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b/data_all_eng_slimpj/shuffled/split/split_finalac/part-19.finalac @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:96ce0304971344e0bce39245cc62202a82c07b600806d7f0f581fa350011d34e +size 12576663042 diff --git a/data_all_eng_slimpj/shuffled/split2/finalzkgo b/data_all_eng_slimpj/shuffled/split2/finalzkgo new file mode 100644 index 0000000000000000000000000000000000000000..1105295615ed7d546e7e544b6df5e7873b875472 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzkgo @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nBoolean network (BN), proposed by Kauffman in 1969 \\cite{Kauffman1969}, is an ideal mathematical model of simulating the gene regulation networks.\nIt quantitates the interactions among genes within cells (or within a particular genome).\nThe expression, replication, transcription and other activities of genes can be directly reflected by system states and functions \\cite{Kauffmanbook1993}.\nBN has prompted many researchers to find and ask for similar models.\nAs a result, a large number of models were born.\nFor example, some genes continuously adjust the glucose consumption of cells and so provide the fuel by which they grow and multiply.\nFor analyzing such a biological system, Boolean control network (BCN) becomes a proper model \\cite{Datta2004,Huang2000}.\nOne of the main tools for studying BNs and BCNs is called the semi-tensor product (STP) of matrices, which was proposed by Prof. Cheng \\cite{Cheng2010BNlinear,Chengbook2011,Chengbook2012}.\nIts basic idea is to describe the system behavior as a discrete time algebra form,\nby which, some classical control ideas are incorporated into the analysis of BNs \\cite{LiangJinBN2020,LiF-IBNS2012,LiRcompleteBN2012,Learning2018,ZhuJBN2015} and the control design of BCNs \\cite{LiHLyapunov2019,LuJstab2018,WuYcontrol2018,Yuregular2019,ZhongJieTrank2019,QZhucontrol2019}, as people have seen in recent years.\n\nMany wild animals carry multiple viruses that have no effect on the animals themselves, but may be both high contagious and deadly to human beings.\nAntiviral immunity plays a key role against virus diseases.\nIts research involves the pathologic manifestations, symptoms and detection technologies of viral disease, which is the major cause of network identification being currently an important topic.\nNetwork identification aims to find the methods or algorithms for constructing the dynamics of systems.\nFor an unknown biological system or an environment where some viruses survive, only input-output data can be obtained, however, their changes can reflect some particular functions and features of a system.\nHence these data are directly used to build the model describing the original complicated network.\nSome early results considered identification of the network transition mappings \\cite{identification2000Akutsu,identification1998Liang,identification2006Nam,identification2005Pal}.\nUnder the framework of STP, the identification of BNs can be equivalently transformed into the identification of related structure matrices, which was proposed in \\cite{ModelConstruction2011} and was extended to BCNs in \\cite{IdentificationofBCN2011}.\n\\cite{IdentificationofBCN2011} noticed that, a BCN is identifiable if and only if it is controllable and O3-observable.\nThis O3-observability originates from one of five branching paths to the development of observability.\nWe list these five definitions of observability below.\n\\begin{definition}\\label{Def}\nA BCN is Oi-observable, {\\rm($i=1,2,3,4,5$)}, if\n\\begin{itemize}\n\\item[{\\rm(O1)}] {\\rm \\cite{OBCN1}} for any two distinct states $x(0)\\neq \\bar{x}(0)$, there exists an input sequence $(u(0), u(1), \\ldots)$, such that the corresponding output sequences are distinct: $(y(0), y(1), $ $\\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O2)}] {\\rm \\cite{OBCN2}} for any a state $x(0)$ there exists an input sequence $(u(0)$, $u(1)$, $\\ldots)$, such that for any $\\bar{x}(0)\\neq x(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O3)}] {\\rm \\cite{OBCN3}} there exists an input sequence $(u(0), u(1),\\ldots)$, such that for any two distinct states $x(0)\\neq \\bar{x}(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O4)}] {\\rm \\cite{OBCN4}} for any two distinct states $x(0)\\neq \\bar{x}(0)$ and for any input sequence $(u(0),u(1),\\ldots)$, the corresponding output sequences are distinct:\n $(y(0),y(1),$ $\\ldots)$$\\neq$ $(\\bar{y}(0), \\bar{y}(1),\\ldots)$;\n\n\\item[{\\rm(O5)}] {\\rm \\cite{OBCN5}} there exists an output-feedback loop $u(t)=f_t(y(t))$ $(u(t)=f(y(t))$ for static control), such that for any two distinct states $x(0)\\neq \\bar{x}(0)$, the corresponding output sequences are distinct: $(y(0), y(1), \\ldots)\\neq (\\bar{y}(0), \\bar{y}(1),\\ldots)$.\n\\end{itemize}\n\\end{definition}\n\nMost of the criteria and methods for judging the first four kinds of observability (O1-O4) are not ideal, some are sufficient conditions, some are too complex to apply.\n\\cite{OBCNZhang20161} proposed a unified approach based on finite automata to determine these four observabilities, and presented corresponding four necessary and sufficient conditions.\nThis automata approach is more suitable for BCNs with fewer state nodes and input nodes due to the high complexity of constructing deterministic finite automata.\n\\cite{OBCNCheng20162} concentrated on the most general observability (O1), and presented a matrix-based approach with lower complexity by STP.\nMathematically speaking, the requirement of input sequences used to recognize the initial state $x(0)$ gradually increases from O1-observability (the most general form) to O4-observability (the sharpest form). Hence, O4 $\\Longrightarrow$ O3 $\\Longrightarrow$ O2 $\\Longrightarrow$ O1, which is shown as a relation diagram in \\cite{OBCNZhang20161}.\nIn particular, determining O3-observability is NP-hard \\cite{OBCN3}.\nO5-observability that recognizes the initial state via output feedback, called output-feedback observability, was first proposed in \\cite{OBCNGuoCCC2017}. This one is much sharper than the first two observabilities.\n\\cite{OBCN5} used paralled interconnected two identical BCNs to determine O5-observability, by converting the observability problem of the original BCN to the set reachability problem of the interconnected BCN.\n\nAs mentioned above, the identification of BCNs requires O3-observability.\nA natural question arises: what about the most general form, O1-observability?\nMotivated by that, we further develop the identification problem for BCNs in this paper.\nMain contributions are summarized as follows:\n\n(1) Three important theoretical results are obtained: (3a) A BN is uniquely identifiable if it is observable; (3b) A BCN is uniquely identifiable if it is O1-observable. It is worth pointing out that O1-observability is the most general one of the existing observability terms. (3c) A BN or BCN may be identifiable, but not observable.\n\n(2) In combination with the phenomena in medical detection, we propose two new concepts: single sample and multiple samples to deal with the identification problem of BCNs.\nBased on them, the identification problem is divided into four situations.\nWe point out that the existing works on identification are actually special cases of these four situations.\n\n(3) By virtue of the observability property, we form a one-to-one correspondence between the state and the output sequence.\nThen four simple criteria to determine the identifiability and four effective algorithms to construct the structure matrices are proposed.\n\n\n\nThe rest of the paper is organized as follows. Section II contains preliminary notations, fundamental definitions and problem formulation.\nSection III presents main results on identification of BNs and BCNs, including several discriminant methods for the identification property, several identification algorithms to construct the structure matrices and illustrative examples. Remarks are given to show some challenging and interesting future research.\nFinally, a table describes the relationships and comparisons of the results obtained in this paper, and Section IV concludes the paper.\n\n\n\\section{Preliminaries}\n\n\\subsection{Semi-tensor product}\nThis section gives some necessary preliminaries. More details can be referred to \\cite{Chengbook2012}.\nFirst, some notations are listed below:\n\\begin{itemize}\n\\item[$\\bullet$] $\\mathbb{N}=\\{0,1,2,\\ldots\\}$: the natural number set.\n\\item[$\\bullet$] $[a,b]_{\\mathbb{N}}$: all the natural numbers from $a$ to $b$.\n\\item[$\\bullet$] $\\delta_n^i$: the $i$th column of the identity matrix $I_n$.\n\\item[$\\bullet$] $\\Delta_n:=\\{\\delta_n^1,\\delta_n^2,\\ldots,\\delta_n^n\\}$.\n\\item[$\\bullet$] $\\mathbf{1}_n:=\\sum_{i=1}^n\\delta_n^i$.\n\\item[$\\bullet$] $[\\delta_n^{i_1}~\\delta_n^{i_2}~\\cdots ~\\delta_n^{i_m}]:=$ $\\delta_n[i_1~i_2 ~\\cdots~i_m]$.\n\\item[$\\bullet$] $(\\delta_n^{i_1},\\delta_n^{i_2},\\ldots,\\delta_n^{i_m}):=$ $\\delta_n(i_1,i_2,\\ldots,i_m)$.\n\\item[$\\bullet$] $Col_i(M)$: the $i$th column of matrix $M$.\n\\item[$\\bullet$] $Col(M)$: the set of all columns of $M$.\n\\item[$\\bullet$] $\\mathcal{L}_{m\\times n}$: $=\\{M| M\\in \\mathbb{R}^{m\\times n},Col(M)\\subseteq \\Delta_{m}\\}$.\n\\item[$\\bullet$] $[M]_{i,j}$: the $(i,j)$th entry of matrix $M$.\n\\item[$\\bullet$] $M^{\\mathrm{T}}$: the transpose of matrix $M$.\n\\item[$\\bullet$] Kronecker product: $A\\otimes B=([A]_{i,j}\\times B)$.\n\\item[$\\bullet$] K-R product: $A\\ast B=C$, $Col_l(C)=Col_l(A)\\otimes Col_l(B)$.\n\n\\end{itemize}\n\n\n\\begin{definition}{\\rm\\cite{Chengbook2012}}\nThe semi-tensor product (STP) of two matrices $A\\in\\mathbb{R}^{m\\times n}$ and $B\\in\\mathbb{R}^{p\\times q}$ is\n\\begin{equation*}\nA\\ltimes B=(A\\otimes I_{\\frac{s}{n}})(B\\otimes I_{\\frac{s}{p}}),\n\\end{equation*}\nwhere $s$ is the least common multiple of $n$ and $p$.\n\\end{definition}\n\nObviously, the STP becomes the conventional matrix product if $n=p$. Hence the symbol $\\ltimes$ is omitted in the sequel.\n\n\\begin{lemma}\\label{Lem1}{\\rm\\cite{Chengbook2012}}\nLet $f(x_1,\\ldots,x_n)$ be a Boolean function, where $x_1,\\ldots,x_n$ are Boolean variables.\nWithin the framework of vector form, $f$ can be converted into $f:\\Delta_{2^n} \\rightarrow \\Delta_2$, and there exists a unique matrix $M_f\\in \\mathcal{L}_{2\\times 2^n}$, called the structure matrix of $f$, such that\n\\begin{equation*}\nf(x_1,\\ldots,x_n)=M_f\\ltimes x_1\\ltimes \\cdots\\ltimes x_n.\n\\end{equation*}\n\\end{lemma}\n\nConsider a BCN with $n$ state nodes, $m$ input nodes and $l$ output nodes as follows:\n\\begin{equation}\\label{BCN1}\n\\left\\{\n\\begin{array}{ll}\nx_i(t+1)=f_i(u_1(t),\\ldots,u_m(t),x_1(t),\\ldots,x_n(t)),\\\\\ny_j(t)=h_j(x_1(t),\\ldots,x_n(t)),\\\\\ni\\in [1,n]_{\\mathbb{N}},~j\\in [1,l]_{\\mathbb{N}},~t\\in \\mathbb{N},\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f_i:\\mathcal{D}^{m+n}\\rightarrow\\mathcal{D}$ and $h_j:\\mathcal{D}^{n}\\rightarrow\\mathcal{D}$ are logical functions, $x_i\\in \\mathcal{D}$, $u_j\\in\\mathcal{D}$ and $y_k\\in\\mathcal{D}$ are the state, input and output of the system, respectively.\n\nFrom Lemma \\ref{Lem1}, each logical function $f_i$ ($h_j$) has unique structure matrix $M_{f_i}$ ($M_{h_j}$), then BCN \\eqref{BCN1} can be equivalently transformed into an algebraic form as follows \\cite{Chengbook2012}:\n\\begin{equation}\\label{BCN2}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fu(t)x(t),\\\\\ny(t)=Hx(t),\n\\end{array}\n\\right.\n\\end{equation}\nwhere $F=M_{f_1}\\ast M_{f_2}\\ast\\cdots\\ast M_{f_n}$, $H=M_{h_1}\\ast M_{h_2}\\ast\\cdots\\ast M_{h_l}$, $x(t)=\\ltimes_{i=1}^nx_i(t)$, $u(t)=\\ltimes_{j=1}^mu_j(t)$, $y(t)=\\ltimes_{k=1}^ly_k(t)$.\nThis form is called the algebraic state space representation of \\eqref{BCN1}.\n\n\n\\subsection{Problem statement}\n\nThe identification problem of BCN \\eqref{BCN2} is to construct two structure matrices $F$ and $H$ via available data. Denote\n\\begin{align*}\nU_i(p_i):=&\\{u_i(t)\\}_{t=0}^{p_i}=(u_i(0),u_i(1),u_i(2),\\ldots,u_i(p_i)), \\\\\nX_i(p_i):=&\\{x_i(t)\\}_{t=0}^{p_i}=(x_i(0),x_i(1),x_i(2),\\ldots,x_i(p_i)), \\\\\nY_i(p_i):=&\\{y_i(t)\\}_{t=0}^{p_i}=(y_i(0),y_i(1),y_i(2),\\ldots,y_i(p_i)),\n\\end{align*}\nand\n\\begin{align*}\n\\{U_i(p_i)\\}:=&\\{u_i(0),u_i(1),u_i(2),\\ldots,u_i(p_i)\\}, \\\\\n\\{X_i(p_i)\\}:=&\\{x_i(0),x_i(1),x_i(2),\\ldots,x_i(p_i)\\}, \\\\\n\\{Y_i(p_i)\\}:=&\\{y_i(0),y_i(1),y_i(2),\\ldots,y_i(p_i)\\}.\n\\end{align*}\n\n\n\\begin{definition}\nA BCN \\eqref{BCN2} is said to be identifiable, if its two structure matrices $F$ and $H$ can be determined via available data: input data $U_1(p_1),U_2(p_2),\\ldots,$ $U_k(p_k)$ and observed data $Y_1(p_1),Y_2(p_2),\\ldots,Y_k(p_k)$.\n\\end{definition}\n\n\nA coordinate transformation $\\omega=Gx$ could convert \\eqref{BCN2} into the following algebraic form:\n\\begin{align}\\label{BCN3}\n\\left\\{\n\\begin{array}{ll}\n\\omega(t+1)=GF(I_{2^m}\\otimes G^{\\mathrm{T}})u(t)\\omega(t)=:\\widehat{F}u(t)\\omega(t), \\\\\ny(t)=HG^{\\mathrm{T}}\\omega(t)=:\\widehat{H}\\omega(t).\n\\end{array}\n\\right.\n\\end{align}\nDue to the arbitrariness of state recognition, \\eqref{BCN2} and \\eqref{BCN3} are considered to be identical in the same input-output data frame, so the set of all possible $(\\widehat{F},\\widehat{H})$ becomes the equivalence class of $(F,H)$.\nA identifiable BCN is also said to be $uniquely$ $identifiable$ in the sense of equivalence.\n\n\n\\begin{assumption}\\label{Assum1}\nThis paper assumes the available data is sufficient.\nIn other words, the input data and the observed data contain all possible situations which the system could generate.\n\\end{assumption}\n\nGenerally speaking, densely populated cities are good places for virus or infectious diseases, which could spread easily from person to person.\nThe Centers for Disease Control and Prevention can collect a large number of samples from different patients infected by the same pathogen.\nHence, Assumption \\ref{Assum1} is reasonable and its implementation requires $multiple$ $samples$ from large numbers of patients (urine sample or blood sample or cheek swab), not a $single$ $sample$ from one patient, since a single sample may exhibit only part of characteristics of the virus.\nMultiple samples mean that the observed data may be generated from different initial states, while, single sample means that the observed data is generated from some initial state.\n\nOn the basis of the statement above, the identification of BNs and BCNs can be divided into four cases:\n\\begin{itemize}\n\\item[Case 1]: the identification process of single sample in the BN records one group of output data $Y(p)$.\n\n\\item[Case 2]: the identification process of multiple samples in the BN records $k$ groups of output data $Y_1(p_1), Y_2(p_2), \\ldots, Y_k(p_k)$.\n\n\\item[Case 3]: the identification process of single sample in the BCN records $r$ groups of input-output data $U_1(p_1),Y_1(p_1),U_2(p_2),Y_2(p_2),\\ldots, U_r(p_r),Y_r(p_r)$.\n\n\\item[Case 4]: the identification process of multiple samples in the BCN records $rk$ groups of input-output data $U_1^i(p_1),Y_1^i(p_1),U_2^i(p_2),Y_2^i(p_2),\\ldots, U_r^i(p_r),Y_r^i(p_r)$, $i\\in [1,k]_{\\mathbb{N}}$.\n\n\\end{itemize}\nBoth Cases 1 and 3 collect the blood sample from only one patient, while Cases 2 and 4 collect from $k$ patients.\nCase 3 divides the blood sample into multiple portions ($r$ portions) for testing with a variety of reagents. That is to say, $r$ groups of input-output data are generated from the same initial state, i.e., $x_1(0)=x_2(0)=\\cdots=x_r(0)$ (in Case 3). Similarly, $x_1^{i}(0)=x_2^i(0)=\\cdots=x_r^i(0), i\\in[1,k]_{\\mathbb{N}}$ in Case 4.\n\n\n\n\n\n\n\\section{Identification of BNs and BCNs}\n\n\n\\subsection{Identification of BNs}\n\\cite{ModelConstruction2011} investigated the identification of the following BN:\n\\begin{align}\\label{BN1}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fx(t),\\\\\ny(t)=x(t),\n\\end{array}\n\\right.\n\\end{align}\nin which the observed data is presented directly by the system state.\nWith a group of observed data $(x_1(0)=\\delta_{2^n}^{i_0},x_1(1)=\\delta_{2^n}^{i_1},\\ldots)$, the $i_0$th column of $F$ can be identified as $Col_{i_0}(F)=\\delta_{2^n}^{i_1}$ and hence the next result is obtained.\n\n\\begin{lemma}\\label{Lem2}{\\rm \\cite{ModelConstruction2011}}\n{\\rm (}Multiple samples{\\rm)} BN \\eqref{BN1} is uniquely identifiable, if and only if the observed data contains all possible states:\n\\begin{align}\n\\{Y_1(p_1)\\}\\cup\\{Y_2(p_2)\\}\\cup\\cdots\\cup\\{Y_k(p_k)\\}=\\Delta_{2^n}.\n\\end{align}\n\\end{lemma}\n\nIt is noted that the observed data considered in \\cite{ModelConstruction2011} may consist of several output sequences (i.e., multiple samples), which is reasonable because a system may contains multiple attractors and multiple attractors mean multiple state trajectories.\nWhen the system state cannot be directly observed, BN \\eqref{BN1} becomes\n\\begin{align}\\label{BN2}\n\\left\\{\n\\begin{array}{ll}\nx(t+1)=Fx(t),\\\\\ny(t)=Hx(t).\n\\end{array}\n\\right.\n\\end{align}\nIn the process of identifying this system, it is the most important to distinguish the states.\n\n\n\\begin{definition}\nIn BN \\eqref{BN2}, a state pair $(x(0),\\bar{x}(0))$, $x(0)\\neq \\bar{x}(0)$ is said to be distinguishable if the corresponding output sequences generated by them are distinct: $(y(0),y(1),\\ldots)\\neq (\\bar{y}(0),\\bar{y}(1),\\ldots)$. \\eqref{BN2} is said to be observable if any state pair is distinguishable.\n\\end{definition}\n\nObservability means that $2^n$ distinct initial states generate $2^n$ distinct groups of observed data, i.e.,\n\\begin{align}\n(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,HF^{2}\\delta_{2^n}^i,\\ldots)\\neq (H\\delta_{2^n}^{i'},HF\\delta_{2^n}^{i'},HF^{2}\\delta_{2}^{i'},\\ldots),~i\\neq i'.\n\\end{align}\nSince each state trajectory will fall into an attractor in $2^n$ steps, the subsequent state trajectory and output trajectory will repeat the previous data. Lemma 1 and Proposition 1 in \\cite{OBCN4} show the following result.\n\\begin{proposition}\\label{Pro3}{\\rm\\cite{OBCN4}}\nBN \\eqref{BN2} is observable if and only if for any $i\\neq i'$,\n\\begin{align}\\label{Pro3-1}\n(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,\\ldots, HF^{2^n-1}\\delta_{2^n}^i)\\neq (H\\delta_{2^n}^{i'},HF\\delta_{2^n}^{i'},\\ldots,HF^{2^n-1}\\delta_{2}^{i'}).\n\\end{align}\n\\end{proposition}\nWe call $(H\\delta_{2^n}^i,HF\\delta_{2^n}^i,\\ldots,HF^{2^n-1}\\delta_{2^n}^i)$ the $effective$ $output$ $sequence$ of state $\\delta_{2^n}^i$.\nAn effective output sequence corresponds to a state and its length is $2^n$ steps.\nUnder the case of Assumption \\ref{Assum1}, if the system is observable, $2^n$ distinct effective output sequences can be found by searching and comparing all $2^n$-step output sequences from sufficient observed data.\n\nAssume that the following $k$ groups of observed data are sufficient,\n\\begin{align}\\label{Th1-data}\nY_j(T_j)=(y_j(0),y_j(1),\\ldots,y_j(T_j)),~j\\in [1,k]_{\\mathbb{N}}.\n\\end{align}\nLet $Y_s^j$ represent the $s$th $2^n$-step output sequence to show up in $Y_j(T_j)$:\n\\begin{align}\\label{Th1-data1}\nY_s^j=&(y_j(s-1),y_j(s),\\ldots,y_j(s+2^n-2)),~s\\in [1,T_j']_{\\mathbb{N}},\n\\end{align}\nwhere $T_j'=T_j-2^n+2$.\nThen by retrieval from \\eqref{Th1-data}, an algorithm (Algorithm \\ref{Alg:A}) to find $2^n$ distinct effective output sequences is established, and this algorithm names the $i$th effective output sequence that occurs in Algorithm \\ref{Alg:A} as $Y_i$, $(i\\in[1,2^n]_{\\mathbb{N}})$.\n\n\\begin{algorithm}[H]\n\\caption{Retrieve all distinct effective output sequences}\n\\label{Alg:A}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithmic}[1]\n \\REQUIRE data \\eqref{Th1-data}.\n \\ENSURE $Y_1,Y_2,\\ldots,Y_{2^n}$.\n \\STATE{set $Y=\\emptyset$ and $i=1$}\n \\FOR{$j=1; j 0$, dependent on how quickly the baseline moves through the corrugations in frequency.\nThe value of $k_{||}$ at which this power is introduced is simply proportional to the ``slope'' of the baseline in the $|u|\\nu$-plane, which is itself proportional to the baseline length or $k_\\perp$. \nFurthermore, as sources closer to the horizon have higher oscillations in their voltage correlations as a function of baseline length, this mode-coupled $k_{||}$ value is also proportional to the distance of the source from phase-centre: $k_{||} \\propto l k_\\perp$. \nSince sources are constrained to be within the horizon, we have $l_{\\rm max} = \\sin \\theta_{k, {\\rm max}} = 1$, and we are able to define the ``horizon limit'' beyond which we do not expect flat-spectrum foreground sources to contribute.\n \nM12 also argues that the wedge is \\textit{fundamental} in low-frequency interferometric observations, and cannot be avoided (at the relevant modes) merely by clever analysis, such as visibility gridding and weighting schemes.\n\n\\citet{Trott2012} (T12) reformulates the description of M12 in the context of a uniform distribution of faint undetected sources, deriving an exact analytical form for the expected foreground power under a set of simplifications.\n\nAlternatively, \\citet[hereafter P12]{Parsons2012} describes the emergence of the wedge in terms of the so-called ``delay transform''. \nThe delay transform considers a single baseline at a time, and associates `delay'-modes -- the fourier-dual of the frequency-track of the baseline -- with the line-of-sight modes $k_{||}$. \nThe delays themselves are simply time-delays between the reception of plane-wave emission at the two antennas composing the baseline (see Fig. 1 of P12 for a clear diagram).\nIn this scheme, the arguments are entirely geometric;\nfor a given source, not at phase centre, a longer baseline will correspond to a higher delay. \nSimilarly, for a given baseline, a source closer to the horizon will correspond to a higher delay -- with a physical maximum at the horizon. \nThis leads to the now familiar equation relating the line-of-sight mode at which power from a single source manifests: $\\tau \\sim k_{||} \\propto l k_\\perp$.\nP12 explains that the delay transform ideally maps a flat-spectrum point source to a delta-function in delay space, but that in practice the non-flat spectral properties of both source and instrument add a (hopefully narrow) kernel which can throw power into modes beyond the ``horizon limit'' (cf. Fig. 1 of P12).\nThey thus suggest that designing instruments with maximal spectral and spatial smoothness, as well as reduced field-of-view, and then ignoring modes below the horizon limit, is a useful way to avoid the foreground problem. \nThis has motivated the design of the PAPER \\citep{,Ali2015} and HERA \\citep{DeBoer2016} experiments. \n\nDespite the simplicity of these intuitive descriptions of the emergence of the wedge, its precise amplitude and shape are dependent on a combination of various complex effects. \nAmongst these are unavoidable sky-based effects such as the (spatially varying) spectrum of sources and diffuse emission, angular distribution of sources \\citep{Bowman2009,Trott2016a,Murray2017}, effects of co-ordinate transformation from curved to flat sky \\citep{Thyagarajan2013,Thyagarajan2015,Ghosh2017} and polarization leakage \\citep{Gehlot2018}, as well as spectral characteristics of the instrument, such as the beam attenuation pattern, bandpass, chromatic baselines and chromatic calibration errors \\citep{Bowman2009,Thyagarajan2015,Pober2015,Trott2016a}.\nDue to the complexity of these effects, they are often investigated either by using simulations or via analytic simplifications which elucidate the effects of some subset of the components. \nSeveral works have been devoted to developing general frameworks to model the foreground wedge in order to mitigate it effectively \\citep[eg.][]{Liu2014,Liu2014a,Pober2015,Ghosh2017}.\n\nDespite the breadth of this research, one aspect which seems to have gained little attention is the layout of the antennas (or correspondingly the baselines) themselves,\nand how they might be used to mitigate the wedge.\nThis is perhaps surprising as several of the seminal works on the topic suggest that one way to alleviate mode-mixing is to employ dense $uv$-sampling so that $uv$-samples overlap at various frequencies \\citep[eg.][]{Bowman2009,Morales2012,Parsons2012}.\nWhile a perfect $uv$-sampling is clearly unachievable, which has perhaps led to this avenue being largely ignored, it would seem advantageous to determine the extent of wedge-suppression possible under reasonable constraints. \n\nThe purpose of this paper is to explore this question, which we attack in two parts. First we approach the question of how the wedge relates to the baseline layout, seeking intuitive semi-analytical understanding of the factors involved. Secondly, we ask the more pointed question of how far the wedge might be suppressed merely by choice of layout, limiting ourselves to layouts which might be realistically achieved. For this latter question, we necessarily turn to simple numerical simulations.\nWe approach the questions from a pedagogical view, making simplifications where necessary in order to elucidate conceptual understanding.\n\n\n\n\nThe layout of the paper is as follows. \\S\\ref{sec:framework} introduces the general equations (and assumptions) used throughout this paper to define the expected 2D PS, and the model simplifications we adopt. \n\\S\\ref{sec:classic} presents the classical form of the wedge by solving the general equation for a suitably sparse layout, which is shown to be equivalent to the delay spectrum.\nThis lays the groundwork for \\S\\ref{sec:weighted}, which considers the same family of $uv$-sampling functions, but with increased density, and thus must account for correlations between baselines.\nIt provides a semi-analytic framework to describe the density of baselines required to mitigate the wedge, and also the effects of deviations of the baseline layout from the assumed perfect regularity.\n\\S\\ref{sec:mitigation:arrays} turns to discussion of the consequences of the preceding results for realistic arrays, and analyses explicit wedge reduction for a series of archetypal layouts.\nFinally, \\S\\ref{sec:conclusions} wraps up with a summary of the key arguments and conclusions throughout the paper, and a prospectus for future work.\n\n\\section{Framework for Expected Foreground Power}\n\\label{sec:framework}\nIn this section we derive a (simplified) general equation that describes the \\textit{expected} 2D PS for a given sky distribution and instrument model, following similar lines as \\citet{Trott2012}, \\citet{Liu2014} and \\citet{Trott2016}\\footnote{Readers familiar with these derivations should be able to skim this section lightly, referring to Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded} and \\ref{eq:power_general} and Tables \\ref{tab:assumptions} and \\ref{tab:models} thereafter.}.\nWe differ from \\citet{Liu2014} in that we express the expected power (closely related to the covariance of visibilities) in the basis of the natural coordinates, ($\\vect{u}$, $\\eta$), rather than baseline vectors and delay (they express their covariance in the latter basis, and reconstitute in cosmologically aligned coordinates via another transformation).\nThis makes sense for our analysis, as we are interested in the conceptual understanding of where foreground power arises from, rather than a numerically efficient power spectrum estimator.\nWe also differ from \\citet{Trott2016} in that we consider correlations between baselines in the expectation of the foreground power, which is necessary in order to properly evaluate the effects of $uv$-sampling. \n\nWe \\textit{a priori} remark that our framework is not fully general -- it does not include all possible factors.\nThis is in the hope of elucidating our primary goal -- the effect of the antenna layout.\nOne simplification we will enforce for this paper is that we only consider the effect of \\textit{point sources}, not Galactic emission, or extended sources (or the negligible EoR signal for that matter). \nThe extension to these other sources of foreground emission is neither conceptually important for this work nor conceptually difficult (though the details of the formulation can be rather involved, eg. \\citet{Trott2017,Murray2017}).\nA second simplification is that we consider the simple case in which the telescope is pointing instantaneously at zenith.\nThis alleviates complications arising from baseline foreshortening (not entirely, though enough for the conceptual understanding aimed for in this work, cf. \\citet{Thyagarajan2015}).\nFurther, for simplicity, we will assume a perfectly co-planar array. \nThe effect of relaxing these assumptions is expected to modulate power within the wedge, and potentially extend its reach to some degree.\nNevertheless, our focus is on examining the fundamental reason for the wedge, and whether it may be suppressed via appropriate $uv$-sampling -- thus focusing on a simple subset is appropriate. \nWe attempt to exhaustively list the various global assumptions and simplifications we have made in Table \\ref{tab:assumptions}.\nWe will outline further model simplifications and choices as we develop the equations within this section.\n\n\\begin{table*}[!ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{ l }\n\t\t\t\\hline\n\t\t\t\\textbf{Assumptions used in framework} \\\\ \n\t\t\t\\hline\n\t\t\tRestriction to extra-galactic point sources \\\\\n\t\t\tZenith-pointing only \\\\\n\t\t\tCo-planar antenna array \\\\\n\t\t\tFlat-sky approximation \\\\\n\t\t\tNaturally-weighted baselines \\\\\n\t\t\tThermal noise values of all baselines drawn from i.i.d Normal distribution, centred on zero \\\\\n\t\t\tFlat-spectrum sources \\\\\n\t\t\t\\hline\n\t\t\t\\hline\t\n\t\t\\end{tabular}\n\t\\end{center}\n\\caption{Summary of universal assumptions and simplifications used in this paper. \\label{tab:assumptions}}\n\\end{table*}\n\n\\subsection{Single-baseline visibility}\nWe begin with the simple visibility equation for a co-planar array, which defines the signal received by any baseline, in the presence of point sources and thermal noise:\n\\begin{equation}\n\\label{eq:simple_vis}\nV_{i}(\\nu, \\vect{u}_i) = \\phi_\\nu \\left[\\mathcal{N}_{i,\\nu} + \\int d\\vect{l} dS \\ n(\\vect{l}, S) SB(\\nu, \\vect{l}) e^{-2\\pi i \\vect{u}_i\\cdot \\vect{l}}\\right],\n\\end{equation}\ni.e. the Fourier-transform over the sky of the emission brightness attenuated by the beam $B$ and a frequency taper $\\phi_\\nu$.\n\nThe vector $\\vect{u}_i$ is the baseline length in units of the observational wavelength:\n\\begin{equation}\n\\vect{u}_i = \\vect{b}_i\/\\lambda,\n\\end{equation}\nwherein lies the chromaticity of the $uv$-sampling.\nHenceforth,\nwe let $\\nu_0$ be a reference frequency (this will later be tied to the mid-point of the frequency band of observation without loss of generality), and define $f = \\nu\/\\nu_0$.\nWe also explicitly let $\\vect{u}_i$ denote the value of $\\vect{u}_i$ \\textit{at the reference frequency}.\nThis is illustrated in Fig.~\\ref{fig:baseline_schematic}, which also shows why the delay approximation -- identifying the vertical shaded regions with the diagonal baseline tracks -- is reasonable for small $u$. \n\nIn this paper, we will exclusively use a Gaussian-shaped beam, and predominantly it will be achromatic\\footnote{For demonstration purposes, \\S\\ref{sec:classic} will also use a chromatic Gaussian beam, for which we have\n\t\\begin{equation}\n\t\\sigma_\\nu = 0.42c\/\\nu D = \\sigma_0\/f. \n\t\\end{equation}\n}. \nThe Gaussian beam is thus\n\\begin{equation}\n\tB_\\nu(l) = e^{-l^2\/2\\sigma^2},\n\\end{equation}\nwhere $\\sigma = 0.42c\/\\nu_0 D$ is the beam width \\citep{Trott2016}.\nThe motivation for using a Gaussian beam is that it is the most realistic analytically-tractable form possible, and its use for conceptual studies has precedent \\citep{Liu2014}.\nWe do note that the choice of a smooth Gaussian, which does not have sidelobes, has desirable effects on the form of the wedge, in that it perfectly suppresses horizon sources. \nThis effectively combats the complexities of baseline foreshortening at the horizon due to our appropriated flat-sky approximation. \n\n\nThe emission brightness is written in Eq.~\\ref{eq:simple_vis} as the sum of the flux density of all point-sources in the sky, where the number counts of these sources are given by $n(\\vect{l},S)$. These differential number counts are in general a statistical quantity, as we will typically consider sources below the confusion limit of an instrument. \nTo simplify the calculations to follow, we have followed common pedagogical practice and assumed that the the spectral shape of each source is entirely flat. \nThis simplification is rather heavy, but it should not affect the \\textit{conceptual} understanding of the following calculations.\n\nFurthermore, the observed frequency window is both physically attenuated by the instrument, and will also be tapered within the analysis in order to suppress frequency side-lobes\\footnote{Note that each sub-band also has its own structure. This may be assumed to be a part of $\\phi$, or may be introduced as a secondary convolution \\citep[cf. $\\gamma$ in][]{Liu2014}. We shall ignore it in this work so as not to complicate the key ideas.}. \nIn this paper, we assume that the bandpass is relatively broad compared to the taper, and that its effect can be safely ignored\\footnote{We note that this is a particularly strong simplification. \n\tThe bandpass will in general introduce smaller-scale oscillations into $\\phi$, which tend to broaden the footprint of the foreground power in $\\omega$. This is in some way countered by our use of a broad $\\omega$-space Gaussian taper.}.\nWe normalize the frequency-taper to $\\phi(0) = 1$ \n\\footnote{This assumption, which we employ for simplicity throughout this paper, ties the central frequency of the bandpass to the reference frequency, $\\nu_0$. \nThis is without loss of generality, as we may always shift all frequency-dependent parameters to a new ``reference'' before any analysis.}, and exclusively use a Gaussian taper here for tractibility:\n\\begin{equation}\n\t\\phi(f-1) = e^{-\\tau^2 (f-1)^2},\n\\end{equation}\nwith $\\tau$ an inverse-width (or precision).\n\nMore common choices for the taper are the Blackman-Harris \\citep{Trott2016a} or its self-convolution \\citep{Thyagarajan2016}.\nThese serve to reduce leakage of power into higher modes, and are better choices than a Gaussian in practice. \nWe utilise the Gaussian for analytic simplicity and note that most qualitative conclusions of the paper are insensitive to this choice (note also that there is precedent for choosing such a taper for theoretical studies, in \\citet{Liu2014}).\nThe primary point of difference is in the definition of the ``brick'' (cf. Table \\ref{tab:simple_summary}), which extends to a higher value of $\\omega$ when using a Gaussian.\nThis also affects the position of the emergence of the wedge.\n\n\nTaking the Fourier transform (over $\\nu$), we arrive at\n\\begin{align}\n\t\\label{eq:delay_vis}\n\t\\tilde{V}_i(\\eta, \\vect{u}_i) = \\nu_0 & \\int df\\ \\ e^{-2\\pi i \\nu_0f \\eta} V_i(\\nu, \\vect{u}_i).\n\\end{align}\nNote that we will make the change of variables $\\omega = \\nu_0 \\eta$ for the remainder of this work, where $\\omega$ is dimensionless and makes for simpler theoretical equations.\nThe square of this particular quantity is called the \\textit{delay spectrum}, and we explore it briefly in \\S\\ref{sec:classic}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{figures\/baseline_schematic}\n\t\\caption{Schematic showing the migration of baseline length (in wavelength units) as a function of frequency. The vertical shaded regions indicate the axis of the frequency Fourier Transform, showing that multiple baselines contribute at different frequencies for high $u$. Orange dots show where each baseline is equivalent to the scale at which it would be equated in the delay approximation. At low $u$, a FT along a baseline is almost equivalent to the true FT, showing the delay approximation to be accurate.}\n\t\\label{fig:baseline_schematic}\n\\end{figure}\n\n\\subsection{Multi-baseline visibility}\nTo increase signal-to-noise, we typically \\textit{grid} the discrete baselines in some fashion.\nThe delay spectrum approach only grids measurements after squaring the visibilities, whereas image-based approaches grid the complex visibilities before squaring to form the power.\nWe follow the latter approach in this work, as it is required in order to utilise the benefits of the $uv$-sampling.\nHowever, the approaches can be forced to align with each other under special array layout conditions, and we explore this briefly in \\S\\ref{sec:classic}.\n\nIn essence, the gridding assigns a weight to each baseline, specifying its contribution to a given UV point (i.e. closer points receive higher weights) \n\\footnote{We are also free to choose a baseline weighting which is a function only of the magnitude of $u_i$. Nevertheless, previous work has shown that an optimal choice for the baseline weighting is to have each baseline weighted equally \\citep{Bowman2009,Parsons2012}, and we follow suit here.}. \nThe total estimated visibility at point $\\vect{u}$ is thus the sum of all weighted visibilities, normalised by the total weight.\nLetting $w_\\nu$ denote the weighting function, and defining the total weight as\n\\begin{equation}\n\\label{eq:wnuu}\nW_\\nu(\\vect{u}) = \\sum_{i=1}^{N_{\\rm bl}}w_\\nu(\\vect{u} - f \\vect{u}_i),\n\\end{equation}\nwe have\n\\begin{equation}\n\\label{eq:vis_gridded}\nV(\\nu, \\vect{u}) = \\frac{1}{W_\\nu(\\vect{u})}\\sum_{i=1}^{N_{\\rm bl}} w_\\nu(\\vect{u} - f\\vect{u}_i) V_i(\\nu,\\vect{u}_i).\n\\end{equation}\nThe Fourier-space visibility is then\n\\begin{align}\n\t\\label{eq:vis_full}\n\t\\tilde{V}(\\omega, \\vect{u}) = \\nu_0 & \\int df\\ \\ e^{-2\\pi i f\\omega} V(\\nu, \\vect{u}),\n\\end{align}\nwhich may be squared to form the grid-based PS.\n\nThe combination of Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded} and \\ref{eq:vis_full} provide the basis for all following work.\n\nIt has been shown in T16 that an unbiased gridding of visibilities is determined by inverting a matrix involving the fourier-transformed primary beam\\footnote{In their work (cf. their Eq. 17) it also involves a matrix $\\vect{G}$ which accounts for sky-curvature and other effects which we ignore here.}.\nIndeed, they find that a good approximation to this matrix inversion, which enhances computability considerably, is to use the diagonalized inversion, which corresponds precisely to a weighted average of baselines, with a weighting function given by the fourier-transform of the beam, $\\tilde{B}(\\vect{u})$. \n\nExplicitly, for the Gaussian beam employed in this work, we have\n\\begin{equation}\nw_\\nu(u) = B(u) = e^{-2\\pi^2 \\sigma^2 u^2}.\n\\end{equation}\n\n\n\\subsection{Statistical properties of the visibility}\nThe visibility is in general a statistical variable, both because of the random thermal noise and the (typically) statistical nature of $n(\\vect{l},S)$.\n$\\tilde{V}$ is in general not Gaussian, nevertheless as we will be dealing with the power spectrum -- a quadratic quantity -- we shall only be required to know up to second-order properties of $\\tilde{V}$ for the purposes of this paper.\nAs long as the thermal noise is independent of the foreground signal, these are simply derived.\n\nOur primary sky model consists of a uniform Poisson process of point sources (uniform in $\\vect{l}$, cf. T16)\\footnote{This is a reasonable approximation for a relatively narrow beam (with no side-lobes) at zenith, where $\\theta \\sim \\vect{l}$. Our adoption of the Gaussian beam means this assumption will have little consequence. In reality, the curved nature of the sky introduces excess brightness towards the horizon, which can result in the ``pitchfork'' structure reported in \\citet{Presley2015,Thyagarajan2015,Thyagarajan2016} (see also \\citet{Kohn2016,Kohn2018})}.\nIn this model, expectation of the frequency-space visibility is obtained simply by replacing the sky emission with the mean flux density,\n\\begin{align}\n\t\\label{eq:su_meanvis}\n\t\\langle V_{i}\\rangle(\\nu, \\vect{u}_i) &= \\phi_\\nu \\int d\\vect{l}\\ \\bar{S} B_\\nu(\\vect{l}) e^{-2\\pi i f \\vect{u}_i \\cdot \\vect{l}}. \\nonumber \\\\\n\t&= \\phi_\\nu \\bar{S} B(f\\vect{u}_i),\n\\end{align}\nand the expected Fourier-space visibility is\n\\begin{align}\n\t\\langle V (\\omega, \\vect{u})\\rangle = \\bar{S} \\nu_0 \\int df \\frac{e^{-2\\pi i f \\omega}\\phi_{\\nu}}{W_\\nu(\\vect{u})} \\sum_{i=1}^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i) B(fu_i). \n\\end{align}\n\nThe variance of the visibility is composed of two parts -- a thermal variance and sky variance -- which are assumed to be independent. \nThe sky variance may be worked out in similar fashion to the procedure outlined in \\cite{Murray2017}.\nThe total covariance (i.e. with thermal noise term) is then\n\\begin{align}\n\\label{eq:su_covvis}\n\\vect{C}_{\\rm bl} &= \\phi_\\nu^2 \\sigma_N^2 \\delta_{ij} \\delta(\\nu-\\nu') \\nonumber \\\\\n& + \\phi_\\nu^2 \\mu_2 \\int d\\vect{l}_1 B_{\\nu}B_{\\nu'} e^{2\\pi i \\vect{l} (f' \\vect{u}_j - f \\vect{u}_i)},\n\\end{align}\nbetween baselines $i$ and $j$, and frequencies $\\nu, \\nu'$, where we have used the assumed flat-spectrum of all sources, and $\\mu_2$ is the second moment of the source count distribution:\n\\begin{equation}\n\\mu_n = \\int dS\\ S^n \\frac{dN}{dS}.\n\\end{equation}\nThus the variance of the Fourier-space gridded visibility is\n\\begin{align}\n\t\\label{eq:general_var}\n\t{\\rm Var}(\\tilde{V}) &=& \\nu_0^2 &\\int df df' \\frac{e^{-2\\pi i \\omega (f-f')}}{W(\\vect{u}) W'(\\vect{u})} \\nonumber \\\\\n\t&& & \\times \\sum_{ij} ^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)B(\\vect{u} - f'\\vect{u}_j) \\vect{C}_{\\rm bl} \\nonumber \\\\\n\t&=& \\nu_0^2 \\sigma_N^2 &\\int df\\frac{ \\phi_\\nu^2 }{W^2_\\nu(\\vect{u})} \\sum_i^{N_{\\rm bl}} B^2(\\vect{u}-f\\vect{u}_i) \\nonumber \\\\\n\t& &\\ + \\nu_0^2 &\\int d\\vect{l} df df' \\frac{e^{-2\\pi i \\omega (f-f')}}{W_\\nu(\\vect{u}) W'_\\nu(\\vect{u})} \\nonumber \\\\\n\t& & & \\times \\sum_{ij} ^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)B(\\vect{u} - f'\\vect{u}_j) \\vect{C}_{\\rm sky}.\n\\end{align}\n\nThe first term of this equation defines the thermal noise variance of a grid-point $(\\omega, \\vect{u})$. \nWe omit the term for all following calculations\\footnote{This can be thought of as going to the limit of a perfectly calibrated instrument, or infinite integration time, and thus setting $\\sigma_N \\rightarrow 0$.}, but we define\n\\begin{equation}\n\t\\label{eq:w_theta}\n\tW^2(\\vect{u}) = \\int df \\frac{ \\phi_\\nu^2 }{W^2_\\nu(\\vect{u})} \\sum_i^{N_{\\rm bl}} B^2(\\vect{u}-f\\vect{u}_i)\n\\end{equation}\nwhich will be used to determine a grid-point's relative weight when averaged to form the 2D PS.\n\n\\subsection{Expected power spectrum}\nArmed with the statistical descriptors of the visibility, we can determine the expected 3D PS:\n\\begin{equation}\n\t\\label{eq:expectation_equation}\n\t\\langle P(\\omega, \\vect{u})\\rangle \\equiv \\langle \\tilde{V}\\tilde{V}^*\\rangle = {\\rm Var}(\\tilde{V}) + |\\langle \\tilde{V} \\rangle|^2.\n\\end{equation}\nNote that for the statistically uniform sky that we primarily adopt (where uniformity is in $\\vect{l}$), the second term is effectively the transfer function of the instrument (i.e. the beam and bandpass), and can typically be neglected on small perpendicular scales when these characteristics are angularly smooth.\n\nThe 2D PS is given by\n\\begin{equation}\n\t\\label{eq:power_general}\n\t\\langle P(\\omega, u) \\rangle = \\frac{\\int_0^{2\\pi} d\\theta \\ \\langle P(\\omega, \\vect{u})\\rangle W^2(\\vect{u})}{\\int_0^{2\\pi} d\\theta \\ W^2(\\vect{u})} , \n\\end{equation}\nwhere $\\theta$ is the polar angle of $\\vect{u}$.\n\nWe provide a synopsis of the meaning of symbols used in this paper in Table \\ref{tab:models}.\n\n\\begin{table*}[!ht]\n\t\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Symbol} & \\textbf{Description} & \\textbf{Models\/Values} \\\\ \n\t\t\t\\hline\n\t\t\t$\\nu$ & Frequency & \\\\\n\t\t\t$f$ & Normalised Frequency, $\\nu\/\\nu_0$ & \\\\\n\t\t\t$\\vect{l}$ & Cosine-angle of sky co-ordinate, $\\cos\\theta$ & \\\\\n\t\t\t$\\vect{b}$ & Baseline length & \\\\\n\t\t\t$\\vect{u}$ & Fourier-dual of $\\vect{l}$, equivalent to $\\vect{b}\/\\lambda$ & \\\\ \n\t\t\t$\\eta$, $\\omega$ & Fourier-dual (and scaled by $\\nu_0$) of $\\nu$ & \\\\\n\t\t\t$k_\\perp$, $k_{||}$ & Cosmologically-scaled $u$ and $\\eta$ respectively & \\\\\n\t\t\t$V$ & Interferometric Visibility as function of frequency& \\\\\n\t\t\t$\\tilde{V}$ & Frequency FT of $V$ & \\\\\n\t\t\t$S$ & Flux density (subscripted for a particular source) & \\\\\n\t\t\t$I(\\nu,\\vect{l})$ & Sky Intensity & \\\\\n\t\t\t\n\t\t\t\\hline\n\t\t\t$\\mu_1 \\equiv \\bar{S}$ & Mean brightness of sky & 1 Jy\/sr \\\\\n\t\t\t$\\mu_2$ & Second moment of source-count distribution & 1 Jy$^2$ \/sr \\\\\n\t\t\t$S_0$ & Flux density of single source in sky & 1 Jy \\\\\n\t\t\t$\\vect{l}_0$ & Position of single source in sky & (1,0) \\\\\n\t\t\t$\\nu_0$ & Reference frequency & 150 MHz \\\\\n\t\t\t$\\sigma$ & Beam-width at $\\nu_0$ & 0.2 rad \\\\\n\t\t\t$\\tau$ & Unitless band-pass precision, $1\/2\\sigma_f^2$ & 100 \\\\\n\t\t\t$D$ & Tile Diameter & 4m \\\\\n\t\t\t$\\phi$ & Frequency Taper & Gaussian \\\\\n\t\t\t$\\phi_B$ & Bandpass & Uniform \\\\\n\t\t\t$\\psi$ & Source spectral shape & Flat \\\\\n\t\t\t\\hline\n\t\t\t$B_\\nu$ & Beam Attenuation & Gaussian (Static; Chromatic) \\\\ \n\t\t\t$w$ & Visibility-gridding weights & Fourier-beam kernel \\\\\n\t\t\t$n(\\vect{l},S)$ & Source count distribution & Single Source; Stochastic Uniform \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\t\\caption{Summary of symbols and models used throughout this paper. Where possible, parameters list their default value used in plots throughout this paper. Any models list all models explored throughout this paper. Any Latin-subscripted perpendicular scale (eg. $\\vect{u}_i$) refers to a particular physical baseline at reference frequency. \\label{tab:models}}\n\t\\end{center}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\\section{Uncorrelated Visibilities}\n\\label{sec:classic}\nThe wedge has been shown to naturally arise in the expected PS when correlations between baselines are either ignored or absent \\citep[eg.][]{Parsons2012,Trott2016}.\nFor example the delay spectrum, which by definition cannot correlate visibility pairs, can be used to provide a simple illustration of the emergence of the wedge. \nTo provide a backdrop for discussion of the effects correlating close baselines, we first turn to this uncorrelated case (within the context of our framework) to illustrate the emergence of the wedge.\n\n\n\nIn order to obtain simple single-baseline measurements of the PS within our framework, we use three conditions: i) baselines sparsely arranged on a set of logarithmic spokes, ii) an artificially narrow gridding kernel, and iii) for simplicity, we evaluate the PS only at reference baseline positions (i.e. $\\vect{u} = \\vect{u}_i$).\n\nAs we shall see, condition (iii) is not really required, but does simplify the procedure slightly.\nCondition (ii) can be more precisely stated as setting the gridding kernel width to approach zero.\nThis is artificial, because we will not enforce the beam width to follow the same limit.\nAlternatively, one may imagine condition (ii) as employing a nearest-baseline weighting method, such that the closest baseline to a point $\\vect{u}$ at a given frequency will be the sole contributor.\nWe shall see that even for standard gridding kernels, this condition will be met for large $u$ if condition (i) is met.\n\nCondition (i) is illustrated in Fig.~\\ref{fig:delay_transform_schematic}.\nThe baselines are arranged in a regular (logarithmic) polar grid, or equivalently, a series of ``spokes\" along which baselines are strung in a logarithmically regular fashion. \nImportantly for this section, the base of the logarithm is large enough such that a single baseline remains the sole contributor to a point co-located with its reference co-ordinate for the entire bandwidth (illustrated by the inset orange bell-curve in Fig.~\\ref{fig:delay_transform_schematic}).\nIn addition, every spoke is equivalent, such that the baselines define a set of concentric rings. \nIn summary then, the layout consists of $N_\\theta \\times N_r$ baselines, with regularly-spaced angular coordinates $\\theta_k = 2\\pi k\/N_\\theta$ and log-spaced radial co-ordinates $u_j = u_{j-1} + \\Delta_u (u_{j-1})$, with $\\Delta_u(u) = u\\Delta$ (with constant $\\Delta$) and arbitrary $u_0$.\nWe shall re-use variants of this simple layout throughout this paper.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{figures\/delay_transform_schematic}\n\t\\caption{Schematic of baseline layout with overlaid averaging\/histogram grid. Each cell of the averaging grid contains a single baseline which defines the \"position\", $\\vect{u}_\\mu$ of the cell at $f=1$. Inset is a figure transforming to frequency space, in which the bandpass\/taper is shown, along with the frequencies at which a particular baseline is at the cell edges and centre. The edges correspond to 3 Gaussian widths of the bandpass.}\n\t\\label{fig:delay_transform_schematic}\n\\end{figure}\n\nThese conditions allow for simple analytic solutions of the expected PS for a range of combinations of sky models and beam shapes.\nWhile emergence of the wedge can in principal be illustrated without specialising to any particular model, we find it illustrative to do so.\nFurthermore, to demonstrate that the wedge arises irrespective of sky model or beam shape, we use two models for each and evaluate the expected PS for three combinations: single-source with static beam, single-source with chromatic beam, and uniform sky with static beam.\nThus in addition to our fiducial choice of static Gaussian beam and uniform point-source sky, we also use a chromatic Gaussian beam (cf. \\S\\ref{sec:framework}) and a single-source sky model. \nThe latter model places a single source of flux density $S_0$ at position $\\vect{l}_0$.\n\nThe solutions are derived in App.~\\ref{app:delay}, and shown in tabulated form in Table \\ref{tab:simple_summary}. \nFor completeness we show the assumed values of various parameters in Table \\ref{tab:models}, and these correspond to the plots in Fig. \\ref{fig:ss_ngp_static}.\n\n\n\\subsection{Discussion of Uncorrelated Examples}\nThe basic form of the three derived solutions is very similar, as can be most clearly seen in the final two columns of Table \\ref{tab:simple_summary}, which provide a schematic view of where the power cuts off in $\\omega$. \nIn each case the standard form of the wedge, in which the power cut-off traces $\\omega \\propto u$, is recovered.\nThe constant of proportionality here (for this choice of coordinates) is given either by $l_0$ in the case of a single source (two leftmost plots, first with $l=1$ and second with $l=0.5$), or the beam width, $\\sigma$ (rightmost plot), in the case of a stochastic uniform sky.\nIn either case, the maximum possible value for these is unity (in fact, for the beam width it must be less than this or else sky-curvature terms become important).\nThus we can draw the standard ``horizon line'' at $\\omega = u$. \n\nWe also find that the lower-left portion of the $\\omega u$-plane forms a ``brick'' whose width is determined primarily --- in our case --- by the taper.\nClearly, the ``brick'' is in general determined by the overall frequency envelope of the instrument and analysis -- i.e. the combination of taper, bandpass, chromaticity of the beam, and spectral structure of the sources. \nIndeed, the effects of the chromaticity of the beam are apparent in the case of a single-source sky; the width of the brick is determined by $p^2 = \\tau^2 + l_0^2\/2\\sigma^2$, which has a dependence on the beam-width. \nIn practice, the taper\/bandpass dominate the spectral response, and $p^2 \\approx \\tau^2$, nevertheless this illustrates that even with a theoretically infinite uniform bandpass, other spectral characteristics of the instrument will limit the ability to sequester power into the lowest $\\omega$ modes. \nIn general, the balancing of the various spectral terms provides the motivation for design criteria on the spectral smoothness of the instrument. \n\nAs has been previously noted, the wedge occupies a greater portion of the $\\omega u$-plane for sources close to the horizon (precisely because their delay transform for the same baseline is larger). \nThus, a beam which is tighter (and doesn't have high-amplitude sidelobes) can effectively attenuate these sources and ameliorate the wedge \\citep{Parsons2012}.\nOf course, such a beam will also attenuate the 21 cm signal, and is therefore not an ideal solution for the problem. \n\nAs has been extensively noted in the literature, many factors affect the precise form of the power in the wedge \\citep[eg.][]{Thyagarajan2013,Thyagarajan2015,Thyagarajan2016,Gehlot2018} --- and most of these we have ignored in this analysis.\nWhile the broad structure remains the same -- a brick with width given by the spectral envelope of the instrument, and linear wedge extending to $\\omega \\approx u$ -- the power within this region may be shifted around or amplified by various factors, and even leaked beyond the horizon line when small-scale spectral features are present in the analysis. \nAn example of this changing of form can be witnessed in Fig.~\\ref{fig:ss_ngp_static} between the single-source and stochastic skies. \nThe smooth attenuation of sources at larger angles causes a smoother cut-off in the wedge.\n\nNevertheless, none of these features can lay claim to being the fundamental reason for the wedge. \nAltering them merely alters the shape or amplitude of the wedge, and not its basic form or existence.\nThe fundamental reason for the wedge is rather the combination of the migration of the baselines with frequency, and the sparsity of the $uv$-sampling. \nWe turn to examining this latter condition for the remainder of this paper.\n\n\n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Sky Dist.} & \\textbf{Beam} & \\textbf{Form of $P(\\omega, u_\\mu)$} & \\textbf{Low $u$} &\\textbf{High $u$} \\\\ \n\t\t\t\\hline\n\t\t\tSingle-Source & Static & \\( \\displaystyle \\frac{1}{N_\\theta} \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right) \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2(\\omega + u_\\mu l_0 \\cos (2\\pi k\/N_\\theta))^2}{\\tau^2} \\right) \\) &$\\tau\/\\pi\\sqrt{2}$ & $ul_0$ \\\\\n\t\t\tSingle-Source & Chromatic & \\(\\displaystyle \\frac{S_0^2 \\nu_0^2 \\pi}{N_\\theta p^2} e^{-\\tau^2l_0^2\/2\\sigma^2p^2} \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2 (\\omega + u l_0 \\cos (2\\pi k\/N_\\theta))^2}{p^2}\\right) \\) &$p\/\\pi\\sqrt{2}$ & $ul_0$ \\\\\n\t\t\tStochastic Uniform & Static & \\(\\displaystyle \\frac{\\nu_0^2 \\pi}{p_u}\\exp\\left(-\\frac{2\\pi^2\\omega^2}{p_u^2}\\right) \\left[ \\frac{\\bar{S}^2}{p_u} \\exp\\left(-\\frac{2\\tau^2\\pi^2\\sigma^2u^2}{p_u^2}\\right) + \\frac{\\mu_2 \\pi^2\\sigma^2}{\\tau}\\right]\\) & $\\tau\/\\pi\\sqrt{2}$ & $u\\sigma$ \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\label{tab:simple_summary} Summary of analytic solutions for the simple discrete polar grid layout with histogram gridding of \\S\\ref{sec:classic}. Final two columns display a schematic representation of where the foreground power cuts off in $\\omega$. For simplicity we list the cutoff such that the power is reduced by a factor of $e$ from the total. For a $\\chi$-order of magnitude suppression, multiply the result by $\\chi \\ln 10$. In the table, $p_u^2 = \\tau^2 + 2\\pi^2\\sigma^2u^2$ and $p^2 = \\tau^2 + l_0^2\/2\\sigma^2$.}\n\t\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth,trim=2cm 0cm 2cm 0cm]{figures\/averaged_gridding_examples}\n\t\\caption{2D PS examples using averaged gridding, and different combinations of beam and sky models. Each displays very similar behaviour, however the wedge is\n\t\tsharper in the case of a single source. In each case, a horizontal ``brick'' line is drawn at the theoretical $10^{th}$ magnitude of suppression (cf. Table \\ref{tab:simple_summary}), and the diagonal line is the ``horizon line'': $\\omega = u$. For each, $\\tau$ remains the same, while the two single-source models have sources at different zenith angles, the first at the horizon and the second at $l=0.5$. The clear difference in amplitude arises due to the difference in number of position of sources in the beam. Changing the position of a single source changes height and slope of the wedge line. A stochastic uniform sky has a softer edge for the wedge.}\n\t\\label{fig:ss_ngp_static}\n\\end{figure*}\n\n\n\\section{Dense Logarithmic Polar Grid}\n\\label{sec:weighted}\n\nMuch can be learned about the effects of including visibility correlations by re-using the logarithmic polar grid baseline layout of \\S\\ref{sec:classic}, and we address this class of problems in this section.\nHere we will dispense with condition (ii) --- that the gridding kernel is arbitrarily narrow --- and use a self-consistent kernel width.\nMore importantly, we will dispense with the condition that the layout be ``sparse'', \nallowing an arbitrary radial density of baselines (i.e. arbitrarily low values of $\\Delta$)\\footnote{The adjustment of angular density trivially has no impact, as each ring measures the same expected PS everywhere}. \nIt is precisely as we modify this density that we will identify the effects of $uv$-sampling.\nWe give a schematic of this layout in Fig. \\ref{fig:weighted_gridding_schematic}, noting the extent of each baseline via the Fourier beam kernel, and also the interplay of this scale with the bandpass shape.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/weighted_gridding_schematic}\n\t\\caption{Schematic of weighted gridding, with logarithmic polar grid of baselines. Extent of Fourier-beam kernels are indicated by blue shaded regions around each baseline. Also shown are representative bandpasses, indicating the weight of a baseline centred on the bandpass at $f=1$ as it moves with frequency. The four bandpasses indicate four regimes which may be considered. Note that bandpass indications on the left hand side adopt a lower value of $\\tau$. The bandpass (as represented in $u$-space) naturally expands at higher $u$.}\n\t\\label{fig:weighted_gridding_schematic}\n\\end{figure}\n\nWe note that the limiting case of this derivation -- namely an arbitrarily densely packed set of baselines (i.e. a smooth continuum of baselines) in the radial direction -- is addressed in App. \\ref{app:radial}. It is in line with the predictions of the \\textit{discrete} polar grid results of this section.\n\nThroughout this section we will use a static beam, and consider the stochastic uniform sky (we have already established that such choices do not greatly impact the qualitative form of the solution). \nFurthermore, due to the fact that the term $|\\langle V \\rangle|^2$ has negligible power on small scales, we consider only the variance term of Eq.~\\ref{eq:power_general}.\nDue to the isotropy of the sky and the fact that we use an angularly symmetric layout, we immediately have that $P(\\omega, u) = {\\rm Var}(\\tilde{V}(\\omega, u))$ without requiring an integral over $\\theta$.\nThus for this section we require only the second term of Eq.~\\ref{eq:general_var}, which can alternatively be written:\n\\begin{subequations}\n\t\\label{eq:wg_master}\n\t\\begin{align}\n\t\t{\\rm Var}(\\tilde{V}) &= \\mu_2\\nu_0^2 \\int d^2\\vect{l}\\ e^{-l^2\/\\sigma^2} |I|^2, \\\\\n\t\tI &= \\int df\\ \\frac{\\phi_\\nu}{W_\\nu(\\vect{u})} \\sum_{i=1}^{N_{\\rm bl}} B(\\vect{u} - f\\vect{u}_i)e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)} \\nonumber \\\\\n\t\t&= \\int df\\ \\frac{\\phi_\\nu \\sum_{i=1}^{N_{\\rm bl}} e^{-2\\pi^2\\sigma^2(\\vect{u}-f\\vect{u}_i)^2} e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}}{\\sum_{i=1}^{N_{\\rm bl}} e^{-2\\pi^2\\sigma^2(\\vect{u}-f\\vect{u}_i)^2}} \n\t\\end{align}\n\\end{subequations}\nThe solution of $I$ here is in general intractable, due primarily to the sum over baselines in the denominator (i.e. $W_\\nu(\\vect{u})$). \n\n\nEq. \\ref{eq:wg_master} may be expanded as follows (with $q^2 = 2\\pi^2 \\sigma^2$):\n\\begin{align}\nI = \\int df\\ \\frac{\\sum_{i=1}^{N_{\\rm bl}} e^{-f^2(\\tau^2 + q^2 u_i^2) -2f(\\tau^2 + q^2 \\vect{u}\\cdot\\vect{u}_i)-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}}{e^{\\tau^2} \\sum_{j=1}^{N_{\\rm bl}} e^{-q^2(f^2 u_j^2 - 2f\\vect{u}\\cdot\\vect{u}_j)}} \n\\end{align}\nEvaluating this in general is here still intractable.\nNevertheless we can appreciate the general characteristics of the solution by considering the two limits of $u$. \n\nAt small $u$, the exponential cut-off of the beam kernel requires that only baselines with small $u_i$ have non-negligible impact in the sum.\nThus, for $u \\ll \\tau\/q$, for which only terms with $u_i \\ll \\tau\/q$ can contribute, all components with a dependency on $u_i$ in the numerator disappear. \nFurthermore, since the denominator is clearly a much broader function of frequency than the numerator (as $q^2\\vect{u}\\cdot\\vect{u}_i \\approx q^2u^2 \\ll \\tau^2$), it can be removed from the frequency integral. \nThus we arrive at\n\\begin{align}\nI_{u\\ll\\tau\/q} = \\frac{\\sum_{i=1}^{N_{\\rm bl}}e^{- q^2 u_i^2}}{\\sum_{j=1}^{N_{\\rm bl}} e^{-q^2(u_j^2 - 2\\vect{u}\\cdot\\vect{u}_j)}} \\int df\\ \\phi_\\nu e^{-2\\pi i f(\\omega + \\vect{l}\\cdot\\vect{u}_i)}. \n\\end{align}\nThe frequency integral is identical to that for the sparse grid (cf. Eq. \\ref{eq:stoch_uniform}).\nIn fact, the solution is a complex sum over terms, all of which are very close to the exact solution of Eq.~\\ref{eq:stoch_uniform}, and therefore the behaviour will be almost identical -- i.e. to produce a ``brick'' feature at $u \\ll \\tau\/q$, with an $\\omega$ cut-off at $\\omega \\approx \\tau\/\\sqrt{2}\\pi$. \nRemember that this is irrespective of the density of baselines and the width of the gridding kernel, though its regime of applicability is determined by both the gridding kernel width and taper width.\n\nConversely, at sufficiently large $u$, the baselines are far enough apart from each other that the weighted sum of visibilities is dominated by the single closest baseline (except in rare cases where we consider scales with two or more equidistant baselines, but the rarity of these will make them negligible in the final angular average)\nFor a given logarithmic separation $\\Delta$, this criterion can be considered to be\n\\begin{equation}\n\tu^2 \\gg \\frac{1}{q^2\\Delta^2}.\n\\end{equation}\nNote that baselines being separated enough to consider just one in the baseline sum is not equivalent to them being radially separated enough to only consider the same baseline over all frequencies.\nIt is entirely possible that the baselines will move enough with frequency that they entirely replace one another, while only ever considering one at a time in the baseline sum (cf. Fig. \\ref{fig:weighted_gridding_schematic}). \nWe denote the baseline closest to $\\vect{u}$ at $f$ as $\\vect{u}_f$ (this is meant to be an identifier, so that the baseline's value of $\\vect{u}$ at $f$ is $f\\vect{u}_f$).\nIn this case, we can rewrite $I$:\n\\begin{equation}\n\t\\label{eq:Iequation}\n\tI_{u\\gg 1\/q^2\\Delta^2} = \\int df \\phi_\\nu e^{-2\\pi i f (\\omega + \\vect{l}\\cdot \\vect{u}_f)}.\n\\end{equation}\n\nWe have already encountered the case in which the baselines are separated enough such that only a single baseline contributes across \\textit{all frequencies} (cf. \\S\\ref{sec:classic}), and this gives the classical form for the wedge. \nThis merely shows that if $\\Delta$ is large enough, there will \\textit{always} be a regime of $u$ such that this classical solution holds for a logarithmic polar grid.\n\nAlternatively, we may consider the limit as $\\Delta \\rightarrow 0$ (for which the $u$ regime is at extremely large $u$). \nIn this case, the closest baseline to $\\vect{u}$ will always have $f\\vect{u}_f \\approx \\vect{u}$ (i.e. there will always be a baseline sitting on $\\vect{u}$).\nThen we have\n\\begin{equation}\nI_{u\\gg 1\/q^2\\Delta^2} = e^{-2\\pi i \\vect{l}\\cdot\\vect{u}} \\int df \\phi_\\nu e^{-2\\pi i f \\omega},\n\\end{equation}\nso that the final power spectrum is\n\\begin{equation}\n\tP(\\omega, u) = \\mu_2 \\nu_0^2 \\tilde{B}(u)\\tilde{\\phi}(\\omega).\n\\end{equation}\nThis separable equation clearly does not contain a wedge, rather containing only the ``brick'' determined by the taper, with an exponential cut-off in $u$.\nThough this was shown only for the fictional region $u \\rightarrow \\infty$ in this case, it is really an example of a continuous distribution of baselines along a radial trajectory, which is shown in detail to omit a wedge in App.~\\ref{app:radial}.\n\nOf course, in most cases, the (radial) density of baselines will lie between these extremes, and a natural question is how dense the baselines must be in order to yield a given level of wedge reduction.\n\\ifanalytic\n{\\color{red} A detailed solution to this question is presented in App.~\\ref{app:logsolution}}, but \n\\fi\nWe address this question semi-empirically following our conceptual interpretation of the next subsection. \n\n\\subsection{Conceptual Interpretation}\n\\label{sec:weighted:conceptual}\nTo gain an intuition for the results of the previous subsection, imagine a point $\\vect{u}$ for which we are evaluating the power, and consider only baselines that are along a spoke passing through $\\vect{u}$ (thus reducing the problem to one dimension). \nFigure~\\ref{fig:wedge_rising} gives a schematic representation of this, similar in form to Fig.~\\ref{fig:baseline_schematic}.\nHere we have chosen a very sparse baseline sampling, akin to the layout chosen for the averaged gridding in the previous section, and show only two points of evaluation (centre of the grey regions), which are concurrent with the baselines at $f=1$. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/wedge_rising}\n\t\\caption{Schematic representation of the mechanics of the emergence of the wedge. Panels should be traced in order of their assigned number. (1.) shows the migration of baselines (black) in $u$ with frequency. Coloured overlaid lines show the ``effective baseline'' traced as an estimate along the points of evaluation, $u$ (grey regions), determined by its weighting kernel. (2.) shows the projection of the effective baseline (coloured) onto the visibility amplitude, accounting for the beam (black line). (3.) shows these results projected (coloured) onto the frequency axis, where they are multiplied by the bandpass (black). The FT of this final curve gives the power spectrum as a function of $\\omega$ for the evaluation point $u$. Wider curves transform to narrower curves.}\n\t\\label{fig:wedge_rising}\n\\end{figure*}\n\nPerforming a FT following the trajectory of a baseline is the delay transform, and at low $u$ this is very close to performing the FT at constant $u$. \nDue to the sparsity of baselines, the effective baseline used in a constant-$u$ transform merely follows the closest baseline.\nThis ``effective baseline'' is illustrated as a coloured shading overlaying the baseline's trajectory. \nAn arrow from this coloured line to the constant-$u$ FT trajectory indicates that it is \\textit{this} value of $u$ that is used in the estimation of the Fourier-space visibility.\n\nThe top panel shows the immediate results of this effective baseline migration. \nThe black curve shows the visibility of the true sky as a function of $u$.\nWe note that as this is a simplified schematic, we show an effective amplitude of the visibility (which is inherently complex). \nThis visibility accounts for the beam attenuation of the instrument, resulting in an exponential curve.\nThe estimated visibility amplitude at any frequency is merely its value as traced vertically from the coloured curve in the lower left panel. \nThat is, in this case, the amplitude is merely traced from left to right as frequency increases, and is indicated by a corresponding coloured line. \nThe greater the value of $u$, the larger the arc-length of this line segment, and therefore the greater the ratio between its minimum and maximum. \n\nThe right-hand panel shows these effects on the frequency axis.\nIn black is the bandpass (or taper). \nTo obtain the frequency-space visibility, this is multiplied by the frequency-dependent sky response from the top panel, and shown as corresponding coloured curves. \nWhile the low-$u$ curve (in blue) is almost constant-amplitude, and therefore barely affects the frequency-space visibility, the high-$u$ curve dramatically suppresses the high-frequency amplitude, effectively causing the frequency-response to be tighter than the natural bandpass (we note that for schematic purposes, we have re-normalised and re-centred the coloured curves).\nThe frequency-space FT of these curves gives the power spectrum for a given $u$ as a function of $\\omega$.\nClearly, tighter curves will transform to wider curves, and hence the ``wedge'' will form when the tightening arising from the effective baseline migration dominates the bandpass (and thence will depend linearly on $u$).\nThis is much the same description as contained in works such as \\citet{Morales2012}, \\citet{Parsons2012} and \\citet{Trott2012}.\n\n\nLet us consider now a very dense array with logarithmically-spaced baselines.\nThis we illustrate in Fig. \\ref{fig:schematic_inf_bl}.\nThis figure is the same in form as Fig. \\ref{fig:wedge_rising}, but clearly has a much larger number of baselines which pass through the point of evaluation, $u$.\nDue to the logarithmic spacing of the baselines, they pass through $u$ at equal intervals of $f$. \nIn this case, the ``effective baseline'', shown as the blue curve in the lower-left panel, periodically swaps from one baseline to another. \nWe recall that this effective baseline is the weighted average position of all baselines, where the weighting kernel is the Fourier-space beam (in this case, a Gaussian).\nSince the baselines are so closely packed, the oscillations created are very small, and it is clear that an infinite number of baselines will yield a truly vertical effective baseline -- corresponding to the true constant-$u$ estimate. \nConsequently, the top-left panel shows that a very small range of visibility amplitudes is covered -- effectively constant over all frequencies.\nThis in turn renders its product with the bandpass to be solely determined by the latter, and therefore the wedge to be completely avoided. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/schematic_inf_bl}\n\t\\caption{The same as Fig. \\ref{fig:wedge_rising}, but for a single evaluation point at high-$u$, with a dense packing of logarithmically-spaced baselines. In this case, the ``effective baseline'' is nearly constant with frequency, and the resulting effect is to negligibly impact the frequency response shape.}\n\t\\label{fig:schematic_inf_bl}\n\\end{figure*}\n\nWhat of a baseline density between the previous two figures?\nThis is shown in Fig. \\ref{fig:schematic_multi_bl}.\nHere the ``effective baseline'' oscillates between true baselines in a much more marked manner, creating a footprint in $u$ which is much wider than in the previous case.\nProjected onto the visibility amplitude, a much wider range is covered, and that range is covered periodically, with period given by the separation of baselines.\nThus it is no surprise to find that the frequency-space product of the response with the bandpass is oscillatory on small scales, with an overall shape given by the bandpass itself. \nThe FT of such a function can be approximated as the combination of a smooth Gaussian with width inverse to the width of the bandpass, and a high-frequency term given by the period of the oscillations. \nIn this case, in place of a pure wedge, one should obtain a ``bar'' in the 2D PS (along with its harmonics), where the position of the bar in $\\omega$-space is inversely proportional to the separation of the baselines, and its amplitude is proportional to this separation.\nThat is, denser baselines will yield a lower-amplitude bar at higher $\\omega$, eventually leading to a negligible bar, and the complete disappearance of the wedge.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 1cm 0cm 0cm]{figures\/schematic_multi_bl}\n\t\\caption{The same as Fig. \\ref{fig:wedge_rising}, but for a single evaluation point at high-$u$, with a semi-dense packing of logarithmically-spaced baselines. In this case, the ``effective baseline'' oscillates widely with frequency. Consequently the $u$-footprint is widened and the range of visibility amplitude is also widened. The effect is to overlay a regular oscillatory component atop the bandpass, which contributes a high-$\\omega$ ``bar'' (and its harmonics) in addition to the basic ``brick'' yielded by the bandpass.}\n\t\\label{fig:schematic_multi_bl}\n\\end{figure*}\n\n\nIn general, baselines will not be regularly logarithmically separated, and various baseline distributions will yield different versions of this oscillatory structure.\nFor instance, linearly-separated baselines should cause the bar's position to be linearly dependent on $u$.\n\nThe exact results of these intuitions are complicated when accounting for baselines in the 2D plane, some of which may significantly contribute to the weight at $\\vect{u}$ without being on its radial trajectory.\nFurthermore, radial distributions which are not regular will also complicate matters.\nWe will continue to explore these issues in the following subsections; nevertheless, the basic intuition will remain the same.\n\n\n\n\n\\subsection{Wedge Mitigation}\n\nWe now turn to considering the properties of wedge mitigation in our simple polar grid baseline layout.\nWe begin by considering the results of our simple schematic representations of the wedge, in Table \\ref{tab:simple_summary}, which suggest that the wedge only emerges from the ``brick'' at $u>\\tau\/q \\equiv \\check{u}$.\nThis defines a region of interest, in which we may hope to mitigate the wedge.\nNote that for $u > 1\/q\\Delta$, the problem can also essentially be considered as 1D, as constructed in \\S\\ref{sec:weighted:conceptual}, regardless of the angular density.\nWe shall assume this condition throughout this section, noting that deviations from the assumption will always be small.\n\nWithin this region, we ask how a wedge may be \\textit{ensured}.\nWe have already seen that if a single, constant baseline contributes to the sum over baselines, for all frequencies in the bandpass, then we arrive at a wedge.\nThus, we may ensure a wedge by considering the integrated contribution of the two baselines closest to $q$.\nIf the one baseline dominates, then we are assured of a wedge. \nIf the second baseline contributes non-negligibly then we cannot rule out a wedge, but open up the possibility of its mitigation.\nWe denote the threshold contribution as $e^{-t}$.\n\nIn effect, our constraints are defined by the inequality\n\\begin{equation}\n\\label{eq:general_wedge_regime}\n\\frac{\\int df \\exp\\left[-\\tau^2 (f-1)^2 - q^2 (u+\\Delta_u - fu)^2\\right]}{\\int df \\exp\\left[-\\tau^2 (f-1)^2 - q^2 (u+\\Delta_u)^2 (1 - f)^2\\right]} < e^{-t}.\n\\end{equation}\nHere we have assumed that the dominant baseline is co-located with $u$ at $f=1$. \nThe qualitative results are insensitive to this assumption.\n\nThe solution to Eq. \\ref{eq:general_wedge_regime} is \n\\begin{equation}\n\\label{eq:solution_wedge}\n\\frac{1}{2}\\ln\\left(\\frac{\\tau^2 + q^2(u+\\Delta_u)^2}{\\tau^2 + q^2 u^2}\\right) - \\frac{\\tau^2 q^2\\Delta_u^2}{\\tau^2+q^2u^2} < -t.\n\\end{equation}\nIf we consider only scales where a wedge is possible (i.e. $u>\\check{u}$), and maintain that at these scales, $\\Delta_u \\ll u$, then we may use the approximation $\\ln(1+\\delta) \\approx \\delta$ to solve for $\\Delta_u$:\n\\begin{equation}\n\t\\label{eq:solution_for_concentric}\n\t\\Delta_u \\gtrsim \\frac{u}{2\\tau^2}\\left(1+\\sqrt{1+4\\tau^2 t}\\right) \\approx \\frac{u\\sqrt{t}}{\\tau},\n\\end{equation} \nwhere the last approximation assumes $t \\gg 1\/\\tau^2$.\nThat is, a wedge is \\textit{ensured} if the baseline separation is larger than $u\\sqrt{t}\/\\tau$.\n\nThe salient features of this equation are \n\\begin{enumerate}\n\t\\item $\\Delta_u$ rises proportionally to $u$, so a regular logarithmic spacing for $u > \\check{u}$ ensures consistency of wedge\/non-wedge for all $u$. \n\t\\item $\\Delta_u$ is inversely proportional to $\\tau$, so that larger bandwidths support larger separations before a wedge is ensured.\n\t\\item $\\Delta_u$ is proportional to the root of the threshold, $t$. This is difficult to assess conceptually, as we are \\textit{a priori} uncertain as to what level the primary baseline must contribute to ensure a wedge.\n\\end{enumerate}\n\nTo get a better sense of the kinds of separations required, we note that \n\\begin{equation}\n\\Delta_u = \\Delta_x\/\\lambda_0 \\approx \\frac{\\Delta_x}{2{\\rm m}},\n\\end{equation}\nwhere $\\Delta_x$ is the difference in baseline lengths in distance units (note that this is \\textit{not} distances between antennae, but differences between these distances).\nExpressing this physical separation in units of the tile diameter, $\\Delta_x = \\chi D$, we can express our results in terms of the parameter $\\chi$.\nWe note first that\nfor a (static) Gaussian beam at $\\nu_0\\approx150$ MHz, the tile diameter can be approximately related to the beam width by\n\\begin{equation}\n D \\approx 1{\\rm m}\/\\sigma,\n\\end{equation}\nThus, we let $\\Delta_u = \\chi D\/2{\\rm m} = \\chi\/2\\sigma$.\nIn this case, we have that\n\\begin{equation}\n\\chi \\gtrsim \\frac{2\\sigma u\\sqrt{t}}{\\tau}\n\\end{equation}\nensures a wedge.\nAs a minimum, at $u= \\check{u}$, we have $\\chi > \\sqrt{t}\/\\pi$. \n\nUnfortunately, it is difficult to exactly specify the value of $t$, as it merely represents an order-of-magnitude estimate of the contribution of secondary baselines.\nFurthermore, even if we could specify it, we do not have a good analytic handle on what happens for baseline separations smaller than that given by $\\chi$ -- we cannot simply assume that the wedge will disappear, though we do expect it to disappear at some small separation.\nThus we turn to a numerical\/empirical approach.\n\nIn Fig. \\ref{fig:dense_log_concentric} we show the numerically-calculated power spectra (see App.~\\ref{app:numerical} for details on the numerical algorithm) for our fiducial set of physical parameters, and a range of logarithmic separations. \nEach panel is titled by the value of $t$ and $\\chi$ which correspond to the baseline separations \\textit{at} $\\check{u}$ (which is marked by the vertical dashed line).\nThis clearly shows that a baseline separation of about half the tile diameter is required (taking the minimum, which is at $\\check{u}$) for the wedge to begin to disappear.\nThis occurs at a threshold value of $t \\sim 0.4$, corresponding to the second baseline contributing $\\sim 60\\%$ of the primary baseline over the range of frequencies.\nThese values are roughly instrument-independent .\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=1cm 1cm 1cm 0cm]{figures\/dense_log_concentric.pdf} \n\t\\caption{2D power spectra for fiducial parameters, with static beam and stochastic sky. The baseline layout is concentric, with logarithmically-increasing separations between each circle. The gridding is weighted according to the Fourier beam kernel. Each panel represents a different regular logarithmic spacing, $\\Delta$. Titles of each panel indicate the order-of-magnitude contribution of the next-closest baseline at $\\check{u}$ (labelled $t$), and the physical separation of baselines as a fraction of the tile diameter at $\\check{u}$. The dashed vertical line marks the scale $\\check{u}$. The colour-scale in each panel is identical, as are the schematic representations of the wedge\/brick shown as black lines (these come from the corresponding row of table \\ref{tab:simple_summary}).}\n\t\\label{fig:dense_log_concentric}\n\\end{figure*}\n\nInterestingly, we also see horizontal ``bars'' as we had predicted from our conceptual consideration of the problem (cf. \\S\\ref{sec:weighted:conceptual}).\nAs predicted, the fundamental bar moves up in $\\omega$ as the baselines become closer, and at some separation we expect the harmonics to effectively disappear.\n\nFinally, we ask whether such a layout is physically feasible.\nIn principle, a perfect polar grid of baselines is unachievable by laying out antennas --- there will always be baselines that are off the grid. \nHowever, the logarithmic polar grid can be achieved by using logarithmic spokes of antennas, and ignoring all baselines that are off the grid (with a great deal of inefficiency).\nIn this case, one cannot physically deploy a layout with $\\chi < 1$, as the antennas will necessarily overlap with themselves. \nWe have found that we require $\\chi \\approx 1\/2$ to mitigate the wedge at $\\check{q}$, rendering this completely infeasible. \nWhile in principle it is possible to design layouts which would enable more closely-spaced baselines, they would come at the cost of reduced layout efficiency, and will be practically infeasible.\nWe will soon (\\S\\ref{sec:mitigation:arrays}) explore how leaving the off-grid baselines in the baseline layout affects results.\n\n\n\\subsubsection{Linear Radial Grid}\n\\label{sec:weighted:linear}\nIt is interesting to consider the case in which radial baselines are regular in linear space. \nEq. \\ref{eq:solution_for_concentric} suggests that in this case, at some point $u>u'$ the separation will become small enough to mitigate the wedge.\nIn fact, letting $\\Delta_u \\equiv \\Delta$, we can explicitly solve for $u'$:\n\\begin{equation}\nu' = \\tau \\Delta\/\\sqrt{t} \\approx 1.5 \\tau \\Delta,\n\\end{equation}\nwhere the last approximation assumes $t=0.4$ defines the transition from wedge to no-wedge, as described above. \nIndeed, if $u' < \\check{u}$, we expect the wedge to be completely mitigated. \nThis is given by the same baseline difference as the logarithmic case, i.e. corresponding to $\\chi \\approx 1\/2$. \nFurthermore, we expect that the bars we saw in the logarithmic case will also be present in the linear case, except that they will not be horizontal, but rather diagonal, as they increase in frequency as $u$ increases.\n\nTo illustrate and check these arguments, we show the linear analogue of Fig. \\ref{fig:dense_log_concentric} in Fig. \\ref{fig:dense_linear_concentric}.\nThe diagonal bars are quite clear in this case. \nWe also see that $t\\approx 0.4$ again roughly delineates the disappearance of the wedge at $\\check{u}$. \nWe note that the vertical lines which appear in the upper panels are due to the fact that in this case, the actual nodes of evaluation, $u_i$, lie at various positions between the radial baselines, rather than being forced to match at $f=1$. \nThis creates oscillatory behavior in $u$, but disappears as the distance between baselines increases. \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=1cm 1cm 1cm 0cm]{figures\/dense_linear_concentric.pdf} \n\t\\caption{Exactly the same as Figure \\ref{fig:dense_log_concentric}, except that the baselines are spaced regularly in \\textit{linear} space.}\n\t\\label{fig:dense_linear_concentric}\n\\end{figure*}\n\nThis kind of layout is \\textit{less} physically feasible than the logarithmic polar grid, as it requires impractical baseline densities for all $u$.\n\n\n\\subsubsection{Effects of Angular Density}\n\nDue to isotropy, the results of this section are not sensitive to the number of ``spokes'' in the layout.\nNevertheless it is clear that a single spoke does not have the same \\textit{covariance} as a layout with a large number, and when performing inference on a given set of observed data, this covariance is crucial.\nAn alternative way to think of this is that while a single spoke will yield the same mean over a large random set of skies, it will not necessarily give a good account of a single sky, whose distribution is randomly deviated from perfect symmetry. \n\n\n\\subsubsection{Radial Irregularities}\n\\label{sec:weighted:irregular}\nThe precise radial alignment (and regularity) of baselines in the simple polar grid lead to it being impractical layout for wedge mitigation.\nIn this section, we consider a relaxation of the ideal assumptions of radial regularity in favour of random radial placement, which will come in two forms: (i) completely random and (ii) a random offset from logarithmic regularity.\n\nThe advantage of a random array is that it may be perfectly efficient in terms of mapping an antenna layout to the baseline layout --- we need not ignore any pairs within a spoke. \nThus we can achieve a much greater overall baseline density for the same cost.\nConversely, however, we shall see that the lack of radial alignment increases the overall required baseline density to achieve wedge mitigation.\n\nIn our ``completely random'' layout, we allow the baselines to be stochastically placed along radial trajectories, with the same overall density as a logarithmic placement. \nWe find that doing so yields a somewhat surprising result, which is illustrated in Fig. \\ref{fig:random_trajectories}.\nIn this plot, we compare the 2D PS of a regular logarithmic layout in which the baseline separation is $\\Delta_u \\approx 0.08\\sqrt{0.4}u\/\\tau$, (i.e. 12.5 times smaller than required to mitigate the wedge), with a layout whose baseline density (and therefore average separation as a function of $u$) is identical, but in which the baselines are stochastically placed. \nWe also show the result of an over-dense random layout.\nFigure \\ref{fig:random_separations} shows the actual separations between baselines as a function of $u$ for each case.\nEven for the random arrangements, \\textit{all} baselines have separations smaller than the wedge-mitigation threshold.\nWhile we might expect all of them to have near-perfect wedge-mitigation, we find that the random layout yields a subdued, but very present, wedge. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1cm 0cm 2cm 0cm]{figures\/random_trajectories}\n\t\\caption{2D PS comparison between baseline layouts for three cases: (i) random radial placement, with mean separation proportional to $u$ and $N=10,000$ (left panel); (ii) the same random placement, but with $N=50,000$ (centre panel), and (iii) regular logarithmic placement of equivalent density to case (i) (right panel). Clearly the introduction of stochastic baseline separations re-introduces a wedge.}\n\t\\label{fig:random_trajectories}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/random_separations}\n\t\\caption{Baseline separations, $\\Delta_u$, as a function of $u$, for three cases: (i) random radial placement, with mean separation proportional to $u$ and $N=10,000$ in green; (ii) the same random placement, but with $N=50,000$ in red, and (iii) regular logarithmic placement of equivalent density to case (i) in orange. The blue line shows the threshold separation below which a regular grid significantly mitigates the wedge (cf. Eq. \\ref{eq:solution_for_concentric}).}\n\t\\label{fig:random_separations}\n\\end{figure}\n\n\nThe explanation for this behaviour arises from the ``bars'' that occur for the regular logarithmic polar grid (cf. Fig. \\ref{fig:dense_log_concentric}).\nA regular grid creates oscillations which sit atop the bandpass in frequency-space (cf. \\S\\ref{sec:weighted:conceptual}). \nIf the grid is logarithmic in $u$-space, these oscillations are linear in frequency space, causing neatly-spaced peaks in the power spectrum. \nWhen the baselines have stochastic separations, the oscillations are irregular, and cause a cacophony of ``bars'' above the main ``brick''.\nEssentially, this haphazard distribution of peaks restores a somewhat subdued wedge. \n\nThe level to which it is subdued will depend on the baseline density, however it clearly requires a significant increase in density to match the regular logarithmic grid. \nWe note that it is not primarily the fluctuating \\textit{minimum} separation of baselines that causes the re-emergence of the wedge.\nThis can be clearly understood from Fig. \\ref{fig:random_separations}, in which for the over-dense random layout, the separation very rarely ventures above that of the regular grid. \nThe issue is rather that the unevenness of the random distribution causes higher-order structure in the oscillations that lie atop the bandpass, which emanate as the smeared peaks within the wedge.\n\nTo determine the extent of this effect, we use the same regular set of 10,000 baselines, and randomly offset them by some fraction of their amplitude, according to a normal distribution. \nWe show the results in Fig.~\\ref{fig:random_offsets}.\nEven when the fractional offset is $\\sim 10^{-5}$, the wedge returns, albeit quite subdued (2-3 orders of magnitude).\nAs the offsets increase in magnitude, the wedge is restored to a greater degree, as expected.\nIt would thus seem that any hopes of mitigating the wedge via regular radial arrays are impractical both due to their high density requirements and their strong dependence on strict regularity. \nNevertheless, it is possible that irregularity between spokes will alleviate some of this, and we will explore this further in the following section.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1.7cm 0cm 3.5cm 0cm]{figures\/random_offsets}\n\t\\caption{2D PS comparison between baseline layouts for a regular logarithmic polar grid, and three cases in which the baselines have been randomly offset from regularity. A subdued wedge clearly returns even for fractional offsets of $10^{-5}$. The amplitude of the wedge increases by a couple of orders of magnitude as the offsets increase in their magnitude.}\n\t\\label{fig:random_offsets}\n\\end{figure}\n\n\\section{Wedge Properties of Real Arrays}\n\\label{sec:mitigation:arrays}\n\n\\begin{table*}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{ l l l }\n\t\t\t\\hline\n\t\t\t\\textbf{Label} & \\textbf{Description} & \\textbf{Varieties} \\\\ \n\t\t\t\\hline\n\t\t\t\\texttt{circle} & Equi-spaced on circumference of circle, diameter $x_{\\rm max}$ & \\\\\n\t\t\t\\texttt{circle\\_filled\\_} & Randomly filled circle of diameter $x_{\\rm max}$ & Uniform (\\texttt{\\_0}), Logarithmic (\\texttt{\\_1}) \\\\\n\t\t\t\\texttt{spokes\\_} & Regular radial\/angular spacing, max $x_{\\rm max}$ & Logarithmic\/Linear, $N_{\\rm spokes}$ \\\\\n\t\t\t\\texttt{rlx\\_boundary} & Equi-spaced on boundary of Reuleaux triangle \\citep[eg.][]{Keto1997} & \\\\\n\t\t\t\\texttt{rlx\\_grid\\_} & Regular concentric Reulaeux triangles & Logarithmic \\\\\n\t\t\t\\texttt{hexagon} & Regular hexagon, width $x_{\\rm max}$ & \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{\\label{tab:baseline_layouts}Summary of antenna layouts used in Figs. \\ref{fig:big_baseline_diagram} -- \\ref{fig:big_layout_std}.}\n\t\\end{center}\n\\end{table*}\n\n\nWe have found that the wedge may in principle be avoided by employing a sufficiently radially dense and regular baseline layout.