
$P_c(x)$,
and the distribution function of CRs at the shock location, $g_s(p)$, at $t/t_o=0.5, 1, 2$ for models
without or with pre-existing CRs: from top to bottom panels,
$M_s=3$ and $T_0=5\times 10^7$K,
$M_s=5$ and $T_0=10^7$K,
$M_s=10$ and $T_0=10^6$K,
and $M_s=100$ and $T_0=10^4$K.
Note that the models with $M_s=3-5$ represent shocks formed in hot ICMs, while those
with $M_s=10$ and 100 reside in filaments and voids, respectively.
The background magnetic field strength corresponds to $B_0=0.63, 0.28, 0.089,$ and $8.9\times 10^{-3}\muG$ for the models with
$M_s=3, 5, 10, $ and 100, respectively, for the fiducial value of $n_{H,0}=10^{-4} \cm3$
(see equation (\ref{b0})).
With our MFA model the postshock field can increase to $B_2 \approx 2-3 \muG$ for all these models,
which is similar to the field strengths observed in radio halos and radio relics.
The postshock CR pressure increases with the sonic Mach number, but saturates at $P_{c,2}/(\rho_0 u_s^2) \approx 0.2$ for $M_s\ga 10$.
One can see that the precursor profile and $g_s(p)$ have reached the time-asymptotic states for $t/t_o\ga 1$
for the $M_s=100$ model, while the lower Mach number models are still approaching to steady state at $t/t_o =2$.
This is because in the $M_s=100$ model, by the time $t/t_o\approx 1$ the CR spectrum has extended to
$p_{\rm max}$ that satisfies the FEB condition.
For strong shocks of $M_s=10-100$, the power-law index,
$q \equiv -\partial \ln f/\partial \ln p$, is about 4.3 - 4.4 at $p\sim m_p c$ instead of $q=4$,
because the Alfv\'enic drift steepens the CR spectrum.
For the models with pre-existing CRs in Figure 2, the pre-existing population is important
only for weak shocks with $M_s\la 5$, because the injected population dominates in shocks with
higher sonic Mach numbers.
As mentioned in the Introduction, the signatures of shocks observed in ICMs through X-ray and
radio observations can be interpreted by low Mach number shocks \citep{markevitch07,feretti12}.
In particular, the presence of pre-existing CRs is expect to be crucial in explaining
the observations of radio relics \citep{krj12}.
Figure 3 shows time-asymptotic values of downstream gas pressure, $P_{g,2}$, and CR pressure, $P_{c,2}$,
in units of $\rho_0u_s^2$,
density compression ratios, $\sigma_1=\rho_1/\rho_0$ and $\sigma_2=\rho_2/\rho_0$,
the ratios of amplified magnetic field strengths to background strength, $B_2/B_0$ and $B_1/B_0$,
and postshock CR number fraction, $\xi=n_{\rm cr,2}/n_2$, as a function of $M_s$
for all the models listed in Table 1.
We note that for the models without pre-existing CRs (left column)
two different values of $T_0$ (and so $u_s$) are considered for each of
$M_s=3,$ 4, 5, 10, 30, and 50 models,
in order to explore the dependence on $T_0$ for a given sonic Mach number.
The figure demonstrates that the DSA efficiency and the MFA factor are determined
primarily by $M_s$ and $M_{A,0}$, respectively, almost independent of $T_0$.
For instance, the two Mach 10 models with $T_0=10^5$K (open triangle) and $10^6$K (filled triangle)
show the similar results as shown in the left column of Figure 3.
But note that the curves for $P_{cr,2}$ and $\xi$ increase somewhat unevenly near $M_s\approx 4-7$
for the models with pre-existing CRs in the right column,
because of the change in $T_0$ (see Table 1).
At weak shocks with $M_s\la 3$, the injection fraction is $\xi \la 10^{-4}$ and the CR pressure is
$P_{c,2}/\rho_0u_s^2 \la 5\times 10^{-3}$ without pre-existing CRs, while both values
depend on $P_{c,1}$ in the presence of pre-existing CRs.
Since the magnetic field is not amplified in the test-particle regime,
these results remain similar to what we reported earlier in \citet{kr11}.
For a larger value of $\epsilon_B$,
the injection fraction and the CR acceleration efficiency would increase.
As shown in \citet{kj07}, however, $\xi$ and $P_{cr,2}/\rho_0u_s^2$ depend sensitively on
the injection parameter $\epsilon_B$ for $M_s\la 5$,
while such dependence becomes weak for $ M_s\ga 10$.
Furthermore, there are large uncertainties in
the thermal leakage injection model especially at weak shocks.
Thus it is not possible nor meaningful to discuss the quantitative dependence of these
results on $\epsilon_B$, until we obtain more realistic pictures of
the wave-particle interactions through PIC or hybrid plasma
simulations of weak collisionless shocks.
In the limit of large $M_s$,
the postshock CR pressure saturates at $P_{c,2}\approx 0.2 \rho_0u_s^2$,
the postshock density compression ratio at $\sigma_2 \approx 5$,
and the postshock CR number fraction at $\xi\approx 2\times 10^{-3}$.
The MFA factors are $B_1/B_0\sim 0.12 M_{A,0} \sim M_s$ and $B_2/B_0\sim 3M_s$ for $M_s\ga 5$,
as expected from equation (\ref{Bpre}).
In \citet{kangetal07} we found that $P_{c,2}\approx 0.55 \rho_0u_s^2$ in the limit of large $M_s$,
when the magnetic field strength was assumed to be uniform in space and constant in time.
Here we argue that MFA and Alfv\'enic drift in the amplified magnetic field steepen the CR spectrum
and reduce the DSA efficiency drastically.
Again, the presence of pre-existing CRs (right column) enhances the injection fraction
and acceleration efficiency at weak
shocks of $M_s\la 5$, while it does not affect the results at stronger shocks.
Since the upstream CR pressure is $P_{c,0} = 0.05 P_{g,0}=(0.03/M_s) \rho_0 u_s^2$ in these models,
the enhancement factor, $P_{c,2}/P_{c,0} \approx 1.5-6$ for $M_s \le 3$.
So the DSA acceleration efficiency exceeds only slightly the adiabatic compression factor,
$\sigma_2^{\gamma_c}$, where $\gamma_c\approx 4/3$ is the adiabatic index of the CR population.
As in \citet{kangetal07}, the gas thermalization and CR acceleration
efficiencies are defined as the ratios of the gas thermal and CR energy fluxes to the shock kinetic energy flux:
\begin{eqnarray}
\delta(M_s) \equiv {{[e_{g,2}-e_{g,0}(\rho_2/\rho_0)^{\gamma_g}]u_2 } \over {(1/2)\rho_0 u_s^3}},~~
\eta(M_s) \equiv {{[e_{c,2}-e_{c,0}(\rho_2/\rho_0)^{\gamma_c}]u_2 } \over {(1/2)\rho_0 u_s^3}},
\label{eff1}
\end{eqnarray}
where $e_{g}$ and $e_{c}$ are the gas thermal and CR energy densities.
The second terms inside the brackets subtract the effect of adiabatic compression occurred at the shock.
Alternatively, the energy dissipation efficiencies not excluding the effect of adiabatic compression
across the shock can be defined as:
\begin{eqnarray}
\delta^{'}(M_s) \equiv {{[e_{g,2}u_2-e_{g,0}u_0] } \over {(1/2)\rho_0 u_s^3}},~~
\eta^{'}(M_s) \equiv {{[e_{c,2}u_2-e_{c,0}u_0] } \over {(1/2)\rho_0 u_s^3}},
\label{eff2}
\end{eqnarray}
which may provide more direct measures of the energy generation at the shock.
Note that $\eta=\eta^{'}$ for the models with $P_{c,0}=0$.
Figure 4 shows these dissipation efficiencies for all the models listed in Table 1.
Again, the CR acceleration efficiency saturates at $\eta \approx 0.2$ for $M_s\ga 10$,
which is much lower than what we reported in the previous studies without MFA \citep{ryuetal03, kangetal07}.
The CR acceleration efficiency is $\eta <0.01$ for weak shocks ($M_s\la 3$) if there is
no pre-existing CRs.
But the efficiency $\eta^{'}$ can be as high as 0.1 even for these weak shocks,
depending on the amount of pre-existing CRs.
The efficiency $\eta$ for weak shocks is not affected by the new models of MFA and
Alfv\'enic drift,
since the magnetic field is not amplified in the test-particle regime.
If we choose a smaller value of $\beta_P$, the ratio $v_{A,0}/c_s$ is larger, leading to
less efficient acceleration due to the stronger Afv\'enic drift effects.
For example, for $\beta_P\sim 1$ (\ie equipartition
fields), which is relevant for the interstellar medium in galaxies,
the CR acceleration efficiency in the strong shock limit reduces to $\eta \approx 0.12$ \citep{kang12}.
On the other hand, if we were to choose a smaller wave drift speed, the CR efficiency $\eta$ will increase slightly.
For example, if we choose $u_w \approx 0.3 v_A$ instead of $u_w \approx v_A$, the value of
$\eta$ in the high Mach number limit would increase to $\sim 0.25$ for the models considered here.
On the other hand, if we choose a smaller injection parameter, for example, $\epsilon_B=0.23$,
the injection fraction reduces from $\xi =2.1\times 10^{-4}$ to $6.2\times10^{-5}$ and
the postshock CR pressure decreases from $P_{c,2}/\rho_0 u_s^2=0.076$ to $0.043$ for
the $M_s=5$ model,
while $\xi =2.2\times 10^{-3}$ to $3.3\times10^{-4}$
and $P_{c,2}/\rho_0 u_s^2=0.18$ to $0.14$ for the $M_s=50$ model.
Considering that the CR injection fraction obtained in these simulations ($\xi > 10^{-4}$)
is in the saturation limit of DSA, the CR acceleration efficiency, $\eta$, for $M\ga 10$
in Figure 4 should be regarded as an upper limit.
\section{SUMMARY}
We revisited the nonlinear DSA of CR protons at cosmological shocks in the LSS,
incorporating some phenomenological models for MFA due to CR streaming
instabilities and Alfv\'enic drift in the shock precursor.
Our DSA simulation code, CRASH, adopts the Bohm-like diffusion and
thermal leakage injection of suprathermal particles into the CR population.
A wide range of preshock temperature, $10^4\le T_0 \le 5\times 10^7$K, is considered
to represent shocks that form in
clusters of galaxies, filaments, and voids.
We found that the DSA efficiency is determined mainly by the sonic Mach number $M_s$,
but almost independent of $T_0$.
We assumed the background intergalactic magnetic field strength, $B_0$,
that corresponds to the plasma beta $\beta_P=100$.
