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+{"text":"\\section{Introduction}\nMany modern particle accelerators are tuned to achieve as small transverse beam emittance as possible. This is due to the fact that most users demand for the highest beam brightness possible. Beam brightness is important for many accelerator applications, such as ultra-fast electron diffraction \\cite{doi:10.1098\/rsta.2005.1735}, free electron lasers \\cite{RevModPhys.88.015006} and the operation of new compact accelerator concepts in general (e.g. \\cite{England:2014bf,KARTNER201624,Giorgianni:2016ds,Nghiem:437316}). A common definition of brightness is \\cite{doi:10.1063\/1.860076}\n\\begin{equation}\n\tB = \\frac{\\eta I}{\\pi ^2 \\varepsilon_x \\varepsilon _y},\n\\end{equation}\nwhere $\\eta$ is a form factor close to unity, $I$ is the beam peak current and $\\varepsilon_{x,y}$ the horizontal and vertical transverse emittance respectively. Hence, in order to maximize $B$, transverse emittance has to be minimal. \n\nThere are multiple methods to characterize the transverse emittance. One of the most common techniques is the phase advance scan technique, where the transverse beam size is recorded on a screen vs.\\ the focusing strength of an upstream quadrupole or solenoid magnet \\cite{ReesRivkin:slac1984,Minty:2003,HACHMANN2016318}. The data can then be fitted based on the beam envelope equation. Space charge effects can be included to some extent \\cite{Hachmann:138874,PhysRevAccelBeams.20.013401}. Instead of scanning the focusing strength of a magnet, also multiple screens can be used to record the beam size vs.\\ the phase advance. Other -- potentially single-shot -- methods involve insertion of masks into the beamline, which then, subsequently, can be imaged on a downstream screen \\cite{Zhang:fermilab1996}. Coupled with advanced reconstruction algorithms these methods are capable of delivering reconstruction of the core 4D phase space \\cite{PhysRevAccelBeams.21.102802}. In this work we concentrate on the phase advance scan technique, as this is the easiest one to implement, only requiring standard beamline components.\n\nOne of the limitations of the phase advance scan technique is that there is no closed description of the beam envelope for space charge dominated beams \\cite{Hachmann:138874,PhysRevAccelBeams.20.013401}. It is therefore difficult to apply the method in this regime. Space charge dominated beams especially occur, for example, in the injector part of high-brightness electron sources, where the beam is still non-relativistic. In order to quantify whether a beam is space charge dominated, the so-called \\emph{laminarity parameter} $\\rho$ can be calculated \\cite{Ferrario:1982426}. This parameter represents the ratio between the space charge term and the emittance term of the beam envelope equation. It is given by\n\\begin{equation}\n \t\\rho = \\frac{I\\sigma^2}{2I_\\text{A}\\gamma \\varepsilon _n^2},\n \t\\label{eq:laminarity}\n \\end{equation}\nwhere $I$ is the peak current of the beam, $I_\\text{A} \\approx \\SI{17}{\\kilo\\ampere}$ is the Alfv\u00e9n current and $\\varepsilon _n = \\beta \\gamma \\varepsilon$ is the normalized emittance with the Lorentz factor $\\gamma$ and $\\beta = v\/c$. In case $\\rho \\gg 1$, the beam can be considered as space charge dominated (laminar flow regime). Otherwise the evolution of the beam envelope is dominated by the emittance pressure (thermal regime).\n\nIn this work we show in simulation that it is possible to successfully analyse phase advance scan data for $\\rho \\gg 1$ beams using a pre-trained fully connected neural network (FCNN). Subsequently, we apply the method to measured data. Machine learning and neural networks in particular have recently been used in the context of accelerators for various purposes. These include, among others, fault detection of machine components \\cite{Solopova:IPAC2019-TUXXPLM2}, machine stability optimization and analysis \\cite{PhysRevX.10.031039,PhysRevLett.121.044801,PhysRevAccelBeams.23.074601}, virtual diagnostics \\cite{Sanchez-Gonzalez2017,PhysRevAccelBeams.21.112802,Ratner:21}, beam quality optimization in plasma accelerators \\cite{PhysRevLett.126.174801,PhysRevLett.126.104801} and orders of magnitude speed-up in multiobjective optimization of accelerator parameters \\cite{PhysRevAccelBeams.23.044601}. Here, we focus on the analysis of otherwise difficult to interpret measurement data.\n\\section{Measurement Technique} \n\\label{sec:MeasurementTechnique}\nThe state of a single particle with respect to a given design, or reference trajectory, is usually defined by the 6D phase space vector\n\\begin{equation}\n\t\\mathbf{X} = (x, x^\\prime, y, y^\\prime, z, \\delta)^\\text{T},\n\\end{equation}\nwhere $x^\\prime = p_x\/p_z$ and $y^\\prime=p_y\/p_z$ are the horizontal and vertical divergence respectively, $x,y,z$ the distances of the particle from the reference trajectory and $\\delta = \\Delta p \/ p_0$ is the relative deviation of the particle's individual momentum from the reference momentum. $p_x, p_y, p_z$ are the three momentum components and $()^\\text{T}$ denotes the transpose of a matrix. We are interested in the evolution of the 4D transverse phase space vector\n\\begin{equation}\n\t \\mathbf{X^\\text{(tr)}} = (x, x^\\prime, y, y^\\prime)^\\text{T}.\n\\end{equation}\nAssuming negligible correlation between the evolution of the phase space coordinates in the $x$ and $y$ planes, it is possible to treat them separately, yielding the two sub-space vectors\n\\begin{equation}\n\t\\mathbf{x} = (x, x^\\prime)^\\text{T}, \\hspace{0.5cm}\\mathbf{y} = (y, y^\\prime)^\\text{T}.\n\\end{equation}\nA common framework to describe the evolution of a charged particle is linear beam optics, where only linear transformations of $\\mathbf{x}$ and $\\mathbf{y}$ are taken into account. Each beamline element is represented by a so-called transfer matrix defined by the relation\n\\begin{equation} \\label{eq:matrixtransformation}\n\t\\mathbf{x} = \\mathbf{M}\\cdot\\mathbf{x_0},\n\\end{equation}\nwhere $\\mathbf{x_0} = (x_0, x^\\prime_0)^\\text{T}$ is the initial phase space coordinate and\n\\begin{equation}\n\t\\mathbf{M} = \n\t\\begin{pmatrix}\n\t\tM_{11} & M_{12}\\\\\n\t\tM_{21} & M_{22}\n\t\\end{pmatrix}.\n\\end{equation}\nThe $2\\times2$ matrix for a simple drift is given by\n\\begin{equation}\n\t\\mathbf{M}_\\text{D}(s) = \n\t\\begin{pmatrix}\n\t\t1 & s\\\\\n\t\t0 & 1\n\t\\end{pmatrix},\n\\end{equation}\nwhere $s$ is the drift distance. Applying this matrix to $\\mathbf{x_0}$ would result in $x = x_0 + x^\\prime s, x^\\prime = x^\\prime_0$, as expected.\n\nIn an experiment only the rms beam size $\\sigma_x = \\sqrt{\\langle x^2 \\rangle}$ is accessible, where $\\langle \\rangle$ denotes the second central moment. Applying this to the general equation\n\\begin{equation}\n\tx = M_{11}x_0 + M_{12}x_0^\\prime\n\\end{equation}\nyields\n\\begin{equation}\n\t\\begin{split}\n\t\\sigma_x^2 = &M_{11}^2 \\sigma_{x,0}^2 \\\\\n\t&+ 2M_{11}M_{12}\\sigma_{x,0}(\\sigma_{x,0})^\\prime \\\\\n\t&+ M_{12}^2 \\left( \\frac{\\varepsilon _x^2}{\\sigma_{x,0}^2} + (\\sigma_{x,0})^{\\prime 2} \\right)\n\t\\end{split}\n\\label{eq:envelope}\n\\end{equation}\nwith $\\varepsilon_x = \\sqrt{\\langle x^2 \\rangle \\langle x^{\\prime 2} \\rangle - \\langle xx^\\prime \\rangle^2}$. Equation~\\ref{eq:envelope} is the so-called rms envelope equation, which can be used to determine the transverse rms emittance $\\varepsilon _x$ using a suitable (i.e. tunable) beam transformation $\\mathbf{M}$. Note that Eq.~\\ref{eq:envelope} does not take any space charge effects into account and is hence only valid in the $\\rho \\approx 1$ regime.\n\nFigure~\\ref{fig:ARES_sketch} shows a sketch of a potential measurement scenario. The elements and distances are chosen according to what is installed at the ARES electron linac at DESY, Hamburg \\cite{instruments5030028}. The transfer matrix of the double solenoid magnet can be written as\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mathbf{M}_\\text{DS} &= \\mathbf{M}_\\text{TL} \\cdot \\mathbf{M}_\\text{D}(l_\\text{D}) \\cdot \\mathbf{M}_\\text{TL} \\\\\n\t\t&=\n\t\t\\begin{pmatrix}\n\t\t\t1 & 0 \\\\\n\t\t\t-\\frac{1}{f} & 1\n\t\t\\end{pmatrix} \n\t\t\\cdot\n\t\t\\begin{pmatrix}\n\t\t\t1 & l_\\text{D}\\\\\n\t\t\t0 & 1\n\t\t\\end{pmatrix}\n\t\t\\cdot\n\t\t \\begin{pmatrix}\n\t\t\t1 & 0 \\\\\n\t\t\t-\\frac{1}{f} & 1\n\t\t\\end{pmatrix}\\\\\n\t\t&=\n\t\t\\begin{pmatrix}\n\t\t\t1-l_\\text{D}\/f & l_\\text{D} \\\\\n\t\t\t(l_\\text{D} - 2f)\/f^2 & 1-l_\\text{D}\/f\n\t\t\\end{pmatrix}, \n\t\\end{split}\n\t\\label{eq:double_solenoid_matrix}\n\\end{equation}\nwhere $l_\\text{D}$ is the drift distance between the two single solenoids and $f$ the focal length of each solenoid. Here the approximation that $f$ is larger than the length of the solenoid was used, i.e. the thin lens approximation. The focal length of a solenoid is given by \\cite{doi:10.1119\/1.3129242}\n\\begin{equation}\n\tf(B_{z,\\text{max}}) = \\left[ \\left( \\frac{q}{2  \\overline p_z } \\right)^2 F_2\\right]^{-1}, \t\n\\end{equation}\nwhere $B_{z,\\text{max}}$ is the peak magnetic field, $q$ the particle charge and $\\overline p_z$ the average longitudinal beam momentum. $F_2 = \\int B_z^2 \\text{d}z \\propto B_{z,\\text{max}}^2$ is the second field integral of the on-axis magnetic field. By inserting the expression $\\mathbf{M}_\\text{D}(l_\\text{S}) \\cdot \\mathbf{M}_\\text{DS}$, where $l_\\text{S}$ is the drift between the solenoid and the screen, in Eq.~\\ref{eq:envelope}, it can be seen that now the $M_{ij}$ elements can be conveniently adjusted in the experiment as $B_{z,\\text{max}}$ is varied. The emittance at the position of the solenoid can thus be determined by fitting the recorded $\\sigma_{x,i}$ vs.\\ $B_{z,\\text{max},i}$ at the screen with Eq.~\\ref{eq:envelope}.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.9\\columnwidth]{FIG_1.pdf}\n  \\caption{Schematic of a suitable beamline layout for transverse emittance measurements using the phase advance scan technique. The distances are based on what is installed at the ARES linac at DESY, Hamburg.}\n  \\label{fig:ARES_sketch}\n\\end{figure}\n\nIt is possible to include transverse space charge forces into the model to some extent. This is done by including a defocusing term in the drift between the focusing element and the screen. Considering a uniformly charged cylindrical bunch with radius $R$ and length $L$, the envelope equation then reads in differential form \\cite{PhysRevAccelBeams.20.013401}\n\\begin{equation}\n\t\\sigma _x ^{\\prime \\prime} - \\frac{P}{4\\sigma_x} G(\\xi, A) - \\frac{\\varepsilon _x ^2}{\\sigma_x ^3} = 0,\n\t\\label{eq:envelope_diff_eq_spch}\n\\end{equation}\nwhere $G(\\xi, A) \\in [0,1]$ is a form factor, which depends on the centered longitudinal intra-bunch coordinate $\\xi$ and the rest frame aspect ratio $A=R\/(\\gamma L)$. $P$ is the so-called generalized perveance given by\n\\begin{equation}\n\tP = \\frac{eQ}{2 \\pi \\epsilon_0 m_e c^2 L \\beta ^2\\gamma^3},\n\\end{equation}\nwhere $Q$ is the total charge of the bunch, $\\epsilon _0$ the vacuum permittivity and $m_e$ the electron rest mass. In principle Eq.~\\ref{eq:envelope_diff_eq_spch} can now be used to construct a similar fit model to Eq.~\\ref{eq:envelope}. There are a number of caveats to take into account, however:\n\\begin{itemize}\n\t\\item The model is only fully valid for a perfectly cylindrical bunch.\n\t\\item The form factor $G$ depends on the aspect ratio, which depends on the transverse beam size and bunch length, both not being constant in the experiment (especially in the over-focused part of the scan).\n\t\\item Since the concept of the envelope equation relies on the emittance being a constant of motion, non-linear space charge forces are intrinsically neglected in this approach.\n\t\\item The perveance term depends on the bunch length, which might not be accessible to sufficient precision in the experiment.\n\t\\item Equation~\\ref{eq:envelope_diff_eq_spch} cannot be solved analytically and has to be approximated by a polynomial series (see \\cite{Hachmann:138874} for a detailed description).\n\\end{itemize}\nAll of these caveats lead to the conclusion, that the envelope equation based data analysis method is not ideal in the $\\rho \\gg 1$ regime. \n\nIndependent of the value of $\\rho$, two more criteria need to be met in order to ensure an accurate fit result. Considering the third term of Eq.~\\ref{eq:envelope}, the purely mathematical criterion\n\\begin{equation}\n\t\\frac{\\varepsilon _x ^2}{\\sigma _{x,0}^2 \\cdot (\\sigma _{x,0})^{\\prime 2}} \\geq 0.01\n\t\\label{eq:fit_criterion_math}\n\\end{equation}\ncan be derived \\cite{Hachmann:138874}. This criterion ensures the numerical significance of $\\varepsilon _x$. The second criterion is based on the fact that the scan needs to include a minimum. The initial beam optics needs to be setup, such that a potential focus of the beam lies behind the screen used to measure the beamsize. At the same time, the focusing element needs to be strong enough to focus the beam onto the screen, which implies a constraint on the distance between focusing element and screen. Using Eq.~\\ref{eq:envelope}, these considerations can be summarized by the criterion\n\\begin{equation}\n\t-\\frac{\\sigma_{x,0} \\cdot (\\sigma_\\text{eff})^\\prime}{f_\\text{max}^2 \\cdot \\left((\\sigma_\\text{eff})^{\\prime 2} + \\frac{\\varepsilon_x ^2}{\\sigma_{x,0}^2}\\right)} \\leq l_\\text{S} \\leq \\frac{\\sigma _{x,0}^2}{2 \\varepsilon_x},\n\\end{equation}\nwhere $(\\sigma_\\text{eff})^{\\prime} = - (\\sigma _{x,0}\/f_\\text{max} - (\\sigma _{x,0})^\\prime)$. In case both of the two criteria are fulfilled, the emittance can be retrieved.\n\nBased on the aforementioned considerations, we propose using an alternative way to analyze phase advance scan data. Specifically, we propose using a pre-trained FCNN to overcome the problem of the incomplete fit model in $\\rho \\gg 1$ cases, as well as the criterion described by Eq.~\\ref{eq:fit_criterion_math}. To this end, we have performed a simulation study, which is presented in detail in the following sections. The resulting FCNN was then subsequently applied to real world data, as shown below.\n\\section{Simulation Study - Methodology}\n\\label{sec:Methodology}\nIt has been shown already in 1989 that neural networks with only one unbounded hidden layer can approximate any Borel measureable function from finite dimensional space to another to arbitrary precision \\cite{Cybenko1989,HORNIK1989359}. More recent research focuses on the expressiveness (approximation accuracy) of both depth (i.e. the number of hidden layers) and width (i.e. the number of artificial neurons in a layer) bounded networks. In \\cite{NIPS2017_32cbf687}, for example, the authors show that any Lebesgue integrable function $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}$ on $n$-dimensional space can be approximated to arbitrary accuracy by a fully connected width-$(d_\\text{in}+4)$ ReLU network with respect to the $\\ell_1$ norm as a measure of approximation quality. In other words, the network represented by the transfer function $F$ satisfies $\\int _{\\mathbb{R}^n} |f(x) - F(x) | \\text{d}x < \\epsilon, \\forall \\epsilon > 0$ (see \\cite{NIPS2017_32cbf687}, Theorem 1). ReLU here refers to the the so-called Rectified Linear Unit neuron activation function, definded by $\\text{ReLU}(x) = \\text{max(0,x)}$ and $d_\\text{in}$ is the input dimensionality. This width boundary $w$ has since been refined and generalized for example in \\cite{hanin2018approximating,Hanin_2019} to be $d_\\text{in} + 1 \\leq w_\\text{min}(d_\\text{in},d_\\text{out}) \\leq d_\\text{in} + d_\\text{out}$, for functions of the form $f:[0,1]^{d_\\text{in}} \\rightarrow \\mathbb{R}^{d_\\text{out}}$, where $d_\\text{in}$ and $d_\\text{out}$ are the input and output dimensionality respectively. Limits on the depth can be estimated in specific cases in terms of the so-called modulus of continuity of $f$, given by $\\omega_f(\\varepsilon) = \\text{sup}\\{|f(x)-f(y)|||x-y|\\leq \\varepsilon\\}$, where $\\varepsilon$ is an arbitrarily small change in the argument of $f$. For continuous functions $f:[0,1]^{d_\\text{in}} \\rightarrow \\mathbb{R}_+$ the depth of a $d_\\text{in}+2$ wide network $\\mathcal N$ can, for example, be expressed as $\\text{depth}(\\mathcal N _\\varepsilon) = 2\\cdot d_\\text{in}! \/ \\omega_f(\\varepsilon)^{d_\\text{in}}$, cf. \\cite{hanin2018approximating}. We note that in practice the specific layout of a neural network is often determined experimentally, as the aforementioned boundaries are merely based on proofs of existence.\n\nIn this study, we aim to map the phase advance scan data to the normalized transverse emittance at the focusing element. Mathematically, this means we assume a connection of the scan data to the physical quantity of the form $f:\\mathbb{R}_+^{d_\\text{in}} \\rightarrow \\mathbb{R}_+$, where the dimensionality $d_\\text{in}$ is given by the number of scan data points. Note that, based on the knowledge of the problem, we can always map (normalize) the input data from $\\mathbb{R}_+^{d_\\text{in}}$ into $[0,1]^{d_\\text{in}}$. The function $f$ operates on the measure space $(\\mathbb{R}_+^{d_\\text{in}}, \\mathcal B, \\lambda)$ with the Borel-$\\sigma$-algebra $\\mathcal B$ and the Lebesgue measure $\\lambda$. It is hence measureable in the mathematical sense. In addition, we expect $f$ to be a continuous function based on the physical background of the problem. We can hence conclude that $f$ is Lebesgue integrable and suitable to be approximated for example by a width bounded ReLU network. \n\nTo validate this approach, we setup a simulation study based on the simple beamline layout shown in Fig.~\\ref{fig:ARES_sketch}. The main simulation study is split into three parts:\n\\begin{enumerate}\n\t\\item Building a large number of data sets (training, validation and test),\n\t\\item Training the FCNN and evaluation of the performance using the test data sets,\n\t\\item Comparison of the FCNN performance to the traditional fit method, as discussed above.\n\\end{enumerate}\nThe first step of the simulation study is to build a large number of data sets. Creating a data set consists of two steps:\n\\begin{itemize}\n\t\\item Numerical tracking of the beam from the cathode to the location of the solenoid,\n\t\\item Numerical simulation of the solenoid scan.\n\\end{itemize}\nFirst, the emittance at the solenoid position is determined by numerical tracking of the particles. In this step the solenoid field is set to zero. In addition to the emittance, other beam parameters, such as the beam size, divergence, or bunch length can be recorded as well. Then, the simulation domain is extended up to the position of the screen, which is used in the experiment to record the beam size vs.\\ the solenoid focusing strength. The experiment is then simulated for $M$ focusing strength settings. It is important to setup the scan range such that the resulting data includes the beam size minimum, i.e. the focus, as it carries most of the information about the emittance at the solenoid position \\cite{Hachmann:138874}. We use the well established code \\textsc{ASTRA} \\cite{ASTRAASpaceChar:LTSRiAsm}, which takes space charge effects into account. The beam size vs.\\ focusing strength scan data functions as the data set to be interpreted by the FCNN. Over the course of the study, a specific way to prepare the input data turned out to yield the best results. For each scan, $M\/2$ scan points centered around the minimum beam size are interleaved with the relative focusing strength difference $\\delta B_i = (B_i - B_\\text{foc})\/B_\\text{foc}$, where $B_\\text{foc}$ is the setting corresponding to the minimal beam size. The data set $S_\\text{in}$ is then of the form\n\\begin{equation}\n\tS_\\text{in} = [\\delta B_1, \\sigma_1, \\delta B_2, \\sigma_2, ..., \\delta B_{(M\/2)}, \\sigma_{(M\/2)}],\n\t\\label{eq:data_set_form}\n\\end{equation}\nwhere $\\sigma_i$ is the $i$th rms beam size. Each of these data sets is labeled with a set of important beam and simulation input parameters. These labels are then used to perform so-called supervised training of the FCNN. After the learning process, the FCNN is able to predict each of these parameters from given scan data, which is prepared according to Eq.~\\ref{eq:data_set_form}. Figure.~\\ref{fig:Methodology_1} summarizes how the data sets are created and what types of data they contain. \n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.72\\columnwidth]{FIG_2.pdf}\n  \\caption{Diagram showing how the FCNN training, validation and test data is produced. $N$ is the total number of data sets. $M$ is the number of ASTRA simulations. $M_\\text{in}$ is the final number of data points to be fed to the FCNN. $M_\\text{out}$ is the length of the output vector, i.e. the number of predicted beam and simulation parameters. The data sets are split into three categories, where in this study the common split of $N_\\text{tra} = 0.6\\cdot N$, $N_\\text{val} = 0.2\\cdot N$ and $N_\\text{tes} = 0.2\\cdot N$ is used.}\n  \\label{fig:Methodology_1}\n\\end{figure}\n\nFor this particular study $N = 16066$ random data sets with $M=40$ were produced. Each data set differs in the three key \\textsc{ASTRA} input parameters \\emph{total charge}, \\emph{laser spot size} and \\emph{cathode emission time}. Table~\\ref{tab:data_set_ranges} summarizes the parameter ranges used for this study, which are losely based on typical settings at the ARES linac at the time. The parameters are varied according to a uniform distribution. The ARES S-band gun was simulated with an, at the time available, peak gradient of \\SI{65}{\\mega \\volt\/\\meter}, resulting in a final $\\gamma = 6.8$. Based on the parameter ranges shown in Table~\\ref{tab:data_set_ranges}, a convergence study in terms of required macro particles in the numerical simulation was performed. For the highest possible charge density, 10000 particles were found to be sufficient.\n\n\\begin{table}[htbp]\n   \\centering\n   \\caption{\\textsc{ASTRA} input parameter ranges for the data set production.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lc}\n   Parameter & Value \\\\\n   \\hline\n   Bunch charge & $[0.01, 2.1]\\,\\SI{}{\\pico\\coulomb}$\\\\\n   Laser spot size (flat top diameter) & $[240, 400]\\,\\SI{}{\\micro\\meter}$\\\\\n   Cathode emission time (rms) & $[60, 100]\\,\\SI{}{\\femto\\second}$\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:data_set_ranges}\n\\end{table}\n\nThe neural network was implemented using the TensorFlow framework \\cite{tensorflow2015-whitepaper}. The input layer has $M$ neurons, corresponding to the length of $S_\\text{in}$. Then one hidden layer with $M$ and two hidden layers with $M\/2$ neurons are added in order to capture non-linearities in the system. The system is then coupled to the output layer of size $M_\\text{out}$, which corresponds to the number of \\textsc{ASTRA} input and simulated beam parameters to be predicted by the network. The overall layout is hence\n\\begin{equation}\n\tM \\rightarrow [M]-[M\/2]-[M\/2] \\rightarrow M_\\text{out}.\n\\end{equation}\nThis particular layout was determined empirically. Each neuron is coupled to every neuron of the following layer, or in other words the layers are fully connected. The neurons are activated using the well established rectified linear activation function (ReLU) \\cite{pmlr-v15-glorot11a,ramachandran2017searching}. Training of the network is performed using a combination of the \\textsc{adam} and \\textsc{adagrad} gradient decent algorithm \\cite{ruder2017overview} with the mean squared error (\\textsc{MSE}) as the loss function. The available $N$ data sets are split into three categories. $N_\\text{tra}$ training sets, $N_\\text{val}$ validation sets and $N_\\text{tes}$ test sets. The training sets are used to adjust the neuron weights during the training procedure, while the performance of the network is judged after each so-called epoch based on the validation sets, which are not used during training. This is done to avoid overfitting the training data. An epoch refers to one forward and backward pass of the entire training data. Finally, the performance of the resulting model is determined using the test sets, which have not been part of the learning procedure at all. We use the common split of $N_\\text{tra} = 0.6\\cdot N$, $N_\\text{val} = 0.2\\cdot N$ and $N_\\text{tes} = 0.2\\cdot N$. The network was trained for $\\sim 10000$ epochs using \\textsc{adam} and another $\\sim 10000$ epochs using \\textsc{adagrad} \\footnote{In our study, the training procedure usually took $<\\SI{1}{\\hour}$ on an Apple M1 processor. Production of the training data sets, however, can take several days, depending on the available compute infrastructure. In order to speed-up data set production, we used the parallelized version of ASTRA}.\n\\section{Simulation Study - Data Set}\nBefore evaluating the prediction performance of the neural network, it is useful to inspect the training data set. Since we are especially interested in analyzing phase advance scan data for space charge dominated beams, the laminarity parameter $\\rho$ at the solenoid position was calculated for each data set (cf. Eq.~\\ref{eq:laminarity}). Figure~\\ref{fig:space_charge_dominance_train_data_sets} shows $\\rho$ vs.\\ the bunch charge. The color scale indicates the laser spot size on the cathode used in the particular simulation. In addition, the distribution of $\\rho$ across the whole data set is shown.\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_3.pdf}\n  \\caption{Top: Scatter plot of the laminarity parameter $\\rho$ at the solenoid according to Eq.~\\ref{eq:laminarity} vs.\\ the bunch charge for all training data sets. $\\gamma = 6.8$ for this population. The color indicates the laser spot size on the cathode for each data set. Bottom: Distribution of the laminarity parameter.}\n  \\label{fig:space_charge_dominance_train_data_sets}\n\\end{figure}\nIt can be seen that all of the data sets lie in the $\\rho \\gg 1$, i.e. space charge dominated, regime ($\\rho_\\text{min} = 17.8$). Also, the higher the charge, the higher the value for $\\rho$, as expected. In addition, the color scale reveals that the smaller the laser spot size on the cathode, the higher the value for $\\rho$. The sensitivity of $\\rho$ on the laser spot size strongly depends on the bunch charge.\n\nAs noted above, the traditional fit method only works if Eq.~\\ref{eq:fit_criterion_math} is satisfied. Figure~\\ref{fig:feasibility_train_data_sets} shows the fit feasibility criterion for each data set, with the same color code as in Fig.~\\ref{fig:space_charge_dominance_train_data_sets}. None of the data sets satisfies the criterion, which leads to the expectation that the traditional fit method should not work well on the training data (and with that in reality for the ARES working point, which is the basis for the parameter space shown in Table~\\ref{tab:data_set_ranges}).\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_4.pdf}\n  \\caption{Top: Scatter plot of the fit feasibility parameter according to Eq.~\\ref{eq:fit_criterion_math} vs.\\ the bunch charge for all training data sets. $\\gamma = 6.8$ for this population. The color indicates the laser spot size on the cathode for each data set. Bottom: Distribution of the fit feasibility parameter. \\emph{Note that the plot shows the values multiplied by 1000 for readability.}}\n  \\label{fig:feasibility_train_data_sets}\n\\end{figure}\n\\section{Results and Comparison} \n\\label{sec:ResultsAndComparison}\nIn this section the performance of the pre-trained FCNN is presented. We also compare its performance against the traditional fit method discussed above. In order to evaluate the performance of the FCNN, we try to predict the labels of the $N_\\text{tes}$ test data sets, which were not used in the supervised training procedure. The main goal of the study is to predict the transverse emittance, but since the labels include a number of other simulation and beam parameters, the FCNN also provides predictions of these. In order to better quantify the prediction performance for the different label components, the relative error between predicition and truth was calculated for each data set. This is shown in Fig.~\\ref{fig:error_dists_perf}. In addition to the error distributions, a radar plot visualizes the prediction performance in terms of number of data sets in \\SI{1}{\\percent}, \\SI{5}{\\percent} and \\SI{10}{\\percent} relative error intervals respectively (see Table~\\ref{tab:pred_perf_table} for the actual percentages). In the ideal case, the heptagon would be filled completely. Inspection of the results reveals that some quantities are predicted much better than others. Specifically, it can be seen that the cathode emission time is predicted particularly bad. This result is somewhat expected, however, because this quantity refers to the longitudinal phase space at emission time, which cannot directly be accessed via a transverse beam size measurement. All quantities, which refer to the transvere phase space at the solenoid show very good prediction performance with $<\\SI{5}{\\percent}$ error. The prediction performance of both bunch charge and bunch length at the solenoid needs to be considered in more detail. In case of the bunch charge, values $\\lesssim\\SI{0.5}{\\pico\\coulomb}$ are predicted much less accurately. This can be explained by the lack of significant space charge effects, which alter the shape of the beam size vs.\\ focusing strength curve, effectively leading to degeneracy w.r.t. the initial bunch charge. Despite being a quantity of the longitudinal phase space, the bunch length is generally predicted with an error $<\\SI{10}{\\percent}$. This is because here the bunch length is directly correlated with the bunch charge and hence space charge effects. The prediction performance decreases towards smaller bunch lengths, which can be explained by the fact that the bunch length at the solenoid actually increases with the initial bunch charge. Therefore the same argument applies as for the bunch charge.\n\\begin{figure*}[htbp]\n  \\centering\n  \\includegraphics[width=0.98\\textwidth]{FIG_5}\n  \\caption{Relative error distributions for all verification data sets and label components. The last plot is a radar plot showing the percentage of data sets within a \\SI{1}{\\percent} (red), \\SI{5}{\\percent} (black) and \\SI{10}{\\percent} (gray) relative error interval respectively. The dashed lines in the distribution plots refer to the same error intervals. $Q:$ Bunch charge, $\\sigma_{t,\\text{las}}:$ Emission time, $\\sigma_{x,\\text{las}}:$ Laser spot size, $\\sigma_{z,\\text{SOL}}:$ Bunch length at the solenoid, $\\sigma_{x^\\prime,\\text{SOL}}:$ Divergence at the solenoid, $\\sigma_{x,\\text{SOL}}:$ Beam size at the solenoid, $\\varepsilon_{\\text{n},x,\\text{SOL}}:$ Normalized emittance at the solenoid.}\n  \\label{fig:error_dists_perf}\n\\end{figure*}\n\nFrom the prediction results and the ground truth a mean relative error was calculated over the whole test data set, yielding a mean prediction error for the emittance at the solenoid of \\SI{1.0}{\\percent}. Beam size and divergence at the solenoid are predicted very accurately with an error less than \\SI{0.1}{\\percent}. In addition to the beam parameters at the solenoid position, \\textsc{ASTRA} input parameters were predicted. The laser spot size is predicted with an error down to \\SI{0.7}{\\percent}. Emission time and bunch charge are predicted with errors of \\SI{32.1}{\\percent} and \\SI{27.1}{\\percent} respectively. The fact that the laser spot size is predicted best out of the three input parameters is due to the fact that it has the strongest effect on the shape of the beam size vs.\\ focusing strength data, especially for low charges (linear dependence of the thermal emittance). These results, as well as the percentage of predictions within a \\SI{1}{\\percent}, \\SI{5}{\\percent} and \\SI{10}{\\percent} relative error interval are summarized in Table~\\ref{tab:pred_perf_table}.\n\n\\begin{table*}[htbp]\n   \\caption{Summary of the prediction performance for the different training label components. Mean error refers to the absolute prediction error and is calculated for the whole verification data set. The three remaining columns refer to the fraction of the verification data set within a certain prediction accuracy interval.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcccc}\n       & Mean Error (\\SI{}{\\percent}) & $<$ \\SI{1}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{5}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{10}{\\percent} Error (\\SI{}{\\percent})\\\\\n      \\hline\n      Laser Spot Size & 0.71 & 76.59 & 99.84 & 100.00\\\\\n      Emission Time & 32.14 & 2.46 & 11.67 & 21.05\\\\\n      Bunch Charge & 27.05 & 10.24 & 52.74 & 76.74\\\\\n      Emittance at the Solenoid & 1.01 & 55.67 & 99.81 & 100.00\\\\\n      Beamsize at the Solenoid & 0.05 & 100.00 & 100.00 & 100.00\\\\\n      Divergence at the Solenoid & 0.05 & 100.00 & 100.00 & 100.00\\\\\n      Bunch Length at the Solenoid & 2.60 & 24.35 & 89.76 & 98.13\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_perf_table}\n\\end{table*}\n\nThe main goal of the study is to find a better way to determine the transverse emittance from phase advance scan data in the $\\rho \\gg 1$ regime, as well as in regimes where Eq.~\\ref{eq:fit_criterion_math} does not hold. It is hence useful to evaluate the emittance prediction performance in form of the relative error versus these two quantities. Figure~\\ref{fig:error_perf_comp_Ideal} shows the result of this analysis using both the FCNN, as well as the traditional fit routine (cf. Sec.~\\ref{sec:MeasurementTechnique}). \n\nAs expected from Fig.~\\ref{fig:space_charge_dominance_train_data_sets} and Fig.~\\ref{fig:feasibility_train_data_sets}, the traditional fit yields inaccurate results across the whole data set. The FCNN, on the other hand, performs much better even for very high values of $\\rho$.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_6.pdf}\n  \\caption{Mean emittance FCNN prediction performance, as well as traditional fit results vs.\\ the laminarity parameter (top) and the fit feasibility criterion (bottom). The data is calculated for 100 equally sized laminarity parameter slices. $\\gamma = 6.8$ for this population. \\emph{Note that the bottom plot shows the fit criterion values multiplied by 1000 for readability.}}\n  \\label{fig:error_perf_comp_Ideal}\n\\end{figure}\n\\section{Extension to Measured Data}\nSo far, the neural network was trained and tested solely with ideal phase advance scan data. This means that the training, validation, as well as the test data sets contain only perfectly evenly spaced data points with flawless beam size values. In addition, all scans were simulated within the same focusing strength range. Although this approach yields very good prediction performance for simulated data, it cannot be applied to experimental data, for several reasons. First, in a measured data set it is not guaranteed that all data points are evenly spaced. It is also not guaranteed that the scan range corresponds to the trained one and that the measured values of the focusing strength are correct \\footnote{This issue is already somewhat alleviated by using $\\delta B_i$, as described in Sec.~\\ref{sec:Methodology}}. Finally, the measured beam sizes are subject to jitter and systematic measurement errors like resolution limitations.\n\nIn order to take all of this into account, the training procedure was modified. Each data set is now created with a slightly different focusing strength scan range. Since the number of scan points is kept constant, the spacing between data points is now slightly different every time, which might also be the case in reality. Furthermore, the first and last focusing setting were added to the range of predicted parameters ($\\rightarrow M_\\text{out} = 9$). Measurement errors were taken into account by generating noisy data sets from the ideal sets by adding normally distributed errors. Both relative and absolute errors on focusing strength and beam size were considered with magnitudes based on experience at ARES. From each data set, $N_\\text{err} = 100$ noisy sets with relative errors and $N_\\text{err}$ noisy sets with absolute errors were generated. In addition, the procedure was repeated, this time enforcing a \\SI{10}{\\micro\\meter} resolution limit on the beam sizes. Including the ideal data, the total number of data sets now increases to $N_\\text{tot} = 2(2N_\\text{err}+1)\\cdot N = 6458532$. To visualize the importance of training the FCNN with noisy data, we fed data sets with increasing artificial noise to both the network trained solely on ideal data, as well as one trained on noisy data. The results are shown in Fig.~\\ref{fig:noise_performance}. It can be seen that the population with less than \\SI{5}{\\percent} decreases significantly with noise level, if the FCNN is not trained on noisy data.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_7.pdf}\n  \\caption{Population with less than \\SI{5}{\\percent} prediction error on the emittance vs.