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GLOSSARY OF SYMBOLS

GLOSSARY OF SYMBOLS

\begin{epigraphs} \qitem {Consistency is the last refuge of the unimaginative.} {Oscar Wilde} \end{epigraphs}

$ \left< x \right>$ The mean value of variable $ x$ [*]
$ \emptyset$ Empty set [*]
$ \,\,\hat{}$ Denotes a unit vector; $ \hat{\bf x}={\bf x}/\vert{\bf x}\vert$ [*]
$ \vert\,\vert$ A vector norm; $ \vert{\bf x}\vert=\sqrt{\sum_{i=1}^{dim({\bf x})} x_i^2}$; also, cardinality of a set [*]
$ \ominus$ Denotes symmetric difference of two sets [*]
$ \otimes$ Denotes vector direct product, a.k.a. dyadic; $ {\bf a} \otimes {\bf b}^T = {\bf c}$, $ c_{i,j}=a_i b_j$ [*]
$ (\,,\,)$ A scalar product of two vectors, a.k.a. dot product; $ ({\bf a},{\bf b})={\bf a}^T\cdot{\bf b}$ [*]
$ \{\,\}$ A set of objects; $ \{{\bf x}_i\}_{1}^n = \{{\bf x}_1,{\bf x}_2,{\bf x}_3,\dots,{\bf x}_n\}$ [*]
$ {\bf0}$ A matrix or a vector filled with zeros [*]
$ A$ One of the two superstates in two-state kinetic model [*]
$ {\bf A}$ Approximation to the inverse Hessian matrix, $ {\bf H}^{-1}$ [*]
$ Adj[i]$ Set of all the nodes adjacent to node $ i$ [*]
$ AdjIn[i]$ Set of all the nodes connected to node $ i$ via incoming edges [*]
$ AdjOut[i]$ Set of all the nodes connected to node $ i$ via outgoing edges [*]
$ B$ One of the two superstates in two-state kinetic model [*]
$ {\rm BLJ}_n$ Binary Lennard-Jones liquid with $ n$ atoms in a periodic cell [*]
$ {\bf C}$ Square matrix, columns of which are eigenvectors [*]
$ C_N$ A chain graph with $ N$ nodes [*]
$ D$ Endpoint separation [*]
$ \Delta_i(j)$ $ i$th displacement in magnitude of an atom between structures $ j-1$ and $ j$ [*]
$ E$ A set of edges [*]
$ \mE _\alpha^G$ Probability of escape from $ G$ starting from $ \alpha $ in a single step [*]
$ E_-$ Energy barrier corresponding to the reverse reaction [*]
$ E_+$ Energy barrier corresponding to the forward reaction [*]
$ F$ Set of all minima connected to the final endpoint [*]
$ \mF $ Frequency distribution function [*]
$ G_N$ An arbitrary graph with $ N$ nodes [*]
$ {\bf H}({\bf x})$ Hessian matrix evaluated at point $ {\bf x}$ [*]
$ I$ A set containing all the minima that do not belong to $ A\cup B$ [*]
$ K_N$ A complete graph with $ N$ nodes [*]
$ L$ Lagrangian function [*]
$ {\rm LJ}_n$ $ n$-atom Lennard-Jones cluster [*]
$ \mho \left({\bf x}\right)$ The set of all possible values of the control variable $ {\bf x}$ [*]
$ N$ Number of atoms; number of nodes in a graph [*]
$ N_c$ Cooperativity index [*]
$ N_f$ Number of frames or points sampled along a path [*]
$ N_i$ Number of images in a band [*]
$ N_p$ Participation index [*]
$ \widetilde{N}$ Participation index evaluated using the endpoints alone [*]
$ O_k$ Displacement overlap evaluated for $ k$ atoms using displacements $ d_i(j)$ [*]
$ \mO (\,)$ $ f(n)=\mO \Bigl(g(n)\Bigr)$ means $ 0 \leq f(n) \leq c g(n)$ holds for some constants $ c > 0$ [*]
$ {\bf P}$ Transition probability matrix [*]
$ P_{i}^{\rm eq}$ Equilibrium occupation probability of state $ i$ [*]
$ P_i(t)$ Occupation probability of state $ i$ at time $ t$ [*]
$ P_{j,i}$ Probability of transition from state $ i$ to state $ j$ [*]
$ \mP _{\xi}$ Pathway probability [*]
$ R_N$ A random graph with $ N$ nodes [*]
$ {\bf R}_\alpha$ $ 3N$-dimensional rotation matrix about axis $ \alpha $, $ \alpha\in\{x,y,z\}$ [*]
$ \mathbb{R}$ The set of all real numbers [*]
$ S$ Set of all minima connected to the starting endpoint [*]
$ \Sigma_\alpha^G$ Total probability of escape from $ G$ if started at node $ \alpha $ [*]
$ \mS _{\alpha,\beta}^G$ Sum of weights of all pathways connecting $ \alpha $ and $ \beta $ and confined to $ G$ [*]
$ T$ Temperature [*]
$ \Theta_k$ Displacement overlap evaluated for $ k$ atoms using displacements $ \Delta_i(j)$ [*]
