diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhkwf" "b/data_all_eng_slimpj/shuffled/split2/finalzzhkwf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhkwf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n \n \nQuantized enveloping algebras of Kac-Moody algebras, and in particular of affine Lie algebras, have been preeminent examples of quantum groups since the 1980's. In the past few years, there has been an increase of research activity focusing on certain coideal subalgebras of those quantized enveloping algebras, for instance on the quantum symmetric pairs associated to finite-dimensional simple Lie algebras (see e.g.~\\cite{KP,BW,BSWW,Ko2,Le3}), which date back to work of G. Letzter in the 1990's \\cite{Le1,Le2}, and on those associated to symmetrizable Kac-Moody algebras (see e.g.~\\cite{BK1,BK2,FLLW1,FLLW2,FL2}), which appeared for the first time in the work of S. Kolb \\cite{Ko1}. See also \\cite{HK,ES,BKLW,FL1}.\n\nImportant examples of coideal subalgebras of quantized enveloping algebras appeared even before the aforementioned work of G. Letzter: around 1990, G. Olshanskii introduced in \\cite{Ol} the twisted Yangians of type AI and AII (that is, those associated to the symmetric pairs $(\\mfgl_N,\\mfso_N)$ and $(\\mfgl_N,\\mfsp_N)$), which are coideal subalgebras of the Yangian of $\\mfgl_N$, the later being one of the two families of quantized enveloping algebras of affine type $A^{(1)}$. These twisted Yangians and their representations have been quite well studied over the years, mainly in the work of A. Molev, M. Nazarov \\textit{et al.}, see e.g. \\cite{Mo1,Mo2,Mo3,Mo4,Mobook,Na,KN1,KN2,KN3,KNP}. In particular, A. Molev obtained in \\cite{Mo1} and \\cite{Mo2} a classification of their finite-dimensional irreducible representations. \n\nIn \\cite{GR}, two of the present authors constructed twisted Yangians for all symmetric pairs $(\\mfg,\\mfg^{\\rho})$ where $\\mfg$ is an orthogonal or a symplectic Lie algebra and $\\mfg^{\\rho}$ is the Lie subalgebra fixed by an involution $\\rho$ of $\\mfg$. These new twisted Yangians of type B-C-D are also coideal subalgebras of the Yangian of $\\mfg$ and their basic properties were established in \\cite{GR}. In \\cite{GRW2}, we initiated a study of their representation theory, proving some general results (e.g. that finite-dimensional irreducible representations are highest weight modules - see Theorem 4.5 in \\textit{loc. cit.}) and obtaining, for twisted Yangians of type CI, DIII and BCD0, a full classification of finite-dimensional irreducible representations in terms of certain polynomials, in analogy with \\cite{Dr} (for the non-twisted Yangians) and \\cite{Mo1,Mo2}: see Theorems 6.2, 6.5 and 6.6 in \\cite{GRW2}. \n\nThe present article is a sequel to our work \\cite{GRW2}. We start addressing the classification problem of finite-dimensional irreducible representations of twisted Yangians for the symmetric pairs $(\\mfg_N,\\mfg_{N-q}\\oplus \\mfg_q)$ of types BI, CII and DI(a) (see Table \\ref{Table:G}), and we obtain complete results for those associated to the symmetric pairs $(\\mfso_N,\\mfso_{N-2}\\op \\mfso_2)$ and $(\\mfso_{2n+1},\\mfso_{2n})$. In Section 2, we recall the relevant preliminary material from \\cite{AMR}, \\cite{GR} and \\cite{GRW2}. In Section 3, we consider the twisted Yangians of orthogonal type when $N=3$ and $N=4$. These twisted Yangians are known to be isomorphic to the twisted Yangian of $(\\mfgl_2,\\mfso_2)$ or a tensor product of these, as was established in \\cite{GRW1}: see the beginning of Subsections \\ref{sec:so4so2so2} and \\ref{sec:so3so2} for precise statements of these isomorphisms. Consequently, the classification of finite-dimensional irreducible representations in these two low rank cases can be translated from the corresponding result of A. Molev established in \\cite{Mo1,Mo2}: see Propositions \\ref{P:so4class} and \\ref{P:so3class}. The latter of these two propositions plays a role in the proofs of two of the main results of this paper, namely Proposition \\ref{P:necessary} and Theorem \\ref{T:BI(b)-Class}.\n\nIn Section \\ref{sec:Nec}, some of the important results established in \\cite{GRW2} are strengthened for the symmetric pairs of types BI, CII and DI(a). Proposition \\ref{P:necessary} is the main result in this section and gives necessary conditions for an irreducible highest weight module of a twisted Yangians to be finite-dimensional. This provides essentially one half of the proofs of Theorems \\ref{T:DI(a)-Class} and \\ref{T:BI(b)-Class}. Its proof relies on a similar result in \\cite{GRW2}, namely Proposition 4.18, and on Theorems 6.5 and 6.6 also from \\cite{GRW2}, which are the classification theorems of finite-dimensional irreducible modules for twisted Yangians of type BCD0. Proposition \\ref{P:int} provides additional restrictions on the complex number $\\al$ which appears in the statement of Proposition \\ref{P:necessary}: its proof boils down to computing the highest weight of a certain highest weight module over $\\mfg_N^{\\rho}$, which is done in Lemma \\ref{L:gtw}. In particular, this proposition shows that, when $\\mfg_q\\ncong \\mfso_2$, the parameter $\\alpha$ can only take certain rational values and must satisfy a specific inequality.\n\nIn order to prove the classification theorem for the twisted Yangians of the pair $(\\mfso_N,\\mfso_{N-2} \\oplus \\mfso_2)$, it is necessary to first construct a family of one-dimensional representations parametrized by $\\C$: this is achieved in Lemma \\ref{L:K-1dim}. It turns out that, up to a twist by an automorphism, the one-dimensional representations provided by this lemma exhaust all of them - see Proposition \\ref{P:1dimso2}. Twisted Yangians can be defined in terms of the reflection equation \\eqref{TX-RE} and the symmetry relation \\eqref{TX-symm}, as recalled in Subsection \\ref{subsec:twYa}. It follows from this that each twisted Yangian admits a one-dimensional representation obtained by sending its matrix of generators to a certain matrix $\\mcG(u)$ which is known explicitly and is part of the definition of the twisted Yangian (Definition \\ref{D:TX}): see the paragraph after \\eqref{grho->X}. We call it the trivial representation. This raises the question of the existence of other one-dimensional representations for twisted Yangians in general. Twisted Yangians of type CI and DIII also admit a family of one-dimensional representations parametrized by $\\C$. It turns out that for twisted Yangians of type BCD0, BI, CII and DI(a) with $\\mfg_{q} \\ncong \\mfso_2$, the one-dimensional representations are all twists by automorphisms of the trivial representation: this is the content of Proposition \\ref{P:no1dim}.\n\nThe last section provides proofs of the classification theorems of finite-dimensional irreducible representations for the twisted Yangians associated to the symmetric pairs $(\\mfso_N,\\mfso_{N-2} \\oplus \\mfso_2)$ and $(\\mfso_{2n+1},\\mfso_{2n})$. These are Theorems \\ref{T:DI(a)-Class} and \\ref{T:BI(b)-Class} respectively. Proofs of analogous results for other twisted Yangians of type BI, CII or DI(a) are expected to be substantially more complicated and will be presented elsewhere. (Please see the end of this introduction for a brief explanation of the extra difficulties in these more general cases.) The classification is in terms of a scalar $\\al$ and certain polynomials which impose conditions on the highest weight of a finite-dimensional irreducible representation. In the $(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)$-case, the proof of the necessity of those conditions is actually given earlier by Proposition \\ref{P:necessary}. The $(\\mfso_{2n+1},\\mfso_{2n})$-case is substantially more complicated. The main differences are due to the role played by the action of an involution of the twisted Yangian on the highest weight of a finite-dimensional irreducible representation (Lemma \\ref{BI:P-psi}). This leads to a new condition on the roots of a certain polynomial, which is the content of Proposition \\ref{P:q=1-nec}. This additional condition is a new feature which is not present in the classification theorems for twisted Yangians of type A \\cite{Mobook,MR} or types CI, DIII or BCD0 \\cite{GRW2}.\n\nFinally, we explain how finite-dimensional irreducible representations can be realized as subquotients of tensor products of fundamental representations along with a one-dimensional representation in the case of $(\\mfso_N,\\mfso_{N-2} \\oplus \\mfso_2)$: see Corollaries \\ref{C:DI(a)fun} and \\ref{C:BI(b)fun}. This is consistent with previously known results for twisted Yangians of type A, CI, DIII and BCD0 (see \\cite{Mobook} and \\cite{GRW2}).\n\n To complete the task of classifying the finite-dimensional irreducible representations of each twisted Yangian studied in \\cite{GRW2}, similar results to those obtained in Section \\ref{sec:main} must be proven for the twisted Yangians which are, in the notation of Table \\ref{Table:G} below, of type CII as well as types BI and DI(a) with $q\\geq 3$. Like the twisted Yangians for the symmetric pairs $(\\mfso_{2n+1},\\mfso_{2n})$ studied in Subsection \\ref{subsec:q=1}, each of these twisted Yangians has the common property that the necessary conditions established in Section \\ref{sec:Nec} are not sufficient for determining precisely when the irreducible quotient of a Verma module is finite-dimensional. It is, however, possible to prove results similar to Proposition \\ref{P:q=1-nec} for each of these quantum algebras, which strengthen significantly the conditions of Section \\ref{sec:Nec}. In particular, for the twisted Yangians of the symmetric pairs $(\\mfsp_{N},\\mfsp_{N-q}\\oplus \\mfsp_q)$ of type CII, this leads to a complete classification of finite-dimensional irreducible modules. These results, which will be presented in \\cite{GRW3}, are notably more complicated to prove than their counterparts in Section \\ref{sec:main}. For instance, their proofs seem to require understanding how to pass representation theoretic information between certain isomorphic presentations of twisted Yangians. \n \n For the twisted Yangians of the symmetric pairs $(\\mfso_N,\\mfso_{N-q}\\oplus \\mfso_q)$ which are of type BI and DI(a) with $q\\geq 3$, there are additional difficulties which arise. One such difficulty involves showing that, in the notation of Definition \\ref{D:assoc}, an irreducible highest weight module which is associated to the scalar $\\alpha=N\/4-1\/2$ and polynomials $P_1(u)=\\cdots=P_n(u)=1$ is finite-dimensional. Lemma \\ref{L:gtw} illustrates that these modules are closely related to the spinor representations of $\\mfso_{2\\ley}$, where\n $\\ley=q\/2$ if $q$ is even and $\\ley=(N-q)\/2$ otherwise. This difficulty, which will be considered in future work, does not arise for the twisted Yangians corresponding to the \n symmetric pairs $(\\mfso_{2n+1},\\mfso_{2n})$ which are studied in the present paper. Indeed, for these twisted Yangians the modules under consideration can be obtained by restricting the (finite-dimensional) spinor representations of the Yangian for $\\mfso_{2n+1}$ which are provided by Lemma 5.18 of \\cite{AMR}. \n \n\n{\\it Acknowledgements.} The first and third named authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada provided via the Discovery Grant Program and the Alexander Graham Bell Canada Graduate Scholarships - Doctoral Program, respectively. Part of this work was done during the second named author's visits to the University of Alberta; he thanks the University of Alberta for the hospitality. The second named author was supported in part by the Engineering and Physical Sciences Research Council of the United Kingdom, grant number EP\/K031805\/1; he gratefully acknowledges the financial support.\n\n\n\n\n \\section{Preliminaries}\n \nThroughout this manuscript we will employ mostly the same notation as in \\cite{GRW2}. Thus in many instances we will try to be concise and will refer to {\\it loc.~cit.} for complete details. We start by recalling some basic definitions.\n\nLet $N=2n$ or $N=2n+1$ with $n\\in\\N$. We will denote by $\\mfg_N$ either the orthogonal Lie algebra $\\mfso_N$ or the symplectic Lie algebra $\\mfsp_N$ (only when $N=2n$). The Lie algebra $\\mfg_N$ can be realized as a Lie subalgebra of $\\mfgl_N$ as follows. We label the rows and columns of matrices in $\\mfgl_N$ by the indices $\\mcI_N=\\{-n,\\ldots,-1,(0),1,\\ldots,n\\}$, where $(0)$ is omitted if $N=2n$. Set $\\theta_{ij}=1$ in the orthogonal case and $\\theta_{ij}=\\mathrm{sign}(i)\\cdot \\mathrm{sign}(j)$ in the symplectic case for $i,j\\in \\mcI_N$. We also introduce a smaller set $\\mcI_N^{+} =\\mcI_N\\cap \\Z_{\\geq 0}$. \n\nFor each $i,j\\in \\mcI_N$, let $E_{ij}$ denote the usual elementary matrix of $\\mfgl_N$. Define the transposition $t$ by $(E_{ij})^{t} = \\theta_{ij} E_{-j,-i}$ and set $F_{ij} = E_{ij} - (E_{ij})^{t}$ so that \n\\[\n[F_{ij},F_{kl}] = \\delta_{jk} F_{il} - \\delta_{il} F_{kj} + \\delta_{j,-l} \\theta_{ij} F_{k,-i} - \\delta_{i,-k} \\theta_{ij} F_{-j,l} \\qu\\text{and}\\qu F_{ij} + \\theta_{ij}F_{-j,-i}=0 .\n\\] \nThen $\\mfg_N$ is isomorphic to $\\mr{span}_{\\C} \\{ F_{ij} \\, : i,j\\in \\mcI_N \\}$ and $\\mfh_N=\\mathrm{span}_{\\C} \\{ F_{ii} \\, : \\, 1 \\le i \\le n \\}$ forms a Cartan subalgebra which will be denoted by $\\mfh_N$. \nGiven a Lie algebra $\\mathfrak{a}$ its universal enveloping algebra will be denoted by $\\mfU\\mathfrak{a}$.\n\n\nIntroduce the permutation operator $P = \\sum_{i,j\\in \\mcI_N} E_{ij} \\ot E_{ji}$ and the one-dimensional projector $Q=P^{t_1}=P^{t_2}$. Let $I$ denote the identity matrix. Then $P^2=I$, $PQ=QP=\\pm Q$ and $Q^2=N Q$, which will be useful below. Here (and further in this paper) the upper sign corresponds to the orthogonal case and the lower sign to the symplectic case. \n \n\nLet tensor products be defined over the field of complex numbers. For a matrix $X$ with entries $x_{ij}$ in an associative algebra $A$ we write\n\\[\nX_s = \\sum_{i,j\\in\\mcI_N} \\underbrace{ I \\ot \\cdots \\ot I}_{s-1} \\ot E_{ij} \\ot I \\ot \\cdots \\ot I \\ot x_{ij} \\in \\End(\\C^N)^{\\ot k} \\ot A .\n\\]\nHere $k \\in \\N_{\\ge 2}$ and $1\\le s\\le k$; it will always be clear from the context what $k$ is.\n\n \n \n\n\n\\subsection{Yangians of type B-C-D and their finite-dimensional irreducible representations} \\label{subsec:Ya}\n\n\nWe introduce elements $t_{ij}^{(r)}$ with $i,j\\in \\mcI_N$ and $r\\in\\Z_{\\ge 0}$ such that $t^{(0)}_{ij}= \\del_{ij}$. Combining these into formal power series $t_{ij}(u) = \\sum_{r\\ge 0} t_{ij}^{(r)} u^{-r}$, we can then form the generating matrix $T(u)= \\sum_{i,j\\in \\mcI_N} E_{ij} \\ot t_{ij}(u)$. \nThe $R$-matrix that we need is $R(u)=I - u^{-1} P + (u-\\ka)^{-1} Q$, where $\\ka = N\/2 \\mp 1$.\n\n\\begin{defn} [\\cite{AACFR}]\\label{D:X}\nThe extended Yangian $X(\\mfg_N)$ is the unital associative $\\C$-algebra generated by elements $t_{ij}^{(r)}$ with $i,j\\in \\mcI_N$ and $r\\in\\Z_{\\ge 1}$ satisfying the relation\n\\eq{ \\label{RTT}\nR(u-v)\\,T_1(u)\\,T_2(v) = T_2(v)\\,T_1(u)\\,R(u-v) .\n}\nThe Hopf algebra structure of $X(\\mfg_N)$ is given by \n\\[\n\\Delta: T(u) \\mapsto T(u)\\ot T(u), \\qq S: T(u)\\mapsto T(u)^{-1},\\qquad \\epsilon: T(u)\\mapsto I. \n\\]\n\\end{defn}\n\nThe expansion of the defining relation \\eqref{RTT} in terms of the generating series $t_{ij}(u)$ can be found in {\\it e.g.}~Definition 3.1 of \\cite{GRW2}. \n\nFor each series $f(u)\\in 1+u^{-1}\\C[[u^{-1}]]$ there is an automorphism $\\mu_f$ of $X(\\mfg_N)$ which is given by the assignment $\\mu_f : T(u) \\mapsto f(u)\\, T(u)$. The Yangian $Y(\\mfg_N)$ is defined as the subalgebra of $X(\\mfg_N)$ consisting of the elements stable under all the automorphisms of the form $\\mu_f$, namely\n\\[\nY(\\mfg_N) = \\{ y \\in X(\\mfg_N) : \\mu_f(y) = y \\text{ for any } f(u) \\in1+u^{-1}\\C[[u^{-1}]] \\}.\n\\]\nBy Theorem 3.1 of \\cite{AACFR} and the proof of Theorem 3.1 from \\cite{AMR}, there is a central series $y(u)=1+\\sum_{r\\geq 1} y_r u^{-r}$ such that \n\\begin{equation*}\n T^t(u+\\ka)T(u)=T(u)T^t(u+\\ka)=y(u)y(u+\\ka)\\cdot I.\n\\end{equation*}\nIt was proven in Corollary 3.2 of \\cite{AMR} that the coefficients $\\tau_{ij}^{(r)}$ of the all of the series $\\tau_{ij}(u)=y(u)^{-1}t_{ij}(u)$ generate the subalgebra $Y(\\mfg_N)$. The corresponding generating matrix $y(u)^{-1}T(u)$ will be denoted by $\\mcT(u)$.\n\nLet us now recall elements of the representation theory of $X(\\mfg_N)$. A representation $V$ of $X(\\mfg_N)$ is a \\textit{highest weight representation} if there exists a nonzero vector $\\xi\\in V$ such that $V=X(\\mfg_N)\\,\\xi$ and the following \nconditions are satisfied:\n\\begin{alignat*}{4}\n & t_{ij}(u)\\,\\xi=0 \\quad &&\\text{ for all } && i0$, that, in terms of the notation introduced in Section 2.2 of \\cite{GRW2}, are of types BI, CII and DI(a). \nThese symmetric pairs are of the form $(\\mfg_N,\\mfg_N^\\rho)$ where $\\rho$ is the involution of $\\mfg_N$ given by $\\rho(X) = \\mcG X \\mcG^{-1}$ for all $X\\in\\mfg_N$ with $\\mcG$ the corresponding matrix in the table below, and $\\mfg_N^\\rho$ is the subalgebra of $\\mfg_N$ fixed by~$\\rho$.\n\n\\begin{table}[H]\n\\centering\n\\caption{Symmetric pairs} \\label{Table:G} \n\n\\begin{tabular}{|c|c|c|c|}\n \\hline \n Type & \\multicolumn{2}{|c|}{Symmetric pair $(\\mfg_N,\\mfg_N^\\rho)$} & $\\mcG$ \\\\ \n \\hline\n BI(a) & $(\\mfso_{2n+1},\\mfso_{2n+1-q}\\oplus \\mfso_q)$ & $q=0\\mod 2$ & \\rule{0pt}{4ex} $\\sum_{i=-\\frac{p-1}{2}}^{\\frac{p-1}{2}} E_{ii}-\\sum_{i=\\frac{p+1}{2}}^{n} (E_{ii} + E_{-i,-i})$ \\\\[1em]\n BI(b) & $(\\mfso_{2n+1},\\mfso_{2n+1-q}\\oplus \\mfso_q)$ & $q=1\\mod 2$ & $-\\sum_{i=-\\frac{q-1}{2}}^{\\frac{q-1}{2}} E_{ii}+ \\sum_{i=\\frac{q+1}{2}}^{n} (E_{ii} + E_{-i,-i})$ \\\\[1em]\n CII & $(\\mfsp_{2n},\\mfsp_{2n-q}\\oplus \\mfsp_q)$ & $q=0\\mod 2$ & $\\sum_{i=1}^{\\frac{p}{2}}(E_{ii}+E_{-i,-i})-\\sum_{i=\\frac{p}{2}+1}^{n}(E_{ii}+E_{-i,-i})$\\\\[1em]\n DI(a) & $(\\mfso_{2n},\\mfso_{2n-q}\\oplus \\mfso_q)$ & $q=0\\mod 2$ & $\\sum_{i=1}^{\\frac{p}{2}}(E_{ii}+E_{-i,-i})-\\sum_{i=\\frac{p}{2}+1}^{n}(E_{ii}+E_{-i,-i})$\\\\[1em]\n \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{rmk}\nNote that we have not included pairs of type DI(b), namely $(\\mfso_{2n},\\mfso_{p}\\op\\mfso_q)$ with both $p$ and $q$ odd. In this case the matrix $\\mcG$ cannot be chosen to be diagonal. Also note that, in type BI(b), when $q=1$, $\\mfso_1 = \\{ 0 \\}$.\n\\end{rmk}\n\nWe will write $g_{ij}$ for the $(i,j)$-th entry of $\\mcG$. We have $\\mcG=\\sum_{i\\in\\mcI_N} g_{ii}E_{ii}$ since $g_{ij}=0$ if $i\\ne j$. Let $\\mcG(u)=(g_{ij}(u))_{i,j\\in \\mcI_N}$ be given by \n\\eq{\n\\mcG(u) = \\frac{d I - u\\,\\mcG}{d - u} \\qu\\text{with}\\qu d=\\frac{p-q}4. \\label{G(u)}\n}\nNotice that $\\mcG(u)=\\mcG$ if $p=q$. \n\n\n\\begin{defn} [{\\cite[Definition~3.1]{GR}}] \\label{D:TX}\nThe extended twisted Yangian $X(\\mfg_N,\\mcG)^{tw}$ is the subalgebra of $X(\\mfg_N)$ generated by the coefficients $s_{ij}^{(r)}$, with $i,j \\in\\mcI_N$ and $r\\in\\Z_{\\ge 1}$, of the entries $s_{ij}(u) = g_{ij} +\\sum_{r=1}^{\\infty} s_{ij}^{(r)} u^{-r}$ of the $S$-matrix\n\\[\nS(u) =\\sum_{i,j\\in\\mcI_N} E_{ij} \\ot s_{ij}(u) = T(u-\\ka\/2)\\,\\mcG(u)\\,T^t(-u+\\ka\/2). \n\\]\n\\end{defn}\n\nThe algebra $X(\\mfg_N,\\mcG)^{tw}$ is a left coideal subalgebra of $X(\\mfg_N)$: $\\Delta(X(\\mfg_N,\\mcG)^{tw}) \\subset X(\\mfg_N) \\otimes X(\\mfg_N,\\mcG)^{tw}$. The $S$-matrix $S(u)$ satisfies the reflection equation \n\\eq{ \\label{TX-RE}\nR(u-v)\\,S_1(u)\\,R(u+v)\\,S_2(v) = S_2(v)\\,R(u+v)\\,S_1(u)\\,R(u-v)\n}\nand the symmetry relation\n\\eq{ \\label{TX-symm}\nS^t(u) = \\,S(\\ka-u) \\pm \\frac{S(u)-S(\\ka-u)}{2u-\\ka} + \\frac{\\Tr(\\mcG(u))\\,S(k-u) - \\Tr(S(u))\\cdot I }{2u-2\\ka} . \n}\nThe above two relations are in fact the defining relations of $X(\\mfg_N,\\mcG)^{tw}$. (Their form in terms of $s_{ij}(u)$ can be found in (4.4) and (4.5) of \\cite{GR}.) By Theorem 4.2 of \\cite{GR}, the algebra generated by the abstract elements $s_{ij}^{(r)}$ subject to those relations, which is called $\\mcB(\\mcG)$ in {\\it loc.~cit.}, is isomorphic to $X(\\mfg_N,\\mcG)^{tw}$. \n\nThe algebra defined in the same way as $\\mcB(\\mcG)$ except with \nthe symmetry relation \\eqref{TX-symm} omitted and $(s_{ij}^{(r)},s_{ij}(u),S(u))$ replaced with generators denoted $(\\wt s_{ij}^{(r)},\\wt s_{ij}(u),\\wt S(u))$ is called the \\textit{extended reflection algebra} of $(\\mfg_N,\\mfg_N^\\rho)$ and is denoted $\\wt X(\\mfg_N,\\mcG)^{tw}$. In Section 5.1 of \\cite{GR}, where $\\wt X(\\mfg_N,\\mcG)^{tw}$ was denoted $\\mathcal{XB}(\\mcG)$, it was shown that there exists a central series $c(u)=1+\\sum_{r\\geq 1}c_r u^{-r}\\in \\wt X(\\mfg_N,\\mcG)^{tw}[[u^{-1}]]$ such that \n\\begin{equation}\n X(\\mfg_N,\\mcG)^{tw}\\cong \\wt X(\\mfg_N,\\mcG)^{tw}\/(c(u)-1) \\label{c(u)}.\n\\end{equation}\nWe refer the reader to {\\it loc.~cit.}~for a more complete treatment of $\\wt X(\\mfg_N,\\mcG)^{tw}$ and return our attention to the twisted Yangian $X(\\mfg_N,\\mcG)^{tw}$. \n\n\n\nBy Proposition 3.1 of \\cite{GR}, the product $S(u)\\,S(-u) = w(u)\\cdot I$ defines a formal power series $w(u) = 1 + \\sum_{r\\ge 1} w_{2r} u^{-2r}$ with coefficients central in $X(\\mfg_N,\\mcG)^{tw}$. The twisted Yangian $Y(\\mfg_N,\\mcG)^{tw}$ is defined as the quotient of $X(\\mfg_N,\\mcG)^{tw}$ by the ideal generated by the coefficients of the {\\it unitary relation} $S(u)\\,S(-u)=I$, that is\n\\eq{\nY(\\mfg_N,\\mcG)^{tw} = X(\\mfg_N,\\mcG)^{tw} \/ (w(u)-1) . \\label{Y=X\/(w-1)}\n}\nBy Theorem~3.1 of \\cite{GR} the algebra $Y(\\mfg_N,\\mcG)^{tw}$ is isomorphic to the subalgebra of $Y(\\mfg_N)$ generated by the coefficients $\\si_{ij}^{(r)}$ with $r\\ge1$ of the matrix entries $\\si_{ij}(u)$ of the $S$-matrix $\\Si(u)$ defined by \n\\[\n\\Si(u) = \\mcT(u-\\ka\/2)\\, \\mcG(u)\\, \\mcT^t(-u+\\ka\/2) . \n\\]\n\nDenote by $Z(\\mfg_N,\\mcG)^{tw}$ the subalgebra of $X(\\mfg_N,\\mcG)^{tw}$ generated by the even coefficients $w_{2r}$ with $r\\ge0$. The subalgebra $Z(\\mfg_N,\\mcG)^{tw}$ is the centre of $X(\\mfg_N,\\mcG)^{tw}$. Moreover, the following tensor decomposition holds\n\\eq{\nX(\\mfg_N,\\mcG)^{tw} \\cong Z(\\mfg_N,\\mcG)^{tw} \\ot Y(\\mfg_N,\\mcG)^{tw} . \\label{X=Z*Y}\n} \n\nGiven $g(u)\\in1+u^{-2}\\C[[u^{-2}]]$, the assignment\n\\eq{\n\\nu_g : S(u) \\mapsto g(u-\\ka\/2)\\,S(u) \\label{nu_g}\n}\nextends to an automorphism $\\nu_g$ of $X(\\mfg_N,\\mcG)^{tw}$, and $Y(\\mfg_N,\\mcG)^{tw}$, viewed as a subalgebra of $X(\\mfg_N)$, is the $\\nu_g$-stable subalgebra of $X(\\mfg_N,\\mcG)^{tw}$: see Corollary 3.1 of \\cite{GR}. \n\nA second family of automorphisms of $X(\\mfg_N,\\mcG)^{tw}$ is provided by conjugating $S(u)$ by certain invertible matrices: let $A\\in GL(N)$ satisfy \n$AA^t=I$ and $A\\mcG A^t=\\mcG$. Then, by Remark 3.2 of \\cite{GR}, the assignment \n\\begin{equation}\n \\alpha_A: S(u)\\mapsto AS(u)A^t \\label{al_A}\n\\end{equation}\n defines an automorphism $\\alpha_A$ of $X(\\mfg_N,\\mcG)^{tw}$ which factors through the quotient $Y(\\mfg_N,\\mcG)^{tw}$.\n\n\nFor each $i,j\\in \\mcI_N$, set $F_{ij}^{\\rho}=(g_{ii}+g_{jj})F_{ij}$ and define $\\bar g_{ij}=(g_{ij}-\\delta_{ij})d$, where we recall that $d=(p-q)\/4$. By Proposition 3.9 of \\cite{GRW2}, the fixed point subalgebra $\\mfU\\mfg_N^\\rho\\subset \\mfU\\mfg_N$ is generated by the $F_{ij}^{\\rho}$, and the assignments\n\\begin{equation}\nF_{ij}^{\\rho}\\mapsto s_{ij}^{(1)}-\\bar{g}_{ij} \\qu \\text{and} \\qu F_{ij}^{\\rho}\\mapsto \\si_{ij}^{(1)}-\\bar{g}_{ij} \\quad \\text{ for all }\\quad i,j\\in \\mcI_N \\label{grho->X}\n\\end{equation}\nextend to injective algebra homomorphisms $\\mfU\\mfg_N^\\rho\\into X(\\mfg_N,\\mcG)^{tw}$ and $\\mfU\\mfg_N^\\rho\\into Y(\\mfg_N,\\mcG)^{tw}$, respectively.\n\n\nThe matrix $\\mcG(u)$ provides the trivial representation $V(\\mcG)$ of $X(\\mfg_N,\\mcG)^{tw}$ defined by the map $\\eps : S(u) \\mapsto \\mcG(u)$, which is the restriction of the counit of $X(\\mfg_N)$ to $X(\\mfg_N,\\mcG)^{tw}$. When $\\mfg_N^\\rho=\\mfg_p\\op\\mfg_q$ with $\\mfg_p\\ncong \\mfso_2\\ncong\\mfg_q$, this is a unique one-dimensional representation of $X(\\mfg_N,\\mcG)^{tw}$, up to twisting by automorphisms of the form $\\nu_g$: see Subsection \\ref{subsec:1dim-gen}. When $(\\mfg_N,\\mfg_N^\\rho)=(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)$ with \n$N\\geq 5$, $V(\\mcG)$ is a special case of a one-parameter family of one-dimensional representations which will be constructed in Subsection \\ref{subsec:1dim-so2}, and if $N=4$ it belongs to a two-parameter family of one-dimensional representations; this will be explained in Subsection \\ref{sec:so4so2so2}. In the case when $(\\mfg_N,\\mfg_N^\\rho)=(\\mfso_3,\\mfso_2)$, $V(\\mcG)$ is again a particular member of a one-parameter family of one-dimensional representations, as will be seen in Subsection \\ref{sec:so3so2}. \n\nWe will also make use of the rational function (see Lemma 2.2 of \\cite{GRW2}) \n\\eq{\np(u) = 1 \\mp \\frac{1}{2u-\\ka} + \\frac{ {\\rm tr}(\\mcG(u))}{2u-2\\ka} \\qq\\text{satisfying}\\qq p(u)\\,p(\\ka-u) = 1 - \\frac{1}{(2u-\\ka)^2}\\,. \\label{p(u)}\n}\nWhen it is necessary to emphasize the dependence of $p(u)$ on the matrix $\\mcG$ we will denote it instead by $p_\\mcG(u)$. \n\nLastly, we remark that we will sometimes write $X(\\mfg_N,\\mfg_N^\\rho)^{tw}$ (resp.~$Y(\\mfg_N,\\mfg_N^\\rho)^{tw}$) instead of $X(\\mfg_N,\\mcG)^{tw}$ (resp.~$Y(\\mfg_N,\\mcG)^{tw}$). \n\n\n\n\n\n\\subsubsection{Highest weight representations} \\label{subsub:HWtw}\n\n\nWe now briefly summarize the relevant results of the highest weight theory for representations of $X(\\mfg_N,\\mcG)^{tw}$ obtained in Section 4 of \\cite{GRW2}.\n\n\nA representation $V$ of $X(\\mfg_N,\\mcG)^{tw}$ is called a \\textit{highest weight representation} if there exists a nonzero vector $\\eta\\in V$ such that $V=X(\\mfg_N,\\mcG)^{tw}\\eta$ and the following conditions are met: \n\\begin{alignat*}{4}\n &s_{ij}(u)\\,\\eta=0 \\quad &&\\text{ for all }\\quad && iTr}\n \\end{equation*}\n\\end{crl}\n\nWe also remark that \\eqref{TrS} holds for any symmetric pair of type B-C-D, and in particular it implies that $\\Tr(S(u))=\\Tr(S(\\ka-u))$ when $(\\mfg_N,\\mfg_N^\\rho)=(\\mfg_N,\\mfg_N)$. \n\n\nWhen the $X(\\mfg_N,\\mcG)^{tw}$ Verma module $M(\\mu(u))$ is non-trivial, it admits a unique irreducible quotient $V(\\mu(u))$, and every finite-dimensional irreducible module is isomorphic to a module of this form.\n\nGiven a representation $V$ of $X(\\mfg_N,\\mcG)^{tw}$, denote by $V^0$ the subspace \n\\begin{equation*}\n V^0=\\{\\xi\\in V\\,:\\,s_{ij}(u)\\,\\xi=0 \\; \\text{ for all }\\; i0$ except in the case when \n$(\\mfg_N,\\mfg_N^\\rho)$ is type BI(b) with $q=1$ ({\\it i.e.}~when $(\\mfg_N,\\mfg_N^\\rho)=(\\mfso_{2n+1},\\mfso_{2n})$). In Subsection 4.3 of \\cite{GRW2}, the pair $(\\ley,\\key)$ was denoted $(\\boldsymbol{\\ell},\\boldsymbol{k})$.\n\nAn important instance of \\eqref{HWT:tensors} occurs when $\\mu(u)$ is taken to be $(g_{ii}(u))_{i\\in \\mcI_N}$, in which case $V(\\mu(u))=V(\\mcG)$ and \\eqref{HWT:tensors}\nprovides formulas for the highest weight $\\ga(u)$ of the $X(\\mfg_N,\\mcG)^{tw}$-module $X(\\mfg_N,\\mcG)^{tw}\\xi \\subset L(\\lambda(u))$. By \\eqref{G(u)}, we have \n\\begin{equation}\n\\wt g_{ii}(u)=2u\\left(\\frac{p-q[\\pm]4u}{p-q-4u} \\right) \\quad \\text{ for all }\\quad \\key+1\\leq i\\leq n, \\label{eq:wtgii1}\n\\end{equation}\nwhile for $i\\in \\mcI_N^+$ satisfying $0\\leq i\\leq \\key$ we have by \\eqref{g(u)} that\n\\begin{equation}\n \\wt g_{ii}(u)=(2u-\\ley)\\left(\\frac{p-q[\\mp]4u}{p-q-4u} \\right) + \\ley\\left(\\frac{p-q[\\pm]4u}{p-q-4u} \\right) =[\\pm]2u\\left(\\frac{p+q-4u}{p-q-4u} \\right)=2u\\cdot\\mathscr{g}(u). \\label{eq:wtgii2}\n\\end{equation}\nHence \\eqref{HWT:tensors} becomes \n\\begin{equation*}\n \\wt \\ga_i(u)=\\begin{cases}\n 2u\\cdot \\mathscr{g}(u)\\la_i(u-\\ka\/2)\\la_{-i}(-u+\\ka\/2) \\; &\\text{ if } \\; 0\\leq i\\leq \\key,\\\\\n 2u\\left(\\frac{p-q[\\pm]4u}{p-q-4u} \\right)\\la_i(u-\\ka\/2)\\la_{-i}(-u+\\ka\/2) \\; &\\text{ if } \\; \\key+1\\leq i\\leq n\n \\end{cases}\n\\end{equation*}\nfor each $i\\in \\mcI_N^+$: see also Corollary 4.11 of \\cite{GRW2}. \n\n\nIn \\cite{GRW2} the finite-dimensional irreducible representations of $X(\\mfg_N,\\mfg_N^\\rho)^{tw}$ were classified for $(\\mfg_N,\\mfg_N^\\rho)$ of type CI, DIII and BCD0. In the present paper the corresponding classification results for the extended twisted Yangians $X(\\mfg_N,\\mfg_N)^{tw}$ of type BCD0 will play an important role, and hence we recall them here. \n\n\nWe emphasize that all the definitions provided in Subsection \\ref{subsec:twYa} still apply for the (extended) twisted Yangians of type BCD0: one must just substitute $(p,q)=(N,0)$. In particular, \nthe definition of $\\mcG(u)$ provided by \\eqref{G(u)} collapses to $\\mcG(u)=I$. The definitions of $\\key$ and $\\ley$ given in \\eqref{ley-key} also extend to include pairs of this type, where we have $(\\key,\\ley)=(n,0)$. \n\n\nLet $\\delta=1$ if $\\mfg_N=\\mfsp_{N}$ and $\\delta=0$ if $\\mfg_N=\\mfso_N$. \n\\begin{thrm}[Theorems 6.5 and 6.6 of \\cite{GRW2}]\\label{T:BCD0-class}\n Suppose that $\\mu(u)=(\\mu_i(u))_{i\\in \\mcI_N^+}$ satisfies the conditions of Proposition \\ref{P:nontriv}. Then the irreducible $X(\\mfg_N,\\mfg_N)^{tw}$-module \n $V(\\mu(u))$ is finite-dimensional if and only if there exists monic polynomials $P_1(u),\\ldots,P_n(u)$ such that \n\\[\n \\frac{\\wt \\mu_{i-1}(u)}{\\wt \\mu_i(u)}=\\frac{P_i(u+1)}{P_i(u)} \\quad \\text{ with }\\quad P_i(u)=P_i(-u+n-i+2) \\; \\text{ for all }\\; 2\\leq i\\leq n, \n\\]\nand $P_1(u)$ satisfies $P_1(u)=P_1(-u+\\ka+2^\\delta)$ together with the relation\n\\begin{align*}\n \\frac{\\wt \\mu_{1}(\\ka-u)}{\\wt \\mu_{2^{1-\\delta}}(u)}&=\\frac{P_1(u+2^\\delta)}{P_1(u)}\\cdot \\frac{\\ka-u}{u} \\; \\text{ if }\\; \\mfg_N=\\mfg_{2n}, \\\\\n \\frac{\\wt \\mu_{0}(u)}{\\wt \\mu_{1}(u)}&=\\frac{P_1(u+\\tfrac{1}{2})}{P_1(u)} \\; \\text{ if }\\; \\mfg_N=\\mfso_{2n+1}.\n\\end{align*}\nMoreover, when $V(\\mu(u))$ is finite-dimensional the associated tuple $(P_1(u),\\ldots,P_n(u))$ is unique.\n\\end{thrm}\nWe end this subsection by noting that the isomorphism \\eqref{X=Z*Y} together with the definition of $Y(\\mfg_N,\\mcG)^{tw}$ implies the following statement:\n\\begin{prop} \\label{P:Y^tw-fd}\nThe isomorphism classes of finite-dimensional irreducible representations of $Y(\\mfg_N,\\mcG)^{tw}$ can be naturally identified with the isomorphism classes of finite-dimensional irreducible $X(\\mfg_N,\\mcG)^{tw}$-modules in which the central series $w(u)$ acts as the identity operator. \n\\end{prop}\n\n\\subsection{Preliminary properties of polynomials}\\hspace{0mm}\n\n\nIn this subsection we prove two elementary results pertaining to polynomials satisfying certain symmetry relations: see Lemmas \\ref{L:poly1} and \\ref{L:poly2}. Both of these lemmas are generalizations of similar results which have appeared in Chapters 3 and 4 of \\cite{Mobook}. As was the case in {\\it loc.~cit.}, these results play a role in classifying the finite-dimensional irreducible representations of twisted Yangians. \n\\begin{lemma}\\label{L:poly1}\n Let $\\alpha,\\beta\\in \\C\\in \\C$, $l\\in \\Z$ and $m\\in \\mathbb{Q}$. Suppose that $P(u)$ and $Q(u)$ are both monic polynomials such that $P(u)=P(-u+l)$ and $Q(u)=Q(-u+l)$. Suppose\n also that $P(\\alpha)\\neq 0\\neq Q(\\beta)$ and that \n \\begin{equation}\n \\frac{P(u+m)}{P(u)}\\cdot \\frac{\\alpha-u}{\\alpha+u-l+m}=\\frac{Q(u+m)}{Q(u)}\\cdot \\frac{\\beta-u}{\\beta+u-l+m}. \\label{poly1}\n \\end{equation}\nThen $P(u)=Q(u)$ and $\\alpha=\\beta$. \n\\end{lemma}\n \n\\begin{proof}\nThis is a generalization of a result proven as part of the proof of Theorem 4.4.3 of \\cite{Mobook} (see in particular (4.58)). There the statement of the lemma was proven in the special case where $l=m=1$. The same argument works in the general case, and we repeat it here for the sake of the reader. \n \nIf $\\alpha=\\beta$ then $\\frac{P(u+m)}{P(u)}=\\frac{Q(u+m)}{Q(u)}$, which implies that the rational function $f(u)=Q(u)\/P(u)$ is periodic. This is impossible unless\n $f(u)$ is constant, and, since $P(u)$ and $Q(u)$ are monic, this is only possible if $f(u)=1$. Hence $P(u)=Q(u)$. \n \nTherefore it suffices to show that an equality of the form \\eqref{poly1} is impossible unless $\\alpha=\\beta$. We prove this by induction on $k$, where\n $k=\\frac{1}{2}(\\deg\\,P(u)+\\deg\\,Q(u))$. If $k=0$ then this follows from the fact that \\eqref{poly1} collapses to $\\frac{\\alpha-u}{\\alpha+u-l+m}=\\frac{\\beta-u}{\\beta+u-l+m}$.\n Suppose inductively that \\eqref{poly1} is impossible whenever $\\alpha\\neq \\beta$ and $k0}, \\\\ \n \\emptyset & \\text{ otherwise. } \n \\end{cases}\n\\end{equation}\n\n\\end{itemize}\n\n\n\n\\section{Representations of low rank twisted Yangians of type BDI} \\label{sec:lowrank}\n \n \n \n \nIn this section, we use the classification results for finite-dimensional irreducible representations of the Olshanskii twisted Yangian $Y^+(2)$ \\cite{Mo1} together with the isomorphisms from \\cite{GRW1} to classify all finite-dimensional irreducible representations of the twisted Yangians $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}$ and $X(\\mfso_3,\\mfso_2)^{tw}$. The classification obtained for $X(\\mfso_3,\\mfso_2)^{tw}$ will provide a necessary step in proving the main results in Subsection \\ref{subsec:q=1}. Here we also obtain explicit formulas for evaluation morphisms $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw} \\onto \\mfU(\\mfso_2\\op\\mfso_2)$ and $X(\\mfso_3,\\mfso_2)^{tw} \\onto \\mfU\\mfso_2$, and study the corresponding evaluation modules. \n\nIn order to distinguish between the generators of $X(\\mfg_N,\\mfg_N^\\rho)^{tw}$ and those of $Y^+(2)$, we shall follow the convention established of \\cite{GRW1} and denote the \ngenerators of $Y^+(2)$ by $s^{\\circ (r)}_{ij}$, where $i,j\\in \\{\\pm1\\}$ and $r\\geq0$. These generators are then arranged as the coefficients of the various series' $s_{ij}^\\circ(u)$, which \nin turn form the $(i,j)^{th}$ entry of the matrix $S^\\circ(u)$. \nSimilarly, the generators of the special twisted Yangian $SY^+(2)$ are denoted by $\\si^{\\circ(r)}_{ij}$, and the\ncorresponding series and matrix are denoted by $\\si^\\circ_{ij}(u)$ and $\\Si^\\circ(u)$, respectively. The defining relations of $Y^+(2)$ are the reflection equation \n\\[ \nR^{\\circ}(u-v)\\, S_1^{\\circ}(u)\\, R^{\\circ t}(-u-v)\\, S_2^{\\circ}(v) = S_2^{\\circ}(v)\\, R^{\\circ t}(-u-v)\\, S_1^{\\circ}(u)\\, R^{\\circ}(u-v) ,\n\\] \nwhere $R^{\\circ}(u) = 1 - u^{-1} \\sum_{i,j\\in\\mcI_2} E_{ij} \\ot E_{ji}$ and the symmetry relation\n\\eq{\nS^{\\circ t}(-u) = S^\\circ(u) + \\frac{S^\\circ(u)-S^\\circ(-u)}{2u} \\,. \\label{Ols:symm}\n}\nThe special twisted Yangian $SY^+(2)$ is the quotient of $Y^+(2)$ by the ideal generated by the coefficients of $\\mathrm{sdet}S^\\circ(u)-1$, where $ \\mathrm{sdet}S^\\circ(u)$ is the Sklyanin determinant \\cite{Mobook}.\n\nWe now recall the classification results for finite-dimensional irreducible representations of $Y^+(2)$ and $SY^+(2)$.\nA representation $V$ of $Y^+(2)$ is called a \\textit{highest weight representation} if there exists a nonzero vector $\\xi\\in V$ such that $V=Y^+(2)\\,\\xi$, $s_{-1,1}^{\\circ}(u)\\,\\xi=0$ and $s^\\circ_{11}(u)\\,\\xi=\\mu^\\circ(u)\\,\\xi$ for some formal series $\\mu^\\circ(u)\\in 1+u^{-1}\\C[[u^{-1}]]$. As usual, we call $\\mu^\\circ(u)$ the highest weight of $V$, and the vector $\\xi$ the highest weight vector. These same definitions can be given in the $SY^+(2)$ setting after replacing $s_{ab}^\\circ(u)$ with $\\sigma^\\circ_{ab}(u)$ for each $a,b\\in \\{-1,1\\}$. \n\nGiven $\\mu^\\circ(u)\\in 1+u^{-1}\\C[[u^{-1}]]$, the Verma module $M(\\mu^\\circ(u))$ for $Y^+(2)$ is defined the same way as for $X(\\mfg_N)$ and $X(\\mfg_N,\\mcG)^{tw}$, and is always non-trivial.