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Rafael Nadal Tells Novak Djokovic Off For Hurling Racket At Olympics
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\section{Motivation}
The recent interest in models of quantum engines and refrigerators stimulates theoretical efforts to precisely formulate fundamental thermodynamical principles and bounds valid on the micro- and nano-scale. In principle these can differ from the standard ones and converge to them only in the limit of macroscopic systems. A sample of references, including both general considerations and particular models, is given in~\cite{Qmachines}.
This paper is about the amount of work that can be extracted from a small quantum mechanical system that is used to store temporarily energy to transfer it from a production to a consumption center. To do so we are not coupling such a quantum battery to external thermal baths in order to drive thermodynamical engines but we address it by controlling its dynamics by external time-dependent fields. The battery comes with its initial state $\rho$ and own internal Hamiltonian $H$. The idealized process of reversible energy extraction is then governed by the system dynamics plus some fields that are only turned on during a certain interval $[0,\tau]$ of time. This leads to a time-dependent unitary dynamics of the battery. We now wonder about the maximal amount of work that can be extracted by such a process.
It has been known for a long time that some states can't deliver work in this way. Such states are called passive \cite{Pusz:1977}, \cite{Lenard:1978}. The maximal amount of work extractable from a battery is then the surplus energy of the initial state with respect to the passive state $\sigma_\rho$ with the same eigenvalues as $\rho$.
As we are dealing with small quantum systems we may wonder whether using processes that entangle two identical copies of a given battery can yield a higher energy extraction. More generally, what happens to a large number of copies?
We numerically demonstrate that the efficiency of energy extraction grows with the number of copies. Next we show rigorously that the maximal amount of extractable energy per battery asymptotically equals the energy difference between the initial state $\rho$ of the battery and the energy of the Gibbs state $\omega_{\overline\beta}$ with the same entropy as $\rho$. We indicate how to construct in principle a unitary that achieves this optimal bound.
\section{General context}
The Hilbert space $\mathcal H$ of wave functions of the battery is for simplicity chosen to be $d$-dimensional and we pick as standard basis for $\mathcal H$ the eigenvectors of the system Hamiltonian
\begin{equation}
H = \sum_{j=1}^d \epsilon_j\, |j\>\<j| \enskip\text{with}\enskip \epsilon_{j+1} > \epsilon_{j}.
\end{equation}
We assume here that the energy levels are non-degenerate which holds for a generic Hamiltonian.
The time-dependent fields that will be used to extract energy from the battery are described by $V(t) = V^\dagger(t)$ where $V(t)$ is possibly only different from zero for $0 \le t \le \tau$.
The initial state of the battery is described by a density matrix $\rho$ and the time evolution of $\rho$ is obtained from the Liouville-von~Neumann equation
\begin{equation}
\frac{d\ }{dt}\, \rho(t) = -i [H + V(t), \rho(t)],\enskip \rho(0) = \rho.
\end{equation}
The work extracted by this procedure is then
\begin{equation}
W = \tr \bigl( \rho H \bigr) - \tr \bigl( \rho(\tau) H \bigr)
\end{equation}
where the state at time $\tau$ is related to the initial state $\rho$ by a unitary transformation
\begin{equation}
\rho(\tau) = U(\tau)\, \rho\, U^\dagger(\tau)
\end{equation}
with $U(t)$ the time-ordered exponential of the total Hamiltonian $H + V(t)$:
\begin{equation}
U(\tau) = \text{Texp} \Bigl( -i \int_0^\tau \!ds\, \bigl( H + V(s) \bigr) \Bigr).
\end{equation}
Note that by a proper choice of controlling term $V$ any unitary $U$ can be obtained for $U(\tau)$. Therefore the maximal amount of extractable work (called \emph{ergotropy} in \cite{Allahverdyan:2004}) can be defined as
\begin{equation}
W_{\text{max}} := \tr \bigl( \rho H \bigr) - \min \tr \bigl( U\, \rho\, U^\dagger H \bigr)
\end{equation}
where the minimum is taken over all unitary transformations of $\mathcal H$.
Following Pusz and Woronowicz~\cite{Pusz:1977} and Lenard~\cite{Lenard:1978}, we call a state $\sigma$ passive if no work can be extracted from $\sigma$, i.e.\ if for all unitaries $U$
\begin{equation*}
\tr \bigl( \sigma H \bigr) \le \tr \bigl( U\, \sigma\, U^\dagger H \bigr).
\end{equation*}
The following theorem then holds:
\begin{thm}[\cite{Pusz:1977,Lenard:1978}]
$\sigma$ is passive if and only if
\begin{equation}
\sigma = \sum_{j=1}^d s_j\, |j\>\<j| \enskip\text{with}\enskip s_{j+1} \le s_{j}
\end{equation}
\end{thm}
In other words, $\sigma$ is passive if and only if it commutes with the system Hamiltonian and its eigenvalues are non-increasing with the energy. Given $\rho$ there is a unique passive state $\sigma_\rho$ minimizing $\tr U\, \rho\, U^\dagger H$. This state is obtained by a unitary rotation of $\rho$ denoted by $U_{\rho}$ and has the form
\begin{equation}
\sigma_\rho = U_{\rho}\,\rho\,U_{\rho}^{\dagger} = \sum_{j=1}^d r_j\, |j\>\<j|
\end{equation}
where $\{r_j\}$ are the eigenvalues of $\rho$ arranged in non-increasing order: $r_{j+1} \le r_{j}$. The corresponding minimal energy is $\sum_{j=1}^d r_j \epsilon_j$ and the maximal amount of extractable work is given by
\begin{equation}
W_{\text{max}} := \tr \bigl( \rho H \bigr) - \tr \bigl( \sigma_\rho\, H \bigr).
\label{wmax}
\end{equation}
\section{A general bound on available work}
We obtain here a bound on $W_{\text{max}}$ by comparing the energies of the passive state $\sigma_\rho$ and of the canonical Gibbs state $\omega_{\overline\beta}$ with the same entropy as $\rho$. Recall that the canonical Gibbs state at inverse temperature $\beta$ is given by
\begin{equation}
\omega_\beta = \frac{\exp(-\beta H)}{\mathcal Z}
\end{equation}
and that its von~Neumann entropy is strictly monotonically decreasing in $\beta$ with range $[0, \log d]$. The von~Neumann entropy $S(\rho)$ of a density matrix $\rho$ is
\begin{equation}
S(\rho) = - \tr \rho \log\rho.
\end{equation}
For any given density matrix $\rho$ on $\mathcal H$ there exists therefore a unique inverse temperature $\overline\beta$ such that $S(\rho) = S(\omega_{\overline\beta})$. The relation between $\rho$ and $\overline\beta$ is, of course, highly non-linear.
We now use the variational principle of statistical mechanics that asserts that the Gibbs canonical density matrix is that which minimizes the free energy:
\begin{equation}
\tr \bigl( \rho H \bigr)- {\overline\beta}^{-1}\, S(\rho) \ge \tr \bigl( \omega_{\overline\beta}\, H \bigr) - {\overline\beta}^{-1}\, S(\omega_{\overline\beta}).
\label{var}
\end{equation}
With our choice of $\overline\beta$ we obtain that
\begin{equation}
\tr \bigl( \rho H \bigr) \ge \tr \bigl( \sigma_{\rho} H \bigr)\ge\tr \bigl( \omega_{\overline\beta}\, H \bigr).
\label{var1}
\end{equation}
and hence the thermodynamical bound on the available work
\begin{equation}
W_{\text{max}} \le \tr \bigl( \rho\, H \bigr) - \tr \bigl(\omega_{\overline\beta}\, H \bigr).
\label{bound}
\end{equation}
Generally, $\omega_{\overline\beta}$ is different from $\sigma_\rho$ as $\omega_{\overline\beta}$ and $\sigma_\rho$ or $\rho$ have different eigenvalues. Note, however, that the two-dimensional case is exceptional because there is a one-to-one correspondence between the entropy of a qubit state and its ordered eigenvalues. Generally it is not true that a product of two independent copies of a passive state still is passive. There is therefore a possibility of improving over~(\ref{wmax}) on the amount of extractable work per copy for several copies of a system. In other words, by using entangling unitaries, one can in principle beat~(\ref{wmax}).
\psfrag{b}{\hspace{-2pt}\raisebox{-3pt}{10}}
\psfrag{c}{\hspace{-2pt}\raisebox{-3pt}{20}}
\psfrag{d}{\hspace{-2pt}\raisebox{-3pt}{30}}
\psfrag{e}{\hspace{-2pt}\raisebox{-3pt}{40}}
\psfrag{n}{$n$}
\psfrag{w}{$e^{(n)}$}
\psfrag{r}{\hspace{-12pt}\raisebox{3pt}{$e^{(\infty)}$}}
\psfrag{g}{\hspace{-9pt}\raisebox{3pt}{$e^{(1)}$}}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.45 \textwidth]{workfig.eps}
\end{center}
\caption{Energy per copy of passive state $\sigma_{\otimes^n \rho}$ associated to $\otimes^n \rho$}
\end{figure}
In Fig.1 the energies $e^{(n)}$ per copy of the passive state $\sigma_{\otimes^n \rho}$ obtained from a product state $\otimes^n \rho$ are plotted as dots for $n=1, 2, \ldots, 40$. The lower line shows the asymptotic value of $e^{(n)}$. The system is a three level battery with energy levels $\{0,0.579,1\}$ and the passive state corresponding to the initial density matrix has eigenvalues $\{0.538,0.237,0.224\}$. The values $e^{(n)}$ have been obtained by rearranging the eigenvalues of $\otimes^n \rho$ and the $n$-copy Hamiltonian $H^{(n)}$, see~(\ref{hamn}). The maximal additional work that can be extracted on top of the single copy extractable work using entangling unitaries is the difference between $e^{(1)}$ and $e^{(\infty)}$. We will compute this value in the next section.
\section{Entangling batteries}
A state $\sigma$ is called completely passive if $\otimes^n \sigma$ is passive for all $n = 1, 2, \ldots$ with respect to the sum Hamiltonian
\begin{equation}
H^{(n)} = \sum_{j=1}^n H_j
\label{hamn}
\end{equation}
where $H_j$ is the $j$-th independent copy of $H$. Thermodynamic equilibrium is equivalent to complete passivity:
\begin{thm}[\cite{Pusz:1977,Lenard:1978}]
$\sigma$ is completely passive if and only if it is a Gibbs state.
\end{thm}
We now consider $n$ independent copies of our battery and apply the general bound~(\ref{bound}) to estimate the maximal amount of available work per battery
\begin{align}
w^n_{\text{max}}
&:= \frac{1}{n}\, \Bigl\{ \tr \bigl( (\otimes^n \rho)\, H^{(n)} \bigr) - \tr \bigl( \sigma_{\otimes^n \rho}\, H^{(n)} \bigr) \Bigr\} \nonumber \\
&\le \tr \bigl( \rho H \bigr) - \tr \bigl( \omega_{\overline\beta}\, H \bigr).
\label{optimal}
\end{align}
It is our aim to show that this bound is actually asymptotically achievable:
\begin{thm}
\begin{equation}
\lim_{n\to\infty} w^n_{\text{max}}= \tr \bigl( \rho H \bigr) - \tr \bigl( \omega_{\overline\beta}\, H \bigr).
\label{ineq}
\end{equation}
\end{thm}
\begin{proof}
The proof is based on the idea of typical configurations. Assume that $\rho$ is diagonal in the eigenbasis of $H$, this can always be achieved by a suitable unitary rotation and let $\{r_j\}$ be the eigenvalues of $\rho$ arranged in non-decreasing order. A configuration of length $n$ is an $n$-tuple $|\mathbf i \> = |i_1, i_2, \ldots, i_n\>$ of indices in $\{1,2,\ldots, d\}$ with corresponding eigenvalue $r_{i_1} r_{i_2} \cdots r_{i_n}$ of $\otimes^n \rho$. Typical configurations will be of the type $|\mathbf i\>$ where the number of times the index $k \in \{1,2,\ldots,d\}$ occurs lies between $(r_k - \epsilon) n$ and $(r_k + \epsilon) n$ for every $k$. The subspace spanned by these typical $|\mathbf i\>$ has approximately dimension $\exp(n S(\rho))$ and each such configuration corresponds to the average energy
\begin{equation}
\frac{1}{n}\, H^{(n)}\, |\mathbf i\> = \Bigl( \sum_{j=1}^d r_j\, \epsilon_j \Bigr) |\mathbf i> + \text{o}(\epsilon).
\label{typical}
\end{equation}
Now we repeat the same construction for the product of $n$ copies of the Gibbs state $\omega_{\overline\beta}$. As $S(\omega_{\overline\beta}) = S(\rho)$, the typical subspaces of $\otimes^n \rho$ and $\otimes^n \omega_{\overline\beta}$ have approximately the same dimension. Moreover, for both product states the probability of finding a system outside the typical subspaces is $\text{o}(\epsilon)$. We can now find a unitary $U^{(\epsilon)}$ on $\otimes^n \mathcal{H}$ that maps one subspace into the other. This unitary is highly non-unique, and generally differs from the optimal reordering given by $U_{\otimes^n\rho}$ but nevertheless produces the state with the energy close to the optimal one, i.e.
\begin{equation}
\bigl| \tr \bigl(U^{(\epsilon)} (\otimes^n \rho)\,{U^{(\epsilon)}}^{\dagger}\, H^{(n)} \bigr) -\tr \bigl( (\otimes^n \omega_{\overline\beta})\, H^{(n)} \bigr) \bigr| \le n\, \text{o}(\epsilon) .
\label{optimal1}
\end{equation}
Using \eqref{var1} we obtain
\begin{align}
\tr \bigl( \otimes^n \omega_{\overline\beta}\, H^{(n)} \bigr) &\leq \tr \bigl( \sigma_{\otimes^n \rho}\, H^{(n)} \bigr)\\
\nonumber
&\le \tr \bigl(U^{(\epsilon)} (\otimes^n \rho)\,{U^{(\epsilon)}}^{\dagger}\, H^{(n)} \bigr),
\label{optimal2}
\end{align}
which combined with~\eqref{optimal1} yields the final estimation
\begin{align}
&\tr \bigl( \rho H \bigr) - \tr \bigl( \omega_{\overline\beta}\, H \bigr)\geq w^n_{\text{max}} \geq\\
\nonumber
&\tr \bigl( \rho H \bigr) - \tr \bigl( \omega_{\overline\beta}\, H \bigr)- \text{o}(\epsilon).
\end{align}
\end{proof}
Remark that a unitary transforming $\otimes^n \rho$ into $\sigma_{\otimes^n \rho}$ cannot be product and must therefore dynamically entangle the $n$ batteries. In the numerical example of Fig.~1 the asymptotic value $e^{(\infty)}$ exactly coincides with $\omega_{\overline\beta}$.
\section{Conclusion}
The notion of maximal reversibly extractable work for a quantum battery motivated by the concept of passivity is discussed. It is applicable to full quantum models of micro- or mesoscopic machines where work is supplied or extracted by a quantum system (`quantum battery', `work reservoir') instead of a time-dependent perturbation of the Hamiltonian. A proper definition of work is important to develop a consistent thermodynamics of small quantum systems which is relevant in nanotechnology and biophysics. Generally, the extractable work is smaller than the thermodynamical bound computed using variational principle for a free energy. Using entanglement one can in general extract more work per battery from several independent copies of a battery and asymptotically reach the thermodynamical bound. However, the optimal procedures of work extraction are generally difficult to implement by realistic control Hamiltonians. An interesting problem for future investigation is to find efficiency bounds when practical restrictions are imposed on the available control mechanisms.
\begin{acknowledgments}
R.A. acknowledges the support by the Polish Ministry of Science and Higher Education, grant NN 202208238 and M.F. the FWO Vlaanderen project G040710N.
\end{acknowledgments}
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Leiophron scitula är en stekelart som först beskrevs av Cresson 1872. Leiophron scitula ingår i släktet Leiophron och familjen bracksteklar. Inga underarter finns listade i Catalogue of Life.
Källor
Bracksteklar
scitula
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Philanthropy & Cause Initiatives
REI Grants Help People Care for and Connect with the Great Outdoors
Co-op Awards $4.6 Million to Nonprofits in 2014 to Enhance Popular Spaces and Promote Outdoor Recreation
Aug 5, 2014 4:30 PM ET
Campaign: REI CSR Related Updates
REI helps take care of the outdoor places our customers and members love.
August 5, 2014 /3BL Media/ - This year, REI will build upon a long-standing commitment to maintain the outdoors and help people connect with recreational opportunities by awarding $4.6 million in grants to more than 300 nonprofit organizations.
The majority of the investments will support local organizations that are enhancing more than 650 parks, trails and waterways across the country – places enjoyed by the co-op's members, employees and other adventurers. Additional grants from REI will help fund national and regional nonprofit programs that support outdoor activities like urban cycling, mountain biking, backcountry skiing and climbing. Through The REI Foundation, the retailer will also partner with several nonprofits to engage younger and more diverse populations in outdoor recreation.
"At REI, we want to share our passion for outdoor recreation far and wide. But to have fun outdoors, you have to have great places to go," said Marc Berejka, director of Government & Community Affairs at REI. "In partnership with our employees and more than 5 million members, we're dedicated to supporting nonprofits that sustain popular outdoor spaces and keep them accessible for generations to come."
Over the past decade, REI and The REI Foundation have awarded more than $37 million in grants to nonprofits. The co-op's 2014 contributions will support dedicated organizations such as:
American Alpine Club – cultivating healthy climbing landscapes and empowering local climbing communities through programs like Cornerstone Conservation Grants;
Atlanta BeltLine Partnership – providing a cleaner, safer environment for the more than 1 million residents and visitors who bicycle, run and hike on the BeltLine Corridor each year;
Evergreen Mountain Bike Alliance – creating and protecting sustainable mountain biking opportunities in Washington state, including at East Tiger Mountain, Duthie Hill Park, Swan Creek Park and beyond;
Friends of the Chicago River – partnering with businesses, community groups, schools, government agencies and others to improve the health of the Chicago River for the benefit of people and wildlife;
Friends of the High Line – maintaining and expanding New York City's High Line, a public park built on a historic freight rail line elevated above the streets on Manhattan's West Side;
Palos Verdes Peninsula Land Conservancy – mobilizing volunteers in Southern California to restore and maintain 1,600 acres of public coastal open space, including over 40 miles of recreational trails;
The Trustees of Reservations – preserving areas of exceptional scenic, historic and ecological value in the state of Massachusetts, including 320 miles of recreational trails;
Volunteers for Outdoor Colorado – enabling outdoor lovers to participate in more than 55 highly diverse service opportunities in some of the most popular outdoor locations across the state; and
Washington Area Bicyclist Association – creating a healthier, more livable Washington, D.C. region by helping to plan and construct a key network of bike trails.
All REI grants are made by invitation and local grantee selection is managed by employees in the co-op's store communities. In addition to making financial contributions, each year REI's store teams help mobilize tens of thousands of local volunteers to build trails, clean up beaches and maintain other outdoor spaces.
For more information on REI's community and environmental efforts, visit rei.com/stewardship. To search for upcoming volunteer service activities, visit REI.com/stores, find the nearest REI location, and select "Stewardship" under "Classes and Events."
REI is a $2 billion national multichannel retail co-op headquartered outside of Seattle. With more than 5 million active members, REI serves the needs of outdoor adventurers through innovative, quality products; inspiring classes and trips; and integrated customer service that allows shoppers to buy great gear and clothing in any way they want. REI has 135 stores in 33 states and REI.com and REI.com/outlet. Anyone may shop with REI, while members pay a one-time $20 fee to receive a share in the co-op's profits through an annual member refund based on patronage. Membership in the co-op also includes special promotions and discounts on REI Adventures trips and REI Outdoor School classes.
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The REI Foundation Awards $410,000 to Connect Younger and More Diverse Groups with Nature
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Apr 8, 2014 1:30 PM ET
Starting at the Source: REI Partners with bluesign technologies ag to Minimize Environmental Impact and Increase Customer Care
Jan 21, 2014 3:00 PM ET
REI and Its Partners Caring for Outdoor Places Around the Country
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{"url":"https:\/\/kalabuslaw.com\/blog\/unax-ugalde-pbv\/viewtopic.php?id=f3fe1f-sklearn-linear-regression-summary","text":"sklearn linear regression summary\n\nInstead of only knowing how to build a logistic regression model using Sklearn in Python with a few lines of code, I would like you guys to go beyond coding understanding the concepts behind. Generalized Linear Models. The method works on simple estimators as well as on nested objects We will be using this dataset to model the Power of a building using the Outdoor Air Temperature (OAT) as an explanatory variable.. Scikit-learn Summary Posted on 2019-04-24 | Edited on 2019-05-03 ... # from sklearn.pipeline import make_pipeline # used when there is no data preprocessing ... sns.regplot- Including a regression line in the scatter plot makes it easier to see linear relationship between two variables. Lasso regression, or the Least Absolute Shrinkage and Selection Operator, is also a modification of linear regression. The following are 30 code examples for showing how to use sklearn.linear_model.LinearRegression().These examples are extracted from open source projects. Fit Summary. sklearn.linear_model.LinearRegression is the module used to implement linear regression. sklearn.linear_model.LinearRegression is the module used to implement linear regression. Linear Regression using Sklearn. It is used to forecast unobserved values. An extension to linear regression involves adding penalties to the loss function during training that encourage simpler models that have smaller coefficient values. These examples are extracted from open source projects. The average unemployment stands at 7771 thousand for the data. Parameters X {array-like, sparse matrix} of shape (n_samples, n_features) The input samples. python - with - sklearn linear regression summary . Also known as Ridge Regression or Tikhonov regularization. sum of squares ((y_true - y_pred) ** 2).sum() and v is the total See Glossary Linear Regression vs Closed form Ordinary least squares in Python (1) I am trying to apply Linear Regression method for a dataset of 9 sample with around 50 features using python. An extension to linear regression involves adding penalties to the loss function during training that encourage simpler models that have smaller coefficient values. An easy way to check your dependent variable (your y variable), is right in the model.summary(). In this step-by-step tutorial, you'll get started with logistic regression in Python. Simple linear regression is a statistical method that allows us to summarize and study relationships between two or more continuous (quantitative) variables. But if it is set to false, X may be overwritten. Will be cast to X\u2019s dtype if necessary. Target values. For example, if \u2026 Code: https:\/\/github.com\/sachinruk\/deepschool.io\/ Lesson 1 First, generate some data that we can run a linear regression on. sklearn.preprocessing.StandardScaler before calling fit on the expected mean value of Y when all X = 0 by using attribute named \u2018intercept\u2019 as follows \u2212. Unlike SKLearn, statsmodels doesn\u2019t automatically fit a constant, so you need to use the method sm.add_constant(X) in order to add a constant. Estimated coefficients for the linear regression problem. You may check out the related API usage on the sidebar. While the X variable comes first in SKLearn, y comes first in statsmodels. 0 Votes 1 Answer when I tried to follow the instruction of the following reg.predict(1740) it shows me it is not a 2D array, how to make it work? Today we\u2019ll be looking at a simple Linear Regression example in Python, and as always, we\u2019ll be using the SciKit Learn library. Logistic Regression. We will use the physical attributes of a car to predict its miles per gallon (mpg). The coefficient R^2 is defined as (1 - u\/v), where u is the residual Linear Regression \u00b6 Linear models with independently and identically distributed errors, and for errors with heteroscedasticity or autocorrelation. The relationship can be established with the help of fitting a best line. contained subobjects that are estimators. In this post, we\u2019ll be exploring Linear Regression using scikit-learn in python. shape = (n_samples, n_samples_fitted), to minimize the residual sum of squares between the observed targets in You can use it to find out which factor has the highest impact on the predicted output and how different variables relate to each other. normalize \u2212 Boolean, optional, default False. Linear Regression\u00b6 Linear models with independently and identically distributed errors, and for errors with heteroscedasticity or autocorrelation. Let\u2019s directly delve into multiple linear regression using python via Jupyter. The predicted regression target of an input sample is computed as the mean predicted regression targets of the trees in the forest. Ex. Elastic-Net is a linear regression model trained with both l1 and l2 -norm regularization of the coefficients. component of a nested object. The normalization will be done by subtracting the mean and dividing it by L2 norm. Independent term in the linear model. Following table consists the parameters used by Linear Regression module \u2212, fit_intercept \u2212 Boolean, optional, default True. the dataset, and the targets predicted by the linear approximation. Setup. fit_intercept = False. This model is available as the part of the sklearn.linear_model module. Sklearn Implementation of Linear and K-neighbors Regression. speedup for n_targets > 1 and sufficient large problems. Importing the necessary packages. In the case considered here, we simply what to make a fit, so we do not care about the notions too much, but we need to bring the first input to \u2026 You have seen some examples of how to perform multiple linear regression in Python using both sklearn and statsmodels. Especially with the help of this Scikit learn library, it\u2019s implementation and its use has become quite easy. We will predict the prices of properties from our test set. This model solves a regression model where the loss function is the linear least squares function and regularization is given by the l2-norm. We see that the resulting polynomial regression is in the same class of linear models we considered above (i.e. (Please check this answer) . from sklearn import linear_model from scipy import stats import numpy as np class LinearRegression(linear_model.LinearRegression): \"\"\" LinearRegression class after sklearn's, but calculate t-statistics and p-values for model coefficients (betas). Ridge regression addresses some of the problems of Ordinary Least Squares by imposing a penalty on the size of the coefficients with l2 regularization. Only available when X is dense. You can vote up the ones you like or vote down the ones you don't like, and go to the original project or source file by following the links above each example. Now, let\u2019s start using Sklearn. Those of us attempting to use linear regression to predict probabilities often use OLS\u2019s evil twin: logistic regression. It is one of the best statistical models that studies the relationship between a dependent variable (Y) with a given set of independent variables (X). Let\u2019s see how we can come up with the above formula using the popular python package for machine learning, Sklearn. Test samples. A summary of a regression model trained with statsmodels. This example uses the only the first feature of the diabetes dataset, in order to illustrate a two-dimensional plot of this regression technique. (such as pipelines). Simple Linear Regression Now, provide the values for independent variable X \u2212, Next, the value of dependent variable y can be calculated as follows \u2212, Now, create a linear regression object as follows \u2212, Use predict() method to predict using this linear model as follows \u2212, To get the coefficient of determination of the prediction we can use Score() method as follows \u2212, We can estimate the coefficients by using attribute named \u2018coef\u2019 as follows \u2212, We can calculate the intercept i.e. The summary provides several measures to give you an idea of the data distribution and behavior. If multiple targets are passed during the fit (y 2D), this sum of squares ((y_true - y_true.mean()) ** 2).sum(). It's a good idea to start doing a linear regression for learning or when you start to analyze data, since linear models are simple to understand. In summary, we\u2019ve presented a tutorial on simple and multiple regression analysis using different libraries such as NumPy, Pylab, and Scikit-learn. LinearRegression fits a linear model with coefficients w = (w1, \u2026, wp) If True, will return the parameters for this estimator and Check out my post on the KNN algorithm for a map of the different algorithms and more links to SKLearn. The latter have parameters of the form Building and training the model Using the following two packages, we can build a simple linear regression model.. statsmodel; sklearn; First, we\u2019ll build the model using the statsmodel package. This will only provide In summary, we learned what linear regression is, introduced ordinary least square to find the line of best fit, and implemented a simple and multiple linear regression. Oftentimes it would not make sense to consider the interpretation of the intercept term. The steps to perform multiple linear regression are almost similar to that of simple linear regression. (L1_wt=0 for ridge regression. See help(type(self)) for accurate signature. __ so that it\u2019s possible to update each You can vote up the ones you like or vote down the ones you don't like, and go to the original project or source file by following the links above each example. Summary. But in logistic regression, the dependent variable is categorical, and hence it \u2026 Linear Regression in Python using scikit-learn. Linear Regression Example\u00b6. Only available when X is dense. To do that, we need to import the statsmodel.api library to perform linear regression.. By default, the statsmodel library fits a line that passes through the origin. sklearn.linear_model.LogisticRegression ... Logistic Regression (aka logit, MaxEnt) classifier. Linear Regression in SKLearn SKLearn is pretty much the golden standard when it comes to machine learning in Python. Regression models a target prediction value based on independent variables. With a team of extremely dedicated and quality lecturers, sklearn linear regression summary will not only be a place to share knowledge but also to help students get inspired to explore and discover many creative ideas from themselves. Simple Linear Regression with sklearn \u2013 Summary Table liqian Zhang 8 months ago. ... sklearn.linear_model.LinearRegression is the module used to implement linear regression. In the multiclass case, the training algorithm uses the one-vs-rest (OvR) scheme if the \u2018multi_class\u2019 option is set to \u2018ovr\u2019, and uses the cross-entropy loss if the \u2018multi_class\u2019 option is set to \u2018multinomial\u2019. Adding a constant, while not necessary, makes your line fit much better. This is an independent term in this linear model. )For now, it seems that model.fit_regularized(~).summary() returns None despite of docstring below. A linear regression approach would probably be better than random guessing but likely not as good as a nonlinear approach. The two variables specifically involve an\u2026 A constant model that always Linear regression produces a model in the form: $Y = \\beta_0 + \\beta_1 X_1 \u2026 It represents the number of jobs to use for the computation. Linear Regression in Python using scikit-learn. In this post, we\u2019ll be exploring Linear Regression using scikit-learn in python. If this parameter is set to True, the regressor X will be normalized before regression. Basic Linear models in sklearn, the machine learning library in python. Linear regression is \u2026 Plot individual and voting regression predictions\u00b6, Ordinary Least Squares and Ridge Regression Variance\u00b6, Robust linear model estimation using RANSAC\u00b6, Sparsity Example: Fitting only features 1 and 2\u00b6, Automatic Relevance Determination Regression (ARD)\u00b6, Face completion with a multi-output estimators\u00b6, Using KBinsDiscretizer to discretize continuous features\u00b6, array of shape (n_features, ) or (n_targets, n_features), {array-like, sparse matrix} of shape (n_samples, n_features), array-like of shape (n_samples,) or (n_samples, n_targets), array-like of shape (n_samples,), default=None, array_like or sparse matrix, shape (n_samples, n_features), array-like of shape (n_samples, n_features), array-like of shape (n_samples,) or (n_samples, n_outputs), Plot individual and voting regression predictions, Ordinary Least Squares and Ridge Regression Variance, Robust linear model estimation using RANSAC, Sparsity Example: Fitting only features 1 and 2, Automatic Relevance Determination Regression (ARD), Face completion with a multi-output estimators, Using KBinsDiscretizer to discretize continuous features. This influences the score method of all the multioutput Notes. The relationship can be established with the help of fitting a best line. I have tried different methodology for Linear Regression \u2026 Linear Regression is a machine learning algorithm based on supervised learning. So, we\u2019ll be using Boston Housing Price dataset from sklearn. Source code linked here.. Table of Contents. Linear Regression is a machine learning algorithm based on supervised learning. The linear regression line is below 0. Generalized Linear Models. ... (Omnibus) is relatively high so the data is somewhat normal, but not altogether ideal. Least Squares (scipy.linalg.lstsq) wrapped as a predictor object. Linear regression is the standard algorithm for regression that assumes a linear relationship between inputs and the target variable. Return the coefficient of determination R^2 of the prediction. Linear Regression with Python Scikit Learn. Internally, its dtype will be converted to dtype=np.float32. Linear regression is only dealing with continuous variables instead of Bernoulli variables. For Multiple linear regression, the beta coefficients have a slightly different interpretation. would get a R^2 score of 0.0. If float, then min_samples_leaf is a fraction and ceil(min_samples_leaf * n_samples) are the minimum number of samples for each node. We will start with simple linear regression involving two variables and then we will move towards linear regression involving multiple variables. In Lasso, the loss function is modified to minimize the complexity of the model by limiting the sum of the absolute values of the model coefficients (also called the l1-norm). We fitted a straight line based on the relationship between the dependent and independent variables. Some of them are support vector machines, \u2026 Import Data. Sklearn Linear Regression. This module allows estimation by ordinary least squares (OLS), weighted least squares (WLS), generalized least squares (GLS), and feasible generalized least squares with autocorrelated AR (p) errors. Linear Regression Equations. I'm trying to generate a linear regression on a scatter plot I have generated, however my data is in list format, and all of the examples I can find of using polyfit require using arange. Basic Linear models in sklearn, the machine learning library in python. It would be a 2D array of shape (n_targets, n_features) if multiple targets are passed during fit. MultiOutputRegressor). If True, the regressors X will be normalized before regression by As already mentioned above, Logistic and Linear Regression are part of a bigger family called Generalized Linear \u2026 Following table consists the attributes used by Linear Regression module \u2212, coef_ \u2212 array, shape(n_features,) or (n_targets, n_features). For instance, in our case, the intercept term has to do with the case where the house has 0 rooms\u2026it doesn\u2019t make sense for a house to have no rooms. It is mostly used for finding out the relationship between variables and forecasting. multioutput='uniform_average' from version 0.23 to keep consistent Find professional answers about \"Simple Linear Regression with sklearn - Summary Table\" in 365 Data Science's Q&A Hub. Brief Introduction. If multiple targets are passed during the fit (y 2D), this is a 2D array of shape (n_targets, n_features), while if only one target is passed, this is a 1D array of length n_features. Linear Regression is a very straight forward and easy to use algorithm. n_jobs \u2212 int or None, optional(default = None). y_train data after splitting. For the prediction, we will use the Linear Regression model. No intercept will be used in the calculation if this set to false. Let us take a step back and try to remember what used to happen in linear regression. We will fit the model using the training data. How to make a single value become a 2D array Thanks. samples used in the fitting for the estimator. Linear regression is sometimes not appropriate, especially for non-linear models of high complexity. Initialize self. The number of jobs to use for the computation. Ordinary least squares Linear Regression. If int, then consider min_samples_leaf as the minimum number. This estimator has built-in support for multi-variate regression (i.e., when y \u2026 The problem of Linear Regression is that these predictions are not sensible for classification since the true probability must fall between 0 and 1, \u2026 the model is linear in $$w$$) and can be solved by the same techniques. Additional attributes available after .fit() are t and p which are of the shape (y.shape[1], X.shape[1]) which is (n_features, n_coefs) This \u2026 # generate regression dataset from sklearn.datasets.samples_generator import make_regression X, y = make_regression(n_samples=100, n_features=1, noise=10) Second, create a \u2026 Scikit Learn - Linear Regression - It is one of the best statistical models that studies the relationship between a dependent variable (Y) with a given set of independent variables (X). Linear regression involving multiple variables is called \"multiple linear regression\". III. First the \"training data\", which should be a 2D array, and second the \"target values\". The limitations of linear regression; The understanding of \u201cOdd\u201d and \u201cProbability\u201d The transformation from linear to logistic regression In this video, we will go over the regression result displayed by the statsmodels API, OLS function. Regression problems want to find the relationship between the input variables and output variables. Python | Linear Regression using sklearn Last Updated: 28-11-2019. Ordinary least squares Linear Regression. For some estimators this may be a data is expected to be centered). Instead, if you need it, there is statsmodels.regression.linear_model.OLS.fit_regularized class. The R2 score used when calling score on a regressor uses Linear Regression is the method of using a straight line to establish a relationship between two variables. The following are 30 code examples for showing how to use sklearn.linear_model.LinearRegression(). It has many learning algorithms, for regression, classification, clustering and dimensionality reduction. scikit-learn 0.23.2 You'll learn how to create, evaluate, and apply a model to make predictions. (i.e. If you wish to standardize, please use Regression models a target prediction value based on independent variables. subtracting the mean and dividing by the l2-norm. to False, no intercept will be used in calculations Before we dive into understanding what logistic regression is and how we can build a model of Logistic Regression in Python, let us see two scenarios and try and understand where to apply linear regression and where to apply logistic regression. But the object has params, summary() can be used somehow. If set This example uses the only the first feature of the diabetes dataset, in order to illustrate a two-dimensional plot of this regression technique. If fit_intercept = False, this parameter will be ignored. Let\u2019s directly delve into multiple linear regression using python via Jupyter. sklearn linear regression summary provides a comprehensive and comprehensive pathway for students to see progress after the end of each module. Linear Regression is one of the simplest machine learning methods. Summary. Fortunately, there are other regression techniques suitable for the cases where linear regression doesn\u2019t work well. Exploring the Dataset. Simple linear regression is an approach for predicting a quantitative response using a single feature (or \"predictor\" or \"input variable\") It takes the following form: y = \u03b2 0 + \u03b2 1 x What does each term represent? The best possible score is 1.0 and it can be negative (because the On the other hand, it would be a 1D array of length (n_features) if only one target is passed during fit. It performs a regression task. Vote Up Vote Down. precomputed kernel matrix or a list of generic objects instead, class sklearn.linear_model. Those of us attempting to use linear regression to predict probabilities often use OLS\u2019s evil twin: logistic regression. Importing the necessary packages. model can be arbitrarily worse). Before applying linear regression models, make sure to check that a linear relationship exists between the dependent variable (i.e., what you are trying to predict) and the independent variable\/s (i.e., the input variable\/s). slr_results.summary() coef: These are the estimates of the factor coefficients. From the implementation point of view, this is just plain Ordinary It is used to estimate the coefficients for the linear regression problem. In this video, we will go over the regression result displayed by the statsmodels API, OLS function. As already mentioned above, Logistic and Linear Regression are part of a bigger family called Generalized Linear \u2026 model = LinearRegression() model.fit(X_train, y_train) Once we train our model, we can use it for prediction. Sklearn, on the other hand, implements linear regression using the machine learning approach and doesn\u2019t provide in-depth summary reports but allows for additional features such as \u2026 predicts the expected value of y, disregarding the input features, First of all, we need some data to apply Linear Regression to it. where n_samples_fitted is the number of By default, it is true which means X will be copied. is a 2D array of shape (n_targets, n_features), while if only New in version 0.17: parameter sample_weight support to LinearRegression. Without much delay, let\u2019s get started. We will use the physical attributes of a car to predict its miles per gallon (mpg). Singular values of X. Join today! In this section we will see how the Python Scikit-Learn library for machine learning can be used to implement regression functions. Linear Regression Example\u00b6. intercept_: array. The Lasso is a linear model that estimates sparse coefficients with l1 regularization. By considering linear fits within a higher-dimensional space built with these basis functions, the model has the flexibility to fit a much broader range of data. Other versions. for more details. The difference lies in the evaluation. Whether to calculate the intercept for this model. As I know, there is no R(or Statsmodels)-like summary table in sklearn. This parameter is ignored when fit_intercept is set to False. Linear Regression Equations. The third line gives summary statistics of the numerical variables. This example uses the only the first feature of the diabetes dataset, in order to illustrate a two-dimensional plot of this regression technique. If you are excited about applying the principles of linear regression and want to think like a data scientist, then this post is for you. (y 2D). regressors (except for Linear regression produces a model in the form:$ Y = \\beta_0 + \\beta_1 X_1 \u2026 We shall use sklearn for model building. The first line of code reads in the data as pandas dataframe, while the second line prints the shape - 574 observations of 5 variables. residuals - sklearn linear regression summary . Linear regression is the standard algorithm for regression that assumes a linear relationship between inputs and the target variable. Regression is a modeling task that involves predicting a numeric value given an input. one target is passed, this is a 1D array of length n_features. Code: https:\/\/github.com\/sachinruk\/deepschool.io\/ Lesson 1 None means 1 unless in a joblib.parallel_backend context. Set to 0.0 if Used to calculate the intercept for the model. -1 means using all processors. Regression is a modeling task that involves predicting a numeric value given an input. The sklearn.LinearRegression.fit takes two arguments. Rank of matrix X. It is one of the best statistical models that studies the relationship between a dependent variable (Y) with a given set of independent variables (X). Variable ( your y variable ), is right in the calculation if this set False... Way to check your dependent variable is categorical, and for errors with heteroscedasticity or autocorrelation specifically! Regression\u00b6 linear models with independently and identically distributed errors, and second the target values.! ) ) and can be used in calculations ( i.e | linear regression and be..., its dtype will be done by subtracting the mean and dividing the... The coefficients the golden standard when it comes to machine learning in python most basic problems regression! Calculations ( i.e regressors ( except for MultiOutputRegressor ) relationships between two or continuous! A target prediction value based on independent variables task that involves predicting a value... How the python scikit-learn library for machine learning library in python if this parameter set. Step back and try to remember what used to happen in linear using. 8 months ago, fit_intercept \u2212 Boolean, optional, default True ( such as pipelines ) y. Prediction, we will move towards linear regression '' a penalty on the relationship be! -Norm regularization of the data distribution and behavior coefficient of determination R^2 of the diabetes dataset, in order illustrate! Summary statistics of the data approach would probably be better than random guessing but likely sklearn linear regression summary as good as predictor! Works on simple estimators as well as on nested objects ( such as )! Score on a regressor uses multioutput='uniform_average ' from version 0.23 to keep consistent with default of... Parameters for this estimator and contained subobjects that are estimators interpretation of the simplest sklearn linear regression summary learning be. Its miles per gallon ( mpg ) most important areas of machine learning library in python linear... Areas of machine learning library in python section we will predict the prices of properties from our set! Forward and easy to use linear regression in python that assumes a linear regression ''.summary sklearn linear regression summary ) can arbitrarily! The physical attributes of a sklearn linear regression summary model trained with both l1 and l2 -norm regularization of factor. A constant, while not necessary, makes your line fit much better X = 0 by using named! Regression involves adding penalties to the loss function during training that encourage simpler models that have smaller values. Can come up with the above formula using the training data steps to perform multiple linear,! To LinearRegression ), is right in the form: \\$ y = \\beta_0 + \\beta_1 \u2026. We train our model, especially for non-linear models of high complexity the part the. Input features, would get a R^2 score of 0.0, will return the parameters for this estimator contained. ).summary ( ) coef: These are the estimates of the problems of regression predict its miles gallon... To standardize, please use sklearn.preprocessing.StandardScaler before calling fit on an estimator with normalize=False,. Mean and dividing by the statsmodels API, OLS function and ceil ( min_samples_leaf * n_samples ) are estimates! Study relationships between two or more continuous ( quantitative ) variables calculations ( i.e and forecasting and it be... See help ( type ( self ) ) for accurate signature objects ( such as ). At 7771 thousand for the linear regression using scikit-learn in python to True, the coefficients... Value based on independent variables and apply a model to make predictions a slightly different interpretation usage the!, it is True which means X will be ignored, default True imposing a penalty the... ( because the model using the training data '', which should be 2D! Line based on the sidebar Updated: 28-11-2019 easy to use linear regression involves adding to. Specifically involve an\u2026 this may have the effect of smoothing the model is available as the minimum.! Other regression techniques suitable for the data the most important areas of machine learning library in.! From the implementation point of view, this parameter will be converted to dtype=np.float32 the regressors will... Get a R^2 score of 0.0 this Scikit learn library, it may overwritten... Be overwritten regression approach would probably be better than random guessing but not... Can be established with the help of fitting a best line with continuous variables instead of variables... First, generate some data that we can use it for prediction one the. ( n_targets, n_features ) the input features, would get a R^2 of. Is categorical, and second the target values '' \u00b6 linear models in sklearn, comes... Else, it seems that model.fit_regularized ( ~ ).summary ( ) and the target variable, for that! Regression '' sklearn linear regression summary while not necessary, makes your line fit much better suitable.","date":"2022-05-28 10:45:10","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\": 0.46840280294418335, \"perplexity\": 1057.0941064649614}, \"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-2022-21\/segments\/1652663016373.86\/warc\/CC-MAIN-20220528093113-20220528123113-00225.warc.gz\"}"}
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Babyboom (en: baby boom, "spädbarnshögkonjunktur"), är en period av kraftigt förhöjda födelsetal, ibland definierad av att de årliga födelsetalen överstiger 2,0 per 100 kvinnor i ett land eller annat område. Personer födda under en sådan period sägs tillhöra en babyboomgeneration.
1940-talet
På engelska åsyftar termen baby boom främst de dramatiskt ökade födelsetalen under och strax efter andra världskriget i stora delar av västvärlden, och som gav upphov till den stora fyrtiotalistgenerationen, i Sverige ibland benämnd efterkrigsbarnen, jätteproppen Orvar eller köttberget, och på engelska "baby boomers". Av dessa fyra benämningar är endast "efterkrigsbarnen" samtida med barnen samt fri från förminskande tendens. Alltså mest autentiskt. "Efterkrigs-" är inte bara en tidsangivelse utan var också en livssituation präglad av svallvågorna efter det världsomfattande kriget, t.ex. ransonering av mat och bränsle mm. Blöjor, varmvatten och kylskåp samt privat tvättmaskin saknade de flesta föräldrar i början, vilket gjorde barnavården mindre privilegierad än den framställts senare.
I Sverige pågick boomen ungefär 1943 till 1949, medan den i USA varade ungefär under åren 1946 till 1964. Den inträffade efter en långvarig period av låga födelsetal i samband med 1930-talets depression. Personer födda under 1950-talet kan även kallas för den "glömda generationen".
Cirka 1965 till 1975
I Sverige pågick en relativt långvarig babyboom mellan ungefär 1965 och 1975, som gav upphov till Generation X (egentligen födda 1965 till 1981), som i huvudsak är barn till 40-talisterna. I USA var det under denna period tvärtom en nedgång i födelsestatistiken, och generationen kallas där ibland även "baby busters" eller "post-peak boomers".
Cirka 1989 till 1993
I Sverige åsyftar begreppet ofta perioden ungefär åren 1989 till 1993, då många i den stora sextiotalistgenerationen fick barn. Denna babyboom inträffade i slutet av en högkonjunktur som senare övergick i en finanskris. Detta infaller ungefär samtidigt som USA:s betydligt längre "echo baby boom", ungefär 1982 till 1994, då generation Y, MTV-generationen eller gratisgenerationen föds. Begreppets genomslag i Sverige vid denna tidpunkt kan måhända förklaras av att svenska biografer 1988 visade filmen Baby Boom från 1987. Magnus Uggla besjöng 1989 fenomenet i sin sång Baby Boom.
Kring år 2010
Sedan antalet födda per kvinna i Sverige var som lägst år 1998–1999 har siffran successivt ökat till 1,98 år 2010, och tangerade därmed gränsen för en babyboom. Barn födda under denna perioden brukar kallas generation Z.
Se även
Generation snöflinga
Demografi
Referenser
Noter
Källförteckning
Sveriges befolkning 1900–2007 – Pressmeddelande från SCB
USA:s födelsestatistik och andra medicinska data enligt CDC
Födda per år i USA enligt CDC
Babyboom (TV-program)
Demografi
Människans fortplantning
Boom
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Vishwadath Kishan Ramsukul (Nieuw-Nickerie, 15 december 1985) is een Surinaams politicus. Hij is sinds 2020 lid van De Nationale Assemblée voor de VHP.
Biografie
Ramsukul studeerde tussen 2004 en 2007 elektrotechniek aan de Anton de Kom Universiteit van Suriname. Van 2008 tot 2010 deed hij nog een vervolgstudie aan de Open Universiteit in software engineering en aansluitend tot 2015 in ICT aan het Institute of Management and Information Technology. Hij sloot zijn studie af met een bachelorgraad in ICT. Daarnaast is hij zijn loopbaan in de ICT-branche begonnen en heeft hij sinds 2015 zijn bedrijf Events in Suriname ernaast, dat een toeristische website beheert. Hij is sinds 2005 lid van de Vooruitstrevende Hervormings Partij (VHP).
In 2011 werd hij gekozen in het bestuur van de VHP-Jongerenraad en in 2013 werd hij tot voorzitter van de VHP-jongeren in Paramaribo gekozen. In aanloop naar de verkiezingen van 2015 leidde hij verschillende Meet The Youth-activiteiten van de partij. Ook kandideerde hij zelf op plaats 13 van de V7-lijst (plaats 3 van de VHP), maar verwierf toen geen zetel.
Tijdens de verkiezingen van 2020 kandideerde hij opnieuw op de VHP-lijst. Hij werd niet direct gekozen, maar nadat enkele VHP-leden De Nationale Assemblée verlieten om deel te nemen aan de regering, kwam een plaats voor hem vrij. Hij werd op 7 augustus 2020 beëdigd.
Surinaams politicus
Surinaams bestuurder
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USHK – UNIVERSITY OF SHKODRA "LUIGJ GURAKUQI"
The University of Shkodra "Luigj Gurakuqi" was founded on 2 September 1957 as a High Pedagogical Institute. On 1991 has changed its status to University. After 2000, the University of Shkodra was engaged in policy making in order to reach the objectives set out in the Bologna Declaration, the inclusion in the European Higher Education Area and the promotion of the European Higher Education System. The University has a full structure organized in 6 faculties and 21 departments. Throughout these years, the University of Shkodra has established cooperative relations with many universities and Higher Education centers.
Why we are Participating in the Project
Higher Education (HE) in Albania has gone through different important stages during its development. The first qualitative step towards international collaboration is when Albania signed Bologna Declarations in 2003. Then this transformation of the HE in Albania, which supported the concept of the internationalization, became a task and was included in the laws and strategies after that. The modernization of teaching cannot be understood without the involvement of technology in this process. Experiences with international HE institutions brought to Albania the introduction of new concepts, such as: e-learning, Lifelong Learning. The implementation of these concepts began with the use of electronic technology in specific subjects or modules.The VALUE-X project is a good possibility for our institution to create a new vision for the teaching and learning process, towards e virtual leaning and virtual mobility.
Our Role in the Project
University of Shkodra "Luigj Gurakuqi" is evolved in many projects that have the focus on the internationalization of the university and the new trends in education, such as: e-learning, distance education and in this project we are involved in a new manner of teaching: virtual learning.
Through this project we need to:
– Increase numbers of incoming and outgoing exchange students and academic staff.
– Integrate European and international elements in the academic programs.
Support required:
– Capacity building in internationalization for academic and administrative staff.
– Expertise in international academic recognition and transfer.
– Development and integration of business-related didactic case studies.
Suzana Golemi
Erard Çurçija
Head of IRO
Mimoza Priku
Prof. of Albanian Language
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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2675 Tolkien eller 1982 GB är en asteroid i huvudbältet som upptäcktes den 14 april 1982 av den brittiske astronomen Martin Watt vid Anderson Mesa Station. Den är uppkallad efter den brittiske författaren J.R.R. Tolkien.
Asteroiden har en diameter på ungefär 10 kilometer och den tillhör asteroidgruppen Flora.
Referenser
Huvudbältesasteroider
Flora-asteroider
Småplaneter namngivna efter personer
Astronomiska upptäckter av M Watt
Astronomiska upptäckter 1982
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{
"redpajama_set_name": "RedPajamaWikipedia"
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import * as React from 'react';
import DocumentTitle from 'react-document-title';
import Banner, { BannerType } from '../components/Banner';
import { Row } from '../components/Page';
const Unathenticated = () => (
<DocumentTitle title='Please Log In'>
<Row>
<Banner bannerType={BannerType.Alert} title='Not Logged In.'>
Please log in to view this page.
</Banner>
</Row>
</DocumentTitle>
);
export default Unathenticated;
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,485
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\section{Introduction}
No other development in physics has changed our view of the world more than
the theory of relativity, introduced by Einstein in 1905. Based on the idea
that motion of a body cannot be defined absolutely, but only relative with
respect to others, he concluded that the laws of physics should look alike in
any two reference systems, moving with respect to each other with constant
velocity. Together with the experimentally confirmed fact that the speed of
light is independent of the reference frame, this led to the conclusion that
time cannot be an absolute quantity, but that time and space constitute a
4-dimensional unit. Temporal and spatial distances both must depend on the
reference system.
The equivalence of all inertial reference systems requires that linear motion
in one system translates into linear motion in another system. Thus the
relation between the coordinates $(x,y,z,t)$ in one system and those in
another system $(x',y',z',t')$, moving with velocity $v$ in the direction
$x$, must be linear. These conditions uniquely define the transformation
equations (the Lorentz transformation)
\begin{equation}
\label{lor}
x'=\gamma (x-\beta t)\qquad y'=y \qquad z'=z \qquad t'=\gamma(-\beta x+t)
\end{equation}
where the time variable is calibrated to an equivalent length by the speed of
light $c$ $(ct\Rightarrow t)$. $\beta$ is the relative velocity as a fraction
of $c$: $\beta=v/c$ and $\gamma =1/\sqrt{1-\beta^2}$. The two systems are
synchronised by the condition that at $x=t=0$ we have $x'=t'=0$.
The fact that the time scale changes with distance appears somewhat strange
to our intuition, but this can be ascribed to the fact that in every days
life we are accustomed only to velocities, which are much less than $c$, so
that relativistic effects are negligible. On the other hand the lack of
imagination has led us to believe in mathematically derived consequences
which can scarcely be proved by experiments.
One of the most discussed consequences of the theory of relativity is the so
called twin paradox, which dates back to first decade after the invention,
but is discussed in numerous scientific papers still today. Basis of the twin
paradox is the dilatation of time, the fact that moving clocks are slowed
down, when observed from the rest system. Thus a clock moving with a
considerable fraction of $c$ measures a shorter time to reach a distant
target than a clock at rest.
The twin paradox is frequently told with the following story: There are twins
Alice and Bob. Alice decides to make a journey to a distant star in a
spaceship capable of moving at a considerable fraction of $c$. When she has
reached the star, she goes back at the same speed. According to time
dilatation a clock moving with her and consequently also Alice herself ages
more slowly than her sibling at home, so that, when she comes back, she finds
Bob as an old man, while for herself the journey has taken only a few years.
In this paper we will show that this interpretation of relativity is
incorrect and that the twin paradox does not exist at all. We will try to
show up, where the mathematical flaws come in, and how they can be corrected.
Subsequently we will consider some real effects of relativity, especially
with respect to the consequences of Lorentz invariance in accelerated
systems, as they are discussed in general relativity.
\section{Invariants of the Lorentz transformation}
The Lorentz transformation describes, how spatial and temporal distances
between events change, when observed from different reference systems, which
are in relative motion with respect to each other, with the additional
condition that the speed of light is independent of the reference system. The
equation $x=t$ transforms into $x'=t'$ (time calibrated by $c$ as in the last
section), independent of the direction of the relative motion of the systems.
This property can be read immediately from the transformation equation
eq.(\ref{lor}), defining the space-time distance between two events by
\begin{equation}
\Delta s=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2-\Delta t^2}.
\end{equation}
or, if the argument of the square root is negative, the proper time
\begin{equation}
\Delta \tau=\sqrt{\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2}.
\end{equation}
Leaving off the $y$ and $z$ coordinates for simplicity, as these are not
affected by the transformation, from eq.(\ref{lor}) we get
\begin{equation}
\label{del}
\Delta \tau=\sqrt{\Delta t^2-\Delta x^2}=\sqrt{\Delta t'^2-\Delta x'^2},
\end{equation}
which reduces to $\Delta \tau=0$ in the case of light. This is the well known
fact that the world lines defined by $\Delta \tau=0$ constitute the limits of
the region, which can be causally connected to some space-time event.
Invariance of the condition $\Delta \tau=0$ means that causality is not
affected by any Lorentz transformation.
But with respect to the twin paradox it is more important that the quantity
$\Delta \tau$ is generally invariant under Lorentz transformations, not only
in the case $\Delta \tau=0$. The space-time distance between two events is a
uniquely defined quantity, independent of the reference system. If two events
take place at the same physical location, we must assume that their spatial
distance is zero in every reference system. As their space-time distance is
uniquely defined, too, the necessary consequence according to eq(\ref{del})
is that the temporal distance is also uniquely defined. It must be
independent of the choice of the reference system.
For Alice and Bob there are two events, where they are physically at the same
spatial position: The first one, when Alice leaves the earth with her
spaceship, and the second one, when she returns from her journey. The
space-time distance between these events is uniquely defined and thus also
the time interval between the events. There is no room for any discrepancy in
the aging of Alice and Bob.
Switching from one reference system to another may change the local time
scale during her journey, but changes of the reference system cannot change
the underlying physics. In the next section we will try to find out, where
the pitfalls are, which lead to positive results for the twin paradox.
\section{False solutions}
False solutions of the twin paradox date back to the very beginning of the
theory of relativity. Even Einstein himself has mentioned time discrepancies
as a peculiar result of his theory \cite{einstein} and the story of the
traveling twins was introduced by Langevin as early as 1911 \cite{langevin}.
Numerous papers on the topic have been published since then, but most of them
give more or less sophisticated explanations, why the twins age differently,
but the existence of the effect is scarcely disputed.
The most simple argument in favour of a different aging runs as follows. The
clock in the moving spaceship is slowed down with respect to a clock on earth
by a factor $1/\gamma =\sqrt{1-\beta^2}$. Thus the time to reach the point of
return is reduced just by this factor. As the return journey is symmetric
with respect to time, the total time is also reduced by this factor compared
to the elapsed time on earth.
This reasoning contains several errors, however. The first one is that it
regards the spatial geometry, which is fixed in the rest frame of the earth,
as fixed also in the comoving frame and only considers the change of the time
scale. The second is that at the return point there is a further change of
the reference system. While the Lorentz transformation between the earth and
the spaceship has been synchronised at the starting point, there is no such
synchronisation at the return point. That means that the zero point of the
time scale is altered. Closely related to the synchronisation problem is the
fact that the Lorentz transformation does not conserve simultaneity.
Simultaneous events at different space points do not remain simultaneous with
changes to a moving coordinate system.
It should be stressed here that a proper time interval is not the change of
some scalar property between two events, but it is the equivalent in a
pseudo-Euclidean metric to the vector length in Euclidean geometry. Thus
adding proper time intervals, measured in different reference frames, must be
done by the rules of vector algebra and not like the addition of scalar data.
Change to a moving reference system is analog to rotation in Euclidean space.
The problem can easily be visualised in Euclidean geometry (see fig.1). We
consider a set of transformations, consisting of a rotation of the $(x,y)$
coordinate system by some angle $\vartheta$ about the zero-point to
$(x',y')$, subsequent shifting the coordinate system in $x'$-direction to
$(x'',y'')$, rotating the system back by $-\vartheta$ to $(x''',y''')$ and
finally shift it back in direction of $x'''$ to $(x'''',y'''')$, so that
$x''''=x$. At the end $y''''$ will not be equal to y. But this operation does
of course not change any distance between points in the $(x,y)$ plane.
\begin{figure}[hbt]\epsfig{file=Zwillingx.eps,width=11cm}
\caption{Coordinate transformations in Euclidean space: rotation, shift
along $x'$ axis, back rotation, back shift along $x'''$ axis}
\end{figure}
It is the fact that we have used rotations with different centers, which
leads to a shift of the y coordinate. In just the similar way the age shift
of the twin paradox results from the fact that the reference frame is changed
twice, first at the starting point, and then a second time at the point of
return, but now with a different center of 'rotation'. It is this change of
velocity at the return point, which causes the supposed age shift. To reach
the velocity of the new comoving reference frame, the spaceship has to be
accelerated. During the acceleration phase the proper time changes
continuously. An instantaneous switch to the new comoving system requires
infinite acceleration and results in a jump of proper time.
But we can try to consider the space trip of Alice without the change of the
reference system at the return point. We compare the situations as seen from
the earth and from the reference system of the spaceship, synchronised at the
starting point.
In the rest frame of the earth the target star of the journey is at a fixed
distance $x_S$, while in the reference system of the spaceship according to
eq.(\ref{lor}) at the state of synchronisation the local time at $x_S$ is
$t_S'=-\gamma\beta x_S$, the position is $x_S'=\gamma x_S$ and the star is
moving towards the spaceship with velocity $-\beta$. The star passes the
spaceship at
\begin{equation}
t_1'=\frac{x_S'}{\beta} +t_S'= \frac{\gamma x_S}{\beta}-\gamma\beta x_S =
\frac{\gamma x_S}{\beta} (1-\beta^2) =\frac{x_S}{\beta\gamma}
\end{equation}
We could, of course, derive the same relation immediately from eq.(\ref{lor})
by setting $t=\beta x_S$ and $x=x_S$. The detour was only to demonstrate the
importance of taking into account missing simultaneity of distant events. In
the comoving system the proper time interval is equal to $t_1'$. In the
earth-bound system we have
\begin{equation}
\Delta\tau = \sqrt{-x_S^2+(x_S/\beta)^2}= \frac{x_S}{\beta\gamma},
\end{equation}
the same value as $t_1'$, as must be expected. The coordinate times are
different, as times at different locations are compared, the proper time is
the same in both systems.
The back journey is a little bit more complicated. Alice in her spaceship
changes the velocity at the turning point. But to keep things simple, we
consider this change only in the system $(x',t')$, as we know that a further
change of the reference system at the turning point would bring in problems
of acceleration and thus of synchronisation. The situation is different now
from the first part of the journey, however. Now for Alice the earth is no
longer at a fixed location, but moving apart at velocity $-\beta$. She is
hunting for a moving target. Thus in her reference system the speed must be
higher than that of the earth. In the non-relativistic case, if the time back
to earth should be equal to $t_1'$, her speed must be $-2\beta$. But if
$\beta > 0.5$, she comes into trouble, as her spaceship is limited to the
speed of light. The back journey will take more time than the first part.
In the limiting case $\beta =1$ the time would even be infinite. No light
pulse can catch up with another pulse, emitted at an earlier time by the same
source. In the relativistic case we must change our problem a little bit. Now
we put the question, when and where will Alice catch up with the moving
earth, when her spaceship now moves with speed $-\beta_2'$, measured in the
reference frame $(x',t')$. The earth is receding at $-\beta$ since $t'=0$,
she herself is moving with $-\beta_2'$ since $t'=t_1'$ Thus she will meet her
twin on earth at $t_2'$, given by
\begin{equation}
-\beta t_2'=-\beta_2'(t_2'-t_1'),
\end{equation}
leading to
\begin{equation}
\label{prime}
t_2'=\frac{\beta_2'}{\beta_2'-\beta}\;t_1', \qquad
x_2'=-\frac{\beta \beta_2'}{\beta_2'-\beta}\;t_1'
\end{equation}
and
\begin{equation}
\Delta\tau_2'=\sqrt{1-\beta^2}\frac{\beta_2'}{\beta_2'-\beta}\;t_1'=\frac{\beta_2'}{\beta}
\frac{1-\beta^2}{\beta_2'-\beta}\;x_S.
\end{equation}
If we insert the values of eq.(\ref{prime}) into eq.(\ref{lor}) and solve for
the time interval in rest frame of the earth, we find $\Delta\tau_2\equiv
t_2=\Delta\tau_2'$. There is no change of the proper time between Alice and
Bob.
This simple example shows that it is only the mixing up time intervals,
measured in different reference frames, which leads to contradictions. If
perceived time intervals depend on distance, it is no longer meaningful to
add time intervals measured in different reference systems, which are in
relative motion. The only quantity, which is of physical relevance, is the
space-time distance of events or the proper time interval, which must not
confounded with the perceived time interval measured in some reference frame.
This topic has already been discussed by Kracklauer \cite{krack} in 2001, but
still there are numerous newer papers, which ignore the invariance of proper
time and insist on the existence of the time discrepancy.
\section{Real effects of Lorentz invariance}
Though there exist no local time discrepancies which depend on the course of
the world line between two events, there are several real effects, which are
explained by the Lorentz invariance of the basic laws of physics. But all
these effects are related to our local observations of physical processes,
generated in systems, which are moving with respect to the local rest frame.
One well known example is the apparent increase of life time of unstable
particles like muons, when they approach the earth at velocities close to the
speed of light.
Another is the red shift of light, emitted from moving sources. Though the
light always reaches us with the velocity $c$, the wavelength is shifted.
There is no change of the relative velocity, as we know it from Doppler
shift, but it is the different time scale at emission, which leads to the
wavelength shift at observation.
The difference between Doppler shift and Lorentz shift clearly shows up in
cosmological observations. The observed red shift of light from distant
objects can be explained by a continuous expansion of space, which is
equivalent to a continuous local acceleration or a recession velocity
proportional to distance. This leads to a change of time scale proportional
to distance between emission and observation. This change does not only
affect the frequency of light, but influences all time dependent processes.
One well observed effect is the dilatation of the time scale of distant
supernovas.
But there is a strong caveat in this interpretation of cosmological red shift
and time dilatation. Contrary to Doppler shift, which occurs only with
motions in the direction of observation, the Lorentz shift of time scale is
independent of the direction of acceleration. A continuous acceleration
perpendicular to the direction of observation and acceleration in this
direction will result in exactly the same time shift. If space is curved,
every geodesic motion must be regarded as accelerated. Thus from red shift or
time dilatation measurements we cannot decide, if space is expanding or if
space is curved.
Only independent measurements, which are affected only by the spatial
component of space-time, like the size distribution of distant galaxies or
clusters, can help to decide if the cause of red shift is expansion or
curvature of space. There is one consequence which remains in both cases. If
Lorentz invariance is valid throughout the entire universe, there is no
sensible definition of a global time scale. Time varies with distance. As a
reasonable definition of time to describe distant events in the universe, one
can only use the running time of light with respect to our local reference
frame.
The persistent discussions of the twin paradox demonstrate that we have not
yet fully understood all the consequences of Lorentz invariance of
interactions as well in the regime of mechanics as in gravity. But in the
last decades a huge amount of measurements and observations has been
accumulated, so that we should be able to decide, if Lorentz invariance is
really the governing principle, not only of electromagnetism, but also of
gravity.
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{"url":"https:\/\/www.cvxpy.org\/examples\/machine_learning\/logistic_regression.html","text":"# Logistic regression with $$\\ell_1$$ regularization\u00b6\n\nIn this example, we use CVXPY to train a logistic regression classifier with $$\\ell_1$$ regularization. We are given data $$(x_i,y_i)$$, $$i=1,\\ldots, m$$. The $$x_i \\in {\\bf R}^n$$ are feature vectors, while the $$y_i \\in \\{0, 1\\}$$ are associated boolean classes.\n\nOur goal is to construct a linear classifier $$\\hat y = \\mathbb{1}[\\beta^T x > 0]$$, which is $$1$$ when $$\\beta^T x$$ is positive and $$0$$ otherwise. We model the posterior probabilities of the classes given the data linearly, with\n\n$\\log \\frac{\\mathrm{Pr} (Y=1 \\mid X = x)}{\\mathrm{Pr} (Y=0 \\mid X = x)} = \\beta^T x.$\n\nThis implies that\n\n$\\mathrm{Pr} (Y=1 \\mid X = x) = \\frac{\\exp(\\beta^T x)}{1 + \\exp(\\beta^T x)}, \\quad \\mathrm{Pr} (Y=0 \\mid X = x) = \\frac{1}{1 + \\exp(\\beta^T x)}.$\n\nWe fit $$\\beta$$ by maximizing the log-likelihood of the data, plus a regularization term $$\\lambda \\|\\beta\\|_1$$ with $$\\lambda > 0$$:\n\n$\\ell(\\beta) = \\sum_{i=1}^{m} y_i \\beta^T x_i - \\log(1 + \\exp (\\beta^T x_i)) - \\lambda \\|\\beta\\|_1.$\n\nBecause $$\\ell$$ is a concave function of $$\\beta$$, this is a convex optimization problem.\n\nimport cvxpy as cp\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n\nIn the following code we generate data with $$n=50$$ features by randomly choosing $$x_i$$ and supplying a sparse $$\\beta_{\\mathrm{true}} \\in {\\bf R}^n$$. We then set $$y_i = \\mathbb{1}[\\beta_{\\mathrm{true}}^T x_i + z_i > 0]$$, where the $$z_i$$ are i.i.d. normal random variables. We divide the data into training and test sets with $$m=50$$ examples each.\n\nnp.random.seed(1)\nn = 50\nm = 50\ndef sigmoid(z):\nreturn 1\/(1 + np.exp(-z))\n\nbeta_true = np.array([1, 0.5, -0.5] + [0]*(n - 3))\nX = (np.random.random((m, n)) - 0.5)*10\nY = np.round(sigmoid(X @ beta_true + np.random.randn(m)*0.5))\n\nX_test = (np.random.random((2*m, n)) - 0.5)*10\nY_test = np.round(sigmoid(X_test @ beta_true + np.random.randn(2*m)*0.5))\n\n\nWe next formulate the optimization problem using CVXPY.\n\nbeta = cp.Variable(n)\nlambd = cp.Parameter(nonneg=True)\nlog_likelihood = cp.sum(\ncp.multiply(Y, X @ beta) - cp.logistic(X @ beta)\n)\nproblem = cp.Problem(cp.Maximize(log_likelihood\/n - lambd * cp.norm(beta, 1)))\n\n\nWe solve the optimization problem for a range of $$\\lambda$$ to compute a trade-off curve. We then plot the train and test error over the trade-off curve. A reasonable choice of $$\\lambda$$ is the value that minimizes the test error.\n\ndef error(scores, labels):\nscores[scores > 0] = 1\nscores[scores <= 0] = 0\nreturn np.sum(np.abs(scores - labels)) \/ float(np.size(labels))\n\ntrials = 100\ntrain_error = np.zeros(trials)\ntest_error = np.zeros(trials)\nlambda_vals = np.logspace(-2, 0, trials)\nbeta_vals = []\nfor i in range(trials):\nlambd.value = lambda_vals[i]\nproblem.solve()\ntrain_error[i] = error( (X @ beta).value, Y)\ntest_error[i] = error( (X_test @ beta).value, Y_test)\nbeta_vals.append(beta.value)\n\n%matplotlib inline\n%config InlineBackend.figure_format = \"svg\"\n\nplt.plot(lambda_vals, train_error, label=\"Train error\")\nplt.plot(lambda_vals, test_error, label=\"Test error\")\nplt.xscale(\"log\")\nplt.legend(loc=\"upper left\")\nplt.xlabel(r\"$\\lambda$\", fontsize=16)\nplt.show()\n\n\nWe also plot the regularization path, or the $$\\beta_i$$ versus $$\\lambda$$. Notice that a few features remain non-zero longer for larger $$\\lambda$$ than the rest, which suggests that these features are the most important.\n\nfor i in range(n):\nplt.plot(lambda_vals, [wi for wi in beta_vals])\nplt.xlabel(r\"$\\lambda$\", fontsize=16)\nplt.xscale(\"log\")\n\n\nWe plot the true $$\\beta$$ versus reconstructed $$\\beta$$, as chosen to minimize error on the test set. The non-zero coefficients are reconstructed with good accuracy. There are a few values in the reconstructed $$\\beta$$ that are non-zero but should be zero.\n\nidx = np.argmin(test_error)\nplt.plot(beta_true, label=r\"True $\\beta$\")\nplt.plot(beta_vals[idx], label=r\"Reconstructed $\\beta$\")\nplt.xlabel(r\"$i$\", fontsize=16)\nplt.ylabel(r\"$\\beta_i$\", fontsize=16)\nplt.legend(loc=\"upper right\")\n\n<matplotlib.legend.Legend at 0x108adedd8>","date":"2020-01-22 12:24:48","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\": 2, \"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\": 0.7924818396568298, \"perplexity\": 2191.6985889646394}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2020-05\/segments\/1579250606975.49\/warc\/CC-MAIN-20200122101729-20200122130729-00030.warc.gz\"}"}
| null | null |
Barkow may refer to:
People
Al Barkow (born 1932), American journalist.
Ben Barkow (born 1956), British librarian.
Frank Barkow (born 1957), American architect.
Jerome H. Barkow, Canadian anthropologist.
Rachel Barkow (born 1971), American law professor.
Sally Barkow, American Olympic sailor.
Places
Mount Barkow, a mountain on the Antarctic.
Barkow (Mecklenburg-Vorpommern), a sub-division of Barkhagen, Mecklenburg-Vorpommern, Germany
|
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| 4,094
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{"url":"https:\/\/recipes-core.com\/if-a-tall-frequency-arresting-is-passing-through-a-capacitor-does-it-signify-if-the-capacitor-is-charged-retort\/","text":"# If a substantial frequency enthralling is passing by means of a capacitor, does it signify if the capacitor is charged? retort\n\nHello expensive customer to our community We will proffer you an answer to this query If a substantial frequency enthralling is passing by means of a capacitor, does it signify if the capacitor is charged? ,and the retort will breathe typical by means of documented data sources, We welcome you and proffer you recent questions and solutions, Many customer are questioning concerning the retort to this query.\n\n## If a substantial frequency enthralling is passing by means of a capacitor, does it signify if the capacitor is charged?\n\nThe expression \u201ccapacitance changes depending on DC color\u201d is a bit misleading. It truly comes from the very fact it\u2019s examined with a DC coloration and a tiny AC voltage added to it to touchstone the capacitance. But in actuality, the capacitance of any capacitor all the time is dependent upon the instantaneous voltage throughout the capacitor, no signify the place this voltage got here from. Usually it\u2019s DC, however in your illustration it\u2019s going to breathe low frequency AC plus substantial frequency AC.\n\nSo you necessity a cap with low $$dC\/dV$$ or capacitance distinction per unit of voltage distinction. That contains C0G ceramics and most varieties of movie caps affection PPS, PP, and many others.\n\nIn your illustration, since mains voltage is , breathe optimistic to prefe a capacitor rated for it. Not simply the AC voltage, however it ought to too breathe rated X. This isn\u2019t about motion pictures, fairly it means it\u2019s propitious to make use of the capacitor throughout AC mains voltage. Mostly that is about not beginning a zeal if there may be an inside brief within the capacitor, so it\u2019s going to maintain options affection self-healing. level to the capacitance worth of those tends to reduce over time as they\u2019re uncovered to voltage spikes which instinct inside arcing, which pokes a indolent within the dielectric and causes a brief. At this level the self-healing function kicks in and the steel across the indolent vaporizes, which implies there isn\u2019t any longer a brief, and prevents zeal. But it too means the capacitance worth has decreased a bit. So if you need long run accuracy, possibly prefe the next voltage rated cap, affection 600V or 1000V.\n\ntoo you must put substantial worth resistors throughout the caps to discharge them as soon as the motif is unplugged, as a result of I speculate this motif goes to carry a masculine mains plug on the stop, and it\u2019s by no means a happy tolerate to seize one among these when there\u2019s a charged capacitor on the opposite stop. The resistors will too equilibrium voltage throughout the caps in order that they each maintain half the mains voltage, in order that they\u2019re each so far as workable from their max voltage score.\n\nSeveral different individuals talked about Class 2 ceramics are absolutely the worst, however as a consequence of having tremendous dC\/dV not solely will the capacitance breathe modulated by the 50Hz AC voltage, it\u2019s going to too breathe modulated by your HF enthralling which is able to create all types of distortion harmonics. They are too fairly lossy, so a few of your enthralling will stop up as losses. They are superior for decoupling energy provides as a consequence of substantial capacitance per quantity, low expense, low inductance, however they\u2019re abominable for all the things else.\n\nwe are going to proffer you the answer to If a substantial frequency enthralling is passing by means of a capacitor, does it signify if the capacitor is charged? query through our community which brings all of the solutions from a number of reliable sources.","date":"2021-03-06 04:22:57","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\": 0, \"katex\": 0, \"math-container\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.27463027834892273, \"perplexity\": 2572.4656100490133}, \"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\/1614178374391.90\/warc\/CC-MAIN-20210306035529-20210306065529-00038.warc.gz\"}"}
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{"url":"https:\/\/ibipti.com\/kundli-chakra-2012-professional-edit\/","text":"# Kundli Chakra 2012 Professional Edit\n\n0\n8\n\nKundli Chakra 2012 Professional Edit\n\nA:\n\nChakra means coiled, it is the thread on the head of a snake.I can\u2019t recommend this software, because it\u2019s commercial and kundali chakra 2012 professional is freeware.\n\nA:\n\nIn many ways, Astrological Charts are similar to an iPhone\u2019s Health app with all the charts already on your device. The difference with the Astrological Chart would be that you\u2019d get to choose which chart to use and which parameters you want to have input to your calculations and yes, it would be professional, if that was your intention to begin with.\nI\u2019m not very familiar with Astrological Charts, but many Google searches lead me to think that they are very basic and do not provide much of value.\nChakra is a bit of a different story. If you think of the human body as a house, then a Chakra is a room of the house. So the acronym Chakra is appropriate. If you don\u2019t think of the human body that way, then you can stick with Kundali.\n\nA:\n\nThere is a free astrology application for Android called WorldStar. Its free on the play store and so is astrology chakra professional.\n\nQ:\n\nExample of nonpolynomial optimization problem\n\nHow to find a suitable example of non-polynomial function that is defined on $\\mathbb{R}^2$ and has only one stationary point which is not a local minimum or maximum (point that is asymptotically stable).\n\nA:\n\nI guess a simple counterexample could be the following function\n$$f(x,y)=\\left(\\left(x^2-1\\right)y^2+1\\right)\\left(x+y-2xy+3\\right)$$\nThere is only one stationary point at $(0,0)$, but this point is neither a local minimum nor a local maximum.\n\nJohn Buckley (Wisconsin politician)\n\nJohn F. Buckley (born 1950) is a politician from The Woodlands, Texas.\n\nBiography\nBuckley was born on December 30, 1950 in Shreveport, Louisiana to Eddie and Helen Buckley. After high school, Buckley left for the University of Texas. While in school, he was a member of the Sigma Chi fraternity. He graduated in 1972 with a B.S. degree in education from Texas A&M\n\nKundli Chakra 2012 \u2013 Green: The Ideal Of The\n\nKundli Chakra 2012. Planet Venus is exalted in Aquarius and is there till December 13. Overall, Libra is a hard sign that would be good for dealing with your chronic impatience\u00c2\u00a0.\nLeading kundali software. Its the basic for kundali calculation software. This is the best software for kundali reports.. and one can also change the time period, sex and date of birth of the person.\nKundali Chakra 2012 \u2013 Name of Dear Professional Indiraji Karma shastra. Chakra aditya astrology sagittarius kommerel agne krishna nyaya vedic astrology in hindi\nUpdateStar is the easiest and fastest way to manage your system with a single. Over 99 of the world\u2019s top programs use UpdateStar to make sure you\u2019re always up to date and secure.Downloads Details Awards, FAQ, Updates. Another of the best tools for creating professional-level websites. Kundali Chakra 2012 Professional is the ideal program for..\nBook Description & Overview \u2013 \u201cKundli Chakra\u201d Chakra is the spot where energy is drawn, the spot where the life energy is drawn for the body, it\u2019s the spot.\nChakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra Chakra.\nKundli Chakra with Pdf pdf. The Suryasti Matrimonial Kundli (vibhuti) is used to establish the matrimonial alliance. It is also called Sura.\nThe chart of your natal chart is the canvas on which you shall work to gain mastery over the many forces which are in charge. The astro-chart is the graphical. \u201cChakras 2012\u201d News of the Day On (MANY) Astrology News.\nRead and hear about every New Moon Chakra In Vedic Astrology: Here you will find many many Vastu Shastra articles.. Its the concept of all the chakras being present at birth,.\nKundli Chakra is the name of the most powerful software for the calculation of the Vedic astrology, that one can use to find all \u00c2\u00a0. \u201cKundali Chakra\u201d Home. \u201cKundali Chakra\u201d includes detailed kundali reports","date":"2022-08-13 08:54:19","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\": 0.19163930416107178, \"perplexity\": 3588.6956719809027}, \"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-2022-33\/segments\/1659882571911.5\/warc\/CC-MAIN-20220813081639-20220813111639-00375.warc.gz\"}"}
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{"url":"https:\/\/www.clutchprep.com\/chemistry\/practice-problems\/68850\/the-decomposition-of-xy-is-second-order-in-xy-and-has-a-rate-constant-of-7-02-x--4","text":"# Problem: The decomposition of XY is second order in XY and has a rate constant of 7.02 x 10-3 M-1\u2022s-1 at a certain temperature.\u00a0a. What is the half life for this reaction at an initial concentration of 0.100M?b. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?c. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?d. If the initial concentration of XY is 0.150 M, what is the concentration of XY after 5.0x101 s? After 5.50x102 s?\n\n\ud83e\udd13 Based on our data, we think this question is relevant for Professor El Ashmawy's class at COLLIN.\n\n###### FREE Expert Solution\n\nThe\u00a0integrated rate law\u00a0for a second-order reaction\u00a0is as follows:\n\n$\\overline{)\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{t}}}{\\mathbf{=}}{\\mathbf{kt}}{\\mathbf{+}}\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{0}}}}$\n\na. What is the half-life for this reaction at an initial concentration of 0.100M?\n\nRecall that\u00a0half-life (t1\/2)\u00a0is the\u00a0time needed for the amount of a reactant to decrease by 50% or one-half\n\nThe half-life of a second-order reaction is given by:\n\n$\\overline{){{\\mathbf{t}}}_{\\mathbf{1}\\mathbf{\/}\\mathbf{2}}{\\mathbf{=}}\\frac{\\mathbf{1}}{\\mathbf{k}{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{0}}}}$\n\n${\\mathbf{t}}_{\\mathbf{1}\\mathbf{\/}\\mathbf{2}}\\mathbf{=}\\frac{\\mathbf{1}}{\\mathbf{k}{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{0}}}$\n\nt1\/2 = 1424.5 s\n\nThe half-life of the reaction is 1424.5 s\u20131.\n\nb. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?\n\n[XY]0 = 0.100 M\n\n[XY]t0.100 M \u00d7 12.5% =\u00a00.0125 M\n\nSolving for\u00a0time:\n\n$\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{t}}}\\mathbf{=}\\mathbf{kt}\\mathbf{+}\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{0}}}$\n\nt =\u00a09971.51 s\n\nIt would take 9971.51 s for XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100 M.\n\n[XY]0 = 0.200 M\n\n[XY]t0.200 M \u00d7 12.5% =\u00a00.025 M\n\nSolving for\u00a0time:\n\n$\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{t}}}\\mathbf{=}\\mathbf{kt}\\mathbf{+}\\frac{\\mathbf{1}}{{\\mathbf{\\left[}\\mathbf{A}\\mathbf{\\right]}}_{\\mathbf{0}}}$\n\nt =\u00a04985.75 s\n\nIt would take 4985.75 s for XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.200 M.\n\nc. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?\n\n###### Problem Details\n\nThe decomposition of XY is second order in XY and has a rate constant of 7.02 x 10-3 M-1\u2022s-1 at a certain temperature.\n\na. What is the half life for this reaction at an initial concentration of 0.100M?\n\nb. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?\n\nc. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?\n\nd. If the initial concentration of XY is 0.150 M, what is the concentration of XY after 5.0x101 s? After 5.50x102 s?","date":"2020-08-13 18:29:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 5, \"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\": 0.8215873837471008, \"perplexity\": 1059.6614031267804}, \"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-2020-34\/segments\/1596439739048.46\/warc\/CC-MAIN-20200813161908-20200813191908-00200.warc.gz\"}"}
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{"url":"https:\/\/forum.kodi.tv\/showthread.php?tid=113824&pid=1039610","text":"\u2022\n\u2022 1\n\u2022 23\n\u2022 24\n\u2022 25(current)\n\u2022 26\n\u2022 27\n\u2022 46\n\u2022\nXBMC on Raspberry Pi - Wonder if this will work out? (Historical Discussion Thread)\nLooks awesome and at this price i'll just buy one to find out how it runs.\nQuick shout out to the devs, I love the way you support all these different devices ipad\\windows\\linux\\pi there is something for everyone. Thanks for all your efforts and can't wait till i get my hands on this little beast.\nOnce there is a working Skype client for the ARM11 i am wondering how easy and cool it would be to have Skype within XBMC. that would make one nice easy unit for someone none-technical to use HTPC and video calling.\nOnce there is a working Skype client for the ARM11 i am wondering how easy and cool it would be to have Skype within XBMC. that would make one nice easy unit for someone none-technical to use HTPC and video calling.\nSince Microsoft owns Skype now i doubt that Skype will ever be ported to arm Linux or updated on Linux at all.\ni donnt think for one seccond that Skype will release one but there are enough dev's and api's to write one (says the man who knows nothing at all)\nCan one of the developers with a beta board report on how well XBMC plays non-H264 video content using software-only playback (since the GPU is not enabled for other content types).\nIn particular - I am especially interested in MPEG2 (DVB-T in UK).\nI can provide a sample recorded from UK TV if that would help.\n\nSince the CPU is reported as being something like an old 900MHz PC then I think that software decoding and playback might just work but perhaps the extra load of copying the data over the ethernet\/USB interface will make it too slow for good frame rates.\nPaul Webster Wrote:Can one of the developers with a beta board report on how well XBMC plays non-H264 video content using software-only playback (since the GPU is not enabled for other content types).\nIn particular - I am especially interested in MPEG2 (DVB-T in UK).\nI can provide a sample recorded from UK TV if that would help.\n\nSince the CPU is reported as being something like an old 900MHz PC then I think that software decoding and playback might just work but perhaps the extra load of copying the data over the ethernet\/USB interface will make it too slow for good frame rates.\n\nit's fine\nThanks - that is good news.\n\nI have another question ... has there been any progress on getting HDMI CEC support or is it still awaiting info from raspberrypi folks with an interface?\njhsrennie Wrote:I think the Raspberry Pi is great, and I was one of those who experienced a frustrating morning trying to order one, but outside the third world why would anyone use one in anger (as opposed to just for fun)?\n\nThe difference in cost between a RBPi and a Revo 3700 (in the UK) is about \u00a3150. Now that's not a trivial amount, but we're talking about something that's an important part of your life (well, my life anyway :-). The 3700 is **much** more powerful than the RBPi, and both are a lot cheaper than the average LCD TV.\n\nIts pretty pathetic, but that difference for me is too much (even more so if i point out the Pi is a luxury for me at \u00a330), and I would just continue to either do without, or continue doing what I currently do which is move a home server from one room to another as thats where my XBMC resides.\n\nHowever theres also the issue of needs, if XBMC can play quite happily without complication on a \u00a330 Pi, whats the point of buying a product thats (in comparison) overkill at 5x the price, even if you consider that to be not excessively priced. Its a good price, but its like all these school mums rolling up in Land Rovers, Nissan Pathfinders and whatnot to pick up little Timmy to do the 2 mile round trip through town.\nTheres people who need them, and theres people who dont. Your paying a premium to buy them, and a premium to then run them... its easy to argue thats rather wasteful.\n\nI'll hopefully get mine on Monday or thereabouts, and once im done playing with it its eventual use will see it'll being on 24\/7, for a couple of hours a day for XBMC, and on avg 1 hours a week for web browsing (cue counter-arguement that thats wasteful :o it is, but it'd be more wasteful with less efficient hardware).\nI dont need anything more than that, not for a secondary or light use location, and I suspect from a regular Joe consumer POV, most people buying a Pi wont need it for much more than that. For heavy usage it does make more sense buying a more competent\/purpose built device, and thats what makes the Pi special. Regular folks should be able to try things and experiment with very capable hardware without the need for paying too much for it. Some might be able to spend a couple hundred on a fun project which may or may not work, I suspect plenty would get put off by that.\nI'm hoping the raspberry pi will make the price of cubox and pandaboard come down to around $50 \/$60\nOr the raspberry pi will evolve to higher specs before the year is over\n(2012-01-07, 16:44)gimli Wrote:\ns7mx1 Wrote:My wild guess would be gstreamer based player which is extremely popular with embedded devices.\n\nYour guess is wrong. OMX is used on the PI.\n\nI would have thought you would have used gstreamer-omx with the Raspberry Pi. Do you have a separate implementation instead ?\n\nrelevant news: http:\/\/www.bbc.co.uk\/news\/technology-17547764\nAgreed this would make a great addition to a satellite unit in my Kids room, soon my pretties, soon\nStill nobody has got one? when do people get them?\n\nhow can a product get so much hype then not deliver?! classic example of how not to launch a product.\n(2012-04-03, 17:55)krish_2k4 Wrote: Still nobody has got one? when do people get them?\n\nhow can a product get so much hype then not deliver?! classic example of how not to launch a product.\n\n...but I bet a lot of people now want one although they had not heard of it before the launch!\n\n\u2022\n\u2022 1\n\u2022 23\n\u2022 24\n\u2022 25(current)\n\u2022 26\n\u2022 27\n\u2022 46\n\u2022","date":"2018-01-18 04:01:42","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\": 0.2259940654039383, \"perplexity\": 2688.698403377426}, \"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-2018-05\/segments\/1516084887065.16\/warc\/CC-MAIN-20180118032119-20180118052119-00598.warc.gz\"}"}
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The online application for NAHB committee and council board of trustees appointments , Oct. 15-21.
Remember, current committee and council members interested in serving another year must resubmit their applications: There are no automatic multiyear appointments.
You must be logged in to .
Review the criteria for serving on a committee or council.
Select up to six committees or councils and rank them according to preference, explaining why you are interested in serving, whether you have served before and whether someone has recommended you to serve on that committee or council.
Review the application and submit. You will receive a confirmation that your application was received.
If you want to make changes to your application, simply go back to the application link and make your edits.
Questions? Contact Cyndi McKinley at 800-368-5242 x8346.
Thanks for your interest in applying! To do so, follow the application link and you can access the application and instruction information.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 795
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\section{Introduction}
Interfaces of water are the most important subjects not only
because water is widely involved in physical, chemical,
environmental as well as biological processes, but also because
water is so far the most mysteries molecule in the
universe.\cite{FranksBook,science-Luecke,TobiasPRL2002,RicePNAS1999}
Among them, air/water interface has been intensively investigated
theoretically or experimentally over the last decades.
Spectroscopy, molecular structure and dynamics at air/water
interface is studied with theoretical analysis such as \textit{ab
initio} calculation or molecular dynamics
simulation,\cite{Mundy-science,
BenjaminPRL1994,HynesCP2000,HynesJPCB2002,MooreJCP2003,ChandraCPL2003,
ChandraCPL2004,RiceJCP1991,BenjaminCR1996} or experimental
techniques such as X-ray
reflection,\cite{PershanPRL1985,PershanPRA1988} Stimulated Raman
Scattering (SRS),\cite{SawadaCPL2002} Near-edge X-Ray Adsorption
Fine Structure (NEXAFS),\cite{SaykallyJPCM2002} Second Harmonic
Generation (SHG),\cite{EisenthalJPC1988,FreyMP2001} as well as Sum
Frequency Generation, etc.\cite{ShultzIRPC2000,
RichmondARPC2001,Richmond:cr102:2693,Shen-science,richmond:science,Shen-prl1994,
DuQuanPRL1993,Richmond-jpca2000,WeiXingPRL2001} Among these
experimental techniques, Second Harmonic Generation and Sum
Frequency Generation are the most important methods for molecular
interface studies because of their surface sensitivity and
specificity.\cite{RichmondSHGReview,ShenANRP1989,CornHigginsReview1994,
ShenMirandaJPCBReview,shen:nature:review,eisenthal:review,Shen-5CT,somorjai:ap:review}
With these investigations, the properties of the water molecules
at the interface, such as the surface density, surface structure,
surface potential as well as surface dynamics, have been
intensively discussed.
However, conclusions on the surface molecule species at air/water
interface are still under discussion.\cite{DuQuanPRL1993,
Richmond-jpca2000,WeiXingPRL2001,SaykallyJPCM2002} With SFG-VS
experimental studies, the following interfacial water species have
been reported in literatures, namely, water molecules straddle at
the interface with one OH bond hydrogen bonded to neighboring
molecules in liquid phase (singly bonded OH) and another OH bond
free from hydrogen bonding (free OH) in gas
phase;\cite{Shen-prl1994,ShultzIRPC2000,Richmond-jpca2000,richmond:jpcb1998}
water molecules with both OH bonds symmetrically hydrogen bonded
in a tetrahedral network (ice-like and liquid-like
structures);\cite{Shen-prl1994,
ShultzIRPC2000,Richmond-jpca2000,richmond:jpcb1998} and water
molecules in gas phase with both OH bonds not hydrogen bonded
pointing into the liquid
phase.\cite{Richmond-jpca2000,richmond:science} With NEXAFS
measurement and \textit{ab initio} molecular dynamics simulation,
water molecules with both OH bonds not hydrogen bonded pointing
out of the interface was also
proposed.\cite{SaykallyJPCM2002,Mundy-science} The latter case is
particularly controversial because NEXAFS is not strictly a
surface specific technique.\cite{SaykallyJPCM2002} With
polarization SFG-VS measurement, Wei \textit{et al.} discussed the
absence of SFG spectra in some polarization combinations and
proposed an explanation through fast orientational motion in a
broad range of about $102^{\circ}$ in a time scale comparable or
less than 0.5 \textit{ps}.\cite{WeiXingPRL2001} However, puzzle
still remains because some of the experimental studies suggests
ordered and slow dynamics for interfaces of hydrogen bonding
liquids, while some experimental investigations suggested a more
dynamic and less ordered picture for the liquid interfaces,
air/water interface included.\cite{ShenMirandaJPCBReview} In
addition, whether the surface orientation relaxation is fast or
slow than the bulk water molecules is also an issue under
discussion in the recent
literatures.\cite{BakkerScience,ChandraCPL2004,BenjaminJPCBASAP}
Besides SHG and SFG-VS experimental
studies,\cite{Eisenthal1992ACR,ShenMirandaJPCBReview} Structure
and dynamics of water molecules at the air/water interface have
also been intensively discussed with theoretical
simulations.\cite{Mundy-science,BenjaminPRL1994,HynesCP2000,HynesJPCB2002,
ChandraCPL2003,ChandraCPL2004,MooreJCP2003,RiceJCP1991,BenjaminCR1996}
Even though with so much efforts and progresses both by
experimentalists and theoreticians, our detailed understanding of
air/water interface is still limited. Just as indicated by B. C.
Garrett recently,\cite{GarrettScience} `...(direct) experiments
are difficult to perform because the liquid interface is
disordered, dynamic, and small (typically only a few molecules
wide) relative to the bulk'.
Actually, direct measurement of the liquid interface is not as
difficult as suggested as above. It has been known that along with
SHG, SFG-VS can provide direct measurement on liquid interface no
other technique can
match.\cite{Eisenthal1992ACR,ShenMirandaJPCBReview} As pointed out
by Miranda and Shen, `SFG is currently the only technique that can
yield a vibrational spectrum for a neat liquid
interface'.\cite{ShenMirandaJPCBReview} In fact, with the advances
of ultrafast laser and detection technology in the past decade and
especially recent few
years,\cite{RichterOL1998,AllenAnaSci2001BroadBandSFG,RichmondApplySpec2004,
JohnsonPCCP2005} particularly with commercial systems designed for
SFG-VS measurement,\cite{EKSPLA&EROSACAN} SFG-VS, as well as SHG,
experiments have come from easier to routine.\cite{ShenApplyPhys}
The real difficulty lies on the fact that quantitative analysis
and interpretation of the SFG-VS, as well as SHG, data had been
not as well developed and widely performed until
recently.\cite{Shen-5CT,WeiXingPRL2001,weixing:pre2000,WHFRaoJCP2003,Lurong1,Lurong2,
HongfeiCJCPPaper,Lurong3,ChenhuaJPCBacetone,
ChenhuaJPCBmethanol,ChenhuaCPLacetone,GanweiCPLNull,HongfeiIRPCreview}
Therefore, conclusions in many previous reports on the
investigations of air/water interface, as well as other liquid
interfaces, with SFG-VS are subjected to different
interpretations.
As we have demonstrated in a series of recent publications,
systematically quantitative treatment to SFG-VS data is not only
possible, but also very effective for obtaining detailed
spectroscopic, structural and thermodynamic properties of liquid
interfaces.\cite{WHFRaoJCP2003,Lurong1,Lurong2,
HongfeiCJCPPaper,Lurong3,ChenhuaJPCBacetone,ChenhuaJPCBmethanol,
ChenhuaCPLacetone,GanweiCPLNull,HongfeiIRPCreview} In these works,
we not only developed methodology for quantitative polarization
and experimental configuration analysis in SFG-VS and SHG, we also
tested accuracy and sensitivity of some of the methodology. We
have applied them to elucidated the anti-parallel double layered
structure and thermodynamics of some organic liquid aqueous
solution interfaces. In addition, we also demonstrated that a set
of polarization selection rules (or guidelines) in SFG-VS can be
developed for vibrational spectrum assignment through symmetry
analysis of the SFG-VS spectral features.\cite{Lurong2,Lurong3}
This latter approach is extremely useful for discerning complex
SFG-VS spectrum with unidentified or controversial assignments.
Recently, based on polarization analysis, Ostroverkhov \textit{et
al.} demonstrated a phase-sensitive interference analysis of SFG
polarization spectra of water/quartz
interface.\cite{Shen2005PRLWaterQuartz} With these development, in
this report we intend to apply these analysis methodologies to the
study of air/water interface.
In this work, we examined SFG-VS spectra at air/water interface
measured in different polarizations under four experimental
configurations with polarization analysis method and experimental
configuration analysis. With these analysis, detailed new
information are obtained for understanding of the spectroscopy,
structure and dynamics of the air/water interface. In the
following sections, after a brief introduction of the theoretical
background and experimental conditions, we first discuss the
motion of the interfacial water molecules at the air/water
interface, which was previously suggested experiencing rapidly
motion over a broad angular range in the vibrational relaxation
time; then we use polarization and symmetry analysis of the SFG-VS
spectral features for assignment of the SFG-VS spectra peaks; in
the end, we shall discuss the structure and orientation of the
water molecules at the air/water interface.
\section{Polarization and Experimental Configuration Analysis in SFG-VS}
Quantitative polarization analysis and experimental configuration
analysis can provide rich and detailed information of
spectroscopy, structure and dynamics of molecular
interfaces.\cite{Shen-5CT,WeiXingPRL2001,WHFRaoJCP2003,Lurong2,Lurong3,GanweiCPLNull}
Generally, the SFG intensity in the reflective direction
is,\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
I(\omega)&=&\frac{{8\pi ^3 \omega ^2sec^2\beta
}}{{c^{3}n_{1}(\omega)n_{1}(\omega_{1}})n_{1}(\omega_{2})}\left|\chi
_{eff}^{(2)}\right|^2 I(\omega_{1})I(\omega_{2})\label{all}
\end{eqnarray}
\noindent
\noindent in which $\omega$, $\omega_{1}$ and $\omega_{2}$ are the
frequencies of the SFG signal, visible and IR laser beam,
respectively. $n_{i}(\omega_{i})$ is the refractive index of bulk
medium $i$ at frequency $\omega_{i}$, and $n'(\omega_{i})$ is the
effective refractive index of the interface layer at $\omega_{i}$.
$\beta_{i}$ is the incident or reflection angle from interface
normal of the $i$th light beams; $I(\omega_{i})$ is the intensity
of the SFG signal or the input laser beam. $\chi_{eff}^{(2)}$ is
the effective second order susceptibility for an interface. The
notations and the experiment geometry have been described in
detail previously.\cite{Lurong2,Shen-5CT}
$\chi_{eff}^{(2)}$ for the four generally used independent
polarization combinations can be deduced from the 7 nonzero
macroscopic susceptibility tensors for an achiral rotationally
isotropic interface ($C_{\infty v}$).\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
\chi_{eff}^{(2),ssp}&=&
L_{yy}(\omega)L_{yy}(\omega_{1})L_{zz}(\omega_{2})sin\beta_{2}\chi_{yyz}\label{ssp}\nonumber
\\
\chi_{eff}^{(2),sps}&=&L_{yy}(\omega)L_{zz}(\omega_{1})L_{yy}(\omega_{2})sin\beta_{1}\chi_{yzy}\label{sps}\nonumber
\\
\chi_{eff}^{(2),pss}&=&L_{zz}(\omega)L_{yy}(\omega_{1})L_{yy}(\omega_{2})sin\beta\chi_{zyy}\label{pss}\nonumber
\\
\chi_{eff}^{(2),ppp}&=&
-L_{xx}(\omega)L_{xx}(\omega_{1})L_{zz}(\omega_{2})
cos\beta{cos\beta_{1}}sin\beta_{2}\chi_{xxz}\nonumber\\
&&-L_{xx}(\omega)L_{zz}(\omega_{1})L_{xx}(\omega_{2})cos\beta{sin\beta_{1}}cos\beta_{2}\chi_{xzx}\nonumber\\
&&+L_{zz}(\omega)L_{xx}(\omega_{1})L_{xx}(\omega_{2})sin\beta{cos\beta_{1}}cos\beta_{2}\chi_{zxx}\nonumber\\
&&+L_{zz}(\omega)L_{zz}(\omega_{1})L_{zz}(\omega_{2})sin\beta{sin\beta_{1}}sin\beta_{2}\chi_{zzz}\nonumber\\
\label{ppp}
\end{eqnarray}
\noindent It is so defined that the $xy$ plane in the laboratory
coordinates system $\lambda(x,y,z)$ is the plane of interface; all
the light beams propagate in the $xz$ plane; $\textit{p}$ denotes
the polarization of the optical field in the $xz$ plane, with $z$
as the surface normal, while $\textit{s}$ the polarization
perpendicular to the $xz$ plane. The consecutive superscript, such
as \textit{ssp}, represents the following polarization
combinations: SFG signal \textit{s} polarized, visible beam
\textit{s} polarized, IR beam \textit{p} polarized, and so forth.
$L_{ii}$ ($i=x,y,z$) is the Fresnel coefficient determined by the
refractive indexes of the two bulk phase and the interface layer,
and the incident and reflected angles.\cite{Lurong2,Shen-5CT}
$\chi_{ijk}^{(2)}$ tensors are related to the microscopic
hyperpolarizability tensor $\beta_{i'j'k'}^{(2)}$ of the molecules
in the molecular coordinates system $\lambda'(a,b,c)$ through the
ensemble average over all possible molecular orientations.
\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{i'j'k'}\langle{R_{ii'}R_{jj'}R_{kk'}\rangle}\beta_{i'j'k'}^{(2)}
\label{hyper}
\end{eqnarray}
\noindent where $R_{\lambda\lambda'}(\theta,\phi,\psi)$ is the
matrix element of the Euler rotational transformation matrix from
the molecular coordination $\lambda'(a,b,c)$ to the laboratory
coordination $\lambda$(\textit{x,y,z}); $\beta_{i'j'k'}^{(2)}$ is
the microscopic (molecular) hyperpolarizability
tensor.\cite{GoldsteinBook,HongfeiCJCPPaper,HongfeiIRPCreview}
Here $N_{s}$ is the molecular number density at the interface.
$\langle A \rangle$ represents orientational average of property
$A(\theta,\phi,\psi)$ over the orientational distribution function
$f(\theta,\phi,\varphi)$.
\begin{eqnarray}
\langle A
\rangle=\frac{\int^{\pi}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}A(\theta,\phi,\psi)
f(\theta,\phi,\psi)\sin\theta d\theta\ d\phi d\psi}
{\int^{\pi}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}f(\theta,\phi,\psi)\sin\theta
d\theta d\phi d\psi}\label{OAverage}
\end{eqnarray}
For SFG-VS, $\beta^{(2)}$ is IR frequency ($\omega_{2}$)
dependent,
\begin{eqnarray}
\beta^{(2)}_{i'j'k'}&=&\beta_{NR,i'j'k'}^{(2)}+\sum_{q}\frac{\beta_{q,i'j'k'}}
{\omega_{2}-\omega_{q}+i\Gamma_{q}}\label{spectrum}
\end{eqnarray}
Thus, $\chi_{ijk}^{(2)}$ can be expressed into,
\begin{eqnarray}
\chi_{ijk}^{(2)}&=&\chi_{NR,ijk}^{(2)}+\sum_{q}\frac{\chi_{q,ijk}}
{\omega_{2}-\omega_{q}+i\Gamma_{q}}\label{spectrum1}
\end{eqnarray}
Therefore, SFG-VS measures the vibrational spectroscopy of
molecular interfaces. For dielectric interfaces, such as liquid
interfaces, the non-resonant term $\beta_{NR,i'j'k'}^{(2)}$ or
$\chi_{NR,ijk}^{(2)}$ is generally negligible compare with the
resonant terms.
Recently, we have found that the following formulation is very
effective in quantitative polarization and orientation analysis of
SFG and SHG data. It can be generally shown that in surface SFG
and SHG for an interface with orientational order, the effective
second order susceptibility $\chi_{eff}^{(2)}$ can be simplified
into the following form.\cite{WHFRaoJCP2003}
\begin{equation}
\chi_{eff}^{(2)} = N_{s}\ast\textit{d}\ast(\langle \cos \theta
\rangle - \textit{c}\ast \langle \cos ^3\theta \rangle
)=N_{s}\ast\textit{d}\ast \textit{r}(\theta) \label{chi}
\end{equation}
\noindent $r(\theta)$ is called the \textit{orientational field
functional}, which contains all molecular orientational
information at a given SFG experimental configuration; while the
dimensionless parameter \textit{c} is called the \textit{general
orientational parameter}, which determines the orientational
response $r(\theta)$ to the molecular orientation angle $\theta$;
and $\textit{d}$ is the susceptibility strength factor, which is a
constant in a certain experimental polarization configuration with
a given molecular system. The $d$ and $c$ values are both
functions of the related Fresnel coefficients including the
refractive index of the interface and the bulk phases, and the
experimental geometry.
The key for quantitative analysis is that both \textit{d} and
\textit{c} can be explicitly derived from the expressions of the
$\chi_{eff}^{(2)}$ in relationship to the macroscopic
susceptibility and microscopic (molecular) hyperpolarizability
tensors for a particular molecular vibrational
modes,\cite{Lurong2,HongfeiIRPCreview} as shown for the water
molecules with $C_{2v}$ symmetry in the appendix. With the
parameters $c$ and $d$, the polarization dependence and the
orientation dependence of the SFG/SHG signal for a certain
interface at certain experimental configuration can be analyzed
and calculated with clear physical picture on molecular
orientation and orientational distribution.\cite{WHFRaoJCP2003}
Reciprocally, information on the molecular symmetry, molecular
orientation and dynamics can be obtained from the analysis on the
SFG intensity relationships measured in different polarization
combinations and experimental
configurations.\cite{Lurong2,Lurong3,HongfeiIRPCreview,GanweiCPLNull}
The orientational average in Eq.\ref{hyper} is only the static
average on molecular orientations, without considering fast
molecular motion effects. Recently Wei \textit{et al.} discussed
the fast and slow limit of the time average over orientational
motion for $\chi_{eff}^{(2)}$, and they also applied this
treatment to analysis the polarization dependence of SFG
measurement of the OH stretching vibrational spectra for the
air/water interface.\cite{WeiXingPRL2001} In the fast motion
limit, the orientational motion is faster than the vibrational
relaxation time scale $1/\Gamma_{q}$ of the $q$th vibrational
mode; while in the slow motion limit, the orientational motion is
much slower than $1/\Gamma_{q}$.
According to Wei \textit{et al.},\cite{WeiXingPRL2001} the slow
motion limit gives,
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{q}\sum_{i'j'k'}
\frac{\beta^{(2)}_{q,i'j'k'}}{\omega_{2}-\omega_{q}+i\Gamma_{q}}
\langle R_{ii'}R_{jj'}R_{kk'}\rangle \label{SlowAverage}
\end{eqnarray}
\noindent while the fast motion gives,
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{q}\sum_{i'j'k'}
\frac{\beta^{(2)}_{q,i'j'k'}}{\omega_{2}-\omega_{q}+i\Gamma_{q}}
\langle R_{ii'}R_{jj'}\rangle \langle R_{kk'}
\rangle\label{FastAverage}
\end{eqnarray}
\noindent in which
$R_{\lambda\lambda'}(t)=\hat{\lambda}\cdot\hat{\lambda'}(t)$ is
the time-dependent direction Euler transformation matrix from
$\lambda'(a,b,c)$ to $\lambda(x,y,z)$ coordinates system. Because
of the molecular orientational motion, the molecular coordinates
$\lambda'(a,b,c)$ is time-dependent. Eq.\ref{SlowAverage} is
equivalent to Eq.\ref{spectrum1}, which is obtained by insertion
of Eq.\ref{spectrum} into Eq.\ref{hyper}.
\section{Experiment}
The details of the laser system has been described in our previous
reports.\cite{Lurong2,ChenhuaJPCBacetone,ChenhuaJPCBmethanol}
Briefly, the 10Hz and 23 picosecond SFG spectrometer laser system
(EKSPLA) is in a co-propagating configuration. The efficiency of
the detection system has been improved for the weak SFG signal of
air/water interface. A high-gain low-noise photomultiplier
(Hamamatsu, PMT-R585) and a two channel Boxcar average system
(Stanford Research Systems) are integrated into the EKSPLA system.
The voltage of R585 was 1300V in the measurement for air/water
interface, and 900V for the Z-cut quartz surface. The wavelength
of the visible is fixed at 532nm and the full range of the IR
tunability is $1000cm^{-1}$ to $4300cm^{-1}$. The specified
spectral resolution of this SFG spectrometer is $<6cm^{-1}$ in the
whole IR range, and about $2cm^{-1}$ around $3000cm^{-1}$. Each
scan was with a $5cm^{-1}$ increment and was averaged over 300
laser pulses per point. Each spectrum has been repeated for at
least several times. Moreover, for \textit{sps} polarization, each
spectrum has been repeated for more than a dozen times and
averaged. The energy of visible beam is typically less than
300$\mu J$ and that of IR beam less than 150$\mu J$ around
$3000cm^{-1}$ and $3700cm^{-1}$, and less than 100$\mu J$ in the
region in between. These are comparable to literature reported
values for measurement of air/water interface.\cite{DuQuanPRL1993}
All measurements were carried out at controlled room temperature
($22.0\pm0.5^{\circ}C$) and humidity (40$\%$) . The sample used
was ultrapure water from standard Millipore treatment (18.2
M$\Omega \cdot cm$). The whole experimental setup on the optical
table was covered in a plastic housing to reduce the air flow. No
detectable evaporation effect was observed for SFG spectrum during
each scan.
The normalization procedure of the SFG signal in different
experimental configurations need to be specifically discussed. The
detail of the normalization procedure for a single experimental
configuration was presented in Xing Wei's Ph.D.
dissertation.\cite{Wei:Thesis} However, the difference of coherent
length and Fresnel factors with different incident angles in the
quartz SFG signal measurements has to be corrected when comparing
SFG signal in different experimental configurations. Therefore,
the measured spectrum is firstly normalized with the energy of the
incident laser beams, and then normalized to the SFG signal of
Z-cut quartz (also normalized by the energy of the incident
lasers). Then it times with a converting factor between different
experimental configurations. This factor contains the influence of
the coherent length of Z-cut quartz,\cite{Wei:Thesis} the Fresnel
coefficients,\cite{Wei:Thesis} the $\chi_{ijk}$ value for Z-cut
quartz, and the factor $sec^{2}\beta$ for each experimental
configuration. Therefore, the end result is directly proportional
to the SFG intensity in Eq.\ref{all}. If the spectrum in Fig.
\ref{allSpectra} is divided by the factor $sec^{2}\beta$ and the
factor of the PMT efficiency between 1300V and 900V, which is
determined as 24.1 in our detection system, and then times the
unit factor $1\times 10^{-40}V^{4}m^{-2}$ which we left out for
simplicity of graph presentation, it will give the value for
$|\chi^{(2)}_{eff}|^{2}$. For example, the peak at about
$3700cm^{-1}$ in the \textit{ssp} spectra of Config.2 in
Fig.\ref{allSpectra} is about 0.23 unit. After above conversion it
gives $|\chi^{(2)}_{eff}|^{2}=4.7\times 10^{-40}V^{4}m^{-2}$,
matching satisfactorily with the reported value for less than
$10\%$ difference.\cite{WeiXingPRL2001}
Even though the normalized intensities are generally consistent
with each other, there can be possibly other sources of errors
when intensities in different experimental configurations need to
be compared. Because the visible and IR beams have different
coherent lengthes in the Z-cut crystal, and because these coherent
lengthes vary with different experimental incident angles, one of
the most likely error might come from the different focusing
parameters with different beam overlapping quality of the visible
and IR beams in the Z-cut quartz crystal with different
experimental configurations. Therefore, quantitative comparison of
the SFG spectral intensities in different polarizations with the
same experimental configuration can be more accurate than
comparison intensities between different experimental
configurations. Even though the latter is a good solution to
reduce such relative error associated with different experimental
configurations need to be developed.
\section{Results and Discussion}
\subsection{Polarization SFG Spectra of the air/water interface}
Firstly we would like to present the polarization SFG spectra of
the air/water interface measured in four different experimental
configurations.
We have demonstrated recently that the change of the SFG spectra
in different polarizations by varying the experimental
configurations can be used for quantitative polarization analysis
and orientational analysis.\cite{HongfeiIRPCreview,GanweiCPLNull}
Here we present in Fig.\ref{allSpectra} the SFG spectra in the
\textit{ssp}, \textit{ppp} and \textit{sps} polarizations on the
air/water interface at four experimental configurations with
different incident angles for the visible and IR laser beams. They
are, Config.1: Visible=39$^{\circ}$, IR=55$^{\circ}$; Config.2:
Visible=45$^{\circ}$, IR=55$^{\circ}$; Config.3:
Visible=48$^{\circ}$, IR=57$^{\circ}$; Config.4:
Visible=63$^{\circ}$, IR=55$^{\circ}$.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=15cm,width=15cm]{allSpectra.eps}
\caption{SFG spectra of air/water interface in different
polarization combination and experimental configurations. All
spectra are normalized to the same scale. The solid lines are
globally fitted curves with Lorentzian line shape function in Eq.
\ref{spectrum1}. Note the different error bars for graphs in
different scales.}\label{allSpectra}
\end{center}
\end{figure*}
\begin{table}[h!]
\caption{The fitting results of the SFG spectra at air/watrer
interface. The spectra are fitted with Lorentzian line shape
function as Eq.\ref{spectrum1}. The peak position of the
vibrational modes $\omega_{q}$, the peak width $\Gamma_{q}$ and
the oscillator strength factor $\chi_{eff,q,ijk}$ of the
vibrational modes are listed. The first column is the fitted value
for $\chi_{NR,eff,ijk}$. The relative error in fitting of
\textit{sps} is larger because of the small signal strength for
\textit{sps} spectra.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline $\omega_{q}(cm^{-1})$ & & & 3281$\pm$5 & 3446$\pm$3 & 3536$\pm$6 & 3693$\pm$1& $$ \\
$\Gamma_{q}(cm^{-1})$ & & & 89$\pm$9 & 103$\pm$7 & 77$\pm$11 & 17$\pm$1 \\
\hline
& ssp & 0.17 & -6.7$\pm$0.6 & -20.1$\pm$1.3 & -5.2$\pm$1.2 & 6.8$\pm$0.2 \\
Config.1 & ppp &-0.04 & 3.2$\pm$0.5 & 2.6$\pm$0.7 & 5.0$\pm$0.4 & 1.1$\pm$0.1 \\
& sps & -0.01 & -0.1$\pm$0.1 & -0.2$\pm$0.1 & -3.5$\pm$0.5 & 0.9$\pm$0.2 \\
\hline
& ssp & 0.19 & -8.3$\pm$0.6 & -24.0$\pm$3.5 & -5.0$\pm$3.5 & 8.5$\pm$0.1 \\
Config.2 & ppp &-0.02 & 1.1$\pm$0.5 & 0.0$\pm$0.8 & 6.9$\pm$0.3 & 2.4$\pm$0.6 \\
& sps & 0.02 & -0.1$\pm$0.1 & -0.3$\pm$0.1 & -4.5$\pm$0.5 & 1.6$\pm$0.1 \\
\hline
& ssp & 0.22 & -10.1$\pm$0.7 & -22.8$\pm$1.5 & -7.0$\pm$2.0 & 8.8$\pm$0.2 \\
Config.3 & ppp &-0.01 & 2.4$\pm$0.6 & 0.9$\pm$0.7 & 6.6$\pm$0.3 & 2.8$\pm$0.1 \\
& sps & 0.01 & -0.2$\pm$0.1 & -0.3$\pm$0.1 & -3.4$\pm$0.7 & 1.4$\pm$0.2 \\
\hline
& ssp& 0.21 & -8.8$\pm$0.7 & -23.3$\pm$1.4 & -5.0$\pm$1.5 & 9.2$\pm$0.2 \\
Config.4 & ppp & 0.15 & -1.0$\pm$0.8 & -3.0$\pm$1.3 & -9.0$\pm$0.7 & 9.3$\pm$0.2 \\
& sps & 0.01 & -0.2$\pm$0.1 & -0.4$\pm$0.1 & -6.6$\pm$0.8 & 3.1$\pm$0.2 \\
\end{tabular}\label{fittingResults}
\end{center}
\end{table}
There are four apparent peaks can be identified in the SFG spectra
in Fig.\ref{allSpectra}. They are around $3700cm^{-1}$,
$3550cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$, respectively. The
$3700cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$ peaks has been
extensively discussed in the SFG
literature.\cite{Shen-prl1994,ShultzIRPC2000,
Richmond-jpca2000,richmond:jpcb1998,Shen-science} However, the
$3550cm^{-1}$ peak has been observed, but not yet clearly
identified or assigned.\cite{WeiXingPRL2001} The results of global
fit of these spectra with four Lorentzian peaks in
Eq.\ref{spectrum1} are listed in Table \ref{fittingResults}. From
the fitting results we can see that the peak bandwidths of the
$3550cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$ peaks are
$77\pm11cm^{-1}$, $103\pm 7cm^{-1}$ and $89\pm 9cm^{-1}$,
respectively. Such broad bandwidths indicate that they all belong
to different hydrogen bonded O-H stretching vibrational modes.
However, the bandwidth of the $3693cm^{-1}$ peak width is only
$17cm^{-1}$, consistent with the symmetric stretching (ss)
vibrational mode of the free O-H bond.\cite{DuQuanPRL1993} The
signs in Table \ref{fittingResults} contain the information of the
relative phase and interference effects of the different
vibrational modes. Here the phase of the $3693cm^{-1}$ peak is
held positive in each fit. Altering the relative phases of the
peaks on the same spectrum can not give a reasonable fit. Because
we used global fitting with all the spectra, these relative phases
can be determined accurately. They can be used to determine the
symmetry properties of each vibrational mode in Section IV.C.
According to Eq.\ref{ppp}, the \textit{ssp} spectra in different
experimental configurations should have the same features from the
$\chi_{yyz}$ term. As shown in Fig.\ref{TryOverlap}, all
\textit{ssp} curves overlap quit well when normalized to the
$3693cm^{-1}$ peak. Calculation of the Fresnel factors with
different incident angles can quantitatively explain the relative
intensities in all four configurations.\cite{JohnsonJPCBpaper}
Because the SFG spectral intensity from the air/water interface in
the OH region is usually several times smaller than that of the
C-H region from other air/liquid interfaces, the air/water
interface SFG spectra are usually very hard to measure
experimentally. Therefore, the well overlapping of the
\textit{ssp} spectra in different experimental configurations is a
proof for the quality of our SFG-VS data. Furthermore, the spectra
we obtained agree very well with these in the
literatures.\cite{WeiXingPRL2001,AllenJPCB2004}
In principle, the \textit{sps} spectra in different experimental
configurations should also overlap with each other when
normalized. However, consistent with the calculations of the
corresponding Fresnel factors, the \textit{sps} signal level for
Config. 1, 2 and 3 are very close to the noise level, and features
in the \textit{sps} spectra can not be clearly identified except
for the spectra of Config.4. Therefore, such normalization and
comparison for \textit{sps} spectra is not as meaningful as the
\textit{ssp} spectra.
Different from the \textit{ssp} and \textit{sps} spectra, the
features in the \textit{ppp} spectra in Fig.\ref{allSpectra}
changed drastically with different experimental configurations.
This is because that the \textit{ppp} spectra is determined by
combination of four different $\chi_{ijk}$ tensors. Detailed
polarization analysis and experimental configuration analysis of
these changes in the \textit{ppp} spectra can provide symmetry
properties for each spectral features, as well as orientation and
structure information of the interfacial molecular groups, as
shall be shown later.\cite{HongfeiIRPCreview,Lurong2,Lurong3} We
shall show that analysis of the \textit{ppp} spectra in different
experimental configurations is very informative. However, this
advantage of \textit{ppp} spectra analysis has not been well
utilized in the previous literatures.
\begin{figure}[h!]
\begin{center}
\includegraphics[height=5cm,width=6cm]{TryOverlap.eps}
\caption{Overlap of the normalized \textit{ssp} spectra of the
air/water interface in different experimental
configurations.}\label{TryOverlap}
\end{center}
\end{figure}
\subsection{Orientation and Motion of the Free OH Bond}\label{IVA}
Now with the knowledge of the SFG vibrational spectra of the
air/water interface, we can discuss the orientation and motion of
the free O-H bond at the air/water
interface.\cite{watershortpaper}
The sharp peak around $3700cm^{-1}$ was generally accepted as the
free OH bond protruding out of the liquid
water,\cite{DuQuanPRL1993,Richmond-jpca2000,WeiXingPRL2001,ShenLeePaper}
and it has been treated with $C_{\infty v}$ symmetry in
polarization analysis.\cite{DuQuanPRL1993,WeiXingPRL2001} Wei
\textit{et al.} studied the polarization dependence of the
intensity of this peak in the \textit{ssp}, \textit{ppp}, and
\textit{sps} polarizations measured with experimental
configuration of Visible$=45^{\circ}$,
IR=$57^{\circ}$.\cite{WeiXingPRL2001} Their SFG-VS data are
quantitatively very close to our data with Config.2 as expected.
Therefore, the \textit{ssp} intensity of the $3693cm^{-1}$ peak is
about 10 times of that of \textit{ppp}, and the \textit{sps}
intensity is essentially close zero. Wei \textit{et al.} realized
that using the step orientational distribution function in
Eq.\ref{StepAverage}, as well as other distribution functions,
such as Gaussian, centered at the surface normal, can not explain
such \textit{ssp}, \textit{ppp} and \textit{sps} intensity
relationships with the slow motion average in
Eq.\ref{SlowAverage}. On the other hand, the fast motion average
centered at the interface normal, as shown in Eq.\ref{FastAverage}
with $\theta_{M}=51^{\circ}$, can fairly well explain the observed
intensity relationships. Thus, it was concluded that the
orientation of the free OH bond of the interfacial water molecule
varies over a very broad angular range ($\theta_{M}=51^{\circ}$)
within the vibrational relaxation time as short as
$0.5ps$.\cite{WeiXingPRL2001}
\begin{eqnarray}
f(\theta)&=&cost\ \ \ \ for \ \ \ 0\leq \theta
\leq\theta_{M}\nonumber\\
f(\theta)&=&0\ \ \ \ \ \ \ \ for \ \ \ \theta \geq
\theta_{M}\label{StepAverage}
\end{eqnarray}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8cm]{FourMotionSimulation.eps}
\caption{SFG intensity of the free OH bond simulated with both
slow motion limit (solid curves) and fast motion limit (dotted
curves) following the procedure and parameters as Wei \textit{et
al.} \cite{WeiXingPRL2001}. $\theta_{M}$ is the range of
orientational motion of the free OH bond. All the curves presented
include the factor of $\sec^{2}\beta$, and all intensities are
normalized to the \textit{ssp} intensity in Config.4 with
$\theta_{M}=0^{\circ}$. The vertical lines indicate the
distribution width suggested by Wei \textit{et
al.}}\label{FourMotionSimulation}
\end{center}
\end{figure}
As shown in Fig.\ref{FourMotionSimulation}, Wei \textit{et al.}'s
treatment predicts clearly zero intensity for the \textit{sps}
spectra at $3693cm^{-1}$ with the assumption of fast orientational
motion centered at the surface normal. Using exactly the same
parameters, our calculation of Config.2 gives the same results as
that by Wei \textit{et al.} as it should have
been.\cite{WeiXingPRL2001} It is clear that the simulation results
in Fig.\ref{FourMotionSimulation} can fairly well explain the data
in Config. 1, 2 and 3, because all of them have relatively very
small \textit{sps} spectral intensity at $3693cm^{-1}$. However,
even though the fast orientational motion picture can explain the
relative intensity between the \textit{ssp} and \textit{ppp}
polarization in Config.4, it is clear that it can not explain the
apparently non-zero \textit{sps} intensity at $3693cm^{-1}$ with
Config.4. As long as the orientation distribution or orientational
motion is assumed to be centered to the interface
normal,\cite{ShenPrivate} orientational distribution functions
other than the step function in Eq.\ref{StepAverage} give the same
conclusion. Since the slow motion limit is already not an
option,\cite{WeiXingPRL2001} alternative description of the motion
and orientation at the air/water interface has to be invoked.
Because the air/water interface is rotationally isotropic around
the interface normal, now we assume that the molecular orientation
is centered around the tilt angle $\theta_{0}\neq 0$, instead of
the interface normal ($\theta_{0}=0$). If the Gaussian
distribution function is assumed, we have
\begin{eqnarray}
f(\theta)&=&\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(\theta-\theta_{0})^{2}/2\sigma^{2}}\label{Gaussian}
\end{eqnarray}
\noindent in which $\sigma$ is the standard deviation of the
angular distribution. We shall show in the followings that by
using this distribution function, the $3693cm^{-1}$ peak in
different polarization and experimental configurations in Fig.
\ref{allSpectra} can be quantitatively analyzed.
\begin{table}[h!]
\caption{The general orientational parameter \textit{c} and the
strength factor \textit{d} for the vibrational stretching mode of
free OH bond in different experimental configurations. The
\textit{d} value bear the unit $\beta_{ccc}$ of single OH bond.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline & & d-ssp & c-ssp & d-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & 0.274 & 0.515 & 0.112 & 1 & -0.154 & 1.53 \\
Config.2 & & 0.256 & 0.515 & 0.118 & 1 & -0.120 & 2.05 \\
Config.3 & & 0.248 & 0.515 & 0.118 & 1 & -0.104 & 2.43 \\
Config.4 & & 0.176 & 0.515 & 0.107 & 1 & -0.035 & 6.55 \\
\end{tabular}\label{CandDvalueForC3v}
\end{center}
\end{table}
Because the $3693cm^{-1}$ peak belongs to the \textit{ss} mode of
the free O-H bond at the air/water interface, it has been treated
with $C_{}\infty v$ symmetry. Now we calculate the general
orientational parameter \textit{c} and the strength factor
\textit{d} for the \textit{ssp}, \textit{sps} and \textit{ppp}
polarizations in all four experimental configurations with the
same parameters of the air/water interface as those used by Wei
\textit{et al.}\cite{WeiXingPRL2001} The details of the
calculation of $c$ and $d$ can be found
elsewhere.\cite{Lurong2,Lurong3,HongfeiIRPCreview} It is clear
from Table \ref{CandDvalueForC3v} that the $c$ values for the
\textit{ssp} and \textit{sps} polarizations are the same for all
four experimental configurations; whereas the $c$ values of the
\textit{ppp} polarization differ significantly for different
experimental configurations.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8cm]{FourC3vSimulation.eps}
\caption{The simulated SFG intensity of vibrational stretch mode
for free OH bond at different orientation angle $\theta$ assuming
$\sigma=0^{\circ}$. The factor $sec^{2}\beta$ in Eq.\ref{all} is
also included for comparison of SFG intensity in different
experimental configurations. All curves are normalized to the
\textit{ssp} intensity in Config.4 with $\theta_{0}=0^{\circ}$.
The vertical lines indicate the orientation which quantitatively
explains the observed SFG data.}\label{FourC3vSimulation}
\end{center}
\end{figure}
As we have demonstrated
previously,\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
the $[d\ast r(\theta)]^{2}$ vs. $\theta$ plot with $\sigma=0$ in
different polarizations can provide direct first look of the
physical picture for polarization analysis of SFG-VS data. Here we
plot $[d\ast r(\theta)\ast \sec\beta]^{2}$ vs. $\theta$ in
Fig.\ref{FourC3vSimulation} in order to compare data in different
experimental configurations. Thus, the relative intensity for the
$3693cm^{-1}$ peak in experimental Config.1, 2, 3 and 4 can be
used to calculate the orientation angle of the free O-H bond.
Using the known
procedures\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
and parameters,\cite{WeiXingPRL2001} they give the following four
values, i.e. $28.7\pm1.2^{\circ}$, $32.6\pm0.5^{\circ}$,
$34.6\pm0.7^{\circ}$ and $35.8\pm1.0^{\circ}$, respectively. These
values agree with each other quite well. However, the value from
Config.1, whose \textit{ppp} and \textit{sps} intensities are both
very weak, is not as reliable as the other three configurations.
Averaging over these values gives $\theta=33^{\circ}\pm
1^{\circ}$.
It is clear that $\sigma=0^{\circ}$ is not physically possible for
the liquid interface. However, the apparent success of the
quantitative explanation of the observed SFG spectra of the free
O-H bond in different experimental configurations using
$\sigma=0^{\circ}$ indicates that the actual $\sigma$ value can
not be very broad. Simulation of the $3693cm^{-1}$ peak in
different polarizations in each of the four experimental
configurations using the Gaussian distribution function in
Eq.\ref{Gaussian} concludes that $\sigma$ has to be smaller than
$15^{\circ}$ to satisfy the measured $3693cm^{-1}$ peak
intensities in all four experimental configurations.
$\sigma=15^{\circ}$ is the largest distribution width allowed by
the SFG experiment data for a Gaussian orientational distribution
function. With $\sigma=15^{\circ}$, we have
$\theta_{0}=30^{\circ}$. This indeed confirms that the orientation
of the free O-H bond is within a relatively narrow range (between
$30^{\circ}$ to $33^{\circ}$), with a relatively small
distribution width ($\sigma\leq 15^{\circ}$). Calculation with
both Eq.\ref{SlowAverage}, i.e. slow average limit, and
Eq.\ref{FastAverage}, i.e. fast average limit, gives
indistinguishable results with $\sigma$ as small as $\leq
15^{\circ}$ if $\theta_{0}$ is around $30^{\circ}$. This is
because that with a small distribution width, fast and slow motion
average should be the same according to Eq.\ref{FastAverage} and
Eq.\ref{SlowAverage}. Using a step distribution function around
$\theta_{0}\neq 0^{\circ}$ also give very close orientation angle
and distribution width.
Thus, our conclusion of the free O-H orientation and distribution
at the air/water interface is drastically different from the
conclusion given by Wei \textit{et al.}, which concluded that the
free O-H bond orientation varies in a broad range as big as
$102^{\circ}$ and as fast as $0.5$ picosecond, which is the
relaxation time for the O-H stretching
vibration.\cite{WeiXingPRL2001} It is clear that our conclusion is
based on the successful explanation of the observed polarized SFG
spectral intensities in different experimental configurations,
especially the relatively small but clearly non-vanishing SFG
spectral intensity at $3693cm^{-1}$ in the \textit{sps}
polarization. Our conclusion explicitly supports ultrafast
libratory motions with a relatively narrow angular range. As we
have known, the dynamics libratory motion of the hydrogen bonding
can be as fast as 0.1
picosecond.\cite{FeckoScience2003,Chandler1996PRL} Even with such
ultrafast dynamics, the air/water interface is nevertheless well
ordered. This is consistent with the generally well ordered
picture of the air/liquid and air/liquid mixture interfaces.
Recent quantitative analysis of data in SFG vibrational
spectroscopy have suggested that vapor/liquid interface are
generally well ordered, and sometimes even with anti-parallel
double-layered structures
\cite{ShenMirandaJPCBReview,ShenLinJCPAcetone2001,JohnsonJPCBpaper,ChenhuaJPCBacetone,
ChenhuaJPCBmethanol,ChenhuaCPLacetone}.
It has been generally accepted that liquid interface with strong
hydrogen bonding between molecules should be well ordered
\cite{ShenMirandaJPCBReview}. Our analysis here not only confirmed
this conclusion, but also provided solid and direct experimental
measurement of the orientation and motion at the air/water
interface.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=7cm]{ssC2vsimulation.eps}
\hspace{1.5cm}%
\includegraphics[width=7cm]{asC2vsimulation.eps}
\caption{The simulated SFG intensity for symmetric stretching
(\textit{ss}) mode (left) and asymmetric stretching (\textit{as})
mode (right) of water molecule with $C_{2v}$ symmetry. All curves
presented include the factor of $\sec^{2}\beta$. The intensity of
\textit{ss} mode is normalized to the \textit{ssp} intensity in
Config.4 with $\theta=0^{\circ}$. The intensity for \textit{as}
mode is normalized to the \textit{sps} intensity in Config.4 with
$\theta_{0}=0^{\circ}$. The units between plots of the \textit{ss}
and \textit{as} modes differ by 9.11 times according to the
$\beta_{ccc}$ and $\beta_{aca}$ values in the
appendix.}\label{C2vsimulation}
\end{center}
\end{figure*}
\subsection{Polarization Analysis and Determination of Spectral Symmetry Property}
Here we try to apply polarization analysis for identifying the
symmetry property and for assignment of the SFG vibrational
spectra of the air/water interface.
The assignment of the SFG spectra of air/water interface in the
range of 3000 to 3800$cm^{-1}$ has been discussed
intensively.\cite{Shen-prl1994,RamanSpectroscopyBook,ShultzIRPC2000,
Richmond-jpca2000,richmond:science,RichmondCPL2004,
richmond:jpcb1998,Shen-science,richmondJPCB2003p546paper,RichmondARPC2001,Richmond:cr102:2693}
Richmond recently reviewed the current understanding of the
bonding and energetics, as well as the SFG spectra assignment, of
various aqueous interfaces, including the air/water
interface.\cite{RichmondARPC2001,Richmond:cr102:2693} The SFG
spectral assignments heavily relied on band fitting of IR and
Raman peak positions of bulk water or water cluster
spectra,\cite{Richmond:cr102:2693} as well as based on theoretical
calculations.\cite{TobiasJPCB2005,JPCA2000Buch,HynesCP2000,HynesJPCB2002}
The sharp peak at about 3700$cm^{-1}$ has been unanimously
assigned to the free O-H stretching vibration mode. The broad
peaks around 3250$cm^{-1}$ and 3450$cm^{-1}$ undoubtedly belong to
the hydrogen bonded O-H stretching modes, but their assignments
are not as unanimous as the 3700$cm^{-1}$ peak. The spectrum
around 3250$cm^{-1}$ was assigned to a continuum of O-H symmetric
stretches(ss), $\nu_{1}$ of water molecules in a symmetric
environment (ss-s), and was generally referred as "ice-like"
region because of its similarity in energy to O-H bonds in bulk
ice. The broad band around 3450$cm^{-1}$ was assigned to more
weakly correlated hydrogen bonded stretching modes, and was called
the "liquid-like" hydrogen-bonded region, where water molecules
reside in a more asymmetrically bonded (as) water
environment.\cite{RichmondARPC2001,Richmond:cr102:2693} The broad
peak around 3550$cm^{-1}$ appeared clearly in the \textit{ppp} SFG
spectra has been identified once and it has not been clearly
assigned so far.\cite{WeiXingPRL2001} Shultz \textit{et al.}
pointed out that these broad peak should also include the
asymmetric stretching mode of water molecules in a symmetry
environment and the bending overtone.\cite{ShultzIRPC2000}
Richmond \textit{et al.} also suggested that the intensity at
about 3450$cm^{-1}$ include the contribution of donor O-H
bond.\cite{richmond:science}
Recent progresses on SFG-VS have made it possible to determine the
symmetry properties of SFG-VS vibrational spectral features
through comparison of SFG spectra in different polarizations and
experimental
configurations.\cite{Lurong2,Lurong3,GanweiCPLNull,HongfeiIRPCreview}
The key idea of this development is from the commonsense of
molecular spectroscopy that vibrational modes of molecular groups
with different symmetry properties have different polarization
dependence on the interacting optical
fields.\cite{MichlBook,SymmetrySpectroscopyBook} Applying these
ideas to polarization analysis of SFG spectra has led to a set of
polarization selection rules for different stretching vibrational
modes of molecular groups with different molecular symmetry
properties, such as stretching vibrational modes of the $CH_{3}$
($C_{3v}$), $CH_{2}$ ($C_{2v}$) and $CH$ ($C_{\infty v}$)
groups.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Many of these
selection rules are independent from molecular orientation and
orientational distribution. Therefore, they can be directly used
to identify symmetry property of SFG stretching vibrational band.
These progresses make it possible to analyze SFG vibrational
spectra \textit{in situ}, instead of rely only on the assignments
from Raman and IR studies of the bulk phases, which can be called
\textit{ex situ}. Because SFG spectra usually has more features
than those from IR and Raman measurement, some confusions and
errors in the previous spectral assignments have also been
clarified.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Even though SFG
is naturally a polarized spectroscopy and the interfacial
molecular groups are usually ordered, this idea has not been
systematically explored until very
recently.\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
Water molecule possesses $C_{2v}$ symmetry. If the two O-H bond of
a water molecule are asymmetrically bonded, both O-H bond has to
be treated separately with $C_{\infty v}$ symmetry. This
classification of the water molecule symmetry is generally true no
matter it is hydrogen-bonded or not, in cluster, in bulk or at the
interface. Therefore, the symmetry property of the SFG vibrational
spectra features of the air/water interface can all be classified
accordingly. Thus, there are three kind of stretching vibrational
modes for us to deal with, namely, the symmetric (ss) and
asymmetric (as) stretching modes for $C_{2v}$ symmetry, and the
stretching mode for $C_{\infty v}$ symmetry. It is fairly easy to
distinguish these three stretching vibrational modes from the
polarization selection rules for SFG spectra of $CH_{2}$ and $CH$
groups.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Because the bond
angle of the $CH_{2}$ group is slightly larger than that of water
molecule, the polarization dependence of the $C_{2v}$ water
molecule are slightly different. The key difference is that even
when the water molecule at the interface rotate freely around its
symmetry axis, the \textit{sps} spectral intensity of its
\textit{ss} mode does not vanish as that for $CH_{2}$. However,
this fact does not make the polarization selection rules different
for the interfacial $CH_{2}$ group and the $C_{2v}$ water
molecule.
Two of the major selection rules for the $C_{2v}$ group at a
dielectric interface are: \textit{(a) $\textit{ssp}$ intensity is
always many times of that of $\textit{ppp}$ for \textit{ss} mode.}
and \textit{(b) $\textit{ppp}$ intensity for \textit{as} mode is
always several times of that of $\textit{ssp}$. That is to say, if
there is any peak which is stronger in the $\textit{ssp}$ than
$\textit{ppp}$ spectra, it can not be from the $\textit{as}$
mode.}\cite{Lurong2,Lurong3} These two rules are independent from
molecular orientation and orientational distributions at a
rotationally isotropic interface.
It is clear from these selection rules, the sharp peak around
$3693cm^{-1}$ in Fig.\ref{allSpectra} does not belong to the
$C_{2v}$ symmetry, because its intensity in \textit{ppp}
polarization in Config. 1,2,3 is smaller than that in \textit{ssp}
polarization; while larger in Config.4. On the other hand, these
all fits well with the simulations in Fig.\ref{FourC3vSimulation}.
Therefore, $3693cm^{-1}$ peak is with $C_{\infty v}$ symmetry as
the free O-H stretching mode. Dissimilarly, both the broad peaks
around $3250cm^{-1}$ and $3450cm^{-1}$ in Fig.\ref{allSpectra} are
very strong only in the \textit{ssp} spectra in all experimental
configurations. They fit well with the \textit{ss} mode of the
$C_{2v}$ symmetry, and can not belong to the \textit{as} mode of
the $C_{2v}$ symmetry, or the $C_{\infty v}$ symmetry.
It is not so easy to determine the symmetry property of the broad
$3550cm^{-1}$ peak, because it appears to be buried in the high
frequency tail of the broad $3450cm^{-1}$ peak. It is not so
straight forward to read its relative intensity in \textit{ssp}
and \textit{ppp} polarizations from Fig.\ref{allSpectra}. However,
it appears significantly bigger in \textit{sps} polarization in
Config.4 than that in Config.1,2,3. Therefore, it appears to fit
with the simulations in Fig.\ref{FourC3vSimulation}. In order to
exclude the possibility that it may belong to a $C_{2v}$ mode (ss
or as), detailed simulation of the $C_{2v}$ modes in different
polarizations and experimental configurations is now called upon.
As describe in the appendix, the parameter \textit{c} and
\textit{d} of the $C_{2v}$ vibrational stretching modes are
calculated for different polarizations and experimental
configurations. Plots of $[d\ast r(\theta)\ast \sec\beta]^{2}$ vs.
tilted angle $\theta$ of the water molecule $c$ axis from the
interface normal using these $c$ and $d$ values are presented in
Fig.\ref{C2vsimulation}. These plots again confirm that the
$3693cm^{-1}$ can not belong to any $C_{2v}$ mode, especially with
the one order of magnitude increase of this peak in \textit{ppp}
polarization.
Clearly, the polarization dependence of the broad $3550cm^{-1}$
peak does not fit to the \textit{ss} mode of the $C_{2v}$
symmetry. Otherwise, according to Fig.\ref{C2vsimulation}, its
\textit{ppp} intensity in Config.3 has to be about one order of
magnitude weaker than that in the observed spectra. This peak can
not be the \textit{as} mode of the $C_{2v}$ symmetry either.
According to the \textit{c} and \textit{d} values for the
\textit{as} mode of the $C_{2v}$ symmetry, the phases in
\textit{ssp} and \textit{ppp} polarizations have to be with
opposite signs in all four experimental configurations. However,
fitting of the \textit{ssp} and \textit{ppp} spectra indicates
that in Config.4, the oscillator strengths had the same signs for
\textit{ssp} and \textit{ppp} polarizations, even though in
Congfig. 1,2,3, the oscillator strengths of this peak do possess
opposite phases these two polarizations. This indicates that the
$3550cm^{-1}$ peak is not $C_{2v}$ symmetry, and it appears to
have $C_{\infty v}$ symmetry.
Because in the bulk phase there is no observation of free O-H
bond, and because this broad peak $3550cm^{-1}$ appears to be
hydrogen-bonded, there is only one possibility that it is the
other O-H bond of the interfacial water molecule which has a free
O-H bond extruding away from the liquid bulk phase. According to
Fig.\ref{FourC3vSimulation}, the $C_{\infty v}$ O-H stretching
mode in \textit{sps} polarization is about twice as large in
Cogfig.4 as that in the other configurations. This is fully
consistent with the SFG spectra data in Fig.\ref{allSpectra} and
the fitting results in Table \ref{fittingResults}. Furthermore, in
Table \ref{fittingResults}, the phase of the broad $3550cm^{-1}$
peak is just opposite to that of the $3693cm^{-1}$ peak in both
\textit{ssp} and \textit{sps} polarizations, indicating these two
O-H pointing to opposite directions. The phase of the \textit{ppp}
polarization of the broad $3550cm^{-1}$ peak changes signs with
different experimental configuration. This is because the
orientational angle of the two O-H bonds are some times on the
same side of the minimum on the \textit{ppp} curves in
Fig.\ref{FourC3vSimulation}, and sometime on the different side of
the minimum, just as predicted with the experimental configuration
analysis. These detail features indicated the ability to
understand very subtle dependence of the SFG spectra on
experimental configurations and the parameters used for the
spectra calculations. Further study shall be reported elsewhere.
Further support for the assignment of the broad $3550cm^{-1}$ peak
came from the IR spectra measurement of the water dimer clusters,
where the stretching frequency for the donor O-H bond is just at
about $3550cm^{-1}$.\cite{RamanSpectroscopyBook,
PimentalWaterDimerPaper,NixonWaterDimerPaper,ShenLeePaper} This
assignment is a good support for our assignment of the peak at
3550$cm^{-1}$ in \textit{ppp} spectra to the single
hydrogen-bonded water molecule at the interface. Furthermore, the
two O-H stretching vibrations for the methanol dimer are at
3574$cm^{-1}$ and 3684$cm^{-1}$.\cite{MethanolJCP1991} The donor
O-H stretching mode is also in the same region of 3550$cm^{-1}$.
There is no observable spectra features in Fig.\ref{allSpectra}
for the \textit{as} mode of the $C_{2v}$ water molecules, neither
hydrogen bonded nor non-hydrogen bonded. According to
Fig.\ref{C2vsimulation}, for the \textit{as} modes corresponding
to the ss mode around $3250cm^{-1}$ and $3450cm^{-1}$, their
intensities have to be at least one order of magnitude weaker than
that of their corresponding \textit{ss} modes. It is
understandable that we do not observe them. Above discussion also
throw doubts on the existence of interfacial water molecules with
two free O-H bonds, as suggested somewhat less convincingly by
some recent studies.\cite{Richmond-jpca2000,richmond:science,
RichmondCPL2004,SaykallyJPCM2002,Mundy-science} According to the
polarization selection rules and the calculation for the
polarization dependence of the $C_{2v}$ water molecules, no
detectable spectral features satisfying the $C_{2v}$ symmetry in
the 3600$cm^{-1}$ and 3800$cm^{-1}$ has been observed in the SFG
spectra.
Here we clearly see that how polarization selection rules,
quantitative polarization and experimental configuration analysis
can help determine the symmetry property of the observed spectra
features. The importance of studying of spectral interference has
been demonstrated in recent
reports.\cite{Richmond-jpca2000,Shen2005PRLWaterQuartz,DaviesInterferenceJPCB2004}
Analysis in this work also demonstrated that, in order to discern
spectral details, it is useful and effective to analyze the
spectral interference of different spectral features through
global fitting of SFG spectra in different polarizations and
experimental configurations, and to compare fitting results with
the prediction from the calculated $c$ and $d$ values. This also
indicates the usefulness of the formulation of total SFG signal
with functions of $c$ and $d$ parameters in Eq.\ref{chi}.
\subsection{Molecular Structure at Air/Water Interface}
With the analysis of the orientation and motion, vibrational
spectral symmetry of the water molecules at the air/water
interface in above sections, we can have more understanding of the
molecular structure of the air/water interface.
In Section IV.B, we have determined that the free O-H oriented
around 30$^{\circ}$ away from the interface normal with a
orientational distribution narrower than $\sigma=15^{\circ}$, and
in Section IV.C, we have identified the spectral feature around
$3550cm^{-1}$ of the other hydrogen bonded O-H bond of this water
molecule. If the plane of this interfacial water molecule is close
to perpendicular to the interface, the orientation of the
hydrogen-bonded O-H should point into the liquid phase with a
orientation around 135$^{\circ}$ away from the interface normal.
This orientation is fully consistent with the calculation of the
polarization and experimental configuration dependence of the
broad $3550cm^{-1}$ peak with a $C_{\infty v}$ symmetry with the
observed SFG intensities, detail to be reported elsewhere. Such
orientation makes the dipole of this water molecule points around
97$^{\circ}$ from the interface normal. This picture is fully
consistent with conclusions in many previous experimental and
theoretical studies,
\cite{DuQuanPRL1993,Mundy-science,JaqamanJCP2004,LAAKSONENMolecularPhysics,
LAAKSONENJCP1997,TILDESLEYMolecularPhysics,HynesCP2000,GarrettJPC1996,RiceJCP1991,
BerkowitzCPL1991,WilsonJPC1987,TownsendJCP1985} but certainly
different from some.\cite{WeiXingPRL2001}
From Section IV.C, the broad spectral features between
3100$cm^{-1}$ to 3500$cm^{-1}$ are determined to be symmetric
stretching modes of the $C_{2v}$ symmetry. Because the peaks are
broad, and their energies is in the range of hydrogen-bonded O-H
stretching range, they can only come from the water molecules with
two donor O-H bonds, whose oxygen atom can accept either two, one
or zero hydrogen atom from other water molecules as hydrogen
donors. Certainly, the water molecule with the oxygen atom forming
two hydrogen-bonds is tetrahedral in shape and is "ice-like". This
is consistent with the previous assignment of the broad
$3250cm^{-1}$ peak.
The water molecule with no hydrogen bond for the oxygen atom is
obviously with $C_{2v}$ symmetry. However, the water molecule with
only one hydrogen bond for the oxygen atom may or may not preserve
the $C_{2v}$ symmetry. However, if this hydrogen bond perturbation
to the water structure is limited, this water molecule can still
be treated as with $C_{2v}$ symmetry. The last two kinds of water
molecules are certainly not "ice-like", but "liquid-like". The two
"liquid-like" species may have slightly different O-H vibrational
frequencies. However, only two apparently broad peaks in the
3100$cm^{-1}$ to 3500$cm^{-1}$ region have been identified in the
literatures.\cite{Shen-prl1994,RamanSpectroscopyBook,ShultzIRPC2000,
Richmond-jpca2000,richmond:science,RichmondCPL2004,
richmond:jpcb1998,Shen-science,richmondJPCB2003p546paper,RichmondARPC2001,Richmond:cr102:2693}
More studies on the possible hydrogen-bonded species are certainly
warranted in the future.
Here we confirm the conclusion by Brown \textit{et al.} that these
$C_{2v}$ water species all have their dipole vector point out of
the bulk liquid phase, i.e. with both hydrogen atoms point into
the bulk liquid phase.\cite{Richmond-jpca2000} It is clear in
Table \ref{fittingResults}, the signs of the \textit{ssp}
polarization oscillator strength factors of the $C_{2v}$ water
species are all in opposite phase to that of the free O-H peak at
$3693cm^{-1}$ in all experimental configurations. The signs and
values of the $c$ and $d$ parameters of the $C_{2v}$ and
$C_{\infty v}$ in Table \ref{CandDforC2vWater} and Table
\ref{CandDvalueForC3v}, respectively, indicate that the $c$ axis
of the $C_{2v}$ species has to be in opposite direction to the $c$
axis of the free O-H bond at the interface. Therefore, as defined
as in the appendix, the $C_{2v}$ species have to have their dipole
vector point out of the bulk liquid phase. The calculation of the
phase of the \textit{ppp} as well as \textit{sps} spectral
features are all consistent with this picture.
However, because the SFG spectral intensities of the \textit{ppp}
and \textit{sps} polarizations are generally in the noise level in
the 3100$cm^{-1}$ to 3500$cm^{-1}$ region (Fig.\ref{allSpectra}),
it is difficult to determine the range of the orientation angle
$\theta$ of these hydrogen-bonded species relative to the
interface normal. The orientational distribution of these $C_{2v}$
species can be quite broad, different from that for the
interfacial water molecules with the free O-H bond. From our
simulations, it appears to us that SFG measurement may not be very
effective to determine the orientational angle of the $C_{2v}$
species at the air/water interface, even though it can do very
well with the $C_{\infty v}$ O-H bonds as shown above. However,
our recent analysis of the SHG measurement of the neat air/water
interface showed that SHG measurement might be able to help
determine the orientational angle of the $C_{2v}$ species, but not
the $C_{\infty v}$ O-H bonds. Recent SHG results indicated that
the average orientation of the interfacial $C_{2v}$ water
molecules is about 40$^{\circ}$ to 50$^{\circ}$ from the surface
normal.\cite{wkzhangSHGwaterPaper}
The molecular structure, orientation and dynamics at nonpolar
material/water interfaces have been studied by \textit{ab initio}
calculation, MD simulation, or them
combined.\cite{Mundy-science,BenjaminPRL1994,HynesCP2000,
HynesJPCB2002,MooreJCP2003,ChandraCPL2003,ChandraCPL2004,RiceJCP1991,BenjaminCR1996,
JedlovskyWaterDCE,JedlovskyWaterCCl4} It appears that some
different conclusions were drawn on the molecular orientation and
structure of the air/water interface in different
studies.\cite{JedlovskyWaterCCl4,BenjaminPRL1994,MatsumotoJCP1987}
Nevertheless, many of these studies concluded that the dipole
vector of the interfacial water molecules prefers lying parallel
to the interface and have one of the O-H bond protrude out of the
liquid phase. The majority of the conclusions from theoretical
calculations agree satisfactorily with the experimental analysis
of ours and previous studies, but all the simulation results were
with significantly broader orientational
distributions.\cite{Mundy-science,JaqamanJCP2004,LAAKSONENMolecularPhysics,
LAAKSONENJCP1997,TILDESLEYMolecularPhysics,HynesCP2000,GarrettJPC1996,RiceJCP1991,
BerkowitzCPL1991,WilsonJPC1987,TownsendJCP1985} There were reports
concluded that some interfacial water molecules have their two O-H
bonds projecting into the vapor phase and with oxygen atoms in the
liquid phase.\cite{Mundy-science,GrayCondencedMatter1994,
RobinsonJPC1991,CroctonPhysica1981} However, we have not found
explicit spectroscopic evidence for such species at the air/water
interface. These all indicate that detailed comparison of the
theoretical calculations and the experimental analysis is
certainly an important subject in the future studies.
\section{Conclusion}
Detailed understanding of the air/water interface is important,
and can be used for the general understanding of the liquid water
structure. In this work, we presented detailed analysis of the SFG
vibrational spectra of the air/water interface taken in different
polarizations and experimental configurations. Polarization and
experimental configuration analysis have provided detailed
information on the orientation, structure and dynamics of the
water molecules at the air/water interface. The success of these
analysis indicated the effectiveness and ability of SFG-VS as a
uniquely interface specific spectroscopic probe of liquid
interfaces and other molecular interfaces. It also indicates that
for the neat air/water interface, as has been studied in the
literature for some other simple air/liquid interfaces, the
contribution from the interface region dominates the SFG
spectra.\cite{ShenApplyPhys,WeiXinJPCBBulkVsSurface,WeiPRBBulkvsSurface,Shen-methanolPRL1991}
Here are major conclusions we have reached for the air/water
interface. Firstly, we concluded that the motion of the
interfacial water molecules can only be in a limited angular
range, instead rapidly varying over a broad angular range in the
vibrational relaxation time suggested previously. Secondly,
because different vibrational modes of different molecular species
at the interface has different symmetry properties, polarization
and symmetry analysis of the SFG-VS spectral features can help
assignment of the SFG-VS spectra peaks to different interfacial
species. These analysis concluded that the narrow $3693cm^{-1}$
and broad $3550cm^{-1}$ peaks belong to $C_{\infty v}$ symmetry,
while the broad $3250cm^{-1}$ and $3450cm^{-1}$ peaks belong to
the symmetric stretching modes with $C_{2v}$ symmetry. Thus, the
$3693cm^{-1}$ peak is assigned to the free OH, the $3550cm^{-1}$
peak is assigned to the single hydrogen bonded OH stretching mode,
and the $3250cm^{-1}$ and $3450cm^{-1}$ peaks are assigned to
interfacial water molecules as two hydrogen donors for hydrogen
bonding (with $C_{2v}$ symmetry), respectively. Thirdly, analysis
of the SFG-VS spectra concluded that the singly hydrogen bonded
water molecules at the air/water interface have their dipole
vector direct almost parallel to the interface, and is with a very
narrow orientational distribution. The doubly hydrogen bond donor
water molecules have their dipole vector point away from the
liquid phase. Finally, we did not find any observable evidence for
interfacial water molecules with doubly free O-H bonds at the
air/water interface.
Many of the conclusions in this work agree well with previous
reports, with much more detailed understandings. The conclusion of
the narrow range motion of the free O-H bond is different from the
literature. The explicit assignment of the broad $3550cm^{-1}$
peak and determination of the symmetry property of the
hydrogen-bonded O-H stretching modes in the 3100$cm^{-1}$ to
3500$cm^{-1}$ region are based on firm evidences. These
conclusions as a whole provided a detailed and general picture of
the spectroscopy, structure and dynamics of the air/water
interface, which can be used for understanding chemical and
biological problems related to the ubiquitous water molecule in
general. The concepts and approaches used in the analysis in this
report can be applied to studying on more complex molecular
interfaces.
Recently, extensive efforts with SFG-VS, as well as SHG,
experimental studies and theoretical simulations have been devoted
to the renewed interests on ion adsorption and the Jones-Ray
effect at the air/aqueous solution
interfaces.\cite{TobiasJPCB2005,ShultzIRPC2000,RichmondJPCB2004,ShultzJPCB2002,
AllenJPCB2004,SaykallyCPL2004,SaykallyCPL2004-2,SaykallyJPCB2005,MuchaJPCB2005}
We suggest that detailed polarization and experimental
configuration analysis of the SFG vibrational spectra be applied
to these interfaces.
\vspace{0.8cm}
\noindent \textbf{Acknowledgment.} This work was supported by the
Chinese Academy of Sciences (CAS, No.CMS-CX200305), the Natural
Science Foundation of China (NSFC, No.20425309) and the Chinese
Ministry of Science and Technology (MOST, No.G1999075305). We
thank Bao-hua Wu for help derive the bond polarizability
derivative model expressions. H.F.W. acknowledges Y. R. Shen for
helpful discussions.
\section*{Appendix: Calculation of \textit{d} and \textit{c} Parameters for $C_{2v}$ Molecule}
Here we present the expressions to calculate the parameter
\textit{c} and \textit{d} for water molecules with C$_{2v}$
symmetry using the bond polarizability derivative model first used
by Hirose \textit{et al}.\cite{hirose:jcp1992,hirose:jpc1993} The
detailed re-derivation of the complete expressions and the
effectiveness of the model can be found in a recent
review.\cite{HongfeiIRPCreview}
The relationship between the Raman depolarization ratio $\rho$ and
the bond polarizability \textit{r} for a molecule group with
C$_{2v}$ symmetry was:\cite{HongfeiIRPCreview}
\begin{eqnarray}
\rho&=\frac{3}{4+20\frac{(1+2r)^2}{(1-r)^2(1+3\cos^2\tau)}}\label{rCH2definition}
\end{eqnarray}
\noindent in which $\tau$ is the H-O-H bond angle between the two
OH bonds of a water molecule. With the Raman depolarization ratio
measured as about 0.03,\cite{Murphy-r} the bond polarizability
\textit{r} for OH bond in water molecule can be deduced to be
0.32, as used by Du \textit{et al.}\cite{DuQuanPRL1993}
The 7 hyperpolarizability tensor elements of water molecule with
$C_{2v}$ symmetry are as the followings.
\begin{eqnarray}
\nonumber\beta_{aac}&=&\frac{G_{a}\beta_{OH}^{0}}{\omega_{a1}}\left[(1+r)-(1-r)\cos\tau\right]\cos(\frac{\tau}{2})\\
\nonumber\beta_{bbc}&=&\frac{2G_{a}\beta_{OH}^{0}}{\omega_{a1}}r\cos(\frac{\tau}{2})\\
\nonumber\beta_{ccc}&=&\frac{G_{a}\beta_{OH}^{0}}{\omega_{a1}}\left[(1+r)+(1-r)\cos\tau\right]\cos(\frac{\tau}{2})\\
\nonumber\beta_{aca}&=&\beta_{caa}=\frac{G_{b}\beta_{OH}^{0}}{\omega_{b1}}\left[(1-r)\sin\tau\right]\sin(\frac{\tau}{2})\\
\beta_{bcb}&=&\beta_{cbb}=0\label{c2vbond}
\end{eqnarray}
\noindent Where G$_{a}=(1+\cos\tau)/M_{O}+1/M_{H}$ and
G$_{b}=(1-\cos\tau)/M_{O}+1/M_{H}$ are the inverse effective mass
for the symmetric ($a_{1}$) and asymmetric ($b_{1}$) normal modes,
with $M_{O}$ and $M_{H}$ as the atomic mass of O and H atoms,
respectively. $\omega_{a1}$ and $\omega_{b1}$ are the vibrational
frequencies of the respective modes.
$\beta_{OH}^{0}=\frac{1}{2\varepsilon_{0}}\alpha'_{\zeta\zeta}\mu'_{\zeta}$,
as defined by Wei \textit{et al.}\cite{weixing:pre2000} The water
molecule are fixed in the molecular coordination $\lambda'(a,b,c)$
with the O atom at the coordination center, the molecule plane in
\textit{ac} plane, and the bisector from the oxygen to the two
hydrogen atoms side is the \textit{c} axis.
For the achiral rotationally isotropic ($C_{\infty v}$) liquid
interface, the symmetric stretching (\textit{ss}, $a_{1}$)
vibrational modes have,\cite{HongfeiIRPCreview}
\begin{eqnarray}
\chi_{xxz}^{(2),ss}&=&\chi_{yyz}^{(2),ss}\nonumber\\
&=&\frac{1}{2}N_{s}[\langle\cos^{2}\psi\rangle\beta_{aac}+
\langle\sin^{2}\psi\rangle\beta_{bbc}+\beta_{ccc}]\langle\cos\theta\rangle\nonumber\\
&+&\frac{1}{2}N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}-
\beta_{ccc}]\langle\cos^{3}\theta\rangle\nonumber\\
\chi_{xzx}^{(2),ss}&=&\chi_{zxx}^{(2),ss}=\chi_{yzy}^{(2),ss}=\chi_{zyy}^{(2),ss}\nonumber\\
&=&-\frac{1}{2}N_{s}[\langle\cos\theta\rangle\nonumber-\langle\cos^{3}\theta\rangle]\nonumber\\
&&[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}
-\beta_{ccc}]\nonumber\\
\chi_{zzz}^{(2),ss}&=&N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}]
\langle\cos\theta\rangle\nonumber\\
&-&N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}-
\beta_{ccc}]\langle\cos^{3}\theta\rangle\nonumber\\
\label{ssofC2Vpsi}
\end{eqnarray}
\noindent And the asymmetric stretching (\textit{as}, $b_{1}$)
vibrational modes have,
\begin{eqnarray}
\chi_{xxz}^{(2),as}&=&\chi_{yyz}^{(2),as}=-N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle
[\langle\cos\theta\rangle-\langle\cos^{3}\theta\rangle]\nonumber\\
\chi_{xzx}^{(2),as}&=&\chi_{zxx}^{(2),as}=\chi_{yzy}^{(2),as}=\chi_{zyy}^{(2),as}\nonumber\\
&=&\frac{1}{2}N_{s}\beta_{aca}[\langle\cos^{2}\psi\rangle-\langle\sin^{2}\psi\rangle]
\langle\cos\theta\rangle\nonumber\\
&+&N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle\langle\cos^{3}\theta\rangle\nonumber\\
\chi_{zzz}^{(2),as}&=&2N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle
[\langle\cos\theta\rangle-\langle\cos^{3}\theta\rangle]\label{asofC2Vpsi}
\end{eqnarray}
The $b_{2}$ asymmetric mode are SFG inactive since the
hyperpolarizability tensors $\beta_{bcb}$ and $\beta_{cbb}$ are
zero.
The Euler angel $\psi$ can be integrated if the H-X-H plane of the
XH$_{2}$ group can rotate freely around its symmetry axis $c$. For
water molecules with both OH bond hydrogen bonded to neighboring
molecules in liquid phase, the Euler angel $\psi$ should not be a
fixed value. Assuming a random $\psi$ distribution we have the
following non-vanishing tensor elements for the
symmetric-stretching mode.,\cite{Lurong2,HongfeiIRPCreview}
\begin{eqnarray}
\chi_{xxz}^{(2),ss}&=&\chi_{yyz}^{(2),ss}=\frac{1}{4}N_{s}(\beta_{aac}
+\beta_{bbc}+2\beta_{ccc})\langle{cos\theta}\rangle\nonumber\\
&+&\frac{1}{4}N_{s}(\beta_{aac}+\beta_{bbc}-2\beta_{ccc})
\langle{cos^3\theta}\rangle\nonumber\\
\chi_{xzx}^{(2),ss}&=&\chi_{zxx}^{(2),ss}=\chi_{yzy}^{(2),ss}=\chi_{zyy}^{(2),ss}\nonumber\\
&=&-\frac{1}{4}N_{s}(\beta_{aac}+\beta_{bbc}-2\beta_{ccc})
(\langle{cos\theta}\rangle-\langle{cos^3\theta}\rangle)\nonumber
\\\chi_{zzz}^{(2),ss}&=&\frac{1}{2}N_{s}(\beta_{aac}+\beta_{bbc})
\langle{cos\theta}\rangle\nonumber\\
&-&\frac{1}{2}N_{s}(\beta_{aac}+\beta_{bbc}
-2\beta_{ccc})\langle{cos^3\theta}\rangle\label{ssofC2V}
\end{eqnarray}
\noindent And the non-vanishing tensor elements for water
asymmetric-stretching modes are,
\begin{eqnarray}
\chi_{xxz}^{(2),as}&=&\chi_{yyz}^{(2),as}=-\frac{1}{2}N_{s}\beta_{aca}
(\langle{cos\theta}\rangle-\langle{cos^3\theta}\rangle)\nonumber\\
\chi_{xzx}^{(2),as}&=&\chi_{zxx}^{(2),as}=\chi_{yzy}^{(2),as}=\chi_{zyy}^{(2),as}
=\frac{1}{2}N_{s}\beta_{aca}\langle{cos^3\theta\rangle}\nonumber\\
\chi_{zzz}^{(2),as}&=&
N_{s}\beta_{aca}(\langle{cos\theta}\rangle-\langle{cos^3\theta\rangle})\label{asofC2V}
\end{eqnarray}
\begin{table}[t]
\caption{The general orientational parameter \textit{c} and the
strength factor \textit{d} for \textit{ss} mode and \textit{as}
mode of water molecule with $C_{2v}$ symmetry in different
polarization combinations. The \textit{d} values bear the unit
$\beta_{ccc}$.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline
\textit{ss} mode& & d-ssp & c-ssp & d-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & 0.400 & 0.038 & 0.012 & 1 & -0.146 & 0.174 \\
Config.2 & & 0.374 & 0.038 & 0.013 & 1 & -0.079 & 0.338 \\
Config.3 & & 0.362 & 0.038 & 0.013 & 1 & -0.046 & 0.589 \\
Config.4 & & 0.257 & 0.038 & 0.012 & 1 & 0.066 & -0.378 \\
\hline
\textit{as} mode& & d-ssp & c-ssp & d$\ast$c-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & -0.154 & 1 & -0.122 & $\infty$ & 0.262 & 0.98 \\
Config.2 & & -0.144 & 1 & -0.129 & $\infty$ & 0.272 & 0.99 \\
Config.3 & & -0.139 & 1 & -0.128 & $\infty$ & 0.277 & 0.99 \\
Config.4 & & -0.099 & 1 & -0.117 & $\infty$ & 0.250 & 1.01 \\
\end{tabular}\label{CandDforC2vWater}
\end{center}
\end{table}
For CH$_{2}$ group, there is a general relationship
$\beta_{aac}+\beta_{bbc}-2\beta_{ccc}\cong 0$, because $\tau
=109.5^{\circ}$.\cite{Lurong2,HongfeiIRPCreview} This relationship
makes $\chi_{xzx}^{(2),ss}$=$\chi_{zxx}^{(2),ss}$=
$\chi_{yzy}^{(2),ss}$=$\chi_{zyy}^{(2),ss}\cong 0$, which means
that the \textit{ss} vibrational mode should vanish in the
\textit{sps} and \textit{pss} polarizations according to Eq.
\ref{ssofC2V}. For water molecule, $\tau=104.5^{\circ}$. Then
hyperpolarizability tensors of the water molecule are as the
followings: $\beta_{aac}=1.296$; $\beta_{bbc}=0.557$;
$\beta_{ccc}=1$; $\beta_{aca}=\beta_{caa}=0.741$;
$\beta_{bcb}=\beta_{cbb}=0$. Here all value are normalized to
$\beta_{ccc}=1$. Then,
$\beta_{aac}+\beta_{bbc}-2\beta_{ccc}=-0.147$. This value is not
$0$, but is very small. So the \textit{ss} vibrational mode
spectra in the \textit{sps} and \textit{pss} polarizations should
vanish as the $CH_{2}$ group mentioned above. However, they have
to be very small comparing with in \textit{ssp} spectra. This is
fully consistent with the small intensities in the \textit{sps}
SFG spectra for the $C_{2v}$ water modes in Fig.\ref{allSpectra}.
With above deduction, and following the procedure in previous
report,\cite{Lurong3} the general orientational parameter
\textit{c} and strength factor \textit{d} for the symmetric
stretching (\textit{ss}) mode and asymmetric stretching
(\textit{as}) mode of water molecule in different polarizations
and experimental configurations can be calculated (see Table
\ref{CandDforC2vWater}). The parameters used in the calculation
are $n_{1}(\omega)$=$n_{1}(\omega_{1})$=$n_{1}(\omega_{2})$=1;
$n_{2}(\omega)$=$n_{2}(\omega_{1})$=1.34;
$n_{2}(\omega_{2})$=1.18; $n'(\omega)$=$n'(\omega_{1})$=1.15;
$n'(\omega_{2})$=1.09, respectively. These parameters are the same
as the dielectric constants used for calculation of the air/water
interface by Wei \textit{et al.}\cite{WeiXingPRL2001} As we have
discussed in our reports,\cite{Lurong2,Lurong3} polarization
analysis with the co-propagating experimental geometry is
insensitive to the value of the dielectric constants of the IR
frequency.\cite{GanweiCPLNull,HongfeiIRPCreview} Therefore, we
used the same refractive constants for the IR frequencies across
the whole $3100cm^{-1}$ to $3800cm^{-1}$ region, and this does not
appear to affect our analysis. These \textit{c} and \textit{d}
values are used to calculate the polarization and orientation
dependence of the SFG intensity, as well as the interference
(phase) of different spectral features in different experimental
configurations. These calculations can satisfactorily explain the
detailed changes of the observed spectral features, as discussed
in the main text.
It is to be noticed that in the above discussion we only used
single water molecule parameters. When there is association and
clustering of water molecules, as long as the $C_{2v}$ symmetry
preserves, and the H-O-H bond angle does not change significantly,
above expressions dictated by symmetry properties should still be
valid.
|
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| 2,614
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"Holiday" is the day the your employer's business is closing for business on any holiday.
Generally, hours worked on Holidays, and weekends are treated like other hours worked on any other work day of the week. California law does not require that your employer provide you and your co-workers with paid holidays. As defined earlier that is the day your employer's business close its business on any holiday. In addition your employer is not obligated that you or your co-workers be given the day off for any particular holiday.
There is nothing in the law that requires such a practice. Additionally, there is nothing in the law that mandates an employer pay an employee a special premium for work performed on a holiday, Saturday, or Sunday, other than the overtime premium required for work performed in excess of eight hours in a workday or 40 hours in a workweek.
3. Pursuant to the terms of an employment agreement between you and employer.
|
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| 3,455
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{"url":"https:\/\/publications.hse.ru\/en\/preprints\/152751506","text":"\u2022 A\n\u2022 A\n\u2022 A\n\u2022 ABC\n\u2022 ABC\n\u2022 ABC\n\u2022 \u0410\n\u2022 \u0410\n\u2022 \u0410\n\u2022 \u0410\n\u2022 \u0410\nRegular version of the site\nThe main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to osp(1,2) and the\u00a0 nonsymmetric Macdonald polynomials of types $A_2^2$ and ${A_2}^{2\\dagger}$\u00a0 . We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules","date":"2021-10-25 03:43:50","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\": 0.7613793015480042, \"perplexity\": 923.4385778061898}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2021-43\/segments\/1634323587623.1\/warc\/CC-MAIN-20211025030510-20211025060510-00219.warc.gz\"}"}
| null | null |
La tabernera del puerto é uma zarzuela em 3 atos, com livreto de Federico Rosemary Sarachaga e Guillermo Fernández-Shaw Iturralde e música de Pablo Sorozábal. Estreou pela primeira vez em 6 de abril de 1936 no Teatro Tivoli de Barcelona.
Sinopse
A ação se desenvolve durante os anos 30 do século de XX no porto da cidade imaginária de Cantabreda, no norte de Espanha.
Neste porto está a taverna de Marola. Ninguém sabe a origem de Marola, só se sabe que a taverna foi financiada pelo bandido Juan de Eguía que todos acreditam ser seu marido.
O marinheiro Simpson fala de Leandro (pescador apaixonado com Marola). Marola e Leandro se apaixonam. Abel, um adolescente, intérprete de acordeon, também se apaixona por Marola mas não é correspondido.
Um grupo de mulheres da cidade entram em conflito com Marola por enlouquecer a todos os homens, mas ela defende os recriminando pois elas não prestam atenção a seus maridos. Juan de Eguía fala a Marola da impotência de Abel.
No dia seguinte Abel conta a todos sobre o ocorrido, isso faz com que o olhar de todos os presentes, se voltem para Juan para lhe pedir explicações. Leandro conversa com Juan que admite que Marola é sua filha, onde convence Leandro para que fuja com sua amante.
O Juan de Eguía aparece novamente promete para Leandro a mão de Marola em troca disso, que entre com um fardo de cocaína na cidade. Leandro aceita e acompanhado por sua amante até uma arte pequena caverna que só se tem acesso por mar, carregando o fardo de cocaína, mas eles são surpresos por uma tempestade e desaparecem.
Abel canta à desaparecida Marola, enquanto entristecido, Juan conta para todos os presentes que na realidade, Marola é sua filha. Em seguida o marinheiro Simpson traz as boas notícias de que Leandro e Marola estão vivos e se dirigem ao porto amparados pelas autoridades alfandegárias. Juan admite para ser o verdadeiro culpado de tudo aquilo que aconteceu e é detido, enquanto Leandro e Marola ganham a liberdade.
Ligações externas
Composições musicais
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 4,505
|
You are here: Home / Sports / Grand Rapids native inks deal with Arizona
Grand Rapids native inks deal with Arizona
June 23, 2016 By Minnesota News Network
Goligoski when he played at the U
The Arizona Coyotes have addressed one of their biggest offseason needs, signing defenseman Alex Goligoski to a five-year, $27.375 million deal.
Arizona acquired Goligoski's negotiating rights in a trade with Dallas last week and brought him out for a visit.
Goligoski is a Grand Rapids native, who played college hockey for the Golden Gophers.
Goligoski liked what he saw and so did the Coyotes, taking one of the top free agent defensemen off the market.
"We are thrilled to sign Alex to a long term contract," Coyotes general manager John Chayka said in a statement. "Alex is a great skater and a smart, efficient defenseman who moves the puck well and makes his teammates better. He will help solidify our defense and support the growth of our young players."
The 30-year-old Goligoski has been a steady defensemen in eight NHL seasons with Pittsburgh and Dallas. He played all 82 games with the Stars last season, finishing with five goals and 32 assists. He also had four goals and three assists in 13 playoff games.
Goligoski, who has 55 goals and 222 assists in 562 career NHL games, agreed to the deal after spending two days in Arizona last week.
"I am very happy to join the Coyotes," he said in a statement. "Arizona is a great place to live and play and I'm excited to join an up and coming team that has a ton of talent and a very bright future."
The Coyotes became an exciting offensive team with the additions of Max Domi and Anthony Duclair, but needed help on the blue line. Though a bit undersized at 5-foot-11, 185 pounds, Goligoski should give them a boost as a top-4 defenseman.
|
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| 3,160
|
.class public final Landroid/webkit/ClientCertRequestHandler;
.super Landroid/os/Handler;
.source "ClientCertRequestHandler.java"
# instance fields
.field private final mBrowserFrame:Landroid/webkit/BrowserFrame;
.field private final mHandle:I
.field private final mHostAndPort:Ljava/lang/String;
.field private final mTable:Landroid/webkit/SslClientCertLookupTable;
# direct methods
.method constructor <init>(Landroid/webkit/BrowserFrame;ILjava/lang/String;Landroid/webkit/SslClientCertLookupTable;)V
.locals 0
.parameter "browserFrame"
.parameter "handle"
.parameter "host_and_port"
.parameter "table"
.prologue
.line 42
invoke-direct {p0}, Landroid/os/Handler;-><init>()V
.line 43
iput-object p1, p0, Landroid/webkit/ClientCertRequestHandler;->mBrowserFrame:Landroid/webkit/BrowserFrame;
.line 44
iput p2, p0, Landroid/webkit/ClientCertRequestHandler;->mHandle:I
.line 45
iput-object p3, p0, Landroid/webkit/ClientCertRequestHandler;->mHostAndPort:Ljava/lang/String;
.line 46
iput-object p4, p0, Landroid/webkit/ClientCertRequestHandler;->mTable:Landroid/webkit/SslClientCertLookupTable;
.line 47
return-void
.end method
.method static synthetic access$000(Landroid/webkit/ClientCertRequestHandler;)I
.locals 1
.parameter "x0"
.prologue
.line 33
iget v0, p0, Landroid/webkit/ClientCertRequestHandler;->mHandle:I
return v0
.end method
.method static synthetic access$100(Landroid/webkit/ClientCertRequestHandler;)Landroid/webkit/BrowserFrame;
.locals 1
.parameter "x0"
.prologue
.line 33
iget-object v0, p0, Landroid/webkit/ClientCertRequestHandler;->mBrowserFrame:Landroid/webkit/BrowserFrame;
return-object v0
.end method
# virtual methods
.method public cancel()V
.locals 2
.prologue
.line 88
iget-object v0, p0, Landroid/webkit/ClientCertRequestHandler;->mTable:Landroid/webkit/SslClientCertLookupTable;
iget-object v1, p0, Landroid/webkit/ClientCertRequestHandler;->mHostAndPort:Ljava/lang/String;
invoke-virtual {v0, v1}, Landroid/webkit/SslClientCertLookupTable;->Deny(Ljava/lang/String;)V
.line 89
new-instance v0, Landroid/webkit/ClientCertRequestHandler$4;
invoke-direct {v0, p0}, Landroid/webkit/ClientCertRequestHandler$4;-><init>(Landroid/webkit/ClientCertRequestHandler;)V
invoke-virtual {p0, v0}, Landroid/webkit/ClientCertRequestHandler;->post(Ljava/lang/Runnable;)Z
.line 94
return-void
.end method
.method public ignore()V
.locals 1
.prologue
.line 77
new-instance v0, Landroid/webkit/ClientCertRequestHandler$3;
invoke-direct {v0, p0}, Landroid/webkit/ClientCertRequestHandler$3;-><init>(Landroid/webkit/ClientCertRequestHandler;)V
invoke-virtual {p0, v0}, Landroid/webkit/ClientCertRequestHandler;->post(Ljava/lang/Runnable;)Z
.line 82
return-void
.end method
.method public proceed(Ljava/security/PrivateKey;[Ljava/security/cert/X509Certificate;)V
.locals 5
.parameter "privateKey"
.parameter "chain"
.prologue
.line 53
invoke-interface {p1}, Ljava/security/PrivateKey;->getEncoded()[B
move-result-object v2
.line 56
.local v2, privateKeyBytes:[B
:try_start_0
invoke-static {p2}, Lorg/apache/harmony/xnet/provider/jsse/NativeCrypto;->encodeCertificates([Ljava/security/cert/Certificate;)[[B
move-result-object v0
.line 57
.local v0, chainBytes:[[B
iget-object v3, p0, Landroid/webkit/ClientCertRequestHandler;->mTable:Landroid/webkit/SslClientCertLookupTable;
iget-object v4, p0, Landroid/webkit/ClientCertRequestHandler;->mHostAndPort:Ljava/lang/String;
invoke-virtual {v3, v4, v2, v0}, Landroid/webkit/SslClientCertLookupTable;->Allow(Ljava/lang/String;[B[[B)V
.line 58
new-instance v3, Landroid/webkit/ClientCertRequestHandler$1;
invoke-direct {v3, p0, v2, v0}, Landroid/webkit/ClientCertRequestHandler$1;-><init>(Landroid/webkit/ClientCertRequestHandler;[B[[B)V
invoke-virtual {p0, v3}, Landroid/webkit/ClientCertRequestHandler;->post(Ljava/lang/Runnable;)Z
:try_end_0
.catch Ljava/security/cert/CertificateEncodingException; {:try_start_0 .. :try_end_0} :catch_0
.line 71
.end local v0 #chainBytes:[[B
:goto_0
return-void
.line 63
:catch_0
move-exception v1
.line 64
.local v1, e:Ljava/security/cert/CertificateEncodingException;
new-instance v3, Landroid/webkit/ClientCertRequestHandler$2;
invoke-direct {v3, p0}, Landroid/webkit/ClientCertRequestHandler$2;-><init>(Landroid/webkit/ClientCertRequestHandler;)V
invoke-virtual {p0, v3}, Landroid/webkit/ClientCertRequestHandler;->post(Ljava/lang/Runnable;)Z
goto :goto_0
.end method
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,205
|
UBB Services
Subs on 1-Gig speed tiers spike in Q3 – OpenVault
Jeff Baumgartner, Senior Editor, Light Reading, 11/16/2020
The percentage of "power" broadband users – defined as those who consume at least 1 terabyte of data per month – and the number of customers who take 1-Gig speed tiers jumped in Q3 2020, according to OpenVault's latest "Broadband Insights" report.
The report, based on anonymized usage data from various ISPs, found a 110% increase in power users, to 8.8% of all subs. It also found a 172% jump in power users who gobble down 2TB or more, to 1% of all subs.
(Source: OpenVault)
OpenVault characterized this surge as a revenue opportunity for ISPs. While increased data usage, particularly during the pandemic, has caused some consumers to upgrade to higher-level broadband tiers, it can also have a financial impact on ISPs that utilize controversial data caps and usage-based broadband data policies.
Of note, OpenVault found that 80% of customers on usage-based data plans subscribe to faster speed tiers with higher average revenue per user (ARPU), while 49% more customers on flat-rate billing plans take lower-cost speed plans. The study also found that ISPs with usage-based plans had about 25% more subs on gigabit speed tiers than operators with flat-rate billing plans during the quarter.
There's no clear uniformity on usage-based policies among ISPs.
Comcast recently restored its residential broadband data plan and raised the monthly limit to 1.2TB – about 200 gigabytes higher than the 1TB limit that was in place before the COVID-19 outbreak. Sales of Cable One's unlimited data option, which runs an extra $40 per month, have been down of late, as the operator temporarily gave unlimited usage away for a few months during the early stages of the pandemic. However, unlimited data remains the "major contributor" to Cable One's ARPU growth, Cable One president and CEO Julie Laulis said on the company's recent Q3 earnings call.
Charter Communications currently does not implement usage-based data policies. But the MSO has asked the FCC to lift conditions tied to its acquisitions of Time Warner Cable and Bright House Networks that prevent Charter from implementing such policies.
More consumers going for a Gig
OpenVault's study also saw a 124% rise in the number of broadband customers provisioned for gigabit speeds – to 5.6%.
Other ISPs are seeing 1-Gig adoption tick up. Altice USA, for example, reported last month that 60% of its new fiber-to-the-premises (FTTP) gross broadband subs took the operator's gigabit broadband product, while 29% of gross-add subs getting service via FTTP or the operator's hybrid fiber/coax (HFC) network opted for 1-Gig.
Average data usage rises nearly 40%
Among other findings in the Q3 report, OpenVault found that the top 10% of broadband subs consumed more than 34% of total downstream traffic and 54% of all upstream traffic. Additionally, the top 1% accounted for almost one fifth (a little more than 19%) of total upstream usage.
Additionally, the monthly weighted average data consumed by measured subs in Q3 was 383.8GB, up nearly 40% from the year-ago period. The average household used 359GB of downstream data and 25GB of upstream data in the period.
The "sweet spot" for bandwidth usage versus speed tier is in the range of 100 Mbit/s to 200 Mbit/s. In that range, 36% of all subs consume about 36% of all bandwidth, OpenVault said.
— Jeff Baumgartner, Senior Editor, Light Reading, special to Broadband World News
More Blogs from Jeff Baumgartner
More from Jeff Baumgartner
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,379
|
This is a placeholder page for David Norman, which means this person is not currently on this site. We do suggest using the tools below to find David Norman.
You are visiting the placeholder page for David Norman. This page is here because someone used our placeholder utility to look for David Norman. We created this page automatically in hopes David Norman would find it. If you are not David Norman, but are an alumni of Lincoln County High School Stanford, KY, register on this site for free now.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,004
|
<?php
/**
* A list of links to display on the homepage. This is really only here to demonstrate
* that PHP is being executed.
*/
$links = ['contact.php'];
?>
<h1>Welcome to the Homepage!</h1>
<ul>
<?php foreach ($links as $link): ?>
<li>Go to <a href="/<?= $link ?>"><code><?= $link ?></code></a></li>
<?php endforeach ?>
<li>This page will (correctly) 404: <code><a href="/homepage.php">homepage.php</code></a></li>
</ul>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,884
|
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="com.kaozgamer.easypermissions">
<application
android:allowBackup="true"
android:label="@string/app_name"
android:supportsRtl="true">
</application>
</manifest>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,638
|
'use strict';
/**
* Requirements
* @ignore
*/
const BaseCommand = require('./BaseCommand.js').BaseCommand;
const Context = require('../application/Context.js').Context;
const ModelSynchronizer = require('../watch/ModelSynchronizer.js').ModelSynchronizer;
/**
* @memberOf command
*/
class BrowserSyncCommand extends BaseCommand
{
/**
*
*/
constructor(context, options)
{
super(context);
this._name = 'server';
this._options = options || {};
this._browsersync = require('browser-sync').create();
}
/**
* @inheritDoc
*/
static get injections()
{
return { 'parameters': [Context, 'command/ServerCommand.options'] };
}
/**
* @inheritDocs
*/
static get className()
{
return 'command/BrowserSyncCommand';
}
/**
* @inheritDocs
*/
get help()
{
const help =
{
name: this._name,
description: 'Adds browsersync to the development server',
actions:
[
]
};
return help;
}
/**
* @inheritDocs
* @returns {Promise<Server>}
*/
start(parameters)
{
const scope = this;
const logger = this.createLogger('command.browsersync');
const section = logger.section('Starting BrowserSync');
// create server
const workServer = logger.work('Starting proxy');
try
{
const port = this._options.port || 3000;
this._options.port = port - 1;
let serverAddress = ((this._options.http2 || false) ? 'https' : 'http') + '://localhost';
serverAddress+= ':' + (this._options.port);
logger.info('Starting proxy for ' + serverAddress);
scope._browsersync.init(
{
proxy: serverAddress,
port: port,
https: this._options.http2 || false
});
}
catch(e)
{
logger.error(e);
}
logger.end(workServer);
// Add watcher
const workSyncer = logger.work('Adding synchronizer');
const synchronizer = this.context.di.create(ModelSynchronizer);
synchronizer.signals.invalidated.add(function(synchronizer, invalidations)
{
if (invalidations.extensions.some(ext => ['.md', '.js', '.j2'].indexOf(ext) >= 0))
{
logger.info('Full reload');
scope._browsersync.reload();
}
if (invalidations.extensions.some(ext => ['.css'].indexOf(ext) >= 0))
{
logger.info('CSS reload');
scope._browsersync.reload('*.css');
}
});
synchronizer.start();
logger.end(workSyncer);
logger.end(section);
return Promise.resolve(scope._browsersync);
}
/**
* @inheritDocs
* @returns {Promise<Server>}
*/
doExecute(parameters)
{
return this.start();
}
}
/**
* Exports
* @ignore
*/
module.exports.BrowserSyncCommand = BrowserSyncCommand;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,863
|
\section{Introduction\wrlab{sect-intro}}
P.~Olver's freeness conjecture (in his words) asserts: ``If a Lie
group acts effectively on a manifold, then, for some $n<\infty$, the
action is free on [a nonempty] open subset of the jet bundle of order $n$.''
In this note, we provide a counterexample
to one interpretation of this conjecture:
In \tref{thm-ctrx}, we show that a $C^\omega$ counterexample exists
for the additive group $\Z$ of integers.
Using Lemma 17.3 of \cite{a1:cinftyfreeness},
and following the proof of Theorem 18.1 in \cite{a1:cinftyfreeness},
it is possible to induce that $\Z$-action to get a $C^\omega$~counterexample for any
connected Lie group with noncompact center.
Conversely, according to Theorem 19.1 of \cite{a1:cinftyfreeness},
the conjecture holds
(in its $C^\infty$ form and, consequently, in its $C^\omega$ form)
for any connected Lie group with compact center.
A $C^\infty$ counterexample for an action of $\Z$
can be found in the proof of Theorem 18.1 of \cite{a1:cinftyfreeness}.
In that counterexample, $\Z$ acts on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^4$, and it seems likely that
a $C^\omega$ counterexample could also be constructed on~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^4$, or, possibly,
even on a lower dimensional Euclidean space.
The best partial result I know of in this direction is \cite{morris:tosandp},
due to D.~Morris, and involves a $C^\omega$ action on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$
of the additive semigroup of positive integers.
However, the rigidity of $C^\omega$ actions makes this kind
of argument technically challenging.
In the proof of \tref{thm-ctrx},
we avoid many technical issues through the use of topology.
To wit, we construct a $C^\omega$~counterexample on a $3$-dimensional manifold
through a kind of dynamical surgery:
Following an iterative recipe, we form a subset $M$ of~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
and give it a locally Euclidean topology $\tau$.
This topology $\tau$ is chosen so as to reproduce the effect of gluing
certain pairs of submanifolds together,
and then passing to a fundamental domain.
The resulting topological space $(M,\tau)$
has infinitely generated fundamental group,
so is significantly more complicated, topologically,
than a contractible space like~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^4$.
Also constructed iteratively is
a maximal $C^\omega$ atlas $\scra$ on $(M,\tau)$,
along with a $C^\omega$ vector field $V$
on the $C^\omega$ manifold $(M,\tau,\scra)$.
The flow of $V$ is an ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$-action that,
when restricted to $\Z$, provides the desired counterexample.
This writeup is not intended for publication.
\section{Miscellaneous notation and terminology\wrlab{sect-notation}}
Let ${\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}:=\{1,2,3,\ldots\}$.
For any sets $A,B$ and function $f:A\to B$,
let $\hbox{dom\,}[f]:=A$ be the domain of $f$,
and let $\hbox{im\,}[f]:=f(A)\subseteq B$ be the image of $f$.
For any set $A$, let $\hbox{Id}_A:A\to A$ be the identity map on~$A$.
A subset of a topological space is {\bf meager}
(a.k.a.~{\bf of first category})
if it is a countable union of nowhere dense sets.
A subset of a topological space is {\bf nonmeager}
(a.k.a.~{\bf of second category}) if it is not meager.
A subset of a topological space is {\bf comeager}
(a.k.a.~{\bf residual}) if its complement is meager.
A subset of a topological space is {\bf locally closed}
if it is the intersection of an open set with a closed set.
A subset of a topological space is {\bf constructible}
if it is a finite union of locally closed sets.
The collection of constructible sets is exactly the
Boolean algebra of sets generated from the topology.
Let $M$ be a set, let $\tau$ be a topology on $M$
and let $M_0$ be a $\tau$-open subset of $M$.
We define $\tau|M_0:=\{U\in\tau\,|\,U\subseteq M_0\}$.
For any maximal $C^\omega$ atlas $\scra$ on $(M,\tau)$, we define
$\scra|M_0:=\{\phi\in\scra\,|\,\hbox{dom\,}[\phi]\subseteq M_0\}$.
Let $M$ be a $C^\omega$ manifold.
Let $V$ be a complete $C^\omega$ vector field on~$M$.
For all $t\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$, we denote the time $t$ flow of $V$ by $\Phi_t^V:M\to M$.
By the Cauchy-Kowalevski Theorem,
$(\sigma,t)\mapsto\Phi_t^V(\sigma):M\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\to M$ is $C^\omega$.
For any $A\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$, for any $B\subseteq M$, let $\Phi_A^V(B):=\{\Phi_a^V(b)\,|\,a\in A,b\in B\}$.
For any $A\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$, for any $b\in M$, let $\Phi_A^V(b):=\{\Phi_a^V(b)\,|\,a\in A\}$.
For any~$a\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$, for any $B\subseteq M$, let $\Phi_a^V(B):=\{\Phi_a^V(b)\,|\,b\in B\}$.
Let $M$ be a $C^\omega$ manifold, let $V$ be a $C^\omega$ vector field on $M$,
let $k\ge0$ be an integer and let $\sigma\in M$.
We say $(V,\sigma)$ is {\bf periodic to order $k$} if, for some integer $T\ne0$,
the map $\Phi_T^V:M\to M$ agrees with the identity map
$\hbox{Id}_M:M\to M$ to order $k$ at $\sigma$.
Let $\tau_\#$ denote the standard topology on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
For all $S\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$, let $S^\circ$~be the $(\tau_\#)$-interior in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$ of $S$.
Let $\scra_\#$ denote the standard maximal $C^\omega$~atlas on $({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3,\tau_\#)$.
Let $E$ be the $C^\omega$ vector field on $({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3,\tau_\#,\scra_\#)$ represented by the constant map
$(x,y,z)\mapsto(1,0,0):{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Then, for all $x,y,z,t\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$, we have $\Phi_t^E(x,y,z)=(x+t,y,z)$.
Define functions $\pi:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ and $\Pi:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ by
$\pi(x,y,z)=x$ and $\Pi(x,y,z)=(y,z)$.
A topology will be called a {\bf manifold topology} if it is
Hausdorff, second countable and locally Euclidean.
For any integer $N\ge1$, let
\begin{gather*}
L_N\,:=\,(-\infty,-N+1)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2,\qquad
R_N\,:=\,(N-1,\infty)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\\
\hbox{and}\qquad S_N\,:=\,(-N,N)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2.
\end{gather*}
A {\bf displayed system} consists of
\begin{itemize}
\item an integer $N\ge1$,
\item a $(\tau_\#)$-constructible subset $M$ of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
\item a manifold topology $\tau$ on $M$,
\item a maximal $C^\omega$ atlas $\scra$ on $(M,\tau)$ \qquad\qquad and
\item a complete $C^\omega$ vector field $V$ on $(M,\tau,\scra)$
\end{itemize}
such that
\begin{itemize}
\item$M^\circ$ is $\tau$-open and $\tau$-dense in $M$,
\item$\tau\,|\,(M^\circ)\,\,=\,\,(\tau_\#)\,|\,(M^\circ)$,
\item$\scra\,|\,(M^\circ)\,\,=\,\,(\scra_\#)\,|\,(M^\circ)$,
\item$M\cap S_N$ is $\tau$-open in $M$,
\item$L_N\,\cup\,R_N\,\,\subseteq\,\,M^\circ$ \qquad\qquad\qquad and
\item there exists a function $\theta:M^\circ\to[0,1]$ such that
\begin{itemize}
\item[]$\theta=1$ on $L_N\cup R_N$ \quad\qquad and \quad\qquad $V=\theta E$ on $M^\circ$.
\end{itemize}
\end{itemize}
Let $D=(N,M,\tau,\scra,V)$ be a displayed system.
We say that $\sigma\in M$ is {\bf$D$-flat} if there exists $v\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ such that
$(-N,v),(N,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$.
Let $\scrf(D)$ denote the set of $D$-flat points in $M$.
We say that $D$ is {\bf generically flat} if $\scrf(D)$ is $\tau$-comeager in $M$.
Let $D=(N,M,\tau,\scra,V)$ and $D'=(N',M',\tau',\scra',V')$ be displayed systems.
We say $D'$ is an {\bf extension of $D$} if all of the following hold:
\begin{itemize}
\item$N'\,\,\ge\,\,N+1$,
\item$M\cap S_N$ is $\tau'$-open in $M'$,
\item$M'\cap S_N\quad=\quad M\cap S_N$,
\item$\tau'\,|\,(M\cap S_N)\quad=\quad\tau\,|\,(M\cap S_N)$,
\item$\scra'\,|\,(M\cap S_N)\quad=\quad\scra\,|\,(M\cap S_N)$ \qquad\qquad and
\item$V'\,|\,(M\cap S_N)\quad=\quad V\,|\,(M\cap S_N)$.
\end{itemize}
\section{Coincidence of jets\wrlab{sect-facts-real-an}}
\begin{lem}\wrlab{lem-match-jet}
Let $d\ge1$ and $k\ge0$ be integers.
Let $w_0\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d$ and let $W$~be an open neighborhood in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d$ of $w_0$.
Let $\lambda:W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ be $C^k$
and assume that $2\le\lambda(w_0)\le3$.
Then there exists a $C^\omega$ function
$\lambda_0:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d\to(1,4)$
such that $\lambda_0$ agrees with $\lambda$ to order $k$ at $w_0$.
\end{lem}
\begin{proof}
Let ${\bf 0}:=(0,\ldots,0)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d$.
We assume, without loss, that $w_0={\bf 0}$.
Define $\mu:=\lambda-2$.
Then $0\le\mu({\bf 0})\le1$.
Let $P:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ be a polynomial such that $P$ agrees with $\mu$ to order $k$ at ${\bf 0}$.
Define $Q:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ by $Q(x_1,\ldots,x_d)=x_1^{2k+2}+\ldots+x_d^{2k+2}$.
Then $Q\ge0$.
Also, $Q$ vanishes to order $2k+1$ at ${\bf 0}$.
For all $a>0$, let $\mu_a:=Pe^{-aQ}:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^d\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$;
then $\mu_a$~agrees with $P$ to order $2k+1$ at ${\bf 0}$,
so $\mu_a$ agrees with $\mu$ to order $k$ at ${\bf 0}$.
We have $P({\bf 0})=\mu({\bf 0})$, so $0\le P({\bf 0})\le1$.
Choose $a_0>0$ so large that $-1<\mu_{a_0}<2$.
Let $\lambda_0:=2+\mu_{a_0}$.
\end{proof}
\section{Facts about displayed systems\wrlab{sect-facts-disp-syst}}
Let $D=(N,M,\tau,\scra,V)$ be a displayed system.
\begin{lem}\wrlab{lem-unambig-nw-dense}
Let $Z\subseteq M$. Then
\begin{itemize}
\item[]$Z$ is $\tau$-nowhere dense \qquad \hbox{iff} \qquad $Z$ is $(\tau_\#)$-nowhere dense.
\end{itemize}
\end{lem}
\begin{proof}
Let $Z'=Z\cap(M^\circ)$.
Then $\,\,Z'\,\,\subseteq\,\,Z\,\,\subseteq\,\,[Z']\cup[M\backslash(M^\circ)]$.
Since $M^\circ$~is $\tau$-open and $\tau$-dense in $M$,
it follows that $M\backslash(M^\circ)$ is $\tau$-nowhere dense in $M$.
Then
\begin{itemize}
\item[(a)]$Z$ is $\tau$-nowhere dense \qquad iff \qquad $Z'$ is $\tau$-nowhere dense.
\end{itemize}
Since $Z'\subseteq M^\circ$ and since $\tau|(M^\circ)=(\tau_\#)|(M^\circ)$, it follows that
\begin{itemize}
\item[(b)]$Z'$ is $\tau$-nowhere dense \qquad iff \qquad $Z'$ is $(\tau_\#)$-nowhere dense.
\end{itemize}
Since $M$ is a $(\tau_\#)$-constructible subset of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
it follows that $M\backslash(M^\circ)$~is $(\tau_\#)$-nowhere dense in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Then
\begin{itemize}
\item[(c)]$Z$ is $(\tau_\#)$-nowhere dense \qquad iff \qquad $Z'$ is $(\tau_\#)$-nowhere dense.
\end{itemize}
The result now follows from (a), (b) and (c).
\end{proof}
\begin{cor}\wrlab{cor-unambig-nw-dense}
Let $Z\subseteq M$. Then
\begin{itemize}
\item[]$Z$ is $\tau$-meager \qquad iff \qquad $Z$ is $(\tau_\#)$-meager.
\end{itemize}
\end{cor}
\begin{lem}\wrlab{lem-one-start-end}
Let $\sigma\in M$. Then both of the following are true:
\begin{itemize}
\item[(i)]The set $[\,\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)\,]\,\,\cap\,\,[\,\{-N\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,]$ has at most one element.
\item[(ii)]The set $[\,\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)\,]\,\,\cap\,\,[\,\{N\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,]$ has at most one element.
\end{itemize}
\end{lem}
\begin{proof}
We prove only (ii); the proof of (i) is similar.
Let $w',w''\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ and assume that $(N,w'),(N,w'')\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$.
We wish to show that $w'=w''$.
Choose $t',t''\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ such that $(N,w')=\Phi_{t'}^V(\sigma)$ and $(N,w'')=\Phi_{t''}^V(\sigma)$.
Fix $t\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ such that $t\ge t'$ and $t\ge t''$.
Since $V=E$ on $R_N$,
we get both $\Pi(\Phi_t^V(\sigma))=w'$ and $\Pi(\Phi_t^V(\sigma))=w''$.
Then $w'=w''$, as desired.
\end{proof}
\begin{lem}\wrlab{lem-Q-facts}
Let $Q:=M\cap S_N$. Then all of the following are true:
\begin{itemize}
\item[(i)]$Q^\circ=Q\cap(M^\circ)$.
\item[(ii)]$Q^\circ$ is $\tau$-open and $\tau$-dense in $Q$.
\item[(iii)]$Q^\circ$ is $\tau$-open in $M$.
\item[(iv)]$\tau|(Q^\circ)=(\tau_\#)|(Q^\circ)$.
\end{itemize}
\end{lem}
\begin{proof}
Since $S_N$ is $(\tau_\#)$-open in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
we get $(M\cap S_N)^\circ=(M^\circ)\cap S_N$.
Since $M^\circ\subseteq M$, we have $M^\circ=(M^\circ)\cap M$.
Then
$$Q^\circ\,=\,(M\cap S_N)^\circ\,=\,(M^\circ)\cap S_N\,=\,(M^\circ)\cap M\cap S_N\,=\,(M^\circ)\cap Q,$$
proving (i).
From the definition of displayed system,
we know both that $Q$~is $\tau$-open in $M$
and that $M^\circ$~is $\tau$-open and $\tau$-dense in $M$.
Then the intersection $Q\cap(M^\circ)$ is $\tau$-open and $\tau$-dense in $Q$,
and so (ii) follows from (i).
By (ii), $Q^\circ$~is $\tau$-open in $Q$, so, as $Q$ is $\tau$-open in $M$,
it follows that $Q^\circ$~is $\tau$-open in $M$, proving (iii).
From the definition of displayed system,
we know that $\tau|(M^\circ)=(\tau_\#)|(M^\circ)$.
So, because $Q^\circ\subseteq M^\circ$,
we conclude that $\tau|(Q^\circ)=(\tau_\#)|(Q^\circ)$, proving (iv).
\end{proof}
\begin{lem}\wrlab{lem-flatpt-density}
Let $Q:=M\cap S_N$.
Assume that $D$ is generically flat.
Then $[\scrf(D)]\cap[Q^\circ]$
is $(\tau_\#)$-dense in $Q^\circ$.
\end{lem}
\begin{proof}
By \lref{lem-Q-facts}(iii--iv),
$Q^\circ$ is $\tau$-open in $M$ and
$\tau|(Q^\circ)\!=\!(\tau_\#)|(Q^\circ)$.
Since $D$ is generically flat, it follows that
$\scrf(D)$ is $\tau$-comeager in~$M$,
so $[\scrf(D)]\cap[Q^\circ]$ is $\tau$-comeager in $Q^\circ$.
So, because $\tau|(Q^\circ)=(\tau_\#)|(Q^\circ)$,
we see that $[\scrf(D)]\cap[Q^\circ]$ is $(\tau_\#)$-comeager in $Q^\circ$.
Then, by the Baire Category Theorem,
$[\scrf(D)]\cap[Q^\circ]$ is $(\tau_\#)$-dense in $Q^\circ$.
\end{proof}
\begin{lem}\wrlab{lem-periodic-contained}
Let $\sigma\in M$ and assume, for some $T\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\backslash\{0\}$,
that $\Phi_T^V(\sigma)=\sigma$.
Then $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)\subseteq S_N$.
\end{lem}
\begin{proof}
Since $V=E$ on $L_N\cup R_N$,
we see that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)]\cap[L_N\cup R_N]=\emptyset$.
Then $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)\subseteq[{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3]\backslash[L_N\cup R_N]\subseteq S_N$.
\end{proof}
\section{The iteration\wrlab{sect-iteration}}
There is a graphic at the end of the paper, following the
references, which should help the reader in understanding the
following lemma.
\begin{lem}\wrlab{lem-iteration-lem}
Let $D=(N,M,\tau,\scra,V)$ be a generically flat displayed system.
Let $\sigma_0\in\scrf(D)$.
Let $k\ge1$ be an integer.
Then there exists a displayed system
$D'=(N',M',\tau',\scra',V')$ such that
\begin{itemize}
\item[(a)] \quad $D'$ is an extension of $D$,
\item[(b)] \quad $D'$ is generically flat,
\item[(c)] \quad $(V',\sigma_0)$ is periodic to order $k$ \qquad and
\item[(d)] \quad $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\sigma_0)\quad\subseteq\quad S_{N'}$.
\end{itemize}
\end{lem}
\begin{proof}
By definition of a displayed system,
$\tau|(M^\circ)=(\tau_\#)|(M^\circ)$
and $\scra|(M^\circ)=(\scra_\#)|(M^\circ)$.
Give $M^\circ$ this common topology and maximal $C^\omega$ atlas.
Give ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ its standard topology and $C^\omega$ maximal atlas.
Give every open subset of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ its standard topology and $C^\omega$ maximal atlas.
Let $H_0^@$ be the $C^\omega$ vector field on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ represented by the linear map
$(y,z)\mapsto(y,-z):{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Then $H_0^@$ is complete and vanishes at~$(0,0)$.
Moreover, for all $t\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$,
$\Phi_t^{H_0^@}(y,z)=(e^ty,e^{-t}z)$.
We let $C_0^@:=\{(y,z)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,|\,y^2+z^2=1\}$ denote the circle
of radius $1$ in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ centered at $(0,0)$.
Define $\zeta_0^@:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ by $\zeta_0^@(y,z)=y^2+z^2-1$.
Then the function $\zeta_0^@$ is $C^\omega$ and vanishes on and only on~$C_0^@$.
We let $B_0^@:=\{(y,z)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,|\,y^2+z^2\le1\}$ be the closed disk
of radius $1$ in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ centered at $(0,0)$.
Let $Z_H^@:={\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times\{0\}\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ and let $Z_V^@:=\{0\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Then $C_0^@$, $Z_H^@$ and $Z_V^@$ are all nowhere dense in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
The sets $\{(0,0)\}$, $Z_H^@$ and $Z_V^@$ are all invariant under the flow of $H_0^@$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_V^@)$,
we have: $\Phi_t^{H_0^@}(v)$ leaves compact sets in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $t\to\infty$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_H^@)$,
we have: $\Phi_{-t}^{H_0^@}(v)$ leaves compact sets in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $t\to\infty$.
Let $q_0:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ be defined by $q_0(y,z)=1-[\hbox{exp}(-y^2-z^2)]$.
Then $q_0$~is $C^\omega$ and vanishes to order $1$ at $(0,0)$.
Moreover, for all $v\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\backslash\{(0,0)\}$, we have $0<q_0(v)<1$.
Let $H_*^@:=q_0^kH_0^@$.
Then $H_*^@$~is complete and $H_*^@$ vanishes to order~$2k$ at $(0,0)$.
Let $\psi^@:=\Phi_1^{H_*^@}:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Then $\psi^@:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ is a $C^\omega$ diffeomorphism
which agrees with the identity $\hbox{Id}_{{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2}:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ to order~$2k$ at~$(0,0)$.
Also, $\psi^@(0,0)=(0,0)$ and $\psi^@(Z_H^@)=Z_H^@$ and $\psi^@(Z_V^@)=Z_V^@$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_V^@)$,
we have: $(\psi^@)^m(v)$ leaves compact sets in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $m\to\infty$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_H^@)$,
we have: $(\psi^@)^{-m}(v)$ leaves compact sets in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $m\to\infty$.
As $\sigma_0\in\scrf(D)$, fix $w_0\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ such that
$(-N,w_0),(N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma_0)$.
We define a translation $\scrt:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ by $\scrt(v)=v+w_0$.
We then define $\psi:=\scrt\circ(\psi^@)\circ(\scrt^{-1}):{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Then $\psi:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ is a $C^\omega$~diffeomorphism
that agrees with $\hbox{Id}_{{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2}:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ to order $2k$ at~$w_0$.
Then $\psi(w_0)=w_0$.
Let $Z_H:=\scrt(Z_H^@)$ and $Z_V:=\scrt(Z_V^@)$.
Then we have both $\psi(Z_H)=Z_H$ and $\psi(Z_V)=Z_V$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_V)$,
we have: $\psi^m(v)$ leaves compact sets in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $m\to\infty$.
For all $v\in({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(Z_H)$,
we have: $\psi^{-m}(v)$ leaves compact sets in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$, as $m\to\infty$.
We define $B_0:=\scrt(B_0^@)$ and $C_0:=\scrt(C_0^@)$.
Then $w_0\in(B_0)\backslash(C_0)$.
Let $\zeta_0:=(\zeta_0^@)\circ(\scrt^{-1}):{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Then $\zeta_0$ is $C^\omega$ and vanishes on and only on~$C_0$.
Let $B_*:=\psi(B_0)$ and $C_*:=\psi(C_0)$.
Then $w_0\in(B_*)\backslash(C_*)$.
Let $\zeta_*:=(\zeta_0)\circ(\psi^{-1}):{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Then $\zeta_*$ is $C^\omega$ and vanishes on and only on~$C_*$.
The sets $C_0$, $C_*$, $Z_H$ and $Z_V$ are all nowhere dense in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Define $\alpha,\beta,\gamma:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ and $f,g,h:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ by
\begin{eqnarray*}
\alpha(x)&=&(x+N+7)(x+N+6),\\
\beta(x)&=&(x-N-2)(x-N-3)(x-N-4)(x-N-5),\\
\gamma(x)&=&(x-N-6)(x-N-7),\\
f(x,y,z)\,\,&=&\,\,1\,\,-\,\,[\,\hbox{exp}\,(\,-\,[\alpha(x)]^2\,-\,[\zeta_*(y,z)]^2\,)\,],\\
g(x,y,z)\,\,&=&\,\,1\,\,-\,\,[\,\hbox{exp}\,(\,-\,[\beta(x)]^2\,-\,[\zeta_0(y,z)]^2\,)\,]\quad\hbox{and}\\
h(x,y,z)\,\,&=&\,\,1\,\,-\,\,[\,\hbox{exp}\,(\,-\,[\gamma(x)]^2\,-\,[\zeta_*(y,z)]^2\,)\,].
\end{eqnarray*}
Then $f,g,h$ are $C^\omega$ and $0\le f<1$ and $0\le g<1$ and $0\le h<1$.
Also,
\begin{itemize}
\item$f$ vanishes on and only on $\{-N-7,-N-6\}\times C_*$,
\item$g$ vanishes on and only on $\{N+2,N+3,N+4,N+5\}\times C_0$,
\item$h$ vanishes on and only on $\{N+6,N+7\}\times C_*$.