\\ different artificial noise levels. The FCNN layout is the same in both shown cases. \\emph{Note that in order to enable a direct comparison, the FCNNs used here were trained on the data sets with fixed scan range and constant spacing. This explains the generally better performance compared to Table~\\ref{tab:pred_perf_table_2}.}}\n  \\label{fig:noise_performance}\n\\end{figure}\n\nWe performed the same general analysis for the new network, as described above, and saw the same overall behaviour. The results are summarized in Table~\\ref{tab:pred_perf_table_2}. Compared to the network based on ideal data, the performance is slightly worse, but still for the majority of data sets the emittance is predicted with less than $\\SI{5}{\\percent}$ error.\n\n\\begin{table*}[htbp]\n   \\caption{Summary of the prediction performance of the modified network for the different training label components. Mean error refers to the absolute prediction error and is calculated for the whole verification data set. The three remaining columns refer to the fraction of the verification data set within a certain prediction accuracy interval.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcccc}\n       & Mean Error (\\SI{}{\\percent}) & $<$ \\SI{1}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{5}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{10}{\\percent} Error (\\SI{}{\\percent})\\\\\n      \\hline\n      Laser Spot Size & 2.76 & 22.25 & 87.85 & 98.83\\\\\n      Emission Time & 15.97 & 4.25 & 21.08 & 43.48\\\\\n      Bunch Charge & 14.13 & 6.44 & 30.89 & 67.20\\\\\n      Emittance at the Solenoid & 2.51 & 29.43 & 89.17 & 97.51\\\\\n      Beamsize at the Solenoid & 0.38 & 96.34 & 100.00 & 100.00\\\\\n      Divergence at the Solenoid & 0.38 & 96.63 & 100.00 & 100.00\\\\\n      Bunch Length at the Solenoid & 4.48 & 12.74 & 60.03 & 95.02\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_perf_table_2}\n\\end{table*}\n\n\\section{Measurements at the ARES linac}\nAs a real world test, we conducted emittance measurements using the phase advance scan technique at the ARES linac at DESY. The layout of the measurement setup is shown in Fig.~\\ref{fig:ARES_sketch}. We took data for several bunch charges by adjusting an attenuator in the cathode laser beamline. The charge was measured both with a Faraday cup, which can be inserted into the beamline instead of the scintillating screen and a cavity based charge monitor $\\sim \\SI{0.7}{\\meter}$ downstream of the screen \\cite{Lipka:166172}. All measurements shown here were performed according to the procedure introduced in Sec.~\\ref{sec:MeasurementTechnique}. Transverse beam sizes were determined from camera images of a scintillating Ce:GAGG (Cerium doped Gadolinium Aluminium Gallium Garnet) screen. The spatial resolution of the system is specified to be $\\sim \\SI{10}{\\micro\\meter}$ \\cite{Wiebers:166567}.\n\nFigure~\\ref{fig:emittance_measurement_ARES} shows the emittance values obtained from the measurements using both the FCNN, as well as the traditional fit method. The data points are compared to an ASTRA simulation including space charge based on the machine settings at the day of the measurements, including uncertainty. It can be seen that the FCNN results are much closer to the expected values than the results obtained from the envelope equation fit. It is interesting to note that Fig.~\\ref{fig:emittance_measurement_ARES} reproduces the expected behaviour shown in Fig.~\\ref{fig:error_perf_comp_Ideal}, as the fit underestimates the emittance for charges $<\\SI{0.5}{\\pico\\coulomb}$ and overestimates them for higher charges. The FCNN result follows the ASTRA curve much closer, mostly staying within the uncertainty of the simulation.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_8.pdf}\n  \\caption{Emittance measurements conducted at the ARES linac at DESY using the phase advance scan technique for different bunch charges. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters.}\n  \\label{fig:emittance_measurement_ARES}\n\\end{figure}\n\nIn order to cross-check the obtained results, we performed additional emittance measurements using a grid mask based method, as described in \\cite{PhysRevAccelBeams.21.102802}, using the same machine setup. Three different grids were used for the measurements: An SPI G200TH TEM grid \\footnote{SPI G200TH TEM Grid, available at \\url{https:\/\/www.2spi.com\/item\/202hc-xa}, \\emph{last access: 22nd April 2022}}, an SPI G300 TEM grid \\footnote{SPI G300 TEM Grid, available at \\url{https:\/\/www.2spi.com\/item\/2030c-xa}, \\emph{last access: 22nd April 2022}} and a custom made pepper pot \\footnote{Custom made Tungsten pepper pot (laser drilled). Hole diameter: \\SI{15}{\\micro\\meter}, pitch: \\SI{85}{\\micro\\meter}, thickness: \\SI{20}{\\micro\\meter}}. Measurements were performed for two different charge settings, \\SI{1}{\\pico\\coulomb} and \\SI{2}{\\pico\\coulomb}. These values were deliberately chosen to be in the strongly space charge dominated regime, where the traditional fit yields particularly bad results. At ARES, the grid masks are installed at the same $z$-position as the screen used to record the phase advance scan data. This means that the emittance obtained from the grid measurements will always be different from the phase advance scan result, as the phase advance scan yields the emittance at the position of the focusing element. The emittance is furthermore expected to be different, because in order to image the grid, the beam needs to be focused slightly before the grid, which can lead to emittance growth. Nevertheless, it is still possible to compare the measured values to ASTRA simulations, which would show that the ARES setup depicted in Fig.~\\ref{fig:ARES_sketch} can be well simulated with ASTRA. This would validate the FCNN results indirectly. Figure~\\ref{fig:emittance_measurement_ARES_grid_sim} shows an ASTRA simulation of the grid measurement using a \\SI{1}{\\pico\\coulomb} bunch with $\\gamma = 6.8$. It can be seen that the emittance is strongly affected by focusing the beam down. The measurement results from all three grids, as well as the expected values from the ASTRA simulation are summarized in Table~\\ref{tab:grid_meas_comp}. The results are very close to the expected value in the \\SI{1}{\\pico\\coulomb} case. The measurement of the \\SI{2}{\\pico\\coulomb} beam shows a slightly higher than expected emittance value, which is in line with the high uncertainty of the high charge results shown in Fig.~\\ref{fig:emittance_measurement_ARES}. We hence conclude that ASTRA simulates the ARES beamline shown in Fig.~\\ref{fig:ARES_sketch} well.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_9.pdf}\n  \\caption{ASTRA simulation of the evolution of the transverse normalized emittance between gun and grid for a \\SI{1}{\\pico\\coulomb} bunch with $\\gamma = 6.8$. The orange dashed line marks the position of the solenoid, which is used to perform the phase advance scan measurements. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line shows the evolution of the beam size \u2013 the beam reaches its focus slightly before the grid.}\n  \\label{fig:emittance_measurement_ARES_grid_sim}\n\\end{figure}\n\\begin{table}[htbp]\n   \\caption{Summary of the grid based measurements, used to verify the FCNN results. The measurements are compared to the ASTRA simulation result.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcc}\n      Method & $\\varepsilon _\\text{n}$ (\\SI{}{\\nano\\meter}) @ \\SI{1}{\\pico\\coulomb} & $\\varepsilon _\\text{n}$ (\\SI{}{\\nano\\meter}) @ \\SI{2}{\\pico\\coulomb}\\\\\n      \\hline\n      \\textbf{ASTRA Simulation} & $\\mathbf{105.5 \\pm 1.2}$ & $\\mathbf{152.9 \\pm 3.0}$ \\\\\n      \\hline\n      SPI G200TH TEM Grid & $107.4 \\pm 5.8$ & $165.4 \\pm 15.7$ \\\\\n      SPI G300 TEM Grid & $102.7 \\pm 3.8$ & $156.0 \\pm 15.6$ \\\\\n      Pepper Pot & $105.6 \\pm 11.9$ & $156.1 \\pm 27.4$ \\\\\n      \\textbf{Mean (Measurements)} & $\\mathbf{105.2 \\pm 8.0}$ & $\\mathbf{160.5 \\pm 20.3}$\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:grid_meas_comp}\n\\end{table}\n\\subsection{Prediction of other parameters}\nAs discussed above, the FCNN also predicts fixed machine parameters and other charge dependent beam parameters to varying accuracy (see Table~\\ref{tab:pred_perf_table_2}). Table~\\ref{tab:pred_fixed_params} summarizes the predicted fixed machine parameters in comparison to what was used in the experiment. It can be seen that both the mean laser spot size and pulse length are predicted to be larger than the expected values. Since measuring the laser spot size on the cathode directly is very difficult in the ARES setup, the predicted values fall within the uncertainty. As discussed in Sec.~\\ref{sec:ResultsAndComparison}, the prediction of the laser pulse length should be treated with caution. The solenoid scan range is predicted well within the uncertainties.\n\\begin{table}[htbp]\n   \\caption{Prediction results for fixed machine parameters.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcc}\n      Parameter & Prediction & Experiment\\\\\n      \\hline\n      Laser Spot Size (\\SI{}{\\micro\\meter}) & $338.5 \\pm 0.5$ & $320 \\pm 30$ \\\\\n      Laser Pulse Length (\\SI{}{\\femto\\second}, rms) & $87.34 \\pm 0.04$ & $76 \\pm 8$ \\\\\n      Solenoid Field -- Start (mT) & $130.516 \\pm 0.001$ & $130.5 \\pm 0.1$ \\\\\n      Solenoid Field -- End (mT) & $150.783 \\pm 0.001$ & $150.7 \\pm 0.1$ \\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_fixed_params}\n\\end{table}\n\nFigure~\\ref{fig:pred_variable_params} shows the prediction results for the beam charge, as well as the charge dependent beam parameters \\emph{bunch length}, \\emph{beam size} and \\emph{beam divergence} at the solenoid. As in Fig.~\\ref{fig:emittance_measurement_ARES}, the data points are compared to an ASTRA simulation including space charge based on the machine settings at the day of the measurements, including uncertainty. It can be seen that the beam charge prediction fits the measured values well. Both beam size and beam divergence follow the ASTRA curve well, albeit at the lower end of the uncertainty, denoted by the shaded area. The bunch length follows a more linear charge dependence than expected from the ASTRA simulation, which might be attributed to either the not fully known temporal and spatial laser pulse shape at the cathode, as well as the overall prediction performance of parameters of the longitudinal phase space (see Sec.~\\ref{sec:ResultsAndComparison}). \n\nIn order to cross-validate the prediction results, an ASTRA simulation using the mean predicted laser spot size and pulse length (see Table~\\ref{tab:pred_fixed_params}) was performed. The results are shown in Fig.~\\ref{fig:pred_variable_params} as the blue dashed line. Indeed, a larger spot size and longer pulse length lead to results closer to the lower end of the uncertainty in all three cases, which can be explained by the reduced charge density. Remaining discrepancies might be explained by the not fully known temporal and spatial laser pulse shape at the cathode.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_10.pdf}\n  \\caption{Prediction results for other charge dependent beam parameters. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line corresponds to an ASTRA simulation using the mean predicted laser spot size and pulse length (see Table~\\ref{tab:pred_fixed_params}).}\n  \\label{fig:pred_variable_params}\n\\end{figure}\n\\section{Conclusion and Outlook}\nWe have shown in simulation that a pre-trained fully connected neural network can be used to predict the transverse emittance from phase advance scan data even in the $\\rho \\gg 1$ regime and in case a traditional envelope equation based fit is mathematically not feasible. We have optimized the network for real-world measurement data and achieved $< \\SI{5}{\\percent}$ error for the majority of the test data set population (\\SI{89.2}{\\percent}), resulting in a mean relative error of \\SI{2.5}{\\percent}. We have applied our method to measurements conducted at the ARES linac at DESY and compared the predictions to numerical simulations using the well benchmarked code ASTRA, as well as results obtained from the traditional fit method. As expected from the simulation study, the FCNN predictions are much closer to what is expected from the numerical simulation. We have furthermore cross-validated the results using additional emittance measurements based on a grid mask based method.\n\nIn addition to the transverse emittance, the network also predicts other key beam and machine parameters to varying accuracy. While quantities directly tied to the transverse phase space are predicted as accurate or better than the emittance, quantities tied to the longitudinal phase space, such as the bunch length, are predicted less accurate, as expected. It should be noted, that in our study the gun setting is not a variable in the process of training the FCNN. This means that for each gun setting (gradient and phase) a separate FCNN has to be trained. Inclusion of these two parameters could be part of a future study. Furthermore, difficult to directly access parameters, such as the thermal emittance could be added. In conclusion, we have demonstrated that pre-trained FCNNs can be a powerful tool for the analysis of previously difficult to interpret data sets.\n\\section{Model Availability}\nThe FCNN models are available from the corresponding author upon request in TensorFlow format.\n\\begin{acknowledgments}\nWe acknowledge support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. Specifically, we would like to thank the DESY beam diagnostics group for their support during the measurement runs at ARES. We also thank the DESY Maxwell team for providing the compute resources we used to generate the training data sets in a reasonable time. Finally, we thank R.~Mayet (Halodi Robotics AS) for NN-related consulting.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn a recent paper, Arditi et al.~\\cite{Arditi2016} stated that a proper patch model of population dynamics must obey a basic logical property: ''If two patches are linked by migration, and if the migration rate becomes infinite, the two patches become perfectly mixed among each other, and the system must behave as a one-patch model for the total population.'' To illustrate the issue, they studied the following model:\n\n\\begin{equation}\n\\label{eq:growth}\n\\begin{aligned}\n\\frac{dN_1}{dt} &= r_1N_1\\left(1-\\frac{N_1}{K_1}\\right)\n+\\beta\\left(N_2-N_1\\right) \\\\\n\\frac{dN_2}{dt} &= r_2N_2\\left(1-\\frac{N_2}{K_2}\\right)+\\beta\\left(N_1-N_2\\right),\n\\end{aligned}\n\\end{equation}\n\n\\noindent where $N_i$ with $i={1,2}$ being population size in patch $i$, $r_i$ the local intrinsic per capita growth rate in patch $i$, $K_i$ the local carrying capacity in patch $i$ and $\\beta$ the migration rate constant from and to any patch in the population. Note that each equation is the classical formula for logistic growth plus a term describing migration between patches.\n\nArditi et al.~\\cite{Arditi2016} found that the asymptotic dynamics of system \\eqref{eq:growth} in the case of perfect mixing (i.e. with $\\beta\\rightarrow \\infty$) is different from the asymptotic dynamics of the sum of the two populations in isolation (i.e. with $\\beta = 0$). In particular, they showed that the equilibrium population size of the system with perfect mixing is different (either larger or smaller) that the sum of equilibrium sizes of the isolated populations. In the limiting but plausible case that the local populations differed in the value of their carrying capacities $K_i$ but not in the values of $r_i$, merging two patches in a single one showed to be always detrimental for equilibrium population size.\n\nAlthough the analysis is correct, it is valid to ask whether the particular choice for describing migration in \\eqref{eq:growth} was the best one for studying such a general ecological phenomenon. Apparently, the choice for migration model in\\cite{Arditi2016} was made because of two main reasons: 1) this system was analyzed previously \\cite{Freedman77} \\cite{DeAngelis1979} \\cite{HOLT1985181} \\cite{hanski1999metapopulation} \\cite{DeAngelis20143087} \\cite{Arditi201545} thus it has some tradition within the ecological literature, and 2) Arditi et al.~\\cite{Arditi2016} considered this model as a ``natural way'' to represent a two-patch system with logistic growth.\n\nAll other things being equal, a well known and widely used model should be favored over its competitors. However this is only valid until we consider a model presenting some objective advantage (e.g. better match with empirical observations) without compromising any substantial aspects (such as number of parameters, mathematical tractability, etc.). In our opinion, model~\\eqref{eq:growth} is neither the most natural nor the best way to extend the logistic growth model to a two-patch scenario. Furthermore, we will show below that the paradoxical results reported by Arditi et al.~\\cite{Arditi2016} are only a consequence of using the specific model~\\eqref{eq:growth} and should not be considered to be a general fact.\n\n\\section*{A  more biologically plausible model}\n\nModel~\\eqref{eq:growth}, used in~\\cite{Arditi2016} to present the ``perfect mixing paradox'' contains as a key component a passive migration rate from patch $i$ to patch $j$, $\\beta(N_i-N_j)$. This formulation of passive migration rate assumes that there will be a positive flux of migrants from patch $i$ to patch $j$ whenever the absolute population size in patch $i$ is larger than the absolute population size in patch $j$, no matter the differences in patch size or quality. This means that, given equal patch quality, it is possible to have a flux of migrants from a path with greater absolute population size but with lesser population density (with a very large patch size) to a small and more dense patch which possesses a lower absolute population size (Fig.~\\ref{fig:fig1}a). This feature of model~\\eqref{eq:growth} represents an assumption of limited biological realism. Under the same scenario, a more reasonable assumption is that migrants should pass from the patch with higher population density (absolute population size divided by patch size) to the patch with lower population density (Fig.~\\ref{fig:fig1}b).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.6\\textwidth]{fig1.pdf}\n\\caption{\\label{fig:fig1}Graphical representation of the flux of migrants in a two-patch population dynamics model. a): biologically unrealistic assumption of model\\eqref{eq:growth}, where the net flux of migrants occurs from the less dense (with higher absolute population size but with a much larger patch size) to the denser patch (b): more realistic assumption, with migrant flux from the more dense to the less dense patch}\n\\end{figure}\n\n\n\nWe propose to re-evaluate the perfect mixing paradox using a slightly different system. This model is both amenable for analysis and contains a more realistic assumption about the direction of the net flux of migrants.\n\n\\begin{equation}\\label{eq:growth2}\n\\begin{aligned}\n\\frac{dN_1}{dt} &=& r_1N_1\\left(1-\\frac{N_1}{K_1}\\right)\n+\\beta\\left(\\frac{N_2}{K_2}-\\frac{N_1}{K_1}\\right) \\\\\n\\frac{dN_2}{dt} &=& r_2N_2\\left(1-\\frac{N_2}{K_2}\\right)\n+\\beta\\left(\\frac{N_1}{K_1}-\\frac{N_2}{K_2}\\right)\n\\end{aligned}\n\\end{equation}\n\n\nIn this model the flux of migrants is governed by the differences between the ratios $N_i\/K_i$. We will refer to the ratio $N_i\/K_i$ as the \\emph{saturation of patch $i$}, which is a balance between the local population size at time $t$ and the local equilibrium population size $K_i$. The value of $K_i$ depends on the quantity and quality of resources in patch $i$. The direction of the net flux of migrants in this model captures the intuition described in Fig.~\\ref{fig:fig1}b. As shown below, this model does not exhibit the paradox presented in~\\cite{Arditi2016}.\n\nFirst, note that in isolation (i.e., with $\\beta = 0$), the system converges to ${N_1}^* = K_1,\\ \\ {N_2}^* = K_2$. This equilibrium is the same as the one of model~\\eqref{eq:growth}.\nUsing the same reasoning used in~\\cite{Arditi2016}, if we assume perfect mixing of local populations (i.e. with $\\beta\\rightarrow\\infty$) in\nmodel~\\eqref{eq:growth2}, it can be shown that for all $t>0$\n\n\\begin{equation}\\label{ratios}\n\\begin{aligned}\n\\frac{N_1}{K_1}=\\frac{N_2}{K_2}\n\\end{aligned}\n\\end{equation}\n\n\\noindent and therefore, for calculating the saturation of both patches combined:\n\n\\begin{equation}\\label{ratios2}\n\\begin{aligned}\n\\frac{N_1+N_2}{K_1+K_2} = \\frac{N_1\\frac{K_1}{K_1}+N_1\\frac{K_2}{K_1}}{K_1+K_2} = \\frac{N_1}{K_1} = \\frac{N_2}{K_2}\n\\end{aligned}\n\\end{equation}\n\nThis shows that total population saturation under perfect mixing is equal to each of the local population saturations.\nNow, let us check whether the main paradoxical property presented in~\\cite{Arditi2016} holds for our model~\\eqref{eq:growth2}. This implies checking whether or not the long term total population size under perfect mixing is equal to total population size in isolation. Adding both equations of system~\\eqref{eq:growth2} and using the equalities~\\eqref{ratios2} which are valid for the perfect mixing scenario, yields: \n\n\\begin{align}\n\\frac{dN_T}{dt} = \\frac{dN_1}{dt} + \\frac{dN_2}{dt} &= (r_1 N_1+r_2 N_2)\\left(1-\\frac{N_T}{K_T}\\right) \\nonumber\\\\\n&= \\frac{r_1 K_1+r_2 K_2}{K_T}\\left(1-\\frac{N_T}{K_T}\\right)N_T \\nonumber \\\\\n&= \\bar{r}\\left(1-\\frac{N_T}{K_T}\\right)N_T\\label{our N_T}\n\\end{align}\n\n\\noindent where \n$N_T=N_1+N_2$,\\ \\ $K_T=K_1+K_2$ and $\\bar{r}=\\dfrac{r_1 K_1+r_2 K_2}{K_T}$.\n\nIt is clear that, at equilibrium, the total population size under perfect mixing (i.e. with $\\beta\\rightarrow\\infty$) is $K_T = K_1+K_2$. Thus, using the more realistic model~\\eqref{eq:growth2} resolves the main paradoxical behavior presented in~\\cite{Arditi2016} for mixed patches. Note also that in the logistic equation for $N_T$ the total intrinsic growth rate $\\bar{r}$ is the weighted average of the local intrinsic growth rates, with weights $K_1$ and $K_2$. In the case that the patches differ only in their intrinsic growth rates $r_i$ and do not differ in their carrying capacities (i.e., $K_1 = K_2$), the total intrinsic growth rate reduces to $\\bar{r}=(r_1+r_2)\/2$. Also, if $r_1=r_2$ then $\\bar{r}=r_1=r_2$.\n\nAnother issue presented by Arditi et al.~\\cite{Arditi2016} is what they call an ``apparent spatial dependency'' of the equation parameters when the dynamics of the total population is represented by the Verhulst equation. The undesirable model property in a multi-patch context is that the value of the self-interference coefficient in the quadratic term decreases with number of patches $S$:\n\n\\begin{equation}\\label{Verhulst1}\n\\frac{dN_T}{dt}=\\bar{r}N_T-\\frac{\\bar\\alpha}{S} {N_T}^2\n\\end{equation}\n\nTo solve this issue, Arditi et al.\\cite{Arditi2016} suggest to treat population size as density, in terms of mean population size per patch $\\bar{N}=N_T\/S$. When doing so, Eqn.\\eqref{Verhulst1} becomes\n\n\\begin{equation}\\label{Verhulst2}\n\\frac{d\\bar{N}}{dt}=\\bar{r}\\bar{N}-\\bar{\\alpha}\\bar{N}^2\n\\end{equation}\n\n\\noindent which follows the Verhulst equation. Thus, the form of the equation is invariant in the number of patches in the metapopulation system and their parameters ($\\bar{r}$ and $\\bar{\\alpha}$) are simply the average of the corresponding local patch parameter values.\n\nIn our case, and under the same reasoning, considering the average population  in $S$ well-mixed patches, $\\bar{N}=N_T\/S$, Eqn.~\\eqref{eq:growth2} becomes:\n\n\\begin{equation}\\label{invariant2}\n\\frac{d\\bar{N}}{dt}=\\bar{r}\\left(1-\\frac{\\bar{N}}{\\bar{K}}\\right)\\bar{N}\n\\end{equation}\n\n\\noindent with $\\bar{K}=K_T\/S$. That is, the carrying capacity of the average population is the average of the local carrying capacities. Like Eqn.~\\eqref{Verhulst2}, our Eqn.~\\eqref{invariant2} is also invariant in the number patches, and their parameters ($\\bar{r}$ and $\\bar{K}$) are the weighted and aritmetic means, respectively, of the corresponding local patch parameters. Therefore there is no reason to favor the Verhulst's logistic equation over the classical formulation with the familiar $r-K$ parameterization in a multi-patch context, as argued by Arditi et al.~\\cite{Arditi2016}.\n\n\\section*{Discussion}\n\nThe paper by Arditi et al.\\cite{Arditi2016} argued that the logistic equation, in its usual $r-K$ parameterization, presents some undesirable properties when used in a multiple patch context. These properties configure what those authors called the ''perfect mixing paradox.'' Arditi et al.~\\cite{Arditi2016} also showed that the Verhulst's formulation of the logistic growth model $\\sfrac{dN}{dt}=rN-\\alpha N^2 $ is less prone to these paradoxical features, as compared with the familiar Lotka formulation $\\sfrac{dN}{dt}=rN(1-\\sfrac{N}{K})$, when generalized to a multi-patch environment. They conclude, on the basis of the analysis of these models extended to a metapopulation context by including a specific migration function, that the Verhulst formulation should be favored over the Lotka one, and that the term ``carrying capacity'' is misleading and should be abandoned in favor of the more correct ``equilibrium density.''\n\nThe supposedly paradoxical behavior of the metapopulation version of the Lotka-Gause model rests, according to Arditi et al.~\\cite{Arditi2016}, on two main features that were exemplified considering a two-patch environment as a study case. The first undesirable property is that the total mixed population equilibrium $K_T$ is in general different from the sum of the equilibria in the isolated patches $K_1 + K_2$. This major shortcoming of the analyzed model led Arditi et al.~\\cite{Arditi2016} to state that using the term ``carrying capacity'' is incorrect except in specific contexts. The second undesirable feature is the parameter dependence on the number of patches in the system, exhibited by the Verhulst form of the logistic growth model for the total population size. However, when population size is expressed as mean (per patch) abundance the model parameters can be calculated as the average of the local parameters and do not depend on the number of patches. Nevertheless, Arditi et al.~\\cite{Arditi2016} claim that this scale-invariance is only exhibited by the Verhulst model and this gives it an advantage over the Lotka-Gause model.\n\nIn this paper, we show that the paradoxical behaviors presented by Arditi et al.~\\cite{Arditi2016} belong only to the specific variant of the Lotka-Gause model they analyzed. Also, we suggest that the model used by Arditi et al.~\\cite{Arditi2016} is not the best choice regarding biological realism. In fact, we present a model as simple as the one they used (two state-variables, five parameters) that is more realistic and is free of the alluded paradoxes exhibited by the Arditi's extensions to both the Lotka-Gause and the Verhulst logistic models. \n\nThe most remarkable advantage of our model \\eqref{eq:growth2} is that, unlike both logistic forms used by Arditi et al.~\\cite{Arditi2016} in their analysis, total population size at equilibrium of a perfectly mixed metapopulation is equal to the sum of local equilibria. This feature immediately invalidates the criticism posed over the meaning and usefulness of the carrying capacity term. In our model \\eqref{our N_T}, global intrinsic growth rate of the metapopulation is not the arithmetic average of  local growth rates but it is equal to the weighted average of the local growth rate parameters. This is very reasonable, since under perfect mixing among patches, the ratios $N_i\/K_i$ are equated while their absolute abundances are not. So, it is possible to have patches with contrasting amount of resources (e.g. space or nutrients) and therefore  with unequal population abundances, say $3$ individuals in patch $1$ and $1000$ individuals in patch $2$. Under this scenario, global intrinsic growth rate could not be the arithmetic mean of the local growth rates but it should be closer to the parameter value of the larger population. In the case of the Arditi's model, the absolute population abundances tend to be the same under perfect mixing and so the arithmetic mean and weighted mean are the same. Regarding the second issue stressed by Arditi et al.~\\cite{Arditi2016}, we showed that our model \\eqref{our N_T} does not suffer from a lack of scale-invariance and that the dynamics of the per patch mean size of the metapopulation is fully consistent with the well known logistic dynamics within a single patch.\n\nIn sum, we show here that the criticisms posed by Arditi et al.~\\cite{Arditi2016} to the familiar form of the logistic equation attributed to Lotka and Gause are only valid for the particular way in which those authors extended that equation to the multi-patch scenario. We also suggest that their model is not the best choice among other plausible models of the same complexity, and that their criticisms against the usefulness of the carrying capacity as a measure of patch size or richness is not well justified. However, the paper by Arditi et al. has the value of highlighting that modeling population, metapopulation or community dynamics requires more attention than is usually given to and that models should not be applied to any scenario without a rigorous theoretical analysis of their properties.\n\n\\section*{Acknowledgments}\nThis study was supported by FONDECYT grant 1150348.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\\setcounter{equation}{0}\nA connected simply connected Lie group $N$\nwith center $Z$ is called {\\em square integrable} if it has unitary\nrepresentations $\\pi$ whose coefficients $f_{u,v}(x) = \n\\langle u, \\pi(x)v\\rangle$ satisfy $|f_{u,v}| \\in L^2(N\/Z)$.  \nC.C. Moore and the author worked out the structure and representation\ntheory of these groups \\cite{MW1973}.  If $N$ has one \nsuch square integrable representation then there is a certain polynomial\nfunction ${\\rm Pf}\\,(\\lambda)$ on the linear dual space $\\mathfrak{z}^*$ of the Lie algebra of\n$Z$ that is key to harmonic analysis on $N$.  Here ${\\rm Pf}\\,(\\lambda)$ is the\nPfaffian of the antisymmetric bilinear form on $\\mathfrak{n} \/ \\mathfrak{z}$ given by\n$b_\\lambda(x,y) = \\lambda([x,y])$.  The square integrable\nrepresentations of $N$ are the \n$\\pi_\\lambda$ where $\\lambda \\in \\mathfrak{z}^*$ with ${\\rm Pf}\\,(\\lambda) \\ne 0$,\nPlancherel  almost irreducible unitary representations of $N$ are square\nintegrable, and up to an explicit constant \n$|{\\rm Pf}\\,(\\lambda)|$ is the Plancherel density of the unitary\ndual $\\widehat{N}$ at $\\pi_\\lambda$.  \nThis theory has proved to have serious analytic consequences.  For example,\nfor most commutative nilmanifolds $G\/K$, i.e. Gelfand pairs $(G,K)$ \nwhere a nilpotent subgroup $N$ of $G$ acts transitively on $G\/K$, the\ngroup $N$ has square integrable representations \\cite{W2007}.\nAnd it is known just which maximal parabolic subgroups of semisimple Lie groups\nhave square integrable nilradical \\cite{W1979}.\n\\medskip\n\n\nIn \\cite{W2012} and \\cite{W2013} the theory of square integrable nilpotent\ngroups was extended to ``stepwise\nsquare integrable'' nilpotent groups.\nThey are the connected simply connected nilpotent Lie groups\nof (\\ref{setup}) just below.  We use $L$ and $\\mathfrak{l}$ to avoid conflict of\nnotation with the $M$ and $\\mathfrak{m}$ of minimal parabolic subgroups.\n\\begin{equation}\\label{setup}\n\\begin{aligned}\nN = &L_1L_2\\dots L_{m-1}L_m \\text{ where }\\\\\n &\\text{(a) each factor $L_r$ has unitary representations with coefficients in\n$L^2(L_r\/Z_r)$,} \\\\\n &\\text{(b) each } N_r := L_1L_2\\dots L_r \\text{ is a normal subgroup of } N\n   \\text{ with } N_r = N_{r-1}\\rtimes L_r \\text{ semidirect,}\\\\\n &\\text{(c) decompose }\\mathfrak{l}_r = \\mathfrak{z}_r + \\mathfrak{v}_r \\text{ and } \\mathfrak{n} = \\mathfrak{s} + \\mathfrak{v}\n        \\text{ as vector direct sums where } \\\\\n &\\phantom{XXXX}\\mathfrak{s} = \\oplus\\, \\mathfrak{z}_r \\text{ and } \\mathfrak{v} = \\oplus\\, \\mathfrak{v}_r;\n    \\text{ then } [\\mathfrak{l}_r,\\mathfrak{z}_s] = 0 \\text{ and } [\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{v}\n        \\text{ for } r > s\\,.\n\\end{aligned}\n\\end{equation}\nDenote\n\\begin{equation}\\label{c-d}\n\\begin{aligned}\n&\\text{(a) }d_r = \\tfrac{1}{2}\\dim(\\mathfrak{l}_r\/\\mathfrak{z}_r) \\text{ so }\n        \\tfrac{1}{2} \\dim(\\mathfrak{n}\/\\mathfrak{s}) = d_1 + \\dots + d_m\\,,\n        \\text{ and } c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!\\\\\n&\\text{(b) }b_{\\lambda_r}: (x,y) \\mapsto \\lambda([x,y])\n        \\text{ viewed as a bilinear form on } \\mathfrak{l}_r\/\\mathfrak{z}_r \\\\\n&\\text{(c) }S = Z_1Z_2\\dots Z_m = Z_1 \\times \\dots \\times Z_m \\text{ where } Z_r\n        \\text{ is the center of } L_r \\\\\n&\\text{(d) }{\\rm Pf}\\,: \\text{ polynomial } {\\rm Pf}\\,(\\lambda) = {\\rm Pf}\\,_{\\mathfrak{l}_1}(b_{\\lambda_1})\n        {\\rm Pf}\\,_{\\mathfrak{l}_2}(b_{\\lambda_2})\\dots {\\rm Pf}\\,_{\\mathfrak{l}_m}(b_{\\lambda_m}) \\text{ on } \n\t\\mathfrak{s}^* \\\\\n&\\text{(e) }\\mathfrak{t}^* = \\{\\lambda \\in \\mathfrak{s}^* \\mid {\\rm Pf}\\,(\\lambda) \\ne 0\\} \\\\\n&\\text{(f) } \\pi_\\lambda \\in \\widehat{N} \\text{ where } \\lambda \\in \\mathfrak{t}^*:\n    \\text{ irreducible unitary rep. of } N = L_1L_2\\dots L_m\n\\end{aligned}\n\\end{equation}\nThe basic result for these groups is\n\n\\begin{theorem}\\label{plancherel-general} {\\rm \\cite[Theorem 6.16]{W2013}}\nLet $N$ be a connected simply connected nilpotent Lie group that\nsatisfies {\\rm (\\ref{setup})}.  Then Plancherel measure for $N$ is\nconcentrated on $\\{\\pi_\\lambda \\mid \\lambda \\in \\mathfrak{t}^*\\}$.\nIf $\\lambda \\in \\mathfrak{t}^*$, and if $u$ and $v$ belong to the\nrepresentation space $\\mathcal{H}_{\\pi_\\lambda}$ of $\\pi_\\lambda$,  then\nthe coefficient $f_{u,v}(x) = \\langle u, \\pi_\\nu(x)v\\rangle$\nsatisfies\n\\begin{equation}\\label{sq-orthogrel}\n||f_{u,v}||^2_{L^2(N \/ S)} = \\frac{||u||^2||v||^2}{|{\\rm Pf}\\,(\\lambda)|}\\,.\n\\end{equation}\nThe distribution character $\\Theta_{\\pi_\\lambda}$ of $\\pi_{\\lambda}$ satisfies\n\\begin{equation}\\label{def-dist-char}\n\\Theta_{\\pi_\\lambda}(f) = c^{-1}|{\\rm Pf}\\,(\\lambda)|^{-1}\\int_{\\mathcal{O}(\\lambda)}\n        \\widehat{f_1}(\\xi)d\\nu_\\lambda(\\xi) \\text{ for } f \\in \\mathcal{C}(N)\n\\end{equation}\nwhere $\\mathcal{C}(N)$ is the Schwartz space, $f_1$ is the lift\n$f_1(\\xi) = f(\\exp(\\xi))$ of $f$ from $N$ to $\\mathfrak{n}$, \n$\\widehat{f_1}$ is its classical Fourier transform,\n$\\mathcal{O}(\\lambda)$ is the coadjoint orbit ${\\rm Ad}\\,^*(N)\\lambda = \\mathfrak{v}^* + \\lambda$,\n$c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)},\nand $d\\nu_\\lambda$ is the translate of normalized Lebesgue measure from\n$\\mathfrak{v}^*$ to ${\\rm Ad}\\,^*(N)\\lambda$.  The Fourier inversion formula on $N$ is\n\\begin{equation}\nf(x) = c\\int_{\\mathfrak{t}^*} \\Theta_{\\pi_\\lambda}(r_xf) |{\\rm Pf}\\,(\\lambda)|d\\lambda\n        \\text{ for } f \\in \\mathcal{C}(N).\n\\end{equation}\n\\end{theorem}\n\\begin{definition}\\label{stepwise2}\n{\\rm The representations $\\pi_\\lambda$ of (\\ref{c-d}(f)) are the\n{\\it stepwise square integrable} representations of $N$ relative to\nthe decomposition (\\ref{setup}).}\\hfill $\\diamondsuit$\n\\end{definition}\n\nOne of the main results of \\cite{W2012} and \\cite{W2013} is that nilradicals\nof minimal parabolic subgroups of finite dimensional real reductive Lie\ngroups are stepwise square integrable.  Even the\nsimplest case, the case of a minimal parabolic in $SL(n;\\mathbb{R})$, was a big\nimprovement over earlier results on the group of strictly upper triangular\nreal matrices.  Here we extend the construction of stepwise square integrable\nrepresentations to a class of locally nilpotent groups that are direct\nlimits in a manner that respects the basic setup (\\ref{setup}) of the\nfinite dimensional case, and we show how this applies to the nilradicals\nof direct limit minimal parabolic subgroups of the real and complex finitary\nreductive Lie groups, including $GL(\\infty;\\mathbb{F})$, $SL(\\infty;\\mathbb{F})$, $U(p,q;\\mathbb{F})$\nand $SU(p,q;\\mathbb{F})$ ($\\mathbb{F} = \\mathbb{R}, \\mathbb{C}$ or $\\mathbb{H}$ and $p+q = \\infty$), $Sp(\\infty;\\mathbb{F})$ \n($\\mathbb{F} = \\mathbb{R}$ or $\\mathbb{C}$), and $SO^*(2\\infty)$.\n\\medskip\n\nIn Section \\ref{sec2} we examine strict direct systems $\\{N_n, \\varphi_{m,n}\\}$\nof finite dimensional connected and simply connected nilpotent Lie groups\nthat satisfy (\\ref{setup}) in a manner that respects the maps\n$\\varphi_{m,n}: N_n \\to N_m$\\, $(m \\geqq n)$.  We show how this \nleads to sequences $\\{\\pi_{\\gamma_n}\\}$ of closely related stepwise square\nintegrable representations of the groups $N_n$\\,,  and then to their\nunitary representation limits $\\pi_\\gamma = \\varprojlim \\pi_{\\gamma_n}$\\,.\n\\medskip\n\nIn Section \\ref{sec3}\nwe prove stepwise Frobenius-Schur orthogonality relations and restriction\ntheorems for the coefficients of the representations $\\pi_{\\gamma_n}$\\,.