\nIt admits a unique irreducible quotient $V(\\mu^\\circ(u))$, and any irreducible highest weight module with the highest weight $\\mu^\\circ(u)$ is isomorphic to $V(\\mu^\\circ(u))$. We will also employ the notation $M(\\mu^\\circ(u))$ and $V(\\mu^\\circ(u))$ for the $SY^+(2)$ Verma module of highest weight $\\mu^\\circ(u)$ and its irreducible quotient, respectively. In the $SY^+(2)$ case, the Verma module $M(\\mu^\\circ(u))$ is non-trivial provided that $\\mu^\\circ(-u)\\mu^\\circ(u-1)=1$. The distinction between $SY^+(2)$ and $Y^+(2)$ will always be clear from context. \n\nThe following classification result is a restatement of Theorems 4.4 and 5.4 of \\cite{Mo1} (see also Theorems 4.3.3 and 4.4.3 of \\cite{Mobook}): the irreducible $Y^+(2)$-module $V(\\mu^\\circ(u))$ is finite-dimensional if and only if there exists a scalar $\\ga^\\circ\\in \\C$ together with a monic polynomial $P^\\circ(u)$ such that $P^\\circ(u)=P^\\circ(-u+1)$, $P^\\circ(\\ga^\\circ)\\neq 0$, and \n\\eq{\n\\frac{\\mu^\\circ(-u)}{\\mu^\\circ(u)}= \\frac{2u+1}{2u-1} \\cdot \\frac{P^\\circ(u+1)}{P^\\circ(u)}\\cdot \\frac{u-\\ga^\\circ}{u+\\ga^\\circ} \\,. \\label{Y+2:findim}\n}\nIn this case, the pair $(P^\\circ(u),\\ga^\\circ)$ is unique. The same classification result holds if $Y^+(2)$ is replaced with $SY^+(2)$, and in this case it establishes a bijective correspondence between finite-dimensional irreducible representations and pairs of the form $(P^\\circ(u),\\ga^\\circ)$: see \\cite[Corollary 4.4.5]{Mobook}.\n \n \n\n\\subsection{Twisted Yangians for the symmetric pair \\texorpdfstring{$(\\mfso_{4},\\mfso_{2}\\op\\mfso_2)$}{}} \\label{sec:so4so2so2}\n \n \nThe isomorphism between the twisted Yangian $X(\\mfso_4,\\mfso_2\\oplus\\mfso_2)^{tw}$ and the tensor product of Olshanskii twisted Yangians $SY^+(2)\\ot Y^+(2)$ was established in Proposition 4.16 and Corollary 4.17 of \\cite{GRW1}. Let $K=E_{11}-E_{-1,-1} \\in \\End(\\C^2)$ and let $\\Sigma^\\circ(u)$ denote the $S$-matrix of $SY^+(2)$ and $S^\\bullet(u)$ that of $Y^+(2)$. Set $V=\\C^2\\otimes \\C^2$ with ordered basis given by $v_{-2}=e_{-1}\\otimes e_{-1}$, $v_{-1}=e_{-1}\\otimes e_{1}$, $v_1=e_1\\otimes e_{-1}$ and $v_2=-e_1\\otimes e_1$. By identifying $V$ with $\\C^4$ equipped with basis $\\{v_{-2},v_{-1},v_1,v_2\\}$, we can consider $S(u)$ as an element of $\\End\\,V\\otimes X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}[[u^{-1}]]$. Then the map\n\\eq{\n\\chi \\;:\\; S(u) \\mapsto -\\Sigma_1^\\circ(u-1\/2) K_1 S^\\bullet_2(u-1\/2) K_2 \\label{iso:so4}\n}\ndefines an isomorphism $X(\\mfso_4,\\mfso_2\\oplus\\mfso_2)^{tw} \\cong SY^+(2)\\ot Y^+(2)$: see Section E in \\textit{loc.~cit.} for the precise meaning of the right-hand side of \\eqref{iso:so4}. The sign difference between \\eqref{iso:so4} and (4.59) of \\cite{GRW1} is due to the fact that the matrix $\\mcG$ that we use equals the matrix $-\\mcG'$ used in \\textit{loc.~cit.} We will use this result to obtain a complete description of the finite-dimensional irreducible representations of $X(\\mfso_4,\\mfso_2\\oplus\\mfso_2)^{tw}$ using those of $SY^+(2)$ and $Y^+(2)$ as recalled above.\n\n\n\n\n\\begin{prop}\\label{P:so4class}\nLet the components of $\\mu(u) = (\\mu_1(u),\\mu_2(u))$ satisfy the conditions of Proposition \\ref{P:nontriv} so that the irreducible $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}$-module $V(\\mu(u))$ exists. Then $V(\\mu(u))$ is finite-dimensional if and only if there exists \na tuple $(Q(u),P(u),\\alpha,\\beta)$, where $\\alpha,\\beta\\in \\C$ and $P(u)$, $Q(u)$ are monic polynomials in $u$, such that $P(u)=P(-u+2)$, $Q(u)=Q(-u+2)$, $P(\\alpha)\\neq 0$, $Q(\\beta)\\neq 0$, and \n\\[\n \\frac{\\widetilde{\\mu}_1(u)}{\\widetilde{\\mu}_2(u)}=\\frac{P(u+1)}{P(u)}\\cdot \\frac{\\alpha-u}{\\alpha+u-1} , \\qq\n \\frac{\\widetilde{\\mu}_1(1-u)}{\\widetilde{\\mu}_2(u)}=\\frac{u}{1-u}\\cdot\\frac{Q(u+1)}{Q(u)}\\cdot \\frac{\\beta-u}{\\beta+u-1} .\n\\]\nMoreover, when they exist, the pair $(Q(u), P (u))$ and the scalars $\\al,\\beta$ are uniquely determined.\n\\end{prop}\n\n\\begin{proof} The proof of this proposition is very similar to that of Proposition 5.4 of \\cite{GRW2}.\nWe begin by showing that the $X(\\mfso_4,\\mfso_2\\oplus\\mfso_2)^{tw}$-module $V(\\mu(u))$, viewed as a $SY^+(2)\\ot Y^+(2)$-module via the isomorphism $\\chi$, is isomorphic to $ V(\\mu^\\circ(u))\\ot V(\\mu^\\bullet(u))$, where the pair $(\\mu^\\circ(u),\\mu^\\bullet(u))$ is completely determined by the relations \n\\begin{equation}\n\\wt{\\mu}_1(u)=(2u-2)\\cdot\\mu^\\circ(\\wt u)\\,\\mu^\\bullet(-\\wt u), \\qq \n\\wt{\\mu}_2(u)=-2u\\cdot\\mu^\\circ(\\wt u)\\,\\mu^\\bullet(\\wt u) \\quad \\text{ with } \\quad \\wt u=u-1\/2\\label{so4:class:4}.\n\\end{equation}\n\nWriting the map \\eqref{iso:so4} explicitly we have that \n\\eq{\n\\chi \\; : \\; \\begin{cases}\\; \n\\begin{aligned}\ns_{11}(u) &\\mapsto \\si^\\circ_{11}(\\wt u)\\,s^\\bullet_{-1,-1}(\\wt u) , \\qu & s_{-1,2}(u) &\\mapsto \\si^\\circ_{-1,1}(\\wt u)\\,s^\\bullet_{11}(\\wt u) , \\\\\ns_{22}(u) &\\mapsto -\\si^\\circ_{11}(\\wt u)\\,s^\\bullet_{11}(\\wt u) , & s_{1,-2}(u) & \\mapsto -\\si^\\circ_{1,-1}(\\wt u)\\,s^\\bullet_{-1,-1}(\\wt u),\n\\end{aligned}\n\\end{cases}\n\\label{so4:class:1}\n}\n\nMoreover, the computation at the beginning of the proof of \\cite[Corollary 4.17]{GRW1} shows that \n\\[\n\\chi \\; : \\; s_{1,-2}(-\\wt{u})\\,s_{-1,2}(\\wt{u}) - s_{11}(-\\wt{u})\\,s_{22}(\\wt{u}) \\mapsto s^\\bullet_{-1,-1}(-u)\\,s^\\bullet_{11}(u-1) .\n\\]\nLetting $\\xi\\in V(\\mu(u))$ denote the highest weight vector, this in turn implies that \n\\[\ns^\\bullet_{-1,-1}(-u)\\, s^\\bullet_{11}(u-1)\\,\\xi=-\\mu_1(-\\wt{u})\\,\\mu_2(\\wt{u})\\,\\xi.\n\\] \nUsing the symmetry relation \\eqref{Ols:symm} of $Y^+(2)$, we can rewrite the equality above as \n\\[\n \\left(s^\\bullet_{11}(u)+\\frac{s^\\bullet_{11}(u)-s^\\bullet_{11}(-u)}{2u} \\right)s^\\bullet_{11}(u-1)\\,\\xi=-\\mu_1(-\\wt{u})\\,\\mu_2(\\wt{u})\\,\\xi.\n\\]\nBy induction on the coefficients $s^{\\bullet(r)}_{11}$ of $s^\\bullet_{11}(u)$, this implies that there exists $\\mu^\\bullet(u)\\in 1+u^{-1}\\C[[u^{-1}]]$ satisfying \n\\eq{\ns^\\bullet_{11}(u)\\,\\xi=\\mu^\\bullet(u)\\,\\xi. \\label{so4:class:3}\n}\nMoreover, $\\mu^\\bullet(u)$ is uniquely determined by the relation \n\\eq{\n \\left(\\mu^\\bullet(u)+\\frac{\\mu^\\bullet(u)-\\mu^\\bullet(-u)}{2u} \\right)\\mu^\\bullet(u-1) = - \\mu_1(-\\wt{u})\\,\\mu_2(\\wt{u}) . \\label{so4:class:2}\n}\nCombining \\eqref{so4:class:3} with \\eqref{so4:class:1} implies that $\\xi$ is also an eigenvector for the action of $\\si^\\circ_{11}(u)$ with weight $\\mu^\\circ(u)$ defined by $\\mu_2(u) = -\\mu^\\circ(\\wt u)\\,\\mu^\\bullet(\\wt u)$ or equivalently by\n\\[\n\\mu_1(u) = \\mu^\\circ(\\wt u)\\,\\Big(\\mu^\\bullet(-\\wt u)+\\frac{\\mu^\\bullet(\\wt u) - \\mu^\\bullet(-\\wt u)}{2\\wt u}\\Big),\n\\]\nwhere we have used that $\\mu^\\circ(-u)^{-1}=\\mu^\\circ(u-1)$: see the proof of \\cite[Proposition 5.4]{GRW2}.\nUsing the notation introduced in \\eqref{tilde-mu(u)} the last two equalities can be rewritten in the equivalent form \\eqref{so4:class:4}. Finally, since $[\\si^\\circ_{ij}(u),s^\\bullet_{kl}(v)]=0$, it follows immediately from the definition of $\\xi$ and the two formulas $\\chi(s_{-1,2}(u))=\\si^\\circ_{-1,1}(\\wt u)\\,s^\\bullet_{11}(\\wt u)$ and $\\chi(s_{12}(u))=\\sigma_{11}^\\circ(\\wt u)s_{-1,1}^\\bullet(\\wt u)$ that $\\si^\\circ_{-1,1}(u)\\,\\xi=s^\\bullet_{-1,1}(u)\\,\\xi=0$. Thus, by the irreducibility of $V(\\mu(u))$ we can conclude that\n\\[\nV(\\mu(u))\\cong V(\\mu^\\circ(u))\\ot V(\\mu^\\bullet(u)). \n\\]\n\nWe can now use the isomorphism above to determine exactly when $V(\\mu(u))$ is finite-dimensional. As recalled above \\eqref{Y+2:findim}, the module $V(\\mu^\\circ(u))\\ot V(\\mu^\\bullet(u))$ is finite-dimensional if and only if there exists a tuple ($P^\\circ(u),P^\\bullet(u),\\ga^\\circ,\\ga^\\bullet)$, where $\\ga^\\circ,\\ga^\\bullet\\in \\C$ and $P^\\circ(u),Q^\\bullet(u)$ are monic polynomials in $u$ such that $P^\\circ(u)=P^\\circ(-u+1)$, $P^\\bullet(u)=P^\\bullet(-u+1)$, $P^\\circ(\\ga_1)\\ne0$, $P^\\bullet(\\ga_2)\\ne0$, and the following equations hold: \n\\begin{equation}\n \\frac{\\mu^\\circ(-u)}{\\mu^\\circ(u)}=\\frac{2u+1}{2u-1}\\cdot\\frac{P^\\circ(u+1)}{P^\\circ(u)}\\cdot\\frac{u-\\ga^\\circ}{u+\\ga^\\circ} \n %\n\\qu\\text{and}\\qu \n\\frac{\\mu^\\bullet(-u)}{\\mu^\\bullet(u)}=\\frac{2u+1}{2u-1}\\cdot\\frac{P^\\bullet(u+1)}{P^\\bullet(u)}\\cdot \\frac{u-\\ga^\\bullet}{u+\\ga^\\bullet} . \\label{LRR:DI(a).6}\n\\end{equation}\nSet $P(u)=P^\\bullet(\\wt u)$, $Q(u)=P^\\circ(\\wt u)$, $\\al=\\ga^\\bullet+\\tfrac{1}{2}$ and $\\beta=\\gamma^\\circ+\\tfrac{1}{2}$. Substituting $u\\mapsto \\wt u$, the above relations become\n\\[\n \\frac{\\mu^\\circ(-\\wt u)}{\\mu^\\circ(\\wt u)}=\\frac{2u}{2u-2}\\cdot \\frac{P(u+1)}{P(u)}\\cdot \\frac{u-\\alpha}{u+\\alpha-1}\n\\qu\\text{ and }\\qu\n\\frac{\\mu^\\bullet(-\\wt u)}{\\mu^\\bullet(\\wt u)}=\\frac{2u}{2u-2}\\cdot \\frac{Q(u+1)}{Q(u)}\\cdot \\frac{u-\\beta}{u+\\beta-1},\n\\]\nrespectively. By relations \\eqref{so4:class:4}, \n\\begin{equation*}\n \\frac{\\wt \\mu_1(u)}{\\wt \\mu_2(u)}=\\frac{2-2u}{2u}\\cdot \\frac{\\mu^\\bullet(-\\wt u)}{\\mu^\\bullet(\\wt u)}\n\\qu\\text{ and }\\qu\n\\frac{\\wt \\mu_1(1-u)}{\\wt \\mu_2(u)}=\\frac{\\mu^\\circ(-\\wt u)}{\\mu^\\circ(\\wt u)}.\n\\end{equation*}\nTherefore, \\eqref{LRR:DI(a).6} is equivalent to \n\\begin{equation*}\n \\frac{\\wt \\mu_1(u)}{\\wt \\mu_2(u)}= \\frac{P(u+1)}{P(u)}\\cdot \\frac{\\alpha-u}{u+\\alpha-1}\\quad \\text{ and } \\quad \\frac{\\wt \\mu_1(1-u)}{\\wt \\mu_2(u)}=\\frac{u}{1-u}\\cdot \\frac{Q(u+1)}{Q(u)}\\cdot \\frac{\\beta-u}{u+\\beta-1}.\n\\end{equation*}\nMoreover, $P(u)=P^\\bullet(u-1\/2)=P^\\bullet(-u+1\/2+1)=P(-u+2)$, and the same is true for $Q(u)$. Finally, $P(\\alpha)=P^\\bullet(\\alpha-1\/2)=P^\\bullet(\\ga^\\bullet)\\neq 0$. Similarly, $Q(\\beta)\\neq 0$.\n\\end{proof}\n\n\n \n\nWe now construct the evaluation morphism $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}\\onto \\mfU(\\mfso_2\\op\\mfso_2)$. The fixed point subalgebra $\\mfU(\\mfso_2\\op\\mfso_2) \\subset \\mfU\\mfso_4$ is generated by the elements $F_{11}$ and $F_{22}$. We recall that $\\mcG = {\\rm diag}(-1,1,1,-1)$ in this case.\n\n\n\\begin{prop} \\label{P:so4:evhom} \nThe assignment \n\\begin{equation}\ns_{ij}(u)\\mapsto g_{ij}+2g_{ij}F_{ij}u^{-1}+\\delta_{ij}(F_{11}^2-F_{22}^2)u^{-2} \\label{so4:evhom}\n\\end{equation}\ndefines a surjective algebra homomorphism $\\mathrm{ev}:X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}\\onto \\mfU(\\mfso_2\\op\\mfso_2)$. \n\\end{prop}\n\n\\begin{proof}\nThe Lie algebra $\\mfso_2$ is one-dimensional, that is $\\mfso_2 \\cong \\C F^\\circ_{11}$, since $F^\\circ_{1,-1}=F^\\circ_{-1,1}=0$ and $F^\\circ_{-1,-1}=-F^\\circ_{11}$. Recall the evaluation homomorphism given by Proposition 3.11 of \\cite{MNO}:\n\\eq{\n{\\rm ev}^\\circ : Y^+(2) \\onto \\mfU\\mfso_2 , \\qu s^\\circ_{ij}(u) \\mapsto \\del_{ij} + (u+ 1\/2)^{-1} F^\\circ_{ij} . \\label{Y+2:evhom}\n}\n\nLet $\\Phi$ be the isomorphism $\\mfso_2\\op\\mfso_2 ( = \\C F^\\circ_{11} \\op \\C F^\\bullet_{11} ) \\iso \\mfso_4^\\rho ( = \\C F_{11} \\op \\C F_{22} )$ given by $F^\\circ_{11} \\mapsto F_{11}+F_{22}$ and $F^\\bullet_{11} \\mapsto F_{22}-F_{11}$. The map $\\Phi$ induces an isomorphism $\\wh\\Phi : \\mfU\\mfso_2\\ot\\mfU\\mfso_2\\iso \\mfU\\mfso_4^\\rho = \\mfU(\\mfso_2\\op\\mfso_2)$, and so the composition $\\wh\\Phi({\\rm ev}^\\circ \\ot {\\rm ev}^\\bullet)$ yields a surjective homomorphism $Y^+(2)\\ot Y^+(2) \\onto \\mfU(\\mfso_2\\op\\mfso_2)$ (here ${\\rm ev}^\\bullet$ denotes the evaluation homomorphism \\eqref{Y+2:evhom} for the second copy of $Y^+(2)$ in the tensor product).\n\n\nFinally, by composing the resulting map with the embedding $\\wt\\chi : X(\\mfso_4,\\mfso_2\\op\\mfso_2) \\into Y^+(2) \\ot Y^+(2)$ given by $S(u) \\mapsto - S_1^\\circ(\\wt u)K_1 S^\\bullet_2(\\wt u) K_2$ (see Proposition 4.16 of \\cite{GRW1}) we obtain precisely \\eqref{so4:evhom}.\n\\end{proof}\n\n\nThe morphism $\\rm ev$ allows us to extend $\\mfso_2\\oplus\\mfso_2$-modules to $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}$-modules. As usual, modules obtained this way are called evaluation modules. Let $V(\\mu_1,\\mu_2)$ denote the irreducible $\\mfso^\\rho_4$-module with the highest weight $(\\mu_1,\\mu_2)$; this is the one-dimensional representation of $\\mfso_4^\\rho$ in which $F_{ii}$ acts as multiplication by $\\mu_i\\in \\C$. The corollary below follows directly from the formula \\eqref{so4:evhom}.\n\n\\begin{crl}\\label{C:so4}\nThe evaluation module $V(\\mu_1,\\mu_2)$ with $\\mu_1,\\mu_2 \\in \\C$ is isomorphic to the $X(\\mfso_4,\\mfso_2\\op\\mfso_2)^{tw}$-module $V(\\mu(u))$ with \n\\[\n\\mu_i(u)=g_{ii}+2g_{ii}\\mu_i u^{-1}+(\\mu_1^2-\\mu_2^2)u^{-2} \\quad \\text{ for } \\quad 1\\leq i\\leq 2.\n\\]\n\\end{crl}\n\nThe collection $\\{V(\\mu_1,\\mu_2)\\}_{\\mu_1,\\mu_2\\in \\C}$ provides a two-parameter family of one-dimensional representations of $X(\\mfso_4,\\mfso_2\\oplus \\mfso_2)^{tw}$. Note that \nthe trivial representation $V(\\mcG)$ may be recovered in the special case where $(\\mu_1,\\mu_2)=(0,0)$. In Remark \\ref{R:1dim}, it will be explained that these are essentially all of the one-dimensional representations of $X(\\mfso_4,\\mfso_2\\oplus \\mfso_2)^{tw}$.\n\n \n \n \n \n\n\\subsection{Twisted Yangians for the symmetric pair \\texorpdfstring{$(\\mfso_{3},\\mfso_{2})$}{}} \\label{sec:so3so2}\n\n\nThe isomorphism between the twisted Yangian $X(\\mfso_3,\\mfso_{2})^{tw}$ and the Olshanskii twisted Yangian $Y^+(2)$ was established in Proposition 4.3 of \\cite{GRW1}. Here we recall the necessary details of this isomorphism, which will allows us to obtain a complete description of the finite-dimensional irreducible representations of $X(\\mfso_3,\\mfso_2)^{tw}$ using those of~$Y^+(2)$.\n\n \nLet the standard basis of $\\C^2$ be given by vectors $e_{-1}$ and $e_1$ and let $V$ be the three--dimensional subspace of $\\C^2\\ot\\C^2$ spanned by vectors $v_{-1}=e_{-1}\\ot e_{-1}$, $v_{0}=\\tfrac{1}{\\sqrt{2}} (e_{-1}\\ot e_1+e_1\\ot e_{-1})$, $v_{1}=-e_1\\ot e_1$. Upon identifying $V$ with $\\C^3$ we may view the matrix $S(u)$ as an element of $\\End V \\ot X(\\mfso_3,\\mfso_{2})^{tw}[[u^{-1}]]$. Moreover, the operator $\\tfrac{1}{2}R^\\circ(-1)=\\tfrac{1}{2}(I+P) \\in \\End(\\C^2\\ot\\C^2)$ is a projector of $\\C^2\\ot\\C^2$ onto the subspace $V$ and the mapping\n\\eqa{\n& \\varphi \\;:\\; X(\\mfso_3,\\mfso_{2})^{tw} \\to Y^{+}(2) , \\qu \nS(u) \\mapsto \\tfrac{1}{2} R^\\circ_{12}(-1)\\, S^\\circ_1(2u-1)\\,R^\\circ_{12}(-4u+1)^{t}\\, S^\\circ_2(2u) K_1 \\, K_2. \\label{iso:so3so2} \n}\nwhere $K=E_{11}-E_{-1,-1}$, is an isomorphism of algebras whose restriction to the subalgebra $Y(\\mfso_3,\\mfso_{2})^{tw}$ induces a isomorphism between $Y(\\mfso_3,\\mfso_{2})^{tw}$ and $SY^\\pm(2)$. \nDenoting $u-1\/2$ by $\\wt{u}$ the map \\eqref{iso:so3so2} explicitly reads as\n\\spl{ \\label{map:so3so2}\ns_{-1,-1}(u) &\\mapsto s^\\circ_{-1,-1}(2\\wt{u})\\, s^\\circ_{-1,-1}(2 u) + \\tfrac{1}{4 u-1}\\, s^\\circ_{-1,1}(2\\wt{u})\\, s^\\circ_{1,-1}(2 u),\n\\\\\ns_{-1,0}(u) &\\mapsto - \\tfrac{1}{\\sqrt{2}}\\, s^\\circ_{-1,-1}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u) - \\tfrac{1}{\\sqrt{2} (4 u-1)} \\big( s^\\circ_{-1,1}(2\\wt{u}) s^\\circ_{11}(2 u) + 4u s^\\circ_{-1,1}(2\\wt{u})\\, s^\\circ_{-1,-1}(2 u) \\big) ,\n\\\\\ns_{-1,1}(u) &\\mapsto \\tfrac{4 u }{1-4 u}\\, s^\\circ_{-1,1}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u),\n\\\\\ns_{0,-1}(u) &\\mapsto \\tfrac{1}{\\sqrt{2}}\\, s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{-1,-1}(2 u) + \\tfrac{1}{\\sqrt{2}(4 u-1)} \\big( s^\\circ_{11}(2\\wt{u})\\,s^\\circ_{1,-1}(2 u) + 4u\\,s^\\circ_{-1,-1}(2\\wt{u})\\, s^\\circ_{1,-1}(2 u) \\big),\n\\\\\ns_{00}(u) &\\mapsto -\\tfrac{1}{2} \\big(s^\\circ_{-1,1}(2\\wt{u})\\, s^\\circ_{1,-1}(2 u) + s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u) \\big) \n\\\\\n& \\qu\\; -\\tfrac{1}{8 u-2} \\big( (s^\\circ_{-1,-1}(2\\wt{u}) + 4u\\, s^\\circ_{11}(2\\wt{u}))\\, s^\\circ_{-1,-1}(2 u) + (4u\\, s^\\circ_{-1,-1}(2\\wt{u}) + s^\\circ_{11}(2\\wt{u}))\\, s^\\circ_{11}(2 u)\\big) ,\n\\\\\ns_{01}(u) &\\mapsto -\\tfrac{1}{\\sqrt{2} (4 u-1)} s^\\circ_{-1,-1}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u) -\\tfrac{1}{\\sqrt{2}} s^\\circ_{-1,1}(2\\wt{u})\\, s^\\circ_{11}(2 u) -\\tfrac{4u}{\\sqrt{2} (4 u-1)} s^\\circ_{11}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u) ,\n\\\\\ns_{1,-1}(u) &\\mapsto \\tfrac{4 u}{1-4 u}\\, s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{1,-1}(2 u),\n\\\\\ns_{10}(u) &\\mapsto \\tfrac{1}{\\sqrt{2}}\\, s^\\circ_{11}(2\\wt{u})\\, s^\\circ_{1,-1}(2 u) + \\tfrac{1}{\\sqrt{2} (4 u-1)} \\big( 4u\\, s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{11}(2 u) + s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{-1,-1}(2 u)\\big),\n\\\\\ns_{11}(u) &\\mapsto s^\\circ_{11}(2\\wt{u})\\, s^\\circ_{11}(2 u) + \\tfrac{1}{4 u-1}\\, s^\\circ_{1,-1}(2\\wt{u})\\, s^\\circ_{-1,1}(2 u),\n}\nwhich can be deduced using the formulas in the proof of Proposition 5.8 of \\cite{GRW2}.\n\n\\begin{prop}\\label{P:so3class}\nLet $\\mu(u) = (\\mu_0(u),\\mu_1(u))$ satisfy the conditions of Proposition \\ref{P:nontriv} so that the irreducible $X(\\mfso_3,\\mfso_2)^{tw}$-module $V(\\mu(u))$ exists. Then $V(\\mu(u))$ is finite-dimensional if and only if there exists a monic polynomial $P(u)$ in $u$ and a scalar $\\al \\in \\C$ such that $P(\\al)\\ne 0$ and $P(u)=P(-u+3\/2)$ and\n\\eq{\n\\frac{\\wt\\mu_0(u)}{\\wt\\mu_1(u)} = \\frac{P(u+1\/2)}{P(u)} \\cdot \\frac{\\al - u}{\\al + u - 1} . \\label{so3so2:findim}\n}\nMoreover, when they exist, the polynomial $P(u)$ and the scalar $\\al$ are uniquely determined.\n\\end{prop}\n\n\\begin{proof}\n\nThe proof is very similar to that of Proposition 5.8 of \\cite{GRW2}. By Proposition \\ref{P:nontriv} the components $\\wt\\mu_0(u)$ and $\\wt\\mu_1(u)$ are required to satisfy \n\\eq{\n\\wt\\mu_0(1\/2-u) = \\frac{1\/2-u}{u}\\cdot \\frac{4u+1}{4u-3} \\cdot \\wt\\mu_0(u) ,\\qq\n\\wt\\mu_0(u)\\,\\wt\\mu_0(-u+1) = \\wt\\mu_1(u)\\,\\wt\\mu_1(-u+1) . \\label{so3so2:mu}\n}\nWe may choose $\\mu^\\circ(u) \\in 1 + u^{-1}\\C[[u^{-1}]]$ such that\n\\eq{\n\\wt\\mu_0(u) = -2 u\\cdot\\frac{4 u-3}{4u-1} \\cdot\\mu^\\circ(-2\\wt{u})\\,\\mu^\\circ(2 u) , \\qq \\wt\\mu_1(u) = 2u\\cdot\\mu^\\circ(2\\wt{u})\\,\\mu^\\circ(2u) , \\label{so3so2:mu01}\n}\nso that both equalities in \\eqref{so3so2:mu} are satisfied. (See Lemma 5.7 of \\cite{GRW2}.) Recall that $\\wt\\mu_0(u) = (2u-1)\\,\\mu_0(u) + \\mu_1(u)$ and $\\wt\\mu_1(u)=2u\\,\\mu_1(u)$. Hence\n\\eq{\n\\mu_0(u) = -\\frac{1}{2\\wt{u}} \\left( \\frac{2u(4u-3)}{4u-1}\\, \\mu^\\circ(-2\\wt{u}) + \\mu^\\circ(2\\wt{u})\\right) \\mu^\\circ(2u) , \\qq \\mu_1(u) = \\mu^\\circ(2\\wt{u})\\,\\mu^\\circ(2u) . \\label{so3so2:mu1}\n}\n\nLet $V(\\mu^\\circ(u))$ denote the irreducible highest weight $Y^+(2)$ module with the highest weight $\\mu^\\circ(u)$ and let~$\\xi$ denote its highest weight vector. $V(\\mu^\\circ(u))$ may be viewed as a $X(\\mfso_3,\\mfso_2)^{tw}$-module via the isomorphism~$\\varphi$. It is immediate from \\eqref{map:so3so2} that $s_{ij}(u)\\,\\xi=0$ for all $iX}).\n\\begin{lemma}\\label{L:gtw}\nSuppose that $\\mu(u)$ is associated to $(\\alpha,P_1(u),\\ldots,P_n(u))$, and let $\\xi$ be a highest weight vector of $V(\\mu(u))$. Then the $\\mfg_N^\\rho$-module $\\mfU\\mfg_N^\\rho\\xi$ is a highest weight module with highest weight $\\mu=(\\mu_i)_{i=1}^n$ given~by\n\\begin{equation} \\label{gtw-action}\n\\mu_i= -\\tfrac{1}{2}A(\\mathbf{P},u)-\\tfrac{1}{2}\\sum_{a=2}^i\\deg\\, P_a(u) +\\delta_{i>\\key}(\\al-\\tfrac{N}{4}) \\qu \\text{for all}\\qu 1\\leq i\\leq n, \n\\end{equation}\nwhere $A(\\mathbf{P},u)$ is as in \\eqref{Ag_N} with $\\mathbf{P}=(P_1(u),\\ldots,P_n(u))$ and $i\\mapsto \\delta_{i>\\key}$ is the indicator function of the set $\\{\\key+1,\\ldots,n\\}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\mu_\\al(u)=(\\mu_{\\al,i}(u))_{i\\in \\mathcal{I}_N^+}$ be the $X(\\mfg_N,\\mcG)^{tw}$-highest weight determined by\n \\begin{equation}\n \\wt \\mu_{\\al,i}(u)= 2u \\cdot g(u) \\; \\text{ for all }\\; 0\\leq i\\leq \\key , \\quad \\wt \\mu_{\\al,i}(u)=2u\\cdot g(u)\\left(\\frac{\\ley-\\al-u}{u-\\al} \\right) \\; \\text{ for }\\; \\key+1\\leq i\\leq n, \\label{mu-al}\n \\end{equation}\nwhere the index $i=0$ is omitted when $N=2n$. Since $P_i(u)=P_i(-u+n-i+2)$ for all $i\\geq 2$, there exists monic polynomials $Q_2(u),\\ldots,Q_n(u)$ such that \n\\begin{equation*}\n P_i(u)=(-1)^{\\deg\\, Q_i(u)}Q_i(u-\\ka\/2)Q_i(-u+n-i+2-\\ka\/2),\n\\end{equation*}\nand similarly since $P_1(u)=P_1(-u+\\ka+2^\\delta)$, there is a monic polynomial $Q_1(u)$ such that $P_1(u)=(-1)^{\\deg\\, Q_1(u)}Q_1(u-\\ka\/2)Q_1(-u+\\ka\/2+2^\\delta)$. Let \n$L(\\lambda(u))$ be a finite-dimensional irreducible $X(\\mfg_N)$-module with Drinfeld polynomials $Q_1(u),\\ldots,Q_n(u)$, and suppose $\\xi\\in L(\\lambda(u))$ and $\\eta \\in V(\\mu_\\al(u))$ are highest weight vectors. Then, as a consequence of Lemmas \\ref{L:unique} and \\ref{L:QxP=QP}, there is an even series $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$ such that $V(\\mu(u))^{\\nu_g}$ is isomorphic to the irreducible quotient of \n$X(\\mfg_N,\\mcG)^{tw}(\\xi\\otimes \\eta)$. As $\\mfg^\\rho_N$ acts identically in $V(\\mu(u))^{\\nu_g}$ and $V(\\mu(u))$, we can assume without loss of generality that $g(u)=1$.\n\nSince $\\tfrac{1}{2u}\\wt \\mu_{\\al,i}(u)$ has $u^{-1}$ coefficient $\\tfrac{1}{2}(2\\cdot \\mu_{\\al,i}^{(1)}-\\ley g_{ii}+[\\mp]\\ley)$, $g(u)=[\\pm]1+[\\mp]\\tfrac{q}{2}u^{-1}+O(u^{-2})$, and $\\mfrac{\\ell-\\al-u}{u-\\al}=-1+(\\ley-2\\al)u^{-1}+O(u^{-2})$, we obtain from \\eqref{mu-al} the relations\n\\begin{equation}\n \\mu_{\\al,i}^{(1)}=[\\mp](\\tfrac{q}{2}-\\ell) \\; \\text{ for }\\; 0\\leq i\\leq \\key, \\quad \\mu_{\\al,i}^{(1)}=[\\pm](\\tfrac{q}{2}+\\ley-2\\al) \\; \\text{ for }\\; i\\geq \\key+1. \\label{mu-al:2}\n\\end{equation}\nSince $F_{ii}^{ \\rho}=2g_{ii}F_{ii}$ and the embedding $\\mfU\\mfg_N^\\rho\\into X(\\mfg_N,\\mcG)^{tw}$ sends $F_{ii}^{ \\rho}$ to $s_{ii}^{(1)}-(g_{ii}-1)\\tfrac{p-q}{4}$ (see \\eqref{grho->X}), we obtain \n\\begin{equation*}\n 2g_{ii}\\mu_{\\al,i}+(g_{ii}-1)\\tfrac{p-q}{4}=\\mu_{\\al,i}^{(1)} \\; \\text{ for all }\\; i\\in\\mathcal{I}_N^+,\n\\end{equation*}\nwhere $\\mu_{\\al,i}$ denotes the $F_{ii}$-weight of $\\eta$. Combining this with \\eqref{mu-al:2} yields \n\\begin{equation*}\n \\mu_{\\al,i}=0 \\; \\text{ for }\\; 0\\leq i\\leq \\key,\\quad \\mu_{\\al,i}=\\al-\\tfrac{N}{4}\\;\\text{ for }\\; \\key+1\\leq i\\leq n.\n\\end{equation*}\nSince $F_{ii}(\\xi\\otimes \\eta)=F_{ii}\\xi\\otimes \\eta +\\mu_{\\al,i}(\\xi\\otimes \\eta)$, \\eqref{gtw-action} now follows immediately from the formulas\n\\eqref{g-action} of Corollary \\ref{C:g-action} and the fact that $\\deg\\, P_i(u)=2\\deg\\, Q_i(u)$ for each $i$.\n\\end{proof}\n\n\n\\begin{rmk}\n Lemma \\ref{L:gtw} (and its proof) also applies for the twisted Yangians of type BCD0. In this case, $\\alpha$ should be removed from the tuple \n $(\\alpha,P_1(u),\\ldots,P_n(u))$. Note that, since $\\key=n$, the term $\\delta_{i>\\key}(\\al-\\tfrac{N}{4})$ does not actually make an appearance in \\eqref{gtw-action}, and that the definition of $\\mu_\\al(u)$ given in \\eqref{mu-al} reduces to $(g_{ii}(u))_{i\\in \\mcI_{N}^+}$, and hence $V(\\mu_\\al(u))\\cong V(\\mcG)$.\n\\end{rmk}\n\n\n\\begin{prop}\\label{P:int}\nSuppose that $N\\geq 4$ if $\\mfg_N=\\mfsp_N$ and $N\\geq 5$ with $q\\neq 2$ if $\\mfg_N=\\mfso_N$. Then $2^{1-\\delta}(\\alpha-\\tfrac{N}{4})$ is an integer satisfying\n\\begin{equation*}\n 2^{1-\\delta}(\\alpha-\\tfrac{N}{4})\\leq A(\\mathbf{P},u)+\\sum_{a=2}^{\\key+1}\\deg\\, P_a(u)+(1-2^{\\delta-1})\\deg\\, P_{\\key+2}(u),\n\\end{equation*}\nwhere we recall that $\\delta=1$ if $\\mfg_N=\\mfsp_{2n}$ and $\\delta=0$ if $\\mfg_N=\\mfso_N$. \n\\end{prop}\n\\begin{proof}\nSince $V(\\mu(u))$ is finite-dimensional, so is the $\\mfg_{2\\ell}$ highest weight module $\\mfU\\mfg_{2\\ell}\\xi\\subset \\mfU\\mfg_N^\\rho \\xi$. The highest weight \nof this module is $(\\mu_{\\key+1},\\ldots,\\mu_n)$ with each $\\mu_i$ as in the relation \\eqref{gtw-action} of Lemma \\ref{L:gtw}. \n\nIf $\\mfg_N=\\mfsp_{2n}$, then $\\mfg_{2\\ell}=\\mfsp_{2\\ell}$ and \\eqref{g-class} implies that $-\\mu_{\\key+1}\\in \\Z_{\\geq 0}$. Otherwise, $\\mfg_{2\\ell}=\\mfso_{2\\ell}$ and \n\\eqref{g-class} yields $-\\mu_{\\key+1}-\\mu_{\\key+2}\\in \\Z_{\\geq 0}$. Taking into account that $\\mu_{\\key+1}$ and $\\mu_{\\key+2}$ are given by \\eqref{gtw-action} we obtain the statement of the proposition.\n\\end{proof}\n\n\n\\section{Classification of one-dimensional representations} \\label{sec:1dim} \n\n\nIn this section we classify the one-dimensional representations of $X(\\mfg_N,\\mfg_p\\oplus\\mfg_q)^{tw}$ and $Y(\\mfg_N,\\mfg_p\\oplus\\mfg_q)^{tw}$ when $(\\mfg_N,\\mfg_p\\oplus\\mfg_q)$ is a symmetric pair of type BDI or CII: see Propositions \\ref{P:no1dim} and \\ref{P:1dimso2}. In fact, Proposition \\ref{P:no1dim} also holds for the twisted Yangians of type BCD0. These results prove that $X(\\mfg_N,\\mfg_p\\oplus\\mfg_q)^{tw}$ (and thus $Y(\\mfg_N,\\mfg_p\\oplus\\mfg_q)^{tw}$) admits non-trivial one-dimensional representations if and only if $\\mfg_p\\oplus\\mfg_q$ does (which occurs precisely when $\\mfg_p$ or $\\mfg_q$ is isomorphic to the one-dimensional Lie algebra $\\mfso_2$). \n\nThe main difficulty in proving Proposition \\ref{P:1dimso2} is the construction of a one-parameter family $\\{V(a)\\}_{a\\in \\C}$ of one-dimensional representations \nfor $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ when $N\\geq 5$: this is proven in Lemma \\ref{L:K-1dim}, and stated explicitly in Corollary \\ref{C:V(a)}. This one-parameter family of representations will play a crucial role in the proof of the classification of finite-dimensional irreducible representations of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ given in Theorem \\ref{T:DI(a)-Class} of the next section. \n\n\n\\subsection{Twisted Yangians for the symmetric pairs \\texorpdfstring{$(\\mfg_N,\\mfg_p\\op\\mfg_q)$}{} when \\texorpdfstring{$\\mfg_p,\\mfg_q\\ncong \\mfso_2$}{}}\\label{subsec:1dim-gen} \n\n\nWe begin by classifying the one-dimensional representations of $X(\\mfg_N,\\mfg_N^\\rho)^{tw}$ in all cases where $\\mfg_N^\\rho=\\mfg_p\\oplus \\mfg_q$ with $\\mfg_p \\ncong \\mfso_2 \\ncong \\mfg_q$. \nWe also relax the requirement that $q>0$ so as to include the twisted Yangians of type BCD0, which correspond to $q=0$. \n\\begin{prop}\\label{P:no1dim} Assume that the symmetric pair $(\\mfg_N,\\mfg_{p}\\oplus \\mfg_q)$ is of type BCD0, BI, CII, or DI(a), and in addition that $\\mfg_p\\ncong \\mfso_2\\ncong \\mfg_q$. Then, a representation $V$ of $X(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$ is one-dimensional if and only if $V\\cong V(\\mcG)^{\\nu_g}$ for some $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$. \n\\end{prop}\n\\begin{rmk}\n The proof of the Proposition exploits the relationship between $X(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$ and the Molev-Ragoucy reflection algebra $\\mcB(n,\\ley)$ which was studied in Subsection 4.3 of \\cite{GRW2}. For the definition and main properties of $\\mcB(n,\\ley)$, we refer the reader to \\cite{MR} and Subsection 3.6 of \\cite{GRW2}. \n\\end{rmk}\n\\begin{proof}\nSuppose that $V$ is a one-dimensional representation of $X(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$. By Proposition \\ref{P:necessary} (if $q\\neq 0$) and Theorem \\ref{T:BCD0-class} (if $q=0$), $V$ can be associated to a tuple \n$(\\alpha,P_1(u),\\ldots,P_n(u))$, where the scalar $\\alpha$ should be omitted if $q=0$. If $q=1$ or $q=0$, then $\\mfg_p\\oplus \\mfg_q=\\mfg_p$. Otherwise, both $\\mfg_p$ and $\\mfg_q$ are complex semisimple Lie algebras. In either case, $\\mfg_p\\oplus\\mfg_q$ is semisimple and thus admits no nontrivial one-dimensional representations. Consequently, $V$ is isomorphic to the trivial representation of $\\mfg_p\\oplus \\mfg_q$ when viewed as a module of this Lie algebra. Therefore, relation \\eqref{gtw-action} of Lemma \\ref{L:gtw} becomes equivalent to\n\\begin{equation*}\n A(\\mathbf{P},u)+\\sum_{a=2}^i\\deg\\, P_a(u)=\\delta_{i>\\key}(2\\al-\\tfrac{N}{2}) \\quad \\text{ for all }\\quad 1\\leq i\\leq n,\n\\end{equation*}\nfrom which it can be deduced that $P_a(u)=1$ for all $a\\neq \\key+1$, and $\\deg\\, P_{\\key+1}(u)=2\\al-\\tfrac{N}{2}$. In the $q=0$ case, this completes the proof \nas $(P_1(u),\\ldots,P_n(u))$ is equal to $(1,\\ldots,1)$, the Drinfeld tuple corresponding to the trivial representation. \nTo complete the proof in the $q\\neq 0$ case, it suffices to show that $\\deg\\, P_{\\key+1}(u)=0$, as this will imply $(\\alpha,P_1(u),\\ldots,P_n(u))=(\\tfrac{N}{4},1,\\ldots,1)$. Since $V(\\mcG)$ is also associated to this tuple, the desired conclusion will follow from Lemma \\ref{L:unique}. \n\n\\noindent\\textit{Case 1}: $\\key>0$.\n\nSuppose first that $\\key>0$, and set $M=N-2\\ley+2$. Consider the one-dimensional representation $V_{\\ley-1}$ of $X(\\mfg_{M},\\mfg_{M-2}\\oplus \\mfg_2)^{tw}$ furnished by Proposition \\ref{P:ind}. By Corollary \\ref{C:poly-low}, this module is associated to $(\\gamma,Q_1(u),\\ldots,Q_{\\key+1}(u))$, where $\\gamma=\\al-\\frac{\\ley-1}{2}$, \n$Q_a(u)=1$ for $1\\leq a\\leq \\key$, and $Q_{\\key+1}(u)=P_{\\key+1}(u+\\tfrac{\\ley-1}{2})$. The central series $w(u)$ of $X(\\mfg_{M},\\mfg_{M-2}\\oplus \\mfg_2)^{tw}$ (see \\eqref{Y=X\/(w-1)}) operates as multiplication by a scalar series $\\mathsf{w}(u)$ in $V_{\\ley-1}$. Let $\\mathsf{q}(u)\\in 1+u^{-1}\\C[[u^{-1}]]$ be the unique series such that \n$\\mathsf{w}(u)=\\mathsf{q}(u)\\mathsf{q}(u+\\ka)$. Since $\\mathsf{w}(u)$ is even, the uniqueness of this expansion forces the relation $\\mathsf{q}(u)=\\mathsf{q}(\\ka-u)$. In particular, the series $g(u)=\\mathsf{q}(u+\\ka\/2)^{-1}$ is even and in the twisted module $W=(V_{\\ley-1})^{\\nu_g}$ the series $w(u)$ operates as $g(u-\\ka\/2)g(u+\\ka\/2)\\mathsf{w}(u)=1$. Since $W$ is one-dimensional, Proposition 4.14 and Remark 4.15 of \\cite{GRW2} imply that $W$ can be regarded as a representation of the Molev-Ragoucy reflection algebra $\\mcB(\\key+1,1)$, which is necessarily associated to the tuple $(\\gamma,Q_2(u),\\ldots,Q_{\\key+1}(u))$ in the sense of Theorem 4.6 of \\cite{MR} (see also (4.74) -- (4.76) of \\cite{GRW2}). \n\nFor $1\\leq i,j\\leq \\key+1$, we let $b_{ij}(u)$ denote the standard generating series of $\\mcB(\\key+1,1)$, and for $1\\leq i,j\\leq 2$ we let $b_{ij}^\\circ(u)$ denote the standard generating series of $\\mcB(2,1)$ (see \\cite[Definition 3.25]{GRW2}). Since $W$ is one-dimensional, it inherits the structure of a $\\mcB(2,1)$-module by allowing $b_{ij}^\\circ(u)$ to operate as $b_{i+\\key-1,j+\\key-1}(u)$ for $i,j\\in \\{1,2\\}$. This can be verified directly, but it also follows from a more general result observed in the first part of the proof of Theorem 4.6 of \\cite{MR}. The resulting $\\mcB(2,1)$-module has Drinfeld tuple $(\\gamma, Q_{\\key+1}(u))$. The reflection algebra $\\mcB(2,1)$ is isomorphic to the twisted Yangian $Y^+(2)$ (see \\cite[Proposition 4.3]{MR} as well as the remarks concluding Section 4.2 of \\textit{loc. cit.}), hence $W$ can also be viewed as a one-dimensional representation of $Y^+(2)$. The arguments used to prove \\cite[Proposition 4.4]{MR} show that this irreducible $Y^+(2)$-module corresponds to the pair $(\\gamma-\\tfrac{1}{2},Q_{\\key+1}(u+\\tfrac{1}{2}))$ (see \\eqref{Y+2:findim}). \nOn the other hand, by Corollary 4.4.5 of \\cite{Mobook}, as a $SY^+(2)$-module $W$ must be isomorphic to the module\n$V(\\gamma-1)$, which is the one-dimensional representation of $Y^+(2)$ with the highest weight \n\\begin{equation*}\n \\gamma(u)=\\frac{1+(\\gamma-1\/2)u^{-1}}{1+1\/2u^{-1}},\n\\end{equation*}\nand may be viewed as a $SY^+(2)$ module by restriction (see equation (4.21) in \\cite{Mobook}). Since this module corresponds to the pair $(\\gamma-\\tfrac{1}{2},1)$, we obtain \n$\\deg P_{\\key+1}(u)=\\deg Q_{\\key+1}(u)=0$.\n\n\n\\noindent \\textit{Case 2}: $\\key=0$. \n\nIn this case, $V_{\\ley-1}$ is a one-dimensional representation of $X(\\mfso_3,\\mfso_2)^{tw}$ which, by Corollary \\ref{C:poly-low}, is associated to the pair $(\\gamma,Q_1(u))=(\\alpha-\\frac{\\ley-1}{2}, P_{1}(u+\\tfrac{\\ley-1}{2}))$. Moreover, it can be made into a $Y^+(2)$-module via the isomorphism \\eqref{iso:so3so2}, and the proof of Proposition \\ref{P:so3class} shows that, as a $Y^+(2)$-module, $V_{\\ley-1}$ corresponds to the pair $(P^\\circ(u),\\gamma^\\circ)=(2^{\\deg Q_1(u)} Q(\\tfrac{u+1}{2}),2\\gamma-1)$. Repeating the last part of the argument of Case 1, we are able to conclude that $\\deg P_1(u)=\\deg P^\\circ(u)=0$, which completes the proof of the Proposition. \\qedhere\n\\end{proof}\nThe below corollary of Proposition \\ref{P:no1dim} now follows immediately from the fact that $Y(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$ is the $\\nu_g$-stable subalgebra of $X(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$.\n\\begin{crl}\n Let $(\\mfg_N,\\mfg_{p}\\oplus \\mfg_q)$ satisfy the conditions of Proposition \\ref{P:no1dim}. Then, up to isomorphism, $V(\\mcG)$ is the unique one-dimensional representation \n of $Y(\\mfg_N,\\mfg_p\\oplus \\mfg_q)^{tw}$. \n\\end{crl}\n\n\n\n\\subsection{Twisted Yangians for the symmetric pairs \\texorpdfstring{$(\\mfg_N,\\mfg_p\\op\\mfg_q)$}{} when \\texorpdfstring{$\\mfg_q\\cong \\mfso_2$}{}} \\label{subsec:1dim-so2}\n\n\nWe now turn to the twisted Yangians associated to pairs of the form $(\\mfg_N,\\mfg_N^\\rho)=(\\mfso_N,\\mfso_{N-2}\\oplus\\mfso_2)$ with $N\\geq 5$. \n \nThe following technical lemma provides a one-parameter family of matrix solutions to \\eqref{TX-RE} and \\eqref{TX-symm} associated to the pair $(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_{2})$.\n\n\\begin{lemma} \\label{L:K-1dim}\nLet $a\\in\\C$ and let $\\mcG$ be of type BI or DI with $q=2$ and $N\\geq 5$. Then the matrix\n\\eq{\nK(u;a) = k(u) \\left( I - \\frac{2u}{u-a} E_{-n,-n} - \\frac{2u}{u+a-2d} E_{nn} \\right), \\qu\\text{where}\\qu k(u)=\\frac{(u-a)(u+a-2d)}{(u-d)^2} \\label{K-1dim}\n}\nand $d=N\/4-1$, is a one-parameter solution of the reflection equation \\eqref{TX-RE}. Moreover, it satisfies the symmetry relation \n\\eq{\nK^t(u;a) = K(\\ka-u;a) + \\frac{K(u;a)-K(\\ka-u;a)}{2u-\\ka} + \\frac{\\Tr(\\mcG(u))\\,K(\\ka-u;a) - \\Tr(K(u;a))\\cdot I }{2u-2\\ka}. \\label{K-1dim:Sym}\n}\nwhere $\\mcG(u)$ is the is the matrix defined in \\eqref{G(u)}.\n\\end{lemma}\n\n\n\\begin{proof}\nWe begin by showing that $K(u;a)$ satisfies \\eqref{TX-RE}, that is\n\\[\n R(u-v)\\, K_1(u;a)\\, R(u+v)\\, K_2(v;a) = K_2(v;a)\\,R(u+v) \\,K_1(u;a)\\,R(u-v). \n\\]\nNotice that we only need to show this for $k(u)^{-1} K(u;a)$. Denote \n\\eq{\nG(u) = - \\frac{2u}{u-a} E_{-n,-n} - \\frac{2u}{u+a-2d} E_{nn}, \\label{L:K-1dim:a0}\n}\nso that $K(u;a)=k(u)(I + G(u))$. Since $I$ is a solution to \\eqref{TX-RE}, our task is to show that \n\\spl{\n& R(u-v)\\,G_1(u)\\,R(u+v) + R(u-v)\\, R(u+v)\\,G_2(v) + R(u-v)\\,G_1(u)\\, R(u+v)\\,G_2(v) \\\\\n& \\qq = G_2(v)\\,R(u+v)\\,R(u-v) + R(u+v)\\,G_1(u)\\,R(u-v) + G_2(v)\\,R(u+v)\\,G_1(u)\\,R(u-v). \\label{L:K-1dim:a}\n}\nWe first show that \n\\spl{\n& \\left(1-\\frac{P}{u-v}\\right)\\left( G_1(u)\\left(1-\\frac{P}{u+v}\\right) + \\left(1-\\frac{P}{u+v}\\right)G_2(v) + G_1(u)\\left(1-\\frac{P}{u+v}\\right)G_2(v) \\right) + H(u,v)\n\\\\\n& \\qq = \\left( G_2(v)\\left(1-\\frac{P}{u+v}\\right)+ \\left(1-\\frac{P}{u+v}\\right)G_1(u) + G_2(v)\\left(1-\\frac{P}{u+v}\\right)G_1(u) \\right) \\left(1-\\frac{P}{u-v}\\right) , \\label{L:K-1dim:b}\n}\nwhere \n\\eq{\nH(u,v) = \\frac{8 u v (a-d)}{(u-a) (v-a) (u+a-2d) (v+a-2d)} \\left( E_{-n,n}\\ot E_{n,-n} - E_{n,-n}\\ot E_{-n,n}\\right ). \\label{L:K-1dim:b1}\n}\nThe equality \\eqref{L:K-1dim:b} reduces to\n\\[\n\\left[ P , 2v\\,G_2(u) - 2u\\,G_2(v) - (u-v)\\,G_2(u)\\,G_2(v) \\right] = 0 .