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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 λ eigenvalue of a matrix, or a Lyapunov exponent α , β real, imaginary parts of an eigenvalue λ = α + β 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 φ t ( x ) solution of x = f ( x ) with x ( ) = x φ 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 Γ ( t ) a parametrized curve, Γ : R → R n 404 guide to commonly used notation 405 Symbol Usual Meaning n a normal vector to a parametrized curve r , θ radius and angle (polar coordinates) μ a bifurcation parameter σ Lyapunov number (associated with Hopf bifurcations) τ 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 γ feedback gain parameter in the tdas algorithm L linear operator, or length of the interval [ 0, L ] φ ( x ) initial condition for a pde ∂ Ω boundary of the domain Ω κ positive-valued diffusion constant (heat equation) c wave speed (wave/transport equations) ψ ( x ) initial velocity (wave equation) S φ convolution of functions S and φ S ( x , t ) (one-dimensional) heat kernel φ odd , φ even odd, even extensions of a function φ X ( x ) , Y ( y ) , T ( t ) separated solutions of pde s (separation of variables) R ( r ) , Θ ( θ ) separated solutions of pde s (polar coordinates) β abbreviation for square roots of eigenvalues λ 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 Δ Laplace operator v ( r ) a radial solution of Laplace’s or Poisson’s equation h ( θ ) boundary condition for Laplace’s equation on a disc References [ 1 ] G. Bachman, L. Narici, and E. Beckenstein,] G....

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