text:
\section{Introduction}\label{Introduction}
In the previous work \cite{VKP} we have considered the BRST quantization of the motion on a hypersurface $V_N$ embedded in the N dimensional Euclidean space $R^N$ based on Batalin-Fradkin-Fradkina-Tyutin (BFFT) Abelianized Batalin-Fradkin-Vilkovisky (BFV) and Batalin-Vilkovisky (BV) formalisms. It is a well known result in differential geometry that any $N$ dimensional Riemann manifold can be locally embedded in the $\frac{N(N+1)}{2}$ dimensional Eucliden space but can not be embedded in $(N+1)$ dimension generally \cite{YAS}. In this sense, the considerations in the previous manuscript \cite{VKP} should be extended to general Riemannian manifolds. This is the motivation of the present manuscript \cite{VKP2}.

It is well known in the literature that the quantization of the system in curved space has been extensively studied about the ordering problem using two different approaches, canonical and path integral \cite{HEWCRS,RS,TK,TKRS,TORS1,TORS2,TKTORS,MOHS,FSTK,FKST,TKW,HKTK,JLGAN,MO,BSKK}. At the same time, the quantization of dynamical systems constrained to curved manifolds embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories \cite{DWOS,AFFR,NDBPD,LPDT,IBMGSL1,IBMGSL2,IGSLAS}. Here, a non-relativistic particle constrained to move on a curved surface embedded in the higher dimensional Euclidean space \cite{NOKFAK,NOKFCK} has been taken. These systems and their various properties they possess, have been investigate by many authors \cite{NOG,INTT,STTT,CFFM,NO,FOUC,CFK,AS,LHL,NOG1,NOG2,NOMN,DPO,FGK,AVG,KFNO,NOM1,NOM2,MN,PM1,PM2,NCAR1,NCAR2,NCAR3,DOBSRT}.  This has motivated us to extend our previous work\cite{VKP} to more general class of systems discussed in \cite{NOKFCK}.

                                    Becchi-Rout-Stora-Tyutin (BRST) quantization
\cite{CRS1,CRS2,CRS3,IVT} is on of the most significant technique to deal with a system with constraints. It has also been found to be symmetry of general class of constrained systems \cite{PAMD1,JLAPGB,PAMD2,KS,MHCT,MH}. In this quantization method, we enlarge the total Hilbert space of the gauge system under study and bring back the gauge symmetry of the gauge fixed action in the extended phase space, keeping the physical contents of the theory unchanged. BRST symmetry plays very significant role in renormalization of spontaneously broken gauge theories like, standard model and hence is of very high significance for different kind of systems. To the best of our knowledge, there is no literature available which studied the BRST symmetry for a particle moving on a hypersurface $V_L (1\leq L < N)$ embedded in the Euclidean space $R_N$. This motivates us in the study of BRST symmetry for this system. We will do the constraints analysis of this sytem using the Dirac's technique. The system is shown to contain second-class constraints. The BFFT method will be used to convert the second class constraints to first class one \cite{LFSS,IABESF1,IABESF2,BFFR,IABIVT,EERM,RBBN,SG,CBAPMT,IBVKAR,ILBAAR,AAR,IBAR,VPRT,MPT}. Then the BRST charge and symmetries will be constructed for this BFFT Abelianized system using BFV method \cite{ESFGV,IABGV,IABESF} with the help of Faddeev-Senjanovic technique \cite{LDFAD,PSENJ}. In the limit $L\rightarrow (N-1)$ the system will return to sytem in \cite{VKP}. The results developed in the manuscript have been verified using a model of particle on torus knot \cite{VS,PDSG,VKPBPM,VPBM,ASSG}. At the end BV-BRST quantization of the BFFT Abelianized sytem will be investigated \cite{IABGAV1,IABGAV2,JGJPSS,RART}. Recently Lagrangian Abelianization procedure for constrained systems has also been developed \cite{SLL, VAASLL}.

This is the second and final part of the work. In this part we will discuss BRST quantization of embedding $V_L$ in Euclidean space $R_N$ where $1\leq L < N$. The paper has been organized in the following way. In the second section, we have reviewed motion in curved space and also calculated all the possible constraints of the theory using Dirac's constraints analysis method. In the third section, we have reviewed BFFT formalism. In the section four, we have constructed first class constraints and Hamiltonians for the general class of systems under study. In the next section we will construct BRST symmetry for this system based on BFV Formalism. In the section six, we have shown consistency of our results in the limit $L \rightarrow (N-1) $. In the section seven, we have given a simple example of this kind of systems. In the section eight, we have discussed BV quantization of this system based on BFFT formalism. In the next section concluding remarks have been made. In the end we have discussed some important calculations in the appendix.

\section{Classical Mechanics on $V_L$ in $R^N$}
\label{Classical Mechanics on $V_L$ in $R^N$}
Let us consider an N dimensional Euclidean space ${R_N}$, a point in which is specified by a set of Cartesian coordinates
\begin{eqnarray}
X^A : \{x^1,x^2,...,x^N\}.
\end{eqnarray}

Further consider in $R_N$ an $L$ dimensional Riemann subspace, $V_L (1\leq L < N)$, a point of which is specified by a set of coordinates $q^k$ \cite{NOKFCK},
\begin{eqnarray}
q^k :  \{ q^1, q^2, ..., q^L\}
\label{qkgnc}
\end{eqnarray}
The metric of this system is defined as $g_{ij} (q^k)$. We can construct in $R_N$ a set of curvilinear coordinates including $(q^1, q^2, ..., q^L)$,
\begin{eqnarray}
q^{\mu} :  \{ q^1, q^2, ..., q^L, q^{L+1}, ..., q^N\}
\label{qalp}
\end{eqnarray}
Let us assume that $\{q^{L+a}\} (a: 1\sim {N-L})$ are the intrinsic coordinates normal to $V_L$ \cite{NOKFCK}. We can also use the notation
\begin{eqnarray}
Q^a \equiv q^{L+a}, \quad  a: 1 \sim {N-L}
\label{qal}
\end{eqnarray}
Then the subspace $V_L$ can be defined as
\begin{eqnarray}
Q^a = q^{L+a} = 0
\label{Qa}
\end{eqnarray}
The metric for the curvilinear coordinates $q^{\mu}$ in $R_N$ is defined as
\begin{equation}
{\tilde g}_{\mu\nu} =
\begin{pmatrix}
{\tilde g}_{ij} & N_{ib} \\
N_{ja} & {\tilde G}_{ab}\\
\end{pmatrix}
\label{tigmn}
\end{equation}
where $N_{ia}$ and ${\tilde G}_{ab}$ are defined as
\begin{eqnarray}
N_{ia} \equiv {\tilde g}_{\mu = i, \nu = L+a}, \quad {\tilde G}_{ab} \equiv {\tilde g}_{\mu = L+a, \nu = L+b}
\label{NLGg}
\end{eqnarray}
It is worth notice that the metric $g_{ij} = {\tilde g}_{ij} (q^k, Q^a = 0)$ is induced metric on $V_L$ and the metric $G_{ab} = {\tilde G}_{ab} (q^k, Q^a = 0)$ can be defined as some function on $V_L$. Using this assumption, we will get $N = 0$ when $Q = 0$. So the metric on $V_L$ have form
\begin{equation}
g_{\mu\nu} =
\begin{pmatrix}
{\tilde g}_{ij} & 0 \\
0 & {\tilde G}_{ab}\\
\end{pmatrix}
\label{gGmn}
\end{equation}
and the inverse matrix is defined as
\begin{equation}
g^{\mu\nu} =
\begin{pmatrix}
g^{ij} & 0 \\
0 & G^{ab}\\
\end{pmatrix}
\label{invgmn}
\end{equation}
which implies that
\begin{eqnarray}
g^{\mu\zeta}.g_{\zeta\nu} = {\delta^\mu}_\nu
\label{gmznd}
\end{eqnarray}
which can further be written as,
\begin{eqnarray}
g^{ij}\cdot g_{jk} = \delta^i_k, \quad G^{ab}\cdot G_{bc} = {\delta^a}_c
\label{gijg}
\end{eqnarray}
We know that ${\tilde g}_{\mu\nu}$ can also be written as
\begin{eqnarray}
{\tilde g}_{\mu\nu} = (\frac{\partial x}{\partial q^\mu})\cdot (\frac{\partial x}{\partial q^\nu}).
\label{qmqn}
\end{eqnarray}
From here, we can obtain following relations \cite{NOKFCK}
\begin{eqnarray}
0 &=& (\frac{\partial x}{\partial q^k})\cdot (\frac{\partial x}{\partial Q^a})|_{Q = 0} = \sum_{A} e_k^A\cdot h_a^A\nonumber\\
g_{ij}(q^k) &=& (\frac{\partial x}{\partial q^i})\cdot (\frac{\partial x}{\partial q^j})|_{Q = 0} = \sum_{A} e_i^A\cdot e_j^A\nonumber\\
G_{ab}(q^k) &=& (\frac{\partial x}{\partial Q^a})\cdot (\frac{\partial x}{\partial Q^b})|_{Q = 0} = \sum_{A} h_a^A\cdot h_b^A
\label{gGQqA}
\end{eqnarray}
where $e_i^A$ and $h_a^A$ are defined as
\begin{eqnarray}
e_i^A (q^k) \equiv (\frac{\partial x^A}{\partial q^i})|_{Q = 0}, \quad h_a^A (q^k) \equiv (\frac{\partial x^A}{\partial Q^a})|_{Q = 0}
\label{eAqk}
\end{eqnarray}
Here, $e_i^A$ are called the natural frame and gives the induced metric on $V_L$.
The inverse metric of ${\tilde g}_{\mu\nu}$ is given by
\begin{eqnarray}
{\tilde g}^{\mu\nu} = \nabla q^{\mu}\cdot \nabla q^{\nu}
\label{nqmnqn}
\end{eqnarray}
where $\nabla \equiv \frac{\partial}{\partial x}$.
From this, we obtain
\begin{eqnarray}
g^{\mu\nu} = {\tilde g}^{\mu\nu}|_{Q = 0} = \nabla q^{\mu}\cdot \nabla q^{\nu}|_{Q = 0}.
\label{tignq}
\end{eqnarray}

The Lagrangian for the particle motion on $V_L$ is defined as \cite{NOKFCK},
\begin{eqnarray}
L = \frac{1}{2}\cdot{\dot x}^A{\dot x}_A - V(x) + \lambda_a Q^a(x)
\label{Lagpm}
\end{eqnarray}
Here $``A"$ varies between 1 to $N$ and $``a"$ from 1 to $N-L$. The metric for the coordinate $x^A$ is $\delta_{AB}$. $\lambda_a'S$ are variables independent of $x^A$ and the dot denotes the time derivative.
The canonical momentum conjugate to $x^A$ and $\lambda^a$ can be written as
\begin{eqnarray}
P_A &\equiv& \frac{\partial L}{\partial {\dot x}^A} = {\dot x}^A \nonumber\\
\Pi^a &\equiv & \frac{\partial L}{\partial {\dot \lambda}_a} \approx 0
\label{PAPia}
\end{eqnarray}
Hamiltonian corresponding to Lagrangian in eqn (\ref{Lagpm}) can be written as,
\begin{eqnarray}
H _0 = \frac {1}{2}\cdot P_A P^A + V(x) - \lambda_a Q^a(x)
\label{H0}
\end{eqnarray}

\subsection{Hamiltonian Analysis}\label{Hamiltonian Analysis}
The primary constraint for the system under study is defined as,
\begin{equation}
\Pi_a \approx 0
\label{pia}
\end{equation}
After including the primary constraint, the new Hamiltonian is written as,
\begin{equation}
H_T = \frac {1}{2}\cdot P_A P^A + V(x) - \lambda_a Q^a(x) + u_a \Pi^a
\label{HT}
\end{equation}
where $u_a'S$ are a set of Lagrange multipliers for the system.
Now, we will perform the constraint analysis of the given system using the Dirac's technique of constraints analysis \cite{PAMD1,JLAPGB,PAMD2,KS,MHCT,MH}. All the constraints of the theory can be calculated in the following manner \cite{NOKFCK},
\begin{eqnarray}
{\dot \Pi}^a & = & \{ \Pi^a, H_T \}_P = Q^a \nonumber\\
{\ddot \Pi}^a & = & \{ Q^a, H_T \}_P = P^A \cdot {\partial_A Q^a} \nonumber\\
{\Pi^a}^{(3)} &=& \{ DQ^a, H_T \}_P = D^2Q^a - \nabla Q^a\cdot \nabla (V - \lambda_d Q^d)
\label{ddPia}
\end{eqnarray}
The constraint $\Pi^{a(4)} = 0$ determines the $u_a'S$ and the procedure is over. So, the explicit form of the constraints are,
\begin{eqnarray}
\Phi_1^a &=& \Pi^a \approx 0 \nonumber\\
\Phi_2^a &=& Q^a \approx 0 \nonumber\\
\Phi_3^a &=& DQ^a \approx 0 \nonumber\\
\Phi_4^a &=& P^AP^B\partial_A\partial_B Q^a  - \nabla Q^a\cdot{\nabla}(V - \lambda_d Q^d) = D^2 Q^a - \nabla Q^a \cdot{\nabla}\Phi \approx 0
\label{Phia}
\end{eqnarray}
Here D and $\Phi$ are defined as, $D = P^A \partial_A$, $\Phi = (V - \lambda_a Q^a)$. Also, the product of partial derivatives is defined as, $\nabla f\cdot\nabla g \equiv \sum_A \partial_A f\cdot \partial_A g$.