\n\\]\nThis follows from the following computations:\n\\eqn{\n2v\\,G(u) - 2u & \\,G(v) - (u-v)\\,G(u)\\,G(v) \\\\\n& \\qq = -\\left(\\frac{4uv}{u-a} - \\frac{4uv}{v-a}\\right) E_{-n,-n} - \\left( \\frac{4uv}{u+a-2d} - \\frac{4uv}{v+a-2d}\\right) E_{nn} \\\\\n& \\qq\\qq - \\frac{4uv\\,(u-v)}{(u-a)(v-a)}\\, E_{-n,-n} - \\frac{4uv\\,(u-v)}{(u+a-2d)(v+a-2d)} E_{nn} = 0,\n}\nthus implying \\eqref{L:K-1dim:b}. Next we use\n\\[\nQ^2 = NQ, \\qu (1-u^{-1}P)\\,Q = (1-u^{-1})\\,Q, \\qu Q\\,G(u)\\,Q=g(u)\\,Q \\qu\\text{where}\\qu g(u)= - \\frac{2u}{u-a} - \\frac{2u}{u+a-2d} \\label{L:K-1dim:b2}\n\\]\nand subtract \\eqref{L:K-1dim:b} from \\eqref{L:K-1dim:a}. Then \\eqref{L:K-1dim:a} holds if and only if the following equality is verified:\n\\eqn{\n& \\frac{G_1(u)\\,Q + Q\\,G_2(v) + G_1(u)\\,Q\\,G_2(v)}{u+v-\\ka} - \\frac{G_2(v)\\,Q + Q\\,G_1(u) + G_2(v)\\,Q\\,G_1(u)}{u+v-\\ka} \\\\\n& -\\frac{ G_2(u)\\,Q + Q\\,G_2(v) + G_2(u)\\,Q\\,G_2(v)}{(u-v)(u+v-\\ka)} + \\frac{G_2(v)\\,Q + Q\\,G_2(u) + G_2(v)\\,Q\\,G_2(u)}{(u-v)(u+v-\\ka)} \\\\\n& +\\frac{Q\\,( G_1(u)+ G_2(v)+G_1(u)\\,G_2(v))}{u-v-\\ka} - \\frac{(G_2(v) + G_1(u) + G_2(v)\\,G_1(u))\\,Q}{u-v-\\ka} \\\\\n& - \\frac{Q\\,(G_2(u) + G_2(v) + G_2(u)\\,G_2(v))}{(u+v)(u-v-\\ka)} + \\frac{(G_2(v)+ G_2(u) + G_2(v)\\,G_2(u))\\,Q}{(u+v)(u-v-\\ka)}\\\\\n& + \\frac{ g(u)\\,Q + N\\,Q\\,G_2(v) + g(u)\\, Q\\,G_2(v) }{(u-v-\\ka)(u+v-\\ka)} - \\frac{N\\,G_2(v)\\,Q + g(u)\\,Q + g(u)\\,G_2(v)\\,Q}{(u-v-\\ka)(u+v-\\ka)} = H(u,v) . \n}\nSince $[G(u),G(v)]=0$ we can simplify the equation above to\n\\spl{\n& \\frac{[G_1(u)-G_2(v),Q] + G_1(u)\\,Q\\,G_2(v)-G_2(v)\\,Q\\,G_1(u)}{u+v-\\ka} +\\frac{[Q, G_1(u)+ G_2(v)+G_1(u)\\,G_2(v)]}{u-v-\\ka} \\\\\n& + \\frac{[Q,G_2(u)-G_2(v)] + G_2(v)\\,Q\\,G_2(u)-G_2(u)\\,Q\\,G_2(v)}{(u-v)(u+v-\\ka)} - \\frac{[Q,G_2(u) + G_2(v) + G_2(u)\\,G_2(v)]}{(u+v)(u-v-\\ka)} \\\\\n& + \\frac{ (N+ g(u))\\, [Q,G_2(v)] }{(u-v-\\ka)(u+v-\\ka)} = H(u,v). \\label{L:K-1dim:d}\n}\nNow observe that $G^t(u) = -u\\,(2d-u)^{-1}G(2d-u)$ which implies that\n\\[\nG_1(u)\\,Q = -\\frac{u}{2d-u}\\,G_2(2d-u)\\,Q, \\qq Q\\,G_1(u) = -\\frac{u}{2d-u}\\,Q\\,G_2(2d-u) .\n\\]\nMoreover,\n\\begin{align}\n\\begin{split}\nG_2(u)\\,Q\\,G_2(v) &= \\frac{4 u v}{(u-a) (v-a)} E_{nn}\\ot E_{-n,-n} + \\frac{4 u v}{(u-a) (v+a-2d)} E_{n,-n}\\ot E_{-n,n} \\\\\n&+ \\frac{4 u v}{(u+a-2d) (v-a)} E_{-n,n}\\ot E_{n,-n} + \\frac{4 u v}{(u+a-2d) (v+a-2d)} E_{-n,-n}\\ot E_{nn}. \\label{L:K-1dim:GQG}\n\\end{split}\n\\end{align}\nUsing \\eqref{L:K-1dim:b1}, \\eqref{L:K-1dim:GQG} and $\\ka=2d+1$ we compute the following identity \n\\eqn{\n& \\frac{ G_1(u)\\,Q\\,G_2(v)-G_2(v)\\,Q\\,G_1(u)}{u+v-\\ka} + \\frac{G_2(v)\\,Q\\,G_2(u)-G_2(u)\\,Q\\,G_2(v)}{(u-v)(u+v-\\ka)} \\\\\n& \\qu = \\frac{4uv}{u+v-\\ka} \\left(\\frac{1}{(u-a) (v-a)}-\\frac{1}{(u+a-2d)(v+a-2d)}\\right) \\left( E_{-n,n}\\ot E_{n,-n} - E_{n,-n}\\ot E_{-n,n} \\right) \\\\\n& \\qu + \\frac{4 u v}{(u-v)(u+v-\\ka)} \\left(\\frac{1}{(u-a) (v+a-2d)}-\\frac{1}{(v-a) (u+a-2d)}\\right) \\left( E_{-n,n}\\ot E_{n,-n} - E_{n,-n}\\ot E_{-n,n} \\right) \\\\\n& \\qu = \\frac{8 u v (a-d)}{(u-a)(v-a) (u+a-2d) (v+a-2d)} \\left( E_{-n,n}\\ot E_{n,-n} - E_{n,-n}\\ot E_{-n,n} \\right) = H(u,v) . \n}\nBy combining the identities above and denoting $\\wt u = u\\,(2d-u)^{-1}$ we rewrite \\eqref{L:K-1dim:d} as\n\\eqn{\n& \\bigg[ Q, \\frac{\\wt u\\,G_2(2d-u)+G_2(v)}{u+v-\\ka} + \\frac{G_2(u)-G_2(v)}{(u-v)(u+v-\\ka)} + \\frac{ (N+ g(u))\\, G_2(v) }{(u-v-\\ka)(u+v-\\ka)} \\\\\n& \\qq -\\frac{\\wt u\\,G_2(2d-u) - G_2(v)+\\wt u\\, G_2(2d-u)\\,G_2(v)}{u-v-\\ka} - \\frac{G_2(u) + G_2(v) + G_2(u)\\,G_2(v)}{(u+v)(u-v-\\ka)} \\bigg] = 0. \n}\nDenoting the commutator above by $[Q,1\\ot F(u,v)]$ we only need to verify that $F(u,v)=0$, which follows by a direct computation using \\eqref{L:K-1dim:a0}, the explicit form of $g(u)$ and $\\ka=2d+1=N\/2-1$, as we now illustrate. After reorganizing the various terms and multiplying by $(u-v-\\ka) (u+v-\\ka)$, we obtain, with $F'(u,v) = (u-v-\\ka) (u+v-\\ka) F(u,v)$:\n\\eqn{\nF'(u,v) &= -\\frac{ 2 \\ka v\\, G(u)}{u^2-v^2} -\\frac{2 u v\\, G(2d-u)}{2d-u} + 2 u \\bigg(1-\\frac{1}{u-a}-\\frac{1}{u+a-2d}+\\frac{\\ka}{u^2-v^2}\\bigg)G(v) \\\\ & \\hspace{7.3cm} -(u+v-\\ka) \\bigg(\\frac{u\\,G(2d-u)\\,G(v)}{2d-u}+\\frac{G(u)\\,G(v)}{u+v}\\bigg)\n\\\\\n& = 4uv\\bigg( -\\bigg(\\frac{1}{u+a-2d} - \\frac{\\ka}{(u-a)(u^2-v^2)} \\bigg) E_{-n,-n} - \\bigg( \\frac{1}{u-a} - \\frac{\\ka}{(u+a-2d)(u^2-v^2)} \\bigg) E_{nn} \\\\\n& \\hspace{1.3cm} - \\bigg(1-\\frac{1}{u-a}-\\frac{1}{u+a-2d}+\\frac{\\ka}{u^2-v^2}\\bigg)\\bigg( \\frac{1}{v-a} E_{-n,-n} + \\frac{1}{v+a-2d} E_{nn} \\bigg) \\\\\n& \\hspace{1.3cm} -\\frac{u+v-\\ka}{(u-a)(u+a-2d)(u+v) }\\bigg( \\frac{(u+a-2d)-(u-a)(u+v)}{v-a} E_{-n,-n} \\\\ & \\hspace{8.5cm}+ \\frac{(u-a)-(u+a-2d)(u+v)}{v+a-2d} E_{nn}\\bigg)\\bigg) \n\\\\\n& = \\frac{2d+1-\\ka}{(v-a)(u+a-2d)} E_{-n,-n} +\\frac{2d+1-\\ka}{(u-a)(v+a-2d)} E_{nn} = 0 . \n}\nThis completes the proof that $K(u;a)$ is a solution to~\\eqref{TX-RE}.\n\n\nWe now turn to proving \\eqref{K-1dim:Sym}. Our work thus far shows that the assignment $\\wt S(u)\\mapsto K(u;a)$ extends to a homomorphism of algebras $\\phi_a:\\wt X(\\mfg_N,\\mcG)^{tw}\\to \\C$ where $\\wt X(\\mfg_N,\\mcG)^{tw}$ is the extended reflection algebra (see Subsection \\ref{subsec:twYa}). By Theorem 5.2 of \\cite{GR}, $K(u;a)$ satisfies \\eqref{K-1dim:Sym} if and only if $\\phi_a(c(u))=1$: see \\eqref{c(u)}. Since $N\\geq 5$, we have $p=N-2>2=q$ and we may therefore apply Corollary \\ref{C:c(u)} which, by \\eqref{g(u):exp}, implies that \n\\begin{equation}\n \\phi_a(c(u))=\\frac{\\mathscr{g}(\\ka-u)}{\\mathscr{g}(u)}\\cdot \\frac{\\Tr(K(u;a))}{\\Tr(K(\\ka-u;a))}=\\bigg(\\frac{u+1-N\/4}{u-N\/4}\\bigg)^2\\frac{\\Tr(K(u;a))}{\\Tr(K(\\ka-u;a))}. \\label{K-1dim:phi}\n\\end{equation}\nBy definition of $K(u;a)$, \n\\[\n \\Tr(K(u;a))=k(u)\\left(N-\\frac{2u}{u-a}-\\frac{2u}{u+a-2d}\\right)=\\frac{N(u-a)(u+a-2d)-2u(2u-2d)}{(u-d)^2}.\n\\]\nLet $P(u)$ be the numerator of the right-hand side. Using that $N=2\\ka+2$ and $2d=\\ka-1$ we find that \n\\[\n P(u)=N(u-a)(u+a-\\ka+1)-2u(2u-\\ka+1)=N(u-a)(u+a-\\ka)-2u(2u-2\\ka)-Na, \n\\]\nand hence $P(u)$ is invariant under the substitution $u\\mapsto \\ka-u$. This implies that \n\\[\n \\frac{\\Tr(K(u;a))}{\\Tr(K(\\ka-u;a))}=\\bigg(\\frac{\\ka-u-d}{u-d} \\bigg)^2=\\bigg(\\frac{u-N\/4}{u+1-N\/4}\\bigg)^2,\n\\]\nwhich by \\eqref{K-1dim:phi} proves that $\\phi_a(c(u))=1$ for any $a\\in \\C$, as required. \n\\end{proof}\n\n\nAs a consequence of this lemma we obtain the existence of a family of one-dimensional representations $\\{V(a)\\}_{a\\in \\C}$:\n\\begin{crl}\\label{C:V(a)}\nLet $a\\in \\C$. Then the assignment $S(u)\\mapsto K(u;a)$ yields a one-dimensional representation $V(a)$ of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_{2})^{tw}$ with the highest weight \n$\\gamma^a(u)$ given by \n\\begin{equation}\n\\gamma_i^a(u)=\\frac{(u-a)((-1)^{\\delta_{in}}u+a-2d)}{(d-u)^2} \\; \\text{ for all }\\; i\\in \\mcI_N^+, \\label{K-1dim:hw}\n\\end{equation}\nwhere $d=N\/4-1$. \n\\end{crl}\n\nThis corollary is an immediate consequence of Lemma \\ref{L:K-1dim}, the formula for each $\\gamma_i^a(u)$ following from \\eqref{K-1dim}. \nNote that \\eqref{K-1dim:hw} implies that the auxiliary tuple $\\wt \\gamma^a(u)$ is determined by \n\\begin{equation*}\n \\wt \\gamma_i^a(u)=2u\\cdot\\frac{(u-a)((-1)^{\\delta_{in}}u+a-2d-1+\\delta_{in})}{(d-u)^2} \\; \\text{ for all }\\; i\\in \\mcI_N^+,\n\\end{equation*}\nand thus that $V(a)$ has Drinfeld tuple $(\\alpha,P_1(u),\\ldots,P_n(u))$ equal to \n\\begin{equation}\n(\\ka-a,1,\\ldots,1) \\label{V(a):tuple} \n\\end{equation}\nbecause $1+2d=\\kappa$. The last two results of this subsection provide classifications of the one-dimensional representations of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ and of $Y(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$, respectively, when $N\\geq 5$. \n\\begin{prop}\\label{P:1dimso2}\n Let $N\\geq 5$. Then a representation $V$ of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ is one-dimensional if and only if $V\\cong V(a)^{\\nu_g}$ for some $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$. \n\\end{prop}\n\\begin{proof}\n By Corollary \\ref{C:V(a)}, for any $a\\in \\C$, $V(a)$ provides a one-dimensional representation of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$. Hence, to prove the proposition we need to establish that, if $V$ is a one-dimensional representation of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$, then it can be associated to a tuple of the form \n \\begin{equation*}\n (\\alpha,P_1(u),\\ldots,P_n(u))=(\\al,1,\\ldots,1). \n \\end{equation*}\nBy Lemma \\ref{L:unique}, this will imply that $V\\cong V(a)^{\\nu_g}$ for some $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$, where $a$ is determined by $\\al=1+2d-a$. \n\nLet $V$ be a one-dimensional representation. Since $N\\geq 5$, $\\mfso_{N-2}$ is semisimple and thus $V$ is equal to the trivial representation when viewed as a $\\mfso_{N-2}$-module. \nSince $\\mfso_2$ is one-dimensional, $F_{11}$ operates in $V$ as multiplication by a scalar $\\gamma$. In particular, the highest weight of $V$ as a $(\\mfso_{N-2}\\oplus \\mfso_{2})$-module is \n$(\\mu_i)_{i=1}^n=(0,\\ldots,0,\\gamma)$. The relation \\eqref{gtw-action} of Lemma \\ref{L:gtw} therefore implies that $P_i(u)=0$ for all $1\\leq i\\leq n-1$ and \n\\begin{equation*}\n \\deg P_n(u)=2\\alpha-\\tfrac{N}{2}-2\\gamma.\n\\end{equation*}\nTo complete the proof, we need to see that $\\deg P_n(u)=0$. This can shown using the same argument as given in the $\\key>0$ case of the proof of Proposition \\ref{P:no1dim}. \n\\end{proof}\n\\begin{crl}\\label{C:1dimso2}\n A representation $V$ of $Y(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ (with $N\\geq 5$) is one-dimensional if and only if there is $a\\in \\C$ such that $V\\cong V(a)$. \n\\end{crl}\n\n\n\\begin{rmk}\\label{R:1dim}\n$ $ \n\\begin{enumerate}[leftmargin=1.8em,topsep=-3pt]\n \\item If $V$ is a one-dimensional representation of $X(\\mfso_4,\\mfso_2\\oplus \\mfso_2)^{tw}$ (resp. of $X(\\mfso_3,\\mfso_2)^{tw}$) then there exists \n $(\\mu_1,\\mu_2)\\in \\C^2$ (resp. $\\mu \\in \\C)$ and $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$ such that $V\\cong V(\\mu_1,\\mu_2)^{\\nu_g}$ (resp. $V\\cong V(\\mu)^{\\nu_g}$): see \n Corollaries \\ref{C:so4} and \\ref{C:so3} for the definitions of $V(\\mu_1,\\mu_2)$ and $V(\\mu)$, respectively. This follows from the isomorphisms \\eqref{iso:so4} and \\eqref{iso:so3so2} together with the fact that the family $\\{V(\\gamma)\\}_{\\gamma\\in \\C}$, which appeared in the proof of Proposition \\ref{P:no1dim}, is a complete list of the isomorphisms classes of one-dimensional representations of $Y^+(2)$ up to twisting by automorphisms of the form $S^\\circ(u)\\mapsto h(u)S^\\circ(u)$ with $h(u)\\in 1+u^{-2}\\C[[u^{-2}]]$. This fact follows from Corollary 4.4.5 of \\cite{Mobook}. \n \n \\item By Lemma 6.1 of \\cite{GRW2}, the twisted Yangians associated to the symmetric pairs $(\\mfg_{2n},\\mfgl_n)$ of type CI and DIII (see \\cite{GR,GRW2}) admit a family\n $\\{V(a)\\}_{a\\in \\C}$ of one-dimensional representations. The arguments used in this section can also be applied for twisted Yangians of these types, yielding analogues of Proposition \\ref{P:1dimso2} and Corollary \\ref{C:1dimso2} for $X(\\mfg_{2n},\\mfgl_n)^{tw}$ and $Y(\\mfg_{2n},\\mfgl_n)^{tw}$, respectively. \n\\end{enumerate}\n\\end{rmk}\n\n\n\n\\section{Classification of finite-dimensional irreducible representations} \\label{sec:main}\n \n\nIn this section, we prove the main results of this paper: the classification of finite-dimensional irreducible modules of the twisted Yangians associated to the pairs $(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)$ with $N\\geq 5$ and $(\\mfso_{2n+1},\\mfso_{2n})$ with $n\\geq 2$. \n \nThe one-dimensional representations $\\{V(a)\\}_{a\\in\\C}$ constructed in Lemma \\ref{L:K-1dim} and Corollary \\ref{C:V(a)} provide the last ingredient \nnecessary to classify the finite-dimensional irreducible representations of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ ($N\\geq 5$) and its unextended counterpart, as is proven in Theorem \\ref{T:DI(a)-Class} and Corollary \\ref{C:DI(a)class}. In particular, Theorem \\ref{T:DI(a)-Class} proves that for these twisted Yangians the necessary conditions of Proposition \\ref{P:necessary} are in fact sufficient.\n \nIn the general setting the necessary conditions of Section \\ref{sec:Nec} are not sufficient for determining exactly when the irreducible module $V(\\mu(u))$ is finite-dimensional. We will illustrate this in Subsection \\ref{subsec:q=1} by classifying the finite-dimensional irreducible representations of $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ and of $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ in Theorem \\ref{T:BI(b)-Class} and Corollary \\ref{C:BI(b)class}, respectively. In the process we will obtain a stronger version of Proposition \\ref{P:int} for these twisted Yangians which is proven without directly using the representation theory of $\\mfso_{2n}$: see Proposition \\ref{P:q=1-nec}. \n\n\n\n\\subsection{Twisted Yangians for the symmetric pairs \\texorpdfstring{$(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_{2})$}{}}\\label{subsec:q=2}\n\n\nThe following theorem provides a classification of the finite-dimensional irreducible representations of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ with $N\\geq 5$. \n\\begin{thrm}\\label{T:DI(a)-Class}\nLet $N\\geq 5$ and suppose that $\\mu(u)$ satisfies the conditions of Proposition \\ref{P:nontriv} so that the Verma module $M(\\mu(u))$ is non-trivial. Then\nthe irreducible $X(\\mfso_{N},\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$-module $V(\\mu(u))$ is finite-dimensional if and only if there are monic polynomials $P_1(u),\\ldots,P_n(u)$ \ntogether with a scalar $\\alpha\\in \\C\\setminus Z(P_n(u))$ such that\n\\begin{equation}\n\\frac{\\wt \\mu_{i-1}(u)}{\\wt \\mu_i(u)}=\\frac{P_i(u+1)}{P_i(u)}\\left(\\frac{\\al-u}{\\al+u-1}\\right)^{\\del_{i,n}} \\;\\text{ with }\\; P_i(u)=P_i(-u+n-i+2) \\label{T:DI(a)-Class.1}\n\\end{equation}\nfor all $2\\leq i\\leq n$, while $P_1(u)$ must satisfy $P_1(u)=P_1(-u+\\tfrac{N}{2})$ and the relation \n\\begin{equation}\\label{T:DI(a)-Class.2}\n \\frac{\\wt \\mu_{\\mathscr{a}}(\\ka-u)}{\\wt \\mu_{\\mathscr{a}+1}(u)}=\\left(\\frac{u+1-\\tfrac{N}{4}}{u-\\tfrac{N}{4}}\\right)^2\\frac{P_1(u+2^{\\mathscr{a}-1})}{P_1(u)}\\cdot \\frac{\\ka-u}{u} \\quad \\text{ where }\\quad \\mathscr{a}=2n+1-N.\n\\end{equation}\nAdditionally, when $V(\\mu(u))$ is finite-dimensional the corresponding tuple $(\\al,P_1(u),\\ldots, P_n(u))$ is unique. \n\\end{thrm}\n\n\\begin{proof}\n$(\\Longrightarrow)$ If $V(\\mu(u))$ is finite-dimensional, then the existence of $\\alpha$ and $P_1(u),\\ldots,P_n(u)$ satisfying the conditions of the theorem is guaranteed by Proposition \\ref{P:necessary}. Here we note that as a consequence of \\eqref{nontriv.2}, the $\\mathscr{a}=0$ version of \\eqref{T:DI(a)-Class.2} is equivalent to the $N=2n+1$ version of \\eqref{nec.2} with $\\key=n-1$. \nMoreover, the uniqueness assertion has been proven in Lemma \\ref{L:unique}. \n \n $(\\Longleftarrow)$ Conversely, assume that $\\mu(u)$ satisfies the conditions of Proposition \\ref{P:nontriv} and there exists a scalar $\\al$ and monic polynomials $P_1(u),\\ldots,P_n(u)$ satisfying the relations of the theorem. Since $P_1(u)=P_1(-u+\\tfrac{N}{2})$ and $P_i(u)=P_i(-u+n-i+2)$ for all $2\\leq i\\leq n$, there exists monic polynomials $Q_1(u),\\ldots,Q_n(u)$ such that \n \\begin{gather*}\n P_1(u)=(-1)^{\\deg Q_1(u)}Q_1(u-\\ka\/2)Q_1(-u+\\tfrac{N}{2}-\\ka\/2),\\\\ P_i(u)=(-1)^{\\deg Q_i(u)}Q_i(u-\\ka\/2)Q_i(-u+n-i+2-\\ka\/2) \\; \\text{ for each }\\; 2\\leq i\\leq n. \n \\end{gather*}\nLet $L(\\lambda(u))$ be a finite-dimensional irreducible $X(\\mfso_N)$-module with Drinfeld polynomials $Q_1(u),\\ldots,Q_n(u)$. It follows from Theorem \\ref{T:X-class} that such a module exists and is uniquely determined up to twisting by automorphisms of the form $\\mu_f:T(u)\\mapsto f(u)T(u)$ (see also Corollary 5.19 of \\cite{AMR}). Let $\\xi\\in L(\\lambda(u))$ be a highest weight vector. Consider the one-dimensional module $V(a)$ from Corollary \\ref{C:V(a)} with $a=\\ka-\\al$, and let $\\eta\\in V(a)$ be any nonzero vector. Since, by \\eqref{V(a):tuple}, $V(\\ka-\\al)$ is associated to the tuple $(\\alpha,1,\\ldots,1)$, Lemma~\\ref{L:QxP=QP} implies that the finite-dimensional $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ highest weight module \n\\begin{equation*}\n X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}(\\xi\\otimes \\eta)\\subset L(\\lambda(u))\\otimes V(a)\n\\end{equation*}\nhas highest weight $\\mu^\\sharp(u)$ which is also associated to $(\\alpha,P_1(u),\\ldots,P_n(u))$. By Lemma \\ref{L:unique}, there exists $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$ such that $V(\\mu^\\sharp(u))\\cong V(\\mu(u))^{\\nu_g}$. Since $V(\\mu^\\sharp(u))$ is finite-dimensional, we can conclude that the same is true for $V(\\mu(u))$.\n\\end{proof}\n \n\\begin{rmk}\nBefore reinterpreting the above theorem as a classification result for finite-dimensional irreducible representations of $Y(\\mfso_{N},\\mfso_{N-2}\\oplus \\mfso_{2})^{tw}$, we give a few remarks on the relation \\eqref{T:DI(a)-Class.2}: \n\\begin{enumerate}[leftmargin=1.8em,topsep=-2pt]\n \\item As has been pointed out in the proof of Theorem \\ref{T:DI(a)-Class}, when $N$ is odd the relation \\eqref{nontriv.2} implies that \\eqref{T:DI(a)-Class.2} is equivalent to the relation \n %\n \\begin{equation*}\n \\frac{\\wt \\mu_0(u)}{\\wt \\mu_1(u)}=\\frac{P_1(u+\\tfrac{1}{2})}{P_1(u)}. \n \\end{equation*}\n %\n \\item If $N=2n$ then the existence of $P_1(u)$ satisfying $P_1(u)=P_1(-u+n)$ and condition \\eqref{T:DI(a)-Class.2} can be replaced by the equivalent requirement that there exists\n a monic polynomial $Q_1(u)$ such that $Q_1(u)=Q_1(-u+n)$, $n\/2\\in Z(Q_1(u))$ and \n \\begin{equation*}\n \\frac{\\wt \\mu_1(\\ka-u)}{\\wt \\mu_2(u)}=\\frac{Q_1(u+1)}{Q_1(u)}\\cdot \\frac{\\ka-u}{u}. \n \\end{equation*}\n \\end{enumerate}\n\\end{rmk}\nBy Theorem 4.5 of \\cite{GRW2}, the isomorphism class of any finite-dimensional irreducible module $V$ has a unique representative of the form $V(\\mu(u))$, and by Theorem \\ref{T:DI(a)-Class} we \ncan assign to this representative a unique tuple $(\\al,P_1(u),\\ldots,P_n(u))$. This assignment is not injective: the second statement of Lemma \\ref{L:unique} shows that $(\\al,P_1(u),\\ldots,P_n(u))$ determines $V(\\mu(u))$ only up to twisting by automorphisms of the form $\\nu_g$. Note, however, that there is a unique series $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$ such that the central series $w(u)$ operates as the identity in $V(\\mu(u))^{\\nu_g}$. Indeed, by Proposition 4.6 of \\cite{GRW2} $w(u)$ acts in $V(\\mu(u))^{\\nu_g}$ as multiplication by the series\n$h(u)h(u+\\ka)\\nu(u)$, where $h(u)=g(u-\\ka\/2)$ and $\\nu(u)=\\mu_n(u)\\mu_n(-u)$. Therefore $w(u)$ will act as the identity in $V(\\mu(u))^{\\nu_g}$ precisely when $h(u)h(u+\\ka)=\\nu(u)^{-1}$. A simple argument shows that there exists a unique series $a(u)\\in 1+u^{-1}\\C[[u^{-1}]]$ such that $a(u)a(u+\\ka)=\\nu(u)^{-1}$, and this series satisfies $a(u)=a(\\ka-u)$ since otherwise $b(u)=a(\\ka-u)$ would be a distinct solution of $b(u)b(u+\\ka)=\\nu(u)^{-1}$ (as $\\nu(u)$ is even). Hence, $w(u)$ will operate as multiplication by $1$ in $V(\\mu(u))^{\\nu_g}$ exactly when $g(u)=a(u+\\ka\/2)$. \n\nThis discussion shows that we have an injective correspondence between isomorphism classes of finite-dimensional irreducible modules\nof $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ and finite sequences $(g(u); \\al,P_1(u),\\ldots,P_n(u))$, where $P_1(u),\\ldots,P_n(u)$ are monic polynomials satisfying \\eqref{P-sym}, $\\alpha$ is a complex number which is not contained in $Z(P_{n}(u))$, and $g(u)\\in 1+u^{-2}\\C[[u^{-2}]]$. A straightforward modification of the $(\\Longleftarrow)$ direction of the proof of Theorem \\ref{T:DI(a)-Class} shows that this is a bijective correspondence. \n\nThe next corollary provides the analogous classification result for finite-dimensional irreducible modules of the twisted Yangians of type BDI(a) with $q=2$ and $N\\geq 5$.\n\\begin{crl}\\label{C:DI(a)class}\nThe isomorphism classes of finite-dimensional irreducible $Y(\\mfso_{N},\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$-modules are parameterized by\ntuples of the form $ (\\al,P_1(u),\\ldots,P_n(u))$, \nwhere $P_1(u),\\ldots,P_n(u)$ are monic polynomials in $u$ satisfying \n \\begin{equation*}\n P_1(u)=P_1(-u+\\tfrac{N}{2}) \\quad \\text{ and }\\quad P_i(u)=P_i(-u+n-i+2) \\; \\text{ for all }\\; 2\\leq i\\leq n,\n \\end{equation*}\nand $\\alpha$ is a complex scalar which is not contained in $Z(P_n(u))$. \n\\end{crl}\n\\begin{proof}\nThe corollary follows from the discussion in the paragraphs preceding its statement combined with Proposition \\ref{P:Y^tw-fd}. It can also be proven using the same arguments as employed to prove Corollary~6.3 of \\cite{GRW2}. \n\\end{proof}\n\n\nRecall from Subsection \\ref{subsec:Ya} the definition of the fundamental representation $L(i:\\alpha)$ of the Yangian $Y(\\mfg_N)$. \nFor each $a\\in \\C$, the one-dimensional irreducible representation $V(a)$ of $X(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ from Corollary \\ref{C:V(a)} can be regarded as an irreducible representation of $Y(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ via restriction. The following result is the analogue of Corollary 6.4 of \\cite{GRW2} for \n$Y(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$ and can be proven in the exact same way. A similar argument is given in the proof of Corollary \\ref{C:BI(b)fun} below. \n\\begin{crl}\\label{C:DI(a)fun}\n Let $V$ be a finite-dimensional irreducible representation of $Y(\\mfso_N,\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$. Then there is $m\\geq 0$, $1\\leq i_1,\\ldots,i_m\\leq n$, and \n $a,\\alpha_{i_1},\\ldots,\\alpha_{i_m}\\in \\C$ such that $V$ is isomorphic to the unique irreducible quotient of the $Y(\\mfso_{N},\\mfso_{N-2}\\oplus \\mfso_2)^{tw}$-module \n \\begin{equation*}\n Y(\\mfso_{N},\\mfso_{N-2}\\oplus \\mfso_2)^{tw}(\\xi_1\\otimes \\cdots \\otimes \\xi_m \\otimes \\eta)\\subset L(i_1:\\al_{i_1})\\otimes \\cdots \\otimes L(i_m: \\al_{i_m})\\otimes V(a),\n \\end{equation*} \n where $\\eta\\in V(a)$ is any nonzero vector and for each $1\\leq k\\leq m$, $\\xi_k\\in L(i_k:\\al_{i_k})$ is a highest weight vector. \n\\end{crl}\n\n\\subsection{Twisted Yangians for the symmetric pairs \\texorpdfstring{$(\\mfso_{2n+1},\\mfso_{2n})$}{}}\\label{subsec:q=1}\n\n\nIn this section we study the finite-dimensional irreducible representations of $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ and $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ with $n\\geq 2$ (which are of type BI(b) with $q=1$), the culmination of our effort being a classification of all such representations in Theorem \\ref{T:BI(b)-Class} and Corollary \\ref{C:BI(b)class}, respectively. Our first step towards proving this theorem is to study how a certain automorphism $\\psi_\\sigma^n$ interacts with highest weight modules of $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$: this will play a critical role in the rest of this section, one that is similar to the role played in the analogous classification for the twisted Yangian of the symmetric pair $(\\mfgl_{2n},\\mfso_{2n})$ by the automorphism (4.69) of \\cite{Mobook}. \n\n\n\\subsubsection{The automorphism $\\psi_\\sigma^n$}\nFor any $n\\geq 1$, let $\\mfS_N$ denote the symmetric group on the $N=2n+1$ symbols $\\{-n,\\ldots,-1,0,1,\\ldots,n\\}$ and let $\\sigma$ be the transposition $(1,-1)$. Set $A_\\sigma=\\sum_{i=-n}^nE_{i,\\sigma(i)}$, and note that $A_{\\sigma}^t=A_\\sigma=A_{\\sigma}^{-1}$. Since we also have $A_\\sigma\\mcG A_\\sigma^t=\\mcG$, \\eqref{al_A} implies that the assignment\n$\\psi_\\sigma^n(S(u))=A_\\sigma S(u)A_\\sigma^t$ extends to an automorphism of $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$.\nOur present goal is to determine the highest weight of the twisted module \n$V(\\mu(u))^{\\psi_{\\sigma}^n}$. To this end, we return to the low rank setting. \n\\begin{lemma}\\label{BI:L-LR}\nSuppose that the $X(\\mfso_3,\\mfso_2)^{tw}$-module $V(\\mu(u))$ is finite-dimensional with associated Drinfeld tuple $(\\al,P(u))$ as in Proposition \\ref{P:so3class}. Then $V(\\mu(u))^{\\psi_{\\sigma}^1}$ is isomorphic to $V(\\mu^\\sharp(u))$, where the components of $\\mu^\\sharp(u)$ are determined by the relations \n\\begin{equation}\\label{BI:sh}\n \\wt\\mu_0^\\sharp(u)=\\wt \\mu_0(u)\\cdot \\frac{3-2u-2\\al}{2\\al-2u}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-1},\\qquad \\wt \\mu_1^\\sharp(u)=\\wt \\mu_1(u)\\cdot \\frac{2u-2\\al+1}{2u+2\\al-2}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-1}.\n\\end{equation}\nIn particular, $V(\\mu(u))^{\\psi_{\\sigma}^1}$ has the Drinfeld tuple $(\\tfrac{3}{2}-\\al,P(u))$. \n\\end{lemma}\n\\begin{proof}\n We appeal to the isomorphism $\\varphi:X(\\mfso_3,\\mfso_2)^{tw}\\to Y^+(2)$ given in \\eqref{iso:so3so2}, which we recall is given by \n \\begin{equation}\n S(u)\\mapsto \\tfrac{1}{2}R_{12}^\\circ(-1)S_1^\\circ(2u-1)R_{12}^\\circ(-4u+1)^{t_+}S_2^\\circ(2u)K_1K_2, \\label{vphi}\n \\end{equation}\nwhere $K=E_{11}-E_{-1,-1}$. Let $A=E_{1,-1}-E_{-1,1}$, and let $\\beta_A$ be the automorphism of $Y^+(2)$ given by $S^\\circ(u)\\mapsto -AS^\\circ(u)A^t$. More explicitly, \n$\\beta_A$ is defined on generators by the assignment $s_{ij}^\\circ(u)\\mapsto (-1)^{\\delta_{i,-1}+\\delta_{j,-1}}s_{-i,-j}^\\circ(u)$ for $i,j\\in \\{-1,1\\}$. \nWe claim that $\\varphi^{-1}\\circ \\beta_A \\circ \\varphi=\\psi_{\\sigma}^1$. \nApplying $\\beta_A$ to the right hand side of \\eqref{vphi} and performing a few straightforward manipulations, we obtain \n\\begin{equation*}\n \\tfrac{1}{2}R_{12}^\\circ(-1)(A_1A_2)S_1^\\circ(2u-1)R_{12}^\\circ(-4u+1)^{t_+}S_2^\\circ(2u)K_1K_2(A_1A_2).\n\\end{equation*}\nHence, $(\\varphi^{-1}\\circ \\beta_A \\circ \\varphi)(S(u))=\\tilde A S(u) \\tilde A$ where $\\tilde A=\\tfrac{1}{4}R_{12}^\\circ(-1)A_1A_2 R_{12}^\\circ(-1)$ (which \nis an element of $\\End(V)\\cong \\End(\\C^3)$). We have $\\tilde A v_{i}=-v_{-i}$ for all $-1\\leq i\\leq 1$, so $\\tilde A=-\\sum_{i=-1}^1 E_{i,\\sigma(i)}=-A_\\sigma$. Since $\\tilde AS(u)\\tilde A=A_{\\sigma}S(u)A_{\\sigma}$, $\\varphi^{-1}\\circ \\beta_A \\circ \\varphi=\\psi_{\\sigma}^1$. \n\nNow let $V(\\mu(u))$ be as in the statement of the lemma. We have already seen in the proof of Proposition \\ref{P:so3class} that there exists $\\mu^\\circ(u)\\in 1+u^{-1}\\C[[u^{-1}]]$ such that \n$\\wt \\mu_0(u)=-2u\\left(\\frac{4u-3}{4u-1}\\right)\\mu^\\circ(2u)\\mu^\\circ(1-2u)$, $\\wt \\mu_1(u)=2u\\mu^\\circ(2u)\\mu^\\circ(2u-1)$, and that viewed as a $Y^+(2)$-module \n$V(\\mu(u))$ is isomorphic to $V(\\mu^\\circ(u))$. Moreover, $V(\\mu^\\circ(u))$ has the Drinfeld tuple $(2\\al-1,Q(u))$, where $Q(u)=2^{\\deg P(u)}P(\\tfrac{u+1}{2})$.\nBy Lemma 4.4.13 of \\cite{Mobook}, the twisted module $V(\\mu^\\circ(u))^{\\beta_A}$ is isomorphic to $V(\\mu^\\bullet(u))$, where $\\mu^\\bullet(u)$ is given by \n\\begin{equation*}\n \\mu^\\bullet(u)=\\mu^\\circ(u)\\cdot \\frac{u-2\\al+2}{u+2\\al-1}.\n\\end{equation*}\nAs a $X(\\mfso_3,\\mfso_2)^{tw}$-module $V(\\mu^\\circ(u))^{\\beta_A}$ is isomorphic to $V(\\mu_0^\\sharp(u),\\mu_1^\\sharp(u))$ where \n\\begin{gather*}\n \\wt \\mu_0^\\sharp(u)=-2u\\left(\\frac{4u-3}{4u-1}\\right)\\mu^\\bullet(2u)\\,\\mu^\\bullet(1-2u)=\\wt \\mu_0(u)\\cdot \\frac{3-2u-2\\al}{2\\al-2u}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-1},\\\\\n \\wt \\mu_1^\\sharp(u)=2u\\cdot\\mu^\\bullet(2u)\\,\\mu^\\bullet(2u-1)=\\wt \\mu_1(u)\\cdot \\frac{2u-2\\al+1}{2u+2\\al-2}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-1}.\n\\end{gather*}\nAs $\\varphi^{-1}\\circ \\beta_A \\circ \\varphi=\\psi_{\\sigma}^1$, we can conclude that $V(\\mu(u))^{\\psi_{\\sigma}^1}$ is isomorphic to $V(\\mu^\\sharp(u))$ with $\\mu^\\sharp(u)$ as in \n\\eqref{BI:sh}. Consequently, we have \n\\begin{equation*}\n\\frac{\\wt \\mu_0^\\sharp(u)}{\\wt \\mu_1^\\sharp(u)}=\\frac{\\wt \\mu_0(u)}{\\wt \\mu_1(u)}\\cdot \\frac{(\\tfrac{3}{2}-\\al)-u}{\\al-u}\\cdot \\frac{\\al+u-1}{(\\tfrac{3}{2}-\\al)+u-1}=\\frac{P_1(u+\\tfrac{1}{2})}{P_1(u)}\\cdot \\frac{(\\tfrac{3}{2}-\\al)-u}{(\\tfrac{3}{2}-\\al)+u-1}. \\qedhere \n\\end{equation*}\n\\end{proof}\n\nWe now consider the case where $n>1$: \n\n\\begin{prop}\\label{BI:P-psi}\n Suppose that $n>1$ and that the $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-module $V(\\mu(u))$ is finite-dimensional with Drinfeld tuple $(\\al,P_1(u),\\ldots,P_n(u))$. Then $V(\\mu(u))^{\\psi_\\sigma^n}$ is isomorphic to \n $V(\\mu^\\sharp(u))$, where $\\mu_i^\\sharp(u)=\\mu_i(u)$ for all $2\\leq i\\leq n$, while $\\mu_0(u)$ and $\\mu_1(u)$ are determined by the relations\n\\begin{gather}\n \\wt\\mu_0^\\sharp(u)=\\wt \\mu_0(u)\\cdot \\frac{N-2u-2\\al}{2\\al-2u}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-N+2}, \\label{mu^sh:0}\\\\ \n \\wt \\mu_1^\\sharp(u)=\\wt \\mu_1(u)\\cdot \\frac{2u-2\\al+1}{2u+2\\al-N+1}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-N+2}. \\label{mu^sh:1}\n\\end{gather}\n\\end{prop}\n\\begin{proof}\nSince $V(\\mu(u))^{\\psi_\\sigma^n}$ is finite-dimensional and irreducible, it is isomorphic to $V(\\mu^\\sharp(u))$ for some $\\mu^\\sharp(u)$. Throughout the proof\nwe fix highest weight vectors $\\xi\\in V(\\mu(u))$ and $\\xi_\\sigma\\in V(\\mu(u))^{\\psi_\\sigma^n}$. \n\n Since $\\psi_\\sigma^n(s_{ij}(u))=s_{\\sigma(i)\\sigma(j)}(u)$ for all $i,j\\in \\mcI_N$, $V(\\mu(u))_{(+,n-1)}$ and $(V(\\mu(u))^{\\psi_\\sigma^n})_{(+,n-1)}$ are equal as subspaces \n of $V(\\mu(u))$ (see \\eqref{V(+,m)}). This implies that the identity map provides a linear isomorphism between the $X(\\mfso_3,\\mfso_2)^{tw}$-modules\n $(V(\\mu(u))_{(n-1)})^{\\psi_\\sigma^1}$ and $(V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$. The first step of our proof is to show that this is also a module homomorphism. \n \n \\noindent \\textit{Step 1: } The identity map $\\mathrm{id}:(V(\\mu(u))_{(n-1)})^{\\psi_\\sigma^1}\\to (V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$ is an isomorphism of $X(\\mfso_3,\\mfso_2)^{tw}$-modules. \n \n To prove that this is the case it suffices to show that, for each $i,j\\in \\mcI_{3}$, the generating series $s_{ij}^{n-1}(u)$ of $X(\\mfso_3,\\mfso_2)^{tw}$ operates the same in both of these modules. Since $s_{ij}^{n-1}(u)$ acts in $V(\\mu(u))_{(n-1)}$ as the operator $h_{n-1}(u)(s_{ij}^{\\circ (n-1)}(u))$ (see \\eqref{scirc} and \\eqref{h_m:triv}), it operates in \n $(V(\\mu(u))_{(n-1)})^{\\psi_\\sigma^1}$ as the operator \n %\n \\begin{equation*}\n h_{n-1}(u)\\left(s_{\\sigma(i)\\sigma(j)}(u+\\tfrac{n-1}{2})+\\frac{\\delta_{ij}}{2u}\\sum_{a=2}^n s_{aa}(u+\\tfrac{n-1}{2})\\right).\n \\end{equation*}\nAs $\\sigma(a)=a$ for all $a\\geq 2$, this is also equal to $h_{n-1}\\psi_\\sigma^n(s_{ij}(u))^{\\circ (n-1)}$ which is precisely the operator by which $s_{ij}^{n-1}(u)$ acts in \n$(V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$. \n \n \\noindent \\textit{Step 2:} $\\xi_\\sigma$ is contained in $V(\\mu(u))_{n-1}$. Moreover $(V(\\mu(u))_{n-1})^{\\psi_\\sigma^1}\\cong (V(\\mu(u))^{\\psi_\\sigma^n})_{n-1}$. \n \n Let $\\xi_{\\sigma}^1$ be a highest weight vector of the irreducible $X(\\mfso_3,\\mfso_2)^{tw}$-module $(V(\\mu(u))_{n-1})^{\\psi_{\\sigma}^1}$. Since this\n is a submodule of $(V(\\mu(u))_{(n-1)})^{\\psi_\\sigma^1}$, Step 1 shows that $\\xi_{\\sigma}^1$ is also contained in $(V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$ and generates a highest weight submodule. By Corollary \\ref{C:nosub}, this submodule must be equal to $(V(\\mu(u))^{\\psi_\\sigma^n})_{n-1}$ and thus $\\xi_{\\sigma}^1$ must be proportional to $\\xi_\\sigma$. \n This implies that $\\xi_\\sigma$, being a scalar multiple of $\\xi_\\sigma^1$, is contained in $V(\\mu(u))_{n-1}$. \n \n Next, Let $W$ be the image of the module $(V(\\mu(u))_{n-1})^{\\psi_{\\sigma}^1}$ under the isomorphism $\\mathrm{id}: (V(\\mu(u))_{(n-1)})^{\\psi_\\sigma^1}\\to (V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$. As $W$ is the irreducible submodule of $(V(\\mu(u))^{\\psi_\\sigma^n})_{(n-1)}$ generated by $\\xi_\\sigma$, it is equal to \n $(V(\\mu(u))^{\\psi_\\sigma^n})_{n-1}$.\n \n \\noindent \\textit{Step 3: } $\\mu^\\sharp_i(u)=\\mu_i(u)$ for all $2\\leq i\\leq n$. \n \n Since $V(\\mu(u))_{n-1}$ is a $X(\\mfso_3,\\mfso_2)^{tw}$ highest weight module, it is generated by monomials of the form \n $(s_{i_1j_1}^{n-1 (r_1)}\\cdots s_{i_c j_c}^{n-1 (r_c)}) \\cdot \\xi$ where $-1\\leq j_a\\tfrac{N}{4}$. Therefore, \n the condition $S(\\al,\\tfrac{N}{2}-\\al)\\cup S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})\\subset Z(P_2(u))$ is vacuous whenever $\\al\\leq \\tfrac{N}{4}$. \n\\end{rmk}\n\n\\begin{proof}[Proof of Proposition \\ref{P:q=1-nec}]\n It was shown in Proposition \\ref{P:int} that $\\al\\in \\frac{1}{2}\\Z+\\tfrac{N}{4}$. However, we will not assume this in our proof. As a consequence of Proposition \\ref{BI:P-psi}\n we have $V(\\mu(u))^{\\psi_\\sigma^n}\\cong V(\\mu^\\sharp(u))$ where $\\mu_{k}^\\sharp(u)=\\mu_k(u)$ for all $2\\leq k\\leq n$ and the pair $(\\mu_0^\\sharp(u),\\mu_1^\\sharp(u))$ is determined from the relations \\eqref{mu^sh:0} and \\eqref{mu^sh:1}. Since $\\mu(u)$ is associated to $(\\al,P_1(u),\\ldots,P_n(u))$, the components of $\\wt \\mu^\\sharp(u)$ satisfy the relations \n %\n \\begin{gather}\\label{sigmaPoly1}\n \\frac{\\wt \\mu_0^\\sharp(u)}{\\wt \\mu_1^\\sharp(u)}=\\frac{P_1(u+\\tfrac{1}{2})}{P_1(u)}\\cdot \\frac{(\\tfrac{N}{2}-\\al)-u}{u+(\\tfrac{N}{2}-\\al)-n}, \\quad \\frac{\\wt \\mu_1^\\sharp(u)}{\\wt \\mu_2^\\sharp(u)}=\\frac{P_2(u+1)}{P_2(u)}\\cdot \\frac{2u-2\\al+1}{2u+2\\al-N+1}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-N+2},\\\\\n \\frac{\\wt \\mu_{i-1}^\\sharp(u)}{\\wt \\mu_i^\\sharp(u)}=\\frac{P_i(u+1)}{P_i(u)}\\quad \\text{ for all }\\quad 3\\leq i\\leq n. \\label{sigmaPoly2}\n \\end{gather}\n %\n On the other hand, since $V(\\mu^\\sharp(u))$ is finite-dimensional Proposition \\ref{P:necessary} implies that $\\mu^\\sharp(u)$ can be associated to a Drinfeld tuple \n $(\\al^\\sharp,Q_1(u),\\ldots,Q_n(u))$. Consequently, from the second relation in \\eqref{sigmaPoly1} we obtain the equality \n %\n \\begin{equation*}\n \\frac{Q_2(u+1)}{Q_2(u)}\\cdot \\frac{(n-\\al)-u}{u+(n-\\al)-n+1}=\\frac{P_2(u+1)}{P_2(u)}\\cdot \\frac{(\\al-\\tfrac{1}{2})-u}{u+(\\al-\\tfrac{1}{2})-n+1}.\n \\end{equation*}\n %\n Applying Lemma \\ref{L:poly2} to both sides (with $m=1$ and $l=n$) we find that there exists monic polynomials $Q_2^\\bullet(u)$ and $P_2^\\bullet(u)$, together with non-negative integers \n $\\ell_P$ and $\\ell_Q$ such that $P_2^\\bullet(u)=P_2^\\bullet(-u+n)$, $Q_2^\\bullet(u)=Q_2^\\bullet(-u+n)$, and \n \\begin{equation*}\n \\frac{Q_2^\\bullet(u+1)}{Q_2^\\bullet(u)}\\cdot \\frac{(n-\\al-\\ell_Q)-u}{u+(n-\\al-\\ell_Q)-n+1}=\\frac{P_2^\\bullet(u+1)}{P_2^\\bullet(u)}\\cdot \\frac{(\\al-\\tfrac{1}{2}-\\ell_P)-u}{u+(\\al-\\tfrac{1}{2}-\\ell_P)-n+1}, \n \\end{equation*}\n with $Q_2^\\bullet(n-\\al-\\ell_Q)\\neq 0$ and $P_2^\\bullet(\\al-\\tfrac{1}{2}-\\ell_P)\\neq 0$. By Lemma \\ref{L:poly1}, we must have $Q_2^\\bullet(u)=P_2^\\bullet(u)$ and \n $n-\\al-\\ell_Q=\\al-\\tfrac{1}{2}-\\ell_P$. The latter relation implies that $2\\al-\\tfrac{N}{2}=\\ell_P-\\ell_Q\\in \\Z$, and thus that $\\al\\in \\frac{1}{2}\\Z+\\tfrac{N}{4}$.