\end{itemize}
As $(-N,w_0),(N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma_0)$,
we get $(N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(-N,w_0)$.
Fix $t_0\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ such that $(N,w_0)=\Phi_{t_0}^{V}(-N,w_0)$.
Since $V=E$ on $L_N$, we get $t_0>0$.
We have $\pi(\Phi_{t_0}^{V}(-N,w_0))=\pi(N,w)=N$.
Choose an open neighborhood $W_0$ in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ of $w_0$
such that, for all $w\in W_0$,
we have $\pi(\Phi_{t_0}^{V}(-N,w))\in(N-0.5,N+0.5)$.
Let $J:=(t_0-0.5,t_0+0.5)\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Then, for all $(t,w)\in J\times W_0$,
we have $\pi(\Phi_t^{V}(-N,w))\in(N-1,N+1)$,
so $\Phi_t^{V}(-N,w)\in(N-1,N+1)\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\subseteq R_N\subseteq M^\circ$.
Moreover,
$$(t,w)\quad\mapsto\quad\Phi_t^V(-N,w)\quad:\quad J\,\times\,W_0\quad\to\quad M^\circ$$
is $C^\omega$.
Then $(t,w)\mapsto\pi(\Phi_t^V(-N,w)):J\times W_0\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ is $C^\omega$.
By the Implicit Function Theorem,
fix an open neighborhood $W_1$ in~$W_0$ of $w_0$
and a $C^\omega$ function $\lambda_1:W_1\to(0,\infty)$
such that $\lambda_1(w_0)=t_0$ and such that, for all $w\in W_1$,
we have $\pi(\Phi_{\lambda_1(w)}^{V}(-N,w))=N$.
Let $W:=W_1\cap[(B_0)\backslash(C_0)]\cap[({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\backslash(C_*)]$.
Then $w_0\in W\subseteq(B_0)\backslash(C_0)$ and $W$~is open in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
We have $W\cap C_*=\emptyset$,
and so we conclude that $f\ne0$ on $[-N-6,-N-5]\times W$.
Also, we have $W\cap C_0=\emptyset$,
and so we conclude that $g\ne0$ on $[N+1,N+2]\times W$.
Also, we have that $h\ne0$ on $[N+1,N+2]\times W$.
Then, using the Implicit Function Theorem,
we define $C^\omega$~functions $\lambda_2,\lambda_3:W\to(0,\infty)$ by:
for all $w\in W$,
\begin{eqnarray*}
\Phi_{\lambda_2(w)}^{fE}\,(\,-N-6\,,w\,)&=&(\,-N-5\,,\,w\,)\qquad\hbox{and}\\
\Phi_{\lambda_3(w)}^{ghE}\,(\,N+1\,,\,w\,)&=&(\,N+2\,,\,w\,).
\end{eqnarray*}
Let $T_0:=[\lambda_1(w_0)]+[\lambda_2(w_0)]+[\lambda_3(w_0)]$.
Then, because we have $\lambda_1(w_0),\lambda_2(w_0),\lambda_3(w_0)\in(0,\infty)$,
it follows that $T_0>0$.
Fix an integer~$T$ such that $T_0+2\le T\le T_0+3$.
Then $2<T$, so $T\ne0$.
Define $\lambda:=T-\lambda_1-\lambda_2-\lambda_3:W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Then $\lambda(w_0)=T-T_0$, so $2\le\lambda(w_0)\le3$.
By \lref{lem-match-jet},
fix a $C^\omega$ function $\lambda_0:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to(1,4)$
such that $\lambda_0$ agrees with $\lambda$ to order $k$ at~$w_0$.
Define $\Lambda:=\lambda_0+\lambda_1+\lambda_2+\lambda_3:W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Then $\Lambda$ agrees with the constant~$T$ to order~$k$ at $w_0$.
Let $N':=N+10$,
\vskip.15in
\begin{itemize}
\item[]$I_1:=(\,-N-8\,,\,-N-5\,]$, \qquad\qquad$I_2:=[\,N+1\,,\,N+8\,)$,
\vskip.17in
\item[]$I_3:=(\,-\infty\,,\,-N-9]$, \qquad\qquad\qquad$I_4:=\,[N+9\,,\,\infty)$,
\end{itemize}
\vskip.1in
\begin{eqnarray*}
X_1&:=&[\,-N-7\,,\,-N-6\,]\,\,\times\,\,B_*,\\
Y_1&:=&\{\,-N-7\,,\,-N-6\,\}\,\,\times\,\,(B_*\backslash C_*),
\end{eqnarray*}
\vskip-.18in
\begin{eqnarray*}
X_2&:=&[\,N+6\,,\,N+7\,]\,\,\times\,\,B_*,\\
Y_2&:=&\{\,N+6\,,\,N+7\,\}\,\,\times\,\,(B_*\backslash C_*),
\end{eqnarray*}
\vskip-.14in
$$X_3\,:=\,[\,N+2\,,\,N+3\,]\,\,\times\,\,B_0,$$
\vskip-.14in
$$X_4\,:=\,[\,N+4\,,\,N+5\,]\,\,\times\,\,B_0,$$
\vskip-.12in
\begin{eqnarray*}
M'_0&:=&M\,\,\cap\,\,S_N,\\
M'_1&:=&[\,\,(I_1\,\,\times\,\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\,\,\backslash\,\,X_1\,\,]\,\,\cup\,\,Y_1,\\
M'_2&:=&[\,\,(I_2\,\,\times\,\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)\,\,\backslash\,\,(X_2\cup X_3\cup X_4)\,\,]\,\,\cup\,\,Y_2,\\
M'_3&:=&I_3\,\,\times\,\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2,\\
M'_4&:=&I_4\,\,\times\,\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\qquad\qquad\hbox{and}\\
M'_5&:=&\{\,(x,v)\,\in\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,\,|\,\,-N-[\lambda_0(v)]\,<\,x\,\le\,-N\,\}.
\end{eqnarray*}
\vskip.1in
Let $M':=M'_0\cup M'_1\cup M'_2\cup M'_3\cup M'_4\cup M'_5$.
For all integers $j\in[1,4]$, we have $M'_j\not\subseteq(M')^\circ$.
By contrast, we have $M'_5\subseteq(M')^\circ$.
In fact, we have
$(M')^\circ=[(M'_0)^\circ]\cup[(M'_1)^\circ]\cup[(M'_2)^\circ]\cup[(M'_3)^\circ]\cup[(M'_4)^\circ]\cup[M'_5]$.
Define $\rho_1,\rho_2,\rho_3,\rho_4:(-0.5,0.5)\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to M'$ by
\begin{eqnarray*}
\rho_1(t,v)&=&\begin{cases}
\Phi_t^E(-N-9,v),&\hbox{if }-0.5<t\le0,\\
\Phi_t^{fE}(-N-8,v),&\hbox{if }0<t<0.5,
\end{cases}\\
\rho_2(t,v)&=&\begin{cases}
\Phi_t^{fE}(-N-5,v),&\hbox{if }-0.5<t\le0,\\
\Phi_t^E(-N-[\lambda_0(v)],v),&\hbox{if }0<t<0.5,
\end{cases}\\
\rho_3(t,v)&=&\begin{cases}
\Phi_t^E(N,v),&\hbox{if }-0.5<t<0,\\
\Phi_t^{ghE}(N+1,v),&\hbox{if }0\le t<0.5,
\end{cases}\\
\rho_4(t,v)&=&\begin{cases}
\Phi_t^{ghE}(N+8,v),&\hbox{if }-0.5<t<0,\\
\Phi_t^E(N+9,v),&\hbox{if }0\le t<0.5.
\end{cases}
\end{eqnarray*}
Define $\rho_5,\rho_6,\rho_7,\rho_8:(-0.5,0.5)\times(B_0\backslash C_0)\to M'$ by
\begin{eqnarray*}
\rho_5(t,v)&=&\begin{cases}
\Phi_t^{fE}(-N-7,\psi(v)),&\hbox{if }-0.5<t\le0,\\
\Phi_t^{ghE}(N+3,v),&\hbox{if }0<t<0.5,
\end{cases}\\
\rho_6(t,v)&=&\begin{cases}
\Phi_t^{ghE}(N+2,v),&\hbox{if }-0.5<t<0,\\
\Phi_t^{fE}(-N-6,\psi(v)),&\hbox{if }0\le t<0.5,
\end{cases}\\
\rho_7(t,v)&=&\begin{cases}
\Phi_t^{ghE}(N+4,v),&\hbox{if }-0.5<t<0,\\
\Phi_t^{ghE}(N+7,\psi(v)),&\hbox{if }0\le t<0.5,
\end{cases}\\
\rho_8(t,v)&=&\begin{cases}
\Phi_t^{ghE}(N+6,\psi(v)),&\hbox{if }-0.5<t\le0,\\
\Phi_t^{ghE}(N+5,v),&\hbox{if }0<t<0.5.
\end{cases}
\end{eqnarray*}
For all integers $j\in[1,8]$, let
$\tau'_j:=\{\rho_j(U)\,|\,U\in\tau_\#\hbox{ and }U\subseteq\hbox{dom\,}[\rho_j]\}$.
Let
$\tau'_0:=([\tau_\#]|[(M')^\circ])\cup(\tau|[M\cap S_N])\cup\tau'_1\cup\cdots\cup\tau'_8$.
Then $\tau'_0$ is a basis for a manifold topology $\tau'$ on~$M'$.
The set $(M')^\circ$ is then $\tau'$-open and $\tau'$-dense in $M'$.
For all integers $j\in[1,8]$,
the map $\rho_j$ is injective and $\hbox{dom\,}[\rho_j]\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$;
let $R_j:=\hbox{im\,}[\rho_j]\in\tau'_j\subseteq\tau'$ and
let $\kappa_j:R_j\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$ be the inverse of $\rho_j$.
Let
$\scra'_0:=([\scra_\#]|[(M')^\circ])\cup(\scra|[M\cap S_N])
\cup\{\kappa_1,\ldots,\kappa_8\}$.
Then $\scra'_0$~is a $C^\omega$ atlas on $(M',\tau')$.
Let $\scra'$ be the unique maximal $C^\omega$~atlas on~$(M',\tau')$
such that $\scra'_0\subseteq\scra'$.
Then $\tau'|[(M')^\circ]=[\tau_\#]|[(M')^\circ]$
and $\scra'|[(M')^\circ]=[\scra_\#]|[(M')^\circ]$.
Give $(M')^\circ$ this common topology and maximal $C^\omega$ atlas.
Define a $C^\omega$ vector field $V'_\circ$ on $(M')^\circ$ by
$$V'_\circ=
\begin{cases}
V&\hbox{on }[(M'_0)^\circ]\cup[(M'_3)^\circ]\cup[(M'_4)^\circ]\cup[M'_5],\cr
fE&\hbox{on }(M'_1)^\circ,\cr
ghE&\hbox{on }(M'_2)^\circ.
\end{cases}$$
By construction of $\tau'$ and $\scra'$,
let $V'$ be the unique $C^\omega$ vector field on~$(M',\tau',\scra')$
such that $V'|[(M')^\circ]=V'_\circ$.
Then $V'=V$ on $M'_0$.
Also $V'=V=E$ on $[(M'_3)^\circ]\cup[(M'_4)^\circ]\cup[M'_5]\cup[L_N\cap S_N]\cup[R_N\cap S_N]$.
Let $D':=(N',M',\tau',\scra',V')$.
Then $D'$~is a displayed system, and from the construction above,
we know that $D'$ is an extension of~$D$.
Therefore (a)~of \lref{lem-iteration-lem} holds.
It remains to prove (b), (c) and (d).
Let $Z'_0:=[M'_0]\backslash[\scrf(D)]$,
\begin{eqnarray*}
Z'_1&:=&\{N+7.5\}\,\,\times\,\,C_*,\\
Z'_2&:=&\{N+3.5\}\,\,\times\,\,C_0,\\
Z'_3&:=&\{-N-5.5\}\,\,\times\,\,[C_*\cup Z_H],\\
Z'_4&:=&\{N+5.5\}\,\,\times\,\,[C_0\cup Z_V],\\
Z'_5&:=&\{N+1.5\}\,\,\times\,\,[C_0\cup Z_V],\\
Z'_6&:=&\{N+5.5\}\,\,\times\,\,C_*,\\
Z'_7&:=&\{-N'\}\,\,\times\,\,C_*,\\
P_1&:=&\big(\,\,[N+7,N+8)\,\times\,B_*\,\,\big)\quad\cap\quad M',\\
P_2&:=&(N+3,N+4)\,\times\,B_0,\\
P_3&:=&\big(\,\,[-N-6,-N-5)\,\times\,B_*\,\,\big)\quad\cap\quad M',\\
P_4&:=&(N+5,N+6)\,\times\,B_0,\\
P_5&:=&(-N-8,-N-7)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2,\\
P_6&:=&[N+1,N+2)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2 \qquad\qquad\qquad\qquad \hbox{and}\\
P_7&:=&[N+9,N+10)\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2.
\end{eqnarray*}
For all integers $j\in[0,7]$,
let $Z_j^*:=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(Z'_j)$.
Let $Z^*:=Z_0^*\cup\cdots\cup Z_7^*$.
Let $P_*:=\{\,\mu\in M'\,\,|\,\,
\exists\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)\,\,\hbox{ s.t. }\,\pi(\mu_*)<(\pi(\mu))-0.5\,\}$.
\vskip.02in
{\it Claim 1:}
We have $[P_1]\backslash[Z_1^*]\subseteq P_*$.
{\it Proof of Claim 1:}
Suppose that $s\in[N+7,N+8)\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$
and that $v\in B_*\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Let $\mu:=(s,v)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Assume that $\mu\in M'$ and that $\mu\notin Z_1^*$.
We wish to prove that $\mu\in P_*$.
We define $\mu_*:=(N+3.5,\psi^{-1}(v))$.
Then $\pi(\mu_*)=N+3.5<s-0.5$,
so $\pi(\mu_*)<(\pi(\mu))-0.5$,
so it remains to prove that
$\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
We have $\mu\notin Z_1^*$, so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_1=\emptyset$.
So, as $(N+7.5,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$, we get $(N+7.5,v)\notin Z'_1$, so $v\notin C_*$.
Then, by construction of $V'$,
we have $(N+3.5,\psi^{-1}(v))\in\Phi_{(-\infty,0)}^{V'}(\mu)$.
Then $\mu_*=(N+3.5,\psi^{-1}(v))\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$,
as desired.
{\it End of proof of Claim 1.}
{\it Claim 2:}
We have $[P_2]\backslash[Z_2^*]\subseteq P_*$.
{\it Proof of Claim 2:}
Suppose that $s\in(N+3,N+4)\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$
and that $v\in B_0\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Let $\mu:=(s,v)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Assume that $\mu\notin Z_2^*$.
We wish to prove that $\mu\in P_*$.
We define $\mu_*:=(-N-7,\psi(v))$.
Then we have $\pi(\mu_*)=-N-7<s-0.5$,
so $\pi(\mu_*)<(\pi(\mu))-0.5$,
so it remains to prove that
$\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
We have $\mu\notin Z_2^*$, so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_2=\emptyset$.
So, as $(N+3.5,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$, we get $(N+3.5,v)\notin Z'_2$, so $v\notin C_0$.
Then, by construction of $V'$, we have
$(-N-7,\psi(v))\in\Phi_{(-\infty,0)}^{V'}(\mu)$.
Then $\mu_*=(-N-7,\psi(v))\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$,
as desired
{\it End of proof of Claim 2.}
{\it Claim 3:}
We have $[P_3]\backslash[Z_0^*\cup Z_3^*]\subseteq P_*$.
{\it Proof of Claim 3:}
Suppose that $s\in[-N-6,-N-5)\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$
and $v\in B_*\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Let $\mu:=(s,v)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Assume $\mu\in M'$, $\mu\notin Z_0^*$ and $\mu\notin Z_3^*$.
We wish to prove that $\mu\in P_*$.
We have $\mu\notin Z_3^*$, and it follows that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_3=\emptyset$.
So, as $(-N-5.5,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$, we get $(-N-5.5,v)\notin Z'_3$, so $v\notin Z_H$.
Fix an integer $m\ge1$ such that
$$v,\psi^{-1}(v),\ldots,\psi^{-m+1}(v)\in B_*\qquad\hbox{and}\qquad\psi^{-m}(v)\notin B_*.$$
Define $\mu_*:=(-N-7,\psi^{-m}(v))$.
Then $\pi(\mu_*)=-N-7<s-0.5$,
so $\pi(\mu_*)<(\pi(\mu))-0.5$,
so it remains to prove that
$\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
We have $\mu\notin Z_0^*$, so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_0=\emptyset$,
so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap M'_0\subseteq\scrf(D)$.
Also, as $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_3=\emptyset$,
we get $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap[\{-N-5.5\}\times C_*]=\emptyset$.
Then, working in reverse time, starting at $\mu=(s,v)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(-\infty,0)}^{V'}(\mu)$:
\begin{itemize}
\item[]$(-N-6,v)$,
\item[]$(N-0.5,\psi^{-1}(v))$, \quad $(-N+0.5,\psi^{-1}(v))$, \quad $(-N-6,\psi^{-1}(v))$,
\item[]$(N-0.5,\psi^{-2}(v))$, \quad $(-N+0.5,\psi^{-2}(v))$, \quad $(-N-6,\psi^{-2}(v))$,
\item[]$\cdots\cdots\cdots\cdots$
\item[]$(N-0.5,\psi^{-m}(v))$, \quad $(-N+0.5,\psi^{-m}(v))$, \quad $(-N-6,\psi^{-m}(v))$,
\item[]$(-N-7,\psi^{-m}(v))$.
\end{itemize}
Then $\mu_*=(-N-7,\psi^{-m}(v))\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
{\it End of proof of Claim 3.}
{\it Claim 4:}
We have $[P_4]\backslash[Z_4^*]\subseteq P_*$
{\it Proof of Claim 4:}
Suppose that $s\in(N+5,N+6)\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$
and that $v\in B_0\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
Let $\mu:=(s,v)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Assume that $\mu\notin Z_4^*$.
We wish to prove that $\mu\in P_*$.
We have $\mu\notin Z_4^*$, and it follows that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_4=\emptyset$.
So, as $(N+5.5,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$,
we conclude that $(N+5.5,v)\notin Z'_4$, so $v\notin Z_V$.
Fix an integer $m\ge1$ such that
$$v,\psi(v),\ldots,\psi^{m-1}(v)\in B_0\qquad\hbox{and}\qquad\psi^m(v)\notin B_0.$$
Define $\mu_*:=(N+4,\psi^m(v))$.
Then we have $\pi(\mu_*)=N+4<s-0.5$,
so $\pi(\mu_*)<(\pi(\mu))-0.5$,
so it remains to prove that
$\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
Because $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap Z'_4=\emptyset$,
we get $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)]\cap[\{N+5.5\}\times C_0]=\emptyset$.
Then, working in reverse time, starting at $\mu=(s,v)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(-\infty,0)}^{V'}(\mu)$:
\begin{itemize}
\item[]$(N+6,\psi(v))$, \quad $(N+5.5,\psi(v))$,
\item[]$(N+6,\psi^2(v))$, \quad $(N+5.5,\psi^2(v))$,
\item[]$\cdots\cdots\cdots\cdots$
\item[]$(N+6,\psi^m(v))$, \quad $(N+5.5,\psi^m(v))$,
\item[]$(N+4,\psi^m(v))$.
\end{itemize}
Then $\mu_*=(N+4,\psi^m(v))\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
{\it End of proof of Claim~4.}
{\it Claim 5:}
$[M'_0]\backslash[Z'_0]\subseteq P_*$
{\it Proof of Claim 5:}
Let $\mu\in M'_0$,
and assume that $\mu\notin Z'_0$.
We wish to show that $\mu\in P_*$.
Since $\mu\notin Z'_0=[M'_0]\backslash[\scrf(D)]$
and since $\mu\in M'_0$,
we get $\mu\in\scrf(D)$.
So fix $v\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ such that $(-N,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\mu)$.
Let $\mu_*:=(-N-1,v)$.
Because $\mu\in M'_0\subseteq S_N$,
it follows that $-N<\pi(\mu)$.
Then we have $\pi(\mu_*)=-N-1<(\pi(\mu))-0.5$,
so it remains to show that $\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
From the construction of $V'$,
we see that $(-N-1,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,v)$.
We have $(-N,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\mu)$, so, since $V'=V$ on $M'_0$,
we conclude that $(-N,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$, and, therefore,
that $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,v)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
Then $\mu_*=(-N-1,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,v)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu)$.
{\it End of proof of Claim~5.}
{\it Claim 6:} Let $\mu_0\in M'$.
Assume $\mu_0\notin Z_0^*\cup\cdots\cup Z_4^*$.
Then there exists $\mu_*\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$
such that $\pi(\mu_*)<(\pi(\mu_0))-0.5$.
{\it Proof of Claim 6:}
By definition of $P_*$, we wish to show that $\mu_0\in P_*$.
Let $P':=P_1\cup\cdots\cup P_4\cup M'_0$.
We have $\mu_0\notin Z_1^*$, $\mu_0\notin Z_2^*$,
$\mu_0\notin Z_0^*\cup Z_3^*$ and $\mu_0\notin Z_4^*$.
Also, $\mu_0\notin Z_0^*\supseteq Z'_0$.
Then, by Claims 1--5, we are done if $\mu_0\in P'$,
so we assume that $\mu_0\in(M')\backslash(P')$.
Let $P'':=P_1\cup\cdots\cup P_7\cup M'_0\cup M'_5$.
By construction of~$V'$, for all~$\mu=(s,v)\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
we have:
\begin{eqnarray*}
\mu\in(M')\backslash(P'')&\Rightarrow&(s-0.6,v)\,\in\,\Phi_{(-\infty,0)}^{V'}(\mu),\\
\mu\in P_5&\Rightarrow&(-N-9,v)\,\in\,\Phi_{(-\infty,0)}^{V'}(\mu),\\
\mu\in P_6&\Rightarrow&(N-0.5,v)\,\in\,\Phi_{(-\infty,0)}^{V'}(\mu),\\
\mu\in P_7&\Rightarrow&(N+7.5,v)\,\in\,\Phi_{(-\infty,0)}^{V'}(\mu) \qquad \hbox{and}\\
\mu\in M'_5&\Rightarrow&(-N-5,v)\,\in\,\Phi_{(-\infty,0)}^{V'}(\mu).
\end{eqnarray*}
Therefore $(M')\backslash(P'')$, $P_5$, $P_6$, $P_7$ and $M'_5$ are all subsets of $P_*$.
Then $\mu_0\in(M')\backslash(P')\subseteq[(M')\backslash(P'')]\cup P_5\cup P_6\cup P_7\cup M'_5\subseteq P_*$,
as desired.
{\it End of proof of Claim 6.}
{\it Claim 7:} Let $\mu_0\in M'$ and assume that $\mu_0\notin Z_0^*\cup\cdots\cup Z_4^*$.
Then there exists $\mu_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$
such that $\pi(\mu_1)=-N'$.
{\it Proof of Claim~7:}
By, if necessary, repeatedly applying Claim 6,
we arrive at~$\mu_+\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$
such that $\pi(\mu_+)\le-N'$.
Let $a:=-(\pi(\mu_+))-N'$.
Let $\mu_1:=\Phi_a^{V'}(\mu_+)$.
Then $\mu_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_+)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$,
and, since $V'=E$ on $(M'_3)^\circ$,
we conclude that $\pi(\mu_1)=(\pi(\mu_+))+a=-N'$,
as desired.
{\it End of proof of Claim 7.}
{\it Claim 8:} Let $v_1\in(B_0)\backslash(B_*)$ and $\mu_1:=(-N',v_1)$.
Assume that $\mu_1\notin Z_0^*\cup Z_5^*\cup Z_6^*$.
Then $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it Proof of Claim~8:}
We have $\mu_1\notin Z_0^*$, so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap Z'_0=\emptyset$,
so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap M'_0\subseteq\scrf(D)$.
Working in forward time, starting at $\mu_1=(-N',v_1)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(0,\infty)}^{V'}(\mu_1)$:
\begin{itemize}
\item[]$(-N-9,v_1)$, \qquad $(-N-5,v_1)$,
\item[]$(-N+0.5,v_1)$, \qquad $(N-0.5,v_1)$, \qquad $(N+1.5,v_1)$.
\end{itemize}
We have $\mu_1\notin Z_5^*$, and it follows that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap Z'_5=\emptyset$.
So, because $(N+1.5,v_1)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)$, we get $(N+1.5,v_1)\notin Z'_5$, so $v_1\notin Z_V$.
Fix an integer $m\ge1$
such that $v_1,\psi(v_1),\ldots,\psi^{m-1}(v_1)\in B_0$
and $\psi^m(v_1)\notin B_0$.
For all integers $j\in[1,m]$,
we have $\psi^j(v_1)\in\psi(B_0)=B_*$.
Then:
\begin{itemize}
\item$v_1\quad\in\quad(B_0)\backslash(B_*)$,
\item$\psi(v_1)\,\,,\,\,\ldots\,\,,\,\,\psi^{m-1}(v_1)\quad\in\quad B_0\cap B_*$ \qquad\qquad and
\item$\psi^m(v_1)\quad\in\quad(B_*)\backslash(B_0)$.
\end{itemize}
Recall that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap M'_0\subseteq\scrf(D)$.
Also, as $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap Z'_5=\emptyset$,
we see that $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap[\{N+1.5\}\times C_0]=\emptyset$.
Working in forward time, starting at $(N+1.5,v_1)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(0,\infty)}^{V'}(\mu_1)$:
\begin{itemize}
\item[]$(-N-6,\psi(v_1))$, \qquad $(N+1.5,\psi(v_1))$,
\item[]$(-N-6,\psi^2(v_1))$, \qquad $(N+1.5,\psi^2(v_1))$,
\item[]$\cdots\cdots\cdots\cdots$
\item[]$(-N-6,\psi^m(v_1))$, \qquad $(N+1.5,\psi^m(v_1))$,
\item[]$(N+3.5,\psi^m(v_1))$, \qquad $(N+5.5,\psi^m(v_1))$.
\end{itemize}
We have $\mu_1\notin Z_6^*$,
and so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap Z'_6=\emptyset$.
That is, we have $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap[\{N+5.5\}\times C_*]=\emptyset$.
Working in forward time, starting at $(N+5.5,\psi^m(v_1))$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(0,\infty)}^{V'}(\mu_1)$:
\begin{itemize}
\item[]$(N+6,\psi^m(v_1))$,
\item[]$(N+5.5,\psi^{m-1}(v_1))$, \qquad $(N+6,\psi^{m-1}(v_1))$,
\item[]$(N+5.5,\psi^{m-2}(v_1))$, \qquad $(N+6,\psi^{m-2}(v_1))$,
\item[]$\cdots\cdots\cdots\cdots$
\item[]$(N+5.5,v_1)$, \qquad $(N+6,v_1)$,
\item[]$(N+7.5,v_1)$, \qquad $(N+9,v_1)$, \qquad $(N',v_1)$.
\end{itemize}
In particular, $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it End of proof of Claim 8.}
{\it Claim 9:} Let $v_1\in B_*$ and let $\mu_1:=(-N',v_1)$.
Assume that $\mu_1\notin Z'_7$.
Then $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it Proof of Claim 9:}
As $(-N',v_1)=\mu_1\notin Z'_7$, we get $v_1\notin C_*$,
and so $\psi^{-1}(v_1)\notin\psi^{-1}(C_*)=C_0$.
Working in forward time, starting at $\mu_1=(-N',v_1)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(0,\infty)}^{V'}(\mu_1)$:
\begin{itemize}
\item[]$(-N-9,v_1)$, \qquad $(-N-7,v_1)$, \qquad $(N+3.5,\psi^{-1}(v_1))$,
\item[]$(N+7,v_1)$, \qquad $(N+9,v_1)$, \qquad $(N',v_1)$.
\end{itemize}
In particular, $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it End of proof of Claim 9.}
{\it Claim 10:} Let $v_1\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ and define $\mu_1:=(-N',v_1)$.
We assume that $\mu_1\notin Z_0^*\cup Z_5^*\cup Z_6^*\cup Z'_7$.
Then $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it Proof of Claim~10:}
By Claim~8, we are done if $v_1\in(B_0)\backslash(B_*)$.
By Claim 9, we are done if $v_1\in B_*$.
Then we assume $v_1\notin[(B_0)\backslash(B_*)]\cup B_*=B_0\cup B_*$.
We have $\mu_1\notin Z_0^*$, so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap Z'_0=\emptyset$,
so $[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)]\cap M'_0\subseteq\scrf(D)$.
Working in forward time, starting at $\mu_1=(-N',v_1)$,
we see, from the construction of $V'$, that all of the following
are elements of $\Phi_{(0,\infty)}^{V'}(\mu_1)$:
\begin{itemize}
\item[]$(-N-9,v_1)$, \qquad $(-N-5,v_1)$, \qquad $(-N+0.5,v_1)$,
\item[]$(N-0.5,v_1)$, \qquad $(N+1,v_1)$, \qquad $(N+9,v_1)$, \qquad $(N',v_1)$.
\end{itemize}
In particular, $(N',v_1)\in\Phi_{(0,\infty)}^{V'}(\mu_1)$.
{\it End of proof of Claim 10.}
{\it Claim 11:} $[M']\backslash[Z^*]\subseteq\scrf(D')$.
{\it Proof of Claim 11:}
Fix $\mu_0\in M'$.
Assume that $\mu_0\notin Z^*$.
We wish to prove that $\mu_0\in\scrf(D')$.
We have $\mu_0\notin Z^*\supseteq Z_0^*\cup\cdots\cup Z_4^*$.
By Claim 7, fix $\mu_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$
such that $\pi(\mu_1)=-N'$.
Let $v_1:=\Pi(\mu_1)$.
Then $(-N',v_1)=\mu_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$.
By definition of $\scrf(D')$,
it now suffices to prove that
$(N',v_1)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$.
The set $Z^*$ is $V'$-invariant.
So, since $\mu_0\notin Z^*$,
we get $\mu_1\notin Z^*$.
Then $\mu_1\notin Z^*\supseteq Z_0^*\cup Z_5^*\cup Z_6^*$
and $\mu_1\notin Z^*\supseteq Z_7^*\supseteq Z'_7$.
Then, by Claim 10, $(N',v_1)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)$.
So, since $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_1)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$,
we get $(N',v_1)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\mu_0)$, as desired.
{\it End of proof of Claim~11.}
{\it Claim 12:} $Z^*$ is $\tau'$-meager in~$M'$.
{\it Proof of Claim 12:}
Because $D$~is generically flat,
it follows that $M\backslash[\scrf(D)]$ is $\tau$-meager in $M$.
So, as $M'_0\subseteq M$,
we see that $Z'_0$ is $\tau$-meager in~$M$.
Then, by \cref{cor-unambig-nw-dense},
$Z'_0$ is $(\tau_\#)$-meager in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
So, because $Z'_0\subseteq M'$,
it follows, from \cref{cor-unambig-nw-dense},
that $Z'_0$ is $\tau'$-meager in~$M'$.
So, since the set $\Q$ of rational numbers is countable,
we see that $\Phi_\Q^{V'}(Z'_0)$ is $\tau'$-meager in $M'$.
Let $M''_0:=M\cap([-N+0.5,N-0.5]\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2)$ and
let $Z''_0:=(M''_0)\backslash(\scrf(D))$.
Because $V'=E$ on $L_N\cap S_N$,
we see that $Z'_0=\Phi_{(-0.5,0.5)}^{V'}(Z''_0)$.
Then, because $\Q+(-0.5,0.5)={\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}={\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}+(-0.5,0.5)$,
we conclude that $\Phi_\Q^{V'}(Z'_0)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(Z''_0)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(Z'_0)$.
Then $\Phi_\Q^{V'}(Z'_0)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(Z'_0)=Z_0^*$.
Then $Z_0^*$ is $\tau'$-meager in $M'$.
It remains to prove: $Z_1^*,\ldots,Z_7^*$ are all $\tau'$-meager in $M'$.
We will only handle $Z_1^*$; proofs for $Z_2^*,\ldots,Z_7^*$ are similar.
Let $Z_1^+:=\Phi_{(-0.5,0.5)}^{V'}(Z'_1)$.
Then $Z_1^+\subseteq(N+7,N+8)\times C_*$,
so $Z_1^+$ is $(\tau_\#)$-nowhere dense in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
So, by \lref{lem-unambig-nw-dense},
$Z_1^+$ is $\tau'$-nowhere dense in $M'$.
We have $\Q+(-0.5,0.5)={\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$,
so $\Phi_\Q^{V'}(Z_1^+)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(Z'_1)$.
That is, $\Phi_\Q^{V'}(Z_1^+)=Z_1^*$.
So, as $\Q$ is countable
and $Z_1^+$~is $\tau'$-nowhere dense in~$M'$,
we see that $Z_1^*$ is $\tau'$-meager in $M'$.
{\it End of proof of Claim~12.}
By Claim 11 and Claim 12,
$D'$ is generically flat, proving (b) of \lref{lem-iteration-lem}.
By \lref{lem-periodic-contained}, (d) follows from (c),
so it only remains to prove (c).
That is, it remains to show: $(V',\sigma_0)$ is periodic to order $k$.
Recall that $(-N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma_0)$.
Recall that $W$, $W_1$ and $W_0$ are open subsets of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$,
that $w_0\in W\subseteq W_1\subseteq W_0\subseteq{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$
and that $W\subseteq(B_0)\backslash(C_0)$.
Recall the $C^\omega$ functions
$\lambda_0:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\to(1,4)$ and $\lambda_1:W_1\to(0,\infty)$
and $\lambda_2,\lambda_3:W\to(0,\infty)$
and $\Lambda=\lambda_0+\lambda_1+\lambda_2+\lambda_3:W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$.
Recall that,
\begin{itemize}
\item[$(+)$]\qquad for all $w\in W_1$, \qquad $\pi(\,\Phi_{\lambda_1(w)}^V(-N,w)\,)\,\,=\,\,N$.
\end{itemize}
Recall that $T$ is an integer and that $T\ne0$.
Recall that $\Lambda$ agrees with the constant $T$ to order $k$ at~$w_0$.
In particular, we have $T=\Lambda(w_0)$.
{\it Claim 13:} For all $w\in W_1$,
we have $\Phi_{\lambda_1(w)}^{V}(-N,w)=(N,w)$.
{\it Proof of Claim 13:}
Let $U':=(-N,-N+1)\times W_1$ and $U'':=[\scrf(D)]\cap U'$.
Because $\tau|(M^\circ)=(\tau_\#)|(M^\circ)$,
because $U'\subseteq L_N\subseteq M^\circ$ and
because $U'$ is $(\tau_\#)$-open in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
it follows that $U'$~is $\tau$-open in $M$ and that $\tau|(U')=(\tau_\#)|(U')$.
Because $\scrf(D)$ is $\tau$-comeager in $M$,
it follows that $U''$~is $\tau$-comeager in $U'$.
Then $U''$ is $(\tau_\#)$-comeager in $U'$.
So, by the Baire Category Theorem, $U''$ is $(\tau_\#)$-dense in $U'$.
So, since $\Pi(U')=W_1$, we conclude that $\Pi(U'')$ is dense in~$W_1$.
Recall that
$$(t,w)\,\,\mapsto\,\,\Phi_t^V(-N,w)\quad:\quad J\,\times\,W_0\,\,\to\,\,M^\circ$$
is $C^\omega$.
Then $w\mapsto\Pi(\Phi_{\lambda_1(w)}^V(-N,w)):W_1\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ is $C^\omega$, hence $C^0$.
Then it suffices to prove,
for all $w\in\Pi(U'')$, that $\Phi_{\lambda_1(w)}^{V}(-N,w)=(N,w)$.
Fix $\sigma\in U''$ and
let $w:=\Pi(\sigma)$.
We wish to prove: $\Phi_{\lambda_1(w)}^{V}(-N,w)=(N,w)$.
Let $x:=\pi(\sigma)$.
We have $(x,w)=\sigma\in U''\subseteq U'\subseteq L_N$.
So, since $V=E$ on~$L_N$,
it follows that $(-N,w)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(x,w)$.
Then we have $(-N,w)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(x,w)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$,
and so $\Phi_{\lambda_1(w)}^V(-N,w)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$.
Also, $w=\Pi(\sigma)\in\Pi(U')=W_1$,
so, by $(+)$, we get $\pi(\Phi_{\lambda_1(w)}^V(-N,w))=N$,
{\it i.e.}, $\Phi_{\lambda_1(w)}^V(-N,w)\in\{N\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$.
So, as $\Phi_{\lambda_1(w)}^V(-N,w)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$, we get
\begin{itemize}
\item[$(*)$] \qquad $\Phi_{\lambda_1(w)}^V(-N,w)\quad\in\quad[\,\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)\,]\,\,\cap\,\,[\,\{N\}\,\times\,{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2\,]$.
\end{itemize}
As $\sigma\in U''\subseteq\scrf(D)$,
fix $v\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$ such that $(-N,v),(N,v)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)$.
Then $(-N,w),(-N,v)\in[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)]\cap[\{-N\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2]$.
So, by \lref{lem-one-start-end}(i), $w=v$.
Then $(N,w)=(N,v)\in[\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma)]\cap[\{N\}\times{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2]$.
So, by $(*)$ and \lref{lem-one-start-end}(ii),
$\Phi_{\lambda_1(w)}^V(-N,w)=(N,w)$.
{\it End of proof of Claim 13.}
{\it Claim 14:} Let $w\in W$.
Then $\Phi_{\Lambda(w)}^{V'}(-N-6,w)=(-N-6,\psi(w))$.
{\it Proof of Claim 14:}
By the definitions of $\lambda_2$ and $\lambda_0$ and $V'$, we have:
\begin{itemize}
\item[(i)] \qquad $\Phi_{\lambda_2(w)}^{V'}\,(\,-N-6\,,\,w\,)\quad=\quad(\,-N-5\,,\,w\,)$ \qquad\qquad and
\item[(ii)] \qquad $\Phi_{\lambda_0(w)}^{V'}\,(\,-N-5\,,\,w\,)\quad=\quad(\,-N\,,w\,)$.
\end{itemize}
Since $w\in W\subseteq W_1$, by Claim 13 and the definition of $V'$, we have:
\begin{itemize}
\item[(iii)] \qquad $\Phi_{\lambda_1(w)}^{V'}\,(\,-N\,,w\,)\quad=\quad(\,N+1\,,\,w\,)$.
\end{itemize}
Since $w\in W\subseteq(B_0)\backslash(C_0)$,
by definitions of $\lambda_3$ and $V'$, we have:
\begin{itemize}
\item[(iv)] \qquad $\Phi_{\lambda_3(w)}^{V'}\,(\,N+1\,,\,w\,)\quad=\quad(\,-N-6\,,\,\psi(w)\,)$.
\end{itemize}
Since $\Lambda(w)=\lambda_3(w)+\lambda_1(w)+\lambda_0(w)+\lambda_2(w)$,
the result follows from (i), (ii), (iii) and (iv).
{\it End of proof of Claim~14.}
Let $U_0:=(0,1)\times W$ and $U_1^*:=(-N-6,-N-5)\times W$.
Then $U_0$~is $(\tau_\#)$-open in~${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$
and $U_1^*$ is open in $(M')^\circ$.
We give $U_0$ the topology $(\tau_\#)|(U_0)$
and we give $U_1^*$ the topology $(\tau_\#)|(U_1^*)=(\tau')|(U_1^*)$.
We give $U_0$ the maximal $C^\omega$ atlas $(\scra_\#)|(U_0)$
and we give $U_1^*$ the maximal $C^\omega$~atlas $(\scra_\#)|(U_1^*)=(\scra')|(U_1^*)$.
We define a map $\Gamma:U_0\to U_1^*$ by
$\Gamma(t,w)=\Phi_t^{V'}(-N-6,w)$.
Let $U_1:=\Gamma(U_0)$.
Then, by the Inverse Function Theorem,
the set $U_1$ is open in $U_1^*$.
We give $U_1$ the topology $(\tau_\#)|(U_1)=(\tau')|(U_1)$
and the maximal $C^\omega$ atlas $(\scra_\#)|(U_1)=(\scra')|(U_1)$.
By the Inverse Function Theorem,
$\Gamma:U_0\to U_1$ is a $C^\omega$ diffeomorphism.
Let $\xi_0:=(0.5,w_0)\in U_0$ and let $\xi_1:=\Gamma(\xi_0)\in U_1$.
By definition of~$\Gamma$, we have $\xi_1=\Phi_{0.5}^{V'}(-N-6,w_0)$.
By construction of $V'$, we see that $(-N-6,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,w_0)$.
Therefore $\xi_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,w_0)$.
Because $(-N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^V(\sigma_0)$, and because $V'=V$ on~$M'_0$,
it follows that $(-N,w_0)\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\sigma_0)$.
Then $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,w_0)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\sigma_0)$.
Then $\xi_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N,w_0)=\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(\sigma_0)$,
so it suffices to show that
$(V',\xi_1)$ is periodic to order $k$.
Since $T$~is an integer and $T\ne0$,
it suffices to show that $\Phi_T^{V'}$~agrees with $\hbox{Id}_M$
to order $k$ at $\xi_1$.
By Claim 14, since $T=\Lambda(w_0)$ and since $\psi(w_0)=w_0$,
we conclude that $\Phi_T^{V'}(-N-6,w_0)=(-N-6,w_0)$.
So, because $\xi_1\in\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V'}(-N-6,w_0)$,
we get $\Phi_T^{V'}(\xi_1)=\xi_1$.
Let $U'_1$ be an open neighborhood in $U_1$ of $\xi_1$
such that $\Phi_T^{V'}(U'_1)\subseteq U_1$.
Give $U'_1$ the topology $(\tau_\#)|(U'_1)=(\tau')|(U'_1)$
and the maximal $C^\omega$ atlas $(\scra_\#)|(U'_1)=(\scra')|(U'_1)$.
Let $\chi_1:=\Phi_T^{V'}|U'_1:U'_1\to U_1$ be
the restriction of $\Phi_T^{V'}$ to $U'_1$.
Let $\iota_1:U'_1\to U_1$ be the inclusion.
We wish to show that $\chi_1$ agrees with $\iota_1$
to order $k$ at $\xi_1$.
Let $U'_0:=\Gamma^{-1}(U'_1)$.
Then $U'_0$ is open in $U_0$.
Give $U'_0$ the topology $(\tau_\#)|(U'_0)$
and the maximal $C^\omega$ atlas $(\scra_\#)|(U'_0)$.
We have $\xi_0\in U'_0$.
Let $\Gamma':=\Gamma|U'_0:U'_0\to U'_1$ be
the restriction of $\Gamma$ to $U'_0$.
Then $\Gamma':U'_0\to U'_1$ is a $C^\omega$ diffeomorphism.
Let $\chi_0:=[\Gamma^{-1}]\circ\chi_1\circ\Gamma':U'_0\to U_0$.
Define $F:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$ by
$$F(t,w)\quad=\quad(\,\,T\,-\,[\Lambda(w)]\,+\,t\,\,,\,\,\psi(w)\,\,).$$
Recall that $\xi_0=(0.5,w_0)$.
Recall that $\Lambda$ agrees with the constant~$T$ to order $k$ at~$w_0$
and that $\psi$ agrees with $\hbox{Id}_{{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2}$ to order $2k$ at $w_0$.
Then, using $\tau_\#|({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W)$ and $\scra_\#|({\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W)$ on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W$,
and using $\tau_\#$ and $\scra_\#$ on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
it follows that $F:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$~agrees with the inclusion ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$
to order $k$ at $\xi_0$.
In particular, $F(\xi_0)=\xi_0$.
Let $U''_0:=[U'_0]\cap[F^{-1}(U_0)]$.
The $U''_0$ is an open neighborhood in $U'_0$ of $\xi_0$
and $F(U''_0)\subseteq U_0$.
{\it Claim 15:} $F|U''_0=\chi_0|U''_0$.
{\it Proof of Claim 15:}
Let $t\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}$ and $w\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^2$
and assume that $(t,w)\in U''_0$.
We wish to prove that $F(t,w)=\chi_0(t,w)$.
Since $(t,w)\in U''_0\subseteq U'_0$,
it follows that $\Gamma'(t,w)\in U'_1$.
Also, since $(t,w)\in U''_0\subseteq F^{-1}(U_0)$,
it follows that $F(t,w)\in U_0$.
From the definitions of $\Gamma'$ and $\Gamma$, we conclude that
$\Gamma'(t,w)=\Gamma(t,w)=\Phi_t^{V'}(-N-6,w)$.
Because $(t,w)\in U'_0\subseteq U_0=(0,1)\times W$, we get $w\in W$.
So, by Claim~14,
$$\Phi_{\Lambda(w)}^{V'}\,(\,-N-6\,,\,w\,)\quad=\quad(\,-N-6\,,\,\psi(w)\,).$$
We apply $\Phi_t^{V'}$ to this last equation;
since $[\Phi_t^{V'}]\circ[\Phi_{\Lambda(w)}^{V'}]=[\Phi_{\Lambda(w)}^{V'}]\circ[\Phi_t^{V'}]$
and since $\Phi_t^{V'}(-N-6,w)=\Gamma'(t,w)$, we get
$$\Phi_{\Lambda(w)}^{V'}\,(\,\Gamma'(t,w)\,)\quad=\quad\Phi_t^{V'}(\,-N-6\,,\,\psi(w)\,).$$
Applying $\Phi_{T-[\Lambda(w)]}^{V'}$ to this last equation yields
$$\Phi_T^{V'}\,(\,\Gamma'(t,w)\,)\quad=\quad\Phi_{T\,-\,[\Lambda(w)]\,+\,t}^{V'}\,(\,-N-6\,,\,\psi(w)\,),$$
which, by definition of $\chi_1$ and $\Gamma$ and $F$, gives
$\chi_1(\Gamma'(t,w))=\Gamma(F(t,w))$.
Then $\chi_0(t,w)=([\Gamma^{-1}]\circ\chi_1\circ\Gamma')(t,w)=F(t,w)$, as desired.
{\it End of proof of Claim~15.}
Let $\iota_0:U'_0\to U_0$ be the inclusion.
As $F:{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$~agrees
with the inclusion ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}\times W\to{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$
to order $k$ at $\xi_0$,
it follows, from Claim~15,
that $\chi_0$~agrees
with $\iota_0$
to order $k$ at $\xi_0$.
So, since $\chi_1=\Gamma\circ\chi_0\circ[(\Gamma')^{-1}]$
and since $\iota_1=\Gamma\circ\iota_0\circ[(\Gamma')^{-1}]$
and since $\Gamma'(\xi_0)=\Gamma(\xi_0)=\xi_1$,
it follows that $\chi_1$ agrees with~$\iota_1$
to order $k$ at $\xi_1$, as desired.
\end{proof}
\section{The counterexample\wrlab{sect-counterex}}
\begin{thm}\wrlab{thm-ctrx}
There exists a $C^\omega$ manifold $M$
and a complete $C^\omega$ vector field $V$ on $M$
and a sequence $\sigma_1,\sigma_2,\ldots$ in $M$
such that $\{\sigma_1,\sigma_2,\ldots\}$ is dense in $M$ and
such that,
\begin{itemize}
\item[]for all integers $k\ge1$, \qquad
$(V,\sigma_k)$ is periodic to order $k$.
\end{itemize}
\end{thm}
\begin{proof}
Let $\{\omega_1,\omega_2,\ldots\}$ be a countable $(\tau_\#)$-dense subset of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
We denote the standard norm on ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$ by $|\,\bullet\,|$.
Let $N_0:=1$.
Let $M_0:={\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Let $\tau_0:=\tau_\#$.
Let $\scra_0:=\scra_\#$.
Let $V_0:=E$.
Let $D_0:=(N_0,M_0,\tau_0,\scra_0,V_0)$.
Then $D_0$ is a generically flat displayed system.
Let $J_0:={\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}$.
Let $Q_0:=M_0\cap S_{N_0}$.
Then $Q_0=S_{N_0}=S_1$, so $Q_0^\circ=S_1$, and so we have $Q_0^\circ\ne\emptyset$.
Let $j_1:=\min\{j\in J_0\,|\,\omega_j\in Q_0^\circ\}$.
By \lref{lem-flatpt-density}, fix $\sigma_1\in[\scrf(D_0)]\cap Q_0^\circ$ such that
$|\sigma_1-\omega_{j_1}|<1$.
By \lref{lem-iteration-lem}, fix
\begin{itemize}
\item[]a generically flat displayed system $D_1:=(N_1, M_1,\tau_1,\scra_1,V_1)$
\end{itemize}
such that
\begin{itemize}
\item$D_1$ is an extension of $D_0$,
\item$(V_1,\sigma_1)$ is periodic to order $1$ \qquad and
\item$\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V_1}(\sigma_1)\quad\subseteq\quad S_{N_1}$.
\end{itemize}
Let $J_1:={\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}\backslash\{j_1\}$.
Let $Q_1:=M_1\cap S_{N_1}$.
By definition of extension, we have
$M_0\cap S_{N_0}=M_1\cap S_{N_0}$ and $N_0<N_1$.
Then
$$Q_0\quad=\quad M_0\cap S_{N_0}\quad=\quad M_1\cap S_{N_0}\quad\subseteq\quad M_1\cap S_{N_1}\quad=\quad Q_1,$$
so $Q_0^\circ\subseteq Q_1^\circ$.
So, since $Q_0^\circ\ne\emptyset$,
we conclude that $Q_1^\circ\ne\emptyset$.
Let $j_2:=\min\{j\in J_1\,|\,\omega_j\in Q_1^\circ\}$.
By \lref{lem-flatpt-density}, fix $\sigma_2\in[\scrf(D_1)]\cap Q_1^\circ$ such that
$|\sigma_2-\omega_{j_2}|<1/2$.
By \lref{lem-iteration-lem}, fix
\begin{itemize}
\item[]a generically flat displayed system $D_2=(N_2,M_2,\tau_2,\scra_2,V_2)$
\end{itemize}
such that
\begin{itemize}
\item$D_2$ is an extension of $D_1$,
\item$(V_2,\sigma_2)$ is periodic to order $2$ \qquad and
\item$\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V_2}(\sigma_2)\quad\subseteq\quad S_{N_2}$.
\end{itemize}
Let $J_2:={\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}\backslash\{j_1,j_2\}$.
Let $Q_2:=M_2\cap S_{N_2}$.
By definition of extension, we have
$M_1\cap S_{N_1}=M_2\cap S_{N_1}$ and $N_1<N_2$.
Then
$$Q_1\quad=\quad M_1\cap S_{N_1}\quad=\quad M_2\cap S_{N_1}\quad\subseteq\quad M_2\cap S_{N_2}\quad=\quad Q_2,$$
so $Q_1^\circ\subseteq Q_2^\circ$.
So, since $Q_1^\circ\ne\emptyset$,
we conclude that $Q_2^\circ\ne\emptyset$.
Let $j_3:=\min\{j\in J_2\,|\,\omega_j\in Q_2^\circ\}$.
By \lref{lem-flatpt-density}, fix $\sigma_3\in[\scrf(D_2)]\cap Q_2^\circ$ such that
$|\sigma_3-\omega_{j_3}|<1/3$.
By \lref{lem-iteration-lem}, fix
\begin{itemize}
\item[]a generically flat displayed system $D_3=(N_3,M_3,\tau_3,\scra_3,V_3)$
\end{itemize}
such that
\begin{itemize}
\item$D_3$ is an extension of $D_2$,
\item$(V_3,\sigma_3)$ is periodic to order $3$ \qquad and
\item$\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V_3}(\sigma_3)\quad\subseteq\quad S_{N_3}$.
\end{itemize}
Continue {\it ad infinitum}, obtaining a sequence $D_0,D_1,D_2,\ldots$
of displayed systems and a sequence $\sigma_1,\sigma_2,\ldots$ in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$.
Also constructed are subsets $Q_0\subseteq Q_1\subseteq Q_2\subseteq\cdots$ of ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$
and subsets $J_0\supseteq J_1\supseteq J_2\supseteq\cdots$ of ${\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}$.
Also constructed is a sequence $j_1,j_2,\ldots$ of distinct positive integers.
Let $M:=Q_0\cup Q_1\cup Q_2\cup\cdots$.
For all integers $k\ge0$, $D_k$ is a displayed system,
and so $Q_k$ is $\tau_k$-open in $M_k$.
Also, for all integers $k\ge0$, for all integers $k'\ge k+1$,
we know that $D_{k'}$ is an extension of $D_k$,
and it follows that
$Q_k$ is $\tau_{k'}$-open in $M_{k'}$, and, moreover, that
$$\tau_{k'}|Q_k=\tau_k|Q_k,\qquad\scra_{k'}|Q_k=\scra_k|Q_k\qquad\hbox{and}\qquad V_{k'}|Q_k=V_k|Q_k.$$
Let $\tau$ be the unique topology on $M$ such that,
for all integers $k\ge0$, we have $\tau|Q_k=\tau_k|Q_k$.
Let $\scra$ be the unique maximal $C^\omega$ atlas on $(M,\tau)$ such that,
for all integers $k\ge0$, we have $\scra|Q_k=\scra_k|Q_k$.
Give $M$ the topology $\tau$ and the
maximal $C^\omega$ atlas $\scra$.
Let $V$ be the unique $C^\omega$ vector field on $M$ such that,
for all integers $k\ge0$, we have $V|Q_k=V_k|Q_k$.
For all integers $k\ge0$,
$Q_k\in\tau_k|Q_k=\tau|Q_k\subseteq\tau$,
so $Q_k$ is open in~$M$.
For all integers $k\ge1$, $(V_k,\sigma_k)$ is periodic to order~$k$;
so, as $\Phi_{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^{V_k}(\sigma_k)\subseteq M_k\cap S_{N_k}=Q_k$
and as $V=V_k$ on~$Q_k$,
we see that $(V,\sigma_k)$~is periodic to order $k$.
Let $\Sigma:=\{\sigma_1,\sigma_2,\ldots\}$.
It remains to show: $\Sigma$ is dense in~$M$.
Since $M=Q_1\cup Q_2\cup\cdots$,
it suffices to show, for all integers $k\ge1$,
that $\Sigma\cap Q_k$ is $\tau$-dense in $Q_k$.
Fix an integer $k\ge1$.
Since $\tau_k|Q_k=\tau|Q_k$,
we wish to prove that
$\Sigma\cap Q_k$ is $\tau_k$-dense in $Q_k$.
Let $\Sigma':=\Sigma\cap Q_k^\circ$.
By \lref{lem-Q-facts}(ii),
$Q_k^\circ$ is $\tau_k$-dense in $Q_k$,
and it therefore suffices to prove that $\Sigma'$ is $\tau_k$-dense in $Q_k^\circ$.
By \lref{lem-Q-facts}(iv), $\tau_k|(Q_k^\circ)=(\tau_\#)|(Q_k^\circ)$,
so it suffices to show: $\Sigma'$ is $(\tau_\#)$-dense in $Q_k^\circ$.
Fix a nonempty $(\tau_\#)$-open subset $U$ of $Q_k^\circ$.
We wish to prove that $\Sigma'\cap U\ne\emptyset$.
Fix a nonempty $(\tau_\#)$-open subset $U_0$ of $U$
and fix $\varepsilon>0$ such that,
\begin{itemize}
\item[$(*)$]for all $\omega\in U_0$,
for all $\sigma\in{\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$, \qquad
if $|\sigma-\omega|<\varepsilon$, \,\, then $\sigma\in U$.
\end{itemize}
Fix an integer $m_0\ge k$ such that $1/m_0<\varepsilon$.
Let $j':=\max\{j_1,\ldots,j_{m_0}\}$.
By density of $\{\omega_{j'+1},\omega_{j'+2},\ldots\}$ in ${\mathbb R}} \def\C{{\mathbb C}} \def\T{{\mathbb T}^3$,
fix an integer $j_*\ge j'+1$
such that $\omega_{j_*}\in U_0$.
Since $j_*>j'$, we have $j_*\notin\{j_1,\ldots,j_{m_0}\}$.
Let $n:=\min\{m\in{\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}\,|\,(m\ge m_0+1)\hbox{ and }(j_m\ge j_*)\}$.
Then we have $n\ge m_0+1$ and $j_n\ge j_*$.
Also, by minimality of $n$, we conclude that $j_*\notin\{j_{m_0+1},\ldots,j_{n-1}\}$.
So, since $j_*\notin\{j_1,\ldots,j_{m_0}\}$,
it follows that $j_*\in{\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}\backslash\{j_1,\ldots,j_{n-1}\}$.
That is, $j_*\in J_{n-1}$.
Since $k\le m_0\le n-1$, we have $Q_k\subseteq Q_{n-1}$,
so $Q_k^\circ\subseteq Q_{n-1}^\circ$.
Then
$$\omega_{j_*}\quad\in\quad U_0\quad\subseteq\quad
U\quad\subseteq\quad Q_k^\circ\quad\subseteq\quad Q_{n-1}^\circ.$$
Let $J':=\{j\in J_{n-1}\,|\,\omega_j\in Q_{n-1}^\circ\}$.
By definition of $j_n$, $j_n=\min J'$.
So, since $j_*\in J'$, we get $j_n\le j_*$.
So, as $j_n\ge j_*$, we get $j_n=j_*$.
Then $\omega_{j_n}=\omega_{j_*}\in U_0$.
So, as
$|\sigma_n-\omega_{j_n}|<1/n<1/m_0<\varepsilon$,
we conclude, from~$(*)$, that $\sigma_n\in U$.
So, as $U\subseteq Q_k^\circ$, we get $\sigma_n\in Q_k^\circ$.
So, since $\sigma_n\in\Sigma$, we get $\sigma_n\in\Sigma\cap Q_k^\circ=\Sigma'$.
Then $\sigma_n\in\Sigma'\cap U$,
and so $\Sigma'\cap U\ne\emptyset$.
\end{proof}
|
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| 4,977
|
{"url":"https:\/\/together.jolla.com\/question\/58315\/wiki-how-to-transfer-files-between-jolla-and-mac-os-x-over-the-network-lan\/","text":"# [Wiki] How to transfer files between Jolla and Mac OS X over the network (LAN)\n\nThis post is a wiki. Anyone with karma >75 is welcome to improve it.\n\nHow to transfer files between Jolla and Mac OS X over the network (LAN)\n\nThis method will allow your Mac to mount the Jolla's filesystem as a network volume.\n\nFor this to work, you need to enable developer mode on the Jolla and install sshfs on the Mac.\n\nWARNING: You can cause serious damage to the phone if you accidentally delete or modify a critical system file. I can't emphasize this enough.\n\nOn the phone:\n\n1. Developer mode must be enabled on the phone: Settings->System->Developer mode\n\n2. Allow remote connection as well (same menu in settings as (1) above)\n\n4. Note down your IP address which should be visible here under Networking. You will need this IP address later on the Mac to establish the connection.\n\nOn the Mac:\n\n1. Install MacPorts from here:\n\nhttps:\/\/www.macports.org\/install.php\n\nMacports has some prerequisites to install; follow their detailed instructions. Note that they may differ slightly depending on the OS X version that you are running. You should also keep macports up-to-date using their instructions\n\n2. Start the OS X Terminal.app located in \/Applications\/Utility\/Terminal.app\n\n3. In the terminal, execute the following command:\n\n$sudo port install sshfs If all goes well, you will now be able to mount the Jolla's file system using its IP address which you noted down earlier (say 192.168.0.18). 4. Create the mountpoint on OS X:$ mkdir \/Volumes\/jolla\n\n5. Now to connect to Jolla. sshfs will ask for the password you entered into the phone. The example below mounts the entire filesystem from root. Be careful, you can do some serious damage if you're careless.\n\n$sshfs nemo@192.168.0.18:\/ \/Volumes\/jolla On the Finder desktop, you should now see a network drive which is your phone's filesystem. Once again, if you delete\/modify a critical file, you can break your phone. I am NOT responsible. To mount just the home directory, which is what you would usually need: $ mkdir \/Volumes\/nemo\n\\$ sshfs nemo@192.168.0.18:\/home\/nemo \/Volumes\/nemo\n\n\nTODO\n\n1. Since OS X deletes the mount point directory when you unmount the network drive, you will have to create it every time you connect to the Jolla. There should be an easier way.\nedit retag close delete\n\nHi @SanjayMehta , I would be very pleased to have your detailed instructions. As didactic as possible,not everyone is familiar with sshfs (me for example :-p). That will also help others I believe. Thank you in advance!\n\n( 2014-10-09 12:55:02 +0300 )edit\n\n@damourti: done. Let me know if anything's unclear.\n\n( 2014-10-09 13:39:59 +0300 )edit\n1\n\nGreat post, but suggesting the following for clearance:\n\n\u2022 Option 1: Make this post a wiki\n\u2022 Option 2: Form your question as a simple question (How can I transfer files between Jolla and Mac OS) and offer this Finder solution as one answer to your own question.\n( 2014-10-09 17:52:45 +0300 )edit\n\nCleaned up and converted to wiki. I need some help with the formatting - the list numbering keeps resetting back to 1.\n\n( 2014-10-10 05:49:52 +0300 )edit\n\nGreat! I've edited the numbering, is it ok now? Count starts from the beginning if there's a line starting from very beginning in between, but using space in front of the line moves it along to the numbered list.\n\n( 2014-10-10 09:43:28 +0300 )edit\n\nSort by \u00bb oldest newest most voted\n\nFor those less familiar with terminals, MacFusion+osxfuse is a pretty nice GUI on top of sshfs for OS X.\n\nmore\n\nThis would be a better and easier solution than mine. Unfortunately I can't get it to work. I get authentication failures when I try to access the Jolla. I can ssh though.\n\nIs there some particular flow to follow while trying this out? I installed osxfuse, sshfs and then macfuse in that order.\n\nThis was on a different Mac from the one which has MacPorts, so no chance of a conflict.\n\n( 2014-10-10 08:13:16 +0300 )edit\n\nDoes the osxfuse version of sshfs work through a terminal for you? Also, have you looked at: https:\/\/github.com\/osxfuse\/osxfuse\/wiki\/SSHFS ?\n\n( 2014-10-10 11:01:34 +0300 )edit\n1\n\n@hobarrera: it took a while, but I finally got around to testing your suggestion. Yes, osxfuse's version of sshfs works from the terminal, but not from the GUI. Will update the wiki to suit. Will do that after I try re-installing macfuse etc. I suspect a conflict with an obsolete version of the original macfuse when it was maintained by google.\n\n( 2014-10-14 08:42:34 +0300 )edit\n1\n\nHave a look at this wiki https:\/\/github.com\/osxfuse\/osxfuse\/wiki\/SSHFS#macfusion It tells you how to replace the sshfs binary from macfusion with a more uptodate version of sshfs that is provided by the osxfuse guys.\n\nSome hints on the configuration and usage of Macfusion: You need to set the home dir, try \/home\/nemo. If you do not set it, you will get auth errors.\n\nSometimes, it simply fails to auth and mount. In these cases, I wasn't able to ssh manually from a terminal either. So it seems that sshd on the Jolla becomes unreachable.\n\nNota bene: Starting with OSX 10.7\/Lion, you need to use osxfuse instead of macfuse. Check the \"use macfuse compatibilty layer\".\n\n( 2014-10-23 23:00:54 +0300 )edit","date":"2019-05-26 20:13:59","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\": 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\": 0.19915428757667542, \"perplexity\": 4093.9308358791736}, \"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-2019-22\/segments\/1558232259452.84\/warc\/CC-MAIN-20190526185417-20190526211417-00438.warc.gz\"}"}
| null | null |
Q: Program Termination error while printing the array #include <stdio.h>
int main() {
int n, i;
char arr[20];
clrscr();
printf("Enter size of array(<=20)");
scanf("%d", &n);
printf("Enter array");
for (i = 0; i < n; i++) {
scanf("%s", &arr[i]);
}
for (i = 0; i < n; i++) {
printf("%s", arr[i]);
}
getch();
return 0;
}
The program does not prints the array
and instead shows
Program termination message
The image shows the program termiantion message
A: The problem is with the line
printf("%s", arr[i]);.
If you change this line to
printf("%c", arr[i]);
then it will work because %s is used with character arrays that contain strings
I am just giving the solution for your program termination. Still we can do some modification in your code.
Thanks
|
{
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| 4,181
|
Top 10 Gadgets To Satiate Your Gaming Hunger!
Helena, January 25, 2020
Gamester Direct is a long-standing store that was founded over a decade ago by Dream Games Sdn. And, by the looks of things, it is supposed to play Gameboy games. Everything from the games themselves to the way the controllers feel in your hands are exactly as you would remember from the 1990s. These gadgets come embedded with various attractive and exciting games which are basically very interactive as they enthral the gamers throughout the gaming process.
This device is specially designed for gamers who love to have not only a great visual experience but also an amazing audio experience. Amazon offers digital codes for Xbox , PlayStation, and the Nintendo Switch. Elsewhere, Rocket league recently launched to rave reviews – playing just as well as on other consoles.
If you were born in the late 70s, chances are you've played one of the most iconic gaming console systems in the period. The nifty charging dock can handle up to 4 Switch Joy-Con controllers at the same time complete with individual LED charging indicator lights so you'll know which of the 4 controllers are already fully charged.
Other PC gadgets include surround sound speakers, flight joysticks for flying games and even USB gadgets such as external storage devices and hubs to conveniently connect other devices to the personal computer or laptop. Some of these can be gaming controllers for consoles to help with furious game play.
The gaming industry has brought a sense of familiarity and broadened the role of technology in our lives. Online gaming Popular online games such as Dota 2 and World of Warcraft is regularly played by a huge audience. The PS4, Xbox One and Nintendo Switch have all amassed stellar game libraries, and with no shortage of hot new controllers, PCs and peripherals on the market, there are plenty of exciting ways to boost your existing setup.
The Intercept is especially designed for very active computer gamers, allowing them to sleep better as this gaming glasses can help reduce eye strain. This nostalgia-inducing, Game Boy-inspired case actually has functional buttons and a screen on it for playing video games.
For those who already have the Nintendo Switch, you might want to get this charging dock especially designed for the Nintendo Switch Joy-Con controllers. Its ergonomic design makes the gaming experience more enjoyable, and the keyboard is backlit. If you are looking for amazing gifts for gamers, then this product is one you should consider.
The Best Gaming Gadgets
An enthusiast gamer always loves to buy gaming accessories, gadgets and also they tend to buy a gaming PC If you are a newbie gamer and looking for the suggestion which gadgets should you buy then this is the right place for you. These game consoles are expensive investments, therefore proper care and maintenance is required to ensure it is able to function well for years. Keyboards give you more options when playing a game, and lets you control the game with more depth. That being said, this gadget can take your gaming desk to the next level and enhance your favorite games in ways you might not have considered.
The graphics, effects, and sounds are made so realistic and intense which add to the unique experience in gaming by the players. Connect the keyboard to the back of the Xbox one wireless controller, and then insert the 2.4G receiver to any USB interface of the Xbox one console , then you can start the text message and email in the games to perform text input function.
If you or your loved one haven't jumped on the PS4 bandwagon yet, this is the perfect way to do so. The Red Dead Redemption 2 PS4 Pro bundle combines Sony's premium 4K console with Rockstar's wildly anticipated open-world epic, giving you endless hours of gameplay.
I create videos about games, movies and nerd culture. Instead of relying on just a single monitor, gamers can consider opting for a dual-screen setup Dual screen monitors are highly favored by artists, graphic designers, video editors, and even writers. Some are used as accessories for mobile phones, entertainment systems, and computers while some have been crafted especially for gamers of different PC games.
This enables users to charge while playing games and reduces the chances of damaging the charger port. The racing synthetic leather chair is suitable for both gamers and those who use PC for other works. Here, in no particular order, are seven great gadgets and accessories that will take your gaming experience up a level.
Top 15 Cheap Gaming Gadgets 2018
Gamester Direct is a long-standing store that was founded over a decade ago by Dream Games Sdn. Spider-Man will appeal to gamers of all ages. The gaming gadgets usually offer interactive games which are accompanied with various attractive illustrations and animations. Some gamers would prefer to have a speaker set with booming bass, while others may prefer clear sound above all else.
This is for the PlayStation 3 system provides the most sensitive game play experience with pressure sensors in each achievement button and the insertion of the highly sensitive SIXAXIS motion sensing technology. Best Geek Gamers Gadgets. For instance, some of the games are only playable through download from the PC, and if the port is damaged, it could be difficult to transfer files into it.
He started his career as a blogger, reviewing games, gaming consoles and usually works remote while exploring the different cultures around the world.
January 25, 2020 Helena programming languages gadgets, gaming, hunger, satiate
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 563
|
Bibliotheca Buddhica (Библиотека Буддика) — многотомная серия переводных и оригинальных буддийских текстов, основанная в 1897 году академиком С. Ф. Ольденбургом в Санкт-Петербурге при Российской академии наук. С момента основания в её составе вышло в свет 40 томов в более, чем ста выпусках. Вплоть до настоящего времени остаётся одним из самых авторитетных изданий среди буддологов мира.
История
Проект многотомной серии был разработан академиком С. Ф. Ольденбургом и принят Академией наук. Проектом предполагалось научное издание текстов на санскрите, китайском, тибетском и монгольском языках; с самого начала он планировался как международный. В его осуществлении приняли участие крупнейшие русские ориенталисты Ф. И. Щербатской, В. В. Радлов, , С. Е. Малов, О. О. Розенберг, А. А. Сталь-Гольштейн, Е. Е. Обермиллер и другие. Вместе с ними над проектом работали выдающиеся западные учёные , , , Л. де ла Вале-Пуссен, Х. Керн и А. Грюнведель, а также такие крупнейшие буддологи Японии как Нандзё Бунъю и Огихара Унрай.
Издание «Bibliotheca Buddhica» началось с публикации памятника буддийской литературы «Cikshasamuccaya» (), которую по просьбе Ольденбурга подготовил английский востоковед Сесил Бендалл. В письме Ольденбургу Бендалл писал: «Мне остаётся лишь выразить чувство благодарности выдающейся Академии, так много сделавшей для исследования Востока, за честь, которую она мне оказала, избрав меня издателем первого тома». Впоследствии ряд томов серии подготовил сам Ольденбург.
Первый выпуск серии сразу же получил высокую оценку научного сообщества. Так, положительные отзывы уже в 1898 году были опубликованы в журналах «», «Journal des savants» и «», а Парижский конгресс ориенталистов выразил благодарность Академии наук за полезное издание.
С 1897 по 1937 вышло в общей сложности тридцать томов серии в более, чем 100 выпусках. Редактором издания был С. Ф. Ольденбург, о стиле работы которого в этом качестве писал: «ни один лист не вышел без его внимательной и иногда даже придирчивой корректуры».
В связи с репрессиями среди советских востоковедов издание книг серии было приостановлено, хотя к моменту приостановки издания к печати было подготовлено (в основном тибетологом А. И. Востриковым) ещё 14 томов. Последним, выпущенным перед прекращением издания серии, стал XXX том, в котором в 1936 году Ф. И. Щербатской опубликовал работу «Мадхьянта-вибханга».
Издание «Мадхьянта-вибханга» стало поводом для резких нападок как на Ф. И. Щербатского лично, так и на серию в целом. В кампанию включились и некоторые коллеги Ф. И. Щербатского по Институту востоковедения. Так и в своей рецензии на серию «Bibliotheca Buddhica» заявили о том, что:
Первое — продолжение издания серии «Библиотека Буддика» в настоящем её виде дело политически вредное. Обслуживать буддистов теологическими текстами, а буржуазную науку кантианско-махистианскими откровениями ак. Щербатского не к лицу Академии наук СССР. Отсюда второе — издание «Библиотека Буддика» прекратить.
Решением Президиума АН СССР от 1937 года деятельность научно-издательской серии «Bibliotheca Buddhica» была прекращена.
Первым с инициативой возобновления издания «Bibliotheca Buddhica» выступил вернувшийся в 1957 году в СССР Ю. Н. Рерих. Ему удалось добиться начала практической реализации инициативы, но затем возник ряд трудностей, вызванных нежеланием властей поддерживать «пропаганду буддизма». Одним из тех, кто помог Рериху преодолеть возникшие препятствия, стал его давний знакомый профессор Малаласекера, бывший в то время послом Цейлона в СССР.
Благодаря Рериху в качестве XXXI тома серии в 1960 году был издан текст афоризмов Будды Шакьямуни — перевод В. Н. Топорова с пали — «Дхаммапада», являющаяся настольной книгой южных буддистов и частью палийского тхеравадинского канона. Следующим, XXXII, томом серии стала книга А. И. Вострикова «Тибетская историческая литература».
Однако после выхода в свет двух томов издание серии вновь прекратилось. Лишь значительно позже, уже в 1980-е, серия при содействии Г. М. Бонгард-Левина продолжила своё существование. На этом этапе Bibliotheca Buddhica стала издаваться в рамках серии книг «Памятники письменности Востока», причём в выходных сведениях указывался как номер тома серии «Памятники письменности Востока», так и номер тома «Bibliotheca Buddhica». Первым изданным после перерыва стал выпущенный в 1985 году том XXXIII «Памятники индийской письменности из Центральной Азии», подготовленный Г. М. Бонгард-Левиным и М. И. Воробьевой-Десятовской. В то время в обществе рос интерес к буддизму и его источникам и книги серии способствовали его удовлетворению.
Всего за время, истекшее с момента основания, в составе серии было подготовлено и издано 40 томов. Том XL — «Памятники индийской письменности из Центральной Азии. Вып. 3» — вышел в свет в 2004 году. В книгах серии не только воспроизводятся оригинальные тексты и публикуются их переводы, но и приводится научный аппарат, включающий вступительные статьи, комментарии к текстам, словари и т. д.
Серия всегда пользовалась широким международным признанием специалистов. Так, например, первые 30 томов неоднократно переиздавались в Индии. Репринтное переиздание серии было предпринято индийским издательским концерном Motilal Banarsidass в 1997 году.
См. также
Памятники письменности Востока
Примечания
Ссылки
Институт восточных рукописей РАН РФ (Санкт-Петербург)
Оцифрованные книги Bibliotheca Buddhica
Буддийская литература
Книжные серии
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Our Girl is quickly turning into our favourite thing ever and we're not ashamed to admit that we're totally obsessed.
Despite being gripped by the explosive humanitarian mission and tear-jerking love story that's developing, it's actress Michelle Keegan that's got everyone talking.
Critics and fans have all showered the former Corrie star with praise for her flawless performance as Lance Corporal Georgie Lane in the BBC hit drama alongside co-star, Luke Pasqualino.
But they're not the only ones, as it turns out that Michelle's sister-in-law, and former The Only Way Is Essex star, Jess Wright is also a massive fan of the show and NEVER misses an episode.
Gushing over Michelle's performance, Jess told The Sun Online: 'It's such a great programme and Michelle is doing such an amazing job in it. I'm really enjoying it.
Since returning from filming in South Africa, Michelle has been spending a lot of time with husband Mark Wright and his family.
She recently celebrated Jess' 31st birthday in London by partying at the exclusive 100 Wardour club, and was pictured enjoying a big Wright dinner in Chigwell at the weekend along with Mark's parents Carol and Mark.
Considering how busy Mich' has been recently, it's great to see her enjoying some down time with the family.
Now excuse us while we go and mentally prepare ourselves for tonight's episode of Our Girl.
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\section{Introduction}
\label{sec:Introduction}
Boundary conditions (BCs) in quantum field theory give rise to nontrivial modifications of the system. A renowned example is the Casimir effect in quantum electrodynamics~\cite{Casimir:1948dh}, where the effect of the BC imposed by conductors is observable as the force acting on them~\cite{Lamoreaux:1996wh,Mohideen:1998iz}. In the Matsubara formalism for thermal field theory, nonzero temperature ($T$) is introduced as the BC along the imaginary-time direction in the Euclidean spacetime. Various phenomena in thermal systems, such as phase transitions, thus can be viewed as those provoked by the BC.
In $SU(N)$ Yang-Mills (YM) theory, a BC introduces a new symmetry of the action, that is the center symmetry of the gauge group, $Z_N$, through the twist at the BC. YM theory at nonzero $T$ thus has a $Z_N$ symmetry. The deconfinement phase transition at nonzero $T$ is characterized by the spontaneous breaking of this symmetry in the high temperature phase~\cite{Svetitsky:1982gs}. Observables that are not invariant under $Z_N$ transformation are order parameters of this phase transition, and the Polyakov loop is a conventional choice among them~\cite{Polyakov:1978vu}.
When BCs along spatial directions are imposed into YM theory at nonzero $T$, the action has additional $Z_N$ symmetries corresponding to the individual BCs. In such systems, therefore, phase transitions associated with the spontaneous breaking of individual $Z_N$ symmetries are expected to occur with variations of $T$ and spatial extent along the BCs. These phenomena, which are regarded as generalized Casimir effects in thermal systems, have been investigated from various motivations~\cite{Simic:2010sv,Tiburzi:2013vza,Flachi:2013bc,Fraga:2016oul,Karabali:2018ael,Mogliacci:2018oea,Ishikawa:2018yey,Santos:2019xlx,Ishikawa:2019dcn,Ishikawa:2020ezm,Ishikawa:2020icy,Dudal:2020yah,Inagaki:2021yhi,Guo:2022mdi} including numerical simulations~\cite{Chernodub:2012em,Chernodub:2016owp,Chernodub:2017mhi,Chernodub:2017gwe,Chernodub:2018pmt,Chernodub:2018aix,Kitazawa:2019otp,Chernodub:2019kon,Chernodub:2022izt}.
Recently, thermodynamics of $SU(3)$ YM theory with a periodic boundary condition (PBC) along one spatial direction has been investigated in the lattice numerical simulation~\cite{Kitazawa:2019otp} using a technique based on the gradient flow~\cite{Suzuki:2013gza,Asakawa:2013laa,Kitazawa:2016dsl,Iritani:2018idk}. With the BC, the pressure becomes anisotropic due to the violation of the rotational symmetry~\cite{Brown:1969na}. The numerical results in Ref.~\cite{Kitazawa:2019otp}, however, show that in comparison with the free field theory the pressure anisotropy is remarkably suppressed until the spatial extent becomes significantly small near the critical temperature $T_c$. This result implies that the non-perturbative nature of the gauge field plays a crucial role in determining the response against the BC near $T_c$. It thus is also expected that the response can be used as a sensitive probe to understand the properties of the system.
In the present study, we investigate the phase transitions and thermodynamics of YM theory at nonzero $T$ with BCs using an effective model. To be specific, throughout this paper we take the spatial dimension to be three and impose the PBC for one direction, say $x$, with the length $L_x$. Since the BC for the temporal direction is also periodic for gauge fields, this system is defined on a Euclidean manifold $\mathbb{T}^2\times \mathbb{R}^2$ with two PBCs of length $L_\tau=1/T$ and $L_x$. This system has two $Z_N$ symmetries corresponding to two PBCs, and the phase transitions associated with their spontaneous breakings are expected to occur by varying $L_\tau$ and $L_x$. To describe these phase transitions simultaneously in an effective-model approach, the model must contain two order parameters.
For the YM theory on an infinite spatial volume without BCs, i.e. the theory on $\mathbb{S}^1\times \mathbb{R}^3$, an effective model including the Polyakov loop as the order parameter has been proposed in Ref.~\cite{Meisinger:2001cq}. In this model, the gauge field has a constant background field corresponding to the non-trivial expectation value of the Polyakov loop that is determined by the minimization of the free energy. It has been shown that thermodynamic quantities obtained on the lattice are qualitatively reproduced in the model. This result indicates that the Polyakov loop plays an important role in describing thermal properties of YM theory near $T_c$~\cite{Pisarski:2000eq}. The idea has been improved in the literature by incorporating various effects, and also applied to QCD with dynamical fermions; see Ref.~\cite{Fukushima:2017csk} as a review and references therein.
In the present study, to describe the theory on $\mathbb{T}^2\times \mathbb{R}^2$ we extend these models by introducing two ``Polyakov loops'' along two compactified directions. Using this model, we investigate their roles in thermal properties of the system, especially the emergence of the anisotropic pressure. As the first trial of such an attempt, we consider a simple model composed of massless gauge field on the background field and a simple trial form for the Polyakov-loop potential. We investigate the phase diagram on the $L_\tau$--$L_x$ plane and thermodynamics for $N=2$ and $3$.
We show that an interesting phase structure with the second- and first-order phase-transition lines emerges for $N=2$ and $3$, respectively. We also calculate thermodynamic quantities on $\mathbb{T}^2\times \mathbb{R}^2$ and compare them with the lattice results in Ref.~\cite{Kitazawa:2019otp}. In this analysis, we find that the thermodynamic quantities are significantly affected by the two Polyakov loops. We, however, find that our model fails to reproduce the lattice results even qualitatively. While the result obviously shows that our approach needs to be improved, we argue that such a modification is possible by changing the form of the potential term. Such a description of the YM theory on $\mathbb{T}^2\times \mathbb{R}^2$ will in turn give us a deeper understanding of the theory on $\mathbb{S}^1\times \mathbb{R}^3$. Although we do not attempt further modifications of the model in this exploratory study, possible directions will be discussed.
This article is organized as follows. In Sec.~\ref{sec:model}, we introduce the effective model for the YM theory on $\mathbb{T}^2\times \mathbb{R}^2$ including two types of the Polyakov loops. Besides, we explain its properties and present concrete expressions for $N=2$ and $N=3$. In Sec.~\ref{sec:PhaseDiagram}, we show our numerical results of the phase diagram on the $L_\tau$--$L_x$ plane for $N=2$ and $N=3$. In Sec.~\ref{sec:Thermodynamics} our model is compared to the lattice data of thermodynamic quantities obtained in Ref.~\cite{Kitazawa:2019otp} and some discussions on the role of the Polyakov loops are provided. Then, our present work is concluded in Sec.~\ref{sec:Conclusions}.
\section{Model}
\label{sec:model}
In the present study we introduce an effective model for describing the YM theory on the $\mathbb{T}^2\times \mathbb{R}^2$ Euclidean manifold with PBCs. This system physically corresponds to a thermal system with the PBC along one spatial direction additionally, where the temporal length is related to temperature as $L_\tau=1/T$. We suppose that the PBC is imposed along $x$ direction and define the spatial extent by $L_x$, while the lengths along $y$ and $z$ directions are taken to be infinite.
\subsection{Polyakov loops}
The YM theory on $\mathbb{T}^2\times \mathbb{R}^2$ is invariant under the center symmetries, which are denoted as $Z_N^{(\tau)}$ and $Z_N^{(x)}$. These symmetries can be spontaneously broken with variations of $L_\tau$ and $L_x$. To describe these phase transitions, we employ the Polyakov loops\footnote{
The quantity that is conventionally called the Polyakov loop is $P_\tau$. In this manuscript, however, we refer to both $P_\tau$ and $P_x$ as the Polyakov loops for simplicity.
}
\begin{eqnarray}
P_\tau(x,{\bm r}_L) &=& \frac1N {\rm Tr} \left[ \hat{P}_\tau(x,{\bm r}_L) \right]\ , \nonumber\\
P_x(\tau,{\bm r}_L) &=& \frac1N {\rm Tr} \left[\hat{P}_x(\tau,{\bm r}_L) \right] \ , \label{PDefinition}
\end{eqnarray}
for the order parameters, where Tr means the trace over the gauge space and the Polyakov-loop matrices are defined by
\begin{eqnarray}
\hat{P}_\tau (x,{\bm r}_L) &=& \mathscr{P}\, {\rm exp}\left(i\int_0^{L_\tau}{A}_\tau(\tau,x,{\bm r}_L) d\tau \right)\ , \nonumber\\
\hat{P}_x (\tau,{\bm r}_L) &=& \mathscr{P}\, {\rm exp}\left(i\int_0^{L_x}{A}_x(\tau,x,{\bm r}_L) dx \right)\ . \label{PHatDef}
\end{eqnarray}
In Eq.~(\ref{PHatDef}), $A_\mu(\tau,x,\bm{r}_L)$ is the YM gauge field [$\mu=(x,y,z,\tau)$], $\mathscr{P}$ stands for the path-ordering integral, and ${\bm r}_L=(y,z)$ represents the uncompactified two spatial variables.
The Polyakov loops $P_\tau$ and $P_x$ are not invariant under $Z_N^{(\tau)}$ and $Z_N^{(x)}$, respectively, and are order parameters of the corresponding symmetries. It is also known that the thermal expectation value $\langle P_\tau \rangle$ is related to the free energy $F_q(x,{\bm r}_L)$ of a static test quark as~\cite{Rothe:1992nt}
\begin{eqnarray}
{\rm e}^{-L_\tau F_q(x,{\bm r}_L)} = N \langle P_\tau(x,{\bm r}_L) \rangle \ .
\end{eqnarray}
Therefore, $\langle P_\tau \rangle=0$ corresponds to the confined phase in which $F_q$ becomes infinite. In YM theory on $\mathbb{S}^1\times\mathbb{R}^3$, this phase is realized as a low-temperature phase, while the deconfined phase with $\langle P_\tau \rangle\ne0$ is realized at high $T$. These phases are separated by the second- and first-order phase transition for $N=2$ and $N\ge3$, respectively~\cite{Svetitsky:1982gs,Yaffe:1982qf,Svetitsky:1985ye,Sannino:2002wb,Ruggieri:2012ny,Sasaki:2012bi,Sasaki:2013xfa,Fukushima:2017csk}.
\subsection{Model construction}
For a simultaneous description of the spontaneous breakings of $Z_N^{(\tau)}$ and $Z_N^{(x)}$ on $\mathbb{T}^2\times \mathbb{R}^2$, we construct an effective model including $P_\tau$ and $P_x$ for their order parameters. To this end, we extend the ``model-B'' introduced in Ref.~\cite{Meisinger:2001cq} that deals with a single Polyakov loop $P_\tau$ on $\mathbb{S}^1\times \mathbb{R}^3$.
In Ref.~\cite{Meisinger:2001cq}, to represent a non-trivial value of $P_\tau$ the ``mean-field'' assumption is imposed, where the gauge field has a constant background field. The free energy of the system consists of two parts; the first one is the one-loop perturbative contribution from the gauge field with the background field, and the other is a potential term that models non-perturbative effects leading to the confinement. The expectation value of $P_\tau$ is determined so as to minimize the free energy. It has been shown that the model can reproduce $T$ dependence of thermodynamic quantities measured in lattice simulations qualitatively.
By extending this idea, we assume that the gauge field on $\mathbb{T}^2\times \mathbb{R}^2$ has a constant background field
\begin{eqnarray}
A_\tau(\tau,x,\bm{r}_L) &=& \frac{1}{L_\tau} {\rm diag}\Big[(\theta_\tau)_1, (\theta_\tau)_2, \cdots, (\theta_\tau)_{N} \Big]\ ,
\label{ABGAnsatz_t}\\
A_x(\tau,x,\bm{r}_L) &=& \frac{1}{L_x} {\rm diag}\Big[(\theta_x)_1, (\theta_x)_2, \cdots, (\theta_x)_{N} \Big]\ , \label{ABGAnsatz_x}
\end{eqnarray}
corresponding to the expectation values of $P_\tau$ and $P_x$.
Substituting Eqs.~(\ref{ABGAnsatz_t}) and~(\ref{ABGAnsatz_x}) into Eq.~(\ref{PHatDef}), one obtains
\begin{eqnarray}
(\hat{P}_c)_{jk} = \exp( i(\theta_c)_j ) \delta_{jk} \ ,
\end{eqnarray}
for $c=\tau,~x$ and
\begin{eqnarray}
P_c = \frac1N \sum_{j=1}^N \exp( i(\theta_c)_j ) \ .
\end{eqnarray}
Since $\hat{P}_c$ is an element of $SU(N)$, the phase variables $\vec\theta_c=( (\theta_c)_1, \cdots, (\theta_c)_{N} )$ satisfy
\begin{eqnarray}
\sum_{j=1}^N (\theta_c)_j = 0 \quad \mbox{(mod $2\pi$)}\ .
\label{theta=0}
\end{eqnarray}
In Eqs.~(\ref{ABGAnsatz_t}) and~(\ref{ABGAnsatz_x}), the background field is assumed to have diagonal forms following Ref.~\cite{Meisinger:2001cq}. In Ref.~\cite{Meisinger:2001cq}, the background field is introduced only for $A_\tau(\tau,x,\bm{r}_L)$ because $P_x$ is irrelevant on $\mathbb{S}^1\times \mathbb{R}^3$. In this case, $A_\tau(\tau,x,\bm{r}_L)$ can be diagonalized only with the gauge transformation. In contrast, the simultaneous diagonalization of $A_\tau(\tau,x,\bm{r}_L)$ and $A_x(\tau,x,\bm{r}_L)$ as in Eqs.~(\ref{ABGAnsatz_t}) and~(\ref{ABGAnsatz_x}) by using gauge degrees of freedom is not possible in general. However, since the purpose of the present study is to explore qualitative effects of $P_\tau$ and $P_x$ on $\mathbb{T}^2\times \mathbb{R}^2$, we employ the ansatz~(\ref{ABGAnsatz_t}) and (\ref{ABGAnsatz_x}), which is a simple choice to realize non-trivial values of $P_\tau$ and $P_x$.
For later use, it is convenient to parametrize the phase variables $\vec\theta_\tau$ and $\vec\theta_x$ for $N=2$ and $N=3$ within the constraint~(\ref{theta=0}). For $N=2$, each phase is parametrized by a single parameter $\phi_c$ as
\begin{eqnarray}
\vec\theta_c = ( \phi_c , -\phi_c ) \ .
\label{AnsatzSU2}
\end{eqnarray}
In this parametrization the Polyakov loops are given by
\begin{eqnarray}
P_c = \cos\phi_c \ .
\end{eqnarray}
The non-perturbative vacuum where the center symmetries are restored is given by $\phi_c=\pi/2$. On the other hand, the completely perturbative vacuum is realized with $\phi_c=0$. We thus impose the range of $\phi_c$ as $0\leq \phi_c \leq \pi/2$ for $N=2$.
For $N=3$, $\vec{\theta}_c$ has two degrees of freedom. The symmetric phase with $P_c=0$ is realized, for example, at $\vec\theta_c=(2\pi/3,0,-2\pi/3)$, while the perturbative vacuum corresponds to $\vec\theta_c=\vec0$. To connect these values, following Ref.~\cite{Meisinger:2001cq,Fukushima:2017csk}, we parametrize the phases as
\begin{eqnarray}
\vec\theta_c = (\phi_c, 0, -\phi_c ) \ ,
\label{AnsatzSU3}
\end{eqnarray}
resulting in
\begin{eqnarray}
P_c = \frac{1}{3}( 1+2\cos\phi_c) \ ,
\end{eqnarray}
with $0\leq\phi_c\leq2\pi/3$.
The one-loop free energy per unit volume with the background field~(\ref{ABGAnsatz_t}) and~(\ref{ABGAnsatz_x}) is calculated to be~\cite{Sasaki:2012bi}
\begin{eqnarray}
&& f_{\rm pert}( \vec\theta_\tau,\vec\theta_x ; L_\tau,L_x )
\nonumber \\
&&= \frac{2}{L_\tau L_x}\sum_{j,k=1}^{N}\left(1-\frac{\delta_{jk}}{N}\right)\sum_{l_\tau,l_x}\int\frac{d^2p_L}{(2\pi)^2} \nonumber\\
&& \times {\rm ln}\left[\left(\omega_\tau-\frac{(\Delta\theta_\tau)_{jk}}{L_\tau}\right)^2 + \left(\omega_x+\frac{(\Delta\theta_x)_{jk}}{L_x}\right)^2 + {\bm p}^2_L\right]\ , \nonumber\\
\label{FPert}
\end{eqnarray}
where the mass of the gauge field is assumed to vanish. In Eq.~(\ref{FPert}), $\omega_\tau=2\pi l_\tau/L_\tau$ and $\omega_x=2\pi l_x/L_x$ are the respective ``Matsubara modes'' with $l_\tau$ and $l_x$ being integers. We have defined momenta for the uncompactified directions by ${\bm p}_L=(p_y,p_z)$, and the phase differences $(\Delta\theta_c)_{jk}=(\theta_c)_j-(\theta_c)_k$.
For a given set of $(L_\tau,L_x)$, the minimum of Eq.~(\ref{FPert}) is at $\vec\theta_\tau=\vec\theta_x=\vec0$ (mod $2\pi$), which gives $P_\tau=P_x=1$. Defining the expectation values of $P_\tau$ and $P_x$ as the minimum of Eq.~(\ref{FPert}), therefore, the system is always in the deconfined phase. To model the confinement phase transition, we introduce a potential term of the Polyakov loops $f_{\rm pot}( \vec\theta_\tau,\vec\theta_x; L_\tau , L_x )$, such that the total free energy per unit volume reads
\begin{align}
f( \vec\theta_\tau,\vec\theta_x; L_\tau , L_x )
=& f_{\rm pert}( \vec\theta_\tau,\vec\theta_x; L_\tau , L_x )
\nonumber \\
&+ f_{\rm pot}( \vec\theta_\tau,\vec\theta_x; L_\tau , L_x ) \ .
\label{F}
\end{align}
Here, $f_{\rm pot}$ describes the non-perturbative effects that would have a dominant contribution at large $L_\tau$ or $L_x$. The specific form of $f_{\rm pot}$ will be determined later. The values of $\vec\theta_c$, and hence $P_c$, are determined so as to minimize Eq.~(\ref{F}).
\subsection{One-loop free energy}
With the parametrizations~(\ref{AnsatzSU2}) and (\ref{AnsatzSU3}), the one-loop free energy~(\ref{FPert}) is transformed into the form that is more suitable for numerical analyses. After the manipulations summarized in Appendix~\ref{sec:FPert} with the regularization explained in Appendix~\ref{sec:ZetaFunction}, one obtains
\begin{eqnarray}
f_{\rm pert} &=& - \frac{\pi^2}{15L_\tau^4} + \frac{4\phi_\tau^2(\phi_\tau-\pi)^2}{3\pi^2L_\tau^4} -\frac{\pi^2}{15L_x^4} + \frac{4\phi_x^2(\phi_x-\pi)^2 }{3\pi^2L_x^4} \nonumber\\
&&-\frac{4}{\pi^{2}}\sum_{l_\tau,l_x=1}^\infty\frac{1+2\cos(2\phi_\tau l_\tau)\cos(2\phi_xl_x)}{X_{l_\tau,l_x}^4} \ , \label{FPertSU2}
\end{eqnarray}
for $N=2$, while for $N=3$ one has
\begin{eqnarray}
f_{\rm pert} &=& -\frac{8\pi^2}{45L_\tau^4}+ \frac{8\phi_\tau^2(\phi_\tau-\pi)^2+\phi_\tau^2(\phi_\tau-2\pi)^2}{6\pi^2L_\tau^4} \nonumber\\
&&- \frac{8\pi^2}{45L_x^4} + \frac{8\phi_x^2(\phi_x-\pi)^2+\phi_x^2(\phi_x-2\pi)^2}{6\pi^2L_x^4} \nonumber\\
&&-\frac{8}{\pi^{2}}\sum_{l_\tau,l_x=1}^\infty\frac{1}{X_{l_\tau,l_x}^4} \Big[1+ 2\cos(\phi_\tau l_\tau)\cos(\phi_xl_x) \nonumber\\
&&+ \cos(2\phi_\tau l_\tau) \cos(2\phi_xl_x)\Big] \ , \label{FPertSU3}
\end{eqnarray}
with
\begin{eqnarray}
X_{l_\tau,l_x} \equiv \sqrt{(l_\tau L_\tau )^2+(l_xL_x)^2} \ .
\label{X}
\end{eqnarray}
In the limit $L_x\to\infty$, only the first two terms in Eqs.~(\ref{FPertSU2}) and (\ref{FPertSU3}) that do not include $1/L_x$ survive. The form of $f_{\rm pert}$ in this limit reproduces that for $\mathbb{S}^1\times\mathbb{R}^3$~\cite{Meisinger:2001cq}.
The subleading terms with respect to $1/L_x$ come from the last double-sum term in Eq.~(\ref{FPertSU2}) or (\ref{FPertSU3}). As discussed in Appendix~\ref{sec:FPert}, the subleading term starts at the order $(L_\tau/L_x)^3$;
\begin{eqnarray}
f_{\rm pert} &=& \hat{f}_{\infty}(\vec\theta_\tau) \frac1{L_\tau^4}
+ \mathcal{O}(L_x^{-3}) \ .
\label{Finf_exp}
\end{eqnarray}
Note that odd-order terms of $L_\tau/L_x$ appear in this expansion, while Eqs.~(\ref{FPertSU2}) and (\ref{FPertSU3}) apparently depend on $L_\tau/L_x$ only through $(L_\tau/L_x)^2$; see Appendix~\ref{sec:FPert}.
\subsection{Potential term}
Next, we determine the form of $f_{\rm pot}$. To this end let us first consider general properties of this term.
First, since the YM theory on $\mathbb{T}^2\times \mathbb{R}^2$ is invariant under the exchange of the $\tau$ and $x$ axes, $f_{\rm pot}$ should satisify
\begin{eqnarray}
f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau , L_x )
= f_{\rm pot}( \vec\theta_x,\vec\theta_\tau ; L_x , L_\tau ) \ .
\label{Fsym}
\end{eqnarray}
Second, in the $L_\tau\to\infty$ ($T\to0$) limit the system with the PBC would be in the confined phase irrespective of the value of $L_x$. We thus have
\begin{eqnarray}
P_\tau = 0 \quad (L_\tau\to\infty) \ .
\label{PtLt->inf}
\end{eqnarray}
By exchanging the $\tau$ and $x$ axes, one also obtains
\begin{eqnarray}
P_x = 0 \quad (L_x\to\infty) \ .
\label{PxLx->inf}
\end{eqnarray}
It is worth emphasizing that the limiting value of $P_c$ in these limits is not the perturbative one $P_c=1$. However, the values of $P_\tau$ and $P_x$ in these limits are irrelevant in the sense that they do not affect the property of the system since effects of the BC should be negligible. In fact, the background field $A_c$ in Eqs.~(\ref{ABGAnsatz_t}) and~(\ref{ABGAnsatz_x}) vanishes in the $L_c\to\infty$ limit irrespective of the value of $\vec\theta_c$.
Third, since the system approaches $\mathbb{S}^1\times \mathbb{R}^3$ in the limit $L_x\to\infty$, the potential term should approach the one on $\mathbb{S}^1\times \mathbb{R}^3$, i.e.
\begin{eqnarray}
f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau , L_x )
\underset{L_x\to\infty}\longrightarrow f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta_\tau ; L_\tau ) \ ,
\label{Fpot_lim1}
\end{eqnarray}
where $f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta ; L )$ is the potential term for the effective model for $\mathbb{S}^1\times \mathbb{R}^3$ with a single Polyakov loop $P_\tau$.
Equation~(\ref{Fpot_lim1}) means that $\vec\theta_x$ dependence in $f_{\rm pot}$ exists at the subleading order of $L_\tau/L_x$,
and this contribution leads to Eq.~(\ref{PxLx->inf}). This implies that the $\vec\theta_x$ dependence in $f_{\rm pot}$ must surpass the perturbative contributions $f_{\rm pert}$ which is of order $(L_\tau/L_x)^3$ as in Eq.~(\ref{Finf_exp}), and hence the former should be weaker than the power of $(L_\tau/L_x)^3$. From Eq.~(\ref{Fsym}), one obtains the same conclusion for the limit $L_\tau\to\infty$, i.e.
\begin{eqnarray}
f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau , L_x )
\underset{L_\tau\to\infty}\longrightarrow f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta_x ; L_x ) \ ,
\label{Fpot_lim2}
\end{eqnarray}
with $\vec\theta_\tau$ dependence satisfying Eq.~(\ref{PtLt->inf}).
When one constructs an effective model on $\mathbb{T}^2\times \mathbb{R}^2$ as an extension of that developed for $\mathbb{S}^1\times \mathbb{R}^3$, Eqs.~(\ref{Fpot_lim1}) and (\ref{Fpot_lim2}) are constraints for $f_{\rm pot}$.
In the model-B of Ref.~\cite{Meisinger:2001cq}, as an effective model on $\mathbb{S}^1\times \mathbb{R}^3$ the form of the potential term motivated by the Haar measure in the strong-coupling expansion
\begin{eqnarray}
f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta_\tau ; L )
= -\frac{1}{L R^3}{\rm ln}\bigg[\prod_{j<k}\sin^2\left(\frac{(\Delta \theta_\tau)_{jk}}{2}\right)\bigg] \ ,
\label{FHaarinf}
\end{eqnarray}
has been employed in combination with $f_{\rm pert}$ for the massless gauge field. Here, the quantity $R$ having a mass dimension of $-1$ is understood as a typical length scale for the confinement; when $R\ll L_\tau$, the potential term dominates over $f_{\rm pert}$ and the confined phase is realized.
In the present study, we employ Eq.~(\ref{FHaarinf}) as the form of $f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}$ and introduce $f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau , L_x )$ so as to satisfy Eqs.~(\ref{Fpot_lim1}) and (\ref{Fpot_lim2}). There are, of course, infinitely many possible forms of $f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau , L_x )$ within the constraint. Among them, in this exploratory study, we employ a simple seprable ansatz
\begin{eqnarray}
\lefteqn{ f_{\rm pot}( \vec\theta_\tau,\vec\theta_x ; L_\tau,L_x ) }
\nonumber \\
&=&
f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta_\tau ; L_\tau ) + f_{\rm pot}^{\mathbb{S}^1\times\mathbb{R}^3}( \vec\theta_x ; L_x ) \nonumber \\
&=&-\frac{1}{L_\tau R^3}{\rm ln}\bigg[\prod_{j<k}\sin^2\left(\frac{(\Delta \theta_\tau)_{jk}}{2}\right)\bigg] \nonumber\\
&&- \frac{1}{L_x R^3}{\rm ln}\bigg[\prod_{j<k}\sin^2\left(\frac{(\Delta \theta_x)_{jk}}{2}\right)\bigg]\ . \label{FHaar}
\end{eqnarray}
We note that Eq.~(\ref{FHaar}) satisfies all the conditions~(\ref{Fsym})--(\ref{Fpot_lim2}).
With the phase variables in Eqs.~(\ref{AnsatzSU2}) and~(\ref{AnsatzSU3}), Eq.~(\ref{FHaar}) is reduced to
\begin{eqnarray}
f_{\rm pot} &=& -\frac{1}{L_\tau R^3}{\rm ln}(\sin^2\phi_\tau)-\frac{1}{L_x R^3}{\rm ln}(\sin^2\phi_x) \ ,\nonumber\\
\label{FHaarSU2}
\end{eqnarray}
for $N=2$ and
\begin{eqnarray}
f_{\rm pot} &=& -\frac{1}{L_\tau R^3}{\rm ln}\left[\left(\sin^4\frac{\phi_\tau}{2}\right)\left(\sin^2\phi_\tau\right)\right] \nonumber\\
&&- \frac{1}{L_xR^3}{\rm ln}\left[\left(\sin^4\frac{\phi_x}{2}\right)\left(\sin^2\phi_x\right)\right]\ , \label{FHaarSU3}
\end{eqnarray}
for $N=3$.
\section{Phase diagram}
\label{sec:PhaseDiagram}
In this section, we investigate the model introduced in the previous section focusing on the behavior of the order parameters and the phase diagram on the $L_\tau$--$L_x$ plane for $N=2,3$.
\subsection{$N=2$}
\label{sec:ResultsSU2}
\begin{figure}[t]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.58]{FiniteTSU2.pdf}
\caption{$L_\tau$ dependence of $P_\tau$ (blue) and $P_x$ (red) at $L_x\to\infty$ for $N=2$. There exists a second-order phase transition at $L_\tau T_c^\infty=1$ with the critical temperature $T_c^\infty\approx 1/(0.874R)$. The spatial Polyakov loop $P_x$ is always zero.}
\label{fig:FiniteTSU2}
\end{figure}
\begin{figure}[hbtp]
\begin{center}
\begin{tabular}{cc}
\begin{minipage}[c]{0.47\hsize}
\centering
\hspace*{-2.5cm}
\includegraphics*[scale=0.56]{3DPsi4SU2.pdf}\\
\end{minipage}
\\
\\
\\
\begin{minipage}[c]{0.4\hsize}
\centering
\hspace*{-2.6cm}
\includegraphics*[scale=0.56]{3DPsiXSU2.pdf}\\
\end{minipage}
\end{tabular}
\caption{$L_\tau$ dependences of $P_\tau$ (top panel) and $P_x$ (bottom panel) at several $L_x$ for $N=2$.}
\label{fig:3DPsiSU2}
\end{center}
\end{figure}
\begin{figure}[t]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.58]{SymSU2.pdf}
\caption{Polyakov loops $ P(\equiv P_\tau=P_x)$ for the symmetric case $L=L_\tau=L_x$ for $N=2$ in the vicinity of the transition point. The second-order transition is found at $LT^\infty_c\approx 0.871$. The vertical line corresponds to the analytic solution for the second-order transition evaluated in Eq.~(\ref{AnalSU2}).}
\label{fig:SymSU2}
\end{figure}
\begin{figure}[t]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.54]{PhaseSU2.pdf}
\caption{Phase diagram on the $L_\tau$--$L_x$ plane for $N=2$. The blue (red) line separates $P_\tau=0$ and $P_\tau\neq0$ ($P_x=0$ and $P_x\neq0$). The dashed gray line stands for $L_\tau=L_x$, and the black dot is the transition point in Fig.~\ref{fig:SymSU2}. All phase transitions are of second order. The dotted line shows the asymptotic formula~(\ref{crit}).}
\label{fig:PhaseSU2}
\end{figure}
\begin{figure}[tb]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.58]{FiniteTSU3.pdf}
\caption{$L_\tau$ dependence of $P_\tau$ (blue) and $P_x$ (red) at $L_x\to\infty$ for $N=3$. The temporal Polyakov loop $P_\tau$ shows a first-order phase transition at $T_c^\infty \approx 1/(0.733R)$.}
\label{fig:FiniteTSU3}
\end{figure}
\begin{figure}[hbtp]
\begin{center}
\begin{tabular}{cc}
\begin{minipage}[c]{0.47\hsize}
\centering
\hspace*{-2cm}
\includegraphics*[scale=0.54]{3DPsi4SU3.pdf}\\
\end{minipage}
\\
\\
\\
\begin{minipage}[c]{0.4\hsize}
\centering
\hspace*{-2.1cm}
\includegraphics*[scale=0.54]{3DPsiXSU3.pdf}\\
\end{minipage}
\end{tabular}
\caption{$L_\tau$ dependences of $P_\tau$ (top panel) and $P_x$ (bottom panel) at several $L_x$ for $N=3$.}
\label{fig:3DPsiSU3}
\end{center}
\end{figure}
\begin{figure}[hbtp]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.58]{SymSU3.pdf}
\caption{$L (\equiv L_\tau=L_x)$ dependence of the Polyakov loop $P(\equiv P_\tau=P_x)$ for $N=3$ in the vicinity of the phase transition point. The first-order transition occurs at $LT^\infty_c\approx 0.991$.}
\label{fig:SymSU3}
\end{figure}
\begin{figure}[hbtp]
\centering
\hspace*{-0.5cm}
\includegraphics*[scale=0.54]{PhaseSU3.pdf}
\caption{Phase diagram on the $L_\tau$--$L_x$ plane for $N=3$. The solid line shows the first-order phase transition.}
\label{fig:PhaseSU3}
\end{figure}
In this subsection we explore the case for $N=2$. In Fig.~\ref{fig:FiniteTSU2} we first show the $L_\tau$ dependence of the Polyakov loops at $L_x\to\infty$ for $N=2$ in order to check the behavior on $\mathbb{S}^1\times \mathbb{R}^3$~\cite{Meisinger:2001cq}. The blue and red curves represent the temporal and spatial Polyakov loops $P_\tau$ and $P_x$, respectively. As can be seen, $P_\tau$ experiences a phase transition. As in Appendix~\ref{sec:GLAnalysis}, it is shown from the Ginzburg-Landau analysis that this is a second-order phase transition with the critical temperature $T_c^\infty=1/((2/3)^{1/3}R) \approx 1/(0.874R)$. The $Z_2^{(\tau)}$ symmetry is broken for $L_\tau T_c^\infty\leq1$ corresponding to the deconfinement. On the other hand, $P_x$ is always zero in $L_x\to\infty$ limit as in Eq.~(\ref{PxLx->inf}).
In Fig.~\ref{fig:3DPsiSU2} we display the $L_\tau$ dependence of $P_\tau$ (top panel) and $P_x$ (bottom panel) for several values of $L_x$. The upper panel indicates that the deconfinement phase transition on $\mathbb{S}^1\times \mathbb{R}^3$ persists even for finite $L_x$. The critical value of $L_\tau$ first decreases with decreasing $L_x$, but it suddenly starts increasing at the symmetric point $L_\tau=L_x$. Meanwhile, the bottom panel shows that a phase with the spontaneous breaking of $Z_2^{(x)}$ symmetry appears at small $L_\tau$, and the critical value of $L_\tau$ for this phase transition becomes larger with decreasing $L_x$. Moreover, $Z_2^{(x)}$ symmetry is mostly broken for sufficiently small $L_x$ ($L_xT_c^\infty\lesssim 0.9$).
While it is analytically shown that the phase transition at $L_x\to\infty$ is of second order, it is difficult to obtain a definite conclusion analytically for finite $L_x$. Our numerical results, however, strongly suggest that the order parameters change continuously and thus the phase transitions are of second order for any value of $L_x$. To see this, in Fig.~\ref{fig:SymSU2} we focus on the symmetric case $L_x=L_\tau=L$ and show the $L$ dependence of the Polyakov loops that behave $P =P_\tau=P_x$ in this case. As shown in Appendix~\ref{sec:GLAnalysis}, provided that the phase transition is of second order it occurs at
\begin{eqnarray}
L T_c^\infty=(2\ln2/\pi)^{1/3}RT_c^\infty=(3\ln2/\pi)^{1/3}\approx0.871\ . \label{AnalSU2}
\end{eqnarray}
The figure suggests that the order parameter becomes nonzero at this point without discontinuity. The analytic solution~(\ref{AnalSU2}) is denoted by the vertical line in the figure.
These results are well summarized as the phase diagram on the $L_\tau$--$L_x$ plane shown in Fig.~\ref{fig:PhaseSU2}. The blue (red) curve separates $P_\tau=0$ and $P_\tau\neq0$ ($P_x=0$ and $P_x\neq0$). The dashed gray line stands for $L_\tau=L_x$, and the black dot represents the transition point on this line.
Finally, let us investigate the fate of the red line that separates $P_x\ne0$ and $P_x=0$ in the limit $(L_x,L_\tau)\to(\infty,0)$. As discussed in Appendix~\ref{sec:GLAnalysis}, provided the second-order transition the limiting behavior of this line is given as in Eq.~(\ref{crit}). In Fig.~\ref{fig:PhaseSU2}, we show Eq.~(\ref{crit}) by the dotted line. The line agrees well for $L T_c^\infty\gtrsim1.3$.
\subsection{$N=3$}
\label{sec:ResultsSU3}
Next, we explore the $N=3$ case. Depicted in Fig.~\ref{fig:FiniteTSU3} is the $L_\tau$ dependence of $P_\tau$ and $P_x$ at $L_x\to\infty$. The blue and red curves represent $P_\tau$ and $P_x$, respectively. One sees from the figure that the deconfined phase is realized for $L_\tau=1/T_c^\infty$, with the critical temperature numerically estimated as $T_c^\infty\approx1/(0.733R)$. Unlike $N=2$, the numerical result shows that $P_\tau$ has a clear discontinuity at this point, which means that the phase transition is of first order. As discussed in Appendix~\ref{sec:GLAnalysis}, this result is analytically confirmed by the Ginzburg-Landau analysis.
In Fig.~\ref{fig:3DPsiSU3} we display the $L_\tau$ dependence of $P_\tau$ (top panel) and $P_x$ (bottom panel) at several $L_x$. At larger $L_x$, the top panel implies that the first-order phase transition of $P_\tau$ observed at $L_x\to\infty$ persists to $L_xT_c^\infty\sim1$. One also finds that both $P_\tau$ and $P_x$ change discontinuously at the identical transition point. The first-order phase transition then ceases to exist at adequately small $L_x$ ($L_xT_c^\infty\lesssim 1$).
To see the order of the phase transition, in Fig.~\ref{fig:SymSU3} we show the $L(=L_\tau=L_x)$ dependence of $P(=P_\tau=P_x)$ for the symmetric case $L_\tau=L_x$ around the transition point. As shown in the figure, $P$ changes discontinuously at $LT_c^\infty\approx0.991$ meaning that the transition is of first order even at the symmetric point. Thus, we conclude that the transition is always of first order for $N=3$.
Finally, we show the phase diagram on the $L_\tau$--$L_x$ plane in Fig.~\ref{fig:PhaseSU3}. As explained above the first-order phase transitions of $P_\tau$ and $P_x$ occur simultaneously. This transition is shown by the purple line. The dashed gray line stands for $L_\tau=L_x$, and the black dot represents the transition point on this line. Note that $P_x$ approaches zero for $L_x\to \infty$ such that the condition in Eq.~(\ref{PxLx->inf}) is satisfied, while the discontinuity of $P_x$ does exist at the first-order transition point.
\begin{figure}[hbtp]
\centering
\includegraphics*[width=0.46\textwidth]{EnergyComp.pdf}
\includegraphics*[width=0.46\textwidth]{PxComp.pdf}
\includegraphics*[width=0.46\textwidth]{PzComp.pdf}
\caption{Comparison of energy density $\epsilon$ and pressures $p_x$ and $p_z$ between our model (solid lines) and the lattice data in Ref.~\cite{Kitazawa:2019otp} with $N_t=16$ (circles) and $12$ (squares). The dashed lines show the results with fixed values of $P_x$ and $P_t$ given in the $L_x\to\infty$ limit. The dotted lines are the results in the massless free theory and their limiting values for $L_xT\to\infty$ are shown by the arrows.}
\label{fig:Comparison}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics*[width=0.46\textwidth]{comp_pxpz.pdf}
\caption{Pressure ratio $p_x/p_z$. The meanings of the symbols are the same as Fig.~\ref{fig:Comparison}.}
\label{fig:pxpz}
\end{figure}
\section{Thermodynamics}
\label{sec:Thermodynamics}
In this section, we investigate $L_\tau$ and $L_x$ dependence of thermodynamic quantities on $\mathbb{T}^2\times\mathbb{R}^2$ and study the impact of $P_\tau$ and $P_x$ on them.
An important feature of the thermodynamics on $\mathbb{T}^2\times\mathbb{R}^2$ is that the pressure becomes anisotropic because of the violation of rotational symmetiry due to the BC~\cite{Kitazawa:2019otp}. In such systems, the stress tensor $\sigma_{ij}$ with $i,j=x,y,z$ should be used in place of the pressure to represent the force acting on the surface. The stress tensor is equivalent with the spatial components of the energy-momentum tensor $T^\mu_\nu$ up to the overall sign. The spatial components of $T^\mu_\nu$, and thus $\sigma_{ij}$, have off-diagonal components in general. However, in our setting that imposes a PBC along $x$ direction, due to the parity symmetry the energy-momentum tensor in the Minkowski space is given by the diagonal form as
\begin{eqnarray}
T^\mu_\nu = {\rm diag} (\epsilon,p_x,p_y,p_z) \ ,
\label{Tmunu}
\end{eqnarray}
where $\epsilon$ is the energy density and $p_i$ represents the pressure for each direction. Because of the rotational symmetry around $x$ axis $p_y=p_z$ is satisfied, but $p_x$ can be different from $p_y$ and $p_z$.
The values of $\epsilon$, $p_x$ and $p_z$ in our model are obtained from the free energy as
\begin{eqnarray}
\epsilon &=& \frac{L_\tau}{ {\cal V}}\frac{\partial}{\partial L_\tau} {\cal V}f \ , \nonumber\\
p_x &=& -\frac{L_x}{ {\cal V}}\frac{\partial}{\partial L_x} {\cal V}f \ , \nonumber\\
p_z &=& -\frac{L_z}{ {\cal V}}\frac{\partial}{\partial L_z} {\cal V}f \ ,
\end{eqnarray}
where we have introduced the Euclidean four dimensional volume ${\cal V}=L_\tau L_xL_yL_z$ and temporary assumed that the lengths for $y$ and $z$ directions, $L_y$ and $L_z$, are finite.
In Fig.~\ref{fig:Comparison}, we show the $L_xT(=L_x/L_\tau)$ dependence of $\epsilon$, $p_x$ and $p_z$ obtained in our model for $N=3$ at $T/T_c^\infty=1/(L_\tau T_c^\infty)=2.10$ by the solid blue lines. In the figure, same quantities calculated in the massless free theory are shown by the dotted lines for comparison. Their limiting values for $L_xT\to\infty$ are shown by the arrows. In the figure, we also plot these quantities in $SU(3)$ YM theory on $\mathbb{T}^2\times\mathbb{R}^2$ for the same $T$ obtained in Ref.~\cite{Kitazawa:2019otp} for comparison. The circle and square symbols show the results obatined for $N_\tau=16$ and $12$, respectively, where $N_\tau$ is the number of lattice sites along the temporal direction which is related to the lattice spacing $a$ as $N_\tau=(aT)^{-1}$.
From Fig.~\ref{fig:Comparison}, one sees that our model results do not show good agreement with the lattice ones even qualitatively. In particular, while the lattice data show that the values of $p_x$ and $p_z$ hardly change from the one in the $L_x\to\infty$ limit for $L_xT\gtrsim1.5$, this behavior is not reproduced in the model calculation. Similar results are obtained for other values of $T/T_c$ investigated in Ref.~\cite{Kitazawa:2019otp}. Figure~\ref{fig:pxpz} compares these results in term of the pressure ratio $p_x/p_z$. From the figure one sees that the ratio obtained in our model is closer to unity than the massless-free result for $L_xT\gtrsim2.0$, but the modification is not enough to reproduce the lattice results.
These results, however, show that the non-trivial expectation values of $P_\tau$ and $P_x$ give non-negligible contribution to the behavior of thermodynamics on $\mathbb{T}^2\times\mathbb{R}^2$. In order to gain insights into their effects, we perform an additional analysis, where the values of $P_x$ and $P_\tau$ are fixed by hand to those at $L_x\to\infty$; $P_\tau\approx0.973$ and $P_x=0$. The obtained results for $\epsilon$, $p_x$ and $p_z$ are shown in Figs.~\ref{fig:Comparison} and~\ref{fig:pxpz} by the dashed lines. As can be seen, these results are insensitive to $L_xT$ and give consistent behavior with the lattice data for $L_xT\gtrsim1.5$. These results show that $P_x$ and $P_\tau$ affect thermodynamic quantities on $\mathbb{T}^2\times\mathbb{R}^2$ significantly. Therefore, although we have failed in reproducing the lattice data in the present study with a simple model, the modification of the model, especially the potential term, would give a consistent result with the lattice data. Such a description of the lattice results on $\mathbb{T}^2\times\mathbb{R}^2$ will in turn give us deeper understanding on the non-perturbative aspects of YM theory near $T_c^\infty$ not only on $\mathbb{T}^2\times\mathbb{R}^2$ but also $\mathbb{S}^1\times\mathbb{R}^3$.
\section{Conclusion and outlook}
\label{sec:Conclusions}
In this paper, we have investigated the phase structure and thermodynamics of the pure Yang-Mills (YM) theory on $\mathbb{T}^2\times\mathbb{R}^2$ with the PBC by means of an effective model. The model has two Polyakov loops along the compactified directions, $P_\tau$ and $P_x$, as the order parameters and thus is capable of describing the phase transitions associated with two $Z_N$ symmetries. As a first investigation of such an effective model, we have employed a simple form for the potential term $f_{\rm pot}$ given by an extension of Ref.~\cite{Meisinger:2001cq}. We have found that a rich phase structure on the $L_\tau$--$L_x$ plane can manifest itself due to two phase transitions, and the phase structure is qualitatively dependent on $N$.
The energy density and anisotropic pressure are also calculated in the model and are compared with the lattice results in Ref.~\cite{Kitazawa:2019otp}. Although we have found that our model fails in reproducing the lattice results, we have also found that thermodynamics on $\mathbb{T}^2\times\mathbb{R}^2$ is sensitive to $P_\tau$ and $P_x$ in the model. Therefore, while the present model with a simple ansatz is not satisfactory, the modification of the model would be able to reproduce the lattice results. The analysis in the previous section that fixes the values of $P_\tau$ and $P_x$ by hand would be used for a guide for such a study. We also note that the model can be improved toward other directions, for example, introduction of the quasi-particle mass of the gauge field and other mean fields~\cite{Gorenstein:1995vm,Peshier:1995ty,Carter:1998ti,Sannino:2002wb,Brau:2009mp,Begun:2010eh,Ruggieri:2012ny,Sasaki:2012bi,Sasaki:2013xfa} especially mimicking the magnetic condensates in the deconfined phase~\cite{Agasian:2003yw}. Comparison with the stress tensor obtained in the AdS/CFT correspondence~\cite{Balasubramanian:1999re,Myers:1999psa} is also interesting to investigate the role of the strong-coupling nature.
The investigation of the field theory with the BC can also be extended to various directions. An example is the use of other BCs such as the anti-periodic BC in place of the PBC. Although we have limited our attention only to $N=2,3$, the study of $N$ dependence for $N\ge4$ is a straightforward extension of the present study. Similar analysis in QCD with dynamical fermions is another important subject. Since the measurement of thermodynamics in QCD on $\mathbb{T}^2\times\mathbb{R}^2$ is possible using the technique developed in Refs.~\cite{Makino:2014taa,Taniguchi:2016ofw,Taniguchi:2020mgg}, the comparison with the lattice data is possible.
\section*{Acknowledgement}
The authors thank Kouji Kashiwa, Makoto Natsuume and Yuya Tanizaki for giving us useful comments. This work is supported by the RIKEN special postdoctoral researcher program (D.~S.), and JSPS KAKENHI Grant Numbers JP19H05598, JP20H01903, JP22K03619 (M.~K.).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,219
|
package com.ivanceras.db.server.util.generators;
import com.ivanceras.db.api.ModelDef;
import com.ivanceras.commons.conf.Configuration;
import com.ivanceras.commons.writer.FileUtil;
import com.ivanceras.commons.writer.SourceWriter;
import com.ivanceras.commons.writer.StringSourceWriter;
public class TableColumnGenerator {
public void start(ModelDef[] modelList, Configuration conf){
generateTableColumnNames(modelList, conf.metaDataPackageName, conf.metaDataDirectory, "TableColumns");
}
private void generateTableColumnNames(ModelDef[] modelList, String packageName, String dir, String className){
SourceWriter sw = new StringSourceWriter();
sw.lnprint("package "+packageName+";");
sw.lnprint();
sw.lnprint("public class "+className+"{");
sw.lnprint();
for(ModelDef model : modelList){
sw.lnTabPrint("public class "+model.getTableName()+"{");
sw.lnTabPrint("");
String[] attributes = model.getAttributes();
for(String att : attributes){
sw.lnTabPrint("\tpublic static final String "+att+" = \""+model.getTableName()+"."+att+"\";");
}
sw.lnprint("");
sw.lnTabPrint("}");
sw.lnprint("");
}
sw.lnprint("}");
FileUtil.writeToFile(sw.toString(), dir, className+".java");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,877
|
#ifndef _INC_VDSLUN
#define _INC_VDSLUN
typedef struct _VDS_INTERCONNECT {
VDS_INTERCONNECT_ADDRESS_TYPE m_addressType;
ULONG m_cbPort; BYTE* m_pbPort;
ULONG m_cbAddress;
BYTE *m_pbAddress;
} VDS_INTERCONNECT;
typedef struct _VDS_LUN_INFORMATION {
ULONG m_version;
BYTE m_DeviceType;
BYTE m_DeviceTypeModifier;
WINBOOL m_bCommandQueueing;
VDS_STORAGE_BUS_TYPE m_BusType;
char* m_szVendorId;
char* m_szProductId;
char* m_szProductRevision;
char* m_szSerialNumber;
GUID m_diskSignature;
VDS_STORAGE_DEVICE_ID_DESCRIPTOR m_deviceIdDescriptor;
ULONG m_cInterconnects;
VDS_INTERCONNECT *m_rgInterconnects;
} VDS_LUN_INFORMATION;
#endif /*_INC_VDSLUN*/
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,446
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Les Éditions du Nord sont une maison d'édition belge fondée en 1920 par Albert Parmentier et active jusqu'en 1966.
Elles étaient spécialisées dans les livres illustrés par des dessinateurs réputés tels que Falké, Touchet, Dignimont, Nelly Degouy, Hermine David, Maurice Langaskens, Luc De Jaegher, Sylvain Sauvage, Pierre Gandon, Philippe Swyncop, Joris Minne, Désiré Acket, Constant Montald, Gus Bofa, Henri Cassiers.
Histoire
Les Éditions du Nord sont issues de la librairie créée par Ch. Demuylder en 1876, spécialisée en patrons de mode et revues de mode. La librairie sera reprise en 1908 par sa belle-soeur Marie Eulalie Herbillon (née Crombé).
Albert Parmentier (1891-1966), gendre de Marie Eulalie Herbillon, reprit la direction de la maison Herbillon-Crombé et créa les Editions du Nord afin de publier des éditions précieuses d'ouvrages littéraires.
Au décès d'Albert Parmentier, la position dominante d'Herbillon-Crombé sur le marché belge des patrons de mode a justifié le rachat des Patrons Herbillon, des Editions du Nord et la Librairie du Nord par la Société American Can Company (à l'époque, propriétaire de Butterick Fashion Marketing Company, une société de patrons de mode célèbre aux USA dont le siège était à New-York, 161 Sixth Avenue).
Vu le manque d'intérêt des repreneurs pour la Librairie du Nord, celle-ci a été reprise par la belle-fille d'Albert Parmentier, Marie-Louise. Les Editions du Nord, elles, disparaissent, et sans celles-ci la Librairie du Nord n'a pas survécu longtemps. Elle a été définitivement fermée en 1970.
Albert Parmentier
Les Éditions du Nord s'identifient à leur fondateur et directeur, Albert Parmentier : plusieurs ouvrages seront édités sous le nom d'Éditions du Nord - Albert Parmentier. Né le 1er aout 1891 à Gand, fils d'un fabricant de pianos, il épouse Catherine Yvonne Lentz, fille de Marie-Eulalie Crombé et de Gustave Lentz.
Il reprend la direction de la maison Herbillon-Crombé en 1920 et la scinde en trois entités : la Librairie du Nord, les Éditions du Nord et les Patrons Herbillon. Bibliophile passionné, il édite la série Les gloires littéraires, puis la série Le panthéon du bibliophile. Avec ces complices, l'imprimeur Charles De Bruycker et Paul Angenot en charge du texte, il éditera des auteurs français et belges (Verhaeren, Streuvels, De Coster) et donnera l'occasion à des illustrateurs belges de développer leur talent d'illustrateur (e.a. Swyncop, Degouy, De Jaegher, Cassiers, Langaskens) .
Pendant la seconde guerre mondiale, malgré les restrictions de papier, il continue à éditer des ouvrages destinés aux bibliophiles et il lance les collections Electa et Flamma Tenax, moins luxueuses, mais toujours dédiées à l'illustration d'ouvrages littéraires.
Albert Parmentier a publié sous son nom, à ses éditions : Quelques artistes de France et de Belgique. Cassiers, Hémard, Dignimont, Montald, Martin, Swyncop, Falké. Portraits et croquis inédits par eux-mêmes.
Il a également publié une revue trimestrielle de bibliophilie: l'Écureuil, dont au moins quatre numéros ont paru entre 1932 et 1935.
Publications
Parmi les nombreux livres édités par les Éditions du Nord, on citera :
Collection Electa
1 Maxence van der Meersch, L'Empreinte du Dieu, bois en couleurs de Maurice Langaskens, 241 pp.
2 Pierre Benoit, Axelle, bois en couleurs de Maurice Langaskens, 1943, 283 pp.
3 Maxence van der Meersch, Maria, fille de Flandre, bois en couleurs de Lucien Dejaegher, 1943, 243 pp. .
4 Jean Martet, Marion des neiges, bois en couleurs de Lucien Dejaegher, 1943, 224 pp.. .
5 Pierre Benoit, Lunegarde, bois en couleurs de Désiré Acket, 1944, 249 pp.
6 Pierre Benoit, L'Atlantide, bois en couleurs de Nelly Degouy, 1944, 284 pp.
7 Romain Rolland, Colas Breugnon, bois en couleurs de Roméo Dumoulin, 1944, 272 pp.
Collection Les Gloires littéraires
1 Honoré de Balzac, Trois contes drôlatiques, illustré par Joseph Hémard
2 Émile Verhaeren, Les Flamandes et les bords de la route, illustré par Henri Cassiers
3 René Boylesve, La Leçon d'amour dans un parc, illustré par Alexandre Benois
4 Jules Renard, Poil de Carotte. Illustré par Pierre Falké
5 Francis Carco, De Montmartre au quartier latin, 60 illustrations d'André Dignimont
6 Paul Morand, Rien que la terre, 70 illustrations de Pierre Falké
7 Vincente Blasco Ibanez, La Femme nue de Goya, 2 volumes, 80 aquarelles de Philippe Swyncop
8 André Maurois, Les Silences du colonel Bramble, 80 illustrations de Charles Martin
10 Paul-Jean Toulet, Mon amie Nane, Illustrations par Chas Laborde
19 et 20 Rudyard Kipling, Le Livre de la jungle, édition illustrée de bois en couleurs de Pierre Falké, 1934
24 Anatole France, Les Dieux ont soif, édition illustrée par Sylvain Sauvage
25 Louis Pergaud, La Guerre des boutons, Illustrations en couleurs de Jacques Touchet
Collection le panthéon du bibliophile
1 Jonathan Swift, Les voyages de Gulliver, 30 eaux fortes et 36 gravures hors-texte de Gus Bofa, deux volumes, 1929
2 Charles de Coster, La Légende et les aventures héroïques, joyeuses et glorieuses d'Ulenspiegel et de Lamme Goedzak au pays de Flandres et ailleurs, 100 aquarelles de Constant. Montald, 2 volumes, 1931
3 Stijn Streuvels, Sous le ciel de Flandre, illustrations de Henri Cassiers
Collection Flamma Tenax
1 Maxence Van der Meersch, Corps et âmes, illustrations de Charles Fouqueray
2 Pierre Louÿs, La Femme et le Pantin, illustrations de Philippe Swyncop
3 Henri De Régnier, La Pécheresse, illustrations d'Umberto Brunelleschi
Notes et références
Liens externes
Maison d'édition ayant son siège en Belgique
Édition indépendante
Bibliophilie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,716
|
{"url":"https:\/\/mathsmadeeasy.co.uk\/gcse-maths-revision\/standard-form-gcse-maths-revision-worksheets\/","text":"## What you need to know\n\nStandard form is a shorthand of expressing very large or very small numbers. It is sometimes called \u201cscientific notation\u201d because often it is used across all kinds of sciences.\n\nThere are strict rules that determine when a number is in standard form. We say a number is in standard when it is written like\n\n$A\\times10^n$\n\nWhere $n$ must be a whole number, and $1\\leq A<10$, or in other words, $A$ must be greater than or equal to 1 but strictly less than 10. For example,\n\n$4.2\\times10^5$\n\nis a number in standard form. But what does it actually mean? Well, it\u2019s just a normal number written in an unusual way, so the question really is: what number is it?\n\nNote: decimal notation is the technical term for numbers written in the usual, familiar way.\n\nExample: Write $4.2\\times10^5$ in decimal notation.\n\nFirstly, recall that $10^5=10\\times10\\times10\\times10\\times10$. Then, we get\n\n$4.2\\times10^5=4.2\\times10\\times10\\times10\\times10\\times10$\n\nTo multiply a number by 10 once, we move the decimal point one space to the right. Here, we\u2019re multiplying by 10 five times, so we must the decimal point 5 spaces to the right (if it helps, think of 4.2 like 4.20000). Doing this, we get\n\n$4.2\\times10\\times10\\times10\\times10\\times10=420,000$\n\nTherefore, we have concluded that $4.2\\times10^{5}=420,000$.\n\nExample: Write $2.8\\times10^{-4}$ in decimal notation.\n\nThis is different because the power is negative, but it\u2019s actually no harder. Firstly, recall that\n\n$10^{-4}=\\dfrac{1}{10^4}$\n\nNow, we know $10^4=10\\times10\\times10\\times10$. Having this on the bottom of the fraction means that rather than multiplying by 10 four times, we must divide by 10 four times. Dividing by 10 means moving the decimal point to the left, so moving it 4 spaces left, we get\n\n$2.8\\times\\dfrac{1}{10\\times10\\times10\\times10}=0.00028$\n\nTherefore, we have concluded that $2.8\\times10^{-4}=0.00028.$\n\nSo, if the power is a positive number, it\u2019s a big number, and if the power is negative, it\u2019s a small number. Now we\u2019re going to have a look at how to go from decimal notation to standard form.\n\nExample: Write 56,700,000 in standard form.\n\nIt says above that the in standard form, your number should be between 1 and 10. So, what we need to do to write this in standard form is: move the decimal point left until this very big number becomes a number between 1 and 10, and then the number of spaces you had to move the decimal point will become the power of 10.\n\nSo, if we move the decimal point from the end of 56,700,000 nine spaces to the left, it becomes 5.67, which is between 1 and 10. Therefore, this number written in standard form is\n\n$5.67\\times10^{9}$\n\nRemember, it\u2019s a big number, so the power should be positive.\n\nExample: Write 0.0000099 in standard form.\n\nJust like in the last example, we need to move the decimal point until we get a number that falls between 1 and 10, and then count the number of times we moved it. The difference is, the power will be the negative of that number, since it is a very small value we\u2019re dealing.\n\nSo, if we move the decimal point in 0.0000099 six spaces to the right, it becomes 9.9, which is between 1 and 10. Therefore, this number written in standard form is\n\n$9.9\\times10^{-6}$\n\nNow we\u2019ve seen what standard form is and how to get to\/from it, we\u2019re going to take a look at how to multiply\/divide. two numbers together without taking them out of standard form. We\u2019re going to be applying the laws of indices; click here (https:\/\/mathsmadeeasy.co.uk\/gcse-maths-revision\/rules-indices-gcse-maths-revision-worksheets\/) for more information on them.\n\nExample: Find the standard form value of $(3\\times10^8)\\times(7\\times10^4)$.\n\nWe could change both these numbers out of standard form and then do the multiplication then put the answer back into standard form, but that\u2019s a lot of work with no calculator. Instead, we\u2019re just going to change the order around of the things being multiplied. We get\n\n$(3\\times10^8)\\times(7\\times10^4)=3\\times7\\times10^8\\times10^4$\n\nThere\u2019s no reason why we can\u2019t do this. We can multiply $2\\times5\\times6$ and get the same answer as $5\\times6\\times2$ \u2013 when multiplying numbers, order doesn\u2019t matter. We know that $3\\times7=21$, and using the multiplication law of indices, we also know that $10^8\\times10^4=10^{12}$. So, we get that\n\n$3\\times7\\times10^8\\times10^4=21\\times10^{12}$\n\nGreat! Well, almost. This answer is not in standard form (21 is not between 1 and 10), and we need it to be. Fortunately, if we recognise that $21=2.1\\times10$, then we get that\n\n$21\\times10^{12}=2.1\\times10\\times10^{12}=2.1\\times10^{13}$\n\n2.1 is between 1 and 10, so we have successfully completed the multiplication in standard form.\n\nExample: Find the standard form value of $(8\\times10^{-5})\\div(2\\times10^6)$.\n\nThis is the exact same idea as the last example. We\u2019re going to break it up and divide the first numbers, 8 and 2, and the powers of 10 separately. Using the division law of indices this time, we get $10^{-5}\\div10^6=10^{-5-6}=10^{-11}$, and so we have\n\n$\\dfrac{8\\times10^{-5}}{2\\times10^6}=\\dfrac{8}{2}\\times\\dfrac{10^{-5}}{10^6}=4\\times10^{-11}$\n\n4 is between 1 and 10, so this answer is in standard form, and so we are done.\n\n## Example Questions\n\nBecause the power is negative, this is going to be a very small number. As the power of ten is -6, we want to divide the number 1.15 by 10 six times, and so we will move the decimal point to the left six spaces. Doing so, we get\n\n$1.15\\times10^{-6}=0.00000115$.\n\nBecause this is a big number, the power of 10 is going to be positive. In standard form, the number must be between 1 and 10, so we will move the decimal point to the left until we have a number between 1 and 10, and we will count the number of spaces we moved. That number will be the power of 10.\n\nSo, if we move the decimal point in 5,980,000 to the left six spaces it becomes 5.98. Therefore, we get that\n\n$5,980,000=5.98\\times10^{6}$\n\nWe will split up this multiplication, multiplying the initial numbers together and the powers of 10 together separately. Firstly,\n\n$2.5\\times6=15$.\n\nSecondly, using the multiplication law of indices,\n\n$10^4\\times10^{-9}=10^{4+(-9)}=10^-5$\n\nSo, we get\n\n\\begin{aligned}(2.5\\times10^{4})\\times(6\\times10^{-9})&=2.5\\times6\\times10^4\\times10^{-9}\\\\&=15\\times10^{-5}\\end{aligned}\n\nWe\u2019re almost done, but this isn\u2019t in standard form since 15 isn\u2019t between 1 and 10. However, recognising that $15=1.5\\times10$, and again applying the multiplication law of indices, we get\n\n$15\\times10^{-5}=1.5\\times10\\times10^{-5}=1.5\\times10^{-4}$\n\nThis is in standard form, and so we are done.\n\n## Standard Form Teaching Resources\n\nFor teachers and tutors looking for standard form resources and revision materials then the worksheets and revision tests above should be exactly what you are looking for. With a range of worksheets to test all abilities and questions that progressively more difficult, hopefully you will find the Maths Made Easy standard form revision resource a useful addition to your portfolio of resources.","date":"2019-05-23 23:03:45","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\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 39, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8562537431716919, \"perplexity\": 220.7713903188237}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2019-22\/segments\/1558232257432.60\/warc\/CC-MAIN-20190523224154-20190524010154-00106.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathhelpforum.com\/advanced-algebra\/178287-notation-question-vertical-bar.html","text":"# Thread: Notation Question - Vertical Bar\n\n1. ## [SOLVED] Notation Question - Vertical Bar\n\nI'm pretty sure this is a basic thing I'm just not aware of, but I can provide the whole text if necessary...(it's from a matrix-based proof for the existence of strongly regular graphs.)\n\nIf A is a matrix and j is an eigenvector of A, what does the vertical bar in A|j^\u22a5 mean?\n\n2. The whole text surrounding the instance of the notation would be a big help (a text without a context is a pretext, for all Derrida's idiotic post-modern deconstructionism ). My guess is that the notation here means the restriction of A to the set of vectors perpendicular to j. That is, you're artificially restricting the domain of A. But I'd need to see the context in order to confirm.\n\n3. Hm, that sounds right, given the context. Here's the full text:\n\nGiven a graph G with adjacency matrix A, G is regular if and only if the all-1 vector j is an eigenvector of A; the corresponding eigenvalue is the valency. Since is symmetric, j^\u22a5 is A-invariant. Then G is strongly regular if and only if A|j^\u22a5 has just two distinct eigenvalues. (We have already seen the 'only-if' part of this. Conversely, if (A-rho_1*I)(A-rho_2*I)|j^\u22a5=0, then (A-rho_1*I)(A-rho_2*I)=alpha*J for some alpha, whence A^2 (in) <I,J,A> and G is strongly regular.\nA quote that long probably deserves a citation - it's on p16 of Graph Theory, Coding Theory, and Block Designs by Cameron and Van Lint.\n\nPS - Please pardon my lack of LaTeX skills.\n\n4. Yeah, it looks like my hunch is correct.\n\nYour lack of LaTeX skills is a bit academic at this point, with LaTeX not working right now, though there are work-arounds. MathGuru, the owner of the site, is, as we speak, working on restoring LaTeX, so yay!","date":"2017-05-26 06:14:17","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\": 0.95087069272995, \"perplexity\": 914.3403946264951}, \"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-2017-22\/segments\/1495463608642.30\/warc\/CC-MAIN-20170526051657-20170526071657-00496.warc.gz\"}"}
| null | null |
School Board sworn in during virtual ceremony
Alexandria School Board members W. Christopher Harris, Abdel-Rahman Elnoubi, Meagan Alderton, Tammy Ignacio, Kelly Carmichael Booz, Ashley Simpson Baird, Michelle Rief, Jacinta Greene and Willie Bailey were sworn in on Tuesday night.
The incoming Alexandria School Board took office in a virtual induction ceremony on Tuesday night.
The evening began with each member, separated by voting district, taking oath. District A includes newcomer Willie Bailey and incumbents Jacinta Greene and Michelle Rief. District B includes newcomers Ashley Simpson Baird, Kelley Carmichael Booz and Tammy Ignacio. District C includes previous Chair Meagan Alderton as well as two newcomers, Abdel-Rahman Elnoubi and W. Christopher Harris.
Subsequently, Harris nominated Alderton, who served as chair on the board this past fall, as chair. In nominating Alderton, Harris highlighted her experience as an 18-year educator and praised her advocacy for ACPS students, communication skills and leadership through various challenges.
Booz seconded the nomination and called attention to Alderton's expertise in transitioning large, new groups of board members.
"I'm glad to have Meagan, who has really guided this board during some really challenging times, help transition for this next year of our work we have to do, and we have a lot of work to do," Booz said.
Alderton was sworn in unanimously.
Ignacio nominated Greene, who also served on the previous board, as vice chair. Ignacio said Greene's service on the Commission for Women, passion for student equity and service on the Youth and Family Commission led to her decision.
"Ms. Greene is a dedicated School Board member. Serving is her passion, and I know that she will continue to fight for all of Alexandria's children," Ignacio said.
Greene was sworn in unanimously.
Terri Mozingo was appointed as designee to the division superintendent, Susan Neilson as the clerk of the board and Shanel Hill as deputy clerk of the board.
As required by the city, each board member then took turns reading sections from the Code of Ethics and Standards of Conduct.
Some of the 12 Code of Ethics clauses include putting loyalty to the welfare of the children and the School Division as a whole above loyalty to individuals, voting districts, particular schools or other special interest groups; ensuring the integrity of the actions of the School Board by avoiding granting special favors or unfair privileges to anyone or any entity; and reporting through appropriate means and channels, corruption, misconduct or neglect of duty whenever discovered.
The induction ceremony, which doubled as a brief School Board operations meeting, also consisted of the approval of rules of order and by-laws and superintendent's delegation of authority for financial functions.
Finally, the board approved the 2022-2023 School Board meeting calendar.
"You're all officially welcomed to the Alexandria City School Board. Thank you for putting yourself out there, putting yourself forward to do what I believe to be the most important work on the planet, and that is educating young people," Alderton said in closing. "It won't always be easy, but it will be 100% worth it."
[Read more: Alexandria recovers after receiving almost 10 inches of snow]
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| 7,881
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The Arctic's Singing Whale
Categories: Innovation & Research | Tags: Anatomy & Neurobiology, Hearing Research, Musculoskeletal Research
When George Crumb composed Vox Balaenae (Voice of the Whale), he called for the small group of performers to wear black masks as they played, to symbolize "the powerful impersonal forces of nature (nature dehumanized)."
This was in 1971, not during the COVID-19 pandemic. The score attracted international attention and has frequently been recorded, including by the International Contemporary Ensemble, heard here in a performance with commentary at the Chicago Humanities Festival.
Crumb's score (and masks!) would fit right in for an upcoming virtual event hosted by the Department of Anatomy and Neurobiology: "The Arctic's Singing Whale," a one-hour presentation by Kate Stafford, Ph.D., senior principal oceanographer at the University of Washington in Seattle.
Dr. Stafford's research, as presented in a recent TED talk, focuses on "the acoustic environment of the Arctic and how declining sea ice and increasing industrial use affect marine animals," according to Ingalls-Brown Professor of Anatomy Hans Thewissen, Ph.D.
The oceanographer/bioacoustician studies the music of cetaceans (marine mammals, including whales, dolphins and porpoises). Dr. Stafford will discuss her work Thursday, March 11, from noon-1 p.m. in a virtual presentation.
What whale song teaches us about mammalian brains
Dr. Thewissen, who teaches in the Department of Anatomy and Neurobiology and invited Dr. Stafford to speak, says that studying the music of whales also adds to our understanding of how the brain works:
"Similar to humans, but unlike most other mammals, bowhead and humpback whale songs evolve over time: Whales listen to each other and add new elements to their repertoire by imitating motives invented by another individual. Thus, the songs that whales in one region sing change from year to year, and are different from the songs of whales that live elsewhere. This is in contrast to closely related right whales (a species of baleen whales), which do not sing, and blue and fin whales, which have simple songs that are similar across the species and stable over many decades.
"Bowhead whales have brains roughly twice as large as those of humans, but the organization of their brain mirrors that of the hoofed animals they are related to, not those of primates. Understanding whale song teaches us about the plasticity of the auditory part of the mammalian brain."
Dr. Stafford's presentation relates to the research on hearing, social communication, and brain plasticity being conducted within the Department of Anatomy and Neurobiology's Hearing Research Focus Area, as well as to the work in evolutionary development advanced by the Musculoskeletal Research Focus Area in the same department. NEOMED researchers are partners in the Kent State University's Brain Health Research Institute.
To request a link to Dr. Stafford's presentation, call 330.325.6293.
Ignite Web Extras
Admissions Alumni Alzheimer's Disease Anatomy & Neurobiology BeST Center Bio-Med Science Academy College of Medicine College of Pharmacy Commencement Community-Based Mental Health Research Community Health Diabetes Obesity & Metabolism Research Diversity Equity & Inclusion Family & Community Medicine Featured Faculty Featured Students Geriatrics Global Engagement Health System Pharmacy Administration Hearing Research Heart & Blood Vessel Diseases Research I-Corps@Ohio Integrated Pharmaceutical Medicine Integrative Medical Sciences Internal Medicine LCME MEDCAMP Mental Health Modern Anatomical Sciences Musculoskeletal Research Neurodegenerative Disease & Aging Research Northeast Ohio Area Health Education Center Office of Faculty Enrichment & Engagement Parkinson's Disease Pharmaceutical Sciences Pharmacy Practice Psychiatry REDIZone Research Rural Medical Education (RMED) Pathway Scholarships & Grants strategic-planning Student-Run Free Clinic Student Organizations VITALS
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,253
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The world wakes up to a Nescafe cup of coffee so it's vital that the brand has a social media presence in every market from America to Zambia. Each social media profile serves up localized content, tailor- made to its audience. Take a look at Nescafe's social media strategies across Facebook, Twitter, Instagram, YouTube, Pinterest and LinkedIn to see how they engage their audience.
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{
"redpajama_set_name": "RedPajamaC4"
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| 4,326
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{"url":"https:\/\/crypto.stackexchange.com\/questions\/64856\/do-i-need-to-provide-entropy-to-secp256k1-ecdsa-sign","text":"# Do I need to provide entropy to secp256k1_ecdsa_sign() ?\n\nusing secp256k1_ecdsa_sign() I noticed the same data signed multiple times, coming back with the same signature. I always thought that signatures are different because random data is somehow involved.\n\nlooking at my (production) code, this is my call:\n\n secp256k1_ecdsa_signature sign(const pb::sha256 &in) {\nsecp256k1_ecdsa_signature sig;\nif ( 1 != secp256k1_ecdsa_sign(pb::TheCTX::instance()->CTX(),\n&sig, in.data, key_data, NULL, NULL) ) {\nqDebug() << \" error sig\";\n}\nreturn sig;\n}\n\n\nThis is the function signature:\n\nint secp256k1_ecdsa_sign(const secp256k1_context* ctx,\nsecp256k1_ecdsa_signature *signature,\nconst unsigned char *msg32, const unsigned char *seckey,\nsecp256k1_nonce_function noncefp, const void* noncedata)\n\n\nQuestion1: Where is randomness coming from? The default secp256k1_nonce_function, when i pass in NULL is nonce_function_rfc6979.\n\nQuestion2: Did I do something wrong? Can my private-keys be derived from my signed messages?\n\n\u2022 you said yourself that it isn't random, so why are you asking where the randomness is coming from, if there is none? did you read rfc6979? it makes the nonce dependent on the message (it is a hash), so that you won't reuse a nonce with different messages and get a playstation3 situation. \u2013\u00a0Janus Troelsen Dec 14 '18 at 0:03\n\nQuestion1: Where is randomness coming from? The default secp256k1_nonce_function, when i pass in NULL is nonce_function_rfc6979.\n\nObviously, there is no randomness involved when RFC 6979 is used. The title of that RFC is \"Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)\". In this case identical messages will generate the same signature value.\n\nAs the RFC reads: \"Even slight biases in that process may be turned into attacks on the signature schemes.\" So the deterministic signature scheme makes sure that the replacement of the random value doesn't have any bias. Fortunately, this is exactly what a pseudo-random-generator (PRG) is able to generate when the hash over the message is used as key. In the RFC a HMAC algorithm is used as PRG. So the true random value is replaced by a pseudo random value that depends on the message.\n\nQuestion2: Did I do something wrong? Can my private-keys be derived from my signed messages?\n\nNo, as long as the security considerations in section 4 of the RFC holds you should be fine.\n\nNotes:\n\n\u2022 Generating the same signature value for message input is not necessarily a problem and it is not a generic requirement for signature schemes. For instance, the PKCS#1 v1.5 signature scheme is fully deterministic. However, for ECDSA, it is possible to calculate the private key value using simple mathematical equations if no (pseudo-)random value is present.\n\u2022 You should consider if deterministic signature generation is OK for your specific protocol. For instance, if sign-and-encrypt is used then a deterministic signature may give information about the plaintext message, breaking the confidentiality requirement (although if the signature is verifiable, this is already the case if the signature and public key is available to an adversary).\n\u2022 Other API's may choose to use some kind of default secure random generator instead of deterministic signature generation. For instance, Java will use the highest priority secure random generator if no specific random generator is explicitly indicated during initialization of signature generation algorithm;","date":"2019-01-24 08:13:23","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\": 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\": 0.310452401638031, \"perplexity\": 1967.4845771488629}, \"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-2019-04\/segments\/1547584519757.94\/warc\/CC-MAIN-20190124080411-20190124102411-00502.warc.gz\"}"}
| null | null |
Cortes de Alcalá es la denominación historiográfica de las reuniones de las Cortes de Castilla que tuvieron lugar en la ciudad de Alcalá de Henares.
Las Cortes de Alcalá de 1345.
Especial importancia tuvieron las Cortes de Alcalá de 1348, donde se originó el Ordenamiento de Alcalá, una recopilación legislativa trascendental para la Historia del Derecho en España.
En una de esas dos reuniones de Cortes se originó el famoso dicho Por Castilla hablaré yo, atribuido al rey Alfonso XI, con el que zanjó la disputa de prelación entre los procuradores de Toledo y Burgos.
Posteriormente hubo una convocatoria de Cortes cuyas reuniones tuvieron lugar en Madrid y Alcalá de Henares, conocidas como Cortes de Madrid-Alcalá de Henares de 1503 (véase también Cortes de Madrid).
Referencias
Historia de Alcalá de Henares
Cortes de Castilla
|
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| 3,279
|
Granuloma inguinale ist eine bakterielle Erkrankung, die charakteristische ulzerierende Läsionen im Genitalbereich aufweist, auch bekannt unter dem Namen Donovanosis nach dem Entdecker der pathognomonischen histologischen Strukturen, oder Granuloma genitoinguinale. 1905 hat der Tropenarzt Charles Donovan die bakterielle Erkrankung zum ersten Mal beschrieben. Der Erreger ist Klebsiella granulomatis (früher Calymmatobacterium granulomatis).
Sie ist in vielen Entwicklungsländern endemisch. Aufgrund mehrerer Probleme (die für fast alle STD gelten), bleibt das Granuloma inguinale dort oftmals unbehandelt und kann destruierend wachsen.
Übertragen wird das Bakterium durch engen Hautkontakt. Sichtbar ist die Geschlechtskrankheit im Bereich der äußeren Geschlechtsorgane sowie im Bereich des Afters. Bis die ersten Symptome auftreten, können wenige Tage oder mehrere Wochen vergehen.
Die schmerzlosen Ulzerationen, welche mit Syphilis verwechselt werden können, schreiten letztendlich in die Zerstörung von Gewebe und ausgeprägte Einblutungen fort. Diese Oberflächenzersetzung führt zu einem erhöhten Risiko für weitere mikrobielle Superinfektionen und kann auch zur Verstümmelung führen.
Diagnose
Die Diagnose basiert auf der Anamnese und der klinischen Untersuchung, welche ein schmerzloses Ulkus mit einem charakteristisch gerollten Rand von Granulationsgewebe zeigt. Anders als bei Ulzerationen, die bei Syphilis auftreten, gibt es hier normalerweise keine Lymphknotenbeteiligung. Gewebebiopsien und eine Wright-Giemsa-Färbung helfen bei der Diagnoseerhebung. Hier zeigen sich in der Färbung die Donovankörperchen tiefpurpur als stabförmige ovale Organismen im Zytoplasma von Makrophagen oder Histiozyten.
Therapie
Die Therapie kann mit Cotrimoxazol, Tetracyclinen oder Makroliden durchgeführt werden. Selbst nach der Behandlung mit Antibiotika kann es bis zu einem Zeitraum von 18 Monaten zu einem Rückfall kommen. Betroffen können auch Sexualpartner sein, die bis zu 40 Tage vor Ausbruch der Erkrankung Geschlechtsverkehr mit der betroffenen Person hatten. Bis zur kompletten Abheilung sollte Geschlechtsverkehr vermieden werden.
Weblinks
Einzelnachweise
Sexuell übertragbare Erkrankung
|
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{"url":"https:\/\/blender.stackexchange.com\/questions\/35438\/use-the-bevel-tool-from-python-script","text":"# Use the bevel-tool from python script\n\nI want to use the bevel-tool to add a face between two other faces. The bevel-tool should use the selected edge(s) as geometry.location is given via a selected edge.\n\nThis is what I have\n\nThis is what I want\n\nAnd how can I set a enum-value? I tried offset_type='OFFSET' but I get the error that int was expected.\n\nScript:\n\nimport bpy, bmesh\nfrom mathutils import Vector\nobj = bpy.context.object\nbm = bmesh.new()\nbm.from_mesh(obj.data)\nbm.verts.ensure_lookup_table()\n\nselected_edge = None\nfor edge in bm.edges:\nif edge.select:\nselected_edge = edge\nprint(\"edge\", edge)\n\nprint(\"selected_edge\", selected_edge)\n\nbpy.ops.object.mode_set(mode='OBJECT') # only working in object mode\nfaces = bmesh.ops.bevel(bm,\ngeom=[selected_edge],\noffset=0.2,\noffset_type='OFFSET',\nsegments=1,\nprofile=0.5,\nvertex_only=False,\nclamp_overlap=True,\nmaterial=-1)\n\nprint(\"faces\", faces)\n\nbm.to_mesh(obj.data)\n\n\u2022 Question edited. \u2013\u00a0Hamburml Aug 10 '15 at 16:06\n\u2022 Bmesh operators use integer constants instead of string enum keys. They are defined in the C code, and there's currently no other way than to find the definition in the C code and use the found values in the Python script. You may want to define some variables to make the numbers more descriptive. See here: developer.blender.org\/diffusion\/B\/browse\/master\/source\/blender\/\u2026 \u2013\u00a0CoDEmanX Aug 11 '15 at 0:03\n\nYou can find out the answer to this in several ways. Here's what I did:\n\n1. Use the UI tool (mark the middle edge then press CtrlB to bevel).\n2. Go to the info panel (above the 3D view on the default layout), and open it up a little bit. Not always, but often, using the UI commands and operators will print out the command used in the info panel. Here we can see that the bpy.ops.mesh.bevel operator was used.\n3. To further explore this command and its documentation, open the python console. Type in:\n\nbpy.ops.mesh.bevel(\n\nThen press CtrlSpace to see the autocompleted documentation for this operator. In this case:\n\nbevel()\nbpy.ops.mesh.bevel(\noffset_type = 'OFFSET',\noffset = 0,\nsegments = 1,\nprofile = 0.5,\nvertex_only = False,\nclamp_overlap = False,\nloop_slide = True,\nmaterial = -1\n)\n\n\nThis shows you all the parameters and their defaults, you can then experiment with other values and search for this operator in the official docs to review any additional information you might need.\n\nThe parameter \"geom\" must be a list of edges AND their vertices, compare:","date":"2019-10-18 11:26:44","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\": 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\": 0.2300388365983963, \"perplexity\": 3170.806779496647}, \"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-2019-43\/segments\/1570986682037.37\/warc\/CC-MAIN-20191018104351-20191018131851-00174.warc.gz\"}"}
| null | null |
<?xml version = '1.0' encoding = 'UTF-8'?>
<TableProxyOraclev11g class="oracle.dbtools.crest.model.design.storage.oracle.v11g.TableProxyOraclev11g" name="PTNT" directorySegmentName="seg_0" id="84F899BA-F0AE-4549-924B-D4B092D6E2A9">
<createdBy>ali</createdBy>
<createdTime>2016-04-11 21:11:32 UTC</createdTime>
<ownerDesignName>CDM</ownerDesignName>
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<logging>YES</logging>
<partitionedRowMovement>DISABLE</partitionedRowMovement>
<segment>CCECCEE0-B4DE-0921-6597-E105DD13A604</segment>
<tableSpace>F5DDBBF6-5836-969A-E8FD-2176B7B3EFC9</tableSpace>
<temporaryChanged>false</temporaryChanged>
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<rejectLimit></rejectLimit>
<columnProxies itemClass="oracle.dbtools.crest.model.design.storage.oracle.v12c.ColumnProxyOraclev12c">
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<ownerDesignName>CDM</ownerDesignName>
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<createdBy>ali</createdBy>
<createdTime>2016-04-11 21:11:33 UTC</createdTime>
<ownerDesignName>CDM</ownerDesignName>
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<PrimaryKey class="oracle.dbtools.crest.model.design.storage.oracle.v10g.PKProxyOraclev10g" name="PTNT_PK" id="DA824586-E9EA-A5D9-63B9-416354DBC8D9">
<createdBy>ali</createdBy>
<createdTime>2016-04-11 21:11:33 UTC</createdTime>
<ownerDesignName>CDM</ownerDesignName>
<schemaObject>1BDDDB71-E132-5384-9C40-40F410BEADCF</schemaObject>
<segment>DEACDE3B-5A6A-E161-715E-EBBE3FEBCBF4</segment>
<tableSpace>F5DDBBF6-5836-969A-E8FD-2176B7B3EFC9</tableSpace>
<indexSegment>DEACDE3B-5A6A-E161-715E-EBBE3FEBCBF4</indexSegment>
<indexSort>SORTED</indexSort>
<usingIndex>BY INDEX NAME</usingIndex>
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<?xml version="1.0" encoding="UTF-8"?>
<!--L
Copyright SAIC, SAIC-Frederick.
Distributed under the OSI-approved BSD 3-Clause License.
See http://ncip.github.com/caadapter/LICENSE.txt for details.
L-->
<web-app version="2.4"
xmlns="http://java.sun.com/xml/ns/j2ee"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://java.sun.com/xml/ns/j2ee
http://java.sun.com/xml/ns/j2ee/web-app_2_4.xsd">
<context-param>
<param-name>gov.nih.nci.security.configFile</param-name>
<param-value>/WEB-INF/conf/ApplicationSecurityConfig.xml</param-value>
</context-param>
<!-- HibernateUtil Transaction per Request Management -->
<!--
<filter>
<filter-name>hibernate-filter</filter-name>
<filter-class>gov.nih.nci.common.persistence.hibernate.HibernateFilter</filter-class>
</filter>
<filter-mapping>
<filter-name>hibernate-filter</filter-name>
<url-pattern>/*</url-pattern>
</filter-mapping>
-->
<!-- setting filter inorder for response not to flush out while exporting data -->
<filter>
<filter-name>ResponseOverrideFilter</filter-name>
<filter-class>org.displaytag.filter.ResponseOverrideFilter</filter-class>
</filter>
<filter-mapping>
<filter-name>ResponseOverrideFilter</filter-name>
<url-pattern>*.do</url-pattern>
</filter-mapping>
<filter-mapping>
<filter-name>ResponseOverrideFilter</filter-name>
<url-pattern>*.jsp</url-pattern>
</filter-mapping>
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<servlet-name>action</servlet-name>
<servlet-class>org.apache.struts.action.ActionServlet</servlet-class>
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<param-name>application</param-name>
<param-value>ApplicationResources</param-value>
</init-param>
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<param-name>config</param-name>
<param-value>/WEB-INF/conf/struts-config.xml</param-value>
</init-param>
<init-param>
<param-name>debug</param-name>
<param-value>2</param-value>
</init-param>
<init-param>
<param-name>detail</param-name>
<param-value>2</param-value>
</init-param>
<init-param>
<param-name>validate</param-name>
<param-value>true</param-value>
</init-param>
<load-on-startup>2</load-on-startup>
</servlet>
<!-- Servlet: Add Scenario -->
<servlet>
<servlet-name>FormulaCalculateService</servlet-name>
<servlet-class>gov.nih.nci.cbiit.cdms.formula.ws.FormulaCalculateService</servlet-class>
<load-on-startup>1</load-on-startup>
</servlet>
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<servlet>
<servlet-name>AddNewScenario</servlet-name>
<servlet-class>gov.nih.nci.caadapter.ws.AddNewScenario</servlet-class>
<load-on-startup>1</load-on-startup>
</servlet>
-->
<!-- Servlet: Validate User -->
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<servlet>
<servlet-name>validateUser</servlet-name>
<servlet-class>gov.nih.nci.caadapter.ws.validateUser</servlet-class>
<load-on-startup>1</load-on-startup>
</servlet>
-->
<!-- Servlet: Axis Engine -->
<servlet>
<servlet-name>AxisServlet</servlet-name>
<servlet-class>org.apache.axis.transport.http.AxisServlet</servlet-class>
</servlet>
<servlet-mapping>
<servlet-name>AxisServlet</servlet-name>
<url-pattern>/ws/AxisServlet</url-pattern>
</servlet-mapping>
<servlet-mapping>
<servlet-name>AxisServlet</servlet-name>
<url-pattern>*.jws</url-pattern>
</servlet-mapping>
<servlet-mapping>
<servlet-name>AxisServlet</servlet-name>
<url-pattern>/ws/*</url-pattern>
</servlet-mapping>
<servlet-mapping>
<servlet-name>FormulaCalculateService</servlet-name>
<url-pattern>/FormulaCalculateService</url-pattern>
</servlet-mapping>
<!--
<servlet-mapping>
<servlet-name>AddNewScenario</servlet-name>
<url-pattern>/AddNewScenario</url-pattern>
</servlet-mapping>
<servlet-mapping>
<servlet-name>validateUser</servlet-name>
<url-pattern>/validateUser</url-pattern>
</servlet-mapping>
-->
<servlet-mapping>
<servlet-name>action</servlet-name>
<url-pattern>*.do</url-pattern>
</servlet-mapping>
<welcome-file-list>
<welcome-file>index.jsp</welcome-file>
<welcome-file>index.html</welcome-file>
</welcome-file-list>
<!-- Struts Tag Library Descriptors -->
<jsp-config>
<taglib>
<taglib-uri>/tags/struts-bean</taglib-uri>
<taglib-location>/WEB-INF/tld/struts-bean.tld</taglib-location>
</taglib>
<taglib>
<taglib-uri>/tags/struts-html</taglib-uri>
<taglib-location>/WEB-INF/tld/struts-html.tld</taglib-location>
</taglib>
<taglib>
<taglib-uri>/tags/struts-logic</taglib-uri>
<taglib-location>/WEB-INF/tld/struts-logic.tld</taglib-location>
</taglib>
<taglib>
<taglib-uri>/tags/tiles</taglib-uri>
<taglib-location>/WEB-INF/tld/struts-tiles.tld</taglib-location>
</taglib>
</jsp-config>
</web-app>
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,954
|
Russia causing disruptions in Baltic Sea - Lithuanian Ambassador to NATO
According to Vytautas Lesevicius, Lithuania's Permanent Representative to NATO, Russia's military activity is causing incidents in the...
Lithuanian political experts dismiss Belarusian elections
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Grybauskaite extends congratulations to Germany on 25 years of reunification
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Prime Minister's adviser criticises Grybauskaite's speech at UN
Lithuanian Prime Minister Algirdas Butkevicius' adviser, Raimundas Lopata, has criticised the the foreign policy objectives of the country&r...
Former KGB spy given Lithuanian residence permit
According to Lithuanian weekly, Veidas, Yuri Sagaidak who was deported from Great Britain back in 1989 as a KGB spy has received a Lithuanian re...
Lithuania's 2016 budget planned to boost slowing economy
Lithuanian Finance Minister, Rimantas Sadzius, says the 2016 budget has been drafted to give a boost to the country's slowing economy....
P.M. Butkevicius questions Russia's military action in Syria
On Thursday October 1, 2015, Lithuanian Prime Minister, Algirdas Butkevicius, criticised Russia's airstrikes in Syria. ...
Savchenko calls for Russia to be stripped of veto powers at UN
Ukrainian Air Force pilot Nadiya Savchenko, who is imprisoned in Russia, has urged the international community to "strip" Russia of it...
Grybauskaite rebukes
Lithuanian President Dalia Grybauskaite has slammed her Russian counterpart, Vladimir Putin, for delivering a "neo-Stalinist" speech...
Putin halves population of Lithuania during interview
During a recent interview with United States' TV channel, CBS, Russian President, Vladimir Putin, claimed only 1,4 million people live in ...
|
{
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| 3,723
|
Q: Alamofire 5 swift 5 wait for the result of the alamofire query to show the initial view The current situation is I have an initial view of login (A) and a second view when (B) has already logged in.
I want that when I open the application and the user has the values stored in the UserDefaults.standard, check that the values match those of the server and if they are correct enter the B view.
The problem is that alamofire is asynchronous, and the A-view is loaded before you get the answer.
SceneDelegate.swift
First attempt before we knew that alamofire was launched asynchronously
func scene(_ scene: UIScene, willConnectTo session: UISceneSession, options connectionOptions: UIScene.ConnectionOptions) {
//Code to open default viewA
...
//Code to try open viewB
let logAcepted = openViewB(url, param)
if logAcepted //<-- this execute before query in func before
{
if let windowScene = scene as? UIWindowScene
{
let window = UIWindow(windowScene: windowScene)
window.rootViewController = UIHostingController(rootView: Principal())
self.window = window
window.makeKeyAndVisible()
}
}
}
I have seen that there is a completion field that could help to solve this problem but I think that I have not applied it well.
func scene(_ scene: UIScene, willConnectTo session: UISceneSession, options connectionOptions: UIScene.ConnectionOptions) {
//Code to open default viewA
...
//Code to try open viewB
openViewB(url, param, completion: {logAcepted in
if logAcepted
{
if let windowScene = scene as? UIWindowScene
{
let window = UIWindow(windowScene: windowScene)
window.rootViewController = UIHostingController(rootView: Principal())
self.window = window
window.makeKeyAndVisible()
}
}
})
}
How can I make this function run synconically? or have the main code wait for the result of the query?
func openViewB(_ url: String, _ param: [String : String], completion : @escaping (Bool)->()) {
var ret : Bool = false
AF.request(url, parameters: param).responseJSON {response in
switch response.result {
case .success(let value):
if let JSON = value as? [String: Any]
{
let status = JSON["status"] as! String
switch status
{
case "Ok":
ret = true
completion(ret)
default:
ret = false
completion(ret)
}
}
case .failure( _):
ret = false
completion(ret)
}
}
}
I found a lot of related information but none that works in this version of swift/alamofire
A: Don't wait, never wait.
Your openViewB method has already a completion handler, use it.
func scene(_ scene: UIScene, willConnectTo session: UISceneSession, options connectionOptions: UIScene.ConnectionOptions) {
//Code to open default viewA
...
//Code to try open viewB
openViewB(url, param) { logAccepted in
if logAccepted
{
if let windowScene = scene as? UIWindowScene
{
let window = UIWindow(windowScene: windowScene)
window.rootViewController = UIHostingController(rootView: Principal())
self.window = window
window.makeKeyAndVisible()
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,658
|
Design Tiny Homes
The Ecocapsule, the Egg-Shaped Tiny Home That Can Go Off-Grid and Off-Pipe, Is Real!
We have seen so many photoshopped dreams that it is wonderful to see one turn into reality.
Remember that scene in Galaxy Quest, when Captain Nesmith finally contacts Brandon and admits that it's not just a set and a TV show, "It's real!" Well, by Grabthar's Hammer, the same can now be said about the Ecocapsule, a prefab off-grid solar and wind powered tiny home.
© It's real! People in the ecocapsule via InhabitatWhen we first covered the Ecocapsule, designed by Bratislava-based Nice Architects, I had doubts that it would ever be anything but vaporware. Looking at their website it is still hard to tell what is real or really good photoshop, but Inhabitat shows photos of real people climbing in and out of it, and a helicopter dropping it on a Bratislava roof. Ecocapsule describes the idea:
Ecocapsule is a self-sustainable smart house powered solely by solar and wind energy. It allows you to live off-the-grid, with the luxury of a hotel room. Ecocapsule is your design way to independent housing. It can serve as a cottage, pop-up hotel or even as a charging station for electric cars. We have engineered the product from scratch to be as self-sufficient, practical and functional as possible. With Ecocapsule, you will achieve a new level of freedom. Now, for the first time ever, we are opening orders and pre-orders. Join us and change the world - starting with yours.
© Well, the cows are real in this photo, don't know about the Ecocapsule
When I first wrote about it I wondered how it could live up to its claims of not requiring any supporting infrastructure and bringing "civilization's standards into the wilderness". I concluded:
Everything needs supporting infrastructure; waste tanks have to be emptied, gas bottles for cooking have to be filled. Truly going off grid is hard work, a lot more than just airdropping an egg. But hey, it is lovely to look at.
Now that it is for sale (cheap at €79,900 or US$98,193, just US$1,115.82 for each of its 88.26 square feet) we can really see its specifications too, which are actually pretty impressive.
© Ecocapsule via Inhabitat
The body is "made from high-capacity insulated fiberglass shells overlaid on an aluminum framework." It comes with a 750 watt wind turbine and 880 watts of solar, connected to a 10kWh battery. Water is collected from the roof and put through a reverse osmosis filter; the toilet is a waterless composting and urine separating, likely something like the Separett. They say they are working on an incinerating toilet for the next version; that's a bad idea because they are noisy and need a lot of fuel.
It is very light, with a dry weight of only 2976 pounds; I could tow it with my Miata on its custom trailer or its little wheels, and the shape would have low drag. There are even hooks on the roof for moving by helicopter. It is sized so that two units will fit in a standard 40' shipping container so it can be transported anywhere at reasonable cost.
Imagine. A warm mountain cabin in the great outdoors. A beach cabana on a remote island. A quiet studio in the countryside. A geeky hotel pod near your house. A spot for your loud teenager... Imagine an opportunity to getaway and enjoy an authentic life in nature.
Ask anyone from Jay Shafer, who pretty much started the tiny home movement, to me, who is still paying for my prototype, and you will find that it is really hard to get these things built and to create a business around it. It requires grit and perseverance and money and far more time than you ever think it would take.
I still have a few problems with the design of the Ecocapsule but to me, the wonder of it is that it exists at all and is for sale. For that, the Ecocapsule team deserves congratulations and the best of luck.
Ecocapsule Is the Egg-Shaped Tiny Home That Can Go Off-Grid and Off-Pipe
The Tiny House Is Finding a Home
Family of Three Builds Their Own Off-Grid Tiny House in Hawaii (Video)
Top 10 Tiny Houses of 2018: Less Is Indeed More
Couple Travels Full-Time With Off-Grid School Bus Tiny Home
Traveling Family Raises the Roof on This Brilliant Off-Grid Bus Conversion (Video)
Head for the Hills With the Track Tvan Camper Trailer
Off Grid Couple Answers Questions About Their Mortgage Free Life
Let's Go Camping! A Tour of Teardrop Trailers
Net-Zero Solar Powered Tiny House Could Help Ease City's Affordable Housing Crisis (Video)
Invisible Sustainability Meets the Disappear Retreat
Graham Hill and LifeEdited Go Off-Grid in Maui
Mirrored Glass Prefab From ÖÖD Is an Instant AirBnB
Professional Bus Homebuilder Is at Home -- On a Converted Bus (Video)
Let's Go Camping! A Tour of Trailers and RVs
Artistic Family Travels in Beautiful Short School Bus Conversion, Selling Art
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,161
|
Ajax in browser is limited to communicate with servers which are in the same domain from which current page in browser originates. Whenever you want to read data from server, which is in another domain, object Ajax must run on server, where this limitation is not applied. Application which is running on intranet server behind the firewall needs also permission on firewall for communication with given remote server.
Czech Crown (CZK) is presented because it is local currency in my country. Currency rate originates from Czech National Bank and is official rate for next day (days in case of holidays). Rates from other resources (Google) may be slightly different. It is not floating rate from Stock Exchange which changes every minute.
By clik on table row select conversion type, then drag pointer on pseudo-logarithmic scale to desired value (in left form field). Right field displays matching converted value. It is possible to enter digital value directly to the input fields (left, or right). Then click on mark "equal" to calculate converted value.
Object XMLHTTPRequest on server can read any file from remote web server, but is designed for reading .xml, or .txt files. Real sense of reading .html file is very limited. Automatic analysis of html code is difficult, just small change in source code breaks up analyse and data are invalid, or none at all.
Practical utilization of XMLHTTPRequest object on server is for reading web data, which changes in time. For example currency rate, weather forecast, or production data on intranet. It is also possible to use that object for test whether web server is currently running (ping is not always sufficient).
Web server of Czech National Bank publishes daily rate of exchange, formated in standard .html code for display in browser and the same data in .txt file, formated as ASCII table. That .txt file is right for reading by XMLHttpRequest object. ASCII table contains 35 different types of rate. Each row contains data for country, currency, quantity, code and rate. Single items are separated by character "pipe".
In code example is part of server code of just displayed page, written in VBScript for classic.asp which runs in MS IIS server only. That function is not portable to other server type. ServerXMLHTTP object is other than the object used in browser for Ajax, but provides the same service.
HTTP.Open "GET", Addr, false, "", ""
Used variables are declared at first. Addr contains URL address of page on remote server, variable HTTP is for object XMLHTTP. This object is opened with values: method of request, URL address, method of processing (here false means synchronous processing). More parameters are used only in case of login name and password demand. Next, response header is set up. Here simple text file is used, therefore text/plain is set up, and finally request is sent. At this point program stops its execution until response is delivered. As soon as program restarts we have to test if required file was really delivered. It is ensured by object status HTTP.status = 200 and data processing is done only in case error free transmission. If any problem occures, default data are used.
Using scale and slider for setting values was described in article forms, date and time, in case of need, check source code of this page please.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,049
|
\section{Introduction}
Classical theory of Loewner evolution gives a one-to-one correspondence between scalar continuous drivers (no smoothness assumptions) and families of continuously growing compact sets in the complex upper half-plane $\mathbb{H}$. There is much interest in the case where these sets admit a continuous trace (or even better: are given by a simple curve in $\mathbb{H}$).
The famous {\it Rohde--Schramm theorem} \cite{BasicSLE}
asserts that Brownian motion with diffusivity $\kappa \neq 8$ a.s. gives rise to a continuous trace (simple when $\kappa \le 4$), better known as SLE($\kappa$)-curves.\footnote{We shall make no attempt here to review the fundemental importance of SLE theory within probability and statistical mechanics. See e.g. \cite{lawlerbook} and the references therein.} The trace also exists for SLE$(8)$ but the proof follows indirectly from the convergence of uniform spanning tree to SLE$(8)$. (We note that the proofs are probabilistic in nature -- ultimately an application of the Borel-Cantelli lemma -- and gives little insight about the exceptional set.)
Deterministic aspects were subsequently explored by Marshall, Rohde, Lind, Huy Tran, Johannson Viklund and
others (see e.g. \cite{Joh15} and the references therein). We observe a number of similarities between the {\it It\^o map}, which
takes a Brownian driver to a diffusion path) with the {\it Schramm--L\"owner map} which takes a Brownian driver to SLE trace $\gamma$ (a ``rough", in the sense of non-smooth, path in $\mathbb{H}$),
$$
\Phi_{SL}: \sqrt{\kappa} B (\omega) \mapsto \gamma (\omega).
$$
In both cases, there is a ``Young" regime (case of drivers with H\"older exponent better than $1/2$) in which case one can fully rely on deterministic theory.\footnote{The analogy is not perfect: Young differential equations of form $dY=f(Y)dX$ are
invariant under reparametrization, hence most naturally formulated in a $p$-variation, $p<2$, context, whereas Loewner evolution is classically tied to parametrization by half-plane capacity. }
Also, in both the cases, Brownian motion does not fall in the afore-mentioned ``Young" regime, and yet both the It\^o- resp. (Schramm-)L\"owner map are well-defined measurable maps. While the It\^o map, also thanks to rough path theory,
is now very well understood, the afore-mentioned proof of the Rohde-Schramm theorem - despite being state of art - is not fully satisfactory. For instance, the case $\kappa=8$ still resists a direct analysis.
Even robustness in the parameter $\kappa$ turns out to be a decisively non-trivial issue, only recently settled in \cite{MR3229999} under the (technical) restriction of $\kappa < 8(2-\sqrt{3}) $.
\vspace{0.2cm}
To some extent, the ``pathwise" theory of Loewner evolutions, concerning existence of trace, has been settled by Rohde--Schramm in form of the following
\begin{theorem}\label{RS} \cite{BasicSLE} Loewner evolution with driver $U$, a continuous real valued path with $U_0 = 0$, admits a continuous trace if and only if \[ \gamma_t := \lim\limits_{y \rightarrow 0+} f_t(iy + U_t)\]exists and is continuous in $t$. In this case, $\gamma$ is the trace.
\end{theorem}
In the above theorem, following standard notation, $f_t(z) = g_t^{-1}(z)$ and for each $z \in \bar{\mathbb{H}}\setminus 0$, let $g_t(z)$ denote the solution of the LDE (Loewner's differential equation) \begin{equation}\label{LDE} \dot{g}_t(z) = \frac{2}{g_t(z)- U_t}, \hspace{2mm} g_0(z)= z. \end{equation}
Evenso, it is a non-trivial matter, and the essence of the afore-mentioned Rohde--Schramm theorem, to see that this applies to a.e. Brownian sample path. It is here that one has to work with Whitney-type boxes and a subtle Borel-Cantelli argument which in fact misses the case $\kappa=8$, subsequently handled with different methods. Readers familiar with the details of the proof may observe that a harmless finite-energy perturbation of the driver will already cause some serious complications, whereas it is a priori clear from the Cameron--Martin theoerem that SLE driven by $\sqrt{\kappa} B +h$, where $h$ is $\int_0^.$ of some $L^2$-function, does produce a continuous trace. (Such perturbations are relatively harmless for the It\^o-map, essentially because integration against $dh=\dot{h}dt$ is deterministic and SDEs driven by $B+h$ can be dealt with via flow decompositions.)
In fact, we observed with some surprise that Loewner evolution driven by finite-energy paths, despite being the ``skeleton" of Wiener-measure, has not been analyzed.
With regard to Lind's ``$1/2$-H\"older norm $< 4$" condition, we note that a finite-energy path $h$ is indeed $1/2$-H\"older (by a simple Sobolev embedding),
but may have arbitrarily large $1/2$-H\"older norm. Evenso, there is a ``poor man's argument" that allows to see that such $h$ generate a simple curve trace:
the remark is that $h$ is {\it vanishing $1/2$-H\"older} in the sense $\sup_{s,t:|t-s|< \varepsilon} \frac{|h_t - h_s|}{|t-s|^{1/2}} \to 0$ with $\varepsilon \to 0$, so that $\gamma$
is given by suitable concatenation of (conformally deformed) simple curves. A better understanding of the situation is given by Theorem \ref{thm:CM} below. To state it define
$\mathcal{H}_t$ as the space of absolutely continuous $h:[0,t] \to \mathbb{R}$, with finite energy i.e. square-integrable derivative $\dot{h}$, and norm-square
$$
|| h ||^2_t := \int_0^t |\dot{h_s}|^2 ds.
$$
Also, $C_T^\alpha$ is the space of paths defined $[0,T]$, with H\"older exponent $\alpha \in (0,1]$
Write $C_T^{p-var}$ for the space of continuous paths on $[0,T]$ of finite $p$-variation; note that $\alpha$-H\"older imflies finite $1/\alpha$-variation. At last, $C^{\dots}_{0,T}$ indicates paths on $[0,T]$ which are started at $0$. For instance, given a standard Brownian motion $B$, with probability one $\sqrt{\kappa} B (\omega) \in
C^\alpha_{0,T}$ for any $\alpha < 1/2$.
\pagebreak
\begin{theorem} \label{thm:CM}
Let $T>0$ and $U \in \mathcal{H}_T$.
\noindent (i) The following estimate holds for all $y>0$ and $t \in [0,T]$,
\begin{equation}\label{eq:CM} |f_t'(iy+U_t)| \leq \exp \biggl[\frac{1}{4} || U ||_t^2\biggr]. \end{equation}
\noindent (ii) The Loewner-trace $\gamma=:\Phi_{SL} (U)$ exists and is a simple curve.
\noindent (iii) The trace is uniformly $1/2$-H\"older in the sense that, for some constant $C$,
\begin{equation} || \gamma ||_{1/2} \le C \exp \biggl[ C ||U||_T^2 \biggr] \end{equation}
\noindent (iv) The map $t \mapsto \gamma(t^2)$ is Lipschitz continuous on $[0,T]$. As a consequence, the trace is of bounded variation and Lipschitz away from $0+$.
\noindent (v) On bounded sets in $\mathcal{H}_T$, the Schramm--Loewner map is continuous from $C[0,T]$ to $C^{1/2-\epsilon}([0,T],\bar{\mathbb{H}})$, any $\epsilon >0$.
\noindent (vi) The Schramm--Loewner map is continuous from $\mathcal{H}_T$ to $C^{(1+\epsilon)-var}$, any $\epsilon > 0$.
\end{theorem}
Our second contribution is a {\it pathwise} inequality that is well-suited to obtain existence of trace for stochastic drivers beyond Brownian motion. To state it, let us say that $U: [0,T] \to \mathbb{R}$ has {\it finite quadratic-variation in sense of F\"ollmer} if (along some {\it fixed} sequence of partitions $\pi=(\pi_n)$ of $[0,T]$, with mesh-size going to zero)
$$
\exists \lim_{n\to\infty} \sum_{ [r,s] \in \pi_n } (U_{s \wedge t} - U_{r \wedge t})^2 =: [U]^\pi_t
$$
and defines a continuous map $t \mapsto [U]^\pi_t \equiv [U]_t$. A function $V$ on $[0,T]$ is called {\it F\"ollmer-It\^o integrable} (against $U$, along $\pi$) if
$$
\exists \lim_{n\to\infty} \sum_{ [r,s] \in \pi_n } V_s (U_{s \wedge t} - U_{r \wedge t}) =: \int_0^t V d^\pi U.
$$
(F\"ollmer \cite{Foe81} shows that integrands of gradient form are integrable in this sense and so defines pathwise integrals of the form $\int \nabla F(U) d^\pi U$.) If the bracket is furthermore Lipschitz, in the sense that
\begin{equation} \label{LipFB}
\sup_{0 \le s < t \le T} \frac{[U]_t - [U]_s}{t-s} \le \kappa < \infty,
\end{equation}
write $U \in \mathcal{Q}_T^{\pi,\kappa}$. For instance, whenever $\pi$ is nested, a martingale argument shows that $\sqrt{\kappa} B (\omega) \in \mathcal{Q}_T^{\pi, \kappa}$ with probability one. We insist again that the following
result is entirely deterministic and highlights the role of the (pathwise) property (\ref{LipFB}) relative to existence of SLE($\kappa$) trace.
\begin{theorem} \label{thm:main} Let $T>0$ and $U \in C^\alpha_{0,T} \cap \mathcal{Q}_T^{\pi,\kappa}$ for some $1/3 < \alpha < 1/2$ and $\kappa < 2$.
For fixed $t \in [0,T]$ set $\beta_s := U_t - U_{t-s}$
and then, for arbitrarily chosen $A \in \mathcal{H}_t$, consider the decomposition\footnote{One could write $\beta^{(t)} = N^{(t)} + A^{(t)}$ to emphasize dependence on $t$.}
$$
\beta = N + A.
$$
Then there exists a continuous function $\dot{G}$, F\"ollmer integrable against $N$, so that for some $b>2, p>1$ and $\epsilon > 0$,
depending only on $\kappa$, we have, for all $y>0$ and $t \in [0,T]$,
\begin{equation} \label{key1}
|f_t'(iy + U_t)|^b \leq \exp\biggl( b\int_0^t \dot{G}_rd^\pi N_r - \frac{pb^2}{2}\int_0^t \dot{G}_r^2d[N]^\pi_r\biggr) \exp\biggl( \frac{b}{4\epsilon} || A ||_t^2 \biggr).
\end{equation}
\end{theorem}
Several remarks are in order.
\begin{remark} $U \in \mathcal{Q}_T^{\pi,\kappa}$ iff $N \in \mathcal{Q}_T^{\pi,\kappa}$ since $[N]_t = [\beta]_t = [U]_T - [U]_{T-t}$.
\end{remark}
\begin{remark} An explicit form of $\dot G$ is found in (\ref{eq:Gdot}). Remark that $\dot G_s$ is obtained as function of $(\beta_u: 0 \le u \le s)$, and in fact is controlled by $\beta$ in the sense of Gubinelli \cite{Gub04} or \cite[Ch. 4]{FH14}, which is a technical aspect in the proof.
\end{remark}
\begin{remark} We believe the restriction $\kappa < 2$ to be of technical nature.
\end{remark}
Write $C^{w,1/2}_{0,T}$ for ``weakly" 1/2-H\"older paths on $[0,T]$, started at zero. Following \cite{JVL11}, this means a modulus of continuity of the form $\omega(r) = r^{1/2} \varphi(1/r)$ for a ``subpower" function $\varphi$ (that is, $\varphi(x) = o(x^\nu)$ for all $\nu>0$, as $x\to \infty$). Thanks to L\'evy's modulus of continuity, with probability one, $\sqrt{\kappa} B (\omega) \in C^{{w, 1/2}}_{0,T} \subset C^\alpha_{0,T}$ for any $\alpha < 1/2$. (A general Besov--L\'evy modulus embedding appears e.g. in \cite[p.576]{FV10}.)
\begin{corollary} \label{cor:1} Let $T>0$ and consider random $U=U(\omega)$ with $U(\omega) \in C^{w,1/2}_{0,T} \cap \mathcal{Q}_T^\kappa$ for $\kappa < 2$ a.s.
For fixed $t \in [0,T]$, define $\beta$ as before and assume $\beta$ is a continuous semimartingale w.r.t. to some filtration, with canonical decomposition
$\beta = N + A$ into local martingale $N$ and bounded variation part $A \in \mathcal{H}_t$, so that $ || A ||_t^2 $ has sufficiently high (depending only on $\kappa$) exponential moments finite uniform in $t$. Then the Loewner-trace $\gamma =: \Phi_{SL} (U)$ exists.
\end{corollary}
\begin{proof} By assumption, we can apply Theorem \ref{thm:main} to a fixed realization of $U=U(\omega)$ in a set of full measure. Moreover, in view of the assumed semimartingale structure of $\beta = N +A$, our interpretation of the right-hand side of (\ref{key1}) can now be in classical It\^o-sense. (In the semimartingale case,
the F\"ollmer's integral and quadratic variation, along suitable sequences of partitions, coincide with It\^o's notion.)
With $b>2$ and then $p>1,\epsilon>0$ as in Theorem \ref{thm:main} , let $q$ be the H\"older conjugate of $p$. H\"older's inequality gives
$$
\mathbb{E} [ |f_t'(iy + U_t)|^b] \le \mathbb{E}\biggl[ \mathcal{E} \biggl( pb\int_0^t \dot{G}dN \biggr) \biggr]^{\frac{1}{p}} \mathbb{E} \biggl[\exp\biggl( \frac{qb}{4\epsilon} \int_0^t \dot{A}_r^2dr\biggr)\biggr]^{\frac{1}{q}}
$$
where $\mathcal{E} (...)$ denotes the stochastic exponential. Since $\dot{G}$ is adapted to $\beta$, its integral against $N$, is again a local martingale and so it the stochastic exponential. By positivity it is also a super-martingale, started at $1$, and thus of expectation less equal one.
Hence, for $b>2$, have $\mathbb{E} [ |f_t'(iy + U_t)|^b] < \infty$, uniformly in $t \in [0,T]$ and $y>0$. Together with $U \in C^{w,1/2}_{0,T}$ a.s. this is well-known (cf. appendix) to imply existence of trace.
\end{proof}
Again, some remarks are in order.
\begin{remark} We cannot make a semimartingale asumption for the Loewner driver $U$ since the time-reversal of a semimartingales can fail to be a semimartingale. That said, time-reversal of diffusion was studied by a number of authors including Millet, Nualart, Sanz, Pardoux ... and sufficient conditions on ``diffusion Loewner drivers" could be given by tapping into this literature.
\end{remark}
\begin{remark} As revealed by the above proof of the corollary, the only purpose of the semimartingale assumption of $\beta$ is to get good concentration of measure for $\int \dot{G} d N$. Recent progress on concentration of measure for pathwise stochastic integrals \cite[Ch.11.2]{FH14}, also {\rm [Ch. 5]} for some Gaussian examples of finite QV in F\"ollmer sense, suggest a possibility to study random Loewner evolutions without martingale methods.
\end{remark}
Theorem \ref{thm:main} and its corollary have little new to say about existence of trace for SLE$_{\kappa}$, especially with its restriction $\kappa < 2$. However, it is capable of dealing with non-Brownian drivers, including situations with non-constant $\kappa$, and $\mathcal{H}$-perturbations thereof.
\begin{example} {\bf (Classical SLE$_\kappa$)} As a warmup, consider Loewner driver $U = \sqrt{\kappa} B$ with fixed $\kappa < 2$. Since, for fixed $t$, $\beta_s := U_t - U_{t-s}$ defines another Brownian motion, we can trivially decompose with $N = \beta, A \equiv 0$ and thus obtain a.s. existence of trace for SLE$_\kappa$ immediately from the above corollary.
\end{example}
\begin{example} {\bf (non-constant $\kappa$)} Consider measurable $\kappa: [0,T] \to [0,\bar\kappa]$, with $\bar\kappa<2$, and then
$$
U_t = \int_0^t \sqrt{\kappa(s)} dB_s.
$$
A.s. existence of trace for ``SLE$_\kappa$ with non-constant $\kappa$" follows immediately from the above corollary. Remark that for piecewise constant $\kappa$, given by $(\kappa_i)$ on a finite partition of $[0,T]$, this conclusion can also by given by a suitable concatenation argument, relying on a.s. existence of trace for each classical SLE$_{\kappa_i}$.
\end{example}
\begin{example} {\bf ($\mathcal{H}$-perturbations)} Consider, with $h \in \mathcal{H}_T$,
$$
U_t = \sqrt{\kappa} B_t + h_t
$$
Then $\beta_s :=\sqrt{\kappa} (B_t - B_{t-s}) + h_t - h_{t-s}$. The corollary applies with Brownian motion, $N_t = \sqrt{\kappa}( B_t - B_{t-s} )$, and deterministic $A_t = h_t - h_{t-s}$. Remark that a.s. existence of trace, for $\kappa >0$, is also obtained as consequence of existence of trace for classical SLE$_{\kappa}$ and the Cameron--Martin theorem. Modifying the example to
$$
U_t = \int_0^t \sqrt{\kappa(s)} dB_s + h_t,
$$
without imposing a lower positive bound on $\kappa$, rules out the Cameron--Martin argument, but Corollary \ref{cor:1} still applies and yields existence of trace a.s.
\end{example}
\begin{example} {\bf (Ornstein--Uhlenbeck drivers)} Consider $U_t = Z_t - Z_0$ where $Z$ is a standard OU process, say with dynamics $dZ = - \lambda Z dt + \sqrt{\lambda} dB_t$ started in its invariant measure. By reversibility of this process, the time-reversed driver $\beta$ has the same law. Existence of trace (for SLE driven by such OU processes) is then a consequence of Corollary \ref{cor:1}.
\end{example}
\begin{corollary}\label{F(B)} Consider
$$
U_t = F(t, B_t).
$$
with
$$ F=F(t,x) \in C^{1,2}, F(0,0)=0 \text{ and } |F'(t,x)|^2 \leq {\kappa} < 2.$$
Assume furthermore, for $\alpha $ large enough (depending only on $\kappa$)
\begin{equation} \label{momentofF} \mathbb{E}\biggl[ \exp\biggl( \alpha \int_0^T \biggl\{\dot{F}(r, B_{r}) - \frac{1}{2}F''(r, B_{r}) + F'(r, B_{r})\frac{B_{r}}{r} \biggr\}^2dr \biggr)\biggr] < \infty.
\end{equation}
Then the Loewner-trace $\gamma =: \Phi_{SL} (U)$ exists.
\end{corollary}
\begin{example} Fix $p>0$ and consider, for the sake of argument on $[0,T]$ with $T \le 1$,
$$ U_t = t^p B_t.
$$
We insist that there is no ``cheap" way to such results. In particular, there is no ``comparison result" for SLE that would yield existence of trace based on $t^p \le 1$ on $[0,1]$ and existence of trace for SLE$_1$, say. (A related question by O. Angel was negatively answered in \cite{LMR10}.)
This is a special case of $U_t = F(t,B_t)$. To apply Corollary \ref{F(B)} one needs to check condition (\ref{momentofF}) which boils down to exponential moments for
$$
Z_T := \alpha \int_0^T \{ r^{p-1} B_r \}^2 dr.
$$
with fixed large $\alpha$, depending on $\kappa$. Note that $\mathbb{E} (Z_T) < \infty$. The centered random variable
$ Z^c_T := Z_T - \mathbb{E} (Z_T)$ lives in the second homogenous chaos over Wiener-space. Exponential moments
of $Z^c_T$ are then guaranteed, e.g. by using results from \cite[Ch. 5]{Ledoux}, provided that the second moment of $Z_T^c$ is small enough. But this can be achieved by
choosing $T>0$ small enough.
\end{example}
\begin{example} Consider
$$ U_t = t log( 1 + B_t^2).$$
we leave it to the reader to check that Corollary \ref{F(B)} applies and yields existence of trace on $[0,T]$ with $T$ small enough.
\end{example}
\begin{acknowledgement}
A. S. acknowledges support from the Berlin Mathematical School (BMS). P. F. received funding from the European Research Council under the European
Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 258237. Both authors would like to thank
S. Rohde for numerous discussions.
\end{acknowledgement}
\section{Some exact presentations of $f'$.}
\begin{lemma}
For each fixed $t\geq 0$ and $U \in C[0,t]$, define $ \beta_s = U_t - U_{t-s}$, $0 \le s \le t$. Then
\begin{equation}\label{crucialformula}\log|f_t'(z+U_t)| = \int_0^t \frac{2(X_r^2 - Y_r^2)}{(X_r^2 + Y_r^2)^2}dr
\end{equation}
where $z = x + iy$ and $(X_s, Y_s), s \in [0,t]$ is the solution of the ODE
\begin{align}\label{XYODE} dX_s & = d\beta_s - \frac{2X_s}{X_s^2 + Y_s^2} ds , \hspace{2mm} X_0 = x \\
dY_s &= \frac{2Y_s}{X_s^2 + Y_s^2}ds, \hspace{2mm} Y_0 = y .
\end{align}
Moreover, $G_s := \beta_s-X_s$ defines a $C^1$-function with,
\begin{equation} \label{eq:Gdot}
\dot{G}_s = \frac{2X_s}{X_s^2 + Y_s^2}.
\end{equation}
\end{lemma}
\begin{proof}For each $ z \in \mathbb{H}$, the path $ g_{t-s}(f_t(z))$ joins $z$ to $f_t(z)$ as $s $ varies from $0$ to $t$. It is then easy to see that \[f_t(z + U_t) = P_t(z) + U_t\] where $P_s(z)$ for $ s \in [0, t]$ is the solution of ODE
\[\dot{P}_s(z) = \frac{-2}{P_s(z) + \beta_s}, \hspace{2mm} P_0(z) = z \]
Writing in polar form, $ P_s' = r_se^{i\theta_s} $, we see that
\[ Re(\frac{|P_s'|}{P_s'} \partial_s P_s' ) = Re(e^{-i\theta_s}( e^{i\theta_s} \partial_s r_s + i r_s e^{i\theta_s}\partial_s \theta_s ) ) = \partial_s r_s \]
So it follows that, \[ \partial_s log|P_s'| = Re( \frac{1}{P_s'} \partial_s P_s') \]
Noting that $ \partial_s P_s' = (\partial_s P_s)' $,
\[ \partial_s log|P_s'| = Re( \frac{1}{P_s'} (\frac{-2}{ P_s + \beta _s } )' ) = 2Re( (P_s + \beta_s)^{-2} )\]
\[ \implies log|P_s'| = 2 \int_0^s Re( (P_r + \beta_r)^{-2} )dr \]
and the claim follows. \\
\end{proof}
\begin{proposition}\label{singular-nonsingular} Fix $t\geq 0$. Let $U \in C^\alpha$ with $\alpha \in (1/3,1/2]$. With $G,\beta$ as in the previous lemma,
\begin{equation}
\log |f_t'( z +U_t)| = M_t - \int_0^t \dot{G}_r^2 dr + \log(\frac{Y_t}{y}) -\log (\frac{X_t^2 + Y_t^2}{ x^2 + y^2}) \label{e:13}
\end{equation}
where
$M_t$ is given as rough integral
\begin{equation} \label{equ:MRP} M_t = \lim_n \sum_{[s,t] \in \pi_n} \dot{G}_s (\beta_t - \beta_s) + \dot{G}'_s \frac{1}{2}(\beta_t - \beta_s)^2
\end{equation}
with the Gubinelli derivate $$\dot{G}'_s := {\dot{Y}_s}/{Y_s} - \dot{G}_s^2 .$$
If in addition, $U$ (equivalently: $\beta$, as defined in the previous lemma) has continuous finite quadratic-variation in sense of F\"ollmer (along $\pi$) then
\begin{equation}
\log |f_t'( z +U_t)| = M^\pi_t + \frac{1}{2} \int_0^t \dot{G}'_s d[\beta]^\pi_s - \int_0^t \dot{G}_r^2 dr + \log(\frac{Y_t}{y}) -\log (\frac{X_t^2 + Y_t^2}{ x^2 + y^2}) \label{e:13}
\end{equation}
with (deterministic) F\"ollmer--It\^o integral
\begin{equation} \label{IFI}
M^\pi_t = \lim_n \sum_{[u,v] \in \pi_n} \dot{G}_u (\beta_v - \beta_u) =: \int_0^t \dot{G} d^\pi \beta . \end{equation}
\end{proposition}
\begin{remark}
When $U \in \mathcal{H}$, i.e. in the case of finite energy driver, $[\beta] \equiv 0$ and $M \equiv M^\pi$ reduces to a classical Riemann-Stieltjes integral.
\end{remark}
\begin{proof}
Consider first the case of $U$ (equivalently: $\beta$) in $C^1$. Then
\begin{align*}
& \qquad \dot{G}_rd\beta_r - \frac{1}{2}\dot{G}_r^2dr + \frac{Y_rdY_r}{X_r^2 + Y_r^2}
- \frac{2X_rdX_r + 2Y_rdY_r}{X_r^2 + Y_r^2} \\ & = \frac{2X_r}{X_r^2 + Y_r^2}d\beta_r - \frac{2X_r^2}{(X_r^2 + Y_r^2)^2}dr -\frac{2X_rdX_r}{X_r^2 + Y_r^2} -\frac{Y_rdY_r}{X_r^2 + Y_r^2} \\ &= \frac{2X_r}{X_r^2 + Y_r^2}d(\beta_r - X_r)
- \frac{2X_r^2}{(X_r^2 + Y_r^2)^2}dr - \frac{2Y_r^2}{(X_r^2 + Y_r^2)^2}dr \\ &= \frac{4X_r^2}{(X_r^2 + Y_r^2)^2}dr - \frac{2X_r^2}{(X_r^2 + Y_r^2)^2}dr - \frac{2Y_r^2}{(X_r^2 + Y_r^2)^2}dr \\ &= \frac{2(X_r^2-Y_r^2)}{(X_r^2 + Y_r^2)^2}dr
\end{align*}
Next note that \[ \frac{1}{2}\dot{G}_r^2dr + \frac{1}{2}\dot{Y}_r^2dr = \frac{\dot{Y}_r}{Y_r}dr\]and \[\frac{Y_rdY_r}{X_r^2 + Y_r^2} = \frac{1}{2}\dot{Y}_r^2dr = \frac{\dot{Y}_r}{Y_r}dr - \frac{1}{2}\dot{G}_r^2dr \]
Putting all together, we get
\[ \frac{2(X_r^2-Y_r^2)}{(X_r^2 + Y_r^2)^2}dr = \dot{G}_rd\beta_r - \dot{G}_r^2dr + \frac{\dot{Y}_r}{Y_r}dr - \frac{2X_rdX_r + 2Y_rdY_r}{X_r^2 + Y_r^2} \]and integrating both side, the claim follows with $M_t = \int_0^t \dot{G_s} d\beta_s$.
In the case of rough driver, meaning $U$ (equivalently: $\beta$) in $C^\alpha$ with $\alpha > 1/3$ the idea is to exploit a cancellation between
$ \dot{G}_rd\beta_r $ and $- \frac{2X_rdX_r}{X_r^2 + Y_r^2}$. We can in fact rely on basic theory of controlled rough path to see existence of the rough integral $M_t$. It suffices
to note that the integrand $\dot{G}$ is controlled by the integrator $\beta$. To see this, write
\[ \dot{G}_s = \frac{2X_s}{X_s^2 + Y_s^2} =: \varphi (X_s, Y_s) \]
and since $\varphi$ is smooth and well-defined (as long as $y>0$ fixed), and $Y$ plainly Lipschitz, it is easy to see that (or just apply directly Exercise 7.8 in \cite{FH14})
\[ \dot{G}_s - \dot{G}_r = \partial_x \varphi (X_s, Y_s) ({X}_s - {X}_r) + O (|s-r|^{2\alpha}) =\partial_x \varphi (X_s, Y_s) ({\beta}_s - {\beta}_r) + O (|s-r|^{2\alpha}) \]
which guarantees existence of (\ref{equ:MRP}) as rough path integral (Theorem 4.10 in \cite{FH14}). The second part concerning the splitting into It\^o--F\"ollmer integral and quadratic variation part, is similar to \cite[Ch. 5.3]{FH14}, using in particular {\rm Lemma 5.9.}.
\end{proof}
\section{A deterministic estimate on $f'$.}
\begin{theorem} \label{thm:keyest} In the context of Proposition \ref{singular-nonsingular}, with continuous finite quadratic-variation in sense of F\"ollmer so that
$ d [\beta]^\pi_s / ds \le \kappa < 2 $ one has the following estimate
\begin{align*}
|f_t'( iy + U_t)| & \leq \exp \biggl[ M^\pi_t - \int_0^t \dot{G}_r^2 d(r + \tfrac{1}{2}[\beta]_r) \biggr]
\end{align*}
where $\int_0^t \dot{G}_r d^\pi \beta_r = M^\pi_t$ is the It\^o-F\"ollmer type integral introduced in (\ref{IFI}).
\end{theorem}
\begin{proof}
From (\ref{e:13}) and definition for $G'$, also taking $x=0$ so that $z = iy$ (i.e. $x=0$),
\begin{align*}
\log| f_t'(iy + U_t)| &= \int_0^t \dot{G}_rd\beta_r -\int_0^t \dot{G}_r^2dr - \frac{1}{2}\int_0^t \dot{G}_r^2d[\beta]_r \\
& + \log(\frac{Y_t}{y}) - \log(\frac{X_t^2 + Y_t^2}{y^2}) + \frac{1}{2}\int_0^t \frac{\dot{Y}_r}{Y_r}d[\beta]_r
\end{align*}
Using positivity of $\dot{Y}_r / {Y_r}$,
\begin{align*}\label{dropterm}
&\log(\frac{Y_t}{y}) - \log(\frac{X_t^2 + Y_t^2}{y^2}) + \frac{1}{2}\int_0^t \frac{\dot{Y}_r}{Y_r}d[\beta]_r \\ & \leq \log(\frac{Y_t}{y}) - 2\log(\frac{Y_t}{y}) + \frac{\kappa}{2} \int_0^t \frac{\dot{Y}_r}{Y_r}dr \\ &= (\frac{\kappa}{2}- 1)\log(\frac{Y_t}{y}) \\ & \leq 0.
\end{align*}
and the desired estimate follows.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}
As immediate corollary of Theorem \ref{thm:keyest}, noting
\begin{align*}
\int_0^t \dot{G}_r^2 dr \geq \frac{1}{\kappa} \int_0^t \dot{G}_r^2 d[\beta]^\pi_r ,
\end{align*}
we obtain the (still pathwise) estimate
\begin{equation} \label{estLM}
|f_t'( iy + U_t)| \leq \exp \biggl[ \int_0^t \dot{G}_rd^\pi \beta_r - \biggl( \frac{1}{2} + \frac{1}{\kappa} \biggr)\int_0^t \dot{G}_r^2d[\beta]^\pi_r \biggr].
\end{equation}
and then, with $b := 2 (\frac{1}{2} + \frac{1}{\kappa} )$
\begin{align*}
|f_t'( iy + U_t)|^b & \leq \exp \biggl[ b \int_0^t \dot{G}_rd^\pi \beta_r - \frac{b^2}{2} \int_0^t \dot{G}_r^2d[\beta]^\pi_r \biggr].
\end{align*}
Note $b>2$, a consequence of $\kappa <2$. We now assume that, for some $A \in \mathcal{H}_t$,
$$
\beta = N + A
$$
It is then immediate, using $[\beta] = [N]$, that
$$ |f_t'( iy + U_t)| \leq \exp \biggl[ \int_0^t \dot{G}_r dN^\pi_r + (*) - \int_0^t \dot{G}_r^2 d(r + \tfrac{1}{2}[N]^\pi_r) \biggr]
$$
where
$$
(*) = \int_0^t \dot{G}_r d A_r = \int_0^t \dot{G}_r\dot{A}_rdr \leq \epsilon \int_0^t \dot{G}_r^2dr + \frac{1}{4\epsilon} \int_0^t \dot{A}_r^2 dr.
$$
As a consequence, arguing exactly like in obtaining (\ref{estLM})
$$ |f_t'( iy + U_t)| \leq \exp \biggl[ \int_0^t \dot{G}_r dN^\pi_r - \biggl( \frac{1-\epsilon}{\kappa} + \frac{1}{2} \biggr) \int_0^t \dot{G}_r^2 d[N]_r) \biggr] . \exp \biggl[\frac{1}{4\varepsilon} || A ||_{t}^2\biggr].
$$
\section{Finite energy drivers, proof of Theorem \ref{thm:CM} }
We show (i) estimate (\ref{eq:CM}), (ii) existence of Loewner-trace $\gamma$ as a simple curve, (iii) uniform $1/2$-H\"older regularity of $\gamma$, (iv) Lipschitz continuity of $ \gamma(t^2)$, (v) continuity of the Schramm--Loewner in uniform topology, on bounded sets in $\mathcal{H}_T$ and (vi) continuity of the trace in $1+ \epsilon$-variation topology w.r.t. Cameron-Martin topology on the driver.
(i) The proof of estimate (\ref{eq:CM}) is a straight-forward consequence of Theorem \ref{thm:keyest}.
Indeed, let $U$ of finite energy on $[0,t]$, note that $U$ and $\beta$ (with $\beta_s := U_t - U_{t-s}$ here) have zero quadratic variation. Then
\begin{align*}
\log|f_t'( iy + U_t)| & \leq M_t - \int_0^t \dot{G}_r^2 dr
\end{align*}
and conclude with \begin{align*} M_t = \int_0^t \dot{G}_r d\beta_r & = \int_0^t \dot{G}_r\dot{\beta}_rdr \\ &\leq \int_0^t \dot{G}_r^2dr + \frac{1}{4} \int_0^t \dot{\beta}_r^2dr.
\end{align*}
In fact, from Proposition \ref{singular-nonsingular}, since $\beta$ has zero quadratic variation,
\begin{equation*}
\log |f_t'( z +U_t)| = \int_0^t \dot{G}_rd \beta_r - \int_0^t \dot{G}_r^2 dr + \log(\frac{Y_t}{y}) -\log (\frac{X_t^2 + Y_t^2}{ x^2 + y^2}) \label{e:13}
\end{equation*}
and using the same argument as above, we obtain a better bound
\begin{equation}\label{finerCMbound}
|f_t'(x + iy + U_t)| \leq \frac{y}{Y_t}\biggl( 1 + \frac{x^2}{y^2}\biggr)\exp\biggl[\frac{1}{4}||U||_t^2\biggr]
\end{equation}
which implies $|f_t'(z + U_t)|$ remains bounded if $z$ remains in a cone $\{z | |Re(z)| \leq M Im(z)\}$
(ii) This is clear from part (i) and Lemma \ref{decaylemma} in the appendix, where it is shown
\[ \gamma_t = \lim\limits_{y \rightarrow 0+ } f_t(iy + U_t)\] exists as a continuous limit. The fact that $\gamma$ is simple follows e.g. from \cite{Lind} or \cite{LIPGRAPHhuy}.