The Poisson brackets between the constraints are defined as \cite{NOKFCK},
\begin{eqnarray}
\{\Phi_1^a, \Phi_4^b \}_P &=& - \nabla Q^a\cdot \nabla Q^b \equiv - \alpha^{ab}, \nonumber\\
\{\Phi_2^a, \Phi_3^b \}_P &=& \nabla Q^a\cdot \nabla Q^b \equiv \alpha^{ab},\nonumber\\
\{\Phi_2^a, \Phi_4^b \}_P &=& 2\nabla Q^a\cdot (\nabla DQ^b)\equiv -\beta^{ab}, \nonumber\\
\{\Phi_3^a, \Phi_4^b \}_P &=& 2\nabla (D Q^a)\cdot \nabla (D Q^b) - \nabla Q^a\cdot \nabla \Phi_4^b \equiv - \gamma^{ab} ,\nonumber\\
\{\Phi_3^a, \Phi_3^b \}_P &=& \nabla (DQ^a)\cdot \nabla Q^b - \nabla Q^a\cdot \nabla (DQ^b)\equiv \rho^{ab}\nonumber\\
\{\Phi_4^a, \Phi_4^b \}_P &=& 2\left[\nabla \Phi_4^a\cdot \nabla (DQ^a) - \nabla \Phi_4^b\cdot \nabla (DQ^a)\right]\equiv \epsilon^{ab}
\label{PhiPhi}
\end{eqnarray}
Other Poisson brackets vanish. It is worth notice that
\begin{eqnarray}
\alpha^{ab} &=& \alpha^{ba}, \mathrm{(symmetric)}  \nonumber\\
\rho^{ab} &=& - \rho^{ba}, \mathrm{(antisymmetric)} \nonumber\\
\epsilon^{ab} &=& - \epsilon^{ba}, \mathrm{(antisymmetric)}
\label{syasym}
\end{eqnarray}
Thus the matrix $\Delta_{ij}^{ab}$ between the constraints has the form
\begin{equation}
\Delta_{ij}^{ab} \equiv \{{\Phi_i}^a, {\Phi_j}^b\}_P =
\begin{bmatrix}
0 & 0 & 0 & - {\alpha}^{ab} \\
0 & 0 & {\alpha}^{ab} & -{\beta}^{ab} \\
0 & -{\alpha}^{ab} & {\rho}^{ab} & -{\gamma}^{ab} \\
{\alpha}^{ab} & {\beta}^{ba} & {\gamma}^{ba} & {\epsilon}^{ab}
\end{bmatrix}
\label{Dijab}
\end{equation}
It is worth note here that $a, b = 1,....,(N-1)$. Hence each element of the matrix $\Delta_{ij}^{ab}$ is a $(N-1)\times (N-1)$ matrix. Thus, the matrix $\Delta_{ij}^{ab}$ is a $4(N-1)\times 4(N-1)$ matrix.
\section{BFFT Formalism}\label{BFFT Formalism}
In this section we will discuss the main results of BFFT technique, which is used to Abelianize the second class constraint systems.
                             The basic idea behind the scheme is to introduce additional phase space variables $\Theta_m^n$, besides the existing physical degrees of freedom $(q, p)$ of the system such that all the constraints in the extended space of the system are first class. This means that the original constraints and Hamiltonian have to be modified accordingly by putting BFFT-extension terms in them. To achieve this, we will use the results discussed in \cite{IABIVT,RBBN,SG}. Let us consider a set of constraints $(\Phi_n^m, \Lambda_j)$ and an Hamiltonian operator $H$. We know from the Dirac's constraint analysis that second-class constraints of a constrained system satisfy an open algebra. These constraints and Hamiltonian satisfy following algebra
\begin{eqnarray}
\{\Phi_n^m (q, p), \Phi_r^s (q, p)\} \approx \Delta_{nr}^{ms}(q,p) \neq 0,\nonumber\\
\{\Phi_n^m (q, p), \Lambda_r^j (q, p)\} \approx 0,\nonumber\\
\{\Lambda^j (q, p), \Lambda_t (q, p)\} \approx 0,\nonumber\\
\{\Lambda^j (q, p), H (q, p)\} \approx 0
\label{secal}
\end{eqnarray}
$``\approx"$ means that the equality holds on the constraint surface.
The additional fields satisfy the symplectic algebra,
\begin{eqnarray}
\{\Theta_m^n, \Theta_s^r\} = \omega_{ms}^{nr}
\label{pofex}
\end{eqnarray}
where $\omega_{ms}^{nr}$ is a constant quantity and $\det\omega_{ms}^{nr} \neq 0$. The constraints are now defined in terms of auxiliary field $\Theta_m^n$ as
\begin{eqnarray}
{\tilde\Phi}_n^m  = {\tilde \Phi}_n^m (q, p; \Theta_m^n ),
\label{tithnm}
\end{eqnarray}
This modified constraint satisfies the boundary condition
\begin{eqnarray}
{\tilde \Phi}_n^m(q, p; 0)  =  \Phi_n^m (q, p),
\label{bdccn}
\end{eqnarray}
These modified constraints should satisfy first class constraints algebra. So the Poisson bracket between the constraints are defined as
\begin{eqnarray}
\{{\tilde \Phi}_n^m, {\tilde \Phi}_r^s\} = 0
\label{mfcal}
\end{eqnarray}
The solution of eqn (\ref{mfcal}) can be achieved by considering an expansion of $\Phi_n^m$, as
\begin{eqnarray}
{\tilde \Phi}_n^m = \sum_{k = 0}^{\infty} {\tilde\Phi}_n^{m(k)},
\label{mdccn}
\end{eqnarray}
where ${\tilde\Phi}^{m(k)} \approx O(\Theta^k)$.
The first order correction in the field is \cite{IABIVT,RBBN,SG}
\begin{eqnarray}
{\tilde\Phi}_n^{m(1)} = X_{nr}^{ms}(q,p)\Theta_{s}^{r}
\label{exton}
\end{eqnarray}
Putting the expression of eqn (\ref{exton}) in eqn (\ref{mfcal}) and using the boundary condition given in eqns (\ref{bdccn}), (\ref{secal}) as well as eqn (\ref{pofex}), we get
\begin{eqnarray}
\Delta_{nr}^{ms} + X_{nc}^{md}\omega_{df}^{ce}X_{re}^{sf} = 0
\label{DeXoX}
\end{eqnarray}
We notice that the eqn (\ref{DeXoX}) does not give a single solution for $X_{ij}^{ab}$, because there is still unknown matrix $\omega_{ab}^{ij}$. We can make choices for $\omega_{ab}^{ij}$ in such a way that the newly defined variables are unconstrained in nature. Using the value of the matrix $\omega_{ab}^{ij}$, we can calculate the possible of $X_{ij}^{ab}$ from the eqn (\ref{DeXoX}). Using the value of $X_{ij}^{ab}$ we can obtain ${\tilde\Theta}_n^{m(1)}$. If ${\Theta_n^m} + {\tilde\Theta}_n^{m(1)}$ is strongly involutive in nature, then series will end otherwise it will continue in the same way till we don't get strongly involutive constraints.
The explicit expression of higher order corrections in the field $\Phi$ is
\begin{eqnarray}
{\tilde\Phi}_n^{m(k+1)} = - \frac{1}{k+2}\Theta_b^c X_{ce}^{bd}\omega_{df}^{eg}B_{gn}^{fm(k)} \quad ; \quad k\geq 1
\label{gexth}
\end{eqnarray}
where $B_{mn}^{ba}$ is defined as
\begin{eqnarray}
B_{rs}^{ba(k)} &=& \sum_{l = 0}^{k}\{{\tilde\Phi}_r^{b(k-l)},{\tilde\Phi}_s^{a(l)}\}_{(q,p)} + \sum_{l = 0}^{k-2}\{{\tilde\Phi}_r^{b(k-l)},{\tilde\Phi}_s^{a(l+2)}\}_{(\Theta)}, \quad k\geq 2\nonumber\\
B_{rs}^{ba(1)} &=& \{{\tilde\Phi}_r^{b(0)},{\tilde\Phi}_s^{a(1)}\}_{(q,p)} - \{{\tilde\Phi}_r^{a(0)},{\tilde\Phi}_s^{b(1)}\}_{(q,p)}
\label{Bmnba}
\end{eqnarray}
In the above expressions, we have defined
\begin{eqnarray}
X_{mn}^{ab}X_{bc}^{nr} = \omega_{mn}^{ab}\omega_{bc}^{nr} = \delta_m^r\delta_c^a
\label{XmnabX}
\end{eqnarray}
Another important part of the BFFT formalism is that any dynamical variable $f(q, p)$ has also to be modified in the same way as discussed above in order to be strongly involutive with the modified constraints ${\tilde\Phi}_n^m$. Denoting the modified quantity by $f(q,p;\Theta)$, we then have
\begin{eqnarray}
\{{\tilde\Phi}_n^m, {\tilde f}\} = 0
\label{tiThf}
\end{eqnarray}
Apart from that, modified variable ${\tilde f}$ must also satisfy the boundary condition given below,
\begin{eqnarray}
{\tilde f}(q, p; 0)  =  f (q, p)
\label{tifqp}
\end{eqnarray}
To obtain $\tilde f$ as an analogous expansion to eqn (\ref{mdccn}), we will consider
\begin{eqnarray}
{\tilde f} = \sum_{k = 0}^{\infty} {f}^{(k)}
\label{tifTh}
\end{eqnarray}
where ${\tilde f}^{(k)}$ is also a term which is of the order n in $\Theta^{'}s$.
The expression in eqn (\ref{tiThf}) above gives us ${\tilde f}^{(1)}$
\begin{eqnarray}
{\tilde f}^{(1)} = - \Theta_n^a\omega_{ab}^{no}X_{om}^{bc}(q,p)\{{\tilde\Phi}_c^m , f \},
\label{fchmt}
\end{eqnarray}
where $\omega_{ab}^{mn}$ and $X_{mn}^{ab}$ are the inverses of $\omega_{mn}^{ab}$ and $X_{ab}^{mn}$.

The corrections in the physical variable $f$ can be written in the more general form as,
\begin{eqnarray}
{\tilde f}^{(k+1)} = - \frac{1}{k+1}\Theta_n^a\omega_{ab}^{no}X_{om}^{bc}(q,p){G(f)_c^m}^{(k)},
\label{nocHm}
\end{eqnarray}
where
\begin{eqnarray}
G_a^{b(k)} = \sum_{l = 0}^{k}\{{\tilde\Phi}_r^{b(k-l)}, f^{(l)}\}_{(q,p)} + \sum_{l = 0}^{(k-2)}\{{\tilde\Phi}_r^{b(k-l)}, f^{(l+2)}\}_{(\Theta)} + \{{\tilde\Phi}_r^{b(k+1)}, f^{(1)}\}_{(\Theta)}
\end{eqnarray}
In the similar way, we can find the involutive form of other variables using the BFFT method described above.

Let us take the initial fields as $q$ and $p$. Then the involutive form of these fields ($\tilde q$ and $\tilde p$), will satisfy the following relations.
\begin{eqnarray}
\{{\tilde \Phi}, {\tilde q}\} = \{{\tilde \Phi}, {\tilde p}\} = 0
\label{minfl}
\end{eqnarray}
Similarly, any function of the physical variables $\tilde q$ and $\tilde p$ will also satisfy the strong involution relation, since
\begin{eqnarray}
\{{\tilde \Theta}, {\tilde F}({\tilde q}, {\tilde p})\} = \{{\tilde \Theta}, {\tilde q}\}\frac{\partial {\tilde F}}{\partial {\tilde q}} + \{{\tilde \Theta}, {\tilde p}\}\frac{\partial {\tilde F}}{\partial {\tilde p}} = 0
\label{tiThF}
\end{eqnarray}
So, if we are taking any dynamical variable in the original phase space, it can be written in involutive form as

\begin{eqnarray}
F (q,p) \rightarrow F({\tilde q}, {\tilde p}) = \tilde F ({\tilde q}, {\tilde p})
\label{Ftqtp}
\end{eqnarray}
It is very much obvious that the initial boundary condition in the BFFT formalism, namely, the reduction of the involutive physical variables to the original physical variables, when the new fields are set to zero, remains preserved.

\section{Construction of the first class constraint Theory}\label{Construction of the first class constraint Theory}
We can easily observe that all the constraints of the theory in eqn (\ref{Phia}) are of second class in nature. To change them in the first class constraints we will introduce $4L$ set of possible BFFT fields $\Theta^{a(1)}, \Theta^{a(2)}, \Theta^{a(3)}, \Theta^{a(4)}$. Here each set of newly introduced fields will correspond to a set of constraints. We will define some relation between these BFFT fields which will help us in the Abelianization of the constraints of the theory. Using the relation between newly introduced fields we will define $\omega$ which will give us the possible solution of the eqn (\ref{DeXoX}). Here we will discuss the Abelianiaztion of constraints and Hamiltonian for the Particle motion on the surface $V_L (1\leq L<N)$ in the Riemann manifold $R_N$ (based on \cite{VPRT}.

Our choice of Poisson Bracket between the fields $\Theta^{a(1)}, \Theta^{a(2)}, \Theta^{a(3)}, \Theta^{a(4)},\ (a = 1,...,N-1)$ are
\begin{eqnarray}
\{ \Theta^{a(1)}, \Theta^{b(3)} \} &=& I^{ab}, \quad \{ \Theta^{a(2)}, \Theta^{b(4)} \} = I^{ab}
\label{ph12341}
\end{eqnarray}
where $I^{ab}$ is an $(N-1)\times(N-1)$ unitary matrix.

From the relation between the fields $\Theta$, we can find matrix $\omega_{ab}^{ij}$ as,
\begin{equation}
\omega^{aibj} =
\begin{bmatrix}
0 & 0 & I^{ab} & 0 \\
0 & 0 & 0 & I^{ab} \\
-I^{ab} & 0 & 0 & 0 \\
0 & -I^{ab} & 0 & 0
\end{bmatrix}
\label{omabij1}
\end{equation}

Using the matrix $\omega^{aibj}$ defined above and the matrix $\Delta_{ij}^{ab}$ between the constraints in the eqn (\ref{Dijab}), we can calculate the possible value of matrix $X_{ij}^{ab}$.
Now, using the matrix $X_{ij}^{ab}$, we can write the modified constraints as,
\begin{eqnarray}
{\tilde\Phi}_1^a &=& \Pi^a - \Theta^{a(3)}, \quad {\tilde\Phi}_2^a =  Q^a + \Theta^{a(2)},\quad {\tilde\Phi}_3^a = (P^A - \partial^A {\bar Q}_b\Theta^{b(4)})\partial_A {\bar Q}^a\nonumber\\
{\tilde\Phi}_4^a &=& (P^A - \partial^A {\bar Q}_b\Theta^{b(4)}) (P^B - \partial^B {\bar Q}_c\Theta^{c(4)})\partial_A\partial_B{\bar Q}^a\Theta^{b1} - \partial_A {\bar V} \cdot \partial^A {\bar Q}^a \nonumber\\ &+& \lambda_d\partial_A {\bar Q}^a \partial^A {\bar Q}^d  + \partial_A {\bar Q}^a \partial^A {\bar Q}_e \Theta^{e(1)}
\label{mdcnt1}
\end{eqnarray}
It is worth mention here that, all the barred quantities defined in eqn (\ref{mdcnt1}) are function of coordinates $x^k$ and fields $\Theta^{a(2)}$ and will take the form of the original unbarred quantities in the limit $\Theta^{a(2)} \rightarrow 0$. Here, any field $\bar f(x^k, \Theta^{a(2)})$ will be written as \cite{VPRT}
\begin{eqnarray}
{\bar f (x^k, \Theta^{a(2)})} = \sum_{n = 0}^{\infty} \frac{f^{(n)}_a}{n!}\Theta^{a(2)}_{(n)}
\label{baxth}
\end{eqnarray}
also, the partial differentiation of field $\bar f$ wrt. any field $x^k$ can be written as \cite{VPRT}
\begin{eqnarray}
{\bar f_{,i}} = Q_{ai}\{\bar f, \Theta^{a(4)}\}
\label{baxth}
\end{eqnarray}
Then the Poisson bracket between these modified set of constraints is,
\begin{eqnarray}
\{{\tilde\Phi}_i^a, {\tilde\Phi}_j^a\} = 0
\label{tiPia1}
\end{eqnarray}
where $i, j = 1,2,3,4,$ and $a, b = 1, 2,..., N-1$.
We can conclude from eqn (\ref{tiPia1}) that the modified constraints are involutive in nature. Hence we have successfully converted the second class constraints of the theory into first class constraints.

