Dynamical systems respond differently to initial perturbations depending on their stability properties.
The top section shows Lyapunov stability, where a trajectory starting within δ stays forever inside the ε-wide tube along time.
The middle section illustrates asymptotic stability: the path approaches the equilibrium as the allowable band ε shrinks to zero at large t.
The bottom section depicts exponential stability with the tube narrowing rapidly from the start, reflecting faster decay.
Satellite attitude control systems use these distinctions to ensure reliable pointing accuracy after disturbances like solar pressure or thruster firings.
Explorations in Monte Carlo Methods (Undergraduate Texts in Mathematics): https://t.co/QNOJH53tyA
From the Back Cover:
"Monte Carlo methods are among the most used and useful computational tools available today, providing efficient and practical algorithims to solve a wide range of scientific and engineering problems. Applications covered in this book include optimization, finance, statistical mechanics, birth and death processes, and gambling systems."
"Explorations in Monte Carlo Methods provides a hands-on approach to learning this subject. Each new idea is carefully motivated by a realistic problem, thus leading from questions to theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. Each chapter ends with a large collection of problems illustrating and directing the material."
"This book is suitable as a textbook for students of engineering and the sciences, as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability and for a more specialized course in Monte Carlo methods. Topics include probability distributions, counting combinatorial objects, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, sampling, and random number generation."
Introducing Lattice Deduction Transformers: An 800k-parameter looped transformer that reasons like a SAT solver achieves 100% on Sudoku-Extreme with only 15 minutes of training.
A collaboration between @axiommathai, @AmherstCollege and @BarnardCollege.
FREE math book.
Wow, 2200-page book. (Imagine if we could download such deep knowledge directly to our brains.) "Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning" by Gallier & Quaintance.
Link: https://t.co/26S7MeJFWn
Physicist Stephen Wolfram's exploration of a theory of everything, and simulating the universe. [Select image to magnify.]
Source:
Stephen Wolfram, "Finally We May Have a Path to the Fundamental Theory of Physics… and It's Beautiful," Stephen Wolfram Writings. https://t.co/fKbZP4iXwt
@CamiloYounes@Mineducacion Cuando tiene unos profesores universitarios que en su vida han generado riqueza, pero que sí han vivido de vampirizar al Estado y explotar estudiantes para hacer publicaciones. Uno deberia cuestionarse el salario que deben tener y los beneficios que tienen
@dr_lopezperez@CamiloPrietoVal Cuando uno tiene profesores universitarios que en su vida han generado riqueza, pero que sí han vivido de vampirizar al Estado y explotar a los estudiantes para hacer publicaciones, debería cuestionarse el salario y los beneficios que reciben.
Bacteria move around using a molecular machine called the flagellar motor that rotates faster than the flywheel of a race car engine and switches directions in an instant. After 50 yrs, scientists have finally figured out how it works. “My lifelong quest is now fulfilled.” Link⤵️
PhD Skills for Claude Code:
@fcakyon (1300+ citations, 7 patents) shares his battle-tested PhD Skills for Claude Code.
github repo link in reply below:
Turing contributed to Enterprise Ops Gym, ServiceNow’s new enterprise agent benchmark.
We designed 1,000 prompts across 8 enterprise scenarios spanning HR, CSM, ITSM, Email, Calendar, Drive, Teams, and hybrid cross domain workflows.
Tasks range from 7 to 30 steps with stateful system updates, expert reference traces, and deterministic verification scripts to evaluate correctness and policy compliance.
A rigorous step forward for long horizon enterprise agent evaluation.
Learn more:
Paper: https://t.co/JULVtkQLUK
Data Set: https://t.co/mGAlgvIiwM
Website: https://t.co/N56iNyPvkN
Code: https://t.co/DBC3SsvK6W
Esta entrevista de @estoescambio con Paloma Valencia y Juan Daniel Oviedo deja en evidencia que hay temas en los que ambos están completamente distanciados y que así será muy difícil gobernar juntos.
Esta pieza periodística los desnudó y aunque digan que van a construir desde la diferencia, el panorama es muy complicado porque en últimas es el presidente el que manda y el vice ¿es una figura de adorno?
Véanla, está buenísima 👇👇
Si Paloma Valencia es la primera presidenta de Colombia, llegará al poder como hija.
Recuerden su grito «Uribe es mi papá» y su fidelidad a ultranza.
La hija devota y garante de la autoridad paterna es el pivote del patriarcado.
Nada tiene de feminista esta elección.