(iii) We show that \[ ||\gamma||_{\frac{1}{2}} \leq g ( ||U||_{T})\] for some continuous function $ g : [0, \infty) \rightarrow (0 , \infty)$. (In fact, in the proof below reveals the possible choice $g(x) = Ce^{Cx^2}$.) %
Define \[ v(t,y) := \int_0^y |f_t'(ir + U_t)|dr\]Note that, \[|\gamma(t) - f_t(iy + U_t)| \leq v(t,y)\]
and by an application of Koebe's one-quater Theorem,
\begin{equation}\label{koebe}
v(t,y) \geq \frac{y}{4} |f_t'(iy + U_t)|
\end{equation}
In the proof below, we will choose $ y = \sqrt{t-s}$. Now,
\begin{align*}
|\gamma(t) - \gamma(s)| \leq & |\gamma(t) - f_t(U_t + iy) | \\ & + |\gamma(s) - f_s(U_s + iy) | \\
& + |f_t(U_s + iy) - f_s(U_s+ iy)| \\ & + | f_t(U_t + iy) - f_t(U_s + iy)|
\end{align*}
The first two terms are bounded by $ v(t,y)$ and $v(s,y)$ respectively. For the third term, Lemma $3.5$ in \cite{JVL11} and (\ref{koebe}) implies,
\[ |f_t(U_s + iy) - f_s(U_s+ iy)| \leq C v(s,y)\]
For the fourth term, \[ | f_t(U_t + iy) - f_t(U_s + iy)| \leq |U_t - U_s| \sup_{r \in [0,1]}|f_t'(rU_t + (1-r)U_s + iy )|\]
Note that \[|U_t - U_s| \leq y ||U||_{\frac{1}{2}} \] and by Lemma $3.6 $ in \cite{JVL11}, there exist constant $C$ and $ \alpha $ such that
\begin{align*}
|f_t'(rU_t + (1-r)U_s + iy )| & \leq C \max\biggl\{1,\biggl(\frac{|U_t - U_s|}{y}\biggr)^\alpha\biggr\} |f_t'(iy + U_t)| \\ & \leq
C \max\biggl\{1, ||U||_{\frac{1}{2}}^{\alpha}\biggr\} |f_t'(iy + U_t)|
\end{align*}
and using (\ref{koebe}) again,
\[ | f_t(U_t + iy) - f_t(U_s + iy)| \ \leq C ||U||_{\frac{1}{2}} \max\biggl\{1, ||U||_{\frac{1}{2}}^\alpha \biggr \} v(t,y)\]
Finally, from part (i)
\[v(t,y) \leq y \exp\biggl\{\frac{1}{4}||U||_{T}^2 \biggr\} \] \[ ||U||_{\frac{1}{2}} \leq ||U||_{T} \]
giving us
\[ |\gamma_t - \gamma_s| \leq \sqrt{t-s} g(||U||_{T})\]
completing the proof. \\
(iv) We will use the results from \cite{LIPGRAPHhuy} for the proof of this part. In particular, we recall from Theorem $3.1$ in \cite{LIPGRAPHhuy} that if $||U||_{\frac{1}{2}} < 4$, then there exist a $\sigma, c > 0$ such that for all $y > 0$, \begin{equation}\label{huyestimate} \sigma \sqrt{t} \leq Im(f_t(i y + U_t)) \leq \sqrt{y^2 + 4t}
\end{equation}
and from Lemma $2.1$ in \cite{LIPGRAPHhuy} \[ |Re(f_t(i y + U_t))| \leq c \sqrt{t} \]
so that trace $\gamma$ lies inside a cone at 0 and $ |f_t(i \sqrt{t} + U_t )| \leq c\sqrt{t}$. We first assume that $||U||_{\frac{1}{2}, [0,T]} <4$. From the proof of part (iii), we have \[ |\gamma_t - \gamma_s | \lesssim v(t, \sqrt{t-s}) + v(s, \sqrt{t-s}) \]If $ s,t \geq \epsilon$, using bound \ref{finerCMbound}
\[ v(t, \sqrt{t-s}) + v(s, \sqrt{t-s}) \lesssim (\frac{1}{Y_t} + \frac{1}{Y_s}) (t-s) \lesssim \frac{1}{\sqrt{\epsilon}}(t-s)\]which implies $\gamma$ is Lipchitz on $[\epsilon, T]$. For proving $|\gamma_{t^2} - \gamma_{s^2}| \lesssim |t-s|$, note that we can assume $s= 0$, for otherwise we can consider the image of $\gamma$ under conformal map $g_{s^2} - U_{s^2}$ whose derivative of the inverse $ f'(. + U_{s^2})$ remains bounded in a cone. Finally again using \ref{finerCMbound} and \ref{huyestimate},
\begin{align*}
|\gamma_{t^2}| &\leq |\gamma_{t^2} - f_{t^2}(it + U_{t^2})| + |f_{t^2}(it + U_{t^2})| \\ & \lesssim
v(t^2, t) + t \\ & \lesssim \frac{t^2}{Y_{t^2}} + t \\ & \lesssim t
\end{align*}
Finally, if $||U||_{\frac{1}{2}, [0,T]} \geq 4$, we split $ [0,T] = \cup_{k =0}^{n-1} [\frac{kT}{n}, \frac{(k+1)T}{n}]$ such that for each $k$, \\$||U||_{\frac{1}{2}, [\frac{kT}{n}, \frac{(k+1)T}{n}]} < 4$. Note again that \[ \gamma[0,T] = \gamma[0,T/n]\cup f_{\frac{T}{n}}g_{\frac{T}{n}}( \gamma[T/n, T])\]
From above, $\gamma(t^2)$ is Lipchitz on $[0,T/n]$. The chain $ g_{T/n}(\gamma_{T/n + t}) - U_{T/n}, t \in [0, T/n]$ is driven by $ U_{T/n+ t} - U_{T/n}$ and since $f_{T/n}'(. + U_{T/n})$ is bounded (from \ref{finerCMbound} and the fact that trace remains in a cone), $\gamma(t^2)$ is also Lipchitz on $[T/n, 2T/n]$. Iterating this argument then completes the proof.
(v) Consider If $U^n $ is a sequence of Cameron-Martin paths with $||U^n-U||_\infty \rightarrow 0$ and \[ \sup_n ||U^n||_{T} + ||U||_{T} < \infty\] We need to show that
for any $ \alpha < \frac{1}{2}$, \[||\gamma^n - \gamma||_{\alpha} \rightarrow 0.\]
We have,
\begin{align*}
|\gamma^n(t) - \gamma(t)| \leq & |\gamma^n(t) - f_t^n(iy + U_t^n)| \\ &+ |f_t(iy + U_t) - \gamma(t)| \\ &+ |f_t^n(iy + U_t^n) - f_t(iy + U_t)|
\end{align*}
Note that for fixed $y > 0$, \[ \lim\limits_{ n \rightarrow \infty}
|f_t^n(iy + U_t^n) - f_t(iy + U_t)| = 0 \]uniformly in $t $ on compacts. From part (i)
\begin{align*}
|\gamma^n(t) - f_t^n(iy + U_t^n)| + |f_t(iy + U_t) - \gamma(t)| & \leq v^n(t, y) + v(t,y) \\ & \leq y \biggl( \exp\biggl\{\frac{1}{4}||U^n||_{T}^2 \biggr\} + \exp\biggl\{\frac{1}{4}||U||_{T}^2 \biggr\} \biggr )
\end{align*}
Thus,
\[ \lim\limits_{y \rightarrow 0+ } |\gamma^n(t) - f_t^n(iy + U_t^n)| + |f_t(iy + U_t) - \gamma(t)| = 0 \]uniformly in $n$ and $t$. Since $y$ can be chosen arbitrarily small, \[ \lim\limits_{n \rightarrow \infty}
||\gamma^n - \gamma||_{\infty} = 0\]
Finally note that from part (iii)
\[\sup_n ||\gamma^n||_{\frac{1}{2}} < \infty \] and standard interpolation argument concludes the proof.
(vi) Let $U^n$ is a sequence with $||U^n - U||_{T} \rightarrow 0$ as $n \to \infty$. From part (v), we have $||\gamma^n - \gamma||_{\infty} \to 0 $. We claim that $\sup_n ||\gamma^n||_{1-var} < \infty$, which together with standard interpolation argument implies $||\gamma^n - \gamma||_{1+ \epsilon-var} \to 0$ as $n \to \infty$. From the proof of part (iv), we see that if $||U||_{\frac{1}{2}} < 4 - \delta $ (and thus $||U^n||_{\frac{1}{2}} < 4 - \delta$ for $n$ large enough), then $\gamma^n(t^2)$ is Lipchitz uniformly in $n$ and thus $\sup_n ||\gamma^n||_{1-var} < \infty$. \\
If $||U||_{\frac{1}{2}} \geq 4 $, we choose a $m$ large enough and dissect $ [0,T] = \cup_{k=0}^{m-1}[\frac{kT}{m}, \frac{(k+1)T}{m}]$ such that for all $n$ and $ k \leq m-1$, $||U^n||_{\frac{1}{2}, [\frac{kT}{m}, \frac{(k+1)T}{m}]} < 4 - \delta$
and similar iteration argument as in proof of part (iv) again implies $\sup_n ||\gamma^n||_{1-var} < \infty$, completing the proof.
\section{Proof of Corollary \ref{F(B)} }
We consider Loewner drivers of the form $U_t = F(t, B_t)$ where $B$ is a standard Brownian motion. For a fixed time $t > 0$, the process $\beta_s = \beta_s^t = U_{t} - U_{t-s}$ is the time reversal of $U$. Note that $ W_s = B_t - B_{t-s}$ is another Brownian motion and a martingale w.r.t. to its natural completed filtration $\mathcal{F}_s $ satisfying usual hypothesis. We recall the following classical result on expansion of filtration. See \cite[Ch. 6]{Pro90} for detaills.
\begin{theorem}
Brownian motion $W$ remains a semimartingale w.r.t. expanded filtration $\tilde{\mathcal{F}}_s := \mathcal{F}_s \vee \sigma(W_t) = \mathcal{F}_s \vee \sigma(B_t) $. Moreover, \[ W_s = \tilde{W}_s + \int_0^s \frac{W_t - W_r}{t-r} dr \]
where $\tilde{W}$ is a Brownian motion adapted to the filtration $\tilde{\mathcal{F}}$.
\end{theorem}
We now prove that $\beta_s$ is a semimartingale w.r.t. to filtration $\tilde{\mathcal{F}}$ and provide its explicit decomposition into martingale and bounded variation part. More precisely, we claim
\begin{align*}
\beta_s &= \int_0^s F'(t-r, B_{t-r})dW_r + \int_0^s \biggl(\dot{F}(t-r, B_{t-r}) - \frac{1}{2}F^{''}(t-r, B_{t-r})\biggr)dr \\
&= \int_0^s F'(t-r, B_{t-r})d\tilde{W}_r \\ & \qquad+ \int_0^s \biggl(\dot{F}(t-r, B_{t-r}) - \frac{1}{2}F^{''}(t-r, B_{t-r}) + F'(t-r, B_{t-r})\frac{B_{t-r}}{t-r} \biggr)dr
\end{align*}
To see this, just use It\^o's formula, \[\beta_s = \int_{t-s}^t F'(r, B_r )dB_r + \int_0^s \biggl(\dot{F}(t-r, B_{t-r}) + \frac{1}{2}F^{''}(t-r, B_{t-r})\biggr)dr\]
and note that by computing the difference between forward (It\^o) and backward stochastic integral, \[ \int_{t-s}^t F'(r, B_r )dB_r = \int_0^s F'(t-r, B_{t-r})dW_r - \int_0^s F^{''}(t-r, B_{t-r}) dr. \]
The proof of corollary \ref{F(B)} is the completed by application of Theorem \ref{thm:main}.
\begin{remark}
If function $F'(t,x)$ is not space depedent, e.g. $F(t, x) = t^px $ or $ F(t, x) = \sqrt{\kappa}x $, we can apply the formula \[ \int_{t-s}^t F'(r)dB_r = \int_0^s F'(t-r)dW_r\]Note that right-hand side is indeed a martingale w.r.t. the filtration $\mathcal{F}$ and we do not have to work with expanded filtration $\tilde{\mathcal{F}}$. In this case the canonical decomposition of $\beta$ is given by \[ \beta_s = \int_0^s F'(t-r)dW_r + \int_0^s \dot{F}(t-r, B_{t-r})dr\]and Theorem \ref{thm:main} again can be applied considering $\beta$ as a semimartingale w.r.t. the filtration $\mathcal{F}$.
\end{remark}
\section*{Appendix}
We collect some variations on familiar results concerning existence of trace via moments of $f'$.
\begin{lemma}\label{decaylemma} Suppose there exist a $\theta < 1$ and $ y_0 > 0$ such that for all $ y \in (0 , y_0] $
\begin{equation}\label{neededbound} \sup_{ t \in [0,T]} |f_t'(iy + U_t )| \leq y^{-\theta}
\end{equation}
then the trace exists.
\end{lemma}
\begin{proof}Note that for $ y_1 < y_2 < y_0$,
\[| f_t(iy_2 + u_t) - f_t(iy_1 + U_t) | = | \int_{y_1}^{y_2} f_t'( ir + U_t) dr| \leq \int_{y_1}^{y_2} r^{-\theta} dr = \frac{1}{1-\theta}(y_2^{1-\theta} - y_1^{1 - \theta} ) \]which implies that $f_t(iy + U_t)$ is Cauchy in $y $ and thus \[ \gamma_t = \lim\limits_{y \rightarrow 0+ } f_t(iy + U_t)\]exists. For continuity of $\gamma$, observe that
\[ |\gamma_t - f_t(iy + U_t) | \leq \frac{y^{1-\theta}}{1-\theta} \]
Now,
\begin{align*}
|\gamma_t - \gamma_s| & \leq |\gamma_t - f_t(iy + U_t) | + |\gamma_s- f_s(iy + U_s) | + | f_t(iy + U_t) - f_s(iy + U_s)| \\
& \lesssim y^{1 -\theta} + | f_t(iy + U_t) - f_s(iy + U_s)|
\end{align*}
It is easy to see that for $ y > 0$, \[ \lim\limits_{s \rightarrow t } | f_t(iy + U_t) - f_s(iy + U_s)| = 0\]and since $y$ was arbitrary, this concludes the proof.
\end{proof}
\begin{lemma}\label{momentbound}If $U$ is weakly $\frac{1}{2}$-Holder and there exist constant $ b > 2 $, $ \theta < 1$ and $ C < \infty$ such that for all $t \in [0,T]$ and $y > 0$
\[ \mathbb{P}[ |f_t'(iy + U_t)| \geq y^{-\theta}] \leq Cy^b\] then the trace exists.
\end{lemma}
\begin{proof} By using of Borel-Cantelli lemma, it is easy that almost surely for $n$ large enough,
\[ |f_{k2^{-2n}}'( i 2^{-n} + U_{ k 2^{-2n}})| \leq 2^{ n \theta} \]
for all $ k = 0, 1, .., 2^{2n} - 1$. Now applying results in section $3$ of \cite{JVL11} (Lemma $3.7$ and distortion Theorem in particular ) completes the proof.
\end{proof}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Karim Ziad né en 1966 à Alger, en Algérie, est un batteur, chanteur et compositeur algérien de world music, de jazz et de pop. Ziad sami layounne
A beautiful baby kid
Biographie
Il commence à jouer de la musique aux environs de l'âge de 4 à 5 ans et était membre, avec le guitariste, Khliff Miziallaoua, de Khindjar, un groupe de hard rock à Alger, formation par laquelle passeront également le batteur Nasser Menia et le guitariste Sami Ben M'hidi, puis, avec le pianiste Mustapha Mataoui, dans Sweet Jazz. Ces deux groupes passeront en 1987 à la salle Ibn Zeydoun, une importante salle de concert d'Alger.
Il commence à jouer pour les mariages, ce qui lui permet de payer ses études universitaires en biologie et part à Paris, en France, en 1988, où il s'inscrit au conservatoire, à l'École supérieure de batterie Emmanuel Boursault. Il est principalement influencé par Chick Corea et Keith Jarrett.
En France, il joue avec Jeff Gardner, Khaled, Safy Boutella, ainsi qu'avec Khliff Miziallaoua et Youcef Boukella de l'Orchestre national de Barbès, huit ans avec Cheb Mami, et en 1999 fait une tournée de 150 concerts avec Joe Zawinul, puis avec Nguyên Lê dans Maghreb & Friends.
Il forme le groupe Ifrikya avec qui il enregistre un album éponyme en 2001. On retrouve sur cet album, Nguyên Lê, Bojan Zulfikarpasic au piano, Michel Alibo à la guitare basse et Jean-Philippe Rykiel aux claviers. Karim Ziad s'y mêle aux rythmes gnaoua, dont il est un grand défenseur de la culture, du mâalem Abdelkebir Merchane de Marrakech et à la flûte peule d'Aly Wagué.
En 2001, il devient directeur artistique, aux côtés de Abdeslam Alikane du Festival d'Essaouira Gnaoua - Musiques du monde.
En , il participe au festival « Boulevard de jeunes musiciens », à Casablanca, au Maroc
En 2009, il joue avec le groupe Ifrikya, au Trafó, à Budapest, en Hongrie.
Il joue en trio dans une tournée internationale depuis 2015 avec Amazigh Kateb et Ptit Moh, tous deux anciens de Gnawa Diffusion , notamment au festival Oct-Loft jazz de Shenzhen, en Chine, et doit sortir avec celui-ci un album en .
Discographie
2001 : Ifrikya (Feldafing)
2004 : Chabiba (Label Sauvage)
2009 : Dawi (Naxos Digital Services/Intuition)
2010 : Yobadi avec Hamid Kasri (Accords Croises)
2014 : Jdid (JMS/Sphinx distribution)
20?? : Gnawi, avec safi, Gnawa Diffusion
20?? : El kolektor, avec Cheikh Sidi Bémol, Mugar (Beida)
Annexes
Notes et références
Liens externes
, entretien.
, entretien
Batteur algérien
Chanteur algérien
Artiste d'ACT Music
Naissance en 1966
Naissance à Alger
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Going to the medicine cabinet and grabbing another aspirin or ibuprofen to knock out a headache is second nature for many of us. However, if you grab too many you're setting yourself up for another headache that could come with serious consequences. Rebound headaches, also known as medication overuse headaches or analgesic rebounds, are caused by taking too many over-the-counter (OTC) or prescription pain relievers (analgesics).
YOU MAY ALSO LIKE: Headache Locations: What Do They Mean?
A new national survey found that while nearly all consumers (97%) say they feel confident when choosing which OTC pain reliever to take, many are in fact disregarding important safety considerations.
The frequent usage of pain relievers rewires the pain pathways in your brain and tells it, "I need more medicine to make me feel better." According to a report from Medical Daily, about 50 percent of migraines and 25 percent of all headaches are linked to pain medication overuse.
Overuse of migraine or headache medication can also cause the medication to stop working and damage the liver and kidneys, according to the National Headache Foundation.
Medical Daily reports that some people with specific types of serotonin profiles are more susceptible to medication overuse, which puts them at a greater risk for rebound headaches.
If you take pain medication for your migraines and headaches consistently (everyday or every other day) you're at risk.
According to the Migraine Trust people who have a family history of rebound headaches are more likely to get them. They generally don't occur in people who take pain medication for anything other than headaches or migraines.
The headaches often subside when you take the pain-relief medication, but returns thereafter when the medication wears off.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
The Ising model is a classic paradigm of statistical mechanics, and
continues to find powerful application in diverse areas of modern
physics.\cite{Lenz,mccoy,mccoy1} It also reveals unique and generic universal
behavior associated with boundaries.~\cite{bariev,cardy84,cardy} In its quantum
1-D chain version, the critical boundary Ising model (BIM) reads
\begin{eqnarray}
\label{bim}
H =- \sum_{i=0}^{\infty} [\sigma^z_i \sigma^z_{i+1}+\sigma^x_i ]- h_B \sigma^z_0.
\end{eqnarray}
The uniform field along $x$ is fixed such that the bulk
system is at the critical point between Ising order and the disordered
phase. The symmetry $\sigma^z \leftrightarrow - \sigma^z$ is broken
when a finite magnetic field $h_B \ne 0$ is applied at the point
boundary. Such a boundary field cannot lead to a finite \emph{bulk}
magnetization. Importantly however, it does cause a renormalization
group flow from a free boundary condition $h_B=0$ to a fixed boundary
condition $h_B\to\pm \infty$.
The renormalization group flow associated with this BIM has been shown
to be at the heart of boundary critical phenomena occurring in a
surprising variety of low-dimensional correlated electron systems,
such as Luttinger liquids containing an impurity,\cite{Leclair}
coupled bulk and edge states in non-abelian fractional quantum Hall
states,\cite{rosennow,Bishara} as well as quantum dots near the two
impurity Kondo\cite{CFT2IKM,SelaAffleck} or the two channel
Kondo\cite{SelaMitchellFritz,Mitchell2011} critical
points.
In the continuum, the BIM is in fact integrable,\cite{ghosal} both in
the massless bulk critical case, and also in the massive regime away
from the critical point. Certain correlation functions can then be
calculated exactly using Form Factor methods;\cite{konik,schuricht}
although in the bulk critical case relevant to Eq.~(\ref{bim}), many
important quantities cannot be easily obtained due to the
proliferation of many-particle excitations.
On the other hand, Chatterjee and Zamolodchikov\cite{cz} (CZ) showed
that conformal field theory imposes linear differential equations
which fully determine correlation functions in this limit.
Of course, conformal field theory has been used
previously for systems with conformal-invariant boundary
conditions.\cite{cardy84,cardy} The remarkable feature of the result
of CZ is that the correlation functions are still determined by
differential equations even for non-conformal invariant boundary
conditions obtained at finite boundary field.
The method of CZ was applied to the calculation of magnetization as a
function of distance from the boundary, $x$. On the semi-infinite
plane, equivalent to the quantum 1-D model, Eq.~(\ref{bim}), at zero
temperature, their result reads\cite{cz}
\begin{eqnarray}
\label{cz}
\langle \sigma(x) \rangle_h = 2^{13/8} \sqrt{\pi} h x^{3/8} \Psi(1/2,1;8 \pi h^2 x),
\end{eqnarray}
where $\Psi$ is a degenerate hypergeometric function. Here $\langle
\sigma(x) \rangle$ has the standard field-theory normalization, which
we emphasize is only \emph{proportional} to $\langle\sigma^z_j\rangle$
of a particular lattice model, such as Eq.~(\ref{bim}). Indeed, $x
\propto j$, and $h\propto h_B$ provided that $h_B \ll 1$. At short
distance one thus obtains,
\begin{equation}
\label{shortd}
\langle \sigma(x) \rangle = -2^{13/8}h x^{3/8} [\ln (x) + \mathcal{O}(1)]+
\mathcal{O}(x^{7/8}) \qquad \text{for} ~x \ll 1,
\end{equation}
and at long distances $\langle \sigma(x) \rangle \rightarrow
(2/x)^{1/8}$, corresponding exactly to the result for fixed boundary
condition, obtained from boundary conformal field
theory.\cite{cardyle} As such, the exact function, Eq.~(\ref{cz}),
captures the full crossover behavior between two fixed points where
conformal invariant boundary conditions do hold.
Since the problem for finite $h$ does not in general possess
conformal invariance at the boundary, it is not possible to generalize
Eq.~(\ref{cz}) to other geometries by means of a simple conformal
mapping. However the method of CZ can be applied directly to other
geometries, yielding a new set of differential equations
(this was demonstrated explicitly for the 2-D disk geometry by
CZ\cite{cz}). Similarly, Leclair, Lesage and Saleur\cite{Leclair}
(LLS) applied the method to the geometry of a semi-infinite
cylinder. In the present paper we shall be concerned with this
semi-infinite cylinder geometry, whose boundary consists of a circle
with circumference $\beta$, at which the boundary field $h$ is
applied. This classical 2-D Ising model is equivalent to the \emph{quantum}
chain model Eq.~(\ref{bim}) at \emph{finite} temperature $T\propto \beta^{-1}$ (see e.g. Ref.\onlinecite{cardybook}).
We re-examine the result of LLS for the local magnetization in
Sec.~\ref{se:saleurformula}. Whereas those nonperturbative results
give the full $x$-dependence of the local magnetization for any $h$
and $\beta$, we find that a more general ansatz for the local
magnetization allows for an additional multiplicative factor $f(2\beta
h^2)$. The physical meaning of this missing factor is then
explained. The full scaling function for the local magnetization is
determined in Sec.~\ref{se:f}; while the lattice model Eq.~(\ref{bim})
is studied directly in Sec.~\ref{se:num}. The local magnetization on
the lattice is calculated numerically, and the results compared with
the refined analytic solution, showing excellent agreement. The paper
ends with a short summary, where implications and applications of the
results are discussed.
\section{Refinement of earlier results}
\label{se:saleurformula}
LLS considered a classical Ising model on the half-cylinder in the continuum
limit.\cite{Leclair} They calculated the local magnetization as a
function of the distance $x$ from the circular boundary of
circumference $\beta$, which was conveniently written in the form\cite{Leclair}
\begin{equation}
\label{sig}
\langle \sigma(x) \rangle = \left ( \frac{1}{\sinh \frac{2 \pi
x}{\beta} }\right)^{1/8} g(X),
\end{equation}
with $X= (1-\coth \frac{2 \pi x}{\beta} )/2$ and where $\langle
\sigma(x) \rangle$ is independent of $\tau \in (0,\beta)$ due to
translation symmetry
along the boundary. LLS derived a linear differential equation for $g(X)$,
which reads\cite{Leclair}
\begin{equation}
\label{difeq}
\left( (X - X^2)\frac{d^2}{(dX)^2}+\left( 1+\frac{\Lambda}{2}-2X \right) \frac{d}{dX} - \frac{1}{4}\right) g(X)=0,
\end{equation}
parametrized in terms of $\Lambda=2\beta h^2$. Their solution is\cite{Leclair}
\begin{equation}
\label{sigma}
\langle \sigma(x) \rangle_{LLS} = \left( \frac{\frac{4 \pi}{\beta}}{\sinh \frac{2 \pi x}{\beta}} \right)^{1/8}
~_{2}F_1\left(\frac{1}{2},\frac{1}{2};1+2 \beta h^2,\frac{1-\coth \frac{2 \pi x}{\beta}}{2} \right),
\end{equation}
where $_{2}F_1(a,b,c,z)$ is the Gauss hypergeometric function. Below we will use its integral representation
\begin{eqnarray}
\label{intrep}
_{2}F_1(a,b,c,z) = \frac{\Gamma[c]}{\Gamma[b]\Gamma[c-b]} \int_0^1 dt \frac{t^{b-1}(1-t)^{c-b-1}}{(1- t z)^a},
\end{eqnarray} where $\Gamma(y)$ is the gamma function. This
result is normalized with an overall constant such that at long
distances $\langle \sigma(x) \rangle \to \left( \frac{\frac{4
\pi}{\beta}}{\sinh \frac{2 \pi x}{\beta}} \right)^{1/8}$
recovers the expected result for fixed boundary conditions (taking
$\beta\rightarrow \infty$ then yields $\langle \sigma(x) \rangle \rightarrow
(2/x)^{1/8}$, consistent with Ref.~\onlinecite{cardyle}).
In this paper we point out that the differential equation
Eq.~(\ref{difeq}) leaves a freedom which goes beyond an overall
normalization constant. Unlike the zero-temperature case
(corresponding to the semi-infinite plane, $\beta\rightarrow \infty$),
here the normalization of Eq.~(\ref{sigma}) can itself be a scaling
function of $\Lambda$. We thus replace Eq.~(\ref{sigma}) by the more
general ansatz,
\begin{eqnarray}
\label{sigmaf}
\langle \sigma(x) \rangle_{h , \beta} =f(2\beta h^2) \times \left( \frac{\frac{4 \pi}{\beta}}{\sinh \frac{2 \pi x}{\beta}} \right)^{1/8}
~_{2}F_1\left(\frac{1}{2},\frac{1}{2};1+2 \beta h^2,\frac{1-\coth
\frac{2 \pi x}{\beta}}{2} \right),
\end{eqnarray}
which depends explicitly on the function $f(2\beta h^2)$,
determined in Sec.~\ref{se:f}, below. Eq.~(\ref{sigmaf}) implies $f (\infty)=
1$, so that $\langle \sigma(x) \rangle_{h , \beta}$ recovers
asymptotically the behavior of the fixed boundary condition fixed point.
We note that the same subtlety occurs with other geometries, as
highlighted by CZ in the case of the disk.\cite{cz} In that case, the
additional scale in the problem is the disk radius $R$; and an
additional scaling function of $R h^2$ (analogous to our
$\Lambda=2\beta h^2$) appears in the expression for the local
magnetization. As in the present case, this function is not fixed by
the linear differential equations.\cite{cz}
Finally, we comment briefly upon the physical significance of the
scaling function $f(2\beta h^2)$. It describes the dependence on the
additional thermal scale influencing the renormalization group flow at
$T \ne 0$. In accord with physical expectation, the renormalization
group flow is cut off at the external scale given by $\max \{ T ,
x^{-1} \}$. Since $h$ grows under renormalization (and has scaling
dimension $1/2$),~\cite{cardy84,cardy} we should consider two regimes depending on the ratio
between this external scale and the field-induced scale $\sim h^2$:
\begin{eqnarray}
\max \{ T , x^{-1} \} & \gg & h^2:{\rm{~free~boundary~condition}}, \nonumber \\
\max \{ T , x^{-1} \} & \ll & h^2:{\rm{~fixed~boundary~condition}}.
\end{eqnarray}
These regimes are illustrated in Fig.~1. The important consequence
following from this is that at finite temperatures, the fixed boundary
condition fixed point is not always reached on taking $x \to
\infty$. The single scaling function $f(2\beta h^2)$ thus describes the
crossover from free to fixed boundary condition at a given $x$, upon
decreasing temperature. Obviously its effect is most apparent at large
$x$, since there is no crossover at small $x$. However, as suggested
by Fig.~1, the system is always close to the free boundary condition
fixed point at small $x$, and this fact will prove useful in
determining $f(2\beta h^2)$, as considered in the next section.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=70mm]{fig1.eps}
\caption{\label{fig:pd} Schematic phase diagram of the BIM as a
function of temperature and distance from the boundary. The dashed
line denotes the crossover between free and fixed boundary
conditions occurring when $\max \{ T , x^{-1} \} \sim h^2$.}
\end{center}
\end{figure}
\section{Determination of the scaling function $f(2 \beta h^2)$}
\label{se:f}
In this section we find the function $f(2 \beta h^2)$ appearing
in Eq.~(\ref{sigmaf}). Since this function is a scaling function of
$\beta h^2$ and does not depend on distance $x$, it could in principle be
determined at any given $x$. While its influence is most pronounced at
large $x$, where the system undergoes a crossover as function of $T$
(see Fig.~1), here we determine $f(2 \beta h^2)$ by exploring the
small $x$ behavior, where the system remains close to free
boundary condition fixed point. Importantly, the resulting behavior at
small $x$ is \emph{perturbative} in $h$ regardless of $\beta h^2$, as
shown explicitly below.
First we note that at both large and small $\Lambda$, the
short-distance behavior of $\langle \sigma(x) \rangle$ is linear in
$h$. As $\Lambda\rightarrow \infty$, one sees this directly from the small
$x$ expansion of the exact $T=0$ result of CZ, Eq.~(\ref{shortd}).
In the opposite limit $\Lambda\rightarrow 0$, the behavior is by
definition perturbative in $h$, and so the leading
correction to magnetization is of course also linear in $h$.
In the next subsection, we perform first-order perturbation theory
in the boundary field $h$, with respect to the free boundary condition
fixed point. The key point is that its short-distance behavior yields
\emph{precisely} Eq.~(\ref{shortd}), implying that
\begin{equation}
\label{shortdT}
\langle \sigma(x) \rangle_{h ,\beta} = -2^{13/8}h x^{3/8} [\ln (x) +
\mathcal{O}(1)] +
\mathcal{O}(x^{7/8}) \qquad \text{for} ~x \ll \beta, h^{-2}
\end{equation}
holds at short distances $x \ll \beta, h^{-2}$ for \emph{any}
$\Lambda$. Naively one might think that the coefficient of the
$x^{3/8} [\ln (x) +\mathcal{O}(1)]$ term could be renormalized by
higher orders in $h$. But the scaling form of the problem implies that
every power of $h$ is accompanied by a power of $\sqrt{\beta}$ [or
$\sqrt{x}$ which gives a subleading $x$ dependence to
Eq.~(\ref{shortdT})]. Such terms diverge as $T \to 0$, and so this
renormalization is not consistent with the exact
nonperturbative $T=0$ result, Eq.~(\ref{cz}), which is
well-behaved at short distances, Eq.~(\ref{shortd}).
Finally, we consider the short distance expansion of our ansatz
Eq.~(\ref{sigmaf}), which using Eq.~(\ref{intrep}) is found to be
\begin{eqnarray}
\label{seriesG}
\langle \sigma(x) \rangle_{h ,\beta} = - f(2 \beta h^2) \frac{2^{9/8}}{\sqrt{\beta } } \frac{ \Gamma[1+2 \beta h^2] }{ \Gamma[\tfrac{1}{2}+2 \beta h^2] } x^{3/8} [\ln (x)+\mathcal{O}(1)]+\mathcal{O}(x^{7/8}) \qquad \text{for} ~x \ll 1.
\end{eqnarray}
Comparing
Eqs.~(\ref{shortdT}) and (\ref{seriesG}) we now obtain the scaling function
\begin{equation}
\label{fresult}
f(\Lambda) =\sqrt{\Lambda} \frac{\Gamma[\tfrac{1}{2}+\Lambda] }{ \Gamma[1+\Lambda] }.
\end{equation}
This function increases monotonically as shown in Fig~ 2, and has asymptotic limits $ f(\Lambda \ll 1) \approx \sqrt{\pi \Lambda} $ and $ f(\Lambda \gg 1) \approx 1- \frac{1}{8 \Lambda} $.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=60mm]{fig2.eps}
\caption{\label{fig:ff} Plot of Eq.~(\ref{fresult}).}
\end{center}
\end{figure}
We note that a similar perturbative method was used by CZ to fix the
scaling function of $R h^2$ for the disk geometry.\cite{cz}
\subsection{Perturbation theory in the boundary field $h$}
\label{ptnfl}
In this subsection we show that the form of Eq.~(\ref{shortdT}) indeed
follows from perturbation theory around the free boundary condition
fixed point. We obtain the full $x/ \beta$ dependence of the
magnetization at small $h$, recovering perturbatively the
$\Lambda\rightarrow 0$ limit of Eq.~(\ref{sigmaf}).
The continuum limit of the critical classical 2-D Ising model
is described by a $c=1/2$ conformal field theory, which admits a
Lagrangian formulation in terms of the free massless
Majorana Fermi field ($\psi$,$\bar{\psi}$), with the action
\begin{equation}
S_{0} = \frac{1}{2 \pi} \int d^2 z [\psi \partial_{\bar{z}} \psi + \bar{\psi} \partial_z \bar{\psi}].
\end{equation}
Here $(z,\bar{z}) = (\tau+i x,\tau-i x)$ are complex coordinates and $d^2 z = d\tau d x$.
In the presence of a boundary $\mathcal{B}$ with a magnetic field $h$,
the action can be decomposed into a bulk part and a boundary part,
\begin{equation}
\label{deltaS}
S = \frac{1}{2 \pi} \int_{\mathcal{D}} d^2 z [\psi \partial_{\bar{z}} \psi + \bar{\psi} \partial_z \bar{\psi}] +h \int_{\mathcal{B}} \sigma_{B}.
\end{equation}
The boundary operator $\sigma_{B}$ was identified in
Refs.~\onlinecite{cardy84,cardy} with a dimension $1/2$ operator
$\sigma_{B}(\tau) \sim \psi(\tau,x=0)$, associated with the fermion
field at the boundary. In our case $\mathcal{B} = \partial
\mathcal{D}$ is a circle
parametrized by $\tau \in [0,\beta]$ and $\mathcal{D}$ is the
semi-infinite cylinder. Following Cardy's method of
images~\cite{cardy84} the one point function of the magnetization is
$\sigma(z_1,z_2) = \langle \sigma_L(z_1) \sigma_L(z_2) \rangle$,
where $\sigma_L(z)$ is a dimension $1/16$ left moving field living in
the geometry of the \emph{infinite} cylinder. We then obtain conformal
invariant boundary conditions, with the `boundary' at $x=0$. The
boundary field $h$ is now considered as a perturbation to the free
boundary condition fixed point. To first order in $h$,
\begin{eqnarray}
\sigma^{(1)}(z_1,z_2) = h \int_0^\beta d \tau \langle \sigma(z_1) \sigma_{B}(0,\tau) \sigma(z_2) \rangle.
\end{eqnarray}
The 3-point function appearing in the integrand is fully determined by conformal invariance, and one obtains up to a normalization constant $N$
\begin{eqnarray}
\label{tpf}
\sigma^{(1)}(z_1,z_2) = h N \left(\frac{\sin \frac{\pi}{\beta} (z_1 -z_2)}{\frac{\pi}{\beta}} \right)^{3/8}
\int_0^\beta d \tau ~\frac{\frac{\pi}{\beta}}{\left[\sin \left( \frac{\pi}{\beta} (\tau -z_1) \right) \sin \left( \frac{\pi}{\beta} (\tau -z_2) \right) \right]^{1/2}}.
\end{eqnarray}
The physical magnetization is obtained
by setting $\langle \sigma(x) \rangle = \langle \sigma(z_1=i x,z_2 = -
i x) \rangle$. We now take $z_1=i x$, $z_2 = - i x$ in Eq.~\ref{tpf} and use the trigonometric identity,
\begin{eqnarray}
\label{trig}
2 \sin \left( \frac{\pi}{\beta} (\tau -z_1) \right) \sin \left( \frac{\pi}{\beta} (\tau -z_2) \right)=\cos \left( \frac{2 i \pi x}{\beta} \right)
- \cos \left( \frac{2 \pi \tau}{\beta} \right).
\end{eqnarray}
The integral in Eq.~(\ref{tpf}) then becomes
\begin{eqnarray}
\label{integral}
\int_0^{2 \pi} \frac{d \theta}{\sqrt{\frac{w^{1/2}+w^{-1/2}}{2} -\cos \theta}} = \frac{2 \pi}{\sqrt{\sinh \frac{2 \pi x}{\beta}}}
~_{2}F_1\left(\frac{1}{2},\frac{1}{2};1,\frac{1-\coth \frac{2 \pi
x}{\beta}}{2} \right),
\end{eqnarray}
in terms of $w=e^{- 4 \pi x/\beta }$ and $\theta =2 \pi \tau
/\beta$. The constant $N$ was carefully accounted for by CZ.\cite{cz}
Using this and Eq.~(\ref{integral}), first-order
perturbation theory in the boundary field $h$ yields
\begin{eqnarray}
\label{pt}
\langle \sigma(x) \rangle_{ \beta}^{(1)} =h \sqrt{2 \pi \beta } \left( \frac{\frac{4 \pi}{\beta}}{\sinh \frac{2 \pi x}{\beta}} \right)^{1/8}
~_{2}F_1\left(\frac{1}{2},\frac{1}{2};1,\frac{1-\coth \frac{2 \pi x}{\beta}}{2} \right)+ \mathcal{O}(h^2).
\end{eqnarray}
It is interesting to compare this with the full result of LLS,
Eq.~(\ref{sigma}). At small $h$, both carry the same $x/\beta$
dependence; however LLS miss the overall \emph{linear} dependence on
$h$, accounted for by the function $f(2\beta h^2)$ in
Eq.~(\ref{sigmaf}).
The short distance behavior of Eq.~(\ref{pt}) is precisely Eq.~(\ref{shortdT}).
\section{Demonstration with numerical solution}
\label{se:num}
In this section we demonstrate the validity of Eq.~(\ref{sigmaf}) as
the continuum limit of the lattice magnetization
\begin{eqnarray}
\label{sigmalattice}\langle
\sigma(j,h_B,T) \rangle \equiv \frac{{\rm{Tr}} (e^{- H/T} \sigma^z_j )}{{\rm{Tr}} (e^{- H/T})},
\end{eqnarray}
where $H$ is the Hamiltonian of the lattice model,
Eq.~(\ref{bim}). The quantum boundary Ising chain can be solved by
applying a Jordan-Wigner transformation, which yields a quadratic
fermionic Hamiltonian. The magnetization is nonlocal in terms of these
fermions: calculation of $\sigma(j,h_B,T) \rangle$ is then equivalent to
evaluation of the determinant of a matrix whose elements are fermionic
correlation functions.\cite{sachdev} We construct these analytically,
but ultimately evaluate them numerically. Details of this calculation
follow in Sec.~\ref{app:num}. Here we pre-empt that discussion and
present our numerical results, comparing to the refined exact
expression, Eq.~(\ref{sigmaf}).
The continuum limit expression Eq.~(\ref{sigmaf}) admits the scaling form
\begin{eqnarray}
\langle \sigma(x,h,T) \rangle= T^{1/8} \mathcal{F}[x/\beta,2\beta h^2].
\end{eqnarray}
For this function to be a continuum limit of the lattice magnetization, there should exist nonuniversal constants $c,c_x,c_h$ such that
\begin{equation}
\label{fit}
\langle \sigma(j,h_B,T) \rangle= c T^{1/8} \mathcal{F}[c_x j/\beta,2 c_h \beta h_B^2]
\end{equation}
is satisfied for all $j, h_B$ and $T$, as long as distances are large compared to the lattice constant, $j \gg 1 $, and the energy scales $h_B$ and $T$ are small compared to the lattice cutoff scale, $h_B,T \ll 1$.
The constant $c_x$ is related to the velocity $v$ of bulk excitations via $c_x = v^{-1}$. This follows from the requirement that the exponential decay at long distances $j \gg \beta$ is given by~\cite{Giamarchi} $\langle \sigma (x=j) \rangle = \langle \sigma(z_1= i x) \sigma(z_2 = -i x) \rangle \to e^{-(2 \nu) \pi (2 j) / (v \beta) }$, with $\nu = 1/16$ being the scaling dimension of the chiral $\sigma$ field. In our model we obtain $c_x = 1/2$ exactly. $c$ is an overall factor relating the lattice magnetization to the field theory one, and $c_h$
relates the (squared) boundary field, $h_B$, in the lattice model to
$h$ appearing in the continuum action, Eq.~(\ref{deltaS}).
We determine $c$ and $c_h$ by demanding that the ratio
\begin{equation}
\frac{\langle \sigma(j,h_B,T) \rangle}{ c T^{1/8} \mathcal{F}[c_x j/\beta,2 c_h \beta h_B^2]}
\end{equation}
is equal to unity for all $h_B$. The best fit from our numerical data
was obtained for $c_h\simeq 0.161$ and $c\simeq 0.729$.
As shown in Fig.~3, we obtain essentially perfect agreement between
numerical calculations and field theoretical predictions for the
magnetization as a full function of distance, over a wide range of
$2 h^2 / T = 2 c_h h_B^2 / T$.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=100mm]{fig3.eps}
\caption{\label{fig:num} Comparison of analytical result using Eq.~(\ref{sigmaf}) (full lines) and numerical results (dashed lines) for fixed temperature $T = 0.5 ~10^{-3}$ and varying boundary field $h_B = 10^{-3+n/4} / \sqrt{2}$, $n=0,1,...,10$ increasing from bottom to top [explicitly $h_B = 0.000707, 0.00125, 0.00223, 0.00397, 0.00707,
0.0125, 0.0223, 0.0397, 0.0707, 0.125, 0.223$].}
\end{center}
\end{figure}
Fig.~3 should be seen as confirmation that the $x$ dependence of
the magnetization is described by the LLS result,
Eq.~(\ref{sigma}). However, the full dependence on $h$, $x$ and $T$ ---
capturing the evolution from the $T \ll h^2$ result of CZ, Eq.~(\ref{cz}),
to the perturbative $T \gg h^2$ result, Eq.~(\ref{pt}) --- is only
recovered on inclusion of the factor $f(2\beta h^2)$ appearing in
Eq.~(\ref{sigmaf}).
\subsection{Modified Jordan-Wigner transformation and construction of
the magnetization determinant}
\label{app:num}
We now describe the calculation of the magnetization using a fermionic
representation of the transverse field quantum Ising chain,
Eq.~(\ref{bim}). We start from a finite lattice with $L$ sites,
\begin{eqnarray}
\label{bimL}
H_L =- \sum_{i=0}^{L-2} \sigma^z_i \sigma^z_{i+1}- \sum_{i=0}^{L-1} \sigma^x_i - h_B \sigma^z_0,
\end{eqnarray}
with boundary field $h_B$ at site $j=0$; and with free boundary
conditions at site $j=L-1$. Ultimately we will take the $L \to \infty$
limit to avoid finite size effects.
Consider first the usual Jordan-Wigner representation of the Pauli matrices $\tau_j$ $(j=0,...,L-1)$,
\begin{eqnarray}
\tau^x_j &=& i \gamma_{B,j} \gamma_{A,j},\nonumber \\
\tau^z_j &=& - \left(\prod_{\ell=0}^{j-1} i \gamma_{A,\ell} \gamma_{B,\ell} \right) \gamma_{B,j},
\end{eqnarray}
in terms of self-Hermitian (Majorana) lattice fermions $\gamma_{A(B),j}$, satisfying $\{ \gamma_{A,j} , \gamma_{A,j'} \} = 2 \delta_{jj'}$, $\{ \gamma_{B,j} , \gamma_{B,j'} \} = 2 \delta_{jj'}$, $\{ \gamma_{A,j} , \gamma_{B,j'} \} =0$. Here, $\tau^y_j$ can be obtained from $i \tau^y_j = \tau^z_j \tau^x_j$.
Employing this representation for Eq.~(\ref{bimL}), one obtains a
linear term involving a single fermionic operator representing the boundary spin
operator, $\tau^z_0=- \gamma_{B,0}$. This proves to be inconvenient
in the following, and so we use a modified fermionic representation of
the spins to eliminate this linear term from the Hamiltonian.
Specifically, we introduce an extra boundary Majorana fermion
$\gamma$ (with $\gamma^2=1$), which anticommutes with all other
fermions $ \gamma_{A,j}$ and $\gamma_{B,j}$. It can be checked that,
if $[\tau_j^a,\tau_{j'}^b] =2 i \epsilon^{abc} \delta_{jj'} \tau^c_j$,
then $\{\sigma_j^x,\sigma_j^y,\sigma_j^z \} \equiv \{\sigma_j^x, i
\gamma \sigma_j^y, i \gamma \sigma_j^z \}$ also satisfy
$[\sigma_j^a,\sigma_{j'}^b] =2 i \epsilon^{abc} \delta_{jj'}
\sigma^c_j$. Thus we work with the modified Jordan-Wigner
representation
\begin{eqnarray}
\label{jw}
\sigma^x_j &=& i \gamma_{B,j} \gamma_{A,j},\nonumber \\
\sigma^z_j &=& -i \gamma \left(\prod_{\ell=0}^{j-1} i \gamma_{A,\ell} \gamma_{B,\ell} \right) \gamma_{B,j}.
\end{eqnarray}
This is formally equivalent to embedding the spins in a larger Hilbert
space. The model Eq.~(\ref{bimL}) now becomes a tight binding model of
Majorana fermions, containing only \emph{quadratic} terms:
\begin{eqnarray}
\label{Hbp}
H_L = \sum_{j=0}^{L-2} i \gamma_{A,j } \gamma_{B,j+1 }+ \sum_{j=0}^{L-1} i \gamma_{A,j} \gamma_{B,j} + h_B i \gamma \gamma_{B,0}.
\end{eqnarray}
The model can be straightforwardly diagonalized by introducing the fermionic modes
\begin{equation}
A_n = \frac{1}{2} \sum_{j=0}^{L-1} \left( g_n(j+\tfrac{1}{2}) \gamma_{A,j}+i g_n(j) \gamma_{B,j} \right),~~~(n=1,2,...,L)
\end{equation}
with
\begin{eqnarray}
g_n(j) = \sqrt{\frac{2}{L+\tfrac{1}{2}}} \sin\left( \frac{\pi n}{L+\tfrac{1}{2}} (j+\tfrac{1}{2}) \right)
\end{eqnarray}
satisfying the completeness relation $\sum_{n=1}^L g_n(j) g_n(j') = \sum_{n=1}^L g_n(j+\tfrac{1}{2}) g_n(j'+\tfrac{1}{2})=\delta_{j j'} $. This gives $\{A_n,A_{n'} \} = \delta_{n,n'}$, and
\begin{eqnarray}
\label{gammaA}
\gamma_{A,j} &=& \sum_{n=1}^L g_n(j+\tfrac{1}{2}) (A_n + A_n^\dagger), \nonumber \\
i \gamma_{B,j} &=& \sum_{n=1}^L g_n(j) (A_n - A_n^\dagger).
\end{eqnarray}
The Hamiltonian thus becomes,
\begin{eqnarray}
\label{Hf}
H_L=\sum_{n=1}^L E_n A_n^\dagger A_n + h_B \sum_{n=1}^L g_n(0) \gamma
(A_n - A_n^\dagger),
\end{eqnarray}
with $E_n=4 \cos \frac{\pi n}{2L+1}$, which consists of a band of fermionic levels coupled to a Majorana impurity.
Using Eq.~(\ref{jw}) the magnetization is given by
\begin{equation}
\langle \sigma^z_j \rangle =- i^{j+1} \langle \gamma ~ \gamma_{B,0} \gamma_{A,0} \gamma_{B,1} \gamma_{A,1} ... \gamma_{B,j-1} \gamma_{A,j-1} \gamma_{B,j} \rangle.
\end{equation}
One proceeds using Wicks theorem,\cite{sachdev} applicable for the
quadratic Hamiltonian Eq.~(\ref{Hf}). Due to the bipartite structure
in Eq.~(\ref{Hbp}) it follows that $\langle \gamma_{A,j}
\gamma_{A,j'} \rangle = \langle \gamma \gamma_{A,j'} \rangle =
\langle \gamma_{B,j} \gamma_{B,j'} \rangle = 0$. All nonzero contractions,
including relative signs, are then captured by the determinant
\begin{equation}
\label{det}
\langle \sigma^z_j \rangle = -
\left| \left(
\begin{array}{cccc}
i\langle \gamma \gamma_{B,0} \rangle & i\langle \gamma \gamma_{B,1} \rangle & \cdots & i\langle \gamma \gamma_{B,j} \rangle \\
i\langle \gamma_{A,0} \gamma_{B,0} \rangle & i\langle \gamma_{A,0} \gamma_{B,1} \rangle & \cdots & i\langle \gamma_{A,0} \gamma_{B,j} \rangle \\
\vdots & \vdots & \ddots & \vdots \\
i\langle \gamma_{A,j-1} \gamma_{B,0} \rangle & i\langle \gamma_{A,j-1} \gamma_{B,1} \rangle & \cdots & i\langle \gamma_{A,j-1} \gamma_{B,j} \rangle \\
\end{array} \right)
\right|.
\end{equation}
The calculation of the fermionic correlators in Eq.~(\ref{det}) can
be done by exact Green function resummation, treating the problem
as a noninteracting impurity model.\cite{hewson} In Eq.~(\ref{Hf})
we have a quasi-continuum of modes labeled by $n$ coupled to a
localized impurity state $\gamma$. The Green functions for $n$-modes
and for the localized state are defined as
\begin{eqnarray}
\hat{G}_{nn'}(\tau) &=& - \langle \mathcal{T} \left(
\begin{array}{c}
A_n(\tau) \\
A_n^\dagger(\tau) \\
\end{array}
\right) \left(
\begin{array}{cc}
A_{n'}^\dagger & A_{n'} \\
\end{array}
\right)
\rangle , \nonumber \\ G_{\gamma}(\tau) &=& - \langle \mathcal{T} \gamma(\tau) \gamma \rangle,
\end{eqnarray}
where $O(\tau) =e^{H_L \tau} O e^{- H_L \tau}$, and $\mathcal{T}$ is
Wick's time-ordering operator. We now construct a perturbative
expansion of the Green functions in $h_B$, with $G(i \omega_m) =
\int_0^\beta d \tau e^{i \omega_m \tau} G(\tau)$ in terms of the
Matsubara frequencies $\omega_m = \pi T (1+2m)$. The zeroth-order
Green functions are given by
\begin{eqnarray}
\hat{G}_{nn'}^{(0)}(i \omega_m) =\delta_{nn'} \left(
\begin{array}{cc}
i \omega_m - E_n & 0 \\
0 & i \omega_m + E_n \\
\end{array}
\right)^{-1},~~~ G_{\gamma}^{(0)}(i \omega_m) =\frac{2}{i \omega_m}.
\end{eqnarray}
The full impurity Green function can then be written as $G_{\gamma}(i
\omega_m) =[(G_{\gamma}^{(0)}(i \omega_m) )^{-1} -
\Sigma( i \omega_m)]^{-1}$. Writing the boundary term in the
Hamiltonian as $H_L|_{h_B} = h_B \sum_{n=1}^L g_n(0) \gamma
\left( \begin{array}{cc} A_n^\dagger & A_n \\ \end{array}
\right)\left( \begin{array}{c} -1 \\ 1 \\ \end{array} \right) $,
the exact self energy follows as
\begin{eqnarray}
\Sigma(i \omega_m) = h_B^2 \sum_{n=1}^L g^2_n(0) \left( \begin{array}{cc} 1 & -1 \\ \end{array} \right)\hat{G}_{nn}^{(0)}(i \omega_m) \left( \begin{array}{c} 1 \\ -1 \\
\end{array} \right)=- h_B^2 \sum_{n=1}^L g^2_n(0)\frac{2 i \omega_m}{(\omega_m)^2+E_n^2}.
\end{eqnarray}
We can now calculate the fermionic correlators entering the
determinant Eq.~(\ref{det}). With $G(\tau) = T \sum_{\omega_m} e^{- i
\omega_m \tau} G(i \omega_m)$, the correlators involving the
impurity fermion are given by
\begin{eqnarray}
i \langle \gamma \gamma_{B,x} \rangle = i \langle \mathcal{T} \gamma(\tau = 0^+) \gamma_{B,x} \rangle = i T \sum_{\omega_m} e^{-i \omega_m 0^+} \langle \gamma \gamma_{B,x} \rangle_{\omega_m} \nonumber \\
=T \sum_{\omega_m} e^{-i \omega_m 0^+} \sum_{n=1}^L g_n(x) \langle \gamma (A_n-A_n^\dagger) \rangle_{\omega_m},
\end{eqnarray}
where we used Eq.~(\ref{gammaA}) in the last equality. Proceeding with
first order perturbation theory in $h_B$, we have
\begin{eqnarray}
i \langle \gamma \gamma_{B,x} \rangle = T \sum_{\omega_m} e^{-i \omega_m 0^+} \sum_{n=1}^L g_n(x) G_\gamma(i \omega_m) h_B g_n(0) \left( \begin{array}{cc} 1 & -1 \\ \end{array} \right)\hat{G}_{nn}^{(0)}(i \omega_m) \left( \begin{array}{c} 1 \\ -1 \\ \end{array}
\right) = \nonumber \\ =- T h_B \sum_{\omega_m} G_\gamma(i \omega_m) e^{-i \omega_m 0^+} \sum_{n=1}^L g_n(x) g_n(0) \frac{2 i \omega_m}{(\omega_m)^2+E_n^2}.
\end{eqnarray}
Using the exact expression for $G_\gamma$, this becomes
\begin{eqnarray}
\label{gammagammaB}
i \langle \gamma \gamma_{B,x} \rangle = - T h_B \sum_{\omega_m} e^{-i \omega_m 0^+} \frac{\sum_{n=1}^L \frac{g_n(x) g_n(0)}{(\omega_m)^2+E_n^2}}{\frac{1}{4}+h_B^2 \sum_{n=1}^L \frac{(g_n(0))^2}{(\omega_m)^2+ E_n^2}} .
\end{eqnarray}
Similarly, the $\langle \gamma_A \gamma_B \rangle$ correlators are given by
\begin{eqnarray}
i \langle \gamma_{A,x} \gamma_{B,x'} \rangle = - T \sum_{\omega_m} e^{-i \omega_m 0^+} \sum_{n,n'=1}^L g_n(x+\tfrac{1}{2}) g_{n'}(x') \left( \begin{array}{cc} 1 & 1 \\ \end{array} \right)\hat{G}_{nn'}(i \omega_m) \left( \begin{array}{c} 1 \\ -1 \\ \end{array} \right).
\end{eqnarray}
Using standard impurity Green function methods, the full
$\hat{G}_{nn'}(i \omega_m) $ Green function is seen to contain two terms,
\begin{eqnarray}
\hat{G}_{nn'}(i \omega_m)= \delta_{nn'} \hat{G}_{nn}^{(0)}(i \omega_m) + h_B^2 \hat{G}_{nn}^{(0)}(i \omega_m) \left( \begin{array}{c} -1 \\ 1 \\\end{array} \right) g_n(0) G_\gamma(i \omega_m) g_{n'}(0) \left( \begin{array}{cc} 1 & -1 \\ \end{array} \right) \hat{G}_{n'n'}^{(0)}(i \omega_m). \end{eqnarray}
Explicitly, the desired correlator can be expressed as
\begin{equation}
\label{gammaAgammaB}
i \langle \gamma_{A,x} \gamma_{B,x'} \rangle = \sum_{n=1}^L g_n(x+\tfrac{1}{2}) g_n(x') \tanh\left(\frac{E_n}{2T} \right) +2 h_B^2 T \sum_{\omega_m} e^{-i \omega_m 0^+} \frac{\left( \sum_{n=1}^L \frac{g_{n}(x + \frac{1}{2}) g_{n}(0) E_n}{(\omega_m)^2+E_{n}^2} \right) \left( \sum_{n'=1}^L \frac{g_{n'}(x') g_{n'}(0)}{(\omega_m)^2+E_{n'}^2} \right)}{\frac{1}{4}+h_B^2 \sum_{n''=1}^L \frac{(g_{n''}(0))^2}{(\omega_m)^2+ E_{n''}^2}}.
\end{equation}
All these expressions are exact for the model Eq.~(\ref{bimL})
containing two boundaries. We are interested in the effect of the
boundary $j=0$, but not on the boundary at $j=L-1$. Thus we proceed by
taking the limit $ L \to \infty$, which reproduces the desired
semi-infinite chain. Replacing discrete summations over $n$ by
integrals, $\frac{2}{L+1/2} \sum_{n=1}^L \rightarrow \frac{4}{\pi}
\int_0^{\pi/2} d \theta$, with $\theta = \frac{\pi n}{2L+1}$,
Eqs.~(\ref{gammagammaB}) and (\ref{gammaAgammaB}) become
\begin{eqnarray}
\label{matrixelements}
i \langle \gamma \gamma_{B,j} \rangle &=& - T h_B \sum_{\omega_m} e^{-i \omega_m 0^+} \frac{\sigma_1(\omega_m, j)}{\frac{1}{4}+h_B^2 \sigma_1(\omega_m,0)}, ~~(j=0,1,2,...) \nonumber \\
i \langle \gamma_{A,j} \gamma_{B,j'} \rangle &=& \sigma(j+j'+1) - \sigma(j-j') +2 h_B^2 T \sum_{\omega_m} e^{-i \omega_m 0^+} \frac{\sigma_2(\omega_m,j) \sigma_1(\omega_m,j') }{\frac{1}{4}+h_B^2 \sigma_1(\omega_m,0)}, ~~(j,j'=0,1,2,...)
\end{eqnarray}
where
\begin{eqnarray}
\label{moreints}
\sigma_1(\omega,j)&=& \frac{4 }{\pi} \int_0^{\pi/2} d \theta ~\frac{\sin(\theta) \sin[(2j+1)\theta]}{\omega^2 + (4 \cos \theta)^2}, \nonumber \\
\sigma_2(\omega,j)&=& \frac{4 }{\pi} \int_0^{\pi/2} d \theta ~\frac{4\sin(\theta) \sin[(2j+2)\theta] \cos(\theta)}{\omega^2 + (4 \cos \theta)^2}, \nonumber \\
\sigma(j)&=& -\frac{2 }{\pi} \int_0^{\pi/2} d \theta ~\cos[(1+2j) \theta] \tanh \left( \frac{4 \cos (\theta)}{2 T}\right).
\end{eqnarray}
The magnetization due to a field $h_B$ applied at the single boundary of
a semi-infinite chain at finite temperatures is thus given
exactly by Eqs.~(\ref{det}) and (\ref{matrixelements}). In practice
we evaluate the integrals in Eq.~(\ref{moreints}) numerically, yielding
the results presented in Fig.~3. It would be interesting to rederive the
field theoretical results by analytic evaluation of the determinant
Eq.~(\ref{det}) in the continuum limit following the methods of Ref.~\onlinecite{bariev}.
\section{Conclusions}
The crossover physics evinced by the boundary Ising model has been
shown to play a key role in a surprisingly diverse range of physical
problems.\cite{Leclair,rosennow,Bishara,CFT2IKM,SelaAffleck,SelaMitchellFritz,Mitchell2011}
Analysis of the exact universal crossover from free to fixed boundary
conditions in such problems at finite temperatures thus requires the
corresponding scaling functions of the boundary Ising model to be
known exactly.
In this paper we obtained the full scaling function for the
magnetization of the boundary Ising chain at finite temperature.
Among the potential applications of our results, one example is
calculation of finite-temperature conductance crossovers in two-channel or two-impurity
Kondo quantum dot systems.\cite{2ckfiniteT} The crossover from
non-Fermi liquid to Fermi liquid physics in such systems is
characterized by the same renormalization group flow as occurs in the
boundary Ising chain.\cite{CFT2IKM} We plan to extend our earlier
work\cite{SelaMitchellFritz} at $T=0$ in this area to finite
temperatures, employing the results of this paper.\cite{2ckfiniteT} We
note in this regard that without the function $f(2\beta h^2)$, the
conductance near the non-Fermi liquid fixed point is unphysical;\cite{2ckfiniteT} but
using the main result of this paper, Eq.~(\ref{sigmaf}), exact
results\cite{CFT2IKM} at both non-Fermi liquid and Fermi liquid fixed
points are recovered precisely.
\begin{acknowledgments}
We thank H. Saleur for discussions. This work was
supported by the A.~v.~Humboldt Foundation (E.S.) and by the DFG through
SFB608 and FOR960 (A.K.M).
\end{acknowledgments}
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| 513
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{"url":"https:\/\/calculator.academy\/muzzle-energy-calculator\/","text":"Enter the mass of the bullet and the exit velocity to determine the muzzle energy of the gun. This can also be known as a bullet energy calculator.\n\n## Muzzle Energy Formula\n\nThe muzzle energy of a gun is determined by the physics equation for kinetic energy which is displayed as follows.\n\nKE = 1\/2*m*v^2\n\nFor guns, the mass of the bullet is typically measured in grains. One ounce is equal to 437.5 grains. Velocity is usually measured in meters per second or feet per second. For the calculator above we use grains and feet per second. The resulting muzzle energy unit is foot-pounds. (lbf)\n\n## What is bullet energy and why does it matter?\n\nBullet energy, also known as muzzle energy, is a measure of the energy a bullet has as it leaves a gun. It\u2019s one of many factors that affect both stopping power and recoil. Stopping power is the ability of a bullet to penetrate material or an object. Higher muzzle energy typically means a higher stopping power, but that\u2019s not the only factor. The aerodynamics of the bullet itself also factor into that stopping power.\n\nThe bullet energy also affects the recoil of a gun. In short, this is also governed by one of Newton\u2019s laws of motion. That is that every motion has an equal and opposite motion. This means that the greater the muzzle energy the greater the recoil. This is an important factor to keep in mind.\n\nAbove is a table that outlines a few different size bullets and their corresponding bullet energies, velocity, and masses. You can use the table to check the calculator you are using above. Note the size is denoted by the size in mm.\n\n## How to calculate muzzle energy\n\nLet\u2019s take a look at how you might calculate the muzzle energy of an unknown gun and bullet.\n\n1. First, we need to weigh the bullet. Most often this is done using a gram scale. Using the conversion noted above we find that the weight is equal to 500 grains.\n2. The next step is to measure the bullet velocity as it\u2019s leaving the gun. This is typically done using a device similar to a speedometer. Let\u2019s say for this example we find the velocity to be 200 ft per second.\n3. Finally, we need to enter all of the information into the formula above. KE = 500*.5*200^2 = 100,000.\n4. Last, check the answer with the table above.","date":"2021-06-23 05:58:50","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\": 0.8557742238044739, \"perplexity\": 589.092816060718}, \"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-2021-25\/segments\/1623488534413.81\/warc\/CC-MAIN-20210623042426-20210623072426-00405.warc.gz\"}"}
| null | null |
Le district de Yuquan ( ; ) est une subdivision administrative de la région autonome de Mongolie-Intérieure en Chine. Il est placé sous la juridiction de la ville-préfecture de Hohhot dont il couvre la partie sud-ouest.
Principaux monuments:
Le temple des cinq pagodes
Le temple Dazhao, premier bâtiment de la ville dont la construction a commencé en 1557.
La tombe de Zhaojun, une des quatre beautés de la Chine antique
Notes et références
Yuquan
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,071
|
module Deis
module Commands
class Enable < Struct.new :app
include Helpers
def run
status "Enabling App: #{app}"
get_units!
if units.any? { |_, v| v > 0 }
status "App Already enabled!"
return
end
scale app, units.keys.each_with_object({}) { |k, h| h[k] = 1 }
status "App enabled!"
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 543
|
Q: Display all surveys using content query web part I am new to sharepoint development. I have created a subsite and called it Surveys. This will contain all the surveys created in the portal. Now, I want to display the latest survey on the homepage. I though I would be able to do that using a content query web part, sort the CQWP descendently on survey creation date and restrict the number of displayed items to one.
Unfortunately, this doesnt work at all! Is my solution achievable? It seems to me that the Survey content types isn't a normal one as I can't see it in the content types gallery. How could my requirement be fulfilled (without code)?
Thanks for any help
A: OOTB Survey List does not contain any Content Types and it has the following structure:
*
*Response entry is represented by List Item
*Response entry consist of questions and answers, where Question correspond to Field and Answer to Item value respectively.
For querying Survey List via CQWP the following properties could be used
*
*BaseType <Lists BaseType="4">
*ServerTemplate <Lists ServerTemplate="102">
Example
To retrieve all the responses from Survey lists, specify List Type property as shown on picture
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,955
|
Магистрату́ра:
Магистратура — общее название государственных должностей в Древнем Риме.
Магистратура — курс наук в высших учебных заведениях.
Магистратура — судейский корпус.
См. также
Магистрат
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,354
|
/**
* Exercise 28
*/
package com.ciaoshen.thinkinjava.chapter21;
import java.util.concurrent.*;
import java.io.*;
public class Exercise28 {
private static BlockingQueue<LiftOff28> rockets;
public static class LiftOffRunner implements Runnable {
public LiftOffRunner(BlockingQueue<LiftOff28> queue) {
rockets = queue;
}
public void run() {
try {
while(!Thread.interrupted()) {
LiftOff28 rocket = rockets.take();
rocket.run(); // Use this thread
}
} catch(InterruptedException e) {
System.out.println("Waking from take()");
}
System.out.println(Thread.currentThread()+"Exiting LiftOffRunner");
}
}
public static class LiftOffFiller implements Runnable{
private int times=0;
public LiftOffFiller(int num){times=num;}
public void add(LiftOff28 lo) {
try {
rockets.put(lo);
} catch(InterruptedException e) {
System.out.println("Interrupted during put()");
Thread.currentThread().interrupt();
}
}
public void run(){
for(int i=0;i<times;i++){
if(Thread.interrupted()){
break; //保证当interrupt时,跳出循环。否则下一次BlockingQueue的put()会继续阻塞,线程无法退出。
}
add(new LiftOff28(5));
}
System.out.println(Thread.currentThread()+"Exiting LiftOffFiller!");
}
}
static void getkey() {
try {
new BufferedReader(new InputStreamReader(System.in)).readLine();
} catch(java.io.IOException e) {
throw new RuntimeException(e);
}
}
static void getkey(String message) {
System.out.println(message);
getkey();
}
static void test(String msg, BlockingQueue<LiftOff28> queue) throws InterruptedException{
System.out.println(msg);
LiftOffRunner runner = new LiftOffRunner(queue);
Thread t = new Thread(runner);
t.start();
Thread f=new Thread(new LiftOffFiller(5));
f.start();
getkey("Press 'Enter' (" + msg + ")");
t.interrupt();
f.interrupt();
System.out.println("Finished " + msg + " test");
}
public static void main(String[] args) throws InterruptedException{
test("LinkedBlockingQueue", new LinkedBlockingQueue<LiftOff28>());
test("ArrayBlockingQueue", new ArrayBlockingQueue<LiftOff28>(3));
test("SynchronousQueue", new SynchronousQueue<LiftOff28>());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,087
|
{"url":"https:\/\/ftp.aimsciences.org\/article\/doi\/10.3934\/dcds.2011.31.1197","text":"# American Institute of Mathematical Sciences\n\nDecember\u00a0 2011,\u00a031(4):\u00a01197-1218. doi:\u00a010.3934\/dcds.2011.31.1197\n\n## On the regularization of the collision solutions of the one-center problem with weak forces\n\n 1 BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park, 48160 Derio, Bizkaia,, Spain 2 Universit\u00e0 di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano\n\nReceived\u00a0 January 2010 Revised\u00a0 March 2010 Published\u00a0 September 2011\n\nWe study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with transmission trajectories, is continuous, though not differentiable, with respect to the initial data.\nCitation: Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1197-1218. doi: 10.3934\/dcds.2011.31.1197\n##### References:\n [1] S. J. Aarseth, Dynamical evolution of clusters of galaxies I, Monthly Notices of the Royal Astronomical Society, 126 (1963), 223-255.\u00a0Google Scholar [2] V. Barutello, D. L. Ferrario and S. Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn 069, 78 pp. \u00a0Google Scholar [3] G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two-body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi:\u00a010.3934\/cpaa.2003.2.323. \u00a0Google Scholar [4] E. De Giorgi, Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. doi:\u00a010.1215\/S0012-7094-96-08114-4. \u00a0Google Scholar [5] C. C. Dyer and P. S. S. Ip, Softening in N-body simulations of collisionless systems, Astrophysical Journal, 409 (1993), 60-67. doi:\u00a010.1086\/172641. \u00a0Google Scholar [6] R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99. \u00a0Google Scholar [7] D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362. doi:\u00a010.1007\/s00222-003-0322-7. \u00a0Google Scholar [8] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi:\u00a010.2307\/2373993. \u00a0Google Scholar [9] L. Hernquist and J. E. Barnes, Are some n-body algorithms intrinsically less collisional than others?, Astrophysical Journal, 349 (1990), 562-569. doi:\u00a010.1086\/168343. \u00a0Google Scholar [10] P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi:\u00a010.1515\/crll.1965.218.204. \u00a0Google Scholar [11] T. Levi-Civita, Sur la r\u00e9gularisation du probl\u00e8me des trois corps, Acta Math., 42 (1920), 99-144. doi:\u00a010.1007\/BF02404404. \u00a0Google Scholar [12] R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi:\u00a010.1007\/BF02566226. \u00a0Google Scholar [13] J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi:\u00a010.1002\/cpa.3160230406. \u00a0Google Scholar [14] C. L. Siegel and J. K. Moser, \"Lectures on Celestial Mechanics,\" Classics in Mathematics, Springer-Verlag, Berlin, 1995.\u00a0Google Scholar [15] C. Stoica and A. Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi:\u00a010.1088\/0305-4470\/36\/28\/302. \u00a0Google Scholar [16] V. G. Szebehely, \"Theory of Orbits -- The Restricted Problem of Three Bodies,\" Academic Press, New York, 1967.\u00a0Google Scholar [17] J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932.\u00a0Google Scholar [18] E. T. Whittaker, \"A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies,\" 4th edition, Cambridge University Press, New York, 1959.\u00a0Google Scholar\n\nshow all references\n\n##### References:\n [1] S. J. Aarseth, Dynamical evolution of clusters of galaxies I, Monthly Notices of the Royal Astronomical Society, 126 (1963), 223-255.\u00a0Google Scholar [2] V. Barutello, D. L. Ferrario and S. Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn 069, 78 pp. \u00a0Google Scholar [3] G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two-body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi:\u00a010.3934\/cpaa.2003.2.323. \u00a0Google Scholar [4] E. De Giorgi, Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. doi:\u00a010.1215\/S0012-7094-96-08114-4. \u00a0Google Scholar [5] C. C. Dyer and P. S. S. Ip, Softening in N-body simulations of collisionless systems, Astrophysical Journal, 409 (1993), 60-67. doi:\u00a010.1086\/172641. \u00a0Google Scholar [6] R. Easton, Regularization of vector fields by surgery, J. Differential Equations, 10 (1971), 92-99. \u00a0Google Scholar [7] D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362. doi:\u00a010.1007\/s00222-003-0322-7. \u00a0Google Scholar [8] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi:\u00a010.2307\/2373993. \u00a0Google Scholar [9] L. Hernquist and J. E. Barnes, Are some n-body algorithms intrinsically less collisional than others?, Astrophysical Journal, 349 (1990), 562-569. doi:\u00a010.1086\/168343. \u00a0Google Scholar [10] P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi:\u00a010.1515\/crll.1965.218.204. \u00a0Google Scholar [11] T. Levi-Civita, Sur la r\u00e9gularisation du probl\u00e8me des trois corps, Acta Math., 42 (1920), 99-144. doi:\u00a010.1007\/BF02404404. \u00a0Google Scholar [12] R. McGehee, Double collisions for a classical particle system with nongravitational interactions, Comment. Math. Helv., 56 (1981), 524-557. doi:\u00a010.1007\/BF02566226. \u00a0Google Scholar [13] J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi:\u00a010.1002\/cpa.3160230406. \u00a0Google Scholar [14] C. L. Siegel and J. K. Moser, \"Lectures on Celestial Mechanics,\" Classics in Mathematics, Springer-Verlag, Berlin, 1995.\u00a0Google Scholar [15] C. Stoica and A. Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi:\u00a010.1088\/0305-4470\/36\/28\/302. \u00a0Google Scholar [16] V. G. Szebehely, \"Theory of Orbits -- The Restricted Problem of Three Bodies,\" Academic Press, New York, 1967.\u00a0Google Scholar [17] J. Touma and S. Tremaine, A map for eccentric orbits in non-axisymmetric potentials, MNRAS, 292 (1997), 905-932.\u00a0Google Scholar [18] E. T. Whittaker, \"A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies,\" 4th edition, Cambridge University Press, New York, 1959.\u00a0Google Scholar\n [1] G. Bellettini, G. Fusco, G. F. Gronchi. Regularization of the two-body problem via smoothing the potential. Communications on Pure & Applied Analysis, 2003, 2 (3) : 323-353. doi: 10.3934\/cpaa.2003.2.323 [2] Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934\/cpaa.2013.12.1363 [3] Vasile Mioc, Ernesto P\u00e9rez-Chavela. The 2-body problem under Fock's potential. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 611-629. doi: 10.3934\/dcdss.2008.1.611 [4] Nicolas Forcadel, Cyril Imbert, R\u00e9gis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 785-826. doi: 10.3934\/dcds.2009.23.785 [5] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934\/cpaa.2020042 [6] Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934\/cpaa.2018068 [7] Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3343-3376. doi: 10.3934\/dcds.2015.35.3343 [8] Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934\/dcdss.2011.4.51 [9] Younghun Hong. Strichartz estimates for $N$-body Schr\u00f6dinger operators with small potential interactions. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934\/dcds.2017233 [10] Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934\/proc.2011.2011.1001 [11] David G\u00f3mez-Castro, Juan Luis V\u00e1zquez. The fractional Schr\u00f6dinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7113-7139. doi: 10.3934\/dcds.2019298 [12] Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schr\u00f6dinger operators with a singular potential. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934\/dcds.2015.35.5827 [13] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934\/mcrf.2016.6.95 [14] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations & Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934\/eect.2018006 [15] Yu Su. Ground state solution of critical Schr\u00f6dinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934\/cpaa.2021108 [16] Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934\/dcdsb.2003.3.313 [17] Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934\/dcdss.2011.4.833 [18] Yulia Karpeshina and Young-Ran Lee. On polyharmonic operators with limit-periodic potential in dimension two. Electronic Research Announcements, 2006, 12: 113-120. [19] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934\/jimo.2010.6.259 [20] Nikolaos S. Papageorgiou, Vicen\u0163iu D. R\u0103dulescu, Du\u0161an D. Repov\u0161. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934\/dcds.2017111\n\n2020\u00a0Impact Factor:\u00a01.392","date":"2021-12-01 03:43:23","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\": 0.8033719658851624, \"perplexity\": 4773.178092560961}, \"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-2021-49\/segments\/1637964359082.78\/warc\/CC-MAIN-20211201022332-20211201052332-00506.warc.gz\"}"}
| null | null |
\section{The PANDA Experiment at FAIR}
\label{sec:intro}
The new international FAIR accelerator complex near GSI in Darmstadt, Germany, is currently under construction.
The heart of the facility is an underground ring accelerator
with a circumference of 1100 meters, which is being excavated.
The existing GSI accelerator complex will serve as an injector for FAIR. The experimental
pillars are NUSTAR, CBM, APPA, and PANDA (antiProton Annihilation at DArmstadt).
The latter is using antiproton beams with unprecedented intensity and quality, stored and
accelerated in the High Energy Storage Ring (HESR). The antiproton beam with the momentum range from 1.5 to 15 GeV/c
annihilates with a fixed target. The experiments will address questions of QCD
in an energy region where perturbation theory is still valid, but the influence of strong QCD cannot
be neglected.
The luminosity of up to $2 \cdot 10^{32} cm^{-2}s^{-1}$, and the momentum resolution of
the antiproton beam down to \mbox{$\Delta$p/p = 4$\cdot10^{-5}$} enable high precision
spectroscopy. The energy range allows accessing resonances above the
threshold for open charm mesons. Thus, the detection of kaons plays an important role for
the identification of the reaction channel. The PANDA detector consists of two parts,
a hermetic target spectrometer and a forward spectrometer for polar angles up to $5^\circ$ and
$10^\circ$ in vertical and horizontal directions, respectively. The target spectrometer includes,
beside a tracking system, an electromagnetic lead tungstate calorimeter, a muon range system, and
ring imaging Cherenkov counters for charged particle identification (PID).
The PID inside the calorimeter has to fulfill the detector requirements
of limited space and limited mass.
Therefore, two Cherenkov counters using the DIRC principle were chosen.
In a DIRC detector the radiator acts also as a lightguide.
The lightguide preserves the
information of the Cherenkov angle of the charged particle over many internal photon reflections.
The Barrel DIRC
covers polar angles between $22^\circ$ and $140^\circ$ and will achieve a pion-kaon
separation of 3 standard deviations (s.d.) up to 3.5 GeV/$c$.
At smaller polar angles an Endcap Disc DIRC \cite{Etzelmueller19} and
a focusing aerogel RICH \cite{Kononov19} provide charged PID.
The focus of the following sections will be on the
Barrel DIRC, its design, the results of the test beams, and selected prototype tests.
\section{The Design of the Barrel DIRC}
\label{sec:design}
\begin{figure}[tbh]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=.8\textwidth,trim=0 0 0 0,clip]{setup2.png}
\caption{\label{fig:1} Schematic of the PANDA Barrel DIRC.}
\end{figure}
The design of the PANDA Barrel DIRC (Figure \ref{fig:1}) is inspired by the successful BaBar DIRC and
the R\&D for the SuperB FDIRC with innovative improvements.
The readout volume is a prism. It is smaller and made from synthetic fused silica
to minimize the influence of background radiation from the accelerator.
The use of pixelated Microchannel Plate Photomultiplier
Tubes (MCP-PMTs) allows the operation of the readout within a magnetic field of 1 Tesla inside the
solenoid magnet of the target spectrometer. The fast detector response and the frontend electronics
aim for a timing precision of
the readout chain of about 100 ps \cite{TDR,Schwiening18}.
Focusing optics are needed due to the compact readout volume.
The positioning of the photodetectors in the magnetic field and space
limitations favored the use of a lens system. A picture of three lens triplets
built by Befort Wetzlar Optics \cite{Befort}
with the width of 53 mm are shown in Figure \ref{fig:3}.
The radiator barrel consists of 16 bar boxes containing three radiator bars each. The radiator bars
are made from synthetic fused silica, 17 mm thick, 53 mm wide, and 2400 mm long. Two 1200 mm long
pieces are
glued end-to-end to form one radiator bar.
With wider bars the needed number of entities in the barrel becomes smaller.
The chosen width is the result of a cost optimization without deteriorating the required performance.
At each downstream end a mirror is attached, reflecting the photons back towards the readout volume.
A lens is glued to the upstream end of the bar and a RTV-silicone cookie couples the lens system
to a 300 mm-long
prism with an opening angle of $33^\circ$.
An array of $2\times4$ MCP-PMTs, each with an $8\times8$ anode grid with a pixel pitch of $6.5$~mm,
is coupled to the prism using a RTV-silicone cookie.
\begin{figure}[bth]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=.65\textwidth,trim=0 0 0 0,clip]{lens_triplet1_parta.jpg}
\caption{\label{fig:3} Three focusing lens triplets, each a lanthanum crown glass lens between
two fused silica pieces with
flat outer surfaces, focus the light from the radiator bar on the back side of the prism.}
\end{figure}
\section{Experiments with Test Beams}
The Barrel DIRC prototypes were tested in several experiments with particle beams from 2008 to 2018.
The measured photon number and single photon angular
resolution of the setups were compared to predictions from GEANT4 \cite{geant4} simulations.
The baseline design was validated in the campaign at CERN-PS in 2015. The following
campaigns studied the possible reduction of the number of MCP-PMTs for cost optimization,
the performance of a wide radiator plate,
different optical couplings, and housing and cable routing of the frontend electronics.
The prototypes consisted of a
single radiator bar or plate coupled to a prism with or without a focusing lens system and were read out by
up to 15 MCP-PMTs.
In the beam tests MCP-PMTs like the XP85012/A1-Q from PHOTONIS \cite{photonis} were used.
The latest tubes are expected to survive 10 years of the operation of the PANDA experiment \cite{lehmann:mcp}.
The electronics for the readout of the MCP-PMTs in the prototype was based on the
Trigger Readout Board Version 3 (TRB3) of the
HADES collaboration, which uses fast TDC channels implemented in FPGAs \cite{trb3-jinst}.
The boards measure the time of
arrival and the time over threshold of logical signals coming from PADIWA discriminator boards
\cite{cardinali:padiwa} plugged onto the MCP-PMTs.
Between the radiator bar and the prism is the focusing lens system. Initial versions with a
focusing plano-convex lens on the radiator and an air gap between the prism and the lens allowed an easy
separation of the expansion volume from the radiator for maintenance.
However, many photons were reflected back by internal reflection from the lens surface.
In addition, the single refracting surface caused a parabolic-shaped focal plane. Therefore, the
air gap was filled with synthetic fused silica and the lens changed to a material
with a high refraction index and
sufficient transmission for UV-photons, lanthanum-crown glass LaK33.
The lens consists of a defocusing and a focusing surfaces to yield a sufficiently flat focal plane.
The transmission loss
of LaK33 at a wavelength of 420 nm is 1.3~\% for 1 Gy X-rays \cite{eRD14_18}
and the anticipated radiation for the lens is 4 Gy within 10 years of operation \cite{TDR}.
\begin{figure}[tbh]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=.7\textwidth,trim=0 0 0 0,clip]{sep_power.png}
\caption{\label{fig:2} Separation power for pion/proton at particle momenta of 7 GeV/c
coming from the timing reconstruction method. This corresponds to
the separation power for pion/kaon at momenta of 3.5 GeV/c.}
\end{figure}
The geometry of the radiator, narrow bars or a single wide plate, requires different
reconstruction methods to identify the particle type \cite{Dzhygadlo19}.
The result of a maximum likelihood test using the time of arrival of photons for
different particle types is
shown in Figure \ref{fig:2} for the test beam campaign in 2018 where pions and protons were
cleanly tagged by a time-of-flight system.
The separation between pions and protons with the momentum of $7$~GeV/c is equivalent to the
separation between pions and kaons at $3.5$~GeV/c,
which is the designed upper momentum limit of the Barrel DIRC in PANDA.
\section{Finalizing the R\&D and Construction}
\begin{figure}[b]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=.615\textwidth,trim=0 0 0 0,clip]{dirich4_parta.jpg}
\includegraphics[width=.35\textwidth,trim=0 0 0 0,clip]{dirich1_parta.jpg}
\caption{\label{fig:x} The DiRICH system: a power card, a DiRICH card, and a data concentrator
(left side) are plugged in a backplane together with the MCP-PMT (right side).}
\end{figure}
Following the successful performance validation with particle beams and the completion of
the technical design report \cite{TDR}, the PANDA Barrel DIRC project has entered the construction phase.
The Barrel DIRC components with the longest production time,
the radiators, have been ordered from Nikon \cite{nikon} and the tendering of the MCP-PMTs
is at an advanced stage.
Several important R\&D topics are still being investigated.
The latest generation of the readout electronics, the DiRICH system \cite{traxler:dirich} comprises a backplane,
a PC board with plugs for MCP-PMTs on one side and three types of electronic cards on the other side (Figure \ref{fig:x}).
The electronic cards are
a power supply, a data concentrator, and a DiRICH card. The latter contains 32 FPGA-based
discriminators and TDC channels. The discriminator input stage, originally designed for multianode PMTs,
still needs to be adapted to the fast signals of the
MCP-PMTs.
The coupling between the bar boxes and prisms and between the MCP-PMTs and the prism
will be done with RTV-silicone cookies and is subject of
ongoing tests. Cookies with layers of different elasticity are produced to
achieve the optimum contact between the optical surfaces. A firmer outer layer with a slightly convex
shape helps to prevent the appearance of air bubbles connecting the parts.
The material of the bar boxes can be aluminum or carbon fiber reinforced polymer (CFRP). While the
CFRP minimizes the material budget, the long-term outgassing of the material and its effect
on the surfaces of the radiator are not well known.
Therefore, a setup was built to test the long-term outgassing behaviour of materials
which may be used in the construction of the bar boxes.
The radiator is sitting in a chemically cleaned stainless-steel tube with an attached steel
container which contains the possibly pollutant material.
A flow of nitrogen gas transports any possible outgassing pollutant across the surfaces of the radiator.
The material to test
can be heated to accelerate the outgassing.
The internal reflection coefficient of the possible polluted bar surface is
measured at regular intervals, ranging from a few days to several weeks,
with a laser scanning setup and compared to the reference bar.
The setup with four testing tubes is shown in Figure \ref{fig:4}.
With the arrival of the first radiator bars in 2020 the Barrel DIRC has entered the construction phase.
The experimental hall for the PANDA detector will be ready to move in and to install first basic elements,
like the solenoid, in 2022. The Barrel DIRC will then be installed in 2023/2024.
\begin{figure}[bh]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=.63\textwidth,trim=0 0 0 0,clip]{ausgas_carsten_01.png}
\includegraphics[width=.34\textwidth,trim=0 0 0 0,clip]{long_stability_setup.jpg}
\caption{\label{fig:4} Schematic of the setup for the measurement of the impact of outgassing
on the bar surfaces (left side).
The heatable volumes are connected to the tubes with the the radiator bars.
The photograph shows the volumes for the outgassing materials (right side).
}
\end{figure}
\section*{Acknowledgments}
This work was supported by
HGS-HIRe,
HIC
for FAIR,
BNL, and eRD14.
We thank the GSI and CERN staff for the opportunity to use
the beam facilities and for their on-site support.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,665
|
{"url":"https:\/\/www.physicsoverflow.org\/15193\/two-different-notions-of-ir-relevant-and-marginal-operators","text":"# Two different notions of (ir)relevant and marginal operators ?\n\n+ 3 like - 0 dislike\n6421 views\n\nOn p10 of these EFT lecture notes, the \"relevance\" of operators in a Lagrangian is determined by comparing their mass dimension to the spacetime \"d\" one considers such that an operator is\n\n\u2022 Relevant if its dimension is $< d$\n\u2022 Marginal if its dimension is $= d$\n\u2022 And irrelevant if its dimension is $> d$\n\nThis means for example for the action of e scalar field in $d=4$$S[\\phi] = \\int d^d x\\left( \\frac{1}{2} \\partial_{\\mu}\\phi \\partial^{\\mu} \\phi - \\frac{1}{2} m^2\\phi^2- \\frac{\\lambda}{4!} \\phi^4 - \\frac{\\tau}{6!} \\phi^6 + ...\\right)$\n\nthat the mass term is relevant, the $\\phi^4$ coupling is marginal, and the $\\phi^6$ coupling is irrelevant, etc\n\nHowever, when analyzing the RG flow around a fixed point $S^{*}[(\\phi)]$, the (ir)rrelevance of an operator is determined by linearizing the RG flow equation around this fixed point ($M$ is the linearized right hand side of the for example Wilson RG flow equation, $t$ is the RG time),\n\n$\\frac{\\partial S}{\\partial t} = M S^{*}[\\phi]$\n\nsolving the corresponding Eigenvalue problem\n\n$M O_i(\\phi) = \\lambda_i O_i(\\phi)$\n\nand looking at the sign of each Eigenvalue $\\lambda_i$.\n\nThe action around the fixed point can then be approximated as\n\n$S[\\phi] = S^{*}[\\phi] + \\sum\\limits_i \\alpha_i e^{\\lambda_i t} O_i(t)$\n\nand the operator $O_i$ (or direction in coupling space) is said to be\n\n\u2022 Relevant, if $\\lambda_i > 0$ (leads away from the fixed point)\n\u2022 Marginal, if $\\lambda_i = 0$\n\u2022 Irrelevant, if $\\lambda_i < 0$ (these operators describe the fixed point theory)\n\nSo my question is: What is the relationship between these two notions \/ definitions of (ir)relevant and marginal operators in an effective field theory? Are they they equivalent, and if so how can I see this (mathematically) ?\n\nedited Apr 15, 2014\n\n+ 4 like - 0 dislike\n\nA scale transformation changes $dx$ to $dx(s)=e^{s}dx$ and\u00a0$\\phi$ to $\\phi(s)=e^{-s}\\phi$, hence $S=\\int dx^d \\sum_k g_k \\phi^k$ to\u00a0$S(s)=\\int dx^d \\sum_k g_k e^{(d-k)s}\\phi^k$.\u00a0Thus your $\\lambda_k$ is $d-k$, which is positive if $k<d$ zero if $k=d$, and negative if $k>d$.\n\nanswered Apr 15, 2014 by (15,458 points)\n\nHa thanks Arnold, feeling that my question was a bit too stupid now ...\n\nIf the answer to a question provides new insight, it is never stupid.\n\n Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the \"link\" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)\u00a0\u00a0 Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\\varnothing$ in the following word:p$\\hbar$ysic$\\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.","date":"2021-01-23 11:10:02","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\": 2, \"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\": 0.9625977873802185, \"perplexity\": 5312.2998214194085}, \"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-2021-04\/segments\/1610703537796.45\/warc\/CC-MAIN-20210123094754-20210123124754-00528.warc.gz\"}"}
| null | null |
# THE BYZANTINE ART OF WAR
MICHAEL J. DECKER
WESTHOLME
Yardley
This book is dedicated to my wife Katy, Will, and Joy
Frontispiece: Sixth century relief showing Roman troops of the V Macedonian legion. ( _Rheinisches Landesmuseum, Trier, Germany_ )
©2013 Michael J. Decker
Maps by T. D. Dungan. ©2013 Westholme Publishing
All rights reserved under International and Pan-American Copyright Conventions. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review.
Westholme Publishing, LLC
904 Edgewood Road
Yardley, Pennsylvania 19067
Visit our Web site at www.westholmepublishing.com
ISBN: 978-1-59416-560-3
Also available in hardback.
Produced in the United States of America.
## CONTENTS
_List of Maps_
_Introduction_
1. HISTORICAL OVERVIEW
2. LEADERSHIP
3. ORGANIZATION, RECRUITMENT, AND TRAINING
4. EQUIPMENT AND LOGISTICS
5. STRATEGY AND TACTICS
6. ENEMIES OF BYZANTIUM
7. THE BYZANTINE ARMY AT WAR
8. THE BYZANTINE ART OF WAR
_Glossary_
_Notes_
_Bibliography_
_Index_
## List of Maps
1. The Balkans
2. The Eastern Frontier, Fourth–Seventh Centuries
3. The Empire in the Sixth Century with Regional Armies
4. Rome's Desert Frontier
5. The Themes ca. 688 and ca. 900
6. The Empire ca. 780
7. The Themes ca. 1025
8. The Empire of the Komenoi
9. Medieval Italy and the Balkans
10. Successor States ca. 1218
11. The Road Network in Anatolia
## INTRODUCTION
ON MAY 29,1453, the twenty-one-year-old Ottoman Sultan Mehmed II led his 80,000-man army through the breach in the walls of the ancient capital of eastern Rome, Constantinople, where many of the 7,000 defenders lay dead. The dramatic assault, made possible by one of the earliest and most impressive displays of gunpowder artillery, punched through the hitherto impregnable fortifications, and led to the death of the last Byzantine emperor, Constantine XI Palaiologos, who perished in the assault; his body was never found. Greek legend holds that at the end of days, Constantine will return and rise from the floor of the great cathedral of Hagia Sophia in Istanbul to lead the Greek nation to final victory and the restoration of God's Roman Empire on earth—the Christian Byzantine state. In the spring of 1453, though, the noose that finally strangled the last vestiges of Greek independent political life from the Balkans was long in the tightening. Since 1356 the Ottoman Turks had made Edirne, in eastern Thrace, their capital and steadily strengthened their hold on the empire's former European lands. The Byzantines had been fatally weakened by the sacking of the capital of Constantinople two and half centuries prior in 1204 by the Christian crusaders from the West. This date marks the effective end of Byzantium as a major military and pan-Mediterranean power; after the sack of the capital, Latin warlords and their Venetian allies partitioned the empire while disparate Byzantine rulers regrouped and attempted to mount an effective resistance. Either date, 1204 or 1453, is arguably an appropriate one for marking the end of the Roman Empire. Although for centuries prior to either conquest the vast majority of the empire's inhabitants spoke Greek, and we refer to them as Byzantines, they called themselves Romans and viewed their empire as the state once ruled by Augustus or Trajan. After all, they were the direct inheritors of the Roman Empire's territory in the eastern Mediterranean, continued its administrative and legal framework without interruption, and, most important for our purposes, relied on the military apparatus that evolved from the old Roman legionary armies of antiquity.
The most striking thing about the Byzantine military, and Byzantine society at large, was its remarkable longevity. These medieval Romans, with their Greek speech and Christian faith, clung tenaciously to their culture in the face of constant internal and external pressures. Warfare, although never embraced by the majority of Byzantines as a virtue in the way that many western peoples viewed it, was nonetheless an essential component of the Byzantine experience. Foreign enemies were constantly at the door and they came from all directions, especially at the end of the empire's existence, when westerners threatened the shrinking borders of the state as much as did eastern and northern peoples. It is impossible to find anything like the _pax Romana_ of the emperor Augustus and his successors, when Rome presided over one of the more tranquil periods of European history, having slaughtered most serious foes and bloodily dispatched of entire races in the process. However it was won, no parallel time of quietude ever descended on the Byzantine realm. Although the citizens of the empire probably fully expected a period of peace following the end of the brutal, apocalyptic struggle with Persia in the 620s, their hopes were sorely dashed when the ravenous armies of Arabia descended on the eastern provinces and rent them from the imperial grasp forever. In a matter of decades the Arab foes and bearers of the kernel of a new religion, Islam, were battering at the gates of Constantinople itself, and the empire had lost most of its territory to the Arabs or other invaders.
The survival of the embattled state and its much-reduced armed forces is one of the miracles of history. Far outclassed in terms of manpower and wealth and subjected to military challengers on multiple fronts, the Roman Empire of Byzantium nevertheless survived the assaults they received in the Dark Ages and emerged with a transformed state and society. The army, for which the bureaucracy and its tax system existed, both absorbed the blows of its enemies and dealt more shocks through rebellions and internal discord that marked the seventh through ninth centuries. Despite the upheavals, societal trauma, and the loss of so much territory and manpower, the Byzantine army adapted and fought on. By the time the Macedonian dynasty, the greatest of the medieval empire, came to power in the form of the usurper Basil I (867–86) the Byzantines were poised to embark on a two centuries-long program of expansion. Their reformed armies pushed the frontier into the borderlands of the caliph and reestablished Byzantium as the predominant power in the Mediterranean world. No state in European history absorbed such losses, survived, and revived to such prominence. At the center of this revival was the army, and the collective action of society, emperors, commanders, and soldiery make for one of the more compelling stories in world history.
In the pages that follow, I provide an overview of the basics of the medieval Roman army, including organization, logistics, armament, tactics, and strategy as well as delve into how these were employed. Although it is doubtful that the Byzantines ever thought of war in terms of grand strategy or professed military doctrines based on perceived universal experiences in war, one can clearly detect patterns to their approach to warfare with the benefit of hindsight. I call this the Byzantine Art of War.
THIS BOOK IS WRITTEN for a nonspecialist audience and students of military history and has been spurred by my own interest in the subject and by the enthusiasm for which my lectures on the topic at the University of South Florida have been received. In its crafting I am greatly indebted to the work of outstanding scholars the world over, especially John Haldon, who has pioneered much work on the Byzantine army, Timothy Dawson, Walter Kaegi, James Howard-Johnston, Taxiarches Kolias, Eric McGeer, Philip Rance, Dennis Sullivan, Warren Treadgold, and a host of other accomplished academics too numerous to mention. The reader wishing to know more will find the references necessary to pursue specific topics at their leisure—for this reason, and because I anticipate an audience whose primary language is English, I have endeavored to supply as many English-language secondary sources and translations as possible. I reference these in the notes. Original language sources may be found in the Abbreviations and Bibliography. Finally, in an effort to produce a text as unencumbered as possible, I have limited diacritics in transliterating foreign-language names, terms, and sources and restricted the number of notes. I trust that those who wish to explore the subject further will find the bibliography an adequate gateway into a vast and growing body of literature on Byzantine warfare.
## ONE
## HISTORICAL OVERVIEW
EARLY PERIOD (FOURTH TO SEVENTH CENTURIES)
After six years of construction, the shining new capital city of Constantinople was consecrated on May 11, 330. By the time the city was completed, its founder, the emperor Constantine, was a hardy and hale emperor fifty-eight years old. Constantine had built a magnificent metropolis on the narrow straits that divided Europe from Asia and was the gateway into the vast hinterland of Anatolia and the Near East. Roman builders largely demolished and remodeled the old Greek fishing town on the site, Byzantium, into a city worthy of being capital of the greatest empire on earth. Thus, for many modern historians, the year 330 marks the beginning of the "Byzantine" or "East Roman" Empire. For their part, the Romans gathered on that spring day on the shores of the Bosphoros had no sense of a break with the past, rather they viewed with satisfaction the achievements and continued power of eternal Rome under their vigorous leader. The inhabitants of the Byzantine Empire called themselves Romans until the destruction of their state by the Ottoman Turks in the spring of 1453.
Constantine, like many of his successors, would find that the new capital was convenient for campaigns northward, across the Danube and against the Sarmatians and Goths. In 322, prior to his becoming sole emperor, Constantine attacked the Iranian Sarmatian tribes north of the river and won a major victory, claiming the conquest of Sarmatia, _Sarmatia Devicta_ on coins issued in 323–24. Both the Sarmatians and Germanic Goths provided troops to Licinius, the emperor in the east and the main rival of Constantine. In 332, Constantine ordered the old bridge of the emperor Trajan to be restored across the Danube, a symbolic act intended to convey to the neighboring peoples that the Romans would return to Dacia, which had been conquered by Trajan but abandoned by the emperor Aurelian during the period of near anarchy that braced much of the third century.
Constantine advanced with his Sarmatian allies against the tribal confederation that the Romans called Goths, a disparate mix of people of uncertain origin with a core Germanic element whose exact complexion and identity still remain open to debate. The Goths lived in a broad belt of territory across eastern Europe, namely present-day Romania eastward to the southern Ukraine and the Crimean steppe. Since the third century Gothic tribesmen had raided Roman territory and from nearly the same period some served in the Roman army. Despite the Goths' considerable numbers and military capacity, Constantine's forces defeated those of their king Ariaric, whose people suffered tremendously from the war and the cold—one source states that 100,000 died. While exaggerated, the figure underscores the bloody contests between Romans and Goths along the northern frontier. The Gothic clans accepted Roman overlordship and remained at peace until the end of their rule. In the closing years of his reign, Constantine again campaigned against the Sarmatians, resettling thousands of them in Thrace, Scythia, Italy, and Macedonia. So thorough was the emperor's pacification of the Danubian frontier that no disturbances are known during the remainder of his rule.
Constantinople provided a valuable strategic location for wars in the east, whence the emperor could march against the most serious threat—the Persian Empire ruled by the Sasanian dynasty, whose ascent to power a century prior had led to increasingly serious hostilities and major Roman setbacks, most notably the collapse of the Roman eastern defenses in the 250s. For centuries, the Romans had battled Iranian peoples in the east, first the Parthians, and then their Sasanian successors. Even at its height the empire proved incapable of digesting Mesopotamia—Hadrian disgorged the conquests of Trajan and beat a hasty retreat and this in spite of the dominance of Roman arms and the discomfiture of the Parthian enemy. These facts betray the lack of a Roman answer for their eastern question: they could rarely win decisive victory over the civilized power on their Syrian border and in those rare instances when they did so, they seemed to prefer a Parthian or Persian enemy to their own hegemony east of the Tigris and Euphrates. The nadir of Roman power in the east came in 260 when the Roman emperor Valerian confidently advanced east to meet the upstart Iranians in Roman Mesopotamia only to meet disaster at the Battle of Edessa (modern Urfa in southeast Turkey) and fall prisoner to the mighty Sasanian "King of Kings" (Shahanshah) Shapur I (ca. 240–ca. 270). The death of Shapur I around 270 led to internal bickering among the Persians that allowed the Romans to seize the initiative. During his brief reign (282–83) the emperor Carus marched in force through Assyria and down the Tigris to southern Mesopotamia and the Sasanian capital of Ctesiphon (about 35 kilometers south of modern Baghdad). This type of campaign, which saw the Roman army march deep into Mesopotamia against the Sasanian capital, was to be repeated several times in later centuries, and in each later excursion there is a sense of déjà vu—once the Romans got there they did not seem to know what to do about the place. Even if they did capture Ctesiphon, as allegedly did Galerius in 298, they did not stay. Perhaps it was the size of the city (really an agglomeration of settlements clustered on the Tigris and along various canal branches), the stubbornness of Persian defenses, or the difficulty of maneuver in a complex, conurbated landscape cluttered with canals. Perhaps it was the unmercifully hot and pestilential land that stymied the Romans. Equally likely, the propaganda value of having reached Ctesiphon far outweighed the difficulties of capturing or administering an occupation. In July or August 283, the sudden death of Carus forced the Romans to withdraw under their new emperor, Numerian—one among many such failures. The youthful Numerian himself died in November 284, when the former duke ( _dux_ ) of Moesia on the Danube, Diocles, assumed the imperial power and became Diocletian.
Diocletian stitched the Roman Empire whole after a half century (235–84) of military anarchy, economic trauma, and civil unrest. He made far-reaching changes in the civil administration, the military, and attempted to stabilize the economy. Although not revolutionary (Carinus had associated relatives in his rule as a fellow Augustus and Caesar), Diocletian formulated a bold solution to the succession crises and attendant chaos that had gripped the state in recent decades. By 293 he established a scheme based on the "rule of four" (Tetrarchy). The Tetrarchic system divided the empire into two zones governed by an emperor (Augustus) each with a subordinate (Caesar) who would take power once the senior emperors stepped aside voluntarily. As Constantine, among others, would prove, this system was effective only if men were willing to give up power, something that has occurred only rarely in history. In military matters, the most important changes were a considerable expansion of the army. During the tumultuous years of the military anarchy, the ceaseless civil and foreign wars had led to critical degrading of the empire's military forces. Diocletian inherited an army of about 389,000 men and, through a great conscription program, nearly doubled its size to somewhere over a half million men. There was an increase in the proportion of cavalry units in order to provide more offensive capabilities and match horsed units of their northern and eastern opponents.
Our best evidence suggests that Diocletian and Constantine molded a Roman army considerably different from their predecessors. The aim of this program was to stabilize the frontiers and to ensure internal security which had been shattered in previous decades. Despite their bellicosity and propaganda, the Romans entertained no serious intention of annexing lands beyond their great river boundaries—the Rhine, Danube, Tigris, and Euphrates. But as the third century unfolded the policing of these permeable frontiers had become increasingly problematic, with multiple threats posed by barbarians who had become gradually more sophisticated and militarily capable. Roman frontier management with its frequent punitive raids, the infrequent large-scale invasion, and the complexities of trade and recruitment from among the tribes and neighbors who were often the target of aggression was both stimulated by and reacted to the shifting conditions of the vast borderlands. Diocletian's determination to keep the barbarians out is best viewed today in the massive fortifications of the east at places like Lejjun in Jordan and Resafa in Syria. In these places, rather standardized, large-scale legionary encampments embedded frontier troops in a line of defense. The troops that garrisoned these forts were called _limitanei_ , border troops or frontier guardsmen who were regular soldiers and not, as some have speculated, a kind of militia. The frontier forces were strong enough to handle internal policing and local disturbances; in Syria the aggressors were often Bedouin tribal raiders. In the case of full-scale invasion, the frontier fortresses were meant to hold the line long enough for the arrival of the recently created mobile field army ( _comitatus_ ) comprised of elite cavalry and infantry units initially drawn from loyal and seasoned legions, especially on the Danube frontier. The limitanei also formed part of the expeditionary armies on major campaigns, but without backing from the mobile imperial field army frontier garrisons lacked strategic initiative. When the enemy arrived in force, as did the Persians at Nisibis in 337 and the Goths on the Danube in 376, they faced strongly manned hard points that they could not risk bypassing.
In 336 war broke out with Persia. Constantine sent east his nineteen-year-old son, the Caesar Constantius, to prepare for the brewing conflict. Constantius had mixed success while his father spent the year 337 preparing for landing a knockout blow against the Sasanians that he hoped would deliver peace to the Roman eastern flank. But the emperor was never to undertake the campaign. Constantine fell ill around Easter of 337 and traveled across the Marmara straits to take the hot waters at Helenopolis in Bithynia (modern Hersek). Sensing his end was near, he summoned his clergy and sought baptism, which had been postponed by the emperor following the common Christian belief of the day that the sacrament cleansed one of all sins committed to that point. On May 22, 337, the emperor died and with him hopes of punishing the Sasanians.
Constantine had divided the empire among his three sons: Constantine II, Constans I, and Constantius II. Constantine II ruled the territories in the far west including Spain, Gaul, and Britain. Constans II ruled the central portion including Italy and North Africa, while the east fell to Constantius (337–61). In addition, their cousins Dalmatius and Hannibalianus served as Caesars. It is difficult to conceive how Constantine envisioned such a brew of power sharing would work in practice, given that he had himself single-handedly overturned the Tetrarchy. In any case, the situation did not long survive him. In 337, Dalmatius and Hannibalianus were butchered along with other family members at the instigation of Constantius. Constantine II met his end in an ambush in Aquileia in 340 and his elder brother Constans in 350 fell victim to the rebel Magnentius. This dynastic strife distracted Constantius from his task of defending the eastern frontier against the Sasanians who had aggressively renewed the war.
Constantius proved a vigorous, if yeoman, commander. His loss of Amida (359) was a terrible blow to Roman prestige and underscored Persian might, but throughout his reign Constantius fought aggressively to defend Roman interests in the east. After the bloodletting of the succession was over, the young emperor faced a Persian siege of the city of Nisibis (today Nusaybin, Syria) on the upper Mesopotamian plain, an ancient city that was the linchpin of Roman defenses in the region. In either 337 or 338, the Persians battered the city in a grueling siege led by the young, vigorous King of Kings Shapur II (307–79) himself. Pitted against the shah was the local Syrian bishop of the city, Jacob of Nisibis, who organized the defenses and bolstered the morale of the citizens. Confronted over seventy days with a stubborn defense that confounded assaults using mobile towers and efforts to undermine the walls, Persian engineers dammed the local river Mygdonius and diverted it, unleashing the power of the pent-up waters against the city walls, one portion of which gave way beneath the rush of the torrent. The Persians delayed their attack as the waters had turned the approach to the breach into a quagmire. The next morning, the Sasanians were shocked to find the breach filled with rubble to the height of the previous wall and defended by the soldiers and citizens of Nisibis, urged on by their omnipresent bishop. Shapur's last assault failed and the Persians were forced to decamp.
The rest of the war between Constantius and Shapur is muddled in our sources; it seems that there were numerous large-scale clashes—rare for the day—between the Romans and Persians, including two more major sieges of Nisibis and two encounters at the salient of Singara, in what is now western Iraq. Probably in the 340s, Singara fell to the Sasanians. In most of these battles, the Romans were bested, though in their assault on Nisibis in 346, the Persians failed to take the town. Their third attempt, in 350, saw the Sasanians mount a colossal four-month effort in which they once again flooded the plain around the city with waters diverted from the Mygdonius River. According to one account, they assaulted the city on boats—surely an amazing sight in what was once the midst of the desert steppe—but were repulsed by the valiant efforts of the defenders. The war ground to a draw.
An uneasy calm settled between the two antagonists since internal rebellions against Constantius made it difficult to devote men and material to fighting the Sasanians. The emperor appointed his cousin Gallus to command of the eastern front in 351. The young Caesar, perhaps twenty-five at the time, was effective militarily but unpopular among the local elites at Antioch; he was executed in 354 for alleged treason. More than by Gallus's abilities, the Sasanians were largely restrained from offensive operations because of conflict on their own eastern front in Central Asia with the Chionites, a group of uncertain origin, but probably Iranian-speakers whom Shapur defeated and absorbed into his armies. Roman writers call the Chionites "Huns," but their ethnic makeup and way of life are unknown. Whatever the case, Shapur integrated large numbers of Chionites into his army and again turned his eye westward after peace negotiations broke down. By 359, the shah with his new Chionite troops under their king Grumbates probed Roman defenses along the Euphrates, bypassing Nisibis and seeking a passage across the flooded river. A high-level Roman deserter, who had fallen into debt and could not pay his taxes, the protector Antoninus, guided the Persians. Antoninus was well placed to be a spy, likely with wide-ranging access to imperial intelligence including the order of battle of the eastern armies and their logistical situation. His information was vital to Shapur, who on account of it attacked Amida, which fell after a bitter siege of seventy-three days of fighting vividly depicted by the Roman historian Ammianus Marcellinus, including a night attack by the Romans that nearly overwhelmed the Persian camp and the final herculean efforts of the Sasanians to carry their siege mounds to the walls. Shapur sacked the city and deported its inhabitants to Khuzestan in what is now southwest Iran.
In the wake of the serious defeat, Constantius reshuffled his high command. More critically, he ordered his cousin Julian, the Caesar in the west, to dispatch Gallic troops to reinforce the east. Julian refused this order on the grounds that his troops were mutinous and declined to serve away from home. Instead, the Gallic legions proclaimed Julian emperor, whereupon they happily marched east to confront Constantius. Upon hearing the news of his cousin's rebellion, Constantius was apoplectic—his rage, coupled with the strain of years of campaigning and the heavy defeat at Amida—killed him, probably of an embolism, November 3, 361, in Cilicia. Julian, now uncontested, donned the imperial purple and immediately set about reversing what he saw as the pillars of the decadent house of Constantine: a devout pagan, Julian offered sacrifices personally to the old gods, ordered the temples reopened, and actively legislated against Christians. He was careful, however, to avoid outright persecution so as not to create more martyrs. Nevertheless, the animosity Christians held against the apostate emperor who had departed the true faith and had risen to power to destroy it knew no bounds—one Christian bishop dreamed that he saw a vision of the popular military saint, Merkourios, spearing the emperor.
Julian was an effective leader and, despite his rather frail frame and awkward manner, a fine soldier. Unlike most commanders, Julian personally fought in engagements, an act that won him widespread admiration among his soldiers, but betrayed a recklessness that would be his undoing. Perhaps his greatest strength was his zeal and devotion to the idea of Roman greatness as well as a personal identification with Alexander. Along with the desire to avenge recent defeats, these ideals drove the emperor to strike a decisive blow against the Persians, something that neither Constantine nor Constantius could do. A victory by the pagan emperor over the feared Sasanians would further undermine the Christian faith that Constantine and his sons had thrust upon the empire. In March 363 Julian left Antioch at the head of a large army that moved down the banks of the Euphrates, accompanied by a river supply fleet. Julian ordered the Roman client king of Armenia, Arsaces, to form a second invasion column and invade from the north. The emperor's forces made good speed and encountered only sporadic resistance on the way to Ctesiphon, which Roman forces reached in April. After defeating the garrison of Ctesiphon and sensing that the Persians were in his hands, Julian rebuffed Shapur's peace overtures, but he was unable to force his way into the Sasanian capital. Instead, as the weather grew hotter and Persian sabotage of the irrigation complex around the sprawling metropolis of Ctesiphon created a fetid quagmire, the Roman high command made the fateful decision to burn the supply fleet and strike inland.
Shapur II shadowed the Roman army as Julian moved northward along the banks of the Diyala River, then the Tigris on his way back to Syria. The Sasanians practiced scorched earth and launched constant harassing attacks that turned the march into a running battle across northern Mesoptamia. Exhausted, suffering from heat, thirst, and starvation, the Roman forces were ground down by desert combat. On June 26, the emperor fought in a major engagement against a large Persian force. Because of the searing heat he rushed to battle without his armor. A Sasanian cavalryman thrust him through with his spear and Julian died in his tent the same day.
The army elected Jovian emperor, the compromise choice selected not on merit but because he posed no threat to imperial elites. By this time, disaster threatened to overwhelm the entire Roman field army, caught as it was far from home and facing a potent enemy who was starving the Romans to death. Jovian proposed peace, and the terms that he accepted were devastating. Nisibis, the powerful Roman bridgehead and thorn in the side of the Persians, was unceremoniously handed over along with the territorial gains made long before by Diocletian. The strategic balance tipped toward the Sasanians but Jovian did not live long beyond the ink drying on the disastrous treaty; the emperor died in western Asia Minor in the winter of 364. His successor, Valentinian (364–75), chose his brother Valens (364–78) as co-emperor in March 364. Like many military men in the late antique empire, the brothers were Pannonians (a region on the central Danube) and among the last effective soldier-emperors of late antiquity. Valentinian assumed control of affairs in the west and Valens governed the eastern half of the state from Constantinople. In 364 the brothers divided the army into eastern and western forces and then left one another to stand on their own resources. Valens marched east and was in the central Anatolian city of Caesarea in Cappadocia when news came of the rebellion of Prokopios, a male relative of Julian and, as a member of the house of Constantine, a serious rival to the upstart emperor. Divisions among the conspirators led to the defeat and execution of Prokopios, who had Gothic military support.
With the east quiet for the moment, Valens turned his attention to punishing the Goths. In a three-year war he humbled the Gothic tribes north of the Danube. When the war wound down in 370 concessions on both sides led to relative quiet in the north for the next five years and established an equilibrium. Neither side could foresee the maelstrom that would destroy the Gothic polity and drive the tribes into a headlong collision with Rome.
Like many nomadic powers, the Huns appeared to have spontaneously generated in the vast steppe that swelled from the Black Sea to China. In the famous and oft-quoted fourth-century description of Ammianus Marcellinus, they were half-beasts who stitched together garments from the skins of field mice and who led a life of savagery forever on horseback. The truth is obscured and cannot be recovered; probably there was a much longer time horizon of Gothic-Hun contact and warfare than Ammianus leads us to believe. Huns, whose ethnic origin is widely debated and uncertain, were most probably a mixed group of steppe warriors of Turkic language. Nomad armies commonly integrate the conquered into their ranks and by the fifth century, the Huns included Chinese, Germanic, and Iranian elements along with the original Hun groups.
The Huns burst onto the European scene in 375 and smashed through the Gothic communities stretching from the Crimea to Transylvania, forcing the flight of many to the banks of the Danube where they sought to enter Roman territory as terrified refugees. Valens, who esteemed the Goths as good troops who had long provided serviceable recruits for Rome as well as formidable enemies, allowed the thousands of beleaguered people to cross the river. Once the Goths were out of immediate danger in Roman territory, the Romans struggled to maintain order and neglected to provide supplies to the mass of people whose precise numbers are unknown. When posing the question Ammianus, quoting the Roman poet Virgil, noted one might as well ask how many grains of sand were in the Libyan desert, so great was the host.
Roman officials took advantage of the precarious state of the Goths and demanded high prices for food, exchanging dogs for children and treating their guests with contempt. After a riot in which Romans and Goths battled while Gothic and Roman leaders feasted at the city of Marcianople (today Devnya, Bulgaria) the Goths rose in revolt. They were led by Fritigern, the tribal head of a group of Goths known as the Tervingi. Other Goths from Roman army units in Thrace joined Fritigern, who disavowed the agreement with the Romans and began pillaging. A sharp encounter with local Roman troops ended in the Goths victorious and rampaging throughout Thrace.
By 377 Valens was alarmed—he ceased hostilities against Persia and prepared a strike against the Goths who ran amok in Thrace and Moesia. Gothic elements had formed themselves into a keen fighting force, well equipped with captured Roman arms and well provisioned. By 378, the Goths were hemmed in by troops from the western half of the empire under the emperor Gratian (367–83), son of Valentinian, who moved to assist his uncle Valens in the east. Though Gratian's advance forces advised Valens to wait for the full western field army to arrive before giving battle, the eastern emperor was impatient for a major victory that would bring glory and legitimacy to him and give the Romans a free hand to deal with their eastern question. Fritigern's army moved past Adrianople to the northeast and awaited reinforcements from the Greutungi Goths to whom he appealed as allies. Valens had been assured by his scouts that the Goths numbered only ten thousand, while the eastern field army was probably three times larger. Fritigern sued for peace, but Valens rebuffed his overtures and attacked the Gothic position on August 9, 378. Only the brief account of Ammianus survives, and due to the fact he was not an eyewitness and more concerned with the events surrounding the battle, we have only the faintest view of what happened that momentous day.
The Greutungi made a sudden appearance in the nick of time to reinforce Fritigern's Tervingi. These reinforcements put the Goths close to numerical parity with the Romans. The Goths took up position on a hill, surrounding a wagon laager in which their families sheltered. The Gothic cavalry were away from the laager, burning the fields to hinder the Roman advance through the morning hours. The Romans arrived in mid-afternoon in the heat of the day in some disorder. Roman elite troops, too eager for battle, advanced before the rest of the army was fully ready and were easily repulsed, while the Roman cavalry on the left advanced to the laager beyond the support of their infantry where they were surrounded by the Gothic cavalry and infantry and routed. The Goths now attacked the Roman left flank and pressed the Roman ranks in a vice. By late afternoon the Roman infantry broke and fled and the slaughter was on. Valens himself was killed and his body never found. Adrianople was a disaster that rivaled Cannae in its significance, with two-thirds of the eastern army killed. The arrival of Gratian did little to halt the losses, as the young western emperor was reluctant to shed his own troops' blood in a risky confrontation with a menacing foe. Gratian recalled a disgraced senior commander, the Spaniard Theodosius, out of forced retirement and elevated him to Augustus. The western emperor provided some men and materiel for the unnerving task of staunching the open wound of the Gothic War.
Although many Goths attacked the Romans after Adrianople, some were induced to serve the empire. Increasingly, it was Gothic troops recruited into imperial service (labeled here as elsewhere by historians "Byzantines" or "East Romans" and later on "Greeks" due to the primary language of the empire) who formed the rank-and-file and officer corps of the eastern army. The Byzantines struggled to integrate their Gothic troops and failed to blend them fully into imperial society. The increasing "barbarization" of the army and officer corps, which endured for about a century from Adrianople, paralyzed the eastern state at a critical time and contributed to eastern passiveness and ineffectiveness against the Huns and other enemies. Fortunately, although there were some sporadic hostilities, the fifth century was generally quiet on the Persian front. This calm was due primarily to conditions within Sasanian Persia, whom the Hunnic Hephthalites had humbled when they killed the Sasanian Shah Peroz and captured his son, Kavad. Kavad ascended the throne in 488 and consolidated his authority against formidable internal enemies, then turned against Byzantium. The war of 502–6 marked the first hostilities between the empires for sixty years. Along with Hephthalite troops, Kavad captured the major Byzantine fortress cities of Theodosiopolis, Martyropolis, and Amida while Arab auxiliaries under the fearsome Arab chief Nu'man pillaged Mesopotamia. Roman bungling and a divided command led to the war dragging on until 506.
By 527, Kavad (488–531) and the Sasanians once again waged war against the Romans. This time war erupted because of disputes over the Caucasus, coupled with the alleged refusal of the emperor Justin I (518–27) to adopt Kavad's son and heir, Kosrow (531–79). From 527–31, Byzantines and Persians fought along the fortified frontiers of Armenia and in a series of lightning raids executed by Mundhir, the Arab king and Sasanian proxy. Mundhir's opponent, the Roman-sponsored antagonist the Arab Harith, fought a series of bitter contests against the Persian Arabs. During these wars, both sides won and lost many battles. The general effectiveness (or ineffectiveness) of the soldiery and their commanders, along with strategic and logistical limitations, made these wars of attrition in which neither side could (or perhaps wished) to deal a knockout blow against the other. For all their propaganda and history of hostility, these ancient states appreciated the known and valued their ability to negotiate and manage their biggest enemies. When Kavad died in 531, Kosrow negotiated the "Endless Peace" out of necessity to deal with internal problems. Justinian (527–65) used the breathing room he gained to embark on recovering former Roman lands in the western Mediterranean. By the end of his reign he had recovered part of North Africa, southern Spain, and Italy.
The terrible consequences of all-out warfare between Byzantium and Persia unfolded in the first decades of the seventh century. When the emperor Maurice was assassinated in 602, his Sasanian counterpart, Kosrow II (590–628) (whom the Romans had helped regain his throne during a civil war of 590–91), warred against Byzantium, ostensibly to avenge the killing of Maurice. In complexion, Kosrow II's war was remarkably different than past encounters. While at first the Sasanians seem not to have intended to permanently occupy Byzantine lands, the total collapse of Roman resistance opened the possibility and the Sasanians quickly adapted their strategy.
The Roman-Persian War of 602–28 was an epochal struggle, which one historian has aptly called "the last great war of antiquity." The two powers ceased sparring and now grappled for supremacy. The coup launched by the usurper Phokas (602–10) divided the Byzantine command and sparked resistance to the regime, which faced invasion from the east and internal rebellion. From the outset things went badly for the Romans; by 609–10 their defenses in Syria and Upper Mesopotamia collapsed, allowing Persian forces access to the Anatolian plateau and Syria, Palestine, and Egypt. These eastern provinces formed the rich, weakly defended underbelly of the Byzantine state where a combination of military defeat, religious dissension, and civil war made them low-hanging fruit plucked by Sasanian hands. In 609 the Byzantine governor of far-away Carthage in Byzantine North Africa equipped a fleet and land army in revolt against the emperor Phokas. In 610, the fleet arrived at Constantinople and deposed Phokas whereupon Heraclius, son of the African governor, ascended to the throne.
Heraclius's initial efforts against the Persians were disastrous. His heavy losses near Antioch in 613 led to the Sasanian conquest of Damascus in the same year, and in 614 the Persians sacked the holy city of Jerusalem, carrying away the "True Cross," the holiest of relics and a potent symbol of the discomfiture of the Byzantines and Christianity. By 619 Alexandria was betrayed to the Persians, and Egypt, the bread basket and most populous region in the empire, fell into Sasanian hands. Backed by the church and employing a highly religiously charged propaganda, Heraclius retrained and reformed the shattered Byzantine army. In 624, the emperor struck into Persian Armenia and Azerbaijan and sacked several cities there, then frightened off Kosrow with a bold strike against the shah and his army. When Kosrow fled, his army disintegrated and left the Byzantines to plunder extensively. In the following year, the Persians dispatched three armies against the Byzantines, but Heraclius outmaneuvered these forces and defeated them in turn. The year 626 brought the climax of the war. The Byzantines drew into alliance with the powerful Western Turk empire that lay astride the north and east of the Persian frontier. A two-pronged Roman-Turk offensive was a strategic nightmare for the Sasanians. For their part, the Persians allied with the new power north of the Danube, the nomadic Avar khaganate, and sought to envelop the Roman state. Persian troops ranged against the great city of Constantinople across the Bosphoros straits to the east, while Avar troops besieged the city. The Persians' allies were to ferry across the Sasanian troops to complete the siege force, but the Byzantine fleet thwarted these efforts. In the meantime, the emperor Heraclius, who had made a colossal gamble in leaving his capital on its own to face the ponderous weight of Persian attack, renewed the offensive in the east. Buoyed by Turkic steppe nomads and Christian allies from among the principalities of the Caucasus, the emperor boldly drove into the heart of the Sasanian Empire. By January 627 the Romans ravaged the fertile heart of Persian Mesopotamia, scarring the land black with their burning.
Inside the capital, disaffected elements of higher Persian society acted to salvage the state and staged a coup on February 23. The Shahanshah was first imprisoned, then executed after his son Kavad II was crowned. The young Kavad lasted less than a year and the Persians descended into dynastic and political chaos. By 630, the Sasanian generalissimo, Shahrvaraz, had taken control, but the fate of both Near Eastern empires was about to be sealed.
About the same year that Shahrvaraz ascended the Sasanian throne and the Romans continued their return to their recently regained provinces, a charismatic Arabian preacher named Muhammad seized his home city of Mecca. Born about 570 and having had a series of revelations beginning around 610, the best guess of historians is that Muhammad had been actively preaching and recruiting converts for the two decades prior to 630. The identity forged by the reception of Muhammad's message fostered a confident and aggressive spirit among the early community of believers who followed the nascent religion of Islam. The core of Muslim believers, the Companions (Arabic _sahabi_ , pl. _sahaba_ ) carried the small body of new co-religionists to the conquest of the Arabian Peninsula and thence to Syria. The invasion of Roman Syria was a natural arena in which to expand; the Arabs of Mecca had extensive trade and property networks there, and the Romans were weak from decades of fighting.
The early Arab armies owed their success in part to Arab warfare experiences in the peninsula and in service of the great powers, in part to religious inspiration and apocalyptic vision, and in part to greed. Despite the decades of warfare, most areas remained relatively unscathed—their territories had only rarely, if ever, been traversed by armies or witnessed sustained sieges or battles, and their populous cities and numerous inhabitants promised rich pickings.
The first Arab attacks on Byzantine Syria had, in fact, preceded the concluding acts of the Persian War. The minor skirmish at Mu'ta, in what is today Jordan, ended in a Byzantine victory and was immortalized in Muslim memory as a heroic engagement in which several prominent early Arabian heroes became martyrs. It was also a battle in which appears Khalid b. al-Walid, outstanding commander and critical leader of the great conquests. When the higher commanders were slain, leadership fell to Khalid, and he is credited with executing the withdrawal of the Muslim forces. As with all of the Muslim conquests, our sources are much later, often piecemeal, or even contradictory, and any reconstruction set forth is our best available interpretation.
When Muhammad died in 632, the Muslim community chose a caliph to be the spiritual and political head of the body. The Muslim invasion of Syria gathered momentum during the reign of Muhammad's successor, the caliph Abu Bakr (ca. 632–34). In 634, the Arab commander Amr b. al-As led a small Muslim army against Gaza, where he defeated the local Byzantine garrison. Probably in the same year, Khalid b. al-Walid led a raiding party from Iraq across the Syrian desert along the fringes of Roman occupation. Khalid's Muslims attacked the Ghassanid Arabs on the Christian holy feast of Easter. The Ghassanids were Arab Christian allies of the Byzantines and march wardens of the empire's desert frontiers. Khalid's attack on the Ghassanids led to the surrender of the important nearby city of Damascus, whose citizens capitulated and agreed to pay tax in exchange for Muslim protection.
In summer of 634, the Arabs again encountered Byzantine forces, this time at Ajnadayn. We have no contemporary information regarding the battle, but we know that the Muslims were once more victorious. The remnants of the Byzantine forces withdrew to Damascus and there faced a Muslim siege. The relief forces that Heraclius sent from his command center of Homs were defeated en route. The details provided of the Muslim siege of Damascus reflect later traditions which claim that one half of the city surrendered peacefully while the other half was stormed. Such incongruities in the sources allow us to only sketch the events of the conquest. What is generally agreed is that Heraclius mobilized a sizable force and marched them south to relieve Muslim pressure on Damascus and to end the Arab threat. It is doubtful at this time that the Byzantines understood that they were dealing with a new religious movement, nor could they recognize any difference between the Muslim Arabs and other Arab groups, of whom the Romans were often disdainful. Clearly by 635–36 the Byzantine high command comprehended that provincial forces had failed utterly in containing the threat and that decisive action was needed.
In 636 Heraclius, now old and ailing, dispatched a large army, variously estimated at 15,000–20,000 strong. They assembled in the Golan Heights, in the traditional pasturelands of their Ghassanid Arab allies. The Muslims prepared to meet the enemy field army with a force that seems to have been slightly larger and was commanded by prominent believers, including Abu Ubayda, Khalid b. al-Walid, Amr b. al-As, and Yazid b. Abi Sufyan, the brother of the future caliph Mu'awiya. No contemporary accounts of the subsequent Battle of Yarmuk survive; therefore, reconstructing the course of individual battles or campaigns cannot be done with confidence. Though the details are opaque, the outcome of the conflict is clear—the Arab army won a crushing victory that ejected Heraclius and the Byzantines from Syria. Roman forces regrouped behind the Taurus Mountains on the Anatolian plateau as they had done during the dark years of the recent Sasanian wars. The emperor must have looked on his forces with a mix of emotions as he recalled the days spent in the highlands drilling his warriors to battle readiness before their epic encounter with the Persians more than a decade previously. But the emperor, now a sexagenarian, and his empire were exhausted by the decades-long war with Persia. Men and resources were strained to the limit and morale was catastrophically low. The imperial authorities were neither nimble enough nor capable enough to resist the determined and skilled Muslim advance. City after city fell.
The rich land of Syria was not the only prize sought by the Muslims. In 639 Amr b. al-As led a raid across the Sinai to Pelusium, the gateway to Egypt, which fell in early 640. Muslim reinforcements spearheaded a full-scale invasion of Egypt. The key fortress of Babylon on the Nile (near modern Cairo) fell in 640, and Alexandria surrendered the following year. By 647 the Muslims were on the offensive against Byzantine North Africa, where they crushed the local commander Gregory at Sbeitla in a battle that broke the backbone of Byzantine resistance. The first Arab civil war (656–61) that ended with the establishment of Mu'awiya ibn Abi Sufyan (661–80) as caliph afforded the Romans in Africa some breathing space in which to regroup. Despite the promising start, Africa did not fall easily—the conquest would not be complete until the Byzantines were driven from Carthage in 698.
DARK AGES AND MIDDLE PERIOD (SEVENTH TO TWELFTH CENTURIES)
Further east, Heraclius and his successors managed to withdraw the remnants of their shattered armies to Anatolia where the strategic triage included settling soldiers in the countryside in scattered garrisons and holding the line along the Taurus. This highland frontier (Map 4), roughly the same that Heraclius had managed to maintain in the dark days of the Persian War, was constantly pressured by the new Umayyad caliphate. On numerous occasions Muslim flying columns penetrated the plateau, and from an early date raids became a regular feature of life in the uplands of Byzantine Anatolia. Like the later English _chevauchées_ of the Hundred Years' War, Muslim raids were generally fast-moving plundering forays that aimed to keep the empire off-balance and weaken its social and economic fabric. However, once Mu'awiya constituted a Muslim fleet, the Mediterranean became a battleground where the empire fought for its life as the Islamic tide rose to the capital itself. The caliph launched a vast expeditionary army against Constantinople and established an operational base in the Sea of Marmara in 674. This sustained series of attacks were only defeated in 678 by determined defenders aided by the early use of a new weapon: "Greek fire", a naptha-based incendiary (see Chapter 8). In 717 the Muslims renewed their efforts to destroy the Byzantine state with a massive attack on the capital and once more they suffered heavy losses and defeat. Such was the weakness of the Byzantine army that the victorious emperor, Leo III (717–41), made no effort to go on the offensive.
Although they could not know it at the time, the empire had faced the worst of the storm. Yet the disruption of these military encounters was total. Cities shrank and virtually disappeared. The money economy faltered to near collapse. Literary society and high culture declined. Never before had an empire absorbed such unremitting punishment at the hands of an enemy, lost so much territory, and remained intact. Over the dark decades at the end of the seventh and beginning of the eighth century, the Byzantines adapted themselves to the new situation: they were much poorer in money and men than their enemy. The collapse of cities and the fiscal structure forced extreme economies on the army. The exact complexion of these changes remains in debate, but it seems that troops were settled in the countryside and provided with the barest of wages. Initially five regional army commands (Maps 4 and 5) were constituted from the remnants of the armies settled in the district—thus the billeting lands of the old Army of the East (Anatole) came to be called the Anatolik. These regional commands were called _thema_ , a word of unknown origin. The thematic army was both qualitatively and quantitatively the inferior of its late Roman predecessor. Equipment and training suffered, but the army remained a professional fighting force with recognizable units and regular drill. The Dark Age thematic army was a defensive force led by commanders who rarely risked pitched battle. Instead, Byzantine commanders preferred to harass and wear out enemies who already had to traverse great distances from their bases in Syria to reach the populated places of the plateau. The low intensity conflict of raid and counter-raid, punctuated by the occasional large-scale imperial or Arab expedition, became the norm from the later seventh through the ninth century. During this period, the frontier dwellers on the Anatolian plateau of Byzantium developed a military caste of families whose fortunes were linked to war. By 750, the eastern Roman state was recovering somewhat demographically and economically—the first glimmers of a revival may be seen during the reign of Constantine V (741–75) who survived a bitter revolt by several themes. Around 743, Constantine formed a permanent body of professional cavalry regiments ( _tagmata_ ) stationed in and near the capital where they could quickly muster to the emperor's aid. The tagmata provided a more professional, loyal, and disciplined core on campaign than the provincial armies of the themes. After decades of fitful defense, in 745 Constantine led the tagmata and thematic troops against the caliphate while the latter was hobbled by the Third Civil War.
In 750, the fall of the Umayyad dynasty in the Third Civil War provided some respite to the Byzantines who nonetheless remained weakened by internal political and religious dissent. The new Islamic state, the 'Abbasid dynasty, based itself in Iraq, and though ideologically still committed to _jihad_ the 'Abbasids favored persistent raiding rather than the massive assaults that had failed the Umayyads and contributed to their downfall. It would be wrong to view the 'Abbasids as less aggressive than their predecessors, however. When the opportunity arose, 'Abbasid commanders were keen to polish their jihad credentials in warring against the Romans, and some invasions, like that of 838, penetrated the heart of the empire and could have laid the groundwork for outright conquests that political realities otherwise forestalled. For each large invasion and investment of Byzantine cities, there were scores of minor raids and spates of violence across the long, sinuous frontier. There, in the mountain passes and the high dusty hill country of Anatolia, endemic warfare and the weakened state led to the rise of the _akritai_ , the border lords immortalized in the medieval Greek epic _Digenis Akritis_ ( _Two-blooded Border Lord_ ) which demonstrates the intimacy, respect, and violence of frontier elite warrior castes of the eighth through tenth centuries.
More than the caliphate itself, 'Abbasid border emirates mustered large forces for full-scale attacks. One of these expeditions provides modern historians blessed with hindsight a turning point in the Byzantine-Arab wars. In 863, Amr al-Aqta, the emir of Melitene (modern Malatya in eastern Turkey), and Ja'far, probably the emir of Tarsos, invaded the empire. Ja'far's troops advanced through the eastern Byzantine region of Cappadocia where they were defeated at a place called Bishop's Meadow, apparently by the emperor Michael III and the forces of the imperial tagmata. The second Arab raiding column, led by Amr al-Aqta, continued its raid, capturing Amisos (Samsun) on the Black Sea. The Byzantine commander-in-chief ( _domestikos ton scholon_ ) Petronas, using elements of the thematic armies and the imperial tagmata, surrounded al-Aqta in the mountains of Anatolia near the banks of the Lalakaon River where, on September 3, the Byzantines dealt a devastating defeat to the army of Melitene and its Paulician allies, the latter a sect of Christian heretics in eastern Anatolia. The victory at Lalakaon marked the end of massive Arab raids to the heart of Anatolia and opened the way for the Byzantine destruction of the Paulician homeland in eastern Asia Minor. It also proved that the Byzantine army of the Amorion dynasty (811–67) could match Arab armies in the field; thus by the end of Michael III's reign in 867, the Byzantines were on the offensive in the east, a drive they would sustain for more than a century.
While the Byzantines sought to maintain the core of the state—Anatolia and Constantinople and its hinterland—the losses in the provinces mounted. In Italy, by 750 the Byzantines had lost most of their territory to the Lombards, save some possessions in the south. Over a seventy-five-year period, beginning in 826, Sicily fell to the North African Muslims. Crete fell to Muslim raiders around 827 and became a raiding emirate founded on piracy. Perhaps the worst disaster to befall the army and state occurred not on the Arab front, but at the hands of the Bulgars, whose power challenged the empire in the north. Over the course of the seventh and eighth centuries, the former Balkan provinces had been mostly lost by the empire to Slavic tribes and the nascent Bulgarian khanate. The Bulgar khanate subjugated many of the Slavic tribes across the Danube and by the ninth century emerged as a major foe. In the spring of 811, the emperor Nikephoros I (802–11) led a large army of professionals and conscripts north across the Danube where the Bulgar khan Krum (ca. 802–14) sued for peace. Nikephoros brushed him aside and on July 20, 811, burned and pillaged the Bulgar capital of Pliska. The emperor then withdrew south, but Krum trapped the imperial forces in a mountain pass. In a dawn attack, the Bulgars killed Nikephoros and mortally wounded his son and designated successor. Krum's forces routed the Roman army and seized the imperial treasury, and along with the emperor perished "an infinite number of soldiers so that the flower of Christendom was destroyed"; the Bulgar khan made a drinking vessel from the emperor's skull. As a result of his victory Krum expanded his power to the south into Byzantine Thrace and set the stage for further conflict. Michael I (811–13) campaigned against the Bulgars but in June 813 after failed peace negotiations, Krum routed the Byzantines near Adrianople (modern Edirne) and pushed to the walls of Constantinople itself, where the new emperor, Leo V (813–20), attempted to assassinate the khan. Frustrated by his inability to breach the massive defenses of the capital, Krum withdrew; on the way he devastated Thrace and captured the major city of Adrianople. Only Krum's death the following year saved the Byzantines further humiliation.
Recruitment and maintenance of paid professionals expanded during the era of the Macedonian dynasty (867–1022), a period that marks the apogee of medieval Byzantine military power. Political fragmentation in the caliphate and improved economic and demographic conditions inside the empire allowed the Byzantines to regain much of their lost Balkan territories from Slavic tribes and to hold at bay their bellicose Bulgar neighbors. Wars against the latter were frequent and bitter. Under their ambitious and capable Symeon (893–927)—who betrayed his imperial designs by adopting the title of _tsar_ (Caesar)—the Bulgars responded to a trade dispute with an invasion of Thrace where they successfully captured Adrianople. In the spring of 896, the Byzantines sent the combined eastern and western thematic armies and tagmata against Symeon, who inflicted a heavy defeat on them at the fortress of Bulgarophygon. Unappeased by being installed within the Byzantine hierarchy, Symeon warred against the Romans who allied with nomadic Pechenegs and the Serbs and sent an army under the _domestikos_ (marshal) Leo Phokas, an easterner from a prominent military family. Romanos Lekapenos, the future Byzantine emperor, commanded the Byzantine fleet that was to ferry the Pechenegs across the river. At Acheloos (Anchialos), Symeon intercepted Leo's forces before his allies could join him and dealt him a decisive defeat—the historian Leo the Deacon commented that the piles of bleached bones of the Roman fallen could be seen in his own day, seventy-five years after the battle. Symeon pressed his advantage and moved south, where Leo Phokas confronted him with hastily raised forces. In the autumn in Thrace not far from the Byzantine capital, Symeon again swept Leo's forces from the field. He broke off his war against the Romans to punish his rebellious Serbian vassal, and wars against the Serbs and Croats occupied the tsar until the end of his life in 927, when his son Peter made peace.
If the wars with the Bulgars demonstrate a certain Roman military futility, they underscore the capabilities of the enemies of New Rome, who were sophisticated, organized, and aggressive. They also demonstrate why the Byzantines preferred proxy warfare and negotiations to full-blown confrontations that were chancy even when the balance of forces seemed to favor the empire. Finally, these conflicts, in which the Byzantines were bloodied as often as victorious, proved the resilience of the Roman army, which could not be destroyed in any one "decisive" battle any more than could their Bulgar or Arab neighbors.
By the tenth century, the Arab attackers who had taught many harsh lessons to the Byzantines were themselves weakening. Byzantine commanders increasingly took the fight to the Muslim states on their borders. This eastern push coincided with the ascendance of powerful military clans in Anatolia, especially the families of Phokas, Skleros, and others like them who won their spurs fighting the Arabs and rose in the imperial hierarchy until they occupied the highest military commands. The prestige, salaries, and access to power the military provided fueled the war effort in the east, which offered plunder and honor. Once the weakness of their neighbors was exposed, the Byzantines were deliberate in their advance. In 934, the Byzantines captured Melitene, the heart of one of the two major Arab emirates on their eastern flank. The _domestikos ton scholon_ , Nikephoros Phokas captured Crete from the Arabs in 961. Two years later, Nikephoros seized the imperial throne and continued to lead armies in person. In 965 he destroyed the raiding emirate of Tarsos and Byzantine armies advanced into Syria. In the Balkans the military successes also mounted. In 970 the Byzantines under Bardas Phokas confronted near Arkadiopolis (modern Lüleburgaz in European Turkey) a Kievan Rus' prince Sviatoslav I (ca. 942–72), who refused to leave Bulgaria. The following year, the emperor John Tzimiskes (969–76) arrived across the Danube with his eastern field forces, which crushed the Rus' at Dorostolon.
The army of the Macedonian era relied increasingly on professional mercenaries. Though apparently eager to serve, the thematic troops were progressively called on to commute their services to cash payments so that the emperor and his generals could recruit paid professionals. These standing forces included not only Greek elements, but also more and more foreign contingents. Especially well-known were the Scandinavian Rus' recruited into the famed Varangian Guard of Basil II (976–1025), but the army continued to be dominated by troops recruited from Anatolia until the eleventh century. The reign of Basil marks another pivotal point in Byzantine military history, for he largely sidelined the eastern military families who challenged his rule on two occasions and who had come near to dethroning him. The eastern campaigns were largely suspended in favor of Basil's project against Bulgaria. Basil II's Bulgarian struggle was war as the Byzantines were best able to wage it—a long, patient trial by fire. The strategy of incremental advance and attrition, rather than master strokes of large pitched battle, proved that Basil adhered to the Byzantine preference for the avoidance of decisive combat in favor of longer term, but ultimately less risky approaches to warfare. By the time the Bulgarian war ended in 1018, the frontier of the Roman Empire lay on the Danube. The price for the attention the emperor paid to the Balkans was the alienation of a large swathe of the eastern military families and their marginalization within the command structure. Along with them probably also went large bodies of experienced, able troops. By the time of the last Macedonian leader, the aged empress Theodora (1055–56), Roman arms were running down. The emperor Constantine IX (1042–55) famously commuted the military service of thousands of Caucasian Iberian thematic troops into cash payments to the treasury, which he wasted on a lavish court.
Over the course of the eleventh century, Turkish tribes moved from the Aral Sea region into Persia and South Russia. A group of these, the Uz confederation, invaded the recently incorporated Bulgarian provinces along the Danube. Constantine X (1059–67) allegedly could muster only 150 men to oppose them. In the east in 1068, the Seljuk Turks under Sultan Alp Arslan (1064–72) seized the large city of Ani in Byzantine Armenia. The new emperor, Romanos IV Diogenes (1068–71), was among the last of the Anatolian military families to hold power. He tried to repair the thematic armies, but ultimately was forced to rely on professional mercenaries drawn from the empire and abroad. His position insecure, Romanos IV sought to land a hammer blow against the Seljuks and thus to stabilize the empire's rich eastern flank. In the summer of 1071, the emperor led to the east what some consider the largest army the Byzantines ever mustered, a polyglot force of Uz, Pecheneg, Norman, Greek, Iberian, and Armenian soldiers. In mid-August, on the road to the town of Mantzikert (today Malazgirt) by Lake Van in eastern Anatolia, the army met the outnumbered forces of the Seljuks under Alp Arslan. Romanos divided his forces and sent those under a Norman adventurer Roussel de Balleul and the Byzantine commander Joseph Tarchaneiotes to seize the fortress of Chliat (today Ahlat) on Lake Van, but the soldiers fled when fighting began. Nevertheless, the emperor's troops acquitted themselves well, absorbing the Seljuk counterattack that followed the Turks' feigned retreat—an ancient tactic of the steppe nomads and one that destroyed many armies throughout history. The fight was bitter and prolonged and lasted into the second day, when Romanos was betrayed by the prominent nobleman Andronikos Doukas and captured by the enemy. Alp Arslan released Romanos, whose return to the empire triggered a civil war that allowed the Turks to continue their inroads.
The Seljuks and Turkmen nomads who ranged over the east posed an acute threat to the empire's stability. By the end of the eleventh century the Seljuks or independent Turkmen raiders had overrun most of Anatolia. Upland Asia Minor, with its vast lands, mineral resources, and pool of military recruits, was largely lost to the empire and the Seljuks had seized Nicaea, a mere 70 kilometers from Constantinople (Map 7). Even more serious than the Seljuk menace was the growing threat from the west. The appearance of the Normans in the Mediterranean would forever change the delicate balance of power there and produce a new, bitter enemy whose capacities were unmatched. Norman adventurers had arrived in south Italy around the year 1000 where they found the fragmented, chaotic political situation to their liking. Norman soldiers in the pay of local Lombard princes fought against Byzantine troops during the Lombard revolt of 1009–22, and over the next decades Norman adventurers were frequently paid by both the Lombard princes and Byzantine commanders in south Italy. Norman troops—among them William "Iron Arm" of the Hauteville family—fought with distinction during the unsuccessful Sicilian campaign of the Byzantine general George Maniakes (1038–40). Following the failure of Maniakes's expedition, the Normans turned against their former Byzantine paymasters and ravaged most of south Italy. Iron Arm allied with Duke Guiamar IV, and under William and his successors the Normans steadily nibbled on Byzantium's south Italian possessions. Robert Guiscard, another of the Hauteville family, conquered Sicily (1061–91) and drove the Byzantines out of the Italian Peninsula with the capture of Bari in 1071. But the Normans had grander designs—the conquest of the empire itself.
The emperor who found himself sorely tried by the Normans was Alexios I Komnenos (1081–1118), scion of a military family with connections to the powerful Doukas clan whose members had defected from the emperor at Mantzikert a decade earlier. Although just twenty-five years old, the emperor was an experienced commander, having fought in civil wars throughout the empire. He seized the throne as the emperor Nikephoros III (1078–81) prepared to meet the Norman onslaught. Alexios rushed to meet Robert Guiscard and his son Bohemund who had invaded in the spring of 1081 and laid siege to the important port of Dyrrachium (modern Durrës in Albania) on the Adriatic coast. In October the imperial forces lay within striking distance. Although advised by his local commander to avoid battle and to wear out the enemy, Alexios pressed for confrontation, probably because of his weak political footing. He screened the front of his force with the Varangian Guards (by now counting in their ranks many Anglo-Saxons who had fled the Norman conquest of England) supported by a unit of archers, and these units advanced against the Norman camp. The emperor commanded the Byzantine center, while the experienced general Gregory Pakourianos commanded the left and Nikephoros Melissenos, another battle-tried Anatolian commander led the right. Guiscard also divided his forces into three battles, commanding the center opposite Alexios while his son Bohemund commanded the left and a Norman count called Amiketas held the right. Much later Anna Komnene, Alexios's daughter, wrote an account of the battle and noted that a Varangian unit attacked the Norman camp through a salt marsh as the garrison of Dyrrachium made a sally. Guiscard sent a detachment of Norman cavalry against the Byzantine center in a feigned retreat—a tactic which the Byzantines themselves knew well. When this failed, the two sides began a general skirmish. As the two armies closed, however, the Norman right under Amiketas clashed with the Varangians; the Normans fled to the shore where Guiscard's wife, Gaita, allegedly rallied them. Guiscard sent a strong detachment of infantry against the tired and isolated Varangians, who were surrounded and broke. Those who ran away took refuge in a church, which Anna accuses the Normans of burning down with the men trapped inside. In the general engagement that followed the Normans punched through the Byzantine lines and killed several prominent commanders. The empire's Turkish mercenaries fled and another ally, the King of Diokleia, refused to assist Alexios. The emperor bolted the field along with the rest of the army and made a dramatic escape by cutting down Guiscard's second-in-command and narrowly avoiding death several times. By winter the Normans had captured Dyrrachium and dug in, in preparation to push east toward the capital.
Alexios maintained himself through the war by emergency confiscations of church plate and bought a costly alliance with the Venetians, who viewed the Normans as serious maritime rivals. The emperor also sent a huge shipment of gold to the Holy Roman emperor, Henry IV (1084–1105), who pressured the Norman homelands and Pope Gregory VII (1073–85), with whom Robert Guiscard had allied himself. Guiscard hurried back to Italy and campaigned against the Germans in the spring of 1082, and Alexios tested Bohemund, who inherited the strategic city of Dyrrachium and the Norman lands in Illyria. At Ioannina in western Greece, the emperor tried to break the ferocious Norman cavalry charge by using wagons, but Bohemund easily thwarted the effort and drove the Romans from the field. In a subsequent engagement, Alexios laced the Byzantine front with caltrops (an iron ball with four sharpened spikes so that one side always pointed up, to pierce the feet of men and horses), but the Normans once more discovered the plan, outflanked the Roman army, and pressed both the right and left flanks. Bohemund laid siege to Larissa in Byzantine Thessaly, which held out for six months until the emperor led a relieving army against him. Alexios decided to employ the feigned retreat—given the Norman success against him a general retreat was certainly believable. As Bohemund pursued the fleeing Roman main force, the emperor sprang his ambush and overran the enemy camp while another force attacked the Norman rear. Bohemund withdrew, and although the following year Robert Guiscard renewed the war with an attack on the island of Corcyra, the old warlord died and his sons hurried to Italy to lay claim to their inheritance.
The first Norman war underscores how threatening new western powers could prove to Byzantine interests. Nor were the Normans the only rivals to be dealt with; in Anatolia the Seljuk Turks ranged unchecked. The penny-packets of Byzantine troops stood no chance, and fortresses and cities succumbed throughout the former hinterland of the empire, leaving Alexios with only scraps of territory along the coast. Pecheneg nomads from the south Eurasian steppe raided Thrace in force—a series of Byzantine victories and defeats drained the treasury and bogged down the empire when their precious resources were needed elsewhere. By the 1090s the emperor had managed to repair relations with the papacy. As is well known, it was Alexios's request to the pope for western mercenaries after the tribulations of his Pecheneg wars that yielded fruit of an entirely unexpected kind: the First Crusade.
When the crusaders arrived in Constantinople late in 1096, Alexios confronted the sour fact that the Norman prince Bohemund was among their leaders. The savvy emperor extracted oaths of allegiance from the westerners, then sent an army and offered logistical support as far as Antioch, where the Greek-Crusader alliance fell apart. The First Crusade did at least disrupt the Seljuks and regained Nicaea for the empire, but it nurtured western and Greek hostilities toward one another, and in the tangled relationship that followed, the Byzantines increasingly alienated western powers who were eager for a share of the eastern Mediterranean. While Byzantine military power remained formidable under the Komnenoi dynasty that Alexios founded, a major defeat at the hands of the Seljuks in 1176 at Myriokephalon in Phrygia spelled the end of Roman efforts to wrest control of Anatolia from the Turks. Ever after the Greeks were largely relegated to the coastlands and under pressure.
During the era of the Komnenoi the themes remained as administrative districts only while the army in the land was replaced by native professionals and foreign mercenaries, especially Frankish knights, Turks, and Pechenegs. Though these professionals were arguably of higher quality than their thematic predecessors, they were expensive and therefore never very numerous, and the Komnenoi apparently never had more than 20,000 soldiers on any campaign. The native element in the army was increasingly comprised of cavalrymen who held _pronoia_ grants; these grants supported soldiers on revenues from the land tax in the locales where they were stationed. Initially such grants could not be inherited and, unlike the fiefs held in the medieval west, the state never relinquished its claims to the land from which the pronoiars were supported. While fiscally attractive, since it spared the state immediate cash outlays, the system fueled the regionalism and factionalism that plagued the later empire.
LATE PERIOD (THIRTEENTH TO FIFTEENTH CENTURIES)
The end for the empire as a major Mediterranean military power came in the spring of 1204 when Venice, the former imperial ally, diverted the army of the Fourth Crusade and trained it on Constantinople. In April 1204, after months of complicated political maneuvering and diplomatic failures, the western crusaders stormed the capital and for three days slaughtered inhabitants and burned and pillaged the greatest city in Christendom. The failure of Byzantine arms was total; inept command, a lack of funding, and the poor quality of the army that had degraded in the decades after the death of Manuel Komnenos in 1180 all contributed to the catastrophe. The military collapse led to a cultural tragedy rarely matched in history. In the ashes of the ruined Byzantine state the Franks and Venetians cobbled together a dysfunctional "empire" while rival Greek regional centers galvanized resistance in the provinces. The loss of power and prestige and the cultural winter of sixty years of foreign occupation rendered the Byzantine state that the emperor Michael VIII Palaiologus (1259–82) led a regional power.
Michael's dynasty, the Palaiologans, clung to power in Greece and portions of Asia Minor and the Balkans for nearly two centuries, wracked by factionalism and crippling self-interest. Their armies were pathetic compared with those of their predecessors and their enemies, and at no time after 1204 did a Byzantine campaign army ever total more than 5,000 soldiers, and it is doubtful that this number was ever fielded. The broken reed of Byzantium fell into a familiar contradiction; without soldiers the empire could not capture more territory whose resources could support more troops for further security and recovery of lost territory. Although history presented opportunities for revival—the Mongol smashup of the Seljuks of Anatolia in the mid-thirteenth century for instance, or the heavy defeat of their Ottoman successors by the Timurids at the Battle of Ankara in 1402—the Byzantines could do nothing to reverse their political weakness. By the spring of 1453 when the Ottoman sultan Mehmed II (1444–46 and 1451–81) led his Turkish army to invest Constantinople, the defenders of the city that had once overawed the world numbered a paltry 7,000. They faced 80,000 determined Muslim opponents armed with gunpowder and cannons that bashed to rubble the Herculean walls of the city. On May 29, a mere thirty-nine years before Columbus landed in the New World, the Old World's longest-lived empire fell. Bereft of its resources and the arms that they sustained, Byzantium vanished into history, replaced by a Muslim empire that eventually grew to parallel its predecessor—stretching from Libya to the Danube and locked in an eerily similar struggle for survival on all fronts. Less remarkable than the slow march into twilight of the Byzantine army and its empire is the incredible fact of its thousand-year existence, one sustained in large measure by the resilience, adaptability, and professionalism of its fighting men.
## TWO
## LEADERSHIP
THROUGH MUCH OF THEIR HISTORY the Byzantines were exceptional not for the brilliance of the military commanders that they produced—after all, they suffered many defeats (see Chapter 1)—but for the general competence of their leadership that allowed them to defend their empire even after such setbacks. Though there does not appear to have been anything akin to West Point in Byzantium, a training campus where soldiers could learn military science and absorb the lessons of others, the Byzantines differed from most of their neighbors by writing down their thoughts on the science of warfare. There were numerous exceptions throughout the long history of the empire, but commanders tended to be professional soldiers with some degree of military competence. Given the value of the men and resources with which they were entrusted, they were held to a high standard.
One must be cautious in generalizing about a military institution and a culture of warfare that evolved tremendously over a millennium. In its most basic form, the ability to lead meant an understanding of strategy, tactics, and a combination of courage tempered by an acute sense of risk and reward. Neither the writings of Vegetius (fourth to fifth centuries) nor the _Strategikon_ of the emperor Maurice (582–602) discussed the qualities of a general in detail, but the requisite knowledge and makeup that emerge from them show that an understanding of logistics was key, as was individual valor. A firm grasp of the morale and condition of one's forces were vital. In the _Taktika_ of the emperor Leo VI (886–912) the traits of the general are enumerated and few will surprise the modern reader. A commander was to be self-controlled, serious, sober, and incorruptible. Further, he was to be intelligent and neither too young nor too old. Physical strength and endurance were prized. He had to have the ability to earn the respect of his men and to be a good public speaker. More surprising to a reader today were his spiritual qualities of exceptional piety and his parental status: Leo thought that men who had children were more motivated, and men of the aristocratic classes were preferred over those of obscure origin.
These traits were thought essential to becoming _strategos_ (Greek "general," pl. _strategoi_ ) and in holding other senior commands. Rarely do our sources tell us about the battlefield actions of lower officers and individuals in the rank-and-file, and even more rarely are the equivalent of today's NCOs or other lower ranks shown in pivotal leadership roles. The stress on professionalism and drill made soldiers who were tactically flexible and capable, but the lower ranks seem to have lacked initiative. There is no Byzantine _Anabasis_ where an army bereft of its leaders pushed its way to safety, no Byzantine "soldiers' battles" where a group stripped of their high command fought to victory. The loss of a general or emperor in command of the host usually meant its defeat and dispersal. This picture is attributable partly to the nature of our sources, whose authors stress the deeds and heroics of men of their own elite class. The army remained professional at its core (although the standards of Dark Age thematic armies are debatable in this regard), so the inability of Byzantine armies to recover from the field losses of their high command cannot have been an attribute of army organization. The Byzantine army officer structure was of considerable depth and was quite advanced for the medieval era (see Chapter 3). And since junior officers seem to have gained their positions primarily through experience and merit, the apparent lack of ready response to battlefield crisis may be attributed to other factors.
Social influences clearly shaped soldiers' attitudes and responses. All men were not equal, and a poor recruit or even the best low-ranking fighter could not compare in worth to a well-born general. One's birth, wealth, and social standing endowed the elite not only with the connections to rise to high station, but also with an aura of superiority in all areas of life. One fulfilled one's role in the universe, and though not all well-born men accomplished great deeds, such expectations help explain why in the chaos of impending defeat it would not have occurred to the lowly soldier of the line to seize the standard and rally a crumbling army to victory even if such heroics were physically possible. While some common soldiers did rise to positions of authority, Byzantine society remained highly stratified. Many commanders of the early Byzantine era were elites, and most senior officers in the middle and late Byzantine periods were members of the aristocracy whose associates formed a hereditary military caste. This is in no way to say that such men were without merit. For example, Damian Dalassenos (d. 998) was one of many gifted leaders whose sons followed in their footsteps and, probably because they were known to imperial officialdom and high-ranking courtiers, found it easier to access high commands. Status, however, had to be paired with success for one to keep imperial favor and maintain command, and while Byzantium had a number of incompetent commanders whose ineptness led to disaster, on balance the army was capably led throughout the empire's existence.
Although disciplined, drilled, and—as the handbooks and historical reality demonstrate—often capable of aggressive support of their comrades and complicated battlefield tactics, there is no sense that the Byzantines prized individual initiative from below. The tactical realities of ancient and medieval warfare played a role in the apparent lack of junior officers rallying their troops or assuming command in place of fallen superiors. Throughout its history the empire faced aggressive and sophisticated enemies. With their range of movement, tactics, and weaponry, nomadic steppe warriors such as the Huns, Avars, and Cumans challenged the most skillful armies. Horse-mounted archery with its combination of striking power, range, and mobility combined with ages-old tactics of steppe warfare to create a dangerous tactical environment for imperial armies. Byzantine commanders were aware of the slender line between victory and defeat; an apparently beaten opponent could regroup quickly and inflict a reverse on troops disordered in pursuit or who stopped to strip the dead of valuables. The stress on discipline and the need to follow protocol made initiative among lower-ranking soldiers positively undesirable. The soldier who became an individual in either victory or defeat broke the cohesion of the unit and exposed his comrades to peril.
When we understand the expectations of a rank-and-file attuned to signs and symbols, the roles of leaders and those they led become clearer. The Byzantines, like most pre-industrial era peoples, were keen observers of omens. The final outcome of any endeavor belonged to God. The battle merely revealed God's plan: soldiers fought bravely because they were soldiers and thus such action was their calling, not necessarily because their bravery and individual skill would decisively affect the outcome. Of course this oversimplifies, but this distinctly fatalistic strand of Byzantine culture played a central role in military encounters. The perceived skill, piety, and well-being of the commander formed a major part of troop morale. Any sign of weakness, ill health, or other omens could panic the troops. Vegetius noted that the general who retreated from the line prior to battle fatally eroded the confidence of his troops. In 917, when Leo Phokas's riderless horse bolted through the ranks and led the men to believe their general had been killed, the army panicked. Just as the heroic encounters that often preceded battle were signs of God's favor or displeasure and hinted at the final outcome, the deeds of a heroic general were important spiritual markers. When in 623–24 Heraclius struck down the giant Persian soldier on the bridge over the Saros River it inspired his troops to victory, not because the emperor cleared the bridge, but rather because the deed's perceived spiritual significance vitally boosted his men's confidence.
Thus, leadership flowed from the top; the general was perceived as both a superior person to his soldiers and a superior soldier, and had a spiritual aura about him. He was usually more experienced in strategy and tactics than most of his men. In essence the strategos and his staff council of high-ranking officers formed the nerve center of the army. Command resided with the general or generals who enforced iron discipline and tightly held authority; this was critical in the campaign environment throughout the empire's history, when soldiers were prone to disorder and when communications were cumbersome (signaling relied on messengers, flags, and musical instruments). The best army was unwieldy, and command and control almost nonexistent once battle was joined. If a general fell or disengaged, there was an abrupt loss of command and control and cohesion dissolved almost immediately as the elite fighters around the general perished or fled. Conversely, in engagements hard-fought over days, with the commanders in the fray, Byzantine armies could demonstrate resilience, as in 971 at Dorostolon, though such encounters were rare.
There are exceptions to the lack of leadership apparent among lower ranks, especially in the early period. Perhaps the best example of a junior officer seizing control in a military crisis is that of Phokas, who was allegedly a lowly _kentarchos_ (centurion, commander of eighty to one hundred men) in a tagma (a unit of about three hundred) under the command of the general Philippikos. Facing a mutiny over pay and conditions of service and unable to assuage his soldiers' wrath, Philippikos fled the encampment. The kentarchos Phokas, who was apparently of low birth, seized control of the situation and was raised on a shield by the troops, who proclaimed him emperor. Phokas led the mutineers to Constantinople where they eventually seized the city and killed the emperor Maurice. Such a usurpation of command by such lowly men during crisis was almost nonexistent outside the context of military unrest.
Combat leadership was most often learned by experience. The sixth-century historian Prokopios mentions many leaders who apparently rose through the ranks based on their abilities as soldiers. But in the early Byzantine period many leaders came from military families whose members followed in the footsteps of successful officers, such as John, nephew of the sometime rebel Vitalian (d. 520), or were barbarian elites drawn to imperial service, like Mundus, the son of a Gepid king who served Justinian loyally in both the Balkans and the eastern frontier.
EARLY PERIOD (FOURTH TO SEVENTH CENTURIES)
Until the disaster at Adrianople in 378, emperors often campaigned in person. Such soldier-emperors were often of low origin: Diocletian was from an obscure family and rose through the military ranks to the throne. Constantine I's father, Constantius Chlorus (ca. 250–306), was of humble origin, and men like him found opportunity, high honors, and power through military service. Constantine I, his son Constantius, and his nephew Julian were active soldiers who marched at the head of their forces. Into the sixth century, the rank-and-file continued to have the opportunity to achieve high positions. By the fifth and sixth centuries emperors no longer led their armies in person and commanders seemed to have been chosen more for their loyalty than their brilliance. Merit still played a key role in selecting leaders, however. Many of those whom the historian Prokopios mentions as leading men to battle apparently were of common birth and rose to authority via service. Once more the imperial palace provides the best story: the illiterate peasant Justin left his farm in Illyricum to join the army, whence he rose to service in the imperial guard and thence to the purple.
In the fourth century the _protectores_ comprised an unknown number of corps under the magister officiorum. A _comes_ (count) officered units of uncertain size and disposition. There were ordinary protectores as well as the domestici who, as their name implies, formed a corps in attendance on the emperor. Men normally selected for the protectores had distinguished themselves early in their careers as particularly promising and loyal. Promotion was based on seniority. Both the regular protectores and domestici provided staff officers—typically adjutants to the _magistri militum_ marshals who seconded them on a variety of special duties, such as rounding up recruits, overseeing military depots, and inspecting fortresses. The emperor commissioned the protectores in person in a ritual of obedience, and thus such men were personally known to the ruler and his high command. Unsurprisingly, the combination of merit and loyalty often led to rapid advancement. Gratian, the father of the emperor Valentinian I, was an expert wrestler and was promoted to protector as a ranker.
In the sixth century, both the emperor's bodyguard and the household men of senior commanders continued to be an important incubator for military commanders. The three-hundred-man palace guard of the Excubitors, a unit raised by the emperor Leo I (457–74), provided numerous officers as it drew to its ranks men of the greatest loyalty, ambition, and fighting abilities. _Boukellarioi_ (named after the sort of bread they ate) comprised the private bodyguards of state officials and powerful private men. Unsurprisingly, men who served in the bodyguard of Justinian when he was magister militum (520–27) rose to high command once he became emperor in 527. Among them were Belisarios and Sittas. When Justinian ascended to the throne, he created a new army command in Armenia and named Sittas its chief officer. After a long career of loyal and distinguished service, Sittas fell in battle in Armenia in 539.
Sittas's comrade Belisarios likewise rose to prominence when Justinian came to power. By 529 Belisarios was Master of Soldiers of the East ( _Magister militum per Orientem_ ). His position allowed him to accumulate seven thousand boukellarioi; the illustrious general was responsible for their pay and maintenance. Some were common native soldiers who had proved themselves capable fighters. Boukellarioi were often given command of detachments of regular troops or sent on special missions, and leaders like Belisarios relied heavily on them. During his African campaign (533–34), Belisarios sent one of his bodyguard, Diogenes, with twenty-two other boukellarioi, to reconnoiter outside of the former Vandal capital of Carthage. The Vandals surprised the detachment and nearly destroyed it and Diogenes was wounded in the battle. In 549, when Belisarios prepared to depart Italy, he left Diogenes in charge of the three-thousand-man garrison of Rome.
Though less common than in the early days of the empire, commands were sometimes given to high-born Romans. These seldom went to men lacking experience. When the Persian War of 502–6 caught the emperor Anastasios flat-footed, he dispatched four generals, among them his nephew Hypatios. The latter apparently had earned campaign experience during the wars against the Isaurian highland rebels in the 490s, but his lack of skill and daring let the emperor down and he was replaced and recalled in 503.
In addition to native sons, the Romans relied on officers of barbarian origin. A number of warlike neighbors surrounded the empire and provided fertile recruiting grounds for not only the rank-and-file but also their leaders. During the fourth to seventh centuries commanders of Germanic, Armenian, and Persian origin were common. The Gepid (a Germanic tribe living in Illyricum) Mundus was the son of a king and nephew of another. He entered Roman service in the late 520s and served as Master of Soldiers of Illyricum ( _Magister militum per Illyricum_ ) and served with distinction. Mundus later replaced Belisarios as _Magister militum per Orientem_ after the Romans were routed in the 531 debacle at Callinicum.
DARK AGES, MIDDLE AND LATE PERIODS (EIGHTH TO FIFTEENTH CENTURIES)
The emperor Heraclius revived the practice of the emperor campaigning in person. Not all of his successors would lead their troops into battle, but many did. After the crisis of the Arab invasions, military men dominated the throne and emperors often campaigned in person. High officers were often drawn from the _spatharii_ (sword men) who formed the emperor's bodyguard. The emperor Leo III (717–41) had once served as a spatharios under the emperor Justinian II. The state required leaders who understood military affairs and who could handle the army. When the last member of the Heraclian dynasty, Justinian II (685–95, 705–11), fell in 711, it was at the hands of a military coup led by an Armenian thematic commander named Philippikos Bardanes. Theophilos (828–42) was trapped with his army and barely escaped the battle of Anzen, while in 863, the emperor Michael III led an army into Anatolia to intercept the raid in force of the emir of Melitene.
The high point of the soldier-emperors was the tenth century, when warriors like Nikephoros II Phokas and John Tzimiskes cultivated an image of imperial triumph and valor in arms as they personally led their men from victory to victory. By the reign of Basil II (976–1025), the grip of the military elite on the levers of power was such that the young emperor had not only to wrest it from their control through two painful and devastating civil wars but also to cast himself as the logical replacement, the vigorous, blessed soldier-emperor who led his troops to victory. Throughout his reign, Basil served in the army in person, often under the wing of more experienced generals.
After Basil, emperors increasingly distanced themselves from the camps until the reign of Romanos IV Diogenes (1068–71), whose capture and humiliation at Mantzikert did not deter his later successors from campaigning at the head of their armies. Alexios I Komnenos (1081–1118), a usurper, personally led his troops, and though he experienced several reversals his successes on the battlefield were numerous and notable. By the time his grandson Manuel I Komnenos had followed his father's footsteps, the age of the great Byzantine soldier-emperors had come to an end. Though during the Palaiologan dynasty fewer emperors led troops in person and the armies progressively dwindled in size until the institution failed, in 1280–82 Michael VIII (1259–82) led troops into Asia Minor against the Turks when in his late fifties and thus very near the end of his life.
Leaders in high commands in the middle and late eras of Byzantium were nearly always well-born men. The ninth to eleventh centuries were the era of the great Anatolian military aristocracy. Among the highest officers were several prominent foreigners, including the general and emperor John Tzimiskes whose family was of Armenian extraction and the Armenian-born Melias. Commanders from the family of the empress Theophano (wife of Theophilos) were also Armenians. The Persian (or Kurd) Theophobos served under Theophilos, Roussel de Bailleul (d. 1078) was a Norman, while during the First Crusade (1096–99) the Roman general Tatikios was a Turk. But during the heyday of the imperial resurgence of the ninth and tenth centuries, most of the senior commanders were native Romans (though many were descendants of immigrants). Families like the Argyroi, Phokades, Skleroi, Maleinoi, Melissenoi, Doukai, and Diogenai held estates in Anatolia and benefited directly from the recovery of vast territories seized by the Muslims over the prior three centuries. These Anatolian families produced some of the most skilled and capable commanders in imperial history, notably Bardas and Nikephoros Phokas, John Tzimiskes, and George Maniakes. Ultimately, their ambitions to seize and dominate the throne led to their downfall and the collapse of the eastern defenses in the face of the Turkish advance.
In the middle period, higher officers often took part in the fighting. After a banquet in which the emperor Romanos I roused his commanders to action, one of them named Saktikios led a dawn attack on the Bulgars encamped against Constantinople and was subsequently killed. In 921 during the Bulgar siege of Adrianople, the city commander Leo was nicknamed Leo the Fool due to his rashness in personally exposing himself to battle. Anatolian commanders frequently fought against the Muslims; in 953 Bardas Phokas (ca. 878–968) was surrounded and wounded by Sayf ad-Dawla's men in a defeat near Marash. In pre-industrial war, before bullets caused generals to hunker behind the lines, the personal nature of war and the bravado of commanders were everywhere evident in Byzantium.
Following the fall of Constantinople in 1204 and the fracturing of the empire, leadership of the army remained with the emperor and elite-born commanders. Some, like Michael VIII (1259–82), campaigned in person at the head of what forces were left to them and even showed a talent for tactics and strategy. The career of the emperor John VI Kantakouzenos (1347–54) demonstrates that ideologically the emperor had to actively resist the Turks. John was himself a competent officer, but after his day the importance of the soldier-emperor faded as Byzantine decay deepened.
BIOGRAPHIES
Byzantium produced its share of brilliant commanders, though few in the modern world have heard of them. Each of the men portrayed here shared the ability to act decisively in times of crisis, and they exhibited the key qualities esteemed among eastern Roman leaders: cool-headedness under duress, caution in the face of the enemy, and a thorough understanding of strategy, tactics, operations, and logistics. As compared with the more blunt tactics of leaders in neighboring societies, the Byzantine generals acted more like surgeons than butchers, with measured gains and a clear appreciation for the delicate instrument of the army in their hands.
Belisarios
The most famous Byzantine general and the one best known to western students of military history is Belisarios, who served the emperor Justinian faithfully over a long, distinguished career. He was born in Germania (today Sapareva Banya in modern southwest Bulgaria), a city in Thrace on the border of Illyria. The western regions of the empire were a rich recruiting ground and produced many late antique commanders and soldiers. Though we remain uncertain of his precise ancestry, Belisarios was apparently the scion of a local Thracian or Illyrian family probably of some means, as indicated by his ability to raise and pay a substantial cohort of personal household troops. Belisarios gained prominence as a commander after service in the bodyguard of Justinian when the latter was the right-hand man and magister militum of his uncle Justin I (518–27).
Despite his role as joint commander of a campaign that ended in defeat at the hands of the Persians in Persian Armenia, by 526 Belisarios had ascended to the office of dux of the province of Mesopotamia. The dux (Greek, _doux_ ) was the senior local commander of a mix of professional garrison troops and frontier guards ( _limitanei_ ) of varying quality. He was thus headquartered at the city of Constantina (modern Viranşehir in southeastern Turkey) or Dara—both were key fortress-cities on the eastern frontier with the Persians. A document belonging to the late fourth or early fifth century, the _Notitia Dignitatum_ provides some insight into what assets the dux had at his disposal; it lists four units (apparently cohorts of about five hundred men) of elite cavalry, all originally raised in Illyria, and six further units of local cavalry (probably cavalry _alae_ of about a hundred men each, though these may have numbered up to five hundred each), along with two infantry legions of about 1,000 men. The total number of troops under this frontier command then would have been at most 7,000, though units were not apparently uniform in number and they were commonly understrength.
In 528, Belisarios led elements of these units against the Persians in Upper Mesopotamia where they faced an invading Persian force. The Romans were attempting to construct a fortified city on the frontier and the Persians aimed to destroy the works. Belisarios was in overall command, but the young general (who was in his late twenties at the time) was joined by other duces of the east with their respective units and Arab allies. The combined forces joined battle with the Sasanians at Tannuris (Tel Thounenir) in the Khabur Valley of northern Syria. The Sasanians employed hidden trenches and pits, into which the rapidly advancing Roman lines fell headlong and a prominent leader, Coutzes, was killed. Belisarios fled back to Dara along with the Roman cavalry and left the infantry to be destroyed, but in the aftermath held onto his command as blame apparently fell on Coutzes. Three years later Justinian elevated Belisarios to the supreme command of the eastern army, the post in which he was to win fame and glory.
When Belisarios assumed command of the eastern forces, morale was low and Roman preparedness lacking. Since the era of Diocletian, the Romans had relied on strategic depth and linear defenses as a bulwark against the menace of their eastern neighbors. That the Byzantines had managed to hold the line was due more to good luck than to good management: within living memory the Persians had battered a large Roman army in the war of 502–6 that had few strategic implications but which served to expose deep fissures in the command structure and the feeble tactical punch of the Roman army.
In the wake of the debacle of 502–6, the government took decisive action. The emperor Anastasios built a massively fortified new city on the village of Dara (modern Oguz). Dara served as a forward staging post for imperial armies, a supply depot, and springboard against the Sasanian homeland. As dux of Mesopotamia, Belisarios made his headquarters there. Seated at the foot of the rough highlands of the Tur Abdin, Dara stared across the hot northern Mesopotamia plains toward the Persian stronghold of Nisibis, a mere 25 kilometers away. Nisibis had been ceded to the Persians after the ruinous campaign of Julian in 363; its loss left a gap in Roman defenses and provided the enemy with a powerful salient. In 527 the Romans had suffered a defeat north of Nisibis at Mindouos while attempting to construct another fortified city to counter it.
When war resumed in 530, Belisarios and the general Hermogenes led a large Roman army of perhaps 25,000 toward Dara. The Persians under their general Firouz commanded a superior Persian force of around 30,000. The Romans decided not to endure a siege but instead arrayed their forces in a strong defensive position outside the walls of Dara. The fortifications shielded their rear, while to the front of the army they dug a lattice work of trenches with alleys that permitted the Romans to blunt the weight of enemy numbers while still permitting them to maneuver. These works shielded Belisarios's untried infantry (fig. 2.1). Belisarios and his bodyguard elite of boukellarioi stationed themselves behind the main body of Roman infantry in the center, while on the left flank were Herul (a Germanic people related to the Goths) cavalry under the command of Pharas and Roman units under Bouzes. The Roman right comprised Hun auxiliaries backed by a larger Roman cavalry force. Hun cavalry also provided a pivot force in the angles of the trenches that fronted the Roman lines. The historian Prokopios was an eyewitness and provides a description of the encounter that allows for a solid reconstruction. On the first day of the engagement, the Persians advanced in lines drawn up in standard fashion—two strong lines and flanking forces—and attacked the Roman left, which gave ground and exposed the Sasanians to a flanking attack by the Huns in the pivot point of the Roman force. The Sasanians withdrew with minor losses, and single combats followed in which the Roman hero Andreas prevailed over two Persian challengers. Given the role ascribed to fate in ancient warfare, these duels sharply raised Roman morale and proved to many of the untested that the Sasanians were not their betters after all.
The next day the two sides parlayed and the Persians received reinforcements from Nisibis. The 10,000 additional Persians must have been the whole of the garrison in that city; their mobilization indicates that, based on his probes of the previous day, Firouz doubted the outcome. Battle recommenced (fig. 2.2), as the Sasanians attempted to soften the enemy lines with missile fire and the Romans returned their volleys which were more effective since the wind favored them. Firouz then ordered an assault across the line. Once more the Romans left under Bouzes gave ground, but as the Sasanians advanced, Belisarios sprung the trap—Pharas and 300 Heruls had hidden themselves behind the cover of a nearby hill and emerged on the Persian right as the Huns swept in from the Persian left. In the resulting crush the Sasanians absorbed heavy losses. Prokopios states that 3,000 Sasanians fell in the rout.
Firouz then ordered his reserve into action, the elite Immortals regiment (named after their illustrious Achaemenid Persian ancestors) against John on the Roman right. Belisarios and Hermogenes sent 600 Massagetae (an Iranian nomadic group) to reinforce John. John's forces buckled under the onslaught of the Immortals and regulars, but once again the Romans in the angles of the trenches assaulted the Sasanian flanks in a vicious attack that split the Persians in two—the larger portion on the flanking forces' right and thus facing John's retreating cavalry, who saw their enemy falter and regrouped and counterattacked, surrounding a major portion of the Sasanian force. When Firouz comprehended the peril of his shock troops, he threw the remainder of his army into the fray along the whole Roman front. The Romans absorbed the charge and held, while their officer Sunicas killed the one-eyed Sasanian general Baresmanas, who was second in command (fig. 2.3). The Persians panicked and broke but could not escape; the Romans surrounded the greater part of their army and killed 5,000 of the enemy. The wretched Sasanian infantry threw down their shields and bolted, only to be cut down in swaths by the pursuing Roman riders (fig. 2.4). For a Roman army that had not witnessed a major victory in the soldiers' lifetimes, and for the young magister Belisarios, the Battle of Dara heralded a critical shift. The Romans proved that they could take the field against a powerful opponent and defeat them, and Belisarios and his commanders exhibited outstanding tactics and leadership.
Though the Roman general went down in defeat at the Battle of Callinicum in 531, he regained Justinian's full confidence in his handling of the Nika Riot of 532, and was awarded the senior command of the emperor's grand expedition against the Vandal kingdom of North Africa. A century prior, this barbarian kingdom had rooted itself in the rich lands of former Roman Africa and battered the Romans in both the west and the east. In 468 the emperor Leo had launched an enormous force under his generalissimo Stilicho that went down in bloody defeat and alleged treachery. Belisarios landed his expeditionary force at Caput Vada on the eastern coast of what is today Tunisia. In two battles, Ad Decimum (September 13, 533) and Tricamarum (December 15, 533), Belisarios broke Vandal power (see Chapter 7). Belisarios displayed superb abilities in the Vandal War; he relied heavily on Hunnic and Roman horse archers against the Vandal lancers, who were helpless against the ranged weapons of their adversaries. He also maintained discipline among his forces and showed a keen understanding of the need to maintain good relations with the Romano-African locals on whose cooperation Byzantine success in Africa depended.
In 535, buoyed by his success in restoring Africa to the empire, Justinian dispatched Belisarios against the Ostrogothic kingdom in Italy, where the Byzantines quickly took Sicily, then Naples, and Rome. The Gothic counterattack led to the brutal siege of Rome in 537–38. Belisarios managed to break the siege of Rome by sending a flying column to the north that defeated an Ostrogoth force near their capital in Ravenna, which Belisarios besieged and seized in 540. By the time the first phase of the Gothic War ended, Belisarios held most of Italy for the empire. The outbreak of war with Persia required his presence in the East, for which he departed in June. In 541 he was once more at Dara on the frontier and the following year he repelled a major Persian invasion without a battle, his prior mastery of them having made the Sasanians wary of the Roman general's strategems. A plot against the emperor Justinian, who had fallen ill of the plague, implicated Belisarios, who lived in disgrace until 544 when he returned to Italy. In 549, once more under suspicion and starved of men and materiel, Belisarios fought in Italy with a force of only 4,000 men.
Belisarios was recalled to service in the twilight of his life when in 559 an invasion of Kutriger Huns threatened Constantinople which was largely bare of troops. Rallying rustics and town guardsmen to his side, he managed to end the threat. Enemies at court had apparently done their damage, however, as his ten-year hiatus in command demonstrates. He died in 565, the same year as his master, Justinian. In his career he had nearly doubled the size of the empire, handled his men with extraordinary skill, and exercised the cautious command that marked him as a brilliant general. Had he been given more with which to work, the war in Italy would have likely ended much sooner, and the imperial conquests strengthened and deepened.
John Tzimiskes
Probably the most vibrant commander in an age replete with fine leaders, John Tzimiskes oversaw the peak of the Byzantine revival in the eastern marches of Syria and Anatolia and personally commanded the decisive Byzantine victory over the Kievan Rus' at Dorostolon. A Byzantine historian described Tzimiskes as "enormously strong...possessed of a heroic soul, fearless and intrepid, displaying supernatural courage in so small a body." In his portrait of John before his ascent to the throne, the Byzantine historian Leo the Deacon has one observer describe John as "ambitious, extremely aggressive, and good in warfare." Tzimiskes was short but powerfully built and personally brave in battle to the point of recklessness. He was a vigorous soldier and leader, skilled with bow and javelin, and a good horseman. He was also a murderer and usurper who rose to power by killing his uncle, the great soldier and emperor Nikephoros II (963–69).
John Tzimiskes was born in northern Anatolia around 925, scion of the Kourkouas clan, a distinguished Armenian family who had settled in Byzantine territory and produced the general John Kourkouas whom his patron, the emperor Romanos I, named domestikos ton scholon (commander-in-chief) in 921. John Kourkouas achieved notable success on the eastern frontier of the empire, including the capture of the important cities of Melitene (934) and Edessa (944), but his star faded with the deposing of Romanos I in 948. John Tzimiskes's mother was the sister of Nikephoros Phokas, and his first wife Maria was the sister of the magistros Bardas Skleros. Tzimiskes was thus related by blood or marriage to the most powerful elite families of Anatolia who dominated the military establishment for the better part of the tenth century.
In 958 John Tzimiskes led a major invasion to the eastern frontier against the ruler of Aleppo, the Hamdanid emir Sayf ad-Dawla (945–67), who was a worthy and vigorous opponent of Byzantine power in Mesopotamia and Syria. Not far from Amid (ancient Amida, modern Diyarbakir), John encountered Hamdanid forces commanded by the Circassian general Naja at the head of an army of 10,000. Tzimiskes's forces utterly destroyed the Muslim army, killing 5,000 and taking 3,000 prisoners along with all the baggage. In autumn Tzimiskes captured Samosata, an important and wealthy Muslim city on the Euphrates.
When he seized power in a military coup, Nikephoros II elevated his talented nephew to the office of domestikos ton scholon. During the 964 offensive against the emirate of Tarsos, Tzimiskes led the Byzantine left in the engagement outside the city of Tarsos in which the Byzantines outflanked the emir's men who broke and ran to the safety of the walls. By 965, however, Tzimiskes was disgraced and cashiered for reasons unknown, and his forced retirement eventually led to his conspiring against his uncle, whom he murdered in December 969. Upon his accession John held the spear tip of the finest army in Europe and southwest Asia. He inherited from his uncle and forebears a veteran force hardened by frequent campaigning built on a core of professional, heavy cavalry ( _kataphraktoi_ —for more on these see Chapter 5). Unlike many of its ancestors, the Byzantine army of Tzimiskes was built for the attack.
Before he could return to the east, Tzimiskes had to deal with Prince Sviatoslav of the Kievan Rus' (945–72). After assisting Nikephoros II Phokas against Boris of Bulgaria (969–71), whom in 968 in a swift campaign Sviatoslav made a vassal, the Rus' ruler expanded his domain immensely—notably southwest to the Danube. The Byzantines watched with growing alarm as Sviatoslav ensconced himself there and transferred his power center to Pereyaslavets on the Danube some 600 kilometers south of his former seat at Kiev and far too close for Roman comfort. After failed negotiations in 970, a powerful force of Rus' and Pechenegs (a Turkic nomadic steppe people) invaded Thrace. Despite being heavily outnumbered, a Roman force under Bardas Skleros employed a feigned retreat and lured the enemy into an ambush in which the Romans killed thousands of the northerners. The following year the Romans helped to persuade the Pechenegs to withdraw their support from Sviatoslav, when the emperor himself led a massive force of up to 30,000 against the Bulgar capital of Preslav. After a brief assault, Tzimiskes took Preslav, slaughtering many of the 7,000 Rus' and Bulgars who held out in the royal palace.
Tzimiskes then advanced against Sviatoslav, who awaited him with a large Rus' army said to number 60,000. At Dorostolon (modern Silistra) on the Danube in northern Bulgaria the two sides clashed (fig. 2.5). Leo the Deacon, an eyewitness to the battle, described both sides as extremely motivated—the Rus' feared their loss of honor and reputation as invincible warriors and the Romans could not concede victory to a horde of barbarian infantry. The description Leo provides of the helter-skelter wild charge of the Rus', which was met with the cool discipline of the Romans, seems like an echo from an earlier era. Late in the day, after hours of fighting, Tzimiskes ordered his kataphraktoi heavy cavalry into the fray against the Rus' left. The kataphraktoi crashed into their infantry line, which Sviatoslav swiftly reinforced, only to be countered by Tzimiskes and his Immortals, whose repeated charges finally smashed the Rus'. The survivors fled behind the walls of Dorostolon.
The following day the Romans brought up their siege equipment and constructed a palisaded encampment, and the next day they assaulted the city walls but were repulsed. The Rus' attacked at daybreak in the early hours of the third morning. Over a number of days the Rus' repeatedly sallied against the Romans. In one encounter the Rus' slipped through the naval cordon and sailed upstream where they managed to slaughter many grooms as they tended the Byzantine mounts. The final climactic confrontation came on a hot summer day in June or July when Sviatoslav again ordered a major assault against the Romans and led the charge in person. The Rus' forces made contact along a narrow front where the Byzantine cavalry could not maneuver—many horses and men were killed by the Kievan archers and throwing-spears. In the heat of the day, the Roman heavy infantry suffered from thirst and the emperor ordered wine mixed with water to be provided by rotation to the troops at the front. The deadlock was broken when the Romans executed a feigned retreat and lured the Rus' into open ground where the cavalry on the Roman wings shattered the Rus' shield wall and drove them back toward Dorostolon, but there the Rus' found their retreat cut off by Bardas Skleros and his eastern cavalry. The Rus' army dissolved in rout and the vast majority of them were butchered as they scattered across the plain. As many as 15,000 fell, according to Leo the Deacon. Sviatoslav sued for peace and abandoned his conquests.
In 972 Tzimiskes turned his soldiers' blades against the crumbling power of the Muslim princes of the eastern frontier. Roman incursions had battered and bruised Hamdanid power since the reign of Constantine VII, and Tzimiskes aimed to finish it. Tzimiskes apparently raided northward, since the emperor led his troops into the north of Mesopotamia, where he burned Nisibis. In 974 John again marched east and brought Amid to terms in return for heavy tribute. He then advanced 70 kilometers eastward to Mayyafarakin (ancient Martyropolis), another key Muslim stronghold in the Diyar Bakir region of eastern Anatolia: In Leo's account, "This is a famous and splendid town, superior in wealth and livestock of the other cities of the same region. And he brought it to terms and carried off numerous beautiful gifts in gold and silver and cloth woven with gold, which he demanded from the inhabitants; then he went to Nisibis." John found Nisibis deserted and he apparently swung south to menace Baghdad, but the expedition stalled in the Syrian Desert—the fate of Roman armies since antiquity.
In 975 several of the Hamdanid towns rebelled against Byzantine authority, and the emperor again marched to Syria where he brought to heel the cities of the Syrian coast and marched on Damascus, which submitted. In a letter to the Armenian king Ashot III, John boasted that he would retake Jerusalem. Certainly the Muslim caliphate in Baghdad quaked—no Muslim force could stand before John and the Byzantine armies. On January 10, 976, at the age of perhaps fifty, John died—a victim of disease or the poisoner. With his death, the prospects of Byzantine expansion in the east vanished.
John II Komnenos
The son of Alexios I Komnenos, John has deservedly earned the reputation as one of the last outstanding leaders of the Byzantine army. The multifront invasions and tenuous nature of his father's rule as a usurper limited his effectiveness, yet Alexios nonetheless passed on a stable empire that had weathered the worst of its immediate storms. Despite the threat of rebellion from within the ranks of the aristocracy, John campaigned in Asia Minor in 1119 and captured Laodikeia-on-the-Lykos in Phrygia (near modern Denizli, Turkey) and thereby set the tone for the reign. John was a more measured commander than either his father Alexios or son and successor, Manuel. John inherited a well-disciplined and experienced army, as evidenced at the siege of Sozopolis (modern Uluborlu, Turkey) in Phrygia. In 1120, John marched into Asia Minor against the Seljuk Turks, who had increased their territory steadily since the Battle of Mantzikert nearly fifty years prior. Sozopolis was well fortified and could not be overcome using siege artillery, so the emperor directed his commander Paktarios to attack the walls with missiles (fig. 2.6). The Turks sallied to drive away the Byzantine archers, who fled in a feigned retreat. Despite their own persistent use of this steppe tactic, the Turks fell victim to the emepror's trap. Some distance from the city the Byzantines sprang their ambush, cut off the garrison from the city, and destroyed the enemy force. The victory at Sozopolis is striking because of John's tactical judgment and the discipline and coordination required for his men to execute his strategy demonstrates that his forces were far from declining.
Late in 1121 nomadic Pecheneg raiders bent on plunder swarmed across the Danube. Once a major steppe power, these Turkic nomads had been pushed from their homeland in southern Russia by the Kipchak hordes. The Pechenegs nevertheless remained a formidable power—in John's day the Byzantines still told stories of devastating Pecheneg attacks on Thrace in the reign of his father and earlier. In the winter of 1121–22, John marched north to meet the invasion and took advantage of the seasonal lull in fighting to bribe contingents of the enemy to his side. In the spring of 1122 John advanced to meet the Pecheneg army which he found arrayed around a wagon-laager, their families and animals inside the protective ring of oxhide-covered carts. The emperor ordered a dawn attack and the two forces fought bitterly to a draw; throughout the course of the fight the nomads retreated to the safe cover of their laager when they tired or the Romans successfully bloodied them. John himself suffered an arrow wound in the leg, but at the critical moment dismounted and led his Varangian Guard on foot against the laager, which his ax-men hacked through. The rout was complete and the emperor's men seized thousands of nomads captive and settled them in the Byzantine Balkans. Short, sharp engagements with Hungary and Serbia followed the defeat of the Pechenegs, but the emperor was able to raise a force to invade Hungary, with whom he established peace.
In 1122 a Western crusading force spearheaded by Venice, the preeminent maritime power of the Mediterranean, laid siege to the Byzantine fortress on Corfu. The Venetians aimed to pressure John into renewing their lucrative trading privileges. John was forced to concede to the Venetians and restore their dominant position within the trade networks of the empire. The uneasy relations between the empire and Venice would simmer to a boil and lead to the disaster of the Fourth Crusade in 1204, a tragedy for which the Komnenoi bear some responsibility. The emperors of the dynasty were unwilling or unable to restore their fleet to a dominant position and seriously challenge the naval superiority of their ambitious western rivals.
Unlike both his predecessor and successors, John was most concerned with the Turkish threat from the east and, with the western front quiet, John turned his attention to Asia Minor. From 1130 to 1137 the emperor led campaigns against the Danishmends (a Turkoman dynasty) in northern and eastern Anatolia. Unlike their neighbors at Ikonion, the Danishmends took seriously the obligation of jihad and embraced its opportunities. In the power vacuum left in the wake of the Byzantine collapse on the high plains of Anatolia, the coalescence of another powerful raiding emirate thwarted the reunification of the coastal zones, much of which remained at least under nominal Byzantine control. John forced the surrender of Kastamon (Kastamonou), a Paphlagonian town where the Komnenoi family had previously owned land. In 1135 the emperor recovered the city of Gangra (today Çankiri about 140 kilometers north of Ankara). Elsewhere in Asia Minor, John sought to protect and expand his coastal positions in the southeast and to form a viable bridge to northern Syria, a relatively rich territory where a Latin principality exercised control and the Byzantines maintained some influence. In 1137, at Anazarbos on the Cilician plain (today Anavarza in southeastern Turkey), John employed a number of counterweight trebuchets, some of which the defenders burned by casting red-hot iron projectiles into the machinery. The Byzantines remedied this weakness by building brickworks around their artillery. A relatively new weapon to which some cities had not adapted their defenses, the trebuchets smashed the walls of the city, whose citizens promptly surrendered. Not much later, after a difficult siege, the Turks once more seized Anazarbos. This failure underscores the strain on imperial resources and the problems inherent in the emperor's strategy of piecing together a ribbon of fortified urban centers without securing the countryside from the Danishmends or the Seljuks based in Ikonion.
In 1137 the emperor again struck eastward and cowed his crusader neighbors. John's arrival at the head of a strong army intimidated the Franks and forced Raymond of Poitiers (1136–49) to pay him homage. By agreement, Antioch was to be ceded to the emperor if John could seize territory outside of Latin control, namely the capture of Aleppo (Halab), Shaizar (Shayzar), and Homs (Hims). By seizing these strategic points, John hoped to undermine the growing power of Imad ad-Din Zengi (1127–46), ruler of Mosul and Aleppo who was tightening the noose around the exposed gullet of the Crusader States in the north—especially the County of Edessa ruled by Joscelin II (1131–59). In the spring of 1138, at Buz 'ah (called Piza by the Latins), a town one day's march north of Aleppo, John's army encountered a strong Muslim garrison. The historian Choniates (d. 1216) stated that a sally of defenders drove back the Byzantine vanguard, but this was likely another feigned retreat, since John arrived on the scene with his elite Varangians and threw the Muslims back to the citadel. There the Muslims lay trapped as the Byzantine engines pounded the walls to rubble. The emperor seized immense plunder and committed it to a subordinate named Thomas. As John pressed on to Aleppo, Thomas marched back to Antioch but Zengi's men ambushed the Roman force and seized their spoils.
John withdrew from Aleppo and turned south where he captured Kefar Tab and the city of Hama then wheeled northwest. Although he could not know it at the time, the decisive moment in Syria for John came in the spring of 1138 when he and his forces invested the important Muslim fortress of Shaizar. Raymond and Joscelin undermined the emperor by their delays and John had to withdraw in the face of a march-in-force by Zengi. The emperor did not wish to risk a major engagement with his powerful Arab rival; a loss would prove disastrous and unravel the strands of imperial policy in the east, while a victory would free the eastern Franks from the immediate Muslim threat that forced them so reluctantly into the arms of the empire. With Antioch brought to heel for the moment, John hoped to put one more stone in an arch of control from north to south that linked imperial territory along a defensible eastern line. Thus, in winter of 1139 the emperor again marched east, this time to Pontos in northern Anatolia, some 700 kilometers from Constantinople. There he threw a sizable army against the Turkish Danishmends and their stronghold of Neaocaesarea (Niksar) which commanded a fertile hinterland and access to the Black Sea coast, a region that had devolved from imperial control to that of the semi-independent Constantine Gabras. Determined Turkish defense and the bitter cold thwarted the emperor, who had to settle for seizing minor strongholds and captive-taking.
When John learned that Raymond of Antioch rose in rebellion, he collected a large army—the sources for his reign rarely provide details—and in 1142 marched to Syria once more. The emperor's death the following year, aged fifty-six, was due to a hunting accident or murder. His son and successor Manuel paid less attention to Anatolia, and Byzantine hopes there waned.
John's record as a general is good but far from stellar, and his career underscores the problems facing the empire following the fall of the Anatolian plateau. Their route to the rich cities of Syria, which honor and economic interest demanded they possess, was effectively severed. The emperor was an indefatigable campaigner who understood geography, strategy, and tactics. It was a mistake, however, not to direct his full energies against the neighboring Seljuk Sultanate of Rum, which occupied the heart of the plateau. The loss of the recruiting grounds, resources, and strategic depth offered by central Asia Minor dampened the resources available for counterattack, and their continued possession by the Turks threatened the fertile coastal belt of the Aegean. Moreover, John misjudged his Latin crusader neighbors and consequently was checked by Raymond and his western allies. John embodied Byzantine caution. Unlike his father he suffered no heavy defeats but he likewise won no decisive victories because he sought no decisive battles; the dangers posed by defeat were too great. Instead he aimed to grind down his enemies and, by a combination of siege warfare and overwhelming shows of imperial force, to intimidate his opponents into cooperation or quiet. John is nonetheless accused of driving his men too hard; he led his forces on grueling campaigns year on year that netted plunder, but few permanent gains. The emperor's strategy of city-taking was a sound one to a degree—John wished to deprive the Turks of secure bases and gain permanent bridgeheads of his own. Thus his siege warfare maximized his forces and greatly reduced his risks. His aggressive campaigns in Anatolia disrupted the Danishmends and bullied the Seljuks, but did nothing to break the foundations of their power. In the final tally, John's leadership stabilized the Byzantine state; when he died he left a stronger empire than the one he inherited.
## THREE
## ORGANIZATION, RECRUITMENT, AND TRAINING
THE BYZANTINES MAINTAINED A professional standing army for most of their thousand-year history. During the early period, from the fourth into the early seventh centuries, there were large standing forces and elite units available for campaigning. The middle period army began its existence as a shattered remnant of this impressive late antique institution. While the precise nature of its forces in the turbulent era of the Dark Ages is unknown, over the course of the later seventh and eighth centuries the thematic system grew from the kernel of field forces that were consolidated and billeted in the provinces. Sustained offensive operations returned in the ninth century, by which time the state employed a mix of professional mercenaries and local thematic forces. The disposition of these troops, their recruitment, and supplement by thematic armies continued until the reign of Basil II. By the end of his reign in 1025 the thematic armies had run down, replaced by the tagmata, mobile armies stationed around the capital. The greater reliance on foreigners in imperial service and the corresponding decline of native troops is not a straightforward issue. The Byzantines always depended on foreign auxiliaries, and while there is little doubt that their role increased markedly from the eleventh century, we should not immediately disparage the loyalty or quality of such men. Native Roman soldiers served for pay in one form or another and were themselves mercenaries in the broadest sense—paid professionals. In order to pay the soldiers and manage their deployment and to achieve their defensive aims, the state relied on a developed bureaucracy and military hierarchy.
ORGANIZATION
Like all armies, ancient and modern, the Byzantines arranged their military apparatus hierarchically. The handbooks portray deep organizational structures, inherited from the Romans and persisting until the fall of the empire, with clearly delineated ranks to the level of five or four soldiers. The overall commander of the army was, of course, the emperor. In all cases emperors were expected to uphold the façade of military competence—even the most pacific possessed a smattering of training, could ride, wield weapons, and were literate in strategy and the structure of their forces. In many instances, the emperors were military men and possessed firsthand experience in the affairs of war. Since no head of state could manage security alone, even when he took the field himself, all relied heavily on practiced commanders.
Early Period
Constantine appears to have made radical structural changes in military organization; he removed the prefects from command and made theirs an administrative post. He further removed the troops stationed in garrison, the frontier guards ( _limitanei_ or _ripenses_ ), from the emperor's guard units ( _protectores_ ) and the field army ( _comitatenses_ ), which he expanded in size. Units were uprooted and pulled from their old third-century bases. The Master of Infantry ( _magister peditum_ ) and Master of Cavalry ( _magister equitum_ ) commanded those branches of individual field armies. We would equate the various magistri with marshals in more modern military parlance, with control over armies in a given theater. After Constantine the empire was once more divided between emperors in east and west and some mobile units transferred to the frontiers where they formed the core of campaign armies and an effective active defense supplemented by the limitanei. Such mobile regional field forces were under the command of a Master of Cavalry who commanded both the infantry and horse.
Prior to Constantine, Diocletian replaced the old Praetorian Guard—which had become infamous through fractiousness, rank insubordination, and regicide—with a new imperial bodyguard. Constantine further increased the new regiments, the _Scholae_ (Latin: schools, group), which totaled twelve units, each with 500 men divided evenly between eastern and western halves of the empire. The _magister officiorum_ (Master of Offices) led them. These units formed an elite guard for the emperor on campaign through the time of Theodosius I (379–95), but most units gradually declined to a civilian honor guard by the later fifth century. By the sixth century a count ( _comes domesticorum_ ) commanded units of the scholae.
In the fifth century, the strategic disposal of forces and consequently the high command settled into the form it would resemble through the reign of Justinian (fig. 3.1). There were two imperial armies attached to the emperor's person led by the _magister militum praesentalis_ (Master of Soldiers of the Emperor's Presence). These praesental armies comprised elite troops and mobile field forces that would form the core of any imperial expeditionary force. Five regional field armies (two praesental armies, Illyricum, Thrace, and the East) and their supporting frontier forces were under the command of the _magister utriusque militiae_ (Master of Combined Forces [meaning of horse and foot]). His lieutenant, the _vicarius_ , is known from the fifth century onward. There were frontier commands directed from the office of _comes rei militaris_ (military counts) in Egypt and Isauria in mountainous and restive southern Asia Minor and thirteen dukes along the Danube, eastern frontier, and Libya. The magister commanded his field forces and also held authority over the armies under control of the _comites_ and _duces_. The _legatus_ (legate) or prefects held the reins of individual infantry legions. Infantry cohorts (regiments) of 500–600 still existed in the fourth century and their cavalry equivalent was formed of vexillations ( _vexillatio_ ) or _alae_ of up to 500 troopers. _Tribune_ was the most common title for officers handling regiment-sized units, whether cavalry or infantry, but we also find the prefect in command of the cavalry vexilliations, _alae_ , and among the limitanei. Another _vicarius_ (hence our word "vicar") was the lieutenant commander of the regiment whose duties and authority increased throughout this period. While much of the army underwent serious changes in organization and deployment, certain areas, such as Egypt, retained older structures and ranks.
Promotion within the ranks was a matter of service time or, not uncommonly, graft. St. Jerome (d. 420) provides a clear hierarchy of grades for enlisted men and noncommissioned officers in the early Byzantine period. He lists from lowest to highest grade: _tiro_ ; _eques/pedes_ ; _circitor_ ; _biarchus_ ; _centenarius_ ; _ducenarius_ ; _senator_ ; and _primicerius_.
A recruit was a _tiro_ (pl. _tirones_ ) until he was trained, and such men did not draw full pay or rations. The anonymous author of a late fourth-century document, the _De Rebus Bellicis (Military Affairs_ ), recommended that cohorts maintain fifty or a hundred tirones so that losses could be quickly and cheaply replaced. Soldiers of the line were _pedes_ (infantry) or _eques_ (cavalryman). The _semissalis_ seems to have been a senior ranker but below what we would consider noncommissioned officer status. At the base of the noncommissioned officer ladder of that time, the _circitor_ at one time inspected sentries but little else is known of his authority or responsibilities. By the fourth century he may have been a junior _biarchus_ , (mess-leader; sometimes called _decanus_ or _dekarch_ , "leader of ten," even though he led eight soldiers, including himself) who commanded the _contubernium_ , the squad or mess-group, which comprised eight to ten men who shared a tent and, as the name suggests, took meals together. By the fourth and fifth centuries the century numbered around eighty men, ten contubernia, commanded by the centurion with the rank of _centenarius_. The _ducenarius_ , rather than commanding two centuries, was probably a higher-ranking centurion, since Vegetius stated that these men formerly commanded two hundred, an indicator that the title no longer reflected its old order. As historian Warren Treadgold argues, the _senator_ likewise was probably a senior kind of noncommissioned officer with specialist duties, such as _adjutor_ (clerk or scribal assistant), _campidoctor_ (a centurion who drilled rankers and recruits), or _actuarius_ (regimental quartermaster). Each regiment also had an _optio_ (quartermaster), a surgeon, two heralds, two standard bearers, _draconarii_ —named for the dragon-headed pennons known in the fourth century, a cape bearer, a trumpeter, and a drummer.
The lower command structure then looked something like fig. 3.2.
The five regional field armies possessed an extensive administration that handled correspondence, pay, logistics, and judicial matters. These large staffs, numbering up to three hundred, mirrored their civilian counterparts in the provinces. Military tribunals were more or less the same throughout the staffs of the magister militum, the dux, or the comes. The army judiciary was staffed by a princeps assisted by a commentariensis and an adiutor and a libellis; the latter dealt with judicial petitions. Deputy assistants ( _subadiuva_ ) and shorthand writers ( _exceptores_ ) handled the judicial clerking. Another bureau headed by a _princeps_ with his assistant, the _primiscrinius_ , two _numerarii_ (principal accountants), and their support staff of _scriniarii_ (clerks) dealt with financial and supply matters.
Scholars debate the tenor and role of the frontier forces (limitanei) who are sometimes characterized as "static" forces or even as "soldier-farmers" whose quality deteriorated in the fifth and sixth centuries. In a much-cited passage written no later than the year 550, Prokopios criticized Justinian for his elimination of their pay. While the loss of payment in coin may be true, frontier garrisons staffed by local troops continued to exist in some areas of the empire. An Egyptian known as Flavius Patermuthis (the name "Flavius" was taken upon entry into imperial service from the reign of Constantine to show one's joining the imperial "family") served as a soldier in Elephantine (modern Aswan, Egypt) from at least 585–613. Patermuthis and his comrades were prominent locals, indicating that in some places the limitanei had come to resemble local self-help forces rather than disciplined professionals. Elsewhere, the picture is somewhat different. Isaac argues that the limitanei were not soldier-farmers but simply the troops under the command of the duces of the provinces and as such they were mobilized for police duties and patrols, manned the frontier posts, and joined the field army on campaign. From papyri recovered in Nessana (modern Nitzana in southern Israel) we know of a _numerus_ of _dromedarii_ (camel riders) who patroled the desert routes around Gaza; these men appear as landowners and prominent members of the community until around 590, when the unit was either disbanded or transferred. Their duties were then probably assumed by allied Arab forces of the great confederation of Ghassan.
Federate soldiers ( _foederati_ ) remained prominent in the Roman military structures of the fourth to seventh centuries. These troops served under a treaty ( _foedus_ ) between the empire and tribes on the frontier. During the time of Diocletian and Constantine, federate troops served under their own commanders and were paid lump sums with which to provide for their soldiers' needs. They also received _annona_ : payment in kind of foodstuffs and fodder. By the sixth century, some tribal groups served under their own leaders in this fashion, such as the Ghassanid Arabs who guarded the eastern frontier from the Euphrates to the Red Sea. Others federates were enrolled in regular military units that appear to have been mixed Roman-barbarian contingents under the command of Roman officers. When not in the field these units were under the authority of the comes foederatorum, but for tactical purposes while on campaign they served under the magistri.
In 528, in light of new strategic realities in which the contest with Persia increasingly centered on Armenia and the Caucasus, Justinian divided the eastern command formerly under the _magister militum per Orientem_. He created a new command, the _magister militum per Armeniam_ , headquartered at Theodosiopolis (modern Erzurum) whose army was drawn from both praesental units and the mobile forces of the old duces and comites of the frontier districts. Following their successful conquests, Africa, Italy, and Spain gained their own regional commands as well, which raised the number of army corps to nine, though there does not seem to have been a commensurate increase in troop numbers.