Now, we will construct the involutive Hamiltonian for this system.

Corrections in the Hamiltonian due to different fields $\Theta$ can be calculated as follows.
We will start it by calculating the inverse of the matrices $\omega^{aibj}$ and $X_{ij}^{ab}$. The inverse of the matrix $\omega^{aibj}$ can be easily written as,
\begin{equation}
\omega_{ij}^{ab} =
\begin{bmatrix}
0 & 0 & - I^{ab} & 0 \\
0 & 0 & 0 &  - I^{ab}\\
I^{ab} & 0 & 0 & 0 \\
0 & I^{ab} & 0 & 0
\end{bmatrix}
\label{finvom}
\end{equation}

The total Hamiltonian with corrections due to BFFT field can be written as,
\begin{eqnarray}
{\tilde H} = \frac {1}{2}\cdot (P_A - \partial_A {\bar Q}_b\Theta^{b(4)}) (P^A - \partial^A {\bar Q}_c\Theta^{c(4)}) + \bar V(x) - (\lambda_a + \Theta_{a(1)}) (Q^a(x) + \Theta^{a(2)})
\label{fmodhm}
\end{eqnarray}
Here $\alpha, \beta, \gamma, \rho, \epsilon$ are $(N-1)\times(N-1)$ matrices and $\Theta^{a(i)}(i = 1,2,3,4)$ takes $4(N-1)$ possible values.

Now, by calculating the Poisson bracket between the modified constraints and the Hamiltonian $\tilde H$ it can be easily verified that modified Hamiltonian is involutive in nature.
\begin{eqnarray}
\{\tilde H, \tilde \Phi_i^a\} = 0
\label{tiHtiP}
\end{eqnarray}
where $i = 1,2,3,4$ and $a = 1,2,..., N-1$.

\section{Hamiltonian BRST Quantization}\label{Hamiltonian BRST Quantization}
\subsection{Charge and Symmetry}\label{Charge and Symmetry}
In this section we will construct BRST symmetry for the Particle motion on the surface $V_L (1\leq L<N)$ in the Riemann manifold $R_N$. To construct the we will use the Hamiltonian BRST formalism also called BFV-BRST formalism \cite{MHCT,MH,ESFGV,IABGV,IABESF}.

In the BFV-BRST technique associated to a general class of system with first class constraints, we introduce two canonical set of ghost and anti-ghost fields  $(C,\bar P)$ with ghost number 1 and -1 respectively and $(P,\bar C)$ with ghost number -1 and 1 respectively with Lagrange multiplier fields $(N,B)$ for each set of constraints.
As there are $4(N-1)$ set of constraints, we will introduce two $4(N-1)$ sets of canonical ghost and anti-ghost fields  $(C^{ka},{\bar P}_k^a)$, $(P^{ka},{\bar C}_k^b)$ and Lagrange multiplier fields $(N^{ka},B_k^a)$.
These fields and corresponding momenta satisfy following super algebra,
\begin{eqnarray}
\{C^{ka},{\bar P}_l^b\} = \{P^{ka},{\bar C}_l^b\} = \{N^{ka},B_l^b\} =\delta^k_l I^{ab}
\label{ghagha}
\end{eqnarray}
Now, using the BFV formalism we can write the general expression for nilpotent BRST charge, gauge-fixing fermion and BRST invariant Hamiltonian for the system under study as,
\begin{eqnarray}
Q_{BRST} = \int dx^N (C^k_a{\tilde\Omega}_k^a + P^k_a B_k^a)
\label{gnbrch}
\end{eqnarray}
\begin{eqnarray}
\Psi = \int dx^N({\bar P}_k^a N^k_a + {\bar C}^k_a \chi_k^a )
\label{genpsi}
\end{eqnarray}
\begin{eqnarray}
H_U = H_P + H_{BF} - \{Q_{BRST}, \Psi\}
\end{eqnarray}
In the BFV-BRST formulation the generating functional is doesn't depend on gauge fixing fermion \cite{ESFGV,IABESF,IABGV,LDFAD,PSENJ}, hence one has freedom to choose it in the convenient form. It is also worth notice here, that gauge-fixing fermion $\chi_k^a$ is Hermitian in nature and of the similar Grassmann parity as of $\tilde\Phi^a$. They also satisfy
\begin{eqnarray}
\det|\{\chi_k^a,\tilde\Phi^l_b\}| \neq 0
\end{eqnarray}

We will apply the general results obtained above to the particle on surface $V^L$ embedded in $R^N$.
\begin{eqnarray}
S_{eff} = \int dt \big[ P_A\dot x^{A} + \Pi_{\Theta a}^k \dot\Theta^a_k + B_k^a{\dot N}^k_a + {\dot P}^k_a {\bar C}_k^a  + {\dot C}_a^k {\bar P}_k^a - H_P - H_{BF} + \left[Q_{BRST}, \Psi \right] \big]
\label{eapstk}
\end{eqnarray}
For the system under study, the modified constraints ${\tilde \Phi}_k^a$ have been calculated in eqn (\ref{mdcnt1}).

Our choice of gauge condition $\chi_k^a$ for the motion on the surface $V_L$ in the Riemann manifold $R_N$ is $\Phi_k^a$ in eqn (\ref{Phia}).

Here $\tilde H = H_P + H_{BF}$ is taken as the BFFT modified Hamiltonian for this system obtained in eqn (\ref{fmodhm}).

We will define the canonical brackets for all dynamical variables as
\begin{eqnarray}
 [x^{A}, P_{B}] = {\delta^A}_B;\quad [\Theta^a_i, \Pi^{jb}_{\Theta} ] = [\lambda^a_i, \Pi^{bj} ] = \delta^j_i I^{ab};\quad \{{\bar C}_k^a, {\dot C}^{bl}\} = i\delta^l_k I^{ab}; \quad\{C_k^a, {\dot{\bar C}}^{bl}\} = - i \delta^l_k I^{ab}
\label{cbfadb}
\end{eqnarray}
Nilpotent BRST transformations corresponding the action in eqn (\ref{eapstk}) can be easily constructed using the relation $S_{\mathrm{BRST}}\Gamma = - [Q_{\mathrm{BRST}}, \Gamma]_{\pm}$. It's relation with infinitesimal BRST transformation $S_{\mathrm{BRST}}$ can be established as $\delta_{\mathrm{BRST}} \ = S_{\mathrm{BRST}} \Gamma \delta \Lambda$. $\delta \Lambda$ defined here is an infinitesimal BRST parameter. Here `$-$' and `$+$' sign are defined for bosonic and fermionic variables respectively. The BRST transformation for the particle motion on general class of Riemann surfaces embedded in higher dimensional Euclidean spaces is written as,
\begin{eqnarray}
&&S_{\mathrm{BRST}} N^{ak} = P^{ak}, \quad S_{\mathrm{BRST}} {\bar P}^{ak} = {\tilde \Phi}^{ak}, \quad S_{\mathrm{BRST}} {\bar C}^{ak} = -B^{ak}\nonumber\\
&& S_{\mathrm{BRST}} P^A = S_{\mathrm{BRST}}  C^{ak} = S_{\mathrm{BRST}} \Pi^{ak} = S_{\mathrm{BRST}} {P}^{ak} = 0
\label{brtrf1}
\end{eqnarray}
It can be easily verified that these transformations are nilpotent in nature.

Using the expressions for BRST charge $Q_{BRST}$ and gauge-fixing fermion $\Psi$, effective action in eqn (\ref{eapstk}) can be written as
\begin{eqnarray}
S_{eff} = \int dt \big[ P_A{\dot x}^A + \Pi_{\Theta a}^k \dot\Theta_k^a + B_k^a{\dot N}^k_a + {\dot P}^k_a {\bar C}_k^a  + {\dot C}^k_a {\bar P}_k^a - \tilde H - P^k_a{\bar P}_k^a + N_k^a \tilde\Phi^k_a + B_k^a\chi^k_a + {\bar C_k^a}C^k_a \big ]
\label{eleps}
\end{eqnarray}
The generating functional for this effective action can be written  as
\begin{eqnarray}
Z_\Psi &=& \int [D \varphi] e^{(iS_{eff})}
\label{gnf}
\end{eqnarray}
The Liouville measure $D\varphi$ for the generating functional is defined as,
\begin{eqnarray}
D\varphi =\prod_i d \Xi_i
\label{Lmes}
\end{eqnarray}
Here $\Xi_i$ are all dynamical variables $(P_A, x^A, \Pi_{\Theta a}^k, \Theta^a_k, N^k_a, B_k^a, {\bar C}_a^k, P^a_k, C_a^a, {\bar P}^a_k)$ of the theory.
Now performing the integration over ghost and antighost momenta $P^a_k$ and ${\bar P}^a_k$, we will get
\begin{eqnarray}
&&{Z_\psi} =  \int D \varphi' \exp \big[i\int dt \big[P_A {\dot x}^A  + \Pi_{\Theta a}^k \dot\Theta^a_k + B_k^a\dot N^k_a + \dot C_k^a{\dot {\bar C}^k_a} - \tilde H + N_k^a\tilde \Phi^k_a - C^k_a \bar C_k^a - B_k^a\chi^k_a \big] \big]
\label{gfaioppb}
\end{eqnarray}
Here $D\varphi'$ is the new path integral measure for the effective action after the performance of integral over the fields $P$ and $\bar P$.
Further performance of integral over fields $B^k_a$ will give us the new effective generating functional as
\begin{eqnarray}
&&{Z_\psi} = \int D \phi'' \exp \big[i\int dt \big[ P_A {\dot x}^A  + \Pi_{\Theta a}^k \dot\Theta^a_k + \dot C_k^a{\dot {\bar C}^k_a} - \tilde H + N_k^a\tilde\Phi^k_a - C^k_a \bar C_k^a - \frac{\{\dot{N_k^a} - \chi_k^a\}^2}{2}  \big] \big]
\label{gfaiopl}
\end{eqnarray}
where $D\varphi''$ is the new path integral measure which corresponds to all the dynamical variables left out after integration. Now, the new BRST transformation for the modified action in eqn (\ref{gfaiopl}) is written as
\begin{eqnarray}
&&S_{\mathrm{BRST}} N^{ak} = {\dot C}^{ak},  \quad S_{\mathrm{BRST}} {\bar C}^{ak} = -{\dot N}^{ak} - \chi^{ak}\nonumber\\
&& S_{\mathrm{BRST}} P^A = S_{\mathrm{BRST}}  C^{ak} = S_{\mathrm{BRST}} B^{ak} = S_{\mathrm{BRST}} {P}^{ak} = 0
\label{brtrf2}
\end{eqnarray}
It is well known in the literature that BRST charges are nilpotent in nature. We also know that the operation of these charges on the states of total Hilbert space gives us the physical subspace of the system.
\begin{eqnarray}
{Q_{\mathrm{BRST}}} |\mathrm{phys}\rangle = 0, \quad |\mathrm{phys}\rangle \neq {Q_{\mathrm{BRST}}} |....\rangle
\label{bcos}
\end{eqnarray}
which can be further written in more explicit form for this system as,
\begin{eqnarray}
iC_k^a\tilde\Phi^k_a|\mathrm{phys}\rangle  = 0, \quad i\dot{\bar C}_k^a N^k_a |\mathrm{phys}\rangle  = 0
\label{bcost}
\end{eqnarray}
The result of eqn (\ref{bcost}) implies that the first class constraints of the system under study will annihilate the physical subspace of the total Hilbert space of the system.

\section{Consistency with Previous result}\label{Consistency with Previous result}
We can check the consistency of the results of this system in the limit of $N-L = 1$ \cite{VKP}.
In this limit, elements of the matrix will transform as,
\begin{equation}
\alpha^{ab} \rightarrow \alpha,
\quad \beta^{ab} \rightarrow \beta,
\quad \gamma^{ab} \rightarrow \gamma
\label{albgab}
\end{equation}
From the antisymmetry (\ref{syasym})
\begin{equation}
\rho^{ab}\rightarrow 0, \quad \epsilon^{ab}\rightarrow 0
\label{rhepab}
\end{equation}
The inverse matrix elements will transform under this limit as
\begin{equation}
A \rightarrow - \frac{\gamma}{\alpha^2}, \quad B \rightarrow \frac{\beta}{\alpha^2},\quad C \rightarrow \frac{\gamma}{\alpha^2}, \quad D \rightarrow -\frac{\beta}{\alpha^2}, \quad {E, F} \rightarrow 0
\label{ABCD}
\end{equation}
Here $E, F$ vanishes due to asymmetry.

