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Nitric Oxide Reductase (NOR) is an integral membrane protein performing the reduction of NO to N2O. NOR is composed of two subunits: the large one (NorB) is a bundle of 12 transmembrane helices (TMH). It contains a b type heme and a binuclear iron site, which is believed to be the catalytic site, comprising a heme b and a non-hemic iron. The small subunit (NorC) harbors a cytochrome c and is attached to the membrane through a unique TMH. With the aim to perform structural and functional studies of NOR, we have immunized dromedaries with NOR and produced several antibody fragments of the heavy chain (VHHs, also known as nanobodies (TM)). These fragments have been used to develop a faster NOR purification procedure, to proceed to crystallization assays and to analyze the electron transfer of electron donors. BIAcore experiments have revealed that up to three VHHs can bind concomitantly to NOR with affinities in the nanomolar range. This is the first example of the use of VHHs with an integral membrane protein. Our results indicate that VHHs are able to recognize with high affinity distinct epitopes on this class of proteins, and can be used as versatile and valuable tool for purification, functional study and crystallization of integral membrane proteins.

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The development of more efficient power converters is the most important and challenging task for Power Electronics specialists. In the same time, many currently existing or yet to appear future applications require full mechanical independence between the transmitter and receiver of the electrical energy. This contactless form of energy transfer is the concern of the presented work. The work is based on the study of the Series Loaded Series Resonant converter which prove to be the best suitable for the contactless energy transfer. The work investigates the idealized Series Resonant Power Converter with the objective to find the best efficiency zones of operation. Generalized expressions obtained are original and useful. Based on the magnetic parameters of the loosely coupled transformer (magnetic link), the characteristics of the contactless power converter are described in approximated form. The approximation permits easier and faster calculation of the converter variables, thus predicting a shift of the maximum efficiency zone compared to the ideal converter case. The approximated form of the equations permitted to present a new instantaneous form of regulation which combines the frequency and pulse width modes which is free from the previously known defects. The method is based on calculating the energy portions supplied to the load during each half period. Measurements performed on industrial converters and on the laboratory experimental converter, confirm the predicted theoretically behaviour of the converter.

We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.

We prove higher-order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized Hölder spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of Böttcher and Silbermann and generalize their results for symbols in standard Hölder spaces.

A proposal for a reference curriculum for teaching Collaborative Networks at university level is introduced. This curriculum is based on the experience of the authors in teaching and disseminating corresponding concepts in the context of several international projects as well as on the findings of a survey conducted worldwide. A set of teaching units and the corresponding content are introduced. Guidelines for the application of the curriculum are given. A set of experiments and projects are also suggested as a support for the accompanying hands-on lab work. Finally, one concrete experience of application of the proposed curriculum is described.

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We find Fredholm criteria and a formula for the index of an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. These results ``localize'' the Gohberg-Krupnik Fredhohn theory with respect to the variable exponent.

We prove asymptotic formulas for block Toeplitz matrices with symbols admitting right and left Wiener-Hopf factorizations such that all partial indices are equal to some integer number. We consider symbols and Wiener-Hopf factorizations in Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Our results complement known formulas for Holder continuous symbols due to Bottcher and Silbermann.

We prove asymptotic formulas for determinants and traces of finite block Toeplitz matrices with symbols belonging to Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. The remainders in these formulas depend on the weights and go rapidly to zero for very smooth symbols. These formulas refine or extend some previous results by Szegö, Widom, Bottcher, and Silbermann.

We prove asymptotic formulas for Toeplitz determinants generated by functions with sequences of Fourier coefficients belonging to weighted Orlicz sequence classes. We concentrate our attention on the case of nonvanishing generating functions with nonzero Cauchy index.

We give a survey on generalized Krein algebras \(K_{p,q}^{\alpha,\beta}\) and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that \(K_{2,2}^{1/2,1/2}\) is a Banach algebra. Subsequently, Widom proved the strong Szeg\H{o} limit theorem for block Toeplitz determinants with symbols in \((K_{2,2}^{1/2,1/2})_{N\times N}\) and later two of the authors studied symbols in the generalized Krein algebras \((K_{p,q}^{\alpha,\beta})_{N\times N}\), where \(\lambda:=1/p+1/q=\alpha+\beta\) and \(\lambda=1\). We here extend these results to \(0<\lambda<1\). The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.

Suppose \(\Gamma\) is a Carleson Jordan curve with logarithmic whirl points, \(\varrho\) is a Khvedelidze weight, \(p:\Gamma\to(1,\infty)\) is a continuous function satisfying \(|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|\) for \(|\tau-t|\le 1/2\), and \(L^{p(\cdot)}(\Gamma,\varrho)\) is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with \(N\times N\) matrix piecewise continuous coefficients are Fredholm on \(L_N^{p(\cdot)}(\Gamma,\varrho)\).

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