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In this paper we study the generalization of the Brocard congurations. The denition of q-period broken line is given and then the new necessary and sucient condition about the existence of two conjugate Brocard points of the q􀀀period loop broken line is investigated. Based on the past research, a geometric method for the construction of all possible loop broken lines with conjugate Brocard points is developed.

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We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules dened over the polynomial ring Fq[\theta]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.

We study spectral and scattering properties of the Laplacian$H$^{(σ)}=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that$H$^{(σ)}is unitarily equivalent to the Neumann Laplacian$H$^{(0)}. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of$H$^{(σ)}under mild assumptions on σ.