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+{"text":"\\section{Introduction}\nThe standard stochastic multi-armed bandit framework captures the exploration-exploitation trade-off in sequential decision making problems under partial feedback constraints. The objective is to actively identify the best member, or members, of a community comprising stochastic sources, termed arms, while suffering the relative loss of non-ideal choices. Often, the arms yield rewards that belong to a known probability distribution with hidden parameters, and upon choosing an arm, the decision maker observes the reward from that arm directly. For ergodic reward distributions, \\cite{gittins1979bandit} shows that the  dynamic programming solution takes the form of an index policy\\footnote{The orignial formulation of \\cite{gittins1979bandit} was in the Bayesian framework.}, called dynamic allocation indices, which motivated the rich body of work that led to the arm-selection rules that eventually achieved the asymptotic regret lower bounds of \\cite{lai1985asymptotically}. Alternatively, when the probability law that governs the rewards is defined conditionally with respect to a hidden state that represents the ``changing world'', restless bandit framework of \\cite{whittle1988restless} leads to arm-selection policies that are often computationally demanding. A key challenge that we address here is to develop index policies that identify the best source in an environment where the underlying state of the world changes erratically and hence, the observations are generated from different probability distributions at each point in time.  \n\nSpecifically, we consider the case where each arm represents a stochastic expert providing opinions on changing tasks and thus, upon consulting an expert, the decision maker observes an opinion, rather than a direct reward. Stochastic experts are sources of subjective information that might fail but not purposefully deceive, as discussed in \\cite{cesa2006prediction}, and often, expert suggestions, or opinions, are used with the aid of side-information: Feedback from past states of the world is used in boosting, \\cite{schapire2012boosting}, models of expert stochasticity, or direct information of expert reliability, or competence, are often used in the Bayesian framework, \\cite{poor2013introduction}. In the absence of \\textit{any} side information, the decision maker operates in a regime that can be termed \\textit{unsupervised}, relying solely on the information in the opinions. Unsupervised opinion aggregation methods such as expectation maximization (EM) \\cite{welinder2010online}, belief propagation (BP) \\cite{karger2011iterative}, and spectral meta-learner (SML) \\cite{parisi2014ranking} exhibit an interesting phenomenon: The reliability of experts are inferred as side-product of the underlying optimization for estimating past states based on a block of opinions. On the other hand, joint sampling and consultation of experts without supervision, or blind exploration and exploitation (BEE) as termed here, requires instantaneously available statistics that would allow reliable inference of expert reliabilities at any and all states of the world. \n\nWe propose a method that relies solely on opinions to infer the competence of an expert by re-defining the notion competence as the probability of agreeing with peers rather than being objectively correct. The proposed method does not only allow empirical inference of competence without any supervision but also enables the use of index policies to efficiently address exploration and exploitation dilemma when the underlying task changes at random. We show that standard, or supervised, exploration-exploitation (SEE) strategies extend their uses to the BEE problem by consulting multiple experts for each task, equivalent to sampling multiple arms in the standard framework. Specifically, we consider the index rules that rely on posterior sampling, \\cite{thompson1933likelihood}, upper-confidence bounds such as UCB1, \\cite{auer2002finite}, and KL-UCB, \\cite{garivier2011klucb}, minimum empirical Kullback-Leibler divergence, in particular, IMED, \\cite{honda2015imed}, and minmax rule MOSS of \\cite{audibert2009minimax}. We investigate two operational regimes: First,\na fixed number of experts are consulted for each task and the opinion of the expert who is believed to be most-reliable at that time is chosen. Second, upon consulting a group of experts, a decision is formed by aggregating their opinions without further supervision. We empirically compare the performance of different BEE index rules and demonstrate that exploration-exploitation-based choice of experts leads to comparable results to those of the original algorithms in the unsupervised framework. \n\nThe organization of this paper is as follows: We summarize the notation used in this paper, provide a background on stochastic experts, and define the BEE problem formally in Section \\ref{sec:probdef}. We discuss the motivation, formal definition, and properties of our technique for unsupervised reliability inference in Section \\ref{sec:pseudocomp}. Then, we discuss the fundamental properties of the BEE index rules in Section \\ref{sec:bee}. The experiments for comparing different BEE algorithms as well as comparing them to their SEE counterparts are in Section \\ref{sec:experiments}. The proofs are deferred to the appendix.   \n\n\\section{Notation, Background, and Problem Formulation}\n\\label{sec:probdef}\nWe begin with a brief overview of the notation used in this paper. Then, we formally define the key concepts regarding stochastic experts. We conclude this section by defining the BEE problem.    \n\\subsection{Notation}\nA probability space is a triplet $\\yay{\\Omega, \\mathscr{F}, \\mathbb{P}}$, where $\\Omega$ is the event space, $\\mathscr{F}$ is the sigma-field defined on $\\Omega$, and $\\mathbb{P}$ is the probability measure. Random variables are denoted by capital letters with the corresponding samples being denoted by lowercase letters: $(X,x)$. A random process is an indexed collection of random variables: $\\myset{Y(t): t\\in \\mathbb{T}}$, where $\\mathbb{T}$ is the index set. Independent random variables $\\yay{X_1, X_2}$ are denoted by $X_1 \\perp X_2$ and conditionally independent random variables $\\yay{X_1, X_2}$ conditioned on $Y$ are denoted by $X_1 - Y - X_2$. Expectation, conditional expectation, and conditional probability operators are denoted by $\\expt{\\cdot}$, $\\condexpt{\\cdot}{\\cdot}$, and $\\condprob{\\cdot}{\\cdot}$ respectively. The indicator function is denoted by $\\ind{\\cdot}$, where domain is to be understood from context. We use  $[T] \\triangleq \\myset{1,\\cdots,T}$ to denote the positive natural numbers up to a finite limit $T<\\infty$. All logarithms $\\yay{\\log}$ are taken with respect to the natural base. We use big $O$ notation when necessary. \n\\subsection{Background}\nConceptually, stochastic experts are honest-but-fallible computational entities that do not deceive the decision maker deliberately. Here, we consider experts that do not collaborate while generating their opinions; \\cite{cesa2006prediction} provides a detailed discussion. The goal of this paper is to propose techniques that identify the best stochastic experts, while dynamically consulting others on varying tasks. In that context, consulting an expert on a task is equivalent to pulling an arm in the standard multi-armed bandit framework. The true reward, however, remains hidden.     \n\nFormally, let us begin with a random process $\\myset{Y(t): t\\in [T]}$ that represents binary states of the world, or tasks with binary labels: $Y(t) \\in \\myset{-1,1}$, $\\forall t\\in[T]$. We allow the nature to generate tasks independently:\n\\begin{equation}\n\t\\label{independent_tasks}\n\tY\\yay{t_1} \\perp Y\\yay{t_2},~\\forall t_1\\neq t_2 \\in [T].\n\\end{equation}\nFurthermore, let the random process $Y(t)$ that governs the evolution of tasks  maximize the uncertainty: \n\\begin{equation}\n\t\\label{unif_distr}\n\t\\prob{Y(t) =1} = \\prob{Y(t)=-1} = \\nicefrac{1}{2},~\\forall t\\in [T].\n\\end{equation} \nIt is worth noting that any bias from non-uniform task generation can either be estimated directly from labeled data, or inferred without supervision via methods such as \\cite{jaffe2016unsupervised}. Furthermore, while independence assumption appears to be restrictive, it is common in stochastic multi-armed bandit formulations, \\cite{bubeck2012regret}.   \n\nFormal characterization of stochastic experts involves the reliability of their opinions and statistical dependence to the others. The probability with which the opinion of an expert identifies the true state of the world correctly determines the reliability, or competence, of that expert: \n\\begin{equation}\n\t\\label{static_competence}\n\tp_i \\triangleq \\prob{X_i(t) =Y(t)},~\\forall t\\in[T].\n\\end{equation} \nHere, the reliability of an expert does not depend on the underlying state of the world \\footnote{A notable exception to this model is the ``two-coin'' model from \\cite{dawid1979maximum}, where conditionally static competences are discussed.}. \nWe further allow that experts generate opinions $\\myset{X_i(t): i\\in [M]}$ independently from one another for every task $t\\in[T]$. Formally: \n\\begin{equation}\n\t\\label{independent_generation}\n\tX_i(t) - Y(t) - X_j(t),~\\forall i\\neq j\\in [M],~\\forall t\\in[T].\n\\end{equation} \nConceptually, it makes sense that for meaningful inference, two different opinions on the same task should never be statistically independent. Furthermore, experts having conditionally independent opinions is equivalent to independence of rewards in the standard framework.\n\nGiven the probability law defined by eq.~\\eqref{independent_tasks}-\\eqref{independent_generation}, we can formally discuss why SEE algorithms requires a toolset to address the impact of the underlying uncertainty. Observe that:\n\\begin{equation}\n\t\\label{motivation}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau =1}^{t} X_i(t) = 0,~\\forall p_i\\in\\brac{0,1},\n\\end{equation} \nwhich follows from the law of total probability, see appendix \\ref{app:average_opinion}. Conceptually, eq.~\\eqref{motivation} indicates that the average opinion does not reflect the competence of an expert, which is the true reward, posing a challenge for joint exploration and exploitation in the context of sequentially consulting stochastic experts, which we formally define next.  \n\n\\subsection{Problem Definition}\nThe first objective of the BEE problem is to identify the best expert in a population while actively consulting members of that group on tasks that change from one consultation to another. The following notion of regret, written here in normalized form, formally captures this phenomenon:  \n\\begin{equation}\n\t\\label{real_regret}\n\tR_T =  \\frac{1}{T}\\sum_{t=1}^{T} \\ind{X^{*}(t)=Y(t)} - \\frac{1}{T}\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}.\n\\end{equation}\nHere $X^{*}$ is the opinion of the most competent expert; $X^{*} = X_{i^{*}}$, where $i^{*} = \\argmax_{i\\in[M]} p_i$ and $I_t\\in[M]$, $\\forall t\\in[T]$ is the expert chosen at time $t$. Observe that the regret, as defined in eq.~\\eqref{real_regret} depends on the sample path of opinions and hence, it is difficult to analyze rigorously. Nonetheless, it simplifies asymptotically:\n\\begin{equation}\n\t\\label{real_regret_asymptotic}\n\t\\lim\\limits_{T\\rightarrow \\infty}R_T = \\max_{i\\in[M]} p_i - \\lim\\limits_{T\\rightarrow \\infty}\\frac{1}{T}\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}.\n\\end{equation}\nThe first term is a direct consequence of the ergodicity of the processs $\\ind{X^{*}(t)=Y(t)}$, which follows directly from eq.~\\eqref{independent_tasks}-\\eqref{static_competence}. Conceptually, this amounts to the fact that one can measure the true reliability of an expert given sufficiently many labeled tasks, as long as the reliability of the expert does not change across tasks, as is the case here. \n\nMotivated by similar asymptotic behaviors, a notion of \\textit{pseudo regret} often arises in the context of stochastic bandits, see, for instance, \\cite{bubeck2012regret}. In the context of stochastic experts, the pseudo regret is defined as follows:  \n\\begin{equation}\n\t\\label{pseudo_regret}\n\t\\tilde{R}_T = \\max_{i\\in[M]} p_i - \\frac{1}{T}\\expt{\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}}.\n\\end{equation}\nAnother notion of pseudo regret provides a reliable metric for the performance of BEE rules that aggregate opinions after consulting experts. Let a $m<M$ experts be consulted for every task $t$, indexed by $\\comm{t}\\subset[M]$ and let $f: \\myset{\\pm 1}^{\\abs{\\comm{}}}\\rightarrow \\myset{\\pm 1}$ be a known opinion aggregation rule. Define the pseudo regret as: \n\\begin{equation}\n\t\\label{swarm_regret}\n\t\\tilde{R}_T = \\max_{\\substack{\\comm{}\\subset[M]\\\\ \\abs{\\comm{}}=m}} \\prob{f\\yay{\\myset{X_i(t):i\\in\\comm{}}}=Y(t)} - \\frac{1}{T}\\expt{\\sum_{t=1}^{T} \\ind{f\\yay{\\myset{X_i(t):i\\in\\comm{t}}}=Y(t)}}.\n\\end{equation}\nTo emphasize the difference in objectives, we will refer to the rules that aim to minimize eq. \\eqref{pseudo_regret} as BEE rules and to those that sequentially weight and aggregate opinions of reliable members (SWARM) and thus, aim to minimize eq. \\eqref{swarm_regret}, as SWARM rules.   \n\nGiven an arm-selection strategy, it is more intuitive to structure a theoretical analysis based on the pseudo regrets in eq.~\\eqref{pseudo_regret}-\\eqref{swarm_regret}, provided the true rewards $\\myset{\\ind{X_{I_{\\tau}}(\\tau))=Y(\\tau)} : \\tau \\in [t]}$ are known. Nonetheless, the main challenge remains: At any given time $t$, the decision maker does not have access to the past labels $\\myset{Y_{\\tau}: \\tau \\in [t]}$ and thus, does not get to observe the true rewards. We address this challenge by consulting multiple experts for each task and leveraging the increased reliability of a collection of experts for inferring those of the individuals. Next, we formally define and discuss the properties of the competence estimation technique that facilitates joint exploration and exploitation based on opinions alone.    \n\n\\section{Unsupervised Estimation of Competences with Instantaneous Updates}\n\\label{sec:pseudocomp}\nTwo seemingly conflicting statistical phenomena should be reconciled to facilitate sampling and consultation of experts based only on their opinions. On one hand, standard index policies for  exploration and exploitation require observations from a joint probability distribution that does not evolve in time. On the other hand, opinions $\\myset{X_1(t),\\cdots,X_M(t)}$ are based on the hidden state $Y(t)$ and thus, their statistics change randomly in time. Furthermore, an operational challenge exists: Multi-armed bandit problems often allow observations to be available instantly so that the player can decide which arm to pull next. This is in sharp contrast to the unsupervised opinion aggregation strategies such as SML, EM, BP from \\cite{parisi2014ranking,welinder2010online,karger2011iterative} respectively, that yield the reliability of experts as a side-product of an optimization process that is often carried out over a block of opinions.\n\nFortunately, there exists an empirically realizable measure of competence that is only based on opinions. Conceptually, consider a compact measure, for instance, an ideal ruler for measuring the length of a two-dimensional line. The length of any such line can be measured reliably by this ruler and these measurements can be used to determine which line is the longest. Recall that the true competence of an expert can be measured experimentally via:\n\\begin{equation}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau=1}^{t} \\ind{X_i(t)=Y(t)} = p_i. \n\\end{equation}      \nIn other words, state feedback is an ideal ruler in our example. In the absence of such reliable reference, one can \\textit{choose to accept} a random line as a measure of length, replacing the phrase ``this line is $\\ell$ units long'' with ``this line is $\\beta$ \\textit{measuring lines} long''. This strategy cannot, of course, infer the true length of any line however, it can infer the \\textit{relative length} of every other line, which is sufficient for constructing index rules that solve the BEE problem.     \n\nWe introduce an alternative notion of reliability, termed \\textit{pseudo competence}, that allows instantaneous, opinion-based inference of the competences, while preserving the true ordering of the competences. Formally, the pseudo competence of an expert is defined as follows:\n\\begin{equation}\n\t\\label{pseudocompdef}\n\t\\pp{i} = \\prob{X_i(t) = \\V{\\myset{X_j(t): j\\in\\comm{}}}} = \\prob{X_i(t) = \\sign\\brac{\\sum_{j\\in\\mathcal{C}}X_j(t)}}.\n\\end{equation} \nHere, $\\V{\\cdot}$ denotes majority voting (where ties are broken arbitrarily\\footnote{The $\\yay{\\sign}$ operator does not conventionally ``break ties arbitrarily''. However, we allow $\\sign\\brac{0}=Z$, where $\\prob{Z=1}=\\prob{Z=-1}=\\nicefrac{1}{2}$ in eq.~\\eqref{pseudocompdef} for notational clarity.}) over a committee $\\mathcal{C}$ that excludes the $i^{th}$ expert: $\\mathcal{C}\\subset [T]\\setminus\\myset{i}$. Observe that it is possible to estimate pseudo competences empirically: \n\\begin{equation}\n\t\\label{empirical_estimate}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau=1}^{t} \\ind{X_i(t)=\\V{\\myset{X_j(t): j\\in\\comm{}}}} = \\pp{i}. \n\\end{equation}   \nFurther note that similar to the standard formulation with direct rewards, one can easily keep track of number of times an expert is consulted and the number of times that expert agreed with the others. \n\nLet us denote the probability of the committee $\\mathcal{C}$ being correct by:\n\\begin{equation}\n\tp_{\\mathcal{C}} \\triangleq \\prob{\\V{\\myset{X_j(t): j\\in\\comm{}}}}.\n\\end{equation}\nThen, the pseudo competence can be written explicitly as:\n\\begin{equation}\n\t\\label{pseudocomp_explicit}\n\t\\pp{i} = p_i p_{\\mathcal{C}} + \\yay{1-p_i} \\yay{1-p_{\\mathcal{C}}}. \n\\end{equation}\nThe proof of eq.~\\eqref{pseudocomp_explicit} appears in appendix \\ref{app:prop} as a part of the proof for Proposition \\ref{prop}. Conceptually, the pseudo competence of an expert increases during empirical estimation if either both the expert and committee are correct or both are incorrect. \n\nBeyond allowing empirical estimation, the pseudo competence, as a measure of reliability, demonstrates a key property: Given pseudo competences for a set experts, or given sufficiently large number of tasks and the corresponding opinions thanks to eq.~\\eqref{empirical_estimate}, it is possible to rank experts based on their true competence, as discussed next.\n\\begin{prop}\n\t\\label{prop}\n\tFor a committee $\\mathcal{C}\\subset [T]$, the following holds for every $i,j\\notin \\mathcal{C}$: \n\t\\begin{equation}\n\t\t\\label{pushtogether}\n\t\t\\pp{i}-\\pp{j} = \\yay{2p_{\\mathcal{C}}-1}\\yay{p_i-p_j}.\n\t\\end{equation}\n\tTherefore, if the probability of the committee $\\mathcal{C}$ making a correct decision under majority rule satisfies:\n\t\\begin{equation}\n\t\t\\label{condition}\n\t\tp_{\\mathcal{C}} = \\prob{\\V{\\myset{X_j(t): j\\in\\comm{}}}=Y(t)}> \\nicefrac{1}{2},\t\n\t\\end{equation}\n\tthen pseudo competence preserves the ordering of true competences: \n\t\\begin{equation}\n\t\tp_i\\geq p_j \\iff \\pp{i} \\geq \\pp{j}.\n\t\\end{equation}\n\\end{prop}\nThe proof is given in appendix \\ref{app:prop}. It is worth noting that while the condition in eq.~\\eqref{condition} is not too restrictive, it is also unavoidable. Conceptually, this condition amounts to the committee $\\mathcal{C}$ of peers being collectively reliable. Equivalently, it is not possible to infer the reliability of an expert based on opinions from a group that often collectively fails to identify the true state of the world. Formally, we restrict our attention to the case where experts are reasonably reliable: $p_i>\\nicefrac{1}{2}$, $\\forall i\\in[M]$, which implies that eq.~\\eqref{condition} holds for every subset $\\mathcal{C}\\subset[M]$ of experts. We further discuss the definition and the properties of the pseudo competence for the experts that belong to the committee $i,j\\in\\mathcal{C}$ in appendix \\ref{app:morepseudo}.\n\n\n\\section{Fundamental Properties of Different BEE and SWARM Rules}\n\\label{sec:bee}\nThere is a rich literature governing how to jointly sample and consult experts when feedback from the true state of the world $\\myset{Y(\\tau): \\tau\\in[t-1]}$ is available to the player. In this section, we propose unsupervised applications of different exploration-exploitation strategies using pseudo competence as reward. Specifically, we focus on UCB1, \\cite{auer2002finite}, due to its use of the Chernoff-Hoeffding bound, KL-UCB, \\cite{garivier2011klucb}, due its optimality for the Bernoulli rewards, IMED, \\cite{honda2015imed}, due to its near-optimal performance and empirical robustness, MOSS, \\cite{audibert2009minimax}, due to its minmax optimality, and Thompson sampling due to the strong synergy between posterior sampling and pseudo competences. The proposed BEE algorithms have the following form:\n\n\\begin{algorithm}\n\t\\caption{BEE Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE \\textbf{Require:} Choice of $\\Psi_i(\\cdot)$ for exploration-exploitation strategy, as defined in eq. \\eqref{ucb1stat}-\\eqref{thompson_stat}.\n\t\t\\STATE \\textbf{Initialize:} \n\t\t\\bindent\n\t\t\\STATE Consult every expert on the first task, \n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{1}=[M]$.\n\t\t\\eindent\n\t\t\\STATE \\textbf{Loop:}  \n\t\t\\bindent\n\t\t\\STATE $\\comm{t}\\leftarrow$ Pick top $m$ (bottom for IMED) experts ranked by $\\Psi_i(t)$,\n\t\t\\STATE $I_t \\leftarrow \\argmax_{i\\in[\\comm{t}]} \\Psi_i(t)$ ($\\argmin$ for IMED),\n\t\t\\STATE Commit to $X_{I_t}$ as the player decision,\\\\\n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{t}$.\n\t\t\\eindent\n\t\\end{algorithmic}\n\\end{algorithm}\n\nAs discussed in Section \\ref{sec:pseudocomp}, the notion of pseudo competence is built upon the availability of peer opinions therefore, we will consider consulting multiple experts for each task $t\\in[T]$, equivalent to playing multiple arms, see, for instance, \\cite{komiyama2015optimal}. Let $\\comm{t}$ denote the set of experts consulted for task $t\\in[T]$ and define the reward from consulting an expert as whether that expert agrees with the collective decision of the peers for that task: \n\\begin{equation}\n\t\\label{unsupervised_reward}\n\tR_i(t;\\comm{t}) = \\ind{X_i(t) = \\V{\\myset{X_j\\yay{t}:~j\\in\\comm{t}\\setminus\\myset{i}}}}.\n\\end{equation} \nObserve that contrary to the state feedback, opinions $\\myset{X_j:j\\in\\comm{t}}$ are available from any subset $\\comm{t}\\subset [M]$ of experts therefore, the empirical pseudo competence estimate for an expert at time $t$ takes the form:  \n\\begin{equation}\n\t\\label{empirical_pseudo}\n\t\\pp{i,T_i(t-1)} \\triangleq  \\frac{1}{T_{i}(t-1)}\\sum_{\n\t\t\\tau\\in[t-1]:~i\\in \\comm{\\tau}} R_i(\\tau;\\comm{\\tau}).\n\\end{equation}\t\nWe will now use the statistics on agreements among experts as defined in eq. \\eqref{unsupervised_reward} and the concomitant empirical competence estimation in eq. \\eqref{empirical_pseudo} to form the relevant statistics $\\Psi_i(t)$ for sampling and consulting experts in the unsupervised setup.  \n\nTo begin with the upper-confidence bound strategies, UCB1 leads to consulting experts that have highest ranking in:\n\\begin{equation}\n\t\\label{ucb1stat}\n\t\\Psi_i^{\\text{UCB1}}(t) = \\pp{i,T_i(t-1)}+\\sqrt{\\frac{2\\log t}{T_i(t-1)}}.\n\\end{equation}\nParticularly in the Bernoulli case, several refinements to UCB1 algorithm exist: Notably, \\cite{garivier2011klucb} shows that Kullback-Leibler upper confidence bound (KL-UCB) algorithm achieves the regret lower bound of \\cite{lai1985asymptotically}. In the unsupervised setup, experts with the highest ranking according to the following (operational)\\footnote{\\cite{garivier2011klucb} remark that the threshold $\\log\\yay{t}$ should be taken as $\\log\\yay{t}+c\\log\\log\\yay{t}$ for analytical purposes, where they recommend $c=0$ in practice. Furthermore, we experimentally implement its more robust counterpart KL-UCB+, see \\cite{honda2019note}.} statistics are consulted:\n\\begin{equation}\n\t\\label{klstat}\n\t\\Psi_i^{\\text{KL-UCB}}(t) = \\max\\yay{q\\in\\brac{0,1}:~ T_i(t-1)~ d\\yay{\\pp{i,T_i(t-1)}, q}\\leq \\log\\yay{\\frac{t}{T_i\\yay{t-1}}}},\n\\end{equation} \nwhere Kullback-Leibler divergence $d\\yay{\\cdot,\\cdot}$ is defined as:\n\\begin{equation*}\n\td\\yay{p,q} = p\\log\\frac{p}{q} + \\yay{1-p}\\log\\frac{1-p}{1-q}.\n\\end{equation*}\nAlbeit more computationally demanding, the performance of the KL-UCB algorithm makes it an appealing choice for binary opinions. Alternatively, using indexed minimum empirical divergence (IMED) algorithm of \\cite{honda2015imed}, we consult the experts that have the lowest ranking according to:\n\\begin{equation}\n\t\\label{imed_stat}\n\t\\Psi_i^{\\text{IMED}}(t) = T_i(t-1)~d\\yay{\\pp{i,T_i(t-1)}, \\max_{i\\in[M]}\\pp{i,T_i(t-1)}} + \\log T_i(t-1).\n\\end{equation}\nAnother key idea is that of minmax optimality, which is achieved by MOSS in the supervised framework and can be applied to the unsupervised framework by selecting experts with highest ranking in: \n\\begin{equation}\n\t\\label{moss_stat}\n\t\\Psi_i^{\\text{MOSS}}(t) = \\pp{i,T_i(t-1)} + \\sqrt{\\frac{\\max\\yay{\\log\\yay{\\frac{T}{M T_i\\yay{t-1}}} , 0}}{T_i\\yay{t-1}}}.\n\\end{equation}\n\nFinally, Thompson sampling (TS) suggests randomly sampling arms based on the posterior distribution over competences. Specifically, \n\\begin{equation}\n\t\\label{thompson_stat}\n\t\\Psi^{\\text{TS}}_{i}\\yay{t} = Z_i, \\text{ where } Z_i \\sim \\text{Beta}\\yay{\\alpha_i(t),\\beta_i(t)}.\n\\end{equation} \nThe prior is often taken be uniform for every expert, $\\yay{\\alpha_i(0),\\beta_i(0)} = \\yay{1,1}$, $\\forall i\\in[M]$ and, in the Bernoulli case, update rules of the beta distribution have a well-defined form, see, for instance, \\cite{russo2017tutorial}: \n\\begin{equation*}\n\t\\yay{\\alpha_{i}(t), \\beta_{i}(t)} = \\yay{\\alpha_{i}(t), \\beta_{i}}(t) + \\yay{R_{i}\\yay{t,\\comm{t}} + 1-R_{i}\\yay{t,\\comm{t}}}, \\forall i\\in\\comm{t},\n\\end{equation*}\nwhere $R_i\\yay{t,\\comm{t}}$ is defined in eq. \\eqref{unsupervised_reward} and the statistics  $\\yay{\\yay{\\alpha_{i},\\beta_{i}}: j\\in[M]\\setminus \\comm{t}}$ of the non-consulted experts are not updated at that time. \n\nAs a random process, $R_i\\yay{t,\\comm{t}}$, as shown in eq. \\eqref{unsupervised_reward}, is defined conditionally with respect to the choice of experts through $\\Psi_i\\yay{\\cdot}$, $\\forall i\\in[M]$, and the resulting opinions, up to time $t$ of consulting. Therefore, it is challenging to form a complete theoretical analysis of how choosing $R_i\\yay{t,\\comm{t}}$ changes the the difference between the ``rewards'' in the unsupervised setup compared to those that correspond to the true competences. Nevertheless, for a fixed committee $\\comm{}$ of peers, choosing $R_i\\yay{t,\\comm{}}$ pushes the competences together in a way that is as quantified through eq. \\eqref{pushtogether} of Proposition \\ref{prop}, making it possible to observe the impact of using pseudo competences on the regret of the exploration-exploitation strategies. The following result is a corollary to Proposition \\ref{prop} and \\cite[Theorem 2]{audibert2009minimax} for UCB1, \\cite[Theorem 1]{garivier2011klucb} for KL-UCB, \\cite[Theorem 5]{honda2015imed} for IMED, \\cite[Theorem 6]{audibert2009minimax} for MOSS, and \\cite[Theorem 1]{agrawal2013further} for Thompson sampling.\n\n\\begin{lemma}\n\t\\label{main_lemma}\n\tLet the player designate a subset of experts $\\comm{}\\subset[M]$ as peers prior to exploration and exploitation and consult them for every task $\\comm{t}=\\comm{}$, $\\forall t\\in [T]$. Define the difference between competences:\n\t\\begin{equation}\n\t\t\\Delta_i = \\max_{j\\in[M]} p_j - p_i,\n\t\\end{equation}\n\tand a potential function for a given set of competences:\n\t\\begin{equation}\n\t\t\\Phi = \\frac{\\log T}{T}\\sum_{\\substack{i\\in[M]\\setminus\\comm{}\\\\\\Delta_i>0}} \\frac{1}{\\Delta_i}.\n\t\\end{equation}\n\tThen, the regret of a BEE algorithm that picks $I_t = \\argmax_{i\\in[M]\\setminus\\comm{}}\\Psi_i(t)$ ($\\argmin$ for IMED) via statistics for UCB1, eq. \\eqref{ucb1stat}, KL-UCB, eq. \\eqref{klstat}, or IMED, eq. \\eqref{imed_stat}, is bounded by: \n\t\\begin{equation}\n\t\t\\label{theform}\n\t\t\\tilde{R}_{T}\\leq C\\frac{ \\Phi}{2p_{\\comm{}}-1},\n\t\\end{equation}\n\twhere the constant $C=10$ for UCB1, $C=\\nicefrac{1}{2}$ for KL-UCB and IMED. The regret of Thompson sampling, eq. \\eqref{thompson_stat}, also takes the form in eq. \\eqref{theform} with $C=1+\\varepsilon$ but with additional term of order $O\\yay{\\frac{M}{\\varepsilon^2}}$. Furthermore, for MOSS, eq. \\eqref{moss_stat}, the regret is upper bounded by: \n\t\\begin{equation}\n\t\t\\tilde{R}_{T}^{MOSS} \\leq \\frac{23M}{T}\\brac{\\frac{1}{2p_{\\comm{}}-1}\\sum_{\\substack{i\\in[M]\\setminus\\comm{}\\\\\\Delta_i>0}} \\frac{\\max\\yay{\\log\\frac{T  \\yay{\\yay{2p_{\\comm{}}-1}\\Delta_i}^{2}           }{M},1}}{\\Delta_i}}.\n\t\\end{equation}\n\\end{lemma} \nThe proof is given in appendix \\ref{app:mainlemma}. The important takeaways from Lemma \\ref{main_lemma} are twofold: First, using pseudo competences introduces additional regret within a constant factor of the original regret bounds but facilitates exploration and exploitation in the unsupervised regime. Second, the factor $\\nicefrac{1}{\\yay{2p_{\\comm{}}-1}}$ diminishes as the committee $\\comm{}$ collectively becomes more competent, which is important since the proposed BEE and SWARM rules that gradually select more competent experts. \n\nBy consulting multiple experts, a set of opinions on each task is acquired and making an unsupervised decision based on such sparsely sampled opinions is often carried out by rules such as SML, EM, or BP from \\cite{parisi2014ranking,welinder2010online,karger2011iterative} respectively. However, inference of pseudo competences allow an unsupervised opinion aggregation strategy that instantaneously yield a decision:  \n\\begin{equation}\n\t\\label{lpnb}\n\tf\\yay{(X_i(t),\\pp{i,T_i(t)}) : i\\in\\comm{t}} = \\sign\\yay{\\sum_{i\\in\\comm{t}}X_i(t)\\yay{\\pp{i,T_i(t)}-\\nicefrac{1}{2}}}.