Topics in Complex Analysis

This book is an outgrowth of lectures given at several occasions at the University of Goteborg and Chalmers University of Technology during the last ten years. Contrary to most introductory texts on complex analysis, it preassumes knowledge of basic analysis. This makes it possible to move rather quickly through the most fundamental material and to reach within a one-semester course some classical highlights such as Fatou theorems and some Nevanlinna theory, as well as more recent topics, for example the corona theorem and the H1-BMO duality.

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1. A Review of Vector and Matrix Algebra Using Subscript/Summation Conventions. 2. Differential and Integral Operations on Vector and Scalar Fields. 3. Curvilinear Coordinate Systems. 4. Introduction to Tensors. 5. The Dirac Delta-Function. 6. Introduction to Complex Variables. 7. Fourier Series. 8. Fourier Transforms. 9. Laplace Transforms. 10. Differential Equations. 11. Solutions to Laplace's Equation. 12. Integral Equations. 13. Advanced Topics in Complex Analysis. 14. Tensors in Non-Orthogonal Coordinate Systems. 15. Introduction to Group Theory. Appendix A. The Levi-Civita Identitiy. Appendix B. The Curvilinear Curl. Appendix C. The Double Integral Identity. Appendix D. Green's Function Solutions. Appendix E. Pseudovectors and the Mirror Test. Appendix F. Christoffel Symbols and Covariant Derivatives. Appendix G: Calculus of Variation. Errata List. Bibliography. Index.

This paper introduces the Spectral Riemann Conjecture (SRC), a novel reformulation of the classical Riemann Hypothesis (RH) through the lens of spectral breaking theory. The Riemann zeta function ζ(s), central to analytic number theory, possesses trivial zeros at negative even integers and non-trivial zeros conjectured to lie on the critical line Re(s) = 1/2. Building on the verified simplicity of the first 109 non-trivial zeros, the SRC proposes a two-step transformation: decoherence (D), which projects all non-trivial zeros onto Re(s) = 1/2, creating a zero of infinite multiplicity; followed by renormalization (R), which regularizes this multiplicity to one. The resulting function, spectral(s), resides in the spectral space—a domain of functions with real and simple zeros. The paper rigorously defines the operators D and R, demonstrates their action on ζ(s), and proves the convergence of spectral(s) using zeta and Hurwitz regularization techniques. Numerical simulations validate the SRC by approximating ζ(s) with finite products and applying D and R to yield spectral(s), whose zeros are located at {-2, -4, ..., 1/2}. This transformation preserves the infinite zeros, maintaining the arithmetic essence of ζ(s) while simplifying its non-trivial structure. The SRC offers a fresh perspective on RH, aligning with physical regularization analogies and suggesting potential applications in prime number distribution. Future work may extend these simulations and explore deeper implications within analytic number theory and mathematical physics.

Iterations of Rational Functions: Some Algebraic and Arithmetic Aspects

Feynman's prescription for a quantum simulator was to find a Hamitonian for a system that could serve as a computer. The Pólya-Hilbert conjecture proposed the demonstration of Riemann's hypothesis through the spectral decomposition of Hermitian operators. Here we study the problem of decomposing a number into its prime factors, N=xy, using such a simulator. First, we derive the Hamiltonian of the physical system that simulates a new arithmetic function formulated for the factorization problem that represents the energy of the computer. This function rests alone on the primes below sqrt[N]. We exactly solve the spectrum of the quantum system without resorting to any external adhoc conditions, also showing that it obtains, for x≪sqrt[N], a prediction of the prime counting function that is almost identical to Riemann's R(x) function. It has no counterpart in analytic number theory, and its derivation is a consequence of the quantum theory of the simulator alone.

A major theme in proof theory consists in classifying the proof strength of mathematical frameworks for reasoning about mathematics. The resulting phase transitions from provability to unprovability are interesting from the foundational as well as from the mathematical point of view. It is very surprising that during these investigations methods from analytic number theory, combinatorial probability theory, complex analysis and Ramsey theory enter the scene. In this abstract we present the underlying research program. We try to explain our intuition about the nature of phase transition results under discussion. It is our objective to show that there are intriguing interrelations between “usual” mathematics and proof theory. The abstract is intended to be accessible to a general mathematical audience. We therefore concentrate on basic examples and ideas without

