diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkvgd" "b/data_all_eng_slimpj/shuffled/split2/finalzzkvgd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkvgd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{Sec:Intro}\nConsider a dynamic network in which nodes come and go, change abruptly or evolve, alone or in communities, and form connections accordingly. Our goal is to find a vector representation, $\\hat{\\+Y}_i^{(t)} \\in \\mathbb{R}^d$, for every node $i$ and time $t$, which could be used for a diversity of downstream analyses, such as clustering, time series analysis, classification, model selection and more. This problem is known as dynamic (or evolutionary) network (or graph) embedding, and a great number of scalable and empirically successful techniques have been put forward, with recent surveys by \\cite{xie2020survey,xue2021dynamic}. We consider, among these, a subset about which it is reasonable to try to establish a certain statistical guarantee.\n\nThe novelty of this paper is \\emph{not} to propose a new procedure. Instead, it is to demonstrate that an existing procedure, unfolded adjacency spectral embedding (UASE) \\cite{jonesrubindelanchy2021MRDPG}, has two important stability properties. Given a sequence of symmetric adjacency matrices $\\+A^{(1)}, \\ldots, \\+A^{(T)} \\in \\{0,1\\}^{n \\times n}$, where $\\+A^{(t)}_{ij} = 1$ if nodes $i$ and $j$ form an edge at time $t$, UASE computes the rank $d$ matrix factorisation of $\\+A := (\\+A^{(1)}|\\cdots|\\+A^{(T)})$ to obtain $\\+A \\approx \\hat{\\+X} \\hat{\\+Y}^\\top$ using the singular value decomposition (precise details later). The matrix $\\hat{\\+Y} \\in \\mathbb{R}^{nT \\times d}$ contains, as rows, the desired representations $\\hat{\\+Y}_1^{(1)}, \\ldots,\\hat{\\+Y}_n^{(1)}, \\ldots, \\hat{\\+Y}_1^{(T)}, \\ldots,\\hat{\\+Y}_n^{(T)}.$\n\n\nThe original motivation for UASE was to analyse multilayer (or multiplex) graphs under an appropriate extension of the random dot product graph model \\cite{jonesrubindelanchy2021MRDPG}. To evaluate UASE and other procedures on the task of dynamic network embedding, we instead consider the dynamic latent position model\n\\begin{align}\n\t\\+A^{(t)}_{ij} \\overset{ind}{\\sim} \\mathrm{Bernoulli}\\left(f\\left\\{\\+Z^{(t)}_i,\\+Z^{(t)}_j\\right\\}\\right), \\label{eq:latent_position_model}\n\\end{align}\nfor $1 \\leq i < j \\leq n$, $t \\in [T]$, where $\\+Z_i^{(t)} \\in \\mathbb{R}^k$ represents the unknown position of node $i$ at time $t$, and $f: \\mathbb{R}^{k} \\times \\mathbb{R}^k \\rightarrow [0,1]$ is a symmetric function. \n\nThis model is well-established \\cite{sarkar2005dynamic,lee2011latent,hoff2011hierarchical,hoff2011separable,robinson2012detecting,lee2013latent,durante2014nonparametric,sewell2015latent,friel2016interlocking,durante2017bayesian}, with recent reviews by \\cite{kim2018review} and \\cite{turnbull2020advancements}, and inference usually proceeds on the basis of a parametric model for $f$ (e.g. logistic in the latent position distance \\cite{sarkar2005dynamic}) and for the dynamics of $\\+Z_i^{(t)}$ (e.g. a Markov process \\cite{sarkar2005dynamic}). The model also includes the dynamic degree-corrected, mixed-membership and standard stochastic block models as special cases, which were studied in \\cite{xing2010state,yang2011detecting,ho2011evolving,liu2014persistent,xu2014dynamic,xu2015stochastic,matias2015statistical,bhattacharyya2018spectral,pensky2019spectral,keriven2020sparse}.\n\nTo make statistical sense of UASE under this latent position model, we must somehow connect its output $\\hat{\\+Y}_i^{(t)}$ to $\\+Z_i^{(t)}$. To this end we construct a canonical representative of $\\+Z_i^{(t)}$, denoted $\\+Y_i^{(t)}$, that UASE can be seen to estimate. Although we make some regularity assumptions on $f$ and the $\\+Z_i^{(t)}$, they are not modelled in an explicit, parametric way, and UASE can clearly be used in practice without having a specific model in mind. For example, we do not make a Markovian assumption on the evolution of $\\+Z_i^{(t)}$ and UASE can be used to uncover periodic behaviours. \n\n\nThe key purpose of imposing a dynamic latent position model is to allow us to put down certain embedding stability requirements. Using this framework, we can define precisely what we mean by two nodes, $i$ and $j$, behaving ``similarly'' at times $s$ and $t$ respectively. In such cases, ideally we would have $\\hat{\\+Y}_i^{(s)} \\approx\\hat{\\+Y}_j^{(t)}$. Two special cases are: to assign the same position, up to noise, to nodes behaving similarly at a given time (cross-sectional stability) and a constant position, up to noise, to a single node behaving similarly across different times (longitudinal stability).\n\n\nTo achieve both cross-sectional and longitudinal stability is generally elusive. We show that two plausible alternatives, omnibus \\cite{levin2017central} and independent embedding of each $\\+A^{(t)}$, alternately exhibit one form of stability and not the other. More generally, we find existing procedures \\cite{chi2007evolutionary,lin2008facetnet,scheinerman2010modeling,dunlavy2011temporal,zhu2016scalable,deng2016latent,liu2018global,chen2018exploiting,bhattacharyya2018spectral,pensky2019spectral,passino2019link,keriven2020sparse} tend to trade one type of stability off against the other, e.g. via user-specified cost functions. As a side-note, it could be observed that omnibus embedding is not being evaluated on a task for which it was designed, since in the theory of \\cite{levin2017central} the graphs are identically distributed. Because the technique is so different from the others, we still feel it makes an interesting addition.\n\nOur central contribution is to prove that UASE asymptotically provides both longitudinal and cross-sectional stability: for two nodes, $i$ and $j$, behaving similarly at times $s$ and $t$ respectively, we have $\\hat{\\+Y}_i^{(s)} \\approx\\hat{\\+Y}_j^{(t)}$ and, moreover, $\\hat{\\+Y}_i^{(s)}$ and $\\hat{\\+Y}_j^{(t)}$ have asymptotically equal error distribution. We emphasise that these properties hold without requiring any sort of global stability --- the network could vary wildly over time but certain nodes still stay fixed. In the asymptotic regime considered, we have $n \\rightarrow \\infty$, but $T$ fixed so that, for example, our results are relevant to the case of two large graphs. The alternative regime where $T \\rightarrow \\infty$ grows but $n$ is fixed is not easily handled by existing theory and to provide constant updates to UASE (or omnibus embedding) in a streaming context presents significant computational challenges.\n\n\n\nThe remainder of this article is structured as follows. Section~\\ref{Sec:Example} gives a pedagogical example demonstrating the cross-sectional and longitudinal stability of UASE in a two-step dynamic stochastic block model, while highlighting the instability of omnibus and independent spectral embedding. In Section~\\ref{Sec:Setup}, we prove a central limit theorem for UASE under a dynamic latent position model, and demonstrate that the distribution satisfies both stability conditions. In Section~\\ref{Sec:Comp}, we review the stability of other dynamic network embedding procedures.\nSection~\\ref{Sec:Data} presents an example of UASE applied to a dynamic network of social interactions in a French primary school, with a dynamic community detection example given in the main text and a further classification example provided in the Appendix. Section~\\ref{Sec:Concs} concludes.\n\n\n\\section{Motivating example}\n\\label{Sec:Example}\nSuppose it is of interest to uncover dynamic community structure, including communities changing, merging or splitting. The dynamic stochastic block model \\cite{yang2011detecting,xu2014dynamic} provides a simple explicit model for this, in which two nodes connect at a certain point in time with probability only dependent on their current community membership. Suppose that at times 1 and 2, we have the following inter-community link probability matrices,\n\\begin{equation*}\n \\+B^{(1)} = \\left(\n \t\\begin{matrix}\n \t0.08 & 0.02 & 0.18 & 0.10 \\\\\n \t0.02 & 0.20 & 0.04 & 0.10 \\\\\n \t0.18 & 0.04 & 0.02 & 0.02 \\\\\n \t0.10 & 0.10 & 0.02 & 0.06\n \\end{matrix} \\right), \\quad\n \\+B^{(2)} = \\left(\n \\begin{matrix}\n \t0.16 & 0.16 & 0.04 & 0.10 \\\\\n \t0.16 & 0.16 & 0.04 & 0.10 \\\\\n \t0.04 & 0.04 & 0.09 & 0.02 \\\\\n \t0.10 & 0.10 & 0.02 & 0.06\n \\end{matrix} \\right).\n\\end{equation*}\nAt time 1 there are four communities present, for example, a node of community 1 connects with a node of community 3 with probability 0.18. At time 2, this matrix changes so that communities 1 and 2 have merged, community 3 has moved, whereas community 4 is unchanged. For simplicity, in this example, the community membership of each node is fixed across time-steps.\n\nWe simulate a dynamic network from this model over these two time steps, on $n = 1000$ nodes, equally divided among the four communities and investigate the results of three embedding techniques: UASE, omnibus, and independent spectral embedding, displayed in Figure~\\ref{Fig:Embed_Comparison}. Note that the different techniques produce embeddings of different dimensions; UASE embeds into $d=4$ dimensions, omnibus embedding $\\tilde d=7$, while independent spectral embedding has $d_1=4$ and $d_2=3$. For visualisation, we show the leading two dimensions for each embedding. Given the dynamics described above, we contend that the following properties would be desirable:\n\\begin{enumerate}\n\\item \\emph{Cross-sectional stability}: The embeddings for communities 1 and 2 at time 2 are close.\n\\item \\emph{Longitudinal stability}: The embeddings for community 4 at times 1 and 2 are close.\n\\end{enumerate}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.80\\textwidth]{Embedding_Comparison.pdf}\n\t\\caption{First two dimensions of the embeddings for the adjacency matrices $\\+A^{(1)}$ and $\\+A^{(2)}$ using three different techniques: UASE, omnibus and separate embedding. The points are coloured according to true community membership, with the black dots showing the fitted community centroid and the ellipses a fitted 95\\% level Gaussian contour.}\n\t\\label{Fig:Embed_Comparison}\n\\end{figure}\nFigure~\\ref{Fig:Embed_Comparison} illustrates that UASE has both properties: the blue and orange point clouds merge at time 2, and the red point cloud has the same location \\emph{and} shape over the two time points. On the other hand, omnibus embedding only has longitudinal stability (the blue and orange point clouds don't merge at time 2), whereas independent spectral embedding only has cross-sectional stability (the red point cloud is at different locations over the two time points).\n\n\\section{Theoretical background and results}\n\\label{Sec:Setup}\nWe consider a sequence of random graphs distributed according to a dynamic latent position model, in which the latent position sequences $(\\+Z^{(t)}_i)_{t \\in [T]}$ (where we use the shorthand $[T] = \\{1,\\ldots,T\\}$) are independent of each other, and identically distributed according to a joint distribution $\\mathcal{F}$ on $\\mathcal{Z}^T$, for a bounded subset $\\mathcal{Z}$ of $\\mathbb{R}^k$. We emphasise that for a fixed $i$ the latent positions $\\+Z^{(t)}_i$ and $\\+Z^{(t')}_i$ may have different distributions --- we are imposing exchangeability over the nodes (invariance to node labelling), but not over time. \n\nBy extending the function $f$ to be zero outside of its domain of definition, we may view it as an element of $L^2(\\mathbb{R}^k \\times \\mathbb{R}^k)$, and consequently may define a compact, self-adjoint operator $A$ on $L^2(\\mathbb{R}^k)$ by setting \n\\begin{align}\n Ag(x) = \\int_{\\mathbb{R}^k} f(x,y)g(y)dy\n\\end{align}\nfor all $g \\in L^2(\\mathbb{R}^k)$. The operator $A$ has a---possibly infinite---sequence of non-zero eigenvalues $\\lambda_1 \\geq \\lambda_2 \\geq \\ldots$ with corresponding orthonormal eigenfunctions $u_1, u_2, \\ldots \\in L^2(\\mathbb{R}^k)$, such that $Au_j = \\lambda_ju_j$ (for further details, see for example \\cite{rubindelanchy2021manifold}). We shall make the simplifying assumption that $A$ has a \\emph{finite} number of non-zero eigenvalues, in which case $f$ admits a canonical eigendecomposition \n\\begin{align}\n\tf(\\+x,\\+y) = \\sum_{i=1}^D \\lambda_i u_i(\\+x)u_i(\\+y),\n\\end{align}\nassumed to hold everywhere, where after relabelling we may assume that $|\\lambda_1| \\geq \\ldots \\geq |\\lambda_D|$. \n\nSeveral families of functions satisfy the finite rank assumption $D < \\infty$, such as the multivariate polynomials \\cite{rubindelanchy2021manifold}. Moreover, several existing statistical models can be written as dynamic latent position network models in which $D < \\infty$, including the dynamic mixed membership, degree-corrected, and standard stochastic block models. To assume $D < \\infty$, more generally, is tantamount to a claim that, for large $n$, $\\+A^{(t)}$ has `low' approximate rank: an overwhelming proportion of its eigenvalues are close to zero. Large matrices with low approximate rank are routinely encountered across numerous disciplines, and the study \\cite{udell2019datamatrices} provides a hypothesis for this ``puzzling'' general observation. However, given a real network, we might reject the hypothesis that $f$ has low rank (we must then also reject any type of stochastic block model), for example on the basis of triangle counts \\cite{seshadhri2020triangles}. In such a setting, we anticipate that UASE is still consistent and stable if $d$ is allowed to grow sufficiently slowly with $n$ and there exist asymptotic results for adjacency spectral embedding, for the single graph case, showing convergence in Wasserstein distance under assumptions on eigenvalue decay which can be related to the smoothness of $f$ \\cite{lei2021network}. However, even if we could obtain such results for UASE, they would not be as powerful as those we present here for the finite rank case, where we show uniform consistency with asymptotic Gaussian error.\n\n\\begin{theorem}\\label{TRDPG_to_MRDPG} The adjacency matrices $\\+A^{(1)}, \\ldots, \\+A^{(T)}$ are jointly distributed according to a multilayer random dot product graph model.\n\\end{theorem}\n\nThe multilayer random dot product graph model (MRDPG, see \\cite{jonesrubindelanchy2021MRDPG}) is a multi-graph analogue of the generalised random dot product graph model \\cite{rubindelanchyetal2020GRDPG}, and is characterised by the existence of matrices $\\+\\Lambda^{(t)} \\in \\mathbb{R}^{d \\times d_t}$ and random matrices $\\+X \\in \\mathbb{R}^{n \\times d}$ and $\\+Y^{(t)} \\in \\mathbb{R}^{n \\times d_t}$ (generated according to some joint distribution $\\mathcal{G}$) such that \n\\begin{align}\n \\+A^{(t)}_{ij} \\overset{ind}{\\sim} \\mathrm{Bernoulli}\\bigl(\\+X_i\\+\\Lambda^{(t)}\\+Y^{(t)\\top}_j\\bigr)\n\\end{align}\nfor each $t \\in [T]$ and $i,j \\in [n]$, where $\\+X_i$ and $\\+Y^{(t)}_j$ denote the $i$th and $j$th rows of $\\+X$ and $\\+Y^{(t)}$ respectively. \n\nWe refer the reader to the supplemental material for full details of the proof of Theorem \\ref{TRDPG_to_MRDPG}, but note that given knowledge of the function $f$ and the underlying distribution $\\mathcal{F}$ we can construct explicit maps $\\varphi: \\mathbb{R}^{Tk} \\to \\mathbb{R}^d$ and $\\varphi_t: \\mathbb{R}^k \\to \\mathbb{R}^{d_t}$, producing row vectors, and matrices $\\+\\Lambda^{(t)} \\in \\mathbb{R}^{d \\times d_t}$ such that \n\\begin{align}\n f(\\+Z^{(t)}_i,\\+Z^{(t)}_j) = \\+X_i \\+\\Lambda^{(t)} \\+Y_j^{(t)\\top},\n\\end{align}\nwhere $\\+X_i = \\varphi(\\+Z_i), \\+Y^{(t)}_j = \\varphi_t(\\+Z_j^{(t)})$ and $\\+Z_i = (\\+Z^{(1)}_i|\\cdots|\\+Z^{(T)}_i) \\in \\mathbb{R}^{Tk}$, for each $t \\in [T]$ and $i, j \\in [n]$. Consequently, each Gram matrix \n\\begin{align}\n \\+P^{(t)} = \\bigl(f(\\+Z^{(t)}_i,\\+Z^{(t)}_j)\\bigr)_{i,j \\in [n]}\n\\end{align}\nadmits a factorisation into a product of low-rank matrices, in which the matrix whose rows are the vectors $\\varphi(\\+Z_i)$ appears as a common factor. The dimensions $d_t$ and $d$ are precisely the ranks of the matrices $\\+P^{(t)}$ and their concatenation $\\+P = (\\+P^{(1)}|\\cdots|\\+P^{(T)})$ respectively. In the theory that follows we will assume that the dimension $d$ is known and fixed, whereas in practice we would usually have to estimate it (for example by using profile likelihood \\cite{zhu2006automatic}). \n\nWe can extend our model to incorporate a range of sparsity regimes by scaling the function $f$ by a sparsity factor $\\rho_n$, which we assume is either constant and equal to $1$, or else tends to zero as $n$ grows, corresponding to dense and sparse regimes respectively. When this factor is present, the maps $\\varphi$ and $\\varphi_t$ are scaled by a factor of $\\rho_n^{1\/2}$. \n\nRealising our model as an MRDPG allows us to make precise statements about the asymptotic behaviour of the point clouds obtained through UASE. In particular, it is the existence of a common factor in the decomposition of each of the Gram matrices $\\+P^{(t)}$ that gives rise to the stability properties previously demonstrated in the embeddings $\\hat{\\+Y}^{(t)}$, as this matrix in a sense acts as an anchor for the individual point clouds.\n\nIn \\cite{jonesrubindelanchy2021MRDPG} UASE is used to obtain \\emph{two} multi-graph embeddings; the algorithm below retains only one --- the `dynamic' component.\n\\vspace*{-0.1cm}\n\\begin{algorithm}[H]\n \\caption{Unfolded adjacency spectral embedding for dynamic networks}\\label{alg:spec}\n\\begin{algorithmic}[1]\n \\Statex \\textbf{input} symmetric adjacency matrices $\\+A^{(1)}, \\ldots, \\+A^{(T)} \\in \\{0,1\\}^{n \\times n}$, embedding dimension $d$\n \\State Form the matrix $\\+A = (\\+A^{(1)}|\\cdots|\\+A^{(T)}) \\in \\{0,1\\}^{n \\times Tn}$ (column concatenation)\n \\State Compute the truncated singular value decomposition $\\+U_\\+A\\+\\Sigma_\\+A\\+V_\\+A^\\top$ of $\\+A$, where $\\+\\Sigma_\\+A$ contains the $d$ largest singular values of $\\+A$, and $\\+U_\\+A, \\+V_\\+A$ the corresponding left and right singular vectors\n \\State Compute the right embedding $\\hat{\\+Y} = \\+V_\\+A\\+\\Sigma_\\+A^{1\/2} \\in \\mathbb{R}^{Tn \\times d}$ and create sub-embeddings $\\hat{\\+Y}^{(t)} \\in \\mathbb{R}^{n \\times d}$ where $\\hat{\\+Y} = (\\hat{\\+Y}^{(1)};\\ldots;\\hat{\\+Y}^{(T)})$ (row concatenation)\n \\Statex \\Return node embeddings for each time period $\\hat{\\+Y}^{(1)},\\ldots,\\hat{\\+Y}^{(T)}$\n\\end{algorithmic}\n\\end{algorithm}\n\\vspace*{-0.1cm}\n\nReplacing $\\+A$ with $\\+P$ in this construction yields the \\emph{noise-free} embeddings $\\tilde{\\+Y}^{(t)}$, whose rows are known to be linear transformations of the vectors $\\varphi_t(\\+Z^{(t)}_i)$ (see the supplemental material for further details). A desirable property of UASE is that since the matrices $\\+A^{(t)}$ follow an MRDPG model there exist known asymptotic distributional results for the embeddings $\\hat{\\+Y}^{(t)}$. In order to ensure the validity of these results, we will make the assumption that the sparsity factor $\\rho_n$ satisfies $\\rho_n = \\omega(n^{-1} \\log^c(n))$ for some universal constant $c > 1$.\n\nIn the results to follow, UASE is shown to be consistent and stable in a uniform sense, i.e. the maximum error of any position estimate goes to zero, under the assumption that the average network degree grows polylogarithmically in $n$. To achieve this for less than logarithmic growth would be impossible, by any algorithm, because it would violate the information-theoretic sparsity limit for perfect community recovery under the stochastic block model \\cite{abbe2017sbm}. In practice this means that the embedding will have high variance when the graphs have few edges. Several approaches have been proposed to make single graph embeddings more robust (e.g. to sparsity or heterogeneous degrees), such as based on the regularised Laplacian \\cite{amini2013pseudo} or non-backtracking matrices \\cite{krzakala2013spectral}, but it is an open and interesting question how to extend them to the dynamic graph setting to achieve both cross-sectional and longitudinal stability.\n\nThe first of our distributional results states that after applying an orthogonal transformation---which leaves the structure of the resulting point cloud intact---the embedded points $\\hat{\\+Y}^{(t)}_i$ converge in the Euclidean norm to the noise-free embedded points $\\tilde{\\+Y}^{(t)}_i$ as the graph size grows:\n\n\\begin{proposition}\\label{consistency} There exists a sequence of orthogonal matrices $\\tilde{\\+W} \\in \\mathrm{O}(d)$ such that \n\\begin{align}\n \\max_{i \\in \\{1,\\ldots,n\\}} \\bigl\\|\\hat{\\+Y}^{(t)}_i \\tilde{\\+W}- \\tilde{\\+Y}^{(t)}_i\\bigr\\| = \\mathrm{O}\\Bigl(\\tfrac{\\log^{1\/2}(n)}{\\rho_n^{1\/2}n^{1\/2}}\\Bigr)\n\\end{align} \nwith high probability for each $t$.\n\\end{proposition}\n\nThe matrices $\\tilde{\\+W}$ are unidentifiable in practice, but are defined to be the solution to the one-mode orthogonal Procrustes problem\n\\begin{align}\n\t\\tilde{\\+W} = \\argmin_{\\+Q \\in \\mathrm{O}(d)} \\|\\+U_\\+A \\+Q - \\+U_\\+P\\|_F^2 + \\|\\+V_\\+A \\+Q- \\+V_\\+P\\|_F^2,\n\\end{align} where $\\lVert \\cdot \\rVert^2_F$ denotes the Frobenius norm. We emphasise that the matrix $\\tilde{\\+W}$ is \\emph{the same} for each embedding, and so similar behaviour in the noise-free embeddings $\\tilde{\\+Y}^{(t)}$ is captured by UASE.\n\nOur second result states that after applying a \\emph{second} orthogonal transformation the above error converges in distribution to a fixed multivariate Gaussian distribution:\n\n\\begin{proposition}\\label{CLT} Let $\\zeta = (\\zeta_1|\\cdots|\\zeta_T) \\sim \\mathcal{F}$, and for $\\+z \\in \\mathcal{Z}$ define\n\\begin{align}\n\t\\+\\Sigma_t(\\+z) = \\left\\{\\begin{array}{cc}\\mathbb{E}_\\zeta\\Bigl[f(\\+z,\\zeta_t)\\bigl(1-f(\\+z,\\zeta_t)\\bigr) \\cdot \\varphi(\\zeta)^\\top\\varphi(\\zeta)\\Bigr]&\\mathrm{if~}\\rho_n = 1\\\\\n\t\\mathbb{E}_\\zeta\\Bigl[f(\\+z,\\zeta_t) \\cdot \\varphi(\\zeta)^\\top\\varphi(\\zeta)\\Bigr]&\\mathrm{if~}\\rho_n \\to 0.\\end{array}\\right..\n\\end{align} \nThen there exists a deterministic matrix $\\+R_* \\in \\mathbb{R}^{d \\times d}$ and a sequence of orthogonal matrices $\\+W \\in \\mathrm{O}(d)$ such that, given $\\+z \\in \\mathcal{Z}$, for all $\\+y \\in \\mathbb{R}^d$ and for any fixed $i \\in [n]$ and $t \\in [T]$, \n\\begin{align}\n\t\\mathbb{P}\\Bigl(n^{1\/2}(\\hat{\\+Y}^{(t)}_i \\tilde{\\+W} - \\tilde{\\+Y}^{(t)}_i)\\+W \\leq \\+y ~|~ \\+Z^{(t)}_i = \\+z\\Bigr) \\to \\Phi\\bigl(\\+y,\\+R_*\\+\\Sigma_t(\\+z)\\+R_*^\\top\\bigr).\n\\end{align}\n\\end{proposition}\n\nAs in our previous results, the matrices $\\+W$ and $\\+R_*$ can be explicitly constructed given knowledge of the underlying function $f$ and distribution $\\mathcal{F}$ (again, we defer full details to the supplemental material). Note that as in Proposition \\ref{consistency}, both constructed matrices are common to \\emph{all} embeddings.