phd thesis in complex analysis

Complex Analysis

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• 2019 August

Advancements in Complex Analysis

From Theory to Practice

• © 2020
• Daniel Breaz 0 ,
• Michael Th. Rassias 1

“1 Decembrie 1918”, University of Alba Iulia, Alba Iulia, Romania

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Institute of Mathematics, Institute for Advanced Study, University of Zurich, Moscow Institute of Physics and Technology, Program in Interdisciplinary Studies, Institutskiy per, d.9, 141700, Dolgoprudny, Moscow, Russia, 1 Einstein Dr, Princeton, NJ, USA, Zurich, Switzerland

• Covers state-of-the-art research and problem solving techniques for a broad spectrum of topics in complex analysis
• Topics covered range from pure to applied and interdisciplinary
• Valuable source for graduate students and researchers in mathematics as well as others benefitting from the interdisciplinarity of the topics covered

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About this book

The contributions to this volume are devoted to a discussion of state-of-the-art research and treatment of problems of a wide spectrum of areas in complex analysis ranging from pure to applied and interdisciplinary mathematical research. Topics covered include:  holomorphic approximation, hypercomplex analysis, special functions of complex variables, automorphic groups, zeros of the Riemann zeta function, Gaussian multiplicative chaos, non-constant frequency decompositions, minimal kernels, one-component inner functions, power moment problems, complex dynamics, biholomorphic cryptosystems, fermionic and bosonic operators. The book will appeal to graduate students and research mathematicians as well as to physicists, engineers, and scientists, whose work is related to the topics covered.

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Computable Complex Analysis

Some Problems

Carleson Measures and Toeplitz Operators

• Complex Analysis
• Fourier analysis
• analytic number theory
• Riemann zeta function
• Riemann Hypothesis
• algebraic geometry
• theoretical Physics

Table of contents (14 chapters)

Front matter, a theory on non-constant frequency decompositions and applications.

• Qiuhui Chen, Tao Qian, Lihui Tan

One-Component Inner Functions II

• Joseph Cima, Raymond Mortini

Biholomorphic Cryptosystems

• Nicholas J. Daras

Third-Order Fermionic and Fourth-Order Bosonic Operators

• Chao Ding, Raymond Walter, John Ryan

Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan

• John Erik Fornæss, Franc Forstnerič, Erlend F. Wold

A Potapov-Type Approach to a Truncated Matricial Stieltjes-Type Power Moment Problem

• Bernd Fritzsche, Bernd Kirstein, Conrad Mädler, Tatsiana Makarevich

Formulas and Inequalities for Some Special Functions of a Complex Variable

• Arcadii Z. Grinshpan

On the Means of the Non-trivial Zeros of the Riemann Zeta Function

• Mehdi Hassani

Minimal Kernels and Compact Analytic Objects in Complex Surfaces

• Samuele Mongodi, Giuseppe Tomassini

On the Automorphic Group of an Entire Function

• Ronen Peretz

Integral Representations in Complex Analysis: From Classical Results to Recent Developments

• Michael Range

On the Riemann Zeta Function and Gaussian Multiplicative Chaos

• Eero Saksman, Christian Webb

Some New Aspects in Hypercomplex Analysis

• Wolfgang Sprößig

Some Connections of Complex Dynamics

• Alexandre De Zotti

Editors and Affiliations

Daniel Breaz

Michael Th. Rassias

About the editors

Michael Th. Rassias   is a Latsis Foundation Senior Fellow at the University of Zürich, a visiting researcher at the Institute for Advanced Study, Princeton, as well as a visiting Assistant Professor at the Moscow Institute of Physics and Technology. He obtained his PhD in Mathematics from ETH-Zürich in 2014. During the academic year 2014-2015, he was a Postdoctoral researcher at the Department of Mathematics of  Princeton University and the Department of Mathematics of ETH-Zürich, conducting research at Princeton. While at Princeton, he prepared with John F. Nash, Jr. the volume  "Open Problems in Mathematics", Springer, 2016. He has received several awards in mathematical problem-solving competitions, including a Silver medal at the International Mathematical Olympiad of 2003 in Tokyo. In 2014 he was awarded with the Notara Prize by the Academy of Athens. He has authored and edited several books with Springer and has published numerous research papers. His current research interests lie in mathematical analysis, analytic number theory, zeta functions, the Riemann Hypothesis, approximation theory, functional equations and analytic inequalities.