\n\\end{equation}   \nThe opinion-aggregation rule in eq. \\eqref{lpnb} is a linearized variant of the na\\\"{i}ve Bayes decision rule, which is robust to the variance of empirical averaging in \\eqref{empirical_pseudo}, as discussed in \\cite{berend2015finite}. In Section \\ref{sec:experiments}, we compare the BEE algorithm to their SEE counterparts and we demonstrate the performance of the SWARM rule that uses linearized na\\\"{i}ve Bayes decision rule with pseudo competence-based weights, comparing them to their state feedback-based counterparts. \n\n\\begin{algorithm}[t]\n\t\\caption{SWARM Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE \\textbf{Require:} \n\t\t\\bindent\n\t\t\\STATE Choice of $\\Psi_i\\yay{\\cdot}$ for exploration-exploitation strategy, as defined in eq. \\eqref{ucb1stat}-\\eqref{thompson_stat}.\n\t\t\\STATE Choice of $f\\yay{\\cdot}$ for unsupervised opinion aggregation, as defined \\eqref{lpnb}\n\t\t\\eindent \t\n\t\t\\STATE \\textbf{Initialize:} \n\t\t\\bindent\n\t\t\\STATE Consult every expert on the first task, \n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{1}=[M]$,\n\t\t\\STATE Commit to majority vote: $\\V{X_i:i\\in[M]}$ as the first decision. \n\t\t\\eindent\n\t\t\\STATE \\textbf{Loop:}  \n\t\t\\bindent\n\t\t\\STATE $\\comm{t}\\leftarrow$ Pick top $m$ (bottom for IMED) experts ranked by $\\Psi_i(t)$,\n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{t}$.\n\t\t\\STATE Commit to $f\\yay{(X_i(t),\\pp{i,T_i(t)}) : i\\in\\comm{t}}$ as the player decision.\\\\\n\t\t\\eindent\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn this section, we discuss empirical performance of BEE and SWARM algorithms that rely on pseudo competences for reliability inference and UCB1, KL-UCB, IMED, MOSS, or Thompson sampling for exploration and exploitation. For experiments, we chose $M=100$ experts with competences chosen uniformly at random from $\\brac{0.5,0.75}$ being consulted on $T=10^5$ tasks. The reason for such a competence interval is to avoid powerful experts in a community that would render the lack of supervision obsolete, similar to the use of weak classifiers in the boosting framework, \\cite{schapire2012boosting}. For BEE  experiments, we consider the true normalized regret as defined in eq. \\eqref{real_regret}, where for SWARM experiments, we consider the pseudo normalized regret in eq. \\eqref{swarm_regret}, allowing the opinion-aggregation rule over the best committee to have direct access to the true competences. In other words, BEE algorithms are compared against supervised rules with state feedback, where SWARM algorithms compete against rules that explore and exploit with state feedback and aggregate opinions with complete information. Figure \\ref{fig:bee_overall} provides an overview of BEE algorithms and Figure \\ref{fig:swarm_overall} provides one for SWARM rules. \n\n\\subsection{BEE}\nWe allow subcommittee sizes $m\\in\\brac{2,24}$, even numbers, where $m$ is the number of experts that are consulted for each task. Even number of experts chosen to make sure peers for every expert can reach a deterministic decision. \n\nFigure \\ref{fig:bee_overall} summarizes the performance of BEE algorithms that adaptively choose their subcommittees. We note that MOSS- and TS-based policies perform better than the rest for small subcommittee sizes, where KL-UCB-based BEE rule outperforms all for larger subcommittee sizes $\\yay{m>8}$. Figure \\ref{fig:bee_reference} illustrates the performance of UCB1, KL-UCB, Thompson sampling, MOSS, and IMED with multiple plays for reference. Interestingly, BEE algorithms that use upper-confidence bound-based exploration and exploitation strategies, as seen in Figures \\ref{fig:klucbbee}-\\ref{fig:ucb1bee}, perform almost identically to their SEE counterparts for large subcommittee sizes. On the other hand, the impact of supervision is clearer across the board for IMED, in Figure \\ref{fig:imedbee}, MOSS, in Figure \\ref{fig:mosssbee}, and Thompson sampling, as seen in Figure \\ref{fig:thompsonbee}. Importantly, Figures \\ref{fig:imedbee}-\\ref{fig:klucbbee} indicate that BEE rules that rely on IMED and KL-UCB sequentially consult experts that on average provide opinions within $0.5\\%$ of the probability of correctness of the best available expert without any supervision upon consulting modest number $\\yay{m=8}$ of experts for each task.  \n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth,draft=false]{swarm_bee_overall.jpg}\n\t\t\\caption{Performance Comparison of \\\\ Different BEE Algorithms}\n\t\t\\label{fig:bee_overall}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth,draft=false]{pnbee_overall.jpg}\n\t\t\\caption{Performance Comparison of \\\\ Different SWARM Algorithms}\n\t\t\\label{fig:swarm_overall}\n\t\\end{minipage}%\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\t\t\t\t\t\n\t\t\\includegraphics[width=\\linewidth,draft=false]{swarm_see_overall.jpg}\n\t\t\\caption{Overview of Multi-play SEE Algorithms}\n\t\t\\label{fig:bee_reference}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{imed_swarm.jpg}\n\t\t\\caption{IMED}\n\t\t\\label{fig:imedbee}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{kl_ucb_swarm.jpg}\n\t\t\\caption{KL-UCB}\n\t\t\\label{fig:klucbbee}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{ucb1_swarm.jpg}\n\t\t\\caption{UCB1}\n\t\t\\label{fig:ucb1bee}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{moss_swarm.jpg}\n\t\t\\caption{MOSS}\n\t\t\\label{fig:mosssbee}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{thompson_swarm.jpg}\n\t\t\\caption{Thompson Sampling}\n\t\t\\label{fig:thompsonbee}\n\t\\end{minipage}\n\t\\caption{Overview of the BEE Experiments: Performance Comparison to Supervised Counterparts}\n\\end{figure}\n\n\\subsection{SWARM}\nSubcommittee sized up to $m=20$ appears to be sufficient for SWARM experiments as all of the tested rules demonstrate lower than $1\\%$ normalized pseudo regret for $m>10$. Figure \\ref{fig:swarm_overall} outlines the performance of the SWARM algorithms, with their supervised counterpart given in Figure \\ref{fig:swarm_reference}, showing that MOSS-, IMED- and TS-based rules perform the with the additional opinion-aggregation objective. It might be expected for TS, as shown in Figure \\ref{fig:thompsonwarm}, considered the synergy between consulting experts based on the posterior distribution on the competences and using an approximation of the maximum a posteriori rule that is na\\\"{i}ve Bayes for opinion aggregation. It is worth noting however, that the performance of IMED-based and MOSS-based consultation, which have overall robust performance in the BEE framework as shown in Figures \\ref{fig:imedbee},\\ref{fig:mosssbee}, stand out in the SWARM framework, as seen in Figures \\ref{fig:mossswarm},\\ref{fig:imedswarm}. Furthermore, Figures \\ref{fig:ucb1swarm},\\ref{fig:klucbswarm} indicate a similar phenomenon for the upper-confidence bound-based strategies as the BEE framework, where the performance difference between supervised and unsupervised rules diminish almost entirely. Importantly, SWARM rules achieve probability of correctness within $0.5\\%$ of the opinion aggregation rule that has direct access to true competences and uses the opinions from the best available experts.  \n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\t\t\t\t\t\n\t\t\\includegraphics[width=\\linewidth,draft=false]{pnbee_reference.jpg}\n\t\t\\subcaption{Overview Multi-play SEE \\\\ with Opinion Aggregation}\n\t\t\\label{fig:swarm_reference}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{thompson_pnbee.jpg}\n\t\t\\subcaption{Thompson Sampling}\n\t\t\\label{fig:thompsonwarm}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{moss_pnbee.jpg}\n\t\t\\subcaption{MOSS}\n\t\t\\label{fig:mossswarm}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{imed_pnbee.jpg}\n\t\t\\subcaption{IMED}\n\t\t\\label{fig:imedswarm}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{ucb1_pnbee.jpg}\n\t\t\\subcaption{UCB1}\n\t\t\\label{fig:ucb1swarm}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{kl_ucb_pnbee.jpg}\n\t\t\\subcaption{KL-UCB}\n\t\t\\label{fig:klucbswarm}\n\t\\end{minipage}\n\t\\caption{Overview of the SWARM Experiments: Performance Comparison to Supervised Counterparts}\n\\end{figure}\n\n\n\\section{Conclusions}\nIn this paper, we proposed techniques that allow exploration-exploitation algorithms to operate in a regime where the rewards are observed indirectly from opinions on dynamically changing tasks. The proposed measure of competence not only allows instantaneous inference, which commonly requires either state feedback or block processing, but also enables the use of a near-optimal opinion aggregation rule in real time. Furthermore, we point out that the formal analysis of the convergence and competence ordering for BEE and SWARM algorithms are exciting open problems with the numerical evidence presented here indicating their strong potential. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Catalan numbers have a rich history. Its ubiquity in mathematics underscores the importance of this sequence. The Catalan sequence has appeared time and again in the works of numerous mathematicians. There are so many objects counted by the Catalan numbers, there's a whole book dedicated to exploring them \\cite{stanCat}. Some of those combinatorial objects are triangulations of convex polygons with $n+2$ vertices (Fig. \\ref{fig:triangulation}), binary trees with $n+2$ vertices, plane trees with $n+1$ vertices, ballot sequences of length $2n$, parenthetizations, and Dyck paths of length $2n$.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=6cm]{Triang_RegPoly.PNG}\n    \\caption{Triangulations of convex polygons with 3, 4, 5, and 6 vertices}\n    \\label{fig:triangulation}\n\\end{figure}\n\nThe modern way we describe Catalan numbers stems from Eug\\`ene Catalan's interest in the triangulation of polygons problem \\cite{stanCat}. Catalan numbers are defined as follows,\n\n\\begin{equation*} \\label{CatChoose}\n    C_n = \\frac{1}{n+1} \\binom{2n}{n}\n\\end{equation*}\n\nWe can also define Catalan numbers in terms of its generating function,\n\\begin{equation*}\\label{CatGenFunc}\n    C(x) = \\sum_{n\\geq 0} = 1+x+2x^2+5x^3+14x^4+\\dots,\n\\end{equation*}\nwhere the $n^{th}$ coefficient is the $n^{th}$ Catalan number. \n\nEuler and Goldbach studied the triangulation of convex polygons which gives us the following recursive definition,\n\\begin{equation*} \\label{FundRec}\nC_{n+1} = \\sum_{k=0}^{n} C_n C_{n-k},\\hspace{1em} C_0=1.\n\\end{equation*}\n\nMany objects enumerated by the Catalan numbers exhibit the same recursive property of the numbers themselves. That is, given a Catalan object, a rule exists that describes how to produce an object from the objects that came before it.\n\nWe introduce a new Catalan object called the \\emph{externally ordered poset of unit interval positroid} and explore its potential recursion. The elements of this poset are bases of a special matroid called a Unit Interval Positroid (UIP), a Catalan object introduced by Chavez--Gotti \\cite{chavez}. The partial order $\\leq_{\\mbox{Ext}}$ is the external order of matroid bases as defined by Las Vergnas \\cite{lvergnas}. We denote such a poset as $\\mbox{Ex}(\\mathcal{P})$, where $\\mathcal{P}$ is a positroid. Our main result is the following theorem:\n\\begin{thm:main} Let $\\mathcal{P}_n = ([2n],\\mathcal{B})$ be the trivial UIP and $\\mbox{Ex}(\\mathcal{P}_n)$ be the poset of externally ordered bases of $\\mathcal{P}_n$. Let $\\gamma$ be the algorithm defined in Section \\ref{algorithm}. Then $\\gamma(\\mbox{Ex}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$, where the partial order induced by $\\gamma(\\mbox{Ex}(\\mathcal{P}_n))$ is the external order.\n\\end{thm:main}\nOur main result shows recursion can be described nicely for the poset of the trivial UIP as well as enumerate the bases of the positroid. This is the first step towards describing the complete recursion for these special posets.\n\nThis paper is organized as follows. In section 2 we provide the background material necessary for this paper. In Section 3 we introduce an algorithm to generate the externally ordered bases of a rank $n+1$ trivial UIP from a rank $n$ trivial UIP. In section 4 we present our main theorem which proves the algorithm preserves ordering. In section 5 we discuss future work.\n\n\\section{Background} \\label{background}\n\n\n\\subsection{Matroids}\n\nMatroids capture the essence of dependence as we know it in linear algebra and graph theory. Though several definitions of a matroid exist, we utilize the basis definition in our work. \n\n\\begin{definition}A \\emph{matroid} $\\mathcal{M}$ is a pair $(E,\\mathcal{B})$ consisting of a finite set $E$ and a nonempty collection of subsets $\\mathcal{B}=\\mathcal{B}(\\mathcal{M})$ of $E$ that satisfy the following properties:\n\t\\begin{enumerate}\n\t\t\\item[(B1)] $\\mathcal{B}\\neq\\emptyset$\n\t\t\\item[(B2)] (Basis exchange axiom) If $B_1,B_2\\in\\mathcal{B}$ and $b_1\\in B_1-B_2$, then there exists an element $b_2\\in B_2-B_1$ such that $B_1-\\{b_1\\}\\cup\\{b_2\\}\\in\\mathcal{B}$.\n\t\\end{enumerate} \n\tThe element $B\\in \\mathcal{B}$ is called a \\emph{basis} and $|B|$ is the rank of $\\mathcal{M}$. A minimally dependent set $C$, that is $C-e$ is independent for any $e\\in C$, is called a \\emph{circuit}. The finite set $E$ is called the \\emph{ground set.}\n\\end{definition}\n\n\\begin{example}\\label{ex:p2_positroid}\nLet $E=\\{1,2,3,4\\}$ be the columns of the following matrix,\n\\vspace{-1em}\n\\begin{center}\n         \\[A=\n    \\bordermatrix{ & \\phantom{-}1 & 2 & \\phantom{-}3 & \\phantom{-}4 \\cr\n       & \\phantom{-}1 & 0 & -1 & -1 & \\cr\n       & \\phantom{-}0 & 1 & \\phantom{-}1 & \\phantom{-}1 }.\n    \\]\n    \\end{center}\nThen the set of bases $\\mathcal{B}$ of the matroid over $E$ is the collection of all maximally independent sets of $E$. That is, $\\mathcal{B} = \\{12,13,14,23,24\\}$. Considering all the sets of minimally dependent sets of the columns of $A$, we get that the set of circuits of $\\mathcal{M}$ are $\\mathcal{C} = \\{123,124,34\\}$. One can easily see that $\\mathcal{B}$ satisfies the bases axioms above.\n\\end{example}\n\nThe matroid described in Example \\ref{ex:p2_positroid} is part of a larger class of matroids called realizable. A matroid $\\mathcal{M}$ whose bases are in bijection with the sets of maximally independent columns of some matrix $A$ over some field $\\mathbb{F}$ is called \\emph{realizable} and denoted $\\mathcal{M}(A)$.\n\nA matrix $A$ is called \\emph{totally nonnegative} if all of its maximal minors are nonnegative. Thus, one can impose this restriciton to the family of realizable matroids and arrive at the following definition due to Postnikov \\cite{pos}.\n\n\\begin{definition}\nA \\emph{positroid} of rank $r$ over $[n]$ is a realizable matroid such that the associated full rank $r\\times n$ matrix $A$ is totally nonnegative. \n\\end{definition}\n\nThe matroid $\\mathcal{M}(A)$ given in example \\ref{ex:p2_positroid} is a positroid of rank 2. In fact, $\\mathcal{M}(A)$ is a unit interval positroid, which we discuss in the next section.\n\n\\subsection{Unit Interval Positroids}\n\nRecall that a \\emph{partially ordered} set, $P$ or \\emph{poset}, is a set with a relation $\\leq$ that satisfies reflexivity, antisymmetry, and transitivity \\cite{stan}. We can represent the poset $(P,\\leq)$ as a Hasse diagram that shows the elements of $P$ with the cover relations. See figure \\ref{fig:UIO rank 6}.\n\n\\begin{definition}\nA \\emph{poset} $P$ is a \\emph{unit interval order} if there exists a bijective map $i \\mapsto [q_i,q_{i}+1]$ from $P$ to a set $S=\\{ [q_i,q_{i}+1] | 1\\leq i \\leq n, q_i \\in \\mathbb{R} \\} $ of closed unit intervals of the real line such that for $i,j\\in P, i <_P j$ if and only if $q_i+1 < q_j$. We then say that $S$ is an \\emph{interval representation} of $P$.\n\\end{definition}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{images\/UIOrder.png}\n    \\caption{Hasse diagram of a unit interval order with 6 elements and its unit interval representation.}\n    \\label{fig:UIO rank 6}\n\\end{figure}\n\nAssociated to every unit interval order is an antiadjacency matrix, the key to describing unit interval positroids.\n\n\\begin{definition}\nFor an $n$-labeled poset $P$, the \\emph{antiadjacency matrix} of $P$ is the $n \\times n$ binary matrix $A=(a_{i,j})$ with $a_{i,j} =0$ if and only if $i <_P j$.\n\\end{definition}\n\nSkandera--Reed \\cite{skandera} proved that by labeling the unit interval order appropriately, every minor of the corresponding antiadjacency matrix $A$ is nonnegative. That is, the determinant of every submatrix of $A$ is 0 or a positive number. This fact, combined with the following result, leads us to our positroid of interest.\n\n\\begin{lemma}\\label{lem:induced_positroid}(Postnikov, \\cite{pos} Lem 3.9)\n\nFor an $n \\times n$ real matrix $A=(a_{i,j})$, consider the \n$n \\times 2n$ matrix $B = \\psi(A)$ where\n\\small{\n\\begin{equation*}\n\\arraycolsep=3pt\n\\left( \\begin{array}{ccc}\na_{1,1}& \\cdots & a_{1,n}\\\\\n\\vdots& \\ddots& \\vdots\\\\\na_{n-1,1}& \\cdots& a_{n-1,n}\\\\\na_{n,1}& \\cdots& a_{n,n}\n\\end{array} \\right)\n\\overset{\\psi}{\\longmapsto}\n\\left( \\begin{array}{cccccccc}\n1&\\cdots&0&0&(-1)^{n-1}a_{n,1}& \\cdots &(-1)^{n-1}a_{n,n} \\\\\n\\vdots&\\ddots&\\vdots&\\vdots& \\vdots & \\ddots & \\vdots\\\\\n0 & \\cdots & 1 & 0  & -a_{2,1} &\\cdots & -a_{2,n}\\\\ \n0&\\cdots&0 &1 & \\phantom{1}a_{1,1} & \\cdots & \\phantom{1}a_{1,n}\n\\end{array}\\right).\n\\end{equation*}\nFor each pair $(I,J)$ with $I,J \\subseteq [n]$ and $|I|=|J|$, define the set \n\\begin{equation*}\n    K = K(I,J) = \\{n+1-k|k\\in [n]\\backslash I\\}\\cup \\{n+j|j\\in J\\}.\n\\end{equation*}}\nThen we have $\\Delta_{I,J}(A)=\\Delta_{[n],K}(B)$.\n\\end{lemma}\n\nLemma \\ref{lem:induced_positroid} gives us a way to encode the same information from the $n\\times n$ antiadjacency matrix into a corresponding $n\\times 2n$ matrix. Explicitly, Lemma \\ref{lem:induced_positroid} associates the determinants of the submatrices of the antiadjacency matrix to the maximal minors of the $n\\times 2n$ matrix. The antiadjacency matrix of a properly labeled unit interval order generates a totally nonnegative matrix, that is, a positroid. A positroid on $[2n]$ induced by a unit interval order is a \\emph{unit interval positroid} or UIP.\n\nSuppose we have a unit interval positroid $\\mathcal{P}$ and we let $C$ be a circuit such that $C\\in \\mathcal{C(P)}$. If we take an element $e\\in C$ from circuit $C\\in \\mathcal{C}(P)$ and $e$ is the smallest, then $C - \\{e\\}$ is a \\emph{broken circuit}. Bases that do not contain broken circuits are called \\emph{atoms}. However, if $B$ is a broken circuit, then there is an element $e$ such that $B \\cup \\{e\\}$ is a circuit and $e$ is the smallest. This notion of broken circuits is captured by the ordering discussed in the next section.\n\n\\subsection{External Activities}\nIn \\cite{lvergnas} Las Vergnas introduced the notion of active orders, a collection of related orders on bases of a matroid using broken circuits. We use only the external order to define our poset of interest, though one can derive the results for other orders from this one.\n\n\\begin{definition}\nLet $M$ be a matroid on a linearly ordered set $E$, and let $A\\subseteq E$. \nWe say an element $e\\in E$ is \\emph{M-active} with respect to $A$ if there is a circuit $C$ of $M$ such that $e\\in C \\subseteq A \\cup \\{e\\}$ and $e$ is the smallest element of $C$. We denote by $\\mbox{Act}_{M}(A)$ the set of \\emph{M-active} elements with respect to $A$.\n\\end{definition}\n\nTo determine the active elements for our positroids we look at those subsets $A\\subseteq E$ where $A$ is a basis. Then those elements $e$ are exactly the elements which make or break a circuit.\n\n\\begin{definition}\nThe \\emph{external set} of an element $A$ is obtained by setting $\\mbox{Ext}_{M}(A)=\\mbox{Act}_{M}(A)\\backslash A$.\n\\end{definition}\n\nThe following proposition defines the external order of a set of bases of a matroid.\n\n\\begin{proposition}\\label{lem:lasvergnas_def} (\\cite{lvergnas}, Proposition 3.1)\nLet $A$, $B$ be two bases of an ordered matroid $M$. The following properties are equivalent:\n\\begin{enumerate}\n    \\item $A \\leq_{\\mbox{Ext}} B$;\n    \\item $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$;\n    \\item $A \\cup \\mbox{Ext}_{M}(A) \\subseteq B \\cup \\mbox{Ext}_{M}(B)$\n   \n    \\item B is the greatest, for the lexicographic ordering, of all bases of $M$ contained in $A \\cup B$.\n\\end{enumerate}\n\n\\end{proposition}\n\nLet $\\mathcal{P}$ be a unit interval positroid generated by the poset where all elements are incomparable. Then we call $\\mathcal{P}$ \\emph{trivial}. We utilize Proposition \\ref{lem:lasvergnas_def}(2) on the trivial UIP in order to determine the externally ordered poset of its bases. For an example of generating the matrix representing the trivial UIP of rank 3, see Figure \\ref{fig:trivialUIP3}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{interval_to_matrix.png}\n    \\caption{A trivial UIP of rank 3}\n    \\label{fig:trivialUIP3}\n\\end{figure}\n\\pagebreak\n\\begin{example}\\label{Ex: calculating external}\nConsider the trivial UIP of rank 3, $\\mathcal{P}_3$. We will determine the external ordering on its bases. Using proposition \\ref{lem:lasvergnas_def}, we first determine the active elements for every basis $B\\in\\mathcal{B}$, that is $\\mbox{Act}_{\\mathcal{P}_3}(\\mathcal{B})$. The bases and circuits of $\\mathcal{P}_3$ are described as follows,\n\n\\begin{align*}\n\\mathcal{C}_{\\mathcal{P}_{3}}  = \\{&(1,2,3,4),(1,2,3,5),(1,2,3,6),(4,5),(4,6),(5,6)\\},\\\\\n\\mathcal{B}_{\\mathcal{P}_{3}}  = \\{&(1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,3,4),(1,3,5),(1,3,6),\\\\   \n    & (2,3,4),(2,3,5),(2,3,6)\\}.\n\\end{align*}\n\n\nThe following tables show necessary calculations.\n\\begin{center}\n\\begin{multicols}{2}\n\\begin{tabular}{ |p{0.5cm}|p{1.7cm}|p{2.3cm}|  }\n\\hline\n\\multicolumn{3}{|c|}{Calculating M-Active Elements} \\\\\n\\hline\n$e$& $c\\in\\mathcal{C_{P}}$&$A\\cup \\{e\\}$ \\\\\n\\hline\n1 & $(1,2,3,4)$ & $(2,3,4) \\cup \\{1\\}$ \\\\\n & $(1,2,3,5)$ & $(2,3,5) \\cup \\{1\\}$ \\\\\n & $(1,2,3,6)$& $(2,3,6) \\cup \\{1\\}$ \\\\\n2 & none  & \\\\\n3 & none &  \\\\\n4 & $(4,5)$ & $(1,2,5)\\cup \\{4\\}$ \\\\\n & &$(1,3,5)\\cup \\{4\\}$ \\\\\n & &$(2,3,5)\\cup \\{4\\}$ \\\\\n & $(4,6)$ &$(1,2,6)\\cup \\{4\\}$ \\\\\n & &$(1,3,6)\\cup \\{4\\}$ \\\\\n & &$(2,3,6)\\cup \\{4\\}$ \\\\\n5 & $(5,6)$& $(1,2,6)\\cup \\{5\\}$  \\\\\n & & $(1,3,6)\\cup \\{5\\}$  \\\\\n & & $(2,3,6)\\cup \\{5\\}$  \\\\\n6 & none & \\\\\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |p{1.4cm}|p{1.5cm}|p{1.5cm}|  }\n\\hline\n\\multicolumn{3}{|c|}{Calculating the External Set} \\\\\n\\hline\n$A$& $\\mbox{Act}_{M}(A)$&$\\mbox{Ext}_{M}(A)$ \\\\\n\\hline\n$(1,2,3)$ & none & $\\emptyset$ \\\\\n$(1,2,4)$ & none & $\\emptyset$\\\\\n$(1,2,5)$ & 4   & 4 \\\\\n$(1,2,6)$ & 4,5 & 4,5 \\\\\n$(1,3,4)$ & none & $\\emptyset$ \\\\\n$(1,3,5)$ & 4 & 4 \\\\\n$(1,3,6)$ & 4,5 & 4,5\\\\\n$(2,3,4)$ & 1   & 1 \\\\\n$(2,3,5)$ & 1,4 & 1,4 \\\\\n$(2,3,6)$ & 1,4,5 & 1,4,5 \\\\\n\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{center}\n\nTo form the externally ordered poset, we use Proposition \\ref{lem:lasvergnas_def}(2) in order to obtain the relations described by \\ref{lem:lasvergnas_def}(1). For example, for bases $A=(1,2,3)$ and $B=(2,3,4)$, we check if the containment $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$ is satisfied. We can observe that $(1,2,3)\\subseteq (2,3,4) \\cup \\{1\\}$, and thus $A \\leq_{\\mbox{Ext}} B$. Continue this calculation for all pairs of bases until all bases are ordered. The resulting object, which is a set of bases ordered by the relation $\\leq_{\\mbox{Ext}}$, is the externally ordered poset of the bases of $\\mathcal{P}_3$.\n\\end{example}\nRecall, we denote the poset of the trivial UIP bases with the external order as $\\mbox{Ex}(\\mathcal{P}_n)$. Continuing with Example \\ref{Ex: calculating external}, we will show how to compute $\\mbox{Ex}(\\mathcal{P}_2)$. We compare the bases of $\\mathcal{P}_2$ pairwise using Proposition \\ref{lem:lasvergnas_def}(2). That is, for every $A, B\\in \\mathcal{B}_{\\mathcal{P}_2}$, check if $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$ is satisfied. Then we can draw the following poset (figure on the right) given the information from the table of $\\mbox{Ex}(\\mathcal{P}_2)$. Using the information in the table below, and the condition just mentioned, we produce the Hasse diagram for $\\mbox{Ex}(\\mathcal{P}_2)$.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.50\\linewidth]{images\/p2posetandhasse.png}\n    \\caption{$\\mbox{Ex}(\\mathcal{P}_2)$ and Hasse diagram.}\n    \\label{fig:p2 and poset}\n\\end{figure}\n\nComparing every pair of bases to find the ordering is easy whenever the rank of the positroid is low. Quickly, this becomes a difficult task for higher ranks. We now introduce our algorithm which streamlines this process.\n\n\n\\section{Algorithm}\\label{algorithm}\n\nTo prove $\\mbox{Ex}(\\mathcal{P})$ exhibits Catalan recursion we first describe an algorithm that generates a set of elements from the set of bases of $\\mathcal{P}_n$ and defines an order induced by $\\mbox{Ex}(\\mathcal{P}_n)$. In the next section we prove this set is precisely the set of bases of $\\mathcal{P}_{n+1}$ and the induced order preserves the external order of the bases of $\\mathcal{P}_{n+1}$. \n\nLet $\\mathcal{B}_{\\mathcal{P}_n}$ be the bases of the trivial UIP $\\mathcal{P}_n$. Then we produce a set of elements in the following way. \n\n\\begin{enumerate\n\\item \\textcolor{blue}{``Reinforce\"}: for $B\\in \\mathcal{B}_{\\mathcal{P}_n}$, add 1 to all elements of $B$ and adjoin $\\min{B}$. The original order is kept.\n\\item \\textcolor{red}{``Build Up\"}: if $2n+1\\in B$ from step 1, then form $B\\backslash\\{2n+1\\}\\cup\\{2(n+1)\\}$ and it covers $B$.\n\\item \\textcolor{olive}{``Grow Spine\"}: if $2\\in B$ from steps 1 and 2, then form $B\\backslash\\{2\\}\\cup\\{1\\}$ and is covered by $B$.\n\n\\end{enumerate}\n\nThis completes the procedure for the algorithm. Figure \\ref{fig:P2_to_p3} illustrates how the algorithm is applied to the external poset of the trivial UIP of rank 2. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{steps_algorithm.png}\n    \\caption{The algorithm applied to the trivial UIP $\\mathcal{P}_2$ and the  resulting ordered set.}\n    \\label{fig:P2_to_p3}\n\\end{figure}\n\n\\section{Main Theorem}\\label{}\n\nDenote the algorithm above by $\\gamma$; that is, $\\gamma(\\mathcal{B}_{\\mathcal{P}_n})$ is the ordered set of elements produced by $\\gamma$. To prove our main result, we show $\\gamma(\\mathcal{B}_{\\mathcal{P}_n})$ is precisely the externally ordered poset of the bases of the rank $n+1$ trivial UIP. That is, we show $\\gamma$ produces the set of bases expected and the bases are externally ordered. \nWe begin by proving the bases and circuits of the trivial UIP have a nice description.\n\n\\begin{lemma}\\label{lem:bases_description} \n\nLet $\\mathcal{P}$ be the trivial unit interval positroid of rank $n$ and $B\\in \\mathcal{B}$ a basis of $\\mathcal{P}$. Then $B$ satisfies one of the following\n\n\\begin{enumerate}\n    \\item $B = (1,\\hdots,n)$,\n    \\item $B = (1,\\hdots,\\hat{j},\\hdots,n)\\bigcup \\{j\\}$ for $j\\in [n+1,\\hdots,2n]$,\n\\end{enumerate}\nwhere $\\hat{j}$ indicates $j$ is not included in the basis $B$. Moreover, $|\\mathcal{B}| = n^2+1.$\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lem:induced_positroid}, we know that $P$ is represented by the $n \\times 2n$ matrix,\n\n\\[\\begin{bmatrix}\n1&0&\\cdots &0&(-1)^{n-1}& \\cdots &(-1)^{n-1} \\\\\n0& 1 &\\cdots &0 & \\vdots & \\ddots & \\vdots\\\\\n\\vdots & \\ddots & 1 & 0  & (-1) &\\cdots & (-1)\\\\\n0&\\cdots&0 &1 & 1 & \\cdots & 1\\\\\n\\end{bmatrix},\\]\\\\\nwhere the first $n$ columns form the $n\\times n$ identity matrix and the last $[n+1,\\hdots, 2n]$ columns are equal. \n\nSince a basis corresponds to a non-zero $n\\times n$ maximal minor, we need only describe which submatrices are non-singular. The $n\\times n$ identity matrix is non-singular, which is the first basis description.\n\nSince the $[n+1,\\hdots,2n]$ columns are all the same, any non-singular maximal submatrix can include at most one of these columns. Thus, all other bases are formed by maximal submatrices with one column from $[n+1,\\hdots,2n]$ and the remaining a subset of the columns $[n]$, which is the second description.\n\nWe use our description to enumerate the bases. To form a basis of the second kind, we remove an element from $\\{1,2,\\dots,n\\}$ and replace it with any of the $n$ elements from $[n+1,\\dots,2n]$. This is done in $\\binom{n}{1}n$ ways. Including the basis $\\{1,2,\\dots,n\\}$, we get $$\\binom{n}{1} n + 1 =n^2 + 1$$ bases, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{circuit_description}\nLet $\\mathcal{P}_n$ be the trivial UIP and $\\mathcal{C}_{\\mathcal{P}_n}$ its set of circuits. Then any $C\\in \\mathcal{C}_{\\mathcal{P}_n}$ must either be $I\\cup i$ where $i\\in [n+1,2n]$ or $(i,j)$ for $i,j\\in[n+1,2n]$ and $i\\neq j$.\n\\end{lemma}\n\n\\begin{proof}\nNote that $I\\cup i$ where $i\\in [n+1,2n]$ is a circuit since the removal of any element produces a basis as described in Lemma \\ref{lem:bases_description}. Similarly, the pair $i,j\\in[n+1,2n]$ and $i\\neq j$ is a circuit since every singleton $i\\in[2n]$ is independent. Since any other subset of $[2n]$ contains one of these sets, there are no other circuits of $\\mathcal{P}_n$.\n\\end{proof}\n\nWe state the following observation about bases of trivial UIPs, though it is not used in any subsequent theorem. \n\\begin{lemma}\\label{min_elts_description}\nLet $B\\in \\mathcal{B}(\\mathcal{P}_n)$, where $\\mathcal{P}_{n}$ is a trivial unit interval positroid. Then $\\epsilon(B)=0$ if and only if $B$ is a minimally ordered basis.\n\\end{lemma}\n\n\\begin{proof}\n$(\\Rightarrow)$ Assume $\\epsilon(B)=0$ (i.e. Ext$(B)=\\emptyset$) and there exists $A\\in \\mathcal{B}$ such that $A\\lessdot B$. Then $A\\subset B\\cup \\text{Ext}(B) = B$, which implies $A=B$.\n\n$(\\Leftarrow)$ Assume $B$ is minimal and $\\text{Ext}(B)\\neq\\emptyset.