\n \n If in addition $\\al>\\tfrac{N}{4}$, then $\\ell_P\\geq \\ell_P-\\ell_Q=2\\al-\\tfrac{N}{2}>0$. Since $(\\ell_P,P_2^\\bullet(u))$ is the pair $(\\ell_{\\al-1\/2}^1,P_{\\al-1\/2}^1(u))$ from Lemma \\ref{L:poly2} (where $P(u)=P_2(u)$), $P_2^\\bullet(u)$ is equal to $P_2(u)$ divided by the polynomial $Q(u)$ from \\eqref{poly1'} with $m=1$ and $\\al$ replaced by $\\al-1\/2$. Therefore\n $P_2(u)$ is divisible by the polynomial \n %\n \\begin{equation}\\label{P-gamma}\n P_\\al(u)=\\prod_{k=0}^{2\\al-\\tfrac{N}{2}-1}(u-\\al+1\/2+k)(u-\\tfrac{N}{2}+\\al-k)=\\prod_{k=0}^{2\\al-\\tfrac{N}{2}-1}(u-\\al+1\/2+k)(u-\\al+1+k).\n \\end{equation}\n %\n The proof of the proposition is completed by observing that the roots of $P_\\al(u)$ are precisely the elements of $S(\\al,\\tfrac{N}{2}-\\al)\\cup S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})$.\n\\end{proof}\n \\begin{rmk}\nThe statement of Proposition \\ref{P:q=1-nec} is much stronger than that of Proposition \\ref{P:int} (in the case $(\\mfg_N,\\mfg_N^\\rho)=(\\mfso_{2n+1},\\mfso_{2n})$). The latter \ntells us that $\\al\\in \\frac{1}{2}\\Z+\\tfrac{N}{4}$ and that \n$\n\\al-\\tfrac{N}{4}\\leq \\tfrac{1}{4}(\\deg P_1(u)+\\deg P_2(u))\n$\nbut says nothing about the roots of $P_2(u)$. In fact, since the strings $S(\\al,\\tfrac{N}{2}-\\al)$ and $S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})$\nare disjoint and both have length $2\\al-\\tfrac{N}{2}$, Proposition \\ref{P:q=1-nec} implies that $\\al-\\tfrac{N}{4}\\leq \\tfrac{1}{4}\\deg P_2(u)$. \n\\end{rmk}\n\n\n\nProvided $\\al>\\tfrac{N}{4}$, the polynomial $P_\\al(u)$ from \\eqref{P-gamma} satisfies the relation \n\\begin{equation}\n \\frac{P_\\al(u)}{P_\\al(u+1)}= \\frac{2u-2\\al+1}{2u+2\\al-N+1}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-N+2}. \\label{P-gamma.2}\n\\end{equation}\nIf instead $\\al\\leq \\tfrac{N}{4}$, let $P_\\al^-(u)$ be the polynomial \n\\begin{equation*}\n P_\\al^-(u)=\\prod_{k=0}^{\\frac{N}{2}-2\\al-1}(u-n+\\al+k)(u-\\al-k)=\\prod_{k=0}^{\\frac{N}{2}-2\\al-1}(u-\\al-\\tfrac{1}{2}-k)(u-\\al-k),\n\\end{equation*}\nwhere the equality $P_\\al^-(u)=1$ is understood to hold if $\\al=\\tfrac{N}{4}$. Then $P_\\al^-(u)$ satisfies the relation \n\\begin{equation*}\n \\frac{P_\\al^-(u+1)}{P_\\al^-(u)}= \\frac{2u-2\\al+1}{2u+2\\al-N+1}\\cdot \\frac{2u-2\\al+2}{2u+2\\al-N+2}.\n\\end{equation*}\nThese observations together with the relations \\eqref{sigmaPoly1} and \\eqref{sigmaPoly2} imply the following corollary. \n\\begin{crl}\\label{C:Vtw-poly}\n Suppose that $V(\\mu(u))$ is finite-dimensional with Drinfeld tuple $(\\al,P_1(u),\\ldots,P_n(u))$. Then the Drinfeld tuple of the finite-dimensional irreducible \n module $V(\\mu(u))^{\\psi_\\sigma^n}$ is $(\\tfrac{N}{2}-\\al,P_1^\\sharp(u),\\ldots,P_n^\\sharp(u))$, where $P_i^\\sharp(u)=P_i(u)$ for all $i\\neq 2$ and \n %\n\\begin{equation}\\label{P2-sharp}\nP_2^\\sharp(u)=\n \\begin{cases}\n P_2(u)P_\\al^-(u) \\; & \\text{ if }\\; \\al\\leq \\tfrac{N}{4},\\\\\n P_2(u)\/P_\\al(u) \\; & \\text{ if }\\; \\al> \\tfrac{N}{4}.\n \\end{cases}\n\\end{equation}\n\\end{crl}\n\nObserve that these formulas together with those of Proposition \\ref{BI:P-psi} imply that, under the assumption that $V(\\mu(u))$ finite-dimensional, \nthe isomorphism $V(\\mu(u))\\cong V(\\mu(u))^{\\psi_{\\sigma}^n}$ will hold if and only if the scalar $\\al$ corresponding to $V(\\mu(u))$ is equal to $\\tfrac{N}{4}$. \n\n\n\n\\subsubsection{Classification}\n\n\nWith Proposition \\ref{P:q=1-nec} at our disposal we can now classify the finite-dimensional irreducible representations of the extended twisted Yangian \n$X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$. \n\\begin{thrm}\\label{T:BI(b)-Class}\nSuppose that $\\mu(u)$ satisfies the conditions of Proposition \\ref{P:nontriv} so that the Verma module $M(\\mu(u))$ is non-trivial. Then\nthe irreducible $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-module $V(\\mu(u))$ is finite-dimensional if and only if there are monic polynomials $P_1(u),\\ldots,P_n(u)$ in $u$, with\n\\begin{equation}\n P_i(u)=P_i(-u+n-i+2) \\; \\text{ for all }\\; i\\geq 2 \\; \\text{ and }\\; P_1(u)=P_1(-u+\\tfrac{N}{2}), \\label{T:BI(b).0}\n\\end{equation}\ntogether with a scalar $\\al\\in \\tfrac{1}{2}\\Z+\\tfrac{N}{4}$ such that the following relations are satisfied:\n\\begin{gather}\n\\al \\notin Z(P_1(u)),\\quad S(\\al,\\tfrac{N}{2}-\\al)\\cup S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})\\subset Z(P_2(u)), \\label{T:BI(b).1}\\\\[.5em]\n\\frac{\\wt \\mu_{i-1}(u)}{\\wt \\mu_i(u)}=\\frac{P_i(u+1-\\tfrac{\\delta_{i1}}{2})}{P_i(u)}\\left(\\frac{\\al-u}{\\al+u-n}\\right)^{\\del_{i,1}}\\text{ for all }\\;1\\leq i\\leq n. \\label{T:BI(b).2}\n\\end{gather}\nAdditionally, when $V(\\mu(u))$ is finite-dimensional the corresponding tuple $(\\al,P_1(u),\\ldots, P_n(u))$ is unique. \n\\end{thrm}\n\\begin{proof}\n $(\\Longrightarrow)$ If $V(\\mu(u))$ is finite-dimensional, then it follows from Propositions \\ref{P:necessary} and \\ref{P:q=1-nec} that $\\mu(u)$ can be associated to \n a (Drinfeld) tuple $(\\al,P_1(u),\\ldots,P_n(u))$ satisfying the relations of the theorem. The uniqueness statement has also been proven in Lemma \\ref{L:unique}. \n \n $(\\Longleftarrow)$ Conversely, suppose that $\\mu(u)$ can be associated to a tuple $(\\al,P_1(u),\\ldots,P_n(u))$ as in Definition \\ref{D:assoc} which also satisfies $\\al\\in \\tfrac{1}{2}\\Z+\\tfrac{N}{4}$ and the relation \\eqref{T:BI(b).1}. We will show that $V(\\mu(u))$ is finite-dimensional, splitting our proof into two cases. \n \n \\noindent \\textit{Case 1:} $\\al\\leq \\tfrac{N}{4}$. \n \n Let $P_i^\\circ(u)=P_i(u)$ for all $i\\neq 1$ and set $P_1^\\circ(u)$ to be the polynomial obtained by multiplying $P_1(u)$ with $\\prod_{k=0}^{N\/2-2\\al-1}(u-\\tfrac{N}{4}+\\frac{k}{2})(u-\\tfrac{N}{4}-\\frac{k}{2})$.\n Then $P_1^\\circ(u)$ satisfies $P_1^\\circ(u)=P_1^\\circ(-u+\\tfrac{N}{2})$ and the relation \n \\begin{equation}\n \\frac{P_1^\\circ(u+\\tfrac{1}{2})}{P_1^\\circ(u)}\\cdot\\frac{\\tfrac{N}{4}-u}{\\tfrac{N}{4}+u-n}=\\frac{P_1(u+\\tfrac{1}{2})}{P_1(u)}\\cdot \\frac{\\al-u}{\\al+u-n}. \\label{\\tfrac{N}{4}$.\n\nLet $\\mu^\\sharp(u)=(\\mu^\\sharp_i(u))_{i\\in \\mcI_N^+}$ be the tuple determined by $\\mu_i^\\sharp(u)=\\mu_i(u)$ for all $2\\leq i\\leq n$ and with \n$\\wt \\mu_0^\\sharp(u)$, $\\wt \\mu_1^\\sharp(u)$ given by the formulas \\eqref{mu^sh:0}, \\eqref{mu^sh:1}, respectively. Since $S(\\al,\\tfrac{N}{2}-\\al)\\cup S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})\\subset Z(P_2(u))$, the polynomial $P_\\al(u)$ from \\eqref{P-gamma} divides $P_2(u)$. By \\eqref{sigmaPoly1}, \\eqref{sigmaPoly2} and \\eqref{P-gamma.2}, $\\mu^\\sharp(u)$ is associated to $(\\tfrac{N}{2}-\\al,P_1^\\sharp(u),\\ldots,P_n^\\sharp(u))$, where $P_i^\\sharp(u)=P_i(u)$ for all $i\\neq 2$ and \n$P_2^\\sharp(u)=P_2(u)\/P_\\al(u)$. \n\nSince $\\tfrac{N}{2}-\\al<\\tfrac{N}{4}$, the argument of Case 1 implies that $V(\\mu^\\sharp(u))$ is finite-dimensional. By Proposition 4.7, \n$V(\\mu^\\sharp(u))^{\\psi_{\\sigma}^n}$ is isomorphic to $V(\\mu(u))$, and thus $V(\\mu(u))$ is also finite-dimensional. \n\\end{proof}\n\n\nA similar argument to that given after Theorem \\ref{T:DI(a)-Class} shows that, as a consequence of Theorem \\ref{T:BI(b)-Class}, the isomorphism classes of \nfinite-dimensional irreducible $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-modules are in bijective correspondence with tuples \n$(g(u); \\al,P_1(u),\\ldots,P_n(u))$, where \n\\begin{enumerate}\n \\item $P_1(u),\\ldots,P_n(u)$ are monic polynomials satisfying \\eqref{T:BI(b).0}, \\label{Xclassi}\n \\item $\\al\\in \\tfrac{1}{2}\\Z+\\tfrac{N}{4}$ is not a root of $P_1(u)$, \\label{Xclassii}\n \\item $S(\\al,\\tfrac{N}{2}-\\al)\\cup S(\\al+\\tfrac{1}{2},\\tfrac{N}{2}-\\al+\\tfrac{1}{2})\\subset Z(P_2(u))$, \\label{Xclassiii}\n \\item $g(u)$ is an element of $1+u^{-2}\\C[[u^{-2}]]$. \\label{Xclassiv}\n\\end{enumerate}\nMore explicitly, the correspondence assigns to $V(\\mu(u))$ the finite sequence $(g(u); \\al,P_1(u),\\ldots,P_n(u))$ where $(\\al,P_1(u),\\ldots,P_n(u))$ is the Drinfeld tuple \nassociated to $\\mu(u)$ and $g(u)$ is the unique series in $1+u^{-2}\\C[[u^{-2}]]$ such that the central series $w(u)$ acts as the identity operator in $V(\\mu(u))^{\\nu_g}$. \n\nThe next corollary provides a classification of finite-dimensional irreducible $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-modules, and it follows from the above classification of finite-dimensional irreducible $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-modules together with Proposition \\ref{P:Y^tw-fd}.\n\\begin{crl}\\label{C:BI(b)class} The isomorphism classes of finite-dimensional irreducible $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-modules are parameterized by\ntuples $(\\al,P_1(u),\\ldots,P_n(u))$ satisfying conditions \\eqref{Xclassi}-\\eqref{Xclassiii}.\n\\end{crl}\n\nWe now turn towards obtaining a result analogous to Corollary \\ref{C:DI(a)fun} for $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$. \n\nSince the automorphism $\\psi_\\sigma^n$ of $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ fixes $w(u)$, it induces an automorphism of the Yangian $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ which is given by the assignment $\\Sigma(u)\\mapsto A_\\sigma \\Sigma(u) A_\\sigma^t$ - see \\eqref{al_A}. We also denote this automorphism by $\\psi_\\sigma^n$. It is not difficult to see that the restriction of the irreducible $X(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-module $V(\\mu(u))^{\\psi_\\sigma^n}$ to the subalgebra $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ is isomorphic to $V^{\\psi_\\sigma^n}$, where \n$V$ is $V(\\mu(u))$ viewed as a $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$-module. \n\nGiven $m\\geq 0$, $1\\leq i_1,\\ldots,i_m\\leq n$, and $\\al_{i_1},\\ldots,\\al_{i_m}\\in \\C$, we can consider the tensor product of $Y(\\mfso_{2n+1})$ fundamental representations \n\\begin{equation}\n L(i_1:\\al_{i_1})\\otimes \\cdots \\otimes L(i_m:\\al_{i_m}). \\label{Y-fun}\n\\end{equation}\nIf $m=0$ then we we will identify this tensor product with the trivial representation of $Y(\\mfso_{2n+1})$. For each $1\\leq k\\leq m$, let \n$\\xi_i$ be a highest weight vector of $L(i_k:\\al_{i_k})$ and set \n\\begin{equation*}\n \\boldsymbol{\\xi}=\\xi_1\\otimes \\cdots \\otimes \\xi_{m}. \n\\end{equation*}\nIf $m=0$ the vector $\\boldsymbol{\\xi}$ is understood to be equal to $1\\in \\C$, where $\\C$ is viewed as the space of the trivial representation of $Y(\\mfso_{2n+1})$. We can then consider the \n$Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ highest weight module \n\\begin{equation}\n Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}\\boldsymbol{\\xi}\\subset L(i_1:\\al_{i_1})\\otimes \\cdots \\otimes L(i_m:\\al_{i_m}). \\label{BI(b)fun.1}\n\\end{equation}\n\n\\begin{crl}\\label{C:BI(b)fun}\t\n Let $V$ be a finite-dimensional irreducible representation of $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}$ with Drinfeld tuple $(\\al,P_1(u),\\ldots,P_n(u))$. Then \n \\begin{enumerate}[label=(\\alph*)]\n \\item $V$ is isomorphic to the unique irreducible quotient of a module of the form \\eqref{BI(b)fun.1} if and only if $\\al\\leq \\tfrac{N}{4}$, \\label{(a)}\n \\item $V^{\\psi_\\sigma^n}$ is isomorphic to the unique irreducible quotient of a module of the form \\eqref{BI(b)fun.1} if and only if $\\al \\geq \\tfrac{N}{4}$. \\label{(b)}\n \\end{enumerate}\n\\end{crl}\n\\begin{proof}\n If $V$ is isomorphic to the irreducible quotient of the module \\eqref{BI(b)fun.1}, Lemma \\ref{L:QxP=QP} and Remark \\ref{R:QxP=QP} with $V(\\mu(u))=V(\\mcG)$ imply that $\\al=\\tfrac{N}{4}-\\tfrac{\\ell_\\al}{2}$ where $\\ell_\\al$ is non-negative integer. This proves the \n$(\\Longrightarrow)$ direction of \\ref{(a)}. \n\nSuppose now that $V^{\\psi_\\sigma^n}$ is isomorphic to the irreducible quotient of the module \\eqref{BI(b)fun.1}. By Corollary \\ref{C:Vtw-poly}, $V^{\\psi_\\sigma^n}$ associated to the Drinfeld tuple $(\\tfrac{N}{2}-\\al,P_1^\\sharp(u),\\ldots,P_n^\\sharp(u))$ where $P_i^\\sharp(u)=P_i(u)$ for all $i\\neq 2$ and $P_2^\\sharp(u)$ is given by \\eqref{P2-sharp}. Hence, the same argument as given in the previous paragraph shows that $\\tfrac{N}{2}-\\al\\leq \\tfrac{N}{4}$, thus proving the $(\\Longrightarrow)$ direction of \\ref{(b)}. \n\nNow let us turn to proving the $(\\Longleftarrow)$ direction of \\ref{(a)} and \\ref{(b)}, beginning with the former. \nAssume that $\\al\\leq \\tfrac{N}{4}$. Then the argument used to treat Case 1 of the $(\\Longleftarrow)$ direction of Theorem \\ref{T:BI(b)-Class}'s proof shows that there is a tuple of monic polynomials $\\mathbf{Q}=(Q_1(u),\\ldots,Q_n(u))$ such that $V$ is isomorphic to the irreducible quotient of $Y(\\mfso_{2n+1},\\mfso_{2n})^{tw}\\xi_\\mathbf{Q}\\subset L(\\mathbf{Q})$, where $L(\\mathbf{Q})$ is the unique up to isomorphism $Y(\\mfso_{2n+1})$-module with the Drinfeld tuple $\\mathbf{Q}$ and $\\xi_\\mathbf{Q}\\in L(\\mathbf{Q})$ is a highest weight vector: see Theorem \\ref{T:X-class} and the two paragraphs immediately following it. The conclusion that $V$ is isomorphic to the unique irreducible quotient of a module of the form \\eqref{BI(b)fun.1} now follows from the well-known result that $L(\\mathbf{Q})$ must be isomorphic to a subquotient of a $Y(\\mfso_{2n+1})$-module of the form \\eqref{Y-fun}: see \\cite[Lemma 5.17]{AMR} and \\cite[Corollary 12.1.13]{CP}.\n\nIf instead $\\al \\geq \\tfrac{N}{4}$, then the argument of the previous paragraph can be used after replacing $V$ by $V^{\\psi_\\sigma^n}$ and using instead Case 2 of the $(\\Longleftarrow)$ direction of the proof of Theorem \\ref{T:BI(b)-Class}. This yields that $V^{\\psi_\\sigma^n}$ is isomorphic to the irreducible quotient of a module of the form \\eqref{BI(b)fun.1}. \n\\end{proof}\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\begin{figure*}\n\\centering\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=1em]\n Neo, \\& the \\& One, \\& is \\& STATE \\& a \\& hero, \\& for \\& chasing \\& this \\& army \\& of \\& Robots \\\\\n \\end{deptext}\n \n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{1}{3}{appos}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{3}{2}{det}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{7}{6}{det}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{11}{10}{det}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{13}{12}{case}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{9}{11}{dobj}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{11}{13}{nmod}\n \\depedge[edge below, label style={draw=black}, edge style={black, solid}]{9}{8}{mark}\n \\depedge[edge below, edge height=6ex, label style={draw=red}, edge style={red,densely dashed}]{7}{4}{cop}\n \\depedge[edge below, edge height=10ex, label style={draw=red}, edge style={red,densely dashed}]{7}{1}{nsubj}\n \\depedge[edge below, label style={draw=red}, edge style={red,densely dashed}]{7}{9}{advcl}\n \\depedge[edge height=12ex]{9}{3}{nsubj}\n \\storelabelnode\\secondlab\n \\depedge[edge height=9ex]{5}{1}{nsubj}\n \\depedge{5}{3}{nsubj}\n \\storelabelnode\\thirdlab\n \\depedge{5}{4}{ev}\n \\depedge{5}{7}{xcomp}\n \\depedge[edge height=9ex]{5}{9}{advcl}\n \\depedge[edge height=15ex]{9}{1}{nsubj}\n \\storelabelnode\\firstlab\n \\depedge{9}{13}{dobj}\n \\storelabelnode\\forthlab\n \\depedge{11}{13}{compound}\n \\storelabelnode\\fifthlab\n \\node (silly1) [above of = \\firstlab, yshift=-0.65cm, font=\\small] {\\S\\ref{sec:nested}};\n \\node (silly2) [above of = \\secondlab, yshift=-0.65cm, font=\\small] {\\S\\ref{sec:nested}};\n \\node (silly3) [above of = \\thirdlab, yshift=-0.65cm, font=\\small] {\\S\\ref{sec:parallel}};\n \\node (silly4) [above of = \\forthlab, yshift=-0.65cm, font=\\small] {\\S\\ref{sec:parallel}};\n \\node (silly5) [above of = \\fifthlab, yshift=-0.65cm, font=\\small] {\\S\\ref{sec:alternation}};\n \\node (silly6) [below of = \\wordref{1}{5}, yshift=0.65cm, font=\\small] {\\S\\ref{sec:state}};\n \\end{dependency}}\n \\caption{Representation of \\emph{Neo, the One, is a hero, for chasing this army of Robots}. The arcs above the sentence are \\textsc{BART}~ additions. The ones below are EUD. Red arcs are removed in \\textsc{BART}~ while black are retained.}\n \\label{fig:3}\n\\end{figure*}\n\nOwing to neural-based advances in parsing technology, NLP researchers and practitioners can now accurately produce syntactically-annotated corpora at scale. However, the use and empirical benefits of the dependency structures themselves remain limited. Basic syntactic dependencies encode the functional connections between words but lack many functional and semantic relations that exist between the content words in the sentence. \nMoreover, the use of strictly-syntactic relations results in structural diversity, undermining the efforts to effectively extract coherent semantic information from the resulting structures.\n\nThus, human practitioners and applications that ``consume'' these syntactic trees are required to devote substantial efforts to processing the trees in order to identify and extract the information needed for downstream applications, such as information and relation extraction (IE). Meanwhile, semantic representations \\cite{banarescu2013abstract, palmer2010semantic, abend2013universal, oepen-etal-2014-semeval} are harder to predict with sufficient accuracy, calling for a middle ground.\n\nIndeed, \\citet{de2008stanford} introduced \\emph{collapsed} and {\\em propagated} dependencies, in an attempt to make some semantic-like relations more apparent. The Universal Dependencies (UD) project\\footnote{\\url{universaldepdenencies.org}} similarly embraces the concept of \\emph{Enhanced Dependencies} \\cite{nivre-etal-2018-enhancing}), adding explicit relations that are otherwise left implicit. \n\\citet{schuster2016enhanced} provide further enhancements targeted specifically at English (Enhanced UD).\\footnote{In this paper we do not distinguish between the Universal Enhanced UD and \\citet{schuster2016enhanced}'s {Enhanced++} English UD. We refer to their union on English as Enhanced UD.} \\citet{candito-etal-2017-enhanced} suggest further enhancements to address diathesis alternations.\\footnote{Efforts such as PropS \\cite{stanovsky2016getting} and PredPatt \\cite{white-EtAl:2016:EMNLP2016}, share our motivation of extracting predicate-argument structures from treebank-trainable trees, though outside of the UD framework. Efforts such as KNext \\cite{knext} automatically extract logic-based forms by converting treebank-trainable trees, for consumption by further processing. HLF \\cite{hlf}, DepLambda \\cite{deplambda} and UDepLambda \\cite{udeplambda} attempt to provide\na formal semantic representation by converting dependency structures to logical forms. While they share a high-level goal with ours --- exposing functional relations in a sentence in a unified way --- their end result, logical forms, is substantially different from pyBART structures. While providing substantial benefits for semantic parsing applications, logical forms are less readable for non-experts than labeled relations between content words. As these efforts rely on dependency trees as a backbone, they could potentially benefit from pyBART's focus on syntactic enhancements on top of (E)UD.}\n\nIn this work we continue this line of thought, and take it a step further. We present pyBART, an easy-to-use Python library which converts English UD trees to a new representation that subsumes the English Enhanced UD representation and substantially extends it. We designed the representation to be linguistically sound and automatically recoverable from the syntactic structure, while exposing the kinds of relations required by IE applications.\nSome of these modifications are illustrated in Figure \\ref{fig:3}.\\footnote{Some preserved UD relations are omitted for readability.} \nWe aim to make event structure explicit, and cover as many linguistically plausible phenomena as possible.\nWe term our representation \\textsc{BART}~ (The BIU-AI2 Representation Transformation).\n\nTo assess the benefits of \\textsc{BART}~ with respect to UD and other enhancements, \nwe compare them in the context of a pattern-based relation extraction task, and demonstrate that \\textsc{BART}~ achieves higher $F_1$ scores while requiring fewer patterns.\n\nThe python conversion library, \\texttt{pyBART}, integrates with the spaCy\\footnote{\\url{https:\/\/spacy.io}} library, and is available under an open-source Apache license. A web-based demo for experimenting with the converter is also available. \\url{https:\/\/allenai.github.io\/pybart\/}.\n\n\\section{The \\textsc{BART}~ Representation}\nWe aim to provide a representation that will be useful for downstream NLP tasks, while retaining the following key properties. The proposal has to be {\\bf (i) based on syntactic structure} and {\\bf (ii) useful for information seeking applications}. As a consequence of (ii), we also want it to {\\bf (iii) make event structure explicit} and {\\bf (iv) allow favoring recall over precision}.\n\nBeing \\textbf{based on syntax} as the backbone would allow us to capitalize on independent advances in syntactic parsing, and on its relative domain independence. We want our representation to be not only accurate but also \\textbf{useful for information seeking applications}. This suggests a concrete methodology (\\S\\ref{sec:method}) and evaluation criteria (\\S\\ref{sec:eval}): we choose which relations to focus on based on concrete cases attested in relation extraction and QA-corpora, and evaluate the proposal based on the usefulness in a relation extraction task.\n\nIn general, information-seeking applications favor \\textbf{making events explicit}. Current syntactic representations prefer to assign syntactic heads as root predicates, rather than actual eventive verb. In contrast, we aim to center our representation around the main event predicate in the sentence, while indicating event properties such as aspectuality (\\emph{Sam \\underline{started} walking}) or evidentiality (\\emph{Sam \\underline{seems} to like them}) as modifiers of rather than heads. To do this in a consistent manner, we introduce a new node of type STATE for copular sentences, making their event structure parallel to those containing finite eventive verbs (\\S\\ref{sec:state})\n\nFinally, downstream users may prefer to \\textbf{favor recall over precision} in some cases. To allow for this, we depart from previous efforts that refrain from providing any uncertain information. We chose to explicitly expose some relations which we believe to be useful but judge to be uncertain, while clearly marking their uncertainty in the output. This allows users to experiment with the different cases and assess the reliability of the specific constructions in their own application domain. We introduce two uncertainty marking mechanisms, discussed in \\S\\ref{sec:unc}.\n\n\\subsection{Data-driven Methodology}\n\\label{sec:method}\nOur departure point is the English EUD representation \\cite{schuster2016enhanced} and related efforts discussed above, which we seek to extend in a way which is useful to information seeking applications. To identify relevant constructions that are not covered by current representations, we use a data-driven process. We consider concrete relations that are expressed in annotated task-based corpora: a relation extraction dataset (ACE05, \\cite{walker2006ace}), which annotates relations and events, and a QA-SRL dataset \\cite{he-etal-2015-question} which connects predicates to sentence segments that are perceived by people as their (possibly implied) arguments.\nFor each of these corpora, we consider the dependency paths between the annotated elements, looking for cases where a direct relation in the corpus corresponds to an indirect dependency path in the syntactic graph. We identify recurring cases that we think can be shortened, and which can be justified linguistically and empirically. We then come up with proposed enhancements and modifications, and verify them empirically against a larger corpus by extracting cases that match the corresponding patterns and browsing the results. \n\n\\subsection{Formal Structure} \\label{formal}\nAs is common in dependency-based representations, \\textsc{BART}~ structures are labeled, directed multi-graphs whose nodes are the words of a sentence, and the labeled edges indicate the relations between them. Some constructions add additional nodes, such as copy-nodes \\cite{schuster2016enhanced} and STATE nodes (\\S\\ref{sec:state}).\n\nAn innovative aspect of our approach is that each edge is associated with additional information beyond its dependency label. This information is structured as follows:\n\n\\noindent{\\bf \\textsc{Src}}: a field indicating the origin of this edge---either ``UD'' for the original dependency edges, or a pair indicating the type and sub-type of the construction that resulted in the \\textsc{BART}~ edge (e.g., \\{\\textsc{Src=}(conj,and)\\} or \\{\\textsc{Src=}(adv,while)\\}).\n\n\\noindent{\\bf \\textsc{Unc}, \\textsc{Alt}}: optional fields indicating uncertainty, described below.\n\n\\subsection{Embracing uncertainty}\n\\label{sec:unc}\nSome syntactic constructions are ambiguous with respect to the ability to propagate information through them. Rather than giving up on all ambiguous constructions, we opted to generate the edges and mark them with an \\textsc{Unc=True} flag, deferring the decision regarding the validity of the edge to the user:\n\\\\[0.5em]\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n She \\& acted, \\& trusting \\& her \\& instincts \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{3}{1}{nsubj \\{\\textsc{Unc}\\}}\n \\depedge[edge height=2ex, edge below, label style={draw=black}, edge style={black, solid}]{2}{3}{dep}\n \\depedge[edge height=2ex, edge below, label style={draw=black}, edge style={black, solid}]{2}{1}{nsubj}\n \\end{dependency}}\n\nIn some cases, we can identify that one of two options is possible, but cannot determine which. In these cases we report both edges, but mark them explicitly as alternatives to each other. This is achieved with an \\textsc{Alt=X} field on both edges, with \\textsc{X} being a number indicating the pair.\\\\[0.5em]\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=-0.1em]\n You \\& saw \\& me \\& while \\& driving, \\& Sue \\& saw \\& Sam \\& after \\& returning \\& \\\\\n \\end{deptext}\n \\depedge[edge height=3ex]{5}{3}{nsubj\\{\\textsc{Alt=0}\\}} \n \\depedge[edge height=5.5ex]{5}{1}{nsubj\\{\\textsc{Alt=0}\\}}\n \\depedge[edge height=3ex]{10}{8}{nsubj\\{\\textsc{Alt=1}\\}} \n \\depedge[edge height=5.5ex]{10}{6}{nsubj\\{\\textsc{Alt=1}\\}}\n \\end{dependency}}\n\n\\begin{figure}[t]\n\\begin{lstlisting}[language=Python]\n# Load a UD-based english model\nnlp = spacy.load(\"en_ud_model\")\n\n# Add BART converter to spaCy's pipeline\nfrom pybart.api import converter\nconverter = converter( ... )\nnlp.add_pipe(converter, name=\"BART\")\n\n# Test the new converter component\ndoc = nlp(\"He saw me while driving\")\nme_token = doc[2]\nfor par_tok in me_token._.parent_list:\n print(par_tok)\n\n# Output:\n{'head': 2, 'rel':'dobj', 'src':'UD'}\n{'head': 5, 'rel': 'nsubj',\n 'src':('advcl','while'), 'alt':'0'}\n\\end{lstlisting}\n\\caption{Usage example of pyBART's spaCy-pipeline component.}\n\\label{fig:5}\n\\end{figure}\n\n\\section{Python code and Web-demo}\n\\label{sec:software}\n\nThe py\\textsc{BART}~ library provides a Python converter from English UD trees to BART.\npy\\textsc{BART}~ subsumes the enhancements of the EUD Java implementation provided in Stanford Core-NLP,\\footnote{\\url{https:\/\/nlp.stanford.edu\/software\/stanford-dependencies.html}} and extends them as described in \\S\\ref{structures}. While pyBART's default performs all enhancements, it can be configured to follow a more selective behavior. py\\textsc{BART}~ has two modes: (1) a converter from CoNLLU-formatted UD trees to CoNLLU-formatted \\textsc{BART}~ structures;\\footnote{The extra edge information is linearized into the dependency label after a `@` separator.} and (2) a spaCy \\cite{spacy2} pipeline component.\\footnote{This requires a spaCy model trained to produce UD trees, which we provide.} After registering py\\textsc{BART}~ as a spaCy pipeline, tokens on the analyzed document will have a \\texttt{.\\_.parent\\_list} field, containing the list of parents of the token in the \\textsc{BART}~ structure. Each item is a dictionary specifying---in addition to the parent-token id and dependency label---also the extra information described in \\S\\ref{formal}. See Figure \\ref{fig:5} for an illustration of the API. \n\nA web-based demo that parses sentences into both EUD and \\textsc{BART}~ graphs, visualizes them, and compares their outputs, is also provided.\\footnote{The dependency graph visualization component uses the TextAnnotationGraphs (TAG) library \\cite{TAG-2018}.} \n\n\\section{Coverage of Linguistic Phenomena} \\label{structures}\n\n\\textsc{BART}~ conversion consists of four conceptual changes from basic UD. The first type propagates shared arguments between predicates in\n{\\bf nested structures}. \nThe second type shares arguments between {\\bf parallel structures}.\nThe third type attempts to unify {\\bf syntactic alternations} to reduce diversity, making structures that carry similar meaning also similar in structure. Finally, the forth type is designed to make {\\bf event structure} explicit in the syntactic representation, \nallowing finite verbs that indicate event properties to act as event modifiers rather than root predicates. In accordance with that, we further introduce a new STATE node, that acts as the main predicate node for {\\em stative} (copular, verb-less) sentences.\\vspace{-0.15em}\n\n\\subsection{Nested Structures}\n\\label{sec:nested}\nOur first type of conversions propagates an external core argument to be explicitly linked as the subject of a subordinate clause.\n\n\\noindent{\\bf Complement control:}\nThe various EUD representations explicitly indicate the external subjects of \\emph{xcomp} clauses containing a \\emph{to} marker. We embrace this choice and extend it to cover also clauses without a \\emph{to} marker, including imperative clauses and clauses with controlled gerunds.\\vspace{0.5em}\n\\begin{equation}\n\\label{complement-control1}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.4cm]\n Let \\& my \\& people \\& go! \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{4}{3}{nsubj}\n \\depedge[edge height=2ex, label style={draw=black}, edge style={black, solid}]{1}{3}{dobj}\n \\depedge[edge height=2ex, edge below, label style={draw=black}, edge style={black, solid}]{1}{4}{xcomp}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Noun-modifying clauses:} Similarly, EUD links the empty subject of a finite relative clause to the corresponding argument of the external clause. We extend this behavior to also cover \\textbf{reduced relative clauses} (\\ref{rrr-participle}a), and we follow \\citet{candito-etal-2017-enhanced} in also including other relative clauses such as \\textbf{noun-modifying participles} (\\ref{rrr-participle}b).\n\\vspace{0.5em}\n\\begin{equation}a.\\hspace{7em} b.\\hspace{8em}\n\\label{rrr-participle}\n\\end{equation}\n\\scalebox{0.80}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}\n The \\& neon \\& god \\& they \\& made \\hspace{1em} \n \\& A \\& vision \\& softly \\& creeping \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{5}{3}{dobj}\n \\depedge[edge height=2ex]{9}{7}{nsubj}\n \\end{dependency}}\n\n\\noindent{\\bf Adverbial clauses and ``dep'':} Adverbial modifier clauses that miss a subject, often modify the subject of the main clause. We propagate the external subject to be the subject of the internal clause.\\footnote{In external clauses that include a subject and an object, ambiguity may arise as to which is to be modified. We propagate both and mark the edges as alternates (\\textsc{Alt}, (\\S\\ref{sec:unc})).}\n\\vspace{0.5em}\\begin{equation} \\label{advcl}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}\n You \\& shouldn't \\& text \\& while \\& driving \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{5}{1}{nsubj}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\nWe observe that many \\textbf{dep} edges empirically behave like adverbial clauses, and treat them similarly. We mark these edges as ``uncertain''.\n\n\\subsection{Parallel structures}\n\\label{sec:parallel}\nThe second type of conversions identifies parallel structures in which the latter instance is elliptical, and share the missing core argument contributed by the former instance.\n\n\\noindent{\\bf Apposition:} Similarly to the PropS proposal \\cite{stanovsky2016getting}, we share relations across {\\em apposition parts}, making the two, currently hierarchical, phrase, more duplicate-like.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}\n E.T., \\& the \\& Extraterrestrial, \\& phones \\& home \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{4}{3}{nsubj}\n \\depedge[edge below, edge height=2.5ex, label style={draw=black}, edge style={black, solid}]{4}{1}{nsubj}\n \\depedge[edge height=2ex, label style={draw=black}, edge style={black, solid}]{1}{3}{appos}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Modifiers in conjunction:} In modified coordinated constructions, we share prepositional (\\ref{mogly}) and possessive (\\ref{parents}) modifiers between the coordinated parts. Since dependency trees are {\\em inherently} ambiguous between conjoined modification and single-conjunt modification, (e.g, compare (\\ref{mogly}) to ``Mogly was lost and raised by wolves\", or (\\ref{parents}) to ``my Father and E.T.\"), we mark both as \\textsc{Unc}.\n\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n I \\& was \\& taught \\& and \\& raised \\& by \\& wolves \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{5}{7}{nmod(UNC)}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{3}{7}{nmod}\n \\end{dependency}}\n\\end{gathered}\n\\label{mogly}\n\\end{equation}\n\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.4cm]\n My \\& father \\& and \\& mother \\& met \\& here \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{4}{1}{nmod:poss(UNC)}\n \\depedge[edge below, edge height=3ex, label style={draw=black}, edge style={black, solid}]{2}{1}{nmod:poss}\n \\end{dependency}}\n\\end{gathered}\n\\label{parents}\n\\end{equation}\n\n\\noindent{\\bf Elaboration\/Specification Clauses:} For noun nominal modifiers that have the form of an {\\em elaboration} or {\\em specification}, we share the head of the modified noun with its dependent modifier. That is, if the modification is marked by \\emph{like} or \\emph{such as} prepositions, we propagate the head noun to the nominal dependent.\\vspace{0.5em}\n\\begin{equation}\n\\label{elaboration}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n I \\& enjoy \\& fruits \\& such \\& as \\& apples \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{2}{6}{dobj}\n \\depedge[edge height=2ex, edge below, label style={draw=black}, edge style={black, solid}]{2}{3}{dobj}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Indexicals:} the interpretation of locative and temporal indexicals such as \\emph{here}, \\emph{there} and \\emph{now} depends on the situation and the speaker, and often modify not only the predicate but the entire situation. We therefore share the adverbial modification from the noun to the main verb. Due to their situation-specific nature, we mark these as \\textsc{Unc}.