By the time of Maurice's _Strategikon_ in the late sixth or early seventh century, the army had changed considerably. The old guard units, the Scholae, Domestici, Protectores, and Candidati (originally a picked unit of the Scholae) became mostly civilianized but remained intact. The limitanei degraded and Justinian seems to have drawn down some of these frontier forces. The military returned to a purely decimal system of organization, with the main building blocks being the commands of ten and one hundred. A change in terminology reflects the decline of Latin in favor of Greek within the military, which was natural since the latter was the language spoken by most people in the eastern Mediterranean.
Book 1 of the _Strategikon_ lays out the ideal officer structure of the Maurician army at the end of the sixth century. The general, now called by the Greek title _strategos_ , held overall command of a given field army. A _hypostrategos_ (lieutenant general) served as his second in command and led the _meros_ (division) in the center of the battle line; this indicates that tactically the _hypostrategos_ was important, since his forces anchored the army. The handbook also says that armies of medium strength were 5,000–12,000, thus representing groups of one to three _meroi_. A _meros_ (Greek "part,""portion") was a division comprised of around 5,000 men, officered by a _merarch_. The division meros was built from multiple units called _moira_. The moira numbered 2,000–3,000 under the command of a duke, _moirarch_ , or _chiliarch_. The units that replaced the cohorts of the older army were variously called _tagma_ (not to be confused with the imperial mobile army which had taken on the name _tagma_ or _tagmata_ after the Greek for "order" or "ranks"), _arithmos_ , or _bandon_. The tagma and its equivalents numbered 200–400 led by a count or tribune, with his second in command, the _ilarch_ , a higher grade _hekacontarch_ who commanded a hundred men. The hekacontarch then was the successor to the old legionary centurion. The lowest levels of command were the _dekarch, pentarch_ , and _tetrarch_ who commanded ten, five, and four men, respectively (including themselves).
The _Strategikon_ provides the order of march for a 310–man cavalry tagma, probably a common strength (for a number of reasons, unit sizes were not uniform). The commanding officer (tribune or count) held under his command two hekacontarchs (or ilarchs), 27 dekarchs, 29 pentarchs, 31 tetrarchs, a standard bearer, a cape bearer, and a trumpeter, with 217 troopers. Treadgold hypothesizes that the tactical units mentioned in the text, ranging from 200–400, represent deployments from standard, 500 men regiments (tagma or bandon) whose remaining 100–300 men remained in quarters. This is a reasonable interpretation, given that unit sizes seem to have been based on decimal units grouped into thousand-man paper legions whose disposition varied according to the tactical situation.
The _Strategikon_ names among mobile field meroi, the Optimates ("best men"), an elite cavalry regiment (bandon) unit of perhaps 1,000 men. In addition, elite cavalry units clearly owed their names to older Roman forces: the Vexillations, Illyriciani, and Federates, all mobile cavalry divisions that Treadgold estimates numbered around 5,000 each. Haldon sees there being only three elite cavalry units: the Optimates, Boukellarii, and Federates, all formed sometime after 575. These cavalry armies probably replaced the old praesental armies as the core of imperial campaign forces, since the author of the _Strategikon_ envisions deployment of the three in the vanguard of an imperial campaign army.
The Persian War of Heraclius occupied more than a decade and drained the empire of men and resources. By the mid-620s the Romans had rebuilt their forces and attained victory, only to see them swept away by the armies of Islam. The Byzantines adapted to these exigencies by reconstituting their battered forces as best they could and billeting troops throughout the countryside of Asia Minor, the last large territory left in imperial possession. From the settlement of the military corps on the land evolved a new military and administrative apparatus called the _theme_ system. _Thema_ ( _theme_ ) is a word of unknown origin, but may be derived from the army muster rolls or the tax rolls needed to support them. During the Persian campaigns of Heraclius the term simply meant headquarters of an army command. The earliest attested themes seem to date to the mid-or late seventh century. "Theme" as a territorial and army designator probably derived from the association in the minds of administrators with the cataloging of military men and corresponding territory and material needed to sustain them.
Dark Ages and Middle Period
The defeats suffered at the hands of Persians and Avars and the civil war led by Heraclius attrited the field armies considerably, and while it did not destroy them, it deprived many of them of their bases in Armenia and the eastern provinces. Heraclius gathered the mobile armies under his personal command; his immediate subordinate was the _comes Obsequii_ (comes domesticorum), now a unified commander of the praesental armies and no longer simply the leader of the largely honorary Scholae and Excubitors.
What we do know is that the names of the themes as they appear in the eighth century bear names derived from the old sixth-century army corps (see Map 4). The region around Constantinople comprising the Opsikion theme derived its name from Latin _obseqium_ , the praesental army. The Thrakesion theme is attested in a letter of Pope Conon (686–87) and included elements from the old Army of Thrace, now garrisoned throughout western Asia Minor. The Anatolikon theme stretched from Cappadocia in the east to Lykia in the west with its northern and southern boundaries defined by the Halys River valley and the Taurus Mountains, respectively. Its name derived from the Army of the East, that is, those forces under the old command of the _magister militum per Orientem_. The Anatolikon was reckoned as the premier theme in the military hierarchy, unsurprising given its location astride the violent frontier with the Muslims. The Armeniakon theme took its name from the Army of Armenia, formerly headquartered at Theodiosiopolis but in the seventh and eighth centuries headquartered at Amaseia (today Amasya). A short-lived naval theme, the Karabision, derived from _karabos_ (Greek, "ship"), formed a permanent naval command possibly centered on the island of Keos (Chios); this theme was disbanded after repeated failures, their last being the siege of Constantinople in 716–17.
Strategic passes were under _kleisourarches_ (Greek, "guardians of defiles" = Greek _kleisoura_ , pl. _kleisourai_ ). The kleisourai were hard points established by Heraclius and his successors to check the advances of Arab raiding forces in the dark days of the 630s and 640s. Nearly all of these commands lay in the east, among the mountain passes from Mesopotamia and Syria that the Arabs used to gain ingress into the Anatolian plateau. The institution is attested early; in 667–68 an unnamed kleisourarch apprehended an imperial rebel conspiring with the Arabs in his kleisoura of Arabissos (modern Afsin, Turkey).
From the fourth to early seventh centuries, the economic strength and strategic realities of the empire favored the development of horsed units. These could respond to threats on multiple fronts with relative speed and counter the peril posed by nomadic tribes and Persian mailed cavalry; by the time of Maurice, the ratio of horse to foot may have been as high as two to one in elite forces. While nothing is known of the precise compositions of the army in the later seventh and eighth centuries, the loss of resources and impoverishment of the state must have reduced the cavalry arms substantially. Due to the Arab conquest of the east, by the 640s the state had effectively lost three-quarters of its revenue. Although the nature of the army, how it was supported, and its level of professionalism are highly debated, there can be little doubt that the troops suffered a decline in numbers and quality. Though cavalry remain prominent among the fragmentary notices we possess of the army of the Byzantine Dark Ages, the eighth-century army probably comprised a higher proportion of infantry than its predecessor. Treadgold suggests that the thematic cavalry represented one-fifth of the total, not an unreasonable number.
The decimal system of organization of the sixth century seems to have mainly survived and been employed in the thematic structure; civilian sources of the late seventh and eighth centuries mention _chiliarchs_ (also called _droungarios/droungar_ , latinized as _drungar_ ), _komes_ (count, who replaced the old tribune), _hekacontarchs_ (also called _centarchs_ ), _pentecontarchs_ , and _dekarchs_. The new officer here is the pentecontarch who, as the name suggests, commanded fifty men. Evidence is scant, but it seems that the old _merarch_ became an officer called _tourmarch_ , first attested in the eighth century.
Theme Armies
The army regional commander, the theme commander, was after the seventh century called strategos (Greek "general") or _komes_ (count). His office replaced the old magistri militum and strategoi of late antiquity. Apparently the komes vanished from the thematic landscape, meaning that there was no officer intervening between the general and the rank of kentarch, whose command was reduced at some point in the eighth century to command of forty men. In 840, the emperor Theophilos (829–42) created the thematic unit of the bandon to match the organization of the provinces to the tagma and abolished the office of pentecontarch, whose fifty-man units were obsolete due to the reduced command of the kentarchs to forty men. The staff of the strategos also reflected the army's Roman inheritance and its modification; the _komes tes kortes_ ("count of the tent") probably took over the judicial and administrative role of the princeps. A _chartoularios_ and staff domestikos handled the rolls, and financial and supply responsibilities, a _mandator_ held specialist duties, perhaps comparable to the old senator title or provincial seconded protectores who served as regimental cadets or staffers.
By the ninth century the empire had recovered somewhat from the shock of losses to the Muslims and began to salvage a portion of its wealth, power, and confidence. A contemporary account of Byzantine military matters compiled by al-Jarmi, based on his experience as a Byzantine captive in 837–45, provides some insight. Although al-Jarmi's original is lost, his report survives in reduced form in later writers such as Ibn Khurradadhbih, whose work dates from 846–70. According to the latter, the Byzantine command structure was as follows:
The _patrikios_ [a court title given to top commanders, including strategoi] commands 10,000 men; he has two _tourmarchs_ under his command, commanding 5,000 each; each tourmarch has under his orders 5 _droungars_ in charge of 1,000 men each; under the command of each _droungar_ are 5 _komites_ in charge of 200 men each; each komes commands 5 _kentarchs_ with 40 men each, and each _kentarch_ has under his command 4 _dekarchs_ with 10 men each.
Thus, sometime between the sixth century and 845 the dekarch's command fell to forty but otherwise the organization largely mirrors that of the _Strategikon_. Throughout the eighth and ninth centuries the Byzantines never abandoned offensive warfare, but the return to the attack accelerated from the reign of Basil I (867–86). Purely Greek terms continued to replace the older Latin-based titles and professional mercenaries—both native and foreign—increased in importance. The thematic armies—built for defense, easily run down, and probably chronically understrength—gradually fell into a role not dissimilar to the old limitanei units of the sixth century, called upon seldom to campaign and mostly serving as a reserve and garrison force. This was especially true as the old central themes were broken up into smaller divisions.
Leo VI (886–912) again modified the command configuration. The emperor writes of a cavalry theme of 4,000 horsemen commanded by a strategos whose subordinates included two tourmarchs, each leading a tourma of 2,000 men. Below the tourmarch were two droungars (or chiliarchs) commanding a droungos or taxiarchia of 1,000 men, each comprised of five banda officered by a komites leading 200. Leo restored the 100-man units led by the kentarch, replacing their forty-soldier predecessor. This allowed for the restoration of the pentekontarch (tribune) over fifty men—a move which Treadgold links to an expansion of the cavalry. Dekarchs and pentarchs round out the order of command. Tempting as it is to extrapolate that the increase of cavalry was general across all the themes, no evidence suggests an image of uniformity—indeed certain themes were cavalry themes, apparently with a preponderance of horsed units, and others were infantry dominant.
As the thematic commands and their armies shrank, the droungar became a tactical officer. In the themes his role as commander of thousand-man units faded, and the _douk_ took on his provincial role as commander of smaller, more flexible, increasingly professional tagmatic units that formed a standing guard in the frontier regions. Thematic and tagmatic forces were brigaded to provide manpower to imperial expeditions, as seen in the attempt to recapture Crete in 949.
The Tagma
Haldon has shown that the sixth-century praesental army evolved into the tagma (Greek "regiment") of the eighth century and that the foundations for the Byzantine army of the medieval period were laid by Constantine V. The original field army of Heraclius was led by the _comes obsequium_. The Opsikion units descended from praesental armies and others attached to it over the course of the seventh century, including the units mentioned in the _Strategikon_ such as the Boukellarioi and Optimatoi. As early as the 620s and no later than the 680s, the army of the Opsikion was established in western Asia Minor, with headquarters at Ankyra (modern Ankara). As Haldon notes, its composition and position in Asia Minor near the capital indicate that it was both the emperor's army and a strategic reserve to defend the capital. The Opsikion proved fractious and unreliable; it was at the epicenter of five revolts and successfully elevated the usurper Theodosios III (715–17). In 741–43 under the emperor's brother-in-law, Artavasdos, the Opsikion waged a bloody uprising against Constantine V (741–75). Following the two-year conflict, Constantine broke the Opsikion into several themes. He also undertook the recruitment of new palatine troops to protect the capital and the emperor from the weight of the provincial armies.
Two new bodyguard units can be securely dated to his reign, the Scholai and the Exkoubitores. Both were commanded by an officer termed domestikos whose lieutenant commander was called _topoteretes_ , an office that descended from an assistant to the old _doukes_ in the provinces. We also find ranks continuing into the tenth century the command structure of the old cavalry vexillations of the sixth century: komites, kentarchs, doukinator, and specialists like _drakonarioi_ ( _draconarii_ ) are known. The imperial tagmata were apparently a mix of new units and those drawn from older groups stationed in the new provinces around the capital. The empress Irene (780–802) added a new tagma, Vigla (Greek, "watch"), and her successor, Nikephoros I (802–11), raised the Hikanatoi (Greek, "able-ones"). Though Treadgold argues that each imperial tagma numbered 4,000, Haldon believes that in the late eighth and early-ninth centuries the guard divisions of the Scholai and Excubitors totaled around 1,300 while the Vigla and Hikanatoi had slightly double this number; 4,000 men in all is therefore a reasonable number.
When Constantine V divided the Opsikion into three districts on a footing with regular themes, he turned the Optimates, who descended from the cavalry unit of that name during the sixth century, into a permanent supply regiment. The Optimates were responsible for marching with campaign armies and supplying and caring for mounts, weapons, and supplies. Alongside these should be mentioned the infantry guard of the capital, the Numeroi, and the Walls regiments.
Throughout the ninth century emperors recruited new units into the imperial tagmata, such as Nikephoros I and his Hikanatoi and Federates, the latter a unit stationed originally in the Anatolikon theme whence the emperor hailed. Constantine V raised a new bodyguard unit called the Imperials. Leo V (813–20) apparently created a new corps of guardsmen, the Hetaireia (Greek:"companions" or "household cavalry"), which comprised three units and initially was recruited, like the old federates, from barbarian mercenaries. Michael II recruited Fortiers, men who were paid forty gold _nomismata_ (the solid gold coin of the empire) annually for their service. The Hetaireia guarded the palace and accompanied the emperor on campaign and was commanded by hetairarches; one estimate puts these new eighth-to ninth-century guard units at 1,200 men total.
Certainly tagmatic cavalry units seem to have expanded, but so did heavy infantry and other specialist troops. The recruitment of foreign soldiers paid in cash corresponded with the decline of the thematic armies—whether the expansion of the tagmatic forces was a cause or consequence or unrelated remains unknown. What seems likely is that the themes were rendered largely irrelevant by the new strategic situation; the 'Abbasid caliphate waned and the Byzantines aggressively sought to regain lost territory by first nibbling, then wolfing territorial emirates. The task of expansion was more easily conducted by professionals centrally directed from a focused high command.
The domestikos commanded the tagma. His lieutenant generals each commanded 2,000-man brigades, each in turn further divided into ten banda of 200 men officered by a komes (count). Each bandon comprised two kentarchiai under a kentarch leading 100 men.
The emperor Leo VI reformed the kentarchies to comprise 100 men. He also created the 50-man cavalry bandon and the ten-bandon, 500-man cavalry _parataxis_ and the introduction into the _tagma_ of the 1,000-man command under the droungar who had prior to this been a thematic officer. This officer structure among mobile troops prevailed for most of the height of imperial power in the late ninth and tenth centuries. Described here is the administrative organization; these were "paper structures" and had to be adapted in the garrison and the field, where units were drawn from across theaters for action. Tagmatic units were increasingly stationed on the frontiers where the eastward expansion brought new territories to garrison and new scope for offensive operations against the crumbling 'Abbasid state.
The need for tactically capable offensive units meant that tagmatic forces and thematic soldiers were combined under new battlefield commands. The taxiarchia, a unit of 1,000 infantry, appears in the tenth-century military treatises. The taxiarch took over the role of the old infantry legion and is parallel in the themes to the chiliarchs. As Haldon has noted, the struggle for new terminology to describe campaign forces hints that the thematic structures and officers were themselves eroding. By the late ninth century, the domestikos ton scholon, divided into two commands of east and west, was commander-in-chief of the army. Operational officers, such as the eastern and western field marshal called the _stratopedarches_ , served as a proxy for the domestikos ton scholon. Nikephoros II Phokas bestowed the rank on his brother Peter, a eunuch who was therefore ineligible to be domestikos ton scholon.
The themes were apparently not up to scratch in their ability to provide campaign-capable forces, and their overall numbers probably declined throughout the tenth century. The replacement of the droungos of 1,000 with the smaller bandon as the major building block in the themes continued apace throughout the later tenth century; Nikephoros II Phokas noted that a normal cavalry banda numbered 50; but another source notes some banda of 400 strong. The bandon structure provided a more flexible command with units more easily integrated into field armies; it probably also reflected the inability of some themes to provide tactically useful, full-strength droungoi. While the reliance (from Theophilos around 840 onward) on the 50-man bandon as the building block allowed for the creation of smaller themes with garrisons of under 1,000, it also reflects a general weakness in thematic arms and the rise of tagmatic forces. The text attributed to the emperor Nikephoros II Phokas, _On Skirmishing_ , noted that a _large_ army in the common eastern lightning campaigns numbered only 3,000 (though very large forces to counter major _ghazi_ or holy warrior expeditions are encountered elsewhere in the text) and continues:
If you are present with only your own theme, General and the force under your command is a small one, then you should follow the enemy cautiously and at a good distance to avoid being detected by them. You should launch your attacks only against those charging into the villages and spreading out.
The army available for counterraiding of an individual tenth century theme must have been well below 3,000, a situation that reflects both smaller themes and reduced thematic forces.
As the empire took the offensive against the Bulgars in the north and the Arabs in the east, the rise of tagmatic forces is partly reflected in the division of the army into western and eastern commands under the domestikos ton scholon, whose role shifted from being commander of an elite palatine unit to the head of imperial forces. This change occurred by the end of the ninth century at the latest, when we find the Cappadocian Phokas family holding the position over multiple generations. While on the whole successful, the Phokas clan proved too rebellious and the domestikos ton scholon fell out of the hands of the military aristocracy after the rebellions of the Anatolian military magnates in the first half of the eleventh century. Alongside the domestikos were the stratopedarchai and the _ethnarches_ who, like the lapsed comites foederatorum of the fifth and sixth centuries, headed foreign troops; the ethnarches were sometimes themselves foreigners who led their troops in battle.
The establishment of new tagma corps continued. John Tzmiskes raised the Athanatoi (Immortals), a heavy cavalry unit whose gilded armor impressed contemporaries. The Immortals formed an imperial vanguard for the emperor while on campaign. They fought in the victories over the Rus' in Bulgaria and probably also in the Syrian expeditions of John, but their existence was short-lived—they were apparently disbanded when the emperor died in 976 and only revived under Michael VII (1071–81) by his chief minister Nikephoritzes. The Immortals were probably among those units destroyed in the first decade of Alexios I Komnenos's reign in his wars against the Normans or Pechenegs. Other units, such the Satrapai and the Megathymoi, make rare appearances in the literature. The rise of these new mercenary units supports the view that professional, mobile units increasingly replaced the theme forces as the wars of conquest proceeded through the tenth and early eleventh centuries.
The army of the tenth and eleventh centuries was structured for expeditionary action and the organization proved capable of supporting extensive conquests in the Balkans and in the east. As the Turks overthrew the established order in the 1060s, the condition of the thematic forces was deplorable; in his eastern campaigns, Romanos IV Diogenes (1068–71) tried to rally the remnants of these regiments to his banner, but by then the system was impossibly broken. The tagmatic forces did survive both these campaigns and the defeat at Mantzikert in 1071. When he seized power in 1081, Alexios I Komnenos inherited this structure and maintained it, though the field armies suffered severe attrition during the wars of the first decade of his reign and the ranks were increasingly filled with foreign mercenaries. The structure of the eastern and western domestikos commanding mobile armies survived until the fall of the capital to the forces of the Fourth Crusade in 1204. In the provinces, the arrival of the Turks destroyed the old thematic organization. The thematic strategoi, the mountain passes held by their _kleisourarch_ , the dukes in charge of small "bandon" themes, and _katepans_ as governor-dukes mostly eroded, though the Komnenoi used the latter for keeping the scraps of Byzantine eastern holdings around Antioch. Foreign mercenaries often served under their own commanders and were seldom integrated into the structure of the Byzantine army; sometimes conquered foreign elements were absorbed, as were the Pechenegs under Alexios I, but this was rare.
Late Period
After 1204 and, more particularly after the restoration of the capital and Thrace to the rule of the Byzantine Palaiologan dynasty, there was an attempt to re-create some centralized fiscal and military administration in the European provinces. The late imperial army seems to have been a checkerboard of structures. The state reconstituted a theme system whose territorial units were often quite small; these territories were called _katepanikion_ , governed from a _kastron_ (fortress). The kastron was typically a stronghold governing a small district, but may have included villages, a group of islands, or even large towns or cities. The _kephale_ ("head") governed the _kastron_. The _allagion_ (squadron) commanded by an archon formed a core unit like the old bandon; Constantine, the brother of emperor Michael VIII (1259–82), commanded eighteen allagia totaling 6,000 men, but we cannot be certain about how many men commonly comprised this division. The kephale and his subordinates were responsible for the maintenance of his troops, the repair of the walls, and ultimately the security of the kastron. His lieutenant was the _kastrophylax_ ("castle guard"), a position that was often granted in concession to prominent aristocrats who fortified their settlements and received in return lifetime privileges from the emperor. The kastrophylax managed the maintenance, watch, and security of the kastron. Some frontier posts and forts were manned by soldiers enrolled in the _megala allagia_ , the "great allagia" (or "big squadron"), and took their names from their administrative capitals or the theme in which they served. An officer called _tzaousios_ (from Turkish _çavu?_ ) usually commanded the megala allagia. But by the end of the thirteenth century the army was devoid of offensive capability and was outpaced by its neighbors who threatened the absorption of the tattered empire. In the Palaiologan era (1259–1453) the mercenary element, both natives and foreigners, remained prominent. Paid troops frequently served in companies ( _syntrophiai_ ) organized and serving under their own leaders rather than imperial officers. Sometimes such companies were absorbed into the empire's permanent forces via grants of cash, _pronoia_ (proceeds in cash and kind from tax allotments or farms), or land.
RECRUITMENT
Early Period
From the fourth through seventh centuries the Roman state ingested soldiers primarily in four ways: through native volunteers, through enforced hereditary service, by conscription, or by hire of foreign mercenaries. Native volunteers were the mainstay of the army and were generally sufficient to fill the requirements of the state. The hereditary obligation for sons to succeed their father in military service, introduced by Diocletian, was soon after abandoned for recruits to the comitatenses, but maintained among the limitanei. The era of Diocletian and Constantine witnessed annual conscription in the provinces in which state agents levied recruits based on regional resources as assessed in the minute reckoning imperial officials had made; villages and estates had either to furnish a set number of men based on their population and expected agricultural surplus or to buy out of their obligation. Slaves were not accepted. In the troubled years of the fifth century when the eastern army suffered from the aftermath of Adrianople and civil war, supplemental conscriptions fell upon elites who had to provide able-bodied men to serve or a cash payment of 30 solidi (the gold coin struck from 309 on at 72 to the pound)—a steep price, since a worker would have received around 12 solidi maximum annually. Unsurprisingly the draft was unpopular and seems to have been employed only in times of significant stress.
With the exception of the ranks of the limitanei, in which service was hereditary, the practice of conscription was generally abandoned. Justinian allowed slaves to join the army rather than resort to general forced levies, which were unpopular among elites and rustics alike. Limitanei did enroll in the regiments in which their fathers served until the end of their existence; there were incentives on both sides for the frontier guard to be maintained. For the state the provincial soldiery still served a useful role as garrisons and as logistics and police forces, even if those outside Syria and Mesopotamia rarely took part in campaigns. Soldiers still received payment, supplies, and certain tax and status privileges that somewhat offset the risks posed by service, which in places like Egypt was infrequent.
Although Justinian did eventually allow for slaves to be enrolled in the army (and these must have been provided as substitutions during episodic ad hoc conscriptions) volunteers usually staffed the mobile armies and imperial guards units. Justinian and his general Belisarios are good examples of this—both sought service as an escape from provincial obscurity. Volunteers continued to provide the manpower for the army through the reign of Phokas, though Maurice provided that sons of fallen soldiers would succeed their fathers in the comitatenses. This was a privilege rather than a burden that the soldiers welcomed—it assured their families salaries and support. When Heraclius found himself chronically short of manpower in the midst of the Persian War, he restored the old hereditary recruitment of all soldiers, something he managed to accomplish in a time of crisis.
Native recruits generally came from the rural, rough-and-ready regions of the empire. Illyricum (the modern eastern Adriatic coasts and mountains) provided an ample pool of military manpower. Countless troops and officers came from this and other regions south of the Danube from Diocletian's time through the sixth century. Isauria, in the mountain lands of southeastern Anatolia, furnished large numbers of military men from the fifth century onwards, when the emperors were especially active in recruiting them to offset Germanic influence in the army. The rugged upland areas of Paphlagonia, Cappadocia, and Pontos also produced surplus men with martial prowess who helped to fill the legions.
Foreign recruits formed a major component of the army. Armenians provided excellent quality cavalrymen and infantry to both Rome and Sasanian Persia. Armenians dominated the imperial scholae after the fifth century. Hunnic horse archers provided a major tactical advantage for Byzantine armies of the sixth century—they were recruited in groups following a native leader and placed under Roman command. Iranian nomadic elements, such as Massagetae, also called "Huns," and Alans in the sources formed another source of mercenary manpower. They fought as both cavalry and infantry. Three hundred "Hun" or Massagetae horse from Belisarios's boukellarios proved decisive in the opening engagements of the battle of Ad Decimum (September 15, 533) when under the command of the Armenian adjutant John, they slaughtered the 2,000-man Vandal lancer vanguard and killed the king's brother, Ammatas. Captured Sasanian Persian soldiers were brigaded into units that served among Byzantine forces, and some Persians or Armenian-Persians rose to high positions in the military command.
Germanic-speaking peoples also provided excellent warriors for the Roman army up through the sixth century. Among these groups we find the east-Germanic Goths, who dominated the ranks of the eastern field army after Adrianople and were still found in Roman service in the sixth century. The east-Germanic Heruls feature prominently in Prokopios's description of Belisarios's campaigns; they are often seen undertaking special missions and were brave to the point of reckless. Their east-Germanic neighbors, the Gepids, formed another tribal confederation that emerged from the shadow of Attila's Hunnic Empire in the fifth century and also provided troops until their defeat and destruction by the Lombards. The west-Germanic Lombards provided significant manpower in Italy—5,500 of them served the Romans during the 551–54 campaigns of Narses.
The loss of most of the Balkans in the seventh century to Slavs and Avars deprived the Romans of some of their finest soldiery. This recruiting ground was replaced mainly with Anatolian Greek-speakers from the rugged interior. Armenians became especially important; at the beginning of the seventh century, the emperor attempted to transfer 30,000 Armenian troops with their families to Thrace. The army that Heraclius reformed in 621–22 was largely from native Roman troops—since the emperor was in the midst of an empire-wide collection of loaned church plate to melt down to coin money, there was little cash to pay foreigners. It was at this moment when Haldon proposes that the emperor made military service once more hereditary, as it certainly was by the end of the century.
Middle and Late Periods
Anatolia formed the heart of the medieval empire and consequently its most vital recruiting ground. As noted above, the soldiers who survived the defeats of the early and mid-seventh century formed the core of the theme armies. To these we can only guess were added local Anatolians drawn from places like Galatia, Phrygia, Cappadocia, Isauria, Lykaonia, and Pontos—the uplands that produced an abundance of durable men knowledgeable of local terrain and capable of the kind of skulking warfare the authorities would soon adopt to slow down Arab raids. Added to their numbers were Arab defectors—the rump of the Christian Ghassanid Arabs and other tribal elements who had fought in the Syrian campaigns. A few Persians adopted into the ranks during the chaos at the end of the Persian Wars and Armenian elements mitigated Roman losses somewhat.
In the Dark Ages the state relied mostly on native troops and Armenian groups that migrated into the empire or were recruited into service, but the use of barbarian mercenaries never really ceased. In 664–65 Constans settled thousands of Slav prisoners in Anatolia—five thousand of these deserted to the Arab army of 'Abd al-Rahman. Justinian II (685–905 and again 705–11) introduced Slavs into the army in large numbers, most notoriously through a program of capture of thousands of Balkan Slavs and their transferal as soldiers to the eastern front, where as many as 30,000 were shifted. In a spectacular failure of imperial policy most of these troops deserted to the Arabs at the battle of Sebastopolis (Sebaste in Cilicia) that led to a Byzantine rout.
During the reign of Michael II (820–29) the tourmarch of the Foederati of the Anatolikon theme revolted, led by Thomas the Slav whose army is said to have included nearly a dozen different ethnic groups (fig. 3.3). Thomas himself had served under the domestikos ton scholon Bardanes Tourkos ("the Turk," probably a Khazar). Theophilos increased foreign elements into the tagma and palatine units; after 840 a unit of the Hetaireia was at least partly staffed by Turkic Khazar mercenaries from the empire of the south Russian steppes and another of Pharganoi (Iranian or Turkic inhabitants of the Fargana Valley in Central Asia). Occasional immigrations of outsiders who fled the caliphate, as in the Persians who defected to Theophilos and the Arab Banu Habib in the tenth century, added temporarily to the manpower available in the themes and on campaign. But the most plentiful recruiting ground for the Dark Ages and Middle Byzantine period of Byzantine history was Armenia. Armenians were an important element in the rank-and-file of the Anatolian armies and many of their commanders rose to prominence within the military hierarchy. Together with native Romans they formed the bulk of the armies from the seventh to early eleventh centuries.
In the eleventh century, the foreign element increased steadily. The formation of the Varangian Guard, a palatine regiment, during the reign of Basil II was precipitated by the arrival in 988 of 6,000 Rus' mercenaries from Kiev to help the emperor quell the fiery rebellions of the Anatolian military magnate Phokas. By 1034 the Varangians formed the regular palace and imperial bodyguard, replacing the older units noted above. The Varangian Guard was known for its steadfast loyalty to the emperor and their devotion was handsomely rewarded; so lucrative was service in the Byzantine army that one Varangian, Harald Hardrada, bought the throne of Norway largely with loot gained in the east. Though the Varangians were mainly recruited from the Kievan Rus', many Scandinavians served. After 1066 and especially after 1080 there was a strong Anglo-Saxon presence in the guard.
At the end of the eleventh century, with the Turks possessing much of the Anatolian plateau, Alexios I Komnenos and his successors faced the loss of the central recruiting grounds of the empire. The Komnenoi therefore turned to the European core of the empire—Thrace, Macedonia, and Epiros in western Greece. But the reliance on foreign men became especially pronounced; Alexios enrolled many Normans in his service—it was these heavily armed and excellent cavalry that the emperor desired when he asked for aid from Pope Gregory VII, a request that helped spark the First Crusade (1095–99) when Norman adventurers spearheaded the western expedition into the Levant. Turkish horsemen feature prominently in Alexios's campaigns against the Normans and Pechenegs. Alexios's general Tatikios was a Turkopole (Gr. _Tourkopouli_ —"sons of Turks"), a Turkish former mercenary who had converted to Christianity and became part of the emperor's inner circle. In 1081 Tatikios commanded a unit of Vardariotai against the Normans in Greece. The Vardariotai, probably Hungarians, were established in the valley of the Vardar River (in modern western Macedonia near the Serbian border) and continued to provide troops until the Serbs conquered the region in the thirteenth century—after this the Vardariotai continued to exist as a palatine regiment, probably staffed by other foreigners. They were horse archers or light lancers who wore distinctive red dress and carried whips.
After his defeat of the Turkic Pechenegs in 1091, Alexios settled them inside the empire and raised troops from among them. The Cuman (Kipchak) confederacy that replaced the Pechenegs on the south Russian steppes and in Bulgaria posed the same challenges as enemies and opportunities as allies; they were superb horsemen and archers and would later form one of the major sources of Mamluk recruitment for the Egyptian state. In 1241 the emperor of Nicaea (one of the successor states that arose following the crusader sack of Constantinople in 1204) John VIII Vatatzes (1221–54) settled 10,000 Cumans in Thrace; they proved useful but fickle allies.
By the Palaiologan era, one-third of the soldiers in the imperial allagia were ethnically Byzantine recruited from Thrace and Macedonia. Emperors supplemented these native soldiers with mercenaries of opportunity—for example, Andronikos II settled 10,000 Alans in Thrace. The enlistment of the Catalan Grand Company perhaps best underscores the lack of native Byzantine manpower and military competence. In 1304 the Byzantines hired the Catalan Company with its 6,000 mercenaries under its mercenary captain Roger de Flor to fight the Turks in Asia Minor. The Catalan Company affair ended in disaster. The empire had neither the money to pay these unruly professional freebooters nor the military force to contain them; the sad affair ended with the capture of Athens and Catalan dominion there until 1388.
PAY
During the era of the Tetrarchy, soldiers' pay was rendered largely in-kind. This was a result of the rampant inflation that plagued the empire in the third century. Since the time of Septimius Severus (193–211) the empire had levied a tax in-kind to support the troops, the _anonna militaris_ and accompanying _capitus_ to supply animal fodder. The state issued clothing, arms, and horses to soldiers. Pay was measured in annona, rations paid annually to rankers. Prior to Anastasios (491–518) each annona was reckoned at 4 solidi. Officers received multiple annona; the primicerius of the fourth-fifth century legions typically earned five annonae. During the reign of Diocletian annual pay in coin continued but was modest to say the least—perhaps 7,500 denarii a year plus donatives and special payments made on accession dates of the emperor and other imperial holidays. Fourth-century pay has been calculated as equivalent to about 12 solidi plus arms and equipment, but by the mid-fifth century had fallen to the equivalent of 9 solidi. To provide some frame of reference, a stone cutter in contemporary Egypt might earn something less than 12 solidi per year. Upon their accession and in anniversaries of their reign, emperors paid substantial bonuses called donatives; Julian paid 5 solidi and a pound of silver, a standard sum offered through the sixth century. Donatives paid every five years from the emperor's accession were about five solidi for soldiers of the line. But over time, by reckoning arms issuances and equipment in annona, the state deeply cut soldiers' pay while theoretically maintaining their ability to fight. One wonders how such issues worked, since a soldier could have hardly worn out a spear or sword in a normal year; possibly these allowances were convertible to food or fodder.
In the fifth century the cumbersome and easily abused in-kind system was replaced by payments in coin; the stability brought by the fourth-century creation of the gold solidus and economic recovery of the empire permitted a remonetization of military pay. Anastasios seems to have spread the five-year donatives out as annual payments and offered cash instead of supplying arms and equipment; prior to his reign soldiers in the field army received something like 9 solidi plus equipment. Under Anastasios field troops earned 20 solidi annually, an increase of as much as two-thirds; the raise was probably a response to a lack of recruits and the general poor condition of the soldiery. By the beginning of the reign of Justinian, soldiers in the comitatenses were then well paid when compared with the average worker.
Limitanei received far less, perhaps 5 solidi and an equipment allowance. Justinian's pay scale for the African limitanei survives. The _dux_ earned 1,582 solidi, the cavalry _primicerius_ 33 solidi, infantry centurions 20 solidi, and their cavalry counterparts 16.5 solidi while infantry rankers earned 5 solidi and cavalry 9. It is probable that even this modest wage was eventually cut by Justinian and that the state paid frontier troops only annona payments in-kind in equipment and capitus issuance for their mounts. Allied units on the frontier, like the Ghassanid confederacy, received annona in cash and kind. But like their comrades in the mobile armies, limitanei received tax exemptions for certain family members and were exempt from corvée labor, among other burdens.
In response to the fiscal and military crisis sparked by the Persian War, in 616 Heraclius seems to have ended the cash allowances for uniforms and equipment, which amounted to reducing pay by one-half. The state returned to issuing clothing and equipment to the soldiery. Constans II (641–68) apparently cut this salary in half again and probably replaced the lost salary with grants in land from which soldiers could support themselves. Annual base pay for the rank and file was thus around 5 solidi during the Dark Ages. To put into perspective this abysmal remuneration, we should note that a carpenter in eighth-century Egypt might earn 16 solidi per year.
By the tenth century, the situation had improved and cash payments in gold had expanded. Officers in the tagma were well paid by contemporary standards. In the mid-ninth century average pay had doubled to about 10 _nomismata_ (singular _nomisma_ , the Greek term for solidus). A tagmatic commander earned 144 nomismata, a topoteretes 72 nomismata, a pentekontarchos 24 nomismata, and a ranker in the tagma 9 nomismata.
The health of state finances and the fineness of the nomisma declined sharply in the middle of the tenth century. Alexios I replaced the nomisma with the _hyperpon_ (pl. _hyperpyra_ ) a coin inferior in fineness to the solidus/nomisma of the past. As most soldiers were by this time native and foreign mercenary professionals, they earned cash payments and donatives. The limited data suggest that soldiers in service in the late Byzantine period were well paid. In 1272 a soldier in Asia Minor earned 24–36 hyperpyra, well above the salaries of common workers, such as cooks or domestic servants (10 hyperpyra each) or doctors (16 hyperpyra). Even though the currency was further inflated by the fourteenth century, the 288 hyperpyra paid to a Catalan mercenary cavalryman even though he had to equip himself, was exorbitant.
Many of the soldiers of the Palaiologan allagia served on the basis of pronoia grants. The origin of these grants is obscure but, like the settlement of troops in the themes centuries earlier, they served to shift the burden of maintaining troops from the central government to the provinces. Pronoia grants included tax revenues or rents from dependent peasants—a system often likened to the "feudal" customs that supported the landed aristocracy of the West. Unlike the medieval western arrangements, however, the pronoia were at first held for the lifetime of the grantee; they became hereditary under Michael VIII. In contrast to the medieval west, the state remained the owner of the land and in control of the fiscal mechanisms by which the pronoia were administered.
Over the centuries the Byzantines showed a continuous tradition of army organization that evolved from the Roman imperial system but was adapted to the strategic and tactical realities with which the empire was confronted. Until the twelfth century, the organizational structure of the army was relatively conservative—were he to view the army of the eleventh century, the emperor Maurice from some five centuries prior would have recognized many units and their officer structure. There was, however, adaptation and reorganization in response to the defeats at the hands of the Arabs, but the seventh-century wars did not expose the system as completely broken and thus most structures continued, albeit in modified form. There was a generally deep command structure present in the organization, with officers down to the level of four or five soldiers which undoubtedly preserved discipline and offered considerable tactical flexibility.
On the whole the state managed the well-being of the soldiers reasonably well—service was often dreary, unpleasant, and dangerous. Only during times of severe crisis, such as the inflationary era arrested by Diocletian and Constantine, and the seventh-century military collapse faced by Heraclius, did the empire economize at the expense of its troops. Even during the worst of the crisis, cash payments were never halted, though they were apparently sometimes paid in copper and in arrears. Since the military was by far the largest governmental expense, it was frequently the only place that such economies could be enacted. However, once the crisis of the Dark Ages ended, pay rates climbed to an average well above those of most workers.
## FOUR
## EQUIPMENT AND LOGISTICS
"THE HARSH NECESSITY OF WAR has invented the guild of _fabricenses_ , which guards the decrees of the emperors with a kind of immortality...for the guild arms, the guild equips our army.
1. Hence provision has been made that such persons shall be subservient to their own skills, and when they have been exhausted by their labors, they, together with their offspring, shall die in the profession into which they were born."
These words, preserved in the law code compiled under Theodosios II (408–50), intrigue for many reasons. Noteworthy is the attribution to war of causal force, a creative energy that produced a guild of skilled craftsmen. Also striking is the mental image the decree conjures of men wearing down in toil and age, and for the draconian harnessing of the children to their fathers' profession.
The Roman state traditionally equipped its warriors and developed an extensive network with which to supply the soldiery. As with all other ancient inheritances, these institutions evolved over the centuries, yet even at the nadir of Byzantine power in the seventh and eighth centuries, the supply system functioned (albeit at a lower level than previously) and was adapted to the new norms of defensive, skirmishing warfare and the rural disposition of the soldiery. By the ninth and tenth centuries the Byzantines sought more offensive capabilities and developed both these and the supply capacity required to push the borders of the empire to the north and east.
PRODUCTION AND ISSUANCE OF MATERIEL
During the era of the Tetrarchs the department of the _sacrae largitiones_ distributed the shirt, tunic, and cloak that formed the basic uniform of the troops. Boots were requisitioned from the local community as tax-in-kind. By the fifth century soldiers were generally paid cash with which to purchase their uniforms, in part because the imperial linen works could never produce enough clothing to keep up with demand; six solidi for gear seems to have been standard. Soldiers might also opt to spend their uniform money elsewhere, as probably the case with the fourth-century Egyptian ranker Apion, who was glad to be supplied a cloak from his loved one Artemis.
A system of imperial arms factories ( _fabricae_ ) owned and managed by the state was spread throughout the east, where in the fifth century fifteen are attested. The fabricae lay on major routes, close to sources of raw materials, such as wood and iron, and often close to the frontier. The Danubian sector possessed six works: in Thrace there were shield and arms production centers at Adrianople (modern Edirne in Turkey) and Marcianople (Devnya in eastern Bulgaria). In the jurisdiction of the diocese of Illyricum, there were four: armories producing unspecified weaponry were established at Naissus (today Niš in southwest Serbia), Ratiaria (the village of Archar on the Danube in northwest Bulgaria), and Thessaloniki, and a shield works at Horreum Margi (Cuprija in central Serbia). By the year 539 Constantinople had a fabrica as well. In the diocese of Asia, in western Asia Minor, there was an armor and shield maker at Sardis in Lydia, while the Pontic diocese had a general arms works at Nicomedia as well as a _clibanaria_ , which manufactured gear for the heavy cavalry, probably including the tack and scale barding for the mounts. There was an additional _clibanaria_ at Caesarea in Cappadocia, which probably also possessed a general arms factory. In the eastern frontier districts, there was a spear factory ( _hastaria_ ) at Irenopolis in Isauria (in modern southeast Turkey), and general armor-works at Antioch, Edessa, and Damascus. Antioch also possessed a _clibanaria_. From this pattern it can be seen that the empire benefited from a planned network of arms-makers. The absence of imperial factories in Egypt in the earliest period is striking, but with no close enemy it made sense for Egyptian troops to be supplied by sea from Antioch or Irenopolis. By the sixth century there was certainly an arms trade in Alexandria, which was mentioned in Justinian's legal code as being a place where the illicit trade was to be curbed. The state had a monopoly on weapons manufacture and it was illegal to produce or traffic in arms without imperial permission.
The Master of Offices ( _magister officiorum_ ) oversaw the fabricae workshops. Although the workers were civilians, they were organized along military lines. Each group of workers served under a tribune or praepositus with a subordinate primicerius. Workers received annona as soldiers and tended to come from the middle classes. They were bound to continue their service, probably for the twenty years that was standard for army recruits. Advancement was based on years of service, with men who had worked for two years attaining the rank of _protector_. Each worker was responsible for a monthly quota of objects in which they specialized. At Antioch, regulations dated to 374 ordered workers there to produce six bronze helmets and gild eight others every thirty days.
Although settlements like Ratiaria, Naissus, and Horreum Margi suffered sack or were lost to the empire by the mid-seventh century, other major centers, such as Constantinople, Nicomedia, and Thessaloniki, remained unconquered. While these cities witnessed attacks throughout the seventh and eighth centuries there is no reason to suppose that arms-making ended there. In the capital region, the state supplied arms and equipment to the troops of the tagma from its stores.
In the themes the situation is murky. When in 616 Heraclius halved military pay, part of the cut involved withdrawing the annona reckoned to cover arms and equipment; these had to be made up by the state if the army was to function, and all the more so when in the 660s Constans II again halved pay. Prior to this, during the sixth century, there were imperial arms depots as well as urban warehouses for the storage of weaponry which would be distributed to limitanei and citizens in the event of a siege or used to resupply the losses of campaign armies. Given the lack of evidence, scholars are deeply divided about the question of central supply of weaponry and uniforms in the Dark Ages. On the one hand, Hendy and Treadgold have argued that a modified form of issue existed probably based on a system of the _apotheke_ warehouses spread throughout the themes and known from lead seals to have been overseen by officials called _kommerkiarioi_ , who were previously associated with the imperial silk monopoly. Soldiers were issued arms or exchanged copper coins or, after the mid-seventh century, agricultural produce from lands assigned to them. They transacted this exchange either at the imperial warehouses, with private weavers and smiths, or with whatever remained of the imperial arms factories. Haldon, on the other hand, believes that state reverted to the old Diocletianic method of requisition among the provincials and collected arms and clothing as tax which was then distributed at the _apotheke_ warehouses.
In the middle Byzantine period the main arsenal was at Constantinople, and an official called an _archon_ probably oversaw the works where arms and "Greek ire" were made, but nothing is known about the size, capacity, or organization of these works. Maurice built an imperial _armamenta_ (arms workshop, storehouse, or factory) in Constantinople adjacent to the Magnaura palace. The emperor Constantine V is accused of turning a church into an arms depot or factory. The central fisc maintained control over the office of the _archon_ into the period of the Komnenoi, and arms-making, like production of other "forbidden" products, remained a state monopoly; the state maintained a certain level of control over the fabrication and distribution of weaponry.
What of the _fabricae_? In the provinces it is likely, though not certain, that some kind of state supply continued. We have no clear indication of the continuation of major arms-making cities, such as Caesarea in Cappadocia. After the reforms of the seventh century, Cappadocia lay mostly in the Anatolikon theme and was in part garrisoned by elements of the old elite cavalry Federates regiment of Maurice's day. These men would have had at least some regimental smiths who kept their horses shod and their gear in repair. The region of the Anatolikon was rich in iron working, and even throughout the lowest ebb of its power the state worked hard to hold onto these precious assets. The needs of the garrison, the availability of raw materials, and the past history of manufacture in Cappadocia lend weight to the idea of continuity. A tenth-century mention of armorers in Caesarea may indicate that the armory there functioned through the troubled times of the seventh and eighth centuries.
By the 840s, when cash payments to the soldiery resembled sixth-century levels, the troops probably once again purchased their equipment, from either the market or the state. In areas where the tagma was based, western Asia Minor, Thrace, and Macedonia, access to urban markets and specialized production was not problematic. However, requisition remained a part of the supply system, especially for major campaigns. Thessaloniki may have supplied its stores from a still functional state armory there or via purchase from private suppliers, although Haldon prefers to see the system of supply in the themes as based on state requisitions. In certain instances, though, private purchase was used, as the _strategos_ of Samos raised cash to pay for nails to support an expedition.
In the tenth and eleventh centuries, in addition to their salaries, soldiers received a cash allowance for food and personal equipment as well as provender for their mounts. This should remind us once again of the old Diocletianic _annona_ system, and just like its ancient relative, it was equally open to abuse. The thirteenth-century Byzantine historian Niketas Choniates—who wrote with the hindsight afforded by the debacle of the Fourth Crusade—groused that Manuel I Komnenos allowed soldiers to profiteer among the provincials:
He was not aware that he was enfeebling the troops by pouring countless sums of money into idle bellies and mismanaging the Roman provinces. The brave soldiers lost interest in distinguishing themselves in the face of danger, as no one any longer spurred them on to perform glorious exploits, and now the concern of all was to become wealthy. The inhabitants of the provinces, who in the past had to pay the imperial tax-collector, now suffered the greatest horrors as the result of military greed, being robbed not only of silver and obols but also stripped of their last tunic, and sometimes they were dragged away from their loved ones.
On campaigns there were large-scale levies imposed, and imperial officials scoured the countryside to drum up recruits, ox wagons, and provender. In 1153 Manuel I ordered his troops preparing to campaign against Hungary to provide wagons and food to the imperial encampment so that supplies would be on hand when reinforcements arrived; such victuals were no doubt procured by draconian round-ups. In the era of the Second Crusade when the armies of Conrad of Germany bore down on the capital with dubious intent, Manuel ordered imperial cavalrymen to Constantinople where they received issues of mail, horses, and cash. Thus we should consider that the origin and supply of weapons varied across time, but even during periods of prosperity and strong centralization, there was not a single way in which the emperor equipped his soldiers but rather a mix of private purchase and state requisition and supply.
TRANSPORT AND SUPPLY
The supply challenges for the empire were, stated mildly, significant. The frontiers were far from the center—in the east the Arab frontier was 600–800 kilometers and had nearly doubled by the end of the expansion begun under the Macedonian emperors, while to the west the upper reaches of the Danubian frontier stretched for 600 kilometers. The Roman logistical apparatus, overseen up through the sixth century by the _Praetorian prefect_ , managed to maintain large standing forces and garrisons scattered over a vast crescent of the eastern Mediterranean world from the rugged Balkans to the snowy lands of Armenia to the desert wastes of Arabia and Sinai. It was no mean feat to equip, provision, and maintain an army of over a half million men. By the Dark Ages the resources available and the strategic realities of the empire rendered supply more localized, with mounts and weapons procured in the neighborhood of the troops from private or state sources. According to the emperor Leo VI, soldiers were to be supplied during the winter, which must have been impossible for units stationed in Armenia and the mountain passes of Anatolia where snows blocked the routes for months. Units in such places must have taken in large stores of provisions in the warmer months and waited, often in isolation, for contact with their superiors.
Throughout its history the empire sustained a network of roads that bound the capital with its provinces. In the west, the main artery of communication was the Via Egnatia (Map 1). This route crossed the Balkan Peninsula from the second European city of the empire, Thessaloniki, via Pella in Macedonia, Edessa, Herakleia Lynkestis, and Ohrid, reaching the Adriatic Sea at two termini—Dyrrachium and Apollonia. The route continued to be used and thus repaired until the end of the medieval period. A northern trunk, the Via Militaris, built in the first century, linked Constantinople to the western Danubian regions via Adrianople, Philippopolis, Serdica, Naissus (Naissos), and thence on to Viminacium on the Danube in what is today eastern Serbia. Recent archaeological work shows that sections of the Via Militaris were built of large, well-dressed blocks; the road was 8 meters (26 feet) wide. Another route hugged the coast, departing Constantinople and going along the Black Sea via Anchialos, Mesembria (Nesebar, Bulgaria), and Odessos (Varna, Bulgaria), finally reaching the Danube at Noviodunum (Isaccea in Romania).
Eastern links (see Maps and ) were maintained by a series of routes originating from opposite the capital. Abydos was a critical transportation node and customs waypoint where goods flowing to the capital were controlled and assessed and a staging post where east-moving armies gathered. A route from Abydos skirted the western edge of Asia Minor and linked the rich cities of the coastal plain such as Pergamum and Ephesus. From Chrysopolis also ran the great trunk road that crossed the plains and mountains of western Asia Minor and climbed to the plateau via Nicomedia and then on to Ankara. At Ankara the road bifurcated—the northern route led across the highland hills across Galatia and Cappadocia to Sebasteia (Sivas) and then on to Theodosiopolis, where a north-south route linked the city of Melitene on the Euphrates with other garrison cities on the frontier and then Antioch in Syria. The southern trunk road in Anatolia from Ankara crossed the high plateau to Caesarea in Cappadocia and then forked southward, where it crossed the Taurus Mountains via the ancient pass of the Cilician Gates, onto the Cilician Plain and thence to Antioch. In the early period, and once Byzantine control returned to Syria in the tenth century, Antioch was a connecting hub of two major north-south military routes; the Via Maris and the desert route to the Red Sea. The first of these, the Via Maris, followed the course of the old Philistine road along the coast of the Mediterranean, linking the prosperous and important seaports of Phoenicia with Gaza and ultimately, after crossing Sinai and the Nile Delta, Alexandria. The more easterly route ran from Antioch, south via Raphanea to the desert steppe city of Bostra on the Arabian frontier, across the deep rift valley, to Aila on the Red Sea. An outer military road from Bostra to the Euphrates, the Strata Diocletiana (Map 5), ran along the desert margins of Syria via Damascus and then to Palmyra. Alongside these major roads were a host of other major and minor routes, the latter generally not suitable for conveyance of an army and its supplies. By law minor public roads had to be 8 feet wide in open country or 16 feet wide in difficult ground, and such widths, even if observed, would have been of limited use for a large force on the march.
A glimpse of the range of installations along major routes used by the military may be drawn from the example of the legionary base at el-Lejjun, dating to the period of the Tetrarchy and located near the Via Nova Traiana, the military highway later subsumed within the network of the Strata Diocletiana. Lejjun is a fortress measuring 242 by 190 meters and covering 4 hectares. Probably built during the reign of Diocletian, the fortress is a massive structure and symbolized an unmistakable statement of imperial power and might in the desert, but it also fulfilled a practical role of outpost, support base, and logistical node. Despite the effort put into these large strongholds at Lejjun and nearby Udruh, neither was occupied for long—they were abandoned by the late fourth century, probably in the aftermath of a devastating earthquake. Justinian's reign saw a great deal of building activity in Syria, including the construction and refurbishment of forts and fortresses in the _limes_ , such as the fortress ( _kastron_ ) at Androna (modern al-Andarin in Syria), which preserves an urban citadel dated by inscription to 558 A.D. The Andron kastron is about 80 meters on a side and built of large blocks of dressed basalt. It protected a large village and agricultural region on the desert fringe. Watch towers, like that found at Kerratin near the imperial estate and supply depot of Taroutia Emperon, measured less than 10 meters on a side, but were constructed of fine basalt ashlars with a strong sloping talus that shows this is not simply an ordinary, private watch tower.
The imperial post system supported communications across the empire, both in the fast movement of orders and in the slow movement of men and materiel. After Constantine, the post system ( _dromos_ ) was divided into the fast post for officials and urgent business and the regular post for heavy goods mainly drawn by oxcarts, which moved the bulk goods of taxes in-kind, bullion, and raw and finished products requisitioned by the state. Regular rest stations (Lat. _mansiones_ , Gr. _stathmoi_ ) and relays of mounts for the fast post, and draft animals and oxcarts for the slow post, were maintained along major routes of travel. Though the fate of the regular post following the Muslim raids and repositioning of the army is uncertain, the dromos as an institution certainly continued, and corvée labor to maintain roads as well as the burden of supply provender and sometimes animals to the post persisted. By the end of the eleventh century, since most of the lands that supported the _mitata_ (imperial lodgings) and stud farms and imperial holdings on the plateau were in the hands of the Turks, the dromos suffered greatly. However the Komnenoi must have relied on elements of the dromos that survived, or at least the practice of levy that had long been practiced to support it, in order to move their large siege trains around Asia Minor. The Byzantines depended especially on heavy weaponry during the reigns of John II and Manuel I, when large, cumbersome counterweight trebuchets formed an important part of the Byzantine arsenal and siege warfare a cornerstone of imperial strategy and tactics.
The postal system _stathmoi_ were at regular intervals, often a half-day's or full day's journey on foot. Egeria, a fourth-century pilgrim to the Holy Land, left an account of her journey in which she describes Roman troops accompanying pilgrim caravans from station to station along the route from Clysma on the Red Sea to the interior of the Sinai. In addition to forts, such routes possessed _mitata_ (sing. _mitaton_ ) which served as a kind of merchant khan in certain areas but as a military installation in others. Installations at the imperial stud of Malagina are referred to as _mitaton_ and in the middle and late Byzantine era the term is also applied to the burden of billeting soldiers, indicating clear links with military and logistical functions. Though we do not know what the _mitata_ typically looked like, if there was a typical design, we can imagine a complex of stables, barracks, and warehouses. In pre-Arab conquest Syria several _mitata_ are known from inscriptions, at Deir Soleib outside of the important city of Apamea and another at Raphanea, south of Epiphaneia (Hama), where units of the Third Gallic Legion ( _Legio III Gallica_ ) were stationed in the fourth century and probably later.
By the Dark Ages, the state warehouses or _apotheke_ , functioned to receive goods collected as tax, some of which supported the army. Most likely these occupied roadside centers where the _mitata_ or _dromos_ already possessed appropriate infrastructure or on imperial estates where produce and labor were available. As early as the fifth century the term _apotheke_ may have been used in Asia Minor to designate such regional military warehouses; an early inscription is known of an _apothekarios_ (warden of a warehouse) in Phrygia. The apotheke was an accounting system and in-kind warehouse network and which dealt military stores, among other goods. Additionally, there were "marching camps" called _aplekton_ (pl. _aplekta_ ) spread along major routes on which armies traveled and where they could be resupplied. Most famous of these was the imperial stud and warehouse at Malagina in the fertile Sangarios River valley. Malagina was the center of an imperial estate whence came many of the emperor's horses, military remounts, and equids for the imperial post. From at least the eighth century, Malagina served as a mustering point for the themes of the _Anatolikon, Thrakesion_ , and the _Opsikion_.
We have details of only one purely logistical unit, the tagma of the _Optimatoi_ , which was charged with supporting imperial campaigns and fighting units dispatched to frontier wars. In the sixth century this unit had been an elite cavalry division comprised of Goths or their descendants, but it fell victim to Constantine V's breaking of the Opsikion and became infantry escorts and support troops. This logistical regiment was headquartered in Asia Minor, opposite the capital, with its headquarters at Nicomedia. Its 4,000 men were charged with obtaining mules and other pack animals and for moving military supplies where they were required. Some notion of the transport arrangements and capacity of the _Optimatoi_ can be gleaned from Constantine VII's writings on expeditions, where he states that 1,086 pack mules were required for the imperial baggage alone, along with thirty saddled riding horses as tribute or from among beasts raised in the imperial studs. In addition the _Optimatoi_ would have carried the baggage of the certain elements of the tagma while on campaign. By the tenth century, heavy infantry regiments were provided one mule per pair of soldiers on which to carry their shields, spears, and victuals. Even with these provisions, the baggage train and the large numbers of men involved in expeditionary forces, as well as the need to locate a suitable site and fortify a camp each day, made marches ponderously slow. A maximal distance for a fast-moving tenth-century army was 16 miles (25.7 kilometers) per day; something like 12 miles was probably more typical, although forces stripped of their baggage trains or their infantry forces or those making forced marches could obviously as much as double this.
When an emperor's army prepared to campaign, the emperor and his inner circle decided on the target and the scale and route of march, then issued orders to provincials in advance of the army's muster. This allowed officials to purchase foodstuffs and equipment needed by the army and to be deposited at the way stations along the proposed route, or to collect it from the provincials by levy. Often the needs of the army were simply drawn out of the provincial treasury's storehouses where they had deposited taxes received in kind. In tenth-century inventories that survive we can see that planning for imperial expeditions involved numerous state departments and not only drew on large reserves of gold and silver, but also required inland thematic lands to provide food and fodder as well as basic stuffs for things like sail-making and cloth for uniforms. Haldon has shown how immense the needs of a tenth-century expeditionary army could be: he estimates that for a two-week (or at most three-week) march, a 15,000-man force needed about 634,500 pounds (about 288,400 kilograms) of grain for soldiers' rations alone, a figure that excludes drinking water/wine and other foods like cooking fats and provender for horses. Once in enemy territory these needs could be met at least partly by foraging or by purchase from merchants, whom we know often supplied military forces, but nonetheless the burden of any major military undertaking to the state and its citizens was immense.
EQUIPMENT
Clothing
The basic clothing of soldiers was a long-sleeved tunic made from goat hair, rough wool, or linen. Such tunics were frequently colorfully embroidered with medallions on the chest and shoulders and are known from mosaics and other pictorial evidence such as the hunting scenes from the late Roman villa at Piazza Armerina in Sicily. By the late sixth century Goth-type tunics were part of normal Roman military dress; these were longer than traditional Roman versions and fell to the knees. By the late Roman period, breeches seem to have been common. Wool leggings, often bound with laces, were normally worn. Soldiers' shoes, _kampagia_ (from Latin _campagus_ ), were high open toed, heavy-strapped sandals. High black leather boots ( _krepides/hyopdemata_ ) were increasingly common throughout the empire's history, and high-ranking officials preferred tall, white leather boots. Woolen or linen leggings, often decorated in imitation of Sasanian silk versions, afforded some protection for the lower legs, and in the most heavily armed men these were covered with greaves, scale, or chainmail leggings. For inclement weather conditions, soldiers were required to have a heavy cloak, a _sagum_ or _gouna_ that fell to the knees and was spacious enough to cover the fully armed and armored trooper, including his bow. These heavy cloaks served to camouflage soldiers—they were often gray in color and, according to the _Strategikon_ , provided extra protection against arrows. The same text indicates that many infantrymen went to battle without the armor protection available to elite troops, even though they fought in a close-order, heavy infantry phalanx.
By the middle Byzantine era, the _kremasmata_ , a quilted, padded skirt worn beneath one's armor, became a common article of clothing. This garment came to below the waist to protect the rider's legs while mounted. A similar item, the _kabadion_ , owes its origin to Iranian garments which were long and buttoned down the front. Attached at the waist, the kabadion had a skirt with front and rear panels that could protect the legs of the rider, as well as the horse's back.
The _Strategikon_ notes that cavalry had saddles and thick saddle cloths and saddle bags large enough for four days' rations. The saddle probably had a high front, with prominent cantles (rear supports) to provide stability. Cavalry troops carried a lasso, an adoption from the steppe peoples who increasingly influenced Roman arms from the fourth century onward.
The baggage train that formed part of the army is also discussed in the handbooks. Light ox-drawn wagons, each of which the author of the _Strategikon_ instructed should carry a hand mill, axes, hatchets, an adz, two picks, a hammer, shovels, baskets, a scythe, and caltrops. Separate wagons carried the arms of each arithmos. Packhorses were also part of the standard army baggage train, and these could be separated from the main supply train to accompany fast-moving battalions and carry enough rations for eight to ten days, probably for each contubernium. The handbooks stress that the supply train carried spare equipment, such as extra bowstrings and arrows for each campaign; Nikephoros Ouranos stipulated that a supply of 15,000 arrows for each division be carried on pack mules and horses, 100,000–200,000 total for each campaign.
Armor
The equipment used by Roman units varied depending on their function. Since no contemporary surviving examples survive in the archaeological record, it is unknown if use of the muscled cuirass depicted on artwork up through the seventh century continued. There is considerable debate about how heavily armored and equipped and how uniformly outfitted were the legionnaires of the fourth century and after. Unlike the legions of the Republic and Principate, the forces of the fourth to sixth centuries certainly witnessed a downgrade in their panoply. Haldon argues that only the men in the front ranks of the army would have worn the entire defensive battle gear. The fifth-century author Vegetius noted that one of the reasons for Roman failure against the barbarians was that they were no longer heavily armored:
For despite progress in cavalry arms thanks to the examples of the Goths, and the Alans and Huns, the infantry as is well known go unprotected.... Thus with their heads and chests unprotected our soldiers have often been destroyed in engagements against the Goths through the multitude of their archers. Even after so many defeats, which led to the sacking of so many cities, no one has troubled to restore either cataphracts [cuirasses] or helmets to the infantry.
This is only partly true. While the days of each legionnaire being protected by _lorica segmentata_ (the segmented plate-armor cuirasses famous from Hollywood movies) were over by the third century, it seems that the mobile field forces of the comitatenses would have been more heavily armed and armored than many of their adversaries. But there was a decline in infantry protection, especially after the disaster at Adrianople, where massive losses of equipment were difficult to make up. Most costly and problematic to replace were the chain mail tunics of the elite infantry and praetorian units, since such coats required skilled smithies and many man hours to produce. Much equipment was lost simply due to age and retirement rather than battlefield losses, and an emphasis on cavalry tactics was also to the detriment of infantry arms.
The _Strategikon_ offers a view into the kit of officers and elite infantry and cavalry of the late sixth and early seventh centuries:
They should have hooded coats of mail reaching to their ankles, which can be caught up by thongs and rings, along with carrying cases; helmets with small plumes on top, bows suited to the strength of each man...spare bow strings in their saddle bags; quivers holding about thirty or forty arrows; in their baldrics small files and awls; cavalry lances of the Avar type with leather thongs in the middle of the shaft and with pennons; swords, round neck pieces of the Avar type made with linen fringes outside and wool inside.
Most of the rank-and-file infantrymen seem to have worn a heavy wool felt gambeson (padded jacket) called a _thoracomachus_. The author of the _De Rebus Bellicis_ , a fourth-century work advising the emperor on military affairs, stated that the felt _thoracomachus_ should have well-sewn covers of African leather to wear in the rain so that soldiers were not burdened by sodden garments. Since the weight of mail fell mostly on the shoulders, the thoracomachus would have protected the wearer from chafing and injury caused by mail and from impact and penetrative blows from swords and arrows as well as blunt force from maces and clubs against which mail offered inadequate protection.
Quilted coats, _kabadion_ , were adopted from Persian kaftans. These were split at the groin and fell to the knees and were the most common form of basic protection in the middle period. The kabadion was a tightly quilted garment with sleeves slit at the armpit; when not needed the sleeves were buttoned to the back. The strong, heavy kabadion served as the undergarment for armored troops or as the only form of defensive clothing for light-armed soldiers. When made of coarse silk, they would have offered some protection, especially against arrows.
Chain Mail
The _Strategikon_ depicts infantry forces less well equipped than the cavalry arms and indicates only the first two ranks of the infantry were normally equipped as heavy infantrymen with full armor. Heavy infantry in the fourth to sixth centuries wore mail cuirasses ( _cataphracta_ ; Greek _zaba_ ) of mail, developed from an evolution of the older _lorica hamata_ , a chain mail shirt made from a combination of drawn wire and riveted rings, or from rings punched from sheets of iron. Mail was used alone or in conjunction or in composite body coverings with plate, scale, or other protections, and was the better choice for leggings and long sleeves because of its flexibility. Coifs offering protection to the neck or full face coverings like those used by the Sasanians were also adopted by the best-equipped Byzantine troops (fig. 4.1). In the sixth century, the historian Agathias (d. ca. 594) described men in the front ranks as armored in mail down to their feet, and ankle-length mail is depicted in contemporary art. By the end of the sixth century, the _Strategikon_ refers to the _zaba_ , usually a long chain-mail ankle-length coat. Chain mail was best against slashing weapons such as swords, though it could absorb some punishment from piercing thrusts of spears or arrow casts. Modern experiments have shown that mail could sustain multiple arrow strikes without failing—penetration and torn rings did not always result in any harm to the wearer and although limited, such experiments show that some felt-backed mail provided adequate protection against the bodkin-piercing arrows of the medieval era.
Experimental archaeology has also shown that chain-mail production required skills that placed its manufacture closer to the realm of a jeweler than a smith, since ring diameter was normally about 12 millimeters with the smallest rings having a diameter of 3 millimeters. Extant examples demonstrate a high degree of accuracy and standardization. Good quality mail required hammering of the rings to slightly flatten them which increased their hardness, then riveting of these rings together at intervals to form fabric. At a minimum, a mail cuirass comprised 12,000 riveted rings and may have required about 4,800 man hours (1.3 work years) to produce a coat of first quality. Twenty of Maurice's tagmas forming an army of about 6,000 may have had approximately 1,240 such heavily armored men, and while probably not all of them would have worn chain cuirasses, if they had done so, 1,240 coats would be needed for the two front ranks. As much as 1,600 man-years of labor could have been required for making this armor. Even if this figure is cut in half, the production of mail cuirasses was both time consuming and costly. While chain mail seems to have been less costly to produce than the best lamellar (discussed below) it was not cheap; the resources required to produce first-quality body protection for the troops were intensive and consequently armor availability declined as the resources of the state diminished.
The infrastructure, skilled labor pool, and materials required to equip and maintain heavy cavalry and infantry were vulnerable to economic and military disruption. Military losses, wear and tear, and the loss of skilled manufacturers in times of stress meant an immediate and widely felt decline in armor availability, especially during the Dark Ages (seventh to ninth centuries). Nevertheless, chain mail seems to have remained in common use throughout the history of the empire. The advantage of mail's adaptability to hot weather, relatively simple (if time consuming) construction, and adequate protection against the missiles and lighter weapons common among Byzantium's foes made it a logical defense.
Scale and Lamellar
Scale armor had a long history of use in the east prior to the advent of Roman power and provided an alternative to the lorica segmentata plate armor of the early empire. Scale armor was constructed from small scales of bronze, leather, or iron overlapping one another and attached to a fabric backing (fig 4.2). A clear depiction of this type appears on coins of the emperor Aurelian (270–75). Fragments of scale armor have been found, mainly in the Roman West, indicating widespread use. Ammianus Marcellinus left a vivid account of an encounter in 363 with Persian heavy cavalry wearing such equipment:
Moreover, all the companies were clad in iron, and all parts of their bodies were covered with thick plates, so fitted that the stiff joins conformed with those of their limbs; and the forms of human faces were so skillfully fitted to their heads that, since their entire bodies were plated with metals, arrows that fell upon them could lodge only where they could see a little through tiny openings fitted to the circle of the eye.
The fearsome impression left by these formidable foes is obvious, and it seems also from the description that Ammianus was not used to seeing such men among the Roman ranks. Throughout the following centuries, though, especially through their contact with the steppe peoples of the north such as the Sarmatians, the Romans recruited and developed more such horsemen. These heavily armed horsemen appear already on Trajan's Column (AD 113) where they are depicted wearing full-scale armor and riding horses with scale barding (fig. 4.3). A tomb painting of the second century from Kerch, Crimea, is taken to portray Sarmatians, and two of the infantrymen depicted there wore scale cuirasses. The Alan-Sarmatian cavalry noted in the fifth-century _Life of St. Germanus_ calls these men "iron cavalry" indicating their full armor that influenced Roman equipment and tactics. Scale mail coats are also depicted on soldiers in fifth- and sixth-century Egyptian tapestries and wood carvings, signifying that these coverings were in wide use and were stereotypes in the minds of contemporary artists. Like chain mail and lamellar, scale mail seems to have continued in use thoughout the empire's history.
By the seventh century, under the influence of the steppe peoples with whom the Romans had frequent and often violent contacts, the latter adopted lamellar armor, that is, crafted from leather, bone, or metal _lamellae_ (sing. _lamella_ ) sewn together (fig. 4.4). The materials and techniques of construction were very similar in lamellar and scale armor. Lamellar tends to be made of larger metal plates—predominantly iron, as opposed to bronze often used in scale armor—and while some consider that the plates were sewn to one another, Dawson has argued that this form of construction is impractical and that lamellae were instead first sewn onto a backing so as not to shear their bindings when one flexed the torso. Following their attachment to the backing, rows of lamellae with rounded tops and square bottoms were bound to one another. This technique, of affixing rather large plates (those found in excavation at Birka belonging to a long Turkic or Byzantine coat comprised lamellae measuring 27 by 100 millimeters or 1 by 4 inches) onto a thick leather backing rendered a heavy, protective garment. The overlapping lamellae effectively doubled the depth of defense and the leather added additional protection. The defensive quality of such pieces is exemplified by the encounter of the emperor Alexios I with the Normans at Dyrrachium in October 1081 when Norman knights assaulted the the emperor from two sides. The knights' spears lifted the emperor from his saddle but did not pierce his coat; Alexios was able to cut away the spear points that had lodged in his armor and make his escape. From this it is apparent that the best lamellar was impervious to arrows and other light projectiles.
Lamellar cuirasses (fig. 4.5) were heavy, often weighing 5–6 kilograms (11–13 pounds) and offered superb protection. The Byzantines developed a technique of inverting the lamellae so that shoulder pieces had the rounded ends of the plates on segments protecting the limbs at the bottom of the rows. This is clearly depicted in an eleventh-century steatite icon of St. George of Mount Athos. Fashioned this way, lamellar offered improved defense to horsemen against blows from below, which typically came from infantry spears and swords thrusting upward at acute angles. Unlike solid plate, the composite nature of lamellar armor meant that energy from strikes were distributed more evenly across the face of the cuirass, helping to reduce damage to the coat and its wearer. Likewise, the failure of an individual lamella could be contained; even if the plates became dislodged from one another they remained anchored to their backing, or vice versa. This heightened protection and reduced the need for frequent repair. Lamellar coifs and limb protection were probably adopted widely around the period of the sixth-century Maurician reforms when Avar influence on the Byzantine army was strong.
From the tenth century onward, lamellar armor seems to have been the most frequently used type in Byzantine armies. It is the common form depicted in contemporary art. But while Dawson sees in its absence in Prokopios and Maurice a telling omission indicating it was not used in Roman armies from the sixth to tenth centuries, Haldon views it as utilized throughout the empire's history from the sixth century onward. But by the later imperial period, after the sack of Constantinople in 1204, the soldiers' armor came under increasing western influence and lamellar probably declined. The common coat of the late period was apparently the shorter chainmail hauberk ( _hauberjon_ ) that generally fell just below the waist and had short or long sleeves. A mail gorget ( _gorgeré_ ) protected the neck and head. This mail shirt was sometimes worn underneath a mail or lamellar cuirass of Turko-Mongol form.
Splint armor was used in protecting limbs, as depicted in the tenth-or eleventh-century fresco in Hosios Loukas monastery, Greece, where a soldier wears upper arm guards of horizontal strips (fig. 5.4), probably of hardened leather, riveted together. The _Strategikon_ notes that heavily armed cavalry wore arm guards ( _cheiromanika_ ), probably of splint type made of wood, bone, or metal. This equipment probably reflected modifications to classical Greek and Roman patterns based on experiences gained from contacts with the Persians and steppe nomads, especially the Avars. The sixth to tenth centuries Treasure of Nagyszentmiklós, a hoard of gold vessels with a range of Byzantine, Iranian, and steppe influences, depicts a warrior of Avar or Bulgar type with splint forearm bracers, splinted greaves, and a chain-mail coat falling below the knees (fig. 4.6). Leg armor, perhaps lamellar or scale, is also portrayed in the illustrations of the twelfth-century manuscript of the history of John Skylitzes. This panoply probably closely parallels Byzantine cavalry protection from the sixth and seventh centuries, which developed under Avar influence.
The author of the _Strategikon_ stipulated that greaves were to be worn over fabric leggings and were to be as smooth as possible in order to effectively deflect missiles and attacks to the legs. They were not to be so heavy as to impede movement or wear down the soldier, but had to be sturdy enough to sustain punishment on the battlefield. Wooden greaves are also mentioned. The recommendation for the creation of smoothed surfaces does not eliminate the likelihood of splint greaves, but it may indicate that molded metal leg protection common in the Classical period and depicted in the Dura Europos fresco of the crossing of the Red Sea of ca. 250 may have continued in use through to the early seventh century. There are no archaeological finds of splinted greaves from within the eastern Roman provinces, but those found in the sixth-century Vendel period in the Valsgärde 8 burial near Gamla, Uppsala, Sweden, are believed to have been derived from Roman models. By the tenth century, most soldiers wore _toubia_ , padded leggings made of wool, felt, or coarse silk. Chausses (leg guards) generally of chain mail, were worn during the Palaiologan period, sometimes underneath greaves and thigh and knee armor ( _cuisses_ ). The Frankish influence is again here evident.
Helmet design evolved considerably during late antiquity and in the medieval period. Unlike the Gallic-type helmets with sloping neck guard, the whole skull covering beaten from a single sheet of iron or brass, late Roman types show manufacturing shortcuts. Typically the design of these helmets (Greek _kassis_ ) was influenced by the eastern and northern neighbors of the empire, especially the Sarmatians and Sasanians. There were several designs in use, including the ridge helmet and _spangenhelm_. Small, skull-cap kassis-type helmets are known from an excavated example from the Dead Sea region. The spangenhelm (fig. 4.7) consisted of four to six pieces curved into the center of the helmet where a band of metal joined the plates together, often carrying prominent cheek pieces and intricately decorated. The ridge helmet consisted of two metal halves joined at the centerline by a metal ridge that often carried a crest (fig. 4.8). Such helms were often high-peaked. These changes reflect adaptations based on encounters with well-armed neighbors whose technology matched or exceeded the Romans' own, as well as manufacturing increases required by the expansion of the army under the Tetrarchy. The multipieced helmets required less skill and time to turn out but still offered adequate protection. An intriguing example of helmet development comes from the eighth-century Arab estate of Khirbat al-Mafjar in the Jordan valley, where a Byzantine archer is shown wearing a kettle-type helm with a conical peak and a wide, arched brow piece or brim (fig. 4.9). By the twelfth century, the kettle helmet was widely employed over a mail hood. Kettle helms had a flat or angled brim, and usually an aventail (a curtain of leather or mail suspended from the rear of the helmet for neck protection).
The shield ( _skouta_ ) was the soldier's most important body protection. Along with the helmet it formed the basic defense of light-armed troops. Shields were convex and constructed of wood planking and covered with colorful painted insignia (fig. 4.10). Affixed to the back were rope handles and a shoulder strap through which one's arm could be threaded. Shields typically measured 0.75–0.9 meters (2.5–3 feet) wide and 0.9–1.1 meters (37–43 inches) long, while smaller shields akin to later bucklers were used by mounted archers. The Byzantines developed the familiar drop-shaped shield with a rounded top and pointed bottom which was easier to use on horseback, since for the mounted warrior the tapered end rested more easily to the left side than a round or ovoid. In later centuries, perhaps under Frankish influence, they adopted the flat-topped kite-shaped shield, though the drop-shaped variety remained the most common.
ARMS
Close Combat Weapons
Throughout the empire's history the two main arms of the military, infantry and cavalry, were similarly outfitted with a variety of arms. Personal equipment depended partly on one's role as heavy or light infantry or heavy or light cavalry but there was considerable overlap in basic gear, with the spear and sword forming the standard weapons of the line. These were supplemented by a number of secondary and tertiary weapons. To see late antique infantry in the eastern empire as purely swordsmen is probably wrong. The primary weapon of the late Roman legionnaire instead was the spear ( _hasta_ ; Greek _kontarion_ ), about 2.5 meters long (just over 8 feet) and consisting of an iron tip, wooden shaft, and butt spike. Shafts of spears and lances were usually lathe-turned from saplings of cornelian cherry (the choice for the ancient Macedonian _sarissa_ pike famous in the battles of Alexander the Great), myrtle, ash, hazel, willow, poplar, or other durable woods with good strength. A spear found at Qasr Ibrim in Nubia was made of tamarisk, a local wood that indicates that soldiers had to adapt to conditions of local supply and could not count on arms from imperial distributions. Spear heads were socketed and usually triangular and rather broad-headed with a narrow cross-section. Spearheads found at Corinth from the early medieval period were of several types, typically 11–14 centimeters long, and usually square in cross section tapering to a narrow piercing point. Others were broad and triangular, gradually tapered (leaf shaped) or barbed. The cavalry _kentron_ was a longer version of the spear, with a length of about 3–4 meters (11.5–13 feet). By the tenth century, this length was usual for infantry spears, which accords well with the generally heavier weaponry of the Macedonian era foot soldier.
The Middle Period treatises describe troops ( _menaulatoi_ ) who carried the heavy spear, _menaulion_. This weapon measured about 3.5 meters long and carried a socketed metal tip 35–45 centimeters (about 12–18 inches) long, probably resembling in make the ancient Macedonian _sarissa_. Like the sarissa, the menaulion was built from a single sapling trunk, honed down and fitted with its blade tip. The menaulion probably also had a butt spike that served as a backup fighting point should the main tip break off, but which more importantly allowed the weapon to be anchored into the ground to help the user sustain the weight of a cavalry charge. Anastasiadis, however, offers an alternative interpretation of the weapon, which he views as a shorter, heavy thrusting spear whose users served as a tactical support against cavalry, especially if the primary heavy infantry wall was penetrated by enemy horse. Even in the last years of the empire, the Byzantines seem to have eschewed the use of pole arms, though this question has not been adequately investigated.
In certain instances infantry used long lances as well, since the _Strategikon_ ordered that in wooded environments foot were to discard such weapons and substitute the standard infantry spear instead. These shorter spears seem to have been based on Slavic examples. By the time of the _Strategikon_ , unskilled soldiers armed themselves with "Slavic" spears, a shorter weapon closer to a javelin than the kontarion but useful in close combat, especially in woods where fighting space was limited.
The fourth- to fifth-century _spatha_ ( _spathion_ ), the single-handed straight sword of the later Roman Empire, is well known and was used by both infantry and cavalry. The tanged blade was pattern welded, a method that involved forging different metal wires together to form a distinctive design, a longstanding and simple smithing technique with many uses, including providing a softer iron core to an edged weapon. Suppleness and hardness were required in balance, the former to keep the blade from breaking, the latter to keep the blade from bending in action and to hold an edge. The spatha was typically 65–80 centimeters (25–31 inches) long, depending on the type, and about 4.5–7.5 centimeters (1.7–3 inches) wide, with a tapered point and double edge (fig. 4.11). The guard of these weapons was slightly concave and quite narrow, only slightly protruding beyond the width of the blade. Pommels were usually parallel to the guard and flattened. A good example of swords of this general type comes from beyond Roman borders in the north, from the Nydam Mose burials in southern Jutland (250–550), where a cache of blades exhibits these traits and was possibly even made by Roman smiths. By the later sixth century, this spatha type had been replaced by the so-called Herul sword ( _spatha Heruliska_ ), named for the Germanic barbarian mercenary allies of the Justinianic era. Although the exact form of this weapon is uncertain, it likely fit into the broader family of Germanic swords found in burials throughout central and southeastern Europe that seem unremarkable. Rather than a morphological difference, the novelty in the Herul sword perhaps lay in the quality of its metal. There is the possibility that the Herul blades were forged using the Germanic techniques that yielded high-carbon, fine steel blades such as attested archaeologically in northern Germany at Heeten (a region close to the presumed homeland of the Heruls) rather than via Mediterranean traditional iron processing.
The find of a sword from Anatolia indicates probable Avar or Sasanian influence on some Byzantine swords of the later sixth and early seventh centuries. The weapon unearthed at Aphrodisias in central Asia Minor has a long (1.8 meters, ca. 6 feet) narrow, double-edged blade that tapers to a steep point. The utility of this long sword for cavalry is apparent, as are its piercing characteristics designed to defeat heavier armor.
By the tenth century, the curved saber was common. In one famous illustration, the emperor Nikephoros II Phokas is shown with a _paramerion_ , a thin, long blade that curves slightly upward to the tapered point. This paramerion saber had a wider guard than the late antique examples discussed above and is well-suited to the slashing and downward cuts of cavalry warfare conducted by the heavily armed horsemen ( _kataphraktoi_ ) that by Nikephoros's time formed the main offensive arm of the military. Straight, double-edged swords continued in use and Nikephoros ordered his kataphrakts to carry them, as the paramerion was prone to breaking in the thick of battle. The eleventh-century depiction (fig. 4.12) of the military martyr St. Bakchos in the Daphne Monastery in Greece shows the saint holding a straight blade _spathion_ with a wide guard and decorated scabbard similar to types used throughout the Mediterranean. By the later days of the empire, Theodore Palaiologos (1355–1407) described soldiers equipped with the _glaive_ , possibly a Frankish long sword (rather than a pole arm of the later western medieval era) because he mentions it might replace the sword ( _espee_ ). Alternatively, this may be a scribal error for _clava_ , the mace, which was a common soldier's weapon from at least the tenth century onward.
The mace was long used in the ancient Mediterranean—it is mentioned in the _Iliad_ —and such a simple weapon was probably widely distributed, since the mace is simply an improved war club fitted with a symmetrical metal head. The latter is frequently studded to make easier the smashing of armor, bone, and soft tissue. The Romans of the early imperial centuries do not seem to have used it, probably because they only rarely encountered heavily armored enemies. The Sasanians particularly favored the mace, especially for their heavy cavalry, who wielded it with great effect; the Romans probably adopted it amid their confrontations with the Persians. A seventh-century source mentions the maces carried by the bodyguard of the emperor Maurice. In the ninth century, the emperor Basil I was proficient in the use of the mace; he is said to have shattered the legs of a bounding deer with a well-placed throw, a feat he repeated against a wolf. The war club ( _rabdion_ ) was the favorite weapon of the mythic hero, Digenis Akrites, the "two-blooded border lord" of Byzantine medieval epic; with this weapon Digenis overcame the Arab emir who opposed him. The late ninth-early eleventh-century military handbooks of Leo VI, Nikephoros II, and Nikephoros Ouranos demonstrate that the mace made entirely of iron had become part of the regular weaponry of both infantry and cavalry, carried sometimes as a primary battle weapon and sometimes in addition to the sword and spear. Numerous examples of knobbed mace heads from the Balkans confirm that by the middle and later periods of the empire its use was ubiquitous.
Another weapon that gained in popularity was the battle axe. The weapon also was known to the Greeks from Homeric times, although it was not used by the Romans of the Republic or early empire. The Germanic peoples whom the Romans contacted used war axes, primarily single-bit types for close combat or throwing, such as the Frankish _francisca_ , an elegant S-curved blade that tumbled when it struck the ground and skipped at the legs and lower body of its targets. Prokopios described the _francisca_ as able to shatter shields and kill the soldier taking cover behind it. The _Praecepta militaria_ of Nikephoros II attests to infantry carrying the axe ( _tzikourion_ ) as a primary weapon. The Byzantine battle axe was a one-handed, single-bladed infantry weapon with the blade backed by a hammer or spike. These were wielded as primary weapons on the line; the manuals show that tactically armies were a mix of "best weapon" men, who used the tool most suited to their training, experience, and physique. The presence of axe-men among the armies of the tenth and eleventh centuries may also indicate a high number of Scandinavian mercenaries in imperial service. The Vikings and their Rus' descendants, who favored axes, were prominent in the Varangians Guards. These men were famous for their use of the _pelekys_ , the two-handed, double-headed axe that they wielded with great effect in battles around the Mediterranean world.
Missile Weapons
Javelins were a main weapon of light infantry throughout the history of the empire. Vegetius describes two types, a heavy version called the _speculum_ , the replacement of the old _pilum_ , measured 1.7 meters (about 5.5 feet) and carried a 22 centimeter (9 inch) tip. A lighter version was just over 1 meter (3.5 feet) with a 12.7 centimeter (5 inch) head; this weapon was called alternatively _vericulum, verutum_ , or _beretta_. In the sixth and seventh centuries, there were apparently multiple javelin types carried including short Moorish varieties. These were throwing weapons carried by light troops and sometimes by forces unskilled with the bow. By the tenth century, javelins had evolved somewhat into longer, lighter weapons of more than 2 meters, with socketed heads. The Rus' were apparently especially proficient with the javelin, which seems to have been their primary missile weapon.
Darts ( _martzobarboula_ ), with a cast-lead weight on a fletched shaft, were another secondary weapon. Vegetius described palatine legions under Diocletian as especially proficient in the use of the dart; they carried five apiece in slots on their shields. "If soldiers throw them at the right moment, it seems as if shield-bearing infantry were almost to imitate the role of archers"; the range, he states, was similar to that of a javelin. In the _Strategikon_ , darts were to be used by both front-line heavy infantry phalangites and lighter skirmishing troops.
One prominent but overlooked weapon, mentioned frequently from Vegetius's day to the end of the empire, is the sling. The sling is cheap to produce, as it consists of a simple leather thong with central pouch in which is placed a smooth stone or lead projectile weighing 50–75 grams (1.8–2.6 oz). The sling is a humble weapon, its use commonly a skill acquired by shepherds and other rustics, and thus a weapon of the poor. The slinger's range was quite good—modern slingers have a cast of over 400 meters, a distance that rivals or surpasses that of archers. It is therefore interesting that the author of the _Strategikon_ recommended both heavy infantry and light infantry carry slings, which shows something of a combined-arms approach gaining prevalence in the Maurician "new model army" of the later sixth and early seventh centuries. This perhaps also represents an effort to continue or revive older legionary training practices mentioned in Vegetius, who recommended that recruits be trained to use the sling. The tenth-century military handbook _Sylloge Tacticorum_ mentions the use of staff-slings ( _sphendone/sphendobola_ ) 1.4 meters in length (about 4.5 feet). This weapon, which had a pouch affixed to the end of a staff from which rocks and other missiles could be hurled, is attested by Vegetius with the Latin name _fustibalus_. It had a length of about 1.25 meters (ca. 4 feet). Staff slings allowed one to cast heavier bullets or grenades and needed little space to operate, which made them useful on board ships or in sieges.
In numerous instances the tactics of the Justinianic era reflect the prominent place of archery in the warfare of the day. Belisarios relied heavily on native Roman and foreign allied horse archers. Since the arrival of the Huns in the fifth century, the Romans had clearly developed an appreciation for the utility of mounted missile troops and recruited as well as trained their own. It was not the Huns who introduced the composite bow to the Mediterranean and Near East, nor was it simply the Huns' proficiency with that weapon that caused such headaches for opponents. What posed such tactical and strategic challenges for the sedentary empires of Byzantium and Sasanian Persia was the Hunnic proficiency as horse archer swarms; the mobility of the horse and the striking power and speed of delivery of arrow fire from their composite bows were nightmares for Mediterranean and Near Eastern armies.
The bow used by the Romans in the sixth century was a composite, Hun-type bow with projecting ears and built up of wood pieces backed by sinew and horn glued together (fig. 4.13). The draw weights of such bows obviously varied from one piece to another—though this could be controlled reasonably well due to the precision making of these pieces—but 80 lb (36 kg) draws for a composite horse bow is a good estimate. While maximum flight distance may have been 300 meters, effective range against an armored target was within 100 meters. The Byzantines seem to have favored the Mongolian release, which used the thumb with a thumb ring and index and middle fingers to draw the bow. In the sixth century, the Byzantines drew their bowstring to the ear and thus packed considerable force and outranged Persian weaponry. This bow remained relatively unchanged until the end of the empire. A good example is the eighth-century depiction of a Byzantine archer at Khirbat al-Mafjar. The Byzantines relied heavily on archers in the expansion of the middle period—light-armed horse archers as well as kataphraktoi wielded them, and one quarter of the infantry were typically bowmen. The bow and arrow was thus the most important projectile weapon and, along with the sword and spear, of critical importance to the success of imperial forces.
While the crossbow was known to the Romans, its use seems to have been limited and faded away in the east; only after the Fourth Crusade, and the heavy Frankish influence on Byzantine arms, do crossbowmen appear in Byzantine service. Although the Greek word _solenarion_ may have meant a crossbow, it more probably implied an arrow-tube used for shooting short, heavy darts over distances greater than those achieved by conventional archery. The solenarion seems to have been a Byzantine invention from around the time of Maurice—it is mentioned in the _Strategikon_ and in the manuals through the tenth century.
Artillery
The Romans used a variety of artillery in both attack and defense, and their medieval successors continued to employ a number of different missile projecting machinery. A passage from a law of Justinian indicates that the Romans regularly defended cities with war machines:
We also desire that those who are called ballistarii, and whom We have stationed in different cities, and authorized to manufacture weapons, shall only repair and place in good condition those belonging to the government, which are deposited in the public arsenals of each town. Where any workmen have manufactured arms they must surrender them to the ballistarii, to be placed with those belonging to the public, but they must by no means sell them to anyone else.
Ballistarii were, of course, men who operated the _ballista_ , which here probably implies artillery generally. The text shows the emperor's concern not only with the manufacturing of artillery, but informs us that cities typically had such heavy weapons stored in public arsenals and trained crews to operate them. The ballista (fig. 4.14) is a torsion-powered bolt and stone thrower that resembled a heavy framed crossbow with vertical springs at each end consisting of twisted fibers; the operators drew the bow arms with a winch and ratchet. There were several varieties and sizes, built to cast bolts 77 centimeters (2.5 feet) long with 200 gram (7 oz.) heads or stones weighing from 2.5 to 40 kilograms (5.5 to 88 lb.) with an optimal range of 100–170 meters, though they could cast projectiles as much as 450 meters. Late Roman examples of iron-framed ballistae are known from the early second century onward and Belisarios's men used iron-framed ballistae during the Gothic campaigns. The small _cheiroballistra_ was apparently a hand-held torsion-powered bolt and stone thrower used by the Romans, but it is doubtful that it continued in use beyond the Arab conquests. The Romans did not seem to make use of torsion weapons beyond the sixth century.
In the fourth century, Ammianus Marcellinus described the torsion-powered _onager_ , a catapult (fig. 4.15) with a vertical arm drawn back using a windlass. It was a difficult weapon to manufacture, maintain, and use on the battlefield, although when working properly and with a skilled crew it could launch projectiles on average weighing around 32 kilograms (70 lbs). The onager was replaced in the Byzantine military arsenal with the traction trebuchet (fig. 4.16), a machine built on an upside down Λ-shaped timber frame with a cross beam at the top; a horizontal axle also at the top of the frame received the end of a pole whose unanchored end had a sling fitted to it. Using straps attached to the center axle, a crew of men hove on the lines and once the arm snapped forward, the sling released and launched the projectile. These traction trebuchets were simple and easy to construct and maintain. They were equal or superior in effectiveness to ancient torsion powered examples. The seventh-century _Miracles of St. Demetrios_ records details of the Avar siege of the city in 597 in which they assembled fifty trebuchets:
These trebuchets had quadrilateral [trusses] that were wider at the base and became progressively narrower toward the top. Attached to these machines were thick axles plated with iron at the ends, and there were nailed to them pieces of timber like the beams of a large house. Hanging from the back side of these pieces of timber were slings and from the front strong ropes, by which, pulling down and releasing the sling, they propel the stones up high and with a loud noise. And on being discharged they sent up many great stones so that neither earth nor human constructions could withstand the impacts.
The Byzantines also developed the counterweight trebuchet, which we shall discuss further in Chapter 7.
The above thumbnail sketch of Byzantine logistics shows that military officials faced considerable challenges in supplying and maintaining both static and campaign forces but in their mobilization, fielding, and sustaining defensive and offensive campaigns over centuries of near-continuous warfare, Roman logistics personnel achieved a remarkable record. Massive supply failures do not really appear in the sources, and given a culture that seemed to embrace the lessons taught by failure, we should expect to see these had they been a common occurrence. The provisioning and equipping of troops was always a state and private endeavor, with one featuring more prominently than the other depending on the era and the soldier's status and location. The equipment that the state issued or the soldier bought was nothing like a modern uniform—in its appearance the Byzantine armies of any age would have presented a far more colorful and varied aspect to an observer who would have seen many different forms of dress, armor, helmets, and weaponry represented in the ranks. Partly this was due to expediency—the state certainly would have struggled to maintain the depth of supply and uniformity of stocks required at all times, and it was partly due to simple adaptability to the realities that soldiers could be drilled without one standard-issue weapon.
More interesting is that during the time of Maurice, but especially in the ninth and tenth centuries period of reconquest, while one's unit affiliation of heavy cavalry, lancer, or light or heavy infantry was clear, one's tactical role was extremely flexible. Cavalry might be called to dismount and fight like infantry, and infantry forces served both offensive and defensive roles. This flexibility and training that clearly backed it is one prime reason for the success of Byzantine arms.
## FIVE
## STRATEGY AND TACTICS
STRATEGY
The _Oxford English Dictionary_ defines "strategy" thus: a. The art of a commander-in-chief; the art of projecting and directing the larger military movements and operations of a campaign; b. An instance or species of this.
Said the emperor Leo VI: Tactics is the science of movement in warfare...Tactics is the military skill [that is concerned with] battle formations, armament, and troop movements.... Strategy is how good commanders put their military training in practice, their drilling with strategems, and putting together ways of defeating [the enemy].
For Leo, strategy is the application of theory and practice in adapting to the exigencies of warfare. The Greek word _strategia_ , from _strategos_ , the general, implies the military art, military wisdom—the art of war. It is in this all-encompassing sense, covering what modern thinkers conceive of as strategy, operations, tactics, logistics, and geography that Byzantine military planners understood the term. We cannot therefore see clear dividing lines between strategy and tactics, nor to expect the Byzantines to understand geography and the organization of war as each forming separate disciplines. Such compartmentalization, to use the present expression, belongs to the era of modern total war as practiced from the nineteenth century onward.
As Haldon has noted, two pillars of Byzantine military doctrine—even if never expressed in explicit fashion—are known from both statements of imperial ideology and the actual waging of warfare throughout the centuries. Emperors were protectors of the Christian people first and foremost, and while they claimed to rule the universe (Greek _oikumene_ , roughly, the inhabited world) these ideological statements were limited by the belief that the "universe" meant the Roman world of the high empire, or in its most ambitious embrace, the Christian world. When Constantine claimed to be protector of the Christians of Persia, he was making fundamental claims to the dominance of the new religion and the extension of imperial beneficence to those who followed the imperial faith. Such grandiose claims were not pursued in lands traditionally outside of direct Roman political control—the Romans did not wage wars of conquest in Persia, for example, to yoke all Christians under their authority. But the emperors did vigorously pursue the maintenance and recovery of territory in the Christian west and Islamic east that had formerly been ruled by the Roman state, and emperors still considered themselves the only divinely selected universal rulers. All wars were defensive. Even offensive campaigns were considered defensive, in that they aimed to recover land that had been seized from the empire and rightfully belonged to it, and this notion of the "forward defense" or "active defense" was something that the Romans probably imparted to Muslim _jihad_ theorists. Even after Charlemagne was crowned "emperor" on Christmas Day 800, the Byzantines operated under the belief that the real Roman emperor resided in Constantinople; the Franks and all other comers were interlopers, inferior culturally and politically. All of this imperial panache was tempered, however, by a generally clear understanding of the actual military capabilities of the empire. Rarely did Byzantine strategists overreach in their attempts to regain territory or in the defense of their core areas of the Balkans and Anatolia.
One notable example that illustrates the practice inspired by the ideological pillars of aggressive action to recover lost territory and the emperor's role as a Christian was the eastern offensive of Justinian II of the 680s and 690s. Justinian's thrust east was partly buoyed by his success over the Balkan Slavs, tens of thousands of whom he pressed into the army. But the emperor was more than a little inspired by an apocalyptic mood that had seized a considerable portion of the Christian population of the empire and beyond, a mood inspired by the new taxation and Arabization efforts of the caliphate that particularly ruffled the Arabs' Christian subjects. Justinian seized the moment, issuing coins with the inflammatory image of Christ on the obverse as tribute that were expected to circulate in the Muslim empire and in the process incensing the caliph 'Abd al-Malik (685–705). The emperor may even have believed he was a divinely anointed figure, as his ancestor Heraclius had proved against the Sasanians. Whatever the specifics, there were compelling psychological, material, and strategic reasons to pursue what was, in hindsight, a series of debacles that ultimately cost Justinian his throne.
The Byzantines often went to great lengths to avoid armed confrontation. While many empires relied heavily on nonmilitary dealings as a primary tool in advancing self-interest, Byzantine diplomacy was perhaps the deftest in history. This was at least partly due to the fact that through most of its existence, the Byzantines did not have the vast resources at their disposal, a fact which made diplomatic action a logical first response and stands in sharp contrast with Roman ideological expressions of eternal victory and overwhelming force that was rarely achieved.
Byzantine diplomacy encompassed many aims; through exchanges of embassies the emperor forged alliances, gathered intelligence, managed clients, or attempted to negotiate peace. In the embassy of the Byzantine courtier Priskos to the Huns in 449, members of the embassy had a more sinister purpose: to assassinate Attila. Usually, though, Byzantine diplomacy used enticements; rich silks, silver services, plate, embroidered cloths, gifts of imperial rank along with the insignia and cash salaries. These tangible inducements demonstrated the wealth of the emperor and enticed many foreigners to imperial service and kept others neutral. The age-old adage "the enemy of my enemy is my friend" was keenly followed and exemplified by an endless scouring for potential alliances which could threaten the flanks and rear of existing enemies. Heraclius's alliance with the Western Turks brought the Sasanians to heel during the war of the 620s, and Byzantine payments to the Rus' and Pechenegs were used in the ninth and tenth centuries against the Magyars and Bulgars who menaced the empire.
Embassies also gathered information and bribed men to the emperor's side. In the ninth century the Arab Samonas, the right-hand man of the emperor Leo VI, used contacts inside the caliphate to ruin the pretender to the throne, Andronikos Doukas. On another occasion, during the regency period of the child Constantine VII's reign, a Byzantine agent in the exchequer of the caliphate, a Greek "deserter" Nicholas, informed Constantinople that Andronikos Doukas's son, Constantine, planned a revolt which would fail because the Arab authorities would not support it.
The sophisticated diplomacy and defensive posture of the empire contrasts with the marked military expansion of the Macedonian era (867–1056). But to view Macedonian actions in isolation from earlier and later efforts is misguided; even in the dim twilight of the seventh century the emperors waged numerous campaigns to recapture lost territory or, barring this, destroy enemy capabilities along the frontier. Likewise, though they are often criticized by modern scholars for their failures to hold onto or reclaim lands in Asia Minor from the Turks, the Komnenoi expended considerable resources in the region and campaigned frequently there. Clearly by the eleventh century, the Byzantine doctrine of "protect and survive" as Haldon has described it, was anchored on the desire to preserve the richest provinces now forming the imperial center—Constantinople and the Balkan hinterland—from enemy ravage and conquest. Imperial intentions never changed but what did alter was the empire's ability to make and sustain territorial gains.
While there were certainly permanent bureaucracies and institutions in the empire that imparted a mandarin flavor to administration and politics, we know of no standing central command responsible for long-term strategic planning. Rather, individual emperors and their commanders and bureaucrats responsible for supply dealt with military situations as they arose. Any discussion of overall strategy of the empire must be recognized to be our own imposition; the Romans themselves never articulated (or if they did it is lost to us) a comprehensive form of long-term war planning envisioning specific scenarios and reactions intended to preserve their borders for the existence of the empire. It can be argued that such actions were more organic and reflexive, though, and it is certain that a body of practices emerged from the fourth century onward whose general application could be termed strategy. The Romans operated under several fundamental doctrines which can be gleaned from the military handbooks, histories, and writings of emperors over the centuries. As in all historical studies that attempt to sketch more than a millennium of history, we are bound to flatten distinctions and differences that may alter the picture in its particulars at any one moment in time. But, as Luttwak has recently argued, in aggregate these fundamental tenets can be taken to comprise something we may call today a "Grand Strategy" these were deeply embedded doctrines and practices that were passed from one generation of leader to another and subject to minor modification.
Just what, then, were the fundamentals of Byzantine war preparedness and practice? From the historical documents we can discern values and actions stressed by Byzantine authors throughout the centuries.
If You Desire Peace, Prepare for War
This maxim of Vegetius in the fifth century was followed throughout the empire's history. The _Strategikon_ urged leaders to always expect conflict, a sentiment repeated centuries later by the emperor Leo VI:
Always be vigilant and alert against confrontation with the enemy. Do not let a period when hostilities have ceased lull you into a period of carelessness. Do not become negligent before the conclusion of firm peace. Always be on guard against the machinations of the enemy. Be careful and watch out for their unfaithfulness. After you have been injured, regret is not of much help.
The handbooks stress constant training, not allowing the soldiers to be idle, enforcing discipline, looking after supply stores, and paying meticulous attention to the mood and equipment of the troops on which success depended. Both the author of the _Strategikon_ and Leo VI advised that households should possess a bow and at least forty arrows and that men should practice archery throughout the year in preparation for conflict, though we do not know how widely these desires reflected reality.
From the fourth century onward, foreigners attacked the empire with astonishing frequency. The military costs borne by the state were immense; by far the bulk of the budget was spent on maintaining, training, and equipping the troops and fleet. It is thus no exaggeration to view Byzantine society as one geared first and foremost to the defense of its territory. Experience taught the emperors that any period of peace was fleeting; never did this come into such sharp clarity more than in the events of the late 620s and 630s, when Heraclius found himself at the top of the wheel of fortune with his victories over the Persians, symbolized by his triumphant entry into Jerusalem in a spirit of millennial jubilation. The wheel turned, however, and within a decade Arab forces seized the whole of the Levant. Such episodes, and countless others, reinforced among the Byzantine elite that there was no shortage of enemies and that one defeated foe would soon be replaced with another. Therefore the state not only maintained standing, permanent forces, but also attempted to maintain and control production and stocks of arms and materiel. The imperial bureaucracy drew on centuries of experience in the needs of the soldiery in material resources and logistical support on campaign. Although we take written knowledge for granted, the vast collection of state archives in the capital provided the Byzantines an immense advantage over most of their neighbors. Detailed intelligence briefs, battlefield reports, and accounts accumulated over centuries of imperial expeditions provided information on most military challenges that confronted the authorities.
The collecting of ancient military writers, much of whose work was adopted in the military treatises of the seventh to eleventh centuries, demonstrates the importance of literacy in maintaining imperial defense. Moreover, beyond the assumption of the army as an institution implicit in their production, these handbooks show a concern for standardization and replication in military experience. They also demonstrate a keen self-awareness in terms of the cost of warfare and the limits of Byzantine material resources. Thus, as the handbooks suggest, the Byzantine army was trained, drilled, and perpetually prepared for conflicts that always arose.
War Was not Merely Material but also Spiritual and Psychological
The Byzantines regarded themselves as inheritors of a Christian empire, guarded by God and the saints, and their state destined to survive until the end of days. Since the reign of Constantine I, the Romans had understood that the universe was ordered according to the principles of Christianity and the world was a reflection of the unseen cosmos: one God, one faith, one emperor, one empire. Christianity was a vital ingredient in the understanding of the place of the empire in the world, and of the individual in action. War was a sinful space in which activities normally considered impious, such as deceit, were acceptable. Such actions as aggression and duplicity were not exculpated by war, though, and soldiers were enjoined to remain always pious and prayerful so that God would favor the Roman cause. Commanders were urged to purify themselves before campaigns and battles, and to always maintain their piety, as in this directive from the _Taktika_ :
It is necessary to worship the Divinity at all times. Especially, O general, should you offer worship when you plan to enter upon the dangers of war. If at that time, you genuinely worship God, then, when the time is full of terror, you will be confident that you can offer your prayers to him as to a friend and you can seek your salvation with utter confidence.
Especially in conflicts with the Persians and Arabs, when Heraclius cast himself as the biblical David—the weak shepherd boy following the moral right, against the physically superior but heathen Goliath—the psychology of conflict was tinged with Christian notions of just war and its moral virtues. Priests accompanied armies, holy icons served as battle standards, and the _kyrie eleison_ (Lord have mercy) was a battle cry for the army in the seventh century and later.
The Byzantines clearly understood the psychological nature of warfare and the use of psychological means to manage one's forces and to undermine the enemy and there are numerous examples in the handbooks: "The bodies of soldiers who have been killed in battle are sacred, especially those who have been most valiant in the fight on behalf of Christians. By all means, it is necessary to honor them reverently and to dignify them with burial and eternal memory."
To those who observe them, the practice of commemorating the fallen is a powerful reminder of communal bonds, a key ingredient of identity, and a strong inducement to collective action. Just as the early saints of the empire were killed by infidels, the soldiers who perished against the barbarians bore a heroic example to the Christians of the empire. The morale of men was to be safeguarded through provision of adequate provender, equipment, and especially by the careful scrutiny of their needs by commanders. Leo stated, "When you do not provide your army with necessity of supplies and food, even without the enemy attacking, you have been defeated." The dead were to be buried ceremoniously, but at night, so that soldiers could be assured of respect for their sacrifice but to conceal the numbers of dead from the enemy. The wounded were cared for and their comfort and morale overseen by commanders who were instructed to visit them. Families and friends were to be placed in units together in camp and on the battle line, so that the emotional bonds that tied them checked their fear and sparked the desire to protect. Signs of fear were to be observed by careful commanders, and bad omens shrugged off or spun as good signs in order to maintain and elevate morale. These remarkable measures show a clear understanding of the mental and emotional toll of warfare and the dangers of low morale.
Subterfuge, bribery, and disinformation were prized bloodless means to undermine or dissolve enemies and were always preferred to open battle. The military manuals instruct, whenever possible, to bribe enemy commanders. Before campaigns on the frontiers, the general Nikephoros Ouranos (ca. 950–1011) ordered that gifts be sent to the emirs along the border in order for the bearers to collect intelligence and possibly induce the enemy to the Byzantine side or at least inaction in the coming conflict. Sowing dissent within populations under siege was a standard tactic:
You must make this announcement to the fortress, that "all the _Magaritai_ (probably Arabic muhajirun = refugees), Armenians [Christians], and Syrians [Jacobite Christians] in this fortress who do not cross over to us before the fortress is taken will be beheaded." These are the things you will proclaim first to those within the fortress, for it causes disagreement and dissension among them.
Even though imperial officials took war for granted as part of the sinful condition of fallen man, they usually went to great lengths to avoid it. In addition there were strenuous diplomatic efforts, like the payments of thousands of pounds of gold to the Persians and Avars throughout the sixth century, that the Romans made to ensure neighbors did not attack. Though often a fruitless exercise, given the wealth of Byzantine society relative to many of their neighbors and the militant nature of most societies on their borders, emperors paid bribes and subsidies to foreigners to keep them from waging war. Gifts were critical to imperial prestige, to demonstrate the superiority of the empire, and to forge ties with outsiders.
As Maurice's _Strategikon_ observed, the cause of war must be just. Just war was by definition defensive. On those occasions where the Byzantines pursued what modern observers would call offensive warfare, their view was that these were conducted to recover lost territory that legally belonged to the empire, or to punish enemies and thereby discourage them from further attack. Examples of the former are Justinian's western campaigns in Africa and Italy and the expansion of imperial boundaries in Bulgaria, Syria-Mesopotamia, and Armenia in the ninth and tenth centuries. A good, though failed, example of the latter case is Manuel I Komnenos's 1155–58 campaigns in Italy which, while they restored suzerainty over a sliver of former Byzantine lands, their main purpose was to punish the Sicilians for their 1147 attack on the Balkans. In the series of conflicts that pitted the empire against Persia during the early period, war aims were always limited to attaining a favorable negotiated settlement. For instance, in 578, in response to Persian invasions in Mesopotamia, Maurice, then magister militum, led a Byzantine raid in force that was retaliatory but aimed to bolster Roman positions in the peace talks to follow. Later, the Macedonian emperors accepted the submission of Arab and Caucasian princes along the eastern marches rather than attack them, and the Komnenoi, especially the emperor John II, took seriously their treaty obligations as overlords of the Latin crusader state of Antioch, despite the duplicity the latter often exhibited. Throughout the history of the empire, when an enemy wished to negotiate, the Byzantines were quick to oblige them: "When the enemy, after God has granted you victory, should seek terms of peace, do not be rigid, but listen graciously to them and make peace. Keep in mind the uncertainties of war and of fortune."
Seek Allies in Conflict and Turn Enemies into Allies
Time and again the Byzantines turned to neighbors who were neutral or well-disposed at the start of conflict to oppose invading forces. In addition to the recruitment of steppe archers and barbarian cavalry and infantry for the campaigns of the Justinianic era, the Byzantines created a network of dependents among frontier peoples who offered additional manpower and specialist troops in wartime. The Ghassanid Arabs were among the most important and effective clients in the eastern regions, and in the west the Muslim conquest of Africa shows that the Roman-Berber alliances forged there from Justinian's day onward continued to function; imperial titles, gold, and weapons flowed to the Berber confederations who provided manpower for the defense of the North African provinces. The arrangements that Heraclius made with the Western Turk Empire against the Sasanians tipped the scales of power in favor of the empire and confronted the Persians with the possibility of war on multiple fronts that their crumbling political structure could not sustain. In facing the Arabs, the intervention of the Bulgars on behalf of the empire during the siege of 717 was decisive; Bulgarian arms constantly harassed the Muslim encampment and added to the misery of the long winter of the siege. The Byzantines also turned to the Khazar khanate on the southern Russian steppes to pressure the Muslims from the north, and this buffer yielded considerable advantages in the seventh and eighth centuries when Justinian II and Constantine V married Khazar princesses and forged personal bonds with the khagan.
The litany of northern peoples who invaded the empire and were defeated in turn and settled on imperial lands is extensive. The Goths in the fourth century were among those who crossed the Danube and were provided with land; some of them maintained distinctive communities from which soldiers were drawn centuries later. Throughout the Dark Ages, Slavic groups were defeated in the Balkans and drafted into the army or settled in Anatolia. The Pechenegs defeated by Alexios I were settled in Thrace and likewise provided recruits, as did the Cumans after them. In the wake of the First Crusade Alexios also tried, unsuccessfully, through lavish gifts of goods and gold to turn the fractious Norman lord of Edessa, Tancred, into a vassal. The settlement of refugees, such as the Alans from the steppes of the north or refugees from lands formerly belonging to the territory, and recruiting their capable men continued into the Palaiologan era.
The quest for alliances to blunt the asymmetry of forces ranged against them remained a constant until the crumbling of the empire; in 1282 in the last great coup of Byzantine diplomacy, Michael VIII, persuaded the Aragonese King Pedro III (1276–85) to intervene in Sicily against the French, who like their Norman predecessors threatened invasion of the empire. Michael VIII also married his daughter to the Nogai khan of the Golden Horde, who in 1282 provided the emperor with 4,000 Mongols for the invasion of Serbia.
Fight Attritive Wars
In the sixth century, no political group bordering the empire possessed a professional standing army. Apart from Persia, no neighbor could even claim to be a state with a developed bureaucracy and other machinery of government. With rare exception, the lack of standing armies in kingdoms and political groupings remained true until the fall of Byzantium. Unfortunately, this made adjacent powers only slightly less dangerous, as Byzantine forces were not in a position to pursue the use of aggressive, blunt force. Only for a brief moment of its history during the fifty years or so from the accession of Nikephoros II Phokas to the death of Basil II did the empire possess the combination of wealth, skilled commanders, and veteran forces to engage in frontal attacks against strong enemies—the direct assault of Nikephoros Phokas and his immediate successors in Syria broke with some of the cautious and patient strategies of the past. The presence of professional, standing forces in and around the capital and in the themes by the ninth century made it possible for the Byzantines to respond fairly rapidly to threats from without and to wear down an enemy by harassment and clever use of geography. In the histories and military handbooks the conduct of war was consistently described and practices repeated; these habits were germane to the Byzantine art of war, even if they were not seen by contemporaries as part of a grand strategy.
By the later sixth century, strategic realities thrust the empire into a defensive posture. Twenty years after the death of Justinian, his far-flung conquests were under attack on multiple fronts—Germanic groups menaced their scattered holdings in Spain and Italy, while the Avars pressed the Balkan frontier and the Persians waged war against the Armenian and Syrian borders. Despite the considerable wealth and men at their disposal, the Romans lacked the resources to contain every threat and their losses mounted. The collapse of the eastern _limes_ (frontier zone) under Persian assault in the 610s destroyed the defensive integrity of the eastern marches and lay open the route across Asia Minor and the capital. After the seventh century, maintaining hard frontiers was not possible and the Byzantines understood that warfare, fluid by its nature, required a set of flexible responses. Among these was knowledge of the paucity of forces, which could only rarely be tested in full-scale engagements; said the author of the _Strategikon_ : "Well aware of our weakness, we have been motivated solely by devotion to the nation." It was this recognition of the structural limitations of the military that prompted one commentator, obviously weary of the manifold warnings of an anonymous Byzantine writer, to observe:"He has a distinctively defensive mind, and sees clearly what the enemy may do to him more than he has time to think of what he may do to the enemy." Unlike most peoples, the Byzantines prepared for the eventuality of battlefield defeat; these were both inevitable and the fault of the commanding officer who had been prepared in the methodical, cautious, and proper way to conduct war. The cautious, defensive tone changes only during the tenth century, when the work _On Skirmishing_ , attributed to Nikephoros II Phokas, expresses great confidence in confronting and overwhelming the enemy. This was the work of an experienced commander who in his lifetime had witnessed Byzantine military resurgence in the east.
_On Skirmishing_ is not only one of the most interesting works of Byzantine history, it ranks as one of the most interesting works on guerrilla tactics ever written. In it the adaptation to the small war, the raid, skirmish, and running battle, is complete. _On Skirmishing_ depicts a method of warfare generations in the making, in which Arab raiders penetrated the highlands of Asia Minor in annual forays to enslave the inhabitants and rustle cattle. The Byzantine theme strategoi called up their troops, watched the passes, evacuated the inhabitants, and shadowed enemy forces. Night attacks, ambushes, assaults on encampments, and surprise movements are described in detail. In many instances, the thematic soldiers strike raiding columns that have already plundered their targets and were plodding back to Syria loaded with booty. The fatigue and slow pace of the encumbered columns made the _ghazis_ vulnerable to the hit-and-run tactics of the thematic light cavalry. Such a defense was born of failure; their employment is an admission that the empire was not strong enough to stop enemy columns from entering their lands. But the methods that _On Skirmishing_ shows were honed to perfection. Ultimately, as the caliphate fractured into disparate political entities, the raiders of the themes turned to the offensive and from the small war to a full onslaught against the border emirates. Envisioning a confrontation with a strong enemy force in Syria, the author of _On Skirmishing_ describes how the vanguard of a divided column would attack the Muslim lines near sunset,
then charge against them, and by the favor of Christ you will be able to defeat them. But if the enemy commanders present there have a large force, they may be able to hold their ground and will struggle to come back from defeat, which is impossible, for with night already falling nothing untoward will happen to you. If, therefore, you do things in this manner, the enemy will be amazed and terrified of you, and they will not dare to ride from their army without food. Finally, the lack of food will force them to return to their own country.
Defense in Depth
From the period of the reorganization of forces under Diocletian and Constantine, the empire's strategic footing indicates that imperial authorities understood not only their own defensive and logistical challenges, but the obstacles that invasion of their territory posed to enemies. Although we need not see an integrated and centrally planned and managed static line of defense on the order of massive modern linear fortifications, from the fourth to sixth centuries the frontier zones were studded with fortresses and fortress cities with substantial garrisons; strategic routes were guarded, and storehouses supplied armies near the frontier. Certainly the Romans grasped strategic geography; following the conquest of Africa, Justinian's surveyors and military architects conceived of a series of strong points and fortified cities that stretched along the rich corn lands at the foot of the untamed Aures Mountains, home to unvanquished and warlike Berbers; the result was a fortified military road that could serve commerce and be patrolled and defended. Heraclius made his headquarters at the well-placed city of Caesarea of Cappadocia, which controlled routes to the east and south of the Anatolian plateau as well as providing many of the raw materials vital to his war effort against the Persians. The Balkan and Anatolian landscapes of the Dark Ages were studded with watchtowers, fortresses with at least modest garrisons, and refuges where threatened civilian populations took refuge from attackers. Much later John II Komnenos pursued the strategy of gradual recovery of lands in Asia Minor by steady advancement of one fortified post to another, a strategy which, though it failed, showed a clear awareness of strategic geography and the role of fortified cities in Byzantine military planning. Until 1204 the sheer size of the empire exposed the enemy to risk; in the north the Danube was used to land troops to the rear of enemy columns, and the Anatolian passes provided natural barriers and ambush points from which to attack eastern enemies.
Fight Small Wars
Large pitched engagements rarely favored the empire, and even when imperial forces were superior in numbers and arms, chance had too much of a factor in the outcome. Decisive battle in which the field forces were put at risk in one stroke were increasingly avoided after the collapse of resistance to the Muslims during the first decades of the seventh century. Instead, small-scale regional warfare, containment, and raiding were favored in order to punish the enemy and keep them off balance. Managing the theater of action was a vital component; the Byzantines repeatedly had to stall invaders on one front in holding actions while making peace with or dispatching other invaders in order to free up resources required for effective action. In their understanding of their own material weakness and the relative strengths of their opponents, the Byzantines show great pragmatism in the prosecution of warfare and patience exhibited by few states in history. Partly this was because of their perpetually threatened position and the limits of imperial resources and partly because of heavy defeats absorbed over the course of the fourth and fifth centuries. The Battle of Adrianople in 378, the 443 defeat of Roman forces at the hands of Attila, the 468 catastrophe dealt by the Vandals—all at great cost of blood and treasure—served to convince the authorities that payments of even heavy indemnities, such as those doled out to the Huns by the eunuch chamberlain Chrysaphius's government under Theodosius II, might be unpopular, but were often a better option than war. The policy of Heraclius to avoid massive confrontation in decisive battle persisted—the Battle of Yarmuk in 636 demonstrated a tactical failure in the local command, not a flaw in the strategy. Such losses merely reinforced the belief that warfare was best limited; the empire simply could not put its field armies at risk except on occasions when the potential damage to their population allowed no other option. Aided by successful defense of the capital in the 670s and 717, the Byzantines frequently negotiated long truces with the caliphate, despite the empire's relative weakness. Although imperial raids were launched to punish the Arabs for incursions into Byzantine territory in the border region under Constans II, and his successors did no lasting harm to the caliphate, nevertheless they showed the local populations that the empire intended to resist and could do so.
After the glaring failures of the mid- and late seventh century, Byzantine strategy continued earlier philosophies of general avoidance of decisive battle. With the vast bulk of their tax base sundered from imperial control and their army increasingly depleted through battlefield losses and lack of funds, the Byzantines were forced on the defensive. The creation of an Arab fleet under the Muslim governor of Syria, Mu'awiya (governor of Syria 640–61, caliph 661–80), posed an existential threat to Byzantine rule in Cyprus and the Aegean, already under Muslim attack, and threatened Constantinople itself. Constans was forced to engage in direct, massive naval action that ended in defeat of the Byzantine navy at the Battle of Phoenix (655). When civil war broke out in the caliphate, the Byzantines were spared further advances for a time and Constans probably was able to address the fiscal and military crisis that confronted him by settling his troops on the land in the provinces, thereby creating the rudiments of the thematic structure. Constans's departure to the west, where he arrived in 663 or 664, was an attempt to shore up the situation in Africa and Sicily. His relocation may indicate that the themes of Anatolia were already established sometime in the 650s, perhaps during the period of truce that existed between 655 and 661 when the Muslims were embroiled in civil war.
The creation of the first major themes and settlement of the army of the land blunted offensive capabilities and reinforced Byzantine reliance on the small war. Containment, harassment, and raiding governed warfare in the seventh to ninth centuries. _On Skirmishing_ portrays a world of constant raid and counterraid, of flying columns and stealth tactics all honed in more than two centuries of border conflict with the Muslims. Preservation of force and the ability to reply to enemy attack, if only in a symbolic way, were paramount. On those occasions when the Byzantines were forced to mass armies to defend targets whose loss was too heavy to contemplate—Constantinople or the thematic capital and major military base of Amorium in Phrygia—their record was mixed. The successful defense of the capital during the series of attacks of Mu'awiya's forces from 674–78 and those of 'Umar II in 717 highlighted Byzantine strengths: the powerful defenses of their capital city that was virtually impregnable, the adroit use of allies in 717, and the defensive depth that the sheer size of their empire, although much reduced, afforded them. In the disastrous invasion by the Arabs in 838, the Byzantine army collapsed in defeat and the key city of Amorium fell to the Muslims. On numerous other occasions, though, when forced to large-scale confrontation against the Arabs, Rus', Normans, Pechenegs, Cumans, and Magyars (Hungarians), imperial forces successfully held the line or scored victories that established long-lasting peace.
The admonition of the emperor Nikephoros II Phokas, although offered in a tactical context, was applied as strategic practices as well:
If the enemy force far outnumbers our own both in cavalry and infantry, avoid a general engagement or close combats and strive to injure the enemy with stratagems and ambushes. The time to seek general engagements with the enemy is when, with the help of God, the enemy has fled once, twice, or three times and are crippled and fearful.... Avoid not only an enemy force of superior strength, but also equal strength.
Divide and Conquer
Sowing division and attempting to dislodge elements of the enemy from opposing ranks was a standard Byzantine strategy. In war or peace, Byzantine diplomatic efforts sought allies from among potential enemy populations and recruited them either as imperial agents or into the military forces. Lulls in hostility were used to open negotiations with those prone to enticement, usually through material rewards. At the heat of the bitter conflict between Heraclius and the Sasanians, the emperor retained back-door communications with senior officers within the Persian hierarchy that eventually yielded the coup that overthrew Kosrow II. After the upheavals of the war, many prominent Sasanian commanders and their households joined Roman service. The emperor Theophilos negotiated for the services of the Khurramite Persian rebels against the 'Abbasids that for a time provided a boost to the emperor's military capabilities. His tampering in 'Abbasid internal affairs temporarily strengthened the emperor's hand and weakened his most dangerous enemy. In later eras, the Komnenoi used bribes of money, land, and offices with their attendant prestige and salary to bring renegade Romans back to the fold; during his wars against the Normans one of the keys to Alexios's ultimate success was his recruitment of the traitor Bryennios. In his 1122 campaign against the Pechenges, John II used the lull in fighting over the winter to bribe elements of the nomads to his side and thereby substantially weakened the opposing force. In the general engagement that followed at Berroia (Veria in Macedonia), Byzantine forces crushed the remainder.
Diplomatically the strategy of divide and conquer—or perhaps more aptly in this instance "divide and thwart"—was most spectacularly achieved late in the empire's history during the reign of Michael VIII, whose agents helped to foment revolt on the island of Sicily among the subjects of its king, Charles of Anjou (1266–85), who planned a powerful expedition against Byzantium. Charles had seized the kingdom of Sicily by conquest in 1266 and had papal backing for further expansion. Charles's success had come at the expense of Manfred of Sicily (1258–66) the last Hohenstaufen king of Sicily. Manfred's daughter Constance (d. 1302) married Peter III (d. 1285). Michael VIII thus found in Peter a natural ally, and following the liberal dispersal of Byzantine gold throughout the island, a revolt broke out in Palermo on March 30, 1282. Weeks later, a strong Aragonese fleet whose announced target was Muslim Tunis appeared off the western coast of Sicily. The episode of the so-called Sicilian Vespers rising (named for the prayer at sunset marking the start of the night vigil of the Easter Monday holiday) sparked war between Aragon and the French that lasted for twenty years and crushed Charles's designs on the empire.
ESPIONAGE AND INTELLIGENCE
Strategic Intelligence
Though they suffered their share of failure, the Byzantines often excelled at collecting intelligence and were generally superior to their enemies in espionage activities. From Diocletian onward the state maintained close to the emperor's person a cadre of secret police, _agentes in rebus_ , who watched over imperial officials and those suspected of treachery; such agents served as couriers and held passes that allowed them to freely use the imperial post system. These operatives conducted sensitive missions and embassies vital to the interest of the state until the seventh century, when officers under the _logothete_ of the dromos (head of the imperial post) assumed their duties.
Prior to the outbreak of hostilities, the Byzantines relied on a network of spies and scouts for advance warning of enemy intentions. Members of embassies dispatched to foreign lands and permanent "secret friends" of the emperor in the courts and entourages of neighbors passed information to Constantinople. In these exchanges, merchants played a vital role. Spies were ideally to live among the lower strata of society and lead unexceptional lives so as to blend in with the population; fluency in the enemy's language, but few family ties, were prerequisites for agents, many of whom were merchants who passed and received intelligence in the marketplace. On the eastern front, Muslim raiding armies gathered in August in the border emirates to wage jihad; at the start of the raiding season the Romans sent merchants across the passes. No doubt some of these were legitimate businessmen in imperial service while others were professional spies. These men visited the target population to collect news of enemy preparations, assess the mood of the enemy, and estimate the stores of materiel gathered and the quality and number of soldiers on campaign.
As noted previously, the Romans also maintained networks of spies throughout the courts of enemies and potential enemies. Peaceful contacts, such as trading and embassies, provided cover under which such agents could collect and pass on intelligence. At times, the Roman state was well informed of what was happening inside the political centers of their enemies. Justinian and his empress Theodora were abreast of the fast-moving developments inside the Ostrogothic kingdom of Italy—the emperor was quickly aware, for example, that his ally, Queen Amalasuntha, had been deposed and imprisoned not long after the event. Her removal and eventual murder provided the pretext on which the Romans would declare war. Agents of this type continued to be employed. In addition to the Bulgar nobles betrayed in 766 during the reign of Constantine V, Byzantine plants within the caliphate passed regular intelligence to imperial agents.
Battlefield Intelligence
Said Nikephoros II, "It is imperative first to find out the number of the enemy host and above all what equipment they have, by means of spies, deserters, and prisoners." The late eleventh-century book of advice by the general Kekaumenos stresses that without a network of informants in hostile territory, a campaign was bound to fail. Just prior to the march into enemy country, generals were instructed to send "defectors" with misinformation about the route of travel and targets. On the eve of the North African campaign of Justinian, the Vandal king Gelimer imprisoned Roman merchants in Carthage and threatened to kill them because he alleged they had urged the emperor to war against the Vandals. When the imperial army landed in Sicily, the historian Prokopios, then Belisarios's secretary, met a merchant who was probably a Roman spy; the man's servant informed them that the Vandal fleet and army had recently departed to quell an uprising in Sardinia.
The emperor Leo recounted that the Byzantines maintained permanent spies in Cilicia who observed the movements of the emirate's armies; the Muslims there were both seaborne and land raiders. When spies reported the Cilician fleet's sailing, the strategoi of the neighboring themes were ordered to attack by land so as to take advantage of the absence of the enemy troops. Conversely, when the Arabs marched on a land raid, the Byzantine fleet was informed and ordered to attack the shore.
The handbooks stress the need to surprise the enemy. Strategic surprise could be achieved by avoiding enemy agents, by disinformation, and by unexpected marches. The _Strategikon_ warns that to avoid enemy spies armies should take little-used routes and march through uninhabited areas that were less likely to be under surveillance. Commanders were instructed to divulge targets to no one, not even their inner councils, but to spread word via prisoners and deserters that they intended to attack locations other than the true target. About a week before imperial raiders embarked on their mission, spies who operated in the targeted theaters reported to army camps; if agents reported quiet in the region, then fast, light raiding columns swooped in to plunder. Ahead of the main army _doukators_ and light, specialist cavalry troops called _trapezites_ or _tasinarioi_ preceded the main force as scouts. Doukators located and determined the composition and strength of enemy formations, assessed battlefield conditions, and located pasture, water, and suitable encampment sites. Trapezites light cavalry, the ancestors of modern hussars, rounded up enemy prisoners for interrogation, scouted enemy formations, and ravaged the countryside to pressure the inhabitants. They prepared the battlefield for advancing army units, and when the Byzantines retreated from a raid, the trapezites practiced scorched earth to deny the enemy any use of the evacuated regions. Nikephoros II Phokas ordered trapezites to destroy the cropland and vines of cities targeted for siege; this ravaging prevented the enemy from storing up supplies and decimated enemy morale so that strongholds were less likely to endure a siege. These practices led to success in attaining strategic surprise—while the Muslims believed the army to be occupied in Bulgaria, Basil II mounted his infantry on mules and in an astonishing two weeks of forced marches pressed across Anatolia. When he descended into Syria the shocked Fatimid army of Egypt fled. When in 1156 Manuel I Komnenos wished to chastise the fractious Armenian prince Thoros and his ally, Reynault, prince of Antioch, for continually harassing imperial holdings in Cilicia, Manuel avoided detection during his march across Asia Minor and caught the Armenians and Latins unprepared; they quickly capitulated.
Battlefield intelligence was generally well conducted, though there were cases of serious breakdowns in the high command, such as the failure in 708 in which khan Tervel of Bulgaria ambushed the expeditionary army of Justinian II. In 1176, Kilij Arslan's Turks surprised and destroyed the northern army of Andronikos Vatatzes on the road to Amaseia, then the sultan's troops mauled the forces of Manuel Komnenos in the pass of Tzivritze on the route to Ankara. Most commanders were careful, however, and avoided being surprised by attacks. In the spring of 586, prior to the battle of Solachon, the spies of general Philippikos detected the approach of a large Persian expeditionary force. Suspecting a Sunday attack to catch the Romans in their religious services, the general sent scouts to determine enemy movements and as a result the Persians' attempted surprise failed and the Romans inflicted a heavy defeat on the enemy. Nikephoros II Phokas ordered scouts to be dispatched in all directions as an army moved; when on the march, the Macedonian-era army sent a company of 100 light horse to the rear of the column to scout for shadowing foes and to avoid any potential ambush; they were supported in the rear guard by archers and infantry. Scouts and spies were assigned to each tagma of the cavalry units described in the _Strategikon_ ; they worked ahead and on the flanks of the armies and patrolled in relays at regular intervals to transmit intelligence to the main army and set up observation posts with messengers to alert commanders to enemy stratagems. During the 971 campaign against Sviatoslav in Bulgaria, the general Bardas Skleros ordered the doukator John Alakaseos to reconnoiter and determine the whereabouts and strength of the Rus'. The next day Alakaseos's messengers informed Bardas that the enemy was nearby, intelligence that Bardas used to set an ambush to the rear of the advancing Rus' army.
Counter-intelligence was a constant concern, and despite the fortified nature of Roman marching camps, spies were able to infiltrate. The _Strategikon_ ordered silence to be maintained in the camps, so that lurking agents could be more easily detected. Leo repeated this advice, as well as the trick of having soldiers enter their tents at the sounding of a trumpet: those left outside the tents would be spies and captured. If spies were bold enough to enter the soldiers' tents they would be recognized and seized. In order to confound enemy battlefield scouts, multiple banners were used to trick the enemy into thinking more units were present than actually were; units and the depth of lines were varied, because uniformity made it easy for the enemy to accurately ascertain troop numbers.
TACTICS
For a military establishment that survived and changed over a millennium, tactical flexibility was embraced as critical for survival. The Byzantines might be defeated even twice or three times by an enemy, but they viewed no foe as invincible and learned from their mistakes. Based on the principle preached in the _Strategikon_ of understanding their own weaknesses, the willingness to adapt and learn from the enemy was a major contributing factor to the longevity of Byzantium.
Early Period Tactics
In the fifth century Vegetius envisioned a battlefield formation comprised of three divisions, with a center and two flanking wings. Heavy infantry formed the center and provided the anchor of the force and often its primary striking power. Units were drawn up in two or three lines, with elite troops held in reserve in the second line so that they could be moved to support any portion of the line that wavered or to counter enemy encirclement or flanking maneuvers. Heavy cavalry _clibanarii_ , mailed and armed with lances, protected the flanks of advancing footmen, while horse archers and light cavalry skirmishers rode on the wings of the battle line; these light cavalry units harassed and broke up the wings of enemy formations. The enemy center, where the best units were expected, had to be broken by the advance of heavy infantry massed in either a block or a wedge formation. The heavy cavalry formations Vegetius envisioned were vulnerable to attack from enemy infantry units and missile troops, thus they were often deployed in mixed formation, that is with light infantry carrying darts, bows, slings, and javelins that could soften the opponent and support retreating cavalry.
By the sixth century the hybrid cavalry, carrying lance, sword, and bow described by Prokopios were difficult to match on the battlefield, as his famous and oft-repeated description attests:
But the bowmen of the present time go into battle wearing corselets and fitted out with greaves which extend up to the knee. From the right side hang their arrows, from the other the sword. And there are some who have a spear also attached to them and, at the shoulders, a sort of small shield without a grip, such as to cover the region of the face and neck. They are expert horsemen, and are able without difficulty to direct their bows to either side while riding at full speed, and to shoot an opponent whether in pursuit or flight. They draw the bowstring along by the forehead about opposite the right ear, thereby charging the arrow with such an impetus as to kill whoever stands in the way, shield and corselet alike having no power to check its force.
While such hybrid horse archer-lancers ( _hippotoxotai_ ) may have been less heavily armed than the clibanarii or _cataphracti_ described by Vegetius, and some certainly were not as well equipped as Prokopios's ideal elite, some were outfitted as Maurice envisioned in his model cavalry; in the battle waged against the Moor Antalas by John Troglita, the Roman commander Geiserith was an imposing figure:
Girt in shining armor, he bore towering weapons. With his whole body covered in steel, he was a glittering vision, for he had adorned the armor plates with a mesh of gold. And he wore a golden helmet dazzling with inlaid steel whose peak and crest he had decked with a horse's mane. He drew in a belt that gleamed with bejeweled knobs and a sword in an ivory sheath adorned his side. He wore greaves, which a Parthian hide bound with many gold fittings on his legs.
Throughout early Byzantine history cavalry occupied primarily an offensive role. Horsemen provided both frontal assault troops and outflankers who used speed to attempt to encircle enemy lines. The best and heaviest cavalry were therefore stationed in the front two ranks and on the right and left edges of the moira. In battle cavalry broke up enemy infantry formations and sought to drive away light cavalry and infantry skirmishers who threatened them and their Roman infantry complement, who were nearly always present in the field. Unsupported they were not a match for Sasanian heavy cavalry, as two episodes from the sixth century demonstrate. In the defeat at Callinicum on the Euphrates, Roman cavalry fled in the face of the Sasanian cavalry; to resist them, the horsemen dismounted and formed up with the infantry in a phalanx and defended themselves effectively against repeated charges. Even the heaviest cavalry of the day therefore could be defeated by disciplined and well-arrayed spearmen.
Frontal Assault, Outflanking, and Envelopment
In a frontal assault on an enemy formation, cavalry drew up in close formation and advanced at the trot. When the hekakontarch ordered the charge, the front lines, comprised of dekarchs and pentarchs—the most experienced and best equipped troops—leaned forward, covered their heads and necks with their shields, and galloped forward, spears held at shoulder height. The mounted archers in the third through fifth files and beyond opened fire. If the enemy line was longer than the Roman front, the flank guards ( _koursores_ ) extended the formation to match the foe, if shorter, the flankers fanned out into the crescent formation and enveloped their adversaries. If they failed to outflank or break the enemy center, the cavalry retreated to re-form behind the cover of the second line, which usually comprised infantry. If repeated charges failed, the second line advanced to close quarters. The cavalry withdrew to the rear of the second line to regroup, ideally supported by a third Roman line. The opening of ranks to allow retreating horsemen to filter through the infantry was a difficult maneuver that demanded great discipline on the part of both the horsemen, who could not lose cohesion and run headlong into their own troops, and the infantry, who needed to form wide alleys at regular intervals to allow passage while maintaining their formations. At Adrianople, fleeing Roman cavalry collided with the infantry and shattered the Roman lines; the same probably also occurred at Yarmuk.
Roman armies generally operated with a mix of infantry and cavalry; to assume that infantry were tactically irrelevant by the sixth century is incorrect. Although the bulk of the _Strategikon_ deals with cavalry tactics of the new model army of Maurice, Book XII outlines infantry composition, formations, and battlefield actions. Among the recommendations that stand in sharp contrast with the previous era of the late Roman army of Vegetius is that, if possible, half the infantry force should comprise archers or, barring this, one third. The Battle of Taginae, where there were 8,000 foot archers in Narses's army, is the kind of prior experience on which this doctrine is grounded. In the Italian campaigns of Justinian, archers provided the Romans with a decided tactical advantage, and in the battles against the cavalry armies of the Persians and Avars their benefits were enormous; even if they were not stout enough to resist direct attacks these troops were capable of wounding and unhorsing enemy riders and thus breaking enemy formations and effectiveness.
In Maurice's army, foot soldier ranks formed a phalanx sixteen men deep with heavy and experienced soldiers in the four ranks at the front and rear of the formation. Heavy and light troops were usually mixed in the phalanx but sometimes fought in separate units. Likewise, there were cases when the heavy infantry formed the middle eight ranks of the army. This implies that skirmishers and light troops were to front and rear and thus opened the action, with the heavy troopers rotating forward as the armies closed to hand-to-hand fighting. As accounts of fourth- to sixth-century combat indicate, the infantry generally formed the center of the army, with the cavalry stationed on the wings, and the horsemen often had flanking guards comprised of heavy and light footmen to screen them against sudden side attack and ambushes or particularly powerful onslaughts of enemy horse. When the signal came to advance, the infantry front moved forward and formed the _foulkon_ to protect the face of the line from enemy missiles. When within range the archers behind fired and the men of the first line threw their darts or spears. They then drew their spathas and moved to close combat while the second line supported them with their spears and those behind sustained arrow fire.
The infantry needed to show a high level of drill. Maurice demanded the foot be capable of splitting the formation in half with the rear troops wheeling about to meet attack from the rear and thus form the double phalanx with two fighting faces. The close support of cavalry, with the combined-arms approach of missile troops closely integrated in the files, all indicate a well-disciplined, professional infantry force.
The Dark Ages
During the Dark Ages, tactics and professionalism changed and the battle record of the Romans from the seventh to ninth centuries is uninspiring. They did, however, prove capable of defense of major strongpoints, notably Constantinople when the stout walls and artillery served as a force multiplier and rendered the superior quality of veteran Muslim soldiers of limited advantage. In the mountains of Anatolia, cantonments of professional troops remained and some of these were undoubtedly infantry. However, there is little evidence for the persistence of heavy infantry units as frontier warfare came to be dominated by light cavalry, probably supported by substantial numbers of light troops. The Romans still maintained some drill and discipline even among the thematic troops dispersed through the countryside, but the strategy of attritive war with its avoidance of decisive battle underscores Byzantine tactical weakness. Defeat at the hands of Arab armies and by numerically weaker Bulgar forces in 678–79 suggests that tactical capabilities of Roman forces weakened. The reliance on ambush from the mid-seventh century and the practice of evading large-scale confrontation accompanied a decline in equipment and battlefield tactical capabilities. Nevertheless battlefield units were trained, though it seems thematic soldiers drilled mainly in the wintertime and likely this drill varied by region and the abilities of the local commander. As Haldon has noted, Byzantine armies still drew up ordered battle lines, worked in clearly delineated units under a recognizable command structure, and were generally superior in numbers to the Bulgars and Slavs in the Balkans, though they seem inferior to campaign armies of the caliphate. Archery, especially horse archery utilizing the Hunnic release and strong composite bow, largely vanished, depriving the empire of one of its major tactical advantages. The main force for tactical offense seems to have been light horse lancers.
Middle- and Late Period
By the ninth century the emergence of professional campaign mercenary tagmas recruited for campaigns and as standing units in the themes increased, and tactics improved as the Byzantines recovered economically and militarily. The mid-tenth-century treatise called the _Sylloge taktikon_ mentions the _menavlion_ pikemen; their role as heavy defensive troops equipped to defeat the strongest cavalry of the day indicates that infantry had returned to a dominant place in the battle line. The integration of 100 heavy-armed pikemen among the 1,000-man taxiarchies demonstrates adaptation to an increased threat from heavily horse units. The menavlatoi were drawn up in gaps between the front line infantry, where they could rush ahead of the main front and meet enemy cavalry charges intended to strike and break up Byzantine infantry units.
Nikephoros Ouranos's treatise of the early eleventh century, the _Taktika_ , indicates further refinement to the combined arms approach mixing heavy infantry, archers, light infantry, and menavlatoi. On the battlefield, the Byzantines utilized the hollow square comprised of infantry men seven deep; through this "marching camp" concept Roman cavalry sheltered within the square deployed to meet enemy attacks on the battlefront or flanks. This formation was not new, but its revival by the Byzantines shows how refined their military prowess had become by the tenth century. Twelve infantry taxiarchies of 1,000 men—400 heavy infantry, 300 archers, 200 skirmishers, and 100 menavlatoi—comprised the square, with gaps wide enough to permit the charge and withdrawal of light and heavy cavalry and support troops and baggage train sheltered in the corners of the square. In broken or rugged terrain, the Romans deployed in a narrower fronted rectangular formation that permitted the same tactics of swift, responsive cavalry egress and withdrawal to the shelter of backing footmen. The marching square made enemy envelopment nearly impossible, since the square had a double face, with pikemen stationed at the front and behind, and light infantry and cavalry support within the shelter of the phalanx which were free to engage any quadrant that came under attack.
According to the _Sylloge_ , combat began by cavalry maneuver through the gaps in the square:
The cavalry are the first to begin battle by moving out through the largest intervals.... Should they put the enemy to flight, they pursue them with all their might, with the infantry divisions trailing behind. In case they are defeated, they turn to go back to the infantry units once again. They either take their place inside the infantry units by coming in through the intervals—that is, inside the vacant place where they were before—or outside, on the wings of the infantry units, and on both its flanks they fight alongside the infantry formation.
The 200 light infantry, armed with sling, javelin, or bows, blocked enemy attack on the intervals in the square. On the front and rear lines of the square, two ranks deep, stood the menavlatoi; those in the rear could be rushed to the front to double the depth of the pike formation to four deep. They were supported by the light infantry who plugged the gaps as the enemy deployed to assault. Once the enemy cavalry committed to an assault on the infantry spearmen, the menavlatoi received the charge with their pikes and the light infantry skirmishers moved to strike the flanks of the enemy kataphraktoi. A vital change to the composition of this force arose under Nikephoros Phokas, who quadrupled the number of menavlatoi in the square to 1,200 men with a corresponding decrease in the regular heavy-armed infantry phalangites.
As the tenth century progressed and the enemy adapted their tactics, the Byzantines responded first by increasing the number of pikemen, as noted, but also in pinning the menavlatoi to the infantry line, rather than forward of the formation as the _Sylloge_ suggested. Another response was a further deepening of the infantry formation; once the enemy angle of attack was determined, the menavlatoi moved from the rear as before, but every second file moved laterally into the adjacent file, which made the formation twice as deep, including six ranks deep of pikemen, with little loss of breadth. This tactical maneuver offered a dense and impressive front to opposing cavalry, but retained the mix of skirmishing troops and flexible response of Phokas's reformed army.
The double line of cavalry deployment of the _Strategikon_ was revived and modified in the tenth century. The _Sylloge tacticorum_ attests that, as in Maurice's day, the standard cavalry deployment remained that of three front-line units and four units in the second, where the commander stationed himself; this host moved with flank guards and skirmishers along its sides, with an additional third line of horse followed by a rear guard, the _saka_ (an Arab term). The new kataphraktoi heavy cavalry deployed in wedge formation on the front line, behind the light prokoursatores (skirmishers) arrayed in open order who ideally numbered 500, 110 to 120 of which were mounted archers. The regular cavalry, in _banda_ of 50 men, grouped into tactical formations of 500 riders, 100 across and five deep. As in Maurice's day, these men were composite cavalry, with the front ranks bearing lances and those behind serving as mounted archers, with the last line also carrying lances, providing the ability to turn about and present a front of lancers no matter from which direction the enemy threat appeared.
Outflankers stationed on the right, as in the sixth century, now supported the kataphraktoi cavalry, heavy armed and armored riders with barded and shod animals whose equipment made them a much weightier striking force than their predecessors. Flank guards moved to their left, their role was to stave off attack from the enemy right that threatened the kataphraktoi and the regular cavalry stationed behind. The kataphraktoi assembled in a wedge formation, twelve rows deep, with the front row comprised of twenty men and each subsequent line adding four men, so that sixty-four men stood in the last line; a smaller wedge of ten men in front with four additional men stationed in each line so that the rear line contained fifty-four was also used. The kataphrakt wedge mirrored the composite makeup of the regular cavalry formation; the first four ranks carried iron maces, and from the fifth line to the twelfth the wedge comprised a mix of troops; lancers occupied the edge of the formation, with mace or sword bearers inside them, and the core of the unit filled with 150 mounted archers. The two 500-strong regular cavalry units stood on either side of the kataphrakt wedge at intervals that allowed these flanking units to support the heavy horsemen but also for advance of units from behind, or the retreat of units to the safety of the second and third lines.
McGeer traces battle tactics which Nikephoros Phokas and Basil II's general Nikephoros Ouranos envisioned when on the offensive in enemy territory. In the first scenario, the Byzantine mixed infantry and cavalry army confronts a foe with a similar composition of horse and foot. Once scouts reported the location and disposition of the enemy, the prokoursatores moved ahead of the infantry square and its supporting cavalry to attack; if the prokoursatores drew the main weight of the enemy force against them, the general sent two regular cavalry troops to their aid, then thrust forward with the supporting second line of cavalry. If the prokoursatores found themselves pressed by the bulk of the enemy formation, they withdrew to the safety of the infantry and additional cavalry forces deployed from the center of the square to attack, followed by the supporting infantry. When the opposing force fled, the prokoursatores returned to the attack.
Should the enemy phalanx strike in good order against the Byzantine square, the Roman commander ordered the kataphrakt wedge and its cavalry escort to assault the front of the enemy infantry. Swift-moving enemy infantry often tried to deny the Byzantines the ability to deploy their heavy cavalry wedge through the square to the front of the line, whereupon the commander ordered his kataphraktoi through the intervals in the side of the square to attack the opposing spearmen in the flanks. All of these movements were coordinated with supporting archery and missile fire from the light units and close reinforcement from the infantry.
In the second scenario, Roman cavalry operate as a vanguard and seek contact with the enemy. The prokoursatores once more open the engagement, probing the enemy formation and attempting to rout them if they see disorder. In the event the enemy holds firm in good order, the general identifies the enemy commander and his picked troops and the Roman wedge moves forward, slowly and silently. The measured, quiet advance of the iron wall of heavy lancers and mace-wielding kataphraktoi had a tremendous psychological effect on the enemy infantry, who faced the prospect of a devastating cavalry charge. As the kataphraktoi came into archery range of the enemy, their heavy armor protected them from missile fire and their own horse archers returned fire; the ensuing charge aimed for the heart of the enemy infantry line and the opposing general, whose death or flight would seal a Roman victory. In the event the enemy's heavy horsemen moved to strike, the Roman general dispatched three units of regular cavalry to surround and destroy them. If the kataphraktoi failed to rout the enemy, the general detached two of the regular cavalry taxiarchies to the front and remained with elements of the second and third line to react to the battle as it unfolded.
The tactics of the tenth century—sharp discipline, and mixed battle formations of heavy and light infantry fighting in close coordination with light and heavy cavalry units, including horse archers and the heaviest armored and equipped lancers of the day, the kataphraktoi—represent the peak of Byzantine tactics. The army of the tenth century was a nearly unstoppable offensive force whose constant campaigning expanded the empire's borders in the east into Syria and in the west to the Danube, a recovery of territory and prestige unmatched in history.
The decline of the thematic armies, whose service obligations were increasingly commuted to cash payments to the state throughout the eleventh century, led to a drastic decline in the defensive and offensive position of the military. By the time of the Battle of Mantzikert in 1071, the emperor Romanos IV found that most of the theme units had degraded to an unserviceable status. Mantzikert, like most subsequent Roman warfare, was fought with a cobbled together host of professional mercenaries, both Byzantine and foreign. The defeat at Mantzikert and subsequent loss of the heart of Asia Minor in the decades that followed due to civil war and Seljuk Turkish aggression meant that Alexios Komnenos possessed nothing like the army of his predecessors. His forces were professional, but the multiplicity of ethnic groups and lack of standard drill meant that the tactical mettle of the armies of the tenth century could not be entirely duplicated. The army of Manuel Komnenos was, however, professional and capable of sustained defensive and offensive operations. It relied on the same mix of heavy and light armed infantry and cavalry units, but their tactical flexibility and discipline did not match those of the tenth century. Heavy infantry persisted, however, as an important tactical component of the armies of the Komnenoi down to the defeat at Myriokephalon, and kataphraktoi were also present, but overall, the decline in state revenues, attrition, and the increasing reliance on foreigners had a deleterious effect on the tactical capabilities of the Byzantine army.
As Haldon argues, the history of Byzantine tactics in a developed form ends with the Komnenoi. The subsequent sack of Constantinople and the loss of the field army and state apparatus needed to support it resulted in a poor-quality army engaged in civil and defensive wars; tactically the avoidance of decisive open-field engagements was paramount and instead ambushes and harassment were used to fight generally superior enemy forces. Western tactics dominated, with infantry of varying quality supporting knights, shock cavalry whose charges were intended to break up enemy infantry formations. Infantry and cavalry continued to campaign together, with cavalry remaining the dominant offensive arm. The infantry probably resembled the more lightly armored forces of the seventh to ninth centuries than the tenth-century phalangites of Nikephoros Phokas's reformed army. Archers provided skirmishing troops; they were only lightly armed and armored. In 1345, at the Battle of Peritheorion, the emperor John Kantakouzenos divided his forces into three tagmata, with kataphraktoi on the left, the emperor and heavy cavalry in the center, and Turkish allied horse archers stationed on the right flank. In 1305 at the Battle of Apros between imperial forces and the Catalan Company, the Byzantines arrayed in five divisions in two lines, a vanguard and a main battle force; the composition of the divisions was based largely on ethnic affiliation, which usually determined the tactics of mercenary units in the first place.
Siege Warfare
The ability to overcome fortified cities and fortresses was a large part of the Roman art of war. By the fourth century, imperial forces had longstanding experience against opponents who possessed developed engineering skills, especially in the east, where cities of great antiquity were commonly walled and protected by other passive defenses, such as ditches and moats. The Romans utilized a range of siege machinery which required specialists to build, operate, and maintain them. Most of these weapons were developed by Hellenistic engineers in the centuries prior to the rise of Rome. However, though the basic principles of defeating walled targets remained the same, the means used to break walls changed considerably from the fourth to the twelfth century.
Vegetius noted that in siege warfare the first assault was often the most likely to succeed, since inexperienced defenders could be terrified by displays of arms and the appearance of siege machinery. In order to take cities, the Romans either starved the city into submission, or found a way over the top of the walls using machines or earthen ramps or under them with mines. Before the targeted city was besieged, the Byzantines prepared the battlefield by constant raids to destroy the crops and economic base of the land around the town and thus deny its inhabitants food and other supplies. Direct attack efforts in which Roman troops attempted to breach gates or walls were difficult, time-consuming, and expensive in terms of time, materiel, and lives. Throughout their history, the Byzantines preferred to rely on traitors, dissension within the city, or starvation when taking enemy cities. Only when other measures failed was an assault planned. Since the _Strategikon_ does not deal with siege warfare, we have manuals discussing siege warfare only from the late ninth- or early tenth-century _Taktika_ of emperor Leo VI, who indicates that at the opening of hostilities, easy terms were to be offered in order to cause doubt and dissension among the citizens; if a prompt surrender was not forthcoming, the general placed under guard the major and postern gates and organized rotations for workers and attackers. Daily attacks, Leo cautioned, wore out the army and though fighting was to be sustained around the clock, only a portion of the besieging force was in action at a given time. Attacks round the clock were necessary to deprive the enemy of sleep, which destroyed morale and made mistakes on their part more likely. If the town had flammable houses, fire darts or pots filled with incendiaries were cast over the walls with trebuchets to burn the houses and spread panic.
Throughout their history the Byzantines, like many ancient armies and their advanced neighbors such as the Persians and Arabs, tunneled to undermine the foundation of walls, an ancient practice that was the most common way of defeating fortifications. As the tunnels were dug, props supported the tunnel ceiling and eventually the hollow spot was created as stones were removed from the wall. Countermining by the enemy was a constant danger and could best be avoided by digging deep mines. In his _Taktika_ , the great general Nikephoros Ouranos (d. after 1007) noted,
The men of old, in their conduct of siege warfare, constructed many devices, such as rams, wooden towers, scaling ladders with various features, as well as tortoises and all kinds of other things which our generation has never even seen. It has, however, tried all these devices and discovered that of all of them, the more effective way, the one the enemy cannot match, is undermining the foundations.
In besieging cities, the Byzantines employed many devices, especially the traction trebuchet, introduced sometime in the sixth century from the east, probably Persia. The traction trebuchet was effective against many walls and had the advantages of its cheapness to produce and ease of operation. Leo's _Taktika_ mentions the general use of the traction trebuchet, which seems to have been the most common stone-thrower used through the middle period. In practice, however, battering down fortification walls was rarely done; instead, Nikephoros Ouranos recommended general assaults using ladders, combined with _laisai_ , siege pavilions woven from vine stalks or other woody plants, that provided refuges to workmen, archers, and staff slingers who bombarded the battlements and offered covering fire to men advancing with rams and hammers against gateways and weak points along the wall. In assaults, the commander divided his forces into three teams, one of which prosecuted the attack while the other two rested inside the lasai. Nikephoros envisioned direct, sharp attacks of relatively short duration in which artillery, missile fire, and ladder assault was combined with undermining efforts that would collapse the fortification walls. These aggressive, frontal assault tactics differed considerably from earlier practices of long investment and starvation and demonstrate the unique situation of the tenth century, when Byzantine capabilities and confidence were at a high point. The later adoption of the counterweight trebuchet, noted above and discussed further in Chapter 7 below, indicates a groundshift in Roman siege tactics; with the employment of this device even the most impressive Near Eastern fortifications could be pounded to rubble. John II Komnenos made the counterweight trebuchet the main weapon of his seize-and-hold policy of fortress taking throughout Anatolia.
When facing a siege, the general was instructed to see first to the provisioning of the city or fortress to be invested by the enemy. Water and food were to be strictly rationed, and those who entered the fortified refuge had to bring four months' provisions with them. Enemy armies encamped round fortified strongpoints in a circle in order to cut off supplies in communication, and Nikephoros noted that this made certain sectors prone to lax discipline and carelessness; he urged night attacks by Roman infantry against these elements. If the terrain did not permit the enemy to encircle and they encamped in one location, the commander was instructed to destroy enemy horses and food stocks, as well as to deny them provisions and shelter through scorched earth tactics—crops and Roman villages that could shelter the enemy were burned. Night ambushes and harassment would not only wear down the besiegers, but also distract them so that supplies and reinforcements could be inserted into the invested fortress or city.
This brief overview of strategy and tactics has shown that the Byzantines maintained longstanding practices gained by innumerable encounters with a range of enemies. Rarely did the Romans deviate from the strategic concepts of limited warfare, the pillars of which were the general avoidance of decisive battle, harassment, and attrition of the enemy. Wars were planned and fought with the aim of defeating the enemy, but the aim of victory was to secure peace in the short or long term. Wars were considered defensive wars even during the aggressive posture of the empire in the tenth century, when offensive operations allowed the recovery of former territories. Byzantine tacticians adopted enemy practices readily when these had proven effective, and though they suffered numerous defeats, the stabilization of the eastern frontier especially hinged on the successful implementation of lightning warfare conducted by light cavalry, defense of strongholds, and guerrilla tactics. The shift of the 950s was again decisive, with the return of heavy units, complex mixed formations, and the capacity for skilled battlefield maneuver that rendered well-led field armies indomitable. The failure to maintain the level of material commitment and aggressiveness ultimately led to the collapse of the eastern and western fronts under sustained pressure by new, more capable enemies.
## SIX
## ENEMIES OF BYZANTIUM
OVER THE MILLENNIUM OF ITS EXISTENCE, the Byzantines faced a vast array of peoples who threatened its territory and people. Several of these proved militarily superior and dealt heavy defeats on the empire. In the end, however, the Byzantines generally gained the upper hand, often through decades or even centuries of defense, stabilization, assimilation, and counterattack. The Byzantines learned a great deal from their enemies; indeed the ability to adapt to the challenges posed by opponents was one of the great pillars of Byzantine military success.
GERMANIC PEOPLES
The Gothic tribal confederacies posed the most serious challenge to the late antique Roman state of any Germanic group. The Goths comprised coalitions of tribal groups, mostly from the east Germanic peoples who by the third century A.D. inhabited a vast swathe of territory from the Oder and Vistula rivers to the southern steppes of Russia, the Crimea, and the Carpathian basin. East Germanic peoples had posed a significant threat to the eastern provinces of the Roman state from the third century. In 267, Goths and Heruls burst through the Danubian defenses and ravaged Thrace and much of the Balkans, sacking Athens before the emperor Claudius Gothicus dealt them a stinging defeat in 269. Following their defeat by the Huns, large groups of Goths migrated south to the Danube where they were admitted as suppliants to Roman territory. Their provisioning was bungled due to corruption, and an underdeveloped transportation response led to starvation among the Goths and rebellion that culminated in the armed confrontation at Adrianople. At the end of the sixth century, after its recovery from the Goths, the empire had to concede the loss of most of Italy to the newly arrived Lombard confederation, whose grip on the peninsula spread throughout the seventh century. The Byzantines also entered sporadic conflicts with the Franks from the sixth century and even fought against Charlemagne (801–10) for control of the Istrian and Dalmatian coasts.
Organization
The Goths were organized in decimal units with major groupings of "hundreds" (hundafaþs) as were their Germanic relatives, the Anglo-Saxons and their Roman neighbors, whose centurions were well known to the eastern Goths. Gothic mercenaries served in the Roman army throughout the late third century, and by the time of the emperor Constantine, Gothic elements were settled in Transdanubia. By the fourth century Gothic military organization had evolved at least in part under the influence of Roman practice. Gothic tribal raiders crossed into Roman territory and proved a sufficient nuisance to attract the interest of Constantine, who waged multiple campaigns against them. By now the Goths probably included and coexisted alongside elements of several ethnic Iranians (Sarmatians), Slavs, Romano-Dacians, and Getae. The remains of a commonly articulated material culture from the second through fifth centuries (the Chernyakhov culture) indicate broad contact and exchanges; such adaptations were not always peaceful and the transferal of knowledge from one people to another certainly included warfare. According to Maurice, the "fair-haired races," especially the Lombards, grouped themselves not into numerically ordered units but according to kin group.
Methods of Warfare
The Goths fought as both cavalry and infantry. Until the last few decades, historians have viewed the Goths as primarily a cavalry army and attributed to this their shattering victory over the infantry legions in 378 at Adrianople. Their numbers were probably never as numerous as some Roman authors would have us suppose—Heather estimates that in sixth century Italy and Gaul there were about 15,000 Gothic elite males. When the Gothic king Theodoric reigned over the united Gothic territories in Spain, Gaul, and Italy, his Gothic subjects numbered about 200,000 people. However, although we have few contemporary sources, the majority of Goths seem to have often fought as infantry spearmen and swordsmen. Certainly the Goths served in large numbers in the legions as infantry. At Adrianople the Goths had perhaps 5,000 cavalry and probably twice as many infantry. According to Vegetius, the Goths possessed numerous archers, who fought on foot. In the sixth century, Prokopios provided a clearer picture of the Gothic army, which fielded a large cavalry component who fought in massed formations as lancers, while the infantry seem to have been mainly skirmishers armed with javelins and archers. Other infantry fought as spearmen and swordsmen equipped with a spatha and carrying shields. Given the high casualty rate caused by Roman archery among the Goths, it is doubtful that they were more heavily armored than their Roman foes. In fact, the Goths closely resembled their late Roman counterparts.
Byzantine Adaptation
Since after Adrianople the empire was too weak to destroy the Gothic confederacies, the Romans sought to neutralize them by treaty. The emperor Theodosius recruited numerous Goths into the Roman army, as an expedient means to replenish the devastated ranks of the eastern field forces, but also as a way to weaken the Goths, whose presence in the Balkans created a state of emergency. Theodosius recruited numerous Gothic federates who fought loyally for him and of whose lives the Roman high command was apparently none too careful—a contemporary panegyrist acclaims the emperor for using barbarian to fight barbarian, thus bleeding both of them. Nonetheless the Goths formed a sizable but not dominant portion of the eastern field army. By 400 A.D. the Gothic warlord Gainas dominated imperial politics in the capital of Constantinople, but his unpopular policies led to his downfall and a riot of citizens who trapped and massacred 7,000 of his Gothic troops. The Gainas affair marked the apogee of Gothic influence in the imperial center; the Romans countered Germanic elements in the army by recruiting Isaurian highlanders from Asia Minor. Finally, the last major elements not assimilated or settled within the Roman Balkans or Asia Minor were sent to Italy under Theodoric the Amal. Justinian renewed the Gothic conflict, invading Italy and conquering it. In 554 the Roman general defeated a Frankish-Alemanni force at Volturnus through his combined arms approach—horse archery again proved a major Roman tactical advantage over the Frankish infantry force. Though the Byzantines lost most of Italy to the Lombards in the later sixth and early seventh centuries, they created the exarchate of Ravenna with several dukes under its control to check the Lombard advance. The exarch held joint civil and military power and, as viceroy of the emperor, was free to respond to crisis without direct orders from Constantinople. These reorganizations helped the Byzantines maintain territory in portions of Italy until 1071.
PERSIANS
The most sophisticated, rich, and militarily threatening power that the Romans faced in the early part of their existence was the empire of Sasanian Persia. Founded after victory in a civil war in 226 A.D., the Sasanian dynasty ruled territory stretching from Central Asia to the Persian Gulf and Mesopotamia. Their propaganda declared dynastic ties to the Achaemenid Persian Empire destroyed by Alexander the Great and consequently the rights to the former Persian territories of Asia Minor, Egypt, and the Mediterranean coast. While the Sasanians acted on these grand claims on only one occasion, during the mighty conflict that raged with Rome in 603–28, clashes over strategic borderlands and satellite peoples were frequent. The frequency and intensity of these conflicts rose from a simmer to a steady boil by the sixth century, culminating in the Persian conquest of most of the Roman east in the following century.
Organization
The Sasanian shah Kosrow I (531–79) reformed the Persian military and in doing so created several Roman-style structures. Kosrow divided the empire into four army districts in which he stationed army corps under the command of four _spahbeds_ (field marshals). Along the border, the king established margraves, _marzbans_ , who administered sensitive border districts and commanded the frontier forces stationed there. The Eran-ambaragbed, "minister of the magazines of empire," was, like his Roman counterpart, the praetorian prefect, in charge of arming and equipping the troops. The general ( _gund-salar_ ) led individual field armies on campaign; sometimes under the authority of the _spahbed_. By the sixth century the army was largely professionally recruited and paid; there was a professional infantry commander in charge of standing guard units, but in the sixth century the Persians apparently continued to rely on conscripts for a large portion of their rank-and-file infantry. Mailed cavalry units and the royal guard formed the crack troops of the empire; these were generally drawn from the Persian nobility or from aristocratic allied families, such as the Hephthalites and Armenians, with whom the Persians had close contacts.
Methods of Warfare
The proportion of infantry to cavalry in the Sasanian army is unknown, but the Persians relied to a large degree on heavy horsemen, who could both shoot the bow and strike with heavy lances. The Persians favored direct massed cavalry assaults to break up enemy formations; the shock of their horsemen proved decisive against the Romans on several occasions. Normally the Sasanians drew up their forces in three cavalry lines. The Sasanians occasionally employed elephants in combat, but though they made a great psychological impression, they were not an important part of their military. The left of the Persian line was traditionally manned by left-handed archers and lancers who could thus strike effectively across the face of the enemy formation (right-handed mounted archers especially had difficulty shooting to their right). The left of the host formed the defensive anchor, whose role was to avoid enemy flanking maneuvers and to support the offensive right of the formation, where were stationed the best noble cavalry. The Sasanian right typically tried to outflank the enemy left, though the heavily armed kataphracts, covered from head to toe in mail and bearing lances, could be used in frontal assaults on infantry and cavalry groups. Behind the center line of regular cavalry were stationed the infantry formation, which supported the cavalry and sheltered retreating horsemen in case their attacks failed. In addition to their archery and horsemanship, the Sasanians were outstanding siege engineers. From the fourth through seventh centuries they seized some of the best defended and most powerfully built Roman fortress cities.
Byzantine Adaptation
The Sasanians and Byzantines knew one another well and there was considerable exchange of military knowledge and practice across the frontiers. Militarily, each side came to resemble the other. In early twentieth-century excavations at Dura Europus, a Roman frontier city on the middle Euphrates taken by Sasanian assault in the year 256 archaeologists discovered the remains of at least nineteen Romans and one Sasanian attacker. The Sasanian wore chain mail, carried a jade-hilted sword, and wore a pointed ridge-type helmet with a prominent center piece whose rivets joined the two lobes of the helmet together. Such gear was typical in Roman armies by the third century. In 533 at the battle of Dara, Belisarios countered Sasanian superiority by limiting their cavalry and playing to their psychological sense of superiority. In subsequent battles he used the Sasanians' wariness of his stratagems to force their withdrawal by aggressive posturing. The Persians, used to the traps and feigned retreats of their nomad enemies, could be made too cautious by aggressive maneuvers. They could be thwarted by the commander's well-chosen battlefield that cut off the Persians' ability to place their weaker elements on protective rough ground. The poor soldiers among the Sasanians did not fight with spear and shield, but seem to have been mainly skirmishers and archers. They were therefore susceptible to Roman cavalry charges delivered over level ground. The Romans thus relied on strategems, strategic maneuver, tactical coordination, and discipline to defeat the Persians. When Roman commanders selected the battlefield, they were able to neutralize or defeat these stubborn eastern opponents.
NOMADS
Throughout its existence, the empire confronted a vast array of steppe nomad military powers. The Byzantines fought major wars against the Huns, Bulgars, Avars, Khazars, Hungarians, Pechenegs, and Cumans and numerous minor conflicts with a host of other groups. Nomads were generally bent on plunder of imperial territory and rarely sought to settle on lands south of the Danube, only a small portion of which were suitable for the transient, cattle-herding life of pastoralists. However, both the Huns and Avars posed existential threats to the empire, as they sought to dominate the lands south of the Danube and to destroy the Roman power that contained them north of the river. Nomadic confederations formed under charismatic leadership or during periods of environmental or physical stress. Maurice stressed in the _Strategikon_ that nomads typically fought in kin-based tribal or extended family groupings, and this contributed to the nature of their tactics.
Organization
Nomadic society was based on nuclear families and wider, extended kinship ties. Like other tribal societies, blood relation or imagined genealogical connections helped to smooth political dealings and allow for larger groupings or "super tribes" that made massive nomadic military enterprise possible. The Huns under Attila formed an effective monarchy and Maurice stressed that the Avars, unlike many nomads, possessed a kingship. Undoubtedly the power of the central figures within a hierarchy during the Hunnic and Avar episodes of Byzantine history bolstered the barbarians' military effectiveness. After they settled north of the Danube in the late sixth century, the Avars conquered and coopted elements among the Bulgars, Slavs, and Hunnic and Germanic peoples in Transdanubia. The Byzantines portray a grim fate for those whom the Avars conquered, especially the Slavs who served as hard laborers and pressed soldiers during the siege of Constantinople in 626. According to Maurice, the Avars arranged themselves by tribe or kin group while on the march. Their social structure made them vulnerable to desertions and divisions within the ranks, which the Byzantines sought to exploit.
Methods of Warfare
Steppe nomads fought primarily as lightly armed horse archers. Speed and surprise were cornerstones of their strategic and tactical success. Their ability to swarm and the firepower they brought to bear could break up enemy formations and drive the enemy from the field. In the fourth century, when the Romans had little experience dealing with the tactical swarming attacks, war cries, strange appearance, and mobile horse archery of the Huns, these nomads struck terror into the hearts of many soldiers and won numerous victories across the length of the empire. In addition to horse archers, the Huns and Avars deployed heavier lancers who bore a resemblance to the Sasanian hybrid cavalry, armed with bow, sword, and lance. The _Strategikon_ notes that the Avars carried a lance strapped on their back which freed them to operate their bows. In addition to the lance and bow, Avar warriors carried swords; they seem to have been more heavily armed than their Hun predecessors, as Maurice noted that they wore chain mail coats. The Avars wore long coats of mail or lamellar split at the crotch, with panels on each side to protect the leg. The famous Nagyszentmiklós Treasure includes a gold plate depicting what is probably an Avar or Bulgar warrior wearing such a coiffed mail coat, splinted greaves, helmet, and carrying a pennoned lance.
Byzantine Adaptation
The Byzantines relied on diplomatic means to buy off and deflect Hun designs on imperial territory. The defensive posture of the empire throughout the fifth century precluded decisive confrontations against a superior enemy in the open field, and the massive defenses of Constantinople shielded the eastern territories from Hun penetration and conquest, though most of their European possessions were ravaged and slipped from Byzantine control. Although our sources provide no insight into the exact mechanisms of the adoption of steppe nomad tactics and equipment, the Byzantines recruited Hunnic horse archers into their armies and probably from these and deserters derived the knowledge of horseback archery. By the sixth century, the hybrid horse archer and lancer cavalry among the armies of Justinian were the most important tactical elements within the Roman army. The Byzantines adopted the stirrup from the Avars and this provided Roman cavalry with a more stable fighting platform. Maurice's _Strategikon_ notes that the thonged Avar lance and Avar-type tents and riding cloaks were also adopted directly from their steppe enemies. Lamellar cavalry armor also became more prominent in the panoply of Roman soldiers in the sixth and seventh centuries and this, too, indicates that the Byzantines borrowed extensively from nomads. The use of the feigned retreat, while known to classical armies, was a common steppe nomad tactic that the Byzantines perfected under steppe influence and employed throughout their history. The adoption of nomadic equipment, tactics, and strategy were among the most important adaptations of the Byzantine army and proved critical to the long-term survival of the empire.
ARABS
By the time of the rise of Islam in the early seventh century, the Romans possessed extensive military experience with the Arabs. Arab scouts and light troops had served as guides and auxiliaries almost from the beginning of Roman rule in the Near East. By the sixth century, the Roman system of paying subsidies to allied tribal confederations to maintain law and order along the frontier from the Red Sea to the Euphrates was integral to the governance of the eastern provinces. The powerful Christian tribal confederation of Ghassan, which included both settled and tribal elements, largely managed the eastern periphery of the empire, and despite the general hostility of Greek-speaking elites to their Arab allies, these clients were both effective and reliable. Ghassanid auxiliaries defeated their Persian-sponsored counterparts and provided valuable light cavalry raiders and skirmishers to the eastern field armies on campaign in Syria and Mesopotamia. At the Battle of Yarmuk in 636 the Ghassanids fought alongside their Roman masters and though many subsequently converted to Islam and remained in Syria, a sizable group migrated to Roman territory. The Muslim Arab victors at Yarmuk overran the whole of Syria, Mesopotamia, and eventually wrested Egypt, Libya and North Africa from Roman control. Muslim Arab attempts to conquer Constantinople and thereby destroy the remnants of the Roman Empire unfolded in the epochal sieges of the seventh and early eighth centuries in which the empire emerged battered but intact. With the overthrow of the Umayyad dynasty and shift of the locus of Muslim government to Mesopotamia, the threat to the existence of the Roman state diminished, and as the Abbasid caliphate unraveled politically, the Byzantines mounted a sustained counterattack to recover lost territories in the east.
Organization
Arab armies of the conquest were organized along tribal lines, though it is uncertain if these were grouped into units of 10–15 soldiers called _'arifs_ known from just after the conquests. Muslim Arab armies were recruited mainly from Arabic-speaking family and tribal groupings. But soldiers were also raised from among Byzantine and Sasanian deserters, as well as non-Arab clients ( _mawali_ ) dependent on regional Arab lords. Larger tribal groups fought under the banners of their tribal sheikhs in army groups of varying strength, usually numbering 2,000–4,000 men. On rare occasions, as at Yarmuk, combined commands could field as many as 30,000 or 40,000 soldiers. In 661, the Battle of Siffin was fought between the Syrian forces under Mu 'awiya and the Iraqi Arabs led by the Prophet's cousin and son-in-law 'Ali, said to have comprised 150,000 and 130,000 men, respectively; these numbers are inconceivable and could probably safely each be reduced by a factor of ten. During the Umayyad era, when the Syrian army provided the main prop to the caliph's authority, armies of 6,000 Syrian troops are commonly mentioned and these may represent standard field force groupings, not dissimilar in size and equipment from their Byzantine neighbors. In 838, the caliph al-Mu 'tasim (d. 842) led an army of up to 80,000 men against Amorium, a number that represented a large force and among the largest the Byzantines ever confronted.
Methods of Warfare
Although the commonly held perception of early Muslim armies today is of swift-moving horsemen mounted on Arabian chargers, the armies of the conquest era were mainly infantry forces fighting as spearmen and archers. Arab archery was particularly deadly to both the Byzantine and Persian forces encountered during the first campaigns of the conquest. Early Muslim armies generally lacked heavy cavalry, and they eagerly accepted the Sasanian horse who deserted to their ranks following the initial encounters in Mesopotamia. Infantry continued to form an important part of Arab armies up to the end of their military encounters with the Byzantines. Nikephoros Phokas noted that the Arab raiders who penetrated the Byzantine borderlands included a mix of cavalry and infantry; like their Roman counterparts, the infantry formed a _foulkon_ , a dense mass of infantry spearmen, and supported the cavalry who formed the major offensive wing of Arab armies. Regular Arab cavalry fought primarily as lancers, while missile support was provided by foot archers. The Arabs never mastered horseback archery and instead relied on Turkic troops to provide mobile fire. The light cavalry encountered by the Byzantines in their reconquest of northern Syria and Mesopotamia were Bedouin light horse riding swift Arabian mounts. Nikephoros advised to keep them at bay with archery rather than chase them, since even the best Byzantine horses, encumbered as they were with heavily equipped fighting men, would not be able to catch them and the danger of being cut off and overwhelmed was a persistent peril of pursuit. Well led and generally possessing superior numbers, training, and equipment, the Arab armies of the early medieval period repeatedly exposed Byzantine weaknesses. Decisive engagements nearly always ended with Arab victories; only when the empire recovered somewhat economically and demographically while the caliphate began to fragment did the initiative return to the Romans.
Byzantine Adaptation
Given the asymetrical nature of the encounter between the Byzantines and Arabs after the initial clashes of the early and mid-seventh century, Byzantine commanders responded in the only way they could, via a strategy of defense coupled with limited, punitive raids to keep the enemy from settling in the strategic Anatolian highlands and to maintain the appearance of Byzantine power among the populations of the border lands. Imperial troops, seriously degraded through the loss of many men in the defeats in Syria and Egypt, underpaid, poorly equipped, and scattered throughout the provinces, were scarcely a match for caliphal field armies. The Byzantines often found themselves paying tribute to convince the Arabs not to attack them—a humiliating concession that drained both the fisc and morale. But the sieges of 674–78 and 717–18 revealed that without achieving naval dominance the Arabs had to conquer the Anatolian plateau if they were to achieve their objective of outright conquest of the Christian empire. Yet, due to their organization of the themes, whose armies could shadow and harass Muslim raiding columns and sometimes defeat them, the Romans made penetration of their territory hazardous. Stubborn Byzantine forces, although no match for grand caliphal campaign armies, often held their own against raiding columns and themselves raided exposed regions when Arab field forces were engaged elsewhere. By the tenth century, the centuries of incessant warfare had helped to create a warrior caste among the frontiersmen of the eastern marchlands who would remake the Byzantine army based on their experiences fighting the Arabs. Their combined arms approach and their use of psychological terror, scorched earth, and incremental advancement of imperial territory by sieges marked the apogee of the practice of Byzantine arms in the medieval east.
BULGARS
The Turkic Bulgars appeared in the sixth century, first as a rump of the so-called Old Bulgarian Empire, the Kutrigurs, defeated by Belisarios outside Constantinople in 559, settled north of the Danube and were absorbed by the Avars. Following the collapse of Avar power in the eighth century, new Bulgar arrivals and existing elites in Transdanubia gradually formed the Bulgar khanate, which adopted Slavic language and customs. Given their cultural origins in the Eurasian steppe, it is unsurprising that throughout the medieval period the Bulgarian social elite fought mostly as heavy armed cavalry lancers. Bulgaria formed the most important state to the north of the empire. Though there were long stretches of peace between the two peoples and even alliance, Byzantine-Bulgar relations were strained by their fundamental conflicting goals—both empires sought to dominate the Balkans and each considered the presence of the other unacceptable. Thus the Bulgars sought to capture Constantinople or subjugate the Byzantines militarily, while the latter sought to contain or even annex Bulgaria outright.
Organization
Initially the Bulgars organized themselves along the lines of most steppe empires, with "inner" and "outer" tribes whose power relationships were articulated through marriage alliances, genealogies, and material exchange. Beneath the outer tribes in the pecking order were subject groups like Slavs, Greeks, and the mélange of Avar, Hunnic, and Germanic remnants that rendered the rich cultural matrix of the Danube basin. The khan stood at the pinnacle of an increasingly sophisticated hierarchy that developed under steppe and Byzantine influence. Senior "inner" nobles, called _boilas_ (often Anglicized as "boyar"), and junior "outer" nobles, _bagains_ , formed the elite of the Bulgar state and provided both the military leadership and elite troops of the khanate. The Bulgars matched their Byzantine foe with a strong hierarchical military organization with the khan in overall command while his leading generals, the _tarqan_ , commanded his administrative regional center and presumably took the center of the battle line as well. The _targan's_ subordinates included _komites_ (sing. _komes_ ), after Byzantine usage, who commanded the wings of the army. The highest-ranking Bulgar nobles were heavily equipped cavalry with barded mounts and relied on heavy household cavalry and lighter armed horse archers as did their steppe nomad ancestors.
Methods of Warfare
The Bulgars employed mass conscription to fill out the ranks for their armies. Fear was the main tool used to compel men to enlist and show up equipped for the occasion. Khan Boris Michael (d. 907) ordered that men who arrived for muster without proper equipment or unprepared for campaign were to be executed, as were those who deserted before or during battle. The rank and file included many Slavs who fought as light infantry, carrying shields and javelins. Bulgar cavalry resembled both their Byzantine enemy and other steppe nomads. The Bulgars were expert in their use of terrain, relying on ambush and surprise in their confrontation with the enemy. They demonstrated a high level of strategic planning, strong discipline, and military cohesion, and on numerous occasions were able to confront and defeat imperial field armies, as they did at Varbica in 811 when they trapped a large force led by the emperor Nikephoros I and destroyed it by hemming the Byzantines against a wooden palisade and surrounding it. The emperor himself was killed and his heir mortally wounded. The Bulgars were intimately acquainted with Byzantine military strategy and tactics and, unlike the fragmented Arab emirates to the east, formed a more unified foe unbowed by the shock of repeated defeats.
Byzantine Adaptation
The Byzantines dealt with the Bulgars via a full range of economic, diplomatic, and military strategies. Trade was limited by treaty to designated zones and monitored by imperial officials. Spies were maintained at the Bulgar court at Pliska; the Bulgar khan Telerig (768–77) tricked the emperor into revealing the identity of Byzantine agents among the Bulgars by the ruse of his promised defection, then slaughtered those in the pay of the empire. Byzantine failures against the Bulgars were often due to weakness in strategic and battlefield intelligence that resulted in the surprise of imperial field forces. Experienced and cautious commanders found warfare in Bulgaria perilous. Thus, in the ongoing dispute over control of lands in Thrace and Mesembria on the Black Sea coast, the emperor Nikephoros II Phokas mounted a brief campaign in which he found the the Bulgars' skillful use of the mountainous terrain and difficulties of supply and communication hard to overcome. Nikephoros therefore induced Sviatoslav I of Kiev to invade Bulgaria; the Rus' captured scores of Bulgarian towns and fortresses and overwhelmed Bulgar resistance, which led to a direct confrontation between the Rus' and their new Bulgar subjects and Byzantium. John I Tzimiskes's defeat of the Rus' at Dorostolon in 971 opened the way for Byzantine annexation of Bulgaria. The subjugation of Bulgaria took decades, however, with persistent and arduous campaigning by the emperor Basil II, who reduced each quarter of the Bulgar state through sieges and attrition, finally grinding down Bulgar resistance. Bulgaria provided another test for Byzantine strategies of attritive warfare: imperial forces used sieges, scorched earth, and incremental capture-and-hold methods to gradually expand their bases of operations and finally wear out a formidable, skillful, and disciplined opponent. Although the empire possessed a dominant position in Bulgaria by the death of Basil II in 1025, serious resistance continued to the death of the Bulgarian tsar Peter II in 1041. Byzantine control of Bulgaria, won over decades of bitter warfare, lasted for nearly a century and a half.
NORMANS
The Normans arrived in the Byzantine world not as enemies, but as valued mercenaries esteemed for their martial prowess. The settlement of Scandinavian raiders created the duchy of Normandy, when the region was ceded to their war leader Rollo (d. ca. 931) by the Carolingian king Charles the Simple (898–922). Rollo's descendants mingled with the local French population to create the Normans, a people thoroughly Christian, doggedly militaristic, and unfailingly expansionistic. Norman soldiers entered Italy around the start of the eleventh century where they served as mercenaries for various Lombard princes. By the 1050s large numbers of "Franks," as the Byzantines called them, had served as mercenaries in Byzantine armies from Syria to Bulgaria, and Normans served as part of the standing garrison of Asia Minor. In the 1040s the Normans began the conquest of south Italy, establishing several counties in the south and finally invading and conquering Sicily from the petty Muslim dynasts there by 1091. Since the late 1050s the Normans had challenged Roman interests in Italy and Robert Guiscard led a Norman invasion of the Byzantine Balkans in 1081. In the ensuing conflict the Normans defeated Alexios I Komnenos, who expelled them only with great difficulty. Two more major Norman invasions followed over the next century, and the Norman kingdom of Sicily remained a threat to imperial ambitions in the west and to the imperial core until the Hauteville Norman dynasty failed in 1194. By this time all hope for the Byzantine recovery of south Italy and Sicily had vanished, thanks to Norman power.
Organization
The Normans served under captains who rose to prominence due to birth or their fortunes in war. Minor nobility like Tancred of Hauteville, who founded the dynasty that would conquer much of Italy and Sicily, was a minor baron in Normandy and probably the descendant of Scandinavian settlers. The warriors who carved out territory within Byzantine Anatolia seem to have been either petty aristocrats or simply successful soldiers. One such Norman was Hervé Frankopoulos, who in 1057 led 300 Franks east in search of plunder and territory. After initial successes around Lake Van, he was delivered to the emperor and eventually pardoned. Thus, Norman companies were of no fixed numbers, and it seems that each baron recruited men according to his wealth and status. Norman lords in Italy raised the core of their army from men to whom they distributed lands and wealth in exchange for permanent military service. Lords were required to provide fixed numbers of troops, either knights or infantry sergeants. Other Normans served for pay and plunder, including conquered lands to be distributed after successful occupation of enemy territory. The Normans that the Byzantines encountered were a fluid group—some fought for the empire and then against it; their interests were pay and personal advancement rather than any particular ethnic allegiance. In this the Normans who warred against the Byzantines resembled the later free companies of the late medieval period—variable in numbers, generally following a capable, experienced, and charismatic commander, and exceptionally opportunistic. As a warlord's success grew, so did his resources. Thus Robert Guiscard rose from the leader of a band of Norman robbers to be Count and then Duke of Apulia and Calabria; in 1084, following his defeat of Alexios at Dyrrachium, Guiscard marched on Rome with thousands of infantry and more than 2,000 knights, a far cry from the scores or hundreds with which he began his career.
Methods of Warfare
The bulk of the Norman fighting forces were infantry, but they formed a largely defensive force that operated in support of the cavalry. Norman infantry fought generally as spearmen—the Bayeux Tapestry shows many Normans on foot wearing the nasal helm and mail hauberks, but it is unlikely that the majority were so armed. Most were probably unarmored and relied on shields for protection like most of their counterparts throughout Europe. Light infantry archers fought with little or no armor, and missile troops played a role in their Balkan campaigns as well—the Byzantine commander George Palaiologos suffered an arrow wound to his head in battle at Dyrrachium in 1082, but generally the Byzantines relied on superior Turkish archery in order to unhorse the Normans and immobilize the knights. Norman knights wore heavy mail hauberks and mail chausses with in-pointed mail foot guards, which Anna Komnene noted slowed the Norman cavalry down when they were unhorsed. These mounted men carried lances and swords. The weight of their mail made them relatively safe from the archery of the day. Norman knights usually decided the course of battle; it was the shock cavalry charge delivered by the Norman knight that delivered victory in battle after battle. Unlike the Turks and Pechenegs with whom the empire regularly contended and whose weaponry was lighter and who relied on mobility, hit-and-run tactics, and feigned retreat, the Normans preferred close combat. They fought in dense, well-ordered ranks and exhibited exemplary discipline. In an era when infantry were generally of questionable quality, most foot soldiers throughout Europe and the Middle East could not stare down a Norman frontal cavalry charge. Norman horsemen punched holes in opposing formations and spread panic and disorder that their supporting troops exploited. By the end of the eleventh century, Norman prowess on the battlefield yielded them possessions from Syria to Scotland.
Byzantine Adaptation
The Byzantines avidly recruited Normans into their armies. Though critics have unfairly blamed the medieval Romans for not adapting their warfare in light of the new western techniques and technologies to which they were exposed, fully equipped and well-trained kataphraktoi could match the skill and shock power of the Norman knight. What the Byzantines of the Komnenoi era lacked were the disciplined heavy infantry of the Macedonian period and combined arms approach of mounted and dismounted archery that could blunt enemy attack and cover infantry and cavalry tactical operations. Alexios I relied on Turkish and steppe nomad auxiliaries and patchwork field armies assembled from mercenaries drawn from the empire's neighbors. As with other intractable foes, the Byzantines relied on a combination of defense and offense—the Normans were contained in the Balkans allowing space for an imperial recovery and the time to muster new forces following the heavy defeat late in 1081 of the Roman army at Dyrrachium on the Adriatic. Alexios allied with southern Italian nobles and the German emperor Henry IV (1084–1105) who menaced the Norman flanks. The death of Robert Guiscard in 1085 removed the most serious threat to Byzantine rule since the seventh century, but Guiscard's son, the redoubtable Bohemund, renewed war against the empire in 1107–8. Alexios had learned from his twenty years of dealing with the Norman adversary and returned to the traditional Byzantine strategies of defense, containment, and attrition. The Byzantines relied on their Venetian allies to provide naval squadrons on the Adriatic that interfered with Norman shipping and resupply, and Alexios's forces blocked the passes around Dyrrachium; the emperor forbade his commanders to engage in a large-scale confrontation with the Normans. In the skirmishes and running battles against Norman scouting and foraging parties Byzantine archers shot the enemy mounts from beneath their riders and then cut down the beleaguered knights. Hunger, disease, and lack of money undid Bohemund, who was forced to sign a humiliating treaty and return to Italy. Thus the ages-old Byzantine principles of indirect warfare proved triumphant against a stubborn and superior enemy.
## SEVEN
## THE BYZANTINE ARMY AT WAR
THROUGHOUT THEIR HISTORY the Byzantine art of war may be seen in numerous campaigns and individual engagements. Though some have been well studied, contemporary sources describe many more encounters with the enemy in the barest terms, or make no mention of them at all. In what follows, we will view the Byzantine army at war via studies of major campaigns, individual battles, and siege warfare.
CAMPAIGNS: THE VANDAL WAR
In 406 the East Germanic Vandals and their tribal confederates, including Germanic Suebi and Iranian Alans, crossed the Rhine. After an initial defeat at the hands of the Franks, the Vandals enlisted Alan support and smashed their way into Gaul, plundering the countryside mercilessly as they advanced into the south. In the early 420s Roman pressure forced the Vandals into southern Spain where the newcomers faced a Roman-Gothic alliance; this threat the Vandals managed to defeat, but there could be no peace. Under their fearless and brilliant war leader Geiseric (428–77), whose fall from a horse had made him lame, the Vandals sought shelter across the Mediterranean; their long exodus led as many as 80,000 of them to Africa where, they believed, they could shelter themselves from Roman counterattack. They commandeered ships and ferried themselves across the straits to Tangiers, in the Roman province of Mauretania Tingitana.
There the local dux had few men to oppose Geiseric, who swept him aside and, after a year's plundering march, in 410 reached the city of Hippo Regius (modern Annaba in Algeria). There one of the great luminaries of Christian history lay dying: Augustine of Hippo, bishop of the city and church father. The Vandals stormed the city and spread death and sorrow, but Augustine was spared the final horror; he died on August 28, 430, about a year before the Vandals returned and finally overcame the city. By then Vandal aggression had prompted a large-scale imperial counteroffensive led by count Boniface. In 431 an imperial expedition from the east led by the generalissimo Aspar joined forces with Boniface but suffered defeat and had to withdraw in tatters. The future eastern emperor Marcian (d. 457) served in the expedition and fell into Vandal hands. He helped broker the resulting peace, which recognized Vandal possession of much of Roman Numidia, the lands of what is now eastern Algeria. The Romans licked their wounds but could in no way accept barbarians in possession of one of the most productive cornlands and who threatened the richest group of provinces of the whole of the Roman west. In 442 the emperor Theodosius II dispatched a powerful force from the east with the aim of dislodging the Vandals. It too was defeated and in 444 the Romans were forced to recognize Vandal control over the provinces of Byzacena, Proconsularis, and Numidia, the regions today comprising eastern Algeria and Tunisia—rich districts with vast farmland and numerous cities. In 455 the Vandals sacked Rome, the second time the great city had suffered sack in fifty years, having been plundered by Alaric in 410. The eastern emperor Marcian had his own problems to deal with, namely the Huns, and therefore sent no retaliatory expedition.
Instead, Constantinople finally responded in 461 in conjunction with the capable western emperor, Majorian (457–61), but Majorian's crossing to Africa from Spain was frustrated by traitors in his midst who burned the expeditionary ships and undid the western efforts. By this time the Vandals had established a powerful fleet and turned to piracy; they threatened the Mediterranean coastlands as far as Constantinople itself. In 468 the emperor Leo I launched another massive attack against Vandal North Africa under the command of his brother-in-law Basiliskos; Prokopios records that the expedition cost the staggering sum of 130,000 lbs. of gold. The expedition began promisingly enough. Leo sent the commander Marcellinus to Sardinia, which was easily captured, while another army under Heraclius advanced to Tripolis (modern Tripoli) and captured it. Basiliskos, however, landed somewhere near modern Hammam Lif, about 27 miles from Carthage. There he received envoys from Geiseric who begged him to wait while the Vandals took counsel among themselves and determined the course of negotiations. While Basiliskos hesitated, the Vandals assembled their fleet and launched a surprise attack using fire ships and burned most of the anchored Roman fleet to cinders. As his ship was overwhelmed, Basiliskos leaped into the sea in full armor and committed suicide.
The stain on Roman honor from the Basiliskos affair was deep; rumors abounded of his incompetence, corruption, or outright collusion with the enemy. The waste of treasure and the loss of life was so severe that the eastern empire made no more effort to dislodge the Vandals and to recover Africa. As the fifth century deepened and the Hunnic threat receded, the east settled into an uneasy relationship with the former imperial territories of North Africa, trading and exchanging diplomatic contacts, but never allowing the Vandals to think that Africa was rightly theirs. The emperor Zeno established an "endless peace" with the Vandal foe, binding them with oaths to cease aggression against Roman territory. Upon the death of Geiseric, his eldest son Huneric (477–84) ruled over the Vandals; he is remembered as a cruel persecutor of Catholics in favor of the heretical form of Christianity, Arianism, practiced by the Vandals and Alans. Huneric's son with his wife Eudoxia, the daughter of the former western emperor Valentinian III, was Hilderic, who claimed power in Africa in 523. Under Hilderic, relations with Constantinople warmed considerably. Hilderic himself had a personal bond with Justinian from the time the latter was a rising talent and force behind the throne of his uncle, the emperor Justin (518–27), and in a policy designed to appease local Africans and the empire, Catholics were left unmolested; many Vandals converted to the orthodox form of Christianity. The Vandal nobility found their situation threatened, as one of the key components of their identity, Arianism, was under attack; assimilation and disintegration, they reasoned, were sure to follow. When, in 530, Hilderic's younger cousin Gelimer overthrew the aged Vandal king it was with the support of the majority of the elites. Hilderic died in prison as Justinian monitored events from Constantinople with dismay. Roman diplomatic attempts to restore Hilderic failed. But Justinian was unable to act because war with Persia had commenced and his forces were tied down in Syria. By 532, Justinian sealed peace with Persia, freeing his forces and their young general Belisarios, the victor in 530 over the Persian army at Dara, to move west.
On the heels of the signing of the peace with Persia in 532, Justinian announced to his inner circle his intentions to invade the Vandal kingdom. According to a contemporary witness and one in a position to know, the general Belisarios's secretary Prokopios, the news was met with dread. Commanders feared being selected to lead the attack, lest they suffer the fate of prior expeditions, while the emperor's tax collectors and administrators recalled the ruinous expense of Leo's campaign that cost vast amounts of blood and treasure. Allegedly the most vocal opponent was the praetorian prefect John the Cappadocian, who warned the emperor of the great distances involved and the impossibility of attacking Africa while Sicily and Italy were in the hands of the Ostrogoths. Eventually, we are told, a priest from the east advised Justinian that in a dream he foresaw Justinian fulfilling his duty as protector of the Christians in Africa, and that God himself would join the Roman side in the war. Whatever the internal debates and the role of faith, there was certainly a religious element to Roman propaganda; Catholic bishops stirred the pot by relating tales of Vandal atrocities against the faithful. Justinian overcame whatever logistical and military misgivings he possessed through belief in the righteousness of his cause.
It could not have been lost on the high command in Constantinople that Justinian's plan of attack was identical to Leo's, which was operationally sound. Imperial agents responded to (or more likely incited) a rebellion by the Vandal governor of Sardinia with an embassy that drew him to the Roman side. Justinian supported another revolt, this one by the governor of Tripolitania, Prudentius, whose Roman name suggests he was not the Vandal official in charge there. Prudentius used his own troops, probably domestic bodyguards, armed householders, and Moors, to seize Tripoli. He then sent word to Justinian requesting aid and the emperor obliged with the dispatch of a force of unknown size under the tribune Tattimuth. These forces secured Tripoli while the main expeditionary army mustered in Constantinople.
The forces gathered were impressive but not overwhelming. Belisarios was in overall command of 15,000 men and men attached to his household officered most of the 5,000 cavalry. John, a native of Dyrrachium in Illyria, commanded the 10,000 infantry. Foederati included 400 Heruls, Germanic warriors who had migrated to the Danubian region from Scandinavia by the third century. Six hundred "Massagetae" Huns served—these were all mounted archers and they were to play a critical role in the tactics of the campaign. Five hundred ships carried 30,000 sailors and crewmen and 15,000 soldiers and mounts. Ninety-two warships manned by 2,000 marines protected the flotilla, the largest seen in eastern waters in at least a century. The ability of the Romans to maintain secrecy was astonishing, for strategic surprise was difficult to achieve in antiquity; merchants, spies, and travelers spread news quickly. Gelimer was clearly oblivious to the existence of the main Roman fleet; apparently an attack in force was inconceivable to him and he saw the Roman ambitions confined to nibbles at the edge of his kingdom. The Vandal king sent his brother Tzazon with 5,000 Vandal horse and 120 fast ships to attack the rebels and their Roman allies in Sardinia.
It had been seven decades since the Romans had launched such a large-scale expedition into western waters, and the lack of logistical experience told. John the Cappadocian economized on the biscuit; instead of being baked twice, the bread was placed near the furnaces of a bathhouse in the capital; by the time the fleet reached Methone in the Peloponnese, the bread was rotten and 500 soldiers died from poisoning. The water was also contaminated toward the end of the voyage and sickened some. After these difficulties, the fleet landed in Sicily near Mount Aetna. In 533 the island was under the control of the Ostrogothic kingdom of Italy, and through diplomatic exchanges the Ostrogoths had been made aware of the Roman intentions of landing there to procure supplies and use the island as a convenient springboard for the invasion. Prokopios reports the psychological effect of the unknown on the general and his men; no one knew the strength or battle worthiness of their foe, which caused considerable fear among the men and affected morale. More terrifying, though, was the prospect of fighting at sea, of which the vast majority of the army had no experience. The Vandal reputation as a naval power weighed heavily on them. In Sicily, Belisarios therefore dispatched Prokopios and other spies to Syracuse in the southeast of the island to gather intelligence about the disposition of the Vandal navy and about favorable landing spots on the African coast. In Syracuse, Prokopios met a childhood acquaintance from Palestine, a merchant, whose servant had just returned from Carthage; this man informed Prokopios that the Vandal navy had sailed for Sardinia and that Gelimer was not in Carthage, but staying four days' distance. Upon receiving this news, Belisarios embarked his men at once and sailed, past Malta and Gozzo, and anchored unopposed at Caput Vada (today Ras Kaboudia in east-central Tunisia). There the high command debated the wisdom of landing four days' march or more from Carthage in unfamiliar terrain where lack of provisions and water and exposure to enemy attack would make the advance on the Vandal perilous. Belisarios reminded his commanders that the soldiers had openly spoken of their fear of a naval engagement and that they were likely to flee if they were opposed at sea. His view carried the day and they disembarked. The journey had taken three months, rendering it all the more remarkable that news of the Roman expedition failed to reach Gelimer.
The cautious Belisarios followed Roman operational protocol; the troops established a fortified, entrenched camp. The general ordered that the dromons, the light, fast war galleys that had provided the fleet escort, anchor in a circle around the troop carriers. He assigned archers to stand watch onboard the ships in case of enemy attack. When soldiers foraged in local farmers' orchards the next day, they were severely punished and Belisarios admonished the army that they were not to antagonize the Romano-African population, whom he hoped would side with him against their Vandal overlords.
The army advanced up the coastal road from the east toward Carthage. Belisarios stationed one of his boukellarioi, John, ahead with a picked cavalry force. Ahead on the army's left rode the 600 Hun horse archers. The army moved 80 stadia (about 8 miles) each day. About 35 miles from Carthage, the armies made contact; in the evening when Belisarios and his men bivouacked within a pleasure park belonging to the Vandal king, Vandal and Roman scouts skirmished and each retired to their own camps. The Byzantines, crossing to the south of Cape Bon, lost sight of their fleet, which had to swing far to the north to round the cape. Belisarios ordered his admirals to wait about 20 miles distant from the army and not to proceed to Carthage where a Vandal naval response might be expected.
Gelimer had, in fact, been shadowing the Byzantine force for some time, tracking them on the way to Carthage where Vandal forces were mustering. The king sent his nephew Gibamund and 2,000 Vandal cavalry ahead on the left flank of the Roman army. Gelimer's strategy was to hem the Romans between his forces to the rear, those of Gibamund on the left, and reinforcements from Carthage under Ammatas, Gelimer's brother. The plan was therefore to envelop and destroy the Roman forces. Without the 5,000 Vandal troops sent to Sardinia, the Vandal and Roman armies were probably about equal in strength. Around noon, Ammatas arrived at Ad Decimum, named from its location at the tenth milestone from Carthage. In his haste, Ammatas left Carthage without his full complement of soldiers and arrived too early by the Vandals' coordinated attack plan. His men encountered John's boukellarioi elite cavalry (fig. 7.1). Outnumbered, the Vandals fought valiantly; Prokopios states that Ammatas himself killed twelve men before he fell. When their commander perished, the Vandals fled to the northwest back toward Carthage. Along their route they encountered penny packets of their countrymen advancing toward Ad Decimum; the retreating elements of Ammatas's forces panicked these men who fled with them, pursued by John to the gates of the city. John's men cut down the fleeing Vandals in great number, bloody work far out of proportion to his own numbers. About four miles to the southeast, the flanking attack of the 2,000 Vandal cavalry under Gibamund encountered the Hunnic flank guard of Belisarios. Though they were outnumbered nearly four to one, the 600 Huns had the advantage of tactical surprise, mobility, and firepower. The Vandals had never experienced steppe horse archers; terrified by the reputation and the sight of them, Gibamund and his forces panicked and ran; the Huns thus decimated the second prong of Gelimer's attack.
Belisarios had still not been informed of his lieutenant's success when at the end of the day his men constructed the normal entrenched and palisaded camp. Inside he left the baggage and 10,000 Roman infantry, taking with him his cavalry force and boukellarioi with the hopes of skirmishing with the enemy to determine their strength and capabilities. He sent the four hundred Herul foederati as a vanguard; these men encountered Gelimer's scouts and a violent clash ensued (fig. 7.2). The Heruls mounted a hill and saw the body of the Vandal army approaching. They sent riders to Belisarios, who pushed forward with the main army—Prokopios does not tell us, but it seems that this could only have been the cavalry wing, since only they were drawn up for action. The Vandals drove the Heruls from the hill and seized the high point of the battlefield. The Heruls fled to another portion of the vanguard, the boukellarioi of Belisarios, who, rather than hold fast, fled in panic (fig. 7.3).
Gelimer made the error of descending the hill; at the bottom he found the corpses of the Vandals slain by John's forces, including Ammatus. Upon seeing his dead brother, Gelimer lost his wits and the Vandal host began to disintegrate. Though Prokopios does not mention it, there was more in play; the string of corpses on the road to Carthage informed the king that his encirclement plan had failed and he now faced a possible Roman encirclement. He could not be certain that a Roman force did not bar the way to Carthage. Thus, as Belisarios's host approached, the Vandal decision to retreat to the southwest toward Numidia was not as senseless as Prokopios claimed. The fighting, which could not have amounted to much more than running skirmishing as the Vandals withdrew, ended at nightfall (fig. 7.4).
The next day Belisarios entered Carthage in order; there was no resistance. The general billeted his soldiers without incident; the discipline and good behavior of the soldiers was so exemplary that Prokopios remarked that they purchased their lunch in the marketplace the day of their entry to the city. Belisarios immediately started repairs on the dilapidated city walls and sent scouts to ascertain the whereabouts and disposition of Gelimer's forces. Not much later his men intercepted messengers who arrived from Sardinia bearing news of the defeat of the rebel governor at the hands of the Vandal general Tzazon. Gelimer and the Vandal army, which remained intact, were encamped on the plain of Bulla Regia, four days' march south of Carthage. The king sent messengers to Tzazon in Sardinia, and the Vandal army there returned and made an uncontested landing west of Carthage and marched overland to Bulla Regia where the two forces unified. Belisarios's failure to intercept and destroy this element of the Vandal force when it landed was a major blunder that Prokopios passes over in silence.
Once Gelimer and Tzazon unified their forces, they moved on Carthage, cut the main aqueduct, and guarded the roads out of the city. They also opened negotiations with the Huns in Roman service, whom they enticed to desert, and they attempted to recruit fifth columnists in the city to help their cause.
The two armies encamped opposite one another at Tricamarum, about 14 1/2 miles south of Carthage. The Vandals opened the engagement, advancing at lunch time when the Romans were at their meal. The two forces drew up against one another, with a small brook running between the front lines. Four thousand five hundred Roman cavalry arrayed themselves in three divisions along the front; the general John stationed himself in the center, and Belisarios came up behind him with 500 household guards. The Vandals and their Moorish allies formed around Tzazon's 5,000 Vandal horsemen in the center of the host. The two armies stared one another down, but since the Vandals did not take the initiative, Belisarios ordered John forward with picked cavalry drawn from the Roman center. They crossed the stream and attacked the Vandal center, but Tzazon and his men repulsed them, and the Romans retreated. The Vandals showed good discipline in their pursuit, refusing to cross the stream where the Roman force awaited them. John returned to the Roman lines, selected more cavalry, and launched a second frontal assault. This, too, the Vandals repulsed. John retired and regrouped and Belisarios committed most of his elite units to a third attack on the center. John's heroic final charge locked the center in a sharp fight. Tzazon fell in the fighting and the Vandal center broke and fled, joined by the wings of the army as the Romans began a general advance. The Romans surrounded the Vandal palisade, inside which they took shelter along with their baggage and families. In the clash that opened the battle of Tricamarum in mid- December 533, the Romans counted 50 dead, the Vandals about 800.
As Belisarios's infantry arrived on the battlefield, Gelimer understood that the Vandals could not withstand an assault on the camp by 10,000 fresh Roman infantry. Instead of an ordered retreat, though, the Vandal king fled on horseback alone. When the rest of the encampment learned of his departure, panic swept the Vandals, who ran away in chaos. The Romans plundered the camp and pursued the broken force throughout the night, enslaving the women and children and killing the males. In the orgy of plunder and captive taking, the cohesion of the Roman army dissolved completely; Belisarios watched helplessly as the men scattered and lost all discipline, enticed by the richest booty they had ever encountered. When morning came, Belisarios rallied his men, dispatched a small force of 200 to pursue Gelimer, and continued to round up the Vandal male captives. The disintegration of the Vandals was clearly complete, since the leader offered a general amnesty to the enemy and sent his men to Carthage to prepare for his arrival. The initial pursuit of Gelimer failed, and Belisarios himself led forces to intercept the king, whose existence still threatened a Vandal uprising and Moorish alliances against the Roman occupiers. The general reached Hippo Regius where he learned Gelimer had taken shelter on a nearby mountain among Moorish allies. Belisarios sent his Herul foederati under their commander Pharas to guard the mountain throughout the winter and starve out Gelimer and his followers.
Belisarios garrisoned the land and sent a force to Sardinia which submitted to Roman control and sent another unit to Caesarea in Mauretania (modern Cherchell in Algeria). In addition, the general ordered forces to the fortress of Septem on the straits of Gibraltar and seized it, along with the Balearic Islands. Finally he sent a detachment to Tripolitania to strengthen the army of Prudentius and Tattimuth to ward off Moorish and Vandal activity there. Late in the winter, facing deprivation and surrounded by the Heruls, Gelimer negotiated his surrender and was taken to Carthage where Belisarios received him and sent him to Constantinople.
Roman victory was total. The Vandal campaign ended with a spectacular recovery of the rich province of Byzacium and the riches of the African cities and countryside the Vandals had held for nearly a century. Prokopios is reserved in his praise for his general, Belisarios, and for the performance of the Roman army as a whole, laying the blame for Vandal defeat at the feet of Gelimer and the power of Fortune, rather than crediting the professionalism or skill of the army commanders and rank and file. The Romans clearly made several blunders—chief among these the failure to intercept Tzazon's reinforcing column, and Belisarios's inability to maintain discipline in the ranks upon the plundering of the Vandal encampment at Tricamarum. On balance, though, the army and the state had performed well enough. The work of imperial agents in outlying regions of Tripolitania and Sardinia distracted the Vandals and led them to disperse their forces. Experienced Roman soldiers who had just returned from years of hard fighting against the Persians proved superior to their Vandal enemy in hand-to-hand fighting. Indeed, they had proved capable of meeting and destroying much larger enemy contingents. Belisarios's leadership, maintenance of morale, and (apart from the Tricarmarum incident) excellent discipline accompanied his cautious, measured operational decisions that conserved and protected his forces. Roman losses were minimal in a campaign that extended imperial boundaries by more than 50,000 square kilometers (19,300 square miles) and more than a quarter million subjects. The empire held its African possessions for more than a century until they were swept under the rising Arab Muslim tide in the mid-seventh century.
THE EASTERN CAMPAIGNS OF NIKEPHOROS PHOKAS, 964–69
Nikephoros II Phokas rose to the office of domestikon ton scholon, replacing his father in command in 954. His elevation reflected both his reputation and the desire of his sovereign, Constantine VII, to wage war aggressively against the Muslims. The eastern emirates were a perpetual threat to the empire. Since the reign of Basil I, however, the Romans had made considerable gains in the east, destroying the heretical state of the Paulicians and striking against the raiding emirate of Melitene in a series of campaigns that culminated in the 934 sack of the city and the destruction of one of the most important Arab bases (see below) by John Kourkouas. Since the eighth century Muslim holy warriors (ghazis) flocked to Melitene or Tarsos in Cilicia to join the jihad against the Byzantines. Once Kourkouas destroyed one prop of the holy war, Nikephoros set his sights on the southern flank of the empire.