$ First, note $B$ minimal implies $1\\in B$. If not, let $B'=B\\backslash2\\cup 1$. Then Ext$(B')=$ Ext$(B)\\backslash\\{1\\}$, so that $B'\\subset B \\cup $Ext$(B)$, implying $B'\\lessdot B$, a contradiction. Since $\\text{Ext}(B)\\neq\\emptyset,$ there exists $e=\\min(C)$ for some circuit $C$ of $P_n$ such that $C\\subset B\\cup\\{e\\}$. By Lemma \\ref{circuit_description}, $C=ej$ for some $j\\in[n+1,2n]$ and $j\\neq e$. Then Ext$(B)=\\{n+1,\\dots,e\\}$, and $j\\in B$ necessarily. Let $A=B\\backslash \\{j\\} \\cup \\{e\\}$. Then $A\\lessdot B$, a contradiction.\n\\end{proof}\n\nNext we prove $\\gamma$ produces the set of desired bases.\n\n\\begin{theorem}\\label{thm:AlgMakesBases}\nLet $\\mathcal{B}$ be the set of bases of the trivial unit interval positroid $\\mathcal{P}_{n}$ of rank $n$ over ground set $[2n]$. Then, after applying algorithm $\\gamma$ to $\\mathcal{B}$, $\\gamma(\\mathcal{B})$ is the set of bases for the trivial unit interval positroid $\\mathcal{P}_{n+1}$ of rank $n+1$ over ground set $[2(n+1)]$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{B}=\\mathcal{B}(\\mathcal{P}_{2n})$. We will show that for every $B\\in\\mathcal{B}$, $\\gamma(\\mathcal{B})$ produces a $B^{'}\\in\\mathcal{B}(\\mathcal{P}_{2(n+1)})$.\n Let $B=\\{1,\\dots,n\\}.$ Then $\\gamma(B)=\\{1,\\dots,n+1\\}$, which by Lemma \\ref{lem:bases_description}(1) is a basis of $\\mathcal{P}_{2(n+1)}$.\n Let $B=\\{1,2,\\dots,\\hat{j},\\dots,n\\}$, where $\\hat{j}\\in [n+1,\\dots, 2n]$. Applying step 1 of $\\gamma$ we have that $\\gamma(B) = B+1\\cup\\{min(B)\\}\\in\\gamma(\\mathcal{B}).$ Explicitly, this means $$\\{1,2,3,\\dots, \\hat{j}+1,\\dots,n+1\\}\\in\\gamma(\\mathcal{B}),$$ where $$\\hat{j}+1 \\in [n+2,2n+1].$$ Since $[n+2,\\dots,2n+1]\\subset [(n+1)+1,\\dots,2(n+1)]$, then by Lemma \\ref{lem:bases_description} we have that $\\gamma(B)$ is a basis of $\\mathcal{P}_{2(n+1)}$. Moreover, there are $n^2$ bases created at this step.\n \nFor the second step of $\\gamma$, let $\\mathcal{B}_{max}$ be the set of bases such that max$(\\gamma(B))=2n+1$. Then for all $B\\in \\mathcal{B}_{max}$, we have $B\\backslash\\{2n+1\\}\\cup \\{2(n+1)\\}$. All sets generated prior to this step have maximum value $2n+1$. Thus, all sets generated at this step of $\\gamma$ are new elements of $\\gamma({\\mathcal{B}})$. Every new set generated is formed by choosing a value of $[n]$ and replacing it with $2(n+1)$. Thus each set is a basis and we have $n$ new bases.\n \nFor the final step of $\\gamma$, let $\\mathcal{B}_{min}$ be the set of bases such that min$(\\gamma(B))=2$. Then for every $B\\in \\mathcal{B}_{min}$, $B\\backslash\\{2\\}\\cup \\{1\\}\\in\\gamma(\\mathcal{B})$. Note Lemma \\ref{lem:bases_description} implies $3\\in B$ for every $B\\in \\mathcal{B}_{min}$. Then for every set generated at this step, it must contain $1,3$ and not $2$. By construction, all sets generated by $\\gamma$ prior to the third step of $\\gamma$ have that their first two entries are consecutive. Thus, all sets generated in this third step are new and, by Lemma \\ref{lem:bases_description}, a basis. Thus all sets are bases of $\\mathcal{P}_{n+1}$.\n\nTo enumerate the bases generated in this final step of $\\gamma$, let us analyze the number of bases it applies to. By the first step of the algorithm, every basis $B\\in\\mathcal{B}$ such that $\\min(B)=2$ produces a basis in $\\mathcal{B}(\\mathcal{P}_{2(n+1)})$ with minimum element 2. Thus, the number of bases produced equals the number of $B\\in\\mathcal{B}$ such that $\\min(B)=2$, which is $n$. By the second step of the algorithm, the only basis produced with minimum element 2 is the one that also has maximum element $2n$. Thus in total, there are $n+1$ bases with minimum element 2 produced by $\\gamma$. Accounting for the all the bases produced at every step of $\\gamma$, we have that $$|\\gamma(\\mathcal{B}(\\mathcal{P}_{2(n+1)}))|=n^2+n+n+1+1 = (n+1)^2+1,$$ as desired.\n\\end{proof}\n\nUsing the above results, we may now prove our main theorem.\n\n\n\\begin{theorem}\\label{thm:main}\nLet $\\mathcal{B}(\\mathcal{P}_n)$ be the bases of the rank $n$ trivial UIP on $[2n]$. Assume $\\gamma$ is defined to be the algorithm described in Section \\ref{algorithm}. Then $\\gamma(\\mathcal{B}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$, the externally ordered poset of bases of the rank $n+1$ trivial UIP on $[2(n+1)]$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{thm:AlgMakesBases}, we know $\\gamma(\\mathcal{B}(\\mathcal{P}_n))$ produces the set of bases of $\\mathcal{P}_{n+1}.$ It is left to show that $\\gamma$ also induces the external order on $\\mathcal{B}(\\mathcal{P}_{n+1}),$ thus producing $\\mbox{Ex}(\\mathcal{P}_{n+1}).$\nLet $A,B\\in\\mathcal{B}(\\mathcal{P}_n).$ We prove this by cases for each step of the algorithm, confirming that the condition $A \\subseteq B\\cup \\mbox{Ext}(B)$ in Lemma \\ref{lem:lasvergnas_def} is satisfied. Denote $\\hat{A}=\\gamma(A)$ for any $A\\in\\mathcal{B}(\\mathcal{P}_n).$ \nFor the following cases, $\\hat{A}$ denotes the basis produced by $\\gamma$ after performing only the first step of the algorithm. Since every basis in $\\mathcal{B}(\\mathcal{P}_n)$ contains either 1 or 2, we need consider only the following three cases.\n\n\\textbf{Case 1:} $1\\in A$ and $1\\in B$.\\\\\nSince $1\\in A$, the only circuits contained in $A\\cup\\{i\\}$ for $i\\in[2n]$ are those of the form $(i,j)$ for $i,j\\in[n+1,\\dots,2n]$. Thus $\\mbox{Ext}(A)=\\{n+1,\\dots,max(A)-1\\}$. Similarly, $\\mbox{Ext}(B)=\\{n+1,\\dots,max(B)-1\\}$. After applying step one of $\\gamma$, $1\\in\\hat{A}$ and $1\\in\\hat{B}$. Thus $\\mbox{Ext}(\\hat{A})=\\{n+2,\\dots,max(\\hat{A})-1\\}$ and $\\mbox{Ext}(\\hat{B})=\\{n+2,\\dots,max(\\hat{B})-1\\}$. Let $a\\in \\hat{A}\\backslash \\{1\\}$. Then $a = a_i+1$ for $a_i \\in A$. Since  $A\\leq_{\\mbox{Ext}}B$, it follows that $a_i \\in B \\cup \\{n+1,\\dots,max(B)-1\\}$. Thus, $a_i+1\\in B+1\\cup \\{n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\n\\textbf{Case 2:} $1\\in A$ and $1\\notin B$.\\\\ \nBy definition, if $1\\notin B$ then $1\\in \\mbox{Ext}(B)$ for every $B$ of a trivial UIP. Let $a\\in \\hat{A}\\backslash \\{1\\}$. Then $a = a_i+1$ for $a_i \\in A$. We know $a_i \\in B \\cup \\{1,n+1,\\dots,max(B)-1\\}$. Thus, $a_i+1\\in B+1\\cup \\{1,n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\n\\textbf{Case 3:} $1 \\notin A \\mbox{ or } B$.\\\\\nBy assumption we know that $1\\in \\mbox{Ext}(A)$ and $1\\in \\mbox{Ext}(B)$. Then for $a\\in \\hat{A}\\backslash \\{2\\}$, we have that $a = a_i+1$ for $a_i \\in A$. Since $a_i \\in B \\cup \\{1,n+1,\\dots,max(B)-1\\}$, it follows $a_i+1\\in B+1\\cup \\{1,n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\nFor the second step of the algorithm, notice that the only change to a basis $B$ is that the element $2n+1$ is replaced with $2(n+1)$. This means that $\\mbox{Ext}(\\hat{B})=\\mbox{Ext}(B)\\cup\\{2n+1\\}$. Then the cases to check are exactly those done for step one of the algorithm, and proceed in the same way. Similarly, for the third step, $\\mbox{Ext}(\\hat{B})=\\mbox{Ext}(B)\\backslash\\{1\\}$ since this step replaces the element 2 with 1. And then the cases proceed as above.\n\nIn all cases, the external order of $\\mathcal{B}(\\mathcal{P}_{n+1})$ is shown to be induced by the external order of $\\mathcal{B}(\\mathcal{P}_n)$. Thus, $\\gamma(\\mathcal{B}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$ as desired.\n\\end{proof}\n\n\\section{Conclusion and Future Work}\\label{}\n\nThe recursive algorithm for the external poset on the bases of trivial UIPs provides a stepping stone to explore and develop either general or specific algorithms for the other UIPs. Recall that these are Catalan objects and that we have Catalan many to consider. In Figure \\ref{fig:all rank 3} we see the trivial UIP and the remaining UIPs of rank $r=3$. There appears to be some symmetry in the Hasse diagrams, but it is unclear how these five UIPs give rise to the non-trivial UIPs of rank $r=4$. See Figure \\ref{fig:soup} for examples of externally ordered posets for non-trivial UIPs of rank $r=4$. As was done for the recursion for the externally ordered poset for trivial UIPs, a potential first step is to find nice descriptions for the bases and circuits of the remaining UIPs.\n\nAnother approach for describing Catalan recursion for the remaining externally ordered posets is to consider matroid minors. One can check if minors of rank $n+1$ UIPs are isomorphic to rank $n$ UIPs. We have begun to explore this direction and our code is provided below. \n\n Our algorithm provides a new way of producing externally ordered posets. It would be interesting to know how the complexity of Las Vergnas' procedure compares to ours. In particular, how efficient is our algorithm in generating the next rank trivial UIP.\n\n\n\n\\section{Acknowledgements}\\label{}\nThis project would not be possible without the support and encouragement of various people I've been fortunate enough to meet and work with at UC Davis. I would like to thank Dr. Anastasia Chavez for her patience and guidance throughout this project. Thank you for giving me the opportunity to explore mathematics research, and for introducing me to matroid theory. I am also grateful to Dr. Jesus De Loera for supporting this project.\nI am grateful to the McNair scholars program for introducing me to undergraduate research, Esha Datta for so many helpful conversations and introductions, Shannon Allen for listening to me talk about this project many, many times, Calvin Cramer for being my Python help desk, and the friends, family, faculty and staff that saw this project in various stages of completion. \n\n\\begin{figure}[h]\\label{fig:uip r=3}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{all_rank3.png}\n    \\caption{All UIPs with rank $r=3$.}\n    \\label{fig:all rank 3}\n\\end{figure}\n\n\\begin{figure}[h\n    \\centering\n    \\includegraphics[width=1\\linewidth]{some_rank4.png}\n    \\caption{Two of the 14 UIPs with rank $r=4$.}\n    \\label{fig:soup}\n\\end{figure}\n\\pagebreak\n\\begin{lstlisting}[language=Python]\nimport sage.matroids.matroid\ndef UnitIntervalPositroid(Dyck_matrix):\n\n    D = Dyck_matrix\n    rank = D.nrows()\n    D_flip = matrix(rank)\n    for i in range(rank):\n        D_flip[rank-i-1] = (-1)^i*D[i]\n    A = matrix.identity(rank).augment(D_flip)\n    cols = A.ncols()\n    M_A = Matroid(A)\n    \n    return M_A\n\ndef powerset(fullset):\n  listrep = list(fullset)\n  n = len(listrep)\n  return [[listrep[k] for k in range(n) if i&1<<k] for i in range(2**n)]\n  \nPos_list3 = []\nfor d in D_matrices:\n    U = UnitIntervalPositroid(d)\n    Pos_list3.append(U)\nprint Pos_list3\n\nE = (Matrix([[1,1,1,0],[1,1,1,0],[1,1,1,0],[1,1,1,1]]))\n    \nU4 = UnitIntervalPositroid(E)\n\ncontU4 = []\nfor s in [0,1,2,3,4,5]:\n    for t in [0,1,2,3,4,5]:\n        if t==s: break\n        else:\n            V = U4.delete(t).contract(s)\n            contU4.append([V])\n\nfor p in contU4:\n    for m in Pos_list3:\n        if p.is_isomorphic(m)==True:\n            print [b for b in m.bases()],\"yes\",\"\\n\"\n            print 'bases of cont-del positroid are ', [b for b in p.bases()],\"\\n\"\n\n\\end{lstlisting}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe following problem was raised in 2001 by I.~M.~Gelfand: Using graph theory describe up to isometry all finite ultrametric spaces~\\cite{Lem2001}. An appropriate representation of finite, ultrametric spaces by monotone trees was proposed by V.~Gurvich and M.~Vyalyi in~\\cite{GV2012DAM}. A simple geometric description of Gurvich---Vyalyi representing trees was found in \\cite{PD2014JMS}. This description allows us effectively use the Gurvich---Vyalyi representation in various problems associated with finite ultrametric spaces. In particular, it leads to a graph-theoretic interpretation of the Gomory---Hu inequality \\cite{DPT2015}. A characterization of finite ultrametric spaces which are as rigid as possible also was obtained \\cite{DPT2017FPTA} on the basis of the Gurvich---Vyalyi representation. Some other extremal properties of finite ultrametric spaces and related them properties of monotone rooted trees have been found in~\\cite{DP2020pNUAA}. The interconnections between the Gurvich---Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in~\\cite{Dov2019pNUAA} (see also \\cite{Qiu2009pNUAA, Qiu2014pNUAA, DP2018pNUAA, Pet2018pNUAA, Pet2013PoI}).\n\nThe Gurvich---Vyalyi representing trees can be considered as a subclass of finite trees endowed with some special labeling on vertex set. The trees with labeled vertices are studied by many mathematicians and there are a number of interesting results in this directions. In survey~\\cite{GalTEJoC2019}, J.~Gallian writes that over 200 graph labelings techniques have been studied in over 2800 paper during the past 50 years. In this regards, we only note that, in almost all studies of trees with labeled vertices, it is assumed that the trees are finite. The infinite trees endowed with positive real labelings on the set of edges are known as the so-called \\(R\\)-trees (see~\\cite{Ber2019SMJ} for some interesting results related to \\(R\\)-trees and ultrametrics). A description of interrelations between finite subtrees of \\(R\\)-trees and finite, monotone rooted trees can be found in~\\cite{Dov2020TaAoG}. The categorical equivalence of trees and ultrametric spaces was investigated in \\cite{H04} and \\cite{Lem2003AU}.\n\nMotivated by Gurvich---Vyalyi representation of finite ultrametric spaces and some results of Bruhn, Diestel, Halin, K\\\"{u}hn, Pott, Spr\\\"{u}ssel, and Stein \\cite{BDSJGT2005, BDCPC2006, BSCPC2010, DKEJC2004, DieJCTSB2006, DieDM2011, DSAM2011, DSTA2011, DSDM2012, DieAMSUH2017, DPJCTSB2017} on topological aspects of infinite graphs we consider infinite trees whose vertices are labeled by nonnegative real numbers and ultrametric spaces generated by such trees.\n\nThe paper is organized as follows. Section~\\ref{sec2} and Section~\\ref{sec3} contain some necessary concepts and facts from the theory of metric spaces and graph theory, respectively. In Section~\\ref{sec4} we introduce into consideration the ultrametric spaces \\((V(T), d_l)\\) generated by non-degenerate vertex labelings \\(l\\) of arbitrary trees \\(T\\). The first main result of the paper is Theorem~\\ref{t4.7} characterizing, up to isomorphism, the labeled trees \\(T(l)\\) for which the corresponding ultrametric spaces \\((V(T), d_l)\\) are complete. The characterizations of labeled trees generating discrete ultrametrics and totally bounded ones are found in Theorem~\\ref{t4.11} and Theorem~\\ref{t11.8}, respectively. Using these results we describe, up to isomorphism, the labeled trees generating discrete totally bounded ultrametrics in Theorem~\\ref{t11.16}. The final result of Section~\\ref{sec4} is Theorem~\\ref{t4.18} characterizing the labeled trees \\(T(l)\\) for which the ultrametric spaces \\((V(T), d_l)\\) are compact. The last fifth section contains some conjectures and examples related to subject of the paper.\n\n\\emph{Concluding remarks}. The results obtained in the paper indicate a close connection between the combinatorial properties of an infinite tree and the properties of ultrametric spaces generated by labelings on its vertex set.\n\\begin{itemize}\n\\item A tree \\(T\\) is rayless if and only if every ultrametric generated by vertex labeling is complete (Corollary~\\ref{c4.7}).\n\\item \\(T\\) is locally finite if and only if every ultrametric generated by vertex labeling is discrete (Corollary~\\ref{c4.12}).\n\\item \\(T\\) is rayless, at most countable, and has no adjacent vertices of infinite degree if and only if there is vertex labeling generating a compact ultrametric (Theorem~\\ref{t7}).\n\\item \\(T\\) is locally finite if and only if there is vertex labeling generating a discrete totally bounded ultrametric (Corollary~\\ref{c4.18}).\n\\end{itemize}\nIt seems interesting to study similar problems for general infinite connected graphs using the spanning trees technique. Another promising direction of research is the study of ultrametric spaces generated by some special labelings. For example, we can consider the case when the label of a vertex depends on the degree of this vertex.\n\n\\section{Definitions and facts from theory of metric spaces}\\label{sec2}\n\nLet us start from basic concepts. In what follows, we will denote by \\(\\RR^{+}\\) the half-open interval \\([0, \\infty)\\) and write \\(\\NN\\) for the set of all positive integers, \\(\\{1, 2, \\ldots\\}\\).\n\nA \\textit{metric} on a set $X$ is a function $d\\colon X\\times X\\rightarrow \\RR^+$ such that for all \\(x\\), \\(y\\), \\(z \\in X\\)\n\\begin{enumerate}\n\\item $d(x,y)=d(y,x)$,\n\\item $(d(x,y)=0)\\Leftrightarrow (x=y)$,\n\\item \\(d(x,y)\\leq d(x,z) + d(z,y)\\).\n\\end{enumerate}\n\nA metric space \\((X, d)\\) is \\emph{ultrametric} if the \\emph{strong triangle inequality}\n\\[\nd(x,y)\\leq \\max \\{d(x,z),d(z,y)\\}.\n\\]\nholds for all \\(x\\), \\(y\\), \\(z \\in X\\). In this case the function \\(d\\) is called \\emph{an ultrametric} on \\(X\\).\n\n\\begin{definition}\\label{d2.2}\nLet \\((X, d)\\) and \\((Y, \\rho)\\) be metric spaces. A bijective mapping \\(\\Phi \\colon X \\to Y\\) is said to be an \\emph{isometry} if\n\\[\nd(x,y) = \\rho(\\Phi(x), \\Phi(y))\n\\]\nholds for all \\(x\\), \\(y \\in X\\). The metric spaces are \\emph{isometric} if there is an isometry of these spaces.\n\\end{definition}\n\nLet \\((X, d)\\) be a metric space. An \\emph{open ball} with a \\emph{radius} \\(r > 0\\) and a \\emph{center} \\(c \\in X\\) is the set \n\\[\nB_r(c) = \\{x \\in X \\colon d(c, x) < r\\}.\n\\]\n\nWe denote by \\(\\mathbf{B}_X\\) the set of all open balls in \\((X, d)\\).\n\nA sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) is a \\emph{Cauchy sequence} in \\((X, d)\\) if, for every \\(r > 0\\), there is an integer \\(n_0 \\in \\NN\\) such that \\(x_n \\in B_r(x_{n_0})\\) for every \\(n \\geqslant n_0\\). It is easy to see that \\((x_n)_{n \\in \\mathbb{N}}\\) is a Cauchy sequence if and only if \n\\begin{equation*\n\\lim_{n \\to \\infty} \\sup\\{d(x_n, x_{n+k}) \\colon k \\in \\mathbb{N}\\} = 0.\n\\end{equation*}\n\n\\begin{remark}\\label{r2.17}\nHere and later the symbol \\((x_n)_{n\\in \\mathbb{N}} \\subseteq X\\) means that \\(x_n \\in X\\) holds for every \\(n \\in \\mathbb{N}\\).\n\\end{remark}\n\nThere exists a comfortable ``ultrametric modification'' of the notion of Cauchy sequence (see, for example, \\cite[p.~4]{PerezGarcia2010} or \\cite[Theorem~1.6, Statement~(13)]{Comicheo2018}).\n\n\\begin{proposition}\\label{p4.5}\nLet \\((X, d)\\) be an ultrametric space. A sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) is a Cauchy sequence if and only if the limit relation\n\\begin{equation*\n\\lim_{n \\to \\infty} d(x_n, x_{n+1}) = 0\n\\end{equation*}\nholds.\n\\end{proposition}\n\nA sequence \\((x_n)_{n \\in \\mathbb{N}}\\) of points in a metric space \\((X, d)\\) is said to be convergent to a point \\(a \\in X\\),\n\\begin{equation*\n\\lim_{n \\to \\infty} x_n = a,\n\\end{equation*}\nif, for every open ball \\(B\\) containing \\(a\\), it is possible to find an integer \\(n_0 \\in \\NN\\) such that \\(x_n \\in B\\) for every \\(n \\geqslant n_0\\). Thus, \\((x_n)_{n \\in \\mathbb{N}}\\) is \\emph{convergent} to \\(a\\) if and only if \n\\begin{equation*\n\\lim_{n \\to \\infty} d(x_n, a) = 0.\n\\end{equation*}\nA sequence is convergent if it is convergent to some point. It is clear that every convergent sequence is a Cauchy sequence.\n\nThe next proposition follows, for example, from Theorem~6.8.3 in \\cite{Sea2007}.\n\n\\begin{proposition}\\label{p2.7}\nLet \\((X, d)\\) be a metric space and let \\((x_n)_{n \\in \\NN} \\subseteq X\\) be a Cauchy sequence in \\((X, d)\\). Then \\((x_n)_{n \\in \\NN}\\) is convergent if and only if it has a convergent subsequence.\n\\end{proposition}\n\nNow we present a definition of \\emph{total boundedness}.\n\n\\begin{definition}\\label{d2.10}\nA subset \\(A\\) of a metric space \\((X, d)\\) is totally bounded if for every \\(r > 0\\) there is a finite set \\(\\{B_r(x_1), \\ldots, B_r(x_n)\\} \\subseteq \\mathbf{B}_X\\) such that\n\\[\nA \\subseteq \\bigcup_{i = 1}^{n} B_r(x_i).\n\\] \n\\end{definition}\n\nThere exists a simple interdependence between the total boundedness of a set \\(A \\subseteq X\\) and Cauchy sequences in \\(A\\).\n\n\\begin{proposition}\\label{p2.11}\nA subset \\(A\\) of a metric space \\((X, d)\\) is totally bounded if and only if every sequence of points of \\(A\\) contains a Cauchy subsequence. \n\\end{proposition}\n\nSee, for example, Theorem~7.8.2 \\cite{Sea2007}.\n\n\\begin{corollary}\\label{c2.17}\nLet \\((X, d)\\) be a metric space. If \\(A \\subseteq X\\) is totally bounded in \\((X, d)\\) and \\(C\\) is a subset of \\(A\\), then \\(C\\) is totally bounded in \\((A, d|_{A \\times A})\\).\n\\end{corollary}\n\nThe next basic for us concept is the concept of \\emph{completeness}.\n\n\\begin{definition}\\label{d2.6}\nA metric space \\((X, d)\\) is complete if for every Cauchy sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) there is a point \\(a \\in X\\) such that \n\\[\n\\lim_{n \\to \\infty} x_n = a.\n\\]\n\\end{definition}\n\nThus, a metric space is complete if and only if the set of Cauchy sequences coincides with the set of convergent sequences in this space.\n\nAn important subclass of complete metric spaces is the class of \\emph{compact} metric spaces.\n\n\\begin{definition}[Borel---Lebesgue property]\\label{d2.3}\nLet \\((X, d)\\) be a metric space. A subset \\(A\\) of \\(X\\) is compact if every family \\(\\mathcal{F} \\subseteq \\mathbf{B}_X\\) satisfying the inclusion\n\\[\nA \\subseteq \\bigcup_{B \\in \\mathcal{F}} B\n\\]\ncontains a finite subfamily \\(\\mathcal{F}_0 \\subseteq \\mathcal{F}\\) such that \n\\[\nA \\subseteq \\bigcup_{B \\in \\mathcal{F}_0} B.\n\\]\n\\end{definition}\n\nA standard definition of compactness usually formulated as: Every open cover of \\(A\\) in \\(X\\) has a finite subcover.\n\nThe following classical theorem was proved by Frechet and it is a ``compact'' analog of Proposition~\\ref{p2.11}. \n\n\\begin{proposition}[Bolzano---Weierstrass property]\\label{p2.9}\nA subset \\(A\\) of a metric space is compact if and only if every sequence of points of \\(A\\) contains a subsequence which converges to a point of \\(A\\).\n\\end{proposition}\n\nThe next corollary shows that the class of compact metric spaces is the intersection of the class of complete metric spaces with the class of totally bounded ones.\n\n\\begin{corollary}[Spatial Criterion]\\label{c2.9}\nA metric space is compact if and only if this space is complete and totally bounded.\n\\end{corollary}\n\nThis and other criteria of compactness can be found, for example, in \\cite[p.~206]{Sea2007}.\n\nLet \\((X, d)\\) be a metric space and let \\(S \\subseteq X\\). The set \\(S\\) is said to be \\emph{dense} in \\((X, d)\\) if for every \\(a \\in X\\) there is a sequence \\((s_n)_{n\\in \\mathbb{N}} \\subseteq S\\) such that\n\\[\na = \\lim_{n \\to \\infty} s_n.\n\\]\n\nRecall that a point \\(p\\) of a metric space \\((X, d)\\) is \\emph{isolated} if there is \\(\\varepsilon > 0\\) such that \\(d(p, x) > \\varepsilon\\) for every \\(x \\in X \\setminus \\{p\\}\\). If \\(p\\) is not an isolated point of \\(X\\), then \\(p\\) is called an \\emph{accumulation point} of \\(X\\). We say that a set \\(A \\subseteq X\\) is \\emph{discrete} if all points of \\(A\\) are isolated.\n\nIt will be shown in Proposition~\\ref{p4.13} of Section~\\ref{sec4} that every ultrametric space, generated by labeled graph, contains a dense discrete subset.\n\n\n\\section{Definitions and facts from graph theory}\\label{sec3}\n\nA \\textit{graph} is a pair $(V,E)$ consisting of a set $V$ and a set $E$ whose elements are unordered pairs \\(\\{u, v\\}\\) of different points \\(u\\), \\(v \\in V\\). For a graph $G=(V,E)$, the sets \\(V=V(G)\\) and \\(E=E(G)\\) are called \\textit{the set of vertices} and \\textit{the set of edges}, respectively. A graph \\(G\\) is \\emph{finite} if \\(V(G)\\) is a finite set. If \\(\\{x,y\\} \\in E(G)\\), then the vertices \\(x\\) and \\(y\\) are called \\emph{adjacent}. In what follows we will always assume that \\(E(G) \\cap V(G) = \\varnothing\\).\n\nRecall that a graph \\(G_1\\) is a \\emph{subgraph} of a graph \\(G\\) if\n\\[\nV(G_1) \\subseteq V(G) \\quad \\text{and} \\quad E(G_1) \\subseteq E(G).\n\\]\nIn this case we will write \\(G_1 \\subseteq G\\). If \\(\\{G_i \\colon i \\in I\\}\\) is a family of subgraphs of a graph \\(G\\), then, by definition, the union \\(\\bigcup_{i \\in I} G_i\\) is a subgraph \\(G^{*}\\) of \\(G\\) such that \n\\[\nV(G^{*}) = \\bigcup_{i \\in I} V(G_i) \\quad \\text{and} \\quad E(G^{*}) = \\bigcup_{i \\in I} E(G_i).\n\\]\nSimilarly, the intersection \\(\\bigcap_{i \\in I} G_i\\) is a subgraph \\(G_{*}\\) of \\(G\\) with\n\\begin{equation}\\label{e3.4}\nV(G_{*}) = \\bigcap_{i \\in I} V(G_i) \\quad \\text{and} \\quad E(G_{*}) = \\bigcap_{i \\in I} E(G_i).\n\\end{equation}\n\nLet \\(v^*\\) be a vertex of a graph \\(G\\). The \\emph{neighborhood} \\(N(v^*) = N_G(v^{*})\\) is a subgraph of \\(G\\) induced by all vertices adjacent to \\(v^{*}\\). Thus, we have\n\\begin{align*}\nV(N(v^{*})) &= \\{u \\in V(G) \\colon \\{u, v^{*}\\} \\in E(G)\\},\\\\\nE(N(v^{*})) &= \\bigl\\{\\{u, v\\} \\in E(G) \\colon u, v \\in V(N(v^{*}))\\bigr\\}\n\\end{align*}\nfor every graph \\(G\\) and each \\(v^{*} \\in V(G)\\). Let \\(k\\) be a cardinal number. The vertex \\(v\\) of a graph \\(G\\) has \\emph{degree} \\(k\\) if \n\\[\nk = \\card V(N(v)).\n\\]\nThe degree of \\(v\\) will be denoted as \\(\\delta_{G}(v)\\) or simply as \\(\\delta(v)\\).\n\nA \\emph{path} is a finite graph \\(P\\) whose vertices can be numbered without repetitions so that\n\\begin{equation}\\label{e3.3}\nV(P) = \\{x_1, \\ldots, x_k\\} \\quad \\text{and} \\quad E(P) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{k-1}, x_k\\}\\}\n\\end{equation}\nwith \\(k \\geqslant 2\\). We will write \\(P = (x_1, \\ldots, x_k)\\) or \\(P = P_{x_1, x_k}\\) if \\(P\\) is a path satisfying \\eqref{e3.3} and said that \\(P\\) is a \\emph{path joining \\(x_1\\) and \\(x_k\\)}. A graph \\(G\\) is \\emph{connected} if for every two distinct vertices of \\(G\\) there is a path \\(P \\subseteq G\\) joining these vertices.\n\nA finite graph \\(C\\) is a \\textit{cycle} if there exists an enumeration of its vertices without repetition such that \\(V(C) = \\{x_1, \\ldots, x_n\\}\\)  and \n\\[\nE(C) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{n-1}, x_n\\}, \\{x_n, x_1\\}\\} \\quad \\text{with } n \\geqslant 3.\n\\]\n\n\\begin{definition}\\label{d9.8}\nA connected graph \\(T\\) with \\(V(T) \\neq \\varnothing\\) and without cycles is called a \\emph{tree}.\n\\end{definition}\n\nA tree \\(T\\) is \\emph{locally finite} if the inequality \\(\\delta(v) < \\infty\\) holds for every \\(v \\in V(T)\\).\n\nWe shall say that a tree \\(T\\) is a \\emph{star} if there is a vertex \\(c \\in V(T)\\), the \\emph{center} of \\(T\\), such that \\(c\\) and \\(v\\) are adjacent for every \\(v \\in V(T) \\setminus \\{c\\}\\).\n\nAn infinite graph \\(G\\) of the form \n\\[\nV(G) = \\{x_1, x_2 \\ldots, x_n, x_{n+1}, \\ldots\\}, \\quad E(G) = \\{\\{x_1, x_2\\}, \\ldots \\{x_n, x_{n+1}\\}, \\ldots\\},\n\\]\nwhere all \\(x_n\\) are assumed to be distinct, is called a \\emph{ray} \\cite{Diestel2017}. It is clear that every ray is a tree. A graph is \\emph{rayless} if it contains no rays.\n\n\\begin{proposition}\\label{p3.2}\nEvery infinite connected graph has a vertex of infinite degree or contains a ray.\n\\end{proposition}\n\nFor the proof see Proposition~8.2.1 in \\cite{Diestel2017}.\n\nThe following statement is well known for finite trees (see, for example, Proposition~4.1 \\cite{BM2008}) and is usually considered self-evident for infinite trees. \n\n\\begin{lemma}\\label{l9.14}\nIn each tree, every two different vertices are connected by exactly one path.\n\\end{lemma}\n\n\\begin{proof}\nIf \\(T\\) is an infinite tree and \\(v_1\\), \\(v_2\\) are two different vertices of \\(T\\) connected by some paths \\(P_1 \\subseteq T\\) and \\(P_2 \\subseteq T\\), then the graph \\(P_1 \\cup P_2\\) is a finite connected subgraph of \\(T\\). Since \\(T\\) does not have cycles, \\(P_1 \\cup P_2\\) is also acyclic. Hence, \\(P_1 \\cup P_2\\) is a finite tree and \\(P_1\\), \\(P_2\\) are paths connected \\(v_1\\) and \\(v_2\\) in \\(P_1 \\cup P_2\\). Thus, \\(P_1 = P_2\\) holds.\n\\end{proof}\n\nIn the next definition we introduce an analogue of convex hull for arbitrary trees.\n\n\\begin{definition}\\label{d3.6}\nLet \\(T\\) be a tree and let \\(A\\) be a nonempty subset of \\(V(T)\\). A subtree \\(H_A\\) of the tree \\(T\\) is the \\emph{hull} of \\(A\\) if \\(A \\subseteq V(H_A)\\) and, for every subtree \\(T^{*}\\) of \\(T\\), the tree \\(H_A\\) is a subtree of \\(T^{*}\\) whenever \\(A \\subseteq V(T^*)\\).\n\\end{definition}\n\nThus, \\(H_A\\) is the smallest subtree of \\(T\\) which contains \\(A\\). We want to make sure that for every tree \\(T\\) and each nonempty \\(A \\subseteq V(T)\\) the hull \\(H_A\\) is well defined and unique.\n\n\\begin{proposition}\\label{p3.7}\nLet \\(T\\) be a tree, \\(A\\) be a nonempty subset of \\(V(T)\\) and let \\(\\mathcal{F}_A\\) be the set of all subtrees \\(T^{*}\\) of \\(T\\) for which the inclusion \\(A \\subseteq V(T^{*})\\) holds. Then the graph \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is the hull of \\(A\\),\n\\begin{equation}\\label{p3.7:e1}\nH_A = \\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nIt is clear that \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is a subgraph of \\(T^{*}\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). In particular, \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is a subgraph of \\(T\\) because \\(T \\in \\mathcal{F}_A\\). Since \\(T\\) is a tree, the subgraph \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) contains no cycles. Hence, to prove \\eqref{p3.7:e1} it suffices to show that \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is connected.\n\nLet \\(u\\) and \\(v\\) be distinct vertices of \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) and let \\(P_{u, v}\\) be the path joining \\(u\\) and \\(v\\) in \\(T\\). Then \\(u\\) and \\(v\\) belong to \\(V(T^{*})\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). Using Lemma~\\ref{l9.14} we obtain \\(P_{u, v} \\subseteq T^{*}\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). From~\\eqref{e3.4} with \\(\\mathcal{F} = \\mathcal{F}_A\\) it follows that the path \\(P_{u, v}\\) is also a subgraph of \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\). Thus, \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is connected as required.