“There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man’s life to know them the little new that each man gets from life is very costly and the only heritage he has to leave.” - Ernest Hemingway (More…)
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Free Books on Dynamical Systems
Arxiv/ Free Books # 1. Lectures on Neural Dynamics - Francesco Bullo # Chapter 1: Neural circuit models based on firing rates and Hopfield networks: their dynamics, interconnections, and local Hebbian adaptation rules Chapter 2: Stability in dynamic neural networks using Lyapunov methods, multistability, and energy functions Chapter 3: Optimization in neural networks through biologically inspired gradient dynamics and sparse representations. Chapter 4: Unsupervised learning via neural dynamics, linking Hebbian rules to tasks like PCA, clustering, and similarity-based representation learning. 2. Linear Geometry and Algebra - Taras Banakh # Abstract: Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So, Linear Geometry studies liners. Imposing some additional axioms on a liner, we obtain some special classes of liners: regular, projective, affine, proaffine, etc. Linear Geometry includes Affine and Projective Geometries and is a part of Incidence Geometry. The aim of this book is to present a self-contained logical development of Linear Geometry, starting with some intuitive acceptable geometric axioms and ending with algebraic structures that necessarily arise from studying the structure of geometric objects that satisfy those simple and intuitive geometric axioms. We shall meet many quite exotic algebraic structures that arise this way: magmas, loops, ternary-ring, quasi-fields, alternative rings, procorps, profields, etc. We strongly prefer (synthetic) geometric proofs and use tools of analytic geometry only when no purely geometric proof is available. Liner Geometry has been developed by many great mathematicians since times of Antiquity (Thales, Euclides, Proclus, Pappus), through Renaissance (Descartes, Desargues), Early Modernity (Playfair, Gauss, Lobachevski, Bolyai, Poncelet, Steiner, Möbius), Late Modernity Times (Steinitz, Klein, Hilbert, Moufang, Hessenberg, Jordan, Beltrami, Fano, Gallucci, Veblen, Wedderburn, Lenz, Barlotti) till our contempories (Hartshorne, Hall, Buekenhout, Gleason, Kantor, Doyen, Hubault, Dembowski, Klingenberg, Grundhöfer).
Mathematics Books
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Pre-print articles on Difference-of-Convex (DC) Programming
57. Stochastic Difference-of-Convex Optimization with Momentum # Authors: El Mahdi Chayti, Martin Jaggi Abstract: Stochastic difference-of-convex (DC) optimization is prevalent in numerous machine learning applications, yet its convergence properties under small batch sizes remain poorly understood. Existing methods typically require large batches or strong noise assumptions, which limit their practical use. In this work, we show that momentum enables convergence under standard smoothness and bounded variance assumptions (of the concave part) for any batch size. We prove that without momentum, convergence may fail regardless of stepsize, highlighting its necessity. Our momentum-based algorithm achieves provable convergence and demonstrates strong empirical performance.