Daniel Breaz has been working in the academic field since 1998 and currently, he is a professor doctor at the "1 Decembrie 1918" University of Alba Iulia and a PhD coordinator in the field of mathematical studies at the “Babes-Bolyai” University of Cluj-Napoca. He graduated from the Faculty of Mathematics and Computer Science, the mathematical section, at "Babeş-Bolyai" University of Cluj-Napoca where he also earned his Master’s degree. In 2002 he earned his PhD in Mathematics, specialization Mathematical Analysis. Daniel Breaz served as Minister of Culture and National Identity and as Interim Minister of Education but he is also known by his prodigious academic activity. He has published over 230 articles in several specialized publications worldwide, he has been an author or acontributor to several science books and participated in dozens of conferences and specialized seminars. His particular interests and research focus are on mathematical analysis, univalent functions, integral operators and geometric theory of functions.. Among the prizes obtained are the Nishiwaki Prize for The Research of Univalent Function Theory, May 21 of 2010, Kyoto, Japan or the "Academic Merit" Medal, awarded in 2016 by the Romanian Academy, Cluj-Napoca branch.

Bibliographic Information

Book Title : Advancements in Complex Analysis

Book Subtitle : From Theory to Practice

Editors : Daniel Breaz, Michael Th. Rassias

DOI : https://doi.org/10.1007/978-3-030-40120-7

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer Nature Switzerland AG 2020

Hardcover ISBN : 978-3-030-40119-1 Published: 13 May 2020

Softcover ISBN : 978-3-030-40122-1 Published: 13 May 2021

eBook ISBN : 978-3-030-40120-7 Published: 12 May 2020

Edition Number : 1

Number of Pages : VIII, 536

Number of Illustrations : 7 b/w illustrations, 2 illustrations in colour

Topics : Several Complex Variables and Analytic Spaces , Complex Systems , Quantum Computing , Functions of a Complex Variable , Difference and Functional Equations

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UCI Mathematics

Ph.d program, doctor of philosophy (phd) in mathematics.

To earn a PhD in Mathematics one must satisfy the following requirements:

• Completion of all required coursework
• Completion of required written examinations
• Completion of Advancement to Candidacy Oral Examination & Graduate Division paperwork
• Completion of Teaching Experience
• Submission of Doctoral Dissertation & Graduate Division paperwork

When accepted into the doctoral program, the student embarks on a program of formal courses, seminars, and individual study courses to prepare for the Ph.D. written examinations, advancement to candidacy oral examination, and dissertation.

Upon entering the program, students are expected to take Math 210 (Real Analysis), Math 220 (Complex Analysis) and Math 230 (Algebra), which must be passed with a grade of B or better.  Students must complete these sequences by the end of the second year.

By the start of the second year , students must achieve at least two passes at the M.S. level among four exams in Real Analysis, Complex Analysis, Algebra and Applied Mathematics.  

By the start of the third year , students must achieve at least two passes at the Ph.D. level among four exams in Real Analysis, Complex Analysis, Algebra and Applied Mathematics.

To satisfy the exam requirements, students may take the Comprehensive Exam (offered in the Spring of every year) or the Qualifying Exams (offered before the start of the fall quarter) in these areas. Students may not attempt to take an exam in a particular subject area more than 3 times .  A student who passes a Qualifying examination prior to taking the corresponding course will be exempted from taking the course.

Please Note: Corresponding qualifying exam coursework, MATH 210,220, & 230 cannot be used to satisfy both exam and coursework requirements (i.e. you can’t ‘double dip’).

Some students may require additional background prior to entering Math 210.  This will be determined by assessment prior to the start of the students’ first year by the Vice Chair for Graduate Studies, upon consultation with the graduate studies committee.  Such students will be directed into Math 205 during their first year.  These students may pass one Comprehensive Exam in the area of Analysis in lieu of achieving a M.S. pass on the Qualifying Exam, which must be satisfied prior to the start of the students’ second year. The Comprehensive Exam in Analysis will be offered once per year in the Spring quarter.

By the end of the second year, students must declare a major specialization from the following areas:

• Applied & Computational Mathematics
• Geometry & Topology
• Probability

Students are required to take two series of courses from their chosen area (students who later decide to change their area must also take two series of courses from the new area).  Additionally, all students must take two series of courses outside their declared major area of specialization.  Special topics courses within certain areas of specialization and courses counted toward the M.S. degree, (other than MATH 205), will count toward the fulfillment of the major specialization requirement.