\n\\end{proof}\n\n\\begin{example}\\label{ex3.7}\nLet \\(T\\) be an infinite tree, let a ray \\(R\\),\n\\[\nV(R) = \\{r_1, r_2, \\ldots, r_n, r_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{r_1, r_2\\}, \\ldots, \\{r_n, r_{n+1}\\}, \\ldots\\},\n\\]\nbe a subgraph of \\(T\\), and \\(v\\) be a vertex of \\(T\\) such that \\(v \\notin V(R)\\). We claim that there is a unique \\(n(v) \\in \\NN\\) such that \\(r_{n(v)}\\) is the only common vertex of \\(R\\) and of the path \\(P_{r_{n(v)}, v}\\) joining \\(r_{n(v)}\\) and \\(v\\) in \\(T\\),\n\\begin{equation}\\label{ex3.7:e1}\n\\{r_{n(v)}\\} = V(R) \\cap V(P_{r_{n(v)}, v}).\n\\end{equation}\nWe first prove the existence of some \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} and show that the graph \\(R \\cup P_{r_{n(v)}, v}\\) is the hull in \\(T\\) of the set \\(A \\stackrel{\\text{def}}{=} V(R) \\cup \\{v\\}\\),\n\\begin{equation*\nH_A = R \\cup P_{r_{n(v)}, v}.\n\\end{equation*}\nLet \\(v \\in V(T) \\setminus V(R)\\) be fixed. To find \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} it suffices to consider an arbitrary \\(u \\in V(R)\\) and the path \\((u^1, \\ldots, u^m) \\subseteq T\\) with \\(u^1 = u\\) and \\(u^m = v\\). Since \\(u \\in V(R)\\) and \\(v \\notin V(R)\\) hold, there is \\(m_1 \\in \\{1, \\ldots, m-1\\}\\) such that \\(u^{m_1} \\in V(R)\\) and \\(u^{j} \\notin V(R)\\) whenever \\(j \\in \\{m_1+1, \\ldots, m-1\\}\\). Consequently, there is \\(n_1 \\in \\NN\\) such that \\(r_{n_1} = u^{m_1}\\). If we set \\(n(v) \\stackrel{\\text{def}}{=} n_1\\), then \\eqref{ex3.7:e1} holds with \\(P_{r_{n(v)}, v} \\stackrel{\\text{def}}{=} (u^{m_1}, \\ldots, u^{m})\\). Since \\(P_{r_{n(v)}, v}\\) and \\(R\\) are connected and have the common vertex \\(r_{n(v)}\\), the union \\(R \\cup P_{r_{n(v)}, v}\\) is a subtree of \\(T\\). It is also clear that\n\\[\nA \\subseteq V(R \\cup P_{r_{n(v)}, v})\n\\]\nholds. Now Definition~\\ref{d3.6} implies that \\(H_A\\) is a subtree of \\(R \\cup P_{r_{n(v)}, v}\\). If we have \n\\[\nH_A \\neq R \\cup P_{r_{n(v)}, v},\n\\]\nthen there is \\(j \\in \\{m_1+1, \\ldots, m-1\\}\\) such that \\(u^{j} \\notin V(H_A)\\). Lemma~\\ref{l9.14} and \\(u^{j} \\notin V(H_A)\\) imply that \\(H_A\\) is disconnected, contrary to Definition~\\ref{d3.6}. From~Proposition~\\ref{p3.7} it follows that the hull \\(H_A\\) is unique. Consequently, the number \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} is also unique. \n\nIn what follows we will say that \\(R \\cup P_{r_{n(v)}, v}\\) is a \\emph{comb} in \\(T\\), \\(v\\) is a \\emph{tooth} of this comb, and \\(r_{n(v)}\\) is the \\emph{root} of the tooth \\(v\\) (see Figure~\\ref{fig2}).\n\\end{example}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.5cm}\n\\node at (5*\\dx, 3*\\dx) [label= below:\\(T\\)] {};\n\\draw (\\dx, 0) -- (6*\\dx, 0);\n\\draw [dashed] (6*\\dx, 0)  --  (7*\\dx, 0);\n\\node [solid node] at (\\dx, 0) [label= below:\\(r_1\\)] {};\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(r_2\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(r_3\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(r_4\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(r_5\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(r_6\\)] {};\n\n\\draw (3*\\dx, 0) -- (3*\\dx, 2*\\dx);\n\\node [solid node] at (3*\\dx, \\dx) [label= below right:\\(v_2\\)] {};\n\\node [solid node] at (3*\\dx, 2*\\dx) [label= below right:\\(v_5\\)] {};\n\n\\draw (3*\\dx, \\dx) -- ($(3*\\dx, \\dx) + (135:\\dx)$) node [solid node, label= left:\\(v_1\\)] {};\n\\draw (3*\\dx, \\dx) -- ($(3*\\dx, \\dx) + (45:\\dx)$) node [solid node, label= right:\\(v_3\\)] {};\n\\draw (3*\\dx, 2*\\dx) -- ($(3*\\dx, 2*\\dx) + (135:\\dx)$) node [solid node, label= left:\\(v_4\\)] {};\n\\draw (3*\\dx, 2*\\dx) -- ($(3*\\dx, 2*\\dx) + (45:\\dx)$) node [solid node, label= right:\\(v_6\\)] {};\n\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (135:\\dx)$) node [solid node, label= above:\\(v_9\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (45:\\dx)$) node [solid node, label= above:\\(v_{10}\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (-135:\\dx)$) node [solid node, label= below:\\(v_7\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (-45:\\dx)$) node [solid node, label= below:\\(v_{8}\\)] {};\n\\end{tikzpicture}\n\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.5cm}\n\\node at (5*\\dx, 2*\\dx) [label= below:\\(H_A\\)] {};\n\\draw (\\dx, 0) -- (6*\\dx, 0);\n\\draw [dashed] (6*\\dx, 0)  --  (7*\\dx, 0);\n\\node [solid node] at (\\dx, 0) [label= below:\\(r_1\\)] {};\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(r_2\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(r_3\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(r_4\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(r_5\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(r_6\\)] {};\n\n\\draw (3*\\dx, 0) -- (3*\\dx, 2*\\dx);\n\\node [solid node] at (3*\\dx, \\dx) [label= below right:\\(v_2\\)] {};\n\\node [solid node] at (3*\\dx, 2*\\dx) [label= below right:\\(v_5\\)] {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The hull \\(H_A\\) of \\(A = \\{v_5\\} \\cup \\{r_i \\colon i \\in \\NN\\}\\) is comb in \\(T\\), the vertex \\(r_3\\) is the root of the tooth \\(v_5\\) in this comb.} \n\\label{fig2}\n\\end{figure}\n\n\\begin{remark}\\label{r3.8}\nThus, we always assume that each comb has exactly one tooth with exactly one root. The combs with a large number of teeth are more often used in theory of ultrametric spaces and graph theory (see, for example, the Comb representation of compact ultrametric spaces \\cite{LamBr2017} or the Star-Comb Lemma~\\cite[Lemma~8.2.2]{Diestel2017}).\n\\end{remark}\n\nLet us recall the concept of labeled trees.\n\n\\begin{definition}\\label{d2.4}\nA labeled tree is a pair \\((T, l)\\), where \\(T\\) is a tree and \\(l\\) is a mapping defined on the set \\(V(T)\\).\n\nWe say that \\(T\\) is a \\emph{free tree} corresponding to \\((T, l)\\) and write \\(T = T(l)\\) instead of \\((T, l)\\). In what follows, we will consider only the nonnegative real-valued labelings \\(l\\colon V(T)\\to \\RR^{+}\\).\n\\end{definition}\n\nBefore introducing into consideration the concept of isomorphism of labeled trees, it is useful to remind the definition of isomorphism for free trees.\n\n\\begin{definition}\\label{d8.7}\nLet $T_1$ and $T_2$ be trees. A bijection $f\\colon V(T_1)\\to V(T_2)$ is an \\emph{isomorphism} of $T_1$ and $T_2$ if\n\\begin{equation*\n(\\{u,v\\} \\in E(T_1)) \\Leftrightarrow (\\{f(u),f(v)\\} \\in E(T_2))\n\\end{equation*}\nis valid for all $u$, $v \\in V(T_1)$. Two trees are \\emph{isomorphic} if there exists an isomorphism of these trees.\n\\end{definition}\n\nFor the case of labeled trees Definition~\\ref{d8.7} must be modified as follows.\n\n\\begin{definition}\\label{d2.5}\nLet \\(T_i=T_i(l_i)\\) be a labeled tree, \\(i = 1\\), \\(2\\). A mapping $f\\colon V(T_1) \\to V(T_2)$ is an isomorphism of \\(T_1(l_1)\\) and \\(T_2(l_2)\\) if it is an isomorphism of the free trees $T_1$ and $T_2$ and the equality\n\\begin{equation*\nl_2(f(v))=l_1(v)\n\\end{equation*}\nholds for every $v \\in V(T_1)$.\n\\end{definition}\n\n\\section{Ultrametrics generated by labeled trees}\\label{sec4}\n\nLet \\(T = T(l)\\) be a labeled tree. Following~\\cite{Dov2020TaAoG}, we define a mapping \\(d_l \\colon V(T) \\times V(T) \\to \\RR^{+}\\) as\n\\begin{equation}\\label{e11.3}\nd_l(u, v) = \\begin{cases}\n0 & \\text{if } u = v,\\\\\n\\max\\limits_{v^{*} \\in V(P)} l(v^{*}) & \\text{if } u \\neq v,\n\\end{cases}\n\\end{equation}\nwhere \\(P\\) is the path joining \\(u\\) and \\(v\\) in \\(T\\).\n\n\\begin{remark}\\label{r11.10}\nThe correctness of this definition follows from Lemma~\\ref{l9.14}.\n\\end{remark}\n\nTo formulate the first theorem of this section we recall the concept of \\emph{pseudoultrametric space}. Let \\(X\\) be a set. A symmetric mapping \\(d \\colon X \\times X \\to \\RR^{+}\\) is a \\emph{pseudoultrametric} on \\(X\\) if\n\\[\nd(x, x) = 0 \\quad \\text{and} \\quad d(x, y) \\leqslant \\max\\{d(x, z), d(z, y)\\}\n\\]\nhold for all \\(x\\), \\(y\\), \\(z \\in X\\). Every ultrametric is a pseudoultrametric, but a pseudoultrametric \\(d\\) on a set \\(X\\) is an ultrametric if and only if \\(d(x, y) > 0\\) holds for all distinct \\(x\\), \\(y \\in X\\).\n\nThe notion of isometries can be extended on pseudoultrametrics as follows. If \\((X, d)\\) and \\((Y, \\rho)\\) are pseudoultrametric spaces, then a mapping \\(\\Phi \\colon X \\to Y\\) is an isometry of \\((X, d)\\) and \\((Y, \\rho)\\) if \\(\\Phi\\) is bijective and\n\\[\nd(x, y) = \\rho(\\Phi(x), \\Phi(y))\n\\]\nholds for all \\(x\\), \\(y \\in X\\).\n\n\\begin{theorem}\\label{t11.9}\nLet \\(T = T(l)\\) be a labeled tree. Then \\((V(T), d_l)\\) is a pseudoultrametric space. The function \\(d_l\\) is an ultrametric on \\(V(T)\\) if and only if the inequality\n\\begin{equation*\n\\max\\{l(u_1), l(v_1)\\} > 0\n\\end{equation*}\nholds for every \\(\\{u_1, v_1\\} \\in E(T)\\).\n\\end{theorem}\n\nA proof of Theorem~\\ref{t11.9} can be obtained by simple modification of the proof of Proposition~3.2 \\cite{Dov2020TaAoG}.\n\n\\begin{proposition}\\label{l11.12}\nLet \\(T_1 = T_1(l_1)\\) and \\(T_2 = T_2(l_2)\\) be labeled trees and let \\(f \\colon V(T_1) \\to V(T_2)\\) be an isomorphism of these trees. Then the equality\n\\[\nd_{l_1}(u, v) = d_{l_2}(f(u), f(v))\n\\]\nholds for all \\(u\\), \\(v \\in V(T_1)\\).\n\\end{proposition}\n\n\\begin{proof}\nIt directly follows from Definition~\\ref{d2.5} and formula~\\eqref{e11.3}, because if \\((v_1, \\ldots, v_n)\\) is a path joining some distinct \\(v = v_1\\) and \\(u = v_n\\) in \\(T_1\\), then \\(f(u) \\neq f(v)\\) holds and \\((f(v_1), \\ldots, f(v_n))\\) is a path joining \\(f(v)\\) and \\(f(u)\\) in \\(T_2\\) and we have the equality \n\\[\n\\{l_1(v_1), \\ldots, l_1(v_n)\\} = \\{l_2(f(v_1)), \\ldots, l_2(f(v_n))\\}.\n\\]\n\\end{proof}\n\n\\begin{corollary}\\label{c11.6}\nLet \\(T_1 = T_1(l_1)\\) and \\(T_2 = T_2(l_2)\\) be isomorphic labeled trees. Then the pseudoultrametric spaces \\((V(T_1), d_{l_1})\\) and \\((V(T_2), d_{l_2})\\) are isometric.\n\\end{corollary}\n\nThe converse statement is not valid in general. The following example is a modification of Example~3.1 \\cite{Dov2020TaAoG}.\n\n\\begin{example}\\label{ex11.13}\nLet \\(V = \\{v_0, v_1, v_2, v_3, v_4\\}\\) be a five-point set, and let \\(S = S(l_S)\\) and \\(P = P(l_P)\\) be a labeled star with the center \\(v_0\\) and, respectively, a labeled path such that \\(V(S) = V(P) = V\\), \\(l_S(v_0) = l_P(v_0) = 1\\) and\n\\[\nl_S(v_i) + 1 = l_P(v_i) = 1\n\\]\nfor \\(i = 1\\), \\(\\ldots\\), \\(4\\) (see Figure~\\ref{fig11}). Then, for all distinct \\(u\\), \\(v \\in V\\), we have\n\\[\nd_{l_P}(u, v) = d_{l_S}(u, v) = 1.\n\\]\nThus, the ultrametric spaces \\((V, d_{l_P})\\) and \\((V, d_{l_S})\\) coincide, but \\(P(l_P)\\) and \\(S(l_S)\\) are not isomorphic as labeled trees or even as free trees.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1cm}\n\\node at (0, \\dx) [label= above:\\(S(l_S)\\)] {};\n\\node [solid node] at (0, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (\\dx, \\dx) [label= right:\\(0\\)] {};\n\\node [solid node] at (\\dx, -\\dx) [label= right:\\(0\\)] {};\n\\node [solid node] at (-\\dx, \\dx) [label= left:\\(0\\)] {};\n\\node [solid node] at (-\\dx, -\\dx) [label= left:\\(0\\)] {};\n\\draw (0, 0) -- (\\dx, \\dx);\n\\draw (0, 0) -- (\\dx, -\\dx);\n\\draw (0, 0) -- (-\\dx, \\dx);\n\\draw (0, 0) -- (-\\dx, -\\dx);\n\n\\node at (5*\\dx, \\dx) [label= above:\\(P(l_P)\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (7*\\dx, 0) [label= below:\\(1\\)] {};\n\\draw (3*\\dx, 0) -- (7*\\dx, 0);\n\\end{tikzpicture}\n\\end{center}\n\\caption{The star \\(S\\) and the path \\(P\\) are not isomorphic as trees, but the labelings \\(l_S\\) and \\(l_P\\) generate the same ultrametric on \\(V\\).}\n\\label{fig11}\n\\end{figure}\n\\end{example}\n\nExample~\\ref{ex11.13} shows that the properties of ultrametric spaces generated by different labeled trees can be the same. Thus, the following problem naturally arises.\n\n\\begin{problem}\\label{pr4.6}\nLet \\(\\mathcal{UP}\\) be the class of ultrametric spaces with a given property \\(\\mathcal{P}\\). What common properties do the labeled trees \\(T = T(l)\\) generating \\((V(T), d_l) \\in \\mathcal{UP}\\) have?\n\\end{problem}\n\nBelow we will consider this problem in the following cases:\n\\begin{itemize}\n\\item  \\(\\mathcal{P} =\\) completeness,\n\\item \\(\\mathcal{P} =\\) discreteness,\n\\item \\(\\mathcal{P} =\\) total boundedness,\n\\item \\(\\mathcal{P} =\\) discreteness + total boundedness,\n\\item \\(\\mathcal{P} =\\) compactness,\n\\end{itemize}\nand in each of these cases we find the corresponding characteristic properties of generating labeled trees.\n\nLet us start from the completeness. \n\nIn what follows we shall say that a labeling \\(l \\colon V(T) \\to \\mathbb{R}^{+}\\) is \\emph{non-degenerate} if the inequality\n\\[\n\\max\\{l(u), l(v)\\} > 0\n\\]\nholds for every \\(\\{u, v\\} \\in E(T)\\). \n\n\\begin{lemma}\\label{l4.7}\nLet \\(R\\) be a ray, \\(V(R) = \\{v_1, v_2, \\ldots, v_n, v_{n+1}, \\ldots\\}\\),\n\\begin{equation}\\label{l4.7:e1}\nE(R) = \\{\\{v_1, v_2\\}, \\ldots, \\{v_n, v_{n+1}\\}, \\ldots\\},\n\\end{equation}\nand let \\(l \\colon V(R) \\to \\RR^{+}\\) be a non-degenerate labeling. The sequence \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in the ultrametric space \\((V(R), d_l)\\) if and only if the limit relation\n\\begin{equation}\\label{l4.7:e2}\n\\lim_{n \\to \\infty} l(v_n) = 0\n\\end{equation}\nholds.\n\\end{lemma}\n\n\\begin{proof}\nBy Proposition~\\ref{p4.5}, \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(R), d_l)\\) if and only if\n\\begin{equation}\\label{l4.7:e3}\n\\lim_{n \\to \\infty} d_l(v_n, v_{n+1}) = 0.\n\\end{equation}\nUsing \\eqref{e11.3} and \\eqref{l4.7:e1} we can rewrite \\eqref{l4.7:e3} as\n\\[\n\\lim_{n \\to \\infty} \\left(\\max\\{l(v_n), l(v_{n+1})\\}\\right) = 0.\n\\]\nSince all \\(l(v_n)\\) belong to \\(\\RR^{+}\\), \\eqref{l4.7:e2} holds if and only if \n\\[\n\\limsup_{n \\to \\infty} l(v_n) = 0.\n\\]\nSimilarly, \\eqref{l4.7:e3} is equivalent to\n\\[\n\\limsup_{n \\to \\infty} (\\max\\{l(v_n), l(v_{n+1})\\}) = 0.\n\\]\nNow using the equality\n\\[\n\\limsup_{n \\to \\infty} l(v_n) = \\limsup_{n \\to \\infty} (\\max\\{l(v_n), l(v_{n+1})\\})\n\\]\nwe see that \\eqref{l4.7:e2} and \\eqref{l4.7:e3} are equivalent.\n\\end{proof}\n\nThe next lemma will be useful to prove Theorem~\\ref{t4.7}.\n\n\\begin{lemma}\\label{l4.8}\nLet \\(R = R(l)\\) be a labeled ray,\n\\[\nV(R) = \\{v_1, v_2, \\ldots, v_n, v_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{v_1, v_2\\}, \\ldots, \\{v_n, v_{n+1}\\}, \\ldots\\},\n\\]\nwith non-degenerate labeling. Then the sequence \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(R), d_l)\\) if and only if \\((v_n)_{n \\in \\NN}\\) contains a subsequence which is Cauchy in \\((V(R), d_l)\\).\n\\end{lemma}\n\n\\begin{proof}\nIf \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence, then \\((v_n)_{n \\in \\NN}\\) is a Cauchy subsequence of itself. \n\nConversely, let \\((v_{n_k})_{k \\in \\NN}\\),\n\\begin{equation*\n1 \\leqslant n_1 < n_2 < \\ldots < n_k < n_{k+1} < \\ldots,\n\\end{equation*}\nbe a Cauchy subsequence of \\((v_n)_{n \\in \\NN}\\). It is easy to see that, for every \\(m \\geqslant n_1\\), there is the unique \\(k(m) \\in \\NN\\) such that \n\\begin{equation}\\label{l4.8:e2}\nn_{k(m)} \\leqslant m < n_{k(m)+1}.\n\\end{equation}\n\nLet us denote by \\(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}}\\) the path joining \\(v_{n_{k(m)}}\\) and \\(v_{n_{k(m)+1}}\\) in \\(R\\). From \\eqref{l4.8:e2} it follows that \n\\begin{equation}\\label{l4.8:e3}\nv_m \\in V(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}}).\n\\end{equation}\nNow using \\eqref{e11.3} and \\eqref{l4.8:e3} we obtain\n\\begin{equation}\\label{l4.8:e4}\nl(v_m) \\leqslant \\max\\{l(v) \\colon v \\in V(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}})\\} = d_l(v_{n_{k(m)}}, v_{n_{k(m)+1}}).\n\\end{equation}\nIt is clear that the mapping \n\\[\n\\{n_1, n_1+1, \\ldots\\} \\ni m \\mapsto n_{k(m)} \\in \\NN\n\\]\nis increasing and satisfies the limit relation\n\\begin{equation}\\label{l4.8:e6}\n\\lim_{m \\to \\infty} n_{k(m)} = +\\infty.\n\\end{equation}\nProposition~\\ref{p4.5}, \\eqref{l4.8:e4} and \\eqref{l4.8:e6} imply\n\\[\n\\limsup_{m \\to \\infty} l(v_m) \\leqslant \\limsup_{m \\to \\infty} d_l(v_{n_{k(m)}}, v_{n_{k(m)+1}}) = 0.\n\\]\nThus, we have\n\\begin{equation*\n\\lim_{m \\to \\infty} l(v_m) = 0,\n\\end{equation*}\nbecause \\(l(v_m) \\in \\RR^{+}\\) for every \\(n \\in \\NN\\). Using the last limit relation we obtain that \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence by Lemma~\\ref{l4.7}.\n\\end{proof}\n\n\n\\begin{lemma}\\label{l4.6}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\) and let \\(T_1\\) be a subtree of \\(T\\) having the labeling \\(l_1 \\colon V(T_1) \\to \\RR^{+}\\) which is the restriction of \\(l\\) on \\(V(T_1)\\), \\(l_1 = l|_{V(T_1)}\\). Then the labeling \\(l_1\\) is also non-degenerate and the ultrametric \\(d_{l_1}\\) is the restriction of the ultrametric \\(d_l\\) on the set \\(V(T_1)\\), \\(d_{l_1} = d_l|_{V(T_1) \\times V(T_1)}\\).\n\\end{lemma}\n\n\\begin{proof}\nIt follows from formula \\eqref{e11.3}, Lemma~\\ref{l9.14} and the definition of trees and subtrees.\n\\end{proof}\n\n\\begin{theorem}\\label{t4.7}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.7:s1} The ultrametric space \\((V(T), d_l)\\) is complete.\n\\item \\label{t4.7:s2} For every ray \\(R \\subseteq T\\), \n\\begin{equation}\\label{t4.7:e2}\nV(R) = \\{x_1, x_2, \\ldots, x_n, x_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{n}, x_{n+1}\\}, \\ldots\\},\n\\end{equation}\nwe have the inequality\n\\begin{equation}\\label{t4.7:e1}\n\\limsup_{n \\to \\infty} l(x_n) > 0.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.7:s1} \\Rightarrow \\ref{t4.7:s2}\\). Let \\((V(T), d_l)\\) be a complete ultrametric space. We must show that condition~\\ref{t4.7:s2} is satisfied. Suppose contrary that there is a ray \\(R \\subseteq T\\) such that \\eqref{t4.7:e2} and \\(\\limsup_{n \\to \\infty} l(x_n) = 0\\) hold. Since all \\(l(x_n)\\) are nonnegative, the last equality holds if and only if \n\\begin{equation}\\label{t4.7:e3}\n\\lim_{n \\to \\infty} l(x_n) = 0.\n\\end{equation}\nFrom \\eqref{t4.7:e3} and \\eqref{e11.3} it follows that\n\\[\n\\lim_{n \\to \\infty} d_l(x_n, x_{n+1}) = \\lim_{n \\to \\infty} \\max\\{l(x_n), l(x_{n+1})\\} = 0,\n\\]\nbecause \\(x_n\\) and \\(x_{n+1}\\) are adjacent in \\(R\\) and \\(R \\subseteq T\\). Hence, by Proposition~\\ref{p4.5}, the sequence \\((x_n)_{n \\in \\mathbb{N}}\\) is a Cauchy sequence in the space \\((V(T), d_l)\\). \n\nNow, using condition~\\ref{t4.7:s1} and Definition~\\ref{d2.6}, we can find a point \\(v^{*} \\in V(T)\\) satisfying the equality\n\\begin{equation}\\label{t4.7:e4}\n\\lim_{n \\to \\infty} d_l(x_n, v^{*}) = 0.\n\\end{equation}\nIf there is \\(n_0 \\in \\NN\\) such that \\(v^{*} = x_{n_0} \\in V(R)\\), then, for every \\(n \\geqslant n_0 + 1\\), the path \\(P_{x_{n_0}, x_{n}}\\) joining \\(v^{*}\\) and \\(x_n\\) in \\(T\\) contains the edge \\(\\{x_{n_0}, x_{n_0+1}\\} \\in E(R)\\). Since \\(l \\colon V(T) \\to \\RR^{+}\\) is non-degenerate, \\eqref{e11.3} and \\(\\{x_{n_0}, x_{n_0+1}\\} \\in E(P_{x_{n_0}, x_{n_0+1}})\\) imply \n\\[\nd_l(v^{*}, x_n) \\geqslant \\max \\{l(x_{n_0}), l(x_{n_0+1})\\} > 0\n\\]\nfor every \\(n \\geqslant n_0 + 1\\), contrary to \\eqref{t4.7:e4}.\n\nSuppose now that \\(v^{*} \\in V(T) \\setminus V(R)\\). Then the hull \\(H_A\\) of the set\n\\[\nA \\stackrel{\\text{def}}{=} V(R) \\cup \\{v^{*}\\}\n\\]\nis a comb in \\(T\\) with the tooth \\(v^{*}\\) (see Definition~\\ref{d3.6} and Example~\\ref{ex3.7}). Write \\(u^{*}\\) for the root of \\(v^{*}\\) in \\(H_A\\). Since \\(u^{*} \\neq v^{*}\\) and \\(u^{*}\\) is the only common vertex of \\(R\\) and of the path \\(P_{u^{*}, v^{*}}\\) joining \\(u^{*}\\) and \\(v^{*}\\) in \\(T\\), we have \n\\[\nd_l(x_n, v^{*}) \\geqslant d_l(u^{*}, v^{*}) > 0\n\\]\nfor all \\(x_n \\in V(R)\\), contrary to~\\eqref{t4.7:e4}. Condition~\\ref{t4.7:s2} follows.\n\n\\(\\ref{t4.7:s2} \\Rightarrow \\ref{t4.7:s1}\\). Let \\ref{t4.7:s2} hold. We must show that the ultrametric space \\((V(T), d_l)\\) is complete. According to Definition~\\ref{d2.6}, the space is complete if every Cauchy sequence of points of this space is convergent.\n\nLet us consider an arbitrary Cauchy sequence \\((y_n)_{n \\in \\NN} \\subseteq V(T)\\) and define the range \\(A\\) of \\((y_n)_{n \\in \\NN}\\) as:\n\\[\n(v \\in A) \\Leftrightarrow (\\exists n \\in \\NN \\colon y_n = v).\n\\]\nIf \\(A\\) is finite, then the sequence \\((y_n)_{n \\in \\NN}\\) contains an infinite constant subsequence and, consequently, \\((y_n)_{n \\in \\NN}\\) is convergent by Proposition~\\ref{p2.7}. \n\nSuppose that \\(A\\) is infinite and denote by \\(H_A\\) the hull of \\(A\\) in \\(T\\). Let us prove that \\(H_A\\) is a rayless graph. \n\nIndeed, let a ray \\(R\\),\n\\[\nV(R) = \\{r_1, r_2, \\ldots, r_n, r_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{r_1, r_2\\}, \\ldots, \\{r_n, r_{n+1}\\}, \\ldots\\},\n\\]\nbe a subgraph of \\(H_A\\). If the intersection \\(A \\cap V(R)\\) is infinite, then the sequence \\((r_n)_{n \\in \\NN}\\) contains a subsequence \\((r_{n_k})_{k \\in \\NN}\\) which is Cauchy in \\((V(T), d_l)\\) and, consequently, in \\((V(R), d_l|_{V(R) \\times V(R)})\\). Using Lemma~\\ref{l4.8} and Lemma~\\ref{l4.6} we see that the sequence \\((r_n)_{n \\in \\NN}\\) is Cauchy in \\((V(R), d_l|_{V(R) \\times V(R)})\\). Now Lemma~\\ref{l4.6} implies that \\((r_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\). Moreover, the equality\n\\begin{equation*}\n\\lim_{n \\to \\infty} l(r_n) = 0\n\\end{equation*}\nholds by Lemma~\\ref{l4.7}. The last equality contradicts \\eqref{t4.7:e1} with \\((x_n)_{n \\in \\NN} = (r_n)_{n \\in \\NN}\\). \n\nThus, the intersection \\(A \\cap V(R)\\) is finite and \\(A\\) is infinite. We claim that there is an \\emph{infinite} subset \\(A^{*}\\) of the set \\(A \\setminus V(R)\\) which satisfies the condition:\n\\begin{enumerate}\n\\item[\\((i^{*})\\)] If \\(a_1\\), \\(a_2\\) are distinct points of \\(A^{*}\\), and\n\\[\nA_1 \\stackrel{\\text{def}}{=} V(R) \\cup \\{a_1\\}, \\quad A_2 \\stackrel{\\text{def}}{=} V(R) \\cup \\{a_2\\},\n\\]\nand \\(H_{A_1}\\), \\(H_{A_2}\\) are the corresponding hulls of \\(A_1\\) and of \\(A_2\\) in \\(T\\), then the roots \\(r(a_1)\\) and \\(r(a_2)\\) are distinct.\n\\end{enumerate}\n\n(Recall that the hulls \\(H_{A_1}\\) and \\(H_{A_2}\\) in \\(T\\) are some combs in \\(T\\), see Example~\\ref{ex3.7}). To find a desired \\(A^{*} \\subseteq A \\setminus V(R)\\) let us consider a number \\(N \\in \\NN\\) such that \n\\[\nA \\cap V(R) \\subseteq \\{r_1, r_2, \\ldots, r_{N}\\}\n\\]\nand suppose that, for every \\(a \\in A \\setminus V(R)\\), the root \\(r(a)\\) of the tooth \\(a\\) in the comb \\(H_{V(R) \\cup \\{a\\}}\\) belongs to the set \\(\\{r_1, r_2, \\ldots, r_{N}\\}\\). The graph \n\\[\nG_{R, N} \\stackrel{\\text{def}}{=} (r_1, \\ldots, r_{N}) \\cup \\bigcup_{a \\in A \\setminus V(R)} P_{a, r(a)},\n\\]\nwhere \\((r_1, \\ldots, r_{N})\\) is the path joining \\(r_1\\) and \\(r_N\\) in \\(R\\) and \\(P_{a, r(a)}\\) is the path joining \\(a\\) and \\(r(a)\\) in the comb \\(H_{V(R) \\cup \\{a\\}}\\), is a connected subgraph of \\(T\\) satisfying the conditions\n\\begin{equation*\nA \\subseteq V(G_{R, N}) \\quad \\text{and} \\quad r_n \\notin V(G_{R, N}) \n\\end{equation*}\nfor every \\(n \\geqslant N+1\\). Since \\(r_n \\in V(H_A)\\) holds for every \\(n \\in \\NN\\), the inclusion\n\\[\nV(H_A) \\subseteq V(G_{R, N})\n\\]\nis false, contrary to Proposition~\\ref{p3.7}. Hence, there is an element \\(y_{n_1}\\) of the sequence \\((y_{n})_{n \\in \\NN}\\) such that \\(y_{n_1} \\in A \\setminus V(R)\\) and \\(r(y_{n_1}) = r_{N_1}\\) and \\(N_1 > N\\). If for every \\(a \\in A \\setminus V(R)\\) the root \\(r(a)\\) belongs to \\(\\{r_1, \\ldots, r_N, \\ldots, r_{N_1}\\}\\), then repeating the above procedure with the graph \\(G_{R, N_1}\\) gives us \\(y_{n_2} \\in A \\setminus V(R)\\) with \\(r(y_{n_2}) = r_{N_2}\\) such that \\(n_2 > n_1\\) and \\(N_2 > N_1\\) and so on. \n\nLet us consider the sequence \\((y_{n_k})_{k \\in \\NN}\\), whose elements are inductively defined above, and write\n\\[\nA^{*} \\stackrel{\\text{def}}{=} \\{y_{n_k} \\colon k \\in \\NN\\}.\n\\]\nThen condition~\\((i^{*})\\) satisfies with this \\(A^{*}\\). Since \\((y_{n})_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\), the sequence \\((y_{n_k})_{k \\in \\NN}\\) is also Cauchy. It is easy to prove that the inequality\n\\begin{equation}\\label{t4.7:e12}\nd_l(r(y_{n_{k_1}}), r(y_{n_{k_2}})) \\leqslant d_l(y_{n_{k_1}}, y_{n_{k_2}})\n\\end{equation}\nholds for all \\(k_1\\), \\(k_2 \\in \\NN\\) (see Figure~\\ref{fig3}). Consequently, \\((r(y_{n_k}))_{k \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\).\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.3cm}\n\\draw (\\dx, 0) node [solid node, label= below:\\(r_1\\)] {} -- (2*\\dx, 0) node [solid node, label= below:\\(r_2\\)] {};\n\\draw [dashed] (2*\\dx, 0) -- (3*\\dx, 0) node [solid node] {};\n\\draw (3*\\dx, 0) -- (4*\\dx, 0) node [solid node, label= below:\\(r(y_{n_{k_1}})\\)] {} -- (5*\\dx, 0) node [solid node] {};\n\\draw (4*\\dx, 0) -- (4*\\dx, \\dx) node [solid node] {};\n\\draw [dashed] (4*\\dx, \\dx) -- (4*\\dx, 2*\\dx) node [solid node] {};\n\\draw (4*\\dx, 2*\\dx) -- (4*\\dx, 3*\\dx) node [solid node, label= right:\\(y_{n_{k_1}}\\)] {};\n\n\\draw [dashed] (5*\\dx, 0) -- (6*\\dx, 0) node [solid node] {};\n\\draw (6*\\dx, 0) -- (7*\\dx, 0) node [solid node, label= below:\\(r(y_{n_{k_2}})\\)] {} -- (8*\\dx, 0) node [solid node] {};\n\\draw (7*\\dx, 0) -- (7*\\dx, \\dx) node [solid node] {};\n\\draw [dashed] (7*\\dx, \\dx) -- (7*\\dx, 2*\\dx) node [solid node] {};\n\\draw (7*\\dx, 2*\\dx) -- (7*\\dx, 3*\\dx) node [solid node, label= right:\\(y_{n_{k_2}}\\)] {};\n\n\\draw [dashed] (8*\\dx, 0) -- (9*\\dx, 0);\n\\end{tikzpicture}\n\\end{center}\n\\caption{The path \\(\\bigl(r(y_{n_{k_1}}), \\ldots, r(y_{n_{k_2}})\\bigr)\\) is a subgraph of the path \\(\\bigl(y_{n_{k_1}}, \\ldots, r(y_{n_{k_1}}), \\ldots, r(y_{n_{k_2}}), \\ldots, y_{n_{k_2}}\\bigr)\\). It implies inequality~\\eqref{t4.7:e12}.}\n\\label{fig3}\n\\end{figure}\n\nNow using Lemma~\\ref{l4.8} and Lemma~\\ref{l4.6} we can prove that the sequence \\((r_n)_{n \\in \\NN}\\) of all vertices of the ray \\(R\\) is also a Cauchy sequence in \\((V(T), d_l)\\). Hence, by Lemma~\\ref{l4.7}, we have the equality\n\\[\n\\lim_{n \\to \\infty} l(r_n) = 0,\n\\]\nthat contradicts condition \\ref{t4.7:s2}. \n\n\\emph{Thus, the hull \\(H_A\\) is rayless}. Since \\(A\\) is infinite, \\(H_A\\) has a vertex \\(v^{*}\\) of infinite degree by Proposition~\\ref{p3.2}. To complete the proof it suffices to show that \\((y_n)_{n \\in \\NN}\\) converges to the point \\(v^{*}\\) in \\((V(T), d_l)\\),\n\\begin{equation*\n\\lim_{n \\to \\infty} d_l(y_n, v^{*}) = 0.\n\\end{equation*}\n\nLet us consider the subgraph \\(F_{v^{*}}\\) obtained from \\(H_A\\) by deleting the vertex \\(v^{*}\\),\n\\[\nV(F_{v^{*}}) = V(H_A) \\setminus \\{v^{*}\\}, \\quad E(F_{v^{*}}) = \\{\\{u, v\\} \\in E(H_A) \\colon u \\neq v^{*} \\neq v\\}.\n\\]\nSince \\(H_A\\) is a tree (as a subtree of \\(T\\)), \\(F_{v^{*}}\\) is a forest and, for every connected component \\(T'\\) of \\(F_{v^{*}}\\), there is a unique \\(p\\in V(N(v^{*}))\\) such that\n\\begin{equation}\\label{t4.7:e11}\np \\in V(T'),\n\\end{equation}\nwhere \\(N(v^{*})\\) is the neighborhood of \\(v^{*}\\) in \\(H_A\\). We will write \\(T^{p}\\) for the component \\(T'\\) if \\eqref{t4.7:e11} holds with \\(p\\in V(N(v^{*}))\\).\n\nIt is clear that\n\\begin{equation}\\label{t4.7:e13}\nH_A = \\left(\\bigcup_{p \\in V(N(v^{*}))} T^{p}\\right) \\cup S(v^{*})\n\\end{equation}\nholds, where \\(S(v^{*})\\) is the star with the center \\(v^{*}\\) and the vertex set \n\\[\nV(S(v^{*})) = V(N(v^{*})).\n\\]\n\nWe claim that the set \\(V(T^p) \\cap A\\) is nonempty for every \\(p \\in V(N(v^{*}))\\). Indeed, suppose that there is \\(p^{*} \\in V(N(v^{*}))\\) such that\n\\begin{equation}\\label{t4.7:e14}\nV(T^{p^{*}}) \\cap A = \\varnothing.\n\\end{equation}\nLet us denote by \\(S_{p^{*}}\\) the graph which is obtained from \\(S(v^{*})\\) by deleting of the vertex \\(p^{*}\\), i.e., \n\\[\nV(S_{p^{*}}) = V(S(v^{*})) \\setminus \\{p^{*}\\}\n\\]\nholds and\n\\[\n\\bigl(\\{x, y\\} \\in E(S_{p^{*}})\\bigr) \\Leftrightarrow \\bigl(\\{x, y\\} \\in E(S(v^{*})) \\text{ and } x \\neq p^{*} \\neq y\\bigr)\n\\]\nis valid for all \\(x\\), \\(y \\in V(N(v^{*}))\\). Then \\(S_{p^{*}}\\) is a star with the center \\(v^{*}\\). From \\eqref{t4.7:e14} it follows that the union\n\\[\n\\left(\\bigcup_{\\substack{p \\in V(N(v^{*}))\\\\ p \\neq p^{*}}} T^{p} \\right) \\cup S_{p^{*}}\n\\]\nis a subtree of \\(T\\) and the set \\(A\\) is a subset of the vertex set of this subtree. Hence, by Definition~\\ref{d3.6}, we have\n\\[\np^{*} \\notin V(H_A),\n\\]\ncontrary to \\eqref{t4.7:e13}. Using the conditions \n\\[\n\\delta_{H_A} (v^{*}) = \\infty \\quad \\text{and} \\quad V(T^{p}) \\cap A \\neq \\varnothing\n\\]\nfor every \\(p \\in V(N(v^{*}))\\), we can find a subsequence \\((y_{n_m})_{m \\in \\NN}\\) of the sequence \\((y_{n})_{n \\in \\NN}\\) such that for every \\(m \\in \\NN\\) there is \\(p(m) \\in V(N(v^{*}))\\) satisfying \\(y_{n_m} \\in T^{p}\\), and \\(p(m_1) \\neq p(m_2)\\) whenever \\(m_1 \\neq m_2\\). Then for every pair of distinct \\(m_1\\), \\(m_2 \\in \\NN\\) the path \\(P_{y_{n_{m_1}}, y_{n_{m_2}}}\\) joining \\(y_{n_{m_1}}\\) and \\(y_{n_{m_2}}\\) in \\(H_A\\) contains the vertex \\(v^{*}\\). Hence, by definition~\\eqref{e11.3}, we have the inequality\n\\begin{equation}\\label{t4.7:e15}\nd_l\\left(y_{n_{m_1}}, y_{n_{m_2}}\\right) \\geqslant \\max \\left\\{d_l\\left(y_{n_{m_1}}, v^{*}\\right), d_l\\left(v^{*}, y_{n_{m_2}}\\right)\\right\\}\n\\end{equation}\nwhenever \\(m_1\\), \\(m_2 \\in \\NN\\). By Proposition~\\ref{p2.7}, the sequence \\((y_{n})_{n \\in \\NN}\\) is convergent if \\((y_{n_m})_{m \\in \\NN}\\) is convergent. Using Proposition~\\ref{p4.5} and inequality \\eqref{t4.7:e15} with \\(n_{m_1} = n_{m}\\) and \\(n_{m_2} = n_{m+1}\\) we obtain\n\\[\n0 = \\lim_{m \\to \\infty} d_l\\left(y_{n_{m}}, y_{n_{m+1}}\\right) \\geqslant \\limsup_{m \\to \\infty} d_l\\left(y_{n_{m}}, v^{*}\\right),\n\\]\nthat implies \n\\[\n\\lim_{m \\to \\infty} d_l\\left(y_{n_{m}}, v^{*}\\right) = 0.\n\\]\nThus, \\((y_{n_m})_{m \\in \\NN}\\) is convergent to \\(v^{*}\\).