\nIn our previous explorations, we have considered explicit baseline layouts, ignoring the fact that no \\textit{antenna} layout may exactly correspond to such a baseline layout (alternatively, choosing an antenna layout which corresponds to a superset of the desired baseline layout and ignoring the extraneous baselines).\nIn this final exploration, we expand our consideration to several physically-feasible antenna layouts, with full correlation of \\textit{all} antennas.\n\nIn this case, simple ``spoke'' antenna layouts will contain both the regular subset which we have previously considered, and a larger set of irregular baselines. \nThus we will find whether the increased baseline density outweighs the increased irregularity in terms of wedge mitigation (cf. \\S\\ref{sec:weighted:irregular}).\n\nThe kinds of antenna layouts we employ (with their variants) can be found in Table \\ref{tab:baseline_layouts}, and an illustration of each is found in Fig. \\ref{fig:big_baseline_diagram}.\nWe note that for the linear ``spokes'' layouts, to decrease the redundancy, we use regularly-spaced antennae for half of the spoke, and a single antenna at the far end.\nThis does not apply for the logarithmic spoke layouts, for which each spoke necessarily begins at the centre. \n\n\nWe use the same number of antennae, $N_{\\rm ant}$, for each layout (or as close to this number as possible, given the constraints of some), and place all baselines within a set radius $x_{\\rm max} \\approx 2u_{\\rm max}$.\nWe choose $N_{\\rm ant} = 256$ and $u_{\\rm max} = 800$ for the figures in this section.\nEach layout is first checked for overlapping antennae, with antenna diameters of 4m (corresponding roughly to the MWA tiles), so that the final layout is physically possible. \nWith these choices, comparisons of the power spectra from each array are roughly insensitive to the overall density or ``cost'' of the array, and are rather indicative of the form of the layout itself.\nNote that we also use the tile diameter of 4m to calculate the beam width.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 3cm 0cm 3cm]{figures\/big_baseline_diagram}\n\t\\caption{Plots of antenna and baseline layouts for the definitions in Table \\ref{tab:baseline_layouts}. The orange markers indicate antennas, and blue points represent baselines. The axes are in units of $u$ at a frequency of 150 MHz. Antennae are all spaced at least 4m apart, which corresponds to the size of an antenna.}\n\t\\label{fig:big_baseline_diagram}\n\\end{figure*}\n\nDespite these considerations, we must be clear that this is not a test for \\textit{how well the layout would recover an underlying 21 cm signal}. \nSpecifically, two quantities are of interest for foreground mitigation: the total expected (foreground) power, and its covariance, $\\Sigma_P$. \nWe have only addressed the expected power in this paper. \nAssuming a reliable model of this quantity can be subtracted from the observed data, the remaining $\\Sigma_P$ is the key factor in defining which power spectral modes are usable for averaging to a final one-dimensional power spectrum. \nTo a large extent, $\\Sigma_P$ is proportional to $P^2$, so the derivations in this paper are indicative of this quantity.\nNevertheless, both sample variance and thermal noise also play an independent role in the determination of $\\Sigma_P$, and these are dictated largely by the density of baselines (the former explicitly by the \\textit{angular} density).\n\nWhile the overall density of baselines should be similar in each of the layouts we have chosen -- due to our restriction of setting the antennas within a prescribed radius -- their angular density is decidedly \\textit{not}.\nWe thus expect those with lower angular coverage (eg. the various ``spokes'' layouts) to yield a greater value of $\\Sigma_P$, which could hamper 21 cm signal extraction.\nWe do not pursue a rigorous analysis of these considerations in this paper; our goal is to identify the general effects these layouts have on the establishment of the wedge feature -- not the prospects of 21 cm signal extraction.\nNevertheless, we suggest that such an analysis should be simple enough, by comparing a numerically-generated expectation of $\\Sigma_P$ in the presence of both sample variance and thermal noise for each layout.\n\n\\ifanalytic\n\tTo determine the expected PS for each layout, we numerically solve the triple integral given by the combination of Eqs. \\ref{eq:power_general} and \\ref{eq:general_var}.\n\tThis is a non-trivial task, requiring high-precision integration over $\\theta$ in order to yield reasonable estimates after dividing by the weight factor. \n\tDetails of the numerical method can be found in App. \\ref{app:numerical_integration}.\n\\else\n\tTo determine the expected PS for each layout, we evaluate the PS using the technique outlined in App.~\\ref{app:numerical} for the same set of 200 random skies for each layout, taking the mean and standard deviation. \n\\fi\n\nFigure \\ref{fig:big_layout_ps} shows the resulting expected 2D power spectra for each of these layouts.\nAll layouts show a strong wedge feature with similar shape, as expected, and each exhibits the same bandpass limits (the low-$\\omega$ ``brick''). \nTwo peculiar features require some explanation.\nFirst, due to our choice of using a gridding kernel which in principle has infinite extent (though in practice, we limit it to 50-$\\sigma_u$), grid points $\\vect{u}$ which are in extremely sparse UV-sampled locations will tend to evaluate to the same power, as the same distant baseline will be the dominant contributor for all grid points in the region. \nIf this is limited to a small arc of the full polar angle, its effect will be negligible, but some of these layouts are extremely sparse for all angles, especially at low-$u$. \nThis effect presents as a horizontal `smearing' of the power, and is most noticeable in the low-$u$ modes of the \\texttt{spokes\\_log\\_4} layout. \nA related effect produces the many thin vertical ``spikes'' witnessed at high-$u$ in many of the spectra. \nIn this case, however, it seems to be a combination of the irregularity of the baselines with this local sparsity that produces the effect. \nWe emphasize that the vertical features, though they appear `noisy', do not disappear as more realizations are averaged, and are therefore systematic. \n\nIn Figure \\ref{fig:big_layout_compare} we show the ratio of each expected 2D PS against the result of a delay spectrum.\nThis is precisely the result of \\S\\ref{sec:classic} (i.e. the limit of sparse baselines), except that each $u$ is assumed to be exactly obtainable.\nThus comparison to this spectrum is appropriate as the sparse limit of baseline density. \nWe reiterate that the physical layouts here can be non-local, so that the evaluated power is determined by a relatively distant baseline, whereas the reference `delay spectrum' is always exactly local in this comparison. \n\nWith this in mind, we note that most of the layouts produce less foreground power over most modes than a simple delay spectrum. \nThis is to be expected, as the averaging over baselines effectively lowers the amplitude of fluctuations. \nThe single exception to this seems to be the hexagonal layout. \nHowever, on closer inspection, most of the power here is exactly the same as the delay spectrum, as expected from its inherent sparsity. \nAt low-$u$, the \\texttt{hexagon} is in a region of extreme local sparsity, as discussed above, and therefore cannot be trusted (in the same way as the low-$u$ region of \\texttt{spokes\\_log\\_4}).\nThe region in and around the wedge does exhibit significantly more power than the delay spectrum, but this is common to all layouts, and we will discuss this momentarily.\n\nThe most significant reduction of power occurs in the EoR window for the \\texttt{spokes\\_log\\_6} and \\texttt{rlx\\_grid\\_log} layouts, at 2-3 orders of magnitude.\nThese layouts have strong logarithmic regularity at the most polar angles compared to other layouts in our sample. \nThough their radial density is not as high as the log-spoke layouts with fewer spokes, it appears that providing some regularity at more angles (and therefore decreasing overall irregularity) outweighs this deficit.\nHowever, these layouts, along with \\texttt{circle\\_filled\\_1}, also have the highest density of short baselines, so it is difficult to isolate the contribution of any single characteristic.\n\nThe most visually obvious feature of the ratio plots (figure \\ref{fig:big_layout_compare}) is the excess power appearing as irregular vertical stripes protruding from the wedge. \nThis power appears to arise due to sparsity of baselines at these scales, such that for a particular grid-point, baselines ``move through'' the grid-point with frequency and leave nulls in the effective spectrum before another baseline passes through. \nThis creates a ringing in the Fourier-space spectrum, which throws power outside the wedge. \nThis interpretation is supported by the fact that the two layouts which minimize this effect are those with the highest density of baselines at high $u$. \nConversely, the hexagonal layout, with its extremely sparse and regular baselines, maximizes this effect over much of the range.\nThis is a well-known key advantage of the delay spectrum, which in principle limits the foreground power exclusively to the theoretical ``horizon line'' for each baseline (notwithstanding other chromatic effects of the instrument and sky).\nNevertheless, it is unclear how this advantage balances against the reduction of window power offered by the dense regular baseline layouts.\nUltimately, these scales, where the density of baselines is low enough to cause this effect, should be ignored in any analysis.\n\nAnother interesting feature are the diagonal strips at high $u$ in the \\texttt{spokes\\_lin} layouts, which appear to be manifestations of the same effect illustrated in Fig. \\ref{fig:dense_linear_concentric}, i.e. dense linear radial regularity introducing scale-dependent harmonics in the sky response.\nNevertheless, these are muted compared to the purely regular theoretical arrays previously considered.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=1cm 2cm 0cm 2cm]{figures\/big_layout_straight}\n\t\\caption{Average 2D PS (over 200 realizations) for each of the layouts we consider (see Table \\ref{tab:baseline_layouts} for details of the layouts).}\n\t\\label{fig:big_layout_ps}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 2cm 0cm 2cm]{figures\/big_layout_ratio}\n\t\\caption{Ratio of 2D PS of each layout in Table \\ref{tab:baseline_layouts} to a delay spectrum evaluated with one baseline at each grid point (see \\S\\ref{sec:mitigation:arrays} for details).}\n\t\\label{fig:big_layout_compare}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth, trim=0cm 2cm 0cm 2cm]{figures\/big_layout_std}\n\t\\caption{Standard deviation of 2D PS over 200 realizations for each layout in Table \\ref{tab:baseline_layouts}.}\n\t\\label{fig:big_layout_std}\n\\end{figure*}\n\nTo verify that the results of this section are not subject to high statistical uncertainty, we show the ratio of the standard error of the mean (SEM) to the mean of each PS in Fig. \\ref{fig:big_layout_std}.\nThis illustrates that the mean is accurate to within $\\sim10$\\%, which means that statistical uncertainty is of minor concern, and that the conceptual results of this section can be trusted in this regard.\nInterestingly, the regions of excess power have a relatively high uncertainty compared to the rest of the spectrum, indicating that these regions are more sensitive to the exact positions of point-sources on the sky.\nThis supports our interpretation that this excess power arises from a dearth of baselines, which would increase the sensitivity of the measured power at a particular $\\vect{u}$ to the sky realization, and also increase the variance of measurements over polar angles. \n\nIn summary, with the number of antennae considered, the precise layout has only a minimal effect on the expected PS within the wedge and window. \nNevertheless, in accord with our semi-analytic considerations of previous sections, it appears that dense logarithmically regular layouts can improve the spectral smoothness of the array, and mitigate foreground power, at the level of 2-3 orders of magnitude. \nWe expect this to improve with a higher number of antennae, so that layout considerations will become relatively more important in future high-$N$ arrays.\nConversely, gridding baselines, as opposed to delay transforming on a per-baseline basis, produces artifacts at high-$u$ which can throw excess power out of the wedge. \nIt is beyond the scope of this paper to quantitatively assess which method is preferable for measuring the EoR.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\\subsection{Summary}\nUsing a simple formalism to describe the expected 2D power spectrum of point-source foregrounds (Eqs. \\ref{eq:simple_vis}, \\ref{eq:vis_gridded}, \\ref{eq:vis_full} and \\ref{eq:power_general}), we verified the standard schematic `wedge', which has been extensively discussed in the literature.\nWe then used this formalism, which includes the ability to utilise arbitrary $uv$-sampling functions, to examine the effects of such on the wedge, with the primary conclusion that dense, radially regular layouts can diminish the extent and amplitude of the wedge, but that this effect is small for physically achievable layouts.\n\nUsing a semi-analytic approach based on a discrete polar grid $uv$-sampling (largely focusing on logarithmically-separated radial baselines) we find, as suggested in previous works \\citep[eg.][]{Bowman2009,Morales2012,Parsons2012}, that increasing the radial density of baselines tends to decrease the amplitude of the wedge (defined as that part of the foreground signature which emerges from the low-$k_{||}$ ``brick''). \nIndeed, we find that for regular log-spaced baselines, a density may (in principle) be achieved at which the wedge effectively disappears. \nWe explain these results intuitively via radial baseline ``replacement'' with change of frequency (cf. \\S\\ref{sec:weighted:conceptual}).\n\n\nUsing this semi-analytic model, we explored some of the ramifications of these ideas.\nWe found that a characteristic separation can be determined which defines the threshold for wedge emergence.\nThis characteristic separation is proportional to the baseline magnitude $u$ and also to the bandwidth of the observation (cf. Eq. \\ref{eq:solution_for_concentric}).\nWe find that the physical baseline separation (in metres) is approximately $u\/\\tau$, where our default value for $\\tau$ is approximately 100, and in the regime of the wedge, $u > \\tau\/\\sqrt{2}\\pi\\sigma \\approx 150$. \nThe minimum baseline separation to mitigate the entire wedge (which occurs at $u=\\tau\/\\sqrt{2}\\pi\\sigma$, or an antenna separation of $\\sim 300$m) is $1\/3\\sigma \\approx 1.7{\\rm m}$.\nWe concluded that such a baseline density is physically impossible for the most efficient antenna layout corresponding to the polar grid baseline layout.\nWith a less compact array, the baseline density is technically achievable, but highly impractical.\n\nFurther, we found that randomising the radial distribution of baselines tends to re-instate the wedge, as the series of overlaid oscillations is smeared out over $\\omega \\propto \\eta$.\nThus the optimal array for wedge mitigation is both dense \\textit{and} regular.\n\nUpon examination of the expected 2D PS from some simple antenna layouts, we found that in practice both the window and wedge can be reduced by up to 3 orders of magnitude by employing antenna spokes which are regular in log-space and as dense as possible across many angles.\nWe noted that such a layout competes with the requirement of angular baseline density to mitigate sample variance. \n\n\n\\subsection{Future Considerations}\nThe work in this paper paints a rather bleak picture: \nit will be very difficult to combat the wedge via any array design.\nNevertheless, we have shown that in principle, layouts with a higher degree of radial alignment and regularity will serve to reduce the magnitude of the wedge, and therefore potentially yield some increase in the fidelity of future PS estimation.\n\nTo establish this rigorously, one needs to consider not only the expected 2D PS, but also its covariance.\nThese will compete with one another -- the more aligned the layout, the lower the expected wedge, but the higher the overall covariance of the estimate.\nA proper analysis of these quantities, and their effect on the signal-to-noise of a fiducial 21 cm signal, is the most pressing future consideration to arise from this work.\nAlong with this, consideration of non-Gaussian band-pass (or taper) shapes, non-co-planar arrays, non-zenith pointings, and non-flat SED's may be interesting realistic effects to add to the analysis.\n\n\n\n\\begin{acknowledgements}\n\tThe Centre for All-Sky Astrophysics (CAASTRO) is an Australian Research Council Centre of Excellence, funded by grant CE11E0090. \n\tParts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.\n\tThis research has made use of NASA's Astrophysics Data System.\n\tAll plots in this paper were generated using \\textsc{matplotlib}.\n\\end{acknowledgements}\n\n\\clearpage\n\n\\begin{appendix}\n\t\n\t\\section{Derivation of Analytic Examples}\n\t\\label{app:delay}\n\t\n\tIn this section we derive the solutions to the three analytic examples shown in Table \\ref{tab:simple_summary}.\n\tRecall that in this section, we have employed the following conditions:\n\t\\begin{enumerate}\n\t\t\\item Sparse discrete polar grid baseline layout\n\t\t\\item Artificially narrow gridding kernel (width $\\rightarrow 0$)\n\t\t\\item Evaluation at grid points co-located with reference baseline positions.\n\t\\end{enumerate}\n\n\tWe note that in this scheme, only one baseline may contribute to a given $\\vect{u}$ over all frequency. \n\tThus, Eq.~\\ref{eq:vis_gridded} is simplified to $V(\\nu, \\vect{u}) = V_i(\\nu,\\vect{u}_i)$, removing the sum over baselines.\n\t\n\tFinally, since the baselines are arranged symmetrically, the average over $\\theta$ is unweighted -- each baseline contributes to the same arc-length.\n\tThe power can thus be simply expressed as\n\t\\begin{equation}\n\t\\label{eq:power_ngp}\n\t\\langle P(\\omega, u_\\mu) \\rangle = \\frac{1}{N_\\theta} \\sum_{k=1}^{N_\\theta} \n\t{\\rm Var}(\\tilde{V}(\\omega, \\vect{u}_{\\mu, k})) + \\left|\\langle \\tilde{V}(\\omega, \\vect{u}_{\\mu, k})\\rangle \\right|^2,\n\t\\end{equation}\n\twhere the sum is over all baselines in a given ring, and the subscript $\\mu$ is meant to indicate that we evaluate the visibilities at $\\vect{u}_i$, rather than at an arbitrary location.\n\tNote that evaluation of the PS at $u \\neq u_i$ is simple, but will yield the same PS as that evaluated at the closest $u_i$. \n\tThese solutions therefore really represent a series of step-functions in $u$, where each step is centred on $u_i$. \n\t\n\t\n\t\\subsection{Sky Model: Single Source}\n\tWithin this subsection we will consider a single-source sky at $\\vect{l} = \\vect{l}_0$ with $S=S_0$.\n\tDue to the non-stochastic nature of the sky we only require the mean visibility, which is simply:\n\t\n\n\n\n\n\t%\n\t%\n\n\n\t\\begin{align}\n\t\\langle \\tilde{V}(\\omega, \\vect{u}_\\mu) \\rangle = \\nu_0 S_0 \\int df\\ \\phi_\\nu B_\\nu(l_0) e^{-2\\pi i f(\\omega + \\vect{u}_\\mu \\cdot \\vect{l}_0)}.\n\t\\end{align}\n\t\n\t\n\t\\subsubsection{Static Beam} \n\tIf the beam is frequency-independent, it comes out of the integral and we are left with\n\t\\begin{equation}\n\t\\label{eq:tmp0}\n\t\\tilde{V}(\\omega, \\mathbf{u}_\\mu) = S_0 \\nu_0 B(l_0) \\int df e^{-\\tau^2 (f-1)^2} e^{-2\\pi if (\\omega +\\vect{u}_\\mu \\cdot\\vect{l}_0)}.\n\t\\end{equation}\n\t\n\tHere we make use of the following useful identity, and we shall repeatedly do so throughout this section:\n\t\\begin{equation}\n\t\\label{eq:int_of_exp}\n\t\\int_{-\\infty}^{+\\infty} e^{-ax^2 - bx + c}dx = \\sqrt{\\frac{\\pi}{a}}e^{b^2\/4a + c}.\n\t\\end{equation}\n\tRearranging Eq. \\ref{eq:tmp0}, we find \n\t\\begin{align}\n\ta &= \\tau^2,\\nonumber \\\\\n\tb &= 2\\pi i (\\omega + \\vect{u}\\cdot \\vect{l}_0) - 2 \\tau^2, \\nonumber \\\\\n\tc &= -\\frac{l_0^2}{2\\sigma^2} - \\tau^2.\n\t\\end{align}\n\tThis yields\n\t\\begin{align}\n\t\\label{eq:vis_ss_sb}\n\t\\tilde{V}(\\omega, \\mathbf{u}_\\mu) = &\\frac{S_0 \\nu_0 \\sqrt{\\pi}}{\\tau} \\exp\\left(-\\frac{l_0^2}{2\\sigma^2}\\right) \\nonumber \\\\\n\t& \\times\\exp\\left( -\\frac{\\pi^2(\\omega + \\vect{u}_\\mu\\cdot \\vect{l}_0)^2}{\\tau^2} - 2 i\\pi (\\omega + \\vect{u}_\\mu\\cdot \\vect{l}_0)\\right)\n\t\\end{align}\n\t\n\tFurthermore, the power can be written:\n\t\\begin{align}\n\t\\label{eq:ss_ngp_static}\n\tP(\\omega, u_\\mu) = \\frac{1}{N_\\theta} & \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right) \\nonumber \\\\\n\t& \\times \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2(\\omega + u_\\mu l_0 \\cos (2\\pi k\/N_\\theta))^2}{\\tau^2} \\right).\n\t\\end{align}\n\tThis sum has no general closed form solution.\n\tNevertheless, it is not difficult to ascertain its general behaviour. \n\tThe two terms in the exponential will compete for dominance, and since cosine has a maximum of unity, we can determine a line of equal weight: $\\omega = ul_0$.\n\tWhen the $\\omega$ term is dominant, the integrand loses sensitivity to $\\theta$, and the power can be written\n\t\\begin{equation}\n\tP(\\omega \\gg ul_0, u) = \\frac{S_0^2 \\nu_0^2 \\pi}{\\tau^2} \\exp\\left(-\\frac{l_0^2}{\\sigma^2}\\right) \\exp\\left( -\\frac{2\\pi^2 \\omega^2}{\\tau^2} \\right),\n\t\\end{equation}\n\tThus we expect that there will be a (sharp) exponential drop in the power for $\\omega \\gg ul_0$. \n\tWhen $ul_0$ is small, the entire function $P(\\omega)$ (i.e. a vertical line in the 2D PS) will obey this equation, and the cutoff will appear at a characteristic scale $\\omega \\sim \\tau\/\\pi\\sqrt{2}$. \n\tLarger $ul_0$ acts as a buffer, requiring $\\omega$ to overcome it before the exponential drop is realised (at a much sharper rate, due to the increased amplitude of the exponential). \n\tThe exact point at which $\\omega$ overcomes the $ul_0$ term is difficult to obtain in closed form (it can easily be obtained as a power series), but we merely state the empirical result that it is close to $ul_0$. \n\tThus we have a cutoff at $\\omega \\approx {\\rm max}(\\tau\/\\pi\\sqrt{2}, ul_0)$, where the first limit defines a ``brick'' at low $(\\omega,u)$, and the second defines a ``wedge'' at higher $u$.\n\t\n\t\n\t\n\t\n\t\n\t\\subsubsection{Chromatic Beam}\n\t\\citet{Liu2014} have pointed out that regardless of whether the beam is chromatic or not, it is a much broader function of frequency than the taper, and therefore it is a good approximation to bring it outside the integral, and evaluate it at $f=1$. \n\tThis would yield precisely the same result as the achromatic beam of the previous section.\n\tNevertheless, we wish to present an exact formula in this section.\n\t\n\tIn this case, the only aspect that changes from the previous section is that we have $a \\rightarrow \\tau^2 + l_0^2\/2\\sigma^2$ since the beam moves back inside the integral.\n\tThus we achieve\n\n\n\n\n\n\t\\begin{align}\n\tP(\\omega, u_\\mu) = & \\frac{S_0^2 \\nu_0^2 \\pi}{N_\\theta p^2} \\exp\\left[-\\tau^2 \\left(1 - \\frac{\\tau^2}{p^2}\\right)\\right] \\nonumber \\\\\n\t& \\times \\sum_{k=1}^{N_\\theta} \\exp\\left( -\\frac{2\\pi^2 (\\omega + u l_0 \\cos (2\\pi k\/N_\\theta))^2}{p^2}\\right),\n\t\\end{align}\n\twith $p^2 = \\tau^2+l_0^2\/2\\sigma^2$.\n\tThe behaviour of this equation is very similar to the previous static case, except that the effect of $\\tau$ is balanced by the effect of the beam-width. \n\tThat is, setting $\\tau$ arbitrarily small (i.e. very wide band-pass) will no longer yield an arbitrarily tight ``brick'', as the beam-width will have the effect of broadening it.\n\t\n\tIn practice, for instruments targeted at observing the EoR, $\\tau^2 \\gg 1\/2\\sigma^2$, so that the achromatic beam is a reasonable approximation, as expected by the arguments from \\citet{Liu2014}.\n\t\n\t\\subsection{Sky Model: Stochastic Uniform}\n\t\t\n\tWe merely need to solve Eqs. \\ref{eq:su_meanvis} and \\ref{eq:su_covvis} for a Gaussian beam, and then integrate over frequency.\n\tWe evaluate only for a static beam in this case, as we have already seen that a chromatic beam is too broad (in frequency) to have a significant impact on the result.\n\t\n\tThe mean term is simply\n\t\\begin{align}\n\t\t\\langle V(\\omega, u_\\mu) \\rangle &= 2\\pi \\sigma^2 \\nu_0\\bar{S} \\int df\\ e^{-2\\pi i f\\omega} \\phi_\\nu e^{-2\\pi^2 \\sigma_\\nu^2 f^2 u^2}.\n\t\\end{align}\n\n\tUsing the identity Eq.~\\ref{eq:int_of_exp}, and noting that the equation depends only on $u^2$ and therefore needs not be integrated around the annulus, we find\n\t\\begin{align}\n\t|\\langle \\tilde{V}\\rangle|^2 &= \\frac{4 \\bar{S}^2 \\nu_0^2 \\pi^3 \\sigma^4}{\\tau^2 + 2\\pi^2 \\sigma^2 u^2} \\exp\\left[ 2\\left(\\frac{\\tau^4 - \\pi^2\\omega^2}{\\tau^2 + 2\\pi^2\\sigma^2 u^2} - \\tau^2 \\right) \\right].\n\t\\end{align}\n\t\n\tTo begin the variance, we use Eq.~\\ref{eq:su_covvis}, along with the various assumptions we have thus far made, to obtain\n\t\\begin{align}\n\t\t\\label{eq:stoch_uniform}\n\t\t{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 & \\int d\\vect{l}e^{-l^2\/\\sigma^2} \\left| \\int df e^{-2\\pi if(\\omega + \\vect{u}\\cdot \\vect{l})} \\phi_\\nu \\right|^2\n\t\\end{align}\n\tWe use the result of (Eq. \\ref{eq:vis_ss_sb}) directly to obtain\n\t\\begin{align}\n\t\t{\\rm Var}(\\tilde{V}) = \\frac{\\pi \\mu_2 \\nu_0^2}{\\tau^2} \\int d\\vect{l}\\ \\ e^{-l^2\/\\sigma^2} \n\t\t\\exp\\left(-\\frac{2\\pi^2(\\omega+\\vect{u}_\\mu\\cdot\\vect{l})^2}{\\tau^2} \\right) .\n\t\\end{align}\n\tDue to statistical isotropy, we may without loss of generality evaluate the case $\\vect{u} = (u,0)$, and perform the integral over $\\vect{l}$ in 2D Cartesian space, to finally find\n\t\\begin{align}\n\t\\label{eq:sparse_solution}\n\t{\\rm Var}(\\tilde{V}) = \\frac{\\mu_2 \\nu_0^2 \\pi^3 \\sigma^2}{\\tau\\sqrt{\\tau^2 + 2\\pi^2\\sigma^2u^2}} \\exp\\left(-\\frac{2\\pi^2 \\omega^2}{2\\pi^2\\sigma^2u^2+\\tau^2}\\right).\n\t\\end{align}\n\t\n\tNow combining both terms of the power spectrum, we can make some simple observations.\n\tFirstly, for $\\pi u \\sigma \\ll \\tau$, we have\n\t\\begin{equation}\n\tP_{\\pi u \\sigma \\ll \\tau} = \\frac{\\nu_0^2 \\pi^2 \\sigma^2}{\\tau^2} e^{-2\\pi^2 \\omega^2\/\\tau^2} \\left(\\bar{S}^2 \\sigma^2 + \\mu_2 \\pi \\right).\n\t\\end{equation}\n\tThis has a sharp cut-off at $\\omega \\approx \\tau\/\\sqrt{2}\\pi$, creating the familiar lower-left ``brick'' in the 2D PS.\n\tConversely, we have\n\t\\begin{align}\n\tP_{\\pi u \\sigma \\gg \\tau} &= \\frac{\\nu_0^2 \\sigma}{u} e^{-\\omega^2\/2u^2\\sigma^2}\\left[\\frac{ 2\\pi \\sigma \\bar{S}^2 e^{-\\tau^2}}{ u} + \\frac{\\mu_2 \\pi e^{-\\omega^2\/2\\sigma^2 u^2}}{\\sqrt{2}\\tau}\\right] \\nonumber \\\\\n\t&\\approx \\frac{\\nu_0^2\\mu_2 \\pi \\sigma}{\\sqrt{2}\\tau u} e^{-\\omega^2\/u^2\\sigma^2}, \n\t\\end{align}\n\twhere the final line assumes that $\\omega < \\tau^2$, which covers all the reasonable values of $\\omega$.\n\n\tThis clearly has a sharp cut-off at $\\omega = u\\sigma$, creating the wedge (cf. rightmost panel of Fig. \\ref{fig:ss_ngp_static}). \n\t\n\t\n\t\n\n\n\\section{Radially Smooth Layout}\n\\label{app:radial}\nHere we consider a polar grid layout in which the radial spokes are no longer discrete but are of such high density that they may be considered smooth.\nThis will allow us to derive some constraints on how `smooth' the radial distribution of baselines must be to avoid a wedge.\n\nLet $\\rho = \\rho_\\theta \\rho_u$ be the density of baselines as a function of $u$ and $\\theta$.\nThen the sums over baselines in Eq. \\ref{eq:wg_master} reduce to integrals over $\\rho$: \n\\begin{align}\nI = \\int df \\frac{\\phi_\\nu}{W_\\nu} \\int d^2 \\vect{u}_i \\frac{\\rho_\\theta \\rho_u}{u_i} e^{-q^2(\\vect{u} - f\\vect{u}_i)^2} e^{-2if(\\omega +\\vect{l}\\cdot\\vect{u}_i)}.\n\\end{align}\n\nWe may calculate the total weight, performing the integration in polar co-ordinates, making the substitution $u'_i = fu_i$:\n\\begin{align}\nW_\\nu = &\\frac{e^{-2\\pi^2 \\sigma^2u^2}}{2\\pi f^2} \\int_0^{2\\pi} d\\theta\\ \\rho_\\theta \\nonumber \\\\\n&\\times \\int du'_i\\ \\rho_u(u'_i\/f^2) e^{-q^2({u'_j}^2 - 2uu'_i\\cos\\theta )}.\n\\end{align}\nIt is difficult to proceed further without specifying some form for $\\rho_u$. \nNevertheless, we note that $\\rho_u$ will only contribute to the $u'_i$ integral if it is sufficiently sharply peaked -- otherwise it can be treated as a constant and removed from the integral.\nWe let $\\rho_u$ be an arbitrary linear combination of Gaussians, centered around points $u_l$:\n\\begin{equation}\n\\rho_u \\propto \\sum_l a_l \\exp\\left(-\\frac{(u'_j - u_l)^2}{2 \\sigma_l^2}\\right),\n\\end{equation}\nwhere the normalisation constant is irrelevant as it cancels in the final visibility. \n\nThe equation for $W_\\nu$ may thus be re-written as\n\\begin{align}\nW^T_j = &\\frac{e^{-q^2 u^2}}{f^2} \\int d\\theta\\ \\rho_\\theta \\sum_l a_l e^\\frac{-u_l^2}{2f^4\\sigma_l^2} \\nonumber \\\\\n&\\times \\int du'_i\\ \\exp\\left(-{u'}_i^2(q^2 + \\frac{1}{2\\sigma_l^2})\\right. \\nonumber \\\\\n&+ \\left. 2u'_i(q^2 u\\cos \\theta + \\frac{u_l}{2f^2\\sigma_l^2})\\right).\n\\end{align}\nPerforming the $u'_i$ integral, each term in the sum becomes\n\\begin{equation}\na_l \\sqrt{\\frac{\\pi}{q^2 + \\frac{1}{2\\sigma_l^2}}} \\exp\\left(\\frac{q^2\\left[q^2 u^2\\cos\\theta\/2 + \\frac{uu_l\\cos\\theta}{2f^2\\sigma_l^2} - \\frac{u_l^2}{f^4 \\sigma_l^2}\\right]}{2q^2 + \\frac{1}{\\sigma_l^2}}\\right).\n\\end{equation}\nIf $\\sigma_l \\gg 1\/2q = 1\/2\\pi\\sigma$, then we can ignore the $\\sigma_l$ term in both the square root and the denominator of the exponential. \nIn fact, this same inequality also reduces the numerator to its first term (for $f\\sim 1$), such that the form for $W_\\nu$ is\n\\begin{align}\nW_\\nu = &\\sqrt{\\frac{\\pi}{q^2}} \\frac{e^{-q^2u^2}}{f^2} \\sum_l a_l \\int d\\theta\\ \\rho_\\theta \\exp\\left(\\frac{q^2 u^2\\cos\\theta}{4}\\right).\n\\end{align}\n\nThe condition that $\\sigma_l \\gg 1\/2\\pi\\sigma$ for all terms $l$ is thus a well-specified ``smoothness\" bound, though we note that it is a conservative bound;\neven if a term is \\textit{more} peaked than permitted by the bound, if it has a small relative amplitude then its contribution may be ignored. \nThis is important for real arrays, in which the baselines form delta-functions in the UV plane. \nThough every point consists of a ``Gaussian'' which is more peaked than the bound, they may be spaced closely enough that each of them contributes negligible weight, thereby approximating a ``smooth\" array.\n\n\nThis smoothness bound, for a realistic array at $\\nu_0 \\approx 150$ MHz, corresponds to constraining $\\sigma_l \\gg 2D$, where $D$ is the diameter of an array tile.\nBreaking this condition would require quite a peaked baseline density indeed.\n\n\n\n\nWith this in mind, for an arbitrary radially smooth layout, the solution is of the form\n\\begin{equation}\nW_\\nu = g(q)\/f^2.\n\\end{equation}\nWe can use the same procedure to determine the final integral of $I$ (except that it has an extra factor of $2\\pi i \\vect{l}\\cdot\\vect{u}'_i$ in the exponent). \nThis implies that the factors of $f^2$ cancel, so that we have\n\\begin{equation}\nI = \\frac{g'(\\vect{u}, \\vect{l})}{g(\\vect{u})} \\int df \\phi_\\nu e^{-2\\pi if\\omega},\n\\end{equation}\nand it is clear that the solution must be separable in $u$ and $\\omega$. \nThis clearly defines a ``brick'' structure valid for all $u$ (and which again has a cut-off at $\\tau\/\\sqrt{2}\\pi$).\nThus a wedge is precluded for any radially smooth layout.\n\nWe note that this was determined for arbitrary angular density $\\rho_\\theta$. \n\n\\ifanalytic\n\\section{Numerical Integration Algorithm}\n\\label{app:numerical_integration}\nTo determine the expected power spectrum for an arbitrary layout (as is done in \\S\\ref{sec:mitigation:arrays}) requires performing the triple-integral implicit in the combination of Eqs. \\ref{eq:power_general} and \\ref{eq:general_var}.\nThere are several difficulties in doing so, because all integrals must be performed numerically (in general).\n\nThe first difficulty is that the $\\theta$ integral must have zero absolute error tolerance. \nThis is due to the fact that it is normalised by a similar $\\theta$ integral over the weight function. \nSince the absolute value of either integral may be arbitrarily small, a small but constant absolute error will be magnified, and results in artificial vertical stripes in the 2D PS. \nConsequently, the efficiency of the procedure is highly reduced.\n\nThe second difficulty is that the $f$-integrals are highly oscillatory when $\\omega$ is large.\nThese then also must be performed with high precision, and often by breaking the integral into independent chunks. \nEven so, the authors have not been able to find a numerical integration scheme which yields acceptable results at high $\\omega$, and this can result in ``negative'' power at some grid-points. \nThis can be partially overcome by making a change of variable $x = f-f'$ so that only one integral contains the oscillations, rather than two. \nThis increases efficiency by tens of percent, but does not allow accurate computation to arbitrarily high $\\omega$.\n\nWith these considerations, the efficiency of the integration is very poor indeed --- for a layout of $\\sim10000$ baselines, some grid-points can take many hours to days to complete. \nClearly this becomes infeasible when multiple arrays with multiple grid-points are required. \n\nTo increase performance, we perform several optimizations. \nFirst, we quickly reduce the number of baselines required to be summed over for a given $(u, \\omega)$ by determining their weight, $W(u)$, and culling all baselines whose weight is smaller than some threshold by the mean. \nIf a single baseline remains, we simply return the sparse-layout solution, Eq.~\\ref{eq:sparse_solution} which shortcuts the process.\n\nSecond, we increase the \\textit{relative} error tolerance on the $\\theta$ to $10^{-3}$. Doing so will yield a final result which is also nominally accurate to 0.1\\%.\n\nA more aggressive solution to this performance issue is described in App.~\\ref{app:approx_variance}, involving analytic approximations of the integral itself.\n\n\n\\section{An Approximate Solution For The Variance}\n\\label{app:approx_variance}\nIn this appendix we derive an approximate analytical solution for the variance in our fiducial case of static beam and stochastic sky (cf. Eq. \\ref{eq:wg_master}).\nThe approximation we make is to replace the weighted gridding with a nearest-baseline gridding. \nThat is, instead of evaluating the visibility at a given grid point $\\vect{u}$ as the weighted average of visibilities at surrounding baselines, we assume that the \\textit{closest} baseline will contribute the dominant weight \\textit{for a given frequency}, and we neglect the rest of the terms in the sum.\nThis is a reasonable approximation to make, since if any other terms are co-dominant, they must be very close to the dominant point, and their visibility will be very similar anyway.\nIt is least accurate when two baselines are a similar distance from the grid-point, but in opposite directions. \nHowever, such a case will not occupy a large fraction of frequency space, and therefore its effect should be limited.\n\nAssumed in this setup is the fact that different baselines could be the dominant contributors at different frequencies. \nNeglecting this point results in the solutions of App. \\ref{app:delay}, which cannot avoid a wedge. \n\nThese assumptions lead to the frequency-space integral being split into a sum of terms containing a single baseline each, which is the closest to $\\vect{u}$ for that range of frequencies.\nThe frequency range for each term will be labeled $(f_i, f_{i+1})$, and the closest baseline will be labeled $\\vect{u}_i$.\n\n\\subsection{Variance at a grid point}\nUsing Eq. \\ref{eq:wg_master} as a starting point, we ignore the sum over baselines within the integral, and first perform the $\\vect{l}$-integral to achieve\n\\begin{align}\n\t{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 \\sum_{ij} &\\int_{f_i}^{f_{i+1}} \\int_{f_j}^{f_{j+1}} df df' \\phi_\\nu \\phi_{\\nu'} \\\\ \\nonumber\n\t& \\times e^{-2\\pi i \\omega (f-f')} e^{-2\\pi^2 \\sigma^2 (f\\vect{u}_i - f'\\vect{u}_j)^2}.\n\\end{align}\nWe now use a change of variables: $x = f- f'$ to get\n\\begin{align}\n{\\rm Var}(\\tilde{V}) = \\mu_2 \\nu_0^2 \\sum_{ij} &\\int_{f_i - f'_j}^{f'_i - f_j}dx\\ e^{-2\\pi i \\omega x} \\\\ \\nonumber\n& \\int_{f_i - x}^{f'_i - x} df' \\ e^{-\\tau^2 ((x+f'-1)^2 + (f'-1)^2)} \\\\ \\nonumber \n& \\times e^{-2\\pi^2 \\sigma^2 ((x+f')\\vect{u}_i - f'\\vect{u}_j)^2}.\n\\end{align}\nLetting\n\\begin{align}\n\tp_{ij}^2 &= 2\\tau^2 + \\pi^2 \\sigma^2 (\\vect{u}_i - \\vect{u}_j)^2 \\\\\n\tr_{ij} &= \\tau^2 + \\pi^2\\sigma^2 \\vect{u}_i (\\vect{u}_i - \\vect{u}_j),\n\\end{align}\nwe can perform the $f'$-integral ($I_i$) simply, resulting in\n\\begin{align}\n\tI^{(')}_{ij} = \\frac{\\sqrt{\\pi}}{2p} e^{r_{ij}^2 x^2\/p_{ij}^2} {\\rm erf}\\left[\\frac{r_{ij}-p_{ij}^2}{p}x + p_{ij}f^{(')}_i\\right]\n\\end{align}\nwhen evaluated at a given boundary.\nWe perform another change of variables:\n\\begin{align}\n\tz &= \\frac{r-p^2}{p}x + pf_i \\\\\n\tdx &= \\frac{p}{r-p^2} dz,\n\\end{align}\nto find the variance is\n\\begin{align}\n{\\rm Var}(\\tilde{V}) &= \\mu_2 \\nu_0^2 \\sum_{ij} \\mathbb{V}_{i'}^j - \\mathbb{V}_{i}^j, \\ \\ \\ {\\rm with} \\nonumber \\\\\n \\mathbb{V}_i^j &= \\frac{\\sqrt{\\pi}}{2(r-p^2)}\\int_{z_{ij}}^{z'_{ij}}dz\\ \\exp\\left[-2\\pi i \\omega\\left(\\frac{zp - p^2f_i}{r-p^2}\\right)\\right] \\nonumber \\\\\n & \\times \\exp\\left[-\\left(\\frac{zp - p^2f_i}{r-p^2}\\right)^2(\\tau^2 + \\pi^2\\sigma^2 u_i^2 - \\frac{r^2}{p^2})\\right] \\nonumber \\\\ &\\times \\exp\\left[ 2\\tau^2\\left(\\left(\\frac{zp-p^2f_i}{r-p^2}\\right) -1\\right)\\right] {\\rm erf}(z), \n\\end{align}\nnoting that the upper limit term $i'$ applies to all $f_i$ \\textit{within the integrand} and setting\n\\begin{align}\n\tz_{i,j} &= \\frac{r-p^2}{p}(f_i - f'_j) + pf_i, \\\\\n\tz'_{i,j} &= \\frac{r-p^2}{p}(f'_i - f_j) + pf_i.\n\\end{align}\n.\n\nWe now expand ${\\rm erf}(z)$ in its Maclaurin series, noting that this series always converges\\footnote{For large $z$, the number of terms required for convergence is high, however these terms should be adequately suppressed by other factors in the integral to make this a viable procedure.}.\nSetting the following variables:\n\\begin{align}\n\tt^2 &= \\tau^2 + \\pi^2 \\sigma^2 u_i^2 - r^2\/p^2 \\\\\n\ta^2 &= \\left(\\frac{pt}{r-p^2}\\right)^2 \\\\\n\tb_i &= \\frac{2p^3 f_i t^2}{(r-p^2)^2} + \\frac{2\\pi i \\omega p}{r-p^2} \\\\\n\tc_i &= 2\\tau^2 \\frac{p^2(1 - f_i) - r}{r-p^2} - \\frac{2\\pi i \\omega p^2 f_i}{r-p^2} - \\left(\\frac{p^2}{r-p^2}\\right)^2 f_i^2 t^2,\n\\end{align}\nwe have\n\\begin{align}\n\t\\mathbb{V}_i^j &= \\frac{1}{r-p^2} \\int_{z_0}^{z_1}dz\\ e^{-a^2 z^2 + b z + c } \\sum_{n=0}^\\infty \\frac{(-1)^n z^{2n+1}}{n!(2n+1)}.\n\\end{align}\n\nCompleting the square in the exponent, and shifting $z$ such that $z \\rightarrow z - b\/2a^2$,\nwe find\n\\begin{align}\n\\label{eq:full_variance_sblpf}\n\\mathbb{V}_i^j &= \\frac{e^{b^2\/4a^2 + c}}{r-p^2} \\sum_{n=0}^\\infty \\frac{(-1)^n}{n!(2n+1)} \\sum_{k=0}^{2n+1}\\binom{2n+1}{k} \\left(\\frac{b}{2a^2}\\right)^{2n+1-k} \\nonumber \\\\\n& \\times\\left[-\\frac{1}{2} z^{k+1} \\left(\\frac{1}{|az|}\\right)^{k+1} \\Gamma\\left(\\frac{k+1}{2}, a^2 z^2\\right)\\right|^{z'_{ij} - b\/2a^2}_{z_{ij} - b\/2a^2}.\n\\end{align}\nWe note that the solution is the sum of real parts of $\\mathbb{V}$, because the imaginary parts will cancel in the summation when swapping $i$ and $j$.\n\n\\subsection{Circular Average}\nThe solution of the previous subsection needs to be angularly averaged to yield the power at $u$. \nThe angular dependence enters both through the determination of contributing baselines, $\\vect{u}_i$, and the relative weight at each point.\n\nThe latter is computed as \n\\begin{equation}\n\tW(\\vect{u}) = \\int df \\phi_\\nu w_\\nu(\\vect{u} - f\\vect{u}_i).\n\\end{equation}\nAgain, we break the $f$ integral into independent sections, to yield\n\\begin{align}\n\t\tW(\\vect{u}) &= \\sum_{i} \\int_{f_i}^{f'_i} df \\phi_\\nu w_\\nu(\\vect{u} - f \\vect{u}_i) \\\\ \\nonumber\n\t\t&= \\sum_i \\frac{\\sqrt{\\pi}}{2p_i}\\exp\\left(-\\frac{2\\pi^2\\sigma^2\\tau^2d_i^2}{p_i^2}\\right) {\\rm erf}\\left[\\frac{p^2}{p_i^2} - f p_i \\right|^{f'_i}_{f_i},\n\\end{align}\nwith\n\\begin{align}\n\t\\vect{d} &= \\vect{u} - \\vect{u}_i \\\\\n\tp_i^2 &= \\tau^2 + 2\\pi^2 \\sigma^2 u_i^2.\n\\end{align}\nThe integration over $\\theta$ only occurs for each baseline as far as its contribution allows, and cannot be written fully analytically without reference to the layout of all baselines. \n\n\\subsection{Determination of contributing baselines}\nEvaluation of the power in this solution will require numerically summing the terms in the equations (though these sums should be significantly faster than performing a full 3D integration).\nTo accomplish this, precise integration limits must be derived for each term, along with the corresponding contributing baseline. \nThis can in principle be done both in frequency and angle, however we opt to limit ourselves to the frequency limits, as the angular limits do not offer a great deal in terms of computational performance.\n\nOur procedure for determination of the frequency limits in the general case is as follows:\n\\begin{enumerate}\n\t\\item Evaluate $W_i(u) = \\int d\\theta W_i(\\vect{u})$ for each baseline $\\vect{u}_i$ and retain only those whose contribution is greater than $10^{-t} \\bar{W}(u)$. \n\t\\item Determine at what frequency each pair of baselines is equidistant from $\\vect{u}$, saved as matrix $f^{ij}_{\\rm eq}$.\n\t\\item Determine the closest baseline to $\\vect{u}$ at $f = f_{\\rm min}$, with $f_{\\rm min}$ suitably low such that the bandpass\/taper renders it ignorable. Set $f_0 = f_{\\rm min}$ and $\\vect{u}_0$ to this baseline.\n\t\\item Until $f\\geq f_{\\rm max}$:\n\t\\begin{enumerate}\n\t\t\\item $j = {\\rm argmin}(f_{\\rm eq}^{ij} > f_{k-1})$\n\t\t\\item $f_k = f_{\\rm eq}^{ij}$\n\t\t\\item $\\vect{u}_k = \\vect{u}_j$\n\t\t\\item $i=j$\n\t\t\\item $k = k+1$\n\t\\end{enumerate}\n\\end{enumerate}\n\nThe equidistant frequencies are computed as \n\\begin{equation}\n\tf^{ij}_{\\rm eq} \\equiv \tf^{ji}_{\\rm eq} = \\left|\\frac{2(\\vect{u}\\cdot(\\vect{u}_i - \\vect{u}_j))}{u_i^2 - u_j^2}\\right|, \\ \\ i\\neq j.\n\\end{equation}\n\n\n\\section{Semi-analytic logarithmic spoke solution}\n\\label{app:logsolution}\nIn this appendix we derive a semi-analytic solution to the case of a logarithmic spoke layout in the high-$u$ limit. \nThis case is partially solved in \\S\\ref{sec:weighted}, for the limits in which the distances between baselines are either very large or very small. \nHowever, all realistic cases lie between these limits, and it is here that we focus in this subsection.\n\nThe logarithmic spoke layout ensures that the single-baseline-per-frequency limit is obtained for $u^2 \\gg 1\/\\pi^2\\sigma^2 \\Delta^2$, which exists for every array.\nThis is the limit explored in detail in App. \\ref{app:approx_variance}, and so we can use those results here.\n\nWe note further that at sufficiently high $u$, and for a sufficiently low number of radial spokes, we can ignore the angular component of the baselines and interpret Eq. \\ref{eq:full_variance_sblpf} directly as the power spectrum at $(u, \\omega)$.\n\nFor simplicity (and without sacrificing a great deal of accuracy), we will consider a point $u$ which is co-located with a baseline $u_i$ at $f=1$. \nFurthermore, we will consider an infinite array, such that the baselines grow arbitrarily small and large.\nDenoting the co-located baseline by the index 0, and larger baselines with negative indices (and vice versa), we can explicitly write the frequency limits within which a given baseline $i$ singularly contributes:\n\\begin{subequations}\n\t\\begin{align}\n\tf_i &= \\frac{1}{(1+\\Delta)^{i}(1+\\Delta\/2)}, \\\\\n\tf'_i &= \\frac{1}{(1+\\Delta)^{i-1}(1+\\Delta\/2)} \\equiv f_i(1+\\Delta).\n\t\\end{align}\n\\end{subequations}\n\nGiven that our primary point of focus is the wedge structure, we investigate the limit $\\pi\\sigma u \\gg \\tau$, which is where we expect the wedge to emerge.\nIn this case, several simplifications can be made. Firstly, we have \n\\begin{align}\n\tp^2 &\\approx \\pi^2\\sigma^2 u^2 \\left[(1+\\Delta)^i - (1+\\Delta)^j\\right] \\\\\n\tr &\\approx \\pi^2\\sigma^2 u^2 \\left[(1+\\Delta)^{2i} - (1+\\Delta)^{i+j}\\right] \\nonumber \\\\\n\t\\frac{r-p^2}{p} &\\approx \\pi \\sigma u \\frac{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}{(1+\\Delta)^i - (1+\\Delta)^j}.\n\\end{align}\n\nWe also have \n\\begin{align}\n\tt^2 &\\approx \\pi^2 \\sigma^2 u_i^2 \\\\\n\ta^2 &\\approx \\left(\\frac{u_i\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]}{u\\left[(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}\\right]}\\right)^2 \\\\\n\tb_i &\\approx 2\\frac{f_i t^2}{\\pi\\sigma u\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]} \\left(\\frac{(1+\\Delta)^i - (1+\\Delta)^j}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right)^2 \\nonumber \\\\\n\t& + \\frac{2 i \\omega}{\\sigma u} \\left(\\frac{(1+\\Delta)^i - (1+\\Delta)^j}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right) \\\\\n\tc_i &\\approx -2\\pi i \\omega f_i \\frac{\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]^2}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}} \\nonumber \\\\\n\t& - \\left[\\frac{\\left[(1+\\Delta)^i - (1+\\Delta)^j\\right]^2}{(1+\\Delta)^{2j} - (1+\\Delta)^{i+j}}\\right]^2 f_i^2 \\pi^2 \\sigma^2 u_i^2.\n\\end{align}\n\\fi\n\n\\section{Description of Numerical Techniques}\n\t\\label{app:numerical}\n\tThe gridding of visibility data and its transformation into an averaged power spectrum are non-trivial tasks that can require a significant computational effort.\n\tIn this appendix we describe the simple method we have taken in this work to accelerate this process and ensure its accuracy.\n\t\n\tNaively, the application of Gaussian weights from each baseline $\\vect{u}'_j$ to a particular point of interest $\\vect{u}_i$ is an order $N\\times M$ calculation (where $N$ is the number of baselines, and $M$ the number of grid-points at which to evaluate the power spectrum).\n\tSeveral standard algorithms can reduce this calculation to of order $N\\log M$.\n\tThe most popular is to use an FFT-backed convolution.\n\tHowever, we do not choose this route, as it requires the $\\vect{u}_i$ to be arranged on a regular Cartesian grid, which has its own difficulties in terms of angular averages and dynamic range.\n\t\n\tInstead, we use a KD-tree algorithm (from the \\textsc{scikit-learn} Python package) to efficiently determine the baselines within a given radius of every $\\vect{u}_i$, and apply the weights\n\tfrom only these baselines. \n\tThe radius can be arbitrarily set, based on the beam width.\n\tThis allows the $\\vect{u}_i$ to be placed arbitrarily. \n\tSince we require an angular average, it is most convenient to choose the $\\vect{u}_i$ in a polar grid, so that the angular average is merely the average of a particular row in the array. \n\tThis has the dual benefits of simplicity and accuracy -- the average is specified at a particular magnitude of $q$, rather than an average over a complicated distribution of $q$ within an annulus. \n\t\n\tThis algorithm enables the numerical calculation of the 2D PS as an arbitrarily precise quantity.\n\tThat is, if the number of nodes in an angular ring is arbitrarily large, the operation exactly converges to the integral Eq. \\ref{eq:power_general}.\n\tIn practice then, if one simultaneously tests for convergence, this algorithm provides an exact non-gridding solution to the numerical calculation of the 2D PS.\n\tIn this paper we do not formally test for convergence, but rather simply use a number of angular nodes we deem to be sufficient to capture the integral adequately.\n\tIn real-world applications, the extension to formal convergence-monitoring is rather simple, and may provide for quite efficient accurate calculations of the 2D PS.\n\t\n\\end{appendix}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRadio hotspots are bright and compact regions located at the end of\npowerful radio galaxies \n\\citep[FRIIs,][]{fr2} and\nconsidered to be the working surfaces of supersonic jets. In these\nregions,\nthe jet emitted by the active galactic nucleus\n(AGN) impacts on the surrounding ambient medium producing a shock that\nmay re-accelerate relativistic particles transported by the jet and\nenhance the radio emission. \nElectrons responsible for synchrotron emission in the\noptical band must be very energetic (Lorentz factor $\\gamma >\n10^{5}$),\nand therefore with short radiative lifetime.\nConsequently the detection of optical emission from hotspots\nsupports the scenario where the emitting electrons are accelerated\nat the hotspots, possibly by strong shocks generated by the impact of \nthe jet with the ambient medium \\citep{meise89,meise97,gb03}. \nThe detection of X-ray synchrotron counterparts of radio hotspots\n would imply the\n presence of electrons with even higher energies. \n However the main radiation process\n responsible for the X-ray emission seems to differ between high and\n low luminosity hotspots \\citep{hardcastle04}. \n In bright hotspots, like Cygnus A and 3C\\,295, the X-ray\n emission is produced by synchrotron-self Compton (SSC) in the presence\n of a magnetic field that is roughly in equipartition, while in \nlow-luminosity hotspots, like 3C\\,390.3, the emission at such high\n energies is likely due to\n synchrotron radiation \\citep{hardcastle07}.\\\\\nThe discovery of optical emission extended to kpc scale \nquestions the standard shock acceleration model, suggesting that \nother efficient\nmechanisms must take place across the hotspot region.\nAlthough it may seem an uncommon phenomenon due to the\ndifficulty to produce high-energy electrons on large scales, deep\noptical images showed that diffuse\noptical emission is present in a handful of hotspots: 3C\\,33, \n3C\\,111, 3C\\,303, 3C\\,351\n\\citep{valta99}, 3C\\,390.3 \\citep{aprieto97}, \n3C\\,275.1 \\citep{cheung05}, Pictor A \\citep{thomson95}, \nand 3C\\,445 \\citep{aprieto02}. \nA possible mechanism able to keep up the optical emission in the post-shock\nregion on kpc scale is a continuous, relatively efficient, stochastic \nmechanism\\footnote{More recently these stochastic\nmechanisms have been also proposed for\nthe acceleration of ultra-high energy cosmic-rays in the lobes of\nradiogalaxies \\citep{hardcastle09}.}.\\\\\nThe sample of low-power hotspots presented by \\citet{mack09} is\ncharacterized by low magnetic field strengths between 40 and 130\n$\\mu$G, a factor 2\nto 5 lower than that estimated in hotspots with optical counterparts\npreviously studied in the literature. A surprisingly high optical\ndetection rate ($\\geq$ 45\\%)\nof the hotspots in this sample was found, and in most cases \nthe optical counterpart extends on kpc scales. This\nis the case of 3C\\,445 South, 3C\\, 445 North, 3C\\,105 South and\n3C\\,227 West \\citep{mack09}.\\\\\nThis\npaper focuses on a multi-band, from radio to X-rays, \nhigh spatial resolution study of the two\nmost interesting cases among the low-luminosity hotspots from\n \\citet{mack09}, 3C\\,105 South and 3C\\,445 South, in which the\nhotspot regions are resolved into multiple components. \n3C\\,105 is hosted by a narrow-line radio galaxy (NLRG) at redshift\n $z=0.089$ \\citep{tadhunter93}. At this redshift 1$^{\\prime\\prime}$\n corresponds to 1.642 kpc. The radio source 3C\\,105 is about\n 330$^{\\prime\\prime}$ (542 kpc) in size, and the hotspot complex 3C\\,105 South is\n located about 168$^{\\prime\\prime}$ (276 kpc) from the core in the\n south-east direction. 3C\\,445 is hosted by a broad-line radio galaxy\n (BLRG) \nat redshift $z=0.05623$ \\citep{eracleous94}. At this redshift\n1$^{\\prime\\prime}$ corresponds to 1.077 kpc. The radio source 3C\\,445\nis about 562$^{\\prime\\prime}$ (608 kpc) in size, and the hotspot\ncomplex 3C\\,445 South is located 270$^{\\prime\\prime}$ (291 kpc)\nsouth of the core.\n\nThroughout this paper, we assume the following cosmology: $H_{0} =\n71\\; {\\rm km\/s\\, Mpc^{-1}}$, \n$\\Omega_{\\rm M} = 0.27$ and $\\Omega_{\\rm \\Lambda} = 0.73$,\nin a flat Universe. The spectral index\nis defined as \n$S {\\rm (\\nu)} \\propto \\nu^{- \\alpha}$.\\\\ \n\n\n\n\\section{Observations}\n\n\\subsection{Radio observations}\n\nVLA observations at 1.4, 4.8, and 8.4 GHz \nof the radio hotspots 3C\\,445 South and\n3C\\,105 South were carried out in July 2003 (project code AM772)\nwith the array in\nA-configuration. Each source was observed for about half an\nhour at each frequency, spread into a number of scans\ninterspersed with other source\/calibrator scans in order to improve\nthe $uv$-coverage. About 4 minutes were spent on the\nprimary calibrator 3C\\,286, while secondary phase calibrators\nwere observed for 1.5 min about every\n5 min. Data at 1.4 and 4.8 GHz were previously published by \\citet{mack09}.\nThe data reduction was carried out following the standard procedures\nfor the VLA implemented in the NRAO AIPS package.\nFinal images were produced after a few phase-only self-calibration\niterations. The r.m.s. noise level on the image plane is negligible if compared\nto the uncertainty of the flux density due to amplitude calibration\nerrors that, in this case, are estimated to be $\\sim$3\\%.\\\\\nBesides the {\\it full-resolution} images, we also produced {\\it\n low-resolution} images at both 4.8 and 8.4 GHz, \nusing the same $uv$-range, image sampling and restoring beam of the\n1.4 GHz data. These new images were obtained with natural grid \nweighting in order to mitigate the differences in the sampling density \nat short spacing, and to perform a robust spectral analysis. \\\\\n\n\n\n\n\\subsection{Optical observations}\n\nFor both 3C\\,105 South and 3C\\,445 South, VLT high spatial resolution\nimages in standard filters taken with both ISAAC in J-, H-, K-, and FORS\nin I-, R-, B- and U- bands are used in this work. All the images have\nexcellent spatial resolutions in the range of 0.5$^{\\prime\\prime}$ $<$ FWHM $<$\n0.7$^{\\prime\\prime}$. Details on the observations and data reduction are given in\n\\citet{mack09}. The pixel scale of the ISAAC images is 0.14 arcsec\npixel$^{-1}$. In the case of the FORS images the pixel scale is 0.2\narcsec pixel$^{-1}$, with the exception of the I-band where it is\n 0.1 arcsec pixel$^{-1}$.\\\\ \nFurther HST observations on 3C\\,445 South only, were obtained with\nthe ACS\/HRC camera on 7th July, 2005 \nin the filters F814W (I-band, exposure time $\\sim$ 1.5 hr) and\nF475W (B-band, exposure time $\\sim$ 2.3 hr).\\\\ \nFor science analysis we used the ``*drz'' images delivered \nby the HST ACS pipeline. These final images are calibrated, \ncosmic-ray cleaned, geometrically corrected, and drizzle-combined, \nprovided in electrons per sec. The final pixel scale of the \ndrizzled images is 0.025$^{\\prime\\prime}$$\\times$0.025$^{\\prime\\prime}$ per pixel. \nThe flux calibration was done using the standard HST\/ACS procedure\nthat relies on the PHOTFLAM keyword in the respective image headers.\nThe quality of the pipeline-delivered images was adequate for the\npurposes of analyzing the hotspot region. \n\n\n\\begin{table*}\n\\caption{Radio flux density and angular size of the hotspot\n components. \\newline {\\it Note 1}: deconvolved angular sizes from a Gaussian\n fit. \\newline {\\it Note 2}:\nthe angular sizes are derived from the lowest contour on the image\nplane; \\newline\n{\\it Note 3}: The diffuse emission is estimated by subtracting\nthe flux density of SW and SE from the total flux density (see Section\n5.3).}\n\\begin{center}\n\\begin{tabular}{|c|r|r||r|r|r|r|r|r|}\n\\hline\nSource&Comp.&z&scale&S$_{1.4}$&S$_{4.8}$&S$_{8.4}$&$\\theta_{\\rm maj}$&$\\theta_{\\rm min}$\\\\\n & & &kpc\/$^{\\prime\\prime}$ \n&mJy&mJy&mJy&arcsec&arcsec\\\\\n\\hline\n&&&&&&&&\\\\\n3C\\,105&S1\\footnotemark[1]&0.089&1.642&130$\\pm$10&67$\\pm$5&45$\\pm$5&1.0&0.8\\\\\n &S2\\footnotemark[1]& & &1250$\\pm$40&620$\\pm$20&460$\\pm$15&1.30&1.0\\\\\n &S3\\footnotemark[1]& & &1180$\\pm$35&510$\\pm$15&320$\\pm$12&1.5&0.8\\\\\n &Ext& & &174$\\pm$10&75$\\pm$5&50$\\pm$3& & \\\\\n3C\\,445&SE\\footnotemark[2]&0.0562&1.077&290$\\pm$30&98$\\pm$15&65$\\pm$10&3.5&1.0\\\\\n &SW\\footnotemark[2]& & &220$\\pm$25&51$\\pm$10&36$\\pm$6&1.5&0.5\\\\\n &Diff\\footnotemark[3]& & & & &13.0$\\pm$1.1& & \\\\\n&&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\\label{tab_flux_rad}\n\\end{table*}\n\n\\subsection{X-ray observations}\n\n\nThe radio source \n3C\\,105 was observed by {\\it Chandra} on 2007 December 17 (Obs ID 9299)\nduring ``The {\\it Chandra} 3C Snapshot Survey for Sources with z$<$0.3''\n\\citep{massaro10}. \nAn $\\sim$8 ksec \nexposure was obtained with the ACIS-S camera, operating in \nVERY FAINT mode.\nThe data analysis was performed following the standard\nprocedures described in the {\\it Chandra} Interactive Analysis of\nObservations (CIAO) threads and using the CIAO software package v4.2 \n(see Massaro et al. 2009 for more details). The {\\it Chandra} Calibration\nDatabase (CALDB) version 4.2.2 was used to process all files. \nLevel 2 event files were generated using the $acis\\_process\\_events$ task,\nafter removing the hot pixels with $acis\\_run\\_hotpix$. Events were\nfiltered for grades 0,2,3,4,6, and we removed pixel randomization.\\\\\n3C\\,445 South was observed by {\\it Chandra} on 2007 October 18\n\\citep{perlman10}, ACIS chip S3, with an exposure time of 45.6 ksec. \nThe data were retrieved from the archive and\nanalysed following the same procedure as for 3C\\,105 South. This\nre-analysis was necessary in order to achieve a proper alignment with\nthe radio data. \\\\\nWe created 3 different flux maps in the soft, medium, and hard X-ray bands\n(0.5 -- 1, 1 -- 2, and 2 -- 7 keV, respectively) by dividing the data with\nmonochromatic exposure maps with nominal energies = 0.8 keV (soft),\n1.4 keV (medium), and 4 keV (hard).\nBoth the exposure maps and the flux maps were regridded to a \npixel size of 0.25 the size of a native ACIS pixel\n(native=0.492$^{\\prime\\prime}\\times0.492^{\\prime\\prime}$). To obtain\nmaps with brightness units of ergs~cm$^{-2}$~s$^{-1}$~pixel$^{-1}$, we\nmultiplied each event by the nominal energy of its respective band.\\\\\nFor 3C\\,445 South, we measured a flux density consistent with what reported by\n\\citet{perlman10}. The flux density was extracted from {\\it Chandra}\nACIS-S images in which the hotspot was placed on axis. \nBoth hotspots have been detected also by {\\it Swift} in the energy range\n0.3-10 keV (See Appendix A). This is remarkable given {\\it\n Swift}'s survey operation mode and its poor spatial resolution. The\ndetection level is about 7$\\sigma$ and 12$\\sigma$ for 3C\\,105 South\nand 3C\\,445 South, respectively. However, given the large {\\it Swift}\nerrors in the counts-to-flux conversion and its low angular\nresolution, \nwe do not provide any further\nflux estimate.\\\\\n\n\\subsection{Image registration}\n\nThe alignment between radio and optical images was done by the\nsuperposition of the host galaxies with the nuclear component of the\nradio source using the AIPS task LGEOM. This results in a shift of\n3.5$^{\\prime\\prime}$. \nTo this purpose, the optical images\nwere previously brought on the same grid, orientation and coordinate system \nas the radio images by\nmeans of the AIPS task CONV and REGR \\citep[see\n also][]{mack09}. \nThe final overlay of radio and optical images is\naccurate to 0.1$^{\\prime\\prime}$.\\\\\nFor 3C\\,105 South the X-ray image has been aligned with the radio one\nby comparing the core position. Then, the final\noverlay of X-ray contours on the VLT image \nis accurate to 0.1$^{\\prime\\prime}$. In the case of 3C\\,445 the\nshape of the nucleus of the galaxy is badly distorted in the {\\it Chandra} image\nbecause of its location far off axis of {\\it Chandra}.\nThe alignment was then\nperformed using three background sources visible both in X-ray and B\nband, and located around the hotspot. The achieved accuracy with \nthis registration is better than 0.15 arcsec, allowing us to confirm \na shift of about 2$^{\\prime\\prime}$ in declination between the X-rays and\nB-band emission centroids, the X-ray one being the closest to the core\n(Fig. \\ref{fig_3c445}).\\\\ \n\n\n\n\\begin{table*}\n\\caption{Near infrared, optical flux density and X-ray (0.5 - 7 keV) \nflux of hotspot components. In the case of 3C\\,445 the X-ray flux \nis not associated to any of the two\nmain components. The X-ray flux reported refers the total emission measured\non the whole hotspot region. \\newline {\\it Note 1}: units in\n10$^{-15}$ erg cm$^{-2}$ s$^{-1}$; \\newline\n{\\it Note 2}: the X-ray value, in $\\mu$Jy, is from\n\\citet{perlman10}; \\newline\n{\\it Note 3}: The diffuse emission is inclusive of the SC\ncomponent and it is estimated by subtracting from\nthe total flux density those arising from SW and SE (see Section 5.3).} \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nSource&Comp.&S$_{\\rm K}$&S$_{\\rm\n H}$&S$_{\\rm J}$&S$_{\\rm I}$&S$_{\\rm R}$&S$_{\\rm B}$&S$_{\\rm\n U}$&S$_{\\rm I}^{\\rm HST}$&S$_{\\rm B}^{\\rm HST}$&S$_{X}$\\\\\n & &$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&$\\mu$Jy&\\\\\n\\hline\n&&&&&&&&&&&\\\\\n3C\\,105&S1&4.6$\\pm$0.9&4.4$\\pm$1.1&$<$2.5&- &0.5$\\pm$0.1\n&0.2$\\pm$0.1&-&-&-& 7.5$\\pm$2.4\\footnotemark[1]\\\\\n &S2&18.4$\\pm$1.4&12.3$\\pm$1.1&3.4$\\pm$1.0&-&0.7$\\pm$0.1&0.2$\\pm$0.1&-&-&-&\n $<$2.0\\footnotemark[1]\\\\\n &S3&31.9$\\pm$2.8&25.7$\\pm$2.9&4.4$\\pm$1.8&-&0.9$\\pm$0.1&0.3$\\pm$0.1&-&-&-&\n 3.2$\\pm$1.6\\footnotemark[1]\\\\\n &Ext&15.4$\\pm$2.0&5.4$\\pm$2.0&-& -&0.4$\\pm$0.1&0.2$\\pm$0.1&-&-&-& -\\\\\n3C\\,445&SE&8.0$\\pm$1.0&5.6$\\pm$2.0&6.0$\\pm$1.5&2.0$\\pm$0.2&1.3$\\pm$0.2&0.7$\\pm$0.1&0.5$\\pm$0.3&1.7$\\pm$0.2&1.5$\\pm$0.3&9.38$\\times$10$^{-4}$\n\\footnotemark[2]\\\\\n &SW&4.6$\\pm$1.4&3.6$\\pm$1.5&3.0$\\pm$0.4&1.7$\\pm$0.3&1.4$\\pm$0.1&0.7$\\pm$0.1&\n0.5$\\pm$0.2&1.4$\\pm$0.1&0.3$\\pm$0.1&-\\\\\n &SC&- &- &- & - &0.8$\\pm$0.1&0.6$\\pm$0.1&0.4$\\pm$0.1&- & - & -\\\\\n &Diff\\footnotemark[3]& -\n&2.1$\\pm$0.6&3.2$\\pm$1.3&1.2$\\pm$0.2&1.0$\\pm$0.2&0.8$\\pm$0.2& - \\\\\n&&&&&&&&&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center} \n\\label{tab_flux_opt}\n\\end{table*}\n\n\n\\section{Photometry}\n\nTo construct the spectral energy distribution (SED) of individual hotspot\ncomponents, the flux density at the various wavelengths must be\naccurately measured in the same region, avoiding \ncontamination from unrelated features. \nTo this purpose, we produced a cube where each plane consists of radio\nand optical images regridded to the same size and smoothed to the same\nresolution. Then the flux\ndensity was derived by means of AIPS task BLSUM which performs an\naperture integration on a selected polygonal region common to all the\nimages. The values derived in this way were then used to construct the\nradio-to-optical SED, and they are reported in\nTables 1 and 2.\\\\\nIn addition to the low-resolution approach, \nwe derive the hotspot flux densities and\nangular sizes on the full resolution images, in order to better\ndescribe the source morphology.\n\nOn the radio images, we estimate the flux density of each component \nby means of TVSTAT, which is similar to BLSUM, but instead of working\non an image cube it works on a single image. The angular size was\nderived from the lowest contour on the image plane, and it\ncorresponds to roughly twice the size of the full width half maximum\n(FWHM) of a conventional Gaussian covering a similar area. \nIn the case of 3C\\,105 South, the hotspot components are unresolved\nat 1.4 GHz, and we derive the flux density at this frequency by means of\nAIPS task JMFIT, which performs a Gaussian fit in the image plane. \nThe angular size was measured on the images in\nwhich the components were resolved, i.e. in the case of 3C\\,105\nSouth we use the 4.8 and 8.4-GHz images, which provide the same value, while\nfor 3C\\,445 South the components could be reliably resolved in the\nimage at 8.4 GHz only (Table 1).\\\\\nFull-resolution infrared and optical flux densities of hotspot sub-components \nwere measured by means of the IDL-based task ATV using \na circular aperture centred on each component.\nSuch values were compared to those derived from the analysis of the\ncube and they were found to be within the expected uncertainties.\\\\\nFor the X-ray flux we constructed photometric apertures \nto accommodate the {\\it Chandra}\npoint spread function and to include the total extent of the\nradio structures.\nThe background regions, with a total area typically twice that of the\nsource region, have been selected close to the source, and\ncentred on a position where other sources or extended structures are\nnot present. The X-ray flux was measured\nin any aperture with only a small correction for the\nratio of the mean energy of the counts within the aperture to the\nnominal energy for the band. \nWe note that in 3C\\,105 South,\nthe hotspot components \nare well separated (2$^{\\prime\\prime}$), allowing us to accurately\nisolate the corresponding X-ray emission. In\n3C\\,445 South the X-ray emission is not associated with the two main\ncomponents clearly visible in the radio and optical bands, and \nflux was derived by using an aperture large enough to include all of the X-ray\nemission extending over the entire hotspot region. Our estimated value\nis in agreement with the one reported by \\citet{perlman10}.\nAll X-ray flux densities have been corrected for the\nGalactic absorption with the column density N$_H$ =\n1.15$\\cdot$10$^{21}$cm$^{-2}$ given by\n\\citet{kalberla05}. \nX-ray fluxes are reported in Table \\ref{tab_flux_opt}.\\\\ \n\n\\begin{figure*}\n\\begin{center}\n\\special{psfile=9pan_label1.ps voffset=-600 hoffset=-40\n vscale=100 hscale=100 angle=0}\n\\vspace{16cm}\n\\caption{Multifrequency images of 3C\\,105 South. From the left to\n right and top to bottom: Radio images at 1.4, 4.8, 8.4 GHz (VLA\n A-array), NIR\/optical images in K, H, J, R, B bands (VLT), and X-ray 0.5-7\n keV ({\\it Chandra}) contours. Each panel covers 9.5$^{\\prime\\prime}$\n (15.6 kpc) in DEC and 14$^{\\prime\\prime}$ (23 kpc) in RA. In the\n radio images the lowest contours are 0.9 mJy\/beam at 1.4 GHz, 0.20\n mJy\/beam at 4.8 GHz, and 0.18 mJy\/beam at 8.4 GHz, and they\n correspond to 3 times the off-source rms noise level measured on the\nimage plane. Contours increase by a factor of 4. The restoring beam is\n1.3$^{\\prime\\prime}$$\\times$1.1$^{\\prime\\prime}$ at 1.4 GHz,\n0.38$^{\\prime\\prime}$$\\times$0.36$^{\\prime\\prime}$ at 4.8 GHz, and \n0.32$^{\\prime\\prime}$$\\times$0.22$^{\\prime\\prime}$ at 8.4 GHz. In the\noptical images the contour levels are in arbitrary units and increase\nby a factor of 2. The FWHM is about 0.4$^{\\prime\\prime}$,\n0.5$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$, 0.6$^{\\prime\\prime}$,\n0.7$^{\\prime\\prime}$ in K, H, J, R, and B band respectively.\nThe X-ray contours were generated from an 0.5-7 keV image, smoothed\nwith a Gaussian of FWHM=0.72$^{\\prime\\prime}$. Contour levels increase\nlinearly: 0.02, 0.04, 0.06,.. 0.14 counts per 0.123$^{\\prime\\prime}$ pixel. \nThe X-ray contours\nare superposed to the R band image, previously shifted as so to\nalign with X-ray.} \n\\label{fig_3c105}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\special{psfile=13pan445label_1.ps voffset=-350 hoffset=0\n vscale=90 hscale=90}\n\\vspace{15cm}\n\\caption{Multifrequency images of 3C\\,445 South. From the left to\n right and top to bottom: Radio images at 1.4, 4.8, 8.4 GHz (VLA\n A-array), NIR\/optical images in K, H, J, I, R, B, U bands (VLT),\n optical images in I and U bands (HST), \nand X-ray 0.5-7\n keV ({\\it Chandra}) contours. Each panel covers 7.3$^{\\prime\\prime}$\n (7.8 kpc) in DEC and 11.4$^{\\prime\\prime}$ (12.2 kpc) in RA. In the\n radio images the lowest contours are 1.3 mJy\/beam at 1.4 GHz, 0.20\n mJy\/beam at 4.8 GHz, and 0.10 mJy\/beam at 8.4 GHz, and they\n correspond to 3 times the off-source rms noise level measured on the\nimage plane. Contours increase by a factor of 4. The restoring beam is\n1.43$^{\\prime\\prime}$$\\times$0.96$^{\\prime\\prime}$ at 1.4 GHz,\n0.45$^{\\prime\\prime}$$\\times$0.37$^{\\prime\\prime}$ at 4.8 GHz, and \n0.24$^{\\prime\\prime}$$\\times$0.21$^{\\prime\\prime}$ at 8.4 GHz. In the\noptical images the contour levels are in arbitrary units and increase\nby a factor of 2. The VLT FWHM are 0.7$^{\\prime\\prime}$,\n0.6$^{\\prime\\prime}$, 0.5$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$,\n0.6$^{\\prime\\prime}$, 0.6$^{\\prime\\prime}$, 0.7$^{\\prime\\prime}$,\nin K, H, J, I, R, B, and U band respectively. In HST images\neach pixel is 0.025$^{\\prime\\prime}$.\nThe X-ray contours in the last panel are superposed on the B band\nimage. They come from an 0.5-7 keV image, smoothed with a Gaussian of\nFWHM=0.87$^{\\prime\\prime}$. Contour levels increase by a factor of 2;\nthe lowest contour is at a brightness of 0.01 counts per\n0.0615$^{\\prime\\prime}$ pixel.} \n\\label{fig_3c445}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Morphology}\n\n\\subsection{3C\\,105 South}\n\nThe southern hotspot complex of 3C\\,105 shows a curved\nstructure of about 8$^{\\prime\\prime}$$\\times$4.5$^{\\prime\\prime}$ \n($\\sim$13$\\times$7 kpc) in\nsize. It is dominated by three bright components, all resolved at\nradio frequencies,\nconnected by a low surface brightness emission also visible in\noptical and infrared (Fig. \\ref{fig_3c105}). \nThe central component, labeled S2 in Fig. \\ref{fig_3c105}, is the\nbrightest in radio and, when imaged with high spatial resolution, it\nis resolved in two different structures separated by about 1.2 kpc. \n\\citet{leahy97} interpreted \nthis as the true jet termination hotspot, while S1, with an elongated\nstructure of (1.6$\\times$1.3) kpc and located 5.7 kpc\nto the north of S2 is considered as jet emission. The southernmost\ncomponent S3,\nlocated about 4.1 kpc from S2, has a resolved structure of\n(2.4$\\times$1.3) kpc in size, and it is elongated in a direction\nperpendicular to the line leading to S2. Its morphology suggests that\nS3 is a secondary hotspot similar to 3C\\,20 East \\citep{cox91}.\\\\ \nAt 1.4 GHz, \nan extended tail\naccounting for $S_{\\rm 1.4} =$ 608 mJy \nand embedding the jet\nis present to the west of the hotspot complex, in agreement with the\nstructure previously found by \\citet{neff95}. \nAt higher frequencies the lack of the short spacings prevents\n the detection of such an extended structure, and only a hint of the\njet,\naccounting for $S_{\\rm 4.8} \\sim 70$ mJy, is still\nvisible at 4.8 GHz. \\\\\nIn the optical and NIR the hotspot complex is characterized by\nthe three main components detected in radio. In NIR and optical, \nthe southernmost component S3\nis the brightest one, with a radio-to-optical spectral index\n $\\alpha_{r-o}=$0.95$\\pm$0.10. It displays an elongated\nstructure rather similar in shape and size to that \nfound in radio. It is resolved in all bands with the only exception of \nB band, likely due to the lower spatial resolution \nachieved. Component S1 is resolved in all NIR\/optical\nbands, showing a tail extending towards S2. Its radio-to-optical\nspectral index is $\\alpha_{r-o}=$0.95$\\pm$0.10.\nOn the other hand, S2 appears unresolved in all bands, with the\nexception of K and H bands, i.e. those with the highest resolution\nachieved. In these NIR bands S2 is extended in the southern direction, \nresembling what is observed in radio. Its radio-to-optical\nspectral index is $\\alpha_{r-o}=$1.05$\\pm$0.10.\\\\\nDiffuse emission connecting the main hotspot\ncomponents and extending to the southwestern part of the hotspot\ncomplex is\ndetected in most of the NIR and optical images.\\\\\nIn the X-ray band, \nS1 is the brightest component, whereas the emission from \nS3 is very weak (formally detected at only 2$\\sigma$ level).\nFor this reason in the following we will use the\nnominal X-ray flux of S3 as a conservative upper limit. \nFor component S2 only an upper limit could be set. \n\n\n\\subsection{3C\\,445 South}\n\nThe hotspot 3C\\,445 South displays an extended east-west structure of\nabout 9.3$^{\\prime\\prime}$ $\\times$ 2.8$^{\\prime\\prime}$\n(10$\\times$3 kpc) in size in radio\n(Fig. \\ref{fig_3c445}). At 8.4 GHz, the hotspot complex is almost\ncompletely resolved out\nand the two main components, clearly visible in\nNIR\/optical images, are hardly distinguishable.\nWhen imaged with enough resolution, these components display an\narc-shaped structure both in radio and NIR\/optical bands,\nwith sizes of about (3.4$\\times$1.5) kpc and\n(2.1$\\times$1.1) kpc for SE and SW respectively. \nComponent SE is elongated in a direction almost perpendicular to the\nline leading to the source core, while SW forms an angle of about\n-20$^{\\circ}$ with the same line.\\\\ \nIn radio and NIR, the \nSE component is the brightest one, with a flux density ratio\nSE\/SW $\\sim$ 1.6, while in the optical both\ncomponents have similar flux densities. Both components have a\nradio-to-optical spectral index $\\alpha_{r-o}=$ 0.9$\\pm$0.10.\nIn the optical R-, B-, and U-band \nimages a third component (labelled SC in Fig. \\ref{fig_3c445}) \naligned with the jet direction becomes visible between SE\nand SW. \nDespite the good resolution and sensitivity of the radio and \nNIR images, SC is not present at such wavelengths.\nWhen imaged with the high\nresolution provided by HST, both SE and SW are clearly resolved, and\nno compact regions can be identified in the hotspot complex. Trace of\nthe SC component is seen in the B-band, in agreement\nwith the VLT images.\\\\\nIn the VLA and VLT images, \nthe two main components are enshrouded by a diffuse emission, visible\nin radio and NIR\/optical bands. \nThe flux densities of the SE and SW components measured on the HST images\nare consistent (within the errors) with those derived on the VLT images.\\\\ \nThe optical component W located about 2.8$^{\\prime\\prime}$ (3 kpc) \non the northwestern part of\nSW does not have a radio counterpart, as it is clearly shown by\n the superposition of I-band HST and 8.4-GHz VLA images\n (Fig. \\ref{hst_vla}), and thus it is considered an\nunrelated object, like a background galaxy. Another possibility\n is that this is a synchrotron emitting region where the impact of the\n jet produces very efficient particle acceleration. However, its steep\n optical spectrum ($\\alpha \\sim 2$ between I and U\n bands, see Section 5.3, Fig. \\ref{slope_3c445}) together with the\n absence of detected radio emission disfavour this\n possibility. Future spectroscopic information would further unveil \n the nature of this optical region.\\\\\n{\\it Chandra} observations of 3C\\,445 South detected X-ray emission from a\nregion that extends over 6$^{\\prime\\prime}$ in the east-west direction\n(Fig. \\ref{fig_3c445}), and \nit peaks almost in the middle of the hotspot structure, suggesting a\nspatial displacement \nbetween X-ray and radio\/NIR\/optical emission \\citep{perlman10}.\\\\\n\\begin{figure}\n\\begin{center}\n\\special{psfile=hst_i-radio.eps voffset=-245 hoffset=0 vscale=33\n hscale=33}\n\\vspace{8cm}\n\\caption{3C\\,445 South. 8.4-GHz VLA contours are superimposed on the I-band\nHST image.} \n\\label{hst_vla}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105n_units_last.ps voffset=-230 hoffset=10 vscale=33 hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the northern component, S1, of \n3C\\,105 South. The solid lines represent the synchrotron model where\n$\\nu_{\\rm b} =5 \\times 10^{12}$ Hz and $\\nu_{\\rm c} = 2 \\times\n10^{15}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 150 $\\mu$G. The\nshort-dashed line represent a synchrotron model where $\\nu_{\\rm b} = 5\n\\times 10^{12}$ Hz, and $\\nu_{\\rm c} = \\infty$.\nThe long-dashed lines represent the IC-CMB models\ncomputed assuming B=16 (and B=32) $\\mu$G,\n$\\Gamma$=6 ($\\Gamma=4$), $\\theta$=0.1 ($\\theta=0.2$) rad,\nwith or without flattening in the observed synchrotron spectrum\nat $\\nu<$ 60 MHz. The magnetic field is in the rest frame.}\n\\label{fig_spectra_105n}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105c_units_last.ps voffset=-230 hoffset=10 vscale=33\n hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the central component, S2, of \n3C\\,105 South. The solid line represents the synchrotron model where\n$\\nu_{\\rm b} =7.5 \\times 10^{12}$ Hz and $\\nu_{\\rm c} = 3 \\times\n10^{14}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 225 $\\mu$G. The\narrow indicates the X-ray upper limit.}\n\\label{fig_spectra_105c}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c105s_units_last.ps voffset=-230 hoffset=10 vscale=33\n hscale=33}\n\\vspace{6.5cm}\n\\caption{The broad-band SED of the southern component, S3, of \n3C\\,105 South. The solid line represents the synchrotron model where\n$\\nu_{\\rm b} =1.5 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 3 \\times\n10^{14}$ Hz, and the SSC\nmodels computed assuming a magnetic field of 50 and 150 $\\mu$G. The\narrow indicates the X-ray upper limit.}\n\\label{fig_spectra_105s}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=Figura_445sw_new.ps voffset=-240 hoffset=10 vscale=33\n hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the western component, SW, of \n3C\\,445 South. The morphology from {\\it Chandra} image shows that\nX-rays are not associated with the western component.\nThe synchrotron models assume \n$\\nu_{\\rm b} =9.4 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 4.7 \\times\n10^{15}$ Hz ({\\it dotted line}), $\\nu_{\\rm b} =5.5 \\times 10^{13}$ Hz \nand $\\nu_{\\rm c} = 2.2 \\times10^{16}$ Hz ({\\it dashed line}), \n$\\nu_{\\rm b} =4.4 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 1.8 \\times\n10^{18}$ Hz ({\\it solid line}), $\\nu_{\\rm b} =4.4 \\times 10^{13}$ Hz\nand $\\nu_{\\rm c} = \\infty$ ({\\it thick solid line}). }\n\\label{fig_spectra_445w}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=Figura_445se_new.ps voffset=-240 hoffset=10 vscale=33\n hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the eastern component, SE, of \n3C\\,445 South. The morphology from {\\it Chandra} image shows that\nX-rays are not associated with the eastern component.\nThe synchrotron models assume \n$\\nu_{\\rm b} =5.2 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 2.6 \\times\n10^{15}$ Hz ({\\it dotted line}), $\\nu_{\\rm b} =2.4 \\times 10^{13}$ Hz \nand $\\nu_{\\rm c} = 9.4 \\times10^{15}$ Hz ({\\it dashed line}), \n$\\nu_{\\rm b} =1.2 \\times 10^{13}$ Hz and $\\nu_{\\rm c} = 4.7 \\times\n10^{17}$ Hz ({\\it solid line}), $\\nu_{\\rm b} =1.2 \\times 10^{13}$ Hz\nand $\\nu_{\\rm c} = \\infty$ ({\\it thick solid line}). }\n\\label{fig_spectra_445e}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=3c445_res_nuFnu_units.ps voffset=-240 hoffset=10 vscale=33\n hscale=33}\n\\vspace{7cm}\n\\caption{The broad-band SED of the diffuse emission (see text) of \n3C\\,445 South. The morphology from the {\\it Chandra} image does not\nallow us to firmly exclude a connection between the X-rays and the\ndiffuse (including SC component) emission.\nThe synchrotron model assumes\n$\\nu_{\\rm b} =8 \\times 10^{16}$ Hz, $\\nu_{\\rm c} \\gg \\nu_{\\rm b}$ and $p$=2.7.}\n\\label{fig_spectra_445diff}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Spectral energy distribution}\n\n\\begin{table}\n\\caption{Synchrotron parameters. Column 1: Hotspot; Column 2:\n component; Columns 3, 4: spectral index and break frequency as \nderived from the fit to the radio-to-optical SED (Section 5.1); \nColumn 5: equipartition magnetic field, computed following the\napproach presented in Brunetti et al. (2002); Column 6: radiative age\ncomputed using Eq. 2.} \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nSource&Comp.&$\\alpha$&$\\nu_{\\rm b}$&$B_{\\rm eq}$&$t_{\\rm rad}$\\\\\n & & &10$^{13}$ Hz&$\\mu$G&yr\\\\\n\\hline\n&&&&&\\\\\n3C\\,105&S1&0.8&0.50&150&12\\\\\n &S2&0.8&0.75&290&4\\\\\n &S3&0.8&1.5&270&3\\\\\n3C\\,445&SE&0.75&5.2&60&15\\\\\n &SW&0.75&9.4&50&15\\\\\n&&&&&\\\\\n\\hline\n\\end{tabular}\n\\label{tab_fit}\n\\end{center}\n\\end{table}\n\n\\subsection{The broad-band energy distribution}\n\nWe model the broad band energy distribution, from radio to optical, of the\nhotspot regions in order to determine the mechanisms at the basis of\nthe emission. The comparison between the model expectation in the\nX-rays and {\\it Chandra} data sets additional\nconstraints.\nIn the adopted models, the\nhotspot components are described by homogeneous spheres with constant magnetic\nfield and constant properties of the relativistic electron\npopulations. \nThe spectral energy distributions of the emitting electrons are modelled\nassuming the formalism described in \\citet{gb02}. According to\n this model a population of seed electrons (with $\\gamma \\leq\n \\gamma_{*}$) is accelerated at the shock and is injected in the\n downstream region with a spectrum dN($\\gamma$)\/dt $\\propto$\n $\\gamma^{-p}$, for $\\gamma_{*} < \\gamma < \\gamma_{c}$, \n $\\gamma_{c}$ being the maximum energy of the electrons accelerated at the\n shock. Electrons accelerated at the shock are advected in the\n downstream region and age due to radiative losses. Based on\n \\citet{gb02}, the volume integrated spectrum of the electron\n population in the downstream region of size $L \\sim T v_{\\rm adv}$\n ($T$ and $v_{\\rm adv}$ being the age and the advection velocity of\n the downstream region) is given by either a steep power-law $N$($\\gamma$)\n $\\propto \\gamma^{-(p+1)}$ for $\\gamma_{b} < \\gamma < \\gamma_{c}$,\n where $\\gamma_{b}$ is the maximum energy of the ``oldest'' electrons\nin the downstream region, or by $N$($\\gamma$)\n $\\propto \\gamma^{-p}$ for $\\gamma_{*} < \\gamma < \\gamma_{b}$, or by\na flatter shape for $\\gamma_{\\rm low} < \\gamma < \\gamma_{*}$, where\n$\\gamma_{\\rm low}$ is the minimum energy of electrons accelerated at\nthe shock.\\\\\nAs the first step we fit the SED in the\nradio-NIR-optical regimes with a synchrotron model, and we derive the\nrelevant parameters of the synchrotron spectrum \n(injection spectrum $\\alpha$, break frequency\n$\\nu_{\\rm b}$, cut-off frequency $\\nu_{\\rm c}$) \nand the slope of the energy distribution of the electron population\nas injected at the shock (p $= 2 \\alpha +1$). Since hotspots have\n spectra with injection slope $\\alpha$ ranging between 0.5 and 1 (as\n a reference, the classical value from the diffuse particle\n acceleration at strong shocks is $\\alpha = 0.5$, e.g. Meisenheimer\n et al. 1997), we decided to consider the injection spectral index as\n a free parameter.\nSuch constraints allow us to determine the spectrum of the emitting\nelectrons (normalization, break and cut-off\nenergy), once the magnetic field strength has been assumed, and\nto calculate the emission from either synchrotron-self-Compton (SSC) \nor inverse-Compton scattering of the cosmic background radiation \n(IC-CMB) expected from the hotspot (or jet)\nregion \\citep[following][]{gb02}.\nModels described in \\citet{gb02} take also into account the boosting\neffects arising from a hotspot\/jet that is moving at relativistic\nspeeds and oriented at a given angle with respect to our line of\nsight.\\\\\n\n\\subsection{3C\\,105 South}\n\nIn Figures \\ref{fig_spectra_105n} to \\ref{fig_spectra_105s} \nwe show the SED \nfrom the radio band to high energy emission measured\nfor the hotspot components of 3C\\,105 South, together with the model fits.\nSynchrotron models with an injection spectral index $\\alpha$=0.8\nprovide an adequate representation of the SED\nof the central and southern components of\n3C\\,105 South, with break frequencies ranging from\n5$\\times$10$^{12}$ to 1.5$\\times$10$^{13}$ W\/Hz, while the cutoff\nfrequencies are between 3$\\times$10$^{14}$ and 2$\\times$10$^{15}$ W\/Hz.\nIn both components, the upper limit to the X-ray emission does\nnot allow us to constrain the validity of the SSC model\n(Figs. \\ref{fig_spectra_105c} and \\ref{fig_spectra_105s}). \nOn the other hand, the northern component of 3C\\,105 shows a prominent\nX-ray emission.\nA synchrotron model (dashed line in Fig. \\ref{fig_spectra_105n}) may\nfit quite reasonably the radio, NIR and X-ray emission, but\nit completely fails in reproducing\nthe optical data. An additional contribution of the SSC \nis not a viable option since it requires a \nmagnetic field much smaller than that\nobtained assuming equipartition (see Section 5.4) (solid lines), and \nimplying an\nunreasonably large energy budget.\nOn the other hand, the high energy emission is well modelled by\nIC-CMB \\citep[e.g.][]{tavecchio00,celotti01} \nwhere the CMB photons are scattered by relativistic electrons with\nLorentz factor $\\Gamma \\sim 6$, and $\\theta$=5$^{\\circ}$ with a\nmagnetic field of 16 $\\mu$G. \nThis model\nimplies that boosting effects play an important role in the X-ray emission\nof this component, suggesting that S1 is more likely a relativistic\nknot in the jet, rather than a hotspot feature. The weakness of\n this interpretation is that 3C\\,105 is a NLRG and its jets are\n expected to form a large angle with our line of\n sight.\nAlternatively,\n the X-ray emission may be synchrotron from a different population of\nelectrons, as suggested in the case of the jet in 3C\\,273 (Jester et\nal. 2007).\\\\ \n\n\\subsection{3C\\,445 South}\n\nThe analysis of the southern hotspot of 3C\\,445 \nas a single unresolved component was carried out in previous work by\n\\citet{aprieto02,mack09,perlman10}. In this new analysis, \nthe high spatial resolution and\nmultiwavelength VLT and HST data of 3C\\,445 South allow us \nto study the SED of each\ncomponent separately in order to investigate in more detail the\nmechanisms at work across the hotspot region.\nIn Figures \\ref{fig_spectra_445w} to \\ref{fig_spectra_445diff} \nwe show the SED \nfrom the radio band to high energy emission measured\nfor the components of 3C\\,445 South, together with the model fits.\nWe must note that at 1.4 GHz the resolution is not\nsufficient to reliably separate the contribution from the two\nmain components. For this reason,\nwe do not consider the flux density at this frequency in constructing\nthe SED. The X-ray emission\n(Fig. \\ref{fig_3c445}) is misaligned with respect to the radio-NIR-optical\nposition. For this reason, on the SED of\nboth components (Figs. \\ref{fig_spectra_445w} and\n\\ref{fig_spectra_445e}) we plot the total X-ray flux which must be\nconsidered an upper limit. \nFor the components of 3C\\,445 South the synchrotron models with\n$\\alpha$=0.75 reasonably\nfit the data, providing break frequencies in the range of 10$^{13}$\nand $10^{14}$ W\/Hz, and cutoff frequencies from 10$^{15}$ Hz and\n10$^{18}$ Hz.\\\\\nBoth the morphology (Fig. \\ref{fig_3c445}) and the SED\n(Figs. \\ref{fig_spectra_445w} and \n\\ref{fig_spectra_445e}) indicate that the bulk of {\\it Chandra} X-ray\nemission detected in 3C\\,445 is not due to synchrotron emission from\nthe two components (Section 6).\\\\\nAs discussed in Section 4.2, diffuse IR and optical emission\n surrounds the two components SE and SW of 3C\\,445 South, and a third\ncomponent, SC, becomes apparent in the optical. We attempt to evaluate the\nspectral properties of the diffuse emission (including component\nSC). When possible, depending on statistics, we subtract from the\ntotal flux density of the hotspot, the contribution \narising from the two main\ncomponents, obtaining in this way the SED of the diffuse emission\n(inclusive of SC component) of\n3C\\,445 South. In the image we also plot the total X-ray flux. \nAs expected the emission has a hard spectrum ($\\alpha\n\\sim 0.85$) without evidence of a break up to the optical band,\n10$^{15}$ Hz $<$ $\\nu_{b}$ $\\leq$ 8$\\times$10$^{16}$ Hz. We also note\nthat this hard component may represent a significant contribution of\nthe observed X-ray emission, although the X-ray peak appears shifted\n($\\sim$ 1$^{\\prime\\prime}$) from the SC component.\nDue to the extended nature of the emission in this hotspot, we\n created a\npower-law spectral index map \nillustrating the change of the spectral index $\\alpha$\nacross the hotspot region (Fig. \\ref{slope_3c445}). \nThe spectral energy distributions presented in\nFigs. \\ref{fig_spectra_445w}, \\ref{fig_spectra_445e}, and\n\\ref{fig_spectra_445diff} show the\ncurvature of the integrated spectrum for the main\ncomponents and the diffuse emission (see Section 5.1). \nThe spectral map in Fig. \\ref{slope_3c445} attempts to provide\ncomplementary information on the spectral slope for the diffuse\ninter-knot emission. Extracting these maps using the largest \npossible frequency range is complicated as it implies combining images \nfrom different instruments with different scale sampling, \nnoise pattern, etc. These effects sum up to produce very \nlow contrast maps given the weakness of the hotspot signal. \nTo minimise these effects it was decided to extract the slope maps from\nthe optical and -IR images only.\\\\\nThe spectral index map between I- and U-band (Fig. \\ref{slope_3c445}) shows \ntwo sharp edges, at the SW and SE components, with the highest value \n$\\alpha \\sim 1.5 $. \nBetween these two main regions there \nis diffuse emission that is clearly seen\nin the I-\/U-band spectral index map. The slope of this \ncomponent is flatter than that of the two main regions \nand rather uniform all over the hotspot, with $\\alpha \\sim 1$.\\\\ \n\n\n\\subsection{Physical parameters}\n\nWe compute the magnetic field of each hotspot component by \nassuming minimum energy conditions,\ncorresponding to equipartition of energy between radiating\nparticles and magnetic field, \nand following the approach by \\citet{gb97}.\nWe assume for the hotspot components an ellipsoidal volume $V$ with a\nfilling factor $\\phi$=1 (i.e. the volume is fully and homogeneously\nfilled by relativistic plasma). \nThe volume $V$ is computed by means:\\\\\n\n\\begin{equation}\nV = \\frac{\\pi}{6} d_{\\min}^{2} d_{\\max}\n\\end{equation}\n\n\\noindent where d$_{\\min}$ and d$_{\\max}$ are the linear size of the\nminor and major axis, respectively. \nWe consider $\\gamma_{\\rm min} =$100, \nand we assume that the energy densities\nof protons and electrons are equal. \nWe find equipartition \nmagnetic fields ranging from $\\sim$ 50 - 290 $\\mu$G (Table\n\\ref{tab_fit}) that is \nlower than those \ninferred in high-power radio hotspots \nwhich range from $\\sim$ 250 to 650 $\\mu$G\n\\citep{meise97, cheung05}. \nRemarkably, if we compare these results with those from \\citet{mack09},\nwe see that in 3C\\,445 South the value\ncomputed considering the entire source volume is similar to those obtained in\nits individual sub-components, suggesting that compact\nand well-separated emitting regions are not present in the hotspot volume. \nOn the other hand, the magnetic field\naveraged over the whole 3C\\,105 South hotspot complex is much smaller\nthan those derived in its sub-components.