By the beginning of their third year, students must have an advisor specializing in their major area.  With the advisor's aid, one should begin to form a committee for the Advancement to Candidacy PhD oral examination.  This committee will be approved by the Department on behalf of the Dean of Graduate Studies and the Graduate Council and will have five faculty members.  At least one (and at most two), of the committee members must be faculty from outside the Department.  Before the end of the third year, students must have a written proposal, approved by their committee, for the Advancement to Candidacy oral examination.  The proposal should explain the role of at least two series of courses from the student's major area of specialization that will be used to satisfy the Advancement to Candidacy requirements.  The proposal should also explain the role of additional research reading material as well as providing a plan for investigating specific topics under the direction of the student's advisor(s).  Only one of the core courses, MATH 210ABC, 220ABC, and 230ABC may count for the course requirement for Advancement to Candidacy Examinations.

After one meets these requirements, the Graduate Studies Committee recommends to the Dean of Graduate Studies the advancement to candidacy for the PhD. degree.  Students should advance to candidacy by the beginning of their fourth year .  After advancing to candidacy, a student is expected to be fully involved in research toward writing his or her PhD dissertation.  Ideally, a student should keep in steady contact/interaction with their doctoral committee.  Teaching experience and training is an integral part of the PhD program.  All doctoral students are expected to participate in the Department's teaching program, unless otherwise communicated during the admissions process.

The candidate must demonstrate independent, creative research in Mathematics by writing and defending a dissertation that makes a new and valuable contribution to mathematics in the candidate's area of concentration.  Upon advancement to candidacy a student must form a thesis committee, a subcommittee of the advancement examination committee, consisting of at least three total faculty members, chaired by the student's advisor.  The committee guides and supervises the candidate's research, study, and writing of the dissertation; participates in or attends the oral defense of the dissertation; and recommends that the PhD be conferred upon approval of the doctoral dissertation.

The normal time for completion of the PhD is six years , and the maximum time permitted is seven years (please note the department may only provide financial support for a maximum of six years ). 

Completion of the PhD degree must occur within 9 quarters of Advancement to PhD candidacy.

Areas of Specialization and Their Corresponding Advancement to Candidacy Courses

PhD students will choose one specialization from the following six areas, as offered by the Mathematics Department, which determines coursework requirements.  Each area of specialization will have a core course, which the Department will do its best to offer each year.  The department will offer other courses every other year, or more frequently depending on student demands and other department priorities.

Algebra : Math 230ABC (core), Math 232ABC, Math 233ABC, 234ABC, 235ABC, 239ABC

Analysis : Math 210ABC (core), Math 220ABC (core), Math 211ABC, Math 260ABC, Math 295ABC, Math 296

Applied & Computational Mathematics: Math 290ABC (core), Math 225ABC, Math 226ABC, Math 227AB, Math 291ABC, Math 295ABC

Geometry & Topology: Math 218ABC (core), Math 222ABC, Math 240ABC, Math 245ABC, Math 250ABC

Logic : Math 280ABC (core), Math 281ABC, Math 282ABC, Math 285ABC

Probability : Math 210ABC, Math 211ABC, Math 270ABC, Math 271ABC, Math 272ABC, Math 274

*PhD Requirements Summarized*

By the beginning of the 2nd year: Pass at the MS level two exams in real analysis, complex analysis, algebra or applied math.

By end of the 2nd year: (1) Declare a major specialization; (2) complete the course series 210ABC, 220ABC, 230ABC.

By the beginning of the 3rd year : (1) Pass at the PhD level two qualifying exams in real analysis, complex analysis, algebra or applied math; (2) Select an advisor specialist in the major area and form a committee for the Advancement to Candidacy oral exam.

Before the end of the 3rd year: (1) Have a written proposal, approved by the committee, for the PhD Advancement to Candidacy examination.

By the beginning of the 4th year: (1) Advanced to Candidacy at the PhD level; (2) form a thesis committee (that is, a subcommittee of the advancement examination committee)

Completion of the PhD:  Average completion time is 5.8 years ; maximum time permitted is seven years . The Department will not financially support students past their sixth year in the PhD program.  Completion of the PhD degree must occur within 9 quarters (three years) of advancement to PhD candidacy.

Graduate Program in Mathematical and Computation Biology (MCSB)

The graduate program in Mathematical, Computational Systems Biology (MCSB) is designed to meet to meet the interdisciplinary training challenges of modern biology and function in concert with selected department programs, including the Ph.D. in Mathematics.

http://mcsb.uci.edu/

Complex Analysis

Learning outcomes.

Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.

Essential results: Power series, integration along curves, Goursat theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle.

Entire functions: Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem.

The gamma and zeta functions: Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function.

Conformal mappings: Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem.

Elliptic Functions: Liouville's Theorems, poles and zeros of elliptic functions, Weierstrass elliptic functions.

For more detailed information visit th Math 532 Wiki page.