\n\\end{proof}\n\n\n\nCondition~\\ref{t4.7:s2} of Theorem~\\ref{t4.7} is vacuously true for every rayless tree \\(T\\). Moreover, if \\(R \\subseteq T\\) is a ray satisfying \\eqref{t4.7:e2}, then it is easy to construct a non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\) such that \\eqref{t4.7:e1} does not hold. Thus, Theorem~\\ref{t4.7} implies the next corollary.\n\n\\begin{corollary}\\label{c4.7}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.7:s1} The ultrametric space \\((V(T), d_{l})\\) is complete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\item \\label{c4.7:s2} \\(T\\) is rayless.\n\\end{enumerate}\n\\end{corollary}\n\nRecall that a metric space \\((X, d)\\) is \\emph{discrete} if every point of \\(X\\) is isolated.\n\n\\begin{theorem}\\label{t4.11}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.11:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete.\n\\item \\label{t4.11:s2} For every \\(v^{*} \\in V(T)\\) we have either \\(l(v^{*}) > 0\\) or \\(l(v^{*}) = 0\\) and \n\\begin{equation}\\label{t4.11:e1}\n0 < \\inf_{u \\in V(N(v^{*}))} l(u),\n\\end{equation}\nwhere \\(N(v^{*})\\) is the neighborhood of \\(v^{*}\\) in \\(T\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.11:s1} \\Rightarrow \\ref{t4.11:s2}\\). Let \\((V(T), d_{l})\\) be a discrete ultrametric space. If \\ref{t4.11:s2} does not hold, then there is a vertex \\(v^{*}\\) such that \\(l(v^{*}) = 0\\) and \n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) = 0.\n\\]\nHence, there exists a sequence \\((u_n)_{n \\in \\NN} \\subseteq V(N(v^{*}))\\) such that\n\\[\n\\lim_{n \\to \\infty} l(u_n) = 0.\n\\]\nThe last limit relation, the equality \\(l(v^{*}) = 0\\), equality \\eqref{e11.3} and the definition of the neighborhoods of vertices of graph imply that\n\\begin{equation}\\label{t4.11:e0}\n\\lim_{n \\to \\infty} d(v^{*}, u_n) = 0\n\\end{equation}\nholds. Hence, \\(v^{*}\\) is an accumulation point in \\((V(T), d_{l})\\), contrary to statement \\ref{t4.11:s1}. \n\n\\(\\ref{t4.11:s2} \\Rightarrow \\ref{t4.11:s1}\\). Let \\ref{t4.11:s2} hold. Statement \\ref{t4.11:s1} is valid if \\(E(T) = \\varnothing\\). Indeed, in this case the vertex set of \\(T\\) is a single-point set \\(\\{v^{*}\\}\\). Thus, \\(V(N(v^{*})) = \\varnothing\\) holds and, consequently, we have\n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) = \\inf_{u \\in \\varnothing} l(u) = +\\infty,\n\\]\nthat implies \\eqref{t4.11:e1}. (We consider here the empty set \\(\\varnothing\\) as a subset of \\([-\\infty, \\infty]\\) and adopt the standard agreement on the supremum and infimum of empty set.)\n\nLet \\(E(T) \\neq \\varnothing\\) hold. \n\nSuppose that \\(v^{*}\\) is a vertex of \\(T\\) such that \\(l(v^{*}) > 0\\). Then \\eqref{e11.3} implies\n\\[\nd_l(u, v^{*}) \\geqslant l(v^{*})\n\\]\nfor every \\(u \\in V(T) \\setminus \\{v^{*}\\}\\). Hence, \\(v^{*}\\) is an isolated point of \\((V(T), d_{l})\\).\n\nLet us consider now the case when \\(v^{*} \\in V(T)\\) has the zero label, \\(l(v^{*}) = 0\\), and assume that we have \\(\\delta_{T}(v^{*}) < \\infty\\). Since the inequality \\(\\max\\{l(u), l(v^{*})\\} > 0\\) holds for every \\(u \\in V(N(v^{*}))\\), from \\(0 < \\delta_{T}(v^{*}) < \\infty\\) follows the inequality\n\\begin{equation}\\label{t4.11:e3}\n\\min_{u \\in V(N(v^{*}))} l(u) > 0. \n\\end{equation}\nLet \\(u_0 \\in V(T) \\setminus \\{v^{*}\\}\\). Then there is \\(u^{*} \\in V(P_{v^{*}, u_0})\\) such that\n\\begin{equation}\\label{t4.11:e2}\nu^{*} \\in V(N(v^{*})).\n\\end{equation}\nNow from \\eqref{e11.3} and \\eqref{t4.11:e2} it follows that\n\\[\nd_l(v^{*}, u_0) = \\max_{u \\in V(P_{v^{*}, u_0})} l(u) \\geqslant d_l(v^{*}, u^{*}) \\geqslant \\min_{u \\in V(N(v^{*}))} l(u) > 0.\n\\]\nHence, \\(v^{*}\\) is an isolated point of \\((V(T), d_{l})\\). \n\nUsing inequality \\eqref{t4.11:e1} instead of \\eqref{t4.11:e3} and repeating the above arguments, we obtain that \\(v^{*}\\) is also isolated for the case \\(l(v^{*}) = 0\\) and \\(\\delta_{T}(v^{*}) = \\infty\\). Thus, the ultrametric space \\((V(T), d_{l})\\) is discrete. \n\\end{proof}\n\n\\begin{corollary}\\label{c4.12}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.12:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\item \\label{c4.12:s2} The tree \\(T\\) is locally finite.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proposition}\\label{p4.13}\nLet \\(T\\) be a tree. Then the ultrametric space \\((V(T), d_{l})\\) contains a dense discrete subset for every non-degenerate \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{proposition}\n\n\\begin{proof}\nLet \\(l \\colon V(T) \\to \\RR^{+}\\) be non-degenerate. It was shown in the second part of the proof of Theorem~\\ref{t4.11} that \\(v \\in V(T)\\) is an isolated point of the ultrametric space \\((V(T), d_l)\\) if at least one of the conditions:\n\\begin{itemize}\n\\item \\(\\delta(v) < \\infty\\),\n\\item \\(l(v) > 0\\),\n\\item \\(\\delta(v) = \\infty\\), \\(l(v) = 0\\) and \\(\\inf_{u \\in V(N(v^{*}))} l(u) > 0\\)\n\\end{itemize}\nis valid. Arguing as in the first part of the proof of Theorem~\\ref{t4.11}, we can show that, for every \\(v\\) satisfying conditions \\(\\delta(v) = \\infty\\) and \n\\[\nl(v) = 0 = \\inf_{u \\in V(N(v^{*}))} l(u),\n\\]\nthere is a sequence \\((u_n)_{n \\in \\NN} \\subseteq V(N(v))\\) which converges to \\(v\\) (see~\\eqref{t4.11:e0}). Now it suffices to note that all elements of this sequence are isolated points of \\((V(T), d_l)\\) because \\(l(v) = 0\\) holds and \\(l\\) is non-degenerate.\n\\end{proof}\n\n\nThe following result gives us the necessary and sufficient conditions under which the ultrametric space \\((V(T), d_l)\\) is totally bounded.\n\n\\begin{theorem}\\label{t11.8}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t11.8:s1} The ultrametric space \\((V(T), d_{l})\\) is totally bounded.\n\\item \\label{t11.8:s2} The set \n\\begin{equation}\\label{t11.8:e1}\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation}\nis finite for every \\(\\varepsilon > 0\\) and the inequality \\(\\delta_{T}(v) < \\infty\\) holds whenever \\(l(v) > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t11.8:s1} \\Rightarrow \\ref{t11.8:s2}\\). Let \\ref{t11.8:s1} hold. Suppose that the set \\(V_{\\varepsilon}\\) is infinite for some \\(\\varepsilon > 0\\). Using the definition of \\(d_{l}\\) (see~\\eqref{e11.3}) it is easy to prove the inequality\n\\begin{equation*\nd_{l}(v_1, v_2) \\geqslant \\varepsilon\n\\end{equation*}\nfor all distinct \\(v_1\\), \\(v_2 \\in V_{\\varepsilon}\\). Hence, the subspace \\((V_{\\varepsilon}, d_{l}|_{V_{\\varepsilon} \\times V_{\\varepsilon}})\\) of totally bounded ultrametric space \\((V(T), d_{l})\\) is not totally bounded, contrary to Corollary~\\ref{c2.17}. Thus, \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\).\n\nAssume now that \\(T\\) contains a vertex \\(v^*\\) of infinite degree, \\(\\delta (v^*) = \\infty\\), and \\(l(v^{*}) > 0\\) holds.\n\nLet \\(N(v^{*})\\) be the neighborhood of \\(v^{*}\\). The equality \\(\\delta_{T}(v^*) = \\infty\\) implies that \\(V(N(v^{*}))\\) has an infinite cardinality. For all distinct \\(u_1\\), \\(u_2 \\in V(N(v^{*}))\\) the unique path joining \\(u_1\\) and \\(u_2\\) in \\(T\\) has the form \\((u_1, v^{*}, u_2)\\). Hence, \n\\[\nd_l(u_1, u_2) \\geqslant l(v^{*}) > 0\n\\]\nholds by \\eqref{e11.3}. It implies that the ultrametric space \\((V(N(v^{*})), d_l|_{V(N(v^{*})) \\times V(N(v^{*}))})\\) is not totally bounded, contrary to \\ref{t11.8:s1}. \n\n\\(\\ref{t11.8:s2} \\Rightarrow \\ref{t11.8:s1}\\). Let \\ref{t11.8:s2} hold. We must show that \\((V(T), d_{l})\\) is totally bounded. It is clear that \\((V(T), d_{l})\\) is totally bounded for finite \\(T\\). Let us consider the case when \\(T\\) is infinite.\n\nBy Proposition~\\ref{p2.11}, the ultrametric space \\((V(T), d_{l})\\) is totally bounded if every sequence of vertices of \\(T\\) contains a Cauchy subsequence. Let us consider a sequence \\((u_j^0)_{j \\in \\mathbb{N}}\\) of pairwise distinct points of \\(V(T)\\). Let \\((\\varepsilon_i)_{i \\in \\mathbb{N}}\\) be a decreasing sequence of strictly positive real numbers such that \\(\\lim_{i \\to \\infty} \\varepsilon_i = 0\\). Statement~\\ref{t11.8:s2} implies that the set \\(V_{\\varepsilon_{1}}\\) is finite. Write \\(G^{1}\\) for the subgraph of \\(T\\) induced by \\(V(T) \\setminus V_{\\varepsilon_{1}}\\), i.e., \\(V(G^{1}) = V(T) \\setminus V_{\\varepsilon_{1}}\\) and\n\\[\n\\bigl(\\{u, v\\} \\in E(G^{1})\\bigr) \\Leftrightarrow \\bigl(u, v \\in V(G^{1}) \\text{ and } \\{u, v\\} \\in E(T)\\bigr).\n\\]\nEvery connected component of \\(G^{1}\\) is a tree. Since \\(V_{\\varepsilon_{1}}\\) is a finite set, and \\(\\delta_{T}(v) < \\infty\\) holds for every \\(v \\in V_{\\varepsilon_{1}}\\), the number of these components are finite. Since \\(T\\) is an infinite tree, there is an infinite subtree \\(T^{1}\\) of \\(T\\) with \\(l(v^{1}) < \\varepsilon_{1}\\) for all \\(v^{1} \\in V(T^{1})\\) and such that \\((u_{j}^{1})_{j \\in \\mathbb{N}} \\subseteq V(T^{1})\\) holds for an infinite subsequence \\((u_{j}^{1})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{0})_{j \\in \\mathbb{N}} \\subseteq V(T)\\). Write \\(u^{1} = u_1^1\\). \n\nLet us consider the subgraph \\(G^{2}\\) of \\(T^{1}\\) induced by \\(V(T^{1}) \\setminus V_{\\varepsilon_{2}}\\). As above, the finiteness of \\(V_{\\varepsilon_{2}}\\) implies the existence of an infinite tree \\(T^{2} \\subseteq T^{1}\\) and a subsequence \\((u_{j}^{2})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{1})_{j \\in \\mathbb{N}} \\subseteq V(T^{1})\\) for which \\((u_{j}^{2})_{j \\in \\mathbb{N}} \\subseteq V(T^{2})\\) holds. Let us write \\(u^{2} = u_1^2\\). \n\nBy induction, for every \\(i \\geqslant 2\\), we find an infinite connected component \\(T^{i+1}\\) of the subgraph \\(G^{i+1}\\) of \\(T^{i}\\) induced by \\(V(T^{i}) \\setminus V_{\\varepsilon_{i+1}}\\) and a subsequence \\((u_{j}^{i+1})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{i})_{j \\in \\mathbb{N}}\\) such that\n\\begin{equation}\\label{t11.8:e8}\n(u_{j}^{i+1})_{j \\in \\mathbb{N}} \\subseteq V(T^{i+1}).\n\\end{equation}\nWrite \\(u^{i+1}\\) for the first element \\(u_{1}^{i+1}\\) of this sequence.\n\nLet us consider now the sequence \\((u^{i})_{i \\in \\mathbb{N}}\\). It is clear that \\((u^{i})_{i \\in \\mathbb{N}}\\) is a subsequence of the original sequence \\((u_j^0)_{j \\in \\mathbb{N}}\\). From~\\eqref{t11.8:e8} it follows that\n\\[\nl(u^{i+1}) < \\varepsilon_{i+1}\n\\]\nholds for every \\(i \\in \\mathbb{N}\\). The last inequality and the limit relation \\(\\lim_{i \\to \\infty} \\varepsilon_i = 0\\) imply\n\\begin{equation}\\label{t11.8:e9}\n\\lim_{i \\to \\infty} l(u^{i}) = 0.\n\\end{equation}\nMoreover, since for every \\(i \\geqslant 2\\) the points \\(u^{i}\\) and \\(u^{i+1}\\) are vertices of the tree \\(T^{i}\\) and the inequality \\(l(v) \\leqslant \\varepsilon_{i}\\) holds for every \\(v \\in V(T^{i})\\), formula \\eqref{e11.3} implies the inequality\n\\[\nd_{l}(u^{i}, u^{i+1}) \\leqslant l(u^{i-1})\n\\]\nfor every \\(i \\geqslant 2\\). Now using Proposition~\\ref{p4.5} and limit relation~\\eqref{t11.8:e9} we obtain that \\((u^{i})_{i \\in \\mathbb{N}}\\) is a Cauchy sequence.\n\nThe same reasons show that \\((u_j^0)_{j \\in \\mathbb{N}}\\) contains a Cauchy subsequence whenever \\((u_j^0)_{j \\in \\mathbb{N}}\\) contains an infinite subsequence of pairwise distinct members.\n\nTo complete the proof, we note only that if all subsequences of pairwise distinct members of a sequence are finite, then there is an infinite constant subsequence of that sequence and this constant subsequence obviously is a Cauchy sequence. \n\\end{proof}\n\n\\begin{corollary}\\label{c4.8}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. If the ultrametric space \\((V(T), d_l)\\) is totally bounded, then the set \\(V(T)\\) is at most countable.\n\\end{corollary}\n\n\\begin{proof}\nIt suffices to show that the inequality \n\\begin{equation}\\label{c4.8:e1}\n\\delta_{T}(v^*) \\leqslant \\aleph_0\n\\end{equation}\nholds for every vertex \\(v^*\\) of \\(T\\), where \\(\\aleph_0\\) is the cardinality of \\(\\mathbb{N}\\).\n\nLet \\((V(T), d_l)\\) be totally bounded and let \\(v^*\\) be a vertex of \\(T\\). Inequality \\eqref{c4.8:e1} follows directly from Theorem~\\ref{t11.8} if \\(l(v^*) > 0\\). Suppose that \\(l(v^*) = 0\\) holds. Then for every \\(\\{u, v^*\\} \\in E(T)\\) we have the inequality \\(l(u) > 0\\). Consequently, the vertex set \\(V(N(v^{*}))\\) satisfies the inclusion \n\\begin{equation}\\label{c4.8:e2}\nV(N(v^{*})) \\subseteq \\bigcup_{n \\in \\mathbb{N}} V_{1\/n},\n\\end{equation}\nwhere \\(V_{1\/n}\\) is defined by \\eqref{t11.8:e1} with \\(\\varepsilon = 1\/n\\). By Theorem~\\ref{t11.8}, \\(V_{1\/n}\\) is finite for every \\(n \\in \\mathbb{N}\\). Hence, \\(\\bigcup_{n \\in \\mathbb{N}} V_{1\/n}\\) is at most countable. Now inequality \\eqref{c4.8:e1} follows from \\eqref{c4.8:e2}.\n\\end{proof}\n\n\n\\begin{theorem}\\label{t11.16}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\label{t11.16:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete and totally bounded.\n\\item \\label{t11.16:s2} The tree \\(T\\) is locally finite and the set\n\\begin{equation*\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation*}\nis finite for every \\(\\varepsilon > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t11.16:s1} \\Rightarrow \\ref{t11.16:s2}\\). Let \\ref{t11.16:s1} hold. Then the set \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\) by Theorem~\\ref{t11.8}. \n\nAssume now that \\(T\\) contains a vertex \\(v^*\\) of infinite degree, \\(\\delta (v^*) = \\infty\\). Then, using Theorem~\\ref{t11.8} again, we obtain the equality\n\\begin{equation}\\label{t11.16:e4}\nl(v^{*}) = 0.\n\\end{equation}\nBy Theorem~\\ref{t4.11}, equality~\\eqref{t11.16:e4} and discreteness of \\((V(T), d_{l})\\) imply that there is \\(\\varepsilon^{*} > 0\\) such that\n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) \\geqslant \\varepsilon^{*}.\n\\]\nHence, we have the inclusion \\(V(N(v^{*})) \\subseteq V_{\\varepsilon^{*}}\\). It was shown above that \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\). Consequently, \\(V(N(v^{*}))\\) is also finite as a subset of a finite set, contrary to \\(\\delta_T (v^{*}) = \\infty\\).\n\n\\(\\ref{t11.16:s2} \\Rightarrow \\ref{t11.16:s1}\\). Let \\ref{t11.16:s2} hold. Then \\(T\\) is locally finite and, consequently, the ultrametric space \\((V(T), d_{l})\\) is discrete by Corollary~\\ref{c4.12}. To complete the proof, it suffices to note that \\((V(T), d_{l})\\) is totally bounded by Theorem~\\ref{t11.8}.\n\\end{proof}\n\nCorollary~\\ref{c4.12} and Theorem~\\ref{t11.16} imply the following.\n\n\\begin{corollary}\\label{c4.15}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{c4.15:s1} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\item \\label{c4.15:s2} The ultrametric space \\((V(T), d_{l})\\) is discrete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nIf \\ref{c4.15:s1} holds, then \\(T\\) is locally finite by Theorem~\\ref{t11.16}, that implies \\ref{c4.15:s2} by Corollary~\\ref{c4.12}.\n\nConversely, suppose that \\ref{c4.15:s2} holds. Then, using Corollary~\\ref{c4.12} again, we see that \\(T\\) is locally finite. If \\(T\\) is finite, then \\ref{c4.15:s1} is trivially valid. Every infinite locally finite tree has countable vertex set. Thus, all vertices of \\(T\\) can be numbered in a sequence \\((v_n)_{n \\in \\NN}\\) of pairwise different points and we can define a labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) as\n\\[\nl_1(v_n) = \\frac{1}{n}\n\\]\nfor every \\(n \\in \\NN\\). Then \\(l_1\\) is a non-degenerate labeling and \\(T(l_1)\\) satisfies condition \\ref{t11.16:s2} of Theorem~\\ref{t11.16}. Hence, \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\end{proof}\n\nUsing Corollaries~\\ref{c4.12} and \\ref{c4.15} we obtain.\n\n\\begin{corollary}\\label{c4.18}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item\\label{c4.18:s1} \\(T\\) is locally finite.\n\\item\\label{c4.18:s2} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{theorem}\\label{t4.18}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.18:s1} The ultrametric space \\((V(T), d_{l})\\) is compact.\n\\item \\label{t4.18:s2} The tree \\(T\\) is rayless and the set\n\\begin{equation*\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation*}\nis finite for every \\(\\varepsilon > 0\\) and the inequality \\(\\delta_{T}(v) < \\infty\\) holds whenever \\(l(v) > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.18:s1} \\Rightarrow \\ref{t4.18:s2}\\). Let \\((V(T), d_{l})\\) be a compact ultrametric space. Every compact metric space is totally bounded and complete by Corollary~\\ref{c2.9}. Hence, by Theorem~\\ref{t11.8}, for every \\(\\varepsilon > 0\\) the set \\(V_{\\varepsilon}\\) is finite, and \\(\\delta_{T}(v) < \\infty\\) holds for all vertices \\(v\\) with \\(l(v) > 0\\). \n\nSuppose that there is a ray \\(R \\subseteq T\\). Then the completeness of \\((V(T), d_{l})\\) and Theorem~\\ref{t4.7} imply the existence of \\(\\varepsilon^{*} > 0\\) and of a sequence \\((x_n)_{n \\in \\NN}\\) of pairwise distinct vertices of \\(R\\) such that\n\\[\n\\limsup_{n \\to \\infty} l(x_n) \\geqslant \\varepsilon^{*} > 0.\n\\]\nThus, the set \\(\\{v \\in V(R) \\colon l(v) \\geqslant \\frac{1}{2} \\varepsilon^{*}\\}\\) is infinite, contrary to the finiteness of the set \\(V_{\\frac{\\varepsilon^{*}}{2}}\\) which contains \\(\\{v \\in V(R) \\colon l(v) \\geqslant \\frac{1}{2} \\varepsilon^{*}\\}\\).\n\n\\(\\ref{t4.18:s2} \\Rightarrow \\ref{t4.18:s1}\\). Let \\ref{t4.18:s2} hold. Then \\ref{t4.18:s1} follows from the Spatial Criterion (Corollary~\\ref{c2.9}) and Theorems~\\ref{t4.7}, \\ref{t11.8}.\n\\end{proof}\n\n\\begin{theorem}\\label{t7}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t7:s1} \\(T\\) is rayless, and the set \\(V(T)\\) is at most countable, and, for every \\(\\{x, y\\} \\in E(T)\\), at least one from the degrees \\(\\delta(x)\\) and \\(\\delta(y)\\) is finite.\n\\item \\label{t7:s2} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is compact.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t7:s1} \\Rightarrow \\ref{t7:s2}\\). Let \\ref{t7:s1} hold. Let us define the subsets \\(V^F\\) and \\(V^I\\) of \\(V(T)\\) as\n\\begin{gather*}\nV^F \\stackrel{\\text{def}}{=} \\bigl\\{v \\in V(T) \\colon \\delta(v) \\text{ is finite}\\bigr\\}, \\quad\nV^I \\stackrel{\\text{def}}{=} \\bigl\\{v \\in V(T) \\colon \\delta(v) \\text{ is infinite}\\bigr\\}.\n\\end{gather*}\nThe set \\(V(T)\\) is at most countable. Consequently, there is a labeling \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) such that:\n\\begin{itemize}\n\\item the set \\(V_{\\varepsilon} = \\{v \\in V(T) \\colon l_1(v)  \\geqslant \\varepsilon\\}\\) is finite for every \\(\\varepsilon > 0\\); \n\\item the inequality \\(l_1(u) > 0\\) holds for every \\(u \\in V^F\\); \n\\item the equality \\(l_1(w) = 0\\) holds for every \\(w \\in V^I\\). \n\\end{itemize}\n\nThese properties of \\(l_1\\) and statement \\ref{t7:s1} imply the inequality\n\\[\n\\max \\bigl\\{l_1(x), l_1(y)\\bigr\\} > 0\n\\]\nfor every \\(\\{x, y\\} \\in E(T)\\). Hence, \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) is a non-degenerate labeling. Thus, \\((V(T), d_{l_1})\\) is compact by Theorem~\\ref{t4.18}.\n\n\n\\(\\ref{t7:s2} \\Rightarrow \\ref{t7:s1}\\). Let \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) be a non-degenerate labeling for which the ultrametric space \\((V(T), d_{l_1})\\) is compact. Then, from Theorem~\\ref{t4.18} it follows that \\(T\\) is rayless. Moreover, since every compact metric space is totally bounded, the set \\(V(T)\\) is at most countable by Corollary~\\ref{c4.8}.\n\nTo complete the proof it is enough to show that the inequality\n\\begin{equation*\n\\min\\bigl\\{\\delta(u), \\delta(v)\\bigr\\} < \\infty\n\\end{equation*}\nholds for every \\(\\{u, v\\} \\in E(T)\\). Assume to the contrary that there exists \\(\\{u, v\\} \\in E(T)\\) such that\n\\[\n\\delta(u) = \\delta(v) = \\aleph_0.\n\\]\nSince \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) is a non-degenerate labeling, \\(\\{u, v\\} \\in E(T)\\) implies\n\\[\n\\max\\bigl\\{l_1(u), l_1(v)\\bigr\\} > 0.\n\\]\nWithout loss of generality, we may assume that \\(l_1(v) > 0\\). The last inequality, the inequality \\(\\delta(v) > 0\\) and Theorem~\\ref{t4.18} imply that \\((V(T), d_{l_1})\\) is not compact, contrary to the definition of \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\).\n\\end{proof}\n\nCorollary~\\ref{c4.7} and Theorem~\\ref{t7} imply the following.\n\n\\begin{corollary}\\label{c4.23}\nLet \\(T\\) be a tree. If there is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is compact, then \\((V(T), d_{l})\\) is complete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{corollary}\n\n\\begin{remark}\\label{r4.21}\nIt is interesting to compare Corollary~\\ref{c4.23} with the following theorem: ``A metrizable topological space \\((X, \\tau)\\) is compact if and only if every metric generated the topology \\(\\tau\\) is complete.'' This result was proved by Niemytzki and Tychonoff in 1928 \\cite{NTFM1928}. There is also an ultrametric modification of Niemytzki---Tychonoff theorem recently obtained by Yoshito Ishiki~\\cite{Isha2020}.\n\\end{remark}\n\nThe next corollary follows from Proposition~\\ref{p3.2} and Theorems~\\ref{t11.16}, \\ref{t4.18}.\n\n\\begin{corollary}\\label{c4.20}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.20:s1} There are non-degenerate labelings \\(l_1 \\colon V(T) \\to \\RR^{+}\\) and \\(l_2 \\colon V(T) \\to \\RR^{+}\\) such that \\((V(T), d_{l_1})\\) is a compact ultrametric space and \\((V(T), d_{l_2})\\) is a discrete totally bounded ultrametric space.\n\\item \\label{c4.20:s2} \\(T\\) is a finite tree.\n\\end{enumerate}\n\\end{corollary}\n\n\\section{Some examples and conjectures}\n\nLet us consider examples of totally bounded non-complete ultrametric spaces, and compact ultrametric spaces generated by labeled trees having infinitely many vertices of infinite degree. To construct these examples, we use the \\emph{gluing} a given set of labeled trees to a fixed labeled tree. \n\nLet \\(\\mathcal{F} = \\{T_i(l_i) \\colon i \\in I\\}\\) be a nonempty set of labeled trees \\(T_i = T_i(l_{i})\\) for which\n\\begin{equation}\\label{e4.34}\nV(T_{i_1}) \\cap V(T_{i_2}) = \\varnothing\n\\end{equation}\nholds for all distinct \\(i_1\\), \\(i_2 \\in I\\), and let \\(T^{*} = T^{*}(l^{*})\\) be a labeled tree such that \\(V(T_i) \\cap V(T^{*})\\) is a single-point set \\(\\{v_i\\}\\),\n\\begin{equation}\\label{e4.35}\nV(T_i) \\cap V(T^{*}) = \\{v_i\\}\n\\end{equation}\nfor every \\(i \\in I\\). Let us suppose also\n\\begin{equation}\\label{e4.36}\nl^{*}(v_i) = l_{i}(v_i)\n\\end{equation}\nfor every \\(i \\in I\\) if \\(v_i\\) satisfies \\eqref{e4.35}. Then we define the gluing \\(\\mathcal{F}\\) to \\(T^{*}\\) as a labeled graph \\(T = T(l)\\) with \n\\begin{equation}\\label{e4.37}\nV(T) \\stackrel{\\mathrm{def}}{=} V(T^{*}) \\cup \\left(\\bigcup_{i \\in I} V(T_i)\\right), \\quad E(T) \\stackrel{\\mathrm{def}}{=} E(T^{*}) \\cup \\left(\\bigcup_{i \\in I} E(T_i)\\right)\n\\end{equation}\nand \\(l \\colon V(T) \\to \\RR\\) such that\n\\begin{equation}\\label{e4.38}\nl(v) = \\begin{cases}\nl^{*}(v) & \\text{if } v \\in V(T^{*}),\\\\\nl_{i}(v) & \\text{if } v \\in V(T_i) \\text{ for some } i \\in I.\n\\end{cases}\n\\end{equation}\nUsing equalities \\eqref{e4.34}--\\eqref{e4.38} one can simply show that \\(T = T(l)\\) is a well-defined labeled tree for which the labeling \\(l\\) is non-degenerate if and only if all labelings \\(l_i\\), \\(i \\in I\\), and \\(l^{*}\\) are non-degenerate.\n\n\\begin{example}\\label{ex11.10}\nLet \\(R^{*} = R^{*}(l^{*})\\) be a labeled ray such that \\(V(R^{*}) = \\mathbb{N}\\) and\n\\[\n\\bigl(\\{m, n\\} \\in E(R^{*})\\bigr) \\Leftrightarrow \\bigl(|m-n|=1\\bigr)\n\\]\nfor all \\(m\\), \\(n \\in \\mathbb{N}\\) and, let the equality\n\\[\nl^{*}(m) = \\begin{cases}\n\\frac{1}{m} & \\text{if \\(m\\) is odd},\\\\\n0 & \\text{if \\(m\\) is even}\n\\end{cases}\n\\]\nhold for each \\(m \\in \\NN\\). Moreover, for every even \\(m \\in \\mathbb{N}\\) we define a labeled star \\(S_m(l_m)\\) with a center \\(c_m = m\\) and suppose that the following conditions hold:\n\\[\n\\quad V(S_m) \\cap \\NN = \\{m\\}, \\quad l_m(c_m) = 0,\n\\]\nand the restriction \\(l_m|_{V(S_m) \\setminus \\{c_m\\}}\\) of \\(l_m\\) on the set \\(V(S_m) \\setminus \\{c_m\\}\\) is a bijection to the set \\(\\{\\frac{1}{mn} \\colon n \\in \\mathbb{N}\\}\\); and\n\\[\nV(S_{m_1}) \\cap V(S_{m_2}) = \\varnothing\n\\]\nholds for all distinct even \\(m_1\\), \\(m_1 \\in \\mathbb{N}\\). Then we can consider the labeled tree \\(T = T(l)\\) obtained by gluing the set \n\\[\n\\{S_m(l_m) \\colon m \\in \\NN \\text{ and } m \\text{ is even}\\}\n\\]\nto the labeled ray \\(R^{*}(l^{*})\\) (see Figure~\\ref{fig10}). The ultrametric space \\((V(T), d_l)\\) is totally bounded by Theorem~\\ref{t11.8} but not complete by Theorem~\\ref{t4.7}.\n\\end{example}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{2cm}\n\\draw (0, 0) -- (6*\\dx, 0);\n\\node [solid node] at (0, 0) [label= below:\\(\\frac{1}{1}\\)] {};\n\\node [solid node] at (\\dx, 0) [label= below:\\(0\\)] {};\n\\node at (0, \\dx) [label= above:\\(T(l)\\)] {};\n\\draw (\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{2}\\)] {};\n\\draw (\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{4}\\)] {};\n\\draw (\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{6}\\)] {};\n\\draw (\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{2n}\\)] {};\n\\draw (\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (\\dx, \\dx) [label= above:\\(S_1(l)\\)] {};\n\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(\\frac{1}{3}\\)] {};\n\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(0\\)] {};\n\\draw (3*\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{4}\\)] {};\n\\draw (3*\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{8}\\)] {};\n\\draw (3*\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{12}\\)] {};\n\\draw (3*\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{4n}\\)] {};\n\\draw (3*\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (3*\\dx, \\dx) [label= above:\\(S_2(l)\\)] {};\n\n\\node [micro node] at (3.3*\\dx, 0) {};\n\\node [micro node] at (3.5*\\dx, 0) {};\n\\node [micro node] at (3.7*\\dx, 0) {};\n\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(\\frac{1}{2m-1}\\)] {};\n\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(0\\)] {};\n\\draw (5*\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{2m}\\)] {};\n\\draw (5*\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{4m}\\)] {};\n\\draw (5*\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{6m}\\)] {};\n\\draw (5*\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{2mn}\\)] {};\n\\draw (5*\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (5*\\dx, \\dx) [label= above:\\(S_m(l)\\)] {};\n\n\\node [micro node] at (5.3*\\dx, 0) {};\n\\node [micro node] at (5.5*\\dx, 0) {};\n\\node [micro node] at (5.7*\\dx, 0) {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The tree \\(T\\) has \\(\\aleph_0\\) vertices with degree \\(\\aleph_0\\) and the ultrametric space \\((V(T), d_l)\\) is totally bounded but not complete.} \n\\label{fig10}\n\\end{figure}\n\n\\begin{example}\\label{ex4.21}\nLet \\(\\mathbb{P}\\) be the set of all integer prime numbers \\(p \\geqslant 2\\) and let \\(S^{*} = S^{*}(l^{*})\\) be the labeled star with the vertex set \\(V(S^{*}) = \\{p \\in \\mathbb{P}\\} \\cup \\{0\\}\\), and the center \\(c^{*} = 0\\), and the labeling \\(l^{*} \\colon V(S^{*}) \\to \\RR^{+}\\) for which \\(l^*(0) = 0\\) and \\(l^{*} (p) = 1\/p\\) hold for all \\(p \\in \\mathbb{P}\\).\n\nFor every \\(p \\in \\mathbb{P}\\) we define a labeled star \\(S_p = S_p(l_p)\\) with a center \\(c_p\\) such that:\n\\[\nV(S_p) \\setminus \\{c_p\\} = \\{p^{n} \\colon n \\in \\mathbb{N}\\},\n\\]\nand \n\\[\nc_p \\notin \\bigcup_{p \\in \\mathbb{P}} \\bigl(V(S_p) \\setminus \\{c_p\\}\\bigr) \\cup V(S^{*});\n\\]\n\\[\nl_p(v) = \\begin{cases}\n0 & \\text{if } v = c_p,\\\\\np^{-n} & \\text{if } v = p^{n} \\text{ for some } n \\in \\NN;\n\\end{cases}\n\\]\nand \\(c_{p_1} \\neq c_{p_2}\\) for all distinct \\(p_1\\), \\(p_2 \\in \\mathbb{P}\\). Then we obtain \\(V(S^{*}) \\cap V(S_p) = \\{p\\}\\), and \\(l^{*} (p) =  l_p(p) = 1\/p\\), and \\(\\delta_{S^{*}} (c^{*}) = \\delta_{S_p} (c_p) = \\aleph_0\\) for every \\(p \\in \\mathbb{P}\\) .\n\nLet us consider the labeled tree \\(T = T(l)\\) which is obtained by gluing the set \\(\\{S_p(l_p) \\colon p \\in \\mathbb{P}\\}\\) to \\(S^{*}(l^{*})\\) (see Figure~\\ref{fig1}), then the ultrametric space \\((V(T), d_l)\\) is compact by Theorem~\\ref{t4.18}.\n\\end{example}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.2cm}\n\\node at (3*\\dx, 3*\\dx) [label= above:\\(T(l)\\)] {};\n\\node [solid node] at (0, 0) [label= below:\\(0\\)] {};\n\n\\def\\dphi{20}\n\\def\\xx{2}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= above:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{80}\n\\def\\xx{3}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= above left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= left:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below right:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{140}\n\\def\\xx{5}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above right:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{200}\n\\def\\xx{7}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= below left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\node [micro node] at (220:3*\\dx) {};\n\\node [micro node] at (230:3*\\dx) {};\n\\node [micro node] at (240:3*\\dx) {};\n\n\\def\\dphi{280}\n\\def\\xx{p}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= below:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= right:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= left:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= left:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\node [micro node] at (310:3*\\dx) {};\n\\node [micro node] at (320:3*\\dx) {};\n\\node [micro node] at (330:3*\\dx) {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The tree \\(T\\) has \\(\\aleph_0\\) vertices with degree \\(\\aleph_0\\) and the ultrametric space \\((V(T), d_l)\\) is compact.} \n\\label{fig1}\n\\end{figure}\n\nThe following simple example shows that the class of finite ultrametric spaces, which are representable in the form \\((V(T), d_l)\\), is a proper subclass of all finite ultrametric spaces.