Also under the transformation $Q^a \rightarrow f(x)$ the modified constraints in eqn (\ref{mdcnt1}) takes the form of modified constraints of $L = N-1$ case,
\begin{eqnarray}
\tilde\Phi_1 &=& \Pi - \Theta^{(3)}\nonumber\\
\tilde\Phi_2 &=& f(x) + \Theta^{(2)} \nonumber\\
\tilde\Phi_3 &=& (\bar P^k - {\partial^k {\bar f(x)}}\Theta^{(4)})\bar{\partial_k f(x)}  \nonumber\\
\tilde\Phi_4 &=& (\bar P^k - \partial^k \bar f(x)\Theta^{(4)})(\bar P^l - \partial^l \bar f(x)\Theta^{(4)})\partial_k \partial_l \bar f(x) - \partial_k \bar V\cdot \partial^k \bar f(x) \nonumber\\ &&+ \lambda \partial_k \bar f(x)\partial^k \bar f(x) + \partial_k \bar f(x)\partial^k \bar f(x)\Theta^1
\label{modcn1}
\end{eqnarray}
The form of modified Hamiltonian in eqn (\ref{fmodhm}) under these transformations is
\begin{eqnarray}
&&\tilde H = \frac {1}{2}\cdot (\bar P_k - \partial_k\bar f(x)\Theta^{(4)}) (\bar P^k - \partial^k\bar f(x)\Theta^{(4)}) + \bar V(x) - (\lambda + \Theta^{(1)})(f(x) + \Theta^{(2)})
\label{modhm1}
\end{eqnarray}
Similary, BRST charge and BRST symmetry can be written in this limit as
\begin{eqnarray}
&&Q_{\mathrm{BRST}} = C^k{\tilde\Phi}_k + P^k B_k\nonumber\\
&&S_{\mathrm{BRST}} N^k = {\dot C}^k,  \quad S_{\mathrm{BRST}} {\bar C}^k = -{\dot N}^k - \chi^k\nonumber\\
&&S_{\mathrm{BRST}} P^a = S_{\mathrm{BRST}} C^k = S_{\mathrm{BRST}} B^k = S_{\mathrm{BRST}} {P}^k = 0
\label{chsym}
\end{eqnarray}
These are the same constraints, Hamiltonian, BRST charge and symmetry which we have obtained in our previous work \cite{VKP}. This shows that all the results obtained here (\ref{mdcnt1},\ref{fmodhm},\ref{brtrf1},\ref{brtrf2}) using BFFT formalism are consistent with the previous results \cite{VKP} in the limit $L \rightarrow (N-1)$.

\section{ Examples of $L (1\leq L < N)$ Dimensional Embedding in $R^N$}\label{Examples of L Dimensional Embedding in $R^N$}
As an example of $L$ dimensional embedding in $R^N$ we will discuss particle on torus knot \cite{VS,PDSG,VKPBPM,VPBM,ASSG}. We will discuss all the important results developed for general system in this case.
\subsection{Particle on Torus Knot}
Particle on torus knot is a one dimensional surface embedded in three dimensional space. It is a special kind of knot that lies on the surface of un-knotted torus in $R^3$. It is specified by a set of co-prime integers $p$ and $q$. A torus knot of type $(p, q)$ winds $p$ times around the rotational symmetry axis of the torus and $q$ times around a circle in the interior of the torus. The toroidal co-ordinate system is a suitable choice to study this system. Toroidal co-ordinates are related to Cartesian co-ordinates ($x_1, x_2, x_3$) in following ways
\begin{eqnarray}
x_1 = \frac{a \sinh\eta \cos\phi}{\cosh\eta - \cos\theta},\quad x_2 = \frac{a \sinh\eta \sin\phi}{\cosh\eta - \cos\theta}, \quad x_3 = \frac{a \sin\theta}{\cosh\eta - \cos\theta}
\label{ccoo}
\end{eqnarray}
where, $0\leq\eta\leq\infty$, $-\pi\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$. A toroidal surface is represnted by some specific value of $\eta$ (say $\eta_0$). Parameters $a$ and $\eta_0$ are written as $a^2 = R^2 - d^2$ and $\cosh\eta_0 = \frac{R}{D}$ where $R$ and $D$ are major and minor radius of torus respectively.

Lagrangian for a particle constrained to move on the surface of torus knot is
\begin{eqnarray}
L = \frac{1}{2} m a^2\frac{{\dot \eta}^2+ {\dot \theta}^2+ {\sinh^2\eta}{\dot\phi}^2}{(\cosh\eta - \cos\theta)^2} + \lambda (p\theta + q\phi)
\label{lpstk}
\end{eqnarray}
where $(r,\theta,\phi)$ are toroidal co-ordinates for toric geometry and $\lambda$ is the Lagrange multiplier.
The canonical Hamiltonian corresponding to the Lagrangian in eqn (\ref{lpstk})is then written as,
\begin{eqnarray}
H = \frac {(\cosh\eta - \cos\theta)^2}{2m a^2}\left[{p^2_\eta} + {p^2 _\theta}+ \frac{{p^2_\phi}}{{\sinh^2\eta}}\right] - \lambda (p\theta + q\phi)
\label{hpstk}
\end{eqnarray}
where $p_\eta$, $p_\theta, p_\phi$ and ${p}_\lambda$ are the canonical momenta conjugate to the coordinate $\eta$, $\theta$, $\phi$ and $\lambda$ respectively, defined as
\begin{eqnarray}
p_\eta = \frac{ma^2{\dot\eta}}{(\cosh\eta - \cos\theta)^2}, \quad
p_\theta = \frac{ma^2\dot\theta}{(\cosh\eta - \cos\theta)^2}, \quad
p_\phi = \frac{ma^2\sinh^2\eta\dot\phi}{(\cosh\eta - \cos\theta)^2},\quad
{p}_\lambda \approx 0
\end{eqnarray}
The ${p}_\lambda$ is the primary constraint of the theory.

After inclusion of primary constraint our new Hamiltonian has the form
\begin{equation}
H_T = \frac {(\cosh\eta - \cos\theta)^2}{2m a^2}\left[{p^2_\eta} + {p^2 _\theta}+ \frac{{p^2_\phi}}{{\sinh^2\eta}}\right] - \lambda (p\theta + q\phi) + u p_\lambda
\label{hmt}
\end{equation}
Now, using Dirac's method of Hamiltonian analysis, we will calculate all the possible constraints of the theory as,
\begin{eqnarray}
&&{\dot p}_\lambda = \{ {p}_\lambda, H_T \}_P = (p\theta + q\phi) \approx 0 \\
&&{\ddot p}_\lambda =  \{ (p\theta + q\phi), H_T \}_P = \frac {(\cosh\eta - \cos\theta)^2}{ma^2} \left[p p_\theta + \frac{q p_\phi}{\sinh^2\eta} \right] \approx 0 \\
&&{p}_\lambda^{(3)} = \{ \frac {(\cosh\eta - \cos\theta)^2}{ma^2} \left[p p_\theta + \frac{q p_\phi}{\sinh^2\eta} \right], H_T \}_P = \frac {(\cosh\eta - \cos\theta)^2}{ma^2}[\{2p p_\theta \sinh{\eta} + \frac{2q p_\phi}{\sinh{\eta}} \nonumber\\ &&- \frac{2q p_\phi\cosh{\eta}(\cosh\eta - \cos\theta)}{\sinh^3{\eta}}\} p_\eta\frac{(\cosh\eta - \cos\theta)}{ma^2} + 2 \sin{\theta}(p p_\theta + \frac{q p_\phi}{\sinh^2\eta})p_\theta\frac{(\cosh\eta - \cos\theta)}{ma^2} \nonumber\\ &&- p\frac{(\cosh\eta - \cos\theta)}{ma^2}\{\sin{\theta}({p^2_\eta} + {p^2 _\theta}+ \frac{p^2_\phi}{\sinh^2\eta}) - \lambda p\} + \lambda \frac{q^2}{{\sinh^2\eta}} ] \approx 0
\label{ddotp}
\end{eqnarray}
${p}_\lambda^{(4)}$ will vanish and the value of $u$ will be determined from it.
All the constraints can be written as,
\begin{eqnarray}
&&\Phi_1^a = p_{\lambda} \nonumber\\
&&\Phi_2^a = Q^a = (p\theta + q\phi) \nonumber\\
&&\Phi_3^a = D Q^a =  \frac {(\cosh\eta - \cos\theta)^2}{ma^2} \left[p p_\theta + \frac{q p_\phi}{\sinh^2\eta} \right] \nonumber\\
&&\Phi_4^a = P^AP^B\partial_A\partial_B Q^a  - \nabla Q^a\cdot{\nabla}(V - \lambda_d Q^d) = D^2 Q^a - \nabla Q^a \cdot{\nabla}\Phi\nonumber\\ &&= \frac{(\cosh\eta - \cos\theta)^2}{ma^2}[\{2p p_\theta \sinh{\eta} + \frac{2q p_\phi}{\sinh\eta} - \frac{2q p_\phi\cosh{\eta}(\cosh\eta - \cos\theta)}{\sinh^3{\eta}}\} p_\eta\frac{(\cosh\eta - \cos\theta)}{ma^2} \nonumber\\ &&+ 2 \sin{\theta}(p p_\theta + \frac{q p_\phi}{\sinh^2\eta})p_\theta\frac{(\cosh\eta - \cos\theta)}{ma^2} - p\frac{(\cosh\eta - \cos\theta)}{ma^2}\{\sin{\theta}({p^2_\eta} + {p^2 _\theta}+ \frac{p^2_\phi}{\sinh^2\eta}) - \lambda p\} \nonumber\\ &&+ \lambda \frac{q^2}{\sinh^2\eta} ]
\label{cnopt}
\end{eqnarray}
Now, the Poisson brackets between the constraints have following values,
\begin{eqnarray}
&&\{\Phi_1^a, \Phi_4^b \}_P = -\nabla Q^a\cdot \nabla Q^b  = - \frac {(\cosh\eta - \cos\theta)^2}{ma^2}[p^2 + \frac{q^2}{\sinh^2\eta}] \equiv - \alpha^{ab}
\label{Phi14}
\end{eqnarray}
\begin{eqnarray}
&&\{\Phi_2^a, \Phi_3^b \}_P = \nabla Q^a\cdot \nabla Q^b = \frac {(\cosh\eta - \cos\theta)^2}{ma^2}[p^2 + \frac{q^2}{\sinh^2\eta}] \equiv  \alpha^{ab}
\label{Phi23}
\end{eqnarray}
\begin{eqnarray}
&&\{\Phi_2^a, \Phi_4^b \}_P = 2\nabla Q^a\cdot (\nabla D Q^b) = 2 \frac {(\cosh\eta - \cos\theta)^3}{m^2a^4} [p\{ p p_\eta \sinh\eta +  \sin{\theta}(p p_\theta + \frac{q p_\phi}{\sinh^2\eta})\}\nonumber\\ && + q\{\frac{q p_\eta}{\sinh{\eta}}(1 - \frac{\cosh{\eta}(\cosh\eta - \cos\theta)}{\sinh^2{\eta}}) + \frac{\sin{\theta}}{\sinh^2\eta}(q p_\theta - p p_\phi)\}] \equiv - \beta^{ab}
\label{Phi24}
\end{eqnarray}
Similar the Poisson bracket between other constraints $(\Phi_3^a, \Phi_4^b)$ (\ref{phi34}), $(\Phi_3^a, \Phi_3^b)$ (\ref{phi33})and $(\Phi_4^a, \Phi_4^b)$ (\ref{phi44}) has been explicitly calculated in the appendix. All these brackets will be nonzero and will be equal to $\gamma^{ab}, \rho^{ab}$ and $\epsilon^{ab}$. Thus the matrix between the constraints will take exactly the same form of matrix $\Delta_{ij}^{ab}$ in eqn ({\ref{Dijab}}).

As all the constraints of the theory (\ref{cnopt}) are second class, we will follow the method of section IV and introduce four possible fields $\Theta^{a(1)}, \Theta^{a(2)}, \Theta^{a(3)}, \Theta^{a(4)}$ corresponding to each constraint. Relation between these fields will provide us possible value of $\omega^{abij}$.
Our choice of Poisson bracket between the fields will be same as  one taken for the general case.
Hence the matrix $\omega^{abij}$ will have the form of eqn (\ref{omabij1}).

Using the matrix $\omega^{abij}$ and the matrix $\Delta_{ab}^{ij}$, in the eqn (\ref{exton}), one can find many possible value of matrix $X_{ab}^{ij}$.

Now, applying the results developed in section IV we can calculate the modified constraints as
\begin{eqnarray}
\tilde\Phi_1^a &=& p_\lambda - \Theta^{a(3)}\nonumber\\
\tilde\Phi_2^a &=& (p\theta + q\phi) + \Theta^{a(2)} \nonumber\\
\tilde\Phi_3^a &=& \frac {\big(\cosh\eta - \cos{(\theta - \frac{\Theta^{a(2)}}{2p} )}\big)^2}{ma^2}\big[(p_\theta - p\Theta^{(4a)})p + \frac{(p_\phi - q\Theta^{(4a)})q}{\sinh^2\eta}\big] \nonumber\\
\tilde\Phi_4^a &=& \frac{(\cosh\eta - \cos{(\theta - \frac{\Theta^{a(2)}}{2p})})^2}{ma^2}[\big\{2p (p_\theta - p\Theta^{(4a)}) \sinh{\eta} + \frac{2q (p_\phi - q\Theta^{(4a)})}{\sinh\eta} \nonumber\\ &-& \frac{2q (p_\phi - q\Theta^{(4a)})\cosh{\eta}\big(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p})\big)}{\sinh^3{\eta}}\big\}  p_\eta\frac{(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p}))}{ma^2} \nonumber\\ &+& 2 \sin{(\theta - \frac{\Theta^{a(2)}}{2p})}(p (p_\theta - p\Theta^{(4a)}) + \frac{q (p_\phi - q\Theta^{(4a)})}{\sinh^2\eta})(p_\theta - p\Theta^{(4a)})\frac{(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p}))}{ma^2} \nonumber\\ &-& p\frac{(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p}))}{ma^2}\{\sin{(\theta - \frac{\Theta^{a(2)}}{2p})}({p^2_\eta} + {(p_\theta - p\Theta^{(4a)})^2}+ \frac{(p_\phi - q\Theta^{(4a)})^2}{\sinh^2\eta}) - \lambda p\} \nonumber\\ &+& \lambda \frac{q^2}{\sinh^2\eta} + \big(\frac{(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p}))}{ma^2}p^2+\frac{q^2}{\sinh^2\eta}\big)\Theta^{a(1)}]
\end{eqnarray}
The Poisson bracket between these modified constraints vanishes which shows that modified constraints are involutive. Hence we have converted the second class constraints of the theory into first class.