\\\\\nIn the presence of such low magnetic fields \nhigh-energy electrons may have longer radiative lifetime than\nin high-power radio hotspots. \nThe radiative age $t_{\\rm rad}$ is related to the\nmagnetic field and the break frequency by\\footnote{The magnetic field\n energy density in these hotpots are at least an order of magnitude\n higher than the energy density of the cosmic microwave background\n (CMB) radiation. Inverse Compton\n losses due to scattering of CMB photons\n are negligible.}:\\\\\n\n\\begin{equation}\nt_{\\rm rad} = 1610 \\; B^{-3\/2} \\nu_{b}^{-1\/2} (1+z)^{-1\/2}\n\\label{eq_trad}\n\\end{equation}\n\n\\noindent where B is in $\\mu$G, $\\nu_{b}$ in GHz and $t_{\\rm rad}$ in\n10$^{3}$ yr. If in Eq. \\ref{eq_trad} we assume the equipartition\nmagnetic field \nwe find that the radiative ages are just a few years (Table 3). \nAs the hotspots\nextend over kpc distances, it is indicative that a very efficient\nre-acceleration mechanism is operating in a similar way over\nthe entire hotspot region.\\\\\n\n\\begin{figure}\n\\begin{center}\n\\special{psfile=alphaqiu.ps voffset=-270 hoffset=-18 hscale=48 vscale=48}\n\\vspace{6.5cm}\n\\caption{Power-law spectral index map for 3C\\,445 South determined from FORS\nI-band and FORS U-band. Contours are\n1, 1.3, 1.5, 1.6, 1.7. First contour is 3 sigma.}\n\\label{slope_3c445}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Discussion}\n\nThe detection of diffuse optical emission occurring well outside\nthe main shock region and distributed over a large fraction of the\nwhole kpc-scale hotspot structure is somewhat surprising. Deep optical\nobservations pointed out that this is a rather common phenomenon\ndetected in about a dozen hotspots \\citep[e.g.][]{mack09,cheung05,thomson95}.\nFirst-order Fermi\nacceleration alone cannot explain optical emission extending on kpc\nscale and additional efficient mechanisms taking place away from the\nmain shock region should be considered, \nunless projection effects play an important role in smearing compact\nregions where acceleration is still occurring.\\\\\nTheoretically, we can consider several scenarios that are able to reproduce\nthe observed extended structures.\n(1) One possibility is that a very wide jet, with a size comparable to the \nhotspot region, impacts simultaneously into various locations across the\nhotspot generating a complex shocked region that defines an arc-shaped\nstructure. This, combined with projection effects may explain a\nwide (projected) emitting region.\n(2) Another possibility is a narrow jet that impacts into the hotspot in a\nsmall region where electrons are accelerated at a strong shock. In this\ncase the accelerated particles are then transported upstream \nin the hotspot\nvolume where they are continuously re-accelerated by stochastic mechanisms,\nlikely due to turbulence generated by the jet and shock itself.\n(3) Finally, extended emission may be explained \nby the ``dentist's drill'' scenario, in which the jet impacts into\nthe hotspot region in different locations at different times. \\\\\nThe peculiar morphology and the rather high NIR\/optical luminosity \nof 3C\\,105 South and 3C\\,445 South, makes \nthese hotspots ideal targets to investigate the\nnature of extended diffuse emission. \\\\\nIn 3C\\,105 South, the detection of optical emission in both \nprimary and\nsecondary hotspots implies that in these regions there is a continuous \nre-acceleration of particles. The secondary hotspot S3 could be interpreted\nas a splatter-spot from material accelerated in the primary one, S2\n\\citep{williams85}. \nBoth the alignment and the distance between these\ncomponents exclude the jet drilling scenario: \nthe light time between the two components is more than 10$^{4}$ years,\ni.e. much longer than their radiative time (Table 3), suggesting that\nacceleration is taking place in both S2 and S3\nsimultaneously. The secondary hotspot S3 shows some\nelongation, always in the same direction, in all the radio and optical\nimages with adequate spatial resolution. \nThis elongation is expected in a splatter-spot and it\nfollows the structure of the shock generated by the impact of the \noutflow from the\nprimary upon the cocoon wall. \\\\\nThis scenario, able to explain the presence of optical emission from two\nbright and distant components, fails in reproducing the diffuse\noptical emission enshrouding the main features, and the extended\ntail. In this case, an additional contribution from stochastic mechanisms\ncaused by turbulence in the downstream region is necessary. \nAlthough this acceleration mechanism is in general less\nefficient than Fermi-I processes, the (radiative) energy losses of\nparticles are smaller in the \npresence of low magnetic fields, such as those in between S2 and S3, \n(potentially)\nallowing stochastic mechanisms to maintain electrons at high energies. \\\\\nIn 3C 445 South the observational picture is complex.\nThe optical images of 3C 445 South show a spectacular 10-kpc arc-shape\nstructure. High resolution HST images allow a further step since they \nresolve this structure in two elongated components enshrouded by diffuse \nemission. \nThese components may mark the regions where a ``dentist's drill'' jet\nimpacts on the ambient medium, representing the most recent episode\nof shock acceleration due to the jet impact. On the other hand, they\ncould simply trace the locations \nof higher particle-acceleration efficiency from a wide\/complex \ninteraction between the jet and the ambient medium.\nHowever, the transverse extension, about 1 kpc, of the two elongated \ncomponents is much larger than what is derived if the relativistic\nparticles, accelerated at the shock, age in the downstream region (provided \nthat the hotspot advances at typical speeds of 0.05-0.1$c$).\nFurthermore, the diffuse optical emission on larger scale\nsuggests the presence of additional, complex, acceleration mechanisms,\nsuch as stochastic processes, \nable to keep particle re-acceleration ongoing in the \nhotspot region.\nThe detection of X-ray emission with {\\it Chandra} \nadds a new grade of complexity. This emission and its displacement \nare interpreted by \\citet{perlman10} as due to IC-CMB originating in\nthe fast part of the decelerating flow. Their model requires that the\nangle between the jet velocity and the observer's line \nof sight is small. \nHowever, 3C\\,445 is a\n classical double radio galaxy and the jet should form a large angle\n with the line of sight (see also Perlman et al. 2010).\nOn the other hand, \nwe suggest that the X-ray\/optical offset might be the outcome of \n ongoing efficient particle acceleration occurring in the hotspot\n region. An evidence supporting this interpretation\nmay reside on the faint and diffuse blob\nseen in U- and B-bands (labelled SC in Fig. \\ref{fig_3c445}) just\nabout 1$^{\\prime\\prime}$ downstream the X-ray peak. The surface\nbrightness of this\ncomponent decreases rapidly as the frequency decreases, as it is shown in\nFig. \\ref{fig_3c445}: well-detected in U- and\nB-bands, marginally visible in I-band, and absent at NIR and radio\nwavelengths. The SED of the diffuse hotspot emission (including SC\ncomponent and excluding SW and SE) is consistent with synchrotron\nemission with a break at high frequencies, 10$^{15}$ Hz $<$ $\\nu_{b}$\n$\\leq$ 8$\\times$10$^{16}$ Hz, and may significantly contribute to the\nobserved X-ray flux. \nSuch a hard spectrum is in agreement with (i) a\nvery recent episode of particle acceleration (the radiative cooling time of the\nemitting particles being 10$^{2}$-10$^{3}$ yr); (ii) efficient\nspatially-distributed acceleration processes,\nsimilar to the scenario proposed for the western hotspot of Pictor A \n(Tingay et al. 2008, see their Fig.5). \\\\\n\n\n\\section{Conclusions}\n\nWe presented a multi-band, high spatial resolution study of the\nhotspot regions in two nearby radio galaxies,\nnamely 3C\\,105 South and 3C\\,445 South, on the basis of \nradio VLA, NIR\/optical VLT and HST, and X-ray {\\it Chandra}\nobservations. At the sub-arcsec resolution achieved at radio and\noptical wavelengths, both hotspots display\nmultiple resolved components connected by diffuse emission detected\nalso in optical. The hotspot region in 3C\\,105 resolves\nin three major components: a primary hotspot, unresolved and aligned\nwith the jet direction, and a secondary hotspot, elongated in shape,\nand interpreted as a splatter-spot arising from continuous outflow of\nparticles from the primary. \nSuch a feature, together with the extremely short\nradiative ages of the electron populations emitting in the optical,\nindicates that the jet has been impacting\nalmost in the same position for a long period, making the\ndrilling jet scenario unrealistic. \nThe detection of an excess of X-ray\nemission from the northern component of 3C\\,105 South \nsuggests that this region is likely a relativistic knot in the jet\nrather than a genuine hotspot feature.\nThe optical diffuse emission enshrouding\nthe main components and extending towards the tail can\nbe explained possibly assuming additional stochastic mechanisms\ntaking place across the whole hotspot region.\\\\ \nIn the case of 3C\\,445 South the optical observations probe a scenario \nwhere the interaction between jet and the ambient medium is very complex.\nTwo optical components pinpointed by HST observations mark either the locations \nwhere particle acceleration is most efficient or the remnants of the most\nrecent episodes of acceleration.\nAlthough projection effects may play an important role, the morphology \nand the spatial extension of the diffuse optical emission suggest that\nparticle accelerations, such as stochastic mechanisms, \nadd to the standard shock acceleration\nin the hotspot region.\nThe X-rays detected by {\\it Chandra} cannot be the counterpart at higher\nenergies of the two main components. It might be due to\nIC-CMB from the fast part of a decelerating flow.\nAlternatively the X-rays could pinpoint synchrotron emission from\nrecent episodes of efficient particle acceleration occurring in the\nwhole hotspot region, similarly to what proposed in other hotspots, that\nwould make the scenario even more complex.\nA possible evidence supporting this scenario comes from the\n hard spectrum of the diffuse hotspot emission and from the\n appearance of a new component (SC) in the optical images.\n\n\n\n\n\\section*{Acknowledgment}\nWe thank the anonymous referee for the valuable suggestions that improved the manuscript. \nF.M. acknowledges the Foundation BLANCEFLOR Boncompagni-Ludovisi, n'ee\nBildt for the grant awarded him in 2010 to support his research.\nThe VLA is operated by the US \nNational Radio Astronomy Observatory which is a facility of the National\nScience Foundation operated under cooperative agreement by Associated\nUniversities, Inc. This work has made use of the NASA\/IPAC\nExtragalactic Database NED which is operated by the JPL, Californian\nInstitute of Technology, under contract with the National Aeronautics\nand Space Administration. This research has made used of SAOImage DS9,\ndeveloped by the Smithsonian Astrophysical Observatory (SAO). Part of\nthis work is based on archival data, software or on-line services\nprovided by ASI Science Data Center (ASDC). The work at SAO is\nsupported by supported by NASA-GRANT GO8-9114A. \nWe acknowledge the use of public data from\nthe Swift data archive. This research has made use of software\nprovided by the Chandra X-ray Center (CXC) in the application packages\nCIAO and ChIPS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\noindent \\textbf{Nodal profile controllability.}\nThe problem of \\emph{nodal profile controllability} of partial differential equations on networks refers to the task of steering the solution thereof to prescribed profiles on specific nodes. Formally speaking, this amounts to saying that said solution should be controlled to given time-dependent functions (called \\emph{nodal profiles}) over certain time intervals by means of controls actuating at one or several other nodes.\nThis is in contrast to the classical question of exact controllability, wherein one seeks to steer the state, at a certain time, to a given final state on the entire network.\nThe nodes with prescribed profiles are then called \\emph{charged nodes} \\cite{YWang2019partialNP} (or \\emph{object-nodes} \\cite{Zhuang2021}), while the nodes at which the controls are applied are the \\emph{controlled nodes} \\cite{YWang2019partialNP} (or \\emph{control nodes} \\cite{Zhuang2021}).\n\n\nThe notion of exact boundary controllability of nodal profiles was, to our knowledge, first introduced by Gugat, Herty and Schleper in \\cite{gugat10}, motivated by applications in the context of gas transport through pipelines networks. \nTherein, consumers are located at the endpoints of the network and the nodal profiles represent the consumer satisfaction, and the former are sought to be attained by the flow which is controlled by means of a number of compressors actuating at several nodes.\n\n\\medskip\n\n\\noindent Motivated by the abundant practical relevance of such control problems, Tatsien Li and coauthors generalized the aforementioned results to one-dimensional first-order quasilinear hyperbolic systems with nonlinear boundary conditions \\cite{gu2011, li2010nodal, li2016book}. \nFor results on the wave equation on a tree-shaped network with a general topology or the unsteady flow in open canals, we refer the reader to \\cite{kw2011, kw2014, YWang2019partialNP} and \\cite{gu2013}, respectively. \n\nWhilst the exact-controllability of the Saint-Venant equations on networks with cycles is not true in general \\cite{LLS, li2010no}, for certain networks with cycles, the exact nodal profile controllability can be shown by means of a so-called \\emph{cut-off} method \\cite{Zhuang2021, Zhuang2018}.\nIn this regard, in line with intuition, the concept of nodal profile controllability is weaker than that of exact controllability.\nHence, when considering a system defined on a network with cycles, a situation which is encountered in many practical applications, the nodal profile control problem is a rather meaningful and feasible goal to attain.\n\n\\medskip\n\n\\noindent\nThe method used by Li et al. to prove nodal profile controllability is \\emph{constructive} in nature, in the sense that it relies on solving the equation forward in time and sidewise, to build a specific solution which achieves the desired goal, before evaluating the trace of this solution to obtain the desired controls. \nAll this is done in the context of regular $C^1_{x,t}$ solutions for first-order systems, which are \\emph{semi-global} in time -- this means that for any time $T>0$, and for small enough initial and boundary data, a unique solution exists at least until time $T>0$ --, a solution concept originating from \\cite{LiJin2001_semiglob}.\nIn \\cite{LiRao2002_cam, LiRao2003_sicon}, this notion of solution is used for proving local exact boundary controllability of one-dimensional quasilinear hyperbolic systems. In these works, a general framework for a constructive method is proposed, from which all subsequent constructive methods derive.\nThe cornerstone of Li's method is thus the proof of semi-global existence and uniqueness, and, in the case of networks, a thorough study of the transmission conditions at multiple nodes. As solving a sidewise problem entails exchanging the role of the spatial and time variables, \nthis method fundamentally exploits the one-dimensional nature of the system (see also Remark \\ref{rem:controllability_thm} \\ref{subrem:sidewise}).\n\n\nVery recently, in the context of the one-dimensional linear wave equation, the controllability of nodal profiles has also been studied in the context of less regular states and controls spaces, by using the duality between controllability and observability and showing an observability inequality. For star-shaped networks, one may see \\cite{YWang2021_NP_HUM} where the sidewise and D'Alembert Formula is used, and for a single string one may see \\cite{Sarac2021} which relies on sidewise energy estimates.\n\n\n\\medskip\n\n\\noindent \\textbf{Geometrically exact beams.} \nMulti-link flexible structures such as large spacecraft structures, trusses, robot arms, solar panels, antennae \\cite{chen_serial_EBbeams, flotow_spacecraft, LLS} have found many applications in civil, mechanical and aerospace engineering.\nThe behavior of such structures is generally modeled by networks of interconnected beams.\n\nIn this article, we will address the problem of nodal profile controllability for \\emph{networks of beams}, possibly with cycles, a problem which has not yet been considered in the literature. \nThe network in question consists of $N$ beams, indexed by $i \\in \\{1, \\ldots, N\\}$, evolving in $\\mathbb{R}^3$, which are mutually linked via rigid joints. \nThe beams are assumed to be freely vibrating, meaning that external forces and moments, such as gravity or aerodynamic forces, have been set to zero.\n\n\\medskip\n\n\\noindent \nNowadays, there is a growing interest in modern highly flexible light-weight structures -- for instance robotic arms \\cite{grazioso2018robot}, flexible aircraft wings \\cite{Palacios2010aero} or wind turbine blades \\cite{Munoz2020, wang2014windturbine} -- which exhibit motions of large magnitude, not negligible in comparison to the overall dimensions of the object.\nTo capture such a behavior, one has to consider a beam model which is \\emph{geometrically exact}, in the sense that the governing system presents nonlinearities in order to also represent large motions -- i.e., large displacements of the centerline and large rotations of the cross sections. \n\nThis beam model, similarly to the more well-known Euler-Bernoulli and Timoshenko systems, is one dimensional with respect to the spatial variable $x$ and accounts for linear elastic material laws, meaning that the strains (which are the local changes in the shape of the material) are assumed to be small.\nModels for geometrically exact beams account for shear deformation, similarly to the Timoshenko system. \nMoreover, the geometrical and material properties of the beam may vary along the beam (indeed, we will see that the coefficients of the system depend on $x$), and the material may be anisotropic.\nAs a matter of fact, the Euler-Bernoulli and Timoshenko systems can be derived from geometrically exact beam models under appropriate simplifying assumptions \\cite[Section IV]{Artola2021damping}.\n\n\\medskip\n\n\\noindent We will see, in Subsection \\ref{subsec:GEBmodels}, that the mathematical model for geometrically exact beams may be written in terms of the position of the centerline of the beam and the orientation of its cross sections, with respect to a fixed coordinate system. This is the commonly known \\emph{Geometrically Exact Beam model}, or GEB, which originates from the work of Reissner \\cite{reissner1981finite} and Simo \\cite{simo1985finite}. The governing system is quasilinear, consisting of six equations. One may draw a parallel with the wave equation as the GEB model is of second order both in space and time.\n\nOn the other hand, the mathematical model can also be written in terms of so-called \\emph{intrinsic} variables -- namely, velocities and internal forces\/moments, or equivalently velocities and strains -- expressed in a moving coordinate system attached to the beam.\nThis yields the \\emph{Intrinsic Geometrically Exact Beam model}, or IGEB, which is due to Hodges \\cite{hodges1990, hodges2003geometrically}. The governing system then counts twelve equations.\nAn interesting feature of the IGEB model is that it falls into the class of one-dimensional first-order hyperbolic systems and is moreover only semilinear. Therefore, from a mathematical perspective, one gains access to the broad literature which has been developed on such system -- see notably by Li and Yu \\cite{Li_Duke85}, Bastin and Coron \\cite{BC2016} -- beyond the context of beam models.\n\nDue to its less compound nature, the IGEB formulation is used in aeroelastic modelling and engineering, notably in the context of very light-weight and slender aircraft aiming to remain airborne almost perpetually, and that consequently exhibit great flexibility \\cite{Palacios2017modes, Palacios2011intrinsic, Palacios2010aero}; see also \\cite{Artola2020aero, Artola2019mpc, Artola2021damping} where the authors additionally take into account structural damping.\n\n\\medskip\n\n\\noindent On another hand, as pointed out in \\cite[Sec. 2.3.2]{weiss99}, one may see the GEB model and IGEB model as being related by a \\emph{nonlinear transformation} (which we define in \\eqref{eq:transfo}). In this work, we will keep track of this link between both models, studying mathematically the latter, and then deducing corresponding results for the GEB model.\n\nAs commonly done in solid mechanics, both the GEB and IGEB models are \\emph{Lagrangian descriptions} of the beam (as opposed to the \\emph{Eulerian description}), in the sense that the independent variable $x$ is attached to matter ($x$ is a label sticking to the particles of the beam's centerline throughout the deformation history) rather than being attached to an inertial frame of reference.\n\nThe IGEB model can also be seen as the beam dynamics being formulated in the \\emph{Hamiltonian} framework in continuum mechanics (see notably \\cite[Sections 5, 6]{Simo1988}), while the GEB model corresponds to the \\emph{Lagrangian} framework.\nThen, taking into account the interactions of the beam with its environment, one may study the IGEB model from the perspective of \\emph{Port-Hamiltonian Systems} (see \\cite{Maschke1992} for the finite dimension setting and \\cite{Schaft2002} and \\cite[Chapter 7]{Zwart2012bluebook} for the infinite dimensions setting), as in \\cite{Macchelli2007, Macchelli2009} and \\cite[Section 4.3.2]{Macchelli2009book}. See also the case of the Timoshenko model in \\cite{Macchelli2004Timo}.\n\n\n\\subsection{Our contributions}\nIn this article we consider the problem of nodal profile controllability in the context of a specific network of geometrically exact beams containing one cycle.\n\\textcolor{black}{Afterwards, the case of other networks, possibly containing several cycles, is discussed in Section \\ref{sec:conclusion}: we give a few typical examples, together with a brief algorithm (Algorithm \\ref{algo:control}) to realize nodal profile controllability under some requirements.}\n\n\\textcolor{black}{Our main results will be given on IGEB networks (Theorem \\ref{th:controllability}) and GEB networks (Corollary \\ref{coro:controlGEB}) as follows.}\n\\begin{enumerate}\n\\item We first consider a general network of beams whose dynamics are given by the IGEB model (System \\eqref{eq:syst_physical} below). We show, in Theorem \\ref{th:existence}, that there exists a unique semi-global in time $C_{x,t}^1$ solution to \\eqref{eq:syst_physical}. \n\n\nThis theorem is also a necessary step to show Theorem \\ref{th:controllability}, namely, the local exact controllability of nodal profiles for System \\eqref{eq:syst_physical}, in the special case of an A-shaped network (see Fig. \\ref{subfig:AshapedNetwork}).\nMore precisely, we drive the solution to satisfy given profiles at one of the multiple nodes by controlling the internal forces and moments at the two simple nodes.\n\n\n\\item For a general network, via Theorem \\ref{thm:solGEB}, we make the link between the IGEB network (System \\eqref{eq:syst_physical}) and the corresponding system \\eqref{eq:GEB_netw} in which the beams dynamics are given by the GEB model. \nMore precisely, we show that the existence of a unique $C^1_{x,t}$ solution to \\eqref{eq:syst_physical} implies that of a unique $C^2_{x,t}$ solution to \\eqref{eq:GEB_netw}, provided that the data of both systems fulfill some compatibility conditions.\n\nIn particular, Theorem \\ref{thm:solGEB}, permits to translate Theorems \\ref{th:existence} and \\ref{th:controllability} to corresponding results in terms of the GEB model \\eqref{eq:GEB_netw}, which are Corollaries \\ref{coro:wellposedGEB} and \\ref{coro:controlGEB}, respectively.\n\\end{enumerate}\n\n\n\n\n\n\\subsection{Notation}\n\\label{subsec:notation}\n\nLet $m, n\\in \\mathbb{N}$. Here, the identity and null matrices are denoted by $\\mathbf{I}_n \\in \\mathbb{R}^{n \\times n}$ and $\\mathbf{0}_{n, m} \\in \\mathbb{R}^{n \\times m}$, and we use the abbreviation $\\mathbf{0}_{n} = \\mathbf{0}_{n, n}$. The transpose of $M\\in \\mathbb{R}^{m\\times n}$ is denoted by $M^\\intercal$.\nThe symbol $\\mathrm{diag}(\\, \\cdot \\, , \\ldots, \\, \\cdot \\, )$ denotes a (block-)diagonal matrix composed of the arguments.\nWe denote by $\\mathcal{S}_{++}^n$ the set of positive definite symmetric matrices in $\\mathbb{R}^{n \\times n}$.\nThe cross product between any $u, \\zeta \\in \\mathbb{R}^3$ is denoted $u \\times \\zeta$, and we shall also write $\\widehat{u} \\,\\zeta = u \\times \\zeta$, meaning that $\\widehat{u}$ is the skew-symmetric matrix \n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{equation*}\n\\widehat{u} = \\begin{bmatrix}\n0 & -u_3 & u_2 \\\\\nu_3 & 0 & -u_1 \\\\\n-u_2 & u_1 & 0\n\\end{bmatrix}, \n\\end{equation*}\n\\end{linenomath}\n\\endgroup\nand for any skew-symmetric $\\mathbf{u} \\in \\mathbb{R}^{3 \\times 3}$, the vector $\\mathrm{vec}(\\mathbf{u}) \\in \\mathbb{R}^3$ is such that $\\mathbf{u} = \\widehat{\\mathrm{vec}(\\mathbf{u})}$. Finally, $\\{e_\\alpha\\}_{\\alpha=1}^3 = \\{(1, 0, 0)^\\intercal, (0, 1, 0)^\\intercal, (0, 0, 1)^\\intercal\\}$ denotes the standard basis of $\\mathbb{R}^3$. \n\n\n\n\n\n\\subsection{Outline}\n\nIn Section \\ref{sec:model_results}, we present in more detail the GEB and IGEB models (Subsection \\ref{subsec:GEBmodels}) before introducing the corresponding systems which give the dynamics of the beam network (Subsection \\ref{subsec:network_systems}). Then, in Subsection \\ref{subsec:main_results} we presents the main results of this article.\n\n\nSection \\ref{sec:exist} is concerned with the well-posedness of the network system \\eqref{eq:syst_physical}: \nin Subsections \\ref{subsec:hyperbolic} and \\ref{subsec:change_var} we show that the system \\eqref{eq:syst_physical} is hyperbolic and write it in Riemann invariants, we then study the transmission conditions for the diagonalized system in Subsection \\ref{subsec:out_in_info}, and finally, we prove Theorem \\ref{th:existence} in Subsection \\ref{subsec:proof_exist}.\n\n\nIn Sections \\ref{sec:controllability} and \\ref{sec:invert_transfo}, we give the proofs of Theorems \\ref{th:controllability} and \\ref{thm:solGEB}, respectively.\n\n\n\\textcolor{black}{Then, in Section 6, we give generalized considerations on more involved networks, namely, with more than one cycles, or with prescribed profiles on several nodes.}\n\n\n\n\n\n\n\n\n\\section{The model and main results}\n\\label{sec:model_results}\n\n\nAs mentioned in the introduction, the beams' dynamics may be given from different points of view, that we specify in the following subsection.\n\n\n\\subsection{Dynamics of a geometrically exact beam}\n\\label{subsec:GEBmodels}\n\n\n\\begin{figure} \\centering\n\\includegraphics[scale=0.7]{beam_netwC}\n\\caption{Beam $i$ in a straight reference configuration, before deformation and at time $t$. Here, $\\{b_i^\\alpha\\}_{\\alpha=1}^3$ denote the columns of $R_i$.}\n\\label{fig:beam_netw}\n\\end{figure}\n \nLet $i$ be the index of any beam of the network.\nFirst, we consider the mathematical model written in terms of the position $\\mathbf{p}_i$ of the centerline and of a rotation matrix $\\mathbf{R}_i$ whose columns $\\{\\mathbf{b}_i^\\alpha\\}_{\\alpha=1}^3$ give the orientation of the cross sections. Both $\\mathbf{p}_i$ and $\\mathbf{R}_i$ depend on $x$ and $t$, with $x\\in [0, \\ell_i]$ where $\\ell_i>0$ is the length of the beam, and both are expressed in the fixed basis $\\{e_\\alpha\\}_{\\alpha=1}^3$.\nThe former has values in $\\mathbb{R}^3$, while the latter has values in the special orthogonal group $\\mathrm{SO}(3)$.\\footnote{$\\mathrm{SO}(3)$ is the set of unitary real matrices of size $3$ and with a determinant equal to $1$, also called \\emph{rotation} matrices.} \n\n\nThe columns of $\\mathbf{R}_i$ may also be seen as a moving basis of $\\mathbb{R}^3$, attached to the beam, and with origin $\\mathbf{p}_i$; we call it \\emph{body-attached basis} as opposed to the fixed basis $\\{e_\\alpha\\}_{\\alpha=1}^3$. We refer to Fig. \\ref{fig:beam_netw} for visualization.\n\n\nThe corresponding model is called the \\emph{Geometrically Exact Beam} model (GEB) and, for a freely vibrating beam, is set in $(0, \\ell_i)\\times(0, T)$ and reads\n\\begin{linenomath}\n\\begin{equation}\n\\label{eq:GEB_pres}\n\\partial_t \\left( \\begin{bmatrix}\n\\mathbf{R}_i & \\mathbf{0}_{3}\\\\ \\mathbf{0}_{3} & \\mathbf{R}_i\n\\end{bmatrix} \\mathbf{M}_i\n\\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}\n\\right) = \\partial_x \\begin{bmatrix}\n\\phi_i \\\\ \\psi_i \\end{bmatrix} + \\begin{bmatrix}\n\\mathbf{0}_{3, 1} \\\\ (\\partial_x \\mathbf{p}_i) \\times \\phi_i\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nwhere $V_i, W_i, \\phi_i, \\psi_i$ are functions of the unknowns $\\mathbf{p}_i, \\mathbf{R}_i$. More precisely, we introduce the linear velocity $V_i$, angular velocity $W_i$, internal forces $\\Phi_i$ and internal moments $\\Psi_i$ of the beam $i$, all having values in $\\mathbb{R}^3$ and being expressed in the body-attached basis. They are defined by (see Subsection \\ref{subsec:notation})\n\\begin{linenomath}\n\\begin{equation} \\label{eq:single_beam_VWPhiPsi}\n\\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}\n= \\begin{bmatrix}\n\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{p}_i\\\\ \\mathrm{vec}\\left( \\mathbf{R}_i^\\intercal \\partial_t \\mathbf{R}_i \\right)\n\\end{bmatrix}, \\quad \n\\begin{bmatrix}\n\\Phi_i \\\\ \\Psi_i\n\\end{bmatrix}= \\mathbf{C}_i^{-1} \\begin{bmatrix}\n\\mathbf{R}_i ^\\intercal \\partial_x \\mathbf{p}_i - e_1 \\\\ \n\\mathrm{vec}\\left(\\mathbf{R}_i^\\intercal \\partial_x \\mathbf{R}_i - R_i^\\intercal \\tfrac{\\mathrm{d}}{\\mathrm{d}x} R_i\\right)\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nwhile the variables $\\phi_i, \\psi_i$ just correspond to $\\Phi_i, \\Psi_i$ when expressed in the fixed basis instead of the body-attached basis; in other words \n\\begin{linenomath}\n\\begin{equation} \\label{eq:def_smallphipsii}\n\\phi_i = \\mathbf{R}_i \\Phi_i, \\quad \\psi_i = \\mathbf{R}_i \\Psi_i.\n\\end{equation}\n\\end{linenomath}\nIn the above governing system and definitions, \n\\begin{linenomath}\n\\begin{equation} \\label{eq:reg_beampara}\n\\mathbf{M}_i, \\mathbf{C}_i \\in C^1([0, \\ell_i]; \\mathcal{S}_{++}^6), \\quad R_i \\in C^2([0, \\ell_i]; \\mathrm{SO}(3))\n\\end{equation}\n\\end{linenomath}\nare the so-called \\emph{mass matrix} $\\mathbf{M}_i$ and \\emph{flexibility matrix} $\\mathbf{C}_i$ which characterize the material and geometry of the beam $i$, while $R_i$ characterizes the initial form of this beam, as it may be pre-curved and twisted before deformation (at rest). All three are given parameters of the beam.\n\n\n\n\\begin{remark}\nConsider a single beam $i$ described by \\eqref{eq:GEB_pres}, with homogeneous Neumann boundary conditions at each end -- i.e., both $\\phi_i$ and $\\psi_i$ are identically equal to zero on $\\{0\\}\\times(0, T)$ and $\\{\\ell\\}\\times (0, T)$.\nWith appropriate initial conditions, rigid body motions such as defined below are solutions to the GEB model:\n\\begin{linenomath}\n\\begin{align} \\label{eq:rigid_body_motion}\n\\mathbf{p}_i(x,t) = f(t) + \\int_0^x R_i(s)e_1 ds, \\qquad \\mathbf{R}_i(x,t) = K(t)R_i(x)\n\\end{align}\n\\end{linenomath}\nfor all $(x,t) \\in [0, \\ell_i]\\times[0, T]$, where $(f, K) \\in C^2([0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$ \nare such that $\\frac{\\mathrm{d}}{\\mathrm{d}t}f \\equiv f_\\circ$ and $\\mathrm{vec}(K^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}K) \\equiv k_\\circ$ for some fixed $f_\\circ, k_\\circ \\in \\mathbb{R}^3$.\n\\end{remark}\n\n\n\n\\noindent The mathematical model may also be written in terms of intrinsic variables expressed in the body-attached basis, namely, linear\/angular velocities and internal forces\/moments $v_i, z_i \\colon [0, \\ell_i]\\times[0, T] \\rightarrow \\mathbb{R}^6$, respectively. In this case, one considers the unknown state $y_i \\colon [0, \\ell_i]\\times[0, T] \\rightarrow \\mathbb{R}^{12}$ of the form\n\\begin{linenomath}\n\\begin{align} \\label{eq:form_yi}\ny_i = \\begin{bmatrix}\nv_i \\\\ z_i\n\\end{bmatrix}, \\quad \\text{where} \\quad v_i = \\begin{bmatrix}\nV_i \\\\ W_i\n\\end{bmatrix}, \\ z_i = \\begin{bmatrix}\n\\Phi_i \\\\ \\Psi_i\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nWe call the corresponding model the \\emph{Intrinsic Geometrically Exact Beam} model (IGEB), and it reads\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:IGEB_pres}\n\\partial_t y_i + A_i(x) \\partial_x y_i + \\overline{B}_i(x) y_i = \\overline{g}_i(x, y_i),\n\\end{align}\n\\end{linenomath}\nwhere the coefficients $A_i,\\overline{B}_i$ and the source $\\overline{g}_i$ depend on $\\mathbf{M}_i, \\mathbf{C}_i$ and $R_i$. \nMore precisely, $A_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ is defined by (see \\eqref{eq:reg_beampara})\n\\begin{linenomath}\n\\begin{align}\\label{eq:def_Ai}\nA_i = - \\begin{bmatrix}\n\\mathbf{0}_6 & \\mathbf{M}_i^{-1}\\\\\n\\mathbf{C}_i^{-1} & \\mathbf{0}_6\n\\end{bmatrix},\n\\end{align}\n\\end{linenomath}\nand we will see, in Subsection \\ref{subsec:hyperbolic}, that the matrix $A_i(x)$ is hyperbolic for all $x \\in [0, \\ell_i]$ (i.e., it has real eigenvalues only, with twelve associated independent eigenvectors).\n\n\nThe matrix $\\overline{B}_i(x)$ is indefinite and, up to the best of our knowledge, may not be assumed arbitrarily small\nimplying not only that the linearized system \\eqref{eq:IGEB_pres} is not homogeneous, but also that \\eqref{eq:IGEB_pres} cannot be seen as the perturbation of a system of conservation laws. The function $\\overline{B}_i \\in C^1([0, \\ell_i];\\mathbb{R}^{12 \\times 12})$ which depends, just as $A_i$, on the mass and flexibility matrices, also depends on the curvature $\\Upsilon_c^i \\colon [0, \\ell_i] \\rightarrow \\mathbb{R}^3$ of the beam before deformation, and is defined by\n\\begin{linenomath}\n\\begin{align*}\n\\overline{B}_i = \\begin{bmatrix}\n\\mathbf{0}_6 & - \\mathbf{M}^{-1}_i\\mathbf{E}_i\\\\\n\\mathbf{C}_i^{-1}\\mathbf{E}_i^\\intercal & \\mathbf{0}_6\n\\end{bmatrix}, \\quad \\text{with} \\ \\ \\mathbf{E}_i = \\begin{bmatrix}\n\\widehat{\\Upsilon}_c^i & \\mathbf{0}_3\\\\\n\\widehat{e}_1 & \\widehat{\\Upsilon}_c^i\n\\end{bmatrix}, \\quad \\Upsilon_c^i = \\mathrm{vec}\\big(R_i^\\intercal \\tfrac{\\mathrm{d}}{\\mathrm{d}x} R_i \\big).\n\\end{align*}\n\\end{linenomath}\n\nThe function $\\overline{g}_i \\colon [0, \\ell_i]\\times \\mathbb{R}^{12} \\rightarrow \\mathbb{R}^{12}$ is defined by $\\overline{g}_i(x, u) = \\overline{\\mathcal{G}}_i(x, u)u$ for all $x \\in [0, \\ell_i]$ and $u=(u_1^\\intercal, u_2^\\intercal, u_3^\\intercal, u_4^\\intercal)^\\intercal \\in \\mathbb{R}^{12}$ with each $u_j \\in \\mathbb{R}^3$, where the map $\\overline{\\mathcal{G}}_i$ is defined by (see Subsection \\ref{subsec:notation})\n\\begin{linenomath}\n\\begin{align*}\n\\overline{\\mathcal{G}}_i(x,u) = - \n\\begin{bmatrix}\n\\mathbf{M}_i(x)^{-1} & \\mathbf{0}_6\\\\\n\\mathbf{0}_6 & \\mathbf{C}_i(x)^{-1}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\widehat{u}_2 & \\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_3\\\\\n\\widehat{u}_1 & \\widehat{u}_2 & \\widehat{u}_3 & \\widehat{u}_4 \\\\\n\\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_2 & \\widehat{u}_1\\\\\n\\mathbf{0}_3 & \\mathbf{0}_3 & \\mathbf{0}_3 & \\widehat{u}_2\n\\end{bmatrix} \n\\begin{bmatrix}\n\\mathbf{M}_i(x) & \\mathbf{0}_6\\\\\n\\mathbf{0}_6 & \\mathbf{C}_i(x)\n\\end{bmatrix}.\n\\end{align*}\n\\end{linenomath}\nOne sees that $\\overline{g}_i$ is a quadratic nonlinearity (in the sense that its components are quadratic forms on $\\mathbb{R}^{12}$ with respect to the second argument), and that it has the same regularity as the mass and flexibility matrices $\\mathbf{M}_i, \\mathbf{C}_i$ with respect to its first argument, and is $C^\\infty$ with respect to its second argument. Moreover, $\\overline{g}_i(x, \\cdot)$ is locally Lipschitz in $\\mathbb{R}^{12}$ for any $x \\in [0, \\ell_i]$, and $\\overline{g}_i$ is locally Lipschitz in $H^1(0, \\ell_i; \\mathbb{R}^{12})$, but no global Lipschitz property is available.\n\n\n\\medskip\n\n\n\\noindent Finally, as mentionned in the introduction, one may see \\eqref{eq:GEB_pres} and \\eqref{eq:IGEB_pres} as being related by the nonlinear transformation $\\mathcal{T}$ defined by (see \\eqref{eq:single_beam_VWPhiPsi})\n\\begingroup\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{equation} \\label{eq:transfo}\n\\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}) = (\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i))_{i \\in \\mathcal{I}}, \\quad \\text{where} \\quad\n\\mathcal{T}_i (\\mathbf{p}_i, \\mathbf{R}_i) = \n\\begin{bmatrix} V_i\\\\ W_i\\\\ \\Phi_i\\\\ \\Psi_i\\end{bmatrix}.\n\\end{equation}\n\\end{linenomath}\n\\endgroup\n\n\n\n\n\n\n\n\\subsection{Dynamics of the network of beams}\n\\label{subsec:network_systems}\n\n\n\\begin{figure}\n \\begin{subfigure}{0.2\\textwidth}\n \\centering\n \\includegraphics[height=3.25cm]{star_blackC.pdf}\n \\caption{Star-shaped}\n \\end{subfigure}%\n \\begin{subfigure}{0.24\\textwidth}\n \\centering\n \\includegraphics[height=3.25cm]{tree_blackC.pdf}\n \\caption{Tree-shaped} \n \\end{subfigure}%\n \\begin{subfigure}{0.28\\textwidth}\n \\centering\n \\includegraphics[height=3.25cm]{A_pres_numC.pdf}\n \\caption{A-shaped} \n \\label{subfig:AshapedNetwork}\n \\end{subfigure}%\n \\hspace*{\\fill} \n \\begin{subfigure}{0.26\\textwidth}\n \\centering\n \\includegraphics[width=2.75cm]{xinC.pdf}\n\\caption{Orientation of an edge $i$ starting and ending at the nodes $k$ and $n$, respectively.}\n\\label{fig:xin}\n \\end{subfigure}\n\n\\caption{Some oriented graphs representing beam networks, and orientation of the edges.}\n\\label{fig:exple_networks}\n\\end{figure}\n\nLet us now give the systems describing the entire beam network.\n\n\\subsubsection{Network notation}\nTo represent a collection of $N$ beams attached in a certain manner to each other at their tips, we use an oriented graph containing $N$ edges. Any edge $i$ is identified with the interval $[0, \\ell_i]$, which is the spatial domain for the beam model in question (GEB or IGEB). Hence, just as for the beams, the \\emph{edges} are indexed by $i \\in \\mathcal{I} = \\{1, \\ldots, N\\}$, while the \\textit{nodes} are indexed by $n \\in \\mathcal{N} = \\{1, \\ldots, \\#\\mathcal{N}\\}$, where $\\#$ denotes the set cardinality. The set of nodes is partitioned as $\\mathcal{N} = \\mathcal{N}_S \\cup \\mathcal{N}_M$, where $\\mathcal{N}_S$ is the set of indexes of \\emph{simple nodes}, while $\\mathcal{N}_M$ is the set of indexes \\emph{multiple nodes}. \n\nThe former set is in addition partitioned as $\\mathcal{N}_S = \\mathcal{N}_S^D \\cup \\mathcal{N}_S^N$, where $\\mathcal{N}_S^D$ contains the simple nodes with prescribed \\emph{Dirichlet} boundary conditions (i.e., the centerline's position and the cross section's orientation in the case of the GEB model, or the velocities in the case of the IGEB model, are prescribed), while $\\mathcal{N}_S^N$ contains the simple nodes with prescribed \\emph{Neumann} boundary conditions (i.e., the internal forces and moments are prescribed).\n\n\\medskip\n\n\\noindent For any $n\\in\\mathcal{N}$, we denote by $\\mathcal{I}^n$ the set of indexes of edges incident to the node $n$, by $k_n = \\# \\mathcal{I}^n $ the \\emph{degree} of the node $n$, and by $i^n$ the index\\footnote{Defining $i^n$ as the \\emph{smallest} element of $\\mathcal{I}^n$, and not the \\emph{largest} for example, is an arbitrary choice and is of no influence here.}\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_in}\ni^n = \\min_{i\\in \\mathcal{I}^n} i.\n\\end{align}\n\\end{linenomath}\nNote that in the case of a simple node, $\\mathcal{I}^n = \\{i^n\\}$.\n\n\n\n\nThe orientation of each beam is given by the variables $\\mathbf{x}_i^n$ and $\\tau_i^n$ defined as follows.\nFor any $i \\in \\mathcal{I}^n$, we denote by $\\mathbf{x}_i^n$ the end of the interval $[0, \\ell_i]$ which corresponds to the node $n$, while $\\tau_i^n$ is the outward pointing normal at $\\mathbf{x}_i^n$: \n\\begin{linenomath}\n\\begin{align*}\n\\tau_i^n = \n\\left\\{ \n\\begin{aligned}\n-1 \\qquad & \\text{if } \\mathbf{x}_i^n = 0,\\\\\n+1 \\qquad &\\text{if } \\mathbf{x}_i^n = \\ell_i.\n\\end{aligned}\n\\right.\n\\end{align*}\n\\end{linenomath}\nAs described in Fig. \\ref{fig:exple_networks}, each edge $i$ is represented by an arrow and each node $n$ by a circle. The arrowhead is at the ending point $x=\\ell_i$; see Fig. \\ref{fig:xin}.\n\n\n\n\n\n\n\\subsubsection{The network model}\n\n\nLet $T > 0$.\nIf all beams are described by the GEB model \\eqref{eq:GEB_pres}, then the overall network is described by System \\eqref{eq:GEB_netw} below, which gives the dynamics of the unknown state $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$:\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:GEB_netw}}\n\\nonumber\n\\partial_t \\left( \n\\left[\\begin{smallmatrix}\n\\mathbf{R}_i & \\mathbf{0}_3\\\\\n\\mathbf{0}_3 & \\mathbf{R}_i\n\\end{smallmatrix}\\right]\n\\mathbf{M}_i\n\\left[\\begin{smallmatrix}\nV_i \\\\ W_i\n\\end{smallmatrix}\\right]\n\\right) & \\\\\n\\label{eq:GEB_gov}\n\\hspace{1cm}= \\partial_x \\left[\\begin{smallmatrix}\n\\phi_i \\\\ \\psi_i \\end{smallmatrix} \\right] + \\left[\\begin{smallmatrix}\n\\mathbf{0}_{3, 1} \\\\ (\\partial_x \\mathbf{p}_i) \\times \\phi_i\n\\end{smallmatrix} \\right] \n&$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:GEB_continuity_pi}\n\\mathbf{p}_i(\\mathbf{x}_i^n, t) = \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_rigid_angles}\n(\\mathbf{R}_i R_{i}^\\intercal)(\\mathbf{x}_i^n, t) = (\\mathbf{R}_{i^n} R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_Kirchhoff}\n{\\textstyle \\sum_{i\\in\\mathcal{I}^n}} \\tau_i^n \\left[ \\begin{smallmatrix} \n\\phi_i \\\\ \\psi_i\n\\end{smallmatrix} \\right] (\\mathbf{x}_i^n, t) = f_n (t)\n&$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:GEB_condNSz}\n\\tau_{i^n}^n \\left[ \\begin{smallmatrix} \n\\phi_{i^n} \\\\ \\psi_{i^n}\n\\end{smallmatrix} \\right] (\\mathbf{x}_{i^n}^n, t) = f_n (t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:GEB_condNSv_p_R}\n(\\mathbf{p}_{i^n}, \\mathbf{R}_{i^n})(\\mathbf{x}_{i^n}^n, t) = (f_n^\\mathbf{p}, f_n^\\mathbf{R})(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:GEB_IC_0ord}\n(\\mathbf{p}_i, \\mathbf{R}_i)(x, 0) = (\\mathbf{p}_i^0, \\mathbf{R}_i^0)(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:GEB_IC_1ord}\n(\\partial_t \\mathbf{p}_i, \\mathbf{R}_i W_i)(x, 0) = (\\mathbf{p}_i^1, w_i^0)(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$,\n\\end{subnumcases}\n\\end{linenomath}\nwhere we recall that $V_i, W_i, \\phi_i, \\psi_i$ are defined in \\eqref{eq:single_beam_VWPhiPsi}-\\eqref{eq:def_smallphipsii}.\nIn this system, \\eqref{eq:GEB_IC_0ord}-\\eqref{eq:GEB_IC_1ord} describe the initial conditions, with data\n\\begin{linenomath}\n\\begin{align} \\label{eq:reg_Idata_GEB}\n(\\mathbf{p}_i^0, \\mathbf{R}_i^0) \\in C^2([0, \\ell_i]; \\mathbb{R}^3 \\times \\mathrm{SO}(3)), \\quad \\mathbf{p}_i^1, w_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^3), \\quad i \\in\\mathcal{I}.\n\\end{align}\n\\end{linenomath}\nThen, \\eqref{eq:GEB_continuity_pi}-\\eqref{eq:GEB_rigid_angles}-\\eqref{eq:GEB_Kirchhoff}\nare the so-called \\emph{transmission} (or \\emph{interface}) conditions for multiple nodes, while the conditions \\eqref{eq:GEB_condNSz}-\\eqref{eq:GEB_condNSv_p_R} are enforced at simple nodes. The nodal data is\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:reg_Ndata_GEB_N}\nf_n \\in C^1([0, T]; \\mathbb{R}^{6}), \\quad &n \\in \\mathcal{N}_M \\cup \\mathcal{N}_S^N\\\\\n\\label{eq:reg_Ndata_GEB_D}\n(f_n^\\mathbf{p}, f_n^\\mathbf{R}) \\in C^2([0, T]; \\mathbb{R}^3\\times \\mathrm{SO}(3)), \\quad &n \\in\\mathcal{N}_S^D.\n\\end{align}\n\\end{linenomath}\n\n\n\\medskip\n\n\n\\noindent On the other hand, if all beams are described by the IGEB model \\eqref{eq:IGEB_pres}, then for the overall network, the unknown state $(y_i)_{i \\in \\mathcal{I}}$ is described by System \\eqref{eq:syst_physical}, which reads\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:syst_physical}}\n\\label{eq:IGEB_gov}\n\\partial_t y_i + A_i \\partial_x y_i + \\overline{B}_i y_i = \\overline{g}_i(\\cdot,y_i) &$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:IGEB_cont_velo}\n(\\overline{R}_i v_i)(\\mathbf{x}_i^n, t) = (\\overline{R}_{i^n} v_{i^n})(\\mathbf{x}_{i^n}^n, t) &$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:IGEB_Kirchhoff}\n\\sum_{i\\in\\mathcal{I}^n} \\tau_i^n (\\overline{R}_i z_i)(\\mathbf{x}_i^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:IGEB_condNSz}\n\\tau_{i^n}^n z_{i^n} (\\mathbf{x}_{i^n}^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:IGEB_condNSv}\nv_{i^n}(\\mathbf{x}_{i^n}^n, t) = q_n(t) &$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:IGEB_ini_cond}\ny_i(x, 0) = y_i^0(x) &$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$,\n\\end{subnumcases}\n\\end{linenomath}\n$v_i,z_i$ representing the first and last six components of $y_i$, respectively (see \\eqref{eq:form_yi}), and where $\\overline{R}_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6 \\times 6})$ is defined by $\\overline{R}_i = \\mathrm{diag}(R_i, R_i)$ (see \\eqref{eq:reg_beampara}).\nHere, \\eqref{eq:IGEB_ini_cond} gives the initial conditions, with data \n\\begin{linenomath}\n\\begin{align} \\label{eq:reg_Idata_IGEB}\ny_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^{12}), \\quad i \\in \\mathcal{I},\n\\end{align}\n\\end{linenomath}\nthe transmission conditions are \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}, while the conditions \\eqref{eq:IGEB_condNSz}-\\eqref{eq:IGEB_condNSv} are imposed at simple nodes, with data\n\\begin{linenomath}\n\\begin{align}\\label{eq:reg_Ndata_IGEB}\nq_n \\in C^1([0, T]; \\mathbb{R}^{6}), \\quad n \\in \\mathcal{N}.\n\\end{align}\n\\end{linenomath}\n\n\n\n\n\n\n\\subsubsection{Origin of the nodal conditions}\n\nAs the form of transmission conditions is an essential aspect in the proof of nodal profile controllability of hyperbolic systems on networks, let us now explain the origin of these conditions for System \\eqref{eq:GEB_netw} and especially those of System \\eqref{eq:syst_physical}. See also \\cite{R2020} for a more detailed presentation, and for the meaning of the states and coefficients of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical}.\n\n\n\\medskip\n\n\n\\noindent Let $n$ be the index of some multiple node. \nIn this work, we assume that, at all times, the beams incident with this node remain attached to each other. In other words, as imposed by \\eqref{eq:GEB_continuity_pi}, the position of their centerlines must coincide. \nMoreover, we work under the \\emph{rigid joint} assumption, namely, at any node, there is no relative motion between the incident beams. As the orientation of the cross sections before deformation is specified by the (given) function $R_i$, the rigid joint assumption is enforced by the condition \\eqref{eq:GEB_rigid_angles} which states that the change of orientation $\\mathbf{R}_iR_i^\\intercal$ (from the undeformed state of the beam network to its state at time $t$) is the same for all incident beams. See also \\cite[Subsection 2.4]{strohm_dissert}.\n\n\nFor the IGEB model, the condition corresponding to the continuity of the centerline's position and of the change of the cross section's orientation, is the \\emph{continuity} of velocities \\eqref{eq:IGEB_cont_velo}. Indeed, one may differentiate \\eqref{eq:GEB_continuity_pi} and \\eqref{eq:GEB_rigid_angles} with respect to time, and then left-multiply each of the obtained equations by $(R_j\\mathbf{R}_j^\\intercal)(\\mathbf{x}_j^n, t)$ for the corresponding beam index $j$ (thereby using the rigid joint assumption), to obtain\n\\begin{linenomath}\n\\begin{align*}\n(R_i\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{p}_i)(\\mathbf{x}_i^n, t) &= (R_{i^n}\\mathbf{R}_{i^n}^\\intercal \\partial_t \\mathbf{p}_{i^n})(\\mathbf{x}_{i^n}^n, t), \\\\\n(R_i\\mathbf{R}_i^\\intercal \\partial_t \\mathbf{R}_i R_i^\\intercal)(\\mathbf{x}_i^n, t) &= (R_{i^n}\\mathbf{R}_{i^n}^\\intercal \\partial_t \\mathbf{R}_{i^n} R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t),\n\\end{align*}\n\\end{linenomath}\nrespectively. The above equations turn out to equate to \\eqref{eq:IGEB_cont_velo}, by the definition of $V_i$ and $W_i$ (see \\eqref{eq:single_beam_VWPhiPsi}), and by using that the invariance of the cross product in $\\mathbb{R}^3$ under rotation provides the identity $R_i \\widehat{W}_iR_i^\\intercal = \\widehat{R_iW_i}$.\n\n\n\\medskip\n\n\n\\noindent Furthermore, at this multiple node $n$, we require the internal forces $\\phi_i$ and moments $\\psi_i$ exerted by incident beams $i\\in\\mathcal{I}_n$ to be balanced with the external load $f_n$ applied at this node, which reads as \\eqref{eq:GEB_Kirchhoff}, and is also called the \\emph{Kirchhoff} condition. \n\n\nThe corresponding Kirchhoff condition \\eqref{eq:IGEB_Kirchhoff} for the IGEB model is then obtained by left-multiplying each term in the right-hand side of \\eqref{eq:GEB_Kirchhoff} by $(R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$ for the corresponding index $i$ (once again using the rigid joint assumption), left-multiplying $f_n$ by $(R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$ for some $i \\in \\mathcal{I}_n$ (for instance as $i^n$), and recalling the relationship between $\\phi_i, \\psi_i$ and $\\Phi_i, \\Psi_i$ (see \\eqref{eq:def_smallphipsii}).\n\n\n\\medskip\n\n\n\\noindent Similar considerations hold for simple nodes. Here, either $n \\in \\mathcal{N}_S^N$ and an external load $f_n$ is applied at this node, yielding the condition \\eqref{eq:GEB_condNSz}, or $n \\in \\mathcal{N}_S^D$ and the centerline's position and cross section's orientation are prescribed as $f_n^\\mathbf{p}$ and $f_n^\\mathbf{R}$, respectively, for the beam $i^n$ incident with this node, yielding the condition \\eqref{eq:GEB_condNSv_p_R}. \n\n\nFor the IGEB model, this translates to \\eqref{eq:IGEB_condNSz} and \\eqref{eq:IGEB_condNSv}, respectively, when one left-multiplies \\eqref{eq:GEB_condNSz} and \\eqref{eq:GEB_condNSv_p_R} by $(R_{i^n}\\mathbf{R}_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n, t)$ and $\\mathbf{R}_{i^n}^\\intercal(\\mathbf{x}_{i^n}^n, t)$, respectively.\n\n\n\n\n\n\n\\subsubsection{Relationship between the data of both systems}\n\nAs mentioned earlier, the unknowns of the GEB and IGEB models are related by the transformation $\\mathcal{T}$, defined in \\eqref{eq:transfo}.\nThus, the initial data of both models are related as follows: for given $\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1$ and $w_i^0$, one has \n\\begin{linenomath}\n\\begin{align} \\label{eq:rel_inidata}\ny_i^0 = \\begin{bmatrix}\nv_i^0 \\\\ z_i^0\n\\end{bmatrix}, \\quad\nv_i^0 = \\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\mathbf{p}_i^1 \\\\\n(\\mathbf{R}_i^0 )^{\\intercal} w_i^0\n\\end{bmatrix}\n, \\quad\nz_i^0 = \\mathbf{C}_i^{-1} \n\\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_i^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_i^0 - R_i^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_i \\right)\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nSimilarly, the nodal conditions of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical} are connected via $\\mathcal{T}$, and with the help of the above considerations on the nodal conditions, one can observe the following relationships between the nodal data of both systems.\nFor any $n \\in \\mathcal{N}_S^D$, for given $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$ of regularity \\eqref{eq:reg_Ndata_GEB_D}, one has\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_qn_D}\nq_n = \\begin{bmatrix}\n(f_n^\\mathbf{R})^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{p}\\\\\n(f_n^\\mathbf{R})^\\intercal \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{R}\n\\end{bmatrix},\n\\end{align}\n\\end{linenomath}\nwhile for any $n \\in \\mathcal{N}_M \\cup \\mathcal{N}_S^N$,\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_fn}\nf_n = \n\\left\\{\n\\begin{aligned}\n&\\mathrm{diag}\\left((\\mathbf{R}_{i^n} R_{i^n}^\\intercal )(\\mathbf{x}_{i^n}^n, \\cdot), (\\mathbf{R}_{i^n} R_{i^n}^\\intercal) (\\mathbf{x}_{i^n}^n, \\cdot)\\right) q_n &&n \\in \\mathcal{N}_M\\\\\n&\\mathrm{diag}\\big(\\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot), \\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot)\\big) q_n &&n \\in \\mathcal{N}_S^N.\n\\end{aligned}\n\\right.\n\\end{align}\n\\end{linenomath}\n\n\n\n\n\n\n\\subsection{Main results}\n\\label{subsec:main_results}\n\nWe may now present our main results, which are divided in two parts: one is concerned with the well-posedness and controllability of the IGEB network, and the other with showing that the transformation from the GEB to the IGEB network is invertible, by means of which one can deduce corresponding results for the former model. \n\n\\subsubsection{Study of the IGEB model}\n\nLet us define compatibility conditions for System \\eqref{eq:syst_physical}. As for the unknown, we write the initial data $(y_i^0)_{i \\in \\mathcal{I}}$ as\n\\begin{linenomath}\n\\begin{equation*}\ny_i^0 = \\begin{bmatrix}\nv_i^0 \\\\ z_i^0\n\\end{bmatrix} , \\qquad \\text{with } v_i^0, z_i^0 \\colon [0, \\ell_i] \\rightarrow \\mathbb{R}^6.\n\\end{equation*}\n\\end{linenomath}\n\n\\begin{definition}\nWe say that the initial data $y_i^0 \\in C^1([0, \\ell_i]; \\mathbb{R}^{12})$, for all $i\\in \\mathcal{I}$, and boundary data $q_n\\in C^0([0, T]; \\mathbb{R}^6)$, for all $n \\in \\mathcal{N}$, fulfill the first-order compatibility conditions of \\eqref{eq:syst_physical} if\n\\begin{linenomath}\n\\begin{equation} \\label{eq:compat_0} \n\\begin{aligned}\n&(\\overline{R}_i v_i^0)(\\mathbf{x}_i^n) = (\\overline{R}_j v_j^0)(\\mathbf{x}_j^n) \\qquad && i,j \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M\\\\\n&{\\textstyle \\sum_{i\\in\\mathcal{I}^n}} \\tau_i^n (\\overline{R}_i z_i^0)(\\mathbf{x}_i^n) = q_n(0) && n \\in \\mathcal{N}_M\\\\\n& \\tau_{i^n}^n z_{i^n}^0(\\mathbf{x}_{i^n}^n) = q_n(0) && n \\in \\mathcal{N}_S^N\\\\\n& v_{i^n}^0 (\\mathbf{x}_{i^n}^n) = q_n(0) && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\n\\end{equation} \n\\end{linenomath}\nholds and $y_i^1 \\in C^0([0, \\ell_i]; \\mathbb{R}^{12})$, for all $i\\in \\mathcal{I}$, defined by \n\\begin{linenomath}\n\\begin{equation*}\ny_i^1 = - A_i \\frac{\\mathrm{d}y_i^0}{\\mathrm{d}x} - \\overline{B}_i y_i^0 + \\overline{g}_i(\\cdot, y_i^0) = \\begin{bmatrix}\nv_i^1 \\\\z_i^1\n\\end{bmatrix},\n\\end{equation*}\n\\end{linenomath}\nalso fulfills \\eqref{eq:compat_0}, where $v_i^0, z_i^0$ are replaced by $v_i^1, z_i^1$ respectively. \n\\end{definition}\n\n\nIn order to ensure a certain regularity of the eigenvalues and eigenvectors of $A_i$, we will later on make the following assumption.\n\n\n\\begin{assumption} \\label{as:mass_flex}\nFor all $i \\in \\mathcal{I}$, we suppose that\n\\begin{enumerate}\n\\item \\label{eq:assump1_1} $\\mathbf{C}_i, \\mathbf{M}_i \\in C^2([0, \\ell_i]; \\mathcal{S}_{++}^6)$;\n\\item \\label{eq:assump1_2} the function $\\Theta_i \\in C^2([0, \\ell_i]; \\mathcal{S}_{++}^6)$ defined by $\\Theta_i = (\\mathbf{C}_i^{\\sfrac{1}{2}} \\mathbf{M}_i\\mathbf{C}_i^{\\sfrac{1}{2}})^{-1}$, is such that there exists $U_i, D_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6 \\times 6})$ for which\n\\begin{linenomath}\n\\begin{align*}\n\\Theta_i = U_i^\\intercal D_i^2 U_i, \\quad \\text{in }[0, \\ell_i],\n\\end{align*}\n\\end{linenomath}\nwhere $D_i(x)$ is a positive definite diagonal matrix containing the square roots of the eigenvalues of $\\Theta_i(x)$ as diagonal entries, while $U_i(x)$ is unitary.\n\\end{enumerate}\n\\end{assumption}\n\n\nOne may note that, in Assumption \\ref{as:mass_flex}, if \\ref{eq:assump1_1} holds, then \\ref{eq:assump1_2} is readily verified if $\\mathbf{M}_i, \\mathbf{C}_i$ have values in the set of diagonal matrices, or if the eigenvalues of $\\Theta_i(x)$ are distinct for all $x \\in [0, \\ell_i]$ (one may adapt \\cite[Th. 2, Sec. 11.1]{evans2}). Clearly, \\ref{eq:assump1_2} is also satisfied if $\\mathbf{M}_i, \\mathbf{C}_i$ are constant, entailing that the material and geometrical properties of the beam do not vary along its centerline.\n\n\\medskip\n\n\\noindent Our first task is to obtain the existence and uniqueness of semi-global in time solutions to \\eqref{eq:syst_physical} for any network. Henceforth, in the norms' subscripts, when there is no ambiguity, we use the abbreviations $C_x^1 = C^1([0, \\ell_i]; \\mathbb{R}^d)$, $C_t^1 = C^1(I; \\mathbb{R}^d)$ and $C_{x,t}^1 = C^1([0, \\ell_i]\\times I; \\mathbb{R}^d)$ for the appropriate time interval $I$ and dimension $d \\in \\{1, 2, \\ldots\\}$.\n\n\n\\begin{theorem} \\label{th:existence}\nConsider a general network, suppose that $R_i$ has the regularity \\eqref{eq:reg_beampara} and that Assumption \\ref{as:mass_flex} is fulfilled.\nThen, for any $T>0$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$ and for some $\\delta>0$, and all initial and boundary data $y_i^0, q_n$ of regularity \\eqref{eq:reg_Idata_IGEB}-\\eqref{eq:reg_Ndata_IGEB}, and satisfying $\\|y_i^0\\|_{C_x^1} +\\|q_n\\|_{C_t^1} \\leq \\delta$ and the first-order compatibility conditions of \\eqref{eq:syst_physical}, there exists a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i] \\times [0, T]; \\mathbb{R}^{12})$ to \\eqref{eq:syst_physical}, with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon$.\n\\end{theorem}\n\n\n\nThe proof of Theorem \\ref{th:existence}, given in Section \\ref{sec:exist}, consists in rewriting \\eqref{eq:syst_physical} as a single hyperbolic system and applying general well-posedness results \\cite{li2010controllability, wang2006exact}. To do so, one has to write \\eqref{eq:syst_physical} in Riemann invariants, the new unknown state being denoted $(r_i)_{i \\in \\mathcal{I}}$, and verify that the nodal conditions fulfill the following rule: at any node, the components of $r_i$ corresponding to characteristics \\emph{entering} the domain $[0, \\ell_i]\\times [0, +\\infty)$ at this node is expressed explicitly as a function of the components of $r_i$ corresponding to characteristics \\emph{leaving} the domain $[0, \\ell_i]\\times [0, +\\infty)$ at this node (more detail is given in Subsection \\ref{subsec:out_in_info}).\n\n\n\\begin{remark} \nAssuming that $R_i \\in C^2([0, \\ell_i]; \\mathrm{SO}(3))$ guaranties that $\\overline{B}_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$. On the other hand, in Assumption \\ref{as:mass_flex}, the extra regularity for $\\mathbf{M}_i, \\mathbf{C}_i$ ($C^2$, instead of $C^1$ as in \\eqref{eq:reg_beampara}) permits us to ensure that the coefficients of the System \\eqref{eq:syst_physical} written in Riemann invariants, in particular $B_i$ (see Subsection \\ref{subsec:change_var}), are sufficiently regular.\n\\end{remark}\n\n\n\nWe now consider a problem of local exact boundary controllability of nodal profiles, for the specific case of the A-shaped network illustrated in Fig. \\ref{subfig:AshapedNetwork}, consisting of five nodes and five edges and having one cycle. More precisely, we consider the network defined by\n\\begin{linenomath}\n\\begin{align} \\label{eq:A_netw}\n\\begin{aligned}\n&\\mathcal{N}_S = \\mathcal{N}_S^N = \\{4, 5\\}, \\ \\mathcal{N}_M = \\{1, 2, 3\\}, \\ \\mathcal{I} = \\{1, \\ldots, 5\\}\\\\\n&\\mathbf{x}_1^1 = 0, \\ \\ \\mathbf{x}_2^1 = 0, \\ \\ \\mathbf{x}_3^2 = 0, \\ \\ \\mathbf{x}_4^2 = 0, \\ \\ \\mathbf{x}_5^3 = 0\\\\\n&\\mathbf{x}_1^2 = \\ell_1, \\ \\mathbf{x}_2^3 = \\ell_2, \\ \\mathbf{x}_3^3 = \\ell_3, \\ \\mathbf{x}_4^4 = \\ell_4, \\ \\mathbf{x}_5^5 = \\ell_5.\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\nLet us first introduce some notation concerning the eigenvalues $\\{\\lambda_i^k (x)\\}_{k=1}^{12}$ of $A_i(x)$ for $i \\in \\mathcal{I}$ and $x \\in [0, \\ell_i]$, which, as we will see in in Subsection \\ref{subsec:hyperbolic}, are such that $\\{\\lambda_i^k\\}_{k=1}^{12} \\subset C^2([0, \\ell_i])$ under Assumption \\ref{as:mass_flex}, and\n\\begin{linenomath}\n\\begin{align} \\label{eq:sign_eigval}\n\\lambda_i^k(x) <0 \\ \\text{ if } \\ k\\leq 6, \\qquad \\lambda_i^k(x) >0 \\ \\text{ if } \\ k\\geq 7.\n\\end{align}\n\\end{linenomath}\nAlso under Assumption \\ref{as:mass_flex}, and for any $i\\in\\mathcal{I}$, we define $\\Lambda_i \\in C^0([0, \\ell_i]; (0, +\\infty))$ and $T_i>0$ by\n\\begin{linenomath}\n\\begin{align} \\label{eq:def_Lambdai_Ti}\n \\Lambda_i(x) = \\left( \\min_{k \\in \\{1, \\ldots, 6\\}} \\left| \\lambda_i^k(x) \\right| \\right)^{-1} \\quad \\text{and} \\quad T_i = \\int_0^{\\ell_i} \\Lambda_i(x) dx;\n\\end{align}\n\\end{linenomath}\nnote that the minimum ranges over the \\textit{negative} eigenvalues of $A_i(x)$. \nThe latter, $T_i$, corresponds to the transmission (or travelling) time from one end of the beam $i$ to its other end (see Section \\ref{sec:controllability}).\n\n\n\n\\begin{theorem} \\label{th:controllability}\nConsider the A-shaped network defined by \\eqref{eq:A_netw}.\nSuppose that $R_i$ has the regularity \\eqref{eq:reg_beampara} and that Assumption \\ref{as:mass_flex} is fulfilled.\nLet $\\overline{T}>0$ be defined by (see \\eqref{eq:def_Lambdai_Ti})\n\\begin{linenomath}\n\\begin{align} \\label{eq:minT}\n\\overline{T} = \\max \\left\\{T_1, T_2 \\right\\} + \\max \\left\\{T_4, T_5 \\right\\}. \n\\end{align}\n\\end{linenomath}\nThen, for any $T> T^*>\\overline{T}$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$, for some $\\delta, \\gamma>0$, and\n\\begin{enumerate}[label=(\\roman*)]\n\\item for all initial data $(y_i^0)_{i \\in \\mathcal{I}}$ and boundary data $(q_n)_{n \\in \\{1, 2, 3\\}}$ of regularity \\eqref{eq:reg_Idata_IGEB}-\\eqref{eq:reg_Ndata_IGEB}, satisfying $\\|y_i^0\\|_{C_x^1} + \\|q_n\\|_{C_t^1} \\leq \\delta$ and the first-order compatibility conditions of \\eqref{eq:syst_physical}, and\n\\item for all nodal profiles $\\overline{y}_1, \\overline{y}_2 \\in C^1([T^*, T]; \\mathbb{R}^{12})$, satisfying $\\|\\overline{y}_i\\|_{C_t^1} \\leq \\gamma$ and the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the node $n=1$,\n\\end{enumerate}\nthere exist controls $q_4, q_5 \\in C^1([0, T]; \\mathbb{R}^6)$ with $\\|q_i\\|_{C_t^1}\\leq \\varepsilon$, such that \\eqref{eq:syst_physical} admits a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i] \\times [0, T]; \\mathbb{R}^{12})$, which fulfills $\\|y_i\\|_{C_x^1} \\leq \\varepsilon$ and\n\\begin{linenomath}\n\\begin{align}\\label{eq:aim}\ny_i(0, t) = \\overline{y}_i(t) \\quad \\text{for all }i \\in \\{1, 2\\}, \\, t \\in [T^*, T].\n\\end{align}\n\\end{linenomath}\n\\end{theorem}\n\nAs mentionned in Section \\ref{sec:intro}, the proof of Theorem \\ref{th:controllability}, given in Section \\ref{sec:controllability}, relies upon the existence and uniqueness theory of \\emph{semi-global} classical solutions to the network problem (here, Theorem \\ref{th:existence}), the form of the transmission condition of the network, and on a \\emph{constructive method}. The idea of the proof is to construct a solution $(y_i)_{i\\in \\mathcal{I}}$ to \\eqref{eq:syst_physical}, such that it satisfies the initial condition, the nodal conditions, and the given nodal profiles. Substituting this solution into the nodal conditions at the nodes $n \\in \\{4, 5\\}$, one then obtains the desired controls $q_4, q_5$. \\textcolor{black}{Our proof follows the lines of \\cite{Zhuang2018}, where the authors develop a methodology for proving the nodal profile controllability for A-shaped networks of canals governed by the Saint-Venant equations.}\n\n\\begin{remark} \\label{rem:controllability_thm}\nA few remarks are in order.\n\\begin{enumerate}\n\n\\item \\label{subrem:global}\n{\\color{black}\nThe smallness of the initial and nodal data and of the nodal profiles in (i) and (ii) is used to ensure the well-posedness of the mixed initial-boundary value problem for beams described by the IGEB model. This limitation leads to the local nature of the controllability result: in a sufficiently small $C^1$-neighborhood of the zero-steady state, we can construct continuously differentiable controls, which then generate a piecewise continuously differentiable solution on the whole network. \nFurthermore, the study of other equilibrium solutions for the IGEB network is relevant to achieve further objectives. For instance, in the spirit of \\cite{gugatLeugering2003}, supposing that the set of equilibria is connected, one might look to use the result of local exact controllability of nodal profiles as a basis to then prove more global results.\n}\n\n\\item {\\color{black}\nFor the system linearized around the zero-steady state, a global nodal profile controllability result is also achieved, though without any limitation on the size of the of the data and nodal profiles, as it rests on an existence and uniqueness result that does not impose such limitations. Moreover, the `optimal' estimate for the controllability time $T^*$ remains that of in Theorem \\ref{th:controllability}, given in terms of the transmission times \\eqref{eq:def_Lambdai_Ti}.\n}\n\n\\item \nThe \\emph{controllability time} $T^*$ from which one can prescribe nodal profiles, has to be large enough, depending on the lengths of the beams and the eigenvalues of $(A_i)_{i \\in \\mathcal{I}}$ (and thus, it depends on the geometrical and material properties of the beam). As we will see in Section \\ref{sec:controllability}, $\\overline{T}$ is the transmission time from the controlled nodes to the charged node. One may note that $T_i \\leq \\frac{\\ell_i}{|\\lambda_i^*|}$, where the constant $\\lambda_i^*<0$ denotes the maximum over $x$ of the largest \\emph{negative} eigenvalue of $A_i(x)$.\n\n\\item \nOne will observe in the proof of Theorem \\ref{th:controllability} that the controls $q_4, q_5$ are not unique due to the use of interpolation and arbitrary nodal conditions throughout the proof. \n\n\\item \\label{subrem:sidewise}\nIn the proof of Theorem \\ref{th:controllability}, to construct the solution $(y_i)_{i\\in\\mathcal{I}}$, one is led to solve a series of forward and sidewise problems for \\eqref{eq:IGEB_gov} for the different beams $i \\in \\mathcal{I}$ of the network. Solving a sidewise problem for \\eqref{eq:IGEB_gov} entails changing the role of $x$ and $t$, considering a governing system of the form \n\\begin{linenomath}\n\\begin{align*}\n\\partial_x y_i + A_i^{-1}\\partial_t y_i + A_i^{-1}\\overline{B}_i y_i = A_i^{-1}\\overline{g}_i(\\cdot, y_i)\n\\end{align*}\n\\end{linenomath}\nand providing ``boundary conditions'' at $t=0$ and $t = T$, and ``initial conditions'' at $x=0$ (rightward problem) or $x = \\ell_i$ (leftward problem). It is consequently important here that $A_i$ does not have any zero eigenvalue.\n\n\\end{enumerate}\n\\end{remark}\n\n\n\\subsubsection{Study of the GEB model}\n\nIn order to translate Theorems \\ref{th:existence} and \\ref{th:controllability} in terms of the GEB model, we prove the Theorem \\ref{thm:solGEB} below, which yields the existence of a unique classical solution to \\eqref{eq:GEB_netw}, provided that a unique classical solution exists for \\eqref{eq:syst_physical} and that the data of both models fulfill some compatibility conditions. \n\nLet us first introduce the compatibility conditions on the initial and boundary data of the GEB network \\eqref{eq:GEB_netw}, that will be of use in the theorem and corollaries that follow\n\\begin{linenomath}\n\\begin{subequations}\\label{eq:compat_GEB_-1_GEBtransmi}\n\\begin{align} \\label{eq:compat_GEB_-1}\n&(f_n^\\mathbf{p}, f_n^\\mathbf{R})(0) = (\\mathbf{p}_{i^n}^0, \\mathbf{R}_{i^n}^0)(\\mathbf{x}_{i^n}^n), \\quad n\\in \\mathcal{N}_S^D,\\\\\n\\label{eq:compat_GEB_transmi}\n&\\mathbf{p}_i^0(\\mathbf{x}_i^n) = \\mathbf{p}_{i^n}^0(\\mathbf{x}_{i^n}^n), \\quad (\\mathbf{R}_i^0 R_i^\\intercal)(\\mathbf{x}_i^n) = (\\mathbf{R}_{i^n}^0 R_{i^n}^\\intercal)(\\mathbf{x}_{i^n}^n), \\quad i\\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M,\n\\end{align}\n\\end{subequations}\n\\end{linenomath}\nand\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{subequations}\\label{eq:compat_GEB_01}\n\\begin{align}\n\\label{eq:compat_GEB_01_1}\n&\\mathbf{p}_i^1(\\mathbf{x}_i^n) = \\mathbf{p}_{i^n}^1(\\mathbf{x}_{i^n}^n), \\quad w_i^0(\\mathbf{x}_i^n) = w_{i^n}^0(\\mathbf{x}_{i^n}^n), \\quad i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M\\\\\n\\label{eq:compat_GEB_01_2}\n&\\sum_{i \\in \\mathcal{I}^n} \\tau_i^n \\left(\\overline{R}_i \\mathbf{C}_i^{-1} \\begin{bmatrix}\n(\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_i^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_i^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_i^0 - R_i^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_i \\right)\n\\end{bmatrix}\\right)(\\mathbf{x}_i^n) = q_n(0), \\quad n \\in \\mathcal{N}_M,\\\\\n\\label{eq:compat_GEB_01_3}\n&\\tau_{i^n}^n \\left(\\overline{R}_{i^n} \\mathbf{C}_{i^n}^{-1} \\begin{bmatrix}\n(\\mathbf{R}_{i^n}^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{p}_{i^n}^0 - e_1\\\\\n\\mathrm{vec}\\left( (\\mathbf{R}_{i^n}^0)^{\\intercal} \\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{R}_{i^n}^0 - R_{i^n}^{\\intercal}\\frac{\\mathrm{d}}{\\mathrm{d}x} R_{i^n} \\right)\n\\end{bmatrix} \\right)(\\mathbf{x}_{i^n}^n) = q_n(0), \\quad n \\in \\mathcal{N}_S^N,\\\\\n\\label{eq:compat_GEB_01_4}\n&\\mathbf{p}_{i^n}^1(\\mathbf{x}_{i^n}^n) = \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{p}(0), \\quad w_{i^n}^0(\\mathbf{x}_{i^n}^n) = \\frac{\\mathrm{d}}{\\mathrm{d}t}f_n^\\mathbf{R}(0), \\quad n \\in \\mathcal{N}_S^D.\n\\end{align}\n\\end{subequations}\n\\end{linenomath}\n\\endgroup\n\n\\begin{theorem} \\label{thm:solGEB}\nConsider a general network, and assume that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{thm:solGEB_c1} the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i, R_i)$ and initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ have the regularity \\eqref{eq:reg_beampara} and \\eqref{eq:reg_Idata_GEB}, and $y_i^0$ is the associated function defined by \\eqref{eq:rel_inidata},\n\n\\item \\label{thm:solGEB_c2} the Neumann data $f_n = f_n(t, \\mathbf{R}_{i^n})$ are of the form \\eqref{eq:def_fn}, for given functions $q_n$ of regularity \\eqref{eq:reg_Ndata_IGEB},\n\n\\item \\label{thm:solGEB_c3} the Dirichlet data $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$ are of regularity \\eqref{eq:reg_Ndata_GEB_D}, and $q_n$ are the associated functions defined by \\eqref{eq:def_qn_D},\n\n\\item \\label{thm:solGEB_c4} the compatibility conditions \\eqref{eq:compat_GEB_-1_GEBtransmi} hold.\n\\end{enumerate}\nThen, if there exists a unique solution $(y_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i]\\times [0, T]; \\mathbb{R}^{12})$ to \\eqref{eq:syst_physical} with initial and nodal data $y_i^0$ and $q_n$ (for some $T>0$), there exists a unique solution\n$(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^2([0, \\ell_i]\\times [0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$\nto \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and nodal data $f_n$, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$, and $(y_i)_{i\\in \\mathcal{I}} = \\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}})$.\n\\end{theorem}\n\n\\begin{remark}\nWe have the following restriction on the form of the Neumann data $f_n$: it must be possible to express it as a function $q_n = q_n(t)$ in the body-attached basis (see Subsection \\ref{subsec:GEBmodels}).\n\\end{remark}\n\n\nThe proof of Theorem \\ref{thm:solGEB}, given in Section \\ref{sec:invert_transfo}, consists in using the last six equations of \\eqref{eq:IGEB_gov} as compatibility conditions to prove that the transformation $\\mathcal{T}$, defined in \\eqref{eq:transfo}, is bijective on some spaces (see Lemma \\ref{lem:invert_transfo}); this relies on the use of quaternions \\cite{chou1992} to parametrize the rotations matrices, and existence and uniqueness results for (seemingly overdetermined) first-order linear PDE systems. Once that this property of the transformation is established, one recovers notably the governing system \\eqref{eq:GEB_gov} by using the first six equations of \\eqref{eq:IGEB_gov}. The transmission conditions are recovered by first showing that the rigid joint assumption \\eqref{eq:GEB_rigid_angles} is fulfilled and then deducing \\eqref{eq:GEB_continuity_pi}-\\eqref{eq:GEB_Kirchhoff} from \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}.\n\n\nCorollary \\ref{coro:wellposedGEB} below follows from Theorem \\ref{th:existence} and Theorem \\ref{thm:solGEB}.\n\n\n\n\\begin{corollary}\n\\label{coro:wellposedGEB}\nConsider a general network and suppose that the conditions \\ref{thm:solGEB_c1}-\\ref{thm:solGEB_c2}-\\ref{thm:solGEB_c3}-\\ref{thm:solGEB_c4} of Theorem \\ref{thm:solGEB} are fulfilled, suppose that the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i)$ satisfy Assumption \\ref{as:mass_flex}, and that the compatibility conditions \\eqref{eq:compat_GEB_01} hold.\nThen, for any $T>0$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon \\in (0, \\varepsilon_0)$, and for some $\\delta>0$, if moreover $\\|y_i^0\\|_{C_x^1}+ \\|q_n\\|_{C_t^1}\\leq \\delta$, then there exists a unique solution $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^2([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$ to \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and nodal data $f_n$, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$.\n\\end{corollary}\n\n\\begin{remark}\nUnder \\eqref{eq:compat_GEB_-1_GEBtransmi}, the conditions \\eqref{eq:compat_GEB_01} are just an equivalent way of imposing that $y_i^0$ fulfills the first-order compatibility conditions of \\eqref{eq:syst_physical}, but expressed in terms of the data of the GEB model.\n\\end{remark}\n\n\nFinally, from Theorems \\ref{th:controllability} and \\ref{thm:solGEB}, one obtains Corollary \\ref{coro:controlGEB} below.\n\n\n\n\\begin{corollary} \\label{coro:controlGEB}\nConsider the A-shaped network defined by \\eqref{eq:A_netw}, and assume that\n\n\\begin{enumerate}[label=(\\roman*)]\n\\item the beam parameters $(\\mathbf{M}_i, \\mathbf{C}_i, R_i)$ and initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ have the regularity \\eqref{eq:reg_beampara} and \\eqref{eq:reg_Idata_GEB}, the former satisfy Assumption \\ref{as:mass_flex} and the latter fulfill \\eqref{eq:compat_GEB_transmi}, and $y_i^0$ is the associated function defined by \\eqref{eq:rel_inidata},\n\n\\item the Neumann data $f_n = f_n(t, \\mathbf{R}_{i^n})$, for $n \\in \\{1, 2, 3\\}$ are of the form \\eqref{eq:def_fn}, for given functions $q_n$ of regularity \\eqref{eq:reg_Ndata_IGEB},\n\n\n\\item the compatibility conditions \\eqref{eq:compat_GEB_01_1}-\\eqref{eq:compat_GEB_01_2} for all $n \\in \\{1, 2, 3\\}$ hold.\n\\end{enumerate}\nLet $\\overline{T}>0$ be defined by \\eqref{eq:minT}. Then, for any $T>T^*>\\overline{T}$, there exists $\\varepsilon_0>0$ such that for all $\\varepsilon\\in (0, \\varepsilon_0)$, for some $\\delta, \\gamma>0$, and for any nodal profiles $\\overline{y}_1, \\overline{y}_2 \\in C^1([T^*, T]; \\mathbb{R}^{12})$ satisfying $\\|\\overline{y}_i\\|_{C_t^1}\\leq \\gamma$ and the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the node $n=1$, if additionally $\\|y_i^0\\|_{C_x^1} + \\|f_n\\|_{C_t^1} \\leq \\delta$ ($i \\in \\mathcal{I}, \\ n \\in \\{1, 2, 3\\}$), then there exist controls $f_4, f_5 \\in C^1([0, T]; \\mathbb{R}^6)$ with $\\|f_n\\|_{C_t^1}\\leq \\varepsilon$ such that System \\eqref{eq:GEB_netw} with initial data $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, \\mathbf{p}_i^1, w_i^0)$ and boundary data $(f_n)_{n \\in \\{1, 2, 3\\}}$, admits a unique solution $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in \\prod_{i=1}^N C^2([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3))$, and $(y_i)_{i\\in\\mathcal{I}} := \\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}})$ fulfills $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon$ and the nodal profiles \\eqref{eq:aim}.\n\\end{corollary}\n\n\\begin{remark}\nIn Corollary \\ref{coro:controlGEB},\n\\begin{enumerate}\n\\item the profiles given at the node $n=1$ affect the intrinsic variables $\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i)$, for $i \\in \\{1, 2\\}$, and not directly the displacements and rotations $(\\mathbf{p}_i, \\mathbf{R}_i)$;\n\n\\item for $i \\in \\{4,5\\}$ the control $f_i$ is given by \\eqref{eq:def_fn} where $q_i$ is the control provided by Theorem \\ref{th:controllability} for System \\eqref{eq:syst_physical}. The smallness of the $C^1$ norm of $f_i$ comes from a combination of the fact that $q_i$ and $y_i$ (and thus, as can be seen in \\eqref{eq:form_yi}, also the angular velocity $W_i$) have small $C^1$ norms, and that the expression of $f_i$ and $\\frac{\\mathrm{d}}{\\mathrm{d}t}f_i$ involves only the functions $q_i, W_i$ and the unitary matrices $\\mathbf{R}_i, R_i$. Indeed, $f_i = \\mathrm{diag}\\big(\\mathbf{R}_i(\\ell_i, \\cdot), \\mathbf{R}_i(\\ell_i, \\cdot)\\big) q_i$ and one may compute that\n\\begin{linenomath}\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} f_i = \\mathrm{diag}\\big(\\mathbf{R}_i(\\ell_i, \\cdot), \\mathbf{R}_i(\\ell_i, \\cdot)\\big) \\frac{\\mathrm{d}}{\\mathrm{d}t} q_i + \\mathrm{diag}\\big((\\mathbf{R}_i \\widehat{W}_i)(\\ell_i, \\cdot), (\\mathbf{R}_i \\widehat{W}_i)(\\ell_i, \\cdot)\\big) q_i.\n\\end{align*}\n\\end{linenomath}\n\\end{enumerate}\n\\end{remark}\n\n\n\n\n\n\\section{Existence and uniqueness for the IGEB network}\n\\label{sec:exist}\n\n\nWe now turn to the proof of Theorem \\ref{th:existence}.\n\n\\subsection{Hyperbolicity of the system}\n\\label{subsec:hyperbolic}\n\nLet $T >0$, $i\\in \\mathcal{I}$ and $x \\in [0, \\ell_i]$. One may quickly verify that the matrix $A_i(x)$, defined in \\eqref{eq:def_Ai}, has only real eigenvalues: six positive ones which are the square roots of the eigenvalues of $\\Theta_i(x)$ (defined in Assumption \\ref{as:mass_flex}), and six negative ones which are equal to the former but with a minus sign.\nFurthermore, some computations yield the following lemma whose proof is given in \\cite[Section 4]{R2020}. \n\n\n\\begin{lemma}\nSuppose that Assumption \\ref{as:mass_flex} is fulfilled and, for any $i \\in \\mathcal{I}$, let $U_i$, $D_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{6\\times 6})$ be the functions introduced in Assumption \\ref{as:mass_flex}.\nThen, $A_i \\in C^2([0, \\ell_i]; \\mathbb{R}^{12\\times 12})$ may be diagonalized as follows. One has $A_i = L_i^{-1} \\mathbf{D}_i L_i$ in $[0, \\ell_i]$, where $\\mathbf{D}_{i}$, $L_i \\in C^2( [0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ are defined by\n\\begin{linenomath}\n\\begin{equation} \\label{eq:def_bfDi_Li}\n\\mathbf{D}_i = \\mathrm{diag}(-D_{i}, D_{i}), \\qquad L_i = \\begin{bmatrix}\nU_i \\mathbf{C}_i^{-\\sfrac{1}{2}} & D_{i}U_i \\mathbf{C}_i^{\\sfrac{1}{2}} \\\\\nU_i \\mathbf{C}_i^{-\\sfrac{1}{2}} & - D_{i}U_i \\mathbf{C}_i^{\\sfrac{1}{2}}\n\\end{bmatrix},\n\\end{equation}\n\\end{linenomath}\nand the inverse $L_i^{-1} \\in C^2( [0, \\ell_i]; \\mathbb{R}^{12 \\times 12})$ is given by\n\\begin{linenomath}\n\\begin{align} \\label{eq:inverseLi}\nL_i^{-1} = \\frac{1}{2} \\begin{bmatrix}\n\\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal & \\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal \\\\\n\\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1} & - \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1}\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\n\\end{lemma}\n\n\n\n\\subsection{Change of variable to Riemann invariants} \n\\label{subsec:change_var}\n\nNow, we can write \\eqref{eq:syst_physical} in diagonal form by applying the change of variable \n\\begin{linenomath}\n\\begin{align} \\label{eq:change_var_Li}\nr_i(x,t) = L_i(x) y_i(x,t), \\qquad \\text{for all }x \\in [0, \\ell_i], \\ t \\in [0, T], \\ i \\in \\mathcal{I}.\n\\end{align}\n\\end{linenomath}\nThe first (resp. last) six components of $r_i$ correspond to the negative (resp. positive) eigenvalues of $A_i$, thus, for all $i \\in \\mathcal{I}$, we denote\n\\begin{linenomath}\n\\begin{align*}\nr_i = \\begin{bmatrix}\nr_i^-\\\\\nr_i^+\n\\end{bmatrix}, \\qquad r_i^-,\\, r_i^+ \\colon [0, \\ell_i]\\times [0, T] \\rightarrow \\mathbb{R}^6.\n\\end{align*}\n\\end{linenomath}\nIn addition, in order to write the transmission conditions concisely, we introduce the invertible matrix $\\gamma_i^n$ and positive definite symmetric matrix $\\sigma_i^n$\n\\begin{linenomath}\n\\begin{align*}\n\\gamma_i^n &= (\\overline{R}_i \\mathbf{C}_i^{\\sfrac{1}{2}} U_i^\\intercal)(\\mathbf{x}_i^n), \\qquad\n\\sigma_i^n = (\\overline{R}_i \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1} U_i \\mathbf{C}_i^{-\\sfrac{1}{2}} \\overline{R}_i^\\intercal)(\\mathbf{x}_i^n)\n\\end{align*}\n\\end{linenomath}\nfor all $n \\in \\mathcal{N}$ and $i \\in \\mathcal{I}^n$.\nNotice that $\\sigma_i^n \\gamma_i^n = \\overline{R}_i(\\mathbf{x}_i^n) \\mathbf{C}_i^{-\\sfrac{1}{2}} U_i^\\intercal D_i^{-1}$.\n\n\\medskip\n\n\\noindent Then, taking \\eqref{eq:def_bfDi_Li}-\\eqref{eq:inverseLi} into account, the system obtained by applying the change of variable \\eqref{eq:change_var_Li} to System \\eqref{eq:syst_physical} reads\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:syst_diagonal}}\n\\label{eq:r_IGEB_gov}\n\\partial_t r_i + \\mathbf{D}_i \\partial_x r_i + B_i r_i = g_i(\\cdot, r_i), &\\hspace{-0.45cm}$\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}$\\\\\n\\label{eq:r_IGEB_cont_velo}\n\\gamma_i^n (r_i^- + r_i^+)(\\mathbf{x}_i^n, t) &\\nonumber \\vspace{-0.2cm}\\\\\n\\qquad \\quad = \\gamma_{i^n}^n (r_{i^n}^- + r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t), &\\hspace{-0.45cm}$t \\in (0, T), \\, i \\in \\mathcal{I}^n, \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:r_IGEB_Kirchhoff}\n\\sum_{i\\in\\mathcal{I}^n} \\frac{\\tau_i^n}{2} \\sigma_i^n \\gamma_i^n (r_i^- - r_i^+)(\\mathbf{x}_i^n, t) = q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_M$\\\\\n\\label{eq:r_IGEB_condNSz}\n(r_{i^n}^- - r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t) &\\vspace{-0.2cm} \\nonumber\\\\\n\\qquad \\quad = 2 \\tau_{i^n}^n (D_{i^n} U_{i^n}\\mathbf{C}_{i^n}^{\\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_S^N$\\\\\n\\label{eq:r_IGEB_condNSv}\n(r_{i^n}^- + r_{i^n}^+)(\\mathbf{x}_{i^n}^n, t) &\\vspace{-0.2cm} \\nonumber\\\\\n\\qquad \\quad = 2 (U_{i^n} \\mathbf{C}_{i^n}^{- \\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) q_n(t), &\\hspace{-0.45cm}$t \\in (0, T), \\, n \\in \\mathcal{N}_S^D$\\\\\n\\label{eq:r_IGEB_ini_cond}\nr_i(x, 0) = r_i^0(x), &\\hspace{-0.45cm}$x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}$.\n\\end{subnumcases}\n\\end{linenomath}\nIn the governing system \\eqref{eq:r_IGEB_gov}, the coefficient $B_i \\in C^1([0, \\ell_i]; \\mathbb{R}^{12\\times 12})$ is defined by $B_i(x) = L_i(x) \\overline{B}_i(x) L_i(x)^{-1} + L_i(x) A_i(x) \\frac{\\mathrm{d}}{\\mathrm{d}x}L_i^{-1}(x)$, while the source is defined by $g_i(x,u) = L_i(x) \\overline{g}_i(x,L_i(x)^{-1} u)$ for all $i \\in \\mathcal{I}$, $x \\in [0, \\ell_i]$ and $u \\in \\mathbb{R}^{12}$. The corresponding initial data in \\eqref{eq:r_IGEB_ini_cond} for this system is $r_i^0 = L_i y_i^0$.\n\n\n\n\\subsection{Outgoing and incoming information}\n\\label{subsec:out_in_info}\n\n\n\\begin{figure}\n \\begin{subfigure}{0.6\\textwidth}\n \\centering\n \\includegraphics[scale=0.75]{enter-leave-charactC}\n\\caption{Characteristic curves $(\\mathbf{x}(t), t)$ with $\\frac{\\mathrm{d}\\mathbf{x}}{\\mathrm{d}t}(t) = \\lambda(\\mathbf{x}(t))$, where either $\\lambda(s)>0$ or $\\lambda(s)<0$ for all $s \\in [0, \\ell_i]$.}\n\\label{fig:charac}\n \\end{subfigure}%\n \\hspace*{\\fill} \n \\begin{subfigure}{0.4\\textwidth}\n \\centering\n\\includegraphics[scale=0.7]{NM_notationC}\n \\caption{Form of $\\mathcal{I}_n$ at a node $n$.}\n \\label{fig:NM_notation}\n \\end{subfigure}\n\n\\caption{Outgoing and incoming information.}\n\\label{fig:out_in_info}\n\\end{figure}\n\n\n\nFor any $n \\in \\mathcal{N}$, let us denote by $s_n \\in \\{0, \\ldots, k_n\\}$ (resp. by $k_n-s_n$) the number of beams ending (resp. starting) at the node $n$; see Fig. \\ref{fig:NM_notation}. More precisely, we suppose that \n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{I}^n = \\{i_1, \\ldots, i_{k_n}\\} \\quad \\text{with} \\quad i_1 < i_2 < \\ldots < i_{s_n} \\quad \\text{and} \\quad i_{s_n+1} < i_{s_n+2} < \\ldots < i_{k_n},\n\\end{align*}\n\\end{linenomath}\nand that $\\tau_{i_\\alpha}^n = -1$ for all $\\alpha \\in \\{1, \\ldots, s_n\\}$, while $\\tau_{i_\\alpha}^n = +1$ for all $\\alpha \\in \\{s_n+1, \\ldots, k_n\\}$. This is not to be confused with the notation $i^n$ introduced in \\eqref{eq:def_in}.\n\n\n\\medskip\n\n\n\\noindent For any node $n$ and any incident edge $i \\in \\mathcal{I}^n$, we call \\emph{outgoing} (resp. \\emph{incoming}) \\emph{information}, the components of $r_i$ which correspond to characteristics entering (resp. leaving) the domain $[0, \\ell_i]\\times[0, +\\infty)$ at this node (see Fig. \\eqref{fig:charac}). \n\n\nNamely, here, the outgoing (resp. incoming) information at the node $n$ is $r_{i_\\alpha}^-(\\ell_{i_\\alpha}, t)$ (resp. $r_{i_\\alpha}^+(\\ell_{i_\\alpha}, t)$) for all $\\alpha \\in \\{1, \\ldots, s_n\\}$, and $r_{i_k}^+(0, t)$ (resp. $r_{i_k}^-(0, t)$) for all $k \\in \\{s_n+1, \\ldots, k_n\\}$.\nWe then define the functions $r_n^\\mathrm{out}, r_n^\\mathrm{in} \\colon [0, T] \\rightarrow \\mathbb{R}^{6k_n}$ by\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{linenomath}\n\\begin{align*}\nr_n^\\mathrm{out}(t) = \\begin{bmatrix}\nr_{i_1}^-(\\ell_{i_1}, t)\\\\\n\\vdots\\\\\nr_{i_{s_n}}^-(\\ell_{i_{s_n}}, t)\\\\\nr_{i_{s_n+1}}^+(0, t) \\\\\n\\vdots\\\\\nr_{i_{k_n}}^+(0, t)\n\\end{bmatrix}, \\qquad r_n^\\mathrm{in}(t) = \\begin{bmatrix}\nr_{i_1}^+(\\ell_{i_1}, t)\\\\\n\\vdots\\\\\nr_{i_{s_n}}^+(\\ell_{i_{s_n}}, t)\\\\\nr_{i_{s_n+1}}^-(0, t) \\\\\n\\vdots\\\\\nr_{i_{k_n}}^-(0, t)\n\\end{bmatrix}.\n\\end{align*}\n\\end{linenomath}\n\\endgroup\nWe also denote $r_n^\\mathrm{out} = ((r_{n,1}^\\mathrm{out})^\\intercal, \\ldots, (r_{n,k_n}^\\mathrm{out})^\\intercal)$, where $r_{n,\\alpha}^\\mathrm{out}(t) \\in \\mathbb{R}^6$ for all $\\alpha \\in \\{1, \\ldots, k_n\\}$; a similar notation is used for $r_n^\\mathrm{in}$.\n\n\n\\medskip\n\n\n\\noindent Taking into account this notation, and the sign of $\\tau_i^n$, we observe that the Kirchhoff condition \\eqref{eq:r_IGEB_Kirchhoff} is equivalent to\n\\begin{linenomath}\n\\begin{align*}\n-\\sum_{\\alpha =1}^{s_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n (r_{i_\\alpha}^- - r_{i_\\alpha}^+)(0, t) + \\sum_{k=s_n+1}^{k_n} \\sigma_{i_k}^n \\gamma_{i_k}^n (r_{i_k}^- - r_{i_k}^+)(\\ell_{i_k}, t) = 2q_n(t),\n\\end{align*}\n\\end{linenomath}\nwhich can also be written in the form\n\\begin{linenomath} \n\\begin{align*}\n\\sum_{\\alpha=1}^{k_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{out}(t) = \\sum_{\\alpha=1}^{k_n} \\sigma_{i_\\alpha}^n \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{in}(t) + 2q_n(t).\n\\end{align*}\n\\end{linenomath}\nThe continuity condition \\eqref{eq:r_IGEB_cont_velo} is equivalent to\n\\begin{linenomath}\n\\begin{align*}\n\\gamma_{i_1}^n (r_{i_1}^- + r_{i_1}^+)(\\mathbf{x}_{i_1}^n, t) = \\gamma_{i_\\alpha}^n (r_{i_\\alpha}^- + r_{i_\\alpha}^+)(\\mathbf{x}_{i_\\alpha}^n, t) \\quad \\text{for all }\\alpha \\in \\{2, \\ldots, k_n\\}\n\\end{align*}\n\\end{linenomath}\nwhich can be seen to also write as \n\\begin{linenomath}\n\\begin{align*}\n\\gamma_{i_1}^n r_{n,1}^\\mathrm{out}(t) - \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{out}(t) = - \\gamma_{i_1}^n r_{n,1}^\\mathrm{in}(t) + \\gamma_{i_\\alpha}^n r_{n,\\alpha}^\\mathrm{in}(t) \\quad \\text{for all }\\alpha \\in \\{2, \\ldots, k_n\\}.\n\\end{align*}\n\\end{linenomath}\nHence, at any multiple node $n$, the transmission conditions \\eqref{eq:r_IGEB_Kirchhoff}-\\eqref{eq:r_IGEB_cont_velo} are equivalent to the following system:\n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{A}_n \\mathbf{G}_n r_n^\\mathrm{out}(t) = \\mathbf{B}_n \\mathbf{G}_n r_n^\\mathrm{in}(t) + \\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.8}\n\\begin{bmatrix}\n2q_n(t) \\\\ \\mathbf{0}_{6k_n-6, 1}\n\\end{bmatrix},\n\\endgroup\n\\end{align*}\n\\end{linenomath}\nwhere $\\mathbf{A}_n, \\mathbf{B}_n, \\mathbf{G}_n \\in \\mathbb{R}^{6k_n \\times 6k_n}$ are defined by\n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{A}_n = \\begin{bmatrix}\n\\mathbf{a}_n & \\mathbf{b}_n\\\\\n\\mathbf{c}_n & \\mathbf{I}_{6(k_n-1)}\n\\end{bmatrix}, \\quad\n\\mathbf{B}_n = \\begin{bmatrix}\n\\mathbf{a}_n & \\mathbf{b}_n\\\\\n-\\mathbf{c}_n & -\\mathbf{I}_{6(k_n-1)}\n\\end{bmatrix}, \\quad \\mathbf{G}_n = \\mathrm{diag}(\\gamma_{i_1}^n, \\ldots, \\gamma_{i_{k_n}}^n),\n\\end{align*}\n\\end{linenomath}\nthe sub-matrices $\\mathbf{a}_n \\in \\mathbb{R}^{6\\times 6}$, $\\mathbf{b}_n \\in \\mathbb{R}^{6\\times 6(k_n-1)}$ and $\\mathbf{c}_n \\in \\mathbb{R}^{6(k_n-1) \\times 6}$ being defined by $\\mathbf{a}_n = \\sigma_{i_1}^n$, $\\mathbf{b}_n = \\big[ \\sigma^n_{i_2} \\ \\sigma^n_{i_3} \\ \\ldots \\ \\sigma^n_{i_{k_n}}\\big]$ and $\\mathbf{c}_n = - \\big[\\mathbf{I}_6 \\ \\mathbf{I}_6 \\ \\ldots \\ \\mathbf{I}_6 \\big]^\\intercal$.\n\n\nThe matrix $\\mathbf{G}_n$ is clearly invertible and one can check that $\\mathbf{A}_n$ is also invertible (using the same reasoning as \\cite[Lemma 4.4]{R2020}).\nFor any $n \\in \\mathcal{N}$, let us define $\\mathcal{B}_n \\in \\mathbb{R}^{6k_n \\times 6 k_n}$ by\n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{B}_n = \\left\\{\\begin{aligned}\n&\\mathbf{G}_n^{-1} \\mathbf{A}_n^{-1} \\mathbf{B}_n \\mathbf{G}_n && n \\in \\mathcal{N}_M\\\\\n&\\mathbf{I}_6 && n \\in \\mathcal{N}_S^N\\\\\n&-\\mathbf{I}_6 && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\\right.\n\\end{align*}\n\\end{linenomath}\nas well as $\\mathcal{Q}_n \\in \\mathbb{R}^{6k_n \\times 6 k_n}$ and $\\mathbf{q}_n \\in C^1([0, T]; \\mathbb{R}^{6k_n})$ by\n\\begin{linenomath}\n\\begin{align*}\n\\mathcal{Q}_n = \\left\\{\\begin{aligned}\n&2 \\mathbf{G}_n^{-1} \\mathbf{A}_n^{-1} && n \\in \\mathcal{N}_M\\\\\n&2 \\tau_{i^n}^n (D_{i^n} U_{i^n}\\mathbf{C}_{i^n}^{\\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) && n \\in \\mathcal{N}_S^N\\\\\n&2 (U_{i^n} \\mathbf{C}_{i^n}^{- \\sfrac{1}{2}})(\\mathbf{x}_{i^n}^n) && n \\in \\mathcal{N}_S^D,\n\\end{aligned}\\right. \\quad\n\\mathbf{q}_n(t) = \\begin{cases}\n\\begingroup\n\\setlength\\arraycolsep{3pt}\n\\renewcommand*{\\arraystretch}{0.9}\n\\begin{bmatrix}\nq_n(t) \\\\ \\mathbf{0}_{6k_n-6, 1}\n\\end{bmatrix}\\endgroup & n \\in \\mathcal{N}_M\\\\\nq_n(t) & n \\in \\mathcal{N}_S.