Table of contents

CONTENTS Introduction § 1. A new inversion formula for the Stieltjes transforms of spectral functions § 2. An inequality for the difference of the Stieltjes transforms of spectral functions § 3. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the semicircle law § 4. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the Marchenko-Pastur density § 5. The asymptotics of sums of smoothed densities of eigenvalue distributions of symmetric random matrices whose diagonal elements have non-zero mathematical expectations § 6. The asymptotics of Stieltjes transforms of spectral functions of symmetric random matrices whose elements have different dispersions § 7. The asymptotics of Stieltjes transforms of spectral functions of empirical covariance matrices § 8. Limit theorems for boundary eigenvalues of symmetric random matrices § 9. Limit theorems for boundary singular eigenvalues of random matrices References

CONTENTS Chapter I. Evaluation in algebraic systems I.1. The concepts of valuation, global validity, and transfer theorem I.2. Heyting algebras and Stone spaces I.3. Definition and examples of valuations I.4. The connection between deducibility and global validity I.5. The connection between validity and global validity I.6. Cancellation of idempotents and transfer theorems in the case of rings. Expressibility of global validity I.7. Universal valuation. The notion of sheaf on a Heyting algebra Chapter II. Localizations and valuations II.1. Local axiomatizability of a class of algebraic systems II.2. Valuation and model completeness. Boolean absoluteness II.3. Macintyre's problem: a model companion of a locally axiomatizable class II.4. A model companion of a class of localizations. The completeness of the theory of a locally axiomatizable class II.5. The transfer of a local theory into a locally axiomatizable class Chapter III. A natural translation of classical into intuitionistic theory for algebras with metric Chapter IV. Heyting completion of locally compact topological spaces Appendix. Evaluation in Boolean algebras References

CONTENTS Introduction § 1. Green-Stieltjes integrals § 2. Some classes of analytic functions § 3. Periods of harmonic functions conjugate to Green-Stieltjes integrals § 4. Construction of non-singular period matrices § 5. The Schwarz kernel § 6. The Blaschke factor § 7. The Blaschke product and its convergence § 8. Boundary behaviour of the Blaschke product § 9. Division by the Blaschke product § 10. Factorization References