\n\n\\begin{example}\\label{ex11.11}\nLet \\(V = \\{v_1, v_2, v_3, v_4\\}\\) be a four-point set and let an ultrametric \\(d \\colon V \\times V \\to \\RR^{+}\\) satisfy the equalities\n\\begin{gather}\\label{ex11.11:e1}\nd(v_1, v_2) = d(v_3, v_4) = 1\\\\\n\\intertext{and}\n\\label{ex11.11:e2}\nd(v_1, v_3) = d(v_1, v_4) = d(v_2, v_3) = d(v_2, v_4) = 2.\n\\end{gather}\nLet \\(T = T(l)\\) be a labeled tree such that \\(V(T) = V\\). We claim that the ultrametric spaces \\((V, d)\\) and \\((V(T), d_l)\\) are not isometric for any non-degenerate labeling \\(l\\). Indeed, if there is a non-degenerate \\(l \\colon V(T) \\to \\RR^{+}\\) for which \\((V, d)\\) and \\((V(T), d_l)\\) are isomorphic, then from~\\eqref{e11.3} and \\eqref{ex11.11:e1} it follows that\n\\[\n\\max_{1 \\leqslant i \\leqslant 4} l(v_i) \\leqslant 1.\n\\]\nThe last inequality and \\eqref{e11.3} imply that \\(d_l(v, u) \\leqslant 1\\) holds for all \\(u\\), \\(v \\in V\\), contrary to \\eqref{ex11.11:e2}.\n\\end{example}\n\nIt seems to be interesting to get a purely metric characterization of ultrametric spaces generated by labeled trees.\n\n\\begin{conjecture}\\label{con11.1}\nLet \\((X, d)\\) be a discrete nonempty totally bounded ultrametric space. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item\\label{con11.1:s1} There is a labeled tree \\(T = T(l)\\) such that \\((V(T), d_l)\\) and \\((X, d)\\) are isometric.\n\\item\\label{con11.1:s2} For every \\(B \\in \\BB_{X}\\), there are \\(c \\in X\\) and \\(r > 0\\) such that\n\\[\nB = \\{x \\in X \\colon d(x, c) = r\\} \\cup \\{c\\}\n\\]\ni.e., the ball \\(B\\) is the sphere \\(S(c, r) = \\{x \\in X \\colon d(x, c) = r\\}\\) with the added center \\(c\\).\n\\end{enumerate}\n\\end{conjecture}\n\nIn conclusion, we formulate a simple conjecture linking the properties of Cauchy sequences in \\((V(T), d_l)\\) with the structure of the hull of the range sets of these sequences (cf. Lemma~\\ref{l4.8}).\n\n\\begin{conjecture}\\label{con5.2}\nLet \\(T\\) be a tree and let \\((v_n)_{n \\in \\NN}\\) be a sequence of distinct vertices of \\(T\\). Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item The hull of the range set of \\((v_n)_{n \\in \\NN}\\) is a union of a ray with some finite tree.\n\\item For every non-degenerate \\(l \\colon V(T) \\to \\RR\\) the existence of Cauchy subsequence in \\((v_n)_{n \\in \\NN}\\) implies that \\((v_n)_{n \\in \\NN}\\) is also a Cauchy sequence.\n\\end{enumerate}\n\\end{conjecture}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGeometry is central to our understanding of the world. Viewed as something real or\nas a convention to describe reality, it has evolved with our thinking and will continue to do so \\cite{Jammer1954}. After the centuries needed to conceive what we  today call the Euclidean space, the advent of calculus and  non-Euclidean geometries on the one side, and the investigation of electromagnetic phenomena on the other, have produced a radical transformation of our concept of space. The turning point is marked by Riemann's trial lecture \\emph{On the Hypotheses That Lie at The Foundations of Geometry} \\cite{Riemann,Pesic2007}.  In just over half a century, Riemann's vision has become the backbone of the general theory of relativity and therefore of our understanding of spacetime. This has stimulated an enormous interest in differential geometry, aiming to reconcile Einstein's theory of gravity with the other fundamental interactions and with the quantum world. \n\nThe purpose of this work is far less ambitious. \nI am going to investigate a differential geometry based solely on the minimal assumption that the distance of every point from itself is equal to zero. Thus, besides renouncing to the \ninfinitesimal Pythagorean distance formula as in Finsler geometry \\cite{Rund2012,BaoChernShen2012}, I will also drop Riemann's residual hypotheses of homogeneity and symmetry. Surprisingly, this leads to a logically simple construction, very close in spirit and techniques to standard Riemannian geometry.\nIn particular, I will show that the infinitesimal Pythagorean distance formula reemerges, without the need of being postulated, as a first approximation to \\emph{almost every}  geometry that is invariant with respect to direction reversal. More in general, \\emph{almost every} geometry turns out to be approximated by a one-parameter family of homogeneous Riemann-Randers metrics  \\cite{Randers1941}. The parameter is related to a geometric conservation law. Homogeneity is recovered for each value of the parameter, but not as a whole. The connection and the curvature associated to a generic line element  emerge from the identification of geodesic equations with auto parallel equations, leading to a univocal generalisation of the Christoffel-Levi-Civita connection and of the Riemannian curvature tensor.\n\nIf, following the spirit of general relativity we apply this geometry to spacetime, \nwe effortlessly  find ourselves with what could well be the unified theory of the classical electromagnetic and gravitational fields, long sought by Weyl, Kaluza, Eddington, Einstein, Schr\\\"{o}dinger and many others \\cite{Goenner2004,Goenner2014,Tonnelat1966}. Compared to other attempts, the identification of geometrical objects with physical quantities is logical and completely straightforward, like the imposition of the possible field equations. Moreover, some fundamental yet puzzling facts, such as the hierarchy between electromagnetic and gravitational interactions, their different attractive\/repulsive nature and CPT invariance, naturally emerge from the geometrical framework. Whether this geometry will bring any progress in our comprehension of the classical spacetime it is too early to say. \nNonetheless, the conceptual simplicity and economy that it offers in the formulation of both the electromagnetic and gravitational theories seems to me undeniable.\\\\\n\nIn section \\ref{hypotheses} I will review Riemann's hypotheses on geometry together with  their revisions, from the indefiniteness of the line element in pseudo-Riemanninan geometry, to line elements that are arbitrary homogeneous functions of direction in Finsler geometry. A differential geometry based on a line element only constrained by the assumption that the distance of every point from itself equals zero is presented in section \\ref{relax}. Section \\ref{spacetime}\ndescribes the possible applications to spacetime. Two appendices respectively discuss the geometrical interpretation of the gravitational field in general relativity and offer an elementary introduction to nonlinear connections.\n\n\\section{Riemann's hypotheses on geometry}\\label{hypotheses}\nIn his celebrated 1854 trial lecture \\emph{On the Hypotheses That Lie at The Foundations of Geometry}, Riemann introduces a general analytic approach to geometry by extending Gauss's work on curved surfaces to ``$n$-fold extended manifolds\"\\footnote{If not otherwise stated, all the quotations throughout this paper are from Riemann's \\emph{Ueber die Hypothesen, welche der Geometrie zu Grunde liegen} \\cite{Riemann} in the English translation by M. Spivack with slight revisions by P. Pesich \\cite{Pesic2007}.}. These are defined as spaces parameterized by $n$ independent coordinates $x^\\mu$, $\\mu=1,2, ...,n$. \nThe investigation of metric relations of such spaces is approached by measuring the length of lines, so that the distance between two points is implicitly defined as the length of the shortest line joining them, a geodesic line. Primarily, Riemann assumes that as ``measurement requires independence of quantity from position [...] the length of lines is independent of their configuration, so that every line can be measured by every other\", a concept that Einstein will better rephrase by stating that ``if two line segments are found to be equal at one time and at some place, they are equal always and everywhere\"\\cite{Einstein1921}. Next, to set up a mathematical expression for the length of lines, Riemann treats the problem ``only under certain restrictions\" by assuming\n\\begin{description}\n  \\item[\\emph{Differentiability}:] The lines whose length is to be determined are differentiable, so that the problem reduces ``to setting up a general expression for the line element $ds$'' connecting two infinitesimally adjacent points  $x=\\left(x^1, x^2, ..., x^n\\right)$ and $x+dx=\\left(x^1+dx^1, x^2+dx^2, ...,x^n+dx^n\\right)$, ``an expression which will involve the quantities $x$ and the quantities $dx$.'' Therefore,  $ds=f(x,dx)$.\n  \\item[\\emph{Homogeneity}:] If all increments $dx^\\mu$ are  increased by the same ratio, $ds$ must also be increased by the same ratio. Equivalently, $ds$ must be  ``an arbitrary homogeneous function of the first degree in the quantities $dx$''. That is to say, $f(x,k\\, dx)=k f(x,dx)$ with $k>0$.\n  \\item[\\emph{Symmetry}:] The line element ``remains the same when all quantities $dx$ change sign.'' Equivalently, $ds$ is an even function of the increments $dx^\\mu$. \nIn symbolic notation, $f(x,-dx)=f(x,dx)$.\n\\end{description}\nAt this point, in order to find the explicit form for $ds$, Riemann seeks  ``an expression for the $(n-1)$-fold extended manifold which are everywhere equidistant from the origin of the line element, i.e.\\ [...] a continuous function which distinguishes them from one another. This must either decrease or increase in all directions from the origin;\" he assumes that ``it increases in all directions and therefore has a minimum at the origin. Then if  its first and second differential quotients are finite, the first order must vanish and the second order cannot be negative;\" he eventually assumes that ``it is always positive.\" From this he deduces that\n\\begin{description}\n \\item[\\emph{Quadratic restriction}:] The line element $ds$ equals ``the square root of an everywhere positive homogeneous function of the second degree in the quantities $dx$, in which the coefficients are continuous functions of the quantities $x$.''\\footnote{The expression \\emph{quadratic restriction} to describe Riemann's choice of the infinitesimal Pythagorean distance formula \nfor the line element was introduced by Chern in \\cite{Chern1996}.}  \n\\end{description}\nNamely, the infinitesimal expression of the Pythagorean distance formula in arbitrary coordinates. In modern notation\n\\begin{equation}\n\\label{Rds}\nds=\\sqrt{g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}},\n\\end{equation}\nwhere $g_{\\mu\\nu}(x)$ are regular functions of the coordinates and the summation over repeated indices is understood. Clearly, the fourth hypothesis implies the previous three but is not implied by them. In particular, the assumption that the ``second differential quotient[...]'' is finite, does not imply that it is nonvanishing. From this Riemann realised that the minimum can also be produced by an appropriate root of a  biquadratic or higher-order expression. In fact, he notes that ``[t]he next simplest case would perhaps include the manifolds in which the line element can be expressed as the fourth root of a differential expression of the fourth degree.\"\\\\\n\nRemarkably, Riemann makes clear that these hypotheses ``are not logically necessary,\" and ``one can therefore investigate their likelihood, which is certainly very great within the bounds of observation, and afterwards decide on the legitimacy of extending them beyond the bounds of observation\". As is well known, the `certainty within the bounds of observation' soon called for a revision of Riemann's fourth hypothesis. This is implicitly contained in Minkowski's geometrical reformulation of Einstein's theory of special relativity and was later made explicit in the general theory.  The unification of space and time into a single four dimensional continuum that ``sprang from the soil of experimental physics\"\\cite{Minkowski1908} required the revision of the \\emph{quadratic restriction} allowing the indefiniteness of the quadratic form appearing in $ds$ or, equivalently, allowing pseudo-Euclidean geometries besides the Euclidean one on infinitesimal scales.\n\nAnother call for revision came some thirty years later from Randers's observation that ``the most characteristic property of the physical world is the uni-direction of time-like intervals\" so that  ``the perfect symmetry in any direction for any coordinate interval does not seem quite appropriate for the application to physical space-time\"\\cite{Randers1941}. In other words, the simultaneous parity and time reversal symmetry PT is not a fundamental symmetry of nature, becoming such only if accompanied by the simultaneous transformation of charge conjugation C. Accordingly, it would be preferable to have something like CPT symmetry embodied in the fundamental texture of spacetime geometry, rather than PT \\emph{symmetry}.\n\nRiemannian geometry without the \\emph{quadratic restriction}, and also with or without the \\emph{symmetry} hypothesis, has been developed since the 1920s by Finsler, Bernwald, Buselman, Cartan, Rund, Chern and many others \\cite{Rund2012,BaoChernShen2012}. For historical reasons it is known as Finsler geometry. The construction, that requires ``no essential new ideas\"\\cite{Chern1996}, is far from being logically simple, leading to a structure supporting at least three different curvatures and five torsion tensors. \nMoreover, ``while Riemannian geometry can be handled, elegantly and efficiently, by tensor analysis [...] Finsler geometry [...] needs more than one space [...] on which tensor analysis does not fit well\"\\cite{Chern1996}. Starting from the 1930s attempts have been made to apply Finsler geometry to spacetime, both in the effort to extend the theory of relativity or to unify gravity with electromagnetism \\cite{Goenner2004,Goenner2014,Tonnelat1966,Rutz1993,Asanov2012,Pfeifer&Wohlfarth2012}. However, in spite of the extreme richness in structure, it ``seems that this particular generalisation of Riemannian geometry is not able to lead to a correct implementation of the electromagnetic field\"\\cite{Stephenson1957}, nor to provide a logically attractive extension of general relativity. Riemann's observation that the ``investigation of this general class [...] require[s] no essential different principles [...] and throw[s] proportionally little new light on the study of space\" seems to be fully justified, after all. \n\nAt present, \\emph{homogeneity}  is  considered a fundamental technical requirement in Riemannian geometry. In fact, it heuristically justifies the chain of identities\n\\begin{equation*}\n\\int ds=\\int{\\textstyle\\frac{1}{d\\tau}} f(x,dx)d\\tau=\\int\nf\\left(x(\\tau),{\\textstyle\\frac{dx}{d\\tau}(\\tau)}\\right)d\\tau\n\\end{equation*} \nthat allows to explicitly evaluate the line integral of $ds$  and, when extrapolated to $k=0$,  implies that the distance of every point from itself equals zero. This latter really seems  to be an unrenounceable requirement for a distance function. In fact, in the abstract theory of metric spaces it is one of the axioms that a distance function must satisfy. However, nothing like homogeneity is generally required for a distance function. It is therefore tempting to consider the possibility of relaxing Riemann's \\emph{homogeneity} hypothesis to the weaker assumption of\n\\begin{description}\n  \\item[\\emph{Indiscernibility}:] If all increments $dx^{\\mu}$ are vanishing, $ds$ must also vanish. That is to say, $f(x,0)=0$.\\footnote{That the distance of every point from itself is zero, is generally assumed together with its inverse statement (if the distance between two points is zero then the two points coincide) under the name of \\emph{identity of indiscernibles}. By analogy, I will call the direct statement alone \\emph{indiscernibility} as the line element does not allows to perceive a point as two distinct entities. Given the existence of hyperbolic geometry the inverse statement does not  fit as an axiom for geometry. }\n\\end{description}\nTo the best of my knowledge, no one so far has considered Riemannian geometry without the hypotheses of \\emph{homogeneity} and \\emph{symmetry}.\n\n\\section{Relaxing homogeneity and symmetry}\\label{relax}\nHereafter I will consider Riemann-type geometries uniquely based on the hypotheses of \\emph{differentiability} and \\emph{inderniscibility}. For the sake of simplicity I will assume differentiability in its strongest sense, i.e.\\ analyticity.\n\\subsection*{Line element}\nThe keystone that supports the construction of Riemann-type geometries without the hypothesis of \\emph{homogeneity},  is the fact that the line element can again be taken in the general form  \n\\begin{equation}\n\\label{ds}\nds=f\\left(x(\\tau),{\\textstyle\\frac{dx}{d\\tau}(\\tau)}\\right)d\\tau\n\\end{equation} \nwith $f(x,\\dot{x})$ an arbitrary analytic function of the coordinates $x$ and of the directions $\\dot{x}=\\frac{dx}{d\\tau}$ and with the \\emph{proviso} that \\emph{the parameter $\\tau$ is implicitly determined by the geodesic equations}. Shortly, I will show that the geodesic equations do actually determine the geodesic parameter $\\tau$. In complete generality I will proceed by series-expanding $ds$ about the point $\\dot{x}=0$. The line element is then rewritten as\n\\begin{equation}\n\\label{tds}\nds=\\left(g_\\mu\\dot{x}^\\mu+\\frac{1}{2}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu+\n\\frac{1}{3!}g_{\\mu\\nu\\xi}\\dot{x}^\\mu\\dot{x}^\\nu\\dot{x}^\\xi+...+\n\\frac{1}{k!}g_{\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+...\\right)d\\tau\n\\end{equation}\nwhere the zero order term is missing because of the hypothesis of \\emph{indiscernibility}  and the symmetric tensors $g_\\mu(x)=\\frac{\\partial f}{\\partial \\dot{x}^\\mu}(x,0)$, $g_{\\mu\\nu}(x)=\\frac{\\partial^2 f}{\\partial \\dot{x}^\\mu\\partial \\dot{x}^\\nu}(x,0)$, ..., $g_{\\mu_1...\\mu_k}(x)=\\frac{\\partial^k f}{\\partial \\dot{x}^{\\mu_1}...\\partial \\dot{x}^{\\mu_k}}(x,0)$ ..., are regular functions of the coordinates. It is important to emphasize that these tensors are not independent geometrical objects but only a different way of encoding all the information contained in the line element $ds$. The variation of the line integral of $ds$ produces now the geodesic equations in terms of the symmetric tensors $g_{\\mu_1...\\mu_k}(x)$  and of their derivatives. However, we do  not need to write them down explicitly in order to see that they do determine the geodesic parameter. As $ds$ does not depend explicitly on $\\tau$, from the analogy with analytical mechanics we can immediately conclude that the geodesic equations admit the first integral\n\\begin{equation}\n\\label{fi} \ng_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu+ \\frac{2}{3}g_{\\mu\\nu\\xi}\\dot{x}^\\mu\\dot{x}^\\nu\\dot{x}^\\xi+...+ \\frac{2}{k(k-2)!}g_{\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+...=2E,\\end{equation}\nwith $E$ an arbitrary constant. This identity implicitly determines $d\\tau$ in terms of the differentials $g_{\\mu_1...\\mu_k}dx^{\\mu_1}...dx^{\\mu_k}$ and the constant $E$. Consequently, it determines $\\tau$ up to an irrelevant additive constant. \\\\\n\nFor example, if $g_{\\mu\\nu}dx^\\mu dx^\\nu>0$ and $E>0$, as is always the case in standard Riemannian geometry, the series-inversion of equation (\\ref{fi}) yields \\begin{eqnarray}\\label{dtau}\nd\\tau=\\frac{1}{2\\varepsilon}\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}+\n\\frac{g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}}{3g_{\\mu\\nu}dx^\\mu dx^\\nu}+\\hskip118pt\n&&  \\nonumber \\\\[5pt]\n+\\varepsilon \\left(\\frac{ g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}{4\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^3}}-\\frac{\\left(g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}\\right)^2}{3\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^5}}\\right)\n+...&& \n\\end{eqnarray}\nwhere, for simplicity, I have set $\\varepsilon=\\sqrt{E\/2}$. The substitution of \n(\\ref{dtau}) into (\\ref{tds}) yields the explicit form of the line element in terms of  ``the quantities $x$ and the quantities  $dx$\" as sought by Riemann\n\\begin{eqnarray}\n\\label{gdsExp}\n\\frac{ds}{\\varepsilon}=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\n\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}+\n\\hskip193pt\n&&  \\nonumber \\\\[5pt]\n-\\varepsilon^2 \n\\left(\\frac{g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}{6\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^3}}-\\frac{2\\left(g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}\\right)^2}{9\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^5}}\\right)\n+...&& \n\\end{eqnarray}\nQuite surprisingly, the most general line element for  Riemann-type geometries without the hypothesis of \\emph{homogeneity} turns out to be a one-parameter family of homogeneous line elements. \n\nFor geometries that are symmetric under direction reversal, where  $g_{\\mu}$ as well as all odd-order tensors $g_{\\mu_1\\mu_2... \\mu_{2k-1}}$ vanishes identically, the (rescaled) line element reduces in first approximation to the standard Riemannian line element $ds=\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$. If terms of order $\\varepsilon^2$ and higher can be neglected, the geometry turns out to be homogeneous because the line element can always be rescaled by a constant factor. This shows the extraordinary generality of Riemann's assumption. In this limit (\\ref{Rds}) can be derived from the sole hypotheses of \\emph{differentiability}, \\emph{indiscernibility} and \\emph{symmetry}. In particular, the Pythagorean proposition emerges, \\emph{without the need of being postulated}, as the fundamental building block of geometry on infinitesimal scales. \nClearly, it is still possible to construct non-Pythagorean geometries\\footnote{If we repeat the series inversion and the substitution of $d\\tau$ in $ds$ under the assumption of a vanishing $g_{\\mu\\nu}$, we obtain that ``the [rescaled] line element can be expressed as the fourth root of a differential expression of fourth degree'' plus correction of order $\\varepsilon$ and higher,\n $${\\textstyle\\frac{3}{\\varepsilon^{3\/2}}}ds=\\sqrt[4]{g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}+O(\\varepsilon).$$\nThis was lucidly foreseen by Riemann's deep insight.} where $g_{\\mu\\nu}\\equiv0$. However, the number of these geometries equals the number of analytic functions with vanishing Hessian at $\\dot{x}=0$. Consequently, non-Pythagorean geometries only represent a subset of zero measure of the set of all possible Riemann-type geometries.\n\nFor generic geometries, the leading contribution to the line element is given by the one form $g_\\mu dx^\\mu$, which is however insufficient to specify a geometry. Consequently,  geometry is described in first approximation by a one-parameter family of homogeneous Riemann-Randers metrics \\cite{Randers1941}\n\\begin{equation}\n\\label{RRds}\nds=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}.\n\\end{equation}\nAs $\\varepsilon$ can no longer be eliminated by a constant rescaling,  the geometry as a whole is no longer homogeneous. On the other hand, while non-symmetrical by construction, the (rescaled) line element remains the same to all orders, if the transformation $dx^\\mu\\to-dx^\\mu$ is accompanied by the sign reversal of the parameter $\\varepsilon\\to-\\varepsilon$.\\\\\n\nThe general case is not substantially different but requires a careful discussion of different cases in order to express the line element $ds$ as a function of ``the quantities $x$ and the quantities $dx$\". On the other hand, this process is completely unnecessary. The geodesic equations can be more simply and conveniently derived by the direct variation of the line integral of (\\ref{tds}).\n\n\\subsection*{Geodesic equations}\n The Euler-Lagrange equations for the lines giving a stationary value to the line integral of  (\\ref{tds}), the geodesic equations of this geometry, are obtained as\n\\begin{eqnarray}\\label{ge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\ddot{x}^{\\lambda}+\n\\hskip20pt\n&&  \\nonumber \\\\[5pt]\n+\\Gamma_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\Gamma_{\\kappa\\mu\\nu}\\dot{x}^{\\mu}\\dot{x}^{\\nu} \n+\\Gamma_{\\kappa\\mu\\nu\\xi}\\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+... \n+\\Gamma_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0, \\hskip-20pt \n\\end{eqnarray}\nwhere dots indicate differentiation with respect to the parameter $\\tau$ and the symbols $\\Gamma$ are defined as\n\\begin{eqnarray}\\label{Gammas}\n&&\\Gamma_{\\kappa\\mu} = \\partial_\\mu g_\\kappa-\\partial_\\kappa g_\\mu \\nonumber \n\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu\\nu}  = \\frac{1}{2}\\left(\\partial_\\mu g_{\\kappa\\nu}\n+\\partial_\\nu g_{\\mu\\kappa}-\\partial_\\kappa g_{\\mu\\nu}\\right)  \\nonumber\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu\\nu\\xi}  = \\frac{1}{3!}\\left(\\partial_\\mu g_{\\kappa\\nu\\xi}\n+\\partial_\\nu g_{\\mu\\kappa\\xi}+\\partial_\\xi g_{\\mu\\nu\\kappa}\n-\\partial_\\kappa g_{\\mu\\nu\\xi}\\right)  \\\\[5pt]\n&& \\hskip10pt ...\\nonumber\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu_1...\\mu_k}=\\frac{1}{k!}\n\\left(\\partial_{\\mu_1}g_{\\kappa\\mu_2...\\mu_{k-1}}+ ...\n+\\partial_{\\mu_n}g_{\\mu_1...\\mu_{k-1}\\kappa} -\n\\partial_\\kappa g_{\\mu_1...\\mu_k}\n\\right)\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\nonumber\n\\end{eqnarray}\n\nUp to an overall sign, the $\\Gamma_{\\kappa\\mu}$ are related to $g_\\mu$\nin the same way as an electromagnetic field is related to its vector potential. \nAlso, the $\\Gamma_{\\kappa\\mu\\nu}$ correspond to the Christoffel symbols of the first kind associated to the metric $g_{\\mu\\nu}$. In the light of (\\ref{Gammas}) these two relationships appear now as the special cases $k=1$ and $k=2$ of a more general rule that links a symmetric tensor $g_{\\mu_1...\\mu_k}$ of order $k$, to  the symbols $\\Gamma_{\\kappa\\mu_1...\\mu_k}$.\n\n Apart from $\\Gamma_{\\kappa\\mu}$,  that transforms like a tensor, all $\\Gamma$ symbols display non-covariant transformation rules, somehow analogues to that of the Christoffel symbols. Nonetheless, equations (\\ref{ge}) can be given an explicit covariant form by introducing a single symmetric linear connection ${C^\\kappa}_{\\mu\\nu}$. This can possibly, but not necessarily, be chosen as the Christoffel-Levi-Civita connection ${\\Gamma^\\kappa}_{\\mu\\nu}=g^{\\kappa\\lambda}{\\Gamma}_{\\lambda\\mu\\nu}$ associated to $g_{\\mu\\nu}$. In order to maintain full symmetry in the formalism, for the moment I will assume  that  ${C^\\kappa}_{\\mu\\nu}\\neq {\\Gamma^\\kappa}_{\\mu\\nu}$. Nothing of what follows depends in any way on the choice of ${C^\\kappa}_{\\mu\\nu}$ (see Appendix A). \n \n By adding and subtracting identical terms on the left hand side of (\\ref{ge}),  the geodesic  equations can be rewritten in the form\n\\begin{eqnarray}\\label{cge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{C^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip2pt\n&&  \\nonumber \\\\[5pt]\n+{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+{F}_{\\kappa\\mu\\nu} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\n+{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0,  && \n\\end{eqnarray}\nwhere the second derivatives $\\ddot{x}^{\\lambda}$ have been made covariant by the addition of appropriate terms and the non-covariant $\\Gamma$ symbols have been replaced by the covariant tensor fields\n\\begin{eqnarray}\n&&{F}_{\\kappa\\mu}=\\Gamma_{\\kappa\\mu} = \\nabla_\\mu g_\\kappa-\\nabla_\\kappa g_\\mu \\nonumber\\\\[5pt]\n&&{F}_{\\kappa\\mu\\nu}  \n=\\Gamma_{\\kappa\\mu\\nu}-g_{\\kappa\\lambda}C^\\lambda_{\\mu\\nu}\n= \\frac{1}{2}\\left(\\nabla_\\mu g_{\\kappa\\nu}\n+\\nabla_\\nu g_{\\mu\\kappa}-\\nabla_\\kappa g_{\\mu\\nu}\\right)\\nonumber\\\\[5pt]\n&&{F}_{\\kappa\\mu\\nu\\xi}  \n=\\Gamma_{\\kappa\\mu\\nu\\xi}-g_{\\kappa\\lambda(\\mu}C^\\lambda_{\\nu\\xi)}\n= \\frac{1}{3!}\\left(\\nabla_\\mu g_{\\kappa\\nu\\xi}\n+\\nabla_\\nu g_{\\mu\\kappa\\xi}+\\nabla_\\xi g_{\\mu\\nu\\kappa}\n-\\nabla_\\kappa g_{\\mu\\nu\\xi}\\right) \n\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\\\[5pt]\n&&{F}_{\\kappa\\mu_1...\\mu_k}=\\Gamma_{\\kappa\\mu_1...\\mu_k}-\\frac{1}{(k-2)!}g_{\\kappa\\lambda(\\mu_1... \\mu_{k-2}}C^\\lambda_{\\mu_{k-1}\\mu_k)}=\\nonumber\\\\[5pt]\n&&\\hskip40pt =\\frac{1}{k!}\n\\left(\\nabla_{\\mu_1}g_{\\kappa\\mu_2...\\mu_{k-1}}+ ...\n+\\nabla_{\\mu_n}g_{\\mu_1...\\mu_{k-1}\\kappa} -\n\\nabla_\\kappa g_{\\mu_1...\\mu_k}\n\\right)\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\nonumber\n\\end{eqnarray}\nwhere $\\nabla_\\kappa$ is the covariant derivative associated to the linear connection ${C^\\kappa}_{\\mu\\nu}$. We note that the tensor fields $F$ are simply obtained from  (\\ref{Gammas}) by replacing the partial derivatives $\\partial_\\kappa$ with the covariant derivatives $\\nabla_\\kappa$. \n\nThe second order symbols $\\Gamma_{\\kappa\\mu}=F_{\\kappa\\mu}$ are unaffected by this transformation. The Christoffel symbols of the first kind $\\Gamma_{\\kappa\\mu\\nu}$ associated to $g_{\\mu\\nu}$, are instead transformed into the analogues of the covariant gravitational field ${F}_{\\kappa\\mu\\nu}$ of what can be described as the Poicar\\'e-Rosen formalism for general relativity (see Appendix A). \nAll other higher order Christoffel-like symbols are transformed into the covariant fields ${F}_{\\kappa\\mu_1...\\mu_k}$.\nIt is interesting to note that even in this general approach to Riemannian geometry, the quadratic term in the series expansion of $ds$ maintains a unique status. The corresponding field ${F}_{\\kappa\\mu\\nu}$, is the only one that produces local effects indistinguishable from a coordinate transformation. Correspondingly, ${F}_{\\kappa\\mu\\nu}$ can always be set identically equal to zero by choosing  ${C^\\kappa}_{\\mu\\nu}$ equal to the Christoffel-Levi-Civita connection associated to $g_{\\mu\\nu}$. Since this considerably simplifies the formalism, making it analogous to that of standard Riemannian geometry and general relativity, from now on I will choose  ${C^\\kappa}_{\\mu\\nu}= {\\Gamma^\\kappa}_{\\mu\\nu}$. With this choice the geodesic equations are rewritten as \n\\begin{eqnarray}\\label{sge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip2pt\n&&  \\nonumber \\\\[5pt]\n+{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0,  && \n\\end{eqnarray}\nwhere ${F}_{\\kappa\\mu\\nu}$ no longer appears and $g_{\\mu\\nu}$ plays a privileged role with respect to the other fields.\nNonetheless, it is important to stress that all the $g_{\\mu_1...