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This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used…
We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3)…
This article is based on lecture notes for the Marie Curie Training school "Initial Training on Numerical Methods for Active Matter". It provides an introductory overview of modeling approaches for active matter and is primarily targeted at…
Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of…
These lecture notes are derived from a graduate-level course in dynamic optimization, offering an introduction to techniques and models extensively used in management science, economics, operations research, engineering, and computer…
These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
These lecture notes provide an introduction to quantum information and quantum computation, which are strongly related disciplines and subject of intense research. The lecture notes contain only a small selection of topics in these…
This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces…
These lecture notes provide an introduction to free probability theory, with a focus on tools and techniques useful in the study of large random matrices. Topics include freeness, free cumulants, additive and multiplicative free…
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
These lecture notes cover undergraduate textbook topics (e.g. as in Sakurai), and also additional advanced topics at the same level of presentation. In particular: EPR and Bell; Basic postulates; The probability matrix; Measurement theory;…
This note presents reflections drawn from my recent experiences in teaching a course on mathematics and sustainability, with a particular emphasis on raising awareness of the topic and its broader implications. The lectures were structured…
The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including…
This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We…
Lecture notes written for a one-semester course in mathematical relativity aimed at mathematics and physics students. Not meant as an introduction to general relativity, but rather as a complementary, more advanced text.
These lecture notes are intended to cover some introductory topics in stochastic simulation for scientific computing courses offered by the IT department at Uppsala University, as taught by the author. Basic concepts in probability theory…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
For several years, students visit us on different occasions at the university. But how to bridge from the school curriculum to the contents of the university mathematics? And how to find a focal point at which an active contribute, despite…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…