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.3cm]\n He \\& wonders \\& in \\& these \\& woods \\& here \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{2}{6}{advmod(UNC)}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{2}{5}{nmod}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{5}{6}{advmod}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Compounds:} \\citet{shwartz-waterson-2018-olive} show that in many cases, compounds can be seen as having a multiple-head. Therefore, we share the existing relations across the compound parts.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.4cm]\n I \\& used \\& canola \\& oil \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{2}{3}{dobj(UNC)}\n \\depedge[edge below,edge height=2ex, label style={draw=black}, edge style={black, solid}]{2}{4}{dobj}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent As many compounds \\emph{do} have a clear head (e.g. \\emph{I used baby oil}, where \\emph{baby} is clearly not the head), we mark these as uncertain.\n\n\\subsection{Syntactic Alternations}\n\\label{sec:alternation}\nThis type of conversions aim to unify syntactic variability. We identify structures that are syntactically different but share (some) semantic structure, and add arcs or nodes to expose the similarity.\n\n\\noindent{\\bf The Passivization Alternation:}\nFollowing \\citet{candito-etal-2017-enhanced} we relate the \\emph{passive} alteration to its \\emph{active} variant.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}\n The \\& Sheriff \\& was \\& shot \\& by \\& Bob \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{4}{6}{nsubj}\n \\depedge[edge height=2ex]{4}{2}{dobj}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{4}{2}{nsubjpass}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{4}{6}{nmod:by}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Hyphen reconstruction:} Noun-verb Hyphen Constructions (HC) which are modifying a nominal can be seen as conveying the same information as a copular sentence wherein the noun is the subject and the verb is the predicate. To explicitly indicate this, \nwe add to all modifying noun-verb HCs a {\\em subject} and a {\\em modifier} relation originating at the verb-part of the HC.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.4cm]\n A \\& Miami \\& - \\& based \\& company \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{4}{5}{nsubj}\n \\depedge[edge height=2ex]{4}{2}{nmod}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{5}{4}{amod}\n \\depedge[edge below, edge height=2ex, label style={draw=black}, edge style={black, solid}]{4}{2}{compound}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Adjectival modifiers:}\nAdjectival modification can be viewed as capturing the same information as a predicative copular sentence conveying the same meaning (so, ``a green apple'' implies that ``an apple is green''). To explicitly capture this productive implication, we add a subject relation from each adjectival modifier to its corresponding modified noun.\\vspace{0.5em}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n I \\& see \\& dead \\& people \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{3}{4}{nsubj}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Genitive Constructions:} Genitive cases can be alternatively expressed as a compound. We add a compound relation to unify the expression of genitives across {\\em X of Y} and {\\em compound} structures.\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n Army \\& of \\& zombies \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{1}{3}{compound}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\subsection{\\bf Event-Centered Representations}\\vspace{-0.5em}\\label{sec:state}\nIn many sentences, the finite root predicate does not indicate the main event. Instead, a verb in the subordinated clause\nexpresses the event, and the finite verb acts as its modifier.\nFor example, in sentences like ``He started working\", ``He seems to work there\", the main event indicated is ``work\", while the root predicates (``started'', ``seemed'') modify this event.\nHere, we present a chain of changes that puts emphasis on {\\em events} by delegating copular and tense auxiliaries (is, was), evidentials (seem, say) and various aspectual verbs (started, continued) to be clausal modifiers, rather than heads of the sentence.\nThis creates a further challenge, since there is a prevalent discrepancy between predicative sentences such as ``He works\" and copular sentences as ``He is smart\". The UD structure for the latter lacks a node that clearly indicates a {\\em stative} event (in \\citet{vendler}'s terminology). We remedy this by adding a node to represent the STATE and have tense, aspect, modality and evidentiality directly modifying it.\\footnote{Pragmatically, some users prefer to not have non-word nodes. pyBART supports this by providing a mode that treats the copula as the head, retaining the other modifications.}\n\n\\noindent{\\bf Copular Sentences and Stative Predicates:}\nWe added to all copula constructions new node named \\emph{STATE}, which represents the {\\em stative} event introduced by the copular clause. This node becomes the root, and we rewire the entire clause around this \\emph{STATE}. By doing so we unify it with the structures of clauses with finite predicative. Once we added the \\emph{STATE} node, we form a new relation, termed \\emph{ev}, to mark event\/state modifications.\nThe resulting structure is as follows:\\vspace{0.5em} \n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.2cm]\n Tomorrow \\& is \\& STATE \\& another \\& day \\& \\\\\n \\end{deptext}\n \\depedge[edge height=4ex]{3}{1}{nsubj}\n \\depedge[edge height=2ex]{3}{2}{ev}\n \\depedge[edge height=2ex]{3}{5}{xcomp}\n \\depedge[edge below, edge height=4ex, label style={draw=red}, edge style={red,densely dashed}]{5}{1}{nsubj}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{5}{2}{cop}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\noindent{\\bf Evidential reconstructions:}\nWe can now explicitly mark properties of events as dependents of the verbal or stative root by means of the label {\\em ev}. We do so, using verbs' white-lists, for verbs marking evidentiality (\\ref{state-ev}) and for reported-speech (\\ref{state-rep}).\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.85}{\n\\vspace{-0.5em}\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}\n Sam \\& seems \\& to \\& like \\& them. \\& \\& They \\& seem \\& STATE \\& nice. \\\\\n \\end{deptext}\n \\depedge[edge height=4ex]{4}{1}{nsubj}\n \\depedge[edge height=2ex]{4}{2}{ev}\n \n \\depedge[edge height=4ex]{9}{7}{nsubj}\n \\depedge[edge height=2ex]{9}{8}{ev}\n \\depedge[edge height=2ex]{9}{10}{xcomp}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{2}{1}{nsubj}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{2}{4}{xcomp}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{8}{7}{nsubj}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{8}{10}{xcomp}\n \\end{dependency}}\n\\end{gathered}\n\\label{state-ev}\n\\end{equation}\n\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.85}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=-0.05cm]\n The \\& Media \\& reported \\& that \\& peace \\& was \\& achieved \\\\\n \\end{deptext}\n \\depedge[edge height=2ex]{7}{3}{ev}\n \\depedge[edge below, edge style={black, solid}, edge height=2ex, label style={draw=black}]{3}{7}{ccomp}\n \\end{dependency}}\n\\end{gathered}\n\\label{state-rep}\n\\end{equation}\n\n\\noindent{\\bf Aspectual constructions:}\nFinally, we can now also mark aspectual verbs as modifying the complement (matrix) verb denoting the main event. The complement (matrix) verb becomes the root of the dependency structure, and we add the new \\emph{ev} relation to mark the aspectual modification of the event.\n\\vspace{0.5em}\\begin{equation}\n\\begin{gathered}\n\\scalebox{0.8}{\n \\begin{dependency}[theme = default, label style={draw=blue}, edge style={thick, blue, dotted}]\n \\begin{deptext}[column sep=0.4cm]\n He \\& started \\& talking \\& funny \\\\\n \\end{deptext}\n \\depedge[edge height=4ex]{3}{1}{nsubj}\n \\depedge[edge height=2ex]{3}{2}{ev}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{2}{1}{nsubj}\n \\depedge[edge below, edge height=2ex, label style={draw=red}, edge style={red,densely dashed}]{2}{3}{xcomp}\n \\end{dependency}}\n\\end{gathered}\n\\end{equation}\n\n\\section{Evaluation}\n\\label{sec:eval}\nOur proposed representation attempts to target information-seeking applications, but is it effective? We evaluate the resulting graph structures against the UD and Enhanced UD representations, in the context of a relation-extraction (RE) task. \nConcretely, we evaluate the representations on their ability to perform pattern-based RE on the TACRED dataset \\cite{zhang2017tacred}.\n\nWe use an automated and reproducible methodology: for each of the representations, we use the RE train-set to acquire extraction patterns. We then apply the patterns to the dev-set, compute F1-scores, and, for each relation, filter the patterns that hurt F1-score. We then apply the filtered pattern-set to the test-set, and report F1 scores.\n\nTo acquire extraction patterns, we use the following procedure: given a labeled sentence consisting of a relation name and the sentence indices of the two entities participating in the relation, we compute the shortest dependency path between the entities, ignoring edge directions. We then form an extraction pattern from the directed edges on this path. We consult a list of trigger words \\cite{yu-etal-2015-read} collected for the different relations. If a trigger word or its lemma is found on the path, we form an unlexicalized path except for the trigger word (i.e. {\\small\\textsf{E1 $<$nsubj ``founded'' $>$dobj $>$compound E2}}).\nIf no trigger-word is found, the path is lexicalized with the word's lemmas (i.e. {\\small\\textsf{E1 $<$nsubj ``reduce'' $>$dobj ``activity'' $>$compound E2}}).\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{||l c c c||} \n \\hline\n Representation & Precision & Recall & F1 \\\\ [0.5ex] \n \\hline\\hline\n UD & 76.53 & 30.65 & 43.77 \\\\ \n Enhanced UD & 77.63 & 32.37 & 45.69 \\\\\n Ours(w\/o-Enhanced) & 73.96 & 33.48 & 46.09 \\\\\n Ours & 74.62 & 36.65 & {\\bf 49.15} \\\\ [1ex] \n \\hline\n\\end{tabular}}\n\\caption{Effectiveness of the different representations on the TACRED relation extraction task.}\n\\label{table:1}\n\\end{table}\n\nWe use this procedure to compare UD, Enhanced UD (EUD), \nBART without EUD enhancements, and full BART, which is a superset of Enhanced UD (Table \\ref{table:1}). BART achieves a substantially higher F1 score of 49.15\\%, an increase of 5.5 F1 points over UD, and $3.5$ F1 points above Enhanced UD. It does so by substantially improving recall while somewhat decreasing precision.\n\nWe also consider \\emph{economy}: the number of different patterns needed to achieve a given recall level.\nFigure \\ref{fig:1} plots the achieved recall against the number of patterns.\nAs the curves show, Enhanced UD is more economic than UD, and our representation is substantially more economic than both.\nTo achieve 30.7\\% recall (the maximal recall of UD), UD requires 112 patterns, EUD requires 77 patterns, while \\textsc{BART}~ needed only 52 patterns.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/economy_example.png}\n \\caption{Economy comparison: Recall vs number of patterns, for the different representations.} \n \\label{fig:1}\n\\end{figure}\n\n\\section{Conclusion}\nWe propose a syntax-based representation that aims to make the event structure and as many lexical relations as possible explicit, for the benefit of downstream information-seeking applications. We provide a Python API that converts UD trees to this representation, and demonstrate its empirical benefits on a relation extraction task.\n\\section*{Acknowledgements}\nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon2020 research and innovation programme, grant agreement 802774 (iEXTRACT) and grant agreement 677352 (NLPRO).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe time required for stars in the stellar halo to exchange their\nenergy and angular momentum is very long compared to the age of the\nGalaxy. Therefore, such stars preserve memories of their initial\nconditions, and so the structure of the stellar halo is intimately\nlinked to the formation mechanism of the Galaxy itself. This\nfundamental insight was already noted by \\cite{eggen62}. It is the reason why the stellar halo has attracted such\ninterest despite containing only a small fraction of the total stellar\nmass of the Galaxy. Stars diffuse more quickly in configuration space,\nas opposed to energy and angular momentum space. So, the spatial\nstructure of the stellar halo may be smooth, even though it is built\nup from merging and accretion.\n\nThe simplest way of studying the stellar halo is through\nstarcounts. Typically, RR Lyrae or blue horizontal branch stars (BHBs)\nare used as tracers, as they are relatively bright ($M_g \\sim 0.5-0.7$, e.g. \\citealt{sirko04}) and can be detected\nat radii out to $\\sim$ 100 kpc. The gathering of such data is\npainstaking work, and carries the price that sample sizes are often\nsmall. Such studies are consistent with a stellar halo that is round\nin the outskirts (with a minor-to-major axis ratio $q =1$) and more\nflattened in the inner parts with $q \\sim 0.5$ (e.g.,\n\\citealt{hartwick87}; \\citealt{preston91}). Rather than selecting\ntypical halo stars, an alternative approach is to model deep star\ncount data in pencil-beam surveys at intermediate and high galactic\nlatitudes, allowing for contamination of the starcounts by the thin\nand thick disk populations. This was attempted by \\cite{robin00}, who\nfound a best-fit halo density law with flattening $q \\sim 0.76$,\ntogether with a power-law fall-off $\\alpha$ of $2.4$ (that is, $\\rho\n\\sim ({\\rm distance)}^{-\\alpha}$). A similar, slightly later, attempt\nby \\cite{siegel02} using data in seven Kapteyn selected areas\nyielded $q \\sim 0.6$ and $\\alpha \\sim 2.75$.\n\nEfforts to detect variations in the flattening with radius have also\nbeen undertaken. \\cite{preston91} argued that the flattening changes\nfrom strongly flattened ($q=0.5$) at 1 kpc to almost round at $20$\nkpc. However, work by \\cite{sluis98} using a compilation of 340 RR\nLyraes and BHBs found a constant flattening of $q \\sim 0.5$ with no\nevidence for changes with radius, as well as and a power-law index\n$\\alpha \\sim -3.2$. The most recent work by \\cite{propris10}\nutilising 666 BHB stars from the 2dF Quasar Redshift Survey find that\nthe halo is approximately spherical with a power-law index of\n$\\alpha=2.5$ out to $\\sim 100$ kpc. Similarly, \\cite{sesar11}\nstudying Main Sequence Turn-Off (MSTO) stars from the\nCanada-France-Hawaii Telescope Legacy Survey find that the flattening\nis approximately constant at $q \\sim 0.7$ out to $35$ kpc.\n\nThe Sloan Digital Sky Survey (SDSS) has transformed our knowledge of\nthe stellar halo. Although it had been suspected that the stellar halo\nis criss-crossed with streams and substructures ever since the\ndiscovery of the disrupting Sagittarius (Sgr), the SDSS provided a\nmemorable picture of the debris in {\\it The Field of Streams}\n(\\citealt{belokurov06}). A wealth of substructure has now been\nidentified, including the Sagittarius stream, the Virgo Overdensity\nand the Hercules-Aquila Cloud (e.g. \\citealt{ibata95};\n\\citealt{belokurov06}; \\citealt{juric08}; \\citealt{Be07}). This has\nbeen seen as vindication of modern theories of galaxy formation, which\npredict that stellar haloes are built up almost exclusively from the\ndebris of disrupting satellites (e.g. \\citealt{bullock05};\n\\citealt{delucia08}; \\citealt{cooper10}). A number of studies have\nattempted to model the smooth halo component by avoiding these known\nsubstructures (e.g. \\citealt{juric08}). The results are only in very\nrough agreement, suggesting that the density profile of the Milky Way\nhas a power-law slope in the range $2 < \\alpha < 4$ and a flattening\nvarying from $0.4 < q < 0.8$ (e.g. \\citealt{yanny00};\n\\citealt{chen01}; \\citealt{newberg06}; \\citealt{juric08};\n\\citealt{sesar11}).\n\nEven though panoramic photometric surveys like SDSS do provide a large\nsample of stellar halo tracers over a considerable portion of the sky,\nthere is no consensus on the flattening and shape of the stellar halo.\nMSTO stars are commonly used tracers owing to their large numbers and\nthe ease by which they can be photometrically identified\n(e.g. \\citealt{bell08}). The absolute magnitudes of such stars centre\naround $M_r \\sim 4.5$ but have a wide range of values ($\\sigma_{M_r}\n\\sim 0.9$ mag) which limits the accuracy to which the density profile\ncan be estimated. BHB stars are superior\ndistance estimators ($\\sigma_{M_r} \\sim 0.15$), but are significantly\nscarcer than main sequence stars. Moreover, they suffer from\ncontamination by blue straggler (BS) stars due to their similar\ncolours. BHB and BS stars can be distinguished by their Balmer line\nprofiles (e.g. \\citealt{kinman94}; \\citealt{yanny00};\n\\citealt{sirko04}; \\citealt{clewley02}), but this requires\nspectroscopic information. Whilst spectroscopic samples can cleanly\nidentify BHB stars, the variety of results on flattening (e.g.,\n\\citealt{preston91}; \\citealt{sluis98}; \\citealt{propris10}) and density fall-off (e.g. \\citealt{xue08}; \\citealt{brown10}) suggests\nthat the completeness biases are difficult to understand and control.\n\nAn independent constraint on the density profile of the stellar halo\nis provided by the velocity distribution of the halo stars. A\nkinematic analysis by \\cite{carollo07} (see also \\citealt{carollo10})\nsuggests that the stellar halo comprises of two components with\ndifferent density profiles and metallicities. The authors find that\nthe density profile becomes shallower beyond $15-20$ kpc. This is in\nstark contrast to the studies by \\cite{watkins09} and \\cite{sesar10},\nwho find a significantly steeper density profile beyond $\\sim 30$ kpc\nfrom the distribution of RR Lyrae stars in SDSS Stripe 82. A caveat to\nthe interpretation of kinematic studies is that the density\ndistribution is not measured directly but rather modelled by assuming\na dark matter halo potential.\n\nTherefore, the present state-of-play is distressingly inconclusive and\na further attack on the problem of the shape of the stellar halo is\nwarranted. In this study, we introduce a new method to model both BHB\nand BS stars based on photometric information alone. We make use of\nthe SDSS DR8 photometric data release which has now mapped an\nimpressive $\\sim 14,000$ deg$^2$ of sky with both Northern and\nSouthern coverage. In contrast to previous work, we combine both the\nwide sky coverage of the SDSS with the accurate distance estimates\nprovided by the BHB stars to model the density profile of the stellar\nhalo out to $\\sim 40$ kpc.\n\nThe paper is arranged as follows. In \\S2.1, we describe the SDSS DR8\nphotometric data and our selection criteria for A-type stars. The\nremainder of \\S2 introduces the probability distribution for BHB and\nBS membership based on colour alone and outlines the absolute\nmagnitude-colour relations for the two populations. In \\S4, we\ndescribe our maximum likelihood method to determine the density\nprofile of the stellar halo and in \\S4 we present our\nresults. Finally, we draw our main conclusions in \\S5.\n\n\n\\section{A-type stars in SDSS Data Release 8}\n\n\\subsection{Data Release 8 (DR8) Imaging}\n\nThe Sloan Digital Sky Survey (SDSS; \\citealt{york00}) is an imaging\nand spectroscopic survey covering roughly $\\sim 1\/4$ of the\nsky. Imaging data is obtained using a CCD camera (\\citealt{gunn98}) on\na 2.5 m telescope (\\citealt{gunn06}) at Apache Point Observatory, New\nMexico. Images are obtained simultaneously in five broad optical bands\n($ugriz$; \\citealt{fukugita96}). The data are processed through\npipelines to measure photometric and astrometric properties\n(\\citealt{lupton01}; \\citealt{smith02}; \\citealt{stoughton02};\n\\citealt{pier03}; \\citealt{ivezic04}; \\citealt{tucker06}). The SDSS\nDR8 release contains all of the imaging data taken by the SDSS imaging\ncamera and covers over $\\sim14,000$ deg$^2$ of sky (\\citealt{dr8}). We\nselect objects classified as stars with clean photometry. The\nmagnitudes and colours we use in the following sections have been\ncorrected for extinction following the prescription of\n\\cite{schlegel98}.\n\nIn the top panel of Fig. \\ref{fig:dr8}, we show the sky coverage of SDSS data release 8 (DR8) in equatorial\ncoordinates. For comparison, the sky coverage of the SDSS data release\n5 (DR5) is indicated by the darker grey region. The more recent SDSS\ndata release covers both Northern and Southern latitudes. We exclude latitudes $|b| < 30^\\circ$, so as to\nconcentrate on regions well away from the plane of the galaxy. Over\nthe distance range probed by this work, the SDSS footprint ($|b| >\n30^\\circ$) covers approximately 20\\% of the total volume of the\nstellar halo. In this study we use blue horizontal branch (BHB) and\nblue straggler stars (BS) to map the density profile of the stellar\nhalo. These A-type stars are selected by choosing stars in the\nfollowing region in colour-colour space:\n\\begin{eqnarray}\n0.9 < \\umg < 1.4 \\notag\\\\\n-0.25 < \\gmr < 0.0 \n\\end{eqnarray}\nThis selection is similar to other work using A-type stars\n(e.g. \\citealt{yanny00}; \\citealt{sirko04}) and is chosen to exclude\nmain sequence stars, white dwarfs and QSOs. The bottom left hand panel\nof Fig. \\ref{fig:dr8} highlights the colour selection box. Whilst we\nassume that all of our selected stars are BHBs or BSs, there may be a\nnon-negligible contribution by variable stars, such as RR Lyrae. We\nuse multi-epoch stripe 82 data to estimate the fraction of variable\nstars in the same magnitude and colour range as our sample. We use the\nlight curve catalogue compiled by \\cite{bramich08} and classify\nvariable stars according to the criteria outlined in\n\\cite{sesar07}. The resulting fraction of variable stars is $\\sim\n5\\%$. This small fraction of non-BHB and non-BS stars will therefore\nmake little difference to the results of this work.\n\nIn the bottom right hand panel of Fig. \\ref{fig:dr8}, we show the\nerror in $\\umg$ and $\\gmr$ colours as a function of $g$ band\nmagnitude. The photometric errors in $\\umg$ are larger than in $\\gmr$,\nespecially at fainter magnitudes. This can be compared with the\ntypical separation between BHBs and BS stars in $\\umg$ colour (see\nbottom panel of Figure \\ref{fig:ridge}) which ranges from 0.05 to 0.1\nmag. Mean photometric error $\\sigma_{(\\umg)}$ reaches 0.05 at $g \\sim\n18.5$ and beyond that the value rapidly increases. Accordingly, in\nthis study we only select stars in the magnitude range $16< g\n<18.5$. This corresponds to a distance range of $\\sim 4-40$ kpc for\ntypical absolute magnitudes of BHB and BS stars. Note that BHB and BS\nstars have different absolute magnitudes, and so span separate, but\noverlapping, distance ranges (see Section \\ref{sec:absmag}).\n\n\\begin{figure*}\n\\centering\n \\begin{minipage}{\\linewidth}\n \\centering\n \\includegraphics[width=15cm,height=5cm]{footprint.ps}\n \\end{minipage}\\hfill\n \\begin{minipage}{\\linewidth}\n \\centering\n \\includegraphics[width=15cm,height=6cm]{dr8_plot_new.ps}\n \\end{minipage}\n \\caption{\\small Top panel: The SDSS DR8 footprint in equatorial\n coordinates. The solid and dotted lines show $|b| =30^\\circ$ and\n $b=0^\\circ$ respectively. The darker grey area shows the sky\n coverage of the SDSS data release 5 (DR5). The more recent DR8\n sample has both Northern and Southern sky coverage. Bottom left panel: The colour\n selection in $\\umg$ and $\\gmr$ used to select BHB and BS\n stars. Our sample consists of $N=20290$ stars in the magnitude\n range $16\n 0$ are dominated by BHB stars. The ratio between the amplitude of\n the Gaussians gives an estimate of the overall number ratio between\n the two populations.}\n\\label{fig:ug_star}\n\\end{figure}\nWe seek to measure the centroids of the BHB and BS loci in colour\nspace as well as their intrinsic widths. Naturally, this can only be\ndone provided there exists a robust classification of the A-type stars\naccording to their surface gravity. It has been shown\n(e.g. \\citealt{clewley02}; \\citealt{sirko04}; \\citealt{xue08}) that\nBHB and BS stars can be separated cleanly on the basis of their Balmer\nline profiles. We proceed by selecting A-type stars from the spectral\nSDSS data release 7 (DR7) catalogue within the same colour range as\nour DR8 photometric sample. Restricting the sample to high\nsignal-to-noise (S\/N) spectra in the magnitude range $16 < g < 17$, we\nfit the Balmer lines $H_\\gamma$, $H_{\\delta}$ and $H_{\\beta}$ with\nSersic profile of the form,\n\\begin{equation}\ny=1.0-a\\,\\exp -\\left(\\frac{|x-x_0|}{b}\\right)^c,\n\\end{equation}\nwhere $x_0$ and $a$ give the wavelength and the line depth at the line\ncentre respectively. The parameters $b$ (the scale width) and $c$ are\nrelated to the line width and line shape respectively. The relation\nbetween combined line widths and line shapes of the three Balmer lines\n$H_\\gamma$, $H_\\delta$ and $H_{\\beta}$ are shown in the top panel of\nFig. \\ref{fig:ridge}. The BHBs (blue points) and BSs (red points) are\nclearly separated in this diagram and the decision boundary is\nindicated by the dashed black line. We use this spectral\nclassification to pinpoint the loci of the two populations in the\n$\\umg,\\gmr$ colour-colour space. In our analysis, these ``ridgelines'',\ni.e. approximate centres in $\\umg$ as a function of $\\gmr$, are defined\nby third order polynomials:\n\\begin{eqnarray}\n\\label{eq:ridge}\n(\\umg)^{0}_{\\rm BHB}&=& 1.167-0.775(\\gmr)-1.934(\\gmr)^2 \\notag \\\\\n&&+9.936(\\gmr)^3, \\notag \\\\\n(\\umg)^0_{\\rm BS}&=& 1.078-0.489(\\gmr)+0.556(\\gmr)^2 \\notag \\\\\n&&+13.444(\\gmr)^3, \n\\end{eqnarray}\nfor $-0.25 < \\gmr < 0.0$. The ridgelines are shown by the thick blue\nand red lines in the bottom panel of Fig. \\ref{fig:ridge}. The green\nline indicates the approximate boundary between BHBs and BSs in\n$\\umg$, $\\gmr$ space. In addition, we calculate the intrinsic spread\nof the two populations about their ridgelines. We find $\\sigma_{\\rm\n BHB,0}(\\umg)=0.04$ and $\\sigma_{\\rm\n BS,0}(\\umg)=0.045$. Fig. \\ref{fig:ridge} makes it apparent that,\neven for brightest stars, photometric information alone is not enough\nto separate BHB and BS stars. Therefore, for a given star we define\nthe probability of BHB or BS class membership based on its distance in\ncolour-colour space from the appropriate locus. We assume that both\npopulations are distributed in a Gaussian manner about their\nridgelines. We model the conditional probability of measuring $\\umg$\nand $\\gmr$ colours, given the star of each species, as\n\\begin{eqnarray}\n\\label{eq:prob}\np(ugr~|~{\\rm\n BHB})\\propto\\mathrm{exp}\\left(-\\frac{\\left[(\\umg)-(\\umg)_{\\rm\n BHB}^0\\right]^2}{2\\sigma_{\\rm BHB}^2}\\right),\n\\notag\\\\ p(ugr~|~{\\rm\n BS})\\propto\\mathrm{exp}\\left(-\\frac{\\left[(\\umg)-(\\umg)_{\\rm\n BS}^0\\right]^2}{2\\sigma_{\\rm BS}^2}\\right).\n\\end{eqnarray}\nNote that, in fact, these probabilities are also functions of $\\gmr$\nsince the centre of the Gaussian distribution, $(\\umg)^0$ varies with\nthe $\\gmr$ colour (see eqn. \\ref{eq:ridge}). The dispersion about the\nridgeline centre depends on the intrinsic width and the photometric\nerrors in $\\umg$, $\\sigma=\\sqrt{\\sigma^2_{0}+\\sigma_{(\\umg)}^2}$. The\ncolour-based posterior probabilities of class membership are then\n\\begin{eqnarray}\n\\label{eq:prob2}\nP({\\rm BHB}~|~ugr)=\\frac{p(ugr~|~{\\rm BHB})~N_{\\rm BHB}}{p(ugr~|~{\\rm BHB})~N_{\\rm BHB}+p(ugr~|~{\\rm BS})~N_{\\rm BS}} \\notag\\\\ \nP({\\rm BS}~|~ugr)=\\frac{p(ugr~|~{\\rm BS})~N_{\\rm BS}}{p(ugr~|~{\\rm BHB})~N_{\\rm BHB}+p(ugr~|~{\\rm BS})~N_{\\rm BS}}\n\\end{eqnarray}\nThe total numbers of stars $N_{\\rm BHB}$ and $N_{\\rm BS}$ in a given\ncolour range can then be found iteratively by integrating equations\n(\\ref{eq:prob2}). In Table~\\ref{tab:fraction}, we give the fraction of\nBHB and BS stars in five $\\gmr$ colour bins. The fraction ranges from\n$f_{\\rm BHB} \\sim 0.15$ at redder colours to $f_{\\rm BHB} \\sim 0.6$ at\nbluer colours. This is in good agreement with the overall BHB to BS\nratios estimated by \\cite{bell10} and \\cite{xue08} who use similar\nmagnitude ranges.\n\\begin{table}\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.2cm}\n\\renewcommand{\\arraystretch}{0.5}\n\\begin{tabular}{| l l l l |}\n \\hline \n & $N_{\\rm tot}$ & $f_{\\rm BHB}$ & $f_{\\rm BS}$\\\\\n \\\\\n \\hline\n $-0.05 < \\gmr < 0.00$ & 5189 & 0.154 &0.846 \\\\\n \\\\\n $-0.10 < \\gmr < -0.05$ & 4973 & 0.231 & 0.769 \\\\\n \\\\ \n $-0.15 < \\gmr < -0.10$ & 4151 & 0.346 & 0.654 \\\\\n \\\\\n $-0.20 < \\gmr < -0.15$ & 3564 & 0.536 & 0.464 \\\\\n \\\\ \n $-0.25 < \\gmr < -0.20$ & 2413 & 0.613 & 0.387 \\\\\n \\\\ \n \\hline\n \\end{tabular}\n \\caption{\\small The fraction of BHB and BS stars in different colour\n bins. We give the $\\gmr$ colour range, the total number of stars,\n the estimated fraction of BHB stars and the estimated fraction of\n BS stars.}\n\\label{tab:fraction}\n\\end{center}\n\\end{table}\nFigure \\ref{fig:ug_star} demonstrates the evolution of the separation\nbetween the two populations in colour space. For illustration\npurposes, the SDSS $\\umg$ colour is transformed into\n$\\left(\\umg\\right)^\\star$ using the following relation:\n\\begin{equation}\n\\left(\\umg\\right)^\\star=(\\umg)-(\\umg)^0_{\\rm border}\n\\end{equation}\nHere, $(\\umg)^0_{\\rm\n border}=1.223-0.632(\\gmr)-0.689(\\gmr)^2+11.690(\\gmr)^3$ is defined\nby the approximate boundary line between BHB and BS stars shown by the\ngreen line in Fig. \\ref{fig:ridge}. In these new coordinates, the\ncurved shape of the decision boundary becomes a straight line. In\nFig. \\ref{fig:ug_star}, we fit two Gaussian distributions to the\ndistribution in $\\left(\\umg \\right)^\\star$ for five bins in\n$\\gmr$. This two component model fits the overall distribution very\nwell. The ratio between the amplitudes of the two Gaussians varies\nwith $\\gmr$ colour. The fraction of BHB stars increases towards bluer\ncolours, in good agreement with our estimates in\nTable~\\ref{tab:fraction}.\n\n\\subsection{Absolute Magnitudes}\n\n\\label{sec:absmag}\n\\begin{figure*}\n \\centering\n \\begin{minipage}{0.33\\linewidth}\n \n \\includegraphics[width=6.cm,height=5.cm]{gc_bhb_absmag.ps}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.33\\linewidth}\n \n \\includegraphics[width=6.cm,height=5.cm]{sag_bs_absmag.ps}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.33\\linewidth}\n \n \\includegraphics[width=6.cm,height=5.cm]{distance_plot.ps}\n \\end{minipage}\n \\caption[]{\\small Left Panel: The colour-absolute magnitude relation\n for BHB stars derived from star clusters published in\n \\cite{an08}. A polynomial of order 4 is fit to the BHB stars in\n eleven star clusters: M2, M3, M5, M13, M53, M92, NGC2419, NGC4147,\n NGC5053 and NGC5466 (solid blue line). The grey contours indicate\n the density of stars within the colour-absolute magnitude region\n (thicker lines indicate higher densities). The ridgelines for\n individual clusters are shown by the red\/black lines. These are\n colour coded by metallicity: red to black goes from more metal\n rich ([Fe\/H] ~ -1.3) to more metal poor ([Fe\/H] ~ -2.3). The inset\n panel shows the distribution of values around the derived relation\n indicating a small degree of scatter. Middle panel: The density of\n BS stars in Stripe 82 belonging to the Sagittarius stream. The\n solid and dashed red lines give the absolute magnitude colour\n relation and the estimated dispersion ($\\sigma_{M_g} \\sim\n 0.5$). The distance to this portion of the Sagittarius stream is\n estimated from \\cite{watkins09} as $D_{\\rm Sgr} =26.1 \\pm 5.6$\n kpc. The coloured dots show the relation derived by\n \\cite{kinman94} for different metallicity BS stars in star\n clusters. Right panel: The radial distribution of the BHB and BS\n star populations. The blue line shows the distribution for high\n probability BHB stars ($P(\\mathrm{BHB}) > 0.7$). The red shaded\n region shows the distribution for high probability BS stars\n ($P(\\mathrm{BS})> 0.7$) where the uncertainty in the absolute\n magnitudes is taken into account.}\n \\label{fig:abs_mag}\n\\end{figure*} \n\nLet us now derive a relationship between the absolute magnitude and\ncolour of BHB and BS stars. BHB stars are intrinsically brighter and\ntheir absolute magnitude varies little as a function of temperature\n(colour) or metallicity. In comparison, BS are intrinsically fainter\nand span a much wider range in absolute magnitude.\n\nThe absolute magnitudes of BHB stars are calibrated using star\nclusters with SDSS photometry published by \\cite{an08}. Ten star\nclusters have prominent BHB sequences; M2, M3, M5, M13, M53, M92,\nNGC2419, NGC4147, NGC5053 and NGC5466\\footnote{We adopt\n distance moduli for NGC2419 and NGC4147 of 19.8 and 16.32\n respectively. These differ form the values given in Table 1 of\n \\cite{an08}. We find that these revised values are in better\n agreement with the colour magnitude diagrams of the\n clusters.}. The density distribution of absolute magnitudes of\nBHB stars in these clusters is shown as a function of $\\gmr$ colour in\nthe left hand panel of Fig. \\ref{fig:abs_mag}. There is little\nvariation of the absolute magnitude with colour for BHB stars. $M_g$\nchanges by 0.2 (from 0.65 to 0.45) in the $-0.25 < \\gmr < 0$\nrange. The inset panel shows the variation of absolute magnitude about\nthe $M_{g(\\rm BHB)}$ vs. $\\gmr$ trend ($M_g^*$). The spread of this\ndistribution is $\\sim 0.1$, indicating that there is a tight relation\ndescribing the BHB absolute magnitude. The star clusters have\nmetallicities typical of halo stars and ranging from $-2.3 <\n[\\mathrm{Fe\/H}] < -1.3$, but we find no obvious trend with\nmetallicity.\n\nBS stars are less common in globular clusters than BHB stars. Instead,\nto calibrate the absolute magnitudes of BS stars we make use of stars\nin Stripe 82 belonging to the Sagittarius stream. The distance to the\nstream in the right ascension range $25^\\circ < \\alpha^\\circ <\n40^\\circ$ was estimated by \\cite{watkins09} using RR Lyrae stars as\n$D_{\\rm Sgr} =26.1 \\pm 5.6$ kpc. In the middle panel of\nFig. \\ref{fig:abs_mag}, we show the absolute magnitude of stars in\nStripe 82 between $25^\\circ < \\alpha^\\circ < 40^\\circ $ as a function\nof colour. The density contours are constructed by using stars outside\nof the range in right ascension as a background and computing the\ndensity contrast. An obvious plume of BS stars is apparent in the\ndensity plot. This is shown by the contour levels extending off the\nmain sequence. The solid red line shows the estimated absolute\nmagnitude colour relation for these BS stars. For comparison, we show\nthe absolute magnitude versus colour relation for BS stars estimated\nby \\cite{kinman94}. This relation is converted from Johnson-Cousins\nphotometry ($UBV$) to Sloan photometry ($ugr$) using the\ntransformation derived in \\cite{jester05}. The different coloured dots\nshow the relation for different metallicity stars. The absolute\nmagnitude calibration derived for the BS stars in the Sagittarius\nstream is almost identical to the \\cite{kinman94} relation for BS\nstars with metallicity $[\\mathrm{Fe\/H}] = -1.5$. This is in good\nagreement with the metallicity of the stream stars found by\n\\cite{watkins09} ($[\\mathrm{Fe\/H}] = -1.43$).