\\mu_k}$ take part in shaping the metric properties of space, such as the length of lines and the distance between points. In principle all fields can be treated on an equal footing.\\\\\n\nLet us now consider the effect of an arbitrary reparameterization $\\tau\\to\\zeta(\\tau)$ of the geodesic equations. A direct computation shows that  (\\ref{sge}) transforms into\n\\begin{eqnarray}\\label{zrcge}\n\\left[g_{\\kappa\\lambda}\n+...\n+\\frac{{\\dot{\\zeta}}^{k-2}}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi+\\frac{\\ddot{\\zeta}}{{\\dot{\\zeta}}^{2}}\\dot{x}^\\lambda\\right)+\\hskip5pt\n&&  \\nonumber \\\\[5pt]\n+\\dot{\\zeta}^{-1}{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\dot{\\zeta}{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+\\dot{\\zeta}^{k-2}{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0. && \n\\end{eqnarray}\nOn the one side, this confirms that the geodesic equations determine the geodesic parameter $\\tau$ up to an arbitrary additive constant. In fact, the simpler form (\\ref{sge}) can hold only for a particular parameter $\\tau$, distinguished up to the linear transformation $\\tau\\to\\tau+\\tau_0$ with constant $\\tau_0$. This transformation is clearly responsible for the conservation of (\\ref{fi}).\nOn the other side, we learn that the geodesic equations are unaffected by a  simultaneous constant rescaling of the parameter $\\tau$ and of the fields  $g_{\\mu_1...\\mu_k}$. In fact, the equations (\\ref{sge}) are invariant under the combined  transformation\n\\begin{equation}\n\\label{rescaling}\n\\tau\\to\\lambda\\tau+\\tau_0\n\\hskip10pt\\text{and}\\hskip10pt\ng_{\\mu_1...\\mu_k}\\to\\lambda^{k-2}g_{\\mu_1...\\mu_k},\n\\end{equation}\nwith $\\lambda$ a nonzero constant.\nCorrespondingly, the line element (\\ref{tds}) and the first integral (\\ref{fi}) \nare rescaled by an irrelevant factor $\\frac{1}{\\lambda^2}$, \n\\begin{equation}\n\\label{rescaling1}\nds\\to\\frac{1}{\\lambda^2}ds\n\\hskip10pt\\text{and}\\hskip10pt\nE\\to\\frac{1}{\\lambda^2}E.\n\\end{equation}\nIn investigating non-null geodesics, we are therefore free to directly solve the equations (\\ref{sge}), whose solutions are labeled by $E$, or to stipulate a standard gauge on $\\tau$ and solve the rescaled equations (\\ref{zrcge}) for a fixed  value of first integral (\\ref{fi}). This last option is customarily made in standard Riemannian geometry and in general relativity, where the geodesic parameter is conventionally identified with the arc length $s$, so that $d\\tau=\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$ and $g_{\\mu\\nu}\\dot{x}^\\mu \\dot{x}^\\nu=1$.\nThe corresponding gauge in Riemannian geometry without the hypotheses of \\emph{homogeneity} and \\emph{symmetry} corresponds to a rescaling by $\\lambda=\\varepsilon=\\text{sgn}(E)\\sqrt{|E|\/2}$.  This produces the geodesic equations in the form\n\\begin{eqnarray}\\label{rcge}\n\\left[g_{\\kappa\\lambda}\n+...\n+\\frac{\\varepsilon^{k-2}}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip10pt\n&&  \\nonumber \\\\[5pt]\n+\\frac{1}{\\varepsilon}{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\varepsilon{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+\\varepsilon^{k-2}{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0, && \n\\end{eqnarray}\nwith  $d\\tau=\\pm\\sqrt{|g_{\\mu\\nu}dx^\\mu dx^\\nu|}+O(\\varepsilon)$ and the first integral (\\ref{fi}) equal to $\\pm1$. \\\\\n\nEquations (\\ref{sge}) and equivalent ones, generalise the geodesic equations of standard (pseudo-)Riemannian geometry, to which they reduce when all $g_{\\mu_1...\\mu_k}$ except $g_{\\mu\\nu}$ vanish identically. By assuming the invertibility of the second order tensor\n\\begin{equation}\n\\label{ }\n\\bm{g}_{\\mu\\nu}\n= g_{\\mu\\nu}\n+g_{\\mu\\nu\\xi}\\dot{x}^{\\xi}+...\n+\\frac{1}{(k-2)!}g_{\\mu\\nu\\xi_1... \\xi_{k-2}}\\dot{x}^{\\xi_1}...\\ \\dot{x}^{\\xi_{k-2}}+...\\ ,\n\\end{equation}\nthese equations can be resolved with respect to the second derivatives. The local existence and uniqueness theorem for their solutions follows then from  standard ODE-theory.\n\n\\subsection*{Connection and curvature}\nThe standard form of the geodesic equations is also a convenient starting point to identify the connection and the curvature associated to the line element (\\ref{tds}). By series-inverting $\\bm{g}_{\\mu\\nu}$ we obtain the symmetric contravariant tensor $\\bm{g}^{\\mu\\nu}$ fulfilling the identity $\\bm{g}^{\\mu\\kappa}\\bm{g}_{\\kappa\\nu}=\\delta^\\mu_\\nu$. The first few terms are  given by\n\\begin{equation}\n\\bm{g}^{\\mu\\nu}=g^{\\mu\\nu}\n-{g^{\\mu\\nu}}_\\xi\\dot{x}^\\xi\n+\\left[\n{g^{\\mu}}_{\\lambda(\\xi}{g^{\\nu\\lambda}}_{o)}\n-\\frac{1}{2}{g^{\\mu\\nu}}_{\\xi o}\n\\right]\\dot{x}^{\\xi}\\dot{x}^{o}+ ...\\ ,\n\\end{equation}\nwhere \nround brackets indicates symmetrisation  and indices are raised by means of the inverse $g^{\\mu\\nu}$ of $g_{\\mu\\nu}$, $g^{\\mu\\kappa}g_{\\kappa\\nu}=\\delta^\\mu_\\nu$. For example, ${g^{\\mu\\nu}}_\\xi=\ng^{\\mu\\rho}g^{\\nu\\sigma}g_{\\rho\\sigma\\xi}$. By multiplying (\\ref{sge}) by $\\bm{g}^{\\iota\\kappa}$, contracting on $\\kappa$ and renaming indices, we  obtain the geodesic equations in the standard form \n\\begin{equation}\n\\label{gesf}\n\\ddot{x}^\\kappa+\\bm{\\gamma}^\\kappa_\\mu\\dot{x}^\\mu=0\n\\end{equation}\nwith $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ expressed as a series in $\\dot{x}$ as\n\\begin{equation}\n\\label{nlco}\n\\bm{\\gamma}^\\kappa_\\mu={\\gamma}^\\kappa_\\mu\n+{\\gamma}^\\kappa_{\\mu\\nu}\\dot{x}^\\nu\n+{\\gamma}^\\kappa_{\\mu\\nu\\xi}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+...+{\\gamma}^\\kappa_{\\mu\\xi_1...\\xi_n}\\dot{x}^{\\xi_1}...\\ \\dot{x}^{\\xi_n}+... \n\\end{equation}\nThe first few coefficients are given by \n\\begin{eqnarray}\n&&  {\\gamma}^\\kappa_\\mu={F^{\\kappa}}_\\mu\n\\nonumber\\\\[5pt]\n&&  {\\gamma}^\\kappa_{\\mu\\nu}=\n{\\Gamma^\\kappa}_{\\mu\\nu}\n-{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\nu)}\n\\nonumber\\\\[5pt]\n&&  {\\gamma}^\\kappa_{\\mu\\nu\\xi}={F}^{\\kappa}_{\\ \\mu\\nu\\xi}\n-{g^{\\kappa}}_{\\lambda(\\mu}{g^{\\lambda\\iota}}_{\\nu}{F}_{\\xi)\\iota}\n-\\frac{1}{2}{g^\\kappa}_{\\lambda(\\mu\\nu}{{F}^{\\lambda}}_{\\xi)}\n\\nonumber\\\\[5pt]\n&&\\hskip10pt...\\nonumber\n\\end{eqnarray}  \nThe linear coefficient ${\\gamma}^\\kappa_{\\mu\\nu}$ is the only non-covariant one. All the others transform like tensors.\nAs a whole, $\\bm{\\gamma}^\\kappa_\\mu$ transforms like a connection one-form.\nIn fact, in (\\ref{gesf}) we recognise the auto parallel equations associated to the nonlinear connection  $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ (see Appendix \\ref{B}).\nTherefore, we can univocally identify $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ as the connection associated the line element (\\ref{tds}). \n\nAs usual, the corresponding curvature tensor is obtained by means of the formula\n\\begin{equation}\n\\label{curvature}\n\\bm{r}^\\kappa_{\\mu\\nu}=\n\\partial_\\mu\\bm{\\gamma}^\\kappa_\\nu\n-\\partial_\\nu\\bm{\\gamma}^\\kappa_\\mu\n-\\bm{\\gamma}^\\lambda_\\mu{\\textstyle\\frac{\\partial}{\\partial\\dot{x}^\\lambda}}\\bm{\\gamma}^\\kappa_\\nu\n+\\bm{\\gamma}^\\lambda_\\nu{\\textstyle\\frac{\\partial}{\\partial\\dot{x}^\\lambda}}\\bm{\\gamma}^\\kappa_\\mu.\n\\end{equation}\nThis yields the curvature $\\bm{r}^\\kappa_{\\mu\\nu}(x,\\dot{x})$ again as series in $\\dot{x}$\n\\begin{equation}\n\\label{nlcu}\n\\bm{r}^\\kappa_{\\mu\\nu}={r^\\kappa}_{\\mu\\nu}\n+{r^\\kappa}_{\\xi\\mu\\nu}\\dot{x}^{\\xi}\n+...+{r^\\kappa}_{\\xi_1...\\xi_n\\mu\\nu}\\dot{x}^{\\xi_1}...\\, \\dot{x}^{\\xi_n}+...\\, .\n\\end{equation}\nwith the first few coefficients given by\n\\begin{eqnarray}\n&&  {r^\\kappa}_{\\mu\\nu}=\n\\nabla_\\mu{{F}^{\\kappa}}_\\nu\n-\\nabla_\\nu{{F}^{\\kappa}}_\\mu\n-\\frac{1}{2}{g^\\kappa}_{\\mu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\nu}\n+\\frac{1}{2}{g^\\kappa}_{\\nu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\mu}\n\\nonumber\\\\[5pt]\n&&  {r^\\kappa}_{\\xi\\mu\\nu}={R^\\kappa}_{\\xi\\mu\\nu}\n-\\nabla_\\mu{g^{\\kappa}}_{\\lambda(\\nu} {{F}^{\\lambda}}_{\\xi)}\n+\\nabla_\\nu{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\xi)}+\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-{g^{\\lambda}}_{\\iota(\\mu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\nu} {{F}^{\\theta}}_{\\lambda)}\n+{g^{\\lambda}}_{\\iota(\\nu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\mu} {{F}^{\\theta}}_{\\lambda)}\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-2{{F}^{\\lambda}}_{\\mu}{\\gamma}^\\kappa_{\\nu\\lambda\\xi}\n+2{{F}^{\\lambda}}_{\\nu} {\\gamma}^\\kappa_{\\mu\\lambda\\xi}\n\\nonumber\\\\[5pt]\n&&\\hskip25pt...\\nonumber\n\\end{eqnarray}\nwhere ${R^\\kappa}_{\\xi\\mu\\nu}=\\partial_\\mu{\\Gamma^\\kappa}_{\\nu\\xi}-\n\\partial_\\nu{\\Gamma^\\kappa}_{\\mu\\xi}-\n{\\Gamma^\\lambda}_{\\mu\\xi}{\\Gamma^\\kappa}_{\\nu\\lambda}+\n{\\Gamma^\\lambda}_{\\nu\\xi}{\\Gamma^\\kappa}_{\\mu\\lambda}$\nis the Riemann tensor associated to the Christoffel-Levi-Civita connection ${\\Gamma^\\kappa}_{\\mu\\nu}$.\nAs in standard Riemannian geometry, it is possible to construct a contracted curvature tensor, the analogue of the Ricci tensor $R_{\\mu\\nu}={R^\\kappa}_{\\mu\\kappa\\nu}$, as \n\\begin{equation}\n\\label{nRicci}\n\\bm{r}_{\\mu}=\\bm{r}^\\kappa_{\\kappa\\mu}={r}_{\\mu}\n+{r}_{\\xi\\mu}\\dot{x}^{\\xi}\n+...+{r}_{\\xi_1...\\xi_n\\mu}\\dot{x}^{\\xi_1}...\\, \\dot{x}^{\\xi_n}+...\\, .\n\\end{equation} \nwith\n\\begin{eqnarray}\n&&  {r}_{\\mu}=\n\\nabla_\\kappa{{F}^{\\kappa}}_\\mu\n-\\frac{1}{2}{g^\\kappa}_{\\kappa\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\mu}\n+\\frac{1}{2}{g^\\kappa}_{\\mu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\kappa}\n\\nonumber\\\\[5pt]\n&&  {r}_{\\xi\\mu}={R}_{\\xi\\mu}\n-\\nabla_\\kappa{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\xi)}\n+\\nabla_\\mu{g^{\\kappa}}_{\\lambda(\\kappa} {{F}^{\\lambda}}_{\\xi)}+\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-{g^{\\lambda}}_{\\iota(\\kappa} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\mu} {{F}^{\\theta}}_{\\lambda)}\n+{g^{\\lambda}}_{\\iota(\\mu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\kappa} {{F}^{\\theta}}_{\\lambda)}\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-2{{F}^{\\lambda}}_{\\kappa}{\\gamma}^\\kappa_{\\mu\\lambda\\xi}\n+2{{F}^{\\lambda}}_{\\mu} {\\gamma}^\\kappa_{\\kappa\\lambda\\xi}\n\\nonumber\\\\[5pt]\n&&\\hskip10pt...\\nonumber\n\\nonumber\n\\end{eqnarray}\nCorrespondingly, it is also possible to define a curvature scalar by contraction with $\\dot{x}^\\mu$, \n\\begin{equation}\n\\label{curvaturescalar}\n\\bm{r}=\\bm{r}_{\\mu}\\dot{x}^\\mu,\n\\end{equation} \nproducing \nthe symmetrisation of all components ${r}_{\\xi_1...\\xi_n\\mu}$ of $\\bm{r}_{\\mu}$. This curvature scalar should not  be confused with the analogue of the scalar curvature $R=g^{\\mu\\nu}R_{\\mu\\nu}$ of standard (pseudo-)Riemannian geometry, whose generalisation is less straightforward. \n\n\\subsection*{Quadratic restriction as quadratic approximation}\nIf all the fields $g_{\\mu_1...\\mu_k}$ except $g_{\\mu\\nu}$ are set equal to zero,\nthe geodesic equations (\\ref{sge}), \nthe connection (\\ref{nlco}) and the curvature (\\ref{nlcu}), respectively reduce to the geodesic equations, connection and curvature of standard (pseudo-)Riemannian geometry.\n Correspondingly, the line element (\\ref{tds}) reduces to the integrand $\\frac{1}{2}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu d\\tau$ of the variational principle introduced by Levi-Civita to describe  null geodesics in general relativity \\cite{Levi-Civita1929}.  In the context of standard Riemannian geometry, with a positive definite metric tensor, this is completely equivalent to Riemann's \\emph{quadratic restriction}. In fact the solutions of  Levi-Civita's variational principle admit the first integral  $g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu=2E$, so that \n\\begin{equation}\n\\label{L-CvsR}\n\\delta\\int {\\textstyle \\frac{1}{2}}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu d\\tau =\n{\\textstyle \\frac{1}{2}}\\delta\\int \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu}  \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu} d\\tau =\n {\\textstyle\\sqrt{\\frac{E}{2}}}\\delta\\int \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu} d\\tau.\n\\end{equation}\nConversely, when indefinite signatures are considered, Levi-Civita's variational principle becomes slightly more general. In fact, the corresponding geodesic equations coincide with the geodesic equations of standard (pseudo-)Riemannian geometry for $E\\neq0$, but also include the case of null geodesics with the correct parameterisation when $E=0$ \\cite{Schroedinger1956}.  The reading of Levi-Civita's integrand as a line element, is therefore an improvement of Riemann's \\emph{quadratic restriction}. \n\n Under this perspective, standard (pseudo-)Riemannian geometry appears as the quadratic approximation to a more general Riemannian geometry, that can be developed along the very same lines by relaxing the hypothesis of \\emph{homogeneity}, but not the one of \\emph{symmetry}. \n As the quadratic term frequently provides an excellent approximation on sufficiently small scales, this suggests why Riemann's \\emph{quadratic restriction} is so effective in the description of geometry.  It is effective to such an extent, that when complemented with space(time) dimension, metric signature and dynamical equations,  it is one and the same with Einstein's general theory of relativity. It is then natural to wonder whether Riemannian geometry without the hypotheses of \\emph{homogeneity }and \\emph{symmetry} can provide a deeper insight into the study of spacetime.\n\n\n\\section{Applications to spacetime}\\label{spacetime}\nThe quadratic approximation to Riemannian geometry based on the hypotheses of \\emph{differentiability}, \\emph{indiscernibility} and \\emph{symmetry},  thus without that of \\emph{homogeneity}, naturally provides the mathematical framework of general relativity without the need for postulating the explicit form of the line element. If we further renounce the hypothesis of \\emph{symmetry}, thus starting from a geometry uniquely based on \\emph{differentiability} and \\emph{indiscernibility}, we obtain the mathematical framework of a unified theory of the classical electromagnetic and gravitational fields. \nUnlike the classical attempts \\cite{Goenner2004,Goenner2014,Tonnelat1966}, there is no need to introduce new fields, extra dimensions or new geometrical structures.\nA logically simple and economical unified description is obtained simply by renouncing hypotheses within Riemannian geometry.\nUnlike the classical attempts, it is directly evident how to identify the physical fields.\nIn the quadratic approximation a generic line element is equivalent to a one-parameter family of Riemann-Randers line elements $ds=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$. This\ncorresponds to a one-parameter family of Lagrangians for charged particles in the background of electromagnetic and a gravitational fields. The $\\varepsilon$-dependent line element is not postulated for a single value of $\\varepsilon$ as in Rander's proposal\\footnote{See the discussion in the Introduction of \\cite{Tonnelat1966}.}. It emerges by itself in as many copies as we need to describe particles with different charges. Also, in the quadratic approximation the connection (\\ref{nlco}) reduces to the nonlinear connection put forward by Broda and Przanowski \\cite{BrodaPrzanowski1985} for Riemann-Randers spaces.\nTherefore, up to a proportionality constant, the field $g_\\mu$ is naturally identified with the electromagnetic potential, as $g_{\\mu\\nu}$ is identified with the gravitational potential\\footnote{In the original formulation of general relativity the metric tensor $g_{\\mu\\nu}$ was identified with the inertial-gravitational tensor \\emph{potential} and the associated Christoffel symbols ${\\Gamma^\\kappa}_{\\mu\\nu}$ with the corresponding force \\emph{field}, in complete analogy with the electromagnetic vector \\emph{potential}  $\\mathrm{A}_\\mu$ and the force \\emph{field} $\\mathrm{F}_{\\mu\\nu}$. In later years, probably following the attempt to identify $g_{\\mu\\nu}$ and $\\mathrm{F}_{\\mu\\nu}$ with the symmetric and antisymmetric parts of a same tensor, $g_{\\mu\\nu}$ has been increasingly referred to as the gravitational field. Under a different prospective, the analogy between the gauge structures of electromagnetism and Einstein's gravity have encouraged the identification of ${\\Gamma^\\kappa}_{\\mu\\nu}$ with $\\mathrm{A}_\\mu$ and of ${R^\\kappa}_{\\lambda\\mu\\nu}$ with $\\mathrm{F}_{\\mu\\nu}$. The equation of motion of charged particles in a curved background leave little doubt of what the correct physical identification is.}.\nCorrespondingly, the parameter $\\varepsilon$ is proportional to  the mass-to-charge ratio of particles, thus providing a geometrical meaning to electric charge.\n\nAs for standard Riemannian geometry and general relativity, the transition from the mathematical framework to the physical theory requires the specification of spacetime dimension, signature and dynamical equations. While the first two necessarily follow those of general relativity,  the choice of adequate dynamical equations is less constrained. The simplest and most natural generalization of Einstein field equations seems to me \n\\begin{equation}\n\\label{fieldeq}\n\\bm{r}=\\bm{t}\n\\end{equation}\nwith $\\bm{r}$ the curvature scalar (\\ref{curvaturescalar}) and $\\bm{t}=t_\\mu(x)\\dot{x}^\\mu+t_{\\mu\\nu}(x)\\dot{x}\\dot{x}^\\nu+...$ an appropriate analytic function of the $x$ and of the $\\dot{x}$, describing the distribution of charge and matter. As the identity of analytic functions requires the identity of all the respective coefficients in their series expansion, in the limit\nwhere all $g_{\\mu_1...\\mu_k}$ except $g_\\mu$ and $g_{\\mu\\nu}$ are negligible, \n(\\ref{fieldeq}) reduces to the simultaneous Maxwell and Einstein field equations \n\\begin{eqnarray}\n&&  \\nabla_\\kappa{{F}^{\\kappa}}_\\mu=t_\\mu,\\label{Maxwell} \\\\\n&& R_{\\mu\\nu}=t_{\\mu\\nu}.\\label{Einstein}\n\\end{eqnarray}\nI find it really remarkable that the Maxwell field equations (\\ref{Maxwell}) minimally coupled to $g_{\\mu\\nu}$, naturally emerge from the geometry. \nOn the other hand, it should also be noted that  equations  (\\ref{Einstein}) do not correspond to the standard minimal coupling, because the electromagnetic stress-energy tensor  $F_{\\mu\\xi}{F_\\nu}^\\xi-\\frac{1}{4}g_{\\mu\\nu}F_{\\xi o}F^{\\xi o}$  does not appear in their right-hand-side term. This is perfectly consistent with all the experiments conducted so far, where the propagation of light is studied ``along null-geodesics in  a prescribed [...] gravitational field which is a solution of Einstein vacuum equation[s] --- and not of the \\emph{electro}-vacuum equation[s]'' \\cite{HehlObukhov2001}. \n\nThe standard minimal coupling can be obtained in the quadratic approximation from the straightforward generalisation of the standard Einstein-Maxwell Lagrangian density\n\\begin{equation}\n\\label{EM}\n\\mathcal{L}_{EM}=F_{\\mu\\nu}F^{\\mu\\nu}+R+...,\n\\end{equation}\nwhere the unwritten terms for the fields ${F}_{\\mu_1...\\mu_k}$ for $k\\geq4$, can be easily guessed from Rosen's expression for the Lagrangian density of the gravitational field  $F_{\\mu\\nu\\xi}$ \\cite{Rosen1940}.\n\nEquivalently, the standard minimal coupling can be obtained, always in the quadratic approximation, by generalizing the geometrically more appealing Broda-Przanowski Lagrangian density \\cite{BrodaPrzanowski1985} \n\\begin{equation}\n\\label{BP}\n\\mathcal{L}_{BP}=g^\\mu r_\\mu+g^{\\mu\\nu}r_{\\mu\\nu}+g^{\\mu\\nu\\xi}r_{\\mu\\nu\\xi}+...\n\\end{equation}\n\nIn both cases, indices are raised by $g^{\\mu\\nu}$ and the spacetime volume element is chosen as $\\sqrt{|\\det g_{\\mu\\nu}|}d^4x$. Whatever our aesthetic preferences may be, such questions can only be solved experimentally.\\\\\n\nIn order to make contact with reality, it is also necessary to specify the proportionality constants between the geometrical and physical  fields and between the adimensional parameter $\\varepsilon$ and the mass-to-charge ratio of physical particles.\nSince the only unit of electromagnetic potential  that can be constructed in terms of classical fundamental constants is the Planck voltage\n\\begin{equation}\n\\label{PlankV}\n\\sqrt{\\frac{c^4}{4\\pi\\epsilon_0G}}\\simeq1.04\\times10^{27}V,\n\\end{equation}\nwith $c$ the speed of light, $\\epsilon_0$ the electric constant and $G$ the gravitational constant,\nthe only possibility is to set\n \\begin{equation}\n\\label{g to A}\ng_\\mu=-\\sqrt{\\frac{4\\pi\\epsilon_0G}{c^4}}\\mathrm{A}_{\\mu}\n\\end{equation}\nwith $\\mathrm{A}_{\\mu}$  the electromagnetic vector potential.\nCorrespondingly, \n\\begin{equation}\n\\label{epsilon}\n\\varepsilon=\\frac{m}{e}\\sqrt{4\\pi\\epsilon_0G},\n\\end{equation}\nwhere $m$  and $e$ respectively indicate the mass and charge of the particle under consideration. For electrons $\\varepsilon\\simeq0.5\\times10^{-21}$, largely justifying the validity of a quadratic approximation.\\\\\n\nBeyond the quadratic approximation, the geometry imagined by Riemann in 1854  deprived of the hypotheses of \\emph{homogeneity} and \\emph{symmetry}, suggests the existence of infinitely more classical force fields, somehow the higher harmonics of a single fundamental classical field, the infinitesimal line element $ds$.  \nThese fields are generated by an identical rule from symmetric tensor potentials of increasing order in the very same way the electromagnetic and gravitational fields  $F_{\\mu\\nu}$ and $\\Gamma_{\\kappa\\mu\\nu}$ (or $F_{\\mu\\nu\\xi}$) are generated by the respective potentials $g_\\mu$ and $g_{\\mu\\nu}$.\nLike electromagnetic and gravitational fields, these higher order fields  propagate in space as waves of speed $c$, interacting in an increasingly non-linear way as their order increases. Nonetheless, as the ratio of strength between successive order fields equals the one between gravity and electromagnetism, it must be expected that already the field $F_{\\mu\\nu\\xi o}$ produces extremely weak and hard to measure local effects. On the other hand, these extremely weak effects could become relevant when integrated over the enormous spacetime distances  and could perhaps play a role in some of the many unresolved problems in astrophysics and cosmology. \nIf we accept the paradigm shift that Riemann's \\emph{quadratic restriction} is nothing but a quadratic approximation to a generic line element, these fields must somehow play a role in the large scale structure of spacetime. In this context, however, a non-perturbative approach to the dynamics of $ds$ might be more appropriate\\footnote{It is always possible to parametrise the line element as $ds=\\frac{\\partial\\sigma}{\\partial\\dot{x}^\\mu}\\dot{x}^\\mu$, with $\\sigma(x,\\dot{x})$ an arbitrary analytic function of position and direction. The generalisation of Maxwell and Einstein field equations (\\ref{fieldeq}) rewrites then as a single differential equation for the scalar field $\\sigma$.}.\n\n\\section{Conclusion}\\label{TheEnd}\nFor a long time the term `Riemannian geometry' has been used to indicate spherical geometry, probably a sign of how small the impact of Riemann's ideas was originally. \nWith the advent of general relativity the term slowly shifted to current use and some twenty years ago Chern \\cite{Chern1996} suggested that it should also include what we currently call Finsler geometry. In this paper I have used it in its broadest sense, to describe a differential geometry  uniquely based on the hypothesis that the distance of a point from itself is equal to zero. No \\emph{quadratic restriction}, no \\emph{homogeneity}, no \\emph{symmetry}. As a matter of fact, this geometry is much more general than Riemann's original vision. \nIts relationship with standard Riemannian geometry is the same as that between a generic analytical function and its quadratic approximation at a given point. Nonetheless, the quadratic term in the series expansion of a generic line element plays such a unique role to justify the choice widely. In fact, Riemann's assumption of the infinitesimal Pythagorean distance formula reemerges as a first  approximation to \\emph{almost every} geometry which is symmetric with respect to direction reversal, also trowing new light on this ancient fundamental theorem. Analogously, the associated Christoffel-Levi-Civita linear connection keeps on playing a pivotal role in the generalised connection.\nFor generic geometries the first approximation to the infinitesimal distance formulas turns instead into something that Riemann could hardly have foreseen. The lack of homogeneity introduces into geometry an adimensional parameter. \nCorrespondingly, \\emph{almost every} geometry is described in first approximation \nby a one-parameter family of Riemann-Randers metrics.  The ``square root of an [...] homogeneous function of the second degree in the quantities $dx$, in which the coefficients [$g_{\\mu\\nu}$] are continuous functions of the quantities $x$'', is now accompanied by a homogenous function of the first degree in the quantities $dx$ in which the coefficients $g_\\mu$ are regular functions of the quantities $x$, multiplied by an arbitrary constant $\\frac{1}{\\varepsilon}$.  This additional term enters the first approximation to the generalised connection as the nonlinear term introduced by Broda and Przanowski.\\\\\n\nIf following the lines of general relativity we  apply this geometry to spacetime by identifying $g_{\\mu\\nu}$ with the gravitational potential, the additional term $g_\\mu$ enters the geodetic and field equations exactly as an electromagnetic potential, naturally providing a unified description of the two classical fundamental interactions.\nAs $g_\\mu$ and $g_{\\mu\\nu}$ respectively appears as the first and second order terms of the series expansion of the line element $ds$, the hierarchy between electromagnetic and gravitational interactions finds its origin in geometry. \nMoreover,  geometry turns out to be invariant under the simultaneous reversal of direction and of the sign of $\\varepsilon$, so that also CPT invariance finds its origin in spacetime geometry. \nCorrespondingly, as the different parity of subsequent terms in the series expansion of $ds$ requires that the coefficient of ${F^\\kappa}_{\\mu}$ changes its sign under direction reversal, while the one of ${\\Gamma^\\kappa}_{\\mu\\nu}$ does not, also the different attractive\/repulsive nature of electromagnetic and gravitational interactions finds its origin in spacetime geometry. Equivalently, geometry naturally accounts for charge to be positive and negative, while mass only positive. Besides the electromagnetic and the gravitational interactions, this geometry suggests the existence of infinitely more force fields, extremely weak and so far unobserved,  that could possibly play a role in spacetime dynamics on a large scale.\n\n Quite remarkably, the whole kinematical setting --the number and type of potentials, their relation to fields, the hierarchy and properties of the interactions, the existence of electric charge, CPT invariance, the form of the curvature tensor that set equal to zero gives both the Maxwell and Einstein free field equations etc.-- entirely follows from the minimal assumption that the line element $ds$ is a regular function of position and direction, that the distance of a point from itself is equal to zero and, of course, from the geodesic hypothesis.     \nHowever, as Riemann reminds us, the properties of space ``can only be deduced from experience\". Therefore it remains to be seen whether these minimal hypotheses will lead to a better comprehension of the structure of spacetime on galactic and cosmological scales, so that they can be extended ``beyond the bounds of observation [...] in the direction of the immeasurably large\".  On the other hand, as  ``the empirical notions on which the metric determinations of space are based [...] lose their validity in the infinitely small [...] it is [...] quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry[...] and [...] one ought to assume this as soon as it permits a [...] way of explaining phenomena\". At the threshold of the quantum world a completely new idea of geometry is needed.\n\n\\subsection*{Aknowledgements}  I am indebted with J.K.\\! Pachos and J.-P.\\! Zendri for carefully reading the manuscript and related comments.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, massive young stellar objects (MYSOs) have been suggested as gamma-ray sources \\citep{ara07,rome08,bosch10}. Massive stars are formed in dense cores of cold clouds. The processes that take place during the formation of the star are mostly unknown. It is clear, however, that the formation of massive stars involves outflows \\citep{gara99, reip01}. The accumulation of material around the core of the cloud would generate a massive protostar that starts to accrete material from the environment. The accretion is expected to have angular momentum that leads to the formation of an accretion disk. The rotation would twist the strong magnetic fields present in the progenitor cloud around the disk, where a magnetic tower can be formed, giving rise to collimated outflows or jets, as simulations predict \\citep{baner06, baner07}. The observational evidence of outflows comes from methanol masers and from direct detection of thermal radio jets. These jets propagate along distances in a fraction of a parsec \\citep{mar93}. At the jet termination region, interaction with the external medium creates two shocks: a bow shock moving in the interstellar medium (ISM) and a reverse shock in the jet. These shocks can accelerate particles that, in turn, can produce gamma rays trough inverse Compton (IC) scattering of infrared (IR) photons, relativistic Bremsstrahlung or inelastic proton-proton collisions, if protons are accelerated as well. In some cases non~thermal radio lobes and jets have been observed, indicating the presence of relativistic electrons that produce synchrotron radiation \\citep{gara03, carrasco10}.