\n\\end{cases}\n\\end{align*}\n\\end{linenomath}\nThen, System \\eqref{eq:syst_diagonal} also reads\n\\begin{linenomath}\n\\begin{align*}\n\\begin{dcases}\n\\partial_t r_i + \\mathbf{D}_i(x) \\partial_x r_i + B_i(x) r_i = g_i(x, r_i) &\\text{in } (0, \\ell_i)\\times(0, T), \\, i \\in \\mathcal{I}\\\\\nr^\\mathrm{out}_n(t) = \\mathcal{B}_n r^\\mathrm{in}_n(t) + \\mathcal{Q}_n \\mathbf{q}_n(t) & t \\in (0, T), \\, n \\in \\mathcal{N}\\\\\nr_i(x, 0) = r_i^0(x) & x \\in (0, \\ell_i), \\, i \\in \\mathcal{I}.\n\\end{dcases}\n\\end{align*}\n\\end{linenomath}\n\n\n\n\\subsection{Proof of Theorem \\ref{th:existence}}\n\\label{subsec:proof_exist}\n\nRelying upon Subsections \\ref{subsec:hyperbolic}, \\ref{subsec:change_var} and \\ref{subsec:out_in_info}, and \\cite{li2010controllability, wang2006exact}, we now prove Theorem \\ref{th:existence}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:existence}]\n\nThe local and semi-global existence and uniqueness of $C_{x,t}^1$ solutions to general one-dimensional quasilinear hyperbolic systems have been addressed in \\cite[Lem. 2.3, Th. 2.1]{wang2006exact}, which is an extension of \\cite[Lem. 2.3, Th. 2.5]{li2010controllability} to nonautonomous systems.\n\n\nSuch results may be applied to the network system \\eqref{eq:syst_physical}, since it can be written as a single larger hyperbolic system. One needs only to apply the change of variable $\\widetilde{r}_i(\\xi, t) = r_i(\\ell_i \\ell^{-1} \\xi, t)$ for all $i \\in \\mathcal{I}$, $\\xi \\in [0, \\ell]$ and $t \\in [0, T]$ for some $\\ell>0$, in order to make the spatial domain identical for all beams, and consider the larger $\\mathbb{R}^{12N}$-valued unknown $\\widetilde{r} = (\\widetilde{r}_1^{\\,\\intercal}, \\ldots, \\widetilde{r}_N^{\\, \\intercal})^\\intercal$. Then, $\\widetilde{r}$ is governed by\n\\begin{linenomath}\n\\begin{align} \\label{eq:single_hyperb_syst}\n\\begin{cases}\n\\partial_t \\widetilde{r} + \\widetilde{\\mathbf{D}}(\\xi) \\partial_\\xi \\widetilde{r} + \\widetilde{B}(\\xi) \\widetilde{r} = \\widetilde{g}(\\widetilde{r}) & \\text{in }(0, \\ell)\\times(0, T)\\\\\n\\widetilde{r}^\\mathrm{\\, out}(t) = \\widetilde{\\mathcal{B}} \\, \\widetilde{r}^\\mathrm{\\, in}(t) + \\widetilde{Q}\\widetilde{\\mathbf{q}}(t) & t \\in (0, T)\\\\\n\\widetilde{r}(\\xi, 0) = \\widetilde{r}^0(\\xi) & \\xi \\in (0, \\ell),\n\\end{cases}\n\\end{align}\n\\end{linenomath}\nwhere $\\widetilde{\\mathbf{D}}, \\widetilde{B}, \\widetilde{\\mathcal{B}}, \\widetilde{\\mathcal{Q}}, \\widetilde{\\mathbf{q}}, \\widetilde{r}^\\mathrm{out}, \\widetilde{r}^\\mathrm{in}, \\widetilde{r}^0$ and $\\widetilde{g}$ are defined by\n\\begin{linenomath}\n\\begin{align*}\n&\\widetilde{\\mathbf{D}}(\\cdot) = \\ell \\mathrm{diag}\\left(\\ell_1^{-1}\\mathbf{D}_1(\\ell_1 \\ell^{-1} \\cdot), \\ldots, \\ell_N^{-1} \\mathbf{D}_N(\\ell_N \\ell^{-1} \\cdot) \\right),\\\\\n&\\widetilde{B}(\\cdot) = \\mathrm{diag}\\left(B_1(\\ell_1 \\ell^{-1} \\cdot), \\ldots, B_N(\\ell_N \\ell^{-1} \\cdot)\\right),\\\\\n&\\widetilde{\\mathcal{B}} = \\mathrm{diag}\\left(\\mathcal{B}_1, \\ldots, \\mathcal{B}_{\\#\\mathcal{N}}\\right), \\quad \\widetilde{\\mathcal{Q}} = \\mathrm{diag}\\left(\\mathcal{Q}_1, \\ldots, \\mathcal{Q}_{\\#\\mathcal{N}}\\right), \\quad \\widetilde{\\mathbf{q}} = (\\mathbf{q}_1^\\intercal, \\ldots, \\mathbf{q}_{\\# \\mathcal{N}}^\\intercal)^\\intercal,\\\\\n&\\widetilde{r}^\\mathrm{out} = \\left((r_1^\\mathrm{out})^\\intercal, \\ldots, (r_{\\#\\mathcal{N}}^\\mathrm{out})^\\intercal\\right)^\\intercal, \\quad \\widetilde{r}^\\mathrm{in} = \\left((r_1^\\mathrm{in})^\\intercal, \\ldots, (r_{\\#\\mathcal{N}}^\\mathrm{in})^\\intercal\\right)^\\intercal,\\\\\n&\\widetilde{r}^0(\\cdot) = \\left(r^0(\\ell_1 \\ell^{-1} \\cdot)^\\intercal, \\ldots, r^0(\\ell_N, \\ell^{-1} \\cdot)^\\intercal \\right)^\\intercal\\\\\n&\\widetilde{g}(\\cdot, \\mathbf{u}) = \\left(\\widetilde{g}_1(\\ell_1 \\ell^{-1} \\cdot, \\mathbf{u}_1)^\\intercal, \\ldots, \\widetilde{g}_N(\\ell_N \\ell^{-1} \\cdot, \\mathbf{u}_N)^\\intercal \\right)^\\intercal,\n\\end{align*}\n\\end{linenomath}\nwhere we denoted $\\mathbf{u} = (\\mathbf{u}_1^\\intercal, \\ldots, \\mathbf{u}_N^\\intercal)^\\intercal$ with $\\mathbf{u}_i \\in \\mathbb{R}^{12}$ for all $i \\in \\mathcal{I}$.\n\nDue to Subsection \\ref{subsec:out_in_info}, the boundary conditions of \\eqref{eq:single_hyperb_syst} are directly written in such a way that the outgoing information for System \\eqref{eq:single_hyperb_syst} is a function of the incoming information, a sufficient criteria in \\cite{li2010controllability, wang2006exact} to deduce well-posedness of the system.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Controllability of nodal profiles for the IGEB network}\n\\label{sec:controllability}\n\n\nWe now consider the A-shaped network defined by \\eqref{eq:A_netw} and our aim is to prove Theorem \\ref{th:controllability}. As pointed out in Section \\ref{sec:model_results}, we will solve several forward and sidewise problems for \\eqref{eq:IGEB_gov} (see Steps 1.3, 1.4, 1.5). The existence and uniqueness of semi-global in time solutions to these problems is provided by \\cite{li2010controllability, wang2006exact}, as in Section \\ref{sec:exist} for the overall network.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:controllability}]\nThe proof is divided in three steps.\nWe start by constructing a solution satisfying all transmission conditions and the nodal profiles.\nThe choice of $\\overline{T}$ (see \\eqref{eq:minT}), and thus $T^*$, is explained in Step 2. \n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.1 (see Fig. \\ref{fig:A} top-left).}\nConsider the forward problem for the entire network until time $\\overline{T}$, where at the simple nodes $n \\in \\{4,5\\}$, the controls $q_4, q_5$ are replaced by any functions $\\overline{q}_4, \\overline{q}_5 \\in C^1([0; \\overline{T}]; \\mathbb{R}^6)$ satisfying the first-order compatibility conditions of \\eqref{eq:syst_physical}. By Theorem \\ref{th:existence}, for any $\\gamma>0$ small enough, there exists $\\delta>0$ such that \\eqref{eq:syst_physical} admits a unique solution $(y_i^f)_{i\\in \\mathcal{I}} \\in \\prod_{i=1}^N C^1([0, \\ell_i]\\times[0, \\overline{T}]; \\mathbb{R}^{12})$ with $\\|y_i^f\\|_{C_{x,t}^1}\\leq \\gamma$, provided that $\\|y_i^0\\|_{C_x^1}+ \\|q_n\\|_{C_t^1}+\\|\\overline{q}_k\\|_{C_t^1} \\leq \\delta$ for all $i \\in \\mathcal{I}, n \\in \\{1, 2, 3\\}$ and $k \\in \\{4, 5\\}$.\n\n\nSimilarly to the state $y_i$ (see \\eqref{eq:form_yi}), we denote $y_i^f = ((v_i^f)^\\intercal, (z_i^f)^\\intercal)^\\intercal$, and later on, we will also use such a notation for $\\overline{y}_i$, $\\overline{\\overline{y}}_i$, $\\widetilde{y}_i$ and $\\mathbf{y}_i$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.2.}\nAt the node $n=1$, to obtain ``data'' $\\overline{\\overline{y}}_1, \\overline{\\overline{y}}_2 \\in C^1([0, T])$ for the entire time interval with small $C^1$ norm and fulfilling the transmission conditions at this node, we connect $y_i^f$ (from Step 1.1), which is defined on $[0, \\overline{T}]$, to the nodal profiles $\\overline{y}_i$ defined on $[T^*, T]$ (see \\eqref{eq:aim}).\n\n\nWe first find functions $\\overline{\\overline{v}}_1, \\overline{\\overline{z}}_1 \\in C^1([0, T]; \\mathbb{R}^6)$ \\textcolor{black}{with $\\|\\overline{\\overline{v}}_1\\|_{C_t^1} + \\|\\overline{\\overline{z}}_1\\|_{C_t^1} \\leq \\gamma$} and such that\n\\begin{linenomath}\n\\begin{align} \\label{eq:barbar_v1_z1}\n\\overline{\\overline{v}}_1(t) = \\left\\{ \\begin{aligned}\n&v_1^f(0, t) && t \\in [0, \\overline{T}]\\\\\n&\\overline{v}_1(t) && t \\in [T^*, T]\n\\end{aligned}\\right., \\quad \\overline{\\overline{z}}_1(t) = \\left\\{\\begin{aligned}\n&z_1^f(0, t) && t \\in [0, \\overline{T}]\\\\\n&\\overline{z}_1(t) && t \\in [T^*, T]\n\\end{aligned}\\right.,\n\\end{align}\n\\end{linenomath}\ncompleting the gap between via, for example, cubic Hermite splines fulfilling the values and first derivatives prescribed by \\eqref{eq:barbar_v1_z1} at $t = \\overline{T}$ and $t = T^*$. The $C^1$ norm of such functions is bounded by that of $v_i^f, \\overline{v}_i$ and $z_i^f,\\overline{z}_i$, respectively.\n\n\nThen, we define $\\overline{\\overline{v}}_2, \\overline{\\overline{z}}_2 \\in C^1([0, T]; \\mathbb{R}^6)$ by $\\overline{\\overline{v}}_2(t) = (\\overline{R}_2^\\intercal \\overline{R}_1)(0) \\overline{\\overline{v}}_1(t)$ and $\\overline{\\overline{z}}_2(t) = - (\\overline{R}_2^\\intercal \\overline{R}_1)(0) \\overline{\\overline{z}}_1(t)$, so that both the continuity and Kirchhoff conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} are fulfilled.\nSince $\\overline{R}_i$ ($i\\in\\mathcal{I}$) is unitary and independent of time, one has $|\\overline{\\overline{v}}_1| = |\\overline{\\overline{v}}_2|$ and $|\\overline{\\overline{z}}_1| = |\\overline{\\overline{z}}_2|$, as well as $|\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{v}}_1| = |\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{v}}_2|$ and $|\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{z}}_1| = |\\frac{\\mathrm{d}}{\\mathrm{d}t}\\overline{\\overline{z}}_2|$, \\textcolor{black}{implying that $\\|\\overline{\\overline{v}}_2\\|_{C_t^1} + \\|\\overline{\\overline{z}}_2\\|_{C_t^1}\\leq \\gamma$.}\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.3 (see Fig. \\ref{fig:A} top-right).}\nNow that we have $\\overline{\\overline{y}}_i$, we consider the sidewise (rightward) problem on $[0, \\ell_i] \\times [0, T]$ for the edges $i\\in \\{1, 2\\}$ (see Remark \\ref{rem:controllability_thm} \\ref{subrem:sidewise}), where at $x=0$ the ``initial data'' is $\\overline{\\overline{y}}_i$, at $t=0$ the ``boundary condition'' prescribes the velocities as $v_i(x,0) = v_i^0(x)$ (thus using a part of the initial conditions of System \\eqref{eq:syst_physical}), and at $t=T$ we set the artificial ``boundary condition'' $z_i(x,T) = \\overline{q}_i(x)$ for any function $\\overline{q}_i \\in C^1([0, \\ell_i]; \\mathbb{R}^6)$.\nThen, for any $\\varepsilon_1>0$ small enough, there exists $\\delta_1>0$ such that the rightward problem admits a unique solution $y_i \\in C^1([0, \\ell_i]\\times[0, T]; \\mathbb{R}^{12})$ with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon_1$, provided that $\\|\\overline{\\overline{y}}_i\\|_{C_{t}^1} + \\|v_i^0\\|_{C_x^1} + \\|\\overline{q}_i\\|_{C_x^1}\\leq \\delta_1$ for all $i \\in \\{1, 2\\}$.\n\n\n\\begin{figure} \\centering\n\\includegraphics[scale=0.55]{AC}\n\\caption{Steps 1.1, 1.3, 1.4, 1.5 of the construction of the solution (top to bottom, left to right), where ``A.C.'' stands for ``artificial conditions''.}\n\\label{fig:A}\n\\end{figure}\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.4 (see Fig. \\ref{fig:A} bottom-left).}\nUsing $y_1(\\ell_1, \\cdot)$, $y_2(\\ell_2, \\cdot)$ (from Step 1.3) as data, consider the forward problem on $[0, \\ell_3]\\times[0, T]$ for the edge $i=3$, with the initial conditions of \\eqref{eq:syst_physical}, and, as boundary conditions at $x=0$ and $x=\\ell_3$, the velocities prescribed as\n\\begin{linenomath}\n\\begin{align} \\label{eq:pb3full_BC_0_l3}\nv_3 (0, t) = \\overline{R}_3(0)^\\intercal \\overline{R}_1(\\ell_1) v_1(\\ell_1, t), \\qquad \nv_3 (\\ell_3, t) = \\overline{R}_3(\\ell_3)^\\intercal \\overline{R}_2(\\ell_2) v_2(\\ell_2, t),\n\\end{align}\n\\end{linenomath}\nso that the obtained solution $y_3$ together with $y_1, y_2$ (provided by Step 1.3) fulfill the continuity conditions \\eqref{eq:IGEB_cont_velo} at the nodes $n \\in \\{2, 3\\}$.\nThen, for any $\\varepsilon_2>0$ small enough, there exists $\\delta_2>0$ such that this problem admits a unique solution $y_3 \\in C^1([0, \\ell_3]\\times[0, T]; \\mathbb{R}^{12})$ with $\\|y_3\\|_{C_{x,t}^1}\\leq \\varepsilon_2$, provided that $\\|y_3^0\\|_{c_x^1}+\\|v_i(\\ell_i, \\cdot)\\|_{C_t^1}+\\|q_n\\|_{C_t^1} \\leq \\delta_2$ for all $i \\in \\{1, 2\\}$ and $n \\in \\{2, 3\\}$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1.5 (see Fig. \\ref{fig:A} bottom-right).}\nFinally, using $y_1(\\ell_1, \\cdot)$, $y_2(\\ell_2, \\cdot)$ (from Step 1.1) and $y_3(0, \\cdot)$, $y_3(\\ell_3, \\cdot)$ (from Step 1.3) as data, consider the rightward problem on $[0, \\ell_i]\\times[0, T]$ for the edges $i \\in \\{4, 5\\}$ similar to that of Step 1.3 except for the choice of the ``initial data'' at $x=0$, denoted by $\\widetilde{y}_i$, that we define by\n\\begin{linenomath}\n\\begin{align}\\label{eq:v45_x=0}\n\\widetilde{y}_4 &= \\begin{bmatrix}\n\\overline{R}_4(0)^\\intercal (\\overline{R}_1v_1)(\\ell_1, \\cdot)\\\\\n\\overline{R}_4(0)^\\intercal( (\\overline{R}_1z_1)(\\ell_1, \\cdot) - (\\overline{R}_3z_3)(0, \\cdot)-q_2)\n\\end{bmatrix}\\\\\n\\label{eq:z45_x=0}\n\\widetilde{y}_5 &= \\begin{bmatrix}\n\\overline{R}_5(0)^\\intercal (\\overline{R}_2 v_2)(\\ell_2, \\cdot)\\\\\n\\overline{R}_5(0)^\\intercal ((\\overline{R}_2 z_2)(\\ell_2, \\cdot) +(\\overline{R}_3 z_3)(\\ell_3, \\cdot) - q_3)\n\\end{bmatrix}.\n\\end{align}\n\\end{linenomath}\nThen, for any $\\varepsilon_3>0$ small enough, there exists $\\delta_3>0$ such that this problem admits a unique solution $y_i \\in C^1([0, \\ell_i]\\times[0 , T]; \\mathbb{R}^{12})$ with $\\|y_i\\|_{C_{x,t}^1}\\leq \\varepsilon_3$, provided that $\\|v_i^0\\|_{C_x^1}+ \\|\\overline{q}_i\\|_{C_x^1} + \\|q_n\\|_{C_t^1} \\leq \\delta_3$ for all $i \\in \\{4,5\\}$ and $n \\in\\{2, 3\\}$, and $\\|y_k(\\ell_k, \\cdot)\\|_{C_t^1} + \\|y_3(0, \\cdot)\\|_{C_t^1} \\leq \\delta_3$ for all $k\\in \\{1, 2, 3\\}$.\n\nNote that the $\\widetilde{y}_i$ for $i \\in \\{4, 5\\}$ have been chosen in such a way that the solutions $y_4, y_5$ together with $y_1, y_2, y_3$ (provided by Step 1.1 and Step 1.3), necessarily fulfill the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff} at the nodes $n \\in \\{2, 3\\}$. \n\n\n\n\\medskip\n\n\n\\noindent It remains to prove that the solution $(y_i)_{i\\in\\mathcal{I}}$ constructed in Step 1 in fact also fulfills the initial conditions \\eqref{eq:IGEB_ini_cond} of the overall network, by showing that $y_i$ coincides with $y_i^f$ on some domain including $[0, \\ell_i]\\times \\{0\\}$ for all $i \\in \\mathcal{I}$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2.1 (see Fig. \\ref{fig:A_ini} leftmost).}\nFirst, consider the edges $i \\in \\{1, 2\\}$. We will see that not only $y_i$ fulfills \\eqref{eq:IGEB_ini_cond}, but one also has (see \\eqref{eq:def_Lambdai_Ti})\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:coinc12_BC}\ny_i(\\ell_i, t) = y_i^f(\\ell_i, t), \\quad &t \\in [0, \\max\\{T_4, T_5\\}], \\, i \\in \\{1, 2\\}.\n\\end{align}\n\\end{linenomath}\nLet $i \\in \\{1, 2\\}$, and let $\\mathbf{t}_i \\in C^1([0, \\ell_i])$ be the function with derivative $\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{t}_i(x) = \\min_{1\\leq k \\leq 12} \\frac{1}{\\lambda_i^k(x)}$ in $[0, \\ell_i]$, which is also equal to $-\\Lambda_i(x)$ (see \\eqref{eq:sign_eigval}-\\eqref{eq:def_Lambdai_Ti}), and such that $\\mathbf{t}_i(0) = T_i + \\max \\{T_4, T_5\\}$. Then, $\\mathbf{t}_i$ describes a curve in $[0, \\ell_i]\\times[0, T]$ that passes through $(0, T_i + \\max\\{T_4, T_5\\})$ and we may also write\n\\begin{linenomath}\n\\begin{align*}\n \\mathbf{t}_i(x) = T_i + \\max \\{T_4, T_5\\} - \\int_0^x \\Lambda_i(s) ds.\n\\end{align*}\n\\end{linenomath}\nThe definition of $T_i$ in \\eqref{eq:def_Lambdai_Ti} ensures that $[0, \\ell_i]\\times[0, \\max\\{T_4, T_5\\}]$ is a subset of the domain $\\mathcal{R}(i, \\mathbf{t}_i)$ defined by\n\\begin{linenomath}\n\\begin{align}\\label{eq:def_dom_calR}\n\\mathcal{R}(i, \\mathbf{t}_i) := \\{(x,t)\\colon 0 \\leq x \\leq \\ell_i, \\ 0 \\leq t \\leq \\mathbf{t}_i(x)\\}.\n\\end{align}\n\\end{linenomath}\nBoth $y_i$ and $y_i^f$ are by definition solutions to the one-sided sidewise (rightward) problem with ``initial data'' $\\overline{\\overline{y}}_i$ at $x=0$ and boundary data $v_i^0$ at $t=0$.\nThe definition of $\\mathbf{t}_i$ ensures that any characteristic curve\\footnote{By characteristic curves passing by $(x_\\circ,t_\\circ)$, we mean the curves specified by the functions $\\mathbf{t}_i^k$ with derivative $\\frac{\\mathrm{d}}{\\mathrm{d}s} \\mathbf{t}_i^k(s) = \\lambda_i^k(s)^{-1}$ and such that $\\mathbf{t}_i^k(x_\\circ) = t_\\circ$, for $k \\in \\{1, \\ldots, 12\\}$.\n}\nof this problem passing by $(x,t) \\in \\mathcal{R}(i, \\mathbf{t}_i)$ is necessarily entering the domain $\\mathcal{R}(i, \\mathbf{t}_i)$ at $\\{0\\} \\times [0, T_i+\\max \\{T_4, T_5\\}]$ or at $[0, \\ell_i] \\times \\{0\\}$. Thus, by \\cite[Section 1.7]{li2016book} the solution in $C^1(\\mathcal{R}(i, \\mathbf{t}_i); \\mathbb{R}^{12})$ to this sidewise problem is unique, and $y_i \\equiv y_i^f$ in $\\mathcal{R}(i, \\mathbf{t}_i)$.\n\n\n\\begin{figure}\\centering\n\\includegraphics[scale=0.7]{A_iniC}\n\\caption{Recovering the initial conditions: \\textcolor{black}{meaning of the controllability time}.}\n\\label{fig:A_ini}\n\\end{figure}\n\\medskip\n\n\n\\noindent \\textit{Step 2.2 (see Fig. \\ref{fig:A_ini} center).}\nConsider the edge $i=3$. We will show that not only $y_3$ fulfills \\eqref{eq:IGEB_ini_cond}, but also\n\\begin{linenomath}\n\\begin{align} \\label{eq:coinc3_BC}\ny_3(0, t) = y_3^f(0, t), \\quad y_3(\\ell_3, t) = y_3^f(\\ell_3, t), \\quad t \\in [0, \\max \\{T_4, T_5\\}]\n\\end{align}\n\\end{linenomath}\nholds. Indeed, $y_3$ and $y_3^f$ both solve the forward problem\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:pb3}}\n\\label{eq:pb3_gov}\n\\partial_t \\mathbf{y}_3 + A_i \\partial_x \\mathbf{y}_3 + \\overline{B}_i \\mathbf{y}_3 = \\overline{g}_i(\\cdot,\\mathbf{y}_3) &\\hspace{-0.45cm}$\\text{in } (0, \\ell_3)\\times(0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_BC_0}\n\\mathbf{v}_3 (0, t) = \\overline{R}_3(0)^\\intercal (\\overline{R}_1 v_1)(\\ell_1, t) (t) &\\hspace{-0.45cm}$t\\in (0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_BC_l3}\n\\mathbf{v}_3 (\\ell_3, t) = \\overline{R}_3(\\ell_3)^\\intercal (\\overline{R}_2 v_2)(\\ell_2, t) (t) &\\hspace{-0.45cm}$t\\in (0, \\max \\{T_4, T_5\\})$\\\\\n\\label{eq:pb3_ini}\n\\mathbf{y}_3(x,0) = y_3^0(x) &\\hspace{-0.45cm}$x \\in (0, \\ell_i)$,\n\\end{subnumcases}\n\\end{linenomath}\nwhich admits a unique solution in $C^1([0, \\ell_3]\\times[0, \\max\\{T_4, T_5\\}];\\mathbb{R}^{12})$.\nIn fact, $y_3$ fulfills \\eqref{eq:pb3} by definition (see Step 3); concerning $y_3^f$, it fulfills \\eqref{eq:pb3_gov} and \\eqref{eq:pb3_ini} by definition, while \\eqref{eq:coinc12_BC} and \\eqref{eq:pb3full_BC_0_l3} imply that $y_3^f$ fulfills \\eqref{eq:pb3_BC_0} and \\eqref{eq:pb3_BC_l3}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2.3 (see Fig. \\ref{fig:A_ini} rightmost).}\nFinally, consider the edges $i\\in\\{4,5\\}$.\nLet $\\mathbf{t}_i$ be the function defined just as in Step 2.1 except that $\\mathbf{t}_i(0) = T_i$. In other words, \n\\begin{linenomath}\n\\begin{align*}\n\\mathbf{t}_i(x) = T_i - \\int_0^x \\Lambda_i(s)ds\n\\end{align*}\n\\end{linenomath}\nHere, the definition of $T_i$ in \\eqref{eq:def_Lambdai_Ti} ensures that $\\mathbf{t}_i(\\ell_i) = 0$, and therefore the corresponding domain $\\mathcal{R}(i, \\mathbf{t}_i)$ defined by \\eqref{eq:def_dom_calR}, contains $[0, \\ell_i] \\times \\{0\\}$.\nBoth $y_i$ and $y_i^f$ fulfill the following one-sided rightward problem with unknown $\\mathbf{y}_i$:\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:pb45}}\n\\label{eq:pb45_gov}\n\\partial_x \\mathbf{y}_i + A_i^{-1} \\partial_t \\mathbf{y}_i + A_i^{-1} \\overline{B}_i \\mathbf{y}_i = (A_i^{-1} \\overline{g}_i)(\\cdot, y_i) & $\\text{in } \\mathcal{R}(i, \\mathbf{t}_i)$\\\\\n\\label{eq:pb45_BC}\n\\mathbf{v}_i(x,0) = v_i^0(x) & $x \\in (0, \\ell_i)$\\\\\n\\label{eq:pb45_ini}\n\\mathbf{y}_i (0, t) = \\widetilde{y}_i(t) & $t\\in (0, T_i)$,\n\\end{subnumcases}\n\\end{linenomath}\nwhere $\\widetilde{y}_i$ is defined by \\eqref{eq:v45_x=0}-\\eqref{eq:z45_x=0}.\nIndeed, while it is clear that $y_i$ fulfills \\eqref{eq:pb45} and $y_i^f$ fulfills \\eqref{eq:pb45_gov}-\\eqref{eq:pb45_BC} by definition, one also obtains, using \\eqref{eq:coinc12_BC}, \\eqref{eq:coinc3_BC} and the fact that $y_i^f$ satisfies the transmission conditions \\eqref{eq:IGEB_cont_velo}-\\eqref{eq:IGEB_Kirchhoff}, that $y_i^f$ also fulfills \\eqref{eq:pb45_ini}.\nThe definition of $\\mathbf{t}_i$ ensures that any characteristic curve of \\eqref{eq:pb45} passing through $(x,t) \\in \\mathcal{R}(i, \\mathbf{t}_i)$ is necessarily entering this domain at $\\{0\\} \\times [0,T_i]$ or at $[0, \\ell_i] \\times \\{0\\}$.\nHence, similarly to Step 2.1, one can apply \\cite[Section 1.7]{li2016book} to obtain that the solution in $C^1(\\mathcal{R}(i, \\mathbf{t}_i); \\mathbb{R}^{12})$ to \\eqref{eq:pb45} is unique.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 3.}\nFinally, we choose $q_i$ defined by $q_i(t) = z_i(\\ell_i, t)$ for all $t \\in [0, T], i \\in \\{4, 5\\}$. In view of the uniqueness of the solution to \\eqref{eq:syst_physical}, $q_4, q_5$ are controls satisfying the desired conditions of Theorem \\ref{th:controllability}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Relationship between the GEB and IGEB networks}\n\\label{sec:invert_transfo}\n\nAs in Section \\ref{sec:exist}, we now consider a general network, and seek to prove Theorem \\ref{thm:solGEB}. To do so, in Lemma \\ref{lem:invert_transfo} below, we start by inverting, on some specific spaces, the transformation $\\mathcal{T}$ defined in \\eqref{eq:transfo} that relates the states of \\eqref{eq:GEB_netw} and \\eqref{eq:syst_physical}.\nHenceforth, for any functions $(u_i)_{i \\in \\mathcal{I}}$ such that $u_i \\colon [0, \\ell_i]\\times[0, T]\\rightarrow \\mathbb{R}^{12}$, we use the notation $u_i=(u_{i,1}^\\intercal, \\ldots, u_{i, 4}^\\intercal)^\\intercal$, where $u_{i,k} \\colon [0, \\ell_i]\\times[0, T]\\rightarrow \\mathbb{R}^3$ for all $k \\in \\{1, \\ldots, 4\\}$.\nLet us define the spaces\n\\begin{linenomath}\n\\begin{align*}\nE_1 &= \\big\\{(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in {\\textstyle \\prod_{i=1}^N} C^2\\left([0, \\ell_i]\\times[0, T]; \\mathbb{R}^3 \\times \\mathrm{SO}(3)\\right) \\colon \\eqref{eq:GEB_condNSv_p_R}, \\eqref{eq:GEB_IC_0ord} \\text{ hold} \\big\\}\\\\\nE_2 &= \\big\\{(y_i)_{i \\in \\mathcal{I}} \\in {\\textstyle \\prod_{i=1}^N} C^1\\left([0, \\ell_i]\\times[0, T] ; \\mathbb{R}^{12}\\right) \\colon u_i := \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i \\text{ satisfies} \\\\\n&\\qquad \\text{\\eqref{eq:compat_last6eq}-\\eqref{eq:compat_ini}-\\eqref{eq:compat_nod}} \\big\\},\n\\end{align*}\n\\end{linenomath}\nwhere \\eqref{eq:compat_last6eq}-\\eqref{eq:compat_ini}-\\eqref{eq:compat_nod} are the following conditions:\n\\begingroup\n\\setlength{\\tabcolsep}{1pt}\n\\renewcommand{\\arraystretch}{0.75}\n\\begin{linenomath}\n\\begin{align}\n\\label{eq:compat_last6eq}\n&\\begin{aligned}\n&\\text{for all }i \\in \\mathcal{I}, \\text{ in }(0, \\ell_i)\\times (0, T)\\\\\n&\\partial_t \\begin{bmatrix}\nu_{i,3} \\\\ u_{i,4}\n\\end{bmatrix} - \\partial_x \\begin{bmatrix}\nu_{i,1} \\\\ u_{i,2}\n\\end{bmatrix} - \\begin{bmatrix}\n\\widehat{\\Upsilon}_c^i & \\widehat{e}_1 \\\\\n\\mathbf{0}_3 & \\widehat{\\Upsilon}_c^i\n\\end{bmatrix}\\begin{bmatrix}\nu_{i,1} \\\\ u_{i,2}\n\\end{bmatrix} = \n\\begin{bmatrix}\n\\widehat{u}_{i,2} & \\widehat{u}_{i,1}\\\\\n\\mathbf{0}_3 & \\widehat{u}_{i,2}\n\\end{bmatrix} \\begin{bmatrix}\nu_{i,3} \\\\ u_{i,4}\n\\end{bmatrix},\n\\end{aligned}\\\\\n\\label{eq:compat_ini}\n&\\begin{aligned}\n\\text{for all }i \\in \\mathcal{I}, \\text{ in } (0, \\ell_i), \\quad \\tfrac{\\mathrm{d}}{\\mathrm{d}x}\\mathbf{p}_i^0(\\cdot) &= \\mathbf{R}_i^0(\\cdot) (u_{i,3}(\\cdot, 0) + e_1),\\\\\n\\tfrac{\\mathrm{d}}{\\mathrm{d}x} \\mathbf{R}_i^0(\\cdot) &= \\mathbf{R}_i^0(\\cdot)(\\widehat{u}_{i,4}(\\cdot, 0) + \\widehat{\\Upsilon}_c^i(\\cdot)),\n\\end{aligned}\\\\\n\\label{eq:compat_nod}\n&\\begin{aligned}\n\\text{for all } n \\in \\mathcal{N}_S^D, \\text{ in } (0, T),\\quad \\tfrac{\\mathrm{d}}{\\mathrm{d}t} f_n^\\mathbf{p} (\\cdot) &= f_n^\\mathbf{R}(\\cdot) \\widehat{u}_{{i^n},1}(\\mathbf{x}_{i^n}^n, \\cdot),\\\\\n\\tfrac{\\mathrm{d}}{\\mathrm{d}t} f_n^\\mathbf{R} (\\cdot) &= f_n^\\mathbf{R}(\\cdot) \\widehat{u}_{{i^n},2}(\\mathbf{x}_{i^n}^n, \\cdot).\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\n\\endgroup \nThe following result then holds.\n\n\n\n\\begin{lemma} \\label{lem:invert_transfo}\nAssume that $(\\mathbf{p}_i^0, \\mathbf{R}_i^0, f_n^\\mathbf{p}, f_n^\\mathbf{R})$ are of regularity \\eqref{eq:reg_Idata_GEB} and \\eqref{eq:reg_Ndata_GEB_D}, and fulfill \\eqref{eq:compat_GEB_-1}.\nThen, the transformation $\\mathcal{T}\\colon E_1 \\rightarrow E_2$, defined in \\eqref{eq:transfo}, is bijective.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:invert_transfo}]\nOne can easily verify that $(\\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i))_{i \\in \\mathcal{I}}$ belongs to $E_2$ for any given $(\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}} \\in E_1$, and $\\mathcal{T}$ is thus well defined.\n\n\nLet $(y_i)_{i\\in\\mathcal{I}} \\in E_2$. We will now show that, there exists a unique $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ such that $\\mathcal{T}((\\mathbf{p}_i, \\mathbf{R}_i)_{i\\in\\mathcal{I}}) = (y_i)_{i\\in\\mathcal{I}}$.\nConsider $(u_i)_{i \\in \\mathcal{I}}$ defined by $u_i := \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i$.\nLet $i \\in \\mathcal{I}$, and let $n$ be the index of any node such that $i \\in \\mathcal{I}^n$.\n\n\\medskip\n\n\\noindent There exists a unique solution $\\mathbf{R}_i \\in C^2([0, \\ell_i]\\times[0, T]; \\mathrm{SO}(3))$ to \n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:overdetR}}\n\\label{eq:overdetR_govt}\n\\partial_t \\mathbf{R}_i = \\mathbf{R}_i \\widehat{u}_{i,2}\n& $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetR_govx}\n\\partial_x \\mathbf{R}_i = \\mathbf{R}_i (\\widehat{u}_{i,4} + \\widehat{\\Upsilon}_c^i) & $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetR_IBC}\n\\mathbf{R}_i(\\mathbf{x}_i^n, 0) = \\mathbf{R}_i^0(\\mathbf{x}_i^n).\n\\end{subnumcases}\n\\end{linenomath}\nTo prove this, a possible way is to first rewrite \\eqref{eq:overdetR}, whose state has values in $\\mathrm{SO}(3)$, as a system with a $\\mathbb{R}^4$-valued state (using \\cite[Lem. 4.1]{RL2019}) via a parametrization of rotation matrices by quaternions \\cite{chou1992}, and then use \\eqref{eq:compat_last6eq} (last three equations) as compatibility conditions in order to deduce that the obtained system is well-posed (using \\cite[Lem. 4.3]{RL2019}); this procedure is detailed in \\cite[Section 4]{RL2019}.\n\n\\medskip\n\n\\noindent Having found $\\mathbf{R}_i$, consider the following system\n\\begin{linenomath}\n\\begin{subnumcases}{\\label{eq:overdetp}}\n\\label{eq:overdetp_govt}\n\\partial_t \\mathbf{p}_i = \\mathbf{R}_i u_{i,1}\n& $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetp_govx}\n\\partial_x \\mathbf{p}_i = \\mathbf{R}_i (u_{i,3} + e_1) & $\\text{in }(0, \\ell_i) \\times (0,T)$\\\\\n\\label{eq:overdetp_IBC}\n\\mathbf{p}_i(\\mathbf{x}_i^n, 0) = \\mathbf{p}_i^0(\\mathbf{x}_i^n).\n\\end{subnumcases}\n\\end{linenomath}\nNote that \\eqref{eq:overdetp_govt} is equivalent to $\\mathbf{p}_i(x,t) = \\mathbf{p}_i(x,0) + \\int_0^t (\\mathbf{R}_i u_{i,1})(x, \\tau)d\\tau$. \nWithout loss of generality, assume that $\\mathbf{x}_i^n = 0$ (in the alternative case, the end of the proof is the same with each integral $+ \\int_{0}^x$ below replaced by $- \\int_x^{\\ell_n}$).\nBy \\eqref{eq:compat_ini} (first equation) and \\eqref{eq:overdetp_IBC}, in the above expression for $\\mathbf{p}_i(x,t)$, one may express the first term as $\\mathbf{p}_i(x,0) = \\mathbf{p}_i^0(\\mathbf{x}_i^n) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i^0(u_{i,3}^0+e_1))(s)ds$. Also, for any $x \\in [0, \\ell_i]$ and any $\\tau \\in [0, t]$ the integrand in the second term may be expressed as $(\\mathbf{R}_i u_{i,1})(x, \\tau) = (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\tau) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i u_{i,1})(s, \\tau) ds$. Hence, \\eqref{eq:overdetp_govt} and \\eqref{eq:overdetp_IBC} are equivalent to\n\\begin{linenomath}\n\\begin{align} \\label{eq:pi_candidate}\n\\begin{aligned}\n\\mathbf{p}_i(x,t) &= \\mathbf{p}_i^0(\\mathbf{x}_i^n) + \\int_0^t (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\tau)d\\tau\\\\\n&+ \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i^0(u_{i,3}^0+e_1))(s)ds + \\int_0^t \\int_{\\mathbf{x}_i^n}^x \\partial_x (\\mathbf{R}_i u_{i,1})(s, \\tau)d\\tau ds.\n\\end{aligned}\n\\end{align}\n\\end{linenomath}\nOn the other hand, we know that \\eqref{eq:pi_candidate} fulfills $\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, \\cdot) = (\\mathbf{R}_i u_{i,1})(\\mathbf{x}_i^n, \\cdot)$, while by \\eqref{eq:compat_last6eq} (first three equations), one has $\\partial_x (\\mathbf{R}_i u_{i,1}) = \\partial_t(\\mathbf{R}_i(u_{i,3}+e_1))$. The latter two facts, together with \\eqref{eq:compat_ini} (second equation), permit us to deduce that \\eqref{eq:pi_candidate} also writes as $\\mathbf{p}_i(x, t) = \\mathbf{p}(\\mathbf{x}_i^n, t) + \\int_{\\mathbf{x}_i^n}^x (\\mathbf{R}_i(u_{i,3} + e_1))(t,s)ds$.\nThus, \\eqref{eq:pi_candidate} is the unique solution to \\eqref{eq:overdetp}.\n\n\n\nFinally, note that, because of \\eqref{eq:compat_ini}, requiring \\eqref{eq:overdetR_IBC} and \\eqref{eq:overdetp_IBC} is equivalent to imposing the initial conditions \\eqref{eq:GEB_IC_0ord}. Moreover, in the case of $n \\in \\mathcal{N}_S^D$, due to \\eqref{eq:compat_GEB_-1} and \\eqref{eq:compat_nod}, requiring \\eqref{eq:overdetR_IBC} and \\eqref{eq:overdetp_IBC} is equivalent to imposing the nodal conditions \\eqref{eq:GEB_condNSv_p_R}. This concludes the proof of Lemma \\ref{lem:invert_transfo}.\n\\end{proof}\n\n\n\n\n\n\nWe now have the tools to prove Theorem \\ref{thm:solGEB}.\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:solGEB}]\nWe divide the proof in seven steps. Let $(y_i)_{i \\in \\mathcal{I}}$ be as in Theorem \\ref{thm:solGEB}, and let $(u_i)_{i \\in\\mathcal{I}}$ be defined by $u_i = \\mathrm{diag}(\\mathbf{I}_6, \\mathbf{C}_i) y_i$.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 1: inverting the transformation.}\nSince the last six equations in \\eqref{eq:IGEB_gov} hold for $(y_i)_{i\\in\\mathcal{I}}$, we know that \\eqref{eq:compat_last6eq} is fulfilled. On the other hand, the last six equations of the initial conditions \\eqref{eq:IGEB_ini_cond} with initial data \\eqref{eq:rel_inidata} yield \\eqref{eq:compat_ini}. Finally, the definition of the boundary data \\eqref{eq:def_qn_D}, together with the nodal conditions \\eqref{eq:IGEB_condNSv} on velocities, yield \\eqref{eq:compat_nod}. Hence, $(y_i)_{i\\in\\mathcal{I}} \\in E_2$, and by Lemma \\ref{lem:invert_transfo} there exists a unique $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}} \\in E_1$ such that\n\\begin{linenomath} \n\\begin{align} \\label{eq:transfo_inverted}\ny_i = \\mathcal{T}_i(\\mathbf{p}_i, \\mathbf{R}_i), \\quad \\text{for all }i \\in \\mathcal{I}.\n\\end{align}\n\\end{linenomath}\n\nNow, we want to check that this ``candidate'' $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$, satisfies the rest of system \\eqref{eq:GEB_netw}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 2: governing equations.}\nUsing \\eqref{eq:transfo_inverted} and the first six governing equations in \\eqref{eq:IGEB_gov}, one can deduce that $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ satisfies the governing system \\eqref{eq:GEB_gov} after some computations.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 3: conditions at simple nodes.}\nFor $n \\in \\mathcal{N}_S^N$, from \\eqref{eq:transfo_inverted} together with the nodal conditions \\eqref{eq:IGEB_condNSz} on forces and moments and the definition of $f_n$ (see \\eqref{eq:def_fn}), one can directly deduce that the nodal conditions \\eqref{eq:GEB_condNSz} hold.\n\n\nFor $n \\in \\mathcal{N}_S^D$, from \\eqref{eq:transfo_inverted} together with the nodal conditions \\eqref{eq:IGEB_condNSv} on velocities and initial conditions \\eqref{eq:GEB_IC_0ord}, we deduce that $(\\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot), \\mathbf{R}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot))$ satisfies\n\\begin{linenomath}\n\\begin{align} \\label{eq:nodPDE}\n\\left\\{ \\begin{aligned}\n&\\frac{\\mathrm{d}\\beta}{\\mathrm{d}t}(t) = \\beta(t) \\widehat{q_n^W}(t), \\ \\ \\frac{\\mathrm{d}\\alpha}{\\mathrm{d}t}(t) = \\beta(t) q_n^V(t) && \\ \\text{ in }(0, T)\\\\\n&(\\alpha, \\beta)(0) = (\\mathbf{p}_{i^n}^0, \\mathbf{R}_{i^n}^0)(\\mathbf{x}_{i^n}^n),\n\\end{aligned} \\right.\n\\end{align}\n\\end{linenomath}\nof unknown state $(\\alpha, \\beta)$, where we denote $q_n = ((q_n^V)^\\intercal, (q_n^W)^\\intercal)^\\intercal$ with $q_n^V, q_n^W \\in C^1([0, T]; \\mathbb{R}^3)$.\nDue to \\eqref{eq:def_qn_D} and \\eqref{eq:compat_GEB_-1}, $(f_n^\\mathbf{p}, f_n^\\mathbf{R})$ also satisfies \\eqref{eq:nodPDE}.\nOne may see that \\eqref{eq:nodPDE} admits a unique solution in $C^2([0, T]; \\mathbb{R}^{3}\\times \\mathrm{SO}(3))$. Indeed, as in the proof of Lemma \\ref{lem:invert_transfo}, one may replace \\eqref{eq:nodPDE} (first equation) by an equivalent equation whose unknown state is the quaternion \\cite{chou1992} parametrizing the rotation matrix $\\beta = \\beta(t)$ (see \\cite[Section 4]{RL2019} for more detail). Having then only vector valued unknowns, one can use the classical ODE theory. Thus, $(\\mathbf{p}_{i^n},\\mathbf{R}_{i^n})(\\mathbf{x}_{i^n}^n, \\cdot) \\equiv (f_n^\\mathbf{p}, f_n^\\mathbf{R})$, and the nodal conditions \\eqref{eq:GEB_condNSv_p_R} hold.\n \n\n\\medskip\n\n\n\\noindent \\textit{Step 4: remaining initial conditions.}\nOne recovers the initial conditions \\eqref{eq:GEB_IC_1ord} directly from the first six equations in \\eqref{eq:IGEB_ini_cond} and the definition of $y_i^0$ \\eqref{eq:rel_inidata}, together with \\eqref{eq:transfo_inverted}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 5: rigid joint condition.}\nIn order to show that $(\\mathbf{p}_i, \\mathbf{R}_i)_{i \\in \\mathcal{I}}$ fulfills the transmission conditions of \\eqref{eq:GEB_netw}, we start with the rigid joint condition. Let $n \\in \\mathcal{N}_M$. For all $i \\in \\mathcal{I}^n$, let us define $\\Lambda_i \\in C^1([0, T]; \\mathbb{R}^{3 \\times 3})$ by $\\Lambda_i(t) = (R_i \\mathbf{R}_i^\\intercal)(\\mathbf{x}_i^n, t)$. By the continuity condition \\eqref{eq:IGEB_cont_velo} (last three equations),\n\\begin{linenomath}\n\\begin{align} \\label{cont_derivative_angle}\n\\textstyle\n\\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_i \\right) \\Lambda_i^\\intercal = \\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_{i^n} \\right)\\Lambda_{i^n}^\\intercal, \\quad \\text{in }(0, T), \\text{ for all }i\\in\\mathcal{I}^n.\n\\end{align}\n\\end{linenomath}\nLet $F_n := \\left(\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_{i^n} \\right)\\Lambda_{i^n}^\\intercal$ and $a_n := (R_{i^n}{\\mathbf{R}_{i^n}^0}^\\intercal)(\\mathbf{x}_{i^n}^n)$. By \\eqref{cont_derivative_angle}, \\eqref{eq:compat_GEB_transmi} (second equation) and the fact that \\eqref{eq:GEB_IC_0ord} holds (by Step 1), for all $i \\in \\mathcal{I}^n$, $\\Lambda_i$ fulfills\n\\begin{linenomath}\n\\begin{align*}\n\\begin{dcases}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\Lambda_i(t) = F_n(t) \\Lambda_i(t) & \\text{for all }t \\in (0, T)\\\\\n\\Lambda_i(0) = a_n,\n\\end{dcases}\n\\end{align*}\n\\end{linenomath}\nwhich admits a unique $C^1([0, T]; \\mathbb{R}^{3 \\times 3})$ solution (see \\cite[Sec. 2.1 and Th. 4.1.1 or Coro. 2.4.4]{vrabie2004}, for instance). Hence, $\\Lambda_i \\equiv \\Lambda_j$ for all $i,j\\in\\mathcal{I}^n$, and the rigid joint condition \\eqref{eq:GEB_rigid_angles} holds. \n\n\nAs \\eqref{eq:GEB_rigid_angles} holds, we can now deduce the transmission conditions of \\eqref{eq:GEB_netw} that remain.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 6: continuity of the displacement.}\nLet $n \\in \\mathcal{N}_M$.\nBy \\eqref{eq:transfo_inverted} together with the rigid joint condition \\eqref{eq:GEB_rigid_angles} and the continuity condition \\eqref{eq:IGEB_cont_velo} (first three equations), one deduces that \n\\begin{linenomath}\n\\begin{align*}\n\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, t) = \\partial_t \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, t), \\quad \\text{in }(0, T), \\text{ for all } i\\in\\mathcal{I}^n.\n\\end{align*}\n\\end{linenomath}\nUsing additionally \\eqref{eq:GEB_IC_0ord} with \\eqref{eq:compat_GEB_transmi} (first equation), we deduce that for all $i \\in \\mathcal{I}^n$, the function $\\mathbf{p}_i(\\mathbf{x}_i^n, \\cdot)$ fulfills the problem\n\\begin{linenomath}\n\\begin{align} \\label{eq:contdipl_ODE}\n\\begin{dcases}\n\\partial_t \\mathbf{p}_i(\\mathbf{x}_i^n, t) = h_n(t) & \\text{for all }t \\in (0, T)\\\\\n\\mathbf{p}_i(\\mathbf{x}_i^n, 0) = \\alpha_n,\n\\end{dcases}\n\\end{align}\n\\end{linenomath}\nwhere we denote $h_n := \\partial_t \\mathbf{p}_{i^n}(\\mathbf{x}_{i^n}^n, \\cdot)$ and $\\alpha_n := \\mathbf{p}_{i^n}^0(\\mathbf{x}_{i^n}^n)$. Since the $C^1([0, T];\\mathbb{R}^3)$ solution to \\eqref{eq:contdipl_ODE} is unique, we conclude that \\eqref{eq:GEB_continuity_pi} holds.\n\n\n\\medskip\n\n\n\\noindent \\textit{Step 7: Kirchhoff condition.}\nOne recovers the Kirchhoff condition \\eqref{eq:GEB_Kirchhoff} from the rigid joint assumption \\eqref{eq:GEB_rigid_angles} together with \\eqref{eq:IGEB_Kirchhoff} and \\eqref{eq:transfo_inverted}.\n\n\nTo finish, the uniqueness of the solution to \\eqref{eq:GEB_netw} is a consequence of the uniqueness of the solution to \\eqref{eq:syst_physical} (Theorem \\ref{th:existence}) and of the bijectivity of the transformation $\\mathcal{T}$ (Lemma \\ref{lem:invert_transfo}). This concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Concluding remarks and outlook}\n\n\\label{sec:conclusion}\n\nIn this article, we have studied networks, possibly with cycles, of geometrically exact beams. Notably, we considered the representations of such beams in terms of either displacements and rotations expressed in a fixed coordinate system (GEB model), or velocities and internal forces\/moments expressed in a moving coordinate system attached to the beam (IGEB model), reflecting on the advantages and drawbacks of these two points of view, and the relationship between them. \nFor these beam networks, we addressed the problem of local exact controllability of nodal profiles in the special case of a network containing one cycle: the A-shaped network depicted in Fig. \\ref{subfig:AshapedNetwork}.\n\nThe fact that one has the possibility of expressing the beam model as a first-order semilinear hyperbolic system -- the IGEB model -- while keeping track of the link with the GEB model, permits us to give a proof of nodal profile controllability in line with works done on other one-dimensional hyperbolic systems -- e.g., wave equation, Saint-Venant equations, Euler equations \\cite{gu2011, gu2013, gugat10, li2010nodal, li2016book, kw2011, kw2014, YWang2019partialNP, Zhuang2018}. Namely, we used the existence and uniqueness theory of semi-global classical solutions to the network system, combined with a constructive method as in \\cite{Zhuang2018} to obtain adequate controls.\n\n\\medskip\n\n\\noindent \\textbf{Local nature of the results.}\n{\\color{black} \nLet us give some comments about the local nature of the nodal profile controllability result, Theorem \\ref{th:controllability}. This theorem notably implies that even though there might be large displacements and rotations of the beam -- due to the use of a geometrically exact (thus nonlinear) beam model --, we apply controls that subsequently keep these motions small. As noted in Remark \\ref{rem:controllability_thm} \\ref{subrem:global}, Theorem \\ref{th:controllability} focuses on the small data scenario and could possibly be a preliminary step in view of obtaining a global result.\n\nBesides, in the proof of Theorem \\ref{th:controllability}, some ``degree of freedom'' has not been used, as we rely on an existence and uniqueness result which has been established for general one-dimensional first-order quasilinear hyperbolic systems. Since we are considering a very specific model -- the IGEB model -- it would be interesting to establish an appropriate well-posedness result and keep track of the bounds on the initial and boundary data to obtain more quantitative information.\n\nOn another hand, as explained in the introduction, the GEB and IGEB models are valid as long as the strains $s_i(x,t)$ are small enough. The latter being proportional to the the internal forces and moments $z_i(x,t)$ (more precisely, they are given by $s_i = \\mathbf{C}_i z_i$ where we recall that $\\mathbf{C}_i(x)$ denotes the flexibility matrix), comparing this assumption to the smallness of the internal forces and moments required in Theorem \\ref{th:controllability} would also be of interest.\n}\n\n\\medskip\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale = 0.8]{otherControllableNet_colorC.pdf}\n \\caption{Other networks for which local exact controllability of nodal profiles is achievable by following Algorithm \\ref{algo:control} (the numbers refer to the variable \\textsf{step}).}\n \\label{fig:otherControllableNet}\n\\end{figure}\n\n\n\\noindent \\textbf{More general networks.} The A-shaped network is an illustrative example where the controllability of nodal profiles is achievable for a network with a cycle, but let us stress that similar arguments to those used in Section \\ref{sec:controllability} apply for various other networks, and with controls at different locations.\n\n\n\\begin{algorithm}\n\\DontPrintSemicolon\n\n\n\\Input{%\n$\\mathcal{I}, \\, \\mathcal{N}, \\, \\mathcal{N}_S, \\, k_n$\\tcp*{edges, nodes, simple nodes, degrees}\n\\hspace{1.2cm}$\\mathcal{I}^n$ for all $n\\in \\mathcal{N}$\\tcp*{edges incident to the node $n$}\n\\hspace{1.2cm}$\\mathcal{N}^i$ for all $i \\in \\mathcal{I}$\\tcp*{nodes at the tips of the edge $i$}\n\\hspace{1.2cm}$\\mathcal{P}$, $\\mathcal{C}$\\tcp*{charged nodes, controlled nodes}\n\\hspace{1.2cm}$\\mathcal{S}$\\tcp*{edges on control paths}\n}\n\n\n\\setcounter{AlgoLine}{0}\n\\ShowLn\n$J$ $\\leftarrow$ $[\\, 0, \\quad $for $n = 1 \\ldots \\#\\mathcal{N}\\, ]$; \\ \\lFor{all $n \\in \\mathcal{P}$}{($J(n)$ $\\leftarrow$ $k_n - 1$);} \n\\tcp*{amount $J(n)$ of data available at $n$ to solve sidewise}\n\n\\ShowLn\n$\\mathcal{F}$ $\\leftarrow$ $\\emptyset$;\\tcp*{solved edges}\n \n\\ShowLn\n\\textsf{step} $\\leftarrow$ $1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false};\\tcp*{to count the steps}\n\n\\ShowLn\n$\\mathcal{M}$ $\\leftarrow$ $\\mathcal{P}\\cup \\{n \\in \\mathcal{N} \\colon\\text{$n$ not incident with any edge in $\\mathcal{S}$}\\};$\\;\n \n\\ShowLn\n\\While(\\hfill \\tcp*[h]{while entire network not solved}){$\\mathcal{F} \\neq \\mathcal{I}$}{\n \n\\ShowLn\n\\For(\\hfill \\tcp*[h]{Principle 1}){$m =\\#\\mathcal{M}, \\ldots, 3, 2, 1$}{\n\n\\ShowLn\n\\For{all ${\\mathcal{N}}^\\dagger \\subseteq \\mathcal{M}$ such that $\\#\\mathcal{N}^\\dagger = m$}{\n\n\\ShowLn\n\\If{there exists a connected subgraph with nodes ${\\mathcal{N}}^\\dagger$ and edges ${\\mathcal{I}}^\\dagger$, such that ${\\mathcal{I}}^\\dagger \\cap ( \\mathcal{S}\\cup \\mathcal{F}) = \\emptyset$}{\n\n\\ShowLn\n solve forward problem for the network $({\\mathcal{I}^\\dagger}, {\\mathcal{N}}^\\dagger)$;\\;\n \n\\ShowLn\n\\lFor{all $n \\in {\\mathcal{N}}^\\dagger$}{\n ($J(n)$ $\\leftarrow$ $J(n) + 1$);}\n \n\\ShowLn\n$\\mathcal{M}$ $\\leftarrow$ $\\mathcal{M} \\cup {\\mathcal{N}}^\\dagger$; \\, $\\mathcal{F}$ $\\leftarrow$ $\\mathcal{F} \\cup {\\mathcal{I}}^\\dagger$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{true};\\;\n}}}\n\n\\ShowLn\n\\lIf{\\upshape \\textsf{moved} $=$ \\textsf{true} }{(\\textsf{step} $\\leftarrow$ \\textsf{step} $+ 1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false});} \n \n\\ShowLn\n\\For(\\hfill \\tcp*[h]{Principle 2}){all $n \\in \\mathcal{M}$}{\n\n\\ShowLn\n\\If(\\hfill \\tcp*[h]{if enough data at $n$}){$J(n) = k_n - 1$}{\n\n\\ShowLn\n\\For{all $i \\in \\mathcal{I}^n \\cap (\\mathcal{S} \\setminus \\mathcal{F}$)}{\n\n\\ShowLn\nsolve sidewise problem for the edge $i$ with ``initial \\;\n conditions'' at the node $n$;\\;\n\n\\ShowLn\n$\\mathcal{M}$ $\\leftarrow$ $\\mathcal{M} \\cup \\mathcal{N}^i$; \\, $\\mathcal{F}$ $\\leftarrow$ $\\mathcal{F}\\cup \\{i\\}$;\\;\n\n\\ShowLn\n\\lFor{all $m \\in \\mathcal{N}^i$}{\n ($J(m)$ $\\leftarrow$ $J(m) + 1$);}\n } \n \n\\ShowLn \n\\textsf{moved} $\\leftarrow$ \\textsf{true};\\;\n}}\n\n\\ShowLn\n\\lIf{\\upshape \\textsf{moved} $=$ \\textsf{true}}{(\\textsf{step} $\\leftarrow$ \\textsf{step} $+ 1$; \\ \\textsf{moved} $\\leftarrow$ \\textsf{false});} \n}\n\n\\ShowLn \nCompute controls $q_n$ by evaluating the trace at nodes $n\\in\\mathcal{C}$;\\;\n \n\\caption{Steps of controllability proof for other networks}\n\\label{algo:control}\n\\end{algorithm}\n\n\n\n\n\nLet us introduce some more notation. For any given network, we denote by $\\mathcal{P}$ and $\\mathcal{C}$ the set of indexes of the \\emph{charged nodes} and \\emph{controlled nodes} (see Section \\ref{sec:intro}), respectively.\nGiven a charged node $n\\in\\mathcal{P}$ and a controlled node $m\\in\\mathcal{C}$, a \\emph{control path} between $n$ and $m$ \\cite{YWang2019partialNP}, is any connected subgraph (of the current graph representing the beam network) forming a path graph\\footnote{A path graph is an oriented graph without cycle such that two of its nodes are of degree $1$, and all other nodes have a degree equal to $2$.} whose nodes of degree $1$ are $n$ and $m$. In Fig. \\ref{fig:otherControllableNet}, examples of control paths are highlighted by blue arrows.\n\n\n\\medskip\n\n\n\n\n\n\\noindent On a \\emph{tree-shaped} network -- hence without loop --, some conditions were proved to be \\emph{sufficient} for the exact controllability of nodal profiles to be achieved \\cite{gu2011, gu2013, li2016book, kw2011, kw2014, YWang2019partialNP}.\nIn \\cite{YWang2019partialNP} the authors are concerned with the wave equation and provide a controllability result for any given tree-shaped network with possibly several charged nodes. Moreover, in \\cite{YWang2019partialNP}, at a charged node $n \\in \\mathcal{P}$, profiles may be prescribed for only some (rather than all) of the edges incident with $n$, and profiles may be prescribed for only part of the state (which would translate in the case of \\eqref{eq:syst_physical} to prescribing the velocities only, or the internal forces and moments only, for example). This type of problem, which is then called \\emph{partial nodal profile controllability}, is not considered here.\n\n\nThe nodal profile controllability has also been established for the Saint-Venant system \\cite{Zhuang2021, Zhuang2018} for numerous networks \\emph{with cycles}, of various shapes and with several charged nodes.\n\n\\medskip\n\n\\noindent It arises, from these works, a series of conditions on the number and location of the charged nodes, which are sufficient to achieve the respective controllability goals. We refer notably to \\cite[Theorem 5.1]{YWang2019partialNP}, and to \\cite[Sections 7 and 8]{Zhuang2021}.\n\n\nIn the case of the beam networks considered in this article, these conditions become (recall that $k_n$ is defined as the degree of the node $n$)\n\\begin{enumerate}\n\\item The total number of controlled nodes $\\#\\mathcal{C}$ is equal to $\\sum_{n\\in \\mathcal{P}} k_n$.\n\\item For any $n \\in \\mathcal{P}$, there are $k_n$ controlled nodes connecting with it through control paths. These control paths have the charged node $n$ for sole common node. \n\\item The control paths corresponding to different charged nodes do not have any common node.\n\\end{enumerate}\nLet us stress again that we are restricting ourselves to the type of systems presented in Subsection \\ref{subsec:network_systems}. Namely, if a multiple node is controlled, then the control is applied at the Kirchhoff condition, while if a simple node is controlled, then the control is applied at either the first six (velocities) or last six (internal forces and moments) components of the state $y_i$, and at any charged node $n \\in\\mathcal{P}$ profiles are prescribed for all incident beams $i \\in \\mathcal{I}^n$ and for the entire state $y_i$.\n\n\n\\medskip\n\n\\noindent Then, one may use the \\emph{constructive} method as in Section \\ref{sec:controllability}, by following the steps instructed by Algorithm \\ref{algo:control}, for different networks; see Fig. \\ref{fig:otherControllableNet}.\nWe can assert that this algorithm yields a proof of controllability for the networks defined in Fig. \\ref{fig:otherControllableNet}, but not that it constitutes a proof for any given network.\n\nIn Algorithm \\ref{algo:control}, edges belonging to control paths are solved according to the \\emph{Principle 2} -- solving a sidewise problem as in the Steps 1.3 and 1.5 of the proof of Theorem \\ref{th:controllability} -- while the other edges are solved according to the \\emph{Principle 1} -- solving a forward problem similar to the Step 1.4 of the proof of Theorem \\ref{th:controllability}.\n\nAs noted here and in the above cited works, the conditions given to obtain controllability of nodal profiles are only \\emph{sufficient} to ensure the controllability result and the search for necessary and sufficient conditions is open.\n\n\n\n\n\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nWe present a derivation of the spin multiplicities that occur in $n$-fold\ntensor products of spin-$j$ representations, $j^{\\otimes n}$. \\ We make use of\ngroup characters, properties of special functions, and asymptotic analysis of\nintegrals. \\ While previous derivations for some of our results are scattered\nthroughout the literature, especially for specific values of $j$, we provide\nhere a treatment that is self-contained, and valid for any $j$ and for any\n$n$. \\ We emphasize two types of novel features: \\ patterns that arise when\ncomparing different values of $j$, and asymptotic behavior for large $n$. \\ \n\nOur methods and results should be useful for various calculations. \\ In\nparticular, the asymptotic behavior that we obtain should be helpful in the\nanalysis of statistical problems such as the determination of partition\nfunctions. \\ In the last section some other applications are briefly\ndiscussed, including a problem of interest for quantum computing, namely, an\nestimation of the number of entangled states.\n\n\\section*{Basic Theory of Group Characters}\n\nThe \\href{https:\/\/en.wikipedia.org\/wiki\/Character_theory}{character\n$\\chi\\left( R\\right) $ of a group representation $R$} succinctly encodes\nconsiderable information about $R$, as is well-known \\cite{Texts}. \\ For\nirreducible representations the characters are orthogonal\n\\begin{equation}\n\\sum\\hspace{-0.2in}\\int~\\mu~\\chi^{\\ast}\\left( R_{1}\\right) \\chi\\left(\nR_{2}\\right) =\\delta_{R_{1},R_{2}}\\ , \\label{OrthoChar\n\\end{equation}\nwhere the sum or integral is over the group parameter space with an\nappropriate measure $\\mu$.\n\nFor a Kronecker product of $n$ representations, the character is given by the\nproduct of the individual characters\n\\begin{equation}\n\\chi\\left( R_{1}\\otimes R_{2}\\otimes\\cdots\\otimes R_{n}\\right) =\\chi\\left(\nR_{1}\\right) \\chi\\left( R_{2}\\right) \\cdots\\chi\\left( R_{n}\\right) \\ ,\n\\end{equation}\nfrom which follows an explicit expression for the number of times that a given\nrepresentation $R$ appears in the product (e.g., see \\cite{V} Chapter I\n\\S 4.7). \\ This multiplicity is\n\\begin{equation}\nM\\left( R;R_{1},\\cdots,R_{n}\\right) =\\sum\\hspace{-0.2in}\\int~\\mu~\\chi^{\\ast\n}\\left( R\\right) \\chi\\left( R_{1}\\right) \\chi\\left( R_{2}\\right)\n\\cdots\\chi\\left( R_{n}\\right) \\ .\n\\end{equation}\nFor real characters, this is totally symmetric in $\\left\\{ R,R_{1\n,\\cdots,R_{n}\\right\\} $, and it immediately shows that the number of times\n$R$ appears in the product $R_{1}\\otimes\\cdots\\otimes R_{n}$ is equal to the\nnumber of times the trivial or \\textquotedblleft singlet\\textquotedblrigh\n\\ representation appears in the product $R\\otimes R_{1}\\otimes\\cdots\\otimes\nR_{n}$.\n\n\\section*{The $SU\\left( 2\\right) $ Case}\n\nConsider now the Lie group $SU\\left( 2\\right) $. \\ In this case the\nirreducible representations are labeled by angular momentum or spin, $j$ or\n$s$, the classes of the group are specified by the angle of rotation about an\naxis, $\\theta$, and the characters are\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chebyshev_polynomials#Trigonometric_definition}{Chebyshev\npolynomials of the second kind}, $\\chi_{j}\\left( \\theta\\right)\n=U_{2j}\\left( \\cos\\left( \\theta\/2\\right) \\right) $. \\ Explicitly, for\neither integer or semi-integer $j$\n\\begin{equation}\n\\chi_{j}\\left( \\theta\\right) =\\frac{\\sin\\left( \\left( 2j+1\\right)\n\\theta\/2\\right) }{\\sin\\left( \\theta\/2\\right) }\\ .\n\\end{equation}\nThese characters are all real. \\ Therefore the number of times that spin $s$\nappears in the product $j_{1}\\otimes\\cdots\\otimes j_{n}$ i\n\\begin{equation}\nM\\left( s,j_{1},\\cdots,j_{n}\\right) =\\frac{1}{\\pi}\\int_{0}^{2\\pi}\\chi\n_{s}\\left( 2\\vartheta\\right) \\chi_{j_{1}}\\left( 2\\vartheta\\right)\n\\cdots\\chi_{j_{n}}\\left( 2\\vartheta\\right) \\sin^{2}\\vartheta~d\\vartheta\\ ,\n\\label{IntegralForm\n\\end{equation}\nwhere we have taken $\\theta=2\\vartheta$ to avoid having half-angles appear in\nthe invariant measure and the Chebyshev polynomials (e.g., see \\cite{V}\nChapter III \\S 8.1) thereby mapping the $SU\\left( 2\\right) $ group manifold\n$0\\leq\\theta\\leq4\\pi$ to $0\\leq\\vartheta\\leq2\\pi$. \\ To re-emphasize earlier\nremarks, we note that (\\ref{IntegralForm}) is totally symmetric in $\\left\\{\ns,j_{1},\\cdots,j_{n}\\right\\} $ and valid if $s$ or any of the $j$s are\ninteger or semi-integer, and we also note that $M\\left( s,j_{1},\\cdots\n,j_{n}\\right) =M\\left( 0,s,j_{1},\\cdots,j_{n}\\right) $. \\ In general\n$M\\left( s,j_{1},\\cdots,j_{n}\\right) $ will obviously reduce to a finite sum\nof integers through use of the Chebyshev\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chebyshev_polynomials#Products_of_Chebyshev_polynomials}{product\nidentity}, $U_{m}U_{n}=\\sum_{k=0}^{n}U_{m-n+2k}$ for $m\\geq n$.\n\\ Alternatively, the integral form (\\ref{IntegralForm}) for the multiplicity\nalways reduces to a finite sum of hypergeometric functions (e.g. see\n(\\ref{Hyper1}) and (\\ref{Hyper2})\\ to follow).\n\nIn particular, for $j_{1}=\\cdots=j_{n}=j$, the $n$-fold product $j^{\\otimes\nn}$ can yield spin $s$ a number of times, as given b\n\\begin{equation}\nM\\left( s;n;j\\right) =\\frac{1}{\\pi}\\int_{0}^{2\\pi}\\sin\\left( \\left(\n2s+1\\right) \\vartheta\\right) \\left( \\frac{\\sin\\left( \\left( 2j+1\\right)\n\\vartheta\\right) }{\\sin\\left( \\vartheta\\right) }\\right) ^{n}\\sin\n\\vartheta~d\\vartheta\\ , \\label{ProdMultInt\n\\end{equation}\nfor $s,j\\in\\left\\{ 0,\\frac{1}{2},1,\\frac{3}{2},2,\\cdots\\right\\} $. \\ Yet\nagain, we note that $M\\left( j;n;j\\right) =M\\left( 0;n+1;j\\right) $.\n\\ Moreover, the symmetry of the integrand in (\\ref{ProdMultInt}) permits us to\nwrit\n\\begin{equation}\nM\\left( s;n;j\\right) =\\int_{0}^{2\\pi}\\frac{\\exp\\left( 2is\\vartheta\\right)\n}{2\\pi}\\left( \\frac{\\sin\\left( \\left( 2j+1\\right) \\vartheta\\right) \n{\\sin\\left( \\vartheta\\right) }\\right) ^{n}~d\\vartheta-\\int_{0}^{2\\pi\n\\frac{\\exp\\left( 2i\\left( s+1\\right) \\vartheta\\right) }{2\\pi}\\left(\n\\frac{\\sin\\left( \\left( 2j+1\\right) \\vartheta\\right) }{\\sin\\left(\n\\vartheta\\right) }\\right) ^{n}~d\\vartheta\\ .\n\\end{equation}\nEach integral in the last expression reduces to a simple residue\n\\begin{equation}\n\\int_{0}^{2\\pi}\\frac{\\exp\\left( 2is\\vartheta\\right) }{2\\pi}\\left(\n\\frac{\\sin\\left( \\left( 2j+1\\right) \\vartheta\\right) }{\\sin\\left(\n\\vartheta\\right) }\\right) ^{n}~d\\vartheta=\\frac{1}{2\\pi i}\\oint\nz^{2s}\\left( \\frac{z^{2j+1}-z^{-2j-1}}{z-z^{-1}}\\right) ^{n}\\frac{dz\n{z}=c_{0}\\left( s,n,j\\right) \\ ,\n\\end{equation}\nwhere $c_{k}$ are the coefficients in the Laurent expansion of the integrand\n\\begin{equation}\nz^{2s}\\left( \\frac{z^{2j+1}-z^{-2j-1}}{z-z^{-1}}\\right) ^{n}=z^{2s}\\left(\n\\sum_{m=0}^{2j}z^{2\\left( m-j\\right) }\\right) ^{n}=\\sum_{k=-2\\left(\njn-s\\right) }^{2\\left( jn+s\\right) }z^{k}~c_{k}\\left( s,n,j\\right) \\ .\n\\end{equation}\nThat is to say, $c_{0}$ is the coefficient of $z^{-2s}$ (or of $z^{+2s}$) in\nthe Laurent expansion of $(z^{-2j}+z^{-2j+2}+\\cdots\\allowbreak+z^{2j-2\n+z^{2jn})$ \\cite{Katriel}, a coefficient that is easily obtained, e.g. using\neither Maple$^{\\textregistered}$ or Mathematica$^{\\textregistered}$. \\ \n\n\\subsection*{Explicit $SU\\left( 2\\right) $ Results as Binomial Coefficients}\n\nSo then, the multiplicity is always given by a difference,\n\\begin{equation}\nM\\left( s;n;j\\right) =c_{0}\\left( s,n,j\\right) -c_{0}\\left(\ns+1,n,j\\right) \\ , \\label{MIsADifference\n\\end{equation}\nwhere $2s$ is any integer such that $0\\leq2s\\leq2nj$, and where $s=0$ is\nalways allowed when $j$ is an integer but is only allowed for even $n$ when\n$j$ is a semi-integer. \\ To be more explicit, the expansion of $\\left(\nz^{-2j}+z^{-2j+2}+\\cdots+z^{2j-2}+z^{2j}\\right) ^{n}$ involves so-called\n\\textquotedblleft generalized binomial coefficients\\textquotedblright\\ (see\nEqn(3) in \\cite{Bollinger}) which can be written as sums of products of the\nusual binomial coefficients. \\ Eventually (see Lemma 6 in \\cite{Kirillov} and\nthe Appendix in \\cite{Mendonca}) this leads t\n\\begin{equation}\nc_{0}\\left( s,n,j\\right) =\\sum_{k=0}^{\\left\\lfloor \\frac{nj+s\n{2j+1}\\right\\rfloor }\\left( -1\\right) ^{k}\\binom{n}{k}\\binom{nj+s-\\left(\n2j+1\\right) k+n-1}{nj+s-\\left( 2j+1\\right) k}\\ .\n\\end{equation}\nFor example, if $j=1\/2$ the $c_{0}$s reduce to a single binomial coefficient\n\\cite{Bethe}\n\\begin{equation}\nc_{0}\\left( s,n,1\/2\\right) =\\binom{n}{n\/2-s}\\ ,\\ \\ \\ M\\left(\ns;n;1\/2\\right) =\\binom{n}{n\/2-s}-\\binom{n}{n\/2-s-1}\\ , \\label{SpinHalf\n\\end{equation}\nwhere $0\\leq2s\\leq n$, with $s=0$ allowed only for even $n$.\n\n\\subsection*{A Lattice of Multiplicities}\n\nOne may visualize $M\\left( s;n;j\\right) $ as a 3-dimensional semi-infinite\nlattice of points $\\left( s;n;j\\right) $ with integer multiplicities\nappropriately assigned to each lattice point. \\ There are many straight lines\non this lattice such that the multiplicities are polynomial in the line\nparameterization. \\ For example, along some of the lattice diagonals\n\\begin{equation}\nM\\left( n;n;1\\right) =1\\ ,\\ \\ \\ M\\left( n-1;n;1\\right)\n=n-1\\ ,\\ \\ \\ M\\left( n-2;n;1\\right) =\\tfrac{1}{2}~n\\left( n-1\\right) \\ .\n\\end{equation}\nThese are, respectively, the number of ways the highest possible spin (i.e.\n$s=n$), the 2nd highest spin ($s=n-1$), and the 3rd highest spin ($s=n-2$)\noccur in the Kronecker product of $n$ vector (i.e. $s=1$) representations.\n\\ The form for the number of spins farther below the maximum $s=n$, that occur\nin products of $n$ vectors, i\n\\begin{subequations}\n\\begin{align}\nM\\left( n-\\left( 2k+2\\right) ;n;1\\right) & =\\tfrac{1}{\\left(\n2k+2\\right) !}~n\\left( n-1\\right) \\left( n-2\\right) \\cdots\\left(\nn-k\\right) \\times p_{k+1}\\left( n\\right) \\ ,\\\\\nM\\left( n-\\left( 2k+3\\right) ;n;1\\right) & =\\tfrac{1}{\\left(\n2k+3\\right) !}~n\\left( n-1\\right) \\left( n-2\\right) \\cdots\\left(\nn-k\\right) \\times q_{k+2}\\left( n\\right) \\ ,\n\\end{align}\nfor $k=0,1,2,3,\\cdots$, where $p_{k+1}$ and $q_{k+2}$ are polynomials in $n$\nof order $k+1$ and $k+2$, as follows.\n\\end{subequations}\n\\begin{subequations}\n\\begin{align}\np_{k+1}\\left( n\\right) & =n^{k+1}+\\tfrac{1}{2}\\left( k+1\\right) \\left(\n5k-2\\right) n^{k}+\\tfrac{1}{24}\\left( k\\right) \\left( k+1\\right) \\left(\n75k^{2}-205k-134\\right) n^{k-1}+\\cdots\\ ,\\\\\nq_{k+2}\\left( n\\right) & =n^{k+2}+\\tfrac{1}{2}\\left( k\\right) \\left(\n5k+7\\right) n^{k+1}+\\tfrac{1}{24}\\left( k+1\\right) \\left( 75k^{3\n-85k^{2}-410k-168\\right) n^{k}+\\cdots\\ .\n\\end{align}\nAs an exercise, the reader may verify the complete polynomials for orders $2$,\n$3$, $4$, and $5$\n\\end{subequations}\n\\begin{gather}\np_{2}\\left( n\\right) =n^{2}+3n-22\\ ,\\ \\ \\ q_{2}\\left( n\\right)\n=n^{2}-7\\ ,\\\\\np_{3}\\left( n\\right) =n^{3}+12n^{2}-61n-192\\ ,\\ \\ \\ q_{3}\\left( n\\right)\n=n^{3}+6n^{2}-49n+6\\ ,\\nonumber\\\\\np_{4}\\left( n\\right) =n^{4}+26n^{3}-37n^{2}-1622n+120\\ ,\\ \\ \\ q_{4}\\left(\nn\\right) =n^{4}+17n^{3}-91n^{2}-587n+1200\\ ,\\nonumber\\\\\np_{5}\\left( n\\right) =n^{5}+45n^{4}+205n^{3}-5565n^{2\n-17486n+48720\\ ,\\ \\ \\ q_{5}\\left( n\\right) =n^{5}+33n^{4}-23n^{3\n-3393n^{2}+2542n+21000\\ .\\nonumber\n\\end{gather}\nAt the time of writing, the authors have not managed to identify the $p_{k}$\nand $q_{k}$ polynomial sequences with any that were previously studied.\n\n\\subsection*{Tabulating Some Examples}\n\nFor more explicit examples, we tabulate the number of singlets that appear in\nproducts $j^{\\otimes n}$ for $j=1,\\cdots,9$ and for $n=1,\\cdots,10$. \\ The\nTable entries below were obtained just by evaluation of the integrals in\n(\\ref{ProdMultInt}) for $s=0$\n\\\n\\begin{array}\n[c]{cccccccccc\n\\mathsf{M}\\left( \\mathsf{0;n;j}\\right) &\n\\text{\\text{\\href{https:\/\/oeis.org\/A005043}{\\text{j = 1}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A007043}{\\text{j = 2}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A264608}{\\text{j = 3}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A272393}{\\text{j = 4}}}} &\n\\text{\\text{\\href{https:\/\/oeis.org\/A272395}{\\text{j = 5}}}} & \\mathsf{j=6} &\n\\mathsf{j=7} & \\mathsf{j=8} & \\mathsf{j=9}\\\\\n\\mathsf{n=1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\\mathsf{n=2} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n\\mathsf{n=3} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n\\mathsf{n=4} & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19\\\\\n\\text{\\textsf{\\text{\\href{https:\/\/oeis.org\/A005891}{\\text{n = 5}}}}} & 6 &\n16 & 31 & 51 & 76 & 106 & 141 & 181 & 226\\\\\n\\text{\\textsf{\\text{\\href{https:\/\/oeis.org\/A005917}{\\text{n = 6}}}}} & 15 &\n65 & 175 & 369 & 671 & 1105 & 1695 & 2465 & 3439\\\\\n\\mathsf{n=7} & 36 & 260 & 981 & 2661 & 5916 & 11\\,516 & 20\\,385 & 33\\,601 &\n52\\,396\\\\\n\\mathsf{n=8} & 91 & 1085 & 5719 & 19\\,929 & 54\\,131 & 124\\,501 & 254\\,255 &\n474\\,929 & 827\\,659\\\\\n\\mathsf{n=9} & 232 & 4600 & 33\\,922 & 151\\,936 & 504\\,316 & 1370\\,692 &\n3229\\,675 & 6836\\,887 & 13\\,315\\,996\\\\\n\\mathsf{n=10} & 603 & 19\\,845 & 204\\,687 & 1178\\,289 & 4779\\,291 &\n15\\,349\\,893 & 41\\,729\\,535 & 100\\,110\\,977 & 217\\,915\\,579\n\\end{array}\n\\]\n\n\n\\subsubsection*{Nonpolynomial Columns}\n\nThe columns of the Table are \\emph{not} expressible as polynomials in $n$, for\nany fixed $j$, but they may be written as sums of hypergeometric or rational\nfunctions of $n$. \\ For example, the first two columns may be written a\n\\begin{align}\nM\\left( 0;n;1\\right) & =3^{n}\\sum_{k=0}^{n}\\binom{n}{k}\\binom{2k+1\n{k+1}\\left( -\\frac{1}{3}\\right) ^{k}=\\frac{4^{n}}{\\sqrt{\\pi}}\\frac\n{\\Gamma\\left( \\frac{1}{2}+n\\right) }{\\Gamma\\left( 2+n\\right) }\\left.\n_{2}F_{1}\\right. \\left( -n,-1-n;\\frac{1}{2}-n;\\frac{1}{4}\\right)\n\\ ,\\label{Hyper1}\\\\\nM\\left( 0;n;2\\right) & =\\frac{1}{2}\\sum_{k=0}^{n}\\frac{\\left( -6\\right)\n^{k}~\\Gamma\\left( \\frac{1}{2}+k\\right) }{\\Gamma\\left( 1+\\frac{k}{2}\\right)\n\\Gamma\\left( \\frac{3}{2}+\\frac{k}{2}\\right) }\\binom{n}{k}\\left. _{3\nF_{2}\\right. \\left( \\frac{1}{4}+\\frac{k}{2},\\frac{3}{4}+\\frac{k\n{2},k-n;1+\\frac{k}{2},\\frac{3}{2}+\\frac{k}{2};-16\\right) \\ . \\label{Hyper2\n\\end{align}\nTo obtain these and other multiplicities as hypergeometric functions, for\ninteger $s$ and $j$, it is useful to change variables, to $t=\\cos^{2\n\\vartheta$, so that (\\ref{ProdMultInt}) becomes\n\\begin{equation}\nM\\left( s;n;j\\right) =\\frac{2}{\\pi}~4^{s+nj}\\int_{0}^{1}\\left(\n\\prod\\limits_{k=1}^{s}\\left( t-r_{k}\\left( s\\right) \\right) \\right)\n\\left( \\prod\\limits_{l=1}^{j}\\left( t-r_{l}\\left( j\\right) \\right)\n\\right) ^{n}\\sqrt{\\frac{1-t}{t}}~dt\\ . \\label{HyperForm\n\\end{equation}\nThe products here involve the known roots $r_{l}\\left( j\\right) $ of the\nChebyshev polynomials. \\ For integer $j$,\n\\begin{equation}\nU_{2j}\\left( \\cos\\left( \\vartheta\\right) \\right) =4^{j}\\prod\n\\limits_{l=1}^{j}\\left( t-r_{l}\\left( j\\right) \\right) \\ ,\\ \\ \\ t=\\cos\n^{2}\\vartheta\\ ,\\ \\ \\ r_{l}\\left( j\\right) =\\cos^{2}\\left( \\frac{l\\pi\n}{2j+1}\\right) \\ , \\label{ChebPolys&Roots\n\\end{equation}\nwhile for semi-integer $j$, for comparison to the integer case,\n\\begin{equation}\nU_{2j}\\left( \\cos\\left( \\vartheta\\right) \\right) =4^{j}\\sqrt{t\n~\\prod\\limits_{l=1}^{j-\\frac{1}{2}}\\left( t-\\ r_{l}\\left( j\\right) \\right)\n\\ ,\n\\end{equation}\nwith the usual convention that the empty product is $1$.\n\nThe columns of the Table should be compared to the multiplicities of integer\nspins that appear in the product of $2m$ spin $1\/2$ representations. \\ These\nare well-known to be given by the\n\\href{https:\/\/en.wikipedia.org\/wiki\/Catalan's_triangle}{Catalan triangle}\n\\cite{SU(2)q}\n\\begin{equation}\nM\\left( s;2m;1\/2\\right) =\\frac{\\left( 1+2s\\right) \\left( 2m\\right)\n!}{\\left( m-s\\right) !\\left( m+s+1\\right) !}\\ , \\label{Catalan\n\\end{equation}\nas follows from (\\ref{SpinHalf}). \\ As an aside, it is perhaps not so\nwell-known that multiplicities of all $SU\\left( N\\right) $ representations\noccurring in the product of $n$ fundamental $N$-dimensional representations\nare given by \\href{http:\/\/oeis.org\/A005789}{$N$-dimensional Catalan\nstructures} \\cite{MultiCat,SU(N)}. \\ \n\nBe that as it may, this aside suggests an alternate route to obtain and to\nre-express some of the above results, especially for $j=1$, a route that\n\\emph{retraces} [pun intended] many of the logical steps. \\ This other route\nuses the explicit formula \\cite{SU(N)} for products of fundamental triplets of\nthe group $SU\\left( 3\\right) $\\ and the \\textquotedblleft tensor\nembedding\\textquotedblright\\ $SU\\left( 3\\right) \\supset SU\\left( 2\\right)\n$ (where the triplet of $SU\\left( 3\\right) $ is identified with the $s=1$\nvector representation) to deduce the number of $s=0$ singlets appearing in the\nproduct of $n$ vector representations of $SU\\left( 2\\right) $, namely,\n\\begin{equation}\nM\\left( 0;n;1\\right) =\\left( -1\\right) ^{n}\\left. _{2}F_{1}\\right.\n\\left( -n,\\tfrac{1}{2};2;4\\right) \\ .\n\\end{equation}\nThis is in exact agreement with the seemingly different result (\\ref{Hyper1}).\n\\ Combining this with the elementary recursion relation that follows from\n$\\overrightarrow{s}\\otimes\\overrightarrow{1}=\\overrightarrow{s+1\n\\oplus\\overrightarrow{s}\\oplus\\overrightarrow{s-1}$, namely\n\\begin{equation}\nM\\left( s;n;1\\right) =M\\left( s+1;n-1;1\\right) +M\\left( s;n-1;1\\right)\n+M\\left( s-1;n-1;1\\right) \\ ,\n\\end{equation}\none then obtains $M\\left( s;n;1\\right) $ as a sum of Gauss hypergeometric\nfunctions. \\ Relations between contiguous functions then simplify the result\nto a single hypergeometric function\n\\begin{equation}\nM\\left( s;n;1\\right) =\\left( -1\\right) ^{n+s}\\binom{n}{s}\\left. _{2\nF_{1}\\right. \\left( s-n,s+\\frac{1}{2};2+2s;4\\right) \\ .\n\\end{equation}\nFinally, the standard integral representation for $\\left. _{2}F_{1}\\right. $\neventually leads to the same integral form for $M\\left( s;n;1\\right) $ as\ngiven by (\\ref{ProdMultInt}) for $j=1$.\n\n\\subsubsection*{Polynomial Rows}\n\nIn contrast to the columns, the rows of the Table \\emph{are} expressible as\npolynomials in $j$ for any fixed $n$. \\ Starting with $n=3$, the entries in\nthe $n$th row of the Table are polynomials in $j$ of order $n-3$. \\ The fourth\nrow is obviously just the dimension of the spin $j$ representation, and the\nfifth row is less obviously $1+\\frac{5}{2}c_{j}$, where $c_{j}$ is the\nquadratic $su\\left( 2\\right) $ Casimir for spin $j$. \\ In fact, based on the\nnumbers displayed above and some modest extensions of the Table, the row\nentries are seen to be of the form $poly_{\\left( n-3\\right) \/2}\\left(\nc_{j}\\right) $ for odd $n\\geq3$ and $poly_{\\left( n-4\\right) \/2}\\left(\nc_{j}\\right) \\times d_{j}$ for even $n\\geq4$, where $poly_{k}\\left(\nc\\right) $ is a polynomial in $c$ of order $k$. \\ For the\\ last eight rows of\nthe Table these polynomials are given by\n\\begin{gather\n\\begin{array}\n[c]{ccccc\n\\mathsf{n=3} & 1\\medskip & & \\mathsf{n=4} & d_{j}\\medskip\\\\\n\\mathsf{n=5} & 1+\\frac{5}{2}c_{j}\\medskip & & \\mathsf{n=6} & \\left(\n1+2c_{j}\\right) d_{j}\\medskip\\\\\n\\mathsf{n=7} & 1+\\frac{14}{3}c_{j}+\\frac{77}{12}c_{j}^{2}\\medskip & &\n\\mathsf{n=8} & \\left( 1+4c_{j}+\\frac{16}{3}c_{j}^{2}\\right) d_{j}\\medskip\\\\\n\\mathsf{n=9} & 1+\\frac{27}{4}c_{j}+\\frac{73}{4}c_{j}^{2}+\\frac{289}{16\nc_{j}^{3}\\medskip & & \\mathsf{n=10} & \\left( 1+6c_{j}+\\frac{143}{9}c_{j\n^{2}+\\frac{140}{9}c_{j}^{3}\\right) d_{j}\\medskip\n\\end{array}\n\\\\\n\\text{where\\ }d_{j}=1+2j\\ ,\\ \\ \\ \\text{and\\ \\ \\ }c_{j}=j\\left( 1+j\\right)\n\\ .\n\\end{gather}\nThus the\\ ten rows of the Table may be effortlessly extended to arbitrarily\nlarge $j$. \\ Moreover, to obtain the polynomial that gives any row for $n>10$,\nfor arbitrary values of $j$, it is only necessary to evaluate $M\\left(\n0;n;j\\right) $ for $1\\leq j\\leq\\left\\lfloor \\frac{n-1}{2}\\right\\rfloor $.\n\\ Once again, at the time of writing, the authors have not managed to identify\nthis polynomial sequence with any that were previously studied.\n\n\\subsection*{Asymptotic Behavior}\n\nFinally, consider the extension of the columns of the Table to arbitrarily\nlarge $n$, or more generally, consider the asymptotic behavior of $M\\left(\ns;n;j\\right) $ as $n\\rightarrow\\infty$ for fixed $s$ and $j$. \\ This behavior\ncan be determined in a straightforward way, for any $s$ and $j$, by a careful\nasymptotic analysis of the integral in (\\ref{ProdMultInt}). \\ Such\n$n\\rightarrow\\infty$ behavior may be of interest in various statistical problems.\n\nThe simplest illustration is $M\\left( 0;n;1\/2\\right) $ for even $n$. \\ For\nthis particular case, (\\ref{Catalan}) and Stirling's approximation,\n$n!\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2\\pi n}\\left( \\frac{n}{e}\\right)\n^{n}$, give directly the main term in the asymptotic behavior\n\\begin{equation}\nM\\left( 0;2m;1\/2\\right) \\underset{m\\rightarrow\\infty}{\\sim}\\frac{4^{m\n}{m^{3\/2}\\sqrt{\\pi}}\\left( 1+O\\left( \\frac{1}{m}\\right) \\right) \\ .\n\\label{SpinHalfSingAsymp\n\\end{equation}\nOn the other hand, upon setting $t=\\cos^{2}\\vartheta$ the integral\n(\\ref{ProdMultInt}) has a form like that in (\\ref{HyperForm}), namely\n\\begin{equation}\nM\\left( 0;2m;1\/2\\right) =\\frac{2}{\\pi}~4^{m}\\int_{0}^{1}t^{m}\\sqrt\n{\\frac{1-t}{t}}~dt=\\frac{2}{\\pi}~4^{m}B\\left( m+\\frac{1}{2},\\frac{3\n{2}\\right) \\ . \\label{SpinHalfSingInt\n\\end{equation}\nThe $t$ integral is just a beta function, $B\\left( m+\\frac{1}{2},\\frac{3\n{2}\\right) =\\Gamma\\left( m+\\frac{1}{2}\\right) \\Gamma\\left( \\frac{3\n{2}\\right) \/\\Gamma\\left( m+2\\right) $, which leads back to exactly\n(\\ref{Catalan}) for $s=0$. \\ But rather than using Stirling's approximation,\nit is more instructive to determine the asymptotic behavior directly from the\nintegral (\\ref{SpinHalfSingInt}) using\n\\href{https:\/\/en.wikipedia.org\/wiki\/Watson's_lemma}{Watson's lemma}. \\ Thu\n\\begin{equation}\nM\\left( 0;2m;1\/2\\right) \\underset{m\\rightarrow\\infty}{\\sim}2\\sqrt{2\n~\\frac{2^{2m}}{\\left( 2m\\right) ^{3\/2}\\sqrt{\\pi}}\\left( 1-\\frac{9\n{8m}+O\\left( \\frac{1}{m^{2}}\\right) \\right) \\ .\n\\end{equation}\nNaively it might be expected that\\emph{ }the leading asymptotic behavior\n(\\ref{SpinHalfSingAsymp}) follows from a heuristic, saddle-point-Gaussian\nevaluation of the integration in (\\ref{SpinHalfSingInt}). \\ Unfortunately,\nthat expectation is not fulfilled.\\ \\ The correct $m$ dependence is obtained\nfor $M$, but with an incorrect overall coefficient. \\ To obtain the correct\ncoefficient, a more careful analysis of the asymptotic behavior is needed, as\nprovided by Watson's lemma.\n\nSimilarly, for large $n$ the number of singlets occurring in the product of\n$n$ spin $1$ representations behaves a\n\\begin{equation}\nM\\left( 0;n;1\\right) \\underset{n\\rightarrow\\infty}{\\sim}\\frac{3\\sqrt{3}\n{8}~\\frac{3^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left( 1-\\frac{21}{16n}+O\\left( \\frac\n{1}{n^{2}}\\right) \\right) \\ , \\label{SpinOneSingAsymp\n\\end{equation}\nand the number of singlets in the product of $n$ spin $2$ representations\nbehaves a\n\\begin{equation}\nM\\left( 0;n;2\\right) \\underset{n\\rightarrow\\infty}{\\sim}\\frac{1}{8\n~\\frac{5^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left( 1-\\frac{15}{16n}+O\\left( \\frac\n{1}{n^{2}}\\right) \\right) \\ . \\label{SpinTwoSingAsymp\n\\end{equation}\n\n\nIn general, the number of spin $s$ representations occurring in the product of\n$n$ spin $j$ representations for large $n$ has asymptotic behavior \\cite{ADF}\n\\begin{equation}\nM\\left( s;n;j\\right) \\underset{n\\rightarrow\\infty}{\\sim}\\left( 1+2s\\right)\n\\left( \\frac{3}{2j\\left( j+1\\right) }\\right) ^{3\/2}~\\frac{\\left(\n1+2j\\right) ^{n}}{n^{3\/2}\\sqrt{\\pi}}\\left( 1-\\frac{3}{4n}-\\frac{9}{8n\n\\frac{1}{j\\left( j+1\\right) }-\\frac{3}{2n}\\frac{s\\left( s+1\\right)\n}{j\\left( j+1\\right) }+O\\left( \\frac{1}{n^{2}}\\right) \\right) \\ .\n\\label{SpinJSpinSAsymp\n\\end{equation}\nThis is correct for either integer or semi-integer $s$ or $j$, although of\ncourse $n$ must be (odd) even to obtain (semi-)integer $s$ from products of\nsemi-integer $j$, and only integer $s$ are produced by integer $j$.\n\\ Asymptotically then, for integer $j$,\n\\begin{equation}\nM\\left( j;n;j\\right) \/M\\left( 0;n;j\\right) =M\\left( 0;n+1;j\\right)\n\/M\\left( 0;n;j\\right) \\underset{n\\rightarrow\\infty}{\\sim}1+2j+O\\left(\n\\frac{1}{n}\\right) \\ .\n\\end{equation}\nRemarkably, this behavior is approximately seen in the Table, with errors\n$\\lessapprox10\\%$. \\ On the other hand, for semi-integer $j$ and even $n$\n\\begin{equation}\nM\\left( j;n+1;j\\right) \/M\\left( 0;n;j\\right) =M\\left( 0;n+2;j\\right)\n\/M\\left( 0;n;j\\right) \\underset{n\\rightarrow\\infty}{\\sim}\\left(\n1+2j\\right) ^{2}+O\\left( \\frac{1}{n}\\right) \\ .\n\\end{equation}\n\n\nFor integer $j$, the result in (\\ref{SpinJSpinSAsymp}) follows directly,\nalbeit tediously, from an application of Watson's lemma to (\\ref{HyperForm})\nafter switching to exponential variables. \\ In that case the overall\ncoefficient in (\\ref{SpinJSpinSAsymp}) arises as a simple algebraic function\nof the Chebyshev roots in (\\ref{ChebPolys&Roots}), namely, $1\/\\left(\n\\sum_{l=1}^{j}\\frac{1}{1-r_{l}\\left( j\\right) }\\right) ^{3\/2}$. \\ This then\nreduces to the Casimir-dependent expression in (\\ref{SpinJSpinSAsymp}) by\nvirtue of the integer $j$ identity\n\\begin{equation}\n\\sum_{l=1}^{j}\\frac{1}{1-r_{l}\\left( j\\right) }=\\frac{2}{3}~j\\left(\nj+1\\right) \\ . \\label{C(j)\n\\end{equation}\nSimilar statements apply when $j$ is semi-integer leading again to\n(\\ref{SpinJSpinSAsymp}). \\ For semi-integer $j$ the relevant identity i\n\\begin{equation}\n\\frac{1}{2}+\\sum_{l=1}^{j-1\/2}\\frac{1}{1-r_{l}\\left( j\\right) }=\\frac{2\n{3}~j\\left( j+1\\right) \\ ,\n\\end{equation}\nwith the usual convention that the empty sum is $0$.\n\n\\subsection*{All-Order Extensions of the Asymptotics}\n\nThe asymptotic behavior given by (\\ref{SpinJSpinSAsymp}) is useful for fixed\n$s$ and $j$ in the limit as $n\\rightarrow\\infty$. \\ If the resulting spin $s$\nproduced by the $n$-fold product is also allowed to become large in the limit,\ne.g. $s=O\\left( \\sqrt{n}\\right) $, then (\\ref{SpinJSpinSAsymp})\\ is\n\\emph{not} useful. \\ However, in that particular case it is possible to use\nrenomalization group methods \\cite{RG} to sum the series of terms involving\npowers of $\\frac{1}{n}\\frac{s\\left( s+1\\right) }{j\\left( j+1\\right) }$ to\nobtain an exponential, and hence an improved approximation. \\ The result i\n\\begin{equation}\nM\\left( s;n;j\\right) \\underset{n\\rightarrow\\infty}{\\sim}\\left( 1+2s\\right)\n\\left( \\frac{3}{2j\\left( j+1\\right) }\\right) ^{3\/2}~\\frac{\\left(\n1+2j\\right) ^{n}}{n^{3\/2}\\sqrt{\\pi}}~e^{-\\frac{3}{2n}\\frac{s\\left(\ns+1\\right) }{j\\left( j+1\\right) }}~\\left( 1-\\frac{3}{4n}-\\frac{9}{8n\n\\frac{1}{j\\left( j+1\\right) }+O\\left( \\frac{1}{n^{2}}\\right) \\right) \\ .\n\\label{TSvKAsymptotics\n\\end{equation}\nFor large $n$ this last expression gives\n\\href{https:\/\/cgc.physics.miami.edu\/SpinAsymptotics.html}{an excellent\napproximation} out to values\\ of $s$ of order $\\sqrt{n}$ and beyond.\n\\ Moreover, the peak in the distribution of spins $s$ produced by the product\nof $n$ spin $j$s is given for large $n$ by\n\\begin{equation}\ns_{\\text{mult}}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{nj\\left( j+1\\right)\n\/3}\\ . \\label{peak\n\\end{equation}\nThis follows from the exact result (\\ref{Catalan}) for spin $1\/2$, or from\n(\\ref{TSvKAsymptotics}) for any $j$. \\ Alternatively, for specific $j$ the\ndirect numerical evaluation of either (\\ref{ProdMultInt}) or\n(\\ref{MIsADifference}) verifies (\\ref{peak}) upon taking $n$ large, say,\n$n\\approx10^{4}$.\n\nPerhaps some further insight is provided by the asymptotic behavior of the\ncontinuous function that gives the \\emph{normalized number of states with a\ngiven total spin}, $s$, as obtained from (\\ref{TSvKAsymptotics}). \\ This is \\\n\\begin{equation}\n\\frac{\\left( 1+2s\\right) M\\left( s;n;j\\right) }{\\left( 1+2j\\right) ^{n\n}~dj\\underset{n\\rightarrow\\infty}{\\sim}\\left( 1-\\frac{3}{4n}\\left(\n1+\\frac{1}{s\\left( s+1\\right) }\\right) +O\\left( \\frac{1}{n^{2}}\\right)\n\\right) ~P\\left( x\\right) ~dx\\ , \\label{ChiSquared\n\\end{equation}\nwhere, with a suitable choice of the variable $x$, $P\\left( x\\right) $ is\nthe normalized\n\\href{https:\/\/en.wikipedia.org\/wiki\/Chi-squared_distribution}{chi-squared\nprobability distribution function} for \\emph{three} degrees of freedom\n\\begin{equation}\nx\\equiv\\frac{3\\left( 1+2j\\right) ^{2}}{8ns\\left( s+1\\right) \n\\ ,\\ \\ \\ P\\left( x\\right) =\\frac{2}{\\sqrt{\\pi}}~\\sqrt{x}~e^{-x\n\\ ,\\ \\ \\ \\int_{0}^{\\infty}P\\left( x\\right) dx=1\\ .\n\\end{equation}\nIn retrospect, this may not be a total surprise since the underlying rotation\ngroup may be parameterized by \\emph{three} Euler angles. \\ Note that this last\nasymptotic form is correctly normalized to give the total number of states as\n$n\\rightarrow\\infty$, i.e.\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\\frac{1}{\\sqrt{\\pi}}\\int_{0}^{nj}\\left( 1+2s\\right)\n^{2}~\\left( \\frac{3}{2nj\\left( j+1\\right) }\\right) ^{3\/2}e^{-\\frac{3\n{2n}\\frac{s\\left( s+1\\right) }{j\\left( j+1\\right) }}~ds=1\\ .\n\\end{equation}\nAlso note that the expression for the number of states, (\\ref{ChiSquared}),\nhas a maximum at spi\n\\begin{equation}\ns_{\\text{state}}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2}~s_{\\text{mult\n}\\underset{n\\rightarrow\\infty}{\\sim}\\sqrt{2nj\\left( j+1\\right) \/3}\\text{ .\n\\end{equation}\n\\ \n\n\\section*{Some Applications}\n\nIn closing, we stress that spin multiplicities play useful roles in a wide\nrange of fields, too numerous to present in detail here. \\ But we briefly\nsketch a few applications of the results described above.\n\nSome of the $SU\\left( 2\\right) $ results for $s=0$ have been used for\ndecades in elasticity theory \\cite{Ogden} and in quantum chemistry \\cite{AT},\nas well as in nuclear physics, as is evident from the literature we have cited\nupon recognizing that\n\\href{http:\/\/mathworld.wolfram.com\/IsotropicTensor.html}{the number of\nisotropic rank-$n$ tensors} in three dimensions is just $M\\left(\n0;n;1\\right) $.\n\nThe theory of group characters has been widely used in lattice gauge theory\ncalculations for a long time \\cite{BGZ,Creutz} and continues to play an\nimportant role in various strong coupling calculations \\cite{Unger}.\n\\ Characters are also indispensible to determine the spin content of various\nstring theories \\cite{CGGT}$.$\n\nMore generally, generic representation composition results continually find\nnew uses. \\ Recent examples include frustration and entanglement entropy for\nspin chains, with possible applications to black hole physics \\cite{Shor}.\n\nMultiplicities such as those in the Table have also attracted some recent\nattention in the field of quantum computing, ultimately with implications for\ncryptography. \\ In particular, there are so-called \\textquotedblleft\nentanglement witness\\textquotedblright\\ (EW) operators that allow the\ndetection of entangled states \\cite{LKCH,Toth,CHI}. \\ By knowing the\ndegeneracy of the EW eigenstates for an $n$-particle state, one can determine\nthe fraction of all states for which entanglement is \\textquotedblleft\ndecidable\\textquotedblright\\ --- a fraction that is especially of interest in\nthe limit of large $n$. For systems of $n$ spin $j$ particles, with the EW\noperator taken to be the Casimir of the total spin, this fraction of decidable\nstates \\cite{CHI} is denoted $f_{j}\\left( n\\right) $. \\ In this case, from\nthe asymptotic expression given above in (\\ref{ChiSquared}), one readily\nobtains\n\\href{https:\/\/www.researchgate.net\/publication\/304660051_Decidable_States_in_the_Large_N_Limit}{the\nexact result\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}~f_{j}\\left( n\\right) =f_{j}\\left( \\infty\\right)\n=\\operatorname{erf}\\left( \\sqrt{\\frac{3\/2}{s+1}}\\right) -\\sqrt{\\frac{6\/\\pi\n}{s+1}}~\\exp\\left( -\\frac{3\/2}{s+1}\\right) \\ ,\n\\end{equation}\nwhere $\\operatorname{erf}\\left( x\\right) =2\\int_{0}^{x}\\exp\\left(\n-s^{2}\\right) ds\/\\sqrt{\\pi}$ is the conventional\n\\href{https:\/\/en.wikipedia.org\/wiki\/Error_function}{error function}. \\ \n\nMany other statistical applications of spin multiplicities for large $n$ have\nbeen proposed in a recent, independent investigation of this subject\n\\cite{Poly}.\\bigskip\n\n\\textbf{Acknowledgements:} \\ We thank J Katriel and J Mendon\\c{c}a for\npointing out elegant ways to re-express the multiplicity in the general case.\n\\ We also thank A Polychronakos and K Sfetsos for an advance copy of their\npaper. \\ Finally, we thank an anonymous reviewer for bringing \\cite{Kirillov\n\\ to our attention. \\ This work was supported in part by a University of Miami\nCooper Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaomu b/data_all_eng_slimpj/shuffled/split2/finalzzaomu new file mode 100644 index 0000000000000000000000000000000000000000..a62259a15922b5feb2e4374953e3621f913923c9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaomu @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe Minimal Supersymmetric Standard Model (MSSM) is the most attractive framework for the physics beyond the Standard Model.\nIn the MSSM, gauge coupling unification is achieved and the Higgs potential is stabilized. \nDespite these good features, the MSSM has difficulty in the Higgs sector. \nThe MSSM has a $\\mu$ term, $\\mu H_1 H_2$, in the superpotential. To \nmaintain the weak-scale vacuum expectation value (VEV) of a Higgs, \n$|\\mu|$ has to be at the weak scale. However, it is difficult to explain why such a dimensionful parameter\nis much smaller than Plank scale or GUT scale. This problem is the so-called $\\mu$ problem.\n\nA simple way of solving the $\\mu$ problem is to introduce a gauge singlet, and replace the $\\mu$ by the VEV of the gauge singlet field:\n\\begin{eqnarray}\n\\mu H_1 H_2 \\rightarrow \\lambda \\left H_1 \\cdot H_2 \\ .\n\\end{eqnarray}\nThe most famous model to include a gauge singlet is the Next-to-Minimal Supersymmetric Standard Model (NMSSM).\nIn the NMSSM, a new discrete symmetry, $Z_3$, is introduced to forbid the mass term for $S$.\nHowever, $Z_3$ symmetry spontaneously breaks down when electroweak symmetry breaking occurs. \nAt this point, unacceptably large cosmological domain walls appear\\cite{domainwall}.\nIn the Nearly Minimal Supersymmetric Standard \nModel (nMSSM)\\cite{nmssm1}\\cite{nmssm2}\\cite{nmssm3}, the cosmological \ndomain wall problem is solved by tadpoles. The tadpoles are generated by \nsupergravity interactions and explicitly break the discrete symmetry. Therefore, the \ndomain wall problem does not arise.\nThe nMSSM has the same attractive feature of electroweak baryogenesis as \nthe NMSSM has. To achieve successful electroweak baryogenesis,\na strong first-order phase transition is required. Therefore, new sources of CP-violation beyond the CKM matrix have to exist. \nIn the nMSSM, there are additional sources of CP-violation in the singlet sector. Therefore, unlike MSSM, \nnMSSM does not rely on radiative contributions from a light stop for strong first-order phase transition\n\\cite{nmssm_ewbg1}\\cite{nmssm_ewbg2}.\n\nSUSY breaking terms are important in discussing phenomenology, SUSY breaking effects are transmitted to the nMSSM sector from\na hidden sector by one or more mediation schemes.\nOne interesting mediation scheme is anomaly mediation \\cite{am-lisa}\\cite{am-org}\\cite{am-sing}.\nIn anomaly mediation, the supergravity actions of the hidden sector and visible sector are sequestered.\nSUSY breaking effects are transmitted to the visible sector due to the superconformal anomaly. There are studies in which\nsoft breaking terms are derived by anomaly mediation with NMSSM-like models\\cite{am-nmssm}\\cite{am-fat}.\nIn these works, successful electroweak symmetry-breaking is achieved; however, the VEV of $S$ is on the order of a few TeV. \nThis leads to a large higgsino mass: $\\mu_{eff} = \\lambda \\left$.\nTherefore, there are large mass splittings among Higgs (and neutralinos).\n\nTo obtain a moderate value for $\\mu_{eff}$, we consider a deflected anomaly mediation scenario \\cite{dam-org}\\cite{dam-ph}\\cite{dam-pos}, \nwhich introduces an additional messenger sector. The SUSY breaking mass for a messenger is given by a VEV of the gauge singlet field, $X$.\nIn the original deflected anomaly mediation scenario\\cite{dam-org}\\cite{dam-ph},\nthe superpotential of $X$ is extremely flat; therefore, the fermionic \ncomponent of $X$, $\\psi_X$, becomes light and the lightest SUSY particle is $\\psi_X$. \nIn the positively deflected anomaly mediation scenario\\cite{dam-pos}, the superpotential is not flat; therefore, $\\psi_X$ does not have to\nbe light\\cite{dam-recent} and an ordinary SUSY particle can be a candidate for dark matter.\nWe consider the positively deflected anomaly mediation scenario.\nWe also consider SUSY breaking with the Fayet-Iliopoulos D-term. \n\nWe show that when nMSSM and deflected anomaly mediation are combined,\nsuccessful electroweak symmetry breaking occurs for a moderate value of $\\mu_{eff}$.\nWe also show that the lightest neutralino, which is mainly composed of a \nsinglino, is a good candidate for dark matter. We also present \nsparticle mass spectra. \n\nThis paper is organized as follows. In section 2, we introduce the nMSSM Lagrangian and discuss tadpoles. We also discuss the\n direct couplings between nMSSM fields and messenger sector fields.\nIn section 3, we derive the soft SUSY breaking terms of\nthe nMSSM fields in the deflected anomaly mediation scenario.\nSection 4 is devoted to the phenomenology of\nthis scenario. Finally, section 5 presents our conclusions.\n\n\\section{Nearly Minimal Supersymmetric Standard Model}\nIn this section, we discuss tadpoles and direct couplings between \nnMSSM fields and messenger sector fields. First, we introduce the nMSSM\nLagrangian.\n\nThe superpotential and soft breaking terms in the nMSSM are \n\\begin{eqnarray}\nW_{nMSSM} &=& \\lambda \\hat{S} \\hat{H_1} \\cdot \\hat{H_2} + \\frac{m_{12}^2}{\\lambda}\\hat{S} + \ny_u \\hat{Q} \\cdot \\hat{H_2} \\hat{U}^c +\ny_d \\hat{Q} \\cdot \\hat{H_1} \\hat{D}^c \\nonumber \\\\\n&& + y_l \\hat{L} \\cdot \\hat{H_1} \\hat{E}^c \\label{eq:nmssm_sp} \\ ,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n-\\mathcal{L}_{soft} &=& m_S^2 |S|^2 + (a_\\lambda S H_1 \\cdot H_2 + h.c.) + (t_S S + h.c.) \\nonumber \\\\\n&& + \\tilde{m}_{H_1}^2 H_1^\\dagger H_1 + \\tilde{m}_{H_2}^2 H_2^\\dagger H_2 \\nonumber \\\\\n&& + \\tilde{m}_Q^2 \\tilde{Q}^\\dagger \\tilde{Q} + \\tilde{m}_U^2 |\\tilde{u_R}|^2 + \\tilde{m}_D^2 |\\tilde{d_R}|^2 \n+ \\tilde{m}_L^2 \\tilde{L}^\\dagger \\tilde{L} + \\tilde{m}_E^2 |\\tilde{e_R}|^2 \\nonumber \\\\\n&& + (a_u \\tilde{Q} \\cdot H_2 \\tilde{u_R}^* + a_d \\tilde{Q} \\cdot H_1 \n \\tilde{d_R}^* + a_l \\tilde{L} \\cdot H_1 \\tilde{e_R}^* + h.c.) \\ . \\label{eq:nmssm_soft}\n\\end{eqnarray}\n$\\hat{S}$ denotes a gauge singlet chiral superfield, and $S$ is the scalar component of $\\hat{S}$. \nWhen $S$ acquires the VEV, the higgsino mass parameter, $\\mu_{eff} = \\lambda v_s$, is\ngenerated effectively. We take $\\lambda$ to be real positive by suitable redefinitions of $S$, $H_1$ and $H_2$.\nUnlike the NMSSM, there are \nno trilinear terms for the gauge singlet.\n$m_{12}^2 \\hat{S}\/\\lambda$ and $t_S S$ are tadpoles. They are absent at the tree level; \nhowever, they are generated radiatively by supergravity interactions. \nThese terms are on the order of the weak scale, as we describe below.\n\n\\subsection{Tadpoles}\nThe greatest difference between the nMSSM and NMSSM is the existence of \ntadpoles in the former. The tadpoles are generated by supergravity interaction.\nIn the nMSSM, the theory has a global discrete symmetry at tree level.\nThis symmetry guarantees that the generated tadpoles are on the order of the weak scale,\ndespite the fact that supergravity interactions break global symmetries.\nBecause the tadpoles explicitly break the discrete symmetry,\nthe domain wall problem does not appears.\n\nIn nMSSM, the Lagrangian has a discrete R symmetry $Z_{nR'}$ at tree \nlevel. The charge assignment of the fields is shown in \nTable \\ref{table:nmssm_symmetry}.\nThe charge of $Z_{nR'}$, $Q_{nR'}$, is defined as\n\\begin{eqnarray}\nQ_{PQ} + 3 Q_R \\ , \n\\end{eqnarray}\nwhere $Q_{PQ}$ denotes the charge of Peccei-Quinn symmetry, \n$U(1)_{PQ}$, and $Q_R$ denotes the charge of $U(1)_R$.\nUnder $Z_{nR'}$, the nMSSM fields transform as\n\\begin{eqnarray}\n \\Phi_i \\rightarrow \\Phi_i \\exp\\left({i \\frac{Q_{nR'}}{n} \\theta}\\right),\n\\end{eqnarray}\nwhere $\\Phi_i$ denotes nMSSM fields. If the Lagrangian respects $Z_{5R'}$ \nor $Z_{7R'}$ at the tree level, the scale of the generated tadpoles \ncan naturally be the weak scale\\cite{nmssm1}\\cite{nmssm3}.\nWhen the discrete symmetry is $Z_{5R'}$, tadpoles of six-loop order are generated.\n When the discrete symmetry is $Z_{7R'}$, tadpoles of seven-loop order are generated.\nThe tadpoles break the $Z_{5R'}$ or $Z_{7R'}$, and therefore no cosmological domain wall problem exists.