\n\nWe can now demonstrate one of the key advantages that UASE holds over other embedding methods, namely that \\emph{UASE exhibits both cross-sectional and longitudinal stability}, in a sense that we shall now define. We say that two space-time positions $(\\+z,t)$ and $(\\+z',t')$ are \\emph{exchangeable} if $f(\\+z,\\zeta_t) = f(\\+z',\\zeta_{t'})$ with probability one, where $\\zeta = (\\zeta_1|\\cdots|\\zeta_T) \\sim \\mathcal{F}$, and that the positions are \\emph{exchangeable up to degree} if $f(\\+z,\\zeta_t) = \\alpha f(\\+z',\\zeta_{t'})$ for some $\\alpha >0$. Equivalently, $(\\+z,t)$ and $(\\+z',t')$ are exchangeable if conditional on $\\+Z^{(t)}_i = \\+z$ and $\\+Z^{(t')}_j = \\+z'$ the $i$th row of $\\+P^{(t)}$ and $j$th row of $\\+P^{(t')}$ are equal with probability one.\n\n\\begin{definition}\nGiven a generic method for dynamic network embedding, with output denoted $(\\hat{\\+Z}^{(t)}_i)_{i \\in [n]; t \\in [T]}$, define the following stability properties:\n\\begin{enumerate}\n \\item \\emph{Cross-sectional stability:} Given exchangeable $(\\+z,t)$ and $(\\+z',t)$, $\\hat{\\+Z}^{(t)}_i$ and $\\hat{\\+Z}^{(t)}_j$ are asymptotically equal, with identical error distribution, conditional on $\\+Z^{(t)}_i = \\+z$ and $\\+Z^{(t)}_j = \\+z'$.\n \\item \\emph{Longitudinal stability:} Given exchangeable $(\\+z,t)$ and $(\\+z,t')$, $\\hat{\\+Z}^{(t)}_i$ and $\\hat{\\+Z}^{(t')}_i$ are asymptotically equal, with identical error distribution, conditional on\n $\\+Z^{(t)}_i = \\+Z^{(t')}_i = \\+z$.\n\\end{enumerate}\n\\end{definition}\n\n\nThe following result then shows that UASE exhibits both types of stability:\n\n\\begin{corollary}\\label{stability} Conditional on $\\+Z^{(t)}_i = \\+z$ and $\\+Z^{(t')}_j = \\+z'$, the following properties hold:\n\\begin{enumerate}\n \\item If $(\\+z,t)$ and $(\\+z',t')$ are exchangeable then $\\hat{\\+Y}^{(t)}_i$ and $\\hat{\\+Y}^{(t')}_j$ are asymptotically equal, with identical error distribution.\n \\item If $(\\+z,t)$ and $(\\+z',t')$ are exchangeable up to degree then $\\hat{\\+Y}^{(t)}_i$ and $\\alpha\\hat{\\+Y}^{(t')}_j$ are asymptotically equal and, under a sparse regime, their error distributions are equal up to scale, satisfying $\\+\\Sigma_t(\\+z) = \\alpha\\+\\Sigma_{t'}(\\+z')$. \n\\end{enumerate}\n\\end{corollary}\n\n\nWe can gain insight into our theoretical results by applying them in the context of common statistical models. Under the dynamic stochastic block model of Section~\\ref{Sec:Example}, a pair $(\\+z,t)$ and $(\\+z',t')$ are exchangeable if and only if the corresponding rows of $\\+B^{(t)}$ and $\\+B^{(t')}$ are identical. Proposition \\ref{CLT} then predicts that the point cloud obtained via UASE should decompose into a finite number of \\emph{Gaussian} clusters, as we observe in Figure \\ref{Fig:Embed_Comparison}. Moreover, when combined with Corollary \\ref{stability} the equality of the fourth rows of $\\+B^{(1)}$ and $\\+B^{(2)}$ implies that we should expect \\emph{identical} clusters (centre and shape) corresponding to the fourth community at both time points (indicating longitudinal stability) and similarly the equality of the first and second rows of $\\+B^{(2)}$ tells us that the communities should be indistinguishable at the second time point (indicating cross-sectional stability), behaviours we observe in Figure \\ref{Fig:Embed_Comparison}. Under degree-correction \\cite{karrer2011stochastic,liu2014persistent}, Corollary \\ref{stability} indicates that we should at least observe cross-sectional and longitudinal stability along `rays' representing communities, a behaviour that is exhibited in Figure \\ref{Fig:Primary_UASE}.\n\n\\section{Comparison}\n\\label{Sec:Comp}\nIn this section we investigate the stability properties of alternatives to UASE. Table~\\ref{tab:Algorithm_Comp} shows the stability of the three embedding algorithms described in Section~\\ref{Sec:Example} and two wider classes of algorithms, described below. \nFor alternatives to UASE, it will be considered sufficient to establish whether an embedding is stable when applied to the Gram matrices $\\+P^{(1)}, \\ldots, \\+P^{(T)}$. A method found to be unstable in this noise-free condition is not expected to be stable when the matrices are replaced by their noisy observations $\\+A^{(1)}, \\ldots, \\+A^{(T)}$.\n\n\n\\begin{table}[ht]\n \\caption{Classes of dynamic network embedding algorithms: a brief description, the algorithm complexity and its cross-sectional\/longitudinal stability. In the three latter classes, one may replace $\\+A^{(t)}$ with other matrix representations of the graph (e.g. the normalised Laplacian). Further details in main text.}\n \\label{tab:Algorithm_Comp}\n \\centering\n \\begin{tabular}{@{}lp{5.9cm}cc@{}}\n \\toprule\n \\textbf{Algorithm} & \\textbf{Description} & \\textbf{Complexity} & \\textbf{Stability} \\\\\n \\midrule\n \\addlinespace[-0em]\n UASE \\cite{jonesrubindelanchy2021MRDPG} & Embed $\\+A = (\\+A^{(1)}| \\cdots | \\+A^{(T)})$ & $\\mathrm{O}(d T n^2)$ & Both \\\\\n \\hline\n Omnibus \\cite{levin2017central} & Embed $\\tilde{\\+A}$; $\\tilde{\\+A}_{s,t} = (\\+A^{(s)} + \\+A^{(t)})\/2$ & $\\mathrm{O}(\\tilde{d} T^2 n^2)$ & Longitudinal \\\\\n \\hline\n Independent & Embed $\\+A^{(t)}$ & $\\mathrm{O}(\\sum_t d_t n^2)$ & Cross-sectional \\\\\n \\hline\n Separate embedding & Embed $\\bar{\\+A}^{(t)} = \\sum_k w_k \\+A^{(t-k)}$ & & Neither \\\\\n \\cite{scheinerman2010modeling,dunlavy2011temporal,bhattacharyya2018spectral,pensky2019spectral,passino2019link,keriven2020sparse} & & \\\\\n \\hline\n Joint embedding & $\\argmin_{\\hat{\\+Y}^{(1)}, \\ldots, \\hat{\\+Y}^{(T)}} \\alpha \\sum_t \\text{CS}(\\hat{\\+Y}^{(t)}, \\+A^{(t)})$ & & Neither \\\\\n \\cite{lin2008facetnet,deng2016latent,zhu2016scalable,chen2018exploiting,liu2018global} & $\\qquad + (1 - \\alpha) \\sum_t \\text{CT}(\\hat{\\+Y}^{(t)}, \\hat{\\+Y}^{(t+1)})$ & & \\\\\n \\addlinespace[-0em]\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\nThe omnibus method computes the spectral embedding of the matrix $\\tilde{\\+A} \\in \\{0,1\\}^{nT \\times nT}$ where the block $\\tilde{\\+A}_{s,t} \\in \\mathbb{R}^{n \\times n}$ is given by $(\\+A^{(k)} + \\+A^{(\\ell)})\/2 \\in \\mathbb{R}^{n \\times n}$ and we denote by $\\tilde{\\+P} \\in [0,1]^{nT \\times nT}$ the noise-free counterpart of $\\tilde{\\+A}$. If $(\\+z, t)$ and $(\\+z, t')$ are exchangeable, then, conditional on $\\+Z^{(t)}_i =\\+Z^{(t')}_i = \\+z$, the rows $\\tilde{\\+P}_{n(t-1) + i}$ and $\\tilde{\\+P}_{n(t'-1) + i}$ are equal. However, if $(\\+z, t)$ and $(\\+z', t)$ are exchangeable, then, in general, $\\tilde{\\+P}_{n(t-1) + i} \\ne \\tilde{\\+P}_{n(t-1) + j}$ when $\\+Z^{(t)}_i = \\+z$ and $\\+Z^{(t)}_j = \\+z'$. One can demonstrate that two rows of $\\tilde{\\+P}$ are equal if and only if the corresponding nodes' embeddings are too. Therefore, omnibus embedding provides longitudinal but not cross-sectional stability\n\nIndependent adjacency spectral embedding computes the spectral embeddings of the matrices $\\+A^{(t)} \\in \\{0,1\\}^{n \\times n}$. If $(\\+z, t)$ and $(\\+z', t')$ are exchangeable, then, conditional on $\\+Z^{(t)}_i = \\+z$ and $\\+Z^{(t')}_j = \\+z'$, the rows $\\+P^{(t)}_i$ and $\\+P^{(t')}_j$ are equal. If $t=t'$ the embeddings of nodes $i$ and $j$ are equal, however, embeddings between different graphs are subject to (possibly indefinite) orthogonal transformations $\\+Q^{(t)}$ which, if $\\+P^{(t)} \\neq \\+P^{(t')}$, differ in a non-trivial way (i.e. \\emph{beyond} simply reflecting the ambiguity of choosing eigenvectors in the spectral decomposition of $\\+P^{(t)}$). Therefore, independent adjacency spectral embedding provides cross-sectional but not longitudinal stability. The same arguments extend to other independent embeddings\n\nSeparate embedding covers a collection of embedding techniques separately applied to time-averaged matrices, $\\bar{\\+A}^{(t)} = \\sum_k w_k \\+A^{(t-k)}$ where $w_k$ are non-negative weights, and $\\+A^{(t)}$ may be replaced by another matrix representation of the graph such as the normalised Laplacian. The weights may be constant, e.g. $w_k = 1\/t$ for all $k$ \\cite{scheinerman2010modeling,bhattacharyya2018spectral}, exponential forgetting factors $w_k = (1-\\lambda)^k$ \\cite{dunlavy2011temporal, keriven2020sparse}, chosen to produce a sliding window \\cite{pensky2019spectral}, based on a time series model \\cite{passino2019link}, and more. In general, temporal smoothing results in two nodes behaving identically at time $t$ being embedded differently if their past or future behaviours differ, whereas the act of embedding the matrices separately will result in the same issues of alignment encountered in independent adjacency spectral embedding. Therefore, those methods can have neither cross-sectional nor longitudinal stability, except in special cases where one can contrive to have one but not other (e.g., $w_0 = 1$ and $w_k = 0$, reducing to independent embedding).\nJoint embedding techniques generally aim to find an embedding to trade-off two costs: a `snapshot cost' (CS) measuring the goodness-of-fit to the observed $\\+A^{(t)}$, and a `temporal cost' (CT) penalising change over time. These are then combined into a single objective function, for example, as \\cite{liu2018global} (where we have replaced the normalised Laplacian by the adjacency matrix):\n\\begin{equation}\n \\argmin_{\\breve{\\+A}^{(1)}, \\ldots, \\breve{\\+A}^{(T)}} \\alpha \\sum_{t=1}^T \\lVert \\+A^{(t)} - \\breve{\\+A}^{(t)}\\rVert^2_F + (1 - \\alpha) \\sum_{t=1}^{T-1} \\lVert \\breve{\\+A}^{(t)} - \\breve{\\+A}^{(t+1)}\\rVert^2_F, \\label{eq:gse}\n\\end{equation}\nwhere $\\alpha \\in [0,1]$, subject to a low rank constraint on $\\breve{\\+A}^{(1)}, \\ldots, \\breve{\\+A}^{(T)}$, where $\\hat{\\+Y}^{(t)}$ is the spectral embedding of $\\breve{\\+A}^{(t)}$. It is easy to see that a change affecting only a fraction of the nodes will result in a change of all node embeddings, precluding both cross-sectional and longitudinal stability (again apart from contrived cases, e.g. when $\\alpha = 1$). \n\nFor the complexity calculations, we assume a dense regime in which the $k$-truncated singular value decomposition of an $m$-by-$n$ matrix is $\\mathrm{O}(kmn)$ \\cite{gu1996efficient}. Independent embedding is more efficient than UASE, on account of $\\sum_t d_t \\le dT$, but UASE is more efficient than omnibus embedding, because of the linear versus quadratic growth in $T$, while $\\tilde{d} = \\rank(\\tilde{\\+P})$ is often larger than $d$.\n\n\\section{Real data}\n\\label{Sec:Data}\nThe Lyon primary school data set shows the social interactions at a French primary school over two days in October 2009 \\cite{stehle2011high}. The school consisted of 10 teachers and 241 students from five school years, each year divided into two classes. Face-to-face interactions were detected when radio-frequency identification devices worn by participants (10 teachers and 232 students gave consent to be included in the experiment) were in close proximity over an interval of 20 seconds and recorded as a pair of anonymous identifiers together with a timestamp. The data are available for download from the Network Repository website\\footnote{\\url{https:\/\/networkrepository.com}} \\cite{nr-aaai15}.\n\nA time series of networks was created by binning the data into hour-long windows over the two days, from 08:00 to 18:00 each day. If at least one interaction was observed between two people in a particular time window, an edge was created to connect the two nodes in the corresponding network. This results in a time series of graphs $\\+A^{(1)}, \\ldots, \\+A^{(20)}$ each with $n = 242$ nodes. Where a node is not active in a given time window, it is still included in the graph as an isolated node. This is compatible with the theory and method, and the node is embedded to the zero vector at that time point. \n\nGiven the unfolded adjacency matrix $\\+A = (\\+A^{(1)} | \\cdots | \\+A^{(20)})$, an estimated embedding dimension $\\hat{d} = 10$ was obtained using profile likelihood \\cite{zhu2006automatic} and we construct the embeddings $\\hat{\\+Y}^{(1)}, \\ldots, \\hat{\\+Y}^{(20)} \\in \\mathbb{R}^{n \\times 10}$, taking approximately five seconds on a 2017 MacBook Pro. Figure~\\ref{Fig:Primary_UASE} shows the first two dimensions of this embedding to visualise some of the structure in the data. Similar plots and discussion for both individual spectral embedding and omnibus embedding are given in the Appendix.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{Primary_School_Hour_UASE.pdf}\n\t\\caption{First two dimensions of the embeddings $\\hat{\\+Y}^{(1)}, \\ldots, \\hat{\\+Y}^{(20)}$ of the unfolded adjacency matrix $\\+A = (\\+A^{(1)} | \\cdots | \\+A^{(20)})$. The colours indicate different school years while the marker type distinguish the two school classes within each year.}\n\t\\label{Fig:Primary_UASE}\n\\end{figure}\n\nFrom this plot, we observe clustering of students in the same school class. For time windows corresponding to classroom time, for example, 09:00--10:00 and 15:00--16:00, the embedding forms rays of points in 10-dimensional space, with each ray broadly corresponding to a single school class. This is to be expected under a degree-corrected stochastic block model, and the distance along the ray is a measure of the node's activity level \\citep{lei2015consistency,lyzinski2014perfect,sanna2020spectral}. However, not all time windows exhibit this structure, for example, the different classes mix more during lunchtimes (time windows 12:00--13:00 and 13:00--14:00).\n\n\\subsection{Clustering}\nFollowing recommendations regarding community detection under a degree-corrected stochastic block model \\cite{sanna2020spectral}, we analyse UASE using spherical coordinates $\\+\\Theta^{(t)} \\in [0,2\\pi)^{n \\times 9}$, for $t \\in [T]$. Since UASE demonstrates cross-sectional and longitudinal stability, we can combine the embeddings into a single point cloud $\\+\\Theta = (\\+\\Theta^{(1) \\top} | \\cdots | \\+\\Theta^{(T) \\top})^\\top \\in \\mathbb{R}^{nT \\times 9}$ where each point represents a student or teacher in a particular time window. This allows us to detect people returning to a previous behaviour in the dynamic network. We fit a Gaussian mixture model with varying covariance matrices to the non-zero points in $\\+\\Theta$ with 20--50 clusters increasing in increments of 5, with 50 random initialisations, taking approximately five minutes on a 2017 MacBook Pro. Using the Bayesian Information Criterion, we select the best fitting model (30 clusters) and assign the maximum a posteriori Gaussian cluster membership to each student in each time window.\n\nFigure~\\ref{Fig:Primary_Clusters} shows how students in the ten classes move between these clusters over time. Each class has one or two clusters unique to it, for example, the majority of students in class 1A spend their classroom time (as opposed to break time) assigned to cluster 1 or cluster 25. This highlights the importance of longitudinal stability in UASE, as we are detecting points in the embedding returning to some part of latent space. \n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{Primary_School_Clusters_Labels.pdf}\n\t\\caption{Bar chart showing the Gaussian cluster assignment of school classes over time. The height of each coloured bar represents the proportion of students, in that class and at that time, assigned to the corresponding Gaussian cluster, the total available height representing 100\\%. If the coloured bars do not sum to the full available height, the difference represents the proportion of inactive students. For legibility, only bars representing over 35\\% of the class are labelled with the cluster number.}\n\t\\label{Fig:Primary_Clusters}\n\\end{figure}\n\nThere are also instances of multiple school classes being assigned the same cluster at the same time period, for example, on the morning of day 1, classes 5A and 5B are mainly in cluster 28 suggesting they are having a joint lesson, and we see this behaviour again on day 2 with classes 3A and 3B in cluster 20. In the lunchtime periods, particularly on day 1, the younger students (classes 1A--2B) mingle to form a larger cluster, as do the older students (classes 4A--5B), potentially explained by the cafeteria needing two sittings for lunch for space reasons \\cite{stehle2011high}. This highlights the importance of cross-sectional stability in UASE, as it allows the grouping of nodes behaving similarly in a specific time window, irrespective of their potentially different past and future behaviours. \n\n\\section{Conclusion}\n\\label{Sec:Concs}\nWe prove that an existing procedure, UASE, allows dynamic network embedding with longitudinal and cross-sectional stability guarantees. These properties make a range of subsequent spatio-temporal analyses possible using `off-the-shelf' techniques for clustering, time series analysis, classification and more. \n\n\n \\section*{Funding Transparency Statement} Andrew Jones and Patrick Rubin-Delanchy's research was supported by an Alan Turing Institute fellowship. Ian Gallagher's research was supported by an EPSRC PhD studentship. \n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\n \nLet $(X, d_X)$ be a metric space $X$ with a distance function $d_X$. \nSuppose that $X$ has infinite diameter, i.e. the function $d_X$ is unbounded.\nThen one may ask what is the structure of the space $X$ \"at infinity\".\nIntuitively, structure at infinity is what is seen if one looks at\nthe space $X$ from an infinitely far point (see [Gr2]).\nM. Gromov suggested several ways to treat this notion rigorously.\nIn this paper we follow one of them.\n\n\\proclaim Definition 1.1.\n{\\rm Let $(X, d_X)$ be a metric space with an infinite diameter.\nA metric space $(T, d_T)$ can be} isometrically embedded at infinity {\\rm into\nthe space $X$ if for every point $t \\in T$ there exists an\ninfinite sequence $\\{x_t^i\\}$, $i=1,2,..$ of points in $X$,\nsuch that for some fixed sequence of positive $\\varepsilon_i \\to 0$\n$$\\lim_{i\\to\\infty}\\varepsilon_i\\cdot d_X(x_{t_1}^i, x_{t_2}^i)=\nd_T(t_1, t_2) \n\\eqno (1.2)\n$$ \nfor every $t_1,t_2 \\in T$.}\n\nIn other words, to every point of the space\n$T$ we may put in correspondence a sequence of points in $X$ going to infinity,\nsuch that the ``normalized'' pairwise distances between the sequences in $X$\ntend to the distances between the corresponding points in $T$. \n\nThe Definition 1.1. somehow clarifies what the ``structure at infinity''\nmeans but it is still too difficult to work with it.\n\nIn order to proceed we need to define a certain \nclass of ``geometrically simpler'' metric\nspaces --- {\\it geodesic} metric spaces (see [Gr1], [GhH]):\n\n\\proclaim Definition 1.3.\n{\\rm Let $x_0,x_1$ be two points of the metric space $X$ and let \n$a=d_X(x_0,x_1)$ be the distance between them. A} geodesic segment {\\rm in\n$X$ {\\rm connecting\n$x_0$ and $x_1$} is an isometric inclusion $g:\\lbrack 0,a \\rbrack \\to X$ such that\n$g(0)=x_0$, $g(a)=x_1$. The image of this inclusion sometimes is also \ncalled a geodesic segment.\nA metric space $X$ is} geodesic {\\rm if for every two points\n$x_1,x_2 \\in X$ there exists a \n(not necessarily unique) geodesic segment connecting these two points.} \n\nFor example, every complete Riemannian manifold is a geodesic space\ndue to Hopf-Rinow theorem. All metric spaces which appear in this paper \nare geodesic.\n\n\\proclaim Definition 1.4.\n{\\rm A metric space $X_0$ is an} asymptotic subcone {\\rm of the space $X$\nif $X_0$ is geodesic and its every finite subset of points can be\nisometrically embedded at infinity into the space $X$.}\n \nAsymptotic subcones were introduced in [Gr1] in the context of \n{\\it hyperbolic} metric spaces (see [Gr1], [GhH]). There are several\nequivalent formal definitions of a hyperbolic metric space but all of them\ndemand additional non--trivial geometric explanations. \nWe are not presenting any of them \nthem here since throughout this paper we deal with the familiar \nLobachveskian (hyperbolic) plane which is the most well--known\nexample of a hyperbolic metric space. \n\nIn fact, there is another very well--known class of hyperbolic\nspaces~--- these are $0$-hyperbolic spaces (see [Gr1], [GhH] for the\ndefinition of $\\delta$-hyperbolicity) , or {\\it real\ntrees}:\n\n\\proclaim Definition 1.5.\n{\\rm A metric space $X$ is a} real tree {\\rm if it satisfies the following\nconditions:\n\\noindent 1) For every two distinct \npoints of the space there exists a unique geodesic segment joining them.\n\\noindent 2)If two geodesic segments\n$[a,b], [b,c]$ have exactly one endpoint $b$ in\ncommon,\ntheir union is also a geodesic segment.}\n \nReal trees play the key role in the asymptotic geometry of\nthe hyperbolic metric spaces.\nThere is a general theorem that every asymptotic\nsubcone of a hyperbolic space is a real tree (see [GhH]). \nMoreover, as it was stated by Gromov (see [Gr1], [GhH]), \nif every asymptotic subcone of a metric space is a real tree\nthan this space is hyperbolic. Therefore, one may define hyperbolic\nspaces as metric spaces whose all asymptotic subcones are real trees.\nFor the hyperbolic groups Gromov considered this definition as the \nmost intuitive ([Gr1]).\n\nThe property of being a real tree does not give a complete description\nof a metric space. Different real trees can be very much unlike each\nother. Therefore in order to describe asymptotic subcones of a particular\nhyperbolic metric space (a Lobachevskian plane in our case)\nit is not satisfactory to say they are just real trees, a far more \n``explicit'' construction is desirable.\n\nWe found only one example of an asymptotic subcone of a hyperbolic\nplane in the literature --- a star--shaped tree formed by\n$k$ segments with a common vertex (see [GhH]). Clearly, the asymptotic\ngeometry of a hyperbolic plane is much richer.\nHyperbolic plane is a homogeneous metric space. It is quite natural\nto look for asymptotic subcones sharing this property. Such a subcone\nshould be real tree branching at every its point --- already an object \nwhich is quite\ndifficult to imagine. We also want our subcone to contain\nall ``simple'' examples of asymtotic subcones (it follows from the \nDefinition 1.4. that every geodesic subset of an asymptotic subcone \nof the space $X$ is itself an asymptotic subcone of $X$). As a criterion\nof ``simplicity'' we choose countability of the number of vertices of\nthe real tree:\n \n\\proclaim Definition 1.6.\n{\\rm A real tree is } thick {\\rm if it allows an isometric\ninclusion of any real tree with countably many vertices.}\n\nThus, we want to ``materialize'' an asymptotic subcone of a hyperbolic\nplane which is a homogeneous and thick real tree. The surprising fact is\nthat such a substantial part of the ``structure at infinity'' of the\nLobachevskian plane can be described as a certain simple functional\nspace.