This is translated to the ratio of the Alfv\'en speed in the background magnetic field to
the preshock sound speed, $v_{A,0}/c_s=\sqrt{6/5\beta_P} \approx 0.11$.
Then the Alfv\'enic Mach number $M_{A,0}=\sqrt{5 \beta_P/6}\ M_s$ determines the extent of MFA (\ie $B_1/B_0$),
which in turn controls the significance of Alfv\'enic drift in DSA.
Although the preshock density is set to be $n_{H,0}=10^{-4} \cm3$ just to give a characteristic scale to the magnetic field strength in the IGM,
our results for the CR proton acceleration, such as the dissipation efficiencies, do not depend on
a specific choice of $n_{H,0}$.
If one is interested in CR electrons, which are affected by synchrotron and inverse Compton cooling,
the electron energy spectrum should depend on the field strength $B_0$ and
so on the value of $n_{H,0} T_0$ (see equation (\ref{b0})).
The main results of this study can be summarized as follows:
1) With our phenomenological models for DSA,
the injected fraction of CR particles is $\xi\approx 10^{-4}-10^{-3}$
and the postshock CR pressure becomes $ 10^{-3}\la P_{c,2}/(\rho_0 u_s^2) \la 0.2$
for $3\le M_s\le 100$, if there are no pre-existing CRs.
A population of pre-existing CRs provides seed particles to the Fermi process,
so the injection fraction and acceleration efficiency increase
with the amount of pre-existing CRs at weak shocks.
But the presence of pre-existing CRs does not affect $\xi$ nor $P_{c,2}$ for strong
shocks with $M_s\ga 10$, in which the freshly injected particles dominate over the re-accelerated ones.
2) The nonlinear stage of MFA via
plasma instabilities at collisioness shocks is not fully understood yet. So we adopted a model
for MFA via CR streaming instabilities suggested by \citet{capri12}.
We argue that the CR current, $j_{cr}\sim e \xi \sigma_2 n_{H,0} u_s$, is high enough to overcome
the magnetic field tension, so the Bell-type instability can amplify
turbulent magnetic fields at cosmological shocks considered here \citep{zweibel10}.
For shocks with $M\ga 5$, DSA is efficient enough to develop a significant shock precursor
due to the CR feedback,
and the amplified magnetic field strength in the upstream region scales as
$B_1/B_0\approx 0.12 M_{A,0}\approx (\beta_P/100)^{1/2} M_s$.
This MFA model predicts that the postshock magnetic field strength becomes $B_2\approx 2-3 \muG$ for
the shock models considered here (see Table 1).
3) This study demonstrates that if scattering centers drift with the effective Alfv\'en speed
in the local, amplified magnetic field, the CR energy spectrum can be steepened
and the acceleration efficiency is reduced significantly, compared to the cases without MFA.
As a result, the CR acceleration efficiency saturates at $\eta =2e_{c,r}/\rho_0 u_s^3\approx 0.2$
for $M_s \ga 10$,
which is significantly lower than what we reported in our previous study, $\eta \approx 0.55$
\citep{kangetal07}.
We note that the value $\eta$ at the strong shock limit can vary by $\sim 10$ \%,
depending on the model parameters
such as the injection parameter, plasma beta and wave drift speed.
Inclusion of wave dissipation (not considered here) will also
affect the extent of MFA and the acceleration efficiency.
This tells us that detailed understandings of plasma physical processes
are crucial to the study of DSA at astrophysical collisionless shocks.
4) At weak shocks in the test-particle regime ($M_s\la 3$),
the CR pressure is not dynamically important enough to generate significant MHD waves,
so the magnetic field is not amplified and the Alfv\'enic drift effects are irrelevant.
5) Finally, we note that the CR injection and the CR streaming instabilities are
found to be less efficient at quasi-perpendicular shocks \citep[e.g.][]{garat12}.
It is recognized, however, streaming of CRs is facilitated through locally parallel
inclination of turbulent magnetic fields at the shock surface, so the CR injection can be
effective even at quasi-perpendicular shocks in the presence of
pre-existing large-scale MHD turbulence \citep{giacal05,zank06}.
At oblique shocks the acceleration rate is faster and
the diffusion coefficient is smaller due to drift motion of particles along the shock surface
\citep{jokipii87}.
In fact, the diffusion convection equation (\ref{diffcon}) should be valid for quasi-perpendicular
shocks as long as there exists strong MHD turbulence sufficient enough to keep the pitch
angle distribution of particles isotropic.
In that case, the time-asymptotic states of the CR shocks should remain the same
even for much smaller $\kappa(x,p)$, as mentioned in Section 2.5.
In addition, the perpendicular current-driven instability is found to be effective
at quasi-perpendicular shocks \citep{riqu10,schure12}.
Thus we expect that the overall conclusions drawn from this study
should be applicable to all non-relativistic shocks, regardless of the magnetic field inclination
angle, although our quantitative estimates for the CR injection and acceleration efficiencies
may not be generalized to oblique shocks with certainty.
\acknowledgements
HK was supported by Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry
of Education, Science and Technology (2012-001065).
DR was supported by the National Research Foundation of Korea
through grant 2007-0093860.
The authors would like to thank D. Capriloi, T. W. Jones, F. Vazza and
the anonymous referee for the constructive suggestions and comments to the paper.
HK also would like to thank Vahe Petrosian and KIPAC for their hospitality during
the sabbatical leave at Stanford university where a part of the paper
was written.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,450 |
{"url":"http:\/\/www.mathematicsgre.com\/viewtopic.php?f=1&t=1360","text":"## f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nForum for the GRE subject test in mathematics.\nlawliet\nPosts: 2\nJoined: Mon Apr 15, 2013 11:15 am\n\n### f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nI have difficulties to deal with these kinds of questions. So, thank you for answering!\n\nThere is a missing condition: f has the 12th derivative. So f(x)=1\/2*x^2+x does not satisfy the condition. Sorry!\nLast edited by lawliet on Mon Apr 15, 2013 8:25 pm, edited 1 time in total.\n\ntarheel\nPosts: 10\nJoined: Wed Sep 26, 2012 4:37 pm\n\n### Re: f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nlawliet wrote:I have difficulties to deal with these kinds of questions. So, thank you for answering!\n\nIf this is a multiple choice question and there is only one correct answer, then it's simple. Let $f(x)=\\frac{1}{2}x^2+x$, then $g(x)=\\frac{1}{2}x^{20}+x^{10}$. So $g^{(11)}(x)=\\frac{1}{2}\\cdot 20 \\cdot 19...\\cdot 10 x^9$. So $g^{(11)}(0)=0$.\n\nlawliet\nPosts: 2\nJoined: Mon Apr 15, 2013 11:15 am\n\n### Re: f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\ntarheel wrote:\nlawliet wrote:I have difficulties to deal with these kinds of questions. So, thank you for answering!\n\nIf this is a multiple choice question and there is only one correct answer, then it's simple. Let $f(x)=\\frac{1}{2}x^2+x$, then $g(x)=\\frac{1}{2}x^{20}+x^{10}$. So $g^{(11)}(x)=\\frac{1}{2}\\cdot 20 \\cdot 19...\\cdot 10 x^9$. So $g^{(11)}(0)=0$.\n\nSorry, there is a missing condition that f(x) has the 12th derivative. So, do you have other methods?\n\nwaiting512\nPosts: 61\nJoined: Sat Dec 10, 2011 10:41 pm\n\n### Re: f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nHis example does have a 12th derivative.\n\n2help\nPosts: 12\nJoined: Fri Mar 22, 2013 7:29 pm\n\n### Re: f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nIf you try to find out the pattern in which the 10th derivative of g=f(x^10) expands , you will find that the only term that does not become 0 when you calculate g^(11)(0) is the first term which is 10!f'(x^10). However, when you take the derivative for the 11th time , there is a x^9 comes out of the composite part of it , which finally makes it 0. so the answer is 0.\n\ntarheel\nPosts: 10\nJoined: Wed Sep 26, 2012 4:37 pm\n\n### Re: f'(0)=f''(0)=1, g=f(x^10), the 11th derivative of g at x=0?\n\nlawliet wrote:\ntarheel wrote:\nIf this is a multiple choice question and there is only one correct answer, then it's simple. Let $f(x)=\\frac{1}{2}x^2+x$, then $g(x)=\\frac{1}{2}x^{20}+x^{10}$. So $g^{(11)}(x)=\\frac{1}{2}\\cdot 20 \\cdot 19...\\cdot 10 x^9$. So $g^{(11)}(0)=0$.\n\nSorry, there is a missing condition that f(x) has the 12th derivative. So, do you have other methods?\n\nYeah, waiting512 is right. If $f(x)=\\frac{1}{2}x^2+x$, then $f^{(12)}=0,\\, \\forall x$, which exists. Its derivative being zero doesn't mean the derivative doesn't exist.\n\nReturn to \u201cMathematics GRE Forum: The GRE Subject Test in Mathematics\u201d\n\n### Who is online\n\nUsers browsing this forum: No registered users and 3 guests","date":"2017-08-21 13:52:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 14, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.872536838054657, \"perplexity\": 1350.0819395679662}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886108709.89\/warc\/CC-MAIN-20170821133645-20170821153645-00172.warc.gz\"}"} | null | null |
package com.coderanch.mocklocalmethod;
import org.junit.Test;
import static org.hamcrest.CoreMatchers.is;
import static org.junit.Assert.assertThat;
public class ActualThingsToBeDoneTest {
@Test
public void doAllTheThings_shouldCall_doTheFirstThing_and_doTheSecondThing() {
TestableThingsToBeDone thingsToBeDone = new TestableThingsToBeDone();
thingsToBeDone.doAllTheThings();
assertThat(thingsToBeDone.hasTheFirstThingBeenDone(), is(true));
assertThat(thingsToBeDone.hasTheSecondThingBeenDone(), is(true));
}
private class TestableThingsToBeDone extends ActualThingsToBeDone {
private boolean firstThingDone = false;
private boolean secondThingDone = false;
@Override
protected void doTheFirstThing() {
this.firstThingDone = true;
}
@Override
protected void doTheSecondThing() {
this.secondThingDone = true;
}
public boolean hasTheFirstThingBeenDone() {
return firstThingDone;
}
public boolean hasTheSecondThingBeenDone() {
return secondThingDone;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 144 |
import React from 'react';
import { mount } from 'enzyme';
import Badge from './Badge.component';
describe('Badge', () => {
it('should render by default', () => {
// given
const label = 'my label';
// when
const wrapper = mount(<Badge label={label} />);
// then
expect(wrapper.html()).toMatchSnapshot();
});
it('should render the given children', () => {
// when
const wrapper = mount(
<Badge>
<div test-id="testId">children</div>
</Badge>,
);
// then
expect(wrapper.find('div[test-id="testId"]').text()).toBe('children');
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,537 |
Матч на первенство мира по шахматам между чемпионом мира Тиграном Петросяном и победителем соревнования претендентов Борисом Спасским проходил с 11 апреля по 9 июня 1966 года в Москве.