\n\\medskip\n\nIn Section \\ref{sec4} we apply the tools of Section \\ref{sec3} to obtain\ninverse systems, by restriction, of the spaces\n$\\mathcal{A}(\\pi_{\\gamma_n})$ of coefficients of the representations $\\pi_{\\gamma_n}$\\,.\nThen we combine density of $\\mathcal{A}(\\pi_{\\gamma_n})$ in\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$ with the\nrenormalization method of \\cite{W2010} to construct inverse systems,\nin the Hilbert space category, of the\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\\,.  These mirror the\ninverse systems of the $\\mathcal{A}(\\pi_{\\gamma_n})$, resulting in an\ninterpretation of the function space $\\mathcal{A}(\\pi_\\gamma) = \n\\varprojlim \\mathcal{A}(\\pi_{\\gamma_n})$ as\na dense subspace of the Hilbert space $\\mathcal{H}_{\\pi_\\gamma}\n\\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^* =\n\\varprojlim \\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\\,.\nThis is somewhat analogous to the infinite dimensional\nPeter---Weyl Theorem of \\cite[Section 4]{W2009}.\n\\medskip\n\nIn Section \\ref{sec5} we set up the Schwartz space machinery that will allow us \nto carry over the somewhat abstract $\\mathcal{H}_{\\pi_\\gamma}\n\\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^* =\n\\varprojlim \\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\nto an explicit Fourier inversion formulae.  This, incidentally, strengthens\nthe stepwise $L^2$ property for coefficients involving $C^\\infty$ vectors\nfrom $L^2$ to $L^1$.\n\\medskip\n\nIn Section \\ref{sec6} we work out that formula for the direct limit\ngroup $N = \\varinjlim N_n$\\,.  See Theorem \\ref{limit-inversion}.\n\\medskip\n\nIn Section \\ref{sec7} we discuss direct systems $\\{G_n , \\varphi_{m,n}\\}$\nof finite dimensional real reductive Lie groups, and conditions on their\nrestricted root systems $\\Delta(\\mathfrak{g}_n,\\mathfrak{a}_n)$, that lead to an appropriate \nlimit restricted root system\n$\\Delta(\\mathfrak{g},\\mathfrak{a}) = \\varprojlim \\Delta(\\mathfrak{g}_n,\\mathfrak{a}_n)$ of the Lie\nalgebra of $G = \\varinjlim \\{G_n , \\varphi_{m,n}\\}$.  That describes the\nstepwise square integrable structure of the nilradicals of minimal parabolic\nsubgroups.  \n\\medskip\n\nFinally, in Section \\ref{sec8}, we arrive at the goal of this paper,\nTheorem \\ref{inversion-for-ss-limits},\nan explicit Fourier inversion formula for the classical \ndirect limit of the nilradicals of those minimal parabolics. \n\\medskip\n\nI thank Michael Christ for useful discussions of Schwartz spaces related\nto the Heisenberg group.\n\n\\section{Alignment and Construction}\n\\label{sec2}\n\\setcounter{equation}{0}\nFor our direct limit considerations it will be necessary to adjust the\ndecompositions (\\ref{setup}) of the connected simply connected nilpotent\nLie groups $N_n$\\,.  This is so that the adjusted decompositions will\nfit together as $n$ increases.  We do that by reversing the indices and\nkeeping the $L_r$ constant as $n$ goes to infinity.  First, we suppose that\n\\begin{equation}\\label{nil-direct-system}\n\\{N_n\\} \\text{ is a strict direct system of connected simply connected \n\tnilpotent Lie groups,}\n\\end{equation}\nin other words  the connected simply connected nilpotent Lie groups\n$N_n$ have the property that $N_n$ is a closed analytic subgroup of $N_\\ell$\nfor all $\\ell \\geqq n$.  As usual, $Z_r$ denotes the center of $L_r$\\,.\n\t\tFor each $n$, we require that\n\\begin{equation}\\label{newsetup}\n\\begin{aligned}\nN_n = L_1&L_2\\cdots L_{m_n} \\text{ where }\\\\\n \\text{(a) }&L_r \\text{ is a closed analytic subgroup of } N_n \\text{ for }\n\t1 \\leqq r \\leqq m_n\\,; \\\\\n \\text{(b) }&\\text{each factor $L_r$ has unitary representations with \n\tcoefficients in $L^2(L_r\/Z_r)$;} \\\\\n \\text{(x) }&\\text{let $L_{p,q} = L_{p+1}L_{p+2}\\cdots L_q$  ($p < q$)\n\tand $N_{\\ell,n} = L_{m_\\ell +1}L_{m_\\ell +2}\\cdots L_{m_n}\n\t= L_{m_\\ell,m_n}$ ($\\ell < n$);}\\\\\n \\text{(c) }&\\text{each } N_{\\ell ,n} \n\t\\text{ is a normal subgroup of } N_n \n   \\text{ and } N_n = N_r \\ltimes N_{r+1,n} \\text{ semidirect product}; \\\\\n \\text{(d) }&\\text{decompose }\\mathfrak{l}_r = \\mathfrak{z}_r + \\mathfrak{v}_r \\text{ and } \n\t\\mathfrak{n}_n = \\mathfrak{s}_n + \\mathfrak{u}_n\n        \\text{ as vector space direct sums where } \\\\\n &\\mathfrak{s}_n = {\\bigoplus}_{r \\leqq m_n}\\, \\mathfrak{z}_r \\text{ and } \n\t\\mathfrak{u}_n = {\\bigoplus}_{r \\leqq m_n}\\, \\mathfrak{v}_r;\n    \\text{ then } [\\mathfrak{l}_r,\\mathfrak{z}_s] = 0 \\text{ and } [\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{v}\n        \\text{ for } r < s\\,.\n\\end{aligned}\n\\end{equation}\nWith this setup we can follow the lines of the constructions in\n\\cite[Section 5]{W2013}.\n\\medskip\n\nWe have the Pfaffian polynomials on the $\\mathfrak{z}_r^*$ and on $\\mathfrak{s}_n^*$\nas follows.  Given $\\lambda_r \\in \\mathfrak{z}_r^*$,\nextended to an element of $\\mathfrak{l}_r^*$ by $\\lambda_r(\\mathfrak{v}_r) = 0$,\nwe have the antisymmetric bilinear form $b_{\\lambda_r}$ on $\\mathfrak{l}_r\/\\mathfrak{z}_r$\ndefined as usual by $b_{\\lambda_r}(x,y) = \\lambda_r([x,y])$, and\n${\\rm Pf}\\,_r(\\lambda_r)$ denotes its Pfaffian.  If $\\gamma_n = \\lambda_1 +\n\\cdots + \\lambda_{m_n} \\in \\mathfrak{s}_n^*$ with each $\\lambda_r \\in \\mathfrak{z}_r^*$, then\nwe have the product\n\\begin{equation}\nP_n(\\gamma_n) = {\\rm Pf}\\,_1(\\lambda_1){\\rm Pf}\\,_2(\\lambda_2) \\cdots {\\rm Pf}\\,_{m_n}(\\lambda_{m_n})\n\\end{equation}\nand the nonsingular set\n\\begin{equation}\n\\mathfrak{t}_n^* = \\{\\gamma_n \\in \\mathfrak{s}_n^* \\mid P_n(\\gamma_n) \\ne 0\\}.\n\\end{equation}\n\nRecall the construction (\\cite{W2013}) of stepwise square integrable \nrepresentations $\\pi_{\\gamma_n}$ of $N_n$\\,, where $\\gamma_n \\in \\mathfrak{t}_n^*$\\,,\nand where we adjust the indices to our situation.\nIf $m_n = 1$ then $\\pi_{\\gamma_n}$ is just the square integrable representation\n$\\pi_{\\lambda_1}$ of $L_1$ defined by $\\gamma_n = \\lambda_1$\\,.  Now let\n$m_n > 1$ and use $N_n = (L_1L_2\\dots L_{m_n-1}) \\ltimes L_{m_n}\n= L_{0,m_n-1} \\ltimes L_{m_n}$\\,.\nBy induction on $m_n$ we have the stepwise square integrable representation \n$\\pi_{\\lambda_1 + \\cdots + \\lambda_{m_n-1}}$ of $L_{0,m_n-1}$\\,, and we view it\nas a representation of $N_n$ whose kernel contains $L_{m_n}$\\,.  We also\nhave the square integrable representation $\\pi_{\\lambda_{m_n}}$ of $L_{m_n}$\\,.\nWrite $\\pi'_{\\lambda_{m_n}}$ for the extension of $\\pi_{\\lambda_{m_n}}$ to\na unitary representation of $N_n$ on the same Hilbert space \n$\\mathcal{H}_{\\pi_{\\lambda_{m_n}}}$ (the Mackey obstruction vanishes).  Now\n\\begin{equation}\\label{def-sq-reps}\n\\pi_{\\gamma_n} = \\pi_{\\lambda_1 + \\cdots + \\lambda_{m_n-1}} \\widehat{\\otimes}\n        \\pi'_{\\lambda_{m_n}}\\,.\n\\end{equation}\n\\medskip\n\nThe parameter space for our representations of the direct limit Lie group\n$N = \\varinjlim N_n$ will be\n\\begin{equation}\n\\mathfrak{t}^* = \\bigcup_{n > 0} \\left\\{\\gamma = \\sum \\gamma_\\ell \\in \\mathfrak{s}^* \\mid\n\t\\gamma_\\ell \\in \\mathfrak{t}_\\ell^* \\text{ for } \\ell \\leqq n\n\t\\text{ and } \\gamma_\\ell = 0 \\in \\mathfrak{s}_\\ell^*\\text{ for } \n\t\\ell > n\\right\\}\n\\text{ where } \\mathfrak{s}^* := \\sum_{\\ell > 0} \\mathfrak{s}_\\ell^* \\,.\n\\end{equation}\nThe representations $\\pi_\\gamma$ of $N$ are defined in a manner similar \nto that of (\\ref{def-sq-reps}).  Given $\\gamma = \\sum \\gamma_\\ell \\in \\mathfrak{t}^*$ \nwe have the index $n = n(\\gamma)$ defined by\n$\\gamma_\\ell \\in \\mathfrak{t}_\\ell^* \\text{ for } \\ell \\leqq n(\\gamma)\n        \\text{ and } \\gamma_\\ell = 0 \\in \\mathfrak{s}_\\ell^*\\text{ for }\n        \\ell > n(\\gamma)$.\n\\begin{equation}\nN = N_{n(\\gamma)} \\ltimes N_{n(\\gamma),\\infty} \\text{ semidirect product, where }\n\tN_{n(\\gamma),\\infty} = {\\prod}_{q > m_{n(\\gamma)}} L_q\\,.\n\\end{equation}\nIn particular the closed normal subgroup $N_{n(\\gamma),\\infty}$ satisfies\n$N_{n(\\gamma)} \\cong N\/N_{n(\\gamma),\\infty}$, and we denote\n\\begin{equation} \\label{def-sq-lim-reps}\n\\pi_\\gamma \\text{ is the lift to $N$ of the stepwise square integrable\nrepresentation } \\pi_{\\gamma_1 + \\cdots + \\gamma_{m_{n(\\gamma)}}} \\text{ of } \nN_{n(\\gamma)}\\,.\n\\end{equation}\nThe representation space of $\\pi_\\gamma$ is the projective (jointly continuous)\ntensor product\n\\begin{equation}\\label{gamma-decomp}\n\\mathcal{H}_{\\pi_\\gamma} = \\mathcal{H}_{\\pi_{\\gamma_1}} \\widehat{\\otimes} \n\t\\mathcal{H}_{\\pi_{\\gamma_2}}\n\t\\widehat{\\otimes} \\cdots \\widehat{\\otimes} \n\t\\mathcal{H}_{\\pi_{\\gamma_{n(\\gamma)}}}\n\\end{equation}\n\nThese representations $\\pi_\\gamma$ are the {\\sl limit stepwise square \nintegrable} representations of $N$.  We go on the see the extent to which\ntheir coefficients and characters imitate the properties of Theorem\n\\ref{plancherel-general}.\n\n\\section{Coefficient Functions}\n\\label{sec3}\n\\setcounter{equation}{0}\nLet $\\mathcal{H}_{\\pi_\\gamma}$ denote the representation space of $\\pi_\\gamma$ and\n$\\langle\\,\\,\\cdot\\, , \\,\\cdot\\,\\,\\rangle_{\\pi_\\gamma}$ the hermitian inner \nproduct on $\\mathcal{H}_{\\pi_\\gamma}$\\,.  Given $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ we have the\ncoefficient function on $N$ given by\n\\begin{equation}\nf_{\\pi_\\gamma,u,v}(g) = \\langle u, \\pi_\\gamma(g)v\\rangle_{\\pi_\\gamma}\\,.\n\\end{equation}\nWe use the standard $(r(x)f)(g) = f(gx)$ and $(\\ell(y)f)(g) = f(y^{-1}g)$.  \nThese right and left translations commute with each other.  They are \nwell defined on the $f_{\\pi_\\gamma,u,v}$ and satisfy\n\\begin{equation}\n\\ell(x) r(y): f_{\\pi_\\gamma,u,v} \\mapsto f_{\\pi_\\gamma, \\pi_\\gamma(x)u,\n\t\\pi_\\gamma(y)v}\\,.\n\\end{equation}\nBy our construction (\\ref{def-sq-lim-reps}), the value \n$f_{\\pi_\\gamma,u,v}(g)$ depends only on the coset $gN''_{n(\\gamma)}$.  In\nother words it really is a function on $N_{n(\\gamma)} \\cong N\/N''_{n(\\gamma)}$. \nFurther, $|f_{\\pi_\\gamma,u,v}(g)|$ depends only on the coset \n$gS_{n(\\gamma)}N''_{n(\\gamma)}$ where $S_{n(\\gamma)}$ is the quasicenter\n$Z_1Z_2\\cdots Z_{m_{n(\\gamma)}}$\nof $N_{n(\\gamma)} = L_1L_2\\cdots L_{m_{n(\\gamma)}}$\\,.\nBuilding on (\\ref{sq-orthogrel}), we have the following variation on the \nFrobenius-Schur orthogonality relations for finite groups:\n\\begin{proposition}\\label{sq-orthogrel-quo}\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Then\n$||f_{\\pi_\\gamma,u,v}||^2_{L^2(N \/ S_n N''_n)} \n\t= \\frac{||u||_{\\pi_\\gamma}^2||v||_{\\pi_\\gamma}^2}{|P_n(\\gamma)|}$\\,\\,.\n\\end{proposition}\n\n\\begin{proof}\nThis is an induction on $n$.  The case $n = 1$ is (\\ref{sq-orthogrel}).\nNow go from $n$ to $n+1$.  Express $N_{n+1} = N_n \\ltimes N_{n,n+1}$ where\n$$\nN_n = L_1L_2\\cdots L_{m_n} \\text{ and }\n\tN_{n,q} = L_{m_n+1}L_{m_n+2}\\cdots L_{m_q}\n\t\\text{ for } q > n.\n$$\nThen $S_{n+1} = S_n \\times S_{n,n+1}$ where the quasi-centers\n$$\nS_n = Z_1Z_2\\cdots Z_{m_n} \\text{ and }\n\tS_{n,q} = Z_{m_n+1}Z_{m_n+2}\\cdots Z_{m_q}\n\t\\text{ for } q > n.\n$$\nNow let $\\gamma_n \\in \\mathfrak{t}_n^*$ and $\\gamma_{n,n+1} \\in \\mathfrak{t}_{n,n+1}^*$\nwhere, as before, $\\mathfrak{t}^*$ is the nonzero set of the Pfaffian in $\\mathfrak{s}^*$.\nNote that $\\pi_{\\gamma_n} \\in \\widehat{N_n}$ and $\\pi_{\\gamma_{n,n+1}}\n\\in \\widehat{N_{n,n+1}}$ are stepwise square integrable.  Write\n$\\pi'_{\\gamma_{n,n+1}}$ for the extension of $\\pi_{\\gamma_{n,n+1}}$ from\n$N_{n,n+1}$ to $N_{n+1}$\\,.  Let\n$u,v \\in \\mathcal{H}_{\\pi_{\\gamma_n}}$ and $x,y \\in \\mathcal{H}_{\\pi_{\\gamma_{n,n+1}}}$\nso $u\\otimes x, v \\otimes y \\in \\mathcal{H}_{\\pi_{\\gamma_{n+1}}}$\\,.  Let\n$a$ run over $N_n$ and let $b$ run over $N_{n,n+1}$\\,.  Compute\n$$\n\\begin{aligned}\n&||f_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}||^2_{L^2(N_{n+1}\/S_{n+1})} \\\\\n&\\phantom{XX}= \\int_{N_{n+1}\/S_{n+1}} |\\langle u\\otimes x, (\\pi_{\\gamma_n}\n\t\\widehat{\\otimes} \\pi'_{\\gamma_{n,n+1}})(ab)(v\\otimes y)\\rangle|^2\n\tda\\,db\\\\\n&\\phantom{XX}= \\int_{N_{n+1}\/S_{n+1}} |\\langle u\\otimes x,\n\t\\pi_{\\gamma_n}(a)\\pi'_{\\gamma_{n,n+1}}(b)(v) \\otimes\n\t\\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 da\\,db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \\left ( \\int_{N_n\/S_n}\n\t|\\langle u, \\pi_{\\gamma_n}(a)\\pi'_{\\gamma_{n,n+1}}(b)(v)\\rangle |^2\n\tda \\right )\n\t\\cdot |\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \n\t\\frac{||u||^2 ||\\pi'_{\\gamma_{n,n+1}}(b)(v)||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\n\t|\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \n        \\frac{||u||^2 ||v||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\n        |\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\frac{||u||^2 ||v||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\\cdot\n\t\\frac{||x||^2 ||y||^2}{|{\\rm Pf}\\,_{n,n+1}(\\gamma_{n,n+1})|} = \n\t\\frac{||u\\otimes x||^2 ||v \\otimes y||^2}{|{\\rm Pf}\\,_{n+1}(\\gamma_{n+1})|}.\n\\end{aligned}\n$$\nThe proposition follows.\n\\end{proof}\n\nIn the notation of the proof of Proposition \\ref{sq-orthogrel-quo},\n\\begin{equation}\\label{restr}\nf_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}(a) \n        = \\langle u, \\pi_{\\gamma_n}(a)v\\rangle\\cdot \\langle x,y\\rangle\n        = \\langle x,y\\rangle f_{\\pi_{\\gamma_n},u,v}(a)\n        \\text{ for } a \\in N_n\\,.\n\\end{equation}\nIn other words, $f_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}|_{N_n}\n= \\langle x,y\\rangle f_{\\pi_{\\gamma_n},u,v}$\\,.  In particular the case\nwhere $x = e = y$, where $e$ is a unit vector, is\n\\begin{equation}\\label{restr1}\nf_{\\pi_{\\gamma_{n+1}},u\\otimes e,v\\otimes e}|_{N_n} = f_{\\pi_{\\gamma_n},u,v}\n\\end{equation}\n\\medskip\n\nIterating this and combining it with Proposition \\ref{sq-orthogrel-quo}\nwe arrive at\n\n\\begin{proposition}\\label{rescale}\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Let $\\gamma' \\in \\mathfrak{t}^*$\nand $n' = n(\\gamma')$ with $n' > n$ and $\\gamma'|_{\\mathfrak{s}_n} = \\gamma$.  Then \n$\\pi_{\\gamma'}|_{N_n}$ is an infinite multiple of $\\pi_\\gamma$.\nSplit $\\mathcal{H}_{\\pi_{\\gamma'}} = \\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes}\\mathcal{H}''$ where\n$\\mathcal{H}'' = \\mathcal{H}_{\\pi_{\\gamma'_{n+1}}} \\widehat{\\otimes} \\cdots \\widehat{\\otimes}\n\\mathcal{H}_{\\pi_{\\gamma'_{n'}}}$ in the notation of {\\rm (\\ref{gamma-decomp})}.  \nChoose a unit vector $e \\in \\mathcal{H}''$.  Then\n\\begin{equation}\\label{unitfactor}\n\\mathcal{H}_{\\pi_{\\gamma}} \\hookrightarrow \\mathcal{H}_{\\pi_{\\gamma'}} \\text{ by }\nv \\mapsto v\\otimes e\n\\end{equation}\nis a well defined $N_n$--equivariant isometric injection.  If\n$u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ then\n\\begin{equation}\\label{rescale1}\n||f_{\\pi_{\\gamma'},u\\otimes e,v\\otimes e}||^2_{L^2(\n\tN \/ S_{n'} N''_{n'})} \n\t= \\frac{|P_n(\\gamma)|}{|P_{n'}(\\gamma')|}\n\t||f_{\\pi_\\gamma,u,v}||^2_{L^2(N \/ S_n N''_n)}\\,.\n\\end{equation}\n\\end{proposition}\n\n\\section{Hilbert Space Limits}\n\\label{sec4}\n\\setcounter{equation}{0}\nNow we combine the restriction maps of Section \\ref{sec3}.\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Then $\\gamma$ defines a\nunitary character\n\\begin{equation}\\label{defrelchar}\n\\zeta_\\gamma = \\exp(2\\pi i \\gamma) \\text{ by } \\zeta_\\gamma(\\exp(\\xi)y) = \n\te^{2\\pi i \\gamma(\\xi)} \\text{ where } \\xi \\in \\mathfrak{s}_n \\text{ and }\n\ty \\in N''_n\\,.\n\\end{equation}\nThat defines the Hilbert space\n\\begin{equation}\\label{defrelHilbert}\n\\begin{aligned}\nL^2&(N\/S_nN''_n,\\zeta_\\gamma) = \\\\\n&\\{f:N \\to \\mathbb{C} \\mid f(gx) = \\zeta_\\gamma(x)^{-1}\n\tf(g) \\text{ and } |f| \\in L^2(N \/ S_n N''_n) \\text{ for all }\n\tg \\in N \\text{ and } x \\in N''_n\\}.\n\\end{aligned}\n\\end{equation}\nThe finite linear combinations of the coefficients \n$f_{\\pi_\\gamma, u, v}$ (where $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$) form a dense\nsubspace of $L^2(N\/S_nN''_n,\\zeta_n$), and that gives an\n$N \\times N$ equivariant Hilbert space isomorphism\n\\begin{equation}\nL^2(N\/S_nN''_n,\\zeta_\\gamma) \\cong \n\t\\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^*.\n\\end{equation}\n\\medskip\n\nWe know that the stepwise square integrable group $N_n = N\/N_n''$ satisfies\n\\begin{equation}\nL^2(N_n) = L^2(N\/N_n'') \n= \\int_{\\gamma \\in \\mathfrak{t}^*\\,\\,  and\\,\\, n(\\gamma) = n}\n   (\\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^*) |P_n(\\gamma)|d\\gamma.\n\\end{equation}\nIn brief, that expands the functions on $N$ that depend only on the first\n$m(n)$ factors in $N = N_1N_2N_3\\cdots$\\,.  To expand the functions that\ndepend on more factors, say the first $m(n')$ factors in the notation of\nProposition \\ref{rescale}, we would like to inject\n$$L^2(N\/N_n'') = \n  \\int_{\\gamma \\in \\mathfrak{t}_n^*} L^2(N\/S_nN''_n,\\zeta_\\gamma)|P_n(\\gamma)|d\\gamma\n$$\ninto\n$$ \nL^2(N\/N_{n'}'') = \n  \\int_{\\gamma' \\in \\mathfrak{t}_{n'}^*}L^2(N\/S_{n'}N''_{n'},\\zeta_{\\gamma'})\n\t|P_{n'}(\\gamma')|d\\gamma'\n$$\nusing the renormalizations of (\\ref{rescale1}).  However, $\\gamma$ has\nmany extensions $\\gamma'$ with the given $n(\\gamma') = n'$, so this will\nnot work directly.  But we can take the orthogonal projections dual to\nthe injections of (\\ref{rescale1}) and form an inverse system of Hilbert\nspaces.  \n\\medskip\n\nTo start, if $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ and $x, y \\in \\mathcal{H}''$, using\n(\\ref{restr}) and Proposition \\ref{rescale}, \n\\begin{equation}\\label{rescale2}\np_{\\gamma',\\gamma}: f_{\\pi_{\\gamma'},u\\otimes x,v\\otimes y} \\mapsto\n\\langle x, y \\rangle \\left |\\frac{P_n(\\gamma)}{P_{n'}(\\gamma')}\\right |^{1\/2} \nf_{\\pi_\\gamma,u,v}\n\\end{equation}\nis the orthogonal projection dual to the isometric inclusion (\\ref{unitfactor}).\nSince $\\gamma$ is the restriction of $\\gamma'$ from $\\mathfrak{s}_{n(\\gamma')}$ to\n$\\mathfrak{s}_{n(\\gamma)}$ we can reformulate (\\ref{rescale2}) as\n\\begin{equation}\\label{rescale3}\np_{\\gamma',n} : f_{\\pi_{\\gamma'},u\\otimes x,v\\otimes y} \\mapsto\n\t\\langle x, y \\rangle \n\t\\left | \\frac{P_n(\\gamma'|_{\\mathfrak{s}_n})}{P_{n'}(\\gamma')} \\right |^{1\/2}\n\tf_{\\pi_{\\gamma'|_{\\mathfrak{s}_n},u,v}} \\text{ where } n = n(\\gamma).\n\\end{equation}\nThe maps $p_{\\gamma,n}$ of (\\ref{rescale3}) sum to a Hilbert space projection,\nessentially restriction of coefficients,\n\\begin{equation}\\label{rescale4}\np_{n',n} = \\left ( \\int_{\\gamma'\\in\\mathfrak{s}^*_{n'}}p_{\\gamma',n}\\,d\\gamma'\\right ) : \n\tL^2(N_{n'}) \\to\n\tL^2(N_n) \\text{ where } n = n(\\gamma'|_{\\mathfrak{s}_n})\n\t\\text{ and } n' = n(\\gamma') \\geqq n.\n\\end{equation}\nThe maps $p_{n',n}$ of (\\ref{rescale4}) define an inverse system in the\ncategory of Hilbert spaces and partial isometries:\n\\begin{equation}\\label{rescale5}\nL^2(N_1) \\overset{p_{2,1}}{\\longleftarrow} L^2(N_2) \n\t\\overset{p_{3,2}}{\\longleftarrow}  L^2(N_3) \n\t\\overset{p_{4,3}}{\\longleftarrow} \\,\\, ... \\,\\,\\,\\longleftarrow\\,\n\tL^2(N)\n\\end{equation}\nwhere the projective limit $L^2(N) := \\varprojlim \\{L^2(N_n),p_{n',n}\\}$ \nis taken in the category\nof Hilbert spaces and partial isometries.  We now have the Hilbert space\n\\begin{equation}\\label{rescale6}\nL^2(N) := \\varprojlim \\{L^2(N_n), p_{n',n}\\}.\n\\end{equation}\n\n\\section{The Schwartz Spaces}\n\\label{sec5}\n\\setcounter{equation}{0}\nIn order to refine (\\ref{rescale6}) to a Fourier\ninversion formula we must first\nmake it more explicit.  The span $\\mathcal{A}(\\pi_{\\gamma_n})$ of the \ncoefficients of the representation $\\pi_{\\gamma_n}$ is dense in\nthe space of functions on $N_n$ given by\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}$\\,. \nThe idea in the background here is to realize Schwartz class functions\nas wave packets\n$\nf(a) = \\int_{\\mathfrak{s}_n^*} \\varphi(\\gamma_n) f_{\\pi_{\\gamma_n},u(\\gamma_n),\nv(\\gamma_n)}(a)d\\gamma_n\n$\nwhere $\\varphi$ is a Schwartz class function on $\\mathfrak{s}_n$ and where \n$u(\\gamma_n)$ and $v(\\gamma_n)$ are fields of \n$C^\\infty$ unit vectors in the $\\mathcal{H}_{\\pi_{\\gamma_n}}$.\nMore concretely we show that the coefficient\n$f_{\\pi_{\\gamma_n,u,v}}$ belongs to an appropriate Schwartz space (and thus\nan appropriate $L^1$ space) when $u$ and $v$ are $C^\\infty$ vectors for\n$\\pi_{\\gamma_n}$\\,.\n\\medskip\n\nWe first collect some standard facts from Kirillov theory concerning the \nanalog of the Schr\\\" odinger representation of the Heisenberg group.\nLet $L$ be a connected simply connected nilpotent Lie group that has\nsquare integrable representations.  $Z$ is the center of $L$, and\n$\\lambda \\in \\mathfrak{z}^*$ with ${\\rm Pf}\\,_\\mathfrak{l}(\\lambda) \\ne 0$.  Let $\\mathfrak{p}$ and $\\mathfrak{q}$ \nbe totally real polarizations for $\\lambda$, $\\mathfrak{p} = \\mathfrak{z} + \\mathfrak{a}$ and\n$\\mathfrak{q} = \\mathfrak{z} + \\mathfrak{b}$, and suppose that we chose them so that \n$b_\\lambda(x,y) = \\lambda([x,y])$ gives  a nondegenerate pairing of\n$\\mathfrak{a}$ with $\\mathfrak{b}$.  In this setting, the square integrable representation\n$\\pi_\\lambda$ of $L$ is ${\\rm Ind\\,}_P^N(\\exp(2\\pi i \\lambda)$, and it\nrepresents $L$ on $L^2(N\/P) = L^2(B)$.  Further, here $\\pi_\\lambda$\nmaps the universal enveloping algebra $\\mathcal{U}(\\mathfrak{l})$ onto the set of all\npolynomial (in linear coordinates from $\\exp: \\mathfrak{b} \\to B$) differential operators\non $B$.  In particular,\n\\begin{lemma}\\label{sch-vectors-sqint}\nThe $C^\\infty$ vectors for the representation $\\pi_\\lambda$ are the\nSchwartz class functions on $B$.  In other words if $p$ and $q$\nare polynomials on $B$\\,, if $D$ is a constant coefficient differential\noperator on $B$\\,, and if $u: B \\to \\mathbb{C}$ is a $C^\\infty$ vector\nfor $\\pi_\\lambda$\\,, then $|q(x)p(D)u|$ is bounded.\n\\end{lemma}\n\nIn order to extend this to stepwise square integrable representations we\nmust take into account the problem that $S_n$ need not be central in $N_n$\\,.\nWe do this by decomposing \n\\begin{equation}\\label{decomp_n_n}\nN_n \\simeq L_1 \\times \\cdots \\times L_{m(n)}\n\\end{equation}\nwhere $\\simeq$ is the measure preserving real analytic diffeomorphism given\nby the polynomial map\n\\begin{equation}\\label{decomp_exp_n}\n\\exp':\\mathfrak{n}_n \\to N_n \\text{ by } \\exp'(\\xi_1 + \\dots + \\xi_{m(n)})\n        = \\exp(\\xi_1)\\exp(\\xi_2)\\cdots\\exp(\\xi_{m(n)})\n\\text{ where each } \\xi_r \\in \\mathfrak{l}_r\\,.\n\\end{equation}\nUsing the part of (\\ref{newsetup}d) that says $[\\mathfrak{l}_r,\\mathfrak{z}_s] = 0$ for $r < s$\nthe decomposition (\\ref{decomp_n_n}) gives us\n\\begin{equation}\\label{decomp_n_n_mod_s}\n\\begin{aligned}\nN_n\/S_n &= \\{x_{m(n)}\\cdots x_2x_1Z_{m(n)}\\cdots Z_2Z_1\\mid x_r \\in L_r\\}\\\\\n&= \\{x_{m(n)}Z_{m(n)} \\cdots x_2Z_2x_1Z_1\\mid x_r \\in L_r\\}\n= (L_{m(n)}\/Z_{m(n)}) \\times \\cdots \\times (L_1\/S_1) \\\\\n&\\simeq (L_1\/S_1) \\times \\cdots \\times (L_{m(n)}\/Z_{m(n)}).\n\\end{aligned}\n\\end{equation}\nNow let $\\gamma_n = \\lambda_1 + \\cdots + \\lambda_{m(n)} \\in \\mathfrak{t}^*_n$\\,.\nLet $\\mathfrak{p}_r$ and $\\mathfrak{q}_r$ be totally real polarizations on $\\mathfrak{l}_r$ for\n$\\lambda_r$\\,, paired as above by $b_{\\lambda_r}$\\,.  We do not claim\nthat $\\mathfrak{p} = \\sum \\mathfrak{p}_r$ and $\\mathfrak{q} = \\sum \\mathfrak{q}_r$ are polarizations on $\\mathfrak{n}_n$\nfor $\\gamma_n$\\, (we don't know that they are algebras), but still\n$\\mathfrak{p}_r = \\mathfrak{z}_r + \\mathfrak{a}_r$ and $\\mathfrak{q}_r = \\mathfrak{z}_r + \\mathfrak{b}_r$ where $b_{\\lambda_r}$ \npairs $\\mathfrak{a}_r$ with $\\mathfrak{b}_r$\\,, so $b_{\\gamma_n}$ is\na nondegenerate pairing of $\\mathfrak{a} = \\sum \\mathfrak{a}_r$ with $\\mathfrak{b} = \\sum \\mathfrak{b}_r$\\,.\nNow the stepwise square integrable representation $\\pi_{\\gamma_n}$ of $N_n$\nis realized on $L^2(B)$ where $B = \\exp'(\\mathfrak{b})$ in the notation of\n(\\ref{decomp_exp_n}).  Again, in this setting, $\\pi_{\\gamma_n}$\nmaps the universal enveloping algebra of $\\mathfrak{n}_n$ onto the set of all\npolynomial (in linear coordinates from $\\exp': \\mathfrak{b} \\to B$) \ndifferential operators on $B$.  This extends Lemma \\ref{sch-vectors-sqint}\nto\n\\begin{lemma}\\label{sch-vectors}\nIdentify $B = \\exp'(\\mathfrak{b})$ with the real vector space $\\mathfrak{b}$.\nThe $C^\\infty$ vectors for the representation $\\pi_{\\gamma_n}$ are the\nSchwartz class functions on $B$.  In other words if $p$ and $q$\nare polynomials on $B$\\,, if $D$ is a constant coefficient differential\noperator on $B$\\,, and if $u: B \\to \\mathbb{C}$ is a $C^\\infty$ vector\nfor $\\pi_{\\gamma_n}$\\,, then $|q(x)p(D)u|$ is bounded.\n\\end{lemma}\n\nNow consider the Schwartz space analog of the definition (\\ref{defrelHilbert}).\nWe define the {\\em relative Schwartz space} $\\mathcal{C}(N\/S_nN''_n,\\zeta_\\gamma)\n= \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$ to be\n\\begin{equation}\\label{defrelSchwartz}\n\\begin{aligned}\n\\text{ all } f\\in &C^\\infty(N) \\text{ such that}\\\\\n\t& f(xs) = \\zeta_\\gamma(s)^{-1} f(x) \\text{ for all }\n\t\tx \\in N_n \\text{ and } s \\in S_n\\,,\\text{ and } \\\\\n\t& |q(x)p(D)f| \\text{ is bounded for all polynomials } \n        p,q \\text{ on } N_n\/S_n \\text{ and all } D \\in \\mathcal{U}(\\mathfrak{n}_n).\n\\end{aligned}\n\\end{equation}\nIt is a nuclear Fr\\' echet space and is dense in $L^2(N\/S_nN''_n,\\zeta_\\gamma)\n= L^2(N_n\/S_n,\\zeta_{\\gamma_n})$.  \n\\medskip\n\nWe define \n$C_c^\\infty(N\/S_nN''_n,\\zeta_\\gamma)\n= C_c^\\infty(N_n\/S_n,\\zeta_{\\gamma_n})$ as the space of functions\n$f \\in C^\\infty(N)$ such that $f(xs) = \\zeta_\\gamma(s)^{-1} f(x)$ for all\n$x \\in N_n \\text{ and } s \\in S_n$\\,, whose absolute values are compactly\nsupported modulo $S_nN''_n$ for $C_c^\\infty(N\/S_nN''_n,\\zeta_\\gamma)$,\nmodulo $S_n$ for $C_c^\\infty(N_n\/S_n,\\zeta_{\\gamma_n})$. It is\ndense in the corresponding Schwartz space.  Thus we have the expected\ncontinuous inclusions $C_c^\\infty \\hookrightarrow \\mathcal{C} \\hookrightarrow L^2$\nwith dense images.\n\n\\begin{theorem}\\label{coef-sch}\nLet $u$ and $v$ be $C^\\infty$ vectors for the stepwise square integrable\nrepresentation $\\pi_{\\gamma_n}$ of $N_n$\\,.  Define $\\zeta_\\gamma$ and \n$\\zeta_{\\gamma_n}$\nas in {\\rm (\\ref{defrelchar})}, and $A = \\exp'(\\mathfrak{a})$ and $B = \\exp'(\\mathfrak{b})$\nas in the discussion following {\\rm(\\ref{decomp_n_n_mod_s})}.  \nThen the coefficient function\n$f_{\\pi_{\\gamma_n},u,v}$ belongs to the relative Schwartz space\n$\\mathcal{C}(N\/S_nN''_n,\\zeta_\\gamma) = \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$\\,.\n\\end{theorem}\n\n\\begin{proof}\nWrite $f_{u,v}$ for $f_{\\pi_{\\gamma_n},u,v}$ and $\\pi$ for \n$\\pi_{\\gamma_n}$\\,.  So $f_{u,v}(x) = \\langle u, \\pi(x)v\\rangle$.\nThe left\/right action of the enveloping algebra is $Df_{u,v}E = \nf_{\\pi(D)u,\\pi(E)v}$.  View $u \\in \\mathcal{C}(A)$ and $v \\in \\mathcal{C}(B)$.\nHere $\\pi(D)u$ is the image of $u$ under the (arbitrary) polynomial\ndifferential operator $\\pi(D)$ on $A$ and $\\pi(E)v$ is the\nimage of $v$ under the (arbitrary) polynomial differential operator\n$\\pi(D)$ on $B$.  Together they give the image of $f_{u,v}$ under\nthe polynomial differential operator $\\pi(D) \\otimes \\pi(E)$\non $A \\times B = N_n\/S_n$\\,.  Every polynomial differential operator\non $A \\times B$ is a finite sum of such operators $\\pi(D) \\otimes \\pi(E)$.\nSince coefficients are bounded, here\n$|f_{\\pi(D)u,\\pi(E)v}(x)| \\leqq ||\\pi(D)u||\\cdot ||\\pi(E)v||$, and\nsince $f_{\\pi(D)u,\\pi(E)v}(xs) = \\zeta(s)^{-1}f_{\\pi(D)u,\\pi(E)v}(x)$,\nthe coefficient $f_{u,v} \\in \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$.\n\\end{proof}\n\n\\begin{corollary}\\label{l1-coef1}\nLet $u$ and $v$ be $C^\\infty$ vectors for the stepwise square integrable\nrepresentation $\\pi_{\\gamma_n}$ of $N_n$\\,.  Then the coefficient function\n$f_{\\pi_{\\gamma_n},u,v} \\in L^1(N_n\/S_n,\\zeta_{\\gamma_n})$\\,.\n\\end{corollary}\n\n\\begin{corollary}\\label{l1-coef2}\nLet $L$ be a connected simply connected nilpotent Lie group, $Z$ its center, \nand $\\pi$\na square integrable representation of $L$.  Let $\\zeta \\in \\widehat{Z}$\nsuch that $\\pi|_Z$ is a multiple of $\\zeta$.  Let $u$ and $v$ be $C^\\infty$\nvectors for $\\pi$.  Then the coefficient $f_{\\pi,u,v} \\in\nL^1(L\/Z,\\zeta)$.\n\\end{corollary}\n\nAny norm $|\\xi|$ on $\\mathfrak{n}_n$ carries over to a norm $|\\exp(\\xi)| := |\\xi|$\non $N_n$.  We have the standard Schwartz space $\\mathcal{C}(N_n)$, given by the \nseminorms $\\nu_{k,D,E}(f) = \\sup_{x \\in N_n} |(1 + |x|^2)^k (DfE)(x)|$\nwhere $k$ is a positive integer and $D, E \\in \\mathcal{U}(\\mathfrak{n}_n)$ acting on the\nleft and right.  Since $\\exp: \\mathfrak{n}_n \\to N_n$ is a polynomial diffeomorphism\nit gives a topological isomorphism of $\\mathcal{C}(N_n)$ onto the classical\nSchwartz space $\\mathcal{C}(\\mathfrak{n}_n)$.  Fourier transform and\ninverse Fourier transform of Schwartz class functions on $S_n$.\n\n\\begin{remark}\\label{make-relative}\n{\\rm If $\\gamma_n \\in \\mathfrak{s}_n^*$ and $f \\in \\mathcal{C}(N_n)$ define\n$f_{\\gamma_n}(x) = \\int_{S_n} f(xs)\\zeta_{\\gamma_n}(s)ds$.  Then\n$f_{\\gamma_n} \\in \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$.\nLet $z \\in S_n$\\,.  Since $S_n$ is commutative, \n$$f_{\\gamma_n}(xz) = \\int_{S_n} f(xzs)\\zeta_{\\gamma_n}(s)ds\n= \\int_{S_n} f(xsz)\\zeta_{\\gamma_n}(s)ds\n= \\int_{S_n} f(xs)\\zeta_{\\gamma_n}(z^{-1}s)ds\n= \\zeta_{\\gamma_n}(z)^{-1}f_{\\gamma_n}(x).\n$$\nGiven $x \\in N_n$ we view $f_{\\gamma_n}(x)$ as a function on $\\mathfrak{s}_n^*$\nby $\\varphi_x(\\gamma_n) := f_{\\gamma_n}(x)$.  Note that $\\varphi_x$ is\n(a multiple of) the Fourier transform of the left translate\n$(\\ell(x^{-1})f)|_{S_n}$\\,, say $\\mathcal{F}_{S_n}(\\ell(x^{-1})f)|_{S_n}$.  \nThe inverse Fourier $\\mathcal{F}_{S_n}^{-1}(\\varphi_x)$ transform reconstructs\n$f$ from the $f_{\\gamma_n}$\\,.   Each of the $f_{\\gamma_n}$ is a limit\n(in $\\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$) of finite linear combinations of\ncoefficient functions $f_{\\pi_{\\gamma_n}, u,v}$\\,.  Thus every\n$f \\in \\mathcal{C}(N_n)$ is approximated (in $\\mathcal{C}(N_n)$) by wave packets of\ncoefficient functions of stepwise square integrable representations. }\n\\hfill $\\diamondsuit$\n\\end{remark}\n\nProceeding as in Section \\ref{sec4} let $n' \\geqq n$ and consider\n$\\gamma_{n'} \\in \\mathfrak{t}^*_{n'}$ with $\\gamma_{n'}|_{\\mathfrak{s}_n} = \\gamma_n$.  \nFor brevity write $\\gamma = \\gamma_n$ and $\\gamma' = \\gamma_{n'}$\\,.\nWe reformulate\n(\\ref{rescale4}) through (\\ref{rescale6}) for the Schwartz spaces.\n\\begin{equation}\\label{rescale4s}\nq_{n',n}: \\mathcal{C}(N_{n'}) \\to \\mathcal{C}(N_n) \\text{ by } f \\mapsto f|_{N_n}\\,. \n\\end{equation}\nThe maps $q_{n',n}$ of (\\ref{rescale4s}) define an inverse system in the\ncategory of complete locally convex topological vector spaces\n\\begin{equation}\\label{rescale5s}\n\\mathcal{C}(N_1) \\overset{q_{2,1}}{\\longleftarrow} \\mathcal{C}(N_2) \n        \\overset{q_{3,2}}{\\longleftarrow}  \\mathcal{C}(N_3) \n        \\overset{q_{4,3}}{\\longleftarrow}  ... \n\\end{equation}\nWe define the projective limit\n\\begin{equation}\\label{rescale6s}\n\\mathcal{C}(N) := \\varprojlim \\{\\mathcal{C}(N_n),q_{n',n}\\}\n\\end{equation}\nto be the Schwartz space of $N = \\varinjlim N_n$\\,.  This is dual to the \nconstruction of \\cite[(2.20)]{W2010}.  Now we relate it to (\\ref{rescale6}).\nWe scale the natural injections to maps\n\\begin{equation}\\label{rescalecl1}\nr_{n,\\gamma}: \\mathcal{C}(N_n\/S_n,\\zeta_\\gamma) \\to L^2(N_n\/S_n,\\zeta_\\gamma)\n\\text{ by } f \\mapsto |{\\rm Pf}\\,_{\\mathfrak{n}_n}(\\gamma)|^{1\/2}f\\,.\n\\end{equation}\nThey sum to maps\n\\begin{equation}\\label{rescalecl2}\nr_n = \\left ( \\int_{\\mathfrak{s}_n^*} r_{n,\\gamma}\\, d\\gamma\\right ) : \n\t\\mathcal{C}(N_n) \\to L^2(N_n)\n\\end{equation}\nthat are equivariant for the maps $p_{n',n}$ and $q_{n',n}$\\,.  The \narguments leading to \\cite[Proposition 2.22]{W2010} can be dualized\nfrom direct limits to projective limits.  Thus, dual to\n\\cite[Proposition 2.22]{W2010},\n\\begin{proposition}\\label{compare}\nThe maps $r_n$ of {\\rm (\\ref{rescalecl2})} satisfy\n$p_{n',n}\\cdot r_{n'} = r_n\\cdot q_{n',n}$ for $n' \\geqq n$ and send the\ninverse system $\\{\\mathcal{C}(N_n),q_{n',n}\\}$ into the inverse system\n$\\{L^2(N_n),p_{n',n}\\}$.  That defines a continuous $N$--equivariant injection\n$$\nr: \\mathcal{C}(N) \\to L^2(N)\n$$\nwith dense image.  In particular $r$ defines a pre Hilbert space structure on\n$\\mathcal{C}(N)$ with completion isometric to $L^2(N)$.\n\\end{proposition}\n\n\\section{Fourier Inversion for the Limit Group}\n\\label{sec6}\n\\setcounter{equation}{0}\nIn this section we apply the material of Section \\ref{sec5} to extend the\nFourier inversion portion of Theorem \\ref{plancherel-general}\nfrom the $N_n$ to the limit group $N = \\varinjlim N_n$\\,.\nTo set this up recall that\n\\begin{itemize}\n\\item $\\mathfrak{t}^* = \\varprojlim \\mathfrak{t}^*_n$ consists of all\ncollections $\\gamma = (\\gamma_n)$ where each $\\gamma_n \\in \\mathfrak{t}_n^*$ and\nif $n' \\geqq n$ then $\\gamma_{n'}|_{\\mathfrak{s}_n} = \\gamma_n$\\,.  \n\\item given $\\gamma = (\\gamma_n) \\in \\mathfrak{t}^*$ the limit representation\n$\\pi_\\gamma = \\varprojlim \\pi_{\\gamma_n}$ is constructed as in \nSection \\ref{sec2},\n\\item The distribution character $\\Theta_{\\pi_{\\gamma_n}}$ are given by\n(\\ref{def-dist-char}), and\n\\item $\\mathcal{C}(N) = \\varprojlim \\mathcal{C}(N_n)$ consists of all collections\n$f = (f_n)$ where each $f_n \\in \\mathcal{C}(N_n)$ and if $n' \\geqq n$ then\n$f_{n'}|_{N_n} = f_n$\\,.\n\\end{itemize}\nThen the limit Fourier inversion formula is\n\\begin{theorem}\\label{limit-inversion}\nSuppose that $N = \\varinjlim N_n$ where $\\{N_n\\}$ satisfies \n{\\rm (\\ref{newsetup})}.  Then the Plancherel measure for $N$ is\nconcentrated on $\\mathfrak{t}^*$.   Let $f = (f_n) \\in \\mathcal{C}(N)$ and $x \\in N$.\nThen $x \\in N_n$ for some $n$ and\n\\begin{equation}\\label{lim-inv-formula}\nf(x) = c_n\\int_{\\mathfrak{t}_n^*} \\Theta_{\\pi_{\\gamma_n}}(r_xf) \n\t|{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|d\\gamma_n\n\\end{equation}\nwhere $c_n = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)}\nand $m$ is the number of factors $L_r$ in $N_n$. \n\\end{theorem}\n\n\\begin{proof}  Apply Theorem \\ref{plancherel-general} to $N_n$.  That gives\n$f(x) = f_n(x) =\nc_n\\int_{\\mathfrak{t}_n^*} \\Theta_{\\pi_{\\gamma_n}}(r_xf) |{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|\\,\nd\\gamma_n$\\,.\n\\end{proof}\n\n\\begin{remark}{\\rm\nA Plancherel Formula of the sort $||f||_{L^2}^2 = \\int ||\\pi(f)||^2_{HS}\\, d\\pi$\nusually is somewhat easier than a Fourier inversion formula.  This \nin part is \nbecause it usually is easier to prove that operators $\\pi(f)$ are\nHilbert-Schmidt than to prove that (for appropriate functions $f$)\nthey are of trace class.  Thus one might expect that a formula \n$||f||^2_{L^2(N)} = \\lim c_n'\\int_{\\mathfrak{t}_n^*}\n||\\pi_{\\gamma_n}(f|_{N_n})||^2_{HS} |{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|\\,d\\gamma_n$\nwould be easier to prove than (\\ref{lim-inv-formula}).  But it is not\nclear how to relate the Hilbert--Schmidt norms to the limit process, because\nwe have have not yet found an appropriate form of the Frobenius--Schur \northogonality relations.  Thus the ``less delicate'' Plancherel Formula remains\nproblematical.  } \\hfill $\\diamondsuit$\n\\end{remark}\n\n\\section{Nilradicals of Parabolics in Finite Dimensional Groups}\n\\label{sec7}\n\\setcounter{equation}{0}\nIn Section \\ref{sec8} we will specialize our results to nilradicals \nof minimal parabolic subgroups\nof finitary real reductive Lie groups such as the infinite special and general\nlinear groups and the infinite real, complex and quaternionic unitary groups.\nIn order to do that, in this section we review the relevant restricted\nroot structure that gives the finite dimensional case, reversing some\nof the enumerations used in \\cite{W2013} to be appropriate for our direct limit\nsystems.\n\\medskip\n\nLet $G$ be a finite dimensional connected \nreal reductive Lie group.  We recall some structural results\non its minimal parabolic subgroups, some standard and some from \\cite{W2013}.\n\\medskip\n\nFix an Iwasawa decomposition $G = KAN$.  Write $\\mathfrak{k}$ for the Lie \nalgebra of $K$, $\\mathfrak{a}$ for the Lie algebra of $A$, and $\\mathfrak{n}$ for the\nLie algebra of $N$.  Complete $\\mathfrak{a}$ to a Cartan subalgebra $\\mathfrak{h}$ of $\\mathfrak{g}$.\nThen $\\mathfrak{h} = \\mathfrak{t} + \\mathfrak{a}$ with $\\mathfrak{t} = \\mathfrak{h} \\cap \\mathfrak{k}$.  Now we have root systems\n\\begin{itemize}\n\\item $\\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$: roots of $\\mathfrak{g}_\\mathbb{C}$ relative to $\\mathfrak{h}_\\mathbb{C}$ \n(ordinary roots), and\n\n\\item $\\Delta(\\mathfrak{g},\\mathfrak{a})$: roots of $\\mathfrak{g}$ relative to $\\mathfrak{a}$ (restricted roots). \n\n\\item $\\Delta_0(\\mathfrak{g},\\mathfrak{a}) = \\{\\gamma \\in \\Delta(\\mathfrak{g},\\mathfrak{a}) \\mid \n\t2\\gamma \\notin \\Delta(\\mathfrak{g},\\mathfrak{a})\\}$ (nonmultipliable restricted roots).