\n\n\\proclaim Definition 1.7.\n{\\rm Let $S$ be \nthe set of all continuous real functions $f(t)$ \ndefined on a finite interval \n$\\lbrack 0,\\rho \\rbrack$, $0 \\le \\rho< \\infty$ (each function has its\nown $\\rho$), such that \n$f(0)=0$ for all $f \\in S$.\nDefine the following metric on $S$: \n$$d_S(f_1,f_2)=(\\rho_1-s)+(\\rho_2-s),\n\\eqno(1.8)\n$$ \nwhere \n$[0,\\rho_i]$ is the domain of the function $f_i$ , $i=1,2$ and\n$$s=\\sup \\lbrace t|f_1(t')=f_2(t') \n\\quad \\forall t' < t \n\\rbrace.\n$$ \nThis defines the} metric space S.\n{\\rm The number $s$ is called the} moment of segregation \n{\\rm of the functions $f_1(t), f_2(t)$.}\n\nLet us formulate the main result of our paper:\n\n\\proclaim Theorem 1.9.\nThe space $S$ is a thick real tree and a homogeneous metric\nspace. It is an asymptotic subcone of the hyperbolic plane.\n \nWe prove this theorem in the next two sections. In the last section\nwe show that the completion of an asymptotic subcone of a metric\nspace is itself an asymptotic subcone of a metric space\n(motivated by the fact that the space $S$ is non--complete). \nThe paper is completed by two appendices.\n\n\\medskip\n\n\\noindent {\\bf Remark 1.10.}\nMost of the results of the present paper were announced in \n[PSh]. Some constructions introduced here were used in [Sh] to describe\nthe {\\it asymptotic cone}, or the {\\it asymptotic space} (see [Gr0], [Gr2]) \nof the Lobachevskian plane by means of non--standard analysis \n(see [D]). \n\n\\medskip\n\n\\noindent{\\it Acknowledgements.}\nThe authors are grateful to M.~Gromov, L.~Poltero-\n\\noindent vich, V.~Buchstaber and A.~Vershik\nfor helpful discussions and support.\n\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\n\n\n\n\\section{Properties of the metric space $S$}\nIn this section we study some properties of the metric space $S$.\n\n\\proclaim Lemma 2.1.\nThe function $d_S$ defined by {\\rm ( 1.8)} is a metric. \n\n\\noindent {\\bf Proof.}\nWe need to check that the metric (1.8) satisfies the triangle inequality.\nLet $f_1,f_2,f_3$ be three functions in $S$, \n$\\rho_1, \\rho_2, \\rho_3$ be the lenghts of their domains and\n$s_{12}, s_{13},s_{23}$ be their segregation moments. \nWe may always assume that $s_{12} \\le s_{13} \\le s_{23}$\nThen clearly $s_{12}=s_{13}$. Therefore,\n$$d_S(f_1,f_2)+d_S(f_1,f_3)=2\\rho_1+\\rho_2+\\rho_3-4s_{12} \\ge\n\\rho_2+\\rho_3-2s_{12} \\ge \n$$\n$$\n\\ge \\rho_2+\\rho_3-2s_{23} = d_S(f_2,f_3).$$\nThe two other inequalities are proved similarly.\n\n\\medskip\n\n\\proclaim Lemma 2.2.\nThe space $S$ is a real tree.\n\n\\noindent {\\bf Proof.}\nLet $f_1, f_2 \\in S$ be two arbitrary functions, $\\rho_1, \\rho_2$ be their\ndomains and $s$ be their moment of segregation. By (1.8)\n$d_S(f_1,f_2)=\\rho_1+\\rho_2-2s$. \nConsider the following inclusion $g:\\lbrack 0,\\rho_1+\\rho_2-2s \\rbrack \\to X$:\n$$g(x)=\\cases{\n\\lbrace f_1(t),\\, 0\\le t\\le \\rho_1-x \\rbrace, 0\\le x\\le \\rho_1-s; \\cr\n\\lbrace f_2(t),\\, 0\\le t \\le x+2s-\\rho_1 \\rbrace, \\rho_1-s\\le x \\le \n\\rho_1+\\rho_2-2s.\\cr}\n \\eqno (2.3)\n$$\nThis inclusion is isometric and clearly unique with such property, therefore\nthe condition 1) of the Definition 1.5. is verified. In order to check the\ncondition 2) we note that any two geodesic segments $[f,g]$, $[g,h]$ may have \nexactly one point $g$ in common if and only if the function $h$ \nis the extension of the function $g$ and segregates from it not earlier \nthan from the function $f$,\nor, symmetrically,\nif the function $f$ is the extension of the function $g$ and segregates \nfrom it not earlier than \nfrom the function $h$. In both cases the formula (2.3) implies that \n$\\lbrack f,h \\rbrack$ is also a geodesic segment. \nTherefore $S$ is a real tree which completes the proof. \n\n\\medskip\n\n\\proclaim Lemma 2.4.\nThe space $S$ is a thick tree.\n\n\\noindent {\\bf Proof.} \nLet $T$ be an arbitrary real tree with countably many vertices. We ``brush''\nthis tree in the following way. Fix some isomorphism between natural\nnumbers and the set of all vertices. Let $a_{ij}$ be the distance between\nvertices corresponding to the numbers $i$ and $j$, and $\\lbrace k_n \\rbrace$ \nbe an infinite strictly increasing sequence of natural numbers.\nNow we build a mapping from $T$ into $S$. Let the vertex $1$ go to zero\n(by zero we denote the function defined \nand equal to 0 at the single point $0$).\nThe vertex $2$ goes to a linear function $f(t)=k_1t$ defined on the\ninterval $\\lbrack 0, a_{12} \\rbrack$. In order to find the image of the\nvertex $3$ we find from $a_{12}$, $a_{13}$ and \n$a_{23}$ where it branches from $1$ and $2$;\nlet $s_3$ be the abscissa of this point. Therefore on the interval\n$\\lbrack 0, s_3 \\rbrack$ it is already defined and on the interval\n$\\lbrack s_3, a_{13} \\rbrack$ we set it to be linear \nwith the angular coefficient\n$k_2$ (the free term is found from continuity). \nRepeating the same inductive algorithm for all $n$ (if $n-1$ vertices\nare already built we find the abscissa $s_n$ of its point of segregation\nfrom the already built tree and continue the function by setting it linear with\nthe coefficient $k_{n-1}$ on the interval \n$\\lbrack s_n, a_{1n} \\rbrack$) we get an inclusion of $T$ into $S$.\nIt is isometric by construction since the sequence $\\lbrace k_n \\rbrace$ \nis strictly increasing and hence segregation is defined correctly.\n\n\\medskip\n\nNow let us prove that the space $S$ is homogeneous, i.e. \nfor every two its points\nthere exists a one-to-one isometry moving one point to another.\n\n\\proclaim Lemma 2.5.\nThe metric space $S$ is homogeneous. \n\n\\noindent {\\bf Proof.}\nClearly it is sufficient to construct a one-to-one isometry $F$ which moves\nany function to zero. \nDenote the preimage of zero by $f_0(t)$, let $[0,\\rho]$ be \nits domain. Let $f(t)$ be any other function with the domain $[0,a+b]$, where\n$a$ is the moment of segregation of the functions $f_0(t), f(t)$.\nIf $a < \\rho$ then the image of $f$ is given by the function \n$F(f(t))$,such that $F(f(t))=0$ on $\\lbrack 0,\\rho-a\\rbrack$ and \n$F(f(t))=f(t-\\rho+2a)-f_0(a)$ on $\\lbrack \\rho-a,\\rho-a+b \\rbrack$.\nIf $a=\\rho$, i.e. the function $f$ is a ``continuation'' of $f_0$, the\nconstruction is more complicated. Let us choose some infinite sequence\nof continuous functions $\\lbrace g_n(t) \\rbrace$ such that $g_1(t)$\nis identically zero and for any two elements of this sequence\ntheir moment of segregation is zero. For example we can take the sequence\n$$g_n(t)=\\frac{(2^n-1)t}{2^n}, n=0,1,2,....$$\nConsider the function \n$F^*(f(t))=f(t+\\rho)-f_0(\\rho)$\ndefined on $\\lbrack 0,b \\rbrack$. If there exists $0 t^1_{j_2}$ for all $j_1>j_2$,\n$j_1,j_2=1,..\\frac{n(n-1)}{2}$. \nTake an arbitrary sequence \nof positive numbers \n$\\lbrace \\varepsilon_i \\rbrace$, $i=1,2,..$\ntending to zero. \nConsider the following sequences of points of the hyperbolic plane:\n\n$$x^1_{i,k}=(\\rho_k\/\\varepsilon_i, \\varphi^1_{i,k}(\\varepsilon_i)),$$\nwhere\n$$ \\varphi^1_{i,k}(\\varepsilon_i)= \\sum_{j=1}^{J^1_k} e^{-t^1_j\/\\varepsilon_i}\nf_k(t^1_j), \\qquad k=1,..,n ,$$\nwhere $J^1_k$ is the number of points $t^1_j$ which lie in the domain\n$ [0,\\rho_k]$ of the function $f_k$; if there are no such points set \n$\\varphi^1_{i,k}(\\varepsilon_i)\\equiv 0$. \n\nDenote by $g^1_k(t)$, $k=1,...n$, the functions \nbelonging to the discrete subcone\n$D$ such that each $g^1_k(t)$ has the same domain $[0,\\rho_k]$ as $f_k(t)$ and\n$$\ng^1_k(t)=\\cases{f_k(t),\\quad t=t^1_j, j=1,...J^1_k;\\cr\n0,\\quad {\\rm otherwise}.\\cr}\n$$\n \nThen, by lemma 3.6 there exists a number \n$I_1$, such that for all $k_1,k_2=1,..n$:\n$$\n|\\varepsilon_{I_1} d_X (x^1_{I_1,k_1},x^1_{I_1,k_2})-\nd_D(g^1_{k_1},g^1_{k_2})| \\le 1\/2.\n\\eqno (3.9)\n$$ \nTherefore, by the choice of the points $t^1_{k_1k_2}$ we have:\n$$\n|\\varepsilon_{I_1} d_X (x^1_{I_1,k_1},x^1_{I_1,k_2})-\nd_S(f_{k_1},f_{k_2})| \\le \n$$\n$$\n\\le |\\varepsilon_{I_1} d_X (x^1_{I_1,k_1},x^1_{I_1,k_2})-\nd_D(g^1_{k_1},g^1_{k_2})|+|d_D(g^1_{k_1},g^1_{k_2})-d_S(f_{k_1},f_{k_2})|\\le\n$$\n$$\n\\le 1\/2+ 2|t^1_{k_1k_2}-s_{k_1k_2}|<1\/2+2\\cdot 1\/4 = 1.\n$$ \n\nNow we take new points $t^2_{k_1k_2}$ which satisfy the condition (3.8)\nwith $1\/8$ instead of $1\/4$ in the right--hand side and get the new set \nof points $\\lbrace t^2_j \\rbrace$, the new\nsequences $x^2_{i,k}$ and the new functions\n$g^2_k(t) \\in D$, $k=1,..n$.\n\nSimilarly, there exists a number $I_2$, greater than $I_1$, such that\n$$\n|\\varepsilon_{I_2} d_X (x^2_{I_2,k_1},x^2_{I_2,k_2})-\nd_D(g^2_{k_1},g^2_{k_2})| \\le 1\/4\n$$\nfor all $k_1,k_2=1,..,n$, and therefore\n$$\n|\\varepsilon_{I_2} d_X (x^2_{I_2,k_1},x^2_{I_2,k_2})-\nd_S(f_{k_1},f_{k_2})| \\le 1\/4+2|t^2_{k_1k_2}-s_{k_1k_2}|<1\/2\n$$\nfor all $k_1,k_2=1,..,n$.\n\nContinuing this procedure (which is possible due to the Lemma 3.6. and\nthe assumption that $f_k(t)$ are\ncontinuous functions)\nwe get the sequence $\\lbrace I_N \\rbrace$, \n$I_N \\to \\infty$ as $N \\to \\infty$, \nthe sequences $\\lbrace {\\varepsilon_{I_N} \\rbrace \\to 0}$\n(or just $\\lbrace \\varepsilon_N \\rbrace$),\n$\\lbrace x^N_{I_N,k} \\rbrace$ (or just $\\lbrace x_{N,k} \\rbrace$),\nand the functions $g^N_k(t) \\in D$, $k=1,..,n$\nsuch that \n$$\n|\\varepsilon_N d_X (x_{N,k_1},x_{N,k_2})-\nd_S(f_{k_1},f_{k_2})| \\le \n$$\n$$\n\\le\n|\\varepsilon_N d_X (x_{N,k_1},x_{N,k_2})-\nd_D(g^N_{k_1},g^N_{k_2})|+\n2|t^N_{k_1k_2}-s_{k_1k_2}|<\n$$\n$$\n< 1\/2^N+2 \\cdot 1\/2^{N+1}=1\/2^{N-1}\n$$\n\nThus we have proved that there exists a limit\n$$\n\\lim_{N \\to \\infty} \\varepsilon_N d_X(x_{N,k_1},x_{N,k_2})=\nd_S(f_{k_1},f_{k_2})\n$$\nfor all $k_1,k_2= 1,..,n.$\nTherefore $S$ is indeed an asymptotic subcone of the \nhyperbolic plane.\n \n\\smallskip\n\nThe Theorem 3.7. together with the Lemmas 2.2., 2.4. and 2.5.\ncompletes the proof of the Theorem 1.9.\n\nWe call $S$ the {\\it continuous subcone} of the hyperbolic plane.\nIn fact it is an asymptotic subcone of an\ninfinitely narrow neighborhood of a single half-line on the hyperbolic plane,\nas follows from the construction of the discrete subcone $D$.\n\nSome modifications of the construction of the\ncontinous subcone $S$ are considered in the Appendix B.\n\n\\medskip\n\nWe would like to conclude this section with a simple corollary of\nthe main result which however reflects the ``wealth''\nof the space $S$:\n\n\n\\proclaim Corollary 3.10.\nAny real tree with countably many vertices is an asymptotic subcone of \nthe hyperbolic plane.\n\n\\noindent{\\bf Proof.}\nIndeed, every geodesic subspace of an asymptotic subcone \nis itself an asymptotic subcone. Real trees are geodesic spaces\nand by Lemma 2.4 any\nreal tree with countably many vertices can be isometrically inluded into \nthe continuous subcone $S$.\nTherefore every such tree is an asymptotic subcone of the hyperbolic plane.\n \n\\section{Completions of the asymptotic subcones}\n\nThe metric spaces $S$ and $D$ are non--complete \nmetric spaces. For $D$ this is obvious;\nfor $S$ it follows from the following example.\nConsider a sequence $\\{f_k(t)\\}$ of functions, defined on the segment\n$[0,1-2^{-k}]$, and equal to $f_k(t)=\\sin(1\/(1-t))$ on its domain of\ndefinition. Clearly, this sequence has no limit \nin the space $S$ as $k \\to \\infty$.\n\nTherefore, it is reasonable to consider the completion \n$\\bar S$ \nof the \ncontinuous subcone $S$. \nThough it does not have such a simple functional description as the\noriginal one, it is also an asymptotic subcone of the hyperbolic plane\ndue to the following simple general theorem (since we could not find it in\nthe literature we found it appropriate to state it here):\n\n\\proclaim Theorem 4.1.\nA completion of an asymptotic subcone of a metric space is also\nan asymptotic subcone of this metric space.\n\n\\noindent {\\bf Proof.}\nLet $X$ be our metric space, $X_0$ --- its asymptotic subcone and $\\bar X_0$ \n--- the completion of $X_0$.\nLet $f_1,f_2,...,f_k$ be a finite number of points in $\\bar X_0$.\nBy definition, \n$$f_i=\\lim_{n \\to \\infty} \\varphi_i^n,$$ \nwhere $\\varphi_i^n \\in X_0$, $n \\in {\\bf N}$, $i=1,...,k$.\n \nChoose a number $N_1=N_1(1\/4)$ such that \n$$d_{\\bar X_0}(\\varphi_i^n,f_i)<1\/4\n$$ \nfor all $i=1,..,k$, $n \\ge N_1$.\nThen for all $l,m=1,..,k$ and $n \\ge N_1$ we get\n\n$$|d_{\\bar X_0}(f_l,f_m)-d_{X_0}(\\varphi_l^n,\\varphi_m^n)|\\le\nd_{\\bar X_0}(\\varphi_l^n,f_l)+d_{\\bar X_0}(\\varphi_m^n,f_m) <\n$$\n$$\n< \\frac{1}{4}+\\frac{1}{4}=\n\\frac{1}{2}\n\\eqno(4.2)\n$$ \n\nThe space $X_0$ is an asymptotic subcone of $X$. Therefore for its\nfinite subset $\\varphi_i^{N_1}$, $i=1,..,k$ there exists a \nsequence $\\{ \\varepsilon_j \\}$ of positive numbers tending to zero,\nand $k$ sequences of points of the space $X$, \n$\\{ x_j^{i,N_1} \\}$, $i=1,..,k$, such that for some $J_1=J_1(N_1,1\/2)$\n$$ |\\varepsilon_j d_X(x_j^{l,N_1},x_j^{m,N_1}) - \nd_{X_0}(\\varphi_l^{N_1},\\varphi_m^{N_1})| < \\frac{1}{2}\n\\qquad \\forall j\\ge J_1.\n\\eqno (4.3)\n$$\n\nTherefore, by (4.2) and (4.3) we get:\n$$ |\\varepsilon_{J_1} d_X(x_{J_1}^{l,N_1},x_{J_1}^{m,N_1}) - \nd_{\\bar X_0}(f_l, f_m)| < \\frac{1}{2} + \\frac{1}{2}=1\n$$\nfor all $l,m=1,..,k$.\n\n\nNext, we choose $N_2=N_2(1\/8)$ and $J_2=J_2(N_2, 1\/4)$\nand similarly obtain\n$$ |\\varepsilon_{J_2} d_X(x_{J_2}^{l,N_2},x_{J_2}^{m,N_2}) - \nd_{\\bar X_0}(f_l, f_m)| < \\frac{1}{4} + \\frac{1}{4}=\\frac{1}{2}\n$$\nfor all $l,m=1,..,k$.\nLet us note that we can always choose $N_2$ and $J_2$ in such a way that\n$N_2 > N_1$ and $\\varepsilon_{J_2}< \\varepsilon_{J_1}\/2$. \n\nContinuing this procedure analogously we get \nthe sequence $\\{\\varepsilon_{J_r}\\}$ of positive numbers tending to zero \nand $k$ sequences of points in $X$, \n$\\{x_{J_r}^{i,N_r}\\}$, $i=1,..,k$, such that\n\n$$ |\\varepsilon_{J_r} d_X(x_{J_r}^{l,N_r},x_{J_r}^{m,N_r}) - \nd_{\\bar X_0}(f_l, f_m)| < \\frac{1}{2^r} + \\frac{1}{2^r}=\\frac{1}{2^{r-1}}\n$$\nfor all $l,m=1,..,k$, which implies\n\n$$\\lim_{r \\to \\infty}\\varepsilon_{J_r} d_X(x_{J_r}^{l,N_r},x_{J_r}^{m,N_r})=\nd_{\\bar X_0}(f_l, f_m)|, \\quad l,m=1,..,k.\n\\eqno(4.4)\n$$\n \nThe relation (4.4) exactly means that $\\bar X_0$ is an asymptotic subcone\nof the metric space $X$, which completes the proof of the theorem. \n \n\\bigskip\n\n\\section*{Appendix A}\n\nWe have shown that the spaces $S$ and $D$\nare real trees. These\ntrees are branching at every point and the cardinal number of \nthe set of their vertices is continuum (for $D$ this is\nclear and for $S$ it follows from the fact that every \ncontinuous function is defined by its values at the rational points). \nAs a metric space $S$ is ``larger'' than $D$, as it is shown by the following\n\n\\proclaim Lemma A.1.\nThe space $D$ can be isometrically included into the space \n$S$ but is not isometric to it.\n\n\\noindent {\\bf Proof.} \nLet us show that $D$ can be \nisometrically included into $S$.\nLet $\\gamma(t)$ be any element of $D$, \n$Z_\\gamma=\\lbrace a_1\\varepsilon_0$, \nwhich contradicts with the choice of $\\alpha_1(t)$; the similar argument\nis valid for $\\beta_1(t)$. \nAt least one of these inclusions is proper; \nindeed, if $Z_{\\alpha_1}=Z_{\\beta_1}= Z_\\gamma$ then one \nof the functions $\\alpha_1(t)$,\n$\\beta_1(t)$, $\\gamma(t)$ is the extension of the two others which is\nimpossible since $\\Phi$ is an isometry and none of the functions\n$f(t)$, $g_1(t)$, $h_1(t)$\nis the extension of any of the other two.\nLet this proper inclusion be $Z_\\gamma \\subseteq Z_{\\alpha_1}$ .\nThen the set $Z_{\\alpha_1}$ consists of at least $N+1$ points.\n\nRepeat this construction taking $g_1$ instead of $f$, and instead of \n$\\varepsilon_0$ taking \n$0< \\varepsilon_1< \\min(\\varepsilon_0\/2,\\rho_1-m_1)$, where\n$[0,\\rho_1)$ is the domain of $\\alpha_1(t)$ and \n$m_1$ is the maximal element in $Z_{\\alpha_1}$.\nSimilarly, we shall get the new function $\\alpha_2(t) \\in D$ such that\n$Z_{\\alpha_2}$ consists of at least $N+2$ points.\n\nContinuing this procedure we obtain a sequence $\\lbrace g_i(t)\\rbrace$ in \n$S$ and the corresponding sequence $ \\lbrace \\alpha_i(t)\\rbrace$ in $D$ \nsuch that\n$Z_{\\alpha_i}$ consists of at least $N+i$ points.\nWhen $i \\to \\infty$ the functions $g_i(t) \\to g(t)$, where $g(t) \\in S $ since \n$\\varepsilon_i<\\varepsilon_0\/2^i$ for all $i=1,2,..$ and therefore\nthe length of the domain of $g(t)$ is finite --- it is not greater than\n$r+2\\varepsilon_0$, where $r$ is the length of the domain of $f(t)$. \nConsider the image of $g(t)$ under the isometry $\\Phi$;\nit is equal to $\\alpha(t)=\\lim_{i\\to \\infty}\\alpha_i(t)$.\nBy our construction $Z_\\alpha \\supset Z_{\\alpha_i}$ for all $i=1,2,..$ ,\ntherefore $Z_\\alpha$ consists of an infinite number of points. \nBut this contradicts with the fact that $\\alpha(t) \\in D$,\nand hence $S$ and $D$ are non-isometric.\nThis completes the proof of our lemma. \n \n\\section*{Appendix B}\n\nInstead of continuous functions with bounded\ndomain one could take generalized functions of bounded domain with the\ndistance defined by (1.8).\nSuch generalized functions are of finite order (see [GeS]), i.e. they\ncan be represented as finite sums of generalized derivatives of continuous\nfunctions. One may check that integration preserving the condition $f(0)=0$\nis an isometry with respect to our metric\n(compare this with the formula (A.2.) --- \nin fact we have represented each element\nof the discrete subcone $D$ as a finite sum of $\\delta$-functions and \nintergrated twice). Hence every space of all generalized\nfunctions of order less than some finite $m > 0$ is isometric to $S$:\nits isometric inclusion into $S$ is obtained by integrating $m$ times\nevery its element as described above, \nand isometric inclusion of $S$ into such \nspace can be given by an identical map. \nSimilarly, if we take $C^m$-smooth functions,\nwe also get an isometric space. Therefore, all such functional spaces are\nisometric asymptotic subcones of the hyperbolic plane. \n\nThe construction of $S$ can be also generalized for\nthe Lobache\\noindent vskian space of arbitrary dimension $n$. \nIn this case instead of scalar functions one has to take \n$n$-component vector functions. \n\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\medskip\n\n\\centerline{ {\\large\\bf References}}\n\n\\medskip\n\n\\noindent [Be] A. Beardon, The geometry of discrete groups, Springer--Verlag, \n1983.\n\n\\smallskip\n\n\\noindent [D] M.~Davis, Applied Nonstandard Analysis, \nJ. Wiley \\& Sons, New York, 1977\n\n\\smallskip\n\n\\noindent [GeS] I.M. Gelfand, G.E. Shilov, \nGeneralized functions, Vol. 2, Academic Press, 1968. \n\n\\smallskip \n\n\\noindent [GhH] E. Ghys, P. de la Harpe, Sur le groupes hyperboliques apres\nMikhael Gromov, Birkh\\\"auser, 1990. \n\n\\smallskip\n\n\\noindent [Gr0] M. Gromov, Groups of polynomial growth and expanding\nmaps, IHES Math. Publ., N 53, 1981, 53-71.\n\n\\smallskip\n \n\\noindent [Gr1] M. Gromov, Hyperbolic Groups, in: Essays in group theory,\ned. S.M.Gersten, M.S.R.I. Publ. 8 , Springer--Verlag, 1987, 75-263.\n\n\\noindent [Gr2] M. Gromov, Asymptotic invariants of infinite groups, \nGeometric group theory. Vol. 2 (Sussex, 1991), London Math. Soc. Lecture\nNote Ser. 182, Cambridge Univ. Press, 1993, 1-295.\n\n\\smallskip\n\\noindent [PSh] I. Polterovich, A. Shnirelman, An asymptotic subcone \nof the Lobachevskii plane as a space of functions, Russian Math. Surveys, \nv. 52 No. 4, 1997, 842-843. \n \n \n\\noindent [Sh] A. Shnirelman, On the structure of asymptotic space \nof the Loba-\n\n\\noindent chevsky plane, 1-23, to appear in the Amer. J. Math.\n\n\n\\end{document} \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n Carbon-rich stars ([C\/Fe] $\\ge$ +1.0) comprise a significant fraction of \nmetal-poor ([Fe\/H] $\\le$ $-$2.0) stars with estimates ranging from \n14 $\\pm$ 4\\% (Cohen et al. 2005) to 21 $\\pm$ 2\\% (Lucatello et al. 2005); \nthis fraction increases with decreasing metallicity (Rossi, Beers and Sneden\n 1999). A large fraction of carbon-enhanced metal-poor stars exhibit\noverabundances of neutron-capture elements relative to iron. Significant \ninsight into the neutron-capture processes taking place in the early \nGalaxy can be derived from chemical composition studies of metal-poor \ncarbon-stars (Norris et al. 1997a 1997b, 2002; Bonifacio et al. 1998; \nHill et al. 2000; Aoki et al. 2002a,b; Goswami et al. 2006, Aoki et al. 2007).\nHowever, formation mechanisms of these stars still remain poorly understood.\n The prime cause of the origin of C-N stars is believed to be the third \ndredge-up during the AGB evolutionary phase of low to intermediate-mass \nstars; the origin of C-R stars as well as SC-type stars still remains \nunclear (Izzard et al. 2007, Zijlstra 2004). \n\n The population II CH stars, characterized by a strong G-band of CH and \ns-process elements play significant roles in probing the impact of s-process \nmechanisms in early Galactic chemical evolution. These stars are classified \ninto two distinct types, the Late-type and the Early-type. This classification\n is based on their $^{12}$C\/$^{13}$C ratios; stars with a large value of\n $^{12}$C\/$^{13}$C ratio ( $\\ge$ 100) belong to Late-type, and those\nwith values of $^{12}$C\/$^{13}$C ratio ($\\le$ 10) belong to Early-type.\n The two groups follow two distinct evolutionary paths. Late-type CH stars \nare further identified as intrinsic stars that generate s-process elements \ninternally and the early-type CH stars as extrinsic stars; they receive \nthe s-process elements via binary mass transfer. The chemical composition \nof early-type CH stars, if remains unaltered, would bear signatures \nof the nucleosynthesis processes operating in the low-metallicity AGB stars.\n Abundance analysis of such stars can provide observational constraints for \ntheoretical modelling of $s-$process nucleosynthesis at very low-metallicity\n revealing the time of influence of this process on early Galactic Chemical\n Evolution (GCE). \n\nAlthough new large-aperture telescopes has substantially enhanced the number\n of target stars for which high spectral resolution data with high\n signal-to-noise ratio can be obtained, literature survey shows not many \nCH stars have been studied in detail so far. A major difficulty is in \ndistinguishing these objects from other types of carbon stars. In particular,\n Population I C-R, C-N and dwarf carbon stars exhibit remarkably similar \nspectra with those of carbon giants. It is important to distinguish them \nfrom one another and understand the astrophysical implications of each \nindividual class of stellar population. It is with this motivation we have\nundertaken to identify the CH stellar content as well as different types\nin a sample of stars presented by Christlieb et al. (2001b). Using low \nresolution spectral analysis we have classified the stars based on a set of\nspectral classification criteria. The present work led \nto the detection of 36 potential CH star candidates among ninety two objects.