We have 57 Mathematics (complex analysis) PhD Projects, Programmes & Scholarships

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Mathematics (complex analysis) PhD Projects, Programmes & Scholarships

Connections between numerical analysis of differential equations and machine learning, phd research project.

PhD Research Projects are advertised opportunities to examine a pre-defined topic or answer a stated research question. Some projects may also provide scope for you to propose your own ideas and approaches.

Funded PhD Project (UK Students Only)

This research project has funding attached. It is only available to UK citizens or those who have been resident in the UK for a period of 3 years or more. Some projects, which are funded by charities or by the universities themselves may have more stringent restrictions.

UCL SECReT: The International Training Centre for Security and Crime Research Degrees

Funded phd programme (uk students only).

Some or all of the PhD opportunities in this programme have funding attached. It is only available to UK citizens or those who have been resident in the UK for a period of 3 years or more. Some projects, which are funded by charities or by the universities themselves may have more stringent restrictions.

EPSRC Centre for Doctoral Training

EPSRC Centres for Doctoral Training conduct research and training in priority areas funded by the UK Engineering and Physical Sciences Research Council. Potential PhD topics are usually defined in advance. Students may receive additional training and development opportunities as part of their programme.

New frontiers in extreme data analysis

Competition funded phd project (uk students only).

This research project is one of a number of projects at this institution. It is in competition for funding with one or more of these projects. Usually the project which receives the best applicant will be awarded the funding. The funding is only available to UK citizens or those who have been resident in the UK for a period of 3 years or more. Some projects, which are funded by charities or by the universities themselves may have more stringent restrictions.

PhD in Psychoacoustic modelling for complex soundscapes

Stability control of hydraulic foot-legged robot in complex environment, funded phd project (students worldwide).

This project has funding attached, subject to eligibility criteria. Applications for the project are welcome from all suitably qualified candidates, but its funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.

Human emotion analysis and recognition for improving trusted human-robot interaction. Main project focus: AI and Robotics

Self-funded phd students only.

This project does not have funding attached. You will need to have your own means of paying fees and living costs and / or seek separate funding from student finance, charities or trusts.

Fully Funded PhD Positions at the IMT School for Advanced Studies Lucca

Funded phd programme (students worldwide).

Some or all of the PhD opportunities in this programme have funding attached. Applications for this programme are welcome from suitably qualified candidates worldwide. Funding may only be available to a limited set of nationalities and you should read the full programme details for further information.

Italy PhD Programme

An Italian PhD usually takes 3-4 years and consists of some taught units as well as research towards your thesis. This will be examined at a public defence, rather than a private viva voce. Some programmes are taught in English.

Active particles in complex geometry environments

Competition funded phd project (students worldwide).

This project is in competition for funding with other projects. Usually the project which receives the best applicant will be successful. Unsuccessful projects may still go ahead as self-funded opportunities. Applications for the project are welcome from all suitably qualified candidates, but potential funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.

Numerical Algorithms and Analysis for Deterministic and Stochastic Systems

Ontological modelling for data analysis, neural networks for complex dynamical systems, novel causal models for multivariate functional data, projects in mathematical systems and control theory, modelling the impact of diagnostic pathways in cancer and cardiovascular disease - university of swansea (part of health data research uk’s big data for complex disease driver programme), unravelling the genetic & environmental basis of chronic pain.

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2023   2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Reham Alahmadi - Asymptotic Study of Toeplitz Determinants with Fisher-Hartwig Symbols and Their Double-Scaling Limits

Anne Sophie Rojahn –  Localised adaptive Particle Filters for large scale operational NWP model

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

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Topics in Complex Analysis

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• Language: English
• Publisher: De Gruyter
• Copyright year: 2022
• Audience: Researchers in mathematics and graduate students.
• Front matter: 16
• Main content: 276
• Illustrations: 30
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• Keywords: Komplexe Analyse ; Zahlentheorie ; komplexe Variable
• Published: October 24, 2022
• ISBN: 9783110757828
• ISBN: 9783110757699

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Complex Analysis (MAST30021)

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Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include:the topology of the complex plane; convergence of complex sequences and series; holomorphic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

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At the completion of this subject, students should understand the concepts of holomorphic function and contour integral and should be able to:

• apply the Cauchy-Riemann equations
• use the complex exponential and logarithm
• apply Cauchy’s theorems concerning contour integrals
• apply the residue theorem in a variety of contexts
• understand theoretical implications of Cauchy’s theorems such as the maximum modulus principle, Liouville’s Theorem and the fundamental theorem of algebra

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In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

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Work of Ted Kaczynski

I hope this question is not too crazy sounding, but I was wondering if anyone is familiar with the work of Ted Kaczynski (or even has cited/used it before). After reading in Lars Ahlfors' Complex Analysis and Serge Lang's book by the same name, I became interested in some of the historic results in complex analysis.