\n\\end{itemize}\nSometimes we will identify a restricted root\n$\\gamma = \\alpha|_\\mathfrak{a}$, $\\alpha \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ and \n$\\alpha |_\\mathfrak{a} \\ne 0$, with the set \n\\begin{equation}\\label{resrootset}\n[\\gamma] := \n\\{\\alpha' \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \\mid \\alpha'|_\\mathfrak{a} = \\alpha|_\\mathfrak{a}\\}\n\\end{equation}\nof all roots that restrict to it.  Further, \n$\\Delta(\\mathfrak{g},\\mathfrak{a})$ and $\\Delta_0(\\mathfrak{g},\\mathfrak{a})$ are root \nsystems in the usual sense.  Any positive system \n$\\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \\subset \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ defines positive \nsystems\n\\begin{itemize}\n\\item $\\Delta^+(\\mathfrak{g},\\mathfrak{a}) = \\{\\alpha|_\\mathfrak{a} \\mid \\alpha \\in \n\\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \n\\text{ and } \\alpha|_\\mathfrak{a} \\ne 0\\}$ and $\\Delta_0^+(\\mathfrak{g},\\mathfrak{a}) =\n\\Delta_0(\\mathfrak{g},\\mathfrak{a}) \\cap \\Delta^+(\\mathfrak{g},\\mathfrak{a})$.\n\\end{itemize}\n\\noindent We can (and do) choose $\\Delta^+(\\mathfrak{g},\\mathfrak{h})$ so that \n\\begin{itemize}\n\\item$\\mathfrak{n}$ is the sum of the positive restricted root spaces and\n\\item if $\\alpha \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ and $\\alpha|_\\mathfrak{a} \\in\n\\Delta^+(\\mathfrak{g},\\mathfrak{a})$ then $\\alpha \\in \\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$.\n\\end{itemize}\n\\medskip\n\nRecall that two roots are {\\em strongly orthogonal} if their sum and their\ndifference are not roots.  Then they are orthogonal.  We define\n\\begin{equation}\\label{cascade}\n\\begin{aligned}\n&\\beta'_1 \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ is a maximal positive restricted root\nand }\\\\\n& \\beta'_{r+1} \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ is a maximum among the roots of }\n\\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ orthogonal to all } \\beta'_i \\text{ with } i \\leqq r\n\\end{aligned}\n\\end{equation}\nThen the $\\beta'_r$ are mutually strongly orthogonal.  \nNote that each $\\beta'_r \\in \\Delta_0^+(\\mathfrak{g},\\mathfrak{a})$.\nThis is the Kostant cascade coming down from the maximal root.  Denote\n\\begin{equation}\\label{wrong-numbering}\n\\{\\beta'_1, \\dots , \\beta'_m\\}: \\text{ the set of strongly orthogonal roots\nconstructed in (\\ref{cascade}).}\n\\end{equation}\nThe enumeration (\\ref{wrong-numbering}) is not appropriate for the direct\nlimit process, but we need it for some of the lemmas below.  For direct \nlimit considerations we will use the reversed ordering\n\\begin{equation}\\label{right-numbering}\n\\beta_r = \\beta'_{m-r+1} \\text{, so the ordered sets }\n\\{\\beta_1, \\dots , \\beta_m\\} = \\{\\beta'_m, \\dots , \\beta'_1\\}.\n\\end{equation}\n\nFor $1\\leqq r \\leqq m$ define \n\\begin{equation}\\label{layers}\n\\begin{aligned}\n&\\Delta^+_m = \\{\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\mid \\beta_m - \\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})\\} \n\\text{ and }\\\\\n&\\Delta^+_{m-r-1} = \\{\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\setminus \n\t(\\Delta^+_m \\cup \\dots \\cup \\Delta^+_{m-r})\n\t\\mid \\beta_{m-r-1} - \\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})\\}.\n\\end{aligned}\n\\end{equation} \n\n\\begin{lemma} \\label{fill-out} {\\rm \\cite[Lemma 6.3]{W2013}}\nIf $\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})$ then either \n$\\alpha \\in \\{\\beta_1, \\dots , \\beta_m\\}$\nor $\\alpha$ belongs to exactly one of the sets $\\Delta^+_r$\\,.\nIn particular the Lie algebra $\\mathfrak{n}$ of $N$ is the\nvector space direct sum of its subspaces\n\\begin{equation}\\label{def-m}\n\\mathfrak{l}_r = \\mathfrak{g}_{\\beta_r} + {\\sum}_{\\Delta^+_r}\\, \\mathfrak{g}_\\alpha \n\\text{ for } 1\\leqq r\\leqq m\n\\end{equation}\n\\end{lemma}\n\n\\begin{lemma}\\label{layers2}{\\rm \\cite[Lemma 6.4]{W2013}}\nThe set $\\Delta^+_r\\cup \\{\\beta_r\\}  \n= \\{\\alpha \\in \\Delta^+ \\mid \\alpha \\perp \\beta_i \\text{ for } i > r\n\\text{ and } \\langle \\alpha, \\beta_r\\rangle > 0\\}.$\nIn particular, $[\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{l}_t$ where $t = \\max\\{r,s\\}$.\nThus $\\mathfrak{n}$ has an increasing foliation based on the ideals\n\\begin{equation}\\label{def-filtration}\n\\mathfrak{l}_{r,m} = \\mathfrak{l}_{r+1} + \\dots + \\mathfrak{l}_m \\text{ for } 0 \\leqq r < m\n\\end{equation}\nwith a corresponding group level decomposition by normal subgroups \n$L_{r,m}$ where\n\\begin{equation}\\label{def-filtration-group}\nN = L_{0,m} = L_1L_2\\dots L_m \\text{ and } L_{r,m} = L_{r+1} \\ltimes N_{r+1,m}\n\t\\text{ for } 0 \\leqq r < m.\n\\end{equation}\n\\end{lemma}\n\nThe structure of $\\Delta^+_r$, and later of $\\mathfrak{l}_r$, is exhibited by a \nparticular Weyl group element of $\\Delta(\\mathfrak{g},\\mathfrak{a})$ and the negative of\nthat Weyl group element.  Denote\n\\begin{equation}\\label{beta-reflect}\ns_{\\beta_r} \\text{ is the Weyl group reflection in } \\beta_r\n\\text{ and } \\sigma_r: \\Delta(\\mathfrak{g},\\mathfrak{a}) \\to \\Delta(\\mathfrak{g},\\mathfrak{a}) \\text{ by }\n\\sigma_r(\\alpha) = -s_{\\beta_r}(\\alpha).\n\\end{equation}\nHere $\\sigma_r(\\beta_s) = -\\beta_s$ for $s \\ne r$, $+\\beta_s$ if $s = r$.\nIf $\\alpha \\in \\Delta^+_r$ we still have $\\sigma_r(\\alpha) \\perp \\beta_i$\nfor $i > r$ and $\\langle \\sigma_r(\\alpha), \\beta_r\\rangle > 0$.  If\n$\\sigma_r(\\alpha)$ is negative then $\\beta_r - \\sigma_r(\\alpha) > \\beta_r$\ncontradicting the maximality property of $\\beta_{m-r+1}$.  Thus, using \nLemma \\ref{layers2}, $\\sigma_r(\\Delta^+_r) = \\Delta^+_r$.\nThis divides each $\\Delta^+_r$ into pairs:\n\n\\begin{lemma} \\label{layers-nilpotent}{\\rm \\cite[Lemma 6.8]{W2013}}\nIf $\\alpha \\in \\Delta^+_r$ then $\\alpha + \\sigma_r(\\alpha) = \\beta_r$.\n{\\rm (}Of course it is possible that \n$\\alpha = \\sigma_r(\\alpha) = \\tfrac{1}{2}\\beta_r$ when \n$\\tfrac{1}{2}\\beta_r$ is a root.{\\rm ).}\nIf $\\alpha, \\alpha' \\in \\Delta^+_r$ and $\\alpha + \\alpha' \\in \\Delta(\\mathfrak{g},\\mathfrak{a})$\nthen $\\alpha + \\alpha' = \\beta_r$\\,.\n\\end{lemma}\n\nIt comes out of Lemmas \\ref{fill-out} and \\ref{layers2} that the \ndecompositions of (\\ref{layers}), (\\ref{def-m}) and\n(\\ref{def-filtration}) satisfy (\\ref{newsetup}), so\nTheorem \\ref{plancherel-general}\napplies to nilradicals of minimal parabolic subgroups.  In other words,\nas in Theorem \\ref{plancherel-general},\n\\begin{theorem}\\label{iwasawa-layers}{\\rm \\cite[Theorem 6.16]{W2013}}\nLet $G$ be a real reductive Lie group, $G = KAN$ an Iwasawa\ndecomposition, $\\mathfrak{l}_r$ and $\\mathfrak{n}_r$ the subalgebras of $\\mathfrak{n}$ defined in \n{\\rm (\\ref{def-m})} and {\\rm (\\ref{def-filtration})},\nand $L_r$ and $N_r$ the corresponding analytic subgroups of $N$.  \nThen the $L_r$ and $N_r$ satisfy {\\rm (\\ref{newsetup})}.  In particular,\nPlancherel measure for $N$ is\nconcentrated on $\\{\\pi_\\lambda \\mid \\lambda \\in \\mathfrak{t}^*\\}$.\nIf $\\lambda \\in \\mathfrak{t}^*$, and if $u$ and $v$ belong to the\nrepresentation space $\\mathcal{H}_{\\pi_\\lambda}$ of $\\pi_\\lambda$,  then\nthe coefficient $f_{u,v}(x) = \\langle u, \\pi_\\lambda(x)v\\rangle$\nsatisfies\n\\begin{equation}\n||f_{u,v}||^2_{L^2(N \/ S)} = \\frac{||u||^2||v||^2}{|{\\rm Pf}\\,(\\lambda)|}\\,.\n\\end{equation}\nThe distribution character $\\Theta_{\\pi_\\lambda}$ of $\\pi_{\\lambda}$ satisfies\n\\begin{equation}\n\\Theta_{\\pi_\\lambda}(f) = c^{-1}|{\\rm Pf}\\,(\\lambda)|^{-1}\\int_{\\mathcal{O}(\\lambda)}\n        \\widehat{f_1}(\\xi)d\\nu_\\lambda(\\xi) \\text{ for } f \\in \\mathcal{C}(N)\n\\end{equation}\nwhere $\\mathcal{C}(N)$ is the Schwartz space, $f_1$ is the lift\n$f_1(\\xi) = f(\\exp(\\xi))$, $\\widehat{f_1}$ is its classical Fourier transform,\n$\\mathcal{O}(\\lambda)$ is the coadjoint orbit ${\\rm Ad}\\,^*(N)\\lambda = \\mathfrak{v}^* + \\lambda$,\n$c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)},\nand $d\\nu_\\lambda$ is the translate of normalized Lebesgue measure from\n$\\mathfrak{v}^*$ to ${\\rm Ad}\\,^*(N)\\lambda$.  The Fourier inversion formula on $N$ is\n\\begin{equation}\nf(x) = c\\int_{\\mathfrak{t}^*} \\Theta_{\\pi_\\lambda}(r_xf) |{\\rm Pf}\\,(\\lambda)|d\\lambda\n        \\text{ for } f \\in \\mathcal{C}(N).\n\\end{equation}\n\\end{theorem}\n\n\\section{Nilradicals of Parabolics in Infinite Dimensional Groups}\n\\label{sec8}\n\\setcounter{equation}{0}\nWe now look at the classical real forms of the three classical simple locally \nfinite countable--dimensional Lie\nalgebras $\\mathfrak{g}_\\mathbb{C} = \\varinjlim \\mathfrak{g}_{n,\\mathbb{C}}$, and their real forms\n$\\mathfrak{g}_\\mathbb{R}$.  The Lie algebras $\\mathfrak{g}_\\mathbb{C}$ are the classical direct limits,\n$\\mathfrak{sl}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{sl}(n;\\mathbb{C})$,\n$\\mathfrak{so}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{so}(2n;\\mathbb{C}) = \\varinjlim \\mathfrak{so}(2n+1;\\mathbb{C})$, and\n$\\mathfrak{sp}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{sp}(n;\\mathbb{C})$,\nwhere the direct systems are\ngiven by the inclusions of the form\n$A \\mapsto (\\begin{smallmatrix} A & 0 \\\\ 0 & 0 \\end{smallmatrix} )$\nor $A \\mapsto \\left (\\begin{smallmatrix} 0 & 0 & 0 \\\\ 0 & A & 0 \\\\ 0 & 0 & 0 \n\\end{smallmatrix} \\right )$.\nWe often consider the locally reductive algebra\n$\\mathfrak{gl}(\\infty;\\mathbb{C}) = \\varinjlim \\mathfrak{gl}(n;\\mathbb{C})$ along with $\\mathfrak{sl}(\\infty;\\mathbb{C})$.\n\\medskip\n\nLet $G_n$ be a real (this includes complex) simple Lie group of classical \ntype and real rank $n$.  We have just described it as sitting in a direct \nsystem $\\{G_n\\}$ of Lie algebras in the same series.  \nSet $G = \\varinjlim G_n$ as above.  Then we have coherent Iwasawa\ndecompositions $G_n = K_nA_nN_n$ with $K_n \\subset K_\\ell$, \n$A_n \\subset A_\\ell$ and $N_n \\subset N_\\ell$ for $\\ell \\geqq n$.  We need\nto do this so that the direct limit respects the restricted root structures,\nin particular the strongly orthogonal root structures, \nof the $N_n$\\,.   To do that we enumerate the set \n$\\Psi_n = \\Psi(\\mathfrak{g}_n, \\mathfrak{h}_n)$ of nonmultipliable simple restricted\nroots so that, in the Dynkin diagram, for type $A$ we spread from the \ncenter of the diagram.  For types $B$, $C$ and $D$ \n$\\psi_1$ is the \\textit{right} endpoint,\nIn other words for $\\ell \\geqq n$ $\\Psi_\\ell$\nis constructed from $\\Psi_n$ adding simple roots to the \\textit{left} end\nof their Dynkin diagrams.  Thus\n\\begin{equation}\\label{rootorderA}\n\\begin{aligned} \n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } A_{2\\ell+1}$ &\n\\setlength{\\unitlength}{.4 mm}\n\\begin{picture}(180,18)\n\\put(10,2){\\circle{2}}\n\\put(5,5){$\\psi_{-\\ell}$}\n\\put(11,2){\\line(1,0){13}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(33,2){\\circle*{1}}\n\\put(36,2){\\line(1,0){13}}\n\\put(50,2){\\circle{2}}\n\\put(45,5){$\\psi_{-n}$}\n\\put(51,2){\\line(1,0){13}}\n\\put(67,2){\\circle*{1}}\n\\put(70,2){\\circle*{1}}\n\\put(73,2){\\circle*{1}}\n\\put(76,2){\\line(1,0){13}}\n\\put(90,2){\\circle{2}}\n\\put(87,5){$\\psi_0$}\n\\put(91,2){\\line(1,0){13}}\n\\put(107,2){\\circle*{1}}\n\\put(110,2){\\circle*{1}}\n\\put(113,2){\\circle*{1}}\n\\put(116,2){\\line(1,0){13}}\n\\put(130,2){\\circle{2}}\n\\put(128,5){$\\psi_n$}\n\\put(131,2){\\line(1,0){13}}\n\\put(147,2){\\circle*{1}}\n\\put(150,2){\\circle*{1}}\n\\put(153,2){\\circle*{1}}\n\\put(156,2){\\line(1,0){13}}\n\\put(170,2){\\circle{2}}\n\\put(167,5){$\\psi_\\ell$}\n\\end{picture}\n&$\\ell \\geqq n \\geqq 0$\n\\\\\n\\hline\n\\end{tabular}\\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n\\setlength{\\unitlength}{.4 mm}\n$\\Psi_\\ell \\text{ type } A_{2\\ell}\\phantom{i.}$ &\n\\setlength{\\unitlength}{.4 mm}\n\\begin{picture}(180,18)\n\\put(1,2){\\circle{2}}\n\\put(-4,5){$\\psi_{-\\ell}$}\n\\put(2,2){\\line(1,0){13}}\n\\put(18,2){\\circle*{1}}\n\\put(21,2){\\circle*{1}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\line(1,0){13}}\n\\put(41,2){\\circle{2}}\n\\put(36,5){$\\psi_{-n}$}\n\\put(42,2){\\line(1,0){13}}\n\\put(58,2){\\circle*{1}}\n\\put(61,2){\\circle*{1}}\n\\put(64,2){\\circle*{1}}\n\\put(67,2){\\line(1,0){13}}\n\\put(81,2){\\circle{2}}\n\\put(78,5){$\\psi_{-1}$}\n\\put(82,2){\\line(1,0){13}}\n\\put(96,2){\\circle{2}}\n\\put(93,5){$\\psi_1$}\n\\put(97,2){\\line(1,0){13}}\n\\put(113,2){\\circle*{1}}\n\\put(116,2){\\circle*{1}}\n\\put(119,2){\\circle*{1}}\n\\put(122,2){\\line(1,0){13}}\n\\put(136,2){\\circle{2}}\n\\put(134,5){$\\psi_n$}\n\\put(137,2){\\line(1,0){13}}\n\\put(153,2){\\circle*{1}}\n\\put(156,2){\\circle*{1}}\n\\put(159,2){\\circle*{1}}\n\\put(162,2){\\line(1,0){13}}\n\\put(176,2){\\circle{2}}\n\\put(173,5){$\\psi_\\ell$}\n\\end{picture}\n&$\\ell \\geqq n \\geqq 1$\n\\\\\n\\hline\n\\end{tabular}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{rootorderBCD}\n\\begin{aligned}\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } B_\\ell$&\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,13)\n\\put(5,2){\\circle{2}}\n\\put(2,5){$\\psi_{\\ell}$}\n\\put(6,2){\\line(1,0){13}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(34,2){\\line(1,0){13}}\n\\put(48,2){\\circle{2}}\n\\put(45,5){$\\psi_n$}\n\\put(49,2){\\line(1,0){23}}\n\\put(73,2){\\circle{2}}\n\\put(70,5){$\\psi_{n-1}$}\n\\put(74,2){\\line(1,0){13}}\n\\put(93,2){\\circle*{1}}\n\\put(96,2){\\circle*{1}}\n\\put(99,2){\\circle*{1}}\n\\put(104,2){\\line(1,0){13}}\n\\put(118,2){\\circle{2}}\n\\put(115,5){$\\psi_2$}\n\\put(119,2.5){\\line(1,0){23}}\n\\put(119,1.5){\\line(1,0){23}}\n\\put(143,2){\\circle*{2}}\n\\put(140,5){$\\psi_1$}\n\\end{picture}\n&$\\ell\\geqq n \\geqq 2$\\\\\n\\hline\n\\end{tabular} \\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } C_\\ell$ &\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,13)\n\\put(5,2){\\circle*{2}}\n\\put(2,5){$\\psi_{\\ell}$}\n\\put(6,2){\\line(1,0){13}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(34,2){\\line(1,0){13}}\n\\put(48,2){\\circle*{2}}\n\\put(45,5){$\\psi_n$}\n\\put(49,2){\\line(1,0){23}}\n\\put(73,2){\\circle*{2}}\n\\put(70,5){$\\psi_{n-1}$}\n\\put(74,2){\\line(1,0){13}}\n\\put(93,2){\\circle*{1}}\n\\put(96,2){\\circle*{1}}\n\\put(99,2){\\circle*{1}}\n\\put(104,2){\\line(1,0){13}}\n\\put(118,2){\\circle*{2}}\n\\put(115,5){$\\psi_2$}\n\\put(119,2.5){\\line(1,0){23}}\n\\put(119,1.5){\\line(1,0){23}}\n\\put(143,2){\\circle{2}}\n\\put(140,5){$\\psi_1$}\n\\end{picture}\n& $\\ell\\geqq n \\geqq 3$\n\\\\\n\\hline\n\\end{tabular}\\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } D_\\ell$ &\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,20)\n\\put(5,9){\\circle{2}}\n\\put(2,12){$\\psi_{\\ell}$}\n\\put(6,9){\\line(1,0){13}}\n\\put(24,9){\\circle*{1}}\n\\put(27,9){\\circle*{1}}\n\\put(30,9){\\circle*{1}}\n\\put(34,9){\\line(1,0){13}}\n\\put(48,9){\\circle{2}}\n\\put(45,12){$\\psi_n$}\n\\put(49,9){\\line(1,0){23}}\n\\put(73,9){\\circle{2}}\n\\put(70,12){$\\psi_{n-1}$}\n\\put(74,9){\\line(1,0){13}}\n\\put(93,9){\\circle*{1}}\n\\put(96,9){\\circle*{1}}\n\\put(99,9){\\circle*{1}}\n\\put(104,9){\\line(1,0){13}}\n\\put(118,9){\\circle{2}}\n\\put(113,12){$\\psi_3$}\n\\put(119,8.5){\\line(2,-1){13}}\n\\put(133,2){\\circle{2}}\n\\put(136,0){$\\psi_1$}\n\\put(119,9.5){\\line(2,1){13}}\n\\put(133,16){\\circle{2}}\n\\put(136,14){$\\psi_2$}\n\\end{picture}\n& $\\ell\\geqq n \\geqq 4$\n\\\\\n\\hline\n\\end{tabular}\n\\end{aligned}\n\\end{equation}\nWe describe this by saying that $G_\\ell$ {\\em propagates} $G_n$\\,.\nFor types $B$, $C$ and $D$ this is the same as the notion of propagation in\n\\cite{OW2011} and \\cite{OW2014}, but for type $A$ is it s bit different.\nWith the simple root enumeration of (\\ref{rootorderA}) and (\\ref{rootorderBCD})\nthe set $\\{\\beta_1, \\dots , \\beta_m\\}$ of strongly orthogonal positive \nrestricted roots of (\\ref{right-numbering}) is\n\\medskip\n\ntype $A_{2n+1}$: $m = n+1$; $\\beta_1 = \\psi_0$\\,; \n\t$\\beta_2 = \\psi_{-1} + \\psi_0 + \\psi_1$\\,; $\\cdots$ ;\\,\n\t$\\beta_r = \\psi_{-r+1} + \\beta_{r-1} + \\psi_{r-1}$\\,; $\\cdots$\n\\medskip\n\ntype $A_{2n}$: $m = n$; $\\beta_1 = \\psi_{-1} + \\psi_1$\\,;\n\t$\\beta_2 = \\psi_{-2} + \\psi_{-1} + \\psi_1 + \\psi_2$\\,; $\\cdots$ ;\\, \n\t$\\beta_r = \\psi_{-r} + \\beta_{r-1} + \\psi_r$\\,; $\\cdots$\n\\medskip\n\ntype $B_{2n+1}$: $m = 2n+1$; $\\beta_1 = \\psi_1\\,;\n\t\\beta_2 = \\psi_3$ and $\\beta_3 = 2(\\psi_1 + \\psi_2)+\\psi_3$\\,;\n\t$\\cdots$ ; \\hfill\\newline\n\t\\phantom{XXXXXXXXXXXXXXXXXXXXXXx}$\\beta_{2r} = \\psi_{2r+1}$ and\n\t$\\beta_{2r+1} = 2(\\psi_1 + \\dots \\psi_{2r})+\\psi_{2r+1}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $B_{2n}$: $m = 2n$; $\\beta_1 = \\psi_2$ and $\\beta_2 = 2\\psi_1 + \\psi_2$\\,; \n   $\\beta_3 = \\psi_4$ and $\\beta_4 = 2(\\psi_1 + \\psi_2 + \\psi_3) + \\psi_4$\\,;\n   $\\cdots$ ; \\hfill\\newline\n   \\phantom{XXXXXXXXXXXXXi}$\\beta_{2r+1} = \\psi_{2r-1}$ and\n\t$\\beta_{2r} = 2(\\psi_1 + \\dots \\psi_{2r-1}) + \\psi_{2r}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $C_n$: $m = n$; $\\beta_1 = \\psi_1$\\,; $\\beta_2 = \\psi_1 + 2\\psi_2$\\,;\n\t\\,\\,$\\cdots$\\,; $\\beta_r = \\psi_1 + 2(\\psi_2 + \\dots + \\psi_r)$\\,;\n\t\\,\\,$\\cdots$\n\\medskip\n\ntype $D_{2n+1}$: $m = 2n$; $\\beta_1 = \\psi_3$\\,; \n\t$\\beta_2 = \\psi_1 + \\psi_2 + \\psi_3$\\,; \\hfill\\newline\n\t\\phantom{XXXXXXXXXXXXXXi} $\\beta_3 = \\psi_5$ and \n\t\t$\\beta_4 = \\psi_1 + \\psi_2 + 2(\\psi_3 +\n\t\t\\psi_4) + \\psi_5$\\,; $\\cdots$ ; \\hfill\\newline\n        \\phantom{XXXXXXXXXXXXXXi} $\\beta_{2r-1} = \\psi_{2r+1}$ and \n\t$\\beta_{2r} = \\psi_1 + \\psi_2 \n\t  + 2(\\psi_3 + \\dots + \\psi_{2r}) + \\psi_{2r+1}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $D_{2n}$: $m = 2n$; $\\beta_1 = \\psi_1$\\,; $\\beta_2 = \\psi_2$\\,;\n\t$\\beta_3 = \\psi_4$ and $\\beta_4 = \\psi_1 + \\psi_2 + 2\\psi_3 + \\psi_4$\\;\n\t\\hfill\\newline \\phantom{XXXXXXXXXXXXXXXXXXXXXXXXi}\n\t$\\beta_5 = \\psi_6$ and $\\beta_6 = \\psi_1 + \\psi_2 + \n\t\t2(\\psi_3 + \\psi_4 + \\psi_5) + \\psi_6$\\,; $\\cdots$ ; \n\t\\hfill\\newline \\phantom{XXXXXXXXXXXXXXXXXXXXXXXXi}\n\t$\\beta_{2r-1} = \\psi_{2r}$ and $\\beta_{2r} = \\psi_1 + \\psi_2 + \n\t\t2(\\psi_3+ \\dots + \\psi_{2r-1})+\\psi_{2r}$\\,;\\,\\,$\\cdots$\n\\smallskip\n\nIn order to simplify use of these constructions we denote\n\\begin{definition}\\label{well-aligned}{\\rm\nLet $G = \\varinjlim G_n$ be a classical simple locally finite \ncountable dimensional Lie group.  Possibly passing to a cofinal subsequence\nsuppose that we have coherent Iwasawa decompositions $G_n = K_n A_n N_n$ \nsuch that $G_\\ell$ propagates $G_n$ for\n$\\ell \\geqq n$.  Then, again possibly passing to a cofinal subsequence,\nwe can assume that all of the nonmultipliable restricted root \nsystems $\\Delta_0(\\mathfrak{g}_n,\\mathfrak{a}_n)$ are of the same type $A_{2n+1}$,\n$A_{2n}$, $B_{2n+1}$, $B_{2n}$, $C_n$, $D_{2n+1}$ or $D_{2n}$.  Then \nwe will say that the direct system $\\{G_n\\}$ is} well--aligned.\n\\hfill $\\diamondsuit$\n\\end{definition}\n\nThe condition that $\\{G_n\\}$ be well--aligned is exactly what we need for\n$\\{N_n\\}$ to satisfy (\\ref{newsetup}), and given $G$ we have a realization\n$G = \\varinjlim G_n$ for which $\\{G_n\\}$ is well--aligned.  In summary,\n\\begin{theorem}\\label{inversion-for-ss-limits}\nLet $G$ be a classical connected countable dimensional real reductive Lie\ngroup.  Express $G = \\varinjlim G_n$ with $\\{G_n\\}$ well--aligned.\nThen $\\{N_n\\}$ satisfies {\\rm (\\ref{newsetup})}.  In particular\n{\\rm Theorem \\ref{iwasawa-layers}} holds for the \nmaximal locally unipotent subgroup $N = \\varinjlim N_n$ of $G$.\n\\end{theorem}\n\n\\begin{remark}{\\rm\nIn Theorem \\ref{inversion-for-ss-limits} the possibilities for $G$ are\nthe finite dimensional simple Lie groups and the infinite dimensional\n$SL(\\infty;\\mathbb{C})$, $SO(\\infty;\\mathbb{C})$, $Sp(\\infty;\\mathbb{C})$, $SL(\\infty;\\mathbb{R})$,\n$SL(\\infty;\\mathbb{H})$, $SU(\\infty,q)$ with $q \\leqq \\infty$, $SO(\\infty,q)$\nwith $q \\leqq \\infty$, $Sp(\\infty,q)$ with $q \\leqq \\infty$, \n$Sp(\\infty;\\mathbb{R})$ and $SO^*(2\\infty)$.  Further, the normalizer\n$P = MAN$ of $N$ in $G$ is a classical minimal parabolic subgroup\n$\\varinjlim (P_n = M_nA_nN_n)$ where $P_n$ is the minimal\nparabolic in $G_n$ that is the normalizer of $N_n$\\,.}\n\\hfill $\\diamondsuit$\n\\end{remark} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nContinuous logic is an extension of classical first order logic used to study the model theory of structures based on metric spaces.  In this paper, we use continuous logic as presented in \\cite{N} and \\cite{BU} to study the model theory of $\\R$-trees.  \n\nAn $\\R$-tree is a metric space $T$ such that for any two points $a,b \\in T$ there is a unique arc in $T$ from $a$ to $b$, and that arc is a geodesic segment (i.e., an isometric copy of some closed interval in \n$\\R$).  These spaces arise naturally in geometric group theory, for example: the asymptotic cone of a hyperbolic finitely generated group is an $\\R$-tree.\n\nAn $\\R$-tree may be unbounded, while the existing full treatments of continuous model theory are restricted to bounded structures. With this in mind, we consider pointed trees, choose a\nreal number $r>0$, and axiomatize the theory $\\mathbb{R}\\mathrm{T}_r$ of pointed $\\R$-trees of\nradius at most $r$ in a suitable continuous signature. \n\nWe then define the notion of \\textit{richly branching} and axiomatize the theory\n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of the class of richly branching pointed $\\R$-trees with radius $r$.  We prove that the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the existentially closed models of $\\mathbb{R}\\mathrm{T}_r$; thus $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  Next, we investigate some model theoretic properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, showing that it is complete and has quantifier elimination.  In particular, that means $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model completion\nof $\\mathbb{R}\\mathrm{T}_r$.  Further, we prove that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable but not superstable and identify its model-theoretic independence relation. We characterize the principal types of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, and show that this theory has no atomic model. Finally, we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is highly non-categorical.  In fact, for any density character this theory has the maximum possible number of pairwise non-isomorphic models; indeed, the models we construct are pairwise non-homeomorphic. We also give examples of richly branching $\\R$-trees which come from the literature, including some that will be familiar to geometric group theorists.\n\nIn the remainder of this introduction we detail the contents of each section of this paper:\n\nIn Sections 2 and 3 we provide background concerning $\\R$-trees and continuous logic, respectively.  In Section 4 we specify a continuous signature $L$ suitable for the class of pointed $\\R$-trees of radius at most $r$, and axiomatize this class of $L$-structures; the theory of the class is denoted $\\mathbb{R}\\mathrm{T}_r$.\n\nIn Section 5 we discuss definability of certain sets and functions in $\\mathbb{R}\\mathrm{T}_r$.  In Section 6 we show \n$\\mathbb{R}\\mathrm{T}_r$ has amalgamation over substructures.  This plays an important role in many of the primary results in this paper.\n\nIn Section 7 we introduce the class of richly branching pointed $\\R$-trees with radius $r$ and axiomatize this class.  The associated theory is denoted $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  We then show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  The theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the main object of study in this paper.\n\nIn Sections 8 and 9 we verify the main model-theoretic properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  We show that this theory is complete and admits quantifier elimination.  We characterize its types over sets of parameters and use this to show $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable if and only if $\\kappa = \\kappa^\\omega$; hence this theory is strictly stable (stable but not superstable).  We also show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is not a small theory; indeed, its space of $2$-types over $\\emptyset$ has metric density character $2^\\omega$.  (The space of $1$-types over $\\emptyset$ is isometric to the real interval $[0,r]$ with the usual absolute value metric.)  We give a simple geometric characterization of the independence relation of \n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Finally, we show that non-algebraic types have built-in canonical bases (\\textit{i.e.}, these bases are sets of ordinary elements in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and do not require the introduction of  imaginaries).\n\nIn Section 10 we discuss some models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have been constructed within the theory of $\\R$-trees \\cite{DP1,DP2} and some other models that arise in geometric group theory.\n\nIn Section 11 we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has very few isolated $n$-types over $\\emptyset$ and conclude that it has no atomic model (equivalently, it has no prime model).  Then we use amalgamation constructions to build large families of models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and to characterize its isolated types over $\\emptyset$.  For each infinite cardinal $\\kappa$, we show there are  $2^\\kappa$ many pairwise non-isomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\kappa$.  This is the maximum possible number of models, and the models we construct are, in fact, pairwise non-homeomorphic. \n\nIn Section 12 we briefly discuss how the results in this paper could be obtained\nfor the full class of pointed $\\R$-trees (\\textit{i.e.}, without imposing a boundedness\nrequirement).\n\n\\section{$\\R$-trees}\nIn this section we give some background concerning $\\R$-trees.\n\n\\begin{definition}\nA \\emph{geodesic segment} in a metric space $M$ is the image of an\nisometric embedding $\\gamma \\colon [0,r] \\to M$ for some $r \\geq 0$.\nWe say that such a geodesic segment is \\emph{ from $\\gamma(0)$ to $\\gamma(r)$}.  A metric space $M$ is called\n\\emph{geodesic} if for every $a,b \\in M$ there is at least one\ngeodesic segment in $M$ from $a$ to $b$.\n\\end{definition}\n\n   \\begin{fact}\n   \\label{midptfact}\n   \n   A complete metric space $M$ is a geodesic space if and only if for any two points $x,y\\in M$ there\n   exists a midpoint $z$ between $x$ and $y$. That is, there exists $z$ such that:\n   $$d(x,z)=\\frac{d(x,y)}{2}, \\text{ and } d(y,z)=\\frac{d(x,y)}{2}.$$\n   \\end{fact}\n\\begin{proof}\nSee \\cite[Chapter I, 1.4]{BH}.\n\\end{proof}\n\n\\begin{definition}\nAn {\\it $\\R$-tree} is a metric space $M$ such that for any two\npoints $a,b \\in M$ there is a unique arc from $a$ to $b$, and that arc is a geodesic\nsegment.\n\\end{definition}\n\nIn an $\\R$-tree, $[a,b]$ denotes the unique geodesic segment from $a$ to $b$.\nSince metric structures are required to be based on complete metric spaces,\nit is a helpful fact that the completion of an $\\R$-tree is an $\\R$-tree\n(see \\cite[Lemma 2.4.14]{C}).\n\nLet $M$ be an $\\R$-tree and $a\\in M$.  Call the connected components of\n$M\\setminus\\{a\\}$ {\\it branches} at $a$.  Let the \\emph{degree} of a point\n$a\\in M$ be the cardinal number of branches at $a$.  If there are three or more\nbranches at $a\\in M$, then we call $a$ a \\emph{branch point}.\nThe {\\it height} of a branch $\\beta$ at $a$ is $\\sup\\{d(a,x)|x\\in \\beta\\}$ if that\nsupremum exists, and is $\\infty$ otherwise.  A \\emph{subtree} of $M$ is any\nsubspace of $M$ that is itself an $\\R$-tree.\nA \\emph{ray} in an $\\R$-tree is an isometric copy of $\\R^{\\geq 0}$.  If $a\\in M$,\na \\emph{ray at $a$} is a ray so that the image of $0$ under the isometric embedding of $\\R^{\\geq0}$\ninto $M$ is $a$.\n\nThe following lemmas and definitions collect some straightforward facts about $\\R$-trees\nused in this paper. For helpful pictures and more facts about $\\R$-trees see \\cite{C}.\n\n\\begin{lemma}\n\\label{branch}\nIf $M$ is an $\\R$-tree and $a,b,c \\in M$, then\n\\begin{enumerate}\n\\item $d(a,b)+d(b,c)=d(a,c)+2 \\dist(b,[a,c]).$\n\\item $b\\in [a,c]$ if and only if $d(a,c)=d(a,b)+d(b,c)$.\n\\item For $b$ distinct from $a$ and $c$, we have $b\\in [a,c]$ if and only if $a$ and $c$ are in different branches at $b$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nStatement (1) follows from \\cite[Lemma 2.1.2]{C},\nStatement (2) is proved in \\cite[Lemma 1.2.2]{C} and\nStatement (3) comes from \\cite[Lemma 2.2.2]{C}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{!geo}\nIf $M$ is an $\\R$-tree and $E_1$, $E_2$ are disjoint, closed, non-empty subtrees of $M$,\nthen there exists a unique shortest geodesic segment $[u, v]$ with $u\\in E_1$ and $v\\in E_2$.  Moreover, for all $b\\in E_1$ and $c\\in E_2$, the geodesic segment from $b$ to $c$ must contain $[u, v]$.\n\\end{lemma}\n\n\\begin{proof}\nThis is \\cite[Lemma 2.1.9]{C}\n\\end{proof}\n\nThe preceding lemma directly implies the following fact, used often in this paper:\nGiven an $\\R$-tree $M$, a closed subtree $E$ and a point $a\\in M$,\nthere is a unique point $e$ in $E$ closest to $a$, so that $\\dist(a, E)=d(a, e)$,\nand for any point $b$ in $E$, $e$ is on the segment $[a,b].$\n\n\n\\begin{definition}\n\\label{piecewise segment}\nLet $x_0, x_1, x_2, ..., x_n$ be points in an $\\R$-tree $M$ and\n$\\gamma \\colon [0, d(x_0, x_n)] \\rightarrow M$  the isometric embedding\nwith $\\gamma(0)=x_0$ and $\\gamma(d(x_0, x_n))=x_n$ that has image\nequal to the geodesic segment $[x_0, x_n]$. If for each $i=0,...,n$\nwe have $x_i=\\gamma(a_i)$ where $0=a_0\\leq a_1\\leq ...\\leq a_n=d(x_0, x_n)$,\nthen we write\n$[x_0, x_n]=[x_0, x_1, ...., x_n]$, and call $[x_0, x_1, ...., x_n]$ a \\emph{piecewise segment}.\n\nIn other words, if $x_0, x_1,..., x_n$ are elements of $[x_0, x_n]$ listed\nin increasing order of distance from $x_0$, then we  write $[x_0, x_n]=[x_0, x_1, ..., x_n]$\nfor this \\emph{piecewise segment.}\nNote that we also know\n$[x_1, x_n]=\\bigcup_{i=1}^{n-1} [x_i, x_{i+1}]$, and  by Lemma 2.1.4 in \\cite{C} we have that\n$[x_1, x_n]=[x_1,x_2,...,x_n]$ if and only if $d(x_1, x_n)=\\sum_{i=1}^{n-1} d(x_i, x_{i+1})$.\n\n\\end{definition}\n\n\\begin{lemma}\n\\label{differentclosestpoints}\nLet $E$ be a closed subtree of $M$ and let $a,b\\in M$. Let $e_a\\in E$ be the\nunique closest point to $a$, and let $e_b\\in E$ be the unique closest point to $b$. If $e_a\\not=e_b$, then $$d(a,b)=d(a, e_a)+d(e_a, e_b)+d(b, e_b).$$\nThat is, $[a, e_a, e_b, b]$ is a piecewise segment\n\\end{lemma}\n\\begin{proof}\nFollows from Lemma \\ref{branch} and Lemma \\ref{!geo}.\n\\end{proof}\n\n\n\n\\begin{definition}[{{\\it Gromov product\\\/}}]\n\\label{gromovproduct}\nFor a metric space $M$ and $x,y,w \\in M$, define\n$$(x \\cdot y)_w=\\frac{1}{2}[d(x,w)+d(y,w)-d(x,y)].$$\n\\end{definition}\n\nIt follows easily from Lemma \\ref{branch} and Lemma \\ref{!geo} that in an $\\R$-tree, the Gromov product $(x\\cdot y)_w$ computes the distance from $w$ to the geodesic segment $[x,y]$.\n\n\\begin{definition}\n\\label{defhyperbolic}\nLet $\\delta>0$. A metric space $M$ is {\\it $\\delta$-hyperbolic} if,\nfor all $ x,y,z,w\\in M$\n$$\\min\\{(x\\cdot z)_w, (y\\cdot z)_w\\}-\\delta \\leq (x\\cdot y)_w.$$\nA metric space is {\\it 0-hyperbolic} if it is $\\delta$-hyperbolic for all $\\delta>0$.\n\\end{definition}\n\nGiven a metric space $X$, a subset $Z$ of $X$ and $\\delta>0$, we\ndenote the set $\\{x\\in X\\mid \\dist(x, Z)\\leq \\delta\\}$ by\n$Z^\\delta$.\n\n\n\n\\begin{lemma}\nIf $M$ is a geodesic metric space, then $M$ is $\\delta$-hyperbolic\nif and only if, given any $a,b,c \\in M$ and geodesic segments\n$[a,b], [b,c],$ and $[c,a]$, the segment $[a,b]$ is contained in\n$([b,c]\\cup[c,a])^\\delta$. \n\\end{lemma}\n\\begin{proof}\nThis is \\cite[Proposition 1.22]{BH}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{0hyp}\nAny $\\R$-tree is $0$-hyperbolic. Moreover, any $0$-hyperbolic metric space\nembeds isometrically in an $\\R$-tree.\n\\end{lemma}\n\n\\begin{proof}\nThis is \\cite[Proposition 6.13]{R}.\n\\end{proof}\n\n\\begin{definition}\n\\label{Ypoint}\nGiven $a, b, c$ in an $\\R$-tree, there is a unique point $Y$ so that $[a,b]\\cap[a,c]=[a,Y]$.\nIn \\cite{C} after Lemma 2.1.2, they show that this $Y$ is also the unique point so that\n$[b,a]\\cap[b,c]=[b, Y]$ and $[c,a]\\cap[c,b]=[c, Y]$, and that in fact,\n$\\{Y\\}=[a,b]\\cap[b,c]\\cap[a,c]$.\nWe denote this point by $Y(a,b,c)$, or simply by $Y$ when $a, b, c$ are understood.\n\n\\end{definition}\n\n\n\n\n\n\n\\begin{definition}\nIf $A \\subseteq M$ is a subset of the $\\R$-tree $M$, let $E_A$ denote the smallest subtree containing $A$.\nWe call this the \\emph{$\\R$-tree spanned by $A$.}\nNote that\n$$E_A=\\bigcup\\{[a_1,a_2]\\mid a_1, a_2\\in A\\}.$$\nThe closure $\\overline{E_A}$ of $E_A$ is the smallest closed subtree containing $A$.\n\\end{definition}\n\\begin{definition}\nAn $\\R$-tree $M$ is {\\it finitely spanned}\nif there exists a finite subset $A\\subseteq  M$ such that $M=E_A$.\n\\end{definition}\n\nThat an $\\R$-tree is $0$-hyperbolic tells us that for any 3 points $a, b, c$\nthe segment $[a,b]$ is contained in $[b,c]\\cup [c,a]$. Thus, either $a, b, c$ are in a single piecewise segment contained in $[b,c]\\cup [c,a]$, or the subtree spanned by $a, b, c$\nis comprised of the segments $[a, Y], [b,Y]$ and $[c, Y]$, which share only the point\n$Y=Y(a,b,c)$, making that subtree ``Y-shaped\" and explaining the terminology defined above.\nPoints $a, b, c$ are arranged along a piecewise segment in some order if and only\nif $Y(a,b,c)$ is the one of $a, b, c$ that is between the other two. (See \\cite[2.1.2]{C}.)\n\n\n\\begin{definition}\nLet $M$ be an $\\R$-tree.  If $c\\in M$ is such that there do \\emph{not}\nexist $a,b\\in M\\setminus \\{c\\}$ with\n$c\\in [a,b]$, then $c$ is called an {\\it endpoint} of $M$.\nEquivalently, an endpoint is a point with degree one.\n\\end{definition}\n\nNote that if $A$ is finite, $E_A=\\overline{E_A}$ and $E_A$ is complete with\nfinite diameter. \nIf an $\\R$-tree $M$ is finitely spanned, then there is a unique smallest\nset of elements of $M$ that spans $M$, namely the endpoints of $M$.\n\n\n\\begin{lemma}\n\\label{mingenset}\nIf an $\\R$-tree $M$ is finitely spanned and $C$ is the set of endpoints of $M$, then\n\n\\begin{enumerate}\n    \\item if $B$ spans $M$, then $C\\subseteq B$;\n    \\item the set $C$ spans $M$.\n\\end{enumerate}\nThus, $C$ is the unique smallest set that spans $M$.\n\\end{lemma}\n\\begin{proof}\nLet $M$ be a finitely spanned $\\R$-tree. Let $D$ be the diameter of $M$.\nLet $B$ be a set that spans $M$.\\\\\nProof  of (1): Assume there is an endpoint $c\\in M$ not contained in $B$.\nThen there must exist $a, b\\in B$ such that $c\\in [a,b]$.  But, this is a contradiction\nbecause $c$ is an endpoint.\\\\\nProof of (2): Let $a\\in M$.\nLet $S_a$ be the set of all segments $[b, c]\\subseteq M$ such that $a\\in [b,c]$\nand order $S_a$ by inclusion.\nThis is a partial ordering on $S_a$.\nLet $\\{[b_i,c_i]\\mid i\\in \\alpha\\}$ be a chain in this partial ordering, where $\\alpha$ is a cardinal.\nLet $I$ be the closure of $\\bigcup_{i\\in \\alpha}[b_i, c_i]$.  Then $I$ is a geodesic segment in $M$.\nClearly $a\\in I$, and the length of $I$ is at most $D$. Therefore $I\\in S_a$, and\n$I$ is an upper bound for the chain. The chain was arbitrary, so any chain has an upper bound. Therefore, by Zorn's Lemma\nthere exists a maximal element of $S_a$.  Let $[b_a,c_a]$ denote such a maximal element.\nThe elements $b_a$ and $c_a$ must be endpoints of $M$.  Say, for instance,\nthat $b_a$ is not an endpoint.  Then there exist $e, f\\in M$ such that\n$b_a\\in[e,f]$, and either $[e,c_a]$ or $[f,c_a]$ will contain $[b_a, c_a]$.\nThis would mean $[b_a, c_a]$ was not maximal in $S_a$.\nTherefore, for each $a\\in M$, there exist endpoints\n$b_a$ and $c_a$ so that $a\\in [b_a, c_a]$.\nSo, $M$ is spanned by the set of its endpoints, and this spanning\nset is as small as possible by (1).\n\\end{proof}\n\n\n\n\\section{Some continuous model theory}\n\nWe investigate the model theory of $\\R$-trees using\ncontinuous logic for metric structures as presented in \\cite{N}\nand \\cite{BU}. In this section we summarize a few facts about model\ncompanions that are not discussed in those papers.\nFor the rest of this section we fix a continuous first\norder language $L$.\n\nAs explained in \\cite[Section 3]{N}, in continuous model theory it is required that \\emph{structures} and \\emph{models} are metrically complete.  However, formulas and conditions are evaluated more generally in \\emph{pre-structures}, as explained in \\cite[Definition 3.3]{N}. Further, it is shown in \\cite[Theorem 3.7]{N} that the completion of a prestructure is an elementary extension.  