Регламент матча: 24 партии на большинство, при счёте 12:12 чемпион сохраняет звание. Главный арбитр — Альберик О'Келли (Бельгия).
Выиграв со счётом 12½:11½ (+4 −3 =17), Петросян сохранил звание чемпиона. Впервые после 1934 г. чемпиону удалось выиграть матч у претендента (в этот период либо побеждал претендент, либо матч заканчивался вничью).
Таблица матча
Примечательные партии
Спасский — Петросян
1. d4 Кf6 2. Кf3 e6 3. Сg5 d5 4. Кbd2 Сe7 5. e3 Кbd7 6. Сd3 c5 7. c3 b6 8. O-O Сb7 9. Кe5 К:e5 10. de Кd7 11. Сf4 Фc7 12. Кf3 h6 13. b4 g5 14. Сg3 h5 15. h4 gh 16. Сf4 O-O-O 17. a4 c4 18. Сe2 a6 19. Крh1 Лdg8 20. Лg1 Лg4 21. Фd2 Лhg8 22. a5 b5 23. Лad1 Сf8 24. Кh2 (см. диаграмму)
24 …К:e5 25. К:g4 hg 26. e4 Сd6 27. Фe3 Кd7 28. С:d6 Ф:d6 29. Лd4 e5 30. Лd2 f5 31. ed f4 32. Фe4 Кf6 33. Фf5+ Крb8 34. f3 Сc8 35. Фb1 g3 36. Лe1 h3 37. Сf1 Лh8 38. gh С:h3 39. Крg1 С:f1 40. Кр:f1 e4 41. Фd1 Кg4 42. fg f3 43. Лg2 fg+, 0 : 1
Ссылки
Партии в базе Chessgames
Матчи за звание чемпиона мира по шахматам
1966 год в шахматах
1966 год в Москве | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,033 |
Is Franklin DynaTech A (FKDNX) a Strong Mutual Fund Pick Right Now?
Having trouble finding a Sector - Tech fund? Franklin DynaTech A (FKDNX) is a possible starting point. FKDNX possesses a Zacks Mutual Fund Rank of 3 (Hold), which is based on nine forecasting factors like size, cost, and past performance.
FKDNX is part of the Sector - Tech category, which boasts an array of different possible selections. With a much more diversified approach, Sector - Tech mutual funds give investors a way to own a stake in a notoriously risky sector. Tech companies are in various industries like semiconductors, software, internet, and networking, among others.
FKDNX finds itself in the Franklin Templeton family, based out of San Mateo, CA. The Franklin DynaTech A made its debut in January of 1968 and FKDNX has managed to accumulate roughly $3.38 billion in assets, as of the most recently available information. The fund is currently managed by Rupert H. Johnson who has been in charge of the fund since January of 1968.
Obviously, what investors are looking for in these funds is strong performance relative to their peers. This fund has delivered a 5-year annualized total return of 13.43%, and it sits in the top third among its category peers. If you're interested in shorter time frames, do not dismiss looking at the fund's 3-year annualized total return of 21.04%, which places it in the top third during this time-frame.
When looking at a fund's performance, it is also important to note the standard deviation of the returns. The lower the standard deviation, the less volatility the fund experiences. FKDNX's standard deviation over the past three years is 15.6% compared to the category average of 10.54%. Looking at the past 5 years, the fund's standard deviation is 15.68% compared to the category average of 10.4%. This makes the fund more volatile than its peers over the past half-decade.
One cannot ignore the volatility of this segment, however, as it is always important for investors to remember the downside to any potential investment. In FKDNX's case, the fund lost 47.2% in the most recent bear market and outperformed its peer group by 6.11%. This might suggest that the fund is a better choice than its peers during a bear market.
Investors should not forget about beta, an important way to measure a mutual fund's risk compared to the market as a whole. FKDNX has a 5-year beta of 1.17, which means it is likely to be more volatile than the market average. Because alpha represents a portfolio's performance on a risk-adjusted basis relative to a benchmark, which is the S&P 500 in this case, one should pay attention to this metric as well. FKDNX's 5-year performance has produced a positive alpha of 1.09, which means managers in this portfolio are skilled in picking securities that generate better-than-benchmark returns.
This fund's turnover is about 17.22%, so the fund managers are making fewer trades than the average comparable fund.
For investors, taking a closer look at cost-related metrics is key, since costs are increasingly important for mutual fund investing. Competition is heating up in this space, and a lower cost product will likely outperform its otherwise identical counterpart, all things being equal. In terms of fees, FKDNX is a load fund. It has an expense ratio of 0.86% compared to the category average of 1.31%. FKDNX is actually cheaper than its peers when you consider factors like cost.
Overall, Franklin DynaTech A ( FKDNX ) has a neutral Zacks Mutual Fund rank, strong performance, worse downside risk, and lower fees compared to its peers.
Want even more information about FKDNX? Then go over to Zacks.com and check out our mutual fund comparison tool, and all of the other great features that we have to help you with your mutual fund analysis for additional information. Want to learn even more? We have a full suite of tools on stocks that you can use to find the best choices for your portfolio too, no matter what kind of investor you are. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,509 |
using System;
class FibonacciNumbers
{
static void Main(string[] args)
{
//10. Write a program that reads a number n and prints on the console the first members
//of the Fibonacci sequence (at a single line, separated by comma and space - ,)
//: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ….
Console.Write("Please enter an integer: ");
int number = int.Parse(Console.ReadLine());
decimal firstMember = 0m;
decimal secondMember = 1m;
decimal sum;
for (int i = 0; i < number; i++)
{
Console.Write("{0}, ", firstMember);
sum = firstMember + secondMember;
firstMember = secondMember;
secondMember = sum;
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,160 |
\section{Introduction}
Stellar atmospheres are not in hydrostatic equilibrium.
Convection, magnetic fields, rotation, oscillations, and other phenomena
induce time and spatially changing structure on the surfaces
of stars. In the case of late-type stars, surface convection
is responsible for the granulation pattern directly
observed in high-resolution pictures of the Sun.
Granulation
broadens absorption line profiles in observed solar
spectra, making them asymmetric and introducing
a net blueshift in their central wavelengths.
Even though the typical granular velocities reach several
kilometers per second, the net blueshift observed in
spatially-averaged spectra of solar photospheric
lines rarely exceeds 1 km s$^{-1}$.
In solar-type stars, the photosphere is usually located above
the convectively unstable zone and the velocity field weakens
with height. This causes the net convective shift
to be strongest for lines formed in deep layers,
progressively weakening and then disappearing for lines
formed higher up in the photosphere (Allende Prieto
\& Garc\'{\i}a L\'opez 1998; Pierce \& Lopresto 2000;
Ram\'irez, Allende Prieto \& Lambert 2008; Gray 2009).
Stellar spectra are often used to measure the velocities of stars projected
along the line of sight. The interpretation of the measured line wavelength
shifts in terms of Doppler shifts purely associated with bulk stellar motion
results in biased radial velocities owing to the presence of photospheric
velocity fields. Furthermore, motions in stellar atmospheres are not the only
mechanism that can bias radial velocity measurements, and a number of other
effects, most notably gravitational redshifts (e.g., Pasquini et al. 2011),
frustrate the direct derivation of the line-of-sight component of the velocity
of a star's center of mass from the spectrum. Correcting for all these effects
involves knowledge that is usually uncertain and, in most cases, model
dependent, and therefore the IAU recommends using the term
'radial-velocity measure' to refer to the velocity naively inferred from the
spectral shifts under classical Newtonian mechanics, ignoring the corrections
mentioned earlier (Lindegren \& Dravins 2003).
The ESA mission Gaia is scheduled for launch in late 2013 with the
purpose of measuring proper motions, parallaxes, and spectrophotometry
for some 10$^9$ stars down to $V\sim 20$ mag, and spectra for
about 10$^8$ stars down to $V\sim 17$ mag (de Bruijne et al. 2009;
Lindegren 2010). The typical uncertainty
of the radial-velocity measures provided by Radial Velocity
Spectrometer (RVS) onboard Gaia (Katz et al. 2004; Wilkinson et al. 2005)
will typically be in the range 1--20 km s$^{-1}$, but taking the average
for large groups of stars will largely reduce the uncertainties,
and the systematic errors discussed above can become dominant if neglected.
More important, the RVS does not have any onboard calibration source
to rely on for wavelength calibration, and systematic effects need
to be considered, since wavelength calibration is based on
a subset of the very same stellar spectra obtained by RVS (Katz et al. 2011).
As nature favors the formation of stars with low masses,
we focus our attention on late-type stars, for which systematic
errors affecting the translation
from spectral shifts to bulk
velocities are mainly gravitational redshifts,
convective shifts, and the effect of the transversal motion on the
relativistic Doppler effect. With the parallaxes measured by Gaia,
through the comparison of colors and absolute brightness with
stellar evolution theory, it will be possible to estimate masses and
radii to calculate gravitational redshifts to better than 10 \%
(Allende Prieto \& Lambert 1999), or some 50 m s$^{-1}$ for a solar-like
star. Accurate proper motions and parallaxes will also provide accurate
transversal motions. With the exception of the Sun, it is very
difficult to disentangle
convective shifts from the bulk stellar motion,
and therefore it becomes necessary to rely on models.
We have shown
that modern three-dimensional hydrodynamical models of the solar surface
can accurately predict the convective shifts of individual weak and
moderate-strength photospheric iron lines to within
70 m s$^{-1}$ (Allende Prieto et al. 2009). The
same models, however, predict net redshifts, which are not observed, for
very strong iron lines (Allende Prieto et al. 2009; see also Asplund et
al. 2000). Convective shifts need to be evaluated for specific spectral
windows used in the derivation of radial velocities, ensuring that
models are appropriate for those windows.
In this paper, we focus our
attention on the observations to be obtained by the Gaia RVS and
predict convective wavelength shifts for stars of spectral types M
through A ($3500<T_{\rm eff} <7000$ K) from a collection of
hydrodynamical simulations presented by Ludwig et al. (2009).
As such corrections are specifically derived for the Gaia RVS, and in
particular for radial velocity measures derived by cross-correlation of
RVS spectra with template spectra calculated from classical (hydrostatic)
model atmospheres, we will refer to them, indistinctively, as net convective
shifts or effective convective shifts for RVS spectra.