\nCombining this result with our previous studies we find ${\\sim}$ 37\\% \n (of 243) objects are potential CH star candidates.\nThis set of objects would make important targets for detailed chemical \ncomposition studies based on high resolution spectroscopy.\n\nSelection of the program stars is outlined in section 2. Observations \nand data reductions are described in section 3. In section 4 we briefly \ndiscuss the main features and spectral characteristics of C-stars.\n Description of the program stars spectra and results are drawn in \nsection 5. Conclusions are presented in section 6. \n\n\\section {Selection of program stars}\n\nThe program stars belong to the list of 403 Faint High Latitude Carbon (FHLC) \nstars presented by Christlieb et al. (2001b) from the database of \nHamburg\/ESO Survey (HES) described by Wisotzki et al. (2000).\nHamburg\/ESO Survey for carbon stars covers 6400 degree$^{2}$ limited by\n${\\delta}$ $\\le$ +2.5$^{0}$ and $\\|$b$\\|$ $\\ge$ 30$^{0}$. The magnitude limit\nis V ${\\sim}$ 16.5. The wavelength range of the spectra is\n3200 to 5200 \\AA\\, \n at a resolution of 15 \\AA\\, at H$_{\\gamma}$. \nChristlieb et al. found a total of 403 FHLC stars in this survey by\n application of an \nautomated procedure to the digitized spectra. \n\nThe identification of these objects as FHLC stars was based on a \nmeasure of line indices - i.e. ratios of the mean photographic densities\n in the carbon molecular absorption features and the continuum band passes.\n The primary consideration is the presence of strong C$_{2}$ and CN \nmolecular bands shortward of 5200 \\AA\\,; CH bands were not considered.\n\n At high galactic latitudes different kinds of carbon stars such as \nN-type carbon stars, dwarf carbon stars, CH-giants, warm C-R stars\n etc. are known to populate the region (Green et al. 1994). \nGoswami (2005)\nand Goswami et al. (2007) have conducted spectral classification of about 151\nobjects that belong to the FHLC stars sample offered\n by Christlieb et al. (2001b). These studies are enhanced by\nmedium resolution\nspectroscopic analysis of an additional sample of ninety two objects\n observed during 2007 to 2009.\n\n\\section{Observations and Data Reduction}\nObservations were carried out using the 2-m Himalayan Chandra Telescope (HCT)\n at the Indian Astronomical Observatory (IAO), Mt. Saraswati, Digpa-ratsa Ri,\n Hanle. The spectrograph used is the Himalayan Faint Object Spectrograph\nCamera (HFOSC). HFOSC is an optical imager cum a spectrograph for conducting \nlow and medium resolution grism spectroscopy\n(http:\/\/www.iiap.res.in\/centers\/iao).\nThe grism and the camera combination used for observation provided a \nspectral resolution of $\\sim$ 1330( ${\\lambda\/\\delta\\lambda}$ );\nthe observed bandpass ran from about 3800 to 6800 \\AA\\,. All the objects\nlisted in Table 1 and 2 are observed during 2007 - 2009. The B$_{J}$, V, B-V, \nU-B colours listed in the tables are taken from Christlieb et al.(2001b).\nDetermination of these values are described in Wisotzki et al. (2000) \n and Christlieb et al.(2001a). B$_{J}$ magnitudes are accurate to better \nthan ${\\pm}$0.2mag including zero point errors (Wisotzki et al. 2000).\nSpectra of HD~182040, HD~26, HD~5223, HD~209621, Z~PSc, V460~Cyg and RV~Sct\n used for comparison were obtained during earlier observations using the \nsame observational set up.\nA few spectra acquired using the OMR spectrograph at the cassegrain focus\nof the 2.3-m Vainu Bappu Telescope (VBT) at Kavalur, cover a wavelength \nrange 4000 - 6100 \\AA\\, at a resolution of ${\\sim}$1000. With a 600 line\nmm$^{-1}$ grating, we get a dispersion of 2.6 \\AA\\, pixel$^{-1}$.\n\n \nObservations of Th-Ar hollow cathode lamp taken immediately before and\nafter the stellar exposures provide the wavelength calibration. The CCD\ndata were reduced using the IRAF software spectroscopic reduction packages.\nFor each object two spectra were taken \nand combined to increase the signal-to-noise ratio.\n\n\n{\\footnotesize\n\\begin{table*}\n\\centering\n{\\bf Table 1: HE stars with prominent C$_{2}$ molecular bands observed during 2007 - 2009 }\\\\\n\\tiny\n\\begin{tabular}{cccccccccccccc}\n\\hline\nStar No. & RA(2000)$^{a}$ & DEC(2000)$^{a}$& $l$ & $b$ & B$_{J}^{a}$& V$^{a}$ & B-V$^{a}$ & U-B$^{a}$ &J & H & K & &Dt of Obs\\\\\n & & & & & & & & & & & & & \\\\\n\\hline\nHE 0008-1712 & 00 11 19.2 & -16 55 34 & 78.5866 & -76.2106 & 16.5 & 15.2 & 1.78 & 1.64 & 13.630 & 13.069 & 12.975 & & 06.12.2008\\\\\n & & & & & & & & & & & & &11.09.2008\\\\\nHE 0009-1824 & 00 12 18.5 & -18 07 55 & 75.8376 & -77.2654 & 16.5 & 15.7 & 1.09 & 0.58 & 14.080 & 13.724 & 13.665 & & 12.09.2008 \\\\\nHE 0037-0654 & 00 40 02.0 & -06 38 13 & 114.8865 & -69.3303 & 16.4 & 15.5 & 1.19 & 0.58 & 14.146 & 13.708 & 13.724 & & 11.09.2008 \\\\\nHE 0052-0543 & 00 55 00.0 & -05 27 02 & 125.3316 & -68.3057 & 16.5 & 15.0 & 1.95 & 1.74 & 12.952 & 12.241 & 12.086 & & 12.09.2008 \\\\\nHE 0100-1619 & 01 02 41.6 & -16 03 01 & 136.7651 & -78.6185 & 15.9 & 14.7 & 1.54 & 1.28 & 13.114 & 12.537 & 12.476 & & 21.11.2008 \\\\\n & & & & & & & & & & & &\n&12.09.2008\\\\\nHE 0136-1831 & 01 39 01.8 & -18 16 43 & 176.4932 & -75.9157 & 16.9 & 15.6 & 1.72 & 1.43 & 14.216 & 13.679 & 13.532 & & 12.09.2008\\\\\nHE 0217+0056 & 02 20 23.5 & +01 10 39 & 163.6094 & -54.5125 & 16.3 & 14.6 & 2.35 & 2.25 & 11.276 & 10.271 & 9.833 & & 22.11.2008\\\\\nHE 0225-0546 & 02 28 19.4 & -05 32 58 & 174.1738 & -58.4222 & 16.5 & 15.2 & 1.79 & 1.51 & 13.347 & 12.707 & 12.528 & & 06.12.2008 \\\\\nHE 0228-0256 & 02 31 15.5 & -02 43 07 & 171.5942 & -55.8545 & 16.2 & 14.7 & 1.99 & 1.52 & 12.506 & 11.813 & 11.531 & & 18.01.2009\\\\\nHE 0308-1612 & 03 10 27.1 & -16 00 41 & 201.1165 & -55.9582 & 12.5 & -- & -- & -- & 10.027 & 9.475 & 9.331 & & 21.11.2008\\\\\nHE 0420-1037 & 04 22 47.0 & -10 30 26 & 205.0454 & -37.7176 & 15.2 & 14.7 & 1.38 & 0.99 & 12.341 & 11.815 & 11.695 & & 21.11.2008 \\\\\nHE 0945-0813 & 09 48 18.7 & -08 27 40 & 245.0790 & 33.1497 & 16.2 & 15.3 & 1.22 & 1.11 & 13.531 & 13.026 & 12.903 & & 08.04.2007\\\\\n & & & & & & & & & & & & &09.04.2007\\\\\nHE 0954+0137 & 09 57 19.2 & +01 23 00 & 237.1030 & 41.0018 & 16.6 & 15.7 & 1.18 & 0.32 & -- & -- & -- & & 21.11.2008\\\\\nHE 1011-0942 & 10 14 25.0 & -09 57 54 & 251.7699 & 36.8590 & 15.4 & 14.2 & 1.65 & 1.76 & 11.209 & 10.432 & 10.125 & & 22.11.2008\\\\\nHE 1019-1136 & 10 22 14.7 & -11 51 39 & 255.1421 & 36.8087 & 15.2 & 13.9 & 1.84 & 2.29 & 10.005 & 9.031 & 8.488 & & 06.02.2009 \\\\ \nHE 1027-2501 & 10 29 29.5 & -25 17 16 & 266.6832 & 27.4163 & 13.9 & 12.7 & 1.73 & 1.51 & 10.627 & 9.896 & 9.722 & & 22.11.2008\\\\\nHE 1045-1434 & 10 47 44.1 & -14 50 23 & 263.5905 & 38.4049 & 15.5 & 14.6 & 1.23 & 0.96 & 12.935 & 12.449 & 12.244 & & 09.04.2007\\\\\nHE 1051-0112 & 10 53 58.8 & -01 28 15 & 253.5294 & 49.7900 & 17.0 & 16.0 & 1.44 & 0.94 & 14.347 & 13.794 & 13.703 & & 06.12.2008 \\\\\nHE 1102-2142 & 11 04 31.2 & -21 58 29 & 272.5241 & 34.5071 & 16.0 & 14.9 & 1.44 & 0.98 & 13.275 & 12.714 & 12.601 & & 16.03.2009 \\\\\nHE 1110-0153 & 11 13 02.7 & -02 09 28 & 261.1493 & 52.4883 & 16.5 & 15.5 & 1.47 & 1.29 & 12.912 & 12.205 & 12.063 & & 04.04.2009 \\\\\nHE 1116-1628 & 11 19 03.9 & -16 44 50 & 273.2181 & 40.7347 & 16.6 & 15.6 & 1.28 & 1.41 & 13.241 & 12.614 & 12.482 & & 06.02.2009 \\\\\nHE 1119-1933 & 11 21 43.5 & -19 49 47 & 275.7342 & 38.2583 & 12.8 & 14.6 & 1.34 & 0.87 & 13.043 & 12.571 & 12.422 & & 18.03.2009 \\\\\nHE 1120-2122 & 11 23 18.6 & -21 38 33 & 277.1276 & 36.7743 & 12.9 & -- & -- & -- & 9.573 & 8.902 & 8.788 & & 17.03.2009 \\\\\nHE 1123-2031 & 11 26 08.7 & -20 48 19 & 277.4490 & 37.8097 & 16.8 & 15.8 & 1.33 & 1.19 & 13.513 & 12.940 & 12.800 & & 18.03.2009\\\\\nHE 1127-0604 & 11 29 36.8 & -06 20 51 & 269.3427 & 51.1090 & 16.8 & 15.9 & 1.23 & 0.87 & 14.594 & 14.087 & 14.027 & & 18.01.2009 \\\\\nHE 1142-2601 & 11 44 52.9 & -26 18 29 & 284.8485 & 34.2157 & 13.9 & 13.0 & 1.28 & 1.07 & 11.218 & 10.675 & 10.539 & \n& 04.04.2009\\\\\nHE 1145-1319 & 11 48 21.4 & -13 36 38 & 280.3756 & 46.4800 & 16.4 & 15.4 & 1.37 & 1.32 & 13.466 & 12.934 & 12.790 & & 17.03.2009\\\\\n & & & & & & & & & & & & &05.04.2009\\\\\nHE 1146-0151 & 11 49 02.3 & -02 08 11 & 273.2807 & 57.0990 & 14.9 & 14.2 & 0.96 & 0.97 & 12.929 & 12.400 & 12.262 & & 16.03.2009\\\\\nHE 1157-1434 & 12 00 11.5 & -14 50 50 & 284.8854 & 46.2211 & 16.1 & 13.8 & 1.62 & 1.41& 11.792 & 11.178 & 11.019 & & 06.06.2007\\\\\n & & & & & & & & & & & & &07.06.2007\\\\\nHE 1205-0521 & 12 07 53. & -05 37 51 & 283.5026 & 55.5893 & 15.2 & 14.4 & 1.09 &0.48 & 12.434 & 11.988 & 11.806 & & 04.04.2009\\\\\nHE 1205-2539 & 12 08 08.1 & -25 56 38 & 290.8957 & 35.9137 & 13.6 & 15.4 & 1.91 &1.72 & 12.313 & 12.665 & 12.540 & & 17.03.2009 \\\\\nHE 1210-2636 & 12 12 59.7 & -26 53 22 & 292.4334 & 35.1974 & 13.8 & 12.6 & 1.65 &1.52 & -- & -- & -- & & 06.05.2009\\\\\nHE 1228-0417 & 12 31 12.5 & -04 33 40 & 293.4045 & 57.9360 & 14.8 & 14.0 & 1.17 &0.96 & 12.406 & 11.969 & 11.818 & & 04.04.2009\\\\\nHE 1230-0327 & 12 33 24.1 & -03 44 28 & 294.2159 & 58.8251 & 13.7 & 15.1 & 1.02 &0.80 & 13.683 & 13.224 & 13.100 & & 17.03.2009\\\\\nHE 1238-0836 & 12 41 02.4 & -08 53 06 & 298.5709 & 53.8987 & 13.1 & -- & -- & -- & 9.189 & 8.503 & 8.154 & & 08.04.2007\\\\\nHE 1253-1859 & 12 56 38.4 & -19 15 32 & 304.6277 & 43.5957 & 13.7 & 12.9 & 1.09 &1.22 & 10.604 & 9.972 & 9.850 & & 16.03.2009\\\\\nHE 1315-2035 & 13 17 57.4 & -20 50 53 & 311.2275 & 41.5961 & 16.7 & 15.7 & 1.35 &0.93 & 13.678 & 13.236 & 13.047 & & 06.05.2009\\\\\nHE 1331-2558 & 13 34 20.1 & -26 13 38 & 314.7873 & 35.6550 & 13.9 & 16.0 & 1.53 &1.10 & 10.998 & 10.394 & 10.240 & & 18.03.2009\\\\\nHE 1344-0411 & 13 47 25.7 & -04 26 04 & 328.2429 & 55.6614 & 16.3 & 14.9 & 2.02 &2.29 & 11.708 & 10.848 & 10.467 & & 05.04.2009\\\\\nHE 1358-2508 & 14 01 12.3 & -25 22 39 & 322.8426 & 34.4322 & 13.2 & 12.3 & 1.25 &0.84 & 9.521 & 8.802 & 8.532 & & 18.01.2009\\\\\nHE 1400-1113 & 14 03 39.8 & -11 28 04 & 329.7145 & 47.6129 & 16.0 & 15.2 & 1.19 &0.75 & 13.838 & 13.411 & 13.276 & \n& 17.03.2009\\\\\nHE 1404-0846 & 14 06 55.1 & -09 00 58 & 332.3876 & 49.4897 & 15.3 & 14.2 & 1.52 & 1.29 & 12.435 &11.783& 11.681& & 07.06.2007\\\\ \nHE 1405-0346 & 14 07 58.3 & -04 01 03 & 336.5341 & 53.7853 & 14.7 & 13.5 & 1.68 &1.28 & 11.608 & 11.068 & 10.903 & \n& 16.03.2009\\\\\nHE 1410-0125 & 14 13 24.7 & -01 39 54 & 340.6313 & 55.0908 & -- & -- & -- & -- & 10.360 & 9.770 & 9.651 & & 16.03.2009\\\\\n & & & & & & & & & & & & &11.06.2008\\\\\nHE 1418-0306 &14 20 57.1 & -03 19 54 & 341.7366 & 52.6618 & -- & 13.0 & 1.66 & 1.11 & 10.505 & 9.767 & 9.505 & & 09.04.2007\\\\ \nHE 1425-2052 & 14 28 39.5 & -21 06 05 & 331.4023 & 36.3382 & 13.6 & 12.7 & 1.27 &1.29 & 10.043 & 9.446 & 9.273 & & 17.03.2009\\\\\nHE 1428-1950 & 14 30 59.4 & -20 03 42 & 332.6046 & 37.0083 & -- & -- & -- & -- & 9.988 & 9.469 & 9.318 & & 06.02.2009\\\\\nHE 1429-1411 & 14 32 40.6 & -14 25 06 & 336.6322 & 41.7341 & 12.5 & 11.1 &1.99 &1.89 & 7.622 & 6.721 & 6.346 & & 06.02.2009\\\\\nHE 1430-0919 & 14 33 12.9 & -09 32 53 & 340.3560 & 45.7983 & 14.9 & 13.8 &1.46 &0.75 & 12.476 & 11.971 & 11.862 & & 18.03.2009 \\\\\n & & & & & & & & & & & & &09.05.2007\\\\\nHE 1431-0755 & 14 34 32.7 & -08 08 37 & 341.8828 & 46.7820 & 14.6 & 13.5 & 1.51 & 1.44 & 11.283 & 10.605 & 10.422 & & 08.05.2007\\\\ \nHE 1439-1338 & 14 42 26.4 & -13 51 18 & 339.7039 & 40.9588 & 14.5 & 13.5 &1.39 &0.97 & 10.810 & 10.112 & 9.836 & & 25.07.2008\\\\\n & & & & & & & & & & & & &09.04.2007\\\\\nHE~1440-1511 & 14 43 07.1 & -15 23 48 & 338.737 & 39.5765 & 14.6 & 13.8 & 1.13 & 0.87 & 12.238 & 11.757 & 11.606 & & 09.05.2007 \\\\\nHE 1442-0058 & 14 44 48.9 & -01 10 56 & 351.4558 & 50.6883 & 17.8 & 16.2 & 2.15 & 1.80 & 12.287 & 11.268 & 10.751 & & 06.06.2007\\\\ \nHE 1447+0102 & 14 50 15.1 & +00 50 15 & 355.2263 & 51.2262 & 15.6 & 15.0 &0.90 &0.14 & 13.207 & 12.760 & 12.682 & & 12.09.2008\\\\\n & & & & & & & & & & & & &08.04.2007\\\\\nHE 1525-0516 & 15 27 52.2 & -05 27 04 & 358.1110 & 40.1019 & 16.8 & 15.8 &1.29 &1.14 & 13.972 & 13.479 & 13.314 & & 11.09.2008 \\\\\n & & & & & & & & & & & & &07.06.2007\\\\\n & & & & & & & & & & & & &08.06.2007\\\\\nHE 2114-0603 & 21 17 20.8 & -05 50 48 & 45.5467 & -34.9459& 16.7 & 15.4 &1.80 &1.58 & 12.472 & 11.786 & 11.615 & & 11.09.2008 \\\\\n & & & & & & & & & & & & &09.05.2007\\\\\nHE 2144-1832 & 21 46 54.7 & -18 18 15 & 34.6476 & -46.7834& 12.6 & -- & -- & -- & -- & -- & -- & & 06.06.2007\\\\\nHE 2157-2125 & 22 00 25.5 & -21 11 23 & 32.0715 &-50.7274 & 16.8 & 15.9 & 1.29 & 1.02 & 14.389& 13.980 & 13.742 & & 08.06.2007\\\\\n & & & & & & & & & & & & &12.09.2008\\\\\nHE 2211-0605 & 22 13 53.5 & -05 51 06 & 55.3066 & -46.9505& 16.0 & 15.1 &1.21 &1.01 & 13.383 & 12.875 & 12.727 & & 24.07.2008 \\\\\nHE 2213-0017 & 22 15 37.1 & -00 02 59 & 62.3019 & -43.8276& 16.4 & 14.6 &2.38 &2.52 & 10.860 & 9.832 & 9.351 & & 24.07.2008\\\\\n & & & & & & & & & & & & &27.05.2007\\\\\nHE 2216-0202 & 22 18 47.5 & -01 47 36 & 61.0859 & -45.5370& 17.2 & 16.4 &1.01 &0.43 & 14.571 & 14.165 & 14.028 & & 06.12.2008\\\\\n & & & & & & & & & & & & &11.09.1008\\\\\n & & & & & & & & & & & & &24.07.2009\\\\\nHE 2222-2337 & 22 25 38.8 & -23 22 44 & 31.2015 & -56.9371& 17.1 & 15.7 &1.93 &1.84 & 13.535 & 12.837 & 12.664 & & 12.09.2008\\\\\nHE 2225-1401 & 22 28 10.7 & -13 46 23 & 47.4570 & -54.0483& 16.5 & 14.5 &2.72 &2.36 & 11.870 & 10.748 & 9.896 & & 21.11.2008\\\\\nHE 2228-0137 & 22 31 26.2 & -01 21 42 & 64.4957 & -47.7030& 15.8 & 14.7 &1.56 &1.20 & 12.301 & 11.715 & 11.589 & & 11.09.2008\\\\\n & & & & & & & & & & & & &25.07.2009\\\\\nHE 2246-1312 & 22 49 26.4 & -12 56 35 & 53.2930 & -58.1569& 17.0 & 15.9 &1.56 &1.60 & 14.101 & 13.472 & 13.303 & & 11.09.2008\\\\\n & & & & & & & & & & & & &25.07.2008\\\\\nHE 2255-1724 & 22 58 06.8 & -17 08 19 & 47.9045 & -62.0030& 16.1 &15.1 &1.45 &1.05 & 12.931 & 12.374 & 12.310 & & 11.09.2008\\\\\n & & & & & & & & & & & & &25.07.2008\\\\\nHE 2305-1427 & 23 08 10.9 & -14 11 27 & 55.9986 & -62.6875& 16.4 &15.3 &1.41 &0.80 & 13.269 & 12.762 & 12.744 & & 25.07.2008\\\\\n & & & & & & & & & & & & &10.10.2008\\\\\nHE 2334-1723 & 23 37 03.7 & -17 06 34 & 59.3241 & -70.1108& 16.1 & 15.1 &1.38 &1.48 & 13.583 & 13.119 & 12.977 & & 10.10.2008\\\\\n & & & & & & & & & & & & &25.07.2008\\\\\nHE 2347-0658 & 23 49 56.3 & -06 41 55 & 84.5844 & -64.8865& 16.9 &16.0 &1.30 & 0.69 & 14.568 & 14.145 & 14.158 & & 11.09.2008 \\\\\n & & & & & & & & & & & & &24.07.2008\\\\\nHE 2353-2314 & 23 55 44.0 & -22 58 09 & 48.1281 & -76.7257& 16.5 &15.2 &1.82 & 1.53 & 13.064 & 12.403 & 12.199 & & 25.07.2008 \\\\\n\\hline\n\\end{tabular}\n\n$^{a}$ From Christlieb et al. (2001b)\\\\\n\n\\end{table*}\n}\n\n\n{\\footnotesize\n\\begin{table*}\n\\centering\n{\\bf Table 2: HE stars without prominent C$_{2}$ molecular bands observed during 2007 - 2009 }\\\\\n\\tiny\n\\begin{tabular}{cccccccccccccc}\n\\hline\nStar No. & RA(2000)$^{a}$ & DEC(2000)$^{a}$& $l$ & $b$ & B$_{J}^{a}$& V$^{a}$ & B-V$^{a}$ & U-B$^{a}$ &J & H & K & mol. bands &Dt of Obs\\\\\n & & & & & & & & & & & & & \\\\\n\\hline\nHE 0422-2518 & 04 24 38.5 & -25 12 10 & 223.2761 & -42.4854 & 13.9 & -- & -- & -- & -- & -- & -- & CH, CN & 22.11.2008\\\\\nHE 0443-2523 & 04 45 17.7 & -25 17 48 & 224.9946 & -38.0024 & 13.8 & -- & -- & -- & 10.704 & 10.169 & 10.012 & CH, CN & 06.12.2008 \\\\\nHE 0513-2008 & 05 15 14.9 & -20 04 59 & 221.6222 & -29.8139 & 13.2 & -- & -- & -- & 9.398 & 8.810 & 8.677 & CH, CN& 18.01.2009\\\\\n & & & & & & & & & & & & &22.11.2008\\\\\nHE 1027-2644 & 10 29 57.8 & -26 59 51 & 267.8862 & 26.0820 & 14.4 & 13.4 & 1.40 & 1.59 & -- & -- & -- & & 17.01.2009\\\\\nHE 1033-0059 & 10 36 34.0 & +00 43 39 & 246.4726 & 48.2458& 12.9 & 13.0 & 1.10 & 1.29 & 10.709 & 10.138 & 9.951 & CH, CN & 09.04.2007\\\\\nHE 1036-2615 & 10 38 25.9 & -26 30 50 & 269.3336 & 27.5445 & 14.6 & 13.7 & 1.21 & 1.49 & -- & -- & -- & CH, CN& 17.01.2009\\\\\nHE 1037-2644 & 10 40 02.3 & -27 00 36 & 269.9776 & 27.3253 & 14.3 & 13.4 & 1.33 & 1.38 & 11.360 & 10.776 & 10.662 & CH, CN& 06.02.2009 \\\\\nHE 1056-1855 & 10 59 12.2 & -19 11 08 & 269.4843 & 36.2931 & 13.6 & -- & -- & -- & 10.784 & 10.249 & 10.090 & CH, CN& 22.11.2008 \\\\\nHE 1104-1442 & 11 06 30.3 & -14 58 56 & 268.5996 & 40.7912 & -- & 13.7 & 1.31 & 1.16 & 12.217 & 11.568 & 11.470 & \nCH, CN & 09.04.2007\\\\\nHE 1105-2736 & 11 07 44.7 & -27 52 35 & 276.5372 & 29.6271 & 14.1 & 13.2 & 1.30 & 1.23 &11.314 & 10.742 & 10.551 & CH, CN& 17.03.2009\\\\\nHE 1112-2557 & 11 15 14.1 & -26 13 28 & 277.4416 & 31.8419 & 14.5 & 13.6 & 1.29 & 1.45 & 11.548 & 11.006 & 10.849 & CH, CN& 06.05.2009\\\\\nHE 1150-2546 & 11 53 15.5 & -26 03 41 & 286.9462 & 34.9979 & 15.6 & -- & -- & -- & -- & -- & -- & & 18.03.2009\\\\\nHE 1152-2432 & 11 54 35.0 & -24 48 44 & 286.8824 & 36.2813 & -- & -- & -- & -- & 9.413 & 8.833 & 8.686 & CH, CN& 18.03.2009 \\\\\nHE 1229-1857 & 12 31 46.8 & -19 14 02 & 296.5412 & 43.3938 & 14.6 & 14.0 & 0.85 &0.73 & 12.124 & 11.569 & 11.459 & CH, CN& 04.04.2009\\\\\nHE 1236-0036 & 12 39 25.4 & -00 52 58 & 296.5586 & 61.8402 & 16.5 & 13.5 & 0.92 & 0.94 &11.679 & 11.176 & 11.104 & CH, CN & 08.05.2007\\\\ \nHE 1406-2016 & 14 09 44.1 & -20 30 57 & 326.6476 & 38.7233 & 15.1 & 14.2 & 1.28 &1.22 & 12.092 & 11.558 & 11.423 & CH, CN& 16.03.2009 \\\\\nHE 1420-1659 & 14 23 03.1 & -17 12 51 &332.7845 &39.8992 & 13.1 & -- & -- & -- & 9.871 & 9.226 & 9.067 &\nCH, CN & 18.01.2009 \\\\\nHE 1514-0207 & 15 16 38.9 & -02 18 33 & 358.6655 & 44.2895 & 13.6 & -- & -- & -- & 10.335 & 9.703 & 9.536 & CH, CN& 24.07.2008\\\\\nHE 2115-0709 & 21 17 42.1 & -06 57 11 & 44.4121 &-35.5571 & 15.1 & 16.2 & 1.68 & 1.70 & 12.673 & 12.000 & 11.808 & CH, CN & 07.06.2007\\\\\nHE 2121-0313 & 21 23 46.2 & -03 00 51 & 49.5147 & -34.9016& 14.9 & 13.9 &1.35 &1.47 & 11.304 & 10.718 & 10.592 & CH, CN& 11.06.2008 \\\\\nHE 2124-0408 & 21 27 06.8 & -03 55 22 & 49.0907 & -36.0859& 14.8 & 13.9 &1.26 &1.15 & 11.578 & 10.986 & 10.848 & CH, CN& 24.07.2008 \\\\\nHE 2205-1033 & 22 08 29.2 & -10 18 37 & 48.6453 & -48.1654& 12.8 & -- & -- & -- & 9.732 & 9.178 & 8.980 & CH, CN& 24.07.2008\\\\\n\\hline\n\\end{tabular}\n$^{a}$ From Christlieb et al. (2001b)\\\\\n\n\\end{table*}\n}\n\n\n{\\footnotesize\n\\begin{table*}\n\\centering\n{\\bf Table 3: Potential CH star candidates}\\\\\n\\tiny\n\\begin{tabular}{cccccccccccccc}\n\\hline\nStar No. & RA(2000)$^{a}$ & DEC(2000)$^{a}$& $l$ & $b$ & B$_{J}^{a}$& V$^{a}$ & B-V$^{a}$ & U-B$^{a}$ &J & H & K & &Dt of Obs\\\\\n & & & & & & & & & & & & & \\\\\n\\hline\nHE 0008-1712 & 00 11 19.2 & -16 55 34 & 78.5866 & -76.2106 & 16.5 & 15.2 & 1.78 & 1.64 & 13.630 & 13.069 & 12.975 & & 06.12.2008\\\\\n & & & & & & & & & & & & &11.09.2008\\\\\nHE 0052-0543 & 00 55 00.0 & -05 27 02 & 125.3316 & -68.3057 & 16.5 & 15.0 & 1.95 & 1.74 & 12.952 & 12.241 & 12.086 & & 12.09.2008 \\\\\nHE 0100-1619 & 01 02 41.6 & -16 03 01 & 136.7651 & -78.6185 & 15.9 & 14.7 & 1.54 & 1.28 & 13.114 & 12.537 & 12.476 & & 21.11.2008 \\\\\n & & & & & & & & & & & &\n&12.09.2008\\\\\nHE 0136-1831 & 01 39 01.8 & -18 16 43 & 176.4932 & -75.9157 & 16.9 & 15.6 & 1.72 & 1.43 & 14.216 & 13.679 & 13.532 & & 12.09.2008\\\\\nHE 0225-0546 & 02 28 19.4 & -05 32 58 & 174.1738 & -58.4222 & 16.5 & 15.2 & 1.79 & 1.51 & 13.347 & 12.707 & 12.528 & & 06.12.2008 \\\\\nHE 0308-1612 & 03 10 27.1 & -16 00 41 & 201.1165 & -55.9582 & 12.5 & -- & -- & -- & 10.731 & 9.821 & 9.406 & & 21.11.2008\\\\\nHE 0420-1037 & 04 22 47.0 & -10 30 26 & 205.0454 & -37.7176 & 15.2 & 14.7 & 1.38 & 0.99 & 12.341 & 11.815 & 11.695 & & 21.11.2008 \\\\\nHE 1027-2501 & 10 29 29.5 & -25 17 16 & 266.6832 & 27.4163 & 13.9 & 12.7 & 1.73 & 1.51 & 10.627 & 9.896 & 9.722 & & 22.11.2008\\\\\nHE 1045-1434 & 10 47 44.1 & -14 50 23 & 263.5905 & 38.4049 & 15.5 & 14.6 & 1.23 & 0.96 & 12.935 & 12.449 & 12.244 & & 09.04.2007\\\\\nHE 1051-0112 & 10 53 58.8 & -01 28 15 & 253.5294 & 49.7900 & 17.0 & 16.0 & 1.44 & 0.94 & 14.347 & 13.794 & 13.703 & & 06.12.2008 \\\\\nHE 1102-2142 & 11 04 31.2 & -21 58 29 & 272.5241 & 34.5071 & 16.0 & 14.9 & 1.44 & 0.98 & 13.275 & 12.714 & 12.601 & & 16.03.2009 \\\\\nHE 1110-0153 & 11 13 02.7 & -02 09 28 & 261.1493 & 52.4883 & 16.5 & 15.5 & 1.47 & 1.29 & 12.912 & 12.205 & 12.063 & & 04.04.2009 \\\\\nHE 1119-1933 & 11 21 43.5 & -19 49 47 & 275.7342 & 38.2583 & 12.8 & 14.6 & 1.34 & 0.87 & 13.043 & 12.571 & 12.422 & & 18.03.2009 \\\\\nHE 1120-2122 & 11 23 18.6 & -21 38 33 & 277.1276 & 36.7743 & 12.9 & -- & -- & -- & 9.573 & 8.902 & 8.788 & & 17.03.2009 \\\\\nHE 1123-2031 & 11 26 08.7 & -20 48 19 & 277.4490 & 37.8097 & 16.8 & 15.8 & 1.33 & 1.19 & 13.513 & 12.940 & 12.800 & & 18.03.2009\\\\\nHE 1142-2601 & 11 44 52.9 & -26 18 29 & 284.8485 & 34.2157 & 13.9 & 13.0 & 1.28 & 1.07 & 11.218 & 10.675 & 10.539 & \n& 04.04.2009\\\\\nHE 1145-1319 & 11 48 21.4 & -13 36 38 & 280.3756 & 46.4800 & 16.4 & 15.4 & 1.37 & 1.32 & 13.466 & 12.934 & 12.790 & & 17.03.2009\\\\\n & & & & & & & & & & & &\n&05.04.2009\\\\\nHE 1146-0151 & 11 49 02.3 & -02 08 11 & 273.2807 & 57.0990 & 14.9 & 14.2 & 0.96 & 0.97 & 12.929 & 12.400 & 12.262 & & 16.03.2009\\\\\nHE 1157-1434 & 12 00 11.5 & -14 50 50 & 284.8854 & 46.2211 & 16.1 & 13.8 & 1.62 & 1.41 & 11.792 & 11.178 & 11.019 & & 06.06.2007\\\\\n & & & & & & & & & & & & &07.06.2007\\\\\nHE 1205-2539 & 12 08 08.1 & -25 56 38 & 290.8957 & 35.9137 & 13.6 & 15.4 & 1.91 &1.72 & 13.313 & 12.665 & 12.540 & & 17.03.2009 \\\\\nHE 1210-2636 & 12 12 59.7 & -26 53 22 & 292.4334 & 35.1974 & 13.8 & 12.6 & 1.65 &1.52 & -- & -- & -- & & 06.05.2009\\\\\nHE 1228-0417 & 12 31 12.5 & -04 33 40 & 293.4045 & 57.9360 & 14.8 & 14.0 & 1.17 &0.96 & 12.406 & 11.969 & 11.818 & & 04.04.2009\\\\\nHE 1253-1859 & 12 56 38.4 & -19 15 32 & 304.6277 & 43.5957 & 13.7 & 12.9 & 1.09 &1.22 & 10.60 & 9.972 & 9.850 & & 16.03.2009\\\\\nHE 1331-2558 & 13 34 20.1 & -26 13 38 & 314.7873 & 35.6550 & 13.9 & 16.0 & 1.53 &1.10 & 10.998 & 10.394 & 10.240 & & 18.03.2009\\\\\n& 17.03.2009\\\\\nHE 1404-0846 & 14 06 55.1 & -09 00 58 & 332.3876 & 49.4897 & 15.3 & 14.2 & 1.52 &1.29 & 12.435 &11.783 & 11.681& & 07.06.2007\\\\ \nHE 1405-0346 & 14 07 58.3 & -04 01 03 & 336.5341 & 53.7853 & 14.7 & 13.5 & 1.68 &1.28 & 11.608 & 11.068 & 10.903 & & 16.03.2009\\\\\nHE 1410-0125 & 14 13 24.7 & -01 39 54 & 340.6313 & 55.0908 & -- & -- & -- & -- & 10.360 & 9.770 & 9.651 & & 16.03.2009\\\\\n & & & & & & & & & & & & &11.06.2008\\\\\nHE 1425-2052 & 14 28 39.5 & -21 06 05 & 331.4023 & 36.3382 & 13.6 & 12.7 & 1.27 &1.29 & 10.043 & 9.446 & 9.273 & & 17.03.2009\\\\\nHE 1431-0755 & 14 34 32.7 & -08 08 37 & 341.8828 & 46.7820 & 14.6 & 13.5 & 1.51 & 1.44 & 11.283 & 10.605 & 10.422 & & 08.05.2007\\\\ \nHE~1440-1511 & 14 43 07.1 & -15 23 48 & 338.737 & 39.5765 & 14.5 & 13.5 & 1.13 & 0.87 & 12.238 & 11.757 & 11.606 & & 09.05.2007 \\\\\nHE 1447+0102 & 14 50 15.1 & +00 50 15 & 355.2263 & 51.2262 & 15.6 & 15.0 & 0.90 & 0.14 & 13.207 & 12.760 & 12.682 & & 08.04.2007\\\\\nHE 1525-0516 & 15 27 52.2 & -05 27 04 & 358.1110 & 40.1019 & 16.8 & 15.8 &1.29 &1.14 & 13.972 & 13.479 & 13.314 & & 11.06.2008 \\\\\n & & & & & & & & & & & & &07.06.2007\\\\\n & & & & & & & & & & & & &08.06.2007\\\\\nHE 2114-0603 & 21 17 20.8 & -05 50 48 & 45.5467 & -34.9459& 16.7 & 15.4 &1.80 &1.58 & 12.472 & 11.786 & 11.615 & & 11.09.2008 \\\\\n & & & & & & & & & & & & &09.05.2007\\\\\nHE 2211-0605 & 22 13 53.5 & -05 51 06 & 55.3066 & -46.9505& 16.0 & 15.1 &1.21 &1.01 & 13.383 & 12.875 & 12.727 & & 24.07.2008 \\\\\nHE 2228-0137 & 22 31 26.2 & -01 21 42 & 64.4957 & -47.7030& 15.8 & 14.7 &1.56 &1.20 & 12.301 & 11.715 & 11.589 & & 11.09.2008\\\\\n & & & & & & & & & & & & &25.07.2009\\\\\nHE 2246-1312 & 22 49 26.4 & -12 56 35 & 53.2930 & -58.1569& 17.0 & 15.9 &1.56 &1.60 & 14.101 & 13.472 & 13.303 & & 11.09.2008\\\\\n & & & & & & & & & & & & &25.07.2008\\\\\n\\hline\n\\end{tabular}\n\n$^{a}$ From Christlieb et al. (2001b)\\\\\n\n\\end{table*}\n}\n\n\\section { Characteristics of carbon stars and spectral classification}\n\n Spectral classification helps reducing\n the number of stars to be analyzed to a tractable number of\nprototype objects of different groups; each group may be correlated\nwith one or more physical parameters such as luminosity and temperature.