I know that Kaczynski did work in the field of complex analysis and specifically geometric function theory. From what I have gathered, he was actually rather brilliant as a research mathematician. What in specific did he research and how are his results used today? (If at all).

Again, I do not want to know about his political views or his history as the Unabomber. I am only interested in the utility of his mathematics.

• complex-analysis
• soft-question
• math-history

• 3 $\begingroup$ There is something of a bibliography here with summaries. I can't say anything about their quality. A lot of it is in AMS journals and is hence relatively easy to track down online to read for yourself. This kind of thing seems somewhat different from the sort of analysis done in introductory complex analysis texts, though. You can still see his name on the Sumner Myers Award for Best Thesis plaque on the first floor of East Hall in Ann Arbor and my understanding is that he got a job at Berkeley after that, so he was probably pretty good. $\endgroup$ –  Hoot Commented Jun 2, 2015 at 23:30
• 1 $\begingroup$ Assistant professorship at Cal for a couple of years, which he then resigned. $\endgroup$ –  Brian Tung Commented Jun 3, 2015 at 0:14
• 1 $\begingroup$ There's a nice description here, in particular Andres's answer: mathoverflow.net/questions/49395/… $\endgroup$ –  Alex R. Commented Jun 3, 2015 at 2:49
• 1 $\begingroup$ I wonder if complex analysts are somehow predisposed to murder. See André Bloch $\endgroup$ –  Jair Taylor Commented Jun 3, 2015 at 3:15
• 8 $\begingroup$ @Vladhagen: Definitely, Galois was a terrible shooter :). $\endgroup$ –  Alex R. Commented Jun 3, 2015 at 4:36

T. Kaczynski has published 6 papers in 4 years (1965-69) which can be considered as a good beginning of a promising career. According to Mathscinet database, one of his papers was cited 4 times (2 of them by himself), and three were cited one time each (two by himself). Mathscinet does not count all citations but most of them.

I am somewhat familiar with the subject of his work. I would not call it "brilliant", but qualify it as average PhD graduating from a good university.

Remark. That two well known serial killers among mathematicians were both doing complex variables, is a coincidence, on my opinion:-) (The other one was Andre Bloch who was a really brilliant mathematician, but unfortunately, insane.)

• $\begingroup$ Wait, there are two? $\endgroup$ –  Vincent Commented Dec 12, 2018 at 13:05
• $\begingroup$ O wait, the other one is mentioned in the comments on the original post, never mind $\endgroup$ –  Vincent Commented Dec 12, 2018 at 13:06
• $\begingroup$ @Vincent 1: No, the person mentioned in the comments was not known as a mathematician. I added the name of the second one to my answer. $\endgroup$ –  Alexandre Eremenko Commented Dec 12, 2018 at 14:32

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Topological analysis of complex networks using assortativity

Ph.D thesis of Mahendrarajah Piraveenan

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Assortativity quantifies the tendency of nodes being connected to similar nodes in a complex network. Degree Assortativity can be quantified as a Pearson correlation. However, it is insufficient to explain assortative or disassortative tendencies of individual nodes or links, which may be contrary to the overall tendency of the network. A number of 'local' assortativity measures have been proposed to address this. In this paper we define and analyse an alternative formulation for node assortativity, primarily for undirected networks. The alternative approach is justified by some inherent shortcomings of existing local measures of assortativity. Using this approach, we show that most real world scale-free networks have disassortative hubs, though we can synthesise model networks which have assortative hubs. Highlighting the relationship between assortativity of the hubs and network robustness, we show that real world networks do display assortative hubs in some instances, particularly when high robustness to targeted attacks is a necessity.

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A Complex Molecular Symmetry Analysis of Silsesquioxane Catalysts for Inorganic Students

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• Affiliation: College of Arts and Sciences, Department of Chemistry
• Teaching symmetry point groups to introductory inorganic chemistry students is often complicated by the difficulties students face with mental manipulation of complex molecules. Additionally, most textbooks provide practice in the form of random two-dimensional molecules or figures of little significance. As an alternative, an assignment is proposed that analyzes the symmetry of borylated cubic silsesquioxane-based catalysts, which can be modeled as simple cubes with differing vertices that depend on boron loading. This assignment challenges students to explore how adding substituents to a highly symmetric molecule affects the symmetry of the molecule and gives rise to spectroscopically equivalent positions.
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• https://doi.org/10.1021/acs.jchemed.2c00172
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