By Tarantello G.

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3 Special State-Space Notation This book Other sources x x~ x x xk xk x^ x^ k À x^ k x_ xk E hx i x x^ kjkÀ1 x^ kÀ x^ kjk x^ k xt dx=dt De®nition of Notational Usage Vector The kth component of the vector x The kth element of the sequence . . ; xkÀ1 ; xk ; xk1 ; . . 4 Common Notation for Array Dimensions Dimensions Symbol x w u z v Vector Name System state Process noise Control input Measurement Measurement noise Dimensions Symbol n r s ` ` F G Q H R Matrix Name State transition Process noise coupling Process noise covariance Measurement sensitivity Measurement noise covariance Row Column n n r ` ` n r r n ` Common Notation for Array Dimensions.

It combines several ideas due to Ward [233], including setting the algorithm parameters to meet a prespeci®ed error bound. It combines PadeÂ approximation with a technique called ``scaling and squaring'' to maintain approximation errors within prespeci®ed bounds. 1 PadeÂ Approximation of the Matrix Exponential PadeÂ approximations. These approximations of functions by rational functions (ratios of polynomials) date from a 1892 publication [206] by H. 9 They have been used in deriving solutions of differential equations, including Riccati equations10 [69].

1 ON THE NOTATION USED IN THIS BOOK Symbolic Notation The fundamental problem of symbolic notation, in almost any context, is that there are never enough symbols to go around. There are not enough letters in the Roman alphabet to represent the sounds of standard English, let alone all the variables in Kalman ®ltering and its applications. As a result, some symbols must play multiple roles. In such cases, their roles will be de®ned as they are introduced. It is sometimes confusing, but unavoidable.

### An Harnak inequality for Liouville-type equations with singular sources by Tarantello G.

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