\nAbundance anomalies observed in carbon stars cannot be explained on \nthe basis of observed temperature and luminosity of the stars; \n it is therefore difficult to \ndevise a classification scheme for carbon stars based on only \nthese two physical parameters. \nMorgan-Keenan system for carbon star classification (Keenan 1993)\ndivided carbon stars into C-R, C-N and C-H sequence,\n with subclasses running to C-R6, C-N6 and C-H6\naccording to temperature criteria. In the old R-N system, CH stars\nthat were classified as R-peculiar are put in a separate class in the\nnew system. \nIn the following we briefly discuss the main characteristics that place \ncarbon stars into different groups. Detailed discussions on carbon stars are\navailable in literature including Wallerstein and Knapp (1998) and \nreferences therein.\n\nIn contemporary stellar classification schemes assigning stars to\n `morphological groups' is largely in practice. \nCarbon stars are primarily classified based on the strength of carbon\nmolecular bands. \n The C-N stars are characterized by strong depression \nof light in the violet part of the spectrum. The cause of rapidly weakening \ncontinuum below about 4500\\AA\\, is not fully established yet, but believed\nto be due to scattering by particulate matter. Oxygen-rich stars of similar \neffective temperatures do not show such weakening.\n C-N stars are also easily detected for their characteristic \ninfrared colours. The majority of C-N stars show ratios of $^{12}$C\/$^{13}$C \nin the range of 30 to 100 while in C-R stars this ratio ranges from\n 4 to 9 (Lambert et al. 1986). They have lower temperatures and stronger \nmolecular bands than those of C-R stars. They are used as tracers of an \nintermediate age population in extragalactic objects. \n\nCH stars are characterised by the strong G-band of CH in their spectra. They\n form a group of warm stars of equivalent spectral\ntypes, G and K normal giants, but show weaker metallic lines.\nIn general, CH stars are high velocity objects, large radial velocities\nindicating that they belong to the halo population of the Galaxy \n(McClure 1983, 1984, McClure \\& Woodsworth 1990).\n `CH-like' stars, \nwhere CH are less dominant have low space velocities (Yamashita 1975).\nFrom radial velocity survey CH stars are known to be binaries. \nAccording to the models of McClure (1983, 1984) and McClure \\& Woodsworth \n(1990) the CH binaries have orbital characteristics consistent with the\npresence of a white dwarf companion. Early type CH stars are believed to \nhave conserved the products \nof the carbon rich primary received through mass transfer and survived until\nthe present in the Galactic halo.\nCH stars are not a homogeneous group of stars. They consist of two populations, \nthe most metal-poor ones have a spherical distribution and the ones slightly \nricher in metals are characterised by a flattened ellipsoidal distribution \n(Zinn 1985). The ratio of the local density of CH stars was found to be \n ${\\sim}$ 30\\% of metal-poor giants (Hartwick \\& Cowley 1985).\n\n Many C-R stars also show a quite strong G band of CH in their \nspectra. Hence, based only on the strength of the G-band of CH it is \nnot possible to make a distinction between CH and C-R stars.\n In such cases the secondary \nP-branch head near 4342 \\AA\\, serves as a more useful indicator.\nThis is a well-defined feature in CH stars spectra in contrast\n to its appearance in C-R stars spectra (refer Fig 2, 3 of Goswami 2005).\nAnother important diagnostic feature is Ca I at 4226 \\AA\\, which in \ncase of CH stars is weakened by the overlying CH band \nsystems. In C-R star's spectra this feature is quite strong; usually \nthe line depth is deeper \nthan the depth of the CN molecular band around 4215 \\AA\\,. \nThese spectral characteristics allow for an identification of the CH and \nC-R stars even at low resolution. \n\nAbundances of neutron-capture elements can also be used as an useful \nindicator of spectral type. The abundances of s-process elements are nearly \nsolar in C-R stars (Dominy 1984); whereas the CH stars show significantly \nenhanced abundances of the s-process elements relative to iron \n(Lambert et al. 1986, Green and Margon 1994). Although, the C-R as well as \nCH stars have warmer temperatures than those of C-N stars and blue\/violet \nlight is accessible \nto observation and atmospheric analysis, at low dispersion the \nnarrow lines are difficult to estimate and elemental abundances can not be\ndetermined. At low dispersion, therefore, the `abundance criteria'\ncan not be used to distinguish the C-R stars from the CH stars.\n Although the CH and \nC-R stars have similar range of temperatures, the distribution of CH stars \nplace most of them in the Galactic halo. The large radial velocities,\ntypically $\\sim$ 200 km s$^{-1}$ of the CH stars are indicative of their \nbeing halo objects (McClure 1983, 1984).\n \nC-J stars spectra are characterized by strong Merrill-Sanford (M-S) bands \nascribed to SiC$_{2}$ that appear in the wavelength region 4900 - 4977 \\AA\\,.\n The SiC$_{2}$ being a triatomic molecule, M-S bands are expected to be the \nstrongest in the coolest stars. SiC$_{2}$ and C$_{3}$ have similar \nmolecular structures and in many C stars C$_{3}$ molecule is believed to be\nthe cause of ultraviolet depression (Lambert et al. 1986).\n These bands are absent in the spectra of known CH stars. A few warmer C-N stars \nare known to exhibit the presence of M-S bands in their spectra. \n Strength of M-S bands are known to show a distinct correlation with \ncarbon isotopic ratios; i.e., stars with higher $^{12}$C\/$^{13}$C ratios show\nweaker M-S bands. \n WZ Cas, V Aql \\& U Cam are a few exceptions that have low $^{13}$C \nand strong M-S bands (Barnbaum et al. 1996).\nStrong C-molecular bands but a weak CH band characterize the class of\nhydrogen deficient carbon stars. \n\nWe have classified the program stars guided by the above spectral \ncharacteristics. In the present sample of ninety two stars the spectra\nof twenty two objects are found to not exhibit molecular bands of C$_{2}$.\n These objects are listed in Table 2.\n Among the seventy stars that exhibit strong carbon molecular bands\n (Table 1) thirty six of them are found to show spectral characteristics \nof CH stars. \nThese potential CH star candidates are listed in Table 3. \n In the following we discuss the spectral characteristics\nof the individual objects. \n\n\\section{ Results and Discussions}\n\n The spectra of the objects are primarily examined in terms of the following \nspectral characteristics.\\\\\n1. The strength (band depth) of the CH band around 4300 \\AA\\,.\\\\\n2. Prominance of the secondary P-branch head near 4342 \\AA\\,.\\\\\n3. Strength\/weakness of the Ca I feature at 4226 \\AA\\,.\\\\\n4. Isotopic band depths of C$_{2}$ and CN, in particular the Swan bands\n of $^{12}$C$^{13}$C and $^{13}$C$^{13}$C near 4700 \\AA\\,.\\\\\n5. Strengths of the other C$_{2}$ bands in the 6000 -6200 \\AA\\, region. \\\\\n6. The $^{13}$CN band near 6360 \\AA\\, and the other CN bands across the\n wavelength range.\\\\\n7. Presence\/absence of the Merrill-Sandford bands around 4900 - 4977 \\AA\\, \nregion.\\\\\n8. Strength of the Ba II features at 4554 \\AA\\, and 6496 \\AA\\,.\\\\\n\nThe membership of a star in a particular group is established from a \ndifferential analysis of the program stars spectra with the spectra\nof the comparison stars. Spectra of carbon stars available in the \nlow resolution spectral atlas of carbon stars of Barnbaum et al. (1996)\nare also consulted. \nIn Figure 1 we reproduce\n the spectra of the comparison stars in the wavelength region\n4000 - 6800 \\AA\\,; in figure 2 we show one example of HE stars spectra \nfrom the present sample corresponding\nto each comparison star's spectrum in figure 1.\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig1.eps}\n\\caption{ The spectra of the comparison stars in the wavelength \nregion 3860 - 6800 \\AA\\,. Prominent features seen on the spectra are indicated.}\n\\label{Figure 1}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig2.eps}\n\\caption{ An example of each of the HE stars corresponding to the \ncomparison stars presented in Figure 1, in the top to bottom sequence,\nin the wavelength region 3860 - 6800 \\AA\\,. The locations of the\nprominent features seen in the spectra are marked on the figure.\n}\n\\label{Figure 2}\n\\end{figure*}\n\n\\subsection { Location of the candidate CH stars on (J-H) vs (H-K) plot}\n\nWe have used JHK photometry as supplementary diagnostics for stellar \nclassification. Figure 3 shows the locations of the candidate CH stars \nlisted in Table 3 on the (J-H) vs (H-K) plot.\n 2MASS JHK measurements of the stars are available on-line at\nhttp:\/\/www.ipac.caltech.edu\/.\nThe thick box on the lower left represents the location of CH stars and the\nthin box on the upper right represents the location of C-N stars (Totten et\nal. 2000). \n Except two lying outside (shown with open squares in figure 3), the locations \nof the candidate CH stars (shown with open circles) are well within the CH \nbox. This supports their identification with the class of CH stars. \nLocation of the comparison CH stars\nHD~26, HD~5223 and HD~209621, C-R star RV Sct, C-N stars V460~Cyg \nand Z PSc, are shown by solid squares on the (J-H) vs (H-K) plot.\nAs described in the next section, we have also used 2MASS JHK photometry \n to determine the effective temperatures of the objects.\n\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig3.eps}\n\\caption{\n A two colour J-H versus H-K diagram of the candidate CH stars.\nThe thick box on the lower left represents the location of CH stars and the\nthin box on the upper right represents the location of C-N stars (Totten et\nal. 2000). Majority of the candidate CH stars\nlisted in Table 3 (represented by open circles) fall well\n within the CH box. The positions of the two outliers are shown with open\nsquares. C-N stars found in our sample are represented by solid \ntriangles. The location of the \ncomparison stars are labeled and marked with solid squares. Location of the \nthree dwarf carbon stars are indicated by open triangles.}\n\\label{Figure 3}\n\\end{figure*}\n\n\\subsection{ Effective temperatures of the program stars}\nSemiempirical temperature calibrations offered by Alonso et al. \n(1994, 1996, 1998)\nare used to derive priliminary temperature estimates of the program stars.\nThese authors used the infrared flux method to measure temperatures for a \nlarge number of lower main sequence stars and subgiants to derive\n the calibrations. The calibrations relate T$_{eff}$ with Stromgren \nindices as well as [Fe\/H] and colours (V-B), (V-K), (J-H) and (J-K). The\ncalibrations hold within a temperature and\nmetallicity range 4000 ${\\le}$ T$_{eff}$ ${\\le}$ 7000 K and\n-2.5 ${\\le}$ [Fe\/H] ${\\le}$ 0 . The estimated uncertainty in T$_{eff}$ \narising from different sources is ${\\sim}$90 K (Alonso et al. 1996).\nAlonso et al. derived the T$_{eff}$ scales using photometric data \nmeasured on TCS system; 2MASS JHK photometric data are therefore\nconverted to the TCS system using the conversion relations \nof Ramirez and Melendez (2004).\nEstimation of T$_{eff}$ from (J-H) \\&\n (V-K) temperature relations involve a metallicity term; (J-K) calibration\nrelation is independent of metallicity.\nWe have estimated the effective temperatures using adopted metallicities\nshown in parenthesis in Table 4.\n (B-V) calibration, normally used in case of normal stars is not \nconsidered as in the case of carbon stars the colour \n B-V depends on the \nchemical composition and metallicity in addition to T$_{eff}$. B-V colour \noften gives a much lower value than the actual surface temperature of the \nstar due to the effect of CH molecular absorption in the B band.\nWe have assumed that the effects of reddening on the measured colours\nare negligible.\n \n\n\\subsection{ Isotopic ratio $^{12}$C\/$^{13}$C from molecular band depths}\n\n Carbon isotopic ratios $^{12}$C\/$^{13}$C, widely used as mixing \ndiagnostics provide an important probe of stellar evolution. \nThese ratios measured on low resoltuion spectra do not give accurate \nresults but provide a fair indication of evolutionary states of the objects.\n\nWe have estimated these ratios, whenever possible, using the molecular band \ndepths of (1,0) $^{12}$C$^{12}$C ${\\lambda}$4737 and\n(1,0) $^{12}$C$^{13}$C ${\\lambda}$4744.\nFor a majority of the candidate CH stars,\nthe ratios $^{12}$C\/$^{13}$C are found to be ${\\le}$ 10.\nThese ratios for HD 26, HD 5223 and HD 209621 are respectively\n5.9, 6.1 and 8.8 (Goswami 2005).\n\nOur estimated ratios of $^{12}$C\/$^{13}$C indicate that most of the candidate\nCH stars belong to the `early-type' category.\nThe low carbon isotope ratios imply that, in a binary system,\nthe material transferred from the now unseen companion has been mixed into\nthe CN burning region of the CH stars or constitute a minor fraction of the\nenvelope mass of the CH stars. Low isotopic ratios are typical of stars\non their first ascent of the giant branch.\nThe $^{12}$C\/$^{13}$C ratios and the total carbon abundances decrease \ndue to the convection which dredges up the products of internal CNO cycle to \nstellar atmosphere as ascending RGB. If it reaches AGB stage, fresh $^{12}$C \nmay be supplied from the internal He burning layer to stellar surface leading \nto an increase of $^{12}$C\/$^{13}$C ratios again. \n\n\\subsection{ Spectral characteristics of the candidate CH stars } \n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig4.eps}\n\\caption{ A comparison of the spectra of three HE stars in the\nwavelength region 3870 - 5400 \\AA\\, with the spectrum of the comparison star\nHD~26. Prominent features noticed in the spectra are marked on the figure.\n}\n\\label{Figure 4}\n\\end{figure*}\n\n{\\bf HE~0420-1037, ~1102-2142, ~1142-2601, \n~1146-0151, ~1210-2636, ~1253-1859, ~1447+0102.~~ }\nThe spectra of these objects resemble closely the spectrum of HD~26.\nHD~26 is a known classical CH star with effective temperature \n5170 K and log $g$ = 2.2 (Vantures 1992b). \n The temperatures of these objects as measured using JHK\nphotometric data range from 4200 to 5000 K.\n The locations of these objects are well within the CH box in figure 3.\nIn figure 4, we show as an example, a comparison of the spectra of three \nobjects HE~0420-1037, HE~1142-2601 and HE~1146-0151 with the spectrum \nof HD~26.\nWith marginal differences in the strengths of the molecular features, the \nspectra of these \nthree objects show more or less a good match with their counterparts in HD~26. \nThe CN band around 4215 \\AA\\, and the C$_{2}$ band around 5165 \\AA\\, in \nthe spectra\nof HE~0420-1037 and HE~1146-0151 are marginally stronger; the Na I D \nfeature also \nappears stronger. The features due to Ca K and H appear with similar strengths.\nIn the spectrum of HE~1142-2601, the CN band depth around 4215 \\AA\\,\n and Ca II K and H line depths are deeper than their counterparts in HD~26.\nThe Ca I line at 4226 \\AA\\, is detected with line depth weaker than the band \ndepth around 4215 \\AA\\,. In HD~26, the Ca I 4226 \\AA\\, feature is \nnot detected.\nThe object HE~1253-1859 also have very similar spectrum with that of HD~26.\nThe CN band around 4215 \\AA\\, and carbon \nisotopic band around 4730 \\AA\\, are stronger, but the CH band, Ca II K and H \nfeatures are of similar strengths. The molecular bands around 5165 \\AA\\, \nand 5635 \\AA\\,\n show an exact match. The lines due to Na I D, Ba II at 6496 \\AA\\, and\n H$_{\\alpha}$ are seen equally strongly as in HD~26. \nThe Ca I feature at 4226 \\AA\\, could not be detected and the\n secondary P-branch head around 4342 \\AA\\, seems to be marginally \nweaker. \n In the spectrum of HE~1102-2142 \nthe molecular C$_{2}$ bands around 4735, 5165 and 5635 \\AA\\, are slightly \ndeeper than those in HD~26. The CN band around 4215\\AA\\, and the CH band \naround 4310 \\AA\\, \nalso appear marginally stronger in the spectrum of HE~1102-2142. The\nH$_{\\alpha}$ feature and the Ba~II line at 6496 \\AA\\, are marginally weaker;\nthe feature due to Na I D\nappears with similar strength as in HD~26. H$_{\\beta}$ at 4856\\AA\\,\nis clearly seen. The Ca I line at 4226 \\AA\\, is weakly detected.\nIn the spectrum of HE~1210-2636, the CN band around 4215 \\AA\\, and \nthe secondary P-branch head around 4342 \\AA\\, appear slightly stronger than \nin HD~26. The C$_{2}$ molecular band around 5165 \\AA\\, is also slightly \nweaker.\nThe carbon isotopic band around 4733 \\AA\\, is marginally detectable. \nCa I line at 4226 \\AA\\, could not be \ndetected. Features due to Ca II K and H and H$_{\\alpha}$ are clearly detected.\n In the spectrum of HE~1447+0102, the CN band around 4215 \\AA\\, is almost\nabsent. Strong well defined features due to Ca II K and\nH are seen. Molecular bands around 4733, 5165, and 5635 \\AA\\,\n are distinctly seen to be stronger than their counterparts in HD~26. \nIn the redward of \n5700 \\AA\\, no molecular bands are detected. We assign these objects to \nthe CH group. \\\\\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig5.eps}\n\\caption{ A comparison of the spectra of three HE stars in the\nwavelength region 3870 - 5400 \\AA\\, with the spectrum of the comparison star\nHD~209621. Prominent features noticed in the spectra are marked on the figure.\n}\n\\label{Figure 5}\n\\end{figure*}\n{\\bf HE~0136-1831, ~0308-1612, ~1027-2501, ~1051-0112, ~1119-1933, ~1120-2122, \n~1123-2031, ~1145-1319, ~1205-2539, ~1331-2558, ~1404-0846, \n~1425-2052, ~1525-0516, ~2114-0603, ~2211-0605, \n~2228-0137, ~2246-1312.~~ } \nThe spectra of these objects resemble closely the spectrum of HD~209621,\na classical CH star with effective temperature ${\\sim}$ 4400 K \n(Tsuji et al. 1991). The effective temperatures estimated for this set \nof objects using JHK \nphotometry range from 3948 K (HE~2114-0603) to 4675 K (HE~1119-1933). Their\nlocations on the J-H vs H-K plot are well within the CH box in figure 3.\nThree examples, HE~0136-1831, HE~1027-2501 and HE~2228-0137 from this \nset are shown in figure 5 together with the spectrum of HD~209621.\nIn the spectrum of HE~0136-1831, the CN band around 4215 \\AA\\, is \nmarginally weaker and the carbon molecular \nbands around 4733 and 5635 \\AA\\, are marginally stronger than those \nin HD~209621. All other features show a good match. The spectrum of \nHE~1027-2501 also shows a close match with the spectrum of HD~209621. \nExcept for the molecular bands around 4733, 5165 and 5635 \\AA\\, that appear \nmarginally weaker in the spectrum of HE~2228-0137 the spectrum of this\n object bears a close resemblance with the spectrum of HD~209621.\nThe spectrum of HE~0308-1612 shows weaker molecular bands around 5165 \\AA\\, \n and 5635 \\AA\\,. The CN band around 4215 \\AA\\, and the carbon \nisotopic band around 4730 \\AA\\, are of similar strengths. The CH band around \n4300 \\AA\\, and Ca II K and H features are of similar depths. The lines \ndue to Na I D, Ba II at 6496 \\AA\\, and H$_{\\alpha}$ are clearly noticed.\nThe Ca I feature at 4226 \\AA\\, could not be detected. \nThe spectrum of HE~1051-0112 shows a weaker CN band around 4215 \\AA\\,. The \nG-band \nof CH appears with almost the same strength as in HD~209621. The secondary \nP-branch head around 4342 \\AA\\, and the bands around 4730 and 5635 \\AA\\, \nare relatively stronger. Features due to Ca II K and H are barely detectable \nin the spectrum of this object. The molecular band around 5165 \\AA\\, shows an \nexact match with its counterpart in HD~209621. The features due \nto H$_{\\alpha}$ and Na I D are detected distinctly; the Ba II feature at\n 6496 \\AA\\, is marginally detected.\n\nIn the spectrum of HE~1119-1933, Ca II K and H\nappear marginally stronger. The molecular band around 5365\\AA\\, appears \nmarginally \nweaker than in HD~209621. The spectra of HE~1120-2122 and HE~1123-2031 are \nvery similar, both exhibit a weaker CN band around 4215 \\AA\\,.\n All other features show a good match with their counterparts in the spectrum \nof HD~209621.\nCa II K and H appear with almost the same strength in the spectrum of \nHE~1120-2122 as in HD~209621. Ca I line at 4226 \\AA\\, is not detectable.\nThe spectra of HE~1123-2031, HE~1145-1319, HE~1205-2539, HE~1331-2558 \n are noisy shortward of 4100 \\AA\\,\nand the lines due to Ca II K and H could not be clearly detected.\nCa I line at 4226 \\AA\\, is not detectable in these spectra.\nFeatures due to Na I D, H$_{\\alpha}$, and Ba II at 6496 \\AA\\, are detected. \n In the spectrum of HE~1145-1319, C$_{2}$ molecular bands around 5635 and \n4733 \\AA\\, are marginally deeper than those in HD~209621. \nThe features redward of 4200 \\AA\\, in the spectrum of HE~1205-2539 show a good \nmatch with those in HD~209621. The molecular features in the spectrum \n of HE~1331-2558 are also of similar strenths with those in HD~209621. \nExcept for the G-band of CH, all other molecular features are weaker in the \nspectrum of HE~1404-0846. The features due to Ca II K and H are marginally \nstronger. The spectrum of HE~1404-0846 is noisy at the blue end.\n\nIn the spectrum of HE~1425-2052 the G-band of CH and the C$_{2}$ band \naround 5635 \\AA\\, are mildly stronger.\nThe rest of the features show a close match with their counterparts \nin HD~209621.\nThe Ba II feature at 6496 \\AA\\, appears with equal intensity as that of\nH$_{\\alpha}$ feature. The feature due to Na I D is clearly detected.\nIn the spectrum of HE~1525-0516, H$_{\\alpha}$ and Na I D features\n are detected with almost equal strength as in HD~209621. \nThe spectrum of HE~2114-0603 shows a remarkably close match with\nthe features in HD~209621 including those longward of 5700 \\AA\\,. However the\nfeatures due to Ca II K and H that are seen very distinctly in the spectrum \nof HD~209621 could not be detected in the spectrum of HE~2114-0603;\nthe spectrum is noisy blueward of 4000 \\AA\\,.