In this paper we use notation of the form $\\mathcal{M} \\models \\theta$ only when $\\mathcal{M}$ is a structure; in other words, $M$ must be metrically complete.\n\n  Next, some reminders about saturation in the continuous logic setting.\n  A set $\\Sigma(x_1,...,x_n)$ of $L$-conditions (with free variables\n  among $x_1,...,x_n$) is called {\\it satisfiable in $\\mathcal{M}$} if\n  there exist $a_1,...,a_n$ in $\\mathcal{M}$ such that $\\mathcal{M}\\models E[a_1,...,a_n]$\n  for every $E(x_1,...,x_n)\\in \\Sigma$.\n  Let $\\kappa$\n  be a cardinal. A model $\\mathcal{M}$ of $T$ is called {\\it $\\kappa$-saturated} if\n  for any set of parameters $A\\subseteq M$ with cardinality $<\\kappa$\n  and any set $\\Sigma(x_1,...,x_n)$ of $L(A)$-conditions, if every finite subset\n  of $\\Sigma(x_1,...,x_n)$ is satisfiable in $(\\mathcal{M},a)_{a\\in A}$, then the entire set $\\Sigma(x_1,...,x_n)$\n  is satisfiable in $(\\mathcal{M},a)_{a\\in A}$.\n \n  \\begin{proposition}\n  For any countably incomplete ultrafilter $U$ on $I$, the $U$-ultraproduct\n  of a family of $L$-structures $(\\mathcal{M}_i\\mid i\\in I)$ is $\\omega_1$-saturated.\n  \\end{proposition}\n\n\\begin{proof}\nSee \\cite[Proposition 7.6]{N}.\n\\end{proof}\n  Note that any non-principal ultrafilter on $\\N$ is countably incomplete.\n  \n  \\begin{proposition}\n  For any cardinal $\\kappa$, any $L$-structure $\\mathcal{M}$ has a $\\kappa$-saturated elementary extension.\n  \\end{proposition}\n\n\\begin{proof}\nSee \\cite[Proposition 7.10]{N}.\n\\end{proof}\n  Saturated structures have many useful properties.\n  For example, in an $\\omega$-saturated structure\n  all quantifiers are realized exactly.\n  The next proposition captures this idea.  \n\n  \\begin{proposition}\n  \\label{exactquantifiers}\n  Let $\\mathcal{M}$ be an $L$-structure and suppose $E(x_1,...,x_m)$ is the $L$-condition\n  $$(Q^1_{y_1}...Q^n_{y_n}\\phi(x_1,...,x_m, y_1,...,y_n))=0$$\n  where each $Q^i$ is either $\\inf$ or $\\sup$ and $\\phi$ is quantifier free.\n  Let $\\mathcal{E}(x_1,...,x_m)$ be the mathematical statement\n  $$\\widetilde{Q}^1_{y_1}...\\widetilde{Q}^n_{y_n}(\\phi(x_1,...,x_m, y_1,...,y_n)=0)$$\n  where each $\\widetilde{Q}^i$ is $\\exists y_i$ if $Q^i_{y_i}$ is $\\inf_{y_i}$ and is\n  $\\forall y_i$ if $Q^i_{y_i}$ is $\\sup_{y_i}$.\n  If $\\mathcal{M}$ is $\\omega$-saturated, then for any elements $a_1,...,a_m$ of $M$,\n  we have\n  $\\mathcal{M}\\models E[a_1,...,a_m]$ if and only if $\\mathcal{E}(a_1,...,a_n)$ is true in $M$.\n  \\end{proposition}\n\\begin{proof}\nSee \\cite[Proposition 7.7]{N}.\n\\end{proof}\n\n\n\\begin{definition}\nAn \\emph{$\\inf$-formula} of $L$ is a formula of the form\n$$\\inf_{s_1,y_1}...\\inf_{s_n,y_n}\\phi(x_1,...,x_k,y_1,...,y_n)$$\nwhere $\\phi(x_1,...,x_k, y_1,...,y_n)$ is quantifier-free.\n\\end{definition}\nA $\\sup$-formula of $L$ is defined similarly. These $\\sup$-formulas\nare the universal formulas in continuous logic. For an $L$-theory $T$, we use the\nnotation $T_\\forall$ for the set of universal sentences ($\\sup$-formulas with no free variables) \nimplied by the theory $T$.  Note that, as in classical first order logic, $T_\\forall$\nis the theory of the class of $L$-substructures of models of $T$.\n\n\\begin{definition}\nLet $T$ be an $L$-theory and suppose $\\mathcal{M} \\models T$. We say $\\mathcal{M}$ is\nan {\\it existentially closed (e.c.)} model of $T$ if, for any\n$\\inf$-formula $\\psi(x_1,\\dots,x_m)$,\nany $a_1,...,a_k \\in M$, and any $\\mathcal{N} \\models T$ that is an extension of $\\mathcal{M}$,\nwe have $\\psi^\\mathcal{N}(a_1,\\dots,a_m) = \\psi^\\mathcal{M}(a_1,\\dots,a_m)$.\n\\end{definition}\n\nAn $L$-theory $T$ is {\\it model complete} if any embedding between models\nof $T$ is an elementary embedding.\n\n\\begin{proposition}\n\\label{allec}\nThe $L$-theory $T$ is model complete if and only if\nevery model of $T$ is an existentially closed model of $T$.\n\\end{proposition}\n\n\\begin{proof}\nThis is Robinson's Criterion for model completeness.  The proof given in\n\\cite[Theorem 8.3.1]{Ho} for classical first order logic can easily be adapted to the continuous setting.\n\\end{proof}\n\nIn \\cite[Appendix A]{IN} there is some further discussion of $\\inf$- and $\\sup$-formulas and of model completeness.\n\n\\begin{definition}\n\\label{defnmodelcompanion} Let $T$ be an $L$-theory. A {\\it model\ncompanion} of $T$ is an $L$-theory $S$ such that:\n\\begin{itemize}\n\\item every model of $S$ embeds in a model of $T$;\n\\item every model of $T$ embeds in a model of $S$;\n\\item $S$ is model complete.\n\\end{itemize}\n\\end{definition}\n\nNote that the first two criteria in this definition together\nare equivalent to the statement $S_{\\forall}=T_{\\forall}$.\nAs in classical first order logic, if a theory has a model\ncompanion, then that model companion is unique (up to equivalence of\ntheories).\n\nRecall that a theory $T$ is \\emph{inductive} if whenever $\\Lambda$ is a linearly ordered set\nand $(\\mathcal{M}_\\lambda\\mid\\lambda \\in \\Lambda)$ is a chain of\nmodels of $T$, then the completion of the union of $(\\mathcal{M}_\\lambda\\mid\\lambda \\in \\Lambda)$ is a model of $T$.\n\n\\begin{proposition}\n\\label{axiomatizeec}\nLet $T$ be an inductive $L$-theory\nand let $\\mathcal{K}$ be the class of existentially closed models of $T$.\nIf there exists an $L$-theory $S$ so that $\\mathcal{K}=\\Mod(S)$, then\n$S$ is the model companion of $T$.\n\\end{proposition}\n\\begin{proof}\nThe proof from \\cite[Theorem 8.3.6]{Ho} can be adapted to the continuous setting.\n\\end{proof}\n\nWe say the $L$-theory $T$ has \\emph{amalgamation over substructures} if\nfor any substructures  $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$ of models of $T$ and\nembeddings $f_1\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_1$, $f_2\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_2$,\nthere exists a model $\\mathcal{N}$ of $T$  and embeddings\n$g_i\\colon \\mathcal{M}_i\\rightarrow \\mathcal{N}$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\n\n\n\\begin{proposition}\n\\label{modelcompanionplusamalg}\nLet $T_1$ and $T_2$ be $L$-theories such that $T_2$\nis the model companion of $T_1$.  Assume $T_1$ has amalgamation over substructures.\nThen $T_2$ has quantifier elimination.\n\\end{proposition}\n\n\\begin{proof}\nThe corresponding result in classical first order logic is the\nequivalence of (a) and (d) in \\cite[Theorem 8.4.1]{Ho}.  The proof\ngiven there can be adapted to the continuous setting.\n\\end{proof}\n\n\\endinput\n\n\n\\section{The theory of pointed $\\R$-trees with radius at most $r$}\n\\label{theory of R-trees}\nIn this section we first present the continuous\nsignature used in this paper to study $\\R$-trees.  We then give\naxioms for the theory $\\mathbb{R}\\mathrm{T}_r$ of $\\R$-trees with radius $\\leq r$.\n\nLet $r>0$ be a real number. Define the signature $L_r:=\\{p\\}$ where $p$ is a \nconstant symbol and specify\nthat the metric symbol $d$ has values which lie in the interval $[0,2r]$.\nAny pointed metric space $(M, p)$ with radius $\\leq r$ naturally\ngives rise to an $L_r$-prestructure $\\mathcal{M}=(M,d,p)$, in which $d$ is a metric;\n$\\mathcal{M}$ is an $L_r$-structure if the metric space involved is\nmetrically complete.\n\nNext we define a set of axioms $\\mathbb{R}\\mathrm{T}_r$ for the class of complete,\npointed $\\R$-trees of radius $\\leq r$. \nRecall the connective\n$\\mathbin{\\mathpalette\\dotminussym{}}\\colon [0,\\infty)\\times[0,\\infty) \\rightarrow [0,\\infty)$\ndefined by $x\\mathbin{\\mathpalette\\dotminussym{}} y=\\max\\{x-y,0\\}$.\n\n\n\\begin{definition}\n\\label{rtreetheory}\nLet $\\mathbb{R}\\mathrm{T}_r$ be the $L_r$-theory consisting of the following conditions:\n\\begin{enumerate}\n\n    \\item \\label{boundax}\n  \\hfil$\\sup_{x} d(x,p) \\leq r$;\\vspace{0.25cm}\n\n    \\item \\label{midptax} \n    \\hfil{$\\sup_{x}\\sup_{y}\\inf_{z}\\max\\{\n    \\big|d(x,z)-\\frac12 d(x,y) \\big|,\\ \\ \\big| d(y,z)-\\frac12 d(x,y) \\big|\\}=0$;}\\vspace{0.25cm}\n\n    \\item \\label{hypax} \n    \\hfil{$\\sup_{x}\\sup_{y}\\sup_{z}\\sup_{w}\n    \\big( \\min\\{(x\\cdot z)_w, (y\\cdot z)_w\\}\\mathbin{\\mathpalette\\dotminussym{}} (x\\cdot y)_w\\big)=0$.}\n\n\\end{enumerate}\n\\end{definition}\n\nThe next lemma shows that the class of complete pointed $\\R$-trees of\nradius $\\leq r$ is axiomatized by $\\mathbb{R}\\mathrm{T}_r$.\n\n\n\\begin{lemma}\n\\label{modelsofT}\nThe models of $\\mathbb{R}\\mathrm{T}_r$ are exactly the complete, pointed $\\R$-trees of radius $\\leq r$. \n\\end{lemma}\n\\begin{proof}\nFirst we assume $\\mathcal{M}\\models\\mathbb{R}\\mathrm{T}_r$.  Then $(M, d,p)$ is a complete, pointed\nmetric space. Axiom (\\ref{boundax}) guarantees that $M$ has radius $\\leq r$. \nAxiom (\\ref{midptax}) implies that for any $x,y\\in M$ and\nany $\\epsilon >0$ there is $z\\in M$ such that $d(x,z)$ and $d(y,z)$ are within $\\epsilon$\nof $\\frac{1}{2}d(x,y)$. Iterating this process, we \ndefine a map from the set of dyadic rational numbers in $[0,1]$ into $M$ so that $f(0)=x$ and $f(1)=y$.\nBy letting $\\epsilon$ approach $0$ fast enough, we make this map uniformly continuous, so that\nits completion will be a path from $x$ to $y$ in $M$.\nTherefore, $M$ is path-connected. \nUsing Definition \\ref{defhyperbolic}, Axiom (\\ref{hypax}) implies that $M$ is a $0$-hyperbolic metric space.\nBy \\cite[Lemma 2.4.13]{C}, any connected $0$-hyperbolic metric space\nis an $\\R$-tree. Therefore, $\\mathcal{M}$ is a pointed $\\R$-tree with radius $\\leq r$.\nThat a complete, pointed $\\R$-tree with radius $\\leq r$ is a model of $\\mathbb{R}\\mathrm{T}_r$ is clear.\n\\end{proof}\n\n\\begin{remark}\nStructures in continuous logic are required to be metrically complete, while\nin general, $\\R$-trees are not complete.\nA pointed $\\R$-tree $M$ with radius $\\leq r$ can naturally be viewed as an\n$L_r$-prestructure, which is an $L_r$-structure iff it is complete (since the\npseudometric on the prestructure is actually a metric). If $M$ is not complete,\nthen its metric completion is known to be an $\\R$-tree (see \\cite[Lemma 2.4.14]{C}).\nFurther, the completion of a prestructure is known to be an elementary extension,\nand therefore the prestructure and its completion are completely equivalent from a \nmodel-theoretic perspective. (See \\cite[pages 15--17]{N}.)\nNote that this also means any two pointed $\\R$-trees of radius $\\leq r$ that have the\nsame metric completion are indistinguishable from a model-theoretic\nperspective, and that a metrically complete $\\R$-tree can be identified model-theoretically \nwith any of\nits dense sub-prestructures.  (However, those metric sub-prestructures are not\nnecessarily $\\R$-trees. In particular, they are not necessarily geodesic spaces.)\n\\end{remark}\n\n\\bigskip\nWe close this section by noting a property of $\\mathbb{R}\\mathrm{T}_r$ that will be used later.\n\n\\begin{lemma}\n\\label{chainofmodels}\nThe theory $\\mathbb{R}\\mathrm{T}_r$ is inductive. That is, the completion of the union\nof an arbitrary chain of models of $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{lemma}\n\\begin{proof}\nThe proof of \\cite[Lemma 2.1.14]{C} can be modified to show that the\nunion of an arbitrary chain of pointed $\\R$-trees is again a pointed $\\R$-tree.\nAlso, the completion of an $\\R$-tree is an $\\R$-tree.\nSince basepoints are preserved by embeddings\nof models, the radius of the underlying pointed $\\R$-tree for the union of a chain is most $r$. \n\nAlternatively, note that $\\mathbb{R}\\mathrm{T}_r$ is an $\\forall \\exists$-theory, and therefore the\nclass of its models is closed under completions of unions of chains.\n\\end{proof}\n\n\\section{Some Definability in $\\mathbb{R}\\mathrm{T}_r$}\n\\label{definable}\n\nWe now discuss the notion of \\emph{definability} for subsets of and functions\non the underlying $\\R$-tree $M$ of a model $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$.  For background\non definable predicates, sets and functions see \\cite{N}.\nThe first result shows that every closed ball centered at the base point\nis uniformly quantifier-free $0$-definable in models of $\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{lemma}\n\\label{geodballs}\nLet $s\\in [0,r]$. Let $\\phi(x)$ be the quantifier-free formula $d(x,p)\\mathbin{\\mathpalette\\dotminussym{}} s$.  Suppose $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and $(M,d,p)$ is the underlying $\\R$-tree of $\\mathcal{M}$. Then the closed ball\n$B_s(p)\\subseteq M$ is uniformly $0$-definable.  Indeed, for $x\\in M$ we have $\\dist(x, B_s(p))=\\phi(x)^\\mathcal{M}$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to show that $\\phi^\\mathcal{M}(x)$ is equal to the distance function\n$\\dist(x,B_s(p))$.\nFor $x\\in M$ we know $\\phi^\\mathcal{M}(x)=0$ if and only if $d(x,p)\\leq s$,\ni.e. if and only if $x\\in B_s(p)$.\nNow, let $x\\notin B_s(p)$. Then $\\phi^\\mathcal{M}(x)=d(x,p)-s$.\nLet $\\gamma$ be a geodesic segment from $p$ to $x$ with $\\gamma(0)=p$.\nThen\n$$\\dist(x,B_s(p))\\leq d(x,\\gamma(s))=d(x,p)-d(p, \\gamma(s))=d(x,p)-s=\\phi^\\mathcal{M}(x).$$\n\nNow toward a contradiction, assume $d(x,p)-s >\\dist(x, B_s(p)).$\nThen there exists a point in $c\\in B_s(p)$ with $d(c,x)<d(x,p)-s$.  This implies,\n$$d(x,p)\\leq d(p,c)+d(c,x)\\leq s+d(c,x)<s+d(x,p)-s=d(x,p)$$\nso $d(x,p)<d(x,p)$,\na contradiction. Therefore, $\\dist(x, B_s(p))=\\phi(x)^\\mathcal{M}.$\n\\end{proof}\n\nNote that the preceding lemma holds in any metric structure where the underlying\nspace is a pointed geodesic space.  Moreover, it can be easily adapted to show that\nany closed ball is definable over its center.  \n\nNext, we present a short discussion of some specific definable functions,\npoints and sets in models of $\\mathbb{R}\\mathrm{T}_r$.\n\nLet $M$ be an $\\R$-tree and for $s\\in [0,1]$ define the function\n$\\nu_s\\colon M\\times M\\rightarrow M$ by:\n$\\nu_s(x_1,x_2)=$ the point in $[x_1,x_2]$ with distance $sd(x_1, x_2)$ from $x_1$\nand distance $(1-s)d(x_1, x_2)$ from $x_2$. When $s=\\frac{1}{2}$ we refer to the function $\\nu_{1\/2}$ as the definable\nmidpoint function.  \n\n\\begin{lemma}\n\\label{definablemidpoints}\nLet $s\\in [0,1]$. The function $\\nu_s\\colon M\\times M \\rightarrow M$\nis uniformly $0$-definable in models $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$ via a quantifier-free formula.\n\\end{lemma}\n\\begin{proof}\nLet $\\psi$ be the formula\n$$\\max\\{d(x_1,y)\\mathbin{\\mathpalette\\dotminussym{}} sd(x_1,x_2), d(x_2,y)\\mathbin{\\mathpalette\\dotminussym{}} (1-s)d(x_1,x_2)\\}.$$\nLet $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r.$\nIn $\\mathcal{M}$, the distance $d(\\nu_s(x_1,x_2),y)$ is equal to $\\psi^\\mathcal{M}(x_1,x_2,y).$\nSo the function $\\nu_s$ is $0$-definable via this quantifier-free formula in any model of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\nIn Definition \\ref{Ypoint} we defined $Y=Y(a,b,c)$, the unique point so that\n$\\{Y\\}=[a,b]\\cap[b,c]\\cap[a,c]$. Next we show that $Y$ is a definable function.\n\n\n\n\\begin{theorem}\nThe function $Y\\colon M^3\\rightarrow M$ \nthat for inputs $x_1, x_2, x_3\\in M$ returns $Y(x_1, x_2, x_3)$ is uniformly\n$0$-definable in models $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$ via a quantifier-free formula.\n\n\\end{theorem}\n\\begin{proof}\nRecall that in an $\\R$-tree, the Gromov\nproduct $(a \\cdot b)_x$ is equal to the distance from $x$ to the segment $[a,b]$,\nand this distance is realized by a unique closest point on $[a,b]$\n\nLet\n$$\\phi(x_1, x_2, x_3, x)=\\max\\left\\{(x_1\\cdot x_2)_x, (x_1\\cdot x_3)_x, (x_2\\cdot  x_3)_x\\right\\}.$$\nLet $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and $x_1, x_2, x_3\\in M$. We will show that in $\\mathcal{M}$, the distance $d( x, Y(x_1, x_2, x_3))=\\phi^\\mathcal{M}(x_1, x_2, x_3, x)$. \nLet $E$ be the closed subtree of $M$ spanned by $x_1, x_2$ and $x_3$. Abbreviate $Y(x_1, x_2, x_3)$ by $Y$.\nFor $x\\in M$, let $z$ be the unique point closest to $x$ in $E$. Then we have $d(x, Y)=d(x, z)+d(z, Y)$. The point $z$ must lie on at least one of $[x_1, Y]$, $[x_2, Y]$ or $[x_3, Y]$.\nWithout loss of generality, assume $z\\in [x_1, Y]=[x_1, x_3]\\cap[x_1, x_2]$. \n\nSince $z$ is closest in $E$ to $x$  and $z\\in [x_1, x_3]\\cap[x_1, x_2]$ we know $d(x, z)=(x_1\\cdot x_2)_x=(x_1\\cdot x_3)_x$.\nThat $z$ is closest in $E$ to $x$ also implies $(x_2\\cdot x_3)_x\\geq d(x,z)$. \nThis makes $(x_2\\cdot x_3)_x\\geq(x_1\\cdot x_2)_x=(x_1\\cdot x_3)_x$.\nThus, $\\phi^{\\mathcal{M}}(x_1, x_2, x_3, x)=(x_2\\cdot x_3)_x$.\n\nSince $z\\in [x_1, Y]$, the point $Y$ is the closest point to $z$ on $[x_2, x_3]$, and therefore $Y$ is the closest point to $x$ on $[x_2, x_3].$ It follows that, $d(x, Y)=(x_2\\cdot x_3)_x$.\nWe conclude that $\\phi^{\\mathcal{M}}(x_1, x_2, x_3, x)=d(x,Y).$\n\n\n\\end{proof}\n\n\n\\section{Amalgamation} \n\nNext, we discuss amalgamation for the $L_r$-theory $\\mathbb{R}\\mathrm{T}_r$.\nThe following result from \\cite{C} discussing amalgamating over points in $\\R$-trees is then extended to amalgamation of $\\R$-trees over subtrees. This leads to a proof of amalgamation over substructures for $\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{lemma}\n\\label{chiswell add to points}\nLet $(X,d)$ be an $\\R$-tree and let $\\{(X_i, d_i)\\mid i\\in I\\}$ be a family of $\\R$-trees\nsuch that\n$X_i\\cap X=\\{x_i\\}$ for all $i\\in I$.\n$X_i\\cap X_j= \\{x_i\\}$ if $x_i=x_j$ and $X_i\\cap X_j=\\emptyset$ otherwise.\nDefine $N=(\\bigcup_{i\\in I} X_i)\\cup M$ and define $\\hat d\\colon N\\times N \\rightarrow \\R$ by \n\\begin{itemize}\n\\item On $X\\times X$, $\\hat d=d$, and on $X_i\\times X_i$, $\\hat d=d_i$\n\\item If $x\\in X_i$ and $x'\\in X_j$, with $i\\not= j$, then\n$\\hat d(x,x')=\\hat d(x',x)=d_i(x,x_i)+d(x_i, x_j)+d_j(x_j, x')$\n\\item If $x\\in X_i$ and $x'\\in X$, then $\\hat d(x,x')=\\hat d(x',x)=d_i(x,x_i)+d(x_i, x')$\n\\end{itemize}\nThen $(N,\\hat d)$ is an $\\R$-tree.\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 2.1.13]{C}\n\\end{proof}\n\nWe next apply this lemma to prove $\\R$-trees can be amalgamated over subtrees.\n\n\\begin{lemma}\n\\label{rtreeamalg}\nGiven $\\R$-trees $M_0$, $M_1$ and $M_2$ and isometric embeddings\n$f_i\\colon M_0\\rightarrow M_i$ for $i=1, 2$ there exists an $\\R$-tree $N$\nand isometric embeddings $g_i\\colon M_i\\rightarrow N$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume that $M_0\\subseteq M_1$ and $M_0\\subseteq M_2$,\nthat $M_1\\cap M_2=M_0$, and that the $f_i$ are inclusion maps.\nLet $\\{x_k\\mid k\\in |M_0|\\}$ be a list of the elements of $M_0$.\nFor each $k\\in |M_0|$ let $\\alpha_k$ be the cardinality of the set of branches at $x_k$,\nand let $A_k=\\{B_v\\mid v\\in \\alpha_k\\}$ be a list of the distinct branches at $x_k$ in $M_2$.\nThen let $I=\\{(k, v)\\mid k\\in|M_0|, v\\in \\alpha_k\\},$\nand define $\\beta_i=B_v\\cup \\{x_k\\}$ for each $i=(k,v)\\in I$.\nThus, for each $i\\in I$ we get a distinct subtree $\\beta_i$ of $M_2$\nsuch that $(\\bigcup_{i\\in I} \\beta_i)=M_2$. \nIt is straightforward to show that the $\\R$-tree $(M_1, d_1)$ and the family\n$\\{\\beta_i\\mid i\\in I\\}$ fit the hypotheses of Lemma \\ref{chiswell add to points}.\nThus $N=(\\bigcup_{i\\in I} \\beta_i)\\cup M_1$ with the metric $\\hat d$ is an $\\R$-tree, and clearly $N=M_2\\cup M_1$. Define $g_i$ to be the inclusion of $M_i$ in $N$. Then $g_1\\circ f_1=g_2\\circ f_2$ is clear.\n\\end{proof}\n\n\nTo apply these results to models of $\\mathbb{R}\\mathrm{T}_r$, we begin by\nproving any subset of a model of $\\mathbb{R}\\mathrm{T}_r$ gives rise to a unique model of $\\mathbb{R}\\mathrm{T}_r$.\nIn particular, any substructure extends to a unique model.\n\n\\begin{lemma}\n\\label{substructureofrtree}\nLet $\\mathcal{N}\\models \\mathbb{R}\\mathrm{T}_r$. Any subset $A\\subseteq N$ extends to a model $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$\nsuch that any embedding of $A$ into $\\mathcal{W}\\models \\mathbb{R}\\mathrm{T}_r$ extends to an embedding of $\\mathcal{M}$\ninto $\\mathcal{W}$.\n\\end{lemma}\n\\begin{proof}\nLet $(N,d,p)$ be the underlying $\\R$-tree of the model $\\mathcal{N}$, and let $A\\subseteq N$.   \nDefine the structure $\\mathcal{M}=(M, d,p)$ by $M=\\overline{E_{A\\cup \\{p\\}}}$ with the metric and base point from $(N,d,p)$.\nThen $M$ is the smallest closed subtree of $N$ containing $A\\cup\\{p\\}$.\nIt is straightforward to show that $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and is the desired extension of $A$.\n\\end{proof}\n\n\n\n\\begin{theorem}\n\\label{amalg}\nThe $L_r$-theory $\\mathbb{R}\\mathrm{T}_r$ has amalgamation over substructures.\nThat is, if  $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$\nare substructures of models of $\\mathbb{R}\\mathrm{T}_r$ and\n$f_1\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_1$, $f_2\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_2$ are embeddings,\nthen there exists a model $\\mathcal{N}$ of $\\mathbb{R}\\mathrm{T}_r$  and embeddings\n$g_i\\colon \\mathcal{M}_i\\rightarrow \\mathcal{N}$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\\end{theorem}\n\\begin{proof}\nFirst, by Lemma \\ref{substructureofrtree} we may assume that $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$\nare models of $\\mathbb{R}\\mathrm{T}_r$ with underlying complete, pointed $\\R$-trees $(M_i, d_i, p_i)$.\nFurther, we assume that $M_0\\subseteq M_i$ and $p_0=p_i$ for $i=1,2$\nand that the $f_i$ are inclusion maps.\n\nWe use Lemma \\ref{rtreeamalg} to construct the $\\R$-tree $(N, d)$ \nand isometric embeddings $g_i\\colon M_i\\rightarrow N$ such that $g_1\\circ f_1=g_2\\circ f_2$.\nRecall that in this construction, $N=M_1\\cup M_2$.\nDefine the base point of $N$ to be $q=p_1=p_2=p_0$.\nSince every point in each $M_i$ has distance $\\leq r$ from $p_i=q$, we conclude that the pointed $\\R$-tree $(N,d,q)$ has radius $\\leq r$.\nLet $\\mathcal{N}=(N,d,q)$ be the corresponding $L_r$-structure. \nThen $\\mathcal{N}$ is a model of $\\mathbb{R}\\mathrm{T}_r$, and since $g_i$ are isometric embeddings\npreserving the base point, they give rise to embeddings of $L_r$-structures as required.\n\\end{proof}\n\n\\endinput\n\n\\section{The model companion of $\\mathbb{R}\\mathrm{T}_r$}\n\\label{model companion}\n\nIn this section we define what it means for a pointed $\\R$-tree of radius $\\leq r$\nto be {\\it richly branching.}\nWe then show that the theory of richly branching pointed $\\R$-trees\nwith radius $r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.\n\n\n\\begin{definition}\n\\label{richlybranching}\n\nA pointed $\\R$-tree $(M, d, p)$ of radius $\\leq r$ is {\\it richly branching} if the set\n$$B=\\{b\\in M\\mid \\text{ at } b \\text{ there are at least 3 branches of height } \\geq r-d(p, b)\\}$$\nis dense in $M$.\n\\end{definition}\n\n\\begin{remark}\n\\label{general richly branching}\nThe preceding definition and arguments below apply specifically to pointed $\\R$-trees with radius $\\leq r.$  For general $\\R$-trees we define:\n\\emph{an $\\R$-tree $M$ is {\\it richly branching} if the set of points at which there are at least 3 branches of infinite height is dense in $M$.} Note that an $\\R$-tree that is richly branching in this sense must be unbounded.\n\nTo help illustrate this idea, consider a segment $[x,y]$ where $x\\not=y$ in a richly branching $\\R$-tree $M$. \\emph{There must exist a point $z$ in the open segment $(x,y)$ such that there is a branch of infinite height at $z$ containing neither $x$ nor $y$.} (We leave the proof as an exercise for the reader.) Indeed, working inductively, it is easy to see that this property implies that the set of such points $z$ is dense in $(x,y)$.  Therefore, $(x,y)$ contains infinitely many points $z$ at each of which there are at least 3 branches of infinite height.\n\nIf we start with an unbounded richly branching $\\R$-tree $M$, assign an arbitrary basepoint $p$ and select $r>0$, we can make a richly branching $\\R$-tree with radius $\\leq r$ as in Definition \\ref{richlybranching} by taking the closed ball of radius $r$ with center $p$ in $M$. Note that this yields an $L_r$-prestructure\nand the completion is an $L_r$-structure.\n\nConversely, suppose $M$ is an $\\R$-tree and $p$ is any point in $M$.  If the closed ball of radius $r$ with center $p$ in $M$ is richly branching in the sense of \\ref{richlybranching} for an unbounded set of $r>0$, then $M$ is richly branching as a general $\\R$-tree.\n(See Lemma \\ref{branches of sufficient height} below.) \\end{remark}\n\n\n\n\n\nNext we give axioms for the class of complete, richly branching, pointed $\\R$-trees with radius $r$.\n\n\\begin{definition}\n\\label{rbrtaxioms}\nDefine ${\\psi}(x)$ to be the $L_r$-formula\n$$\\inf_{y_1}\\inf_{y_2}\\inf_{y_3}\\max\\left\\{\\max_{i=1,2,3}\\{|d(x,y_i)-(r-d(p,x))|\\},\n\\max_{1\\leq i<j \\leq 3}\\{ d(x,y_i)+d(x,y_j)-d(y_i,y_j) \\}\\right\\}$$\nand let $\\phi=\\displaystyle \\sup_{x} {\\psi}(x)$.\n\\end{definition}\n\n\n\\begin{definition}\n\\label{rbrttheory}\nLet $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r=\\mathbb{R}\\mathrm{T}_r\\cup \\{\\phi=0\\}$.\n\\end{definition}\n\nTo help parse these axioms and picture what they mean, note that\n$d(x,y_i)+d(x,y_j)-d(y_i,y_j)=2(y_i\\cdot y_j)_x$ so in an $\\R-$tree\n$d(x,y_i)+d(x,y_j)-d(y_i,y_j)=2\\dist(x, [y_i, y_j])$. Consider the case when the infima are realized exactly\nin $M$ (e.g. if $M$ an $\\omega$-saturated model.) In this setting, $\\phi=0$ means that for any element $a\\in M$\nwith $d(p, a)<r$, there exist $b_1, b_2, b_3$ each on a separate branch at $a$ and each\nwith distance $r-d(p,a)$ from $a$.\nWe know $b_1, b_2, b_3$ are on distinct branches at $a$  because $2\\dist(a, [b_i, b_j])=0$ for each $1\\leq i<j\\leq 3$. This makes $a$ a branch point.\nIn a general model where the infima are not necessarily realized, an approximate version\nof this is true; given an element $a$ there must be a branch point within $\\epsilon$ of $a$, as we now show.\n\n\n\\begin{lemma}\n\\label{branching point within epsilon}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying pointed $\\R$-tree $(M, d, p)$ and let $a\\in M$. Let $h\\in \\R$ so that $0<h<r-d(p,a)$. For any $\\epsilon>0$ there exists a point\n$b\\in M$ so that $d(a,b)<\\epsilon$ and there are 3 branches at $b$,\neach with height at least $h.$ \n\\end{lemma}\n\\begin{proof}\nLet $l=r-d(p, a)$, and assume $0<4\\epsilon< l-h$.\nSince $\\phi^{\\mathcal{M}}=0$, for $a\\in M$ we know $\\psi(a)^{\\mathcal{M}}=0$.\nThus there exist points $c_1, c_2, c_3\\in M$ so that \n$$|d(a,c_i)-l|<\\epsilon \\text{ and } d(a,c_i)+d(a,c_j)-d(c_i,c_j)<\\epsilon. $$\nIt follows that for $i\\not=j$ $$d(c_i, c_j)>d(a, c_i)+d(a, c_j)-\\epsilon >2l-3\\epsilon>0$$ meaning $c_1, c_2, c_3$ must be distinct. Moreover, $c_1, c_2$\nand $c_3$ cannot lie along a single geodesic segment.\nTo see this assume $[c_i, c_k]=[c_i, c_j, c_k]$  for $\\{i,j,k\\}\\in\\{1, 2, 3\\}$. Then $d(c_i, c_k)=d(c_i, c_j)+d(c_j, c_k)$, implying\n$$d(c_i, c_k)=d(c_i, c_j)+d(c_j, c_k)>4l-6\\epsilon.$$\nHowever, \n$$d(c_i, c_k)\\leq d(a, c_i)+d(a, c_k)<2l+2\\epsilon.$$\nThese inequalities imply $4\\epsilon >l>l-h$, a contradiction.  So, each of\n$c_1, c_2, c_3$ lies on\na different branch at $Y(c_1, c_2, c_3)$.\nThus, there are at least 3 branches at $Y(c_1, c_2, c_3)$.  \n\nNext, let $q=Y(a, c_1, c_2)$,  $r=Y(a, c_2, c_3)$ and $s=Y(a, c_1, c_3)$.\nWe proceed under the assumption that $d(a, q)\\leq d(a, r)\\leq d(a, s)$, or in other\nwords, $\\dist(a, [c_1, c_2])\\leq \\dist(a, [c_2, c_3])\\leq \\dist (a, [c_1, c_3])$.\nThe other possible cases proceed similarly.\nNote that $M$ being $0$-hyperbolic now implies $\\dist(a, [c_1, c_2])=\\dist(a, [c_2, c_3]$).\nLemma 2.1.6 in \\cite{C} and the subsequent discussion\nyield that $q=r$, so the point closest to $a$  on $[c_1, c_2]$ and the point\nclosest to $a$ on $[c_2, c_3]$ are the same point and $s=Y(c_1, c_2, c_3)$.\nBy Lemma \\ref{branch} $2\\dist(a, [c_1, c_3])=d(a,c_1)+d(a,c_3)-d(c_1,c_3)$.\nThus, $d(a, s)=\\dist(a, [c_1, c_3])<\\frac{\\epsilon}{2}<\\epsilon$.\nSo, $s=Y(c_1, c_2, c_3)$ has distance $<\\epsilon$ from $a$.\nLastly, for $i=1, 2, 3$ we have\n$$d(s, c_i)\\geq d(a, c_i)-d(a,s)>l-\\epsilon-\\frac{\\epsilon}{2}$$\n\n$$=l-\\frac{3\\epsilon}{2}>l-4\\epsilon>h.$$\nSo, $b=Y(c_1, c_2, c_3)$ works.\n\\end{proof}\n\nThe next lemma shows that any branch at any point in a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$\nmust have maximum possible height with that height realized by a point with distance $r$ from $p$.  This also\nimplies every model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has radius equal to $r$.\n\n\\begin{lemma}\n\\label{branches of sufficient height}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and let $a\\in M$.\nIn any branch $\\beta$ at $a$, there exists at least one point $b$ so that $d(p, b)=r$. \n\\end{lemma}\n\\begin{proof}\nLet $a\\in M$ and $\\beta$ a branch at $a$. Let $\\delta=r-d(p, a)$. \nWe first assume $p\\notin \\beta$ (including the case where $a=p$.) \nIn what follows, by iterating the use of Lemma \\ref{branching point within epsilon},\nwe build a sequence $b_1, b_2, b_3,....$ of points in $\\beta$\nso that for any $k\\in \\N$, $[p, a, b_1, ..., b_k]$ is a geodesic segment, and\n$|r-d(p, b_i)|=r-d(p, b_i)<\\frac{\\delta}{2^i}$ \n\n\nLet $c\\in \\beta$. So, $d(a,c)>0$. If $c$ is such that $r-d(p, c)<\\frac{\\delta}{2}$, then\nlet $c=b_1$. If $r-d(p,c)\\geq \\frac{\\delta}{2}$, then let $0<\\epsilon<\\frac{d(a, c)}{2}$.\nUse Lemma \\ref{branching point within epsilon} to find $c'$ so that $d(c, c')<\\epsilon$, \nand so that there are at least 3 branches at $c'$ with height at least\n$(1-\\frac{d(a,c)}{2\\delta})(r-d(p, c)).$  So, $|d(a, c)-d(a, c')|<\\epsilon$, and since $\\epsilon<\\frac{d(a, c)}{2}$ we know $c'\\in \\beta$.  Then find a point $b_1$\non a branch at $c'$ other than the one containing $a$. This makes $[p, a, c', b_1]$ a piecewise segment.  Select\n$b_1$ so that $d(c', b_1)=(1-\\frac{d(a,c)}{\\delta})(r-d(p,c))\\geq (1-\\frac{d(a,c)}{\\delta})\\frac{\\delta}{2}=\\frac{\\delta}{2}-\\frac{d(a,c)}{2}$.\nThen\n$$d(a, b_1)=d(a, c')+d(c', b_1)>(d(a, c)-\\epsilon)+d(c', b_1)$$\n$$>d(a,c)-\\frac{d(a,c)}{2}+\\frac{\\delta}{2}-\\frac{d(a,c)}{2}\n=\\frac{\\delta}{2}.$$\nThus $d(a, b_1)>\\frac{\\delta}{2}$. So, $r-d(p, b_1)=r-(d(p, a)+d(a, b_1))=(r-d(p,a))-d(a, b_1)\n=\\delta-d(a, b_1)<\\delta-\\frac{\\delta}{2}=\\frac{\\delta}{2}.$\n\n\nOnce we have $b_i$, find $b_{i+1}$ in a manner analogous to the argument above,\nso that $|r-d(p, b_i)|=r-d(p, b_i)<\\frac{\\delta}{2^i}.$ Then,\ngiven an index $i$, for any $j>i$\nwe have $d(b_i, b_j)=d(p, b_j)-d(p, b_i)<r-(r-\\frac{\\delta}{2^i})=\\frac{\\delta}{2^i}$.\nIt follows that the sequence $(b_n)$ is Cauchy.\nLet $b$ be the limit of the sequence $(b_i)$.  \n\n\nClaim: $b_i\\in [p, b]$ for all $i\\in \\N$. If not, let $i\\in \\N$ be such that $b_i\\notin [p, b]$.\nThen for all $j\\geq i$, $b_j\\notin [p, b]$ because $[p, b_i, b_j]$ is a geodesic segment.\nLet $j>i$ and let $q$ be the closest point to $b$ on $[p, b_i, b_j]$, i.e. $q=Y(p, b_j, b)$. \nEither $q\\in [p, b_i]$ or $q\\in [b_i, b_j]$.  If $q\\in [b_i, b_j]$, then $[p, q]=[p, b_i, q]$\nimplying $b_i\\in [p, b]=[p, b_i, q, b]$, a contradiction. \nThus, $q\\in [p, b_i]$, and $[p, q, b_i, b_j]$ is a geodesic segment.\nUsing these facts and our choice of $q$ we conclude\n$$d(b, b_j)=d(b, q)+d(q, b_j)=d(b, q)+d(q, b_i)+d(b_i, b_j)\\geq d(b, q)+d(q, b_i)=d(b, b_i).$$\nWe have demonstrated that  $d(b, b_j)\\geq d(b, b_i)>0$ for all $j>i$, contradicting\nthat $b$ is the limit of the sequence $(b_i)$.\n\nBecause $b_i\\in [p, b]$, we know, $r\\geq d(p, b)\\geq d(p, b_i)>r-\\frac{\\delta}{2^i}$  for any $i\\in \\N$.\nTherefore, $d(p, b)=r$. And $b\\in \\beta$, since otherwise it would be on a different branch\nat $a$ than $b_1$. This would make $[b_1, a, b]$ a geodesic segment and contradict\nthat $[p,b]=[p, a, b_1, b]$ is a geodesic segment.\n\n\nNow, assume $p\\in \\beta$. Then $[p,a)\\subseteq \\beta$.\nUsing Lemma \\ref{branching point within epsilon}\nwe find a point $a'\\in [p, a)$ with 3 branches at $a'$.\nSelect $\\beta'$ a branch\nat $a'$ so that $p\\notin \\beta'$ and $a\\notin \\beta'$. The latter guarantees $\\beta'\\subseteq \\beta$. Then apply Case I to $a'$.\n\n\n\\end{proof}\n\n\n\nThe following theorem shows that complete richly branching pointed $\\R$-trees\nwith radius $r$ form an elementary class.\n\n\\begin{theorem}\n\\label{rbrt}\nThe models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the complete, richly branching $\\R$-trees with radius $r$.\n\\end{theorem}\n\\begin{proof}\nLet $(M,d,p)$ be a complete, richly branching pointed $\\R$-tree with radius $r$ and let $\\mathcal{M}$\nbe the corresponding $L_r$-structure. Clearly $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$, and it remains\nto verify that $\\phi^\\mathcal{M}=0.$\nLet $a\\in M$. If $d(p, a)=r$, then let $c_1=c_2=c_3=a$\nand note that these witness $\\psi^\\mathcal{M}(a)=0$. \nIf $d(p, a)<r$, let $\\epsilon >0$  be such that $\\frac{\\epsilon}{2}<r-d(p,a)$.\nSince $M$ is richly branching, the set $B$ from Definition \\ref{richlybranching}\nis dense in $M$.  So, there exists $b\\in B$ so that $d(a,b)<\\frac{\\epsilon}{2}.$\nThen since $M$ is complete, there exist $c_1, c_2, c_3\\in M$\nthat are each distance $r-d(b, p)$ from $b$, \nand each of $c_1, c_2, c_3$ is on a different branch at $b$.\n\nThe triangle inequality in $M$ implies $|d(a, c_i)-d(b, c_i)|<\\frac{\\epsilon}{2}$ for\nall $i=1, 2, 3$,\nand $|d(p, a)-d(p, b)|\\leq d(a,b)<\\frac{\\epsilon}{2}$.\nIt follows directly that $|(r-d(p,b))-(r-d(p,a))|<\\frac{\\epsilon}{2}$.\n\n\nClaim: $|d(a,c_i)-(r-d(p,a))|<\\epsilon$ for each $i=1, 2, 3$.\nLet $i$ be one of $1,2$ or $3$. By the triangle inequality in $\\R$\n\\begin{align*}\n|d(a,c_i)-(r-d(p,a))|&\\leq|d(a, c_i)-d(b, c_i)|+|d(b,c_i)-(r-d(p,a))|\\\\\n&=|d(a, c_i)-d(b, c_i)|+|(r-d(p,b))-(r-d(p,a))|\\\\\n&<\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\\\\n\\end{align*}\n\nClaim: $|d(a,c_i)+d(a,c_j)-d(c_i, c_j)|<\\epsilon$ for each $i \\neq j$ in $\\{1,2,3\\}$.\n\nWe know $d(c_i, c_j)=d(c_i, b)+d(c_j, b)$ \nbecause $c_1, c_2, c_3$ are on different branches at $b$.\nSince $|d(a, c_i)-d(b, c_i)|<\\frac{\\epsilon}{2}$\nfor each $i$, we conclude that \n\\begin{align*}\n|d(a, c_i)+d(a, c_j)-d(c_i, c_j)|&=|d(a, c_i)+d(a, c_j)-(d(b, c_i)+d(b, c_j))|\\\\\n&=|(d(a, c_i)-d(b, c_i))+(d(a, c_j)-d(b, c_j))|\\\\\n&\\leq|d(a, c_i)-d(b, c_i)|+|d(a, c_j)-d(b, c_j)|<\\epsilon.\n\\end{align*}\nSo, for each $\\epsilon$, there are $c_1, c_2$ and $c_3$ making\n$$\\max\\left\\{\\max_{i=1, 2, 3}\\{|d(a,c_i)-(r-d(p,a))|\\}, \\max_{1\\leq i<j \\leq 3}\\{ |(d(a,c_i)+d(a,c_j))-d(c_i,c_j)| \\}\\right\\}<\\epsilon.$$\nThus, $\\psi(a)^\\mathcal{M}=0$, and since $a$ was arbitrary, we conclude $\\phi^\\mathcal{M}=0$.\n\n\nNow, assume $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, so $(M, d, p)$ is clearly a complete, \npointed $\\R$-tree. Let $a\\in M$ and $\\epsilon>0$.\nBy Lemma \\ref{branching point within epsilon}\nwe may find $b\\in M$ so that $d(a, b)<\\epsilon$ and there are at least 3\ndistinct branches at $b$.\nBy Lemma \\ref{branches of sufficient height}\neach branch at $b$ contains a point with distance $r$ from $p$,\nso the height of each branch is at least $r-d(p, b)$. It also follows from \\ref{branches\nof sufficient height} that $M$ has radius $r$.\nSince $a\\in M$ and $\\epsilon>0$ were arbitrary, we conclude the\nset $B$ of points with at least 3 branches of height $\\geq r-d(p, b)$\nis dense in $M$.  Therefore, $M$ is richly branching.\n\\end{proof}\n\n\nNext, we turn to a series of lemmas that are needed for our proof that the theory \n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  (See Theorem \\ref{modcomp}.)  \n\n\n\\begin{lemma}\\label{ec implies richly branching}\nEvery existentially closed model of $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{M} \\models \\mathbb{R}\\mathrm{T}_r$ be existentially closed with underlying $\\R$-tree $(M,d,p)$.\nConsider $a \\in M$. If $r-d(p,a)=0$, then ${\\psi}^{\\mathcal{M}}(a)=0$\nis true: make $y_1=y_2=y_3=a$.\nWhen $r-d(p,a)>0$ using Lemma \\ref{chiswell add to points}\n we may construct an extension $\\mathcal{N} \\models \\mathbb{R}\\mathrm{T}_r$ of $\\mathcal{M}$\nwith underlying $\\R$-tree $(N,d,p)$ such that $N$\nhas at least 3 branches of height $r-d(p,a)$ at $a$.\nThen in $N$ there exist $c_1, c_2$\nand $c_3$, each on a different branch at $a$ and each with distance $r-d(p,a)$ from\n$a$.  It follows that ${\\psi}^{\\mathcal{N}}(a)=0$.\nSince $\\mathcal{M}$ is existentially closed ${\\psi}^{\\mathcal{N}}(a)={\\psi}^{\\mathcal{M}}(a)=0$.\nOur choice of $a \\in M$ was arbitrary, implying $\\phi^\\mathcal{M}=0$.\nIt follows that $\\mathcal{M}$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\nThe next lemma connects $\\kappa$-saturation with the number\nof branches at every interior point in a richly branching $\\R$-tree.\nNote that for points on the boundary where $d(p, a)=r$, there is always\nexactly one branch at $a$, namely the branch containing $p$. \nOtherwise, there would be $b\\in M$ on a different branch at $a$,\nmaking $d(p, b)=d(p, a)+d(a,b)>r$.\nThe converse of Lemma \\ref{saturatedrbrt} is also true, and this\ncharacterization of saturation is Theorem \\ref{kappasaturated=kappabranching},\nwhich will be proved later, once we have shown that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ admits quantifier elimination.