Section 2 describes the hydrodynamical simulations.
Section 3 is devoted to the spectral synthesis calculations, and \S
4 describes the results. In \S 5 we comment briefly on
the size of the gravitational
shifts expected for the same stars previously modeled, and Section 6
presents a summary of our findings.
\section{Model atmospheres}
\label{models}
Simulations of stellar surface convection for 83 combinations of surface
gravity, entropy flux at the bottom of the atmosphere,
and chemical composition were computed using the
code CO$^5$BOLD (Freytag, Steffen \& Dorch 2002; Wedemeyer et al. 2004;
Freytag et al. 2012).
The code solves the equations of hydrodynamics in a Cartesian grid with
periodic horizontal boundary conditions and accounting for the effect
of the radiation field on the energy balance.
The models typically have a resolution of $140\times140\times150$ (X-Y-Z) grid
points, corresponding to physical sizes significantly larger than
the typical size of granules, and range from a few Mm for dwarfs to
thousands of Mm for giant stars.
The radiation field applying a multi-group technique
(Nordlund 1982) using 5 groups for models of solar and 6 groups for models
of subsolar metallicity.
The number of groups was chosen by trial-and-error to ensure an acceptable
representation of the radiative cooling or heating effects.
This was done primarily by looking at solar-like stars. It turned out,
however, that also giants, as well as hotter and cooler dwarfs could be
treated in the same way obtaining similar accuracies.
It was found that at subsolar metallicities the treatment of the radiative
transfer in the continuum is more critical, so that an extra group was introduced.
We believe that the typically employed five or six groups
are sufficient for
obtaining accurate line shifts, but this is difficult to
demonstrate in general since only for few cases are models with
more groups available. However,
the line shifts hinge very much on the velocity field which is driven in
deeper layers, where the wavelength dependence of the radiative transfer
becomes of secondary importance. In the few models employing 12 groups
that are available, changes in the level of the temperature fluctuations
are not large. We think that rather spatial resolution and sampling
of the time series dominate the uncertainties in the line shifts.
The equation of state takes into account H and He
ionization and H$_2$ formation, while the opacities are calculated with the
Uppsala package (Gustafsson et al. 2008). The physical size of the
computational box is the same for models with the same surface temperature and
gravity. The upper boundary of the models is open, and extended beyond a
Rosseland optical depth of 10$^{-6}$, while
the lower boundary is well into optically
thick, convectively unstable layers. Each
simulation was run until fully relaxed. Chemical abundances are from Grevesse
\& Sauval (1998), with the exception of CNO, which are updated following
Asplund (2005). More details are provided by Ludwig et al. (2009).
For each simulation, a set of uncorrelated
snapshots was selected for spectral synthesis.
The number of snapshots and the total time span of the simulations varied from
star to star, according to the granular turn-over time scale for each
model. The snapshots were hand chosen in order to be representative of the
whole simulated time series, avoiding statistical anomalies.
We found from tests that about 20 uncorrelated snapshots
are enough to represent a model, but in some cases a smaller
number was adequate.
In order to determine the net convective shift predicted for each 3D model, we
calculated a 1D model atmosphere with the same effective temperature, surface
gravity and metallicity. The effective temperature for each simulation was
determined from the temporally and spatially averaged emergent bolometric
flux. The 1D models were calculated with the LHD package (Caffau \& Ludwig
2007), and share the basic ingredients (most importantly the opacities and
equation of state) with the 3D simulations. A second comparison set of 1D
models was derived by linear interpolation of the ODFNEW model grid by
Castelli \& Kurucz\footnote{kurucz.harvard.edu} (2004), based on the solar
abundances of Grevesse \& Sauval (1998).
\begin{figure*}[t!]
\footnotesize
\centering
\includegraphics[width=13cm,angle=90]{f1_allende.ps}
\caption{Observed (black line) and synthetic (1D in red, 3D in blue)
spectra for Arcturus and the Sun -- all data smoothed with a Gaussian kernel
to have a resolving power of about 11,500 as appropriate for the Gaia RVS.
Both the normalized spectra
and the residuals are shown for each star.
A microturbulence of 1.5 km s$^{-1}$ was adopted
for the 1D models, which were calculated with the LHD package.}
\label{f1}%
\end{figure*}
\section{Spectral synthesis}
\label{synthesis}
\subsection{3D spectral synthesis}
\label{3ds}
We computed synthetic spectra covering the Gaia RVS
wavelength range (847--874 nm) with wavelength steps
equivalent to $\leq 0.1$ km s$^{-1}$ (or a resolving power
$R \equiv \lambda/\delta\lambda \sim 10^6$).
The calculations were
carried out with the 3D synthesis code ASS$\epsilon$T
(Koesterke 2009; Koesterke, Allende Prieto \& Lambert 2008).
The reference solar abundances are from
Asplund, Grevesse \& Sauval (2005),
and while temperature, density and velocity fields
were taken from the model atmospheres, the electron density was recalculated
with an equation of state including the first 99 elements
in the periodic table and 338 molecules (Tsuji 1973, with partition
functions from Irwin 1981).
The number of snapshots used varied between 8 and 21,
but in the vast majority of cases was about 20, and the total time
covered ranged between 0.6 and 19500 hours, spanning at least 10 times the typical lifetimes of convective cells.
The number of pixels considered for
each snapshot was reduced by taking one out of three points in the X- and
Y-directions, and at least the uppermost 90 vertical points.
These choices are based on previous experience and limited testing
with the set of models used in this paper. These tests indicate that
dropping spatial points would lead only to differences at the level of
about 0.2\% for a typical strong Fe I line. For
the continuum and weak lines the differences tend to be much smaller
than that.
The adopted solar abundances are very similar to those
used in the 3D simulations. The small existing inconsistencies are
expected to have a negligible impact in the predicted convective shifts.
Continuous absorption from H and H$^-$ is included.
Line absorption is included in detail from the atomic and molecular
files compiled by Kurucz. Opacities are precomputed in a
temperature-density grid bracketing all the snapshots
with steps of 250 K and 0.25 dex,
and subsequently derived for the grid points using cubic B\'ezier
interpolation.
For each individual snapshot a spectrum is derived in 2 steps. First a
background radiation field is calculated. This is used for the
scattering term in the subsequent spectrum synthesis. Note that actually
2 spectra are calculated for each snapshot. The second one, which is
derived from opacity data without spectral lines, is used for continuum
normalization.
The radiative transfer calculations account for atomic hydrogen Rayleigh
and electron scattering. The mean background radiation field
is calculated from 24 angles (8 azimuthal with linear weights,
and 3 zenith per half-sphere with Gaussian weights)
using a short characteristics scheme, while the emergent flux
is integrated for 21 angles (vertical ray plus 3 zenithal rays
and 8 azimuthal angles again with linear weights,
except that only 4 zenithal angles are considered
for the shallowest rays). The abscissae and
weights for each case can be found
in Abramowitz \& Stegun (1972). This setup evenly covers a sphere
with 8 angles for the azimuth (0-360 deg) and 3-4 angles
for the inclination (0-180 deg). The integration along a
ray was performed from the top
layer down to optical depths of at least $20$. The final emergent flux
is an average of the flux for all the snapshots considered for each
simulation.
Rotation was not accounted for in this step. It is well-known that rotation
affects the observed line asymmetries (Gray 1986), but low rotational
velocities should only have a modest effect on the net shifts derived by
cross-correlation for the entire Gaia RVS spectral window. In part, this is
related to the spectral resolution of the RVS instrument,
corresponding to 26 \mbox{km s$^{-1}$} (de Bruijne 2012).
The calculations were performed on the Longhorn cluster at the Texas
Advanced Computing Center (TACC). The intensities for about 150 million
frequencies were derived across all snapshots. The total computational
effort is roughly equivalent to 3 months on a modern workstation with 4
cores.
Fig. \ref{f1} illustrates the spectra in our 1D and 3D grids that
are closest to the atmospheric parameters for Arcturus and the Sun.
This comparison is done after smoothing the data to a FWHM resolving
power of $R = 11,500$ as appropriate for the RVS;
we adopted a microturbulent velocity of 1.5 km s$^{-1}$ for the 1D calculations.
The agreement between models and observations is significantly better
in the solar case
(observed spectrum from Kurucz et al. 1984) than for Arcturus
(Hinkle et al. 2000). Overall, predicted absorption features are
slightly stronger in 1D than in 3D.
The agreement is reasonable, considering that the models are not
tailored to these stars. Arcturus has an $T_{\rm eff}\sim 4300 $ K,
$\log g \simeq 1.7$ (with $g$ in cm s$^{-2}$) and a metallicity about [Fe/H]
$\simeq -0.5$
(see, e.g., Ram\'{\i}rez \& Allende Prieto 2011), and
the Sun has an
$T_{\rm eff}= 5777 $ K,
$\log g \simeq 4.437$ and, by definition,
[Fe/H]$=0$ (see, e.g. Stix 2004). The model compared here with
Arcturus (d3t40g15mm10n01) has an
$T_{\rm eff}$ of 4040 K,
$\log g = 1.5$ and a metallicity of [Fe/H]$= -1.0$, and
that compared with the Sun (d3t59g45mm00n01) has an
$T_{\rm eff}$ of 5865 K,
$\log g = 4.5$ and solar metallicity.
\subsection{1D spectral synthesis}
\label{1ds}
The 1D counterpart radiative transfer calculations were performed
with the 1D version of ASS$\epsilon$T, and therefore the results
can be directly compared between the 1D and 3D models.
In particular, chemical compositions, opacities, and
radiative transfer calculation for the emergent intensities
are shared by the 1D and 3D branches. The only differences between
the 1D and 3D calculations
are that the opacities are calculated exactly for the grid points
in 1D (every depth in the model atmospheres), while they are interpolated
in 3D (from a precomputed grid in temperature and density, as described
in \S \ref{3ds}),
and the radiative transfer
solver for the calculation of the mean intensities in the grid,
which is based on Feautrier's method (Feautrier 1964)
in 1D and an integral method in 3D.
Our tests nonetheless indicate that these differences have a negligible
effect on the continuum-normalized fluxes.
The difference between the Feautrier and the direct method in 1D
is about 10$^{-5}$. The effect of the interpolation of the opacities
in 3D is estimated to be on the order of 10$^{-4}$.
The different choices for the opacity calculation and the radiation transfer method are made to accommodate the very expensive
calculations in 3D, which are not feasible without interpolating
the opacities and benefit from the faster 1st order method for
the integration of the intensities.