$ \mT _{i}^G$ Mean escape time from graph $ G$ if started at node $ i$ [*]
$ U$ Set of all minima that do not belong to $ S\cup F$ [*]
$ \Upsilon({\bf x},\varepsilon)$ A set of feasible points contained in the neighbourhood $ \varepsilon$ of $ {\bf x}$ [*]
$ V$ Potential energy functional; also, a set of graph nodes [*]
$ \mathcal{V}$ $ 3N$-dimensional vector of velocities[1]This vector and the other vectors defined here are column vectors. [*]
$ \widetilde{V}$ Spring potential [*]
$ W(a,b)$ Weight of the shortest path $ \xi=a\leftarrow b$; $ W(a,b)=-\ln(\mW _\xi)$ [*]
$ \mathbb{W}$ The set of whole numbers; $ \mathbb{W}=\{0,1,2,\dots\}$ [*]
$ \mW _\xi$ Product of branching probabilities associated with path $ \xi$ [*]
$ {\bf X}$ $ 3N$-dimensional vector representing a point in configuration space [*]
$ \Xi$ Pathway ensemble [*]
$ a$ A state that belongs to a superstate $ A$ [*]
$ \alpha $ Pathway nonlinearity index [*]
$ b$ A state that belongs to a superstate $ B$ [*]
$ \beta $ Energy barrier asymmetry index [*]
$ {\bf c}$ Eigenvector [*]
$ \det{\bf M}$ A determinant [33] (a scalar-valued function) of matrix $ {\bf M}$ [*]
$ d_i$ Integrated path length for atom $ i$; also, degree of node $ i$ [*]
$ d_i(j)$ Displacement of atom $ i$ between structures $ j-1$ and $ j$ [*]
$ e_{j,i}$ Directed edge that describes a transition from node $ i$ to node $ j$ [*]
$ \epsilon $ A parameter in LJ potential (the depth of the potential energy well) [*]
$ \varepsilon$ Small positive parameter [*]
$ \eta$ Number of atomic degrees of freedom [*]
$ f({\bf x})$ Objective function of a vector argument $ {\bf x}$ [*]
$ {\bf g}$ $ 3N$-dimensional gradient vector of the true potential [*]
$ \gamma $ Kurtosis of a distribution evaluated using moments about the mean [*]
$ \gamma'$ Kurtosis of a distribution evaluated using moments about the origin [*]
$ \gspar $ Spring gradient vector component parallel to the path [*]
$ \gsper $ Spring gradient vector component perpendicular to the path [*]
$ \widetilde{\bf g}$ $ 3N$-dimensional gradient vector of the spring potential [*]
$ \gtpar $ True gradient vector component parallel to the path [*]
$ \gtper $ True gradient vector component perpendicular to the path [*]
$ i,j,k$ Indices; range and meaning may vary depending on the context [*]
$ k_B$ Boltzmann's constant [*]
$ k_{j,i}$ Rate constant for transitions from state $ i$ to state $ j$ [*]
$ k_{spr}$ Spring force constant [*]
$ l(\xi)$ Length of path $ \xi$ [*]
$ \lambda$ Eigenvalue [*]
$ m$ Atomic mass [*]
$ m_n$ $ n$th moment of a distribution function about the mean [*]
$ m_n'$ $ n$th moment of a distribution function about the origin [*]
$ n$ Time parameter of a discrete-time stochastic process [*]
$ o(\,)$ $ f(n)=o\Bigl(g(n)\Bigr)$ means $ 0 \leq f(n) \leq c g(n)$ holds for all constants $ c > 0$[1]Otherwise known as an upper bound that is not asymptotically tight. [*]
$ {\bf p}$ Search direction vector [*]
$ \pi $ Path length asymmetry index [*]
$ {\bf r}_i(j)$ Three-dimensional Cartesian coordinates vector of atom $ i$ for structure $ j$ [*]
$ s$ Integrated path length [*]
$ \sigma $ A parameter in the LJ potential ( $ 2^{1/6}\sigma$ is the pair equilibrium separation) [*]
$ \tau$ $ 3N$-dimensional tangent vector [*]
$ \tau_i$ Mean waiting time in state $ i$ before escape [*]
$ t$ Time [*]
$ \delta t$ Time integration step [*]
$ \,^T$ A matrix or vector transpose [*]
$ \varpi$ Step size [*]
$ v_i$ $ i$th graph node [*]
$ w(u,v)$ Weight of the undirected edge connecting nodes $ u$ and $ v$ [*]
$ x_i$ $ i$th component of vector $ {\bf x}$ [*]
$ \xi$ A pathway [*]
$ {\bf x},{\bf y},{\bf z}$ Vectors; dimensionality and meaning may vary depending on the context [*]

next up previous contents
Next: INTRODUCTION Up: thesis Previous: LIST OF ABBREVIATIONS   Contents
Semen A Trygubenko 2006-04-10