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**Unformatted text preview: **Guide to Commonly Used Notation Symbol Usual Meaning R the set of real numbers R n n-dimensional Euclidean space t independent variable (time) u , v vectors consisting of real numbers x , y vectors consisting of dependent variables x , y vectors consisting of initial conditions x ( ) , y ( ) x * , y * equilibrium solution of a system of ode s f a vector consisting of functions A , M , P square n n matrices D a diagonal n n matrix N a nilpotent n n matrix tr ( A ) trace of a square matrix A det ( A ) determinant of a square matrix A eigenvalue of a matrix, or a Lyapunov exponent , real, imaginary parts of an eigenvalue = + i E s , E u , E c stable, unstable, and center subspaces W s , W u , W c stable, unstable, and center manifolds A B direct sum of subspaces A and B span { v 1 , v 2 , . . . v n } span of vectors v 1 , v 2 , . . . v n t ( x ) solution of x = f ( x ) with x ( ) = x t flow of a system of ode s f : R n R m a function from R n into R m f gradient of a function from R n into R J f Jacobian matrix of a function from R n into R m u v dot product of vectors u and v v 2 Euclidean norm (length) of a vector v in R n u- v 2 Euclidean distance between points u and v in R n B ( x , ) open ball of radius centered at x V ( x ) a Lyapunov function ( t ) a parametrized curve, : R R n 404 guide to commonly used notation 405 Symbol Usual Meaning n a normal vector to a parametrized curve r , radius and angle (polar coordinates) a bifurcation parameter Lyapunov number (associated with Hopf bifurcations) time delay (for delay differential equations) x ( a + ) the right-hand limit lim t a + x ( t ) x ( a- ) the left-hand limit lim t a- x ( t ) x * a fixed-point of a first-order difference equation Sf ( x ) Schwarzian derivative of a function f : R R feedback gain parameter in the tdas algorithm L linear operator, or length of the interval [ 0, L ] ( x ) initial condition for a pde boundary of the domain positive-valued diffusion constant (heat equation) c wave speed (wave/transport equations) ( x ) initial velocity (wave equation) S convolution of functions S and S ( x , t ) (one-dimensional) heat kernel odd , even odd, even extensions of a function X ( x ) , Y ( y ) , T ( t ) separated solutions of pde s (separation of variables) R ( r ) , ( ) separated solutions of pde s (polar coordinates) abbreviation for square roots of eigenvalues A n , B n Fourier coefficients f , f L 2 different types of norms of the function f f , g inner product of functions f and g S N ( x ) partial sum of an infinite series of functions f ( x- ) , f ( x + ) left and right-hand limits of f ( x ) as x x f ( x- ) , f ( x + ) left and right-hand derivatives of f ( x ) as x x Laplace operator v ( r ) a radial solution of Laplaces or Poissons equation h ( ) boundary condition for Laplaces equation on a disc References [ 1 ] G. Bachman, L. Narici, and E. Beckenstein,] G....

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