The emirate of Tarsos was one of the frontier bastions of Islam (thugur). Tarsian raiders attacked the frontier zone incessantly, and launched major invasions throughout the ninth and tenth centuries. The caliph al-Ma'mun used the city as a staging ground from which to invade Byzantium in 833, a prelude to the massive campaign his successor al-Mu'tasim launched from Tarsos in 838 that ruined the vital Byzantine city of Amorium. Major raids launched from Tarsos in 862 and 878 penetrated Cappadocia and captured several fortresses. The expedition of 878–79 consisted of 3,000 ghazis whom the Byzantines defeated at Herakleia in Cappadocia. In 894, the Muslims of Tarsos mounted another expedition in force as far as Pisidia in Anatolia. In 931, the emir Thamal al-Dulafi led a raid to Amorium and captured a huge number of slaves—women and children—who fetched 136,000 gold dinars in the slave markets. During the governorship of Thamal the frontier fortresses of Adana, Massisa (Yakapinar in Cilicia), and Mar'ash (ancient Germanikeia, modern Kahramanmaras in southeastern Turkey) were repopulated and garrisoned. When Sayf ad-Dawla rose to power in the northern Syrian city of Aleppo, he coordinated his jihad activities with the emir of Tarsos—in 950, Sayf (whose name means "Sword of the State") led a large army into Anatolia that included some 4,000 men from Tarsos. The Byzantines badly mauled this force and the raid ended in disaster.
In the tenth century, Tarsos was populous and rich. It had extensive trade connections and was well situated in the midst of the lush, well-watered Cilician plain. Although the city lay on level ground, it was large and as impressively defended as any city of the Levant. The Kydnos River flowed by the city, providing plenty of water. A near-contemporary Muslim writer, Tarsusi, noted the city had a double wall, the inner wall was of great height and strengthened by one hundred towers and crenellations offering protection to archers and to artillery—traction trebuchets and bolt casting machines. Five gates pierced each wall—those on the outer wall were iron-sheathed wood while those inside were solid iron. The inhabitants of the city included many full-time warriors and seasonal ghazis; its population was fervent in its pursuit of the jihad and had a reputation as skilled horsemen and well-trained warriors. Even boy volunteers were given weapons appropriate for their size and age when the city came under threat. According to Tarsosi, the city had 34,000 houses, two-thirds of which domiciled ghazi warriors who made Tarsos their home in fulfilling their jihad vows. Ibn Hawqal, the Muslim geographer who visited the city before its conquest by the Byzantines, wrote around 988 that the ghazis who packed the city came from all corners of the Muslim world, from places as far-flung as North Africa, Yemen, and Kerman in eastern Iran. Ibn Hawqal's assertion that Tarsos fielded 100,000 cavalry is probably off by a factor of ten, but it nonetheless remained a menace with which Nikephoros wanted to deal once and for all.
In August 963, Nikephoros Phokas seized power in a bloody coup, assuming protection over the young boy emperors Basil and Constantine. The following year, Nikephoros dispatched John Tzimiskes against the Muslims of Cilicia. Tzimiskes arrived on the warm Cilician plain in December 963 or January 964, which meant he led his Cappadocian troops through the mountain passes in bitter winter; this out-of-season attack probably surprised the Muslims. Near Adana the forces of Sayf ad-Dawla, emir of Aleppo and champion of the border wars and struggle against the Christian Romans, appeared. The Tarsian army numbered 15,000; the number of Byzantine troops is unknown. During what became known as the battle of the Mountain of Blood, the Cilician Muslims first routed a section of the Byzantine army—it is uncertain whether this was a feigned retreat, but either Tzimiskes's ambush force or his reserve cut the pursuing Muslim force in half. Consequently 4–5,000 Muslims took refuge on a steep hilltop, inaccessible to cavalry. Tzimiskes dismounted his cavalry and, along with the infantry, fought his way to the summit, massacring every Muslim defender there. This act of extreme brutality certainly lay outside the bounds of the normal conduct of war between the two powers. It shocked and demoralized the Muslims of Cilicia, and the people of Adana abandoned their town and fled to nearby Massisa (ancient Mopsuestia). John's annihilation of his enemy also deprived Sayf, who was sick and near the end of his life, of precious veteran troops and paved the way for the Byzantine assault on their main target, the city of Tarsos itself. Tzimiskes then moved toward Massisa, 20 km (about 12 1/2 miles) east, another one of the thugur cities. Like Tarsos, Massisa was splendidly fortified, and the Roman general assaults on the circuit failed; after a siege of three months, Tzimiskes abandoned the operations as the summer season began. Already a famine gripped Cilicia because of the war and the Romans could not forage enough supplies to feed themselves or their horses. Before he withdrew, Tzimiskes smashed through the defenses of al-Mallun, the port of Massisa, then pillaged and burned his way to Tarsos. Muslim reinforcements from Khurasan (eastern Iran and regions beyond) arrived in large number, but they encountered a devastated countryside and were unable to find enough provisions to maintain themselves—most drifted back home before the invasion of the emperor in the following year.
In November 964, Nikephoros II Phokas himself led an army of Romans along with Iberian (from the Caucasus) and Armenian allies into Cilicia. The emperor's aim was the capture of Tarsos and the destruction of the raiding emirate. Nikephoros divided his forces in two and put his brother Leo in charge of the force sent against Tarsos. Leo's forces were apparently driven from the walls of Tarsos by stubborn defenders. Meanwhile the emperor himself moved against Massisa, another strongly fortified city bifurcated by the Pyramos (today the Ceyhan). The siege dragged on for many months. Finally, in July 965, the historian Leo the Deacon records that Nikephoros inspected the walls of the city and instructed his sappers where to dig, and that in one night they removed enough earth to completely undermine a tower. This seems unlikely; it must have taken many days to mine sufficient material from underneath the foundations. At dawn on July 13, the Byzantines fired the wooden props beneath the tower, which collapsed with considerable loss of life among the defenders. The Romans then stormed the gap and seized half the town. The main battle took place when the Byzantines forced the bridge between the main city and the major suburb of Kafarbayya and drove the Muslim inhabitants out, seizing thousands of prisoners and huge spoils.
Nikephoros then swung the giant maw of his army to the east, against steadfast Tarsos. The Romans encamped around the city and besieged it, cutting down the orchards and destroying the food and fodder throughout the plain. The Tarsians, fractious as ever, decided to challenge the imperial host in the open field. Nikephoros arrayed his army:
The emperor himself led out from camp the bravest and most robust soldiers and arranged the divisions on the battlefield, deploying the ironclad horsemen in the van, and ordering the archers and slingers to shoot at the enemy from behind. He himself took his position on the right wing, bringing with him a vast squadron of cavalrymen, while John Tzimiskes...fought on the left.... When the emperor ordered the trumpets to sound the charge, one could see the Roman divisions move into action with incredible precision, as the entire plain sparkled with the gleam of their armor. The Tarsians could not withstand such an onslaught; forced back by the thrusts and spears and by the missiles of the [archers] shooting from behind, they immediately turned to flight.... They were overwhelmed by a terrible cowardice.
After driving the Tarsians from the field, the emperor settled in for a siege and as the citizens began to starve, they sued for peace. Most departed for Antioch in Syria under a Roman escort, and Nikephoros moved against the remaining cities of Muslim Cilicia, capturing all of them. With Tarsos destroyed and Melitene in imperial hands, the two most important border emirates were dismantled and the Romans possessed a clear path to Syria.
In 966 the emperor returned to the field and plundered northern Mesopotamia. Passing via Melitene, Nikephoros ransacked through the territory of the city of Amida (Diyarbakir), pillaged Dara, Nisibis, and Mayyafarakin (today Silvan in eastern Turkey), then turned south along the Euphrates River, arriving at the Syrian city of Membij, not far west of the river, in October. The citizens of Membij spared their city by handing over a holy tile with a miraculous image of Christ's face on it. The Byzantines then turned south toward Aleppo, where Sayf ad- Dawla resided. Sayf offered to pay tribute to Nikephoros, but the emperor scorned the offer and instead ravaged his way toward Sayf, who fled southward. Nikephoros wasted the land along the route to Antioch, then returned to Roman territory to face the Bulgarians. In 967 Sayf ad-Dawla died and the Muslims lost a vigorous and capable defender whose effectiveness in his last years was blunted by dynastic strife and ill health. Nikephoros only returned to Syria in 968, when he once again descended into Mesopotamia, and plundered as far south as the coast of Lebanon, seizing cities and fortresses and immense plunder as he went. Antioch fell to a Roman force in October the following year and Aleppo arranged tribute during December 969 or January 970. In the deep of the chilly night of December 11 the brilliant commander John Tzimiskes crept through the imperial palace and into his sleeping uncle's bedchamber and struck him down. Phokas, who was fifty-seven years old, had overseen a dramatic overhaul of the Byzantine army, under intense discipline and with tremendous battlefield effectiveness. It was this sharp instrument that Tzimiskes turned at the throats of his neighbors and passed on to his successor, Basil II, who would accomplish the conquest of Bulgaria and push the frontier to the Danube for the first time since the days of Justinian.
THE BATTLE OF KLEIDION, 1014
The empire reached its largest medieval territorial extent under Basil II, who is considered by many to have been the greatest Byzantine emperor. While the view of Basil as a perfect sovereign who was wise in counsel and indomitable in war is largely a function of his effective propaganda, his campaigns against Bulgaria led to the annexation of vast territories in the Balkans and carried Byzantium to the apex of its medieval prestige and glory. He proved to be the bane of the Bulgars, in particular, and though the statement of the historian Skylitzes that he campaigned annually against them is exaggerated, Basil vigorously pursued their subjugation. Since the seventh century, when the Bulgars first settled between the Danube and the Balkan Mountains, the Byzantines and Bulgars had fought one another for control of the region. Severe clashes were interspersed with periods of simmering peace. In 708 Justinian II suffered defeat at Bulgar hands at the first Battle of Acheloos, but Bulgar allies played a critical role in staving off the Muslim attack on Constantinople in 717–18. Although imperial forces scored several important victories throughout the eighth century, the emperors could neither dislodge the Bulgars from their homeland, nor bring them under Byzantine political domination. In 811, the major expedition of the emperor Nikephoros I, the largest in centuries, met with disaster—the Bulgars destroyed the army, killed the emperor, and mortally wounded his heir. Though periodic conflicts followed, peaceful relations between the two powers dominated the ninth century, when the Byzantines were increasingly focused on the east and the Bulgars faced Frankish expansion and threats from the steppe.
Upon his ascent to the throne, the khan Simeon (893–927) pursued hostilities with Byzantium in the hopes of becoming emperor of a unified Byzantine-Bulgar realm. In 917, at the second Battle of Acheloos (Anchialos), Simeon's forces ambushed and crushed the divided military command of Leo Phokas assisted by the fleet of Romanos Lekapenos. Simeon warred against the Romans for the rest of his reign and hostilities continued under his son and successor, Peter I (927–69), who suffered from the Byzantine-Kievan Rus' alliance negotiated by Nikephoros Phokas. The invasion of Sviatoslav, prince of Kiev culminated in heavy Bulgar defeats in 968 and 969. Under John Tzimiskes, the Byzantines drove out their former Rus' allies after their victory at the Battle of Dorostolon in the summer of 971. From this point on the Byzantines claimed rule over Bulgaria, but it would take decades of hard fighting for the empire to wear down their opponents and establish peace.
Following his suppression in 979 of the attempted usurpation of the Anatolian military magnate, Bardas Skleros, the young Basil II (he was just twenty-one at the time) sought to win his spurs against the Bulgars. Basil led a large imperial army northwest and struck Serdica (modern Sofia) and thus cut the Bulgar kingdom in half. The historian Leo the Deacon was present during the expedition in which Basil sieged Serdica for about three weeks but could accomplish nothing, allegedly due to the inexperience of his soldiers and the incompetence of the senior commanders. Clearly Basil was in large measure to blame—in all likelihood he excluded from the campaign seasoned veterans of the eastern wars who had fought for Tzimiskes a decade prior; perhaps these men had backed Bardas Skleros in his rebellion and consequently were stricken from the rolls. Whatever the case, as the army withdrew the Bulgars ambushed the Byzantines and routed them in a defile near present Ihtiman, in western Bulgaria. The imperial forces suffered heavy losses and withdrew. Little was accomplished in the war with the Bulgars since Basil, as a consequence of his internal military policies, faced renewed opposition from the Anatolian magnate families.
Only in 1001–5 could the emperor return to the theater. He made great gains, capturing Serdica in 1001 and besieging Vidin in the northwest of the kingdom at the confluence of the Sava and Danube rivers. In subsequent years Basil methodically campaigned, reorganized the political landscape by establishing Byzantine administrators, and undermined Tsar Samuel (997–1014) by dislodging his followers. In 1005 the Byzantine diplomatic offensive yielded the greatest of the low-hanging fruit of Bulgaria with the handover of Dyrrachium on the Adriatic by the influential Chryselios family who had previously acknowledged the overlordship of Samuel. Basil's efforts in 1001–5 returned to imperial control the major trans-Balkan road, the ancient Via Egnatia, and provided the Byzantines a coherent strategic front on Bulgaria's southern flank.
No sources detail action between 1005 and 1014, but when we next see the emperor in action, in 1014 at Kleidion, Basil faced a Bulgar army that blocked the passage of his army as it marched from the valley of the Strymon River in eastern Thrace to the valley of the Axios (Vardar). Samuel's men had built a series of ramparts that blocked the trunk road between lofty mountains that led from Thessaloniki to Niš. Basil's troops repeatedly assaulted the Bulgar earthworks, but the enemy repulsed these attacks and hurled missiles at the Byzantines from above. Basil was about to give up and depart for Roman territory when Nikephoros Xiphias, Basil's senior commander and active campaigner with the emperor since 1001, hatched a plan: Basil's forces would continue to attack the Bulgar wooden palisades while he picked infantry and led these troops to the south. Xiphias's men pushed through the heavily wooded mountains and, via unknown trackways made their way to the Bulgar rear (fig. 7.5). On July 29, Xiphias fell upon the Bulgars from the heights behind them. Samuel's men broke and fled as the Byzantines dismantled the makeshift fortifications. A vast number of Bulgars, said by contemporary sources to number as many as 15,000, were taken prisoner. The historian Skylitzes states that the emperor blinded these men and sent them back to Samuel with one-eyed leaders for each hundred men. Blinding was a treatment reserved for rebellious subjects, and this incident, apocryphal or not, shows Basil's determination to bring to heel the Bulgar state and reflects the view of the emperor and those who later retold the story: the lands from Thrace to the Danube belonged to the empire. Although the final annexation of Bulgaria came in 1018 only after four years' hard campaigning, the incorporation of the Bulgar realm within Byzantium was given its final impetus by the victory at Kleidion.
THE BATTLE OF SEMLIN, 1167
Manuel I Komnenos (1143–80) preserved a similar lion's image as his predecessor. Like Basil II, he was an indefatigable soldier and statesman who energetically rose to meet every challenge to his empire. These were numerous and came from all quarters. Manuel had to deal not only with the renewed Norman threat from Sicily and various Balkan powers in the west, but the arrival of the Second Crusade (1145–49) as well. Manuel has been blamed on more than one occasion, unjustly, for the stupendous failure of the Crusade, which included the luminous princes of the Europe, including Holy Roman Emperor Conrad II (1138–52) and Louis VII of France (1137–80). The Second Crusade was launched in response to the fall of the most exposed crusader enclave in Frankish Outremer, the county of Edessa that had been established by the crusader adventurer Baldwin of Boulogne in 1098 but which fell in 1144 to the forces of Imad al-Din Zengi (1127–46), atebeg of Mosul and the main protagonist in the counter crusade launched by Muslim forces in Syria. Manuel also faced encroachment in the east by the Seljuk sultanate of Rum, the political entity that thrived on the Anatolian plateau in the old heartlands of former Byzantine Asia Minor, feasting on the carcass of the old Byzantine heartland following the civil wars that rived the empire following the Battle of Mantzikert in 1071. No fewer than five usurpers or warlords had subsequently challenged Michael VII (1071–78), either directly or by carving out independent petty states from the trunk of former imperial lands, greatly eroding the strategic position of the empire and allowing the Turks to settle extensively across the plateau.
In 1144 Manuel contained the fractious prince of Antioch, Raymond of Toulouse (1136–49), who had to acknowledge Byzantine overlordship and give up claims to Cilicia in the face of mounting danger from Zengi. The young emperor skillfully played his hand, exercising a show of force to awe the Antiochian Franks while avoiding direct confrontation with Zengi, who was a useful lever against the Franks in the east. With his hands free, Manuel turned his forces against the Sultanate of Rum, Masud (1116–56); this expedition defeated the Seljuks at Akroinon (Afyon) and pressed on to plunder the environs of Masud's capital of Konya. Having made a show of his zeal against the infidel, Manuel signed a treaty and hurried to receive the western emperor Conrad and his 20,000 Germans. Despite Manuel's relationship with Conrad (he was married to Conrad's sister-in-law Bertha of Salzburg), Byzantine-German dealings were tense. The emperor had the German army ferried across the Bosphoros as quickly as possible. On the dusty high plains of Anatolia, the two German columns met separate but similar fates, being ambushed and routed with heavy losses along the route to Konya.
In the meantime, the Norman Roger II of Sicily (1130–54) had taken advantage of the pandemonium of the Second Crusade to seize Corfu and pillage the mainland cities of Thebes and Corinth where the imperial silk works were plundered and their Jewish weavers removed to Sicily. In 1148 the emperor raised a powerful army comprised of the combined forces of the eastern and western tagmas and foreign mercenaries, as well as a combined Byzantine and Venetian fleet. The historian Choniates numbers this force in the tens of thousands; among them the historian Kinnamos numbered five hundred triremes and one thousand horse transports and supply ships. Before the emperor could cross to Corfu he had to deal with a Cuman (the Cumans were Turks of the Kipchak tribal confederacy) raid across the Danube. Since the loss of Asia Minor, Greek holdings south of the Danube formed the economic core of the state and every threat from the north had to be dealt with swiftly. Late in 1148 the emperor unleashed his attack against the city of Kerkyra, on Corfu, and penned the Norman garrison in the citadel, which he sieged. The attempts on the citadel by Byzantines and their Venetian allies, who fought from siege towers erected on ships, failed when the siege ladders broke under the weight of the troops and plunged them into the sea. The Roman general, Stephanos Kontostephanos, died when the defenders cast a particularly well-aimed trebuchet round and smashed the siege engine he was supervising.
Roger II sent his admiral, George, against Thrace and Constantinople and a portion of the Byzantine squadron pursued the Sicilian vessels, preventing them from doing much damage to the rich suburbs of the capital. Roger had also forged an alliance with the Germans, Serbs, and Hungarians, who were aware that the emperor's forces were tied down in the Ionian Islands. Roger's diplomacy served the interest of Byzantium's neighbors, who chafed under the domination of their powerful neighbor and sought to expand their territories at the expense of the Romans.
By 1149, Manuel's alliance with Conrad II checked Roger II in Italy, where the German emperor had an interest in maintaining a presence and support for the papacy, which was generally opposed to Sicilian ambitions. The emperor thus momentarily abandoned his efforts against the Sicilians and returned to the Balkans, leading a campaign against the župan (count) of Serbia, Uroš II (1145–62), who was supported by Hungary. Manuel attacked Ražanj, 55 kilometers northeast of Niš and pillaged the environs. The emperor took numerous captives and continued his raid in force through the Nišava and Morava valleys. The Serbs defeated a stay-behind detachment and Manuel returned the following year, when his forces advanced up the Drina River where they encountered a Hungarian force allied with the Serbs. This turned out to be the vanguard of a much larger Hungarian army that intended to link up with Uroš Serbs and surround Manuel. The Hungarians and Serbs abandoned the river crossing at the sight of the imperial banner and Manuel personally led the charge that broke their formations—Kinnamos reported the wild chase in which the emperor in his zeal to capture the župan outstripped his supporting troops and fought a series of hand-to-hand engagements with the Hungarians. The Hungarian commander Bagan landed a sword blow across the emperor's cheek, but Manuel's heavy chain-mail mask deflected the blow, and Manuel cut off the Hungarian's hand and took him prisoner. Not long after the battle on the Drina, Uroš II sued for peace and became a vassal of Manuel.
In 1162, the death of King Géza II (1141–62) presented the opportunity for Manuel to interfere in his neighbor's realm. After a failed attempt to install an uncle of the reigning monarch, King Stephen III (1162–73), on the throne, the emperor reached a compromise whereby Géza's youngest son Béla would live at the court in Constantinople and succeed Stephen as king. Béla married one of Manuel's daughters, solidifying a Byzantine dynastic alliance. But Stephen continued to resist Byzantium in the Balkans, allying with the Holy Roman Empire under Frederick I Barbarossa (1155–90), Serbia, and the Russian principalities of Gallicia and Kiev. In violation of the treaty, Stephen designated his own son as his successor. In 1164, Stephen III and Duke Vladislav II of Bohemia marched to confront Manuel, who was stationed with his army on the Danube. Stephen agreed to cede to the empire the rich region of Syrmia, which was a family holding of Prince Béla, in exchange for the empire withdrawing its support for Stephen III's uncle, also named Stephen, who had been fighting with Byzantine assistance to claim the throne. Later in the year, Stephen III seized Sirmium, a blatant act of war against the empire.
Manuel dislodged Frederick I Barbarossa from his Hungarian alliance, and pulled onto his side the Russian principality of Kiev, as well as Venice. Stephen's forces busied themselves with the siege of Zeugminon (part of modern Belgrade, Serbia), which they seized by April 1165. Manuel led his forces northward in June 1165 and laid siege to Zeugminon. Manuel's troops stormed the city on their third attempt and plundered the place mercilessly. In the meantime, Manuel's general John Doukas had cut through Serbia and subdued the coastal cities and fortresses of Dalmatia, which Stephen III had also ceded as part of Béla's holdings. In 1166 the Hungarians defeated Byzantine forces in Dalmatia and at Sirmium.
THE BATTLE OF SIRMIUM, JULY 8,1167
Manuel responded with the dispatch of his nephew, Andronikos Kontostephanos at the head of a strong Roman army, about one-third of which were mercenaries or allied foreigners. Roman scouts captured a Hungarian who revealed that the enemy force numbered 15,000 knights, bowmen, and light infantry. The Byzantine army was probably about equal in numerical strength. Kontostephanos drew up his marching order with Cuman and Turkish horse archers and a handful of western knights in the vanguard. Behind came three divisions of Byzantine regular cavalry and kataphraktoi, followed by units of allied Turkish and western mercenary cavalry. The last line comprised a mixed formation of Roman infantry and archers alongside a battalion of armored Turks, presumably also infantry.
Dénes, count of Bács, commanded the combined Hungarian-German force. Dénes drew up his mailed knights in the front, with infantry support to the rear (fig. 7.6). The historian Choniates noted that the Hungarian battle line was drawn up in a single, dense mass, in the shape of a tower; the cavalry fronted this deep formation. The Hungarian lancers presented an awesome sight—their horses wore frontlets and breastplates (these must have been padded or mail, since plate horse armor was uncommon in Europe prior to 1250) and carried riders mailed from head to foot. In short the Hungarian forces featured the best of modern western arms and equipment. They faced a lighter Byzantine force arrayed with the Turk and Cuman horse archers in the front of the formation. Behind, Andronikos divided his army into three divisions. On the left he stationed the regular Roman cavalry. In the center stood Andronikos, commanding elements of the Varangian Guard, Hetaireia imperial guard cavalry, Serbians, probably mailed cavalry, and Italian mercenary knights. The Roman right consisted of the third element of the line of march, with German mercenary knights and Turkish cavalry and Roman kataphraktoi cavalry. Behind the right and left wings of the army Andronikos stationed supporting troops, which presumably were mainly regular cavalry and infantry flank guards and outflankers who could also support the wings when pressured. That two of these supporting battalions were cavalry seems to be indicated by how the battle unfolded.
Andronikos opened the battle by sending ahead the Turk and Cuman horse archers and presumably the light infantry as well (fig. 7.7). They were instructed to send an arrow storm into the Hungarian cavalry and thus break up the formation. In the face of a Hungarian charge Andronikos instructed them to fan out to left and right and thus sweep to the side of the Byzantine force. The Byzantine left broke in the face of the Hungarian charge and fled toward the river Sava, but two battalions stood fast—these were likely the flank guards stationed behind the left wing. Dénes led a general charge into the Byzantine center, hoping to kill Andronikos; those in the center of the Roman formation sustained the heavy cavalry charge. The Byzantine right attacked the flank of the Hungarian cavalry formation, Andronikos's men in the center of the line drew their iron maces and pressed forward for close combat, and the "routed" Byzantine left that had feigned flight returned to strike the Hungarian right flank (fig. 7.8). This envelopment broke the Hungarians, and thousands perished or were captured in the ensuing rout. Kinnamos reported that 2,000 cuirasses were taken from the dead, and countless shields, helmets, and swords came into Roman hands from the great number of fallen. The Battle of Sirmium was the greatest victory of Manuel's reign; it demonstrated that tactical skill and great discipline were still to be found in the armies of the Komnenoi, as were commanders who were able to conceive and execute complicated battlefield maneuvers. As a result of Sirmium, Hungary became a client, and upon the death of Stephen III in 1172 Manuel easily installed his protégé Béla on the Hungarian throne, which remained at peace with the empire until 1180.
The campaigns of Manuel against Hungary that culminated in the Battle of Sirmium demonstrate that, when properly led, the Byzantine army remained the finest in eastern Europe, capable of defeating heavily armed and armored western knights. But these actions also show that the strategic situation of Byzantium had deteriorated significantly—with the coalescence of larger, more organized, and economically vibrant states on all sides, the empire faced extreme challenges to its territorial integrity. While Belisarios's decisive victory over the Vandals a half millennium in the past had brought Africa under imperial control and established a peace that was largely maintained for a century, the "decisive" victory of Manuel at Sirmium delivered only twenty years of peace. In light of the capabilities of his enemies, it is small wonder that Manuel generally preferred attritive campaigns and small-war actions that wore down his foes and made enemy aggression too costly for them, rather than risking his limited forces in all-or-nothing engagements on the battlefield. In this sense, his failures are more telling than his numerous minor successes, since the emperor removed neither Sicily nor Hungary nor the Seljuks from their menacing positions along the frontiers. Instead, Manuel had to settle for a largely defensive posture in the territory he inherited from his father John.
SIEGE WARFARE
The most famous sieges in Byzantine history were defensive rather than offensive operations. Over its millennial existence as capital of the Byzantine state, Constantinople endured dozens of sieges; only two, the Fourth Crusade of 1204 and the Ottoman siege of 1453, were successful in breaching the massive defenses of the capital, which were established in the fifth century to counter the Hun threat and to expand the defended area of the city. The massive land walls cut the peninsula of the old city of Byzantion from the Golden Horn in the north to the Sea of Marmora in the south, a distance of about 6 kilometers (just under 4 miles); the curtains, whose remains are visible today, are largely the work of the early fifth century; they were completed in nine years, between 404–5 and 413. In 448 an earthquake leveled much of the defenses and exposed the city to attack by Attila the Hun, whose forces bore down on the capital. The praetorian prefect Constantine supervised a Herculean refortification effort that employed thousands of workmen, who repaired or rebuilt long stretches of the wall and fifty-seven damaged towers in just sixty days. The land walls of Anthemios included a 20 meters wide moat up to 10 feet deep whose inner side was crowned by a crenelated parapet 1.5 meters high. A terrace 20 meters wide separated the parapet from the outer wall. The outer wall was constructed of limestone ashlars broken by bands of bricks, each course bonded to a rubble and mortar core, 2 meters thick at its foundations, and rising to a height of about 9 meters; along its length stood more than seventy loop or squared towers (fig. 7.9), each rising to a height of about 14 meters. A courtyard 20 meters across separated the outer from the inner wall. Courses of well-cut limestone ashlars broken by bands of brick that helped to protect the stone from the expansion and contraction caused by weather and earthquakes, formed the shell of the 4.5–6 meters thick rubble-cored wall. The inner wall rose to a height of 12 meters; it was crowned by battlements and strengthened by ninety-six massive towers along its length. The breadth of the four-layer defenses was more than 225 feet, making it nearly impossible for enemies to use engines or mine the walls. The walls of Constantinople thus represented the pinnacle of late Roman defensive engineering and a defensive masterpiece.
The Avar-Sasanian siege of 626 was remembered by the defenders of the city in apocalyptic terms, with divine intervention on the part of the Virgin Mary and the saints saving the capital, whose defense was directed by the Patriarch Sergios. The Byzantines defeated a series of attacks on Constantinople from 667 to 673 launched by Mu'awiya, the governor of Syria; here more worldly defenses secured the safety of the citizens. By 671 at the latest, the Byzantines had developed "Greek fire," the enigmatic substance that burned on water (see Chapter 7) and equipped dromons (light warships with a single bank of oars) with the projection tubes and cooking materials needed to prepare and cast the substance. According to the chronicler Theophanes (d. 817–18), the Romans used Greek fire to burn the ships and crews in 672, and by 673 the enemy fleet withdrew.
From Muslim quarters, the greatest challenge to the empire and the city came with the sustained assaults across the years 717–718, when internal upheavals in the empire complicated the defensive efforts. Leo, the strategos of the Anatolikon theme, usurped power in 717. In 716, the Muslim army led by Maslama, the brother of the caliph Sulayman, traversed Asia Minor and sacked a number of cities and forts on its route of march. The Muslim force camped at Abydos on the Hellespont and waited for Leo to turn over the city, which he refused to do. The Arabs crossed over to the Thracian side of the straits and dug a trench the length of the peninsula and behind it erected a dry stone wall. The massive Arab force was there to stay. Inside the city, it is doubtful that Leo had more than 15,000 men at his disposal, given how depleted the army had become and the practicalities of supplying and billeting any force larger than this.
The Arab fleet arrived in September 717—Theophanes states it was a massive fleet of 1,800 vessels of all kinds—but as they passed through the straits, Leo unleashed the Greek fire from his galleys on the large transports and sent many to the bottom. The naval confrontation cost the Arabs considerable men and the loss of vital supplies. The winter of 717 was bitter and deep; the invaders lost thousands of camels, pack mules, and cavalry horses. In the spring two large Arab relief fleets bearing corn, weapons, and other supplies arrived, one from Egypt of 400 vessels, and a second from North Africa comprised of 360 ships. These large fleets feared to approach the capital due to the Greek fire ships, and they anchored on the Asian side of the straits in a sheltered bay. When the emperor learned of their location, he dispatched dromons and biremes with Greek fire siphons against them and destroyed them. The failure to resupply the besieging land army was devastating, but the Bulgar allies of the empire, who hemmed in the Muslims in Thrace and prevented their foraging, were the death blow. The field forces of the caliphate starved in their encampment, eating their pack animals and suffering from the disease that inevitably descended on the malnourished. A Bulgar attack in force killed thousands. On August 15, 718, a year after their siege began, the defeated Muslims embarked on their transports and sailed through the straits. Most of the ships were scattered or destroyed in a series of storms in the Aegean. The capital and the empire were saved.
Since Byzantium was on the defensive during most of its history, offensive siege operations were less common than defensive engagements. Offensive sieges were always important components of strategy, designed to weaken enemy strongpoints, capture people or plunder, or permanently recover territory. Especially from the later ninth century, when the empire was on the offensive first to regain lost territories from the Arabs and then later across multiple fronts to recover territories lost to the Hungarians, Turks, Arabs, and Bulgarians, the Byzantines frequently besieged cities and fortresses. With their development of the counterweight trebuchet, Roman capabilities to break cities reached their pinnacle.
A momentous event in the revival of imperial military fortunes was the capture of the capital and heart of the raiding emirate of Crete by Nikephoros Phokas, who was domestikos ton scholon during the reigns of Romanos I Lekapenos (920–44), Constantine VII (945–59), and Romanos II (959–63). Crete had fallen into Arab hands around 824, when Andalusian Arab refugees attacked and settled the island under their leader Abu Hafs. They made their capital Chandax (modern Heraklion), from whence they raided and engaged in piracy throughout the Aegean and eastern Mediterranean. The island, astride the major communications routes of the empire, posed a major threat to Byzantium's shipping and the Aegean isles. Unsurprisingly the empire struggled mightily to drive out the Muslims. Michael II (820–29) launched two attacks against the Cretans in 825–26. Both invasions met with defeat. Again in 866 the Muslims destroyed another sizable imperial expedition. In 911 a major fleet was prepared, consisting of 177 warships carrying 5,937 soldiers. In 949, the empire equipped a fleet of 128 vessels and 4,186 men; it too ended in failure. Romanos II (959–63) appointed Nikephoros Phokas as domestikos ton scholon of the east to assault Crete once more. Our sources do not record the invasion force size but likely it was similar to the earlier attempts of 911 and 949, thus around 5,000 men and over 150 warships. On July 13, 960, Nikephoros landed at Almyros, not far west of Chandax, and took the enemy by surprise. Leo the Deacon provides an account of the landing of the Byzantine army on Crete in which the transports, provided with ramps, permitted the swift disgorgement of fully armed men who immediately formed three closely ordered detachments, surprising the Cretans. The Romans drove them from the beach after a short, sharp encounter. However, the majority of sources make no mention of this fight, which in all likelihood occurred a day or two after the landing. An uncontested landing is more likely—such surprise was vital to Byzantine chances for success in avoiding a risky sea engagement or an opposed landing, as prior engagements had shown. Given the network of spies maintained by the Muslims, the frequency of shipping, and the fast spread of news, keeping secret the equipping and target of such a large flotilla was another impressive feat. In all likelihood, disinformation about the target for the expedition (possibly suggesting the Levantine coast) and the knowledge of past Byzantine failures may have been enough to convince the Cretan Muslims that they had nothing to fear.
Phokas's men apparently met scant resistance—there was probably a skirmish not long after his forces disembarked, and within three days he had created a fortified camp. In the face of the powerful Byzantine fleet and its fireships, the Muslim navy apparently withdrew. Leo the Deacon mentions a detachment under the Thrakesian theme commander Pastilas, who led his army on a foraging expedition in the countryside. Pastilas's men lost discipline: after they plundered and got drunk on local wine, an Arab force ambushed them and inflicted heavy losses. Pastilas himself fought valiantly, even after his horse was killed under him, but his death in the fray caused his men to panic and flee.
Nikephoros entrenched his forces around Chandax. The city had a high wall of packed earth and a double moat that rendered attack difficult. The Roman general therefore constructed a palisade between the city and the sea and cut it off from any possibility of maritime resupply. The Byzantine fleet patrolled the coastal waters, wary of a Muslim relief force from one of the major Mediterranean powers, Syria, Egypt, or North Africa. Although 'Abd al-Aziz, the emir of Crete, sent calls for help to rulers throughout the Mediterranean, no Arab reinforcements arrived. Phokas sent strong detachments around the island to subdue the numerous fortresses and cities and these missions proved successful. In the meantime, Leo the Deacon reports a night battle between Phokas and Muslim forces on the island but this is not confirmed by other sources. According to Leo's account, local Christians informed the domestikos of a large enemy force lurking nearby that intended a surprise attack on the besieging Romans. Phokas led his troops on a night march and, when he located the enemy, had his soldiers sleep through the day, then at night he surrounded the hilltop on which the Muslims were encamped and annihilated them; he ordered his men to display some of the severed heads of the fallen before the battlements and for others to be cast over the walls with trebuchets. By the end of 960 Chandax was isolated and most of the island was under Roman control. Nikephoros had launched several frontal assaults, supported by traction trebuchet bombardments, but these direct attacks failed. The Byzantines settled down for a long siege that lasted throughout the autumn of 960 and into the winter of 960–61.
The winter of 960–61 was savage. Both the besiegers and besieged suffered terribly from the cold and from lack of provisions. There seems to have been a dearth of supplies within the empire, and the stormy weather made seaborne resupply of the imperial army difficult. Phokas worked hard to bolster his men's morale and requested additional provisions from Constantinople—despite the winter the ships arrived with supplies from the capital. Just as prescribed in the military handbooks, Phokas began to mine the walls of Chandax. By the night of March 6, 961, a large section had been compromised and the wooden props were fired, probably at dawn on March 7, when the Byzantine troops stormed through the gaping holes in the circuit. Fierce street fighting broke out, and the Byzantines massacred the Muslim inhabitants of the city until Phokas finally got his troops under control and ended the slaughter.
Over the course of his reign the emperor John II Komnenos campaigned with a large siege train that included trebuchets, and we see his armies in city-taking operations throughout the east. Throughout the 1130s the Armenian Prince Leo, whose people had moved into Cilicia in what is today southeastern Mediterranean Turkey, seized imperial holdings there. The rich Cilician plain and its cities remained a cornerstone of Byzantine strategy, viewed as key for eastern communications with Antioch, over which the empire claimed lordship, and with the Levantine seaboard to the south. In 1136–37 Leo threatened Seleukia, an important imperial port, and John II responded with a large-scale counterattack. In 1137 the emperor crossed the Anatolian plateau and descended through the Cilician Gates, seizing Adana and Tarsos. John then moved east against Anazarbos, which was once more a strongly fortified and flourishing city. Choniates describes the place as having a high citadel on a bluff (the remains of which one can see today) and a strong curtain wall protected by artillery. John dispatched two battalions of Turkish mercenaries to test the Armenian defenders, who sallied to meet them and forced the Turks to withdraw. Several taxiarchs of imperial troops rushed forward to support the Turkish vanguard; together they drove the Armenians inside the walls. The Romans then invested the city, erecting their huge stone-throwing machines behind wooden lattice works. The Armenian defenders conducted a stubborn defense, using traction trebuchets mounted on the walls to cast stone and red-hot iron pellets at the Roman besiegers. An Armenian sortie burned the Roman heavy artillery.
John immediately ordered the engines repaired and surrounded their positions with earthworks and clay bricks. The counterweight machines could operate from behind these dugouts, while the Armenian red-hot iron pellets struck the defenses harmlessly. The empire's trebuchets smashed the walls while the defenders despaired of relief and, as the breaches across the curtain wall grew, the Armenians surrendered. After the fall of Anazarbos, the emperor led his forces to the fortress town of Baka (unlocalized), a citadel perched on a high bluff and garrisoned by a strong Armenian force commanded by a certain Constantine. John first attempted to negotiate surrender, but after being rebuffed, the emperor again set his machines to work. The scene of initial resistance then the destruction of the defensive works by Roman artillery was repeated throughout John's campaign into Mesopotamia and northern Syria, where he reduced a number of Muslim fortresses and cities and then, after a two-year campaign, retired to his own territory. Since Nikephoros II Phokas's day nearly two centuries prior, when the reduction of an impressive defensive work using artillery was unheard of, Byzantine siege tactics had changed considerably. Now, artillery bombardment inevitably brought about surrender or a successful assault, since the trebuchet "city breakers" rendered a decisive advantage to the besiegers.
## EIGHT
## THE BYZANTINE ART OF WAR
The Byzantine Empire's existence spanned more than a millennium from the establishment of the new capital, Constantinople, in the east until its final capture by the Ottoman Turks on May 29, 1453. Over the course of its history, the empire fought innumerable wars against a host of different enemies who sought to destroy, plunder, or settle within its borders. Though the Byzantines suffered numerous defeats, their military record is one of the greatest in European and Asian history, maintaining the security of a state that endured continuous challenges to its territory and existence. The massive shock of the Persian conquests of the seventh century, followed by the miraculous imperial recovery led by the emperor Heraclius, who lived to see his gains unravel at the hands of the Muslim enemy, were contests similar to those that had destroyed the western Roman Empire. Despite the defeats and the loss of the greater share of their population, territory, and fiscal resources, the Romans in the eastern Mediterranean fought on for another seven centuries—a feat unparalleled in the military annals of western Eurasia and Europe.
STRENGTHS
The strengths of the empire's military apparatus and fighting men were many. A system of discipline and drill inherited from Rome waxed and waned throughout the history of the Byzantine state. On balance, however, the cultural values of relying on a core of professional soldiers, trained under officers who themselves were usually the product of the military institutions in which they served, meant that lessons from the battlefield could be absorbed and learned from. Numerous defeats show that individual commanders and armies and their execution of strategy and tactics often did not meet the ideals expressed in the military handbooks, yet aspects of the ideal armies based on expert leadership, caution, and flexibility were achieved to varying degrees. As the handbooks repeatedly stress, even when the outcome of battle seemed to favor the Romans, many factors could undermine the effort and steal victory from them. They had learned these lessons through numerous failures and military breakdowns on fields stained with Roman blood from Adrianople to Yarmuk—engagements where the Romans probably outnumbered and outclassed their enemy, but nonetheless suffered tremendous defeats. The caution instilled by these lessons, and Roman knowledge of their own forces' limits and the extreme difficulty of replacing skilled fighting men, shaped the behavior of most commanders. The Byzantines learned early that, deprived of the resources of the unified Roman Mediterranean, they would often be outnumbered. They therefore adapted to the realities of the military parity or superiority of their enemies. Conservation of forces became a central pillar of Byzantine military doctrine, one that proved to be a key to their tremendous endurance in warfare and critical to the survival of Byzantine civilization.
The high point of military culture, when the Byzantine state was at its most bellicose and militarily successful, occurred during the Cappadocian interlude, when powerful warlords with deep family connections and interests among the border-warrior elites of the eastern frontiers dominated politics: Romanos I, Nikephoros II Phokas, and John Tzimiskes. Nikephoros was truly a revolutionary figure, virtually unknown today, who reshaped the field army into a powerful offensive force without peer in the Mediterranean world. The principles of Byzantine warfare, articulated as early as the sixth century in the _Strategikon_ and elaborated and refined against the Muslim foes in the east, demanded patience and indirect confrontation. The defensive depth of the empire, with its frontiers girded by mountainous terrain and the sea, allowed the Byzantine army to harry and wear down determined enemy forces; Roman methods of harassment, disruption of supply, and containment made it difficult for invaders to safely occupy conquered territory. The enduring Byzantine belief in the empire as a state ordained and supported by the Christian God allowed them to absorb shocking defeats that would have broken lesser peoples; it is a testament to the core Byzantine identity that major reverses suffered at the hands of the Arabs, Turks, and Normans did not lead to complete collapse. The Byzantines instead overcame these failures. The Roman abilities to adapt, to learn from the enemy, and to modify their tactics were fundamental to their success. The principles of Byzantine warfare grew from experience gleaned over centuries of confrontation and the confidence that, while the empire was eternal and warfare would always dog the state, no enemy was ordained by God to overthrow them. Only after the fall of Constantinople in 1204 to Christian crusaders was this unshakable faith in Byzantine superiority and destiny eroded. The survival of the Christian Roman Empire as a political and cultural entity over more than a thousand-year span bears ample testimony to the effectiveness of Byzantine strategy and tactics.
WEAKNESSES
In many ways the weaknesses of the Byzantine military were corollaries of its strengths. While the size and topography of the empire, especially the vast area and inhospitable terrain of much of the Anatolian plateau, favored the Roman state, its position astride the meeting point of Europe and Asia rendered it vulnerable to attack from the Eurasian steppe to the north. From the seventh century the Muslim Arab caliphate, animated by a sense of religious destiny unmatched outside of Constantinople, proved a more dangerous enemy than the Sasanians had ever posed, and as Europe awoke from the Dark Ages at the end of the tenth century, the new kingdoms that coalesced around the Mediterranean sometimes gazed upon the riches and opulence of their Christian neighbors with suspicious and greedy eyes. Access to trade routes and the wealth of the east drew many western powers into the Byzantine world, first as adventurers and then as conquerors. As wealth, militarization, and Catholic Christian bigotry increased among the western powers and found their perfect embodiment in the Normans and their successors, the Byzantines found their dominant place in the Balkans and Italy undercut. In the pre-modern era, no state save perhaps China had to sustain conflicts on three frontiers simultaneously, yet this is where the Roman state found itself in the eleventh and twelfth centuries, when steppe peoples, the Catholics of Hungary and Norman Italy, and the Muslim Turks to the east all posed grave challenges to an increasingly beleaguered empire.
Modern critics have been dismissive of obvious Byzantine weaknesses—after Basil II their ability to conduct crushing offensive operations and deliver a knockout blow to opponents when it was most needed was sorely lacking. The Roman predilection for defense and attrition often precluded the types of commanders and soldiers who could meet the challenge of decisive battle. In the eleventh century the Byzantine field army suffered some of its most crippling defeats—at Mantzikert in 1071 and only a decade later against the Normans at Dyrrachium; these battles exposed Roman weaknesses of command and control, but they also destroyed much of the old tagmatic armies and led to an increasing reliance on foreign professionals.
Ironically, while adaptation to the exigencies of warfare was a Byzantine hallmark, their greatest military failures were direct results of an inability to adapt to the new strategic realities of the Mediterranean world. While it would be wrong to see the Norman knight in the revolutionary light that he is often cast, the western mounted warrior was a new phenomenon in eastern warfare and much of his initial success was due to its novelty. The most catastrophic Roman military failing prior to the fall of the empire, the siege of Constantinople in 1204, was dealt by westerners. Not unlike their modern descendants, the Byzantines built their armies to fight the greatest threat; for centuries this had arisen in the steppe lands north of the Danube and the Muslim east. Heavily equipped cavalry had always been a part of Byzantine warfare, but never a dominant one, and the failure to match western knights with equally impressive regiments of kataphraktoi on the scale of their enemies denied the Byzantines the opportunity to end the threat posed by westerners from the eleventh century onward.
Leadership was another area of weakness. In the eleventh century, deepening divisions among the military and civilian aristocracies exposed fractures in society whose existence hindered the effective governance and martialing of resources in a state that had to be on a constant footing for war. The desertion of prominent Byzantine aristocrats to the Normans of Robert Guiscard and Bohemund exposed rifts in elite society on which the state depended and which were never healed, even though the Komnenoi dynasty was able to quell these for more than a century. The increasing fractiousness of the military aristocracy required a vigorous, powerful, and deft ruler at the helm at all times. Threats to the empire from within and without necessitated later Byzantine emperors be superior war leaders who campaigned in person from one end of the empire to another and dealt with the complicated web of foreign and domestic politics in which the empire was embroiled. This need for central control embodied in the person of the emperor developed to the detriment of senior officers, who rarely fought wars with the full confidence and resources of the state in the manner of Belisarios or Nikephoros II Phokas. When Manuel I Komnenos died in 1180 and left his minor son on the throne, no one emerged with the late emperor's talents or energy and the dynasty perished.
WHY DID THE EMPIRE FALL?
After it survived the onslaught of innumerable foes for a thousand years, the Byzantine Empire's fall in 1453 still surprises today. Unlike the earlier fall of the Roman Empire, which has spawned numerous books and an endless number of explanatory theories, the end of Byzantium is today relatively unexplored. Some factors are evident in the decline and fall of the state, namely the rise of a hereditary aristocracy and their takeover of the state and the levers of wealth. Land ownership and upward mobility were constrained and the peasantry largely impoverished during the era of the Komnenoi and after. Earlier scholars have erred in propagating the myth of the robust and happy Byzantine peasantry in earlier eras who formed the backbone of the empire. The truth is that peasants, no matter how well off, were incapable of evading taxation and thwarting the state in the way that aristocrats could. The loss of circulating wealth and the concentration of more and more land into the hands of the powerful starved the state of precious resources and access to manpower it needed to perpetuate itself.
I have noted above that, militarily, the Byzantine high command failed to meet the crusader threat of 1204 that ruined the fortunes of the empire and ensured its destruction. In this moment of crisis, no capable war leader emerged as the empire had produced on countless occasions in the past. This, too, can be partly laid at the feet of the Komnenoi, whose consolidation of power and need for aristocratic support meant fewer chances for advancement from outside this cadre; military men of real ability from humble backgrounds no doubt had limited opportunities under this system.
In tactical terms, the Byzantines suffered from a decline in the quality of local troops. The emergency conditions at the start of Alexios's rule, in which the heavy attrition of the imperial field armies led to reliance on foreign mercenaries, became a permanent feature later in his reign. The main problem with using foreign mercenaries was not their reliability (they generally could be counted on as much as locals) but their cost. Without the Greek professionals to staff the permanent rank-and-file core of the imperial field forces, the pillars of the Byzantine art of war became unsustainable. The loss of local forces meant the disruption of strategy and training regimens. Of course local troops continued to serve in number, but the loss of intensive drill, stamina, and flexibility among local forces coincided with the increased reliance on easily recruited foreign professionals. These men, while capable soldiers, could not be molded into Byzantine soldiers like those who served under Nikephoros II Phokas. By the time the crusaders arrived—jealous of Greek wealth, suspicious of Greek religion and culture, and convinced their mission was just—the Byzantines possessed neither the commander on the spot nor the well-drilled armies of past generations.
When the forces of the Fourth Crusade sacked Constantinople, the nerve center of the empire was ruptured. The parasitical state of the Latin Empire of Constantinople that the Franks formed on the carcass of most former Byzantine lands leached much of the vigor from the Greek elite. The Greek successor states that formed around the Latin Empire were more Greek than Byzantine—led by bickering, petty dynasts who were unable to recover the economic control of the state, nor could they repair the military institutions that were irretrievably broken. The seeds of the fall of the empire in 1453 were therefore sown much earlier, in the social, economic, and military mistakes of the twelfth and thirteenth centuries.
BYZANTINE CONTRIBUTIONS TO WARFARE
In his classic study of the history of warfare in support of his theory of the indirect approach, military analyst Liddell Hart focused on the battles of Belisarios and Narses in the Gothic Wars. In these campaigns, he saw the articulation of the defensive-offensive strategy that would predominate in the Byzantine military approach throughout its history. Liddell Hart viewed Belisarios's and Narses's cautious method (in which the Byzantines tested opponents, exposed their weaknesses, and then acted decisively to exploit those failings) as early manifestations of the "indirect approach." If one subscribes to his view that frontal assaults were generally bound to fail and in order to succeed other means of attack were required, then from the preceding pages it should be obvious that the Byzantines utilized on numerous occasions this kind of strategy. By understanding their enemy's deficiencies, medieval Roman commanders relied on maneuver to time their engagements with superior foes and were often able to select the battleground where large-scale encounters took place. Fuller, in his analysis of decisive battles in history, saw the generalship of both Belisarios at Tricamarum and Narses at Taginae as moments of Roman brilliance, but followed Gibbon and other early modern historians' grim assessment of Byzantine life by asserting that Italy and Africa would have been better off without Justinian's military intervention. Fuller included the siege of Constantinople of 717–18 among his studies of decisive battles in world history which, combined with the victory of the Frankish leader Charles Martel at Poitiers in 732, marked the high-water mark of Muslim expansion in the Mediterranean world and ensured the survival of the Christian successor kingdoms of the Roman Empire in the west, whose future was their own and not in the hands of the caliph. Although such views are today unfashionable, there can be little debate that Muslim forces, unchecked by Byzantium, would have based themselves in Greece with the real possibility of expansion along the northern shores of the Mediterranean; had the Muslims conquered Byzantium, the fate of the nascent papacy and of Christian Italy and France would have been very different.
In the practice of warfare, a full assessment of the direct legacy of Byzantium within western and eastern traditions is difficult. While the Byzantines certainly influenced their neighbors and those who served in their armies, we have no writings that directly indicate what knowledge was absorbed by Roman allies and enemies and either perpetuated or passed on to others. The relationships between late antique and medieval military handbooks in Greek, Persian, and Arabic have not been studied to appreciate their relationship with one another and the means of transmission of the ideas that they contained. While the Latin military writer Vegetius was known in medieval western Europe, Greek military handbooks do not seem to have traveled outside of the empire and were unknown among westerners before the Renaissance. There was probably much more influence on the Arabic tradition, whose equipment and methods of warfare were often quite close to the Romans' own.
Service in the Byzantine army proved a valuable experience for many westerners and of great historical import. The most famous mercenary known to us was Harald Hardrada (d. 1066) who, following the defeat and death of his half-brother, Saint Olaf, king of Norway (1015–28) at the famous Battle of Stiklestad (1030) traveled to Kiev. Sometime in the early 1030s, Harald journeyed to Constantinople where he and his retinue joined the Varangian Guard. The Norseman won fame and glory, fighting throughout the Mediterranean as far afield as Sicily. During his career Harald gained such an immense quantity of loot that he was able to finance his return home and seizure of the throne of Norway. Some of the plunder may well have equipped the great Norse fleet that assaulted the northern shores of England, where Harald landed in the autumn of 1066. The Norse fought the English army on September 25, and though the Anglo-Saxons defeated them and killed Harald, the old king's attack fatally weakened the Anglo-Saxon host and contributed in no small way to the historic victory of William the Bastard of Normandy at Hastings on October 14. While Harald seemed not to have employed Byzantine tactics, his campaigns and subsequent actions were founded upon the wealth he and his following gained in imperial service.
Certain tactics were passed on to the west via mercenaries who served in Byzantine armies. Theotokis noted that the Normans employed the feigned retreat in their invasion of Sicily in 1061, at Hastings in 1066, and again at Dyrrachium in 1081. Though Theotokis suggests that these tactics were learned by Normans in the east from the Seljuks, by the eleventh century the feigned retreat was such a fundament of Byzantine military doctrine that we probably need look no further. Interestingly, the Norman invasion of Sicily in 1061 took the same invasion route used by the Byzantine general George Maniakes in his campaign of 1038, which included Norman mercenaries among the imperial soldiery. The Normans who arrived in Italy possessed no clear military organization, but after their service as Byzantine mercenaries, they adopted the 300-man field unit, modeled on the Byzantine _bandon_ into which they had been grouped, in order to organize their own battle armies. These Norman units were further based on units of ten, also adopted from the Byzantine military norms.
The Byzantines adapted and refined a number of key battlefield technologies and introduced many other Asiatic modes of warfare to their neighbors. Equipment certainly trickled via confrontation, trade, or mercenary service from Byzantium to their neighbors and allies. The stirrup entered the Roman world via the Avars, and by the time the _Strategikon_ was written around the early seventh century the army had adopted this technology. Although Lynn White famously credited the adoption of the stirrup as the first in a chain of events that led to a feudal revolution and the rise of the heavily armored medieval knight, more recent historians have been critical of major elements of his thesis. There is, however, little doubt that the stirrup created a better platform for the mounted warrior and was an important, though probably not decisive, piece of equipment in the history of warfare. Throughout the seventh century, the stirrup likely spread via the Byzantines to several of the surrounding Mediterranean peoples. From the tenth century onward, the warriors of the Kievan Rus' were equipped in a fashion that owed much to Byzantine influence, including their use of the kite-shaped shield. The kite-shaped shield probably spread to the Normans via their service as imperial mercenaries. By the time of the western European return to the military offensive, though, Byzantium had little new to offer them in terms of personal arms.
The Normans adopted other critical Byzantine technologies, especially in the area of logistics and organization. While northern European peoples around the year 1050 had limited knowledge of the large-scale transport of men and supplies by sea, the Byzantines had considerable experience in large-scale naval expeditions, as evidenced by their numerous invasions of Crete, Italy, and Sicily. The fact that in their invasions of Sicily from 1060–64 the Normans used horse transports fabricated in southern ports that were under Byzantine dominion or were culturally Greek argues for an important transferal of knowledge. Bachrach has argued that William the Bastard used men from southern Italy and Sicily to fashion and maneuver his own horse transports prior to his fateful invasion of England in 1066.
Greek Fire
Byzantine adaptations and advances in incendiaries and artillery made a marked impact on warfare in medieval Europe. The most famous Byzantine invention in warfare was "Greek fire" or "liquid fire," the exact composition of which is lost. The aura of the substance as a secret weapon, its decisive role in several famous battles in Byzantine history, and the fact that the recipe for its creation has vanished add to the mystery of Greek fire today. The chronicler Theophanes linked the invention of Greek fire with an individual named Kallinikos, a Syrian from Heliopolis (either today's Ba 'albek in Lebanon or Membij in Syria) which was then under Arab control. According to Theophanes the new weapon was used by the Byzantine navy to destroy the Muslim fleet during the siege of 674–78; the chronicler's notice makes it clear that the weapon was a key part of the Roman victory and was a novel device. Understandably, modern scholars have questioned the timing and novelty of the weapon—incendiary weapons had been used extensively throughout antiquity, including sulfur mixed with pitch that burned on water as well as fire arrows coated in sulfur, resin, asphalt, and pitch mentioned by Vegetius. Greek fire burned on water, which made it all the more terrifying at sea where it was primarily used. The system used to project it created the world's first flamethrower. On ships, bronze or copper cylinders contained the naptharesin Greek fire compound, which was preheated by a fire fueled by slow-burning flax fibers. A simple force pump pushed the highly flammable mixture into a swiveling projection tip; Haldon's modern experiments show that a light petroleum-based liquid and a simple force pump could effectively deliver a devastating wall of fire, smoke, and heat at a range of up to 15 meters. Everything and everyone downwind of the stream of fire would be charred or rendered unfit for service. In combat, Greek ships pressed close to their opponents, and through tubes projecting from the bow, sides, or stern of the vessel elite squads discharged the flaming mixture onto their enemy's ships (fig. 8.1). Since the liquid did not vaporize upon ignition, the stream of burning Greek fire tended to arch down, which rendered it all the more effective against the low-riding galleys used in medieval Mediterranean warfare.
Like most new weapons, the psychological shock of Greek fire amplified its effectiveness. In the section of the _Taktika_ that dealt with naval warfare, the emperor Leo VI noted the use of "prepared fire with thunder and fiery smoke discharged through siphons, blackening them with smoke." Crews unaccustomed to encountering the naptha firestorm at sea were shocked and overwhelmed by the experience. Roman use of the liquid combustible was critical during the Arab sieges of 674–78 and 717–18. When the fleet of the amir of Tarsos attacked the town of Euripos in Greece in 883, the strategos of Hellas and his troops devastated the Muslim fleet using Greek fire discharged from the fortifications. Greek fire once again proved decisive against the Rus attack of 941, when fifteen mothballed galleys equipped with fire-projecting siphons were pressed into service and destroyed much of the enemy fleet.
Byzantine preparations of Greek fire remained a closely held state secret. Although neighboring peoples captured components of the projection siphons, pumps, and the substance itself, neither the Arabs nor Bulgars were able to duplicate the Byzantine delivery system. When in 812 the Bulgars seized the town of Develtos on the Black Sea, thirty-six bronze cauldrons and siphon systems fell into their hands along with some of the fuel to supply them, but the Bulgars were unable to make use of the weapon. Likewise, although the Arabs managed to utilize a combustible mixture like Greek fire that burned on water, and though they captured ships equipped with the siphons and preparation chambers used to cook the mixture and translated Byzantine military literature that spoke of it, we do not know whether the Muslims were able to duplicate the delivery systems and the precise recipe for making true Greek fire. On one occasion the Muslims did use siphons, as during their attack on Thessaloniki by the renegade Leo of Tripoli, whose men fired from bridges mounted on their ship masts. Muslim Arab forces generally seem to have cast their combustible version of Greek fire in grenades or ceramic pots by hand or from trebuchets. Western peoples had direct experience of the Byzantine use of the weapon; Anna Komnene describes an attack against the Pisans launched by her father Alexios I:
On hearing this the Emperor ordered ships to be furnished by all the countries under the Roman sway. He had a number built in the capital itself and would at intervals go round in a monoreme and instruct the shipwrights how to make them As he knew that the Pisans were skilled in sea warfare and dreaded a battle with them, on the prow of each ship he had a head fixed of a lion or other land-animal, made in brass or iron with the mouth open and then gilded over, so that their mere aspect was terrifying. And the fire which was to be directed against the enemy through tubes he made to pass through the mouths of the beasts, so that it seemed as if the lions and the other similar monsters were vomiting the fire.
In 1081, the Venetians defeated a Norman fleet off Dyrrachium who were "skilfully blowing the fire which they call Greek and is not extinguished by water, from hidden passages of tubes beneath the waves, cunningly burned between those same waves of the sparkling sea-top a certain ship of ours [of the Normans]." It seems that in the thirteenth century the Angevin kingdom of Sicily used a combustible that they hurled from catapults, but whether they used the siphon systems of Byzantine origin is unclear.
Grenades
In addition to the ship-mounted projecting tubes, the Byzantines delivered Greek fire, as did their Arab enemies, in pottery grenades, tossed by hand or hurled using the small _cheiromangana_ (hand trebuchet) or sling staffs. Larger clay vessels were cast using traction trebuchets. Clay spheroconic vessels have been found in large numbers in excavations in Corinth, Hama, and Israel. There is considerable debate about whether these vessels are in fact medieval grenades of the kind that were clearly filled with Greek fire or for other substances, such as mercury or precious unguents, but it seems that ceramic grenades of a type similar to these currently known vessels were used in warfare. Spheroconic terracottas recovered from excavations measure about 8–10 centimeters (fig. 8.2) in diameter and have a strong resemblance to modern grenades; several recovered in Danish excavations of the citadel of the city of Hama in the 1930s appear to have come from a workshop where shells and waxy films present in excavated horizons probably indicate two of the major ingredients for the production of incendiaries—lime and naptha. In conjunction with the spheroconic vessels found there, Pentz has argued persuasively for this complex as a Greek fire grenade manufactory. Whatever the case, textual evidence notes that ceramic firebombs containing Greek fire were a standard part of the Byzantine arsenal.
Hand-held Flamethrowers
The Byzantines further refined the projection of Greek fire by producing a hand-held infantry siphon operated by an individual soldier. These were typically used on bridges suspended from ship masts, but they were also employed in city defense and in assaults on enemy personnel and fortifications. That the Romans possessed the first firmly attested hand-held flamethrower is evidenced by Nikephoros II Phokas's work, the _Praecepta militaria_ , in which infantry units were said to be equipped with small hand trebuchets to cast Greek fire, as well as hand pumps and swivel tubes used to project the substance. A manuscript illustration in Biblioteca Apostolica Vaticana MS. Vat. Gr. 1605, a text on siege warfare attributed to Heron of Byzantion, depicts a soldier on a flying bridge assaulting a city wall with a hand-held flamethrower (fig. 8.3), no doubt projecting Greek fire using a pneumatically powered siphon.
Artillery
The Byzantines did not invent the traction trebuchet, but they were certainly key in its diffusion throughout the Mediterranean world. The traction trebuchet, which originated in China, consisted of a wooden frame supporting an axle on which was mounted a pivoting arm that ended in a sling. A Roman engineer allegedly taught the Avars the secret of manufacturing the traction trebuchet prior to their use of the weapon in the siege of Thessaloniki in 587. By the end of the sixth century the Romans employed small trebuchets on wagons accompanying the baggage train as antipersonnel devices, and utilized larger machines as both offensive and defensive weapons. The latter were mounted on towers of fortresses and cities throughout the empire. A crew hove upon the short end of the beam using suspended ropes, and trained crews using average-sized counterweight trebuchets were able to cast projectiles 80–120 meters, according to the twelfth-century Arab authority al-Tarsusi. Depending on the size of the machine and the skill of the crew, traction trebuchets could propel projectiles weighing as much as 110 kilograms (250 pounds), although smaller, smoothed stones of around 5 kilograms were more common; several of these have been recovered in archaeological excavations. Smaller versions are attested in the Byzantine military handbooks, operated by one or two men and used as antipersonnel weapons that could break up enemy formations on the battlefield.
The traction trebuchet has several advantages over the torsion- and tension-powered artillery machines of the Hellenistic and early Roman periods. A simple machine with few components—Tarver's reconstruction used only about a dozen parts—it was easy to construct and portable, as each component could often be carried by one man for rapid assembly on the battlefield. Unlike the complex and dangerous torsion powered devices it replaced, the traction trebuchet was easy to man and maintain, while its range and effectiveness matched or exceeded earlier stone-throwing devices.
Counterweight Trebuchet
Historian Paul Chevedden has convincingly argued for the Byzantine origins of the counterweight trebuchet, the famous massive piece of siege artillery that marked the apogee of siege weaponry prior to the introduction of the cannon. The first historically attested use of the weapon was during the siege of Nicaea in 1097 during the First Crusade. Anna Komnene noted that during this siege operation, her father had constructed "city-takers" ( _helepoleis_ ) that did not follow conventional design. In the following century, Byzantine armies operated these engines in sieges of Laodicaea in 1104, Mylos, Zeugminon (1165), and Nicaea (1184). The scale of these machines and their destructive power led to a revolution in fortification design around 1200, as engineers attempted to counter the advantage of besieging armies, which for the first time could batter down strong walls and towers. Twelfth-century sources refer in awe to the new artillery, frequently describing them as "huge" or "frightful," a clear indication that a novel and impressively effective form of siege weapon was employed in eastern warfare.
The counterweight trebuchet operates on the same fundamental principles as its traction-powered predecessor (fig. 8.4). However, in place of the pulling crew was a massive counterweight, either fixed, such as a mass of lead or other dense substance, or a wooden coffer filled with rocks or other heavy material. A windlass was used to elevate the box, while the machine was cocked by lowering the beam and its sling-end to ground level; an iron pin inserted where the pulleys used to lower the beam provided the trigger for the counterweight trebuchet. Knocking out the pin released the arm and allowed the counterweight to fall vertically toward the ground, propelling the massive arm and its payload through an arc of ninety degrees. The addition of heavy counterweight, rigging, and winch technologies allowed for great striking power, as modern reconstructions have shown. The Danish scholar Hansen built a trebuchet with a 2,000-kilogram (4,400 pounds) counterweight that hurled 15-kilogram cement shot 168 meters, while scholars theorize that larger machines with 9-ton counterweights could fire 140-kilogram stones up to 300 meters.
The devastating capabilities of the counterweight trebuchet against masonry fortification structures are widely attested in contemporary medieval sources and in modern reconstructions. As noted above, the emperor John II Komnenos was particularly adept at siege warfare, and his counterweight trebuchets provided the firepower to batter down any fortified position on which the Romans set their sights.
From the Byzantines the crusader armies of western Europe learned of the manufacture and deployment of these heavy artillery devices. The Normans were the first to make use of them, in 1185 turning counterweight trebuchets against the Byzantines themselves at Thessaloniki. From the Byzantines and crusaders the counterweight trebuchet spread both east and west, to the Muslim Arabs, Turks, and Persians and from there to China where in the thirteenth century the Mongols employed Muslim engineers familiar with the technology. The western Europeans eagerly adopted the device; one of the most famous examples was "War Wolf," the huge machine employed by King Edward I Longshanks during the 1304 siege of Stirling Castle in Scotland. Thus, in a little over a century after its invention by the engineers of Alexios I, the counterweight trebuchet had changed the fundamentals of warfare from one end of the known world to the other.
OVERALL LEGACY
The Byzantine legacy in the history of warfare is difficult to trace. Until the development of modern ideas of strategy following the rise of the centralized state in Europe, the influence of Byzantine developments on strategy and tactics remained informal and indirect. Byzantine handbooks were read and translated within the Islamic states of the Middle East, where they certainly affected the way that commanders thought about and conducted warfare, but no systematic study has ever attempted to trace the strands of medieval Roman influence there or elsewhere. As we have seen, most of the lessons that the Byzantines had to impart in terms of organization, logistics, and the prosecution of military operations took place in the school of the battlefield, where both friend and foe alike learned a great deal about discipline, unit organization, and the movements and maintenance of large bodies of troops. The Normans provide the most obviously affected group: via the imperial armies in which they served or from southern Italians whom they fought or came to rule, the Normans readily adopted many of the strategems and military configurations of their eastern neighbors. According to Anna Komnene, during the First Crusade, when contingents of the western European armies gathered at Constantinople, Alexios I Komnenos held a council of the Latin barons and provided extensive intelligence about the Turks and instructed the crusaders on what to expect in battle, how to form up their own battle lines, how to lay ambushes, and the dangers of pursuing the Seljuks, whose use of the feigned retreat could lure the heavily armed western knights into traps where they could be surrounded and destroyed. How much of his advice the crusaders followed is unclear—certainly the standard narrative of the Battle of Dorylaeum (1097) has the crusaders barely escaping disaster. Given the western bias, no credit for the disposition of any tactical maneuvers that helped them gain the eventual victory would have been afforded their eastern allies.
Among the Turks themselves, Byzantine military methods were transferred in similar ways, both via peaceful contacts as the conquerors of the Anatolian plateau recruited Byzantines from their new territories in their armies and as elite families came into Turkish service. In practical terms we can trace Byzantine influence on the structures of the Ottoman military by the latter's likely adoption of the _pronoia_ system of land-supported cavalry units in the form of the Turkish _timar_. Other Byzantine contributions in the form of logistics, the maintenance of imperial transportation networks, and fortifications were probably adopted piecemeal since a full three centuries passed between the formation of the Ottoman state and absorption of Byzantine elements there and the final conquest of the Balkans in the fourteenth and fifteenth centuries.
The Byzantine practice of combined arms is most directly evidenced in the armies of Belisarios and Maurice onward, in which long-range fire, cavalry mobility, and the weight in defense and attack of well-drilled infantry, were present in the best and most successful Byzantine armies. Belisarios further used his elite boukellarioi as officer cadets and special forces in a surprisingly modern manner, reacting to the exigencies of the battlefield. His legacy and those of his successor Narses were felt through the military development of the Frankish kingdom, whose later reliance on cavalry probably grew in part from their experiences in Italy, where their infantry forces proved no match for Roman missile and cavalry units. In the east, the reformed army of the Macedonians revived this combined arms approach of close ordered heavy infantry supporting armored mixed cavalry formations capable of both speed and shock alongside lighter missile troops and artillery that provided ranged attack. These methods, still in force in the eleventh century when the Normans in the south were first exposed to them, parallel later Norman activities in the north, especially the armies of William the Bastard at Normandy, so closely that it is tempting to see Byzantine influence in them. The Norman and Angevin armies continued these traditions of combined arms that sustained the English in their medieval quest for empire through the end of the Hundred Years' War.
The rediscovery of Byzantine tactics in the early modern and modern eras is likewise difficult to trace, but Byzantine theory and practice foreshadowed many of the major tenets of warfare today. Jomini, whose _Art of War_ was required reading at West Point among those cadets who staffed the officer corps of the Civil War, merely mentioned Belisarios's expedition among the Vandals. No study exists of the influence of classical thought on the generals of the Civil War, but the American public was so steeped in the tradition that a cartoon of 1864 depicting General George McClellan as a "Modern Belisarius" resonated. In his study of amphibious operations, Vagts viewed the operations of Belisarios in Africa and Italy as the pinnacle of late Roman achievement in the arena, which he viewed as a rare high point in an otherwise dismal period. Students of the concept of maneuver warfare, with its emphasis on spatial mastery and the "indirect approach," could have used one of a number of Byzantine commanders as their inspiration. Indeed, Liddell Hart's "indirect approach" in warfare, which influenced British, German, and later Israeli strategists in World War II the and postwar era, depended in part on his studying Byzantine tacticians like Belisarios and Narses. Liddell Hart's friend and correspondent T. E. Lawrence noted in his classic _Seven Pillars of Wisdom_ that he, "like any other man at Oxford," had read of the wars of Belisarios. In fact, with the warfare of our era resembling the small wars fought by Lawrence, it is perhaps time for military planners to revisit the strategy and tactics of the Byzantine art of war.
## GLOSSARY
_agentes in rebus_ – early period special agents in imperial service
_ala_ (plural _alae_ ) – late Roman cavalry unit of 100–500
_annona_ – taxes in-kind used to support officials and soldiers
_bandon_ – unit that varied in number depending on the century and force composition, 50–400 men was common
_biarchus_ – lower rank, possibly junior centurion
_boukellarioi_ – in the early period, private bodyguards usually raised by generals or barbarian chiefs
_chiliarch_ – leader of 1,000
_circitor_ – military rank of uncertain function
_clibanarius_ – see _kataphraktoi_
_comes_ – count
_comitatenses_ – early Byzantine field army or mobile units (contrast with _limitanei_ )
_comitatus_ – imperial bodyguard
_dekarch_ – leader of ten men
_domestikos ton scholon_ – originally commander of the Scholae bodyguards, evolved into marshal commanding eastern or western imperial field armies
_dromos_ – imperial post system
_droungarios (droungar_ ) – commander of 1,000 men, later commander of bandon of 300–600 men
_dux_ (Greek _doux_ , pl. _doukes_ ) – senior commander; duke
_eques_ – horseman
_ethnarch_ – leader of foreign contingent
_foederati_ – allied foreigners recruited via treaty with their chiefs
_foulkon_ – dense infantry formation
_ghazi_ – Islamic holy warrior
_hekacontarch_ (or _ilarch/kentarchos_ ) – rough equivalent to Roman centurion, leader of 80–100 men
_hippotoxotai_ – horse archer
_kastron_ \- fortress
_kataphraktoi_ – heavy armed and armored cavalry
_kentarchos_ – see _hekacontarch_
_kleisourarch_ – guardian of frontier pass
_koursor_ – scout or skirmishing cavalry
_legate (legatus_ ) – Roman commander of a legion
_limitanei_ – troops stationed in frontier regions
_magister militum_ – commander of army division
_menavlatoi_ – infantry armed with menavlion
_menavlion_ – heavy spear/pike
_meros_ – division
_pedes_ – infantry
_pentarch_ – leader of five men
_pentecontarch_ – leader of fifty men
_primicerius_ – palatine guard
_pronoia_ – grant of revenue from state to support a soldier
_protectors_ – group of imperial bodyguards, third to seventh centuries
_saka_ – from Arabic _saqat_ , rearguard
_spatharios_ – (pl. _spatharii_ ) in early period designated a bodyguard, later became a court title
_strategos_ – general
_stratopedarches_ – often equivalent to strategos/general
_tagma_ – regiment (pl. _tagmata_ ) of imperial mobile imperial field regiments in Dark Ages replacing comitatenses; in the later medieval era mercenary regiments
_tasinarioi_ – light cavalry scouts and raiders
_tetrarchs_ – four leaders, called after the rule of four devised by Diocletian, with two senior emperors and two junior colleagues who would accede to the throne after the senior emperors retired
_theme (thema_ ) – medieval Byzantine province
_tiro_ – recruit
_tourma_ – a thematic unit or administrative districts
_tourmarch_ – commander of a turma
_trapezites_ – light skirmishing cavalry scouts
_tribune_ – commander of a cohort, later of a vexillatio or bandon
_vexillatio_ – vexillation, an early Byzantine cavalry unit 300 or 600 strong
_župan_ – Serbian ruler of a district, often translated as "count"
## NOTES
ABBREVIATIONS
_Alexiad_
_Anna Comnenae Alexias_ , ed. D. Reinsch (Berlin: de Gruyter, 2001)
_Anna Komnene, The Alexiad_ , trans. E. R. A. Sewter, rev. ed. (London: Penguin, 2009)
_Ammianus Marcellinus_
_Ammianus Marcellinus_ , 3 vols., trans. J. C. Rolfe (Cambridge, Mass.: Harvard University Press, 1935)
_Anonymous Strategy_
_The Anonymous Byzantine Treatise on Strategy_ , ed. and trans. G. T. Dennis, in _Three Byzantine Military Treatises_ (Washington, D.C.: Dumbarton Oaks, 1985): 1–136.
_Campaign_
_Campaign Organization and Tactics_ , ed. and trans. G. Dennis, in _Three Byzantine Military Treatises_ (Washington, D.C.: Dumbarton Oaks, 1985): 241–328.
_Choniates_
Nicetae Choniatae, _Historia_ , volume 1, ed. I. A. van Dieten (Berlin: de Gruyter, 1975).
_O City of Byzantium, Annals of Niketas Choniates_ , trans. H. J. Magoulias (Detroit: Wayne State University Press, 1984)
_CJ_
_Corpus iuris civilis_ , ed. P. Krueger, T. Mommsen, R. Schöll, W. Kroll (Berlin: Weidemann, 1954)
_Codex Iustinianus_ _http://uwacadweb.uwyo.edu/blume &justinian/Book%2012.asp_
_CTh_
_Codex Theodosianus_ , ed. Th. Mommsen, _Codex Theodosianus 1.2: Theodosiani libri XVI cum Constitutionibus Sirmondi[a]nus_ (Berlin, 1905)—trans. C. Pharr et al., _The Theodosian Code and Novels and the Sirmonidan Constitutions_ (Princeton: Princeton University Press, 1952)
_De Rebus Bellicis_
_Anonymi Auctoris De Rebus Bellicis_ , ed. R. Ireland (Leipzig: Teubner, 1984)
_A Roman Reformer and Inventor: Being a New Text of the De Rebus Bellicis_ , trans. E.A. Thompson (Oxford: Clarendon Press, 1952)
_EI2_
_The Encyclopaedia of Islam_ , 2nd edition, ed. H. A. R. Gibb (Leiden: Brill, 1954–2002)
_IGLS_
_Inscriptions grecques et latines de la Syrie_ , ed. L. Jalabert and R. Mouterde (Paris: Paul Geuthner, 1929)
_Kekaumenos_
_Strategikon_ , ed. and Italian trans. M. D. Spadaro, _Raccomandazioni e consigli di un galatuomo_ (Alexandria: Edizioni dell'Orso, 1998)
_Kinnamos_
_Ioannis Cinnami Epitome rerum ab Ioannine et Alexios Comnenos_ gestarum, ed. A. Meineke (Bonn: E. Weber, 1836)
_Deeds of John and Manuel Comnenus_ , trans. C. M. Brand (New York: Columbia University Press, 1976)
_Leo the Deacon_
_Leonis Diaconi caloënsis Historiae libri decem et Liber de velitatione bellica Nicephori Augustii_ , ed. C. B. Hasii (Bonn: E. Weber, 1828)
_The History of Leo the Deacon: Byzantine Military Expansion in the Tenth Century_ , trans. A. M. Talbot (Washington, D.C.: Dumbarton Oaks, 2005)
_ODB_
_The Oxford Dictionary of Byzantium_ , ed. A. P. Kazhdan and A.-M. Talbot with A. Cutler, T. Gregory, N. P. Ševčenko (New York: Oxford University Press, 1991)
_Ouranos_
_Le "Tactique" de Nicéphore Ouranos_ (Paris: Les Belles lettres, 1937)
_The Taktika of Nikephoros Ouranos, Chapters 56–65_ , ed. and trans. E. McGeer, in _Sowing the Dragon's Teeth: Byzantine Warfare in the Tenth Century_ (Washington, D.C.: Dumbarton Oaks, 1995)
_PLRE_
_The Prosopography of the Later Roman Empire_ , ed.A. H. M. Jones, J. R. Martindale, J. Morris (Cambridge: Cambridge University Press, 1971–1992)
_Praecepta militaria_
_The Praecaepta militaria of the Emperor Nikephoros II Phokas (963–69)_ , ed. and trans. E. McGeer, in _Sowing the Dragon's Teeth: Byzantine Warfare in the Tenth Century_ (Washington, D.C.: Dumbarton Oaks, 1995): 12–60.
_Priskos_
R. C. Blockley, ed. and trans., _The Fragmentary Classicising Historians of the Later Roman Empire: Eunapius, Olympiodorus, Priscus and Malchus_ , vol. 2: _Text, Translation and Historiographical Notes_ (Liverpool: Francis Cairns, 1983).
Prokopios, _Wars_
H. B. Dewing, ed. and trans., _Procopius_ (Cambridge, Mass.: Harvard University Press, 1969), vols. 1–5.
_Sebeos_
_The Armenian History Attributed to Sebeos_ , trans. R.W. Thompson, vols. 1–2 (Liverpool: Liverpool University Press, 1999)
_Skirmishing_
_Skirmishing_ , ed. and trans. G. T. Dennis, in _Three Byzantine Military Treatises_ (Washington, D.C.: Dumbarton Oaks, 1985): 137–240.
_Skylitzes_
_Ioannis Scylitzae Synopsis Historiarum_ , ed. I. Thurn (Berlin: de Gruyter, 1973) _A Synopsis of Byzantine History, 811–1057_ , trans. J.Wortley (Cambridge: Cambridge University Press, 2010)
_Strategikon_
_Das Strategikon des Maurikios_ , ed. and German trans. G. T. Dennis and E. Gamillscheg (Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1981)
G. T. Dennis, trans., _Maurice's Strategikon: Handbook of Byzantine Military Strategy_ (Philadelphia: University of Pennsylvania Press, 1984)
_Taktika_ (of Leo VI)
G. T. Dennis, ed. and trans., _The Taktika of Leo VI_ (Washington, D.C.: Dumbarton Oaks, 2010)
_Theophanes_
_Theophanes Chronographia_ , ed. C. de Boor, vols. 1-2 (Leipzig: Teubner, 1885)
_Theophanes, The Chronicle of Theophanes Confessor_ , AD 284–813, trans. C. A. Mango and R. Scott with G. Greatrex (Oxford: Clarendon Press, 1997)
_Vegetius_
_Epitoma rei militaris_ , ed. M. D. Reeve (Oxford: Oxford University Press, 2004)
_Vegetius, epitome of military science_ , trans. N. P. Milner (Liverpool: Liverpool University Press, 1993)
CHAPTER ONE: HISTORICAL OVERVIEW
. Michael Kulikowski, _Rome's Gothic Wars: From the Third Century to Alaric_ (Cambridge: Cambridge University Press, 2007), 100–105.
. Warren T. Treadgold, _Byzantium and Its Army, 284–1081_ (Stanford, Calif.: Stanford University Press, 1995), 55.
. Benjamin H. Isaac, _The Limits of Empire: The Roman Army in the East_ (Oxford: Clarendon Press, 1990), 169–70.
. Michael H. Dodgeon and Samuel N. C. Lieu, _The Roman Eastern Frontier and the Persian Wars (AD 226–363): A Documentary History_ (London: Routledge, 1991), 170–71.
. _Ammianus Marcellinus_ , XVIII.9–XIX.9.
. Kulikowski, _Rome's Gothic Wars_ , 128.
. Otto Maenchen-Helfen, _The World of the Huns: Studies in Their History and Culture_ (Berkeley: University of California Press, 1973), 441.
. _Ammianus Marcellinus_ , XXXI.4.6.
. _Ammianus Marcellinus_ , XXXI.13.
. J. D. Howard-Johnston, "Heraclius' Persian Campaigns and the Revival of the East Roman Empire, 622–630," _War in History_ 6 (1999).
. _Sebeos_ , p. 67.
. _EI2_ "Muta."
. Fred Donner, _The Early Islamic Conquests_ (Princeton: Princeton University Press, 1981), 131.
. These events cannot be reconstructed with certainty; see Donner, _Early Islamic Conquests_ , for the possibilities.
. Walter Kaegi, _Byzantium and the Early Islamic Conquests_ (Cambridge: Cambridge University Press, 1992) is one such effort. More recently, an overview has been provided by Hugh Kennedy, _The Great Arab Conquests: How the Spread of Islam Changed the World We Live In_ (Philadelphia: Da Capo, 2007).
. _Theophanes_ , AM 6237.
. George L. Huxley, "The Emperor Michael III and the Battle of Bishop's Meadow (A.D. 863)," _Greek, Roman, and Byzantine Studies_ (1975).
. _Theophanes_ , AM 6303.
. _Leo the Deacon_ , II.5.
. _Skylitzes_ , p. 444.
. J.-C. Cheynet, "Mantzikert: un désastre militaire?" _Byzantion_ 50 (1980).
. Marjorie Chibnall, _The Normans_ (Malden, Mass.: Blackwell, 2006), 77–81.
. _Alexiad_ , IV.6.
. John W. Birkenmeier, _The Development of the Komnenian Army: 1081–1180_ (Leiden: Brill, 2002), 62–70.
. Mark C. Bartusis, _The Late Byzantine Army: Arms and Society, 1204–1453_ (Philadelphia: University of Pennsylvania Press, 1992), 262–70.
CHAPTER TWO: LEADERSHIP
. _Taktika_ 1 and 2.
. ODB, "Dalassenos," p. 578.
. _Vegetius_ , III.22.
. _Skylitzes_ , pp. 197–198.
. _Theophanes_ , AM 6116.
. _PLRE_ , "Mundus," pp. 903–5.
. A. H. M. Jones, _The Later Roman Empire, 284–602: A Social Economic and Administrative Survey_ (London: Blackwell, 1964), 637.
. _PLRE_ , "Sittas 1," pp. 1161–63.
. PLRE, "Diogenes 2," pp. 400–401.
. _PLRE_ , "Mundus," pp. 903–5.
. Ioseph Genesios, _On the Reigns of the Emperors_ , trans. Anthony Kaldellis (Canberra: Australian Association for Byzantine Studies, 1998), IV.13.
. Michael Psellus, _Fourteen Byzantine Rulers: The Chronographia of Michael Psellus_ , trans. E. R. A. Sewter (New York: Penguin Books, 1966), 35.
. Bartusis, _The Late Byzantine Army_ , 64.
. Ibid., 210.
. John F. Haldon, _Warfare, State, and Society in the Byzantine World, 565–1204_ (London: UCL Press, 1999), 229; _Skylitzes_ , p. 211.
. _Skylitzes_ , p. 233.
. Savvas Kyriakidis, _Warfare in Late Byzantium, 1204–1453_ (Leiden: Brill, 2011), 31.
. Otto Seeck, _Notitia dignitatum: accedunt Notitia urbis Constantinopolitanae et laterculi prouinciarum_ (Berlin: Weidmannos, 1876), 78.
. Prokopios, _Wars_ , I.xiii.
. _Skylitzes_ , p. 296.
. _Leo the Deacon_ , III.2.
. _Leo the Deacon_ , X.1.
. Birkenmeier, _The Development of the Komnenian Army_ , 89.
. _Choniates_ , p. 11.
. _Choniates_ , p. 16; Kinnamos, Bk. 1.8.
. _Choniates_ , p. 20.
CHAPTER THREE: ORGANIZATION, RECRUITMENT, AND TRAINING
. Jones, _The Later Roman Empire_ , 608–36.
. John F. Haldon, _Byzantine Praetorians: An Administrative, Institutional, and Social Survey of the Opsikion and Tagmata, c. 580–900_ (Bonn: R. Habelt, 1984), 143.
. Jones, _The Later Roman Empire_ , 640.
. Ibid., 633. _De Rebus Bellicis_ , V.5.
. _Vegetius_ , III.8.
. Treadgold, _Byzantium and Its Army_ , 91; Jones, _The Later Roman Empire_ , 567, 634, 26.
. Treadgold, _Byzantium and Its Army_ , 88–89.
. Jones, _The Later Roman Empire, 284–602_ , 597.
. Warren T. Treadgold, _The Early Byzantine Historians_ (New York: Palgrave Macmillan, 2007), 188.
. Isaac, _The Limits of Empire_ , 209.
. Jones, _The Later Roman Empire_ , 664.
. Haldon, _Byzantine Praetorians_ , 164; Isaac, _The Limits of Empire_ , 210. disagrees; Treadgold, _Byzantium and Its Army_ , 93, states that these men were removed from the payroll by Justinian.
. Treadgold, _Byzantium and Its Army_ , 94.
. Ibid., 96.
. Haldon, _Byzantine Praetorians_ , 116.
. Ibid., 173.
. A. N. Oikonomides, "Les premières mentions des thèmes dans la chronique de Théophane," _Zbornik radova Vizantoloskog Instituta_ 16 (1975).
. Haldon, _Byzantine Praetorians_ , 176.
. _ODB_ , "Anatolikon," pp. 89–90; _ODB_ "Armeniakon," 177; _ODB_ "Karabisianoi," pp. 1105–6; Hélène Ahrweiler, _Byzance et la mer: la marine de guerre, la politique et les institutions maritimes de Byzance aux VIIe-XVe siècles_ (Paris: Presses universitaires de France, 1966), 19–31. Many of the issues surrounding the themes are dealt with in John F. Haldon, "Military Service, Military Lands, and the Status of Soldiers: Current Problems and Interpretations," _Dumbarton Oaks Papers_ 47 (1993).
. _Theophanes_ , AM 6159; Jadran Ferluga, "Le Clisure bizantine in Asia Minore," in _Byzantium on the Balkans: Studies on the Byzantine Administration and the Southern Slavs from the VIIth to the XIIth Centuries_ , ed. Jadran Ferluga (Amsterdam: A. M. Hakkert, 1976), 73.
. Treadgold, _Byzantium and Its Army_ , 106–8.
. _ODB_ , "Tourmarches," pp. 2100–2101.
. Treadgold, _Byzantium and Its Army_ , 100–105.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 110. Ibn Khurradadhbih, _Kitab masalik wa al-mamalik_ (Arabic), 111, 82 (French).
. Treadgold, _Byzantium and Its Army_ , 103.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 116.
. _Byzantine Praetorians_ , 191–95.
. Ibid., 196.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 111; Treadgold, _Byzantium and Its Army_ , 102; Hans Joachim Kühn and Johannes Koder, _Die byzantinische Armee im 10. und 11. Jahrhundert: Studien zur Organisation der Tagmata_ (Vienna: Fassbaender, 1991), 73 ff.
. Haldon, _Byzantine Praetorians_ , 280–81.
. Ibid., 224–26.
. Treadgold, _Byzantium and Its Army_ , 110.
. Ibid., 106; Haldon, _Warfare, State, and Society in the Byzantine World_ , 116.
. _Skirmishing_ , §16.
. Nicephorus Bryennius, _Nicéphore Bryennios histoire: Introduction, texte, traduction et notes_ , trans. P. Gautier, Corpus fontium historiae Byzantinae (Brussels: Byzantion, 1975), 282–83.
. _Leo the Deacon_ , VI.11.
. _ODB_ , "Nikephoritzes," p. 1475.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 93.
. Birkenmeier, _The Development of the Komnenian Army_ , 28.
. ODB, "Allagion," pp. 67–68.
. Rodolphe Guilland, _Recherches sur les institutions byzantines_ (Berlin: Akademie-Verlag, 1967), 596–600;Mark C. Bartusis, "The Megala Allagia and the Tzaousios: Aspects of Provincial Military Organization in Late Byzantium," _Revue des études byzantines_ 47 (1989): 203.
. Bartusis, _The Late Byzantine Army_ , 199.
. Jones, _The Later Roman Empire_ , 616; Cécile Morrisson and Jean-Claude Cheynet, "Prices and Wages in the Byzantine World," in _The Economic History of Byzantium: From the Seventh Through the Fifteenth Century_ , ed. Angeliki E. Laiou and Charalampos Bouras (Washington, D.C.: Dumbarton Oaks Research Library and Collection, 2002), 864.
. John F. Haldon, _Recruitment and Conscription in the Byzantine Army c. 550–950_ (Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1979), 26.
. Jones, _The Later Roman Empire_ , 614.
. Prokopios, _Wars_ , III.18; PLRE, "Ioannes 14" vol. 3.A, pp. 635–36;he was an _"optio,"_ here not a quartermaster, but a picked commander.
. Marie Theres Fögen, "Lombards," in _Oxford Dictionary of Byzantium_ , ed. A. P. Kazhdan (New York: Oxford University Press, 1991), 1249.
. _Sebeos_ , p. 31; Richard G. Hovannisian, _The Armenian People from Ancient to Modern Times_ (New York: St. Martins Press, 1997), 110.
. Haldon, _Recruitment and Conscription in the Byzantine Army_ , 37.
. Andreas Nikolau Stratos, _Byzantium in the Seventh Century 3. 642–668_ , trans. Harry T. Hionides (Amsterdam: Hakkert, 1975), 234.
. _Theophanes_ , AM 6184.
. Treadgold, _Byzantium and Its Army_ , 110.
. Peter Charanis, "The Armenians in the Byzantine Empire," _Byzantinoslavica_ 22 (1961).
. Jonathan Shepard, "The English and Byzantium: A Study of Their Role in the Byzantine Army in the Later Eleventh Century," _Traditio_ 29 (1973): 60.
. Bartusis, _The Late Byzantine Army_ , 192–93.
. Treadgold, _Byzantium and Its Army_ , 155.
. Morrisson and Cheynet, "Prices and Wages in the Byzantine World," 864.
. Jones, _The Later Roman Empire_ , 623–24, 34.
. Treadgold, _Byzantium and Its Army_ , 150–54.
. Morrisson and Cheynet, "Prices and Wages in the Byzantine World," 860.
. Treadgold, _Byzantium and Its Army_ , 153–54.
. Ibid., 146–48.
. Morrisson and Cheynet, "Prices and Wages in the Byzantine World," 865.
. Ibid., 862–65.
CHAPTER FOUR: EQUIPMENT AND LOGISTICS
. Simon James, "The _Fabricae_ : State Arms Factories of the Later Roman Empire," in _Military Equipment and the Identity of Roman Soldiers: Proceedings of the Fourth Roman Military Equipment Conference_ , ed. J. C. N. Coulston (Oxford: British Archaeological Reports, 1988), 257.
. Jones, _The Later Roman Empire_ , 624–25; Roger S. Bagnall and Raffaella Cribiore, _Women's Letters from Ancient Egypt, 300 BC–AD 800_ (Ann Arbor: University of Michigan Press, 2006), Letter 288.
. Karen R. Dixon and Pat Southern, _The Roman Cavalry_ (London: Routledge, 1997), 62.
. Jones, _The Later Roman Empire_ , 835.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 132.
. Haldon, "Military Service, Military Lands, and the Status of Soldiers: Current Problems and Interpretations," 17;Haldon, "Some Aspects of Byzantine Military Technology from the Sixth to the Tenth Centuries," _Byzantine and Modern Greek Studies_ 1 (1975): 42.
. Michael Hendy, _Studies in the Byzantine Monetary Economy c. 300–1450_ (Cambridge: Cambridge University Press, 1985), 628–30; John F. Haldon, _Byzantium in the Seventh Century: The Transformation of a Culture_ (Cambridge: Cambridge University Press, 1990), 232–44. See also R.-J. Lilie, "Die Zweihundertjährige Reform: zu den Anfängen der Themenorganisation im 7. und 8. Jahrhundert," _Byzantinoslavica_ 45 (1984): 190–201.
. Norbert Kamp and Joachim Wollasch, _Tradition als historische Kraft: interdisziplinäre Forschungen zur Geschichte des früheren Mittelalters_ (Berlin: de Gruyter, 1982), 254–66.
. Haldon, "Military Service, Military Lands, and the Status of Soldiers," 17.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 328, n. 8.
. _Choniates_ , p. 118.
. _Choniates_ , p. 58.
. Birkenmeier, _The Development of the Komnenian Army_ , 178.
. Walter Kaegi, "Byzantine Logistics: Problems and Perspectives," in _Feeding Mars: Logistics in Western Warfare from the Middle Ages to the Present_ , ed. J. A. Lynn (Boulder: Westview Press, 1993), 41.
. Ibid., 45.
. S. Thomas Parker, _The Roman Frontier in Central Jordan: Final Report on the Limes Arabicus Project, 1980–1989_ (Washington, D.C.: Dumbarton Oaks, 2006), 120.
. Michael Decker, "Towers, Refuges, and Fortified Farms in the Late Roman East," _Liber Annuus_ 56 (2006).
. _IGLS_ , 4.1397.1; 4.1385.
. Louis Robert, "Noms métiers dans des documents byzantins," in _Charisterion eis Anastasion K. Orlandon 1_ , ed. Anastasios K Orlandos (Athens: He en Athenais Archaiologike Hetairea, 1965), 333.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 150.
. Constantine Porphyrogenitus, _Constantine Porphyrogenitus: Three treatises on imperial military expeditions_ , trans. John F. Haldon (Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1990), 99.
. _Praecepta militaria_ , II.1–4.
. John F. Haldon, "The Organisation and Support of an Expeditionary Force: Manpower and Logistics in the Middle Byzantine Period," in To empolemo Byzantio = Byzantium at War (9th–12th c.): International Symposium 4 [of the] Institute for Byzantine Research [Athens, 1996], ed. Nicolas Oikonomides, Kostas Tsiknakes (Athens: Goulandre-Chorn Foundation, 1997), 122–23.
. Ibid., 116–21.
. Ibid., 125.
. P. Grotokowski, _Arms and Armour of the Warrior Saints_ (Leiden: Brill, 2010), 193–98.
. _Strategikon_ , I.2.
. Timothy Dawson, "Kremasmata, Kabbadion, Klibanion: Some Aspects of Middle Byzantine Military Equipment Reconsidered," _Byzantine and Modern Greek Studies_ 22 (1998): 42; Grotowski, _Arms and Armour of the Warrior Saints_ , 166–70; Taxiarchos Kolias, _Byzantinische Waffen: Ein Beitrag zur byzantinischen Waffenkunde von den Anfängen bis zur lateinischen Eroberung_ (Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1988), 54–57.
. _Strategikon_ , I.2.
. _Strategikon_ , XII.8.1–6.
. _Ouranos_ , 56.139–44; pp. 96–97.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 128.
. _Vegetius_ , I.20.
. J. C. N. Coulston, "Late Roman Armour, 3rd–6th Centuries AD," _Journal of Roman Military Equipment Studies_ (1990), 147.
. _Strategikon_ , I.2.
. _Strategikon_ , XII.8.
. _De Rebus Bellicis_ , XV.2.
. Ian P. Stephenson, _Romano-Byzantine Infantry Equipment_ (Stroud, Gloucester: Tempus, 2006), 52.
. On the _zaba_ see Kolias, _Byzantinische Waffen_ , 37–39; Russ Mitchell, "Archery Versus Mail: Experimental Archaeology and the Value of Historical Context," _Journal of Medieval Military History_ 4 (2006).
. _Ammianus Marcellinus_ , XV.i.13.
. S. M. Perevalov, "The Sarmatian Lance and the Sarmatian Horse-Riding Posture," _Anthropology & Archeology of Eurasia_ 41, no. 4 (2003).
. _Alexiad_ , IV.7; Dawson, "Kremasmata, Kabbadion, Klibanion," 46.
. Ibid., 47; Maria G. Parani, _Reconstructing the Reality of Images: Byzantine Material Culture and Religious Iconography (11th–15th Centuries)_ (Leiden: Brill, 2003), 108, pl. 29.
. Dawson, "Kremasmata, Kabbadion, Klibanion," 47; Haldon, "Some Aspects of Byzantine Military Technology from the Sixth to the Tenth Centuries"; Kolias, _Byzantinische Waffen_ , 45–46.
. Bartusis, _The Late Byzantine Army_ , 323–24.
. _Strategikon_ , XII.4.
. Stephenson, _Romano-Byzantine Infantry Equipment_ , 71.
. Timothy Dawson, "Suntagma Hoplôn: The Equipment of Regular Byzantine Troops, c. 950–c. 1204," in _A Companion to Medieval Arms and Armour_ , ed. D. Nicolle (Woodbridge: Boydell Press, 2002), 83.
. Stephenson, _Romano-Byzantine Infantry Equipment_ , 17; Kolias, _Byzantinische Waffen_ , 76.
. Stephenson, _Romano-Byzantine Infantry Equipment_ , 27.
. Dawson, "Suntagma Hoplôn," 83.
. Ibid., 83–84.
. Minor M. Markle, III, "The Macedonian Sarrissa, Spear and Related Armor," _American Journal of Archaeology_ 81, no. 3 (1977): 324; N. Tarleton, "Pastoralem Praefixa Cuspide Myrtum (Aeneid 7.817)," _Classical Quarterly_ 39, no. 1 (1989): 270; Stephenson, _Romano-Byzantine Infantry Equipment_ , 80; Marijke van der Veen and Alison Cox, _Consumption, Trade and Innovation: Exploring the Botanical Remains from the Roman and Islamic Ports at Quseir al-Qadim, Egypt_ (Frankfurt am Main: Africa Magna Verlag, 2011), 220.
. G. R. Davidson and Tibor Horváth, "The Avar Invasion of Corinth," _Hesperia_ 6, no. 2 (1937): 232–34.
. M. P. Anastasiadis, "On Handling the Menavlion," _Byzantine and Modern Greek Studies_ 18 (1994).
. _Strategikon_ , XII.8.
. Conrad Engelhardt, _Nydam mosefund, 1859–1863_ (Copenhagen: I commission hos GEC Gad, 1865), 81–82, plates VI–VII.
. Evelyne Godfrey and Matthijs van Nie, "A Germanic Ultrahigh Carbon Steel Punch of the Late Roman-Iron Age," _Journal of Archaeological Science_ 31 (2004).
. Stephenson, _Romano-Byzantine Infantry Equipment_ , 92.
. _Praecepta militaria_ , II.
. Parani, _Reconstructing the Reality of Images_ , fig. 104.
. Bartusis, _The Late Byzantine Army_ , 329.
. Prokopios, _Wars_ , VI.xxv.1.
. _Praecepta militaria_ , I.25–26, pp. 14–15.
. _Strategikon_ , XI.3.
. _Vegetius_ , I.17.
. _Sylloge_ , 38.10.
. Haldon, "Some Aspects of Byzantine Military Technology," 38–39.
. _Vegetius_ , III.14, p. 91.
. Wallace McLeod, "The Range of the Ancient Bow," _Phoenix_ 19, no. 1 (1965): 7–10.
. Stephenson, _Romano-Byzantine Infantry Equipment_ : fig. 26.
Bartusis, _The Late Byzantine Army_ , 298–99.
. Ibid., 127–32.
. _CJ_ 85.2: http://webu2.upmf-grenoble.fr/Haiti/Cours/Ak/link_en.htm.
. Duncan B. Campbell, _Greek and Roman Artillery 399 BC–AD 363_ (Oxford: Osprey Publishing, 2003), 21, 36.
. Prokopios, _Wars_ , V.xxi.14–22.
. Josephus, _Jewish War_ , V.270 notes that first-century scorpions (one-armed stone throwers similar to the later _onagers_ ) cast one talent (about 32 kg/71 lbs.) about 400 meters.
. Quoted in Paul E. Chevedden, "The Invention of the Counterweight Trebuchet: A Study in Cultural Diffusion," _Dumbarton Oaks Papers_ 54 (2000): 74.
CHAPTER FIVE: STRATEGY AND TACTICS
. "Strategy," _Oxford English Dictionary_. Third edition. <http://www.oed.com>.
. _Taktika_ , 1§1–3.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 34. See the recent work on proposing a unified "Grand Strategy": Edward N. Luttwak, _The Grand Strategy of the Byzantine Empire_ (Cambridge, Mass.: Harvard University Press, 2009).
. _Priskos_ , IV.1–2.
. Francis Dvornik, _Origins of Intelligence Services: The Ancient Near East, Persia, Greece, Rome, Byzantium, the Arab Muslim Empires, the Mongol Empire, China, Muscovy_ (New Brunswick, N.J.: Rutgers University Press, 1974), 132–37, 48.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 67 ff.
. _Taktika_ , 20§37.
. _Taktika_ , 20§47: a repetition of _Strategikon_ 8.2.1, which in turn echoed ancient Greek notions of just war, cf. Onasander, 5.1.
. _Strategikon_ , II.18.
. _Taktika_ , 20§72.
. _Taktika_ 20§63.
. _Taktika_ 4§41; 20§160.
. _Ouranos_ , 65.79–85.
. _Strategikon_ , VIII.2.12.
. Birkenmeier, _The Development of the Komnenian Army_ , 112 ff.
. _Taktika_ , 20§112.
. _Alexiad_ , XIV.2.
. Bartusis, _The Late Byzantine Army_ , 345.
. Ibid., 65.
. _Strategikon, Proem_.
. _Anonymous Strategy_ , §25, n.1.
. _Skirmishing_ , §22.
. Treadgold, _Byzantium and Its Army_ , 21–23.
. _Praecepta militaria_ , IV.195–203.
. _Skylitzes_ , pp. 66–69.
. _Alexiad_ , VI.1.
. _Choniates_ , p. 10.
. Steven Runciman, _The Sicilian Vespers: A History of the Mediterranean World in the Later Thirteenth Century_ (Cambridge: Cambridge University Press, 1992), 214–87.
. _Anonymous Strategy_ , §42.
. Dvornik, _Origins of Intelligence Services_ , 147–48.
. _Praecepta militaria_ , IV.192–95.
. _Kekaumenos_ , II.24.
. Prokopios, _Wars_ , III.xx.1.
. Prokopios, _Wars_ , III.xiv.7.
. _Taktika_ , 20§131–32.
. _Strategikon_ , I.9; _Taktika_ , 19§21.
. _Ouranos_ , 63.12–14.
. _Ouranos_ , 65.165–172.
. _Skirmishing_ , §6; _Campaign_ , §21.
. Yahya-Ibn-Sa'ïd, _Histoire de Yahya-Ibn-Sa'ïd d'Antioche, continuateur de Sa'ïd-Ibn-Bitriq: fascicule II_ , ed. René Graffin and François Nau, trans. I. Kratchkovsky and A. Vasiliev, 2 vols., vol. 2, Patrologia Orientalis (Paris: Firmin-Didot, 1932), 442.
. _Kinnamos_ , IV.17.
. _Choniates_ , pp. 58–59.
. _Skirmishing_ , §6; _Campaign_ , §10.
. _Strategikon_ , II.11.
. _Leo the Deacon_ , VI.12–13.
. _Taktika_ , 17§89.
. Prokopios, _Wars_ , 1.1.12–15.
. Flavius Cresconius Corippus, _The Iohannis, or, De bellis Libycis_ , trans. George W. Shea (Lewiston, N.Y.: Edwin Mellen Press, 1998), 125.
. _Strategikon_ , III.12–16.
. _Strategikon_ , XII.9; Philip Rance, "Narses and the Battle of Taginae (Busta Gallorum) 552:"Procopius and Sixth-Century Warfare," _Historia: Zeitschrift für Alte Geschichte_ 54, no. 4 (2005): 430.
. _Strategikon_ , XII.17.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 210–17.
. Ibid., 210–18; Alphonse Dain and Emperor Leo VI, _Sylloge tacticorum, quae olim "Inedita Leonis tactica" dicebatur_ (Paris: Société d'édition "Les Belles lettres," 1938), 47.16.
. _Ouranos_ , 56.35–77.
. McGeer, _Sowing the Dragon's Teeth_ , 264.
. Ibid., 268–70.
. Ibid., 273.
. Ibid., 276.
. Haldon, _Warfare, State, and Society in the Byzantine World_ , 224–25.
. Bartusis, _The Late Byzantine Army_ , 253–60.
. _Ouranos_ , 65.139–43.
. _Ouranos_ , 65.110–15.
. _Ouranos_ , 65.105–39.
CHAPTER SIX: ENEMIES OF BYZANTIUM
. Peter Heather, _The Goths_ (Oxford: Blackwell, 1996), 274.
. Ibid., 236.
. Simon James, "Stratagems, Combat, and 'Chemical Warfare' in the Siege Mines of Dura-Europos," _American Journal of Archaeology_ 115, no. 1 (2011): 76; "Evidence from Dura Europos for the Origins of Late Roman Helmets," _Syria_ 63, nos. 1-2 (1986). For additional information on Sasanian helmets, see Stephen V. Grancsay, "A Sasanian Chieftain's Helmet," _Metropolitan Museum of Art Bulletin_ 21, no. 8 (1963).
. _Strategikon_ , XI.2.
. Hugh Kennedy, _The Armies of the Caliphs_ (London: Routledge, 2001).
. Michael Lecker, "Siffin," in _The Encyclopaedia of Islam_ , ed. C. E. Bosworth, E. van Donzel, and W. P. Heinrichs (Leiden: Brill, 1997), 552–56.
. Kennedy, _The Armies of the Caliphs_ , 32–34.
. Warren T. Treadgold, _The Byzantine Revival, 780–842_ (Stanford, Calif.: Stanford University Press, 1988), 297.
. Panos, Sophoulis, _Byzantium and Bulgaria, 775–831_ (Leiden: Brill, 2012).
. Ibid., 77.
. Ibid.
. _Theophanes_ , AM 6266.
. Jonathan Shepard, "The Uses of the Franks in Eleventh-Century Byzantium," _Anglo-Norman Studies_ 15 (1993): 288–89.
. Goffredo Malaterra, _The Deeds of Count Roger of Calabria and Sicily and of His Brother Duke Robert Guiscard_ (Ann Arbor: University of Michigan Press, 2005), 165–66.
. _Alexiad_ , IV.4.
. _Alexiad_ , V.6.
. Birkenmeier, _The Development of the Komnenian Army_ , 60–61.
. Ibid., 69–70.
CHAPTER SEVEN: THE BYZANTINE ARMY AT WAR
. Clifford Edmund Bosworth, "The City of Tarsus and the Arab-Byzantine Frontiers in Early and Middle Abbasid Times," _Oriens_ 33 (1992): 274–77.
. Marius Canard, _Histoire de la dynastie des H'amdanides de Jazîra et de Syrie_ (Algiers: Imprimeries "La Typo-litho" et J. Carbonel réunies, 1951): 763–70.
. Bosworth, "The City of Tarsus and the Arab-Byzantine Frontiers," 282.
. Yahya Ibn Sa'id, I. Kratchkovsky, and A. Vasiliev, _Histoire de Yahya-Ibn-Sa'ïd d'Antioche, continuateur de Sa'ïd-Ibn-Bitriq, fasicule 1_ , 3 vols., vol. 1, Patrologia Orientalis (Paris: Firmin-Didot, 1924), 95; Canard, _Histoire de la dynastie des H'amdanides de Jazîra et de Syrie_ , 818–19; _Skylitzes_ , 257.
. _Leo the Deacon_ , III.11.
. _Leo the Deacon_ , IV.4.
. Paul Stephenson, _Byzantium's Balkan Frontier: A Political Study of the Northern Balkans, 900–1204_ (Cambridge: Cambridge University Press, 2000), 65.
. Ibid., 67.
. _Skylitzes_ , 331.
. _Kinnamos, III.2; Choniates_ , pp. 45–46.
. _Kinnamos_ , III.9.
. _Kinnamos_ , V.10; Birkenmeier, _The Development of the Komnenian Army_ , 119.
. _Kinnamos_ , VI.7.
. _Choniates_ , p. 88.
. _Theophanes_ , AM 6165.
. _Theophanes_ , AM 6209; R.-J. Lilie, "Die Zweihundertjährige Reform: zu den Anfängen der Themenorganisation im 7. und 8. Jahrhundert," _Byzantinoslavica_ 45 (1984): 190–201, 128–29.
. Demetres Tsounkarakes and Euangelos K. Chrysos, _Byzantine Crete: From the Fifth Century to the Venetian Conquest_ (Athens: Historical Publications St. D. Basilopoulos, 1988), 64–65.
. _Leo the Deacon_ , I.3.
. _Leo the Deacon_ , I.4.
. _Leo the Deacon_ , I.8.
. _Ouranos_ , 65.20.
. Tsounkarakes and Chrysos, _Byzantine Crete_ , 72.
CHAPTER EIGHT: THE BYZANTINE ART OF WAR
. B. H. Liddell Hart, _The Strategy of Indirect Approach_ (London: Faber and Faber, 1941), 59–62.
. Ibid., 72.
. J. F. C. Fuller, _A Military History of the Western World_ (New York: Funk & Wagnalls, 1954), 339.
. Georgios Theotokis, "The Norman Invasion of Sicily, 1061–1072: Numbers and Military Tactics," _War in History_ 17, no. 4 (2010): 393.
. Bernard Bachrach, "On the Origins of William the Conqueror's Horse Transports," _Technology and Culture_ 26, no. 3 (1985): 513.
. Ibid., 514.
. _Theophanes_ , AM 6165.
. _Vegetius_ , IV.18.
. John F. Haldon, "'Greek Fire' Revisited: Recent and Current Research," in _Byzantine Style, Religion and Civilization: In Honour of Sir Steven Runciman_ , ed. Elizabeth Jeffreys (Cambridge: Cambridge University Press, 2006), 314.
. _Taktika_ , 19.59.
. John H. Pryor and Elizabeth M. Jeffreys, _The Age of the Dromon: The Byzantine Navy ca. 500–1204_ (Leiden: Brill, 2006), 620.
. Ibid., 618.
. Ibid., 609.
. Ibid., 609ff.
. Ibid., 612.
. _Alexiad_ , XI.10.
. Pryor and Jeffreys, _The Age of the Dromon_ , 612.
. Ibid., 613.
. Peter Pentz, "A Medieval Workshop for Producing 'Greek Fire' Grenades," _Antiquity_ 62, no. 234 (1988).
. Maurice Mercier, _Le feu grégeois, les feux de guerre depuis l'antiquité, la poudre à canon_ (Paris: P. Geuthner, 1952), 84–91.
. _Praecepta militaria_ , I.150–55.
. Pryor and Jeffreys, _The Age of the Dromon_ , 619–20.
. George T. Dennis, "Byzantine Heavy Artillery: The Helepolis," _Greek, Roman, and Byzantine Studies_ 39 (1998): 101.
. W. T. S. Tarver, "The Traction Trebuchet: A Reconstruction of an Early Medieval Siege Engine," _Technology and Culture_ 36, no. 1 (1995): 149.
. Ibid., 162.
. Chevedden, "The Invention of the Counterweight Trebuchet," 77.
. _Alexiad_ , X.11.
. S. Vryonis, "The Byzantine Legacy and Ottoman Forms," _Dumbarton Oaks Papers_ 23–24 (1969–70): 273–74.
. Antoine Henri Jomini, baron de, _The Art of War_ , trans. G. H. Mendell and W. P. Craighill (Philadelphia: J. P. Lippincott, 1862), 366.
. Alfred Vagts, _Landing Operations: Strategy, Psychology, Tactics, Politics, from Antiquity to 1945_ (Harrisburg, Pa.: Military Service Pub. Co., 1946), 116.
. Liddell Hart, _The Strategy of Indirect Approach_ , 49–69.
. T. E. Lawrence, _Seven Pillars of Wisdom_ (London: Vintage, 2008), 117.
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## INDEX
Abbasid dynasty, –, –, ,
ad-Dawla, Sayf, , , ,
Adriatic Sea,
akritai,
Akrites, Digenis,
al-Aqta, Amr,
al-As, Amr b., –
al-Aziz, Abd,
al-Dulafi, Thamal,
Alexander the Great, ,
al-Jarmi,
al-Mafjar, Khirbat, ,
al-Malik, Abd,
al-Ma'mun,
al-Mu'tasim, ,
al-Rahman, Abd,
al-Tarsusi,
al-Walid, Khalid b., –
Amisos,
Ammatas, ,
Amorion dynasty,
Anastasios, , , –
Anatolia, road network in,
Anatolik, , –, , , , ,
Andreas,
Androna fortress,
Andronikos II,
Antoninus,
Apion,
Arab civil war (656–61),
Arabs, _see also_ Muslims
Byzantine adaptation and,
Christian tribal confederation of Ghassan and,
methods of warfare and,
organization of armies and, –
Romans paying subsidies to allied tribal confederations and, –
Arab Samonas,
Aral Sea,
Ariaric,
Arsaces,
Arslan, Alp,
Artavasdos,
Artemis,
artillery, vii, , , –, , , , –, , –
_Art of War_ (Jomini),
Ashot III,
Aspar,
Athanatoi (Immortals), –
Attila the Hun, , , , ,
Augustine of Hippo,
Augustus, viii, ,
Aurelian, ,
Avar khaganate,
Avars, , , , , , , , –, , ,
Bakr, Abu,
Baldwin of Boulogne,
Ballista, –
Barbarossa, Frederick I, –
Bardanes, Philippikos, –, ,
Baresmanas,
Basil II, , , , , , , , –, –, ,
Basil I, ix, , ,
Basiliskos, –
Battle at Dyrrachium, –, , –, , , , ,
Battle of Acheloos, ,
Battle of Ad Decimum, , , , –
Battle of Adrianople, , , , , –, , , , –,
Battle of Ankara,
Battle of Apros,
Battle of Callinicum, , ,
Battle of Dara, –, , ,
Battle of Dorostolon, , , , –, ,
Battle of Dorylaeum,
Battle of Edessa, , ,
Battle of Kleidion, –
Battle of Mantzikert, , , , , , ,
Battle of Peritheorion, –
Battle of Phoenix,
Battle of Sebastopolis,
Battle of Semlin, –
Battle of Siffin,
Battle of Sirmium, –
Battle of Solachon,
Battle of Sozopolis, –
Battle of Stiklestad,
Battle of Taginae, ,
Battle of Tricamarum, , –,
Battle of Yarmuk, , , , ,
Bayeux Tapestry,
Béla, –,
Belisarios, –, –, –, , , , , , –, –, , , , –
Bertha of Salzburg,
Biblioteca Apostolica Vaticana, text on siege warfare and,
Black Sea, , , , , , ,
Bohemund, –, , ,
Boniface,
Boris of Bulgaria,
Bryennios,
Bulgarophygon fortress,
Bulgars
attack on Muslims and,
Basil II and, –
Battle of Acheloos and, ,
Bulgar khanate and,
Byzantine adaptation and, –
Byzantine defeat and,
Byzantine payments to Rus' and Pechenegs,
depiction of Bulgar warrior and,
Greek fire and, –
hierarchical military organization and,
John Tzimiskes and,
killing of Nikephoros I and,
methods of warfare and, –
nomads and, –
Old Bulgarian Empire and,
ordered battle lines and,
organization of,
rise of tagmatic forces and,
Roman military futility and,
siege of 717 and,
siege of Adrianople and,
spies and,
Byzantine Empire
Anatolikon theme and, ,
arms factories and,
army judiciary and,
artillery and, vii, , , –, , , , –, , –
avoidance of decisive combat and,
basic clothing of soldiers and, –
battle at Ajnadayn and,
Battle of Kleidion and, –
Battle of Mantzikert and, , , , , , ,
Battle of Semlin and, –
Battle of Sirmium and, –
Battle of Sozopolis and, –
bows and, –
bribery and,
chain mail and, –, , –, –,
command structure and, –
contributions to warfare and, –
counterweight trebuchets and, –
decimal system of organization and,
department of the sacrae largitiones and, –
diplomacy and, –,
early period of, –
eastern campaigns of Nikephoros Phokas and, –
fall in 1453 and, –
Fifth century Byzantine army ranks and,
fighting attritive wars and, –
first Arab attacks on Byzantine Syria and,
First Crusade and, –, , , , ,
fortress at Resafa, Syria and,
Fourth Crusade and, , , , , , , ,
Ghassanids and, ,
"Greek fire" and, , –
grenades and,
hand-held flamethrowers and,
horse-mounted archery and,
Kavad and, –,
legacy in the history of warfare and, –
Lombard revolt of 1009–22 and,
Lombards and, , , ,
military promotion and, , –
military structure in the fifth century and,
military tribunals and,
military weaknesses and, –
offensive warfare and, , , ,
omens and,
Opsikion theme and,
organization of military apparatus and, –
Pecheneg wars and,
proxy warfare and,
psychological nature of warfare and, –
Roman-Persian War of 602–28 and, –
siege warfare and, –
social influences on soldiers' attitudes and, –
soldiers' pay and, –
strengths of the empire's military apparatus and, –
supply challenges and, –
tagma and, –, , , , , , –, , , , , , , , , , , ,
theme armies and, –
Third Civil War and,
Thrakesion theme and,
turning enemies into allies and, –
two pillars of military doctrine and,
use of foreign auxiliaries and, –
using officers of barbarian origin and,
Vandal War and, –
vast collection of state archives and,
victory over the Kievan Rus' at Dorostolon and, , , , –, ,
war and spirituality, –
war of 502–506 and,
westerners serving in army and,
Byzantine Thrace, vii, , , –, , , , , , , , , –, , , , , , –, ,
Carinus,
Carus, –
Catalan Grand Company, , ,
chain mail, –, , –, –,
Charlemagne, ,
Charles of Anjou,
Chevedden, Paul,
Chionites,
Chliat fortress,
Chlorus, Constantius,
Choniates, Niketas, , , ,
close combat weapons, –
Companions, Muslim believers and,
Conrad II, –
Constance,
Constans I, , ,
Constans II, , , ,
Constantine
allagia command and,
annual conscription in the provinces and,
attack on Iranian Sarmatian tribes and, –
Christian faith and,
death of,
defeat of Goths and,
federate troops and,
inflationary era and military pay,
Nikephoros Phokas and,
organization of military apparatus and, –
post system and,
principles of Christianity and,
protector of Christians of Persia and,
reorganization of forces and,
Tetrarchic system and,
walls of Anthemios and,
war with Persian Empire and,
Constantine II,
Constantine IX,
Constantine V, , –, , , ,
Constantine VII, , , , ,
Constantine X,
Constantinople
Anatolikon theme and,
Armeniakon theme and,
Avar-Sasanian siege of 626 and, –
consecration of,
defense of major strongpoints and,
fabrica and,
fall of,
"Greek fire" and,
Harald Hardrada and,
Hilderic and,
invasion of Kutriger Huns and,
Karabision theme and,
main arsenal at,
major military base of Amorium and,
Manuel I and,
massive defenses of,
Mehmed II and,
Muslim Arab attempts to conquer and, ,
Nicolas de Fer sketch of,
Opsikion theme and,
sack of,
siege of, , , ,
soldiers' armor and,
spies and,
strategic location for wars and,
Thomas the Slav's assault on,
Thrakesion theme and,
Valens and,
Constantius, –,
Constantius II,
counterweight trebuchets, , , –, –, , , , –, –
Cuman (Kipchak) confederacy, , , , , , , –
Dalassenos, Damian,
Dalmatius,
Danishmends, –
Danube River, –, –, , , , , , , , , , , , , , , , , –, –, –, –,
Daphne Monastery,
darts, , , , ,
Dawson, Timothy, ix, ,
de Bailleul, Roussel,
de Balleul, Roussel,
de Fer, Nicolas, sketch of Constantinople and,
de Flor, Roger,
Dénes, –
_De Rebus Bellicis_ , military affairs manual and,
_Digenis Akritis (Two-blooded Border Lord)_ ,
Diocletian, –, , , , , , –, , , –, , , , ,
Diogenes, Romanos IV, , –, ,
Diyala River,
Doukas, Andronikos, ,
Doukas, John,
Drina River,
Eudoxia,
Euphrates River, , , , , , , , , , ,
Firouz, –
First Crusade, –, , , , ,
Fourth Crusade, , , , , , , ,
Frankopoulos, Hervé,
Fritigern,
Fuller, J. F. C., Justinian's military intervention and,
Gainas,
Galerius,
Gallus,
Geiseric, –
Geiserith,
Gelimer, –, , –, –
Gepids, , ,
Géza II,
Ghassanid Arabs, , , , , ,
Gibamund,
Golan Heights,
Golden Horde,
Gothicus, Claudius,
Goths
Adrianople battle and, –
captured soldiers and service in Roman army,
Claudius Gothicus and,
defeat by Constantine and,
eastern field army and,
limitanei and,
methods of warfare and, –
organization of,
Prokopios and,
revolt against Roman Empire,
siege of Rome in 537–38 and,
territory of,
Tervingi and,
Gratian, –,
"Greek fire", , –, –
Gregory, Muslim offensive against Byzantine North Africa and,
Gregory VII, ,
grenades, , –
Greutungi Goths,
Grumbates,
Guiamar IV,
Guiscard, Bohemund, –, ,
Guiscard, Gaita,
Guiscard, Robert, –, , –, ,
Habib, Banu,
Hadrian,
Hafs, Abu,
Haldon, John, ix, , –, , –, , , , , , , ,
hand-held flamethrowers,
Hannibalianus,
Hardrada, Harald, ,
Harith,
Hart, Liddell,
Gothic Wars and,
"indirect approach" in warfare and,
T. E. Lawrence and,
Hawqal, Ibn,
helmets, , –, , ,
Henry IV, ,
Heraclius
Abd al-Malik and,
alliance with Western Turks and,
capture of Tripolis and,
casting himself as biblical David and,
crushing victory by Arab army and,
defense in depth and, –
emperor campaigning in person and, –
imperial recovery and,
initial efforts against the Persians and,
Kosrow II and,
leaving his capital to face Persian attack and,
military pay and,
Muslim siege of Damascus and,
Opsikion units and,
Persian War and, , –, –
recruitment of soldiers and, –
striking down giant Persian soldier and,
triumphant entry into Jerusalem and, –
turning enemies into allies and, –
Hermogenes, ,
Heruls, –, , , , , –,
Hetaireia, , ,
Hilderic, –
Hundred Years' War,
Huneric,
Hunnic Empire,
Hunnic Hephthalites
killing of Sasanian Shah Peroz and,
war of 502–506 and,
Huns
Attila the Hun and, , , , ,
"barbarization" of the army and,
Battle of Dara and, –, –, , ,
Chionites and,
description of,
massive defenses of Constantinople and,
mercenary manpower and,
range of movement, tactics, and weaponry,
utility of mounted missile troops and,
Vandal War and, –
Vegetius and,
_Iliad_ , mace and,
Imperials,
Irene,
Jacob of Nisibis,
Ja'far,
jihad, , , , , –
John the Cappadocian, –
Jomini, Antoine Henri, _Art of War_ and,
Joscelin II,
Jovian,
Julian, –, , ,
Justin I, , ,
Justinian, , –, –, –, , –, , , , , , , , , , , , , –, , –, ,
Justinian II, , , , , ,
Kallinikos,
Kantakouzenos, John VI,
Kavad, –,
Kavad II,
kettle helmets,
Khurradadhbih, Ibn,
kleisourai,
Komnene, Anna, , , , ,
Komnenoi dynasty, , –, , , , , , , , , , , –
Komnenos, Alexios I, , , , , , , ,
Komnenos, John II, , , , , , , ,
Komnenos, Manuel, , , , , –, , , , , –, –,
Kontostephanos, Andronikos, –
Kontostephanos, Stephanos,
Kosrow, –,
Kosrow II, ,
Kourkouas, John, –,
Krum,
Kydnos River,
Lake Van, ,
Lalakaon River,
lamellar armor, –, –
Lawrence, T. E.,
Lejjun fortress, ,
Lekapenos, Romanos I, , , , ,
Leo, Armenian Prince,
Leo I, ,
Leo III, ,
Leo of Tripoli,
Leo the Deacon, , , , , , –
Leo V, ,
Leo VI, , , , , , , –, ,
Licinius,
_Life of St. Germanus_ , "iron cavalry" and,
limitanei, border troops and, , , –, –, , , , , –
Lombards, , , ,
Longshanks, Edward I, counterweight trebuchets,
Louis VII,
Luttwak, Edward N., "Grand Strategy" and,
Macedonian dynasty, ix, , , , , , , , ,
Majorian,
Manfred of Sicily,
Maniakes, George, , ,
maps
Balkans and,
Battle of Dara and,
Eastern Frontier, Fourth–Seventh Centuries and,
Empire ca. 780 and,
Empire in the Sixth Century with Regional Armies and,
Empire of the Komenoi and,
Medieval Italy and the Balkans and,
Rome's Desert Frontier and,
Successor States ca. 1218 and,
Themes ca. 668 and ca. 900 and,
Themes ca. 1025 and,
Marcellinus, Ammianus, , , ,
Marcian,
Martel, Charles,
Massagetae, , ,
Master of Offices, fabricae workshops and,
Masud,
Maurice
army organization and,
assassination of, ,
Book XII and,
hybrid horse archer-lancers and,
imperial armamenta and,
lamellar armor and,
Lombards and,
maces and,
practice of combined arms and,
ratio of horse to foot and,
sons of fallen soldiers and, –
_Strategikon_ and, , –, –, –, , , , , , , , –, , , , –, ,
tactics and, ,
tagmas and,
McClellan, George,
Mehmed II, vii,
Merkourios,
Michael I,
Michael II, , ,
Michael III, ,
Michael, Khan Boris,
Michael VII, ,
Michael VIII, , –, , , ,
_Miracles of St. Demetrios_ , Avar siege and,
Mount Aetna,
Mu'awiya, –, ,
Muhammad, ,
Mundhir,
Mundus, ,
Muslims, _see also_ Arabs
Anatolian commanders and,
attacks on Constantinople and,
Basil II and,
battle at Chandax and, –
battle with Heraclius's army and, –
civil war and,
Constantine VII and,
creation of an Arab fleet and,
eastern campaigns of Nikephoros Phokas and, –
fortress of Babylon and,
fortress of Shaizar and,
"forward defense" and,
Greek fire and,
jihad and, , , , , –
John Tzimiskes and,
Khazar khanate and,
methods of warfare and,
organization of armies and, –
post system and,
siege of 674–78 and,
spies and,
sustained assaults (717–718) and, –
victory at Constantinople and,
Mygdonius River, –
Nagyszentiklós
depiction of Bulgar warrior and,
treasure of, ,
Nika Riot of 532,
Nikephoritzes,
Nikephoros I, , –, , ,
Nikephoros II, , –, –, –, –, , –, , , , , , –,
Nikephoros III,
nomads, , , , –, , , –,
Normans
Alexios I Komnenos and,
Alexios's ultimate success and,
appearance in Mediterranean and, –
Byzantine adaptation and, –
counterweight trebuchets and,
Dyrrachium battle and,
failure of Maniakes's expedition and,
feigned retreat in invasion of Sicily and, –
Greek fire and,
Hervé Frankopoulos and,
Immortals and,
kite-shaped shields and,
lamellae armor and,
methods of warfare and, –
outflanked the Roman army and,
prowess on the battlefield and,
Robert Guiscard and,
Roger II of Sicily and,
Tancred and, , –
_Notitia Dignitatum_ , dux commander and,
Nu'man,
Numerian,
Nydam Mose burials,
Oder River,
_On Skirmishing_ (Nikephoros II), , –,
Opsikion military units, , –,
Ouranos, Nikephoros, , , , , ,
_Oxford English Dictionary_ , strategy and,
Pakourianos, Gregory,
Palaiologan dynasty, ,
Palaiologos, George,
Palaiologos, Theodore,
Palaiologus, Michael VIII, , , –, , , ,
Pannonians,
Pastilas,
Patermuthis, Flavius,
Pechenegs, , , , –, , –, , , , ,
Pedro III,
Peroz,
Persian Empire
battle at Dara and, –
Belisarios and,
Byzantine adaptation and, –
captured Persian soldiers and service in Byzantine army,
collapse of the eastern limes and,
Khurramite Persian rebels and,
limitanei and,
methods of warfare and,
organization of, –
quilted coats and, –
Roman-Persian War of 602–28 and, –
sacking of Jerusalem and,
Sasanian dynasty and,
siege of Nisibis and, –
Persian War
fiscal and military crisis sparked by, –
manpower shortage and,
Peter I,
Peter II,
Peter III,
Petronas,
Phokas, Bardas, ,
Phokas, Leo, , ,
Phokas, Nikephoros, –
Phokas, Nikephoros II, , –, –, –, –, , –, , , , , , –,
Pope Conon,
_Praecepta militaria_ (Nikephoros II), hand-held flamethrowers and,
Praetorian Guard,
Prince Sviatoslav of the Kievan Rus', –
Prokopios, , –, –, , , , , , , , , –,
Prudentius, ,
Raymond of Antioch,
Raymond of Poitiers,
Raymond of Toulouse,
Reynault,
Rhine River, ,
Roger II,
Roman Empire
Adrianople battle and, –
armor and, –
artillery and, vii, , , –, , , , –, , –,
basic clothing of soldiers and, –
Basiliskos affair and,
Battle of Ad Decimum and, , , , –
Battle of Callinicum and, , ,
Battle of Dorostolon and, , , , –, ,
Battle of Tricamarum, , –,
burning of Persian Mesopotamia and,
close combat weapons and, –
defeat at Nisibis and,
Diocletian and, –
federate soldiers and, –
frontier of,
Goth revolt and,
loss of Amida and,
missile weapons and, –
nadir of Roman power and,
recruitment of soldiers and, –
Roman-Persian War of 602–28 and, –
Second Crusade and, , –
Shapur II and,
siege of Nisibis and,
supply challenges and, –
war of 502–506 and,
Romanos II, –
Romanos IV,
Saint Olaf,
Saktikios,
Samuel, –
Saros River,
Sasanians, –, –, –, , , , , , , –,
scale armor, –
Sea of Marmora,
Second Crusade, , –
Seljuk Turks, , ,
Septimius Severus,
Sergios,
_Seven Pillars of Wisdom_ (Lawrence),
Shahanshah, ,
Shahrvaraz,
Shapur I,
Shapur II, ,
shields, , , –, –, , , , ,
siege warfare, –, , , , , , , , , –, , , , , , , , , –, –, , , , , , , –, , –, –, , , –, –
Simeon,
Sittas,
Skleros, Bardas, –, , , –
slings, –, , , , , , –
spangenhelm helmets, –
spathas, , ,
St. Bakchos,
Stephen III, –,
St. George of Mount Athos,
Stilicho,
Stirling Castle, siege of,
St. Jerome,
Strata Diocletiana,
_Strategikon_
Avars and, –
Book 1 of, –
cavalry saddles and, –
chain mail and,
darts and,
dekarch's command structure and,
double line of cavalry deployment and,
expecting conflict and, –
just wars and, –
kit of officers and elite infantry, –
knowledge of the paucity of forces and,
nomads and,
Optimates and,
order of march for a 310–man cavalry tagma and,
principles of Byzantine warfare and,
"Slavic" spears and, –
slings and, –
solenarion and,
spies and, –
tactics and, –
wearing of greaves and, –
strategy
Dark Ages and,
defense in depth and,
divide and conquer, –
espionage and, –
fighting attritive wars and, –
fighting small wars and, –
middle and late period, –
overview of, –
preparing for war and, –
siege warfare and, –
spirituality and, –
strategic intelligence and, –
tactics and, –
Strymon River,
Sufyan, Mu'awiya ibn Abi,
Sufyan, Yazid b. Abi,
Sunicas,
Sviatoslav I, , ,
_Sylloge_ , cavalry maneuvers and, –
_Sylloge Tacticorum_ , staff-slings and,
Symeon, –
_Taktika_
combined arms approach and,
commanders maintaining their purity and,
naval warfare and,
siege warfare and, –
Tancred, , –
Tarchaneiotes, Joseph,
Tarsusi, ,
Tarver, W. T. S., traction trebuchets and,
Tatikios, ,
Tattimuth, ,
Taurus Mountains, , ,
Telerig,
Tervel of Bulgaria,
Tervingi,
thematic armies, , –, , , , , , , , –, , , , , ,
Theodora, ,
Theodoric,
Theodosios II,
Theodosios III,
Theodosius I, , ,
Theophano,
Theophilos, –, , , ,
Theophobos,
Thessaloniki, siege of,
Third Gallic Legion,
Thomas the Slav,
Thoros,
Tigris River, , ,
torsion, –,
Tourkos, Bardanes,
traction trebuchets, –
Trajan, viii, , ,
Treadgold, Warren, ix, , , , –,
trebuchets, , , –, –, , , , –, –
Troglita, John,
Tzazon, , , –
Tzimiskes, John
battle of Dorostolon and, , , , –, ,
battle with Tarsians and, –
border-warrior elites of the eastern frontiers and,
death of,
Muslims of Cilicia and, –
seasoned veterans of the eastern wars and,
soldier-emperors and, –
Ubayda, Abu,
Umar II,
Uros II,
Uz confederation,
Valens, –
Valentinian I, –,
Valentinian III,
Valerian,
Vandals
battle of Adrianople and,
Belisarios's decisive victory and, ,
uprising in Sardinia and,
Vandal War and, –
wounding of Diogenes and,
Varangian Guards,
Vardariotai,
Vatatzes, Andronikos,
Vatatzes, John VIII,
Vegetius, , , , , –, , –, , , , ,
Via Egnatia, ,
Via Maris,
Via Militaris, –
Vikings,
Virgil, Roman poet,
Vistula River,
Vitalian,
Vladislav II,
"War Wolf", counterweight trebuchets,
White, Lynn, adoption of the stirrup and,
William "Iron Arm" of the Hauteville family,
William the Bastard, , ,
Xiphias, Nikephoros, –
Zengi, Imad ad-Din, ,
Zeno,
Zeugminon, siege of, ,
|
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"redpajama_set_name": "RedPajamaBook"
}
| 9,604
|
Эре́нж () — коммуна во Франции, находится в регионе Бургундия. Департамент коммуны — Кот-д'Ор. Входит в состав кантона Монбар. Округ коммуны — Монбар.
Код INSEE коммуны — 21248.
Население
Население коммуны на 2010 год составляло 64 человека.
Экономика
В 2010 году среди 42 человек трудоспособного возраста (15—64 лет) 29 были экономически активными, 13 — неактивными (показатель активности — 69,0 %, в 1999 году было 74,4 %). Из 29 активных жителей работали 27 человек (17 мужчин и 10 женщин), безработных было 2 (1 мужчина и 1 женщина). Среди 13 неактивных 4 человека были учениками или студентами, 6 — пенсионерами, 3 были неактивными по другим причинам.
См. также
Список округов Франции
Примечания
Ссылки
Национальный институт статистики — Эренж
Коммуны департамента Кот-д'Ор
|
{
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| 7,523
|
Blockhaus d'Éperlecques (auch: Bunker de Watten) ist der Name einer deutschen Bunkeranlage aus dem Zweiten Weltkrieg in Nordfrankreich im Département Pas-de-Calais. Sie befindet sich im Wald von Éperlecques etwa zehn Kilometer nordwestlich von Saint-Omer.
Die Bunkeranlage mit dem Decknamen "Kraftwerk Nord West" (KNW) sollte zur Herstellung und zum Abschuss von V2-Raketen dienen. Ihr Bau geht auf eine Initiative Walter Dornbergers zurück, der im Dezember 1942 von Rüstungsminister Albert Speer eine solche Anlage in der Nähe der Kanalküste forderte. In den letzten Dezembertagen prüften Offiziere und Ingenieure der Heeresversuchsanstalt Peenemünde mögliche Standorte und entschieden sich für den Wald bei Éperlecques. Die mit dem Bau im März 1943 beauftragte Organisation Todt konnte die Anlage (trotz des Einsatzes von Zwangsarbeitern) nicht fertigstellen, da die Briten über das Vorhaben und seinen Zweck informiert waren. Sie führten am 27. August und am 7. September 1943 erste Bombenangriffe auf den Standort aus. Am 19. Juni 1944 wurde die Anlage dann im Rahmen der Operation Crossbow durch Luftangriffe schwer beschädigt: Bomber der Royal Air Force warfen Tallboy-Bomben ab. Einzig die Anlage zur Herstellung von Flüssigsauerstoff wurde weiter ausgebaut und in Betrieb genommen. Am 27. Juli 1944 überstand sie einen weiteren Tallboyangriff unbeschädigt; in Anbetracht des Heranrückens der alliierten Truppen wurde sie demontiert und nach Deutschland geschafft.
Die Anlage beherbergt heute ein Freilichtmuseum, das ihre Geschichte und Funktion dokumentiert. Neben dem Bunker ist ein Krater einer Tallboy-Bombe zu sehen. Der Krater hat einen Durchmesser von rund 42 Metern und ist etwa 18 Meter tief.
1986 wurde die Anlage vom französischen Kulturministerium als Monument historique registriert.
Literatur
Weblinks
Website des Museums
Beschreibung und Geschichte auf den Museumsseiten von La Coupole
Einzelnachweise
Militärmuseum in Frankreich
Museum über den Zweiten Weltkrieg
Bunker in Frankreich
Ruine in Frankreich
Raketenstartplatz
V-Waffen-Programm
Bauwerk im Département Pas-de-Calais
Organisation (Département Pas-de-Calais)
Monument historique im Département Pas-de-Calais
Erbaut in den 1940er Jahren
Freilichtmuseum in Frankreich
Bauwerk aus Beton
Museum in Hauts-de-France
Monument historique (Militärbauwerk)
Monument historique seit 1986
Blockhaus
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,746
|
\section{Gelfand theory}
To understand noncommutative geometry we must first come back to Gelfand
theory for \emph{commutative} $C^{\ast }$-algebras.
\subsection{$C^{*}$-algebras}
Recall that a $C^{\ast }$-algebra $\mathcal{A}$ is a (unital) Banach algebra
on $\mathbb{C}$ (i.e. a $\mathbb{C}$-algebra which is normed and complete
for its norm) endowed with an involution $x\rightarrow x^{\ast }$ s.t. $%
\left\Vert x\right\Vert ^{2}=\left\Vert x^{\ast }x\right\Vert $. The norm
(the metric structure) is then deducible from the algebraic structure.$\;$%
Indeed, $\left\Vert x\right\Vert ^{2}$ is the spectral radius of the $\geq 0$
element $x^{\ast }x$, that is, the $\func{Sup}$ of the modulus of the
spectral values of $x^{\ast }x$:\footnote{%
In the infinite dimensional case, the spectral values ($x-\lambda I$ is not
invertible) are not identical with the eigenvalues ($x-\lambda I$ has a non
trivial kernel). Indeed non invertibility no longer implies non injectivity
(a linear operator can be injective and non surjective). For instance, if $%
e_{n}$, $n\in \mathbb{N}$, is a countable basis, the shift $\sum_{n}\lambda
_{n}e_{n}\rightarrow \sum_{n}\lambda _{n}e_{n+1}$ is injective but not
surjective and is not invertible.}$\;$%
\[
\left\Vert x\right\Vert ^{2}=\func{Sup}\left\{ \left\vert \lambda
\right\vert :x^{\ast }x-\lambda I\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is not invertible}\right\}
\]%
(where $I$ is the unit of $\mathcal{A}$).\ In a $C^{\ast }$-algebra the norm
becomes therefore a purely \emph{spectral} concept.
An element $x\in \mathcal{A}$ is called self-adjoint if $x=x^{*}$, normal if
$xx^{*}=x^{*}x$, and unitary if $x^{-1}=x^{*}$ ($\left\| x\right\| =1$).
In this classical setting, the mathematical interpretations of fundamental
physical concepts such as a space of states, an observable, or a measure,
are the following:
\begin{enumerate}
\item a space of states is a smooth manifold: the phase space $M$ (in
Hamiltonian mechanics, $M=T^{\ast }N$ is the cotangent bundle of the space
of configurations $N$ endowed with its canonical symplectic structure);
\item an observable is a function $f:M\rightarrow \mathbb{R}$ (interpreted
as $f:M\rightarrow \mathbb{C}$ with $f=\bar{f}$) which measure some property
of states and output a real number;
\item the measure of $f$ in the state $x\in M$ is the evaluation $f(x)$ of $f
$ at $x$; but as $f(x)=\delta _{x}(f)$ (where $\delta _{x}$ is the Dirac
distribution at $x$) a state can be dually interpreted as a continuous
linear operator on observables.
\end{enumerate}
Observables constitute a commutative $C^{\ast }$-algebra $\mathcal{A}$ and
Gelfand theory explains that the \emph{geometry} of the manifold $M$ can be
completely recovered from the \emph{algebraic} structure of $\mathcal{A}$.
\subsection{Gelfand's theorem}
Let $M$ be a topological space and let $\mathcal{A}:=\mathcal{C}(M)$ be the $%
\mathbb{C}$-algebra of continuous functions $f:M\rightarrow \mathbb{C}$ (the
$\mathbb{C}$-algebra structure being inherited from the structure of $%
\mathbb{C}$ via pointwise addition and multiplication).\ Under very general
conditions (e.g. if $M$ is compact~\footnote{%
If $M$ is non compact but only locally compact, then one take $\mathcal{A}=%
\mathcal{C}_{0}(M)$ the algebra of continuous functions vanishing at
infinity but $\mathcal{A}$ is no longer unital since the constant function $1
$ doesn't vanish at infinity.}), it is a $C^{\ast }$-algebra for complex
conjugation $f^{\ast }=\overline{f}$.
The possible values of $f$~-- that is the possible results of a measure of $f
$~-- can be defined in a purely algebraic way as the \emph{spectrum} of $f$
that is
\[
\limfunc{sp}\nolimits_{\mathcal{A}}(f):=\left\{ c:f-cI\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is not
invertible in }\mathcal{A}\right\} ~.
\]
\noindent Indeed, if $f(x)=c$ then $f-cI$ is not invertible in $\mathcal{A}$%
. $\limfunc{sp}\nolimits_{\mathcal{A}}(f)$ is the complementary set of what
is called the \emph{resolvent} of $f$,
\[
r(f):=\left\{ c:f-cI\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is invertible in }\mathcal{A}\right\} ~.
\]
The main point is that the evaluation process $f(x)$~-- that is measure~--
can be interpreted as a \emph{duality} $\left\langle f,x\right\rangle $
between the space $M$ and the algebra $\mathcal{A}$. Indeed, to a point $x$
of $M$ we can associate the \emph{maximal ideal} of the $f\in \mathcal{A}$
vanishing at $x$:
\[
\mathfrak{M}_{x}:=\left\{ f\in \mathcal{A}:f(x)=0\right\} ~.
\]
\noindent But the maximal ideals $\mathfrak{M}$ of $\mathcal{A}$ constitute
themselves a space~-- called the \emph{spectrum} of the algebra $\mathcal{A}$%
. They can be considered as the kernels of the \emph{characters} of $%
\mathcal{A}$, that is of the morphisms (multiplicative linear forms) $\chi :%
\mathcal{A}\rightarrow \mathbb{C}$,
\[
\mathfrak{M=}\chi ^{-1}(0)~.
\]
\noindent A character is by definition a coherent procedure for evaluating
together all the elements $f\in \mathcal{A}$. The evaluation $\chi (f)$ is
also a \emph{duality} $\left\langle \chi ,f\right\rangle $.and its outputs $%
\chi (f)$ belong to $\limfunc{sp}\nolimits_{\mathcal{A}}(f)$.\ Indeed, as
\emph{distributions} (continuous linear forms), the characters correspond to
the Dirac distributions $\delta _{x}$ and if $\chi =\delta _{x}$, then $\chi
(f)=f(x)=c$ and $c\in \limfunc{sp}\nolimits_{\mathcal{A}}(f)$.
The \emph{spectrum} of the $C^{\ast }$-algebra $\mathcal{A}$ (not to be
confused with the spectra $\limfunc{sp}\nolimits_{\mathcal{A}}(f)$ of the
single elements $f$ of $\mathcal{A}$) is by definition the space of
characters $\func{Sp}(\mathcal{A}):=\{\chi \}$ endowed with the topology of
simple convergence: $\chi _{n}\rightarrow \chi $ iff $\chi
_{n}(f)\rightarrow \chi (f)$ for every $f\in \mathcal{A}$. It is defined
uniquely from $\mathcal{A}$ without any reference to the fact that $\mathcal{%
A}$ is of the form $\mathcal{A}:=\mathcal{C}(M)$. It is also the space of
irreducible representations of $\mathcal{A}$ (since $\mathcal{A}$ is
commutative, they are $1$-dimensional).
Now, if $f\in \mathcal{A}$ is an element of $\mathcal{A}$, using duality, we
can associate to it canonically a \emph{function} $\tilde{f}$ on the space $%
\func{Sp}(\mathcal{A})$%
\[
\begin{array}{rll}
\tilde{f}:\func{Sp}(\mathcal{A}) & \rightarrow & \mathbb{C} \\
\chi & \mapsto & \tilde{f}(\chi )=\chi (f)=\left\langle \chi
,f\right\rangle ~.%
\end{array}%
\]
\noindent We get that way a map
\[
\begin{array}{rll}
\symbol{126}:\mathcal{A} & \rightarrow & \mathcal{C(}\func{Sp}\mathcal{(%
\mathcal{A}))} \\
f & \mapsto & \tilde{f}%
\end{array}
\]
\noindent which is called the \emph{Gelfand transform}. For every $f$ we
have
\[
\tilde{f}\left( \func{Sp}(\mathcal{A})\right) =\limfunc{sp}\nolimits_{%
\mathcal{A}}(f)~.
\]
The key result is then:
\textbf{Gelfand-Neimark theorem}.\ If $\mathcal{A}$ is a \emph{commutative} $%
C^{*}$-algebra, the Gelfand transform $\symbol{126}$ is an \emph{isometry}
between $\mathcal{A}$ and $\mathcal{C(}\func{Sp}\mathcal{(\mathcal{A}))}$.
Indeed, the norm of $\tilde{f}$ is the spectral radius of $f$, $\rho \left(
f\right) :=\underset{n\rightarrow \infty }{\lim }\left( \left\|
f^{n}\right\| ^{\frac{1}{n}}\right) $ and we have $\left\| \tilde{f}\right\|
=\rho \left( f\right) =\left\| f\right\| $.\ To see this, suppose first that
$f$ is self-adjoint ($f=f^{*}=\bar{f}$).\ We have $\left\| f\right\|
^{2}=\left\| f.f^{*}\right\| =\left\| f^{2}\right\| $.\ So, $\left\|
f\right\| =\left\| f^{2^{n}}\right\| ^{2^{-n}}$ and as $\left\|
f^{2^{n}}\right\| ^{2^{-n}}\rightarrow \rho \left( f\right) $ by definition
we have $\left\| f\right\| =\rho \left( f\right) $.\ Suppose now that $f$ is
any element of $\mathcal{A}$. Since $f.f^{*}$ is self-adjoint, we have $%
\left\| f\right\| ^{2}=\left\| f.f^{*}\right\| =\rho \left( f.f^{*}\right)
=\left\| \widetilde{f.f^{*}}\right\| $.\ But$\left\| \widetilde{f.f^{*}}%
\right\| =\left\| \widetilde{f}.\widetilde{f^{*}}\right\| =\left\|
\widetilde{f}\right\| ^{2}$ and therefore $\left\| f\right\| ^{2}=\left\|
\widetilde{f}\right\| ^{2}$ and $\left\| f\right\| =\left\| \widetilde{f}%
\right\| $.
Gelfand theory shows that, in the classical case of commutative $C^{*}$%
-algebras $\mathcal{A}:=\mathcal{C}(M)$ ($M$ compact), there exists a
complete \emph{equivalence} between the geometric and the algebraic
perspectives.\
\subsection{Towards a new kinematics}
We think that Gelfand theorem has a deep philosophical meaning. In classical
mechanics kinematics concerns the structure of the configuration spaces $N$
and phase spaces $M:=T^{\ast }N$, and motions and trajectories in them.
Observables and measurements are defined in terms of functions on these
basic spaces directly construed from the geometry of space-time. Gelfand
theorem shows than we can \emph{exchange} the primary geometrical background
with the secondary process of measure, take measure as a primitive fact and
reconstruct the geometric background from it.
\subsection{Towards Noncommutative Geometry}
In Quantum Mechanics, the basic structure is that of the \emph{noncommutative%
} $C^{\ast }$-algebras $\mathcal{A}$ of observables. It is therefore natural
to wonder if there could exist a \emph{geometric }correlate of this
noncommutative algebraic setting.\ It is the origin of Connes'
Noncommutative Geometry (NCG) also called Spectral Geometry or Quantum
Geometry. In NCG the basic structure is the NC $C^{\ast }$-algebra $\mathcal{%
A}$ of obervables: any phenomenon is something which is observable in the
quantum sense, and not an event in space-time. But observables must be
defined for states and are therefore represented in the space of states of
the system, which is an Hilbert space and not the classical space. The
associated NC space is then the space of irreducible representations of $%
\mathcal{A}$.
NCG is a fundamentally new step toward a geometrization of physics. Instead
of beginning with classical differential geometry and trying to develop
Quantum Mechanics on this backgrond, it begins with Quantum Mechanics and
construct a new quantum geometrical framework.\ The most fascinating aspect
of Connes' research program is how he succeeded in reinterpreting all the
basic structures of classical geometry inside the framework of NC $C^{\ast }$%
-algebras operating on Hilbert spaces. The basic concepts remain almost the
same but their mathematical interpretation is significantly complexified,
since their classical meaning becomes a \emph{commutative limit}. We meet
here a new very deep example of the conceptual transformation of physical
theories through mathematical enlargements, as it is the case in general
relativity. As explained by Daniel Kastler \cite{KastlerNCG}:
\begin{quotation}
\noindent
``Alain Connes' noncommutative geometry (...) is a systematic quantization
of mathematics parallel to the quantization of physics effected in the
twenties. (...) This theory widens the scope of mathematics in a manner
congenial to physics.''
\end{quotation}
\section{NCG and differential forms}
Connes reinterpreted (in an extremely deep and technical way) the six
classical levels:
\begin{enumerate}
\item Measure theory;
\item Algebraic topology and topology ($K$-theory);
\item Differentiable structure;
\item Differential forms and De Rham cohomology;
\item Fiber bundles, connections, covariant derivations, Yang-Mills theories;
\item Riemannian manifolds and metric structures.
\end{enumerate}
Let us take as a first example the reinterpretation of the differential
calculus.
\subsection{A universal and formal differential calculus}
How can one interpret differential calculus in the new NC paradigm? One
wants first to define \emph{derivations} $D:\mathcal{A}\rightarrow \mathcal{E%
}$, that is $\mathbb{C}$-linear maps satisfying the \emph{Leibniz rule}
(which is the universal formal rule for derivations):
\[
D(ab)=(Da)b+a(Db)~.
\]
\noindent For that, $\mathcal{E}$ must be endowed with a structure of $%
\mathcal{A}$-bimodule (right and left products of elements of $\mathcal{E}$
by elements of $\mathcal{A}$).\ It is evident that $D(c)=0$ for any scalar $%
c\in \mathbb{C}$ since $D(1.a)=D(1)a+1D(a)=D(a)$ and therefore $D(1)=0$.
Let $\func{Der}(\mathcal{A},\mathcal{E})$ be the $\mathbb{C}$-vector space
of such derivations. In $\func{Der}(\mathcal{A},\mathcal{E})$ there exist
very particular elements, the \emph{inner} derivatives, associated with the
elements $m$ of $\mathcal{E}$, which express the difference between the
right and left $\mathcal{A}$-module structures of $\mathcal{E}$:
\[
D(a):=\func{ad}(m)(a)=ma-am~.
\]
\noindent Indeed,
\begin{eqnarray*}
\func{ad}(m)(a).b+a.\func{ad}(m)(b) &=&(ma-am)b+a(mb-bm) \\
&=&mab-abm \\
&=&\func{ad}(m)(ab)~.
\end{eqnarray*}
\noindent In the case where $\mathcal{E}=\mathcal{A}$, $\func{ad}(b)(a)=%
\left[ b,a\right] $ expresses the non commutativity of $\mathcal{A}$. By the
way, $\func{Der}(\mathcal{A},\mathcal{A})$ is a Lie algebra since $\left[
D_{1},D_{2}\right] $ is a derivation if $D_{1},D_{2}$ are derivations.
Now, it must be stressed that there exists \emph{a universal derivation}
depending only upon the algebraic structure of $\mathcal{A}$ (supposed to be
unital), and having therefore nothing to do with the classical
\textquotedblleft infinitesimal\textquotedblright\ intuitions underlying the
classical concepts of differential and derivation. It is given by
\[
\begin{array}{rll}
d:\mathcal{A} & \rightarrow & \mathcal{A\otimes }_{\mathbb{C}}\mathcal{A}
\\
a & \mapsto & da:=1\otimes a-a\otimes 1~.%
\end{array}%
\]
Let $\Omega ^{1}\mathcal{A}$ be the sub-bimodule of $\mathcal{A\otimes }_{%
\mathbb{C}}\mathcal{A}$ generated by the elements $adb:=a\otimes b-ab\otimes
1$, i.e. the kernel of the multiplication $a\otimes b\mapsto ab$.\footnote{%
For $a\otimes b-ab\otimes 1$ the multiplication gives $ab-ab=0$. Conversely
if $ab=0$ then $a\otimes b=a\otimes b-ab\otimes 1$ and $a\otimes b$ belongs
to $\Omega ^{1}\mathcal{A}$.} $\Omega ^{1}\mathcal{A}$ is isomorphic to the
tensorial product $\mathcal{A\otimes }_{\mathbb{C}}\overline{\mathcal{A}%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,}}$ where $\overline{\mathcal{A}}$ is the quotient $\mathcal{A}/%
\mathbb{C}$ (i.e. $\mathcal{A}=\mathbb{C}1\oplus \overline{\mathcal{A}}$),
with $adb=a\otimes \overline{b}$. It is called the bimodule of \emph{%
universal }$1$\emph{-forms} on $\mathcal{A}$ where \textquotedblleft
universality\textquotedblright\ means that
\[
\func{Der}(\mathcal{A},\mathcal{E})\simeq \limfunc{Hom}\nolimits_{\mathcal{A}%
}\left( \Omega ^{1}\mathcal{A},\mathcal{E}\right)
\]
\noindent i.e. that a derivation $D:\mathcal{A}\rightarrow \mathcal{E}$ is
the same thing as a morphism of algebras between $\Omega ^{1}\mathcal{A}$
and $\mathcal{E}$. If $D:\mathcal{A}\rightarrow \mathcal{E}$ is an element
of $(\mathcal{A},\mathcal{E})$, the associated morphism $\tilde{D}:\Omega
^{1}\mathcal{A}\rightarrow \mathcal{E}$ is defined by
\[
a\otimes b\mapsto aD(b)~.
\]
\noindent So $da=1\otimes a-a\otimes 1\mapsto 1.D(a)-a.D(1)=D(a)$ (since $%
D(1)=0$).
We can generalize this construction to universal $n$-forms, which have the
symbolic form\footnote{$da_{1}...da_{n}$ is the exterior product of $1$%
-forms, classically denoted $da_{1}\wedge ...\wedge da_{n}$.}
\[
a_{0}da_{1}...da_{n}~.
\]
\noindent If $\Omega ^{n}\mathcal{A}:=\left( \Omega ^{1}\mathcal{A}\right)
^{\otimes _{\mathcal{A}}n}$ with $a_{0}da_{1}...da_{n}=a_{0}\otimes
\overline{a_{1}}\otimes ...\otimes \overline{a_{n}}$, the differential is
then
\[
\begin{array}{rll}
d:\Omega ^{n}\mathcal{A} & \rightarrow & \Omega ^{n+1}\mathcal{A} \\
a_{0}da_{1}...da_{n} & \mapsto & da_{0}da_{1}...da_{n} \\
a_{0}\otimes \overline{a_{1}}\otimes ...\otimes \overline{a_{n}} & \mapsto
& 1\otimes \overline{a_{0}}\otimes \overline{a_{1}}\otimes ...\otimes
\overline{a_{n}}~.%
\end{array}%
\]
\noindent Since $d1=0$, it is easy to verify the fundamental cohomological
property $d^{2}=0$ of the graduate differential algebra $\Omega \mathcal{A}%
:=\bigoplus_{n\in \mathbb{N}}\Omega ^{n}\mathcal{A}$. Some technical
difficulties must be overcome (existence of ``junk'' forms) to transform
this framework into a ``good'' formal differential calculus.
\subsection{Noncommutative differential calculus or ``quantized'' calculus}
To use this noncommutative differential in physics, Connes wanted to \emph{%
represent} the universal differential algebra in spaces of physical states.
Let us suppose therefore that the $C^{\ast }$-algebra $\mathcal{A}$ acts
upon an Hilbert space of states $\mathcal{H}$. One wants to interpret in
this representation the universal, formal, and purely symbolic differential
calculus of the previous section. For achieving that, one must interpret the
differential $df$ of the elements $f\in \mathcal{A}$ when these $f$ are
represented as \emph{operators} on $\mathcal{H}$. Connes' main idea was to
use the well-known formula of quantum mechanics
\[
\frac{df}{dt}=\frac{2i\pi }{h}[F,f]
\]
\noindent where $F$ is the Hamiltonian of the system and $f$ any obervable.
Consequently, he interpreted the symbol $df$ as
\[
df:=\left[ F,f\right]
\]
\noindent for an appropriate self-adjoint operator $F$. One wants of course $%
d^{2}f=0.$\ But $d^{2}f=\left[ F^{2},f\right] $ and therefore $F^{2}$ must
commute with all observables.\
The main constraint is that, once interpreted in $\mathcal{H}$, the symbol $%
df$ must correspond to an \emph{infinitesimal}.\ The classical concept of
infinitesimal ought to be reinterpreted in the noncommutative framework.
Connes' definition is that an operator $T$ is infinitesimal if it is \emph{%
compact}, that is if the eigenvalues $\mu _{n}(T)$ of its absolute value $%
\left\vert T\right\vert =\left( T^{\ast }T\right) ^{1/2}$~-- called the
\emph{characteristic values} of $T$~-- converge to $0$, that is if for every
$\varepsilon >0$ the norm $\left\Vert T\right\Vert $ of $T$ is $<\varepsilon
$ outside a subspace of\emph{\ finite} dimension. If $\mu _{n}(T)\underset{%
n\rightarrow \infty }{\rightarrow }0$ as $\frac{1}{n^{\alpha }}$ then $T$ is
an infinitesimal of order $\alpha $ ($\alpha $ is not necessarily an
integer). If $T$ is compact, let $\xi _{n}$ be a complete orthonormal basis
of $\mathcal{H}$ associated to $\left\vert T\right\vert $, $T=U\left\vert
T\right\vert $ the polar decomposition of $T$~\footnote{%
The polar decomposition $T=U\left\vert T\right\vert $ is the equivalent for
operators of the decomposition $z=\left\vert z\right\vert e^{i\theta }$ for
complex numbers. In general $U$ cannot be unitary but only a partial
isometry.} and $\eta _{n}=U\xi _{n}$.\ Then $T$ is the sum
\[
T=\sum_{n\geq 0}\mu _{n}(T)\left\vert \eta _{n}\right\rangle \left\langle
\xi _{n}\right\vert ~.
\]
If $T$ is a positive infinitesimal of order $1$, its trace $\func{Trace}%
\left( T\right) =\sum_{n}\mu _{n}(T)$ has a logarithmic divergence.\ If $T$
is of order $>1$, its trace is finite $>0$. It is the basis for
noncommutative\ integration which uses the \emph{Dixmier trace}, a technical
tool for constructing a new trace extracting the logarithmic divergence of
the classical trace. Dixmier trace is a trick giving a meaning to the
formula $\underset{N\rightarrow \infty }{\lim }\frac{1}{\ln N}%
\sum_{n=0}^{n=N-1}\mu _{n}(T)$. It vanishes for infinitesimals of order $>1$.
Therefore, we interpret the differential calculus in the noncommutative
framework through triples $(\mathcal{A},\mathcal{H},F)$ where $\left[ F,f%
\right] $ is compact for every $f\in \mathcal{A}$.\ Such a structure is
called \emph{a Fredholm module}. The differential forms $a_{0}da_{1}...da_{n}
$ can now be interpreted as operators on $\mathcal{H}$ according to the
formula%
\[
a_{0}da_{1}...da_{n}:=a_{0}\left[ F,a_{1}\right] ...\left[ F,a_{n}\right] ~.
\]
It must be emphasized that the noncommutative generalization of differential
calculus is a wide and wild generalization since it enables us to extend
differential calculus to fractals!
\section{NC Riemannian geometry, Clifford algebras, and Dirac operator}
Another great achievement of Alain Connes was the complete and deep
reinterpretation of the $ds^{2}$ in Riemannian geometry. Classically, $%
ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }$.\ In the noncommutative framework, $dx
$ must be interpreted as $dx=[F,x]$ (where $(\mathcal{A},\mathcal{H},F)$ is
a Fredholm module), and the matrix $\left( g_{\mu \nu }\right) $ as an
element of the $n\times n$ matrix algebra $M_{n}(\mathcal{A})$. The $ds^{2}$
must therefore become a \emph{compact} and \emph{positive} operator of the
form
\[
G=[F,x^{\mu }]^{\ast }g_{\mu \nu }[F,x^{\nu }]~.
\]
\subsection{A redefinition of distance}
Connes' idea is to reinterpret the classical definition of distance $d(p,q)$
between two points $p,q$ of a Riemannian manifold $M$ as the $\func{Inf}$ of
the length $L(\gamma )$ of the paths $\gamma :p\rightarrow q$%
\[
d(p,q)=\underset{\gamma :p\rightarrow q}{\func{Inf}}L(\gamma )
\]
\[
L(\gamma )=\int_{p}^{q}ds=\int_{p}^{q}\left( g_{\mu \nu }dx^{\mu }dx^{\nu
}\right) ^{1/2}~.
\]
\noindent Using the equivalence between a point $x$ of $M$ and the pure
state $\delta _{x}$ on the commutative $C^{*}$-algebra $\mathcal{A}%
:=C^{\infty }\left( M\right) $, an elementary computation shows that this
definition of the distance is equivalent to the dual algebraic definition
using only concepts concerning the $C^{*}$-algebra $\mathcal{A}$%
\[
d(p,q)=\func{Sup}\left\{ \left| f(q)-f(p)\right| :\left\| \func{grad}%
(f)\right\| _{\infty }\leq 1\right\}
\]
\noindent where $\left\Vert ...\right\Vert _{\infty }$ is the $L^{\infty }$
norm, that is the $\func{Sup}$ on $x\in M$ of the norms on the tangent
spaces $T_{x}M$.\footnote{%
Let $\gamma :I=\left[ 0,1\right] \rightarrow M$ be a $C^{\infty }$ curve in $%
M$ from $p$ to $q$. $L(\gamma )=\int_{p}^{q}\left\vert \dot{\gamma}\left(
t\right) \right\vert dt=\int_{0}^{1}g\left( \dot{\gamma}\left( t\right) ,%
\dot{\gamma}\left( t\right) \right) ^{1/2}dt$. If $f\in C^{\infty }\left(
M\right) $, using the duality between $df$ and $\func{grad}f$ induced by the
metric, we find $f(q)-f(p)=\int_{0}^{1}df_{\gamma (t)}\left( \dot{\gamma}%
\left( t\right) \right) dt=\int_{0}^{1}g_{\gamma (t)}\left( \func{grad}%
_{\gamma (t)}f,\dot{\gamma}\left( t\right) \right) dt$. This shows that $%
\left\vert f(q)-f(p)\right\vert \leq \int_{0}^{1}\left\vert \func{grad}%
_{\gamma (t)}f\right\vert \left\vert \dot{\gamma}\left( t\right) \right\vert
dt\leq \left\Vert \func{grad}f\right\Vert _{\infty }L\left( \gamma \right) $%
. Therefore, if $\left\Vert \func{grad}(f)\right\Vert _{\infty }\leq 1$ we
have $\left\vert f(q)-f(p)\right\vert \leq d(p,q).\;$When we take the Sup we
retreive $d(p,q)$ using the special function $f_{p}(x)=d(p,x)$ since $%
\left\vert f_{p}(q)-f_{p}(p)\right\vert =d(p,q)$.}
\subsection{Clifford algebras}
Now the core of the noncommutative definition of distance is the use of the
\emph{Dirac operator}.\ In order to explain this key point, which transforms
the geometrical concept of distance into a quantum concept, the \emph{%
Clifford algebra} of a Riemannian manifold must be introduced.
Recall that the formalism of Clifford algebras relates the differential
forms and the metric on a Riemannian manifold. In the simple case of
Euclidean space $\mathbb{R}^{n}$, the main idea is to encode the isometries $%
O(n)$ in an algebra structure.\ Since every isometry is a product of
reflections (Cartan), one can associate to any vector $v\in \mathbb{R}^{n}$
the reflection $\overline{v}$ relative to the orthogonal hyperplane $v^{\bot
}$ and introduce a multiplication $v.w$ which is nothing else than the
composition $\overline{v}\circ \overline{w}$. We are then naturally led to
the anti-commutation relations
\[
\left\{ v,w\right\} :=v.w+w.v=-2(v,w)
\]
\noindent where $(v,w)$ is the Euclidean scalar product.
More generally, let $V$ be a $\mathbb{R}$-vector space endowed with a
quadratic form $g$.\ Its Clifford algebra $\func{Cl}(V,g)$ is its tensor
algebra $\mathcal{T}(V)=\oplus _{k=0}^{k=\infty }V^{\otimes k}$ quotiented
by the relations
\[
v\otimes v=-g(v)1,\;\forall v\in V
\]
\noindent (where $g(v)=g(v,v)=\left\| v\right\| ^{2}$). In $\func{Cl}(V,g)$
the tensorial product $v\otimes v$ becomes a product $v.v=v^{2}$. It must be
stressed that there exists always in $\func{Cl}(V,g)$ the constants $\mathbb{%
R}$ which correspond to the $0$th tensorial power of $V$.
Using the scalar product
\[
2g(v,w)=g(v+w)-g(v)-g(w)
\]
\noindent one gets the anti-commutation relations
\[
\left\{ v,w\right\} =-2g(v,w)~.
\]
Elementary examples are given by the $\func{Cl}_{n}=\func{Cl}(\mathbb{R}%
^{n},g_{\func{Euclid}}).$
\begin{itemize}
\item $\func{Cl}_{0}=\mathbb{R}$.
\item $\func{Cl}_{1}=\mathbb{C}$ ($V=i\mathbb{R}$, $i^{2}=-1$, $\func{Cl}%
_{1}=\mathbb{R\oplus }i\mathbb{R}$).
\item $\func{Cl}_{2}=\mathbb{H}$ ($V=i\mathbb{R+}j\mathbb{R}$, $ij=k$, $%
\func{Cl}_{2}=\mathbb{R\oplus }i\mathbb{R\oplus }j\mathbb{R\oplus }k\mathbb{R%
}$).
\item $\func{Cl}_{3}=\mathbb{H\oplus H}$.
\item $\func{Cl}_{4}=\mathbb{H}[2]$ ($2\times 2$ matrices with entries in $%
\mathbb{H}$).
\item $\func{Cl}_{5}=\mathbb{C}[4]$.
\item $\func{Cl}_{6}=\mathbb{R}[8]$.
\item $\func{Cl}_{7}=\mathbb{R}[8]\oplus \mathbb{R}[8]$.
\item $\func{Cl}_{n+8}=\func{Cl}_{n}\otimes \mathbb{R}[16]$ (Bott
periodicity theorem).
\end{itemize}
If $g(v)\neq 0$ (which would always be the case for $v\neq 0$ if $g$ is non
degenerate) $v$ is invertible in this algebra structure and
\[
v^{-1}=-\frac{v}{g(v)}~.
\]
\noindent The multiplicative Lie group $\func{Cl}^{\times }(V,g)$ of the
invertible elements of $\func{Cl}(V,g)$ act through inner automorphisms on $%
\func{Cl}(V,g)$.\ This yields the adjoint representation
\[
\begin{array}{rll}
\func{Ad}:\func{Cl}^{\times }(V,g) & \rightarrow & \func{Aut}\left( \func{Cl%
}(V,g)\right) \\
v & \mapsto & \func{Ad}_{v}:w\mapsto v.w.v^{-1}~%
\end{array}%
\]
\noindent But~\footnote{$v.w.v^{-1}=$ $-v.w.\frac{v}{g(v)}=$ $-(-w.v-2g(v,w))%
\frac{v}{g(v)}=$ $w.\frac{v^{2}}{g(v)}+\frac{2g(v,w)v}{g(v)}=$ $-w+\frac{%
2g(v,w)v}{g(v)}.$}
\[
v.w.v^{-1}=-w+\frac{2g(v,w)v}{g(v)}=\func{Ad}_{v}(w)~.
\]
\noindent As $-\func{Ad}_{v}$ is the reflection relative to $v^{\bot }$,
this means that reflections act through the adjoint representation of the
Clifford algebra. The derivative $\func{ad}$ of the adjoint representation
enables to recover the Lie bracket of the Lie algebra $\func{cl}^{\times
}(V,g)=\func{Cl}(V,g)$ of the Lie group $\func{Cl}^{\times }(V,g)$%
\[
\begin{array}{rll}
\func{ad}:\func{cl}^{\times }(V,g)=\func{Cl}(V,g) & \rightarrow & \func{Der}%
\left( \func{Cl}(V,g)\right) \\
v & \mapsto & \func{ad}_{v}:w\mapsto [v,w]%
\end{array}
\]
Now there exists a fundamental relation between the Clifford algebra $(V,g)$
of $V$ and its exterior algebra $\Lambda ^{\ast }V$.\ If $g=0$ and if we
interpret $v.w$ as $v\wedge w$, the anti-commutation relations become simply
$\{v,w\}=0$, which is the classical antisymmetry $w\wedge v=-v\wedge w$ of
differential $1$-forms. Therefore
\[
\Lambda ^{\ast }V=(V,0)~.
\]
\noindent In fact, $(V,g)$ can be considered as a way of \emph{quantizing} $%
\Lambda ^{\ast }V$ using the metric $g$ in order to get non trivial
anti-commutation relations.
Due to the relations $v^{2}=-g(v)1$ which decrease the degree of a product
by $2$, $\func{Cl}(V,g)$ is no longer a $\mathbb{Z}$-graded algebra but only
a $\mathbb{Z}/2$-graded algebra, the $\mathbb{Z}/2$-gradation corresponding
to the even/odd elements. But we can reconstruct a $\mathbb{Z}$-graded
algebra $\mathcal{C}=\bigoplus\limits_{k=0}^{k=\infty }C^{k}$ associated to $%
\func{Cl}(V,g)$, the $C^{k}$ being the homogeneous terms of degree $k$: $%
v_{1}.\cdots .v_{k}$.
\textbf{Theorem}.\ The map of graded algebras $\mathcal{C}%
=\bigoplus\limits_{k=0}^{k=\infty }C^{k}\rightarrow \Lambda ^{\ast
}V=\bigoplus\limits_{k=0}^{k=\infty }\Lambda ^{k}$ given by $v_{1}.\cdots
.v_{k}\rightarrow v_{1}\wedge \cdots \wedge v_{k}$ is a \emph{linear}
isomorphism (but not an \emph{algebra} isomorphism).
We consider now $2$ operations on the exterior algebra $\Lambda ^{\ast }V$ :
\begin{enumerate}
\item The outer multiplication $\varepsilon (v)$ by $v\in V$:
\[
\varepsilon (v)\left( \underset{i}{\wedge }u_{i}\right) =v\wedge \left(
\underset{i}{\wedge }u_{i}\right) ~.\
\]%
We have $\varepsilon (v)^{2}=0$ since $v\wedge v=0$.
\item The contraction (inner multiplication) $\iota (v)$ induced by the
metric $g$:\footnote{%
In the following formula $\widehat{u_{j}}$ means that the term $u_{j}$ is
deleted.}
\[
\iota (v)\left( \underset{i}{\wedge }u_{i}\right)
=\sum\limits_{j=1}^{j=k}(-1)^{j}g(v,u_{j})\;u_{1}\wedge \cdots \wedge
\widehat{u_{j}}\wedge \cdots u_{k}~.
\]
We have also $\iota (v)^{2}=0$. The inner multiplication $\iota (v)$ is a
supplementary structure involving the metric structure.
\end{enumerate}
\noindent One shows that the following anti-commutations relations obtain:
\[
\left\{ \varepsilon (v),\iota (w)\right\} =-g(v,w)1~.
\]
\noindent Let now $c(v)=\varepsilon (v)+\iota (v)$.\ We get the
anti-commutation relations of the Clifford algebra
\[
\left\{ c(v),c(w)\right\} =-2g(v,w)1
\]
\noindent and $\func{Cl}(V,g)$ is therefore generated in $\func{End}_{%
\mathbb{R}}\left( \Lambda ^{*}V\right) $ by the $c(v)$ (identified with $v$).
\subsection{Spin groups}
The isometry group $O(n)$ is canonically embedded in $\func{Cl}(V,g)$ since
every isometry is a product of reflections. In fact $\func{Cl}(V,g)$
contains also the \emph{pin group} $\func{Pin}(n)$ which is a $2$-fold
covering of $O(n)$.\ If we take into account the orientation and restrict to
$SO(n)$, the $2$-fold covering becomes the \emph{spin group} $\func{Spin}(n)$%
. $\func{Spin}(n)$ is generated by the even products of $v$ s.t. $g(v)=\pm 1$%
, $SO(n)$ is generated by even products of $-Ad_{v}$ and the covering $\func{%
Spin}(n)\rightarrow SO(n)$ is given by $v\mapsto -Ad_{v}$. By restriction of
the Clifford multiplication and of the adjoint representation $w\mapsto
v.w.v^{-1}$ to $\func{Spin}(n)$, we get therefore a representation $\gamma $
of $\func{Spin}(n)$ into the spinor space $\mathbb{S}=\func{Cl}(V,g)$.
\subsection{Dirac equation}
We can use the Clifford algebra, and therefore the metric, to change the
classical exterior derivative of differential forms given by
\[
d:=\varepsilon \left( dx^{\mu }\right) \frac{\partial }{\partial x^{\mu }}~.
\]
\noindent We then define the Dirac operator on spinor fields $\mathbb{R}%
^{n}\rightarrow \mathbb{S}$ as
\begin{eqnarray*}
D &:&=c\left( dx^{\mu }\right) \frac{\partial }{\partial x^{\mu }} \\
&=&\gamma ^{\mu }\frac{\partial }{\partial x^{\mu }}~.
\end{eqnarray*}
\noindent where $c$ is the Clifford multiplication,\ and $D$ acts on the
spinor space $\mathbb{S}=\func{Cl}(V,g)$. As $\left\{ c(v),c(w)\right\}
=-2g(v,w)1$, the $\gamma ^{\mu }$ satisfy standard Dirac relations of
anticommutation $\left\{ \gamma ^{\mu },\gamma ^{\nu }\right\} =-2\delta
^{\mu \nu }$ in the Euclidean case.\footnote{%
The classical Dirac matrices are the $-i\gamma ^{\mu }$ for $\mu =0,1,2,3$.}
On can check that $D^{2}=\Delta $ is the Laplacian.
\subsection{Dirac operator}
More generally, if $M$ is a Riemannian manifold, the previous construction
can be done for every tangent space $T_{x}M$ endowed with the quadratic form
$g_{x}$.\ In this way we get a bundle of Clifford algebras $\func{Cl}(TM,g)$%
. If $S$ is a spinor bundle, that is a bundle of $\func{Cl}(TM)$ -modules
s.t. $\func{Cl}(TM)\simeq \func{End}(S)$, endowed with a covariant
derivative ${\Greekmath 0272} $, we associate to it the Dirac operator
\[
D:\mathcal{S}=\Gamma (S)=C^{\infty }(M,S)\rightarrow \Gamma (S)
\]
\noindent which is a first order elliptic operator interpretable as the
``square root'' of the Laplacian $\Delta $, $\Delta $ interpreting itself
the metric in operatorial terms. The Dirac operator $D$ establishes a
coupling between the covariant derivation on $S$ and the Clifford
multiplication of $1$-forms. It can be extended from the $C^{\infty }(M)$%
-module $\mathcal{S}=\Gamma (S)$ to the Hilbert space $\mathcal{H}
=L^{2}(M,S) $.
In general, due to chirality, $S$ will be the direct sum of an even and an
odd part, $S=S^{+}\oplus S^{-}$ and $D$ will have the characteristic form
\begin{eqnarray*}
D &=&\left[
\begin{array}{cc}
0 & D^{-} \\
D^{+} & 0%
\end{array}%
\right] \\
D^{+} &:&\Gamma (S^{+})\rightarrow \Gamma (S^{+}) \\
D^{-} &:&\Gamma (S^{-})\rightarrow \Gamma (S^{-})
\end{eqnarray*}
\noindent $D^{+}$ and $D^{-}$ being adjoint operators.
\subsection{Noncommutative distance and Dirac operator}
In this classical framework, it easy to compute the bracket $[D,f]$ for $%
f\in C^{\infty }(M)$.\ First, there exists on $M$ the \emph{Levi-Civita
connection}:
\[
{\Greekmath 0272} ^{g}:\Omega ^{1}(M)\rightarrow \Omega ^{1}(M)\underset{C^{\infty }(M)}%
{\otimes }\Omega ^{1}(M)
\]
\noindent satisfying the Leibniz rule for $\alpha \in \Omega ^{1}(M)$ and $%
f\in C^{\infty }(M)$:
\[
{\Greekmath 0272} ^{g}(\alpha f)={\Greekmath 0272} ^{g}(\alpha )f+\alpha \otimes df
\]
\noindent (as ${\Greekmath 0272} ^{g}(\alpha )\in \Omega ^{1}(M)\underset{C^{\infty }(M)%
}{\otimes }\Omega ^{1}(M)$, ${\Greekmath 0272} ^{g}(\alpha )f\in \Omega ^{1}(M)\underset%
{C^{\infty }(M)}{\otimes }\Omega ^{1}(M)$ and as $\alpha $ and $df\in \Omega
^{1}(M)$, $\alpha \otimes df\in \Omega ^{1}(M)\underset{C^{\infty }(M)}{%
\otimes }\Omega ^{1}(M)$). There exists also the \emph{spin connection} on
the spinor bundle $S$
\[
{\Greekmath 0272} ^{S}:\Gamma (S)\rightarrow \Omega ^{1}(M)\underset{C^{\infty }(M)}{%
\otimes }\Gamma (S)
\]
\noindent satisfying the Leibniz rule for $\psi \in \Gamma (S)$ and $f\in
C^{\infty }(M)$:
\begin{eqnarray*}
{\Greekmath 0272} ^{S}(\psi f) &=&{\Greekmath 0272} ^{S}(\psi )f+\psi \otimes df \\
{\Greekmath 0272} ^{S}\left( \gamma (\alpha )\psi \right) &=&\gamma \left( {\Greekmath 0272}
^{g}(\alpha )\right) \psi +\gamma (\alpha ){\Greekmath 0272} ^{S}(\psi )
\end{eqnarray*}
\noindent where $\gamma $ is the spin representation.
The Dirac operator on $\mathcal{H}=L^{2}(M,S)$ is then defined as
\[
D:=\gamma \circ {\Greekmath 0272} ^{S}~.
\]
\noindent If $\psi \in \Gamma (S)$, we have (making the $f$ acting on the
left in $\mathcal{H}$)
\begin{eqnarray*}
D\left( f\psi \right) &=&\gamma \left( {\Greekmath 0272} ^{S}(\psi f)\right) \\
&=&\gamma \left( {\Greekmath 0272} ^{S}(\psi )f+\psi \otimes df\right) \\
&=&\gamma \left( {\Greekmath 0272} ^{S}(\psi )\right) f+\gamma \left( \psi \otimes
df\right) \\
&=&fD(\psi )+\gamma \left( df\right) \psi
\end{eqnarray*}
\noindent and therefore $[D,f](\psi )=fD(\psi )+\gamma \left( df\right) \psi
-fD(\psi )=\gamma \left( df\right) \psi $, that is
\[
\lbrack D,f]=\gamma \left( df\right) ~.
\]
In the standard case where $M=\mathbb{R}^{n}$ and $S=\mathbb{R}^{n}\times V$%
, $V$ being a $\func{Cl}_{n}$-module of spinors ($\func{Cl}_{n}=\func{Cl}%
\left( \mathbb{R}^{n},g_{\func{Euclid}}\right) $), we have seen that $D$ is
a differential operator with constant coefficients taking its values in $V$.
\[
D=\sum\limits_{k=1}^{k=n}\gamma ^{\mu }\frac{\partial }{\partial x^{\mu }}
\]
\noindent with the constant matrices $\gamma ^{\mu }\in \mathcal{L}(V)$
satisfying the anti-commutation relations
\[
\left\{ \gamma ^{\mu },\gamma ^{\nu }\right\} =-2\delta ^{\mu \nu }~.
\]
\noindent The fundamental point is that the $\gamma ^{\mu }$ are associated
with the basic $1$-forms $dx^{\mu }$ through the isomorphism
\[
c:\mathcal{C}=\Lambda ^{*}(M)\rightarrow \func{gr}\left( \func{Cl}
(TM)\right)
\]
\[
\lbrack D,f]=\gamma \left( df\right) =c(df)
\]
\noindent and $\left\Vert [D,f]\right\Vert $ is the norm of the Clifford
action of $df$ on the space of spinors $L^{2}(M,S)$. But
\begin{eqnarray*}
\left\Vert c(df)\right\Vert ^{2} &=&\underset{x\in M}{\func{Sup}}%
g_{x}^{-1}\left( d\overline{f}(x),df(x)\right) \\
&=&\underset{x\in M}{\func{Sup}}g_{x}\left( \limfunc{grad}\nolimits_{x}%
\overline{f},\limfunc{grad}\nolimits_{x}f\right) \\
&=&\left\Vert \func{grad}(f)\right\Vert _{\infty }^{2}~.
\end{eqnarray*}
\noindent Whence the definition:
\[
d(p,q)=\func{Sup}\left\{ \left\vert f(p)-f(q)\right\vert :f\in \mathcal{A}%
,\left\Vert [D,f]\right\Vert \leq 1\right\} ~.
\]
In this reinterpretation, $ds$ corresponds to \emph{the propagator of the
Dirac operator} $D.$ As an operator acting on the Hilbert space $\mathcal{H}$%
, $D$ is an unbounded self-adjoint operator such that $[D,f]$ is bounded for
every $f\in \mathcal{A}$ and such that its resolvent $(D-\lambda I)^{-1}$ is
compact for every $\lambda \notin \func{Sp}(D)$ (which corresponds to the
fact that $ds$ is infinitesimal) and the trace $\func{Trace}\left(
e^{-D^{2}}\right) $ is \emph{finite}. In terms of the operator $G=[F,x^{\mu
}]^{\ast }g_{\mu \nu }[F,x^{\nu }]$, we have $G=D^{-2}$.
\section{Noncommutative spectral geometry}
Basing himself on several examples, Alain Connes arrived at the following
concept of noncommutative geometry. In the classical commutative case, $%
\mathcal{A}=C^{\infty }\left( M\right) $ is the commutative algebra of
\textquotedblleft coordinates\textquotedblright\ on $M$ represented in the
Hilbert space $\mathcal{H}=L^{2}(M,S)$ by pointwise multiplication~\footnote{%
If $f\in \mathcal{A}$ and $\xi \in \mathcal{H}$, $\left( f\xi \right)
(x)=f(x)\xi (x)$.} and $ds$ is a symbol non commuting with the $f\in
\mathcal{A}$ and satisfying the commutation relations $\left[ \left[
f,ds^{-1}\right] ,g\right] =0$, for every $f,g\in \mathcal{A}$. Any specific
geometry is defined through the representation $ds=D^{-1}$ of $ds$ by means
of a Dirac operator $D=\gamma ^{\mu }{\Greekmath 0272} _{\mu }$. The differential $df=%
\left[ D,f\right] $ is then the Clifford multiplication by the gradient $%
{\Greekmath 0272} f$ and its norm in $\mathcal{H}$ is the Lipschitz norm of $f$: $%
\left\Vert \left[ D,f\right] \right\Vert =\underset{x\in M}{\func{Sup}}%
\left\Vert {\Greekmath 0272} f\right\Vert $.
These results can be taken as a definition in the general case.\ The
geometry is defined by a spectral triple $\left( \mathcal{A},\mathcal{H}%
,D\right) $ where $\mathcal{A}$ is a noncommutative$\;C^{\ast }$-algebra
with a representation in an Hilbert space $\mathcal{H}$ and $D$ is an
unbounded self-adjoint operator on $\mathcal{H}$ such that $ds=D^{-1}$ and
more generally the resolvent $\left( D-\lambda I\right) ^{-1}$, $\lambda
\notin \mathbb{R}$, is compact, and at the same time all $\left[ D,a\right] $
are bounded for every $a\in \mathcal{A}$ (there is a tension between these
two last conditions). As Connes \cite{Connes2000b} emphasizes
\begin{quotation}
\noindent
``It is precisely this lack of commutativity between the line element and
the coordinates on a space [between $ds$ and the $a\in \mathcal{A}$] that
will provide the measurement of distance.''
\end{quotation}
\noindent The new definition of differentials are then $da=\left[ D,a\right]
$ for any $a\in \mathcal{A}$.
\section{Yang-Mills theory of a NC coupling between $2$ points and Higgs
mechanism}
A striking example of pure noncommutative physics is given by Connes'
interpretation of the Higgs phenomenon.\ In the Standard Model, the Higgs
mechanism was an \emph{ad hoc} device used for confering a mass to gauge
bosons.\ It lacked any geometrical interpretation.\ One of the deepest
achievement of the noncommutative perspective has been to show that Higgs
fields correspond effectively to gauge bosons for a \emph{discrete}
noncommutative geometry.
\subsection{Symmetry breaking and classical Higgs mechanism}
Let us first recall the classical Higgs mechanism.\ Consider e.g. a $\varphi
^{4}$ theory for $2$ scalar real fields $\varphi _{1}$ and $\varphi _{2}$.
The Lagrangian is
\[
\mathcal{L}=\frac{1}{2}\left( \partial _{\mu }\varphi _{1}\partial ^{\mu
}\varphi _{1}+\partial _{\mu }\varphi _{2}\partial ^{\mu }\varphi
_{2}\right) -V\left( \varphi _{1}^{2}+\varphi _{2}^{2}\right)
\]
\noindent with the quartic potential
\[
V\left( \varphi _{1}^{2}+\varphi _{2}^{2}\right) =\frac{1}{2}\mu ^{2}\left(
\varphi _{1}^{2}+\varphi _{2}^{2}\right) +\frac{1}{4}\left\vert \lambda
\right\vert \left( \varphi _{1}^{2}+\varphi _{2}^{2}\right) ^{2}~.
\]
\noindent It is by construction $SO(2)$-invariant.
For $\mu ^{2}>0$ the minimum of $V$ (the quantum vacuum) is non degenerate: $%
\varphi _{0}=(0,0)$ and the Lagrangian $\mathcal{L}_{os}$ of small
oscillations in the neighborhood of $\varphi _{0}$ is the sum of $2$
Lagrangians of the form:
\[
\mathcal{L}_{os}=\frac{1}{2}\left( \partial _{\mu }\psi \partial ^{\mu }\psi
\right) -\frac{1}{2}\mu ^{2}\psi ^{2}
\]
\noindent describing particles of mass $\mu ^{2}$.
But for $\mu ^{2}<0$ the situation becomes completely different. Indeed the
potential $V$ has a full circle (a $SO(2)$-orbit) of minima
\[
\varphi _{0}^{2}=-\frac{\mu ^{2}}{\left\vert \lambda \right\vert }=v^{2}
\]
\noindent and the vacuum state is highly \emph{degenerate}.
One must therefore \emph{break the symmetry} to choose a vacuum state.\ Let
us take for instance $\varphi _{0}=\left[
\begin{array}{l}
v \\
0%
\end{array}%
\right] $ and translate the situation to $\varphi _{0}$:
\[
\varphi =\left[
\begin{array}{l}
\varphi _{1} \\
\varphi _{2}%
\end{array}%
\right] =\left[
\begin{array}{l}
v \\
0%
\end{array}%
\right] +\left[
\begin{array}{l}
\xi \\
\eta
\end{array}%
\right] ~.
\]
\noindent The oscillation Lagrangian at $\varphi _{0}$ becomes
\[
\mathcal{L}_{os}=\frac{1}{2}\left( \partial _{\mu }\eta \partial ^{\mu }\eta
+2\mu ^{2}\eta ^{2}\right) +\frac{1}{2}\left( \partial _{\mu }\xi \partial
^{\mu }\xi \right)
\]
\noindent and describes $2$ particles:
\begin{enumerate}
\item a particle $\eta $ of mass $m=\sqrt{2}\left| \mu \right| $, which
corresponds to radial oscillations,
\item a particule $\xi $ of mass $m=0$, which connects vacuum states. $\xi $
is the \emph{Goldstone boson}.
\end{enumerate}
As is well known, the Higgs mechanism consists in using a cooperation
between gauge bosons and Goldstone bosons to confer a mass to gauge bosons.
Let $\varphi =\frac{1}{\sqrt{2}}\left( \varphi _{1}+i\varphi _{2}\right) $
be the scalar complex field associated to $\varphi _{1}$ and $\varphi _{2}$%
.\ Its Lagrangian is
\[
\mathcal{L}=\partial _{\mu }\overline{\varphi }\partial ^{\mu }\varphi -\mu
^{2}\left\vert \varphi \right\vert ^{2}-\left\vert \lambda \right\vert
\left\vert \varphi \right\vert ^{4}~.
\]
\noindent It is trivially invariant by the global internal symmetry $\varphi
\rightarrow e^{i\theta }\varphi $. If we \emph{localize} the global symmetry
using transformations $\varphi (x)\rightarrow e^{iq\alpha (x)}\varphi (x)$
and take into account the coupling with an electro-magnetic field deriving
from the vector potential $A_{\mu }$, we get
\[
\mathcal{L}={\Greekmath 0272} _{\mu }\overline{\varphi }{\Greekmath 0272} ^{\mu }\varphi -\mu
^{2}\left| \varphi \right| ^{2}-\left| \lambda \right| \left| \varphi
\right| ^{4}-\frac{1}{4}F_{\mu \nu }F^{\mu \nu }
\]
\noindent where ${\Greekmath 0272} $ is the covariant derivative
\[
{\Greekmath 0272} _{\mu }=\partial _{\mu }+iqA_{\mu }
\]
\noindent and $F$ the force field
\[
F_{\mu \nu }=\partial _{\nu }A_{\mu }-\partial _{\mu }A_{\nu }~.
\]
\noindent The Lagrangian remains invariant if we balance the localization of
the global internal symmetry with a change of gauge
\[
A_{\mu }\rightarrow A_{\mu }^{\prime }=A_{\mu }-\partial _{\mu }\alpha (x)~.
\]
For $\mu ^{2}>0$, $\varphi _{0}=0$ is a minimum of $V(\varphi ),$ the vacuum
is non degenerate, and we get $2$ scalar particles $\varphi $ and $\overline{%
\varphi }$ and a photon $A_{\mu }$.
For $\mu ^{2}<0$, the vacuum is degenerate and there is a spontaneous
symmetry breaking.\ We have $\left| \varphi _{0}\right| ^{2}=-\frac{\mu ^{2}%
}{2\left| \lambda \right| }=\frac{v^{2}}{2}$.$\;$If we take $\varphi _{0}=%
\frac{^{v}}{\sqrt{2}}$ and write
\[
\varphi =\varphi ^{\prime }+\varphi _{0}=\frac{1}{\sqrt{2}}(v+\eta +i\xi
)\approx \frac{1}{\sqrt{2}}e^{i\frac{\xi }{v}}(v+\eta )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }\xi \RIfM@\expandafter\text@\else\expandafter\mbox\fi{
and }\eta \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ small,}
\]
\noindent we get for the Lagrangian of oscillations:
\[
\mathcal{L}_{os}=\frac{1}{2}\left( \partial _{\mu }\eta \partial ^{\mu }\eta
+2\mu ^{2}\eta ^{2}\right) +\frac{1}{2}\left( \partial _{\mu }\xi \partial
^{\mu }\xi \right) -\frac{1}{4}F_{\mu \nu }F^{\mu \nu }+qvA_{\mu }\left(
\partial _{\mu }\xi \right) +\frac{q^{2}v^{2}}{2}A_{\mu }A^{\mu }~.
\]
\begin{enumerate}
\item The field $\eta $ (radial oscillations) has mass $m=\sqrt{2}\left| \mu
\right| .$
\item The boson $A_{\mu }$ acquires a mass due to the term $A_{\mu }A^{\mu }
$ and interacts with the Goldstone boson $\xi $.
\end{enumerate}
The terms containing the gauge boson $A_{\mu }$ and the Goldstone boson $\xi
$ write
\[
\frac{q^{2}v^{2}}{2}\left( A_{\mu }+\frac{1}{qv}\partial _{\mu }\xi \right)
\left( A^{\mu }+\frac{1}{qv}\partial ^{\mu }\xi \right)
\]
\noindent and are therefore generated by the gauge change
\begin{eqnarray*}
\alpha &=&\frac{\xi }{qv} \\
A_{\mu } &\rightarrow &A_{\mu }+\partial _{\mu }\alpha ~.
\end{eqnarray*}
We see that we can use the gauge transformations
\[
A_{\mu }\rightarrow A_{\mu }^{\prime }=A_{\mu }+\frac{1}{qv}\partial ^{\mu
}\xi
\]%
for \emph{fixing} the vacuum state.\ The transformation corresponds to the
phase rotation of the scalar field
\[
\varphi \rightarrow \varphi ^{\prime }=e^{-i\frac{\xi }{v}}\varphi =\frac{%
v+\eta }{\sqrt{2}}~.
\]
In this new gauge where the Goldstone boson $\xi $ disappears, the vector
particule $A_{\mu }^{\prime }$ acquires a mass $qv$. The Lagrangian writes
now
\[
\mathcal{L}_{os}=\frac{1}{2}\left( \partial _{\mu }\eta \partial ^{\mu }\eta
+2\mu ^{2}\eta ^{2}\right) -\frac{1}{4}F_{\mu \nu }F^{\mu \nu }+\frac{%
q^{2}v^{2}}{2}A_{\mu }^{\prime }A^{\prime \mu }~.
\]
\noindent The Goldstone boson connecting the degenerate vacuum states is in
some sense ``captured'' by the gauge boson and transformed into mass.
\subsection{Noncommutative Yang-Mills theory of $2$ points and Higgs
phenomenon}
The noncommutative equivalent of this description is the following.\ It
shows that Higgs mechanism is actually the standard Yang-Mills formalism
applied to a \emph{purely discrete noncommutative geometry}.
Let $\mathcal{A}=\mathcal{C}(Y)=\mathbb{C}\oplus \mathbb{C}$ be the $C^{*}$%
-algebra of the space $Y$ composed of $2$ points $a$ and $b$. Its elements $%
f=\left[
\begin{array}{cc}
f(a) & 0 \\
0 & f(b)%
\end{array}
\right] $ act through multiplication on the Hilbert space $\mathcal{H}=%
\mathcal{H}_{a}\oplus \mathcal{H}_{b}$. We take for Dirac operator an
operator of the form
\[
D=\left[
\begin{array}{cc}
0 & M^{*}=D^{-} \\
M=D^{+} & 0%
\end{array}
\right]
\]
\noindent and introduce the ``chirality'' $\gamma =\left[
\begin{array}{cc}
1 & 0 \\
0 & -1%
\end{array}
\right] $ (the $\gamma _{5}$ of the standard Dirac theory). In this discrete
situation we define $df$ as
\[
df=[D,f]=\Delta f\left[
\begin{array}{cc}
0 & M^{*} \\
-M & 0%
\end{array}
\right]
\]
\noindent with $\Delta f=f(b)-f(a).\;$Therefore
\[
\left\| \lbrack D,f]\right\| =\left| \Delta f\right| \lambda
\]
where $\lambda =\left\| M\right\| $ is the greatest eigenvalue of $M$.
If we apply now the formula for the distance, we find:
\begin{eqnarray*}
d(a,b) &=&\func{Sup}\left\{ \left| f(a)-f(b)\right| :f\in \mathcal{A}%
,\left\| [D,f]\right\| \leq 1\right\} \\
&=&\func{Sup}\left\{ \left| f(a)-f(b)\right| :f\in \mathcal{A},\left|
f(a)-f(b)\right| \lambda \leq 1\right\} \\
&=&\frac{1}{\lambda }
\end{eqnarray*}
\noindent and we see that the distance $\frac{1}{\lambda }$ between the two
points $a$ and $b$ has a \emph{spectral} content and is measured by the
Dirac operator.
To interpret differential calculus in this context, we consider the $2$
idempotents (projectors) $e$ and $1-e$ defined by
\begin{eqnarray*}
e(a) &=&1,e(b)=0 \\
(1-e)(a) &=&0,(1-e)(b)=1~.
\end{eqnarray*}
\noindent Every $f\in \mathcal{A}$ writes $f=f(a)e+f(b)(1-e)$, and therefore
\begin{eqnarray*}
df &=&f(a)de+f(b)d(1-e) \\
&=&\left( f(a)-f(b)\right) de \\
&=&-(\Delta f)de \\
&=&-(\Delta f)ede+(\Delta f)(1-e)d(1-e)~.
\end{eqnarray*}
\noindent This shows that $ede$ and $(1-e)d(1-e)=-(1-e)de$ provide a natural
basis of the space of $1$-forms $\Omega ^{1}\mathcal{A}$.\ Let
\begin{eqnarray*}
\omega &=&\lambda ede+\mu (1-e)d(1-e) \\
&=&\lambda ede-\mu (1-e)de
\end{eqnarray*}
\noindent a $1$-form.\ $\omega $ is represented by
\[
\omega =\left( \lambda e-\mu (1-e)\right) [D,e]~.
\]
\noindent But on $\mathcal{H}$ $[D,e]=-\left[
\begin{array}{cc}
0 & M^{\ast } \\
-M & 0%
\end{array}%
\right] $ and therefore
\[
\omega =\left[
\begin{array}{cc}
0 & -\lambda M^{\ast } \\
-\mu M & 0%
\end{array}%
\right] ~.
\]
Let us now construct with Connes the Yang-Mills theory corresponding to this
situation. A vector potential $V$~--- a connection in the sense of gauge
theories~--- is a self-adjoint $1$-form and has the form
\begin{eqnarray*}
V &=&-\overline{\varphi }ede+\varphi (1-e)de \\
&=&\left[
\begin{array}{cc}
0 & \overline{\varphi }M^{\ast } \\
\varphi M & 0%
\end{array}%
\right] ~.
\end{eqnarray*}
\noindent Its curvature is the $2$-form
\[
\theta =dV+V\wedge V
\]
\noindent and an easy computation gives
\[
\theta =-\left( \varphi +\overline{\varphi }+\varphi \overline{\varphi }%
\right) \left[
\begin{array}{cc}
-M^{\ast }M & 0 \\
0 & -MM^{\ast }%
\end{array}%
\right] ~.
\]
The Yang-Mills \emph{action} is the integral of the curvature $2$-form, that
is the trace of $\theta $:
\[
YM(V)=\func{Trace}\left( \theta ^{2}\right) ~.
\]
\noindent But as $\varphi +\overline{\varphi }+\varphi \overline{\varphi }
=\left| \varphi +1\right| ^{2}-1$ and
\[
\func{Trace}\left( \left[
\begin{array}{cc}
-M^{*}M & 0 \\
0 & -MM^{*}%
\end{array}
\right] ^{2}\right) =2\func{Trace}\left( (M^{*}M)^{2}\right)
\]
\noindent we get
\[
YM(V)=2\left( \left\vert \varphi +1\right\vert ^{2}-1\right) ^{2}\func{Trace}%
\left( (M^{\ast }M)^{2}\right) ~.
\]
\subsection{Higgs mechanism}
This Yang-Mills action manifests a pure Higgs phenomenon of symmetry
breaking. The minimum of $YM(V)$ is reached everywhere on the circle $%
\left\vert \varphi +1\right\vert ^{2}=1$ (degeneracy) and the gauge group $%
\mathcal{U}=U(1)\times U(1)$ of the unitary elements of $\mathcal{A}$ acts
on it by
\[
V\rightarrow uVu^{\ast }+udu^{\ast }
\]
\noindent where $u=\left[
\begin{array}{cc}
u_{1} & 0 \\
0 & u_{2}%
\end{array}
\right] $ with $u_{1},u_{2}\in U(1)$.
The field $\varphi $ is a Higgs bosonic field corresponding to a gauge
connection on a noncommutative space of $2$ points. If $\psi \in \mathcal{H}$
represents a fermionic state, the fermionic action is $I_{D}\left( V,\psi
\right) =\left\langle \psi ,\left( D+V\right) \psi \right\rangle $ with
\[
D+V=\left[
\begin{array}{cc}
0 & \left( 1+\overline{\varphi }\right) M^{\ast } \\
\left( 1+\varphi \right) M & 0%
\end{array}%
\right] ~.
\]
\noindent The complete action coupling the fermion $\psi $ with the Higgs
boson $\varphi $ is therefore
\[
YM(V)+I_{D}\left( V,\psi \right) ~.
\]
\section{The noncommutative derivation of the Glashow-Weinberg-Salam
Standard Model (Connes-Lott)}
A remarkable achievement of this noncommutative approach of Yang-Mills
theories is given by Connes-Lott's derivation of the Glashow-Weinberg-Salam
Standard Model. This derivation was possible because, as was emphasized by
Martin \emph{et al.} \cite{Martin} (p.\ 5), it ties
\begin{quotation}
\noindent
``the properties of continuous spacetime with the intrinsic discreteness
stemming from the chiral structure of the Standard Model''.
\end{quotation}
\subsection{Gauge theory and NCG}
It is easy to reinterpret in the noncommutative framework classical gauge
theories where $M$ is a spin manifold, $\mathcal{A}=\mathcal{C}^{\infty }(M)$%
, $D$ is the Dirac operator and $\mathcal{H}=L^{2}(M,S)$ is the space of $%
L^{2}$ sections of the spinor bundle $S$. $\func{Diff}(M)=\func{Aut}(%
\mathcal{A})=\func{Aut}\left( \mathcal{C}^{\infty }(M)\right) $ is the
relativity group (the gauge group) of the theory: a diffeomorphism $\varphi
\in \func{Diff}(M)$ is identified with the $\ast $-automorphism $\alpha \in
\func{Aut}(\mathcal{A})$ s.t. $\alpha \left( f\right) (x)=f\left( \varphi
^{-1}\left( x\right) \right) $. The main problem of quantum gravity is to
reconcile quantum field theory with general relativity, that is non abelian
gauge theories, which are noncommutative at the level of their \emph{internal%
} space of quantum variables, with the geometry of the \emph{external}
space-time $M$ with its group of diffeomorphism $\func{Diff}(M)$. The
noncommutative solution is an extraordinary principled one since it links
the standard \textquotedblleft inner\textquotedblright\ noncommutativity of
quantum internal degrees of freedom with the new \textquotedblleft
outer\textquotedblright\ noncommutativity of the external space.
\subsubsection{Inner automorphisms and internal symmetries}
The key fact is that, in the NC\ framework, there exists in $\func{Aut}(%
\mathcal{A})$ the normal subgroup $\func{Inn}(\mathcal{A})$ of \emph{inner
automorphisms} acting by conjugation $a\rightarrow uau^{-1}$.$\;\func{Inn}(%
\mathcal{A})$ is trivial in the commutative case and constitutes one of the
main feature of the NC case. As Alain Connes \cite{Connes96} emphasized:
\begin{quotation}
\noindent
``Amazingly, in this description the group of gauge
transformation of the matter fields arises spontaneously as a normal
subgroup of the generalized diffeomorphism group $\func{Aut}(\mathcal{A})$.
It is the \emph{non commutativity} of the algebra $\mathcal{A}$ which gives
for free the group of gauge transformations of matter fields as a (normal)
subgroup of the group of diffeomorphisms.''
\end{quotation}
In $\func{Inn}(\mathcal{A})$ there exists in particular the \emph{unitary}
group $\mathcal{U}(\mathcal{A})$ of unitary elements $u^{*}=u^{-1}$ acting
by $\alpha _{u}\left( a\right) =uau^{*}$.
\subsubsection{Connections and vector potentials}
In the noncommutative framework we can easily reformulate standard
Yang-Mills theories. For that we need the concepts of a connection and of a
vector potential.
Let $\mathcal{E}$ be a finite projective (right) $\mathcal{A}$-module. A
connection ${\Greekmath 0272} $ on $\mathcal{E}$ is a collection of morphisms (for
every $p)$
\[
{\Greekmath 0272} :\mathcal{E}\otimes _{\mathcal{A}}\Omega ^{p}\left( \mathcal{A}%
\right) \rightarrow \mathcal{E}\otimes _{\mathcal{A}}\Omega ^{p+1}\left(
\mathcal{A}\right)
\]
\noindent satisfying for every $\omega \in \mathcal{E}\otimes _{\mathcal{A}
}\Omega ^{p}\left( \mathcal{A}\right) $ and every $\rho \in \Omega
^{q}\left( \mathcal{A}\right) $ the Leibniz rule in $\mathcal{E}\otimes _{%
\mathcal{A}}\Omega ^{p+q+1}\left( \mathcal{A}\right) $%
\[
{\Greekmath 0272} \left( \omega \otimes \rho \right) ={\Greekmath 0272} \left( \omega \right)
\otimes \rho +\left( -1\right) ^{p}\omega \otimes d\rho
\]
\noindent where we use the relation $\Omega ^{p+1}\left( \mathcal{A}\right)
\otimes _{\mathcal{A}}\Omega ^{q}\left( \mathcal{A}\right) =\Omega
^{p}\left( \mathcal{A}\right) \otimes _{\mathcal{A}}\Omega ^{q+1}\left(
\mathcal{A}\right) $. ${\Greekmath 0272} $ is determined by its restriction to $\Omega
^{1}\left( \mathcal{A}\right) $
\[
{\Greekmath 0272} :\mathcal{E}\otimes _{\mathcal{A}}\Omega ^{0}\left( \mathcal{A}%
\right) =\mathcal{E}\rightarrow \mathcal{E}\otimes _{\mathcal{A}}\Omega
^{1}\left( \mathcal{A}\right)
\]
\noindent satisfying ${\Greekmath 0272} \left( \xi a\right) ={\Greekmath 0272} \left( \xi \right)
a+\xi \otimes da$ for $\xi \in \mathcal{E}$ and $a\in \mathcal{A}$.
The \emph{curvature} $\theta $ of ${\Greekmath 0272} $ is given by ${\Greekmath 0272} ^{2}:%
\mathcal{E}\rightarrow \mathcal{E}\otimes _{\mathcal{A}}\Omega ^{2}\left(
\mathcal{A}\right) $.\ As
\begin{eqnarray*}
{\Greekmath 0272} ^{2}\left( \xi a\right) &=&{\Greekmath 0272} \left( {\Greekmath 0272} \left( \xi \right)
a+\xi \otimes da\right) \\
&=&{\Greekmath 0272} ^{2}\left( \xi \right) a-{\Greekmath 0272} \left( \xi \right) \otimes
da+{\Greekmath 0272} \left( \xi \right) \otimes da+\xi \otimes d^{2}a \\
&=&{\Greekmath 0272} ^{2}\left( \xi \right) a\;,
\end{eqnarray*}
\noindent ${\Greekmath 0272} ^{2}$ is $\mathcal{A}$-linear. And as $\mathcal{E}$ is a
projective $\mathcal{A}$-module,
\[
\theta ={\Greekmath 0272} ^{2}\in \limfunc{End}{}_{\mathcal{A}}\mathcal{E}\otimes _{%
\mathcal{A}}\Omega ^{2}\left( \mathcal{A}\right) =M\left( \mathcal{A}\right)
\otimes _{\mathcal{A}}\Omega ^{2}\left( \mathcal{A}\right)
\]
\noindent is a matrix with elements in $\Omega ^{2}\left( \mathcal{A}\right)
$.
