Estimating with a Coin
This paper introduces a novel Monte Carlo method for estimating by tossing a coin, which utilizes a new interpretation of derived from Catalan-number series identities.
🔢 Category
math.HO
14 papers
Research projects and Moscow Mathematical Conference for high school students
This paper outlines the educational framework and practical examples of the Moscow Mathematical Conference for High School Students, demonstrating how non-novel research projects can effectively guide adolescents through the complete scientific process—from developing intuition and peer review to receiving formal recognition.
Celebrating Women in Mathematics
Motivated by the global May 12 initiative to celebrate women in mathematics, this paper outlines the initiative's origins and key events while highlighting the pioneering scientific contributions of six influential female mathematicians: Hypatia, Sofia Kovalevskaya, Emmy Noether, Maryam Mirzakhani, Karen Uhlenbeck, and Ingrid Daubechies.
Mathematical Proof
This paper presents a set of course notes designed to introduce undergraduate students to mathematical proofs, covering foundational topics such as logic, proof techniques, induction, set theory, and real analysis, complete with numerous examples and exercises.
On the mechanical creation of mathematical concepts
The paper proposes a model of mathematical problem-solving as a belief-update loop that distinguishes between implicit concept formation, which optimizes search within a fixed vocabulary, and explicit concept creation, which introduces new moves to resolve unsolvable problems and argues that while current AI excels at the former, achieving the latter is essential for machines to replicate the distinctive nature of mathematical discovery.
Semantic Search over 9 Million Mathematical Theorems
This paper introduces a scalable semantic search system for a corpus of 9.2 million mathematical theorems, demonstrating that representing theorems with natural-language descriptions significantly improves retrieval accuracy for both specific theorems and entire papers compared to existing baselines.
Invariants of almost embeddings of graphs in the plane
This paper explores invariants of almost embeddings of graphs in the plane by establishing relations among them, connecting these to the homology of the deleted product, constructing examples, and presenting these topological concepts in an accessible manner while highlighting open conjectures.
A catalog of interesting and useful Lambert series identities
This paper provides a comprehensive catalog of Lambert series identities, detailing their formal properties, combinatorial generalizations, and connections to partition functions, while offering a reference compendium of known formulas for various special cases without focusing on rigorous analytic convergence.
Hyperbolic elliptic parabolic disks approximated by half distance bands
This paper investigates the geometric relationship between hyperbolic, elliptic, and parabolic disks and their supporting half-distance bands within the Beltrami–Cayley–Klein model, aiming to define and quantify their "closeness" through precise approximations of area and circumference.
Peacock's Principle as a Conservative Strategy
This paper argues that Peacock's principle of permanence was not invalidated by Hamilton's non-commutative algebras, but rather correctly understood as a conservative strategy—rooted in Hume's philosophy—that permits exceptions like non-commutativity only when the reasons for violating established laws of reasoning outweigh the reasons for preserving them.
English translation of Sophie Kowalevski's "On the problem of the rotation of a rigid body about a fixed point"
This paper presents an English translation and digitization of Sophie Kowalevski's 1889 French publication on the rotation of a rigid body about a fixed point, which introduced the famous Kovalevskaya Top.
Blackwells Demon: Postdiction and Prediction in Random Walks
This paper introduces "Blackwells Demon," a thought experiment demonstrating that under specific restrictive conditions involving a complex environment, one can predict the direction of a random walk with a success probability greater than 1/2 by exploiting inhomogeneities in a statistically homogeneous system, analogous to how Maxwell's Demon exploits molecular speed variations.
Discovering mathematical concepts through a multi-agent system
This paper presents a multi-agent system that autonomously discovers mathematical concepts, such as homology, by dynamically interweaving conjecture generation, proof attempts, and counterexample analysis, demonstrating that optimizing these local processes effectively aligns with human notions of mathematical interestingness.
Mathematicians in the age of AI
This essay urges mathematicians to stay informed about AI's emerging ability to prove research-level theorems and to proactively adapt their practices to the resulting challenges and opportunities.