Combinatorial group theory concerns itself with the theory of group presentations, that is, it deals with groups specified by a set of generators and corresponding defining relations. The theory has its origins and roots in topology and complex analysis and in particular in the theory of the fundamental groups of combinatorial cell complexes. Because of its nature and origins, combinatorial group theory comes into contact with, and uses, many different areas of mathematics: clearly algebra and topology as mentioned above, but also hyperbolic geometry via the study of the Cayley graph; pure mathematical logic through the study of various decision problems; and computer science through the study of computational aspects of group theory and rewriting systems. These last three areas have been the subject of increased interest and intense study during the 1990s. The ties with hyperbolic geometry have evolved into geometric group theory—especially the theory of the hyperbolic groups—by considering the groups themselves as geometric objects via the Cayley graph. The ties with computation and computer science have evolved into the theory of automatic groups, where rewriting and computability properties are paramount. Finally, there have been extensive studies on the nature of solutions of the various decision problems and on the logical underpinnings of the the whole theory. Relative to the latter there has been a great deal of work on the Tarski conjectures (see below) and their ties to purely group theoretical properties.KeywordsFree GroupCayley GraphFree ProductHyperbolic GroupMaximal Abelian SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

To prove: that the generalized Riemann hypothesis is true, namely, that all the zeta zeros of L -functions are balanced on the one-half number line, using the scale of the harmonic zeta function. Now that the Riemann hypothesis has been proven (in “Summa Characteristica and the Riemann Hypothesis: Scaling Riemann’s Mountain”, Journal of Interdisciplinary Mathematics, Vol. 11 (6) (December 2008)), it can be extended by analytical continuation to prove the generalized hypothesis concerning L-functions. The initial “gene” or generating (one half) principle that proves the Riemann hypothesis also proves that no L-series has a zero with real part on the complex plane other than one half. As a result of this proof, the Birch/Swinnerton-Dyer conjecture is also resolved. Finally, what impact, if any, do these proofs have on security codes based upon prime numbers?

In my talk I will attempt to draw some connections between complex analysis and computational geometry, particularly between conformal mappings, hyperbolic geometry, the medial axis and optimal meshing. The Riemann mapping theorem says that there is a conformal (angle preserving) map of the unit disk, DISK, to the interior Ω of any simple n-gon. How much work is needed to compute this map? [MR2671015] We can compute the conformal map f: DISK -> Ω to within error e in time O(n ⋅ log 1/e log log 1/e).The proof uses an iteration that converges quadratically to f, but it needs a good initial guess that is close to the correct answer. The medial axis allows us to quickly construct a map Ω -> DISK that is close to Riemann's map with precise estimates. This gives our starting point, and the time estimates in the theorem depend on time needed to compute the medial axis (linear by work of Chin, Snoeyink and Wang).Conversely, the proof of Theorem 1 uses ideas from analysis that may be of interest in CG. For example, the dome of a planar domain Ω is the surface in R+3 given by the upper envelope of all hemispheres whose base on R2 is a medial axis disk of Ω. Domes arise in the study of 3-manifolds, and a fundamental result of Dennis Sullivan about convex sets in hyperbolic 3-space lies at the heart of Theorem 1. The proof of Theorem 1 also introduces the thick-thin decomposition of a polygon, inspired by the thick-thin decomposition of a hyperbolic manifold. The thin parts correspond to long, narrow pieces of the polygon (but not in the obvious way; the precise definition uses a conformal invariant called extremal length). This decomposition plays in important role in proving: [MR2671014] Any simple n-gon has a O(n) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O.The sharp upper bound is due to Bern and Eppstein; the lower bound is the novel part here. The result is also true for PSLGs with at most n edges and vertices, although simple examples show the O(n) must be replaced by O(n2): [Bishop-Quad] Every PSLG has an O(n2) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O. Only O(n/e) angles satisfy |θ-90O| > e, for any e >0.

Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry. This book redresses the balance and provides an overview of the geometry of both the complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.

Factorization in Krein Spaces

Mechanics of liquids, an elementary text in hydraulics and fluid mechanics: by Ralph W. Powell. 271 pages, illustrations, tables, 16 × 24 cms. New York, The Macmillan Company, 1940. Price $3.50

The Riemann mapping theorem has been said by some to be the greatest theorem of the nineteenth century. The entire concept of the theorem is profoundly original, and its proof introduced many new ideas. Certainly normal families and the use of extremal problems in complex analysis are just two of the important techniques that have grown out of studies of the Riemann mapping theorem.

In the last two decades a number of different types of wavelets transforms have been introduced in various areas of mathematics, natural sciences, and technology. These transforms can be generated by means of a uniform principle based on the machinery of harmonic analysis. In this way we pass from the affine group to the wavelet transforms, from the Heisenberg group to the Gábor transform. Taking the congruences of the hyperbolic geometry and using the same method we introduced the concept of hyperbolic wavelet transforms (HWT). These congruences can be expressed by Blaschke functions, which play an eminent role not only in complex analysis but also in control theory. Therefore we hope that the HWT will become an adequate tool in signal and system theories. In this paper we give an overview on some results and applications concerning HWT.KeywordsWaveletsHyperbolic geometryRational systemsBlaschke functionsSystem identificationSignal processing