Continued fractions of Laurent series with partial quotients from a given set
1. Introduction. Van der Poorten and Shallit's paper [10] begins: " It is notorious that it is damnably difficult to explicitly compute the continued fraction of a quantity presented in some other form ". The quantity is usually presented either as a power series or as the root of a specific equation. There has been some success in the former case for continued fractions of real numbers, such as Euler's famous continued fraction for e [11] and more recent work [10] on " folded " continued fractions; however, other than the well-known results for quadratic real numbers, the only success with the latter has been for continued fractions of Laurent series rather than real numbers. In this paper we continue this line of investigation. We consider families of continued fractions of Laurent series whose partial quotients all lie in a given set. Following ideas of Baum and Sweet [2], we show that one may describe the zeros of certain collections of equations in terms of such families. The paragraphs that follow introduce the notation and definitions necessary to give a fuller description of our results. Let F q be the finite field with q elements and L q denote the field of formal Laurent series in x
Linear Complexity Profiles and Continued Fractions
The linear complexity, \( \mathcal{L} \)(sn), of a sequence sn is defined as the length of the shortest linear feedback shift-register (LFSR) that can generate the sequence. The linear complexity profile, L sn = L 1 L 2....L n, of s n (where L i = \( \mathcal{L} \)(si), 1 ≤ i ≤ n, denotes the linear complexity of first i digits of sn) provides better insight into the complexity of an individual sequence. By the increment sequence Δsn = Δ1Δ2...Δm in a linear complexity profile, L 1 L 2...L n, we mean the subsequence of positive numbers in the sequence L 1(L 2 − L 1) ... (L n − L n−1). For example, if L 1 ... L 5 = 02223, its increment sequence is \( \Delta _{s^3 } {\text{ = }}\Delta _1 \Delta _{2{\text{ }}} = {\text{21}} \). If we associated a sequence sn over F with an element S(z) in the field of Laurent series over F in the following way 
 
$$ s^n = s_1 s_2 ...s_n \Leftrightarrow {\text{S(}}z) = s_1 z^{ - 1} + s_2 z^{ - 2} + ... + s_n z^{ - n} $$ 
 
, S(z) can then be written as 
 
$$ S\left( z \right) = a_0 \left( z \right) + \frac{1} {{a_1 \left( z \right) + \frac{1} {{a_2 \left( z \right) + {\text{ }}_{_{.{\text{ }}_{.{\text{ }}_{. + \frac{1} {{a_k \left( z \right)}}} } } } }}}}, $$ 
 
where a i(z) ∈ F[z], the ring of polynomials in z over F, for all i ≥ 0. It will be shown that, for a sequence sn, the increment sequence Δsn of the linear complexity profile of sn is as follows. (1) If \( \sum\nolimits_{i = 1}^k {\deg } \left( {a_i (z)} \right) - \deg \left( {a_k (z)} \right) \leqslant n \), then \( \Delta _{s^n } = \deg \left( {a_1 (z)} \right)\deg \left( {a_2 (z)} \right)...\deg \left( {a_k (z)} \right) \). (2) If \(\sum\nolimits_{i = 1}^k {\deg \left( {a_i (z)} \right) - \deg \left( {a_k (z)} \right) > n} \), then \( \Delta _{s^n } = \deg \left( {a_1 (z)} \right)\deg \left( {a_2 (z)} \right)...\deg \left( {a_k (z)} \right) \), where \( k^1 = max\left\{ {j:2.\sum\nolimits_i^j {deg\left( {a_j (z) - \deg \left( {a_j (z} \right)} \right) \leqslant n} } \right\} \).
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...