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We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting…
Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are…
In this book, there are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students…
In this paper we present our experience in using visualization in mathematics education. The experience with our university courses: "Computer tools in matematics" and "Symbolic algebra" provides the basis for mathematics teacher education…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space, but with a non-convex constraint set introduced by model parameterization.…
When applying machine learning to problems in NLP, there are many choices to make about how to represent input texts. These choices can have a big effect on performance, but they are often uninteresting to researchers or practitioners who…
Scientific research involves mathematical modelling in the context of an interactive balance between theory, experiment and computation. However, computational methods and tools are still far from being appropriately integrated in the high…
We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of…
Artificial students -- models that simulate how learners act and respond within educational systems -- are a promising tool for evaluating tutoring strategies and feedback mechanisms at scale. However, most existing approaches rely on…
We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of $2\times 2$ matrices for the whole hierarchy, we construct the…
This paper surveys the machine learning literature and presents in an optimization framework several commonly used machine learning approaches. Particularly, mathematical optimization models are presented for regression, classification,…
Visualisations drive all aspects of the Machine Learning (ML) Development Cycle but remain a vastly untapped resource by the research community. ML testing is a highly interactive and cognitive process which demands a human-in-the-loop…
These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix…
The paper examines the construction of a course in mathematical analysis at a pedagogical university, aimed at developing the ability of future mathematics teachers to detect and solve problems related to finding proofs. Key words: teaching…
Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated…
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was…
Universal learning machine is a theory trying to study machine learning from mathematical point of view. The outside world is reflected inside an universal learning machine according to pattern of incoming data. This is subjective pattern…
Task vector composition has emerged as a promising paradigm for editing pre-trained models, enabling model merging through addition and unlearning through subtraction. Fine-tuning in the tangent space of a pre-trained model (linear…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…