\n\nWe compute the spread of absolute magnitudes for each colour bin as\n$\\sigma_{M_g} \\sim 0.5$. This dispersion takes into account the\ndistance errors ($D_{\\rm Sgr} =26.1 \\pm 5.6$ kpc) and encompasses a\nrange of metallicities (see dashed red lines in\nFig. \\ref{fig:abs_mag}). Hence, we conclude that our calibration for\nBS absolute magnitudes does not have a strong metallicity bias. Note\nthat the `average' BS absolute magnitude is $\\sim 2.5$, approximately\n$2$ magnitudes fainter than the BHB stars. This is entirely consistent\nwith the absolute magnitudes of halo BS stars found by \\cite{yanny00}\nand \\cite{clewley04}.\n\nThe resulting absolute magnitudes of BHB and BS stars as a function of\n$\\gmr$ colour are:\n\\begin{eqnarray}\n\\label{eq:absmag}\nM_{g(\\rm BHB)}&=& 0.434-0.169(\\gmr)+2.319(\\gmr)^2 \\notag\\\\\n&&+20.449(\\gmr)^3+94.517(\\gmr)^4, \\notag\\\\\nM_{g(\\rm BS)}&=& 3.108+5.495(\\gmr),\n\\end{eqnarray}\nwhich are valid over the colour range $-0.25 < \\gmr < 0.0$. For BHB\nstars, this allows us to estimate accurately the distances of BHB\ncandidates. For BS stars, the relation is only approximate and the\nscatter for each colour interval needs to be taken into account for\nall distance estimates.\n\nIn the right hand panel of Fig. \\ref{fig:abs_mag}, we show the\nestimated radial distributions for the BHB and BS populations. We\nselect high probability BHB and BS stars by using the membership\nprobabilities defined in eqn.~(\\ref{eq:prob2}). `High' probability\nBHB\/BS stars are defined as those for which we are $68\\%$ (or\n$1\\sigma$) confident of BHB\/BS membership. The BHB stars probe a\nradial range from $r \\sim 10$ kpc to $r \\sim 45$ kpc. BS stars, which\nhave fainter absolute magnitudes, only probe out to $\\sim 30$\nkpc. However, there is a large degree of overlap between the two\npopulations between $10$ and $30$ kpc.\n\n\n\n\\section{Maximum Likelihood Method}\n\n\\label{sec:method}\n\nIn this section, we outline the maximum likelihood method used to\nconstrain the density profile of the stellar halo. An important\nassumptions in the modelling is that the BHB and BS stars follow the\nsame density distribution, modulo an overall scaling. The number of\nBHB stars and BS stars in a given increment of magnitude and area on\nthe sky is then described by\n{\\setlength\\arraycolsep{0.1em}\n\\begin{eqnarray}\n\\label{eq:probglb}\n\\Delta N_{\\rm BHB}(m_g\\!-\\!M^{\\rm BHB}_g, \\ell, b) &=& \\rho^{0}_{\\rm\n BHB}\\rho(m_g\\!-\\!M^{\\rm BHB}_g,\\ell, b) D^3_{\\rm BHB} \\notag \\\\\n&& \\quad \\times \\, \\frac{1}{5}\\mathrm{ln}10 \\, \\Delta m_g \\, \\mathrm{cos}b \\, \\Delta \\ell \\, \\Delta b \\notag \\\\\n\\Delta N_{\\rm BS}(m_g\\!-\\!M^{\\rm BS}_g, \\ell,b) & =&\n\\rho^{0}_{\\rm BS}\\rho(m_g\\!-\\!M^{\\rm BS}_g,\\ell, b) D^3_{\\rm BS} \\\\\n&& \\quad \\times \\, \\frac{1}{5}\\mathrm{ln}10 \\, \\Delta m_g \\, \\mathrm{cos}b \\, \\Delta \\ell \\, \\Delta b . \\notag\n\\end{eqnarray}\n}\nHere, the distance increment $\\Delta D$ has been converted into the\napparent magnitude increment via the relation $\\Delta\nD=\\frac{1}{5}\\mathrm{ln}10 \\, D \\Delta m_g$. The normalising factors,\n$\\rho^{0}_{\\rm BHB}=N_{\\rm BHB}\/V_{\\rm BHB}$ and $\\rho^{0}_{\\rm\n BS}=N_{\\rm BS}\/V_{\\rm BS}$ are found by performing volume integrals\nover the SDSS DR8 sky coverage and over the required magnitude range\n{\\setlength\\arraycolsep{0.1em}\n\\begin{eqnarray}\nV_{\\rm BHB}(M^{\\rm BHB}_g)&=&\\int \\int \\int \\left[ \\rho(m_g-M^{\\rm\n BHB}_g, \\ell,\nb) D_{\\rm BHB}^3 \\right. \\notag \\\\\n&& \\qquad \\qquad \\qquad \\left. \\times \\, \\frac{1}{5}\\mathrm{ln}10 \\, \\mathrm{d}m_g \\, \\mathrm{cos}b \\, \\mathrm{d}\\ell \\, \\mathrm{d}b \\right]\\notag \\\\\nV_{\\rm BS}(M^{\\rm BS}_g)&=&\\int \\int \\int \\left[ \\rho(m_g-M^{\\rm BS}_g,\\ell, b)\nD_{\\rm BS}^3 \\right. \\notag \\\\\n&& \\qquad \\qquad \\qquad \\left. \\times \\, \\frac{1}{5}\\mathrm{ln}10 \\, \\mathrm{d}m_g \\, \\mathrm{cos}b\\, \\mathrm{d}\\ell \\, \\mathrm{d}b \\right].\n\\end{eqnarray}\n}\nThese normalising integrals depend on the absolute magnitude of the\nstars and hence on the $\\gmr$ colour ($M_g=M_g(\\gmr)$ from eqn.\n\\ref{eq:absmag}). Note that the values of $V_{\\rm BHB}$ and $V_{\\rm\n BS}$ play a minor role in identifying the maximum likelihood model\ngiven the choice of parameters, but are important when evaluating the\nperformances of different model families. We also assume that our sample consists only of BHB and\nBS stars so the total number of stars is the sum of these two\npopulations, $N_{\\rm tot}=N_{\\rm BHB}+N_{\\rm BS}$ where $N_{\\rm\n BHB}=f_{\\rm BHB}N_{\\rm tot}$ and $N_{\\rm BS}=f_{\\rm BS}N_{\\rm\n tot}$. The overall fraction of BHB to BS stars varies as a function\nof $\\gmr$ colour, as shown in the previous section.\n\nCombining equations (\\ref{eq:prob}) and (\\ref{eq:probglb}) gives the\nnumber of stars in a cell of colour, magnitude, longitude and latitude\nspace\n{\\setlength\\arraycolsep{0.1em}\n\\begin{eqnarray}\n \\Delta N &=& p(ugr~|~{\\rm BHB})\\Delta N_{\\rm BHB}+p(ugr~|~{\\rm\n BS})\\Delta N_{\\rm BS} \\notag \\\\\n&=&N_{\\rm tot} \\nu (ugr,l,b) \\Delta (\\gmr)\n \\Delta m_g \\Delta \\ell \\Delta b \\mathrm{cos}b \\frac{1}{5}\n \\mathrm{ln}10,\n\\end{eqnarray}\n}\nwhere the stellar density is\n{\\setlength\\arraycolsep{0.1em}\n\\begin{eqnarray}\n \\nu (ugr,l,b) &=& p(ugr~|~{\\rm BHB})\\frac{f_{\\rm BHB}}{V_{\\rm BHB}}\\rho(m_g\\!-\\!M^{\\rm BHB}_g,\\ell, b) D^3_{\\rm BHB} +\n \\notag\\\\ \n&& p(ugr~|~{\\rm BS})\\frac{f_{\\rm BS}}{V_{\\rm BS}}\\rho(m_g\\!-\\!M^{\\rm BS}_g, \\ell, b) D^3_{\\rm BS}\n\\end{eqnarray}\n}\nHere, each star is assigned a `BHB distance' ($D_{\\rm BHB}$)\n\\textit{as well as} a `BS distance' ($D_{\\rm BS}$). The colour\nprobability functions weight the contribution of each star to the BHB\ndensity or the BS density. For simplicity, we group all the stars into\nfive bins in $\\gmr$ of width 0.05 mags. Thus, stars in each $\\gmr$ bin\nhave the same normalisations ($V_{\\rm BHB}$, $V_{\\rm BS}$), fraction\nof BHB\/BS stars ($f_{\\rm BHB}$, $f_{\\rm BS}$) and absolute magnitudes\n($M^{\\rm BHB}_g$, $M^{\\rm BS}_g$). We take into account the\nuncertainty in the BS absolute magnitude by convolving the BS number\ndensity with a Gaussian magnitude distribution. This distribution is\ncentred on the estimated absolute magnitude ($M^{\\rm BS}_g=M^{\\rm\n BS}_g(\\gmr)$) and has a dispersion of $\\sigma_{M_g}=0.5$ (see\nFig. \\ref{fig:abs_mag}).\n\nThe log-likelihood function can then be constructed from the density\ndistribution,\n\\begin{equation}\n \\mathrm{log}\\mathcal{L}=\\sum_{i=1}^{N_{\\rm tot}} \\mathrm{log} \\, \\left [ \\nu ( m^i_g,ugr^i,\\ell^i,b^i)~\\mathrm{cos}b \\right ].\n\\end{equation}\nThe number of free parameters constrained by the likelihood function\ndepends on the complexity of the model stellar-halo density\nprofile. The log-likelihood is maximised to find the best-fit\nparameters using a brute-force grid search.\n\n\\begin{table}\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.12cm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{| l c c l l r |}\n \\hline \n $N_{\\rm tot}$ & $\\alpha$ & $q$ &$\\mathrm{ln}(\\mathcal{L}) \\times 10^{4}$ & $\\sigma\/\\rm\n tot$ & with V\\&S?\n \\\\\n \\hline\n 20290 & $2.60^{+0.05}_{-0.05}$ & $0.65^{+0.02}_{-0.02}$ & -17.1171\n & $0.38 \\pm 0.01$ & yes\n \\\\\n 15403 & $2.90^{+0.05}_{-0.1}$ & $0.53^{+0.02}_{-0.01}$ & -12.2917\n & $0.22 \\pm 0.02$ & no\n \\\\ \n \\hline\n \\end{tabular}\n \\caption{\\small A summary of our best-fit oblate power-law models\n with and without the Virgo Overdensity and the Sagittarius\n stream. We give the total number of stars, the model parameters,\n the average log-likelihood value for the model and $\\sigma\/\\rm\n tot$.}\n\\label{tab:res}\n\\end{center}\n\\end{table}\n\\section{Results}\n\\label{sec:results}\nIn this section, we outline the results of applying our maximum\nlikelihood method to the sample of A-type stars selected from the SDSS\nDR8. We consider in turn a number of simple density profiles with\nconstant flattening -- single power-law, broken power-law and Einasto\n-- before examining the case for refinements, such as triaxiality,\nradial variations in shape and substructure.\n\n\\begin{figure*}\n \\centering\n \\begin{minipage}{0.48\\linewidth}\n \\centering\n \\includegraphics[width=9cm,height=7cm]{data_model_radec.ps}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.48\\linewidth}\n \\centering\n \\includegraphics[width=9cm,height=7cm]{data_model_lb.ps}\n \\end{minipage}\n \\begin{minipage}{0.48\\linewidth}\n \\centering\n \\includegraphics[width=9cm,height=4cm]{dmm_radec.ps}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.48\\linewidth}\n \\centering\n \\includegraphics[width=9cm,height=4cm]{dmm_lb.ps}\n \\end{minipage}\n\n \\caption{\\small Left panels: Density plots in equatorial\n coordinates. Right panels: Density plots in Galactic\n coordinates. The top and middle panels show the density plots for\n the the data and single power-law model respectively. The bottom panels show the\n residuals of the best-fit single power-law model. The dark red regions show the\n obvious overdensities of Virgo and Sagittarius. The dashed lines\n indicate the regions of sky removed in the maximum likelihood\n procedure. The Virgo Overdensity and parts of the leading tail of\n the Sagittarius stream are apparent in the North Galactic Cap\n whereas another portion of the Sagittarius stream is found in the\n South Galactic Cap. Note that these have been excised when\n calculating the best-fit single power-law model. The regions away from these known\n overdensities are reasonably well fit by a smooth, power-law\n density model.}\n \\label{fig:residuals}\n\\end{figure*} \n\\begin{figure}\n\\centering\n \\includegraphics[width=8cm,height=7cm]{likelihood.ps}\n \\caption{\\small The maximum likelihood contours for the flattening\n $q$ and power-law index $\\alpha$ of our single power-law halo models. The black\n (red) lines show the $1\\sigma$, $2\\sigma$ and $3\\sigma$ contours\n when stars in the region of the Virgo overdensity and\n Sagittarius stream have been included (excluded). The inset\n panel illustrates that the difference in maximum likelihood\n parameters is relatively small whether the overdensities are\n included or excluded.}\n\\label{fig:likelihood}\n\\end{figure}\n\\subsection{Single Power-Law Profile}\n\nFirst, let us consider a simple power-law density model of the form,\n\\begin{equation}\n \\rho(\\rellip) \\propto \\rellip^{-\\alpha},\\qquad\\qquad\n \\rellip^2 =x^2+y^2+z^2q^{-2}\n\\label{eq:rn}\n\\end{equation}\nThe parameters $q$ and $\\alpha$ describe the halo flattening and the\npower-law fall-off in stellar density respectively. Oblate density\ndistributions have $q <1$, spherical $q = 1$ and prolate $q>1$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=14cm,height=5cm]{likelihood_dpl.ps}\n\\caption{\\small Likelihood contours for the broken power-law\n models. Left panel: The contours for the inner ($\\alphain$) and\n outer ($\\alphao$) power laws. The fall off steepens at larger\n radii. Right panel: The contours for the break radius ($\\rb$) and\nflattening ($q$). The maximum likelihood solution favours a flattening\nof $q=0.59$ and a break radius of $\\rb=27$ kpc with inner and outer power-laws of $\\alphain=2.3$ and $\\alphao=4.6$ respectively. }\n\\label{fig:dpl}\n\\end{figure*}\n\nWe summarise the best-fitting single power-law model parameters in Table \\ref{tab:res}.\nUsing the entire sample, we find the maximum likelihood model\nparameters of $\\alpha=2.6$ and $q=0.65$. We repeat the analysis by\nexcising stars in the regions of the Virgo Overdensity and Sagittarius\nstream, which amounts to removing stars in the regions defined by\n\\begin{equation}\n0 < X < 30, \\quad\\quad X=63.63961\\sqrt{2(1 - \\sin b)}.\n\\end{equation}\nThis is the mask introduced by \\cite{bell08} to remove stars belonging\nto the Virgo overdensity (as well as parts of the leading tail of the\nSagittarius stream). Another portion of the Sagittarius stream is\nlocated in the Southern part of the sky (see Fig. \\ref{fig:residuals})\nand is removed by\n\\begin{equation}\n 0^\\circ < \\alpha < 50^\\circ, \\quad\n -30^\\circ < \\delta < 0^\\circ .\n\\end{equation}\nBy discarding these large overdensities, we find a slightly steeper\npower law and a more flattened shape with $\\alpha=2.9$ and\n$q=0.53$. There is also a substantial increase in the maximum\nlikelihood, implying that the fit is indeed affected by the presence\nof these large overdensities. In Fig ~\\ref{fig:likelihood}, we show\nthe maximum likelihood contours for the model parameters $q$ and\n$\\alpha$ for both cases. The contours encompass the $1\\sigma$,\n$2\\sigma$ and $3\\sigma$ regions respectively. We find that our\nlikelihood function has a well defined peak. The maximum likelihood\nmodel parameters ($\\alpha=2.9$, $q=0.53$ excluding overdensities) are\nin good agreement with some of the previous work assuming a single\npower-law model for the stellar halo (e.g. \\citealt{yanny00};\n\\citealt{newberg06}; \\citealt{juric08}). This exercise indicates that\nthe two large overdensities induce a bias in deduced model parameters\nand, therefore, in the subsequent sections, we remove stars in the\nvicinity of the Virgo Overdensity and Sagittarius stream in all our\ncalculations.\n\nWe use our maximum likelihood smooth oblate halo model ($\\alpha=2.9$,\n$q=0.53$) to show the residuals of the data minus model on the sky. In\nFig. \\ref{fig:residuals}, we compare the data and model in both\nequatorial and galactic coordinates. The top and middle panels show\nthe data and model on the sky (the magnitude distribution has been\ncollapsed) whilst the bottom panels show the residuals of the\nmodel. The Virgo overdensity is the most obvious feature located at\n($\\alpha \\sim 190^\\circ$, $\\delta \\sim 30^\\circ$). This overdensity\ncovers a substantial fraction of the North Galactic cap. A portion of\nthe Sagittarius stream can be seen at ($\\alpha \\sim 20^\\circ$, $\\delta\n\\sim -25^\\circ$) in the South Galactic cap. The residuals for these\nfeatures reach up to approximately three times the model values. We\ncan estimate the total fraction of stars residing in these\noverdensities from the excess numbers of stars in these regions of the\nsky. Approximately $\\sim 5.5\\%$ of our total sample ($N_{\\rm\n tot}=20290$) reside in the Northern Virgo+Sagittarius overdensity\nwhilst less than $1\\%$ occupy the Southern portion of the Sagittarius\nstream. The fraction of stars in these overdensities is relatively\nsmall as they are located in the vicinity of the Northern and Southern\nGalactic caps where the density of stars is small. However, as the\nrelative difference (i.e. (data-model)\/model) between the data and\nmodel is large in these regions, these overdensities can influence the\nlikelihood values.\n\n\\subsection{Broken Power-Law Profile}\n\nWe relax our models to allow a change in the steepness of the density\nfall off and consider broken power-law models of the form,\n\\begin{equation}\n\\rho(\\rellip) \\propto \\begin{cases} \\rellip^{-\\alphain} & \\rellip \\le \\rb \\\\\n \\rellip^{-\\alphao} & \\rellip > \\rb. \n\\end{cases}\n\\end{equation}\n\nIn Fig. \\ref{fig:dpl} we show the maximum likelihood contours for the\ninner and outer power-laws (left hand panel) and break radii and\nflattening (right hand panel). A model with break radius $\\rb=27$ kpc\nis preferred with slopes of $\\alphain=2.3$, $\\alphao=4.6$\nrespectively. A steeper power-law is favoured at larger radii while\nthe power-law within the break radius is shallower. Note that the\nbreak radius is an ellipsoidal distance and only corresponds to a\nradial Galactocentric distance in the Galactic plane ($z=0$). The\nmodel is slightly less flattened ($q \\sim 0.6$) than the single\npower-law model but, even with an additional two parameters, an oblate\nhalo model is favoured.\n\nThe maximum likelihood value for a broken power-law model is\nsignificantly larger than for a single power-law model\n($-2\\mathrm{ln}(\\mathcal{L}_{\\rm SPL}\/\\mathcal{L}_{\\rm BPL}) \\sim\n400$) (see Table \\ref{tab:like}). Our broken power-law model is in\nvery good agreement agreement with the results of \\cite{watkins09},\nwho use a sample of RR Lyrae stars to probe the density distribution\nof the stellar halo out to $\\sim 100$ kpc (see also\n\\citealt{sesar10}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=16cm,height=10cm]{gmag_mc.ps}\n\\caption{\\small The magnitude distribution of our DR8 data sample. The\n black points give the distribution of the data where the error bars\n are Poissonian. Top-left panel: Single power-law models. The solid\n and dashed red lines show the best fit oblate and spherical models\n respectively. Error bars incorporate Poissonian uncertainties and\n the spread of absolute magnitudes for BS stars. A flattened model\n provides a much better fit to the data. Top-right panel: The solid\n red, blue and green lines show the distributions for the best-fit\n single power-law, broken power-law and Einasto models\n respectively. The latter two provide a better representation of the\n data. Bottom-left panel: The magnitude distribution for the most\n probable BHB stars (with $P_{\\rm BHB} > 0.7$). Bottom-right panel:\n The magnitude distribution for the most probable BS stars (with\n $P_{\\rm BS} > 0.7$). }\n\\label{fig:gmag}\n\\end{figure*}\n\n\n\\subsection{Einasto Profile}\n\nThe Einasto profile (\\citealt{einasto89}) is often used to describe\nthe density distribution of dark matter haloes\n(e.g. \\citealt{graham06}; \\citealt{merritt06};\n\\citealt{navarro10}). The Einasto model is given by the equation\n\\begin{equation}\n\\mathrm{ln} \\left[\\rho(\\rellip)\/\\rho(\\reff)\\right] = -d_n[(\\rellip\/\\reff)^{1\/n}-1].\n\\end{equation}\nThe shape of the density profile is described by the parameter\n$n$. Density distributions with steeper inner profiles and shallower\nouter profiles are generated by large values of $n$. For example, dark\nmatter haloes with `cuspy' inner profiles typically have values of $n\n\\sim 6$. The parameter $d_n$ is a function of $n$. For $n \\ge 0.5$ a\ngood approximation is given by $d_n= 3n-1\/3+0.0079\/n$\n(\\citealt{graham06}).\n\nThis profile allows for a non-constant fall-off without the need for\nimposing a discontinuous break radii. We find the maximum likelihood\nsolutions for $q$, $n$ and\n$\\reff$. Our best-fit Einasto model has parameters $q=0.58$, $n=1.7$ and $\\reff =20$\nkpc. The slope of the density profile varies rapidly with radius as\nindicated by the relatively small value of $n$.\n\n\n\\subsection{Model Comparisons}\n\nWe now test how accurately our models represent the observed magnitude\ndistribution of the data. Using Monte Carlo methods, we create a\ndistance distribution according to the model density profile. The fake\ndata is given a $\\gmr$ colour distribution drawn from the real data,\nwhich can then be converted into absolute magnitudes (see\neqn. \\ref{eq:absmag}). The ratio of BHB and BS stars in each colour\nbin is chosen to match the values given in Table\n\\ref{tab:fraction}. In the case of the BS stars, absolute magnitudes\nare determined from the $\\gmr$ colour by drawing randomly from a\nGaussian distribution centered on the estimated value (from\neqn. \\ref{eq:absmag}) with a dispersion of 0.5 mags. The resulting\n(apparent) magnitude distributions for our models are compared with\nthe data in Fig. \\ref{fig:gmag}\n\nIn the top-left hand panel, we show the magnitude distribution for our\nsingle power-law oblate model with the solid red line. The error bars\ntake into account the Poisson uncertainty as well as the uncertainty\nspread of absolute magnitudes for BS stars. The distribution of\nmagnitudes for our DR8 A-type star sample is shown by the black\npoints. There is reasonably good agreement but there is a notable\ndeviation at fainter magnitudes. For comparison, we show the maximum\nlikelihood \\textit{spherical} model by the red dashed line (with\n$\\alpha \\sim 2.7$), which provides a very poor fit to the data. The\ntop-right hand panel shows the magnitude distributions for the single\npower-law, broken power-law and Einasto profiles by the solid red,\nblue and green lines respectively. The broken power-law and Einasto\nmodels provide a better representation of the data than a single\npower-law, although there are still discrepancies at the faintest\nmagnitudes.\n\nIn the bottom panels of Fig. \\ref{fig:gmag}, we show the magnitude\ndistributions for the most probable BHB and BS stars. We select stars\nwith $P_{\\rm BHB} > 0.7$ as `BHB' stars and stars with $P_{\\rm BS} >\n0.7$ as `BS' stars (see section \\ref{sec:absmag}). Of course, this is\nfor illustration purposes only, as a clean separation of BHB and BS\nstars requires spectroscopic classification. While the BS stars are\nadequately described by a single power-law model, this is a poor\ndescription of the BHB stars. This is unsurprising, as the BS stars\ncover a smaller distance range and barely populate distances beyond\nthe break radius. The bottom-left panel clearly shows the need for a\nmore steeply declining density law at larger radii.\n\nIn Fig. ~\\ref{fig:n_r}, we show the distribution of (probable) BHB\nstars in spherical shells with the solid black line. Here, we only\nconsider the most likely BHB stars (with $P_{\\rm BHB} > 0.7$) as for\nthese we can accurately estimate their distance. The red, blue and\ngreen lines show the radial distributions for our best fit single\npower-law, broken power-law and Einasto models respectively. The\nsingle power-law is a poorer description of the data, whilst the\nbroken power-law and Einasto models both provide better\nrepresentations of the radial distribution.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=14cm,height=5cm]{n_r_new.ps}\n\\caption{\\small Left panel: The number of BHB stars in spherical\n shells. We select stars with $P(\\rm BHB) > 0.7$ as `BHB' stars and\n show their radial distribution with the thick black line. The solid\n blue, red and green lines show our best-fit broken power-law, single\n power-law and Einasto models respectively. Right panel: The\n residuals for our best fit models.}\n\\label{fig:n_r}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.1cm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{| l l c c c c c c |}\n \\hline \n Model & Parameters & $N_p$ & $\\mathrm{ln}(\\mathcal{L}) \\times 10^{4}$ &\n $-2\\mathrm{ln}(\\mathcal{L}_{\\rm max}\/\\mathcal{L})$ &$\\mathrm{ln}(E\/E_{\\mathrm{max}})$ & $\\sigma\/\\rm tot$ (w\/o V \\& S) & $\\sigma\/\\rm tot$ (V \\& S)\\\\\n \\hline\n SPL - spherical & $\\alpha=2.7^{+0.05}_{-0.05}$, $\\mathbf{q=1.0}$ &\n 1 & -12.5516 & 5606 & -2801 & $0.44 \\pm 0.01$ & $0.49 \\pm 0.01$\n \\\\\n SPL - oblate & $\\alpha=2.9^{+0.04}_{-0.06}$,\n $q=0.53^{+0.02}_{-0.01}$ & 2 & -12.2917 & 408 & -206 & $0.22 \\pm 0.02$\n & $0.38 \\pm 0.01$\n \\\\\n SPL - triaxial & $\\alpha=2.9^{+0.05}_{-0.05}$,\n $q=0.50^{+0.02}_{-0.01}$, $p=0.71^{+0.03}_{-0.03}$ & 3 & -12.2818\n & 210 & -110 & $0.20 \\pm 0.02$ & $0.34 \\pm 0.01$\n \\\\ \n SPL - $q=q(r)$ & $\\alpha=2.9^{+0.05}_{-0.05}$,\n $q=0.53^{+0.02}_{-0.01}$, $r_0 > 10^3$kpc & 3 & -12.2917 & 368 &\n -206 &\n $0.22 \\pm 0.02$ & $0.38 \\pm 0.01$\n \\\\ \n BPL - oblate & $r_{b}=27^{+1}_{-1}$kpc,\n $q=0.59^{+0.02}_{-0.03}$, & 4 & -12.2713 & 0 & 0 &$0.21 \\pm 0.02$ & $0.36\n \\pm 0.01$\n \\\\\n & $\\alphain=2.3^{+0.1}_{-0.1}$, $\\alphao=4.6^{+0.2}_{-0.1}$ & & &\n & & &\n \\\\\n Einasto - oblate & $n=1.7^{+0.2}_{-0.2}$,\n $r_{\\rm eff}=20^{+1.0}_{-1.0}$kpc, $q=0.58^{+0.02}_{-0.02}$ & 3 & -12.2757 & 88\n & -45 & $0.22 \\pm 0.01$ & $0.37 \\pm 0.01$\n \\\\\n \\hline\n \\end{tabular}\n \\caption{\\small A summary of our best-fit models. We give the type\n of model, the best-fit parameters of the model, the number of free\n parameters, the maximum log-likelihood, the difference in\n likelihood relative to the maximum likelihood model, the log\n evidence$^{1}$ ratio relative to the maximum likelihood model and\n $\\sigma\/\\rm tot$ both with and without the Virgo and Sagittarius\n overdensities. Parameters which are kept fixed are highlighted in\n bold.}\n\\label{tab:like}\n\\end{center}\n\\end{table*}\n\n\\footnotetext[1]{The Bayesian evidence is the integral of the\n likelihood values over the parameter space (assuming a uniform\n prior). $E \\approx \\int \\mathcal{L}(\\theta) \\mathrm{d}\\theta$, where $\\theta$\n is the model parameter vector.}\n\n\\subsection{Refinements: Triaxiality and a Radially Dependent Shape}\n\nA natural question to ask is whether further refinements might provide\na still more accurate description of the data. Up to now, we have\nassumed spheroidal halo models with a constant flattening with radius.\n\nFirst, we consider whether triaxiality makes any improvement. The\ndefinition of $\\rellip$ is modified to\n\\begin{equation}\n\\rellip^2 =x^2+y^2p^{-2} + z^2q^{-2}.\n\\end{equation}\nWe fit single power-law triaxial models to the data and obtain maximum\nlikelihood parameters of $q=0.50$ and $p=0.71$. The likelihood value\nincreases relative to an oblate model with the inclusion of this extra\nparameter ($-2\\mathrm{ln}(\\mathcal{L}_{\\rm oblate}\/\\mathcal{L}_{\\rm triaxial}) \\sim\n200$). The magnitude distribution of the triaxial model is largely the\nsame as the oblate model. We inspect the difference between these two\nmodels in equatorial coordinates in Fig. \\ref{fig:triax}. The\ndot-dashed lines indicates the regions of sky with low galactic\nlatitudes $|b| < 40^\\circ$. Regions with latitudes below $|b| <\n30^\\circ$ have been removed. The triaxial model is notably overdense\nrelative to the oblate model in the regions centered on $(\\alpha,\n\\delta)=(125^\\circ,50^\\circ)$ and $(\\alpha,\n\\delta)=(60^\\circ,10^\\circ)$. The latter region, located in the\nSouthern part of the sky, is close to the portion of the Sagittarius\nstream excised in our best fit model. The former is coincident with\nthe Monocerus ring (\\citealt{newberg02}), which can be identified, for\nexample, in Figure 1 of \\cite{belokurov06}. We suggest that the\napparent triaxiality may well be caused by the presence of these\noverdensities. The increased flexibility of the model can cause it to\n`fit' to such substructure and thus the increase in likelihood may be\nan artifact.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=8cm,height=4cm]{triaxial_oblate.ps}\n \\caption{\\small The residuals of our maximum likelihood triaxial\n and oblate models. The dot-dashed region indicates the boundary\n between low ($|b| < 40^\\circ$) and high ($|b| < 40^\\circ$)\n Galactic latitudes. Note the difference in scale that is used\n for this figure to that used for Fig. \\ref{fig:residuals}.}\n\\label{fig:triax}\n\\end{figure}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=15cm,height=3.5cm]{halo_models.ps}\n\\caption{\\small A side view of our maximum likelihood\n models. Greyscale shows density of stars in plane of Galactocentric\n $(x,z)$ at $y=0$ in four maximum-likelihood models. The blue, green,\n yellow and red contours show density levels corresponding to the\n 50th, 90th, 95th and 99th percentiles of the spherical model. The\n white dot marks the location of the Sun.}\n\\label{fig:models}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n \\includegraphics[width=16cm,height=4cm]{sigma_scale.ps}\n \\caption{\\small The scale dependence of substructure. We show the\n $\\sigma\/\\mathrm{tot}$ fraction as a function of superpixel\n size. Individual panels show the relation for our various\n maximum likelihood models. The different line styles indicate\n different magnitude bins. The black lines show the relation for\n all the stars. The red lines show the relation when stars in the\n vicinity of the Virgo overdensity and Sagittarius have been\n removed.}\n\\label{fig:superpix}\n\\end{figure*}\n\nSome earlier investigations have found evidence that the shape of the\nstellar halo changes from a flattened distribution at smaller radii to\nan almost spherical distribution at larger radii\n(e.g. \\citealt{preston91}). We can test this claim by allowing the\nflattening $q$ to vary with radius. Following the reasoning of\n\\cite{sluis98}, we make the following substitution\n\\begin{equation}\n q \\rightarrow q\\sqrt{\\frac{r^2+r^2_0}{q^2r^2+r_0^2}}\n\\end{equation}\nThe halo flattening is still $q$ at small radii but tends to\nsphericity at large radii. The scale radius $r_0$ determines the\nradial range over which this change occurs. For example, for large\nvalues of $r_0$ (e.g. larger than the most distant stars) the\nflattening is approximately constant over the applicable radial range.\nWe fit single power-law models with a varying flattening to the data\nand find that large scale radii are preferred ($r_0 > 10^{3}$ kpc)\nwith an inner flattening of $q=0.53$. This indicates that the\nflattening is approximately constant over the radial range of the data\n(out to $\\sim 40$ kpc). This is in agreement with the deductions of\n\\cite{sluis98} and \\cite{sesar11}, who also found no real evidence for\na varying shape with radius.\n\nIn summary, we find that the data are well described by an oblate\ndensity distribution with a constant flattening of $q \\sim 0.6$ and a more\nsteeply declining profile at larger radii. We summarise our maximum\nlikelihood models by giving a `side view' of the profiles in\nFig. \\ref{fig:models}. A spherical model is strongly disfavoured\nwhereas oblate broken power-law and Einasto models provide a good\nrepresentation. \n\nOur best-fit density distribution can be used to estimate the\n total stellar mass. We find the total number of BHB stars by\n integrating the BHB density profile over all space in the distance range\n $1-40$kpc. The number of BHB stars can be converted\n into a total Luminosity using the relation derived in \\cite{deason11} using\n globular clusters: $N_{\\rm BHB}\/L \\sim 10^{-3}$. Assuming a\n mass-to-light ratio of $M\/L\n \\sim 1-5$, the stellar mass is approximately $2-10 \\times\n 10^{8}M_{\\odot}$. \\cite{bell08} calculate a total stellar mass of\n $\\sim 3.7 \\times 10^{8}M_{\\odot}$ using main sequence stars, in good\n agreement with our estimate. Note that the total stellar halo mass\n is believed to be $\\sim 10^{9}M_{\\odot}$ (e.g. \\citealt{morrison93}) so we are\n probing a significant fraction of the stellar halo ($\\sim 20-100\\%$).\n\n\n\\subsection{The Amount of Substructure}\n\nA rough idea of the amount of substructure present in the data can be\nattained by computing the rms deviation of the models about the data\n($\\sigma\/\\mathrm{tot}$) as defined by equations (2) and (3) in\n\\cite{bell08}:\n\\begin{equation}\n\\langle \\sigma^2 \\rangle= \\frac{1}{n} \\sum_k \\left( D_k-M_k \\right)^2\n- \\frac{1}{n} \\sum_k \\left(M^{'}_{k}-M_k \\right)^2,\n\\label{eq:sigma1}\n\\end{equation}\n\\begin{equation}\n\\frac{\\sigma}{\\mathrm{tot}} ={ \\langle \\sigma^2 \\rangle^{1\/2} \\over \n (1\/n) \\sum_k D_k}.\n\\label{eq:sigma2}\n\\end{equation}\nHere, $D_k$ is the number of stars in bin $k$, $M_k$ is the expected\nnumber from the model, $M^{'}_{k}$ is a Poisson random deviate from\nthe model value and $n$ is the total number of bins (in $l$, $b$ and\n$m_g$). However, with a total of $\\sim 20,000$ stars we can not afford\nto use a fixed bin size over the entire sky: pixels must be\nsimultaneously large enough to contain ample stars and small enough to\nadequately sample the substructure. To circumvent this problem, we use\nthe Voronoi binning method of \\cite{cappellari03} to partition the\ndata in pixels on the sky. This adaptive binning method groups pixels\ntogether, forming `superpixels', with the objective of obtaining a\nconstant signal-to-noise ratio per bin. We choose the signal-to-noise\nratio of $S\/N \\sim 4$ (assuming Poisson noise) to ensure the mean\nnumber of stars in each 2D bin is $\\gtrsim 10$. We also check that\nour results are not affected by the choice of signal-to-noise. The\nstars are split into three $g$ band magnitude ranges, binned into\n$\\sim 65,000$ $1^\\circ \\times 1^\\circ$ pixels for full sky coverage\nwhich are then combined into $\\sim 750$ superpixels using the\nprocedure described above to estimate the overall\n$\\sigma\/\\mathrm{tot}$ values. These are given in the last two columns\nof Table \\ref{tab:like}. As expected, our $\\sigma\/\\mathrm{tot}$\nfraction is significantly reduced when the stars in the region of the\nVirgo Overdensity and Sagittarius stream are removed: the estimated\nfraction of substructure reduces from $40\\%$ to $20\\%$. However,\nfurther refinement of our model makes little difference to the\n$\\sigma\/\\mathrm{tot}$ fraction, even if the likelihood is\nincreased. This is indicative of the resilience of our maximum\nlikelihood method to relatively minor amounts of substructure present\nin the data.\n\nWe illustrate the dependence of $\\sigma\/\\mathrm{tot}$ on spatial scale\nin Fig. \\ref{fig:superpix} by grouping together superpixels with\nsimilar spatial scales and computing $\\sigma\/\\mathrm{tot}$ for each\ngroup. This shows how the fraction of substructure depends on the\nsuperpixel size (in $\\mathrm{deg}^2$) for different maximum likelihood\nmodels. $\\sigma\/\\mathrm{tot}$ is computed both with (black lines) and\nwithout (red lines) the stars in the vicinity of the Virgo Overdensity\nand the Sagittarius stream. The spherical model has relatively large\nvalues of $\\sigma\/\\mathrm{tot}$ over all scales, illustrating the poor\nfit of this model to the data. The single and broken power-law oblate\nmodels show a scale dependence, whereby larger scales (larger number\nof pixels) have an increased $\\sigma\/\\mathrm{tot}$ fraction relative\nto smaller scales. Similar behaviour is also true for the triaxial\nmodel but the $\\sigma\/\\mathrm{tot}$ fractions are slightly lower. As\nalluded to earlier, we suspect that this may be caused by the triaxial\nmodel describing some of the large-scale substructure (namely the\nMonocerus ring or parts of the Sagittarius stream which have not been\nexcised).\n\nDifferent lines in Fig. \\ref{fig:superpix} show the dependence of the\n$\\sigma\/\\mathrm{tot}$ measure on the apparent magnitude of the tracers\nand hence correspond to different distances probed. The slight rise of\nthe rms with magnitude indicates that higher fractions of substructure\nmay be present at larger distances. This can be simply explained with\nthe increase of the substructure lifetime with radius as governed by\nthe orbital period of the infalling Galactic fragments.\n \nWhen interpreting these plots, it is important to bear in mind that\nthe smallest superpixels are biased towards the densest parts of the\nhalo, while the emptiest regions are sampled by the largest\nagglomerations of 2D bins. This bias could potentially explain at\nleast some of the trend with size. On the scales smaller than several\ntens of degrees, $\\sigma\/\\mathrm{tot}$ in our models is in the range\n$0.05 <\\sigma\/\\mathrm{tot} < 0.2$ indicating that the halo is\n\\textit{not dominated} by substructure and is relatively smooth. While\nthere is evidence that the fraction of substructure increases with\nscale, this apparent increase in $\\sigma\/\\mathrm{tot}$ could also be\ncaused by spurious pixels. In sparse regions, the grouping of many\npixels, each containing very few stars, poses a limitation to this\nmethod.\n\n\n\n\n\\section{Conclusions}\n\nWe have developed a new method to simultaneously model the density\nprofile of both blue horizontal branch (BHB) and blue straggler (BS)\nstars based on their broad-band photometry alone. The probability of\nBHB or BS membership is defined by the locus of these stars in $\\umg$,\n$\\gmr$ space. We use these colour-based weights to construct the\noverall probability function for the density of the two\npopulations. When applied to the A-type stars selected from the Sloan\nDigital Sky Survey (SDSS) Data Release 8 (DR8) photometric catalog,\nthe best-fit stellar halo models are identified by applying a maximum\nlikelihood algorithm to the data.