Now, ${\Greekmath 0272} $ defines a connection $\left[ {\Greekmath 0272} ,\bullet \right] $ on $%
\func{End}_{\mathcal{A}}\mathcal{E}$ by
\[
\begin{array}{cccc}
\left[ {\Greekmath 0272} ,\bullet \right] : & \func{End}_{\mathcal{A}}\mathcal{E}%
\otimes _{\mathcal{A}}\Omega ^{p}\left( \mathcal{A}\right) & \rightarrow &
\func{End}_{\mathcal{A}}\mathcal{E}\otimes _{\mathcal{A}}\Omega ^{p+1}\left(
\mathcal{A}\right) \\
& \alpha & \mapsto & \left[ {\Greekmath 0272} ,\alpha \right] ={\Greekmath 0272} \circ \alpha
-\alpha \circ {\Greekmath 0272}%
\end{array}
\]
\noindent and the curvature $\theta $ satisfies the \emph{Bianchi identity} $%
\left[ {\Greekmath 0272} ,\theta \right] =0$.
On the other hand, a vector potential $A$ is a self-adjoint operator
interpreting a $1$-form
\[
A=\sum_{j}a_{j}[D,b_{j}]
\]
\noindent and the associated force is the curvature $2$-form
\[
\theta =dA+A^{2}~.
\]
The unitary group $\mathcal{U}(\mathcal{A})$ acts by gauge transformations
on $A$ and its $2$-form curvature $\theta $
\begin{eqnarray*}
A &\rightarrow &uAu^{\ast }+udu^{\ast }=uAu^{\ast }+u[D,u^{\ast }] \\
\theta &\rightarrow &u\theta u^{\ast }~.
\end{eqnarray*}
\subsection{Axioms for geometry}
There are characteristic properties of classical (commutative) and
noncommutative geometries which can be used to axiomatize them.
$1$. (Classical and NC geometry). $ds=D^{-1}$ is an infinitesimal of order $%
\frac{1}{n}$ ($n$ is the dimension)~\footnote{%
In the NC\ framework, $ds$ and $dx$ are completely different sort of
entities.\ $dx$ is the differential of a coordinate and $ds$ doesn't commute
with it. In the classical case, the order of $ds$ as an infinitesimal is not
$1$ but $1/n$. As we will see later, the Hilbert-Einstein action is the NC
integral of $ds^{n-2}$.} and for any $a\in \mathcal{A}$ integration is given
by $\limfunc{Tr}\nolimits_{Dix}\left( a\left\vert D\right\vert ^{-n}\right) $
(which is well defined and $\neq 0$ since $\left\vert D\right\vert ^{-n}$ is
an infinitesimal of order $1$). One can normalize the integral dividing by $%
V=\limfunc{Tr}\nolimits_{Dix}\left( \left\vert D\right\vert ^{-n}\right) $.
$2$. (Classical geometry). Universal commutation relations: $\left[ \left[
D,a\right] ,b\right] =0$, $\forall a,b\in \mathcal{A}$. So (Jones, Moscovici
\cite{Jones})
\begin{quotation}
\noindent
``while $ds$ no longer commutes with the coordinates, the algebra they
generate does satisfy non trivial commutation relations.''
\end{quotation}
$3$. (Classical and NC geometry). $a\in \mathcal{A}$ is ``smooth'' in the
sense that $a$ and $\left[ D,a\right] $ belong to the intersection of the
domains of the functionals $\delta ^{m}$ where $\delta \left( T\right) =%
\left[ \left| D\right| ,T\right] $ for every operator $T$ on $\mathcal{H}$.
$4$. (Classical geometry). If the dimension $n$ is \emph{even} there exists
a $\widetilde{\gamma }$ interpreting a $n$-form $c\in Z_{n}\left( \mathcal{A}%
,\mathcal{A}\right) $ associated to orientation and chirality (the $\gamma
^{5}$ of Dirac), $\widetilde{\gamma }$ being of the form $a_{0}\left[ D,a_{1}%
\right] \ldots \left[ D,a_{n}\right] $ and s.t. $\widetilde{\gamma }=%
\widetilde{\gamma }^{*}$ (self-adjointness), $\widetilde{\gamma }^{2}=1$, $%
\left\{ \widetilde{\gamma },D\right\} =0$ (anti-commutation relation) and $%
\left[ \widetilde{\gamma },a\right] =0$, $\forall a\in \mathcal{A}$
(commutation relations). $\widetilde{\gamma }$ decomposes $D$ into two parts
$D=D^{+}+D^{^{\_}}$ where $D^{+}=(1-p)Dp$ with $p=\frac{1+\widetilde{\gamma }%
}{2}$. If $e$ is a self-adjoint ($e=e^{*}$) idempotent ($e^{2}=e$) of $%
\mathcal{A}$ (i.e. a projector), $eD^{+}e$ is a Fredholm operator from the
subspace $ep\mathcal{H}$ to the subspace $e(1-p)\mathcal{H}$. This can be
extended to the projectors of $e\in M_{q}\left( \mathcal{A}\right) $
defining finite projective left $\mathcal{A}$-modules $\mathcal{E}=\mathcal{A%
}^{N}e$ (if $\xi \in \mathcal{E}$ then $\xi e=\xi $) with the $\mathcal{A}$%
-valued inner product $\left( \xi ,\eta \right) =\sum_{i=1}^{i=N}\xi
_{i}\eta _{i}^{*}$.
$4$ bis.\ (Classical geometry). If $n$ is odd we ask only that there exists
such an $n$-form $c$ interpreted by $1$: $a_{0}\left[ D,a_{1}\right] \ldots %
\left[ D,a_{n}\right] =1$.
$5$. (Classical and NC geometry). $\mathcal{H}_{\infty }=\bigcap\limits_{m}%
\func{Domain}\left( D^{m}\right) $ is finite and projective as $\mathcal{A}$
-module and the formula $\left\langle a\xi ,\eta \right\rangle =\limfunc{Tr}%
_{Dix}a\left( \xi ,\eta \right) ds^{n}$ ($\left( \xi ,\eta \right) $ being
the scalar product of $\mathcal{H}$ and $\limfunc{Tr}_{Dix}$ the Dixmier
trace of infinitesimals of order $1$) defines an Hermitian structure on $%
\mathcal{H}_{\infty }$.
$6$. (Classical geometry). One can define an \emph{index pairing} of $D$
with $K_{0}\left( \mathcal{A}\right) $ and an \emph{intersection form} on $%
K_{0}\left( \mathcal{A}\right) $~\footnote{%
Remember that $K_{0}\left( \mathcal{A}\right) =\pi _{1}\left( GL_{\infty
}\left( \mathcal{A}\right) \right) $ classifies the finite projective $%
\mathcal{A}$-modules and that $K_{1}\left( \mathcal{A}\right) =\pi
_{0}\left( GL_{\infty }\left( \mathcal{A}\right) \right) $ is the group of
connected components of $GL_{\infty }\left( \mathcal{A}\right) $.}. If $%
\left[ \mathcal{E}\right] \in K_{0}\left( \mathcal{A}\right) $ is defined by
the projector $e$, we consider the scalar product $\left\langle \func{Ind}%
D,e\right\rangle $ which is an integer.\ We define therefore $\left\langle
\func{Ind}D,e\right\rangle :K_{0}\left( \mathcal{A}\right) \rightarrow
\mathbb{Z}$. As $\mathcal{A}$ is commutative, we can take the multiplication
$m:\mathcal{A}\otimes \mathcal{A}\rightarrow \mathcal{A}$ given by $%
m(a\otimes b)=ab$ which induces $m_{0}:K_{0}\left( \mathcal{A}\right)
\otimes K_{0}\left( \mathcal{A}\right) \rightarrow K_{0}\left( \mathcal{A}%
\right) $.\ Composing with $\func{Ind}D$ we get the intersection form
\begin{eqnarray*}
\left\langle \func{Ind}D,m_{0}\right\rangle &:&K_{0}\left( \mathcal{A}%
\right) \otimes K_{0}\left( \mathcal{A}\right) \rightarrow \mathbb{Z} \\
(e,a) &\rightarrow &\left\langle \func{Ind}D,m_{0}(e\otimes a)\right\rangle
~.
\end{eqnarray*}
\emph{Poincar\'{e} duality}: the intersection form is invertible.
$7$. \emph{Real structure} (Classical geometry). There exists an anti-linear
isometry (charge conjugation) $J:\mathcal{H}\rightarrow \mathcal{H}$ which
combines charge conjugation and complex conjugation and gives the $*$%
-involution by algebraic conjugation: $JaJ^{-1}=a^{*}$ $\forall a\in
\mathcal{A}$, and s.t. $J^{2}=\varepsilon $, $JD=\varepsilon ^{\prime }DJ$,
and $J\gamma =\varepsilon ^{\prime \prime }\gamma J$ with $\varepsilon $, $%
\varepsilon ^{\prime }$, $\varepsilon ^{\prime \prime }=\pm 1$ depending of
the dimension $n$ $\limfunc{mod}8$:
\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ \\ \hline
$\varepsilon $ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $1$ & $1$ \\ \hline
$\varepsilon ^{\prime }$ & $1$ & $-1$ & $1$ & $1$ & $1$ & $-1$ & $1$ & $1$
\\ \hline
$\varepsilon ^{\prime \prime }$ & $1$ & & $-1$ & & $1$ & & $-1$ & \\
\hline
\end{tabular}
\]
In the classical case ($M$ smooth compact manifold of dimension $n$), Connes
proved that these axioms define a unique Riemannian spin geometry whose
geodesic distance and the spin structure are those defined by $D$.\
Moreover, the value of the Dixmier trace $\limfunc{Tr}_{Dix}ds^{n-2}$ is the
\emph{Einstein-Hilbert action functional}:
\[
\limfunc{Tr}\nolimits_{Dix}ds^{n-2}=c_{n}\int_{M}R\sqrt{g}%
d^{n}x=c_{n}\int_{M}Rdv
\]
\noindent with $dv$ the volume form $dv=\sqrt{g}d^{n}x$ and $c_{n}=\frac{n-2%
}{12}\left( 4\pi \right) ^{-\frac{n}{2}}\Gamma \left( \frac{n}{2}+1\right)
^{-1}2^{\left[ \frac{n}{2}\right] }$.\ $\limfunc{Tr}\nolimits_{Dix}ds^{n-2}$
is well defined and $\neq 0$ since $ds^{n-2}$ is an infinitesimal of order $%
\frac{n-2}{n}<1$).\ For $n=4$, $c_{4}=\frac{1}{6}\left( 4\pi \right)
^{-2}\Gamma \left( 3\right) ^{-1}2^{2}=\frac{1}{48\pi ^{2}}$.
In the NC case the characteristic properties (2), (6), (7) must be modified
to take into account the NC:
$7^{NC}$. \emph{Real structure} (NC geometry). In the noncommutative case,
the axiom $JaJ^{-1}=a^{*}$ is transformed into the following axiom saying
that the conjugation by $J$ of the involution defines the \emph{opposed}
multiplication of $\mathcal{A}$.\ Let $b^{0}=Jb^{*}J^{-1}$, then $\left[
a,b^{0}\right] =0$ , $\forall a,b\in \mathcal{A}$. By means of this real
structure, the Hilbert space $\mathcal{H}$ becomes not only a (left) $%
\mathcal{A}$-module through the representation of $\mathcal{A}$ into $%
\mathcal{L}(\mathcal{H})$ but also a $\mathcal{A}\otimes \mathcal{A}^{\circ
} $-module (where $\mathcal{A}^{\circ }$ is the opposed algebra of $\mathcal{%
A})$ or a (left-right) $\mathcal{A}$-bimodule through $\left( a\otimes
b^{0}\right) \xi =aJb^{*}J^{-1}\xi $ or $a\xi b=aJb^{*}J^{-1}\xi $ for every
$\xi \in \mathcal{H}$.
$2^{NC}$. The universal commutation relations $\left[ \left[ D,f\right] ,g%
\right] =0$, $\forall f,g\in \mathcal{A}$ become in the NC case $\left[ %
\left[ D,a\right] ,b^{\circ }\right] =0$, $\forall a,b\in \mathcal{A}$
(which is equivalent to $\left[ \left[ D,b^{\circ }\right] ,a\right] =0$, $%
\forall a,b\in \mathcal{A}$ since $a$ and $b^{\circ }$ commute by $7^{NC}$).
$6^{NC}$. $K$-theory can be easily generalized to the NC case.$\;$We
consider finite projective $\mathcal{A}$-modules $\mathcal{E}$, that is
direct factors of free $\mathcal{A}$-modules $\mathcal{A}^{N}$.$\;$They are
characterized by a projection $\pi :\mathcal{A}^{N}\rightarrow \mathcal{E}$
admitting a section $s:\mathcal{E}\rightarrow \mathcal{A}^{N}$ ($\pi \circ
s=Id_{\mathcal{E}}$). $K_{0}\left( \mathcal{A}\right) $ classifies them. The
structure of $\mathcal{A}\otimes \mathcal{A}^{\circ }$-module induced by the
real structure $J$ allows to define the intersection form by $%
(e,a)\rightarrow \left\langle \func{Ind}D,e\otimes a^{\circ }\right\rangle $
with $e\otimes a^{\circ }$ considered as an element of $K_{0}\left( \mathcal{%
\ A}\otimes \mathcal{A}^{\circ }\right) $.
One of the fundamental aspects of the NC case is that inner automorphisms $%
\alpha _{u}\left( a\right) =uau^{*}$, $u\in \mathcal{U}\left( \mathcal{A}%
\right) $ act upon the Dirac operator $D$ via NC gauge connections (vector
potentials) $A$%
\begin{eqnarray*}
\widetilde{D} &=&D+A+JAJ^{-1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with } \\
A &=&u\left[ D,u^{*}\right]
\end{eqnarray*}
\noindent the equivalence between $D$ and $\widetilde{D}$ being given by $%
\widetilde{D}=UDU^{-1}$ with $U=uJuJ^{-1}=u\left( u^{*}\right) ^{\circ }$.
\subsection{The crucial discovery of a structural link between ``external''
metric and ``internal'' gauge transformations}
One can generalize these transformations of metrics to gauge connections $A$
of the form $A=\sum a_{i}\left[ D,b_{i}\right] $ which can be interpreted as
internal perturbations of the metric or as \emph{internal fluctuations of
the spectral geometry} induced by the internal degrees of freedom of gauge
transformations. This coupling between metric and gauge transformations is
what is needed for coupling gravity with quantum field theory. In the
commutative case, this coupling vanishes since $U=uu^{\ast }=1$ and
therefore $\widetilde{D}=D$. The vanishing $A+JAJ^{-1}=0$ comes from the
fact that $A$ is self-adjoint and that, due to its special form $A=a\left[
D,b\right] $, we have $JAJ^{-1}=-A^{\ast }$. Indeed, since $\left[ D,b^{\ast
}\right] =-\left[ D,b\right] ^{\ast }$%
\begin{eqnarray*}
JAJ^{-1} &=&Ja\left[ D,b\right] J^{-1}=JaJ^{-1}J\left[ D,b\right]
J^{-1}=a^{\ast }\left[ D,b^{\ast }\right] \\
&=&-a^{\ast }\left[ D,b\right] ^{\ast }=-\left( a\left[ D,b\right] \right)
^{\ast }=-A^{\ast }~.
\end{eqnarray*}
So the coupling between the \textquotedblleft external\textquotedblright\
metric afforded by the Dirac operator and the internal quantum degrees of
freedom is a purely noncommutative effect which constitutes a breakthrough
for the unification of general relativity and quantum field theory in a
\textquotedblleft good\textquotedblright\ theory of quantum gravity.
\subsection{Generating the Standard Model (Connes-Lott)}
Before concluding this compilation with some remarks on quantum gravity, let
us recall that the first main interest of noncommutative geometry in physics
was to couple classical gauge theories with purely NC such theories.\ This
led to the NC\ interpretation of Higgs fields. Connes' main result is:
\textbf{Connes' theorem.}\ The Glashow-Weinberg-Salam Standard Model (SM)
can be entirely reconstructed from the NC $C^{*}$-algebra
\[
\mathcal{A}=\mathcal{C}^{\infty }(M)\otimes (\mathbb{C}\oplus \mathbb{H}%
\oplus M^{3}(\mathbb{C}))
\]
\noindent where the \textquotedblleft internal\textquotedblright\ algebra $%
\mathbb{C}\oplus \mathbb{H}\oplus M^{3}(\mathbb{C})$ has for unitary group
the symmetry group
\[
U(1)\times SU(2)\times SU(3)~.
\]
The first step is to construct the toy model which is the product $\mathcal{C%
}^{\infty }(M)\otimes (\mathbb{C}\oplus \mathbb{C})$ of the classical Dirac
fermionic model $\left( \mathcal{A}_{1},\mathcal{H}_{1},D_{1},\gamma
_{5}\right) $ and the previously explained, purely NC, $2$-points model $%
\left( \mathcal{A}_{2},\mathcal{H}_{2},D_{2},\gamma \right) $ with $D_{2}=%
\left[
\begin{array}{cc}
0 & M^{*} \\
M & 0%
\end{array}
\right] $:
\[
\left\{
\begin{array}{l}
\mathcal{A}=\mathcal{A}_{1}\otimes \mathcal{A}_{2} \\
\mathcal{H}=\mathcal{H}_{1}\oplus \mathcal{H}_{2} \\
D=D_{1}\otimes 1+\gamma _{5}\otimes D_{2}\;.%
\end{array}
\right.
\]
The second step is to complexify the model and to show that it enables to
derive the complete GWS Lagrangian.
The key idea is to take the product of a $4$-dimensional spin manifold $M$
with a finite NC geometry $\left( \mathcal{A}_{F},\mathcal{H}%
_{F},D_{F}\right) $ of dimension $0$ where $\mathcal{H}_{F}$ is the Hilbert
space with basis the generations of fermions: quarks and leptons. The
particule/antiparticule duality decomposes $\mathcal{H}_{F}$ into $\mathcal{H%
}_{F}=\mathcal{H}_{F}^{+}\oplus \mathcal{H}_{F}^{-}$, each $\mathcal{H}%
_{F}^{\pm }$ decomposes into $\mathcal{H}_{F}^{\pm }=\mathcal{H}_{l}^{\pm
}\oplus \mathcal{H}_{q}^{\pm }$ ($l=$ lepton and $q=$ quark), and chirality
decomposes the $\mathcal{H}_{p}^{\pm }$ ($p=$ particule) into $\mathcal{H}%
_{pL}^{\pm }\oplus \mathcal{H}_{pR}^{\pm }$ ($L=$ left, $R=$ right). The $4$
quarks are $u_{L},u_{R},d_{L},d_{R}$ ($u=$ up, $d=$ down) with $3$ colours ($%
12$ quarks for each generation) and the $3$ leptons are $e_{L},\nu _{L},e_{R}
$, the total being of $2\left( 12+3\right) =30$ fermions for each generation.
The real structure $J$ is given for $\mathcal{H}_{F}=\mathcal{H}%
_{F}^{+}\oplus \mathcal{H}_{F}^{-}$ by $J\left(
\begin{array}{c}
\xi \\
\overline{\eta }%
\end{array}%
\right) =\left(
\begin{array}{c}
\eta \\
\overline{\xi }%
\end{array}%
\right) $ that is, if $\xi =\sum_{i}\lambda _{i}p_{i}$ and $\overline{\eta }%
=\sum_{j}\mu _{j}\overline{p_{j}}$,
\[
J\left( \sum_{i}\lambda _{i}p_{i}+\sum_{j}\mu _{j}\overline{p_{j}}\right)
=\left( \sum_{j}\overline{\mu _{j}}p_{j}+\sum_{i}\overline{\lambda _{i}}%
\overline{p_{i}}\right) ~.
\]
The action of the internal algebra $\mathcal{A}_{F}=\mathbb{C}\oplus \mathbb{%
H}\oplus M^{3}(\mathbb{C})$ is defined in the following way.\ Let $a=\left(
\lambda ,q,m\right) \in \mathcal{A}_{F}$, $\lambda \in \mathbb{C}$ being a
complex scalar acting upon $\mathbb{C}^{2}$ as the diagonal quaternion $%
\left(
\begin{array}{cc}
\lambda & 0 \\
0 & \overline{\lambda }%
\end{array}%
\right) $, $q=\alpha +\beta j\in \mathbb{H}$ being a quaternion written as $%
\left(
\begin{array}{cc}
\alpha & \beta \\
-\overline{\beta } & \overline{\alpha }%
\end{array}%
\right) $ with $j=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0%
\end{array}%
\right) $, and $m\in M^{3}(\mathbb{C})$ being a $3\times 3$ complex matrix.
The element $a=\left( \lambda ,q,m\right) $ acts on quarks, independently of
color, via $au_{R}=\lambda u_{R}$, $au_{L}=\alpha u_{L}-\overline{\beta }%
d_{L}$, $ad_{R}=\overline{\lambda }d_{R}$, $ad_{L}=\beta u_{L}+\overline{%
\alpha }d_{L}$, that is as
\[
\left( \lambda ,q,m\right) \left(
\begin{array}{c}
u_{L} \\
d_{L} \\
u_{R} \\
d_{R}%
\end{array}
\right) =\left(
\begin{array}{cccc}
\alpha & -\overline{\beta } & 0 & 0 \\
\beta & \overline{\alpha } & 0 & 0 \\
0 & 0 & \lambda & 0 \\
0 & 0 & 0 & \overline{\lambda }%
\end{array}
\right) \left(
\begin{array}{c}
u_{L} \\
d_{L} \\
u_{R} \\
d_{R}%
\end{array}
\right) =\left(
\begin{array}{c}
\alpha u_{L}-\overline{\beta }d_{L} \\
\beta u_{L}+\overline{\alpha }d_{L} \\
\lambda u_{R} \\
\overline{\lambda }d_{R}%
\end{array}
\right)
\]
\noindent (the pair $\left( u_{R},d_{R}\right) $ can be considered as an
element of $\mathbb{C}\oplus \mathbb{C}$, while $\left( u_{L},d_{L}\right) $
can be considered as an element of $\mathbb{C}^{2}$). It acts on leptons via
$ae_{R}=\overline{\lambda }e_{R}$, $ae_{L}=\beta \nu _{L}+\overline{\alpha }%
e_{L}$, $a\nu _{L}=\alpha \nu _{L}-\overline{\beta }e_{L}$, that is as
\[
\left( \lambda ,q,m\right) \left(
\begin{array}{c}
e_{R} \\
\nu _{L} \\
e_{L}%
\end{array}%
\right) =\left(
\begin{array}{ccc}
\overline{\lambda } & 0 & 0 \\
0 & \alpha & -\overline{\beta } \\
0 & \beta & \overline{\alpha }%
\end{array}%
\right) \left(
\begin{array}{c}
e_{R} \\
\nu _{L} \\
e_{L}%
\end{array}%
\right) =\left(
\begin{array}{c}
\overline{\lambda }e_{R} \\
\alpha \nu _{L}-\overline{\beta }e_{L} \\
\beta \nu _{L}+\overline{\alpha }e_{L}%
\end{array}%
\right) ~.
\]
\noindent It acts on anti-particules via $a\overline{l}=\lambda \overline{l}$
for antileptons and via $a\overline{q}=m\overline{q}$ for antiquarks where $%
m $ acts upon color.
The internal Dirac operator $D_{F}$ is given by the matrix of Yukawa
coupling $D_{F}=\left(
\begin{array}{cc}
Y & 0 \\
0 & \overline{Y}%
\end{array}
\right) $ where $Y=\left( Y_{q}\otimes 1_{3}\right) \oplus Y_{l}$ (the $%
\otimes 1_{3}$ comes from the $3$ generations of fermions) with
\[
Y_{q}=
\begin{array}{cc}
&
\begin{array}{cccc}
u_{L}\; & d_{L}\; & u_{R}\; & d_{R}\;%
\end{array}
\; \\
\begin{array}{c}
u_{L} \\
d_{L} \\
u_{R} \\
d_{R}%
\end{array}
& \left(
\begin{array}{cccc}
0 & 0 & M_{u} & 0 \\
0 & 0 & 0 & M_{d} \\
M_{u}^{*} & 0 & 0 & 0 \\
0 & M_{d}^{*} & 0 & 0%
\end{array}
\right)%
\end{array}
\]
\noindent and
\[
Y_{l}=
\begin{array}{cc}
&
\begin{array}{ccc}
e_{R} & \nu _{L} & e_{L}%
\end{array}
\\
\begin{array}{c}
e_{R} \\
\nu _{L} \\
e_{L}%
\end{array}
& \left(
\begin{array}{ccc}
0 & 0 & M_{l} \\
0 & 0 & 0 \\
M_{l}^{*} & 0 & 0%
\end{array}
\right)%
\end{array}
\]
\noindent where (Connes \cite{Connes96}) $M_{u}$, $M_{d}$, and $M_{l}$ are
matrices \textquotedblleft which encode both the masses of the Fermions and
their mixing properties\textquotedblright . Chirality is given by $\gamma
_{F}\left( p_{R}\right) =p_{R}$ and $\gamma _{F}\left( p_{L}\right) =-p_{L}$
($p$ being any particule or anti-particule).
Connes and Lott then take the product of this internal model of the
fermionic sector with a classical gauge model for the bosonic sector:
\[
\left\{
\begin{array}{l}
\mathcal{A}=C^{\infty }\left( M\right) \otimes \mathcal{A}_{F}=\left(
C^{\infty }\left( M\right) \otimes \mathbb{C}\right) \oplus \left( C^{\infty
}\left( M\right) \otimes \mathbb{H}\right) \oplus \left( C^{\infty }\left(
M\right) \otimes M^{3}(\mathbb{C})\right) \\
\mathcal{H}=L^{2}\left( M,S\right) \otimes \mathcal{H}_{F}=L^{2}\left(
M,S\otimes \mathcal{H}_{F}\right) \\
D=\left( D_{M}\otimes 1\right) \oplus \left( \gamma _{5}\otimes D_{F}\right)
~.%
\end{array}%
\right.
\]
The extraordinary \textquotedblleft tour de force\textquotedblright\ is that
this model, which is rather simple at the conceptual level (a product of two
models, respectively fermionic and bosonic, which takes into account only
the known fundamental properties of these two sectors), is in fact extremely
complex and generates the standard model in a \emph{principled} way.
Computations are very intricate (see Kastler's papers in the bibliography).\
One has to compute first vector potentials of the form $A=\sum_{i}a_{i}\left[
D,a_{i}^{\prime }\right] $, $a_{i},a_{i}^{\prime }\in \mathcal{A}$ which
induce fluctuations of the metric.\ As $D$ is a sum of two terms, it is also
the case for $A$.\ Its discrete part comes from $\gamma _{5}\otimes D_{F}$
and generates the Higgs bosons.\ Let $a_{i}\left( x\right) =\left( \lambda
_{i}\left( x\right) ,q_{i}\left( x\right) ,m_{i}\left( x\right) \right) $.\
The term $\sum_{i}a_{i}\left[ \gamma _{5}\otimes D_{F},a_{i}^{\prime }\right]
$ yields $\gamma _{5}$ tensored by matrices of the form
\begin{itemize}
\item for the quark sector:
\end{itemize}
\[
\left(
\begin{array}{cccc}
0 & 0 & M_{u}\varphi _{1} & M_{u}\varphi _{2} \\
0 & 0 & -M_{d}\overline{\varphi _{2}} & M_{d}\overline{\varphi _{1}} \\
M_{u}^{*}\varphi _{1}^{\prime } & M_{d}^{*}\varphi _{2}^{\prime } & 0 & 0 \\
-M_{u}^{*}\overline{\varphi _{2}^{\prime }} & M_{d}^{*}\overline{\varphi
_{1}^{\prime }} & 0 & 0%
\end{array}
\right)
\]
with
\[
\left\{
\begin{array}{l}
\varphi _{1}=\sum_{i}\lambda _{i}\left( \alpha _{i}^{\prime }-\lambda
_{i}^{\prime }\right) \\
\varphi _{2}=\sum_{i}\lambda _{i}\beta _{i}^{\prime } \\
\varphi _{1}^{\prime }=\sum_{i}\alpha _{i}\left( \lambda _{i}^{\prime
}-\alpha _{i}^{\prime }\right) +\beta _{i}\overline{\beta _{i}^{\prime }} \\
\varphi _{2}^{\prime }=\sum_{i}\beta _{i}\left( \overline{\lambda
_{i}^{\prime }}-\overline{\alpha _{i}^{\prime }}\right) -\alpha _{i}\beta
_{i}^{\prime }\;.%
\end{array}
\right.
\]
\begin{itemize}
\item and for the lepton sector:
\[
\left(
\begin{array}{ccc}
0 & -M_{d}\overline{\varphi _{2}} & M_{d}\overline{\varphi _{1}} \\
M_{d}^{\ast }\varphi _{2}^{\prime } & 0 & 0 \\
M_{d}^{\ast }\overline{\varphi _{1}^{\prime }} & 0 & 0%
\end{array}%
\right) ~.
\]
\end{itemize}
Let $q=\varphi _{1}+\varphi _{2}j$ and $q^{\prime }=\varphi _{1}^{\prime
}+\varphi _{2}^{\prime }j$ be the quaternionic fields so defined. As $%
A=A^{*} $, we have $q^{\prime }=q^{*}$.\ The $\mathbb{H}$-valued field $q(x)$
is the \emph{Higgs doublet}.
The second part of the vector potential $A$ comes from $D_{M}\otimes 1$ and
generates the gauge bosons. The terms $\sum_{i}a_{i}\left[ D_{M}\otimes
1,a_{i}^{\prime }\right] $ yield
\begin{itemize}
\item the $U(1)$ gauge field $\Lambda =\sum_{i}\lambda _{i}d\lambda
_{i}^{\prime }$;
\item the $SU(2)$ gauge field $Q=\sum_{i}q_{i}dq_{i}^{\prime }$;
\item the $U(3)$ gauge field $V=\sum_{i}m_{i}dm_{i}^{\prime }$.
\end{itemize}
The computation of the fluctuations of the metric $A+JAJ^{-1}$ gives
\begin{itemize}
\item for the quark sector:
\end{itemize}
\[
\begin{array}{cc}
&
\begin{array}{cccc}
u_{L}\;\;\;\;\; & d_{L}\;\;\;\;\; & u_{R}\;\;\;\;\; & d_{R}\;\;\;\;\;%
\end{array}
\\
\begin{array}{c}
u_{L} \\
d_{L} \\
u_{R} \\
d_{R}%
\end{array}
& \left(
\begin{array}{cccc}
Q_{11}1_{3}+V & Q_{12}1_{3} & 0 & 0 \\
Q_{21}1_{3} & Q_{22}1_{3}+V & 0 & 0 \\
0 & 0 & \Lambda 1_{3}+V & 0 \\
0 & 0 & 0 & -\Lambda 1_{3}+V%
\end{array}
\right)%
\end{array}
\]
\noindent which is a $12\times 12$ matrix since $V$ is $3\times 3$,
\begin{itemize}
\item and for the lepton sector:
\[
\begin{array}{cc}
&
\begin{array}{ccc}
e_{R}\;\;\;\;\; & \nu _{L}\;\;\;\;\; & e_{L}\;\;\;\;\;%
\end{array}
\\
\begin{array}{c}
e_{R} \\
\nu _{L} \\
e_{L}%
\end{array}
& \left(
\begin{array}{ccc}
-2\Lambda & 0 & 0 \\
0 & Q_{11}-\Lambda & Q_{12} \\
0 & Q_{21} & Q_{22}-\Lambda
\end{array}%
\right)
\end{array}%
~.
\]
\end{itemize}
\noindent One can suppose moreover that $\func{Trace}V=\Lambda $, that is $%
V=V^{\prime }+\frac{1}{3}\Lambda $ with $V^{\prime }$ traceless, which gives
the correct hypercharges.
The crowning of the computation is that the total (bosonic$~+$ fermionic)
action
\[
\limfunc{Tr}\nolimits_{Dix}\theta ^{2}ds^{4}+\left\langle \left(
D+A+JAJ^{-1}\right) \psi ,\psi \right\rangle =YM\left( A\right)
+\left\langle D_{A}\psi ,\psi \right\rangle
\]
\noindent (where $\theta =dA+A^{2}$ is the curvature of the connection $A$)
enables to derive the complete GWS Lagrangian
\[
\mathcal{L}=\mathcal{L}_{G}+\mathcal{L}_{f}+\mathcal{L}_{\varphi }+\mathcal{L%
}_{Y}+\mathcal{L}_{V}\;.
\]
1. $\mathcal{L}_{G}$ is the Lagrangian of the gauge bosons
\begin{eqnarray*}
\mathcal{L}_{G} &=&\frac{1}{4}\left( G_{\mu \nu a}G_{a}^{\mu \nu }\right) +%
\frac{1}{4}\left( F_{\mu \nu }F^{\mu \nu }\right) \\
G_{\mu \nu a} &=&\partial _{\mu }W_{\nu a}-\partial _{\nu }W_{\mu
a}+g\varepsilon _{abc}W_{\mu b}W_{\nu c}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, } \\
&&\RIfM@\expandafter\text@\else\expandafter\mbox\fi{where }W_{\mu a}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is a }SU(2)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ gauge field (weak isospin)}
\\
F_{\mu \nu } &=&\partial _{\mu }B_{\nu }-\partial _{\nu }B_{\mu }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,
with }B_{\mu }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ a }SU(1)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ gauge field. }
\end{eqnarray*}
2.$\;\mathcal{L}_{f}$ is the fermionic kinetic term
\begin{eqnarray*}
\mathcal{L}_{f} &=&-\sum \overline{f_{L}}\gamma ^{\mu }\left( \partial _{\mu
}+ig\frac{\tau _{a}}{2}W_{\mu a}+ig^{\prime }\frac{Y_{L}}{2}B_{\mu }\right)
f_{L}+ \\
&&\overline{f_{R}}\gamma ^{\mu }\left( \partial _{\mu }+ig^{\prime }\frac{%
Y_{R}}{2}B_{\mu }\right) f_{R}
\end{eqnarray*}
\noindent where $f_{L}=\left[
\begin{array}{l}
\nu _{L} \\
e_{L}%
\end{array}%
\right] $ are left fermion fields of hypercharge $Y_{L}=-1$ and $%
f_{R}=\left( e_{R}\right) $ right fermion fields of hypercharge $Y_{R}=-2$.
3.$\;\mathcal{L}_{\varphi }$ is the Higgs kinetic term
\[
\mathcal{L}_{\varphi }=-\left| \left( \partial _{\mu }+ig\frac{\tau _{a}}{2}%
W_{\mu a}+i\frac{g^{\prime }}{2}B_{\mu }\right) \varphi \right| ^{2}
\]
\noindent where $\varphi =\left[
\begin{array}{l}
\varphi _{1} \\
\varphi _{2}%
\end{array}%
\right] $ is a $SU(2)$ pair of scalar complex fields of hypercharge $%
Y_{\varphi }=1$.
4.$\;\mathcal{L}_{Y}$ is a Yukawa coupling between the Higgs fields and the
fermions
\[
\mathcal{L}_{Y}=-\sum \left( H_{ff^{\prime }}\left( \overline{f_{L}}.\varphi
\right) f_{R}^{\prime }+H_{ff^{\prime }}^{*}\overline{f_{^{\prime }R}}\left(
\varphi ^{+}.f_{L}\right) \right)
\]
\noindent where $H_{ff^{\prime }}$ is a coupling matrix.
5. $\mathcal{L}_{V}$ is the Lagrangian of the self-interaction of the Higgs
fields
\[
\mathcal{L}_{V}=\mu ^{2}\left( \varphi ^{+}\varphi \right) -\frac{1}{2}%
\lambda \left( \varphi ^{+}\varphi \right) ^{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }\lambda >0~.
\]
\section{Quantum gravity, fluctuating background geometry, and spectral
invariance (Connes~- Chamseddine)}
\subsection{Quantum Field Theory and General Relativity}
As we have already emphasized, Alain Connes realized a new breakthrough in
the approaches of quantum gravity by coupling such models with general
relativity. In NCG, quantum gravity can be thought of in a principled way
because it becomes possible to introduce in the models of quantum field
theory the gravitational Einstein-Hilbert action as a direct consequence of
the specific invariance of spectral geometry, namely \emph{spectral
invariance}. As Alain Connes \cite{Connes96} explains:
\begin{quotation}
\noindent
``However this [the previous NC deduction of the standard
model] requires the definition of the curvature and is still in the spirit
of gauge theories.\ (...) One should consider the internal gauge symmetries
as part of the diffeomorphism group of the non commutative geometry, and the
gauge bosons as the internal fluctuations of the metric.\ It follows then
that the action functional should be of a purely gravitational nature.\ We
state the principle of spectral invariance, stronger than the invariance
under diffeomorphisms, which requires that the action functional only
depends on the spectral properties of $D=ds^{-1}$ in $\mathcal{H}$%
.''
\end{quotation}
The general strategy for coupling a Yang-Mills-Higgs gauge theory with the
Einstein-Hilbert action is to find a $C^{*}$-algebra $\mathcal{A}$ s.t. the
normal subgroup $\func{Inn}(\mathcal{A})$ of inner automorphisms is the
gauge group and the quotient group $\func{Out}(\mathcal{A})=\func{Aut}(%
\mathcal{A})/\func{Inn}(\mathcal{A})$ of ``external'' automorphisms plays
the role of $\func{Diff}(M)$ in a gravitational theory. Indeed, in the
classical setting we have principal bundles $P\rightarrow M$ with a
structural group $G$ acting upon the fibers and an exact sequence
\[
Id\rightarrow \mathcal{G}\rightarrow \func{Aut}\left( P\right) \rightarrow
\func{Diff}(M)\rightarrow Id
\]
\noindent where $\mathcal{G}=C^{\infty }\left( M,G\right) $ is the gauge
group.\ The non abelian character of these gauge theories comes solely\ from
the non commutativity of the group of \emph{internal} symmetries $G$. The
total symmetry group $\func{Aut}\left( P\right) $ of the theory is the
semidirect product $\mathfrak{G}$ of $\func{Diff}(M)$ and $\mathcal{G}%
=C^{\infty }\left( M,G\right) $.\ If we want to geometrize the theory
completely, we would have to find a generalized space $X$ s.t. $\func{Aut}%
\left( X\right) =\mathfrak{G}$.
\begin{quotation}
\noindent
``If such a space would exist, then we would have some
chance to actually geometrize completely the theory, namely to be able to
say that it's pure gravity on the space $X$.'' (Connes \cite%
{Connes2000a})
\end{quotation}
But this is\emph{\ impossible }if $X$ is a\emph{\ }manifold since a theorem
of John Mather proves that in that case the group $\func{Diff}(X)$ would be
simple (without normal subgroup) and could'nt therefore be a semidirect
product. But it is possible with a NC\ space $\left( \mathcal{A},\mathcal{H}%
,D\right) $. For then (Iochum, Kastler, Sch\"{u}cker \cite{Iochum})
\begin{quotation}
\noindent
``the metric `fluctuates', that is, it picks up additional degrees of
freedom from the internal space, the Yang-Mills connection and the Higgs
scalar. (...) In physicist's language, the spectral triplet is the Dirac
action of a multiplet of dynamical fermions in a background field.\ This
background field is a fluctuating metric, consisting of so far adynamical
bosons of spin $0$,$1$ and $2$''.
\end{quotation}
If we find a NC\ geometry $\mathcal{A}$ with $\func{Inn}(\mathcal{A})\simeq
\mathcal{G}$, a correct spectral triple and apply the spectral action, then
gravity will correspond to $\func{Out}(\mathcal{A})=\func{Aut}(\mathcal{A})/%
\func{Inn}(\mathcal{A})$. As was emphasized by Martin \emph{et al}. \cite%
{Martin}:
\begin{quotation}
\noindent
``The strength of Connes' conception is that gauge theories are thereby
deeply connected to the underlying geometry, on the same footing as
gravity.\ The distinction between gravitational and gauge theories boils
down to the difference between outer and inner automorphisms.''
\end{quotation}
\noindent Jones and Moscovici \cite{Jones} add that this implies that
\begin{quotation}
\noindent
``Connes' spectral approach gains the ability to reach below the Planck
scale and attempt to decipher the fine structure of space-time''.
\end{quotation}
\noindent So, just as general relativity extends the Galilean or Minkowskian
invariance into diffeomorphism invariance, NCG extends both diffeomorphism
invariance and gauge invariance into a larger invariance, the spectral
invariance.
\subsection{The spectral action and the eigenvalues of the Dirac operator as
dynamical variables for general relativity}
The key device is the bosonic spectral action
\[
\func{Trace}\left( \phi \left( \frac{D^{2}}{\Lambda ^{2}}\right) \right)
\]
\noindent where $\Lambda $ is a cut-off of the order of the inverse of
Planck length and $\phi $ a smooth approximation of the characteristic
function $\chi _{\left[ 0,1\right] }$ of the unit interval. $D^{2}=\left(
D_{M}\otimes 1+\gamma _{5}\otimes D_{F}\right) ^{2}$ is computed using
Lichnerowicz' formula $D^{2}=\Delta ^{S}+\frac{1}{4}R$. As this action
counts the number $N\left( \Lambda \right) $ of eigenvalues of $D$ in the
interval $\left[ -\Lambda ,\Lambda \right] $, the key idea is, as formulated
by Giovanni Landi and Carlo Rovelli \cite{Landi},
\begin{quotation}
\noindent
``to consider the eigenvalues of the Dirac operator as dynamical variables
for general relativity''.
\end{quotation}
This formulation highlights the physical and philosophical significance of
the NC framework: since the distance is defined through the Dirac operator $D
$, the spectral properties of $D$ can be used in order to modify the metric.
The eigenvalues are spectral invariants and are therefore, in the classical
case, automatically $\func{Diff}(M)$ invariant.\
\begin{quotation}
\noindent
``Thus the general idea is to describe spacetime geometry by
giving the eigen-frequencies of the spinors that can live on that spacetime.
[...] The Dirac operator $D$ encodes the full information about the
spacetime geometry in a way usable for describing gravitational
dynamics.'' (Landi-Rovelli \cite{Landi}: the quotation
concerns our $D_{M}$ acting on the Hilbert space of spinor fields on $M$.)
\end{quotation}
This crucial point has also been well explained by Steven Carlip (\cite%
{Carlip}, p.\ 47).\ Due to $\func{Diff}(M)$ invariance, in general
relativity points of space-time loose any physical meaning so that
obervables must be radically non-local.\ This is the case with the
eingenvalues of $D$ which
\begin{quotation}
\noindent
``provide a nice set of non local, diffeomorphism-invariant obervables.''
\end{quotation}
\noindent They yield
\begin{quotation}
\noindent
``the first good candidates for a (nearly) complete set of
diffeomorphism-invariant observables''.
\end{quotation}
Let us look at $N\left( \Lambda \right) $ for $\Lambda \rightarrow \infty $%
.\ $N\left( \Lambda \right) $ is a step function which encodes a lot of
information and can be written as a sum of a mean value and a fluctuation
(oscillatory) term $N\left( \Lambda \right) =\left\langle N\left( \Lambda
\right) \right\rangle +N_{\func{osc}}\left( \Lambda \right) $ where the
oscillatory part $N_{\func{osc}}\left( \Lambda \right) $ is random. The mean
part $\left\langle N\left( \Lambda \right) \right\rangle $ can be computed
using a semi-classical approximation and a heat equation expansion. A
wonderful computation shows that for $n=4$ the asymptotic expansion of the
spectral action is
\[
\func{Trace}\left( \phi \left( \frac{D^{2}}{\Lambda ^{2}}\right) \right)
=\Lambda ^{4}f_{0}a_{0}\left( D^{2}\right) +\Lambda ^{2}f_{2}a_{2}\left(
D^{2}\right) +f_{4}a_{4}\left( D^{2}\right) +O\left( \Lambda ^{-2}\right)
\]
\noindent with
\begin{itemize}
\item $f_{0}=\int_{\mathbb{R}}\phi \left( u\right) udu$, $f_{2}=\int_{%
\mathbb{R}}\phi \left( u\right) du$, $f_{4}=\phi \left( 0\right) $.
\item $a_{j}\left( D^{2}\right) =\int_{M}a_{j}\left( x,D^{2}\right) dv$ ( $%
dv=\sqrt{g}d^{4}x$).
\item $a_{0}\left( x,D^{2}\right) =\frac{1}{\left( 4\pi \right) ^{2}}\func{%
Trace}_{x}\left( 1\right) $.
\item $a_{2}\left( x,D^{2}\right) =\frac{1}{\left( 4\pi \right) ^{2}}\func{%
Trace}_{x}\left( \frac{1}{6}s1-E\right) $.
\item $a_{4}\left( x,D^{2}\right) =\frac{1}{360\left( 4\pi \right) ^{2}}%
\func{Trace}_{x}\left( 5s^{2}1-2r^{2}1+2R^{2}1-60sE+180E^{2}+30R_{\mu \nu
}^{{\Greekmath 0272} }R^{{\Greekmath 0272} \mu \nu }\right) $.
\item $R$ is the curvature tensor of $M$ and $R^{2}=R_{\mu \nu \alpha \beta
}R^{\mu \nu \alpha \beta }$.
\item $r$ is the Ricci tensor of $M$ and $r^{2}=r_{\mu \nu }r^{\mu \nu }$.
\item $s$ is the scalar curvature of $M$.
\item $E$ and $R_{\mu \nu }^{{\Greekmath 0272} }$ come from Lichnerowicz' formula.
\end{itemize}
Let
\[
\mathcal{E}=C^{\infty }(M,S\otimes \mathcal{H}_{F})=C^{\infty }(M,S)\otimes
_{C^{\infty }(M)}C^{\infty }(M,\mathcal{H}_{F})~.\
\]
\noindent The connection on $\mathcal{E}$ is
\[
{\Greekmath 0272} ={\Greekmath 0272} ^{S}\otimes Id_{C^{\infty }(M,\mathcal{H}_{F})}+Id_{C^{\infty
}(M,S)}\otimes {\Greekmath 0272} ^{F}
\]
\noindent and $R_{\mu \nu }^{{\Greekmath 0272} }$ is the curvature $2$-tensor of this
total connection ${\Greekmath 0272} $. If $D=ic^{\mu }{\Greekmath 0272} _{\mu }+\varphi $ with $%
c^{\mu }=\gamma ^{\mu }\otimes Id_{C^{\infty }(M,\mathcal{H}_{F})}$, then $%
D^{2}=\Delta +E$, with
\[
\left\{
\begin{array}{l}
\Delta =-g^{\mu \nu }\left( {\Greekmath 0272} _{\mu }{\Greekmath 0272} _{\nu }-\Gamma _{\mu \nu
}^{\alpha }{\Greekmath 0272} _{\alpha }\right) \\
E=\frac{1}{4}s1-\frac{1}{2}c\left( R^{F}\right) +ic^{\mu }\left[ {\Greekmath 0272}
_{\mu },\varphi \right] +\varphi ^{2} \\
c\left( R^{F}\right) =-\gamma ^{\mu }\gamma ^{\nu }\otimes R_{\mu \nu }^{F}%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ (}R^{F}=\RIfM@\expandafter\text@\else\expandafter\mbox\fi{curvature of }{\Greekmath 0272} ^{F}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{)~.}%
\end{array}%
\right.
\]
The asymptotic expansion of the spectral action is dominated by the first
two terms which can be identified with the Einstein-Hilbert action with a
cosmological term.\ The later can be eliminated by a change of $\phi $.
\textbf{Addendum}. In a forthcoming book, Alain Connes, Ali Chamseddine and
Matilde Marcolli show how the previous results can be strongly improved and
yield a derivation of the standard model minimally coupled to gravity
(Einstein-Hilbert action) with massive neutrinos, neutrino mixing, Weinberg
angle, and Higgs mass (of the order of 170 GeV).\ This new achievement is
quite astonishing.
\section*{Abstract (Not appropriate in this style!)}%
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\def\kappa{{\Greekmath 0114}}%
\def\lambda{{\Greekmath 0115}}%
\def\mu{{\Greekmath 0116}}%
\def\nu{{\Greekmath 0117}}%
\def\xi{{\Greekmath 0118}}%
\def\pi{{\Greekmath 0119}}%
\def\rho{{\Greekmath 011A}}%
\def\sigma{{\Greekmath 011B}}%
\def\tau{{\Greekmath 011C}}%
\def\upsilon{{\Greekmath 011D}}%
\def\phi{{\Greekmath 011E}}%
\def\chi{{\Greekmath 011F}}%
\def\psi{{\Greekmath 0120}}%
\def\omega{{\Greekmath 0121}}%
\def\varepsilon{{\Greekmath 0122}}%
\def\vartheta{{\Greekmath 0123}}%
\def\varpi{{\Greekmath 0124}}%
\def\varrho{{\Greekmath 0125}}%
\def\varsigma{{\Greekmath 0126}}%
\def\varphi{{\Greekmath 0127}}%
\def{\Greekmath 0272}{{\Greekmath 0272}}
\def\FindBoldGroup{%
{\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}%
}
\def\Greekmath#1#2#3#4{%
\if@compatibility
\ifnum\mathgroup=\symbold
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{\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
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\mathchar"#1#2#3#
\fi
\else
\FindBoldGroup
\ifnum\mathgroup=\theboldgroup
\mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
\else
\mathchar"#1#2#3#
\fi
\fi}
\newif\ifGreekBold \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}%
\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
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\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}
\@ifundefined{mathletters}{%
\newcounter{equationnumber}
\def\mathletters{%
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\edef\@currentlabel{\arabic{equation}}%
\setcounter{equationnumber}{\c@equation}
\setcounter{equation}{0}%
\edef\arabic{equation}{\@currentlabel\noexpand\alph{equation}}%
}
\def\endmathletters{%
\setcounter{equation}{\value{equationnumber}}%
}
}{}
\@ifundefined{BibTeX}{%
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\@ifpackageloaded{amstex}%
{\message{amstex already loaded}\makeatother\endinput}
{}
\@ifpackageloaded{amsgen}%
{\message{amsgen already loaded}\makeatother\endinput}
{}
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\def\FN@{\futurelet\next}%
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\def\ints@{\findlimits@\ints@@}%
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\newif\iflimits@
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\def\multint@{\int\ifnum\intno@=\z@\intdots@
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\ifnum\intno@>\tw@\int\intkern@\fi
\ifnum\intno@>\thr@@\int\intkern@\fi
\int
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
\ifnum\intno@>\tw@\intop\intkern@\fi
\ifnum\intno@>\thr@@\intop\intkern@\fi\intop}%
\def\intic@{%
\mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
{\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@
\def\ints@@@{\iflimits@\negintic@
\mathop{\intic@\multintlimits@}\limits
\else\multint@\nolimits\fi
\eat@
\else
\def\ints@@@{\iflimits@\negintic@
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\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
{{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
\def\RIfM@{\relax\protect\ifmmode}
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\RIfM@\expandafter\RIfM@\expandafter\text@\else\expandafter\mbox\fi@\else\expandafter\mbox\fi}
\let\nfss@text\RIfM@\expandafter\text@\else\expandafter\mbox\fi
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi@#1{\mathchoice
{\textdef@\displaystyle\f@size{#1}}%
{\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
{\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
{\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
\glb@settings}
\def\textdef@#1#2#3{\hbox{{%
\everymath{#1}%
\let\f@size#2\selectfont
#3}}}
\newif\iffirstchoice@
\firstchoice@true
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}%
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
\baselineskip\fontdimen10 \scriptfont\tw@
\advance\baselineskip\fontdimen12 \scriptfont\tw@
\lineskip\thr@@\fontdimen8 \scriptfont\thr@@
\lineskiplimit\lineskip
\vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
\def\endSb{\crcr\egroup\egroup\egroup}%
\def\Sp{^\multilimits@}%
\let\endSp\endSb
\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}%
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}%
\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
\noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
\noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
\leftrightarrowfill@#1\crcr
\noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
$\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
$\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}%
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
\hfil#1#2\hfil$\crcr
\noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%
\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}
\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
\hfil$#1\m@th\operator@font lim$\hfil\crcr
\noalign{\nointerlineskip}#2#1\crcr
\noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
$#1\copy\z@\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
$#1\mathord\leftarrow\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
\mkern-6mu\box\z@$}
\def\qopnamewl@{proj\,lim}{\qopnamewl@{proj\,lim}}
\def\qopnamewl@{inj\,lim}{\qopnamewl@{inj\,lim}}
\def\mathpalette\varlim@\rightarrowfill@{\mathpalette\varlim@\rightarrowfill@}
\def\mathpalette\varlim@\leftarrowfill@{\mathpalette\varlim@\leftarrowfill@}
\def\mathpalette\varliminf@{}{\mathpalette\mathpalette\varliminf@{}@{}}
\def\mathpalette\varliminf@{}@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
\hbox{$#1\m@th\operator@font lim$}}}}
\def\mathpalette\varlimsup@{}{\mathpalette\mathpalette\varlimsup@{}@{}}
\def\mathpalette\varlimsup@{}@#1{\mathop{\overline
{\hbox{$#1\m@th\operator@font lim$}}}}
\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\binom#1#2{{#1 \choose #2}}%
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}%
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%
\def\QATOP#1#2{{#1 \atop #2}}%
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}%
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}%
\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}%
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}%
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}%
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}%
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}%
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}%
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
\def\QTABOVED#1#2#3#4#5{{\textstyle
{#4 \abovewithdelims#1#2#3 #5}}}%
\def\QDABOVED#1#2#3#4#5{{\displaystyle
{#4 \abovewithdelims#1#2#3 #5}}}%
\def\tint{\mathop{\textstyle \int}}%
\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%
\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint }}%
\def\diiint{\mathop{\displaystyle \iiint }}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
\def\dcoprod{\mathop{\displaystyle \coprod }}%
\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]
|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]
|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]
|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]
|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]
|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]
|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]
|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]
|endgroup
\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist
\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist
\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist
\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist
\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist
\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist
\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist
\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist
\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist
\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist
\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist
\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist
\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist
\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist
\def\@@eqncr{\let\@tempa\relax
\ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
\else \def\@tempa{&}\fi
\@tempa
\if@eqnsw
\iftag@
\@taggnum
\else
\@eqnnum\stepcounter{equation}%
\fi
\fi
\global\@ifnextchar*{\@tagstar}{\@tag}@false
\global\@eqnswtrue
\global\@eqcnt\z@\cr}
\def\endequation{%
\ifmmode\ifinner
\iftag@
\addtocounter{equation}{-1}
$\hfil
\displaywidth\linewidth\@taggnum\egroup \endtrivlist
\global\@ifnextchar*{\@tagstar}{\@tag}@false
\global\@ignoretrue
\else
$\hfil
\displaywidth\linewidth\@eqnnum\egroup \endtrivlist
\global\@ifnextchar*{\@tagstar}{\@tag}@false
\global\@ignoretrue
\fi
\else
\iftag@
\addtocounter{equation}{-1}
\eqno \hbox{\@taggnum}
\global\@ifnextchar*{\@tagstar}{\@tag}@false%
$$\global\@ignoretrue
\else
\eqno \hbox{\@eqnnum
$$\global\@ignoretrue
\fi
\fi\fi
}
\newif\iftag@ \@ifnextchar*{\@tagstar}{\@tag}@false
\def\@ifnextchar*{\@tagstar}{\@tag}{\@ifnextchar*{\@tagstar}{\@tag}}
\def\@tag#1{%
\global\@ifnextchar*{\@tagstar}{\@tag}@true
\global\def\@taggnum{(#1)}}
\def\@tagstar*#1{%
\global\@ifnextchar*{\@tagstar}{\@tag}@true
\global\def\@taggnum{#1
}
\makeatother
\endinput
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,262
|
Q: React Hot Loader doesn't update view for 'requireAuth' wrapped component I'm diving into 'react-redux-webpack' world for the last couple of days, willing to try it out and create a template for login.
I have injected React Hot Loader for fast view updates to work, but it updates only the view for unwrapped in 'requireAuth' part.
I'm using componentWillUpdate and componentWillMount to detect login changes. So I wonder whats the reason React Hot Loader isn't able to push changes to view? Since I see that new data (after its changed/saved in IDE) is pulled to browser (inside chrome network inspector).
I would appreciate any input, since I tried my thoughts already, but obviously two days in redux world is not yet enough to figure that out fast ))
here is 'require authentication' component I've built with help of tutorial which works for its purpose grate:
import React, {Component} from 'react';
import {connect} from 'react-redux';
export default function (ComposedComponent) {
class Authentication extends Component {
// to receive access to ROUTER >>>
static contextTypes = {
router: React.PropTypes.object
};
componentWillMount() {
if (!this.props.authenticated) {
this.context.router.push('/');
alert("Please, SIGN IN to continue!");
}
};
componentWillUpdate(nextProps) {
if (!nextProps.authenticated) {
this.context.router.push('/');
alert("Come back soon :)");
}
}
render() {
return <ComposedComponent {...this.props} />
}
}
function mapStateToProps(state) {
return {authenticated: state.authenticated};
}
return connect(mapStateToProps)(Authentication);
}
And view with wrapped 'resource' component, which available only after sign in:
import React from 'react';
import ReactDOM from 'react-dom';
import { Provider } from 'react-redux';
import { createStore, applyMiddleware } from 'redux';
import {Router, Route, browserHistory} from 'react-router';
import requireAuth from './components/require_authentication';
import App from './components/app';
import Resources from './components/resources';
import reducers from './reducers';
const createStoreWithMiddleware = applyMiddleware()(createStore);
ReactDOM.render(
<Provider store={createStoreWithMiddleware(reducers)}>
<Router history={browserHistory}>
<Route path="/" component={App}>
<Route path="resources" component={requireAuth(Resources)} />
</Route>
</Router>
</Provider>
, document.getElementById('root'));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,438
|
RUBY_VER="2.3.4"
RAILS_VER="5.0.2"
SQLITE3_VER="1.3.13"
RUBY_INSTALL_VER="0.6.1"
echo "--> BEGIN $0"
set -x
#################
# install ruby
apt-get update
apt-get -y install build-essential
wget -O ruby-install-${RUBY_INSTALL_VER}.tar.gz https://github.com/postmodern/ruby-install/archive/v${RUBY_INSTALL_VER}.tar.gz
tar -xzvf ruby-install-${RUBY_INSTALL_VER}.tar.gz
cd ruby-install-${RUBY_INSTALL_VER}/
make install
ruby-install --system ruby $RUBY_VER -- --disable-install-rdoc
cd ..
rm -r ruby-install-*
set +x
echo "--> END Ruby $RUBY_VER installation"
#################
# install bundler
echo "--> Install bundler"
gem install bundler --no-document
#################
# install sqlite3
echo "--> Install sqlite3"
apt-get install libsqlite3-dev
gem install sqlite3 -v "${SQLITE3_VER}" --no-document
#################
# install rails
echo "--> Install rails $RAILS_VER"
gem install rails --version=${RAILS_VER} --no-document
#################
# install nodejs
echo "--> Install nodejs"
apt-get -y install nodejs
#################
# install git
echo "--> Install git"
apt-get -y install git
#################
# auto-remove some bloat
apt-get autoremove
echo "--> END $0"
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,213
|
By Colleen Shalby
Thousands at abortion-rights rallies across U.S. Oppose Abortion Restrictions
Los Angeles County Board of Supervisors banned official travel to Alabama for one year, in response to a new abortion ban
Mary Leipziger
Young women hold a sign saying, "I am a woman, not a womb" at the Los Angeles Women's March, May 19, 2019.
Thousands of abortion-rights activists rallied across the nation Tuesday in opposition to a recently approved ban on the procedure in Alabama and other legislative efforts seen as challenges to Roe vs. Wade, the landmark U.S. Supreme Court ruling that legalized abortion nationwide.
Organizers behind the #StopTheBan rallies included Planned Parenthood Fund, the American Civil Liberties Union and the Women's March. More than 400 rallies were planned throughout the country, most notably on the steps of the U.S. Supreme Court in Washington, D.C.
At least 20 rallies were planned throughout California in cities including San Diego, Los Angeles, San Luis Obispo and San Francisco.
In downtown Los Angeles, activists gathered in Pershing Square. And in West Hollywood, which was declared a pro-choice city in 1991, crowds united outside City Hall.
"Now is the time for us to join the National Organization for Women, join Supermajority, find your tribe and fight this," Karen Eyers, vice president of Hollywood NOW, told the crowd. "Fight this and let the legislators know that we will not stand for this, that it ends now."
Lindsey Horvath, West Hollywood's mayor pro tem, praised those who rallied, saying they will "lead us in the battle against the forces that seek to divide us, to wear us down, make us give up, give in, throw in the towel. But we will never give up. We will never give in. We will never be silenced, and we won't go back."
The West Hollywood City Council on Monday approved an ordinance barring the city from doing any official business with states that have antiabortion laws on the books.
Similarly, the Los Angeles County Board of Supervisors voted Tuesday to ban official travel to Alabama for one year, in response to the new law. Exceptions would be allowed for an emergency or matters in which the prevention of travel could harm county interests.
"This challenge by Alabama and other states would overturn decades of precedent," Supervisor Hilda Solis said in a statement after the vote. "It is an attack not only confined to the residents of those states, but an act of aggression upon all of us. We must stand in solidarity and in opposition against extremist and unconstitutional laws that put the health and well-being of families at risk."
The Los Angeles Unified School District also is scheduled to consider a resolution in opposition to the law.
Alabama's law, signed by Gov. Kay Ivey last week, is the most restrictive of its type in the nation. It bars virtually all abortions, including in cases of rape and incest, allowing only those necessary to preserve a woman's life. It also includes a penalty against doctors who perform abortions for up to 99 years in prison.
The law is seen as an effort to challenge Roe vs. Wade, with antiabortion forces eyeing the now-conservative-leaning high court as ripe for a challenge to the case.
Women wearing costumes from Handmaids Tale, at the Los Angeles Women's March, May 19, 2019.
Alabama Republican state Sen. Clyde Chambliss, who sponsored the bill in the Alabama Senate, staunchly defended the legislation.
"When God creates the miracle of life inside a woman's womb, it is not our place as human beings to extinguish that life," he said.
Several states have long-standing "trigger" laws on the books to ban or restrict abortion rights if Roe vs. Wade were overturned. More than 1,000 abortion restrictions have been enacted since Roe vs. Wade.
California is one of nine states that has a policy explicitly protecting abortion rights if the law were overturned.
Colleen Shalby is a reporter for the Los Angeles Times. She previously worked at PBS NewsHour in Washington, D.C. She's a graduate of George Washington University and a native of Southern California.
Story Corps Oral History Project Records on SM's Promenade
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section*{Introduction}\label{introduction}
Given a second order differential operator $L$ on a suitable manifold we
consider the Cauchy problem for the associated wave equation
\begin{equation}\label{cauchy}
\big(\partial_\tau^2+L\big)u=0, \quad u\big|_{\tau=0} =f, \quad \partial_\tau
u\big|_{\tau=0}=g.
\end{equation}
This paper is a contribution to the problem of $L^p$
bounds of the solutions at fixed time $\tau$,
in terms of $L^p$ Sobolev norms of the initial data $f$ and $g$.
This problem is well understood if $L$ is the standard Laplacian
$-\Delta$ (i.e. defined as a positive operator) in
$\mathbb R^d$ (Miyachi \cite{miyachi}, Peral \cite{peral}),
or the Laplace-Beltrami operator on a compact manifold
(\cite{SSS}) of
dimension $d$.
In this case \eqref{cauchy} is a strictly hyperbolic problem and
reduces to estimates for Fourier integral operators associates to a
local canonical graph. The known sharp regularity results in this case
say that
if $\gamma(p)=(d-1)|1/p-1/2|$ and the initial data $f$ and $g$
belong to the $L^p$-Sobolev spaces $L^p_{\gamma(p)}$ and
$L^p_{\gamma(p)-1}$, resp., then the solution $u(\cdot,\tau)$ at fixed
time $\tau$ (say $\tau=\pm 1$) belongs to $L^p$.
In the absence of strict hyperbolicity, the classical Fourier
integral operator techniques do not seem available anymore and it is
not even clear how to efficiently construct parametrices for the
solutions;
consequently the $L^p$ regularity problem is largely open.
However some considerable progress has been made for the specific case
of an invariant operator on the
Heisenberg group ${\mathbb {H}}_m$ which is often considered as a model case
for more general situations.
Recall that
coordinates on $\mathbb H_m$ are given by $(z,u)$, with $z=x+iy\in
{\mathbb {C}}^m$, $u\in {\mathbb {R}}$, and the group law
is given by $(z,u)\cdot (z',u')= (z+z', u+u'-\frac 12\operatorname{Im\,} (z\cdot
\overline{z'})$. A basis of left invariant vector
fields is
given by
$X_j=\frac{\partial}{\partial x_j}-
\frac{y_j}{2}\frac{\partial}{\partial u}$, $Y_j=\frac{\partial}{\partial y_j}+
\frac{x_j}{2}\frac{\partial}{\partial u},$ and we consider the Kohn Laplacian
$$L=-\sum_{j=1}^m (X_j^2+Y_j^2).$$
This operator is perhaps the simplest example of a nonelliptic sum of squares
operator
in the sense of H\"ormander \cite{hoermander-hypo}.
In view of the
Heisenberg group structure
it is natural to analyze the corresponding wave group using tools
from noncommutative Fourier analysis. The operator $L$
is essentially selfadjoint on $C^\infty_0(G)$ (this follows from the methods used in \cite{ne-st})
and the solution of
\eqref{cauchy} can be expressed using the spectral theorem in terms of
functional calculus; it
is given by
$$u(\cdot,\tau)= \cos(\tau\sqrt L)f + \frac{ \sin (\tau\sqrt L)}{\sqrt L} g.$$
We are then aiming to prove
estimates of the form
\begin{equation}\label{SobL}
\|u(\cdot,\tau)\|_p \lesssim
\|(I+\tau^2L)^{\frac \gamma 2} f\|_p + \|\tau (I+\tau^2L)^{\frac \gamma 2 -1} g\|_p.
\end{equation}
involving versions of $L^p$-Sobolev spaces defined by the
subelliptic operator $L$.
Alternatively, one can consider equivalent
uniform $L^p\to L^p$ bounds for operators $a(\tau\sqrt L)
e^{\pm i \tau\sqrt{L}}$
where $a$ is a standard (constant coefficient) symbol of order $-\gamma$.
Note that it suffices to prove those bounds for times
$\tau=\pm 1$, after a scaling using the
automorphic dilations $(z,u)\mapsto (rz,r^2 u),\ r > 0$.
A first study about the solutions to \eqref{cauchy} has been
undertaken
by Nachman \cite{nachman} who showed that
the wave operator on ${\Bbb H}_m$ has a fundamental solution
whose
singularities lie on the cone $\Gamma$ formed by the
characteristics
through the origin. He showed
that the singularity set $\Gamma$ has a far more complicated structure
for ${\mathbb {H}}_m$ than
the corresponding cone in
the Euclidean case.
The fundamental solution is given by a series involving Laguerre polynomials and Nachman was able to examine the
asymptotic behavior as one approaches a
generic
singular point on $\Gamma$. However his method does not seem to yield
uniform estimates in a neighborhood of the singular set which are crucial for obtaining $L^p$-Sobolev estimates for solutions to
\eqref{cauchy}.
In \cite{MuSt-wave} the first author and E. Stein were able to derive nearly sharp
$L^1$ estimates (and by interpolation also $L^p$ estimates, leaving open the interesting endpoint bounds).
Their approach
relied on explicit calculations using Gelfand transforms
for the algebra of radial
$L^1$ functions on the Heisenberg group, and the geometry
of the singular support remained hidden in this approach.
Later, Greiner, Holcman and Kannai \cite{ghk}
used
contour integrals and an explicit formula
for the heat kernel
on the Heisenberg group to derive
an integral
formula for the fundamental solution of the wave equation on ${\mathbb {H}}^m$
which exhibits the singularities of the wave kernel.
We shall follow a somewhat different approach,
which allows us to link the geometrical picture to a
decomposition of the joint spectrum of $L$ and the
operator $U$ of differentiation in the central direction
(see also Strichartz \cite{str}); this linkage is crucial to prove optimal $L^p$ regularity estimates.
In order to derive parametrices we will
use a subordination argument based on stationary phase calculations
to write the wave
operator as an integral involving Schr\"odinger operators for which
explicit formulas are available (\cite{gaveau}, \cite{hulanicki}).
This will yield some type of oscillatory integral representation of the
kernels, as in the theory of Fourier integral operators which will be amenable to proving $L^p$ estimates.
Unlike in the classical theory of Fourier integral operators (\cite{hoermander-fio}) our
phase functions
are not smooth everywhere and have substantial singularities; this
leads to considerable complications.
Finally, an important point in our proof is the identification of a
suitable Hardy
space
for the problem, so that $L^p$ bounds can be proved by interpolation
of $L^2$ and Hardy space estimates. We then obtain the
following sharp $L^p$ regularity result which is a direct analogue of the result by Peral \cite{peral} and Miyachi \cite{miyachi} on the wave equation
in the Euclidean setting.
\medskip
\noindent{\bf Theorem.}
{\it Let $d=2m+1$,
$1<p<\infty$, and
$\gamma \ge (d-1)|1/p - 1/2|$.
Then the operators
$ (I+\tau^2 L)^{-\gamma/2} \exp(\pm i\tau\sqrt L)$
extend to bounded operators on $L^p({\mathbb {H}}^m)$.
The solutions $u$ to the initial value problem \eqref{cauchy} satisfy the Sobolev type inequalities
\eqref{SobL}.}
\medskip
Throughout the paper we shall in fact consider the more general situation
of
{\it groups of Heisenberg type}, introduced by Kaplan \cite{kaplan}.
These include groups with center
of dimension $>1$.
The extension of the above result for the wave operator to groups of
Heisenberg type and further results will be formulated in the next section.
\section{The results for groups of Heisenberg type}
\subsection{\it Groups of Heisenberg type}\label{groupsofheis}
Let $d_1$, $d_2$ be positive integers, with $d_1$ {\it even}, and consider a Lie algebra ${\mathfrak {g}}$
of Heisenberg type, where
$\frak g=\frak g_1\oplus\frak g_2,$ with ${\hbox{\roman dim}}\frak g_1=d_1$ and ${\hbox{\roman dim}}\frak g_2=d_2,$
and
$$
[\frak g,\frak g]\subset \frak g_2\subset \frak z(\frak g)\ ,
$$
$\frak z(\frak g)$ being the center of $\frak g$. Now $\frak g$ is
endowed with an inner product
$\langle \ ,\ \rangle$ such that $\frak g_1$ and $\frak g_2$ or orthogonal subspaces and,
and if we define for $\mu\in\frak g_2^*\setminus\{0\}$ the symplectic form $\omega} \def\Omega{\Omega_\mu$ on $\frak g_1$ by
\begin{equation} \label{omegamu}
\omega} \def\Omega{\Omega_\mu(V,W): =\mu\big([V,W]\big)\ ,
\end{equation}
then there is a unique skew-symmetric linear endomorphism $J_\mu$ of $\frak g_1$ such that
\begin{equation}\label{repofomegabyJ}
\omega} \def\Omega{\Omega_\mu(V,W)=
\langle J_\mu(V),W\rangle
\end{equation}
(here, we also used the natural identification of $\frak g_2^*$ with $\frak g_2$ via the inner product). Then on a Lie algebra of Heisenberg type
\begin{equation}\label{Jmusquared}
J_\mu^2=-|\mu|^2 I
\end{equation}
for every
$\mu\in\frak g_2^*$.
As the corresponding connected, simply connected Lie group $G$
we then choose the linear manifold $\frak g,$ endowed with the Baker-Campbell-Hausdorff product
$$(V_1,U_1)\cdot (V_2,U_2):=(V_1+V_2,U_1+U_2+\frac 12 [V_1,V_2]).
$$
As usual, we identify $X\in\frak g$ with the corresponding left-invariant vector field on $G$ given by the Lie-derivative
$$
X f(g):=\frac d{dt} f(g\exp(tX))|_{t=0},
$$
where $\exp:\frak g\to G$ denotes the exponential mapping, which agrees with the identity mapping in our case.
Let us next fix an orthonormal basis $X_1,\dots,X_{d_1}$ of $\frak g_1,$ as well as an orthonormal basis $U_1,\dots,U_{d_2}$ of $\frak g_2.$ We may then
identify $\frak g=\frak g_1+\frak g_2$ and $G$ with $\Bbb R^{d_1}\times\Bbb R^{d_2}$
by means of the basis $X_1,\dots,X_{d_1}$, $U_1,\dots,U_{d_2}$ of $\frak g.$
Then our inner product on $\frak g$ will agree with the canonical
Euclidean product $v\cdot w=\sum_{j=1}^{d_1+d_2}v_jw_j$ on $\Bbb R^{d_1+d_2},$
and $J_\mu$ will be identified with a skew-symmetric $d_1\times d_1$
matrix. We shall also identify the dual spaces of $\frak g_1$ and $\frak g_2$ with $\Bbb R^{d_1}$ and $\Bbb R^{d_2},$ respectively, by means of this inner product.
Moreover, the Lebesgue measure $dx\, du$ on $\Bbb R^{d_1+d_2}$ is a
bi-invariant Haar measure on $G.$
By
\begin{equation}\label{dimension}
d:=d_1+d_2
\end{equation}
we denote the topological dimension of $G.$
The group law on $G$ is then given by
\begin{equation} \label{grouplaw}
(x,u)\cdot (x',u')= (x+x', u+u' + \frac{1}{2} \inn{\vec Jx}{x'})
\end{equation}
where $\inn{\vec Jx}{x'}$ denotes the vector in ${\mathbb {R}}^{d_2}$ with components
$\inn{J_{U_i}x}{x'}$.
Let
\begin{equation} \label {subLaplacian} L:=-\sum_{j=1}^{d_1} X_j^2
\end{equation}
denote the sub-Laplacian corresponding to the basis $X_1,\dots,X_{d_1}$ of $\frak g_1.$
\medskip
In the special case $d_2=1$ we may assume that $J_\mu=\mu J, \mu\in \Bbb R,$ where
\begin{equation}\label{cansympl}
J:=\left ( \begin{array}{cc}
0 & I_{d_1/2}\\
-I_{d_1/2} & 0
\end{array}
\right)
\end{equation}
and $ I_{d_1/2}$ is the identity matrix on $\mathbb R^{d_1/2}$.
In this case $G$ is the {\it Heisenberg group} ${\Bbb H}_{d_1/2},$ discussed
in the introduction.
Finally, some dilation structures and the corresponding metrics
will play an important role in our
proofs; we shall work with both isotropic and nonisotropic dilations.
First, the natural dilations on the Heisenberg type groups are the
automorphic dilations
\begin{equation}\label{automorphic}
{\delta}_r(x,u):=(rx,r^2u),\quad r>0,
\end{equation}
on $G$. We work with the
{\it Koranyi norm}
$$
\|(x,u)\|_{\text{Ko}}:=(|x|^4+|4u|^2)^{1/4}
$$
which is a homogeneous norm with respect to the dilations ${\delta}_r.$
Moreover, if we denote the corresponding balls by
\begin{align*}
Q_r(x,u):=\{(y,v)\in G:\|(y,v)^{-1}\cdot(x,u)\|_{\text{Ko}}<r\},\ (x,u)\in G,\ r>0,
\end{align*}
then the volume $|Q_r(x,u)|$ is given by
$$
|Q_r(x,u)|=|Q_1(0,0)|\, r^{d_1+2d_2}.
$$
Recall that $d_1+2d_2 = d+d_2$
is the {\it homogeneous dimension} of $G.$
We will also have to work
with a variant of the \lq
Euclidean' balls, i.e. 'isotropic balls" skewed by the
Heisenberg translation, denoted by
$Q_{r,E}(x,u)$.
\begin{equation}\label{isotropicballs}
\begin{aligned}Q_{r,E}(x,u)
:&=\{(y,v)\in G:|(y,v)^{-1}(x,u)|_E<r\},\\
&=\big\{ (y,v)\in G:\big| x-y|+| u-v+\frac{1}{2}\inn{\vec
Jx}{y}|<r\big\};
\end{aligned}
\end{equation}
here
$$
|(x,u)|_E:=|x|+|u|
$$
is comparable with the standard Euclidean norm
$(|x|^2+|u|^2)^{1/2}$. Observe that the
balls $ Q_r(x,u)$ and $Q_{r,E}(x,u)$ are the left-translates by $(x,u)$ of the corresponding balls centered at the origin.
\medskip
\subsection{\it The main results}\label{main}
We consider symbols $a$ of class $S^{-\gamma} $, {\it i.e.} satisfying the
estimates
\begin{equation} \label{symbols}\Big|\frac {d^j}{(ds)^j} a(s)\Big |\le c_j (1+|s|)^{-\gamma-j} \end{equation}
for all $j=0,1,2,\dots$.
Our main boundedness
result is
\begin{theorem}\label{main-theorem} Let $1<p<\infty$,
$\gamma(p):=(d-1)|1/p-1/2|$
and
$a\in S^{-\gamma(p)}$. Then for $-\infty<\tau<\infty$,
the operators
$a(\tau\sqrt L) e^{ i\tau\sqrt L}$ extend to bounded operators
on $L^p(G)$.
The solutions $u$ to the initial value problem \eqref{cauchy} satisfy the Sobolev type inequalities
\eqref{SobL}, for $\gamma\ge \gamma(p)$.
\end{theorem}
Our proof also gives sharp $L^1$ estimates
for
operators with symbols supported in dyadic intervals.
\begin{theorem}\label{main-theoremL1} Let $\chi\in C^\infty_c$ supported in $(1/2, 2)$ and let $\lambda\ge 1$.
Then the operators
$\chi(\lambda} \def\La{\Lambda^{-1}\tau\sqrt L) e^{\pm i\tau\sqrt L}$ extend to bounded operators
on $L^1(G)$, with operator norms $O(\lambda} \def\La{\Lambda^{\frac{d-1}{2}})$.
\end{theorem}
In view of the invariance under automorphic dilations it suffices to
prove these results for $\tau=\pm 1$, and by symmetry considerations, we only need to consider $\tau=1$.
An interesting question posed in \cite{MuSt-wave} concerns the
validity of an appropriate result in
the limiting case $p=1$ (such as a
Hardy space bound). Here the situation is more complicated than
in the Euclidean case because of the interplay of isotropic and
nonisotropic dilations. The usual Hardy spaces $H^1(G)$ are defined using
the nonisotropic
automorphic dilations
\eqref{automorphic}
together with the Koranyi balls. This geo\-metry is
not appropriate for our problem; instead the estimates for our kernels
require a Hardy space
that is
defined using isotropic dilations (just as in the Euclidean case) and
yet is compatible with the Heisenberg group structure.
On the other hand we shall use a dyadic decomposition of the spectrum
of $L$ which corresponds to a Littlewood-Paley decomposition
using nonisotropic dilations.
This space $h^1_{\text{\rm iso}}(G)$ is a variant
of the isotropic local or
(nonhomogeneous) Hardy space in the Euclidean setting.
To define it we first introduce the appropriate notion of atoms.
For $0<r\le 1$ we define a $(P,r)$ atom as a function
$b$ supported in the isotropic Heisenberg ball
$Q_{r,E}(P)$ with radius $r$ centered at $P$
({\it cf}. \eqref{isotropicballs}), such that
$\|b\|_2\le r^{-d/2},$ and $\int b =0$ if $r\le 1/2$.
A function $f$ belongs to $h^1_{\text{\rm iso}}(G)$ if
$f=\sum c_\nu b_\nu$ where $b_{\nu}$ is a $(P_\nu, r^\nu)$ atom for
some point $P_\nu$ and some radius $r_\nu\le 1$, and the sequence
$\{c_\nu\}$ is absolutely convergent. The norm on $h^1_{\text{\rm iso}}(G)$ is given by
$$\inf \sum_\nu|c_\nu|$$
where the infimum is taken over representations of $f$ as a sum
$f=\sum_\nu c_\nu b_\nu$ where the $b_\nu$ are atoms.
It is easy to see that $h^1_{\text{\rm iso}}(G)$ is a closed subspace of $L^1(G)$.
The spaces
$L^p(G)$, $1<p<2$,
are complex interpolation spaces for the couple
$(h^1_{\text{\rm iso}}(G), L^2(G))$ (see {\mathcal S} \ref{interpolation}) and by an analytic interpolation argument
Theorem \ref{main-theorem} can be deduced from an $L^2$ estimate and the
following $h^1_{\text{\rm iso}}\to L^1$ result.
\begin{theorem} \label{h1thm}
Let $a\in S^{-\frac{d-1}{2}}$. Then the operators
$a(\sqrt L) e^{\pm i \sqrt L}$ map the isotropic Hardy space
$h^1_{\text{\rm iso}}(G)$
boundedly to $L^1(G)$.
\end{theorem}
The norm in the Hardy space
$h^1_{\text{\rm iso}}(G)$ is not invariant under the automorphic dilations
\eqref{automorphic}.
It is not currently known whether there is
a suitable Hardy space result which can be used for interpolation and
works
for all
$a(\tau\sqrt L) e^{ i\tau \sqrt L}$ with bounds uniform in $\tau$.
\medskip
\subsection{\it Spectral multipliers}\label{specmultipliers}
If $m$ is a bounded spectral multiplier, then clearly the operator $m(L)$ is bounded on $L^2(G).$ An important question is then under which additional conditions on the spectral
multiplier $m$ the operator $m(L)$ extends from $L^2\cap L^p(M)$ to an $L^p$-bounded
operator, for a given $p\ne 2$.
Fix a non-trivial cut-off function $\chi\in C^{\infty}_0(\Bbb R)$ supported in
the interval $[1,2]$; it is convenient to assume that
$\sum_{k\in {\mathbb {Z}}}\chi(2^ks)=1$ for all $s>0$.
Let $L^2_{\alpha}({\mathbb {R}})$ denote the classical Sobolev-space of order $\alpha$.
Hulanicki and Stein (see Theorem 6.25 in \cite{FollSt}),
proved analogs of the classical Mikhlin-H\"ormander multiplier theorem
on stratified groups, namely
the inequality
\begin{equation}\label{locsob}
\|m(L)\|_{L^p\to L^p} \le C_{p,\alpha} \sup_{t>0}\|\chi m(t\cdot)\|_{L^2_\alpha},
\end{equation} for sufficiently large $\alpha$.
By the work of
M.~Christ \cite{christ}, and also Mauceri-Meda \cite{mauceri-meda},
the inequality \eqref{locsob} holds true for $\alpha> (d+d_2)/2$, in fact
they established a more general result for all stratified groups.
Observe that, in comparison to the classical case $G=\Bbb R^d$,
the homogeneous dimension $d+d_2$ takes over the role of the Euclidean
dimension $d$. However, for the special case of the Heisenberg groups
it was shown by
E.M.~Stein and the first author \cite{MuSt-mult} that
\eqref{locsob} holds for the larger range $\alpha>d/2$.
This result, as well as an extension to Heisenberg type groups has
been proved independently by Hebisch \cite{hebisch}, and Martini \cite{martini}
showed that Hebisch's argument can be used to prove a similar result on
M\'etivier groups. Here we use our estimate on the wave equation to prove,
only for Heisenberg type groups, a result that covers a larger
class of multipliers:
\begin{theorem}\label{multipliers}
Let $G$ be a group of Heisenberg type, with topological dimension $d$.
Let $m\in L^\infty({\mathbb {R}})$, let $\chi\in C_0^\infty$ be as above, let
$$
{\mathfrak {A}}_{R}:=\sup_{t>0} \int_{|s|\ge R}\big|{\mathcal {F}}^{-1}_{\mathbb {R}}[\chi m(t\cdot)](s)\big|s^{\frac{d-1}{2}} ds
$$ and assume
\begin{equation} \label{Ftcondition}
\|m\|_\infty + \int_{2}^\infty {\mathfrak {A}}_{R} \frac{dR}{R} <\infty.
\end{equation}
Then the operator $m(\sqrt L)$
is of weak type $(1,1)$ and
bounded on $L^p(G)$,
$1<p<\infty$.
\end{theorem}
\begin{remarks*}
(i)
Let $H^1(G)$ be the standard Hardy space defined using
the automorphic dilations
\eqref{automorphic}.
Our proof shows that under condition
\eqref{Ftcondition},
$m(\sqrt L)$ maps $H^1(G)$ to $L^1(G)$.
(ii) Note that by
an application of the Cauchy-Schwarz inequality and Plan\-cherel's theorem
that the condition
$$\sup_{t>0}\|\chi m(t\cdot)\|_{L^2_\beta}<\infty\,,
\text{ for some $\beta>d/2$}
$$ implies ${\mathfrak {A}}_R\lesssim_\gamma R^{\frac d2-\beta}$ for $R\ge 2$ and thus
Theorem \ref{multipliers}
covers and extends the above mentioned
multiplier results in \cite{MuSt-mult}, \cite{hebisch}.
(iii) More refined results for fixed $p>1$ could be deduced by
interpolation, but such results would likely not be sharp.
\end{remarks*}
\section{Some notation}
\subsection{\it Smooth cutoff functions}\label{cutoffsect}
We denote by $\zeta_0$ an even $C^\infty$ function
supported in $(-1,1)$ and assume that $\zeta_0(s)=1$ for $|s|\le
9/16$.
Let $\zeta_1(s)=\zeta_0(s/2)-\zeta_0(s)$ so that $\zeta_1$ is supported
in $(-2,-1/2)\cup (1/2,2)$.
If we set $\zeta_j(s) = \zeta_1(2^{1-j}s)$ then $\zeta_j$ is supported
in $(-2^{j},-2^{j-2} )\cup(2^{j-2},2^j)$
and we have $1=\sum_{j=0}^\infty \zeta_j(s)$ for all $s\in {\mathbb {R}}$.
Let $\eta_0$ be a $C^\infty$ function supported in $(-\frac {5\pi}8 ,
\frac {5\pi}8 )$
which has the property that $\eta_0(s)=1$ for
$|s|\le \frac {3\pi}8$ and
satisfies
$\sum_{k\in \mathbb Z} \eta_0(t-k\pi)=1$
for all $t\in {\mathbb {R}}$.
For $l=1,2,\dots$ let
$\eta_l(s)=\eta(2^{l-1} s)-\eta_0(2^l s)$ so that $\eta_0(s)=
\sum_{l=1}^\infty \eta_l(s)$ for $s\neq 0$.
\subsection{\it Inequalities}
We use the notation $A\lesssim B$ to indicate
$A\le CB$ for some constant $C$. We sometimes use the notation
$A\lesssim_\kappa B$ to emphasize that the implicit constant depends on the
parameter $\kappa$. We use $A\approx B$ if $A\lesssim B$ and $B\lesssim A$.
\subsection{\it Other notation.}
We use the definition
$$\widehat f(\xi)\equiv {\mathcal {F}} f(\xi)=\int f(y) e^{-2\pi i\inn{\xi}{y}} dy$$ for the
Fourier transform in Euclidean space ${\mathbb {R}}^d$.
The convolution on $G$ is given by
$$f*g(x,u)= \int f(y,v) g(x-y,
u-v+\tfrac 12 \inn{\vec Jx}{y}) \,dy\, dv.$$
\section{Background on groups of
Heisenberg type and the Schr\"odinger group}
For more on the material reviewed here see, e.g., \cite{folland}, \cite{edinburgh} and
\cite{MR-solv}.
\subsection{\it The Fourier transform on a group of Heisenberg type}\label{FTHEIS}
Let us first briefly recall some facts about the unitary representation theory of a Heisenberg type group $G.$ In many contexts, it is useful to establish analogues of the Bargmann-Fock representations of the Heisenberg group for such groups \cite{kaplan-ricci} (compare also \cite{ricci},\cite{damek-ricci}). For our purposes, it will be more convenient to work with Schr\"odinger type representations. It is well-known that these can be reduced to the case of the Heisenberg group ${\Bbb H}_{d_1/2}$ whose product is given by
$(z,t)\cdot (z',t')=(z+z',t+t'+\frac 1 2 \omega} \def\Omega{\Omega(z,z')),$
where $\omega} \def\Omega{\Omega$ denotes the {\it canonical symplectic form}
$\omega} \def\Omega{\Omega(z,w) :=\langle Jz, w\rangle$, with $J$ is as in \eqref{cansympl}.
For the convenience of the reader, we shall outline this reduction to the Heisenberg group.
Let us split coordinates $z=(x,y)\in\Bbb R^{d_1/2}\times\Bbb R^{d_1/2}$ in $\Bbb R^{d_1},$
and consider the associated natural basis of left-invariant vector fields
of the Lie algebra of ${\Bbb H}_{d_1/2}$,
\begin{eqnarray*}
\tilde X_j:=\partial_{ x_j}-\tfrac 1 2 y_j\partial_{ t},\quad \ \tilde Y_j:=\partial_{y_j}+
\tfrac 1 2 x_j \partial_t,
\ \ j=1,\dots,\frac{d_1}{2},\ \mbox{ and } T:= \partial_t\,.
\end{eqnarray*}
For $\tau\in \Bbb R\setminus\{0\}$, the
{\it Schr\"odinger representation} $\rho_\tau$ of ${\Bbb H}_{d_1/2}$ acts on the Hilbert space $L^2(\Bbb R^{d_1/2})$
as follows:
\begin{eqnarray*}
[\rho_\tau(x,y,t)h](\xi):=e^{2\pi i\tau(t+y\cdot \xi +\frac 1 2 y\cdot x)}
h(\xi+x), \quad h\in L^2(\Bbb R^{d_1/2}).
\end{eqnarray*}
This is an irreducible, unitary representation, and every irreducible unitary representation of ${\Bbb H}_{d_1/2}$ which acts non-trivially on the center is in fact unitarily equivalent to exactly one of these, by the Stone-von Neumann theorem (a good reference for these and related results is for instance \cite{folland}; see also \cite{edinburgh}).
Next, if $\pi$ is any unitary representation, say, of a Heisenberg type group $G,$ we denote by
$$
\pi(f):=\int_G f(g)\pi(g)\, dg,\quad f\in L^1(G),
$$
the associated representation of the group algebra $L^1(G).$ For $f\in L^1(G)$ and $\mu\in\frak g_2^*=\Bbb R^{d_2},$ it will also be useful to define the partial Fourier transform $f^\mu$ of $f$ along the center by
\begin{equation}\label{partialFT}
f^\mu(x)\equiv {\mathcal {F}}_2 f(x,\mu):= \int_{\Bbb R^{d_2}} f(x,u) e^{-2\pi i\mu\cdot u}\,du\quad (x\in \Bbb R^{d_1}).
\end{equation}
Going
back to the Heisenberg group (where $\frak g_2^*=\Bbb R$), if $f\in{\mathcal {S}}({\Bbb H}_{d_1/2}),$ then it is
well-known and easily seen that
$$\rho_\tau(f)=\int_{\Bbb R^{{d_1}}}f^{-\tau}( z)\rho_{\tau}(z,0)\, dz$$
defines
a trace class operator on $L^2(\Bbb R^{d_1/2}),$ and its trace is given by
\begin{equation}\label{traceformula}
{\rm tr} (\rho_\tau(f))=|\tau|^{-d_1/2}\,\int_\Bbb R f(0,0,t)e^{2\pi i \tau t}\, dt=|\tau|^{-d_1/2}\,f^{-\tau}(0,0),
\end{equation}
for every $\tau\in\Bbb R\setminus 0.$
\medskip
From these facts, one derives the Plancherel formula for our
Heisenberg type group $G.$ Given $\mu\in\frak g^*_2=\Bbb R^{d_2},\ \mu\ne 0,$
consider the matrix $J_\mu$
as in \eqref{repofomegabyJ}. By \eqref{Jmusquared} we have
$J_\mu^2=-I$ if $|\mu|=1,$ and $J_\mu$ has only
eigenvalues $\pm i$. Since it is orthogonal
there exists an orthonormal basis
$$
X_{\mu,1},\dots,X_{\mu, \frac{d_1}{2}},Y_{\mu,1},\dots,Y_{\mu,\frac{d_1}{2}}
$$
of $\frak g_1=\Bbb R^{d_1}$ which is symplectic with respect to the form
$\omega} \def\Omega{\Omega_\mu,$
i.e., $\omega} \def\Omega{\Omega_\mu$ is represented by the standard symplectic matrix $J$
in
\eqref{cansympl} with respect to this basis.
This means that, for every $\mu\in\Bbb R^{d_2}\setminus\{0\},$ there is
an
orthogonal matrix $R_\mu=R_{\frac{\mu}{|\mu|}}\in O(d_1,\Bbb R)$ such that
\begin{equation}\label{JmufromJ}
J_\mu=|\mu|R_\mu J \,{}^t\! R_\mu.
\end{equation}
Condition \eqref{JmufromJ}
is in fact equivalent to $G$ being of Heisenberg type.
Now consider the subalgebra $L^1_{\text{\rm rad}}(G)$ of $L^1(G),$ consisting of all
\lq radial' functions $f(x,u)$ in the sense that they depend only on $|x|$ and
$u$. As for Heisenberg groups (\cite{folland},\cite{edinburgh}), this algebra is
commutative for arbitrary Heisenberg type groups (\cite{ricci}), i.e.,
\begin{equation}\label{commutativity}
f*g=g*f \quad \mbox { for every } f,g\in L^1_{\text{\rm rad}}(G).
\end{equation}
This can indeed be reduced to the corresponding result on Heisenberg groups by applying the partial Fourier transform in the central variables.
The following lemma is easy to check and establishes a useful link between representations of $G$ and those of ${\Bbb H}_{d_1/2}.$
\begin{lemma}\label{epi}
The mapping $\alpha_\mu:G\to {\Bbb H}_{d_1/2},$ given by
$$
\alpha_\mu(z,u):=(\,{}^t\! R_\mu z,\tfrac {\mu\cdot u}{|\mu|}),\quad
(z,u)\in
\Bbb R^{d_1}\times\Bbb R^{d_2},
$$
is an epimorphism of Lie groups. In particular, $G/\ker \alpha_\mu$ is isomorphic to ${\Bbb H}_{d_1/2},$ where $\ker \alpha_\mu=\mu^\perp$ is the orthogonal complement of $\mu$ in the center $\Bbb R^{d_2}$ of $G.$
\end{lemma}
Given $\mu\in\Bbb R^{d_2}\setminus\{0\},$ we can now define an irreducible unitary representation $\pi_\mu$ of $G$ on $L^2(\Bbb R^{d_1})$ by putting
$$
\pi_\mu:=\rho_{|\mu|}\circ \alpha_\mu.
$$
Observe that then $\pi_\mu(0,u)=e^{2\pi i \mu\cdot u} I.$ In fact, any irreducible representation of $G$ with central character $ e^{2\pi i \mu\cdot u}$ factors through the kernel of $\alpha_\mu$ and hence, by the Stone-von Neumann theorem, must be equivalent to $\pi_\mu.$
One then computes that, for $f\in{\mathcal {S}}(G),$
$$
\pi_\mu(f)=\int_{\Bbb R^{d_1}}f^{-\mu}(R_\mu z)\rho_{|\mu|}(z,0)\, dz,
$$
so that the trace formula \eqref{traceformula} yields the analogous trace formula
$${\rm tr}\, \pi_\mu(f)=|\mu|^{-\frac{d_1}2}\, f^{-\mu}(0)$$
on $G.$ The Fourier inversion formula in $\Bbb R^{d_2}$ then leads to
$$f(0,0)=\int_{\mu\in\Bbb R^{d_2}\setminus\{0\}} {\rm tr}\, \pi_\mu(f)\, |\mu|^{\frac{d_1}2}\, d\mu.$$
When applied to ${\delta}_{g^{-1}}*f,$ we arrive at the Fourier inversion formula
\begin{equation}\label{Fourierinversion}
f(g)=\int_{\mu\in\Bbb R^{d_2}\setminus\{0\}} {\rm tr}\,( \pi_\mu(g)^*\pi_\mu(f))\,|\mu|^{\frac{d_1}2}\, d\mu,\quad g\in G.
\end{equation}
Applying this to $f^* *f$ at $g=0,$ where $f^*(g):=\overline{f(g^{-1})},$ we obtain the Plancherel formula
\begin{equation}\label{plancherel}
\|f\|_2^2=\int_{\mu\in\Bbb R^{n}\setminus\{0\}}\, \|
\pi_\mu(f)\|^2_{HS}\,
|\mu|^{\frac{d_1}2}\, d\mu,
\end{equation}
where $\|T\|_{HS}=({\rm tr}\, (T^*T))^{1/2}$ denotes the Hilbert-Schmidt norm.
\subsection{\it The Sub-Laplacian and the group Fourier transform}\label{SublGrouptr}
Let us next consider the group Fourier transform of our sub-Laplacian $L$ on $G.$
We first observe that $d\alpha_\mu(X)=\,{}^t\! R_\mu X$ for every
$X\in\frak g_1=\Bbb R^{d_1},$ if we view, for the time being, elements of the Lie algebra as tangential vectors at the identity element. Moreover, by
\eqref{JmufromJ},
we see that
$$\,{}^t\! R_\mu X_{\mu,1}, \dots, \,{}^t\! R_\mu X_{\mu,d_1/2},\,{}^t\! R_\mu Y_{\mu,1},\dots,\,{}^t\! R_\mu Y_{\mu,d_1/2}
$$
forms a symplectic basis with respect to the canonical symplectic form
$\omega} \def\Omega{\Omega$ on $\Bbb R^{d_1}.$ We may thus assume without loss of generality
that this basis agrees with our basis $\tilde X_1,\dots,\tilde
X_{d_1/2}, \tilde Y_1,\dots,
\tilde Y_{d_1/2}$ of $\Bbb R^{d_1},$ so that
$$
d\alpha_\mu(X_{\mu,j})=\tilde X_j, \ d\alpha_\mu(Y_{\mu,j})=\tilde Y_j, \quad j=1,\dots, d_1/2.
$$
By our construction of the representation $\pi_\mu,$ we thus obtain for the derived representation $d\pi_\mu$ of $\frak g$ that
\begin{equation}\label{derivedrep}
d\pi_\mu(X_{\mu,j})=d\rho_{|\mu|}(\tilde X_j), \ d\pi_\mu(Y_{\mu,j})=d\rho_{|\mu|}(\tilde Y_j), \quad j=1,\dots, d_1/2.
\end{equation}
Let us define the sub-Laplacians $L_\mu$ on $G$ and $\tilde L$ on ${\Bbb H}_{d_1/2}$ by
$$
L_\mu:=-\sum_{j=1}^{d_1/2} (X_{\mu,j}^2+Y_{\mu,j}^2),\quad
\tilde L:=-\sum_{j=1}^{d_1/2} (\tilde X_j^2+\tilde Y_j^2),
$$
where from now on we consider elements of the Lie algebra again as left-invariant differential operators. Then, by \eqref{derivedrep},
$$
d\pi_\mu(L_\mu)=d\rho_{|\mu|}(\tilde L).
$$
Moreover, since the basis $X_{\mu,1},\dots,X_{\mu,d_1/2},
Y_{\mu,1},\dots,Y_{\mu,d_1/2}$ and our original basis
$X_1,\dots,X_{d_1}$
of $\frak g_1$ are both orthonormal bases, it is easy to verify that the distributions $L{\delta}_0$ and $L_\mu{\delta}_0$ agree. Since $Af=f*(A{\delta}_0)$ for every left-invariant differential operator $A,$ we thus have $L=L_\mu,$ hence
\begin{equation}\label{8.8}
d\pi_\mu(L)=d\rho_{|\mu|}(\tilde L).
\end{equation}
But, it follows immediately from our definition of Schr\"odinger representation $\rho_\tau$ that $d\rho_\tau(\tilde X_j)=\partial_{\xi_j}$ and
$d\rho_\tau(\tilde Y_j)=2\pi i\tau \xi_j$, so that $d\rho_{|\mu|}(\tilde L)=-\Delta_\xi+(2\pi|\mu|)^2|\xi|^2$ is a rescaled Hermite operator ({\it cf}. also \cite{folland}), and an orthonormal basis of $L^2(\Bbb R^{d_1/2})$ is given by the tensor products
$$
h^{|\mu|}_\alpha:=h^{|\mu|}_{\alpha_1}\otimes \cdots\otimes h^{|\mu|}_{\alpha_{d_1/2}},\quad \alpha\in\Bbb N^{d_1/2},
$$
where $h_k^\mu(x):=(2\pi|\mu|)^{1/4}h_k((2\pi|\mu|)^{1/2}x),$ and
$$
h_k(x)=c_k(-1)^k e^{x^2/2}\frac{d^k}{dx^k}e^{-x^2}
$$
denotes the $L^2$-normalized Hermite function of order $k$ on $\Bbb R.$ Consequently,
\begin{equation}\label{dpiL}
d\pi_\mu(L)h^{|\mu|}_\alpha= 2\pi|\mu|(\frac{d_1}{2}+2|\alpha|) h^{|\mu|}_\alpha, \quad \alpha\in
\Bbb N^{d_1/2}.
\end{equation}
It is also easy to see that
\begin{equation}\label{dpiU}
d\pi_\mu(U_j)=2\pi i\mu_j I, \quad j=1,\dots,d_2.
\end{equation}
Now, the operators $L, -iU_1,\dots,-iU_{d_2}$ form
a set of pairwise strongly commuting self-adjoint operators, with joint core
${\mathcal {S}}(G),$
so that they admit a joint spectral resolution, and we can thus give meaning to expressions
like $\varphi(L,-iU_1,\dots,-iU_{d_2})$ for each continuous function $\varphi$ defined on the corresponding joint spectrum. For simplicity of notation we write
$$U:=(-i U_1,\dots, -i U_{d_2}).$$
If $\varphi$ is bounded, then $\varphi(L,U)$ is a bounded,
left invariant operator on $L^2(G),$ so that it is a convolution operator
$$
\varphi(L,U)f=f*K_\varphi, \quad f\in {\mathcal {S}}(G),
$$
with a convolution kernel $K_\varphi\in{\mathcal {S}}'(G)$ which will also be denoted by $\varphi(L,U){\delta}.$ Moreover, if $\varphi\in {\mathcal {S}}(\Bbb R\times\Bbb R^{d_2}),$ then $\varphi(L,U){\delta}\in{\mathcal {S}}(G)$ ({\it cf}. \cite{MRS2}).
Since functional calculus is compatible with unitary representation theory, we obtain in this case from \eqref{dpiL}, \eqref{dpiU} that
\begin{equation}\label{pimuphi}
\pi_\mu(\varphi(L,U){\delta})h^{|\mu|}_\alpha= \varphi\Big(2\pi|\mu|(\frac{d_1}{2}+2|\alpha|),2\pi \mu\Big) h^{|\mu|}_\alpha
\end{equation}
(this identity in combination with the Fourier inversion formula could in fact be taken as the definition of $\varphi(L,U){\delta}$).
In particular, the Plancherel theorem implies then that the operator norm on $L^2(G)$ is given by
\begin{equation}\label{opnorm}
\|\varphi(L,U)\|=\sup\{|\varphi(|\mu|(\frac{d_1}{2}+2q), \mu)|: \mu\in\Bbb R^{d_2}, q\in\Bbb N\}.
\end{equation}
Finally, observe that
\begin{equation}\label{Kphimu}
K_\phi^\mu=\varphi(L^\mu,2\pi \mu){\delta};
\end{equation}
this follows for instance by applying the unitary representation induced from the character $ e^{2\pi i \mu\cdot u}$ on the center of $G$ to $K_\varphi$.
We shall in fact only work with functions of $L$ and $|U|,$ defined by
$$
\pi_\mu(\varphi(L,|U|){\delta})h^{|\mu|}_\alpha= \varphi\Big(2\pi|\mu|(\frac{d_1}{2}+2|\alpha|),2\pi |\mu|\Big) h^{|\mu|}_\alpha.
$$
Observe that if $\varphi$ depends only on the second variable, then
$\varphi(|U|)$ is just the radial convolution operator
acting only in the central variables, given by
\begin{equation}\label{hU}{\mathcal {F}}_{{\mathbb {R}}^{d_2}}[\varphi(|U|) f](x,\mu)
= \varphi(2\pi|\mu|){\mathcal {F}}_{{\mathbb {R}}^{d_2}}f(x,\mu)\quad\text{ for all $\mu\in ({\mathbb {R}}^{d_2})^*$}\,.\end{equation}
\medskip
\subsection{\it Partial Fourier transforms and twisted convolution}\label{PFTtwisedconv}
For $\mu\in\frak g_2^*,$ one defines the $\mu$--{\it twisted convolution} of two suitable functions (or distributions) $\varphi$ and $\psi$ on $\frak g_1=\Bbb R^{d_1}$ by
$$
(\varphi*_\mu \psi)(x):=\int_{\Bbb R^{d_1}} \varphi(x-y)\psi(y)e^{-i\pi\omega} \def\Omega{\Omega_\mu(x,y)}\,dy
$$
where $\omega_\mu$ is as in \eqref{omegamu}.
Then, with $f^\mu$ as in \eqref{partialFT},
$$
(f*g)^\mu=f^\mu *_\mu g^\mu,
$$
where $f*g$ denotes the convolution product of the two functions $f,g\in L^1(G).$
Accordingly, the vector fields $X_j$ are transformed into the $\mu$-twisted first order differential operators $X_j^\mu$ such that $(X_j f)^\mu=X_j^\mu f^\mu,$ and the
sub-Laplacian is transformed into the $\mu$-twisted Laplacian $L^\mu,$ i.e.,
$$
(Lf)^\mu=L^\mu f^\mu=-\sum_{j=1}^{d_1} (X_j^\mu)^2,
$$
say for $f\in{\mathcal {S}}(G)$. A computation shows that explicitly
\begin{equation}\label{Xjmu}
X_j^\mu=\partial_{x_j}+i\pi\omega} \def\Omega{\Omega_\mu(\cdot,X_j).
\end{equation}
\subsection{\it The Schr\"odinger group $\{e^{itL^\mu}$\}}\label{Schrgroup}
It will be important for us that the
Schr\"odinger operators
$e^{itL^\mu}, \
t\in {\mathbb {R}},$ generated by $L^\mu$ can be computed explicitly.
\begin{prop}\label{eit}
(i) For $f\in {\mathcal {S}}(G),$
\begin{equation}\label{twistedschr}
e^{itL^\mu} f=f*_\mu \gamma_t^\mu, \quad t\ge 0,
\end{equation}
where $\gamma_t^\mu\in {\mathcal {S}}'(\Bbb R^{d_1})$ is a tempered distribution.
(ii) For all $t$ such that $\sin (2\pi t |\mu|)\ne 0$ the distribution
$\gamma^\mu_t$ is
given by
\begin{equation}\label{gammatmu}
\gamma_t^\mu(x)=2^{-d_1/2}\Big(\frac {|\mu|}{\sin (2\pi t|\mu|)}\Big)^{d_1/2}\,
e^{-i\frac {\pi}2 |\mu|\cot(2\pi t|\mu|)|x|^2}.
\end{equation}
(iii) For all $t$ such that $\cos (2\pi t |\mu|)\ne 0$ the Fourier transform of $\gamma^\mu_t$ is given by
\begin{equation}\label{gammatmuF}
\widehat{\gamma_t^\mu}(\xi)=\frac 1{(\cos (2\pi t|\mu|))^{d_1/2}}\,
e^{i\frac{2\pi}{|\mu|}\tan(2\pi t|\mu|)|\xi|^2}.
\end{equation}
\end{prop}
Indeed, for $\mu\ne 0,$ let us consider the symplectic vector space $V:=\frak g_1,$ endowed with the symplectic form $\sigma} \def{\Sigma}{\Sigma:=\frac 1{|\mu|}\omega} \def\Omega{\Omega_\mu.$ Notice first that, because of \eqref{Jmusquared},
the volume form $\sigma^{\wedge (d_1/2)},$ i.e., the ${d_1}/{2}$-fold exterior product of $\sigma} \def{\Sigma}{\Sigma$
with itself, can be identified with Lebesgue measure on $\Bbb R^{d_1}.$ As in \cite{dissip}, we then associate to the
pair $(V,\sigma} \def{\Sigma}{\Sigma)$ the Heisenberg group ${\Bbb H}_V,$ with underlying manifold $V\times\Bbb R$ and endowed with the product
$$
(v,u)(v',u'):=(v+v',u+u'+\frac 12 \sigma} \def{\Sigma}{\Sigma(v,v')).
$$
It is then common to denote for $\tau\in\Bbb R$ the $\tau$-twisted convolution
by $\times_\tau$ in place of $*_\tau$ (compare \S5 in \cite{dissip}).
The $\mu$-twisted convolution associated to the group $G$ will then agree
with the $|\mu|$-twisted convolution $\times_{|\mu|}$ defined on the symplectic vector
space $(V,\sigma} \def{\Sigma}{\Sigma).$
Moreover, if we
identify the $X_j\in V$ also with left-invariant vector fields on
${\Bbb H}_V,$ then \eqref{Xjmu} shows that
$$X_j^\mu=\partial_{x_j}+i\pi|\mu|\sigma} \def{\Sigma}{\Sigma(\cdot,X_j)$$ agrees with the
corresponding $|\mu|$-twisted differential operators $\tilde
X_j^{|\mu|}$ defined in \cite{dissip}.
Accordingly, our $\mu$-twisted Laplacian $L^\mu$ will agree with the $|\mu|$-twisted Laplacian
$$\tilde L_S^{|\mu|}=\tilde{\mathcal L}_{-I}^\mu=\sum_{j=1}^{d_1} (\tilde X_j^{|\mu|})^2
$$
associated to the symmetric matrix $A:=-I$ in \cite{dissip}. Here,
$$S=-A\frac 1{|\mu|}J_\mu=\frac 1{|\mu|}J_\mu.
$$
Consequently,
$$ e^{itL^\mu}=e^{it\tilde L_S^{|\mu|}}.
$$
From Theorem 5.5 in \cite{dissip} we therefore obtain that
for $f\in L^2(V)$
$$
\exp (\frac {it}{|\mu|} \tilde L_S^{|\mu|})f=f \times_{|\mu|}\Gamma_{t,iS}^{|\mu|}, \quad
t\ge0,
$$
where, $\Gamma_{t,iS}^{|\mu|}$ is a tempered distribution whose Fourier
transform is given by
$$
\widehat{\Gamma_{t,iS}^{|\mu|}}(\xi)=\frac 1{\sqrt{\det(\cos 2\pi itS)}}
e^{-\frac{2\pi}{|\mu|}\sigma(\xi,\tan(2\pi itS)\xi)}
$$
whenever $\det(\cos (2\pi it S)\ne 0$. Since $S^2=-I$ because of \eqref{Jmusquared}, one sees that
$$\sin(2\pi itS)=i\sin(2\pi t) S,\ \cos(2\pi itS)=\cos(2\pi t)I.
$$
Note also that $\sigma} \def{\Sigma}{\Sigma(\xi,\eta)=\langle S\xi,\eta\rangle.$ We thus see that \eqref{twistedschr} and \eqref{gammatmuF} hold true,
and the formula \eqref{gammatmu} follows by Fourier inversion ({\it cf}. Lemma 1.1 in \cite{MR1}).
\section{An approximate subordination formula}\label{subordination}
We shall use Proposition \ref{eit} and the following subordination formula to
obtain manageable expressions for the wave operators.
\begin{prop} \label{subordop}Let $\chi_1\in C^\infty$ so that $\chi_1(s)=1$ for $s\in [1/4,4]$. Let $g$ be a $C^\infty$ function supported in $(1/2,2)$.
Then there are $C^\infty$ functions $a_\lambda} \def\La{\Lambda$ and $\rho_\lambda} \def\La{\Lambda$,
depending linearly on $g$, with $a_\lambda} \def\La{\Lambda$
supported in $[1/16,4]$, and $\rho_\lambda} \def\La{\Lambda$ be supported in $[1/4,4]$,
so that for all $K=2,3,\dots$, $N_1, N_2\ge 0$, and all $\lambda} \def\La{\Lambda\ge 1$
\begin{equation}\label{alaest}
\sup_s\big|\partial_s^{N_1} \partial_\lambda} \def\La{\Lambda^{N_2} a_\lambda} \def\La{\Lambda(s)\big|\le c(K)
\lambda} \def\La{\Lambda^{-N_2} \sum_{\nu=0}^K\|g^{(\nu)}\|_\infty,
\quad \,\,N_1+N_2< \frac{K-1}2,
\end{equation}
\begin{equation}\label{rholaest}
\sup_s\big|\partial_s^{N_1} \partial_\lambda} \def\La{\Lambda^{N_2} \rho_\lambda} \def\La{\Lambda(s)\big|\le c(K,N_2)
\lambda} \def\La{\Lambda^{N_1+1-K}\sum_{\nu=0}^K\|g^{(\nu)}\|_\infty,
\quad N_1\le K-2\,.
\end{equation}
and the formula
\begin{equation}
g(\lambda} \def\La{\Lambda^{-1}\sqrt L)e^{i\sqrt L}=\chi_1(\lambda} \def\La{\Lambda^{-2} )L)
\sqrt{\lambda} \def\La{\Lambda }\int e^{i\frac{\lambda} \def\La{\Lambda }{4s}} a_\lambda} \def\La{\Lambda(s) \,
e^{i s L/\lambda} \def\La{\Lambda}\,
ds\,+\, \rho_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^{-2}L)
\label{localmultop}
\end{equation}
holds.
For any $N\in {\mathbb {N}}$, the functions $\lambda} \def\La{\Lambda^{N} \rho_\lambda} \def\La{\Lambda$ are uniformly bounded in the topology of the Schwartz-space, and the operators $\rho_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^{-2}L)$
are bounded on $L^p (G)$, $1\le p\le \infty$,
with operator norm $O(\lambda} \def\La{\Lambda^{-N})$.
\end{prop}
For explicit formulas of $a_\lambda} \def\La{\Lambda$ and $\rho_\lambda} \def\La{\Lambda$ see
Lemma \ref{subordlemma} below.
The proposition is essentially an application of the method of stationary phase
where we keep track on how $a_\lambda} \def\La{\Lambda$, $\rho_\lambda} \def\La{\Lambda$ depend on $g$.
We shall need an auxiliary lemma.
\begin{lemma} \label{stationaryphase}
Let $K\in \mathbb N$ and $g\in C^K(\mathbb R)$. Let $\zeta_1\in
C^\infty({\mathbb {R}})$ be
supported in $(1/2,2)\cup(-2,-1/2)$ and $\Lambda\ge 1$ and $\ell\ge 0$.
Then, for all nonnegative integers $M$,
\begin{multline}
\label{fixedellest}
\Big| \int y^{2M} g(y) \zeta_1(\La^{1/2}2^{-\ell} y) e^{i\La y^2}
dy\Big|\ \\
\le C_{M,K} 2^{-2\ell K} \big( 2^{\ell} \La^{-1/2}\big)^{1+2M} \sum_{j=0}^K
(2^\ell \La^{-1/2})^j \|g^{(j)}\|_\infty\,.
\end{multline}
Moreover, for $0\le m<\frac{K-1}2$,
\begin{equation}\label{sumell}
\Big|\Big(\frac{d}{d\Lambda}\Big)^m \int g(y) e^{i\La y^2}
dy\Big|
\le C_{K} \La^{-m-\frac 12} \sum_{j=0}^K
\La^{-j/2} \|g^{(j)}\|_\infty.
\end{equation}
\end{lemma}
\begin{proof}
By
induction on $K$ we prove the following assertion labeled
\medskip
\noindent $({\mathcal {A}}_K)$: \
If $g\in C^K$ then
\begin{multline} \label{indclaim}
\int y^{2M} g(y) \zeta_1(\La^{1/2}2^{-\ell} y) e^{i\La y^2}
dy \\=
\La^{-K}\sum_{j=0}^K \int g^{(j)}(y) \zeta_{j,K,M,\La}(y)
e^{i\La y^2} dy
\end{multline}
where $\zeta_{j,K,M,\La}$
is supported on $\{y: |y|\in
[2^{\ell-1}\La^{-1/2}, 2^{\ell+1}\La^{-1/2}]\}$ and,
for $0\le j\le K$, satisfies the
differential inequalities
\begin{equation}\label{diffineq}
\big|\zeta_{j,K,M,\La}^{(n)}(y)\big|
\le C(j,K,M,n) (2^{-\ell} \La^{1/2})^{n-2M} 2^{-\ell(2K-j)} \La^{K-j/2}\,.
\end{equation}
Clearly this assertion implies
\eqref{fixedellest}.
We set
$ \zeta_{0,0,M,\La}(y) =y^{2M}
\zeta_1(\La^{1/2}2^{-\ell} y)$ and the claim
$({\mathcal {A}}_K)$ is immediate for $K=0$.
It remains to show that the implication
$({\mathcal {A}}_K)\implies ({\mathcal {A}}_{K+1})$, holds for all $K\ge 0$.
Assume
$({\mathcal {A}}_K)$ for some $K\ge 0$ and let $g\in C^{K+1}$.
We let $0\le j\le K$ and examine the $j$th term in the sum in
\eqref{indclaim}.
Integration by parts yields
\begin{multline*}
\int g^{(j)}(y) \zeta_{j,K,M,\La}(y)
e^{i\La y^2} dy\\
=
i \int \Big[ \frac{ g^{(j+1)}(y)}{2y\La} \zeta_{j,K,M,\La}(y)
+ g^{(j)}(y) \frac{d}{dy}\Big(\frac{\zeta_{j,K,M,\La}(y) }{2y\La}\Big)
\,\Big] e^{i\La y^2} dy\,.
\end{multline*}
The sum $\La^{-K}\sum_{j=0}^K \int g^{(j)}(y) \zeta_{j,K,M,\La}(y)
e^{i\La y^2} dy$ can now be rewritten as
\[
\La^{-K-1}\sum_{\nu=0}^{K+1} \int g^{(\nu)}(y) \zeta_{\nu,K+1,M,\La}(y)
e^{i\La y^2} dy
\]
where
\begin{align*}
\zeta_{0,K+1,M,\La}(y) &= i
\frac{d}{dy}\Big(\frac{\zeta_{0,K,M,\La}(y) }{2y}\Big)\,,
\\
\zeta_{\nu,K+1,M,\La}(y) &= i
\frac{d}{dy}\Big(\frac{\zeta_{\nu,K,M,\La}(y) }{2y}\Big)
+i \frac {\zeta_{\nu-1,K,M,\La}(y) }{2y}, \quad 1\le \nu \le K,
\\
\zeta_{K+1,K+1,M,\La}(y) &=i
\frac {\zeta_{K,K,M,\La}(y) }{2y}\,.
\end{align*}
On the support of the cutoff functions we have $|y|\ge 2^{\ell-1}\La^{-1/2}$ and the asserted differential inequalities for
the
functions $\zeta_{\nu,K+1,M,\La}$ can be verified using the Leibniz rule.
This finishes the proof of
the implication $({\mathcal {A}}_{K})\implies
({\mathcal {A}}_{K+1})$ and thus the proof of
\eqref{fixedellest}.
We now prove \eqref{sumell}. Let $\zeta_0$ be an even $C^\infty$ function
supported in $(-1,1)$ and assume that $\zeta_0(s)=1$ for $|s|\le
1/2$.
Let $\zeta_1(s)=\zeta_0(s/2)-\zeta_0(s)$ so that $\zeta_1$ is supported
in $[-2,-1/2]\cup[1/2,2]$, as in the statement of \eqref{fixedellest}.
We split the left hand side of
\eqref{sumell} as $\sum_{\ell=0}^\infty I_{\ell,m}$ where
$$
I_{\ell,m}=
\int (iy^{2})^m g(y) \zeta_1(\La^{1/2}2^{-\ell} y) e^{i\La y^2}
dy, \qquad \text{for } \ell>0\,
$$
and $I_{0,m}$ is defined similarly with
$\zeta_0(\La^{1/2} y)$ in place of $\zeta_1(\La^{1/2}2^{-\ell} y)$.
Clearly $|I_{0,m}|\lesssim \La^{-m-1/2}\|g\|_\infty$ and by
\eqref{fixedellest}
$$I_{\ell,m}\lesssim_{m,K} \sum_{j=0}^K
2^{-\ell (2K-2m-j-1)} \La^{-\frac{1+2m+j}2} \|g^{(j)}\|_\infty.$$
Since $j\le K$ we can sum in $\ell$ if $m<\frac{K-1}2$ and the
assertion \eqref{sumell} follows.
\end{proof}
\begin{lemma}\label{subordlemma}
Let $K\in \mathbb N$ and let $g\in C^K(\mathbb R)$ be supported in $(1/2,2)$, and let $\chi_1\in C^\infty_c({\mathbb {R}})$ so that $\chi_1(x)=1$ on $(1/4,4)$.
Also let $\varsigma$ be a $C^\infty_0({\mathbb {R}})$ function
supported in $[1/9, 3]$ with the property that $\varsigma(s)=1$ on
$[1/8,2]$.
Then
\begin{equation}
g(\sqrt x)e^{i\lambda} \def\La{\Lambda\sqrt x}= \chi_1(x)
\Big[
\sqrt{\lambda} \def\La{\Lambda }\int e^{i\frac{\lambda} \def\La{\Lambda }{4s}} a_\lambda} \def\La{\Lambda(s) \,
e^{i\lambda} \def\La{\Lambda s x}\,
ds\,+\,\widetilde \rho_\lambda} \def\La{\Lambda(x)\Big]
\label{localmult}
\end{equation}
where $a_\lambda} \def\La{\Lambda$ is supported in $[\frac 1{16},4]$,
and
\begin{equation} \label{alambda}
a_\lambda(s) =\pi^{-1}\sqrt{\lambda} \varsigma(s)\int (y+\tfrac 1{2s}) g(y+\tfrac 1{2s})
e^{-i\lambda} \def\La{\Lambda sy^2} \,dy
\end{equation}
and
\begin{equation} \label{rholambda} {\mathcal {F}}[\widetilde{\rho}_\lambda} \def\La{\Lambda](\xi )=
(1- \varsigma(\tfrac{2\pi\xi}{\lambda} \def\La{\Lambda})) {\mathcal {F}}[g(\sqrt{\cdot}) e^{i\lambda} \def\La{\Lambda \sqrt{\cdot}}] (\xi)\,.
\end{equation}
Let $\rho_\lambda} \def\La{\Lambda=\chi_1\widetilde \rho_\lambda} \def\La{\Lambda$.
Then the estimates \eqref{alaest} and \eqref{rholaest}
hold for all $\lambda} \def\La{\Lambda\ge 1$.
\end{lemma}
\begin{proof}
Let $\Psi_\lambda} \def\La{\Lambda$ be the Fourier transform of $x\mapsto g(\sqrt x)
e^{i\lambda} \def\La{\Lambda \sqrt{x}} $, i.e.
\begin{equation} \Psi_\lambda} \def\La{\Lambda(\xi)=\int g(\sqrt x)e^{i\lambda} \def\La{\Lambda \sqrt{x}}
e^{-2\pi i\xi x}\, dx\,
= \int 2sg(s) e^{i(\lambda} \def\La{\Lambda s-2\pi \xi s^2)}\, ds
\label{Psifirst}
\end{equation}
Observe that $g(\sqrt x)
=0$ for $x\notin (1/4,4) $, thus $g(\sqrt x)=\chi_1(x)g(\sqrt x)$.
By the Fourier inversion formula we have
$$g(\sqrt x)e^{i\lambda} \def\La{\Lambda \sqrt{x}} =\chi_1(x)\big(\upsilon_\lambda} \def\La{\Lambda(x)+\rho_\lambda} \def\La{\Lambda(x)\big)$$ where
\begin{equation}\label{vlarhola}
\begin{aligned}
\upsilon_\lambda} \def\La{\Lambda(x)
&=\int \varsigma(\tfrac{2\pi \xi}{\lambda} \def\La{\Lambda})\Psi_\lambda} \def\La{\Lambda(\xi)e^{2\pi ix\xi} d\xi
\\
\widetilde \rho_\lambda} \def\La{\Lambda(x)
&=\int \big(1-\varsigma(\tfrac{2\pi\xi}{\lambda} \def\La{\Lambda})\big)
\Psi_\lambda} \def\La{\Lambda(\xi)e^{2\pi ix\xi} d\xi
\end{aligned}
\end{equation}
so that $\widetilde \rho_\lambda} \def\La{\Lambda$ is as in \eqref{rholambda}.
We first consider $\widetilde \rho_\lambda} \def\La{\Lambda$ and verify that
the inequalities \eqref{rholaest} hold.
On the support of $1-\varsigma(2\pi\xi/\lambda} \def\La{\Lambda)$ we have either $|2\pi\xi|\le
\lambda} \def\La{\Lambda/8$
or $|2\pi\xi|\ge 2\lambda} \def\La{\Lambda$.
Clearly, on the support of $g$ we have
$|\partial_s (\lambda} \def\La{\Lambda s-2\pi \xi s^2)|\ge \lambda} \def\La{\Lambda/2$ if $|2\pi\xi|\le
\lambda} \def\La{\Lambda/8$ and
$|\partial_s (\lambda} \def\La{\Lambda s-2\pi\xi s^2)|\ge |2\pi\xi|/2$ if $|2\pi\xi|\ge 2\lambda} \def\La{\Lambda$.
Integration by parts in \eqref{Psifirst} yields
$$\big|\partial^{M_1}_\xi \partial^{M_2}_\lambda} \def\La{\Lambda
\big[(1-\varsigma(2\pi \xi/\lambda} \def\La{\Lambda))\Psi_\lambda} \def\La{\Lambda(\xi)\big]\big| \le C_{M_1,M_2,K}
\|g\|_{C_K}(1+|\xi|+|\lambda} \def\La{\Lambda|)^{-K} .
$$ Thus, if $N_1\le K-2$,
\begin{align*}
\Big|\Big(\frac{d}{dx}\Big)^{N_1}&\widetilde \rho_\lambda} \def\La{\Lambda(x)\Big|= \Big|\int (2\pi\xi)^{N_1}
(1-\varsigma(2\pi \xi/\lambda} \def\La{\Lambda))\Psi_\lambda} \def\La{\Lambda(\xi) e^{2\pi ix\xi}d\xi\Big|
\\&\le C_{N_1,K} \|g\|_{C^K}\,\int\frac{(1+|\xi|)^{N_1}}{(1+|\xi|+|\lambda} \def\La{\Lambda|)^{K}}
d\xi
\le C_{N_1,K}' \|g\|_{C^K} \lambda} \def\La{\Lambda^{-K+N_1+1}.
\end{align*}
This yields \eqref{rholaest} for $N_2=0$, and the same argument
applies to the $\lambda} \def\La{\Lambda$-derivatives.
It remains to represent the function $\lambda} \def\La{\Lambda^{-1/2} \upsilon_\lambda} \def\La{\Lambda$ as the
integral in \eqref{localmult}.
Let
\begin{equation}\label{wtg}\widetilde g(s) = 2sg(s)\,.\end{equation}
By a change of variable we may
write
\begin{equation}\label{Psisecond}
\Psi_\lambda} \def\La{\Lambda(\xi)= e^{\frac{i\lambda} \def\La{\Lambda^2}{8\pi\xi}}
\int \widetilde g
(y+\tfrac{\lambda} \def\La{\Lambda}{4\pi\xi}) e^{-2\pi i\xi y^2} dy.
\end{equation}
We compute
from \eqref{vlarhola}, \eqref{Psisecond},
$$
\upsilon_\lambda} \def\La{\Lambda(x)
= \lambda} \def\La{\Lambda\int\varsigma(s) e^{i\frac{\lambda} \def\La{\Lambda}{4s}+i \lambda} \def\La{\Lambda sx} \lambda} \def\La{\Lambda^{-1/2}a_\lambda} \def\La{\Lambda(s) ds
$$
where
$$ a_\lambda} \def\La{\Lambda(s) =(2\pi)^{-1}\sqrt \lambda} \def\La{\Lambda \, \varsigma(s)\int \widetilde g
(y+\tfrac{1}{2s}) e^{-i\lambda} \def\La{\Lambda s y^2} dy\,,
$$
i.e. $a_\lambda} \def\La{\Lambda$ is as in \eqref{alambda}.
In order to show the estimate \eqref{alaest} observe
$$2\pi\partial_\lambda} \def\La{\Lambda^{N_2} (\lambda} \def\La{\Lambda^{-1/2} a_\lambda} \def\La{\Lambda(s))=
\, \varsigma(s)\int \widetilde g
(y+\tfrac{1}{2s})
(-isy^2)^{N_2} e^{-i\lambda} \def\La{\Lambda s y^2} dy
$$ and then by the Leibniz rule
$\partial_s^{N_1} \partial_\lambda} \def\La{\Lambda^{N_2} [\lambda} \def\La{\Lambda^{-1/2}a_\lambda} \def\La{\Lambda(s)]
$ is a linear combination of terms of
the form
\begin{equation}\label{typicalterm}
\Big(\frac{d}{ds}\Big)^{N_3}\big[\varsigma(s) s^{N_2}]
\int y^{2N_2} (\lambda} \def\La{\Lambda y^2)^{N_5}
\Big(\frac{d}{ds}\Big)^{N_4}
\big[ \widetilde g
(y+\tfrac{1}{2s}) \big]
e^{i\lambda} \def\La{\Lambda sy^2}
\, dy
\end{equation}
where and $N_3+N_4+N_5=N_1$.
By estimate \eqref{sumell} in Lemma \ref{stationaryphase}
we see that the term
\eqref{typicalterm} is bounded (uniformly in
$s\in [1/9,3]$) by a constant times
$$\lambda} \def\La{\Lambda^{-N_2-\frac 12} \big\|(\tfrac{d}{ds})^{N_4}[ \widetilde g
(\cdot+\tfrac{1}{2s}) \big] \big\|_{C^{K-N_4}}
$$
provided that $N_2+N_5 < (K-N_4-1)/2$.
This condition is satisfied
if $N_1+N_2<(K-1)/2$ and under this condition we get
$$
\sup_s|\partial_s^{N_1} \partial_\lambda} \def\La{\Lambda^{N_2} [\lambda} \def\La{\Lambda^{-1/2}a_\lambda} \def\La{\Lambda(s)]|
\lesssim \lambda} \def\La{\Lambda^{-N_2-\frac 12} \|g\|_{C^K}.
$$
Now \eqref{alaest} is a straightforward consequence.
\end{proof}
\begin{proof}[Proof of Proposition \ref{subordop}]
The identity
\eqref{localmultop} is an immediate consequence of the
spectral resolution $L=\int_{{\mathbb {R}}^+} xdE_x,$
Lemma \ref{subordlemma} (applied with $x/\lambda} \def\La{\Lambda$ in place of $x$) and Fubini's theorem. Note that in
view of the symbol estimates \eqref{rholaest} any Schwartz norm of
$\rho_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^{-2}\,\cdot)$ is $O(\lambda} \def\La{\Lambda^{-N})$ for every $N\in\Bbb N.$
The statement
on the operator norms of $\rho_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^{-2} L)$ follows then from the known multiplier theorems
(such as the original one by Hulanicki and Stein, see \cite{hulanicki}, \cite{FollSt}).
\end{proof}
Thus in order to get manageable formulas for our wave operators it
will be important to get
explicit formulas for the
Schr\"odinger group
$e^{isL}$, $ s\in {\mathbb {R}}$.
\section{Basic decompositions of the wave operator and statements of refined results}
\label{dyadicdecsect}
We consider operators $a(\sqrt L)e^{i\sqrt L}$ where $a\in S^{(d-1)/2}$
(satisfying \eqref{symbols} with
$\gamma=\frac{d-1}{2}$).
We split off the part of the symbol supported near $0$.
Let $ \chi_0\in C^\infty_c({\mathbb {R}})$ be supported in $[-1,1]$; then we observe
that the operator $\chi_0(\sqrt L) \exp (i\sqrt L)$
extends to a bounded operator on
$L^p(G)$, for $1\le p\le \infty$.
To see this we decompose
$\chi_0(\sqrt{\tau})e^{i\sqrt\tau}
=\chi_0(\sqrt \tau)+\sum_{n=0}^\infty \alpha_n(\tau)$,
$\tau>0$,
where $$\alpha_n(\tau)=\chi_0(\sqrt \tau) (e^{i\sqrt \tau}-1)
( \zeta_0(2^{n-1}\tau)-\zeta_0(2^n\tau))$$
where $\zeta_0$ is as in {\mathcal S}\ref{cutoffsect}.
Clearly $\chi_0(\sqrt \cdot)\in C^\infty_0$.
Thus by
Hulanicki's theorem \cite{hulanicki}
the convolution kernel of
$\chi_0(\sqrt L)$
is a Schwartz function and hence $\chi_0(\sqrt L)$ is bounded on $L^1(G)$.
Moreover the functions
$2^{n/2}\alpha_n(2^{-n}\cdot)$ belong to a bounded set of Schwartz functions
supported in $[-2,2]$. By dilation invariance and again
Hulanicki's theorem the convolution kernels of
$2^{n/2}\alpha_n(2^{-n}L)$ are Schwartz functions
and form a bounded subset of the Schwartz space ${\mathcal {S}}(G)$. Thus, by rescaling,
the operator $\alpha_n(L)$ is bounded on $L^1(G)$ with operator norm
$O(2^{-n/2})$. We may sum in $n$ and obtain the desired bounds for
$\chi_0(\sqrt{\tau})e^{i\sqrt\tau}$.
The above also implies that for any $\lambda} \def\La{\Lambda$ the operator
$\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L) \exp (i\sqrt L)$ is bounded on $L^1$ (with a
polynomial and nonoptimal growth in $\lambda} \def\La{\Lambda$).
Thus, in what follows it suffices to consider symbols $a\in S^{-(d-1)/2}$
with the property that $a(s)=0$ in a neighborhood of $0$. Then
\begin{equation}\label{dyadicdec}a(\sqrt L)e^{i\sqrt L}\,=\, \sum_{j>C} 2^{-j\frac{d-1}{2}}
g_j (\sqrt{2^{-2j} L}) e^{i\sqrt L},
\end{equation}
where the $g_j$ form a family of smooth functions supported in $(1/2,2)$ and bounded in the $C^\infty_0$ topology. In many calculations below when $j$
is fixed we shall also use the parameter $\lambda} \def\La{\Lambda$ for $2^j$.
Let $\chi_1$ be a smooth function such that
\begin{subequations}\label{chi1}
\begin{align} \label{chi1support}
&{\hbox{\roman supp}}(\chi_1) \subset (2^{-10}, 2^{10})\,,
\\
\label{chi1=1}
&\chi_1(s)=1 \text{ for $s\in (2^{-9}, 2^9)$.}
\end{align}
\end{subequations}
By Proposition \ref{subordop} and
Lemma \ref{subordlemma} we may thus write
\begin{equation}\label{adydecomposition}a(\sqrt L)e^{i\sqrt L}\,=\,
m_{\text{negl}}(L)+\sum_{j>100} 2^{-j\frac{d-1}{2}}
\chi_1(2^{-2j}L) m_{2^j}(L),
\end{equation}
where the ``negligible'' operator $m_{\text{negl}}(L)$ is a convolution with a Schwartz kernel,
\begin{equation} \label{mj} m_{\lambda} \def\La{\Lambda}(\rho)= \sqrt{\lambda} \def\La{\Lambda} \int e^{i\lambda} \def\La{\Lambda/(4\tau )}
a_\lambda} \def\La{\Lambda(\tau)
e^{i\tau \rho/\lambda} \def\La{\Lambda}
d\tau, \quad \text{ with } \lambda} \def\La{\Lambda =2^j,
\end{equation}
and the $a_\lambda} \def\La{\Lambda$
form a family of smooth functions supported in $(1/16,4)$, bounded in the $C^\infty_0$ topology.
We shall use the formulas \eqref{gammatmu}, which give explicit expressions for the partial Fourier transform in the central variables of the Schwartz kernel of
$e^{it L}.$ In undoing this partial Fourier transform, it
will be useful to
recall from \S3 that if $\rho_1$ denotes the spectral parameter for $L$
then the joint spectrum of the operators $L$ and $|U|$ is contained
in the closure of \begin{equation}\label{jtspec}\{(\rho_1,\rho_2):\rho_2\ge 0, \, \rho_1
=(\tfrac{d_1}{2}+2q)\rho_2 \text{ for some nonnegative
integer $q$}\}\,.\end{equation}
As the phase in \eqref{gammatmu} exhibits periodic singularities
it natural to introduce an equally spaced decomposition in the central
Fourier
variable (i.e., in the spectrum of the operator $|U|$).
Let $\eta_0$ be a $C^\infty$ function such that
\begin{subequations}
\begin{align}
&{\hbox{\roman supp}} (\eta_0)
\subset (-\tfrac {5\pi}8 ,
\tfrac {5\pi}8 )\,,
\label{suppeta0}
\\
\label{eta0equalto1}
&\eta_0(s)=1 \,\text{ for } s\in
(-\tfrac {3\pi}8, \tfrac{3\pi}{ 8})\,,
\\
\label{etapartitionof1}
&\sum_{k\in \mathbb Z} \eta_0(t-k\pi)=1 \text{ for $t\in {\mathbb {R}}$.}
\end{align}
\end{subequations}
We decompose
\begin{equation}\label{kdecomposition}
\chi_1(\lambda} \def\La{\Lambda^{-2} L)m_\lambda} \def\La{\Lambda(L)= \sum_{k=0}^{\infty} \chi_1(\lambda} \def\La{\Lambda^{-2} L)
T^k_\lambda} \def\La{\Lambda,
\end{equation}
where
\begin{equation}\label {Tkla}
T^k_\lambda} \def\La{\Lambda=
\lambda} \def\La{\Lambda^{1/2}\int e^{i\lambda} \def\La{\Lambda/(4\tau)}
a_\lambda} \def\La{\Lambda(\tau) \eta_0 (\tfrac {\tau}{ \lambda} \def\La{\Lambda}|U|- k\pi)
e^{i\tau L/\lambda} \def\La{\Lambda}
d\tau\,.
\end{equation}
The description \eqref{jtspec} of the joint spectrum of $L$ and $|U|$
gives a restriction on the
summation in $k$.
Namely the operator $\eta_0 (\tfrac {\tau}{ \lambda} \def\La{\Lambda}|U|- k\pi)
\chi_1(\lambda} \def\La{\Lambda^{-2} L)$ is identically zero unless there exist positive
$\rho_1$ and $\rho_2$ with $\rho_1\ge \rho_2 d_1/2$ such that
$\frac{\lambda} \def\La{\Lambda^2} 5<\rho_1<5\lambda} \def\La{\Lambda^2$ and
$(k\pi- \tfrac{5\pi}{8})\frac{\lambda} \def\La{\Lambda}{\tau}
<\rho_2 <
(k\pi+ \frac{5\pi}{8})\frac{\lambda} \def\La{\Lambda}{\tau}$ for some
$\tau \in (\tfrac 1{16},4)$.
A necessary condition for these two conditions to hold simultaneously is
of course $\frac{d_1}{2} (k\pi-\frac 58 \pi)\frac{\lambda} \def\La{\Lambda}{ 4} \le 5\lambda} \def\La{\Lambda^2$ and since $d_1\ge 2$ and $\lambda} \def\La{\Lambda\ge 1$ we see that the sum in \eqref{kdecomposition} extends only over $k$ with
\begin{equation}\label{kleeightla}0\le k<8\lambda} \def\La{\Lambda.\end{equation}
We now
derive formulas for the
convolution kernels of $T^{k}_\lambda} \def\La{\Lambda$, which we denote
by
$K^{k}_\lambda} \def\La{\Lambda$.
The identity \eqref{gammatmu} first gives formulas for the partial Fourier transforms
${\mathcal {F}}_{{\mathbb {R}}^{d_2}}K^{k}_\lambda} \def\La{\Lambda$. Applying the Fourier
inversion formula we get
\begin{multline}\label{Kklaexpression}
K^{k}_\lambda} \def\La{\Lambda(x,u) \,=\, \lambda} \def\La{\Lambda^{1/2} \int_{{\mathbb {R}}^{d_2}}
\int_{\mathbb {R}} e^{i\frac{\lambda} \def\La{\Lambda}{4\tau}}
a_\lambda} \def\La{\Lambda(\tau) \eta_0 (2\pi|\mu|\tfrac {\tau}{ \lambda} \def\La{\Lambda}- k\pi)\,\times
\\
\Big(\frac{|\mu|}{2\sin (2\pi|\mu|\tau/\lambda} \def\La{\Lambda)}\Big)^{d_1/2}
e^{-i|x|^2\frac {\pi}{ 2} |\mu|\cot (2\pi |\mu|\tau/\lambda} \def\La{\Lambda)} d\tau
\, e^{2\pi i \inn{u}{ \mu}} d\mu\,.
\end{multline}
We note that the term
$|\mu|\cot(2\pi t|\mu|)$ in \eqref{Kklaexpression}
is singular for $2t|\mu|\in {\mathbb {Z}}\setminus \{0\}$ and therefore we shall treat the operator $T_\lambda} \def\La{\Lambda^0$ separately from $T^k_\lambda} \def\La{\Lambda$ for $k>0$.
We shall see that $T^0_\lambda} \def\La{\Lambda$, and the operators $\sum_j \chi(2^{-2j}L) T^0_{2^j}$ can be handled using known results about Fourier integral operator, while the operators $T^k_{2^j}$ need a more careful treatment due to the singularities of the phase function.
We shall see that the decomposition into the operators $T^k_{2^j}$ encodes useful information on the singularities of the wave kernels.
In {\mathcal S}\ref{Tlaksection}, {\mathcal S}\ref{L1section}
we shall prove the following $L^1$ estimates
\begin{theorem} \label{TjL1thm}
(i) For $\lambda} \def\La{\Lambda\ge 2^{10}$
\begin{equation}\label{Tj0L1} \|T^0_\lambda} \def\La{\Lambda\|_{L^1\to L^1} \lesssim \lambda} \def\La{\Lambda^{(d-1)/2}\,.\end{equation}
(ii) For $\lambda} \def\La{\Lambda \ge 2^{10}$, $k=1,2,\dots$,
\begin{equation} \label{TlakL1}
\|T^k_\lambda} \def\La{\Lambda\|_{L^1\to L^1} \lesssim k^{-\frac{d_1+1}{2}}
\lambda} \def\La{\Lambda^{(d-1)/2}\,.
\end{equation}
\end{theorem}
Note that $d_1\ge 2$ and thus
the estimates \eqref{TlakL1} can be summed in $k$. Hence
Theorem \ref{main-theoremL1}
is an immediate consequence of Theorem \ref{TjL1thm}.
\subsection*{\it Dyadic decompositions}
For the Hardy space bounds we shall need to combine the dyadic pieces in $j$ and also refine the dyadic decomposition in \eqref{adydecomposition}.
Define
\begin{align}
\label{Vjdefinition}
V_j &= 2^{-j(d-1)/2} \chi_1(2^{-2j}L) T^0_{2^j}
\\
\label{Wjdefinition}
W_j&= 2^{-j(d-1)/2}\chi_1(2^{-2j} L) (m_{2^j}(L)- T^0_{2^j})
\end{align}
In section {\mathcal S}\ref{FIO} we shall use standard estimates on Fourier integral operators to prove
\begin{theorem}\label{Vtheorem}
The operator ${\mathcal {V}}=\sum_{j>100}V_j$ extends to a bounded operator from
$h^1_{\text{\rm iso}}$ to $L^1$.
\end{theorem}
We further decompose the pieces $W_j$ in \eqref{Wjdefinition} and let
\begin{equation}\label{Wjndefinition}
\begin{aligned}
W_{j,0} &= \zeta_0(2^{-j}|U|)\,
W_j
\\
W_{j,n} &= \zeta_1(2^{-j-n}|U|) W_j\,;
\end{aligned}
\end{equation}
here again $\zeta_0$, $\zeta_1$ as in {\mathcal S}\ref{cutoffsect}, i.e.
$\zeta_0$ supported
in $(-1,1)$, $\zeta_1$ supported in $\pm(1/2,2)$ so that
$\zeta_0+\sum_j \zeta_1(2^{1-j}\cdot)\equiv 1.$
By the description \eqref{jtspec} of the joint spectrum of $L$ and $|U|$
and the support property \eqref{chi1support}
we also have
$$\chi_1(2^{-2j} L) \zeta_1(2^{-j-n}|U|)= 0 \text{ when } 2^{2j+10} \le
2^{j+n-1}\,,
$$
i.e when $j\le n-11$ and thus
\begin{equation} \label{vanishingterms} W_{j,n}=0 \text{ when $n\ge j+11$ }.\end{equation}
Observe from \eqref{jtspec}, as in the discussion following \eqref{Tkla} that,
for $k=1,2,\dots$,
$$\zeta_0(2^{-j}\rho_2) \eta_0(\frac{\tau}{2^j}\rho_2-k\pi)= 0
\text {
for $\tau \in (\frac 1{16},4)$, $\rho_2\ge 0$, } \text{ if } 2^j\le (k-\tfrac 58)\pi 2^{j}/4\,,
$$
and \begin{multline*}
\zeta_1(2^{j-n}\rho_2)
\eta_0(\frac{\tau}{2^j}\rho_2-k\pi)= 0 \text {
for $\tau \in (\frac 1{16},4)$, $\rho_2\ge 0$, }
\\ \text{ if } 2^{j+n+1} \le 2^j(k-\tfrac 58)\pi/4 \text { or } 16\cdot 2^j (k+\tfrac 58)\pi \le 2^{j+n-1}\,.
\end{multline*}
Thus we have
for $k=1,2,\dots$,
\begin{align*}
&\zeta_0(2^{-j} |U|) T^k_{2^j} = 0 \text{ when } k\ge 2\,,
\\
&\zeta_1(2^{-j-n} |U|) T^k_{2^j} = 0 \text{ when } k\notin
[2^{n-8}, 2^{n+2}]\,.
\end{align*}
Let
\begin{equation}\label{jndef}
{\mathcal {J}}_n=\begin{cases} \{1\}, &n=0\,,\\
\{k: 2^{n-8}\le k\le 2^{n+2}\}, &n\ge 1\,.
\end{cases}
\end{equation}
Then by \eqref{kdecomposition} we have
$m_{2^j}(L)- T^0_{2^j}=\sum_{k=1}^\infty T^k_{2^j}$
and therefore we get
\begin{subequations}
\begin{align} \label{Wj0k}
W_{j,0} &=
2^{-j(d-1)/2} \chi_1(2^{-2j}L) \zeta_0(2^{-j}|U|)\sum_{k\in {\mathcal {J}}_0}
T^k_{2^j}\,,
\\
\label{Wjnk}
W_{j,n} &= 2^{-j(d-1)/2} \chi_1(2^{-2j}L) \zeta_1(2^{-j-n}|U|)
\sum_{k\in {\mathcal {J}}_n}T^k_{2^j}\,.
\end{align}
\end{subequations}
Observe that Theorem \ref{TjL1thm}
implies
\begin{equation}\label{Wjnest}
\|W_{j,n}\|_{L^1\to L^1} \lesssim 2^{-n(d_1-1)/2}
\end{equation}
uniformly in $j$.
Define
for $n=0,1,2,\dots$
\begin{equation} \label{Wnopdefinition}
{\mathcal {W}}_n =\sum_{j>100} W_{j,n}
\end{equation}
Theorem \ref{h1thm} will then be a consequence of Theorem \ref{Vtheorem}
and
\begin{theorem} \label{refinedh1thm}
The operators ${\mathcal {V}}$ and ${\mathcal {W}}_n$ are bounded from $h^1_{\text{\rm iso}}$ to $L^1$; moreover
\begin{equation}\label{Wnhardybd}
\|{\mathcal {W}}_n\|_{h^1_{\text{\rm iso}}\to L^1}\lesssim (1+n)2^{-n(d_1-1)/2}
\end{equation}
\end{theorem}
The proofs will be given in {\mathcal S}\ref{FIO} and {\mathcal S}\ref{hardyspaceestimates}.
\section{Fourier integral estimates}
\label{FIO}
In this section we shall reduce the proof of the estimates for $T^0_\lambda} \def\La{\Lambda$ and ${\mathcal {V}}$ in Theorems \ref{TjL1thm} and \ref{refinedh1thm} to standard bounds for Fourier integral operators in \cite{SSS} or \cite{beals}.
We will prove a preliminary lemma that allows us to add or suppress
$\chi_1(\lambda} \def\La{\Lambda^{-2} L)$ from the definition of $T_\lambda} \def\La{\Lambda^0$.
\begin{lemma} \label{Tla0error}
For $\lambda} \def\La{\Lambda>2^{10} $ we have
$$\|T^0_\lambda} \def\La{\Lambda -\chi_1(\lambda} \def\La{\Lambda^{-2}L) T^0_\lambda} \def\La{\Lambda\|_{L^1\to L^1}\lesssim C_N \lambda} \def\La{\Lambda^{-N}
$$ for any $N$.
\end{lemma}.
\begin{proof}
The operator $T^0_\lambda} \def\La{\Lambda -\chi_1(\lambda} \def\La{\Lambda^{-2}L) T^0_\lambda} \def\La{\Lambda$ can be written as
$b_\lambda} \def\La{\Lambda(|L|, |U|)$ where
$$b_\lambda} \def\La{\Lambda(\rho_1,\rho_2)=
\lambda} \def\La{\Lambda^{1/2} (1-\chi_1(\lambda} \def\La{\Lambda^{-2}\rho_1))
\lambda} \def\La{\Lambda^{1/2} \int a_\lambda} \def\La{\Lambda(\tau) e^{i\varphi(\tau,\rho_1,\lambda} \def\La{\Lambda)}
\eta_0(\tau\rho_2/\lambda} \def\La{\Lambda) d\tau
$$
with $$\varphi(\tau,\rho_1,\lambda} \def\La{\Lambda) = \frac{\lambda} \def\La{\Lambda}{4\tau}+\frac{\tau\rho_1}{\lambda} \def\La{\Lambda}\,.$$
Only the values of $\rho_1\le \lambda} \def\La{\Lambda^{2} 2^{-9}$ and $\rho_1\ge 2^{9}\lambda} \def\La{\Lambda^2$ are relevant.
Now $$\frac{\partial\varphi}{\partial \tau}= -\frac{\lambda} \def\La{\Lambda}{4\tau^2}+\frac{ \rho_1}\lambda} \def\La{\Lambda$$ and $(\partial/\partial\tau)^n \varphi= c_n \lambda} \def\La{\Lambda\tau^{-n-1}$ for $n\ge 2$.
Note that for $\rho_1\ge 2^9 \lambda} \def\La{\Lambda^2$ we have
$|\varphi'_\tau| \ge \rho_1/\lambda} \def\La{\Lambda - (16^2/4)\lambda} \def\La{\Lambda \ge
\rho_1\lambda} \def\La{\Lambda^{-1} (1- 2^{-9}2^6)\ge \rho_1/(2\lambda} \def\La{\Lambda)$.
Similarly for
$\rho_1\le 2^{-9} \lambda} \def\La{\Lambda^2$ we have
$|\varphi'_\tau| \ge \lambda} \def\La{\Lambda/16- 16 \cdot 2^{-9}\lambda} \def\La{\Lambda \ge 2^{-5}\lambda} \def\La{\Lambda$.
Use integrations by parts to conclude
that
$$
\Big| \frac{\partial^{n_1+n_2}[b_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^2 \cdot,\lambda} \def\La{\Lambda \cdot )]
}{(\partial\rho_1)^{n_1}(\partial\rho_2)^{n_2}}
(\rho_1,\rho_2)
\Big| \le C_{n_1,n_2,N}\lambda} \def\La{\Lambda^{-N}
$$
and in view of the compact support of
$ b_\lambda} \def\La{\Lambda(\lambda} \def\La{\Lambda^2 \rho_1,\lambda} \def\La{\Lambda \rho_2)$ the assertion can be deduced from
a result in \cite{MRS2}
(or alternatively from Hulanicki's result \cite{hulanicki} and a Fourier expansion in $\rho_2$).
\end{proof}
\subsection*{\it The convolution kernel for $T_\lambda} \def\La{\Lambda^{0}$} It is given by
\begin{multline*}
K^{0}_\lambda} \def\La{\Lambda(x,u) \,=\, \lambda} \def\La{\Lambda^{1/2} \int_{{\mathbb {R}}^{d_2}}
\int_{\mathbb {R}} e^{i\frac{\lambda} \def\La{\Lambda}{4s }}
a_\lambda} \def\La{\Lambda(s ) \eta_{0} (2\pi|\mu|\tfrac {s }{ \lambda} \def\La{\Lambda})\,\times
\\
\Big(\frac{|\mu|}{2\sin (2\pi|\mu|s /\lambda} \def\La{\Lambda)}\Big)^{d_1/2}
e^{-i|x|^2\frac {\pi}{ 2} |\mu|\cot (2\pi |\mu|s /\lambda} \def\La{\Lambda)} ds
\, e^{2\pi i \inn{u}{ \mu}} d\mu\,.
\end{multline*}
We introduce frequency variables $\theta= (\omega} \def\Omega{\Omega, \sigma)$ on the cone
\begin{equation}\label{Gammacone}\Gamma_\delta= \{\theta=(\omega} \def\Omega{\Omega, \sigma)\in {\mathbb {R}}^{d_2}\times {\mathbb {R}}: \, |\omega} \def\Omega{\Omega|\le ( \pi -\delta)\sigma, \,\,\sigma>0\},\end{equation}
Set
$$\omega} \def\Omega{\Omega=\frac{\pi \mu}2, \quad \sigma= \frac{\lambda} \def\La{\Lambda}{4s }.$$
Note that $\sigma \approx \lambda} \def\La{\Lambda$ for $s \in {\hbox{\roman supp}}(a_\lambda} \def\La{\Lambda)$.
We note that we will consider the case $\delta=\pi/4$ in view of the support of $\eta_0$ but any choice of $\delta\in (0,\pi/4)$ is permissible with some constants below depending on $\delta$.
If we set \begin{equation}\label{gdef}
g(\tau):= \tau \cot\tau,
\end{equation}
the above integral becomes
\begin{equation}\label{oscillintK00}
K^{0}_\lambda} \def\La{\Lambda(x,u) \,=\, \iint e^{i\Psi(x,u,\omega} \def\Omega{\Omega,\sigma)} \beta_\lambda} \def\La{\Lambda(\omega} \def\Omega{\Omega,\sigma) d\omega} \def\Omega{\Omega\,d\sigma
\end{equation}
with
$$\Psi(x,u,\omega} \def\Omega{\Omega,\sigma)=\sigma \big(1-|x|^2 g(|\omega} \def\Omega{\Omega|/\sigma )\big) +\inn {4u}\omega} \def\Omega{\Omega$$
and
$$
\beta_\lambda} \def\La{\Lambda(\omega} \def\Omega{\Omega,\sigma)=
4^{-1}\Big(\frac 2\pi\Big)^{\frac{d_1}{2}+d_2}
\lambda} \def\La{\Lambda^{3/2}
\sigma^{\frac {d_1}{2}-2}
a_\lambda} \def\La{\Lambda (\tfrac{\lambda} \def\La{\Lambda}{4\sigma}) \eta_{0} (\tfrac{|\omega} \def\Omega{\Omega|}{|\sigma})\Big(\frac{\tfrac{|\omega} \def\Omega{\Omega|}{\sigma|}}
{2\sin(\tfrac{|\omega} \def\Omega{\Omega|}{\sigma})}\Big)^{d_1/2} \,.
$$
The $\beta_{\lambda} \def\La{\Lambda}$ are symbols of order $\frac{d_1-1}{2}$
uniformly in $\lambda} \def\La{\Lambda$, and supported in $\Gamma$. The same applies to
$\sum_{k>10} \beta_{2^k}$.
We will need formulas for the derivatives of $\Psi$ with respect to the frequency variables
$\theta=(\omega} \def\Omega{\Omega,\sigma)$:
\begin{equation}
\label{Psinuder}
\begin{aligned}
&\frac{\partial \Psi}{\partial \omega} \def\Omega{\Omega_i} = 4u_i -|x|^2 \frac{\omega} \def\Omega{\Omega_i}{\sigma} \frac{g'(\tfrac{|\omega} \def\Omega{\Omega|}{\sigma}}{\tfrac{|\omega} \def\Omega{\Omega|}{\sigma}}
\\
&\frac{\partial \Psi}{\partial \sigma} =1-|x|^2\big( g(\tfrac{|\omega} \def\Omega{\Omega|}{\sigma}) - \tfrac{|\omega} \def\Omega{\Omega|}{\sigma} g'(\tfrac{|\omega} \def\Omega{\Omega|}{\sigma})\big)
\end{aligned}
\end{equation}
Now $g$ is analytic for $|\tau|<2\pi$ and we have
\begin{subequations}
\begin{align}
\label{gfirstderiv}
g'(\tau)&= \frac{\sin(2\tau)-2\tau}{2\sin^2\tau}
\\
\label{gfsecderiv}
g''(\tau)&=\frac{2(\tau \cos \tau - \sin \tau)}{\sin^3 \tau}
\end{align}
\end{subequations}
Observe that $$g'(\tau)<0 \text { and } g''(\tau)<0 \text{ for $0<\tau<\pi$.}$$
Moreover as $\tau \to 0$,
$$g(\tau)=1-\tau^2/3+O(\tau^4)$$
and hence $g'(0)=0$ and $g''(0)=-2/3$.
The even expression $$g(\tau)-\tau g'(\tau)= 1+ \int_0^\tau (-sg''(s)) ds $$ will frequently occur; from the above we get
\begin{equation}\label{g-tg'}\begin{aligned}
g(\tau)-\tau g'(\tau)\ge 1, \text{ for } 0\le |\tau|<\pi\,,
\\
|g(\tau)-\tau g'(\tau)|\le 10, \text{ for } 0\le |\tau|<3\pi/4\,.
\end{aligned}
\end{equation}
\begin{lemma} \label{decaylemmalargexu} We have
\begin{equation}\label{largexu}|K_{\lambda} \def\La{\Lambda}^{0}(x,u)| \lesssim
\lambda} \def\La{\Lambda^{\frac{d_1+2d_2+1}{2}-N}
(|x|^2+|u|)^{-N},
\text{ $|x|^2+4|u|>2$.}
\end{equation}
and
\begin{equation}\label{xsmall}|K_{\lambda} \def\La{\Lambda}^{0}(x,u)| \lesssim \lambda} \def\La{\Lambda^{\frac{d_1+2d_2+1}{2}-N}
(1+|u|)^{-N} , \text{ $|x|^2\le 1/20.$}\end{equation}
\end{lemma}
\begin{proof} If $|x|\ge \sqrt 2$ we may integrate by parts with respect to $\sigma$ (using \eqref{g-tg'}), and obtain
$$|K_{\lambda} \def\La{\Lambda}^0(x,u)|\lesssim_N \lambda} \def\La{\Lambda^{\frac{d_1+2d_2+1}{2}-N} |x|^{-N}, \quad |x|\ge \sqrt 2\,.
$$
If $|u|\le 10|x|^2$ this also yields \eqref{largexu}.
Since $\max_{|\tau|\le 3\pi/4}|g'(\tau)|\le 3\pi/2$ we have
$|\nabla_\omega} \def\Omega{\Omega \Psi|\ge 4|u|- (3\pi/2)|x|^2$ and hence $|\nabla_\omega} \def\Omega{\Omega \Psi|\ge |u|$ when
$|u|\ge 10|x|^2$.
Thus integration by parts in $\omega$ yields
$$|K_{\lambda} \def\La{\Lambda}^0(x,u)|\lesssim_N \lambda} \def\La{\Lambda^{\frac{d_1+2d_2+1}{2}-N} |u|^{-N},
\quad \text{$|u|\ge 10|x|^2$.}
$$
This proves
\eqref{largexu}.
Since $|g'(\tau)|\le 3\pi$ for $|\tau|\le 3\pi/2$
we have $|\nabla_\omega} \def\Omega{\Omega \Psi|\ge 2|u|$ if $|x|^2 \le 2|u|/3\pi$ and
$|\Psi_\sigma |\ge 1/2$ if $|x|^2\le 1/20$.
Integrations by parts imply \eqref{xsmall}.
\end{proof}
\subsection*{\it Fourier integral operators}
Let $\rho\ll 10^{-2}$. Let $\chi\in C^\infty_c({\mathbb {R}}^d\times {\mathbb {R}}^d)$ so that
$$\chi(x,u,y,v)=0 \text{ for } \begin{cases}
|y|+|v|\ge \rho,\\ |x-y|<1/20, \\ |x-y|^2+ |u-v| \ge 4 .\end{cases}
$$
Let
$$b_\lambda} \def\La{\Lambda(x,y,u,v,\omega} \def\Omega{\Omega,\sigma)= \chi(x,u,y,v) \beta_\lambda} \def\La{\Lambda(\omega} \def\Omega{\Omega,\sigma),$$
let as before $g(\tau)= \tau\cot\tau$, and let
\begin{equation}\label{Phidef}\begin{aligned}
\Phi(x,u,&y,v,\omega} \def\Omega{\Omega,\sigma)\,=\, \Psi(x-y,u-v+
\tfrac 12 \inn {\vec Jx}{y},
\omega} \def\Omega{\Omega,\sigma)\\
&=\sigma \big(1-|x-y|^2 g(|\omega} \def\Omega{\Omega|/\sigma )\big)
+\sum_{i=1}^{d_2}
(4u_i-4v_i-2 x^\intercal J_i y)\omega} \def\Omega{\Omega_i\,.
\end{aligned}
\end{equation}
Let ${\mathfrak {F}}_\lambda} \def\La{\Lambda$ be the Fourier integral operator with Schwartz kernel
\begin{equation} \label{Fourierint}{\mathcal {K}}_\lambda} \def\La{\Lambda(x,u,y,v)=
\iint e^{i\Phi(x,u,y, v, \omega} \def\Omega{\Omega,\sigma)} b_\lambda} \def\La{\Lambda(\omega} \def\Omega{\Omega,\sigma) d\omega} \def\Omega{\Omega\,d\sigma.
\end{equation}
Given Lemma \ref{decaylemmalargexu} it suffices to prove the inequalities
\begin{equation}\label{L1Fla}
\| {\mathfrak {F}}_\lambda} \def\La{\Lambda\|_{L^1\to L^1} \le \lambda} \def\La{\Lambda^{\frac{d-1}{2}}.
\end{equation}
and
\begin{equation}\label{h1F}
\Big\|\sum_{k>C} 2^{-k(d-1)/2} {\mathfrak {F}}_{2^k} \Big\|_{h^1\to L^1} <\infty .
\end{equation}
To this end we apply results in \cite{SSS} on Fourier integral operators associated with canonical graphs and now check the required hypotheses.
\subsection*{\it Analysis of the phase function $\Phi$}
We compute the first derivatives:
\begin{align*}
\Phi_{x_j}&= -2\sigma (x_j-y_j) g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) -2 \sum_{i=1}^{d_2} \omega} \def\Omega{\Omega_i e_j^\intercal J_i y
\\
\Phi_{u_i}&=4\omega} \def\Omega{\Omega_i
\\
\Phi_{\omega} \def\Omega{\Omega_i}&=-|x-y|^2 g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \tfrac{\omega} \def\Omega{\Omega_i}{|\omega} \def\Omega{\Omega|}
+4u_i-4v_i-2 x^\intercal J_i y
\\
\Phi_{\sigma}&=\big(1-|x-y|^2 g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma} )\big) + |x-y|^2 {\tfrac{|\omega} \def\Omega{\Omega|}\sigma} g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\end{align*}
For the second derivatives we have, with $\delta_{jk}$ denoting the Kronecker delta and $J^\omega} \def\Omega{\Omega=\sum_{i=1}^{d_2} \omega} \def\Omega{\Omega_i J_i$
$$ \begin{aligned}
\Phi_{x_jy_k}&= 2\sigma g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})\delta_{jk} - 2 e_j^\intercal J^\omega} \def\Omega{\Omega e_k\,,
\\ \Phi_{x_j v_l}&=0\,,
\\
\Phi_{x_j\omega} \def\Omega{\Omega_l} &=-2(x_j-y_j)g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \tfrac{\omega} \def\Omega{\Omega_l}{|\omega} \def\Omega{\Omega|} - 2 e_j^\intercal J_l y \,,
\\
\Phi_{x_j\sigma} &=2(x_j-y_j) \big(- g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})+ {\tfrac{|\omega} \def\Omega{\Omega|}\sigma} g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})\big)\,,
\end{aligned}
$$ and
$$\Phi_{u_iy_k}=0\,, \quad
\Phi_{u_i v_l}=0\,,\quad \Phi_{u_i \omega} \def\Omega{\Omega_l}= 4\delta_{il}\,, \quad
\Phi_{u_i\sigma}=0\,.
$$
Moreover
$$
\begin{aligned}
&\Phi_{\omega} \def\Omega{\Omega_i y_k}=2(x_k-y_k) g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \tfrac{\omega} \def\Omega{\Omega_i}{|\omega} \def\Omega{\Omega|}- 2 x^\intercal J_ie_k
\\
&\Phi_{\omega} \def\Omega{\Omega_iv_l}=-4 \delta_{il}
\\
&\Phi_{\omega} \def\Omega{\Omega_i\omega} \def\Omega{\Omega_l}= -|x-y|^2 \big( g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\tfrac{\delta_{il}|\omega} \def\Omega{\Omega|^2-\omega} \def\Omega{\Omega_i\omega} \def\Omega{\Omega_l}{|\omega} \def\Omega{\Omega|^3} + g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \tfrac{\omega} \def\Omega{\Omega_i\omega} \def\Omega{\Omega_l}{\sigma|\omega} \def\Omega{\Omega|^2} \big)
\\
&\Phi_{\omega} \def\Omega{\Omega_i\sigma}= |x-y|^2 \tfrac{\omega} \def\Omega{\Omega_i}{\sigma^2} g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\end{aligned}$$ and
$$
\begin{aligned}
&\Phi_{\sigma y_k}=2(x_k-y_k) \big( g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})-{\tfrac{|\omega} \def\Omega{\Omega|}\sigma} g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \big)
\\
&\Phi_{\sigma v_l}=0
\\
&\Phi_{\sigma \omega} \def\Omega{\Omega_l}=|x-y|^2 \tfrac{\omega} \def\Omega{\Omega_l}{\sigma^2}g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\\
&\Phi_{\sigma\sigma}=-|x-y|^2 \tfrac{|\omega} \def\Omega{\Omega|^2}{\sigma^3} g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\end{aligned}
$$
The required
$L^2$ boundedness properties follow if we can show that
associated canonical relation is locally the graph of a canonical transformation;
this follows from the invertibility of the matrix
\begin{equation}\label{PhimixedHess}
\begin{pmatrix}
\Phi_{xy}&\Phi_{xv}&\Phi_{x\omega} \def\Omega{\Omega}&\Phi_{x\sigma}
\\
\Phi_{uy}&\Phi_{uv}&\Phi_{u\omega} \def\Omega{\Omega}&\Phi_{u\sigma}
\\
\Phi_{\omega} \def\Omega{\Omega y}&\Phi_{\omega} \def\Omega{\Omega v}&\Phi_{\omega} \def\Omega{\Omega\om}&\Phi_{\omega} \def\Omega{\Omega\sigma}
\\
\Phi_{\sigma y}&\Phi_{\sigma v}&\Phi_{\sigma \omega} \def\Omega{\Omega}&\Phi_{\sigma\sigma}
\end{pmatrix}\,,
\end{equation}
see \cite{hoermander-fio}.
This matrix is given by
$$
\begin{pmatrix}
2\sigma g I_{d_1}-2 J^\omega} \def\Omega{\Omega &0& (*)_{13} & 2(x-y) (\tau g'-g)
\\
0&0&4I_{d_2}&0
\\
(*)_{31} & -4I_{d_2} &(*)_{33} &(*)_{34}
\\
2(x-y)^\intercal (g-\tau g')&0 &(*)_{43} &-|x-y|^2\sigma^{-1} \tau^2g''
\end{pmatrix}\,,
$$
where
$\tau={\tfrac{|\omega} \def\Omega{\Omega|}\sigma}$, $g, g', g''$ are evaluated at $\tau={\tfrac{|\omega} \def\Omega{\Omega|}\sigma}$, and
$x-y$ is considered a $d_1\times 1$ matrix,
$ (*)_{13}$ is a $d_1\times d_2$-matrix,
$ (*)_{31}$ is a $d_2\times d_1$-matrix,
$ (*)_{33}$ is a $d_2\times d_2$-matrix,
$ (*)_{34}$ is a $d_2\times 1$-matrix, and
$ (*)_{43}=(*)_{34}^\intercal$.
The determinant $D$
of the displayed matrix is equal to
\begin{equation}\label{detD}D=16^{d_2} \det
\begin{pmatrix}
2\sigma g I_{d_1}-2J^\omega} \def\Omega{\Omega & 2(x-y) (\tau g'-g)
\\
2(x-y)^\intercal (g-\tau g')& -|x-y|^2 \sigma^{-1}\tau^2g''
\end{pmatrix}\,.
\end{equation}
To compute this we use the formula
$$
\begin{pmatrix}
I&0\\a^\intercal &1
\end{pmatrix}
\begin{pmatrix} A&-b\\b^\intercal &\gamma
\end{pmatrix}
\begin{pmatrix} I&-a\\0&1
\end{pmatrix}
\,=\,
\begin{pmatrix}A&
-Aa-b
\\a^\intercal A+b^\intercal & -a^\intercal A a- 2a^\intercal b+\gamma
\end{pmatrix}\,.
$$
If $A$ is invertible we can choose
$a= - A^{-1} b$.
Since $b^\intercal S b=0$ for the skew symmetric matrix $S= (A^{-1})^\intercal -A^{-1}$
this choice of $a$ yields the matrix
$$\begin{pmatrix}
A&0\\-b^\intercal (A^{-1})^\intercal A +b^\intercal &-b^\intercal (A^{-1})^\intercal b-2b^\intercal A^{-1} b +\gamma
\end{pmatrix}=\begin{pmatrix} A&0\\ * & \gamma +b^\intercal A^{-1} b\end{pmatrix}
$$
and
hence
\begin{equation} \label{determinantformula}
\det \begin{pmatrix} A&-b\\b^\intercal &\gamma\end{pmatrix} = ( \gamma+ b^\intercal A^{-1} b) \det (A)\,.
\end{equation}
\begin{lemma}\label{skewsymnew} Let $c,\Lambda\in {\mathbb {R}}$, $c^2+\La^2\neq 0$. Let $S$ be a skew symmetric $d_1\times d_1$- matrix satisfying
$S^2=-\La^2 I$. Then $cI+S$ is invertible with
$$(cI+S)^{-1}= \frac{c}{c^2+\Lambda^2} I-\frac{1}{c^2+\Lambda^2} S,$$
and $\det (cI+S)=(c^2+\Lambda^2)^{\frac {d_1}2}.$
\end{lemma}
\begin{proof} $(cI+S)(cI+S)^*=(cI+S)(cI-S)=c^2 I-S^2= (c^2+\Lambda^2)I$.
\end{proof}
In our situation \eqref{detD} we have $A=cI+S,$ with
$$\begin{aligned}
c&=2\sigma g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}), \\ S&=-2J^\omega} \def\Omega{\Omega\,,\end{aligned}$$
moreover,
$$\begin{aligned}
\Lambda&=2|\omega} \def\Omega{\Omega|,\\
\gamma&=-|x-y|^2\sigma^{-1} (\tfrac{|\omega} \def\Omega{\Omega|}{\sigma})^2 g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}),
\\b&=2(x-y) (g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})-{\tfrac{|\omega} \def\Omega{\Omega|}\sigma} g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}))\,.\end{aligned}
$$
In particular, if we recall that $\tau=|\omega} \def\Omega{\Omega|/\sigma,$ we see that
$$\det A=\big((2\sigma} \def{\Sigma}{\Sigma g(\tau))^2+(2|\omega} \def\Omega{\Omega|)^2\big)^{\frac{d_1}2}=(2\sigma)^{d_1}\Big(\frac \tau {\sin \tau}\Big)^{d_1}.
$$
Moreover,
\begin{align*}&\gamma+ b^\intercal A^{-1} b\\
&=|x-y|^2\Big( - \frac{|\omega} \def\Omega{\Omega|^2}{\sigma^3} g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
+ 4\big(g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})-{\tfrac{|\omega} \def\Omega{\Omega|}\sigma} g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})\big)^2 \frac{ 2\sigma g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})}
{4\sigma^2 g({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})^2 + 4|\omega} \def\Omega{\Omega|^2}\Big)\\
&=\frac{|x-y|^2}{\sigma} \Big(-\tau^2 g''(\tau) +2(g(\tau)-\tau g'(\tau))^2 \frac{g(\tau)}{g(\tau)^2+\tau^2} \Big).
\end{align*}
From \eqref{gfirstderiv}, we get $$g(\tau)-\tau g'(\tau)=\Big(\frac
\tau{\sin\tau}\Big)^2,$$ and in combination with
\eqref{gfsecderiv} this implies after a calculation that
$$
\gamma+ b^\intercal A^{-1} b=\frac{|x-y|^2}{\sigma}2\Big(\frac \tau {\sin \tau}\Big)^2.
$$
Thus we see from \eqref{determinantformula} that the determinant of the matrix \eqref{PhimixedHess} is given by
\begin{equation}\label{ detHessian}
D=2^{d_1+4d_2+1} \sigma^{d_1-1}\Big(\frac {\tfrac{|\omega} \def\Omega{\Omega|}\sigma} {\sin {\tfrac{|\omega} \def\Omega{\Omega|}\sigma}}\Big)^{d_1+2}.
\end{equation}
This shows that $D>0$ for ${\tfrac{|\omega} \def\Omega{\Omega|}\sigma}\in [0,\pi),$ and $D\sim \sigma^{d_1-1}$ for ${\tfrac{|\omega} \def\Omega{\Omega|}\sigma}\in [0,\pi-{\delta}],$
for every sufficiently small ${\delta}>0.$ In particular, the matrix \eqref{PhimixedHess} is invertible for
${\tfrac{|\omega} \def\Omega{\Omega|}\sigma}\in [0,\pi-{\delta}].$
\bigskip
\medskip
We now write $${\mathfrak {F}}_\lambda} \def\La{\Lambda f(x)= \int K_\lambda} \def\La{\Lambda(x,y) f(y) dy$$ where $K_\lambda} \def\La{\Lambda$ is given by our oscillatory integral representation \eqref{Fourierint}. In that formula
we have $d_2+1$ frequency variables $d_2+1$, and thus, given any $\alpha\in {\mathbb {R}}$ the operator
convolution with $\sum_{k>C} {\mathfrak {F}}_{2^k} 2^{-k\alpha}$ is a Fourier integral operator of
order $$\frac{d_1-1}{2}-\alpha- \frac{d-(d_2+1)}{2}=-\alpha\,.$$
With these observations we can now apply the boundedness result of \cite{SSS}
and deduce that
$$ \|{\mathfrak {F}}_\lambda} \def\La{\Lambda f\|_1\lesssim \lambda} \def\La{\Lambda^{\frac{d-1}{2} }\|f\|_1$$ and
$$ \Big\|\sum_{k>C} 2^{-k(d-1)/2} {\mathcal {F}}_{2^k} f_\rho\Big \|_1 \lesssim 1$$
for standard $h_1$ atoms supported in $B_\rho$.
But atoms associated to balls centered at the origin are also atoms in our Heisenberg Hardy space
$h^1_{\text{\rm iso}}$. Thus if we also take into account Lemma
\ref{decaylemmalargexu} and use translation invariance under Heisenberg translations we get
$$\Big\|\sum_{k\ge 0} T^0_{2^k} f \Big \|_1\lesssim \|f\|_{h^1_{\text{\rm iso}}}.$$
{\it Remark.} We also have
\begin{multline*}
\begin{pmatrix}
\Phi_{\omega} \def\Omega{\Omega\om}&\Phi_{\omega} \def\Omega{\Omega\sigma}\\ \Phi_{\sigma\omega} \def\Omega{\Omega}&\Phi _{\sigma\sigma}
\end{pmatrix}
\\=|x-y|^2
\begin{pmatrix}
-\big(g'({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})\frac{I_{d_2}|\omega} \def\Omega{\Omega|^2-\omega} \def\Omega{\Omega\om^\intercal}{|\omega} \def\Omega{\Omega|^3}
+g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma}) \frac{\omega} \def\Omega{\Omega\om^\intercal}{\sigma|\omega} \def\Omega{\Omega|^{2}}\big)
& \tfrac{\omega} \def\Omega{\Omega}{\sigma^2} g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\\
\tfrac{\omega} \def\Omega{\Omega^\intercal}{\sigma^2}
g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})& -
\tfrac{|\omega} \def\Omega{\Omega|^2}{\sigma^3} g''({\tfrac{|\omega} \def\Omega{\Omega|}\sigma})
\end{pmatrix} \,
\end{multline*}
which has maximal rank $d_2+1-1=d_2$.
Thus the above result can also be deduced from Beals \cite{beals},
via the equivalence of phase functions theorem.
\section{The operators $T_\lambda} \def\La{\Lambda^k$}
\label{Tlaksection}
We now consider the operator $T^\lambda} \def\La{\Lambda_k$, for $k\ge 1$, as defined in \eqref{Tkla}.
In view of the singularities of $\cot$
we need a further decomposition in terms of the
distance to the singularities.
For $l=1,2,\dots$ let
$\eta_l(s)=\eta_0(2^{l-1} s)-\eta_0(2^l s)$ so that $$\eta_0(s)=
\sum_{l=1}^\infty \eta_l(s)\text{ for $s\neq 0$.}$$
Define
\begin{equation}
\label{Tlakdef}
T^{k,l}_\lambda} \def\La{\Lambda =\lambda} \def\La{\Lambda^{1/2}\int e^{i\frac{\lambda} \def\La{\Lambda}{4\tau}}
a_\lambda} \def\La{\Lambda(\tau) \eta_l (\tfrac {\tau}{ \lambda} \def\La{\Lambda}|U|- k\pi)
e^{i\tau L/\lambda} \def\La{\Lambda}
d\tau;
\end{equation}
then
\begin{equation}\label{decompTlak}
T^k_\lambda} \def\La{\Lambda =\sum_{l=1}^\infty
T^{k,l}_\lambda} \def\La{\Lambda\,.
\end{equation}
From the formula \eqref{Kklaexpression} for the kernels $K_\lambda} \def\La{\Lambda^k$ we get
a corresponding formula for the kernels $K^{k,l}_\lambda} \def\La{\Lambda$, namely
\begin{multline*}
K^{k,l}_\lambda} \def\La{\Lambda(x,u) \,=\, \lambda} \def\La{\Lambda^{1/2} \int_{{\mathbb {R}}^{d_2}}
\int_{\mathbb {R}} e^{i\frac{\lambda} \def\La{\Lambda}{4\tau}}
a_\lambda} \def\La{\Lambda(\tau) \eta_l (2\pi|\mu|\tfrac {\tau}{ \lambda} \def\La{\Lambda}- k\pi)\,\times
\\
\Big(\frac{|\mu|}{2\sin (2\pi|\mu|\tau/\lambda} \def\La{\Lambda)}\Big)^{d_1/2}
e^{-i|x|^2\frac {\pi}{ 2} |\mu|\cot (2\pi |\mu|\tau/\lambda} \def\La{\Lambda)} d\tau
\, e^{2\pi i \inn{u}{ \mu}} d\mu\,.
\end{multline*}
Now we use polar coordinates in ${\mathbb {R}}^{d_2}$ and the fact that the
Fourier transform of the surface carried measure on the unit sphere in
${\mathbb {R}}^{d_2}$ is given by $$(2\pi)^{d_2/2}{\mathscr{J}\ci{d_2}\!}(2\pi|u|), \text{ with }
{\mathscr{J}\ci{d_2}\!}(\sigma):=\sigma^{-\frac{d_2-2}2} {J}_{\frac{d_2-2}{2}} (\sigma)$$
(the standard Bessel function formula, {\it cf}. \cite{stw}, p.154).
Thus
\begin{multline*}
K^{k,l}_\lambda} \def\La{\Lambda(x,u) \,=\,
\lambda} \def\La{\Lambda^{1/2} \int_{0} ^\infty
\int_{\mathbb {R}} e^{i\frac{\lambda} \def\La{\Lambda}{4\tau}}
a_\lambda} \def\La{\Lambda(\tau) \eta_l (2\pi\tau \rho / \lambda} \def\La{\Lambda- k\pi)\,\times
\\
\Big(\frac{\rho}{2\sin (2\pi\tau \rho/\lambda} \def\La{\Lambda)}\Big)^{d_1/2}
e^{-i\frac {\pi}{ 2}|x|^2
\rho\cot (2\pi \rho\tau/\lambda} \def\La{\Lambda)} \,d\tau\,(2\pi)^{d_2/2} {\mathscr{J}\ci{d_2}\!} (2\pi
\rho |u|)\,\rho^{d_2-1} d\rho\,.
\end{multline*}
In this integral we introduce new variables
\begin{equation}\label{stvariables}
(s,t)= \Big( \frac{1}{4\tau}, \frac{2\pi \tau \rho}{\lambda} \def\La{\Lambda}\Big),
\end{equation}
so that $(\tau,\rho)=((4s)^{-1}, 2\lambda} \def\La{\Lambda ts/\pi)$ with $d\tau d\rho= \lambda} \def\La{\Lambda(2\pi s)^{-1} ds dt$. Then we obtain for $k\ge 1$
\begin{multline} \label{Klaklrep}
K^{k,l}_\lambda} \def\La{\Lambda(x,u) = \lambda} \def\La{\Lambda^{d_2+\frac{d_1+1}2}
\quad \times \\\iint
\beta_\lambda} \def\La{\Lambda(s)
\eta_l(t-k\pi)
\Big(\frac{t}{\sin t}\Big)^{d_1/2} t^{d_2-1}
e^{i\lambda} \def\La{\Lambda s \psi(t,|x|) } {\mathscr{J}\ci{d_2}\!} (4s \lambda} \def\La{\Lambda t|u| )
\, ds \, dt
\end{multline}
where
\begin{equation}\label{psiphase}
\psi(t,r)= 1- r^2 t\cot t
\end{equation}
and \begin{equation}\label{betala}\beta_\lambda} \def\La{\Lambda(s)= 2^{\frac{3d_2}2-2}\pi^{-\frac{d_1+d_2}{2}}
a_\lambda} \def\La{\Lambda(\tfrac 1{4s}) \,s^{\frac{d_1}{2}+d_2-2};\end{equation}
thus $\beta_\lambda} \def\La{\Lambda$ is $C^\infty$ with bounds uniform in $\lambda} \def\La{\Lambda$, and $\beta_\lambda} \def\La{\Lambda$ is also supported in $[1/16,4]$.
In the next two sections we shall prove
the
$L^1$ estimates
\begin{equation}\label{L1boundsKlakl}
\sum_{k<8\lambda} \def\La{\Lambda}\sum_{l=0}^{\infty}\iint \lambda} \def\La{\Lambda^{-\frac{d-1}{2}} |K^{k,l}_\lambda} \def\La{\Lambda(x,u)| \, dx\, du = O(1)
\end{equation}
and
Theorem \ref{TjL1thm} and then also
Theorem \ref{main-theoremL1} will follow by summing the pieces.
Moreover we shall give some
refined estimates which will be used in the proof of Theorem
\ref{refinedh1thm}.
\subsection{\it An $L^\infty$ bound for the kernels}
The expression
\begin{equation}
\label{Clakl}
{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}
= \lambda} \def\La{\Lambda^{1+\frac{d_2}{2}} k^{d_2-1}(2^{l}k)^{\frac{d_1}{2}}
\end{equation}
will frequently appear in pointwise estimates,
namely as upper bounds for the integrand in the integral defining
$\lambda} \def\La{\Lambda^{-\frac{d-1}{2}}K^{k,l}_\lambda} \def\La{\Lambda$.
Note that
\begin{equation} \label{Claklinfty}
\|\lambda} \def\La{\Lambda^{-\frac{d-1}{2}}K^{k,l}_\lambda} \def\La{\Lambda\|_\infty
\lesssim 2^{-l}{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}\,;
\end{equation}
the additional factor of $2^{-l}$ occurs since the
integration in $t$ is over the union of two intervals of length $\approx 2^{-l}$.
\subsection{\it Formulas for the phase functions}\label{reductions}
For later reference we gather some formulas for the $t$-derivatives
of the phase $\psi(t,r)=1-r^2 t\cot t$:
\begin{subequations}
\begin{align}
\psi_t(t,r)&= r^2 \Big(\frac{t}{\sin^2t}-\cott\Big)
\label{psit}
\\&= r^2\Big(\frac{2t-\sin(2t)}{2\sin^2 t}\Big);
\label{psitalt}
\end{align}
\end{subequations}
moreover
\begin{align}
\psi_{t\mut}(t,r)&= \frac{2r^2}{\sin^3t}
\big(\sin t-t \cos t\big)\,=\, \frac{2r^2}{\sin^3t}\int_0^t\tau\sin\tau\,d\tau.
\label{psitt}
\end{align}
Observe that $\psi_{tt}=0$ when $\tan t=t$ and $t\neq 0$ and thus
$\psi_{tt}(t,r)\approx r^2$ for $0\le t\le \tfrac {3\pi}{4}$,
namely, we use $\frac{2\sqrt 2}{3\pi}t\le\sin t\le t$ to get the crude estimate
\begin{equation} \label{psittequiv}
\pi^{-1} r^2< \psi_{tt}(t,r)< \pi^3 r^2, \quad 0< t\le \tfrac {3\pi}{4}\,.
\end{equation}
It is also straightforward to establish estimates for
the higher derivatives:
\begin{equation} \label{higherpsidersmallt}
|\partial_t^{n} \psi(t,r)|\lesssim r^2, \qquad |t|\le 3\pi/4
\end{equation}
and
\begin{equation} \label{higherpsider}
\partial_t^{n} \psi(t,r)
=O\Big(\frac{r^2 |t|}{|\sin t|^{n+1}}\Big), \quad
\end{equation}
for all $t$.
\medskip
\subsection{\it Asymptotics in the main case $|u|\gg (k\lambda} \def\La{\Lambda)^{-1}$}
We shall see in the next section that there are straightforward $L^1$ bounds
in the region where $|u|\lesssim (k+1)^{-1}\lambda} \def\La{\Lambda^{-1}$. We therefore concentrate on
the region
$$\{(x,u):|u|\ge C(k+1)^{-1}\lambda} \def\La{\Lambda^{-1}\}$$ where
we have to take into account the oscillation
of the terms
${\mathscr{J}\ci{d_2}\!} (4s \lambda} \def\La{\Lambda t|u| )$.
The standard asymptotics for Bessel functions imply that for
\begin{equation} {\mathscr{J}\ci{d_2}\!}(\sigma) = e^{- i|\sigma|}{\varpi}_1(|\sigma|)+
e^{i|\sigma|}{\varpi}_2(|\sigma|),\qquad |\sigma|\ge 2,
\label{bessel}
\end{equation}
where ${\varpi}_1, {\varpi}_2\in S^{-(d_2-1)/2}$
are supported in $\mathbb R\setminus [-1,1]$.
Thus we may split, for $|u|\gg(k+1)^{-1}\lambda} \def\La{\Lambda^{-1}$,
\begin{equation}
\label{ABsplitkl} \lambda} \def\La{\Lambda^{-\frac{d-1}2} K^{k,l}_\lambda} \def\La{\Lambda(x,u) = A_\lambda} \def\La{\Lambda^{k,l}(x,u)+ B^{k,l}_\lambda} \def\La{\Lambda
(x,u)\end{equation}
where, with ${\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}$ defined in \eqref{Clakl},
\begin{equation}\label{Adefin}
A_\lambda} \def\La{\Lambda^{k,l} (x,u) = {\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} \iint \eta_{\lambda} \def\La{\Lambda,k,l}(s,t)
e^{i\lambda} \def\La{\Lambda s(\psi(t, |x|)- 4t|u|)}
{\varpi}_1(4\lambda} \def\La{\Lambda st|u|)\, dt \,ds\,,
\end{equation}
and
\begin{equation}\label{Bdefin}
B_\lambda} \def\La{\Lambda^{k,l} (x,u) = {\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} \iint \eta_{\lambda} \def\La{\Lambda,k,l}(s,t)
e^{i\lambda} \def\La{\Lambda s(\psi(t, |x|)+4t|u|)}
{\varpi}_2(4\lambda} \def\La{\Lambda st|u|)\, dt \,ds\,;
\end{equation}
here, as before
$\psi(t,r)=1-r^2 t\cot t$ and,
with $\beta_\lambda} \def\La{\Lambda$ as in \eqref{betala},
\begin{subequations}
\label{etadefinitions}
\begin{align}
\label{etadefinitionzero}
\eta_{\lambda} \def\La{\Lambda,0}(s,t)&=\beta_\lambda} \def\La{\Lambda(s)\eta_0(t)
\big(\frac {t}{\sin t}\big)^{d_1/2} t^{d_2-1}\,,
\\
\label{etadefinitionkl}
\eta_{\lambda} \def\La{\Lambda,k,l}(s,t)&= \beta_\lambda} \def\La{\Lambda(s) \eta_l(t-k\pi)
\big(\frac {t/k}{2^l\sin t}\big)^{d_1/2}
(t/k)^{d_2-1}\,.
\end{align}
\end{subequations}
Note that $\|\partial_s^{N_1}\partial_t^{N_2} \eta_{\lambda} \def\La{\Lambda,k,l}\|_\infty \le C_{N_1,N_2} 2^{lN_2}$. Moreover if
\begin{equation} \label{Jkl}
J_{k,l}:=
(k\pi- 2^{-l}\tfrac{5\pi}4, k\pi-2^{-l}\tfrac {3\pi}8]
\cup
[k\pi+ 2^{-l}\tfrac{3\pi}8, k\pi+2^{-l}\tfrac {5\pi}4)
\end{equation}
then
\begin{equation}\label{supportetakl}\eta_{\lambda} \def\La{\Lambda,k,l}(s,t)\neq 0 \implies t\in J_{k,l}\,.
\end{equation}
The main contribution in our estimates comes from the kernels
$A_{\lambda} \def\La{\Lambda}^{k,l}$ while the kernels
$B_{\lambda} \def\La{\Lambda}^{k,l}$ are negligible terms with rather small $L^1$ norm.
The latter will follow from the support properties of $\eta_{\lambda} \def\La{\Lambda,k,l}$ and the observation
that $$\partial_t (\psi(t,|x|)+4t|u|)\neq 0, \quad (x,u)\neq (0,0);$$
{\it cf}. \eqref{psitalt}.
As a consequence
only the kernels $A_{\lambda} \def\La{\Lambda}^{k,l}$ will exhibit the singularities of the kernel away from the origin.
\subsection{\it The phase functions and the singular support}\label{phase-singsupp}
We introduce polar coordinates in ${\mathbb {R}}^{d_1}$ and ${\mathbb {R}}^{d_2}$
(scaled by a factor of $4$ in the latter) and set
$$r=|x|\,,\qquad v= 4|u|\,.$$
We define for {\it all} $v\in {\mathbb {R}}$,
\begin{equation}\label{defphi}
\phi(t,r,v)\,:=\,\psi(t,r)-tv\,=\,1-r^2t\cot t -t v,\,.
\end{equation}
Then from \eqref{psitalt} and \eqref{psit}
\begin{equation} \label{phimualt}
\begin{aligned}
\phi_t(t,r,v)
&= r^2\Big(\frac{2t-\sin(2t)}{2\sin^2 t}\Big)-v
\\&= \frac{r^2t}{\sin^2t}-\frac{1}{t} +
\frac{\phi(t,r,v)}{t} \,.
\end{aligned}
\end{equation}
Moreover $\phi_{tt}=\psi_{tt}$, and we will use the formulas
\eqref{psitt} and \eqref{higherpsider} for the derivatives of $\phi_t$.
\medskip
\medskip
If we set
\begin{equation}\label{rv(t)}\begin{gathered}
r(t)= \Big| \frac{\sin t}{t}\Big|\,,\quad v(t)= \frac 1{t} - \frac{\sin(2t)}{2t^2}
\\
r(0)=1\,,\quad v(0)= 0
\end{gathered}
\end{equation}
then we have
\begin{subequations}\label{phaseversuscurve}
\begin{align}
\label{phitversuscurve}
\phi_t(t,r,v)&= \frac{v(t)}{r^2(t)} r^2 -v\,=
- \Big(v-v(t) -
\, v(t)\frac{r^2-r(t)^2}{r(t)^2}\Big)\,,
\\
\label{phiversuscurve}
\phi(t,r,v)&= \frac{r(t)^2-r^2}{r(t)^2} + t\phi_t(t,r,v)\,.
\end{align}
\end{subequations}
Thus
\begin{equation}\label{system}
\phi(t,r,v)=\phi_t(t,r,v)=0 \quad \iff \quad (r,v)=(r(t),v(t))\,.
\end{equation}
Only the points $(r,v)$ for which there exists a $t$ satisfying \eqref{system} may contribute to the singular support $\Gamma$ of $e^{i\sqrt L} {\delta}_0.$
One recognizes the result
by Nachman \cite{nachman} who showed for the Heisenberg group
that the singular support of the
convolution kernel of $e^{i\sqrt L} $ consists of those $(x,u)$
for which there is a $t>0$ with $(|x|,4|u|)=(r(t),v(t))$.
\begin{figure}
\bigskip
\vspace{0.2cm}
\includegraphics[width=.8\textwidth]{curve.eps}
\vspace{-0.2cm}
\medskip
\caption{$\{\pi(r(t),v(t)): t>0\}$}
\end{figure}
The figure pictures the singular support, including the contribution near $|u|=0$ and $|x|$ near $1$. However we have taken care of the corresponding estimates in {\mathcal S}\ref{FIO}, and thus
we are only interested in the above formulas for $t>3\pi/8$.
For later reference we gather some formulas and estimates for the derivatives
of $r(t)$ and $v(t)$.
For the vector of first derivatives we get, for $t\notin \pi{\mathbb {Z}}$,
\begin{equation} \label{rvprime}
\begin{pmatrix} r'(t)\\ v'(t)\end{pmatrix}=
\frac{\sin t-t\cos t}{t^2}\begin{pmatrix} -{\text{\rm sign}}((\sin t)/t)\\ 2t^{-1}\cos t
\end{pmatrix}
\end{equation}
with $ r'(t)=O(t)$ and $v'(t)-\frac 23=O(t)$ as $t\to 0$.
Hence, for
$t\notin \pi{\mathbb {Z}}$,
\begin{equation} \label{slope}
\frac{v'(t)}{r'(t)}= -{\text{\rm sign}}((\sin t)/t) \frac{2\cos t}t = - 2r(t) \cot t\,.
\end{equation}
Clearly all derivatives of $t$ and $v$ extend to
functions continuous at $t=0$.
Further computation yields for positive $t\notin \pi \Bbb Z$,
$\nu\ge1$,
\begin{subequations}\label{higherrvderform}
\begin{equation}
{\text{\rm sign}}(\frac{\sin t}t)\, r^{(\nu)}(t)=
\sum_{n=1}^{\nu+1} a_{n,\nu} t^{-n} \sin t
+ \sum_{n=1}^\nu b_{n,\nu} t^{-n} \cos t
\end{equation} and
\begin{equation}
v^{(\nu)}(t)=\gamma_\nu t^{-\nu-1} +
\sum_{n=1}^{\nu+1} c_{n,\nu} t^{-n-1} \sin 2t
+ \sum_{n=1}^\nu d_{n,\nu} t^{-n-1} \cos 2t \,;
\end{equation}
\end{subequations}
here
$a_{n,\nu}=c_{n,\nu}=0$ if $n-\nu$ is even, and $b_{n,\nu}=d_{n,\nu}=0$
if $n-\nu$ is odd;
moreover $\gamma_\nu=(-1)^\nu(\nu-1)!$, and
$a_{1,\nu}=(-1)^{\nu/2}$ for $\nu=2,4,\dots$.
For the coefficients in the first derivatives formula we get
$b_{1,1}=1$, $a_{2,1}=-1$, $d_{1,1}=-1$, and $c_{2,1}=1$.
For the second derivatives, we have the coefficients
$a_{1,2}=-1$, $b_{2,2}=-2$, $a_{3,2}=2$,
$c_{1,2}=2$, $d_{2,2}=4$, $c_{3,2}=-3$.
Consequently, for the second derivatives we get the estimates
\begin{equation}\label{secondrvderbound}
|r''(t)|\lesssim t^{-1}|\sin t| + (1+t)^{-2}, \quad
|v''(t)|\lesssim t^{-2}|\sin 2t|+(1+t)^{-3}.
\end{equation}
Also, $|r^{(\nu)}(t)|\lesssim_\nu(1+t)^{-1}$, and $|v^{(\nu)}(t)|\lesssim_\nu (1+t)^{-2}$ for all $t>0$.
\section{$L^1$ estimates}\label{L1section}
In this section we prove the essential $L^1$ bounds needed for the
proof of Theorem \ref{main-theoremL1}. We may assume that $\lambda} \def\La{\Lambda$ is large.
In what follows we frequently need to perform repeated integrations by parts in the presence of oscillatory
terms with nonlinear phase functions and we start with a standard calculus lemma which will be used several times.
\subsection{\it Two preliminary lemmata}
Let $\eta\in C^\infty_0({\mathbb {R}}^n)$ and let
$\Phi\in C^\infty$ so that $\nabla\Phi\neq 0$ in the support of $\eta$.
Then, after repeated integration by parts,
\begin{equation}\label{repeatedinbyparts}
\int e^{i\lambda} \def\La{\Lambda \Phi(y)} \eta(y) \,dy = (i/\lambda} \def\La{\Lambda)^{N} \int e^{i\lambda} \def\La{\Lambda
\Phi(y)} {\mathcal {L}}^N \!\eta(y) \, dy
\end{equation} where the operator ${\mathcal {L}}$ is defined by
\begin{equation} \label{diffopdef}
{\mathcal {L}} a
= \text{div} \big(\frac{a\nabla \Phi}{|\nabla\Phi|^2}\big).
\end{equation}
In order to analyze the behavior of ${\mathcal {L}}^N$ we
shall need a lemma. We use multiindex notation, i.e. for $\beta=(\beta^1,\dots, \beta^n)\in ({\mathbb {N}}\cup\{0\})^n$
we write $\partial^\beta = \partial_{y_1}^{\beta^1}\cdots
\partial_{y_n}^{\beta^n}$ and let $|\beta|=\sum_{i=1}^n \beta^i$ be the order of the multiindex.
\begin{lemma} \label{iterateddiffop}
Let ${\mathcal {L}}$ be as in \eqref{diffopdef}. Then ${\mathcal {L}}^N a$ is a linear
combination of $C(N,n)$ terms of the form
$$\frac{\partial^\alpha a \prod_{\nu=1}^j \partial^{\beta_\nu} \Phi}
{|\nabla\Phi|^{4N}}$$ where $2N\le j\le 4N-1$ and
$\alpha, \beta_1,\dots, \beta_{j}$ are multiindices in $({\mathbb {N}}\cup\{0\})^n$
with $1\le |\beta_\nu|\le |\beta_{\nu+1}|$, satisfying
\begin{enumerate}
\item \ \ $0\le |\alpha|\le N$,
\item \ \ $|\beta_\nu|=1$ for $\nu=1,\dots, 2N$,
\item \ \ $|\alpha|+\sum_{\nu=1}^j|\beta_\nu|=4N$,
\item \ \ $\sum_{\nu=1}^j (|\beta_\nu|-1)=N-|\alpha|$.
\end{enumerate}
\end{lemma}
\begin{proof} Use induction on $N$. We omit the
straightforward details.\end{proof}
{\it Remark:} {\it In dimension $n=1$ we see that ${\mathcal {L}}^N a$ is a linear
combination of $C(N,1)$ terms of the form
$$
\frac {a^{(\alpha)}}{(\Phi')^\alpha}\prod_{\beta\in {\mathfrak {I}}}\frac{\Phi^{(\beta)}}{(\Phi')^{\beta}},
$$
where ${\mathfrak {I}}$ is a set of integers $\beta\in\{2,\dots, N+1\}$ with the property that $\sum_{\beta\in {\mathfrak {I}}}(\beta-1)=N-\alpha$.
If ${\mathfrak {I}}$ is the empty set then we interpret the product as $1$. }
In what follows we shall often use the following
\begin{lemma}\label{integralobs}
Let $\Lambda>0$, $\rho>0$, $n\ge 1$ and $N>\frac{n+1}2$.
Then
\begin{equation*}\int_{-\infty}^\infty \frac{
(1+\La|v|)^{-\frac{n-1}{2}} |v|^{n-1}}
{(1+\La|\rho-v|)^{N}} dv
\lcs{n}
\begin{cases} \La^{-\frac{n+1}{2}} \rho^{\frac{n-1}2}
&\text{ if } \La\rho\ge 1\,,
\\
\La^{-n} &\text{ if }
\La\rho\le 1\,.\qquad
\end{cases}
\end{equation*}
\end{lemma}
We omit the proof. Lemma \ref{integralobs}
will usually be
applied after using integration by parts with respect to the $s$-variable, with the
parameters $n=d_2$ and $\La=\lambda} \def\La{\Lambda k$.
\subsection{\it Estimates for $|u|\lesssim (k+1)^{-1}\lambda^{-1}$}
We begin by proving an $L^1$ bound for the part of the kernels
$K_{\lambda} \def\La{\Lambda}^{k,l}$ for which the terms ${\mathscr{J}\ci{d_2}\!} (4s \lambda} \def\La{\Lambda t|u| )$ have no
significant oscillation, i.e. for the region where $|u|\le C (\lambda} \def\La{\Lambda
k)^{-1}$ (or $|u|\lesssim \lambda} \def\La{\Lambda^{-1}$ if $k=0$).
\begin{lemma} \label{smallulemmakl}Let $\lambda} \def\La{\Lambda\ge 1$, $k\ge 1$, $l\ge 1$. Then
\begin{equation} \label{Kklsmallu}
\iint_{|u|\lesssim (\lambda} \def\La{\Lambda k)^{-1}} |\lambda} \def\La{\Lambda^{-\frac{d-1}{2}}K^{k,l}_\lambda} \def\La{\Lambda(x,u)|
dx\, du
\lesssim
(2^l k)^{-1}\lambda} \def\La{\Lambda^{1-\frac{d}{2}}.
\end{equation}
\end{lemma}
\begin{proof}
First we integrate the pointwise bound
\eqref{Claklinfty}
over the region where
$|x|\le(\lambda} \def\La{\Lambda k 2^l)^{-1/2}$, $|u|\le (\lambda} \def\La{\Lambda k)^{-1}$
and obtain
\begin{multline*}
\iint_{\substack{
|x|\le C (\lambda} \def\La{\Lambda k 2^l)^{-1/2}\\|u|\le C (\lambda} \def\La{\Lambda k)^{-1}}}
|\lambda} \def\La{\Lambda^{-\frac{d-1}{2}}K^{k,l}_\lambda} \def\La{\Lambda(x,u)| dx\, du \\
\lesssim
2^{-l} {\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} (\lambda} \def\La{\Lambda k 2^l)^{-d_1 /2} (\lambda} \def\La{\Lambda k)^{-d_2}
=(2^l k)^{-1}\lambda} \def\La{\Lambda^{1-\frac{d_1+d_2}{2}}.
\end{multline*}
If $|x|\ge C (\lambda} \def\La{\Lambda k 2^{l})^{-1/2}$ then from \eqref{psitalt}, \eqref{higherpsider} we get that $|\psi_t(t,|x|)|
\gtrsim 2^{2l} k |x|^2$ on the support of $\eta_l(t-k\pi)$, moreover
$(\partial/\partial t)^{(n)} \psi(t,|x|)
=O(|x|^2 k 2^{l(n+1)})$.
The $n$th $t$-derivative of $\eta_l(t-k\pi) {\mathscr{J}\ci{d_2}\!}(4s\lambda} \def\La{\Lambda t|u|)$ is
$O(2^{ln})$. Thus an integration by parts gives
$$\lambda} \def\La{\Lambda^{-\frac{d-1}{2}} |K^{k,l}_\lambda} \def\La{\Lambda (x,u)|\le C_N
2^{-l}{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} (\lambda} \def\La{\Lambda 2^l k |x|^2)^{-N}
$$ for
$|x|\ge (\lambda} \def\La{\Lambda k 2^{l})^{-1/2}$ and
$|u|\le (\lambda} \def\La{\Lambda k)^{-1}$.
The bound
$O((2^l k)^{-1}\lambda} \def\La{\Lambda^{1-\frac{d}{2}})$ follows by integration by parts.
\end{proof}
\medskip
\subsection{\it Estimates for $|u|\gg (k+1)^{-1}\lambda^{-1}$}
We now proceed to give $L^1$ estimates for the kernels $A^{k,l}_\lambda} \def\La{\Lambda$
and $B^{k,l}_\lambda} \def\La{\Lambda$ for $k\ge 1$, in the region where $|u|\gg (k\lambda} \def\La{\Lambda)^{-1}$.
\subsubsection{An estimate for small $x$}
As a first application we prove $L^1$ estimates for
$|x|\lesssim (2^l \lambda} \def\La{\Lambda k)^{-1/2}$, $k\ge 1$.
\begin{lemma}\label{smallx}
Let $C\ge 1$. Then
\begin{equation}\label{smallxest}
\iint\limits_{\substack {(x,u):\\|x|\le C
(2^l\lambda} \def\La{\Lambda k)^{-1/2}}}
\big[|A_{\lambda} \def\La{\Lambda}^{k,l}(x,u)| + |B_{\lambda} \def\La{\Lambda}^{k,l}(x,u)|\big]\, dx\, du
\lesssim\cin C
(2^l k)^{-1} \lambda} \def\La{\Lambda^{-\frac {d_1-1}2 }.
\end{equation}
\end{lemma}
\begin{proof}
Integration by parts with respect to $s$ yields
\begin{multline}\label{intbyparts}
|A_{\lambda} \def\La{\Lambda}^{k,l}(x,u)|+|B_{\lambda} \def\La{\Lambda}^{k,l}(x,u)|
\\ \lesssim_N
\sum_\pm
\frac{{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}}{(1+\lambda} \def\La{\Lambda k|u|)^{\frac{d_2-1}{2}}}
\int_{|t-k\pi|\lesssim 2^{-l}}
(1+\lambda} \def\La{\Lambda k\big| \pm |4u|-|x|^2 \cot t +t^{-1}\big|)^{-N} dt.
\end{multline}
We first integrate in $u$.
Notice that by
Lemma \ref{integralobs}
we have for
fixed $t$ and fixed $r\le
(2^l\lambda} \def\La{\Lambda k)^{-1/2}$
$$\int_0^\infty \frac{(1+\lambda} \def\La{\Lambda kv)^{-\frac{d_2-1}{2}}v^{d_2-1}}
{(1+\lambda} \def\La{\Lambda k\big| \pm |v|-r^2 \cot t +t^{-1}\big|)^{N}}
dv \lesssim \lambda} \def\La{\Lambda^{-\frac{d_2+1}2}k^{-d_2}.
$$
We integrate in $x$ over a set of measure
$\lesssim (2^lk\lambda} \def\La{\Lambda)^{-d_1/2}$ and then in $t$ (over an interval of
length $\approx 2^{-l}$) and \eqref{smallxest} follows.
\end{proof}
\subsubsection{$L^1$-bounds for $B_\lambda^{k,l}$}
\begin{lemma}\label{Bkllemma}
For $\lambda} \def\La{\Lambda\ge 1$, $0<k\le 8\lambda} \def\La{\Lambda$ ,
\begin{equation} \label{Bklestimate}
\big\|B_{\lambda} \def\La{\Lambda}^{k,l}\big\|_1 \lesssim
(2^l k)^{-1} \lambda} \def\La{\Lambda^{-\frac {d_1-1}2 } .\end{equation}
\end{lemma}
\begin{proof}
The bound for the region
with $|x|\lesssim(2^l\lambda} \def\La{\Lambda k)^{-1/2}$
(for which there is no significant oscillation in the $t$ integral) is
proved in Lemma \ref{smallx}.
Consider the region where $|x|\approx 2^m (2^l\lambda} \def\La{\Lambda k)^{-1/2}.$
We perform $N_1$ integration by parts in $t$
followed by $N_2$ integrations by parts with respect to $s$.
Denote by ${\mathcal {L}}_t$ the operator defined by
${\mathcal {L}}_t g=\partial_t (\tfrac{g(t)}{\psi_t(t,|x|)+4|u|})$. Then
\begin{multline*}B^{k,l}_\lambda} \def\La{\Lambda(x,u)={\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}(i/\lambda} \def\La{\Lambda)^{N_1} \times
\\
\iint
e^{i\lambda} \def\La{\Lambda s(\psi(t, |x|)+4t|u|)} \frac{(I-\partial_s^2)^{N_2}\big[
s^{-N_1} {\mathcal {L}}_t^{N_1}\{ \eta_{\lambda} \def\La{\Lambda,k,l}(s,t)
{\varpi}_2(4\lambda} \def\La{\Lambda st|u|)\,\}\big]}
{(1+ \lambda} \def\La{\Lambda^2|\psi(t,|x|)+4t|u||^2)^{N_2}}
dt \,ds
\end{multline*}
From \eqref{psitalt},
$$|\partial_t (\psi(t,|x|)+4t|u|)| \gtrsim 2^{2l}k|x|^2+4|u| \gtrsim 2^{2m+l}\lambda} \def\La{\Lambda^{-1}.$$
Moreover, for $\nu\ge 2$, $\partial_t^\nu \psi = O(2^{2m+l\nu}\lambda} \def\La{\Lambda^{-1})$ and
$\nu$ differentiations of the amplitude
produce factors of $2^{l\nu}$.