\n\n\n\\begin{lemma}\n\\label{saturatedrbrt}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Let $\\kappa$ be an infinite cardinal.\nIf $\\mathcal{M}$ is $\\kappa$-saturated, then $(M,d,p)$ has at least $\\kappa$ many branches at every point $a$ such that $d(p,a)<r$.\n\\end{lemma}\n\\begin{proof}\nLet $a\\in M$ with $d(p,a)<r$. Assume toward a contradiction that there are exactly $\\alpha$-many distinct branches at $a$ where $\\alpha<\\kappa.$\nBy \\ref{rbrt}, $(M, d, p)$ is a complete, pointed richly branching $\\R$-tree.\nIndex the branches at $a$ by $i<\\alpha$,\nand by Lemma \\ref{branches of sufficient height} on each of these branches\ndesignate a point $b_i$ such that $d(a, b_i)=r-d(p,a)$. \n\nLet $A=\\{a\\}\\cup \\{b_i\\mid i<\\alpha\\}$\nand note that $A$ has cardinality less than $\\kappa$. Define\n$$\\Sigma=\\{|d(a, x)-r-d(p,a)|=0\\}\\cup\\{|d(b_i, x)-\\left(d(b_i, a)+d(a, x)\\right)|=0\\mid i<\\alpha\\}.$$\nIt is straightforward to show $\\Sigma$ is finitely satisfiable.\nSo, by $\\kappa$-saturation, there exists $b\\in M$ that satisfies all of these conditions.\nBy Lemma \\ref{branch}, $b$ is on a different branch out of $a$ from each $b_i$,\ncontradicting that every branch at $a$ was represented by one of the $b_i$.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{embed}\n Let $\\kappa$ be an infinite cardinal. Assume $\\mathcal{M}$ is a $\\kappa$-saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying $\\R$-tree $(M,d,p)$.\nLet $K$ be a non-empty, finitely spanned $\\R$-tree. Designate a base point $q$ in $K$.\n\nFor any $e\\in M$ such that $r-d(p, e)\\geq \\sup\\{d(q, x)\\mid x\\in K\\}$\nand any collection $\\{\\beta_i \\mid i< \\alpha\\}$ of branches at $e$ with cardinality\n$\\alpha<\\kappa$, there exists an isometric embedding $f$ of $K$\ninto $M$ such that $f(q)=e$ and $f(K)\\cap \\beta_i=\\{e\\}$ for all $i<\\alpha$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{saturatedrbrt},\n$(M,d,p)$ has at least $\\kappa$ many branches at each point $a$ satisfying $d(p,a)<r$. By Lemma \\ref{mingenset}, there is\na minimal set that spans $K$, namely, the set of endpoints of $K$.\nProceed by induction on the size of this minimal spanning set,\nbuilding up the embedding at each step using the fact that there are $\\kappa$-many branches of sufficient height at every interior point.\nThe restriction that $r-d(p, e)\\geq \\sup\\{d(q, x)\\mid x\\in K\\}$\nkeeps the image of the embedding inside $M$. \n\\end{proof}\n\n\\begin{theorem}\n\\label{modcomp}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{theorem}\n\n\\begin{proof}\nSince $\\mathbb{R}\\mathrm{T}_r$ is an inductive theory, by Lemma \\ref{axiomatizeec} it suffices\nto show that the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the existentially closed models of $\\mathbb{R}\\mathrm{T}_r$.\nBy Lemma \\ref{ec implies richly branching} we know every existentially closed model\nof $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nIt remains to show that every model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is an existentially closed model\nof $\\mathbb{R}\\mathrm{T}_r$.  Let $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r.$ Let $\\mathcal{N}\\models \\mathbb{R}\\mathrm{T}_r$ be an\nextension of $\\mathcal{M}$. We may assume $\\mathcal{M}$ and $\\mathcal{N}$ are\n$\\omega_1$-saturated. (This is because we may consider the structure\n$\\big(\\mathcal{M},\\mathcal{N}, \\iota \\big)$ where $\\iota$ is the embedding\nfrom $\\mathcal{M}$ to $\\mathcal{N}$, and take an $\\omega_1$-saturated extension of\nthat structure. If we can verify the definition of existentially\nclosed in that setting, it will be true of $\\mathcal{M}$ and $\\mathcal{N}$.) Let $(M,d,p)$ and $(N,d,p)$\nbe the underlying pointed $\\R$-trees for $\\mathcal{M}$ and $\\mathcal{N}$ respectively.\n\nLet $a=a_1,...,a_k\\in M$, and without loss of generality assume basepoint $p$ is\namong $a_1,...,a_k$.\nWe claim that for any $b_1,...,b_l\\in N$, there exist $c_1,...,c_l\\in M$\nso that $d(b_i, b_j)=d(c_i, c_j)$ for all $i,j\\in \\{1,...,l\\}$ and $d(a_i, b_j)=d(a_i, c_j)$\nfor all $i\\in \\{1,...,k\\}$ and $j\\in \\{1,...,l\\}$.\\\\\nTo prove this claim, let  $b_1,...,b_l\\in N$, and\nlet $E_a\\subseteq M$ be the subtree spanned by $a=a_1,...,a_k$.\nNote that $E_a=\\overline{E_a}$.\nDefine an equivalence relation on $b_1,...,b_l$ by: $b_i\\sim b_j$\nif $b_i$ and $b_j$ have the same closest point in $E_a$. Let $A_1,...,A_m$\nbe the equivalence classes of this equivalence relation, and for $v=1,...,m$\nlet $e_v$ be the unique closest point in $E_a$ common to the members of $A_v$.\n\nFor each $v=1,...,m$\nlet $K_v$ be the $\\R$-tree spanned by $A_v\\cup\\{e_v\\}$ in $N$, with base point $e_v$.\nNote that each $K_v$ is closed and for $u\\not=v$, $K_u\\cap K_v=\\emptyset$.\nSince $(N, d, p)$ has radius $r$ and $p\\in E_a$ and $e_v$ is the unique closest point to $x$ in $E_a$,\nLemma \\ref{branch} gives\n$d(p, e_v)+d(e_v, x)=d(p, x)\\leq r$ for each $x\\in K_v$. Therefore\n$d(e_v, x)\\leq r-d(p, e_v)$ for each $x\\in K_v$. Thus, $\\sup\\{d(e_v, x)\\mid x\\in K_v\\}\\leq r-d(p, e_v).$\nNow, for each $v=1,...,m$, by Lemma \\ref{embed}, there is an isometric embedding\n$f_v\\colon K_v\\rightarrow M$ sending $e_v$ to $e_v$\nsuch that $f_v(K_v)$ does not intersect $E_a$ except at $e_v$.  Note that $e_v$ is the\nunique closest point in $E_a$ for every point in $f_v(K_v)$.\n\nLet $f$ be the union of the functions $f_v$ for $v=1,...,m$. If $b_i$ and $b_j$\nare both in $A_v$, then $d(b_i, b_j)=d(f_v(b_i), f_v(b_j))=d(f(b_i), f(b_j))$.\nIf $b_i$ and $b_j$ are in $A_{u}\\not=A_{v}$ respectively, then using Lemma\n\\ref{differentclosestpoints}\n\\begin{align*}\nd(b_i, b_j)&=d(b_i, e_u)+d(e_u, e_v)+d(e_v, b_j)\\\\\n&= d(f_u(b_i), f_u(e_u))+d(e_u, e_v)+d(f_v(e_v), f_v(b_j))\\\\\n&= d(f(b_i), e_u)+d(e_u, e_v)+d(e_v, f(b_j))\\\\\n&=d(f(b_i), f(b_j)).\n\\end{align*}\nTherefore the function $f$ is an isometric embedding from\n$\\displaystyle\\bigcup_{v=1}^m K_v$ to $M$.\\\\\nLet $c_j=f(b_j)$ for all $j\\in\\{1,...,l\\}$.\nThen clearly $d(b_i, b_j)=d(f(b_i), f(b_j))=d(c_i, c_j)$ for all $i,j\\in \\{1,...,l\\}$.\nNow let $i\\in\\{1,...,k\\}$ and $j\\in \\{1,...,l\\}$.  Let $e_v$ be the closest point to $b_i$\nin $E_a$.  Then\n\\begin{align*}\nd(a_i,b_j)&=d(a_i, e_v)+d(e_v, b_j)\\\\\n&=d(a_i, e_v)+d(f(e_v),f(b_i))\\\\\n&=d(a_i, e_v)+d(e_v, c_j)=d(a_i, c_j).\n\\end{align*}\nThus, the claim is true.\n\nThe values of quantifier free\nformulas in $\\mathcal{M}$ are determined by distances in $M$. (This can be shown\nusing induction on the definition of quantifier free formula, since connectives\nare continuous functions on atomic formulas, which in this case are all of the\nform $d(t_1,t_2)$ for terms $t_1, t_2$.)\nSo, the preceding claim implies that for any quantifier free formula\n$\\phi(x_1,...,x_k, y_1,...,y_l)$\nand any $b_1,...,b_l\\in \\mathcal{N}$,\nthere exist $c_1,...,c_l\\in \\mathcal{M}$ so that\n$$\\phi(a_1,...,a_k,b_1,...,b_l)^\\mathcal{N}=\\phi(a_1,...,a_k,c_1,...,c_l)^\\mathcal{N}=\\phi(a_1,...,a_k,c_1,...,c_l)^\\mathcal{M}.$$\nThen standard arguments about infima imply\n$$\\inf_{y_1}...\\inf_{y_l}\\phi(a_1,...,a_k, y_1,...,y_l)^\\mathcal{M}=\\inf_{y_1}...\\inf_{y_l}\\phi(a_1,...,a_k, y_1,...,y_l)^\\mathcal{N}.$$\nTherefore, $\\mathcal{M}$ is an existentially closed model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\n\\section{Properties of the theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$}\n\\label{propertiesrbrt}\nIn this section we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has\nquantifier elimination and is complete and stable, but not superstable.  We characterize types, and show that\nthe space of 2-types over the empty set has metric density $2^\\omega$. We also characterize definable closure and algebraic closure in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. In the rest of the paper, for convenience of notation, whenever we take a set of parameters $A$ in a model $\\mathcal{M}=(M, d, p)$ we assume $p\\in A$.\n\n\\begin{lemma}\n\\label{qe}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has quantifier elimination.\n\\end{lemma}\n\\begin{proof}\nBy Theorem \\ref{modcomp}, Theorem \\ref{amalg} and Proposition \\ref{modelcompanionplusamalg}.\n\\end{proof}\n\n\\begin{corollary}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete.\n\\end{corollary}\n\\begin{proof}\nIn any model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ we may embed the structure consisting of just the base point.\nThis fact together with quantifier elimination implies that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete.\n\\end{proof}\n\nWe now turn our attention to a discussion of types and type spaces.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Recall that if $b=b_1,...,b_n$ is tuple of elements in $\\mathcal{M}$, $\\tp_\\mathcal{M}(b\/A)$ is the complete $n$-type of $b$ over $A$ in $\\mathcal{M}$.\nThe space of all $n$-types in models of a theory $T$ is denoted $S_n(T)$. The space\nof $n$-types over $A$ is denoted $S_n(T_A)$ or $S_n(A)$ when the theory $T$ is\nclear from context. If $q$ is an $n$-type and $a\\in \\mathcal{M}$ is a tuple such that $\\mathcal{M}\\models q(a)$ we say $a$ realizes $q$, and write $a\\models q$. If $\\phi(x_1,..., x_n)$ is an $L_r$-formula,\nwe write $\\phi(x_1,..., x_n)^q$ for the value of this formula specified by the type $q$. \n\n\n\nThe following lemma gives criteria for when two $n$-tuples have the same type.\n\n\\begin{lemma}\n\\label{determinetypes}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Fix $A\\subseteq M$.\n\\begin{enumerate}\n\\item Let $b,c \\in M$. Then $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if\n$b$ and $c$ have the same unique closest point $e\\in \\overline{E_{A}}$ and\n$d(b,e)=d(c,e)$.\n\\item Let $b=(b_1,\\dots,b_n)$ and $c=(c_1,\\dots,c_n)$ be tuples in $M$. \nThen $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if $\\tp_\\mathcal{M}(b_i\/A)=\\tp_\\mathcal{M}(c_i\/A)$\nfor all $1 \\leq i \\leq n$ and $d(b_i,b_j)=d(c_i,c_j)$ for all $1 \\leq i,j \\leq n$.\n\\item Let $b=(b_1,\\dots,b_n)$ and $c=(c_1,\\dots,c_n)$ be tuples in $M$. Let \n$E_b$ be the subtree of $M$ spanned by $A\\cup b$, and similarly define $E_c$.\nThen $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if there is an isometry $f\\colon E_b\\rightarrow E_c$\nthat fixes $A$ (or equivalently $\\overline{E_A}$) and satisfies $f(b_i)=c_i$ for all $i=1,...,n$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} \nTo show (1), first assume $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$.  This implies\n$d(b,a)=d(c,a)$ for all $a\\in \\overline{E_{A}}$, which means $b$ and $c$ must\nhave the same unique closest point  $e\\in \\overline{E_{A}}$.\nMoreover, $d(b, e)$ must equal $d(c,e)$.\nFor the other direction, assume $b$ and $c$ have the same\nunique closest point  $e\\in  \\overline{E_{A}}$ and that $d(b, e)=d(c,e)$.\nSince $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has quantifier elimination,  and the values quantifier-free formulas\nare determined by the values of atomic formulas, it suffices to show\n$d(a, b)=d(a,c)$ for all $a\\in A$ and $d(p,b)=d(p,c)$.  This follows easily\nfrom our assumptions using Lemma \\ref{branch}, since for any $a\\in A$,\nthe point $e$ must be on both $[a, b]$ and $[a,c]$.\n\\newline\nStatement (2) follows from part (1) and the fact that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ admits quantifier elimination.\nnote that $\\tp_\\mathcal{M}(b_i\/A)=\\tp_\\mathcal{M}(c_i\/A)$\nfor all $1 \\leq i \\leq n$, if and only if $d(b_i, a)=d(c_i, a)$ for all $a\\in A.$ \nAgain, we use that the values of quantifier-free formulas are determined\nby the values of the atomic formulas. The atomic formulas involved are\nall of the form $d(t_1, t_2)$ with terms $t_1$ and $t_2$ involving the variables\n$x_1,...,x_n$ and\/or parameters from $A$.\n\\newline\nStatement (3) follows from (2) and the fact that an isometry between subsets\nof an $\\R$-tree extends uniquely in the obvious way to an isometry\nof the subtrees spanned by those sets.\n\\end{proof}\n\nLemma \\ref{determinetypes} tells us that a type $q\\in S_n(A)$ is uniquely determined\nby the choice of closest points $e_1,..., e_n$ in $\\overline{E_A}$,  the distances \nto these points, and the pairwise distances between the elements of a tuple realizing the type.\nMoreover, given a realization $b_1,..., b_n$ of an $n$-type $q$, the points $b_1, ..., b_n, e_1,..., e_n$\nform a finite $0$-hyperbolic metric space where $d(b_i, e_i)\\leq r-d(p, e_i).$\nThus, $b_1,...,b_n, e_1,...,e_n$ span an $\\R$-tree that extends (perhaps trivially) the $\\R$-tree spanned by $e_1,...,e_n$ in which all points have distance at most $r$ from $p$.\n\n\nIn fact, for any choice of finitely many points $e_1,...,e_n$ in $\\overline{E_A}$\nand values for the formulas $d(x_i, e_i)$ and $d(x_i, x_j)$ satisfying\nconditions guaranteeing $0$-hyperbolicity and boundedness, there exists a unique type\n$q\\in S_n(A)$ realizing this information. \nWe prove this in detail in Lemma \\ref{n types over A} below.\n\n\nIn Definitions \\ref{defhyperbolic} and \\ref{0hyp} we defined the notion of\na $0$-hyperbolic metric space. Definition \\ref{four point condition} is another condition\nthat, in a metric space, is equivalent to $0$-hyperbolicity.  This version is convenient for our description of types.\n\n\\begin{definition}\\label{four point condition}\nLet $X$ be a set, and $d\\colon X\\times X\\rightarrow \\R^{\\geq 0}$ a\nfunction. If for all $x, y, z, t\\in X$\n$$d(x, y)+d(z, t)\\leq \\max\\{d(x, z)+d(y, t), d(y, z)+d(x, t)\\}$$\nwe say $d$ satisfies the \\emph{4-point condition} on $X$.\n\\end{definition}\n\n\n\\begin{lemma}\n\\label{four point condition implies triangle}\nA metric space $(X, d)$ is $0$-hyperbolic if and only if $d$ satisfies the 4-point condition on $X$.\nMoreover, if $X$ is a non-empty set and $d\\colon X\\times X\\to \\R$ satisfies the\nfour point condition on $X$, and satisfies $d(x,y)=0$ if and only if $x=y$ and $d(y, x)=d(x, y)$, then the triangle inequality follows.\n\\end{lemma}\n\\begin{proof}\nFor the first statement, see page 13 in \\cite{C}. For the second statement, let $z=t$ in the 4-point condition.\n\\end{proof}\n\n\n\nIn the next lemma, we focus on $n$-types realized by $n$ distinct\nelements. It is straightforward to extend to the case where the elements realizing the type may not be distinct.\n\n\n\\begin{lemma}\\label{n types over A}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and $A\\subseteq M$. Let $e_1,..., e_n\\in\\overline{E_A}$.\nLet $\\Sigma$ be a set of $L$-conditions over $A$\n$$\\Sigma=\\left\\{d(x_i, x_j)=\\rho_{i,j}, \\, d(x_i, e_j)=d(e_i, e_j)^{\\mathcal{M}}+s_i, \\, d(e_i, e_j)=d(e_i, e_j)^{\\mathcal{M}}\\mid 1\\leq i,j\\leq n\\right\\}$$\nwhere\n\\begin{itemize}\n\\item for each $1\\leq i\\leq n$,  $s_i \\in [0, r-d(p, e_i)^{\\mathcal{M}}]\\subseteq \\R$\n\\item $\\{\\rho_{i,j}\\in \\R^{\\geq 0}\\mid 1\\leq  i,j\\leq n\\}$ are such that \n$\\rho_{i,i}=0$ and $\\rho_{i, j}=\\rho_{j, i}\\not=0$ for all $1\\leq i, j, \\leq n\n\n\\item the values of $d$ specified by $\\Sigma$ satisfy the 4-point condition on the set of symbols $\\{e_1,..., e_n, x_1,..., x_n\\}$.\n\\end{itemize}\n\nThere exists a unique type $q\\in S_n(A)$ so that $\\Sigma\\subseteq q$. \n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $\\kappa> |\\overline{E_A}|$ and let $\\mathcal{N}$ be a $\\kappa$-saturated elementary extension of $\\mathcal{M}$.\nLet $X=\\{x_1,..., x_n\\}$ and note that by Lemma \\ref{four point condition implies triangle}, $d$ is a metric on $X\\cup \\{e_1,..., e_n\\}$.\nDefine an equivalence relation on $X$ by $x_i\\sim x_j$ if and only if $d(e_i, e_j)^{\\mathcal{M}}=0.$\nLet $C_1,..., C_k$ be the equivalence classes. Each of these equivalence classes\ncorresponds to exactly one of the $e_1,..., e_n$.\nFor each $C_l$, $1\\leq l\\leq k$ let $e_l$ be the corresponding element of $\\overline{E_A}$. Then, with the values of $d$ specified in $\\Sigma$, the set $C_l\\cup\\{e_l\\}$\nis a finite $0$-hyperbolic metric space. Thus it spans an $\\R$-tree. Call that $\\R$-tree $K_l$.\nApply Lemma \\ref{embed} to $K_l$ with basepoint $q=e_l$ to embed $K_l$ in $N$\nso that the image of $K_l$ intersects $\\overline{E_A}$ only at $e_l$.\nThat the conditions of Lemma \\ref{embed} on $K_l$ are satisfied follows\nfrom the conditions on the values of $d$ specified in $\\Sigma$. Let $f_l\\colon K_l\\rightarrow N$\ndenote this isometric embedding.\n\nFor each $l$ and each $i$ where $x_i\\in C_l$, let $b_i=f_l(x_i)$. Note that for\n$i\\not=j$ we know $b_i\\not= b_j$, because $d(b_i, b_j)=\\rho_{i,j}\\not=0$.\nLet $q$ be the type\nof $b_1,...,b_n$ over $A$ in $\\mathcal{N}$. \nAn straightforward argument like that in the proof of Theorem \\ref{modcomp} now shows that\n$\\Sigma\\subseteq q$. Uniqueness follows from Lemma \\ref{determinetypes}.\n\\end{proof}\n\n\nLemmas \\ref{determinetypes} and \\ref{n types over A} tell us that types over $A$ correspond\n(up to isometry fixing $\\overline{E_A}$) to\nfinitely-spanned $\\R$-trees with radius at most $r$ that extend\nfinitely-spanned subtrees of $\\overline{E_A}$. \nNext, we give some results about the type spaces as metric spaces.\nFirst, a reminder of the metric on types.\n\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ such that every type in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ is realized in $\\mathcal{M}$ for each $n\\geq 1$.\nAs defined in \\cite{N}, the $d$-metric on $n$-types over the empty set is:\n$$d(q,s)=\\inf_{a\\models q, b\\models s}\\max_{i=1,...,n} d(a_i, b_i)$$\nwhere $a=a_1,...,a_n$ and $b=b_1,...,b_n$ are tuples in $\\mathcal{M}$..\nNote that by compactness, the infimum in the definition is actually realized. This definition can be extended in the obvious way to spaces of types over parameters.\nIn what follows, we will call a choice of a particular $a\\models q$ and $b\\models s$\na  \\emph{configuration.}  \n\n\n\\begin{lemma}\n\\label{how many types}\n\n\\begin{enumerate}\n\\item The space $S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ of $1$-types over the empty set is in\nbijective correspondence with the interval $[0,r]$. In fact, with the $d$-metric\non types, it is isometric to $[0, r]$.\n\n\\item If $A$ is a set of parameters, then $S_1(A)$ is \nin bijective correspondence with the set of ordered pairs\n$$\\{(e,s)\\mid e\\in \\overline{E_A} \\text{ and } s\\in [0,r-d(p,e)]\\subseteq \\R\\}.$$\n\\item Given $t,u\\in S_1(A)$ and $e_t$, $e_u$ the unique\nclosest points in $\\overline{E_A}$ to realizations of $t$, $u$ respectively,\nwe have 2 cases.\n\\begin{enumerate}\n\\item  If $e_t\\not=e_u$, then\n$d(t, u)= \\dist(x, \\overline{E_A})^t+d(e_t, e_u)+\\dist(x, \\overline{E_A})^u$.\n\\item If $e_t=e_u$, then $d(t,u)=|d(x,p)^t-d(x,p)^u|.$\n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nTo prove statement (1), define the function $f\\colon S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)\\rightarrow [0,r]$\nby $f(q)=d(p, b)$ for any $q\\in S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ and any $b\\models q$. This function\nis well defined because the value of the formula $d(p,x)$ will be the same for any\nrealization of the type $q$.\nThe function $f$ is surjective because given $s\\in [0,r]$, in any model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ there\nis always at least one point with distance $s$ from $p$ by Lemmas \\ref{branching point within epsilon} and \\ref{branches of sufficient height}.\nGiven two 1-types $t, u \\in S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$\na configuration minimizing the distance between realizations $b\\models t$\nand $c\\models u$ is the one where $b$ and $c$ are arranged along a piecewise segment.\nIf $d(p,c)\\geq d(p,b)$ then this segment is of the form $[p, b, c]$, and it is of the form\n$[p, c, b]$ otherwise.  This configuration makes\n$d(b, c)=|d(p, b)-d(p, c)|$ which, by the triangle inequality, is the least possible value of $d(b,c)$.\nThus, $d(t, u)=d(b, c)=|d(p, b)-d(p, c)|=|f(t)-f(u)|$, showing $f$ is an isometry.\n\nStatement (2) is a consequence of Lemmas \\ref{determinetypes} and \\ref{n types over A}.\nTo prove (3) first assume $e_t\\not=e_u$. Then for any realizations $b\\models t$ and $c\\models u$,\nby Lemma \\ref{differentclosestpoints}, $[b, e_t, e_u, c]$ is a piecewise segment, and the conclusion\nfollows.  If $e_t=e_u$, then as in the proof of (1), a minimizing configuration will have\n$b\\models t$ and $c\\models u$ along a geodesic segment, and the conclusion follows.\n\\end{proof}\n\nIt becomes complex to give precise description, as in the preceding lemma, of the metric on $S_n(A)$ over the theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ from the data describing the types. This complexity\nonly increases with $n$. Accordingly, we limit ourselves to giving (in the next result) a precise\nstatement of the metric density of the space of 2-types over $\\emptyset$. Its proof illustrates\nsome of the ideas needed to understand these metric spaces more completely.\n\n\\begin{proposition}\nThe space of $2$-types over the empty set has metric density character equal to $2^\\omega$.\n\\end{proposition}\n\\begin{proof}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ such that every type in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ is realized in $\\mathcal{M}$ for each $n\\geq 1$.\nFor each $s\\in [\\frac{r}{2}, r]$, choose a pair of points $a_s, b_s\\in M$ so that\n$d(p, a_s)=2s$, $d(p, b_s)=2s$ and $d(a_s, b_s)=2s$. Let $Y_s=Y(a_s, b_s, p)$. \nWith our choice of distances, $Y_s$ is not equal to $a_s, b_s$ or $p$, so $p, a_s, b_s$ are not\narranged along a piecewise segment.\nThe subtree spanned by $p, a_s, b_s$ consists of 3 branches at $Y_s$,\neach with length $s$, where $p, a_s, b_s$ are the endpoints of the 3 branches.\nFor any $a_s', b_s'\\models \\tp(a_s, b_s)$ in $\\mathcal{M}$,\nthe subtree spanned by $p, a_s', b_s'$ is isometric to the one spanned by\n$p, a_s, b_s$ via an isometry fixing $p$ and matching $a_s$ and $a_s'$, and $b_s$ with $b_s'$.\n\nClaim: If $s,t \\in [\\frac{r}{2}, r]$ and $s>t$, then $d(\\tp(a_s,b_s), \\tp(a_t,b_t))\\geq 2s.$\\newline\nTo determine the distance between these types, we consider\npossible configurations of points $a_s',b_s'\\models \\tp(a_s, b_s)$\nand $a_t',b_t'\\models \\tp(a_t, b_t)$.\nIt is straightforward to see that in a minimizing configuration,\nwe must put $Y_t'$ and $Y_s'$ on the same branch at $p$.\nSpecifically, we must have $Y_t'\\in [p,Y_s']$ making $d(Y_t', Y_s')=s-t$. \n\nSince $a_t', b_t'$ must be on separate branches at $Y_t'$, we know that\nat most one of $a_t'$ or $b_t'$ is on the same branch at $Y_t'$ as $Y_s'$.\nAssume $a_t'$ is on a different branch at $Y_t'$ from $Y_s'$. \nThen $[a_t', Y_t', Y_s', a_s']$ is a piecewise segment, and thus\n\n\\begin{align*}\nd(a_t', a_s')\n&=d(a_t', Y_t')+d(Y_t', Y_s')+d(Y_s', a_s')\\\\  \n&=t+(s-t)+s=2s.\n\\end{align*}\nIf, instead, it is $b_t'$ on a different branch at $Y_t'$ from $Y_s'$,\nthen we get $d(b_t', b_s')=2s$. \nTherefore, $\\max\\{d(a_s', a_t'), d(b_s', b_t')\\}\\geq 2s$ for any configuration.\nSince $s\\in [\\frac{r}{2}, r]$, we know $2s\\geq r$.\nThus, we can make a set of $2$-types, one for each $s\\in [\\frac{r}{2}, r]$,\nso that the distance  between any two of them is $\\geq r$. \nThis implies that the space of $2$-types has metric density character at least $2^\\omega$.  It is straightforward to show using Lemma \\ref{determinetypes} that the cardinality of the space of $2$-types over the empty set is $2^\\omega$, and therefore the metric density character must be exactly\nequal to $2^\\omega$.\n\n\\end{proof}\n\n\n\nNext, we show that the definable closure and the algebraic closure\nof a set of parameters $A$ are the same, and equal to the closed subtree spanned\nby $A$.\n\n\\begin{proposition}\n\\label{L: dcl and acl characterized}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with $(M,d,p)$ as its underlying $\\R$-tree.\nLet $A\\subseteq M$ be a non-empty set of parameters. Then $\\dcl(A)=\\acl(A)=\\overline{E_A}$.\n\\end{proposition}\n\\begin{proof}\nIf $a,b\\in A$, then by Lemma \\ref{definablemidpoints} any point in $[a,b]$ is in $\\dcl(A)$. Then $E_A\\subseteq \\dcl(A)$ because $E_A$ is the union of all such geodesic segments.\nSince $\\dcl(A)$ is closed, $\\overline{E_A}\\subseteq \\dcl(A)$. Combined with the fact that\n$\\dcl(A)\\subseteq \\acl(A)$, this gives $\\overline{E_A}\\subseteq \\dcl(A)\\subseteq \\acl(A)$.\nIt remains to show $\\acl(A)\\subseteq \\overline{E_A}$, which we do in the contrapositive. \n\nIf $c\\notin \\overline{E_A}$, let $e$ be the unique closest point to $c$\nin $\\overline{E_A}$. This $e$ exists by Lemma \\ref{!geo}.\nLet $B$ be the branch at $e$ that contains $c$.\nUsing Lemma \\ref{chiswell add to points} we construct\nan extension $\\mathcal{N}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of $\\mathcal{M}$ which adds an infinite number of branches at $e$, each of which is isometric to $B$.   \nBy Lemma \\ref{determinetypes}, on each of these branches is a realization of $\\tp(c\/A)$. The distance between any two such realizations is $2d(e, c)>0$. This gives us a non-compact set of realizations. Thus, $c\\notin \\acl(A)$. \n\\end{proof}\n\n\n\nThe next result gives a characterization of $\\kappa$-saturated $\\R$-trees, completing the result\npromised in Section \\ref{model companion}.\n\\begin{theorem}\n\\label{kappasaturated=kappabranching}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nLet $\\kappa$ be an infinite cardinal.\nThen $\\mathcal{M}$ is $\\kappa$-saturated if and only if $M$ has at least $\\kappa$ many branches\nat every point $a\\in M$ where $d(p, a)<r$.\n\\end{theorem}\n\\begin{proof}\nThe forward direction is Lemma \\ref{saturatedrbrt}.\nNow, assume $M$ has $\\kappa$-many\nbranches at every point $a\\in M$ where $d(p, a)<r$, and recall\nthat by Lemma \\ref{branches of sufficient height} each of these branches\ncontains at least one point with distance $r$ from $p$.\nLet $A\\subseteq \\mathcal{M}$ have cardinality less than $\\kappa$, and\nwithout loss of generality assume basepoint $p$ is in $A$.\n\nLet $q$ be a $1$-type over $A$.\nBy Lemma \\ref{determinetypes}, this type is determined by a\nclosest point $e\\in \\overline{E_A}$ and a distance $0\\leq s\\leq r-d(p, e)$\nwhere $s$ is the distance between $\\overline{E_A}$ and any realization of $q$.\nIf $s=0$, then the type\n$q$ is satisfied by the point $e\\in \\overline{E_A}$. Now, assume $s>0$.\nBecause there are $\\kappa>2|A|$ branches at $e$, it follows from the definition of $\\overline{E_A}$ that\nthere are branches at $e$ in $M$ that do not intersect $\\overline{E_A}$ except at $e$. On one of these branches take $b$ with distance $s$ from $e$.\nThis $b$ satisfies the type $q$.  Therefore, all $1$-types over $A$ are realized in $\\mathcal{M}$, implying that $\\mathcal{M}$ is $\\kappa$-saturated.\n\\end{proof}\n\nTo finish this section we show $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable, but not superstable. (\\textit{i.e.}, it is strictly stable.)\n\n\\begin{theorem}\n\\label{rbrtstability}\nThe theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable.\nIndeed when $\\kappa$ is an infinite cardinal, $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable\nif and only if $\\kappa$ satisfies $\\kappa^\\omega = \\kappa$.\n\\end{theorem}\n\\begin{proof}\nLet $\\kappa$ be an infinite cardinal.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ be $\\kappa^{+}$-saturated, with underlying $\\R$-tree $(M,d,p)$.\\\\\nFirst, assume $\\kappa=\\kappa^\\omega$.\nLet $|A|=\\kappa$. Then\n$$|E_A|\\leq |A\\times A|2^\\omega=\\kappa^2 2^\\omega\n\\leq \\kappa^\\omega 2^\\omega=\\kappa^\\omega=\\kappa.$$\nThus, $|E_A|=\\kappa$.\nWe count possible $1$-types using Lemma \\ref{how many types}, showing that \\\\\n$$|S_1(A)|\\leq |\\overline{E_A}\\times [0,1]|=|\\overline{E_A}|2^\\omega\\leq\n|E_A|^\\omega 2^\\omega=\\kappa^\\omega 2^\\omega=\\kappa^\\omega=\\kappa.$$\nThus the $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable.\n\nFor the other direction, assume $\\kappa<\\kappa^\\omega$. We construct, via a tree construction, a subset $A$ of $M$ with $|A|=\\kappa$ and\n$|\\overline{E_A}|=\\kappa^\\omega$.\nAt Step 1 choose $\\kappa$-many points $(a_i\\mid i<\\kappa)$ on distinct branches at $p$, each with distance $\\frac{r}{4}$ from $p$.\nWe can do this since there are at least $\\kappa$ branches of sufficient height at every point in $M$ by Theorem \\ref{kappasaturated=kappabranching} and Lemma \\ref{branches of sufficient height}. Note that we only need $\\kappa$-saturation\nto guarantee $\\kappa$-many branches.  \nAt Step 2, for each $a_{i}$ we choose $\\kappa$-many points on distinct branches at $a_i$,\neach with distance $\\frac{r}{8}$ from $a_i$, and distance $\\frac{3r}{8}$ from $p$. \nWe can index all of these points by $(a_{i,j}\\mid i,j<\\kappa)$.\nAt Step $n$ for $n\\geq 2$, we have already designated points $a_{i_1, i_2,...,i_{n-1}}$ each\nof which has distance\n$\\sum_{k=1}^{n-1} \\frac{r}{2^{k+1}}$ from $p$. At each of these points choose $\\kappa$-many\npoints on distinct branches at $a_{i_1, i_2,...,i_{n-1}}$,\neach with distance $\\frac{r}{2^{n+1}}$ from $a_{i_1, i_2,...,i_{n-1}}$, and distance $\\sum_{k=1}^{n} \\frac{r}{2^{k+1}}$ from $p$. \nWe can index all of these points by $(a_{i_1, i_2,...,i_n}\\mid i_1,...,i_n<\\kappa)$. \nLet $A=\\{p\\}\\bigcup_{k=1}^\\infty (a_{i_1,...,i_k}\\mid i_1,...,i_k<\\kappa)$. If we associate $p$\nwith the empty sequence, then the elements of $A$ are in 1-1 correspondence with $\\kappa^{<\\omega}$.\nSo, the cardinality of $A$ is $|\\kappa^{<\\omega}|=\\kappa$.\n\nNow, for each function $f\\colon \\omega \\rightarrow \\kappa$ with $f(0)=0$ there is a unique\nsequence $(b_n)$ of elements of $A$ with $b_0=p$ and $b_n=a_{f(1),...,f(n)}$.\nNote that $b_n$ has distance $\\frac{r}{2^{n+1}}$ from $b_{n-1}$,\nmaking $(b_n)$ a Cauchy sequence. Since $M$ is complete, $b_n$ must converge to a limit with distance $\\frac{r}{2}$ from $p$.\\\\\nLet $f$ and $g$ be two distinct functions from $\\omega$ to $\\kappa$, and let $(b_n)$ and $(c_n)$\nbe their respective associated sequences.   Let $m$ be the first index at which $f(m)$ and $g(m)$ \ndisagree (note that $m\\not=0$). Then $b_m=a_{f(1),...,f(m)}$ and $c_m=a_{g(1),...,g(m)}$\nare on different branches out of \n$a_{f(1),...,f(m-1)}=b_{m-1}=c_{m-1}$. Moreover, for all $k\\geq m$, $b_k$ is in the same branch at\n$b_{m-1}$ as $b_m$ and likewise for the sequence $c_m$.\nThus, the limits of these two sequences must be in\ndifferent branches at $b_{m-1}=c_{m-1}$, so these limits are distinct points.\nThere are $\\kappa^\\omega$-many different functions from $\\omega$ to $\\kappa$\nwith $f(0)=0$.  Thus, $|\\overline{E_A}|=\\kappa^\\omega$.\n\nFor each limit $e\\in \\overline{E_A}$ constructed above,\nchoose $b_e$ on a branch at $e$ that intersects $\\overline{E_A}$\nonly at $e$ with $d(e,b_e)=\\frac{r}{4}$.  We may always\nfind such a branch, because there are $\\kappa$-many branches at the limit $e$, but\nonly one of those branches intersects $\\overline{E_A}$ at any point other than $e$.\nThe branch will be of sufficient height by Lemma \\ref{branches of sufficient height}.\nThe set of such points $b_e$ has\ncardinality $\\kappa^\\omega$, and for any limits $e\\not= f$ in $\\overline{E_A}$\nit is straightforward to show $d(\\tp(b_e\/A), \\tp(b_f\/A))\\geq \\frac{r}{2}$.\nSince $\\kappa^\\omega>\\kappa$, this implies the theory is not $\\kappa$-stable.\n\\end{proof}\n\n\n\\section{The independence relation for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$}\n\nIn this section we characterize the model theoretic independence relation of\n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and show that in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, types have canonical\nbases that are easily-described sets of ordinary (not imaginary) elements.\n\nLet $\\kappa$ be a cardinal so that $\\kappa=\\kappa^\\omega$ and $\\kappa >2^\\omega$.\nIn this section let $U$ be a $\\kappa$-universal domain for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nA subset of $U$ is \\emph{small} if its cardinality is $<\\kappa$.\n\\begin{definition}\n\\label{starindependence}\nLet $A,B$ and $C$ be small\nsubsets of $U$.\nSay A is ${}^*$-independent from B over C, denoted $A\\ind[*]_C B$,\nif and only if for all $a\\in A$ we have $\\dist(a,\\overline{E_{B\\cup C}})=\\dist(a,\\overline{E_C})$.\n\\end{definition}\n{\\bf Note:} In what follows, we will abbreviate unions such as $B\\cup C$ as $BC$.\n\n\\begin{lemma}\n$A\\ind[*]_C B$\nif and only if for all $a\\in A$ the closest point to $a$ in\n$\\overline{E_{BC}}$ is the same as the closest point to $a$ in $\\overline{E_C}$.\n\\end{lemma}\n\\begin{proof}\nAssume $A\\ind[*]_C B$.\nTake an arbitrary $a\\in A$. Let $e_1$ be the unique closest point to $a$ in $\\overline{E_{BC}}$\nand $e_2$ the unique closest point to $a$ in $\\overline{E_C}$.\nWe assumed $\\dist(a,\\overline{E_{BC}})=\\dist(a,\\overline{E_C})$, which implies $d(a, e_1)=d(a, e_2)$.\nSince $e_2\\in \\overline{E_C}\\subseteq \\overline{E_{BC}}$, we know $e_1\\in [a,e_2]$ by Lemma \\ref{!geo}.\nTherefore, $e_1=e_2$. Since $a$ was arbitrary, we know this holds for all $a\\in A$.\nFor the other direction, assume for all $a\\in A$ the closest point to $a$ in $\\overline{E_{BC}}$\nis the closest point to $a$ in $\\overline{E_C}$. Then clearly\n$\\dist(a, \\overline{E_{BC}})=\\dist(a, \\overline{E_C})$ for all $a\\in A$.\n\\end{proof}\n\n\\begin{theorem}\n\\label{starindependencetheorem}\nThe relation $\\ind[*]$ is the model theoretic\nindependence relation for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Moreover, types over arbitrary sets\nof parameters are stationary.\n\\end{theorem}\n\\begin{proof}\nWe will show $\\ind[*]$ satisfies all the properties of a stable independence relation\non a universal domain of a stable theory as given in \\cite[Theorem 14.12]{N}.\nThen by \\cite[Theorem 14.14]{N} we know $\\ind[*]$ is the\nmodel theoretic independence relation for the stable theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\\noindent{\\bf (1) Invariance under automorphisms}\n\nAny automorphism $\\sigma$ satisfies $\\sigma(\\overline{E_A})=\\overline{E_{\\sigma(A)}}$\nand is distance preserving.\n\n\\noindent{\\bf (2) Symmetry:} if $A\\ind[*]_C B$, then $B\\ind[*]_C A$.\n\nAssume $A\\ind[*]_C B$.\nThis means for all $a\\in A$ we have that the closest point in $\\overline{E_{BC}}$\nto $a$ is $e_a\\in \\overline{E_C}$.\nThus, by Lemma \\ref{branch}, for any $a\\in A$, for any $y\\in \\overline{E_{BC}}$ we have\n$[a, y]\\cap \\overline{E_C}\\not=\\emptyset$. It follows that\nfor any $x\\in \\overline{E_A}$, for any $y\\in \\overline{E_{BC}}$\nthere exists a point of $\\overline{E_C}$ on $[x,y]$.\nLet $b\\in B$. Then for any $x\\in \\overline{E_A}$ there is a point of $\\overline{E_C}$\non $[x,b]$. It follows that the closest point in $\\overline{E_{AC}}$ to any $b\\in B$ is in $\\overline{E_C}$.\n\n\\noindent{\\bf (3) Transitivity:} $A\\ind[*]_C BD$ if and only if $A\\ind[*]_C B$ and $A\\ind[*]_{BC} D$.\n\nWe know\n$$E_C\\subseteq E_{BC}\\subseteq E_{BCD}$$\nwhich implies\n$$\\dist(a, \\overline{E_C})\\geq \\dist(a, \\overline{E_{BC}})\\geq \\dist(a, \\overline{E_{BCD}}).$$\nTherefore $\\dist(a, \\overline{E_{BCD}})=\\dist(a, \\overline{E_C})$ if and only if\n$$\\dist(a, \\overline{E_{BC}})=\\dist(a, \\overline{E_C}) \\text{ and }\n\\dist(a, \\overline{E_{BCD}})=\\dist(a, \\overline{E_{BC}}).$$\nHence\n$$A\\ind[*]_C BD \\text{ if and only if } A\\ind[*]_C B \\text{ and } A\\ind[*]_{BC} D.$$\n\n\\noindent{\\bf (4) Finite character:} $A\\ind[*]_C B$ if and only if $a\\ind[*]_CB$ for all finite tuples $a\\in A$.\n\nThis is clear from the definition.\n\n\n\\noindent{\\bf (5) Extension:} for all $A, B, C$ we can find $A'$ such that\ntp$(A\/C)=$ tp$(A'\/C)$ and $A'\\ind[*]_C B$.\n\nBy finite character and compactness, it suffices to show this statement when $A$ is\na finite tuple.  Let $e\\in \\overline{E_C}$ be the unique point closest to $\\overline{E_A}=E_A$.\nLet $\\beta<\\kappa$ be the cardinality of $\\overline{E_B}$.  Then there\nare at most $\\beta$ branches in $\\overline{E_B}$ at any point of $\\overline{E_B}$.\nSince $A$ is finite, use Lemma \\ref{embed} to embed a copy of $\\overline{E_A}=E_A$ on branches\nat $e$ that do not intersect $\\overline{E_B}$ except at $e$.\nThe image of $A$ under this embedding gives us $A'$.\n\n\n\\noindent{\\bf (6) Local Character:} if $a=a_1,...,a_m$ is a finite tuple,\nthere is a countable $B_0\\subseteq B$ such that $a\\ind[*]_{B_0}B$.\n\nLet $e_i$ be the closest point of $\\overline{E_B}$ to $a_i$ for $i=1,...,m$. Let $B_i$\nbe a countable subset of $B$ such that $e_i$ is an element of $\\overline{E_{B_i}}$.\nLet $B_0=\\bigcup_i^m B_i$.\n\n\\noindent{\\bf  (7) Stationarity (over arbitrary sets of parameters):} \nif $\\tp(A\/C)=\\tp(A'\/C)$, $A\\ind[*]_C B$, and $A'\\ind[*]_C B$,\nthen\n$\\tp(A\/BC)=\\tp(A'\/BC)$,\nwhere $C$ is a small submodel of $U$.\n\nBy quantifier elimination, $\\tp(A\/BC)$ is determined by $\\{\\tp(a\/BC)\\mid a\\in A\\}$ plus\nthe information $\\{d(a_1, a_2)\\mid a_1, a_2\\in A\\}$. These distances\n$\\{d(a_1, a_2)\\mid a_1, a_2\\in A\\}$ are fixed by $\\tp(A\/C)$.