\nIn the spectrum of HE~2211-0605 the molecular\n features are weaker than their counterparts in HD~209621 but stronger \nthan those in HD~26. The CH band matches exactly with the one in HD~209621.\n Ca II K and H appear with almost equal strengths as in HD~209621.\nThe spectrum of HE~2246-1312 shows a weaker molecular band around\nCN 4215 \\AA\\,. Other molecular bands appear with almost of equal strengths \nas their counterparts in HD~209621. The spectrum \nblueward of 4100 \\AA\\, is noisy and \n Ca II K and H features could not be detected as well defined features.\nThe spectrum obtained in september 2008 has a better signal. \\\\\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig6.eps}\n\\caption{ A comparison of the spectra of three HE stars in the\nwavelength region 3880 - 5400 \\AA\\, with the spectrum of the comparison star\nHD~5223. Prominent features noticed in the spectra are marked on the figure.\n}\n\\label{Figure 6}\n\\end{figure*}\n\n {\\bf HE~0008-1712, ~0052-0543, ~0100-1619, ~0225-0546, ~1045-1434, \n~1110-0153, ~1157-1434, ~1228-0417, ~1405-0346, ~1410-0125, ~1431-0755,\n ~1440-1511.~~ }\nThe spectra of these objects closely resemble the spectrum of HD~5223, \na well-known classical CH star with effective temperature ${\\sim}$ 4500 K,\n log $g$ = 1.0 and metallicity [Fe\/H] = $-$2.06 (Goswami et al. 2006).\nThe effective temperatures of these objects derived from J-K colour range \nfrom about 3924 K (HE~0052-0543) to 4795 K (HE~1228-0417).\nExcept for the two outliers HE~1045-1434 and HE~1228-0417 (represented \nwith open squares) the locations of this set of \nobjects are well within the CH box in figure 3.\n\nA comparison of the spectra of HE~0008-1712, HE~0100-1619 and HE~1405-0346 \nwith the spectrum of HD~5223 is shown in figure 6.\nThe Ca I line at 4226 \\AA\\, is not detectable in any of these spectra.\nThe CH band as well as other molecular bands show a very good match.\nThe features due to Ca II K and H are seen with equal strength as in HD~5223.\nThe CN band around 4215 \\AA\\, in HE~0100-1619\nis slightly deeper. The bands longward of 5635 \\AA\\, are also marginally \ndeeper.\nThis object HE~0100-1619 is also mentioned as a CH star in \nTotten et al. (2000). Heliocentric radial velocity of HE~0100-1619 as reported\n by Bothun et al. (1991) is $-$142 km s$^{-1}$.\n\n The spectra of HE~0052-0543 and HE~1110-0153 \nshow stronger molecular bands than their counterparts in HD~5223.\nIn the spectrum of HE~0225-0546 the molecular bands are marginally stronger \nthan in HD~5223. The molecular features above 5700 \\AA\\, seen in these two\nspectra are barely noticed in the spectrum of HD~5223.\nThe spectrum of HE~1157-1434 also show a good match with the spectrum of\nHD~5223 except for the molecular band around 5635 \\AA\\, which is distinctly\n weaker in its spectrum.\nThe molecular features redward of 5700 \\AA\\, are also noticed\nweakly in the spectrum of this object.\n The spectrum of HE~1405-0346 shows a stronger CN band around\n4215 \\AA\\, as well as a stronger carbon molecular band around 5635 \\AA\\,. \nThe secondary P-branch head near 4342 \\AA\\, is also stronger\nthan its counterpart in HD~5223. Other molecular bands around 4733 and \n5165 \\AA\\, show a good match. Ca II K and H are seen as strongly as \nin HD~5223.\n The effective temperature of the object from J-K colour is 4391 K, slightly \nlower than the effective temperature of HD~5223. \nIn the spectrum of HE~1410-0125 the molecular features are slightly shallower\nthan their counterparts in HD~5223. The CH band depth is however of similar \nstrength. The feature at Ca I 4226 \\AA\\, is absent; the features due to \n Ca II K and H are of similar strengths. The CN band around 4215 \\AA\\, \nmatches well with the CN feature in HD~5223. The radial velocity\nof this object as quoted by Frebel et al. (2006) is +80 Km s$^{-1}$. \nThe effective temperature estimated using\n(J-K) calibation returns a value 4378 K for this object.\n\nThe spectrum of HE~1431-0755 is noisy blueward of 4000 \\AA\\,; the features of\n Ca II K and H could not be detected. The CH band around 4310 \\AA\\, and\n the CN band around 4215 \\AA\\, appear\n slightly stronger than their counterparts in HD~5223. \nOther C$_{2}$ molecular bands present in the spectrum are narrower than their\ncounterparts in HD~5223. The spectrum redward of 5700 \\AA\\, shows molecular \nfeatures that are barely noticed in the spectrum of HD~5223.\nThe spectrum of HE~1440-1511 shows molecular bands with almost equal depths\nwith those in HD~5223. Ca II K and H features are however stronger than \ntheir counterparts in\nHD~5223. The spectrum shows a distinctly stronger feature due to Na I D. \nFeatures of \nH$_{\\alpha}$ and Ba II at 6496 \\AA\\, are of equal strengths. The spectrum \nredward of 5700 \\AA\\, shows a good match.\n\nIn the spectrum of HE~1228-0417 the Ba II feature at 6496 \\AA\\, \nand H$_{\\alpha}$ are\nseen with equal strengths as in HD~5223. The part of the spectrum redward of\n5700 \\AA\\, shows a very good match. The Ca I line at 4226 \\AA\\, is not detected.\nThe feature due to Na I D is clearly detected. Other carbon molecular features\naround 4730, 5165, and 5635 \\AA\\, appear marginally weaker than their \ncounterparts in HD~5223. The G-band of CH appears with almost equal \nstrength but the CN band around 4215 \\AA\\, is marginally weaker than its \ncounterpart in HD~5223.\n The effective temperature of the object estimated using J-K colour \ncalibration is 4795 K, higher than the effective temperature of \nHD~5223 ${\\sim}$ 4500 K (Goswami et al. 2006). The location of this object\noutside the CH box is not obvious from its low resolution spectra.\\\\\n\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig7.eps}\n\\caption{ The spectra of three dwarf carbon stars in the\nwavelength region 4500 - 6800 \\AA\\,. Prominent features noticed in the \nspectra are marked on the figure.\n}\n\\label{Figure 7}\n\\end{figure*}\n\n{\\bf HE~0009-1824, ~1116-1628, ~1358-2508.~~} \n The spectra of these objects are illustrated in figure 7. \nThese three objects are known dwarf carbon stars.\nThe effective temperatures of HE~0009-1824, HE~1116-1628, HE~1358-2508\n as estimated from (J-K) calibration are respectively 5530 K, 4224 K and \n3623 K. As expected, the molecular band depths are the strongest \nin HE~1358-2508, \nthe coolest of the three objects; and weakest in HE~0009-1824.\nIn the spectrum of HE~0009-1824 the CN band around \n4215 \\AA\\, is completely missing.\nThe features due to Ca II K and H as well as the CN band near 3880 \\AA\\, \nare detected. The G-band of CH is strong but not as strong \nas it appears in CH stars. \nThe secondary P-branch head near 4342 \\AA\\, is seen distinctly. Apart from \nthe absence of the CN band around 4215 \\AA\\, the \nspectrum of this object looks somewhat similar to the spectrum\n of HD~209621. \nThe distance of this object as reported by Mauron et al. (2007) is 300 pc. \n\nThe spectra of HE~1116-1628 and HE~1358-2508 show characteristics of C-R star \nRV Sct with marginal differences in the molecular band depths. \nIn the spectrum of HE~1358-2508, the CH band is marginally stronger \nthan in RV Sct. The C$_{2}$ molecular bands are stronger in the spectrum \nof this object. The CN band around 4215\\AA\\, is clearly detected.\nRatnatunga (1983) first proposed this object HE~1116-1628 to be a \ndwarf carbon\nstar. This object is also present in the list of dwarf carbon stars\nof Lowrance et al. (2003). Mauron et al. (2007) reported the proper motions\n in ${\\alpha}$ and\n${\\delta}$ and their respective 1${\\sigma}$ errors in mas yr$^{-1}$ as\n$-23.5 \\pm 6.7$ and $+29.8 \\pm 4.6$. \n The distances of HE~1116-1628 and HE~1358-2508 \nas reported by Mauron et al. (2007) are 170 pc and 270 pc respectively.\n Totten and Irwin (1998) reported a radial velocity of $-69$ km s$^{-1}$\n for the object HE~1116-1628.\nAll the three objects have total proper motion ${\\mu}$ \n${\\ge}$ 30 mas yr$^{-1}$ (Mauron et al. 2007).\n\n The locations of the three dwarf carbon stars are \n indicated by open triangles in figure 3. Location of HE~0009-1824\nis on the left below the CH box, the location of HE~1358-2508 is on the\nright edge of the CH box and the location of HE~1116-1628 is found to be \nwell inside the CH box. \n Dwarf carbon stars have anomalous infrared colours (Green et al. 1992 and \nWesterlund et al. 1995). In the conventional two colour JHK diagram\nthe locus of dwarf-carbon-stars colours is away from the normal carbon-star\nlocus. The locus defined by dwarf carbon stars is bounded by \n(J-H) ${\\le}$ 0.75\nand (H-K) ${\\ge}$ 0.25 (Westerlund et al. 1995). This condition is \nsatisfied by HE~1358-2508; however\nHE~0009-1824, and HE~1116-1628 both have (H-K) colours less than the lower \nlimit of 0.25 mag set for dwarf carbon stars.\\\\\n\n\n\\begin{figure*}\n\\epsfxsize=14truecm\n\\epsffile{aruna_chstars_fig8.eps}\n\\caption{ A comparison of the spectra of the candidate C-N stars with the\n spectrum of V460 Cyg in the wavelength region 4500 - 6800 \\AA\\,.\n The bandheads of the prominant molecular bands, Na I D and H$_{\\alpha}$ \nare marked on the figure.\n}\n\\label{Figure 8}\n\\end{figure*}\n\n\n{\\bf Candidate C-N stars: HE~0217+0056, ~0228-0256, ~1019-1136, \n~1344-0411, ~1429-1411, ~1442-0058, ~2213-0017, ~2225-1401.~~} \\\\\nThe spectra of these objects show a close resemblance with the spectrum of the\nC-N star Z PSc with similar strengths of CN and C$_{2}$ molecular\nbands across the wavelength regions. \nIn figure 3, the objects HE~0217+0056, HE~1019-1136, HE~1344-0411, \nHE~1429-1411, HE~1442-0058 and HE~2213-0017 represented by solid triangles\n lie well within the CN box.\nThe spectra have low flux below about 4400 \\AA\\,.\n The spectrum of HE~1429-1411 is similar to that of Z PSc's spectrum,\n except that the CN band around 4215 \\AA\\, is marginally weaker in this star. \nThe CH band is weakly detected in the spectrum of HE~1116-1628. The molecular\nbands near 4735, 5135 and 5635 \\AA\\, are noticed distinctly. The feature\ndue to Na I D is strongly\ndetectable. The Ba II line at 6496 \\AA\\, is detectable\n but the H$_{\\alpha}$ feature could not be detected. \nHE~1127-0604 has low flux below about 4200 \\AA\\,. The CH band and \n C$_{2}$ molecular bands \naround 4735, 5165, 5635 \\AA\\, are detected. All the features in the spectrum \nare weaker than their counterparts in Z Psc. While features of \nCa II K and H are detected, the CN band around 4215 \\AA\\, is not clearly seen.\nThe spectrum of HE~2225-1401 has low flux below about 4700 \\AA\\,.\nThe strong C$_{2}$ molecular bands around 5165, 5635 \\AA\\, appear\n stronger than their counterparts in Z Psc. The spectrum of HE~2225-1401 \nalthough have spectral characteristics of C-N stars, its location in figure 3\n is not within the CN box. The spectra of the objects\n HE~2213-0017, HE~1442-0058, HE~1344-0411 compare closest to the\nspectrum of C-N star V460 Cyg as illustrated in figure 8. \n\nThe objects HE~0217+0056, HE~1019-1136, HE~1442-0058, HE~2213-0017 and \nHE~2225-1401 are also mentioned as N-type stars in the \nAPM survey of cool carbon stars in the Galactic halo (Totten \\& Irwin 1998).\nTotten et al. (2000) have provided proper motion measurements for \nthese objects. The distances measured\nby these authors assuming an average M$_{R}$ = $-$3.5 for these objects are \nrespectively 24, 16, 43, 29 and 24 kpc. Heliocentric radial velocities \nestimated by\nTotten \\& Irwin (1998) for these objects are respectively \n$-142 \\pm 3$, $126 \\pm 4$, $37 \\pm 4$, $-44 \\pm 3$, and $-113 \\pm 5$ \n km s$^{-1}$.\nHeliocentric radial velocity of HE~0228-0256 is $-72$ km s$^{-1}$\n (Bothun et al. 1991).\\\\\n\n\n\n{\\bf HE~0945-0813, ~1011-0942, ~1127-0604, ~1205-0521, ~1238-0836, \n ~1418-0306, ~1428-1950, ~1439-1338, ~2222-2337.~~ }\nThe spectra of these objects show characteristics of C-R stars. \n The spectra of HE~1011-0942, HE~1205-0521, HE~1238-0836 match closest to \nthe spectrum of RV~Sct. The effective temperatures of the objects estimated\nusing (J-K) calibration range from 3521 K (HE~1238-0836) to 4875 K\n(HE~1127-0604).\n\nIn the spectra of HE~1238-0836 and HE~1428-1950 the CH band around 4300 \\AA\\, \nis slightly deeper than in RV~Sct. The molecular features in the redward of \n5700 \\AA\\, appear marginally weaker. \nIn the spectra of HE~0945-0813, the CN band around 4215 \\AA\\, is \ndistinctly detected. Ca I line at 4226 \\AA\\, which is generally very\nweak or absent in CH stars appears very strongly in the spectrum\nof this object. The feature due to Na I D appears very strong.\nFeature due to the secondary P-branch head around 4342 \\AA\\, is \n somewhat noticeable in the spectrum of HE~0945-0813. The lines due to \nBa II at 6496 \\AA\\, and\nH$_{\\alpha}$ are detected in both the spectra. The molecular bands around 4733, \n5165 and 5635 \\AA\\, appear strongly in the spectrum of \nHE~0945-0813. \nThe spectrum of HE~2222-2337 has low flux below about 4100 \\AA\\,.\nThe CH band does not appear as strong as it should be in C-R star's spectrum.\nThe CN band around 4215 \\AA\\, is marginally \ndetected. Other molecular bands are of similar \nstrengths. The molecular features redward of 5635 \\AA\\, are slightly weaker. \nWe place these objects in the C-R group.\\\\\n\n{\\bf HE~0037-0654, ~0954+0137, ~1230-0327, ~1400-1113, ~ 1430-0919, \n~1447-0102, ~2157-2125, ~2216-0202, ~2255-1724, ~2305-1427, ~2334-1723,\n ~2347-0658, ~2353-2314.~~}\nThe spectra of these objects are characterized by a weak (or absent) CN band \naround 4215 \\AA\\,. Apart from this feature the spectra are somewhat similar\n to the spectrum of HD~209621.\n\nThe spectrum of HE~1230-0327 shows a strong G-band of CH\nand a distinct secondary P-branch head near 4342 \\AA\\,. Ca I feature\nat 4226 \\AA\\, is not detected. The CN band around 4215 \\AA\\, is almost \nabsent. While\natomic lines of Ca II K, H , H$_{\\alpha}$, Na I D are distinctly seen,\n BaII line at 6496 \\AA\\, is marginally detected.\nThe spectra of HE~0954+0137, HE~1400-1113, \nHE ~1430-0919, HE~2255-1724 and HE~2347-0658 look very similar to the spectrum\n of HE~1230-0327.\nIn the spectra of these objects the feature due to the CN band \naround 4215 \\AA\\\nis marginally detectable. Weak molecular bands noticed in the \nspectrum of HD~209621 upward of 5700 \\AA\\, are not observable in \nthese spectra. Compared to \n HD~209621, the molecular bands around 4733, 5165, and \n5635 \\AA\\, are slightly weaker in the spectra of these objects. \nCa II K and H are detected almost with equal strength as in HD~209621. \n In the spectrum of HE~1430-0919 the secondary P-branch head near 4342 \\AA\\,\n is marginally weaker than in HD~209621. While the molecular band \naround 5165 \\AA\\ shows a good match, the \nbands around 4733 and 5635 \\AA\\, are marginally stronger. The spectrum \nin the redward of 5700 \\AA\\, resembles the spectrum of HD~209621.\nFeatures due to Na I D, H$_{\\alpha}$ and Ba II line at 6496 \\AA\\, are detected.\nIn the spectrum of HE~1447+0102, the CN band around 4215 \\AA\\, is almost\nabsent. Strong well defined features of Ca II K and H are seen. \nThe C$_{2}$ molecular bands around 4733, 5165, 5635 \\AA\\, \n are distinctly present. No other molecular bands are noticed longward of \n5700 \\AA\\,.\nThe spectra of HE~2305-1427 and HE~2334-1723 show the CN band around\n 4215 \\AA\\, with band depth almost half of that in HD~209621. \nAll other molecular features match well with their counterparts in the\nspectrum of HD~209621. Weak\n molecular bands that are noticed in the spectrum of HD~209621 upward \nof 5700 \\AA\\,\nare not noticeable in the spectra of these two objects. The features due to \nNa I D, H$_{\\alpha}$ and Ba II at 6496 \\AA\\, could be detected.\nThe secondary P-branch head at 4223 \\AA\\, is seen as distinctly as \nin HD~209621.\n\nIn the spectrum of HE~2353-2314, the CH band around 4300 \\AA\\, as well as the\n CN band around 4215 \\AA\\,\n are marginally detected. The carbon molecular band around 5165 \\AA\\, is \nclearly detected; the band around 5635 \\AA\\, is much weaker. No other \nmolecular bands or atomic lines are detectable. \nIn the spectrum of HE~2255-1724, the CN band around 4215 \\AA\\ is much \nweaker \n than that in HD~209621. The CH band and Ca II K and H are of\nsimilar depths. The molecular bands around 4733, 5165, 5635 \\AA\\, are slightly \nweaker in this object. Features of Na I D, Ba II line at 6496 \\AA\\,, and\n H$_{\\alpha}$ are distinctly seen. \n The molecular bands longward of 5635 \\AA\\, are not detectable. \nThe spectrum acquired on Sep 11, 2008 have a better signal. In the\n spectrum of HE~2334-1723, the CN band around 4215 \\AA\\, is much weaker than \nin HD~209621; all other molecular bands show a good match.\n The features due to Ca II K and H also show a good match. No molecular \nbands are detectable upward of 5700 \\AA\\,.\nIn the spectrum of HE~2347-0658 the CN band around 4215 \\AA\\, is almost \nabsent. \nCa II K and H features and carbon molecular bands around 4733, 5165, \n5635 \\AA\\\nshow a good match. Molecular bands seen in HD~209621 upward of 5700 \\AA\\,\n are not detectable in the spectrum of this object.\n\n The spectrum of HE~1400-1113 is noisy below about 4220 \\AA\\,. The CN band \naround 4215 \\AA\\,\ncould be marginally detected. Ca II K and H are detected as weak features.\n A strong CH band around 4300 \\AA\\, and the secondary P-branch head \nnear 4342 \\AA\\, are distinctly seen. Other molecular\n features have band depths \nmarginally weaker than their counterparts in HD~209621. Except for \nNa I D, Ba II at 6496 \\AA\\, and \nH$_{\\alpha}$ no other atomic lines are detected redward of 5670 \\AA\\,. \n\nThe spectrum of HE~1430-0919 also shows a very weak CN band around 4215 \\AA\\,. \nThe features\ndue to Ca II K and H are not detected. The G-band of CH around 4300 \\AA\\, is \nhowever very strong in the spectrum.\nThe spectrum of HE~2157-2125 shows the CH band around 4300 \\AA\\, with almost \nequal \nstrength to its counterpart in HD~209621. Features due to Ca II K and H \n and other \nmolecular features are also seen with equal intensities. However, \nthe CN band around 4215 \\AA\\, is much \nweaker than that in HD~209621. The spectrum obtained in October, 2008 has a\nbetter signal than the spectra obtained in June and September, 2008.\n\n The spectrum of HE~0037-0654 looks very similar to the spectrum\nof HD~26; however, molecular bands of C$_{2}$ around 4730, 5165 and \n5635 \\AA\\, are marginally stronger than their counterparts in HD~26. The\nCN band around 4215 \\AA\\, is barely detected, much weaker than in HD~26.\n Ba II line at 6496 \\AA\\, is clearly detected. Strong lines \nof H$_{\\alpha}$ and Na I D are distinctly noticed. \nExcept for the features of Ca II K and H which are much weaker,\nthe spectrum of HE~2216-0202 is very similar to the spectrum of HD~26.\n The secondary \nP-branch head around 4342 \\AA\\, is much stronger than in HD~26. \nThe CN band around 4215 \\AA\\, is not\nobserved. The CH band at 4305 \\AA\\, is not as strong as in HD~26. The molecular\nbands around 4733 \\AA\\, and 5236 \\AA\\, are of similar depths. The carbon \nmolecular band around \n5635 \\AA\\, is weaker than the band around 5165 \\AA\\,. No molecular bands \nlongward of 5700 \\AA\\, are detectable. \n\nThe spectrum of HE~1315-2035 is noisy blueward \nof 4200 \\AA\\,. The G-band of CH and the carbon molecular bands near \n 4733, 5165 and 5635 \\AA\\, are detected in the spectrum. The H$_{\\alpha}$ \nfeature is clearly detected. The effective temperature of this object is \n4639 K as estimated from (J-K) colour calibration.\n\n{\\bf HE~1027-2644.~~} The spectra of HE~1027-2644 do not show presence \nof any carbon molecular bands. The features due to Ca II K and H are \n not detected. The G-band of CH is seen as a weak feature. \nFeatures due to Ca I at 4226 \\AA\\, and Na D I are \n seen as strong features. Ba II line at 6496 \\AA\\, and H$_{\\alpha}$ \nfeature are detected. 2MASS JHK photometry is not available for this object.