\n\n\\TABLE[thbp]{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n& $\\hat{H}_1$ & $\\hat{H}_2$ & $\\hat{S}$ & $\\hat{Q}$ & $\\hat{L}$ & $\\hat{U}^c$ & $\\hat{D}^c$ & $\\hat{E}^c$ & $W$ \\\\\n\\hline\n$U(1)_{PQ}$ & 1 & 1 & -2 & -1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline\n$U(1)_R$ & 0 & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 2 \\\\\n\\hline\n$Z_{nR'}$& 1 & 1 & 4 & 2 & 2 & 3 & 3 & 3 & 6 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Charge assignments of fields}\n\\label{table:nmssm_symmetry}\n}\n\nThe generated tadpoles are given by\\cite{nmssm3}:\n\\begin{eqnarray}\nV_{tad} \\sim \\frac{1}{(16\\pi^2)^l} \\left(M_p M_{susy}^2 S +M_{susy} F_S + h.c. \\right),\n\\end{eqnarray}\nwhere $l$ is the number of the loops at which tadpoles first appear. \n$l=6$ in the $Z_{5R'}$ case and $l=7$ in $Z_{7R'}$ case.\nIn a deflected anomaly mediation scenario as well as an anomaly mediation scenario,\n$M_{susy}$ is $\\mathcal{O}(10 {\\rm \nTeV})$; therefore, $Z_{7R'}$ is favorable. \nAs we describe below, $Z_{7R'}$ also forbids direct couplings between \nnMSSM fields and messenger sector fields.\n\n\\subsection{Direct Couplings to the Messenger sector}\nIn a deflected anomaly mediation scenario, the messenger sector is \nintroduced in addition to the hidden sector, which is the origin of SUSY breaking.\nThe messenger sector contains a gauge singlet chiral superfield and messenger superfields.\nThe messengers transmit the SUSY breaking to the nMSSM sector, and this SUSY breaking is \ncomparable to that of anomaly mediation. \nIn this subsection, we show that direct couplings between the messenger sector fields and the nMSSM fields do not exist.\n\nWe consider the following superpotential in the messenger sector.\n\\begin{eqnarray}\nW_{mess} = \\frac{1}{2} m_X \\hat{X}^2 + \\lambda_X \\hat{X} \\bar{\\Psi}_i{\\Psi^i} \\ ,\\label{eq:mess}\n\\end{eqnarray}\nwhere $\\hat{X}$ is a gauge singlet chiral superfield. $\\bar{\\Psi}_i$ and ${\\Psi^i}$ are the messenger fields \nthat transform ${\\bf \\bar{5}}$ and ${\\bf 5}$ for the \n$SU(5)$ GUT gauge group respectively. \nThe $Z_{nR'}$ charge assignment of the fields is shown in Table \\ref{table:nmssm_gm_charge}.\n\n\\TABLE[thbp]{\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& $X$ & $\\Psi$ & $\\bar{\\Psi}$ & $W_{mess}$ \\\\\n\\hline\n$U(1)_{PQ}$ & 0 & 0 & 0 & 0\\\\\n\\hline\n$U(1)_R$ & 1 & 1\/2 & 1\/2 & 2\\\\\n\\hline\n$Z_{nR'}$& 3 & 3\/2 & 3\/2 & 6\\\\\n\\hline\n\\end{tabular}\n\\caption{Charge assignment of the fields in the messenger sector}\n\\label{table:nmssm_gm_charge}\n}\n\nIn this charge assignment, there are no direct couplings between messenger sector fields and nMSSM fields.\nA direct coupling between the messengers and nMSSM gauge singlet, $S \\bar{\\Psi}_i{\\Psi}^i$, is forbidden by $Z_{nR'}$ symmetry.\n$S X^2$ and $X H_u H_d$ terms are also forbidden. On the other hand, the $S^2 X$ term is forbidden by $Z_{7R'}$ but allowed by $Z_{5R'}$. \nIn the discussion about tadpoles in the previous subsection, we assumed \nthat $Z_{7R'}$ symmetry exists at tree level.\nTherefore, there are no direct couplings between nMSSM fields and the messenger sector fields.\nThe phenomenology of the nMSSM at the weak scale does not depend on the detail of messenger sector.\n\n\\section{SUSY Breaking}\nIn this section, we derive the SUSY breaking terms of the nMSSM in the deflected anomaly \nmediation scenario. We also show the corrections to the soft scalar mass \nwith the Fayet-Iliopoulos D-term.\n\nIn the original deflected anomaly mediation scenario,\n the superpotential of the gauge singlet $\\hat{X}$ is flat; therefore in general,\nthe lightest SUSY particle (LSP) is the fermionic component of $X$, $\\psi_X$.\n Threshold corrections to the sparticle mass squared are negative. \nIn the positively deflected anomaly mediation scenario,\nthe superpotential of $\\hat{X}$ is not flat; \ntherefore, $\\psi_X$ is not necessarily the LSP. Corrections to the sparticle mass squared are positive. \nWe consider the positively deflected anomaly mediation scenario.\nIn Appendix A, we give an \nexplicit example of the positively deflected anomaly mediation scenario \nin which the fermionic partner of $X$ is not the LSP.\n\nWhen $\\hat{X}$ acquires the VEV,\nmessengers obtain SUSY breaking mass, $X + F_X \\theta^2$. This SUSY breaking mass introduces an intermediate threshold \nthat depends on $\\theta^2$.\nIn deflected anomaly mediation, \ncorrections from anomaly mediation to soft breaking terms are generated by the following threshold.\n\\begin{eqnarray}\n\\frac{X+F_X \\theta^2}{\\Lambda \\hat{\\phi}} = \\frac{X}{\\Lambda} \\left[1+\\left(\\frac{F_X}{X}-F_\\phi\\right)\\theta^2 \\right] \\equiv \\frac{X}{\\Lambda}\\left(1+dF_\\phi \\theta^2\\right) , \\label{eq:thre}\n\\end{eqnarray}\nwhere $\\hat{\\phi}$ is the chiral compensator field, $\\hat{\\phi}=1+F_\\phi \n\\theta^2$, and $\\Lambda$ is the ultraviolet cutoff. $d$ is the deflection \nparameter, which denotes the threshold correction to the SUSY breaking. \nIn the positively deflected anomaly mediation scenario, $d$ is positive.\n\n\nMessengers also affect the beta-functions of gauge couplings. The \nbeta-functions of gauge couplings above the scale $|X|$ are written as\n\\begin{eqnarray}\n\\frac{d g_a}{d \\ln \\mu} = -\\frac{g_a^3}{16\\pi^2} (b_a -N_f) ,\n\\end{eqnarray}\nwhere $N_f$ is the number of messengers. \nFor the intermediate threshold and modification of the beta-functions, \nsoft breaking terms in an anomaly mediation are changed to \nthose of a deflected anomaly mediation scenario.\n\nIn a deflected anomaly mediation, the gaugino mass, soft breaking mass and \nscalar trilinear couplings at the scale $\\mu$ are obtained using the following relations\\cite{dam-org}\\cite{dam-pos}.\n\\begin{eqnarray}\n\\frac{m_\\lambda(\\mu)}{g^2(\\mu)} &=& -\\frac{F_\\phi}{2}\\left(\\frac{\\partial}{\\partial \\ln\\mu} -d\\frac{\\partial}{\\partial\\ln |X|}\\right) g^{-2}\\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right) \\nonumber \\\\\n\\tilde{m}_i^2(\\mu) &=& -\\frac{|F_\\phi^2|}{4}\\left(\\frac{\\partial}{\\partial\\ln\\mu}-d\\frac{\\partial}{\\partial\\ln|X|}\\right)^2 \\ln Z_i \\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right) \\nonumber \\\\\n\\frac{a_{ijk}(\\mu)}{y_{ijk}(\\mu)} &=& - \\frac{F_\\phi}{2}\\left(\\frac{\\partial}{\\partial\\ln\\mu}-d\\frac{\\partial}{\\partial\\ln|X|}\\right) \\sum_{l=i,j,k} \\ln Z_l \\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right), \\nonumber \\\\\n\\end{eqnarray}\nwhere $F_\\phi$ is the F-term of the chiral compensator fields and corresponds to the gravitino mass. $|X|$ is the messenger scale and $\\mu < |X|$. $d$ \nis defined in eq. (\\ref{eq:thre}).\n$|X|$ and $d$ can be determined by the superpotential and the soft breaking terms in \nthe messenger sector (see Appendix A). However, we treat them as the parameters of SUSY breaking because we focus on the \nphenomenology at the weak scale.\n\nIn general, the formula for soft breaking terms is complicated. However, by setting the scale as $\\mu = |X|$,\n the soft breaking terms are simplified as\n\\begin{eqnarray}\nm_{\\lambda_a} &=& -\\frac{g_a^2}{(4\\pi)^2}\\left(b_a - dN_f\\right) F_\\phi \n \\ ,\\nonumber \\\\\n\\tilde{m}_i^2 &=& \\frac{|F_\\phi|^2}{2(4\\pi)^4} \\sum_a c_a^i g_a^4 \\left[b_a + d(d+2)N_f\\right] \\nonumber \\\\\n&& - \\frac{|F_\\phi|^2}{4} \\sum_y \\frac{\\partial \\gamma_i(|X|)}{\\partial y} \\beta_y(|X|) \\ ,\\nonumber \\\\\na_{ijk} &=& -\\frac{F_\\phi}{2} \\left[\\gamma_i(|X|) + \\gamma_j(|X|) + \\gamma_k(|X|) \\right] y_{ijk} \\label{eq:formula}.\n\\end{eqnarray}\nHere, $N_f$ is the number of messengers. $b_a$ and $c_a^i$ are the coefficients \nof the gauge coupling beta functions and the anomalous dimensions of the \nfields respectively.\n$b_a = (-33\/5, -1, 3)$, $c_a^L=(3\/5, 3, 0)$, $c_a^{E^c}=(12\/5, 0, 0)$, $c_a^Q=(1\/15, 3, 16\/3)$,\n$c_a^{U^c}=(16\/15, 0,16\/3)$ and $c_a^{D^c}=(4\/15, 0, 16\/3)$.\n\nThe formula for gaugino masses is easily obtained with\n\\begin{eqnarray}\ng_a^{-2}(\\mu) = g_a^{-2}(\\Lambda) + \\frac{b_a}{8\\pi^2} \\ln\\frac{\\mu}{|X|} + \\frac{b_a-N_f}{8\\pi^2}\\ln\\frac{|X|}{\\Lambda}. \\label{eq:gg}\n\\end{eqnarray}\nEquation (\\ref{eq:gg}) can be obtained by integrating the beta-functions explicitly.\nThe derivations of $\\tilde{m}_i^2$ and $a_{ijk}$ are given in Appendix B. \n\nFor the first and second generations of squarks and sleptons, we can\nneglect the contributions from Yukawa couplings. However, for the soft scalar \nmass and the A-term of the third generation of squarks and sleptons,\n the contributions from Yukawa couplings are important. \nFor $\\tilde{m}_S^2$, $\\tilde{m}_{H_1}^2$, $\\tilde{m}_{H_2}^2$ and $a_\\lambda$, contributions from\nYukawa couplings are also important. The anomalous dimensions of $H_1$ and $H_2$ are different from those of the MSSM due to $\\lambda$\nand are given in Appendix C. \nThe anomalous dimensions of the other fields are same as those of the MSSM and are given in \\cite{beta-functions}.\nWhen Yukawa couplings are small,\nwe obtain the results of \\cite{dam-org}\\cite{dam-pos}.\n\nIn a supersymmetric model, there is an additional source of SUSY breaking, the Fayet-Iliopoulos D-term. \nThis term contributes to the square of the scalar mass.\n\nThe Fayet-Iliopoulos D-term is\n\\begin{eqnarray}\n\\mathcal{L} \\ni - \\xi D .\n\\end{eqnarray}\nThe D-term of the Lagrangian is written as\n\\begin{eqnarray}\n\\mathcal{L}_D = \\frac{1}{2} D^2 - g D \\sum_i q_i A_i^\\dagger A_i - \\xi D , \\label{eq:dterm-xi}\n\\end{eqnarray}\nwhere $q_i$ is the U(1) charge of the field $A_i$. After eliminating the \nD-term with the equation of motion,\nthe Lagrangian $\\mathcal{L}_D$ becomes\n\\begin{eqnarray}\n\\mathcal{L}_D = - \\frac{1}{2} \\left( \\sum_i q_i A_i^\\dagger A_i + \\xi \n\t\t\t \\right)^2 \\ .\n\\end{eqnarray}\nThis leads to additional contributions to the scalar mass terms:\n\\begin{eqnarray}\n\\tilde{m}_{ij}^2 \\rightarrow \\tilde{m}_{ij}^2 + q_i \\xi \\delta_{ij} .\n\\end{eqnarray}\n\nIn the Supersymmetric Standard Model, there is only one $U(1)$ gauge group.\nIn the lepton sector, the hypercharge of the $SU(2)$ doublet is $-1$ and the \nhypercharge of the $SU(2)$ singlet is $+2$. Therefore, we can not solve the tachyonic slepton mass problem \nin anomaly mediation using only the $U(1)_Y$ D-term. \n\nSo far, the additional contributions from the D-term to the soft \nbreaking mass of the nMSSM matter fields\nare\n\\begin{eqnarray}\n&& \\delta \\tilde{m}_L^2 = -D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_E^2 = 2D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_Q^2 = \\frac{1}{3} D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_U^2 = -\\frac{4}{3}D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_D^2 = \\frac{2}{3}D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_{H_1}^2 = -D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_{H_2}^2 = D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ , \\label{eq:dtermcont}\n\\end{eqnarray}\nwhere $D_Y$ comes from the common parameter $\\xi$ in eq. (\\ref{eq:dterm-xi}).\n\nWe can now evaluate the soft breaking terms of the nMSSM at the messenger scale using eq. (\\ref{eq:formula}). \nWe solve the renormalization group equations (RGE) using them as the boundary conditions, and then evaluate the soft breaking\nterms at the weak scale. We use RGE codes contained in the NMSSMTools software package\\cite{nmssmtools1}\\cite{nmssmtools2}.\nWe also add the D-term contributions in eq. (\\ref{eq:dterm-xi}) to the soft breaking mass.\nIn the next section, we discuss the phenomenology of the nMSSM with the soft breaking terms obtained by deflected anomaly mediation. \n\n\n\\section{Phenomenology}\nIn this section, we investigate the phenomenological aspects of the nMSSM. \nFirst, we discuss the existence of Landau poles.\nWe demand that $\\lambda$ should not meet the Landau pole up to the scale at\nwhich the tadpoles are generated.\nNext we study the regions of parameter space where successful electroweak breaking occurs, and we evaluate \nthe mass of the lightest Higgs. Subsequently, we discuss the lightest \nneutralino as a dark matter candidate. We evaluate the relic density of the\nlightest neutralino. We also discuss the direct detection of dark matter. Finally, we obtain sparticle mass spectra.\n\n\\subsection{The Landau pole}\nIn the nMSSM, tadpoles generated by supergravity interaction are proportional to powers of $\\lambda$\\cite{nmssm3}.\nTherefore to maintain the tadpoles at the weak scale, $\\lambda$ should not meet \nthe Landau pole up to the scale, at which the tadpoles are generated.\nWe investigate the region of $\\lambda$ and $\\tan\\beta$\n that satisfies the perturbativity condition below the GUT scale.\n\nThe beta-functions of $\\lambda$ and $y_t$ are\n\\begin{eqnarray}\n\\beta_\\lambda = \\frac{1}{16\\pi^2}\\left(4\\lambda^2 + 3 y_t^2 + 3 y_b^2 + y_\\tau^2 -\\frac{3}{5}g_1^2 -3 g_2^2 \\right) \\lambda \\ , \\nonumber \\\\\n\\beta_{y_t} = \\frac{1}{16\\pi^2}\\left(\\lambda^2 + 6 y_t^2 + y_b^2 -\\frac{13}{15}g_1^2 -3 g_2^2 -\\frac{16}{3} g_3^2 \\right) y_t \\ .\n\\end{eqnarray}\nThese beta-functions strongly depend on $\\lambda$ and $\\tan \\beta$ \nthrough the top Yukawa coupling.\nFigure \\ref{fig:lpole} shows the allowed region where the perturbativity \nis satisfied up to the GUT scale. \nThe calculation is performed using the RGE code included in the NMSSMTools package. The current experimental value of the top mass is \n$173.1 \\pm 1.3$ GeV\\cite{topmass}. We take the central value for $m_{\\rm top}$ as $173.1$ GeV. \nThe shaded region is consistent with the perturbativity of $\\lambda$. The result depends on \nthe value of $m_{\\rm top}$ and supersymmetric threshold corrections of $\\alpha_s$.\nTherefore there is small difference among the results of \\cite{nmssm4} and \\cite{nmssm_ewbg1} and our results.\nIn our calculation, the region where $\\tan\\beta \\gtrsim 2.0$ and $\\lambda \\lesssim 0.7$ is allowed.\n\n\\FIGURE[htbp]{\n\\epsfig{file=landaupole.eps, width=0.6\\hsize}\n\\caption{\nThe region that is consistent with the perturbativity of $\\lambda$ up to the GUT scale is shown.\nThe gray shaded region below the solid line is allowed. The region above the solid line is excluded owing to the existence of the Landau pole below the GUT scale.\n}\n\\label{fig:lpole}\n}\n\n\\subsection{Electroweak symmetry breaking}\nIn this subsection, we consider the conditions for electroweak \nsymmetry breaking and evaluate $\\mu_{eff} = \\lambda \\left$. We \nalso evaluate the mass of the lightest Higgs. \n\nAfter obtaining the soft breaking terms at the weak scale, we now evaluate the Higgs potential, $V = V_{tree} + \\Delta V$.\nFrom eqs. (\\ref{eq:nmssm_sp}) and (\\ref{eq:nmssm_soft}), the tree-level Higgs \npotential is written as \n\\begin{eqnarray}\nV_{tree} &=& \\tilde{m}_{H_1}^2 H_1^\\dagger H_1 + \\tilde{m}_{H_2}^2 H_2^\\dagger H_2 + m_s^2 |S|^2 + m_{12}^2 (H_1\\cdot H_2 + h.c.) \\nonumber \\\\\n&+& \\lambda^2 |H_1 \\cdot H_2 |^2 + \\lambda^2 |S|^2 (H_1^\\dagger H_1 +H_2^\\dagger H_2 ) + \\frac{g^2}{2}|H_1^\\dagger H_2|^2 \\nonumber \\\\\n&+& \\frac{\\bar{g}^2}{8}(H_2^\\dagger H_2 - H_1^\\dagger H_1)^2 + (t_s S + \n h.c.) +(a_\\lambda S H_1 \\cdot H_2 + h.c.) \\ , \\label{eq:potential}\n\\end{eqnarray}\nwhere $\\bar{g}^2 = g^2+g'^2$. \n$\\Delta V$ is the one-loop contribution to the effective potential \\cite{cw-potential}:\n\\begin{eqnarray}\n\\Delta V = \\frac{1}{64\\pi^2}\\left(\\sum_b g_b m_b^4 \\left[\\ln\\left(\\frac{m_b^2}{Q^2}\\right)-\\frac{3}{2}\\right]\n- \\sum_f g_f m_f^4 \\left[\\ln\\left(\\frac{m_f^2}{Q^2}\\right)-\\frac{3}{2}\\right]\\right) \\ . \\label{eq:coleman-wein}\n\\end{eqnarray}\n$g_b$ and $g_f$ are the degrees of freedom for bosons and fermions respectively.\nWe determine $\\mu_{eff}\\equiv \\lambda \\left$, $t_s$ and \n$m_{12}^2$ using the stationary conditions of the Higgs potential. \nFrom eqs. (\\ref{eq:potential}) and (\\ref{eq:coleman-wein}), the stationary conditions are\n\\begin{eqnarray}\n \\frac{\\partial V}{\\partial v_1} &=& 2v_1 \\left[\\tilde{m}_{H_1}^2 + (m_{12}^2 + \n\t\t\t\t\t a_\\lambda \t\t\t\t\t \nv_s)\\frac{v_2}{v_1}-\\frac{\\bar{g}^2}{4}(v_2^2-v_1^2)+\\lambda^2(v_2^2+v_s\n^2) + \\frac{1}{2v_1}\\frac{\\partial \\Delta V}{\\partial v_1} \\right] = 0 \\ , \\nonumber \\\\\n \\frac{\\partial V}{\\partial v_2} &=& 2v_2 \\left[\\tilde{m}_{H_2}^2 + (m_{12}^2 + \n\t\t\t\t\t a_\\lambda v_s)\\frac{v_1}{v_2}+\\frac{\\bar{g}^2}{4}(v_2^2-v_1^2)+\\lambda^2(v_1^2+v_s\n^2) + \\frac{1}{2v_2}\\frac{\\partial \\Delta V}{\\partial v_2} \\right] =0 \\ ,\\nonumber \\\\\n \\frac{\\partial V}{\\partial v_s} &=& 2v_s \\left[m_s^2+\\lambda^2(v_1^2+v_2^2)+\\frac{t_s}{v_s}+a_\\lambda \\frac{v_1 v_2}{v_s} + \\frac{1}{2v_s}\\frac{\\partial \\Delta V}{\\partial v_s} \\right]=0 \\label{eq:higgs_kyokuchi} \\ ,\n\\end{eqnarray}\nwhere $v_1=\\left$, $v_2=\\left$ and $v_s = \\left$. \nAs we describe later, there is only a small region\nof parameter space where successful electroweak symmetry breaking \noccurs with $v_s > 0$; therefore, we take $v_s < 0$.\nFrom eq. (\\ref{eq:higgs_kyokuchi}), $\\mu_{eff}$ can be determined by,\n\\begin{eqnarray}\n\\mu_{eff}^2 = -\\frac{M_Z^2}{2} + \\frac{\\tilde{m}_{H_1}^2 + \\frac{1}{2v_1}\\frac{\\partial \\Delta V}{\\partial v_1} - \n\\left(\\tilde{m}_{H_2}^2 + \\frac{1}{2v_2}\\frac{\\partial \\Delta V}{\\partial v_2}\\right) \\tan^2\\beta}{\\tan^2\\beta-1} \\label{eq:mueff}.\n\\end{eqnarray}\n$\\mu_{eff}$, $m_{12}^2$ and $t_s$ are determined from eq. (\\ref{eq:higgs_kyokuchi}). We now evaluate the Higgs mass. We \nexpand $H_1^0$, $H_2^0$ and $S$ as\n\\begin{eqnarray}\nH_1^0 &=& v_1 + \\frac{1}{\\sqrt{2}}\\left(h_1^0 + i a_1\\right), \\nonumber \\\\\nH_2^0 &=& v_2 + \\frac{1}{\\sqrt{2}}\\left(h_2^0 + i a_2\\right), \\nonumber \\\\\nS^0 &=& v_s + \\frac{1}{\\sqrt{2}}\\left(s + i a_s\\right). \n\\end{eqnarray}\nUsing these expanded fields, the CP-even Higgs mass matrix is written as\n\\begin{eqnarray}\n\\left(h_1^0 \\ h_2^0 \\ S \\right) M^2 \\left(\n\\begin{array}{c}\nh_1^0 \\\\\nh_2^0 \\\\\nS\n\\end{array}\n\\right) .\n\\end{eqnarray}\nAt tree level, the components of $M^2$ are\n\\begin{eqnarray}\nM_{11}^2 &=& s_\\beta^2 M_a^2 + c_\\beta^2 M_Z^2 \\ , \\nonumber \\\\\nM_{12}^2 &=& -s_\\beta c_\\beta \\left(M_a^2 + M_Z^2 - 2\\lambda^2 v^2 \\right) \\ ,\\nonumber \\\\\nM_{13}^2 &=& v \\left(s_\\beta a_\\lambda + 2 c_\\beta \\lambda^2 v_s^2 \\right) \\ ,\\nonumber \\\\\nM_{22}^2 &=& c_\\beta^2 M_a^2 + s_\\beta^2 M_Z^2 \\ ,\\nonumber \\\\\nM_{23}^2 &=& v \\left(c_\\beta a_\\lambda 2 + s_\\beta \\lambda^2 v_s \\right) \\ ,\\nonumber \\\\\nM_{33}^2 &=& -\\frac{1}{v_s}\\left(t_s + s_\\beta c_\\beta a_\\lambda v_s \\right),\n\\end{eqnarray}\nwhere $c_\\beta = \\cos\\beta$ and $s_\\beta = \\sin\\beta$.\nThe CP-odd Higgs mass matrix at tree-level is\n\\begin{eqnarray}\n\\left(A^0 \\ a_s \\right)\n\\left[\n\\begin{array}{cc}\nM_a^2 & -a_\\lambda v_s \\\\\n-a_\\lambda v_s & -\\frac{1}{v_s}\\left( t_s + s_\\beta c_\\beta a_\\lambda v^2 \\right)\n\\end{array}\n\\right]\n\\left(\n\\begin{array}{c}\nA^0 \\\\\na_s\n\\end{array}\n\\right) ,\n\\end{eqnarray}\nwhere $M_a^2 = -\\left(m_{12}^2 + a_\\lambda v_s \\right)\/c_\\beta s_\\beta$. \n$A^0 = a_d s_\\beta + a_u c_\\beta$, and its orthogonal combination is absorbed by the Z boson.\n\nWe now present the results of numerical calculations.\nFigure \\ref{fig:nmssm_ewsbok} shows the allowed region of successful electroweak symmetry breaking without tachyonic sleptons.\nWe set the messenger scale to $5 F_\\phi \\simeq 150 \\ {\\rm TeV}$.\nSuccessful electroweak symmetry breaking occurs in the region covered by red squares.\nIn the region covered by blue crosses, the mass of the lightest Higgs \nsatisfies the LEP bound with the electroweak symmetry breaking.\nWhen the number of the messengers, $N_f$ increases, the allowed region of the \ndeflection parameter $d$ is shifted downward. Therefore, in the scenario with small \n$d$ ($d < 1$), two or more messengers have to exist. \nFor simplicity, we assume that there is one messenger in the following analysis. \nAlthough there is a region where successful electroweak symmetry \nbreaking occurs with large $\\tan\\beta$ and $v_s >0$, \nthe region is very small. Therefore we take $v_s < 0$.\n\nFigure \\ref{fig:nmssm_mueff} shows the dependence of $\\mu_{eff}$ on SUSY \nbreaking. \nWe see that $|\\mu_{eff}|$ is a decreasing function of $D_Y$, while it is an increasing function of $d$.\nThis can be understood from eqs. (\\ref{eq:formula}), (\\ref{eq:dtermcont}) and (\\ref{eq:mueff}).\nWhen $D_Y$ increases,\n$m_{H_1}^2$ decreases and $m_{H_2}^2$ increases. This implies that $|\\mu_{eff}|$ decreases as $D_Y$ increases.\nWhen $d$ increases, \n$m_{H_1}^2$ and $m_{H_2}^2$ increase at almost the same rate. This implies that $|\\mu_{eff}|$ increases as $d$ increases.\nIn this scenario, moderate values of $\\mu_{eff}$, $100 < |\\mu_{eff}| < \n550$, are obtained without meeting the Landau pole.\n\nFigure \\ref{fig:higgsmass} shows the dependence of the lightest Higgs mass on $d$ and $D_Y$. \nThe calculation is performed with NMSSMTools, including two-loop corrections. We extend the codes to include tadpoles. \nIn this scenario, the mass of the lightest Higgs can be\nheavier than the LEP bound.\n\n\\FIGURE[htbp]{\n\\hspace*{-6mm}\n\\epsfig{file=ewsb1.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb2.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb3.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb4.eps,width=0.45\\hsize}\n\\caption{\nSuccessful electroweak symmetry breaking occurs in the region covered by red squares, and the region covered by blue crosses satisfies\nthe Higgs mass bound of the LEP. In other regions, the sleptons are tachyonic.\nThe calculation is performed with $\\lambda=0.69$ and $m_0 = F_\\phi\/(4\\pi)^4 = 200 \\ {\\rm GeV}$. The messenger scale is taken to be $5 F_\\phi$.\n$\\tan\\beta$ and the number of messengers $N_f$ are $\\tan\\beta=2$ and $N_f=1$ in the top-left figure, \n$\\tan\\beta=3$ and $N_f=1$ in the top-right figure,\n$\\tan\\beta=2$ and $N_f=2$ in the bottom-left figure and\n$\\tan\\beta=20$ and $N_f=1$ in the bottom-right figure. The bottom-right figure is evaluated with $v_s > 0$. \nThe others are evaluated with $v_s < 0$.\n}\n\\label{fig:nmssm_ewsbok}\n}\n\n\\FIGURE[htbp]{\n\\epsfig{file=mueff_dy.eps,width=0.45\\hsize}\n\\epsfig{file=mueff_d.eps,width=0.45\\hsize}\n\\caption{\nThe values of $|\\lambda v_s|$ are shown. The calculations are performed with $\\lambda=0.69$, $\\tan\\beta=2$ and $m_0=200 \\ {\\rm GeV}$.\nModerate values of $|\\lambda v_s|$ are obtained.\n}\n\\label{fig:nmssm_mueff}\n}\n \n\\FIGURE[htbp]{\n \\epsfig{file=mhiggs_dy.eps,width=0.45\\hsize}\n \\epsfig{file=mhiggs_d.eps,width=0.45\\hsize}\n\\caption{\nThe dependence of the Higgs mass on the SUSY breaking parameter is shown. In the left figure we set $d=2.5$, and in the right figure we set $D_Y = 6$. \nOther parameters are chosen as $\\lambda=0.69, \\tan\\beta=2.0$ and $m_0=200 {\\rm GeV}$ in both figures.\n}\n\\label{fig:higgsmass}\n}\n\n\\subsection{Dark matter}\nIn this scenario, the lightest neutralino is the LSP in the wide range of parameter space. \nTherefore, the lightest neutralino is a candidate for dark matter. In this \nsubsection, we evaluate the relic density of the lightest neutralino, \nwhich is mainly composed of a singlino. We also calculate the the neutralino-proton \nscattering cross section, and discuss the direct detection of dark matter.\n\nIn the nMSSM, the relic density of the lightest neutralino strongly depends \non its mass\\cite{nmssm1}\\cite{nmssm2}. Although the dominant contribution to the\nannihilation cross section is s-channel $Z$ boson exchange, the coupling \nbetween the $Z$ boson and $\\tilde{N}_1$ is significantly small. This is \nbecause the lightest neutralino, $\\tilde{N}_1$ is mainly composed of the \nfermionic component of the nMSSM gauge singlet, $\\hat{S}$. \nThe resonant effect near the $Z$ pole mass is important for the sufficient\nannihilation of the lightest neutralino.\n\nThe neutralino mass matrix is\n\\begin{eqnarray}\n\\left(\\tilde{B} \\ \\tilde{W} \\ \\tilde{H}_1^0 \\ \\tilde{H}_2^0 \\ \\tilde{S}\\right)\n\\left(\n\\begin{array}{ccccc}\nm_{\\lambda_1} & 0 & -c_\\beta s_w M_Z & s_\\beta s_w M_Z & 0 \\\\\n0 & m_{\\lambda_2} & c_\\beta c_w M_Z & -s_\\beta c_w M_Z & 0 \\\\\n-c_\\beta c_w M_Z & c_\\beta c_w M_Z & 0 & \\mu_{eff} & \\lambda v_2 \\\\\ns_\\beta s_w M_Z & -s_\\beta c_w M_Z & \\mu_{eff} & 0 & \\lambda v_1 \\\\\n0 & 0 & \\lambda v_2 & \\lambda v_1 & 0 \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\tilde{B} \\\\\n\\tilde{W} \\\\\n\\tilde{H}_1^0 \\\\\n\\tilde{H}_2^0 \\\\\n\\tilde{S}\n\\end{array}\n\\right)\n,\n\\end{eqnarray}\nwhere $s_\\beta = \\sin\\beta$, $c_\\beta = \\cos\\beta$ and $s_w = \\sin{\\theta_W}$.\n$\\tilde{B}$, $\\tilde{W}$, $\\tilde{H}_{1,2}^0$ and \n$\\tilde{S}$ denote the bino, wino, higgsino and singlino respectively. \nThe mass of the lightest neutralino, $m_{\\chi_1}$, becomes heavier\nas $|\\mu_{eff}|$ decrease. This is because the mixing of the higgsinos becomes small as one can see from eq. (4.10).\nSince $|\\mu_{eff}|$ is a decreasing function of $D_Y$, a larger $D_Y$ leads to a larger $m_{\\chi_1}$. \nThe dependence of the lightest neutralino mass on $D_Y$ is shown in Fig.\\ref{fig:relic}.\nOn the other hand,\na larger $d$ leads to a smaller $m_{\\chi_1}$.\n\nFigure \\ref{fig:relic} shows $m_{\\chi}$ and the relic density of the neutralino, $\\Omega_{\\chi} h^2$. $m_{\\chi}$ and $\\Omega_{\\chi} h^2$ are calculated \nwith NMSSMTools and micrOMEGAs\\cite{microomegas}\\cite{microomegas2}. When $m_{\\chi}$ is large and close to $m_Z$, \n$\\Omega_{\\chi} h^2$ is small. The observed relic density of dark matter is given by\\cite{wmap1}\\cite{wmap2}\n\\begin{eqnarray}\n0.094 < \\Omega_{CDM} h^2 < 0.136 .\n\\end{eqnarray}\nThis condition is satisfied with $m_{\\chi} \\simeq 35$ GeV. With such light dark matter, there are strong limits for the spin-independent\n WIMP-nucleon scattering cross section from CDMS\\cite{cdms} and XENON10\\cite{xenon}. The strongest limit for\n the cross section is for it to be less than $5\\times 10^{-44} {\\rm cm}^2$ for $m_{\\chi_1} \\simeq 30 \\ {\\rm GeV}$.\n\n\\FIGURE[htbp]{\n \\epsfig{file=relic1.eps,width=0.6\\hsize}\n\\caption{\nThe mass and the relic density of the lightest neutralino are shown as functions of the \nSUSY breaking parameter $D_Y$. The other parameters are chosen as $m_0 = 200$ GeV, $d=2.25, \\lambda=0.69$ and $\\tan\\beta=2.0$\n}\n\\label{fig:relic}\n}\n\nThe spin-independent WIMP-nucleon elastic scattering cross section is \nwritten as\n\\begin{eqnarray}\n\\sigma^{\\rm SI} = \\frac{4 m_{\\chi}^2 m_{nucleus}^2}{\\pi (m_{\\chi} + \n m_{nucleus})^2} \\left[Z f_p +(A-Z)f_n\\right]^2\\ .\n\\end{eqnarray}\n$f_{p,n}$ is the coupling between the WIMP and a nucleon given by \\cite{susy_darkmatter}\n\\begin{eqnarray}\nf_{p,n} = \\sum_{q=u,d,s} f_{T_q}^{(p,n)} a_q \\frac{m_{p,n}}{m_q} + \\frac{2}{27} f_{T_G}^{(p,n)} \\sum_{q=c,b,t} a_q \\frac{m_{p,n}}{m_{q}} .\n\\end{eqnarray}\n$a_q$ are the WIMP-quark couplings. We focus on the dark matter-proton scattering cross section in the following discussion. \nThe parameter $f_{T_q}$ is defined by \n\\begin{eqnarray}\nm_p f_{T_q} \\equiv m_q \\left \\equiv m_q B_q ,\n\\end{eqnarray}\nand $f_{TG}=1-\\sum_{q=u,d,s} f_{T_q}$. $f_{T_q}$ can be written as \n\\cite{direct_dm_update}\n\\begin{eqnarray}\nf_{T_u} &=& \\frac{m_u B_u}{m_p} = \\frac{2\\sigma_{\\pi N}}{m_p \\left(1+ \\frac{m_d}{m_u}\\right)\\left(1+ \\frac{B_d}{B_u}\\right)} \\ ,\\nonumber \\\\\nf_{T_d} &=& \\frac{m_d B_d}{m_p} = \\frac{2\\sigma_{\\pi N}}{m_p \\left(1+ \n \\frac{m_u}{m_d}\\right)\\left(1+ \\frac{B_u}{B_d}\\right)} \\ ,\\nonumber \\\\\nf_{T_s} &=& \\frac{m_s B_s}{m_p} = \\frac{ y \\left(\\frac{m_s}{m_d}\\right) \\sigma_{\\pi N} }{m_p \\left(1+ \\frac{m_u}{m_d}\\right)} , \\label{eq:ftq}\n\\end{eqnarray}\nwhere $\\sigma_{\\pi N}$ is the $\\pi$-nucleon sigma term:\n\\begin{eqnarray}\n\\sigma_{\\pi N} = \\frac{1}{2}\\left(m_u + m_d\\right)\\left(B_u + B_d\\right) .\n\\end{eqnarray}\nThe phenomenological value of $\\sigma_{\\pi N}$ is $64\\pm 8$ MeV\\cite{direct_dm_update}.\n$y$ denotes the ratio of the strange quark component in the nucleon, defined as\n\\begin{eqnarray}\ny = \\frac{2 B_s}{B_u + B_d}\\ .\n\\end{eqnarray}\n$y$ can be determined by the relation,\n\\begin{eqnarray}\n\\sigma_{0} = \\sigma_{\\pi N} \\left(1-y\\right) = \\frac{1}{2}\\left(m_u + m_d \\right)\\left(B_u + B_d -2B_s \\right) .\n\\end{eqnarray}\n$\\sigma_0$ can be evaluated from baryon mass spectra using chiral perturbation theory. From \\cite{sigma_0}, $\\sigma_0 = 36 \\pm 7$ MeV.\nThere is large ambiguity for $y$. When $(\\sigma_{\\pi N}, \\sigma_0) = (64, 36)$ MeV, $y=0.44$. \nOn the other hand, according to a recent lattice calculation \\cite{ohki},\n $y$ has a small value such as $0.03$.\n\nThe ratios of the quark mass are taken from \\cite{quark_mass_ratio}.\n\\begin{eqnarray}\n\\frac{m_u}{m_d} = 0.553 \\pm 0.043, \\ \\frac{m_d}{m_s} = 18.9 \\pm 0.8 . \\label{eq:mumd}\n\\end{eqnarray}\nThe ratios of the form factors are written as\n\\begin{eqnarray}\n\\frac{B_d}{B_u} = \\frac{2 + (z-1)y}{2z-(z-1)y} , \\label{bdbu}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nz = \\frac{B_u-B_s}{B_d-B_s} .\n\\end{eqnarray}\n$z$ can be calculated from the baryon mass, and its value is \n1.49\\cite{dm_zparameter}. We can now determine $f_{T_q}$ from eqs. \n(\\ref{eq:ftq}), (\\ref{eq:mumd}) and (\\ref{bdbu}). \nWhen $y=0.44$,\n\\begin{eqnarray}\nf_{T_u} \\approx 0.027, \\ \\ f_{T_d} \\approx 0.039, \\ \\ f_{T_s} \\approx 0.365, \\ \\ f_{TG} \\approx 0.569,\n\\end{eqnarray}\nand when $y=0.03$,\n\\begin{eqnarray}\nf_{T_u} \\approx 0.029, \\ \\ f_{T_d} \\approx 0.036, \\ \\ f_{T_s} \\approx 0.025, \\ \\ f_{TG} \\approx 0.91.\n\\end{eqnarray}\nIn these two cases, $f_{T_s}$ and $f_{TG}$ are very different. This \naffects the spin-independent cross section of the WIMP-nucleon scattering significantly.\n\nWIMP-quark couplings, $a_q$, consist of two parts. One part arises from squark \ns-channel exchange and the other arises from the t-channel exchange of \nthe neutral Higgs. \nThe couplings from squark exchange are given by \\cite{aq_squark}\n\\begin{eqnarray}\na_{q_i}^{\\tilde{q}} = -\\frac{1}{2(\\tilde{m}_{1i}^2-m_{\\chi}^2)} {\\rm Re} \\left[X_i Y_i^* \\right] \n- \\frac{1}{2(\\tilde{m}_{2i}^2-m_{\\chi}^2)} {\\rm Re} \\left[W_i \n\t\t\t\t\t\t V_i^*\\right] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_i &=& \\eta_{11}^* \\frac{g m_{q_i} N_{1,5-i}^*}{2M_w B_i} - \\eta_{12}^* \n e_i g' N_{11}^* \\ ,\\nonumber \\\\\nY_i &=& \\eta_{11}^* \\left(\\frac{y_i}{2} g' N_{11} + g T_{3i} N_{12}\\right) + \\eta_{12}^* \\frac{g m_{q_i} N_{1,5-i}}{2M_w B_i} \\ ,\\nonumber \\\\\nW_i &=& \\eta_{21}^* \\frac{g m_{q_i} N_{1,5-i}^*}{2M_w B_i} - \\eta_{22}^* e_i g' N_{11}^* \\ ,\\nonumber \\\\\nY_i &=& \\eta_{21}^* \\left(\\frac{y_i}{2} g' N_{11} + g T_{3i} N_{12}\\right) + \\eta_{22}^* \\frac{g m_{q_i} N_{1,5-i}}{2M_w B_i},\n\\end{eqnarray}\nand $i=1$ for an up-type quark and $i=2$ for a down-type quark. $\\tilde{m}_{1i}$ and \n$\\tilde{m}_{2i}$ denote a light squark mass and a heavy squark mass respectively. \n$\\eta$ denotes a squark\nmixing such that\n\\begin{eqnarray}\n\\tilde{q}_l = \\eta_{l1} \\tilde{q}_L + \\eta_{l2} \\tilde{q}_R .\n\\end{eqnarray}\n$y_i$, $T_{3i}$ and $e_i$ denote the hypercharge, isospin and electric \ncharge of the quarks respectively. $B_1 = \\sin\\beta$ and $B_2 = \\cos\\beta$.\n\nThe couplings from neutral Higgs exchange in the nMSSM are given by\\cite{nmssm_aq}\n\\begin{eqnarray}\na_{q_i}^h = \\sum_{a=1}^3 \\frac{1}{m_{h_a^0}^2} {C_Y}_a^i {\\rm Re} \n [C_{H}^a] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{C_Y}_a^i &=& -\\frac{g m_{q_i}}{4 M_w B_i} S_{a,3-i} \\ ,\\nonumber \\\\\n{C_{H}^a} &=& \\left(-g N_{12}^* + g' N_{11}^* \\right)\\left(S_{a1}N_{13}^* - S_{a2}N_{14}^* \\right) \\nonumber \\\\\n&& - \\sqrt{2} \\lambda \\left[S_{a3}N_{13}^* N_{14}^* + N_{15}^* \n\t\t \\left(S_{a2}N_{13}^* + S_{a1}N_{14}^* \n\t\t \\right)\\right] \\ .\\label{eq:a_from_higgs} \n\\end{eqnarray}\n$S_{ij}$ denotes Higgs mixing. One can write the mass eigenstate of the Higgs as\n\\begin{eqnarray}\nh_a^0 = S_{a1} h_d^0 + S_{a2} h_u^0 + S_{a3} h_s .\n\\end{eqnarray}\nWhen $\\lambda=0$, eq. (\\ref{eq:a_from_higgs}) agrees with the couplings \nin the MSSM given in \\cite{aq_squark}.\n\nFigure \\ref{fig:si_cross} shows the spin-independent cross section as a function of $d$ and $D_Y$. When $y=0.44$, $\\sigma^{SI}$ is already excluded\nby the current experiments. On the other hand, when $y=0.03$, \n$\\sigma^{SI}$ is smaller than the upper limit from XENON10 in many regions of the parameter space.\nIn this case, $\\sigma^{SI}$ is large enough to be detected or be \nexcluded by the next-generation experiments.\n\n\\FIGURE[htbp]{\n \\epsfig{file=sigma_d.eps,width=0.49\\hsize}\n \\epsfig{file=sigma_dy.eps,width=0.49\\hsize}\n\\caption{\nThe spin-independent cross sections, $\\sigma^{SI}$ are shown. $\\sigma^{SI}$ are calculated with $m_0 = 200$ GeV, $\\lambda=0.69$ and $\\tan\\beta=2.0$.\nThe upper three lines are calculated with $y=0.44$. $y$ is evaluated with chiral perturbation.\nThe lower three lines are calculated with $y=0.03$, which is the result from a recent lattice calculation.\n}\n\\label{fig:si_cross}\n}\n\n\\subsection{Mass spectrum}\nHere we present the sparticle mass spectrum, the relic density of the \nneutralino and the spin-independent cross section of dark matter-proton scattering.\nThe mass spectra are calculated using NMSSMTools, and the relic \ndensities are calculated using micrOMEGAs.\n\nIn this scenario, the gluino is light in a wide range of the parameter space. This is because the contributions from gauge mediation and anomaly mediation\n cancel. From eq. (\\ref{eq:formula}), the gluino mass at the messenger scale \n is written as\n\\begin{eqnarray}\nm_{\\tilde{g}} = -g_3^2 \\left(3-d\\right)m_0 \\label{eq:mass_gl} \\ ,\n\\end{eqnarray}\nwhere $m_0 = \\displaystyle F_\\phi\/(16\\pi^2)$. Particularly in the region where $d=3$, the gluino mass $m_{\\tilde{g}}$ vanishes.\n\nThe results of the numerical calculation are presented in Table \\ref{tab:masses}. When the deflection parameter $d$ changes, \nthe overall scale of the soft breaking terms changes. However, once we impose the condition of the observed relic density, \n$\\mu_{eff}$ is almost determined by $m_{\\chi}$. Therefore the mass \nspectrum of the Higgs does not change significantly. \nTwo samples in Table \\ref{tab:masses} satisfy the current experimental limits from LEPII and XENON10 and also explain the observed\nrelic abundance of dark matter. The mass of the gluino is $m_{\\tilde{g}} \\sim 200\\ {\\rm GeV}$. \nThe lightness of the gluino is the characteristic feature of this scenario.\n\n\\TABLE[htbp]{\n\\caption{Mass spectra}\n\\begin{tabular}{|c|c c c c c c|}\n\\hline\n\\multicolumn{1}{|l|}{} & $m_0$ & $N_f$ & $d$ & $D_Y$ & $\\lambda$ & $\\tan\\beta$ \\\\ \\hline\ninput p1 & 200 & 1 & 2.2 & 4.84 & 0.69 & 2 \\\\ \\hline\np2 & 200 & 1 & 3.5 & 8.74 & 0.69 & 2 \\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|cccccccc|}\n\\hline\n & $|\\mu_{eff}|$ & $m_{H_1^0}$ & $m_{H_2^0}$ & $m_{H_3^0}$ & $m_{A_1^0}$ & $m_{A_2^0}$ & $m_{H^{\\pm}}$ & $m_{\\chi_1^0}$ \\\\ \\hline\noutput p1 & 264.2 & 127.2 & 328.8 & 460.1 & 285.5 & 487.8 & 433.8 & 34.2 \\\\ \\hline\np2 & 270.6 & 127.5 & 313.5 & 590.0 & 286.6 & 601.6 & 582.7 & 34.3 \\\\ \\hline \\hline\n & $m_{\\chi_2^0}$ & $m_{\\chi_3^0}$ & $m_{\\chi_4^0}$ & $m_{\\chi_5^0}$ & $m_{\\tilde{g}}$ & $m_{\\chi_1^{\\pm}}$ & $m_{\\chi_2^{\\pm}}$ & $m_{\\tilde{\\nu}_L}$ \\\\ \\hline\np1 & 198.3 & 310.5 & 336.4 & 403.8 & 262.1 & 197.2 & 350.2 & 174.5 \\\\ \\hline\np2 & 237.1 & 317.0 & 417.9 & 459.2 & 185.1 & 237.7 & 428.9 & 408.9 \\\\ \\hline \\hline\n & $m_{\\tilde{\\nu}_\\tau}$ & $m_{\\tilde{e}_L}$ & $m_{\\tilde{e}_R}$ & $m_{\\tilde{\\tau}_1}$ & $m_{\\tilde{\\tau}_2}$ & $m_{\\tilde{u}_L}$ & $m_{\\tilde{u}_R}$ & $m_{\\tilde{t}_1}$ \\\\ \\hline\np1 & 174.4 & 184.5 & 651.7 & 184.3 & 651.6 & 1231.7 & 991.9 & 829.5 \\\\ \\hline\np2 & 408.8 & 413.1 & 925.1 & 413.1 & 925.1 & 1682.2 & 1347.1 & 1126.7 \\\\ \\hline \\hline\n & $m_{\\tilde{t}_2}$ & $m_{\\tilde{d}_L}$ & $m_{\\tilde{d}_R}$ & $m_{\\tilde{b}_1}$ & $m_{\\tilde{b}_2}$ & $\\Omega h^2$ & $\\sigma_p^{SI} ({\\rm cm}^2)$ & \\\\ \\hline\np1 & 1177.4 & 1233.2 & 1169.1 & 1166.9 & 1170.1 & 0.111 & $3.3\\times 10^{-44}$ & \\multicolumn{1}{l|}{} \\\\ \\hline\np2 & 1605.4 & 1683.3 & 1573.3 & 1573.1 & 1598.9 & 0.131 & $3.1\\times 10^{-44}$ & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\end{tabular}\n\\label{tab:masses}\n}\n\n\\section{Conclusions}\nWe investigated the phenomenology of the nMSSM with a Fayet-Iliopoulos D-term in the positively deflected anomaly mediation scenario.\n\nIn the deflected anomaly mediation scenario, the messenger sector is introduced.\nWe showed that the couplings between the nMSSM fields and the messenger sector fields are forbidden by the discrete symmetry,\nand therefore the phenomenology at the weak scale is not affected by the detail of the\nmessenger sector.\nWe evaluated the soft breaking terms at the messenger scale without assuming small Yukawa couplings,\nand showed that the contributions from Yukawa couplings are the same as those of anomaly mediation. \nThe soft breaking parameters are determined by the deflection parameter $d$, \nthe messenger scale and contributions from the Fayet-Iliopoulos D-term.\n\nWe also discussed the phenomenology of the nMSSM at the weak scale.\nWe found that electroweak symmetry breaking is successful, and moderate values of $\\mu_{eff}$ are obtained.\nThe mass of the lightest Higgs is heavier than the LEP bound.\nWe also obtained sparticle mass spectra, and interestingly, the gluino is light.\n\nWe showed that the lightest neutralino is a good candidate for dark matter. \nThe relic density explains the observed abundance of dark matter.\nThe spin-independent dark matter-proton scattering cross section \nsatisfies the upper limit from XENON10 when we consider a small value of the strange quark content of the nucleon as \nindicated by a recent lattice calculation.\nThe cross section is large enough to be detected or excluded by next-generation experiments of direct detection. \n\nWe consider this scenario phenomenologically viable. If the light gluino is discovered, it may imply\nthat SUSY breaking is mediated by supergravity and messengers,\nand these two effects are comparable.\n\n\\section*{Acknowledgements}\nWe thank H. Ohki for useful discussion on the direct detection of dark matter and informing us of the nucleon sigma term.\nWe also thank M. Ibe for discussion on SUSY breaking effects of an intermediate threshold and DM-nucleon scattering cross section. \nWe would like to thank K. Akina and T. Morozumi for careful reading of the manuscript.\nWe thank A. Masiero for discussion on symmetry breaking terms. \nWe acknowledge M. Okawa, K. Ishikawa and T. Inagaki for support and encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nWeak scale supersymmetry (SUSY) provides an elegant solution to the \nnaturalness problem of the standard model, by invoking a cancellation \nbetween the standard model and its superpartner contributions to the \nHiggs potential. An interesting consequence of this framework is that \nthe three gauge couplings unify at an extremely high energy of order \n$M_U \\approx 10^{16}~{\\rm GeV}$, if a normalization of the $U(1)_Y$ \ngauge coupling is adopted that allows the embedding of the standard \nmodel gauge group into a larger simple symmetry group: $SU(5) \\supset \nSU(3)_C \\times SU(2)_L \\times U(1)_Y$. This suggests the existence \nof some unified physics above this energy scale, which in some form \nutilizes $SU(5)$ or a larger group containing it.\n\nThe simplest possibilities for physics above $M_U$ are four dimensional \n(4D) supersymmetric grand unified theories (GUTs)~\\cite{Dimopoulos:1981zb}. \nIn these theories, physics above $M_U$ is described by 4D supersymmetric \ngauge theories in which the standard model gauge group is embedded \ninto a larger (simple) gauge group. This, however, leads to the \nproblem of doublet-triplet splitting in the Higgs sector, and often \nleads to too rapid proton decay caused by the exchange of colored \ntriplet Higgsinos~\\cite{Murayama:2001ur}. While several solutions \nto these problems have been proposed within conventional 4D SUSY \nGUTs~[\\ref{Witten:1981kv:X}~--~\\ref{Yanagida:1994vq:X}], their \nexplicit implementations often require the introduction of a larger \nmultiplet(s) and\/or specifically chosen superpotential interactions, \nespecially when one tries to make the models fully realistic. This \nloses a certain beauty the simplest theory had, especially if one \nadopts the viewpoint that these theories are ``fundamental,'' arising \ndirectly from physics at the gravitational scale, e.g. string theory.\n\nAn alternative possibility for physics above $M_U$ is that the unified \ngauge symmetry is realized in higher dimensional (semi-)classical \nspacetime~[\\ref{Kawamura:2000ev:X}~--~\\ref{Hall:2002ea:X}]. In this \ncase there is no 4D unified gauge symmetry containing the standard \nmodel gauge group as a subgroup --- the unified symmetry in higher \ndimensions is broken locally and explicitly by a symmetry breaking \ndefect. This structure allows a natural splitting between the doublet \nand triplet components for the Higgs fields, while the successful \nprediction for gauge coupling unification is recovered by diluting \nthe effects from the defect due to a moderately large extra dimension(s). \nDangerous proton decay is suppressed by an $R$ symmetry, arising \nnaturally from the higher dimensional structure of the triplet Higgsino \nmass matrix. The framework also allows for a simple understanding \nof the observed structure of fermion masses and mixings, in terms \nof wavefunction suppressions of the Yukawa couplings arising for \nbulk quarks and leptons~\\cite{Hall:2001zb,Hebecker:2002re}.\n\nIn this paper we study a framework for physics above $M_U$ in \nwhich the standard model gauge group is unified into a simple gauge \ngroup in precisely the same sense as in conventional 4D SUSY GUTs, \nand yet mechanisms and intuitions developed in higher dimensions \ncan be used to address the various issues of unified theories. \nLet us consider that the standard model gauge group is embedded into \na simple unified gauge group, e.g. $SU(5)$, at energies above $M_U$. \nWe assume that the unified gauge symmetry is broken by strong gauge \ndynamics associated with another gauge group $G$, and that this gauge \ngroup has a large 't~Hooft coupling $\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \n\\gg 1$, where $\\tilde{g}$ and $\\tilde{N}$ are the gauge coupling \nand the number of ``colors'' for the gauge group $G$. With these \nvalues of the 't~Hooft coupling for $G$, an appropriate (weakly \ncoupled) description of physics is given in higher dimensional \nwarped spacetime (for $\\tilde{N} \\gg 1$), due to the gauge\/gravity \ncorrespondence~\\cite{Maldacena:1997re}. In the simplest setup where \n$\\tilde{g}$ evolves slowly above the dynamical scale, our theories are \nformulated in 5D anti-de~Sitter (AdS) spacetime truncated by two branes, \nwhere the curvature scales on the ultraviolet (UV) and infrared (IR) \nbranes are chosen to be $k \\approx (10^{17}-10^{18})~{\\rm GeV}$ and \n$k' \\approx (10^{16}-10^{17})~{\\rm GeV}$, respectively. This allows \nus to construct simple ``calculable'' unified theories in which the \nunified gauge symmetry is broken dynamically --- physics above $M_U$ \nis determined simply by specifying parameters in higher dimensional \neffective field theory.\n\nIn this paper we construct realistic unified theories in the \nframework described above. In general, there are many ways to address \nthe issues of unified theories in our framework. In one example, \nwhich we discuss in detail, we use the idea that the Higgs doublets \nof the minimal supersymmetric standard model (MSSM) are pseudo-Goldstone \nbosons of a broken global symmetry~\\cite{Inoue:1985cw,Barbieri:1992yy}. \nSpecifically, we assume that the $G$ sector possesses a global $SU(6)$ \nsymmetry, of which an $SU(5)$ ($\\times U(1)$) subgroup is gauged. \nThe gauged $SU(5)$ group contains the standard model gauge group as \na subgroup. We assume that the dynamics of $G$ breaks the global \n$SU(6)$ symmetry down to $SU(4) \\times SU(2) \\times U(1)$ at the scale \n$M_U$, which leads to the correct gauge symmetry breaking, $SU(5) \n\\rightarrow SU(3)_C \\times SU(2)_L \\times U(1)_Y$, and ensures that \nthe Higgs doublets remain massless after the symmetry breaking, without \nbeing accompanied by their colored triplet partners. The simplest \nrealization of our theory, corresponding to this symmetry structure, \nis then obtained in 5D truncated warped space in which the bulk $SU(6)$ \ngauge symmetry is broken to $SU(5) \\times U(1)$ and $SU(4) \\times SU(2) \n\\times U(1)$ on the UV and IR branes, respectively. Realistic unified \ntheories having this symmetry structure were constructed previously \nin flat space in Ref.~\\cite{Burdman:2002se}, where the symmetry \nbreakings on the two branes are both caused by boundary conditions, \nand in Ref.~\\cite{Cheng:1999fw}, where the breakings are by the \nHiggs mechanism. In our context, we find that the simplest theory \nis obtained if the breakings on the UV and IR branes are caused by \nboundary conditions and the Higgs mechanism, respectively. Note that, \nin the ``4D description'' of the theory, the Higgs breaking on the IR \nbrane corresponds to dynamical GUT breaking, and the low-energy Higgs \ndoublets are interpreted as composite particles of the dynamical \nGUT-breaking sector. This theory thus provides a simple explicit \nrealization of the composite pseudo-Goldstone Higgs doublets, in which \nthe origin of the global $SU(6)$ symmetry can be understood as the \n``flavor'' symmetry of the dynamical GUT-breaking sector. \n\nBelow the GUT-breaking scale $M_U$, our theory is reduced to the \nMSSM (supplemented by small seesaw neutrino masses). The successful \nunification prediction for the low-energy gauge couplings is \npreserved as long as the threshold corrections from the dynamical \nGUT-breaking sector are sufficiently small. Our higher dimensional \ndescription of the theory allows us to estimate the size of these \ncorrections, and we find that this can be the case. Dimension five \nproton decay does not exclude the theory, because of the existence \nof these threshold corrections. Realistic quark and lepton mass \nmatrices can also be reproduced, where the observed hierarchies \nin masses and mixings are understood in terms of the wavefunction \nprofiles of the quark and lepton fields. In the 4D description of \nthe theory, these hierarchies arise through mixings between elementary \nstates and composite states of $G$, which are given by powers of \n$M_U\/M_*$, where $M_*$ is the fundamental scale of the theory, close \nto the 4D Planck scale. Unwanted unified mass relations for the \nfirst two generation fermions do not arise, because of GUT breaking \neffects in the $G$ sector.\n\nWe also discuss other possible theories in our framework. We show \nthat it allows for the construction of large classes of models, \nincluding missing partner type and product group type models. \nIn most of them, the Higgs doublets arise as states localized to \nthe IR brane, corresponding to composite states of the strong $G$ \ndynamics. A 4D scenario related to these theories was discussed \npreviously in Ref.~\\cite{Kitano:2005ez}, based on a supersymmetric \nconformal field theory (CFT), where a possible AdS interpretation \nwas also noted. In all of these theories, our higher dimensional \nframework allows a straightforward implementation of the mechanism \ngenerating the hierarchical fermion masses and mixings, in terms of \nthe wavefunction profiles of matter fields in the extra dimension.\n\nThe organization of the paper is as follows. In the next section we \ndescribe the basic structure of our theory using the 4D description. \nWe describe how the MSSM arises naturally at low energies in this \ntheory. In section~\\ref{sec:model} we construct an explicit model \nin truncated 5D warped space. We show that the model does not suffer \nfrom problems of conventional 4D SUSY GUTs, e.g. the doublet-triplet \nsplitting and dimension five proton decay problems, and also that \nthe observed hierarchies in the quark and lepton mass matrices can \nbe understood in terms of the wavefunction profiles of these fields \nin the extra dimension. In section~\\ref{sec:other}, we discuss \nother possible theories in our framework, including missing partner \ntype and product group type models. Discussion and conclusions \nare given in section~\\ref{sec:concl}, which include a comment \non the possibility of having a theory with $\\tilde{g}^2 \n\\tilde{N}\/16\\pi^2 \\simlt 1$. \n\n\n\\section{Basic Picture}\n\\label{sec:picture}\n\nIn this section we describe our theory using the 4D description. \nHere we focus on the case where the light Higgs doublets of the MSSM \narise as pseudo-Goldstone supermultiplets of the GUT scale dynamics. \nThis has the virtue that the success of the theory is essentially \nguaranteed by its symmetry structure, without relying on specifically \nchosen matter content or interactions. Other possibilities will be \ndiscussed in section~\\ref{sec:other}.\n\nWe consider that the standard model gauge group is embedded into \na simple gauge group $SU(5)$, which is spontaneously broken at \nthe scale $M_U \\approx 10^{16}~{\\rm GeV}$. What is the underlying \ndynamics of this symmetry breaking? A hint will come from considering \nhow the MSSM arises below the symmetry breaking scale $M_U$. In \nparticular, considering how the MSSM matter content naturally appears \nat energies below $M_U$ and why interactions among these particles \n-- the gauge and Yukawa interactions -- take the observed form and \nvalues will provide a guide to the physics of this symmetry breaking. \nThe suppression of certain operators allowed by standard model gauge \ninvariance, e.g. the ones leading to dangerous dimension five proton \ndecay, may also give hints regarding the structure of this physics.\n\nWe focus on the possibility that the unified gauge group, $SU(5)$, is \nspontaneously broken by dynamics associated with another gauge group \n$G$. In this setup, the $G$ sector is charged under $SU(5)$, as it \nbreaks $SU(5)$ dynamically. The setup also allows the existence of other \nfields -- elementary fields -- that are singlet under $G$ and charged \nunder $SU(5)$. Suppose now that the theory has a matter content that \nsatisfies $n_{{\\bf 5}^*} - n_{\\bf 5} = n_{\\bf 10} - n_{{\\bf 10}^*} = 3$ \nand $n_{\\bf r} - n_{{\\bf r}^*} = 0$ (${\\bf r} \\neq {\\bf 5}, {\\bf 10}$), \nwhere $n_{\\bf r}$ represents the number of $SU(5)$ multiplets in \na complex representation ${\\bf r}$. The matter content is arbitrary \notherwise. (Note that this is not a very strong requirement on the \nspectrum --- with $n_{\\bf r} - n_{{\\bf r}^*} = 0$ for ${\\bf r} \\neq \n{\\bf 5}, {\\bf 10}$, the condition $n_{{\\bf 5}^*} - n_{\\bf 5} = n_{\\bf 10} \n- n_{{\\bf 10}^*}$ arises automatically as a consequence of anomaly \ncancellation.) With this assumption, the low energy matter content \nis expected to be just the three generations of quarks and leptons, \nno matter what happens associated with the dynamics of the GUT-breaking \nsector $G$. In general, the gauge dynamics of $G$ will produce an \narbitrary number of split GUT multiplets as composite states, by picking \nup the effect of GUT breaking. These states can then mix with the \nelementary states, so that the low energy states are in general mixtures \nof elementary and composite states and thus a collection of various \nincomplete $SU(5)$ multiplets. Nevertheless, conservation of chirality \nguarantees that we always have three generations of quarks and leptons \nat low energies, although they may not arise simply from three copies \nof $({\\bf 5}^* + {\\bf 10})$. Assuming that all the fields vector-like \nunder the standard model gauge group obtain masses of order $M_U$ \nthrough nonperturbative effects of $G$, the matter content below \n$M_U$ is exactly the three generations of quarks and leptons.\n\nThe above argument shows that we can naturally obtain a low-energy \nchiral matter content that fills complete $SU(5)$ multiplets for chirality \nreasons (although each component in a multiplet may come from several \ndifferent $SU(5)$ multiplets at high energies). It also implies that \nany multiplets that do not fill out a complete $SU(5)$ multiplet must \nbe vector-like. It is interesting that the MSSM has exactly this \nstructure. Unless there is some special reason, however, the vector-like \nstates are all expected to have masses of order $M_U$ from nonperturbative \neffects of $G$. What could the special reason be for the Higgs doublets? \n\nThe lightness of the Higgs doublets can be understood group theoretically \nif we identify these states as pseudo-Goldstone bosons of a broken global \nsymmetry~\\cite{Inoue:1985cw}. Suppose that the $G$ sector possesses \na global $SU(6)$ symmetry, of which an $SU(5)$ ($\\times U(1)$) subgroup \nis gauged and identified as the unified gauge symmetry. We assume that \nthe dynamics of $G$ breaks the global $SU(6)$ symmetry down to $SU(4) \n\\times SU(2) \\times U(1)$ at the dynamical scale $\\approx M_U$ in such \na way that the gauged $SU(5)$ subgroup is broken to the standard model \ngauge group $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ (321). This leads \nto Goldstone chiral supermultiplets, whose quantum numbers under 321 \nare given by $({\\bf 3}, {\\bf 2})_{-5\/6} + ({\\bf 3}^*, {\\bf 2})_{5\/6} \n+ ({\\bf 1}, {\\bf 2})_{1\/2} + ({\\bf 1}, {\\bf 2})_{-1\/2}$. While the \nfirst two of these are absorbed by the broken $SU(5)$ gauge multiplets \n(the massive XY gauge supermultiplets), the last two are left in the \nlow energy spectrum. Although the global $SU(6)$ symmetry of the $G$ \nsector is explicitly broken by the gauging of the $SU(5)$ ($\\times U(1)$) \nsubgroup, the supersymmetric nonrenormalization theorem guarantees that \nthe mass term for $({\\bf 1}, {\\bf 2})_{1\/2} + ({\\bf 1}, {\\bf 2})_{-1\/2}$ \nis not generated without picking up the effect of supersymmetry breaking, \nallowing us to identify these states as the two Higgs doublets of the \nMSSM: $H_u({\\bf 1}, {\\bf 2})_{1\/2}$ and $H_d({\\bf 1}, {\\bf 2})_{-1\/2}$. \nThis provides a complete understanding of the MSSM field content in \nour framework. The MSSM states -- the gauge, matter and Higgs fields \n-- are the only states that could not get a mass of order $M_U$ from \n$G$, because they are protected by gauge invariance, chirality, and \nthe (pseudo-)Goldstone mechanism.\n\nSince the two Higgs doublets arise from the dynamical breaking of \n$SU(6)$, they are composite states of $G$. Suppose now that the \ndynamics of $G$ also produces composite states that have the same 321 \nquantum numbers as the MSSM quarks and leptons, ${\\cal Q}, {\\cal U}, \n{\\cal D}, {\\cal L}$ and ${\\cal E}$. These composite states will then \nhave ``Yukawa couplings'' with the Higgs fields at $M_U$, $W \\approx \n{\\cal Q} {\\cal U} H_u + {\\cal Q} {\\cal D} H_d + {\\cal L} {\\cal E} H_d$, \nwhere the sizes of the couplings are naturally of order $4\\pi$. These \ncouplings, however, disappear at low energies after integrating out all \nthe heavy modes, because the strong $G$ dynamics respects $SU(6)$ and \nthe Higgs doublets are the Goldstone bosons associated with the dynamical \nbreaking of $SU(6)$. Now, suppose that the theory also has several \nelementary fields that transform as ${\\bf 5}^*$ and ${\\bf 10}$ under \n$SU(5)$. In this case the low-energy quarks and leptons, \n$Q, U, D, L$ and $E$, are in general linear combinations of the \nelementary and composite states. The Yukawa couplings for these \nlow-energy fields, $W \\approx Q U H_u + Q D H_d + L E H_d$, can then \nbe nonzero because the elementary states do not respect the full \n$SU(6)$ symmetry. The sizes of the Yukawa couplings are determined by \nthe strengths of the mixings between the elementary and composite states, \nwhich are in turn determined by the dimensions of the $G$-invariant \noperators that interpolate the composite states. This situation is \nanalogous to the case where the standard model Higgs boson is identified \nas a pseudo-Goldstone boson of strong gauge dynamics at the TeV \nscale~\\cite{Contino:2003ve}. By choosing operator dimensions to be \nlarger for lighter generations, we can naturally understand the origin \nof the hierarchical structure for the quark and lepton masses and \nmixings. The unwanted mass relations for the quarks and leptons \ncan be avoided because low-energy quarks and leptons feel the \nGUT-breaking effects in the $G$ sector.\n\nDangerous dimension four and five proton decay can be suppressed if \nthe theory possesses a continuous or discrete $R$ symmetry, under which \nthe low-energy MSSM fields carry the charges $Q(1)$, $U(1)$, $D(1)$, \n$L(1)$, $E(1)$, $H_u(0)$ and $H_d(0)$ (and $N(1)$ if we introduce \nright-handed neutrino superfields $N$). This $R$ symmetry is most \nlikely spontaneously broken by the dynamics of the $G$ sector (unless \nthere is a low-energy singlet field that transforms nonlinearly \nunder this symmetry; see discussion in section~\\ref{sec:other}). \nThe $R$ symmetry should also be broken to the $Z_2$ subgroup, \nthe $R$ parity of the MSSM, in order to give weak scale masses to \nthe gauginos. Supersymmetry breaking produces supersymmetric and \nsupersymmetry-breaking masses for the Higgs doublets, as well as \nmasses for the gauginos, squarks and sleptons, ensuring the stability \nof the desired vacuum. Successful supersymmetric gauge coupling \nunification is preserved if the threshold corrections associated \nwith the $G$ sector are sufficiently small. \n\nWe have depicted the basic picture of the theory in Fig.~\\ref{fig:basic}. \n\\begin{figure}[t]\n\\begin{center} \n\\begin{picture}(310,170)(0,0)\n \\CArc(10,50)(10,180,270) \\CArc(240,50)(10,270,360)\n \\CArc(10,160)(10,90,180) \\CArc(240,160)(10,0,90)\n \\Line(10,40)(240,40) \\Line(10,170)(240,170)\n \\Line(0,50)(0,160) \\Line(250,50)(250,160)\n \\Text(15,152)[l]{gauged $SU(5)$} \\Text(125,152)[]{$\\subset$}\n \\Text(32,130)[l]{${\\bf 5}^*_1, {\\bf 5}^*_2, \\cdots$}\n \\Text(32,114)[l]{${\\bf 10}_1, {\\bf 10}_2, \\cdots$}\n \\Text(32,98)[l]{$\\cdots \\cdots$}\n \\CArc(192,95)(43,0,360) \\Text(228,152)[r]{global $SU(6)$}\n \\Text(192,109)[]{\\Large $G$}\n \\Text(156,89)[l]{\\tiny $SU(6) \\rightarrow$}\n \\Text(160,82)[l]{\\tiny $SU(4) \\!\\times\\! SU(2) \\!\\times\\! U(1)$}\n \\Text(264,161)[l]{$n_{{\\bf 5}^*} - n_{\\bf 5} = 3$}\n \\Text(264,146)[l]{$n_{{\\bf 10}} - n_{{\\bf 10}^*} = 3$}\n \\Text(264,131)[l]{$n_{{\\bf r}} - n_{{\\bf r}^*} = 0$}\n \\Text(285,117)[l]{\\small $({\\bf r} \\neq {\\bf 5}, {\\bf 10})$}\n \\Line(22,140)(22,25) \\Line(22,25)(19,31) \\Line(22,25)(25,31)\n \\Text(22,20)[t]{$V_{SU(3)_C}, V_{SU(2)_L}, V_{U(1)_Y}$}\n \\Line(140,50)(172,72) \\Line(140,50)(76,94)\n \\Line(140,50)(140,25) \\Line(140,25)(137,31) \\Line(140,25)(143,31)\n \\Text(140,20)[t]{$3 \\times (Q,U,D,L,E)$}\n \\Line(220,72)(220,25) \\Line(220,25)(217,31) \\Line(220,25)(223,31)\n \\Text(220,20)[t]{$H_u, H_d$}\n \\Text(253,15)[l]{$\\cdots \\cdots$}\n \\Text(294,16)[l]{MSSM}\n\\end{picture}\n\\caption{The basic picture of the theory in the 4D description.}\n\\label{fig:basic}\n\\end{center}\n\\end{figure}\nHow can we realize this picture in explicit models? It is not so \nstraightforward to construct such models in the conventional 4D framework. \nIn particular, it is not easy to find explicit gauge group and matter \ncontent for the $G$ sector having all the features described above. (The \ndifficulty increases if some of the relevant composite states are excited \nstates of the $G$ sector. We then cannot use beautiful exact results for \n${\\cal N} = 1$ supersymmetric gauge theories~\\cite{Intriligator:1995au}, \nwhich are applicable to lowest-lying modes.) In our framework, however, \nthis problem is in some sense ``bypassed.'' Suppose that the $G$ sector \npossesses a large 't~Hooft coupling, $\\tilde{g}^2 \\tilde{N} \/16\\pi^2 \n\\gg 1$. In this case, the theory is so strongly coupled that the gauge \ntheory description in terms of ``gluons'' and ``quarks'' does not make \nmuch sense. Instead, in this parameter region, the theory is better \nspecified by composite ``hadron'' states, which have a tower structure. \nFor $\\tilde{N} \\gg 1$, these ``hadronic'' tower states are weakly \ncoupled~\\cite{'tHooft:1973jz}, and under certain circumstances \nthey can be identified as the Kaluza-Klein (KK) states of a weakly \ncoupled higher dimensional theory. In particular, if the $G$ \nsector is quasi-conformal ($\\tilde{g}$ evolves very slowly) \nabove its dynamical scale, the corresponding higher dimensional \ntheory is formulated in warped AdS spacetime truncated by \nbranes~\\cite{Maldacena:1997re,Arkani-Hamed:2000ds}. In the next section \nwe construct an explicit unified model in truncated 5D warped spacetime, \nwhich has all the features described in this section. In practice, \nonce we have a theory in higher dimensions, we can forget about the \n``original'' 4D picture for most purposes --- our higher dimensional \ntheory is an effective field theory with which we can consistently \ncalculate various physical quantities. The theory does not require \nany more information than the gauge group, matter content, boundary \nconditions, and values of various parameters, to describe physics \nat energies below the cutoff scale $M_*$ ($\\gg M_U$). \n\n\n\\section{Model}\n\\label{sec:model}\n\n\\subsection{Basic symmetry structure}\n\\label{subsec:symm}\n\nFollowing the general picture presented in the previous section, we \nconsider 5D warped spacetime truncated by two branes: the UV and IR \nbranes. The spacetime metric is given by\n\\begin{equation}\n ds^2 = e^{-2ky} \\eta_{\\mu\\nu} dx^\\mu dx^\\nu + dy^2,\n\\label{eq:metric}\n\\end{equation}\nwhere $y$ is the coordinate for the extra dimension and $k$ denotes the \ninverse curvature radius of the warped AdS spacetime. The two branes \nare located at $y=0$ (the UV brane) and $y=\\pi R$ (the IR brane). This \nis the spacetime considered in Ref.~\\cite{Randall:1999ee}, in which the \nAdS warp factor is used to generate the large hierarchy between the weak \nand the Planck scales by choosing the scales on the UV and IR branes \nto be the Planck and TeV scales, respectively ($kR \\sim 10$). Here \nwe choose instead the UV-brane and IR-brane scales to be $k \\approx \n(10^{17}-10^{18})~{\\rm GeV}$ and $k' \\equiv k\\, e^{-\\pi kR} \\approx \n(10^{16}-10^{17})~{\\rm GeV}$, respectively, so that the IR brane serves \nthe role of breaking the unified symmetry. (A more detailed discussion \non the determination of the scales is provided in later subsections.) \nIn this sense, we may loosely call the UV and IR branes the Planck \nand GUT branes, respectively. \n\nWe consider supersymmetric unified gauge theory on this gravitational \nbackground. We choose the gauge symmetry in the bulk to be $SU(6)$, \ncorresponding to the global symmetry that the dynamical GUT-breaking \nsector possesses in the 4D description of the model. The bulk \n$SU(6)$ gauge symmetry is broken to $SU(5) \\times U(1)$ and $SU(4) \n\\times SU(2) \\times U(1)$ on the UV and IR branes, respectively, \nleaving an unbroken $SU(3) \\times SU(2) \\times U(1) \\times U(1)$ gauge \nsymmetry at low energies. There are two ways to break a gauge symmetry \non a brane: by boundary conditions and by the Higgs mechanism. Let \nus first consider $SU(6) \\rightarrow SU(5) \\times U(1)$ on the UV \nbrane. If this breaking is caused by the Higgs mechanism, then in \nthe corresponding 4D description the fundamental gauge symmetry \nof the theory is $SU(6)$, which is spontaneously broken to $SU(5) \n\\times U(1)$ at a very high energy $E \\gg M_U$. In this case, we \nmust introduce matter fields in representations of $SU(6)$, so that \nthe standard $SU(5)$ embedding of matter fields~\\cite{Georgi:1974sy} \nshould be modified\/extended. On the other hand, if $SU(6) \\rightarrow \nSU(5) \\times U(1)$ on the UV brane is caused by boundary conditions, \nthen in the corresponding 4D description only the $SU(5) \\times \nU(1)$ subgroup of the global $SU(6)$ symmetry is explicitly gauged \n(see Fig.~\\ref{fig:basic}), so that we can employ the standard $SU(5)$ \nembedding for matter fields. We thus adopt the latter option to \nconstruct our minimal model here, although models based on the former \noption can also be accommodated in our framework.\n\n\\begin{figure}[t]\n\\begin{center}\n \\input{figure.tex}\n\\caption{A schematic picture of the model in 5D.}\n\\label{fig:5D}\n\\end{center}\n\\end{figure}\nWhat about the symmetry breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \n\\times U(1)$ on the IR brane? If we break $SU(6)$ to $SU(4) \\times SU(2) \n\\times U(1)$ by boundary conditions on the IR brane, the two massless \nHiggs doublets, whose existence is guaranteed by the general symmetry \nargument presented in the previous section, arise from extra-dimensional \ncomponents of the bulk $SU(6)$ gauge fields. This setup, however, leads \nto extra states lighter than $k' \\approx (10^{16}-10^{17})~{\\rm GeV}$ \nonce matter fields are introduced in the bulk with the zero modes \nlocalized towards the UV brane (such matter fields are used to naturally \nexplain the observed hierarchies in the fermion masses and mixings; \nsee subsection~\\ref{subsec:matter}). These extra states generically \ndo not fill complete $SU(5)$ representations and thus induce large \nthreshold corrections for the standard model gauge couplings. Large \nthreshold corrections can be avoided if we judiciously choose boundary \nconditions for matter fields, but bulk $SU(6)$ gauge invariance then \nstill requires complicated structure for the matter sector to reproduce \nthe observed fermion masses and mixings. These issues do not arise \nif the breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \\times U(1)$ \nis caused by the Higgs mechanism on the IR brane, as we will see later. \nWe therefore adopt the Higgs breaking of $SU(6)$ on the IR brane. \nThis completely determines the basic symmetry structure of our model, \nwhich is depicted in Fig.~\\ref{fig:5D}.\n\n\n\\subsection{Gauge-Higgs sector and scales of the system}\n\\label{subsec:Higgs}\n\nLet us start by describing the gauge-Higgs sector of the model. Using \n4D $N=1$ superfield language, in which the gauge degrees of freedom \nare contained in $V(A_\\mu, \\lambda)$ and $\\Phi(\\phi+iA_5, \\lambda')$, \nthe boundary conditions for the 5D $SU(6)$ gauge supermultiplet are \ngiven by\n\\begin{eqnarray}\n && V:\\: \\left( \\begin{array}{ccccc|c}\n (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \\hline\n (-,+) & (-,+) & (-,+) & (-,+) & (-,+) & (+,+) \n \\end{array} \\right),\n\\label{eq:bc-gauge-1} \\\\\n && \\Phi:\\: \\left( \\begin{array}{ccccc|c}\n (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \\hline\n (+,-) & (+,-) & (+,-) & (+,-) & (+,-) & (-,-) \n \\end{array} \\right),\n\\label{eq:bc-gauge-2}\n\\end{eqnarray}\nwhere $+$ and $-$ represent Neumann and Dirichlet boundary conditions, \nrespectively, and the first and second signs in parentheses represent \nboundary conditions at $y=0$ and $y=\\pi R$, respectively. These boundary \nconditions lead to $SU(6) \\rightarrow SU(5) \\times U(1)$ on the UV \nbrane. Since only $(+,+)$ components have zero modes, we obtain 4D \n$N=1$ $SU(5) \\times U(1)$ gauge supermultiplets as massless fields at \nthis point (coming from the upper left $5 \\times 5$ block and the lower \nright element in $V$). All the other KK modes have masses of order \n$\\pi k'$ or larger.\n\nThe symmetry breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \\times \nU(1)$ on the IR brane is caused by the vacuum expectation value (VEV) \nof a field $\\Sigma({\\bf 35})$ localized to the IR brane, where the \nnumber in the parenthesis represents the transformation property under \n$SU(6)$. We here consider that the $\\Sigma$ field is strictly localized \non the IR brane and has the following superpotential:\n\\begin{equation}\n {\\cal L}_\\Sigma = \\delta(y-\\pi R) \n \\biggl[ \\int\\! d^2\\theta \\Bigl( \\frac{M}{2}\\, {\\rm Tr}(\\Sigma^2) \n + \\frac{\\lambda}{3}\\, {\\rm Tr}(\\Sigma^3) \\Bigr) \n + {\\rm h.c.} \\biggr],\n\\label{eq:IR-Higgs}\n\\end{equation}\nwhere the metric factor is absorbed into the normalization of \nthe $\\Sigma$ field. (We will always absorb the metric factor into \nthe normalizations of fields in similar expressions below, denoted \nby ${\\cal L}$.) The field $\\Sigma$ is canonically normalized in 4D, \nso that natural values of the parameters $M$ and $\\lambda$ are of \norder $M'_* = M_* e^{-\\pi kR}$ and $4\\pi$, respectively. Here, \n$M_*$ is the cutoff scale of the 5D theory. In general, the IR-brane \npotential for $\\Sigma$ also has higher dimension terms suppressed \nby $M'_*$, in addition to Eq.~(\\ref{eq:IR-Higgs}). The presence \nof these terms, however, does not affect the qualitative conclusions \nof our paper. Below, we assume that the parameter $M$ is a factor \nof a few smaller than its naive size, e.g. $M \\sim k'$, to make \nour analysis better controlled. In this case, the effect of higher \ndimension terms are expected to be suppressed even quantitatively.\n\nThe superpotential of Eq.~(\\ref{eq:IR-Higgs}) has the following vacuum:\n\\begin{eqnarray}\n \\langle \\Sigma \\rangle \n &=& {\\rm diag}\\biggl(-\\frac{2M}{\\lambda}, -\\frac{2M}{\\lambda}, \n \\frac{M}{\\lambda}, \\frac{M}{\\lambda}, \\frac{M}{\\lambda}, \n \\frac{M}{\\lambda} \\biggr),\n\\label{eq:Sigma-VEV}\n\\end{eqnarray}\nwhere we have chosen $\\lambda, M > 0$ without loss of generality. \nThe VEV of Eq.~(\\ref{eq:Sigma-VEV}) leads to $SU(6) \\rightarrow \nSU(4) \\times SU(2) \\times U(1)$ on the IR brane, making a part of \nthe $SU(5) \\times U(1)$ gauge multiplet $V$ massive. The remaining \nmassless 4D $N=1$ gauge multiplet is that of $SU(3) \\times SU(2) \n\\times U(1) \\times U(1)$, which we identify as the standard model \ngauge group with an extra $U(1)_X$: $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$.\n\nAn important aspect of the model is that the vacuum of \nEq.~(\\ref{eq:Sigma-VEV}) is a part of a continuum of vacua, \nwhich can easily be seen by studying the excitations. Under \nthe unbroken $SU(3)_C \\times SU(2)_L \\times U(1)_Y \\times \nU(1)_X$ gauge symmetry, the $\\Sigma$ field decomposes as\n\\begin{eqnarray}\n \\Sigma &=& \\Sigma_G({\\bf 8}, {\\bf 1})_{(0,0)}\n + \\Sigma_W({\\bf 1}, {\\bf 3})_{(0,0)}\n + \\Sigma_B({\\bf 1}, {\\bf 1})_{(0,0)}\n\\nonumber\\\\\n && {} + \\Sigma_D({\\bf 3}^*, {\\bf 1})_{(1\/3,2)}\n + \\Sigma_{\\bar{D}}({\\bf 3}, {\\bf 1})_{(-1\/3,-2)}\n + \\Sigma_L({\\bf 1}, {\\bf 2})_{(-1\/2,2)}\n + \\Sigma_{\\bar{L}}({\\bf 1}, {\\bf 2})_{(1\/2,-2)}\n\\nonumber\\\\\n && {} + \\Sigma_X({\\bf 3}, {\\bf 2})_{(-5\/6,0)}\n + \\Sigma_{\\bar{X}}({\\bf 3}^*, {\\bf 2})_{(5\/6,0)}\n + \\Sigma_{S}({\\bf 1}, {\\bf 1})_{(0,0)},\n\\label{eq:Sigma-comp}\n\\end{eqnarray}\nwhere the numbers in parentheses represent the quantum numbers \nunder $SU(3)_C \\times SU(2)_L \\times U(1)_Y \\times U(1)_X$. \nThe normalization of $U(1)_Y$ is chosen to match the conventional \ndefinition of hypercharge, while that of $U(1)_X$ is chosen, when \nmatter fields are introduced, to match the conventional definition \nfor the ``$U(1)_\\chi$'' symmetry arising from $SO(10)\/SU(5)$. \nExpanding the superpotential of Eq.~(\\ref{eq:IR-Higgs}) around the \nvacuum, we find that all the components of $\\Sigma$ obtain masses \nof order $M$ except for $\\Sigma_X$, $\\Sigma_{\\bar{X}}$, $\\Sigma_L$ \nand $\\Sigma_{\\bar{L}}$. Among these four, the first two are absorbed \ninto the massive $SU(5)\/321$ gauge fields, but the last two remain \nas massless chiral superfields, which parameterize the continuous \ndegeneracy of vacua. This degeneracy is a consequence of the \nspontaneously broken $SU(6)$ symmetry, and the massless fields have \nthe quantum numbers of a pair of Higgs doublets. We thus identify \nthese fields as the two Higgs doublets of the MSSM: $H_u$ and $H_d$. \n\nWe have found that the gauge-Higgs sector of our model gives only \nthe 4D $N=1$ gauge supermultiplet for $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$ and the two Higgs doublets $H_u$ and $H_d$ \nbelow the scale of order $M \\sim k'$. The $U(1)_X$ gauge symmetry \ncan be broken at a scale somewhat below $k'$ by the Higgs mechanism. \nFor example, we can introduce the superpotential on the UV brane \n\\begin{equation}\n {\\cal L}_X = \\delta(y) \n \\biggl[ \\int\\! d^2\\theta\\, Y (X \\bar{X} - \\Lambda^2) \n + {\\rm h.c.} \\biggr],\n\\label{eq:UV-X}\n\\end{equation}\nwhere $Y$, $X$ and $\\bar{X}$ are UV-brane localized chiral superfields \nthat are singlet under $SU(5)$ and have charges of $0$, $10$ and \n$-10$ under $U(1)_X$, respectively. The scale $\\Lambda$ is that \nfor $U(1)_X$ breaking, which may be generated by some other dynamics. \nThe superpotential of Eq.~(\\ref{eq:UV-X}) produces the VEVs $\\langle \nX \\rangle = \\langle \\bar{X} \\rangle = \\Lambda$, leading to $U(1)_X$ \nbreaking at the scale $\\Lambda$. The nonvanishing VEV for $\\bar{X}$ \ncan also be used to generate small neutrino masses through the \nconventional seesaw mechanism, as we will see later. This motivates\nthe values of the $X$ and $\\bar{X}$ charges.\n\nVarious scales of our system -- the AdS inverse curvature radius \n$k$, the size of the extra dimension $R$, and the cutoff scale \nof the effective 5D theory $M_*$ -- are constrained by the scale \nof gauge coupling unification, the size of the unified gauge coupling \n$g_U \\simeq 0.7$, and the value of the 4D (reduced) Planck scale \n$M_{\\rm Pl} \\simeq 2.4 \\times 10^{18}~{\\rm GeV}$. In our warped \n5D theory, it is natural to consider that parameters in the bulk and \non the IR brane obey naive dimensional analysis (at least roughly) \nwhile those on the UV brane do not, because the former represent \nstrongly coupled $G$ dynamics while the latter represent the weakly \ncoupled elementary sector. Using naive dimensional analysis in \nhigher dimensions~\\cite{Chacko:1999hg}, we obtain the following \nLagrangian for the graviton and the gauge fields:\n\\begin{equation}\n {\\cal L} \\approx \\delta(y) \n \\Biggl[ \\frac{\\tilde{M}^2}{2} {\\cal R}^{(4)}\n - \\frac{1}{4 \\tilde{g}^2} F^{\\mu\\nu} F_{\\mu\\nu} \\Biggr]\n + \\Biggl[ \\frac{1}{2} \\frac{M_*^3}{16\\pi^3} {\\cal R}^{(5)} \n - \\frac{1}{4} \\frac{C M_*}{16\\pi^3} F^{MN} F_{MN} \\Biggr]\n\\label{eq:gravity-gauge}\n\\end{equation}\nwhere ${\\cal R}^{(4)}$ and ${\\cal R}^{(5)}$ are the 4D and 5D Ricci \ncurvatures, respectively, $M, N = 0,1,2,3,5$, and $C$ is a group \ntheoretical factor, $C \\simeq 6$. This leads to the following relations:\n\\begin{eqnarray}\n \\frac{1}{g_U^2} &\\simeq& \\frac{1}{\\tilde{g}^2} \n + \\frac{C}{16\\pi^2} \\biggl(\\frac{M_*}{\\pi k} \\biggr) \\pi kR,\n\\label{eq:g_U}\\\\\n M_{\\rm Pl}^2 &\\simeq& \\tilde{M}^2 \n + \\frac{k^2}{16} \\biggl(\\frac{M_*}{\\pi k} \\biggr)^3 .\n\\label{eq:M_Pl}\n\\end{eqnarray}\nNow, gauge coupling unification at $M_U \\approx 10^{16}~{\\rm GeV}$ \nimplies that we should choose $M$ to be around this scale, and thus \n$k' = k\\, e^{-\\pi kR} \\approx (10^{16}-10^{17})~{\\rm GeV}$. Then, \nchoosing $M_*\/\\pi k$ to be a factor of a few, e.g. $M_*\/\\pi k \\simeq \n(2\\!\\sim\\!3)$, to make the higher dimensional description trustable, \nwe obtain $k \\simlt 10^{18}~{\\rm GeV}$ from Eq.~(\\ref{eq:M_Pl}) \n(and $\\tilde{M}^2 > 0$). We thus find that the scales of our 5D \ntheory should be chosen as $k \\approx (10^{17}-10^{18})~{\\rm GeV}$ \nand $k' \\approx (10^{16}-10^{17})~{\\rm GeV}$, which implies \n$kR \\sim 1$, with the cutoff scale $M_*$ a factor of a few larger \nthan $\\pi k$. The UV-brane gauge coupling $\\tilde{g}$ is then likely \nto be nonzero, implying that the elementary $SU(5)$ gauge field has \nnonvanishing tree-level kinetic terms in the 4D description. In \nparticular, this implies that elementary $SU(5)$ gauge interactions \nare likely to be weakly coupled at energies $E \\gg M_U$.\n\n\n\\subsection{Matter sector and quark and lepton masses and mixings}\n\\label{subsec:matter}\n\nLet us now include matter fields in the model. In the 4D description \nof the theory, low-energy quark and lepton fields arise from mixtures \nof elementary states, which transform as ${\\bf 10}$'s and ${\\bf 5}^*$'s \nunder the gauged $SU(5)$, and composite states of $G$, which form \nmultiplets of the global $SU(6)$. In the 5D theory, this situation is \nrealized by introducing matter hypermultiplets in the bulk, which are \nrepresentations of $SU(6)$, and by imposing $SU(6)$-violating boundary \nconditions on the UV brane. We here present an explicit realization \nof this picture, leading to realistic phenomenology at low energies.\n\nWe begin by considering the structure of the matter sector for \na single generation. For quarks and leptons that are incorporated \ninto the ${\\bf 10}$ representation of $SU(5)$, $\\{ Q, U, E \\}$, \nwe introduce a bulk hypermultiplet $\\{ {\\cal T}, {\\cal T}^c \\}$ \ntransforming as ${\\bf 20}$ under $SU(6)$:\n\\begin{eqnarray}\n {\\cal T}({\\bf 20})\n &=& {\\bf 10}^{(+,+)}_{1}\n \\oplus {\\bf 10}^{*(-,+)}_{-1},\n\\label{eq:T} \\\\\n {\\cal T}^c({\\bf 20})\n &=& {\\bf 10}^{*(-,-)}_{-1}\n \\oplus {\\bf 10}^{(+,-)}_{1},\n\\label{eq:Tc}\n\\end{eqnarray}\nwhere ${\\cal T}$ and ${\\cal T}^c$ represent 4D $N=1$ chiral superfields \nthat form a hypermultiplet in 5D. (Our notation is such that \n``non-conjugated'' and ``conjugated'' chiral superfields have the \nopposite gauge quantum numbers; see e.g.~\\cite{Arkani-Hamed:2001tb}. \nThey have the same quantum numbers for ${\\bf 20}$ of $SU(6)$ \nbecause ${\\bf 20}$ is a (pseudo-)real representation.) The \nright-hand-side of Eqs.~(\\ref{eq:T},~\\ref{eq:Tc}) shows the \ndecomposition of ${\\cal T}$ and ${\\cal T}^c$ into representations \nof $SU(5) \\times U(1)_X$ (in an obvious notation), as well as the \nboundary conditions imposed on each component (in the same notation \nas that in Eqs.~(\\ref{eq:bc-gauge-1},~\\ref{eq:bc-gauge-2})). With \nthese boundary conditions, the only massless state arising from \n$\\{ {\\cal T}, {\\cal T}^c \\}$ is ${\\bf 10}_1$ of $SU(5) \\times U(1)_X$ \nfrom ${\\cal T}$, which we identify as the low-energy quarks and \nleptons $Q, U$ and $E$.\n\nA bulk hypermultiplet $\\{ {\\cal H}, {\\cal H}^c \\}$ can generically \nhave a mass term in the bulk, which is written as \n\\begin{equation}\n S = \\int\\!d^4x \\int_0^{\\pi R}\\!\\!dy \\, \n \\biggl[ e^{-3k|y|}\\! \\int\\!d^2\\theta\\, \n c_{\\cal H}\\, k\\, {\\cal H} {\\cal H}^c + {\\rm h.c.} \\biggr],\n\\label{eq:bulk-mass}\n\\end{equation}\nin the basis where the kinetic term is given by $S_{\\rm kin} = \\int\\!d^4x \n\\int\\!dy\\, [e^{-2k|y|} \\int\\!d^4\\theta\\, ({\\cal H}^\\dagger {\\cal H} + \n{\\cal H}^c {\\cal H}^{c\\dagger}) + \\{ e^{-3k|y|} \\int\\!d^2\\theta\\, \n({\\cal H}^c \\partial_y {\\cal H} - {\\cal H} \\partial_y {\\cal H}^c)\/2 + \n{\\rm h.c.} \\}]$~\\cite{Marti:2001iw}. The parameter $c_{\\cal H}$ controls \nthe wavefunction profile of the zero mode. For $c_{\\cal H} > 1\/2$ \n($< 1\/2$) the wavefunction of a zero mode arising from ${\\cal H}$ is \nlocalized to the UV (IR) brane; for $c_{\\cal H} = 1\/2$ it is conformally \nflat. (If a zero mode arises from ${\\cal H}^c$, its wavefunction \nis localized to the IR (UV) brane for $c_{\\cal H} > -1\/2$ ($< -1\/2$) \nand conformally flat for $c_{\\cal H} = -1\/2$.) We choose these $c$ \nparameters to take values larger than about $1\/2$ for matter fields. \nFor these values of $c$ parameters, all the KK excited states of \n$\\{ {\\cal T}, {\\cal T}^c \\}$ have masses of order $\\pi k'$ or larger, \nso that the $\\{ {\\cal T}, {\\cal T}^c \\}$ multiplet gives only the \nmassless ${\\bf 10}_1$ state below the energy scale of $k'$.\n\nFor quarks and leptons incorporated into the ${\\bf 5}^*$ representation \nof $SU(5)$, $\\{ D, L \\}$, we introduce a bulk hypermultiplet $\\{ {\\cal F}, \n{\\cal F}^c \\}$ transforming as ${\\bf 70}^*$ under $SU(6)$:\n\\begin{eqnarray}\n {\\cal F}({\\bf 70}^*)\n &=& {\\bf 5}^{*(+,+)}_{-3}\n \\oplus {\\bf 10}^{*(-,+)}_{-1}\n \\oplus {\\bf 15}^{*(-,+)}_{-1}\n \\oplus {\\bf 40}^{*(-,+)}_{1},\n\\label{eq:F} \\\\\n {\\cal F}^c({\\bf 70})\n &=& {\\bf 5}^{(-,-)}_{3}\n \\oplus {\\bf 10}^{(+,-)}_{1}\n \\oplus {\\bf 15}^{(+,-)}_{1}\n \\oplus {\\bf 40}^{(+,-)}_{-1},\n\\label{eq:Fc}\n\\end{eqnarray}\nwhere the right-hand-side again shows the decomposition into \nrepresentations of $SU(5) \\times U(1)_X$, together with the boundary \nconditions imposed on each component.%\n\\footnote{Note that the signs $\\pm$ for the boundary conditions in \nEqs.~(\\ref{eq:F},~\\ref{eq:Fc}) represent the Neumann\/Dirichlet boundary \nconditions in the interval $y: [0, \\pi R]$. In the orbifold picture, \nthe boundary conditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) can be \nobtained effectively as follows. We prepare a hypermultiplet obeying \nthe boundary conditions ${\\cal F}({\\bf 70}^*) = {\\bf 5}^{*(+,+)}_{-3} \n\\oplus {\\bf 10}^{*(-,+)}_{-1} \\oplus {\\bf 15}^{*(-,+)}_{-1} \\oplus \n{\\bf 40}^{*(+,+)}_{1}$ and ${\\cal F}^c({\\bf 70}) = {\\bf 5}^{(-,-)}_{3} \n\\oplus {\\bf 10}^{(+,-)}_{1} \\oplus {\\bf 15}^{(+,-)}_{1} \\oplus \n{\\bf 40}^{(-,-)}_{-1}$, where the first and second signs in the \nparentheses represent transformation properties under the reflection \n$y \\leftrightarrow -y$ and $(y-\\pi R) \\leftrightarrow -(y - \\pi R)$, \nrespectively. We then introduce a UV-brane localized chiral superfield \ntransforming as ${\\bf 40}_{-1}$ under $SU(5) \\times U(1)_X$, and couple \nit to the ${\\bf 40}^{*(+,+)}_{1}$ state from ${\\cal F}({\\bf 70}^*)$. \nThis reproduces the boundary conditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) \nin the limit that this coupling (brane mass term) becomes large. (For \nthe relation between a large brane mass term and the Dirichlet boundary \ncondition, see e.g.~\\cite{Nomura:2001mf}.) The fact that the boundary \nconditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) can be reproduced in \nthe orbifold picture by taking a consistent limit guarantees their \nconsistency. In the 4D description, this corresponds to introducing \nonly a ${\\bf 5}^*_{-3}$ elementary state, which couples to \na component of a $G$-invariant operator transforming as ${\\bf 70}$ \nunder the global $SU(6)$. Similar remarks also apply to other \nfields, e.g. the $\\{ {\\cal N}, {\\cal N}^c \\}$ hypermultiplet \nin Eqs.~(\\ref{eq:N},~\\ref{eq:Nc}).}\nWith these boundary conditions, the only massless state arising from \n$\\{ {\\cal F}, {\\cal F}^c \\}$ is ${\\bf 5}^*_{-3}$ of $SU(5) \\times \nU(1)_X$ from ${\\cal F}$, which we identify as the low-energy quarks \nand leptons $D$ and $L$. All the KK excited states have masses of \norder $\\pi k'$ or larger for $c_{\\cal F} \\simgt 1\/2$.\n\nThe right-handed neutrino $N$ arises from a bulk hypermultiplet \n$\\{ {\\cal N}, {\\cal N}^c \\}$ transforming as ${\\bf 56}$ of $SU(6)$:\n\\begin{eqnarray}\n {\\cal N}({\\bf 56})\n &=& {\\bf 1}^{(+,+)}_{5}\n \\oplus {\\bf 5}^{(-,+)}_{3}\n \\oplus {\\bf 15}^{(-,+)}_{1}\n \\oplus {\\bf 35}^{(-,+)}_{-1},\n\\label{eq:N} \\\\\n {\\cal N}^c({\\bf 56}^*)\n &=& {\\bf 1}^{(-,-)}_{-5}\n \\oplus {\\bf 5}^{*(+,-)}_{-3}\n \\oplus {\\bf 15}^{*(+,-)}_{-1}\n \\oplus {\\bf 35}^{*(+,-)}_{1}.\n\\label{eq:Nc}\n\\end{eqnarray}\nThe zero mode arises only from ${\\bf 1}_5$ in ${\\cal N}$, which is \nidentified as the right-handed neutrino supermultiplet $N$. The other \nKK states are all heavier than of order $\\pi k'$ for $c_{\\cal N} \n\\simgt 1\/2$.\n\nThe Yukawa couplings for the quarks and leptons arise from IR-brane \nlocalized terms\n\\begin{equation}\n {\\cal L}_{\\rm Yukawa} = \\delta(y-\\pi R) \n \\biggl[ \\int\\! d^2\\theta \\Bigl( \n y_{\\cal T} {\\cal T} {\\cal T} \\Sigma\n + y_{\\cal F} {\\cal T} {\\cal F} \\Sigma \n + y_{\\cal N} {\\cal F} {\\cal N} \\Sigma \\Bigr) \n + {\\rm h.c.} \\biggr].\n\\label{eq:IR-Yukawa}\n\\end{equation}\n(The Yukawa couplings also receive contributions from higher dimension \nterms as will be seen later in this subsection.) Note that these \ninteractions, as well as those in Eq.~(\\ref{eq:IR-Higgs}), respect \nthe usual $R$ parity of the MSSM, with $\\Sigma$ even.\n\nThe interactions of Eq.~(\\ref{eq:IR-Yukawa}) give the Yukawa \ncouplings of the quark and lepton chiral superfields, $Q, U, D, L, E$ \nand $N$, with the Higgs doublets, $H_u$ and $H_d$, at low energies \n($W = QUH_u$, $QDH_d + LEH_d$ and $LNH_u$ from the first, second \nand third terms, respectively). Recall that the two Higgs doublets \nof the MSSM, $H_u$ and $H_d$, arise from $\\Sigma$ as pseudo-Goldstone \nchiral superfields of the broken $SU(6)$ symmetry. For matter fields \nwith $|c| > 1\/2$, the Yukawa couplings receive suppressions due \nto the fact that the fields effectively feel only the IR brane \n(strong dynamics) or the UV brane (explicit breaking of $SU(6)$), \nboth of which are needed to generate nonvanishing Yukawa couplings \nat low energies~\\cite{Contino:2003ve}. Then, considering that \n$y_{\\cal T} \\sim y_{\\cal F} \\sim y_{\\cal N} = O(4\\pi^2\/M'_*)$ from \nnaive dimensional analysis, we find that the low-energy Yukawa \ncoupling $y$ arising from the IR-brane term $\\int\\!d^2\\theta\\, \n{\\cal M}_1 {\\cal M}_2 \\Sigma$ (${\\cal M}_1, {\\cal M}_2 \n= {\\cal T}, {\\cal F}, {\\cal N}$) takes a value \n\\begin{equation}\n y \\approx 4\\pi f_1 f_2\\,\n \\biggl( \\frac{\\pi k}{M_*} \\biggr),\n\\label{eq:Yukawa-value}\n\\end{equation}\nwhere $f_i \\simeq (k'\/k)^{|c_{{\\cal M}_i}|-1\/2}$ for $|c_{{\\cal M}_i}| \n> 1\/2$ and $f_i \\simeq 1$ for $|c_{{\\cal M}_i}| < 1\/2$ ($i=1,2$); for \n$|c_{{\\cal M}_i}| \\simeq 1\/2$, $f_i$ receives a logarithmic suppression, \n$f_i \\simeq 1\/(\\ln(k\/k'))^{1\/2}$. This allows us to explain the observed \nhierarchies of fermion masses and mixings by powers of $k'\/k = e^{-\\pi kR} \n= O(0.1)$, by choosing different values of $c_{\\cal T}, c_{\\cal F}$ and \n$c_{\\cal N}$ for different generations. This is similar to the situation \nwhere the hierarchies are explained by overlaps of matter and Higgs \nwavefunctions~\\cite{Gherghetta:2000qt,Hebecker:2002re}, although \nin the present setup the low-energy Yukawa couplings are also suppressed \nfor $c_{{\\cal M}_i} < -1\/2$, where apparent overlaps between matter \nand Higgs fields are large, due to the pseudo-Goldstone boson nature \nof the Higgs doublets. This opens the possibility of localizing the \nfirst two generations to the IR brane, rather than to the UV brane \nas we will do shortly, to generate the observed hierarchies of fermion \nmasses and mixings.\n\nThe right-handed neutrino superfield $N$ can obtain a large mass \nterm through the UV-brane operator\n\\begin{equation}\n {\\cal L}_N = \\delta(y) \n \\biggl[ \\int\\! d^2\\theta\\, \\frac{\\eta}{2} \\bar{X} N^2 \n + {\\rm h.c.} \\biggr],\n\\label{eq:UV-N}\n\\end{equation}\nwhere $\\bar{X}$ is a $U(1)_X$-breaking field, having the VEV \n$\\langle \\bar{X} \\rangle = \\Lambda$ (see Eq.~(\\ref{eq:UV-X})). \nThis gives a small mass for the observed left-handed neutrino \nthrough the conventional seesaw mechanism~\\cite{Seesaw}.%\n\\footnote{An alternative possibility to generate a small neutrino \nmass is to strongly localize the $N$ field to the IR brane by taking \n$c_{\\cal N} \\ll -1\/2$, in which case the neutrino Yukawa coupling \nis strongly suppressed and we can obtain a small Dirac neutrino \nmass. The scale of the neutrino mass, however, is unexplained \nin this case.}\n\nIt is rather straightforward to generalize the analysis so far \nto the case of three generations. We simply introduce a set of \nbulk hypermultiplets $\\{ {\\cal T}, {\\cal T}^c \\}$, $\\{ {\\cal F}, \n{\\cal F}^c \\}$ and $\\{ {\\cal N}, {\\cal N}^c \\}$ for each generation. \nThe couplings $y_{\\cal T}$, $y_{\\cal F}$ and $y_{\\cal N}$ in \nEq.~(\\ref{eq:IR-Yukawa}) and $\\eta$ in Eq.~(\\ref{eq:UV-N}) then \nbecome $3 \\times 3$ matrices. We assume that there is no special \nstructure in these matrices, so that all the elements in $y_{\\cal T}$, \n$y_{\\cal F}$ and $y_{\\cal N}$ are of order $4\\pi^2\/M'_*$, suggested \nby naive dimensional analysis. The observed fermion masses and \nmixings, however, can still be reproduced through the dependence of \nthe low-energy Yukawa couplings on the values of bulk hypermultiplet \nmasses $c_{\\cal T}, c_{\\cal F}$ and $c_{\\cal N}$. Let us take, for \nexample, the bulk masses to be\n\\begin{equation}\n c_{{\\cal T}_1} \\simeq \\frac{5}{2}, \\quad \n c_{{\\cal T}_2} \\simeq \\frac{3}{2}, \\quad \n c_{{\\cal T}_3} \\simeq \\frac{1}{2}, \\quad\n c_{{\\cal F}_1} \\simeq c_{{\\cal F}_2} \\simeq \n c_{{\\cal F}_3} \\simeq \\frac{3}{2}, \\quad\n c_{{\\cal N}_1} \\simeq c_{{\\cal N}_2} \\simeq \n c_{{\\cal N}_3} \\simeq \\frac{1}{2}.\n\\label{eq:c-flavor}\n\\end{equation}\nThen, taking $M_*\/\\pi k$ to be a factor of a few, e.g. $2\\!\\sim\\!3$, \nwe obtain the following low-energy Yukawa matrices from \nEq.~(\\ref{eq:Yukawa-value}):\n\\begin{equation}\n y_u \\approx \n \\pmatrix{\n \\epsilon^4 & \\epsilon^3 & \\epsilon^2 \\cr\n \\epsilon^3 & \\epsilon^2 & \\epsilon \\cr\n \\epsilon^2 & \\epsilon & 1 \\cr\n },\n\\quad\n y_d \\approx y_e^T \\approx\n \\epsilon\n \\pmatrix{\n \\epsilon^2 & \\epsilon^2 & \\epsilon^2 \\cr\n \\epsilon & \\epsilon & \\epsilon \\cr\n 1 & 1 & 1 \\cr\n },\n\\quad\n y_\\nu \\approx\n \\epsilon\n \\pmatrix{\n 1 & 1 & 1 \\cr\n 1 & 1 & 1 \\cr\n 1 & 1 & 1 \\cr\n },\n\\label{eq:y-values}\n\\end{equation}\nwhere $y_u$, $y_d$, $y_e$ and $y_\\nu$ are defined in the low-energy \nsuperpotential by\n\\begin{equation}\n W = (y_u)_{ij} Q_i U_j H_u + (y_d)_{ij} Q_i D_j H_d\n + (y_e)_{ij} L_i E_j H_d + (y_\\nu)_{ij} L_i N_j H_u,\n\\label{eq:y-def}\n\\end{equation}\nwith $i,j, = 1,2,3$, and\n\\begin{equation}\n \\epsilon \\equiv \\frac{k'}{k} \\simeq \\frac{1}{20}\n \\quad {\\rm for} \\,\\,\\, kR \\simeq 1.\n\\label{eq:epsilon}\n\\end{equation}\nTogether with a structureless Majorana mass matrix for the \nright-handed neutrinos, $M_N = \\eta \\langle \\bar{X} \\rangle$, \nthe Yukawa matrices of Eq.~(\\ref{eq:y-values}) well reproduces \ngross features of the observed quark and lepton masses and \nmixings~\\cite{Hall:1999sn}. It is straightforward to make \nfurther refinements on this basic picture; for example, we can \nmake $c_{{\\cal F}_1}$ somewhat larger than $3\/2$ to better reproduce \ndown-type quark and charged lepton masses, as well as the neutrino \nmixing angles $\\theta_{12}$ and $\\theta_{13}$. A schematic picture \nfor the zero-mode wavefunctions (for the $\\{ {\\cal T}, {\\cal T}^c \\}$ \nmultiplets) is depicted in Fig.~\\ref{fig:5D}.\n\nUnwanted $SU(5)$ mass relations for the first two generation \nfermions can be avoided by using higher dimension operators, e.g. \nof the form ${\\cal L} \\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, {\\cal T} \n{\\cal F} \\Sigma^2$. (Violation of $SU(5)$ relations may also come \nfrom $SU(5)$-violating mixings between the matter zero modes and \nthe corresponding KK excited states, arising from the IR-brane \nterms of Eq.~(\\ref{eq:IR-Yukawa}) through the $\\Sigma$ VEV.) Since \nthe effects are higher order in $\\langle \\Sigma \\rangle\/M'_*$, \nwhich we assume somewhat small, $O(1)$ violation in the Yukawa \ncoupling requires a somewhat suppressed coefficient for the leading \n$SU(5)$-invariant piece coming from Eq.~(\\ref{eq:IR-Yukawa}). \nA realistic pattern for the fermion masses and mixings can be \nobtained if (only) the 22 element of the $y_{\\cal F}$ matrix \nis somewhat suppressed~\\cite{Georgi:1979df}.