\\\\\n\nOther possible scenarios have been suggested for the gamma-ray production involving young stars, such as the case of the massive stars with strong winds. In this scenario, gamma-rays could be produced in the interaction between the supersonic winds and the ISM. The terminal shock can accelerate particles and ions up to high energies, which might interact with the ambient matter producing gamma-rays. Whereas the luminosity produced by a single massive star wind should be low, collective effects might be important \\citep{torr04}. \\\\\nGamma-rays can also be produced in the wind interaction region of a WR+OB binary system \\citep{bena03}. In this case, the acceleration region is between the two components of the binary system and is exposed to strong photon fields where IC cooling of the electrons can generate a significant amount of high-energy (HE) non~thermal emission. A source of this class, $\\rm{\\eta-Carinae}$, has been recently detected at $E > 100$ MeV by \\textit{AGILE} \\citep{tava09}.\n\nOf-type stars have strong winds with velocities higher than the escape velocity, which implies a strong mass loss rate ($10^{-6}-10^{-5}$ M$_{\\odot}$ yr$^{-1}$). The action of this wind in the interstellar medium can create hot gas bubbles with expanding boundaries of swept-up material, which might produce gamma rays in a similar way to the case of WR stars. The case of gamma-ray emission in Of-type stars has been discussed in the past, e.g. by \\cite{volk82}. The predicted luminosity, however, is still below the current sensitivity of gamma-ray instruments.\\\\\n\nFinally, OB associations are tracers of a number of galactic objects that can produce gamma rays, such as neutron stars, massive stars with strong winds, young stellar objects (YSOs), etc. They are also thought to be places where acceleration of a significant fraction of galactic cosmic rays (CRs) might occur \\citep[e.g.][]{binns08}. \\\\\n\nThe aim of this work is to find evidence supporting the presence of HE emission coming from massive YSOs and other young galactic sources. To attain this goal we study the spatial coincidence between gamma-ray sources detected by \\textit{Fermi} and samples of young objects, such as YSOs, WR stars, Of-type stars, and OB associations. We also estimate the probability of chance coincidences by using Monte Carlo simulations, and we provide a list of counterpart candidates of the gamma-ray sources.\n\n\\section{Cross-correlation of the First \\textit{Fermi} Catalog with massive young galactic objects}\n\nThere is not observational evidence of any YSO emitting gamma-rays so far. The new generation of gamma-ray telescopes, like \\textit{Fermi}, will make it possible to reach the necessary sensitivity level to detect these faint gamma-ray sources soon, in case the predictions were right. To identify those young objects that might be emitting gamma rays, we first took the recently published First \\textit{Fermi} Catalog \\citep{abdo10} by the \\textit{Fermi} Collaboration and we excluded all known firm identifications, getting a list of 1392 sources. Then we crossed this list with catalogs of confirmed and well characterized YSOs and other type of young stars. We also did a Monte Carlo study to determine the probability of pure chance coincidences between the crossed catalogs. \n\n\\subsection{Catalogs}\n\nThe catalogs used in this study are listed in Table \\ref{tab:catalogs}. Here we describe each one in more detail.\n\nThe \\textit{Fermi Large Area Telescope} First Catalog \\citep{abdo10} contains the detected sources during the first 11 months of the science phase of the mission, which began on 2008 August 4. This catalog contains 1451 gamma-ray sources detected and characterized in the 100 MeV to 100 GeV range with a typical position uncertainty of $\\sim 6^\\prime$. After excluding the firm identifications from the original sample, we get 1392 sources. Most of them are located on the Galactic plane (see Figure \\ref{fig:hist}). \n\\\\\n\nThe Red \\textit{MSX} Source (RMS) survey is an ongoing multi~wavelength (from radio to infrared) observational program with the objective of providing a well-selected sample of MYSOs in the entire Galaxy \\citep{urqu08}. About $\\sim$2000 MYSO candidates have been identified by comparing the colors of \\textit{MSX} and 2MASS point sources (at 8, 12, 14, and 23 $\\mu$m) with those of well known MYSOs. The survey also uses high-resolution radio continuum observations at 6~cm obtained with the VLA in the northern hemisphere and at 3.6~cm and 6cm with ATCA in the southern hemisphere. They help to distinguish between genuine MYSOs and other types of objects, such as ultracompact HII regions, evolved stars, or planetary nebulae, which contaminate the sample. In addition to these targeted observations, archival data of a previous VLA survey of the inner Galaxy were used. This ongoing program has provided a sample of 637 well-identified MYSOs until now, which were used in our work.\n\\\\\n\nThe VIIth catalog of Population I WR stars \\citep{vander01} contains 227 stars, with spectral types and \\textit{bv} photometry. In recent years, the number of WR stars has increased in 71 new stars, respect to the VIth catalog and the coordinates have also been improved. The position uncertainty is close to a fraction of an arcsecond. \n\\\\\n\nThe catalog of Of-type stars is the one of \\cite{cruz74}, which contains 664 stars. The catalog provides $\\rm{m_v}$, B--V, spectral type, radial velocity, radial component of the peculiar velocity, possible multiplicity of the object, and other characteristics for each source. The typical uncertainty in the star position is $\\sim 1^\\prime$.\n\\\\\n\nFinally, the catalog of OB associations is the one by \\cite{melnik95}. This catalog contains 88 associations and provides distances to the association, number of stars, and size of the association along the Galactic latitude and longitude axes. The typical value of the size is $\\sim 20-30$~pc.\n\n\\subsection{Spatial coincidences}\n\nWe crossed the \\textit{Fermi} catalog with the catalogs of young galactic objects mentioned above. We calculated the distance between two sources using the statistical parameter $S$ \\citep{aling82}:\n\\\\\n\\begin{equation*} \nS=\\sqrt{\\frac{(\\Delta \\alpha \\cos \\delta)^2}{\\sigma_{i_\\alpha}^2+\\sigma_{j_\\alpha}^2} + \\frac{\\Delta \\delta^2}{\\sigma_{i_\\delta}^2+\\sigma_{j_\\delta}^2}}\n\\end{equation*}\n\\\\\nwhere $\\Delta\\alpha$ and $\\Delta\\delta$ are the difference between the right ascension and the declination of the two compared sources, respectively, $\\sigma_{a_b}$ is the uncertainty in the position of the source, and $(i,j)$ represent the two sources. The error in the position of the \\textit{Fermi} sources is taken as the 95$\\%$ confidence ellipse. The error in the position of the other compared sources is the precision in the coordinates for YSOs, WR stars and Of-type stars, and the angular size of the association in the case of OB associations. If $S$ is lower than or equal to the unit, it means that the source position (YSO, WR, Of-type, or OB associations) is inside the 95$\\%$ uncertainty ellipse of the \\textit{Fermi} source, within its own position uncertainty, and that case is considered to be a coincidence. Massive YSO and protostar are point-like objects compared with the confidence contours of Fermi sources. The same is valid for WR and Of stars, either in binary systems or isolated. OB associations are large  systems that can contain more than a single gamma-ray source. We note that our study is based on two-dimensional coincidences, since we are comparing the equatorial coordinates of the sources.\n\n\\subsection{Monte Carlo analysis}{\\label{subs:monte}}\n\nTo determine the chance coincidences we used the Monte Carlo method for simulating sets of synthetic gamma-ray sources starting from the \\textit{Fermi} Large Area Telescope First Catalog. We followed a similar criteria to that used by \\cite{rome99} to search for the possible association of unidentified \\textit{EGRET} sources with other type of celestial objects. In this algorithm, the galactic coordinates of a gamma-ray source $\\left( l, b \\right)$ are moved to new ones $\\left( l^\\prime, b^\\prime \\right)$. The new galactic longitude coordinate is calculated by doing $l^\\prime=l + R_1 \\times 360^\\circ$, where $R_1$ is a random number between 0 and 1 that never repeats from source to source or from set to set. Since the distribution of \\textit{Fermi} sources is almost constant in Galactic longitude, we do not impose any constraint on this coordinate in the simulations. The sources in the \\textit{Fermi} catalog have a given distribution in galactic latitude (see Figure \\ref{fig:hist}). In order to constrain the simulations with this distribution, the galactic latitude coordinate is calculated by doing $b^\\prime=b + R_2\\times 1^\\circ$, where again $R_2$ is a random number between 0 and 1. Here, if the integer part of $b^\\prime$ is greater than the integer part of $b$ or the sign of $b^\\prime$ is different than the sign of $b$, then $b^\\prime$ is replaced by $b^\\prime-1^\\circ$.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{histogram_fermi.pdf}}\n  \\caption{Distribution of First \\textit{Fermi} Catalog sources in galactic latitude. \\label{fig:hist}}\n\\end{figure}\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\caption{List of the catalogs used in the study.\\label{tab:catalogs}}        \n    \\scalebox{0.90}[0.90]{\n    \\begin{tabular}{lll}\n    \\hline\n    \\hline\n     Object & Catalog & $\\#$ of \\\\\n\t type   &         & sources \\\\\n     \\hline\n     $\\gamma$-ray sources & First \\textit{Fermi} Catalog\\tablefootmark{a} & 1392\\tablefootmark{f}\\\\\n     YSO & RMS Survey\\tablefootmark{b}& 637\\\\\n     WR & VII$^{th}$ Catalog of Galactic Wolf-Rayet stars\\tablefootmark{c} &  227\\\\ \n     Of-type & A catalog of galactic O stars and&  664 \\\\\n\t    & the ionization of the low density & \\\\\n\t    &  interstellar medium by runaway stars \\tablefootmark{d} & \\\\\n     OB associations & A new list of OB & 88\\\\\n        & associations in our Galaxy\\tablefootmark{e}& \\\\\n     \\hline\n     \\end{tabular}}\n     \\tablefoot{\\tablefoottext{a}{\\cite{abdo10}},\n\t \\tablefoottext{b}{\\cite{urqu08}},\n\t \\tablefoottext{c}{\\cite{vander01}},\n\t \\tablefoottext{d}{\\cite{cruz74}},\n\t \\tablefoottext{e}{\\cite{melnik95}};\n     \\tablefoottext{f}{number of gamma-ray sources after excluding the firm identifications}}\n  \\end{center}\n\\end{table} \n\nWe simulated 1500 sets of synthetic \\textit{Fermi} sources and each set was compared with a fixed set of different kinds of objects: YSOs, WR stars, Of-type, and OB associations. In each simulation, we calculated the distance between the two sources using the statistical parameter \\emph{S}.\n\nFor each kind of compared objects we calculated the average number of coincidences and its standard deviation after all Monte Carlo simulations. We calculated the chance coincidence probability using the actual number of coincidences for each type of object and assuming a Gaussian distribution in the simulations. This probability allows us to know the reliability of our study. \nWe also repeated this process moving the \\textit{Fermi} sources in $2^\\circ$-bins in galactic latitude, i.e. replacing the galactic latitude coordinate by $b^\\prime=b + R_2\\times 2^\\circ$. If the integer part of $b^\\prime$ is greater than the integer part of $b$ or the sign of $b^\\prime$ is different than the sign of $b$, then $b^\\prime$ is replaced by $b^\\prime-2^\\circ$. This binning allows us to keep the initial distribution in galactic latitude as well.\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\caption{Statistical results obtained from simulations. Latitude galactic coordinate has been constrained, while galactic longitude remains free.\\label{tab:stat}}\n    \\scalebox{0.80}[0.80]{\n    \\begin{tabular}{llr@{$\\pm$}llr@{$\\pm$}ll}\n    \\hline\n    \\hline\nObject&Coincident &\\multicolumn{2}{c}{Simulated} &Probability&\\multicolumn{2}{c}{Simulated} &Probability\\\\\ntype&$\\gamma$-ray sources &\\multicolumn{2}{c}{$1^\\circ -bin$} &$1^\\circ -bin$&\\multicolumn{2}{c}{$2^\\circ -bin$} &$2^\\circ -bin$\\\\\n\\hline\nYSO & 12 & 4.4\t& 2.0 & 1.8$\\times 10^{-4}$ & 3.6 & 1.8 & 5.6$\\times 10^{-6}$\\\\\nWR & 2 & 1.3 & 1.1 & 2.9$\\times 10^{-1}$ & 1.2 & 1.1 & 2.9$\\times 10^{-1}$\\\\\nOf-type & 5 & 2.9 & 1.7 & 1.1$\\times 10^{-1}$ & 2.9 & 1.7& 1.1$\\times 10^{-1}$\\\\\nOB assoc. & 107 & 72.5& 8.0& 4.2$\\times 10^{-6}$& 72.8&8.0&5.5$\\times 10^{-6}$\\\\\n \\hline\n \\end{tabular}}\n  \\end{center}\n\\end{table} \n\n\\input{yso_table.tex}\n\n\\section{Results}{\\label{sec:results}}\n\nThe results from our statistical study are shown in Table \\ref{tab:stat}. In this table we list, from left to right, the object type, the number of coincidences between the each compared catalog and the original \\textit{Fermi} catalog, the simulated average number of coincidences, and the chance coincidence probability for each binning in galactic latitude. We find 12 gamma-ray sources spatially coincident with YSOs, 2 with WR stars, 5 with Of-type stars and 107 with OB associations. \n\nFrom the Monte Carlo analysis, we see that there is a strong correlation between gamma-ray sources and YSOs: the catalog cross-check returns 12 coincidences between gamma-ray sources and YSOs. The Monte Carlo simulations, for the case of displacing the \\textit{Fermi} sources in 1$^\\circ$-bins, returns an average of coincidences of 4.4$\\pm$2.2 sources, which is the number of chance coincidences. This means that 7.6 of the 12 coincident \\textit{Fermi} sources ($\\sim 63\\%$ of the total coincidences with a $\\sim 4\\sigma$ confidence level ) should be associated with a probability of chance coincidence of 1.8$\\times 10^{-4}$. Similarly, for displacing the \\textit{Fermi} sources in 2$^\\circ$-bins, we obtain an average of coincidences of 3.6$\\pm$1.8. That result indicates that 8.4 of the 12 coincident sources ($\\sim 70\\%$ of the total coincidences with a $\\sim 5\\sigma$ confidence level) should be associated with a probability of $5.6 \\times 10^{-6}$ of being chance coincidences. In a similar way, the association of \\textit{Fermi} sources with WR and Of-type stars is unclear since the number of actual coincidences and the results of the Monte Carlo simulations are too similar and thus the probability of chance coincidence is too high ($0.29$ and $0.11$ for WR stars and Of-type stars, respectively). The results for WR and Of-type stars are different from those obtained by \\cite{rome99} for \\textit{EGRET} sources. The probability of chance coincidences has increased while the number of candidates decreased in the case of WR stars. In the case of Of-type stars the probability of chance association has increased as the number of coincidences increased as well. The probability of chance coincidence with OB associations is as low as $\\sim 10^{-6}$. In this case, however, the nature of the gamma-ray emission is not clear, as it is discussed in section \\ref{sec:obasso}\n\n\n\\subsection{Young stellar objects}{\\label{subsec:ysos}}\nThe association of gamma-ray sources and massive YSOs has been suggested in the last years after studying a reasonable scenario for the production of non~thermal emission \\citep{ara07}. However, this is the first time that the study of YSOs as gamma-ray sources is carried out in a statistical way, taking advantage of the \\textit{Fermi} catalog. \n\nThe results of our cross-check, shown in Table \\ref{tab:yso}, indicate that 12 gamma-ray sources are positionally coincident with 23 YSOs. In this table we present, from left to right, the \\textit{Fermi} source name, its J2000 equatorial coordinates, its positional uncertainty, the spectral $\\gamma$-ray index, the energy flux (E$>$100 MeV), the YSO name, its J2000 equatorial coordinates, the angular distance between the two compared sources, the distance to the YSO, its IR luminosity, and the mass of the star forming region where it is embedded.\n\nIn what follows we present a case by case discussion of the gamma-ray fields.\\\\\n\n- 1FGL J0541.1+3542. This source is coincident with three YSOs: G173.6328+02.8604, G173.6339+02.8218 and G173.6882+02.7222.  Their luminosities are below 4.8$\\times 10^{4}L_\\odot$. The three objects belong  to the G173.6036+02.6237 complex (in the RMS Survey notation), which is situated at a distance of 1.6 kpc. We also find two Herbig Haro like objects, GGD 5 and GGD 6 \\citep{ggd}, within the error ellipse of the \\textit{Fermi} source. Outside the \\textit{Fermi} error box the complex also harbors three more YSOs and an HII region.\\\\\n\n- 1FGL J0647.3+0031. The YSO G212.0641-00.7395 is the only coincidence with this source. Its kinematic distance is 6.4 kpc and it has a luminosity of 2.5$\\times 10^4 L_\\odot$. This source belongs to the G211.9800-00.9710 complex, together with another YSO that lies outside the error box of the gamma-ray source. \\\\\n\n-  1FGL J1256.9$-$6337. We find G303.5990-00.6524 within the error ellipse of this gamma-ray source. It is a YSO with a bolometric luminosity of $8.3\\times 10^3 L_\\odot$ and located at a kinematic distance of 11.3 kpc. It belongs to the G303.5670-00.6253 complex that also harbors an HII region. \\\\\n\n- 1FGL J1315.0$-$6235. This source is coincident with G305.4840+00.2248, which is a YSO  with a luminosity of $3.8 \\times 10^3 L_\\odot$ and located at a distance of 3.6 kpc.  \\\\\n\n- 1FGL J1651.5-4602.  The source G339.8838-01.2588 is coincident with this gamma-ray source. It is a YSO with a bolometric luminosity of 2.1$\\times 10^4 L_\\odot$. It is located at 2.6 kpc from the Earth. This source has been detected in radio at 8.6 GHz with an integrated flux of 2.6 mJy \\citep{walsh98}. There is no indication of whether the radio flux is non~thermal or not.\\\\\n\n- 1FGL J1702.4$-$4147. This source is coincident with two YSOs: G344.4257+00451B and G344.4257+00451C, which form the G344.4120+00.0492 complex, together with two HII regions. They are located at 5 kpc from the Earth and both have a bolometric luminosity of 1.5$\\times 10^4 L_\\odot$. These gamma-ray source is also near to a cluster of stars \\citep[see][]{dutra03} and a molecular cloud (see \\cite{russeil04}). \\\\\n\n- 1FGL J1846.8$-$0233.  We find a coincidence with the source G030.1981-00.1691. It has a bolometric luminosity of 2.9$\\times 10^4 L_\\odot$ and is located at 7.4 kpc. Within the error ellipse of the \\textit{Fermi} source, there are several other sources, such as dark nebulae and HII regions. \\\\\n\n- 1FGL J1848.1$-$0145. This source is coincident with two YSOs, G030.9726-00.1410 and G030.9959-00.0771, located at 5.7 kpc and with bolometric luminosities of 3.9$\\times 10^3 L_\\odot$ and 5.1$\\times 10^3 L_\\odot$, respectively. Both are part of the G031.1451+00.0383 complex, which hosts five more YSOs (outside the \\textit{Fermi} error box), five diffuse HII regions, and ten HII regions. Within the error ellipse of the gamma-ray emission, there is an unidentified very high-energy gamma-ray source, HESS~J1848-018 \\citep{chaves08}, which is probably the most suitable very high-energy candidate counterpart to the \\textit{Fermi} detection. \\\\\n\n- 1FGL J1853.1+0032. This source has the biggest error ellipse ($\\Delta\\theta_{95\\%} \\sim 0.5^\\circ$). For that reason, there is a high number of sources within its location error box, including pulsars, supernova remnants and X-ray sources. The cross-match of the catalogs yields six coincidences. These six YSOs belong to three different complexes with different distance to the Earth: G032.8205-00.3300 and G033.3891+00.1989 belong to the G033.1844-00.0572 complex, located at 5.1 kpc. The luminosities of these two sources are 1.7 and 1.1 $\\times 10^3 L_\\odot$, respectively; G033.3933+00.0100 and G33.5237+00.0198 belong to the G033.6106+00.0464 complex, located at a distance of 6.8 kpc and the luminosity is 7.9$\\times 10^3 L_\\odot$ for both sources; G034.0126-00.2832 and G034.0500-00.2977 belong to the G034.0313-00.2904 complex. The distance to these objects is 13.3 kpc and their bolometric luminosities are 3.4$\\times 10^4 L_\\odot$ and 2.4$\\times 10^4L_\\odot$, respectively. \\\\\n\n- 1FGL J1925.0+1720. Two YSOs are coincident with it: G052.2025+00.7217A and G052.2078+00.6890. The bolometric luminosities are 1.5$\\times 10^4 L_\\odot$ and 2.0 $\\times10^4 L_\\odot$, respectively. They belong to the G052.2052+00.7053 complex which is located at 10.2 kpc, and hosts also a HII region.\\\\\n\n- 1FGL J1943.4+2340. There is spatial coincidence with the source G059.7831+00.0648, located at 2.2 kpc. It has a bolometric luminosity of 6.8$\\times 10^4 L_\\odot$. This YSO has been detected in radio at 8.6 GHz with an integrated flux of 1.0 mJy \\citep{sridharan02}. The infrared counterpart of this YSO is IRAS 19410+2336. It was observed by Chandra in 2002, which found hard X-ray emission from a number of sources within this high-mass star-forming region \\citep{beut02}. The region has two cores where star formation takes place, with masses of 840 $M_\\odot$ and 190 $M_\\odot$ \\citep{sridharan02,beuther02b}. In the latter paper, it is proposed that the X-ray emission is produced by magnetic reconnection effects between the protostars and their accretion disks. The interaction of several molecular outflows, where the YSO from the RMS survey is located, and the combined effects of the stellar winds make a good scenario that might result in particle acceleration up to relativistic energies.  \\\\\n\n- 1FGL J2040.0+4157. There is spatial coincidence with G081.5168+00.1926. This YSO is located at 1.7 kpc and shows a bolometric luminosity of 7.04$\\times 10^2 L_\\odot$. There is a galaxy (2MASX J20395796+4159152) located at 2.3$^{\\prime \\prime}$ from the position of that YSO. \n\n\n\\subsection{WR and Of-type stars}\n\nWe found two \\textit{Fermi} sources coincident with 15 WR stars in the Galactic center cluster and nine WR stars in the Quintuplet cluster. Some of these stars \\citep[see][]{vander06} show variability and have been detected at X-rays with evidence of nonthermal emission.\nThe results of our study are shown in Table \\ref{tab:wr}.\nWe cannot state that these two coincidences correspond to physical associations given the high chance association probability obtained from the Monte Carlo study. The first one is in the direction of the galactic center, and there are several other sources that introduce confusion. The second one was suggested as potential association with the Pistol Star in \\cite{abdo10} and it is situated in a very crowded field. \n\nIn the case of Of-type stars, which are the evolved state of O stars, and precursors of WR stars, our results show that five \\textit{Fermi} sources are coincident with five stars (see Table \\ref{tab:of}). The probability of chance coincidence is too high to state a physical association. The sources 1FGL J1315.0-6235 and 1FGL J1853.1+0032 also show positional coincidence with YSOs (see Table \\ref{tab:yso}). The case of association with YSOs has a much lower value for the chance probability. The source 1FGL~J1112.1-6041 is coincident with HD~97434, a multiple star system. 1FGL~J1315.0-6235 is located in a regions that harbors dark nebulae and molecular clouds. The Of star coincident with this source is HD~115071, which is a spectroscopic binary. Finally, 1FGL J2004.7+3343 is coincident with HD~227465, and there is also the source G70.7+1.2 inside the error box of the \\textit{Fermi} detection, which might contain a Be star and an X-ray-emitting B star pulsar binary \\citep{cameron07}.\n\n\\input{wr_table.tex}\n\n\\subsection{OB associations}\\label{sec:obasso}\n\nOur study yields 107 \\textit{Fermi} sources positionally coincident with 35 OB associations, which represent $\\sim 41\\%$ of the sample. We list the results in Table \\ref{tab:obassoc}, available in the electronic version. We get this large number of gamma-ray sources because of the size of the OB associations. Most of them have angular sizes of $\\sim 1^\\circ$ in diameter or higher. Our results extend those of \\cite{rome99}, where they found 26 coincidences between \\textit{EGRET} sources and OB associations. Most of the OB associations that they found are present in our results, along with other new candidates, as expected from the higher sensitivity of \\textit{LAT}.  Although the number of sources has increased, the probability of chance associations is approximately the same as in \\cite{rome99}.\n\nMost of the OB associations have fewer than five \\textit{Fermi} sources within their error boxes. There are seven OB associations with five or more \\textit{Fermi} sources within their error boxes. There are four associations with a number of gamma-ray coincidences between five and ten (Ori 1 B, Ori 1 C, Car 2, and Cyg 1,8,9), located at short distances from the Earth (less than 1 kpc) and located at galactic latitudes of $\\sim |15^\\circ|$ (association centroid position). There is an exception, Car 2, which is located at 2.2 kpc and has a galactic latitude of $-0.13^\\circ$.  The OB associations with the large number of gamma-ray coincidences ($> 10$) are those from Scorpius (Sco 2 A, Sco 2 B and Sco 2 D). Those associations are located at $\\sim$ 170 pc on average and have very large angular sizes ($\\sim 9.5^\\circ$ on average). \n\nThere are five \\textit{Fermi} sources that are coincident with OB associations and YSOs at the same time: 1FGL J1256.9-6337, 1FGL~J1315.5-6235, 1FGL~J1702.4-447, 1FGL~J1943.4+2340, and 1FGL~J2040.0+4157. In all cases the OB association overlaps both the \\textit{Fermi} source and the YSO. In all cases but one, however, the distances to the YSO and the OB association are much too different.\n\n\\input{of_table.tex}\n\n\\section{Discussion}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{luminosity.pdf}}\n  \\caption{IR luminosity versus gamma-ray luminosity above 100 MeV of the brightest YSOs coincident with each \\textit{Fermi} source. The shaded area represents the 1-$\\sigma$ confidence interval of a least square fit taking eight data points randomly and repeating the fit one thousand times. It can be roughly  seen that the higher the gamma-ray luminosity, the higher the IR luminosity. \\label{fig:lumi}}\n\\end{figure}\t\n\nThe fact that there are five \\textit{Fermi} sources coincident with both YSOs and OB associations makes us consider which is the chance probability of having coincidence of \\textit{Fermi} sources with YSOs alone. Using the Monte Carlo algorithm again, but taking this constrain into account, we obtain a mean value of coincidences of $2.8\\pm 1.7$, and the chance probability for the seven \\textit{Fermi} sources coincident only with YSOs is $\\sim 1.6\\%$.\n\nThe brightest IR YSOs have more molecular mass available for proton-proton collisions and Bremsstrahlung interactions than those that are faint. It is expected then that the brightest IR YSOs would show the highest gamma-ray luminosity. To test such a trend, we have plotted in Figure \\ref{fig:lumi} the IR luminosity versus the gamma-ray luminosity of the brightest YSOs coincident with each \\textit{Fermi} source. The gamma-ray luminosity was calculated from the gamma-ray flux (see Table \\ref{tab:yso}) assuming that the distance to the gamma-ray source is equal to the distance to the corresponding YSO. We see that, under this assumption, there is a trend toward increasing the gamma-ray luminosity as the IR luminosity increases, too. This trend, however, is obtained using the whole data set, and according to the results from our Monte Carlo simulations, we should have four chance coincidences out of the 12 candidates. To see how this could affect the trend, we selected eight data points randomly and fitted them to a straight line by least squares. We repeated this process one thousand times, getting an average least square fit within one standard deviation. The limits of this fit are plotted in Figure 2. The increasing trend in the data is visible, although with a significant dispersion. This is not surprising considering the broad approach.\n\nThe association of WR stars with gamma-ray sources has been discussed in the past by \\cite{kaul97} and \\cite{rome99}. In both cases, the authors studied the positional coincidence between WR stars and unidentified EGRET sources. In the later work, they find that two WR stars are of special interest: WR 140 and WR 142. The first one is a binary system, composed of a WC 7 plus an O4-5 star where the region of collision of the winds seems to be a good place for particle acceleration and high energy emission. It should be mentioned, however, that WR 140 is a long-period binary with variability expected on time scales of years \\citep{williams87}. In the second one, WR 142, the hard X-ray emission and fast wind may indicate a colliding wind shock that could be explained by a companion close to this star \\citep{sokal10}. No companion, however, has been reported so far.\nIn our work none of these mentioned WR stars appear to be coincident with any \\textit{Fermi} source. The poor statistical correlations found from the simulations does not allow us to be confident with any of the coincidences found. The case of Of-type stars is also unclear since the probability of chance association is high. \n\nFinally, our results for the OB associations are not conclusive. We list the results in Table \\ref{tab:obassoc}, available in the electronic version. The probability of chance coincidence is negligible ($\\sim 10^{-6}$), but we do get several gamma-ray sources for each OB association. Thus, is very difficult to assign a specific counterpart to the gamma-ray emission.\n\n\\section{Conclusions}\n\nWe studied the two-dimensional coincidence between unidentified \\textit{Fermi} sources and catalogs of galactic young objects, such as YSOs, WR stars, Of stars, and OB associations. We have found a statistical correlation between gamma-ray sources and YSOs. The correlation with the early type stars with strong winds remain unclear, since the candidates are located in crowded fields with many other alternatives to the gamma-ray emission, and the probability of chance associations is high. In the case of OB associations, the probability of chance associations is negligible. However, we cannot assign a specific counterpart to the gamma-ray emission because of the high angular size of most of OB associations.\n\nWhat we have presented here is the first statistical evidence for gamma-ray emission from massive YSOs.\n\n\n\\begin{acknowledgements} \nThe authors acknowledge support from the Spanish Ministerio de Ciencia e Innovaci\\'on (MICINN) under grant AYA2010-21782-C03-01. G.E.R. also acknowledges support of PIP 0078 (CONICET).\n\\end{acknowledgements}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}