\section{Effective convective shifts for the Gaia RVS}
We divided the spectra by the pseudo continua, estimated by
iteratively fitting a low order polynomial to the upper envelope
of each spectrum. We then calculated the net offset between the
3D and 1D spectra to derive the overall convective shift
predicted by the hydrodynamical simulations by cross-correlation.
The cross-correlation of two arrays (o spectra)
{\bf T} and {\bf S} of equal and even number of elements $N$ is defined
as a new array {\bf C} with
\begin{equation}
C_i = \sum_{k=1}^{N} T_k S_{k+i-\frac{N}{2}},
\label{Ci}
\end{equation}
\noindent where $i$ runs from $1$ to $N$.
If the spectrum {\bf T} is identical to {\bf S}, but shifted
by an an integer number of pixels $p$, the maximum value in the array {\bf C}
will correspond to its element $i=p+\frac{N}{2}$.
Cross-correlation can be similarly
used to measure shifts that correspond to non-integer numbers.
In this case, finding the location of the maximum value of
the cross-correlation function
can be performed with a vast choice of algorithms.
We calculated the cross-correlation of 1D and 3D model
spectra using the software by Allende Prieto (2007), which
allows fitting the peak of the cross-correlation function
with a parabola, a cubic polynomial, or a Gaussian.
Based on tests adding noise to simulated spectra,
we chose to use parabolic fittings
to the central 3 points around the maximum of
the cross-correlation function.
\begin{figure}[t!]
\footnotesize
\centering
\includegraphics[width=6cm,angle=90]{f2_allende.ps}
\caption{The spectra for a single snapshot of model d3t59g45mm00n01 (black) and its
1D counterpart (black), with near solar atmospheric
parameters, are shown in the left-hand panels for two different values of the
resolving power ($R$). The right-hand panels illustrate how the corresponding
shifts (1D relative to 3D) are obtained
by fitting the peak of the cross-correlation function with a parabola.
}
\label{xc}%
\end{figure}
\begin{figure}[t!]
\footnotesize
\centering
\includegraphics[width=9cm]{f3_allende.ps}
\caption{Derived convective shift for model d3t59g45mm00n01, with near solar atmospheric
parameters, as a function of the resolving power. These shifts are obtained
by cross-correlation using the Gaia RVS spectral window, and
fitting the peak of the cross-correlation function with a parabola. The dashed
vertical line marks the RVS resolving power.}
\label{resolution}%
\end{figure}
The net convective shift derived by cross-correlation
not only depends on the chosen spectral range, but
also on the resolving power of the spectrograph. A degraded
resolution will introduce, in addition to random noise, a
systematic error in the derived velocity shifts. This is illustrated
in Fig. \ref{xc} for a single 3D snapshot and its 1D counterpart at two
different values of the resolving power: resolution does change the
apparent convective shift. Fig. \ref{resolution} shows the derived
shifts (1D relative to 3D) for model d3t59g45mm00n01 (see Table 1 for
reference), which was compared
with the solar spectrum in Fig. \ref{f1}, as a function of the resolving power.
At the resolving power
of the Gaia RVS ($R=$11,500), the inferred convective blueshift is about
0.1 km s$^{-1}$, while that measured when the spectral lines
are fully resolved ($R>200,000$) approaches 0.24 km s$^{-1}$.
Our results for the simulations are included in Table 1.
All the values presented in the table, and discussed
below correspond to the RVS resolution.
In addition to calculating the net convective shifts for the
average spectra from the selected snapshots for each simulation,
we also calculated the shifts for each snapshot and averaged
them out. These correspond to the 'mean' shifts in Table 1.
The variance {\bf ($\sigma^2$)} among snapshots was used to estimate the
intrinsic uncertainties as $\sigma/\sqrt{n}$, with $n$ the number
of snapshots considered, and these are also included in the table.
We believe that the uncertainties calculated in this way are conservative since the subsample
of snapshots selected for spectral synthesis is chosen to represent the
properties of the complete run (e.g., in effective temperature and mean
velocities). Moreover, the separation in time among the snapshots is large
enough that they can be considered as independent despite being selected from
one time series.
The agreement between the shifts of the average spectra
and the average shifts from individual snapshots is excellent,
with an average of 0.0004 km s$^{-1}$ and a standard deviation
of 0.001 km s$^{-1}$.
\begin{figure*}[ht!]
\footnotesize
\centering
\includegraphics[width=16cm]{f4_allende.ps}
\caption{Effective convective shifts
predicted by the 3D hydrodynamical models for Gaia RVS observations.}
\label{cs}%
\end{figure*}
Fig. \ref{cs} shows the convective shifts for all models.
Note that there are models for four different metallicities, color-coded,
and that the resulting effective temperatures of the simulations
are not always exactly the same for any combination of
surface gravity and metallicity, as this is not an input
parameter -- the entropy of the material entering the
simulations box through the open bottom boundary
is a chosen input instead.
The changes of the net convective shifts with the atmospheric
parameters are perhaps most obvious in Fig.~\ref{cs2}.
The net shifts show a mild correlation with the effective
temperature of the models, in the sense that warmer stars tend to exhibit
larger blueshifts. As found in previous studies
(Dravins \& Nordlund 1990a,b; Nordlund \& Dravins 1990),
the net energy flux traversing the atmosphere
is the most important parameter. Surface gravity has a small
effect on the convective shifts predicted by our simulations
-- overall the shifts intensify at lower gravities --, and so has metallicity,
with lower net blueshifts at higher metallicities.
Fig. \ref{cs} shows the results obtained when the LHD 1D models,
which are fully consistent with the 3D simulations in terms of
opacities and the equation of state, are used as reference.
However, very similar results are found when Kurucz models are
used as a reference (see below).
We adopted a micro-turbulence of 2.0 km s$^{-1}$, and a macro-turbulence of
1.5 km s$^{-1}$ for the reference 1D calculations, but our experiments
indicate that these parameters would only lead to a modest impact on the spectral
shifts determined by cross-correlation. For instance, completely neglecting
macro-turbulence leads to a mean correction in the predicted convective
blueshifts of just 0.01 ($\sigma=0.032$) km s$^{-1}$. Adopting a
micro-turbulence of 1.5 km s$^{-1}$ instead of 2.0 km s$^{-1}$ will, on
average, reduce the convective blueshifts by about 0.007 ($\sigma= 0.011$) km
s$^{-1}$. Finally, adopting Kurucz model atmospheres instead of LHD models as
our 1D reference will, on average, enhance the blueshifts by about 0.01
($\sigma=0.03$) km s$^{-1}$.
\begin{figure*}[t!]
\footnotesize
\centering
\includegraphics[width=12cm]{f5_allende.ps}
\caption{Three uppermost panels: Predicted convective shifts and atmospheric
parameters of the 3D hydrodynamical models for Gaia RVS observations. Two
lowermost panels: Convective shifts versus relative horizontal temperature
fluctuations and the rms of the vertical velocity at Rosseland optical depth
$\tau=2/3$. In all panels data points are color coded by metallicity as in Fig. \ref{cs}.
}
\label{cs2}%
\end{figure*}
The smallest (in absolute terms) convective shifts in the grid are for the
coolest dwarfs of low metal content, and amount barely to $<0.1$ km s$^{-1}$.
The most vigorous net velocities are found for low metallicity F-type
subgiants, practically the warmest stars considered in the grid, where they
exceed 0.3 km s$^{-1}$.
There are a few simulations (mostly with [Fe/H]$=-1$)
for which redshifts are predicted.
While perhaps not very evident in Fig.~\ref{cs2} the drop is
clearly apparent in a global analytical fit of the convective shifts of the 3D simulations
(see Appendix~\ref{sec:fittingfunction}).
These redshifts are a clear prediction of the simulations.
They are caused by strong redshifts of the cores of the Ca\,II triplet lines,
but they dominate the overall cross-correlation signal.
The cores get redshifted when the lines are strong and the flow exhibits a
pronounced "reverse granulation" pattern. Still, the overall shift of
the Ca\,II triplet line cores is a superposition of redshifted and
blueshifted contributions and therefore it is difficult to predict a
priori which ones are dominating.
We tried to interpret our findings in terms of the temperature and velocity
fluctuations present in the 3D simulations. We averaged the temperature and
vertical velocity component horizontally and temporally on surfaces of
constant Rosseland optical depth. We took values at Rosseland depth $\tau=2/3$
as representative for the conditions prevailing in the line forming regions --
at least for weak lines. These are shown in the two bottom panels of Fig. \ref{cs2},
where a mild correlation between temperature and velocity fluctuations
is apparent.
Figure~\ref{avgs2} shows how the temperature and velocity fluctuations
depend on effective temperature, surface gravity and metallicity.
Interestingly, we find that the velocity fluctuations depend on metallicity
for $\ensuremath{T_{\mathrm{eff}}}<5000\pun{K}$ but they do not for warmer stars.
Only close to the main-sequence a
systematic dependence appears below 5000\pun{K} with higher metallicity
implying greater fluctuations. For main-sequence stars, $\ensuremath{T_{\mathrm{eff}}}=5000\pun{K}$
also separates regions of different correlation between metallicity and
temperature fluctuations.
Above 5000 K, higher metallicity corresponds to
lower temperature fluctuations but this is reversed below 5000 K. While
Fig.~\ref{avgs2} certainly motivates the increase in convective shifts
towards higher effective temperatures and lower gravity, it cannot explain
all trends of the convective shifts. What is indeed missing is information
on the mutual correlation between temperature and velocity fluctuations, as
well as on changes of the line formation conditions with atmospheric parameters
-- as discussed further below. In particular, Fig.~\ref{avgs2} does not
explain the fast drop of the predicted convective shifts near the highest
effective temperatures in our grid.
\begin{figure}
\footnotesize
\centering
\includegraphics[width=8.0cm]{f6_allende.ps}
\caption{Relative horizontal temperature fluctuation (top panel) and absolute
vertical velocity fluctuations (bottom panel) at Rosseland optical depth
$\tau=2/3$ in the 3D models. The surface gravity is color coded, metallicity
is depicted by different plot symbols.
}
\label{avgs2}%
\end{figure}
Nevertheless, our results resemble earlier high-resolution studies of spectral
line shapes and convective shifts, but the net velocity shifts predicted for
the RVS observations are smaller than for individual, resolved, spectral
lines (see, e.g.,
Bigot \& Thevenin 2006, 2008; Chiavassa et al. 2011).
As expected, the net shift measured by cross-correlation is a weighted
average of the lineshifts from all spectral features. At high metallicity,
many lines contribute, with strong lines giving very small convective shifts,
and weaker features contributing larger shifts. For example, for a solar-like
atmosphere, if we apply cross-correlation by isolating windows around the Ca
II lines, we find very small convective shifts on the order of 0.030 km
s$^{-1}$, while limiting the spectral window to the stretch between the Ca II
lines (857-863 nm), the net shift is about $-0.18$ km s$^{-1}$, double than
what we obtain for the whole RVS spectral range.