\n\nThe three generation model allows IR-brane operators of the form \n${\\cal L} \\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\epsilon^{ij} \n{\\cal T}_i {\\cal T}_j$, where the antisymmetry in the generation \nindices $i,j$ arises from the pseudo-real nature of the ${\\bf 20}$ \nrepresentation. The existence of these operators, however, does \nnot significantly affect predictions of the model.\n\nTo summarize, we have obtained an $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$ gauge theory below the scale of $M \\sim k'$, \nwith three generations of matter, $Q, U, D, L, E$ and $N$, and \ntwo Higgs doublets, $H_u$ and $H_d$. The Yukawa couplings of \nEq.~(\\ref{eq:y-def}) are obtained with realistic patterns for quark \nand lepton masses and mixings. The $U(1)_X$ gauge symmetry is \nspontaneously broken at the scale $\\Lambda$, somewhat below $k'$, \ngiving masses to the right-handed neutrino superfields of order \n$\\Lambda$. We thus have the complete MSSM, supplemented by seesaw \nneutrino masses, below the unification scale $\\sim k'$. We emphasize \nthat the successes of our model depend only on its basic features, \nsuch as the symmetry structure and locations of fields. They are \nthus quite robust. For example, the existence of higher dimension \noperators in the IR-brane potential, e.g. terms of the form ${\\rm \nTr}(\\Sigma^n)$ ($n$: integers $>3$) added to Eq.~(\\ref{eq:IR-Higgs}), \ndoes not destroy these successes.\n\n\n\\subsection{Gauge coupling unification and proton decay}\n\\label{subsec:analysis}\n\nIn this subsection, we present a study on proton decay and gauge \ncoupling unification in our model, to demonstrate that it can \naccommodate realistic phenomenology at low energies. In this \nsubsection we consider matter configurations such that lighter \ngenerations are localized more towards the UV brane, as in the \nexample of Eq.~(\\ref{eq:c-flavor}).\n\nWe first note that the terms in Eqs.~(\\ref{eq:IR-Yukawa}) introduce, \nthrough the VEV of $\\Sigma$, $SU(5)$-violating mass splittings \ninto the KK towers for the matter fields: $\\{ {\\cal T}, {\\cal T}^c \\}$, \n$\\{ {\\cal F}, {\\cal F}^c \\}$ and $\\{ {\\cal N}, {\\cal N}^c \\}$. \nThese splittings, in turn, give threshold corrections to gauge \ncoupling unification. Similar corrections also arise from the \ngauge KK towers. We expect, however, that these corrections are \nnot large. Using the AdS\/CFT correspondence, we estimate the \nsize of the corrections to be of order $(C\/16\\pi^2)(M_*\/\\pi k)$ \nfor $1\/g_a^2$, where $g_a$ are the 4D gauge couplings. Moreover, \nif the value of $\\langle \\Sigma \\rangle$ (and thus $M$) is somewhat \nsuppressed compared with its naive size of $M'_*\/4\\pi$, as we assume \nhere, the threshold corrections receive additional suppressions \nof $O(4\\pi \\langle \\Sigma \\rangle\/M'_*)$ because the spectrum \nof the KK towers becomes $SU(5)$ symmetric for $4\\pi \\langle \\Sigma \n\\rangle\/M'_* \\ll 1$. The contributions from tree-level IR-brane \noperators, such as $\\int\\!d^2\\theta\\, \\Sigma {\\cal W}^\\alpha \n{\\cal W}_\\alpha$, are also sufficiently small, of order $C\/16\\pi^2$ \nfor $1\/g_a^2$ with an additional suppression of $O(4\\pi \\langle \n\\Sigma \\rangle\/M'_*)$ for small $\\langle \\Sigma \\rangle$.\n\nAnother important issue in supersymmetric unified theories is \ndimension five proton decay caused by low-energy operators of the \nform $W \\sim QQQL$, $UUDE$. There are two independent sources for \nthese operators: tree-level operators existing at the gravitational \nscale and operators generated by the GUT (breaking) dynamics. In \nour theory, the former correspond to tree-level operators ${\\bf 10}_1 \n{\\bf 10}_1 {\\bf 10}_1 {\\bf 5}^*_{-3} \\supset QQQL$, $UUDE$ located \non the UV brane, where the subscripts on ${\\bf 10}_1 \\subset {\\cal T}$ \nand ${\\bf 5}^*_{-3} \\subset {\\cal F}$ denote the $U(1)_X$ charges. \nWhile the coefficients of these operators are suppressed by the \nfundamental scale $M_*$, which is larger than the unification scale, \nit is still problematic, especially because we do not have any \nYukawa suppressions in the coefficients. Therefore, to suppress \nthese contributions, we impose a discrete $Z_{4,R}$ symmetry on the \ntheory, whose charge assignment is given in Table~\\ref{table:Z4R} \n(in the normalization that the $R$ charge of the superpotential \nis $2$).\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|cc|c|cccccc|ccc|} \\hline \n & $V$ & $\\Phi$ & $\\Sigma$ & \n ${\\cal T}$ & ${\\cal T}^c$ & ${\\cal F}$ & \n ${\\cal F}^c$ & ${\\cal N}$ & ${\\cal N}^c$ & \n $Y$ & $X$ & $\\bar{X}$ \\\\ \\hline\n $Z_{4,R}$ & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & \n 2 & 0 & 0 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{$Z_{4,R}$ charges for fields.}\n\\label{table:Z4R}\n\\end{table}\nAs is clear from the terms in Eq.~(\\ref{eq:IR-Higgs}), this symmetry \nshould be broken on the IR brane, i.e. broken by the dynamics of \nthe GUT breaking sector $G$. This can be incorporated by introducing \na spurion chiral superfield $\\phi$ with $\\langle \\phi \\rangle \\sim \nM'_*\/4\\pi$ on the IR brane, whose $Z_{4,R}$ charge is $+2$, or \nequivalently introducing fields $\\phi$ and $\\bar{\\phi}$ of $Z_{4,R}$ \ncharges $+2$ and $-2$ with the superpotential giving the VEVs for \nthese fields. This introduces ``$O(1)$'' breaking of $Z_{4,R}$ on \nthe IR brane, keeping $Z_{4,R}$ invariance for the UV-brane terms.%\n\\footnote{The $Z_{4,R}$ symmetry can be gauged in 5D if we cancel \nthe discrete $Z_{4,R}$-$SU(5)^2$ anomaly by the Green-Schwarz \nmechanism~\\cite{Green:1984sg}, by introducing a singlet field $S$ \nthat transforms nonlinearly under $Z_{4,R}$ and couples to the \n$SU(5)$ gauge kinetic term on the UV brane. We consider that the \n$S$ field appears only in front of the kinetic term of the $SU(5)$ \ngauge superfields, and not in UV-brane superpotential terms. Such \nterms would potentially induce dimension five proton decay, although \nthey are suppressed in a certain (broad) region for the $S$ VEV.}\n\nAfter killing the UV-brane operators, the low-energy dimension five \nproton decay operators can still be generated through strong $G$ \ndynamics, since the $Z_{4,R}$ symmetry is spontaneously broken by \nthis dynamics. One source is tree-level dimension five operators \non the IR brane. These operators, however, receive suppressions \nof order the Yukawa couplings in 4D, because the wavefunctions for \nlight generation matter are suppressed on the IR brane due to the \nbulk hypermultiplet masses, and so are not particularly dangerous. \n(In the 4D picture, these suppressions arise from small mixings \nbetween the elementary and composite matter states for light \ngenerations.) The only potentially dangerous contribution to \ndimension five proton decay in our model then comes from the exchange \nof the colored triplet Higgsinos -- composite states of $G$ arising \nas components of $\\Sigma$ -- because the mass of these states can \nbe smaller than $M'_*$. To suppress this contribution, we can simply \nraise the mass of the colored triplet Higgsino states compared with \nthe unification scale; in fact, the mass is expected to be larger \nthan the GUT breaking VEV because the coupling $\\lambda$ in \nEq.~(\\ref{eq:IR-Higgs}) is naturally of order $4\\pi$. Note that \nbecause of the existence of threshold corrections from KK towers \nto gauge coupling unification, there is no tight relation between \nthe mass of the triplet Higgsinos and the low-energy values of \nthe gauge couplings, which excluded the minimal SUSY $SU(5)$ GUT \nin 4D~\\cite{Murayama:2001ur}.\n\n\n\\subsection{Supersymmetry breaking}\n\\label{subsec:SUSY-breaking}\n\nOur model can be combined with almost any supersymmetry breaking \nscenario. If the mediation scale of supersymmetry breaking is \nlower than the unification scale, there are essentially no particular \nimplications from our theory on the pattern of supersymmetry breaking. \nOn the other hand, if the mediation scale is higher, there can be \ninteresting implications, e.g., on the flavor structure of supersymmetry \nbreaking masses. For example, if the supersymmetry breaking sector \nis localized on the IR brane, i.e. arises as a result of the dynamics \nof $G$, the third generation superparticles (presumably only the \nones coming from the ${\\bf 10}$ representation of $SU(5)$) can \nhave different masses than the lighter generation superparticles, \nwhich receive universal masses from the gauginos through loop \ncorrections~\\cite{Kitano:2006}.%\n\\footnote{We thank R.~Kitano for discussions on this issue.}\nThese are consequences of our way of generating hierarchies \nin fermion masses and mixings.\n\nThe supersymmetric mass (the $\\mu$ term) and supersymmetry-breaking \nmasses (the $\\mu B$ term and non-holomorphic scalar squared masses) \nfor the Higgs doublets are both generated through supersymmetry \nbreaking. In the case that the supersymmetry breaking sector is \nlocalized on or directly communicates with the IR brane, these masses \nare generated through IR-brane operators of the form, ${\\cal L} \n\\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\{ Z M \\Sigma^2\/M'_* + \nZ \\Sigma^3\/M'_*\\} + {\\rm h.c.}$ and $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \n\\{ (Z+Z^\\dagger)\\Sigma^\\dagger \\Sigma\/M'_* + Z^\\dagger Z (\\Sigma^2 \n+ \\Sigma^{\\dagger 2})\/M_*^{\\prime 2} + Z^\\dagger Z \\Sigma^\\dagger \n\\Sigma\/M_*^{\\prime 2} \\}$, where $Z$ is a chiral superfield responsible \nfor supersymmetry breaking, $\\langle Z \\rangle = \\theta^2 F_Z$, \nand we have omitted $O(1)$ coefficients. These operators produce \nsupersymmetry breaking terms in the $\\Sigma$ potential, which lead \nto a slight shift of the vacuum from Eq.~(\\ref{eq:Sigma-VEV}) and \nconsequently generate weak scale masses for components of $H_u$ \nand $H_d$. The generated masses respect the relation\n\\begin{equation}\n \\mu B = \\left||\\mu|^2 + m_{H_u}^2 \\right|,\n\\qquad\n m_{H_u}^2 = m_{H_d}^2,\n\\label{eq:Higgs-1}\n\\end{equation}\nreflecting the fact that the scalar potential for $\\Sigma$ still \nhas a global $SU(6)$ symmetry, where $m_{H_u}^2$ and $m_{H_d}^2$ are \nnon-holomorphic supersymmetry breaking squared masses for $H_u$ and \n$H_d$, and we have taken the phase convention that $\\mu B > 0$. Note \nthat, unlike the case where the Higgs fields are non pseudo-Goldstone \nfields~\\cite{Giudice:1988yz}, the K\\\"ahler potential terms of the \nform $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \\Sigma^2$, $\\delta(y-\\pi R) \n\\int\\!d^4\\theta\\, \\{ Z^\\dagger \\Sigma^2\/M'_* + {\\rm h.c.} \\}$ and \n$\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \\{ Z^\\dagger Z \\Sigma^2\/M_*^{\\prime 2} \n+ {\\rm h.c.} \\}$ do not produce a weak scale $\\mu$ term; we need \nsupersymmetry breaking interactions for $\\Sigma$, generated by \nsuperpotential terms or $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \n(Z+Z^\\dagger)\\Sigma^\\dagger \\Sigma\/M'_*$. \n\nAn interesting case arises if supersymmetry is broken in a hidden \nsector that does not have direct interactions with the GUT breaking \nsector. In this case, the Higgs sector supersymmetry breaking \nparameters arise through gravitational effects and obey the \ntighter relation\n\\begin{equation}\n \\mu B = |\\mu|^2, \\qquad m_{H_u}^2 = m_{H_d}^2 = 0.\n\\label{eq:Higgs-2}\n\\end{equation}\nIn the language of the compensator formalism (see \ne.g.~\\cite{Randall:1998uk}), these terms arise from ${\\cal L} \\sim \n\\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\phi (M\/2) \\Sigma^2 + {\\rm h.c.}$, \nwhere $\\phi = 1 + \\theta^2 m_{3\/2}$ is the compensator field \nwith $m_{3\/2}$ the gravitino mass. (Here, we have assumed that \nsupersymmetry breaking in the compensator field is not canceled \nby the conformal dynamics of the GUT breaking sector.) The relations \nof Eqs.~(\\ref{eq:Higgs-1},~\\ref{eq:Higgs-2}) hold at the unification \nscale of $O(k')$, so that their connections to low energy parameters \nmust involve renormalization group effects between the unification \nand the weak scales. It is also possible that there are \nadditional contributions to supersymmetry breaking parameters, \ne.g. $m_{H_u}^2$ and $m_{H_d}^2$, in addition to the ones in \nEqs.~(\\ref{eq:Higgs-1},~\\ref{eq:Higgs-2})\n\nAlternatively, the $\\mu$ and $\\mu B$ terms may be generated below \nthe unification scale. For example, they may be generated associated \nwith the dynamics of $U(1)_X$ breaking~\\cite{Hall:2002up}. In this \ncase there is no trace in the Higgs sector parameters that the Higgs \nfields are pseudo-Goldstone fields of the GUT breaking dynamics.\n\n\n\\section{Other Theories: GUT Engineering on the IR Brane}\n\\label{sec:other}\n\nSo far, we have considered a theory in which the lightness of the \ntwo Higgs doublets is understood by the pseudo-Goldstone mechanism \nassociated with the dynamics of GUT breaking. As we have seen, this \ncan be elegantly implemented in our framework by considering the \nbulk $SU(6)$ gauge symmetry, broken to $SU(5) \\times U(1)$ and \n$SU(4) \\times SU(2) \\times U(1)$ on the UV and IR branes, respectively. \nThe mass of the light Higgs doublets is protected from the existence \nof explicit breaking by localizing these fields ($\\subset \\Sigma$) \non the IR brane, which is geometrically separated from the UV brane \nwhere explicit breaking of $SU(6)$ resides. In the 4D description \nof the model, the global $SU(6)$ symmetry of the GUT breaking sector \nis understood as a ``flavor'' symmetry of this sector, and the extreme \nsuppression of explicit symmetry breaking effects in the $\\Sigma$ \npotential comes from the fact that $\\Sigma$ is a composite field, \nwith the corresponding operator having a (very) large canonical mass \ndimension. An interesting thing about our construction is that it \nallows us to implement these mechanisms in simple and controllable \nways in effective field theory, giving a simple and calculable unified \ntheory above $M_U$ in which the lightness of the Higgs doublets is \nunderstood by a symmetry principle.\n\nLet us now consider if we can construct simpler theories in our warped \nspace framework. Suppose we consider a supersymmetric $SU(5)$ gauge \ntheory in the 5D warped spacetime of Eq.~(\\ref{eq:metric}), and suppose \nthat the bulk $SU(5)$ gauge symmetry is broken to the $SU(3)_C \\times \nSU(2)_L \\times U(1)_Y$ subgroup on the IR brane by boundary conditions. \nIn this case, the doublet Higgs fields may be light without being \naccompanied by their triplet partners if they propagate in the \nbulk with appropriate boundary conditions imposed at the GUT breaking \nbrane~\\cite{Kawamura:2000ev,Hall:2001pg}, or if they are simply \nlocated on that brane~\\cite{Hebecker:2001wq}. In the 4D description, \nhowever, this seems to be simply ``postulating'' particular dynamics \nof the GUT breaking sector that splits the mass of the doublet components \nfrom that of the triplet partners, and it is not clear if this can be \nregarded as a ``solution'' to the doublet-triplet splitting problem. \nFor example, we have a continuous parameter, the tree-level mass of the \nHiggs doublets on the IR brane, that has to be chosen to be very small \nto achieve the splitting. The situation may be better if this parameter \nis forbidden by a symmetry, e.g. an $R$ symmetry~\\cite{Hall:2001pg}. \nThis symmetry may be imposed as a global symmetry in 5D, but in that \ncase it is not entirely clear if such a symmetry is preserved by strong \n$G$ dynamics (5D quantum gravity effects). To avoid this and to give \na non-trivial meaning to the symmetry in the context of gauge\/gravity \nduality, we can gauge the symmetry in higher dimensions (although it \ncan still be broken on the UV brane, eliminating the existence of the \ncorresponding gauge field in 4D). In this case, anomaly cancellation \nconditions become an issue, and we find that for a continuous $U(1)_R$ \nor a discrete $Z_{4,R}$ symmetry (with the charge assignment given \nby $V_{SU(5)}(0)$, $T_{\\bf 10}(1)$, $F_{{\\bf 5}^*}(1)$, $N_{\\bf 1}(1)$, \n$H_{\\bf 5}(0)$, $\\bar{H}_{{\\bf 5}^*}(0)$, assuming the MSSM matter \ncontent at low energies) we need to cancel the low energy anomalies \nvia the Green-Schwarz mechanism~\\cite{Green:1984sg}. This requires the \nintroduction of a singlet field $S$ on the IR brane which transforms \nnonlinearly under the $R$ symmetry and couples to the $SU(3)_C$, $SU(2)_L$ \nand $U(1)_Y$ gauge kinetic terms with appropriate coefficients. (Anomaly \ntransmission across the bulk~\\cite{Callan:1984sa} may also be necessary \nto make the full 5D theory anomaly free, depending on the symmetry \nand matter content.) We assume that the $S$ field appears only in \nfront of the gauge kinetic terms, and not in IR-brane superpotential \nterms, so that a large mass term for the Higgs doublets is not \nregenerated. (This may naturally occur in a UV theory in the absence \nof other gauge groups.) Note that, in the 4D description, this \nsetup corresponds to the situation where the $R$-$SU(5)^2$ anomaly \nis canceled between the elementary-field and $G$-sector contributions.%\n\\footnote{We can show that this construction is not available in \na 4D SUSY GUT theory where the GUT-breaking (Higgs) sector does not \ngive tree-level contributions to the low energy anomalies. Assuming \nthe MSSM matter content below the unification scale, with the $U(1)_R$ \ncharges given by $V_{321}(0)$, $Q(1)$, $U(1)$, $D(1)$, $L(1)$, $E(1)$, \n$H_u(0)$ and $H_d(0)$, we find the low-energy $U(1)_R$-$SU(3)_C^2$, \n$U(1)_R$-$SU(2)_L^2$ and $U(1)_R$-$U(1)_Y^2$ anomalies to be $3$, \n$1$ and $-3\/5$, respectively, which cannot be matched to high \nenergy theories, where these anomalies arise as a $U(1)_R$-$SU(5)^2$ \nanomaly and are thus universal. (Here, the $SU(5)$ normalization \nis employed for the $U(1)_Y$ charges.) This implies that $U(1)_R$ \nshould either be spontaneously broken, or there is explicit \n$SU(5)$-violating physics in the effective field theory. In our case, \nthis conclusion can be avoided because the (dynamical) GUT-breaking \nsector carries the $U(1)_R$-$SU(5)^2$ anomaly, a part of which can \nbe manifested as Green-Schwarz terms at low energies.} \nIn this setup, doublet-triplet splitting seems ``natural,'' at least \nin the higher dimensional picture. Thus, while the theory with the \n$R$ symmetry still seems to correspond to a particular choice of GUT \nbreaking dynamics in the 4D description, we may say that the theory \ndoes not have the problem of doublet-triplet splitting.%\n\\footnote{An interesting feature of this class of theories is \nthat the low energy theory contains an axion field $S$ that \ncouples to the QCD gauge fields with the decay constant of order \nthe unification scale. This can be used to solve the strong $CP$ \nproblem~\\cite{Peccei:1977hh}, although the initial amplitude of \nthis field in the early universe must be (accidentally) small to \navoid the cosmological difficulty of overclosing the universe.}\nAfter all, the ``formulation'' of the doublet-triplet splitting problem \nmay have to be changed in the large 't~Hooft coupling regime, where \nphysics is specified by the ``hadronic'' quantities, i.e. matter \ncontent, location, and boundary conditions in higher dimensions.%\n\\footnote{If we break the bulk $SU(5)$ gauge symmetry by boundary \nconditions at the UV brane, it leads to a theory which is interpreted \nas an $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ gauge theory in the 4D \ndescription. The doublet-triplet splitting problem does not arise \nas the theory is not unified, and yet the successful unification of \ngauge couplings arises at the leading-log level in the limit that \nthe tree-level gauge kinetic terms on the UV brane are small. This \ncorresponds in the 4D description that the 321 gauge couplings \nat the unification scale are dominated by the asymptotically non-free \ncontribution from a strong sector that has a global $SU(5)$ symmetry, \nof which the $SU(3) \\times SU(2) \\times U(1)$ subgroup is gauged \nand identified as the low-energy 321 gauge group. While this theory \nis somewhat outside the framework described in this paper, it is \ninteresting on its own.}\n\nIn these respects, our framework offers many possible ways to \naddress the problems of conventional 4D SUSY GUTs. For example, we \ncan again consider a 5D $SU(5)$ gauge theory in the warped spacetime \nof Eq.~(\\ref{eq:metric}), but then break the bulk $SU(5)$ by the \nVEV of a chiral superfield located on the IR brane, generated by \nan appropriate IR-brane superpotential. Then, if this superpotential \ndoes not have the problem of doublet-triplet splitting, e.g. by having \nthe form of missing partner type models~\\cite{Masiero:1982fe}, then \nwe may say that the problem has been solved. (As discussed before, \nit is better if the IR-brane superpotential is protected by a (discrete) \nsymmetry; otherwise, it would correspond to ``artificially'' choosing \nthe dynamics of the GUT breaking sector. In practice, this may be \ndifficult, since we cannot use the non-universal Green-Schwarz terms \non the IR brane because the GUT breaking there is due to the Higgs \nmechanism. We do not pursue this issue further here.) An advantage \nof this approach over conventional 4D model building is that we need \nnot care about physics above the unification scale when engineering \nGUT breaking physics, i.e. the GUT-breaking Higgs content and \nsuperpotential. In the conventional 4D SUSY GUT framework, theories \nsolving the doublet-triplet and\/or dimension five proton decay \nproblems often have too large matter content, leading to the problem \nof the unified gauge coupling hitting a Landau pole (well) below \nthe gravitational\/Planck scale. In our case, all (possibly large) \nmultiplets located on the IR brane correspond to composite fields \nof the GUT breaking dynamics in the 4D description, and do not \ncontribute to the running of the unified gauge coupling above the \nunification scale, $M_U \\sim k'$ (see e.g.~\\cite{Goldberger:2002cz}). \nPotential complication of this sector may also not bother us, because \nit is the result of ``dynamics'' of the GUT breaking sector. We note \nthat this makes the extension to $SO(10)$ unified theories trivial \n--- we can break $SO(10)$ on the IR brane by arbitrary combinations \nof boundary condition and Higgs breakings with an arbitrary field \ncontent.\n\nThere are many applications of the ideas described above. For instance, \nwe can apply it to product-group theories~\\cite{Yanagida:1994vq,%\nIzawa:1997he,Weiner:2001pv}. Let us once again consider \na supersymmetric $SU(5)$ gauge theory in the 5D warped spacetime \nof Eq.~(\\ref{eq:metric}). We then introduce an additional gauge \ngroup $SU(3) \\times SU(2) \\times U(1)$ on the IR brane, with the Higgs \ndoublets charged under this IR-brane gauge group (and thus without \nbeing accompanied by any partner). Now, we can consider that our \nlow-energy $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ gauge group is \na diagonal subgroup of the bulk $SU(5)$ and the IR-brane $SU(3) \\times \nSU(2) \\times U(1)$. (This breaking can be caused by the VEV of an \nappropriate IR-brane localized field). Then, if the gauge couplings, \n$\\tilde{g}_a$ of the original $SU(3)$, $SU(2)$ and $U(1)$ are large, \n$\\tilde{g}_a \\approx 4\\pi$ ($a = 1,2,3$), the low-energy MSSM gauge \ncouplings are effectively unified at the scale where $SU(5) \\times \nSU(3) \\times SU(2) \\times U(1) \\rightarrow SU(3)_C \\times SU(2)_L \n\\times U(1)_Y$ occurs $\\sim k'$, since the gauge couplings of $SU(3)_C$, \n$SU(2)_L$ and $U(1)_Y$, $g_a$, are given by $1\/g_a^2 = 1\/g_5^2 + \n1\/\\tilde{g}_a^2 \\approx 1\/g_5^2$ at that scale. Here, $g_5$ ($= O(1)$) \nis the coupling of the zero mode of the bulk $SU(5)$ gauge field. \nA problem of the corresponding scenario in 4D~\\cite{Weiner:2001pv} \nis that, since the original $U(1)$ gauge coupling is strong at the \nunification scale, it hits the Landau pole immediately above that scale. \nThere, it is also not clear why the three independent gauge couplings \nof $SU(3)$, $SU(2)$ and $U(1)$ become strong at a single scale, which \nmust also coincide with the scale of diagonal breaking to avoid large \nthreshold corrections. Our theory addresses all of these issues \nnaturally --- since the $SU(3)$, $SU(2)$ and $U(1)$ gauge fields are \ncomposite states of the GUT breaking dynamics, they all have strong \ncouplings, $\\tilde{g}_a \\approx 4\\pi$, at the scale where breaking \nto the diagonal subgroup occurs, and there is no issue of a Landau \npole above this scale. A potentially large mass for the Higgs \ndoublets can be avoided by introducing a (discrete) gauge symmetry \nwith the anomalies canceled by the Green-Schwarz terms on the IR brane.%\n\\footnote{Another possibility for canceling anomalies is to add extra \nmatter fields (in complete $SU(5)$ multiplets) that obtain masses from \nsupersymmetry breaking~\\cite{Kurosawa:2001iq}. We thank N.~Maru for \npointing out this work to us.}\n(To do all of these completely within the regime of effective field \ntheory, the scale of diagonal breaking should be somewhat below the \nIR-brane cutoff, and the $SU(3)$, $SU(2)$ and $U(1)$ gauge couplings \nshould be asymptotically non-free. These can be arranged with an \nappropriate introduction of massive fields on the IR brane. In the \nlimit that the scale of diagonal breaking approaches the IR-brane \ncutoff, this theory is reduced to one of the theories discussed in \nthe second paragraph of this subsection, where the Higgs doublets \nare located on the GUT-breaking IR brane.) Note that since the quarks \nand leptons are introduced in the bulk in representations of the \nbulk $SU(5)$, we are still considering a unified theory of quarks \nand leptons (although it is possible to introduce them on the IR brane \nin representations of $SU(3) \\times SU(2) \\times U(1)$). In particular, \nproton decay from unified gauge boson exchange still exists. The Yukawa \ncouplings of matter to the Higgs fields arise through the IR-brane \nVEV, breaking $SU(5) \\times SU(3) \\times SU(2) \\times U(1)$ down to \n$SU(3)_C \\times SU(2)_L \\times U(1)_Y$. \n\nIn most of the theories described above, the Higgs fields are localized \nto the IR brane, so that they are composite fields of the dynamical \nGUT breaking sector in the 4D description. (This need not be the case. \nOne of the theories described in the second paragraph of this subsection \ncontains Higgs fields that propagate in the bulk, with appropriate \nboundary conditions imposed at the IR brane. We can, however, always \nlocalize them to the IR brane by introducing appropriate hypermultiplet \nmasses.) The observed hierarchies of quark and lepton masses and mixings \ncan then always be explained by wavefunction overlaps between the matter \nand Higgs fields, by appropriately choosing the bulk hypermultiplet \nmasses for the matter fields such that lighter generations are localized \nmore towards the UV brane, as e.g. in Eq.~(\\ref{eq:c-flavor}). (The \noption of localizing lighter generations towards the IR brane is not \navailable unless the Higgs fields are pseudo-Goldstone boson multiplets.)\nWe find it very interesting that our framework of ``holographic grand \nunification'' accommodates many different ideas of solving the problems \nof conventional SUSY GUTs, developed mainly in the 4D context, with \nthe automatic bonus of explaining the observed hierarchies of fermion \nmasses and mixings through the wavefunction profiles of matter fields \nin the extra dimension.\n\n\n\\section{Discussion and Conclusions}\n\\label{sec:concl}\n\nIn this paper we have studied a framework in which grand unification \nis realized in truncated warped higher dimensional spacetime, \nwhere the UV and IR branes set the Planck and unification scales, \nrespectively. In the 4D description, this corresponds to theories \nin which the the grand unified gauge symmetry is spontaneously \nbroken by strong gauge dynamics having a large 't~Hooft coupling, \n$\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \\gg 1$ (and a large number of ``colors'', \n$\\tilde{N} \\gg 1$). In this parameter region, an appropriate (weakly \ncoupled) description of physics is obtained in higher dimensions, and \nphysics above the unification scale is determined by higher dimensional \nfield theories, e.g. by specifying the spacetime metric, gauge group, \nmatter content, boundary conditions, and Lagrangian parameters. This \nallows us to control certain dynamical properties of the GUT breaking \nsector in the regime where effective field theory applies. For example, \nwe can make the size of threshold corrections small by making the \nsymmetry breaking VEV on the IR brane (slightly) smaller than its naive \nvalue. Moreover, the framework allows us to straightforwardly adopt \nintuitions and mechanisms arising from the higher dimensional picture. \nIn particular, we can explain the observed hierarchies in quark and \nlepton masses and mixings in terms of the wavefunction profiles of \nmatter fields in higher dimensions. The generated hierarchies are \nnaturally of the right size, of order $M_U\/M_* \\simeq 1\/20$.\n\nWe have presented several realistic models within this framework. In \none model, on which we have focused the most, the lightness of the Higgs \ndoublets is explained by the pseudo-Goldstone mechanism. The strong \ngauge dynamics sector possesses a global $SU(6)$ symmetry as a ``flavor'' \nsymmetry, of which the $SU(5)$ ($\\times U(1)$) subgroup is gauged and \nidentified as the unified gauge group. When the global symmetry is \nbroken dynamically to $SU(4) \\times SU(2) \\times U(1)$, the unified \ngauge symmetry is broken to the standard model gauge group, and the two \nMSSM Higgs doublets arise as massless pseudo-Goldstone supermultiplets. \nIn our framework, this is realized by postulating a bulk $SU(6)$ gauge \nsymmetry, broken to $SU(5) \\times U(1)$ and $SU(4) \\times SU(2) \\times \nU(1)$ on the UV and IR branes, respectively. One of the difficulties \nin implementing this mechanism in the conventional 4D framework is to \nfind a way to suppress effects of explicit breaking in the potential \ngenerating the spontaneous $SU(6)$ breaking, since such effects would \nreintroduce an unacceptably large mass for the Higgs doublets. In our \ncase, these effects are (exponentially) suppressed by a large mass \ndimension for the operator generating the spontaneous $SU(6)$ breaking. \nSuch an assumption is easy to implement in higher dimensions -- simply \nassume that the Higgs field breaking $SU(6)$ is localized to the IR \nbrane. This provides another example of the ``controllability'' of \nstrong gauge dynamics in the large 't~Hooft coupling regime.\n\nWe have also demonstrated that many ideas for solving the problems of \nconventional 4D SUSY GUTs can be naturally implemented on the IR brane. \nWe have presented several realistic models of this kind, for example, \nones based on missing partner type or product group type scenarios. \nThese models have the interesting feature that the GUT scale physics \non the IR brane does not affect physics at higher energies, since the \nrelevant physics arises as a result of the strong GUT breaking dynamics \n(as composite states) in the 4D description. For example, large \nGUT multiplets, often needed to solve the problems of SUSY GUTs, \ndo not contribute to the evolution of the unified gauge coupling \nat higher energies, and gauge fields having very large gauge couplings \ncan naturally arise at the GUT scale without having the problem of \na Landau pole. These features open up new possibilities for GUT model \nbuilding.\n\nOne can view the ``success'' of the present framework in several \ndifferent ways. For one who is interested in addressing the \nphenomenology of unified theories, such as gauge coupling unification \nand proton decay, models in our framework can be used to give predictions \nof observable quantities. For example, we can explore relations \nbetween the branching ratios of proton decay and matter configurations \nin the extra dimension, as in the case of unified theories in flat \nspace~\\cite{Nomura:2001tn}. Models of fermion masses and mixings, \nas well as models of supersymmetry breaking, can also be developed \nwithin the framework.%\n\\footnote{For example, we can take one of the supersymmetry breaking \nmodels in~\\cite{Goldberger:2002pc}, with the boundary conditions at \nthe UV brane changed to be trivial, and glue that spacetime (the 5D \nwarped spacetime with the scales at the UV and IR branes taken to be the \nPlanck and TeV scales, respectively) to one of our holographic warped \nGUT spacetimes discussed in section~\\ref{sec:other}, at the UV branes \nof both spacetimes. (The consistency of such constructions in effective \nfield theory has been discussed recently in~\\cite{Cacciapaglia:2006tg}.) \nIn the 4D description, this corresponds to the situation where both \nthe unified gauge symmetry and supersymmetry are broken by strong \ngauge dynamics, at the unification scale and the TeV scale, respectively. \nIn practice, this system is analyzed most efficiently by first \nintegrating out the GUT scale physics. Then the low energy effective \ntheory is simply reduced to one of the models in~\\cite{Goldberger:2002pc}, \nbut now we have an understanding of the hierarchical structure of the \nYukawa couplings, located on the UV brane of the effective theory. \nWhile this effective field theory may be at the border of the weak \nand strong coupling regimes in 5D, it may still reproduce gross \nfeatures of physical quantities, e.g. the superparticle spectrum, \nas is the case in higher dimensional formulations of QCD.}\nOn the other hand, one may be interested in exploring possible \n``UV completions'' of models formulated in warped spacetime. It is \npossible, after all, that there may be some nontrivial consistency \nconditions in higher dimensional field theories, which are difficult \n(though not impossible) to catch in effective theory, and one way \nof ensuring the consistency of such theories is to ``derive'' them \nfrom complete UV theories. Such ``UV completions'' may be achieved, \nfor example, by embedding models into string theory, identifying \na ``dual'' 4D theory, or by finding a 4D theory whose infrared \nfixed point has similar features as the original models in warped \nspace~\\cite{Arkani-Hamed:2001ca}. From this perspective, our framework \noffers a guide on which models ``UV theorists'' should aim to reproduce; \nfor example, string theorists may want to reproduce unified theories \nin 5D warped spacetime, with the unified gauge symmetry broken at \nan IR throat, rather than 4D unified theories directly from \ncompactification.\n\nWe finally comment on the possibility that the unified gauge \nsymmetry is broken by strong gauge dynamics whose 't~Hooft coupling \nis large but not extremely large, i.e. $\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \n\\sim 1$. In this case, the picture based on higher dimensional \nspacetime is not fully justified, but even then some properties of \ntheories, especially properties associated with the IR brane physics \n(GUT breaking dynamics), may be effectively described by our higher \ndimensional warped unified theories. In fact, such an approach had \na certain level of successes in describing physics of lowest-lying \nexcitations in QCD~\\cite{Erlich:2005qh}. In this sense, our framework \nmay have a larger applicability than what is naively expected.\n\nIn summary, we have presented a framework in which dynamical GUT \nbreaking models are realized in a regime that has a weakly coupled \n``dual'' picture. Grand unified theories are realized in warped \nhigher dimensional spacetime, with the UV and IR spacetime cutoffs \nproviding the Planck and the unification scales, respectively. \nSeveral types of realistic models are discussed, with interesting \nimplications for quark and lepton masses and mixings. It would \nbe interesting to study further implications of these models, such \nas those on proton decay, precise gauge coupling unification, \nsupersymmetry breaking, and flavor physics.\n\n\n\\section*{Acknowledgments}\n\nThis work was supported in part by the Director, Office of Science, Office \nof High Energy and Nuclear Physics, of the US Department of Energy under \nContract DE-AC02-05CH11231. The work of Y.N. was also supported by the \nNational Science Foundation under grant PHY-0403380, by a DOE Outstanding \nJunior Investigator award, and by an Alfred P. Sloan Research Fellowship.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Statement of Conjectures}\n\nLet $E$ be an elliptic curve defined over the field $\\mathbf{Q}$ of rational numbers. Assume that $E$ does not have complex multiplication. Denote by $N$ the conductor of $E$, so that $E$ has good reduction at primes $p$ not dividing $N$. Let $\\{p_k\\}_{k \\geq 1}$ be the set of primes in ascending order. For $p_k$ not dividing $N$, define as usual the quantity $a_{p_k}:= p_k+1-\\#E(\\mathbf{F}_{p_k})$ (here $\\mathbf{F}_{p_k}$ is the finite field of cardinality $p_k$, and $E(\\mathbf{F}_{p_k})$ is the group of points of $E$ over $\\mathbf{F}_{p_k}$). By the Hasse bound one has $|a_{p_k}| \\leq 2 p_k^{1\/2}$. We define $x_k \\in [0,1]$ for $k \\geq 1$, referred to as the (normalized) Frobenius angle of $E$ at the prime $p_k$, by the condition:\n\\[\na_{p_k} = 2p_k^{1\/2} \\cos(\\pi x_k)\n\\]\nif $p_k$ does not divide $N$, and we simply define $x_k=1\/2$ if $p_k$ divides $N$. The Sato-Tate conjecture, as established by Taylor {\\it et. al.} \\cite{CHT,T,HSBT,BLGHT}, states that the sequence $\\{x_k\\}_{k \\geq 1}$ is uniformly distributed with respect to the Sato-Tate measure on $[0,1]$. Specifically define the Sato-Tate measure $\\mu_{ST}$ on $[0,1]$ (which is a probability measure) by:\n\\[\nd \\mu_{ST} = 2 \\sin^2(\\pi u) \\, du\n\\] \n(where $du$ is the Lebesgue measure on $[0,1]$) and for $x_k$ as above, denote by $\\delta_{x_k}$ the Dirac point mass distribution on $[0,1]$ supported at $x_k$. For any integer $K \\geq 1$ consider the probability distribution on $[0,1]$ given by:\n\\begin{eqnarray}\n\\frac{1}{K} \\sum_{k=1}^K \\delta_{x_k}\n\\end{eqnarray}\nthen the Sato-Tate conjecture states that the probability distribution (1.1) converges weakly to the Sato-Tate measure $\\mu_{ST}$ as $K$ tends to infinity.\n\n\\bigskip\nIn this paper we propose the following refinement of the original Sato-Tate conjecture, that in addition to uniform distribution, we conjecture that the sequence $\\{x_k\\}_{k \\geq 1}$ is pseudorandom, in other words the $x_k$'s are in fact {\\it statistically independently} distributed with respect to the Sato-Tate measure. Specifically for any integer $s \\geq 1$, denote by $\\mu_{ST}^{[s]}$ the probability measure on $[0,1]^s$ given by the product of $s$ (independent) copies of the original one dimensional Sato-Tate measure $\\mu_{ST}$. We consider the joint distributions for $s$ successive terms from the sequence $\\{x_k\\}_{k \\geq 1}$. Thus define $X_k \\in [0,1]^s$ for $k \\geq 1$ to be the $s$-dimensional vector given by:\n\\[\nX_k = (x_k,x_{k+1}, \\cdots,x_{k+s-1})\n\\]\nand denote by $\\delta_{X_k}$ the Dirac point mass distribution on $[0,1]^s$ that is supported at $X_k$.\n\n\\bigskip\nFor integer $K \\geq 1$ consider similarly the probability distribution on $[0,1]^s$ given by:\n\\begin{eqnarray}\n\\frac{1}{K} \\sum_{k=1}^K \\delta_{X_k}\n\\end{eqnarray}\n\n\\bigskip\nWe propose the following:\n\\begin{conjecture}\nFor any integer $s \\geq 1$, the sequence $\\{X_k\\}_{k \\geq 1}$ is uniformly distributed with respect to $\\mu_{ST}^{[s]}$. In other words the probability distribution on $[0,1]^s$ as given by (1.2), converges weakly to $\\mu_{ST}^{[s]}$, as $K$ tends to infinity. \n\\end{conjecture}\n\n\\bigskip\nFor $s=1$ this is the original Sato-Tate conjecture (which is already proved). For $s \\geq 2$ this asserts the statistical independence of the distribution of $s$ successive terms from the sequence $\\{x_k\\}_{k \\geq 1}$ with respect to Sato-Tate measure. Thus in terms of statistical distribution, Conjecture 1.1 asserts that, with respect to the Sato-Tate measure, the sequence $\\{x_k\\}_{k \\geq 1}$ is $\\infty$-distributed in the sense of Knuth, {\\it c.f.} Definition C on page 151 of \\cite{K}. It is in this sense that we say that the sequence $\\{x_k\\}_{k \\geq 1}$ is pseudorandom (with respect to the Sato-Tate measure). We present numerical evidences for Conjecture 1.1 in section 2 below.\n\n\\bigskip\n\\begin{remark}\n\\end{remark}\n\\noindent Let $E$ and $E^{\\prime}$ be elliptic curves over $\\mathbf{Q}$ without complex multiplication, and assume that $E$ and $E^{\\prime}$ are non-isogenous. Let $\\{x_k\\}_{k \\geq 1}$ and $\\{x_k^{\\prime} \\}_{k \\geq 1}$ be the sequences of Frobenius angles associated to $E$ and $E^{\\prime}$ respectively. Consider the two dimensional vectors $(x_k,x_k^{\\prime}) \\in [0,1]^2 $ for $k \\geq 1$. Harris \\cite{H} established that the sequence $\\{ (x_k,x_k^{\\prime})\\}_{k \\geq 1}$ is uniformly distributed with respect to $\\mu_{ST}^{[2]}$. By contrast, the setting of our Conjecture 1.1 concerns the statistical independence of the distribution of Frobenius angles for a single elliptic curve.\n\\bigskip\n\\bigskip\n\nEven more optimistically, we propose the following quantitive refinement of Conjecture 1.1. Firstly define the extreme discrepancy $D^{[s]}_K$ (with respect to the measure $\\mu_{ST}^{[s]}$) for integer $K \\geq 1$ as follows. For any rectangular region $\\mathcal{R} \\subset [0,1]^s$ of the form:\n\\[\n\\mathcal{R} = [a_1,b_1) \\times \\cdots \\times [a_s,b_s)\n\\]\ndefine\n\\[\nA(\\mathcal{R};K) = \\#\\{ 1\\leq k\\leq K \\,\\ | \\,\\ X_k \\in \\mathcal{R} \\}\n\\]\n\\bigskip\nThen define\n\\[\nD^{[s]}_K = \\sup_{\\mathcal{R} } \\Big| \\frac{A(\\mathcal{R};K)}{K} - \\mu_{ST}^{[s]} (\\mathcal{R}) \\Big|\n\\]\nwhere $\\mathcal{R}$ ranges over all rectangular regions in $[0,1]^s$ as above.\n\n\\bigskip\n\n\nIn general for a point $W =(w^{(1)},\\cdots,w^{(s)}) \\in [0,1]^s$, we denote by $\\mathcal{R}_W \\subset [0,1]^s$ the rectangular region:\n\\[\n\\mathcal{R}_W = [0,w^{(1)} ) \\times \\cdots \\times [0,w^{(s)} ) \n\\]\n\n\\bigskip\n\nWe define the star discrepancy $D_K^{*,[s]}$ (with respect to the measure $\\mu_{ST}^{[s]}$) as:\n\\[\nD^{*,[s]}_K = \\sup_{W \\in [0,1]^s } \\Big| \\frac{A(\\mathcal{R}_W;K )}{K} - \\mu_{ST}^{[s]} (\\mathcal{R}_W) \\Big|\n\\]\n\n\\bigskip\nWe have $0 \\leq D_K^{[s]} ,D_K^{*,[s]}\\leq 1$, and the inequalities ({\\it c.f.} p. 93 of \\cite{KN}):\n\\begin{eqnarray}\nD_K^{*,[s]} \\leq D_K^{[s]} \\leq 2^s \\cdot D_K^{*,[s]}\n\\end{eqnarray}\n(remark that in {\\it loc. cit.} the notion of discrepancy with respect to the Lebesgue measure on $[0,1]^s$ is considered, but the same considerations apply verbatim with respect to the measure $\\mu_{ST}^{[s]}$ as well).\n\n\\bigskip\nThe discrepancies $D^{[s]}_K $ and $D_K^{*,[s]}$ quantify the uniformity of distribution of the finite set $\\{ X_k \\}_{k=1}^K$ with respect to the measure $\\mu_{ST}^{[s]}$ on $[0,1]^s$.\n\n\\bigskip\n\\begin{conjecture}\nFor any integer $s \\geq 1$ and $\\epsilon >0$, there exists a constant $C=C(E,s,\\epsilon)$ (depending only on the elliptic curve $E$, $s$ and $\\epsilon$), such that:\n\\[\nD^{[s]}_K \\leq C K^{\\epsilon- \\frac{1}{2}}\n\\]\nfor any integer $K \\geq 1$.\n\\end{conjecture}\n\n\\bigskip\nOf course, Conjecture 1.3 is interesting only when $ \\epsilon < \\frac{1}{2} $; in addition it can also be stated equivalently in terms of the star discrepancy $D_K^{*,[s]}$ instead of $D_K^{[s]}$, by virtue of the inequalities (1.3).\n\n\\bigskip\n\n\nWhen $s=1$, Conjecture 1.3 was originally formulated by Akiyama-Tanigawa (Conjecture 1 of \\cite{AT}), which refines the original Sato-Tate conjecture. In general Conjecture 1.3 is a refinement of Conjecture 1.1; namely that by standard results on uniform distribution ({\\it c.f.} p. 93 of \\cite{KN}), Conjecture 1.1 is equivalent to the assertion:\n\\[\n\\lim_{K \\rightarrow \\infty} D^{[s]}_K =0.\n\\]\n\n\\bigskip\n\n Conjecture 1.3 is a very strong statement. Indeed Akiyama-Tanigawa showed that their conjecture (i.e. Conjecture 1.3 in the case $s=1$) implies the validity of the Riemann Hypothesis for the $L$-function associated to $E$ (that the $L$-function associated to $E$ has analytic continuation is of course the consequence of the modularity of $E$); more generally their conjecture implies the validity of the Riemann Hypothesis for all the higher symmetric power $L$-functions associated to $E$, {\\it c.f.} Poposition 3.5 of \\cite{M} for the precise statement (in {\\it loc. cit.} suitable analytic hypotheses on the higher symmetric power $L$-functions are assumed, which are in any case consequences of the Langlands Functoriality Conjecture with respect to the symmetric power functorial liftings of the modular form associated to $E$. The existence of the symmetric power functorial liftings of the modular form associated to $E$ is established by Newton-Thorne \\cite{NT} under quite general conditions, including all semistable $E$ for instance). \n \n \\bigskip\n Conversely Nagoshi showed (see Theorem 2 of \\cite{N}) that the conjecture of Akiyama-Tanigawa holds (at least) for $ \\epsilon > 1\/4$, if one supposes the validity of the Riemann Hypothesis for all the higher symmetric power $L$-functions associated to $E$ (again assuming suitable analytic hypotheses on the higher symmetric power $L$-functions); see also \\cite{RT} for the explicit version. At this moment we do not know whether our Conjecture 1.3 for $s \\geq 2$ can be approached using the theory of $L$-functions. Nevertheless we present numerical evidences for Conjecture 1.3 in section 3 below.\n\n\\bigskip\n\nIn this paper the computations of the orders of the group of points of elliptic curves over finite fields were performed using the {\\it GP\/PARI} program. The rest of the computations were then performed using {\\it Mathematica 9.0}.\n\n\n\n\\section{Numerical Evidences for Conjecture 1.1}\n\nWith the setting as in Conjecture 1.1, for continuous function $f$ defined on $[0,1]^s$, we would like to test whether:\n\\[\n \\frac{1}{K} \\sum_{k=1}^K f(X_k) \\stackrel{?}{\\rightarrow} \\int_{[0,1]^s} f \\, d \\mu_{ST}^{[s]}\n\\]\n as $K$ tends to infinity. \n\n\\bigskip\nWe consider the following six elliptic curves over $\\mathbf{Q}$ without complex multiplication, taken from the {\\it $L$-functions and modular forms database}, whose affine Weierstrass equations are given as follows, with conductor $N$ and Mordell-Weil rank $r$ as indicated:\n\n\\[\nE_1: y^2+y=x^3-x^2, \\,\\ N=11,\\,\\ r=0\n\\]\n\n\\[\nE_2: y^2+y =x^3-x, \\,\\ N=37, \\,\\ r=1\n\\]\n\n\\[\nE_3: y^2 +xy=x^3+1, \\,\\ N=433, \\,\\ r=2\n\\]\n\n\\[\nE_4:y^2+y=x^3-7x+6, \\,\\ N=5077, \\,\\ r=3\n\\]\n\n\\[\nE_5: y^2+y=x^3-7x+36, \\,\\ N=545723, \\,\\ r=4\n\\]\n\n\\[\nE_6: y^2+y=x^3-79x+342, \\,\\ N=19047851, \\,\\ r=5\n\\]\n\n\\bigskip\n\n\n\n\\bigskip\nWe denote the coordinates on $[0,1]^s$ as $u^{(1)},\\cdots,u^{(s)}$. For testing statistical independence, it is good enough to choose test functions $f$ of the form:\n\\[\nf(u^{(1)},\\cdots,u^{(s)}) = \\prod_{i=1}^s f_i(u^{(i)})\n\\]\nfor continuous functions $f_i$ on $[0,1]$, in which case we have\n\\begin{eqnarray}\n\\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} = \\prod_{i=1}^s \\int_{[0,1]} f_i (u^{(i)}) \\cdot 2 \\sin^2(\\pi u^{(i)}) \\, d u^{(i)}\n\\end{eqnarray}\n\n\\bigskip\nWe first consider the case $s=10$. Define the function $f^{[10]}$ on $[0,1]^{10}$:\n\n\\begin{eqnarray*}\n& & f^{[10]}(u^{(1)},\\cdots,u^{(10)}) \\\\ \n&= & \\ln(2 + u^{(1)}) \\cdot \\ln (3 + u^{(2)}) \\cdot \\exp(-u^{(3)}) \\cdot (1+ u^{(4)})^2 \\cdot (2+ u^{(5)}) \\cdot \\\\\n& & \\sqrt{2 + u^{(6)}} \\cdot \\sqrt{ 3 + u^{(7)}} \\cdot (4+ u^{(8)})^{\\frac{1}{3}} \\cdot (8+ u^{(9)})^{\\frac{1}{4}} \\cdot \\exp( \\sqrt{1+u^{(10)}} )\n\\end{eqnarray*}\n\n\\bigskip\nUsing the command NIntegrate of {\\it Mathematica}, the numerical value of the integral $\\int_{[0,1]^{10}} f^{[10]} \\, d\\mu_{ST}^{[10]}$ is computed as in (2.1) to be:\n\\[ \n\\int_{[0,1]^{10}} f^{[10]} \\, d\\mu_{ST}^{[10]} \\stackrel{.}{=}114.076\n\\]\n\nThe numerical results for $\\frac{1}{K} \\sum_{k=1}^K f^{[10]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$ are tabulated in Figure 1 below.\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 113.87 & 113.753 & 113.903 & 114.009 & 114.032\\\\ \n $E_2$ & 114.196 & 114.08 & 114.074 & 114.128 & 114.181\\\\\n\n $E_3$ & 114.534 & 114.576 & 114.493 & 114.154 & 114.237 \\\\\n\n $E_4$ & 115.375 & 115.011 & 114.683 & 114.441 & 114.248\\\\\n\n $E_5$ & 116.127 & 115.137 & 114.499 & 114.62 & 114.474 \\\\\n $E_6$ & 116.559 & 115.371 & 115.312 & 114.471 &114.519\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K f^{[10]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\nWe next consider examples with larger values of $s$. For $s \\geq 1$ define the function $g^{[s]}$ on $[0,1]^s$ given by:\n\\begin{eqnarray*}\n g^{[s]}(u^{(1)},\\cdots,u^{(s)}) = 100 \\cdot \\prod_{i=1}^s \\exp ( - u^{(i)} \/i)\n\\end{eqnarray*}\n\n\\bigskip\nWe have:\n\\begin{eqnarray}\n& & \\int_{[0,1]^s} g^{[s]} \\, d \\mu_{ST}^{[s]} \\\\ &=& 100 \\cdot \\prod_{i=1}^s \\int_{[0,1]} \\exp (-u^{(i)}\/i) \\cdot 2 \\sin^2(\\pi u^{(i)})\\, d u^{(i)} \\nonumber \\\\\n &=& 100 \\cdot \\prod_{i=1}^s \\left( \\big( 1 - \\exp (-1\/i) \\big)\n \\cdot \\frac{4 \\pi^2 i^3}{1+ 4\\pi^2 i^2} \\right) \\nonumber\n\\end{eqnarray}\n\n\\bigskip\nThe numerical values of $\\int_{[0,1]^s} g^{[s]} \\, d \\mu_{ST}^{[s]}$ for $s=500$, $s=1000$, and $s=2000$, are computed as in (2.2) to be:\n\n\n\\begin{eqnarray*}\n\\int_{[0,1]^{500}} g^{[500]} \\, d \\mu_{ST}^{[500]} \\stackrel{.}{=}\n3.44034\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1000}} g^{[1000]} \\, d \\mu_{ST}^{[1000]} \\stackrel{.}{=}\n2.43333\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{2000}} g^{[2000]} \\, d \\mu_{ST}^{[2000]} \\stackrel{.}{=}\n1.72086\n\\end{eqnarray*}\n\n\\bigskip\nWhile the numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[s]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$, in the cases $s=500$, $s=1000$, and $s=2000$ respectively, are tabulated in Figures 2, 3, 4 below.\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n$E_1$& 3.4513 & 3.4541 & 3.45074 & 3.4417 & 3.44034 \\\\\n $E_2$&3.4208 & 3.43017 & 3.43423 & 3.43228 & 3.42744 \\\\\n $E_3$&3.4013 & 3.38951 & 3.40058 & 3.43055 & 3.42734 \\\\\n $E_4$&3.33751 & 3.36335 & 3.38431 & 3.40763 & 3.42225 \\\\\n $E_5$&3.27776 & 3.35011 & 3.40524 & 3.3932 & 3.40618 \\\\\n $E_6$&3.24535 & 3.33898 & 3.33186 & 3.4024 & 3.39838 \\\\\n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[500]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n \n $E_1$&2.4422 & 2.44414 & 2.44149 & 2.43447 & 2.43337 \\\\\n $E_2$&2.41925 & 2.42568 & 2.42922 & 2.42732 & 2.42337 \\\\\n $E_3$&2.40409 & 2.39574 & 2.40298 & 2.426 & 2.42349 \\\\\n $E_4$&2.35616 & 2.37465 & 2.39071 & 2.40813 & 2.4195 \\\\\n $E_5$&2.31073 & 2.3628 & 2.40653 & 2.39708 & 2.40692 \\\\\n $E_6$&2.28672 & 2.35609 & 2.3498 & 2.40406 & 2.40116 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[1000]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n \n $E_1$&1.72737 & 1.72919 & 1.72717 & 1.72179 & 1.72091 \\\\\n $E_2$&1.71064 & 1.71501 & 1.71777 & 1.71642 & 1.71329 \\\\\n $E_3$&1.69752 & 1.6939 & 1.69775 & 1.71531 & 1.71333 \\\\\n $E_4$&1.66527 & 1.6769 & 1.68915 & 1.70186 & 1.71047 \\\\\n $E_5$&1.63056 & 1.66685 & 1.70142 & 1.6933 & 1.70075 \\\\\n $E_6$&1.61208 & 1.66286 & 1.65744 & 1.69837 & 1.69624 \\\\ \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[2000]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\n\n\\bigskip\n\n\n\n\nIn a similar way for $s \\geq 1$ define the function $h^{[s]}$ on $[0,1]^s$ given by:\n\\begin{eqnarray*}\n& & h^{[s]}(u^{(1)},\\cdots,u^{(s)}) \\\\ & =& 100 \\cdot \\prod_{i=1}^s \\cos\\left(\\frac{ \\pi u^{(i) }}{2 i^{1\/2}} \\right)\n\\end{eqnarray*}\n\n\\bigskip\n\nWe have:\n\\begin{eqnarray}\n& & \\int_{[0,1]^s} h^{[s]} \\, d \\mu_{ST}^{[s]} \\\\ &=& 100 \\cdot \\prod_{i=1}^s \\int_{[0,1]} \\cos\\left(\\frac{\\pi u^{(i)}}{2 i^{1\/2}}\\right) \\cdot 2 \\sin^2(\\pi u^{(i)})\\, d u^{(i)} \\nonumber \\\\\n &=& 100 \\cdot \\prod_{i=1}^s \\left( \\frac{2 i^{1\/2}}{\\pi} \\cdot \\sin\\left( \\frac{\\pi}{2 i^{1\/2}}\\right) \\cdot \\frac{16 i}{16i-1} \\right) \\nonumber\n\\end{eqnarray}\n\n\\bigskip\nThe numerical values of $\\int_{[0,1]^s} h^{[s]} \\, d \\mu_{ST}^{[s]}$ for $s=500$, $s=1000$, $s=1500$, and $s=2000$, are computed as in (2.3) to be:\n\n\\begin{eqnarray*}\n\\int_{[0,1]^{500}} h^{[500]} \\, d \\mu_{ST}^{[500]} \\stackrel{.}{=} 8.814\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1000}} h^{[1000]} \\, d \\mu_{ST}^{[1000]} \\stackrel{.}{=} 6.92239\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1500}} h^{[1500]} \\, d \\mu_{ST}^{[1500]} \\stackrel{.}{=}6.0099 \n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{2000}} h^{[2000]} \\, d \\mu_{ST}^{[2000]} \\stackrel{.}{=} 5.43635\n\\end{eqnarray*}\n\n\\bigskip\n\nWhile the numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[s]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$, in the cases $s=500$, $s=1000$, $s=1500$, and $s=2000$ respectively, are tabulated in Figures 5, 6, 7, 8 below.\n\n\n\\bigskip\n\\bigskip\n\n\nSumming up: the numerical results of this section supply evidences for the validity of Conjecture 1.1; this conjecture can be described as saying that, the sequence $\\{x_k\\}_{k \\geq 1}$ is a pseudorandom sequence in $[0,1]$ with distribution law given by the Sato-Tate measure. \n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$&$8.84564$&$ 8.8739$&$ 8.85475$&$ 8.82749$& $8.81591$\\\\\n $E_2$&$8.74331$&$ 8.7891$&$ 8.80022$&$ 8.78814$&$ 8.7736$\\\\\n $E_3$&$8.70232$&$ 8.63423$&$ 8.67626$&$ 8.79667$&$ 8.77114$\\\\\n $E_4$&$8.49599$&$ 8.54784$&$ 8.62823$&$ 8.69612$&$ 8.76489$\\\\\n $E_5$&$8.27357$&$ 8.53379$&$ 8.69698$&$ 8.67127$&$ 8.70012$\\\\\n $E_6$&$8.18894$&$ 8.50111$&$ 8.45486$&$ 8.70109$&$ 8.69085$\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[500]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n$E_1$&$6.9513$&$ 6.97341$&$ 6.95763$&$ 6.93425$&$ 6.9241$\\\\\n $E_2$&$6.86542$&$ 6.90182$&$ 6.91247$&$ 6.90096$&$ 6.8879$\\\\\n $E_3$&$6.82891$&$ 6.77459$&$ 6.80619$&$ 6.90821$&$ 6.8863$\\\\\n $E_4$&$6.65655$&$ 6.69763$&$ 6.76587$&$ 6.822$&$ 6.88$\\\\\n $E_5$&$6.47181$&$ 6.67989$&$ 6.82385$&$ 6.80146$&$ 6.8251$\\\\\n $E_6$&$6.40304$&$ 6.65958$&$ 6.61765$&$ 6.82616$&$ 6.8182$\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[1000]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$&$ 6.03675 $&$ 6.0559 $&$ 6.04206 $&$ 6.02086 $&$ 6.01151 $\\\\\n $E_2$&$ 5.96016 $&$ 5.99199 $&$ 6.00129 $&$ 5.99081 $&$ 5.97872 $\\\\\n $E_3$&$ 5.9214 $&$ 5.87986 $&$ 5.90473 $&$ 5.9971 $&$ 5.97718 $\\\\\n $E_4$&$ 5.77546 $&$ 5.80653 $&$ 5.86934 $&$ 5.91933 $&$ 5.97276 $\\\\\n $E_5$&$ 5.60953 $&$ 5.7892 $&$ 5.92279 $&$ 5.90058 $&$ 5.92183 $\\\\\n $E_6$&$ 5.54555 $&$ 5.77334 $&$ 5.73411 $&$ 5.92231 $&$ 5.91537 $\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[1500]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$ & $ K=10000$ & $ K=20000$ & $K=50000 $ & $K=100000$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$&$5.46061$&$ 5.47935$&$ 5.46676$&$ 5.4467$&$ 5.4379$\\\\\n $E_2$&$5.39035$&$ 5.41982$&$ 5.42836$&$ 5.41881$&$ 5.4073$\\\\\n $E_3$&$5.3513$&$ 5.31787$&$ 5.33875$&$ 5.42458$&$ 5.4058$\\\\\n $E_4$&$5.2241$&$ 5.24905$&$ 5.30693$&$ 5.35259$&$ 5.4022$\\\\\n $E_5$&$5.06843$&$ 5.23096$&$ 5.35741$&$ 5.33487$&$ 5.3545$\\\\\n $E_6$&$5.01111$&$ 5.21909$&$ 5.18121$&$ 5.35458$&$ 5.3483$\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[2000]}(X_k)$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\\section{Numerical Evidences for Conjecture 1.3, part I}\n\nFor numerics related to Conjecture 1.3 in the case $s=1$ (i.e. the original conjecture of Akiyama-Tanigawa), we refer to \\cite{AT} and \\cite{St}. To test Conjecture 1.3 directly one would need to evaluate the values the $D_K^{[s]}$ or $D_K^{*,[s]}$ for $K$ large. But in higher dimensions $s$ there are serious combinatorial difficulties in computing (even just numerically) the values of $D_K^{[s]}$ or $D_K^{*,[s]}$; this is the well known phenomenon known as the {\\it Curse of Dimensionality}. \n\n\\bigskip\n\nWe first note that, Conjecture 1.3 is obviously equivalent to the statement:\n\\[\n\\liminf_{K \\rightarrow \\infty} - \\frac{\\ln D^{[s]}_K}{\\ln K} \\geq \\frac{1}{2}\n\\]\n(and similarly with $D_K^{[s]}$ being replaced by $D_K^{*,[s]}$).\n\n\\bigskip\n\\begin{proposition}\nLet $f$ be a function defined on $[0,1]^s$, which is of bounded variation in the sense of Hardy and Krause. Then Conjecture 1.3 implies:\n\\[\n\\liminf_{K \\rightarrow \\infty} - \\frac{\\ln \\big| \\frac{1}{K}\\sum_{k=1}^K f(X_k) - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} \\big|}{\\ln K} \\geq \\frac{1}{2}\n\\]\n\\end{proposition}\n\\begin{proof}\nThis is an immediate consequence of the Koksma-Hlawka inequality ({\\it c.f.} p. 151 of \\cite{KN} and p. 967 of \\cite{Ni2}):\n\\[\n\\Big| \\frac{1}{K}\\sum_{k=1}^K f(X_k) - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} \\Big| \\leq V(f) \\cdot D_K^{*,[s]}\n\\]\nwhere $V(f)$ is the total variation of $f$ in the sense of Hardy and Krause (the version of the Koksma-Hlawka inequality stated in {\\it loc. cit.} is with respect to the Lebesgue measure on $[0,1]^s$, but the same proof works verbatim with respect to the measure $\\mu_{ST}^{[s]}$ on $[0,1]^s$).\n\\end{proof}\n\n\\bigskip\nIn view of Proposition 3.1. we may then test Conjecture 1.3 indirectly as follows. With $f$ defined on $[0,1]^s$ as above (of bounded variation in the sense of Hardy and Krause), denote the relative error:\n\n\\[\n\\OP{RelErr}(f,K) = \\frac{\\frac{1}{K}\\sum_{k=1}^K f(X_k) - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} }{ \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} }\n\\]\n\n\\bigskip\n\\noindent (assuming that the integral is nonzero). \n\n\\bigskip\n\nThen we evaluate:\n\n\\[\n-\\frac{\\ln |\\OP{RelErr} (f,K)|}{\\ln K}\n\\]\nwith $K$ being large. By virtue of Proposition 3.1, Conjecture 1.3 implies that\n\n\\[\n\\liminf_{K \\rightarrow \\infty} \\frac{-\\ln |\\OP{RelErr}(f,K) |}{\\ln K} \\geq \\frac{1}{2}\n\\]\n\n\\bigskip\n\\bigskip \n\nIn the following numerical examples the dimensions $s$ and the test functions $f$ on $[0,1]^s$ are chosen as in section 2, namely: \n\n\\[\nf^{[10]},g^{[500]},g^{[1000]},g^{[2000]},h^{[500]},h^{[1000]},h^{[1500]}, h^{[2000]}\n\\]\n\n\\bigskip\nThe results are tabulated in Figures 9 -16 below.\n\n\\bigskip\n\\bigskip\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.69949&0.679386&0.630927&0.66101&0.691336 \\\\ \n $E_2$ & 0.633969&0.581946&0.687151&0.641671&0.670621\\\\\n\n $E_3$ & 0.573905&0.57178&0.585748&0.576866&0.592804\\\\\n\n $E_4$ & 0.523103&0.542605&0.531523&0.566703&0.525969\\\\\n\n $E_5$ & 0.514324&0.516659&0.513771&0.50841&0.504189\\\\\n $E_6$ & 0.494667&0.524282&0.505647&0.490742&0.491219\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (f^{[10]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.638723& 0.648979& 0.631947& 0.862372& 0.808676 \\\\ \n $E_2$ & 0.575948&0.505878&0.633176&0.552754&0.543251 \\\\\n\n $E_3$ & 0.487867&0.496663&0.514921&0.501386&0.498532 \\\\\n\n $E_4$ & 0.444677&0.462&0.462372&0.493716&0.460063 \\\\\n\n $E_5$ & 0.431359&0.436573&0.438725&0.445434&0.44006 \\\\\n $E_6$ & 0.404953&0.436986&0.430817&0.421285&0.426719 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[500]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\\newpage\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.631169&0.645835&0.630361&0.712336&0.706233 \\\\ \n $E_2$ & 0.567706&0.497695&0.617072&0.544564&0.534925 \\\\\n\n $E_3$ & 0.480746&0.489362&0.507584&0.494107&0.492241 \\\\\n\n $E_4$ & 0.437471&0.455814&0.455769&0.487224&0.453975 \\\\\n\n $E_5$ & 0.424289&0.429748&0.432216&0.439139&0.434175 \\\\\n $E_6$ & 0.39818&0.430663&0.424525&0.414925&0.420545 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[1000]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.627366& 0.638355&0.622921&0.675519&0.66696 \\\\ \n $E_2$ & 0.559385&0.490252&0.605272&0.53634&0.526713 \\\\\n\n $E_3$ & 0.473656&0.482631&0.500549&0.487117&0.485837 \\\\\n\n $E_4$ & 0.430963&0.449776&0.449301&0.480237&0.447863\\\\\n\n $E_5$ & 0.418162&0.423543&0.426154&0.4331&0.428402 \\\\\n $E_6$ & 0.391767&0.424534&0.418375&0.408966&0.414847 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[2000]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.606447& 0.634717&0.596622&0.769728&0.72325 \\\\ \n $E_2$ & 0.592215&0.483297&0.643853&0.573784&0.538485 \\\\\n\n $E_3$ & 0.469397&0.46504&0.491604&0.482104&0.506944 \\\\\n\n $E_4$ & 0.428098&0.451091&0.44635&0.467237&0.44823 \\\\\n\n $E_5$ & 0.417505&0.421578&0.424573&0.429395&0.423905\\\\\n $E_6$ & 0.398343&0.426324&0.412744&0.399427&0.40276 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[500]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\\newpage\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.599239&0.632049&0.594434&0.699701&0.677617 \\\\ \n $E_2$ & 0.58362&0.4755&0.618896&0.561171&0.528059 \\\\\n\n $E_3$ & 0.462718&0.458196&0.484627&0.474634&0.499336 \\\\\n\n $E_4$ & 0.421188&0.445555&0.440249&0.46107&0.442045\\\\\n\n $E_5$ & 0.410704&0.414996&0.418179&0.423116&0.41825\\\\\n $E_6$ & 0.391969&0.420306&0.406692&0.393337&0.396837 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[1000]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.596406 & 0.624722 & 0.588404 & 0.675777 & 0.659381 \\\\ \n $E_2$ & 0.578855 & 0.471502 & 0.612237 & 0.555763 & 0.523419 \\\\\n\n $E_3$ & 0.458443 & 0.454467 & 0.480825 & 0.470964 & 0.495892 \\\\\n\n $E_4$ & 0.41748 & 0.442316 & 0.436737 & 0.457416 & 0.438645 \\\\\n\n $E_5$ & 0.40721 & 0.411428 & 0.414829 & 0.419795 & 0.415218 \\\\\n $E_6$ & 0.388235 & 0.416779 & 0.403099 & 0.390014 & 0.393701 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[1500]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\\bigskip\n\\bigskip\n\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$ & $ K= 10^6$ & $ K= 2 \\times 10^6$ & $K=5 \\times 10^6 $ & $K=10^7$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.594196&0.620986&0.584018&0.670973&0.654911 \\\\ \n $E_2$ & 0.574501&0.468545&0.607293&0.55214&0.520405 \\\\\n\n $E_3$ & 0.456099&0.452267&0.478601&0.468548&0.493642\\\\\n\n $E_4$ & 0.415082&0.440149&0.434365&0.45504&0.436376\\\\\n\n $E_5$ & 0.404872&0.409085&0.412494&0.417499&0.413082\\\\\n $E_6$ & 0.385872&0.414499&0.40085&0.387842&0.391689 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[2000]},K)|}{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\\section{Numerical Evidences for Conjecture 1.3, part II}\n\n\n\n\nFinally we test Conjecture 1.3 directly for the dimensions $s=2$ and $s=3$. We first recall the following general result of Niederreiter \\cite{Ni1} in order to compute the star discrepancy $D_K^{*,[s]}$ of $\\{X_k\\}_{k=1}^K \\subset [0,1]^s$ (with respect to the measure $\\mu_{ST}^{[s]}$).\n\n\\bigskip\n\nFor $1 \\leq i \\leq s$, denote by $0= \\beta^{(i)}_1 < \\cdots <\\beta^{(i)}_{n_i}=1$ the set of distinct values of the set of $i$-th coordinates of the points $\\{X_k\\}_{k=1}^K$, with the values $0$ and $1$ being included. Denote by $\\mathfrak{q}$ the collection of rectangular regions $Q \\subset [0,1]^s$ of the form:\n\\[\nQ = \\prod_{i=1}^s ( \\beta^{(i)}_{j_i} , \\beta^{(i)}_{j_i +1} ], \\,\\ 1 \\leq j_i < n_i \\mbox{ for } 1 \\leq i \\leq s\n\\]\nwhich thus forming a partition of $(0,1]^s$. For $Q$ as above, denote:\n\\[\nY(Q) = (\\beta^{(1)}_{j_1 +1},\\cdots, \\beta^{(s)}_{j_s +1} ) \\in [0,1]^s\n\\]\nthe upper end point of $Q$, and\n\\[\nZ(Q) = (\\beta^{(1)}_{j_1},\\cdots, \\beta^{(s)}_{j_s} ) \\in [0,1]^s \n\\]\nthe lower end point of $Q$. \n\n\n\n\n\\bigskip\nWe then have:\n\\begin{proposition} The star discrepancy $D_K^{*,[s]}$ is equal to:\n\\begin{eqnarray*}\n\\max_{Q \\in \\mathfrak{q}} \\Big(\\max \\Big( \\Big| \\frac{A( \\mathcal{R}_{Y(Q)};K) }{K} - \\mu_{ST}^{[s]}( \\mathcal{R}_{Y(Q)} ) \\Big|, \\Big| \\frac{A(\\mathcal{R}_{Z(Q)} ;K)}{K} - \\mu_{ST}^{[s]}( \\mathcal{R}_{Z(Q)} ) \\Big| \\Big) \\Big) \n\\end{eqnarray*}\n\\end{proposition}\n\\begin{proof}\nThis is Theorem 2 of \\cite{Ni1}. In {\\it loc. cit.} it is stated with respect to the Lebesgue measure on $[0,1]^s$, but the argument works verbatim with respect to the measure $\\mu_{ST}^{[s]}$.\n\\end{proof}\n\n\\bigskip\n\\begin{remark}\n\\end{remark}\n\\noindent To compute $D_K^{*,[s]}$ using Proposition 4.1 ({\\it i.e.} Theorem 2 of \\cite{Ni1}) requires the evaluation of $O(K^s)$ terms. Although there are algorithms that improve upon that of \\cite{Ni1} for computing the star discrepancy ({\\it c.f.} for example \\cite{DE}), the time complexity of the known algorithms is still exponential in terms of the dimension $s$; this is an instance of the {\\it Curse of Dimensionality}. In fact it is known that the computation of star discrepancy belongs to the class of NP-hard problems \\cite{GSW}.\n\n\\bigskip\n\\bigskip\n\nBelow we use Proposition 4.1 to compute the numerical values of $D_K^{*,[s]}$ and hence test Conjecture 1.3 (with respect to $D_K^{*,[s]}$), in the cases $s=2$ and $s=3$. The results are tabulated in Figures 17, 18 below.\n\n\\bigskip\nSumming up: the numerical results of section 3 and section 4 supply evidences for the validity of Conjecture 1.3. It is a refinement of Conjecture 1.1, and can be regarded as a qualitative form, of the Law of Iterated Logarithm for random numbers ({\\it c.f.} Chapter 7 of \\cite{Ni3}). In particular, Conjecture 1.3 implies that, with respect to the Sato-Tate measure, the sequence of Frobenius angles of an elliptic curve over $\\mathbf{Q}$ without complex multiplication forms a pseudorandom sequence in $[0,1]$ with strong randomness property, (at least) as far as statistical distribution is concerned.\n\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^3$ & $ K= 10^4$ & $ K= 2 \\times 10^4$ & $K=5 \\times 10^4 $ & $K=10^5$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.513743 & 0.481735 & 0.493825 & 0.508233 & 0.506597\\\\ \n $E_2$ & 0.506241 & 0.511887 & 0.468917 & 0.494157 & 0.492688\\\\\n\n $E_3$ & 0.484667 & 0.483204 & 0.484577 & 0.526097 & 0.487613\\\\\n\n $E_4$ & 0.442152 & 0.418237 & 0.434046 & 0.440515 &0.467903\\\\\n\n $E_5$ & 0.423393 & 0.443996 & 0.46427 & 0.441762 & 0.42277\\\\\n $E_6$ & 0.413569 & 0.421813 & 0.419844 & 0.458051 &0.426849\\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln D_K^{*,[2]} }{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering \n\t\n\t\\begin{tabular}{c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^3$ \\\\\n\t\t\n\t\t\\midrule \n $E_1$ & 0.495306 \\\\ \n $E_2$ & 0.479892 \\\\\n\n $E_3$ & 0.472156 \\\\\n\n $E_4$ & 0.413948 \\\\\n\n $E_5$ & 0.405477 \\\\\n $E_6$ & 0.39623 \\\\\n \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n \\caption{Numerical results for $-\\frac{\\ln D_K^{*,[3]} }{\\ln K}$} \n \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Final Remarks}\n\nIn this paper we propose conjectures that refine the Sato-Tate conjecture, specifically we conjecture that the Frobenius angles of a given elliptic curve over $\\mathbf{Q}$ without complex multiplication, are statistically independently distributed with respect to the Sato-Tate measure, including the more quantitative version involving the discrepancy of joint distributions. Numerical evidences are presented to support the conjectures. \n\n\\bigskip\n\nTaylor {\\it et. al.} \\cite{CHT,T,HSBT,BLGHT} had established the Sato-Tate conjecture for elliptic curves over totally real fields without complex multiplication, and more generally \\cite{BLGG} established the Sato-Tate conjecture for Hilbert modular forms over totally real fields ({\\it c.f.} \\cite{ACC+} for the latest results on the Sato-Tate conjecture for automorphic forms over number fields). Thus it is natural to expect that our Conjectures 1.1 and 1.3 extend to the more general setting as well. \n\n\\bigskip\nIt would be intriguing to find possible connections between Conjecture 1.3 in the case $s \\geq 2$ and properties of $L$-functions.\n\n\\bigskip\n\nFinally and most interestingly, as observed experimentally from the numerics, the rate of convergence to the measure $\\mu_{ST}^{[s]}$, is slower in the case of curves with higher Mordell-Weil ranks (in the one dimensional case $s=1$ this was already observed in \\cite{St}). Heuristically, in accordance with the original form of the Birch and Swinnerton-Dyer conjecture, this can be seen as due to the fact that, for curves of high Mordell-Weil rank, there is a Chebyshev bias for the quantities $a_{p_k}$ towards being negative, {\\it c.f.} \\cite{M}, \\cite{S}, \\cite{KM}. It would be important to understand the rate of convergence to the measure $\\mu_{ST}^{[s]}$ in a more precise form (for example along the lines suggested in \\cite{St} in the case $s=1$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{State Counting in Fermi Liquids}\n\nIn a non-interacting system, the number of low-energy addition states\nper electron per spin is equal to one. Should the number of low-energy\naddition states per electron per spin exceed unity, Fermi liquid\ntheory fails and new electronic states emerge at low energy that\ncannot be constructed from the non-interacting system. To show that\nthis state of affairs obtains in a doped Mott insulator, we compare\nthe number of electrons per site ($n_h$) that can be added to the\nholes created by the dopants with the number of single-particle\n addition states per site at low energy,\n\\begin{eqnarray}\\label{dos}\nL=\\int_\\mu^\\Lambda N(\\omega)d\\omega,\n\\end{eqnarray}\ndefined as the integral of the single-particle density of states\n($N(\\omega)$) from the chemical potential, $\\mu$, to a cutoff energy\nscale, $\\Lambda$, demarcating the IR and UV scales. \nConsider first\nthe case of a Fermi liquid or non-interacting system. As illustrated\nin Fig. (\\ref{fig1}), the total weight of the valence band is 2, that\nis, there are 2 states per site. The integrated weight of the valence\nband up to the chemical potential determines the filling.\nConsequently, the unnocupied part of the spectrum, which determines $L$,\nis given by $L=2-n$. The number of electrons that can be added to the\nempty sites is also $n_h=2-n$ (see Fig. (\\ref{fig2})). Consequently, the number\nof low-energy states per electron per spin is identically unity. The\nkey fact on which this result hinges is that the total weight of the\nvalence band is a constant independent of the electron density. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9.0cm]{mhtransfer.eps}\n\\caption{Evolution of the single-particle density of states from\n half-filling to the one-hole limit in a doped Mott insulator\n described by the Hubbard model. Removal of an electron results\n in two empty states at low energy as opposed to one in the\n band-insulator limit. The key difference with the Fermi liquid is\n that the total weight spectral weight carried by the lower Hubbard\n band (analogue of the valence band in a Fermi liquid) is not a\n constant but a function of the filling. }\n\\label{fig2}\n\\end{figure}\n\n\\section{Doped Mott Insulators: Not just electrons}\n\nFor a doped Mott insulator, the situation is quite different as a\ncharge gap splits the spectrum into lower and upper Hubbard bands (LHB\nand UHB, hereafter)\ndepicted in Fig. (\\ref{fig2}). At half-filling the chemical potential\nlies in the gap. The sum rule that 2 states exist per site \napplies only to the combined weight of both bands. At any finite\ndoping,\nthe weight in the LHB and UHB is determined by the density. For\nexample, at half-filling each carries half the spectral weight. Even\nin the atomic limit, the spectral weights in the LHB and UHB are\ndensity dependent as shown in Fig. (\\ref{fig2}).\nNonetheless, for a doped Mott insulator in the atomic limit, i.e. one\nelectron per site with infinite on-site repulsion $U$, it is still true that $L\/n_h=1$, because creating a hole leaves behind an empty site which\ncan be occupied by either a spin-up or a spin-down electron. Hence,\nwhen $x$ electrons (see Fig. (\\ref{dos})) are removed, $L=n_h=2x=2-2n$\nin the atomic limit. Recall for a Mott system, $x=1-n$ as the hole\ndoping occurs relative to half-filling. Hence, for a Mott system in\nthe atomic limit, the total weight in the LHB increases from 1\/2 at\nhalf-filling to $1-x+2x=2-n$ as the system is doped. This result\nillustrates that the total weight in the LHB goes over smoothly to the\nnon-interacting limit when $n=0$. That is, 2 states exist per site at\nlow energy entirely in the LHB. The change from half the spectral\nweight at $n=1$ to all the spectral weight residing in the LHB at\n$n=0$ is a consequence of spectral weight transfer. The atomic limit,\nhowever, only captures the static (state counting) part of the\nspectral weight. In fact the $2x$ sum rule,in which $L\/n_h=1$, is\ncaptured by the widely used\\cite{lee,lee1,lee2} $t-J$ model of a doped\nMott insulator in which no doubly occupied sites are allowed. However, real Mott systems are not in the\natomic limit. Finite hopping with matrix element $t$ creates double\noccupancy, and as a result empty sites with weight $t\/U$. Such empty sites with fractional weight contribute to $L$. Consequently,\nwhen $01$. Consequently, in contrast to a Fermi liquid, simply counting\nthe number of electrons that can be added does {\\it not} exhaust the\navailable phase space to add an electron at low energy. Thus,\nadditional degrees of freedom at low-energy, not made out of the elemental\nexcitations, must exist. They arise from the hybridization with the doubly occupied sector\nand hence must emerge at low energy from a collective charge 2e\nexcitation. The new charge $e$ state that emerges at low energy must correspond to a\nbound state of the collective charge $2e$ excitation and the hole that\nis left behind. It is the physics of this new charge $e$ state that\nmediates the non-Fermi liquid behaviour in a doped Mott insulator.\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.0cm,angle=-90]{dos.eps}\n\\caption{a) Integrated low-energy spectral\nweight, $L$, defined in Eq. (\\ref{dos}), as a function of the\nelectron filling, n: 1) the dashed line is the non-interacting limit, vanishing on-site\ninteraction ($U=0$), in which $L=2-n$, 2) atomic limit\n(blue line) of a doped\nMott insulator, $U=\\infty$, in which $L=2(1-n)=2x$, $x$ the doping\nlevel and 3) a real Mott insulator in which $02x$ away from the\natomic limit. (b) Hopping processes mediated by the $t\/U$ terms in\nthe expansion of the projected transformed operators in terms of the bare electron operators (see Eq. (\\ref{trans})). As a\nresult of the $t\/U$ terms in Eq. (\\ref{eq:sc1}), the low-energy\ntheory in terms of the bare fermions does not preserve double\noccupancy. The process\nshown here illustrates that mixing between the high and low-energy\nscales obtains only if double occupancy neighbours a hole. In\nthe exact low-energy theory, such processes are mediated by the new\ndegree of freedom, $\\varphi_i$, the charge $2e$ bosonic field which\nbinds a hole and produces a new charge $e$ excitation, the collective\nexcitation in a doped Mott insulator. }\n\\label{dos}\n\\end{figure}\n\nIn a series\\cite{lowen1} of recent papers, we found the collective charge mode in a doped Mott insulator by integrating out exactly\nthe high-energy scale in the Hubbard model, thereby obtaining an exact description of the IR physics. Consistent with the physical argument presented above, the collective mode is a\ncharge $2e$ bosonic field which mediates new electron dynamics at low\nenergy and is not made out of the elemental excitations in the UV. In\nan attempt to clarify this theory, we establish the relationship\nbetween the standard perturbative approach and the physics mediated by the charge $2e$ boson. By comparing how the operators\ntransform in both theories, we are able to unambiguously associate\nthe charge $2e$ boson with dynamical (hopping-dependent) spectral weight\ntransfer across the Mott gap\\cite{diag,eskes,slavery}. Further, the\nspectral weight transfer is mediated by a bound excitation of the\ncharge $2e$ boson and a hole in accord with the physical argument\npresented above.\n\n\\section{Standard Approach}\n\nOf course the standard approach for treating the fact that $L\/n_h>1$\nin a doped Mott insulator is through perturbation theory, not by explicitly constructing the missing degree of freedom. The goal of the perturbative approach is to bring the Hubbard model\n\\begin{eqnarray}\nH&=&-t\\sum_{i,j,\\sigma} g_{ij}\na^\\dagger_{i,\\sigma}a_{j,\\sigma}+\nU\\sum_i a^\\dagger_{i,\\uparrow}a^\\dagger_{i,\\downarrow}a_{i,\\downarrow}a_{i,\\uparrow}\n\\end{eqnarray}\ninto block diagonal form in which each block has a fixed number of `fictive' doubly occupied sites. Here $i,j$ label lattice sites,\n$g_{ij}$ is equal to one iff $i,j$ are nearest neighbours and $a_{i,\\sigma}$\nannihilates an electron with spin $\\sigma$ on lattice site $i$. We say `fictive' because the operators which make\ndouble occupancy a conserved quantity are not the physical electrons but\nrather a transformed (dressed) fermion we call $c_{i\\sigma}$\ndefined below.\nFollowing Eskes et al.\\cite{eskes},\nfor any operator $O$, we define $\\tilde O$ such that\n$ O\\equiv {\\bf O}(a)$ and $\\tilde{O}\\equiv {\\bf O}(c)$,\nsimply by replacing the Fermi operators $a_{i\\sigma}$ with the\ntransformed fermions $c_{i\\sigma}$. Note that $O$ and $\\tilde O$ are only\nequivalent in the $U=\\infty$ limit. The procedure which makes the Hubbard model\nblock diagonal is now well known\\cite{eskes,spalek,sasha,girvin,anderson}.\nOne constructs a similarity\ntransformation $S$ which connects sectors that differ by at most one\n`fictive' doubly occupied site such that\n\\begin{eqnarray}\nH=e^S\\tilde H e^{-S}\n\\end{eqnarray}\nbecomes block diagonal, where $\\tilde H$ is expressed in terms of the transformed fermions. In the new basis,\n$[H,\\tilde V]=0$,\nimplying that double occupation of the transformed fermions\nis a good quantum number, and all of the eigenstates\ncan be indexed as such. This does not mean that $[H,V]=0$. If it\nwere, there would have been no reason to do the similarity transformation in\nthe first place. $\\tilde V$, and\nnot $V$, is conserved. Assuming that $V$ is the conserved\nquantity results in a spurious local SU(2)\\cite{lsu21,lsu22}\nsymmetry in the strong-coupling limit at\nhalf-filling.\n\nOur focus is on the relationship between the physical and `fictive'\nfermions. To leading order\\cite{eskes} in $t\/U$, the bare fermions,\n\\begin{eqnarray}\na_{i\\sigma}&=&e^Sc_{i\\sigma}e^{-S}\n\\simeq c_{i\\sigma}-\\frac{t}{U}\\sum_{\\langle j, i\\rangle}\n\\left[(\\tilde n_{j\\bar\\sigma}-\\tilde n_{i\\bar\\sigma})c_{j\\sigma}\\right.\\nonumber\\\\\n&-&\\left.c^\\dagger_{j\\bar\\sigma}c_{i\\sigma}c_{i\\bar\\sigma}+c^\\dagger_{i\\bar\\sigma}c_{i\\sigma}c_{j\\bar\\sigma}\\right],\n\\end{eqnarray}\nare linear combinations of\nmultiparticle states in the transformed basis as is expected in\ndegenerate perturbation theory\nBy inverting this relationship, we find that to leading order, the transformed operator is simply,\n\\begin{eqnarray}\nc_{i\\sigma}\\simeq a_{i\\sigma}+\\frac{t}{U}\\sum_{j} g_{ij}X_{ij\\sigma}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_{ij\\sigma}=\\left[(n_{j\\bar\\sigma}-n_{i\\bar\\sigma})a_{j\\sigma}-a^\\dagger_{j\\bar\\sigma}a_{i\\sigma}a_{i\\bar\\sigma}+a^\\dagger_{i\\bar\\sigma}a_{i\\sigma}a_{j\\bar\\sigma}\\right].\n\\end{eqnarray}\nWhat we would like to know is what do the transformed fermions look\nlike in the lowest energy sector. We accomplish this by computing\nthe projected operator\n\\begin{eqnarray}\n(1-\\tilde n_{i\\bar\\sigma})c_{i\\sigma}&\\simeq &(1-n_{i\\bar\\sigma})a_{i\\sigma}+\\frac{t}{U}\\sum_{j}g_{ij}\\left[\n(1-n_{i\\bar\\sigma})X_{ij\\sigma}\\right.\\nonumber\\\\\n&&\\left.-X^\\dagger_{ij\\bar\\sigma}a_{i\\bar\\sigma}a_{i\\sigma}-a^\\dagger_{i\\bar\\sigma}X_{ij\\bar\\sigma}a_{i\\sigma}\\right].\n\\end{eqnarray}\nSimplifying, we find that\n\\begin{eqnarray}\\label{trans}\n(1-\\tilde n_{i\\bar\\sigma})c_{i\\sigma}&\\simeq &(1-n_{i\\bar\\sigma})a_{i\\sigma}+\\frac{t}{U}V_\\sigma\na_{i\\bar\\sigma}^\\dagger b_i\\nonumber\\\\\n&+&\\frac{t}{U}\\sum_{j}g_{ij}\\left[\nn_{j\\bar\\sigma}a_{j\\sigma}+n_{i\\bar\\sigma}(1-n_{j\\bar\\sigma})a_{j\\sigma}\\right.\\nonumber\\\\\n&&\\left.+(1-n_{j\\bar\\sigma})\\left(a_{j\\sigma}^\\dagger\na_{i\\sigma}-a_{j\\sigma}a^\\dagger_{i\\sigma}\\right)a_{i\\bar\\sigma}\\right].\n\\end{eqnarray}\nHere $V_\\sigma=-V_{\\bar\\sigma}=1$ and\n$b_i= \\sum_{j\\sigma} V_\\sigma c_{i\\sigma}c_{j\\bar\\sigma}$ where $j$ is summed over the nearest neighbors of $i$.\nAs is evident, the projected `fictive' fermions involve the projected\nbare fermion, $(1-n_{i\\bar\\sigma})a_{i\\sigma}$, which yields the $2x$ sum\nrule plus admixture with the doubly occupied sector\nmediated by the $t\/U$ corrections. These $t\/U$ terms, which are\nentirely local and hence cannot be treated at the mean-field level, generate the $>2x$ or\nthe dynamical part of the spectral weight transfer. This physics (which has been shown to play a significant role even at half-filling\\cite{trem}) is absent from projected models such as the standard implementation\\cite{lee,lee1,lee2} of the $t-J$ model\nin which double occupancy is prohibited. As we have pointed out in the introduction, the physics left out by projecting out double occupancy is important because it tells us immediately that $L\/n_h>1$; that is, new degrees of freedom must be present at low energy. Put another way, the operator in Eq. (\\ref{trans}) is not a free excitation and but rather describes a non-Fermi liquid ($L\/n_h>1$). A process mediated by the\nlast term in Eq. (\\ref{trans}), depicted in Fig. (\\ref{fig1}), obtains only\nif a doubly occupied and empty site are neighbors. This underscores\nthe fact that in Mott systems, holes can be heavily dressed by the upper\nHubbard band. It is this dressing that generates dynamical\nspectral weight transfer.\n\nBefore we demonstrate how a single collective degree of freedom describes\nsuch dressing, we focus on the low-energy Hamiltonian in the bare\nelectron basis. The answer in the transformed basis\nis well-known\\cite{eskes} and involves\nthe spin-exchange term as well as the three-site hopping term. Our\ninterest\nis in what this model corresponds\nto in terms of the bare electron operators which do not preserve\ndouble occupancy. To accomplish this, we simply undo the\nsimilarity transformation after we have projected the transformed\ntheory onto the lowest energy sector. Hence, the quantity of interest\nis $H_{sc}=e^{-S}P_0e^SHe^{-S}P_0e^S$. Of course, without projection, the\nanswer in the original basis at each order of perturbation theory\nwould simply be the Hubbard\nmodel. However, the question at hand is what does the low-energy\ntheory look like in the original electron basis. Answering this\nquestion is independent of the high energy sectors in the transformed\nbasis because all such subspaces lie at least $U$ above the $m=0$\nsector. Hence, it is sufficient to focus on $P_0e^SHe^{-S}P_0$. To\nexpress $P_0e^SHe^{-S}P_0$ in the bare electron operators, we substitute\nEq. (\\ref{trans}) into the first of Eqs. (14) of Eskes, et\nal.\\cite{eskes} to obtain\n\\begin{widetext}\n\\begin{eqnarray} H_{sc}&=&e^{-S}P_0e^SHe^{-S}P_0e^S\\nonumber\\\\\n &=& -t\\sum_{\\langle i,j\\rangle}\\xi_{i\\sigma}^{\\dagger}\\xi_{j\\sigma}-\\frac{t^{2}}{U}\\sum_{i}b_{i}^{(\\xi)\\dagger}b_{i}^{(\\xi)}\n-\\frac{t^{2}}{U}\\sum_{\\langle i,j\\rangle,\\langle\ni,k\\rangle,\\sigma}\\left\\{\n\\xi_{k\\sigma}^{\\dagger}\\left[(1-n_{i\\bar{\\sigma}})\\eta_{j\\sigma}+\\xi_{j\\bar{\\sigma}}^{\\dagger}\\xi_{i\\bar{\\sigma}}\\eta_{i\\sigma}+\\xi_{i\\bar{\\sigma}}^{\\dagger}\\xi_{i\\sigma}\\eta_{j\\bar{\\sigma}}\\right]+h.c.\\right\\}\n\\label{eq:sc1}\n\\end{eqnarray}\n\\end{widetext}\nas the low-energy theory in terms of the original electron\noperators. Here, $\\xi_{i\\sigma}=a_{i\\sigma}(1-n_{i\\bar\\sigma})$ and\n$\\eta_{i\\sigma}=a_{i\\sigma}n_{i\\bar\\sigma}$. The first two terms correspond to the $t-J$ model in\nthe bare electron basis plus 3-site hopping. However, terms in the bare-electron basis\nwhich do not preserve the number of doubly\noccupied sites explicitly appear. As\nexpected, the\nmatrix elements which connect sectors which differ by a single doubly\noccupied site are reduced from the bare hopping $t$ to $t^2\/U$. All\nsuch terms arise from the fact that the transformed and bare electron\noperators differ at finite $U$. Hence, Eq. (\\ref{eq:sc1}) makes\ntransparent that the standard\nimplementation\\cite{lee,lee1,lee2} of the $t-J$ model in which the transformed and bare\nelectron operators are assumed equal is inconsistent because the terms\nwhich are dropped are precisely of the same order, namely $O(t^2\/U)$,\nas is the spin-exchange.\n\n\\section{New Approach}\n\nExplicitly integrating out\\cite{lowen1} the high-energy scale in the\nHubbard model uncloaks the collective degree of freedom that accounts\nfor the key difference between the bare and\ntransformed electrons and the ultimate origin of the breakdown of\nFermi liquid theory.\nThe central element of this theory is an elemental field, $D_i$, which we associated with the\ncreation of double occupation via a constraint. Our approach is in the spirit of Bohm and Pines\\cite{bohm} who also extended the Hilbert space with a constrained field to decipher the collective behaviour of the interacting electron gas. A Lagrange multiplier\n$\\varphi_i$, the charge $2e$ bosonic field, enters the action much the way the\nconstraint $\\sigma$ does in the non-linear $\\sigma$ model. The\ncorresponding Euclidean Lagrangian is\n\\begin{eqnarray}\\label{LE}\n{\\cal L}&&=\\int d^2\\theta\\left[\\bar{\\theta}\\theta\\sum_{i,\\sigma}(1- n_{i,-\\sigma}) a^\\dagger_{i,\\sigma}\\dot a_{i,\\sigma} +\\sum_i D_i^\\dagger\\dot D_i\\right.\\nonumber\\\\\n&&+U\\sum_j D^\\dagger_jD_j- t\\sum_{i,j,\\sigma}g_{ij}\\left[ C_{ij,\\sigma}a^\\dagger_{i,\\sigma}a_{j,\\sigma}\n+D_i^\\dagger a^\\dagger_{j,\\sigma}a_{i,\\sigma}D_j\\right.\\nonumber\\\\\n&&+\\left.\\left.(D_j^\\dagger \\theta a_{i,\\sigma}V_\\sigma a_{j,-\\sigma}+h.c.)\\right]+H_{\\rm con}\\right]\n\\end{eqnarray}\nwhere\n$C_{ij,\\sigma}\\equiv\\bar\\theta\\theta\\alpha_{ij,\\sigma}\\equiv\\bar\\theta\\theta(1-n_{i,-\\sigma})(1-n_{j,-\\sigma})$\nand $d^2\\theta$ represents a complex Grassman integration. The constraint Hamiltonian $H_{\\rm con}$ is taken to be\n\\begin{eqnarray}\\label{con}\nH_{\\rm con} = s\\bar{\\theta}\\sum_j\\varphi_j^\\dagger (D_j-\\theta a_{j,\\uparrow}a_{j,\\downarrow})+h.c.\n\\end{eqnarray}\nThe Grassman variable $\\theta$ is needed to\nfermionize double occupancy so that it can properly be associated with the\nhigh energy Fermi field, $D_i$. The constant $s$ has been inserted to\ncarry the units of energy. The precise value of $s$ will be determined\nby comparing the low-energy transformed electron with that in\nEq. (\\ref{trans}). This Lagrangian was constructed so that if we solve\nthe constraint, that is, integrate over $\\varphi$ and then $D_i$, we\nobtain exactly $\\int d^2\\theta \\bar\\theta\\theta L_{\\rm Hubb}=L_{\\rm Hubb}$, the\nLagrangian of the Hubbard model.\n\nThe advantage of this construction,\nhowever, is that we have been able to coarse-grain cleanly over the\nphysics on the scale $U$. That is, all the physics on the scale $U$ appears as the mass term of the new fermionic degree of freedom, $D_i$. It makes sense to integrate out $D_i$ as it\nis a massive field in the new theory. The low-energy theory to\n$O(t^2\/U)$,\n\\begin{eqnarray}\n\\label{HIR-simp}\nH_{\\rm eff}&=&-t\\sum_{i,j,\\sigma}g_{ij} \\alpha_{ij\\sigma}a^\\dagger_{i,\\sigma}a_{j,\\sigma}\\nonumber\\\\\n&&-\\frac{t^2}U \\sum_{j} b^\\dagger_{j} b_{j}-\\frac{s^2}U\\sum_{i}\\varphi_i^\\dagger \\varphi_i\\nonumber\\\\\n&&-s\\sum_j\\varphi_j^\\dagger a_{j,\\uparrow}a_{j,\\downarrow}-\\frac{ts}U \\sum_{i}\\varphi^\\dagger_i\nb_{i}+h.c.\\;\\;.\n\\end{eqnarray}\ncontains explicitly the charge $2e$ boson, $\\varphi_i$. Here $b^{(a)}_i=\\sum_{\\sigma j}V_{\\sigma}a_{i\\sigma}a_{j\\bar\\sigma}$ where $j$ is the nearest-neighbour of $i$. To fix the\nenergy scale $s$, we determine how the electron operator transforms in\nthe exact theory. As is standard, we add a source term to the\nstarting Lagrangian which generates the canonical electron operator\nwhen the constraint is solved. For hole-doping, the appropriate\ntransformation that yields the canonical electron operator in the UV\nis\n\\begin{eqnarray}\n{\\cal L}\\rightarrow {\\cal L}+\\sum_{i,\\sigma} J_{i,\\sigma}\\left[\\bar\\theta\\theta(1-n_{i,-\\sigma} ) a_{i,\\sigma}^\\dagger + V_\\sigma D_i^\\dagger \\theta a_{i,-\\sigma}\\right] +\nh.c.\\nonumber\n\\end{eqnarray}\nHowever, in the IR in which we only integrate over the heavy degree of\nfreedom, $D_i$, the electron creation operator becomes\n\\begin{eqnarray}\\label{cop}\na^\\dagger_{i,\\sigma}&\\rightarrow&(1-n_{i,-\\sigma})a_{i,\\sigma}^\\dagger\n+ V_\\sigma \\frac{t}{U} b_i a_{i,-\\sigma}\\nonumber\\\\\n&+& V_\\sigma \\frac{s}{U}\\varphi_i^\\dagger a_{i,-\\sigma}\n\\end{eqnarray}\nto linear order in $t\/U$. This equation bares close resemblance to the\ntransformed electron operator in Eq. (\\ref{trans}), as it should. In\nfact, the first two terms are identical. The last term in\nEq. (\\ref{trans}) is associated with double occupation. In\nEq. (\\ref{cop}), this role is played by $\\varphi_i$. Demanding that Eqs. (\\ref{trans}) and (\\ref{cop}) agree requires that $s= t$, thereby eliminating\nany ambiguity associated with the constraint\nfield. Consequently, the complicated interactions appearing in\nEq. (\\ref{eq:sc1}) as a result of the inequivalence between\n$c_{i\\sigma}$ and $a_{i\\sigma}$ are replaced by a single charge $2e$ bosonic field\n$\\varphi_i$ which generates dynamical spectral weight transfer across the\nMott gap. The interaction in Fig. (\\ref{fig1}), corresponding to the\nsecond-order process in the term $\\varphi_i^\\dagger b_i$, is the key physical process that enters the dynamics at low-energy. That the dynamical spectral weight transfer can be captured\nby a charge $2e$ bosonic degree of freedom is the key outcome of the\nexact integration of the high-energy scale. This bosonic field represents a\ncollective excitation of the upper and lower Hubbard bands. However, we should not immediately conclude that $\\varphi_i$ gives rise to a propagating charge $2e$ bosonic mode, as it does not have canonical kinetics; at the earliest, this could be generated at order $O(t^3\/U^2)$ in perturbation theory. Alternatively, we believe that $\\varphi$ appears as a bound degree of freedom. Since the\ndominant process mediated by $\\varphi_i$ requires a hole and a doubly\noccupied site to be neighbours (see Fig. (\\ref{fig1})) we identify $\\varphi_i^\\dagger a_{i\\bar\\sigma}$ as a new\ncharge $e$ excitation responsible for dynamical spectral weight\ntransfer. It is the appearance of this state at low-energy that\naccounts for the breakdown of Fermi liquid theory in a doped Mott\ninsulator. Physically, $\\varphi_i$ is the dressing of a hole by the\nhigh-energy scale. We have\npreviously shown\\cite{lowen1} that the formation of this bound state can produce the experimentally observed\nbifurcation\nof the electron dispersion below the chemical potential seen in\nPbBi2212\\cite{graf}, the mid-infrared band in the optical\nconductivity and the pseudogap\\cite{lowen2}. Further, the breakup of\nthe bound state beyond a critical doping leads to $T-$ linear \nresistivity\\cite{lowen2}.