Now, we will construct first class Hamiltonian for this system using the results in section V.

The total involutive Hamiltonian for this system will take the form as \cite{VPRT},
\begin{eqnarray}
\tilde H &=& \frac {(\cosh\eta - \cos(\theta - \frac{\Theta^{a(2)}}{2p}))^2}{2m a^2}\left[{p^2_\eta} + {(p_\theta - p\Theta^{(4a)})^2}+ \frac{(p_\phi - q\Theta^{(4a)})^2}{\sinh^2\eta}\right] \nonumber\\ &-& (\lambda + \Theta^{a(1)})(p\theta + q\phi + \Theta^{a(2)})
\end{eqnarray}
It can be easily verified that the Hamiltonian $\tilde H$ is involutive by computing it's Poisson bracket with modified constraints of the theory.
\begin{eqnarray}
\{\tilde H, \tilde \Phi_i^a\} = 0
\label{tiHPhtk}
\end{eqnarray}
where $i = 1,2,3,4$.

BRST charge for this first class system can be written using above expression, as
\begin{eqnarray}
Q_{BRST} = i C^i_a\tilde\Phi_i^a + i P^i_a B_i^a
\label{brchtr}
\end{eqnarray}

and corresponding BRST symmetry transformation can be written as
\begin{eqnarray}
&&S_{\mathrm{BRST}} N^{ak} = P^{ak}, \quad S_{\mathrm{BRST}} {\bar P}^{ak} = {\tilde \Phi}^{ak}, \quad S_{\mathrm{BRST}} {\bar C}^{ak} = -B^{ak}\nonumber\\
&& S_{\mathrm{BRST}} P^A = S_{\mathrm{BRST}}  C^{ak} = S_{\mathrm{BRST}} \Pi^{ak} = S_{\mathrm{BRST}} {P}^{ak} = 0
\label{brsttk}
\end{eqnarray}
This shows that result obtained in section IV is true for any $L(1\leq L < N)$ dimensional surface embedded in $R^N$.

\section{Batalin - Vilkovisky Quantization}\label{Batalin - Vilkovisky Quantization}
In the current section of the manuscript, we are going to discuss the quantization of the Particle motion on the surface $V_L (1\leq L<N)$ in the Riemann manifold $R_N$ using the field-antifield formalism \cite{IABGAV1,IABGAV2,JGJPSS} developed for BFFT systems in \cite{RART}. We will start by introducing $4(N-1)$ set of antifields $\varpi^{k\star}_\mu = (x^{\star}_A, \Theta^{k\star}_a, \lambda^{k\star}_a, C^{k\star}_a)$ corresponding to the fields $\varpi^\mu_k = (x^A, \Theta^a_k, \lambda^a_k, C^a_k)$. Here, fields $ x^A, \Theta^{ak}$ and $\lambda^{ak}$ are bosonic in nature and have ghost number zero whereas the ghost fields $C^{ak}$ are fermionic in nature and have ghost number one. Antifields corresponding to these fields have opposite Grassmann parity and their ghost numbers are given by minus the ghost number of the corresponding fields minus one.

BV-action for this system in terms of fields and antifields is written as
\begin{eqnarray}
S & = & S_0 + \int dt \big[ x^{\star}_A \{ x^A, {\tilde\Phi}_a^k \} C^a_k + {\Theta}^{\star}_{bk} \{ \Theta^{bk}, {\tilde\Phi}_a^l \} C^a_l + \lambda^{k\star}_a {\dot C}^a_k \big ]
\label{acbff}
\end{eqnarray}
where action $S_0$ is defined as
\begin{eqnarray}
S_0 & = &\int dt \big[ P_A{\dot x}^A + \Pi_a^k {\dot\Theta}^a_k - \lambda^a_k {\tilde\Phi}_a^k - {\tilde H}\big ]
\label{ginac}
\end{eqnarray}
Here ${\tilde\Phi^a_i}$ and ${\tilde H}$ are modified constraints and modified Hamiltonian in eqn (\ref{mdcnt1}) and eqn (\ref{fmodhm}) respectively.
The BV-action defined in eqn (\ref{acbff}) satisfies the classical master equation
\begin{eqnarray}
\frac{1}{2}(S,S) = 0
\label{clmeq}
\end{eqnarray}
where the antibracket between any two dynamical variables $X[\varpi, \varpi^{\star}]$ and $Y[\varpi, \varpi^{\star}]$ is defined as
\begin{eqnarray}
(X,Y) = \frac{\delta_r X}{\delta \varpi^\mu}\frac{\delta_l Y}{\delta \varpi^{\star}_\mu} - \frac{\delta_r X}{\delta\varpi^{\star}_\mu}\frac{\delta_l Y}{\delta \varpi^\mu }
\label{antib}
\end{eqnarray}
Here, de Witt's notation of sum and integration over intermediary variables will be assumed, whenever necessary.
The BRST differential in is the BV formalism can be introduced using the relation $sX = (X,S)$ for any local functional $X[\varpi, \varpi^{\star}]$ of fields.
Nilpotentcy of the BRST operator $s$ can be proved using classical master equation and Jacobi identity. So, the BV action satisfying the master equation is equivalent to it's BRST invariance.

To fix gauges, we will extend the Hilbert space to introduce $4(N-1)$ pairs of ghost-antighost fields and corresponding momenta $({\bar C}_a^k, P_a^k)$, $(\bar {C^{k\star}_a}, P^{k\star}_a)$, as well as gauge-fixing fermions $\Psi$. These antifields can be easily eliminated by choosing $\varpi^{\star}_\mu = \frac{\partial \Psi}{\partial \varpi^\mu}$. One of the possible forms of $\Psi$, we can choose for the given system is,
\begin{eqnarray}
\Psi = {\bar C}_a^k \Theta^a_k
\label{gffer}
\end{eqnarray}
We have liberty to make other possible choices also.
Now, we will extend the BV action defined above to a nonminimal action,
\begin{eqnarray}
S \rightarrow S_{nm} = S + \int dt P_a^k {\bar C}^{\star a}_k
\label{nnmac}
\end{eqnarray}
in order to implement the many set of gauge fixing conditions introduced by $\Psi$. Now, the generating functional for gauge-fixed action is defined as
\begin{eqnarray}
{Z_\psi} & = & \int [d\varpi^\mu ][d\omega]^{-\frac{1}{2}} [df]^{-\frac{1}{2}} \exp {\frac{i}{\hbar}S_{nm}[\varpi^\mu, \varpi^{\star}_\mu = \frac{\partial \Psi}{\partial {\varpi^\mu}}]}
\label{vacfn}
\end{eqnarray}

Now, we will replace the original classical field–antifield action $S$ by some quantum action $\Sigma$ which is expressed as a local functional of fields and antifields and also satisfy a new equation called quantum master equation defined as
\begin{eqnarray}
\frac{1}{2}(\Sigma,\Sigma) - i{\hbar}\Delta \Sigma = 0
\label{qnmeq}
\end{eqnarray}
then the gauge symmetries of the extended action are not obstructed at quantum level. Here $\Delta$ act as an operator and is defined as
\begin{eqnarray}
\Delta \equiv (\frac{\delta_r}{\delta \varpi^\mu})(\frac{\delta_l}{\delta \varpi^{\star}_\mu})
\label{drdpA}
\end{eqnarray}
It is also assumed here that the quantum action $\Sigma$ can be expanded in powers of $\bar h$ in the following manner,
\begin{eqnarray}
\Sigma[\varpi^\mu,\varpi^{\star}_\mu] = S[\varpi^\mu,\varpi^{\star}_\mu] + \sum_{p = 1}^{\infty}{\hbar}^p M_p[\varpi^\mu,\varpi^{\star}_\mu]
\label{qnact}
\end{eqnarray}
The first two term of the quantum master eqn (\ref{qnmeq}) reads as,
\begin{eqnarray}
(S,S) = 0\nonumber\\
(M_1,S) = i\Delta S
\label{qnaeq}
\end{eqnarray}
It can be easily observed that if $\Delta S$ is non-zero and gives a nontrivial result, then there exists some $M_1$ which can be expressed in terms of local fields such that eqn (\ref{qnaeq}) is satisfied.
Using the cohomological arguments, it can be easily shown that the quantum master equation for the first order systems with pure second class constraints converted to first class by the use of the BFFT procedure, can always be solved. BRST transformations for the fields and antifields for the BFFT Abelianized system can be written as follows,
\begin{eqnarray}
&&S_{\mathrm{BRST}} N^{k\nu} = {\dot C}^{k\nu}, \quad S_{\mathrm{BRST}}  C^{k\nu} = 0, \quad S_{\mathrm{BRST}} {\bar C}^{k\nu} = P^{k\nu}, \quad S_{\mathrm{BRST}} P^{k\nu}  = 0\nonumber\\
&&S_{\mathrm{BRST}} x^{\star}_A = -\frac{\partial S}{\partial x^A}, \quad S_{\mathrm{BRST}} \Theta^{k\star}_\nu = -\frac{\partial^k S}{\partial {\Theta}^\nu},\quad S_{\mathrm{BRST}} N^{k\star}_\nu = {\tilde\Phi^k}_\nu, \quad S_{\mathrm{BRST}} {\bar P}^{{k\star}\nu} =  {\bar C}^{{k\star}\nu}\nonumber\\&&
S_{\mathrm{BRST}} C^{k\star}_\nu = - x^{\star}_A \{ x^A, {\tilde\Phi}_\nu^k \} - {\Theta}^{i\star}_l \{ \Theta^l_i, {\tilde\Phi}_\nu^k \} - {\dot N}^{k \star},
\quad S_{\mathrm{BRST}} \bar{C^{k\star}_\nu} = 0
\label{bvbrst}
\end{eqnarray}
The symmetry transformations obtained in the eqn (\ref{bvbrst}) are identical to the one obtained in eqn (\ref{brtrf1}) using BFV formalism. Also, we can easily show on the basis of argument given in \cite{RART} that the enlarged symmetries due to the compensating fields (BFFT variables) are non-anomalous in nature. These BFFT fields also plays very significant role at the quantum level because of the existence of a counterterm, by modifying the expectation values of the relevant physical quantities.

\section{Result and Discussion}\label{Result and Discussion}
The BRST symmetry for a particle moving in a curved space $V_{L}(1\leq L < N) $ embedded in a Euclidean space $R_N$ is investigated in both Hamiltonian and Lagrangian formalism. All the constraints of the system have been calculated using Dirac's Hamiltonian analysis. Using the algebra of constraints, we have found that all the constraints of the system are second class. To construct a gauge invariant theory, we have used the BFFT technique. Using this technique, all the second class constraints of the system are converted into first class constraints and corresponding Hamiltonian is also constructed explicitly. Using the involution of Hamiltonian with first class constraints, this Hamiltonian is shown to be first class. In the limit of $\Theta \rightarrow 0$ the constraints and Hamiltonian returns to original second class constraints and Hamiltonian. Now, using this gauge invariant system, we have constructed BRST charge, symmetries and the BRST invariant action. For constructing BRST symmetry from the first class constraint system, BFV formalism is used. These BRST charges acting on the stetes of the total Hilbert space, will annihilate the physical subspace of it. From it, we have deduced that first class constraints operating on total Hilbert space of the system will annihilate its physical subspace which can be used as a physicality criteria for the BRST invariant system. We have shown that the general results derived here for any surface embedded in $R^N$ is consistent with the results of previous work \cite{VKP}. We have also discussed particle motion on the torus knot surface as an example of this kind of system. In this example, we have explicitly calculated all the constraints of this system and converted them to first class constraints. It has been found that diagonal elements $\rho$ and $\epsilon$ of matrix (of the Poisson's bracket between the constraints) are nonzero if we take two different torus knot systems. In the limit of $a\rightarrow b$, the $\rho, \epsilon$ will vanish and we will achieve the commutative case of \cite{VKP}. We have also constructed the first class Hamiltonian, BRST charge and symmetry for this system. It has been shown that all the results deduced for the general system are consistent with this system. At the end we have discussed Batalin - Vilkovisky quantization of this system based on BFFT formalism. Here also we have explicitly calculated the BRST symmetry for the general system which is consistent with the symmetry derived from Hamiltonian formalism. This again proves the equivalence between the Hamiltonian and Lagrangian formalism. Recently a more general technique of Lagrangian Abelianization has been developed {\cite{SLL,VAASLL}}. It will be interesting to apply this technique to the motion in Riemann manifold.

\section{Appendix}\label{Appendix}
The calculation of Poisson bracket between some of the constraints for particle on the torus knot model is straightforward but cumbersome process. We have calculated these brackets explicitly here.

