Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

http://www.journalofinfection.com/article/S016344532200247X/pdf

SARS-CoV-2 Omicron sublineage BA.2 replaces BA.1.1: Genomic surveillance in Japan from September 2021 to March 2022

http://www.journalofinfection.com/article/S0163445322002511/pdf

Remdesivir use in COVID-19 patients might predispose bacteremia, matched case-control analysis

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9033627

High rate of reinfection with the SARS-CoV-2 Omicron variant

Heavy-tailed fractional Pearson diffusions

Aim To investigate how age, sex, and comorbidities affect the survival of hospitalized coronavirus disease 2019 (COVID-19) patients. Methods We retrospectively analyzed the records of 4014 consecutive adults hospitalized for COVID-19 in a tertiary-level institution from March 2020 to March 2021. Results The median age was 74 years. A total of 2256 (56.2%) patients were men. The median Charlson-comorbidity-index (CCI) was 4 points; 3359 (82.7%) patients had severe or critical COVID-19. A significant interaction between age, sex, and survival (P < 0.05) persisted after adjustment for CCI. In patients <57 years, male sex was related to a favorable (odds ration [OR] 0.50, 95% confidence interval [CI] 0.29-0.86), whereas in patients ≥57 years it was related to an unfavorable prognosis (OR 1.19, 95% CI 1.04-1.37). Comorbidities associated with inferior survival independently of age, sex, and severe/critical COVID-19 on admission were chronic heart failure, atrial fibrillation, acute myocardial infarction, acute cerebrovascular insult, history of venous thromboembolism, chronic kidney disease, major bleeding, liver cirrhosis, mental retardation, dementia, active malignant disease, metastatic malignant disease, autoimmune/rheumatic disease, bilateral pneumonia, and other infections on admission. Conclusion Among younger patients, female sex might lead to an adverse prognosis due to undisclosed reasons (differences in fat tissue distribution, hormonal status, and other mechanisms). Patient subgroups with specific comorbidities require additional considerations during hospital stay for COVID-19. Future studies focusing on sex differences and potential interactions are warranted.

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The associations of age, sex, and comorbidities with survival of hospitalized patients with coronavirus disease 2019: data from 4014 patients from a tertiary-center registry

We propose a refined version of the stochastic SEIR model for epidemic of the new corona virus SARS-Cov-2, causing the COVID-19 disease, taking into account the spread of the virus due to the regular infected individuals (transmission coefficient β ), hospitalized individuals (transmission coefficient l β , l > 0 ) and superspreaders (transmission coefficient β ′ ). The model is constructed from the corresponding ordinary differential model by introducing two independent environmental white noises in transmission coefficients for above mentioned classes - one noise for infected and hospitalized individuals and the other for superspreaders. Therefore, the model is defined as a system of stochastic differential equations driven by two independent standard Brownian motions. Existence and uniqueness of the global positive solution is proven, and conditions under which extinction and persistence in mean hold are given. The theoretical results are illustrated via numerical simulations.

A two diffusion stochastic model for the spread of the new corona virus SARS-CoV-2.

https://doi.org/10.1016/j.ajem.2022.04.051

Ivan Papić

Papers

Remdesivir use in COVID-19 patients might predispose bacteremia, matched case-control analysis

Heavy-tailed fractional Pearson diffusions

The associations of age, sex, and comorbidities with survival of hospitalized patients with coronavirus disease 2019: data from 4014 patients from a tertiary-center registry

A two diffusion stochastic model for the spread of the new corona virus SARS-CoV-2.

Atrial fibrillation in COVID-19 patients receiving remdesivir, matched case-control analysis