\n\nThe essential problem of Mottness is that in a hole-doped Mott\ninsulator, empty sites can arise from doping or from hopping processes\nwhich mix the upper and lower Hubbard bands. Both contribute to $L$.\nHowever, the spectral weight on the empty sites resulting from mixing\nwith the high-energy scale is proportional to $t\/U$. Hence, such\nempty sites effectively represent holes with fractional charge $-e(t\/U)$ not \n$-e$ as is the the case with the holes resulting from doping. Consequently,\nthey make no contribution to $n_h$, thereby giving rise to $L\/n_h>1$\nfor a doped Mott insulator and a general breakdown of the standard\nFermi liquid theory of metals. At half-filling, such fractionally charged\nsites still persist. Adding an electron to such a system at low\nenergies would require adding it coherently to a number of sites equal to $U\/t\\gg 1$. Such coherent addition of an electron at low energies has vanishing probability. The result is a gap for charge $e$ but not for\n$e(t\/U)$\nexcitations. \nAt finite doping, holes in a Mott insulator are linear superpositions\nof both kinds of empty sites. As a result, holes in the\nhard-projected\\cite{lee,lee1,lee2} $t-J$ model, in which $L=2x$, are not equivalent to holes in the Hubbard model. Approximations which prohibit explicit\ndouble occupancy, such as the standard treatment\\cite{lee,lee1,lee2} of the $t-J$ model in\nwhich the operators are not transformed, miss completely\\cite{lh1,prelovsek,haule} the localizing\\cite{lh2,kotliar,choy} physics\nresulting from the orthogonality between charge $e$ excitations and\nthe sites with spectral weight $t\/U$. In the exact theory, the\nphysics associated with a finite length scale for double occupancy is\ncontained straightforwardly in a charge $2e$ bosonic field, instead of being buried\nin complicated interaction terms in\nEq. (\\ref{eq:sc1}).\n\nAs Polchinski\\cite{polchinski} (as well as others\\cite{shankar}) have emphasized that from the point of\nview of the renormalization group, $T-$linear resistivity in the\ncuprates makes a Fermi liquid description untenable. We believe that\nour low energy theory containing the charge $2e$ bosonic field is in\nthis sense a suitable replacement for Fermi liquid theory as it can\nexplain\\cite{lowen2} $T-$linear resistivity. \nWe have shown above that the bosonic field accounts for what would be\na consequence of complicated non-linear dependences on electron\noperators in projected models. What is clear from\nPolchinski's\\cite{polchinski} arguments is that projected models do\nnot give a good basis upon which to build a theory -- they mask the\nubiquitous physics of strong coupling, namely that new degrees of freedom emerge at low energy.\n\n\\acknowledgements This work was initiated at the Kavli Institute for\nTheoretical Physics and funded partially through PHY05-51164. We thank George Sawatzky for several conversations\nthat initiated this work, R. Bhatt, M. Hastings, R. Shankar, M. Sobol,\nA. Chernyshev, A. -M. Tremblay, and O. Tchernyshyov for helpful discussions and the NSF, Grant Nos. DMR-0605769.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAg$_3$Co(CN)$_6$ has attracted a lot of attention due to its colossal positive and negative thermal expansion~\\cite{GoodwinAgCoCN2008,Conterio2008}, and also because of its giant negative linear compressibility~\\cite{GoodwinAgCoCNnlc2008}. The negative thermal expansion (NTE) along the $c$-axis and the positive thermal expansion (PTE) along the $a$($b$) axes are an order of magnitude larger than that observed in many other crystalline solids. The material also shows negative linear compressibility (NLC), namely along the $c$-axis, that is several times greater than the typical value found in crystals. As shown in Fig.~\\ref{fig:primitive}, the ambient-pressure phase of Ag$_3$Co(CN)$_6$ has a trigonal structure with space group $P\\bar{3}1m$. The structure consists of layers of Kagome sheets of Ag atoms in the $(001)$ crystal plane at height $z=1\/2$, with Co--CN--Ag--NC--Co chains along the $\\langle011\\rangle$ lattice directions linking [Co(CN)$_6$]$^{3-}$ octahedra. These chains are hinged together in a way that gives the structure a high degree of flexibility; expansion in the trigonal $(001)$ plane is accompanied by a shrinkage in the orthogonal direction in a way that does not change the relevant bond lengths.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{PrimitiveKagome2.pdf}\n\\end{center}\n\\caption{\\label{fig:primitive} Ambient phase $P\\bar{3}1m$ of Ag$_3$Co(CN)$_6$: (a) unit cell with silver in red, cobalt in blue, carbon in black and nitrogen in white grey; (b) looking down the $\\left[0,\\,0,\\,1\\right]$ direction with Ag atoms (red) in a Kagome sheet connected to the octahedra [Co(CN)$_6$]$^{3-}$ anions (blue) above and below.}\n\\end{figure}\n\n\nPrevious \\emph{ab initio} density functional theory (DFT) calculations were unable to reproduce the correct ground-state structure and the high-pressure phase of the material~\\cite{Calleja2008,Hermet2013,Mittal2012}. Whilst these studies were able to reproduce the lengths of the Co--C, C--N and N--Ag bonds which characterise the structure, the predicted lattice parameters differ considerably from the experimental values. The key interatomic distance that changes as the structure flexes is the Ag\\ldots Ag distance, which is equal to half the value of the $a$ lattice parameter. The first of the DFT studies~\\cite{Calleja2008} showed that a \\textit{post hoc} correction for dispersive interactions between the Ag cations was sufficient to shift the equilibrium DFT structure into good agreement with the experimental crystal structure. The same study also showed that there is no significant covalent bonding between neighbour Ag atoms; it was this factor, combined with the fact that DFT calculations on the structural analogue in which hydrogen or deuterium atoms replace the Ag atoms are in excellent agreement with experiment, that suggested an important role for dispersive Ag$\\ldots$Ag interactions.\n\nOn this basis, it would be useful to see if a DFT calculation that explicitly includes a correction for the long-range dispersive forces will reproduce the ground state and the high-pressure phase of Ag$_3$Co(CN)$_6$ correctly. If so, it should then be possible to obtain reliable phonons via such calculation in order to better understand the exotic behaviour of this material.\n\nModern implementations of DFT now include a correction for the long-range dispersive interactions~\\cite{Grimme2004,Grimme2006,Dobson2006,Dion2004,Thonhauser2007,Tkatchenko2009,Roman-Perez2009}. One widely used method is called `DFT+D2'~\\cite{Grimme2006} where a dispersive interaction that is dampened at short range to avoid double counting of energy is added to the DFT energy from the generalised-gradient approximation (GGA) calculation. Semi-empirical parameters in such a dispersive interaction are provided in Ref.~\\onlinecite{Grimme2006} for most elements in the periodic table. The method has been successfully applied to various materials in which the dispersive interactions are important. One good example is the recent work on cesium halides by Zhang et al.~\\cite{Zhang2013}, where the DFT+D2 formalism gives both an improved agreement between the optimised and experimental crystal structures and a correct prediction of the ground-state phases.\n\nIn this work, we have carried out DFT+D2 calculations for Ag$_3$Co(CN)$_6$. This has confirmed that the inclusion of dispersive forces give the correct ground state structure, as anticipated in the first DFT study of this material~\\cite{Calleja2008}. It is also shown that the DFT+D2 model correctly gives the structure of the high-pressure phase; without the dispersive interaction DFT gives a structure without the interdigitation found experimentally~\\cite{GoodwinAgCoCNnlc2008}. On the basis of these successes it is now reasonable to investigate the lattice dynamics of Ag$_3$Co(CN)$_6$, from which we have been able to study a number of physical and thermodynamic properties. These form the focus of this paper.\n\n\n\\section{Methods}\n\n\\subsection{DFT calculations}\n\nThe DFT calculations were performed using the CASTEP code~\\cite{Segafll2002}. For comparison, we used both local-density approximation (LDA) and GGA of Perdew-Burke-Ernzerhof (PBE)~\\cite{Perdew1996} for the exchange-correlation functional. Optimized norm conserving pseudopotentials generated using the RRKJ method~\\cite{Rapper1990} as implemented in the OPIUM package and with parameters from the Rappe and Bennett library~\\cite{link} were used in various calculations. A plane-wave basis set was used with the cut-off energy of 1800~eV. Sampling of the Brillouin zone was performed on a $6\\times6\\times6$ Monkhorst-Pack (MP)~\\cite{Monkhorst1976} grid.\n\nThe geometries of all structures were optimised using the BFGS method to achieve a convergence of less than $10^{-6}$ eV per atom change in energy per cycle and a force residual of $5\\times10^{-4}$~eV\/{\\AA}. At different pressures, tolerance for accepting convergence of the maximum stress component during unit cell optimization is $5\\times10^{-3}$ GPa.\n\n\n\n\\begin{table*}[t]\n\\caption{\\label{tab:groundstate} Calculated ground-state structures (from GGA+D, GGA and LDA), including the unit-cell edges ($a=b$ and $c$), fractional coordinates of C and N, and the nearest-neighbouring ion distances. $V$ is the volume of one formula unit (note that there is one formulate unit per unit cell). The Ag--Ag distance is equal to $a\/2$. $\\Delta_\\mathrm{GGA+D}$, $\\Delta_\\mathrm{GGA}$ and $\\Delta_\\mathrm{LDA}$ represent the deviations of the different calculations compared to experiment at a temperature of 10~K from reference \\onlinecite{Conterio2008}.}\n\\begin{tabular}{@{\\extracolsep{8pt}}c|ccc|c|ccc}\n\\hline & LDA & GGA & GGA+D & Experiment & $\\Delta_\\mathrm{GGA+D}$ & $\\Delta_\\mathrm{GGA}$& $\\Delta_\\mathrm{LDA}$ \\\\\n\\hline\n$a\\left(=b\\right)$ (\\AA) & $6.118$ & $7.629$ & $6.664$ & $6.754$ & $-1.3$\\% & $+13$\\% & $-9$\\% \\\\\n$c$ (\\AA) & $7.626$ & $6.621$ & $7.416$ & $7.381$ & $+0.5$\\% & $-10$\\% & $+3$\\%\\\\\n$V$ (\\AA$^3$) & $247.2$ & $333.7$ & $285.2$ & $291.6$ & $-2$\\% & $+14$\\% & $-15$\\% \\\\\nC$_x$ & $0.238$ & $0.202$ & $0.225$ & $0.220$ & $+0.003$ & $-0.020$ & $+0.016$ \\\\\nC$_z$ & $0.154$ & $0.171$ & $0.158$ & $0.153$ & $+0.002$ & $+0.015$ & $-0.002$ \\\\\nN$_x$ & $0.364$ & $0.321$ & $0.347$ & $0.342$ & $+0.008$ & $-0.018$ & $+0.025$ \\\\\nN$_z$ & $0.269$ & $0.282$ & $0.270$ & $0.266$ & $+0.006$ & $+0.018$ & $+0.005$ \\\\\nC--N (\\AA) & $1.170$ & $1.164$ & $1.164$ & $1.170$ & $-0.5$\\% & $-0.5$\\% & $0$\\% \\\\\nAg--N (\\AA) & $1.948$ & $1.988$ & $1.983$ & $2.034$ & $-2.5$\\% & $-2.3$\\% & $-4$\\%\\\\\nCo--C (\\AA) & $1.868$ & $1.914$ & $1.906$ & $1.865$ & $+2.2$\\% & $+2.6$\\% & $+0.2$\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{DFT+D2 calculations}\n\nThe dispersive contribution was directly added to the DFT GGA energy using a semi-empirical form introduced by Grimme~\\cite{Grimme2006},\n\\begin{eqnarray}\\label{g06}\nE_\\mathrm{disp} = - s_6 \\sum\\limits_{i = 1}^{N - 1} {\\sum\\limits_{j = i + 1}^N {\\frac{{C_6^{ij} }}{{R_{ij}^6 }}} } f_\\mathrm{damp} \\left( {R_{ij} } \\right)\n\\end{eqnarray}\n\n\\noindent where $N$ the number of atoms in the system. $C_6^{ij}$ is the dispersion coefficient of atomic pair $\\left(i,j\\right)$ that can be computed from the dispersion coefficient of the individual atoms as\n\\begin{eqnarray}\\label{dispcoefficient}\nC_6^{ij} = \\sqrt {C_6^i C_6^j }\n\\end{eqnarray}\n\n\\noindent where $R_{ij}$ is the distance between the two atoms, and $R_r$ is the sum of the atomic van der Waals radii of the pair. The dampening factor $f_\\mathrm{damp}$ is defined as\n\\begin{eqnarray}\\label{dampenfactor}\nf_\\mathrm{damp} \\left( {R_{ij} } \\right) = \\frac{1}{{1 + \\exp \\left[ { - d\\left( {R_{ij} \/R_r - 1} \\right)} \\right]}}\n\\end{eqnarray}\n\n\\noindent with $d=20$. $s_6$ is a scaling factor dependent on the functional used in the calculation; for PBE, $s_6=0.75$. This method has been implemented in CASTEP for geometry optimisation. In what follows we will refer to this method as `GGA+D'; calculations without the dispersion correction will simply be labelled as `LDA' or `GGA' as appropriate.\n\n\n\n\\subsection{Lattice dynamics with DFPT+D2}\\label{section:DFTPD2methods}\n\n\nDensity functional perturbation theory (DFPT)~\\cite{Baroni2001,Refson2006} was used to calculate phonons on a $5\\times5\\times5$ grid of wave vectors, and frequencies for phonons of other wave vectors were then obtained using interpolation~\\cite{Baroni2001}. Phonon density of states (DoS) were calculated using a $25\\times25\\times25$ MP grid~\\cite{Monkhorst1976} corresponding to a total of 1470 independent wave vectors.\n\nAt the present time CASTEP can only support a DFT+D2 calculation for phonons using the supercell method of finite displacement~\\cite{Karki1997}, which turns out to be too expensive to be feasible for Ag$_3$Co(CN)$_6$. Therefore, we first carried out a regular DFPT phonon calculation using CASTEP to get the corresponding dynamical matrices of different wave vectors. We then used the dispersive interaction of Eq.~\\eqref{g06} implemented in the lattice simulation program GULP~\\cite{Gale1997} to calculate its contribution to the dynamical matrices separately, all based on the same optimised structure from GGA+D. The dynamical matrices from the two codes are added together using a combination of Python scripts and the use of MATLAB, and the combined dynamical matrix was diagonalised to give the phonon frequencies with effects of the dispersive interaction included. For future convenience, we call this the `DFPT+D' method.\n\nTo check the accuracy of our scripts for the DFPT+D method, we performed a benchmark phonon calculation for NaI, chosen because it has a large refractive index (the largest among alkali halides~\\cite{Li1976}) and hence likely to have a significant dispersive energy term. This material has a simple structure with only 2 atoms in the primitive cell, so that it was feasible to carry out a DFT+D2 phonon calculation using the supercell method in CASTEP (here called the `supercell+D' method). By comparing the calculated phonon frequencies from DFPT+D and supercell+D, we found the two agree with each other extremely well, with a mean relative discrepancy less than $2\\%$ (see phonon dispersion curves in the Supplemental Material~\\cite{supplemental}).\n\nWith the calculated phonon frequencies, the linear Gr\\\"{u}neisen parameter $\\gamma_{ab}$ is calculated by varying the $a$ and $b$ dimensions of the unit cell by $0.005\\%$ with fixed $c$ dimension,\n\\begin{eqnarray}\\label{gammaa}\n\\gamma_{ab}=\\left(-\\partial \\ln\\omega\/\\partial \\ln a\\right)_c\n\\end{eqnarray}\nand the linear Gr\\\"{u}neisen parameter $\\gamma_{c}$ is calculated by varying the $c$ dimension of the unit cell by $0.005\\%$ with fixed $a$ and $b$ dimensions,\n\\begin{eqnarray}\\label{gammac}\n\\gamma_{c}=\\left(-\\partial \\ln\\omega\/\\partial \\ln c\\right)_{ab}.\n\\end{eqnarray}\nWe will show later how these two quantities determine the coefficients of linear thermal expansion $\\alpha_a = \\partial \\ln a\/\\partial T$ and $\\alpha_c =\\partial \\ln c\/\\partial T$.\n\n\\section{Ground-state properties of Ag$_3$Co(CN)$_6$}\\label{groundstate}\n\n\\subsection{Crystal structure}\n\nThe details ground-state structures of Ag$_3$Co(CN)$_6$ optimised using GGA, with and without the dispersive interaction, and using LDA are reported in Table~\\ref{tab:groundstate}, where they are compared to the experimental values~\\cite{Conterio2008}. It is clear that, without the dispersive interaction, the calculated ground-state structure is wrong. Inclusion of the dispersive interaction results in the correct structure with small deviations from experiment.\n\nIt is worth remarking on the role the Ag$\\ldots$Ag dispersive interaction has on the structure. The dispersive interaction is a weak attractive interaction, which opposing the repulsive Coulomb interaction, Thus the effect of the dispersive interaction is to reduce the overall Ag$\\ldots$Ag interaction. On this basis, addition of the dispersive interaction to the GGA model enables the structure to relax with a shorter Ag$\\ldots$Ag distance and hence a smaller value of the $a$ lattice parameter, as see in the results in Table \\ref{tab:groundstate}. On the other hand, the well-known tendency of LDA to overbind already results in a shorter Ag$\\ldots$Ag distance.\n\nWe can quantify this point. The DFT calculations give an approximate value for the charge of the Ag cation of $+0.65|e|$\\cite{Segall1996}, where $e$ is the electronic charge. Calculation of the Ag$\\ldots$Ag forces due to the Coulomb and dispersive interactions (taking $f_\\mathrm{damp}=1$ in Eq.~\\ref{g06}) over the range of distances 3.3--3.5~\\AA\\ shows that the dispersive interaction reduces the net force between neighbouring Ag ions by nearly a factor of 2.\n\n\\subsection{Elasticity}\n\nThe GGA+D computed elastic compliances are given in Table~\\ref{tab:compliance}. The linear compressibilities along the $a$($b$) and $c$ crystal axes were calculated using the elastic compliances as\n\\begin{eqnarray}\\label{linearcompressibility}\n\\beta _{ab} = -\\partial \\ln a\/\\partial p =s_{11} + s_{12} + s_{13}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{linearcompressibility2}\n\\beta _c = -\\partial \\ln c\/\\partial p = 2s_{13} + s_{33},\n\\end{eqnarray}\nrespectively. The volume compressibility was calculated as the sum\n\\begin{eqnarray}\\label{linearcompressibility3}\n\\beta = -\\partial \\ln V\/\\partial p =2\\beta_{ab} + \\beta_c\n\\end{eqnarray}\nThe linear elastic moduli $B_{ab}$ along $a$ and $b$ axes as well as $B_c$ along $c$ axis are the inverse of the $\\beta_{ab}$ and $\\beta_c$, respectively. Their relations with the elastic constants are given in the supplemental material~\\cite{supplemental}.\n\nAs shown in Table~\\ref{tab:compliance}, the GGA+D calculated $s_{33}$ and $s_{13}$ have almost the same magnitude but with opposite sign, showing that the $c$ dimension would response equivalently to a stress acting on the $a$ or $b$ dimension and a tension directly acting on the $c$ dimension. This shows the effectiveness of the hinging mechanism in the material. In comparison, the small value of $s_{12}$ shows that the change in dimension $a$ (or $b$) is barely correlated to the change in $b$ (or $a$) dimension.\n\nNegative values of $\\beta_c$ and $B_c$ correspond to the NLC of the material, namely the material will \\textit{elongate} in the $c$ dimension under hydrostatic compression. The bulk modulus and its first derivative were calculated as $B=15.8(8)$~GPa and $B^\\prime=-4.9(8)$, respectively~\\cite{supplemental}. Using the 3rd-order Birch-Murnaghan (BM) equation of state (EoS)~\\cite{Birch1947} to fit to the calculated isotherm data from 0 to 0.6~GPa also results in a negative value of $B^\\prime$ of $-3(2)$. These results predict that the material will have pressure-induced softening~\\cite{Fangzeolite2013,Fangexp2013,Fangexpression2014} at low pressures.\n\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{4pt}\n\\caption{\\label{tab:compliance} Calculated compliances at different pressures for the ambient phase of Ag$_3$Co(CN)$_6$ obtained from calculating the change of energy corresponding to a set of given strains $\\varepsilon _{ij}$ generated according to the trigonal symmetry. Results were obtained using GGA+D, and the LDA results are from Ref.~\\onlinecite{Hermet2013}.The compliances of a trigonal phase have the symmetry~\\cite{Nye1985} $s_{ij}=s_{ji}, s_{22}=s_{11}, s_{55}=s_{44}, s_{23}=s_{13}, s_{24}=-s_{14}, s_{66}=2(s_{11}-s_{12})$. The corresponding elastic constants and elastic moduli are given in the Supplemental Material~\\cite{supplemental}. The linear compressibility $\\beta_{ab}$ and $\\beta_c$ as well as the volume compressibility $\\beta$ are calculated from the compliances using Eqs~\\eqref{linearcompressibility} to~\\eqref{linearcompressibility3}.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline Compliance (TPa$^{-1}$) & 0.0 GPa & 0.04 GPa & 0.1 GPa & LDA \\\\\n\\hline\n$s_{11}$ & 61(3) & 62(3) & 64(4) & 85 \\\\\n$s_{33}$ & 22(1) & 21.4(9) & 23(2) & 16\\\\\n$s_{44}$ & 38.5(9) & 37.7(7) & 44(3) & 73 \\\\\n$s_{12}$ & 2(1) & 1(1) & 3(2) & $-22$ \\\\\n$s_{13}$ & $-21(1)$ & $-21(1)$ & $-23(2)$ & $-17$ \\\\\n$s_{14}$ & $15(1)$ & $15(1)$ & $17(2)$ & $-41$ \\\\\n$\\beta_{c}$ & $-21(2)$ & $-21(2)$ & $-23(4)$ & $-19$ \\\\\n$\\beta_{ab}$ & $42(4)$ & 42(4) & 44(5) & 45 \\\\\n$\\beta$ & $63(6)$ & 63(6) & 65(8) & 72 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nThe calculated bulk modulus at 0 K, as the inverse of $\\beta$ in Eq.~\\ref{linearcompressibility3}, is 15.8(8)~GPa which is significantly larger than the experimental value of $B=6.5(3)$~GPa at 300~K~\\cite{GoodwinAgCoCNnlc2008}. This apparent overestimation of the calculation may actually be due to a considerable softening of the material on heating, as will be discussed later in Section~\\ref{softening}. The same idea can be used to explain the apparent large underestimation of the compressibilities: the calculated values $\\beta_{ab}=42(4)$~TPa$^{-1}$ and $\\beta_c=-21(2)$~TPa$^{-1}$ are much lower than the experimental values of $\\beta_{ab}=115(8)$~TPa$^{-1}$ and $\\beta_c=-79(9)$~TPa$^{-1}$ at 300~K~\\cite{GoodwinAgCoCNnlc2008}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phaseIInews.pdf}\n\\end{center}\n\\caption{\\label{fig:phaseII} Structures of the high-pressure phase of Ag$_3$Co(CN)$_6$ (space group $C2\/m$) optimised using (a) GGA+D , and (b) either GGA or LDA without a correction for the dispersion energy. The experimentally observed interdigitated structure, characterised by the indented Ag atoms, can be seen only when dispersion corrections are used.}\n\\end{figure}\n\n\n\\section{High-pressure phase of Ag$_3$Co(CN)$_6$}\n\n\\subsection{Crystal structure of the high-pressure phase}\nAg$_3$Co(CN)$_6$ undergoes a structural phase transition at $0.19$ GPa to a monoclinic phase~\\cite{GoodwinAgCoCNnlc2008} and denoted as Phase-II. The phase transition involves displacements of Ag atoms in alternative rows, which cause the high-pressure phase to possess an interdigitated structure as seen by viewing down the $\\left[0,\\,0,\\,1\\right]$ direction. This is indicated in Fig.~\\ref{fig:phaseII}(a) by the indented Ag atoms.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{phasetransition2.pdf}\n\\end{center}\n\\caption{\\label{fig:phasetransition2} The calculated enthalpy of the high-pressure phase relative to that of the ambient phase using GGA+D. An overestimated phase-transition pressure of $2.5$~GPa is predicted.}\n\\end{figure}\n\nOur calculations show that neither LDA or GGA without the dispersive interaction can give the correct optimised high-pressure phase with the interdigitated structure~\\cite{GoodwinAgCoCNnlc2008}, as shown by Fig.~\\ref{fig:phaseII}(b). It is only by including the dispersive interaction in the GGA+D calculation that the interdigitated structure of the high-pressure phase can be reproduced, as shown in Fig.~\\ref{fig:phaseII}(a).\n\nFig.~\\ref{fig:phasetransition2} shows the difference in enthalpy between the two phases as calculated using the GGA+D method. The predicted phase-transition pressure of about 2.5 GPa overestimates the experimental value of 0.19 GPa~\\cite{GoodwinAgCoCNnlc2008}. Although this appears to be a large discrepancy, it is magnified by the fact that the experimental transition pressure is so low. Phase transition pressures are hard to calculate; we attribute the discrepancy to an accumulation of small errors associated with a number of approximations in the DFT method and the dispersion correction. The calculated relative change of the cell volume at the phase transition is $11\\%$, smaller than the experimental value of $16\\%$~\\cite{GoodwinAgCoCNnlc2008}. Table~\\ref{tab:highpphase} compares the optimised structure with the $C2\/m$ space group in GGA+D with the experiment values at 0.23 GPa.\n\n\n\\begin{table}[t]\n\\caption{\\label{tab:highpphase} Comparison of optimised (GGA+D2) and experimental \\cite{GoodwinAgCoCNnlc2008} crystal structures of the high-pressure phase (space group $C2\/m$) at a pressure of 0.23 GPa. $\\Delta_\\mathrm{GGA+D}$ represents the differences between the two. $V$ is the volume of one formula unit (note that there are 2 formula units in the unit cell). The fractional coordinates of Ag1 are $(1\/2, 0, 1\/2)$. }\n\\begin{tabular}{@{\\extracolsep{10pt}}c|ccc}\n\\hline & GGA+D & Experiment & $\\Delta_\\mathrm{GGA+D}$ \\\\\n\\hline\n$a$ (\\AA) & $6.485$ & 6.693 & $-3.1$\\%\\\\\n$b$ (\\AA) & $11.144$ & 11.539 & $-3.4$\\% \\\\\n$c$ (\\AA) & $6.658$ & 6.566 & $+1.4$\\%\\\\\n$\\beta$ ($^\\circ$) & $101.84$ & $101.48$ & $+0.36$ \\\\\n$V$ (\\AA$^3$)& 235.6 & 248.5 & $+5.2$\\% \\\\\nC1$_x$ & 0.790& 0.825 & $-0.035$ \\\\\nC1$_z$ & 0.163& 0.182 & $-0.019$ \\\\\nN1$_x$ & 0.664& 0.715 & $-0.051$ \\\\\nN1$_z$ & 0.264& 0.302 & $-0.038$ \\\\\nC2$_x$ & 0.145& 0.163 & $-0.019$ \\\\\nC2$_y$ & 0.123& 0.119 & $+0.004$ \\\\\nC2$_z$ & 0.177& 0.157 & $+0.0209$ \\\\\nN2$_x$ & 0.241& 0.258 & $-0.017$ \\\\\nN2$_y$ & 0.197& 0.185 & $+0.012$ \\\\\nN2$_z$ & 0.280& 0.259 & $+0.021$ \\\\\nAg2$_y$ & 0.243& 0.240 & $+0.002$ \\\\\nC1--N1 (\\AA) & 1.161 & 1.183 & $-1.8$\\%\\\\\nC2--N2 (\\AA) & 1.170 & 1.126 & $+3.9$\\%\\\\\nAg1--N1 (\\AA)&2.069 & 2.123 & $-2.5$\\%\\\\\nAg2--N2 (\\AA)&2.097 & 2.199 & $-4.6$\\%\\\\\nCo--C1 (\\AA) &1.907 & 1.830 & $+4.2$\\%\\\\\nCo--C2 (\\AA) &1.922 & 1.924 & $-0.1$\\%\\\\\nAg--Ag(1) (\\AA)&2.868&2.996 & $-4.3$\\%\\\\\nAg--Ag(2) (\\AA)&5.407&5.548 & $-2.5$\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nOriginally, it was found~\\cite{GoodwinAgCoCNnlc2008} that the high-pressure phase of the material has a space group of $C2\/m$. However, recently, it was proposed~\\cite{Hermet2013} that the high-pressure phase should have the lower symmetry of space group $Cm$, because a structure with this symmetry can be obtained as a subgroup of the space group of the ambient-pressure phase, $P\\bar{3}1m$, whereas a structure with space group $C2\/m$ cannot. Our calculations indicate that the optimised structures starting from both space groups $C2\/m$ and $Cm$ have exactly the same enthalpy up to a pressure of 7~GPa (the highest we examined), with relaxed structures that differ only by a small origin offset. We conclude that the structure of the high-pressure phase has the originally-proposed $C2\/m$ structure.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{celldistance2.pdf}\n\\end{center}\n\\caption{\\label{fig:phaseIIparameters} Upper panel: various lattice parameters of the monoclinic phase of Ag$_3$Co(CN)$_6$ at different pressures using GGA+D. The calculated (solid lines) and the experimental (symbol) values of each parameter are in the same colour. Lower panel: the calculated nearest Ag$\\ldots$Ag distance in two phases. In the high-pressure phase, the GGA+D result (symbol line) shows the correct trend of two types of Ag$\\ldots$Ag distances changing with compression, while the GGA\/LDA (dashed line) result does not.}\n\\end{figure}\n\n\\subsection{Elasticity}\n\nA fit of the 3rd-order BM EoS to the calculated isotherm of the high-pressure phase yields $B=17(6)$~GPa and $B^\\prime=17(7)$; experimental values are $B=11.8(7)$~GPa and $B^\\prime=13(1)$, respectively~\\cite{GoodwinAgCoCNnlc2008}. Thus, unlike the ambient-pressure phase, which has pressure-induced softening at low pressures, the high-pressure phase of the material quickly becomes harder under compression.\n\nThe calculated change of lattice parameters of the high-pressure monoclinic phase-II are presented in Fig.~\\ref{fig:phaseIIparameters}, and compared to the experimental values. The agreement between the two are good with the largest relative deviation below $10\\%$. By fitting to a 3rd-order polynomial of pressure $\\left(p-p_\\mathrm{c}\\right)$ with the phase-transition pressure $p_\\mathrm{c}=2.5$~GPa, the linear compressibilities of $a_\\mathrm{II}$, $b_\\mathrm{II}$ and $c_\\mathrm{II}$ were obtained at different pressures. Their averaged values over $2.5$--$8.0$~GPa are $19(1)$, $6.9(4)$ and $-4.1(3)$~TPa$^{-1}$, respectively. These values are in good agreement with experimental values~\\cite{GoodwinAgCoCNnlc2008} of $15.9(9)$, $9.6(5)$ and $-5.3(3)$~TPa$^{-1}$.\n\nAs pointed out in Ref.~\\onlinecite{GoodwinAgCoCNnlc2008}, the relatively small compressibility along $b_\\mathrm{II}$ is due to the interdigitation in the high-pressure phase. Upon compression, the structure becomes more indented (Fig.~\\ref{fig:phaseII}(b)), resulting in the Ag$\\ldots$Ag(1) distance between the indented Ag atom and its nearest neighbour increases with pressure, while the Ag$\\ldots$Ag(2) distance between the two indented Ag atoms at the opposite sites decreases. This behaviour of the Ag$\\ldots$Ag distances under pressure is seen in the GGA+D calculated results shown in the lower panel of Fig.~\\ref{fig:phaseIIparameters}.\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{3pt}\n\\caption{\\label{tab:raman} The calculated Raman and infrared spectrums (in THz) of Ag$_3$Co(CN)$_6$ using DFPT+D compared to the experimental values at 80 K (Raman)~\\cite{Rao2011} and 295 K (Infrared)~\\cite{Hermet2013}. $\\Delta_\\mathrm{DFPT+D}$ is the deviation of the DFPT+D calculated frequencies compared to the experiment (in THz). The first derivative of the frequency with respect to pressure is in unit of THz\/GPa.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline Raman & $\\omega_\\mathrm{DFPT+D}$ & $\\Delta_\\mathrm{DFPT+D}$ & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{Exp.}$~\\cite{Rao2011} & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{DFPT+D}$ \\\\\n\\hline\n$2.6$ & 2.9 & 0.3 & 0.3 & 0.4 \\\\\n$4.2$ & 4.3 & 0.1 & 0.6 & 0.4 \\\\\n$4.9$ & 5.0 & 0.1 & 0.3 & 0.6 \\\\\n$9.7$ & 9.8 & 0.1 & $-0.3\\footnotemark[1]$&$-0.04$ \\\\\n$14.2$ & 13.8 & $-0.4$ & 0.1 &$-0.006$ \\\\\n$14.2$ & 13.9 & $-0.3$ & 0.1 &$-0.04$ \\\\\n$15.6$ & 16.1 & 0.5 & 0.7 & 0.06 \\\\\n$15.6$ & 16.1 & 0.5 & 0.7 & 0.2 \\\\\n$65.5$ & 65.2 & $-0.3$ & 0.2 & 0.1 \\\\\n$66.1$ & 66.0 & $-0.1$ & 0.3 & 0.1 \\\\\n\\hline Infrared & $\\omega_\\mathrm{DFPT+D}$ & $\\Delta_\\mathrm{DFPT+D}$ & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{Exp.}$~\\cite{Hermet2013} & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{DFPT+D}$ \\\\\n\\hline\n1.2 & 1.4 & 0.2 & -- & 0.2 \\\\\n1.4 & 1.5 & 0.1 & -- & $0.1$ \\\\\n1.6 & 2.2 & 0.6 & -- & $-0.3$ \\\\\n4.0 & 4.2 & 0.2 & $-0.2$ & $0.01$ \\\\\n5.3 & 5.6 & 0.3 & -- & $-0.4$ \\\\\n5.5 & 5.7 & 0.2 & -- & $-0.2$ \\\\\n8.0 & 8.6 & 0.6 & -- & 0.3 \\\\\n8.0 & 8.8 & 0.8 & -- & 0.2 \\\\\n13.0 & 12.8 & $-0.2$ & -- & $-0.03$ \\\\\n14.5 & 14.8 & 0.3 & $-0.02$ & $0.01$ \\\\\n14.8 & 14.9 & 0.1 & 0.03 & 0.01 \\\\\n17.6 & 17.8 & 0.2 & -- & 0.3 \\\\\n-- & 18.0 & -- & -- & 0.2 \\\\\n-- & 65.1 & -- & -- & $0.1$ \\\\\n-- & 65.2 & -- & -- & 0.1 \\\\\n\\hline\n\\end{tabular}\n\\footnotetext[1]{From non-hydrostatic experiment~\\cite{Rao2011}.}\n\\end{table}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.97\\textwidth]{phonons2.pdf}\n\\end{center}\n\\caption{\\label{fig:phonon} (a) DFPT+D calculated phonon dispersion curves along the high-symmetry directions in the Brillouin zone. (b) and (c) are dispersion curves coloured according to the values of linear Gr\\\"{u}neisen parameters along the $a$($b$) axes ($\\gamma_{ab}$) and $c$ axis ($\\gamma_c$), respectively, with values $\\leq-20$ in red gradually passing to values $\\geq+20$ in blue.}\n\\end{figure*}\n\n\n\\section{Lattice dynamics calculations}\n\nThe phonon calculations were performed using the DFPT+D method as discussed in Section~\\ref{section:DFTPD2methods}. Table~\\ref{tab:raman} shows that the calculated Raman and infrared spectra are in good agreement with the experiment~\\cite{Rao2011,Hermet2013}. The phonon dispersion curves along the high-symmetry directions in the Brillouin zone for frequencies up to 18~THz are presented in Fig.~\\ref{fig:phonon}(a).\n\nWe have studied the eigenvectors of different vibrational modes as shown by the animations in the Supplemental Materials~\\cite{supplemental}. We found that the infrared-active modes at $1.4$--$1.5$~THz showing negative linear Gr\\\"{u}neisen parameters $\\gamma_{ab}$ and positive linear Gr\\\"{u}neisen parameters $\\gamma_c$ correspond to the rotation of Ag-triangle pairs against each other in the Kagome sheet about their shared apex. The Raman-active mode at $2.9$~THz, having positive $\\gamma_{ab}$ and negative $\\gamma_c$, corresponds to the rotations of CoC$_6$ octahedra that pulls the connected layers of Ag atoms along the $c$ axis closer together. The Raman-active modes at $4.3$ and $5.0$~THz correspond to similar type of vibrations but with CoC$_6$ octahedra deforming, and these also show positive $\\gamma_{ab}$ and negative $\\gamma_c$.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Amode1s.pdf}\n\\end{center}\n\\caption{\\label{fig:Amode1} Vibration corresponds to the first mode at point A $\\left(0,\\,0,\\,1\/2\\right)$ from its eigenvector looking down the $\\left[1,\\,0,\\,0\\right]$ direction. Each Ag atom (red) is connected to two [Co(CN)$_6$] octahedra (blue) in the upper and lower layers via the Co--CN--Ag--NC--Co linkages. Arrows show the transverse motion of the nearly-rigid bridging group CN--Ag--NC resulted from the concerted rotation of the octahedra. The dashed square shows the unit cell.}\n\\end{figure}\n\n\nThe dispersion curves are also shown in Fig.~\\ref{fig:phonon}(b) and (c) with colours that reflect the calculated values of $\\gamma_{ab}$ and $\\gamma_c$ as given by Eqs~\\ref{gammaa} and~\\ref{gammac}, respectively. One can see that it is almost the same set of low-frequency modes that contribute to the PTE along the $a$($b$) axes and NTE along the $c$ axis, i.e. their values of $\\gamma_{ab}$ and $\\gamma_c$ show similar magnitudes but are opposite in sign. This is directly related to the hinging structure in the material where any level of expansion in the $a$($b$) axes would transfer into a similar level of contraction in the $c$ axis via the Co--CN--Ag--NC--Co linkage. Modes around the wave vector A $\\left(0,\\,0,\\,1\/2\\right)$ and around the middle point along the H$\\left(-1\/3,\\,2\/3,\\,1\/2\\right)$$\\rightarrow$K$\\left(-1\/3,\\,2\/3,\\,0\\right)$ direction have the lowest frequencies ($< 1.0$ THz) and hence have the most extreme values of Gr\\\"{u}neisen parameters, The first two degenerate modes at A correspond to concerted rotations of rigid Co(CN)$_6$ octahedra together with the nearly-rigid CN--Ag--NC linkages moving sideways~\\cite{supplemental}, as shown by its eigenvector in Fig.~\\ref{fig:Amode1}. The first mode at the middle point $\\left(-1\/3,\\,2\/3,\\,1\/4\\right)$ along the H$\\rightarrow$K direction corresponds to the Ag atoms vibrating along the $c$ axis, producing a transverse wave passing through each Kagome sheet~\\cite{supplemental}.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.97\\textwidth]{dosgp.pdf}\n\\end{center}\n\\caption{\\label{fig:dosgp} Calculated DoS of modes with frequencies $\\leq 9$ THz. At pressures (a) 0.0 GPa, (b) 0.04 GPa and (c) 0.1 GPa, the DoS in the upper panel is coloured according to the averaged value of $\\gamma_{ab}$ and the DoS in the lower panel is coloured according to the averaged value of $\\gamma_{c}$ around each energy. Values $\\leq -10$ are in red and $\\geq 10$ are in blue. $\\gamma_{ab}$ of the low-frequency modes, especially the modes below 1.0 THz, decrease largely upon compression and even change their signs at 0.1 GPa as indicated by the change of colour from blue to red. (d) Coloured DoS according to the average value of relative frequency change with pressure (in GPa$^{-1}$) around each energy bin. The upper panel shows the frequency change from 0.0 to 0.04 GPa and the lower panel shows that from 0.04 to 0.1 GPa. Stiffened phonons ($\\partial \\ln \\omega\/\\partial p \\geq +0.1$) are in blue and softened phonons ($\\partial \\ln \\omega\/\\partial p \\leq -0.1$) are in red.}\n\\end{figure*}\n\nThe picture shown in Fig.~\\ref{fig:phonon} is reflected in plots of the vibrational densities of states (DoS), which are shown in Fig.~\\ref{fig:dosgp}. These were calculated from the full set of DFPT+D vibrations computed on a $25\\times25\\times25$ grid (corresponding to a total of 1470 wave vectors in the Brillouin zone). Plots of the DoS are plotted for three pressures and coloured according to the averaged value of $\\gamma_{ab}$ and $\\gamma_c$ of the modes around each energy. The plots for vibrations at ambient pressure (Fig.~\\ref{fig:dosgp}(a)) show that the same low-frequency modes contribute positively to $\\gamma_{ab}$ and negatively to $\\gamma_c$. This situation changes under pressure, as we will now discuss.\n\n\n\\section{Effect of compression on thermal expansion}\n\n\\subsection{Increase of linear thermal expansion on compression}\n\nFrom the calculated Gr\\\"{u}neisen parameters and the compliances given in Table~\\ref{tab:compliance}, the linear coefficients of thermal expansion of Ag$_3$Co(CN)$_6$ along the $a$($b$) and the $c$ axes were calculated within the quasi-harmonic approximation as~\\cite{Barron1980}\n\\begin{eqnarray}\\label{linearcteab}\n\\alpha _{ab} &=& \\frac{1}{\\Omega }\\sum\\limits_{s,\\textbf{k}} {\\left\\{ {c_{s,\\textbf{k}} \\left[ {\\frac{{\\left( {s_{11} + s_{12} } \\right)\\gamma _{ab}(s,\\textbf{k}) }}{2} + s_{13} \\gamma _c(s,\\textbf{k}) } \\right]} \\right\\}} \\nonumber \\\\\n&=& \\frac{1}{\\Omega }\\left[ {\\frac{{\\left( {s_{11} + s_{12} } \\right)\\overline \\gamma _{ab} }}{2} + s_{13} \\overline \\gamma _c } \\right]\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{linearctec}\n\\alpha _c &=& \\frac{1}{\\Omega }\\sum\\limits_{s,\\textbf{k}} {\\left\\{ {c_{s,\\textbf{k}} \\left[ {s_{13} \\gamma _{ab}(s,\\textbf{k}) + s_{33} \\gamma _c(s,\\textbf{k}) } \\right]} \\right\\}} \\nonumber \\\\\n&=& \\frac{1}{\\Omega }\\left[ {s_{13} \\overline \\gamma _{ab} + s_{33} \\overline \\gamma _c } \\right],\n\\end{eqnarray}\nrespectively, where\n\\begin{eqnarray}\\label{specificheat}\nc_{s,\\textbf{k}} = \\hbar \\omega _{s,\\textbf{k}} \\frac{{\\partial n_{s,\\textbf{k}} }}{{\\partial T}}\n\\end{eqnarray}\nis the contribution of the normal-mode $\\left\\{s,\\textbf{k}\\right\\}$ to the specific heat with $n_{s,\\textbf{k}} = \\left[ {\\exp \\left( {\\hbar \\omega _{s,\\textbf{k}} \/k_BT } \\right) - 1} \\right]^{ - 1}$, and $\\Omega$ is the volume of the unit cell. The overall Gr\\\"{u}neisen parameters are defined as\n\\begin{eqnarray}\\label{overallgamma}\n\\overline \\gamma_{ab}&=&\\sum\\limits_{s,\\textbf{k}} {c_{s,\\textbf{k}} \\gamma_{ab}(s,\\textbf{k})} \\nonumber \\\\\n\\overline \\gamma_{c}&=&\\sum\\limits_{s,\\textbf{k}} {c_{s,\\textbf{k}} \\gamma_{c}(s,\\textbf{k})}.\n\\end{eqnarray}\nThe volume CTE is calculated as\n\\begin{eqnarray}\\label{linearctev}\n\\alpha _V = 2\\alpha _{ab} + \\alpha _c.\n\\end{eqnarray}\n\nThe calculated values of $\\alpha_{ab}$ and $\\alpha_c$ at different temperatures and pressures are shown in Fig.~\\ref{fig:cote}. The averaged values of $\\alpha_{ab}$ and $\\alpha_c$ over 50--500 K are $+127$~MK$^{-1}$ and $-101$~MK$^{-1}$, respectively. These exceptionally large values are in reasonable agreement with the experimental values~\\cite{GoodwinAgCoCN2008} of $\\alpha_{ab}=+135$~MK$^{-1}$ and $\\alpha_c=-131$~MK$^{-1}$. The hinging mechanism of the material as discussed previously results in similar magnitude of the PTE along the $a$($b$) axes and the NTE in the $c$ axis.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{cotenew.pdf}\n\\end{center}\n\\caption{\\label{fig:cote} DFPT+D calculated coefficients of thermal expansion at different temperatures for pressures of 0.0 (solid line), 0.04 (dashed line) and 0.1 GPa (dotted line) using quasi-harmonic approximation, compared to the experiment at ambient pressure (in open circle)~\\cite{GoodwinAgCoCN2008}. A plot from GGA calculated phonons based on the correct structure optimised using the GGA+D method is given in the Supplemental Material~\\cite{supplemental} for comparison.}\n\\end{figure}\n\nIn addition to reproducing the experimentally-observed~\\cite{GoodwinAgCoCN2008} colossal PTE and NTE of Ag$_3$Co(CN)$_6$, an interesting finding from Fig.~\\ref{fig:cote} is that $\\partial \\alpha_c \/ \\partial p > 0$, that is $\\alpha_{c}$, which has a negative value, becomes less negative on compression. This is opposite to the usual behaviour that $\\partial \\alpha\/ \\partial p<0$ as found in most PTE materials such as metals, metal oxides and alkali halides~\\cite{Fangmetal2010,Zhang2007,Song2012,Sun2013} and also in many isotropic NTE materials~\\cite{Chapman2005,Fangmd2013,Fangzeolite2013,Fangmodel2014}.\n\nAccording to the standard thermodynamic relation~\\cite{Fangexp2013}\n\\begin{eqnarray}\\label{linearwarmhardening}\n\\left( {\\frac{{\\partial B_c }}{{\\partial T}}} \\right)_p = B_c^2 \\left( {\\frac{{\\partial \\alpha _c }}{{\\partial p}}} \\right)_T,\n\\end{eqnarray}\na positive value of $\\partial \\alpha_c \/ \\partial p$ means a corresponding positive value of $\\partial B_c \/ \\partial T$. If $B_c$ were positive as would usually be the case, this would give the unusual property of the material becoming harder at higher temperature~\\cite{Fangmodel2014}, but in this case $B_c$, as the inverse of $\\beta_c$ is negative (see Table~\\ref{tab:compliance}), and thus $B_c$ becomes less negative on heating with $\\beta_c$ becoming more negative. Hence higher temperatures enhance NLC.\n\nTo understand this, we note that the values of $\\alpha_{ab}$ and $\\alpha_c$ depend on $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ weighted by the compliances, as given in Eqs~\\ref{linearcteab} and~\\ref{linearctec}. Since the compliances listed in Table~\\ref{tab:compliance} change little with pressure, any significant change of the CTE with pressure must be due to the change of the overall Gr\\\"{u}neisen parameters.\n\nIn the temperature range of 0--500 K, only contributions from the low-frequency modes ($\\leq 9$ THz) (Fig.~\\ref{fig:dosgp}(a)) are important. At zero pressure, contributions from the low-frequency modes result in positive $\\overline \\gamma_{ab}$ and negative $\\overline \\gamma_c$ as shown in Fig.~\\ref{fig:overallgamma}. Since $s_{11}$ and $s_{12}$ are positive and $s_{13}$ is negative, both $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ would contribute constructively to the positive value of $\\alpha_{ab}$ according to Eq.~\\ref{linearcteab}. Similarly, since $s_{13}$ is negative and $s_{33}$ is positive, $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ would also contribute constructively to the negative value of $\\alpha_{c}$. Thus, the increase of $\\overline \\gamma_{ab}$ (becoming more positive) and decrease of $\\overline \\gamma_{c}$ (becoming more negative) would enhance the linear PTE and NTE, while the decrease of $\\overline \\gamma_{ab}$ and increase of $\\overline \\gamma_{c}$ would reduce the linear PTE and NTE of the material.\n\nFig.~\\ref{fig:overallgamma} shows that there is a significant decrease of $\\overline \\gamma_{ab}$ and a smaller decrease of $\\overline \\gamma_{c}$ on compression. According to Eqs~\\ref{linearcteab} and~\\ref{linearctec}, the first effect is more dominant and results in the large decrease in the magnitude of both $\\alpha_{ab}$ and $\\alpha_c$ with pressure, corresponding to the conventional decrease of elastic moduli on heating ($\\partial B_{ab}\/\\partial T \\propto \\partial \\alpha_{ab}\/\\partial p < 0$) and the heat enhancement of NLC ($\\partial B_{c}\/\\partial T \\propto \\partial \\alpha_{c}\/\\partial p > 0$), respectively.\n\nIt is interesting to note this enhancement of NLC on heating could not happen without the hinging mechanism in the structure working efficiently, because it is this mechanism that gives almost the same magnitudes to $s_{13}$ and $s_{33}$ (as discussed in Section~\\ref{groundstate}), which in turn provide the same weighting of $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ in their contributions to $\\alpha_c$. If we had the case where the hinging is not effective, a much smaller value of $s_{13}$ compared to $s_{33}$ would make the decrease of $\\overline \\gamma_c$ dominate, resulting in a decrease of $\\alpha_c$ on compression (corresponding to $\\partial B_{c}\/\\partial T \\propto \\partial \\alpha_{c}\/\\partial p < 0$); in this case the enhancement of NLC on heating would not be observed.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{averagegammanew.pdf}\n\\end{center}\n\\caption{\\label{fig:overallgamma} Calculated overall Gr\\\"{u}neisen parameters $\\overline \\gamma_{ab}$ (in blue) and $\\overline \\gamma_{c}$ (in red) by Eq.~\\ref{overallgamma}. $\\overline \\gamma_{ab}$ at 0.0, 0.04 and 0.1~GPa correspond to solid, dashed and dotted lines, respectively. $\\overline \\gamma_{c}$ at the same pressures corresponds to solid, dashed and dotted lines, respectively. From 0.0 to 0.1~GPa, $\\overline \\gamma_{ab}$ decreases significantly and even becomes negative at 0.1~GPa, while $\\overline \\gamma_c$ decreases much less.}\n\\end{figure}\n\n\n\n\n\\subsection{Exceptionally large $\\partial \\alpha_V\/\\partial p$}\\label{softening}\n\n\nAnother interesting finding in Fig.~\\ref{fig:cote} is the exceptionally large reduction in $\\alpha_V$ on compression. The magnitude of $\\partial \\alpha_V \/ \\partial p$ is found to be about 1125~MK$^{-1}$\/GPa from 0.0 to 0.04~GPa and 2083~MK$^{-1}$\/GPa from 0.04 to 0.1~GPa, values that are more than 10 times larger than what is normally considered as a large value~\\cite{Cetinkol2008} (\\textit{ca} 100~MK$^{-1}$\/GPa) and more than 10$^4$ times larger than that of a hard metal~\\cite{Fangmetal2010}.\n\nFrom 0.0 to 0.1~GPa, the linear CTE of the material is reduced from its colossal value to a more moderate value of about $\\pm25$~MK$^{-1}$ which is similar to the values found in the NTE metal cyanides~\\cite{Goodwin2005,Fangmd2013}. As discussed in the previous section, such significant reduction in the magnitudes of $\\alpha_{ab}$ and $\\alpha_c$ is due to the large decrease of $\\overline \\gamma_{ab}$. In particular, when $\\overline \\gamma_{ab}$ becomes negative at 0.1 GPa, it begins to contribute to $\\alpha_{ab}$ and $\\alpha_c$ (Eqs~\\ref{linearcteab} and~\\ref{linearctec}) with opposite sign to that of $\\overline \\gamma_{c}$.\n\nThe significant decrease on compression of $\\overline \\gamma_{ab}$ is attributed to the large decrease in $\\gamma_{ab}$ of most low-frequency modes ($\\leq9$~THz), especially the modes with frequencies $<1.0$~THz (as those at wave vector A in Fig.~\\ref{fig:phonon}). On one hand, such a decrease is related to the increase of mode frequencies (see Eq.~\\ref{gammaa}) upon hydrostatic compression as shown in Fig.~\\ref{fig:dosgp}(d). On the other hand, the sign change of $\\gamma_{ab}$ at 0.1 GPa is indicated in Figs~\\ref{fig:dosgp}(a) to (c) by the coloured DoS according to the values of $\\gamma_{ab}$ at different pressures.\n\nThe sign change of $\\gamma_{ab}$ of the low-frequency modes under pressure can be explained with the help of Fig.~\\ref{fig:Amode1}. As discussed previously, the transverse vibration of the CN--Ag--NC bridge of such modes can pull the connected Co closer hence contract the $c$ dimension of the crystal. With relaxed Co--CN--Ag--NC--Co linkage at zero pressure, reducing the $a$ and $b$ dimensions of the unit cell tends to extend the $c$ dimension due to the hinging mechanism. This would make the transverse vibration that contracts the dimension more difficult and result in positive $\\gamma_{ab}$ in Eq.~\\ref{gammaa}. However, at high hydrostatic pressures, large elongation in the $c$ dimension (due to the giant NLC of the material) would largely extend the Co--CN--Ag--NC--Co linkage. This time, reducing the $a$ and $b$ dimensions with fixed $c$ of the unit cell can accommodate part of the extension in the linkage and make the linkage less taut. This would in turn make it easier for the CN--Ag--NC linkage to vibrate transversely, which would result in negative $\\gamma_{ab}$ in Eq.~\\ref{gammaa}.\n\nThe scissor-like behaviour of the change of linear CTE seen in the upper panel of Fig.~\\ref{fig:cote}, namely the decrease of $\\alpha_{ab}$ accompanied by the increase of $\\alpha_c$ upon compression, makes the combined $\\alpha_V$ in Eq.~\\ref{linearctev} close to zero at high pressure. The large value of $s_{11}$ due to the weak interaction between Ag atoms in the $a$--$b$ plane makes sure that the contribution from $\\overline \\gamma_{ab}$ to $\\alpha_{ab}$ in Eq.~\\ref{linearcteab} dominates, so that $\\alpha_{ab}$ would decrease largely according to the decrease of $\\overline \\gamma_{ab}$. On the other hand, as discussed in the previous section, the effective hinging mechanism guarantees the similarly large increase of $\\alpha_c$. Thus, it is the dispersive interaction together with the hinging mechanism that make $\\alpha_{ab}$ and $\\alpha_c$ change with pressure like a scissor.\n\nAccording to the relation~\\cite{Fangexp2013}\n\\begin{eqnarray}\\label{bsoftening}\n\\left( {\\frac{{\\partial B_V }}{{\\partial T}}} \\right)_p = B_V^2 \\left( {\\frac{{\\partial \\alpha _V }}{{\\partial p}}} \\right)_T,\n\\end{eqnarray}\nthe giant reduction of $\\alpha_V$ with pressure implies a giant decrease of $B$ on heating. From Eq.~\\ref{bsoftening}, $B(T)$ can be calculated as\n\\begin{eqnarray}\\label{bulkmodulustemperature}\nB(T) = \\left( {\\frac{1}{{B_{T = 0} }} - \\int_0^T {\\frac{{\\partial \\alpha }}{{\\partial p}}} dT} \\right)^{ - 1},\n\\end{eqnarray}\nand is shown in Fig.~\\ref{fig:bt}. From 0.0 to 300 K, $B$ is reduced by $75\\%$ which is much larger than the observed giant softening ($\\sim 45\\%$) of the isotropic NTE material ZrW$_2$O$_8$~\\cite{Pantea2006} on heating. Such softening results in a value of $B$ in much better agreement with the experimental value of 6.5(3) GPa at room temperature~\\cite{GoodwinAgCoCN2008}, as shown in Fig.~\\ref{fig:bt}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{BT3.pdf}\n\\end{center}\n\\caption{\\label{fig:bt} Calculated temperature dependence of the bulk modulus $B$ of Ag$_3$Co(CN)$_6$ at zero pressures using Eq.~\\eqref{bulkmodulustemperature}. The great softening of $B$ on heating brings the calculated value in better agreement to the experimental one at room temperature.}\n\\end{figure}\n\n\n\\section{Conclusions}\n\n\nBy including the dispersive correction in the DFT GGA calculation, we are now able to correctly reproduce the ground state of Ag$_3$Co(CN)$_6$ as well as the the high-pressure phase of the material having the interdigitated structure.\n\nWe found that, by using the DFPT+D calculated phonons, it is almost the same set of low-frequency modes that contribute to both linear PTE and NTE of the material with their linear Gr\\\"{u}neisen parameters showing similar magnitudes but with opposite sign. Such modes, as those around the wave vector A and the middle point along the H$\\rightarrow$K, correspond to the transverse vibrations of the CN--Ag--NC bridge within the Co--CN--Ag--NC--Co linkage that can transfer the expansion in the $a$($b$) dimension to the contraction in the $c$ dimension.\n\nFrom the DFPT+D results, we have predicted that the value of $\\alpha_c$ of Ag$_3$Co(CN)$_6$ increases on compression, contrary to what is normally seen in PTE and NTE materials. In turn this suggests that the NLC of Ag$_3$Co(CN)$_6$ will be enhanced on heating. We also predicted an exceptionally large reduction in volume CTE on compression, which corresponds to the change of sign of the linear Gr\\\"{u}neisen parameters under pressure together with the right elasticity of the material. The latter is based on the weak interactions between Ag atoms in the $a$--$b$ plane and the effective hinging mechanism in the structure. This property also suggests a giant softening of the material on heating with a reduction in the bulk modulus of about $75\\%$ from 0--300 K.\n\nThe method and results presented in this work would be able to apply to other framework materials, such as KMn[Ag(CN)$_2$]$_3$ and Zn[Au(CN)$_2$]$_2$, that have atoms (e.g. Ag and Au) with large dispersive interactions and show large anisotropic properties of PTE\/NTE as well as NLC~\\cite{Cairns2012,Kamali2013,Cairns2013,Gatt2013}. It would be interesting in a future study to see if the phenomena of heat enhancement of NLC and giant reduction of volume CTE on compression predicted for Ag$_3$Co(CN)$_6$ can also be found in these other materials. It would be also interesting to use other schemes to include the van der Waals dispersion correction (such as the use of non-local Langreth-Lundqvist functional~\\cite{Dion2004} in the DFT) in calculating properties of these materials and compare the results.\n\n\n\\begin{acknowledgements}\nWe gratefully acknowledge financial support from the Cambridge International Scholarship Scheme (CISS) of the Cambridge Overseas Trust and Fitzwilliam College of Cambridge University (HF). We thank the CamGrid high-throughput environment of the University of Cambridge. We thank the UK HPC Materials Chemistry Consortium, funding by EPSRC (EP\/F067496), to allow us to use the HECToR\/ARCHER national high-performance computing service provided by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and Technology through EPSRC's High End Computing programme.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzazuc b/data_all_eng_slimpj/shuffled/split2/finalzzazuc new file mode 100644 index 0000000000000000000000000000000000000000..bc4267dae4ae1b213853b88bd9eab9a1c9e5fe6f --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzazuc @@ -0,0 +1,5 @@ +{"text":"\\section{Basic Terminology and Notation}\n\\label{notation}\n\nWe assume an introductory knowledge in formal language and automata theory \\cite{Hopcroft}, including\nthe definitions of finite automata, context-free languages, Turing machines etc. Next, some notations are given.\nAn {\\em alphabet} is a finite set of symbols. A {\\em word} $w$ over $\\Sigma$ is any finite sequence of\nsymbols from $\\Sigma$. Given an alphabet $\\Sigma$, then $\\Sigma^*$ is the\nset of all words over $\\Sigma$, including the empty word $\\epsilon$, and $\\Sigma^+$ is the set\nof non-empty words over $\\Sigma$. A {\\em language} $L$\nis any subset of $\\Sigma^*$. The {\\em complement} of a language $L \\subseteq \\Sigma^*$ with respect to $\\Sigma$ is\n$\\overline{L} = \\Sigma^* - L$. Given a word $w \\in \\Sigma^*$, $w^R$ is the reverse of $w$,\n $w[i]$ is the $i$'th character of $w$, and $|w|$ is the length of $w$.\nGiven an alphabet $\\Sigma = \\{a_1, \\ldots, a_m\\}$ and $a \\in \\Sigma$, $|w|_a$ is the number of $a$'s in $w$. The {\\em Parikh map} of $w$ is\n$\\psi(w) = ( |w|_{a_1}, \\ldots, |w|_{a_m})$, which is extended to the\nParikh map of a language $L$, $\\psi(L) = \\{\\psi(w) \\mid w \\in L\\}$.\nAlso, $\\alp(w) = \\{ a \\in \\Sigma \\mid |w|_a>0\\}$. \nGiven languages $L_1,L_2$, the {\\em left quotient} of $L_2$ by $L_1$, $L_1^{-1}L_2 = \\{ y \\mid xy \\in L_2, x \\in L_1\\}$, and the {\\em right quotient} of $L_1$ by $L_2$ is $L_1 L_2^{-1} = \\{x \\mid xy \\in L_1, y \\in L_2\\}$.\n\n\n\nFor a language $L$, let $f_L(n)$ be the number of strings of length $n$ in $L$. \nA language $L$ is called {\\it counting-regular} if there exists a regular language $L'$ such that for all integers $n \\geq 0$, $f_L(n)$ = $f_{L'}(n)$. Furthermore, $L$ is called {\\it strongly counting-regular} if, for any regular language $L_1$, $L \\cap L_1$ is counting-regular.\nLet $k \\ge 1$. \nA language $L$ is $k$-slender if $ f_L(n) \\le k$ for all $n$, and $L$ is thin if is is $1$-slender. Furthermore, $L$ is slender if it is $k$-slender for some $k$.\n\n\nA language $L\\subseteq \\Sigma^*$ is {\\em bounded} if there exist (not necessarily distinct)\nwords $w_1, \\ldots, w_k \\in \\Sigma^+$ such that $L \\subseteq w_1^* \\cdots w_k^*$;\nit is also {\\em letter-bounded} if each of $w_1, \\ldots, w_k$ are letters. \n\nLet $\\mathbb{N}$ be the set of positive integers and $\\mathbb{N}_0 = \\mathbb{N} \\cup \\{0\\}$. \nA {\\em linear set} is a set\n$Q \\subseteq \\mathbb{N}_0^m$ if there exist $\\vec{v_0},\n\\vec{v_1}, \\ldots, \\vec{v_n}$ such that\n$Q = \\{\\vec{v_0} + i_1 \\vec{v_1} + \\cdots + i_n \\vec{v_n} \\mid\ni_1, \\ldots, i_n \\in \\mathbb{N}_0\\}$. The vector\n$\\vec{v_0}$ is called the {\\em constant}, and \n$\\vec{v_1}, \\ldots, \\vec{v_n}$ are called the {\\em periods}. We also say that $Q$ is the\nlinear set generated by constant $\\vec{v_0}$ and periods $\\vec{v_1}, \\ldots, \\vec{v_n}$.\nA linear set is called {\\em simple} if the periods form a basis.\nA {\\em semilinear set} is a finite union of linear sets.\nAnd, a semilinear set is {\\em semi-simple} if it is the finite disjoint union of simple sets\n\\cite{Flavio,Sakarovitch}.\n\n\n\n\n\nA language $L \\subseteq \\Sigma^*$ is {\\em semilinear} if $\\psi(L)$ is a semilinear set.\nEquivalently, a language $L$ is semilinear if and only if there is a regular language\n$L'$ with the same Parikh map (they have the same commutative closure) \\cite{harrison1978}.\nThe {\\em length set} of a language $L$ is the set $\\{n \\mid w \\in L, n = |w|\\}$.\nA language $L$ is {\\em length-semilinear} if the {\\em length set} is a semilinear set;\ni.e.\\ after mapping all letters of $L$ onto one letter, the Parikh map is semilinear, which is equivalent to it being regular.\n\n\nA language $L \\subseteq \\Sigma^+$ is a {\\em code} if $x_1 \\cdots x_n = y_1 \\cdots y_m, x_i, y_j \\in L$,\nimplies $n=m$ and $x_i = y_j$, for $i$, $1 \\leq i \\leq n$. Also, $L$ is a prefix code if\n$L \\cap L \\Sigma^+ = \\emptyset$, and a suffix code if $L \\cap \\Sigma^+ L = \\emptyset$. \nSee \\cite{CodesHandbook} for background on coding theory.\n\nA language family ${\\cal L}$ is said to be {\\em semilinear} if all $L \\in {\\cal L}$ are semilinear. It is said that language family ${\\cal L}$ is a {\\em trio}\nif ${\\cal L}$ is closed under inverse homomorphism, $\\epsilon$-free homomorphism, and intersection with regular languages. \nIn addition, ${\\cal L}$ is a full trio if it is a trio closed under homomorphism; and a full AFL is a full trio closed\nunder union, concatenation, and Kleene-*.\nMany well-known families form trios, such as each family of the Chomsky hierarchy \\cite{Hopcroft}.\nThe theory of these types of families is explored in \\cite{G75}.\nWhen discussing a language family that has certain properties, such as a semilinear trio, we say that the family has {\\em all properties effective} if all these properties provide effective constructions. For semilinearity, this means that \nthere is an effective construction to construct the constant and periods from each linear set making up the semilinear set.\n\n\n\nA pushdown automaton $M$ is $t$-reversal-bounded if $M$ makes at most\n$t$ changes between non-decreasing and non-increasing the size of its pushdown on every input, and it is reversal-bounded if it is $t$-reversal-bounded for some $t$. A pushdown automaton $M$ is unambiguous if, for all $w\\in \\Sigma^*$, there is at most one accepting computation of $w$ by $M$. More generally, $M$ is $k$-ambiguous if there are at most $k$ accepting computations of $w$. \n\nLet $\\DPDA$ ($\\PDA$) denote the class of deterministic (nondeterministic) pushdown automata (and languages). We also use $\\CFL = \\PDA$, the family of context-free languages.\n\nWe make use of one particular family of languages, which we will only describe intuitively (see \\cite{ibarra1978} for formal details). Consider a nondeterministic machine with a one-way input and $k$ pushdowns, where each pushdown only has a single symbol plus a bottom-of-stack marker. Essentially, each pushdown operates like a counter, where each counter contains some non-negative integer, and machines can add or subtract one, and test for emptiness or non-emptiness of each counter. \nWhen $k = 1$, we call these nondeterministic (and deterministic) one counter machines. \nAlthough such a machine with two counters has the same power as a Turing machine \\cite{Hopcroft}, if the counters are restricted, then the machine can have positive decidability properties. Let $\\NCM(k,t)$ be the family of $k$ counter machines where the counters are $t$-reversal-bounded counters, and let $\\DCM(k,t)$ be the deterministic subset of these machines. Also, let $\\NCM$ be $\\bigcup_{k,t\\geq 1} \\NCM(k,t)$ and $\\DCM$ be $\\bigcup_{k,t \\geq 1} \\DCM(k,t)$. The class of $\\PDA$'s or $\\DPDA$'s augmented with reversal-bounded counter machines is denoted by $\\NPCM$ or $\\DPCM$ \\cite{ibarra1978}.\nIt is known that $\\NCM$ has a decidable emptiness problem, and $\\DCM$ also has a decidable containment problem, with both being closed under intersection \\cite{ibarra1978}. Furthermore, $\\NCM$ \nand $\\NPCM$ are semilinear trios.\n\n\\section{Counting-Regular Languages}\n\\label{sec:countingregular}\n\nObviously every regular language is counting-regular, and so are many non-regular languages. For example, $L_{Sq}$ = $\\{ w w \\mid w \\in \\{a, b\\}^*\\}$ is counting-regular since $L'$ = $(a(a+b))^*$ has the same number of strings of length $n$ as $L_{Sq}$ for all $n$. It is a simple exercise to show that $L_{Sq}$ is actually strongly counting-regular.\nIt is also easy to exhibit languages that are not counting-regular, e.g., $L_{bal}$ = $\\{ w \\mid w$ is a balanced parentheses string$\\}$ over alphabet $\\{ [, ]\\}$. The reason is that there is no regular language $L$ such that the number of strings of length $2n$ in $L$ is the $n$'th Catalan number, as will be seen from the characterization theorem stated below.\n\nOur goal in this section is to explore general families of languages that are counting-regular. \nWe briefly note the following:\n\\begin{theorem}\nAny family ${\\cal L}$ that contains some non-length-semilinear language $L$ is not counting regular.\n\\end{theorem}\n\\begin{proof}\nGiven such an $L$, then examine the set of all $n$ with $f_L(n) >0$. But, as every regular language $R$\nis length-semilinear, the set of all $n$ with $f_R(n) >0$ must be different.\n\\qed \\end{proof}\nThus, it is immediate that e.g.\\ any family that contains some non-semilinear unary language\nmust not be counting-regular. This includes such families as those accepted by checking stack\nautomata \\cite{CheckingStack}, and many others. Of interest are exactly what families of length-semilinear languages\nare counting-regular. We will investigate these questions here.\n\n\n\nThe following result is due to Berstel \\cite{Berstel2}.\n\n\\begin{theorem}\n\\label{regularcharacterization}\nLet $L$ be a regular language and let $f_L(n)$ denote the number of strings of length $n$. Then, one of the following holds:\n\\begin{enumerate}\n\\item[(i)] $f_L(n)$ is bounded by a constant $c$.\n\n\\item[(ii)] There is an integer $k > 0$ and a rational $c > 0$ such that $limsup_{n \\rightarrow \\infty} {{f_L(n)} \\over {n^k}} = c$.\n\n\\item[(iii)] There exists an integer $k \\geq 0$ and an algebraic number $\\alpha$ such that \\\\\n$limsup_{n \\rightarrow \\infty} {{f_L(n)} \\over {\\alpha^n n^k}} = c$ (where $c \\neq 0$ is rational).\n\\end{enumerate}\n\\end{theorem}\n\nWe also need the following theorem due to Soittola \\cite{Berstel}. We begin with the following definitions. \nA sequence $s$ = $\\{s_n\\}$, $n \\geq 0$, is said to be the {\\it merge} of the sequences $\\{s^{(0)}\\}, \\ldots , \\{s^{(p-1)}\\}$, where $p$ is a positive integer, if \n$s_n^{(i)}$ = $s_{i+np}$ for $0 \\leq i \\leq p-1$. A sequence $\\{s_n\\}$ is said to be {\\it regular} if there exists a regular language $L$ such that $f_L(n)$ = $s_n$ for all $n$.\n\n\nNext we define a $\\mathbb{Z}$-rational sequence as follows: A sequence $\\{s_n\\}$, $n \\geq 0$, is $\\mathbb{Z}$-rational if there is a matrix $M$ of order $d \\times d$, a row vector $u$ of order $1 \\times d$, and a column vector $v$ of order $d \\times 1$ such that $s_n$ = $u \\ M^n v$. All the entries in $M$, $u$ and $v$ are over $\\mathbb{Z}$.\nA $\\mathbb{Z}$-rational sequence is said to have a {\\it dominating pole} if its generating function $s(z)$ = $\\sum_{n=0}^{\\infty} s_n z^n$ \ncan be written as a rational function $s(z)$ = $p(z)\/q(z)$ where $p$ and $q$ are relatively prime polynomials, and $q$ has a simple root $r$ such that \n$r' > r$ for any other root $r'$.\n\n\nSoittola's theorem \\cite{Berstel} can be stated as follows.\n\n\\begin{theorem}\n\\label{Soittola}\nA $\\mathbb{Z}$-rational sequence with non-negative terms is regular if and only if it is the merge of $\\mathbb{Z}$-rational sequences with a \ndominating pole.\n\\end{theorem}\n\nWe will also need the following theorem due to B\\'{e}al and Perrin \\cite{Beal}.\n\\begin{theorem}\n\\label{Beal}\nA sequence $s$ is the generating sequence of a regular language over a $k$-letter alphabet if and only if both the sequences $s$ = $\\{s_n\\}$, $n \\geq 0$ \nand $t$ = $\\{(k^n - s_n)\\}$, $n \\geq 0$ are regular.\n\\end{theorem}\n\nWe now show the main result of this section. This result can be viewed as a strengthening\n of a result of Baron and Kuich \\cite{Kuich} that $L(M)$ has a rational generating function if $M$ is an unambiguous finite-turn $\\PDA$. \n \n \\begin{comment}\n First a normal form is required.\nA $\\PDA$ is in {\\em normal form} if\n\\begin{enumerate}\n\\item the stack starts with $Z_0$, always pushes some letter, and accepts by final state and empty stack (ending at $Z_0$),\n\\item each transition either pushes one symbol onto the stack (a push move) or pops one off the stack (a pop move).\n\\end{enumerate}\nThus, there can be no moves applied that do not change the size of the stack.\n\\begin{lemma}\n\\label{normalform}\nLet $t \\geq 1$. Given a one-way unambiguous $t$-reversal-bounded $\\PDA$ $M$, there\nis a one-way unambiguous $t$-reversal-bounded $\\PDA$ $M'$ in normal form such that $L(M) = L(M')$.\n\\end{lemma}\n\\begin{proof}\nIt is clear that the first condition can be assumed. \nIt can also be assumed that each transition that replaces the topmost stack symbol with another symbol does not change the stack (i.e. it is a `stay' transition) as follows:\nafter a push, nondeterminism is used to guess (and verify) the final value of the topmost\nsymbol before either a pop or push occurs. After a pop, instead of changing the topmost symbol, it guesses whether the next\nchange is another pop or a push. If it is a pop, the simulated top symbol can be stored in the finite control. If it is a push, then the new machine pushes a new tagged symbol associated with the new top of the stack (which when popped, pops the untagged symbol beneath).\n\n\nGiven $M$ satisfying these conditions, construct $M'$ in normal form as follows: $M'$ simulates every push transition exactly as in $M$, but immediately after simulating a push transition (and at the beginning of the computation), it guesses whether or not the next change to the stack will be a push or a pop (at a reversal). If it guesses a push, then it simulates all stay transitions by pushing a new symbol $\\#_1$ for each such stay transition applied. Then, immediately before simulating the next push transition, it pushes arbitrarily many copies of the symbol $\\#_2$ onto the stack (on $\\epsilon$ transitions), then immediately continues the simulation of the next push transition. If it guesses the next transition is a pop (i.e.\\ a reversal occurs), then $M'$ guesses whether an even or odd number of stay transitions are to be applied. If it guesses even, it simulates some number of stay transitions while pushing $\\#_1$, then nondeterministically switches to simulating while popping $\\#_1$, then continuing to simulate $M$'s pop moves. If it guesses odd, it applies one push on $\\#_1$, then proceeds as with the even case.\nIn a similarly fashion, $M'$ simulates each pop transition exactly as in $M$, and then simulates each stay transition but it must pop a $\\#_2$ for each such stay transition simulated. Before the next pop (or push if there is a reversal) transition simulated, it pops all $\\#_1$ symbols first (on $\\epsilon$ input). It is clear that $L(M) = L(M')$, and that $M'$ is $t$-reversal-bounded. \n \n For unambiguity, for each word $w \\in L(M)$, there is exactly one accepting computation of $M$ on $w$. For each symbol $c$ pushed onto the pushdown, there is some some unique number of stay transitions applied, $l$, say, before another transition is simulated that changes the stack. Assume first that the next move that changes the stack is a push. Then later in the computation before the next time this same symbol $c$ reaches the top of the stack again, there is some pop move followed by a sequence of $r$ say, stay transitions with $c$ on top of the stack before another move is simulated that changes the stack. Since $l$ and $r$ are unique, when $M'$ simulates $M$, in any accepting computation, it must push exactly $l$ copies of $\\#_1$ during the stay transitions, and then must push exactly $r$ copies of $\\#_2$ on $\\epsilon$ input since later in the computation $r$ is unique. Similarly after the corresponding pop transition, since $r$ is unique, it must pop $r$ copies of $\\#_2$ during the stay transitions simulated, and it must then pop $l$ copies of $\\#_1$ on $\\epsilon$ input. Next, assume the next move that changes the stack is a pop (at a point of reversal from non-decreasing to non-increasing), then the number of stay transitions $l$ is unique, and if $l$ is even then only the even strategy above (pushing $l\/2$ copies of $\\#_1$ following by popping them) leads to acceptance, and similarly with the odd case. Hence, there is only one accepting computation of $M'$ on $w$.\n\\qed\n\\end{proof}\n\nThis normal form is helpful in showing the following result:\n\\begin{theorem}\n\\label{main2}\nLet $M$ be an unambiguous reversal-bounded $\\PDA$ over a $k$ letter alphabet $\\Sigma$.\nThen, $L(M)$ is strongly counting-regular where the regular language is over a $k+1$ letter alphabet. \n\\end{theorem}\n\\begin{proof}\n\nIt is already known that all languages in $\\DCM(1,t)$ are counting-regular \\cite{R}, but here the result is shown to work for reversal-bounded pushdown automata, and with nondeterminism as long as the $\\PDA$'s are unambiguous.\nMoreover, an important improvement we make here is the size of the alphabet associated with this construction. Specifically, the proof presented in \\cite{R} used an alphabet of size $3k$ for the regular language $L'$ such that $f_{L'}$ = $f_{L(M)}$. Here we show that there is a regular language $L'$ over an alphabet of size $(k+1)$ such that $f_{L'}$ = $f_{L(M)}$.\n\nLet $M$ be $t$-reversal-bounded, $t \\geq 1$.\nFirst we note that it is enough to show that $L(M)$ is counting-regular. The stronger claim that $L(M)$ is strongly counting-regular can be seen as follows: for an arbitrary $\\DFA$ $M_1$, using the standard proof of closure of $\\PDA$'s under intersection with regular languages \\cite{Hopcroft}, the $\\PDA$ constructed accepting $L(M_1) \\cap L(M)$ is unambiguous and $t$-reversal-bounded. Hence, proving counting-regularity of the language accepted by each $t$-reversal-bounded unambiguous $\\PDA$ is enough.\n\n\nWe will present the proof in detail for the case $t$ = 1 and at the end, describe how to extend the result for a general $t$.\nWe assume without loss of generality, by Lemma \\ref{normalform}, that $M = (Q,\\Sigma,\\Gamma,\\delta,q_0,Z_0,F)$ is in normal form.\n\nIn the following, we consider a specific input string $w = w_1 w_2\\ \\cdots\\ w_n \\in L(M)$,\nwhere $w_i \\in \\Sigma, 1 \\leq i \\leq n$, and the computation described below refers to the unique accepting computation of $M$ on $w \\neq \\epsilon$ (the empty string can be handled as a special case).\nConsider an accepting (and unique) computation of $M$ on $w$:\n\\begin{equation}\n\\label{acceptingcomp}\n(q_0,x_0,\\gamma_0) \\vdash (q_1,x_1,\\gamma_1) \\vdash \\cdots \\vdash (q_m,x_m,\\gamma_m),\n\\end{equation}\nwhere $q_i \\in Q, x_i \\in \\Sigma^*, \\gamma_i \\in \\Gamma^*, 0 \\leq i \\leq m, w = x_0, \\gamma_0 = \\gamma_m = Z_0$ (the bottom-of-stack symbol), $x_m = \\epsilon, \\ q_m \\in F$.\nFor each $i$, $0 \\leq i \\leq m$, let $c_i$ be the top of the pushdown $\\gamma_i$.\nSince every transition applied before the reversal pushes a character, then after the reversal, each transition applied pops a character, it is immediate that the first $m\/2$ transitions applied must be push transitions, the last $m\/2$ must be pop transitions, and for each $i$, $0 \\leq i \\leq m\/2, c_i = c_{m-i}$.\nNote that some of these moves could be $\\epsilon$ moves (that do not depend on the input symbol scanned by the input head and do not advance the input head), and others depend on the input symbol and result in advancing the input head on the tape. \n\n\nLet $e$ be a new symbol ($e$ used in place of $\\epsilon$), \nand let $\\Sigma_e = \\Sigma \\cup \\{e\\}$. For $1 \\leq i \\leq m\/2$, let $\\sigma_i \\in \\Sigma_e$ be the letter \nread in Equation (\\ref{acceptingcomp}) when the $i$'th symbol gets pushed on the stack, with an $e$ in place of every $\\epsilon$ transition.\nSimilarly, for $1 \\leq i \\leq m\/2$, let $\\tau_i \\in \\Sigma_e$ be the letter read when the $i$'th symbol above $Z_0$ in the stack gets popped, with $e$ in \nplace of $\\epsilon$.\nLet $z = \\sigma_1 \\sigma_2 \\cdots \\sigma_{m\/2} \\tau_{m\/2} \\tau_{m\/2-1} \\cdots \\tau_1$ be over $\\Sigma_e^*$,\nwhich is the word read in Equation (\\ref{acceptingcomp}) with $e$ in place of $\\epsilon$.\n\nLet $\\Sigma_1 = \\{a^{+}, a^{-}, a^{+,\\epsilon}, a^{-,\\epsilon} \\mid a \\in \\Sigma\\}$.\nFor a pair $a,b \\in \\Sigma_e$, let \n$$f(a,b) = \\begin{cases}\n a^{+} b^{-} & \\mbox{if~} a,b \\in \\Sigma,\\\\\na^{+,\\epsilon} & \\mbox{if~} a \\in \\Sigma, b = e,\\\\\nb^{-,\\epsilon} & \\mbox{if~} b \\in \\Sigma, a = e,\\\\\n\\epsilon & \\mbox{if~} a = b = e.\n\\end{cases}$$\nFor $1 \\leq i \\leq m\/2$, let $y_i = f(\\sigma_i,\\tau_i)$, and let $y = y_1 \\cdots y_{m\/2}$.\n\n\nHence, $y \\in \\Sigma_1^*$ satisfies the condition that $|y|$ = $|w|$. Let $\\code(w)$ denote the string $y$ as defined above. Define the language $L' = \\{\\code(w) \\ | \\ w \\in L(M)\\}$. \n\nTo complete the proof, we will show the following: (a) $L'$ is regular and (b) $f_L(n)$ = $f_{L'}(n)$ for all $n$. \n\n\nTo show (a), we create an $\\NFA$ (with $\\epsilon$ transitions and multiple initial states) $M'$ accepting $L'$.\nIntuitively, $M'$ simulates $M$ on the $+$ and $+,\\epsilon$ marked symbols, and in parallel, $M'$ simulates\n$M$ ``in reverse'' on the $-$ and $-,\\epsilon$ marked symbols, also verifying that when there are consecutive symbols, one marked with $+$, and the next marked with $-$, they correspond to manipulating the same stack symbol; i.e., that the $i$'th symbol pushed matches the same symbol popped.\nThe state set of $M'$ is $Q' = Q \\times Q \\times \\Gamma \\cup Q \\times Q \\times \\Gamma \\times \\{-\\}$. Here, the first and third component is the state and topmost stack symbol in the forward simulation, and the second component is the state in the reverse simulation with the same topmost symbol as the third component, and the last optional component is a\nsignal that the next transition must only apply to the reverse simulation. \nThe initial state set is $\\{q_0\\} \\times F \\times \\{Z_0\\}$, and the final state set $F'= \\{(q,q,c) \\mid q \\in Q,c \\in \\Gamma\\}$. The transition function $\\delta'$ is\nconstructed as follows:\n\\begin{enumerate}\n\\item For all pairs of transitions, $(q',push(c))) \\in \\delta(q,a,d)$ and $(p',pop) \\in \\delta(p,b,c)$, where $p,q,p',q' \\in Q, a,b \\in \\Sigma, c,d \\in \\Gamma$, add\n$$(q',p',c,-) \\in \\delta'((q,p',d),a^+)\\mbox{~and~} (q',p,c) \\in \\delta'((q',p',c,-), b^-).$$\n\\item For all pairs of transitions $(q',push(c)) \\in \\delta(q,a,d)$ and $(p',pop) \\in \\delta(p, \\epsilon,c)$, where $p,q,p',q' \\in Q, a \\in \\Sigma, c,d \\in \\Gamma$, add\n$$(q',p,c) \\in \\delta'((q,p',d), a^{+,\\epsilon}).$$\n\\item For all pairs of transitions $(q',push(c)) \\in \\delta(q,\\epsilon,d)$ and $(p',pop) \\in \\delta(p,b,c)$, where $p,q,p',q' \\in Q, b \\in \\Sigma, c,d \\in \\Gamma$, add\n$$(q',p,c) \\in \\delta'((q,p',d) , b^{-,\\epsilon}).$$\n\\item For all pairs of transitions $(q',push(c)) \\in \\delta(q,\\epsilon,d)$ and $(p',pop) \\in \\delta(p, \\epsilon,c)$, where $p,q,p',q' \\in Q, c,d \\in \\Gamma$, add\n$$(q',p,c) \\in \\delta'((q,p',d),\\epsilon).$$\n\\end{enumerate}\n\nGiven the accepting computation of $w$ in Equation (\\ref{acceptingcomp}), $M'$\naccepts $\\code(w) = y = y_1 \\cdots y_{m\/2}$ as follows. \nFirst, $M'$ starts in $(q_0,q_m, Z_0)$. We will show by induction that for each $i$,\n$0 \\leq i \\leq m\/2$, $$(q_i, q_{m-i},c_i) \\in \\hat{\\delta}((q_0,q_m,Z_0), y_1 \\cdots y_i),$$ (where $i=0$ implies $y_1 \\cdots y_i = \\epsilon$). The base case is true as when $i = 0$, $(q_0,q_m,Z_0) = (q_i, q_{m-i},c_i)$.\nAssume it is true for $i$, $0 \\leq i < m\/2$. First assume that from\n$(q_i,x_i,\\gamma_i)$ to $(q_{i+1},x_{i+1},\\gamma_{i+1})$, a transition that reads $a\\in \\Sigma$ is applied, and from $(q_{m-i-1},x_{m-i-1},\\gamma_{m-i-1})$ to\n$(q_{m-i},x_{m-i},\\gamma_{m-i})$, a transition that reads $b \\in \\Sigma$ is applied. Then $y_{i+1} = a^+ b^-$, and using the sequence of two transitions created in step 1,\n$(q_{i+1}, q_{m-i-1},c_{i+1}) \\in \\hat{\\delta}(q_i,q_{m-i},c_i),y_{i+1})$. Similarly for the other three cases, and hence the induction follows. In particular,\n$(q_{m\/2},q_{m\/2},c_{m\/2}) \\in \\hat{\\delta}((q_0,q_m,Z_0), \\code(w)) \\cap F'.$\n\n\n\nConversely, given an accepting computation of some word $y \\in \\Sigma_1^*$, the first component must change as in $M$ while pushing is occurring, and the second component must \nchange as in $M$ in reverse from the final state on the same sequence of stack symbols. Thus, there exists an accepting computation of $h_+(y) h_-(y)^R$ in $M$ where $h+$ is a homomorphism mapping each letter marked with $+$ or $+,\\epsilon$ to the corresponding letter of $\\Sigma$, and erasing all others, and $h_-$ is a homomorphism that takes each letter marked with $-$ or $-,\\epsilon$ and maps it onto the corresponding letter of $\\Sigma$, and erases all others.\n\n\n\n\nFinally, we note that the mapping from $w \\rightarrow \\code(w)$ is bijective. The mapping in the forward direction is well-defined since $M$ (an unambiguous $\\PDA$) has a unique accepting sequence on any input $w \\in L(M)$. Thus the coding follows a deterministic procedure and $\\code(w)$ is well-defined and is unique for a given $w \\in L(M)$. Conversely, given $y$ accepted by $M_1$, it can be decoded in to a unique string $w \\in L(M)$ such that $y$ = $\\code(w)$ (using \n$h_+(y) h_-(y)^R$ as above). \n\nNext, we provide a sketch of how the above proof can be generalized to unambiguous $\\PDA$'s $M$ that reverse the stack a finite number of times. The coding is similar to the one described above. Suppose, the number of reversals is $t$. We can align the input positions corresponding to the same symbol getting pushed and popped by dividing the input string into several segments where in each segment, $M$ only pushes or only pops. As an example, (see Figure \\ref{revboundedfigure}) suppose the $\\PDA$ makes two reversals and suppose the height of the stack corresponding to the input position is as shown in the figure. In this case, the input $w$ is divided into $w_1 \\cdots w_6$ and the segments aligned to create $\\code(w)$ will be $(w_1, w_6^R)$, followed by $(w_2, w_3^R)$, followed by $(w_4, w_5^R)$. Thus, in this case, the states of $M'$ have four state components where it starts by deriving $w_1$ and $w_6^R$ in the first and last component until it reaches line $l$ in the figure, where it continues to derive $w_2$ and $w_3^R$ using the first and second component until those reach the same state, then it derives $w_4$ and $w_5^R$ using the third and fourth component.\nMore generally, in order to globally check that the simulation is correct, the states associated with all the segments must be kept in a vector of states by the simulating $\\NFA$ with $t+1$ components to carry out the simulation and check the `global' correctness of the simulation. The $\\NFA$ will guess when each block starts and ends, and keeps track of the states associated with the blocks in the global vectors of states, and verifies the start and end states of the blocks.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{graph.pdf}\n\\caption{A $2$-reversal-bounded $\\PDA$ and its pattern of stack change.}\n\\label{revboundedfigure}\n\\end{center}\n\\end{figure}\n\n\nWe have shown that, for any unambiguous $\\PDA$ $M$ that reverses the stack a finite number of times, there is a regular language $L(M_1)$ such that $f_{L(M_1)}(n)$ = $f_{L(M)}(n)$ for all $n$. The size of the alphabet over which $L(M_1)$ is defined is $4s$ where $s$ = $|\\Sigma|$. \n\nSurprisingly, we can reduce the size of the alphabet from $4s$ to $s+1$. More precisely, let $M$ be an unambiguous $\\PDA$ that reverses the stack a finite number of times and $s$ be the size of the alphabet over which $M$ is defined. We will show that there is a regular language $L_1$ over an alphabet of size $s+1$ such that $f_{L(M)}(n)$ = $f_{L_1}(n)$ for all $n$. As we have shown above, there is a regular language $L(M_1)$ such that $f_{L(M_1)}(n)$ = $f_{L(M)}(n)$ for all $n$. Thus, by Theorem \\ref{Soittola}, the generating function $a(z)$ associated with the sequence $(f_{L(M)}(n))$, $n \\geq 0$, is a rational function $p(z)\/q(z)$ that satisfies the conditions of Theorem \\ref{Soittola}. Let $b_n$ be defined as $b_n$ = $(k+1)^n - a_n$. The generating function $b(z)$ for the sequence $(b_n)$, $n \\geq 0$, is ${p(z)} \\over {(1-(k+1)z)\\ q(z)}$. This is a rational function with dominating pole $1 \\over{k+1}$ and hence it satisfies Theorem \\ref{Soittola}. Since both $(a_n)$ and $(b_n)$ = $((k+1)^n - a_n)$ are regular sequences, by Theorem \\ref{Beal}, there is a regular language $L_1$ over a $k+1$ letter alphabet such that $f_{L_1}(n)$ = $f_{L(M)}(n)$ for all $n$.\n\\qed \\end{proof}\n\n\\end{comment}\n\n\nHere, an $\\NTM$ is considered to have a one-way read-only input tape, plus a two-way read\/write worktape that uses blank symbol $\\blank$. Such a machine is said to be {\\em reversal-bounded} if there is a bound on the number of changes in direction between moving left and right and vice versa on the worktape.\nAn $\\NTM$ is said to be in {\\em normal form} if,\nwhenever the worktape head moves left or right (or at the beginning of a computation) to\na cell $c$, then the next transition can change the worktape letter in cell $c$, but then the cell does not change again until after the worktape head moves.\nThis essentially means that if there is a sequence of `stay' transitions (that do not\nmove the read\/write head) followed by a transition that moves, then only the first such transition can change the tape contents.\n\\begin{lemma}\nGiven an unambiguous reversal-bounded $\\NTM$ $M$, there exists an unambiguous reversal-bounded\n$\\NTM$ $M'$ in normal form such that $L(M) = L(M')$.\n\\end{lemma}\n\\begin{proof}\nGiven $M$, an $\\NTM$ $M'$ is constructed as follows:\nAfter a transition that moves the read\/write head (or at the first move of the computation),\ninstead of simulating a `stay' transition directly, $M'$ guesses the final value to be\nwritten on the cell before the head moves, and writes it during the first `stay' transition.\n$M'$ then continues to simulate the sequence of stay transitions without changing the\nvalue in the cell but remembering it in the finite control. Then, during the last transition of this sequence\nthat moves the tape head, $M'$ verifies that it guessed the final value correctly.\nCertainly, $M'$ is reversal-bounded if and only if $M$ is reversal-bounded. Further,\nas $M$ is unambiguous, there is only one computation that is accepting on every word in $L(M)$. And, in $M'$ therefore, there can only be one value guessed on each sequence of `stay' transitions that leads to acceptance. Thus, $M'$ is unambiguous. \n\\qed\\end{proof}\n\nIt will be shown that every such $\\NTM$ is counting regular.\n\\begin{theorem}\n\\label{main2}\nLet $M$ be an unambiguous reversal-bounded $\\NTM$ over a $k$ letter alphabet. Then $L(M)$ is strongly counting regular, where the regular language is over a $k+1$ letter alphabet. \n\\end{theorem}\n\\begin{proof}\nFirst we note that it is enough to show that $L(M)$ is counting regular, as the class of languages accepted by unambiguous reversal-bounded $\\NTM$s are closed under\nintersection with regular languages.\n\nLet $M= (Q,\\Sigma,\\Gamma,\\delta,q_0,F)$ be an unambiguous $\\NTM$ that is $t$-reversal-bounded such that\n$M$ is in normal form. Also, assume without loss of generality that $t$ is odd.\n\nIntuitively, the construction resembles the construction that the store languages (the language\nof all contents of the worktape that can appear in an accepting computation) of every such\n$\\NTM$ is a regular language \\cite{StoreLanguages}. \nA $2\\NFA \\ M'$ is constructed that has $t+1$ ``tracks'', and it uses the first track for simulating $M$\nbefore the first reversal, the second track for simulation of $M$ between the first and the second reversal, etc. Thus the input to $M'$ is the set of strings in which the first track contains an input string $x$ of $M$, and the other tracks are annotated with the contents of the read-write tape during moves of $M$ between successive reversals. Formal details are given below.\n\nA $2\\NFA$ $M'= (Q',\\Sigma',\\delta',q_0',F')$ is constructed as follows:\nLet $C = [(Q \\times (\\Sigma \\cup \\{\\epsilon\\})\\times Q \\times \\Gamma) \\cup \\{\\blank\\}]^{t+1}$\n(the $t+1$ tracks; each track is either a blank, or some tuple in $Q \\times (\\Sigma \\cup \\{\\epsilon\\}) \\times Q \\times \\Gamma$). Also, let $C_i$ have $t+1$ tracks where the $i$'th track, for $1 \\leq i \\leq t+1$, contains an element from $ Q \\times (\\Sigma \\cup \\{\\epsilon\\}) \\times Q \\times \\Gamma$, and\nall other tracks contain new symbol $\\#$.\nLet $\\Sigma' = C \\cup C_1 \\cup \\cdots \\cup C_{t+1}$.\nTo simulate moves between the $(i-1)$st reversal and the $i$'th reversal, $M'$ will\nexamine track $i$ of a letter of $C$ to simulate the first transition after the tape head\nmoves to a different cell (or at the first step of a computation), and track $i$ of letters\nof $C_i$ to simulate any stay transitions that occur before the tape head moves again, followed by the transition that moves.\n\nLet $X = C_{t+1}^* \\cdots C_4^* C_2^* C C_1^* C_3^* \\cdots C_t^*$. Let $h_i$ be a homomorphism that maps each string in $(\\Sigma')^*$ to the $i$'th track for symbols in $C \\cup C_i$, and erases all symbols of $C_j, j \\neq i$. Also, $\\bar{h_i}$ is a homomorphism that maps each string in $(\\Sigma')^*$\nto the $i$'th track if it is not $\\blank$, and $\\epsilon$ otherwise, for symbols in $C \\cup C_i$, and erases all symbols of $C_j$, $j \\neq i$.\nThen $M'$ does the following:\n\\begin{enumerate}\n\\item $M'$ verifies that the input $w$ is in $X^*$, and that no letter of $[\\blank]^{t+1}$ is used in $w$.\n\\item $M'$ verifies that for each $i$, $h_i(w) \\in \\blank^* (Q \\times (\\Sigma \\cup \\{\\epsilon\\}) \\times Q \\times \\Gamma)^* \\blank^*$, so blanks can only occur at the ends.\n\\item $M'$ verifies that $w$ represents an accepting computation of $M$ as follows: $M'$ goes to the first symbol of $C$ with a non-blank in the first track. Say \n$\\bar{h_1}(w) = (p_1,a_1,p_1',d_1) \\cdots (p_m,a_m,p_m',d_m), m \\geq 1$. For $j$ from $1$ to $m$,\nwe say that $j$ is from $C$ if $(p_j,a_j,p_j',d_j)$ is from a symbol of $C$ and not $C_1$, and we say $j$ is from $C_1$ otherwise. It verifies from \nleft-to-right on $w$ that for each $j$ from $1$ to $m$, there is a transition of $M$ that switches from $p_j$ to $p_j'$ while reading $a_j \\in \\Sigma \\cup \\{\\epsilon\\}$ as input on worktape letter $\\blank$ if $j$ is in $C$, and $d_{j-1}$ otherwise, replacing it with $d_j$, that:\n\\begin{itemize}\n\\item moves right on the worktape, if $j$ where $R$ is a regular language, $C$ a system of linear constraints and $\\mu$, a length-preserving morphism that is injective on $R \\cap [C]$. The specific language we denote by $L_{\\RCM}$\n is defined as follows:\n$$L_{\\RCM} = \\{w \\in \\{a,b\\}^* \\ |\\ w[|w|_a] = b\\}.$$\n(From the definition, it is clear that $a^n$ is not in $L_{\\RCM}$. But the definition does not specify the status of the string $b^n$ since there is no position 0 in the string. We resolve this by explicitly declaring the string $b^n$ to be in $L_{\\RCM}$ for all $n$.) \n\nWe first observe that $L_{\\RCM}$ is in $\\NCM(1,1)$ and hence is context-free: We informally describe a nondeterministic 1-reversal-bounded 1-counter machine $M$ for $L_{\\RCM}$. Let $w$ = $w_1 \\cdots w_n \\in L_{\\RCM}$ and suppose $|w|_a$ = $i$ and\n$|w_1 w_2 \\cdots w_{i-1}|_a$ = $j$. Then, it is clear that $w_i$ = $b$, $j < i$ and $|w_{i+1} \\cdots w_n|_a$ = $i-j$. Now we describe the operation of $M$ on input string $w$ (not necessarily in $L_{\\RCM}$) as follows. $M$ increments the counter for every $b$ until it reaches position $i$. $M$ guesses that it has reached position $i$, and checks that the symbol currently scanned is $b$, then switches to decrementing phase in which it decrements the counter for each $a$ seen. When the input head falls off the input tape, if the counter becomes 0 it accepts the string. If $w \\in L_{\\RCM}$, it is clear that when $M$ switches from incrementing to decrementing phase, the counter value is $i-j$ and hence when it finishes reading the input, the counter will become 0. The converse is also true. Thus it is clear that $L(M)$ = $L_{\\RCM}$.\n\nAn interesting aspect of the next result is that it applies the simulation technique of Theorem \\ref{main2} twice --- by first bijectively (and in length-preserving way) mapping $L_{\\RCM}$ to a language accepted by $\\DCM(1,1)$, then applying Theorem \\ref{main2} to map it to a regular language. \n\n\\begin{theorem}\n$L_{\\RCM}$ is strongly counting-regular.