\nThus, it suffices to show the conclusion in the case when $A=\\{a\\}$ and $A'=\\{a'\\}$.\nIf $a$ or $a'$ is in $C$ the conclusion is obvious, so assume $a, a' \\notin C$.\nThe type of $a$ (or $a'$) over $BC$ is determined by two parameters,\nthe unique point in $\\overline{E_{BC}}$ that is closest to $a$, and the distance from $a$\nto that point.  Since $a\\ind[*]_C B$, it follows that the closest point in $\\overline{E_C}$\nto $a$ is the same as the closest point in $\\overline{E_{BC}}$ to $a$,\nand the same is true for $a'$.\nSince tp$(a\/C)=$tp$(a'\/C)$, we know $a$ and $a'$ have the same\nclosest point $e$ in $\\overline{E_C}$ and $d(a, e)=d(a',e)$.  Since $e$ is also the closest point\nin $\\overline{E_{BC}}$ to $a$ and $a'$, we know that tp$(a\/BC)=$tp$(a'\/BC)$\nby Lemma \\ref{determinetypes}.\n\\end{proof}\n\n\\noindent \\textbf{Canonical Bases} \\medskip\n\nA canonical base of a stationary type is a minimal set of parameters over which that type is\ndefinable.  However, to avoid a discussion of definable types, we here use an\nequivalent definition of canonical base, as given in \\cite{IAW}.  As in that paper, we\nhere take advantage of the fact (Lemma \\ref{L: dcl and acl characterized} and part (7)\nof the proof of Theorem \\ref{starindependencetheorem}) that every type over an\narbitrary set of parameters is stationary.\n\nFor stable theories in general, canonical bases exist as sets of imaginary elements, \nhowever, in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, they are sets of ordinary elements.  That is,\nthe theory has built-in canonical bases.  Indeed, in this setting they are very simple.\n\nFor sets $A\\subseteq B\\subseteq U$,\nand $q\\in S_n(A)$ we say $q'\\in S_n(B)$ is a \\emph{non-forking extension} of $q$ if\n$b\\models q'$\nimplies $b\\models q$ and $b\\ind_A B$.\nBy the definition of independence, the condition $b\\ind_A B$ implies that the points\n$e_1,...,e_n$ in $\\overline{E_A}$ closest to $b_1,...,b_n$ respectively must\nalso be the closest points to $b_1,...,b_n$ in $\\overline{E_B}$.\nBecause $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable and all types are stationary, non-forking extensions are unique.\nDenote the unique non-forking extension of $q$ to the set $B$ by $q\\upharpoonright^B$. \nGiven a type $q$ over a set $A\\subseteq U$ and an automorphism $f$ of $U$,\n$f(q)$ denotes the set of $L_r$-conditions over $f(A)$ corresponding to the\nconditions in $q$, where each parameter $a\\in A$ is replaced by its image $f(a)$.\n\n\\begin{definition}[{\\cite[Definition 6.1]{IAW}}]\nA \\emph{canonical base} $Cb(q\/A)$ for a type $q\\in S_n(A)$ is a subset $C\\subseteq U$ such that\nfor every automorphism $f\\in \\Aut(U)$, we have:\n$q\\upharpoonright^U= f(q)\\upharpoonright^U$ if and only if $f$ fixes each member of $C$.\n\\end{definition}\n\nThe following result describes canonical bases in $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. \n\n\\begin{theorem}\nLet $b=(b_1,...,b_n)\\in U^n$ and $A\\subseteq U$ a set of parameters. Let $q \\in S_n(A)$\nbe the type over $A$ of the tuple $b$. Then a canonical base of $q$ is given by the set\n$\\{e_i \\mid 1\\leq i\\leq n\\}$, where $e_i\\in \\overline{E_A}$ is the closest point to $b_i$ in $\\overline{E_A}$.\nNote that this set depends only on $q$.\n\\end{theorem}\n\\begin{proof}\nLet $b$,  $A\\subseteq U$ and $q \\in S_n(A)$\nbe as described in the statement of the theorem. Let $C=\\{e_i \\mid 1\\leq i\\leq n\\}$\nwhere $e_i\\in \\overline{E_A}$ is the closest point to $b_i$ in $\\overline{E_A}$.\nFirst, assume $f$ is an automorphism of $U$ fixing $C$ pointwise.\nLet $c=(c_1,...,c_n)$ be a realization of $q\\upharpoonright^U$ (in some extension of $U$).\nThen $c\\models q$ and $c\\ind_{A} U$. To show $f(q)\\upharpoonright^U$=$q\\upharpoonright^U$\nit suffices to show that $c\\models f(q)$ and $c\\ind_{f(A)} U$, because\nthen $q\\upharpoonright^U$ is the unique non-forking extension of $f(q)$ to $U$.\nBy Lemma \\ref{determinetypes}, an $n$-type over a set $A$ is determined by the values it assigns to the formulas\n$d(x_i, x_j)$ and $d(x_i, a)$ for $a\\in A$.  Note that in $f(q)$, the parameter-free\n$L_r$-conditions are the same as in the type $q$. So,\nfor example, $d(x_i, x_j)$ must have the same value in $f(q)$ as in $q$.  \nThus, to show that $c\\models f(q)$ we just need to show\nthat $d(c_i,a)=d(c_i, f(a))$ for all $a\\in A$ and $i\\in \\{1,...,n\\}.$\n\nWe know $c\\models q$. Thus Lemma \\ref{determinetypes} implies that $e_i$ must be the closest\npoint to $c_i$ in $\\overline{E_A}$.\nAlso, $c\\ind_A U$ implies that $e_i$ is also the closest point in $U$ to $c_i$.\nSince $f(\\overline{E_A})\\subseteq U$ we know the closest point in $f(\\overline{E_A})$ to $c_i$ is $e_i=f(e_i)$.\nTherefore by Lemmas \\ref{!geo} and \\ref{branch}\nwe know $d(c_i, a)=d(c_i, e_i)+d(e_i, a)$ and $d(c_i, f(a))=d(c_i, e_i)+d(e_i, f(a))$ for any $a\\in A$. Thus, \n\\begin{eqnarray*}\nd(c_i, a)&=&d(c_i, e_i)+d(e_i, a)\\\\\n&= &d(c_i, e_i)+d(f(e_i), f(a))\\\\\n&=&d(c_i, e_i)+d(e_i, f(a))=d(c_i, f(a))\n\\end{eqnarray*}\nestablishing that $c\\models f(q)$.\n\nSince $f$ is an isometry, clearly $f(e_i)=e_i$ is the closest point to $c_i$ in $f(\\overline{E_A})$.\nThe closed subtree $f(\\overline{E_A})$ is equal to the closed subtree $\\overline{E_{f(A)}}$\nsince $f([a,b])=[f(a), f(b)]$ for all $a, b\\in U$.\nThis implies that $c\\ind_{f(A)} U$.   We conclude that $f(q)\\upharpoonright^U$=$q\\upharpoonright^U$.\n\n\n\nFor the other direction, assume $f$ is an automorphism of $U$\nthat does not fix all of the elements of $C$. Without loss of generality,\nassume $f(e_1)\\not=e_1$. Let $(c_1,...,c_n)\\models q\\upharpoonright^U$.\nThen the closest point in $U$ to $c_1$ is $e_1$, which is also the closest point to $c_1$ in $\\overline{E_A}$.\nThen, since $f(e_1)\\in U$, the point $e_1$ must be\non the geodesic segment joinging $f(e_1)$ and $c_1$, so $d(f(e_1), c_1)=d(f(e_1), e_1)+d(e_1,c_1)$.\nThus, $d(f(e_1), e_1)=d(f(e_1), c_1)-d(e_1, c_1)$.\nSince $d(f(e_1), e_1)\\not=0$, then $d(f(e_1), c_1)\\not=d(e_1, c_1)$.\nBut, $d(e_1, c_1)$ is the value of the formula $d(e_1, x_1)$ in $q\\upharpoonright^U$,\nand by the definition of $f(q)$, the value of $d(f(e_1), x_1)$ in $f(q)$ must equal the value of\n$d(e_1, x_1)$ in $q$. So, the $L$-condition $|d(f(e_1), x_1)-d(e_1, c_1)|=0$ is in the type $f(q)$,\nand therefore in the type $f(q)\\upharpoonright^U$.\nThus, the tuple $c=(c_1,...,c_n)$ cannot be a realization of $f(q)\\upharpoonright^U$, and\ntherefore $q\\upharpoonright^U\\not= f(q)\\upharpoonright^U$\n\\end{proof}\n\n\\endinput\n\n\n\\section{Models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$: Examples}\n\nIn this section we discuss examples of models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ from the literature.  Our first examples\ncome from the explicitly described universal $\\R$-trees that are treated in \\cite{DP2}.  We show that\nthey give exactly the (fully) saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Our second examples come from\nasymptotic cones of hyperbolic finitely generated groups.  They give exactly the saturated\nmodel of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density $2^\\omega$.\n\nWe begin with a lemma about the density of a $\\kappa$-saturated model.\n\n\\begin{lemma}\n\\label{kappabranch}\n\\label{saturateddensity}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying $\\R$-tree $(M,d,p)$, and let $\\kappa$\nbe an infinite cardinal.\n\\newline\n(1) If there exists $a\\in M$ with degree $\\kappa$, then\nthe density character of $M$ is at least $\\kappa$.\n\\newline\n(2) If  $\\mathcal{M}$ is $\\kappa$-saturated, then the density character of $\\mathcal{M}$ \nis at least $\\kappa^\\omega$.\n\\end{lemma}\n\\begin{proof}\n(1) If $d(p,a)=r$, then there is a single branch at $a$, namely the branch\ncontaining $p$. So, we must have $d(p, a)<r$.\nUsing Lemma \\ref{branches of sufficient height} and taking points on different branches at $a$\neach with distance $r$ from $p$, we find a collection of\n$\\kappa$-many points such that the distance between any two of them is $2(r-d(p,a))$.\nThus, the density character of $M$ must be at least $\\kappa$.\n\n(2) By Theorem \\ref{kappasaturated=kappabranching}, at each point in the \nunderlying $\\R$-tree of $M$ of $\\mathcal{M}$ there are at least $\\kappa$-many branches.\nThe tree construction from Theorem \\ref{rbrtstability} then yields at least\n$\\kappa^\\omega$-many\ndistinct points with pairwise distances at least $\\frac{r}{2}$.\n\\end{proof}\n\n\n\n\n\n\\noindent \\textbf{Universal $\\R$-trees} \\medskip\n\nNext, we include the description of the universal $\\R$-trees from \n\\cite{DP2} and relate them to saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{definition}[{See \\cite{MNO}}]\nLet $\\mu$ be a cardinal. An $\\R$-tree $M$ is called {\\it $\\mu$-universal}\nif, for any $\\R$-tree $N$ with $\\leq \\mu$ branches at every point, there\nis an isometric embedding of $N$ into $M$.\n\\end{definition}\n\n\\begin{example}[{See \\cite[Lemma 2.1.1]{DP2}}]\n\\label{universalrtrees}\nFor each $\\mu \\geq 2$ let $C_{\\mu}$ be\na set with cardinality $\\mu$ if $\\mu$ is infinite, and cardinality\n$\\mu-1$ if $\\mu$ is finite. Consider the set of functions from intervals in $\\R$ to $C_{\\mu}$\n\n$$f\\colon (-\\infty, \\rho_f)\\to C_{\\mu}$$\nwhere $f$ is such that\n\\begin{enumerate}\n\\item there exists $\\tau_f\\leq \\rho_f$ so that $f=0$ for all $t\\in (-\\infty, \\tau_f).$\n\\item The function $f$ is piecewise constant from the right. That is, for\nany $t\\in (-\\infty, \\rho_f)$ there exists $\\delta>0$ so\nthat $f$ is constant on $[t, t+\\delta]$.\n\\end{enumerate}\n\nOn this set of functions define the metric\n$$d(f, g)=(\\rho_f-s)+(\\rho_g-s) \\text{ where } s=\\sup\\{t\\mid f(t')=g(t') \\, \\forall t'<t\\}.$$\n\nThe authors of \\cite{DP2} show that $A_\\mu$ is a complete $\\R$-tree with $\\mu$-many branches of infinite height at every point. They also show that $A_\\mu$ is the unique (up to isometry) $\\R$-tree with $\\mu$ branches at each point and that it is homogeneous and $\\mu$-universal.\nIn our terminology, for $\\mu\\geq 3$ the space $A_\\mu$ is a complete, unbounded richly branching $\\R$-tree.\nLet $\\mathcal{M}_\\mu$ be the model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying space equal to\nthe closed $r$-ball of $A_\\mu$ with an arbitrary choice of base point. \nFor infinite $\\mu$, Theorem \\ref{kappasaturated=kappabranching} shows $\\mathcal{M}_\\mu$\nis a $\\mu$-saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\nClassical model theory suggests that since $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete and $\\kappa$-stable\nexactly when $\\kappa=\\kappa^\\omega$, there should be a unique saturated model\nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with density $\\kappa$ exactly when $\\kappa=\\kappa^\\omega$.\nWe verify here that these unique saturated models are exactly those described in the preceding\nparagraph\nfor $\\mu=\\mu^\\omega$.\n\n\nSince $\\mathcal{M}_\\mu$ is $\\mu$-saturated, $\\mathcal{M}_\\mu$ (and $A_\\mu$) have density character at least $\\mu^\\omega$\nby Lemma \\ref{saturateddensity}.\nAn examination of the construction reveals that the cardinality of $A_\\mu$ is\nat most $\\mu^\\omega$. Conditions (1) and (2) in Example \\ref{universalrtrees}\nimply that each $f\\in A_{\\mu}$ can only change values at (at most) countably many points.\nMore specifically, we can recursively build a sequence $\\{t_n\\}_{n\\in \\N}$\nrecording where $f$ changes value as follows. For $f\\in A_{\\mu}$, let $t_0=\\tau_f$.\n\nGiven $t_n$, let $t_{n+1}=\\sup\\{t\\in (-\\infty, \\rho_f)\\mid f(t)=f(t_n)\\}$\nif this supremum is $<\\rho_f$. If $\\sup\\{t\\in (-\\infty, \\rho_f)\\mid f(t)=f(t_n)\\}=\\rho_f$,\nthen set $t_{n+i}=\\rho_f$ for all $i\\in \\N$.\n\\begin{itemize}\n\\item if $t_n=t_{n+1}$, then $t_{n+i}=t_n=\\rho_f$ for all $i\\in \\N$\n\\item $f$ is constant on $[t_n, t_{n+1})$ for each $n\\in \\N$\n\\end{itemize}\nFor each $n\\in \\N$, let $\\alpha_n\\in C_{\\mu}$\nbe the value of $f$ on $[t_n, t_{n+1})$.\nThis gives us a sequence $\\{\\alpha_n\\}$ of elements of $C_{\\mu}$.\nThe sequences $\\{t_n\\}$ and  $\\{\\alpha_n\\}$ determine $f$.\nThus, the cardinality of $A_{\\mu}$ is at most $2^\\omega \\cdot \\mu^\\omega=\\mu^\\omega$.\nIt follows that the metric density of $A_{\\mu}$ is exactly $\\mu^\\omega$.\nThus, in the case that $\\mu=\\mu^\\omega$, the model $\\mathcal{M}_{\\mu}$ is a\nsaturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, and uniqueness follows from the uniqueness\nof $A_{\\mu}$.  \nIf $\\mu<\\mu^\\omega$, then there is no $\\mu$-saturated model\nwith density $\\mu$, by Part (2) of Lemma \\ref{saturateddensity}.\n\nIn the case that $\\mu=\\mu^\\omega$, we outline an alternative\nmodel-theoretic argument that $A_{\\mu}$ is the unique complete $\\R$-tree\nwith $\\mu$ branches at every point. \nGiven any two complete $\\R$-trees $M_1$ and $M_2$ with $\\mu$-many branches every point,\nselect a basepoint in each.\nThen for each $r>0$ the closed $r$-balls in $M_1$ and $M_2$ (centered at their respective basepoints) are saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.   Hence those $r$-balls are isomorphic\nby the fact that saturated models of a complete theory with the same density are isomorphic.\nA back-and-forth argument can be used to build an isomorphism\nfrom $M$ to $N$, where\neach time we extend the partial isomorphism we take its distance from the basepoint\ninto account and work in a large enough closed $r$-ball.\n\n\n\n\n\\end{example}\n\n\\noindent \\textbf{Asymptotic Cones} \\medskip\n\nA finitely generated group is $\\emph{hyperbolic}$ if its Cayley graph is a $\\delta$-hyperbolic\nmetric space for some $\\delta>0$.\nA \\emph{non-elementary} hyperbolic group is one that has no cyclic subgroup of finite index.\n\n\n\\begin{definition}\nLet $(M, d, p)$ be a metric space. Let $U$ be a non-principal ultrafilter on $\\N$ and let\n$(\\nu_m)_{m\\in \\N}$ be a sequence of positive integers\nsuch that $\\lim_{m\\to \\infty} \\nu_m=\\infty$.\nThe asymptotic cone of $(M, d, p)$ with respect to $(\\nu_m)_{m\\in \\N}$ and $U$ is\nthe ultraproduct of pointed metric spaces $\\prod_U (M, \\frac{d}{\\nu_m}, p)$. \nDenote this asymptotic cone by $\\mathrm{Con}_{U, (\\nu_m)}(M, d,p)$.  Elements of\n$\\mathrm{Con}_{U, (\\nu_m)}(M, d,p)$ are denoted $[a_n]$ where $a_n\\in M$ for each $n$.\n\\end{definition}\nThere are broader versions of this definition that allow, for example, a different\nchoice of base point in each factor. Keeping the same base point is sufficient\nfor our discussion.\n\n\n\\begin{example}\nAn asymptotic cone $\\mathrm{Con}_{U, (\\nu_m)}(G)$ of a finitely generated\ngroup is defined to be the asymptotic cone of its Cayley graph with base point $e$ and some designated word metric on $G$.\nIt is a fact that any asymptotic cone of a hyperbolic group is an $\\R$-tree and is homogeneous (see \\cite{DW} or \\cite{D}).\nIn fact, in the case of a non-elementary hyperbolic group, all asymptotic cones are homogeneous with $2^\\omega$ branches at every point (see \\cite[Proposition 3.A.7]{D}) and are thus are isometric to $A_{2^\\omega}$ from Example \\ref{universalrtrees}.   The next result gives\na proof of this fact.\n\\end{example}\n\n\n\\begin{fact}\\label{switch gen sets}\nSay $B$ and $C$ are both generating sets for the hyperbolic group $G$ and $U$ a non-principal ultrafilter.\nThe word metrics $d_B$ and $d_C$ are Lipschitz equivalent (and the corresponding\nCayley graphs are quasi-isometric.) It follows that the asymptotic cones $\\mathrm{Con}_{U, (\\nu_m)}(G, d_B, e)$ and $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$ are homeomorphic. \n\\end{fact}\n\n\\begin{lemma}\nLet $G$ be a non-elementary hyperbolic group. Let $U$ be a non-principal ultrafilter and let\n$\\{\\nu_m\\}_{m\\in \\N}$ be a sequence of positive integers such that $\\lim_{m\\to \\infty} \\nu_m=\\infty$.\nThen for any generating set $C\\subseteq G$ the asymptotic cone $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$  is a homogeneous richly branching $\\R$-tree with $2^\\omega$-many branches at every point.\n\\end{lemma}\n\\begin{proof}\nSince $G$ is finitely generated, we know that $G$ is countable.\nTherefore, any asymptotic cone of $G$ has cardinality (and therefore density)\nat most $2^\\omega$. By Lemma \\ref{kappabranch} we conclude that any point in the cone\ncan have at most $2^\\omega$ branches.\nWe also know $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$ is a homogeneous $\\R$-tree.\nSince it is non-elementary $G$ contains a free subgroup $F$ with 2 generators (See \\cite{BH}).\nBy Fact \\ref{switch gen sets} we may assume without loss of generality that\nthe generators of $F$ are also generators\nof $G$ and that $C$ is a minimal set of generators. In $F$,\nfor each $m\\in \\N$, we can find a finite set $C_m$ such that $d(e,a)\\geq \\nu_m$ for any $a\\in C_m$\nand so that $(a\\cdot b)_e\\leq \\sqrt{\\nu_m}$ for distinct $a, b\\in C_m$.\nThe Cayley graph of $F$ is a subgraph of the Cayley graph of $G$, and the Cayley graph of a free group is an $\\R$-tree, thus we use $\\R$-tree terminology to describe how to find $C_m.$\n\nGiven $m\\in \\N^{>0}$ let $n_m$ be the largest integer so that $n_m\\leq \\sqrt{\\nu_m}$.\nNote that $n_m\\to\\infty$. There are $4\\cdot3^{{n_m}-1}$ elements of $F$ with distance $n_m$ from $e$. Let $c$ denote such an element. There are 3 branches at $c$ in the Cayley graph of $F$ that do not not contain $e$.\nOn each of these branches, choose a point $a$ with $d(a, e)\\geq \\nu_m$.\nRepeat this process for each $c$ with distance $n_m$ from $e$.\nLet $C_m$ be the collection of all the points $a$. Then $|C_m|=4\\cdot3^{n_m}$, and since $n_m\\to \\infty$\nwe know $|C_m|\\to\\infty$. For any distinct $a, b\\in C_m$, the distance from $e$ to $[a,b]$\nis at most $n_m\\leq \\sqrt{\\nu_m}$, because $[a,b]$ will always contain at least one of the points\nin $F$ with distance $n_m$ from $e$.\n\n\nSince $|C_m|\\to \\infty$ and $U$ is a non-principal ultrafilter on $\\N$,\nwe know that $\\displaystyle \\Pi_m C_m\/U$ is a set of points in the asymptotic cone with cardinality $2^\\omega$.\nMoreover, in the asymptotic cone $d([a_m],[e])=\\lim_U \\frac{d_C(a_m, e)}{\\nu_m}\\geq 1$.\nSince $(a_m\\cdot b_m)_e\\leq \\sqrt{\\nu_m}$, we know the distance from\n $[e]$ to the geodesic segment connecting $[a_m]$ and $[b_m]$ is\n$$([a_m]\\cdot [b_m])_{[e]}=\\lim_U \\frac{(a_m\\cdot b_m)_e}{\\nu_m}=0.$$\nThus, $[e]\\in \\left[[a_m],[b_m]\\right]$, putting $[a_m]$ and $[b_m]$ on separate branches\nat $[e]$.  This gives us $2^\\omega$ many distinct\nbranches at $[e]$ in $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$. Therefore,\nthere must be $2^\\omega$ many branches at every point in the cone.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $G$ be a non-elementary hyperbolic group.\nLet $\\mathcal{M}$ be the model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying space\nequal to the closed $r$-ball of  $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$.\nThen $\\mathcal{M}$ is the unique saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density $2^\\omega$.\n\\end{corollary}\n\\begin{proof}\nBy the preceding lemma and Lemma \\ref{kappabranch}, we know\n$\\mathcal{M}$ has density $2^\\omega$ and by Lemma \\ref{kappasaturated=kappabranching} $\\mathcal{M}$ is $2^\\omega$-saturated.\n\\end{proof}\n\n\n\\section{Models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$: Constructions and non-categoricity}\\label{noncat}\n\nIn this section we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has the maximum number of\nmodels of density character $\\kappa$ for every infinite cardinal $\\kappa$.\nIndeed, for each $\\kappa$ we construct a family of $2^\\kappa$-many such models such that\nno two members of the family are  homeomorphic.  (Two models \nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are homeomorphic if their underlying $\\R$-trees are homeomorphic \nby a map that takes base point to base point.  Note that non-homeomorphic models\nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are necessarily non-isomorphic.)  First we treat \nseparable models, and the amalgamation techniques used\nin that case also allow us to characterize the principal types of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and\nto show that this theory has no atomic model.  Then we use simple amalgamation\nconstructions to handle nonseparable models.\n\n\\begin{lemma}\n\\label{separabletreewitharbbranching}\nLet $S$ be a non-empty set of integers, each of which is $\\geq 3$.\nThere exists a separable richly branching $\\R$-tree $M$ such that\n\\begin{enumerate}\n\\item for each $k\\in S$ the set $\\{x\\in M\\mid x \\text{ has degree }k\\}$ is dense in $M$\n\\item given a branch point $x\\in M$ the degree of $x$ is an element of $S$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $(k_j\\mid j\\in \\N)$\nbe a sequence such that every element of $S$ appears infinitely many times in the sequence,\nand every term of the sequence is an element of $S$.\nWe construct an increasing sequence $\\mathcal{N}_0\\subseteq \\mathcal{N}_1\\subseteq ... \\subseteq \\mathcal{N}_j... $ of \nseparable $\\R$-trees as follows.\n\nLet $N_0$ be the $\\R$-tree $\\R$ with base point $0$.\nLet $A_0$ be a countable, dense subset of $N_0$. \nUse Lemma \\ref{chiswell add to points} to add $k_0-2$ distinct rays (copies of $\\R^{\\geq 0}$) at each\npoint in $A_0$, bringing the number of branches of infinite length at each point in $A_0$ up to $k_0$.\nCall the resulting $\\R$-tree $N_1$. Note that $N_0\\subseteq N_1$.\nThe $\\R$-tree $N_1$ is separable, since it is a countable union of separable spaces.\nNote also that all the points in $N_1\\setminus A_0$ only have 2 branches,\nand it is straightforward to show $N_1\\setminus A_0$ is uncountable and dense in $N_1$.\n\nOnce $N_j$ has been constructed, to construct $N_{j+1}$\nlet $\\displaystyle A_{j}\\subset N_j\\setminus (\\cup_{i=0}^{j-1} A_j)$ be a countable, dense subset of $N_j$. \nThis is possible since $\\displaystyle  N_j\\setminus (\\cup_{i=0}^{j-1} A_j)$ is dense in $N_j$.\nUse Lemma \\ref{chiswell add to points} to add $k_j-2$ rays at each\npoint in $A_j$, bringing the number of branches of infinite length at each point in $A_j$ up to $k_j$.\nThe resulting $\\R$-tree is $N_{j+1}$.\nNote that $N_{j+1}$ is separable, since it is a countable union of separable spaces.\nNote also that all the points in $\\displaystyle N_{j+1}\\setminus \\cup_{j=0}^{j} A_j$ still only have 2 branches,\nand that this set is uncountable and dense in $N_{j+1}$.\nLastly, it is clear that given $x\\in N_{j+1}$ the number of branches at $x$ must be\neither $2$ (in which case $x$ is not a ``branch point\") or one of $\\{k_0, ..., k_j\\}$.\nThis is because at the $j$th step, the only points at which we add rays are those in $A_j$,\nand then in subsequent steps we do not add rays at any of those points.\n\nLet $\\displaystyle M=\\cup_{j\\in\\N} N_j$ be the union of this countable chain of separable $\\R$-trees. Then $M$ is a separable $\\R$-tree (see \\cite[Lemma 2.1.14]{C}.)\nSince for each $N_j$ the number of branches at each branch point is an element\nof $S$, this will also be true in $M$.\nLet $k\\in S$. Let $\\displaystyle J(k)=\\{j\\in \\N\\mid k_j=k\\}$. By how we chose the sequence $(k_j)$ the set $J(k)$\nis infinite.  The set of points in $M$ which have exactly $k$ branches is $\\cup_{j\\in J(k)} A_j$.\nWe will show this set is dense in $M$.\nLet $x\\in M$. Let $j_x\\in \\N$ be the smallest integer such that $x\\in N_{j_x}$. Let $j^*\\in J(k)$ be\nsuch that $j^*>j_x$. We know that $A_{j^*}$ is dense in $N_{j^*}$, and that $\\displaystyle x\\in N_{j_x}\\subseteq N_{j^*}$.\nTherefore, there are points in $\\displaystyle A_{j^*}\\subseteq  \\cup_{j\\in J(k)} A_j$ arbitrarily close to $x$.\nOur choice of $x\\in M$ was arbitrary, therefore $\\displaystyle \\cup_{j\\in J(k)} A_j$ is dense in $M$.\nBecause we chose a non-empty $S$ with members all $\\geq 3$\nthe set of branch points with at least 3 branches of infinite length is dense in $M$. Therefore, $M$\nis a richly branching $\\R$-tree.\n\\end{proof}\n\n\\begin{remark}\nIn the preceding proof, we did not use the fact that we are considering homeomorphisms\nof \\emph{pointed} topological spaces. The $\\R$-trees constructed in Lemma \\ref{separabletreewitharbbranching}\nare in fact non-homeomorphic even when we are not required to preserve the base point.\n\\end{remark}\n\n\\begin{remark}\nThe $\\R$-trees constructed in Lemma \\ref{separabletreewitharbbranching} all have\nat least 2 branches at every point. It is straightforward to modify this construction so that\nthere are some points with degree 1 as well. For example, begin the construction with the\n$\\R$-tree $[0, 1]$, and always exclude $0$ from the set of points where rays are added. This\nwill result in a richly branching tree where we know there is a single branch at that point.\nOne could also start with a richly branching tree and ``trim\" away all but one branch at some points.\n\\end{remark}\n\n\\begin{theorem}  \n\\label{number of separable models}\nThere exist $2^{\\omega}$-many pairwise non-homeomorphic\n(hence non-isomorphic) separable models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{theorem}\n\n\\begin{proof}\nAny homeomorphism $g$ between models $\\mathcal{M}$ and $\\mathcal{N}$ of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is a homeomorphism\non the underlying $\\R$-trees which must preserve branching.  In particular, given $n\\in \\N^{\\geq 3}$,\nif there is a point with degree $n$ in $M$, then there must be a point with degree $n$ in $N$.\nChoose two different subsets $S$ and $S'$ of $\\N^{\\geq 3}$, and construct\na richly branching tree for each as in Lemma \\ref{separabletreewitharbbranching}. Let\n$\\mathcal{M}$ and $\\mathcal{M}'$ be the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ based on the completions of their closed $r$-balls, respectively.\n(Note that taking the completion here can only add points of degree 1.)\nIt follows that $\\mathcal{M}$ and $\\mathcal{M}'$ cannot be homeomorphic.\nSince there are $2^\\omega$-many different such sets $S$,\nthere are $2^\\omega$-many different non-homeomorphic, separable models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\n\nRecall that given a continuous theory $T$, a type $q\\in S_n(T)$ is \\emph{principal}\nif for every model $\\mathcal{M}$ of $T$, the set $q(\\mathcal{M})$ of realizations of $q$ in $\\mathcal{M}$ is definable\nover the empty set.  As in classical first order logic, given a complete theory in a countable signature, there is\na Ryll-Nardzewski theorem stating the equivalence between $\\omega$-categoricity and the fact\nthat every type is principal.  (See \\cite[Theorem 12.2]{N}.)   Furthermore, a type $q$ is principal\nif and only if $q$ is realized in every model of $T$.  (See \\cite[Theorem 12.6]{N}.)\n\nTheorem \\ref{number of separable models} obviously implies that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is not $\\omega$-categorical, \nand thus not every type of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is principal.   Our next result gives a characterization of the principal  types in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$. In particular,\na principal type is the type of a tuple of points that all lie along a single piecewise segment\nwith $p$ as an endpoint. Thus, there are very few of them. \nAs a consequence, we\nconclude that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ does not have a prime model (equivalently, does not have an atomic model, one in which only principal types are realized).\n\nFor a clear and comprehensive treatment of separable models in continuous model theory,\nwe refer the reader to Section 1 in \\cite{BUdfinite}.  Note that where we and \\cite{N} have the word \\emph{principal}, the authors of \\cite{BUdfinite} use \\emph{isolated,} which is now the standard terminology.\nIn (\\cite[Theorem 1.11]{BUdfinite}) they prove an omitting types theorem, and as a corollary (\\cite[Corollary 1.13]{BUdfinite}) show that principal types can be omitted.\nFurther, it follows from \\cite[Definition 1.7]{BUdfinite} and properties of definable sets in continuous model theory, that every principal type is realized in every model, and this is implicit in the discussion following that definition.\n\n\\begin{theorem}\n\\label{principal types}\nLet $q\\in S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$. The following are equivalent\n\\begin{enumerate}\n\\item The type $q$ is principal.\n\\item If $j\\in\\{1,...,n\\}$ is such that $d(p, x_j)^q\\geq d(p, x_i)^q$ for all $i\\in \\{1,..,n\\}$,\nthen $d(p, x_j)^q=d(p, x_i)^q+d(x_i, x_j)^q$ for all $i\\in \\{1,...,n\\}$.\n\\item For any model $\\mathcal{M}$ and $b_1,..., b_n\\in M$ that realize $q$,\nif $j\\in\\{1,...,n\\}$ is such that $d(p, b_j)\\geq d(p, b_i)$ for all $i\\in \\{1,..,n\\}$,\nthen $b_i$ is on the segment $[p, b_j]$ for all $i\\in \\{1,...,n\\}$. \n\\item  For any model $\\mathcal{M}$ and $b_1,..., b_n\\in M$ that realize $q$,  the points\n$p, b_1,...,b_n$ are arranged (in some order) along a piecewise segment with $p$ as\nan endpoint.\n\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}\nThat (2) and (3) are equivalent follows from Lemma \\ref{branch}.\nThat (3) and (4) are equivalent follows from the definition of a piecewise segment.\nWe show (1) implies (3) by proving the contrapositive.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and $b_1, ..., b_n\\in M$ realizing $q$ and let\n$j\\in\\{1,...,n\\}$ be such that $d(p, b_j)\\geq d(p, b_i)$ for all $i\\in \\{1,..,n\\}$.\nAssume there exists $i\\in \\{1,...,n\\}$ so that $b_i$ is not on $[p, b_j]$.\n\nThen let $e$ be the closest point to $b_i$ on the\nsegment $[p, b_j]$. If $e=b_k$ for some $k\\in \\{1,...,n\\}$, with $b_k\\not=b_i$, then\n$$d(p, b_i)=d(p, e)+d(e, b_i)=d(p, b_k)+d(b_k, b_i)>d(p, b_k).$$\nThe last inequality gives us a contradiction.\nThis leaves two possibilities.\n\nCase 1: Assume $e=p$. Then $b_i$ and $b_j$ are on different branches at $p$.\nUsing techniques as in\n\\ref{separabletreewitharbbranching} we can construct $\\mathcal{N}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ so that there\nis only a single branch at $p$. Then clearly $q$ is not realized in $\\mathcal{N}$.\n\nCase 2: Assume $e\\not=p$.\nThen $p$ and $b_j$ are on different branches at $e$,\nand Lemma \\ref{!geo} implies that $e\\in [p, b_j]$ and $e\\in [b_i, b_j]$. So,\n$p$ and $b_j$ are on different branches at $e$, as are $b_i$ and $b_j$.\nThus, there are at least 3 branches at $e$ in the subtree spanned\nby $p, b_1,...,b_n$. \nLet $\\alpha=d(p, e)$, which is a value determined\nby the type $q$.\\\\\nBuild a model $\\mathcal{N}$ so that there are no branching points at distance\n$\\alpha$ from $p$. Then $q$ is not realized in $\\mathcal{N}$, implying that $q$ is not\na principal type.\n\nLastly, we show (2) implies (1).\nLet $q\\in S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ be a type satisfying the condition in (2)\nand let $j\\in \\{1,...,n\\}$ be such that $d(p, x_j)^q\\geq d(p, x_i)^q$ for all $i\\in \\{1,..,n\\}$.\nTake any model $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Then there is at least one branch\nat the base point $p$, and by Lemma \\ref{branches of sufficient height} along this branch we can find a point $b$ so that $d(p, b)=d(p, x_j)^q$. Let $b=b_j$. We can find realizations\nof the other $x_i$'s at appropriate distances along the segment $[p,b_j]$, and prove that\n$b_1,...,b_n$ realize $q$ by Lemma \\ref{determinetypes}.\nThus, if $q$ is a type satisfying the conditions in (2), we know $q$ is a principal type.\n\\end{proof}\n\n\\begin{corollary}\nThe $L$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has no prime model.\n\\end{corollary}\n\\begin{proof}\nAssume $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has a prime model $\\mathcal{M}$ with underlying $\\R$-tree\n$(M, d,p)$. Then $\\mathcal{M}$ is atomic and the type\nof any tuple $b_1,...,b_n$ must be principal. By the preceding theorem, this means\nthat for any pair $b_1, b_2$ in $M$, either $b_1\\in [p, b_2]$ or $b_2\\in [p, b_1]$.\nIt follows that $M$ consists of a piecewise segment with endpoint $p$.  In particular, \n$\\mathcal{M}$ is not richly branching, which is a contradiction.\n\\end{proof}\n\nWe finish this section by showing that when $\\kappa$ is uncountable,\nthen the number of different models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ having density character\nequal to $\\kappa$ is also the maximum possible, namely $2^\\kappa$.\nAs in the case $\\kappa = \\omega$, which was treated in the first part of this\nsection, we produce large sets of models that are not only non-isomorphic,\nbut in fact have underlying $\\R$-trees which are non-homeomorphic\n(as pointed topological spaces).\n\nWe will carry out the construction by induction on $\\kappa$, and we begin with a useful lemma. \n\\begin{lemma}\n\\label{getting many nonseparable models}\nLet $\\kappa$ be an uncountable cardinal.  \nThe following conditions are equivalent:\n\\newline\n(1) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\leq \\kappa$ is at least $\\kappa$.\n\\newline\n(2) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\leq \\kappa$ that have just one branch at the base point is at least $\\kappa$.\n\\newline\n(3) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $= \\kappa$ is $2^\\kappa$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\kappa$ be an uncountable cardinal. Clearly, (3) implies (1).\nTo show (2) implies (3), assume (2) and  let $(B_\\alpha \\mid \\alpha <  \\kappa)$ be a list of the pairwise non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, each with density character $\\leq \\kappa$ and exactly one branch at the base point.  Given a subset $S\\subseteq \\kappa$ of cardinality $= \\kappa$, take the collection of $B_\\alpha$\nfor $\\alpha\\in S$ and glue them all together at their base points\nusing Theorem \\ref{amalg}.  Call this amalgam $M_S$,\nand let $p$ be the point in $M_S$ at which the $B_\\alpha$ are all glued together. \nMake $p$ the base point of the $L_r$-structure $\\mathcal{M}_S$.   Note that the density character of $\\mathcal{M}_S$ is exactly $\\kappa$, since each branch of \nits underlying $\\R$-tree $M_S$ at $p$ has density $\\leq \\kappa$\nand height $r$, and there are exactly $\\kappa$ many branches at $p$.\nMoreover, it is easy to check that $\\mathcal{M}_S$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\nBy this construction, if $B$ ranges over the branches of $M_S$ at $p$, the\nhomeomorphism type of $B \\cup \\{p\\}$ (with $p$ as distinguished element)\nranges bijectively over the homeomorphism types of $B_\\alpha$ (also with $p$ as distinguished element) as $\\alpha$ ranges over $S$.  It follows that the homeomorphism type of $\\mathcal{M}_S$ determines $S$.\nTherefore the family $\\{ \\mathcal{M}_S \\mid S \\subseteq \\kappa \\text{ and } S \\text{ has cardinality } = \\kappa \\}$\nverifies condition (3), since $\\kappa$ has $2^\\kappa$ many subsets of cardinality $= \\kappa$.\n\nFinally, we prove that (1) implies (2).\nFor each model $\\mathcal{M}$ of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, let $b(\\mathcal{M})$ denote the number of branches of $M$ at its base point; we take this to be a positive integer or $\\infty$, where $b(\\mathcal{M}) = \\infty$ means that there are infinitely many branches.  Since $\\kappa$ is uncountable, condition (1) yields a class $\\mathcal{K}$ of at least $\\kappa$ many non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, each of density character $\\leq \\kappa$, such that $b(\\mathcal{M})$ has a constant value $b$ as $\\mathcal{M}$ ranges over $\\mathcal{K}$.  We may assume $b \\neq 1$, since otherwise condition (2) is satisfied by the models in $\\mathcal{K}$.\n\nLet $a$ be an integer $\\geq 3$ that is different from $b+1$.  ( Note $a\\not=b+1$ is  automatically true if $b$ is $\\infty$.)  Using a method similar to that in the proof of Lemma \\ref{separabletreewitharbbranching}, we may take $\\mathcal{N}=(N, d, p)$ to be a separable model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with the following properties: (1) for all $x$ in $N$, the number of branches in $N$ at $x$ is $1$ or $2$ or $a$; (2) $N$ has a single branch at its base point $p$; and (3) \n$N$ has a single branch at some point $y$, where $d^{\\mathcal{N}}(p, y)=\\frac{r}{2}$.\nTo get single branches at 2 points with a given distance as required here,\ninstead of starting the construction with\nthe $\\R$-tree $\\R$ as in the proof of Lemma \\ref{separabletreewitharbbranching}, start with the interval $[0, \\frac{r}{2}]\\subseteq \\R$, and in\nsubsequent steps always exclude $0$ and $\\frac{r}{2}$ from the sets of points where rays\nare added. \n\n\nNow consider an arbitrary $\\mathcal{M} \\in \\mathcal{K}$, and denote the base point of $\\mathcal{M}$ by $q$.\nScale the metric on $\\mathcal{M}$ down by a factor of 2, resulting in an $L_r$-structure with a radius of \n$\\frac{r}{2}$.\nWe construct a larger $\\R$-tree $\\mathcal{M}^*$ by amalgamating the scaled-down $\\mathcal{M}$ and $\\mathcal{N}$ in the way that identifies $q$ and $y$; we will denote this point of $M^*$ by $qy$.  We take the base point of $\\mathcal{M}^*$ to be the base point $p$ of $\\mathcal{N}$.  