Thus
we obtain the bound
\begin{multline*}|B_{\lambda} \def\La{\Lambda}^{k,l}(x,u)|\lesssim
\frac{{\mathfrak {C}}_{\lambda} \def\La{\Lambda,l,k}}
{(1+4\lambda} \def\La{\Lambda k|u|)^{\frac{d_2-1}{2}}}
2^{-2m N_1}\quad \times\\
\int_{|t-k\pi|\lesssim 2^{-l}}
(1+\lambda} \def\La{\Lambda k\big||t^{-1}-|x|^2 \cot t +4|u||)^{-2N_2} \,dt.
\end{multline*}
From Lemma \ref{integralobs} (with $n=d_2$, $\La=\lambda} \def\La{\Lambda k$, $\rho\lesssim k^{-1}
\max\{1, 2^{2m}\lambda} \def\La{\Lambda^{-1}\}$)
\begin{multline}\label{vintegralfixedt}
\int_{v=0}^\infty\frac{(1+\lambda} \def\La{\Lambda kv)^{-\frac{d_2-1}{2}}v^{d_2-1}}
{(1+\lambda} \def\La{\Lambda k\big|v-|x|^2 \cot t +t^{-1}\big|)^{N}}
dv\\
\lesssim \lambda} \def\La{\Lambda^{-\frac{d_2+1}{2}} k^{-d_2} \max\{ 1,
(2^{2m}\lambda} \def\La{\Lambda^{-1})^{\frac{d_2-1}{2}}\}.
\end{multline}
We integrate in $t$ over an interval of length $O(2^{-l})$ and in $x$
over the annulus $\{x: |x|\approx 2^m (2^l\lambda} \def\La{\Lambda k)^{-1/2}\}$.
This gives
\begin{align}\label{Bklinannulus}
&\iint_{\substack{(x,u):\\
|x|\approx 2^{m}(2^l\lambda} \def\La{\Lambda k)^{-1/2} }}
|B_{\lambda} \def\La{\Lambda}^{k,l}(x,u)| dx\, du\,
\\& \lesssim \, 2^{-2m N} \,2^{-l} \Big(\frac{2^{m}}{\sqrt{ 2^l \lambda} \def\La{\Lambda k}}\Big)^{d_1}
{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l}
\lambda} \def\La{\Lambda^{-\frac{d_2+1}{2}} k^{-d_2} \max\{ 1,
(2^{2m}\lambda} \def\La{\Lambda^{-1})^{\frac{d_2-1}{2}}\}
\notag
\\
&\lesssim
(2^{l}k)^{-1} \lambda} \def\La{\Lambda^{-\frac {d_1-1}2 } 2^{-m(2N-d_1)}\max\{ 1,
(2^{2m}\lambda} \def\La{\Lambda^{-1})^{\frac{d_2-1}{2}}\}
\notag
\end{align}
and choosing $N$ sufficiently large the lemma follows by summation in $m$.
\end{proof}
\subsubsection{$L^1$-bounds for
$A^{k,l}_\lambda} \def\La{\Lambda$, $2^lk\ge10^5\lambda} \def\La{\Lambda$}
\begin{lemma}\label{kgrlalemma}
For $k\le 8\lambda} \def\La{\Lambda$, $2^l\ge 10^5\lambda} \def\La{\Lambda/k$,
\begin{equation} \label{kgrla}
\big\|A_{\lambda} \def\La{\Lambda}^{k,l}\big\|_1 \lesssim
(2^l k)^{-1} \lambda} \def\La{\Lambda^{-\frac {d_1-1}2 } .\end{equation}
\end{lemma}
\begin{proof}
We use Lemma \ref{smallx} to obtain the appropriate $L^1$ bound in the region
$\{(x,u): \,|x|\le C_0
(2^l\lambda} \def\La{\Lambda k)^{-1/2}\}$. Next, consider the region where
\begin{equation} \label{gain2m}
2^m (2^l\lambda} \def\La{\Lambda k)^{-1/2}\le |x|\le 2^{m+1} (2^l\lambda} \def\La{\Lambda k)^{-1/2}
\end{equation} for large $m$.
This region is then split into two subregions,
one where $4|u|=v\le 10^{-2} 2^{2m+l}\lambda} \def\La{\Lambda^{-1}$
and the complementary region.
For the region with small $v$ we proceed as in Lemma \ref{Bkllemma}.
From formula \eqref{psitalt} we have $|\psi_t|\ge kr^22^{2l}/20$
and hence $|\psi_t|\ge 2^{2m+l-5}\lambda} \def\La{\Lambda^{-1}$.
Thus if $v\le 10^{-2}2^{2m+l}\lambda} \def\La{\Lambda^{-1}$
then
$|\phi_{t}|\approx k2^{2l} r^2 \approx 2^{2m+l}\lambda} \def\La{\Lambda^{-1}$.
Moreover $\partial_t^\nu \phi = O(
2^{2m+l\nu}\lambda} \def\La{\Lambda^{-1})$ for $\nu\ge 2$.
Therefore, if we perform integration by parts in $t$
several times, followed by integrations by parts on $s$,
we obtain the bound
\begin{multline*}|A_{\lambda} \def\La{\Lambda}^{k,l}(x,u)|\lesssim
\frac{{\mathfrak {C}}_{\lambda} \def\La{\Lambda,l,k}}
{(1+\lambda} \def\La{\Lambda k|u|)^{\frac{d_2-1}{2}}}
2^{-2m N}\quad \times\\
\int_{|t-k\pi|\lesssim 2^{-l}}
(1+\lambda} \def\La{\Lambda k\big||x|^2 \cot t -t^{-1}-4|u|\big|)^{-N} dt.
\end{multline*}
In the present range $|x|^2|\cot t|\approx
2^{2m}(\lambda} \def\La{\Lambda k)^{-1}$ and $t^{-1}\approx k^{-1}$ and thus
we see from Lemma \ref{integralobs} that inequality \eqref{vintegralfixedt}
in the proof of Lemma \ref{Bkllemma} holds. From this we proceed as in
\eqref{Bklinannulus} to bound
\begin{multline*}\iint_{\substack{
|x|\approx 2^{m}(2^l\lambda} \def\La{\Lambda k)^{-1/2}
\\
4|u|\le 10^{-2} 2^{2m+l} \lambda} \def\La{\Lambda^{-1} }}
|A_{\lambda} \def\La{\Lambda}^{k,l}(x,u)| dx du\,\\
\lesssim
(2^{l}k)^{-1}\lambda} \def\La{\Lambda^{-\frac {d_1-1}2 } 2^{-m(2N-d_1)}\max\{ 1,
(2^{2m}\lambda} \def\La{\Lambda^{-1})^{\frac{d_2-1}{2}}\} \,.
\end{multline*}
For large $N_1$ we can sum in $m$ and obtain the
bound
$C(2^{l}k)^{-1}\lambda} \def\La{\Lambda^{-\frac {d_1-1}2 }$.
Next assume that
$v\ge 2^{2m+l}\lambda} \def\La{\Lambda^{-1}/100$
(and still keep \eqref{gain2m}).
Then
\begin{equation} \label{philowerbdlargev}
|tv+r^2 t\cot t-1|\ge k|v|
\text{ for $t\in {\hbox{\roman supp}}( \eta_{\lambda} \def\La{\Lambda,k,l})$}\,.\end{equation}
Indeed, we have $tv\ge 2^{2m} 2^lk\lambda} \def\La{\Lambda^{-1}/100 \ge 10^3$
and
$$r^2t|\cot t|\le 2^{2m+2}(2^l\lambda} \def\La{\Lambda k)^{-1}
t[\sin(\tfrac{3\pi}{8}2^{-l})]^{-1} \le
2^{2m+6}\lambda} \def\La{\Lambda ^{-1}\le \frac{2^{2m+l}}{100\lambda} \def\La{\Lambda} 2^5 10^2 2^{-l}
$$
where we used \eqref{gain2m} and $\sin \alpha>2\alpha/\pi$ for $0\le \alpha\le \pi/2$.
By our assumptions $2^l\ge 10^5 \lambda} \def\La{\Lambda/k>10^4$ and thus the right hand side of the display is $\le v/10$.
Now
\eqref{philowerbdlargev}
is immediate by the triangle inequality.
We use \eqref{philowerbdlargev} to get from an $N_1$-fold integration by
parts in $s$
$$|A^{k,l}_\lambda} \def\La{\Lambda(x,u)|
\lesssim 2^{-l}{\mathfrak {C}}_{\lambda} \def\La{\Lambda,l,k}
(\lambda} \def\La{\Lambda kv)^{-N_1-\frac{d_2-1}{2}}.
$$
Then
\begin{align*}
&\iint_{\substack{|x|\approx 2^m (2^l \lambda} \def\La{\Lambda k)^{-1/2}\\
4|u|\ge 10^{-2} 2^{2m+l}\lambda} \def\La{\Lambda^{-1} }}|A^{k,l}_\lambda} \def\La{\Lambda(x,u)| \,du \,dx
\\
&\quad \lesssim 2^{-l} {\mathfrak {C}}_{\lambda} \def\La{\Lambda,l,k} \Big(\frac{2^m}{\sqrt{\lambda} \def\La{\Lambda 2^l k}}\Big)^{d_1}
(\lambda} \def\La{\Lambda k)^{-N_1-\frac{d_2-1}{2}} \Big(\frac{2^{2m+l}}{\lambda} \def\La{\Lambda}\Big)^{-N_1+ \frac{d_2+1}{2}}
\\
&\quad \lesssim
\lambda} \def\La{\Lambda^{1-\frac{d_1}2-\frac{d_2}{2}}2^{-l(N_1-\frac{d_2-1}{2})}k^{\frac{d_2-1}{2}-N_1}
2^{m(d_1+d_2+1-2N_1)}\,.
\end{align*}
For $N_1$ large we may sum in $m$ to finish the proof.
\end{proof}
\subsubsection{Estimates for $A_\lambda} \def\La{\Lambda^{k,l}$, $2^l\lesssim\lambda} \def\La{\Lambda/k$.}
In the early approaches to prove $L^p$ boundedness for Fourier integral operators the oscillatory integral were analyzed using the method of stationary
phase (\cite{peral}, \cite{miyachi}, \cite{beals}).
This creates some difficulties in our case at points where $\phi$, $\phi_t$ and $\phi_{tt}$ vanish simultaneously, namely at positive $t$ satisfying
$\tan t=t$.
To avoid this difficulty we use a decomposition in the spirit of \cite{SSS}.
In what follows we assume
$k\le 8\lambda} \def\La{\Lambda$ and $2^l\le C_0\lambda} \def\La{\Lambda/k$ for large $C_0$ chosen independently of
$\lambda} \def\La{\Lambda, k,l$. The choice $C_0=10^{10}$ is suitable. We decompose the interval
$J_{k,l}$ into smaller subintervals of length $\varepsilon \sqrt{\frac {k}{2^l\lambda} \def\La{\Lambda}}$ (which is $\lesssim 2^{-l}$ in the range under consideration), here $\varepsilon \ll 10 ^{-100}$ (to be chosen
sufficiently small but independent of $\lambda} \def\La{\Lambda, k,l$).
This decomposition is motivated by the following considerations: according to \eqref{splitting},
$\lambda} \def\La{\Lambda \phi(t,r,v)$ contains the term $-\lambda} \def\La{\Lambda(r-r(t))^2 t\cot t$ depending entirely on $r$ and $t$. For $t\in J_{k,l},$ this is of size $\lambda} \def\La{\Lambda k 2^l |r-r(t)|^2,$ hence of order $O(1)$ if $|r-r(t)|\lesssim (\lambda} \def\La{\Lambda k 2^l)^{-1/2}.$
Moreover, on a subinterval $I$ of $J_{k,l}$ on which $r(t)$ varies by at most a small fraction of the same size, the term
$-\lambda} \def\La{\Lambda(r-r(t))^2 t\cot t$ is still $O(1)$ and contributes to no oscillation in the integration with respect to $s.$ Since $|r'(t)|\sim 1/k$ by $\eqref{rvprime}, $ this suggests to choose intervals $I$ of length $\ll k(\lambda} \def\La{\Lambda k 2^l)^{-1/2}=\sqrt{k2^{-l}\lambda} \def\La{\Lambda^{-1}}.$ Similarly, the first term of $\lambda} \def\La{\Lambda \phi(t,r,v)$ in \eqref{splitting} is of size $\lambda} \def\La{\Lambda k|w(t,r,v)|$ and does not contribute to any oscillation in the integration with respect to $s$ if $|w(t,r,v)|\lesssim (\lambda} \def\La{\Lambda k)^{-1}.$ These considerations also motivate our later definitions of the set ${\mathcal {P}}_0$ and the sets ${\mathcal {P}}_m, m\ge 1$, {\it cf}. \eqref{cPm}.
\medskip
As before we denote by $\eta_0$ a $C^\infty_0({\mathbb {R}})$ function so that $\sum_{n\in{\mathbb {Z}}} \eta_0(t-\pi n)=1$ and ${\hbox{\roman supp}}(\eta_0)\subset (-\pi,\pi)$.
Define, for
$b\in \pi\varepsilon\sqrt{k2^{-l}\lambda} \def\La{\Lambda^{-1}}\,{\mathbb {Z}}$,
\begin{equation}\label{defetaklb} \eta_{\lambda} \def\La{\Lambda,k,l,b}(s,t)= \eta_{\lambda} \def\La{\Lambda, k,l}(s,t)
\eta_0 \big(\varepsilon^{-1}\sqrt{\tfrac{\lambda} \def\La{\Lambda 2^l}{k}}(t-b)\big).
\end{equation}
Then we may split
\begin{equation}\label{bdecomp}A_{\lambda} \def\La{\Lambda}^{k,l}=\sum_{b\in {\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}} A_{\lambda} \def\La{\Lambda, b}^{k,l}
\end{equation}
where
${\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}\subset
\pi\varepsilon\sqrt{ k2^{-l}\lambda} \def\La{\Lambda^{-1}}\,
{\mathbb {Z}} \cap J_{k,l}$ ({\it cf}.\eqref{Jkl}),
$\#{\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}=O(\varepsilon^{-1}\sqrt{\lambda} \def\La{\Lambda 2^{-l} k^{-1}} )$, and
\begin{multline}\label{Blakb}
A_{\lambda} \def\La{\Lambda,b}^{k,l}(x,u)=
\\
{\mathfrak {C}}_{\lambda} \def\La{\Lambda,l,k}
\iint \chi(s)
\eta_{\lambda} \def\La{\Lambda,k,l,b}(t)
e^{i\lambda} \def\La{\Lambda s(1-|x|^2t\cot t -t|4u|)} {\varpi}_1(\lambda} \def\La{\Lambda st |4u|) dt ds\,.
\end{multline}
\medskip
We now give some formulas relating the phase
$\phi(t,r,v)=1-r^2t\cot t -t v$
to the geometry of the curve $(r(t), v(t))$ ({\it cf}.\eqref{rv(t)}).
By \eqref{system} and \eqref{slope},
\begin{align}
&\frac{\phi(t,r,v)}{t}=
\frac{\phi(t,r,v)-\phi(t, r(t),v(t))}{t}
\notag
\\
&=(r(t)^2-r^2) \cot t +v(t)-v
\notag
\\
&= v(t)-v -\big(r-r(t)\big) 2 r(t) \cot t
-(r-r(t))^2 \cot t
\notag
\end{align}
and, setting
\begin{equation}\label{wdefin}w(t,r,v)= v-v(t)- \frac{v'(t)}{r'(t)}(r-r(t))\,,\end{equation}
we get
\begin{equation}
\label{splitting}
\frac{\phi(t,r,v)}{t}= - w(t,r,v)-(r-r(t))^2 \cot t\,.
\end{equation}
Moreover,
\begin{align} \notag
\phi_t(t,r,v)&=
\frac{\phi(t,r,v)}{t}+
\frac{r^2 t}{\sin^2 t}- \frac 1t
\\&=\frac{\phi(t,r,v)}{t}+\frac{t}{\sin^2 t} (r+r(t))(r-r(t))
\label{phit-splitting}
\end{align}
We shall need estimates describing how $w(t,r,v)$ changes in $t$.
Use
\eqref{splitting} and the expansion
\begin{align*}
&w(t,r,v)- w(b,r,v)\,=\,
- \big[ v(t)-v(b)- \frac{v'(b)}{r'(b)} (r(t)-r(b))\big]
\\& \qquad - \Big[ \frac{v'(t)}{r'(t)}
- \frac{v'(b)}{r'(b)}\Big](r-r(b))
+ \Big[ \frac{v'(t)}{r'(t)} - \frac{v'(b)}{r'(b)}\Big](r(t)-r(b)).
\notag
\end{align*}
From
\eqref{secondrvderbound}
we get $|r''|+k|v''|\lesssim 2^{-l}k^{-1}+k^{-2}$ on $J_{k,l}$, thus the first term
in the displayed formula is $\lesssim (2^{-l}k^{-2}+k^{-3} )|t-b|^2$.
Differentiating in \eqref{slope} we also get
$(v'/r')'=
O(2^{-l}k+ k^{-2})$ on $J_{k,l}$,
and see that the second term in the display is
$\lesssim (2^{-l}k^{-1}+k^{-2})
|t-b| |r-r(b)|$ and the third is
$\lesssim ( 2^{-l}+k^{-1})k^{-2}(t-b)^2$.
Hence
\begin{equation}\label{wdiff2}
|w(t,r,v)- w(b,r,v)|\lesssim
(2^{-l}+k^{-1})|t-b|\Big(\frac{|t-b|}{k^{2}}+ \frac{|r-r(b)|}{k}\Big)\,.
\end{equation}
We now turn to the estimation of $A^{k,l}_{\lambda} \def\La{\Lambda,b}$ with $k\ge 1$ and
$b\in {\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}$.
Let, for $b>1/2$, $l=1,2,\dots$, and $m=0,1,2,\dots$,
\begin{multline}\label{cPm}
{\mathcal {P}}_m\equiv{\mathcal {P}}_m(\lambda} \def\La{\Lambda,l,k;b):=\big\{(r,v)\in (0,\infty)\times (0,\infty):\\
\quad
v\ge (\lambda} \def\La{\Lambda k)^{-1},\,\,
|r-r(b)|\le 2^m (\lambda} \def\La{\Lambda k 2^l)^{-1/2}, \,\, |w(b,r,v)|\le 2^{2m}
(\lambda} \def\La{\Lambda k)^{-1} \big\}
\end{multline}
and let
\begin{equation} \label{Omm}
\Omega_m\equiv\Omega_m(\lambda} \def\La{\Lambda,l,k;b):=
\begin{cases}
&\{(x,u):\, (|x|, 4|u|)\in {\mathcal {P}}_0\}\,,\text{ if } m=0,
\\
&\{(x,u):\, (|x|, 4|u|)\in {\mathcal {P}}_m\setminus {\mathcal {P}}_{m-1}\} \text{ if } m>0\,.
\end{cases}
\end{equation}
For later reference we note that in view
$2^l \le \lambda} \def\La{\Lambda/k$, $|t-b|\le \varepsilon \sqrt{ \frac{k}{\lambda} \def\La{\Lambda 2^l}}$
and the upper bound $|r'(t)|\le 2t^{-1}$ we have
$r(t)-r(b)= O\big(\frac{\varepsilon}{\sqrt{k\lambda} \def\La{\Lambda 2^l}}\big),$
and, by \eqref{wdiff2},
\begin{equation}\label{wdiffestimate}
|w(t,r,v)- w(b,r,v)|\lesssim \varepsilon 2^m (\lambda} \def\La{\Lambda k)^{-1},\quad (r,v)\in {\mathcal {P}}_m.
\end{equation}
Moreover it is easy to check that, still for
$|t-b|\le \varepsilon \sqrt{ \frac{k}{\lambda} \def\La{\Lambda 2^l}}$,
\begin{equation}\label{sectermdiff}
|(r-r(t))^2\cot t- (r-r(b))^2\cot b|
\lesssim \varepsilon 2^{2m} (\lambda} \def\La{\Lambda k)^{-1}.\end{equation}
\medskip
\begin{prop} \label{vert-hor}
Assume that $1\le k\le8\lambda} \def\La{\Lambda$, $l=1,2,\dots$, and $2^l\le C_0 \lambda} \def\La{\Lambda/k$
(and let $\varepsilon$ in the definition \eqref{defetaklb}
be $\le C_0^{-1} 10^{-100}$).
Let $b\ge 1$ and $b\in {\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}$.
Then
\begin{align} \label{corridorzero}
&\iint_{\Omega_0(\lambda} \def\La{\Lambda, l,k;b)} |A^{k,l}_{\lambda} \def\La{\Lambda,b}(x,u)|
\, dx du \, \lesssim\, (2^lk)^{-\frac{d_1+ 1}2}
\sqrt{\tfrac{2^lk}{\lambda} \def\La{\Lambda}}
\\ \label{corridorm}
&\iint_{\Omega_{m}(\lambda} \def\La{\Lambda, l,k;b)} |A^{k,l}_{\lambda} \def\La{\Lambda,b}(x,u)| \, dx du \,
\lcs{N}\, 2^{-m N} (2^lk)^{-\frac{d_1+ 1}2}
\sqrt{\tfrac{2^lk}{\lambda} \def\La{\Lambda}}\,.
\end{align}
\end{prop}
\begin{proof}
Note that, for fixed $k\ge 1$, $l\ge 1$, $b\in {\mathcal {T}}_{\lambda} \def\La{\Lambda,k,l}$,
\begin{equation} \label{rvinPm}
(r,v)\in {\mathcal {P}}_m \,\implies r\lesssim 2^m (2^l k)^{-1} \text {and } v\lesssim
2^{2m}k^{-1}\,.
\end{equation}
This is immediate in view of $2^lk\lesssim\lambda} \def\La{\Lambda$, $r(b)\approx (2^{l}k)^{-1}$,
$v(b)\approx k^{-1}$
and thus
\begin{equation}\label{rvsizebounds}
\begin{aligned}&r\lesssim (2^l k)^{-1}(1+ 2^m \sqrt{\tfrac{k2^l}{\lambda} \def\La{\Lambda}})\,
\lesssim 2^m (2^l k)^{-1}\,,
\\&v\lesssim k^{-1}(1+2^{2m}\lambda} \def\La{\Lambda^{-1})\,\lesssim 2^{2m}k^{-1}\,.
\end{aligned}
\end{equation}
Also recall that $v=4|u|\ge (\lambda} \def\La{\Lambda k )^{-1}$ for $(x,u)\in \Omega_m(\lambda} \def\La{\Lambda,l,k;b)$.
A crude size estimate yields
\begin{equation}\label{basicsizeest}
\iint_{(|x|,4|u|)\in {\mathcal {P}}_m} |A^{k,l}_{\lambda} \def\La{\Lambda,b}(x,u)|
\, dx \,du \, \lesssim\,
2^{m(d_1+d_2+1)} (2^{l}k )^{-(d_1+1)/2} \sqrt {\tfrac{2^l k}{\lambda} \def\La{\Lambda}}.
\end{equation}
Indeed, the left hand side is $\lesssim \varepsilon \sqrt{\tfrac{k}{2^l\lambda} \def\La{\Lambda}}\,{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} \,{\mathcal {I}}$ where
$${\mathcal {I}}:= \iint_{\substack{|r-r(b)|\lesssim 2^m(2^l \lambda} \def\La{\Lambda k)^{-1/2}\\|w(b,r,v)|\lesssim 2^{2m}(\lambda} \def\La{\Lambda k)^{-1}}}
(\lambda} \def\La{\Lambda k v)^{-\frac{d_2-1}{2}}
v^{d_2-1} r^{d_1-1} \,dvdr
$$ is
$\lesssim
\frac{2^m}{\sqrt{\lambda} \def\La{\Lambda 2^l k}}
\big(\frac{2^m}{2^l k}\big)^{d_1-1}
\frac{2^{2m}}{\lambda} \def\La{\Lambda k} \big(\frac{2^{2m} k^{-1}}{\lambda} \def\La{\Lambda k}\big)^{\frac{d_2-1}{2}},$ in view of \eqref{wdefin} and \eqref{rvsizebounds}.
This yields \eqref{basicsizeest}.
In regard to its dependence on $m$ this bound is nonoptimal and
will be used for $2^m\le C(\varepsilon)$.
\medskip
We now derive an improved $L^1$ bound for the region $\Omega_m$ when $m$ is large.
For $(r,v)\in {\mathcal {P}}_m\setminus {\mathcal {P}}_{m-1}$ we
distinguish two cases $I$, $II$ depending on the size of
$|\phi(b,r,v)|$ and define for $m>0$, and fixed $k,l,b$,
$$\begin{aligned}
{\mathcal {R}}_m^I&= \{(r,v)\in {\mathcal {P}}_m\setminus {\mathcal {P}}_{m-1}:
|\phi(b,r,v)| > 2^{l-100}(r-r(b))^2\,\}\,,
\\
{\mathcal {R}}_m^{II}&= \{(r,v)\in {\mathcal {P}}_m\setminus {\mathcal {P}}_{m-1}:
|\phi(b,r,v)| \le 2^{l-100}(r-r(b))^2\,\}\,.
\end{aligned}
$$
We also have the corresponding decomposition $\Omega_m=\Omega_m^I+\Omega_m^{II}$ where $\Omega_m^I$
and $\Omega_m^{II}$
consist of those $(x,u)$ with $(|x|, 4|u|)\in {\mathcal {R}}_m^I$
and $(|x|, 4|u|)\in {\mathcal {R}}_m^{II}$,
respectively.
\medskip
\noindent{\it Case I: $|\phi(b,r,v)| \ge 2^{l-100}k(r-r(b))^2$.}
We shall show that
\begin{equation}\label{philowbd}
|\phi(t,r,v)| \gtrsim c 2^{2m}\lambda} \def\La{\Lambda^{-1},\quad \text{ for }(r,v)\in
{\mathcal {R}}_m^{I},
\quad|t-b|\le \varepsilon
\sqrt{\tfrac{k}{2^l\lambda} \def\La{\Lambda}}\,.
\end{equation}
with $c>0$ if $0<\varepsilon\ll 10^{-100}$ is chosen sufficiently small. Given
\eqref{philowbd} we can use an $N_2$-fold integration by parts in $s$ to obtain a gain of
$2^{-2m N_2}$ over the above straightforward size estimate \eqref{basicsizeest}, which
leads to
\begin{equation}\label{OmIest}
\iint_{\Omega_m^{I}} |A^{k,l}_{\lambda} \def\La{\Lambda,b}(x,u)|
\, dx \,du \, \lesssim_{\varepsilon,N_2}
2^{m(d_1+d_2+1-2N_2)} (2^lk)^{-\frac{d_1+1}2}
\sqrt{\tfrac{2^lk}{\lambda} \def\La{\Lambda}}\,.
\end{equation}
It remains to show \eqref{philowbd}.
We distinguish between two subcases. First
if $|r-r(b)|\ge 2^{m-5}(\lambda} \def\La{\Lambda k2^l)^{-1/2}$ then by the {\it Case I}
assumption
we have
$|\phi(b,r,v)|\ge 2^{l-100}k 2^{2m-10} (\lambda} \def\La{\Lambda k2^l)^{-1} = 2^{2m-110}\lambda} \def\La{\Lambda^{-1},$
and by \eqref{splitting},
\eqref{wdiffestimate} and \eqref{sectermdiff} we also get
\eqref{philowbd} provided that $\varepsilon\ll 2^{-200}$.
For the second subcase we have $|r-r(b)|\le 2^{m-5}(\lambda} \def\La{\Lambda k2^l)^{-1/2}$.
Since $(r,v)\notin {\mathcal {P}}_{m-1}$ this implies that
$|w(b,r,v)|\ge 2^{2m-2}(\lambda} \def\La{\Lambda k)^{-1}$, and since the quantity
$b(r-r(b))^2 |\cot b|$ is bounded by $2^{l+4}b(r-r(b))^2 \le 2^{2m-6} (b/k)\lambda} \def\La{\Lambda^{-1}$ we also get
$|\phi(b,r,v)|\ge 2^{2m-3}\lambda} \def\La{\Lambda^{-1}$, by \eqref{splitting}.
Now by \eqref{splitting}, \eqref{wdiffestimate} and \eqref{sectermdiff} we also get
$|\phi(t,r,v)|\ge 2^{2m-4}\lambda} \def\La{\Lambda^{-1}$, if $\varepsilon$ is sufficiently small.
Thus
\eqref{philowbd} is verified and
\eqref{OmIest} is proved.
\medskip
\noindent{\it Case II: $|\phi(b,r,v)| \le 2^{l-100}k(r-r(b))^2
$.}
We show
\begin{multline} \label{phiderivlowerbound}
|\phi_t(t,r,v)|\ge 2^{m-20} 2^{3l/2} k^{1/2} (r+r(b))\lambda} \def\La{\Lambda^{-1/2}\\
\text{ if } (r,v)\in {\mathcal {R}}^{II}_m\,,\quad
|t-b|\le \varepsilon\sqrt{\tfrac{\lambda} \def\La{\Lambda 2^l}{k}}\,.
\end{multline}
and this will enable us to get a gain when integrating by parts in $t$.
To prove \eqref{phiderivlowerbound} we first establish
\begin{equation}\label{rminusrblwbd}
|r-r(b)|\ge 2^{m-10} (\lambda} \def\La{\Lambda k2^l)^{-1/2}\,\text{ for }
(r,v)\in {\mathcal {R}}^{II}_m\,.
\end{equation}
Note that if $|w(b,r,v)|\le 2^{2m-3}(\lambda} \def\La{\Lambda k)^{-1}$ then
$|r-r(b)|\ge 2^{m-1}
(\lambda} \def\La{\Lambda k2^l)^{-1/2}$ since ${\mathcal {R}}_m^{II}\subset {\mathcal {P}}_{m-1}^\complement$. Thus to verify
\eqref{rminusrblwbd} we may assume
$|w(b,r,v)|\ge 2^{2m-3}(\lambda} \def\La{\Lambda k)^{-1}$. In this case we get from
\eqref{splitting}, $(r,v)\in{\mathcal {P}}_m$ and the {\it Case II} assumption
\begin{align*}&(r-r(b))^2 |\cot b|\,\ge\,
|w(b,r,v)|- b^{-1}|\phi(b,r,v)|\\
&\,\ge\, 2^{2m-3}(\lambda} \def\La{\Lambda k)^{-1}- b^{-1} k2^{l-100}2^{2m}(\lambda} \def\La{\Lambda k2^{l})^{-1}\,\ge\, 2^{2m-4}(\lambda} \def\La{\Lambda k)^{-1}
\end{align*}
and hence $(r-r(b))^2 2^{l+4} \ge 2^{2m-4}(\lambda} \def\La{\Lambda k)^{-1}$ which implies
\eqref{rminusrblwbd}.
In order to prove
\eqref{phiderivlowerbound} we use \eqref{phit-splitting} and
\eqref{rminusrblwbd} to estimate
\begin{align*}
|\phi_t(b,r,v)|&\ge \frac{b}{\sin^2 b} (r+r(b))|r-r(b)|-2^{l-100} \frac kb
(r-r(b)^2
\\&\ge \frac{|r-r(b)|}{b} \Big(\frac{r+r(b)}{r(b)^2}- \frac{2^l k}{2^{100}}|r-r(b)|\Big)\ge \frac{(r+r(b))|r-r(b)|}{2b r(b)^2}
\\&\ge2^{2l-4} k(r+r(b)) \frac{2^{m-10}}{\sqrt{\lambda} \def\La{\Lambda k2^l}}
\ge 2^{m-15} k^{1/2} 2^{3l/2} (r+r(b)) \lambda} \def\La{\Lambda^{-1/2}
\end{align*}
which yields \eqref{phiderivlowerbound} for $t=b$.
We need to show the lower bound for
$|t-b| \le \varepsilon \sqrt{k/(2^l\lambda} \def\La{\Lambda)}$.
By \eqref{psitt} we have $|\phi_{tt}(t',r,v)|\le r^2 b 2^{3l+4}$ for
$|t'-b| \le \varepsilon \sqrt{\frac{b}{2^l\lambda} \def\La{\Lambda}}$ and thus
$$|\phi_t(t,r,v)-\phi_t(b,r,v)|\le 2^6 r^2 2^{3l} k\varepsilon
\sqrt{\tfrac{k}{2^l\lambda} \def\La{\Lambda}} \le 2^{m-30} 2^{3l/2} k^{1/2}\lambda} \def\La{\Lambda^{-1/2}(r+r(b))
$$
if $\varepsilon \ll 2^{-100}$.
The second inequality in the last display is easy to check. If $r\le 2r(b)$
then use $r\lesssim (2^l k)^{-1}\approx r+r(b)$ and if $r>2r(b)$ then use
$r-r(b)\approx r+r(b)\approx r$. In both cases the asserted inequality holds for small $\varepsilon$ and thus \eqref{phiderivlowerbound} holds for
$|t-b| \le \varepsilon \sqrt{k/(2^l\lambda} \def\La{\Lambda)}$.
We note that under the condition \eqref{rminusrblwbd}
the range $r\le 2 r(b)$ corresponds to $2^m \lesssim \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$
and the range $r\ge 2 r(b)$ corresponds to $2^m \gtrsim \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$.
We now estimate the $L^1$ norm over the region
where $(r,v)\in {\mathcal {R}}_m^{II}$.
Let ${\mathcal {L}}_t$ be the differential operator defined by ${\mathcal {L}}_t g= \frac{\partial}{\partial t}( \frac{g}{\phi_t})$.
By $N_1$ integration by parts in $t$ we get
(with $|x|=r$, $4|u|=v$)
\begin{multline*}
A_{\lambda} \def\La{\Lambda,b}^{k,l} (x,u) \,= \,
i^{N_1}\lambda} \def\La{\Lambda^{-N_1}{\mathfrak {C}}_{\lambda} \def\La{\Lambda,k,l} \,\times \\
\iint
e^{i\lambda} \def\La{\Lambda s\phi(t,|x|,4|u|)}
s^{-N_1}{\mathcal {L}}_t^{N_1} [\eta_{\lambda} \def\La{\Lambda,k,l,b}(s,t) {\varpi}_1(\lambda} \def\La{\Lambda stv)]
\, dt \,ds\,.
\end{multline*}
To estimate the integrand use the lower bound on $|\phi_t|$,
\eqref{phiderivlowerbound}.
Moreover we have the upper bounds
\eqref{higherpsider} for the higher derivatives of $\psi$ (and then $\phi$)
which give $\partial_t^n\phi=O(2^{l(n+1)}br^2)$ for $n\ge 2$.
Each differentiation of the cutoff function produces a factor of
$(\lambda} \def\La{\Lambda 2^l k^{-1})^{1/2}$.
By the one-dimensional version of Lemma \ref{iterateddiffop} described in the subsequent remark the expression
$\lambda} \def\La{\Lambda^{-N_1}(\lambda} \def\La{\Lambda bv)^{(d_2-1)/2} |{\mathcal {L}}_t^{N_1}
[\eta_{\lambda} \def\La{\Lambda,k,l,b}(s,t) {\varpi}_1(\lambda} \def\La{\Lambda st v)]|$ can be estimated by a sum of $C(N_1)$ terms of the form
\begin{equation} \label{factorsintbyparts}
\lambda} \def\La{\Lambda^{-N_1}\frac{ (\lambda} \def\La{\Lambda 2^l/k)^{\alpha/2}}
{(2^m2^{3l/2}k^{1/2}(r+r(b))\lambda} \def\La{\Lambda^{-1/2})^\alpha}
\prod_{\beta \in {\mathfrak {I}}} \frac{2^{l(\beta+1)}kr^2}{(2^m 2^{3l/2}k^{1/2} (r+r(b))\lambda} \def\La{\Lambda^{-1/2})^\beta}
\end{equation}
where $\alpha\in \{0,\dots, N_1\}$, ${\mathfrak {I}}$ is a set of integers $\beta\in \{2,\dots, N_1+1\}$
with the property that $\sum_{\beta\in {\mathfrak {I}}}(\beta-1)=N_1-\alpha$.
If ${\mathfrak {I}}$ is the empty set then we interpret the product as $1$.
We observe that for $(r,v)\in {\mathcal {R}}_m^{II}$ we have $|r-r(b)|\approx 2^m(\lambda} \def\La{\Lambda k 2^l)^{-1/2}$. Thus if $2^m \le \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$
we have
$r\lesssim (2^lk)^{-1}$ and $r+r(b)\approx (2^lk)^{-1}$ while for
$2^m > \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$ we have
$r\approx r-r(b) \approx r+r(b)\approx 2^m (\lambda} \def\La{\Lambda k2^l)^{-1/2}$.
A short computation which uses these observations shows that in the case
$2^m \le \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$
the terms
\eqref{factorsintbyparts} are $\lesssim
2^{-m\alpha} \prod_{\beta\in {\mathfrak {I}}} \big[ 2^{-m\beta} (2^lk/\lambda} \def\La{\Lambda)^{\beta/2-1}\big]$.
In the case $2^m > \sqrt{\lambda} \def\La{\Lambda(2^l k)^{-1}}$
the terms
\eqref{factorsintbyparts} are dominated by a constant times
$ (\la2^{-l} k^{-1})^{\alpha/2}2^{-2m\alpha} \prod_{\beta\in {\mathfrak {I}}}2^{-m(\beta-1)}$.
In either case the terms \eqref{factorsintbyparts} are $\lesssim 2^{-mN_1}$
(since $\alpha+\sum_{\beta\in{\mathfrak {I}}}\beta\ge N_1$).
This means that we gain a factor of $2^{-mN_1}$ over the size estimate
\eqref{basicsizeest}.
Consequently,
\begin{equation}\label{OmIIest}
\iint_{\Omega^{II}_m}
|A^{k,l}_{\lambda} \def\La{\Lambda,b}(x,u)| \, dx \,du \, \lesssim 2^{m(d_1+d_2+1-N_1)} (2^lk)^{-\frac{d_1+1}2}
\sqrt{\tfrac{2^lk}{\lambda} \def\La{\Lambda}}\,.
\end{equation}
The assertion of the proposition follows then from \eqref{OmIest} and
\eqref{OmIIest}.
\end{proof}
\subsection{\it $L^1$ estimates for $T^k_\lambda} \def\La{\Lambda$ and $W_{j,n}$}
\begin{proof} [Proof of \eqref{TlakL1}]
Let us recall that $k\le 8\lambda} \def\La{\Lambda.$
If we sum the bounds in Proposition
\ref{vert-hor} in $b\in {\mathcal {T}}_{2^j,k,l}$ we get
$$\|A^{k,l}_{2^j}\|_{L^1} \lesssim (2^l k)^{-\frac{d_1+1}{2}},
\quad 2^l\lesssim \frac{2^{j}}k\,.$$
We also have
\begin{equation}
\label{KklminusAkl}
\|2^{-j\frac{d-1}{2}}K^{k,l}_{2^j}-A^{k,l}_{2^j}\|_{1}
\lesssim (2^l k)^{-1} 2^{-j\frac{d_1-1}{2}};
\end{equation}
for the part of $ K^{k,l}_{2^j}$ where $|u|\lesssim 1/{k\lambda} \def\La{\Lambda}$ this follows from Lemma \ref{smallulemmakl}, and for the remaining part from Lemma \ref{Bkllemma}.
Combining these two estimates, we find that
\begin{equation}
\label{Kkllsmall}
\|2^{-j\frac{d-1}{2}}K^{k,l}_{2^j}\|_{1}
\lesssim (2^l k)^{-\frac{d_1+1}{2}}, \quad 2^l\lesssim \frac{2^{j}}k.
\end{equation}
Moreover, by
Lemma \ref{Bkllemma} and Lemma \ref{kgrlalemma}, we have
\begin{equation}
\label{Kkllarge}
\|2^{-j\frac{d-1}{2}}K^{k,l}_{2^j}\|_{1}
\lesssim (2^l k)^{-1} 2^{-j\frac{d_1-1}{2}}, \quad 2^l\ge 10^6 \frac{2^{j}}k.
\end{equation}
Altogether this leads to
\begin{equation}
\label{Tlaklbounds}
2^{-j(d-1)/2}\|T_{2^j}^{k,l}\|_{L^1\to L^1} \lesssim (2^l k)^{-\frac{d_1+1}{2}}\,.
\end{equation}
and \eqref{TlakL1} follows if we sum in $l$.
\end{proof}
\subsection{\it An estimate away from the singular support} \label{awayestim}
For later use in the proof of Theorem \ref{multipliers} we need the following
observation.
\begin{prop}\label{junkaway} Let $\lambda} \def\La{\Lambda \ge 1$, $K_\lambda} \def\La{\Lambda$ be the convolution kernel for the
operator $\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L) e^{i\sqrt L},$ where $\chi\in{\mathcal {S}}(\Bbb R),$ and let $R\ge 10$.
Then, for every $N\in\Bbb N,$
$$\int_{\max\{|x|,|u|\} \ge R} |K_\lambda} \def\La{\Lambda(x,u)| \, dx\, du \le C_N (\lambda} \def\La{\Lambda R)^{-N}\,.
$$
Moreover, the constants $C_N$ depend only on $N$ and a suitable Schwartz norm of $\chi.$
\end{prop}
\begin{proof} This estimate is implicit in our arguments above, but it is easier to establish it as a direct consequence of the finite propagation speed of solutions to the wave equation \cite{melrose}. Indeed, write
$$
\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L) e^{i\sqrt L}=\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L) \cos{\sqrt L}
+i\lambda} \def\La{\Lambda\tilde\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L)\frac {\sin{\sqrt L}}{\sqrt L},
$$
with $\tilde\chi(s)=s\chi(s),$ and denote by ${\varphi}_\lambda} \def\La{\Lambda$ and ${\mathcal {P}}$ the convolution kernels for the operators $\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L)$ and $\cos{\sqrt L},$ respectively. Then ${\mathcal {P}}$ is a compactly supported distribution (of finite order). Indeed, ${\mathcal {P}}$ is supported in the unit ball with respect to the optimal control distance associated to the H\"ormander system of vector fields $X_1,\dots,X_{d_1},$ which is contained in the Euclidean ball of radius $10.$ Moreover, by homogeneity, ${\varphi}_\lambda} \def\La{\Lambda(x,u)=\lambda} \def\La{\Lambda^{d_1+2d_2}{\varphi}(\lambda} \def\La{\Lambda x,\lambda} \def\La{\Lambda^2 u),$ with a fixed Schwartz function ${\varphi}.$ Note also that by Hulanicki's theorem \cite{hulanicki}, the mapping taking $\chi$ to ${\varphi}$ is continuos in the Schwartz topologies. Since the convolution kernel $K_\lambda} \def\La{\Lambda^c$ for the operator $\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L) \cos{\sqrt L}$ is given by ${\varphi}_\lambda} \def\La{\Lambda *{\mathcal {P}},$ it is then easily seen $K_\lambda} \def\La{\Lambda^c(x,u)$ can be estimated by $C_N\lambda} \def\La{\Lambda^M(\lambda} \def\La{\Lambda|x|+\lambda} \def\La{\Lambda^2|u|)^{-N}$ for every $N\in\Bbb N,$ with a fixed constant $M.$ A very similar argument applies to $\tilde\chi(\lambda} \def\La{\Lambda^{-1}\sqrt L)\frac {\sin{\sqrt L}}{\sqrt L},$ and thus we obtain the above integral estimate for $K_\lambda} \def\La{\Lambda.$
\end{proof}
\section{$h_{\text{\rm iso}}^1\to L^1$ estimates for the operators ${\mathcal {W}}_n$}\label{hardyspaceestimates}
In this section we consider the operatots
${\mathcal {W}}_n=\sum_j W_{j,n}$ and prove the relevant estimate
in Theorem \ref{refinedh1thm}.
In the proof we shall use a simple $L^2$ bound which follows from the spectral theorem,
namely for $j_0>0$
\begin{equation} \label{WjnL2lemma}
\Big\| \sum_{j\ge j_0} W_{j,n}\Big\|_{L^2\to L^2} \lesssim
2^{-j_0(d-1)/2}\,.
\end{equation}
\subsection*{\it Preliminary considerations}
Let $\rho \le 1$ and let $f_\rho$ be an $L^2$-function satisfying
\begin{equation} \label{atomprop}
\begin{gathered}\|f_\rho\|_2 \le\rho^{-d/2}, \\
{\hbox{\roman supp}}(f_\rho)\subset Q_{\rho,E}:=\{(x,u): \max \{|x|,|u|\} \le \rho\}\,,
\end{gathered}
\end{equation}
and we also assume that
\begin{equation} \label{cancel} \iint f_\rho(x,u) dx\,du=0\,, \text{ if } \rho\le 1/2.
\end{equation}
In what follows we also need notation for the expanded
Euclidean "ball"
\begin{equation}\label{expandedball}
Q_{\rho,E, *}=\{(x,u): \max \{|x|,|u|\} \le C_*\rho\}\,,
\end{equation} where $C_*= 10(1+d_2\max_i\|J_i\|)$.
We begin with the situation given by \eqref{cancel}. By translation invariance and the definition of $h^1_{\text{\rm iso}}$ it will
suffice to
check that
\begin{equation}\label{atomest}
\|{\mathcal {W}}_n f_\rho\|_{L^1} \lesssim (1+n)2^{-n(d_1-1)/2}\,.
\end{equation}
We work with dyadic spectral decompositions for the operators $|U|$ and $\sqrt{L}$ and need to discuss how they act on the atom $f_\rho$.
For $j> 0$, $n\ge 0$, let $H_{j,n}$ be the convolution kernel
defined by
$$\chi_1(2^{-2j}L)\zeta_1(2^{-j-n}|U|)f= f*H_{j,n}.$$
From \eqref{jtspec} we see that
$$H_{j,n}=0 \text{ when } n> j+11\,.$$
\begin{lemma} \label{Hjnatomlem}
Let $\rho\le 1$, and $f_\rho$ be as in \eqref{atomprop}.
Then
(i) $\|f_\rho* H_{j,n}\|_1\lesssim 1$ and
\begin{equation} \label{Hjnatomaway}
\|f_\rho* H_{j,n}\|_{L^1(Q_{\rho,E,*}^{\complement})} \, \lesssim_N(2^{j}\rho)^{-N}.
\end{equation}
(ii) If $f_\rho$ satisfies \eqref{cancel} then
\begin{equation} \label{Hjnatom}
\|f_\rho* H_{j,n}\|_1 \lesssim \min \{ 1,\, 2^{j+n}\rho\}\,.
\end{equation}
\end{lemma}
\begin{proof}
By Hulanicki's theorem \cite{hulanicki} the convolution kernel of $\chi_1(L)$ is a Schwartz function $g_1$ on
${\mathbb {R}}^{d_1+d_2}$. The convolution kernel of $\zeta_1(|U|)$ is $\delta\otimes g_2$
where $\delta$ is the Dirac measure in ${\mathbb {R}}^{d_1}$ and $g_2$
is a Schwartz function on ${\mathbb {R}}^{d_2}$. Then
\begin{equation}\label{hjn} H_{j,n}(x,u) =
\int 2^{j(d_1+2d_2)}
g_1 (2^j x, 2^{2j}w) 2^{(j+n)d_2}g_2(2^{j+n}(u-w)) \, dw\end{equation}
Clearly $\|H_{j,n}\|_1 =O(1)$ uniformly in $j,n$ and since
$\|f_\rho\|_1\lesssim 1$ we get from Minkowski's inequality
$\|f_\rho*H_{j,n}\|_1\lesssim 1$.
For the proof of
\eqref{Hjnatomaway} we may thus assume
$2^j\ge 1/\rho$ and it suffices to verify that for every
$(y,v)\in Q_{\rho,E}$ the $L^1(Q_{A\rho,E}^\complement)$ norm of $(x,u)\mapsto$
$$
\int \frac{2^{j(d_1+2d_2)}}{(1+2^j|x-y|+2^{2j}|w|)^{N_1}}
\frac{2^{(j+n) d_2}} {(1+2^{j+n}|u-v-w+\frac 12\inn{\vec Jx}{y}|)^{N_1}}
dw
$$
is bounded by $C(2^j\rho)^{-N}$ if $N_1\gg N+ d_1+2d_2$. This is straightforward.
For the proof of \eqref{Hjnatom} we observe that \eqref{hjn} implies
$$
2^{-j}\| \nabla_x H_{j,n}\|_1 +2^{-j-n}\|\nabla_u H_{j,n}\|_1 = O(1)\,.
$$
Moreover $2^{-n}\||x|\nabla_u H_{j,n}\|_1 = O(1)$.
By the cancellation condition \eqref{cancel}
\begin{align*}
&f* H_{j,n}(x,u) \\
&\quad= \int f_\rho(y,v)\big[ H_{j,n}(x-y, u-v+ \tfrac 12 \inn{\vec Jx}{y})
-H_{j,n}(x, u)\big] \, dy\, dv
\\
&\quad=
-\int f_\rho (y, v) \Big(\int_0^1
\biginn{y}{\nabla_x H_{j,n}(x-sy, u-sv+\tfrac s2 \inn{\vec Jx}{y})}
\\
&\qquad\qquad+ \biginn {v+\tfrac 12 \inn{\vec Jx}{y}}{\nabla_u H(x-sy, u-sv+\tfrac s2 \inn{\vec Jx}{y})}\,ds\Big) dy\, dv\,.
\end{align*}
We also use $\inn{\vec Jx}{y})=\inn{\vec J(x-sy)}{y})$ and a change of
variable to estimate
$$\|f_\rho*H_{j,n}\|\lesssim \|f_\rho\|_1\,\rho\,
\big[ \|\nabla_x H_{j,n}\|_1
+ \|\nabla_u H_{j,n}\|_1+ \||x|\nabla_u H_{j,n}\|_1\big]\,,
$$
and \eqref{Hjnatom} follows.
\end{proof}
\begin{proof}[Proof of \eqref{atomest}] For $n>0$ split
$$
\begin{aligned}
{\mathcal {W}}_n f_\rho&=
\sum_{\substack {j\ge n-11\\2^j \rho< 2^{-10 n}}} W_{j,n}f_\rho
+
\sum_{\substack {j\ge n-11\\2^{-10 n}\le 2^j \rho\le 2^{10 n}}}
W_{j,n}f_\rho
+
\sum_{2^{10 n}< 2^j \rho}
W_{j,n}f_\rho
\\&:=
I_{n,\rho}+II_{n,\rho}+III_{n,\rho}\,.
\end{aligned}
$$
The main contribution comes from the middle term and by \eqref{Wjnest} and the estimate $\|f_\rho\|_1\lesssim 1$ we immediately get
\begin{equation}\label{twonrho}\|II_{n,\rho}\|_1 \lesssim (1+n)2^{-n(d_1-1)/2} \,.\end{equation}
Let ${\mathcal {J}}_n$ be as in \eqref{jndef}, so that $\#({\mathcal {J}}_n)=O(2^n)$.
We use the estimate
\eqref{Tlaklbounds}
in conjunction with
\eqref{Hjnatom} and estimate
\begin{align*}
\|I_{n,\rho}\|_1&\le \sum_{2^j \rho< 2^{-10 n}} \sum_{k\in{\mathcal {J}}_n}
\sum_{l=1}^\infty
\|2^{-j(d-1)/2}T_{2^j}^{k,l} (f_\rho*H_{j,n})
\|_1
\\&\lesssim\sum_{2^j \rho< 2^{-10 n}} \sum_{k\in{\mathcal {J}}_n}\sum_{l=1}^\infty
(2^l k)^{-\frac{d_1+1}2} 2^{n+j}\rho
\lesssim 2^{-n(9+ \frac{d_1-1}{2})}.
\end{align*}
We turn to the estimation of the term $III_{n,\rho}$.
Let ${\mathfrak {T}}_{\rho,n}$ be a maximal $\sqrt{\varepsilon \rho}$ separated set of
$[2^{n-6}, 2^{n+6}]$.
For each $\beta \in {\mathfrak {T}}_{\rho,n}$ let, for large $C_1\gg 1$,
\begin{equation}\label{nghood}
{\mathcal {N}}_{n,\rho}(\beta)=
\{(x,u): \big||x|-r(\beta)\big|\le \sqrt{C_1\rho},\quad|w(\beta, x, 4|u|)|\le
C_1\rho\}
\end{equation}
and
$${\mathcal {N}}_{n,\rho}=\bigcup_{\beta\in {\mathfrak {T}}_{\rho,n}} {\mathcal {N}}_{n,\rho}(\beta)\,.$$
Observe that ${\text{\rm meas}}({\mathcal {N}}_{n,\rho}(\beta))\lesssim_{C_1}
2^{-n(d_1+d_2-2)}\rho^{3/2}$ (by \eqref{rv(t)} and \eqref{slope}) and thus
${\text{\rm meas}}({\mathcal {N}}_{n,\rho})\lesssim_{C_1}\rho$.
We separately estimate the quantity $III_{n,\rho}$ on ${\mathcal {N}}_{n,\rho}$ and
its complement.
First, by the Cauchy-Schwarz inequality and \eqref{WjnL2lemma}
(with $2^{j_0}\approx 2^{10 n}\rho^{-1}$)
$$\| III_{n,\rho}\|_{L^1({\mathcal {N}}_{n,\rho})} \lesssim \rho^{1/2}
\|III_{n,\rho}\|_2
\lesssim (2^{-10 n}\rho)^{\frac{d-1}{2}}\rho^{1/2} \|f_\rho\|_2
$$
and, since $\rho^{d/2}\|f_\rho\|_2\lesssim 1$,
\begin{equation}\label{estonexcset}
\| III_{n,\rho}\|_{L^1({\mathcal {N}}_{n,\rho})} \lesssim 2^{-5(d-1)n}\,.
\end{equation}
In the complement of the exceptional set ${\mathcal {N}}_{n,\rho}$ we split the
term
$III_{n,\rho}$ as
$$III_{n,\rho}= \sum_{2^j \rho> 2^{10 n}}\sum_{k\in{\mathcal {J}}_n}\sum_{l=1}^\infty
(III_{n,\rho,j}^{k,l}+IV_{n,\rho,j}^{k,l})
$$
where
$$\begin{aligned}
& III_{n,\rho,j}^{k,l}=
2^{-j\frac{d-1}{2}} T^{k,l}_{2^j} [
(f_{\rho}*H_{j,n})\chi_{Q_{\rho,E,*}}]
\\
& IV_{n,\rho,j}^{k,l}=
2^{-j\frac{d-1}{2}} T^{k,l}_{2^j} [
(f_{\rho}*H_{j,n})\chi_{Q_{\rho,E,*}^\complement}]
\end{aligned}
$$ and $Q_{\rho,E,*} $ is as in \eqref{expandedball}.
From \eqref{Hjnatomaway} and \eqref{Tlaklbounds}
we immediately get
$\| IV_{n,\rho,j}^{k,l}\|_1\lesssim_N (2^lk)^{-(d_1+1)/2}
(2^j \rho)^{-N} $ and thus
$$ \sum_{2^j \rho> 2^{10 n}}\sum_{l=1}^\infty
\sum_{k\in{\mathcal {J}}_n} \| IV_{n,\rho,j}^{k,l}\|_1
\lesssim 2^{-10 nN}\,.
$$
It remains to show that
\begin{equation}\label{offexcset}
\sum_{l=1}^\infty\sum_{k\in{\mathcal {J}}_n} \sum_{2^j\rho>2^{10 n}}
\| III_{n,\rho,j}^{k,l}\|_{L^1({\mathcal {N}}_{n,\rho}^\complement)} \lesssim 2^{-n(d_1-1)/2}\,.
\end{equation}
Let $F_{j,n,\rho}=(f_\rho*H_{j,n})\chi_{Q_{\rho,E,*}}$, so that
$\|F_{j,n,\rho}\|_1\,\lesssim 1$.
We shall show that for $k\approx 2^n$
\begin{equation}\label{offexcsetmain}
\|F_{j,n,\rho}*A^{k,l}_{2^j}\|_{L^1({\mathcal {N}}_{n,\rho}^\complement)}
\lesssim_N (2^{j-n}\rho)^{-N} 2^{-(l+n)
\frac{d_1}{2}},
\quad 2^l\le 10^{8} 2^{j-n}\,,
\end{equation}
and \eqref{offexcset} follows by combining
\eqref{offexcsetmain} with the estimates \eqref{KklminusAkl} and
\eqref{Kkllarge}.
\medskip
{\it Proof of \eqref{offexcsetmain}.}
We split $A_{2^j}^{k,l}=\sum_{b\in {\mathcal {T}}_{2^j,k,l}}
A_{2^j,b}^{k,l}$
as in \eqref{bdecomp}.
For each $b\in {\mathcal {T}}_{2^j,k,l} $ we may assign a $\beta(b)\in
{\mathfrak {T}}_{\rho,n}$
so that $|\beta(b)-b|\le \sqrt{\varepsilon\rho}$.
Let
$ {\mathcal {T}}_{2^j,k,l}^\beta$
be the set of $b\in{\mathcal {T}}_{2^j,k,l}$ with
$\beta(b)=\beta$.
Then $\# {\mathcal {T}}_{2^j,k,l}^\beta\lesssim 2^{-n/2}\sqrt{ 2^{l+j}\rho}$.
In order to see \eqref{offexcsetmain} it thus suffices to show that
for $2^l\le 10^8 2^{j-n} $, $|\beta-b|\le\rho$,
\begin{equation} \label{offexcsetbeta}
\|F_{j,n,\rho}*A^{k,l}_{2^j,b}\|_{L^1(({\mathcal {N}}_{n,\rho}(\beta))^\complement)}
\lcs{N_1} (2^{j-n}\rho)^{-N_1} 2^{-(l+n)\frac{d_1+1}{2}} 2^{(n+l-j)/2}.
\end{equation}
To prove this we verify the following
\begin{equation}\label{geomclaim}
\text{{\it Claim: }}
\begin{aligned}
&\text{If } (\widetilde x,\widetilde u)\in
Q_{\rho,E, *}, \,
(x,u)\in ({\mathcal {N}}_{n,\rho}(\beta))^\complement
\text{ and } 2^{2m-j+n}\le \rho
\\& \text{then }
\big(|x-\widetilde x|, 4|u-\widetilde u+\tfrac12 \inn{\vec Jx}{\widetilde
x}|\big)\notin
{\mathcal {P}}_m(2^j, l,k; b) \,;
\end{aligned}
\end{equation}
${\mathcal {P}}_m(2^j,l,k;b)$ was defined in \eqref{cPm}.
Indeed the claim implies
$$
\big\|F_{j,n,\rho}*A^{k,l}_{2^j,b}\big\|_{L^1(({\mathcal {N}}_{n,\rho}(\beta))^\complement)}
\lesssim
\int_{\substack{(|x|, 4|u|)\notin \\{\mathcal {P}}_m(2^j, l,k; b)}}
|A^{k,l}_{2^j,b}(x,u)|dx\,du
$$
since
$\|F_{j,n,\rho}\|_1 =O(1)$
and \eqref{offexcsetbeta} follows from Proposition
\ref{vert-hor}.
To verify the claim \eqref{geomclaim} we pick $(x,u)
\notin {\mathcal {N}}_{n,\rho}(\beta)$
and distinguish two cases:
\begin{align*}
&\text{\it Case 1: $||x|-r(\beta)|\ge \sqrt{C_1\rho}$.}
\\
&\text{\it Case 2: $|w(\beta, |x|, 4|u|)| \ge C_1\rho $ and
$||x|-r(\beta)|\le \sqrt{C_1\rho}$.}
\end{align*}
It is clear that the conclusion of the claim holds if we can show that
under the assumption that $C_1$ in the definition \eqref{nghood} is chosen sufficiently large
(depending only on $\vec J$ and the dimension $d$)
we have for all $(\widetilde x,\widetilde u)\in Q_{\rho,E,*}$
\begin{align} \label{case1conclusion}
&\big||x-\widetilde x|-r(b)\big| \,\ge \, \frac{\sqrt {C_1\rho}}{2}
&\text{ in Case 1},
\\
\label{case2conclusion}
&\big|w(b, |x-\widetilde x|, 4|u-\widetilde u+ \tfrac 12
\inn{\vec Jx}{\widetilde x}|)\big| \ge \frac{C_1\rho}{2}
&\text{in Case 2}.
\end{align}
The Case 1 assumption implies for $(\widetilde x, \widetilde u)\in Q_{\rho,E,*}$ (and sufficiently large $C_1$)
\begin{align*}
&\big||x-\widetilde x|-r(b)\big| \,\ge \,
||x|-r(\beta)| -|\widetilde x|- |r(b)-r(\beta)|
\\
&\ge C_1 \rho^{1/2} - C_*\rho - C |b-\beta|2^{-n} \ge
\frac{\sqrt {C_1\rho}}{2}
\end{align*}
which is \eqref{case1conclusion}.
Now assume that $(x,u)$ satisfies the Case 2 assumption.
We then have for all
$(\widetilde x, \widetilde u)\in Q_{\rho,E,*}$
\begin{align*}
&\big|w(b, |x-\widetilde x|, 4|u-\widetilde u+ \tfrac 12
\inn{\vec J x}{\widetilde x}|)
- w(\beta,|x|, 4|u|)
\big|
\\
\,&\le\,\big|w(b, |x|, 4|u|)-w(\beta,|x|, 4|u|)\big|
\\&\quad+ \,\big |w(b, |x-\widetilde x|, 4|u-\widetilde u+\tfrac 12
\inn{\vec J x}{\widetilde x}) - w(b, |x|, 4|u|)\big|
\end{align*}
The first term on the right hand side can be estimated using
\eqref{wdiff2}
(with $(t,b)$ replaced by $(b,\beta)$), and we see that it is
$\le (C+\sqrt C_1)\rho$ under the present Case 2 assumption. The
second term on the right hand side is equal to
$$
\Big|4|u|-4|u-\widetilde u+\tfrac 12 \inn{\vec Jx}{ \widetilde x}|
-\frac{v'(b)}{r'(b)} (|x|-|x-\widetilde x|)\Big|
$$
and since the Case 2 assumption implies $|x|=O(1)$ we see that the
displayed expression is $O(\rho)$. Thus, if $C_1$ in the definition
is sufficiently large
we obtain \eqref{case2conclusion}. This concludes the proof of the
claim
\eqref{geomclaim} and thus the estimate \eqref{offexcsetmain}.
\medskip
We finally consider the case where $1/2<\rho\le 1,$ in which condition \eqref{cancel} is not required. This case can easily be handled by means of Proposition \ref{junkaway}. To this end, we decompose
$$
a(\sqrt{L})e^{i\sqrt{L}}=\sum_{j\ge 10}2^{-\frac{d-1} 2 j}g_j(2^{-j}\sqrt{L})e^{i\sqrt{L}},
$$
with $g_j(s)=2^{\frac{d-1} 2 j} a(2^{j}s)\chi_1(s).$ The family of functions $g_j$ is uniformly bounded in the Schwartz space. If $K_j$ denotes the convolution kernel for the operator $g_j(2^{-j}\sqrt{L})e^{i\sqrt{L}},$ we thus obtain from Proposition \ref{junkaway} the uniform estimates
$$\int_{\max\{|x|,|u|\} \ge 100} |K_j(x,u)| \, dx\, du \le C_N 2^{-jN}\,.
$$
This implies that $$\int_{\max\{|x|,|u|\} \ge 200}|(a(\sqrt{L})e^{i\sqrt{L}}f_\rho)(x)|dxdu\lesssim \|f\|_1\lesssim 1.$$
And, by H\"older's inequality, $$\int_{\max\{|x|,|u|\} \le 200}|(a(\sqrt{L})e^{i\sqrt{L}}f_\rho)(x)|dx\lesssim \|(a(\sqrt{L})e^{i\sqrt{L}}f_\rho)\|_2\lesssim\|f_\rho\|_2\lesssim 1.$$
This concludes the proof of Theorem \ref{refinedh1thm}.
\end{proof}
\section{Interpolation of Hardy spaces}\label{interpolation}
By interpolation for analytic families
Theorem \ref{main-theorem} can be deduced from the Hardy space estimate
if we show that $L^p(G)$ is an interpolation space for the couple
$(h^1_{\text{\rm iso}}(G), L^2(G))$, with respect to
Calder\'on's
complex $[\cdot,\cdot]_\vartheta$ method.
\begin{theorem} \label{interpolthm} For $1<p\le 2$,
\begin{equation}\label{interpol} [h^1_{\text{\rm iso}}(G), L^2(G)]_\vartheta = L^p(G), \quad
\vartheta =2-2/p,
\end{equation} with equivalence of norms.
\end{theorem}
\begin{proof}
We deduce \eqref{interpol} from an analogous formula for the Euclidean
local Hardy spaces $h^1_E,$ more precisely, the vector-valued extension
\begin{equation}\label{interpolvect}
[\ell^1(h^1_E), \ell^2(L^2)]_\vartheta = \ell^p(L^p), \quad
\vartheta =2-2/p.\end{equation}
Here $\ell^p\equiv \ell^p({\mathbb {Z}}^{d_1+d_2})$.
To do this one uses the method of retractions and coretractions
({\it cf}. \cite{triebelinterpol}); \eqref{interpolvect} follows from the
definition of the complex interpolation method if operators
\begin{align*}
R: \ & h^1_{\text{\rm iso}}+L^2
\to \ell^1(h^1_E) +\ell^2(L^2)
\\
S: \ &\ell^1(h^1_E) +\ell^2(L^2) \to h^1_{\text{\rm iso}}+L^2
\end{align*}
can be constructed such that
\[R: \begin{cases} h^1_{\text{\rm iso}}
\to \ell^1(h^1_E)
\\ L^2 \to \ell^2(L^2)
\end{cases}
\qquad S:\begin{cases} \ \ell^1(h^1_E)
\to h^1_{\text{\rm iso}}
\\
\ell^2(L^2) \to L^2 \end{cases}
\]
are bounded
and
$$SR= I, $$
the identity operator on $L^p$ or $h^1_{\text{\rm iso}}$.
To define $R$ and $S$ let $\varphi_1\in C^\infty_0({\mathbb {R}}^{d_1})$,
$\varphi_2\in C^\infty_0({\mathbb {R}}^{d_2})$ supported in
in $(-1,1)^{d_1}$ and $(-1,1)^{d_2}$ respectively
and such
that
for all $ x\in {\mathbb {R}}^{d_1}$, $u\in {\mathbb {R}}^{d_2}$
\begin{equation} \label{unity}\sum_{X\in {\mathbb {Z}}^{d_1}}
\varphi^2_1(x+X) =1, \quad
\sum_{U\in {\mathbb {Z}}^{d_2}}
\varphi^2_2(u+U) =1.
\end{equation}
We define $\varphi(x,u)=\varphi_1(x)\varphi_2(u)$ and set
\begin{multline*} Rf= \{R_{X,U} f\}_{(X,U)\in {\mathbb {Z}}^{d_1}\times{\mathbb {Z}}^{d_2}} \\
\text{ where }
R_{X,U} f(x,u)= \varphi(-x,-u) f(x+X, u+U+ \tfrac 12\inn{\vec J X}{x});
\end{multline*}
moreover for $H=\{H_{X,U}\}_{(X,U)\in {\mathbb {Z}}^{d_1}\times{\mathbb {Z}}^{d_2}}\in \ell^1(h^1_E) $
we set
\begin{multline*} SH(x,u) =\\
\sum_{(X,U)\in
{\mathbb {Z}}^{d_1}\times{\mathbb {Z}}^{d_2}} \varphi(X-x, U-u-\tfrac 12\inn{\vec J X}{x})
H_{X,U} (x-X, u-U- \tfrac 12\inn{\vec J X}{x})
\end{multline*}
One verifies quickly from \eqref{unity} that $SR$ is the identity.
We now examine the boundedness properties of $R$ and $S$.
For the $ h^1_{\text{\rm iso}}
\to \ell^1(h^1_E) $
of $R$ we consider a (Heisenberg-)$(P,\rho)$ atom $a$ with $P=(x_P,u_P)$ and
$\rho\le 1$.
Note that
$\varphi(-x,-u) a(x+X, u+U+ \inn{\vec J X}{x})$ is then supported on the
set of $(x,u)\in (-1,1)^{d_1+d_2}$ such that
$$|x_P-X-x|^2 +
|u_P-U-u -\inn{\vec J(X-x_P)}{x}|^2\le \rho^2.$$
Thus $R_{X,U} f$ is not identically zero only when $|X-x_P|+|U-u_P|\le
C_d$ some absolute constant $C_d$. And, since
$\inn{\vec J(X-x_P)}{X-x_P}=0$ we also see that in this case
the function \begin{equation}\label{atomchanged}(x,u)\mapsto a(x+X, u+U+ \tfrac 12\inn{\vec
J X}{x})
\end{equation} is supported in a
Euclidean ball of radius $C \rho$ with center $(x_P-X, u_P-U)$. Since the
cancellation
property (if $\rho\le 1/2$) is not affected by the change of variable we see
that the function \eqref{atomchanged} is equal to $c_b b$ where
$b$ is a Euclidean atom
and $|c_b|\lesssim 1$. Thus this function is in
$h^1_E$ with norm $\lesssim 1$. We also use that multiplication with
$\varphi(-x,-u)$ defines an operator which is bounded on the local
Hardy-space $h^1_E$.
Now it follows quickly that $R$ is
bounded as an operator from
$ h^1_{\text{\rm iso}}$ to $ \ell^1(h^1_E) $.
Indeed if $f=\sum_{c_\nu} a_{\nu}$ where $a_{\nu} $ are
$(P_\nu,r_\nu)$ atoms for suitable $r_\nu\le 1$ and $P_\nu$ then
\begin{align*}
&\|R f\|_{\ell^1(h^1_E)}
= \sum_{X,U}\big\|R_{X,U}\sum_\nu c_\nu a_{P,\nu}\big\|_{h^1_E}
\\ &\le C
\sum_{X,U} \sum_{\substack{\nu: |x_{P_\nu} -X|\le C_d\\ |u_{P_\nu}-U|\le C_d}}
|c_\nu| \le C' \sum_\nu |c_\nu|.
\end{align*}
This completes the proof of the
$h^1_{\text{\rm iso}}
\to \ell^1(h^1_E)$ boundedness of $R$.
We now show that $S$ maps
$\ell^1(h^1_E)$ boundedly to $h^1_{\text{\rm iso}}$.
We first recall that the operation of multiplication with a smooth bump function maps $h^1_E$ to itself ({\it cf}. \cite{goldberg}), thus $$\|\varphi(-\cdot) G_{X,U} \|_{h^1_E}\le C
\|G_{X,U} \|_{h^1_E}.$$
Using the atomic decomposition of $h^1_E$ functions we can
decompose
$$\varphi(-\cdot) G_{X,U} = \sum_{\nu} c_{X,U,\nu} a_{X,U,\nu}$$
where $\sum_{X,U,\nu} |c_{X,U,\nu}| \lesssim \|G\|_{\ell^1(h^1_E)}$ and the
$a_{X,U,\nu}$ are Euclidean atoms
supported in a ball
$$\{(x,u): |x-x_P|^2+ |u-u_P|^2 \le r^2\}\subset[-3,3]^{d_1+d_2};$$
with $P=P(X,U,\nu)$ and $r=r(X,U,\nu).$
Fix such an atom $a=a_{X,U,\nu}$. The function
\begin{equation} \label{movedatom}
\widetilde a_{X,U,\nu}: (x,u)\mapsto a_{X,U,\nu}(x-X, u-U-\tfrac 12\inn{\vec JX}{x})
\end{equation}
is supported in
\[\{(x,u): (|x-X-x_P|^2+ |u-U -\tfrac 12\inn{\vec JX}{x}
-u_P|^2)^{1/2} \le r\}\]
which is contained in the set of $(x,u)$ such that
$$\big(|x-(X+x_P)|^2+ |u-U - \tfrac 12\inn{\vec J(X+x_P)}{x} -u_P+
\tfrac 12 \inn{\vec Jx_P}{X+x_P}
|^2\big)^{\frac 12} $$
is $\le (1+\tfrac 32\sqrt{ d_1}) r$.
Here we have used that $|\inn{\vec J x_P}{ x-(X+x_P)}| \le |x_P| r $ and
$|x_P|\le 3\sqrt{d_1}$. The inclusion shows
that there is an constant independent of $X,U,\nu$ so that
function $\widetilde a_{X,U}/C$ is a Heisenberg atom associated with
a cube centered at $(X+x_P, U+u_P+\tfrac 12 \inn{\vec Jx_P}{X+x_P}$.
This statement holds at least if $r\le 1/(4d_1)$. If $r$ is close to
one then we can express $\widetilde a_{X,U}$ as a finite sum of
$6^d$ atoms supported in cubes of sidelength $1$.
Thus we see that the function in \eqref{movedatom} has $h^1_{{\text{\rm iso}}}$
norm $\lesssim 1$. This implies the
$ \ell^1(h^1_E)\to h^1_{\text{\rm iso}}$ boundedness of $S,$ since
it follows that
$$\|SG\|_{h^1_{{\text{\rm iso}}}}\lesssim \sum_{(X,U)}
\sum_{\nu} |c_{X,U,\nu}|
\|\widetilde a_{X,U,\nu}\|_{h^1_{{\text{\rm iso}}}}\lesssim \sum_{X,U,\nu}
|c_{X,U,\nu}|
\lesssim \|G\|_{\ell^1(h^1_E)}.
$$
Finally the
$ L^2
\to \ell^2(L^2)$ boundedness of $R$ and the
$ \ell^2(L^2)
\to L^2$ boundedness of $S$ are even more straightforward and
follow by modifications of the arguments.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main-theorem}]
By duality we may assume $1<p<2$. By scaling and symmetry
we may assume $\tau=1$. Let $a\in S^{-(d-1)(1/p-1/2)}$.
Consider the analytic family of operators
$${\mathcal {A}}_z= e^{z^2}\sum_{j=0}^\infty 2^{-jz\frac{d-1}{2}}
2^{j(d-1)(\frac 1p-\frac 12)}
\zeta_j(\sqrt L) a(\sqrt L)e^{i\sqrt{L}}.$$
We need to check that ${\mathcal {A}}_z$ is bounded on $L^p$ for $z=(2/p-1)$.
But for $\operatorname{Re\,}(z)=0$ the operators ${\mathcal {A}}_z$ are bounded on $L^2$;
and
for $\operatorname{Re\,}(z)=1$ we have shown that ${\mathcal {A}}_z$ maps $h^1_{\text{\rm iso}}$ boundedly to $L^1,$ by
Theorem
\ref{h1thm}.
We apply the abstract version of the
interpolation theorem for
analytic families in conjunction with
Theorem \ref{interpolthm} and the corresponding standard version
interpolation result for $L^p$ spaces; the result is that
${\mathcal {A}}_\theta$ is bounded on $L^p$ for $\theta= 2/p-1$. This proves Theorem
\ref{main-theorem}.
\end{proof}
\section{Proof of Theorem \ref{multipliers}}
We decompose $m=\sum_{k\in {\mathbb {Z}}} m_k$ where $m_k$ is supported in
$(2^{k-1}, 2^{k+1})$ and where $h_k= m_k(2^k\cdot)$ satisfies
$$\sum_{\ell>1} \sup_k\int_{2^\ell}^{\infty}|\widehat h_k(\tau)|
\tau^{\frac{d-1}2}\, d\tau \,\le\, A \,.
$$
By the translation invariance and the usual
Calder\'on-Zygmund arguments (see, e.g., \cite{steinharmonic})
it
suffices to prove that for all
$\rho>0$ and for all $L^1$ functions $f_\rho$ supported in the
Koranyi-ball $Q_\rho:=Q_\rho(0,0)$ and satisfying
$\int f_\rho \,dx=0$ we have
\begin{equation}\label{CZL1}
\sum_k \iint_{Q_{10\rho}^\complement}
|m_k(\sqrt{L})f_\rho| \,dx \lesssim A+\|m\|_\infty
\end{equation}
Let $\chi_1\in C^\infty_0$ be supported in $(1/5,5)$ so that $\chi_1(s)=1$ for $s\in [1/4,4]$. Then for each $k$ write
$$m_k(\sqrt L)= h_k(2^{-k}\sqrt L)\chi_1(2^{-k}\sqrt L)
\,=\, \int \widehat {h_{k}}(\tau)
\chi_1(2^{-k}\sqrt L) e^{i 2^{-k} \tau\sqrt{L}}
d\tau
\,.
$$
By scale invariance and Theorem \ref{main-theoremL1},
the $L^1$ operator norm of the operator
$\chi_1(2^{-k}\sqrt L) e^{i 2^{-k} \tau\sqrt{L}}$ is
$O(1 +|\tau|)^{(d-1)/2}$ and thus
\begin{equation}\notag
\|m_k(\sqrt L)\|_{L^1\to L^1}
\lesssim
\int_{-\infty}^\infty
|\widehat {h_{k}}(\tau)| (1+|\tau|)^{\frac{d-1}{2}}
d\tau \,.
\end{equation}
Also observe that since the convolution kernel of $\chi_1(\sqrt L) $
is a Schwartz kernel we can use the cancellation and
support properties of
$f_\rho$ to get, with some $\varepsilon>0$,
$$\|\chi_1(2^{-k}\sqrt L) f_\rho\|_1 \lesssim \min\{1, (2^k\rho)^{\varepsilon}\}
\|f_\rho\|_1\,.
$$
Thus the two preceding displayed inequalities yield
\begin{align}\notag
\sum_{k: 2^k\rho \le M}
\|m_k(\sqrt L)f_\rho\|_{1}
&\le C_M
\sup_k \int_{-\infty}^\infty
|\widehat {h_{k}}(\tau)| (1+|\tau|)^{\frac{d-1}{2}}
d\tau \, \|f_\rho\|_1
\\&\lcs{M} (\|m\|_\infty + {\mathfrak {A}}_2) \, \|f_\rho\|_1\,
\label{firstgeomsum}
\end{align}
where for the last estimate we use
$|\widehat {h_{k}}(\tau)|\le \|h_k\|_\infty \lesssim\|m\|_\infty$ when $|\tau|\le 2$.
We now consider the terms for $2^k\rho\ge M$ and $M$ large,
in the complement of the expanded Koranyi-ball $Q_{\rho,*}= Q_{C\rho}$
(for suitable large $C\gg 2$).
By a change of variable and an application of
Proposition \ref{junkaway},
\begin{align*}
&\big\|e^{i2^{-k}\tau \sqrt L} \chi_1(2^{-k}\sqrt L)
f_\rho \big\|_{L^1(Q_{\rho,*}^\complement)}=
\big\|e^{i\sqrt L} \chi_1(\tau^{-1}\sqrt L)
f_\rho^{2^k/\tau} \big\|_{L^1(Q_{C_*\tau^{-1}2^k \rho}^\complement)}
\\ &\lesssim (2^k\rho \tau^{-1})^{-N}\quad \text{ if $2^k\rho \gg \tau$}\,,
\end{align*}
where $f_\rho^{2^k/\tau}$ is a re-scaling of $f_\rho$ such that $\|f_\rho^{2^k/\tau}\|_1=\|f_\rho\|_1\lesssim 1.$
Hence if $M$ is sufficiently large then for $2^k\rho>M$
\begin{multline*}
\|m_k(\sqrt L)f_\rho\|_{L^1(Q_{\rho,*}^\complement)}\,\lesssim_N\,\|f_\rho\|_1
\Big[\int_{|\tau|>2^k\rho}
|\widehat {h_k}(\tau)|(1+|\tau|)^{\frac{d-1}{2}} d\tau
\\+(2^k\rho)^{-N}
\int_{|\tau|\le 2^k\rho}|\widehat {h_k}(\tau)|(1+|\tau|)^{-N} d\tau\Big]
\end{multline*} and thus
\begin{equation}\label{largekcontribution}
\sum_{2^k\rho>M} \|m_k(\sqrt L)f_\rho\|_{L^1(Q_{\rho,*}^\complement)}
\lesssim \|m\|_\infty + \sum_{k: 2^k\rho>M} {\mathfrak {A}}_{2^k\rho}\,.
\end{equation}
The theorem follows from \eqref{firstgeomsum} and
\eqref{largekcontribution}.
\smallskip\hfill Q.E.D.\medskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,647
|
\section{Structural overview}
This work focusses on the recent developments in the area of superradiant scattering, primarily in asymptotically global Anti-de Sitter spacetimes, aiming at reviewing what the authors consider significant advances on the topic, in a manner that should be accessible to most readers with background in General Relativity. The work starts in section \ref{sec:sec2} with a brief motivation for the interest behind the phenomenon of superradiance in asymptotically global Anti-de Sitter spacetimes, followed by a short presentation of global AdS itself and its interesting properties. The section finishes off with an exact definition of an asymptotically AdS space, given in a few different ways. Section \ref{sec:sec3} is devoted to an introduction to the basic concepts behind superradiance illustrated by a simple example and an overview of the methods for calculating superradiant modes in different spacetimes. Chapters \ref{sec:sec4} and \ref{sec:sec5} are committed to reviewing the work done on superradiance in the specific cases of Reissner-Nordstr\"om-AdS and Kerr-AdS, respectively, with an emphasis on its effect on the stability of the two spacetimes. Finally, concluding remarks are gathered in the conclusion, followed by a list of references.
\thispagestyle{empty}
\pagenumbering{gobble}
\clearpage
\newpage
\pagenumbering{arabic}
\setcounter{page}{1}
\section{Introduction}\label{sec:sec2}
\subsection{Motivation}
Even if General Relativity was discovered just a bit more than hundred years ago, it still has not ceased to surprise us. After finding a particular solution to Einstein equation, the most tempting and logical thing to do is to investigate its behaviour under perturbations. It is in this way that one might hope to uncover the complete analytical beauty of the theory and understand more about the structure of spacetime. Moreover, there is no system in nature that is truly isolated from external influences, thus it is highly likely that the results of perturbation theory might be relevant to astrophysical observations. Following this line of thoughts, one usually starts from the simplest model there is and builds slowly on complexity. In General Relativity this corresponds to the vacuum Einstein equations with constant curvature. From the three different solutions in this case, determined by the curvature's sign, Anti-de Sitter space (with negative curvature), which is the main background spacetime in this work, stands out with a crucial difference - its conformal boundary is timelike. This implies that in order to have a well-defined Cauchy problem, one has to impose boundary conditions at infinity, with the physically relevant ones turning out to be acting like a reflecting wall. This is why AdS becomes important in the study of the other main aspect of this report - superradiance - the phenomenon in which one can extract energy from a rotating or charged black hole by scattering waves off of its horizon, depending on a certain condition satisfied by their frequency - a generalisation of the Penrose process for particles draining rotational energy from a Kerr black hole. It is Teukolsky and Press who first conjectured in \cite{press1972floating} that if the Kerr black hole is confined in a reflecting box, then the process of superradiant scattering will go on indefinitely, resulting in an exponentially growing instability. Nevertheless, black holes enclosed by perfectly reflecting walls are not something one expects to observe in nature - and even if massive fields can lead to confining potentials with trapping regions for the scattered waves - it is AdS that is the perfect system for the study of superradiance due to its reflective boundary conditions, which provide a natural confining mechanism for the radiation.\\
\hspace*{5mm}However, the importance of analysing superradiance in asymptotically AdS spacetimes does not come only due to the possibility of extending the conclusions to astrophysical systems\cite{dolan2007instability,witek2013superradiant} by juxtaposing them with the scenario of a massive field creating a confining potential around a black hole with a characteristic lengthscale similar to the radius of curvature of an AdS system. It also has implications on the stability of the spacetime - whether a solution is stable to a generic perturbation or not is vital, not only because this determines the actual significance of the theoretical construction, but also because it enables one to assess one's understanding of the phase space of the system under consideration. As it will be presented in the late part of this review - in the case of Kerr-AdS investigating its stability subject to superradiance has lead to the discovery that it is not the only stationary solution in asymptotically AdS spacetimes in four dimensions. Furthermore, there is growing evidence that its superradiant instability might have an endpoint that contradicts one of the cosmic censorship hypotheses - a result that will definitely change the way we look at General Relativity in four dimensions. On the other hand, even if not one of the main aspects of this work - the famous AdS/CFT correspondence should not be omitted. The significance of superradiance in this context comes from the fact that the effects of perturbations on the classical side can be translated into dynamical behaviour of thermal fluctuations on the field theory side. With this side remark we go back to the two points made about Kerr-AdS, as they represent some of the main results of the research in the area in recent years and them we would like to address in this essay. With this aim in mind, we take on a brief tour of the physics and mathematics behind these statements, starting from the definition of the first key ingredient in the study - pure AdS.
\subsection{Pure Anti-de-Sitter spacetime}
Anti-de-Sitter (${\rm AdS}_d$) is uniquely defined as the maximally symmetric solution of the vacuum Einstein equation with constant\footnote{With the only other two solutions with constant curvature (0 and positive) being Minkowski and De-Sitter space} negative cosmological constant $\Lambda$ in $d$ dimensions
\begin{align}
R_{ab}=\frac{2\Lambda}{d-2}g_{ab},
\end{align}
where
\begin{align}
\Lambda=-\frac{(d-1)(d-2)}{2L^2},\quad\mbox{and}\quad R_{abcd}=\frac{R}{d(d-1)}\left(g_{ac}g_{bd}-g_{ad}g_{cb}\right),\label{eq:RiemannAdS}
\end{align}
with $L$ being the radius of curvature and the characteristic lengthscale for ${\rm AdS}_d$. The second equation above implies a vanishing Weyl tensor $C_{abcd}=0$ and the symmetry group of the space is $O(d-1,2)$. The most intuitive way to visualise Anti-de-Sitter space is by embedding it in Euclidean space $\mathbb{R}^{2,d-1}$ as a hyperboloid defined by the equation
\begin{align}
X_0+X_d-\sum\limits_{i=1}^{d-1}X_i^2=L^2,\label{eq:Hyperboloid}
\end{align}
which is readily solved in coordinates $(\tau,\rho,\theta_1,...,\theta_d-3,\phi)$ by
\begin{equation*}
\begin{aligned}[c]
X_0&=L\cosh\rho\cos \tau\notag\\
X_d&=L\cosh\rho\sin \tau\notag\\
X_i&=L\sinh\rho\hat{\Omega}_i\notag\\
&\hspace*{-4.5mm}\sum\limits_{i=1}^{d-1}\hat{\Omega}_i=1\notag\\
\end{aligned}
\qquad\qquad
\begin{aligned}[c]
\hat{\Omega}_1&=\rho\cos\theta_1\notag\\
\hat{\Omega}_2&=\rho\sin\theta_1\cos\theta_2\notag\\
\vdots
\\
\hat{\Omega}_{d-2}&=\rho\sin\theta_1...\sin\theta_{d-4}\cos\theta_{d-3}\notag\\
\hat{\Omega}_{d-1}&=\rho\sin\theta_1...\sin\theta_{d-4}\sin\theta_{d-3}\notag,
\end{aligned}
\end{equation*}
where $\rho\in[0,\infty)$ and $\tau\in[0,2\pi)$, while the $\hat{\Omega}_i$'s parametrise an $S^{d-2}$ sphere with $\theta_1,...,\theta_{d-4}\in[0,\pi]$ and $\theta_{d-3}\in[0,2\pi)$\footnote{For $d=4$ one usually denotes $\theta_{d-3}$ by $\phi$}. In this way the metric for ${\rm AdS}_d$ acquires the form
\begin{align}
ds^2=L^2\left(-\cosh^2\hspace*{-0.5mm}\rho\,d\tau^2+d\rho^2+\sinh^2\hspace*{-0.5mm}\rho\, d\Omega_{d-2}^2\right),\label{eq:AdSGlobal1}
\end{align}
whereby intuitively looking at the $\rho\rightarrow0$ limit, the topology of the space can be inferred to be $S^1\times \mathbb{R}^{d-1}$, as the metric behaves like $ds^2\approx L^2\left(-d\tau^2+d\rho^2+\rho^2d\Omega_{d-2}^2\right)$.
However, due to the periodicity of $\tau$ closed timelike curves are allowed to exist in the spacetime, leading to the violation of causality. The usual approach to get around this problem (in the above way the space is also not simply connected) is to consider the universal cover of the space by effectively unrolling the circle $S^1$ and extending the limits of $\tau$ to $\tau\in(-\infty,\infty)$ (corresponding to infinitely many loops around the hyperboloid), which eliminates the possibility for closed timelike curves and changes the topology to that of $\mathbb{R}^d$. This gives the definition of biggest interest to physicists of ${\rm AdS}_d$ in global (as it covers the whole space) coordinates. There are coordinate singularities at $\rho=0$ and $\theta_i=0,\pi$, with the latter being the usual ones for spherical coordinates. Continuing in this setting, one can make the change of variables (and swapping $\tau$ for $t$)
\begin{align}
\tan\chi=\sinh\rho,\quad\mbox{with}\quad\chi\in\left(0,\frac{\pi}{2}\right),
\end{align}
leading to the metric form
\begin{align}
ds^2=\frac{L^2}{\cos^2\chi}\left(-dt^2+d\chi^2+\sin^2\chi\,d\Omega^2_{d-2}\right),
\end{align}
\begin{figure}[b]
\centering
\subfloat[\textit{Global ${\rm AdS}_d$ \eqref{eq:AdSGlobal1} (the solid-lines cylinder) which is conformally equivalent to one half ($0\leq\chi\leq\pi/2$) of the Einstein static universe (dashed cylinder).}\label{fig1a:AdScylinder}]{ %
\includegraphics[width=0.32\linewidth]{AdScylinder}}\hfill
\subfloat[\textit{${\rm AdS}_4$ as given by \eqref{eq:AdSPenroseMetric} on the hyperboloid \eqref{eq:Hyperboloid} - covering only a part of it.}\label{fig1b:AdShyper}]{ %
\includegraphics[width=0.37\linewidth]{AdShyper}}\hfill
\subfloat[\textit{Penrose diagram of ${\rm AdS}_4$ as given by \eqref{eq:AdSPenroseMetric} after conformal compactification, whereby each point represents a 2-sphere.}\label{fig1c:AdSPenrose}]{ %
\includegraphics[width=0.25\linewidth]{AdSPenrose2}}
\caption{}
\label{fig1:AdSDiagrams}
\end{figure}
\hspace*{-2mm}which is conformally equivalent to one half of the Einstein Static Universe due to the limits of $\chi$. There are two properties of ${\rm AdS}_d$, evident from the above metric, key to for the main discussion of this work. Firstly, the conformal boundary $\mathcal{I}$ (figure \ref{fig1a:AdScylinder}), corresponding to $\chi=\pi/2$ ($\rho=\infty$), is a timelike hypersurface (in contrast to Minkowski and De-Sitter, where it is null- and spacelike, respectively), given by
\begin{align}
d\tilde{s}^2=-dt^2+d\Omega^2_{d-2},\quad\mbox{with topology }\mathbb{R}\times S^{d-2},
\end{align}
which is clearly by itself conformally flat. Its timelike character implies that Anti-de-Sitter is not globally hyperbolic - there does not exist a complete Cauchy surface in the space. Whatever family of spacelike surfaces one takes, there will always be a null geodesic that does not intersect a given such surface anywhere - e.g. surfaces of $t=const$ cover the space completely, but it is straightforward to observe that taking a null geodesic coming out from $\mathcal{I}$, at a point above the surface itself, proves the above statement in that case. This hints that to have a well-defined Cauchy problem in AdS, one must not only specify the initial data on a surface, but one must also impose appropriate boundary conditions at the conformal boundary. In fact, this was rigorously demonstrated in 1995 in \cite{friedrich1995einstein} and will be discussed in more detail in a short while. The second interesting feature of this spacetime is that null geodesics reach $\mathcal{I}$ in finite coordinate time\footnote{Whereas timelike ones never do.}, which is easily shown in another very often utilised set of coordinates for global ${\rm AdS}_d$, derived by making the following transformation in \eqref{eq:AdSGlobal1}
\begin{align}
r=L\sinh\rho,\quad\mbox{and}\quad t=L\tau
\end{align}
resulting in
\begin{equation}
ds^2=-\left(1+\frac{r^2}{L^2}\right)dt^2+\left(1+\frac{r^2}{L^2}\right)^{-1}dr^2+r^2d\Omega^2_{d-2}.\label{eq:globalAdS}
\end{equation}
By taking the normalisation condition for a radial null geodesic $g_{ab}u^au^b=0$, with $u^a=dx^a/d\tau$ - the tangent vector to the geodesic - a straightforward integration shows that
\begin{align}
\vartriangle\hspace{-1mm}t=\int\limits_{0}^{\infty}\frac{dr}{1+\frac{r^2}{L^2}}=\frac{\pi L}{2},
\end{align}
where $\vartriangle\hspace{-1mm}t$ is some finite time interval, while $r\rightarrow\infty$ corresponds to $\rho\rightarrow\infty$ where the conformal boundary $\mathcal{I}$ is located. A similar calculation for timelike geodesics leads to a divergent integral, indicating that they never reach $\mathcal{I}$.\\
\hspace*{5mm}In order to obtain the Penrose diagram of Anti-de-Sitter space, it is worth considering the $d=4$ case in yet another set of coordinates which represent a solution of \eqref{eq:Hyperboloid} - namely
\begin{align}
\begin{array}{rcl}
\left.\begin{array}{@{}l@{}}
X_0=L\sin t\vspace{1mm}\\[\jot]
X_1=L\cos t\sinh\rho\cos\theta\vspace{1mm}\\[\jot]
X_2=L\cos t\sinh\rho\sin\theta\cos\phi\vspace{1mm}\\[\jot]
X_3=L\cos t\sinh\rho\sin\theta\sin\phi\vspace{1mm}\\[\jot]
X_4=L\cos t\cosh\rho
\vspace{1mm}\end{array}\right\}\Rightarrow ds^2=L^2\left[-dt^2+\cos^2t\left(d\rho^2+\sinh^2\rho\,d\Omega^2\right)\right]\\
\end{array}\label{eq:AdSPenroseMetric}
\vspace*{7.5mm},
\end{align}
where $t\in(-\infty,\infty)$, $\rho\in[0,\infty)$, $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$ with apparent singularities at $t=\pm\frac{\pi}{2}+n\pi$, $n\in\mathbb{Z}$ and $\mathcal{I}$ is approached at $\rho\rightarrow\infty$. The above metric does not cover the whole hyperboloid, as it does not extend along the curving bits of the manifold, as seen in figure \ref{fig1b:AdShyper}, but by pulling a conformal factor of $L^2\cos^2t$ and making the transformation $t\rightarrow\tan t$ one can easily obtain an illuminating Penrose diagram for ${\rm AdS}_4$ presented in \ref{fig1c:AdSPenrose}. The worldlines of $\rho,\theta,\phi=const$ correspond to timelike geodesics (normal to surfaces of constant $t$) and as is evident from the figure - they all emanate from the same point (which without loss of generality can be taken to be $t=\pi/2$ as pure AdS is a homogeneous space) and converge at a point distance $\pi$ in $t$ away, just to defocus again and reconverge at another point further up by $\pi$ along $t$. This way it is easily observed that for a given event there are regions of space in its future light cone that cannot be reached by any timelike geodesic, including the conformal boundary $\mathcal{I}$. Therefore the infinite chain of diamonds, two of which are given in \ref{fig1c:AdSPenrose}, represents the set of points, reachable from our chosen spacetime event by timelike geodesics. This can ultimately be used as another way of seeing that a Cauchy surface in AdS cannot be found and that boundary conditions play a vital role in doing physics in AdS.\\
\hspace*{5mm}In the next section we will move on from the pure Anti-de-Sitter space and define what one means by asymptotically AdS spacetimes - which will enable us to delve later into the world of superradiance in RN-AdS and Kerr-AdS.
\subsection{Asymptotically global AdS spacetimes}\label{sec:AAdS}
The arguments in this section will be presented for $d=4$, but in general they apply for all $d\geq4$.\\
\hspace*{5mm}As mentioned in the previous section, AdS is the maximally symmetric solution of the vacuum Einstein equation with negative cosmological constant, henceforth, for $\Lambda<0$ it plays the role that Minkowski plays for flat spacetimes. It is therefore not only natural to think about the concept of being asymptotically AdS (by which in this work we mean exclusively asymptotically global AdS), but it is also needed when one wants to explore more thoroughly the properties and dynamics of black holes and matter in Anti-de Sitter. This is most often carried out with the tools of perturbation theory which implies that one should find a proper way of introducing perturbations, such that they are generalised enough in order to reveal new things about the system, while keeping the spacetime well-defined and preserving its structure - by which, as in the case of asymptotically flat spacetimes, it is understood the asymptotic one. Therefore, a definition of an asymptotically AdS spacetime is required and it is supplied by \cite{henneaux1985asymptotically} in the form of three requirements on the imposed boundary conditions at spatial infinity:
\begin{enumerate}[label=\protect\color{black}$\blacklozenge$]
\setlength\itemsep{-1mm}
\item They should be invariant under the global AdS symmetry group $O(3,2)$
\item They should make the surface integrals associated with the generators of the AdS group $O(3,2)$ finite
\item They should include the asymptotic behaviour of the Kerr-AdS metric
\end{enumerate}
The first of these is straightforward - if it were not the case, then a symmetry transformation could take an allowed set of conditions to one which is not, making the whole procedure meaningless. The second requirement is based on the canonical formulation of the problem - wherein one rewrites the otherwise vanishing Hamiltonian of the theory by adding surface integrals corresponding to the generators of the $O(3,2)$ group, which make the variational derivatives of the canonical variables well defined, thus enabling the exploration of the dynamics of the system. If these surface terms are not finite, the newly written Hamiltonian will diverge, hence the second condition. The last requirement is what ensures that the boundary conditions are not too restrictive in the sense that they allow for metrics that are of interests to physicists to be considered and Kerr-AdS, as in the case of $\Lambda=0$ and pure Kerr, is what is reasonably expected to be the configuration to which isolated systems asymptote in AdS settings. By considering possible perturbations that obey these three points (most simply achieved for gravitational ones by acting with $O(3,2)$ on the metric of Kerr-AdS itself, as it has been defined to be asymptotically AdS, and looking at the decays of the components at spatial infinity), it can be shown that $O(3,2)$ will be realised as the asymptotic symmetry group at spatial infinity and given that `reflective' boundary conditions are imposed there, then $\mathcal{I}$ will be conformally flat. The last bit can be ensured by looking at the Weyl tensor and its asymptotic behaviour, but involves some technicalities and will not be presented here - a more detailed discussions can be found in \cite{henneaux1985asymptotically,ashtekar2000asymptotically} and references therein - crucially, certain requirements on the decay of the components of the Weyl tensor are derived. On a further note, reflective in this context implies that allowed perturbations of Anti-de-Sitter should be described as standing waves with a node at the conformal boundary $\mathcal{I}$. Combining these observations with the second condition above implies that the asymptotic structure of AdS is conserved - that is the boundary metric is preserved.\\
\hspace*{5mm}The notion of an asymptotically AdS space can also be formulated in the spirit of the textbook definition of asymptotic flatness based on conformal compactification \cite{ashtekar2000asymptotically}, with the obvious difference that spatial infinity is required to approach that of AdS, rather than Minkowski - i.e. have an $\mathbb{R}\times S^2$ topology and vanishing fluxes across it. However, this will not be laid out here and the interested reader is referred to the above article for a good presentation of the topic.\\
\hspace*{5mm}Finally, a more physically intuitive elucidation of being asymptotically AdS, in light of our definition of Anti-de-Sitter \eqref{eq:RiemannAdS}, and motivated by conformal compactification comes from \cite{skenderis2002lecture} - namely: \textit{`An Asymptotically AdS metric is a conformally compact Einstein metric'}. This is actually fairly straightforward to deduce - taking a conformally compact manifold\footnote{For a discussion on the definition of a conformally compact manifold the reader is redirected to \cite{penrose1988spinors}.} $M$ with metric $g$, such that
\begin{align}
\tilde{g}=z^2g,\label{eq:ConfMetric}
\end{align}
where $\tilde{g}$ is the conformal metric and $z$ a smooth positive function on $M$, and working to leading order in $z$ for $z\rightarrow0$ (where the conformal boundary is located), by just plugging \eqref{eq:ConfMetric} into the definition of the Christoffel symbols and then taking the leading order contribution and inserting it in the definition of $R_{\mu\nu\rho\sigma}$ in terms of $\Gamma^\lambda_{\mu\nu}$, it is found that
\begin{align}
R_{\mu\nu\rho\sigma}(g)=\tilde{g}^{\tau\lambda}\partial_{\tau}z\partial_{\lambda}z\left(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}\right)+\mathcal{O}\left(z^{-3}\right),\label{eq:RiemannConformal}
\end{align}
Furthermore, by demanding that $g$ is a solution to the vacuum Einstein equation with negative cosmological constant, it can be derived that $\tilde{g}^{\tau\lambda}\partial_{\tau}z\partial_{\lambda}z=1/L^{2}$ and thus one sees in the limit $z\rightarrow0$ that equation \eqref{eq:RiemannConformal} approaches \eqref{eq:RiemannAdS}, hence the above definition.\\
\hspace*{5mm}
Having given a brief introduction to Anti-de-Sitter space, in the next section the main topic of this work - superradiance - will be introduced, but firstly in a heuristic fashion by using an example in the flat case of Kerr, while later its natural extension to black holes in AdS will be considered.
\section{Superradiance}\label{sec:sec3}
\subsection{Simple example and discussion}\label{sec:SimpleEx}
The idea of superradiance or superradiant scattering is usually introduced as a generalisation of the Penrose process to waves, but this will not be the approach taken here\footnote{For a good description of the Penrose process the reader is referred to \cite{wald2010general}.}. It will be rather illustrated with a simple example in the Kerr spacetime, followed by a short discussion of its appearance in different contexts and its implications on the studied systems.\\
\hspace*{5mm}Consider the Kerr spacetime with two spacelike surfaces $\Sigma$ and $\Sigma'$, both stretching from $i^0$ to $\mathcal{H^+}$, with $\Sigma'$ being entirely to the future of $\Sigma$. Furthermore, define $H$ and $H'$ as the intersections of $\Sigma $ and $\Sigma'$ with the future event horizon and take $\mathcal{N}$ to be the part of $\mathcal{H}^+$ from $H$ to $H'$. Moreover, as Kerr is a stationary spacetime, total energy of matter on a spacelike hypersurface can be defined naturally as
\begin{align}
E(\Sigma)=-\int_{\Sigma}\star J,
\end{align}
where $J_a=-T_{ab}k^b$ is the conserved energy-momentum 4-vector and $k^b$ is the timelike Killing vector field. Using this, it is easily shown that
\begin{align}
E(\Sigma')-E(\Sigma)=\int_{\mathcal{N}}\star J,\label{eq:HorizonFlux}
\end{align}
which gives a definition of the flux across the horizon as the difference between the energies of the two spacelike hypersurfaces. To continue, take matter to be given by a massless scalar field with stress-energy tensor $T_{ab}=\partial_a\psi\partial_b\psi-\frac{1}{2}g_{ab}\partial^c\psi\partial_c\psi$. As the spacetime is stationary and axisymmetric with corresponding commuting Killing vector fields - $\partial_t$ and $\partial_\phi$ - the scalar field can be decomposed as $\psi(t,r,\theta,\phi)=Re\left[\psi_0\left(r,\theta\right)e^{-i\omega t}e^{im\phi}\right]$, where $\omega$ is a frequency and $m$ - an integer - the azimuthal quantum number. By a simple brute force calculation it can be quickly shown that for $0<\omega<m\Omega_H$ the right hand side of equation \eqref{eq:HorizonFlux} is positive. To this end, take the 4D Kerr metric in Kerr coordinates $(v,r,\theta,\chi)$ and note that $\mathcal{N}$ is a three-dimensional manifold, hence in order to carry out the integration of the three form $\star J$ one just needs to specify
\begin{equation}
\left(\star J_{\nu\theta\chi}\right)_{r=r_+}=\sqrt{-\det g}\,\epsilon_{\nu\theta\chi\mu}J^\mu=\sqrt{-\det g}\,g^{r\mu}J_\mu,
\end{equation}
where the determinant and the inverse metric can be found by straightforward computations or using \textit{Mathematica} and look as $\det g=-\Sigma^2\sin^2\theta$ and $g^{r\mu}\partial_\mu=\frac{1}{\Sigma}\left[\Delta\partial_r+(r^2+a^2)\left(\partial_v+\Omega_H\partial_\chi\right)\right]$, with $\Sigma=r^2+a^2\cos^2\theta$, $\Omega_H=\frac{a}{r^2+a^2}$, $\Delta=r^2-2Mr+a^2=(r-r_+)(r-r_r)$ and $a$ the rotation parameter. Using the fact that $J_\mu=g(J,\partial_\mu)=\langle J,\partial_\mu\rangle$ and the definition of the horizon generating Killing vector $\xi^a=k^a+\Omega_Hm^a=\left((\partial_v)^a+\Omega_h(\partial_\chi)^a\right)$ one obtains the required quantity
\begin{equation}
\left(\star J_{\nu\theta\chi}\right)_{r=r_+}=\sin\theta(r_+^2+a^2)\xi^aJ_a.
\end{equation}
Henceforth, to determine the sign of \eqref{eq:HorizonFlux} one just needs to look at $\xi^aJ_a=-\xi^aT_{ab}k^b=-\xi^a\partial_a\psi\partial_b\psi k^b$, where the second term that would come from the given stress-energy tensor vanishes due to $\xi\cdot k=0$ on $\mathcal{H^+}$, as a consequence of the horizon invariance under the isometries of the spacetime, implying that KVFs should be tangent to it and thus orthogonal to its generators (the horizon is Killing). Finally, a little bit of differentiation of the given scalar field leads to the final answer, which takes the form
\begin{equation}
\omega\left(m\Omega_H-\omega\right)\geq0,
\end{equation}
giving the condition $0<\omega<m\Omega_H$ for \eqref{eq:HorizonFlux} to be positive. This simple results has the remarkable implication that energy can be extracted from the black hole by scattering waves off of it - the phenomenon of superradiance. Now, it is only natural for a theoretical physicists to try to enclose the superradiant object in question with a reflecting surface, so that the waves can go on scattering back and forth indefinitely, thus draining all the energy of the black hole. This can be achieved by surrounding the object with a giant mirror for example - which was first proposed in \cite{zel1972amplification} for the case of electromagnetic waves impinging upon a conducting rotating cylinder. A more realistic pathway towards achieving superradiance in Kerr, which has also recently started to attract more attention in the astrophysics community\footnote{For an interesting read on the topic the following two papers are recommended \cite{dolan2007instability,witek2013superradiant}}, is by considering a massive scalar field instead of a massless one. The addition of the mass term leads to a potential in the Klein-Gordon equation for the field that exhibits a local minimum between the event horizon of the black hole and spatial infinity, wherein scattered waves can get `trapped' and reflected backwards, so as to be amplified again due to superradiance \cite{cardoso2005superradiant}. By looking at the asymptotic behaviour of the potential one sees that this is always the case in $d=4$, as long as $\mu<\omega$, where $\mu$ is the mass of the scalar field. Of course, there is a separate condition on the frequency $\omega$ itself for when the wave modes are superradiant that depends on the rotation parameter and the radius of the black hole (for fixed $\mu$ and $m$).\\
Naturally, as the reader might have already guessed there is an obvious candidate to investigate superradiance in and this is asymptotically AdS spaces, due to the timelike nature of spatial infinity and the reflective boundary conditions there. These imply that Anti-de Sitter acts just like a confining box and any waves (moving at the speed of light) scattered from the bulk outwards will eventually reach spatial infinity (as eluded to earlier) and get reflected backwards there. Depending on the type of perturbations (scalar, electromagnetic or gravitational) and the spacetime under consideration, the situation may be quite different. For Schwarzschild-AdS\cite{cardoso2001quasinormal,michalogiorgakis2007low,dias2013boundary} it has been shown that superradiance does not occur\footnote{Which can be expected once one has looked in more detail into the requirements for the appearance of superradiance - which will be done later in the section.}, while for RN-AdS and Kerr-AdS there are both quasinormal(QNM) and superradiant modes present. The former are usually defined in a physics context as wave solutions which are purely outgoing at spatial infinity $\psi\approx e^{-i\omega(t-r^*)}$ and solely ingoing near the horizon $\psi\approx e^{-i\omega(t+r^*)}$, where $r^*$ is the usual tortoise coordinate ($dr^*=dr/f(r)$ for Schwarzschild-AdS) and $\omega\in\mathbb{C}$. Mathematically, QNMs appear when the two solutions of the wave equation under consideration are linearly dependent, with the coefficient of proportionality being the complex QNM frequency. It is interesting to note\cite{kokkotas1999quasi} that an analysis of the behaviour of the QNM eigenfunctions shows that their decay in time depends on the asymptotic properties of the potential and for it to be exponential in nature the potential has to be vanishing outside a certain region centred at the origin. Therefore, the usual identification of QNMs with exponentially decaying perturbations might be a bit naive. Nevertheless, for the spacetimes investigated in this work the potential is always asymptotically vanishing and thus the QNMs at large distances from the origin will be dying off with time. On the other hand, the superradiant modes which we introduced as growing in time and which are the main focus of this work, seem to cause the RN- and Kerr-AdS systems to take on two at first similar paths that later split in opposite directions.\\
\hspace*{5mm}It is the stability of the spacetimes under investigation that is being referred to at the end of the last paragraph and it is one of the main reasons why superradiant scattering is interesting. Stability is important from the point of view that an instability with a timescale that is not comparable with the age of the universe will most probably lead to a very small number of representatives of the system in question in Nature. Moreover, superradiance in an astrophysical context might lead to observable gravitational wave emissions and may be used to constrain certain beyond-the-standard-model-physics models\cite{arvanitaki2011exploring,pani2012black}. Also, back to AdS and relating to the AdS/CFT correspondence - perturbations of black holes in the bulk are linked to ones on the boundary CFT, thus the time evolution of the quasinormal and superradiant modes is dual to the evolution of fluctuations of the field theory.\\
\hspace*{5mm}Elucidating more on the instabilities - the defining prerequisite for their existence in a spacetime is the presence of growing in time perturbations - that is, superradiant modes - and the exciting consequences thereof are that they might lead to the transition of the system to a different state, the formation of new objects or redistribution of energy between the excited modes of the perturbation - all of which are interesting possibilities that might uncover some new black hole physics, which is why, in recent years, a lot of effort has been put in understanding the effect of superradiant scattering in asymptotically AdS spacetimes.\\
\hspace*{5mm}As promised earlier, a brief discussion on the requirements for the presence of superradiant modes will be now presented\cite{hawking1999charged}. Take the first law of black hole mechanics for a rotating black hole
\begin{align}
dM=\frac{\kappa}{8\pi}dA+\Omega_HdJ,
\end{align}
which relates the change in mass and angular momentum, due to a linearised perturbation, to the change in the horizon area. Then, consider a scalar field (the same argument can be generalised to any electromagnetic or gravitational perturbation in a straightforward way) with a stress energy tensor given as before by
\begin{align}
T_{ab}=\partial_a\psi\partial_b\psi-\frac{1}{2}g_{ab}\partial^c\psi\partial_c\psi.
\end{align}
Looking at the $\tensor{T}{^r_t}$ and $\tensor{T}{^r_\phi}$ components, corresponding to the net radial flux of energy and angular momentum, respectively, it is easy to show that the ratio of mass to angular momentum carried in the black hole by the wave results in
\begin{align}
\frac{dM}{dJ}=\frac{\omega}{m},
\end{align}
where as before the scalar field has been decomposed according to the isometries of the spacetime - $\psi(t,r,\theta,\phi)=\psi_0\left(r,\theta\right)e^{-i\omega t}e^{im\phi}$. Consequently, referring to the second law of black hole mechanics, which informs us that classically
\begin{align}
dA\geq0,
\end{align}
for a field scattering off the black hole, given that it obeys the dominant energy condition, it is straightforward to derive that energy can be extracted from the black hole under the condition that
\begin{align}
\omega<m\Omega_H.
\end{align}
Reasoning along exactly the same lines for a charged black hole, where the angular momentum is replaced by the electrostatic potential of the black hole, leads to the analogous conclusion that superradiance appears given that
\begin{align}
\omega<q\Phi,
\end{align}
where $q$ is the charge of the scalar field and $\Phi$ the electrostatic potential difference between the horizon and infinity. This can also be written as $\omega<q\frac{Q}{r_+}$, treating the extracting superradiant mode as being just outside the black hole, thus permitting the usual approximation for a homogeneous spherically symmetric lump of charge ($r_+$ is the radius of the black hole).\\
\hspace*{5mm}The above conditions do not actually imply that a given rotating or charged black hole will suffer from superradiant instabilities. In order to show this, one has to go through the linearised Einstein equations for a given perturbation and show that modes with the desired frequencies actually exist, which is definitely not a straightforward process. However, in the case of asymptotically AdS spacetimes it has very recently\cite{green2015superradiant} been proven mathematically that any asymptotically AdS black hole with a Killing horizon, whose corresponding Killing field becomes spacelike in some region of space\footnote{Which defines it as an ergoregion.}, will be linearly unstable due to superradiance of gravitational perturbations\footnote{Technically, what is shown in the paper is that the system does not go back to equilibrium, which does not rule out the case of oscillations around it with a constant amplitude. Even though this will clearly not lead to an instability, it is a special case that usually can be ruled out with befitting confidence by numerical results.}. This result is not going to be rederived here, but one of its immediate implications is that Kerr-AdS is linearly unstable to superradiance, which will be covered in great detail later in this work. It should be noted that the above theorem can not be straightforwardly implied to charged black holes in AdS, even if a notion of a generalised ergoregion can be introduced for them - a little discussions about this with a reference is given at the end of \cite{green2015superradiant}.\\
\hspace*{5mm}Having presented the phenomenon of superradiance in a short manner, we are going to move on and illustrate in the next section some of the popular methods for actually calculating the superradiant modes of different black hole spacetimes subject to various perturbations. This is also a good place to refer the reader to a long review on the subject that is extremely helpful in obtaining references - \cite{brito2015superradiance}.
\subsection{Calculating quasinormal and superradiant modes\\in asymptotically AdS spacetimes}
It should be made clear that from this point onwards only asymptotically AdS spacetimes in 4D are investigated - in particular Reissner-Nordstr\"{o}m-AdS (RN-AdS) and Kerr-AdS, with an occasional reference to Schwarzschild-AdS.
\subsubsection{Scalar fields and wave equations}\label{subsec:Scalar}
Generally, perturbations are devised in three types - scalar, electromagnetic and gravitational, and in this section it will be the simplest type of perturbations - scalar fields - that will be considered first\cite{cardoso2004small,uchikata2009scalar,uchikata2011quasinormal,bosch2016nonlinear}. The reason being that in this way the reader will be gently introduced to the logical flow behind this type of calculations, which are fairly similar in character, even if they differ quite a lot in the complexities of their specifics. A further simplification in this case, for asymptotically AdS spacetimes, comes from the fact that it is sufficient to consider massless fields due to the reflective nature of the boundary at infinity. Therefore, one starts from the Klein-Gordon equation for the scalar field
\vspace*{4.5mm}
\begin{align}
\begin{array}{rcl}
\nabla_\mu\nabla^\mu\Phi=
\smash{\left\{\begin{array}{@{}l@{}}
\frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}g^{\mu\nu}\partial_\nu\Phi\right)=0,\quad\hfill\mbox{for Kerr-AdS}\vspace{3mm}\\[\jot]
\left(\nabla_\mu-ieA_\mu\right)\left(\nabla^\mu-ieA^\mu\right)\Phi=0,\quad\mbox{for RN-AdS}\vspace{1mm}\\[\jot]
\end{array}\right.}\\\vspace*{0.5mm}
\end{array},\label{eq:waveEq}
\end{align}
where $\Phi$ is the scalar field and $A_\mu$ is the Maxwell gauge field. Afterwards, a separation ansatz can be imposed
\begin{align}
\Phi(t,r,\theta,\phi)=e^{-i\omega t}e^{im\phi}R(r)S(\theta),\label{eq:ScalarField}
\end{align}
where as before the field has been decomposed into Fourier modes taking advantage of the isometries of the background, corresponding to the Killing vectors $\partial_t$ and $\partial_\phi$, with $\omega$ a complex frequency and $m$ an integer. For RN-AdS, due to spherical symmetry, the $e^{im\phi}$ and $S(\theta)$ parts can be combined into the usual spherical harmonics $Y_{lm}(\theta,\phi)$ with $l$ and $m$ the usual angular momentum and azimuthal quantum numbers. By plugging the decomposition \eqref{eq:ScalarField} back into \eqref{eq:waveEq} two equations are obtained - a radial and an angular one, whereby the separation constant (which can be shown to be the same for both equations) corresponds to the eigenvalue of the angular equation. Continuing analytically at this point is usually done by defining a near-horizon region $r-r_+\ll1/\omega$, where $r_+$ is the location\footnote{It is the largest root of $\Delta$ in the usual notation for the metrics of RN-AdS and Kerr-AdS.} of the future event horizon $\mathcal{H^+}$, and a far region $r-r_+\gg r_+$. In the former the contribution of the Cosmological constant can be neglected and $r\approx r_+$, which leads to a number of simplifications of the radial equation, which after a suitable transformation can be turned into a standard hypergeometric differential equation, whose general solution is readily available. Of course, it should not be forgotten that the relevant boundary conditions have to be imposed and near the horizon this implies that only ingoing waves are allowed, as one does not expect perturbations to be coming out of the black hole (in the classical picture). Deducing which coefficient should be set to zero in the general solution in terms of hypergeometric functions can be done by performing a Frobenius analysis of the radial equation around the horizon\footnote{That is - expanding $R(r)$ in the appropriate power series of the form $r^k\sum\limits_{n=0}^{\infty}c_nr^n$, where $c_n$ are some coefficients.}. Afterwards, turning to the far-region - there the effects of the black hole can be neglected and the radial equation reduces to that of pure AdS with the subtle difference that the inner boundary in that case is at $r_+$ and not at $r=0$. Nevertheless, progress is achieved in the same way as in the near-horizon region and after a suitable substitution, the equation reduces to a standard hypergeometric equation. Then again the relevant boundary conditions have to be taken into account, however, in this situation more caution is required. As it was already mentioned in the section on asymptotically AdS spacetimes, at spatial infinity perturbations have to behave like a standing wave with a node there, meaning that in general both the ingoing and outgoing waves have to be considered. Nevertheless, after undertaking a Frobenius analysis near $\mathcal{I}$ it is seen that for a scalar field this is straightforward as, in order to avoid the field diverging, one of the coefficients has to be set to zero and the surviving part of the solution meets the requirements on the decay of the Weyl tensor from \cite{henneaux1985asymptotically}. Although at present completing the solution analytically for the whole phase space of black holes is not possible, a restriction to $r\omega\ll1$ provides a way out\footnote{Plus further taking $\frac{a}{r_+}\ll1$ for Kerr-AdS}. In this regime the near- and far-regions overlap in the zone $r_+\ll r-r_+\ll1/\omega$ and it can be shown that the condition $r_+\omega\ll1$ is equivalent to working in the regime $\frac{r_+}{L}\ll1$ - i.e. small black holes. This equivalence will be derived here for Kerr-AdS, but it is analogous and simpler in the case of RN-AdS. So starting from $\frac{r_+}{L}\ll 1$, taking the condition for extremality for small radii
\begin{align}
a\leq r_+\sqrt{\frac{3r_+^2+L^2}{L^2-r_+^2}},\quad\mbox{for }r_+<\sqrt{3}L,\label{eq:aExtremal}
\end{align}
and expanding in series for small $r_+$ to get $a\leq r_++\mathcal{O}(r_+^3)$, it is immediately obvious that $\frac{a}{L}\ll1$. Afterwards, arguing that $\frac{r_+}{L}\ll 1$ means that the real part of the frequencies would be of the order of those in pure AdS which are calculated in \cite{dias2013boundary} and behave as $\omega L\approx\mathcal{O}(1)$, it is easily observed that one also gets $r_+\omega\ll1$ and $a\omega\ll1$ (which are also used in the simplification of the radial equation and are needed for the Kerr-AdS condition $a/r_+\ll1$). This argument is sensible, as for a tiny black hole, one would generally expect the effect on the spacetime to be fairly negligible throughout most of it. Therefore, for small black holes the near-horizon and far-region solutions can be matched asymptotically in the intermediate, overlapping zone. This requires deriving the asymptotic behaviour of the former for large $r$ and of the latter for small, which can be straightforwardly achieved by using the properties of hypergeometric functions. The result of this procedure is a quantised spectrum for the frequency $\omega$, whereby it turns out that the sign of the imaginary part depends on a condition on the real part - that is
\begin{align}
&\mbox{Re}(\omega)-q\frac{Q}{r_+}<0\quad\Rightarrow\quad\mbox{Im}(\omega)>0,\quad\mbox{for RN-AdS}\label{eq:ineqRN}\\
&\mbox{Re}(\omega)-m\Omega_H<0\quad\Rightarrow\quad\mbox{Im}(\omega)>0,\quad\mbox{for Kerr-AdS},\label{eq:ineqKerr}
\end{align}
where for an equality in the conditions for the real part of the frequencies (that is - Re$(\omega)=q\frac{Q}{r_+}$ or Re$(\omega)=m\Omega_H$) the imaginary part vanishes, while the reversed inequalities expectedly lead to Im$(\omega)<0$. The sign of the imaginary part is a clear indicator for the nature of the mode under consideration, as evident from equation \eqref{eq:ScalarField} - for Im$(\omega)<0$ the wavefunction is exponentially decaying in time - thus the mode is damped and it is identified as a QNM. For Im$(\omega)>0$ the scalar field perturbation has an exponential time growth and thus corresponds to a superradiant mode as it bounces back and forth between the horizon and the conformal boundary. It is important to note that the above relations can be obtained without actually deriving the frequency spectrum. This can be achieved in a similar way to the example with the real scalar field at the beginning of the previous section. Define, as before, the total flux through a hypersurface as
\begin{align}
\mathcal{F}(\Sigma)=-\int_{\Sigma}\star J,
\end{align}
where $J_a=-\mathcal{T}_{ab}\xi^b$ is the conserved 4-current associated with a given Killing vector field $\xi^b$, which for RN-AdS and Kerr-AdS can be either $\partial_t$ or $\partial_\phi$ in which case $J$ represents the energy or angular momentum 4-vector, respectively, and $F(\Sigma)$ - the energy or angular momentum flux through $\Sigma$. $\mathcal{T}_{ab}$ is the stress-energy tensor of the perturbation, which from the linearised Einstein equation can be shown to be proportional to the Landau-Lifschitz pseudotensor\cite{landau1971classical}, which most importantly is expressible only in terms of metric components - which in linearised theory are the ones of the perturbation. Of course, for RN-AdS there is no notion of angular momentum (hence the above expression will always vanish for $\partial_\phi$) and one instead defines the electric charge on a hypersurface in an analogous way
\begin{align}
\mathcal{Q}(\Sigma)=-\int_{\Sigma}\star j,
\end{align}
where $d\star F=-4\pi\star j$ - with $F$ the usual Maxwell field-strength tensor. Computing these integrals is a rather long and not very exciting task\footnote{For Kerr-AdS one can take directly $\xi=\partial_t+\Omega_H\partial_\phi$, which is both the horizon generator and the normal to it.}, usually done in ingoing Eddington-Finkelstein cooridnates, but the final answers reduce to the inequalities presented above - which is a good indication that the perturbative expansions in the two regions and the asymptotic matching in the overlapping zone provide a good approximation in the regime of small black holes. Confirming this analysis and exploring the rest of the phase space for black holes is then done numerically - \cite{cardoso2004small,cardoso2006classical,uchikata2009scalar,cardoso2014holographic,uchikata2011quasinormal,dias2012hairy,bosch2016nonlinear}.
\subsubsection{The Newman-Penrose and Teukolsky's formalisms}\label{subsec:NPFormalism}
A different approach to the above calculations, which nevertheless in the case of a scalar field perturbation reduces to what is laid out above, but is applicable to all types of perturbations, is given by the so called Teukosly's formalism\cite{teukolsky1973perturbations,chandrasekhar1998mathematical,kramer1980exact}, which is based on the Newman-Penrose (NP) tetrad formalism, which will be given a brief introduction here, but is very well presented in \cite{chandrasekhar1998mathematical} and in almost any other textbook on General Relativity. Technically, Teukolsky's approach was initially devised for rotating black holes, but it applies equally well to the Schwarzschild spacetime in the limit of a vanishing rotation parameter $a\rightarrow0$ and can then straightforwardly be generalised to RN. To begin with the basics - a tetrad formalism uses a tetrad basis - four linearly independent vector fields $\tensor{e}{_{(a)}^i}$, $a=\{1,2,3,4\}$ - which set up at each point of the spacetime a basis of four vectors that satisfy
\begin{align}
\tensor{e}{_{(a)}^i}\tensor{e}{^{(b)}_i}=\tensor{\delta}{_{(a)}^{(b)}},\quad\mbox{and}\quad \tensor{e}{_{(a)}^i}\tensor{e}{^{(a)}_j}=\tensor{\delta}{^i_j},\quad\mbox{where }\tensor{e}{_{(a)i}}=\tensor{g}{_{ij}}\tensor{e}{_{(a)}^j},
\end{align}
and $\tensor{e}{^{(b)}_i}$ is the matrix inverse of $\tensor{e}{_{(a)}^i}$. The bracketed indices indicate the tetrad components, whereas the ones without a bracket are the usual tensor indices. Also part of the definition is
\begin{align}
\tensor{e}{_{(a)}^i}\tensor{e}{_{(b)}_i}=\tensor{\eta}{_{(a)(b)}},
\end{align}
with $\tensor{\eta}{_{(a)(b)}}$ a constant symmetric matrix, which is used to lower and raise the tetrad indices. The general idea behind the adoption of a tetrad basis is that with an appropriate choice it should be possible to get a better handle of the underlying symmetries of the system under consideration. Of course, this implies that choosing the tetrad vectors is not a trivial process. With the above definitions it is a simple exercise to show that
\begin{align}
\tensor{e}{_{(a)i}}\tensor{e}{^{(a)}_j}=\tensor{g}{_{ij}}.\label{eq:MetricCompsTetrad}
\end{align}
The idea of the tetrad formalism is to project all the quantities of interest onto it and solve the relevant equations for them in this basis, whereby the projections are defined as
\begin{align}
\tensor{T}{_{(a)(b)}}=\tensor{e}{_{(a)}^i}\tensor{e}{_{(b)}^j}\tensor{T}{_{ij}}=\tensor{e}{_{(a)}^i}\tensor{T}{_{i(b)}},\\
\tensor{T}{_{ij}}=\tensor{e}{^{(a)}_i}\tensor{e}{^{(b)}_j}\tensor{T}{_{(a)(b)}}=\tensor{e}{^{(a)}_i}\tensor{T}{_{(a)j}},
\end{align}
with the obvious generalisation for tensors of any rank. From here onwards brackets around indices will be omitted - adopting the convention that earlier letters in the Latin alphabet correspond to tetrad components, while the later ones designate tensor indices. Furthermore, by choosing a coordinate basis, the tetrad can be written as linear combinations of tangent vectors
\begin{align}
\tensor{e}{_{a}}=\tensor{e}{_{a}^i}\partial_i,
\end{align}
which identifies them as directional (with respect to the tetrad basis vectors) derivatives and additionally implies that differentiating with respect to the tetrad indices can be expressed in terms of the usual partial and covariant derivatives of tensor quantities
\begin{align}
\tensor{A}{_{a,b}}=\tensor{e}{_{a}^j}\tensor{e}{_{b}^i}\nabla_iA_j+\gamma_{cab}A^c,\label{eq:TetradDerv}
\end{align}
where $\gamma_{cab}$ are called the Ricci-rotation coefficients, defined by
\begin{align}
\gamma_{cab}=\tensor{e}{_c^k}\tensor{e}{_b^i}\nabla_i\tensor{e}{_{ak}}\quad\mbox{and}\quad\gamma_{cab}=-\gamma_{acb},
\end{align}
and are the second key ingredient, after the tetrad basis vectors, of a given tetrad formalism as will become clearer in a bit, when the Petrov classification of spacetimes is reviewed. The rotation coefficients can be viewed alternatively as a connection in this basis, as is easily identifiable from the following definition
\begin{align}
\nabla_i\tensor{e}{_{ak}}=\tensor{e}{^c_k}\gamma_{cab}\tensor{e}{^b_i}\quad\Rightarrow\quad\nabla_i\tensor{e}{_a^k}=\tensor{\gamma}{_a^k_i},
\end{align}
which makes it possible to rewrite equation \eqref{eq:TetradDerv} as
\begin{align}
\tensor{e}{_{a}^j}\tensor{e}{_{b}^i}\nabla_iA_j=\tensor{A}{_{a,b}}-\eta^{cd}\gamma_{cab}A_d=A_{a|b}
\end{align}
where the RHS of the equation has been identified with the \textit{intrinsic derivative} of $A_a$ in the direction $e_b$. This quantity will not be explicitly needed here, but it is an essential part of the tetrad formalism and is used in many derivations of interest, hence its mentioning. The above definitions provide all the necessary tools to project all the relevant quantities - like the Riemann, Weyl and Ricci tensors onto the tetrad basis and obtain the Ricci- and Bianchi-identities in terms of tetrad components. These will not be presented here, as the expressions are quite space-consuming, but they can be readily found in many textbooks - \cite{chandrasekhar1998mathematical} with the mostly negative convention or \cite{kramer1980exact} for the predominantly positive one. Lastly, before formally introducing the Newman-Penrose choice of tetrad basis, we will mention that quite often\footnote{Especially in simpler calculations and in university courses on General Relativity} the natural choice for a tetrad basis is an orthonormal one in which case $\eta_{ab}$ takes the form of the Minkowski metric. For example, for asymptotically flat Schwarzschild one can take $e^1=(1-2M/r)^{1/2}dt$, $e^2=(1-2M/r)^{-1/2}dr$, $e^3=rd\theta$ and $e^4=r\sin\theta d\phi$, where it is sometimes easier to define the tetrad basis in terms of covectors.\\
\hspace*{5mm}The Newman-Penrose formalism consists in a special choice of the tetrad basis vectors, based on the belief of Roger Penrose that the causal structure of a spacetime is one of its key elements, which is also evident from the Penrose diagrams he introduced. Therefore, unsurprisingly, the NP tetrad basis consists of four null vectors: $\textbf{l}, \textbf{n}, \textbf{m}$ and $\bar{\textbf{m}}$, where the former two are real, while the latter are complex conjugates of each other. They satisfy
\begin{gather}
\textbf{l}\cdot\textbf{m}=\textbf{l}\cdot\bar{\textbf{m}}=\textbf{n}\cdot\textbf{m}=\textbf{n}\cdot\bar{\textbf{m}}=0\\
\textbf{l}\cdot\textbf{n}=-1\quad\mbox{and}\quad\textbf{m}\cdot\bar{\textbf{m}}=1
\end{gather}
where the latter two relations are not strictly necessary, but in most cases simplify computations significantly as one does not need to worry about various coefficients arising while raising and lowering indices and playing with directional and intrinsic derivatives in tensor notation. In this formalism both the directional derivatives and the rotation coefficients, which are called spin coefficients now, are given special symbols
\begin{gather}
D=\textbf{l}^k\partial_k,\quad \tilde{\Delta}=\textbf{n}^k\partial_k,\quad \delta=\textbf{m}^k\partial_k,\quad \delta^*=\bar{\textbf{m}}^k\partial_k\\
\kappa=-\gamma_{311},\;\sigma=-\gamma_{313},\;\lambda=\gamma_{424},\;\nu=\gamma_{422},\;\rho=-\gamma_{314},\;\mu=\gamma_{423},\;\tau=-\gamma_{312},\; \pi=\gamma_{421}\notag\\
\epsilon=\frac{1}{2}(\gamma_{341}-\gamma_{211}),\quad\gamma=\frac{1}{2}(\gamma_{342}-\gamma_{212}),\quad \alpha=\frac{1}{2}(\gamma_{344}-\gamma_{214}),\quad \beta=\frac{1}{2}(\gamma_{343}-\gamma_{213}).
\end{gather}
It should be pointed out that as a general rule - the complex conjugate of any quantity can be obtained by interchanging the indices 3 and 4 in any expression. Furthermore, the Riemann tensor can be split into a trace-free part (the Weyl tensor $C_{abcd}$, with $\eta^{ad}C_{abcd}=0$) and a trace part - given by the Ricci tensor ($R_{ac}=\eta^{bd}R_{abcd}$) and Rici scalar ($R=\eta^{ab}R_{ab}=2(R_{34}-R_{12})$). To define these the NP formalism firstly supplies five complex scalars, which completely determine the ten\footnote{This only holds in four dimensions} independent components of the Weyl tensor - $\Psi_0,...,\Psi_4$ -
\begin{gather}
\Psi_0=C_{1313}=C_{abcd}\textbf{l}^a\textbf{m}^b\textbf{l}^c\textbf{m}^d,\quad\Psi_1=C_{1213}=C_{abcd}\textbf{l}^a\textbf{n}^b\textbf{l}^c\textbf{m}^d,\quad\Psi_2=C_{1342}=C_{abcd}\textbf{l}^a\textbf{m}^b\bar{\textbf{m}}^c\textbf{n}^d\notag\\
\Psi_3=C_{1242}=C_{abcd}\textbf{l}^a\textbf{n}^b\bar{\textbf{m}}^c\textbf{n}^d,\quad\Psi_0=C_{2424}=C_{abcd}\textbf{n}^a\bar{\textbf{m}}^b\textbf{n}^c\bar{\textbf{m}}^d,
\end{gather}
and secondly, three more complex scalars and four real ones for the ten independent components of the Ricci tensor
\begin{gather}
\Phi_{00}=\bar{\Phi}_{00}=\frac{1}{2}R_{44},\quad\Phi_{01}=\bar{\Phi}_{10}=\frac{1}{2}R_{41},\quad\Phi_{02}=\bar{\Phi}_{20}=\frac{1}{2}R_{11},\quad\Phi_{11}=\bar{\Phi}_{11}=\frac{1}{4}\left(R_{43}\right),\notag\\
\Phi_{12}=\bar{\Phi}_{21}=\frac{1}{2}R_{31},\quad\Phi_{22}=\bar{\Phi}_{22}=\frac{1}{2}R_{33}.
\end{gather}
This is all that is needed in order to specify everything else - the Riemann tensor\footnote{$C_{abcd}=R_{abcd}-\left(\eta_{a[c}R_{d]b}-\eta_{b[c}R_{d]a}\right)+\frac{1}{3}R\eta_{a[c}\eta_{d]b}$ in tetrad components.}, the Ricci and Bianchi identities. These, as before, will not be presented here as they are rather long and not extremely illuminating, but an extra line will be given just to specify the components of the Maxwell field-strength tensor in terms of complex scalars, as it is needed when electromagnetic perturbations of the spacetime are investigated
\begin{align}
\phi_0=-F_{ab}\textbf{l}^a\textbf{m}^b,\quad\phi_1=-\frac{1}{2}F_{ab}\left(\textbf{l}^a\textbf{n}^b-\textbf{m}^a\bar{\textbf{m}}^b\right),\quad\phi_2=F_{ab}\textbf{n}^a\bar{\textbf{m}}^b.
\end{align}
By the above definitions it is not at all obvious why the NP-tetrad formalism should be any more special than a straightforward choice of an orthonormal tetrad. However, the real power of such a null-tetrad becomes clear once the Petrov classification of the Weyl tensor and the Goldberg-Sachs theorem have been considered. These will not be fully covered here, as detailed proofs are available in the already mentioned references, nevertheless a brief overview of the logic behind them will be presented. Clearly, the null frame (or any other tetrad frame) can be subjected to Lorentz transformations, which provide six degrees of freedom (corresponding to the six specifying parameters of the Lorentz group in 4D) to rotate the frame. These can be devised in such a way as to make a general Lorentz transformation be comprised of three types of rotations that act differently on the different tetrad basis vectors and hence on all other quantities. Moreover, in this work and in many others it is usually solutions to the vacuum Einstein equations that are investigated\footnote{With matter often introduced as a perturbation, as in the case with the massless scalar field in the previous subsection.}, in which case the Riemann curvature and Weyl tensors coincide. The latter is described by the five complex scalars introduced earlier and these are exactly the focus of the Petrov classification, which basically explores how many of them can be set to zero by a suitable orientation of the tetrad frame with the help of a Lorentz transformation. This is achieved by combining all five of them in a fourth-order equation for the parameter of one of the classes of rotations discussed just above and then looking at the possibilities in terms of the roots and Lorentz transformations. This leads to organisation of different spacetimes into five Petrov types - I, II ,III ,D and N. Remarkably, it turns out that black hole solutions of General Relativity are all of type D, which very fortunately turns out to have only one of the five Weyl scalars non-vanishing and this is $\Psi_2$. The story is not over yet, nonetheless, as by choosing the vectors $\textbf{l}$ to form a null-congruence of geodesics and referring to the Goldberg-Sachs theorem\footnote{Which applies to the Petrov type II, but leads to a corollary for type D}, it can be shown that the spin coefficients $\kappa$, $\sigma$, $\nu$ and $\lambda$ also vanish. Finally, the null geodesics in the congruence can always be chosen to be affinely parametrised which in addition also sets $\epsilon=0$. All these quite remarkable conclusions are what makes the Newman-Penrose formalism so special and the reader is encouraged to go over a detailed analysis of all this.\\
\hspace*{5mm}Moving on to what we are really interested in - perturbing the spacetime. From everything aforementioned - the most general perturbation of a type D spacetime - like Kerr-AdS or RN-AdS - will split in two parts - changes in the quantities that vanish in the unperturbed background - $\delta\Psi_0,\;\delta\Psi_1,\;\delta\Psi_3,\;\delta\Psi_4,\;\delta\kappa,\;\delta\sigma,\;\delta\lambda,\;\delta\nu$ and changes in all the rest, which do not vanish in the background (including the three complex scalars specifying the Maxwell field-strength tensor). This is worked out in excruciating detail in \cite{chandrasekhar1998mathematical} for the cases of Schwarzschild, RN and Kerr black holes. What is astonishing in all the cases is that it is possible to go on solving for the first group of quantities, listed a few sentences ago, without having to refer to any of the other perturbed variables, and successfully do so. Moreover, it turns out that the system of equations can always be reduced to a set of two equations for the Weyl scalars $\delta\Psi_0$ and $\delta\Psi_4$\footnote{$\delta\Psi_1$ and $\delta\Psi_3$ can be made to vanish by an infinitesimal coordinate rotation - this type of transformation provides four more degrees of freedom, in addition to the six due to Lorentz transformations. $\Psi_0$ and $\Psi_4$ are invariant under gauge transformations in zeroth and first order.}, which in turn allow to be separated in radial and angular parts, whereby as in the case for the massless scalar field, discussed earlier, the separation constants in both equations can be shown to be the same. Not only this, but the radial equations are complex conjugates to each other, while the angular ones are linked through a simple relation, making it sufficient to solve for only one set of them. Continuing in this fashion and fixing the normalisation of the angular solutions, the only ingredient left undetermined is the relative normalisation of the radial parts of the solution. Notwithstanding, this obstacle can also be overcome, this time with the help of the Starobinsky-Teukolsky identities, which are a small group of theorems providing a very useful set of functional transformations between the differential operators involved in the radial and angular equations. Finally, to top off all the amazing results in this computation, it has been shown by Chandrasekhar in his book\cite{chandrasekhar1998mathematical} that the rest of the perturbed quantities are also fully determined by the solutions for $\delta\Psi_0$ and $\delta\Psi_4$. This, combined with the fact that the metric components can be expressed in terms of the tetrad basis vectors \eqref{eq:MetricCompsTetrad}, implies that the most general perturbations of the metric (scalar, electromagnetic or gravitational) in the case of Schwarzschild, RN or Kerr, can be obtained from solving two separable differential equations for the two Weyl scalars $\delta\Psi_0$ and $\delta\Psi_4$ in the Newman-Penrose tetrad formalism. Fortunately, in the derivation of this result the asymptotic character of the background spacetime plays no role, as it should be, since actually solving the equations for the Weyl scalars has not been attempted yet, thus absolutely straightforwardly the above conclusions would hold for Schwarzschild-AdS, RN-AdS and Kerr-AdS. The only difference comes when one tries to solve for $\delta\Psi_0$ and $\delta\Psi_4$ and has to be cautious with what boundary conditions are imposed at spatial infinity.\\
\hspace*{5mm}In order to complete this discussion we feel that the Teukolsky master equation in its most general form for Kerr-AdS, applicable to any type of perturbation, should be given explicitly. Therefore, the Kerr-AdS metric in four dimensions, discovered by Carter\cite{carter1968hamilton} will be introduced first in the usual Boyer-Lindquist coordinates $\{\hat{t},\hat{r},\theta,\hat{\phi}\}$
\begin{align}
ds^2=-\frac{\Delta_r}{\Sigma^2}\left(d\hat{t}-\frac{a}{\Xi}\sin^2\theta\,d\hat{\phi}\right)^2+\frac{\Sigma^2}{\Delta_r}d\hat{r}^2+\frac{\Sigma^2}{\Delta_\theta}d\theta^2+\frac{\Delta_\theta}{\Sigma^2}\sin^2\theta\left(ad\hat{t}-\frac{\hat{r}^2+a^2}{\Xi}d\hat{\phi}\right)^2,\label{eq:KerrAdS}
\end{align}
where
\begin{align}
\Delta_r=\left(\hat{r}^2+a^2\right)\left(1+\frac{\hat{r}^2}{L^2}\right)-2M\hat{r},\quad\Xi=1-\frac{a^2}{L^2},\quad\Delta_\theta=1-\frac{a^2}{L^2}\cos^2\theta,\quad\Sigma^2=\hat{r}^2+a^2\cos^2\theta.
\end{align}
The solution is asymptotically AdS, as mentioned earlier, with ADM mass and angular momentum $M/\Xi^2$ and $Ma/\Xi^2$, respectively, while the event horizon is located at $\hat{r}=r_+$, where $r_+$ is the largest root of $\Delta_r$. By a suitable transformation it can be checked that the above metric is asymptotic to global ${\rm AdS}_4$ in a rotating frame with angular velocity $\Omega_\infty=-a/L^2$.\\
One way of achieving this is by first doing a slight change of variables in \eqref{eq:KerrAdS} by introducing
\begin{align}
T=\Xi\hat{t}\quad\mbox{and}\quad\chi=a\cos\theta,
\end{align}
in order to get
\begin{align}
ds^2=-&\frac{\Delta_r}{(\hat{r}^2+\chi^2)\Xi^2}\left(dT-\frac{a^2-\chi^2}{a} d\hat{\phi}\right)^2+\left({\hat{r}^2+\chi^2}\right)\left(\frac{d\hat{r}^2}{\Delta_r}+\frac{d\chi^2}{\Delta_\chi}\right)+\notag\\
+&\frac{\Delta_\chi}{(\hat{r}^2+\chi^2)\Xi^2}\left(dT-\frac{a^2+\hat{r}^2}{a}d\hat{\phi}\right)^2,\label{eq:KerrAdSv2}
\end{align}
where $\Delta_\chi=(a^2-\chi^2)(1-\frac{chi^2}{L^2})$ with the angular velocity at infinity becoming $\Omega_\infty=-a/(L^2\Xi)$, followed by another coordinates transformation:
\begin{align}
&t=\frac{T}{\Xi},\hfill \hspace*{28mm}R=\frac{\sqrt{L^2(a^2+\hat{r}^2)-(L^2+\hat{r}^2)\chi^2}}{L\sqrt{\Xi}},\notag\\
&\phi=\hat{\phi}+\frac{a}{L^2}\frac{T}{\Xi},\hspace*{15mm}\cos\Theta=\frac{L\sqrt{\Xi}\hat{r}\chi}{a\sqrt{L^2(a^2+\hat{r}^2)-(L^2+\hat{r}^2)\chi^2}}.
\end{align}
One does not need to find the metric explicitly - rather only the asymptotic behaviour, as $\hat{r}\rightarrow\infty$ (and respectively $R\rightarrow\infty$), is of interest - thus working to next-to-leading order in $\hat{r}$ (or $R$) is enough. The $t$ and $\phi$ components are straightforward to handle, while for $R$ and $\Theta$ it is easier to invert their expressions for $\hat{r}^2$ and $\chi^2$ and then proceed by brute force. The result is the global ${\rm AdS}_4$ metric \eqref{eq:globalAdS} in terms of the coordinates $\{t,R,\Theta,\phi\}$.\\
Going back to Kerr-AdS, in order to move to a non-rotating frame at infinity, one can introduce the new coordinates $\{\hat{t},\hat{r},\theta,\hat{\varphi}\}=\{\hat{t},\hat{r},\theta,\hat{\phi}+\frac{a}{L^2}\hat{t}\}$, wherein the angular velocity of the horizon with respect to an observer at spatial infinity is given by
\begin{align}
\Omega_H=\frac{a}{r_+^2+a^2}\left(1-\frac{a^2}{L^2}\right),
\end{align}
which can be easily derived by finding the equation of $\hat{\varphi}$ in terms of $\hat{t}$ on the integral curves of the horizon generating Killing vector field $\xi$\footnote{One way of doing this would be to take $\xi$ as a vector field and act on the difference between its $\hat{\varphi}$ and $\hat{t}$ components.}. The expression for the metric will not be rewritten, but it is a simple task to obtain it as it just requires the replacement of $\hat{\phi}$ in the two brackets. The rotation parameter has to be bounded by $a<L$ as is evident from the expressions for the ADM mass and energy, because for a fixed horizon radius $r_+$ they diverge in the limit $a\rightarrow L$. Furthermore, the Hawking temperature of the black hole is given by
\begin{align}
T_H=\frac{r_+}{2\pi}\left(1+\frac{r_+^2}{L^2}\right)\frac{1}{r_+^2+a^2}-\frac{1}{4\pi r_+}\left(1-\frac{r_+^2}{L^2}\right),
\end{align}
which can be used to arrive at expressions for $a$ \eqref{eq:aExtremal} and $M$ at extremality, where $T_H=0$ and $\Delta_r(r_+)=0$:
\begin{align}
a_{ext}=r_+\sqrt{\frac{3r_+^2+L^2}{L^2-r_+^2}},\quad\mbox{and}\quad M_{ext}\frac{r_+\left(1+r_+^2/L^2\right)^2}{1-r_+^2/L^2}.
\end{align}
Getting Teukolsky master equation requires the introduction of a tetrad basis - which will be provided as the extension of the original tetrad used by Teukolsky to AdS spaces - known as Kinnersly's tetrad,
\begin{align}
&\textbf{l}^\mu\partial_\mu=\frac{1}{\Delta_r}\left(\left(\hat{r}^2+a^2\right)\partial_{\hat{t}}+\Delta_r\partial_{\hat{r}}+a\left(1+\frac{\hat{r}^2}{L^2}\partial_{\hat{\varphi}}\right)\right)\notag\\
&\textbf{n}^\mu\partial_\mu=\frac{1}{2\Sigma^2}\left(\left(\hat{r}^2+a^2\right)\partial_{\hat{t}}-\Delta_r\partial_{\hat{r}}+a\left(1+\frac{\hat{r}^2}{L^2}\partial_{\hat{\varphi}}\right)\right)\notag\\
&\textbf{m}^\mu\partial_\mu=\frac{\sin\theta}{\sqrt{2\Delta_\theta\left(\hat{r}+ia\cos\theta\right)}}\left(ia\partial_{\hat{t}}+\frac{\Delta_\theta}{\sin\theta}\partial_\theta+\frac{i\Delta_\theta}{\sin^2\theta}\partial_{\hat{\varphi}}\right).
\end{align}
There is an important subtlety, worth noting, concerning the application of boundary conditions to the resulting equations. The components of the metric perturbations, which can be obtained from the solution to Teukolsky master equation by what is called the Hertz map\cite{kegeles1979constructive}, will clearly depend on the picked tetrad basis, implying that matching these with the requirements for their decay rates at $\mathcal{I}$, as discussed in \ref{sec:AAdS}, is also dependent on this choice. Unfortunately, for Kerr-AdS this has not been achieved in the aforementioned Kinnersly tetrad, but in the Chambers-Moss one\cite{chambers1994stability,dias2013boundary}, where Teukolsky equation takes on a different form than the one presented here. We will glance over this issue and hope that in the near future someone will derive the required boundary conditions. The non vanishing Weyl-scalar is given by $\Psi_2=-M(r-ia\cos\theta)^{-3}$ and we are working in vacuum. The perturbations are naturally decomposed as
\begin{align}
\Psi^{(s)}=e^{-i\omega\hat{t}}e^{im\hat{\varphi}}R^{(s)}_{lm\omega}(\hat{r})S^{(s)}_{lm\omega}(\theta),
\end{align}
where $s=0$ corresponds to scalar perturbations with $\Psi^{0}=\Psi$, $s=1$ designates electromagnetic waves with $\Psi^{(1)}=\delta\phi_0$ and $\Psi^{(-1)}=\left(-\Psi_2\right)^{-\frac{2}{3}}\delta\phi_2$, while $s=2$ denotes gravitational perturbations with $\Psi^{(2)}=\delta\Psi_0$ and $\Psi^{(-2)}=\left(-\Psi_2\right)^{-\frac{4}{3}}\delta\Psi_4$. The equation is also valid for $s=\pm\frac{1}{2}$, which is the case of massless fermions but will not be given here. With all these definitions, the radial and angular parts of the Teukolsky master equation can be presented:
\begin{gather}
\Delta_r^{-s}\partial_{\hat{r}}\left[\Delta_4^{s+1}\partial_{\hat{r}}R_{lm\omega}^{(s)}(\hat{r})\right]+\Bigg\{\frac{K_T^2-is\Delta_r'K_T}{\Delta_r}+2isK_T'-|s|(|s|-1)(2|s|-1)(2|s|-7)\frac{\hat{r}^2}{3L^2}+\notag\\
+\frac{s+|s|}{2}\Delta''-|s|\left(|s|-2\right)(4s^2-12|s|+11)\frac{a^2}{3L^2}-\hat{\lambda}^{(s)}_{lm\omega}\Bigg\}R^{(s)}_{lm\omega}(\hat{r})=0,
\end{gather}
where
\begin{align}
K_T(\hat{r})=\omega(\hat{r}^2+a^2)-ma\left(1+\frac{\hat{r}^2}{L^2}\right)\quad\mbox{and}\quad\hat{\lambda}^{(s)}_{lm\omega}=\hat{\Lambda}^{(s)}_{lm\omega}-2ma\omega+a^2\omega^2+(s+|s|),
\end{align}
and $\hat{\Lambda}^{(s)}_{lm\omega}$ is the separation constant which will get more elaboration after the angular part of the equation has been shown:
\begin{gather}
\frac{1}{\sin\theta}\partial_\theta\left(\sin\theta\Delta_\theta\partial_\theta S^{(s)}_{lm\omega}(\theta)\right)+\left[\left(a\omega\cos\theta\right)^2\frac{\Xi}{\Delta_\theta}-2sa\omega\cos\theta\frac{
\Xi}{\Delta_\theta}+s+\hat{\Lambda}^{(s)}_{lm\omega}-\right.\notag\\
\left.-\left(m+s\cos\theta\frac{\Xi}{\Delta_\theta}\right)^2\frac{\Delta_\theta}{\sin^2\theta}-2\delta_s\frac{a^2}{L^2}\sin^2\theta\right]S^{(s)}_{lm\omega}(\theta)=0,
\end{gather}
where $\delta_s=1$ for $|s|=\{1/2,1,2\}$ and $\delta_s=0$ if $s=0$. The eigenfunctions $e^{im\hat{\varphi}}S^{(s)}_{lm\omega}(\theta)$ are the so called spin-weighed AdS spheroidal harmonics - a generalisation of their flat counterparts, with $l$ - a positive integer identified with the number of zeroes along the polar direction, which is given by the relation $l-\mbox{max}\{|m|,|s|\}$. The separation constants $\hat{\Lambda}^{(s)}_{lm\omega}$ are their associated eigenvalues and can be determined numerically, with the leading order contribution in the regime $a/L\ll1$ (which was discussed earlier in the case of the massless scalar field) proportional to $l$ and $s$ only, which will be seen later in the section on Kerr-AdS. Similar to the case of the ordinary spherical harmonics, regularity requires that $-l\leq m\leq l$ and $m\in\mathbb{Z}$. Furthermore, as mentioned before $\Phi^{(s)}_{lm\omega}(\hat{r})$ is the complex conjugate of $\Phi^{(-s)}_{lm\omega}(\hat{r})$, hence their differential equations are complex conjugates as well, while the angular solutions are related by $S^{s}_{lm\omega}(\theta)=S^{-s}_{lm\omega}(-\theta)$ and can be freely normalised by
\begin{align}
\int_{0}^{\pi}\left(S^{(s)}_{lm\omega}\right)^2d\theta=1.
\end{align}
Taking $L\rightarrow\infty$, corresponding to a vanishing Cosmological constant, reduces the above equations to the case of the flat Kerr solution. Moreover, in the limit of $a\rightarrow0$ the radial and angular equations take on the form appropriate for Schwarzshild-AdS (or Schwarzschild if $L\rightarrow\infty$ has already been taken), from where the RN-AdS form of the equations can be deduced by changing $\left.\Delta_r\right|_{a\rightarrow0}$ to $\left.\Delta_r\right|_{a\rightarrow0}+e^2$, where $e^2=\sqrt{Q^2+P^2}$, with $Q$ and $P$ representing the electric and magnetic charges of the black hole, respectively. In addition, setting $M=0$ in the Schwarzschild equations produces the global ${\rm AdS}_4$ ones. Having the radial and angular parts of Teukolsky master equation readily available, in order to study some type of perturbations of Kerr-AdS (or Schwarzschild- or RN-AdS), one just needs to take the appropriate value for $s$ and start solving. The approach is the same as for the massless scalar field - a complete analytical solution is not known currently, but a perturbative expansion near the horizon and at large radial distances, where attention has to be paid in applying the correct boundary conditions to ensure that $\mathcal{I}$ acts as a reflecting wall, combined with an asymptotic matching procedure in an intermediate region, with the same assumptions as before on the parameters of the black hole, provides a very good approximation in the regime of small black holes. Afterwards, a numerical analysis can be performed in the whole phase space in order to confirm the perturbative results and investigate what happens for large black holes. The conclusions of such types of investigations of the QNMs and superradiant modes of RN-AdS and Kerr-AdS will be presented in the next two sections, where a particular focus will be paid to the superradiance and its effect on the stability of the spacetimes under question, but as one might expect the inequalities \eqref{eq:ineqRN} and \eqref{eq:ineqKerr} for the real and imaginary parts of the frequencies of the perturbative modes will show up again.
\section{Superradiance in Reissner-Nordstr\"om-AdS and its stability}\label{sec:sec4}
Reissner-Nordstr\"om metrics are not expected to represent black holes of particular importance to astrophysics, due to the charge neutrality of the universe, which implies that large charge imbalances are unlikely to occur. Furthermore, a charged black hole would definitely attract particles of opposite charge and will eventually lose most of its charge. Nevertheless the RN-AdS metric is a manageable toy model for superradiance (which does not occur in Schwarzschild-AdS) and from the fairly simple arguments put out in subsection \ref{sec:SimpleEx}, one might expect that it should be possible to make some analogies between RN-AdS and Kerr-AdS. There definitely are similarities between the phenomenon in the two spacetimes, although this might be more due to the fact that it is the same problem being investigated. As it turns out, translating the results for charged black holes into conclusions for rotating black holes is clearly not straightforward.\\
\hspace*{5mm}The RN-AdS metric in 4D is given by
\begin{align}
ds^2=-\frac{\Delta}{r^2}dt^2+\frac{r^2}{\Delta}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2),
\end{align}
where (taking the magnetic charge $P=0$)
\begin{align}
\Delta=\frac{r^4}{L^2}+r^2-2Mr+Q^2,
\end{align}
while the Hawking temperature and the mass (in terms of the horizon radius) of the black hole are evaluated to be
\begin{align}
T_H=\frac{1}{4\pi r_+}\left(1-\frac{Q^2}{r_+^2}+\frac{3r_+^2}{L^2}\right)\quad\mbox{and}\quad M=\frac{1}{2}\left(r_++\frac{r_+^3}{L^2}+\frac{Q^2}{r_+}\right),
\end{align}
which as before can be used to get the parameters at extremality, leading to
\begin{align}
Q_{ext}=r_+\sqrt{1+\frac{3r_+^2}{L^2}}.
\end{align}
As the black hole is charged and posses no angular momentum, the simplest way of achieving superradiance is through a massless charged scalar field which was investigated analytically in \cite{uchikata2011quasinormal} and then fully numerically in \cite{bosch2016nonlinear}. It should be noted that all the considerations laid out in the subsection on scalar fields \ref{subsec:Scalar} apply here - one just needs to give the field a charge, which we will take to be designated by $e$ from here onwards. Moreover, a complete phase space diagram of static, charged, asymptotically AdS solutions in 5D in terms of the charge $Q$ and the mass $M$ ($x$- and $y$-axis, respectively) of the solution is available and was obtained in the microcanonical ensemble in \cite{dias2012hairy}. It consists of static charged solitons, RN-AdS black holes and `hairy' black holes. Technically, there are more possible solutions - excited solitons or excited hairy black holes - but these should not be important as long as the charge of the scalar field is not very large. The first of the three solutions is basically a static blob of charged condensate with zero entropy in global ${\rm AdS}_5$, which for a given value of $e$ is entirely determined by its charge, whereby in the limit of the latter being infinitesimally small, the condensate reduces to the lowest energy linear perturbation of ${\rm AdS}_5$ by the scalar field. Skipping the second solution, as it is well-known, the third one represents a black hole, which is not entirely depleted of charge with a charged scalar condensate around it\footnote{It can be said that the condensate is an orbiting hair, but it should not be forgotten that the orbit is static - that is - nothing rotates around the black hole, as there is no angular momentum in the system.}. Depending on the value of $e$\footnote{The exact numerical values can be looked up in \cite{dias2012hairy}.} there are three different possible configurations of the phase space. For small charges of the scalar field there are only RN-AdS black holes and charged solitons, with the former always being the dominant phase from an entropy point of view, while the latter exist only up to some finite value of the charge $Q_{crit}$ and there are no instabilities in the system. However, in an intermediate range of values for the charge $e$, the RN-AdS black holes become unstable near extremality, with a condition on their charge $Q$ as a function of the scalar field one $e$. As expected from perturbative analysis, at the onset curve of this instability, the hairy black hole solutions branch off. The numerical construction in \cite{dias2012hairy} has shown that they do exist below the extremality curve for RN-AdS black holes and are the thermodynamically preferred solution whenever they appear. Solitons are also present up to some finite charge $Q_{crit}$, but for a given mass and charge are never the dominant stable solution in the phase diagram. Finally, in the third possible regime, for $e$ higher than some numerically found critical value (but not analytically justified yet), the RN-AdS black holes are unconditionally unstable near extremality. Moreover, this time the solitonic solution always exist, with masses below the extremal curve and it represents the ground state of the system for black hole charges below a certain transitioning value $Q_{c_2}$, whereas for $Q>Q_{c_2}$ it is again the hairy black holes that become thermodynamically favoured, but this time reducing in their zero mass limit to an infinite temperature soliton (they still branch off at the onset of instability). All these considerations have been deduced for the five-dimensional static charged vacuum solution of Einstein equations with negative cosmological constant, but one expects that the behaviour in four dimensions will not be qualitatively different. It should also be mentioned that the analysis in \cite{dias2012hairy} takes into account two types of mechanisms leading to instability - the first applies for black holes with radius much larger than the AdS curvature and is the result of the violation of the Breitenl\"ohner-Freedman (BF) bound of the near horizon extremal geometry (with topology ${\rm AdS}_2$) by the mass of the scalar field.\\
A detailed discussion will not be presented, but an intuitive description of the BF bound can be obtained in a simple way. Take a massive scalar field in pure ${\rm AdS}_d$ space and examine its Klein-Gordon equation $\left(\nabla_\mu\nabla^\mu-\mu^2\right)\Phi=0$ - by decomposing the scalar field according to the symmetries of the spacetime, as usual, and carrying out certain algebraic manipulations, a Schr\"odinger-like equation is derived. By consequently studying its potential, a condition on the stability of the solutions can be obtained, which gives the BF bound on the mass of the scalar field. One can perform a similar procedure for a massive scalar field in an asymptotically ${\rm AdS}_d$ spacetime that satisfies the Klein-Gordon equation $\left(\nabla_\mu\nabla^\mu-\mu^2\right)\Phi=0$, whereby doing a Frobenius analysis at infinity and requiring that the powers of the resulting coefficients of the solution are real provides a bound on the mass of the scalar field. Afterwards, looking at an extremal, asymptotically AdS black hole solution and considering the near horizon region, where the topology contains ${\rm AdS}_2$, whose BF bound is above the one of the background AdS space, it is realised that for scalar fields with masses between the two BF values, the near horizon geometry will be unstable, while the asymptotic space will not be.\\ The second kind of instability is due to superradiance and is applicable in the case of small black holes and is what interests us mainly. As already mentioned, in four dimensions the first detailed investigation of the superradiant modes and the instability of RN-AdS is carried out in \cite{uchikata2011quasinormal}, where the authors first perform the analytical analysis as outlined in \ref{subsec:Scalar} and find that the frequencies\footnote{The authors work with a few different frequency definitions, differing by a constant that is coming from the potential difference due to charge of the black hole - we have listed all the results in terms of the original $\omega$, used in the decomposition of the scalar field. The superradiance condition itself does not depend on that constant as one might expect.} are quantised as follows
\begin{gather}
\mbox{Re}(\omega)=\frac{2n+l+3}{L}\\
\mbox{Im}(\omega)=-\sigma_0\frac{(l!)^2(l+2+n)!2^{l+3}(2l+1+2n)!!}{(2l+1)!(2l)!n!(2l-1)!!(2l+1)!!(2n+3)!!}\frac{(r_+-r_-)^{2l+1}}{\pi L^{2l+2}}\prod_{k=1}^{l}(k^2+\sigma_0^2),
\end{gather}
where
\begin{align}
\sigma_0=\left(\omega_0-e\frac{Q}{r_+}\right)\frac{r_+^2}{r_+-r^-}\quad\mbox{with}\quad\omega_0=\frac{2n+l+3}{L}=\mbox{Re}(\omega),
\end{align}
where $\omega_0$ represents the QNM frequencies of the pure ${\rm AdS}_4$ spacetime, which are derived from Teukolsky equation in \cite{dias2013boundary}, with $n$ a non-negative integer called the radial overtone, which gives the number of nodes along the radial direction (of the radial eigenfunction). Clearly, $\sigma_0$ determines the sign of the imaginary part of the frequency, which in turn dictates whether the mode is superradiant or quasinormal. For $\sigma_0<0$ one gets Im$(\omega)>0$ and hence the mode is exponentially growing in time, whereas for $\sigma_0>0$, when Im$(\omega)<0$, the modes are quasinormal. As seen from the formula for $\sigma_0$, these statements are equivalent to
\begin{align}
\mbox{Re}(\omega)-e\frac{Q}{r_+}<0\;\Rightarrow\;\mbox{Im}(\omega)>0,\quad\mbox{and}\quad\mbox{Re}(\omega)-e\frac{Q}{r_+}>0\;\Rightarrow\;\mbox{Im}(\omega)<0
\end{align}
which is exactly the condition for superradiance that was presented earlier in section \ref{subsec:Scalar}. The authors of \cite{uchikata2011quasinormal} also provide a numerical investigation that supports their claims based on the perturbative analysis and show that indeed the imaginary part of the frequency changes sign, when the superradiant condition is met, however they do not do it at the full non-linear level, which would enable one to accurately say what is the endpoint of the superradiant instability. Fortunately, this is done in \cite{bosch2016nonlinear}. The simulations that have been carried out by the authors confirm the analytical results described just above as long as the perturbation remains small - which is expected, as for significant perturbations the non-linear effects and the backreaction on the spacetime become important. Furthermore, it is found that for small charges $e$ of the scalar field there are no unstable modes present - agreeing with what is discovered in \cite{dias2012hairy}. In the presence of superradiant modes, both mass and charge are extracted from the black hole by them, with the ratio between the two depending on the initial value of $e$ - the larger it is, the more charge and less mass is drained. The resulting charged scalar hair `orbits' the black hole with its distance from the black hole increasing with increasing $e$. The evolution of this instability proceeds in the following way - firstly, based on the initial data there will be a mix of QNMs and superradiant modes. The former will quickly decay, whereas the latter will steadily grow with time. Nonetheless, while the extraction is in progress, the charge $Q$ of the black hole will decrease, while the horizon radius $r_+$ will be increasing, consistent with the second law of black hole mechanics, meaning that the superradiance condition will be getting more and more stringent. This implies that gradually the superradiant modes (starting from large $n$) will cease being such and turn into QNMs and eventually decay and get absorbed by the black hole, restoring a bit of its mass and charge, but not enough to compensate for the extraction (the fundamental $n=0$ mode is the most effective at extracting). This goes on until only the fundamental mode $n=0$ is left - neither growing, nor decaying (this is seen from the simulations) - as a charged scalar condensate `orbiting' the black hole. Therefore, the endpoint of the instability, due to superradiance, for RN-AdS is at a hairy black hole, whereby the hair consists in a charged scalar condensate `orbiting' the black hole at a distance - the four-dimensional equivalent of the hairy black holes constructed in \cite{dias2012hairy}. A few remarks are in order here. After the fundamental mode has reached zero growth rate, it starts oscillating harmonically, which implies that the scalar field stress-energy tensor becomes time-independent and there are no more changes in the metric. Moreover, it is observed that the higher the value of $e$ is, the faster the whole evolution proceeds and as already mentioned much more charge than mass is extracted and the scalar hair condensates further away from the black hole. In the limit of very large scalar charge $e$ it is expected that the resulting configuration will be a Schwarzschild-AdS black hole with a scalar condensate very far away. Finally, we mentioned before that one might be tempted to make analogies between what happens in RN-AdS and what might happen in Kerr-AdS, by naively looking at the conditions for superradiance derived earlier (the one for Kerr-AdS is also confirmed in the literature)
\begin{align}
\mbox{Re}(\omega)-e\frac{Q}{r_+}<0\quad\mbox{and}\quad\mbox{Re}(\omega)-m\Omega_H<0.
\end{align}
This, unfortunately is not possible, due to the fact that in the first case the scalar field charge $e$ is held fixed at a given value, whereas for Kerr-AdS $m$ is allowed to take on any integer value. corresponding to the active superradiant modes, resulting in more complicated dynamics for the instability. Nevertheless, one might speculate that similar to the situation just discussed, the condition for superradiance will be getting stronger and stronger until only a single superradiant mode is left excited. An evolution at the fully non-linear level has not been carried out for Kerr-AdS yet, but there are a lot of results that point to very interesting possibilities for the endpoint of its instability, as it will be shown in the next section.
\vspace{-1mm}
\section{Kerr-AdS, Superradiance and the problem with instability}\label{sec:sec5}
\vspace{-1mm}
The previous section started with the remark that Reissner-Nordstr\"om black holes are not particularly relevant to astrophysical observations, but a similar comment can be made about Kerr-AdS solutions, as according to cosmological observations our Universe is almost flat. Nonetheless, as eluded to earlier, one can imagine a massive scalar field creating a trapping potential at a distance comparable to the radius of curvature of AdS, which would make it possible to compare the two situations. This is not the only reason why Kerr-AdS is attracting attention recently (as is evident from the growing number of papers on the topic) - from what has been done up to now in terms of research in the area it is currently not clear what the endpoint of its superradiant instability is. This is a rather delicate question with regards to the cosmic censorship conjectures as will become clearer soon, as we present the work that has been carried out on the subject until now.\\
\hspace*{5mm}The metric for Kerr-AdS was given in \ref{subsec:NPFormalism} and it will not be presented here again. The first analytical study of its QNM and superradiant modes was carried ot in \cite{cardoso2004small}. The type of perturbation considered was again a massless scalar field (uncharged) and the path of the analysis was similar to the one laid out in section \ref{subsec:Scalar} and subsequently repeated in the previous section on RN-AdS. The calculation starts from the wave equation for the field $\nabla_\mu\nabla^\mu\Phi=0$ and proceeds through the same decomposition of the field according to the background symmetries and then finishes with an asymptotic matching procedure in a zone where the near-horizon and far away regions overlap for the range of parameters $a/r_+\ll1$ and $r_+\omega\ll1$, corresponding to small black holes. Unfortunately, the authors did not impose the appropriate reflecting boundary conditions at $\mathcal{I}$ that will preserve the boundary metric and hence the asymptotic AdS structure. This was corrected in \cite{uchikata2009scalar}, where a numerical study of the problem was also supplemented in order to confirm the perturbative analysis and show that indeed small Kerr-AdS black holes are unstable to superradiant modes. We pause to say that a detailed derivation of the required boundary conditions at $\mathcal{I}$ for a general perturbation, corresponding to the definition of asymptotically AdS, given in section \ref{sec:AAdS}, is available in \cite{dias2013boundary}. Back to the scalar field perturbations - as for RN-AdS the analytically determined expression for the frequencies of the modes will be displayed, as found in \cite{uchikata2009scalar}:
\begin{align}
\omega&=\omega_0+i\delta,\quad\mbox{where}\quad\omega_0=\frac{2n+l+3}{L}=\mbox{Re}(\omega)\\
\delta&\approx-\sigma\left(\omega_0-m\Omega_H\right)\frac{(r_+^2+a^2)(r_+-r_-)^{2l}}{\pi L^{2(l+1)}},
\end{align}
where
\begin{gather}
\sigma=\frac{(l!)^2(l+2+n)!}{(2l+1)!(2l)!n!}\frac{2^{l+3}(2l+1+2n)!!}{(2l-1)!!(2l+1)!!(2n+3)!!}\prod_{k=1}^{l}\left(k^2+4\varpi^2\right),\\
\mbox{with}\quad\varpi=\left(\omega_0-m\Omega_H\right)\frac{r_+^2+a^2}{r_+-r_-},
\end{gather}
The results look of qualitatively the same form as for RN-AdS - the real part of the frequency is again equal to the normal modes of global ${\rm AdS}_4$, with $n$ a non-negative integer that denotes the radial overtone as before and the structure of the imaginary part is very similar. Its sign is determined by what is called the superradiant factor $\varpi$ (through the combination $\omega_0-m\Omega_H$) and the superradiance condition takes the form
\begin{align}
\mbox{Re}(\omega)-m\Omega_H<0\;\Rightarrow\;\mbox{Im}(\omega)>0,\quad\mbox{and}\quad\mbox{Re}(\omega)-m\Omega_H>0\;\Rightarrow\;\mbox{Im}(\omega)<0,
\end{align}
where the former implies that the mode is exponentially growing in time, hence superradiant, while the latter designates a QNM. As we already mentioned, the numerical results presented in \cite{uchikata2009scalar} confirm the above formulae and show that indeed there are superradiant modes present in the spectrum of massless scalar field perturbation of Kerr-AdS, depending on the value of the rotation parameter. Unfortunately, the authors do not manage to clarify the nature of this dependence - but the numerics suggest that for smaller black holes faster rotation implies more superradiance (possibly up to some critical value of $a$). As briefly remarked in the introduction of the section, there are no fully-nonlinear simulations performed for Kerr-AdS yet, thus we move on to another type of spacetime probing, which will be followed by a tour of the possible evolution of the system.\\
\hspace*{5mm}The analogue of the aforementioned analysis in the case of gravitational perturbations is performed in \cite{cardoso2014holographic} (The task was firts attempted in \cite{cardoso2006classical}, but unfortunately with incorrect boundary conditions at infinity). The approach to the perturbative calculations is very similar - it just starts from the Teukolsky equation for the case of gravitational perturbations - $s=\pm2$. Afterwards, approximating solutions in terms of hypergeometric functions are obtained in the near-horizon and far regions and then matched in an overlapping zone as before, while being cautious to impose the correct boundary conditions as prescribed by \cite{dias2013boundary}. The quantised spectrum of the perturbations' frequency will not be shown here, as it is much longer than the ones given before and contains hypergeometric functions, which makes it rather less illuminating. It is important to note that due to the restriction on $l$ that was given in the remarks following Teukolsky master equations, the smallest value it can take is $l=|s|=2$, implying that the Teukolsky formalism misses out two of the modes of the perturbations. Fortunately, in \cite{dias2013algebraically} it was proven that these modes only shift the mass and angular momentum of the solution, hence only correspond to deformations within the Kerr-AdS family. The perturbation sector under question is separated in two - scalar and vector gravitational perturbations and it is to be noted that the authors of \cite{cardoso2014holographic} concentrate on modes of the type $l=m$, but this should not have any effects on the qualitative results presetned. The analytical investigation indicates that the real part of the frequency in both sectors is very close to the values of the corresponding normal modes in global ${\rm AdS}_4$, which are given in \cite{dias2013boundary} (produced by solving Teukolsky equation with $a=0$ and $M=0$ for $s=2$). The imaginary parts can be either calculated numerically or expanded in series of the rotation parameter and horizon radius, both divided by the AdS curvature radius, in accordance with the parameter regime for Kerr-AdS that we introduced earlier and that is used in the paper for the analytical computations - $r_+/L\ll1$ and $a/L\ll1$. The expressions for Im$(\omega)$ resulting from these series expansions show that for $a=0$ the modes are always quasinormal and thus decaying, agreeing with results from Schwarzschild-AdS, where there are no superradiant modes\cite{dias2013boundary}. Furthermore, it is seen that for Im$(\omega)=0$ one gets Re$(\omega)-m\Omega_H\approx0$, while for Re$(\omega)-m\Omega_H>0$ the modes are damped with Im$(\omega)<0$ and if Re$(\omega)-m\Omega_H<0$, then Im$(\omega)>0$, indicating superradiant modes and confirming again the familiar condition for superradiance. Interestingly, the authors find that the Im$(\omega)$ increases with faster rotation, similar to the results of \cite{uchikata2009scalar}. The consequently presented numerical investigation of the modes, apart from confirming the analytical results, demonstrates that they posses a few interesting properties. Firstly, plotting the onset of the superradiant instability (where the imaginary part vanishes and $\omega=m\Omega_H$) as a function of the angular velocity and the (gauge invariant) radius of the black hole (that is - a contour plot as a function of $\Omega_H/L$ and $R_+/L$, where $R_+=\frac{\sqrt{r_+^2+a^2}}{\sqrt{\Xi}}$), it is discovered that all the onset curves lie above the line $\Omega_HL=1$, which was first conjectured in \cite{kunduri2006gravitational}, but in the limit of $R_+/L\rightarrow\infty$ - approach it gradually (in a different way for scalars and vectors). Furthermore, in the scalar sector, for small horizon radii, the larger $l=m$ is, the lower the onset curve of the mode starts as a function of the rotation - that is for small black holes the $l=m=2$ mode is the last to go unstable, as all modes with higher $l=m$ numbers will turn on at a lower rotation parameter. However, at larger radii, things seems to start reversing and the onset curves of the modes begin to cross, such that there are regions where the $l=m=2$ mode will switch on at a lower angular velocity than the $l=m=3$ mode, for example. Nonetheless, it should be noted that in the limit of $l=m\rightarrow\infty$, the corresponding mode will be an almost horizontal line, infinitesimally close to $\Omega_HL=1$, hence these modes will always be the first to become unstable. Lastly, all the modes asymptote to the $\Omega_HL=1$ line as the black hole radius goes to infinity. For the gravitational vector modes the picture is rather different - the first modes to go unstable are still the largest ones in terms of the numbers $l=m$, but this time there are no crossings whatsoever between the onset curves. Moreover, all the onset curves end at the extremality curve for Kerr-AdS, whereby modes with higher $l=m$ end up reaching it for even larger radii, such that they are slowly approaching the $\Omega_HL=1$ line (as the extremality curve is asymptotic to it). It can be deduced that the $l=m\rightarrow\infty$ mode will only reach the extremal curve in the limit $R_+/L\rightarrow\infty$, where it should also asymptote to the $\Omega_HL=1$ line (becoming again almost horizontal). Finally, two remarks regarding both sectors - it is observed in the numerical data that for a black hole of a fixed size the highest growth rate for a superradiant mode is always close to extremality, with this being much more the case for vector modes (probably due to the fact that their onset curves end up at the extremality curve). Secondly, the strength of the gravitational perturbations seems to be higher than that of the massless scalar field up to a few orders of magnitude in some cases.\\
\hspace*{5mm}Even though it was argued that the RN-AdS model does not allow for a straightforward generalisation to rotating black holes, looking at the story there, one might expect that, in the current scenario, at the onset of superradiance there might be a new family of stationary black holes merging or bifurcating with Kerr-AdS. This was actually proposed for the first time in \cite{kunduri2006gravitational}, based on the observation that the zero mode corresponding to the onset curves - $\omega=m\Omega_H$ and Im$(\omega)=0$ - is invariant under the horizon-generating Killing vector field $k=\partial_{\hat{t}}+\Omega_H\partial_{\hat{\varphi}}$. Hence it might be reasonable to expect the existence of black holes with a single helical KVF $k=\partial_{\hat{t}}+\Omega_H\partial_{\hat{\varphi}}$, which are neither time-symmetric, nor axisymmetric. Such a type of black holes, coupled to a matter field, were constructed perturbatively and fully numerically in the case of five-dimensional AdS background and a scalar field perturbation in \cite{dias2011black}, while the formulation as a solution to the Einstein equation with negative cosmological constant was accomplished in \cite{dias2015black}, numerically. This achievement is similar in nature to what was presented above in the situation of RN-AdS from \cite{dias2012hairy}, with the notable difference that these single KVF black holes represent a second unique solution in the system under question, together with the Meyers-Perry-AdS black holes, which are the five-dimensional generalisation of the Kerr-AdS solution\footnote{The uniqueness theorems are not violated, as the helical KVF is generating the horizon, thus is normal to it, which is in contrast with the assumptions of the theorems.}. Likewise, in this configuration the solitons are replaced by what are called rotating boson stars - smooth horizonless geometries with harmonic time dependence, parametrised by the amplitude of the scalar field (instead of its charge, as for RN-AdS), whereas the hairy black holes\footnote{Which are the single KVF black holes} have decided to change hairstyles and have opted for a chargless rotating scalar condensate. We are not going to investigate this phase space in great detail, instead the focus will be shifted towards four dimensions and gravitational perturbations (they were also shown to be stronger). Similar constructions have been devised both perturbatively and numerically in \cite{dias2015black,dias2012gravitational,horowitz2014geons}. The analogue of the charged solitons from the previous section and the aforementioned rotating boson stars are the so called geons - blobs of gravitational energy with harmonic time dependence - smooth and horizonless - they are the single mode, non-linear generalisations of some of the linearised gravitational perturbations of global ${\rm AdS}_4$ and posses helical symmetry. They have been analysed both analytically and numerically in \cite{dias2012gravitational,horowitz2014geons} with relevance to the stability of global AdS\footnote{Interestingly, global AdS is linearly stable, but non-linearly unstable due to a high number of resonances between modes, which are equidistantly spaced in the linearised theory. This will not be discussed here, but the above cited papers are a good read on the topic.}. On the other hand, the four-dimensional single Killing field black holes in this purely gravitational set up are first investigated analytically in \cite{cardoso2014holographic}, where their thermodynamic properties are derived to leading order, and then numerically in \cite{dias2015black} where they got named black resonators by the authors. This is due to the fact that these single KVF black holes branch off at the onset of superradiance for a specific superradiant mode in Kerr-AdS and thus select out its particular frequency, meaning that they are not unstable to perturbations by that mode. It is also proposed by the authors that the definition of stationary should be extended to include these solutions as well, even though they do not posses a timelike KVF, as they are still periodic in a sense, due to their helical KVF. Nevertheless, they are still unstable to modes with higher $m$ numbers as argued in the paper and as mathematically proven by the results in \cite{green2015superradiant} (with the slight caveat on this result, as elucidated earlier in \ref{sec:SimpleEx}), since the single KVF (which also generates the horizon) is not everywhere timelike, thus implying the existence of an ergoregion. This can also be expected from the results of \cite{cardoso2014holographic}, presented above, for small Kerr-AdS black holes, because it turns out that small black resonators can be approximated by small Kerr-AdS BHs centered at a geon. This is an interesting construction and the idea behind the perturbative formulation of the black resonators. The Kerr-AdS black hole is placed at the core of the geon and the angular velocities of the two are matched\footnote{So that there are no unwanted fluxes across the horizon.}, whereby the former controls the entropy and the temperature of the resulting object (geons have zero entropy and undefined temperature), whereas the latter is responsible for the single helical KVF nature of the resonators. In the limit of zero size they become geons with the picked out frequency corresponding to a normal mode of global ${\rm AdS}_4$. Henceforth, it can be said that the black resonators connect the onset of superradiance in Kerr-AdS to the horizonless geons. There are a few remarks, though, that have to be made. Firstly, the above analysis is done only for a single mode $l=m=2$, but other $l=m$ modes are expected to behave qualitatively the same, while $l\neq m$ ones are subject to investigations at the moment. Secondly, the black resonators have higher entropy than the Kerr-AdS black holes, thus whenever the two solutions coexist, the former will be favoured entropically (at the same asymptotic charges $E$ and $J$, that is). Furthermore, for a fixed energy $E$ and angular momentum $J$ the entropy of the black resonators is an increasing function of $m$, therefore progression towards superradiant modes with higher azimuthal numbers is preferred. These last two results should play an important role in the evolution of the superradiant instability. Starting with some initial data for Kerr-AdS with a mixture of QNMs and superradiant modes, the former will quickly decay, whilst the latter will go on bouncing back and forth between the horizon and the conformal boundary, extracting mass and angular momentum from the black hole on every cycle. This clearly leads to the decrease of $\Omega_H$ and should eventually result in a black hole with lumpy gravitational hair that is co-rotating with the black hole, which is invariant under only a single helical KVF - $k=\partial_{\hat{t}}+\Omega_H\partial_{\hat{\varphi}}$ - a black resonator that has branched off at the onset of superradiance. However, this cannot be the endpoint of the instability, as most likely during the evolution higher $m$ superradiant modes will be activated and while the newly formed black resonator is stable against the mode, from whose onset curve it had emerged, it is still unstable to perturbations with higher azimuthal numbers $m$. One might assume then that the system will continue evolving towards configurations with higher and higher $m$ - perhaps a mixture of a black resonator (or maybe a few) and co-rotating gravitational hair. In the limit of $m\rightarrow\infty$, as was already explained, the onset curve becomes a horizontal line, infinitesimally close to $\Omega_HL=1$ with all other onset curves lying above it, hence a configuration that is stable against this limiting mode will be stable against all other modes. Therefore the superradiance phenomenon will cease and one might expect that the resulting black hole will be a limiting black resonator with $m\rightarrow\infty$. There is a slight caveat however - it was explained earlier that in their zero size limit black resonators should represent geons - but in \cite{niehoff2015towards} the authors argue on the basis of perturbation theory and supersymmetry that such a geon does not exist, as it should be a minimum energy solution to the vacuum Einstein equations with negative cosmological constant. Unfortunately, in a supersymmetric setting AdS is the only such solution. This argument relies on the assumption of smoothness but it is also possible to envisage, as the endpoint of the instability, a limiting black resonator, whose zero size limit is a singular geon, which sticks well with the idea that singularities should be enclosed by event horizons. It turns out, though, that in perturbation theory the geons' curvature gets smaller and smaller with increasing azimuthal number $m$, thus ruling out the singularity scenario for the limiting geon. Of course, it should not be forgotten that these arguments are not backed by a full non-linear evolution of the system as in the case of RN-AdS, thus there might be other factors that will come into play as the system evolves. Finally, there is one more point worth mentioning. The numerical investigation in \cite{cardoso2014holographic} confirmed that only black holes with $\Omega_HL<1$ are stable to superradiance, implying that a logical expectation will be that single KVF black hole solutions with $\Omega_HL<1$ will be the endpoint of the superradiant instability. Unfortunately, in the very similar in nature configuration in five dimensions, which was analysed in \cite{dias2011black}, none of the fully numerically constructed single KVF black holes has $\Omega_HL<1$.\\
\hspace*{5mm}Summarising the above discussion, without a full numerical simulation, it seems that there is no regular solution that comes out at the endpoint of the superradiant instability. The system either settles down to a singularity in a finite time - violating the weak cosmic censorship, or it goes on evolving indefinitely towards configurations with even higher azimuthal number $m$ and therefore also higher entropy. This implies that eventually it would be necessary to consider physics on such small scales that the effects of quantum theory might become important, which is not what is expected from the point of view of the strong cosmic censorship, since the initial system was well defined classically.
\section{Conclusion}
In this essay we examined the phenomenon of superradiance in asymptotically AdS spacetimes, giving priority to its effect on the stability of the involved spaces. Due to its timelike boundary at spatial infinity, AdS provides us with a natural way of working in a confining box with reflecting walls, given that the correct boundary conditions at spatial infinity - keeping the boundary metric fixed - are defined. In this set up one only needs to take advantage of the well-established Newman-Penrose-Teukolsky formalism in order to study perturbations of any type in the given spacetime by directly going on solving the Teukolsky master equation. This has been done for a plethora of configurations and we presented the results for charged (RN-AdS) and rotating (Kerr-AdS) black holes, with the obvious absence of Schwarzschild-AdS, because it does not posses any superradiant modes. Even though the former two spacetimes share many similarities in their QNM and superradiant spectra, the little difference between the condition for superradiance in both cases - $\mbox{Re}(\omega)-q\frac{Q}{r_+}<0$ and $\mbox{Re}(\omega)-m\Omega_H<0$ - that is, the fixed value of the charge of the external perturbation $q$ for RN-AdS, versus the freedom of the azimuthal number $m$ to take on any integer value in Kerr-AdS - leads to conceptually different outcomes. While for RN-AdS a fully non-linear evolution of the system has confirmed that there is an endpoint for the superradiant instability at a black hole with static charged scalar condensate around it, for Kerr-AdS this seems unlikely due to conjectured progress of the system towards configurations with even higher $m$ modes. According to the presented in this work research in the area, at the onset of superradiance in Kerr-AdS a second stationary\footnote{With a discussion in the main text on its meaning in this case.} solution branches off, which represents the so called black resonators - black holes with a single helical KVF that is also a generator of the horizon and which are connected in their zero-size limit to smooth horizonless solutions of the vacuum Einstein equations in AdS. Unfortunately, by a mathematical result in \cite{green2015superradiant}, the spacelike nature of the single KVF in some regions implies the existence of an ergoregion, thus rendering the resonators unstable to superradiance as well. By naively following one's nose towards the limit of $m\rightarrow\infty$ one reaches the conclusion that what might be the endpoint of the instability in the form of the limiting black resonator seems to be not well-defined, as its limiting geon is proven not to exist. The conclusion that is drawn from the situation is that there are two possibilities - either the instability leads to a singularity, which violates the weak cosmic censorship, or the system evolves towards configurations which require considerations at even smaller scales, making it necessary to take quantum effects into account - going against the spirit of the strong cosmic censorship, as the system that was started from is classically well-defined. All this conclusions were derived on the basis of perturbation thery, as a fully non-linear simulation in the case of Kerr-AdS has not been carried out yet, but the above propositions only make it more exciting until the complete answer is uncovered.
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Fitz Hugh Lane (* 18. Dezember 1804 in Gloucester, Massachusetts; † 13. August 1865 ebenda), vor seiner offiziellen Namensänderung im Jahr 1831 Nathaniel Rogers Lane, war ein amerikanischer Maler und Lithograf, der vor allem für seine Schiffsporträts bekannt wurde. Er gilt als einer der zentralen Maler des Luminism, den er etwa zur gleichen Zeit entwickelte wie die Maler der Hudson River School.
Leben und Werk
Fitz Hugh Lane wurde 1804 in Gloucester an der Küste von Massachusetts geboren. Er musste, wahrscheinlich aufgrund einer Erkrankung an der Kinderlähmung im Alter von zwei Jahren, bereits als Kind an Krücken gehen. Als Jugendlicher begann er mit dem Malen und Zeichnen und 1832 arbeitete er für kurze Zeit bei einem Lithografen in Gloucester. Im gleichen Jahr begann er eine Ausbildung in Boston bei William S. Pendleton, der das größte Lithografieunternehmen Bostons betrieb, und blieb dort bis 1837. Während dieser Anstellung fertigte er Lithografien für Noten und Landschaftsbilder an.
Lane lernte in Boston das Werk des englischen Marinemalers Robert Salmon kennen, der ebenfalls hier lebte und bei Pendleton arbeitete. Er begann 1840 selber damit, Ölbilder nach Salmons Vorbild zu malen, und konnte 1841 sein Werk Scene at the Sea (Verbleib unbekannt) im Bostoner Athenaeum ausstellen. Dieses Museum zeigte Lanes Bilder ab 1845 regelmäßig. Bis Mitte der 1840er Jahre konzentrierte sich Lane auf Hafenansichten, Landschaftsbilder und Schiffsporträts, wobei er sowohl als Maler wie auch, gemeinsam mit John W. Scott, als Lithograf tätig war. 1848 erfolgte ein erster Verkauf an die Art Union in New York City, die später weitere Bilder von ihm kaufte. Im gleichen Jahr reiste er erstmals nach Maine, dessen Landschaften vor allem um Cape Ann neben Gloucester später das Zentrum seines Werks ausmachten.
Im Jahr 1848 baute Lane gemeinsam mit seiner Schwester und ihren Mann ein Haus in Gloucester und kehrte an seinen Geburtsort zurück. Während der 1850er und 1860er Jahre entwickelte er seinen Stil weiter. In den 1850er Jahren malte er unter anderem eine Serie von Bildern des Bostoner Hafens. Seine Werke der 1850er Jahre zeichnen sich durch eine ruhige Komposition mit deutlich strahlenden Licht- und atmosphärischen Effekten aus, wie sie in den luministischen Werken der Hudson River School vorhanden sind, wobei ein äußerer Einfluss auf das Werk von Lane insgesamt kaum merkbar ist und sich Lane selbst kaum mit den Werken anderer Maler beschäftigte und auch nicht in Künstlerkreisen verkehrte. In den 1860er Jahren konzentrierte er sich wieder auf die Landschaften und Meeresansichten von Gloucester sowie Cape Ann in Maine, wobei er das Format der Bilder deutlich reduzierte und die Darstellung präziser gestaltete.
1864 verschlechterte sich der Gesundheitszustand und im August jenes Jahres erlitt er einen Herzinfarkt oder einen Schlaganfall, wahrscheinlich infolge eines Sturzes. Er starb am 13. August des Jahres. Zu seinen Lebzeiten gelangte er eigentlich nur zu lokaler Bedeutung, was dazu führte, dass er später weitgehend vergessen wurde. Erst mit dem neu erwachten Interesse an der amerikanischen Malerei des 19. Jahrhunderts in den 1940er Jahren sowie durch eine große Bilderspende des Sammlers Maxim Karolik 1949 an das Museum of Fine Arts in Boston mit zahlreichen Werken Lanes erwachte auch das Interesse an diesem Künstler neu.
Bildauswahl
Literatur
Stephan Koja: America. Die Neue Welt in Bildern des 19. Jahrhunderts. Prestel, München 1999, ISBN 3-7913-2051-3, S. 265.
Matthew Baigell: Dictionary of American Art. Harper & Row New York u. a. 1979; S. 201–203. ISBN 0-06--433254-3.
Thomas W. Gaethgens: Bilder aus der Neuen Welt. Amerikanische Malerei des 18. und 19. Jahrhunderts Prestel, München 1988, ISBN 3-7913-0879-3, S. 312.
Weblinks
Fitz Hugh Lane in der Artcyclopedia
Einzelnachweise
Maler (Vereinigte Staaten)
Marinemaler
US-Amerikaner
Geboren 1804
Gestorben 1865
Mann
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Rockin'1000 est un groupe d'un millier de musiciens rock provenant principalement d'Italie et de quelques autres pays comme la France, le Canada, le Mexique, l'Angleterre, l'Autriche, la Bosnie-Herzégovine et l'Allemagne.
Historique
Comme son nom l'indique, le groupe réunit environ mille musiciens qui se produisent simultanément lors des prestations qu'ils effectuent. L'organisation est basée à Cesena en Italie, initialement assemblée en pour faire venir jouer les Foo Fighters dans sa ville en reprenant Learn to Fly ce qui a finalement eu l'effet escompté en .
Ils sont aujourd'hui le « plus grand groupe au monde ».
Ils ont été constitués à l'origine via un financement participatif organisé par Fabio Zaffagnini . Leur prestation initiale en 2015 a été dirigée par Marco Sabiu.
Ils ont joué un concert de 18 chansons le à Cesena, en Italie dans le stade Orogel devant un public d'environ personnes.
Le , le groupe a joué un concert de 19 chansons dans le Stade de France à Saint-Denis près de Paris auquel personnes ont assisté.
Le , Rockin'1000 a joué 18 chansons à la Commerzbank Arena de Francfort, en Allemagne. .
Le de la même année, Rockin'1000 a joué un concert de 18 chansons à l'aéroport de Milan Linate pour célébrer la réouverture de l'aéroport au Milano Linate Airshow devant personnes.
Le 30 octobre 2020 le Rockin'1000 établit un record du monde auprès du Guinness World Records™ pour "Le plus de vidéos dans un medley musical" à Dubaï lors du Global GIG 2020 avec plus de 2 500 chanteurs et musiciens venant de plus de 80 pays.
Le , soit près de 3 ans après le dernier show, Rockin'1000 a joué un concert de 21 chansons au Stade de France à Saint-Denis (93) devant plus de personnes.
Un hommage spécial au regretté batteur des Foo Fighters Taylor Hawkins a été rendu avec l'interprétation de la chanson "My Hero".
Références
Lien externe
Groupe de rock
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Let $\pi(x)$ denotes the number of primes not exceeding $x$. Since there are infinitely many primes, we have $\pi(x) \to \infty$ for $x \to \infty$. In 1793, Gau{\ss} \cite{gauss} stated a conjecture concerning an asymptotic behavior of $\pi(x)$, namely
\begin{equation}
\pi(x) \sim \text{li}(x) \quad\quad (x \to \infty), \tag{1.1} \label{1.1}
\end{equation}
where the \emph{logarithmic integral} $\text{li}(x)$ defined for every real $x \geq 0$ as
\begin{equation}
\text{li}(x) = \int_0^x \frac{dt}{\log t} = \lim_{\varepsilon \to 0} \left \{ \int_{0}^{1-\varepsilon}{\frac{dt}{\log t}} + \int_{1+\varepsilon}^{x}{\frac{dt}{\log t}} \right \} = \int_2^x \frac{dt}{\log t} + 1.04516\ldots. \tag{1.2} \label{1.2}
\end{equation}
The asymptotic formula \eqref{1.1} was proved independently by Hadamard \cite{hadamard1896} and by de la Vall\'{e}e-Poussin \cite{vallee1896} in 1896, and is known as the \textit{Prime Number Theorem}. In his later paper \cite{vallee1899}, where he proved the existence of a zero-free region for the Riemann zeta-function $\zeta(s)$ to the left of the line $\text{Re}(s) = 1$, de la Vall\'{e}e-Poussin also estimated the error term in the Prime Number Theorem by showing
\begin{equation}
\pi(x) = \text{li}(x) + O(x \exp(-a\sqrt{\log x})), \tag{1.3} \label{1.3}
\end{equation}
where $a$ is a positive absolute constant. The work of Korobov \cite{korobov1958} and Vinogradov \cite{vinogradov1958} implies a much better result, namely that there is a positive absolute constant $c$ so that
\begin{displaymath}
\pi(x) = \text{li}(x) + O \left( x \exp \left( - c (\log x)^{3/5} (\log \log x)^{-1/5} \right) \right).
\end{displaymath}
In 1901, von Koch \cite{koch1901} deduced under the assumption that the Riemann hypothesis is true a remarkable refinement of the error term, namely
\begin{equation}
\pi(x) = \text{li}(x) + O(\sqrt{x} \log x). \tag{1.4} \label{1.4}
\end{equation}
In 2000, Panaitopol \cite[p. 55]{pan3} gave another asymptotic formula for the prime counting function by showing that for each positive integer $m$, we have
\begin{equation}
\pi(x) = \frac{x}{ \log x - 1 - \frac{k_1}{\log x} - \frac{k_2}{\log^2x} - \ldots - \frac{k_m}{\log^m x}} + O \left( \frac{x}{\log^{m+2}x} \right), \tag{1.5} \label{1.5}
\end{equation}
where the positive integers $k_1, \ldots, k_m$ are defined by the recurrence formula
\begin{displaymath}
k_m + 1!k_{m-1} + 2!k_{m-2} + \ldots + (m-1)!k_1 = m \cdot m!.
\end{displaymath}
For instance, we have
\begin{displaymath}
k_1 = 1, \, k_2 = 3, \, k_3 = 13, \, k_4 = 71, \, k_5 = 461, \, k_6 = 3\,441.
\end{displaymath}
Hence, the asymptotic formula \eqref{1.5} implies that the inequality that
\begin{equation}
\pi(x) > \frac{x}{\log x - 1 - \frac{1}{\log x} - \frac{3}{\log^2 x} - \ldots - \frac{k_n}{\log^n x}} \tag{1.6} \label{1.6}
\end{equation}
holds for every positive integer $n$ and all sufficently large values of $x$. The first result in this direction is from 1962 and is due to Rosser and Schoenfeld \cite[Corollary 1]{rosser1962}. They showed that the inequality
\begin{equation}
\pi(x) > \frac{x}{\log x} \tag{1.7} \label{1.7}
\end{equation}
holds for every $x \geq 17$. In 1998, Dusart \cite[Th\'{e}or\`{e}me 1.10]{pd1} obtained that
\begin{displaymath}
\pi(x) > \frac{x}{\log x - 1}
\end{displaymath}
for every $x \geq 5393$. The current best result concerning an upper bound which corresponds to
the first terms of \eqref{1.5} is given in \cite[Korollar 1.24]{axler2013} and states that
\begin{displaymath}
\pi(x) > \frac{x}{\log x - 1 - \frac{1}{\log x}}
\end{displaymath}
for every $x \geq 468\,049$. In the following theorem, we make a first progress in finding the smallest positive integer $N_0$ so that the inequality \eqref{6.2} holds for $n=2$ and every $x \geq N_0$.
\begin{thm} \label{thm101}
The inequality
\begin{displaymath}
\pi(x) > \frac{x}{\log x - 1 - \frac{1}{\log x} - \frac{3}{\log^2 x}}
\end{displaymath}
holds for every $x$ such that $65\,405\,887 \leq x \leq 2.73 \cdot 10^{40}$ and every $x \geq e^{580044/13}$.
\end{thm}
Integration of parts in \eqref{1.3} implies that the asymptotic expansion
\begin{equation}
\pi(x) = \frac{x}{\log x} + \frac{x}{\log^2 x} + \frac{2x}{\log^3 x} + \frac{6x}{\log^4 x} + \ldots + \frac{(m-1)! x}{\log^mx}+ O \left( \frac{x}{\log^{m+1} x} \right) \tag{1.8} \label{1.8}
\end{equation}
holds for each positive integer $m$, which implies that there exists a smallest positive integer $g_1(n) \geq 2$ so that the inequality
\begin{displaymath}
\pi(x) > \frac{x}{\log x} + \frac{x}{\log^2 x} + \frac{2x}{\log^3 x} + \frac{6x}{\log^4 x} + \frac{24x}{\log^5 x} + \ldots + \frac{(n-1)!x}{\log^nx}
\end{displaymath}
holds for every positive integer $n$ and every $x \geq g_1(n)$. Again, the inequality \eqref{1.7}, obtained by Rosser and Schoenfeld \cite[Corollary 1]{rosser1962}, was the first result concerning an upper bound which corresponds to the first terms of \eqref{1.8}. Dusart \cite[Th\'{e}or\`{e}me 1.10]{pd1} found in 1998 that $g_1(2) = 599$. In 2010, he \cite[Theorem 6.9]{dusart2010} improved his own result by showing that $g_1(3) = 88\,783$. In the following theorem, we go one step further by finding an upper bound for the smallest positive integer $g_1(4)$.
\begin{thm} \label{thm102}
The inequality
\begin{displaymath}
\pi(x) > \frac{x}{\log x} + \frac{x}{\log^2 x} + \frac{2x}{\log^3 x} + \frac{6x}{\log^4 x}
\end{displaymath}
holds for every $x$ such that $10\,384\,261 \leq x \leq 2.73 \cdot 10^{40}$ and every $x \geq e^{6719}$.
\end{thm}
As an application of the estimates for the prime counting function which hold for all sufficiently large values of $x$, we consider an inequality established by Ramanujan. In one of his notebooks (see Berndt \cite{berndt1994}), Ramanujan used \eqref{1.8} with $n = 5$ to find that
\begin{displaymath}
\pi(x)^2 - \frac{ex}{\log x} \pi \left( \frac{x}{e} \right) = - \frac{x^2}{\log^6x} + O \left( \frac{x}{\log^7 x} \right)
\end{displaymath}
and concluded that the inequality
\begin{equation}
\pi(x)^2 < \frac{ex}{\log x} \pi \left( \frac{x}{e} \right) \tag{1.9} \label{1.9}
\end{equation}
holds for all sufficiently large values of $x$. The inequality \eqref{1.9} is called \textit {Ramanujan's prime counting inequality}. The problem arose to find the smallest integer $H_0$ so that the inequality \eqref{1.9} holds for every real $x \geq H_0$. Under the assumption that the Riemann hypothesis is true (RH), Hassani \cite[Theorem 1.2]{hassani2012} has given the upper bound
\begin{displaymath}
RH \; \Rightarrow \; H_0 \leq 138\,766\,146\,692\,471\,228.
\end{displaymath}
In 2015, Dudek and Platt \cite[Lemma 3.2]{dudekplatt} refined Hassani's result by showing
\begin{equation}
RH \; \Rightarrow \; H_0 \leq 1.15 \cdot 10^{16}. \tag{1.10} \label{1.10}
\end{equation}
Wheeler, Keiper and Galway (see Berndt \cite[p. 113]{berndt1994}) attempted to determine the value of $H_0$, but they failed. Nevertheless, Galway found that the largest prime up to $10^{11}$ for which the inequality \eqref{1.9} fails is $x=38\,358\,837\,677$. Hence
\begin{displaymath}
H_0 > 38\,358\,837\,677.
\end{displaymath}
Dudek and Platt \cite[Theorem 1.3]{dudekplatt} showed by computation that $x = 38\,358\,837\,682$ is the largest integer counterexample below $10^{11}$ and that there are no more failures at integer values before $1.15 \cdot 10^{16}$. Hence the inequality \eqref{1.9} holds unconditionally for every $x \in I_0$, where $I_0 = [38\,358\,837\,683, 1.15 \cdot 10^{16}]$. Together with \eqref{1.10},
\begin{equation}
RH \; \Rightarrow \; H_0 = 38\,358\,837\,683. \tag{1.11} \label{1.11}
\end{equation}
Based on a result of Büthe \cite[Theorem 2]{buethe}, we extend the interval $I_0$, in which the inequality \eqref{1.9} holds unconditionally by showing the following theorem.
\begin{thm} \label{thm103}
Ramanujan's prime counting inequality \eqref{1.9} holds unconditionally for every $x$ such that $38\,358\,837\,683 \leq x \leq 10^{19}$.
\end{thm}
In addition, Dudek and Platt \cite[Theorem 1.2]{dudekplatt} claimed to give an upper bound for $H_0$ which does not depend on the assumption that the Riemann hypothesis is true, namely
\begin{equation}
H_0 \leq e^{9658}. \tag{1.12} \label{1.12}
\end{equation}
After the present author raised some doubts about the correctness of the proof of \eqref{1.12}, one of the authors confirmed (email communication) that the proof of \eqref{1.12} given in \cite{dudekplatt} is not correct. This motivated us to write this paper, where we prove the following even stronger result. In our proof, explicit estimates for the prime counting function which hold for all sufficiently large values of $x$ play an important role.
\begin{thm} \label{thm104}
Ramanujan's prime counting inequality \eqref{1.9} holds unconditionally for every real $x \geq e^{9032}$; i.e.
\begin{displaymath}
H_0 \leq e^{9032}.
\end{displaymath}
\end{thm}
In Section 7, we use Theorem \ref{thm103}, Theorem \ref{thm104} and \eqref{1.11} to establish a result concerning a generalized inequality of Ramanujan's prime counting inequality \eqref{1.9}.
\section{On Chebyshev's $\vartheta$-function}
In order to prove Theorem \ref{thm101} and Theorem \ref{thm102}, we first consider Chebyshev's $\vartheta$-function, which is defined by
\begin{displaymath}
\vartheta(x) = \sum_{p \leq x} \log p,
\end{displaymath}
where $p$ runs over primes not exceeding $x$. The prime counting function and Chebyshev's $\vartheta$-function are connected by the well-known identities
\begin{equation}
\pi(x) = \frac{\vartheta(x)}{\log x} + \int_{2}^{x}{\frac{\vartheta(t)}{t \log^{2} t}\ dt}, \tag{2.1} \label{2.1}
\end{equation}
and
\begin{equation}
\vartheta(x) = \pi(x) \log x - \int_{2}^{x}{\frac{\pi(t)}{t}\ dt}, \tag{2.2} \label{2.2}
\end{equation}
which hold for every $x \geq 2$ (see, for instance, Apostol \cite[Theorem 4.3]{ap}). Using \eqref{2.2}, it is easy to see that the Prime Number Theorem is equivalent to
\begin{equation}
\vartheta(x) \sim x \quad\quad (x \to \infty). \tag{2.3} \label{2.3}
\end{equation}
By proving the existence of a zero-free region for the Riemann zeta-function $\zeta(s)$ to the left of the line $\text{Re}(s) = 1$ , de la Vall\'{e}e-Poussin \cite{vallee1899} was abled to bound the error term in \eqref{2.3} by proving
\begin{equation}
\vartheta(x) = x + O(x \exp(-a\sqrt{\log x})), \tag{2.4} \label{2.4}
\end{equation}
where $a$ is a positive absolute constant. In this direction, we give the following result.
\begin{prop}\label{prop201}
Let $R = 5.573412$. Then,
\begin{equation}
|\vartheta(x) - x| < \frac{\sqrt{8}}{\sqrt{\pi \sqrt{R}}} \, x (\log x)^{1/4} e^{- \sqrt{(\log x)/R}} \tag{2.5} \label{2.5}
\end{equation}
for every $x \geq 3$.
\end{prop}
\begin{proof}
By Mossinghoff and Trudgian \cite[Theorem 1]{mossing}, there are no zeros of the Riemann zeta fuction $\zeta(s)$ for $| \text{Im}(s)| \geq 2$ and
\begin{displaymath}
\text{Re}(s) \geq 1 - \frac{1}{R \log | \text{Im}(s)|}.
\end{displaymath}
Applying this to \cite[Theorem 1.1]{dusart2016}, we get that the required inequality holds for every $x \geq e^{390}$. Further, Trudgian \cite[Theorem 1]{trud} showed that the inequality
\begin{displaymath}
|\vartheta(x) - x| < \frac{\sqrt{8}}{\sqrt{17\pi \sqrt{6.455}}} \, x (\log x)^{1/4} e^{- \sqrt{(\log x)/6.455}}
\end{displaymath}
holds for every $x \geq 149$. We conclude for the case $149 \leq x \leq e^{390}$ by comparing the right hand side of the last inequality with the right hand side of \eqref{2.5}. For the remaining case $3 \leq x \leq 149$, we check the desired inequality with a computer.
\end{proof}
Now, we use Proposition \ref{prop201} to obtain the following result concerning an explicit estimates for the distance between $x$ and $\vartheta(x)$, which we use in the proof of Theorem \ref{thm101}.
\begin{kor} \label{kor202}
For every $x \geq 2$, we have
\begin{displaymath}
\vert \vartheta(x) - x \vert < \frac{580115x}{\log^5 x}.
\end{displaymath}
\end{kor}
\begin{proof}
We use Proposition \ref{prop201} to get that the required inequality
holds for every $x \geq e^{5801.149}$.
In \cite[Proposition 2.5]{axler2017}, it is shown that the inequality $\vert \vartheta(x) - x \vert < 100 x/\log^4 x$ holds for every $x \geq 70\,111$, which implies the validity of the required inequality for every $70\,111 \leq x \leq e^{5801.15}$. For the remaining cases, we use a computer.
\end{proof}
\section{Proof of Theorem \ref{thm101}}
Let $k$ be a positive integer, $\eta_k$ and $x_1(k) \geq 2$ positive real numbers so that
\begin{equation}
|\vartheta(x) - x| < \frac{\eta_kx}{\log^k x} \tag{3.1} \label{3.1}
\end{equation}
for every $x \geq x_1(k)$ (The existence of such parameters is guaranteed by \eqref{2.4}). By \eqref{2.1}, we have
\begin{displaymath}
\pi(x) = \pi(x_1(k)) - \frac{\vartheta(x_1(k))}{\log x_1(k)} + \frac{\vartheta(x)}{\log x} + \int_{x_1(k)}^{x}{\frac{\vartheta(t)}{t\log^{2} t}\ dt}.