\n\nBased on the data from regions with $30^\\circ < |b| < 70^\\circ$, we\nfind that the stellar halo is not spherical in shape, but\nflattened. Spherical models cannot reproduce the distribution of the\nA-type stars in our sample and are discrepant with the data at both\nbright and faint magnitudes. Our best-fit models suggest that the\nstellar halo is oblate with a flattening (or minor axis to major axis\nratio) of $q = 0.59$ with a typical uncertainty of $\\sigma_q\\sim\n0.1$. As a simple representation of the stellar halo, we advocate the\nuse of a broken power-law model with an inner slope $\\alphain = 2.3$\nand an outer slope $\\alphao = 4.6$, the break radius occurring at\nabout 27 kpc. This gives the formula\n\\begin{equation}\n\\rho(\\rellip) \\propto \\begin{cases} r_q^{-2.3} & \\rellip \\le 27\\ {\\rm kpc}, \\\\\n r_q^{-4.6} & \\rellip > 27\\ {\\rm kpc},\n\\end{cases}\n\\end{equation}\nwith $r_q^2 = R^2 + z^2\/0.59^2$. For those who prefer Einasto models,\nan equally good law for the stellar halo density is\n\\begin{equation}\n \\rho(\\rellip) \\propto \\exp \\left[-4.77[(\\rellip\/ 20 {\\rm\n kpc})^{1\/1.7}-1] \\right].\n\\end{equation}\nThese two formulae are fundamental results of our paper, and can be\nsummarised as the Milky Way stellar halo is {\\it squashed and broken}.\nThe results are qualitatively similar to those obtained by Sesar et\nal. (2011) using main-sequence stars. There is no evidence in the\ndata for variation of the flattening with radius. There is some mild\nevidence for a triaxial shape, but the apparent triaxiality might be\ndue to the presence of the Monocerus ring at low latitudes ($|b| <\n40^\\circ$) and regions of the Southern Sagittarius stream.\n\nThe root-mean-square deviation of the data around the maximum\nlikelihood model $\\sigma\/\\mathrm{tot}$ typically ranges between $5\\%$\nand $20\\%$. This indicates that the Milky Way stellar halo, or at\nleast the component traced by the A-type stars in the SDSS DR8, is\n{\\it smooth} and not dominated by unrelaxed substructure.\n\nThis finding is discrepant with the conclusions of\n\\cite{bell08}. These authors use main sequence turn-off stars selected\nfrom SDSS data release 5 (DR5) to model the stellar halo density. They\nargue that no smooth model can describe the data and conclude that the\nstellar halo is dominated by substructure (with $\\sigma\/\\mathrm{tot}\n\\ge 0.33$). We suggest that these contrasting results may be due to\nthe different methods and tracers used. \\cite{bell08} search for the\nlowest $\\sigma\/\\mathrm{tot}$ fraction models. However, we find that\nthe $\\sigma\/\\mathrm{tot}$ values very little between different density\nmodels even if the likelihood is substantially increased. Furthermore,\na lower $\\sigma\/\\mathrm{tot}$ could also indicate that a model is\n`fitting' to any substructure (e.g. triaxial models both in this work\nand in \\citealt{bell08}). \\cite{bell08} also use main sequence stars\nas tracers which are more numerous than A-type stars, but are much\npoorer distance indicators. The adopted absolute magnitude scale for\nmain sequence stars is dependent on metallicity and\ncolour. \\cite{bell08} assume a median absolute magnitude of $M_r \\sim\n4.5$ with a scatter of $\\sigma_{M_r} =0.9$; an appropriate choice for\na halo population with metallicity $[\\mathrm{Fe\/H}] \\sim -1.5$. The\nscatter, even with an assumed metallicity, is greater than the spread\nin absolute magnitude for BS stars ($\\sigma_{M_g} \\sim 0.5$ per colour\nbin). Uncertainties in the distances to stars could lead to compact\nsubstructures becoming more blurred out over a wider range of\ndistance. It is possible that \\cite{bell08} see a somewhat\nyounger and\/or metal-richer component of the stellar halo with the\nmain sequence tracers. This component could be more unrelaxed than the\none traced by A type stars.\n\nOur conclusion that the stellar halo is composed of a smooth\nunderlying density, together with some additional substructures such\nas the Virgo Overdensity and the Sagittarius Stream, is very\nreassuring. If the stellar halo were merely a hotch-potch of tidal\nstreams and unrelaxed substructures, then modelling and estimation of\ntotal mass and potential would be much more difficult. Many of the\ncommonly-used tools of stellar dynamics -- such as the steady-state\nJeans equations -- implicitly assume a well-mixed and smooth\nequilibrium. This raises the hope that a full understanding of the\nspatial and kinematic properties of stars in the smooth, yet squashed\nand broken, stellar halo can yield the gravitational potential and\ndark matter profile of the Galaxy itself.\n\n\n\\section*{Acknowledgements}\nAJD thanks the Science and Technology Facilities Council (STFC) for\nthe award of a studentship, whilst VB acknowledges financial support\nfrom the Royal Society. We thank the anonymous referee for many\nhelpful suggestions.\n\n\n\\label{lastpage}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Conclusion}\nIn this study, we showed that drones can be detected and distinguished from birds using an object detection model based on a CNN. The trained network generalizes well as it can achieve high precision and recall values at the same time. \n\\par\nFor future work we plan to consider time domain to improve the performance even further. Since collecting such data is not easy, we plan to devise an algorithm that generates random flight videos instead of randomly generated images.\n\\section{Experiments}\n\\label{exp}\nThis section describes training details and the conducted experiments on the artificial dataset and the real dataset provided by the organizers of the challenge. The former ones are evaluated quantitatively whereas the others are evaluated qualitatively due to the lack of ground truth information.\n\\par\n\n\\textbf{Training details:} In order to apply fine tuning mentioned in Section \\ref{method}, we have started with the pre-trained weights using the ImageNet dataset \\cite{russakovsky2015imagenet} for image classification problem. Then the dataset provided by the challenge organizers and the created one are divided into training (85\\%) and validation (15\\%) parts. The training part of the former one is duplicated four times before combining them to training sets since it is too scarce compared to the artificially created, large scale one. Then, the network is fine-tuned for 10,000 iterations with 128 as batch size and batch normalization after all convolutional layers.\n\\par\nAfter the training phase is completed, we combined the two validation sets to evaluate the resulting network. Although we use $480 \\times 480 \\times 3$ as input size in training (see Figure \\ref{yolomodel}), we increase the resolution to $800 \\times 800 \\times 3$ in testing configuration. This is applicable since the network is fully convolutional. This increase is helpful in detecting small sized targets.\n\\par\n\n\\textbf{Evaluation metrics:} We use precision-recall curves to evaluate the network. The curves are constructed by changing the detection threshold. The precision metric is defined as $\\frac{tp}{tp+fp}$, where $tp$ is the number of true positives and $fp$ is the number of false positives. Recall is then defined as $\\frac{tp}{tp+fn}$, where $fn$ is the number of false negatives. We count a predicted bounding box as true positive if the area of the overlap of the predicted bounding box with the ground truth is greater than half of the area of their union.\n\nAnother metric that we used is the prediction penalty, which is basically the area of smallest rectangle that includes both the ground truth and predicted bounding boxes divided by the area of ground truth bounding box.\n\n\\textbf{Results:} Figure \\ref{prcurve} presents the performance of the method with different detection thresholds in the range [0,1]. The closer the Precision-Recall (PR) curve to the top right corner the better the performance of the method. We can understand from the curve that precision and recall can be achieved to be approximately 0.9 at the same time. This shows that the approach performs well in detecting the correct bounding boxes.\n\\begin{figure}\n\\centering\n\t\\includegraphics[width=\\columnwidth]{images\/PR}\n \\caption{Precision-Recall (PR) curve showing the performance of the approach on the outdoor test videos.}\n \\label{prcurve}\n\\end{figure}\n\\par\nFigure \\ref{penalty} shows the change of the average penalty with respect to detection threshold. The reason for higher penalties is that when the threshold increases detection rate decreases. When a drone cannot be found, the top-left pixel is reported as prediction, which results in a huge penalty. Hence, we have chosen the smallest possible threshold (which is zero) for quantitative evaluation on the test video of the challenge. Although this threshold hurts precision in the artificial dataset, it works well in the provided test video except its detecting the bird as drone when the bird is closer to the camera and in specific poses that cannot be easily distinguished from a drone by human eye. Another observation is that when the drone and the bird are too close to each other, the network supposes that the bird is a part of the drone, and outputs a bounding box enclosing both of them. The predictions are provided online\\footnote{\\url{http:\/\/user.ceng.metu.edu.tr\/~cemal\/predictions.mp4}} as a video rendered in 15 fps.\n\\begin{figure}\n\\centering\n\t\\includegraphics[width=\\columnwidth]{images\/penalty}\n \\caption{Change of prediction penalty with respect to detection threshold.}\n \\label{penalty}\n\\end{figure}\n\n\n\\section{Introduction}\nDrone, as a general definition, is the name coined for the unmanned vehicles. However, in this paper the term will refer to a specific type, namely unmanned aerial vehicles (UAV). With the rapid development in the field of unmanned vehicles and technology used to construct them, the number of drones manufactured for military, commercial or recreational purposes increases sharply with each passing day. This situation poses crucial privacy and security threats when cameras or weapons are attached to the drones. Hence, detecting the position and attributes, like speed and direction, of drones before an undesirable event, has become very crucial.\n\\par\nUnpredictable computer controlled movements, speed and maneuver abilities of drones, their resemblance to birds in appearance when observed from a distance make it challenging to detect, identify and correctly localize them. In order to solve this problem, one can think of various types of sensors to perceive the presence of a drone in the environment. These may include global positioning systems, radio waves, infrared, and audible sound or ultrasound signals. However, it has been reported that they have many limitations for this problem, and suggested that computer vision techniques be used \\cite{gokcce2015vision}. Although deep learning methods have been shown to be very powerful in computer vision tasks, the studies to detect UAVs have not taken advantage of it by placing deep learning methods at the core of the approach. To this end, this study is the first to evaluate the success of convolutional neural networks (CNN) as a standalone approach on drone detection.\n\\par\nIn this study we have used an end-to-end object detection method based on CNNs to predict the location of the drone in the video frames. In order to be able to train the \nnetwork, we created an artificial dataset by combining real drone and bird images with different background videos. The results show that the variance and the scale of the dataset make it possible to perform well on drone detection problem. With this method, we have participated in the Drone-vs-Bird Detection Challenge\\footnote{\\url{https:\/\/wosdetc.wordpress.com\/challenge}} organized within the International Workshop on Small-drone\nSurveillance, Detection and Counteraction Techniques, and our trained network ranked third in terms of lowest prediction penalty described in Section~\\ref{exp}.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{images\/found3}\n \\caption{Detection samples from the created dataset where the green rectangles show the bounding boxes of the drones.}\n\\end{figure}\n\\section{Method} \\label{method}\nOur solution is based on a single shot object detection model, YOLOv2 \\cite{DBLP:journals\/corr\/RedmonF16}, which is the follow-up study of YOLO. We adapt and fine-tune this model to detect objects of two classes (\\ie, drone and bird). Although the problem is detecting drones in the scene, we have included the bird class so that the network can learn robust features to distinguish them too. In order to achieve high accuracy with such deep models, one needs a large scale dataset that includes many scenarios of the problem, to get better generalization. To this end, we created an artificial dataset including real drones, real birds and real backgrounds. The following paragraphs first describe the approach in YOLOv2, the dataset creation approach, training and testing details.\n\\par\n\\subsection{The deep network} YOLOv2 tries to devise an end-to-end regression solution to the object detection problem. Former layers of the fully convolutional architecture that can be seen in Figure \\ref{yolomodel} are trained to extract high level features. Then the two highest level features are combined to get the final feature map of the image. Then it is divided into an $S \\times S$ grid where the duty of each grid cell is predicting bounding boxes of the form $(x,y,w,h,c)$. In this output, $x$ and $y$ are the coordinates of the centers of the boxes with respect to the grid cell, $w$ and $h$ are the width and height in proportion to the whole image, and $c$ is the confidence that an object is in the bounding box. The final task of a grid cell is computing conditional class probabilities given the probability that the corresponding bounding boxes have objects in them. While predicting those bounding boxes, the model utilizes some prior information computed by K-means clustering on width and heights of ground truth bounding boxes. The final output size for a grid cell is:\n\\[Output\\:Size = (N_{cls}+N_{coord}+1)\\times N_{anc},\\]\nwhere $N_{cls}$ is the number of classes, $N_{coord}$ is the number of coordinates, $N_{anc}$ is the number of anchor bounding boxes used as prior knowledge and the 1 in the parenthesis is for the confidence value. In our approach, grid size is set to 15, number of classes is two, number of coordinates is four and number of anchor boxes is five. Hence, the final output is of the shape $15\\times 15 \\times 35$.\n\n\n\\begin{figure*}[t]\n\t\\includegraphics[width=\\textwidth]{images\/yolo_model}\n \\caption{Our adaptation of the YOLOv2 network. All layers are fine-tuned with the dataset collected in the paper.}\n \\label{yolomodel}\n\\end{figure*}\n\n\\subsection{Dataset Preparation}\nHaving mentioned the model details, we can now come to the most important part of the study which is dataset preparation. Since drone flights have limitations due to inadequate battery technology, weather conditions and legislative regulations, there is no publicly available large scale dataset for training deep networks. However, our approach requires immense amount of data to learn useful features. One possible solution to this is creating an artificial dataset. For this end, we have collected public domain pictures of drones and birds, and videos of coastal areas. After subtracting the background of drones and birds, they are randomly placed on the frames of the videos. The overall process is summarized in Algorithm \\ref{dataset}. The details of the dataset can be found in the Table \\ref{datasettable}. As can easily be seen, the dataset needs a huge storage size when all of the configurations are used. Hence, we eliminated some portion of the configurations with probability \n\\[p=1-\\frac{Max.\\:allowed\\:size}{Total\\:size\\:for\\:all\\:configurations},\\]\nto reduce the size of the dataset to reasonable amounts. Samples drawn from the resulting dataset can be seen in the Figure \\ref{samples}. These samples show that although they are created artificially, they look like real images of flying drones and birds.\n\n\\begin{algorithm}[!h]\n\\caption{The algorithm for preparing the dataset.}\n\\label{dataset}\n\\SetAlgoLined\n\\DontPrintSemicolon\n\n$S \\gets \\text{predefined size intervals}$\\;\n$D \\gets \\text{foregrounds of drone images}$\\;\n$B \\gets \\text{foregrounds of bird images}$\\;\n$V \\gets \\text{background videos}$\\;\n$R \\gets \\text{\\# of rows that the image will be divided into}$\\;\n$C \\gets \\text{\\# of columns that the image will be divided into}$\\;\n$G \\gets R\\times C \\text{ grid}$\\;\n\n\n\\ForEach{$(d,g,s,v) \\in D\\times G \\times S \\times V$}\n{\n ignore this configuration with probability $p=1-\\frac{Max.\\:allowed\\:size}{Total\\:size\\:for\\:all\\:configurations}$, and continue\\;\n\n draw a random position $p_0$ in $g$\\;\n draw a random size $s_0$ for smaller edge of the drone from $s$\\;\n draw a random frame $f_0$ from $v$\\;\n resize $d$ with respect to $s_0$\\;\n overlay $f_0$ with $d$ in position $p_0$\\;\n\n draw $(p_1,s_1,f_1)$ in the same way\\;\n draw a random bird $b_0$ from $B$\\;\n draw $(p_{b,0},s_{b,0})$ for bird where $s_{b,0}$ is drawn from smaller half of $S$\\; \n resize $d$ with respect to $s_1$\\;\n overlay $f_1$ with $d$ in position $p_1$\\;\n resize $b_0$ with respect to $s_{b,0}$\\;\n overlay $f_1$ with $b_0$ in position $p_{b,0}$\\;\n\n draw $(p_2,s_2,f_2)$ in the same way\\;\n draw a random bird $b_1$ from $B$\\;\n draw $(p_{b,1},s_{b,1})$ for bird where $s_{b,1}$ is drawn from greater half of $S$\\;\n resize $d$ with respect to $s_2$\\;\n overlay $f_2$ with $d$ in position $p_2$\\;\n resize $b_1$ with respect to $s_{b,1}$\\;\n overlay $f_1$ with $b_1$ in position $p_{b,1}$\\; \\hspace*{5cm}\n \n save $f_0, f_1, f_2$ into the dataset\n}\n\n\\end{algorithm}\n\n\n\\begin{table}[hbt!]\n\t\\begin{center}\n\t\t\\caption{Details of the dataset.\n \\label{datasettable}}\n\t\t\\vspace{-0.2cm}\n\t\t\\begin{tabularx}{\\columnwidth}{l l}\n \t\\hline\n\t\t\t\\textbf{Aspect} & \\textbf{Information}\\\\\n\t\t\t\\hline \\hline\n\t\t\t\\# drones & 89\\\\\n\t\t\t\\# birds & 126\\\\\n\t\t\t\\# background videos & 11\\\\\n\t\t\t\\# rows in grid & 12\\\\\n\t\t\t\\# columns in grid & 10\\\\\n\t\t\t\\# size intervals & 19\\\\\n\t\t\tsize intervals & in [5,160] \\\\\n \t\t\t\t\t\t\t\t\t &(bias towards smaller values) \\\\\n image resolution & $850 \\times 480$\\\\ \n\t\t\t\\hline\n\t\t\t\\# resulting images & 676,534\\\\\n \\hline\n\t\t\\end{tabularx}\n\t\\end{center}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=\\columnwidth]{images\/dataset_samples2}\n \t\\caption{Samples from the artificial dataset which represent various scenarios with different backgrounds and bird inclusion. Although the dataset includes very small objects, the bigger ones have been chosen for better visibility. (best viewed in color).}\n \\label{samples}\n\\end{figure}\n\n\\par\nThe last things to mention about our approach are the training and prediction procedures. After creating the artificial dataset, we have applied a commonly used technique called fine-tuning. In this technique the network is first trained with a different and more general dataset for a similar problem. This provides us with better initial points than random for the parameters of the network. Then, training is continued with the actual dataset for the actual problem. This technique is useful especially when the training data is scarce. After training, there comes the prediction for unseen data. Since the network is trained with two classes, the bird detections are eliminated after getting all predicted bounding boxes from the last layer. Than a threshold, which can be determined according to accuracy on a validation set (our approach is explained in Section \\ref{exp}), is applied on the confidence values for objectness. If this operations eliminate all predicted bounding boxes, it means that the frame does not include a drone or it is not clear enough to detect. Otherwise, the one that has the highest confidence is selected as the prediction. We choose the best prediction to report since the aforementioned challenge requires to detect the only drone in the scene. However, the algorithm can easily be extended to multi-drone situations with more intelligent thresholding strategies. One possible problem with this approach is encountered when the network mixes a bird up with the drone. If the objectness confidence of it is higher than that of the drone, it is selected as the prediction. In order to decrease the number of such misinterpretations, we propose a \\textit{limited ignorance approach}. After determining the bounding box that the network is most confident, we control its intersection with the rectangle having same center, three times the width and height as the predicted bounding box in the previous frame, assuming that the drone cannot move more than its height or width in a single frame. If the rectangles intersect, we can accept the newly predicted one. Otherwise, we ignore the current prediction and report the previous one if the limit has not been exceeded yet. After exceeding the limit, we reset it and cancel the technique for the same number of frames. During this period, we report the current predictions directly. Likewise, when there is no predicted bounding box in the previous frame, we directly report the prediction in current frame.\n\n\\section{Related Work}\nIn this section, we review the related studies in two parts. \n\n\\subsection{Object Detection Methods with Computer Vision}\nThe task of object detection is to decide whether there are any predefined objects in a given image or not, and report the locations and dimensions of the smallest rectangles that bind them if they exist. Early attempts for this task involves the representations of objects using handcrafted features whereas the state of the art techniques utilizes deep learning.\n\n\\textbf{Detection with Handcrafted Features:}\nThe most successful approaches using handcrafted features require bag of visual words (BoVW) \\cite{sivic2003video} representations of the objects with the help of local feature descriptors such as scale invariant feature transform (SIFT) \\cite{lowe1999object}, speeded-up robust features (SURF) \\cite{bay2008speeded}, and histogram of oriented gradients (HOG) \\cite{dalal2005histograms}. After training a discriminative machine learning model, \\eg, support vector machines (SVM) \\cite{cortes1995support}, with such representations, the images are scanned for occurrence of learned objects with the sliding window technique generally. These methods have two crucial drawbacks. The first one is that the features have to be crafted well for the problem domain to highlight and describe the important information in the image. The second one is the computational burden of the exhaustive search done by the sliding window technique.\n\n\\textbf{Detection with Deep Networks:}\nWith the remarkable achievements of the deep learning methods in the image classification tasks, similar approaches have started to be used for attacking the object detection problem. These techniques can be divided into two simple categories; region proposal based and single shot methods. The approaches in the first category differs from the traditional methods by using features learned from data with CNNs and selective search or region proposal networks to decrease the number of possible regions \\cite{girshick2015fast, girshick2014rich, ren2015faster}. In the single shot approach, the aim is to compute bounding boxes of the objects in the image directly instead of dealing with regions in the image. A method for this is extracting multi-scale features using CNNs and combining them to predict bounding boxes \\cite{he2014spatial, liu2016ssd}. Another one, named YOLO, divides the final feature map into a 2D grid and predicts a bounding box using each grid cell \\cite{redmon2016you}.\n\n\\subsection{UAV Detection Methods with Computer Vision}\nAlthough the problem of detecting UAVs is not a well studied subject, there are some attempts to mention. Mejias \\etal utilized morphological pre-processing and Hidden Markov Model filters to detect and track micro unmanned planes \\cite{mejias2010vision}. G\\\"{o}k\\c{c}e \\etal used cascaded boosted classifiers along with some local feature descriptors \\cite{gokcce2015vision}. In addition to this pure spatial information based methods, spatio-temporal approaches exist. Rozantsev \\etal propose a method that first creates spatio-temporal cubes using sliding window method at different scales, applies motion compensation to stabilize spatio-temporal cubes, and finally utilizes boosted tree and CNN based regressors for bounding box detection \\cite{EPFL-ARTICLE-218330}.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\bf Introduction}\n\nPayne and Weinberger \\cite{PW60} proved for arbitrary convex\ndomains that\n\\[\n\\text{$\\mu_1 D^2$ is minimal for the degenerate rectangular box,}\n\\]\nwhere $\\mu_1$ is the first nonzero eigenvalue of the Neumann\nLaplacian and $D$ is the diameter of the domain. Our main result is a stronger inequality for triangular domains in the plane:\n\\[\n\\text{$\\mu_1 D^2$ is minimal for the degenerate acute isosceles\ntriangle.}\n\\]\nWe prove our result by first stretching to an isosceles triangle and then\nbisecting and stretching repeatedly to approach the degenerate\ncase. Payne and Weinberger's method of thinly slicing an arbitrary\ndomain does not apply, since the slices would not be triangular.\n\nA corollary is an optimal Poincar\\'{e} inequality for triangles, namely that\n\\[\n\\int_T v^2 \\, dA < \\frac{D^2}{j_{1,1}^2} \\int_T |\\nabla v|^2 \\, dA\n\\]\nwhenever the function $v$ has mean value zero over the triangle\n$T$. Here $j_{1,1} \\simeq 3.8317$ denotes the first positive root of the Bessel function $J_1$.\n\nOur proof relies on symmetry properties of isosceles triangles.\nWe show by our ``Method of the Unknown Trial Function''\n(Section~\\ref{isec3}) that the first nonconstant Neumann mode of\nan isosceles triangle is symmetric when the aperture of the\ntriangle is less than $\\pi\/3$. We similarly prove antisymmetry\nwhen the aperture exceeds $\\pi\/3$. In that case the nodal curve\nlies on the shortest altitude. \n\nOur companion paper \\cite{LS09a} \\emph{maximizes} $\\mu_1$ among\ntriangles, under perimeter or area normalization, with the minimizer being\nequilateral. We know of no other papers in the literature that\nstudy sharp isoperimetric type inequalities for Neumann\neigenvalues of triangles. Note the Neumann eigenfunctions of triangles were investigated for the ``hot spots'' conjecture, by Ba\\~{n}uelos and Burdzy \\cite{BaBu}, and the approximate location of the nodal curve for non-isosceles triangles was studied in recent work of Atar and Burdzy \\cite{AtBu}, using probabilistic methods. \n\nDirichlet eigenvalues of triangles have received considerable\nattention \\cite{AF06,F06,fresiu,S07,S09}. Particularly interesting is the Dirichlet gap conjecture for triangles, due to Antunes and Freitas \\cite{AF08}, which claims $(\\lambda_2 - \\lambda_1)D^2$ is minimal for the equilateral triangle; some progress has been made recently by Lu and Rowlett \\cite{LR08}. Dirichlet eigenvalues of degenerate domains have also\nbeen investigated lately \\cite{BF09,F07}. Some of these Dirichlet triangle results are discussed in our companion paper \\cite[Section~10]{LS09a}.\n\nFor broad surveys of isoperimetric eigenvalue\ninequalities, see the paper by Ashbaugh \\cite{A99}, and the\nmonographs of Bandle \\cite{B79}, Henrot \\cite{He06}, Kesavan\n\\cite{K06} and P\\'{o}lya--Szeg\\H{o} \\cite{PS51}.\n\n\n\\section{\\bf Notation}\n\\label{notation}\n\nThe Neumann eigenfunctions of the Laplacian on a bounded plane\ndomain $\\Omega$ with Lipschitz boundary satisfy $-\\Delta u = \\mu u$ with\nnatural boundary condition $\\partial u \/ \\partial n = 0$. The\neigenvalues $\\mu_j$ are nonnegative, with\n\\[\n0 = \\mu_0 < \\mu_1 \\leq \\mu_2 \\leq \\dots \\to \\infty .\n\\]\nCall $\\mu_1$ the \\textbf{fundamental tone}, since\n$\\sqrt{\\mu_1}$ is proportional to the lowest frequency of vibration\nof a free membrane over the domain. Call the eigenfunction $u_1$ a\n\\textbf{fundamental mode}.\n\nThe Rayleigh Principle says\n\\[\n\\mu_1 = \\min_{\\int_\\Omega v \\, dA = 0} R[v]\n\\]\nwhere \n\\[\nR[v] = \\frac{\\int_\\Omega |\\nabla v|^2 \\, dA}{\\int_\\Omega v^2 \\, dA}\n\\]\nis the Rayleigh quotient of $v \\in H^1(\\Omega)$. Sometimes we write $R_\\Omega[v]$ to emphasize the domain over which we take the Rayleigh quotient.\n\nFor a triangular domain with side lengths $l_1 \\ge l_2 \\ge l_3 > 0$, we denote:\n\\begin{itemize}\n \\item[] $D=l_1 =$ diameter,\n \\item[] $L=l_1 + l_2 + l_3 =$ perimeter,\n \\item[] $A=$ area.\n \\end{itemize}\nWrite $j_{0,1} \\simeq 2.4048$ and $j_{1,1} \\simeq 3.8317$ for the first positive roots of the Bessel functions $J_0$ and $J_1$, respectively.\n\n\\section{\\bf Results}\n\nFirst we develop symmetry and antisymmetry properties of\nthe fundamental mode of an isosceles triangle. \n\\begin{definition}\\rm\nThe \\textbf{aperture} of an isosceles triangle is the angle between\nits two equal sides. Call a triangle \\textbf{subequilateral} if it is isosceles with\naperture less than $\\pi\/3$, and \\textbf{superequilateral} if it is\nisosceles with aperture greater than $\\pi\/3$.\n\\end{definition}\n\\begin{theorem} \\label{sharpsymmetric} Every fundamental mode of a subequilateral triangle is symmetric\nwith respect to the line of symmetry of the triangle.\n\\end{theorem}\nSee Figure~\\ref{symmfig}(a), where the nodal curve is sketched. The theorem is plausible because the main variation of a fundamental mode should take place in the ``long'' direction of the triangle.\n\nWe will use this symmetry result when proving the lower bound on the fundamental tone, in Theorem~\\ref{th:tld2} below. \n\nNext we state an antisymmetry result for superequilateral triangles.\n\\begin{theorem} \\label{bluntantisymmetric} Every fundamental mode of a superequilateral triangle is\nantisymmetric with respect to the line of symmetry of the triangle.\n\\end{theorem}\n\\begin{figure}[t]\n \\begin{center}\n \\hspace*{\\fill}\n \\subfloat[subequilateral triangle]{\n\\begin{tikzpicture}[scale=0.4]\n \\path (0,-7) -- (12,7);\n \\draw (0,0) -- (12,-3) -- (12,3) -- cycle;\n \\clip (0,0) -- (12,-3) -- (12,3) -- cycle;\n \\draw (0,0) -- (12,0);\n \\draw[dashed] (7.2,-3*7.2\/12) .. controls +(3\/12*1.1,1.1) and +(3\/12*1.1,-1.1) .. (7.2,3*7.2\/12);\n \\end{tikzpicture}\n }\n \\hspace*{\\fill}\n \\subfloat[superequilateral triangle]{\n \\begin{tikzpicture}[scale=0.4]\n \\path (0,-7) -- (12,7);\n \\draw (4,0) -- (8,7) -- (8,-7) -- cycle;\n \\draw[dashed] (4,0) -- (8,0);\n \\end{tikzpicture}\n }\n \\hspace*{\\fill}\n \\end{center}\n \\caption{Nodal curves (dashed) for the fundamental mode of an isosceles triangle. The fundamental mode satisfies a Neumann condition on each solid line, and a Dirichlet condition on each dashed curve.} \\label{symmfig}\n\\end{figure}\n\n\\medskip Now we develop lower bounds on the fundamental tone, under diameter normalization.\nThe sharp lower bound of Payne and Weinberger \\cite{PW60} says that for\nconvex domains in all dimensions,\n\\begin{equation} \\label{PW}\n \\mu_1 D^2>\\pi^2.\n\\end{equation}\nThe bound is asymptotically correct for rectangular boxes that\ndegenerate to an interval. \n\nFor triangles we will prove a better lower bound:\n\\begin{theorem} \\label{th:tld2}\nFor all triangles,\n\\[\n \\mu_1 D^2 > j_{1,1}^2\n\\]\nwith equality asymptotically for degenerate acute isosceles\ntriangles.\n\\end{theorem}\nThe theorem improves considerably (for triangles) on Payne and\nWeinberger's inequality, because $j_{1,1}^2 \\simeq 14.7$ is\ngreater than $\\pi^2 \\simeq 9.9$.\n\n\\begin{corollary}[Optimal Poincar\\'{e} inequality for triangular\ndomains] For all triangles $T$, one has\n\\[\n\\frac{\\int_T |\\nabla v|^2 \\, dA}{\\int_T v^2 \\, dA} >\n\\frac{j_{1,1}^2}{D^2}\n\\]\nwhenever $v \\in H^1(T)$ has mean value zero.\n\\end{corollary}\nThe corollary follows immediately, by the Rayleigh characterization of the fundamental tone.\n\n\\subsubsection*{Remarks on the literature.} Payne and Weinberger's inequality \\eqref{PW} has been generalized to geodesically convex domains on surfaces with nonnegative Gaussian curvature by Chavel and Feldman \\cite{CF77}, who adapted Payne and Weinberger's idea of slicing the domain into thin strips. A different approach is to employ a ``$P$-function'' and the maximum principle \\cite[Theorem 8.13]{S81}. This approach extends to manifolds of any dimension. It yields only $\\mu_1 D^2 \\geq \\pi^2\/4$, but the unwanted factor of $4$ disappears when the fundamental mode is known to have maximum and minimum values of opposite sign and equal magnitude. We do not know whether our inequality for triangles in Theorem~\\ref{th:tld2} can be proved by a $P$-function method. \n\n\\medskip Next we deduce lower bounds in terms of perimeter $L$. Since $L>2D$, the Payne--Weinberger lower bound \\eqref{PW} implies for all convex, bounded plane domains that\n\\begin{equation} \\label{eq:pwl}\n \\mu_1 L^2>4\\pi^2 ,\n\\end{equation}\nwith equality holding asymptotically for rectangles that degenerate to a segment. We\ndeduce a stronger inequality for triangles from\nTheorem~\\ref{th:tld2}.\n\\begin{corollary} \\label{co:tll}\nFor all triangles,\n\\[\n \\mu_1 L^2>4j_{1,1}^2\n\\]\nwith equality asymptotically for degenerate acute isosceles\ntriangles.\n\\end{corollary}\nThe constant $4j_{1,1}^2 \\simeq 58.7$ for triangles exceeds the\nvalue $4\\pi^2 \\simeq 39.5$ for general convex domains in \\eqref{eq:pwl}.\n\nIncidentally, the area cannot provide a lower bound on the Neumann\nfundamental tone, because for a sequence of triangles degenerating\nto a line segment, one finds $\\mu_1$ is bounded while the area $A$\napproaches zero; thus $\\mu_1 A$ can be arbitrarily close to $0$.\n\n\\medskip Lastly we examine \\textit{upper} bounds on the fundamental tone in terms of diameter. (For upper bounds in terms of area and perimeter, see our paper \\cite{LS09a}.) Cheng \\cite[Theorem~2.1]{cheng} gave an upper bound for general convex domains\nthat complements Payne and Weinberger's lower bound; it says in two\ndimensions that\n\\begin{equation} \\label{Cheng}\n \\mu_1 D^2 < 4j_{0,1}^2 \\simeq 23.1.\n\\end{equation}\nA slightly more general result was proved by Ba\\~nuelos and Burdzy\n\\cite[Proposition 2.2]{BaBu} using probabilistic methods. See also\nthe non-sharp inequality proved using different methods by Smits\n\\cite[Theorem~4]{S96}.\n\nOur contribution is to obtain a complementary lower bound for all\nisosceles triangles of aperture greater than $\\pi\/3$.\n\\begin{proposition}\\label{1D}\nFor superequilateral triangles of aperture $\\alpha \\in (\\pi\/3,\\pi)$, one has\n\\[\n 4j_{0,1}^2 \\sin^2(\\alpha\/2) \\leq \\mu_1 D^2< 4j_{0,1}^2.\n\\]\n\\end{proposition}\nLetting $\\alpha \\to \\pi$ yields equality asymptotically in Proposition~\\ref{1D}, for degenerate obtuse isosceles triangles. Thus Cheng's upper bound \\eqref{Cheng} is best possible even in the restricted class of triangular domains, a fact that has been observed previously in the literature \\cite[p.~10]{BaBu}.\n\nOur lower bound in Proposition~\\ref{1D} can be improved to $2j_{0,1}^2 (\\pi-\\alpha) \\tan(\\alpha\/2)$ by combining antisymmetry of the fundamental mode (Theorem~\\ref{bluntantisymmetric}) with a sectorial rearrangement result \\cite[p.