For a solar 3D model not included in the grid discussed here, we found in a
previous paper (Allende Prieto et al. 2009) a convective shift of $-0.26$ km
s$^{-1}$ , in excellent agreement with the value directly derived from the
solar flux atlas of Kurucz et al. (1984). These two values correspond to
observations at much higher spectral resolution, and can be compared with
those we find (see Fig. 2) for the model in the grid with parameters closest
to solar, d3t59g45mm00n01, namely $\simeq -0.24$ km s$^{-1}$ when the
resolving power is $R>200,000$ but note that changes to $-0.09$ km s$^{-1}$ at $R=11,500$.
From a theoretical standpoint, the main effect of a reduced gravity is a lower
surface pressure, as well as an increase in both the vertical and horizontal
scales. At a given effective temperature the flow is constrained
to transport the
same energy flux so that a reduced pressure and density (since the temperature
is almost fixed) has to be compensated for by an increase in the velocity
and/or the temperature fluctuations (Dravins \& Nordlund 1990b; Dravins et
al. 1993; Collet, Asplund \& Trampedach 2007; Gray 2009). As a result,
convective blueshifts should strenghten -- as long as the surface area
fractions covered by up-flows and down-flows do not change. As illustrated
in Fig. \ref{avgs2}, our
detailed simulations roughly support this simple picture.
At lower metallicity we expect the reduced continuum opacity to make visible
deeper atmospheric layers with more vigorous convection (Allende Prieto et
al. 1999), and the simulations indicate that the convective shifts of
individual lines become stronger, but the span of the line bisectors weakens
dramatically with metallicity.
This is illustrated for the strong Fe I line at 868.86 nm in solar-like
($T_{\rm eff}\simeq 5900$ K and $\log g =4.5$) stars in Fig.
\ref{bisector}. In this figure, we show line profiles (top panel) and
the corresponding bisectors (lower panel), with the solid lines for
the calculations based on 3D models and the dashed lines for
1D models. This analysis is performed at full resolution,
since line bisectors are erased at the RVS resolution, using
a micro-turbulence of 2 km s$^{-1}$ and no macro-turbulence.
As the feature weakens, the range of atmospheric heights
sampled from the wings to the core is reduced, which is a likely explanation for
the smaller bisector spans. For this line, we also find that the predicted
convective shift is $-0.00$, $-0.10$, $-0.12$, and $-0.13$ km s$^{-1}$ at
[Fe/H]$=0, -1, -2$, and $-3$, respectively. Note that these bisectors are
calculated using the standard definition, i.e. subtracting the velocity
in the blue wing from that at the same normalized flux level in the
red wing, and thus the net line shifts are not included and they
approach zero velocity in the line core (Gray 1992).
In addition to the characteristics
of surface convection changing as a function of metallicity,
the strengthening of the line blueshifts
with decreasing metallicity is related to the line formation
shifting deeper into the photosphere as the line weakens. This is
consistent with our findings for the overall net blueshift for
Gaia RVS observations.
\begin{figure}[h!]
\footnotesize
\centering
\includegraphics[width=9cm]{f7_allende.ps}
\caption{Line profiles and velocity bisectors for the Fe I transition
at 868.6 nm. The filled circles in the upper panel correspond to solar observations.
Solid lines are for 3D simulations and broken lines for 1D models. Note that
the 1D calculation at solar-metallicity for this line (broken solid line)
shows a non-zero velocity bisector, due to overlapping CN transitions included
in all our calculations that distort the otherwise perfectly symmetric
1D profiles.}
\label{bisector}%
\end{figure}
The filled circles in the upper panel of Fig. \ref{bisector} show the renormalized
observations for the Sun of Kurucz et al. (1984). A significant improvement
in the agreement between theory and observation is obvious for the line core
when going from 1D to 3D models (dashed and solid black lines, respectively).
A similar effect is found using a Kurucz model,
instead of its LHD counterpart,
for the 1D spectrum. It is also noticeable the extreme discrepancies between
the 1D and 3D cases at [Fe/H]$=-2$, as it is the large
reduction in strength for the line calculated in 3D
between [Fe/H]$=-2$ and $-3$. An exhaustive check of these
predictions against high quality observations of metal-poor stars
is very much needed.
Our calculations are likely least reliable for cool giants, and in fact the
predicted convective shifts for stars approaching 4000 K are unlikely to be
realistic. Gray, Carney \& Yong (2008) examined bisectors for a number of red
giant stars with low metallicity, finding that those with temperatures of
about 4100 K or less tend to exhibit reversed bisectors (with an 'inverse C'
shape, rather than with a 'C'-like shape seen in solar-type stars). More
recent results for the supergiant $\gamma$\,Cyg confirm the same trend (Gray
2010). These stars also tend to exhibit radial velocity 'jitter' (see, e.g.,
Carney et al. 2008, and the theoretical predictions by Chiavassa et al. 2011).
We have
identified an expression that can reproduce the net convective blueshifts
theoretically predicted for the RVS observations in Table 1 with an rms of
0.021 km s$^{-1}$; and a maximum deviation of 0.074 km s$^{-1}$.
We have coded this expression into an IDL function
provided in the appendix.
\section{Gravitational redshifts}
Photons escaping from a stellar photosphere to infinity are redshifted
as a result of climbing up the gravitational field by
$Gc M/R$, where $G$ is the Gravitational constant, $c$ the speed of light,
$M$ the stellar mass, and $R$ the stellar radius. Using solar units for
the stellar mass and radius, the gravitational redshift amounts to
0.6365 $M/R$ km s$^{-1}$, and for light intercepted at 1 AU
0.6335 $M/R$ km s$^{-1}$ (Lindegren \& Dravins 2003).
The same effect acts when photons
enter in the Earth's gravitational field, which reduces the redshift
by about 0.2 m s$^{-1}$, but this small correction can be ignored for most purposes.
With the parallaxes and spectrophotometry
provided by Gaia, it should be possible to obtain fairly accurate
customized predictions for the gravitational redshifts for individual Gaia
targets.
Gravitational shifts for late-type dwarfs amount to about 0.7-0.8 km s$^{-1}$
at $\log g \sim 4.5$, but they dramatically decrease with surface gravity down
to 0.02--0.03 km s$^{-1}$ for the giants in our grid ($\log g \sim 1.5$).
\section{Discussion and Summary}
The radial velocities inferred from the wavelength shifts
measured in stellar spectra are systematically
offset from the true space velocities of stars projected
along the line of sight
due to a number of effects. Quantifying the impact of such
effects is model dependent, and therefore the IAU has introduced
the term 'radial-velocity measure' (Lindegren \& Dravins
2003) to refer to the radial velocity naively inferred
from the relative wavelength shifts of lines
$c \delta \lambda/\lambda$,
and encourages observers to keep measurements disentangled
from subsequent corrections\footnote{We note that the 'radial-velocity measure' does not have a unique definition, since the value derived depends
on the spectral features and the actual method used. Therefore, such measures
must be accompanied by a detailed description of the procedure used to
derive it.}.
Nevertheless, for most stars, two relatively clean corrections
are dominant: photospheric gravitational redshifts and
convective blueshifts. In the case of stars moving at high
speed in the direction perpendicular to the observer, the relativistic
contribution to the Dopper shift needs to
be considered at a precision level of hundred of meters
per second; the spectrum of a star with a transverse
velocity of 300 km s$^{-1}$ and no radial velocity
will be redshifted as much as that of a star with a receding
radial velocity of 0.15 km s$^{-1}$.
We have used three-dimensional radiative-hydrodynamical simulations of
surface convection in late-type stars to evaluate the impact of
convective shifts on the spectra from the Radial Velocity Spectrometer
onboard Gaia. The net velocity shifts derived by cross-correlation
depend both on the wavelength range and the spectral resolution of the
observations. Our models predict very small net convective shifts
for the RVS observations of K-type stars, which increase with effective
temperature to reach bout 0.3 km s$^{-1}$ for F-type dwarfs. The
predicted shifts depend as well on metallicity and surface gravity.
In the case of the Gaia mission, the corrections discussed above, which are
$<1$ km s$^{-1}$ are small in comparison with the typical
uncertainties in the space velocities determined for individual stars,
but can cause systematic errors when average values are derived
for ensembles of stars. Even more important, since the Gaia RVS
is a self-calibrated instrument, in the sense that the wavelength solution is
derived from stellar observations, systematic offsets from star to
star such as those related to convective shifts, need to be accounted
for in the data reduction. In this paper, we used
a grid of 3D hydrodynamical simulations to estimate
convective blueshifts for stars with spectral types
F to M, at different metallicities and evolutionary stages.
We note that from data provided
by Gaia itself, it will be possible to constrain the stellar masses and
radii for individual stars, and therefore the gravitational
redshifts.
Gravitational redshifts dominate convective
blueshifts predicted for the Gaia window for dwarfs of all temperatures,
in particular for later spectral types (K and M), where the convective
shifts are weakest. Gravitational redshifts become weaker
for stars with lower gravities. The opposite tendency, if less
pronounced, is expected for the convective blueshifts, in such a way
that gravitational and convective shifts nearly cancel each other
for stars with intermediate gravities.
When adding together convective and gravitational shifts, we
expect the largest velocity offset for the coolest dwarfs
in our study, with effective temperatures around 4000 K,
where the net effect, dominated by gravitational redshifts,
amounts up to nearly 0.6 km s$^{-1}$.
By using the corrections provided here and custom-made
estimates of the gravitational shifts, these sources
of systematic error should be reduced significantly to a level
ranging from about $0.1$ km s$^{-1}$ for F-type dwarfs, to
a few tens of meters per second for late-K-type stars.
In addition to calculated convective shifts, we provide
FITS files including the 1D and 3D model fluxes we
have computed. These may turn
useful to build radial velocity templates for derivation
of radial velocities from observations.
We end with a word of caution. The theoretical calculations presented
here only have been checked against a limited set of observations,
mainly for the Sun. An exhaustive check against high-quality observed
spectra for warmer and cooler temperatures, especially at very low
metallicity, is still needed before one can ensure that the predicted
shifts are completely trustworthy.
\begin{acknowledgements}
We are indebted to Ivan Hubeny for assistance computing radiative opacities.
Carlos Allende Prieto is thankful to
Mark Cropper, David Katz, and Fr\'ed\'eric Th\'evenin for
fruitful discussions.