The radius of  $\\mathcal{M}^*$ is $r$, because the radius of $\\mathcal{N}$ was $r$, and by our amalgamation construction, for every point $x\\in M^*$ residing in the scaled down copy of $M$, \n$$d^{\\mathcal{M}^*}(p,x)=d^{\\mathcal{M}^*}(p, qy)+d^{\\mathcal{M}^*}(qy, x)=\\frac{r}{2}+\\frac{d^{\\mathcal{M}}(y, x)}{2}$$\nwhich attains its maximum value $r$ as $x$ ranges over the scaled down copy of $M$.\n\nIt is straightforward to check that $\\mathcal{M}^*$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, has density $\\leq \\kappa$, and has a single branch at its base point.  Note that the branches of $M^*$ at the amalgamated point $qy$ consist of the branches of $q$ in $M$ together with the tree that results from $N$ by removing $y$.  In particular, this means that $M^*$ has $b+1$ many branches at $qy$.\n\nWe claim that the class $\\mathcal{K}^* = \\{ \\mathcal{M}^* \\mid \\mathcal{M} \\in \\mathcal{K} \\}$ verifies condition (3); it remains only to show that no two members of this class are homeomorphic.  (Recall that the homeomorphisms we consider must take base point to base point.)  The key to this is the fact that the point $qy$ can be topologically identified in $\\mathcal{M}^*$, given that we know the base point $p$.  To do this, note first that the segment $X=[p,qy)$ in $\\mathcal{M}^*$ is identical to the segment $[p,y)$ in $\\mathcal{N}$, and every point in $X$ has the same number of branches in $\\mathcal{M}^*$ as in \n$\\mathcal{N}$. Therefore every point $x$ of $X$ has  $1$, $2$, or $a$ many branches in $\\mathcal{M}^*$, and thus the number of branches at $x$ is different from the number of branches at $qy$.  From this we conclude that for any $\\mathcal{M}_1,\\mathcal{M}_2 \\in \\mathcal{K}$, any homeomorphism of $\\mathcal{M}_1^*$ onto $\\mathcal{M}_2^*$ that takes base point to base point must map the scaled version of $\\mathcal{M}_1$ onto the scaled version of $\\mathcal{M}_2$.  Since this can only happen when $\\mathcal{M}_1 = \\mathcal{M}_2$, by assumption on $\\mathcal{K}$, we conclude that $\\mathcal{M}_1^* = \\mathcal{M}_2^*$, as desired.\n\\end{proof}\n\n\n\\begin{theorem}\nLet $\\kappa$ be an uncountable cardinal. The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character equal to $\\kappa$ is $2^\\kappa$.\n\\end{theorem}\n\n\\begin{proof}\nWe assume that $\\sigma$ is the least uncountable cardinal at which there are strictly fewer than $2^\\sigma$ many non-homeomorphic models of density character equal to $\\sigma$, and derive a contradiction.  Using Theorem \\ref{number of separable models}, we see that condition (1) in Theorem \\ref{getting many nonseparable models} holds when $\\kappa = \\omega_1$; condition (3) in that result yields that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has $2^{\\omega_1}$ many non-homeomorphic models of density character equal to $\\omega_1$.  Thus $\\sigma > \\omega_1$.  Now suppose $\\sigma$ is a successor cardinal; say it is the next cardinal bigger than $\\lambda$, which must be uncountable.  Our choice of $\\sigma$ ensures that there must be $2^\\lambda \\geq \\sigma$ many non-homeomorphic models of density character $\\lambda$.  Applying Lemma \\ref{getting many nonseparable models} with $\\kappa = \\sigma$ gives a contradiction; indeed, we have verified condition (1), while condition (3) is false.\nSo $\\sigma$ must be a limit cardinal.  Let $\\tau$ be the number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have density character $\\leq \\sigma$; our treatment of $\\omega_1$ shows that $\\tau$ is uncountable.  Furthermore, Lemma \\ref{getting many nonseparable models} applied to $\\kappa = \\sigma$ yields $\\tau < \\sigma$.   Our choice of $\\sigma$ ensures that there are $2^ \\tau > \\tau$ many non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have density character $\\tau$, contradicting the definition of $\\tau$.\n\\end{proof}\n\n\\section{Unbounded $\\R$-trees}\n\nAs noted in the Introduction, we have chosen to treat \\emph{bounded} pointed $\\R$-trees\nin this paper, because many of the model-theoretic ideas and tools we need from continuous\nfirst order logic are only documented for bounded metric structures in the literature.\n\nHowever, it would certainly be a natural research topic to study the model theory of unbounded\n(\\textit{i.e.}, not necessarily bounded) pointed $\\R$-trees.  The most immediately available setting\nfor doing this would be to consider a pointed $\\R$-tree $(M,p)$ as a many-sorted metric structure\nin which each sort is one of the (closed) bounded balls of $(M,p)$ (centered at $p$), and the union\nof the family of distinguished balls is all of $M$.  \nEverything done in this paper can easily be carried over to that setting.  The disadvantages of\ndoing so are the technical awkwardness of the many-sorted framework and the need for imposing\nan arbitrary family of radii for the bounded balls into which the full tree is stratified.\n\nIt is certainly more mathematically natural to consider pointed $\\R$-trees on their own, without\nimposing a many-sorted stratification.  There are suitable\nlogics for doing model theory with such unbounded structures.  For example, a version of continuous first\norder logic for unbounded metric structures is described in \\cite{BY}.  Also, a logic based\non \\emph{positive bounded formulas} and an associated concept of \\emph{approximate satisfaction}\nis presented in Section 6 of \\cite{DI}.  However, for neither of these approaches are the ideas and tools\nof model theory developed as we need them in this paper.\n\nIn each of these three available settings for treating arbitrary pointed $\\R$-trees, the arguments in this paper\ncan be used easily to demonstrate: (1) the class of pointed $\\R$-trees is axiomatizable and (2) for each $r>0$, the \nball $\\{ x \\mid d(x,p) \\leq r \\}$ is a definable set (over $\\emptyset$, uniformly in all pointed $\\R$-trees).  \nTogether with what is developed in \\cite{N}, \\cite{BY}, and \\cite{DI}, this quickly yields that the model theoretic \nframeworks for pointed $\\R$-trees provided by these three settings are completely equivalent.  \nIn particular, this approach yields a model completion \nfor the theory of pointed $\\R$-trees whose models are exactly the\nrichly branching $\\R$-trees (\\textit{i.e.}, the complete pointed $\\R$-trees described in Remark \\ref{general richly branching}).  Furthermore, this\nmodel completion has suitably stated versions of all the properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that are proved in this paper.\n\n\\endinput\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe commutative division algebras constructed by Dickson \\cite{Dic} yield proper semifields of even dimension over finite fields. They have been subsequently studied in many papers, for example in \\cite{Bur}, \\cite{Bur2}, \\cite{HTW}, \\cite{Thompson19}. Knuth recognised that Dickson's commutative division algebras also appear as a special case of another family of semifields \\cite{Knu}: A subalgebra $L$ of a division algebra $S$ is called a \\textit{weak nucleus} if $x(yz)-(xy)z=0$, whenever two of $x,y,z$ lie in $L$.  Semifields which are quadratic over a weak nucleus are split into two cases; Case I semifields contain Dickson's construction as the only commutative semifields of this type. Due to this, Case I semifields are also called \\textit{generalized Dickson semifields}. Their construction is as follows: given a finite field $K=GF(p^n)$ for some odd prime $p$, define a multiplication on $K\\oplus K$ by $$(u,v)(x,y)=(uv+c\\alpha(v)\\beta(y), \\sigma(u)y+vx),$$ for some automorphisms $\\alpha$, $\\beta$, $\\sigma$ of $K$ not all the identity automorphism and $c\\in K\\setminus K^2$. This construction produces a proper semifield containing $p^{2n}$ elements. Further work on semifields quadratic over a weak nucleus was done in \\cite{Ganley} and \\cite{Cohen}.\\\\\nIn this paper, we define a doubling process which generalizes Knuth's construction in \\cite{Knu}: for a central simple associative algebra $D\/F$ or finite field extension $K\/F$, we define a multiplication on the $F$-vector space $D\\oplus D$ (resp. $K\\oplus K$) as $$(u,v)(x,y)=(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x))$$ for some $c\\in D^{\\times}$ and $\\sigma_i\\in Aut_F(D)$ for $i=1,2,3,4$ (resp. $c\\in K^{\\times}$ and $\\sigma_i\\in Aut_F(K)$). This yields an algebra of dimension $2dim_F(D)$ or $2[K:F]$ over $F$. \nOver finite fields, our construction yields examples of some Hughes-Kleinfeld, Knuth and Sandler semifields (for example, see \\cite{Cor}) and all generalized Dickson and commutative Dickson semifields \\cite{Knu}\\cite{Dic}. Hughes-Kleinfeld, Knuth and Sandler semifield constructions were studied over arbitrary base fields in \\cite{BrownSteelePump}. Dickson's commutative semifield construction was introduced over finite fields in \\cite{Dic} and considered over any base field of characteristic not 2 when $K$ is a finite cyclic extension in \\cite{Bur}. This was generalized to a doubling of any finite field extension and central simple algebras in \\cite{Thompson19}.\\\\\nAfter preliminary results and definitions, we define a doubling process for both a central simple algebra $D\/F$ and a finite field extension $K\/F$; we recover the multiplication used in Knuth's construction of generalized Dickson semifields when $\\sigma_4=id$. We find criteria for them to be division algebras. We then determine the nucleus and commutator of these algebras and examine both isomorphisms and automorphisms. The results of this paper are part of the author's PhD thesis written under the supervision of Dr S. Pumpl\\\"{u}n.\n\n\\section{Definitions and preliminary results}\n\nIn this paper, let $F$ be a field. We define an $F$-algebra $A$ as a finite dimensional $F$-vector space equipped with a (not necessarily associative) bilinear map $A\\times A\\to A$ which is the multiplication of the algebra. $A$ is a \\textit{division algebra} if for all nonzero $a\\in A$ the maps $L_a:A\\to A$, $x\\mapsto ax$, and $R_a:A\\to A$, $x\\mapsto xa$, are bijective maps. As $A$ is finite dimensional, $A$ is a division algebra if and only if there are no zero divisors \\cite{Sch}.\\\\\nThe \\textit{associator} of $x,y,z\\in A$ is defined to be $[x,y,z]:=(xy)z-x(yz).$ Define the \\textit{left, middle and right nuclei} of $A$ as $\\text{Nuc}_l(A):=\\lbrace x\\in A \\mid [x,A,A]=0\\rbrace,$ $\\text{Nuc}_m(A):=\\lbrace x\\in A \\mid [A,x,A]=0\\rbrace,$ and $\\text{Nuc}_r(A):=\\lbrace x\\in A \\mid [A,A,x]=0\\rbrace.$ The left, middle and right nuclei are associative subalgebras of $A$. Their intersection $\\text{Nuc}(A):=\\lbrace x\\in A \\mid [x,A,A]=[A,x,A]=[A,A,x]=0\\rbrace$ is the \\textit{nucleus} of $A$. The \\textit{commutator} of $A$ is the set of elements which commute with every other element, $Comm(A):=\\lbrace x\\in A\\mid xy=yx \\:\\forall y\\in A\\rbrace.$ The \\textit{center} of $A$ is given by the intersection of $\\text{Nuc}(A)$ and $Comm(A)$, $Z(A):=\\lbrace x\\in \\text{Nuc}(A)\\mid xy=yx\\: \\forall y\\in A\\rbrace.$ For two algebras $A$ and $B$, any isomorphism $f:A\\to B$ maps $\\text{Nuc}_l(A)$ isomorphically onto $\\text{Nuc}_l(B)$ (similarly for the middle and right nuclei).\\\\\nAn algebra $A$ is \\textit{unital} if there exists an element $1_A\\in A$ such that $x1_A=1_Ax=x$ for all $x\\in A$. \nA form $N:A \\to F$ is called \\textit{multiplicative} if $N(xy)=N(x)N(y)$ for all $x,y\\in A$ and \\textit{nondegenerate} if we have $N(x)=0$ if and only if $x=0$. Note that if $N:A\\to F$ is a nondegenerate multiplicative form and $A$ is a unital algebra, it follows that $N(1_A)=1_F$. Every central simple algebra admits a uniquely determined nondegenerate multiplicative form, called the \\textit{norm} of the algebra.\\\\\n\n\n\n\n\\section{A doubling process which generalizes Knuth's construction}\\label{GCDD_Field}\n\n\nLet $D$ be a central simple associative division algebra over $F$ with nondegenerate multiplicative norm form $N_{D\/F}:D\\to F$. Given $\\sigma_i\\in Aut_F(D)$ for $i=1,2,3,4$ and $c\\in D^{\\times}$, define a multiplication on the $F$-vector space $D\\oplus D$ by $$(u,v)(x,y)=(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x)).$$ We denote the $F$-vector space endowed with this multiplication by $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$. We can also define an analogous multiplication on $K\\oplus K$ for a finite field extension $K\/F$ for some $c\\in K^{\\times}$ and $\\sigma_i\\in Aut_F(K)$. We similarly denote these algebras by $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4).$ This yields unital $F$-algebras of dimension $2dim_F(D)$ and $2[K:F]$ respectively. When $\\sigma_4=id$, our multiplication is identical to the one used in the construction of generalized Dickson semifields. For every subalgebra $E\\subset D$ such that $c\\in E^{\\times}$ and $\\sigma_i\\mid_E=\\phi_i\\in Aut_F(E)$ for $i=1,2,3,4$, it is clear that $\\text{Cay}(E,c,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ is a subalgebra of $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$.\n\n\\begin{theorem}\\label{KDivision Algebras} \n\\begin{enumerate}[(i)]\n\\item If $N_{D\/F}(c)\\not\\in N_{D\/F}(D^{\\times})^2$, $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra.\n\\item If $K$ is separable over $F$ and $N_{K\/F}(c)\\not\\in N_{K\/F}(K^{\\times})^2$, then $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\n(i) Suppose $(0,0)=(u,v)(x,y)$ for some $u,v,x,y\\in D$ such that $(u,v)\\neq(0,0)\\neq(x,y)$. This is equivalent to \\begin{align}\nux+c\\sigma_1(v)\\sigma_2(y)=&0,\\label{Div1K}\\\\\n\\sigma_3(u)y+v\\sigma_4(x)=&0.\\label{Div2K}\n\\end{align}\nAssume $y=0$. Then by (\\ref{Div1K}), $ux=0$, so $u=0$ or $x=0$ as $D$ is a division algebra. As $(x,y)\\neq (0,0)$, we must have $x\\neq 0$ so $u=0$. Then by (\\ref{Div2K}), $v\\sigma_4(x)=0$ which implies $v=0 \\mbox{ or } x=0$. This is a contradiction, thus it follows that $y\\neq 0$. By (\\ref{Div2K}), $v\\sigma_4(x)=-\\sigma_3(u)y.$ Let $N=N_{D\/F}:D\\to F$. Taking norms of both sides, we have \\begin{align*}\n&N(v)N(x)=-N(u)N(y)\\\\\n\\implies &N(u)= -N(v)N(x)N(y)^{-1},\n\\end{align*}\nsince $y\\neq 0$. Substituting this result into (\\ref{Div1K}) implies \\begin{align}\n0=&N(u)N(x)+N(c)N(v)N(y)\\nonumber\\\\\n=&(-N(v)N(x)N(y)^{-1})N(x)+N(c)N(v)N(y) \\nonumber\\\\\n=&N(v)[(N(x)N(y)^{-1})^2-N(c)]. \\label{Div3K}\n\\end{align}\nIf $N(v)=0$, then $v=0$ so by (\\ref{Div1K}) $ux=0$ implies $x=0$ (else $(u,v)=(0,0)$).  Thus (\\ref{Div3K}) implies $N(c)=0\\not\\in F^{\\times},$ which cannot happen as $c\\neq 0$.\nThus we must have $N(v)\\neq 0$ and $(N(x)N(y)^{-1})^2=N(c)$. Thus we conclude $N(c)\\in N(D^{\\times})^2$.\\\\\n(ii) The proof follows analogously as in (i); we require $K$ to be separable over $F$ so that $N_{K\/F}(\\sigma(x))=N_{K\/F}(x)$ for all $\\sigma\\in Aut_F(K)$ and $x\\in K$.\n\\end{proof}\n\n\\begin{remark}\nIf $F=\\mathbb{F}_{p^s}$ and $K=\\mathbb{F}_{p^r}$ is a finite extension of $F$, then $Aut_F(K)$ is cyclic of order $r\/s$ and is generated by $\\phi^s$, where $\\phi$ is defined by the Frobenius automorphism $\\phi(x)=x^p$ for all $x\\in K$. Then $A=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra if and only if $c$ is not a square in $K$. The proof of this is analogous to the one given in \\cite[p. 53]{Knu}.\n\\end{remark}\n\n\n\\subsection{Commutator and nuclei}\n\nUnless otherwise stated, we will write $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$.\n\n\\begin{proposition}\\label{Commutator of Cay(K)} If $\\sigma_1=\\sigma_2$ and $\\sigma_3=\\sigma_4$, $Comm(A_D)=F\\oplus F$ and $A_K$ is commutative. Otherwise, $Comm(A_D)=F$ and $Comm(A_K)=\\lbrace (u,0)\\mid \\sigma_3(u)=\\sigma_4(u)\\rbrace\\subseteq K.$\n\\end{proposition}\n\\begin{proof}\nWe compute this only for $A_D$ as the computations for $A_K$ follow analogously. By definition, $(u,v)\\in Comm(A_D)$ if and only if for all $x,y\\in D$, $(u,v)(x,y)=(x,y)(u,v).$ This is equivalent to \\begin{align*}\nux+c\\sigma_1(v)\\sigma_2(y)=&xu+c\\sigma_1(y)\\sigma_2(v),\\\\\n\\sigma_3(u)y+v\\sigma_4(x)=&\\sigma_3(x)v+y\\sigma_4(u),\n\\end{align*}\nfor all $x,y\\in D$. The first equation implies $u\\in F$ and either $\\sigma_1=\\sigma_2$ or $v=0$ . Additionally, the second equation implies $v\\in F$ and $\\sigma_3=\\sigma_4$ or $v=0$. The result follows immediately.\n\\end{proof}\n\n\\begin{proposition}\\label{Cay(B) left nuc}\n\\begin{enumerate}[(i)]\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item $\\sigma_2\\circ\\sigma_4\\neq id$,\n\\item $\\sigma_1\\circ\\sigma_4\\neq \\sigma_2\\circ\\sigma_3$,\n\\item $\\sigma_4\\circ\\sigma_1\\neq \\sigma_3\\circ\\sigma_2$.\n\\end{itemize}\nThen $\\text{Nuc}_l(A_D)=\\lbrace x\\in D\\mid \\sigma_1\\circ\\sigma_3(x)=c^{-1}xc\\rbrace\\subset D$ and $\\text{Nuc}_l(A_K)=\\text{Fix}(\\sigma_1\\circ\\sigma_3)\\subset K$.\n\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item there exists some $x\\in D$ (resp. $K$) such that $\\sigma_1\\circ\\sigma_3(x)\\neq c^{-1}xc$,\n\\item $\\sigma_2\\circ\\sigma_4\\neq id$,\n\\item for all $v\\in D$, there exists some $x\\in D$ (resp. $K$) such that $\\sigma_3(c)\\sigma_3\\circ\\sigma_1(x)\\sigma_3\\circ\\sigma_2(v)\\neq x\\sigma_4(c)\\sigma_4\\circ\\sigma_1(v)$.\n\\end{itemize}\nThen $\\text{Nuc}_m(A)=\\text{Fix}(\\sigma_3^{-1}\\circ\\sigma_2^{-1}\\circ\\sigma_1\\circ\\sigma_4)$ for both $A=A_D$ and $A=A_K$.\n\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item  there exists some $x\\in D$ (resp. $K$) such that $\\sigma_1\\circ\\sigma_3(x)\\neq c^{-1}xc$,\n\\item $\\sigma_1\\circ\\sigma_4\\neq\\sigma_2\\circ\\sigma_3$,\n\\item there exists some $x\\in D$ (resp. $K$) such that $\\sigma_3(c)\\sigma_3\\circ\\sigma_1(x)\\neq x\\sigma_4(c)$.\n\\end{itemize}\nThen $\\text{Nuc}_r(A)=\\text{Fix}(\\sigma_2\\circ\\sigma_4)$ for both $A=A_D$ and $A=A_K$. \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nWe show the proof for (i) since (ii) and (iii) follow analagously. First consider all elements of the form $(k,0)$ for $k\\in D$. Then $(k,0)\\in \\text{Nuc}_l(A_D)$ if and only if we have $((k,0)(u,v))(x,y)=(k,0)((u,v)(x,y))$ for all $u,v,x,y\\in D$. Computing this directly, we obtain the equations \\begin{align*}\nkux+c\\sigma_1(\\sigma_3(k)v)\\sigma_2(y)=&kux+kc\\sigma_1(v)\\sigma_2(y),\\\\\n\\sigma_3(ku)y+\\sigma_3(k)v\\sigma_4(x)=&\\sigma_3(k)\\sigma_3(u)y+\\sigma_3(k)v\\sigma_4(x).\n\\end{align*}\nThese hold for all $u,v,x,y\\in D$ if and only if $c\\sigma_1\\circ\\sigma_3(k)=kc$, i.e. we have $\\sigma_1\\circ\\sigma_3(k)=c^{-1}kc$. The same calculations yield that this holds for all $u,v,x,y\\in D$ if and only if $\\sigma_1\\circ\\sigma_3(k)=k$.\\\\\nTo show that there are no other elements in the left nucleus, it suffices to check that there are no elements of the form $(0,m)$, $m\\in D$, in $\\text{Nuc}_l(A_D)$. This is because the associator is linear in the first component:\n$[(k,m),(u,v),(x,y)]=[(k,0),(u,v),(x,y)]+[(0,m),(u,v),(x,y)].$ If $(0,m)\\in \\text{Nuc}_l(A_D)$, then for all $u,v,x,y\\in D$ we have $((0,m)(u,v))(x,y)=(0,m)((u,v)(x,y)).$ This holds for all $u,v,x,y\\in D$ if and only if $$c\\sigma_1(m)[\\sigma_2(v)x+\\sigma_1(\\sigma_4(u))\\sigma_2(y)]=c\\sigma_1(m)[\\sigma_2(v)\\sigma_2(\\sigma_4(x))+\\sigma_2(\\sigma_3(u))\\sigma_2(y)],$$ $$m\\sigma_3(c\\sigma_2(v))y=m\\sigma_4(c\\sigma_1(v)\\sigma_2(y)).$$ In order for this to be satisfied for all $u,v,x,y\\in D$, we have either $m=0$ or all the following must hold:\\begin{itemize}\n\\item $\\sigma_2\\circ\\sigma_4=id$,\n\\item $\\sigma_1\\circ\\sigma_4=\\sigma_2\\circ\\sigma_3$,\n\\item $\\sigma_4\\circ\\sigma_1=\\sigma_3\\circ\\sigma_2$.\n\\end{itemize}\nIf $m\\neq 0$, this contradicts the assumptions we made, so this yields $m=0$. The same argument also gives $m=0$ in the field case.\n\\end{proof}\n\n\n\\begin{corollary} $A_K$ is associative if and only if $A_K=\\text{Cay}(K,c,\\sigma,\\tau,\\sigma^{-1},\\tau^{-1})$ for some $\\tau,\\sigma\\in Aut_F(K)$ such that $(\\sigma\\circ\\tau)^2=id$ and $c\\in \\text{Fix}(\\sigma\\circ\\tau)$.\n\\end{corollary}\n\n\nAs the center of $A$ is defined as $Z(A)=Comm(A)\\cap \\text{Nuc}_l(A)\\cap \\text{Nuc}_m(A)\\cap \\text{Nuc}_r(A),$ we see that $Z(A_K)\\subset K$ unless $\\sigma_1=\\sigma_2=\\sigma$ and $\\sigma_3=\\sigma_4=\\sigma^{-1}$. If $A_K=\\text{Cay}(K,c,\\sigma,\\sigma,\\sigma^{-1},\\sigma^{-1})$ for some $\\sigma\\in Aut_F(K)$, then $A_K$ is a commutative, associative algebra.\n\n\n\\subsection{Isomorphisms}\n\n\\begin{theorem}\\label{ConstructingIsomorphismsDtoD'}\nLet $D$ and $D'$ be two central simple $F$-algebras (respectively, $K$ and $L$ finite field extensions of $F$) and $g,h:D\\to D'$ be two $F$-algebra isomorphisms. Let $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_{D'}=\\text{Cay}(D',g(c)b^2,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in F^{\\times}$ (resp. $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_L=\\text{Cay}(L,g(c)\\phi_1(b)\\phi_2(b),\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in K^{\\times}$). If \n\\begin{align}\n\\phi_i&=g\\circ\\sigma_i\\circ h^{-1} \\mbox{ for } i=1,2, \\label{Equation 1}\\\\\n\\phi_i&=h\\circ\\sigma_i\\circ g^{-1} \\mbox{ for } i=3,4, \\label{Equation 2}\n\\end{align}\nthen the map $G:A\\to B,$  $G(u,v)=(g(u),h(v)b^{-1})$ defines an $F$-algebra isomorphism.\n\\end{theorem}\n\\begin{proof}\nWe show the proof in the central simple algebra case. It follows analogously when we take field extensions $K$ and $L$. Clearly $G$ is $F$-linear, additive and bijective. It only remains to show that $G$ is multiplicative; that is, $G((u,v)(x,y))=G(u,v)G(x,y)$ for all $u,v,x,y\\in D$. First we have \\begin{align*}\nG(u,v)G(x,y)=&(g(u),h(v)b^{-1})(g(x),h(y)b^{-1})\\\\\n=&(g(u)g(x)+g(c)b^2\\phi_1(h(v)b^{-1})\\phi_2(h(y)b^{-1}),\\phi_3(g(u))h(y)b^{-1}+h(v)b^{-1}\\phi_4(g(x)))\\\\\n=&(g(ux)+g(c)\\phi_1(h(v))\\phi_2(h(y)),[\\phi_3(g(u))h(y)+h(v)\\phi_4(g(x))]b^{-1}).\n\\end{align*}\nIt similarly follows that \\begin{align*}\nG((u,v)(x,y))=&G(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x))\\\\\n=&(g(ux+c\\sigma_1(v)\\sigma_2(y)),h(\\sigma_3(u)y+v\\sigma_4(x))b^{-1})\\\\\n=&(g(ux)+g(c)g(\\sigma_1(v))g(\\sigma_2(y)),[h(\\sigma_3(u))h(y)+h(v)h(\\sigma_4(x))]b^{-1}).\n\\end{align*}\nBy (\\ref{Equation 1}) and (\\ref{Equation 2}), we obtain equality and thus $G$ is an $F$-algebra isomorphism.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{ConstructingIsomorphismsKtoK}\nLet $g,h\\in Aut_F(D)$ (resp. $Aut_F(K)$) and $b\\in F^{\\times}$ (resp. $b\\in K^{\\times})$. Let $B_D=\\text{Cay}(D,g(c)b^2,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ (resp. $B_K=\\text{Cay}(K,g(c)\\phi_1(b)\\phi_2(b),\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in K^{\\times}$). If \n\\begin{align*}\n\\phi_i&=g\\circ\\sigma_i\\circ h^{-1} \\mbox{ for } i=1,2,\\\\\n\\phi_i&=h\\circ\\sigma_i\\circ g^{-1} \\mbox{ for } i=3,4,\n\\end{align*}\nthen the map $G:A\\to B,\\quad G(u,v)=(g(u),h(v)b^{-1})$ defines an $F$-algebra isomorphism.\n\\end{corollary}\n\n\\begin{corollary} Every generalised Dickson algebra $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is isomorphic to an algebra of the form $\\text{Cay}(D,c,\\sigma_1',\\sigma_2',\\sigma_3', id)$ (analogously for the algebras $A_K$).\n\\end{corollary}\n\\begin{proof}\nConsider the map $G:D\\oplus D\\to D\\oplus D$ defined by $G(u,v)=(u,\\sigma_4^{-1}(v))$. By Theorem \\ref{ConstructingIsomorphismsDtoD'}, this yields the isomorphism $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)\\cong \\text{Cay}(D,c,\\sigma_1\\circ\\sigma_4,\\sigma_2\\circ\\sigma_4,\\sigma_4^{-1}\\circ\\sigma_3,id).$\n\\end{proof}\n\n\\begin{remark} If $Comm(A_D)\\neq F$ or $Comm(A_K)\\not\\subset K$, then $\\sigma_1=\\sigma_2$ and $\\sigma_3=\\sigma_4$ by Lemma \\ref{Commutator of Cay(K)}. Via the map $G(u,v)=(u,\\sigma_3^{-1}(v))$, Corollary \\ref{ConstructingIsomorphismsKtoK} yields that every such algebra is isomorphic to the generalisation of commutative Dickson algebras as defined in \\cite{Thompson19}.\n\\end{remark}\n\nIn certain cases, the maps defined in Theorem \\ref{ConstructingIsomorphismsDtoD'} and Corollary \\ref{ConstructingIsomorphismsKtoK} are the only possible isomorphisms between two algebras constructed via our generalised Cayley-Dickson doubling:\n\n\\begin{theorem}\\label{IsomorphismRestrictToKtoL}\nLet $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_L=\\text{Cay}(L,c',\\phi_1,\\phi_2,\\phi_3,\\phi_4)$. Suppose that $G:A_K\\to B_L$ is an isomorphism that restricts to an isomorphism $g:K\\to L$. Then $G$ is of the form $G(x,y)=(g(x),h(y)b)$ for some isomorphism $h:K\\to L$ such that $\\phi_i\\circ h=g\\circ\\sigma_i$ for $i=1,2$ and $\\phi_i\\circ g=h\\circ\\sigma_i$ for $i=3,4$ and some $b\\in L^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{theorem}\n\\begin{proof}\nSuppose $G$ is an isomorphism from $A_K$ to $B_L$ such that $G\\!\\mid_K=g:K\\to L$ is an isomorphism. Then for all $x\\in K$, we have $G(x,0)=(g(x),0).$ Let $G(0,1)=(a,b)$ for some $a,b\\in L$. As $G$ is multiplicative, this yields\\begin{align*}\nG(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&(g(x),0)+(g(\\sigma_3^{-1}(y)),0)(a,b)\\\\\n=&(g(x)+g(\\sigma_3^{-1}(y))a,\\phi_3(g(\\sigma_3^{-1}(y)))b),\n\\end{align*}\nand \n\\begin{align*}\nG(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n=&(g(x),0)+(a,b)(g(\\sigma_4^{-1}(y)),0)\\\\\n=&(g(x)+g(\\sigma_4^{-1}(y))a,b\\phi_4(g(\\sigma_4^{-1}(y)))).\n\\end{align*}\nIt follows that either $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}$ or $b=0$. However, if $b=0$ this would imply that $G$ was not surjective, which is a contradiction to the assumption that $G$ is an isomorphism. Thus it follows that $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}.$ Additionally, we have either $g\\circ\\sigma_3^{-1}=g\\circ\\sigma_4^{-1}$ or $a=0$.\\\\\nConsider $G((0,1)^2)=G(0,1)^2$. This gives $(a^2+c'\\phi_1(b)\\phi_2(b),\\phi_3(a)b+b\\phi_4(a))=(g(c),0).$ As we have established that $b\\neq 0$, this implies that $\\phi_3(a)=-\\phi_4(a).$ If $a\\neq 0$, we obtain $g\\circ\\sigma_3^{-1}=g\\circ\\sigma_4^{-1}$. Substituting this into the condition $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}$, we conclude that $\\phi_3=\\phi_4$. This contradicts $\\phi_3(a)=-\\phi_4(a)$. Thus we must in fact have $a=0$ and $G(x,y)=(g(x),h(y)b)$ where $h=\\phi_3\\circ g\\circ\\sigma_3^{-1}$ and $g(c)=c'\\phi_1(b)\\phi_2(b)$. Computing $G(u,v)G(x,y)=G((u,v)(x,y))$ gives the remaining conditions.\n\\end{proof}\n\nThis proof does not hold when we consider the algebras $A_D$, as we rely heavily on the commutativity of $K$.\n\n\\begin{corollary}\\label{IsomorphismRestrictToK}\n Suppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism $g$ of $K$. Then $G$ is of the form $G(x,y)=(g(x),h(y)b)$ for $g,h\\in Aut_F(K)$ such that $\\phi_i\\circ h=g\\circ\\sigma_i$ for $i=1,2$ and $\\phi_i\\circ g=h\\circ\\sigma_i$ for $i=3,4$ and some $b\\in K^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{corollary}\n\nIf $\\text{Nuc}_l(A)=\\text{Nuc}_l(B)=K$, all isomorphisms from $A\\to B$ must restrict to an automorphism of $K$; similar considerations are true for restrictions to the middle and right nuclei. It follows that we can determine precisely when two such algebras are isomorphic by Corollary \\ref{IsomorphismRestrictToK}.\n\n\\begin{corollary}\\label{IsomorphismRestrictToKCommutes}\nSuppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism $g$ of $K$. If $\\sigma_i=\\phi_i=id$ for any $i=1,2,3,4$, $G$ must be of the form $$G(x,y)=(g(x),g(y)b)$$ for $g\\in Aut_F(K)$ such that $\\phi_i\\circ g=g\\circ\\sigma_i$ for $i=1,2$ and $\\sigma_i\\circ g=g\\circ\\phi_i$ for $i=3,4$ and some $b\\in K^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{corollary}\n\\begin{proof}\nFrom Theorem \\ref{IsomorphismRestrictToK}, we see that $\\phi_i\\circ h=g\\circ\\sigma_i.$ If $\\phi_i=\\sigma_i=id$ for some $i=1,2,3,4$, we conclude that $g=h$ and the result follows.\n\\end{proof}\n\n\\begin{corollary}\\label{IsomorphismRestrictToKN}\nSuppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism of $K$. If $K$ is a separable extension of $F$, we must have $N_{K\/F}(cc'^{-1})=N_{K\/F}(b^2)$ for some $b\\in K^{\\times}$.\n\\end{corollary}\n\\begin{proof}\nSuppose $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism of $K$. By Theorem \\ref{IsomorphismRestrictToK}, we have $g(c)=c'\\phi_1(b)\\phi_2(b).$ Applying norms to both side, we obtain $$N_{K\/F}(g(c))=N_{K\/F}(c'\\phi_1(b)\\phi_2(b)).$$ As $K$ is a separable extension of $F$, it follows that $N_{K\/F}(g(x))=N_{K\/F}(x)$ for all $x\\in K$, $g\\in Aut_F(K)$. This yields $N_{K\/F}(c)=N_{K\/F}(c'b^2).$ As $c'\\in K^{\\times}$ and $N_{K\/F}$ is multiplicative, we conclude that $N_{K\/F}(cc'^{-1})=N_{K\/F}(b^2)$.\n\\end{proof}\n\n\\begin{example} Let $F=\\mathbb{Q}_p$ ($p\\neq 2$) and $K$ be a separable extension of $\\mathbb{Q}_p$. It is well known that $(\\mathbb{Q}_p^{\\times})^2\/\\mathbb{Q}_p=\\lbrace [1],[u],[p],[up]\\rbrace$ for some $u\\in\\mathbb{Z}_p\\setminus \\mathbb{Z}_p^2$. If $N_{K\/F}(c)$ and $N_{K\/F}(c')$ do not lie in the same coset of $(\\mathbb{Q}_p^{\\times})^2\/\\mathbb{Q}_p$, there does not exist an isomorphism that restricts to $K$ such that $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)\\cong \\text{Cay}(K,c',\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ by Corollary \\ref{IsomorphismRestrictToKN}.\n\\end{example}\n\n\n\\subsection{Automorphisms}\n\n\\begin{theorem}\\label{CSA Construction Automorphisms}\nLet $g,h\\in Aut_F(D)$ (resp. $Aut_F(K)$) such that $g\\circ f=f\\circ h$ for $f=\\sigma_1,\\sigma_2,\\sigma_3^{-1},\\sigma_4^{-1}$ and let $b\\in F^{\\times}$ (resp. $b\\in K^{\\times}$) such that $g(c)=b^2c$ (resp. $g(c)=\\sigma_1(b)\\sigma_2(b)c$). Then the map $G:A\\to A$ defined by $G(u,v)=(g(u), h(v)b)$ is an automorphism of $A_D$ (resp. $A_K$).\n\\end{theorem}\nThis is easily checked via some long calculations.\n\n\\begin{theorem}\\label{GeneralAutomorphisms}\nSuppose that at least one of $\\text{Nuc}_l(A_K)$, $\\text{Nuc}_m(A_K)$, $\\text{Nuc}_r(A_K)$ is equal to $K$. Then $G:A_K\\to A_K$ is an automorphism of $A_K$ if and only if $G$ has the form stated in Theorem \\ref{CSA Construction Automorphisms}.\n\\end{theorem}\n \\begin{proof}\n Let $A=A_K$. Suppose $G\\in Aut_F(A)$ and $\\text{Nuc}_l(A)=K$. As automorphisms preserve the nuclei of an algebra, $G$ restricted to $\\text{Nuc}_l(A)$ must be an automorphism of $K$; that is, $G\\mid_K= g\\in Aut_F(K)$ and so we have $G(x,0)=( g(x),0)$ for all $x\\in K$.\\\\\nIf $\\text{Nuc}_l(A)\\neq K$, by our assumptions one of $\\text{Nuc}_m(A)$ or $\\text{Nuc}_r(A)$ are equal to $K$. In either case, we can use an identical argument by restricting $G$ to $\\text{Nuc}_m(A)$ or $\\text{Nuc}_r(A)$ respectively. As automorphisms preserve the nuclei of an algebra, $G$ restricted to $\\text{Nuc}_m(A)$ (respectively $\\text{Nuc}_r(A)$) must be an automorphism of $K$. Let $G(0,1)=(a,b)$ for some $a,b\\in K$. Then \\begin{align*}\n G(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&( g(x)+ g\\sigma_3^{-1}(y)a,\\sigma_3 g\\sigma_3^{-1}(y)b),\n \\end{align*}\nand also \\begin{align*}\n G(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n=&( g(x)+ g\\sigma_4^{-1}(y)a,\\sigma_4 g\\sigma_4^{-1}(y)b)\n\\end{align*}\nfor all $x,y\\in K$. Hence we must have $ g\\sigma_3^{-1}(y)a= g\\sigma_4^{-1}(y)a$ for all $y\\in K$, which implies either $\\sigma_3=\\sigma_4$ or $a=0$. Additionally we have $\\sigma_3 g\\sigma_3^{-1}(y)b=\\sigma_4 g\\sigma_4^{-1}(y)b$. If $b=0$, this would imply $G(x,y)=( g(x)+ g\\sigma_4^{-1}(y)a,0)$, which is a contradiction as it implies $G$ is not surjective. Thus we must in fact have $\\sigma_3 g\\sigma_3^{-1}(y)=\\sigma_4 g\\sigma_4^{-1}(y)$ for all $y\\in K$.\\\\\nNow we consider $G((0,1)^2)=G(0,1)^2$. This gives $(a,b)(a,b)=( g(c),0),$ which implies \\begin{align*}\na^2+c\\sigma_1(b)\\sigma_2(b)=& g(c),\\\\\n\\sigma_3(a)b+b\\sigma_4(a)=& 0.\n\\end{align*}\nIf $\\sigma_3\\neq\\sigma_4$, we already know that $a=0$. On the other hand if $\\sigma_3=\\sigma_4$, we obtain $2\\sigma_3(a)b=0$. As $K$ has characteristic not 2 and $b\\neq 0$, this implies $a=0$. In either case, we obtain $c\\sigma_1(b)\\sigma_2(b)= g(c)$ and $G(u,v)=( g(u), h(v)b),$ where $ h=\\sigma_3\\circ g\\circ\\sigma_3^{-1}=\\sigma_4\\circ g\\circ\\sigma_4^{-1}.$ We note that this definition of $ h$ implies that $ h\\circ\\sigma_3=\\sigma_3\\circ g$ and $ h\\circ\\sigma_4=\\sigma_4\\circ g$.\\\\\nFinally we consider $G(u,v)G(x,y)=G((u,v)(x,y))$ for all $u,v,x,y\\in K$. We obtain $( g(u), h(v)b)( g(x), h(y)b)=( g(uv+c\\sigma_1(v)\\sigma_2(y)), h(\\sigma_3(u)y+v\\sigma_4(x))b)$ which gives the equations \\begin{align*}\nc\\sigma_1( h(v)b)\\sigma_2( h(y)b)=& g(c) g(\\sigma_1(v)\\sigma_2(y)),\\\\\n\\sigma_3( g(u)) h(y)b+ h(y)\\sigma_4( g(x))b=& h(\\sigma_3(u)y+v\\sigma_4(x))b.\n\\end{align*}\n As $ h\\circ\\sigma_3=\\sigma_3\\circ g$ and $ h\\circ\\sigma_4=\\sigma_4\\circ g$, the second equation holds for all $u,v,x,y\\in K$. Substituting $ g(c)=c\\sigma_1(b)\\sigma_2(b)$ into the first equation, we obtain $\\sigma_1(h(v))\\sigma_2(h(y))= g(\\sigma_1(v)) g(\\sigma_2(y))$ for all $v,y\\in K$. This implies $\\sigma_1\\circ h= g\\circ\\sigma_1$ and $\\sigma_2\\circ h= g\\circ\\sigma_2$. Hence if $G$ is an automorphism of $A$ we must have $G(u,v)=( g(u), h(v)b)$ for some $ g, h\\in Aut_F(K)$, such that $ g\\circ f=f\\circ h$ for $f=\\sigma_1,\\sigma_2,\\sigma_3^{-1},\\sigma_4^{-1}$ and some $b\\in K^{\\times}$, such that $ g(c)=\\sigma_1(b)\\sigma_2(b)c.$\n\\end{proof}\n\n \n\\begin{corollary}\\label{KDoublingAutGroup}\nSuppose that at least one of $\\text{Nuc}_l(A_K)$, $\\text{Nuc}_m(A_K)$, $\\text{Nuc}_r(A_K)$ is equal to $K$ and $Aut_F(K)=\\langle\\sigma\\rangle$. Then $G:A_K\\to A_K$ is an automorphism of $A_K$ if and only if $G(u,v)=(\\sigma^i(u),\\sigma^i(v)b)$ for some $i\\in \\mathbb{Z}$ and $b\\in K^{\\times}$ satisfying $\\sigma^i(c)=c\\sigma^{\\alpha_2}(b)\\sigma^{\\beta_2}(b).$ \n\\end{corollary}\n\n\nIn the case when doubling a central simple algebra, we obtain a partial generalisation of Theorem \\ref{GeneralAutomorphisms}:\n\n\\begin{lemma}\\label{CSAAutomorphismsScalars} Let $G\\in Aut(A_D)$ be such that $G\\!\\mid_D=g\\in Aut_F(D)$. Then there must exist some $a,b\\in D$, $b\\neq 0$, such that for all $y\\in D$, $$a g\\circ\\sigma_4^{-1}(y)=g\\circ\\sigma_3^{-1}(y)a,$$ $$b\\sigma_4\\circ g\\circ\\sigma_4^{-1}(y)=\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b.$$\n\\end{lemma}\n\\begin{proof}\nSuppose $G\\mid_D=g\\in Aut_F(D)$. Then for all $x\\in D$, we obtain $G(x,0)=(g(x),0)$. Let $G(0,1)=(a,b)$ for some $a,b\\in D$. It now follows that \\begin{align*}\n G(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&( g(x)+ g\\circ\\sigma_3^{-1}(y)a,\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b),\n \\end{align*}\nand also \\begin{align*}\n G(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n\n=&( g(x)+ ag\\circ\\sigma_4^{-1}(y),b\\sigma_4\\circ g\\circ\\sigma_4^{-1}(y)).\n\\end{align*}\nSetting these two equivalent expressions for $G(x,y)$ equal to each other yields the result. Note that if $b=0$, $G$ would no longer be surjective, which would contradict our assumption that $G\\in Aut(A_d)$. \n\\end{proof}\n\n\\begin{theorem}\\label{CSAAutomorphisms3=4} Let $G\\in Aut(A_D)$ be such that $G\\!\\mid_D=g\\in Aut_F(D)$. If $\\sigma_3=\\sigma_4$, then $G:A_D\\to A_D$ must have the form as stated in Theorem \\ref{CSA Construction Automorphisms}.\n\\end{theorem}\n\\begin{proof}\nSuppose $G\\!\\mid_D=g\\in Aut_F(D)$. Substituting $\\sigma_3=\\sigma_4$ into Lemma \\ref{CSAAutomorphismsScalars}, we see that $G(0,1)=(a,b)$ for some $a,b\\in D$ such that $$a g\\circ\\sigma_3^{-1}(y)=g\\circ\\sigma_3^{-1}(y)a,$$ $$b\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)=\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b.$$ This is satisfied for all $y\\in D$ if and only if $a,b\\in F$ and so $G(x,y)=(g(x)+g\\circ\\sigma_3^{-1}(y)a,\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b).$ The remainder of this proof follows almost exactly the same to Theorem \\ref{GeneralAutomorphisms}:\n\nNow we consider $G((0,1)^2)=G(0,1)^2$. This gives $(a,b)(a,b)=( g(c),0),$ which implies \\begin{align*}\na^2+c\\sigma_1(b)\\sigma_2(b)=& g(c)\\\\\n\\sigma_3(a)b+b\\sigma_4(a)=0.\n\\end{align*}\nAs $a,b\\in F$, the second equation is equivalent to $2ab=0$. As $F$ has characteristic not 2, this implies $a=0$ or $b=0$. If $b=0$, $G$ would not be surjective, which contradicts our assumption that $G$ is an isomorphism. Thus we must have $a=0$ and so we obtain $g(c)=cb^2$ and $G(u,v)=( g(u), h(v)b),$ where $ h=\\sigma_3\\circ g\\circ\\sigma_3^{-1}.$ We note that this definition of $ h$ implies that $ h\\circ\\sigma_3=\\sigma_3\\circ g$.\\\\\nFinally we consider $G(u,v)G(x,y)=G((u,v)(x,y))$ for all $u,v,x,y\\in D$. We obtain $( g(u), h(v)b)( g(x), h(y)b)=( g(uv+c\\sigma_1(v)\\sigma_2(y)), h(\\sigma_3(u)y+v\\sigma_4(x))b),$ which gives the equations \\begin{align*}\nc\\sigma_1( h(v)b)\\sigma_2( h(y)b)=& g(c) g(\\sigma_1(v)\\sigma_2(y)),\\\\\n\\sigma_3( g(u)) h(y)b+ h(y)\\sigma_4( g(x))b=& h(\\sigma_3(u)y+v\\sigma_4(x))b.\n\\end{align*} \nAfter substituting $cb^2=g(c)$, we conclude that this is satisfied for all $x,y,u,v\\in D$ if and only if we have $\\sigma_1\\circ h=g\\circ \\sigma_1$ and $\\sigma_2\\circ h=g\\circ \\sigma_2$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\printbibliography\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}