\n\\end{theorem}\n\\begin{proof} Consider $L'$ = $\\{ [a_1, b_1] [a_2, b_2] \\cdots [a_n, b_n]\\ |\\ a_1 a_2 \\cdots a_n \\in L_{\\RCM}$, and $b_i$ = $c$ for all $1 \\leq i \\leq k$, and $b_i$ = $d$ for all $k+1 \\leq i \\leq n$ where $k$ is the number of $a$'s in $a_1 a_2 \\cdots a_n\\}$. Thus, $L'$ is defined over $\\Sigma_1$ = $\\{[a,c], [b,c], [a,d],[b,d]\\}$. As an example, the string $[a, c][b,c][a, d]$ is in $L'$ since $aba \\in L_{\\RCM}$. It is easy to see that there is a bijective mapping between strings in $L_{\\RCM}$ and strings in $L'$. Given a string $w \\in L_{\\RCM}$, there is a unique string $w'$ in $L'$ whose upper-track consists of $w$ and the lower-track consists of $c^k d^{n-k}$ where $|w|$ = $n$ and $|w|_a$ = $k$. Since $k$ and $n$ are uniquely specified for a given string $w$, the string $w'$ is uniquely defined for a given $w$. Conversely, the projection of a $w' \\in L'$ to its upper-track string uniquely defines a string in $L_{\\RCM}$. \n\nLet $R$ be a regular language over $\\{a, b\\}$. Define a language $R' = \\{[a_1, b_1] [a_2, b_2] \\cdots [a_n, b_n]\\ |\\ a_1 \\cdots a_n \\in L' \\cap R\\}$. It is clear that there is a 1-1 correspondence between strings in $R'$ and $L_{RCM} \\cap R$. Finally, we note that there is a $\\DCM(1,1)$ $M$ that accepts $R'$: $M$ simulates the $\\DFA$ for $R$ on the upper-track input and at the same time, performs the following operation. For every input $[b,c]$, the counter is incremented, and for every $[a,d]$ the counter is decremented. It also remembers the previous symbol scanned, and checks that the previous symbol scanned before the first occurrence of $[b,d]$ is $[a,c]$. Finally, when the counter value becomes 0, the string is accepted. Note that since all $d$'s in the second track occur after the $c$'s, the counter reverses at most once.\n\nSince all $\\DCM(1,1)$'s are $1$-reversal-bounded $\\DPDA$'s, then by Corollary \\ref{cortomain}, $R'$ is counting-regular. Hence, $L_{\\RCM} \\cap R$ is counting-regular.\n\\qed\n\\end{proof}\n\nIn this section, we have shown that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are strongly counting-regular. We also showed that some natural extensions of this class fail to be counting-regular. We presented some relationships between counting-regular languages and the class $\\RCM$. However, our understanding of which languages in $\\DCM$, $\\NCM$, or $\\CFL$ are (strongly) counting-regular is quite limited at this time.\n\n\\section{Bounded Semilinear Trio Languages are Counting-Regular}\n\\label{bounded}\n\nIn this section, we will show that all bounded languages in any semilinear\ntrio are counting-regular.\n\n\nFirst, the following is known \\cite{CIAA2016} (follows from results in \\cite{ibarra1978}, and the fact that all bounded $\\NCM$\nlanguages are in $\\DCM$ \\cite{IbarraSeki}):\n\n\\begin{lemma}\nLet $u_1, \\ldots, u_k \\in \\Sigma^+$, and let $\\phi$ be a function from $\\mathbb{N}_0^k$ to $u_1^* \\cdots u_k^*$ which associates\nto every vector $(l_1, \\ldots, l_k)$, the word $\\phi((l_1, \\ldots, l_k)) = u_1^{l_1} \\cdots u_k^{l_k}$.\nThen the following are true:\n\\begin{itemize}\n\\item given a semilinear set $Q$, then $\\phi(Q) \\in \\DCM$,\n\\item given an $\\NCM$ (or $\\DCM$) language $L \\subseteq u_1^* \\cdots u_k^*$, then $IND(L) = \\{(l_1, \\ldots, l_k) \\mid \\phi((l_1, \\ldots, l_k)) \\in L\\}$ is a semilinear set.\n\\end{itemize} Moreover, both are effective.\n\\label{phi}\n\\end{lemma}\n\n\nRecall that two languages $L_1$ and $L_2$ are called commutatively equivalent if there is a Parikh-map preserving bijection between them. Therefore, a language $L$ being commutatively equivalent to some regular language is stronger than saying it is counting-regular.\nSplit across three papers in \\cite{FlavioBoundedSemilinearpaper1,FlavioBoundedSemilinearpaper2,FlavioBoundedSemilinear}, it was shown that all bounded semilinear languages ---\nwhich are all bounded languages where $IND(L)$ is a semilinear set --- are commutatively\nequivalent to some regular language, and are therefore counting-regular. Recently, it was shown that\nall bounded languages from any semilinear trio are in $\\DCM$ \\cite{CIAA2016} (and are therefore\nbounded semilinear by Lemma \\ref{phi}). This enables us to conclude that all bounded languages\nin any semilinear trio are commutatively equivalent to some regular language, and thus counting-regular.\nHowever, as the proof that all bounded semilinear languages are commutatively equivalent to some regular language is quite lengthy, we provide a simple alternate proof that all bounded languages in any semilinear trio (all bounded semilinear languages) are counting-regular. The class $\\DCM$ plays a key role in this proof.\n\n\n\\begin{lemma}\nLet $u_1, \\ldots, u_k \\in \\Sigma^+$, $L\\subseteq u_1^* \\cdots u_k^*$ be a bounded $\\DCM$ language, and let $\\phi$ be a function from Lemma \\ref{phi}.\nThere exists a semilinear set $B$ such that $\\phi(B) = L$ and $\\phi$ is injective on $B$. Also, the construction of $B$ is effective.\n\\label{injective}\n\\end{lemma}\n\\begin{proof}\nLet $A = \\{a_1, \\ldots, a_k\\}$, and consider the homomorphism $h$ that maps $a_i$ to $u_i$, for $1 \\leq i \\leq k$. It is known that\nthere exists a regular subset $R$ of $a_1^* \\cdots a_k^*$ that $h$ maps bijectively from $R$ onto $u_1^* \\cdots u_k^*$ \n(the Cross-Section Theorem of Eilenberg \\cite{Flavio}).\nLet $L' = h^{-1}(L) \\cap R$. Then $L'$ is in $\\DCM$ since $\\DCM$ is closed under inverse homomorphism and intersection with regular languages \\cite{ibarra1978}. Hence, there is a semilinear set $B = IND(L')$ from Lemma \\ref{phi}(2).\nThen $\\phi(B) = L$ since, given $(l_1, \\ldots, l_k) \\in B$, then\n$u_1^{l_1} \\cdots u_k^{l_k} \\in L$, and given $w \\in L$, by the bijection $h$,\nthere exists a string $a_1^{l_1} \\cdots a_k^{l_k}$ of $R$ such that $a_1^{l_1} \\cdots a_k^{l_k} = h^{-1}(w)$,\nand so $w \\in \\phi(B)$. Also, $\\phi$ is injective on $B$, as given two distinct elements\n$(l_1, \\ldots, l_k)$ and $(j_1, \\ldots, j_k)$ in $B$, then both\n$a_1^{l_1} \\cdots a_k^{l_k}$ and $a_1^{j_1} \\cdots a_k^{j_k}$ are in $R$, which means that $h$ maps them onto\ndifferent words in $u_1^* \\cdots u_k^*$ since $h$ is a bijection.\n\\qed \\end{proof}\n\n\nThe proof of the next result uses similar techniques as the proof that all bounded context-free languages are counting-regular from \\cite{Flavio}. But because there are key\ndifferences, we include a full proof for completeness.\n\n\\begin{lemma}\nLet $L \\subseteq u_1^* \\cdots u_k^*$ be a bounded $\\DCM$ language\nfor given words $u_1, \\ldots, u_k$. Then there exists an effectively constructible bounded\nregular language $L'$ such that, for every $n \\geq 0$, $f_L(n) = f_{L'}(n)$.\n\\end{lemma}\n\\begin{proof}\n\nLet $\\phi$ be a function from $\\mathbb{N}_0^k$ to $u_1^* \\cdots u_k^*$ such that \n$\\phi((l_1, \\ldots, l_k)) = u_1^{l_1} \\cdots u_k^{l_k}$. By Lemma \\ref{phi}, there exists a semilinear\nset $B$ of $\\mathbb{N}_0^k$ such that $\\phi(B) = L$ \\cite{ibarra1978}. Let\n$B = B_1\\cup \\cdots \\cup B_m$, where $B_i$, $1 \\leq i \\leq m$ are linear sets. \nLet $L_1 = \\phi(B_1) \\in \\DCM$ (by Lemma \\ref{phi}), \n$L_2 = \\phi(B_2) - L_1 \\in \\DCM$ (by Lemma \\ref{phi} and since $\\DCM$ is closed under intersection\nand complement \\cite{ibarra1978}), etc.\\ until $L_m = \\phi(B_m) - (L_1 \\cup \\cdots \\cup L_{m-1}) \\in \\DCM$ (inductively, by Lemma \\ref{phi}, by closure of $\\DCM$ under intersection, complement, and union). Then $L_1 \\cup \\cdots \\cup L_m = L$, and also $L_1, \\ldots, L_m$\nare pairwise disjoint, and therefore\nby Lemma \\ref{injective}, there is a semilinear\nset $B_i'$ such that $\\phi(B_i') =L_i$, and $\\phi$ is injective on $B_i'$.\nIt is known that, given any set of constants and periods generating a semilinear set $Q$, \nthere is a procedure to effectively construct another set of constants and periods that forms a semi-simple set,\nalso generating $Q$ \\cite{Flavio,Sakarovitch} (this is phrased more generally in both works, to say that the rational sets of a commutative monoid\nare semi-simple; but in our special case it amounts to constructing a new set of constants and periods generating the same\nsemilinear set such that the linear sets are disjoint, and the periods generating each linear set form a basis). \nHence, each $B_i'$ must also be semi-simple as well (generated by a possibly different set of constants\nand periods). Let $B' = B_1' \\cup \\cdots \\cup B_m'$. Since each word in $L$ is only in exactly one language of $L_1, \\ldots, L_m$,\nit follows that for each $(l_1, \\ldots, l_k)$, $\\phi((l_1, \\ldots, l_k))$ is in at most one language of $L_1, \\ldots, L_m$.\nAnd, since $\\phi$ is injective on each $B_i'$, it \ntherefore follows that $\\phi$ is injective on $B'$. Also, $\\phi(B') = L = \\phi(B)$.\n\nThe rest of the proof then continues just as Theorem 10 of \\cite{Flavio} (starting at the second paragraph) which we describe. That is, define an alphabet $A = \\{a_1, \\ldots, a_k\\}$, and let $\\psi$ be the Parikh map of $A^*$ to $\\mathbb{N}_0^k$.\nFor every linear set $B''$ making up any of the semilinear sets of some $B_i'$, let\n$B''$ have constant $b_0$ and \nperiods $b_1, \\ldots, b_t$. Define the regular language $R_{B''} = v_0 v_1^* \\cdots v_t^*$\nwhere $v_0, \\ldots, v_t$ are any fixed words of $A^*$ such that, for every $i$,\n$\\psi(v_i) = b_i$. Thus, $\\psi(R_{B''}) = B''$, for each $B''$. Let $R$ be the union of all $C_{B''}$ over all the linear\nsets making up $B_1', \\ldots, B_m'$. Certainly $R$ is a regular language.\n\nIt is required to show that $\\psi$ is injective on $R$. Indeed, consider\n$x, y$ be two distinct elements in $R$. If $x,y$ are constructed from two different linear sets\nfrom distinct semilinear sets $B_i', B_j', i \\neq j$, then $\\psi(x) \\neq \\psi(y)$ since the semilinear\nsets $B_i'$ and $B_j'$ are disjoint. If $x,y$ are constructed from two different linear sets\nmaking up the same semilinear set $B_i'$, then since $B_i'$ is semi-simple,\nthe linear sets must be disjoint, and hence $\\psi(x) \\neq \\psi(y)$.\nIf $x,y$ are in the same linear set, then $\\psi(x)\\neq \\psi(y)$ since the linear\nset must be simple, its periods form a basis, and therefore, there is only one\nlinear combination giving each. Hence, $\\psi$ is injective on $R$.\n\nConsider the map such that, for every $i$, $1 \\leq i \\leq k$, $a_i$ maps to\n$a_i^{|u_i|}$, and extend this to a homomorphism $\\chi$ from $A^*$ to $A^*$.\nSince $\\chi(A)$ is a code, $\\chi$ is an injective homomorphism of $A^*$ to\nitself. Let $L' = \\chi(R)$. Then $L'$ is a regular language.\n\nNext, it will be shown that $f_L(n) = f_{L'}(n)$. Consider the relation\n$\\zeta = \\phi^{-1} \\psi^{-1} \\chi$. Then, when restricting $\\zeta$ to \n$L$, this is a bijection between $L$ and $L'$, since $\\phi$ is a bijection\nfrom $B'$ to $L$, $\\psi$ is a bijection of $R$ to $B'$, and $\\chi$ is a bijection\nof $R$ to $L'$. It only remains to show that, for each $u \\in L$,\n\\begin{equation}\n|u| = |\\zeta(u)|,\n\\label{conc}\n\\end{equation}\nwhich therefore would imply $f_L(n) = f_{L'}(n)$.\nFor each $u \\in L$, then\n$u = u_1^{l_1} \\cdots u_k^{l_k} = \\phi((l_1, \\ldots, l_k)) = \\phi(\\psi(x))$,\nwhere $x$ is in $R$ and $\\psi^{-1}((l_1, \\ldots, l_k))$.\nSince $|x| = \\sum_{1 \\leq i \\leq k}|x|_{a_i} = \\sum_{1 \\leq i \\leq k} l_i$,\nthen $$|\\chi(x)| = \\sum_{1 \\leq i \\leq k} |x|_{a_i} |\\chi(a_i)| = \\sum_{1 \\leq i \\leq k} l_i|\\chi(a_i)| = \\sum_{1 \\leq i \\leq k} l_i|u_i| = |u|.$$ Thus, \\ref{conc} is true, and\nthe theorem follows.\n\\qed \\end{proof}\nWe should note that if the bounded language $L \\subseteq a_1^* \\cdots a_k^*$, where\n$a_1, \\ldots, a_k$ are distinct symbols, then there is a simpler proof of the theorem\nabove as follows: Let $L \\subseteq a_1^* \\cdots a_k^*$ where the symbols are distinct.\nThen the Parikh map of $L$ is semilinear, and therefore a regular language $L'$\ncan be built with the same Parikh map, and in this language $f_L(n) = f_{L'}(n)$. \nBut when the bounded language $L$ is not of this form, this simpler proof does not work. \n\\begin{comment}\n{\\bf I'm tempted to remove this part. What do you think.\nFor example, $L = \\{(ab)^i c^i, c^i (ab)^i \\mid i \\geq 0\\} \\subseteq a^* c^* a^*$.\nThen, even if we apply an inverse homomorphism that maps $d$ to $ab$ and $c$ to $c$ to get\n$\\{d^i c^i, c^i d^i \\mid i \\geq 0\\}$, then the regular language\n$(dc)^* + (dc)^*$ has the same Parikh map, and re-applying the homomorphism\ngives $(abc)^* + (abc)^* = (abc)^*$, but the latter language does not have the same counting function.}\n\\end{comment}\n\nOur next result is a generalization of the previous result.\n\n\\begin{theorem} \nLet $L \\subseteq u_1^* \\cdots u_k^*$, \nfor words $u_1, \\ldots, u_k$ where $L$ is in any \nsemilinear trio ${\\cal L}$. \nThere exists a bounded\nregular language $L'$ such that, for every $n \\geq 0$, $f_L(n) = f_{L'}(n)$.\nMoreover, $L$ is strongly counting-regular. Furthermore, if $u_1, \\ldots, u_k$ are given,\nand all closure properties are effective in ${\\cal L}$, then $L'$ is effectively constructible.\n\\label{cor9}\n\\end{theorem}\nAgain, this follows from \\cite{CIAA2016} since it is known that every\nbounded language from any such semilinear trio where the closure properties are effective\ncan be effectively converted\ninto a $\\DCM$ language. Strong counting-regularity follows since intersecting\na bounded language in a trio with a regular language produces another bounded language\nthat is in ${\\cal L}$, since trios are closed under intersection with regular languages.\n\nAlso, since the family of regular languages is the smallest semilinear trio \\cite{G75}, it follows that the counting functions for the bounded languages in every semilinear trio are identical.\n\\begin{corollary}\nLet ${\\cal L}$ be any semilinear trio. The counting functions for the bounded\nlanguages in ${\\cal L}$ are identical to the counting functions for the bounded regular languages.\n\\end{corollary}\n\n\n\nThis works for many semilinear full trios. We will briefly discuss some in the next example.\n\\begin{example}\n\\label{semilinearfulltrioexamples}\nThe families accepted\/generated from the following grammar\/machine models form semilinear full trios (the closure properties and semilinearity are effective):\n\\begin{enumerate}\n\\item the context-free languages, $\\CFL$s, \n\\item one-way nondeterministic reversal-bounded multicounter machines, $\\NCM$s, \\cite{ibarra1978},\n\\item finite-index $\\ETOL$ systems ($\\ETOL$ systems where the number of non-active\nsymbols in each derivation is bounded by a constant) \\cite{RozenbergFiniteIndexETOL}.\n\\item $k$-flip $\\NPDA$s ($\\NPDA$s with the ability to ``flip'' their pushdown up to $k$ times) \\cite{Holzer2003}. \n\\item one-way reversal-bounded queue automata (queue automata with a bound on the number of switches between enqueueing and dequeueing) \\cite{Harju}.\n\\item $\\NTM$s with a one-way read-only input tape and a finite-crossing worktape \\cite{Harju},\n\\item uncontrolled finite-index indexed grammars (a restricted version of indexed grammars, where\nevery accepting derivation has a bounded number of nonterminals), \\cite{LATA2017}.\n\\item multi-push-down machines (a machine with multiple pushdowns where the machine can simultaneously push to all pushdowns, but can only\npop from the first non-empty pushdown) \\cite{multipushdown}.\n\\end{enumerate}\nMoreover, all of these machine models can be augmented by reversal-bounded counters and\nthe resulting machines are semilinear full trios \\cite{Harju,fullaflcounters}.\n\\end{example}\n\n\n\\begin{corollary}\nLet $L \\subseteq u_1^* \\cdots u_k^*$, be a bounded language\nfor given words $u_1, \\ldots, u_k$, such that $L$ is from any of the \nfamilies listed in Example \\ref{semilinearfulltrioexamples}.\nThen there exists an effectively constructible bounded\nregular language $L'$ such that, for every $n \\geq 0$, $f_L(n) = f_{L'}(n)$.\n\\end{corollary}\nNote that it is not assumed for these models that the machines are unambiguous, like in\nTheorem \\ref{main2}.\n\n\nThe results in this section assumed that the words $u_1, \\ldots, u_k$ \nsuch that $L \\subseteq u_1^* \\cdots u_k^*$ are given. However, it is an open problem whether, given a language $L$ in an arbitrary semilinear trio ${\\cal L}$, it is possible to determine whether\n$L \\subseteq u_1^* \\cdots u_k^*$ for some words $u_1, \\ldots, u_k$.\n\n\\section{Closure Properties for Counting-Regular Languages}\n\\label{sec:closure}\n\nIn this section, we will address the closure properties of counting-regular languages, and also\ncounting-regular $\\CFL$'s. \n\nFirst, it is immediate that counting-regular languages are closed under reversal (and since the $\\CFL$s\nare closed under reversal, so are the counting-regular $\\CFL$s). Next Kleene-* will be addressed.\n\\begin{theorem}\n\\label{code}\nIf $L$ is counting-regular and $L$ is a code, then $L^*$ is counting-regular.\n\\end{theorem}\n\\begin{proof}\nSince $L$ is a code, for each word $w \\in L^*$, there is a unique decomposition of\n$w = u_1 \\cdots u_k$, where each $u_i \\in L$. Since $L$ is counting-regular, there is\nsome regular language $R$ with the same counting function. From $R$, make\n$R'$ where the first letter of each word is tagged with a prime, and all other letters are unmarked.\nNow, $R'$ is a code because of the tagged letters, and $R'$ has the same counting function\nas $R$.\n\nMoreover, $(R')^*$ has the same counting function as $L^*$. Indeed, let $n \\geq 0$. Consider all sequences $u_1, \\ldots, u_k$ such that $n = |u_1| + \\cdots + |u_k|$. Then for each\n$u_i$, $L$ has the same number of words of length $|u_i|$ as does $R'$. Since $R'$ is a code,\nit follows that there are the same number of such sequences using elements from $R'$.\n\\qed \\end{proof}\n\n\nA similar relationship to codes exists for concatenation.\n\\begin{theorem}\n\\label{prefixcode}\nIf $L_1, L_2$ are counting-regular and either $L_1$ is a prefix code or $L_2$ is a suffix code, then $L_1 L_2$ is counting-regular.\n\\end{theorem}\n\\begin{proof}\nAssume first that $L_1$ is a prefix code, so that $L \\cap L \\Sigma^+ = \\emptyset$. Let $w = uv, u \\in L_1, v \\in L_2$. Then this decomposition is unique since $L_1$ is a prefix code. Let $R_1, R_2$ be regular\nlanguages with the same counting functions as $L_1, L_2$ respectively. Let $R_1'$ be obtained from $R_1$ by tagging the last letter with a prime. Then $R_1'$ is also a prefix code. Further,\nthe counting function for $R_1'R_2$ is equal to that of $L_1L_2$. \n\nThe case is similar for $L_2$ being a suffix code.\n\\qed \\end{proof}\n\n\n\\begin{corollary}\n\\label{corcode}\nIf $L_1, L_2 \\subseteq \\Sigma^*$ are counting-regular, and $\\$,\\#$ are new symbols, then $L^R,\nL_1 \\$L_2, \\$L_1 \\cup \\#L_2$, and $(L\\$)^*$ are counting-regular.\n\\end{corollary}\n\\begin{proof}\nThe first was discussed above. The second follows from Theorem \\ref{prefixcode} and since\n$L_1 \\$$ is a prefix code. The fourth\nfollows from Theorem \\ref{code} and since $L\\$$ is a code. For the third, since $L_1,L_2$ are\ncounting-regular, this implies there exist regular languages $R_1, R_2$ with the same\ncounting functions, and $\\$L_1 \\cup \\#L_2$ has the same counting function as \n$\\$R_1 \\cup \\#R_2$.\n\\qed\n\\end{proof}\n\n\nThis means that even though e.g.\\ non-reversal-bounded $\\DPDA$s can accept non-counting-regular\nlanguages but reversal-bounded $\\DPDA$s cannot, if a $\\DPDA$ was reversal-bounded but\nreading a $\\$$ caused a ``reset'' where the pushdown emptied, and another reversal-bounded computation was then possible, then this model would only accept counting-regular languages.\nThis is also the case with say $\\DTM$s where the worktape was reversal-bounded, but reading\na $\\$$ caused a reset, where more reversal-bounded computations were again possible.\nThis is quite a general model for which this property holds.\n\nThe next questions addressed are whether these are true when removing the $\\$$ and $\\#$\n(or removing or weakening the coding properties).\n\n\n\\begin{comment}\nClearly counting-regular $\\CFL$'s are not closed under intersection since $\\{a^n b^n c^m \\mid n, m \\geq 1\\}$ and \n$\\{a^n b^m c^m \\mid n, m \\geq 1\\}$ are counting-regular $\\CFL$'s, but their intersection is not. (It is not a $\\CFL$ although it is counting-regular.) Similarly, counting-regular $\\CFL$'s are not closed under complement since $\\overline{L_{Sq}}$ (recall that $L_{Sq}$ = $\\{ w w \\mid w \\in \\{a, b\\}^*\\}$) is a counting-regular $\\CFL$ since it has the same counting function as the regular language \n$\\{a_1b_1 \\cdots a_n b_n \\mid a_i,b_i \\in \\Sigma, a_1 \\cdots a_n \\neq b_1 \\cdots b_n\\}$, but $L_{Sq}$ is not a $\\CFL$ and is therefore not a counting-regular $\\CFL$. These are trivial (non)-closure properties inherited from the super-class of the class of $\\CFL$'s. Similarly it follows trivially that counting-regular $\\CFL$'s are closed under reversal. However, a more interesting fact is that counting-regular $\\CFL$'s are {\\it not} closed under union or intersection with regular sets, as we show next. \n\\end{comment}\n\n\\begin{theorem}\nThe counting-regular languages (and the counting-regular $\\CFL$'s) are not closed under union or intersection with regular languages.\n\\end{theorem}\n\\begin{proof} Recall the $\\DFA$ $M_2$ presented in Figure \\ref{fig3}. Since $L_{\\MAJ}$ is a counting-regular $\\CFL$, and $L_{MAJ} \\cap L(M_2)$ is not, the non-closure under intersection with regular sets follows.\n\nFor non-closure under union with regular sets, we show that $L_{\\MAJ} \\cup L(M_2)$ is not counting-regular by explicitly computing the generating function for this language using the fact that there is a 1-1 mapping between strings of $L_{\\MAJ}$ and $\\overline{L_{MAJ}}$ and between strings of $L(M_2)$ and $\\overline{L(M_2)}$. Thus the generating functions for both $f_{L_{MAJ}}(n)$ and $f_{L(M_2)}(n)$ are ${1-z} \\over {1-2z}$, from which it follows that the generating function for $L_{MAJ} \\cup L(M_2)$ is ${{1-2z-z^2} \\over {1-2z}}-$ ${{z^2+z} \\over {\\sqrt {1-4z^2}}}$. Since this is not a rational function, the claim follows.\n\\qed \\end{proof}\nThus, Corollary \\ref{corcode} cannot be weakened to remove the marking from the marked union.\n\n\nIt is an open question as to whether Theorem \\ref{code} can be weakened to remove the code\nassumption, but we conjecture that it cannot.\nHowever, for concatenation, we are able to show the following:\n\\begin{theorem}\nThe counting-regular languages (and counting-regular $\\CFL$'s) are not closed under concatenation with regular languages.\n\\end{theorem}\n\\begin{proof}\nLet $S_1$ = $\\{ w \\ |\\ w = a^n b v a^n$ for some $v \\in \\{a, b\\}^*\\}$ and let \n$S$ = $\\{ w \\ |\\ w = a^n b v_1 a^n v_2$ for some $v_1, v_2 \\in \\{a, b\\}^*\\}$ (as in Theorem \\ref{counter-examples}).\nIt is easy to see that $S$ = $S_1 (a+b)^*$. We already showed that $S$ is not counting-regular. It is easy to show that $S_1$ is a counting-regular $\\CFL$. In fact, we can explicitly exhibit $f_{S_1}(n)$ as follows: $f_{S_1}(0)$ = 0 and, for $n \\geq 1$, $f_{S_1}(n)$ = $\\sum_{j=0}^{\\lfloor (n-1)\/2 \\rfloor} 2^{n-2i-1}$. From this, it is easy to see that $f_{S_1}(n)$ = $f_L(n)$ for the regular language $L$ with regular expression $(aa)^*b(a+b)^*$. \n\\qed \\end{proof}\nHence, Corollary \\ref{corcode} and Theorem \\ref{prefixcode} cannot be weakened to remove the marking with marked concatenation or the coding properties. It is an open problem as to whether\nTheorem \\ref{prefixcode} is true when $L_1$ or $L_2$ are codes (the set of\nsuffix codes together with the set of prefix codes is a strict subset of the set of codes \\cite{CodesHandbook}).\n\n\n\n\\begin{theorem}\n\\label{rightquotient}\nThe counting-regular languages (and counting-regular $\\CFL$'s) are not closed under right quotient with a single symbol, and are not closed under left quotient with a symbol.\n\\end{theorem}\n\\begin{proof} (sketch) \nFirst, it will be shown for right quotient.\nThe language $L_{\\MAJ}$ used in Theorem \\ref{MAJ} is counting-regular. In fact, it has exactly $2^{n-1}$ strings of length $n$ for all $n \\geq 1$. We will outline an argument that $L$ = $L_{\\MAJ}\\{0\\}^{-1}$ is not counting-regular. The number of strings of length $n$ in $L$ is \n$2^{n-1} - {n \\choose {n-1 \\over 2}}$ (for odd $n$), $2^{n-1} - {n-2 \\choose {n \\over 2}}$ (for even $n$). Using a technique similar to the proof of Theorem \\ref{MAJ}, we can show that the generating function for $L$ is not rational.\n\nNext, it will be shown for left quotient.\nSince $L_{\\MAJ}$ is counting regular, and the reversal of every counting\nregular language has the same counting function, then $L_{\\MAJ}^R$ is also counting-regular. \nBut, as in in proof for right quotient, $\\{0\\}^{-1}L_{\\MAJ}^R$ is not \ncounting-regular. \n\\qed \\end{proof}\nThe language $L_{MAJ}$ used is a deterministic one counter language with no reversal bound.\nThe following however provides a contrast, as it follows from closure properties of deterministic\nmachines.\nWhen the languages are accepted by reversal-bounded machines, we have: \n\\begin{theorem} \n\\begin{enumerate}\n\\item If $L$ is accepted by a reversal-bounded \n$\\DTM$, and $R$ is a regular language, then \n$LR^{-1}$ is counting-regular. \n\\item If $L$ is accepted by an unambiguous reversal-bounded $\\NTM$, and \n$x$ is a string, then $L \\{x\\}^{-1}$ is also counting-regular.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} Part 1 follows from the fact that the languages accepted by \nreversal-bounded $\\DTM$'s are closed under \nright-quotient with regular languages \\cite{StoreLanguages}. \n\nFor Part 2, clearly, if $L$ is accepted by an unambiguous reversal-bounded \n $\\NTM$ $M$, we can construct \nan unambiguous reversal-bounded \n$\\NTM$ $M$ accepting $L\\{x\\}^{-1}$. \n\\qed \\end{proof}\nThis result also holds for all machine models in Corollary \\ref{cortomain}\n(that is, all deterministic models listed there work with right quotient with\nregular languages \\cite{StoreLanguages}, and the unambiguous nondeterministic models there work\nwith right quotient with a word). It is an open question as to whether\nunambiguous nondeterministic $\\NTM$s with a reversal-bounded worktape\nare closed under right quotient with regular languages, which would allow\npart 2 to be strengthened.\n\n\nPart 1 of of the next theorem contrasts Part 1 of the previous theorem. \n\\begin{theorem}\n\\begin{enumerate}\n\\item There is a counting-regular language $L$ accepted by a $\\DCM(1,1)$ and distinct symbols $\\$$ and $\\#$ such that $\\{\\$,\\#\\}^{-1}L$ is not counting-regular. \n\\item If $L$ is accepted by a reversal-bounded $\\DPDA$ (resp., reversal-bounded \nunambiguous $\\NPDA$, reversal-bounded unambiguous $\\NTM$), \nand $x$ is a string, then $\\{x\\}^{-1} L$ is also counting-regular. \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nFor Part 1, let \n$L_1 = \\{x 1 ^n \\mid x \\in (a+b)^+, n = |x|_a \\}$ and\n$L_2 = \\{x 1 ^n \\mid x \\in (a+b)^+, n = |x|_b \\}$. \nThen $L_1$ and $L_2$ can each be accepted by a $\\DCM(1,1)$. Let $L = L_1 \\cup L_2$, shown\nin Theorem \\ref{counter-examples} to not be counting-regular.\nLet $L' = \\$L_1 \\cup \\#L_2$, which can also be accepted by a $\\DCM(1,1)$, hence it is counting-regular. However, $\\{\\$,\\#\\}^{-1}L' = L$ is not counting-regular. \nPart 2 is obvious.\n\\qed \\end{proof}\n\n\n\n\nIt may seem obvious that for any counting-regular language $L$, $\\overline{L}$ is counting-regular because of the following putative reasoning: If there is a regular language $L'$ whose counting function equals that of the counting function of $L$, the complement of $L'$ (which is regular) has the same counting function as that of $\\overline{L}$. The fallacy in this argument is as follows. Suppose that the size of the alphabet over which $L$ is defined is $k$. The size of alphabet $k_1$ over which $L'$ is defined may be larger, i.e., $k_1 > k$. Thus, the complement of $L'$ has a counting function ${k_1}^n - f_{L'}(n)$ which is not the same as the counting function $k^n-f_L(n)$ of $\\overline{L}$. \n\n\nIn fact, the following result shows that the exact opposite of the fallacy is actually true.\n\\begin{theorem}\nThere is a counting-regular language $L$ (that is in $\\P$, i.e., $L$ is deterministic polynomial time computable) such that $\\overline{L}$ is not counting-regular.\n\\end{theorem}\n\\begin{proof} The proof relies on a result presented in B\\'{e}al and Perrin \\cite{Beal}. B\\'{e}al and Perrin \\cite{Beal} provide an example of a sequence $\\{r_n\\}$, $n \\geq 0$ and an integer $k$ such that $\\{r_n\\}$ is not a counting function of any regular language, but $\\{k^n - r_n\\}$ is the counting function of a regular language. Specifically, it is shown in \\cite{Beal} that the sequence $r_n$ = $b^{2n} \\cos^2(n\\theta)$, with $cos\\ \\theta$ = $a \\over b$, where the integers $a$, $b$ are such that $b \\neq 2a$ and $0 < a < b$, and $k$ such that $b^2 < k$, satisfies the properties stated above.\n\nDefine a language $L$ over an alphabet of size $k$ as follows: arrange the strings of length $n$ over the alphabet $\\{0, 1, \\ldots , k-1\\}$ lexicographically. A string $w$ of length $n$ is defined to be in $L$ if and only if the rank of $w$ (in the lexicographic order) is greater than $r_n$. (Rank count starts at 1.) Clearly, the number of strings of length $n$ in $L$ is exactly $s_n$ = $k^n - r_n$. As shown in \\cite{Beal}, $L$ is counting-regular and $\\overline{L}$ is not counting-regular. \n\nFinally, we provide a sketch of the proof that there is a deterministic polynomial time algorithm for $L$. Given a string $w$ of length $n$, and an integer $T$ (in binary) where the number of bits in $T$ is $O(n)$, it is easy to see that there is a deterministic algorithm that determines in time polynomial in $n$ if the rank of $w$ is greater than $T$. (This algorithm simply converts $T$ from binary to base $k$ and compares the resulting string to $w$ lexicographically. Base conversion can be shown to have complexity no more than that of integer multiplication.)\nTo complete the algorithm, we need to show how to compute $r_n$ (in binary), given $n$, in time polynomial in $n$. Note that $r_n$ is given by $b^{2n} \\cos^2(n\\theta)$. Clearly, $b^{2n}$ can be computed in polynomial time by repeated multiplication by $b$ . We don't even need repeated squaring to achieve a polynomial bound since the time complexity is measured in terms of $n$, not $log\\ n$. Also, $\\cos \\ (n \\ arccos (a\/b))$ can be computed as follows: $\\cos \\ (n \\ arccos (a\/b))$ is the well-known Tchebychev polynomial $T_n(a\/b)$ which is explicitly given by the series \\cite{Mason}:\n$$\\sum_{m=0}^{\\lfloor n\/2 \\rfloor} {n \\choose {2m}} \\left ( a \\over b \\right )^{n-2m} \\left( \\left ( a \\over b \\right )^2 -1 \\right)^m. $$\n\nSince each of the terms in the above series can be computed in time polynomial in $n$, and since there are $\\lfloor {n \\over 2} \\rfloor$ terms in the series, it is clear that $r_n$ can be computed in time polynomial in $n$. \n\\qed \\end{proof}\n\nIt is evident from Corollary \\ref{cortomain} and closure of reversal-bounded $\\DPDA$'s\nunder complement that counting-regular reversal-bounded $\\DPDA$'s are closed under complement. For $\\DPDA$'s generally,\nit is open whether counting-regular $\\DPDA$'s are closed under complement.\n At the end of the proof of Theorem \\ref{main2}, we observed that the mapping that we used to map the strings from a language $L$ accepted by an unambiguous reversal-bounded $\\NTM$ to a regular language increased the size of the alphabet from $k$ to $k+1$. If for every counting-regular $\\DPDA$, the size of the alphabet of the simulating $\\NFA$ is $k$, then it will follow that counting-regular $\\DCFL$'s are closed under complement.\n\nIn fact, we can show the following stronger claim.\n\n\\begin{theorem}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item For every counting-regular $\\DCFL$ over a $k$-letter alphabet, there is a regular language $L'$ over a $k$-letter alphabet such that $f_L(n)$ = $f_{L'}(n)$ for all $n$.\n\n\\item Counting-regular $\\DCFL$'s are closed under complement.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} (1) $\\Rightarrow$ (2) is immediate from the above discussion.\n\nTo show that (2) $\\Rightarrow$ (1), let $L$ be a counting-regular $\\DCFL$ over a $k$-letter alphabet. This means $f_L(n)$ is a regular sequence. By (2), the complement of $L$ is counting-regular, so $k^n - f_L(n)$ is also a regular sequence. From Theorem \\ref{Beal}, it follows that there is a regular language $L'$ over a $k$-letter alphabet such that $f_L(n)$ = $f_{L'}(n)$ for all $n$. \n\\qed \\end{proof} \n\n\n\nOur conclusion is that the class of counting-regular languages (or counting-regular $\\CFL$s) are very fragile in that it is not closed under basic operations such as union or intersection with regular languages. We conjecture that the counting-regular $\\CFL$s are also not closed under Kleene star.\n\n\n\\section{Some Decision Problems Related to Counting-Regularity}\n\\label{sec:decidability}\n\nIn this section, some decision problems in regards to counting-regularity are addressed.\nIn particular, we will show the following:\n(1) It is undecidable,\ngiven a real-time $1$-reversal-bounded $2$-ambiguous $\\PDA$ $M$,\nwhether $L(M)$ (resp.\\ $\\overline{L(M)}$)\nis counting-regular;\n(2) It is undecidable, given a real-time $\\NCM(1,1)$\n $M$, whether $L(M)$ (resp.\\ $\\overline{L(M)}$)\nis counting-regular. \n\nWe begin with the following result:\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\n\\label{undecide1}\nIt is undecidable, given two real-time $1$-reversal-bounded $\\DPDA$'s $M_1$ and $M_2$ accepting strongly counting-regular languages, whether $L(M_1) \\cap L(M_2)$ is counting-regular.\n\\end{theorem}\n\\begin{proof} \n\nLet $Z$ be a single-tape $\\DTM$ working on an initially blank tape. We assume that if $Z$ halts on blank tape, it makes $2k$ steps for some $k \\ge 2$.\nWe assume that $Z$ has one-way infinite tape and does not write blank. Let\n$$L_1' = \\{ \\begin{array}[t]{l} ID_1 \\# ID_3 \\# \\cdots \\# ID_{2k-1} \\$ ID_{2k}^R \\# \\cdots \\# ID_4^R \\# ID_2^R \\mid k \\ge 2, \\mbox{~each~} ID_i\\\\\n\\mbox{is a configuration of~} Z, ID_1 \\mbox{~is initial~} ID \\mbox{~of~} Z \\mbox{~on blank tape,} \\\\\nID_{2k} \\mbox{~is a unique halting~} ID, ID_i \\Rightarrow ID_{i+1} \\mbox{~for~} i = 1, 3, \\ldots, 2k-1 \\},\\end{array}$$\n$$L_2' = \\begin{array}[t]{l} \\{ ID_1 \\# ID_3 \\# \\cdots \\# ID_{2k-1} \\$ ID_{2k}^R \\# \\cdots \\# ID_4^R \\# ID_2^R \\mid k \\geq 2, \\mbox{~each~} ID_i\\\\\n\\mbox{is a configuration of~} Z, ID_1 \\mbox{~is initial~} ID \\mbox{~of~} Z \\mbox{~on blank tape}, \\\\\nID_{2k} \\mbox{~is a unique halting~} ID, ID_i \\Rightarrow ID _{i+1} \\mbox{~for~} i = 2, 4, \\ldots, 2k-2 \\},\\end{array}$$\nwith $L_1',L_2' \\subseteq \\Sigma^*$.\nClearly, $L_1'$ and $L_2'$ can be accepted by real-time $1$-reversal-bounded $\\DPDA$'s $M_1'$ and $M_2'$. Moreover, $L_1' \\cap L_2'$ is empty or\na singleton (if and only if $Z$ accepts, which is undecidable).\nLet $a$, $b$, $1$ be three new symbols. Let\n\\begin{eqnarray*}\nL_1 &=& \\{ x w 1^n\\ |\\ x \\in (a+b)^+, \\ w \\in L(M_1'),\\ |x|_a = n \\},\\\\\nL_2 &=& \\{x w 1^n\\ |\\ x \\in (a+b)^+, \\ w \\in L(M_2'),\\ |x|_b = n \\}.\n\\end{eqnarray*}\nWe can construct a real-time $1$-reversal-bounded $\\DPDA$ $M_1$ accepting $L_1$\nas follows: $M_1$ reads $x$ and stores the number of $a$'s in the stack.\nThen it simulates the $1$-reversal-bounded $\\DPDA$ $M_1'$ on $w$, and\nfinally checks (by continuing to pop the stack) that the number\nof $a$'s stored in the stack is $n$. Similarly, we can construct a \nreal-time $1$-reversal-bounded $\\DPDA$ $M_2$ accepting $L_2$.\nBy Corollary \\ref{cortomain}, $L_1$ and $L_2$ are\nstrongly counting-regular.\n\nClearly, if $L(M_1') \\cap L(M_2')$ = $\\emptyset$,\nthen $L_1 \\cap L_2$ = $\\emptyset$,\nhence $L_1 \\cap L_2$ is counting-regular.\n\nOn the other hand, if $L(M_1') \\cap L(M_2')$ is not empty, the intersection\nis a singleton $w$ and thus $L_1 \\cap L_2$ is given by:\n$$L_1 \\cap L_2 = \\{ x w 1^n\\ | \\ x \\in (a+b)^+ ,\\ |x|_a = |x|_b = n\\}.$$\n\nAs in the proof of Theorem \\ref{counter-examples}(ii), we can show that\n$L_1 \\cap L_2$ is not counting-regular. In fact, if $|w|$ = $t$, then the \nnumber of strings of length $3n+t$ is ${2n}\\choose{n}$ from which we can explicitly construct the generating function $f(z)$ for $L_1 \\cap L_2$ as:\n$$f(z) = {1 \\over 3} \\left ( {1 \\over {\\sqrt{1-4z^3}}} + {1 \\over {\\sqrt{1-4{\\omega}z^3}}} + {1 \\over {\\sqrt{1-4{\\omega^2}z^3}}} \\right ) z^t.$$\nClearly $f(z)$ is not rational and the claim follows.\n\\qed \\end{proof}\n\nThis result is used within the next proof:\n\n\\begin{theorem} \\label{thm16}\nIt is undecidable, given a $\\PDA$ $M$ that is real-time $2$-ambiguous and $1$-reversal-bounded,\nwhether $\\overline{L(M)}$ is counting-regular.\nAlso, it is undecidable, given such a machine $M$,\nwhether $L(M)$ is counting-regular.\n\\end{theorem}\n\\begin{proof}\nConsider the languages $L_1$ and $L_2$ in the proof of Theorem \\ref{undecide1}. Since $L_1$ and $L_2$\ncan be accepted by real-time $1$-reversal-bounded $\\DPDA$'s,\n$\\overline{L_1}$ and $\\overline{L_2}$ \ncan also be accepted by real-time $1$-reversal-bounded $\\DPDA$'s.\nHence $L = \\overline{L_1} \\cup \\overline{L_2}$ \ncan be accepted by a real-time $1$-reversal-bounded $2$-ambiguous $\\PDA$ $M$.\nIt follows that $\\overline{L(M)}$ is\ncounting-regular if and only if\n$L_1 \\cap L_2$ is counting-regular, which is undecidable\nfrom Theorem \\ref{undecide1}.\n\nNow consider $L(M)$. If $Z$ does not halt on blank tape,\nthen $L(M) = \\Sigma^*$, which is counting-regular.\nIf $Z$ halts on blank tape,\n then we will show that $L(M)$ is not counting-regular as follows:\nthe generating function $g(z)$ of $L(M)$ is given by \n${1 \\over {1-sz}} - f(z)$ where $f(z)$ is the generating function of $\\overline{L(M)}$, and $s$ is the size of the alphabet over which $M$ is defined. If $L(M)$ is counting-regular, then $g(z)$ is rational, then so is $f(z)$ = ${1 \\over {1-kz}} - g(z)$, contradicting the fact that $f(z)$ is not rational.\nHence, $L(M)$ is counting-regular if and only if $Z$ does not halt.\n This completes the proof. \n\\qed \\end{proof}\n\n\nNext, we will consider\n$\\NCM(1,1)$ ($1$-reversal-bounded $1$-counter machines).\n\\begin{theorem}\nIt is undecidable, given a real-time $\\NCM$(1,1) $M$, whether $\\overline{L(M)}$ is counting-regular. Also, it is undecidable, given such a machine $M$, whether $L(M)$ is counting-regular.\n\\end{theorem}\n\\begin{proof} \nAgain, we will use the undecidability of the halting problem for a $\\DTM$ $Z$ \non an initially blank tape. As before, assume that $Z$ has a one-way infinite tape\nand does not write blank symbols. Represent each $ID$ (configuration) of $Z$ with \nblank symbols filled to its right, since for the languages we will define below, we\nrequire that all $ID_i$'s have the same length. So, e.g, $ID_1 = q_0 B \\cdots B$ \n(where $B$ is the blank symbol). We also require that the halting $ID$ have all non-blanks in state $f$, which is unique. Clearly since \n$Z$ does not write any blank symbols, if $Z$ halts on the blank tape, the lengths of the $ID$'s in\nthe halting sequence of $ID$'s do not decrease in length. Assume that if $Z$ halts, \nit halts after $k$ steps for some $k \\ge 2$. \n\nLet\n$$L_1 = \\begin{array}[t]{l} \\{ ID_1 \\# ID_3 \\# \\cdots \\# ID_{2k-1} \\$ ID_{2k}^R \\# \\cdots \\# ID_4^R \\# ID_2^R \\mid k \\ge 2, \\mbox{~each~} ID_i\\\\\n\\mbox{is a configuration of~} Z, ID_1 \\mbox{~is initial~} ID \\mbox{~of~} Z \\mbox{~on blank tape,~} \nID_{2k} \\mbox{~is the}\\\\ \\mbox{halting~} ID, ID_i \\Rightarrow ID_{i+1} \\mbox{~for~} i = 1, 3, \\ldots, 2k-1, |ID_1| = \\cdots = |ID_{2k}|\\},\\end{array}$$\n$$L_2 = \\begin{array}[t]{l}\\{ ID_1 \\# ID_3 \\# \\cdots \\# ID_{2k-1} \\$ ID_{2k}^R \\# \\cdots \\# ID_4^R \\# ID_2^R \\mid k \\ge 2, \\mbox{~each~} ID_i \\mbox{~is} \\\\\n\\mbox{~a configuration of~} Z, ID_1 \\mbox{~is initial~} ID \\mbox{~of~} Z \\mbox{~on blank tape}, ID_{2k} \\mbox{~is the}\\\\\n\\mbox{halting~} ID, ID_i \\Rightarrow ID _{i+1} \\mbox{~for~} i = 2, 4, \\ldots, 2k-2, |ID_1| = \\cdots = |ID_{2k}|\\},\\end{array}$$\nwith $L_1,L_2 \\subseteq \\Sigma^*$.\nNote that $ID_{2k}$ must have all non-blanks in\nstate $f$.\n\nLet $a, b, 1$ be new symbols. We can construct a real-time\n$\\NCM$(1,1) $M_1$ which operates as follows. When given input string $z$,\n$M_1$ nondeterministically selects one of the following tasks to execute:\n\n\\begin{enumerate}\n\\item $M_1$ checks and accepts if $z$ is not a string of the form $x w 1^n \\in (a+b)^+ \\Sigma^+ 1^+$.\n(This does not require the use of the counter.)\n\n\\item $M_1$ checks and accepts if $z$ is of the form $x w 1^n \\in (a+b)^+ \\Sigma^+ 1^+$\nbut $|x|_a \\ne n$. (This requires only one counter reversal.)\n\n\\item $M_1$ checks that $z$ is of the form $x w 1^n \\in (a+b)^+ \\Sigma^+ 1^+$, but $w$ \nis not a string of the form in $L_1$. $M_1$ does not check the lengths of the\n$ID$'s and whether $ID_{i+1}$ is a successor of $ID_i$. (This does not require\na counter.)\n\n\\item $M_1$ checks that $z$ is of the form $x w 1^n \\in (a+b)^+ \\Sigma^+ 1^+$\nand $$w = ID_1 \\# ID_3 \\# \\ \\cdots \n\\# ID_{2k-1} \\$ ID_{2k}^R \\# \\ \\cdots \\ \\# ID_4^R \\# ID_2^R,$$ for\nsome $k \\ge 2$ but $|ID_i| \\ne |ID_j|$ for some $i \\ne j$, or $ID_1$ is not the initial $ID$,\nor $ID_{2k}$ is not the halting $ID$. (This requires one counter reversal.)\n\n\\item $M_1$ assumes that $z$ is of the form $x w 1^n \\in (a+b)^+ \\Sigma^+ 1^+$\nand $$w = ID_1 \\# ID_3 \\# \\cdots \n\\# ID_{2k-1} \\$ ID_{2k}^R \\# \\cdots \\# ID_4^R \\# ID_2^R,$$ for\nsome $k \\ge 2$ , $|ID_1| = \\cdots = |ID_{2k}|$, $ID_1$ is the initial $ID$, $ID_{2k}$\nis the halting $ID$, and accepts if $ID_{i+1}$ is not the successor of $ID_i$ for\nsome $i$ = $1, 3, \\ldots , 2k-1$. Since all the $ID$'s are assumed to have the same length,\n$M_1$ needs to only use a counter that reverses once to check one of the conditions.\n\\end{enumerate}\nSimilarly, we can construct a real-time $\\NCM$(1,1) $M_2$ as above using $L_2$.\nLet $M$ be a real-time $\\NCM(1,1)$ accepting $L(M_1) \\cup L(M_2)$ and consider $\\overline{L(M)}$.\n\nIf $Z$ does not halt on blank tape,\nthen $\\overline{L(M)} = \\overline{L(M_1) \\cup L(M_2)} = \\overline{L(M_1)}\n\\cap \\overline{L(M_2)} = \\emptyset$, which is counting-regular.\n\nIf $Z$ halts on blank tape, then\n$\\overline{L(M)}$ = $\\{ x w 1^n\\ |\\ x \\in (a+b)^+ ,\\ |x|_a = |x|_b = n\\}$ for some $w$\nwhich is not counting-regular (as shown in Theorem \\ref{undecide1}). \n\nNow consider $L(M)$.\nIf $Z$ does not halt on blank tape, then $L(M)$ = $\\Sigma^*$, hence is counting-regular. If $Z$ halts on blank tape, then\n$L(M)$ = $L(M_1) \\cup L(M_2)$, which we will show to be not counting-regular as follows: the generating function $g(z)$ of $L(M)$ is given by \n${1 \\over {1-sz}} - f(z)$ where $f(z)$ is the generating function of $\\overline{L(M)}$, and $s$ is the size of the alphabet over which $M$ is defined. If $L(M)$ is counting-regular, then $g(z)$ is rational, then so is $f(z)$ = ${1 \\over {1-kz}} - g(z)$, contradicting the fact that $f(z)$ is not rational. \n\\qed \\end{proof}\n\n\n\n\n\n\n\n\n\\noindent\nNote that the machine $M$ constructed in the proof above\nis 1-reversal-bounded but not finitely-ambiguous.\nNext, it is shown to be undecidable for $2$-ambiguous machines but without the reversal-bound.\n\n\\begin{theorem} \\label{thmXX}\nIt is undecidable, given a one-way 2-ambiguous nondeterministic one counter machine \n$M$,\nwhether $\\overline{L(M)}$ is counting-regular.\nAlso, it is undecidable, given such a machine \n$M$,\nwhether $L(M)$ is counting-regular.\n\\end{theorem}\n\\begin{proof}\nIt is known that it is undecidable, given two deterministic\none counter machines (with no restriction on counter reversals) $M_1$ and\n$M_2$, whether $L(M_1) \\cap L(M_2) = \\emptyset$ (shown implicitly in \\cite{undecidablehartmanis}). Moreover, if\nthe intersection is not empty, it is a singleton.\nLet $\\Sigma$ be the input alphabet of $M_1$ and $M_2$. Let $a, b, 1$ be three new symbols. Let\n\\begin{eqnarray*}\nL_1 &=& \\{ w x 1^n\\ |\\ w \\in L(M_1), \\ x \\in (a+b)^+, \\ |x|_a = n \\},\\\\\nL_2 &=& \\{ w x 1^n\\ |\\ w \\in L(M_2), \\ x \\in (a+b)^+, \\ |x|_b = n \\}.\n\\end{eqnarray*}\nClearly, we can construct deterministic one counter machines\naccepting $L_1$ and $L_2$. Hence, \n$\\overline{L_1}$ and $\\overline{L_2}$ can be accepted by \ndeterministic one counter machines. It follows that \n$\\overline{L_1} \\cup \\overline{L_2}$\ncan be accepted by a $2$-ambiguous nondeterministic one counter machine $M$. Then, as in the\nproof of Theorem \\ref{thm16}, $\\overline{L(M)}$ (resp., $L(M)$)\nis counting-regular if and only if $L(M_1) \\cap L(M_2) = \\emptyset$,\nwhich is undecidable.\n\\qed\n\\end{proof}\n\n\nIn view of the above theorems, it is an interesting\nopen question whether the undecidability holds for reversal-bounded\nfinitely-ambiguous $\\NCM$ machines.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Slender Semilinear and Length-Semilinear Languages and Decidability Problems}\n\\label{sec:slender1}\n\nA topic closely related to counting functions of formal languages\nis that of slenderness. \nDecidability and closure properties of context-free languages ($\\CFL$s) have\nbeen investigated in \\cite{Ilie,matrix,Salomaa,Ilie2,Honkala1998}.\nFor example, \\cite{Ilie}\nshows that it is decidable whether a $\\CFL$ is slender, and\nin \\cite{matrix}, it is shown\nthat for a given $k \\geq 1$, it is decidable whether a language generated by\na matrix grammar is $k$-slender (although here, the $k$ needs to be provided as input\nin contrast to the $\\CFL$ result).\n\n\nIn this section, we generalize these results to arbitrary language families that\nsatisfy certain closure properties. These generalizations would then\nimply the known results for context-free languages and matrix languages,\nand other families where the problem was open.\n\n\nFirst, we discuss the bounded language case. \nThe constructive result of Theorem \\ref{cor9} above, plus\ndecidability of slenderness for regular languages implies the following:\n\\begin{corollary} \\label{cor13}\nLet ${\\cal L}$ be a semilinear trio (with all properties effective). \nThen, it is decidable, given $L$ bounded in ${\\cal L}$ and words \n$u_1, \\ldots, u_k$ such that $L \\subseteq u_1^* \\cdots u_k^*$,\nwhether $L$ is slender.\n\\end{corollary}\nIn \\cite{Honkala1998}, the similar result was shown that \nit is decidable whether or not a given bounded semilinear language $L$\nis slender. \n\nIn our definition of a semilinear family of languages ${\\cal L}$, we only\nrequire that every language in ${\\cal L}$ has a semilinear Parikh map.\nHowever, it is known that in every semilinear trio ${\\cal L}$, all\nbounded languages are bounded semilinear \\cite{CIAA2016}, and therefore\nthe result of \\cite{Honkala1998} also implies Corollary \\ref{cor13}. Conversely, all\nbounded semilinear languages are in the semilinear trio $\\NCM$\n\\cite{ibarra1978,CIAA2016}; hence, given any bounded semilinear \nlanguage (in any semilinear family\nso long as we can construct the semilinear set), slenderness is decidable.\nThis method therefore also provides an alternate proof to the result\nin \\cite{Honkala1998}.\n\n\nNext, we will examine the case where $L$ is not necessarily bounded.\nOne recent result is quite helpful in studying $k$-slender languages. In \\cite{fullaflcounters}, the following was shown.\n\\begin{theorem}\\cite{fullaflcounters}\nLet ${\\cal L}$ be any semilinear full trio where the semilinearity and intersection with regular languages properties are effective. Then the smallest full\nAFL containing intersections of languages in ${\\cal L}$ with $\\NCM$,\ndenoted by $\\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$, is effectively semilinear.\nHence, the emptiness problem for \n$\\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ is decidable.\n\\label{fullAFL}\n\\end{theorem}\nWe make frequent use of this throughout the proofs of the next two sections.\n\nFirst, decidability of $k$-slenderness is addressed.\n\\begin{theorem}\n\\label{kslender}\nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nIt is decidable, given $k$ and $L \\in {\\cal L}$, whether $L$ is a k-slender language.\n\\end{theorem}\n\\begin{proof} Let $L \\subseteq \\Sigma^*$, and let $\\#$ be a new symbol. \nFirst construct an $\\NCM$ $M_1$\nwhich when given $x_1 \\# x_2\\# \\ldots \\# x_{k+1}$, $x_i \\in \\Sigma^*, 1 \\leq i \\leq k+1$,\naccepts if $|x_1| = \\cdots = |x_{k+1}|$, and $x_i\\neq x_j$ are different, for\nall $i \\neq j$. To do this, $M_1$ uses (many) counters to verify the lengths,\nand it uses counters to guess positions and verify \nthe discrepancies between $x_i$ and $x_j$, for each $i \\ne j$.\nLet ${\\cal L}' = \\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ (or just ${\\cal L}$ in the second case), which is semilinear\nby Theorem \\ref{fullAFL}, (or length-semilinear in the second case, by assumption).\nConstruct $L_2 \\in {\\cal L}'$ \nwhich consists of all words of the form \n$x_1 \\# x_2\\# \\ldots \\# x_{k+1}$, where $x_i \\in L$ (every full AFL is closed\nunder concatenation). \nThen $L_3 = L(M_1) \\cap L_2 \\in {\\cal L}'$.\nClearly, $L$ is not $k$-slender if and only if $L_3$ is not empty, \nwhich is decidable by Theorem \\ref{fullAFL}.\n\\qed \\end{proof}\n\n\nThere are many known semilinear full trios listed in Example \\ref{semilinearfulltrioexamples}. Plus,\nit is known that languages generated by matrix grammars form a length-semilinear (but not semilinear, in general) full\ntrio closed under concatenation and intersection with $\\NCM$ (\\cite{matrix},\nwhere is it is shown that the languages are closed under intersection with\nBLIND multicounter languages, known to be equivalent to $\\NCM$ \\cite{G78}). Therefore, the\nresult is implied for matrix grammars as well, although this is already known.\n\\begin{corollary}\n\\label{allcorollaries}\nLet ${\\cal L}$ be any of the families listed in Example \\ref{semilinearfulltrioexamples}. \nThen, the problem, ``for $k \\geq 1$ and $L \\in {\\cal L}$, is $L$ a $k$-slender language?''\nis decidable.\n\\end{corollary}\n\n\nAll the machine models used in Example \\ref{semilinearfulltrioexamples} have one-way inputs, however with a two-way input,\nthe problem is more complicated.\nNow let $2\\DCM(k)$ (resp., $2\\NCM(k)$) be a two-way $\\DFA$ (resp., two-way\n$\\NFA$) with end-markers on both sides of the input, augmented with $k$ reversal-bounded counters.\n\\begin{theorem} It is decidable, given $k$ and $2\\DCM(1)$ $M$, whether $M$ accepts \na $k$-slender language.\n\\label{deckslender}\n\\end{theorem}\n\\begin{proof} We may assume that $M$ always halts \\cite{IbarraJiang}. Given $M$,\nconstruct another $2\\DCM(1)$ $M'$ with a $(k+1)$-track tape. First for each\n$1 \\le i < k+1$, $M'$ checks that the string in track $i$ is different from\nthe strings in tracks $i+1, \\ldots, k+1$. Thus, $M$ needs to make multiple\nsweeps of the $(k+1)$-track input.\n\nThen $M'$ checks that the string in each track is accepted. Clearly, $L(M)$ is \nnot $k$-slender if and only if $L(M')$ is not empty. The result follows, since\nemptiness is decidable for $2\\DCM(1)$ \\cite{IbarraJiang}.\n\\qed \\end{proof}\n\nThe above result does not generalize for $2\\DCM(2)$:\n\n\\begin{theorem} The following are true:\n\\begin{enumerate} \n\\item It is undecidable, given $k$ and a $2\\DCM(2)$ $M$, whether $M$ accepts a\n$k$-slender language, even when $M$ accepts a letter-bounded language\nthat is a subset of $a_1^* \\cdots a_r^*$ for given $a_1, \\ldots, a_r$.\n\\item It is undecidable, given $k$ and a $2\\DCM(2)$ $M$, whether $M$ accepts a slender\nlanguage, even when $M$ accepts a letter-bounded language.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} It is known \\cite{ibarra1978} that it is undecidable, given a $2\\DCM(2)$\n$M$ accepting a language that is in $b_1^* \\cdots b_r^*$ for given $b_1, \\ldots, b_r$, \nwhether $L(M) = \\emptyset$. Let $c$ and $d$ be new symbols.\nConstruct another $2\\DCM(2)$ $M'$ which when given\na string $w = b_1^{i_1} \\cdots b_k^{i_r} c^i d^j$, simulates $M$ on \n$b_1^{i_1} \\cdots b_k^{i_r}$, and when $M$ accepts, $M'$ accepts $w$.\nThen $L(M')$ is not $k$-slender for any given $k$ (resp., not slender) if and only if $L(M)$\nis not empty, which is undecidable.\n\\qed \\end{proof}\n\n\nWhether or not Theorem \\ref{kslender} holds for $2\\NCM(1)$ $M$ is open.\nHowever, we can prove a weaker version using the fact that it is\ndecidable, given a $2\\NCM(1)$ $M$ accepting a bounded language\nover $w_1^* \\cdots w_r^*$ for given $w_1, \\ldots, w_r$, whether\n$L(M) = \\emptyset$ \\cite{DangIbarra}.\n\n\\begin{theorem} It is decidable, given $k$ and $2\\NCM(1)$ $M$ that accepts a \nlanguage over $w_1^* \\cdots w_r^*$ for given $w_1, \\ldots, w_r$, whether $M$ accepts\na $k$-slender language.\n\\end{theorem}\n\\begin{proof}\nWe construct from $M$ another $2\\NCM(1)$ $M'$ which, when given \n$x_1 \\# x_2\\# \\cdots \\# x_{k+1}$, where each\n$x_i$ is in $w_1^* \\cdots w_r^*$ first checks that all $x_i$'s \nare different. For each $i$, and $j = i+1, \\ldots, k+1$, $M'$ guesses the position\nof discrepancy between $x_i$ and $x_j$ and records this position in the counter\nso that it can check the discrepancy. (Note that only one reversal-bounded\ncounter is needed for this.) Then $M'$ checks that each $x_i$ is accepted. \nClearly, $M$ is not $k$ slender if and only if $L(M')$ is not empty, which is \ndecidable, since the language accepted by $M'$ is bounded.\n\\qed \\end{proof}\n\nWe can also prove some closure properties. Here is an example:\n\n\\begin{theorem}\nThe following are true:\n\\begin{enumerate} \n\\item Slender (resp., thin) $\\NCM$ languages are closed under intersection.\n\\item Slender (resp., thin) $2\\DCM(1)$ languages ($2\\NCM(1)$ languages) are \n closed under intersection.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nStraightforward since the families of languages above are\nclosed under intersection.\n\\qed \\end{proof}\n\nDeciding if an $\\NCM$ language (or anything more general than $\\CFL$s) is $k$-slender for some $k$ (where $k$ is not part of the input) is open,\nalthough as we showed in Theorem \\ref{deckslender}, for a given $k$, we can \ndecide if an $\\NCM$ or $\\NPCM$ accepts a $k$-slender language. We conjecture\nthat every $k$-slender $\\NPCM$ language is bounded. If \nthis can be proven, we will also need an algorithm to determine words $w_1, \\ldots,\nw_r$ such that the language is a subset of $w_1^* \\cdots w_r^*$, which \nwe also do not yet know how to do. \n\n\n\nLet $c \\ge 1$. A $2\\DCM(k)$ ($2\\NCM(k))$ $M$ is $c$-crossing if the number of times\nthe input head crosses the boundary of any two adjacent cells of the input is\nat most $c$. Then $ M$ is finite-crossing if it is $c$-crossing for some $c$. It is known\nthat a finite-crossing $2\\NCM(k)$ can can be converted to an $\\NCM(k')$ for some\n$k'$ \\cite{Gurari1981220}. Hence every bounded language accepted by any\nfinite-crossing $2\\NCM(k)$ is counting-regular. The next result shows that this is\nnot true if the two-way input is unrestricted:\n\\begin{theorem} The letter-bounded language $L = \\{a^i b^{ij} ~|~ i, j \\ge 1\\}$\nis accepted by a $2\\DCM(1)$ whose counter makes only $1$ reversal, but $L$ is not\ncounting-regular.\n\\end{theorem}\n\\begin{proof}\nA $2\\DCM(1)$ $M$ accepting $L$ operates as follows, given input $a^ib^k$ ($i, k \\ge 1$):\n$M$ reads and stores $k$ in the counter. Then it makes multiple sweeps on $a^i$\nwhile decrementing the counter to check that $k$ is divisible by $i$.\n\nTo see that $L$ is not counting-regular, we note \nthat, for any $n \\geq 2$, the number of strings of length $n$ in $L$ is $\\phi(n)$, Euler's totient function \nis equal to the number of divisors of $n$. In fact, let the divisors of $n$ be\n$d_1, d_2, \\ldots, d_m$. Then, there are exactly $m$ strings of length $n$, namely: $a^{d_1} b^{(n\/d_1 - 1)d_1}, a^{d_2} b^{(n\/d_2 - 1)d_2}, \\ldots , \\\\\na^{d_m} b^{(n\/d_m - 1)d_m}$. Conversely, for each string $w$ of length $n$ in $L$, there is a unique divisor of $n$ (namely the number of $a$'s in $w$) associated with the string. This means the generating function of $L$ is $f(z)$ = $\\sum_{n \\geq 2} \\phi(n) z^n$. \n\nIt can be shown that $f(z)$ = $\\sum_{n \\geq 2} a_n {z^n \\over {1-z^n}}$ where $a_n$ = $\\sum_{d|n} \\phi(d) \\mu(n\/d)$ where $\\mu$ is the Mobius function.\nFrom this expression, it is clear that every solution to $z^n=1$ is a pole of $f(z)$ and so the $n$'th root of unity is a pole of $f(z)$\nfor each positive integer $n \\geq 2$. Since a rational function can have only a finite number of poles, it follows that $f(z)$ is not\nrational and hence $L$ is not counting-regular.\n\\qed \\end{proof}\nHowever, it is known that unary languages accepted by $2\\NCM(k)$s are\nregular \\cite{IbarraJiang}; hence, such languages are counting-regular.\n\n\nThis is interesting, as $2\\DCM(1)$ has a decidable emptiness and slenderness problem, yet\nthere is a letter-bounded language from the theorem above accepted by a $2\\DCM(1)$\nthat is not counting-regular.\n\nThe following result provides an interesting contrast, as it\ninvolves a model with an undecidable membership\n(and emptiness) problem, but provides an example of a (non-recursively enumerable) \nlanguage that is all of slender, thin,\nbounded, semilinear, but also counting-regular. \n\n\\begin{theorem}\nThere exists a language $L$ that is letter-bounded and semilinear and thin and counting regular but is not recursively enumerable. Moreover, we can effectively construct a \n$\\DFA$ $M$ such that\n$f_{L(M)}(n) = f_L(n)$. \n\n\\end{theorem}\n\\begin{proof}\nLet $L \\subseteq a^*$ be a unary language that is not recursively enumerable, which is known to exist \\cite{Minsky}. Assume without loss of generality that the empty word is not in $L$, \nbut the letter $a$ itself is in $L$.\n\nLet $L'$ be the language consisting of, for each $n \\geq 1$, the single word\nwith all $a$'s except for one $b$ in the position of the largest $m \\leq n$ such that\n$a^m \\in L$.\n\n\nThen $L'$ has one word of every length, and is therefore thin. Also, it has one word of every length and exactly one $b$ in every word, so it has the same Parikh map as $a^* b$, so is semilinear. Also, it is clearly bounded in $a^*b a^*$. It is also not recursively enumerable, otherwise if it were, then make a gsm \\cite{Hopcroft} that outputs $a$'s for every $a$ until a $b$, then it outputs $a$. Then, for every remaining $a$, it outputs the empty word. Applying this to $L'$ gives $L$ (the position of the $b$ lets the gsm recover the words of $L$). But the recursively enumerable languages are closed under gsm mappings, a contradiction.\n\\qed \\end{proof}\n\n\n\\section{Characterization of $k$-Slender Semilinear and Length-Semilinear Languages}\n\\label{sec:slender2}\n\n\nThis section discusses decidability properties (such as the problem of testing whether two\nlanguages are equal, or one language is contained in another) for $k$-slender languages in\narbitrary families of languages satisfying certain closure properties. \nIt is known that the equivalence problem for $\\NPCM$ languages \nthat are subsets of $w_1^* \\cdots w_r^*$ for given $w_1, \\ldots , w_r$ \nis decidable. However, as mentioned above, we do not know yet \nif $k$-slender languages are bounded and even if they are, we do not\nknow yet how to the determine the associated words $w_1, \\ldots, w_r$.\nHence, the definition and results below are of interest.\n\nThe following notion is useful for studying decidability properties\nof slender languages.\nLet $k \\ge 1$ be given. A language $L$ is $k$-slender effective if we \ncan effectively construct a $\\DFA$ over a unary alphabet $\\{1\\}$ with $k+1$\ndistinguished states $s_0, s_1, \\ldots, s_k$ (other states can exist) \nwhich, when given an input\n$1^n$ where $n \\ge 0$, halts in state $s_i$ if $f_L(n) = i$, where \n$0 \\le i \\le k$. (Hence, the $\\DFA$ can determine the number of strings \nin $L$ of length $n$, for every $n$.)\n\nFor example, consider the language\n $$L = \\{a^i b^i \\mid i \\ge 1\\} \\cup \\{c^i a^i d^i \\mid i \\ge 1\\}.$$\nThen,\n$$f_L(n) = \n\\begin{cases} 0 & \\mbox{if $n = 0$ or $n=1$ or not divisible by $2$ and $3$,}\\\\\n 1 & \\mbox{if $n \\ge 2$, is divisible by $2$, and not divisible by $3$,}\\\\\n 1 & \\mbox{if $n \\ge 3$, is divisible by $3$, and not divisible by $2$,}\\\\\n 2 & \\mbox{if $n \\ge 2$, is divisible by $2$ and divisible by $3$}.\n\\end{cases}$$\nClearly, $L$ is $2$-slender effective.\n\n\nNext, we will discuss which $k$-slender languages are $k$-slender effective.\nFirst, we say that the length-semilinear property is effective if it is possible\nto effectively construct, for $L \\in {\\cal L}$, a $\\DFA$ accepting $\\{1^n \\mid f_L(n) \\geq 1 \\}$.\n\\begin{theorem}\n\\label{finiteunion} Let ${\\cal L}$ be an effective length-semilinear trio.\nA finite union of thin languages in ${\\cal L}$ is $k$-slender effective.\n\\end{theorem}\n\\begin{proof}\nLet $L$ is the finite union of $L_1, \\ldots, L_k \\in {\\cal L}$, each thin. Then construct\n$h(L_i)$, where $h$ maps $L_i$ onto the single letter $1$. Then $h(L_i) = \\{1^n \\mid f_{L_i}(n) = 1\\}$. Then since ${\\cal L}$ is length-semilinear, each of $h(L_i)$ are regular, and can be accepted by a $\\DFA$ $M_i$. Thus, make another $\\DFA$ $M'$ such that, on input $1^n$, $M'$ runs each $M_i$ in parallel on $1^n$, and then switches to distinguished state $s_j$ if there are $j$ of the $\\DFA$s $M_1, \\ldots, M_k$ that are accepting.\n\\qed \\end{proof}\n\n\nIt will be seen next that for all $k$-slender languages in ``well-behaved''\nfamilies, they are $k$-slender effective. First, the following lemma is needed.\n\\begin{lemma} \n\\label{makeeffective}\nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nLet $ k \\ge 1$ and $L \\in {\\cal L}$, a $k$-slender\nlanguage such that $f_{L}(n)$ is equal to $0$ or $k$ for every $n$. \nThere\nis a $\\DFA$ that can determine $f_L(n)$. Hence $L$ is a $k$-slender\neffective language.\n\\end{lemma}\n\\begin{proof}\nLet ${\\cal L}' = \\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ (or just ${\\cal L}$ in the second case), which is semilinear\nby Theorem \\ref{fullAFL}, (or length-semilinear by assumption).\n \nConsider\n$L' = \\{ x_1 \\# \\cdots \\# x_k \\mid x_1, \\ldots, x_k \\in L\\} \\in {\\cal L}'$.\nCreate $L''$ by intersecting $L'$ with an $\\NCM$ language that enforces that all words\nof the form \n$ x_1 \\# \\cdots \\# x_k $ have\n$|x_1| = \\cdots = |x_k|$, and $x_i \\ne x_j$ for each $i \\ne j$.\nThus $L'' = \\{ x_1 \\# \\cdots \\# x_k \\mid x_1, \\ldots, x_k \\in L,\n|x_1| = \\cdots = |x_k|, x_i \\ne x_j \\mbox{~for each~} i \\ne j\\}$.\nHence, $L''\\in {\\cal L}'$. Let $L'''$ be the language obtained from $L''$ by\nhomomorphism that projects onto the single letter $1$. \nSince $L'''$ is length-semilinear, it can be accepted by a $\\DFA$. Moreover, the length $n$ of a word\n$x_1\\# \\cdots \\# x_k \\in L''$ can be transformed into $|x_1|$ via $\\frac{n-(k-1)}{k}$. Given the $\\DFA$,\nthen another \n$\\DFA$ can be built that can determine $f_L(n)$.\n\\qed \\end{proof}\n\nThen, the following is true.\n\\begin{theorem}\nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, union, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nIf $L \\in {\\cal L}$ be a $k$-slender language $L$, then\n$L$ is $k$-slender effective.\n\\label{thm22}\n\\end{theorem}\n\\begin{proof} The case $k = 1$ is true by Theorem \\ref{finiteunion}. \n Assume by induction that\nthe theorem is true for $k \\geq 1$.\n\nNow consider an $L \\in {\\cal L}$ that is a $(k+1)$-slender language,\n$k \\ge 1$. Hence $f_{L}(n) \\leq (k+1)$ for each $n \\geq 0$. \n\nLet ${\\cal L}' = \\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ (or just ${\\cal L}$ in the second case), which is semilinear\nby Theorem \\ref{fullAFL}, (or length-semilinear by assumption).\nLet $A = \\{x_1 \\# \\cdots \\# x_{k+1} \\mid x_1, \\ldots,x_{k+1} \\in L\\} \\in {\\cal L}'.$\nThen intersect\n$A$ with an $\\NCM$ that enforces that all words\n$x_1 \\# \\cdots \\# x_{k+1}$ have\n$|x_1| = \\cdots = |x_{k+1} |$, and $x_i \\ne x_j$ for each $i \\ne j$.\nLet $A'$ be the resulting language.\nThen\n$A' = \\{x_1 \\# \\cdots \\# x_{k+1} \\mid x_1, \\ldots,x_{k+1}\\in L \\mbox{~such that~} \n |x_1| = \\cdots = |x_{k+1} |, x_i \\ne x_j \\mbox{~for each~} i \\ne j\\}.$\nBy Lemma \\ref{makeeffective}, a $\\DFA$ accepting $\\{1^n \\mid f_L(n) = k+1\\}$\ncan be effectively constructed. Thus, a $\\DFA$ accepting\n$\\{1^n \\mid f_L(n) \\neq k+1\\}$ can also be constructed. Furthermore,\n$B = \\{w \\mid w\\in L, f_L(|w|) \\neq k+1\\} \\in {\\cal L}'$.\nThen, $B$ is $k$-slender and, hence $k$-slender effective by induction hypothesis.\nHence, $L$ is $(k+1)$-slender effective.\n\\qed \\end{proof}\n\n\n\n\n\nThe proof of Theorem \\ref{thm22} actually shows the following:\n\\begin{corollary} \nLet ${\\cal L}$ be a full trio closed under concatenation, union, and intersection with $\\NCM$, and is length-semilinear\nwith all properties effective.\nLet $k \\ge 1$. A language $L \\in {\\cal L}$ is a $k$-slender language \nif and only if $L = L_1 \\cup \\cdots \\cup L_k$, where for $1 \\le i \\le k$, $L_i$ is\nan $i$-slender effective language such that $f_{L_i}(n)$ is equal to $0$ or $i$ for each $n$.\n\\end{corollary}\n\n\nNext, decidability of containment is addressed.\n\\begin{theorem} \nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nIt is decidable, given $L_1, L_2 \\in {\\cal L}$ with $L_2$\nbeing a $k$-slender language, whether \n$L_1 \\subseteq L_2$.\n\\label{containment}\n\\end{theorem}\n\\begin{proof} Then $L_2$ is $k$-slender effective by Theorem \\ref{thm22}. Without loss of generality, assume that the input alphabet \nof both $L_1$ and $L_2$ is $\\Sigma$. Let $1, \\#$, and $\\$$ be new symbols. \nLet ${\\cal L}' = \\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ (or just ${\\cal L}$ in the second case), which is semilinear\nby Theorem \\ref{fullAFL}, (or length-semilinear by assumption).\nWe will construct a sequence of machines and languages below.\n\\begin{enumerate}\n\\item First, let $M_1'$ (resp.\\ $M_2'$) be the unary $\\DFA$ accepting all words $1^n$ where\na word of length $n$ is in $L_1$ (resp.\\ in $L_2$). Let $A_1 = L(M_1) - L(M_2)$. (This is\nempty if and only if all lengths of words in $L_1$ are lengths\nof words in $L_2$. This language is regular.\n\n\\item Construct $A_2 \\in {\\cal L}'$ consisting of all words\n$w = 1^n \\$ x \\$ y_1\\# \\cdots y_r\\$$, where \n$x \\in L_1$ and each $y_j \\in L_2$.\n\n\\item Construct an $\\NCM$ $A_3$ which, when given \n$w = 1^n\\$x\\$y_1\\# \\cdots y_r\\$$, accepts $w$ if the following is true:\n\\begin{enumerate}\n\\item $r = f_{L_2}(n)$ (which can be tested since $L_2$ is $k$-slender effective). \n\\item $|x| = |y_1| = \\cdots = |y_r| = n$.\n\\item $y_i \\ne y_j$ for each $i \\ne j$.\n\\item $x \\ne y_i$ for each $i$.\n\\end{enumerate}\nNote that $A_3$ needs multiple reversal-bounded counters to carry out the four\ntasks in parallel.\n\\item Construct $A_4 = A_2 \\cap A_3 \\in {\\cal L}'$.\n\\item Finally construct an $A_5 = A_4 \\cup A_1 \\in {\\cal L}'$ (full trios are closed\nunder union with regular languages \\cite{G75}).\n\\end{enumerate}\n\nIt is easy to verify that $L_1 \\not\\subseteq L_2$ if and only \nif $A_5$ is not empty, which is decidable, since emptiness is decidable.\n\\qed \\end{proof}\n\\begin{corollary}\nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nIt is decidable, given $L_1,L_2 \\in {\\cal L}$ that are \n$k$-slender languages, whether $L_1 = L_2$.\n\\end{corollary}\n\n\nThere are many semilinear full trios in the literature for which the properties in this section hold. \n\\begin{corollary}\nLet ${\\cal L}$ be any of the families from Example \\ref{semilinearfulltrioexamples}.\nThe following are decidable:\n\\begin{itemize}\n\\item For $L_1, L_2$ with $L_2$ being $k$-slender, is $L_1 \\subseteq L_2$?\n\\item For $L_1, L_2$ being $k$-slender languages, is $L_1 = L_2$?\n\\end{itemize}\n\\end{corollary}\n\n\n\n\nFurthermore, matrix grammars are an example of a length-semilinear \\cite{DassowHandbook} full trio closed under concatenation, union, and intersection with $\\NCM$ (although they do accept non-semilinear languages). We therefore\nget all these properties for matrix grammars as a consequence of these proofs. However, this result is already known \\cite{matrix}.\n\nUsing the ideas in the constructive proof of the theorem above,\nwe can also show:\n\\begin{theorem} Let ${\\cal L}$ be a union and concatenation closed length-semilinear full trio with all properties effective\nthat is closed under intersection with $\\NCM$.\nLet $L_1, L_2 \\in {\\cal L}$ with $L_2$ a $k$-slender language.\nThen $L_1-L_2 \\in {\\cal L}$. Hence, the complement of any $k$-slender language\nin ${\\cal L}$ is again in ${\\cal L}$.\n\\label{difference}\n\\end{theorem}\n\\begin{proof} Let $\\Sigma$ be the (without loss of generality) joint alphabet of $L_1$ and\n$L_2$, and let $\\Sigma'$ be the set of the\nprimed versions of the symbols in $\\Sigma$. Let $\\#$, and $\\$$ be new symbols. \nConsider input \n\\begin{equation}\nw = x\\$y_1\\# \\cdots \\#y_r,\n\\label{w}\n\\end{equation} where $x$ is in $(\\Sigma')^*$ and $y_1, \\ldots, y_r$ \nare in $\\Sigma^*$, for some $0 \\leq r \\leq k$. \nBy Theorem \\ref{thm22}, $L_2$ is $k$-slender effective. \nLet $M'$ be this unary $\\DFA$ accepting all words of lengths in $L_2$.\nBuild an $\\NCM$ $M''$ that on input $w$, verifies:\n\\begin{enumerate}\n\\item $r = f(n)$.\n\\item $|x|=|y_1| = \\cdots = |y_r|$.\n\\item $y_i \\ne y_j$ for each $i \\ne j$.\n\\item $h(x) \\ne y_i$ for each $i$, where $h(a') = a$ for each $a' \\in \\Sigma'$.\n\\end{enumerate}\n\nConsider $L''' \\in {\\cal L}$ \nconsisting of all words of the form of $w$ in Equation \\ref{w},\nwhere $x \\in L_1$, and each $y_i \\in L_2$.\nThis is in ${\\cal L}$ since ${\\cal L}$ is closed under concatenation.\n\nNow define a homomorphism $h_1$ which maps $\\#, \\$$, and symbols in $\\Sigma$ to\n$\\epsilon$ and fixes letters in $\\Sigma'$. Clearly, $h_1(L''' \\cap L(M''))$ is \n$L_1 - L_2$, and it is in ${\\cal L}$.\n\\qed \\end{proof}\n\nThis holds for not only the matrix languages, but also concatenation and union-closed\nsemilinear full trios closed under intersection with $\\NCM$. Some examples are:\n\\begin{corollary}\nLet ${\\cal L}$ be any family of languages that are\naccepted by a machine model in Example \\ref{semilinearfulltrioexamples} \nthat are augmented by reversal-bounded counters.\nGiven $L_1, L_2 \\in {\\cal L}$ with $L_2$ being $k$-slender, then $L_1 - L_2 \\in {\\cal L}$.\nFurthermore, the complement of any $k$-slender language in ${\\cal L}$ is again in ${\\cal L}$.\n\\end{corollary}\n\nNext, decidability of disjointness for $k$-slender languages will be addressed.\n\\begin{theorem}\nLet ${\\cal L}$ be a full trio which is either:\n\\begin{itemize}\n\\item semilinear, or\n\\item is length-semilinear, closed under concatenation, union, and intersection with $\\NCM$,\n\\end{itemize}\nwith all properties effective.\nGiven $L_1,L_2 \\in {\\cal L}$ being $k$-slender languages,\nit is decidable whether $L_1 \\cap L_2 = \\emptyset$.\n\\end{theorem}\n\\begin{proof}\nLet ${\\cal L}' = \\hat{{\\cal F}}({\\cal L} \\wedge \\NCM)$ (or just ${\\cal L}$ in the second case), which is semilinear\nby Theorem \\ref{fullAFL}, (or length-semilinear by assumption).\n\nNotice that $L_1 \\cap L_2 = (L_1 \\cup L_2) - ((L_1 - L_2) \\cup (L_2-L_1))$.\nBy Theorem \\ref{difference}, $L_1 - L_2 \\in {\\cal L}$ and $L_2 - L_1 \\in {\\cal L}$, and\nboth must be $k$-slender since $L_1$ and $L_2$ are both $k$-slender.\nCertainly $(L_1 - L_2) \\cup (L_2-L_1) \\in {\\cal L}$, and is also $2k$-slender.\nAlso, $L_1 \\cup L_2 \\in {\\cal L}$.\nHence, by another application of Theorem \\ref{difference},\n$(L_1 \\cup L_2) - ((L_1 - L_2) \\cup (L_2-L_1))\\in {\\cal L}$. Since emptiness\nis decidable in ${\\cal L}$, the theorem follows.\n\\qed \\end{proof}\nThis again holds for all the families in Example \\ref{semilinearfulltrioexamples} plus the languages accepted by matrix grammars.\n\n\n\nAn interesting open question is whether every $k$-slender $\\NCM$ language (or other more\ngeneral families)\ncan be decomposed into a finite disjoint union of\nthin $\\NCM$ languages.\nAlthough we have not been able to show this, we can give a related result. \nTo recall, in \\cite{Harju}, the model $\\TCA$ is introduced consisting of a nondeterministic Turing machine with a one-way read-only input tape, a finite-crossing read\/write tape, and reversal-bounded counters. It is shown that this model only accepts semilinear languages, and indeed, it is a full trio. Clearly, the model is closed under intersection with $\\NCM$ by adding more counters. Although we do not know whether it is possible to decompose $\\NCM$ slender languages into thin $\\NCM$ languages, we can decompose them into thin $\\TCA$ languages.\n\n\\begin{theorem}\nEvery $k$-slender $\\NCM$ language $L$ is a finite union of thin $\\TCA$ languages.\n\\end{theorem}\n\\begin{proof}\nLet $M$ be an $\\NCM$ accepting $L$. Since $L$ is $k$-slender, for each $n$, there are either exactly $k$ words of length $n$, or $k-1$ words of length $n$, etc.\\ or $0$ words of length $n$. \nLet $A_k = \\{ x_1 \\# \\cdots \\# x_k \\mid x_1, \\ldots, x_k \\in L(M), |x_1| = \\cdots = |x_k|$, $x_1 < \\cdots x_1$. From that point on, it replaces $x_1$ on the tape with $x_2$. It then\nrepeats up to $x_k$.\n\nLet $G_i$ be a gsm that extracts the $i$'th ``component'' of $A_k$. Then $G_1(A_k), \\ldots, G_k(A_k)$ are all thin languages.\nAs they are thin, there is a $\\DFA$ $M_k$ accepting all these lengths of words. \nNext, let $A_{k-1} = \\{ x_1 \\# \\cdots \\# x_{k-1} \\mid x_1, \\ldots, x_{k-1} \\in L(M), |x_1| = \\cdots = |x_{k-1}|$, $x_1 < \\cdots \\nonumber \\\\\n& +\\log \\pi(\\boldsymbol{\\theta})\n\\end{array}\n\\end{equation}\nwhere the brackets $<.>$ imply expectation with respect to $\\hat{p}(\\boldsymbol{x}_{1:L} \\mid \\boldsymbol{\\theta}^{(k-1)}, \\boldsymbol{y}_{1:L})$ as in \\refeq{eq:batch}. In order to maximize $\\hat{Q}(\\boldsymbol{\\Theta}^{(1:k-1)}, \\boldsymbol{\\Theta})$ as in \\refeq{eq:mstepsa} one needs to solve the system of equations arising from $\\frac{\\partial \\hat{Q}(\\boldsymbol{\\theta}^{(1:k-1)}, \\boldsymbol{\\theta}) }{ \\partial \\boldsymbol{\\theta} }=\\boldsymbol{0}$\nThese equations equations with respect to $\\boldsymbol{\\theta}$ are solved with fixed point iterations. They depend on the following $7$ sufficient statistics $\\boldsymbol{\\Phi}=\\{\\Phi_j\\}_{j=1}^7$:\n\\begin{equation}\n\\begin{array}{l}\n \\Phi_1=<\\boldsymbol{x}_1> \\\\\n\\Phi_2= < \\boldsymbol{x}_1 \\boldsymbol{x}^T_1 > \\\\\n\\Phi_3=\\left< \\sum_{t=2}^L\\boldsymbol{x}_{t-1} \\right> \\\\\n\\Phi_4= \\left< \\sum_{t=2}^L \\boldsymbol{x}_t-\\boldsymbol{x}_{t-1} \\right> \\\\\n\\Phi_5= \\left< \\sum_{t=2}^L \\boldsymbol{x}_{t-1} \\boldsymbol{x}_{t-1}^T \\right> \\\\\n\\Phi_6= \\left< \\sum_{t=2}^L (\\boldsymbol{x}_t-\\boldsymbol{x}_{t-1})\\boldsymbol{x}_{t-1}^T \\right> \\\\\n\\Phi_7= \\left< \\sum_{t=2}^L (\\boldsymbol{x}_t-\\boldsymbol{x}_{t-1})(\\boldsymbol{x}_t-\\boldsymbol{x}_{t-1})^T \\right> \n\\end{array}\n\\end{equation}\n\n\\subsection*{ Sufficient statistics for parameters appearing in the likelihood}\n\nThe process a bit more involved in the case of the parameters appearing in the likelihood \\refeq{eq:like} i.e. the projection matrices $\\{ \\boldsymbol{P}^{(m)} \\}_{m=1}^M$ of dimension $d \\times K$ and the covariance $\\boldsymbol{\\Sigma}$ which is a (positive definite) matrix of $d \\times d$. In order to retain scalability in high-dimensional problems (i.e. $d>>1$) we assume a diagonal form of $\\boldsymbol{\\Sigma}=diag(\\sigma_1^2, \\sigma_2^2, \\ldots, \\sigma_d^2)$ which implies learning $d$ parameters rather than $d(d+1)\/2$.\n\nDenoting now by $\\boldsymbol{\\theta}=(\\{ \\boldsymbol{P}^{(m)} \\}_{m=1}^M, \\{\\sigma_j^2\\}_{j=1}^d )$ , $\\pi(\\boldsymbol{\\theta})$ the prior and according to Equations (\\ref{eq:sdlpost}) and (\\ref{eq:batch}) we have that:\n\\begin{equation}\n\\label{eq:likess}\n\\begin{array}{ll}\n\\hat{Q}(\\boldsymbol{\\theta}^{(k-1)}, \\boldsymbol{\\theta})& =\\left< \\sum_{t=1}^L - \\frac{1}{2} \\log \\mid \\boldsymbol{\\Sigma} \\mid -\\frac{1}{2} (\\boldsymbol{y}_t-\\boldsymbol{W}_t \\boldsymbol{X}_t )^T \\boldsymbol{\\Sigma}^{-1} (\\boldsymbol{y}_t-\\boldsymbol{W}_t \\boldsymbol{X}_t) \\right> \\\\\n& + \\log \\pi(\\boldsymbol{\\theta})\n\\end{array}\n\\end{equation}\nDifferentiation with respect to $ \\boldsymbol{P}^{(m)} $ reveals that the stationary point must satisfy:\n\\begin{equation}\n\\boldsymbol{A}^{(m)}=\\sum_{n=1}^M \\boldsymbol{P}^{(n)} \\boldsymbol{B}^{(n,m)}\n\\end{equation}\nwhere the sufficient statistics are:\n\\begin{equation}\n\\label{eq:ap1}\n\\underbrace{\\boldsymbol{A}^{(m)}}_{d \\times K}=\\left< \\sum_{t=1}^L z_{t,m} \\boldsymbol{y}_t (\\boldsymbol{x}_t^{(m)})^T \\right>, \\quad m=1,2,\\ldots, M\n\\end{equation}\nand:\n\\begin{equation}\n\\underbrace{ \\boldsymbol{B}^{(n,m)} }_{K \\times K} =\\left< \\sum_{t=1}^L z_{t,n} z_{t,m} \\boldsymbol{x}_t^{(n)} (\\boldsymbol{x}_t^{(m)})^T \\right>\n\\end{equation}\nIn the absence of a prior $\\pi(\\boldsymbol{\\theta})$ and if $\\boldsymbol{P}_j^{(m)}$ and $\\boldsymbol{A}^{(m)}_j$ represent the $j^{th}$ rows ($j=1,\\ldots,d$) of the matrices $\\boldsymbol{P}^{(m)}$ and $\\boldsymbol{A}^{(m)}$ respectively, then \\refeq{eq:ap1} implies:\n\\begin{equation}\n\\begin{array}{ll}\n\\underbrace{ \\left[ \\begin{array}{llll} \\boldsymbol{A}^{(1)}_j & \\boldsymbol{A}^{(2)}_j & \\ldots & \\boldsymbol{A}^{(M)}_j \\end{array} \\right] }_{\\boldsymbol{A}_j: (1 \\times K~M)}= &\n\\underbrace{ \\left[ \\begin{array}{llll} \\boldsymbol{P}^{(1)}_j & \\boldsymbol{P}^{(2)}_j & \\ldots & \\boldsymbol{P}^{(M)}_j \\end{array} \\right] }_{\\boldsymbol{P}_j: (1\\times K~M)} \\\\\n& \\underbrace{ \\left[ \\begin{array}{llll} \\boldsymbol{B}^{(1,1)} & \\boldsymbol{B}^{(1,2)} & \\ldots & \\boldsymbol{B}^{(1,M)} \\\\ \\boldsymbol{B}^{(2,1)} & \\boldsymbol{B}^{(2,2)} & \\ldots & \\boldsymbol{B}^{(2,M)} \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ \\boldsymbol{B}^{(M,1)} & \\boldsymbol{B}^{(M,2)} & \\ldots & \\boldsymbol{B}^{(M,M)} \\end{array} \\right] }_{\\boldsymbol{B}: ( MK\\times MK)}\n\\end{array}\n\\end{equation}\nThis leads to the following update equations for $\\boldsymbol{P}_j^{(m)}, ~\\forall j,m$:\n\\begin{equation}\n\\boldsymbol{P}_j=\\boldsymbol{A}_j \\boldsymbol{B}^{-1}\n\\label{eq:ap2}\n\\end{equation}\nNote that the matrix $\\boldsymbol{B}$ to be inverted is \\underline{ independent } of the dimension of the observables $d$ ($d>>1$) and the inversion needs to be carried out once for all $j=1,\\ldots,d$. Hence the {\\em scaling of the update equations for $\\boldsymbol{P}^{(m)}$ is $O(d)$} i.e. linear with respect to the dimensionality of the original system.\n\n\nFurthermore, in the absence of a prior $\\pi(\\boldsymbol{\\theta})$, differentiation with respect to $\\sigma_j^{-2}$ ($j=1,\\ldots,d$) leads to the following update equation:\n\\begin{equation}\n\\begin{array}{ll}\nL~\\sigma_j^2 & = \\sum_{t=1}^L y_{t,j}^2-2 \\boldsymbol{A}_j ~\\boldsymbol{P}_j^T+\\boldsymbol{P}_j~\\boldsymbol{B} \\boldsymbol{P}_j^T\n\\end{array}\n\\end{equation}\n In summary the sufficient statistics needed are the ones in Equations (\\ref{eq:ap1}) and (\\ref{eq:ap2}).\n\nIn the numerical examples in this paper a diffuse Gaussian prior was used for $\\boldsymbol{P}^{(m)}$ with variance $100$ for each of the entries of the matrix. This leads to the addition of the term $1\/100$ in the diagonal elements of the $\\boldsymbol{B}$ in \\refeq{eq:ap2}. No priors were used for $\\sigma_j^2$.\n\n\\subsection{From static-linear to dynamic-nonlinear dimensionality reduction}\n\\label{sec:genintro}\n\nThe inherent assumption of all multiscale analysis methods is the existence of a lower-dimensional parameterization of the original system with respect to which the dynamical evolution is more tractable at the scales of interest.\nIn some cases these slow variables can be identified a priori and the problem reduces to finding the necessary closures that will give rise to a consistent dynamical model.\nIn general however one must identify the reduced space $\\mathcal{\\hat{Y}}$ as well as the dynamics within it.\n\nA prominent role in these efforts has been held by Principal Component Analysis (PCA) -based methods. With small differences and depending on the community other terms such as Proper Orthogonal Decomposition (POD) or Karhunen-Lo\\`eve expansion (KL), Empirical Orthogonal Functions (EOF) have also been used. PCA finds its roots in the early papers by Pearson \\cite{pea01lin} and Hotelling \\cite{hot33ana} and was originally developed as a {\\em static} dimensionality reduction technique. It is based\non projections on a reduced basis identified by the leading eigenvectors of the covariance matrix $\\boldsymbol{C}$. In the dynamic case and in the absence of closed form expressions for the actual covariance matrix, samples of the process $\\boldsymbol{y}_{t} \\in \\mathbb R^d$ at $N$ distinct time instants $t_i$ are used in order to obtain an estimate of the covariance matrix:\n\\begin{equation}\n\\label{eq:cov}\n \\boldsymbol{C} \\approx \\boldsymbol{C}_N=\\frac{1}{N-1} \\sum_{i=1}^N (\\boldsymbol{y}_{t_i} -\\boldsymbol{\\mu}) (\\boldsymbol{y}_{t_i}-\\boldsymbol{\\mu})^T\n\\end{equation}\nwhere $\\boldsymbol{\\mu}=\\frac{1}{N} \\sum_{i=1}^N \\boldsymbol{y}_{t_i} $ is the empirical mean. If there is a spectral gap after the first $k$ eigenvalues and $\\boldsymbol{V}_K$ is the $d\\times K$ matrix whose columns are the $K$ leading normalized eigenvectors of $\\boldsymbol{C}_N$ then the reduced-order model is defined with respect to $\\boldsymbol{\\hat{y}}_t=\\boldsymbol{V}_K \\boldsymbol{y}_t$. The reduced dynamics can be identified by a Galerkin projection (or a Petrov-Galerkin projection) of the original ODEs in \\refeq{eq:master}:\n\\begin{equation}\n\\label{eq:pca}\n\\frac {d \\boldsymbol{\\hat{y}}_t}{d t}=\\boldsymbol{V}^T_K \\boldsymbol{f}(\\boldsymbol{V}^T_K \\boldsymbol{\\hat{y}}_t)\n\\end{equation}\nHence the reduced space $\\mathcal{\\hat{Y}}$ is approximated by a hyperplane in $\\mathcal{Y}$ and the projection mapping $\\mathcal{P}$ linear (Figure \\ref{fig1a}). While it can be readily shown that the projection adopted is optimal in the mean square sense for stationary Gaussian processes, it is generally not so in cases where non-Gaussian processes or other distortion metrics are examined. The application of PCA-based techniques, to high-dimensional, multiscale dynamical systems poses several modeling limitations. Firstly, the reduced space $\\mathcal{\\hat{Y}}$ might not be sufficiently approximated by a hyperplane of dimension $K<>1$) and large datasets ($N>>1$) as the $K$ leading eigenvectors of large matrices (of dimension proportional to $d$ or $N$) need to be evaluated.\nThis effort must be repeated, if more samples become available (i.e. $N$ increases) and an update of the reduced-order model is desirable. Recent efforts have concentrated on developing online versions \\cite{war06onl} that circumvent this problem.\n\n\n\n\\begin{figure}\n\\subfigure[PCA]{\n\\psfrag{x1}{\\tiny $y^{(1)}$}\n\\psfrag{x2}{\\tiny $y^{(2)}$}\n\\psfrag{px}{\\color{red} $\\boldsymbol{\\hat{y}=Py}$}\n\\label{fig1a}\n\\includegraphics[width=0.30\\textwidth,height=3.5cm]{FIGURES\/pod.eps}} \\hfill\n\\subfigure[Nonlinear PCA]{\n\\psfrag{x1}{\\tiny $y^{(1)}$}\n\\psfrag{x2}{\\tiny $y^{(2)}$}\n \\psfrag{px}{\\tiny \\color{red} PCA:$\\boldsymbol{\\hat{y}=Py}$}\n\\psfrag{npx}{\\color{green} $\\boldsymbol{\\hat{y}=P(y)}$}\n\\label{fig1b}\n\\includegraphics[width=0.30\\textwidth,height=3.5cm]{FIGURES\/npod.eps}} \\hfill\n\\subfigure[Mixture PCA (SLDS)]{\n\\psfrag{x1}{\\tiny $y^{(1)}$}\n\\psfrag{x2}{\\tiny $y^{(2)}$}\n\\psfrag{px1}{\\color{red} $\\boldsymbol{\\hat{y}=P^{(1)}y}$}\n\\psfrag{px2}{\\color{red} $\\boldsymbol{\\hat{y}=P^{(2)}y}$}\n\\psfrag{px3}{\\color{red} $\\boldsymbol{\\hat{y}=P^{(3)}y}$}\n\\label{fig1c}\n\\includegraphics[width=0.30\\textwidth,height=3.5cm]{FIGURES\/hmm_pca.eps}} \\\\\n\\caption{ \\em The phase space is assumed two-dimensional for illustration purposes i.e. $\\boldsymbol{y}_t=(y^{(1)}_{t}, y^{(2)}_{t})$. Each black circle corresponds to a realization $\\boldsymbol{y}_{t_i}$. $\\mathcal{P}: \\mathcal{Y} \\to \\mathcal{\\hat{Y}} $ is the projection operator from the original high-dimensional space $\\mathcal{Y}$ to the reduced-space $\\mathcal{\\hat{Y}}$. }\n\\end{figure}\n\n\n\nThe obvious extension to the linear projections of PCA is nonlinear dimensionality reduction techniques. These have been the subject of intense research in statistics and machine learning in recent years (\\cite{sch97ker,Roweis:2000,Tenenbaum:2000,Donoho:2003, Shi:2000,Bach:2006,AzranG06}) and fairly recently have found their way to computational physics and multiscale dynamical systems (e.g. \\cite{Coifman:2005,Lafon:2006,Nadler:2006,bas08non}).\nThey are generally based on calculating eigenvectors of an affinity matrix of a weighted graph. While they circumvent the limiting, linearity assumption of standard PCA, they still assume that the underlying process is stationary (Figure \\ref{fig1b}). Even though the system's dynamics might be appropriately tracked on a lower-dimensional subspace for a certain time period, this might not be invariant across the whole time range of\n interest. The identification of the dynamics on the reduced-space $\\mathcal{\\hat{Y}}$ is not as straightforward as in standard PCA and in most cases, a deterministic or stochastic model is fit directly to the projected data points \\cite{Coifman:2008,Erban:2007,fra08hid}. More importantly since the inverse mapping $\\mathcal{P}^{-1}$ from the manifold $\\mathcal{\\hat{Y}}$ to $\\mathcal{Y}$ is not available analytically, approximations have to be made in order to find pre-images in the data-space \\cite{DBLP:conf\/dagm\/BakirZT04,Erban:2007}. From a computational point of view, the cost of identifying the projection mapping is comparable to standard PCA as an eigenvalue problem on a $N \\times N$ matrix has to be solved. Updating those eigenvalues and the nonlinear projection operator in cases where additional data become available implies a significant computational overhead although recent efforts \\cite{sch07fas} attempt to overcome this limitation.\n\n\n\n\n\nA common characteristic of the aforementioned techniques is that even though the reduced coordinates are learned from {\\em a finite amount of simulation data}, there is no {\\em quantification of the uncertainty} associated with these inferences. This is a critical component not only in cases where multiple sets of reduced parameters and coarse-grained models are consistent with the data, but also for assessing errors associated with the analysis and prediction estimates. It is one of the main motivations for adopting a {\\em probabilistic approach} in this project. Statistical models can naturally deal with stochastic systems that frequently arise in a lot of applications. Most importantly perhaps, even in cases where the fine-scale model is deterministic (e.g. \\refeq{eq:master}), a stochastic reduced model provides a better approximation that can simultaneously quantify the uncertainty arising from the information loss that takes place during the coarse-graining process \\cite{fat04com,kou07sto}. \n\n\n\n\n\n\n\n\n\n\n\n\n\nA more general perspective is offered by latent variable models where the observed data (experimental or computationally generated) is augmented by a set of hidden variables \\cite{bis99lat}. {\\em In the case of high-dimensional, multiscale dynamical systems, the latent model corresponds to a reduced-order process that evolves at scales of practical relevance.} Complex distributions over the observables can be expressed in terms of simpler and tractable joint distributions over the expanded variable space. Furthermore, {\\em structural characteristics} of the original, high-dimensional process $\\boldsymbol{y}_t$ can be revealed by interpreting the latent variables as generators of the observables.\n\n\n\nIn that respect, a general setting is offered by Hidden Markov Models (HMM, \\cite{gha01int}) or more generally State-Space Models (SSM) \\cite{cap01ten,gha04uns,Horenko:2007}. These assume the existence of an {\\em unobserved (latent)} process $\\boldsymbol{\\hat{y}}_t \\in \\mathbb R^K$ described by a (stochastic) ODE:\n\n\\begin{equation}\n\\label{eq:ssm1}\n\\frac{d \\boldsymbol{\\hat{y}}_t}{dt}=\\boldsymbol{\\hat{f}}(\\boldsymbol{\\hat{y}}_t; \\boldsymbol{w}_t) \\quad (\\textrm{transition equation})\n\\end{equation}\nwhich gives rise to the observables $\\boldsymbol{y}_t \\in \\mathbb R^d$ as: \n\n\\begin{equation}\n\\label{eq:ssm2}\n\\boldsymbol{y}_t=\\boldsymbol{h}(\\boldsymbol{\\hat{y}}_t, \\boldsymbol{v}_t) \\quad (\\textrm{emission equation})\n\\end{equation}\nwhere $\\boldsymbol{w}_t$ and $\\boldsymbol{v}_t$ are unknown stochastic processes (to be inferred from data) and $\\boldsymbol{\\hat{f}}: \\mathbb R^K \\to \\mathbb R^K$, $\\boldsymbol{h}: \\mathbb R^K \\to \\mathbb R^d$ are unknown measurable functions. The transition equation defines a prior distribution on the coarse-grained dynamics whereas the emission equation, the mapping that connects the reduced-order representation with the observable dynamics. The object of Bayesian inference is to learn the unobserved (unknown) model parameters from the observed data. Hence the coarse-grained model and its relation to the observable dynamics are inferred from the data. \n\nThe form of Equations (\\ref{eq:ssm1}) and (\\ref{eq:ssm2}) affords general representations. Linear and nonlinear PCA models arise as special cases by appropriate selection of the functions and random processes appearing in the transition and emission equations. Note for example that the transition equation (\\refeq{eq:ssm1}) for $\\boldsymbol{\\hat{y}}_t$ in the case of the PCA-based models reviewed earlier is given by \\refeq{eq:pca} and the {\\em emission equation} (\\refeq{eq:ssm2}) that relates latent and observed processes is linear, deterministic and specified by the matrix of $K$ leading eigenvectors $\\boldsymbol{V}_K$. \n\n\n An extension to HMM is offerered by switching-state models \\cite{har76bay,cha78sta,ham89new,shu91dyn} which can be thought of as dynamical mixture models \\cite{DBLP:journals\/tsmc\/ChaerBG97,DBLP:journals\/neco\/GhahramaniH00}. The latent dynamics consist of a discrete process that takes $M$ values, each corresponding to a distinct dynamical behavior. This can be represented by an $M$-dimensional vector $\\boldsymbol{z}_t$ whose entries are zero except for a single one $m$ which is equal to one and represents the active mode\/cluster. Most commonly, the time-evolution of $\\boldsymbol{z}_t$ is modeled by a first-order stationary Markov process:\n\\begin{equation}\n\\label{eq:slds}\n\\boldsymbol{z}_{t+1}=\\boldsymbol{T} \\boldsymbol{z}_t\n\\end{equation}\nwhere $\\boldsymbol{T}=[T_{m,n}]$ is the transition matrix and $T_{m,n}=Pr[z_{m,t+1}=1 \\mid z_{n,t}=1]$.\n In addition to $\\boldsymbol{z}_t$, \n $M$ processes $\\boldsymbol{x}_t^{(m)} \\in \\mathbb R^K, ~m=1,\\ldots,M$ parameterize the reduced-order dynamics (see also discussion in section \\ref{sec:pmhmm}). Each is activated when $z_{m,t}=1$. In the linear version (Switching Linear Dynamic System, SLDS \\footnote{sometimes referred to as jump-linear or conditional Gaussian models}) and conditioned on $z_{m,t}=1$, the observables $\\boldsymbol{y}_t$ arise by a projection from the active $\\boldsymbol{x}_t^{(m)}$ as follows:\n\\begin{equation}\n\\label{eq:obs_slds}\n\\boldsymbol{y}_t= \\boldsymbol{P}^{(m)} \\boldsymbol{x}_{t}^{(m)}+\\boldsymbol{v}_t, \\quad \\boldsymbol{v}_t \\sim N(\\boldsymbol{0}, \\boldsymbol{\\Sigma})~(i.i.d)\n\\end{equation}\nwhere $\\boldsymbol{P}^{(m)}$ are $d\\times K$ matrices ($K<