B.F. acknowledges financial support from
the {\sl Agence Nationale de la Recherche} (ANR), and the
{\sl ``Programme National de Physique Stellaire''} (PNPS) of
CNRS/INSU, France.
HGL acknowledges financial support by the Sonderforschungsbereich SFB\,881
``The Milky Way System'' (subproject A4) of the German Research Foundation
(DFG).
\end{acknowledgements}
| {
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Valmet TS in Beijing Gaobeidian Wastewater Treatment Plant serving 2.4 million people
The principal goal of a sludge process is water removal, performed as efficiently and economically as possible. Sludge pumping control, based on reliable total solids measurement, to optimize sludge quality early in the process is vital for the whole sludge handling procedure.
In the Beijing Gaobeidian Wastewater Treatment Plant, sludge total solids from the primary clarifier varied from below 1 to 60 g/l, which means that from time to time only water is pumped to the sludge thickening tank. The Valmet TS*) total solids transmitter was installed in the outlet of the primary clarifier to control pumping.
Beijing Gaobeidian Wastewater Treatment Plant is one of the largest wastewater treatment plant in China. It serves a population of 2.4 million people with a daily capacity of 100,000 m3, processing about 40% of Beijing Municipality's total wastewater volume. When the plant was built in the 1990's, national standards for phosphorous and nitrogen in treated effluent had not yet been set and, because of high levels of nitrogen and phosphorous, treated wastewater did not meet criteria for even the national level V (not suitable for use in agriculture) surface water standard. Since then, the plant's secondary treatment facilities are undergoing modifications which will allow approximately 530,000 m3/day to be treated to national 4 standard for surface water (suitable for industrial use and other uses not in direct contact with humans) and 470,000 m3/day to be treated to national 3 standard for surface water (suitable for aquaculture and recreation purposes). Mostly all of WWTPs' effluents meet the US Level II standard for surface water.
In addition to its primary and secondary treatment facilities, Beijing Gaobeidian WWTP also boasts facilities to process and collect methane gas produced during the anaerobic digestion processes. According to the plant's operating conditions, a portion of the generated biogas is used onsite to make the Gaobeidian's wastewater treatment processes even more energy efficient, for example as a natural gas substitute to heat the wastewater digesting processes, the rest is sold to Beijing's energy grid.
The Valmet TS sensor was installed in the outlet of Primary Clarifier in September 2012. Since then personnel at the plant have been very satisfied with the unit and its capability for control over the range of 1 g/l to 60 g/l total solids in the system. Compared to an optical sensor accuracy has improved from ±1500 to ±500 mg/l with a correlation to the laboratory measurement of 0.996, and unlike the optical sensor, there is no need for maintenance.
Valmet Total Solid meter installed to the outlet of the primary clarifier.
*) Company and product name Metso until March 31, 2015.
Total Solids Measurement
Valmet Total Solids Measurement (Valmet TS) is a reliable and accurate microwave real time measurement used to control and optimize the solids treatment process, giving operators information on the total solids 24/7. | {
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ADJ' s myDMX 3. View and Download American DJ Verti Color user instructions online. 0 now available in the " Downloads" section of this page.
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/docs.stack-assessment.org\/en\/CAS\/Real_Intervals\/","text":"# Real intervals and sets of real numbers\n\nSTACK has a simple system for representing and dealing with real intervals and sets of real numbers.\n\nSimple real intervals may be represented by the inert functions oo(a,b), oc(a,b), co(a,b), and cc(a,b). Here the character o stands for open end point, and c for a closed end point. So oc(-1,3) is the interval $\\{ x\\in\\mathbb{R} | -1 < x \\mbox{ and } x\\leq 3.\\}$, and is displayed as $(-1,3]$ with mismatching brackets in the tradition of UK mathematics.\n\nThe Maxima function union requires its arguments to be sets, and intervals are not sets. You must use the %union function (from the package to_poly_solve) to join simple intervals and combine them with discrete sets. E.g. %union(oo(-2,-1),oo(1,2))\n\nNote that the %union function sorts its arguments (unless you have simp:false), and sort puts simple intervals of the form oo(-inf,a) out of order at the right hand end. So, some sorting functions return lists of intervals, not %union as you might expect, to preserve the order.\n\nAs arguments, the %union command can take both simple intervals and sets of discrete real numbers, e.g.\n\n%union(oo(-inf,0),{1},oo(2,3));\n\n\nSimilarly, STACK provides %intersection to represent an intersection of intervals (which the package to_poly_solve does not have).\n\nPredicate functions\n\n1. intervalp(ex) returns true if ex is a single simple interval. Does not check ex is variable free, so oo(a,b) is a simple interval. {}, none, all and singleton sets are not considered \"intervals\" by this predicate, use realsetp instead. The primary purpose of this predicate is to detect intervals oo, oc etc within code.\n2. inintervalp(x, I) returns true if x is an element of I and false otherwise. x must be a real number. I must be a set of numbers or a simple interval of the form oo(a,b) etc.\n3. trivialintervalp(ex) returns true if ex is a trivial interval such as $(a,a)$.\n4. unionp(ex) is the operator a union?\n5. intersectionp(ex) is the operator an intersection?\n6. realsetp(ex) return true if ex represents a definite set of real numbers, e.g. a union of intervals. All end points and set elements must be real numbers, so oo(a,b) is not a realset. If you want to permit variables in sets and as endpoints use realset_soft_p instead.\n7. interval_disjointp(I1, I2) establishes if two simple intervals are disjoint.\n8. interval_subsetp(S1, S2) is the real set S1 contained within the real set S2?\n9. interval_containsp(I1, S2) is the simple interval I1 an explicit sub-interval within the real set S2? No proper subsets here, but this is useful for checking which intervals a student has.\n\nBasic manipulation of intervals.\n\n1. interval_simple_union(I1, I2) join two simple intervals.\n2. interval_sort(I) takes a list of intervals and sorts them into ascending order by their left hand ends. Returns a list.\n3. interval_connect(S) Given a %union of intervals, checks whether any intervals are connected, and if so, joins them up and returns the ammended union.\n4. interval_tidy(S) Given a union of sets, returns the \"canonical form\" of this union.\n5. interval_intersect(S1, S2) intersect two two simple intervals or two real sets, e.g. %union sets.\n6. interval_intersect_list(ex) intersect a list of real sets.\n7. interval_complement(ex) take a %union of intervals and return its complement.\n8. interval_set_complement(ex) Take a set of real numbers, and return the %union of intervals not containing these numbers.\n9. interval_count_components(ex) Take a set of real numbers, and return the number of separate connected components in the whole expression. Simple intervals count as one, and sets count as number number of distinct points in the set. Trivial intervals, such as the empty set, count for 0. No simplification is done, so you might need to use interval_tidy(ex) first if you don't want to count just the representation.\n\n## Natural domains, and real sets with a variable.\n\nThe function natural_domain(ex) returns the natural domain of a function represented by the expression ex, in the form of the inert function realset. For example natural_domain(1\/x) gives\n\nrealset(x,%union(oo(0,inf),oo(\u2212inf,0)));\n\n\nThe inert function realset allows a variable to be passed with a set of numbers. This is mostly for displaying natural domains in a sensible way. For example, where the complement of the intervals is a discrete set, the realset is displayed as $x\\not\\in \\cdots$ rather than $x \\in \\cdots$ which is normally much easier to read and understand.\n\nrealset(x,%union(oo(0,inf),oo(-inf,0)));\n\n\nis displayed as $x \\not\\in\\{0\\}$.\n\nStudents must simply type union (not %union) etc.\nValidation of students' answer has a very loose sense of \"type\". When we are checking the \"type\" of answer, if the teacher's answer is a \"set\" then the student's answer should also be a \"set\" (see setp). If the teacher's answer is acually a set in the context where an interval should be considered valid, then the teacher's answer should be the inert function %union, e.g. %union({1,2,3}), to bump the type of the teacher's answer away from set and into realset.\nValidation does some simple checks, so that mal-formed intervals such as oo(1) and oo(4,3) are rejected as invalid.\nThe algebraic equivalence answer test will apply interval_tidy as needed and compare the results. Currently the feedback in this situation provided by this answer test is minimal.","date":"2021-02-26 09:16:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 6, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6321820616722107, \"perplexity\": 1869.4765126139498}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178356456.56\/warc\/CC-MAIN-20210226085543-20210226115543-00625.warc.gz\"}"} | null | null |
'use strict'
import { TestNumber } from "./TestNumber.js"
/**
Stores the number and name of the
test.
*/
export class TestInformation {
constructor(name) {
this.number = new TestNumber()
this.name = name
}
}
| {
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{"url":"https:\/\/www.statsmodels.org\/stable\/generated\/statsmodels.stats.multivariate.test_cov_blockdiagonal.html","text":"# statsmodels.stats.multivariate.test_cov_blockdiagonal\u00b6\n\nstatsmodels.stats.multivariate.test_cov_blockdiagonal(cov, nobs, block_len)[source]\n\nOne sample hypothesis test that covariance is block diagonal.\n\nThe Null and alternative hypotheses are\n\n$\\begin{split}H0 &: \\Sigma = diag(\\Sigma_i) \\\\ H1 &: \\Sigma \\neq diag(\\Sigma_i)\\end{split}$\n\nwhere $$\\Sigma_i$$ are covariance blocks with unspecified values.\n\nParameters\ncovarray_like\n\nCovariance matrix of the data, estimated with denominator (N - 1), i.e. ddof=1.\n\nnobsint\n\nnumber of observations used in the estimation of the covariance\n\nblock_lenlist\n\nlist of length of each square block\n\nReturns\nresinstance of HolderTuple\n\nresults with statistic, pvalue and other attributes like df\n\nReferences\n\nRencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher\/Methods. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. https:\/\/doi.org\/10.1002\/9781118391686.\n\nStataCorp, L. P. Stata Multivariate Statistics: Reference Manual. Stata Press Publication.","date":"2021-04-12 07:23:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.675349771976471, \"perplexity\": 5091.148426013548}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038066613.21\/warc\/CC-MAIN-20210412053559-20210412083559-00371.warc.gz\"}"} | null | null |
Q: Shortcut of finding a tangent line and tangent point to a curve that's explicitly or implicitly defined Any shortcut command of finding a tangent line and tangent point to a curve that's explicitly or implicitly defined? The focus is on implicit curves.
An implicit curve could be sth like $(x/exp(y)-1)^2-y^2=1$
A: Take a look at the Applications section in the documentation for ImplicitD. Note that ImplicitD has been available only since version 13.1.
