On unitary interpolants and Fredholm infinite block Toeplitz matrices

Conditions for the existence and uniqueness of unitary n × n matrix valued functions f on the unit circle with prescribed Fourier coefficients f_j for j 0 are given (in terms of infinite block Hankel matrices based on the prescribed coefficients f₀,f₁, ) for a natural class of functions. A unitary function belongs to this class if and only if it admits a generalized factorization (in a sense which will be made precise in the paper) or equivalently if and only if any one (and hence both) of the two Toeplitz operators defined by the function are Fredholm. In particular this class includes all continuous unitary n × n matrix valued functions. It is shown that the nonnegative factorization indices of every such unitary f are uniquely determined by f₀,f₁, and formulas for them are given.     

Electrokinetic characterization of superparamagnetic nanoparticle–aptamer conjugates: design of new highly specific probes for miniaturized molecular diagnostics

With the view of designing new nanoparticle (NP)–aptamer conjugates and proving their suitability as biorecognition tools for miniaturized molecular diagnostics, new maghemite–silica core–shell NP–aptamer conjugates were characterized for the first time in terms of grafting rate and colloidal stability under electrophoretic conditions using capillary electrophoresis. After the grafting rate (on the order of six to 50) of the lysozyme-binding aptamer had been estimated, the electrophoretic stability and peak dispersion of the resulting oligonucleotide–NP conjugates were estimated so as to determine the optimal separation conditions in terms of buffer pH, ionic strength and nature, as well as temperature and electric field strength. The effective surface charge density of the NPs was close to zero for pH lower than 5, which led to some aggregation. The NPs were stable in the pH range from 5 to 9, and an increase in electrophoretic mobility was evidenced with increasing pH. Colloidal stability was preserved at physiological pH for both non-grafted NPs and grafted NPs in the 10–100 mM ionic strength range and in the 15–60 °C temperature range. A strong influence of the nature of the buffer counterion on NP electrophoretic mobility and peak dispersion was evidenced, thus indicating some interactions between buffer components and NP–aptamer conjugates. Whereas an electric field effect (50–900 V cm^?1) on NP electrophoretic mobility was evidenced, probably linked to counterion dissociation, temperature seems to have an appreciable effect on the zeta potential and aptamer configuration as well. This information is crucial for estimating the potentialities of such biorecognition tools in electrophoretic systems.     

Targeting an Interaction Between Two Disordered Domains by Using a Designed Peptide

Intrinsically disordered regions in proteins (IDRs) mediate many disease-related protein–protein interactions. However, the unfolded character and continuous conformational changes of IDRs make them difficult to target for therapeutic purposes. Here, we show that a designed peptide based on the disordered p53 linker domain can be used to target a partner IDR from the anti-apoptotic iASPP protein, promoting apoptosis of cancer cells. The p53 linker forms a hairpin-like structure with its two termini in close proximity. We designed a peptide derived from the disordered termini without the hairpin, designated as p53 LinkTer. The LinkTer peptide binds the disordered RT loop of iASPP with the same affinity as the parent p53 linker peptide, and inhibits the p53–iASPP interaction in vitro. The LinkTer peptide shows increased stability to proteolysis, penetrates cancer cells, causes nuclei shrinkage, and compromises the viability of cells. We conclude that a designed peptide comprising only the IDR from a peptide sequence can serve as an improved inhibitor since it binds its target protein without the need for pre-folding, paving the way for therapeutic targeting of IDRs.     

An abstract version of a limit theorem of Szegö

The theory of factorization with respect to chains of orthogonal projections is used to deduce an abstract version of a well-known limit theorem of Szegö. The theory is then specialized to matrix Fredholm operators and finally to matrix Wiener-Hopf operators. The well-known continuous analogues of Kac and Achiezer emerge as special cases.     

Criteria for the strong regularity of J-inner functions and γ-generating matrices

The class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A₂) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented.     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.