The Poisson bracket between constraints $(\Phi_3^a, \Phi_4^b)$ is written as,
\begin{eqnarray}
&&\{\Phi_3^a, \Phi_4^b \}_P = 2\nabla (D Q^a)\cdot \nabla (D Q^b) - \nabla Q^a\cdot \nabla \Phi_4^b = 2\frac {(\cosh\eta - \cos\theta)^4}{m^3a^6}\{ \sinh\eta(p p_\theta + \frac{q\phi}{\sinh^2\eta}) \nonumber\\ && - \frac{q p_\phi\cosh\eta (\cosh\eta - \cos\theta)}{\sinh^3{\eta}}\}\cdot \{p p_\theta \sinh\eta + \frac{q p_\phi}{\sinh\eta} - \frac{q p_\phi\cosh\eta (\cosh\eta - \cos\theta)}{\sinh^3{\eta}} - p p_\eta \sin\theta\} \nonumber\\ &&+ 4 \frac{(\cosh\eta - \cos\theta)^4}{m^3a^6} \sin\theta (p p_\theta + \frac{q\phi}{\sinh^2\eta})\{p p_\eta \sinh\eta + \sin\theta (p p_\theta + \frac{q\phi}{\sinh^2\eta})\}- p\frac{(\cosh\eta - \cos\theta)^2}{ma^2}\nonumber\\ &&\big[2\sin\theta\frac{(\cosh\eta - \cos\theta)}{ma^2}\cdot\{ \{2p p_\theta \sinh{\eta} + \frac{2q p_\phi}{\sinh{\eta}} - \frac{2q p_\phi\cosh{\eta}(\cosh\eta - \cos\theta)}{\sinh^3{\eta}}\} p_\eta\frac{(\cosh\eta - \cos\theta)}{ma^2} \nonumber\\ &&+ 2 \sin{\theta}(p p_\theta + \frac{q\phi}{\sinh^2\eta})p_\theta\frac{(\cosh\eta - \cos\theta)}{ma^2} - p\frac{(\cosh\eta - \cos\theta)}{ma^2}\{\sin{\theta}({p^2_\eta} + {p^2 _\theta}+ \frac{p^2_\phi}{\sinh^2\eta}) - \lambda p\} \nonumber\\ &&+ \lambda \frac{q^2}{{\sinh^2\eta}} \} +\frac{(\cosh\eta - \cos\theta)^2}{ma^2} \{ -2q \sin\theta p_\phi\frac{\cosh\eta}{\sinh^3\eta}\cdot p_\eta\frac{(\cosh\eta - \cos\theta)}{ma^2}\} + \{2p p_\theta \sinh{\eta}  \nonumber\\ &&+ \frac{2q p_\phi}{\sinh{\eta}} - \frac{2q p_\phi\cosh{\eta}(\cosh\eta - \cos\theta)}{\sinh^3{\eta}}\}\frac{p_\eta \sin\theta}{ma^2} + 2p_\theta \cos\theta (p p_\theta + \frac{q p_\phi}{\sinh^2\eta})\frac{(\cosh\eta - \cos\theta)}{ma^2}\nonumber\\ &&+ 2\sin\theta (p p_\theta + \frac{q p_\phi}{\sinh^2\eta})\frac{p_\theta\sin\theta}{ma^2} - \frac{p_\theta\sin\theta}{ma^2}\{\sin\theta({p^2_\eta} + {p^2 _\theta}+ \frac{p^2_\phi}{\sinh^2\eta}) - \lambda p\} - p\cos\theta\nonumber\\ && \frac{(\cosh\eta - \cos\theta)}{ma^2}({p^2_\eta} + {p^2 _\theta}+ \frac{p^2_\phi}{\sinh^2\eta})\big]\equiv - \gamma^{ab}
\label{phi34}
\end{eqnarray}

Poisson bracket $(\Phi_3^a, \Phi_3^b)$ is written as,
\begin{eqnarray}
&&\{\Phi_3^a, \Phi_3^b \}_P = \nabla (D Q^a)\cdot \nabla Q^b - \nabla Q^a\cdot \nabla (D Q^b)\nonumber\\ && = \{2\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}(p {p_{\theta^a}} + \frac{q\phi^a}{\sinh^2\eta^a})\cdot p'\frac{(\cosh\eta^b - \cos\theta^b)^2}{ma^2} \nonumber\\ &&- p\frac{(\cosh\eta^a - \cos\theta^a)^2}{ma^2}\cdot2\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}(p' p_{\theta^b} + \frac{q'\phi^b}{\sinh^2\eta^b})\}\equiv \rho^{ab}
\label{phi33}
\end{eqnarray}
Similarly for $(\Phi_4^a, \Phi_4^b)$ can be written as,
\begin{eqnarray}
&&\{\Phi_4^a, \Phi_4^b \}_P = 2\left[\nabla \Phi_4^a\cdot \nabla (D Q^a) - \nabla \Phi_4^b\cdot \nabla (D Q^a)\right] = 2\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}\sinh\eta^a[\{2p p_{\theta^a} \sinh{\eta^a} \nonumber\\&& + \frac{2q p_{\phi^a}}{\sinh\eta^a} - \frac{2q p_{\phi^a}\cosh{\eta^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^3{\eta^a}}\} p_{\eta^a}\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} + 2 \sin{\theta^a}(p p_{\theta^a} + \frac{q\phi^a}{\sinh^2{\eta^a}})p_\theta\nonumber\\&&\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} - p\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}\{\sin{\theta^a}({p^2_{\eta^a}} + {p^2 _{\theta^a}}+ \frac{p^2_{\phi^a}}{\sinh^2{\eta^a}}) - \lambda^a p\} + \lambda^a \frac{q^2}{\sinh^2{\eta^a}} ] \nonumber \\&&+ \frac{(\cosh\eta^a - \cos\theta^a)^2}{ma^2}[\{2p p_{\theta^a} \cosh{\eta^a} - \frac{4q p_{\phi^a} \cosh{\eta^a}}{\sinh^2{\eta^a}} - \frac{2q p_{\phi^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^2{\eta^a}} \nonumber\\&& + 6\frac{q p_{\phi^a}\cosh^2{\eta^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^4{\eta^a}}\} p_{\eta^a} \frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} + \frac{p_{\eta^a} \sinh\eta^a}{ma^2} \{2p p_{\theta^a} \sinh{\eta^a} + \frac{2q p_{\phi^a}}{\sinh{\eta^a}} \nonumber\\&& - \frac{2q p_{\phi^a}\cosh{\eta^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^3{\eta^a}}\} - 4 \sin\theta^a(\frac{2q p_{\phi^a}\cosh\eta^a}{\sinh^3{\eta^a}})\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}p_{\theta^a} + 2\sin\theta^a(p p_{\theta^a} \nonumber\\&&+ \frac{q p_{\phi^a}}{\sinh^2\eta^a}) \frac{\sinh\eta^a}{ma^2}p_\theta - p\frac{\sinh\eta^a}{ma^2}\{\sin{\theta^a}({p^2_{\eta^a}} + {p^2 _{\theta^a}} + \frac{p^2_{\phi^a}}{\sinh^2{\eta^a}}) - \lambda^a p\} + 2p\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} \nonumber\\&&\{\sin\theta^a\frac{p^2_{\phi^a} \cosh\eta^a}{\sinh^3{\eta^a}}\} - 2 \lambda^a \frac{q^2 \cosh\eta^a}{\sinh^3{\eta^a}}] \cdot 2\frac{(\cosh\eta^b - \cos\theta^b)^3}{m^2a^4}\{p p_{\theta^b} \sinh\eta^b + \frac{q p_{\phi^b}}{\sinh\eta^b} \nonumber\\&&- \frac{q p_{\phi^b}\cosh\eta^b (\cosh\eta^b - \cos\theta^b)}{\sinh^3{\eta}}  - p p_{\eta^b} \sin\theta^b\} - 2\frac{(\cosh\eta^a - \cos\theta^a)^3}{m^2a^4}\{p p_{\theta^a} \sinh\eta^a + \frac{q p_{\phi^a}}{\sinh\eta^a} \nonumber\\&&- \frac{q p_{\phi^a}\cosh\eta^a (\cosh\eta^a - \cos\theta^a)}{\sinh^3{\eta^a}} - p p_{\eta^a} \sin\theta^a \} \cdot 2\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}\sinh\eta^b[\{2p p_{\theta^b} \sinh{\eta^b} \nonumber\\&&+ \frac{2q p_{\phi^b}}{\sinh\eta^b} - \frac{2q p_{\phi^b}\cosh{\eta^b}(\cosh\eta^b - \cos\theta^b)}{\sinh^3{\eta^b}}\} p_{\eta^b}\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} + 2 \sin{\theta^b}(p p_{\theta^b} + \frac{q\phi^b}{\sinh^2{\eta^b}})p_{\theta^b}\nonumber\\&&\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}- p\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}\{\sin{\theta^b}({p^2_{\eta^b}} + {p^2 _{\theta^b}} + \frac{p^2_{\phi^b}}{\sinh^2{\eta^b}}) - \lambda^b p\} + \lambda^b \frac{q^2}{\sinh^2{\eta^b}} ] \nonumber\\&&+ \frac{(\cosh\eta^b - \cos\theta^b)^2}{ma^2}[\{2p p_{\theta^b} \cosh{\eta^b} - \frac{4q p_{\phi^b} \cosh{\eta^b}}{\sinh^2{\eta^b}} - \frac{2q p_{\phi^b}(\cosh\eta^b - \cos\theta^b)}{\sinh^2{\eta^b}} \nonumber\\&&+ 6\frac{q p_{\phi^b}\cosh^2{\eta^b}(\cosh\eta^b - \cos\theta^b)}{\sinh^4{\eta^b}}\} p_{\eta^b} \frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} + \frac{p_{\eta^b} \sinh\eta^b}{ma^2}\{2p p_{\theta^b} \sinh{\eta^b} + \frac{2q p_{\phi^b}}{\sinh{\eta^b}} \nonumber\\&&- \frac{2q p_{\phi^b}\cosh{\eta^b}(\cosh\eta^b -\cos\theta^b)}{\sinh^3{\eta^b}}\} - 4 \sin\theta^b(\frac{2q p_{\phi^b}\cosh\eta^b}{\sinh^3{\eta^b}})\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}p_{\theta^b} + 2\sin\theta^b(p p_{\theta^b} \nonumber\\&&+ \frac{q p_{\phi^b}}{\sinh^2\eta^b}) \frac{\sinh\eta^b}{ma^2}p_{\theta^b} - p\frac{\sinh\eta^b}{ma^2}\{\sin{\theta^b}({p^2_{\eta^b}} + {p^2 _{\theta^b}} + \frac{p^2_{\phi^b}}{\sinh^2{\eta^b}}) - \lambda^b p\} + 2p\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} \nonumber\\&&\{\sin\theta\frac{p^2_{\phi^b} \cosh\eta^b}{\sinh^3{\eta^b}}\}  - 2 \lambda^b \frac{q^2 \cosh\eta^b}{\sinh^3{\eta^b}}] + [2\sin\theta^a\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}\cdot\{ \{2p p_{\theta^a} \sinh{\eta^a} + \frac{2q p_{\phi^a}}{\sinh{\eta^a}} \nonumber\\&&- \frac{2q p_{\phi^a}\cosh{\eta^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^3{\eta^a}}\} p_{\eta^a}\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} + 2 \sin{\theta^a}(p p_{\theta^a} + \frac{q\phi^a}{\sinh^2\eta^a}) p_{\theta^a}\nonumber\\&&\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} - p\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}\{\sin{\theta^a}(p^2_{\eta^a} + p^2 _{\theta^a}+ \frac{p^2_{\phi^a}}{\sinh^2\eta^a}) - \lambda^a p\} + \lambda^a \frac{q^2}{{\sinh^2\eta^a}} \} \nonumber\\&&+\frac{(\cosh\eta^a - \cos\theta^a)^2}{ma^2} \{ -2q \sin\theta^a p_{\phi^a}\frac{\cosh\eta^a}{\sinh^3\eta^a}\cdot p_{\eta^a}\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}\} + \{2p p_{\theta^a} \sinh{\eta^a} + \frac{2q p_{\phi^a}}{\sinh{\eta^a}} \nonumber\\ &&- \frac{2q p_{\phi^a}\cosh{\eta^a}(\cosh\eta^a - \cos\theta^a)}{\sinh^3{\eta^a}}\} \frac{p_{\eta^a} \sin\theta^a}{ma^2} + 2p_{\theta^a} \cos\theta^a (p p_{\theta^a} + \frac{q p_{\phi^a}}{\sinh^2\eta^a})\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2} \nonumber\\&&+ 2\sin\theta^a (p p_{\theta^a} + \frac{q p_{\phi^a}}{\sinh^2\eta^a})\frac{p_{\theta^a}\sin\theta^a}{ma^2} - \frac{p_{\theta^a}\sin\theta^a}{ma^2}\{\sin\theta^a(p^2_{\eta^a} + p^2 _{\theta^a} + \frac{p^2_{\phi^a}}{\sinh^2\eta^a}) - \lambda^a p\} - p\cos\theta^a \nonumber\\&&\frac{(\cosh\eta^a - \cos\theta^a)}{ma^2}(p^2_{\eta^a} + p^2 _{\theta^a} + \frac{p^2_{\phi^a}}{\sinh^2\eta^a})\big]\cdot 2\frac{(\cosh\eta^b - \cos\theta^b)^3}{m^2a^4}\{p p_{\eta^b}\sinh\eta^b + p p_{\theta^b} + \frac{q p_{\phi^b}\sin\theta^b}{\sinh^2\eta^b}\} \nonumber\\&&- 2\frac{(\cosh\eta^a - \cos\theta^a)^3}{m^2a^4}\{p p_{\eta^a}\sinh\eta^a + p p_{\theta^a} + \frac{q p_{\phi^a}\sin\theta^a}{\sinh^2\eta^a}\}\cdot \big[2\sin\theta^b\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}\cdot\{ \{2p p_{\theta^b} \nonumber\\ &&\sinh{\eta^b} + \frac{2q p_\phi^b}{\sinh\eta^b} - \frac{2q p_{\phi^b}\cosh{\eta^b}(\cosh\eta^b - \cos\theta^b)}{\sinh^3{\eta^b}}\} p_\eta^b\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} + 2 \sin{\theta^b}(p p_{\theta^b} + \frac{q\phi^b}{\sinh^2\eta^b})\nonumber\\ &&p_\theta^b\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} - p\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}\{\sin{\theta^b}(p^2_{\eta^b} + p^2 _{\theta^b}+ \frac{p^2_{\phi^b}}{\sinh^2\eta^b}) - \lambda^b p\} + \lambda^b \frac{q^2}{{\sinh^2\eta^b}} \} \nonumber\\ &&+\frac{(\cosh\eta^b - \cos\theta^b)^2}{ma^2} \{ -2q \sin\theta^b  p_{\phi^b}\frac{\cosh\eta^b}{\sinh^3\eta^b}\cdot p_{\eta^b}\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}\} + \{2p p_{\theta^b} \sinh{\eta^b}  + \frac{2q p_{\phi^b}}{\sinh{\eta^b}} \nonumber\\ &&- \frac{2q p_{\phi^b}\cosh{\eta^b}(\cosh\eta^b - \cos\theta^b)}{\sinh^3{\eta^b}}\}\frac{p_{\eta^b} \sin\theta^b}{ma^2} + 2p_{\theta^b} \cos\theta^b (p p_{\theta^b} + \frac{q p_{\phi^b}}{\sinh^2\eta^b})\frac{(\cosh\eta^b - \cos\theta^b)}{ma^2} \nonumber\\ && + 2\sin\theta^b (p p_{\theta^b} + \frac{q p_{\phi^b}}{\sinh^2\eta^b})\frac{p_{\theta^b}\sin\theta^b}{ma^2} - \frac{p_{\theta^b}\sin\theta^b}{ma^2}\{\sin\theta^b(p^2_{\eta^b} + p^2 _{\theta^b} + \frac{p^2_{\phi^b}}{\sinh^2\eta^b}) - \lambda^b p\}  \nonumber\\ &&- p\cos\theta^b \frac{(\cosh\eta^b - \cos\theta^b)}{ma^2}(p^2_{\eta^b} + p^2 _{\theta^b}+ \frac{p^2_{\phi^b}}{\sinh^2\eta^b})\big] \equiv \epsilon^{ab}
\label{phi44}
\end{eqnarray}