(* Define your curve *)
curve = (x - Exp[y] - 1)^2 - y^2 == 1;
(* Calculate the appropriate partial derivative *)
slope = ImplicitD[curve, y, x];
(* Find points on the curve at x = -1 and x = 4 *)
points = FindInstance[curve && (x == -1 || x == 4), {x, y}, Reals, 4];
(* Define tangent lines *)
tangents = InfiniteLine[{x, y}, {1, slope}] /. points;
(* Plot the curve and tangent lines *)
Show[ContourPlot @@ {curve, {x, -5, 5}, {y, -5, 5}},
Graphics[{{Red, PointSize[Medium], Point[{x, y}] /. points}, {Orange,
Dashed, tangents}}]]
A: Edit
*
*For the implicit form of curve f[x,y]==0, the normal of the tangent line is Grad[f[x,y],{x,y}], so the tangent line of the f[x,y]==0 is
({x, y} - {x0, y0}) . Grad[f[x, y], {x, y}] == 0
Here {x0,y0} is the arbitary pont on the tangent line.
*
*We use ImplicitRegion and DiscretizeRegion to solve the equation.(since NSolve,Reduce or FindRoot does not work for this case)
Clear["Global`*"];
f[x_, y_] = (x - Exp[y] - 1)^2 - y^2 - 1;
{x0, y0} = {-5, -3};
reg = ImplicitRegion[{({x, y} - {x0, y0}) . Grad[f[x, y], {x, y}] ==
0, f[x, y] == 0}, {x, y}];
pts = MeshPrimitives[DiscretizeRegion[reg, {{-10, 10}, {-10, 10}}],
0][[;; , 1]]
ContourPlot[f[x, y] == 0 // Evaluate, {x, -10, 10}, {y, -10, 10},
Epilog -> {Blue, Point[{x0, y0}], Red, Point[pts], Red,
Arrowheads[{{.05, .8}}], Arrow[{{x0, y0}, #}] & /@ pts}]
*
*We can test the pont {x0, y0} = {1, 1}, there are three tangent lines through {x0,y0}={1,1}.( so there are three tangent points)
*
*animation.
Clear["Global`*"];
Manipulate[
Module[{f, reg, dreg, pts},
f[x_, y_] = (x - Exp[y] - 1)^2 - y^2 - 1;
{x0, y0} = pt;
reg = ImplicitRegion[{({x, y} - {x0, y0}) . Grad[f[x, y], {x, y}] ==
0, f[x, y] == 0}, {x, y}];
pts = MeshPrimitives[DiscretizeRegion[reg, {{-10, 10}, {-10, 10}}],
0][[;; , 1]];
ContourPlot[f[x, y] == 0 // Evaluate, {x, -10, 10}, {y, -10, 10},
Epilog -> {Blue, Point[{x0, y0}], Red, Point[pts], Red,
Arrowheads[{{.05, .8}}], Arrow[{{x0, y0}, #}] & /@ pts},
PerformanceGoal -> "Quality"]], {{pt, {0, 0}}, Locator}]
Original
It seems that it is not easy to find all of the tangent lines for any point {x0,y0} outside the curve. Here we only plot one tangent line from a point does not on the curve.
Clear["Global`*"];
f[x_, y_] = (x - Exp[y] - 1)^2 - y^2 - 1;
{x0, y0} = {-5, -3};
sol = FindInstance[{({x, y} - {x0, y0}) . Grad[f[x, y], {x, y}] == 0,
f[x, y] == 0}, {x, y}, Reals, 1]
pts = {x, y} /. sol // N
ContourPlot[f[x, y] == 0 // Evaluate, {x, -10, 10}, {y, -10, 10},
Epilog -> {Blue, Point[{x0, y0}], Red, Point[pts], Red,
Arrow[{{x0, y0}, #}] & /@ pts}]
A: f[x_, y_] = (x - Exp[y] - 1)^2 - y^2 - 1;
df = ImplicitD[f[x0, y0] == 0, y0, x0];
pts = NSolve[f[x0, y0] == 0 && (x0 == -1 || x0 == 4), {x0, y0}, Reals]
{{x0 -> -1., y0 -> -1.9024}, {x0 -> 4., y0 -> -2.76129},
{x0 -> 4., y0 -> 0.60499}, {x0 -> 4., y0 -> 1.58371}}
tangent = (x - x0)*df + y0 /. pts // Expand
{-0.935624 + 0.966771 x, 1.79955 - 1.14021 x,
-1.09796 + 0.425737 x, 0.590528 + 0.248296 x}
Show[Plot[tangent, {x, -3, 6}, PlotStyle -> Dashed,
Epilog -> {Red, PointSize -> Large, Point[{x0, y0} /. pts]}],
ContourPlot[f[x, y] == 0, {x, -5, 6}, {y, -5, 5}, ContourStyle -> Thick], AspectRatio -> 1]
For versions below 13.1 you can replace ImplicitD with the following version:
df = -D[f[x0, y0], x0]/D[f[x0, y0], y0] // Simplify
A shortcut command of finding a tangent line is given by ResourceFunction:
tangent = ResourceFunction["TangentLine"];
The function is bijective at x = -1.
tangent[(x - Exp[y] - 1)^2 - y^2 == 1, {x, -1}, y] // N // Dataset
At x = 4 the function is surjective. You get only one tangentline at [4, 0.60499].
tangent[(x - Exp[y] - 1)^2 - y^2 == 1, {x, 4}, y] // N // Dataset
There is a note in the description:
If only one coordinate of the intersection point is given, the other
coordinate is inferred. For expressions that are multivalued at the
given value of x or y, information on only one of potentially several
tangent lines is returned.
| {
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Listed companies' analysis | Ranking | Industry ratios | Ratios (SECT) | Statements (SECT)
SECTOR 10, Inc. (SECT) financial statements (2023 and earlier)
Business Address 14553 S 790 WEST SUITE C
Fiscal Year End March 31
SIC 509 - Miscellaneous Durable Goods (benchmarking)
Deferred tax assets ✕ ✕ ✕ 1,113 1,028 952 859
Total current assets: 1,113 1,028 952 859
TOTAL ASSETS: 1,113 1,028 952 859
Accounts payable and accrued liabilities, including: 9,676 10,056 11,592 10,234 8,929 7,708 6,568
Other undisclosed accounts payable and accrued liabilities 9,676 10,056 11,592 10,234 8,929 7,708 6,568
Debt 164 321 804 804 804 804 724
Total current liabilities: 9,840 10,376 12,396 11,038 9,732 8,511 7,292
Total liabilities: 9,840 10,376 12,396 11,038 9,732 8,511 7,292
Stockholders' equity attributable to parent, including: (9,840) (10,376) (12,396) (11,038) (9,732) (8,511) (7,292)
Additional paid in capital 6,148 6,148 6,148 6,148 6,148 6,148 6,148
Accumulated deficit (15,989) (16,525) (18,544) (17,186) (15,881) (14,660) (13,441)
Other undisclosed stockholders' equity attributable to parent 0
Total stockholders' equity: (9,840) (10,376) (12,396) (11,038) (9,732) (8,511) (7,292)
Operating expenses 526 (49) (832) (832) (801) (855) (828)
Operating income (loss): 526 (49) (832) (832) (801) (855) (828)
(Other Nonoperating income) 576 2,116
Interest and debt expense 2,451 2,068 (526) (473) (421) (364) (309)
Other undisclosed income from continuing operations before equity method investments, income taxes 2,116
Income (loss) from continuing operations before equity method investments, income taxes: 5,668 4,135 (1,358) (1,305) (1,221) (1,219) (1,137)
Other undisclosed loss from continuing operations before income taxes (2,122) (2,116)
Net income (loss) available to common stockholders, diluted: 3,546 2,020 (1,358) (1,305) (1,221) (1,219) (1,137)
Net income (loss): 3,546 2,020 (1,358) (1,305) (1,221) (1,219) (1,137)
Comprehensive income (loss), net of tax, attributable to parent: 3,546 2,020 (1,358) (1,305) (1,221) (1,219) (1,137) | {
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Луи Венсан Леон Пальер (; 19 июля 1787, Бордо — 29 декабря 1820, там же) — французский художник.
Биография
Луи Венсан Леон Пальер родился в Бордо. В ранней юности уехал в Париж, где учился в Национальной школе изящных искусств у Франсуа Андре Венсана. С 1805 года участвовал в конкурсах на Римскую премию, которую выиграл в 1812 году с картиной «Одиссей и Телемах, убивающие женихов жены Одиссея». С 1813 по 1816 год Пальер проживал в Риме на вилле Медичи на стипендию, полученную в рамках Римской премии. Его другом стал художник Франсуа Эдуар Пико, совместно с которым они выполнили росписи для римской церкви Сантиссима-Тринита-дей-Монти.
В 1819 году Пальер с большим успехом выставлялся на Парижском Салоне, где получил медаль первого класса. В этом и следующем году он получает много заказов от парижских церквей, таких как Сен-Эсташ и Сен-Северин, на создание картин религиозного содержания.
На пике своей известности и славы Пальер скончался в 1820 году. Его незаконченную картину для церкви Святого Петра в Бордо (фр.) завершил его друг Пико.
Пальер был женат на художнице Франсуазе Виргинии Леже (1797—1880). После смерти Пальера она вышла замуж за его друга, художника Жана Ало, и стала известна под именем Фанни Ало (фр.).
За свою весьма недолгую жизнь Пальер сумел стать видной фигурой в художественных кругах своего времени и создать достаточно большое число картин, до сих пор представленных в целом ряде церквей и музеев.
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Примечания
Художники Франции XIX века | {
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Many of us dont understand the importance of the choices we make or the power that we have over them. More often than not it is said that choices have consequences.
Coincidences mean youre on the right path shannon l.
Quotes about choices and consequences. Dont ever stray from yoursel. Sometimes courage is the quiet voice at the end of the day saying i will try again tomorrow mary anne radmacher. But not knowing which decision to take can sometimes be. This is my hand crafted collection of stephen covey quotes. The big question is.
It is a quote collection like no other. Search by author topic or a couple words you know. Read on to. Courage doesnt always roar. Browse through hundreds of lds quotes by church leaders and the scriptures.
While we are free to choose our actions we are not free to. Too often people attribute what is good to external causes. Stephen covey was a master of words and a maker of meaning. Environmental quotes quotations about the environment categorized by topics such as pollution agriculture food biodiversity chemicals mining nuclear. Choices quotes a man is happy so long as he chooses to be happy and nothing can stop him as.
Does this assertion stand or hold any essential truth. 62 quotes have been tagged as right path. 384 quotes have been tagged as decision making. If youve some time today i invite you to join me in this self discovery journey as we go through this 50 wonderful motivational or.
That Was Luxury Quotes About Choices And Consequences, Hopefully it's useful and you like it. | {
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