\n\n\n\\begin{table}\n\\centering\n{\\bf Table 4: Estimated effective temperatures (T$_{eff}$) from \nsemi-empirical relations}\n\n\\begin{tabular}{|c|c|c|}\n\\hline\\noalign{\\smallskip}\nStar Name & Teff(J-K) & Teff(J-H) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nHE0008-1712 & 4556.52& 4378.94(-0.5) \\\\\n & & 4446.43(-2.5)\\\\ \n & & \\\\\nHE0009-1824 & 5530.27& 5377.40(-0.5) \\\\ \n & & 5443.08(-2.5) \\\\\n & & \\\\\nHE0037-0654 & 5496.71& 4912.74(-0.5) \\\\ \n & & 4980.33(-2.5) \\\\\n & & \\\\\nHE0052-0543 & 3924.82 &3831.70(-0.5) \\\\ \n & &3896.64(-2.5) \\\\\n & & \\\\\nHE0100-1619 & 4615.21 &4305.94(-0.5) \\\\\n & &4373.23(-2.5) \\\\\n & & \\\\\nHE0113+0110 & 4130.72 &3973.94(-0.5) \\\\\n & & 4039.78(-2.5) \\\\\n & & \\\\\nHE0136-1831& 4459.39& 4493.10(-0.5) \\\\\n & & 4560.81(-2.5) \\\\\n & & \\\\\nHE0217+0056& 2794.61& 3053.31(-0.5) \\\\\n & & 3110.53(-2.5) \\\\\n & & \\\\\nHE0225-0546& 4051.67& 4086.82(-0.5) \\\\ \n & & 4153.25(-2.5) \\\\\n & & \\\\\nHE0228-0256 & 3655.89& 3916.80(-0.5) \\\\ \n & & 3982.30(-2.5) \\\\\n & & \\\\\nHE0308-1612 & 4420.27& 4428.17(-0.5) \\\\\n & & 4495.77(-2.5) \\\\\n & & \\\\\nHE0341-0314& 4119.26& 4336.67(-0.5) \\\\\n & & 4404.05(-2.5) \\\\\n & & \\\\\nHE0420-1037& 4587.42& 4534.65(-0.5) \\\\ \n & & 4602.42(-2.5) \\\\\n & & \\\\\nHE0945-0813 & 4650.36& 4629.30(-0.5) \\\\\n & & 4697.14 (-2.5) \\\\\n & & \\\\\nHE1011-0942 & 3417.52& 3646.15(-0.5) \\\\\n & & 3709.67(-2.5) \\\\\n & & \\\\\nHE1019-1136 & 2690.13 &3140.84(-0.5) \\\\\n & & 3199.16(-2.5)\\\\ \n & & \\\\\nHE1028-2501 & 3824.73 & 3769.05(-0.5) \\\\ \n & & 3833.54(-2.5) \\\\\n & & \\\\\nHE1045-1434 &4436.49& 4738.21(-0.5) \\\\ \n & & 4806.03(-2.5) \\\\\n & & \\\\\nHE1051-0112 & 4594.34& 4411.69(-0.5) \\\\\n & & 4479.26(-2.5) \\\\\n & & \\\\\nHE1102-2142 &4492.46& 4383.27(-0.5) \\\\\n & & 4450.77(-2.5)\\\\\n & & \\\\\n\\noalign{\\smallskip}\\hline \n\\end{tabular}\n\n\\end{table}\n\n\\begin{table}\n\\centering\n{\\bf Table 4: Estimated effective temperatures (T$_{eff}$) from \nsemi-empirical relations (continued)}\n\n\\begin{tabular}{|c|c|c|}\n\\hline\\noalign{\\smallskip}\nStar Name & Teff(J-K) & Teff(J-H) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nHE1110-0153 & 3969.89& 3842.74(-0.5) \\\\\n & & 3907.75(-2.5) \\\\\n & & \\\\\nHE1116-1628 & 4224.43 &4125.85(-0.5) \\\\ \n & & 4192.46(-2.5)\\\\\n & & \\\\\nHE1119-1933 & 4675.25& 4790.28(-0.5)\\\\ \n & & 4858.06(-2.5) \\\\\n & & \\\\\nHE1120-2122 & 4148.01& 3961.57(-0.5)\\\\\n & & 4027.33 (-2.5)\\\\\n & & \\\\\nHE1123-2031 & 4365.87& 4339.85(-0.5) \\\\\n & & 4407.24(-2.5) \\\\\n & & \\\\\nHE1127-0604 & 4875.50& 4604.60(-0.5) \\\\\n & & 4672.42(-2.5) \\\\\n & & \\\\\nHE1142-2601 & 4475.87 & 4464.64(-0.5) \\\\ \n & & 4532.31(-2.5) \\\\\n & & \\\\\nHE1145-1319 &4485.81 &4514.13(-0.5) \\\\\n & & 4581.87(-2.5) \\\\\n & & \\\\\nHE1146-0151 & 4515.88& 4525.81(-0.5) \\\\ \n & & 4593.57(-2.5) \\\\\n & & \\\\\nHE1157-1434 & 4182.97 &4181.08(-0.5) \\\\\n & & 4247.93(-2.5)\\\\\n & & \\\\\nHE1205-0521 & 4650.36 &4927.12(-0.5) \\\\\n & & 4994.67(-2.5)\\\\\n & & \\\\\nHE1205-2539& 4182.97 & 4046.42(-0.5) \\\\\n & & 4112.64(-2.5) \\\\\n & & \\\\\nHE1228-0417 & 4795.85& 4964.09(-0.5) \\\\ \n & & 5031.55(-2.5)\\\\\n & & \\\\\nHE1230-0327 & 4814.61& 4846.74(-0.5) \\\\\n & & 4914.44(-2.5) \\\\\n & & \\\\\nHE1238-0836& 3521.23 & 3954.13(-0.5) \\\\\n & & 4019.85(-2.5) \\\\\n & & \\\\\nHE1253-1859 & 4239.41& 4105.00(-0.5) \\\\\n & & 4171.51(-2.5) \\\\\n & & \\\\\nHE1315-2035 & 4639.76 &4949.26(-0.5) \\\\\n & & 5016.77(-2.5) \\\\\n & & \\\\\nHE1318-1657 & 4423.50& 4612.48(-0.5)\\\\\n & & 4680.31(-2.5) \\\\\n & & \\\\\nHE1331-2558 & 4227.41& 4218.84(-0.5) \\\\\n & &4285.84(-2.5) \\\\\n & & \\\\\nHE1344-0411 & 3118.34 &3414.55(-0.5) \\\\\n & &3475.93(-2.5) \\\\ \n & & \\\\\n\\noalign{\\smallskip}\\hline \n\\end{tabular}\n\n\\end{table}\n\n\\begin{table}\n\\centering\n{\\bf Table 4: Estimated effective temperatures (T$_{eff}$) from \nsemi-empirical relations (continued)}\n\n\\begin{tabular}{|c|c|c|}\n\\hline\\noalign{\\smallskip}\nStar Name & Teff(J-K) & Teff(J-H) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nHE1358-2508& 3623.67& 3826.04(-0.5) \\\\\n & & 3890.93(-2.5) \\\\\n & & \\\\\nHE1400-1113 & 4894.81& 5010.94(-0.5) \\\\\n & & 5078.27(-2.5) \\\\\n & & \\\\\nHE1404-0846 & 4239.40& 4027.25(-0.5)\\\\\n & & 4093.38(-2.5) \\\\\n & & \\\\\nHE1405-0346 & 4391.32& 4484.40(-0.5) \\\\\n & & 4552.09(-2.5) \\\\\n & & \\\\\nHE1410-0125& 4378.56 &4266.53(-0.5) \\\\\n & & 4333.70(-2.5) \\\\\n & & \\\\\nHE1418-0306 & 3598.71& 3761.98(-0.5) \\\\\n & & 3826.41(-2.5) \\\\\n & & \\\\\nHE1425-2052& 4191.80 & 4250.49(-0.5) \\\\\n & & 4317.60 (-2.5) \\\\\n & & \\\\\nHE1428-1950 & 4505.82 &4573.18 (-0.5)\\\\\n & & 4640.98(-2.5) \\\\\n & & \\\\\nHE1429-1411 & 3057.76 & 3302.55(-0.5) \\\\\n & & 3362.74(-2.5) \\\\\n & & \\\\\nHE1430-0919 & 4700.38& 4625.80(-0.5) \\\\\n & & 4693.64(-2.5)\\\\\n & & \\\\\nHE1431-0755 & 3937.98 & 3950.23(-0.5)\\\\\n & & 4015.92(-2.5) \\\\\n & & \\\\\nHE1432-2138 & 5074.94 & 5075.25(-0.5)\\\\\n & & 5142.38(-2.5) \\\\\n & & \\\\\nHE1439-1338 &3658.21 & 3898.38(-0.5)\\\\\n & & 3963.76(-2.5) \\\\\n & & \\\\\nHE1440-1511 & 4636.24 &4747.85(-0.5)\\\\\n & & 4815.66 (-2.5)\\\\\n & & \\\\\nHE1442-0346 & 4622.20 & 4655.40(-0.5) \\\\\n & & 4723.24(-2.5)\\\\\n & & \\\\\nHE1447+0102 & 5042.06 & 4893.55 (-0.5)\\\\\n & & 4961.18(-2.5) \\\\\n & & \\\\\nHE1525-0516 & 4546.30 &4695.10 (-0.5)\\\\\n & & 4762.94(-2.5) \\\\\n & & \\\\\nHE2114-0603 & 3948.56 & 3919.97(-0.5) \\\\\n & & 3985.48(-2.5) \\\\\n & & \\\\\nHE2157-2125 & 4583.97 & 5135.81(-0.5)\\\\\n & & 5202.71(-2.5)\\\\ \n & & \\\\\nHE2211-0605 & 4553.10 &4621.90(-0.5) \\\\ \n & & 4689.73(-2.5) \\\\ \n & & \\\\\n\\noalign{\\smallskip}\\hline \n\\end{tabular}\n\\end{table}\n\n{\\footnotesize\n\\begin{table}\n\\centering\n{\\bf Table 4: Estimated effective temperatures (T$_{eff}$) from \nsemi-empirical relations (continued)}\n\n\\begin{tabular}{|c|c|c|}\n\\hline\\noalign{\\smallskip}\nStar Name & Teff(J-K) & Teff(J-H) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nHE2213-0017 & 2701.10 & 3005.84(-0.5)\\\\ \n & & 3062.45(-2.5) \\\\ \n & & \\\\\nHE2216-0202 & 4969.45& 5122.15(-0.5) \\\\ \n & & 5189.11(-2.5) \\\\\n & & \\\\\nHE2222-2337 & 3911.73 & 3879.01(-0.5) \\\\ \n & & 3944.26(-2.5) \\\\ \n & & \\\\\nHE2225-1401 & 2169.18 & 2845.70(-0.5)\\\\ \n & & 2900.11(-2.5) \\\\ \n & & \\\\\nHE2228-0137 & 4369.03 & 4284.06(-0.5) \\\\ \n & & 4351.28(-2.5) \\\\ \n & & \\\\\nHE2246-1312 & 4110.70& 4125.95(-0.5) \\\\ \n & & 4192.57(-2.5)\\\\ \n & & \\\\\nHE2255-1724 & 4675.26& 4388.73(-0.5)\\\\ \n & & 4456.25(-2.5) \\\\ \n & & \\\\\nHE2305-1427 & 5042.06& 4594.23(-0.5) \\\\ \n & & 4662.05(-2.5) \\\\ \n & & \\\\\nHE2334-1723 &4729.39 &4827.13 (-0.5) \\\\ \n & & 4894.87(-2.5) \\\\ \n & & \\\\\nHE2347-0658 & 5554.46 &4989.70(-0.5) \\\\ \n & & 5057.10(-2.5) \\\\ \n & & \\\\\nHE2353-2314 & 3927.44 &4014.92(-0.5)\\\\ \n & & 4080.98(-2.5)\\\\ \n\\noalign{\\smallskip}\\hline \n\\end{tabular}\n\nThe numbers inside the parentheses indicate the adopted metallicities [Fe\/H] \\\\\n\\end{table}\n}\n\n{\\footnotesize\n\\begin{table}\n\\centering\n{\\bf Table 5: Stars with radial velocity estimates }\n\\begin{tabular}{|c|c|c|}\n\\hline\\noalign{\\smallskip}\nStar Name & $v_{\\rm r}$ km s$^{-1}$ & Reference \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nHE~0100$-$1629 & $-$142.0 & 1 \\\\\nHE~0217$+$0056 & $-$142.0 ${\\pm}$ 3 & 2 \\\\\nHE~0228$-$0256 & $-$72.0 & 1 \\\\\nHE~1019$-$1136 & $-$126.0 ${\\pm}$ 4 & 2 \\\\\nHE~1105$-$2736 & $-$36.0 ${\\pm}$ 1.3 & 3 \\\\\nHE~1116$-$1628 & $-$69.0 & 2 \\\\\nHE~1152$-$0355 & $+$431.3 ${\\pm}$ 1.5 & 4 \\\\\nHE~1305$+$0007 & $+$217.8 ${\\pm}$ 1.5 & 4 \\\\\nHE~1410$-$0125 & $+$88.0 ${\\pm}$ 3 & 5 \\\\\nHE~1429$-$1411 & $-$90.0 ${\\pm}$ 1.5 & 6 \\\\\nHE~1442$-$0058 & $-$37.0 ${\\pm}$ 4 & 2 \\\\\nHE~2213$-$0017 & $-$44.0 ${\\pm}$ 3 & 2 \\\\\nHE~2225$-$1401 & $-$113.0 ${\\pm}$ 5 & 2 \\\\\nHD~26 & $+$217.8 ${\\pm}$ 1.5 & 6 \\\\\nHD~5223 & $-$244.9 ${\\pm}$ 1.5 & 4 \\\\\nHD~209621 & $-$390.5 ${\\pm}$ 1.5 & 6 \\\\\n\\noalign{\\smallskip}\\hline \n\\end{tabular}\n\nReferences: 1: Bothun et al. (1991), 2: Totten \\& Irwin (1998), \n3: Zwitter et al. (2008), 4: Goswami et al. (2006), 5: Frebel et al. (2006), \n6: Goswami et al. (in preparation\n\\end{table}\n}\n\n\n\n\\section{Concluding Remarks}\n\nAn accurate assessment of the fraction of CH stars can significantly aid \nour understanding of formation and evolution of heavy elements at low \nmetallicity. Another important issue is the role of low to intermediate-mass\nstars of the halo in the early Galactic chemical evolution. Thus large samples\nof faint high latitude stars such as the one reported by Christlieb et al. \n(2001b) that contain different types of carbon stars need to be analyzed to \nunderstand the astrophysical implications of each individual type of \nstellar population. Our objective in this study has been to identify the \nCH stars (as well as different type of stellar objects) in a selected sample \nof high Galactic latitude field stars. The sample is based on our \non-going observational programs with HCT and VBT on cool stars. During \n2007 and 2009 we have acquired low resolution spectra for a large number of \nstars that included about ninety two objects from the Hamburg survey of \nChrisltieb et al. (2001b).\n\nThe spectral classification criteria are those presented in Goswami (2005).\nAmong the ninety two objects, the spectra of seventy objects are \ncharacterized by the presence of strong C$_{2}$ molecular bands. The spectra\nof twenty two objects show only a weak or moderate G-band of CH and a CN \nband \naround 4215 \\AA\\,. One object, HE~1027-2644 does not show presence of any \nmolecular bands in its spectrum. The spectral analysis led to the \ndetection of \nthirty six potential CH star candidates. Their locations on the two\ncolor J-H versus H-K diagram, estimated effective temperatures, and carbon\nisotopic ratios are in support of their classification with this class\nof objects. This set of objects will make important targets for subsequent\nchemical composition studies based on high resolution spectroscopy and for \nconfirmation of these objects with this class of identification.\n\n While identification of C-N and C-J type stars are relatively easy, \nseparating C-R stars from CH stars is not so straightforward.\nThe two main properties, presence or absence of $s-$process elements and\nbinarity that differentiate early-R stars from CH stars, can\nbe known only through detailed abundance studies that require high\nresolution spectroscopy and from long-term radial velocity monitoring.\n The faintness of these objects makes high resolution spectroscopic\nstudies arduous and time-consuming. As such, the method described in\nGoswami (2005) to distinguish a C-R from a CH star proved quite\nuseful (Goswami et al. 2006).\nHigh resolution spectra will also allow for an accurate \nmeasurement of $^{12}$C\/$^{13}$C ratios. Based on medium resolution spectra,\n for the potential CH star candidates, we find a low (${\\le}$ 10) \ncarbon isotopic ratios, indicating that they belong \nto the group of early-type CH stars.\n\nAbia et al. (2002) have shown that CH stars cannot be formed above a \nthreshold metallicity, around Z $\\sim$ 0.4Z$_{\\odot}$. According to \nDominy (1984), the metallicities of C-R stars are either solar or slightly \nsub-solar. C-R stars are believed to be Core Helium Burning (CHeB) \ncounterparts of CH stars in which $s-$process elements are either absent \nor not detectable (Izzard et al. 2007). These authors have predicted an \nearly-R\/CH ratio $\\sim$ 7\\%, at [Fe\/H] = -2.3, a metallicity typical of the\nGalactic halo. This ratio is derived considering only CHeB CH stars, if \nCH giants and dwarfs are also considered, this ratio is likely to get much\nlower.\n\nWesterlund et al. (1995) defined dwarf carbon stars as having J-H ${\\le}$ 0.75,\n H-K ${\\ge}$ 0.25 mag. Among the three dwarf carbon stars in our sample the\n objects HE~1358-2508 occupies a region defined by these limits on J-H, H-K \nplane. HE~0009-1824 and HE~1116-1628 however do not follow the JHK \ndefinition of dwarf carbon stars offered by Westerlund et al. (1995). It \nseems, these limits on J-H and H-K may not be very tight. Proper motions\nof these objects have been estimated by Mauron et al. (2007) and have \nplaced them as dwarf carbon stars.\n\n The temperature estimates of the program stars derived using JHK-temperature\n calibration relations of Alonso et al. (1996), although varying over a wide \nrange, provide a preliminary temperature check for the program stars and \ncan be \nused as starting values in deriving atmospheric parameters from high \nresolution spectra using model atmospheres. \n\nImportant information such as kinematic properties of the stars can be \nderived from radial velocity estimates.\nFrom low resolution spectra, radial velocities are generally computed using\n Fourier cross-correlation method. This method,\nwidely employed, uses the spectrum of a radial velocity standard as the \ntemplate spectrum. Unfortunately, we could not acquire spectra of radial \nvelocity standards that are usable for the present set of program stars \nunder study. \nEstimated radial velocities using a few atomic lines,\ndetectable on the low resolution spectra, did not return consistent results.\nHowever, a systematic estimation of radial velocities\nof the program stars using appropriate radial velocity standard templates\nwould be a worthwhile future program.\nRadial velocities of a few stars belonging to our program star list,\nthat are available in literature, are listed in Table 5.\n\nThe primary focus of this work is CH stars; however, a detailed discussion on\nthe objects of other spectral types is also necessary and is under progress.\\\\\n\n {\\it Acknowledgement}\\\\\n We thank the staff at IAO, VBO and at the remote control station at CREST,\nHosakote for assistance during the observations. This work made use of the\nSIMBAD astronomical database, operated at CDS, Strasbourg, France, and the\nNASA ADS, USA. Ms Drisya K. is a JRF in the DST project NO. SR\/S2\/HEP-09\/2007; \nfunding from the above mentioned project is greatfully acknowledged. \n\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nBiochemical reactions in cellular microdomains involve, in general,\na small number of molecules that can bind to agonist molecules,\nconfined in some subregions. A microdomain $\\Omega$ is defined as a\nbounded domain where a large fraction of the boundary $\\partial\\Omega_r$\nis reflective and a small part $\\partial\\Omega_a$ of it is absorptive,\nwhich allows molecules to enter or\/and exit. Under normal\nphysiological conditions, a molecule can be trapped inside a\nmicrodomain for a period long enough, compared to other time scales\nsuch as free diffusion or binding time. Because a molecule can be\ntrapped for a long time, many chemical bounds can be made, before it\nexits. From a physiological stand point, an interesting property of\nsuch microdomain is that depending on the number of chemical bounds\nmade by a molecule, a cascade of chemical reactions can be\ninitiated, which ultimately affects some physiological properties.\nOur interest here is to estimate the number of chemical bounds made\nby a Brownian interacting molecule inside $\\Omega$. This number\ndepends on the geometry of the microdomain and the distribution of\nagonist molecules. We also provide an estimation of the mean time\nspent (Dwell time) by a molecule inside the microdomain, denoted by\n$E(\\tau_D)$. For that purpose, we derive an asymptotic formula for\nthe Dwell time, the mean and the variance of the number of bounds\nmade before the molecule exits, as the ratio\n$\\frac{|\\partial\\Omega_a|}{|\\partial\\Omega_r|}$ tends to zero. Finally, we\nderive a formula for the Dwell time of a molecule inside $\\Omega$,\nwhen a steady state flux of molecules is maintained fixed at the\nabsorbing boundary. The role of the flux is to fix the number of\nfree binding sites. Using the present method, we also obtain an\nestimate of the forward binding rate constant. Historically, the\ntheory of chemical reactions at a molecular level limited by\ndiffusion has been developed by many authors, to quote but a few\n\\cite{Wilemski,Perico,Agmon,Szabo1,Szabo2}: using the classical\ntheory of diffusion and interactions with binding sites, various\nrate constants were computed. Recently, using averaged equations,\nchemical reactions in microdomains have been described in\n\\cite{BereSmalhole,Berepartialflux}. Chemical reactions in closed\nmicrodomains were studied in \\cite{JChemPhys}, where we obtain, in\nparticular, some estimates on the mean of the variance of the\nnumber of bound molecules in a steady state regime. In \\cite{HS},\nan asymptotic estimate of the mean time it takes for a Brownian\nmolecule to escape an empty domain through small openings located\non the boundary, was obtained using a new type of singular\nperturbation problem. More specifically, if D denotes the\ndiffusion constant, $|\\Omega|$ the volume of the domain $\\Omega$\nand $\\varepsilon=\\frac{|\\partial \\Omega_a|}{|\\partial \\Omega|}<<1$ is the\nratio of the absorbing to the total boundary, then for\n$\\varepsilon$ small, the leading order term of the mean escape\ntime $\\tau(\\mbox{\\boldmath$x$})$ (for a molecule starting at position $\\mbox{\\boldmath$x$}$, far\nfrom the entrance) is given by\n\\begin{eqnarray}\\label{HolcSchuss}\n\\tau(\\mbox{\\boldmath$x$}) =\n\\frac{|\\Omega|}{\\pi D}\\log(\\frac1{\\varepsilon})+O(1).\n\\end{eqnarray}\nIn the first approximation, the mean time $\\tau(\\mbox{\\boldmath$x$})$ does not depend\non the initial position $\\mbox{\\boldmath$x$}$ and will be denoted by $\\tau$. In this\narticle, we obtain an explicit asymptotic estimation of the Dwell\ntime $E(\\tau_D)$ as a function of the characteristic sizes of the\ndomain $\\Omega$, the size of the small openings $\\partial \\Omega_a$, the\nnumber of the binding molecules. More specifically, $E(\\tau_D)$ is\ngiven by expression (\\ref{eq:formula_mean_mean}), which depends on\nthe mean time $\\mean{\\tau}$ to exit when no binding occurs, the mean\ntime $\\mean{T}$ to enter into the binding site area, $m_{\\delta}$\nthe mean probability to bind before exit and the backward binding\nrate $k_{-1}$. We get\n\\begin{equation}\\label{eq:formula_mean_meanintro}\nE(\\tau_D) = \\mean{\\tau}+\\frac{1-m_{\\delta}}{m_{\\delta}}\\left(\n\\mean{T}+\\frac{1}{k_{-1}}\\right).\n\\end{equation}\nIt is well known from the theory of chemical reactions that the\nbackward binding rate $k_{-1}$ depends only on the local\ninteractions between two interacting molecules. If $\\Delta E$\ndenotes the activation barrier, $kT_e$ is the energy due to the\ntemperature, the Arrhenius law states that: \\begin{eqnarray} k_{-1} = C\ne^{-\\frac{\\Delta E}{kT_e}},\n \\end{eqnarray}\nwhere C is a constant that depends on the temperature $T_e$, the\nelectrostatic potential barrier $\\Delta E$ generated by the\nbinding molecule and the friction coefficient \\cite{Schuss}. In\nthe first part of the paper, we derive equation\n(\\ref{eq:formula_mean_meanintro}) by counting the number of bounds\nbetween the Brownian molecule and the agonist molecules, before\nthe Brownian molecule exits the domain. In the second part, we\nderive some asymptotic estimates of the quantities $\\mean{T}$,\n$\\mean{\\tau}$ and $m_{\\delta}$ as a function of the geometry, when\nthe radius $\\delta$ of the binding site tends to zero. Although\nthe present computations are carried out in two dimensions, they\ncan be extended to dimension 3 by using the techniques developed\nin \\cite{SSHE}. Finally in the last part, we apply the present\ncomputations to study chemical reactions occurring in synaptic\nmicrodomains: the chemical reactions are the binding of receptors\nwith the scaffolding molecules. It is indeed of great interest to\nanalyze the mechanism that regulates the number and the type of\nreceptors at a synapse, because receptors control the synaptic\nweight. Any fluctuations of the number results in a variation of\nthe synaptic weight and affects the reliability of the synaptic\ntransmission. Moreover, certain experimental protocols have lead\nto a Long Term Potentiation of a synapse, a mechanism which is\nassociated with a change of the number and the type of certain\nreceptors \\cite{Bredt,Nicoll}. The regulation of synaptic\nplasticity is a fundamental process underlying learning and memory\n\\cite{Bredt,Nicoll} and recently, single molecule tracking has\nrevealed that the number of postsynaptic receptors, which\nparticipate to the synaptic transmission, is not fixed but it\nchanges due to constant traffick of receptors on the surface of\nneurons. Receptors move in and out from synaptic regions\n\\cite{Choquet} \\cite{Triller} and following these observations,\nmany questions have been raised: in particular, what determines\nthe time spent by a receptor inside a synapse? How receptors can\nbe stabilized inside a synapse? How long they stay inside synaptic\nmicrodomains? Such questions are partially answered in the present\npaper. In particular, our computation of the Dwell time of a\nreceptor inside a specific microdomain, called the Postsynaptic\ndensity (PSD) takes into account the interaction with the\nscaffolding molecules, which was ignored in \\cite{HS}.\n\n\\subsection{Molecular dynamics in a microdomain}\n{The dynamics of a molecule or a protein moving on the surface of\na cell is usually described in the Smoluchowsky limit (large\nfriction) of the Langevin equation \\cite{Schuss}: for a molecule\nof mass $m$, described by its position $X$ at time $t$, with a\nfriction coefficient $\\gamma$, moving inside a potential well $V$,\nthe Smoluchowsky limit of the Langevin equation is \\begin{eqnarray}\\label{SL}\n\\gamma \\dot{X}+\\nabla V(X)=\\sqrt{2\\gamma \\varepsilon_e}\\dot{w},\n \\end{eqnarray}\nwhere $\\varepsilon_e=\\frac{kT_e}{m}$ and $w$ is a Gaussian random\nvariable with variance 1 and mean 0. In a microdomain $\\Omega$,\nwhere a large fraction of the boundary is reflective $\\partial \\Omega_r$\nand a small part of it is absorptive $\\partial \\Omega_a$, the probability\ndensity function (pdf) $p$ to find $X$ at time $t$ in the surface\nelement $\\mbox{\\boldmath$x$}+d\\mbox{\\boldmath$x$}$ satisfies the Fokker-Planck Equation (FPE)\n\\begin{eqnarray}\\label{FPE} \\frac{\\partial p(\\mbox{\\boldmath$x$},t)}{ \\partial t} &=& D \\Delta p(\\mbox{\\boldmath$x$},t) -\n\\nabla\\cdot (\\nabla V(\\mbox{\\boldmath$x$})p(\\mbox{\\boldmath$x$},t)) \\hbox{ for } \\mbox{\\boldmath$x$} \\in \\Omega\\\\\n& & \\nonumber\\\\\n\\mbox{\\boldmath$J$}(\\mbox{\\boldmath$x$},t)\\cdot\\mbox{\\boldmath$n$}&=&0\\hbox{ for } \\mbox{\\boldmath$x$} \\in \\partial \\Omega_r \\\\\n& & \\nonumber\\\\\np(\\mbox{\\boldmath$x$},t)&=&0 \\hbox{ for } \\mbox{\\boldmath$x$} \\in \\partial \\Omega_a\n \\end{eqnarray}\nwhere $D=\\gamma \\varepsilon_e$ is the diffusion constant, $\\mbox{\\boldmath$n$}$ is\nthe external normal at the boundary, the flux $\\mbox{\\boldmath$J$}$ is given by\n\\begin{eqnarray} \\mbox{\\boldmath$J$}(\\mbox{\\boldmath$x$},t)=-D\\nabla p(\\mbox{\\boldmath$x$},t) +\\nabla V(\\mbox{\\boldmath$x$})p(\\mbox{\\boldmath$x$},t).\n \\end{eqnarray}\nWe denote by $t^{x}$ the first time the molecule arrives at the\nabsorbing boundary $\\partial \\Omega_a$, when it started at position $\\mbox{\\boldmath$x$}$.\nIt is well known \\cite{Schuss} that the mean first passage is the\nexpectation of the time $t^{x}$ and is given by\n\\begin{eqnarray*}\nE^{\\mbox{\\boldmath$x$}}(t^{x})&=& \\int_{0}^{\\infty} t\\frac{d}{dt}\n Pr\\{ t^xt\\}dt\\\\\n &=& \\int_0^{\\infty} \\int_{\\Omega} p(\\mbox{\\boldmath$y$},t|\\mbox{\\boldmath$x$})d\\mbox{\\boldmath$y$} dt,\n\\end{eqnarray*}\nwhere $p(\\mbox{\\boldmath$y$},t|\\mbox{\\boldmath$x$})$ is the pdf of the process $X$, conditioned on the\ninitial position $\\mbox{\\boldmath$x$}$, that is, as $t$ goes to zero, \\begin{eqnarray}\np(\\mbox{\\boldmath$y$},t|\\mbox{\\boldmath$x$})\\rightarrow \\delta(\\mbox{\\boldmath$x$}-\\mbox{\\boldmath$y$}),\n \\end{eqnarray}\nwhere $\\delta$ is the Delta-Dirac function. In equation (\\ref{FPE}),\n$V$ represents the potential wells generated by the binding\nmolecules inside the domain $\\Omega$. It is, in fact, the sum of the\nlocal potential wells, supported inside a ball of finite radius\ngenerated by the binding molecules. Usually the binding molecules\nare scattered inside the domain $\\Omega$, but in the present model\nwe replace the scattered distribution of binding molecules by a\nsimplified distribution, where we imagine that all the binding\nmolecules are located inside a compartment $D(\\delta)$ in $\\Omega$.\nIn the present description, the potential $V$ becomes an effective\npotential defined in $D(\\delta)$, whose characteristics should be\nsuch that the classical chemical reaction theory is recovered. More\nspecifically, we can define the microdomain $\\Omega$ containing the\nbinding domain $D(\\delta)$, which replaces the domain with many\nscattered binding sites: this simplified domain made of two\ncompartments is called the homogenized microdomain and is described\nas (see figure 1)\n\\begin{enumerate}\n\\item An internal compartment, which is described as a disk\n$D(\\delta)$ of radius $\\delta$. This disk represents the region\nwhere the binding sites are located. Instead of using the dynamics\nassociated with equation (\\ref{SL}), we describe the entrance and\nthe exit of a molecule inside $D(\\delta)$ using a Poissonnian\ndescription, where the mean can be related to the properties of\nthe potential well. For that purpose, we recall that a chemical\nreaction with a binding molecule is described as the arrival of a\nBrownian molecule inside the disk $D(\\delta)$. The release process\nis modeled as the escape of the molecule from the potential well\n$V$ and is described by the chemical reaction \\begin{eqnarray} \\label{chem}\nR+S\n\\begin {array}{c}\n{k_1}\\\\\n\\rightleftharpoons \\\\\n{k_{-1}} \\end{array} RS \\end{eqnarray} where $\\frac{1}{k_{-1}}$ is the mean\nbinding time and depends only on the local potential well, generated\nby the binding molecules \\cite{Tier84} \\cite{Schuss}.