\section{References}\label{References}

redpajama_set_name: RedPajamaArXiv


Crowdfunding Commercial Real Estate: Panacea or Path to Disaster?
As seen in the March 14, 2016 edition of the Providence Business News.
Mogul have sprung up to provide what Selequity calls "the leading edge CRE investment platform." However, has crowdfunding been the panacea for developers and owners it was touted as being – and how have investors fared?
Crowdfunding is seen as providing greater flexibility to CRE owners. Online investing websites claim that they benefit owners seeking access to capital, liquidity for sale adverse owners, and maintenance of control over a highly valued asset. Such sites also claim that they benefit new investors seeking to include CRE ownership in their asset mix without the need for specialized industry knowledge.
The founder and CEO of crowdfunding portal RealtyShares, Nav Athwal, wrote in the December 2, 2015, issue of Forbes that "real estate crowdfunding is rapidly reshaping the way individuals find and invest in properties… [and] has brought benefits not only for investors but also for real estate companies." There is some proof in the market place that crowdfunding a CRE project can work for real estate developers and owners. For example, the developers of a new extended stay project in New York City, the AKA United Nations, successfully used crowdfunding to fill gaps in financing the project's costs. However, is there any evidence to date that crowdfunding brings benefits to individual investors?
The very short history of crowdfunding and the web portals that promote it indicates that investors should only proceed with a firm grasp of the potential high risks. The United States Securities and Exchange Commission ("SEC") has yet to issue any regulations governing crowdfunding and the portals that raise equity from investors. However, there are already calls from consumer advocates calling for the SEC to investigate the portals, claiming that some are charging excessively high fees or are otherwise taking advantage of investors. It is certain that at some point the SEC will investigate crowdfunding and the portals that promote it. The results of any SEC investigation may include additional rules and regulations that could significantly alter the crowdfunding landscape. And, don't forget that additional rules and regulations will only be necessary if the SEC deems the current environment unfair to investors, which is not the ideal environment in which to invest!
Individual investor liquidity is also an evolving issue. Portals usually carry warnings that investments made through crowdfunding are made in private entities, and once made, the investor is a minority owner with no control over how his money is used. As a result, it is incumbent upon each investor to carefully consider the project's time frame as stated in the developer's business plan. Each investor must further consider her needs for the invested amount that she may have during that time frame as there will be no way to recover the invested money until the occurrence of one of the liquidity events set forth in the business plan. Although the JOBS Act requires that investors be accredited, the standards are based on net worth outside the value of a primary residence or a minimum annual income – neither of which guaranties any degree of sophistication in reading and understanding a business plan designed to entice people to part with their money. Until a secondary market develops CRE crowdfunding investors will be stuck with an illiquid share in a project that may or may not succeed.
One final risk that has not yet been fleshed out in the marketplace: what happens to an investor who places money for a project with a portal that fails? A corollary to that is what happens to the CRE developer or owner who has raised money through a portal that fails before releasing the money? The law is evolving very slowly with respect to CRE crowdfunding and is still largely undeveloped; one sure bet is that these scenarios will benefit only the law firms handling the resultant litigation.
Crowdfunding for commercial real estate projects thus may be both a panacea and a path to disaster, depending entirely on the project and the portal through which one invests. It is likely that as the market evolves, the SEC sets some rules, and the crowdfunding process matures, it will become a more common means of raising capital. However, until that time one can only urge the investor to pay heed to the Latin warning: caveat emptor.


I was wondering if someone might have the knowledge to port the auto trendline detection indicator into NT7 beta?
I have become used to watching this indicator and am keen to use it in the beta 7.
Since you are asking about NT7, I will have someone reply to you on Monday. Thank you for your patience.
Hopefully that can provide you with some guidance.
I have NO skill in code stuff. But I have had a go of adjusting the old 6.5 trendline automatic and have some issues with it before compiling.
I have attached it for some kind person to have a look for me.
The lines that create the problem are the 2 DrawRay functions.
I shouldn't sound so surprised but I am... after trying to understand the error messages and having not a single clue what to do- it seems amazing that it works now.
I guess when NT7 is out of beta the indicator section will be filled with specially tweaked ones for people to download... Looking forward to it.


text:
Rachael Ray to reunite men with puppies they rescued in August flood
Today's episode of talk show host Rachael Ray's show will feature Baton Rouge area men who saved several animals' lives during Louisiana's August flood, according to The Advocate. They eventually gave two pitbull puppies they found to Companion Animal Alliance. On the show, Mike Anderson and Darrell Watson will be reunited with puppies they rescued during the flooding.
Ray's pet organization, Rachael's Resuce, sent a truck to pick up abandoned animals in the aftermath of the flood.
Also on the show will be singer Kelly Clarkson, who is an avid dog lover.
The show airs at 11 a.m. on WVLA.
Where to watch the National Championship game on Monday
Anticipation is growing as the 2020 College Football National Championship game approaches where LSU will take on Clemson in a Tigers versus Tigers showdown at 7 pm. Those of us unable to head to New Orleans to watch...
The Benefits of Plyometric Training
Fournette insured for two $10 million policies, report says
LSU Board approves raise and extension for D-D Breaux
LSU ranked No. 24 in Preseason Poll
Update: Joe Alleva Confirmed to Step Down, Will Take on New Role for LSU

redpajama_set_name: RedPajamaCommonCrawl


Are you looking for the best e-cig shop out there? Well, you've found it. We have the widest selection of premium e-cig's in all Goodyear. We only carry the products that we'd use ourselves. That means we have the top brands and most popular models as well as all the things you need to go along with it.
When you come into our store, you'll be met with professionals who are completely passionate about what they do. We're here to answer any of your questions about vaping and vape products. Come in and chat with our team. Let us find you the perfect products!
The Vapor Vault © All Rights Reserved.


Bruce Rauner Pushes Forward With "Conservationists Coalition"
We've long held that being 'for' the environment can and should be an issue 'owned' by just about every Republican candidate up and down the ballot. So, we're glad to see Bruce Rauner come out with his own group of Conservationists supporting him.
Gubernatorial candidate Bruce Rauner and running mate Evelyn Sanguinetti today announced the launch of the Conservationists for Rauner Coalition.
The coalition consists of a broad range of community and political leaders who endorse Rauner for Governor.
The group includes: Mary Jo Trimble, Mike Kepple, Larry Closson, Brent Manning, Neal Graves, Wayne Rosenthal, Tim Sickmeyer, Richard Mottershaw, Glenn Sanders.


it was just to be able to compare speed between substring search and regex.
IMHO, it is a bad explanation. The only reason your client is downloaded so many time is probably because it is better than the standard nmdc client and because both are available on windoz. It has nothing to do with the capabilities of the system. If we should have waited for a good network protocol from windoz, we still would run netbeui, not TCP/IP. If a feature must be added, a possible missing function is a bad reason, just add it.
but not everyone share as many files as the test share file. Moreover, currently, the problem is not the CPU speed, it is the bandwidth (especially upstream). If performing 40 searches per second use all your upload bandwidth then I think we will just have something like ed2k (good to find something, awful to obtain it).


text:
How healthy is New South Wales?
Tips, reports and trends for Australians
New South Wales has one of the lowest proportions of smokers in the countryi, a large and growing population, and death rates from major causes of disease below the national average.
Consuming more fruit and vegetables, smoking less, and exercising more have contributed to the positive health trends in NSW, but it seems that more needs to be done to counter the state-wide problem of expanding waistlines. Compared to the other states NSW is lagging behind on physical activity, with more than a quarter of the NSW residents leading sedentary lifestyles i.
Health snapshot
New South Wales is home to a third of Australia's population and remains the country's most populous state despite slowing population growth rates: in 2011, 7.21 million people were living in NSWii. New South Wales' locals enjoy a high life expectancy rate that continues to increase: in 2010, life expectancy for males was 79.6 years and 84.1 years for females; these life expectancies were an increase of 2.7 years and 1.7 years respectively since 2001iii.
Death rates from major causes of disease have also fallen significantly. Death rates for New South Wales are below the national averageiv, and can be attributed to people making healthier lifestyle choices - fewer people are smoking and gains have been made on adequate fruit and vegetable consumption.
However, not enough is being done to offset the state's growing overweight and obese population. In 2007, cardiovascular disease, encompassing coronary heart disease (accounting for 7,818 deaths), stroke (4,135), and heart failure (1,032), was the leading cause of death in New South Walesv.
Risky behaviour
While people living in New South Wales are leading more active lifestyles compared to 10 years ago, the latest Australian Health Survey reveals the state has the highest proportion of people who are inactive in Australiai. In 2011-2012, 38% of the adult population were classified as leading sedentary lifestylesi. Furthermore, New South Wales (94.4%) has the fourth largest proportion of people that do not eat the recommended daily intake of fruit and vegetables, trailing closely behind the Northern Territory (96%), Queensland (95%), and Victoria (94.8%)i.
In addition to physical inactivity and unhealthy eating, smoking behaviours and alcohol misuse are having negative impacts on the health of people living in New South Wales. While smoking rates are among the lowest in the nation, smoking remains the leading cause of preventable disease in the statevi and alcohol-induced conditions cause more than 1,220 deaths and 48,000 hospitalisations each yearvii. In 2011-2012, 19% of adults drink alcohol at levels that put their health in jeopardyi.
There is an increasing prevalence of chronic illnesses as a result of these unhealthy lifestyle behavioursviii. In 2007, lung cancer was the leading cause of cancer death in New South Wales, with 91% of lung cancer deaths attributed to smokingix.
Health system challenges
New South Wales has the largest health system in the country and often performs above average compared with the overall Australian performancex. However, the rising incidence of chronic illness is putting pressure on the healthcare system and may reverse the state's longstanding trend of increased life expectancyxi. Other significant challenges facing New South Wales' health care system are an expanding and ageing population, and people leading busier but more sedentary lives.
In 2011-2012, the 65 years and older age group had the largest number of people leading inactive lifestylesi. Furthermore, they were the second largest group to be overweight and obesei. Promotion of healthy lifestyle choices in earlier stages of life is paramount to lowering the numbers of avoidable deaths and preventable hospitalisations in the state.
In addition to this, changes to people's lifestyles, such as a blurring of work hours and family time is increasing the likelihood of heightened levels of stress, which could have significant impacts on both physical and mental wellbeingxii.
iAustralian Bureau of Statistics, 2012, Australian Health Survey: First Results 2011-2012, http://www.abs.gov.au/AUSSTATS/abs@.nsf/DetailsPage/4364.0.55.0012011-12?OpenDocument
iiAustralian Bureau of Statistics, 2011, Population by Age and Sex, Regions of Australia, 2011, http://www.abs.gov.au/ausstats/abs@.nsf/Latestproducts/3235.0Main%20Features52011?opendocument&tabname=Summary&prodno=3235.0&issue=2011&num=&view=
iiiChief of Health Officer NSW, 2010, The health of the people of New South Wales, NSW Government, http://www0.health.nsw.gov.au/pubs/2010/pdf/chorep_summary_2010.pdf, p.iii
ivAustralian Bureau of Statistics, 2012, State and Territory Statistical Indicators, 2012, http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/1367.0~2012~Main%20Features~Mortality%20Rate~2.3
vHealth Statistics NSW, Deaths by disease type, NSW Government, http://www.healthstats.nsw.gov.au/Indicator/cvd_typedth
viNSW Ministry of Health, 2012, NSW Tobacco Strategy, NSW Government, http://www0.health.nsw.gov.au/pubs/2012/pdf/nsw_tobacco_strategy_2012.pdf, p.2
viiChief of Health Officer NSW, 2010, The health of the people of New South Wales, NSW Government, http://www0.health.nsw.gov.au/pubs/2010/pdf/chorep_summary_2010.pdf, p.54
viiiNSW Department of Health, 2007, A new direction for NSW, NSW Government, http://www0.health.nsw.gov.au/pubs/2007/pdf/state_health_plan.pdf, p.9
ixChief of Health Officer NSW, 2010, The health of the people of New South Wales, NSW Government, http://www0.health.nsw.gov.au/pubs/2010/pdf/chorep_summary_2010.pdf, p.94
xNSW Department of Health, 2009, Annual Report 2008-09, NSW Government, http://www0.health.nsw.gov.au/pubs/2009/pdf/annualreport09_27-82.pdf, p.28
xiNSW Department of Health, 2007, Future Directions for Health in NSW, NSW Government, http://www0.health.nsw.gov.au/pubs/2007/pdf/future_directions_8-15.pdf, p.6
xiiNSW Department of Health, 2007, Future Directions for Health in NSW, NSW Government, http://www0.health.nsw.gov.au/pubs/2007/pdf/future_directions_8-15.pdf, p.7


The Trimble Quantm system is ideal for any road project, from small bypasses to major highways, new projects and realignments. Quantm comprehensively integrates design standards, terrain, geological, and hydrological data, environmental areas, property ownership, and cost information in a simultaneous optimization process. This delivers a range of alignment alternatives that can be analyzed to address project owner and stakeholder concerns and maintain the momentum of the road project.


