Related papers: A short course on Tensor algebra and analysis, wit…
Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present…
This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced…
An introduction to applied mathematics written for students in engineering and science. Focus is on a rigorous presentation that also builds understanding by discussion, analogy, and examples. Discussion of concepts involved in modeling…
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a…
Tensor analysis provides a frame-invariant foundation for continuum mechanics, yet numerical implementations rely on matrix representations expressed in user-selected bases. When these bases are non-Cartesian and non-orthonormal, additional…
In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential…
Introduction of curvilinear coordinates might be very convenient in many cases. Theoretically, tensor analysis would be most suited. However, tensor notation can't be used in numerical procedure. For example, the strict discrimination of…
We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
These are lecture notes for the course "Analysis and X-ray tomography". The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical…
The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four…
Regression analysis is a key area of interest in the field of data analysis and machine learning which is devoted to exploring the dependencies between variables, often using vectors. The emergence of high dimensional data in technologies…
Dense and sparse tensors allow the representation of most bulk data structures in computational science applications. We show that sparse tensor algebra can also be used to express many of the transformations on these datasets, especially…
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
Lecture notes to a one-term course on operator algebras and their application in physics. Very brief and basic introduction to the subject of Banach- and C-star algebras complemented with their appearance in physics. The course is intended…
In this chapter, I discuss teaching mathematical tools specifically tailored for economics students. A typical one-semester course in this area seeks to blend a range of topics: from foundational elements of subjects such as linear algebra…
1. Connection between integration and differentiation/ Gauss theorem. Coordinate-free definitions: gradient, divergence, curl. Stokes theorem. 2. Elements of continuum mechanics/ Continuity equation. Stress tensor. Euler equation.…