\n\\item An external compartment separated from the rest of the\nbiological environment by the boundary $\\partial \\Omega_r$, containing\nsmall openings $\\partial \\Omega_a$. Molecules can enter or exit through\nthe openings and thus can be exchanged with the rest of the\ncellular medium (see figure 1). The dynamics of a molecule in that\ncompartment is described as pure Brownian until it escapes.\n\\end{enumerate}\n\\begin{figure}[htbp]\\label{fig1}\n\\centerline{\\includegraphics[width=2.0in,height=1.9in]{Dwell_fig2.eps}}\n \\caption{Model of a cellular microdomain in two\ndimensions. The domain is a disk $D(R)$ of radius $R$, made of two\ncompartments: an inner disk $D(\\delta)$ of radius $\\delta$ and the\nannulus $A_\\delta=D(R)-D(\\delta)$. A molecule moves by Brownian\nmotion inside $A_\\delta$ until it hits $D(\\delta)$ or the\nabsorbing boundary $\\partial \\Omega_a$. When the Brownian molecule\nenters into $D(\\delta)$, which represents the domain of chemical\nreactions, while it reacts with an effective binding molecule\nduring a mean time (the inverse of the backward binding rate) its\nmovement in $D(\\delta)$ is frozen. The molecule is then released\nuniformly inside the annulus $A_\\delta=D(R)-D(\\delta)$. This\nscenario repeats until the molecule hits the absorbing boundary\n$\\partial \\Omega_a$, where the molecule is finally removed.}\n\\end{figure}\n\n\\subsection{Time spent by a molecule inside a disk containing a small opening}\nWe estimate asymptotically the Dwell time $E(\\tau_D)$ of a molecule\ninside a domain $\\Omega$, when the domain $\\Omega$ is a disk $D(R)$\nof radius $R$. The case of a general domain is left open. By\ndefinition, $E(\\tau_D)$ is the mean time to exit, averaged over an\ninitial uniform distribution. We make the following assumptions: the\nratio $\\varepsilon$ of the absorbing to the reflective boundary of\n$D(R)$ is small $<<1$. The absorbing (resp. reflecting) part is\ndenoted $\\partial\\Omega_a$ (resp. $\\partial\n\\Omega_r$). The computation of the Dwell time will be made using\nthe homogenization version of the domain, described in the\nprevious paragraph: the binding molecules are located in the small\nconcentric disk D($\\delta$), where $\\delta<T_{\\mbox{\\boldmath$x$}}^S $ satisfies an elliptic partial\ndifferential equation \\cite{Karlin}, with mixed boundary conditions\ngiven by \\begin{eqnarray} \\label{eq5}\n \\Delta q_{\\delta}&=&0\\mbox{ on }A_{\\delta} , \\\\\n \\frac{\\partial q_{\\delta}}{\\partial n}(\\mbox{\\boldmath$x$})&=&0\\mbox{ on }\\partial \\Omega _r , \\nonumber \\\\\n q_{\\delta}(\\mbox{\\boldmath$x$})&=&0\\mbox{ on }\\partial \\Omega_a ,\\nonumber\\\\\n q_{\\delta}(\\mbox{\\boldmath$x$})&=&1\\mbox{ on }\\partial D(\\delta), \\nonumber\n\\end{eqnarray} where $\\partial \\Omega _a $ is the small opening located on the\nexternal boundary of $\\Omega$, $\\partial \\Omega _r $ is the\nremaining part of the external boundary, which is reflecting. In\npolar coordinates $(r,\\theta)$, the portion of the boundary $\\partial\n\\Omega_a$ is parameterized by $|\\theta-\\pi|\\leq\\varepsilon$. By using the\nparticular geometry of the annulus $A_{\\delta}$, an explicit\n estimation of the solution can be obtained by using spectral methods, developed in the context\n of mixed boundary value problems \\cite{Sneddon,Fabrikant1,Fabrikant2}. Using the method of\nseparation of variables, the solution $q_{\\delta}$ of problem\n(\\ref{eq5}) has the general form \\begin{eqnarray}\n&&q_{\\delta}(r,\\theta)=\\frac{a_0}{2}+\\sum_{n=1}^{\\infty}\n\\Big[a_n\\Big( \\frac r R\\Big)^n+b_n\\Big(\\frac \\delta\nr\\Big)^n\\Big]\\cos(n\\theta)+\\alpha \\log\\Big(\\frac r\n\\delta\\Big).\\label{eq:series} \\end{eqnarray} We wish now to estimate the\ncoefficients $a_n$ and $b_n$. We denote $\\beta=\\frac{\\delta}{R}$.\nFrom the boundary conditions on\n $\\partial D(\\delta)$ and on $r=R$, we get\n\\begin{eqnarray}\n \\frac{a_0}{2}+\\sum_{n=1}^\\infty\n\\Big[a_n\\Big(\\frac{\\delta}{R}\\Big)^n+b_n\\Big]\\cos(n\\theta)&=1,& \\label{bc1}\\\\\n\\sum_{n=1}^\\infty\nn\\Big[\\frac{a_n}{R}-\\frac{b_n}{R}\\beta^{n}\\Big]\\cos(n\\theta)+\n\\frac{\\alpha}{R}&=0&\\mbox{ for $|\\theta-\\pi|>\\varepsilon$}\\label{eq:ser1p}\\\\\n1+\\sum_{n=1}^\\infty \\Big[a_n+b_n\\beta^{n}\\Big]\\cos(n\\theta)-\\alpha\n\\log(\\beta)&=0&\\mbox{ for $|\\theta-\\pi|\\leq\\varepsilon$}\\label{eq:ser2p}.\n \\end{eqnarray}\nUsing equation (\\ref{bc1}), we get the following identities:\n \\begin{eqnarray*}\n&&a_0=2\\\\\n&& b_n=-a_n\\beta^n \\mbox{ for $n\\geq 1$}. \\end{eqnarray*} Using the\nidentities above and (\\ref{eq:ser1p}) and (\\ref{eq:ser2p}), we\nobtain the double series equations\n \\begin{eqnarray}\n&\\sum_{n=1}^\\infty\nn\\Big[\\frac{a_n}{R}+\\frac{a_n}{R}\\beta^{2n}\\Big]\\cos(n\\theta)+\n\\frac{\\alpha}{R}=0,&\\mbox{ for $|\\theta-\\pi|>\\varepsilon$}\\label{eq:ser1}\\\\\n&1+\\sum_{n=1}^\\infty\n\\Big[a_n-a_n\\beta^{2n}\\Big]\\cos(n\\theta)-\\alpha\n\\log(\\beta)=0,&\\mbox{ for $|\\theta-\\pi|\\leq\\varepsilon$.}\\label{eq:ser2}\n \\end{eqnarray}\nSubstituting $c_n=a_n(1+\\beta^{2n})$ and $H_n=\\frac{2\\beta^{2n}}{1-\\beta^{2n}}$\n equations (\\ref{eq:ser1}),(\\ref{eq:ser2}) have the following\n form\n \\begin{eqnarray}\n&\\frac{c_0}{2}+\\sum_{n=1}^\\infty \\frac{c_n}{1+H_n}\n\\cos(n\\theta) =0,&\\theta\\in [\\pi,\\pi-\\varepsilon]\\label{eq:gen1}\\\\\n&&\\nonumber\\\\\n &\\alpha+\\sum_{n=1}^\\infty n c_n\n\\cos(n\\theta)=0,&\\theta\\in[0,\\pi-\\varepsilon],\\label{eq:gen2} \\end{eqnarray} where\n\\begin{eqnarray}\\label{eq:c0} \\frac{c_0}{2}=1-\\alpha\\log(\\beta). \\end{eqnarray} The\nasymptotic solution of equations (\\ref{eq:gen1})-(\\ref{eq:gen2})\nuses the double series expansion, developed in\n\\cite{Sneddon,Fabrikant1} and used in \\cite{HS} in the context of\na small opening asymptotic. In Appendix A, the general solution is\ngiven. By using these results, we have the following expression\nfor the coefficient $c_0$\n \\begin{eqnarray}\\label{eq:c0p}\n c_0=-2\\alpha\\Big[2\\log{\\frac 1\n \\varepsilon}+2\\log2+4\\beta^2+O(\\beta^2,\\varepsilon)\\Big].\n \\end{eqnarray}\nUsing equation (\\ref{eq:c0}) and (\\ref{eq:c0p}), we get the\nexpression \\begin{eqnarray} \\label{alpha}\n\\alpha=-\\frac1{\\Big(\\log(\\frac1{\\beta})+\\Big[2\\log{\\frac 1\n\\varepsilon}+2\\log2+4\\beta^2+O(\\beta^2,\\varepsilon)\\Big]\\Big)}. \\end{eqnarray} To remember\nthat $\\alpha$ depends on $\\beta$ and $\\varepsilon$, we denote it\n$\\alpha(\\beta,\\varepsilon)$.\nFor $\\epsilon$ fixed and $\\delta$ small, the other coefficients are\nestimated by using the expression of $c_n$ (given in the appendix)\nby \\begin{eqnarray} c_n=\\alpha(\\beta,\\varepsilon) \\tilde{c}_n=O(\\alpha(\\beta,\\varepsilon)),\n\\end{eqnarray} where $\\tilde{c}_n$ depends only on $n$ and \\begin{eqnarray}\na_n &= & \\frac{c_n}{1+\\beta^{2n}} \\sim O(\\alpha(\\beta,\\varepsilon)) \\\\\nb_n &=&-c_n\\beta^{n} \\sim O(\\alpha(\\beta,\\varepsilon)\\beta^{n}). \\end{eqnarray}\nThese estimates show for $\\delta$ small that the leading term of\n$q$ is given by : \\begin{eqnarray}\\label{eq:asymptotic q} q_{\\delta}(r,\\theta)\n= \\left\\{\n\\begin{array}{cc}\n1+\\alpha(\\beta,\\varepsilon) \\log(r\/\\delta) +O(\\beta) & \\hbox{ for } r \\sim \\delta \\\\\n& \\label{midd}\\\\\n1+\\alpha(\\beta,\\varepsilon) \\log(r\/\\delta)+ O(\\alpha) & \\hbox{ for } r \\sim\nR.\n\\end{array}\n\\right. \\end{eqnarray} The probability $p_{\\delta}(\\mbox{\\boldmath$x$})$ that $T_x^a0$ and this is exactly the Fichera conditions,\n discussed in \\cite{Fichera,Nirenberg}, where\nboundary conditions cannot be given.\n\\subsection{Asymptotic expansion of the mean first passage time}\nWhen the radius $\\delta $ of the inner circle is small, we obtain\nan explicit asymptotic expansion of the mean conditional time\n$t^A$, solution of equation (\\ref{eqfd}). We first derive an\nasymptotic solution by gluing two solutions: 1) when the initial\npoint $\\mbox{\\boldmath$x$}$ is far from the inner-circle, the solution looks like\nthe mean exit time when the drift term is set to zero. This\nsolution is called the outer-solution and has been estimated in\n\\cite{SSH2} with similar boundary conditions. 2) When the initial\npoint $\\mbox{\\boldmath$x$}$ is chosen near the inner-circle, the solution can be\napproximated by a radial function. The approximation is valid in a\nboundary layer and has to\n match the radial part of the outer-solution at least $C^1$ .\n\\subsection{Outer-solution}\nWe now provide a construction of the outer-solution to equation\n(\\ref{eqfd}). We start with the expansion of $p$, which depends on\nthe parameter $\\delta$.\n \\begin{eqnarray} p_{\\delta}(r,\\theta)=\n1-\\alpha(\\delta,\\epsilon)\\phi_{\\delta}(r,\\theta). \\end{eqnarray} By using\nthe appendix, we obtain the following expression \\begin{eqnarray}\n\\phi_{\\delta}(r,\\theta) =\\Big( \\sum_{n=1}^{\\infty}\n\\frac{\\tilde{c_n}}{1+\\beta^{2n}}\\Big[\\Big(\\frac r R\\Big)^n\n-\\Big(\\frac{\\delta \\beta} r\\Big)^n\\Big]\\cos(n\\theta)+\n\\log\\Big(\\frac rR\\Big)+4\\log(\\frac1 \\varepsilon)\\Big). \\end{eqnarray}\nEquation (\\ref{eqfd}) can be written outside the boundary layer:\n$U_{\\delta}=\\{ r| r_{\\delta} \\leq r \\leq R\\}$, \\begin{eqnarray}\\label{eqfdp1}\n& D (1-\\alpha(\\delta,\\epsilon)\\phi_{\\delta}(r,\\theta) ) \\Delta t^A\n-2\\alpha(\\delta,\\epsilon) \\nabla \\cdot\\phi_{\\delta}(r,\\theta)\n\\nabla t^A =-(1-\\alpha(\\delta,\\epsilon)\\phi_{\\delta}(r,\\theta) )\n\\mbox{ on\n} U_{\\delta} ,\\nonumber\\\\&\\\\\n& \\frac{\\partial }{\\partial n}t^A(\\mbox{\\boldmath$x$})=0 \\mbox{ on }\\partial \\Omega _r ,\n\\mbox{ and }\n t^A (\\mbox{\\boldmath$x$})=0\\mbox{ on }\\partial \\Omega_a. \\nonumber\n\\end{eqnarray} We look for a regular asymptotic expansion of the solution :\n\\begin{eqnarray} \\label{expa} t^A(\\mbox{\\boldmath$x$})=u(\\mbox{\\boldmath$x$}) -\\alpha(\\delta,\\epsilon) u_{1}(\\mbox{\\boldmath$x$}) +\nO(\\alpha^2(\\delta,\\epsilon) ). \\end{eqnarray} Using expression (\\ref{expa})\nand\n the behavior of $p$ as $\\delta$ goes to zero, in the closed domain $D(R)$ (see appendix), we\nobtain from equation (\\ref{eqfdp1}) that $u$ satisfies\n\\begin{eqnarray}\\label{eqfdp3}\nD \\Delta u &=&-1\\mbox{ on } D(R) ,\\\\\n \\frac{\\partial u }{\\partial n}&=&0\\mbox{ on }\\partial \\Omega _r ,\\nonumber \\\\\n u&=&0\\mbox{ on }\\partial \\Omega _a.\\nonumber \\end{eqnarray}\nWe can now use the result of \\cite{HS,SSH2} to compute the leading\norder term of $u$. For $\\mbox{\\boldmath$x$}$ that does not belong to a boundary\nlayer near the absorbing boundary $\\partial\\Omega _a$, the asymptotic\nexpansion of $u$ is given by \\begin{eqnarray} u(\\mbox{\\boldmath$x$})= \\frac{R^2}{D} \\big(\n\\ln(\\frac{1}{\\varepsilon})+ \\ln2 \\big)+ O(\\varepsilon) \\mbox{ on }\nU_{\\delta}. \\end{eqnarray}\n We conclude that $u$ does not depend on the variable $r$ and $\\theta$ at the first order\nin $\\delta$ and $\\epsilon$. Because $u$ is a solution of equation\n(\\ref{eqfdp3}), outside a boundary layer of $\\partial\\Omega _a$, the\nderivatives $\\frac{\\partial u}{\\partial r }$ and $\\frac{\\partial u}{r\\partial \\theta }$\nare small at the first order in $\\delta$ and $\\epsilon$. We now\nconsider the first order term in $\\alpha$ in equation\n(\\ref{eqfdp1}), which satisfies the equation \\begin{eqnarray*} & D \\Delta u_1\n-2\\nabla \\phi\\nabla u = 0\\\\ &\\frac{\\partial u_1 }{\\partial\nn}(\\mbox{\\boldmath$x$})=0\\mbox{ on }\\partial \\Omega _r, \\mbox{ and }\n u_1 (\\mbox{\\boldmath$x$})= 0\\mbox{ on }\\partial \\Omega _a.\n \\nonumber\n\\end{eqnarray*} Outside the boundary layer near $\\partial \\Omega _a$, the\nleading order term is given by $ u_1 (\\mbox{\\boldmath$x$})= o(1)$ (for the variable\n$\\delta$). We get that \\begin{eqnarray} \\label{ext}\nt^A(\\mbox{\\boldmath$x$})&=& u(\\mbox{\\boldmath$x$}) +o(\\alpha(\\delta,\\epsilon) ) \\mbox{ on } U_{\\delta} \\nonumber \\\\\n &=& \\frac{R^2}{D} \\big( \\ln(\\frac{1}{\\varepsilon})+ \\ln2 \\big)\n +o(\\alpha(\\delta,\\epsilon) ) \\mbox{ on } U_{\\delta}.\n\\end{eqnarray}\n\\subsection{Asymptotic solution inside the boundary layer }\nWe now provide an asymptotic expansion of the mean time $t^A$\ninside the\n boundary layer $BL_{\\delta}=\\{\\delta \\delta$ by\\begin{eqnarray} (t^A)'(\\mbox{\\boldmath$x$})\n=\\frac{1}{r(r-\\delta)^2}\\left(C\n-\\frac{1}{D}(\\frac{r^2\\delta^2}{2}-2\\delta\n\\frac{r^3}{3}+\\frac{r^4}{4}) \\right) \\end{eqnarray} where the constant is\nchosen such that the function $t^A$ is integrable. Thus the\nnumerator vanishes for $r=\\delta$. Actually $r=\\delta$ has to be a\nthird order zero of the numerator and \\begin{eqnarray} (t^A)'(\\mbox{\\boldmath$x$})\n=-\\frac{r-\\delta}{4Dr} \\left(r+\\delta\/3\\right). \\end{eqnarray} Thus, \\begin{eqnarray}\nt^A(\\mbox{\\boldmath$x$}) =-\\frac{1}{4D} \\left(\\frac{r^2}{2}\n-\\frac{2r\\delta}{3}-\\frac{\\delta^2}{3} \\ln(r) +C \\right), \\end{eqnarray}\nwhere the constant $C$ is determined by the matching condition:\n\\begin{eqnarray} t^A(r_{\\delta}) =\\frac{R^2}{D} \\big(\n\\ln(\\frac{1}{\\varepsilon})+ \\ln2 \\big)+O(\\alpha^2(\\delta,\\epsilon)\n). \\end{eqnarray} We get \\begin{eqnarray} C \\approx\n-4R^2\\left(\\ln(\\frac{1}{\\varepsilon})+ \\ln2\\right)\n-\\frac{\\delta^2}{2\\alpha^2}, \\end{eqnarray} and for $\\mbox{\\boldmath$x$} \\in BL_{\\delta}$,\n\\begin{eqnarray} \\label{BL}t^A(\\mbox{\\boldmath$x$}) &=&\\frac{1}{D}\n\\left(R^2\\left(\\ln(\\frac{1}{\\varepsilon})+ \\ln2\n\\right)+\\frac{\\delta^2}{8\\alpha^2}- \\frac{r^2}{8}\n+\\frac{r\\delta}{6}+\\frac{\\delta^2}{12} \\ln(r) \\right).\\nonumber\\\\\n\\end{eqnarray}\n\\subsection{Asymptotic estimate of the average Mean time $\\mean{\\tau}$}\nWe now compute asymptotically the mean time $\\mean{\\tau}$ defined in\nequation (\\ref{eq:tau}) by \\begin{eqnarray}\\label{eq:taur} \\mean{\\tau}=\n\\frac{\\int_{A_{\\delta}} t^A(\\mbox{\\boldmath$x$}) p_{\\delta}(\\mbox{\\boldmath$x$})\nd\\mbox{\\boldmath$x$}}{\\int_{A_{\\delta}}p_{\\delta}(\\mbox{\\boldmath$y$})d\\mbox{\\boldmath$y$}} =\n\\frac{\\mean{\\tau}_{A_{\\delta}}}{m_{\\delta}}, \\end{eqnarray}\n where we recall that $\n\\mean{\\tau}_{A_{\\delta}} = \\frac{1}{Vol({A_{\\delta}})}\n\\int_{A_{\\delta}} t^A(\\mbox{\\boldmath$x$}) p_{\\delta}(\\mbox{\\boldmath$x$}) d\\mbox{\\boldmath$x$}.$\n$Vol({A_{\\delta}})=\\pi(R^2-\\delta^2)$. To compute\n$\\mean{\\tau}_{A_{\\delta}}$, we decompose\n the domain $A_{\\delta}= BL_{\\delta} \\cup \\left(A_{\\delta}-BL_{\\delta}\\right)$. Using\nthe previous computations for the outer solution (\\ref{ext}) and\nthe boundary layer solution (\\ref{BL}), we compute each term\nseparately and we get: \\begin{eqnarray*} \\mean{\\tau}_{A_{\\delta}} =\nm_{\\delta}\\frac{R^2}{D} \\big( \\ln(\\frac{1}{\\varepsilon})+ \\ln2\n\\big)+\\frac{R^2\\beta^4}{9\\alpha^2D}+\n\\frac{R^2}{8D(-\\alpha)^3}\\ln(\\frac{-1}{\\alpha})R^2\\beta^4+\\frac{-\\delta^2\\alpha}{12DR^2}\n\\ln(\\frac{-\\delta}{\\alpha})\\ln(\\frac{-1}{\\alpha}).\\end{eqnarray*}\n By taking into account the leading order term only, using the expressions of\n$m_{\\delta}$ and $p$, we obtain after some computations that \\begin{eqnarray}\n\\label{tau}\\mean{\\tau}\\approx\\frac{R^2}{D} \\big(\n\\ln(\\frac{1}{\\varepsilon})+ \\ln2 \\big) -\n\\frac{R^2}{8D}\\ln(\\ln(\\frac{1}{\\beta}))\\ln^3(\\frac{1}{\\beta})\\beta^4\n.\\end{eqnarray} We conclude that the boundary layer has very little influence\non the leading order term at the order $\\alpha$.\n\\subsection{Computation of the mean time $\\mean{T}$ for a molecule to hit the boundary $\\partial D(\\delta)$ before exit}\nWe now provide an estimate of mean time it takes for a Brownian\nmolecule to enter into the binding site domain $D(\\delta)$, before\nexit. To estimate the mean time $\\mean{T}$, we observe that\n$E\\{T^S_{\\mbox{\\boldmath$x$}}|T^S_{\\mbox{\\boldmath$x$}}0$ such that for all $r,\\theta \\in K$,\n \\begin{eqnarray*}\n|p_{\\delta}(r,\\theta)-1| &\\leq& C \\alpha(\\beta,\\varepsilon) \\\\ \\\\\n|\\nabla p_{\\delta}(r,\\theta)| &\\leq& C \\alpha(\\beta,\\varepsilon)\n \\end{eqnarray*}\nwhere $\\alpha(\\beta,\\varepsilon)$ is defined by (\\ref{alpha}) and \\begin{eqnarray}\n0<\\alpha(\\beta,\\varepsilon)=\\alpha(\\frac{\\delta}{R},\\varepsilon)\\leq\n\\frac{C}{\\ln(\\frac1{\\delta})} \\end{eqnarray} which tends to zero when\n$\\delta$ goes to zero.\n\\end{prop}\n{\\noindent \\bf Proof.}\nTo estimate $p_{\\delta}$ we consider expression (\\ref{eq:series}),\nwhich can be written as \\begin{eqnarray*}\n&&-p_{\\delta}=q_{\\delta}(r,\\theta)-1=\\sum_{n=1}^{\\infty}\n\\frac{c_n}{1+\\beta^{2n}}\\Big[\\Big( \\frac r\nR\\Big)^n-\\Big(\\frac{\\delta \\beta}\nr\\Big)^n\\Big]\\cos(n\\theta)+\\alpha(\\beta,\\varepsilon) \\log\\Big(\\frac r\n\\delta\\Big).\\label{eq:seriesp} \\end{eqnarray*} where $\\alpha(\\beta,\\varepsilon)$ is\ngiven by expression (\\ref{alpha}) and we recall that \\begin{eqnarray}\n\\label{coef2} &c_n& = \\frac{1 + H_n}{\\sqrt 2} \\int_0^{\\pi-\\varepsilon}\nh(t) P_n(\\cos (t)) + P_{n-1}(\\cos(t))] dt. \\end{eqnarray} By definition\n$H_n=O(\\beta^{2n})$ and $P_n$ is the Legendre polynomial. For\n$x\\in [-1,1]$ and all n, $|P_{n}(x)| \\leq 1$. Moreover, using\nformula (\\ref{expand}), we obtain that \\begin{eqnarray} \\label{fomzp}\nh(t)=z(t)+O(\\beta^2)=-\\frac{2\\alpha}{\\pi}\\frac{d}{dt}\\int_0^t\\frac{u\n\\sin u}{\\sqrt{\\cos u-\\cos t}}du. \\end{eqnarray} Thus we conclude from the\nexplicit formula (\\ref{coef2}) that there exists a constant $C>0$\nsuch that for all $n$ \\begin{eqnarray} |c_n| \\leq C \\alpha(\\beta,\\varepsilon) \\end{eqnarray}\nwhich is independent of $n$. We denote\n$c_n=\\tilde{c_n}\\alpha(\\beta,\\varepsilon).$ We can now obtain the desired\nestimates. For $r_0 2190 \\, bytes$ then $IW = 2 * SMSS \\, bytes$ and must not be greater than 2 segments\n \\item If $(SMSS > 1095 \\, bytes) \\, and \\, (SMSS \\leq 2190 \\, bytes)$ then $IW = 3 * SMSS \\, bytes$ and must not be more than 3 segments\n \\item If $SMSS \\leq 1095 \\, bytes$ then $IW = 4 * SMSS \\, bytes$ and should not be more than 4 segments\n\\end{itemize}\n\nThe initial value of \\emph{ssthresh} should be set to the size of the largest possible window (advertised window), but \\emph{ssthresh} should be reduced in response to congestion. The slow start algorithm is used when $cwnd < ssthresh$, while the congestion avoidance algorithm is used when $cwnd > ssthresh$. In the event of a tie, the issuer can choose which algorithm to use. During the slow start phase, TCP increments \\emph{cwnd} by at most \\emph{SMSS bytes} and slow start stops when \\emph{cwnd} reaches or exceeds \\emph{ssthresh}. It is recommended to augment \\emph{cwnd} in the following way: \\[cwnd \\mathrel{+}= min(N, SMSS)\\] with \\emph{N} being the number of bytes that were acknowledged in the last ACK.\n\nDuring the congestion avoidance phase, \\emph{cwnd} is increased by approximately one segment per RTT (Round-Trip Time) and the algorithm continues its work until congestion is detected. It is recommended to increase \\emph{cwnd} as follows and adjust each time an ACK is received: \\[cwnd \\mathrel{+}= SMSS \\times \\frac{SMSS}{cwnd}\\]\n\nWhen a transmitter detects the loss of a segment using the retransmission timer and this segment has not been retransmitted, the value of \\emph{ssthresh} must not be set beyond the value given by the equation: \\[ssthresh=max(\\frac{FlightSize}{2}, 2\\times SMSS)\\] with \\emph{FlightSize} being the amount of data waiting in the network.\n\nA receiver, in TCP, should immediately send a duplicate ACK when an out-of-order segment arrives, with the objective of informing the sender of what happened and requesting sending the segment again. The sender should use the fast retransmit algorithm to detect and repair the loss based on incoming duplicate ACKs. The algorithm uses the arrival of three duplicate ACKs as an indication that a segment has been lost. From then on, TCP performs a retransmission of the lost segment without waiting for the expiration of the retransmission timer. After transmission of the lost segment, the fast recovery algorithm takes over the transmission of new data until a non-duplicate ACK arrives. The implementation of the fast retransmit and fast recovery algorithms follows rules for positioning values for \\emph{cwnd} and \\emph{ssthresh}.\n\nIn a simplified way, the TCP throughput is calculated with the following formula: \\[Throughput = \\frac{Window Size}{RTT}\\]\n\nStudies have shown that it is necessary to take into account other parameters such as MSS (Maximum Segment Size) and packet loss. As explained in \\cite{4}, assuming that the network has a packet loss probability called \\emph{p}, the sender will be able to send an average of $\\frac{1}{p}$ packets before a packet loss. According to this model, \\emph{cwnd} will never exceed a maximum \\emph{W} (window expressing the number of segments that can be sent during an RTT), because at approximately $\\frac{1}{p}$ packets, a new packet loss will cause the division by two from \\emph{cwnd}. Therefore, the total quantity of data delivered at each cycle of $\\frac{1}{p}$ packets is given by: \\[\\frac{1}{p} = \\left( \\frac{W}{2} \\right) ^2 + \\frac{1}{2} \\left( \\frac{W}{2} \\right) ^2 = \\frac{3}{8} W^2\\]\n\nTherefore, $MSS \\times \\frac{3}{8} W^2$ bytes are emitted every $RTT \\times \\frac{W}{2}$ cycle.\n\nAs $W = \\sqrt{\\frac{8}{3p}}$ the bit rate is therefore:\n\n\\begin{align}\n\tThroughput &= \\frac{MSS \\times \\frac{3}{8} W^2}{RTT \\times \\frac{W}{2}}\\nonumber\\\\\n\t&= \\frac{\\frac{MSS}{p}}{RTT \\sqrt {\\frac{2}{3p}}}\\nonumber\\\\\n\t&= \\frac{MSS \\times \\sqrt{\\frac{3}{2}}}{RTT \\times \\sqrt{p}}\\nonumber\\\\\n\t&= \\frac{MSS \\times 1.22}{RTT \\times \\sqrt{p }}\\nonumber\n\\end{align}\n\nA study \\cite{5} compared different TCP implementations and highlighted the performance differences in situations where RTT and loss probability values vary. As a result, on a wired network, the TCP Hybla and TCP CUBIC implementations achieve very good performance while TCP Reno also performs well, but TCP-LP achieves the worst results. On a wireless network, the cards are redistributed and the TCP Reno, TCP BIC and TCP-LP implementations obtain the best results. On long distance networks with delay, such as satellite networks, the TCP CUBIC implementation performs best. Finally, the implementation that seems to behave homogeneously in all circumstances remains TCP Reno.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}