text:
How to watch Motherwell v Ross County
Storylines: Motherwell v Ross County
There's a live stream available of Motherwell v Ross County on Saturday.
The Scottish Premiership match can be watched in two ways.
Racism has no place in football. Racism has no place in society.
We are proudly continuing our support for Show Racism the Red Card, as they continue their Fortnight of Action. The education charity have spread messages of tolerance and inclusion within football for many years.
Now more than ever, the importance of this fortnight cannot be overstated.
Our fight to end racism exists every day. This weekend against Ross County – and across all Scottish football matches – we as clubs will united to call for an end to racial discrimination and inequality throughout society.
Show Racism the Red Card work all year round with schools, workplaces, grassroots, youth football clubs and the prison service to educate young people and adults on challenging prejudice and stereotypes, using the power of football as a positive way to instigate conversation on anti-racism.
Covid-19 and events overseas have shone a light on the growing inequalities that some of us unfairly face. Still today, many experience this disproportionately due to the colour of their skin.
Last year in Scotland, 81% of young people felt more confident in identifying and safely challenging racism after working directly with Show Racism the Red Card.
Scottish football is committed to eradicating racism and discrimination in all its forms from our game. We call on our supporters to continue to challenge racism. It is the responsibility of us all to ensure that the anti-racism message is carried out beyond the Fortnight of Action.
"It is so important that we continue to push the message that racism has no place in our sport or society," Motherwell FC manager Stephen Robinson said.
"Sadly, we continue to see incidents of racism around the world. Whether they are occurring on our doorstep or thousands of miles away, the connected world we live in means that these despicable moments are brought close to home for all.
"We must never be complacent. We must never stop calling it for what it is. We must stand against racism."
We stand with everyone who has had to endure racial prejudice in all areas of their lives. We condemn all forms of racism and any racist behaviour will not be tolerated at our club.
Football is a sport that is loved by people across the world and football matches should be an enjoyable experience for all, whatever your background. Education is the key. We support Show Racism the Red Card to ensure our future generation are equipped with the tools to challenge and eradicate racism from football and society.
For useful resources and information, please visit theredcard.org or follow the organisation on Facebook, Instagram or Twitter.
Motherwell and Ross County face for the second time this season in the Scottish Premiership at the weekend.
Here are the storylines going into Saturday's match.
The return of Jermaine Hylton
Having started the season at Fir Park, the forward swapped ML1 for Dingwall at the start of October.
Part of the squad which roared to third spot in the Premiership in the 2019/20 campaign, Hylton contributed four goals in 41 appearances for the Steelmen.
With first team opportunities limited in 2020/21, the 28-year-old took the chance to head north in search of action and to add to the Staggies' forward options.
The back and forth of results
Neither side has been able to settle on a consistent run of results in the games between the two in recent years.
Since Motherwell won on the five on the bounce across 2013 and 2014, neither team has won more than two consecutively against the other.
Ross County were winners last time out in Dingwall, with Motherwell winning the previous time at Fir Park.
The scramble up the table
With both sides harbouring ambitions of a top six finish, there's three valuable points on offer on Saturday.
Ross County currently sit four points ahead of Motherwell in seventh spot, albeit having played two matches more.
A win for Stuart Kettlewell's side could lift them to fifth in the Premiership, while Motherwell could jump to eighth.
A chance to get some playing action
Motherwell's last competitive game came on 27 September at home to Rangers, meaning almost a month has passed.
With games against Kilmarnock and St Mirren cancelled due to Covid-19 issues with our opponents, there has been significant time spent in cold storage for Stephen Robinson's men.
In the same time period, County have played four matches – losing away to Rangers, drawing with Montrose, beating Arbroath and drawing at home to Hibernian.


text:
In a fitting Memorial Day tribute, Danbury unveiled a monument yesterday to the 90 African-Americans from the city who fought in the Civil War, making it one of the few memorials in the state to mention the important role of black Americans in fighting for their freedom., The New York Times
Black troops from the state made up the 29th and 30th Connecticut Volunteer Infantries. The 29th was the first Union infantry regiment to enter Richmond, Va., after the Confederate withdrawal. Troops from both units witnessed Lee's surrender.
Northeastern states rarely consider in much depth their own roles in slavery, preferring to focus on what they did to fight against that institution rather than examine how they helped perpetuate it.
Connecticut sanctioned the keeping of slaves for years before finally outlawing the practice. But the state took a big step toward fairness when African-Americans began to enlist in the Union Army. At the time, the federal government paid black soldiers at a lower salary than white troops, so the legislature made up the difference, ensuring that for Connecticut troops at least, everyone's salary was equal.
Freedom is often spoken of as a gift. But for African-Americans, liberty was no gift. They bled for it; they died for it.
Remembering these men is a reminder of that sacrifice. Good for Danbury for establishing this tribute.


I remember when I first started submitting my short stories to literary magazines in my early thirties, I'd been working on my craft for years. There was one particular magazine that I was dying to get into called Glimmer Train. I admired them immensely. Plus they were actually one of the kinder magazines because they posted a list of 20 finalists whenever they published anybody.
The first time I ever looked up the results, I was so disappointed As I searched for my name in the list of finalists, the text blurred as I realized I didn't even make it in the top 20. Then a certain numbness came over me and I stopped working for weeks.
That first experience hung over me like a heavy fog, until I started to investigate the arena I was stepping into. Over a thousand stories were submitted to all of these magazines every single month. That meant that I had less than a 0.1% chance of being published and a 2% chance of making the top 20.
Instead of being intimidated by those odds, I let them give me the courage to laugh at rejection. Did I really expect to walk into a room of over a thousand people and be immediately chosen and exulted? That attitude was laughable. Thus my ability to laugh at rejection was born. AMEN!
I kept submitting to magazines, and as the rejections piled up, they really weren't that painful anymore. Just stepping stones along the path. In fact I submitted to Glimmer Train twice after that first attempt (along with many other magazines) and never made the top twenty.
However, when I received my first acceptance from a magazine of equal stature, something ironic happened. My eyes blurred as I read the word, "Congratulations…" and I thought, that's kind of a sarcastic opening for a rejection letter.
It took a minute before I realized, this was not a rejection letter.
After all that time and effort, I truly understood what an honor it was to be published in a literary magazine, something that would have washed right over me if I would have been immediately successful. I'd paid my dues, done the work, and had a palpable lived experience of what it was like to be one in a thousand.
Sometimes people believe they get an easy out of the rejection lottery by choosing self-publishing. Not true. I love working with self-published authors because it's actually a better way to achieve a variety of goals. However you have to understand that every time you set up a goal, you also set up the possibility of failure. And the higher the goal the more failure you might have to learn from, which is great!
One of my favorite spiritual teachers, Jeffrey Allen once explained that sometimes we receive a lot of rejection because our guides are on the other side of that door holding it shut with their backs and shoulders trying to get us to go down a different path.
In the case of writing, I would say our guides hold their backs against the door until we're worthy and understand the significance of success. This may not be true for everybody, but in my case, I wouldn't have it any other way. There's nothing more fulfilling than working to accomplish something difficult.


These candles that Daddy keeps on buying and very lovely smelling, I've never really taken the time before to appreciate their wonderful beauty.
The smell almost puts you in a trance. When River invited me out to play I happily trotted over.
Not too sure what's going on here, I'm so chilled, I think I have definitely been put under some sort of a spell.
It must be a spell I'm transfixed by my ball and I never am. Think I should keep away from candle sniffing, bit overcome by the perfume. Trippy!


text:
Brundidge gears up for annual parade
Published 8:29 pm Thursday, June 4, 2009
Brundidge is a red, white and blue community filled with patriotic pride.
Not many communities of less than 2,500 people can boast of a longstanding Main Street parade that celebrates the country's independence. But, on the last Saturday in June each year, the Brundidge Business Association rolls out the red carpet for the Independence Day Parade.
This year's parade will begin at 9 a.m. on June 27 and follow a Main Street route through town. The annual parade honors those whose military service has earned and preserved the freedoms that all Americans enjoy.
"The Brundidge Independence Day Parade has a proud history of nearly 30 years, and it's a tradition that we hope will endure for many generations," said Jennifer Garrett, BBA president.
Entries for the parade are being accepted and the only requirement is that each entry has a patriotic theme.
The Fort Rucker Military Band will lead the parade ,and the band alone draws a big crowd, Garrett said
"We welcome floats and everything on wheels from tricycles to limousines," Garrett said. "Those who would rather walk than ride are encouraged to show their patriotic spirit and join us."
To place an entry in the Independence Day Parade, visit Rue's Antiques on Main Street in Brundidge or call 334-735-3125.
The 2009 Independence Day Parade will be the highlight of a day of fun-filled activities in Brundidge. Following the parade, the queens from cities around the area will be introduced at the Knox Ryals Pavilion followed by the Queens Tea which will be hosted by the BBA.
"Entertainment for the day will be provided by Don Fuda and The Classics and Walter Wilson, formerly of Brundidge, who is a gospel/country singer.
"We'll have hotdogs for sale on the grounds of the Bass House and also drawings for several really great prizes," Garrett said. "Don Fuda has donated a guitar ,and we'll also have a 32-inch flat-screen TV to give to some lucky person. Tickets for the drawing are a dollar and will be available at the parade."
Garrett said everyone is encouraged to get the Fourth of July holidays off to an early start by being a part of the Independence Day Parade in Brundidge on June 27.
June Jubilee at TH& Sunday


text:
Rewards for the best business blueprints
WINNERS of the coveted Blueprint Awards have been unveiled after a hard fought competition between the finalists from the region's five universities.
Karen McLauchlan
And once again Teesside University had cause for celebration, taking first and third place in the competition's Business Awards Category.
David Hazeldine, who recently graduated in business management, won first place with his business Drive Green Taxi Services - an innovative green taxi service using a fleet of electric vehicles.
Meanwhile Little Lost Buttons, an accessories business set up by four fashion enterprise undergraduates, took third place.
Now in its eighth year, The Blueprint Awards, in association with Newcastle Science City, saw students and academic staff from the Universities of Durham, Newcastle, Northumbria, Sunderland and Teesside compete against each other to present the most compelling business plans to some of the region's leading business experts.
The Blueprint Awards have already helped a host of pre-start-up companies become commercial success stories.
This year, with the additional help and mentoring support offered to the finalists from some of the region's leading business people, the judges acknowledged that the compelling business plans emerging from the competition would result in a number of new and exciting businesses.
All the winners walked away with cash prizes to boost their business as well as additional in kind support from sponsors, ranging from IP consultancy to incubator work space.
Commenting on his win David said: "The competition has helped to cement my business idea and given me the confidence to take it further. Some of the best business brains in the region challenged my thinking and have helped me to map out a business plan with huge potential."
Brian Tanner, dean of knowledge transfer at Durham University and co-founder of Blueprint, said: "This year's entries were excellent across the board and stand testament to an enterprise culture shared by all the region's universities.
"Whilst the prizes and accolade are a huge part of the competition, it is the journey that the competitors take to reach the final that is most important. In the build up to the grand final, the finalists are given access to some of the best business brains in the region as mentors and advisors and are introduced to some hugely influential networks, which will no doubt prove invaluable as their businesses develop."
Blueprint 2011 winners:
Business Award Category
Drive Green Taxi Services - David Hazeldine - Teesside University
Creativity and Design
The Tea Shed - Jules Quinn - Northumbria University
Social Entrepreneurship Award
Film Central (Jam Jar Cinema) - Daniel Ellis - Newcastle University
Knowledge Transfer Award
Brock Fine Chemicals - Graham Sandford - Durham University


However, in a new geographic and psychic location.
The physical geography, I am sure of: Vashon Island, Cascadia, Gaia. Looking out on Mount Rainier, a massive stratovolcano about 50 miles away. 14, 411 feet high. Considered one of the most dangerous volcanoes in the world.
I live on a sweet rock along with twelve thousand music loving souls, in relative rural harmony, plus some chickens, goats, Jersey cows, horses and donkeys.
Dozens of colors of rhododendrons.
In terms of my new inner location, I am exploring.
Il faut être absolument moderne. One must be absolutely modern.
Wildness is a preoccupation. Volcanic wildness.
So I can sit at the Red Bicycle and blog?


We all know that, to remain competitive, businesses need to be able to evolve, and that organisations which are best equipped to manage change, in a way that is not disruptive to current ongoing operations, will create and maintain a competitive edge.
Changing your business costs money, of course, hence all change initiatives are investments which, as with all investments, should not be undertaken unless they will provide a 'return on investment' (ROI). This is usually evaluated as a business case, based on predicted benefit and cost data, which then enables you to establish the Affordability of multiple selected change initiatives in relation to annual budgets.
But Affordability is not the only important factor in evaluating the Attractiveness of a change initiative. An evolving organisation should create and maintain a business plan to achieve its strategic objectives. Change initiatives which are evaluated to offer the best Alignment to the organisation's strategic objectives are therefore of greater value.
Projects and programmes have varying levels of complexity, depending upon the environment and circumstances within which they are to be delivered. There will, therefore, be different levels of risks to delivery of the projects and programmes. By evaluating the risks of each change initiative, their Achievability can be established and taken into account when prioritising and selecting which initiatives to deliver.
The first of two aspects of portfolio management is to ensure that you are 'doing the right projects and programmes' by prioritising and selecting the change initiatives which are the most Attractive, based on Affordability, Alignment and Achievability. If you are not optimising your change portfolio then you are not investing your money as wisely as you could be.
After you have decided which change initiatives to take forwards, it is then important that they are successfully delivered as projects and programmes, which achieve the expected benefits within the predicted costs (set as budgets). This is the other aspect of portfolio management; 'doing the projects and programmes right'.
Organisations should create the capability to select and deliver the optimum portfolio of projects and programmes, on an ongoing and ever improving basis, by continuously developing their people, processes and technology to evolve their P3M maturity. By doing so, they will become well equipped to manage change and maintain a competitive edge.


ANKARA (AA) - An appeal by dozens of Russian athletes hoping to compete in the Winter Olympics in South Korea starting on Friday has been dismissed.
The 47 athletes appealed to the Court of Arbitration for Sport (CAS) against their expulsion from the games by the International Olympic Committee (IOC) over doping during the 2014 Sochi Olympics.
"The Applicants challenged the IOC decision refusing to invite them to participate in the 2018 Olympic Winter Games and requested that CAS overturn the IOC decision," the court said in a statement.
The statement said the court's arbitrators considered that the process created by the IOC to establish an invitation list of Russian athletes to compete as Olympic Athletes from Russia (OAR) could not be described as a "sanction" but rather as an "eligibility" decision.
Late 2017, the IOC barred Russian athletes from competing under the country's flag in PyeongChang, South Korea, although those with a clean record of drug-testing will be able to compete as neutrals.
For its part, the Kremlin on Friday rejected the court's decision to bar Russian athletes from the Olympics, which lasts through Feb. 25.
"Regarding the CAS decision, we regret it and we take into account the decision of the court and, of course, we will continue to help our athletes to defend their rights," Kremlin spokesman Dmitry Peskov was quoted as saying by Russia's official news agency TASS.
Peskov said it was not time for "loud statements" on global sport reform, adding: "The main thing is to support our athletes".