\end{displaymath}
Now, we use \eqref{3.1} to derive
\begin{equation}
J_{k,-\eta_k,x_1(k)}(x) \leq \pi(x) \leq J_{k,\eta_k,x_1(k)}(x) \tag{3.2} \label{3.2}
\end{equation}
for every $x \geq x_1(k)$, where
\begin{align}
J_{k,\eta_k,x_1(k)}(x) & = \pi(x_1(k)) - \frac{\vartheta(x_1(k))}{\log x_1(k)} + \frac{x}{\log x} + \frac{\eta_k x}{\log^{k+1} x} + \int_{x_1(k)}^{x}{\left( \frac{1}{\log^{2} t} + \frac{\eta_k}{\log^{k+2} t} \ dt \right)}. \tag{3.3} \label{3.3}
\end{align}
The function $J_{k,\eta_k,x_1(k)}$ given in \eqref{3.3} was already introduced by Rosser and Schoenfeld \cite[p.81]{rosser1962} (for the case $k=1$) and Dusart \cite[p. 9]{dusart2010} and plays an important role in the following proof of Theorem \ref{thm101}
\begin{proof}[Proof of Theorem \ref{thm101}]
First, we verify the validity of the required inequality, i.e.
\begin{equation}
\pi(x) > \frac{x}{\log x - 1 - \frac{1}{\log x} - \frac{3}{\log^2 x}}, \tag{3.4} \label{3.4}
\end{equation}
for every $x \geq e^{580044/13}$. For this, let $k=5$, $x_1 = 10^{13}$ and
\begin{displaymath}
f(x) = \frac{x}{\log x - 1 - \frac{1}{\log x} - \frac{3}{\log^2x} - \frac{13}{\log^3 x} + \frac{580044}{\log^4 x}}.
\end{displaymath}
Further, we set $g(x) = J_{5, -580115, x_1}(x) - f(x)$. Then,
\begin{displaymath}
g'(x) = \frac{s(\log x)}{(\log^5x - \log^4x - \log^3x - 3\log^2x - 13\log x + 580044)^2\log^7x},
\end{displaymath}
where
\begin{align*}
s(y) & = 580\,576y^{10} - 6\,381\,045y^9 - 4\,060\,210y^8 - 15\,661\,259y^7 - 336\,607\,082\,789y^6 \\
& \phantom{\quad\quad} + 4\,037\,979\,215\,095y^5 - 2\,691\,881\,529\,325y^4 - 1\,345\,840\,694\,825y^3 \\
& \phantom{\quad\quad} - 1\,345\,478\,703\,065y^2 - 195\,224\,040\,181\,960\,440y + 975\,901\,480\,963\,513\,200.
\end{align*}
Since $s(y) > 0$ for every $y \geq \log x_1 \geq 28$, we get that
\begin{equation}
J'_{5, -580115, x_1}(x) \geq f'(x) \tag{3.5} \label{3.5}
\end{equation}
for every $x \geq x_1$. By Dusart \cite[Table 6.1]{dusart2010}, we have $\vartheta(x_1) \leq 9\,999\,996\,988\,294$. Since $\pi(x_1) = 346\,065\,536\,839$, we use \eqref{3.3} to get $J_{5, -580115, x_1}(x_1) - f(x_1) > 3 \cdot 10^8$. Together with \eqref{3.5}, we obtain that $J_{5, -580115, x_1}(x) > f(x)$ for every $x \geq x_1$. Now, we use \eqref{3.2} and Corollary \ref{kor202} to get that the inequality $\pi(x) \geq f(x)$ holds for every $x \geq x_1$, which implies the validity of \eqref{3.4} for every $x \geq e^{580044/13}$.
In the second step, we show that the inequality \eqref{3.4} is fulfilled for every $10^{12} \leq x \leq 2.73 \cdot 10^{40}$. In \cite[Theorem 3.8]{axler2017}, it is shown that
\begin{displaymath}
\pi(t) > \frac{t}{\log t - 1 - \frac{1}{\log t} - \frac{2.85}{\log^2t} - \frac{13.15}{\log^3t} - \frac{70.7}{\log^4t} - \frac{458.7275}{\log^5t} - \frac{3428.7225}{\log^6t}}
\end{displaymath}
for every $t \geq 19\,033\,744\,403$. A comparsion of the last right hand side with the right hand side of \eqref{3.4} implies that the desired inequality \eqref{3.4} holds for every $19\,033\,744\,403 \leq x \leq 2.73 \cdot 10^{40}$.
To complete the proof, we check with a computer that $\pi(p_n) > s(p_{n+1})$ for every $\pi(65\,405\,887) \leq n \leq \pi(19\,033\,744\,403) + 1$.
\end{proof}
Using a result of Schoenfeld \cite[Corollary 1]{schoenfeld1976}, we obtain the following result.
\begin{prop} \label{prop301}
Under the assumption that the Riemann hypothesis is true, the inequality \eqref{3.4} holds for every $x \geq 65\,405\,887$.
\end{prop}
\begin{proof}
We denote the right hand side of \eqref{3.4} by $g(x)$ and set $h(x) = - \log^8x + 208\pi \sqrt{x}\log^2x + 96\pi \sqrt{x}\log x + 144\pi \sqrt{x}$. Then, $h(x) > 0$ for every $x \geq 233\,671\,227\,509$. Further, we define $f(x) = \text{li}(x) - \sqrt{x} \log x /(8\pi) - g(x)$. Then, $f'(x) \geq h(x)/(16\pi\sqrt{x}(\log^3 x - \log^2x - \log x - 3)^2 \log x) > 0$ for every $x \geq 233\,671\,227\,509$. In addition, we have $f(10^{12}) > 0$. So,
\begin{equation}
\text{li}(x) - \frac{\sqrt{x}}{8\pi} \, \log x > \frac{x}{\log x - 1 - \frac{1}{\log x} - \frac{3}{\log^2 x}} \tag{3.6} \label{3.6}
\end{equation}
for every $x \geq 10^{12}$. Under the assumption that the Riemann hypothesis is true, Schoenfeld \cite[Corollary 1]{schoenfeld1976} showed that the inequality $\pi(x) > \text{li}(x) - \sqrt{x} \log x /(8\pi)$ holds for every $x \geq 2\,657$. We conclude by applying \eqref{3.6} and Theorem \ref{thm101}.
\end{proof}
\section{Proof of Theorem \ref{thm102}}
In this section, we give a proof of Theorem \ref{thm102}. Let $n$ be a positive integer and $R = 5.573412$. Proposition \ref{prop201} implies that the inequality
\begin{equation}
|\vartheta(x) - x| < \frac{a_n(x)x}{\log^n x} \tag{4.1} \label{4.1}
\end{equation}
holds for every $x \geq 3$, where the function $a_n: [2, \infty) \to (0, \infty)$ is defined by
\begin{displaymath}
a_n(x) = \frac{\sqrt{8}}{\sqrt{\pi \sqrt{R}}} \, (\log x)^{n + 1/4} e^{-\sqrt{(\log x)/R}}.
\end{displaymath}
A straightforward calculation shows that the function $a_n(x)$ has a global minimum at $x_0 = e^{(4n+1)^2R/4}$.
For the proof of Theorem \ref{thm104}, we need the following inequality involving the function $a_n(x)$.
\begin{prop} \label{prop401}
For every $x \geq 851$, we have
\begin{displaymath}
\int_3^x \frac{a_n(t)}{\log^{n+2}t} \, dt \leq \frac{\sqrt{2}}{\sqrt{\pi \sqrt{R}}} \cdot \frac{x}{(\log x)^{3/4} e^{\sqrt{\log x/R}}}.
\end{displaymath}
\end{prop}
\begin{proof}
Let $x \geq 851$. From the definition of $a_n(t)$, we have
\begin{displaymath}
\int_3^x \frac{a_n(t)}{\log^{n+2}t} \, dt = \frac{\sqrt{8}}{\sqrt{\pi \sqrt{R}}} \int_3^x (\log t)^{-7/4} e^{-\sqrt{\log t/R}} \, dt.
\end{displaymath}
The substitution $t = e^{Ry}$ gives
\begin{equation}
\int_3^x \frac{a_n(t)}{\log^{n+2}t} \, dt = \frac{\sqrt{8}}{R\sqrt{\pi}} \int_{\log 3/R}^{\log x/R} \frac{e^{Ry}}{y^{7/4} e^{\sqrt{y}}} \, dy. \tag{4.2} \label{4.2}
\end{equation}
For convenience, we write $b = \log 3/R$ and $c = \log x/R$, and define $f : [3,c] \to (0, \infty), y \mapsto e^{Ry}/(y^{7/4}e^{\sqrt{y}})$. It is easy to see that the function $f$ is convex on the interval $[b,c]$. Hence,
\begin{equation}
\int_b^c f(y) \, dy \leq \frac{c-b}{2} (f(b) + f(c)). \tag{4.3} \label{4.3}
\end{equation}
The function $g : [3,\infty) \to (0, \infty), y \mapsto y/(y^{11/4}e^{\sqrt{y/R}})$ is strictly increasing for every $x \geq 22.75$ and fulfilled $g(851) \geq g(3)$. Hence $g(y) \geq g(3)$ for every $y \geq 851$, which is equivalent to $bf(c) \geq cf(b)$. Applying this inequality to \eqref{4.3}, we get
\begin{displaymath}
\int_b^c f(y) \, dy \leq \frac{cf(c)}{2},
\end{displaymath}
since $bf(b) \geq 0$. Together with \eqref{4.2} and the definition of the function $f$, we conclude the proof.
\end{proof}
Now, we use the identity \eqref{2.1} and Proposition \ref{prop401} to obtain the following estimates for the prime counting function.
\begin{prop} \label{prop402}
Let $c = 3\sqrt{2}/\sqrt{\pi \sqrt{R}}$. For every $x \geq 2$, we have
\begin{equation}
\pi(x) > \emph{li}(x) - \frac{cx}{(\log x)^{3/4}e^{\sqrt{\log x/R}}} \tag{4.4} \label{4.4}
\end{equation}
and
\begin{equation}
\pi(x) < \emph{li}(x) + \frac{cx}{(\log x)^{3/4}e^{\sqrt{\log x/R}}} - \emph{li}(2) + \frac{2}{\log 2}. \tag{4.5} \label{4.5}
\end{equation}
\end{prop}
\begin{proof}
First, let $x \geq 851$. Since $\vartheta(t)/(t \log^2 t) > 0$ for every $t \geq 2$, we use the identity \eqref{2.1} to get
\begin{displaymath}
\pi(x) > \frac{\vartheta(x)}{\log x} + \int_{3}^x \frac{\vartheta(t)}{t \log^2 t} \, dt.
\end{displaymath}
Applying \eqref{4.1}, we obtain that the inequality
\begin{displaymath}
\pi(x) > \frac{x}{\log x} - \frac{a_n(x)x}{\log^{n+1} x} + \int_3^x \frac{dt}{\log^2 t} - \int_3^x \frac{a_n(t)}{\log^{n+2} t} \, dt
\end{displaymath}
holds. Together with Proposition \ref{prop401} and the identity
\begin{displaymath}
\int_3^x \frac{dt}{\log^2 t} = \text{li}(x) - \frac{x}{\log x} - \text{li}(3) + \frac{3}{\log 3},
\end{displaymath}
we obtain the inequality
\begin{displaymath}
\pi(x) > \text{li}(x) - \frac{a_n(x)x}{\log^{n+1} x} - \text{li}(3) + \frac{3}{\log 3} - \frac{\sqrt{2}}{\sqrt{\pi \sqrt{R}}} \cdot \frac{x}{(\log x)^{3/4}e^{\sqrt{\log x/R}}},
\end{displaymath}
which implies \eqref{4.4} for every $x \geq 851$, since $3/\log 3 - \text{li}(3) > 0$. For smaller values of $x$, we check the inequality \eqref{4.4} with a computer.
The identity \eqref{2.1} gives that the identity
\begin{equation}
\pi(y) - \text{li}(y) = \frac{\vartheta(y) - y}{\log y} + \frac{2}{\log 2} - \text{li}(2) + \int_2^y \frac{\vartheta(t) - t}{t \log^2 t} \, dt \tag{4.6} \label{4.6}
\end{equation}
holds for every $y \geq 2$. First we consider the case $x \geq 851$. By Büthe \cite[Theorem 2]{buethe}, we have $\vartheta(t) < t$ for every $1 \leq t \leq 10^{19}$. Hence, by \eqref{4.6} and \eqref{4.1},
\begin{displaymath}
\pi(x) - \text{li}(x) < \frac{a_n(x)x}{\log^{n+1} x} + \frac{2}{\log 2} - \text{li}(2) + \int_3^x \frac{a_n(t)}{\log^{n+2} t} \, dt.
\end{displaymath}
Using Proposition \ref{prop201}, we get
\begin{displaymath}
\pi(x) - \text{li}(x) < \frac{a_n(x)x}{\log^{n+1} x} + \frac{2}{\log 2} - \text{li}(2) + \frac{\sqrt{2}}{\sqrt{\pi \sqrt{R}}} \cdot \frac{x}{(\log x)^{3/4} e^{\sqrt{\log x/R}}}.
\end{displaymath}
Substituting the definition of $a_n(x)$, we get that the inequality \eqref{4.5} holds for every $x \geq 851$. Again, we check the required inequality for smaller values of $x$ with a computer.
\end{proof}
The function $x \mapsto x/ \log^{n+2}x$ is strictly increasing for every $x > e^{n+2}$ and tends to infinity as $x \to \infty$. Therefore, there exists a positive integer $A_0(n) \geq 2$ so that
\begin{displaymath}
\frac{x}{\log^{n+2}x} \geq \frac{1}{(n+1)!}\sum_{k \leq n+1} \frac{2(k-1)!}{\log^k2}
\end{displaymath}
for every $x \geq A_0(n)$ and we get the following proposition.
\begin{prop} \label{prop403}
Let $c = 3\sqrt{2}/\sqrt{\pi \sqrt{R}}$. Then, for every $x \geq \max \{27, A_0(n) \}$, we have
\begin{equation}
\pi(x) > \sum_{k =1}^{n+1} \frac{(k-1)!x}{\log^kx} - \frac{cx}{(\log x)^{3/4}e^{\sqrt{\log x/R}}} \tag{4.7} \label{4.7}
\end{equation}
and for every $x \geq 4$, we have
\begin{equation}
\pi(x) < \sum_{k=1}^n \frac{(k-1)!x}{\log^kx} + \frac{n!\sqrt{x}}{\log^{n+1}2} + \frac{n!2^{n+1}x}{\log^{n+1}x} + \frac{cx}{(\log x)^{3/4}e^{\sqrt{\log x/R}}} + d, \tag{4.8} \label{4.8}
\end{equation}
where $d = - \emph{li}(2) + 2/\log 2$.
\end{prop}
\begin{proof}
We start with the proof of \eqref{4.7}. Let $x \geq \max \{ 27, A_0(n) \}$. We use \eqref{1.2} to get
\begin{displaymath}
\text{li}(x) \geq \sum_{k=1}^{n+1} \frac{(k-1)!x}{\log^kx} + (n+1)! \int_2^3 \frac{dt}{\log^{n+2}t} + (n+1)! \int_3^x \frac{dt}{\log^{n+2}t} - \sum_{k=1}^{n+1} \frac{2(k-1)!}{\log^k2}.
\end{displaymath}
Notice that the function $t \mapsto 1/\log^mt$ is strictly decreasing on the interval $[2,x]$ for every positive integer $m$. Hence
\begin{equation}
\text{li}(x) \geq \sum_{k=1}^{n+1} \frac{(k-1)!x}{\log^kx} + \frac{(n+1)!}{\log^{n+2}3} + \frac{(x-3) \cdot (n+1)!}{\log^{n+2}x} - \sum_{k=1}^{n+1} \frac{2(k-1)!}{\log^k2}. \tag{4.9} \label{4.9}
\end{equation}
We have $1/\log^{n+2} 3 \geq 3/\log^{n+2}t$ for every $t \geq 27$. Applying this to \eqref{4.9}, we get
\begin{displaymath}
\text{li}(x) \geq \sum_{k=1}^{n+1} \frac{(k-1)!x}{\log^kx} +\frac{(n+1)! x}{\log^{n+2}x} - \sum_{k=1}^{n+1} \frac{2(k-1)!}{\log^k2}.
\end{displaymath}
Since $x \geq A_0(n)$, wo obtain that the inequality
\begin{displaymath}
\text{li}(x) \geq \sum_{k=1}^{n+1} \frac{(k-1)!x}{\log^kx}
\end{displaymath}
holds. Now use \eqref{4.4} to complete the proof of \eqref{4.7}.
Next, we check the validity of \eqref{4.8}. Let $x \geq 4$. Again, we use \eqref{1.2} and integration by parts to get
\begin{displaymath}
\text{li}(x) \leq 1.05 + \sum_{k=1}^n \frac{(k-1)!x}{\log^kx} + n! \int_2^x \frac{dt}{\log^{n+1}t} - \sum_{k=1}^n \frac{2(k-1)!}{\log^k2}.
\end{displaymath}
Since $1.05 \leq 2/\log 2$, we get that the inequality
\begin{equation}
\text{li}(x) \leq \sum_{k=1}^n \frac{(k-1)!x}{\log^kx} + n! \int_2^x \frac{dt}{\log^{n+1}t} \tag{4.10} \label{4.10}
\end{equation}
holds. In the first part of the proof, we note that the function $t \mapsto 1/\log^{n+1}t$ is strictly decreasing on the interval $[2,x]$. Therefore
\begin{displaymath}
\int_2^x \frac{dt}{\log^{n+1}t} = \int_2^{\sqrt{x}} \frac{dt}{\log^{n+1}t} + \int_{\sqrt{x}}^x \frac{dt}{\log^{n+1}t} \leq \frac{\sqrt{x}}{\log^{n+1}2} + \frac{2^{n+1}x}{\log^{n+1}x}.
\end{displaymath}
Together with \eqref{4.10} and \eqref{4.5}, we obtain that the required inequality \eqref{4.8} holds.
\end{proof}
Now, we give the proof of Theorem \ref{thm102} in which Proposition \ref{prop403} plays an important role.
\begin{proof}[Proof of Theorem \ref{thm102}]
In the first step, we verify that the inequality
\begin{equation}
\pi(x) > \frac{x}{\log x} + \frac{x}{\log^2 x} + \frac{2x}{\log^3 x} + \frac{6x}{\log^4 x} \tag{4.11} \label{4.11}
\end{equation}
holds for every $x \geq e^{6719}$. Let $n= 4$. It is easy to see that we can choose $A_0(4) = 132\,718\,993$.
Further, we set $A_1(4) = e^{6719}$. Then,
\begin{displaymath}
\frac{cx}{(\log x)^{3/4}e^{\sqrt{\log x/R}}} \leq \frac{n!x}{(\log x)^{n + 1}}
\end{displaymath}
for every $x \geq A_1(4)$, where $R = 5.573412$ and $c = 3\sqrt{2}/\sqrt{\pi \sqrt{R}}$. Now we apply the last inequality to \eqref{4.7} and get that the inequality \eqref{4.11} holds for every $x \geq e^{6719}$.
Next, we verify that the inequality \eqref{4.11} is valid for every $10\,384\,261 \leq x \leq 2.73 \cdot 10^{40}$. We denote the right hand side of the inequality \eqref{4.11} by $U(x)$. For $y > 0$ let $R(y) = U(y)\log y/y$ and $S(y) = (y^4 - y^3 - y^2 - 3y)/y^3$. We have $S(t) > 0$ for every $t > 2.14$ and $y^5R(y)S(y) = y^6 - T(y)$, where $T(y) = 11y^2 + 12y + 18$. Then, by Theorem \ref{thm101},
\begin{equation}
\pi(x) > \frac{x}{S(\log x)} > \frac{x}{S(\log x)} \left( 1 - \frac{T(\log x)}{\log^6 x} \right) = U(x), \tag{4.12} \label{4.12}
\end{equation}
which completes the proof for every $65\,405\,887 \leq x \leq 2.73 \cdot 10^{40}$. Finally, we use a computer to check that $\pi(p_n) > U(p_{n+1})$ for every positive integer $n$ such that $\pi(10\,384\,261) \leq n \leq \pi(65\,405\,887)$.
\end{proof}
Finally, we use Proposition \ref{prop301} to obtain the following result concerning \eqref{4.11}.
\begin{prop} \label{prop404}
Under the assumption that the Riemann hypothesis is true, the inequality \eqref{4.11} holds for every $x \geq 10\,384\,261$.
\end{prop}
\begin{proof}
We assume that the Riemann hypothesis is true. By \eqref{4.12} and Proposition \ref{prop301} we get that the inequality \eqref{4.11} is valid for every $x \geq 65\,405\,887$. Finally, it suffices to apply Theorem \ref{thm102}.
\end{proof}
\section{The proof of Theorem \ref{thm103}}
In the following proof of Theorem \ref{thm103}, we use a recent result of Büthe \cite[Theorem 2]{buethe} and an explicit estimate for the prime counting function $\pi(x)$ obtained in \cite[Korollar 1.24]{axler2013}.
\begin{proof}[Proof of Theorem \ref{thm103}]
First, we check that the inequality \eqref{1.9} holds for every real $x$ such that $1.62 \cdot 10^{12} \leq x \leq 10^{19}$. By Büthe \cite[Theorem 2]{buethe}, we have
\begin{equation}
\pi(t) < \text{li}(t) \tag{5.1} \label{5.1}
\end{equation}
for every $t$ such that $2 \leq t \leq 10^{19}$. Further, we use \cite[Theorem 2]{buethe} to get that $\pi(t) > \text{li}(t) - 2.1204\sqrt{t}/\log t$ for every $t$ such that $5.94 \cdot 10^{11} \leq t \leq 10^{19}$. Together with \eqref{5.1}, we obtain that
\begin{displaymath}
\pi \left( \frac{x}{e} \right) - \frac{\pi(x)^2\log x}{ex} > \text{Ram}(x),
\end{displaymath}
where
\begin{displaymath}
\text{Ram}(x) = \text{li} \left( \frac{x}{e} \right) - \frac{2.1204\sqrt{x/e}}{\log(x/e)} - \text{li}(x^2) \frac{\log x}{ex}.
\end{displaymath}
We show that $\text{Ram}(x)$ is positive. In order to prove this, we first show that the derivative of $\text{Ram}(t)$ is positive for every $1.06 \cdot 10^{12} \leq t \leq 10^{19}$. A straightforward calculation gives
\begin{equation}
\text{Ram}'(t) = \frac{(\text{li}(t)\log(t/e) - t)^2}{et^2\log(t/e)} - \frac{1.0602(\log t - 3)}{e\log^2(t/e)\sqrt{t/e}}. \tag{5.2} \label{5.2}
\end{equation}
From \eqref{5.1} and the lower bound for the prime counting function given in \cite[Korollar 1.24]{axler2013}, it follows that $\text{li}(t)\log(t/e) - t > t/(\log t \log(t/e))$ for every $t$ such that $468\,049 \leq t \leq 10^{19}$. Combined with \eqref{5.2}, we obtain that the inequality
\begin{displaymath}
\text{Ram}'(t) > \frac{1}{e\log^2t \log^3(t/e)} - \frac{1.0602(\log t - 3)}{e\log^2(t/e)\sqrt{t/e}}
\end{displaymath}
holds for every $t$ such that $468\,049 \leq t \leq 10^{19}$. Since $\sqrt{y} \geq 1.0602\sqrt{e}\log^4 y$ for every $y \geq 1.06 \cdot 10^{12}$, we conclude that the derivative of $\text{Ram}(t)$ is positive for every $1.06 \cdot 10^{12} \leq t \leq 10^{19}$. Together with $\text{Ram}(1.62 \cdot 10^{12}) > 85.86$, we get that $\text{Ram}(x)$ is positive, which implies that Ramanujan's prime counting inequality \eqref{1.9} holds unconditionally for every $1.62 \cdot 10^{12} \leq x \leq 10^{19}$. It remains to show that the inequality \eqref{1.9} holds for every $38\,358\,837\,683 \leq x \leq 1.62 \cdot 10^{12}$ as well. Dudek and Platt \cite[Theorem 1.3]{dudekplatt} showed by computation that $x = 38\,358\,837\,682$ is the largest integer counterexample below $10^{11}$ and that there are no more failures at integer values before $1.15 \cdot 10^{16}$. Since $t \mapsto t/\log t$ is a strictly increasing function for every $t > e$, we get that the inequality \eqref{1.9} holds for every $x$ such that $38\,358\,837\,683 \leq x \leq 1.62 \cdot 10^{12}$ as well and conclude the proof.
\end{proof}
\section{The proof of Theorem \ref{thm104}}
Now we use Proposition \ref{prop403} to prove our second main result concerning Ramanujan's prime counting inequality, which is stated in Theorem \ref{thm104} .
\begin{proof}[Proof of Theorem \ref{thm104}]
First, let $R = 5.573412$ and let $a$ be a positive real number. Since there is a positive integer $A_1(n,a) \geq 2$ so that
\begin{displaymath}
e^{\sqrt{\log x/R}} \geq \left( \frac{\log x}{a} \right)^{n + 1/4}
\end{displaymath}
for every $x \geq A_1(n,a)$, Proposition \ref{prop403} implies that
\begin{equation}
\pi(x) > \sum_{k=1}^n \frac{(k-1)!x}{\log^k x} + \frac{(n!-ca^{n+1/4})x}{\log^{n+1}x}, \tag{6.1} \label{6.1}
\end{equation}
for every $x \geq \max \{ 27, A_0(n), A_1(n,a) \}$, and
\begin{equation}
\pi(x) < \sum_{k=1}^n \frac{(k-1)!x}{\log^k x} + \frac{x}{\log^{n+1}x} \left(\frac{n! \log^{n+1}x}{\sqrt{x}\log^{n+1}2} + n!2^{n+1} + ca^{n+1/4} + \frac{d\log^{n+1}x}{x} \right) \tag{6.2} \label{6.2}
\end{equation}
for every $x \geq \max \{4, A_1(n,a) \}$, where $d = - \text{li}(2) + 2/\log 2$.
Now, let $n = 6$ and let $x_0 = e^{9031}$. It is easy to show that $A_0(6) = 1\,657\,493\,059\,174$ is a suitable choice for $A_0(6)$.
Further, we set $a = 14.4086$. Then the function
\begin{displaymath}
t \mapsto t - R\left( n + \frac{1}{4} \right)^2 \left( \log t + \log \left (\frac{1}{a} \right) \right)^2
\end{displaymath}
is positive for every $t \geq 9\,031$ and we can choose $A_1(6, 14.4086) = x_0$. Using \eqref{6.1} and \eqref{6.2}, we get
\begin{displaymath}
\sum_{k = 1}^6 \frac{(k-1)!x}{\log^kx} - \frac{27158494x}{\log^7 x} < \pi(x) < \sum_{k = 1}^6 \frac{(k-1)!x}{\log^kx} + \frac{27251374x}{\log^7x}
\end{displaymath}
for every $x \geq x_0$. Using these inequalities we conclude that the inequality
\begin{equation}
\frac{ex}{\log x} \pi \left( \frac{x}{e} \right) - \pi(x)^2 > \frac{x^2 f(\log x)}{\log^{14} x (\log x - 1)^7} \tag{6.3} \label{6.3}
\end{equation}
holds for every $x \geq ex_0$, where
\begin{align*}
f(y) & = y^{15} + 7y^{14} - 81\,660\,454y^{13} + 327\,013\,544y^{12} - 872\,039\,437y^{11} + 1\,199\,056\,017y^{10} \\
& \phantom{\quad\q} - 1\,308\,062\,388y^9 - 1\,199\,031\,244y^8 - 742\,610\,678\,698\,880y^7 + 5\,198\,360\,646\,460\,072y^6 \\
& \phantom{\quad\q} - 15\,595\,195\,794\,997\,976y^5 + 25\,992\,104\,849\,073\,228y^4 - 25\,992\,179\,953\,690\,916y^3 \\
& \phantom{\quad\q} + 15\,595\,340\,608\,417\,428y^2 - 5\,198\,455\,153\,885\,372y + 742\,637\,384\,887\,876.
\end{align*}
Now, it is easy to verify that $f(y) > 0$ for every $y \geq 9\,032$. Applying this to \eqref{6.3}, we get that Ramanujan's prime counting inequality \eqref{1.9} holds for every $x \geq ex_0 = e^{9032}$, as desired.
\end{proof}
\begin{rema}
Recently, Platt and Trudgian announced that they have fixed the error in the proof of \eqref{1.12} and even managed to improve the result in Theorem \ref{thm104} by showing
\begin{displaymath}
H_0 \leq e^{8801.037}.
\end{displaymath}
\end{rema}
\section{On a generalization of Ramanujan's prime counting inequality}
Let $n$ be a positive integer and let $\Xi_n : (1, \infty) \to \mathds{R}$ be given by
\begin{displaymath}
\Xi_n(x) = \prod_{k=1}^n \left( 1 - \frac{k-1}{\log x} \right)^{2^{n-k}}.
\end{displaymath}
In 2013, Hassani \cite[Theorem 1]{hassani2013} defined
\begin{displaymath}
R_n^{\Xi}(x) = \frac{e^n}{\Xi_n(x)} \left( \frac{x}{\log x} \right)^{2^n-1} \pi \left( \frac{x}{e^n} \right) - \pi(x)^{2^n}.
\end{displaymath}
and showed by induction that $R_n^{\Xi}(x) > 0$ for every $x \geq e^{n-1}x_R$, whenever Ramanujan's prime counting inequality \eqref{1.9} holds for every $x \geq x_R$ (For $n=1$, the inequality $R_1^{\Xi}(x) > 0$ is equivalent to the inequality \eqref{1.9}). Together with Theorem \ref{thm103}, Theorem \ref{thm104} and \eqref{1.11}, respectively, we obtain the following result.
\begin{prop} \label{prop701}
Let $n$ be a positive integer. Then the following hold:
\begin{enumerate}
\item[(i)] The inequality $R_n^{\Xi}(x) > 0$ holds for every $x$ such that $38\,358\,837\,683e^{n-1} \leq x \leq 10^{19}e^{n-2}$ and for every $x \geq e^{9031 + n}$.
\item[(ii)] Under the assumption that the Riemann hypothesis is true, we have $R_n^{\Xi}(x) > 0$ for every $x \geq 38\,358\,837\,683e^{n-1}$.
\end{enumerate}
\end{prop}
\begin{proof}
For (i), we follow the proof of Theorem 1 in \cite[p. 150]{hassani2013} and use Theorems \ref{thm103} and \ref{thm104}, respectively. Analogously, by using \eqref{1.11}, we conclude the proof of (ii).
\end{proof}
\subsection*{Acknowledgements}
I would like to thank David Platt for the fruitful conversations on this subject.
Furthermore, I would like to thank Mehdi Hassani for drawing my attention to the present subject.
|
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}
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Kaliman ist der Name von
Kaliman I. Assen (1234–1246), von 1241 bis 1246 bulgarischer Zar aus der Dynastie Asen
Kaliman II. Assen († 1256), für kurze Zeit im Jahr 1256 bulgarischer Zar aus der Dynastie Asen
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"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/mathhelpboards.com\/threads\/evaluate-a-b-c.8516\/","text":"# Evaluate a+b+c\n\n#### anemone\n\n##### MHB POTW Director\nStaff member\nHi MHB,\n\nI have solved the problem as stated below but I don't know if it's an unique solution and even if it is, I have no idea how to prove that would be the case.\n\nCan anyone show me how to approach the problem correctly?\n\nFor the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$, all of its roots are positive real numbers. Evaluate the sum of $a+b+c$.\n\nAttempt:\n\nIt's quite obvious from the values of the product of all 5 roots and the sum of them reveal that the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$ has roots of 2 and 4, of which 2 is the repeated root of multiplicity 4, since\n\n$2+2+2+2+4=12$ and $2^4(4)=64$,\n\nThen there are many ways to find the values for $a, b, c$ and at last, after the values of $a, b, c$ are known, we can conclude that $a+b+c=56+144-128=72$.\n\n#### mente oscura\n\n##### Well-known member\nHi MHB,\n\nFor the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$, all of its roots are positive real numbers. Evaluate the sum of $a+b+c$.\nHello.\n\n$$Let \\ r_1, \\ r_2, \\ r_3, \\ r_4, \\ r_5 \\ roots \\ of: \\ \/$$\n\n$$\/ \\ x^5-12x^4+ax^3+bx^2+c^x-64=(x-r_1)(x-r_2)(x-r_3)(x-r_4)(x-r_5)$$\n\n$$a=r_1r_2+r_1r_3+r_1r_4+r_1r_5+r_2r_3+r_2r_4+r_2r_5+r_3r_4+r_3r_5+r_4r_5=C^5_2$$\n\n$$b=-(r_1r_2r_3+r_1r_2r_4+r_1r_2r_5+r_1r_3r_4+r_1r_3r_5+r_1r_4r_5+r_2r_3r_4+r_2r_3r_5+r_2r_4r_5+r_3r_4r_5)=C^5_3$$\n\n$$c=r_1r_2r_3r_4+r_1r_2r_3r_5+r_1r_2r_4r_5+r_1r_3r_4r_5+r_2r_3r_4r_5=C^5_4$$\n\nTo include all combinations in the factors $$r_i$$, guarantees us that the solution is unique. Since you get all numerical products between 2, 3 and 4 factors, with the same result.\n\nI do not know if I understood your question correctly.\n\nregards.\n\n#### anemone\n\n##### MHB POTW Director\nStaff member\nHello.\n\n$$Let \\ r_1, \\ r_2, \\ r_3, \\ r_4, \\ r_5 \\ roots \\ of: \\ \/$$\n\n$$\/ \\ x^5-12x^4+ax^3+bx^2+c^x-64=(x-r_1)(x-r_2)(x-r_3)(x-r_4)(x-r_5)$$\n\n$$a=r_1r_2+r_1r_3+r_1r_4+r_1r_5+r_2r_3+r_2r_4+r_2r_5+r_3r_4+r_3r_5+r_4r_5=C^5_2$$\n\n$$b=-(r_1r_2r_3+r_1r_2r_4+r_1r_2r_5+r_1r_3r_4+r_1r_3r_5+r_1r_4r_5+r_2r_3r_4+r_2r_3r_5+r_2r_4r_5+r_3r_4r_5)=C^5_3$$\n\n$$c=r_1r_2r_3r_4+r_1r_2r_3r_5+r_1r_2r_4r_5+r_1r_3r_4r_5+r_2r_3r_4r_5=C^5_4$$\n\nTo include all combinations in the factors $$r_i$$, guarantees us that the solution is unique. Since you get all numerical products between 2, 3 and 4 factors, with the same result.\nThanks, mente oscura for the reply.\n\nBut...I don't quite get you especially the part when you mentioned the way to guarantee the only set values for all the 5 real positive roots (that I obtained via eyeballing) is the unique set of solution for solving the equation $x^5-12x^4ac^3+bx^2+cx-640=0$.\n\nYes, I know $a$ consists of the sum of $5\\choose2$ terms, $b$ consists of the sum of $5\\choose3$ terms and last, $c$ consists of the sum of $5\\choose4$ terms, but how does one relate it to the number of sets of solution that we could get based on the only known values for the sum\/product of roots?\n\n#### Opalg\n\n##### MHB Oldtimer\nStaff member\nHi MHB,\n\nI have solved the problem as stated below but I don't know if it's an unique solution and even if it is, I have no idea how to prove that would be the case.\n\nCan anyone show me how to approach the problem correctly?\n\nFor the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$, all of its roots are positive real numbers. Evaluate the sum of $a+b+c$.\n\nAttempt:\n\nIt's quite obvious from the values of the product of all 5 roots and the sum of them reveal that the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$ has roots of 2 and 4, of which 2 is the repeated root of multiplicity 4, since\n\n$2+2+2+2+4=12$ and $2^4(4)=64$,\n\nThen there are many ways to find the values for $a, b, c$ and at last, after the values of $a, b, c$ are known, we can conclude that $a+b+c=56+144-128=72$.\nThat is not the only solution, and the value of $a+b+c$ is not unique. The equation $\\displaystyle x^5 - 12x^4 + \\frac{509}9x^3 - \\frac{1174}9x^2 + \\frac{440}3x - 64 = 0$ has roots $\\displaystyle \\frac43,\\,2,\\,\\frac83,\\,3,\\,3$, and the sum of coefficients $a+b+c$ is $\\displaystyle \\frac{509}9 + \\frac{440}3 - \\frac{1174}9 = \\frac{655}9 \\ne 72.$\n\n#### anemone\n\n##### MHB POTW Director\nStaff member\nThat is not the only solution, and the value of $a+b+c$ is not unique. The equation $\\displaystyle x^5 - 12x^4 + \\frac{509}9x^3 - \\frac{1174}9x^2 + \\frac{440}3x - 64 = 0$ has roots $\\displaystyle \\frac43,\\,2,\\,\\frac83,\\,3,\\,3$, and the sum of coefficients $a+b+c$ is $\\displaystyle \\frac{509}9 + \\frac{440}3 - \\frac{1174}9 = \\frac{655}9 \\ne 72.$\nThank you so much Opalg for your reply and also showing me the counter example (I have been trying very hard to find for another solution set by the help of wolfram, after I tried the combinations such as $\\displaystyle \\frac12,\\,4,\\,\\frac52,\\,r_4,\\,r_5$ for which the remaining effort to look for the perfect candidates for all those 5 roots seemed no easy task for me.). Now I would just discard this question.\n\n#### Opalg\n\n##### MHB Oldtimer\nStaff member\nI tried the combinations such as $\\displaystyle \\frac12,\\,4,\\,\\frac52,\\,r_4,\\,r_5$\nYou are told that $r_1 + r_2 + r_3 + r_4 + r_5 = 12$ and $r_1r_2r_3r_4r_5 = 64$. The GM-AM inequality applied to the numbers $r_1,r_2,r_3,r_4,\\sqrt{r_5},\\sqrt{r_5}$ shows that $2 \\leqslant \\frac16(r_1 + r_2 + r_3 + r_4 + 2\\sqrt{r_5})$, from which $r_5 \\leqslant 2\\sqrt{r_5}$, with equality only if $r_1=r_2=r_3=r_4=\\sqrt{r_5}$. So if one of the roots is $4$ then the others must all be $2$. When I realised that, I tried putting three of the roots equal to $2,\\,3$ and $3$, and I was surprised to find that I could then get a solution for the other two roots.\n\n#### anemone\n\n##### MHB POTW Director\nStaff member\nYou are told that $r_1 + r_2 + r_3 + r_4 + r_5 = 12$ and $r_1r_2r_3r_4r_5 = 64$. The GM-AM inequality applied to the numbers $r_1,r_2,r_3,r_4,\\sqrt{r_5},\\sqrt{r_5}$ shows that $2 \\leqslant \\frac16(r_1 + r_2 + r_3 + r_4 + 2\\sqrt{r_5})$, from which $r_5 \\leqslant 2\\sqrt{r_5}$, with equality only if $r_1=r_2=r_3=r_4=\\sqrt{r_5}$. So if one of the roots is $4$ then the others must all be $2$. When I realised that, I tried putting three of the roots equal to $2,\\,3$ and $3$, and I was surprised to find that I could then get a solution for the other two roots.\nThank you again Opalg for your patience and willingness to teach me more about how to look for other possible solution for this problem. I really appreciate your help!\n\nI like it how you made the six terms up $r_1,r_2,r_3,r_4,\\sqrt{r_5},\\sqrt{r_5}$ and then applied the AM-GM inequality for those numbers! I learn a great deal from you today!","date":"2021-06-23 06:42:39","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\": 0.8637754321098328, \"perplexity\": 185.87782947310342}, \"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-2021-25\/segments\/1623488534413.81\/warc\/CC-MAIN-20210623042426-20210623072426-00627.warc.gz\"}"}
| null | null |
'''
A package for each of the different layout managers.
'''
# Some helper utils...
def exit_pos(ninfo):
x,y = ninfo.get('position')
xsize, ysize = ninfo.get('size', (0,0))
return (x + xsize/2, y + ysize)
def entry_pos(ninfo):
x,y = ninfo.get('position')
xsize, ysize = ninfo.get('size', (0,0))
return (x + xsize/2, y)
def center_pos(ninfo):
x,y = ninfo.get('position')
xsize, ysize = ninfo.get('size', (0,0))
return (x + (xsize/2), y + (ysize/2))
class GraphLayout:
'''
A graph layout uses several graph meta properties and node properties
to communicate with a renderer which is expected to display the graph:
size = ( width, height ) - Set by the renderer
position = ( x, y ) - Set by the layout
repr = <display text> - A fallback for what to display on a node
'''
def __init__(self, graph):
self.graph = graph
def layoutGraph(self):
'''
Layout the graph nodes and edges
'''
raise Exception('%s must implement layoutGraph()!' % self.__class__.__name__)
def getLayoutSize(self):
raise Exception('%s must implement getLayoutSize()!' % self.__class__.__name__)
def renderGraph(self, rend):
'''
Render the graph to the given renderer.
'''
rend.setNodeSizes(self.graph)
self.layoutGraph()
width, height = self.getLayoutSize()
self.graph.setMeta('size', (width, height) )
rend.renderGraph()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,094
|
Guide to using theme\_mpr()
================
David H. Montgomery
December 29, 2017
Hadley Wickham's `ggplot` package is a powerful and flexible way to make graphics ([learn the basics here](https://github.com/dhmontgomery/r-data-for-beginners/blob/master/r_for_beginners.md)), but its standard output doesn't conform to MPR News' house style for graphics. All that power and flexibility doesn't mean much if graphics require extensive customization in Illustrator before they can be used.
Fortunately, `ggplot` supports custom themes that can radically change the appearance of its graphs. I wrote one such theme, `theme_mpr`, designed to apply MPR News' graphics style to `ggplot` graphs. This is a guide for how to apply that style.
(The actual theme belongs to MPR and is not shared here; the following guide was produced for MPR News and is shared here with permission as an example of my work there.)
Beginning
---------
Let's say we want to make a simple graph of a basic dataset, approval ratings for four politicians in a KSTP poll:
``` r
# Create the data
kstp <- data.frame(
"candidate" = c("Sen. Al Franken","Sen. Amy Klobuchar","Gov. Mark Dayton","Pres. Donald Trump"),
"approval" = c(0.36,0.56,0.42,0.31),
"party" = c("Democrat", "Democrat", "Democrat", "Republican")) %>%
mutate(candidate = candidate %>% as.factor() %>% fct_relevel("Pres. Donald Trump", "Sen. Al Franken", "Gov. Mark Dayton", "Sen. Amy Klobuchar")) # Reorder the politicians
kstp
```
## candidate approval party
## 1 Sen. Al Franken 0.36 Democrat
## 2 Sen. Amy Klobuchar 0.56 Democrat
## 3 Gov. Mark Dayton 0.42 Democrat
## 4 Pres. Donald Trump 0.31 Republican
Here's how we'd make that graph in basic `ggplot`:
``` r
library(tidyverse) # Load necessary libraries
library(scales)
library(forcats)
# Create the plot from the data source `kstp` with candidates on the X-axis and approval rating on the Y-axis
ggplot(kstp, aes(candidate, approval)) +
geom_col() + # Make it a column graph
scale_y_continuous(labels = percent) + # Format the Y-axis labels as percents
# Add labels
geom_text(aes(label = paste0(round(approval * 100), "%")), # Define labels as approval * 100, with a % sign
vjust = 1.3, color = "white") + # Move the labels down vertically, color them white
# Add labels
labs(title = "Minnesotans' approval of politicians in KSTP survey", # Title
subtitle = "Source: KSTP-TV/SurveyUSA. Survey conducted Nov. 20, 2017.", # Subtitle
caption = "Graphic by David H. Montgomery", # Caption
x = "", y = "Percent of Minnesotans") # Axis titles, with the X-axis title blank
```
That's a perfectly fine graphic, but it doesn't match MPR's style.
Adding theme\_mpr
-----------------
You can get most of the way to MPR's style by just adding a single line of code to what we had above.
Well, two lines of code, since first we have to load the theme from our disk. Use the `source()` function and point it to where `theme_mpr.R` is saved on your computer. In my case, it's in the folder `graphics`:
``` r
source("graphics/theme_mpr.R")
```
Once you've done that, then the function `theme_mpr()` has been added to your R session (along with a few other things we'll get to in a bit).
So now let's run that same code as above, but add a line calling `theme_mpr()`:
``` r
ggplot(kstp, aes(candidate, approval)) +
geom_col() +
scale_y_continuous(labels = percent) +
geom_text(aes(label = paste0(round(approval * 100), "%")), vjust = 1.3, color = "white") +
labs(title = "Minnesotans' approval of politicians in KSTP survey",
subtitle = "Source: KSTP-TV/SurveyUSA. Survey conducted Nov. 20, 2017.",
caption = "Graphic by David H. Montgomery",
x = "", y = "Percent of Minnesotans") +
theme_mpr() # Add theme_mpr
```
Adding that one line, `theme_mpr()`, changed a whole host of things:
- Fonts
- Font size
- Font color
- Bar color
- Plot background
- Gridlines
- Label justification
- Outlined the graphic
Further tweaks
--------------
There are a few parts of MPR News' style that `theme_mpr()` can't change. Principally, this involves anything that reflects data and not just appearance. Even though MPR News' style calls for axis labels to be capitalized, `theme_mpr()` can't do that for us. Let's do that now. (We'll also move the Source line down to the caption.)
``` r
kstp$candidate <- toupper(kstp$candidate) # Modify the original data to put candidate names in all caps
ggplot(kstp, aes(candidate, approval)) +
geom_col() +
scale_y_continuous(labels = percent) +
geom_text(aes(label = paste0(round(approval * 100), "%")), vjust = 1.3, color = "white") +
labs(title = "Minnesotans' approval of politicians in KSTP survey",
subtitle = "Survey conducted Nov. 20, 2017",
caption = "SOURCE: KSTP-TV/SURVEYUSA GRAPHIC BY DAVID H. MONTGOMERY",
x = "", y = "PERCENT OF MINNESOTANS") +
theme_mpr() # Add theme_mpr
```
Don't worry about the title running off the end of the page for now.
Just as we have to manually change text content, while `theme_mpr()` can set a new default color for bars, bar color is treated as data by `ggplot`, so if we want to do any fine-tuning to bar color, we have to do it manually.
Here we'll color the bars based on the party of the politician. We'll do so by calling another function contained in `theme_mpr.R`. In `ggplot`, the `scale_fill_` functions define color schemes when we're assigning color or fill according to a variable. I've added a custom function, `scale_fill_bluered`, which colors a graph with a blue-red color scheme ideal for Democrats and Republicans. (There's also a `scale_fill_redblue`, which just reverses the order of the colors; use it if you're getting red Democrats and blue Republicans.)
Here, I've changed four things:
- Added `fill = party` to `aes()` at the top, to tell `ggplot` to determine fill color by the party variable
- Added `scale_fill_bluered()`
- Updated the `labs()` to include a label for the legend
- I put "Democrat" and "Republican" from the original data frame into all caps by running `kstp$party <- toupper(kstp$party)`.
``` r
kstp$party <- toupper(kstp$party)
ggplot(kstp, aes(candidate, approval, fill = party)) +
geom_col() +
scale_y_continuous(labels = percent) +
geom_text(aes(label = paste0(round(approval * 100), "%")), vjust = 1.3, color = "white") +
labs(title = "Minnesotans' approval of politicians in KSTP survey",
subtitle = "Survey conducted Nov. 20, 2017",
caption = "SOURCE: KSTP-TV/SURVEYUSA GRAPHIC BY DAVID H. MONTGOMERY",
x = "", y = "PERCENT OF MINNESOTANS",
fill = "PARTY AFFILIATION: ") +
theme_mpr() + # Add theme_mpr
scale_fill_bluered()
```
Adding the logo
---------------
The final piece of the puzzle is to add MPR's logo to the chart. (This will also conveniently fix the issue with the headline running off screen.)
To do that, we only need to add a single extra line of code, thanks to another function included in `theme_mpr.R`: `addlogo()`.
``` r
p <- ggplot(kstp, aes(candidate, approval, fill = party)) +
geom_col() +
scale_y_continuous(labels = percent) +
geom_text(aes(label = paste0(round(approval * 100), "%")), vjust = 1.3, color = "white") +
labs(title = "Minnesotans' approval of politicians in KSTP survey",
subtitle = "Survey conducted Nov. 20, 2017",
caption = "SOURCE: KSTP-TV/SURVEYUSA GRAPHIC BY DAVID H. MONTGOMERY",
x = "", y = "PERCENT OF MINNESOTANS",
fill = "PARTY AFFILIATION: ") +
theme_mpr() + # Add theme_mpr
scale_fill_bluered()
addlogo(p)
```
(We also changed the code slightly, storing the original graph as an object `p` and then calling that object as the variable for `addlogo()`. This is necessary.)
Sometimes you'll make a graph that just doesn't fit in the standard rectangular 16:9 ratio. `theme_mpr` can handle that! Instead of `addlogo(p)`, you'll want to run `addlogo(p, square = TRUE)`.
Other elements
--------------
The `theme_mpr.R` file also has a few other useful features that can help you customize graphs made with `ggplot` to fit MPR News' style.
- One thing `theme_mpr()` does is change R's default color to an MPR blue. This will stay until you restart R or RStudio -- or until you call the function `theme_reset()` contained in `theme_mpr.R`. That turns the default color (and font) back to basic `ggplot`.
- The file also contains several color palettes you can call: `mpr_blues` and `mpr_greys`. Each is a list of five colors that you can add to your graphs by indicating which element of the list you want. For example, `color = mpr_blues[2]` will set the color to the second color in `mpr_blues`.
- There are also three specific colors that you can call by name: `mpr_red`, `mpr_orange` and `mpr_brown`
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,045
|
/**
* An experimental (and propably very inefficient) implementation of
* the string formatting proposal from ECMAScript 6.
* Proposal: http://wiki.ecmascript.org/doku.php?id=strawman:string_format_take_two
*
* It's not quite complete; some number formatting isn't yet there:
* - The '#' flag isn't supported
* - e/E format specifier isn't supported
* - g/G format specifier isn't supported
* - Capitalization is incorrect with f/F
*
* (c) 2012 Rob Brackett (rob@robbrackett.com)
* This code is free to use under the terms of the accompanying LICENSE.txt file
*/
String.prototype.format = function (data) {
var args = arguments;
// regex for separating the various parts of an identifier from each other
var identifierIdentifier = /^(?:\[([^\.\[\]]+)\]|\.?([^\.\[\]]+))(.*)$/
// convert an identifier into the actual value that will be substituted
var findByPath = function (path, data, top) {
var identifiers = path.match(identifierIdentifier);
if (!identifiers) {
throw "Invalid identifier: " + path;
}
var key = identifiers[1] || identifiers[2];
// For the first identifier, named keys are a shortcut to "0.key"
if (top && !isFinite(key)) {
data = data[0];
}
value = data[key];
// recurse as necessary
return identifiers[3] ? findByPath(identifiers[3], value) : value;
};
// replace expression matches things inside {thisIsAToken:withSpecifier} brackets and "{{" and "}}"
return this.replace(/(?:(^|[^{])\{([^{].*?)\}(?!\}))|(\{\{|\}\})/g, function (match, before, token, doubleBrackets) {
// if we found double brackets, they're just an escape sequence for single brackets
if (doubleBrackets) {
return doubleBrackets[0];
}
// separate the identifier (index 0) and the format specifier (index 1)
var parts = token.split(":");
var value = findByPath(parts[0], args, true);
var specifier = parts[1];
// if a specifier is an identifier itself, do the replacement
if (specifier && specifier[0] === "{" && specifier.slice(-1) === "}") {
specifier = findByPath(specifier.slice(1, -1), args, true);
}
// format the value
var result = "";
if (value) {
result = value.toFormat ? value.toFormat(specifier) : value.toString();
}
return before + result;
});
};
Number.prototype.toFormat = function (specifier) {
var value = this;
if (!specifier) {
return value.toString();
}
var formatters = specifier.match(/^([\+\-#0]*)(\d*)(?:\.(\d+))?(.*)$/);
var flags = formatters[1],
width = formatters[2],
precision = formatters[3],
type = formatters[4];
var repeatCharacter = function (character, times) {
var result = "";
while (times--) {
result += character;
}
return result;
}
var applyPrecision = function (result) {
if (precision) {
var afterDecimal = result.split(".")[1];
var extraPrecision = precision - afterDecimal;
if (isNaN(extraPrecision)) {
extraPrecision = precision;
}
if (extraPrecision > 0) {
if (result.indexOf(".") === -1) {
result += ".";
}
for (; extraPrecision > 0; extraPrecision--) {
result += "0";
}
}
}
return result;
}
var result = "";
switch (type) {
case "d":
result = Math.round(value - 0.5).toString(10);
result = applyPrecision(result);
break;
case "x":
result = Math.round(value - 0.5).toString(16);
break;
case "X":
result = Math.round(value - 0.5).toString(16).toUpperCase();
break;
case "b":
result = Math.round(value - 0.5).toString(2);
break;
case "o":
result = Math.round(value - 0.5).toString(8);
break;
// TODO: e,E,g,G types
// not quite clear on whether g/G ignores the precision specifier
case "f":
case "F":
// TODO: proper case for NaN, Infinity
// proposal talks about INF and INFINITY, but not sure when each would be used :\
default:
result = value.toString(10);
result = applyPrecision(result);
}
if (~flags.indexOf("+")) {
if (value >= 0) {
result = "+" + result;
}
}
if (width && result.length < width) {
// "-" flag is right-fill
if (~flags.indexOf("-")) {
result += repeatCharacter(" ", width - result.length);
}
else {
var padding = repeatCharacter(~flags.indexOf("0") ? "0" : " ", width - result.length);
if (~flags.indexOf("0") && (result[0] === "+" || result[0] === "-")) {
result = result[0] + padding + result.slice(1);
}
else {
result = padding + result;
}
}
}
// TODO: # flag
return result;
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,238
|
Australia is an ideal nation for business and venture. The solid economy, positive and favorable Government arrangements, perfect work conditions make it a favorable place for everyone. These elements make Australia Investment Visa as a country that help, support and energize the specialists, business visionaries, financial specialists and bosses and business transients. Aside from this, being a propel country, Australia offers useful condition as far as framework, transportation, Workplace laws, and so forth which are essential for the development and improvement of business and ventures.
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This permit has two classes, Business Innovation stream, and Investor stream. The first is for those people who are anxious to claim and direct a crisp or available organization in the nation, and the second one is for those competitors who are anxious to make a named interest in an Australian region or state, and who seek to bear on the business and venture doings in the country, in the aftermath of the original investment's maturing.
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\section{Background}
\section{Introduction}
\label{section_intro}
Agents' expectation formation has long been a fundamental building block of many macroeconomic models. The economies we live in is constantly evolving in ways that are not perfectly understood by neither private agents nor policy makers. How an economic agent adjust its expectations facing these non-stationarities is imperative in understanding where the economy is heading and assessing relevant policies. This paper builds on the long line of research in modelling economic agents' expectation formation process, particularly those that study deviations from full information rational expectation assumption. It proposes a plausible model for expectation formation that agents face constraints in processing information and do not know how to make model-consistent beliefs. To test this expectation formation model, an experiment is conducted. Inspired by the early literature on accelerationist controversy, it observes how this agent adjusts its subjective belief in an environment, in which a government changes its inflation target.
Borrowing from the artificial intelligence (AI) literature, how an agent forms expectation is modelled through the actor-critic framework of reinforcement learning. AI algorithms produced many successes in fields outside of economics. For example, WaveNet is used for google assistant and latest android devices for voice recognition \citep{Wavenet}; Deepmind AI reduces Google data centre cooling bill by 40\% \citep{Googlecooling}. The key motivation for its application in this paper is how the algorithm naturally incorporates bounded rationality, and how it is connected closely to psychology and neural science research on human decision making.
In this setting, the agent first needs to interact with an environment by taking random actions and receiving corresponding reward signals, which is inspired by learning through trail and error in the psychology of animal learning \citep{SB2018}. The randomness of the action is linked to how exploratory the agent is at trying different options in an action space, which can be viewed as a hardwired trait in the agent's brain. The agent has to exploit what it has already experienced in order to obtain rewards, but it also has to explore in order to make better action selections in the future. Moreover, exploration is directly linked to the agent's experience. With a high level of exploration, the agent could experience a wide range of possibilities. This ensures that the agent has sufficient amount of experience to learn that the environment is undergoing a change (e.g., a monetary policy target shift). The experience collected is processed with the goal of finding a decision-making strategy that maximises expected future return. This expected future return is a subjective belief of this learning agent and it evolves based on past experience. Lastly, both a decision-making strategy and a value function are learnt from randomly initialised neural networks. This implies that the agent is not learning about a particular (group) of model parameters, nor how to process a certain set of information, but learning in general how to make decisions based on its past experience, and adjust its subjective beliefs about the world.
To highlight its fitness in modelling economic agents' expectation formation process, I employ an environment where a government changes its inflation target to `fool' the agent in this economy. This is motivated by the early literature in accelerationist controversy. Accelerationist or backward looking Phillips curve with adaptive expectation permits a trade-off between inflation and unemployment. It was then argued that a government could exploit such an opportunity to maintain a low rate of unemployment through accelerating the money supply process. As the agent in this AI algorithm also learns from past experience, it is a form of adaptive expectation. Would this AI agent also make systematic errors, and if not, how does it adjust to the new policy regime? In other words, can this AI agent learn to converge from one rational expectation equilibrium to another, and hit the moving target of monetary policy?
\vspace{1.0cm}
\textbf{Related Literature}
This paper builds on the vast existing literature on modelling agents' expectation formation process. The importance of agents' expectations can be traced back to Keynes on his idea of how expectation determines output and employment \citep{Keynes:1936}. Fast forward two decades later, \cite{Cagan1956} and \cite{Friedman1957} formalised the idea of adaptive expectation. In combination with the Phillips curve, it generated a large debate on if and how a government could exploit a possible negative relationship between inflation and unemployment. However, it was criticised for assuming that an agent would make an inflation forecast the same as past period inflation. What came as an alternative and soon revolutionised macroeconomics was the rational expectation hypothesis \citep{LUCAS1972103, LUCAS197619, Sargent1971, Sargent19733}. This is when agents are assumed to make model-consistent beliefs, and they know the ins and outs of an economy. In other words, the agents go from very naive (adaptive expectation) to very smart. It has many advantages, and one of which is its usefulness in thinking about policy experiments in a relatively stationary environment. It also has, as most methods are, its disadvantages. One of which is providing convincing dynamics of inflation in response to shocks. Many techniques are proposed to model an agent that is neither as naive as an adaptive expectation agent, nor smart enough to utilise full information available and make model-consistent beliefs. These methods aim at deviating from the full information rational expectation assumption, and can be broadly viewed as two groups of literature.
One group pursues the implications of information rigidities, and this includes sticky information \citep{MankiwReis2002, Balletal2005}, noisy information \citep{Woodford2001} and rational inattention \citep{SIMS2003665}. The main focus is that agents are constrained to obtain or process information, and thus only use a portion of the full information to make `optimal' decisions, i.e., still hold model-consistent beliefs but with less than full information. Similar to this literature, this paper argues that agents are constrained on the amount of information they can both collect and process at any given time. What makes it different from this line of research is how the constraints are integrated, and moreover, agents' subjective beliefs about the world constantly evolve.
The other group that is also closely linked to this work focuses on bounded rationality \citep{Sargent1993} and adaptive learning \citep{EvansHonkapohja1999}. \cite{Schorfheide2005}, \cite{OzdenWouter2021}, \cite{Airaudo2021} also look at combining adaptive learning with Markov switching specifications to model learning agents with policy regime change. The main idea is that agents are believed to be as smart as econometricians, and thus learn about model parameters through running a regression with past data or applying Bayesian updating. Similarly, the AI learning agent updates its subjective belief based on past experience. Hence AI learning is a form of adaptive learning. What it contributes to the existing research is to show, drawing inspirations from psychology and neural science research, how this past experience is gathered and what information is used for updating its belief.
The methodology adopted is closely related to the fast-moving AI literature, and belongs to a class of algorithms called deep reinforcement learning (DRL) algorithms. The pioneer algorithm is called deep Q network algorithm \citep{mnih-atari-2013}, which is capable of human level performance on many Atari video games using unprocessed pixels for input. However, it can only handle discrete action spaces. Economic decision-making processes often involve continuous action space, and thus this paper applies \cite{lillicrap2015drl}'s algorithm, namely deep deterministic policy gradient (DDPG). The application of DRL algorithms in macroeconomic models represents a new branch of research. In a companion paper, \cite{Shi2021learning} adopts a DRL algorithm in a stochastic growth model environment to highlight how an AI agent can learn from no information on its environment and own preference and its ability in adapting to transitory and permanent income shocks. \cite{Shi2021deep} apply a DRL algorithm in a model with different monetary and fiscal policy regimes, and show evidence that DRL agent can locally learn and converge to neighbouring regions of all equilibria in the model. This paper adopts similar methodology to both \cite{Shi2021learning} and \cite{Shi2021deep}, however, with the key difference that it accentuates the adaptability feature of AI agents when faced with a monetary policy regime change. Will this agent notice the shift in policy target and hence adapt it's decision rule accordingly?
In the following sections, I first introduce the economic model adopted in this exercise. This is followed by the methodology section that consists of details on how an AI algorithm is implemented in the economic environment. Simulation experiments and results are then presented.
\section{An Economic Model}
In this section, I present an economic model with a representative household that follows the rational expectation assumption. In the following section, I illustrate how an AI learning agent is modelled, and what happens if it was living in this economic environment presented here.
\subsection{A Representative Household}
A representative household determines its consumption level and real money balance holding each period to maximise its expected lifetime utility,
\begin{equation}
\max_{c_t, m_t} E_0 \sum_{t = 0}^{\infty} \beta^t u \left(c_t, m_t\right)
\end{equation}
subject to the nominal period budget constraint,
\begin{equation}
P_t c_t + M_t + B_t \leq P_t y_t + M_{t-1} + i_{t-1} B_{t-1} + P_t \tau_t
\label{NBC}
\end{equation}
where $P_t$ is the price level of period $t$, $c_t$ is the consumption level, $M_t$ is nominal money balance, $B_{t-1}$ is the stock of nominal bond that a household enters period $t$ with, and they pay out gross nominal interest rate $i_{t-1}$, $y_t$ is the endowment or income of the agent. $\tau_t$ is the government transfer at $t$.
In real terms, Equation \ref{NBC} is,
\begin{equation}
c_t +m_t + b_t \leq y_t +\frac{m_{t-1} + i_{t-1} b_{t-1}}{\pi_t} + \tau_t
\label{RBC}
\end{equation}
where $m_t \equiv \frac{M_t}{P_t}$ refers to real money balances, and $b_t \equiv \frac{B_t}{P_t}$ is real bond holding. Inflation is defined as $\pi_t \equiv \frac{P_t}{P_{t-1}}$
A Lagrangian for this household is:
\begin{equation}
L = E_0\sum_{t = 0}^{\infty} \beta^t \left[ u \left(c_t, m_t\right) + \lambda_t \left(y_t + ß \frac{m_{t-1} + i_{t-1} b_{t-1}}{\pi_t} + \tau_t - c_t - m_t - b_t \right)\right]
\end{equation}
The first-order conditions with respect to consumption, real money holding, and real bond holding are as follows. The utility function with a superscript refers to the derivative of this utility function with respect to the superscript variable, for example, $u^c(c_t, m_t)$ represents the derivative of the utility function with respect to consumption, $c$.
\begin{equation}
\frac{\partial \mathcal{L}}{\partial c_t} = 0 \iff u^c (c_t, m_t) - \lambda_t = 0 \iff u^c (c_t, m_t) = \lambda_t
\label{FOCC}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{L}}{\partial m_t} = 0 \iff u^m (c_t, m_t) - \lambda_t + \beta E_t \frac{\lambda_{t+1}}{\pi_{t+1}} = 0
\label{FOCM}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{L}}{\partial b_{t}} = 0 \iff - \lambda_t + \beta E_t \lambda_{t+1} \frac{i_t}{\pi_{t+1}} = 0
\label{FOCB}
\end{equation}
Equation \ref{FOCC} and \ref{FOCB} give the consumption Euler equation.
\begin{equation}
u^c (c_t, m_t) = \beta E_t u^c (c_{t+1}, m_{t+1}) \frac{i_t}{\pi_{t+1}}
\label{Euler}
\end{equation}
It states that utility lost from consumption today equals utility from consuming tomorrow adjusted for the (real) gain from keeping bonds.
Equation \ref{FOCC}, \ref{FOCM}, and \ref{Euler} give the money demand equation of the agent, which equates the marginal rate of substitution between real money and consumption to their relative price.
\begin{equation}
\frac{u^m (c_t, m_t)}{u^c (c_t, m_t) } = 1 - \frac{1}{i_t}
\label{MDI}
\end{equation}
\subsection{Government: fiscal and monetary policy}
The monetary authority follows an interest rate rule, specified as equation \ref{Tay}.
\begin{equation}
i_t =\frac{\hat{\pi}}{\beta} \left(\frac{\pi_t}{\hat{\pi}}\right)^{1+\lambda}
\label{Tay}
\end{equation}
where $\hat{\pi}$ denotes the inflation target, $\lambda$ is a parameter that governs how responsive the monetary authority is at a deviation from the inflation target. When $\lambda<0$, it is a passive monetary policy, whereas when $\lambda>0$, it is an active one.\footnote{The analyses presented in this paper are based on the specification of a passive monetary policy. However, the results equally hold for the case of an active monetary policy, which is available upon request.}
Bond is in 0 net supply. The government incurs no consumption, and its budget is balanced every period, i.e.,
\begin{equation}
\tau_t = m_t - \frac{m_{t-1}}{\pi_t}.
\end{equation}
\section{An AI Learning Model}
In this section, I introduce the algorithm adopted and how to apply it to the economic environment specified in the previous section. For a comprehensive review of reinforcement learning, please see \cite{SB2018}.
\subsection{AI Learning Framework: actor-critic model}
The deep reinforcement learning algorithm adopted here was first introduced by \cite{lillicrap2015drl}, namely deep deterministic policy gradient (DDPG). Its core follows the actor-critic model of reinforcement learning, and it uses the formal framework of a Markov decision process to define the interaction between a learning agent and its environment in terms of states, actions, and rewards (Figure \ref{MDP}).
\vspace{1.5cm}
\begin{figure}[H]
\caption{The agent-environment interaction in a reinforcement learning setting}
\centerline{\includegraphics[width=12cm,height=5cm]{fig1.png}}
\label{MDP}
Source: \cite{SB2018}
\end{figure}
\vspace{1.5cm}
State $s$ is a random variable from a bounded and compact set of state space\footnote{Latest research on reinforcement learning also investigates the setting with unbounded state space, e.g., \cite{shah2020stable}.}, i.e., $s \in \mathcal{S}$. Taking an action $a$, which belongs to an action space $\mathcal{A}$, $a\in\mathcal{A}$, is how the agent interacts with an environment. The state evolves through time following a probability function, $p: \mathcal{S} \times \mathcal{S} \times \mathcal{A} \rightarrow [0,1]$, which is defined as,
\begin{equation}
p(s'|s, a) \equiv Pr \{s_t = s' | s_{t-1} = s, a_{t-1} = a \}.
\end{equation}
It shows the probability for the random variable state $s'$ occurring at time $t$, given the preceding values of state, $s$, and action, $a$.
Reward is a random variable and can be generated from a reward function, $r: \mathcal{S} \times \mathcal{A} \rightarrow \mathcal{R}$.
Return from a state is defined as the sum of discounted future reward,
\begin{equation}
G_t \equiv r_{t} + r_{t+1} + ... = \sum_{k=0}^{\infty} \beta^k r_{t+k},
\end{equation}
where $\beta$ is the discount factor.
In a standard setup of reinforcement learning, an agent's behaviour is described by a policy (also known as an actor) that maps states to probabilities of selecting each possible action. In the actor-critic framework adopted here, a deterministic policy is adopted, that is the policy is a function $\mu: \mathcal{S} \rightarrow \mathcal{A}$, which maps a state from the state space to an action from the action space.
A value function\footnote{In reinforcement learning literature, two types of value functions are defined. What is defined here is normally referred to as an action value function. To not complicate the matter, when talking about value function in this paper, it means action value function in the reinforcement learning literature.} (known as critic) shows the `expected' return of taking an action in a state and thereafter following policy $\mu$. Expectation here is a subjective belief that depends on past experience. The value function is defined as,
\begin{equation}
Q^{\mu}(s,a) \equiv E^\mu [G_t|s_t = s, a_t = a],
\end{equation}
where $Q^\mu$ means the action value function follows policy $\mu$, and $E^\mu$ reflects that it's a subjective belief that depends on a policy $\mu$ that is formed by past experience\footnote{This forms of notation, e.g., $E^\mu$, largely follows the handbook for reinforcement learning by \cite{SB2018}.}.
Many approaches in reinforcement learning makes use of the recursive relationship known as Bellman equation,
\begin{equation}
Q^{\mu}(s,a) = r(s, a) + \beta E^\mu Q(s', a'),
\end{equation}
where $a' = \mu(s')$.
Reinforcement learning methods accentuate how the agent's policy and value functions change as a result of its experience. The DDPG algorithm makes use of two neural networks to approximate policy and value functions respectively: the actor network is denoted as $\mu(s|\theta^\mu)$, where $\theta^\mu$ represents parameters of the neural network; the critic network is denoted as $Q(s, a|\theta^Q)$, and $\theta^Q$ is its parameters. $\theta^\mu$ and $\theta^Q$ are updated during learning, and can be viewed as the coefficients of two functions and the probabilities involved in making subjective expectations. Two neural networks are updated with respect to each other. In the following passages, I highlight key elements on how the actor and critic networks are updated. The full algorithm is presented in Section \ref{fullalgo}.
The goal of this learning agent is to continuously update its subjective belief about the world based on experience, and to form a decision-making strategy (approximated by the actor network) that produces the highest discounted future return (approximated by the critic network). The actor network is updated with the goal of maximising the corresponding critic network. In other words, the actor network is updated based on what the agent believes, at that time, to be a strategy that produces high `expected' returns. `at that time' means that the critic network evolves - what the agent follows as the critic network at period $t$ is most possibly different from what it is at $t+1$. Expectation here is the learning agent's subjective belief that is formed from past experience.
The critic network, in a nutshell, is updated with the goal of minimising, what is named, a TD error (a temporal difference error).\footnote{The full algorithm is in the next section.} The TD error follows the form,
\begin{equation}
\label{TD error}
y - Q(s, a|\theta^Q),
\end{equation}
where $y$ is called a TD target, and it is the addition of reward based on a state-action pair and the discounted value of next state and action, i.e.,
\begin{equation}
y = r(s, a) + \beta Q(s',a'|\theta^Q)
\end{equation}
and the next period action $a'$ is assumed to follow the actor network $\mu(s|\theta^\mu)$ at that time \footnote{It needs not be the same as the true policy.}.
\begin{equation}
a' = \mu(s'|\theta^\mu).
\end{equation}
This TD target means that if following the subjective belief formed at that time, what would the best outcome be given a state-action pair chosen.
Learning agent's value function is updated to minimise the TD error. In neural science research, it is revealed that dopamine neuron firing rate in the brain resembles the TD error sequence during learning \citep{Botvinick2019}. This also inspires further research in neural science to model decision-making in connection with reinforcement learning algorithms.
As highlighted in Section \ref{section_intro} Introduction, exploration plays a crucial rule ensuring that the learning agent collects a wide range of information. To ensure that the agent explores its environment and tries out new actions, an exploratory policy is adopted and takes the following form,
\begin{equation}
\label{OU}
\mu'(s_t) = \mu(s_t|\theta^\mu) + \mathcal{N}_t.
\end{equation}
This shows that the final action the agent takes, i.e., what $\mu'(s_t)$ outputs, depends on the actor network output $\mu(s_t|\theta^\mu)$, and a random variable sampled from a noise process $\mathcal{N}_t$. Following \cite{lillicrap2015drl}, $\mathcal{N}_t$ is sampled from a discretised Ornstein-Uhlenbeck (OU) process.\footnote{There is a strain of literature in computer science solely focus on different exploration strategies to achieve the best performance for a given task. It is out of the scope of this current exercise, and not discussed in details.} This exploratory policy produces a random action. The randomness decreases over time (by design) but it never disappears in this paper. The implication is that in a stationary environment, the policy network becomes closer to the true underlying policy as it learns but it is never identical to it. However, in a non-stationary environment, it allows the policy network to adjust and be flexible facing changes in the environment. Connect it with economic concepts, it means that the policy will converge to a close region of the rational expectation solution (if exists), but will not be identical to it. In an economic model that is subject to structural breaks or regime changes, this exploratory policy allows the learning agent to adjust its expectation and adapt its policy to a new regime.
As a solution method, reinforcement learning algorithms are connected to dynamic programming that is widely employed in macroeconomics. Most reinforcement learning algorithms attempt to achieve similar results as dynamic programming but with less computation and without assuming for a perfect model of the environment \citep{SB2018}. This paper focuses on the dimension of the learning process rather than adopting the framework as a solution method, and it allows the agent to never stop exploring (as we do in real life).
On the dimension of learning from past data, reinforcement learning algorithms are also similar and connected to the adaptive learning methods. Both methods update some desired parameters with past data. Reinforcement learning algorithms could provide a plausible and flexible expectation formation model that is connected to psychology and neural science. Moreover, it is computationally flexible given the use of neural networks.
\subsection{Connecting to the Economic Model}
To implement this expectation formation framework in an economic setting, I first translate the economic model into the aforementioned components within a Markov decision process, which are presented in Table \ref{RL1}.
Assume for a logarithmic utility function of the form $u(c_t, m_t) = ln(c_t) + \gamma ln(m_t)$, where $\gamma$ is a preference parameter. Assume for $c_t=y_t=1$ for all $t$ and $\gamma=1$ for simplicity.
\begin{table}[H]
\centering
\caption{RL components and the economic environment}
\label{RL1}
\begin{tabular}{@{}lll@{}}\\ \hline \hline
\textbf{Terminologies} & \textbf{Description} & \textbf{\specialcell{Representation in the\\ economic environment}} \\ \hline
\vspace{0.5cm}
\textbf{State, $s_t$} & \specialcell{A random variable from a state space, \\ $s_t \in \mathcal{S}$} & $\pi_{t-1}, E^A_{t-1}\pi_t, m_{t-1}$\\
\vspace{0.5cm}
\textbf{Actions, $a_t$} & \specialcell{A random variable from an action space, \\ $a_t \in \mathcal{A}$} & inflation belief, $E^A_t\pi_{t+1}$ \\
\vspace{0.5cm}
\textbf{Rewards, $r_t$} & A function of state and action & \specialcell{a function of the forecast error, \\$ -|E^A_{t-1} \pi_t - \pi_t|$} \\
\vspace{0.5cm}
\textbf{\specialcell{Policy function,\\ $\mu(s|\theta^\mu)$}} & \specialcell{A mapping from state to action, \\ $\mu: \mathcal{S} \rightarrow \mathcal{A}$} & \specialcell{Approximated by a neural network,\\ ie., actor network; \\ parameterised by $\theta^\mu$ \\to be updated during learning} \\
\vspace{0.5cm}
\textbf{\specialcell{Value function, \\$Q(s,a|\theta^Q)$}} &\specialcell{the `expected' (subjective belief) \\return of taking an action in a state} &\specialcell{Approximated by a neural network,\\ ie., critic network; \\parameterised by $\theta^Q$ \\to be updated during learning}\\ \hline \hline
\end{tabular}
\end{table}
As Table \ref{RL1} shows, the state of this economy contains past period inflation, inflation belief, and real money holding. The action of this AI agent is to form an inflation belief, denoted by $E^A_t\pi_{t+1}$, where $E^A$ is this AI agent's subjective belief based on past experience. The reward of this agent correlates negatively to its forecast errors. How this reward is generated is unknown to the AI agent. Policy and value functions are both approximated by neural networks. As the AI agent does not have any information on the environment and its own preference, it must gather information by taking an action each period and observes its corresponding rewards. The agent also does not know how state transitions after it takes an action at each period. This transitional dynamics involves Equation \ref{Euler}, \ref{Tay}, and \ref{MDI}. Given an agent's inflation belief, Equation \ref{Euler}, $u^c (c_t, m_t) = \beta E_t u^c (c_{t+1}, m_{t+1}) \frac{i_t}{\pi_{t+1}}$, gives the interest rate that is consistent with the agent's optimal intertemporal allocation. Given this interest rate, Equation \ref{Tay}, $i_t =\frac{\hat{\pi}}{\beta} \left(\frac{\pi_t}{\hat{\pi}}\right)^{1+\lambda}$, provides the actual inflation that leads to central bank's nominal interest rate decision. The job of this agent is to learn about its preference and the aforementioned state transition dynamics as best as possible, so as to come up with a decision-making strategy, i.e., a policy function, that maximises the value function (according to the agent's subjective belief) in states of relevance.
\subsection{Full Algorithm and Sequence of Events}
\label{fullalgo}
The full algorithm consists of three main steps:
\vspace{1cm}
Step I: Initialisation
\begin{itemize}
\item Sep up two neural networks: an actor network $\mu(s|\theta^\mu)$ takes the argument of state and outputs an action; a critic network $Q(s,a|\theta^Q)$ takes the argument of a state-action pair and outputs a value.
\item $\theta^\mu$ and $\theta^Q$ represent the parameters of the two networks respectively. Both are initialised randomly. Both parameters update during the learning process so that the networks will move towards the true policy and value functions.
\item Define a replay buffer $\mathcal{B}$ (called transitions in the DRL literature), which is a memory that stores information that is collected by a DRL agent during the agent-environment interactive process. A transition is characterised by a sequence of variables $(s_t, a_t, r_t, s_{t+1})$.
\item Define a length of $N$, which is the size of a mini-batch. A mini-batch refers to a sample from the memory.
\item Define the total number of episodes $E$ and simulation periods $T$ per episode. Each episode contains $T$ simulation periods. The higher the episodes, the longer the learning periods.\footnote{In the DRL literature, AL agent is usually set to learn a particular task or an Atari game. An episode, thus, represents re-starting the game or task, and it ends with a terminal state (i.e., the end result of a game). In an economic environment, however, a clear terminal state can be difficult to specify. Therefore, the concept of episodes only correlates to how long an agent has been learning.}
\end{itemize}
For each episode, loop over step II and III.\\
Step II: The AI agent starts to interact with the environment.
\begin{itemize}
\item The agent observes the current state real money holding of previous period $m_{t-1}$, previous period realised inflation $\pi_{t-1}$, and its last period inflation belief $ E^A_{t-1}\pi_t$. The agent then forms an inflation belief $E^A_{t}\pi_{t+1}$ according to its actor network, i.e., $a_t = E^A_{t}\pi_{t+1} = \mu(s_t|\theta^{\mu})+\mathcal{N}_t$, which consists of the current policy $\mu(s_t|\theta^{\mu})$ and exploration noise $\mathcal{N}_t$.
\item Execute action $a_t$, and observe a reward $r_t = -|E^A_{t-1} \pi_t - \pi_t|$. The state transitions to the next, i.e., $ s_{t+1} = \{m_{t}, \pi_{t}, E^A_{t}\pi_{t+1}\}$. The state transition dynamics are as follows.
\begin{itemize}
\item Given the agent's inflation expectation, an Euler equation (Equation \ref{Euler}) consistent interest rate can be obtained, $\beta i_t = E^A_t\pi_{t+1}$.
\item Real money holding follows the money demand equation (Equation \ref{MDI}), $\frac{c_t}{m_t} = 1 - \frac{1}{i_t}$.
\item Lastly, given the interest rate, to be consistent with the central bank's interest rate rule (Equation \ref{Tay}), realised inflation is $\pi_t = \left(\frac{i_{t}\beta}{\hat{\pi}}\right)^{\frac{1}{1+\lambda}} \hat{\pi}$.
\end{itemize}
This transitional dynamics illustrate how private agent's expectation and macroeconomic variables (e.g., inflation, and nominal interest rate) are linked.
\item Store a transition $(s_t, a_t, r_t, s_{t+1})$ in the memory $\mathcal{B}$.
\end{itemize}
Step III: Training the AI agent (when the AI agent starts to learn) for period $N \leq t \leq T $.
\begin{itemize}
\item Sample a random mini-batch of N transitions $(s_i, a_i, r_i, s_{i+1})$ from the memory $\mathcal{B}$.
\item Calculate a value $y_{i}$ for each transition $i$ following
\begin{equation}
y_{i} = r_i + \beta Q^{\mu}(s_{i+1},\mu(s_{i+1}|\theta^{\mu})|\theta^{Q})
\end{equation}
for all $i\in N$, where $Q^{\mu}(s_{i+1},\mu(s_{i+1}|\theta^{\mu})|\theta^{Q})$ is a prediction made by the critic network with state-action pair $(s_{i+1},\mu(s_{i+1}|\theta^{\mu}))$, and $\mu(s_{i+1}|\theta^{\mu})$ is a prediction made by the actor network with input $s_{i+1}$.
\item Obtain $Q(s_i, a_i|\theta^Q)$ from the critic network with input state-action pair $(s_i, a_i)$
\item Calculate the average loss for this sample of $N$ transitions
\begin{equation}
L = \frac{1}{N}\sum_i\big(y_i-Q(s_i,a_i|\theta^Q)\big)^2.
\end{equation}
\item Update the critic network with the objective of minimising the loss function $L$.\footnote{This involves applying back propagation and gradient descent procedures.}
\item For the policy function, i.e., the actor network, the objective is to maximise its corresponding value function. This means that a value function $Q(s,a|\theta^Q)$ that follows a particular policy. In other words, the input action $a$ of $Q$ function is from the policy $\mu$, $a = \mu(s|\theta^\mu)$. Define the objective function as,
\begin{equation}
J(\theta^{\mu}) = Q^\mu(s_i, \mu(s_i|\theta^{\mu})|\theta^Q).
\end{equation}
\item This objective function could also be rephrased as minimising $-J(\theta^\mu)$. Update the actor network parameters with the objective of minimising $-J(\theta^\mu)$.\footnote{Similar to the critic network, the specific steps of updating ANN's parameters by minimising an objective function involves back propagation and gradient descent.}
\end{itemize}
\section{Experiments}
Motivated by the early literature on accelerationist controversy, I set up a price stationary environment, and then shift it to an inflation-stationary one. This is to investigate how the AI agent reacts to the monetary authority's action in increasing inflation (perhaps with the goal of exploiting a long-run trade off between inflation and a real variable). More specifically, the AI agent in this exercise first lives in an environment where the inflation target is $1$, i.e., it learns to form inflation expectation in this price stationary environment. The inflation target is then shifted to $1.1$ (i.e., an inflation stationary environment). The agent does not know this change is occurring. The aim is to observe its behaviour in response to this unforeseen target change, and how the economy transitions. I do not dive into the reason to this change or the probability of its occurrence. The main focus is if the AI agent can adapt its inflation belief with respect to the change of monetary policy regime. The steady state values under two targets are summarised in Table \ref{2target}.
\begin{table}[H]
\centering
\caption{Steady State Values under Two Policy Targets}
\label{2target}
\begin{tabular}{@{}lll@{}}\\ \hline \hline
\textbf{Steady State Values} & \textbf{Target I} & \textbf{Target II} \\ \hline
\vspace{0.5cm}
\textbf{Inflation Target} &$\hat{\pi} = 1.0$ & $\hat{\pi} = 1.1$ \\
\vspace{0.5cm}
\textbf{Inflation} &$\pi = 1.0$ & $\pi = 1.1$ \\
\vspace{0.5cm}
\textbf{Interest Rate} & $i = 1.25$& $i = 1.375$ \\
\vspace{0.5cm}
\textbf{Real Money Holdings}& $m = 5$ & $m = 3.67$ \\ \hline \hline
\end{tabular}
\end{table}
Table \ref{2target} shows that in the first regime with the $1.0$ inflation target, the steady state interest rate is $1.25$ and the real money holding value is 5.\footnote{As the importance here is to show if and how the economy converges from one to the other steady state, these values are not designed to match the reality. } In the inflation stationary environment, the nominal interest rate increases to $1.375$, and the real money holding reduces to $3.67$. These steady state values are used as a benchmark to observe which steady state the economy filled with an AI agent (gradually) converges to.
The main parameters are presented in Table \ref{param}. In this setup, a passive monetary policy is adopted with $\lambda < 0$. The case of active monetary policy is also considered, and it does not affect the main findings in this paper.\footnote{Results generated from an active monetary policy are available upon request.} Discount factor chosen as $0.8$ for computational simplicity.\footnote{A more realistic value such as $\beta = 0.99$ was also considered, and it does not change the main findings of this paper.} Exploration level of the baseline agent is 0.2, this means that the standard deviation of the noise added to the policy function (i.e., Equation \ref{OU}) is 0.2.
\begin{table}[H]
\centering
\caption{Main Parameters}
\label{param}
\begin{tabular}{@{}ll@{}}\\ \hline \hline
\textbf{Parameters} & \textbf{Baseline Agent} \\ \hline
\vspace{0.2cm}
\textbf{Policy rule parameter $\lambda$} & - 0.5 \\
\vspace{0.2cm}
\textbf{Discount Factor $\beta$} &0.8 \\
\vspace{0.2cm}
\textbf{Exploration Level} & 0.2 \\ \hline \hline
\end{tabular}
\end{table}
\section{Results}
This section highlights two main findings: 1. an AI agent that is not expecting a change in the policy target has the ability to adapt to this change, and this is reflected by its inflation forecasts. Following its adaptive behaviours in forming inflation expectations, based on state transitional dynamics (described by Equation \ref{Euler}, \ref{MDI}, and \ref{Tay}), the corresponding inflation, nominal interest rate, and real money holding move from one steady state to the other. This result depends crucially on agents' ability to explore and continuously learn; 2. how well the economy converges to the new steady state, quantified by the distance between the actual steady state values and simulation results of inflation and nominal interest rate, depends on the past experience of an AI agent. With more experience in an environment with a changing target, holding everything else constant, the agent makes inflation forecasts that are more closely related to the realised inflation. Hence the economy converges to the new steady state better. The amount of past experience an AI agent has, in this setting, depends on how long it has been living in an environment.
This section presents simulation results during a policy regime change. Results showing how an AI agent learns from no information when the inflation target is 1.0 is available in Appendix \ref{appendixA}.
\subsection{With and Without Exploration}
Figure \ref{Pihat0} and \ref{Pihat0rm} plot simulation paths of an agent who cannot explore and learn when there is a shift in the policy target. During the transitional period of a policy target change, Figure \ref{Pihat0} plots the simulation paths of the agent's inflation belief and realised inflation when it does not explore its environment. The x-axis denotes simulation periods. The vertical black dashed line shows the timing of this target change. At period 0 in the figure, the inflation target changes from 1.0 (i.e., price stationary) to 1.1 (i.e., inflation stationary). However, given this agent does not explore its environment and learn from it, its inflation belief barely changes in response to the new target. The corresponding nominal interest rate and real money holdings, as plotted in Figure \ref{Pihat0rm}, are both off their new steady state values of 3.67 and 1.375, respectively. Given the monetary policy rule equation \ref{Tay}, an almost constant nominal interest rate plus an increasing inflation target correspond to a decrease in the realised inflation, as shown in Figure \ref{Pihat0}.
\begin{figure}[H]
\caption{Inflation and Expected Inflation, without exploration}
\centerline{\includegraphics[width=19cm,height=8cm]{Pihat0.png}}
\label{Pihat0}
\end{figure}
\begin{figure}[H]
\caption{Real Money Holdings and Nominal Interest Rate, without exploration}
\centerline{\includegraphics[width=19cm,height=8cm]{pihat0rm.png}}
\label{Pihat0rm}
\end{figure}
In contrast, Figure \ref{Pihat10} to \ref{Pihat10i} show simulation behaviours when the AI agent has the ability to explore and learn from its experience gained through exploration. Its transitional behaviours different significantly from when it has no ability to explore. Figure \ref{Pihat10} plots inflation belief and realised inflation during the simulation periods when inflation target changes. As shown in the figure, at period 0 when the inflation target changes, the agent responds with a gradual increase in its inflation expectation. A delay in the change of inflation belief can be observed right after period 0. This can be explained by how an agent learns in this algorithm. The agent only changes its behaviours or decision-making strategy once it has gained relevant experience. In this circumstance, once the inflation target changes, the only way for the agent to be aware of this change is by making an inflation forecast in this environment, and observe how the reward and the next state are different from its own past experience when making the same forecast. This delayed response is also consistent with the temporary dip of actual inflation in Figure \ref{Pihat10}, which gradually increases to the new steady state value of 1.1 at around period 8. Given the agent changes its inflation expectation, the corresponding real money holdings, as plotted in Figure \ref{Pihat10rm}, also converges to a value that is close to the new steady state 3.67. Figure \ref{Pihat10i} shows that the nominal interest rate, in response to the realised inflation, moves close to the new steady state value 1.375.
\begin{figure}[H]
\caption{Inflation and Expected Inflation, with exploration}
\centerline{\includegraphics[width=19cm,height=8cm]{Pihat10.png}}
\label{Pihat10}
\end{figure}
\begin{figure}[H]
\caption{Real Money Holdings, with exploration}
\centerline{\includegraphics[width=19cm,height=8cm]{pihat10rm.png}}
\label{Pihat10rm}
\end{figure}
\begin{figure}[H]
\caption{Nominal Interest Rate, with exploration}
\centerline{\includegraphics[width=19cm,height=8cm]{pihat10i.png}}
\label{Pihat10i}
\end{figure}
This result attests that with exploration and constant learning, this AI agent adjusts its inflation belief with respect to the new regime (with a delay). Given the general equilibrium setup of the economic environment, its behaviours result in that the economy shifts from the neighbourhood of price stationary rational expectation equilibrium to the inflation stationary one. One natural question to ask is what could impact how well the economy converges to the new steady state. If this agent were to experience and learn in this environment for longer period, would it lead to a better convergence of the economy to the new rational expectation equilibrium? This is illustrated in the following section.
\subsection{More or Less Experience}
In this subsection, I present results showing that the more an AI agent experiences and learns in a given environment, the better it is at making a decision that maximises its reward, which corresponds to the economy converging closer to the new rational expectation equilibrium.
\begin{figure}[H]
\caption{Inflation expectation during an inflation target change}
\centerline{\includegraphics[width=19cm,height=8cm]{pihat01epi.png}}
\label{pihat01epi}
\end{figure}
\begin{figure}[H]
\caption{Inflation during an inflation target change}
\centerline{\includegraphics[width=19cm,height=8cm]{pihat01pi.png}}
\label{pihat01pi}
\end{figure}
Figure \ref{pihat01epi} plots the simulated paths of inflation forecasts for four agents with different experience. All agents learnt separately. Their learning environments (i.e., the economic model and initial conditions) are identical. The x-axis plots simulation periods. The vertical black dashed line shows the timing of the inflation target change. The blue line labelled as ep20 (i.e., episode 20) represents the agent who has been learning for the longest, whereas the red line (episode 5) represents the agent who has been experiencing for the shortest among of time. It can be observed that all four agents, facing the inflation target change, shift their inflation belief (with a delay) to the neighbouring region of the new steady state under target II. However, the difference is that the blue line agent, the one who has more experience in changing environment, shifts its inflation expectation closer to the target of 1.1. The agent who has been learning for the shortest amount of period settles down at a level that is close of 1.08, which is the furthest to the 1.1 target among the four agents. Corresponding inflation is plotted in Figure \ref{pihat01pi}. It shows that the inflation path corresponding to the blue line agent converges to the new target the best among the four. This shows that, the longer an agent interacts with an environment and the more experience it gains, the better it learns to make a decision that maximises its long term rewards. Under the current setup, this translates into a better convergence to the new rational expectation equilibrium.
This result agrees with the argument made by \cite{MalmendierNagel2016}. They show that individuals of different ages disagree significantly in their inflation expectations, and this can be explained by differences in their lifetime experiences of inflation. In the simulation experiments here, as an AI agent gains more experience in an environment with a change in the monetary policy target, it learns and forms a better policy, and makes inflation forecasts that are more similar to the realised inflation. On the contrary, if an agent only has been experiencing in this changing environment for a short amount of time (e.g., similar to the red-line agent in Figure \ref{pihat01epi} and \ref{pihat01pi}), it adapts its policy based on this limited experience and its inflation forecasts are thus less accurate.
\section{Discussions}
In addition to providing a plausible transitional dynamic of an economy facing an inflation target change, this exercise offers an explanation on how expectation is formed, and how an agent is adaptive and becoming aware of changes in its environment. An AI Agents under the DDPG algorithm becomes aware of any policy changes through interacting with the economic environment that it lives in. This means making a decision given a current state, and observing the next state and the reward signal corresponding to its decision. When it observes that its decision-making strategy does not generate a high reward like it used to, the agent starts to make changes of its policy so that it can obtain a higher reward in the long run, and this is how an agent adapts its behaviours with respect to a monetary policy target change. This offers a way to explain what \cite{Cavalloetal2017} observes in their experiments. They provide evidence showing that private agents are more likely to adjust their inflation expectations with respect to supermarket price changes than actual inflation statistics. Price changes in supermarkets are likely to have a direct impact on consumers' welfare than the observed inflation statistics or a central bank announcement on changes in an inflation target.
One main criticism of DRL algorithms is the speed of learning. In this paper, it takes a significant amount of simulation periods for the agent to learn the solution of the model. This can be accelerated through modifying several training parameters. However, it remains a lengthy simulation periods (could be a lifetime) before the agent converges to a steady state solution.\footnote{ As explained by \cite{Botvinick2019}, the slow learning is mainly led by the incremental parameter adjustment and weak inductive bias within the algorithm. However, as it is a fast-evolving literature, many new DRL algorithms are proposed to mitigate this and speed up the learning process, for example, inspired by \cite{Gershman2017}, DRL algorithms with episodic memory are being developed. } This criticism matters more if the goal of applying the algorithm is for a learning agent to converge to a rational expectation solution. This involves the debate on if it is reasonable or sensible to assume that an economy is operating on a steady state. The economy might also be on a learning path that is away from an equilibrium.
\section{Summary}
In this exercise, I present a plausible expectation formation model, and show how a learning agent adjusts its subjective belief with respect to a monetary policy inflation target change. This agent's expectation formation process is modelled with an innovative AI algorithm. More specifically, this agent is born in an unknown environment, and learns through interacting with the environment. This involves taking exploratory actions and observing corresponding stimulus signals. The experience is then being processed by artificial neural networks with the goal of forming a decision-making strategy that maximises the agent's expected (subjective belief) future return. This subjective belief also evolves based on the agent's past experience.
With this algorithm, I highlight that the AI agent notices and adapts to a monetary policy regime shift. In a money-in-utility model with an interest rate rule, I observe this AI agent's behaviour once the inflation target of the monetary authority changes from 1.0 (i.e., price stationary) to 1.1 (i.e., inflation stationary). The AI agent living in this world recognises this monetary policy target change, and adapts its decision-making strategy so as to achieve high long-term rewards in the new environment. I argue that this result depends crucially on the agent's ability to explore its environment. More specifically, with the exploration property, the AI agent can collect a wide range of information, and this may include those that help it recognise a change in its environment. This result becomes apparent when comparing an exploring agent with an non-exploring one. Without exploration, the agent does not collect new information (i.e., does not gain new experience) and still behave following the policy that is formed in the old regime. With exploration, however, the agent collects and processes new information, which feeds into its subjective belief updates. This allows the agent to adjust its policy, and move towards the new equilibrium. I also present simulation results that show when an AI agent has more experience, it learns to make inflation forecasts that contribute to a better convergence of the economy to the new regime steady state.
This exercise offers a plausible view on how expectation is formed, and how an AI agent is adaptive and becoming aware of changes in its environment. An AI agent becomes aware of any environment changes (include inflation target changes) through interacting with the economic environment that it lives in. This means that given a state, it makes a decision, and observes the next state and the stimulus signal corresponding to its decision. When it notices that its decision-making strategy does not generate a high reward like it used to, the agent starts to make changes to its policy so that it can obtain a higher reward in the long run, and this is how an agent adapts its behaviours with respect to a monetary policy target change. One implication of this is that it could undermine the effectiveness of forward guidance. If what is communicated to the public cannot be reflected onto issues that directly impact their welfare (i.e., not reflected onto their past experience), private agents may delay their actions that are desired by the policy makers.
\newpage
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\section{Introduction}
Highly spin-polarized ferromagnetic oxides, such as CrO$_2$, Fe$_3$O$_4 $, La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ (LSMO)
have been the focus of recent fundamental and technological studies in the field
of spin electronics. Using these materials, various devices, such as giant
magnetoresistance (GMR) junctions \cite{vandijken04} and tunnel
magnetoresistance (TMR) junctions \cite{seneor99} have been fabricated and
studied. To fabricate a magnetoresistive device based on a junction, usually two
ferromagnetic layers are separated by a thin non magnetic layer (the spacer).
The nature of the spacer is chosen in order to control the spin-dependent
transport mechanism at the interface: metallic spacer (GMR) or insulating spacer
(TMR).
In the usual case of transition metal electrodes, the thickness of the spacer is
chosen in order to magnetically decouple the magnetic layers (\emph{i.e.}
thickness larger than a few atomic planes to break the direct coupling exchange
path, and to prevent indirect coupling such as the RKKY one). The transport
across the spacer must conserve the spin information, thus
the spacer thickness must be kept thinner
than a few mean free paths (current-in-plane cip-GMR) or spin diffusion lengths
(current-perpendicular-to-plane cpp-GMR) or a few 1/$\rm{k_F}$ (tunnel
probability in TMR junction).
A spacer is not necessary if it is possible to weaken the magnetic coupling
between both electrodes. Such devices have already been proposed: ballistic
junctions \cite{garcia99} or break junctions \cite{gabureac04}. These junctions
were designed to break the exchange coupling between two transition metals. To
achieve this, both electrodes have to be mechanically separated, which is a
difficult step, source of non reproducibilities, and sensitive to parasitic
phenomena such as magnetostriction \cite{egelhoff04}. In this paper we propose a
simple solid state structure, adapted to collective fabrication.
In oxides, magnetic coupling is due to indirect exchange (3d ion - oxygen - 3d
ion) and it is very sensitive to the atomic details of such a bond. For example
it is possible to weaken the coupling by changing the bond angle (manganite's
$\rm{T_C}$ varies as a function of the Mn-O-Mn bond angle \cite{hwang95}). Thus,
tuning the interfacial magnetic coupling is achievable in oxides.
The spin-polarization of a material is positive if the majority spin at the
Fermi level is parallel to the magnetization and negative if the minority spin
at the Fermi level is parallel to the magnetization. Half metallic ferromagnets
have a spin polarization of 100~\% (only one spin direction is present at the
Fermi level). Magnetite (Fe$_3$O$_4 $) stands out as a predicted half metallic
ferromagnet (ferrimagnet in fact) with negative spin polarization
\cite{yanase84} and a remarkably high Curie temperature ($\rm{T_C}$) of 858~K.
La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ is predicted to have 100~\% positive polarization \cite{livesay99} with
$\rm{T_C}$ of 350~K. A junction between two such half metallic ferromagnetic
compounds would in theory behave as an ideal magnetic-field-controlled switch with 100~\%
negative magnetoresistance (MR).
As mentioned above, recent efforts have been made to fabricate Fe$_3$O$_4 $/I/La$_{0.7}$Sr$_{0.3}$MnO$_3 $\
junctions, where I is SrTiO$_3$, CoCr$_2$O$_4$ \cite{ogale98,hu03},
but a Fe$_3$O$_4 $/La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ junction without a spacer has never been proposed.
\section{Experimental Details}
The Fe$_3$O$_4 $/La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ bilayers were grown on (001)-oriented SrTiO$_3$ (STO) substrates using pulsed
laser deposition. First, the LSMO layer was grown at 1023 K under 40 Pa of
O$_2$, then the Fe$_3$O$_4 $\ layer was grown at 623~K under 5.10$^{-4}$ Pa of O$_2$.
The thickness of LSMO was 50 nm whereas the Fe$_3$O$_4 $\ was grown with two different
thicknesses, 15 nm and 50 nm, estimated \textit{in situ} by optical
reflectometry. Prior to the deposition, the substrate was heated in oxygen up to
the deposition temperature.
To study the magnetotransport properties of the junction, 50~nm Au was deposited
upon the 15/50~nm Fe$_3$O$_4 $/LSMO/STO structure at room temperature using the
sputtering technique and subsequently, junctions of 500 $\times$~500~$\mu$m and 140
$\times$~140~$\mu$m were fabricated by photolithography and Ar ion etching process. All
transport measurements were carried out with cpp geometry and applied magnetic
field parallel to the plane.
X-ray diffraction (XRD) study was carried out to examine the structural
properties of the bilayers. Despite the 1~\% lattice mismatch between LSMO and
STO, the LSMO growth is pseudomorphic up to a critical thickness (100 nm) larger
than the thickness chosen for these bilayers \cite{ranno02}. The LSMO film is
epitaxially-strained (0.18$^{\circ}$ FWHM rocking-curve) and a large
epitaxially-induced magneto-elastic anisotropy is present \cite{ranno02}. The
Fe$_3$O$_4 $\ film on LSMO is textured with multiple orientations (diffraction peaks
corresponding to [001] and [011] directions, but not [311]), whereas Fe$_3$O$_4 $\ films
on SrTiO$_3$ deposited under similar conditions, were grown textured along the
[001] direction with 1$^{\circ}$ FWHM rocking curve. The details of the
deposition of films of LSMO and Fe$_3$O$_4 $\ on STO have been reported
elsewhere~\cite{ranno02, carvello04}.
\section{Results}
To check for magnetic coupling between both oxide layers, M(H) hysteresis loops
of the unpatterned 50/50~nm bilayer structure were measured in the temperature
range 10-350~K and up to 3 Tesla using a VSM and a SQUID magnetometer.
\begin{figure}[b!]
\includegraphics[width=9cm]{Bilayer_MH_50K}
\caption{\label{fig:MH}Hysteresis loops of unpatterned bilayer film at 50~K}
\end{figure}
Fig.~1
shows the typical hysteresis loop from an unpatterned bilayer
structure at 50~K, with the magnetic field applied along the substrate [110]
axis, which is the easy direction of the La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ layer. The hysteresis loop clearly
shows two distinct coercive fields (about 5~mT and 100~mT), which correspond to
LSMO and Fe$_3$O$_4 $, respectively. The temperature dependence of Fe$_3$O$_4 $\ coercivity
shows the Verwey transition at 110~K ($\rm{T_V}$~=~122~K in single crystals).
To study the magnetic coupling between these layers, minor loops of the softer
layer (LSMO) were measured at 10~K. No shift of the loops was detected.
Thus no exchange bias field larger than 3~mT exists.
The large coercivity difference between layers and squareness of
the LSMO hysteresis loop create well-defined parallel and antiparallel magnetic
configurations.
As far as transport is concerned, we have measured I(V) characteristics from 5
to 300~K. They are linear up to the point where heating effects come into play.
The evolution of the resistance with temperature
(Fig.~2)
can be
divided into 3 regimes. The high temperature regime, above 90~K, exhibits the
well-known resistance and CMR (colossal magnetoresistance) of manganites, and
can thus be attributed to the LSMO electrode, which dominates the transport at
these temperatures due to geometrical reasons.
Between 30 and 90~K, the
transport is thermally activated, due to the increasing Fe$_3$O$_4 $\ dominance when
temperature decreases. Below 30~K, a plateau is observed, which is surprising in
a Fe$_3$O$_4 $-dominated regime. This remains to be investigated, but could be explained
by the onset of a hot electron transport mechanism due to the high electric
field (40~kV/cm), such as the one observed in \cite{chern92}.
\begin{figure}[t!]
\includegraphics[width=9cm]{Bilayer_R_GMR}
\caption{\label{fig:RT} Resistance and GMR of a 140 $\times$~140~$\mu$m junction
measured with a current of 100~$\mu$A}
\end{figure}
At high field (1 to 6~T range), the junction shows a negative
magnetoresistance. This high field MR is large below 40~K (over -1~\%/T,
consistent with Fe$_3$O$_4 $\ thin films), smaller between 40 and 100~K, and increases
to high values (-2~\%/T) at high temperature (LSMO CMR). In the intermediate
regime, though, this negative slope is only visible above 2~T. Under that field,
Fe$_3$O$_4 $\ is poorly saturated, and GMR dominates (see below), giving a positive
slope.
\begin{figure}[b!]
\includegraphics[width=9cm]{Bilayer_GMR_55K}
\caption{\label{fig:GMR} R(H) magnetotransport measurement of a 140 $\times$ 140
$\mu$m junction for T~=~55~K }
\end{figure}
In the intermediate temperature range, the magnetic field dependent transport
measurement shows a characteristic inverse GMR behavior
(Fig.~3, the applied field is sweeped as the arrows indicate).
The magnetic fields at which the junction resistance changes the most abruptly
correspond to the coercivities of both oxide layers and the junction resistance
is lower when the magnetizations of both layers are antiparallel to each other.
The GMR was measured at $\pm$~100~$\mu$A constant current as a function of
temperature
(Fig.~2).
$R_{\uparrow \downarrow}$ is measured at 20~mT
and $R_{\uparrow \uparrow}$ at 400~mT. This magnetoresistance diminishes in
absolute value both above 90~K and below 30~K. The maximum GMR of -~5.2~\% is
found at 55~K.
\section{Discussion}
The two keypoints which have to be discussed are the origin of the magnetic
decoupling of the La$_{0.7}$Sr$_{0.3}$MnO$_3 $\ and Fe$_3$O$_4 $\ electrodes and the transport mechanism
responsible for the large magnetoresistance. The ferromagnetic coupling
mechanism is double exchange (Mn$^{3+}$-O-Mn$^{4+}$ bonds) in the case of LSMO,
and is superexchange (Fe$^{3+}$(A)-O-Fe$^{2+/3+}$(B) bonds) as well as double
exchange (Fe$^{3+}$(B)-O-Fe$^{2+}$(B) bonds) in the case of Fe$_3$O$_4 $. To
magnetically decouple the two layers, these nearest-neighbour mechanisms have to
be weakened. The interface between LSMO and Fe$_3$O$_4 $\ is a structurally
disordered layer due to the 6.7~\% lattice mismatch, which prevents
heteroepitaxy. In oxides, due to the localised character of electrons, weakening
the exchange is much easier than in transition metals where electrons are more
delocalised (RKKY coupling range can reach a few nanometers). So one disordered
layer due to the lattice mismatch
between a perovskite and a spinel ferromagnet is enough to reduce the exchange
coupling and to decouple both layers. This is a general statement since the
lattice mismatch between a spinel and a perovskite structure will always be a
few~\%.
We claim that our structure is a bilayer. However the presence of an intermixed
layer between the electrodes has to be ruled out to support this claim. In our
system any intermixing layer would be made of Fe and Mn ions and therefore it
would be magnetic. Since transport in conducting ferromagnetic oxides is based
on a nearest-neighbour hopping mechanism, any magnetic layer depolarizes the
current. Thus, the characteristic type of magnetoresistance we can measure rules out the presence
of an intermixing layer. Furthermore since the Fe$_3$O$_4 $\ layer is deposited at low
temperature (623~K), the spinel/perovskite interface is expected to be stable.
We have recently conducted a TEM study on the SrTiO$_3$~/~Fe$_3$O$_4 $\ interface, which
showed that the perovskite / spinel mismatch can be accomodated through a
regular array of dislocations \cite{carvello05} and confirms that no intermixing takes place.
As the MR is small (-5~\%), spin disorder in the interfacial plane, leading to a
partial depolarisation, cannot be ruled out. However, this cannot be called a
distinct magnetic layer, it is better characterized as an interfacial disorder.
The two electrodes are in direct electrical contact, so the nature of the
transport mechanism is related to the presence or absence of an electrical
barrier. The cpp transport characteristics are ohmic, and the R(T) exhibits no
regime that could be interpreted as a tunnel transport.
Since a significant MR has been measured and the characteristic fields of this MR are
the coercive fields of both electrodes a spin-coherent mechanism has to be
proposed. Through the interface, transport could still be based on a hopping
mechanism. Interface disorder only reduces hopping integrals (also impacting magnetism by
reducing the double-exchange coupling). Thus the MR mechanism is closer to the cpp-GMR
than to the TMR mechanism and the only significant interfacial resistance is the
GMR itself.
The value of the magnetoresistance is difficult to interpret in a quantitative
manner, given that the resistance of the interface is not dominant compared to
that of the electrodes. Thus the -5.2~\% GMR ratio is not intrinsic, and could
be enhanced through an optimized junction pattern.
It is also worth noting
that it is difficult to obtain a real parallel or antiparallel state with Fe$_3$O$_4 $.
Because of structural defects present in all Fe$_3$O$_4 $\ thin films (antiphase
boundaries), the remanence is less than 80~\%, and full saturation is not
achieved even at large fields. Lastly, the decay of the GMR at high temperature
can be explained by the decrease of LSMO polarisation well below $\rm{T_C}$, as
is known from other studies (\cite{favre01}). Nonetheless, our measurements
evidence negative spin polarization of Fe$_3$O$_4 $, as predicted, down to the interface
with LSMO.
For these reasons, we propose that the mechanism responsible for the exchange
weakening and the large MR in our bilayer is fundamentally different from the
TMR and spacer-assisted cpp-GMR mecanisms reported in other studies. It is
purely interfacial (i.e. spacerless) and specific to oxides.
\begin{acknowledgments}
One of the authors (MPS) would like to thank EGIDE and the French Foreign
Ministry, for providing the postdoctoral fellowship.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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ERROR: type should be string, got "https://www.wax-lyrical.com/love-candles-heart 18498 LOVE❤ https://www.wax-lyrical.com/media/catalog/product/a/l/almumv_web1.jpg 12 15 GBP InStock /Our Collections /Candles /Our Collections/Alphabet Collection /Candles/Alphabet Collection /Promotions/Mother's Day/Mother's Day Alphabet Perfect for placing on any mantelpiece, coffee table or stylish shelf, these votive candles feature a wonderfully fruity fragrance that will suit any home. Show Mum how much she means to you with a personalised set of Alphabet candles. Perfect for placing on any mantelpiece, coffee table or stylish shelf, these votive candles feature a wonderfully fruity fragrance that will suit any home and burn for up to 14 hours per candle, meaning you can continue to enjoy the fine fragrance long after the big day.\nPlus, save 20% when you buy a 5 candle set – regular price £15, our special price now only £12! GBP https://www.wax-lyrical.com/media/catalog/product/a/l/almumv_web1.jpg add-to-cart Alphabet Collection Votive Candle Raspberry and Vanilla Contains: Ethyl methylphenylglycidate. May produce an allergic reaction."
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,813
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\section{Introduction}
\label{sec:intro}
On-shell recursion is a procedure to determine all scattering amplitudes in a theory recursively from a finite set of ``seed'' amplitudes. It plays a central part in the modern $S$-matrix program where physical and mathematical properties of scattering amplitudes are used to construct the $S$-matrix directly without the aid of a Lagrangian. Originally developed in the context of gauge theory by Britto, Cachazo, Feng and Witten (BCFW)~\cite{BCFW}, on-shell recursion was soon generalized to gravity theories~\cite{Gravity}, string theory~\cite{String}, generic renormalizable and some nonrenormalizable theories~\cite{(non)ren}. More recently, there has also been progress towards an on-shell formulation of scattering amplitudes in effective field theories (EFTs)~\cite{NLSMrecursion,CheungRecursion,LuoWen,Periodic}.
Beyond providing an efficient tool for calculating scattering amplitudes, recursion relations have also been successfully utilized as a framework to explore and classify the landscape of possible EFTs~\cite{Periodic,Elvang,SoftSmatrix,Shifman}. This connects to the newly emerging paradigm that seeks to \emph{define} quantum (effective) field theory without reference to a Lagrangian. While the basic principles underlying this program are mere locality and unitarity, the bulk of work done so far has focused on the sector of Lorentz-invariant field theories.\footnote{The only exceptions we are aware of include several recent works in a cosmological context, limited to EFTs with Lorentz-invariant kinematics but Lorentz-breaking interactions~\cite{boostlessbootstrap}, and a specific application of recursion techniques to scattering of phonons in Navier-Stokes fluids~\cite{CheungNavier}.} Yet, recent years have witnessed the EFT framework claiming a much larger territory than originally conceived. The range of novel applications of quantum field theory without Lorentz invariance now stretches from nonrelativistic gravity~\cite{Horava} and spacetime geometry~\cite{Son} to previously unthinkable exotic phases of quantum matter (see e.g.~Refs.~\cite{fractonGromov,fractonSeiberg} and references therein).
Should the modern scattering amplitude program provide new fundamental insight into the very nature of quantum field theory, it therefore seems mandatory to extend the scope of discussion by giving up on Lorentz invariance altogether. The aim of the present letter is to initiate the exploration of this new terra incognita. Our main result is that the existing on-shell recursion approach to EFT can be modified to nonrelativistic EFTs with rotationally-invariant gapless kinematics, where energy is proportional to an in principle arbitrary (integer) power of momentum. We demonstrate this by explicit examples of EFTs for a complex Schr\"odinger scalar and a real Lifshitz scalar.
The plan of the text is as follows. In the remainder of this section, we first briefly overview the BCFW recursion approach and its modification applicable to EFTs, and then outline the landscape of nonrelativistic EFTs relevant to our discussion. Sections~\ref{sec:deformation} and~\ref{sec:recursion} constitute the core of this letter, showing how to set up the recursion procedure for EFTs with nonrelativistic kinematics. An integral part of the text is section~\ref{sec:example} where we work out three examples.
\subsection{BCFW on-shell recursion}
\label{subsec:BCFWrecursion}
A central idea of the on-shell recursion technology is to promote $n$-particle on-shell amplitudes $A_n$ to meromorphic functions by complexifying external momenta in a way that preserves both on-shellness and conservation of energy and momentum. In the BCFW recursion, two selected external momenta, $p_i$ and $p_j$, are shifted,
\begin{align}
\label{BCFWshift}
\hat{p}_i\equiv p_i+zq, \qquad \hat{p}_j\equiv p_j-zq, \qquad z\in \mathbb{C}.
\end{align}
(Shifted quantities are denoted with a hat.) The auxiliary momentum $q$ must satisfy the on-shell conditions $q^2=p_i\cdot q=p_j\cdot q=0$. In four spacetime dimensions, it is thus fixed up to rescaling. At tree level, the complexified amplitude $\hat A_n(z)$ is a rational function of $z$. The original, physical amplitude $A_n=\hat A_n(0)$ can be recovered by
\begin{align}
A_n=\frac{1}{2\pi\text{i}}\oint dz\frac{\hat{A}_n(z)}{z},
\end{align}
where the integration contour is an infinitesimal circle enclosing the origin of the complex plane. Cauchy's theorem and factorization then relate the physical amplitude $A_n$ to lower-point amplitudes in the following way,
\begin{align}
A_n&=-\sum_I\Res_{z=z_I}\frac{\hat{A}_n(z)}{z}+B_n\nonumber \\
&=\sum_I\frac{\hat{A}_L^{(I)}(z_I)\hat{A}_R^{(I)}(z_I)}{P_I^2}+B_n.
\end{align}
The sum runs over all factorization channels $I$ where the lower-point amplitudes $\hat{A}_L^{(I)}$ and $\hat{A}_R^{(I)}$ contain one of $\hat p_i$, $\hat p_j$ each. Moreover, $P_I$ is the intermediate momentum evaluated at $z=0$, and $z_I$ is fixed by the on-shell condition $\hat P_I^2(z_I)=0$ to $z_I=-P_I^2/(2P_I\cdot q)$. Finally, $B_n$ denotes the contribution of the residue of the pole at $z=\infty$. The validity of the recursion relies on the latter either vanishing or being calculable.\footnote{Calculating $B_n$ is a challenging problem that has been considered in several contexts~\cite{Boundary}.}
The above approach does not extend straightforwardly to low-energy EFTs. Technically, the problem is that the derivative couplings of EFTs imply polynomial growth of scattering amplitudes at large $z$, and thus preclude the standard recursion procedure. A different kind of complexification of the kinematical phase space is needed.
\subsection{On-shell recursion for EFTs}
\label{subsec:EFTrecursion}
The deeper reason why BCFW recursion fails for EFTs is that factorization alone is not sufficient to relate higher-point EFT amplitudes to lower-point ones; more information is needed. Since the form of an EFT is largely dictated by symmetries, it is hardly surprising that the additional input comes from symmetry (breaking).
Spontaneous symmetry breaking constrains the scattering amplitudes of the associated Nambu-Gold\-sto\-ne (NG) boson(s) in the ``(single) soft limit,'' in which the momentum of one of the particles participating in the scattering process vanishes. This limit can be probed by rescaling the momentum of the chosen particle, $p_i$, as $p_i\to \epsilon p_i$, and taking the scaling parameter $\epsilon$ to zero. The asymptotic behavior of the amplitude $A_n$ is characterized by a single scaling exponent,
\begin{align}
\label{scaling}
A_n\propto \epsilon^{\sigma_i}, \qquad \epsilon\to 0.
\end{align}
As a rule, albeit not without exceptions~\cite{Shifman}, spontaneous symmetry breaking ensures that $\sigma_i\geq1$; this fact is known as ``Adler's zero.'' Theories where $\sigma_i$ is larger than naively expected from counting derivatives in the Lagrangian are dubbed ``exceptional.'' The landscape of Lorentz-invariant exceptional EFTs is very strongly constrained~\cite{CheungEFT,Periodic,Bogers}. Single-flavor scalar exceptional EFTs were the first effective theories shown to be on-shell constructible~\cite{CheungRecursion} by a modification of the BCFW recursion procedure known as ``soft recursion.''
In the soft recursion procedure, \emph{all} external momenta are shifted,
\begin{align}
\label{softrel}
\hat{p}_i&\equiv p_i(1-a_iz), \qquad z\in \mathbb{C},\\
\label{relcon}
\sum_{i=1}^na_ip_i&=0,
\end{align}
where Eq.~(\ref{relcon}) is imposed by energy and momentum conservation. Nontrivial solutions for the coefficients $a_i$ exist for generic kinematical configurations when $n\geq D+2$, where $D$ is the spacetime dimension. The soft limit for the $i$-th particle can then be accessed by taking $z\to1/a_i$.
In order to be able to apply Cauchy's theorem, one modifies the behavior of the complexified amplitude $\hat A_n(z)$ at large $z$ by dividing it by the factor
\begin{align}
\label{Fdef}
F_n(z)\equiv\prod_{i=1}^n(1-a_iz)^{\sigma_i}.
\end{align}
For exceptional EFTs, this is sufficient to ensure vanishing of the boundary term $B_n$~\cite{CheungRecursion}. At the same time, the scaling~(\ref{scaling}) of the amplitude in the soft limit guarantees that adding $F_n(z)$ does not create any new poles in $\hat A_n(z)$. One can then reconstruct the physical amplitude $A_n=\hat A_n(0)$ similarly to the BCFW recursion,
\begin{align}
\label{softAn}
A_n=\frac{1}{2\pi\text{i}}\oint dz\frac{\hat{A}_n(z)}{zF_n(z)}=-\sum_I\Res_{z=z_I^{\pm}}\frac{\hat{A}_n(z)}{zF_n(z)},
\end{align}
where each factorization channel $I$ now gives rise to two poles $z_I^{\pm}$ corresponding to solutions of the shifted on-shell condition $\hat{P}^2_I(z)=0$. These are given explicitly by
\begin{align}
\label{zpm}
z_I^\pm=\frac1{Q_I^2}\Bigl[P_I\cdot Q_I\pm\sqrt{(P_I\cdot Q_I)^2-P_I^2Q_I^2}\Bigr],
\end{align}
where $P_I\equiv\sum\limits_{i\in I}p_i$ and $Q_I\equiv\sum\limits_{i\in I}a_ip_i$. Factorization together with Eq.~(\ref{softAn}) then imply the recursion formula~\cite{CheungRecursion}
\begin{align}
\label{eq10}
A_n=\sum_I\frac{\hat{A}_L^{(I)}(z_I^-)\hat{A}_R^{(I)}(z_I^-)}{P_I^2\left(1-\frac{z_I^-}{z_I^+}\right)F_n(z_I^-)}+(z_I^-\leftrightarrow z_I^+).
\end{align}
\subsection{Nonrelativistic EFTs}
\label{subsec:NREFTs}
The theories we will focus on in this letter live in a flat spacetime of $D\equiv d+1$ dimensions. They enjoy invariance under spacetime translations and $d$-dimensional spatial rotations. This is a fairly general setup that admits, if desired, a variety of kinematical algebras~\cite{Bacry}. The latter include the static (or Aristotelian) algebra containing no boosts whatsoever, and the Poincar\'e, Galilei (and its central extension, Bargmann) and Carroll algebras featuring different implementations of the relativity principle.
The NG modes stemming from spontaneous breakdown of global symmetry in such theories can be classified into two families, referred to as type $A_m$ and type $B_{2m}$ with positive integer $m$~\cite{HoravaShift}. A NG mode from the first family is described by a real scalar field with dispersion relation $\omega^2\propto\boldsymbol{p}^{2m}$. A NG mode from the second family, on the other hand, is described by two real scalar fields (or one complex scalar) forming a canonically conjugated pair with dispersion relation $\omega\propto \boldsymbol{p}^{2m}$.
Whether or not NG modes belonging to the $A_m$ and $B_{2m}$ families can exist in a given spatial dimension $d$ is constrained by the nonrelativistic version of the Cole\-man-Hohenberg-Mermin-Wagner (CHMW) theorem~\cite{HoravaNatural,WatanabePRX}. In short, at zero temperature, a NG boson of type $A_m$ may exist only if $m<d$. For fixed $m$, this in turn gives a lower bound on the dimension of space $d$. On the contrary, type $B_{2m}$ NG modes are not constrained at all and can exist, at zero temperature, for any positive $d$ and $m$.
It was observed early on~\cite{CheungEFT} that the enhanced scaling~\eqref{scaling} of scattering amplitudes in exceptional EFTs is a consequence of hidden symmetry. Motivated by this observation, one of us mapped in Ref.~\cite{NRLie} the landscape of nonrelativistic EFTs that admit such a hidden symmetry. We will show in a forthcoming paper~\cite{us} that unlike in the Lorentz-invariant case, this is in fact not sufficient to guarantee that a given EFT is exceptional. The catalogue of candidate EFTs compiled in Ref.~\cite{NRLie} will nevertheless serve as a useful guide for construction of explicit examples of nonrelativistic EFTs via recursion in section~\ref{sec:example}. We will thus be able to give examples of theories of the $A_1$, $A_2$ and $B_2$ type. Before doing so, we however first need to establish the soft recursion procedure for nonrelativistic EFTs. This is the subject of the next two sections.
\section{Momentum deformation in nonrelativistic EFTs}
\label{sec:deformation}
In this section, we introduce the momentum shifts nee\-ded for soft recursion. In contrary to the relativistic momentum shift in Eq.~(\ref{softrel}), we first shift the spatial momenta $\boldsymbol{p}_i$ only, and then use the on-shell condition to define an appropriate shift of the energies.
\subsection{Soft shifts for type $B_{2m}$ theories}
\label{subsec:B2m}
The following shifts respect the on-shell condition for type $B_{2m}$ theories,
\begin{align}
\boldsymbol{\hat{p}}_i&\equiv \boldsymbol{p}_i(1-a_iz),\\
\hat{p}^0_i&\equiv\boldsymbol{\hat{p}}_i^{2m}=\boldsymbol{p}_i^{2m}(1-a_iz)^{2m}.
\end{align}
Momentum and energy conservation then impose respectively the following constraints on the $a_i$ coefficients,
\begin{align}
\label{B2m1}
\sum_{i=1}^na_ie_i\boldsymbol{p}_i&=0, \\
\label{B2m2}
\sum_{i=1}^n(1-za_i)^{2m}e_i\boldsymbol{p}_i^{2m}&=0.
\end{align}
Here $e_i$ denotes a sign, chosen so that $e_i=+1$ for particles in the final state and $e_i=-1$ for particles in the initial state. Similarly to the relativistic case reviewed in section~\ref{subsec:EFTrecursion}, the existence of nontrivial solutions to Eq.~(\ref{B2m1}) requires $n\geq d+2$. Equation~(\ref{B2m2}) then imposes $2m$ additional constraints. Only amplitudes with $n\geq d+2+2m$ may therefore be reconstructed using soft recursion. For given $d$ and $m$, this tells us how many seed amplitudes we need to initiate the recursion procedure.
\subsection{Soft shifts for type $A_m$ theories}
\label{subsec:Am}
For type $A_{m}$ theories we define analogously
\begin{align}
\boldsymbol{\hat{p}}_i&\equiv \boldsymbol{p}_i(1-a_iz),\\
\hat{p}^0_i&\equiv |(\boldsymbol{p}_i^{2m})^{1/2}|(1-a_iz)^m,
\end{align}
which preserves on-shellness and yields the following constraints from momentum and energy conservation,
\begin{align}
\label{ca1}
\sum_{i=1}^na_ie_i\boldsymbol{p}_i&=0, \\
\label{ca2}
\sum_{i=1}^n (1-za_i)^me_i|(\boldsymbol{p}_i^{2m})^{1/2}|&=0.
\end{align}
Analogously to the type $B_{2m}$ case, the existence of nontrivial solutions for $a_i$ requires $n\geq d+2+m>2+2m$, where the last inequality follows from the nonrelativistic CHMW theorem. For the special case of $m=1$, which includes the family of Lorentz-invariant theories, the above constraints become equivalent to Eq.~(\ref{relcon}) and we recover the relativistic bound $n\geq d+3=D+2$.
Note that for both type $A_m$ and type $B_{2m}$ theories, the manifold of solutions for the $a_i$ coefficients is invariant under overall rescaling, $a_i\to\lambda a_i$, and overall shift, $a_i\to a_i+c$. This guarantees that in the special case of type $A_1$ theories where all the constraints on $a_i$ are linear, possible solutions for $a_i$ span an affine space.
\section{Soft recursion}
\label{sec:recursion}
We argued in section~\ref{subsec:EFTrecursion} that for relativistic exceptional EFTs, recursion relations among scattering amplitudes may be set up using Eq.~(\ref{softAn}). Since the argument only depends on the assumed soft behavior of $A_n$, factorization and vanishing of the boundary term, it can be generalized to any theory with these properties. Specifically, for theories of type $A_m$ and $B_{2m}$ we obtain
\begin{align}
\label{NRrec}
A_n=-\sum_I\sum_{i=1}^{2m}\Res_{z=z_I^{i}}\frac{\hat{A}_n(z)}{zF_n(z)}.
\end{align}
Here $z_I^i$, $i=1,\dotsc,$ $2m$ are solutions to the on-shell condition, which is of algebraic order $2m$ in $z$,
\begin{align}
\bigl(\hat{P}_I^0\bigr)^2-\hat{\boldsymbol{P}}_I^{2m}&=0 && \text{for } A_m, \\
\hat{P}_I^0-\hat{\boldsymbol{P}}_I^{2m}&=0 && \text{for } B_{2m},
\end{align}
for a given factorization channel $I$, where compared to Eq.~(\ref{eq10}), $P_I$ is now defined with the appropriate signs $e_i$ where necessary. Factorization then implies that the amplitude~(\ref{NRrec}) can be expressed in terms of lower-point amplitudes,
\begin{align}
\label{Angeneral}
A_n=-\sum_I\sum_{i=1}^{2m}\Res_{z=z_I^{i}}\frac{\hat{A}_L^{(I)}(z)\hat{A}_R^{(I)}(z)}{zF_n(z)D^{(I)}(z)},
\end{align}
where
\begin{align}
D^{(I)}(z)&=\bigl(\hat{P}_I^0\bigr)^2-\hat{\boldsymbol{P}}_I^{2m} && \text{for } A_m, \\
D^{(I)}(z)&=\hat{P}_I^0-\hat{\boldsymbol{P}}_I^{2m} && \text{for } B_{2m}.
\end{align}
Notice that the contribution from factorization channel $I$ in Eq.~(\ref{Angeneral}) matches the residue at $z=z_I^i$ of the following meromorphic function
\begin{align}
\label{analyticf}
\frac{\hat{A}_L^{(I)}(z)\hat{A}_R^{(I)}(z)}{zF_n(z)D^{(I)}(z)}.
\end{align}
This function can also have nonvanishing residues at $z=1/a_i$ and $z=0$. This follows from the fact that the intermediate propagator $D^{(I)}(z)$, hence also the subamplitudes $\hat{A}_L^{(I)}(z)$ and $\hat{A}_R^{(I)}(z)$, is off-shell for $z\neq z_I^i$. The on-shell argument implying that the soft behavior of the amplitudes dictated by Eq.~(\ref{scaling}) cancels the zeros of $F_n(z)$ is then no longer valid. In the special case where $\hat{A}_L^{(I)}(z)$ and $\hat{A}_R^{(I)}(z)$ are both local functions of momenta (that is, they have no poles) we can apply Cauchy's theorem to the meromorphic function in Eq.~(\ref{analyticf}) and recast the amplitude~(\ref{Angeneral}) in terms of a sum over residues at $z=0$ and $z=1/a_i$,
\begin{align}
A_n={}&\sum_I\frac{\hat{A}_L^{(I)}(0)\hat{A}_R^{(I)}(0)}{D^{(I)}(0)}\nonumber\\
&+\sum_I\sum_{i=1}^n\Res_{z=1/a_i}\frac{\hat{A}_L^{(I)}(z)\hat{A}_R^{(I)}(z)}{zF_n(z)D^{(I)}(z)}\nonumber \\
\label{master}
\equiv{}& A_n^{\text{ch}}+A_n^{\text{ct}}.
\end{align}
This expression is particularly useful for concrete applications. In terms of Feynman diagrams, the first term corresponds to the sum over diagrams with an internal propagator, whereas the second (double) sum encodes contributions from $n$-point contact operators. The two different types of contributions are distinguished by the notation introduced in the last line of Eq.~(\ref{master}).
\subsection{Validity criterion}
\label{subsec:validity}
Thus far we have simply assumed that the boundary term $B_n$ vanishes. A sufficient condition for this to happen is that $\hat{A}_n(z)/F_n(z)\to 0$ as $z\to\infty$. A criterion for the latter was in turn given by Elvang et al.~in Ref.~\cite{Elvang}. Their argument only relies on dimensional analysis, the soft behavior of $A_n$, the analytic structure of tree-level amplitudes, and the freedom to shift all $a_i$ by an overall constant. Since the latter property survives in all type $A_m$ and $B_{2m}$ theories, as shown in section~\ref{sec:deformation}, it is easy to adapt the argument of Ref.~\cite{Elvang} for our purposes.
We start with a generic expression for the $n$-point tree-level amplitude,
\begin{align}
A_n=\sum_j\Bigl(\prod_kg_k^{n_{jk}}\Bigr)M_j,
\end{align}
where $M_j$ are functions of momenta and $g_k$ are coupling constants associated with fundamental operators in the Lagrangian. Fundamental operators are defined in turn as the lowest-dimension operators whose on-shell matrix elements are needed to derive, at the leading-order in the low-energy expansion, any tree-level amplitude in the theory by recursion. Following the line of reasoning of Ref.~\cite{Elvang} then leads to the generalized validity criterion
\begin{align}
\label{criterion}
[A_n]-\min_j\Bigl(\sum_k n_{jk}[g_j]\Bigr)-\sum_{i=1}^n\sigma_i<0,
\end{align}
where square brackets indicate scaling dimension with respect to a uniform rescaling of all the momenta $\boldsymbol{p}_i$. It is easy to check that the criterion~(\ref{criterion}) is satisfied by all the example theories presented in the next section.
\section{Example calculations}
\label{sec:example}
We will now work out three simple analytical examples of recursive reconstruction of scattering amplitudes in theories of type $B_2$, $A_1$ and $A_2$, respectively. All three sample theories feature tree-level amplitudes with soft scaling $\sigma_i=2$. Yet, each of the theories possesses Lagrangian representations with less than two derivatives per field, which means that they possess enhanced soft limits. We will show in a forthcoming paper~\cite{us} that the enhanced scaling of scattering amplitudes in these theories is a consequence of an interplay of spontaneously broken symmetry and dispersion relations of NG bosons. Each of the three theories contains just one physical NG mode. Since we no longer have to distinguish different $\sigma_i$ for different particles participating in the scattering process, we introduce a shorthand notation replacing Eq.~(\ref{Fdef}),
\begin{align}
F_n^{(\sigma)}(z)\equiv \prod_{i=1}^n(1-a_iz)^{\sigma}.
\end{align}
\subsection{$B_2$: Schr\"odinger-DBI theory}
\label{subsec:SDBI}
Our first example features a complex scalar field $\Phi$ endowed with the action
\begin{align}
\label{SDBI}
S={}&\int\text{d}t\,\text{d}^d\boldsymbol{x}\, \bigl(\Phi^{\dagger}\text{i}\partial_0\Phi+\sqrt{G}-1\bigr), \\
\label{Gmetric}
G\equiv{}&1-2\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi^{\dagger}+\bigl(\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi^{\dagger}\bigr)^2\nonumber \\
&-\bigl(\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi\bigr)\bigl(\boldsymbol\nabla\Phi^{\dagger}\cdot\boldsymbol\nabla\Phi^{\dagger}\bigr).
\end{align}
This is a minimal nonrelativistic modification of one of the very few relativistic single-flavor exceptional theories~\cite{CheungEFT}: the Dirac-Born-Infeld (DBI) theory. We therefore name it the ``Schr\"odinger-DBI'' (SDBI) theory.
Our SDBI theory can be interpreted as describing fluctuations of a $d$-dimensional brane embedded in a $(d+2)$-dimensional Euclidean space. The symmetry of the SDBI action~(\ref{SDBI}) is accordingly $\mathbb{R}\times\text{ISO}(d+2)$, with the first factor of $\mathbb{R}$ corresponding to time translations~\cite{NRLie}. This symmetry is spontaneously broken down to $\mathbb{R}\times\text{ISO}(d)\times\text{SO}(2)$ by the presence of the brane, and the real and imaginary parts of $\Phi$ correspond to NG fields of spontaneously broken translations in the two extra dimensions. The term in Eq.~(\ref{SDBI}) with a single time derivative is only invariant under the full symmetry up to a surface term. It is thus an example of a Wess-Zumino-Witten (WZW) term.
The action~(\ref{SDBI}) fixes all tree-level amplitudes. We will now demonstrate that the recursion formula~(\ref{master}) correctly reproduces the six-point amplitude starting from the seed four-point amplitude. In fact, the argument of section~\ref{subsec:B2m} limits the validity of the recursion for $n=6$ to $d\leq2$ spatial dimensions. However, the amplitudes $A_n$ as functions of the momenta $\boldsymbol{p}_i$ do not depend explicitly on $d$. Whatever analytic relations between the amplitudes we find will therefore be independent of $d$ as well. One may think of this as carrying out the recursive step from $A_4$ to $A_6$ in $d=2$ dimensions, and then analytically continuing the result to any value of $d$ of interest.
To make the calculation transparent, we first explicitly list the relevant parts of the Lagrangian,
\begin{align}
&\mathcal{L}_2=\Phi^{\dagger}(\text{i}\partial_0+\boldsymbol\nabla^2)\Phi, \\
\label{SL4}
&\mathcal{L}_4=-\frac{1}{2}\bigl(\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi\bigr)\bigl(\boldsymbol\nabla\Phi^{\dagger}\cdot\boldsymbol\nabla\Phi^{\dagger}\bigr), \\
\label{L6}
&\mathcal{L}_6=-\frac{1}{2}\bigl(\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi\bigr)\bigl(\boldsymbol\nabla\Phi\cdot\boldsymbol\nabla\Phi^{\dagger}\bigr)\bigl(\boldsymbol\nabla\Phi^{\dagger}\cdot\boldsymbol\nabla\Phi^{\dagger}\bigr).
\end{align}
Charge conservation dictates that the numbers of incoming and outgoing Schr\"odinger scalars must match in any scattering process. We use the convention that the particles labeled $1,\dotsc,n/2$ are incoming, whereas the particles $n/2,\dotsc,n$ are outgoing. The seed on-shell four-point amplitude then follows immediately from Eq.~(\ref{SL4}) as
\begin{align}
A_4=2(\boldsymbol{p}_1\cdot\boldsymbol{p}_2)(\boldsymbol{p}_3\cdot \boldsymbol{p}_4).
\end{align}
We are now ready to derive the six-point amplitude by recursion. We will use the indices $a$, $b$, $c$ to label a permutation of the incoming particles and $d$, $e$, $f$ a permutation of the outgoing particles such that $a$, $b$, $f$ are on the same side of the factorization channel. We can then identify the nine factorization channels in terms of $c$ and $f$ alone,
\begin{align}
\nonumber
I=\{(c,f)\}=\{&(14),(15), (16), (24),(25), (26),\\
&(34), (35), (36) \}.
\end{align}
Energy and momentum conservation fix the parameters of the intermediate propagator for each factorization channel,
\begin{align}
\boldsymbol{P}_I&\equiv\boldsymbol{p}_a+\boldsymbol{p}_b-\boldsymbol{p}_f=\boldsymbol{p}_d+\boldsymbol{p}_e-\boldsymbol{p}_c, \\
\frac{1}{2}\left(P_I^0-\boldsymbol{P}_I^2\right)&=-\boldsymbol{p}_a\cdot\boldsymbol{p}_b-\boldsymbol{p}_f\cdot\boldsymbol{p}_f+\boldsymbol{p}_a\cdot\boldsymbol{p}_f+\boldsymbol{p}_b\cdot\boldsymbol{p}_f\nonumber \\
&=-\boldsymbol{p}_d\cdot\boldsymbol{p}_e-\boldsymbol{p}_c\cdot\boldsymbol{p}_c+\boldsymbol{p}_d\cdot\boldsymbol{p}_c+\boldsymbol{p}_e\cdot\boldsymbol{p}_c.
\nonumber
\end{align}
The channel contribution $A_6^{\text{ch}}$ as defined by Eq.~(\ref{master}) reads
\begin{align}
\label{channeldbi}
&A_6^{\text{ch}}=4\sum_I\frac{(\boldsymbol{p}_a\cdot\boldsymbol{p}_b)(\boldsymbol{p}_d\cdot\boldsymbol{p}_e)(\boldsymbol{p}_c\cdot\boldsymbol{P}_I)(\boldsymbol{p}_f\cdot\boldsymbol{P}_I)}{P_I^0-\boldsymbol{P}_I^2}\\
&=\smashoperator{\sum_{\sigma,\rho\in S_3}}\frac{(\boldsymbol{p}_{\sigma(1)}\cdot\boldsymbol{p}_{\sigma(2)})(\boldsymbol{p}_{\rho(4)}\cdot\boldsymbol{p}_{\rho(5)})(\boldsymbol{p}_{\sigma(3)}\cdot\boldsymbol{k}_{\sigma\rho})(\boldsymbol{p}_{\rho(6)}\cdot\boldsymbol{k}_{\sigma\rho})}{k_{\sigma\rho}^0-\boldsymbol{k}_{\sigma\rho}^2},\nonumber
\end{align}
where $\sigma$ and $\rho$ denote respectively permutations of $\{1,2,3\}$ and $\{4,5,6\}$, and we have used the shorthand notation
\begin{align}
\boldsymbol{k}_{\sigma\rho}\equiv \boldsymbol{p}_{\sigma(1)}+\boldsymbol{p}_{\sigma(2)}-\boldsymbol{p}_{\rho(6)}.
\end{align}
The second line of Eq.~(\ref{channeldbi}) is manifestly equal to the Feynman diagram expression one obtains from Eq.~(\ref{SL4}).
Similarly, the contact contribution to the six-point amplitude follows from Eq.~(\ref{master}) as
\begin{align}
\label{contact1}
A_6^{\text{ct}}&=4\sum_I\sum_{i=1}^6\Res_{z=z_i}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{P}}_I)(\boldsymbol{\hat{p}}_f\cdot\boldsymbol{\hat{P}}_I)}{zF_6^{(2)}(z)\bigl(\hat{P}_I^0-\boldsymbol{\hat{P}}_I^2\bigr)} \nonumber \\
&\equiv -2\sum_I\sum_{i=1}^6f(z_i).
\end{align}
The residues at $z_i\equiv1/a_i$ for a given factorization channel can be rewritten as
\begin{align}
f(z_a)&=\Res_{z=z_a}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_f-\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_b)}{zF_6^{(2)}(z)}, \nonumber\\
f(z_b)&=\Res_{z=z_b}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_f-\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_a)}{zF_6^{(2)}(z)}, \nonumber\\
f(z_c)&=\Res_{z=z_c}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_d+\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_f+\boldsymbol{\hat{p}}_e\cdot\boldsymbol{\hat{p}}_f)}{zF_6^{(2)}(z)}, \nonumber\\
f(z_d)&=\Res_{z=z_d}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_f-\boldsymbol{\hat{p}}_e\cdot\boldsymbol{\hat{p}}_f)}{zF_6^{(2)}(z)}, \nonumber\\
f(z_e)&=\Res_{z=z_e}\frac{(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_b)(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_c\cdot\boldsymbol{\hat{p}}_f-\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_f)}{zF_6^{(2)}(z)}, \nonumber\\
f(z_f)&=\Res_{z=z_f}\frac{(\boldsymbol{\hat{p}}_d\cdot\boldsymbol{\hat{p}}_e)(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_c+\boldsymbol{\hat{p}}_b\cdot\boldsymbol{\hat{p}}_c)(\boldsymbol{\hat{p}}_a\cdot\boldsymbol{\hat{p}}_f+\boldsymbol{\hat{p}}_b\cdot\boldsymbol{\hat{p}}_f)}{zF_6^{(2)}(z)}.\nonumber
\end{align}
After substituting the expressions above into Eq.~(\ref{contact1}), collecting the contributions to the residue at each $z_i$ from all factorization channels, and using (shifted) momentum conservation, we obtain
\begin{align}
A_6^{\text{ct}}=&-\frac{1}{2}\sum_{i=1}^6\Res_{z=z_i}\frac1{zF_6^{(2)}(z)}\\
&\times\smashoperator{\sum_{\sigma,\rho\in S_3}}(\boldsymbol{\hat p}_{\sigma(1)}\cdot\boldsymbol{\hat p}_{\sigma(2)})(\boldsymbol{\hat p}_{\sigma(3)}\cdot\boldsymbol{\hat p}_{\rho(4)})(\boldsymbol{\hat p}_{\rho(5)}\cdot\boldsymbol{\hat p}_{\rho(6)}).\nonumber
\end{align}
A final application of Cauchy's theorem yields
\begin{align}
A_6^{\text{ct}}=\frac{1}{2}\smashoperator{\sum_{\sigma,\rho\in S_3}}(\boldsymbol{p}_{\sigma(1)}\cdot\boldsymbol{p}_{\sigma(2)})(\boldsymbol{p}_{\sigma(3)}\cdot\boldsymbol{p}_{\rho(4)})(\boldsymbol{p}_{\rho(5)}\cdot\boldsymbol{p}_{\rho(6)}),
\end{align}
which is manifestly equal to the contribution from the contact term in Eq.~(\ref{L6}).
\subsection{$A_1$: spatial Galileon}
\label{subsec:galileon}
Our second example includes a whole class of La\-gran\-gians of a real scalar field $\phi$,
\begin{align}
\label{spatialGalileon}
\mathcal{L}=\frac12(\partial_\mu\phi)^2+\sum_{n=3}^{d+1}c_n\phi G_{n-1},
\end{align}
where $c_n$ are real coupling constants and $G_n$ is a polynomial of order $n$ in the second spatial derivatives of $\phi$,
\begin{align}
G_n\equiv{}&\frac1{(d-n)!}\epsilon^{i_1\dotsb i_nk_{n+1}\dotsb k_d}\epsilon^{j_1\dotsb j_n}_{\phantom{j_1\dotsb j_n}k_{n+1}\dotsb k_d}\nonumber\\
&\times(\partial_{i_1}\partial_{j_1}\phi)\dotsb(\partial_{i_n}\partial_{j_n}\phi).
\end{align}
This is a nonrelativistic version of another type of a relativistic single-flavor exceptional theory~\cite{CheungEFT}: the Galileon. As opposed to the usual, Lorentz-invariant Galileon theory~\cite{Gal}, the interaction part of Eq.~(\ref{spatialGalileon}) contains only spatial derivatives of $\phi$. We therefore dub it ``spatial Galileon.'' The action~(\ref{spatialGalileon}) is invariant under polynomial shifts of $\phi$ of first order in spatial coordinates, $\phi\to\phi+\alpha+\boldsymbol\beta\cdot\boldsymbol x$. This spatial version of the usual Galileon symmetry is a special case of a class of ``multipole algebras'' that have recently attracted attention in the context of fracton physics~\cite{fractonGromov}. All interaction terms in Eq.~(\ref{spatialGalileon}) as well as the spatial part of the kinetic term are of the WZW type~\cite{Goon}.
Since the spatial Galileon is a type $A_1$ theory, the validity of the recursion is limited to $n$-point amplitudes with $n\geq d+3$, as shown in section~\ref{subsec:Am}. For illustration, we will now restrict Eq.~(\ref{spatialGalileon}) to the quartic interaction term and show how to reconstruct the six-point amplitude. This requires setting $d=3$, since for $d<3$ the quartic spatial Galileon interaction does not exist.
It is convenient to express the Feynman rule for the $n$-point spatial Galileon vertex as~\cite{GalDual}
\begin{align}
\label{VnGal}
V_n(\boldsymbol p_1,\dotsc,\boldsymbol p_n)=c_n'\smashoperator{\sum_{\sigma\in Z_n}}G(\boldsymbol p_{\sigma(1)},\dotsc,\boldsymbol p_{\sigma(n-1)}),
\end{align}
where $G(\boldsymbol p_1,\dotsc,\boldsymbol p_{n-1})$ is the Gram determinant, that is the determinant of the $(n-1)\times(n-1)$ matrix with entries $\boldsymbol p_i\cdot\boldsymbol p_j$. Importantly, the Gram determinant is a symmetric, homogeneous polynomial of order two in all its arguments,
\begin{align}
\label{gscaling}
G(\lambda\boldsymbol p_1,\dotsc,\boldsymbol p_{n-1})=\lambda^2G(\boldsymbol p_1,\dotsc,\boldsymbol p_{n-1}).
\end{align}
Due to momentum conservation in the vertex, all the contributions to the sum in Eq.~(\ref{VnGal}) are then equal and we can write $V_n=nc_n'G(\boldsymbol p_1,\dotsc,\boldsymbol p_{n-1})$.
The six-point amplitude is now determined in terms of the four-point seed amplitude by Eq.~(\ref{master}),
\begin{align}
\label{galA6}
A_6=\sum_I\Biggl\{&\frac{A_{4L}^{(I)}A_{4R}^{(I)}}{(P_I^0)^2-\boldsymbol{P}_I^2}\nonumber\\
&+\sum_{i=1}^6\Res_{z=1/a_i}\frac{\hat{A}_{4L}^{(I)}(z)\hat{A}_{4R}^{(I)}(z)}{zF^{(2)}_6(z)\bigl[(\hat{P}_I^0)^2-\boldsymbol{\hat{P}}^2_I\bigr]}\Biggr\}.
\end{align}
For a generic permutation $\sigma$ of the external momenta, the numerator in the last term can be cast as
\begin{align}
&V_4(\boldsymbol{\hat p}_{\sigma(1)},\boldsymbol{\hat p}_{\sigma(2)},\boldsymbol{\hat p}_{\sigma(3)},\boldsymbol {\hat P}_I)V_4(\boldsymbol{\hat p}_{\sigma(4)},\boldsymbol{\hat p}_{\sigma(5)},\boldsymbol{\hat p}_{\sigma(6)},\boldsymbol {\hat P}_I)\nonumber\\
&=(4c_4')^2G(\boldsymbol{\hat p}_{\sigma(1)},\boldsymbol{\hat p}_{\sigma(2)},\boldsymbol{\hat p}_{\sigma(3)})G(\boldsymbol{\hat p}_{\sigma(4)},\boldsymbol{\hat p}_{\sigma(5)},\boldsymbol{\hat p}_{\sigma(6)}).
\end{align}
The scaling property~(\ref{gscaling}) of the Gram determinant then ensures that the denominator factor $F^{(2)}_6(z)$ in Eq.~(\ref{galA6}) is canceled. Thus, all the residues inside the second sum in Eq.~(\ref{galA6}) vanish and only the first, ``channel'' term therein survives. This is manifestly equal to the expression for $A_6$ one obtains using Feynman diagrams.
\subsection{$A_2$: Lifshitz scalar with polynomial shift symmetry}
\label{subsec:lifshitz}
Our final example is a so-called $z=2$ Lifshitz theory, which possesses the following kinetic term,
\begin{align}
\mathcal{L}_2=\frac{1}{2}\left(\partial_0\phi\right)^2-\frac{1}{2}\left(\boldsymbol\nabla^2\phi\right)^2.
\end{align}
This Lagrangian is strictly invariant under the spatial Ga\-li\-leon symmetry.\footnote{Lifshitz scalars with polynomial shift symmetries have been classified in Refs.~\cite{HinteShift,HoravaShift} and shown to exhibit rich and surprising features that shed new light on the concept of naturalness in nonrelativistic quantum field theory~\cite{HoravaShift,HoravaNatural}.} We can thus add the spatial Ga\-li\-leon interactions in Eq.~(\ref{spatialGalileon}) to it. The ensuing theory can be viewed as a fine-tuned version of the spatial Galileon where the usual kinetic term proportional to $(\boldsymbol\nabla\phi)^2$ is set to zero.
This is a type $A_2$ theory, so the validity of the recursion is limited to $n$-point amplitudes with $n\geq d+4$ as shown in section~\ref{subsec:Am}. At the same time, the CHMW theorem requires that $d>2$. We thus cannot reconstruct the six-point amplitude by recursion. We can however consider a seed five-point vertex and use recursion to reconstruct the eight-point amplitude. This requires setting $d=4$, since for $d<4$ the quintic spatial Galileon does not exist.
Following the same steps as in the previous example, Eq.~(\ref{master}) then gives the following result for the eight-point amplitude,
\begin{align}
A_8=\sum_I\frac{A_{5L}^{(I)}A_{5R}^{(I)}}{(P_I^0)^2-\boldsymbol{P}_I^4},
\end{align}
which agrees with the Feynman diagram expression.
\section{Outlook}
\label{sec:outlook}
We have derived recursion relations for nonrelativistic EFTs with enhanced soft limits. To the best of our knowledge, this is the first time that on-shell constructibility for theories without Lorentz invariance has been shown.
Beyond providing a new tool for calculating explicit tree-level amplitudes in specific field theories, soft recursion is a key ingredient in the ``soft bootstrap'' program, which explores and classifies the space of possible EFTs. In a paper soon to appear~\cite{us}, we will carry out a more detailed classification of possible seed amplitudes. When combined with soft recursion, this will allow us to perform a scan of the landscape of nonrelativistic EFTs, improving on our previous symmetry-based study~\cite{NRLie}.
Our recursion relations can also be applied to theories with universal albeit not necessarily vanishing soft behavior by following the line of reasoning in Ref.~\cite{LuoWen}. This would require new soft theorems for NG boson amplitudes~\cite{Shifman}, an avenue we leave open for future work.
\section*{Acknowledgements}
T.B.~would like to thank Andreas Helset for a discussion on a related subject. M.A.M.~acknowledges the hospitality of the University of Stavanger, where the majority of the work was done. This work has been supported by the grant no.~PR-10614 within the ToppForsk-UiS program of the University of Stavanger and the University Fund.
|
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| 9,489
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{"url":"http:\/\/mathhelpforum.com\/differential-geometry\/78676-fundamental-theorem-algebravproof.html","text":"# Thread: Fundamental Theorem of AlgebravProof\n\n1. ## Fundamental Theorem of AlgebravProof\n\nWhy do many books go through \"downsizing\" arguments to prove the Fundamental Theorem of Algebra? Isn't this the simplest proof?\n\nTheorem. Every non-constant polynomial with complex coefficients has a zero in $\\mathbb{C}$.\n\nProof. Let $P(z)$ be any polynomial. If $P(z) \\neq 0$ for all $z \\in \\mathbb{C}$, $f(z) = 1\/P(z)$ is an entire function. Furthermore, if $P$ is non-constant, $P \\to \\infty$ as $z \\to \\infty$ and $f$ is bounded. By Liouville's Theorem, $f$ is constant and so $P$, contrary to our assumption.\n\nYet most books make this so complicated. Why?\n\n2. Originally Posted by manjohn12\nWhy do many books go through \"downsizing\" arguments to prove the Fundamental Theorem of Algebra? Isn't this the simplest proof?\n\nTheorem. Every non-constant polynomial with complex coefficients has a zero in $\\mathbb{C}$.\n\nProof. Let $P(z)$ be any polynomial. If $P(z) \\neq 0$ for all $z \\in \\mathbb{C}$, $f(z) = 1\/P(z)$ is an entire function. Furthermore, if $P$ is non-constant, $P \\to \\infty$ as $z \\to \\infty$ and $f$ is bounded. By Liouville's Theorem, $f$ is constant and so $P$, contrary to our assumption.\n\nYet most books make this so complicated. Why?\nthere are many proofs of the fundamental theorem of algebra [most of them require knowledge of topics outside of abstract algebra].\n\nthe proof learned in algebraic topology is, in my opinion, the most straightforward. It is actually a very quick proof.\n\n3. Originally Posted by manjohn12\nWhy do many books go through \"downsizing\" arguments to prove the Fundamental Theorem of Algebra? Isn't this the simplest proof?\n\nTheorem. Every non-constant polynomial with complex coefficients has a zero in $\\mathbb{C}$.\n\nProof. Let $P(z)$ be any polynomial. If $P(z) \\neq 0$ for all $z \\in \\mathbb{C}$, $f(z) = 1\/P(z)$ is an entire function. Furthermore, if $P$ is non-constant, $P \\to \\infty$ as $z \\to \\infty$ and $f$ is bounded. By Liouville's Theorem, $f$ is constant and so $P$, contrary to our assumption.\n\nYet most books make this so complicated. Why?\nIn order to understand this proof you need to be familar with complex analysis while this result is an algebraic result. Thus, some books feel that algebraic results deserve algebraic proofs. I am familar with another proof that uses Galois theory, however it is little bit more involved, but still nice.\n\n4. Originally Posted by ThePerfectHacker\nIn order to understand this proof you need to be familar with complex analysis while this result is an algebraic result. Thus, some books feel that algebraic results deserve algebraic proofs. I am familar with another proof that uses Galois theory, however it is little bit more involved, but still nice.\n\nBut most complex analysis books still use a really algebraic approach to proving the Fundamental Theorem of Algebra.\n\n5. Originally Posted by manjohn12\nBut most complex analysis books still use a really algebraic approach to proving the Fundamental Theorem of Algebra.\nHow? What books?\n\n6. Originally Posted by ThePerfectHacker\nHow? What books?\nBak and Newman use above proof. I don't recall the other books which use the complicated proofs.\n\nBut I know triangle inequalities and stuff like that.\n\n7. Originally Posted by manjohn12\nBak and Newman use above proof. I don't recall the other books which use the complicated proofs.\n\nBut I know triangle inequalities and stuff like that.\nThat proof is analytic! You are using a fact about analytic functions on the complex plane which is proven using the knowledge of contour integration in the complex plane. This is not in any way an algebraic proof, it is 100% analytic.\n\n8. What you post is a \"complicated\" proof: its complications are just hidden in \"Liouville's Theorem\".\n\nThe simplest proof that I know only requires knowing that a complex number can be written in exponential form. Of course it takes quite a lot computation. Is that what you mean by \"complicated\"?\n\n9. Originally Posted by ThePerfectHacker\nThat proof is analytic! You are using a fact about analytic functions on the complex plane which is proven using the knowledge of contour integration in the complex plane. This is not in any way an algebraic proof, it is 100% analytic.\n\nTheorem. Let $P_{n}(z) = a_{z}z^{n}+ a_{n-1}z^{n-1} + \\ldots + a_{1}z + a_0$. Then $P_{n}(z)$ has at least one zero.\n\nProof. Assume to the contrary. Then $f(z) = \\frac{1}{P_{n}(z)}$ is a bounded entire function, hence constant by Liouville's Theorem. Contradiction. We have:\n\n$P_{n}(z) = a_{n}z^{n}+ a_{n-1}z^{n-1} + \\cdots + a_{1}z + a_0$\n\n$\\therefore |P_{n}(z)| = |z^n| \\cdot \\left|a_{n}+ \\frac{a_{n-1}}{z} + \\frac{a_{n-2}}{z^2} + \\cdots + \\frac{a_{1}}{z^{n-1}} + \\frac{a_0}{z^n} \\right|$\n\n$\\geq |z^n| \\left( |a_n| - \\left|\\frac{a_{n-1}}{z} + \\frac{a_{n-2}}{z^2} + \\cdots + \\frac{a_{1}}{z^{n-1}} + \\frac{a_0}{z^n} \\right| \\right)$\n\n$\\geq |z^n| \\left( |a_n|- \\left(\\underbrace{\\frac{|a_{n-1}|}{|z|} + \\frac{|a_{n-2}|}{|z^2|} + \\cdots + \\frac{|a_{1}|}{|z^{n-1}|} + \\frac{|a_0|}{|z^n|}}_{Q} \\right) \\right)$\n\nConsider two regions in the complex plane. The region inside and on the circle $|z| = R$ and the region outside the circle $|z| = R$. We choose $R$ so large such that $Q \\leq \\frac{|a_n|}{2}$ for all $z$.\n\n$\\geq |z^n| \\left(|a_n|- \\frac{|a_n|}{2} \\right) = |z^n| \\frac{|a_2|}{2}$\n\n$\\therefore |f(z)| = \\frac{1}{P_{n}(z)|} \\leq \\frac{1}{|z^n| \\frac{|a_n|}{2}} \\leq \\frac{1}{R^{n} \\frac{|a_n|}{2}} = M$\n\nSo $f(z)$ is a bounded entire function and constant.\n\nWhy don't Bak and Newman do this?\n\n10. Originally Posted by manjohn12\nTheorem. Let $P_{n}(z) = a_{z}z^{n}+ a_{n-1}z^{n-1} + \\ldots + a_{1}z + a_0$. Then $P_{n}(z)$ has at least one zero.\n\nProof. Assume to the contrary. Then $f(z) = \\frac{1}{P_{n}(z)}$ is a bounded entire function, hence constant by Liouville's Theorem. Contradiction. We have:\n\n$P_{n}(z) = a_{n}z^{n}+ a_{n-1}z^{n-1} + \\cdots + a_{1}z + a_0$\n\n$\\therefore |P_{n}(z)| = |z^n| \\cdot \\left|a_{n}+ \\frac{a_{n-1}}{z} + \\frac{a_{n-2}}{z^2} + \\cdots + \\frac{a_{1}}{z^{n-1}} + \\frac{a_0}{z^n} \\right|$\n\n$\\geq |z^n| \\left( |a_n| - \\left|\\frac{a_{n-1}}{z} + \\frac{a_{n-2}}{z^2} + \\cdots + \\frac{a_{1}}{z^{n-1}} + \\frac{a_0}{z^n} \\right| \\right)$\n\n$\\geq |z^n| \\left( |a_n|- \\left(\\underbrace{\\frac{|a_{n-1}|}{|z|} + \\frac{|a_{n-2}|}{|z^2|} + \\cdots + \\frac{|a_{1}|}{|z^{n-1}|} + \\frac{|a_0|}{|z^n|}}_{Q} \\right) \\right)$\n\nConsider two regions in the complex plane. The region inside and on the circle $|z| = R$ and the region outside the circle $|z| = R$. We choose $R$ so large such that $Q \\leq \\frac{|a_n|}{2}$ for all $z$.\n\n$\\geq |z^n| \\left(|a_n|- \\frac{|a_n|}{2} \\right) = |z^n| \\frac{|a_2|}{2}$\n\n$\\therefore |f(z)| = \\frac{1}{P_{n}(z)|} \\leq \\frac{1}{|z^n| \\frac{|a_n|}{2}} \\leq \\frac{1}{R^{n} \\frac{|a_n|}{2}} = M$\n\nSo $f(z)$ is a bounded entire function and constant.\n\nWhy don't Bak and Newman do this?\nThis is my third time telling you, this is still not an algebraic proof. You are using the theory of entire functions - analytic results. Bak and Newman do this, not exactly in the way you wrote it, but their proof is standard.\n\n11. Originally Posted by ThePerfectHacker\nThis is my third time telling you, this is still not an algebraic proof. You are using the theory of entire functions - analytic results. Bak and Newman do this, not exactly in the way you wrote it, but their proof is standard.\nOk. So I guess they dont to the trouble to show that it is bounded.\n\n12. The interesting thing about the fundamental theorem of algebra is that it is not so fundamental.\n\n13. Originally Posted by ThePerfectHacker\nThe interesting thing about the fundamental theorem of algebra is that it is not so fundamental.\nIt is fundamental in the sense like the fundamental theorem of arithmetic. E.g. the building blocks of numbers are primes.","date":"2017-07-25 21:18:15","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\": 63, \"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\": 0.8420611023902893, \"perplexity\": 216.90134037679812}, \"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-2017-30\/segments\/1500549425381.3\/warc\/CC-MAIN-20170725202416-20170725222416-00309.warc.gz\"}"}
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We provide comprehensive advice in all areas of European and Austrian competition law, in which Ferdinand Graf and his team have extensive experience. We represent domestic and foreign companies in proceedings before the Austrian Cartel Court, the Austrian Federal Competition Authority and European Institutions. Our areas of expertise include: merger control filings, abuse of market power dominance proceedings, and advice on drafting contractual agreements & structuring of distribution systems.
We have industry-specific experience and know-how in the following areas: construction, timber and sawmill, pharmaceutical, media and technology, banking, food and beverage, transport and logistics and renewable energy. Our most recent work includes advising the buyer on the acquisition of the majority shareholding in the publishing group Verlagsgruppe News from Gruner + Jahr and representation in the highly publicized proceedings concerning the "Construction Cartel" (alleged forbidden cartel agreements relating to tramway construction).
We represented clients from the timber and sawmill industry in proceedings concerning illegal price agreements, as well as regional banks in market power dominance proceedings. Furthermore, we advised in high profile merger proceedings involving the acquisition of a solar panels manufacturer and involving the acquisition of the mineral water producer Römerquelle. In terms of private enforcement actions, we represented injured parties with regard to damage claims following the "Elevator Cartel" case.
Our out of court work includes advising clients on contracts and distribution structures with respect to compliance with Austrian and European cartel law and developing individual compliance programs for clients along with employee training and compliance guidelines.
Ferdinand Graf is 'an excellent negotiator', who has 'a good understanding of the energy sector'.
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David Soto Rodríguez (Irún, 7 de junio de 1990) es un abogado y político vasco, Diputado en el Parlamento Vasco, portavoz de Podemos en el Parlamento Vasco y Secretario de Organización de Podemos Ahal Dugu.
Biografía
Se licenció en Derecho en la Universidad del País Vasco. Hizo un Máster en Asesoría Fiscal en la Universidad de Deusto y Máster en Abogacía en la Universidad del País Vasco.
Fue cabeza de lista y candidato a la alcaldía de Podemos al Ayuntamiento de Irún en las elecciones municipales de España de 2015, saliendo elegido concejal. Como concejal fue el portavoz del Grupo Municipal Podemos en el Ayuntamiento. Fue elegido Secretario General de Podemos en Irún. Repitió como candidato a la alcaldía en las elecciones municipales de España de 2019, siendo elegido concejal por segunda vez.
En el año 2020 se unió al equipo de Miren Gorrotxategi, después de que esta ganase las primarias de Podemos Ahal Dugu, y fue nombrado Secretario de Organización de Podemos Ahal Dugu, con Pilar Garrido como coordinadora.
Fue cabeza de lista de Elkarrekin Podemos por Guipúzcoa en las elecciones al Parlamento Vasco de 2020, saliendo elegido como diputado en el Parlamento Vasco.
Referencias
Alumnado de la Universidad del País Vasco
Alumnos de Derecho de la Universidad del País Vasco
Podemos
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Frontiers in Plant Science
Open Supplemental Data
Methods ARTICLE
Front. Plant Sci., 02 May 2011 | https://doi.org/10.3389/fpls.2011.00013
Real-time imaging of leaf apoplastic pH dynamics in response to NaCl stress
Christoph-Martin Geilfus and Karl H. Mühling*
Institute of Plant Nutrition and Soil Science, Christian Albrechts University, Kiel, Germany
Knowledge concerning apoplastic ion concentrations is important for the understanding of many processes in plant physiology. Ion-sensitive fluorescent probes in combination with quantitative imaging techniques offer opportunities to localize, visualize, and quantify apoplastic ion dynamics in situ. The application of this technique to the leaf apoplast is complicated because of problems associated with dye loading. We demonstrate a more sophisticated dye loading procedure that enables the mapping of spatial apoplastic ion gradients over a period of 3 h. The new technique has been used for the real-time monitoring of pH dynamics within the leaf apoplast in response to NaCl stress encountered by the roots.
The apoplast is of major significance for the nutrition of higher plants (Sattelmacher et al., 1998). Knowledge of the apoplastic ion concentrations is necessary for the understanding of many transport processes across the plasma membrane, of membrane potentials, and of cell expansion. The activity of extracellular enzymes, the binding of ligands to receptors, and the wall structure are all likely to be affected by the apoplastic environment, which is known to be a polarized microenvironment for selective ion binding (Grignon and Sentenac, 1991; Canny, 1995; Mühling and Sattelmacher, 1995; Fricker et al., 1999). Because of the quantitative importance of ions and the impact of their apoplastic distribution on metabolic regulation, the localization, and quantification of ions is important for the understanding of the nutritional physiology of higher plants (Grignon et al., 1997).
The combination of ion-sensitive fluorescent probes with quantitative imaging techniques provides opportunities to localize, visualize, and quantify ion dynamics in situ via the technique of ratio imaging (Fricker et al., 1997a, 2008). Use of ratiometric analysis allows high temporal and spatial resolution with a minimum of perturbation to be achieved (Fricker et al., 1999). Although ratio imaging has been successfully employed to measure cytoplasmic calcium and pH in plants (as summarized by Fricker et al., 1994), little information is available on the application of this technique to the measurement of apoplastic ion concentrations (Taylor et al., 1996). The method has however been successfully used to measure apoplastic pH (Hoffmann et al., 1992; Hoffmann and Kosegarten, 1995; Mühling et al., 1995; Taylor et al., 1996; Bibikova et al., 1998; Mühling and Läuchli, 2000; Yu et al., 2001; Pitann et al., 2009) and, to a lesser extent, apoplastic potassium (Mühling and Sattelmacher, 1997; Mühling and Läuchli, 2000), calcium (Mühling et al. 1998), and sodium (Mühling and Läuchli, 2002). To the authors' knowledge, no ratiometric, non-transgenic in planta measurements are available that show leaf apoplastic ion dynamics by using ratio imaging. This is mainly attributable to problems associated with gaining access to the intact leaf apoplast (Fricker et al., 1994). The major difficulty here remains the development of appropriate loading protocols to introduce the fluorescent ion probe into the apoplast of a living plant without destroying the cell membranes, which in consequence would allow the dye to diffuse into the symplast, leading to an erroneous quantification of cytosolic pH signals. However, in planta analysis is important because it assures that the apoplast operates in a correct physiological context and hence includes the effects of cell–cell interactions and the mechanical, ionic, and physiological effects of the extracellular matrix of cell–wall interactions (Errington et al., 1997; Fricker et al., 1997b).
Here we present a non-invasive approach for loading a fluorescent ion indicator into the leaf apoplast of an intact plant (Vicia faba L.). In combination with camera-based inverse fluorescence microscopy, ratiometric in planta mapping of ion dynamics in various apoplastic components was achieved over a period of hours.
Cultivation of Plant Material
Vicia faba L., minor cv. Scirocco (Saaten-Union GmbH, Isernhagen, Germany) was grown under hydroponic culture conditions in a climate chamber (14/10 h day/night; 20/15°C; 50/60% humidity). Seeds were soaked in an aerated CaSO4 solution (0.5 mM) for 1 day at 25°C and subsequently placed into quartz sand moistened with CaSO4 (1 mM). After 7 days of germination, seedlings were transferred to plastic pots containing one-quarter-strength aerated nutrient solution. Following 2 days of cultivation, the concentration of nutrients was increased to half-strength and, after 4 days of cultivation, to full-strength. The nutrient solution had the following composition: 0.1 mM KH2PO4, 1.0 mM K2SO4, 0.2 mM KCl, 2.0 mM Ca(NO3)2, 0.5 mM MgSO4, 60 μM Fe–EDTA, 10 μM H3BO4, 2.0 μM MnSO4, 0.5 μM ZnSO4, 0.2 μM CuSO4, 0.05 μM (NH4)6Mo7O24. The solution was changed every third day to avoid nutrient depletion. After 30 days of plant cultivation, intact growing leaves with an average leaf area of 7 cm2 were sampled for in situ pH recording.
Dye Loading
For the purpose of ratiometric in planta measurements, the fluorescent pH indicator was loaded into the leaf apoplast of an intact plant. For this, the opening of a syringe was carefully pressed onto the abaxial leaf side (no needle was attached on the top of the syringe). By means of gentle pressure, 50 μL of 25 μM Oregon Green 488 dextran (Invitorgen GmbH, Darmstadt, Germany; dissolved in deionized water) were fed into the apoplast, whereby not the complete, but only a small part of the apoplast was loaded. The loading procedure could easily be monitored because the loaded area appeared darker than its surroundings (see Figure A1 in Appendix). In order to avoid measurement artifacts caused by the possible impact of the loading procedure on leaf physiology, images were collected (1) adjacent to the area that had been loaded and (2) at the adaxial leaf side. This was possible since the dye proved to be mobile within the apoplast (described in detail in section Results and Discussion).
Inverse Microscopy Imaging
A Leica inverted microscope (DMI6000B; Leica Microsystems, Wetzlar, Germany) connected to a DFC-camera (DFC 360FX; Leica Microsystems, Wetzlar, Germany) via a 20-fold magnification, 0.4 numerical aperture, dry objective (HCX PL FLUOTAR L, Leica Microsystems, Wetzlar, Germany) was used for image collection. An HXP lamp (HXP Short Arc Lamp; Osram, München, Germany) was used for illumination at excitation wavelengths of (ex) 440/20 and 495/10 nm. Exposure time was 25 mS for both channels. Light intensity and camera gain were coupled for both channels, ensuring identical excitation conditions. Excitation filters were switched by means of a filter wheel. The dye fluorescence at both excitation channels was collected by using a 535/25 nm emission band-pass filter (BP 535/25; ET535/25M; Leica Microsystems, Wetzlar, Germany) and a dichromatic mirror (LP518; dichroit T518DCXR BS, Leica Microsystems, Wetzlar, Germany). Time series were collected with a time interval of 2 or 5 min. During the measurements, leaves were not detached from the plant and the plants were supplied with aerated nutrient solution. The use of microscopy-based techniques for mapping ion concentrations in the leaves of living plants over several hours involves the problem that, from time to time, the specimen shifts out of the focal plane. This can be ascribed to the growth of the stem or the leaf; the latter growth can attain 4–6.25 mm2/h (Dennett et al., 1978). In order to avoid any shifting of the specimen, care must be taken that the region of interest is firmly attached to the measuring device.
Ratiometric Analysis
Fluorescence ratio analysis corrects quantitative fluorescence imaging with regard to artifacts in signal strength associated with sample path length, dye distribution, leakage, and photobleaching (Bright et al., 1989; Gilroy, 1997). As a measure of pH, the fluorescence ratio F495/F440 (Pitann et al., 2009) was obtained by using the pH-sensitive fluophore Oregon Green 488 that is conjugated to 10 kDa dextran. The F440 signal was captured because this fluorescence is almost insensitive to protons, whereas the F495 signal highly depends on protons. An analysis of a time series was carried out by using Leica Application Suite Advanced Fluorescence, (LAS AF software, version 2.3.5, Leica Microsystems, Wetzlar, Germany). Ratio images were calculated on a pixel-by-pixel basis as F495/F440. The background noise values were subtracted at each channel. For pseudo-color display, the ratio was coded by hue on a spectral color scale ranging from purple (no signal), over blue (lowest pH signal; pH 3.9), to pink (highest pH signal; pH 6.3), with the limits being set by an in situ calibration. Ratios below 1.1 (corresponds to pH ≤ 3.9) and above 3.6 (corresponds to pH ≥ 6.3) were not considered, because they proved to lie outside the linear range of the in situ calibration (Figure A2 in Appendix). Quantitative measurements were calculated as the ratio of the mean intensity for user-defined regions of interest (ROIs).
In situ Apoplastic pH Calibration
For converting fluorescence ratio data taken from living plants into apoplastic pH values, an in situ calibration procedure was performed. Hence, 25 μM Oregon Green dye solutions buffered with 100 mM (2-[N-morpholino]ethanesulfonic acid, MES) to a pH ranging from 3 to 7 (steps of 0.5 pH units) were loaded into the leaf apoplast. The Boltzmann fit was chosen for fitting sigmoidal curves to calibration data (Figure A2 in Appendix) as described by Schulte et al. (2006). Fitting was performed by using Origin 7.0 (OriginLab Corp., Northampton, MA, USA).
Confocal Laser Scanning Microscopy
To demonstrate that no Oregon Green 488 dextran had entered the cytosol unintentionally, CLSM-imaging via a Leica TCS SP1 confocal laser scanning system (Leica Microsystems, Wetzlar, Germany) was carried out. For dye excitation, the 488 nm beam line of the Argon laser was chosen. The dye fluorescence was collected by using an emission bandwidth at 514–556 nm (green channel). A planapochromatic objective (HC PL APO CS 10.0 × 0.40; Leica Microsystems, Wetzlar, Germany) was used for image collection. The three-dimensional XYZ-image stack was created on the basis of 30 xy sections. Step size was 2.04 μm.
Gas Exchange and Spad Readings
Gas-exchange parameters such as stomatal conductance to H2O (mol H2O m−2 s−1) were measured with an open-flow gas-exchange system (portable photosynthesis system; LI-COR Biosystems GmbH, Bad Homburg, Germany) with an integrated fluorescence chamber head (LI-COR Biosystems GmbH, Bad Homburg, Germany). Leaves were placed across a 2 × 3 cm leaf cuvette. The conditions for the measurements inside the chamber were equal to the outside conditions in the climate chamber. Light was provided by an LED red light source built into the top of the leaf chamber (100 μmol quanta m−2 s−1) and the CO2 concentration was controlled by a Li-Cor LI-6400 CO2 injection system. Stomatal conductance was calculated by the internal software. The relative chlorophyll concentration of the leaves was measured with a portable chlorophyll meter (SPAD-502, Minolta, Japan). At least fifteen SPAD-502 readings for each leaf were averaged by the internal software of the SPAD-502 meter and were taken as a single data point for each biological replicate.
Leaf Area
Leaf area was calculated by means of measuring the length and maximum breadth of the leaflet, whereas lengths and breadths were converted to area by multiplying their product by 0.764 according the formula given by Dennett et al. (1978). Leaf area (cm2) was chosen as a measure for leaf growth.
Data Evaluation
The t-test and the paired t-test, as outlined by Köhler et al. (1984), were used to test for differences between mean.
Dye Loading Procedure
In planta measurements of apoplastic ion dynamics by using microscopy-based ratio analysis require that the ion indicator is inserted into the leaf apoplast of an intact leaf. Traditional dye loading procedures, such as the time-consuming transpiration-driven loading technique via the petiole or the pressure loading technique of cut leaf discs, do not allow the in situ monitoring of ion concentrations over an extended period of time. In both cases, such monitoring is not possible because the leaves are detached from the plant. However, by gently feeding the ion indicator into the leaf apoplast with the above-described method (see section Materials and Methods), the dye can be inserted into an intact plant system. A prerequisite for in situ studies of the apoplastic microenvironment is that the ion indicator is linked to a dextran molecule that avoids compartmentalization of the indicator into the symplast. Inverse fluorescence microscopy images at ex 495 and ex 440 nm (Figure 1) and CLSM images at ex 488 nm (XYZ-image stack; Movie S1 in Supplementary Material) indicate that the dye does not enter the cells. If the dye had entered the tissue unintentionally, then the dye signals would be emitted from the cells. However, the dye signals solely have their origin in the apoplastic space, whereas the tissue appears dark, indicating that no fluorescence signal is emitted from the cells themselves (Figures 1A,B; Movie S1 in Supplementary Material). Immediately after the dye is loaded, the flooded apoplastic area is filled with water. This represents an artificial surrounding for the plant because, under normal physiological conditions, the apoplastic fluid is only a thin film (Felle, 2001). For this reason, measurements should not be started before the water has disappeared through the stomata of the intact leaves. Evidence that the loaded water can exit the plant apoplast is demonstrated in Figure 2 and in Figure A1 in Appendix. The water droplets on the bright-field images in Figures 2B-i and C-i strongly suggest that the water has left the apoplast through the stomata. The corresponding fluorescent images in Figures 2B-ii and C-ii demonstrate that no dye exits the apoplast together with the water droplets. Otherwise, these droplets would appear in pseudo-green when illuminated with ex 495 nm. The finding that the apoplastic space becomes free of excess water is important for the ratio analysis, because (1) the presence of water within the apoplast causes an acute lack of oxygen to which all the involved cells respond and (2) the water would dilute the apoplastic ion concentration (Felle and Hanstein, 2002). Another point that needs to be considered is the potential mechanical stress to which the leaf is exposed during dye loading. Although only small volumes (max. 50 μL/leaf apoplast) are gently fed into a part of the leaf apoplast, the fluid might interact through adhesive forces with the cell walls and plasma membranes and might thus stretch-activate channels that then release ions and organic acids (Felle and Hanstein, 2002). Hence, images are collected (1) at the earliest at 1.5 h after loading and (2) not at the dye-loaded area itself but adjacent to the loaded area. The latter is possible because the dye is mobile within the leaf apoplast of the living plant and is carried to the faces of the leaf edges (Figure 3). The mobility of the dye suggests possible alterations of the initial dye concentration (25 μM) within the apoplast. We have tested whether differences in dye concentration ranging from at least 10–150 μM Oregon Green 488 dextran still produce stable F495/F440 ratios. At all concentrations, the ratios are constant at 2.5 (Table 1). Moreover, changes in dye concentration do not represent a major problem, because differences in local indicator concentrations are corrected by mean of the ratio analysis (Gilroy, 1997).
Figure 1. Distribution of the dextran-conjugated dye Oregon Green 488 within the leaf apoplast of Vicia faba L. directly after loading. (A) Cross-section. Overlay of ex 495 nm pseudo-green fluorescence image and ex 440 nm pseudo-red fluorescence image. a, adaxial epidermis cell; b, palisade cell; c, spongy cell; d, abaxial epidermis cell; e, stomatal apparatus. (B) Adaxial leaf face. Pseudo-green fluorescence image at ex 495 nm. a, stomatal cavity; b, palisade cell. Tissue in (A) and (B) appears dark indicating that no fluorescence signals are emitted. This demonstrates that no dye has entered the cells. In contrast, the apoplastic space appears in pseudo-color because of the dye emission.
Figure 2. Excessive water exits the apoplast through the stomata. Bright-field images of the adaxial leaf face (gray-scaled; A-i, B-i, and C-i) and corresponding pseudo-green fluorescence images at ex 495 nm (A-ii, B-ii, and C-ii). Images were collected (A) immediately after dye loading and at (B) 2 min and (C) 10 min after loading. (A-i, -ii) Immediately after loading, no water droplets are found to be located onto the stomata. (B-i) Water droplets that had left the apoplast through the stomata 2 min after dye insertion are indicated by #. (B-ii) The corresponding fluorescence image at ex 495 nm reveals that these droplets did not contain any dye, as otherwise fluorescence signals would be emitted at ex 495 nm and appear in pseudo-green. (C-i and C-ii) After 10 min, more water had left the apoplastic space as indicated by #. a, stomatal cavity; b, palisade cell.
Figure 3. Distribution of the dextran-conjugated dye Oregon Green 488 at 1.5 h after the loading event. Adaxial leaf face. Overlay of pseudo-green fluorescence image at ex 495 nm and pseudo-red fluorescence image at ex 440 nm. Images were collected at the leaf edges adjacent to the area that was loaded with the dye. The detection of dye emission signals at an area that was not loaded with the dye itself indicates that the dye is mobile within the leaf apoplast. This allows apoplastic pH to be measured in areas of the leaf apoplast not affected by the dye loading. a, stomatal apparatus; b, epidermal apoplast; c, palisade cell (dark structure; no dye signal); d, apoplast surrounding palisade mesophyll.
Table 1. Effects of various Oregon Green 488 concentrations on the F495/F440 ratio.
In planta Real-Time Imaging of Leaf Apoplastic pH Dynamics
To demonstrate that the improved dye loading procedure enabled in planta monitoring and the quantification of apoplastic pH dynamics in real-time, an intact Vicia faba leaf was prepared with the pH indicator. Subsequently, the intact leaf was placed on the measuring device. Throughout the experiment, the plant itself was well supplied with aerated nutrient solution. An effect of the leaf apoplastic pH was induced by adding 50 μL of a 10 mM NaCl solution onto the surface of a Vica faba leaf. Within 5 min, leaf pH within the stomatal cavity, the epidermal apoplast, and the apoplast surrounding the palisade tissue transiently increased, whereas the pH within the palisade apoplast became the most alkaline (Figure 4). After 25 min, the pH began to normalize and finally reached the range at which the alkalinzation had started before the leaves were treated with NaCl stress. This time course demonstrated the suitability of this dye loading approach for quantitatively monitoring leaf apoplastic pH dynamics. In general, such monitoring is also possible with ion-sensitive micro-electrodes. However, micro-electrodes take measurements at a single point, i.e., the stomatal cavity, whereas microscopy-based techniques enable quantitative data to be averaged over a larger area of the leaf apoplast. Furthermore, the high resolution of these microscopy-based imaging analyses allows spatial ion gradients to be discriminated in various apoplastic components (Figure 4). Therefore, the informative value of a single measurement now increases since the data provide more information about the spatial concentration and the temporal course of the ion of interest. For this reason, a single measurement enables for a more detailed and accurate detection of the stress responses, which in consequence, facilitates the physiological interpretation of the data.
Figure 4. Ratiometric real-time quantitation of leaf apoplastic pH in response to the addition of 50 μL of a 10 mM NaCl solution onto the leaf. (A) pH as recorded at the adaxial face of Vicia faba leaves is plotted over time. Black arrow indicates the time of the addition of the 10 mM NaCl stress stimulus onto the surface of the leaf. Leaf apoplastic pH was discriminated within three apoplastic components, viz. in the stomatal cavity (n = 10 ROI; green kinetic; mean ± SE of ROIs), in the epidermal apoplast (n = 20 ROI; red kinetic; mean ± SE of ROIs) and in the apoplast surrounding the palisade mesophyll (n = 20 ROI; blue kinetic; mean ± SE of ROIs). Ratiometric images (B–D) show the time series of apoplastic leaf pH at (B) 23, (C) 31, and (D) 43 min after measurement had started (time of image acquisition is presented in the upper right corner of the ratio images). Ratios were color-coded on a spectral color scale (see lookup-table as inset in B). Representative kinetics of five equivalent recordings of plants gained from independent experiments.
In a second experiment, we demonstrated that the new dye loading procedure also enabled in planta monitoring of leaf apoplastic pH dynamics in response to a NaCl stress treatment applied to the roots. NaCl was added to the nutrient solution yielding a concentration of 20 mM (Figure 5). The pH transiently increased over a period ranging from 60 to 70 min, with an alkalinization occurring in the three apoplastic compartments, viz. the stomatal cavity, the epidermal apoplast, and the palisade apoplast, starting circa 20 min after NaCl was added to the roots. Similar to the transient alkalinization described in Figure 4A, the pH was the most alkaline within the apoplast built by the palisade tissue. To the authors' knowledge, this is the first time that a physiological stress response that was induced by adding NaCl to the roots has been ratiometrically monitored within the intact leaf apoplast in a resolution that allows spatial ion gradients to be discriminated in various apoplastic components over a period of 3 h in planta.
Figure 5. Ratiometric real-time quantitation of spatial leaf apoplastic pH gradients in response to 20 mM NaCl stress added to the roots. (A) Leaf apoplastic pH response was discriminated within three apoplastic components, viz. in the stomatal cavitiy (n = 9 ROI; green kinetic; mean ± SE of ROIs), in the epidermal apoplast (n = 20 ROI; red kinetic; mean ± SE of ROIs) and in the apoplast surrounding the palisade mesophyll (n = 20 ROI; blue kinetic; mean ± SE of ROIs). Black arrow indicates the time of the addition of the 20 mM NaCl stress stimulus into the nutrient solution. Ratiometric images (B–E) show the time series of apoplastic leaf pH at (B) 25, (C) 50, (D) 110 min, and (E) 170 min after measurement had started (time of image acquisition is presented in the upper right corner). Ratios were color-coded on a spectral color scale (see lookup-table as inset in B). The ratio images under the most alkaline conditions suggest that fluorescent signals come directly from the palisade cells as indicated by the # in (D). This phenomenon is explained in section "Discussion." Representative kinetics of four equivalent recordings of plants gained from independent experiments.
The pseudo-color ratios presented in Figures 4 and 5 require additional comment. The ratio images at the most alkaline conditions suggest that fluorescent signals come directly from within the palisade cells (indicated by # in Figure 5D). This implies that, in addition to the apoplastic pH signals, cytosolic pH signals have been quantified erroneously. Such an idea can however be rejected for several reasons. (1) A XYZ-image stack showing the leaf apoplastic space that was labeled with Oregon Green 488 dextran was created using CLSM. Dye excitation with an Argon laser line at 488 nm (Movie S1 in Supplementary Material) clearly demonstrated that the dye signals were solely emitted from the apoplast. The mesophyll cells appeared dark, providing evidence that no dye had unintentionally entered the cells. Thus the appearance of cytosolic or cellular pH signals can be rebutted. (2) The pH probe is covalently linked to a 10 kDa dextran, which would require pressure microinjection for loading into the cytoplasm (Fricker et al., 1999). (3) The cytosolic pH is known to be around 7.2 (Schwarzländer et al., 2008; D'Onofrio and Lindberg, 2009), whereas the quantified pH-values shown in Figures 4A and 5A do not exceed a value of pH 5.4 and 5.6, respectively. (4) Even supposing the dye had entered the cytosol unintentionally, the apoplastic pH response shown in Figures 4 and 5 could not drop back to its initial level at pH 4.2, because the cytosol never reaches such an acid milieu. Nevertheless, why do the ratio images at the most alkaline conditions suggest that fluorescent signals come directly from within the palisade cells (indicated by # in Figure 5D)? Immediately after dye was loaded into the apoplast, the inserted water exits the leaf through the stomata (shown in Figure 2). However, the dye has been demonstrated to remain within the apoplast (Figures 2A-ii, B-ii, and C-ii). We presume that the dye is dissolved within the apoplastic fluid, which is a thin film that is attached to the surface of the cells (Felle and Hanstein, 2002). Consequently, the fluorescent dye is closely attached around the palisade cells, thus, light that is emitted by the dye mimics the shape of the cells. However, in response to a physiological alkalinization of the apoplastic fluid that surrounds the palisade cells, the intensity of the emitted light raises, because the emission triggered by the F495 channel increases with decreasing [H+]. This strong F495 fluorescence erroneously conveys (see above) the impression that these signals arise within the cell (indicated by # in Figure 5D).
Potential artifacts
Several potential artifacts might occur during ratio imaging experiments and, if not considered, might introduce errors in quantitation. A point that needs to be considered during imaging is the level of background signals, viz. autofluorescence coming from the measuring devices (i.e., lens elements), the specimen (i.e., cell wall or chloroplasts), the shot noise associated with sampling of the signal (Fricker et al., 1997b, 2001), and the noise arising from residual light in the laboratory (i.e., computer LEDs, monitor screens). For the testing of the level of autofluorescence, the specimen without the dye is illuminated (background signal intensity) and compared with the specimen plus dye (total signal intensity). Only negligible noise has been detected (0.1% of the weakest fluorescence signals and 0.03‰ of the strongest; Figure 6). The background noise values are subtracted at each channel. In addition to the occurrence of background fluorescence, confirmation is required that the NaCl molecules themselves do not influence the F495/F440 ratio and thus the quantification. Such an effect might occur because the NaCl that is added to the nutrient solution might be transported through the xylem into the leaf apoplast where it might interact with the fluorophore. In order to exclude such an effect, we have demonstrated that NaCl concentrations ranging from 0 to 125 mM do not affect the Oregon Green 488 ratio (Figure 7). Moreover, we have tested whether the presence of NaCl within the leaf apoplast increases background noise signal intensity when illuminated with ex 495 and ex 440 nm. After the loading of NaCl solution directly into the leaf apoplast, no additional unspecific noise signals have been detected (Figure A3 in Appendix). Another problem relates to the physiological response that the excitation or emission wavelengths used for imaging might trigger in the specimen (Fricker et al., 1994). To check whether the illumination regime itself influences the apoplastic pH, a time series of ex 495 and ex 440 nm images has been collected. No effects on apoplastic pH have been detected (Figure 8) supporting the proposal that the transient pH increases demonstrated in Figures 4 and 5 are solely attributable to the salt treatment. Furthermore, the dye loading procedure represents a potential event of (local) mechanical stress and the presence of the dye itself within the apoplast might have an impact on the apoplastic space. Hence, leaf growth, stomatal conductance, and chlorophyll concentrations have been compared in loaded leaves and in leaves not loaded with ion indicator (control leaves). None of the tested parameters is influenced (Figure 9). The monitoring of the physiological ion dynamics as demonstrated in Figure 5 requires 3 h. Altogether, 36 fluorescence images at each channel have been captured at an exposure time of 25 mS/image, yielding in an accumulated exposure time of 1800 mS. Obviously, a probe is required that is not prone to (photo-)bleaching. Fortunately, dextran-linked Oregon Green 488 is extremely photostable under circumstances of up to 2.4 × 105 mS of continuous F495-light exposure (Figure 10).
Figure 6. Specific dye signals versus unspecific noise signals. (A) Adaxial leaf face. Overlay of ex 495 nm pseudo-green fluorescence image, ex 440 nm pseudo-red fluorescence image, and the corresponding bright-field image (gray-scaled). The apoplast was partially loaded with the dye (lower right part of the image). Dye-loaded areas appear light green when illuminated, because pseudo-red and pseudo-green are mixed within the overlay. The gray area was not loaded with the dye (upper left part of the image). Such a specimen represents a suitable object to compare the amount of unspecific signals (noise) to specific signals being emitted from the dye. The intensity of gray values was chosen as a measure for signal intensity. A profile of the gray-value intensity was taken from the area tagged with the scale bar (1.15 mm length; white line in A) and is presented in (B). This profile is displayed for both fluorescence excitation channels (ex 495 and ex 440 nm). Negligible signals were emitted from the area of the leaf without dye (i.e., detail 1). The signals were much higher in the dye-loaded areas (i.e., detail 2 and 3; in both cases, signal intensity at stomatal cavity).
Figure 7. Dependency of the Oregon Green 488 F495/F440 ratio on NaCl. Ratios are displayed against NaCl concentration. NaCl ranging from 0 to 125 mM was mixed into the dye solution (25 μM Oregon Green). Ratios were stable and thus not affected by NaCl. Values represent the mean ± SE (n = 4 biological replicates). t-Test revealed no significant mean differences at p ≤ 0.05.
Figure 8. Ratiometric real-time quantitation of leaf apoplastic pH in response to the given illumination regime used for inverse microscopy image acquisition. pH as recorded at the adaxial face of Vicia faba leaves is plotted over time. Leaf apoplastic pH was discriminated within three apoplastic components, viz. in the stomatal cavity (n = 6 ROI; green kinetic; mean ± SE of ROIs), in the epidermal apoplast (n = 20 ROI; red kinetic; mean ± SE of ROIs) and in the apoplast surrounding the palisade mesophyll (n = 20 ROI; blue kinetic; mean ± SE of ROIs). No effects on apoplastic pH were detected that were attributable to the illumination of the specimen with the excitation or emission wavelengths used for imaging. Representative kinetics of four equivalent recordings of plants gained from independent experiments.
Figure 9. Leaf growth (A), stomatal conductance (B), and relative chlorophyll concentration (C) as influenced by the dye and the loading procedure. To test whether these parameters were affected, a paripinnate leaf was used. One leaflet was loaded with the dye, whereas the second was not loaded with the dye and was used as control. Leaves were tested (A) prior to loading and at 1 and 2 days after loading or (B,C) prior to loading and 2, 3, and 4 h after loading. The parameters were not influenced by the presence of the dye or the loading procedure. The data are mean of four biological replicates gained from independent experiments ± SE. Paired t-test revealed no significant (ns) mean differences at p ≤ 0.05.
Figure 10. Photostability of the fluorescent ion indicator. (A) To test whether the dye was prone to bleaching, a selected area in the middle of the specimen (presented in pseudo-red) was continuously excited by 495 nm illumination over a period of 2 min. The outer edges of the specimen were not illuminated because the field diaphragm foreclosed the illumination of this area, thus representing the control area of the leaf (control area = dark area within the image). (B) After 2 min continuous excitation, the field diaphragm was opened for collecting an image at ex 495 nm (exposure time was 25 mS). The image is presented in pseudo-green and contains the part of the specimen that was continuously illuminated plus the adjacent edges of the specimen that were not exposed to the continuous illumination at ex 495 (control). Image (C) presents the merged overlay of (A, B). The pseudo-orange area (mixing pseudo-red and pseudo-green yields orange) represents the part of the leaf with the possibly bleached dye, whereas the pseudo-green area represents the control part of the leaf. Image (B) was used to create a profile of the gray-value intensity from the area tagged by the white line as a measure for the dye signal intensity after 2 min of continuous excitation at 495 nm (dark-gray profile) and after 4 min of continuous excitation (light-gray profile). The gray-value intensity profiles are presented in (D). The dotted rectangle in (D) flags the gray-value intensity from the region that was exposed to continuous illumination. The gray-value intensity outside of the rectangle represents the control values. A comparison of the signal intensities between the possibly bleached part of the specimen and the control reveals that no dye-bleaching had actually occurred after 2 min of continuous illumination. This was also confirmed after 4 min.
Our new dye loading procedure has been demonstrated to be suitable for ratiometric in planta mapping and the quantification of apoplastic pH dynamics. The spatial resolution of the microscopy-based imaging allows the discrimination of spatial gradients within the leaf apoplastic ion milieu. The more sophisticated dye loading procedure in combination with camera-assisted ratio imaging enables the real-time mapping of spatial apoplastic ion gradients over a period of 3 h in situ. Moreover, we have used this technique to monitor leaf apoplastic pH dynamics in response to NaCl stress events.
Christoph-Martin Geilfus is the grateful recipient of a grant from the Friedrich-Ebert-Foundation. We thank Dr. Christoph Plieth (Center of Biochemistry and Molecular Biology, University of Kiel) for giving advice with regard to fitting calibration data to a sigmoidal Boltzmann fit and for providing help with CLSM-imaging. Many thanks are also due to Christina Neuhaus for giving critical comments on the manuscript.
The Movies S1 for this article can be found online at http://www.frontiersin.org/plant_nutrition/10.3389/fpls.2011.00013/abstract/
Video S1|Confocal XYZ-image stack showing abaxial leaf apoplast of Vicia faba as labeled with Oregon Green 488 dextran. Excitation with an Argon laser line at 488 nm; emission at 514–556 nm (green channel); Leica TCS SP confocal laser scanning system; HC PL APO CS 10.0 × 0.40 planapochromatic objective. Dye signals are emitted from the apoplast (green color), whereas the mesophyll cells appear dark. This establishes that no Oregon Green 488 dextran had entered the cytosol unintentionally, as otherwise signals would be detectable from the cells.
Bibikova, T. N., Jacob, T., Dahse, I., and Gilroy, S. (1998). Localized changes in apoplastic and cytoplasmic pH are associated with root hair development in Arabidopsis thaliana. Development 125, 2925–2934.
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Fricker, M. D., Plieth, C., Knight, H., Blancaflor, E., Knight, M. R., White, N. S., and Gilroy, S. (1999). "Fluorescent and luminescent techniques to probe ion activities in living plant cells," in Fluorescent and luminescent probes, ed. W. T. Mason (London: Academic Press), 569–596.
Fricker, M. D., Tlalka, M., Ermantraut, J., Obermeyer, G., Dewey, M., Gurr, S., Patrick, J., and White, N. S. (1994). Confocal fluorescence ratio imaging of ion activities in plant cells. Scanning Microsc. 8, 391–405.
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Hoffmann, B., and Kosegarten, H. (1995). FITC-dextran for measuring apoplast pH and apoplastic pH gradients between various cell types in sunflower leaves. Physiol. Plant 3, 327–335.
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Mühling, K. H., and Läuchli, A. (2000). Light-induced pH and K+ changes in the apoplast of intact leaves. Planta 212, 9–15.
Mühling, K. H., and Läuchli, A. (2002). Determination of apoplastic Na+ in intact leaves of cotton by in vivo fluorescence ratio imaging. Funct. Plant Biol. 29, 1491–1499.
Mühling, K. H., Plieth, C., Hansen, U. P., and Sattelmacher, B. (1995). Apoplastic pH of intact leaves of Vicia faba as influenced by light. J. Exp. Bot. 46, 377–382.
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Mühling, K. H., and Sattelmacher, B. (1997). Determination of apoplastic K+ in intact leaves by ratio imaging of PBFI fluorescence. J. Exp. Bot. 48, 1609–1614.
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Pitann, B., Kranz, T., and Mühling, K. H. (2009). The apoplastic pH and its significance in adaptation to salinity in maize (Zea mays L.): comparison of fluorescence microscopy and pH-sensitive microelectrodes. Plant Sci. 176, 497–504.
Sattelmacher, B., Mühling, K. H., and Pennewiß, K. (1998). The apoplast – its significance for the nutrition of higher plants. J. Plant Nutr. Soil Sci. 161, 485–498.
Schulte, A., Lorenzen, I., Böttcher, M., and Plieth, C. (2006). A novel fluorescent pH probe for expression in plants. Plant Methods 2, 7.
Schwarzländer, M., Fricker, M. D., Müller, C., Marty, L., Brach, T., Novak, J., Sweetlove, L. J., Hell, R., and Meyer, A. J. (2008). Confocal imaging of glutathione redox potential in living plant cells. J. Microsc. 231, 299–316.
Taylor, D. P., Slattery, J., and Leopold, A. C. (1996). Apoplastic pH in corn root gravitropism: a laser scanning confocal microscopy measurement. Physiol. Plant 97, 35–38.
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Keywords: apoplastic pH, live cell imaging, apoplastic ions, abiotic stress, salinity, pH regulation, fluorescence ratio imaging
Citation: Geilfus CM and Mühling KH (2011) Real-time imaging of leaf apoplastic pH dynamics in response to NaCl stress. Front. Plant Sci. 2:13. doi: 10.3389/fpls.2011.00013
Received: 02 March 2011; Accepted: 16 April 2011;
Published online: 02 May 2011.
Jan Kofod Schjoerring, University of Copenhagen, Denmark
Doug Van Hoewyk, Coastal Carolina University, USA
Alexander Schulz, University of Copenhagen, Denmark
Copyright: © 2011 Geilfus and Muehling. This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.
*Correspondence: Karl H. Mühling, Institute of Plant Nutrition and Soil Science, Christian Albrechts University, Hermann-Rodewald-Str. 2, 24118 Kiel, Germany. e-mail: khmuehling@plantnutrition.uni-kiel.de
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Amid labor strife at South Africa's gold, diamond, palladium and platinum mines, the country's rand currency has been plunging. Standard & Poor's paring South Africa's credit rating by one notch to BBB with a negative outlook last week did not help matters for the commodity currency. Over the past month, the rand has been one of the worst-performing of the so-called exotic currencies. The WisdomTree Dreyfus South African Rand Fund (NYSE:SZR) is off about 5.1 percent since September 17.
The faltering rand is not just bad news for SZR. Rand woes have plagued previously resurgent precious metals mining ETFs. The iShares MSCI South Africa Index Fund (NYSE:EZA) has outperformed the currency fund by about 200 basis points over the past month, but EZA is not a pure materials/mining play. While those sectors are vital to South Africa's economic output, this Africa's largest economy. To that end, EZA offers some level of sector diversity as financials account for 25.5 percent of the fund's weight.
At 18.4 percent, materials names barely outpace the allocation given to consumer discretionary issues. Pure mining ETFs have not been so lucky during South Africa's labor woes.
As Index Universe astutely points out, mining ETFs with significant weights to South Africa have been decked by the rand's demise.
The Market Vectors Gold Miners (NYSE:GDX), the largest gold miners ETF with almost $9.7 billion in assets under management, features an allocation of 11.2 percent to South Africa. That fund has lost 2.4 percent in the past month. The new iShares MSCI Global Gold Miners Fund (NYSE:RING) devotes nearly 10 percent of its country weight to South Africa and that ETF has slipped 4.7 percent in the past 30 days.
Should South African labor strife continue and show no signs of abating, currency traders will be all the more inclined to punish the rand. That proposition does not bode well for the likes of GDX and RING, but those are not the only ETFs under pressure by way of the faltering rand.
The Global X Pure Gold Miners ETF (NYSE:GGGG) features a 14.6 weight to South Africa. As a result, that ETF has slid almost three percent in the past month. However, neither GDX, GGGG nor RING are the most victimized ETFs due to the rand's decline. SZR, the ETF that tracks the currency, is not, either.
For all the talk about the rand's impact on shares of gold miners and the corresponding ETFs, it cannot be forgotten that South Africa is the world's largest platinum producer and the second-largest producer of palladium behind Russia. To that end, it probably is not surprising that the First Trust ISE Global Platinum Index Fund (NASDAQ:PLTM) has plunged 14.7 percent in the past 30 days.
PLTM focuses on companies that mine metals from the platinum group, which includes platinum, palladium, osmium, iridium, ruthenium and rhodium. These firms are often smaller by market value and feature higher betas than gold miners. The median market cap of PLTM's holdings is just $365 million, but the fund has a weight of 38.4 percent to South Africa.
Those two factors are enough to make PLTM vulnerable to ongoing negative headlines from South Africa.
For more on South Africa and ETFs, click here.
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{"url":"https:\/\/crypto.stackexchange.com\/questions\/61334\/is-it-safe-to-use-a-counter-as-iv-for-aes256-gcm","text":"# Is it safe to use a counter as IV for AES256-GCM\n\nWhat if I will use 96-bit integer counter starting from 0 as IV for AES256-GCM cipher. Is it safe? Is there any other ways to ensure uniqueness of IV for GCM cipher?\n\n\u2022 Yes GCM was designed to work with simple counters. \u2013\u00a0SEJPM Aug 6 '18 at 14:50\n\u2022 Just be careful to NEVER, EVER reuse the counter - the IV is a nonce. Safely implementing a counter isn't as easy as it might seem at first. \u2013\u00a0Swashbuckler Aug 20 '18 at 15:04\n\nThere is no difference in terms of security as long as no AES input block (the counter\/IV not the plaintext) is repeated. (And you're not doing something extremely silly like making the input $AES_k^{-1}(ctr)$.)\n\nYou probably don't want\/need to use a 96 bit IV if you can use a counter. Using a larger IV is good when you need to use random IVs to prevent repeats. But you're not using random IVs and it only leaves 32 bits for the other counter. You probably won't encrypt $2^{96}$ messages but you could want to encrypt something larger than 64 GB. (Or $2^{32} * (128 \/ 8)$ bytes.)\n\n\u2022 Thank you for the answer. Unfortunately I don't understand what \"making the input $AES_k^{-1}(ctr)$\" means. \u2013\u00a0igor.sol Aug 6 '18 at 16:28\n\u2022 No I don't think I will encrypt something bigger than 64GB \u2013\u00a0igor.sol Aug 6 '18 at 16:29\n\u2022 @igor.sol The inverse of AES block encryption (decryption) using key $k$ and some value $ctr$ short for counter. But $ctr$ could be from any non-repeating pattern. $AES_k(AES_k^{-1}(x)) = x$ just as $AES_k^{-1}(AES_k(y)) = y$. It's unrealistically silly, but it was a necessary warning if we want to be pedantic. \u2013\u00a0Future Security Aug 6 '18 at 16:36\n\u2022 Well, I probably won't encrypt $2^{96}$ messages. On the other side there are some difficulties with maintaining last used counter value between sessions. I need very fast generation of IV values so I cannot store counter value into durable memory after each increment. So I will reserve counter range blocks of some size (maybe something like $2^{15}$). This is why I will need at least 64 bits for counter. Anyway I will need to estimate these numbers \u2013\u00a0igor.sol Aug 6 '18 at 16:39\n\u2022 Ah, ok now I see what $AES_k^{-1}(ctr)$ means. This can happen randomly in a very rare case. \u2013\u00a0igor.sol Aug 6 '18 at 16:43\n\nSee page 20 of the NIST recommendations (28 in the pdf, 20 on the page) - basically this is the deterministic construction (assuming you're doing less than $2^{64}$ messages), so yes this should be fine - assuming (!) that you don't reuse IV (even across restarts \/ crashes).\n\nAlternatively, you can use the RBG construction, and use a secure random source to generate the IV.","date":"2019-11-21 19:06:36","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\": 0.27994516491889954, \"perplexity\": 1285.348956770391}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2019-47\/segments\/1573496670948.64\/warc\/CC-MAIN-20191121180800-20191121204800-00379.warc.gz\"}"}
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Invisalign® and Invisalign® Teen work with your schedule. There is minimal time required in our office for treatment as there are no wires and brackets to fix.
For more information about Advantages of Invisalign or to schedule a consult with Dr. Tendler, please use our Appointment form or call our office in Boca Raton, FL at Tendler Orthodontics Boca Raton Office Phone Number 561-826-7955.
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{"url":"https:\/\/www.physicsforums.com\/threads\/tangent-line-to-curve-q.121276\/","text":"# Tangent line to curve Q\n\nCan anybody help me out with this Q?\n\n\"A curve R in space has vector equation:\n\n$$x = (sin(\\pi u), u^2 - 1, u^2 + 3u + 3)$$\n\nu is a real number. Find a vector equation of the tangent line to R at the point (0, 0, 1)\"\n\nRelated Calculus and Beyond Homework Help News on Phys.org\nsiddharth\nHomework Helper\nGold Member\nWhat are your ideas or thoughts on how to solve this problem? You need to show some work to get help.\n\nWell, I originally thought of taking the gradient of x and then plugging in the values of (0,0,1). Not sure if this is the right way to go about it though.\n\n0rthodontist","date":"2020-07-15 19:30:16","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\": 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\": 0.4968050718307495, \"perplexity\": 343.7765470297379}, \"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-2020-29\/segments\/1593657170639.97\/warc\/CC-MAIN-20200715164155-20200715194155-00114.warc.gz\"}"}
| null | null |
Thulani Rudolf Maseko (* 1. März 1970; † 21. Januar 2023 in Luhleko, Manzini) war ein Menschenrechtsanwalt und Oppositionspolitiker aus Eswatini sowie Mitgründer der Sektion der Lawyers for Human Rights in Eswatini. 2011 erhielt er den Vera Chirwa Prize für seine Verdienste um die Menschenrechte.
Leben
Thulani Maseko stammte aus Luhleko, einem Ort bei Bhunya in der Region Manzini.
Am 18. März 2014 wurde Thulani zusammen mit dem Journalisten Bheki Makhubu wegen Missachtung des Gerichts verurteilt und inhaftiert, nachdem er in Zeitungsartikeln das Justizsystem des Landes kritisiert hatte.
Im April 2014 wurde auch der Generalsekretär des People's United Democratic Movement (PUDEMO) Mlungisi Makhanya inhaftiert, weil er ein T-Shirt trug, mit dessen Aufdruck die Partei gegen die Inhaftierung der beiden Aktivisten Maseko und Makhubu protestierte.
Im August 2014 schrieb Maseko an den US-Präsidenten Barack Obama und bat ihn um Intervention vor dem United States–Africa Leaders Summit 2014. Im März 2015 wurde er daraufhin in Einzelhaft genommen.
Maseko wurde am 30. Juli 2015 aus dem Gefängnis entlassen. Er war von Amnesty International als "Prisoner of Conscience" anerkannt worden.
Thulani war seit 2008 mit Tanele Maseko verheiratet.
Tod
Am 21. Januar 2023 wurde er von Unbekannten in seinem Haus, etwa 50 Kilometer von der Hauptstadt Mbabane entfernt, erschossen. Die Tat ereignete sich nur wenige Stunden, nachdem König Mswati III. Aktivisten, die die Abschaffung der Monarchie forderten, gewarnt hatte, sie sollten "sich nicht darüber beschweren, dass sie von Söldnern umgebracht werden". Laut Sikelela Dlamini, Generalsekretär des von Maseko gegründeten Swaziland Multistakeholder Forum (SMF), hätten der oder die Täter ihn von außen durch ein Fenster des Hauses erschossen, in dem er sich mit seiner Familie aufhielt.
Reaktionen
Der Hohe Kommissar der Vereinten Nationen für Menschenrechte, Volker Türk, forderte Eswatini am 23. Januar 2023 auf, eine "unverzügliche, unabhängige, unparteiische und wirksame Untersuchung" der Ermordung des prominenten Oppositionspolitikers zu gewährleisten und "alle Verantwortlichen in fairen Verfahren zur Rechenschaft zu ziehen". Er erklärte:
Werke
Artikel im Nation Magazine:
"Speaking my mind." Februar 2014.
"Where the law has no place." März 2014. (in Bezug auf eine Verurteilung von Bansthana Vincent Gwebu durch den damaligen Chief Justice of Swaziland, Michael Ramodibedi.)
Literatur
Weblinks
Einzelnachweise
Rechtsanwalt
Bürgerrechtler
Gefangener
Opfer eines ungeklärten Tötungsdelikts
Kriminalfall 2023
Swasi
Geboren 1970
Gestorben 2023
Mann
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