~114]{B79}. The improvement is substantial when the aperture $\\alpha$ is close to the equilateral value $\\pi\/3$. On the other hand, sectorial rearrangement is nontrivial to prove, whereas the lower bound in Proposition~\\ref{1D} uses only antisymmetry and domain monotonicity.\n\n\n\\section{\\bf The equilateral triangle and its eigenfunctions} \\label{equilateral}\n\nThis section gathers together the first three Neumann eigenfunctions\nand eigenvalues of the equilateral triangle, which we use later to\nconstruct trial functions for close-to-equilateral triangles.\n\nThe modes and frequencies of the equilateral triangle were derived\ntwo centuries ago by Lam\\'{e}, albeit without a proof of\ncompleteness. We present the first few modes below. For proofs,\nsee the recent exposition (including completeness) by McCartin\n\\cite{M02}, building on work of Pr\\'{a}ger \\cite{Pr98}. A\ndifferent approach is due to Pinsky \\cite{Pi80}.\n\nConsider the the equilateral triangle $E$ with vertices at $(0,0)$,\n$(1,0)$ and $(1\/2,\\sqrt{3}\/2)$. Then $\\mu_0=0$, with eigenfunction\n$u_0 \\equiv 1$, and\n\\[\n\\mu_1 = \\mu_2 = \\frac{16 \\pi^2}{9}\n\\]\nwith eigenfunctions\n \\begin{align*}\n u_1(x,y) & = 2 \\Big[ \\cos \\big( \\frac{\\pi}{3}(2x-1) \\big) + \\cos \\big( \\frac{2\\pi y}{\\sqrt{3}} \\big) \\Big] \\sin \\big( \\frac{\\pi}{3} (2x-1)\n \\big) , \\\\\n u_2(x,y) & = \\cos \\big( \\frac{2\\pi}3(2x-1) \\big) - 2\\cos\\big( \\frac\\pi3(2x-1) \\big) \\cos\\big( \\frac{2\\pi y}{\\sqrt{3}} \\big) .\n \\end{align*}\nClearly $u_1$ is antisymmetric with respect to the line of symmetry\n$\\{ x=1\/2 \\}$ of the equilateral triangle, since\n$u_1(1-x,y)=-u_1(x,y)$, whereas $u_2$ is symmetric with respect to\nthat line.\n\nWe evaluate some integrals of $u_1$ and $u_2$, for later use:\n\\begin{align*}\n\\int_E u_1^2 \\, dA & = \\int_E u_2^2 \\, dA = \\frac{3\\sqrt{3}}{8} , \\\\\n\\int_E \\Big( \\frac{\\partial u_1}{\\partial x} \\Big)^{\\! 2} \\, dA & = \\int_E \\Big( \\frac{\\partial u_2}{\\partial y} \\Big)^{\\! 2} \\, dA = \\frac{32\\pi^2 + 243}{32\\sqrt{3}} , \\\\\n\\int_E \\Big( \\frac{\\partial u_1}{\\partial y} \\Big)^{\\! 2} \\, dA &\n= \\int_E \\Big( \\frac{\\partial u_2}{\\partial x} \\Big)^{\\! 2} \\, dA\n= \\frac{32\\pi^2 - 243}{32\\sqrt{3}} .\n\\end{align*}\n\n\n\\section{\\bf Isosceles triangles}\n\nIn this section we focus on isosceles triangles, establishing bounds that will help show symmetry\nof the fundamental mode for subequilateral triangles (in\nSection~\\ref{isec3}) and antisymmetry of the fundamental mode for\nsuperequilateral triangles (in Section~\\ref{isec4}).\n\n\\subsection{Bounds for sub- and super-equilateral triangles} \\label{isec1}\n\nFirst we bound the fundamental tone of an isosceles triangle, by\ntransplanting it to a sector. Write the polar coordinates as $(r,\\theta)$, let $l>0$, and define\n\\[\nS(\\alpha) = \\{ (x,y) : 0 2.68$. \n\nIt is known that $j_{\\nu,1}^\\prime$ is a strictly increasing function of $\\nu$; one can consult the original proof in \\cite[p.~510]{W52}, or the more elementary proof in \\cite{L90}. Since $j_{2.68,1}^\\prime \\simeq 3.8384 > 3.8317 \\simeq j_{1,1}$, we conclude $j_{\\nu,1}^\\prime > j_{1,1}$ when $\\nu > 2.68$, which proves the lemma.\n\nIncidentally, more precise numerical work reveals that the transition occurs at $\\nu=2.6741$, to four decimal places.\n\\end{proof}\n\nNext consider the isosceles triangle $T(\\alpha)$ having aperture\n$0<\\alpha<\\pi$, equal sides of length $l$, and vertex at the origin.\nAfter rotating the triangle to make it symmetric about the\npositive $x$-axis, it can be written as\n\\[\nT(\\alpha) = \\{ (x,y) : 0 \\int_{T(\\alpha)} (v+C)^2 \\, rdrd\\theta && \\text{since $\\rho < 1$} \\\\\n& \\geq \\int_{T(\\alpha)} v^2 \\,dA\n\\end{align*}\nsince $\\int_{T(\\alpha)} v \\, dA = 0$ and $C^2 \\geq 0$.\n\nFor the numerator of the Rayleigh quotient, we first apply the chain rule:\n\\[\n(v \\circ \\sigma)_r = (v_r \\circ \\sigma) \\cdot \\rho , \\qquad (v \\circ \\sigma)_\\theta = (v_r \\circ \\sigma)\\cdot r \\rho^\\prime + (v_\\theta \\circ \\sigma) .\n\\]\nHence the numerator is\n\\begin{align}\n& \\int_{S(\\alpha)} |\\nabla ( (v+C)\\circ \\sigma)|^2 \\, dA \\notag \\\\\n& = \\int_{S(\\alpha)} \\left[ (v_r \\circ \\sigma)^2 \\rho^2 + \\frac{1}{r^2}\\left( (v_r \\circ \\sigma)\\cdot r \\rho^\\prime + (v_\\theta \\circ \\sigma) \\right)^2 \\right] \\, r drd\\theta \\notag \\\\\n& = \\int_{T(\\alpha)} \\left[ v_r^2+\\frac{1}{r^2}\\Big( v_r \\frac{r}{\\rho} \\rho^\\prime + v_\\theta \\Big)^{\\! 2} \\right] \\, r drd\\theta \\qquad \\text{by $r \\mapsto r\/\\rho(\\theta)$} \\label{eq:varchange} \\\\\n& \\leq \\int_{T(\\alpha)} \\left[ v_r^2\\left(1+\\left(\\frac{\\rho^\\prime}{\\rho}\\right)^{\\! 2}+\\left|\\frac{\\rho^\\prime}{\\rho}\\right|\\right)+\\frac{v_\\theta^2}{r^2}\\left(1+\\left|\\frac{\\rho^\\prime}{\\rho}\\right| \\right)\\right] dA \\notag \\\\\n& \\leq [1+\\tan(\\alpha\/2)+\\tan^2(\\alpha\/2)] \\int_{T(\\alpha)} |\\nabla v|^2 \\, dA \\notag\n\\end{align}\nsince $|\\rho^\\prime\/\\rho|=|\\tan(\\theta)| \\leq \\tan(\\alpha\/2)$. Combining the numerator and denominator, we see\n\\[\n \\mu_1 \\big( S(\\alpha) \\big) \\leq \\frac{\\int_{S(\\alpha)} |\\nabla ( (v+C)\\circ \\sigma)|^2 \\, dA}{\\int_{S(\\alpha)} |(v+C)\\circ \\sigma|^2 \\, dA} < [1+\\tan(\\alpha\/2)+\\tan^2(\\alpha\/2)] \\mu_1(\\alpha).\n\\]\nRecalling that $\\mu_1 \\big( S(\\alpha) \\big) = (j_{1,1}\/l)^2$ and $D=l$, we deduce the lower bound in Lemma~\\ref{boundsiso}.\n\n\\smallskip\n(b) To get an upper bound, we transplant the eigenfunction of the sector to yield a trial\nfunction for the triangle. Write\n\\[\n v(r)=J_0 \\left( j_{1,1}\\frac{r}{l} \\right)\n\\]\nfor the fundamental mode of the sector $S(\\alpha)$. Notice $v$ is also a radial mode of the disk ${\\mathbb D}(l)$ of radius $l$, satisfying $-\\Delta v = (j_{1,1}\/l)^2 v$ with normal derivative $\\partial v\/\\partial r = 0$ at $r=l$ (using that $J_0^\\prime = -J_1$). Hence $\\int_{{\\mathbb D}(l)} v \\, dA = 0$ by the divergence theorem. Thus the transplanted eigenfunction $v \\circ \\sigma^{-1}$ integrates to $0$ over the triangle $T(\\alpha)$:\n\\begin{align*}\n\\int_{T(\\alpha)} (v \\circ \\sigma^{-1}) \\, dA\n& = \\int_{S(\\alpha)} v \\, \\rho^2 rdr d\\theta \\qquad \\text{since $\\sigma^{-1}(r,\\theta)=(r\/\\rho(\\theta),\\theta)$} \\\\\n& = \\int_{-\\alpha\/2}^{\\alpha\/2} \\rho^2 \\, d\\theta \\int_0^l v \\, rdr \\\\\n& = \\frac{1}{2\\pi} \\big( 2\\sin (\\alpha\/2) \\cos (\\alpha\/2) \\big) \\int_{{\\mathbb D}(l)} v \\, dA \\\\\n& = 0 .\n\\end{align*}\nSimilarly, the denominator of the Rayleigh quotient for $v \\circ \\sigma^{-1}$ evaluates to\n\\[\n\\int_{T(\\alpha)} (v \\circ \\sigma^{-1})^2 \\, dA\n= \\frac{1}{2\\pi} (2\\sin \\alpha\/2 \\cos \\alpha\/2) \\int_{{\\mathbb D}(l)} v^2 \\, dA .\n\\]\nThe numerator of the Rayleigh quotient equals\n\\begin{align*}\n& \\int_{T(\\alpha)} |\\nabla (v \\circ \\sigma^{-1})|^2 \\, dA \\\\\n& = \\int_{S(\\alpha)} \\left[ v_r^2+\\frac{1}{r^2}\\left(v_r r\\rho (1\/\\rho)^\\prime + v_\\theta\\right)^{\\! 2} \\right] \\, r drd\\theta && \\text{by arguing like for \\eqref{eq:varchange}} \\\\\n& = \\int_{-\\alpha\/2}^{\\alpha\/2} \\big( 1 + (\\rho^\\prime\/\\rho)^2 \\big) \\, d\\theta \\int_0^l v_r^2 \\, rdr && \\text{since $v_\\theta \\equiv 0$} \\\\\n& = \\frac{1}{2\\pi} \\big( 2 \\tan (\\alpha\/2) \\big) \\int_{{\\mathbb D}(l)} |\\nabla v|^2 \\, dA .\n\\end{align*}\nWe conclude\n\\begin{align*}\n\\mu_1(\\alpha)\n& \\leq R[v \\circ \\sigma^{-1}] \\\\\n& = \\frac{2 \\tan (\\alpha\/2)}{2\\sin (\\alpha\/2) \\cos (\\alpha\/2)} \\, \\frac{\\int_{{\\mathbb D}(l)} |\\nabla v|^2 \\, dA}{\\int_{{\\mathbb D}(l)} v^2 \\, dA} \\\\\n& = \\frac{1}{\\cos^2 (\\alpha\/2)} (j_{1,1}\/l)^2 ,\n\\end{align*}\nwhich gives the upper estimate in Lemma~\\ref{boundsiso}.\n\\end{proof}\n\nNext we give a proof for superequilateral triangles of Cheng's bound \\eqref{Cheng}.\n\n\\begin{lemma} \\label{chengsupereq}\nWhen $\\pi\/3 < \\alpha < \\pi$, the superequilateral triangle $T(\\alpha)$ satisfies\n\\[\n\\mu_1(\\alpha) D^2 < 4 j_{0,1}^2 .\n\\]\nHere the diameter is $D=2 l \\sin(\\alpha\/2)$, which is the length of the vertical side.\n\\end{lemma}\n\n\\begin{proof}\nWe simply specialize Cheng's method to triangles. Denote the upper and lower vertices of $T(\\alpha)$ by\n\\[\nz_\\pm = \\big( l \\cos(\\alpha\/2) , \\pm l \\sin(\\alpha\/2) \\big) ,\n\\]\nso that the diameter is $D=|z_+ - z_-|$. Write\n\\[\nv_0(z)=J_0 \\! \\left( j_{0,1}\\frac{|z|}{D\/2} \\right)\n\\]\nfor the fundamental mode of the disk of radius $D\/2$. Define a trial function for $z=(x,y)$ in $T(\\alpha)$ by\n\\[\nv(z) =\n\\begin{cases}\n+v_0(z-z_+) , & \\text{if $|z-z_+|\\beta$ and\n\\[\n \\frac{\\int_{T(\\beta)} w_y^2 \\, dA}{\\int_{T(\\beta)} (w_x^2+w_y^2) \\, dA}\n > \\sin^2(\\alpha\/2)+\\frac{\\sin^2(\\alpha\/2)\\cos^2(\\alpha\/2)}{\\sin^2(\\beta\/2)-\\sin^2(\\alpha\/2))}G(\\beta) .\n\\]\n\\end{lemma}\n\n\\begin{proof}\nObserve\n\\[\n\\frac{\\sin^2(\\beta\/2)}{\\sin^2(\\alpha\/2)} - \\frac{\\cos^2(\\beta\/2)}{\\cos^2(\\alpha\/2)} = \\frac{\\sin^2(\\beta\/2) - \\sin^2(\\alpha\/2)}{\\sin^2(\\alpha\/2) \\cos^2(\\alpha\/2)} .\n\\]\nMultiplying these expressions on the left and right of (i), respectively, implies that\n\\begin{equation} \\label{eq:G}\n\\frac{\\cos^2(\\beta\/2)}{\\cos^2(\\alpha\/2)} (1-\\kappa ) + \\frac{\\sin^2(\\beta\/2)}{\\sin^2(\\alpha\/2)} \\kappa < 1+G(\\beta)\n\\end{equation}\nwhere $\\kappa = \\int_{T(\\beta)} w_y^2 \\, dA \\Big\/ \\int_{T(\\beta)} (w_x^2+w_y^2) \\, dA$. The same holds for (ii). Therefore\n\\begin{align*}\n\\mu(\\alpha) & \\leq R[w \\circ \\tau] \\\\\n& = \\frac{\\int_{T(\\alpha)}\\left(\\frac{\\cos^2(\\beta\/2)}{\\cos^2(\\alpha\/2)}w_x^2+\\frac{\\sin^2(\\beta\/2)}{\\sin^2(\\alpha\/2)}w_y^2\\right)\\circ \\tau \\, dA}{\\int_{T(\\alpha)}|w\\circ \\tau|^2 \\, dA} \\\\\n& = \\frac{\\int_{T(\\beta)} \\left(\\frac{\\cos^2(\\beta\/2)}{\\cos^2(\\alpha\/2)}w_x^2+\\frac{\\sin^2(\\beta\/2)}{\\sin^2(\\alpha\/2)}w_y^2\\right) \\, dA}{\\int_{T(\\beta)} w^2 \\, dA} \\qquad \\text{by changing variable} \\\\\n& = \\left( \\frac{\\cos^2(\\beta\/2)}{\\cos^2(\\alpha\/2)} (1-\\kappa) + \\frac{\\sin^2(\\beta\/2)}{\\sin^2(\\alpha\/2)} \\kappa \\right) R[w] \\\\\n& < \\big( 1+G(\\beta) \\big) R[w] \\label{mucomp}\n\\end{align*}\nby \\eqref{eq:G}. Since $w$ is an eigenfunction for $\\mu(\\beta)$ we have $R[w]=\\mu(\\beta)$, and so the proof is complete.\n\\end{proof}\n\nThe lemma simplifies considerably when the number $G(\\beta)$ is zero:\n\\begin{corollary}\\label{corcomp}\nLet $\\mu(\\alpha)$ and $\\mu(\\beta)$ be eigenvalues of the triangles $T(\\alpha)$ and $T(\\beta)$ respectively, for some $\\alpha, \\beta \\in (0,\\pi)$. Let $w$ be a nonconstant eigenfunction belonging to $\\mu(\\beta)$, and assume $w\\circ\\tau$ can be used as a trial function for $\\mu(\\alpha)$, meaning $\\mu(\\alpha) \\leq R[w \\circ \\tau]$. Then $\\mu(\\alpha) < \\mu(\\beta)$ if\n\\begin{align*}\n\\text{(i)} \\qquad \\alpha<\\beta & \\quad \\text{and} \\quad \\frac{\\int_{T(\\beta)} w_y^2 \\, dA}{\\int_{T(\\beta)} w_x^2 \\, dA} < \\tan^2(\\alpha\/2) \\\\\n\\text{or \\ (ii)} \\qquad \\alpha>\\beta & \\quad \\text{and} \\quad \\frac{\\int_{T(\\beta)} w_y^2 \\, dA}{\\int_{T(\\beta)} w_x^2 \\, dA} > \\tan^2(\\alpha\/2) .\n\\end{align*}\n\\end{corollary}\n\n\n\\section{\\bf Proof of Theorem~\\ref{sharpsymmetric}: symmetry of the fundamental mode for subequilateral triangles} \\label{isec3}\n\nEigenfunctions of an isosceles triangle can be assumed either symmetric or antisymmetric. Indeed, any eigenfunction $v$ can be decomposed into the sum of its symmetric part $(v+v^r)\/2$ and antisymmetric part $(v-v^r)\/2$, where $v^r$ denotes the reflection of $v$ across the line of symmetry of the triangle. Each of these two parts is itself an eigenfunction, unless it is identically zero (as happens when the eigenfunction is already symmetric or antisymmetric).\n\nWe will show that for a subequilateral triangle, the fundamental tone satisfies $\\mu_1 D^2 < 16\\pi^2\/9$, whereas the smallest eigenvalue $\\mu_a$ having an antisymmetric eigenfunction satisfies $\\mu_a D^2 > 16\\pi^2\/9$. (See Figure~\\ref{mustarfig}.) It follows that every fundamental mode of the triangle is symmetric.\n\n\\begin{figure}[t]\n \\begin{center}\n\\begin{tikzpicture}[smooth,xscale=5,yscale=0.4]\n \\draw[<->] (1.2,12) node [below] {\\tiny $\\alpha$} -- (0,12) -- (0,25);\n \\clip (-0.3,11) rectangle (1.2,25);\n \\draw (0,12) -- +(0,-0.12) node [below] {\\tiny $0$};\n \n \\draw (1.0472,12) -- +(0,-0.12) node [below] {\\tiny $\\pi\/3$};\n \\draw[loosely dotted] (pi\/3,12) -- +(0,16*pi^2\/9-12);\n \n \n \n \n \n \n \\draw (0,12) -- +(-0.02,0) node [left] {\\tiny $12$};\n \\draw (0,16) -- +(-0.02,0) node [left] {\\tiny $16$};\n \\draw (0,20) -- +(-0.02,0) node [left] {\\tiny $20$};\n \\draw (0,24) -- +(-0.02,0) node [left] {\\tiny $24$};\n \\draw (0,14.682) -- +(-0.02,0) node [left] {\\tiny $j_{1,1}^2$};\n \\draw (0,17.546) -- +(-0.02,0) node [left] {\\tiny $16\\pi^2\/9$};\n \\draw[loosely dotted] (0,17.546) -- +(pi\/3,0);\n \n \\draw[densely dotted,domain=0:0.33] plot (\\x,{3.8317^2\/(1+tan(\\x*90\/3.1416)^2+tan(\\x*90\/3.1416))});\n \\draw[densely dotted,domain=0:0.9] plot (\\x,{3.8317^2\/(cos(\\x*90\/3.1416)^2)});\n \n \\draw plot coordinates {\n( 1.0472,17.5460)( 1.0001,17.2927)( 0.9529,17.0484)(\n0.9058,16.8145)( 0.8587,16.5920)( 0.8116,16.3816)( 0.7645,16.1837)(\n0.7173,15.9985)( 0.6702,15.8261)( 0.6231,15.6665)( 0.5760,15.5195)(\n0.5288,15.3850)( 0.4817,15.2629)( 0.4346,15.1530)( 0.3875,15.0551)(\n0.3403,14.9689)( 0.2932,14.8943)( 0.2461,14.8312)( 0.1990,14.7793)(\n0.1518,14.7386)( 0.1047,14.7089)(0.05,14.688)(0,14.682) }; \\path\n(0.7645,16.1837) node [below=-1pt] {\\tiny $\\mu_1 D^2$};\n\\draw[densely dashed] plot coordinates {\n( 1.0472,17.5460)( 1.0001,18.8398)( 0.9529,20.3180)( 0.9058,22.0180)( 0.8587,23.9871)( 0.8116,26.2863)( 0.7645,28.9953)( 0.7173,32.2201\n}; \\path ( 0.9529,20.3180) node [left=-1pt] {\\tiny $\\mu_a D^2$};\n \\end{tikzpicture}\n \\end{center}\n \\caption{Numerical plot of the smallest Neumann eigenvalue $\\mu_1$ and the smallest Neumann eigenvalue $\\mu_a$ with antisymmetric eigenfunction, for subequilateral triangles with aperture $\\alpha$. The eigenvalues are normalized by multiplying by the square of the diameter. Dotted lines show the bounds from Lemma \\ref{boundsiso}, converging to the asymptotic value $j_{1,1}^2$.} \\label{mustarfig}\n\\end{figure}\n\nTake the equal sides of the isosceles triangle $T(\\alpha)$ to have length $l=1$. Assume $\\alpha<\\pi\/3$ so that $T(\\alpha)$ is subequilateral with diameter $D=1$, and take $\\beta=\\pi\/3$ so that $T(\\beta)$ is equilateral. Let $w$ be the eigenfunction of the equilateral triangle $T(\\beta)$ that is symmetric with respect to the $x$-axis (meaning $w$ is obtained from the eigenfunction $u_2$ in Section~\\ref{equilateral} by first translating $E$ to shift its vertex $(1\/2,\\sqrt{3}\/2)$ to the origin and then rotating by $\\pi\/2$ counterclockwise).\n\nRecall the linear transformation $\\tau$ from Section~\\ref{isec2}, which maps $T(\\alpha)$ onto $T(\\beta)$. Note $w \\circ \\tau$ has mean value zero over $T(\\alpha)$, and hence can be used as a trial function for the fundamental mode $\\mu_1(\\alpha)$. Condition (i) in Corollary~\\ref{corcomp} is equivalent to\n\\[\n 0.130 \\simeq \\frac{32\\pi^2-243}{32\\pi^2+243} = \\frac{\\int_E u_{2,x}^2 \\, dA}{\\int_E u_{2,y}^2 \\, dA} < \\tan^2(\\alpha\/2) ,\n\\]\nwhere the integrals of $u_2$ were evaluated in\nSection~\\ref{equilateral}. Therefore by\nCorollary~\\ref{corcomp}(i),\n\\begin{equation} \\label{mustarup}\n \\mu_1(\\alpha) < \\mu_1(\\pi\/3)=\\frac{16\\pi^2}{9}\n\\end{equation}\nif $\\tan^2(\\alpha\/2) \\gtrsim 0.130$. On the other hand, the upper bound from Lemma~\\ref{boundsiso} gives \\eqref{mustarup} whenever\n\\[\n \\cos^2(\\alpha\/2) > \\frac{9j_{1,1}^2}{16\\pi^2} ,\n\\]\nwhich is equivalent to $\\tan^2(\\alpha\/2) < (16\\pi^2\/9j_{1,1}^2)-1 \\simeq 0.195$.\nThus \\eqref{mustarup} holds for all $\\alpha<\\pi\/3$.\n\nNote that our proof of \\eqref{mustarup} relies on transplanting\ntrial functions from both the equilateral triangle (for ``large''\n$\\alpha$, near $\\pi\/3$) and the sector (for ``small'' $\\alpha$,\nwhen we call on Lemma~\\ref{boundsiso}).\n\nNow change notation and take $\\alpha=\\pi\/3$ and $\\beta<\\pi\/3$. Consider the smallest eigenvalue of $T(\\beta)$ that has an antisymmetric eigenfunction; call this eigenvalue $\\mu_a(\\beta)$ and its corresponding antisymmetric eigenfunction $v$. Note $v \\circ \\tau$ has mean value zero, and hence can be used as a trial function for the fundamental tone $\\mu_1(\\alpha)$ of the equilateral triangle. By Corollary~\\ref{corcomp}(ii) we see\n\\begin{equation} \\label{eq:antisym}\n \\mu_a(\\beta) > \\mu_1(\\pi\/3) = \\frac{16\\pi^2}{9}\n\\end{equation}\nif\n\\[\n \\frac{\\int_{T(\\beta)} v_y^2 \\, dA}{\\int_{T(\\beta)} v_x^2 \\, dA}\n > \\tan^2(\\pi\/6)=\\frac{1}{3} .\n\\]\nAssume, on the other hand, that this last condition does not hold. Then $\\int_{T(\\beta)} v_x^2 \\, dA \\geq 3 \\int_{T(\\beta)} v_y^2 \\, dA$. Write $h=\\cos(\\beta\/2)$ for the width of $T(\\beta)$ and $\\gamma=\\tan(\\beta\/2)$ for the slope of its upper side. Then\n\\[\n\\mu_a(\\beta) = \\frac{\\int_{T(\\beta)} |\\nabla v|^2 \\, dA}{\\int_{T(\\beta)} v^2 \\, dA}\n\\geq \\frac{4 \\int_0^h \\int_{-\\gamma x}^{\\gamma x} v_y^2 \\, dy dx}{\\int_0^h \\int_{-\\gamma x}^{\\gamma x} v^2 \\, dy dx} .\n\\]\nNotice that for each fixed $x$, the function $y \\mapsto v(x,y)$ has mean value zero, by the antisymmetry. Hence $y \\mapsto v(x,y)$ is a valid trial function for the fundamental tone of the one dimensional Neumann problem on the interval $[-\\gamma x, \\gamma x]$. That fundamental tone equals $(\\pi\/2\\gamma x)^2$, and so\n\\[\n\\frac{\\int_{-\\gamma x}^{\\gamma x} v_y^2 \\, dy}{\\int_{-\\gamma x}^{\\gamma x} v^2 \\, dy} \\geq \\Big( \\frac{\\pi}{2\\gamma x} \\Big)^{\\! 2} .\n\\]\nSince $x \\leq h$, we conclude from above that\n\\[\n\\mu_a(\\beta) \\geq 4 \\Big( \\frac{\\pi}{2\\gamma h} \\Big)^{\\! 2} = \\frac{\\pi^2}{\\sin^2(\\beta\/2)} .\n\\]\nThen because $\\beta<\\pi\/3$ we deduce $\\mu_a(\\beta) > 4\\pi^2$, which is certainly greater than $16\\pi^2\/9$. Hence \\eqref{eq:antisym} holds for all $\\beta<\\pi\/3$.\n\nWe have shown that\n\\[\n\\mu_1(\\alpha) < \\frac{16\\pi^2}{9} < \\mu_a(\\beta)\n\\]\nwhenever $\\alpha,\\beta < \\pi\/3$. The proof is complete.\n\n\\subsection*{Method of the Unknown Trial Function} Our proof above uses the\nantisymmetric eigenfunction to construct antisymmetric trial\nfunctions for the two ``endpoint'' situations: the equilateral\ntriangle and the narrow isosceles triangle (via the interval in\nthe $y$-direction, above). We know those two endpoint eigenvalues\nexactly, and so we obtain the lower bound \\eqref{eq:antisym} on\nthe antisymmetric eigenvalue.\n\nWe call this approach the ``Method of the Unknown Trial\nFunction'', because we do not know the antisymmetric eigenfunction\nexplicitly, and for a given aperture $\\beta$ we do not even know\n\\textit{which} of the two endpoint situations will give the\nlower bound \\eqref{eq:antisym}.\n\nThe method will be used again in the proof of\nTheorem~\\ref{bluntantisymmetric}, using different endpoint cases.\n\n\n\\section{\\bf Proof of Theorem~\\ref{bluntantisymmetric}: antisymmetry of the fundamental mode for superequilateral triangles} \\label{isec4}\n\nWe will show that for a superequilateral triangle, the fundamental tone $\\mu_1$ is smaller than the smallest eigenvalue $\\mu_s$ having a symmetric eigenfunction. See Figure~\\ref{fig:supereq}. It follows that every fundamental mode of the triangle is antisymmetric.\n\n\\begin{figure}[t]\n \\begin{center}\n\\begin{tikzpicture}[smooth,xscale=3,yscale=0.1]\n \\draw[<->] (3.3,16) node [below] {\\tiny $\\beta$} -- (pi\/3,16) -- (pi\/3,62);\n \\draw (pi\/2,16) -- +(0,-0.5) node [below] {\\tiny $\\pi\/2$};\n \\draw (pi\/3,16) -- +(0,-0.5) node [below] {\\tiny $\\pi\/3$};\n \\draw (pi,16) -- +(0,-0.5) node [below] {\\tiny $\\pi$};\n \\draw[loosely dotted] (pi\/3,23.18) -- (pi,23.18);\n \\draw[loosely dotted] (pi\/3,58.727) -| (pi,16);\n \\draw[loosely dotted] (pi\/2,16) |- (pi\/3,4*pi^2);\n \n \n \\draw (pi\/3,4*pi^2) -- +(-0.02,0) node [left] {\\tiny $4\\pi^2$};\n \\draw (pi\/3,58.727) -- +(-0.02,0) node [left] {\\tiny $4j_{1,1}^2$};\n \\draw (pi\/3,23.18) -- +(-0.02,0) node [left] {\\tiny $4j_{0,1}^2$};\n \\draw[densely dotted,domain=2:pi] plot (\\x,{23.18*sin(deg(\\x\/2))^2});\n \\draw (pi\/3,17.546) -- +(-0.02,0) node [left] {\\tiny $16\\pi^2\/9$};\n \\draw[densely dashed] plot coordinates {\n(1.0472,17.5460)( 1.0996,19.4768)( 1.1519,21.5119)( 1.2043,23.6435)( 1.2566,25.8595)( 1.3090,28.1419)( 1.3614,30.4660)( 1.4137,32.8003)( 1.4661,35.1073)( 1.5184,37.3464)( 1.5708,39.4785)( 1.7237,44.8402)( 1.8766,48.8091)( 2.0296,51.6706)( 2.1825,53.7772)( 2.3354,55.3675)( 2.4883,56.5804)( 2.6412,57.4933)( 2.7942,58.1511)( 2.9471,58.5981)\n} -- (pi,58.7279);\n\\draw plot coordinates {\n( 1.0472,17.5460)( 1.0996,17.7879)( 1.1519,18.0243)( 1.2043,18.2556)( 1.2566,18.4818)( 1.3090,18.7031)( 1.3614,18.9196)( 1.4137,19.1314)( 1.4661,19.3386)( 1.5184,19.5412)( 1.5708,19.7392)\n( 1.7237,20.2918)( 1.8766,20.8052)( 2.0296,21.2783)( 2.1825,21.7085)( 2.3354,22.0927)( 2.4883,22.4262)( 2.6412,22.7038)( 2.7942,22.9191)( 2.9471,23.0656)( 3.1000,23.1767)} -- (pi,23.18);\n\\draw (2.0296,21.2783) node[below=-1pt] {\\tiny $\\mu_1 D^2$};\n\\draw (2.0296,51.6706) node[below right=-3pt] {\\tiny $\\mu_s D^2$};\n \\end{tikzpicture}\n \\end{center}\n \\caption{Numerical plot of the smallest Neumann eigenvalue $\\mu_1$ and the smallest Neumann eigenvalue $\\mu_s$ having a symmetric eigenfunction, for superequilateral triangles with aperture $\\beta$. The eigenvalues are normalized by multiplying by the square of the diameter. The dotted line shows the bound from Proposition~\\ref{1D}, converging to the asymptotic value $4j_{0,1}^2$.} \\label{fig:supereq}\n\\end{figure} \n\nOur proof will rely on Theorem~\\ref{th:tld2}, but there is no\ndanger of logical circularity because\nTheorem~\\ref{bluntantisymmetric} plays no role in proving Theorem~\\ref{th:tld2}; it is used only in the proof of Proposition~\\ref{1D}.\n\nLet us continue to assume that the equal sides of the isosceles triangle\n$T(\\beta)$ have length $l=1$. Assume $\\pi\/3 < \\beta < \\pi$, so that the triangle is superequilateral with diameter $D=2\\sin(\\beta\/2)$.\n\nLet $\\mu_s(\\beta)$ denote the smallest positive eigenvalue of $T(\\beta)$\nthat has a symmetric eigenfunction $w$. (The nodal domains for this symmetric eigenfunction are sketched in Figure~\\ref{fig:symmmode}, based on numerical work.) Cut $T(\\beta)$ in half along its line of symmetry, and call the upper half right triangle $U(\\beta)$. Then $w$ has mean value zero over each half of $T(\\beta)$, and thus is a valid trial function for $\\mu_1 \\big( U(\\beta) \\big)$. Hence\n\\[\n\\mu_s(\\beta) = R_{T(\\beta)}[w] = R_{U(\\beta)}[w] \\geq \\mu_1 \\big( U(\\beta) \\big) > j_{1,1}^2\n\\]\nby Theorem \\ref{th:tld2}, since $U(\\beta)$ has diameter $1$. Further, Cheng's bound \\eqref{Cheng} gives an upper bound\n\\[\n \\mu_1(\\beta)<\\frac{4j_{0,1}^2}{D^2} = \\frac{j_{0,1}^2}{\\sin^2(\\beta\/2)}.\n\\]\nCombining the two estimates, we deduce that if\n\\[\n\\sin^2(\\beta\/2) \\geq \\frac{j_{0,1}^2}{j_{1,1}^2} \\simeq 0.39\n\\]\nthen $\\mu_1(\\beta)<\\mu_s(\\beta)$. In particular, for obtuse isosceles triangles ($\\pi\/2 \\leq \\beta < \\pi$) we deduce the fundamental mode must be antisymmetric. \n\n\\begin{figure}[t]\n \\hspace{\\fill}\n \\subfloat[\\text{acute triangle}]{\n \\begin{tikzpicture}[scale=1.5]\n \\path (60:2) -- (-60:2);\n \\draw[clip] (0,0) -- (37:2) -- (-37:2) -- cycle;\n \\draw[dashed] (0,0) circle (1.1);\n \\draw (0,0) -- (2,0);\n \\end{tikzpicture}\n }\n \\hspace{\\fill}\n \\subfloat[right triangle]{\n \\begin{tikzpicture}[scale=1.5]\n \\path (60:2) -- (-60:2);\n \\draw[clip] (0,0) -- (45:2) -- (-45:2) -- cycle;\n \\draw (0,0) -- (2,0);\n \\draw[dashed] (45:1) -- ({sqrt(2)},0) -- (-45:1);\n \\end{tikzpicture}\n }\n \\hspace{\\fill}\n \\subfloat[obtuse triangle]{\n \\begin{tikzpicture}[scale=1.5]\n \\path (-0.5,0) -- (1.5,0);\n \\draw[clip] (0,0) -- (60:2) -- (-60:2) -- cycle;\n \\draw (0,0) -- (2,0);\n \\draw[dashed] (60:2) circle (1.15);\n \\draw[dashed] (-60:2) circle (1.15);\n \\end{tikzpicture}\n }\n \\hspace{\\fill}\n \\caption{Nodal curves (dashed) for the lowest symmetric modes of isosceles triangles. The symmetric mode satisfies a Neumann condition on each solid line, and a Dirichlet condition on each dashed curve.}\n\\label{fig:symmmode}\n\\end{figure}\n\nSuppose from now on that $\\pi\/3 < \\beta < \\pi\/2$. We have\n\\begin{equation} \\label{eq:mu1upper}\n \\mu_1(\\beta) <\n \\frac{16\\pi^2}{3S^2}=\\frac{16\\pi^2}{12\\sin^2(\\beta\/2)+6},\n\\end{equation}\nwhere $S^2=l_1^2+l_2^2+l_3^2$ is the sum of squares of side lengths of the triangle,\nby Theorem~3.1 in our companion paper \\cite{LS09a}. \n\nTo show that $\\mu_s(\\beta)$ exceeds this last value, we employ our\nMethod of the Unknown Trial Function, this time with the\n``unknown'' function being the symmetric eigenfunction $w$, and\nwith certain equilateral and right triangles providing the\nendpoint cases.\n\nLet\n\\[\n\\kappa = \\frac{\\int_{T(\\beta)} w_y^2 \\, dA}{\\int_{T(\\beta)} (w_x^2+w_y^2) \\, dA} .\n\\]\nThe proof will divide into two cases, depending on whether $\\kappa < 1\/2$ or $\\kappa \\geq 1\/2$.\n\nTake $\\alpha=\\pi\/3$ so that $T(\\alpha)$ is equilateral. Recall $\\mu_s(\\alpha)=16\\pi^2\/9$ from Section~\\ref{equilateral}. Note $w \\circ \\tau$ has mean value zero over $T(\\alpha)$, and is symmetric, and so can be used as a trial function for $\\mu_s(\\alpha)$. Let $G(\\beta)= \\big( 4\\sin^2(\\beta\/2) -1 \\big)\/3$. Since $\\alpha=\\pi\/3 < \\beta$, Lemma~\\ref{lemcomp}(i) implies that if\n\\[\n\\kappa < \\frac{1}{4} + \\frac{(1\/4)(3\/4)}{\\sin^2(\\beta\/2)-(1\/4)} \\frac{4\\sin^2(\\beta\/2) - 1}{3} = \\frac{1}{2}\n\\]\nthen\n\\[\n\\frac{16\\pi^2}{9} = \\mu_s(\\pi\/3) < \\frac{4\\sin^2(\\beta\/2) + 2}{3} \\mu_s(\\beta) ,\n\\]\nwhich can be rewritten as\n\\begin{equation} \\label{eq:rewrite}\n\\frac{16\\pi^2}{12\\sin^2(\\beta\/2)+6} < \\mu_s(\\beta) .\n\\end{equation}\nThus if $\\kappa < 1\/2$ then $\\mu_1(\\beta)<\\mu_s(\\beta)$ by \\eqref{eq:mu1upper} and \\eqref{eq:rewrite}, and so the fundamental mode of $T(\\beta)$ must be antisymmetric.\n\nNext take $\\alpha=\\pi\/2$, so that $T(\\alpha)$ is a right isosceles\ntriangle. Its smallest positive eigenvalue with symmetric\neigenfunction is $\\mu_s(\\pi\/2)=2\\pi^2$ (with eigenfunction\n$\\cos(\\sqrt{2}\\pi x)+\\cos(\\sqrt{2}\\pi y)$, which gives the nodal domains in Figure~\\ref{fig:symmmode}(b)). \n\nNote $w \\circ \\tau$ has mean value zero over\n$T(\\alpha)$, and is symmetric, and so can be used as a trial\nfunction for $\\mu_s(\\alpha)$. Let $G(\\beta)= \\big(\n6\\sin^2(\\beta\/2) -1 \\big)\/4$. Notice $G(\\beta)>0$ because\n$\\beta>\\pi\/3$. Putting $\\alpha=\\pi\/2$ and $\\beta<\\pi\/2$ into\nLemma~\\ref{lemcomp}(ii), we see that if $\\kappa \\geq 1\/2$ then\n\\[\n2\\pi^2 = \\mu_s(\\pi\/2) < \\frac{6\\sin^2(\\beta\/2) + 3}{4} \\mu_s(\\beta) .\n\\]\nThis last expression is equivalent to \\eqref{eq:rewrite}, completing the proof when $\\kappa \\geq 1\/2$.\n\n\n\\section{\\bf Proof of Theorem \\ref{th:tld2}: the lower bound on $\\mu_1 D^2$} \\label{sec:bisec}\n\nFirst we reduce to subequilateral triangles.\n\\begin{proposition} \\label{pr:sharp}\nGiven any triangle, there exists a subequilateral or equilateral triangle of the\nsame diameter whose fundamental tone is less than or equal\nto that of the original triangle. The inequality is strict,\nunless the original triangle is itself subequilateral or equilateral.\n\\end{proposition}\n\\begin{proof}\nLinearly stretch the given triangle in the direction perpendicular\nto its longest side, until one of the other sides has the same\nlength as the longest side. This new triangle is isosceles by\nconstruction, with the same diameter (\\emph{i.e.}, longest side\nlength) as the original triangle. The new triangle is\nsubequilateral or equilateral, since its third side is at most as\nlong as the two equal sides.\n\nWe will show that the stretching procedure reduces\nthe fundamental tone, by assuming the longest side of the triangle lies along\nthe $x$-axis and applying the following general argument.\n\nLet $\\Omega$ be a planar Lipschitz domain. For each $t > 1$, let\n$\\Omega_t = \\{ (x,ty) : (x,y) \\in \\Omega \\}$ be the domain obtained\nby stretching $\\Omega$ by the factor $t$ in the $y$ direction. Given\nany trial function $u \\in H^1(\\Omega)$ we have the trial function\n$v(x,y)=u(x,y\/t)$ in $H^1(\\Omega_t)$, with Rayleigh quotient\n\\begin{align}\nR[v] & = \\frac{\\int_{\\Omega_t} \\big( u_x(x,y\/t)^2+\nu_y(x,y\/t)^2\/t^2 \\big) \\, dA}{\\int_{\\Omega_t} u(x,y\/t)^2 \\, dA} \\notag \\\\\n& = \\frac{\\int_\\Omega \\big( u_x(x,y)^2+u_y(x,y)^2\/t^2 \\big) \\,\ndA}{\\int_\\Omega u(x,y)^2 \\, dA} \\notag \\\\\n& \\leq R[u] \\label{eq:stretch}\n\\end{align}\nsince $t > 1$. In addition, if $u$ has mean value zero over $\\Omega$, then so does $v$ over $\\Omega_t$. Hence taking $u$ to be a fundamental mode $u_1$ for $\\Omega$ implies that $\\mu_1(\\Omega_t) \\leq \\mu_1(\\Omega)$, by the Rayleigh Principle.\n\nThe inequality of Rayleigh quotients in \\eqref{eq:stretch} is strict\nunless $u_y \\equiv 0$. Thus the only possibility for the fundamental tone to\nremain unchanged by the stretching is for the fundamental mode $u_1$ to\ndepend only on $x$. That cannot occur for a triangle, since\non sides of the triangle that are not parallel to the $x$-axis the Neumann boundary condition would force $\\partial u_1 \/\\partial x \\equiv 0$, so that $u_1 \\equiv \\text{(const.)}$ on the whole triangle, contradicting that $u_1$ is orthogonal to the constant mode.\nHence for triangles, the stretching procedure strictly reduces\n$\\mu_1$, when $t>1$.\n\\end{proof}\n\nThe point of Proposition~\\ref{pr:sharp} is that when proving\nTheorem~\\ref{th:tld2}, we need only consider subequilateral and equilateral\ntriangles.\n\nRecall the isosceles triangle $T(\\alpha)$ with aperture $\\alpha$\nand side length $l$, and fundamental tone $\\mu_1(\\alpha)$. Assume\n$0 < \\alpha \\leq \\pi\/3$, so that the triangle is subequilateral or\nequilateral, with diameter $D=l$. Our task is to prove\n\\begin{equation} \\label{eq:isos}\n\\mu_1(\\alpha) > \\frac{j_{1,1}^2}{D^2} , \\qquad \\alpha \\in (0,\\pi\/3] .\n\\end{equation}\nEquality holds asymptotically for degenerate acute isosceles triangles, since $\\lim_{\\alpha \\to 0} \\mu_1(\\alpha) = j_{1,1}^2\/D^2$ by Lemma~\\ref{boundsiso}.\n\nNumerical work suggests that $\\mu_1(\\alpha)$ is strictly increasing on\n$(0,\\pi\/3]$, as shown in Figure~\\ref{mustarfig}, but we have not been able to prove such monotonicity. Instead we bisect and stretch, as follows.\n\nCutting $T(\\alpha)$ along its line of symmetry yields two right\ntriangles. Let $U(\\alpha)$ be one of them.\n\nThe fundamental mode $v$ of $T(\\alpha)$ is symmetric in the subequilateral case $\\alpha \\in (0,\\pi\/3)$, by Theorem~\\ref{sharpsymmetric}, and it can be chosen to be symmetric in the equilateral case $\\alpha=\\pi\/3$, by Section~\\ref{equilateral}. Since $v$ has mean value zero over\n$T(\\alpha)$, it also has mean value zero over $U(\\alpha)$. It follows from the Rayleigh Principle and symmetry that\n\\[\n\\mu_1 \\big( U(\\alpha) \\big) \\leq R_{U(\\alpha)}[v] = R_{T(\\alpha)}[v] = \\mu_1(\\alpha) .\n\\]\n\nNow linearly stretch the right triangle $U(\\alpha)$ in the direction\nperpendicular to its longest side. After some amount of stretching, we\nobtain a subequilateral triangle $T(\\alpha_1)$ with the same diameter and with aperture determined by $\\cos \\alpha_1 = \\cos^2 (\\alpha\/2)$, as some simple trigonometry reveals (see Figure~\\ref{trigfig}).\nHence $\\sin(\\alpha_1\/2) = \\sin(\\alpha\/2)\/\\sqrt{2}$.\n\nThe stretching strictly reduces the fundamental tone, by Proposition~\\ref{pr:sharp} and its proof, and so\n\\[\n\\mu_1(\\alpha_1) < \\mu_1 \\big( U(\\alpha) \\big) \\leq \\mu_1(\\alpha) .\n\\]\n\n\\begin{figure}[t]\n \\begin{center}\n\\beginpgfgraphicnamed{triangles5_pic4}\n\\begin{tikzpicture}[scale=1]\n \\pgfmathsetmacro{\\x}{10}\n \\pgfmathsetmacro{\\y}{40}\n \\coordinate (a) at (\\x,0);\n \\draw[dotted] (a) arc (0:\\y:\\x) coordinate (b);\n \\fill[lightgray] (0,0) -- ($(a)!0.5!(b)$) coordinate (c) {node[black,pos=0.8,below right=3pt] {$U(\\alpha)$}} {node [below,pos=0.19,black] {$\\alpha$}} {node [below,pos=0.36,black] {$\\alpha_1$}} -- (a) -- cycle;\n \\draw (0,0) -- (a) node[below,pos=0.5] {$D$}-- (b) -- (0,0) {node [above,pos=0.5] {$D$}} node [below right=3pt,pos=0.2] {$T(\\alpha)$};\n \\draw[loosely dashed] ($(0,0)!(c)!(a)$) coordinate (d) -- (c);\n \\draw[scale=0.15] (\\x,0) arc (0:\\y:\\x);\n \\clip (0,0) -- (a) arc (0:\\y:\\x) -- cycle;\n \\node (cir) at (0,0) [circle through=(a)] {};\n \\draw[clip] (0,0) -- (intersection 2 of cir and c--d) coordinate (e) node [below right=2pt,pos=0.8] {$T(\\alpha_1)$} -- (a);\n \\draw[scale=0.3] (\\x,0) arc (0:\\y:\\x);\n \\draw[loosely dashed] (c) -- (e);\n \\end{tikzpicture}\n\\endpgfgraphicnamed\n \\end{center}\n \\caption{Illustration of the bisection and stretching algorithm in Section~\\ref{sec:bisec}, for the subequilateral triangle $T(\\alpha)$ (which has been rotated here for clarity). } \\label{trigfig}\n\\end{figure}\n\n\nContinuing in this fashion, we deduce\n\\[\n\\mu_1(\\alpha_n) < \\cdots < \\mu_1(\\alpha_1) < \\mu_1(\\alpha)\n\\]\nwhere the apertures $\\alpha_n < \\cdots < \\alpha_1 < \\alpha$ satisfy\n\\[\n\\sin(\\alpha_n\/2) = \\frac{\\sin(\\alpha_{n-1}\/2)}{\\sqrt{2}} .\n\\]\nThus $\\lim_{n \\to \\infty} \\alpha_n = 0$, so that $\\lim_{n \\to\n\\infty} \\mu_1(\\alpha_n) = j_{1,1}^2\/D^2$ by\nLemma~\\ref{boundsiso}. Hence $(j_{1,1}^2\/D^2) < \\mu_1(\\alpha)$,\nwhich proves \\eqref{eq:isos}.\n\n\n\\section{\\bf Proof of Proposition \\ref{1D}}\nBy rescaling and rotating we can suppose the superequilateral triangle is $T(\\alpha)$ for some $\\pi\/3<\\alpha<\\pi$.\n\nThe upper bound in the Proposition is just Cheng's inequality \\eqref{Cheng}, which was proved directly for superequilateral triangles in Lemma~\\ref{chengsupereq}.\n\nFor the lower bound, first recall from Theorem~\\ref{bluntantisymmetric} that\nthe fundamental mode $v$ of the superequilateral triangle $T(\\alpha)$ is antisymmetric, and hence vanishes along the $x$-axis. Write $U(\\alpha)$ for the upper half of $T(\\alpha)$, so that $v$ satisfies a Dirichlet condition on the bottom edge of $U(\\alpha)$. Let $z_+= \\big( l\\cos(\\alpha\/2),l\\sin(\\alpha\/2) \\big)$ be the upper vertex of $U(\\alpha)$.\n\nConsider the sector with center at $z_+$ and sides of length $l$ running from $z_+$ to the origin and from $z_+$ to $z_+ - (0,l)$, and with its arc running from the origin to $z_+ - (0,l)$. We are interested in the fundamental tone of the Laplacian on this sector, when Dirichlet boundary conditions are imposed on the arc and no conditions are imposed on the two sides. Defining $w$ to equal $v$ on $U(\\alpha)$ and zero outside it, we see that $w$ is a Sobolev function in the sector and equals zero on the arc. Hence $w$ is a valid trial function for the fundamental tone of the sector. That fundamental tone equals $(j_{0,1}\/l)^2$ (with fundamental mode $J_0(j_{0,1}|z-z_+|\/l)$), and so\n\\[\n\\Big( \\frac{j_{0,1}}{l} \\Big)^{\\! 2} \\leq R[w] = R_{U(\\alpha)}[v] = R_{T(\\alpha)}[v] = \\mu_1(\\alpha) .\n\\]\nSince $T(\\alpha)$ has diameter $D=2l\\sin(\\alpha\/2)$, the Proposition follows immediately.\n\n\\section*{Acknowledgments} We are grateful to Mark Ashbaugh and G\\'{e}rard Philippin for guiding us to relevant parts of the literature. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}