The palindrome formed by 6/28/26 corresponds to a reciprocal polynomial with roots at angles τ/3 and a symmetric pair.
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An integer palindrome is a self-reciprocal polynomial evaluated at its base, so its roots are symmetric about the unit circle -- where the coordinate is angle, in turns of $\tau$. Read this way, the date $\texttt{6/28/26}\to 62826$ secretly contains the primitive cube roots of unity -- at angle $\tau/3$ -- along with one further pair of roots on the circle.
The maps for chained extensions recover the direct map up to permutations, letting one 6x6 matrix over F_2 show both F_64/F_8/F_2 and F_64/F
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Finite fields are important algebraic structures that have a wide range of applications in fields such as coding theory and cryptography. But the standard construction of finite field extensions through polynomial quotients is computationally opaque, especially when we want to identify a degree-$2$ extension of $F_8$ and a degree-$3$ extension of $F_4$.
In this short note, we present a coherent family of representations by matrices $\rho_q^n\colon F_{q^n} \to F_q^{n\times n}$ for all prime powers $q$ and all degrees $n \ge 1$. These maps are chosen so that concatenating $\rho_{q^n}^m$ and $\rho_q^n$ recovers $\rho_q^{nm}$ up to row and column permutations. As a consequence, the images of $\rho_2^6$ can be partitioned into four $3 \times 3$ blocks or nine $2 \times 2$ blocks to visualize the subfield chains $F_{64} / F_8 / F_2$ and $F_{64} / F_4 / F_2$ at the same time. A variant $\varrho$ is also discussed, wherein the Frobenius automorphism is represented by a cyclic shift of rows and columns.
From an educational point of view, these rhos give explicit and self-contained mental models of finite fields; subfields, trace, norm, minimal polynomial, and Frobenius all become visible through matrix algebra accessible to most students. From a theoretical point of view, the construction exhibits structural implications of Conway polynomials and the normal basis theorem.
Number trains reason to recognize order, completing its work only when the mind reaches the divine.
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In 1973, Russian mathematician I.R. Shafarevitch delivered a lecture to the G\"ottingen Academy of Sciences on the purpose of mathematics. The conclusion he reached in his address is that the ultimate purpose of mathematics must be religious. In this talk, we will explore a possible way in which this claim can be justified by understanding the purpose that mathematics served within a person's intellectual formation according to Plato. To place Plato's view into a Christian perspective, we will then investigate the thought of St. Augustine of Hippo, the great fifth century theologian and bishop. Augustine's insight on the role that number plays in the development of reason sheds light on how knowledge of mathematics conduces to knowledge of God.
When researchers need a mathematical model for a research problem, they face a fragmented landscape: relevant formulas, quantities, assumptions, and model variants are scattered across publications and domain-specific conventions. The Mathematical Models Database (MathModDB) addresses this challenge by providing a curated knowledge graph for mathematical models, deployed on the MaRDI Portal as part of the German National Research Data Infrastructure (NFDI). Building on ontology designs presented in earlier work, this paper focuses on MathModDB as a publicly available service. It addresses researchers who use mathematical models in their work -- whether in applied mathematics, engineering, or the natural sciences. We describe its deployment on the Wikibase-powered MaRDI Portal, report on its current scale, and demonstrate its practical use through a walkthrough of an electric discharge modeling use case from plasma physics. We further discuss the ecosystem around MathModDB, including its connection to the MathAlgoDB knowledge graph for numerical algorithms and the MaRDMO documentation tool.
We study the use of an educational proof assistant in an introductory proof course through a quasi-experiment in a varied setting: multiple teachers, students with different study programs, and a mixed Dutch-English language environment. First-year university students are known to struggle with writing proofs. Waterproof is a proof assistant that is designed to support the transfer of skills to paper proofs by working with controlled natural language. We focus on the students' ability to construct valid mathematical proofs, and on their learning process. We study this through in-class observation, surveys, and analysis of student performance and proof structure. We present evidence that effects of using an educational proof assistant carry over to the pen-and-paper context,even when the assistant is English and the proof is given in Dutch. We also present evidence that suggests students in the Mathematics-Computer Science program achieve higher grades when using Waterproof. Our most important conclusion is that an educational proof assistant can help students be more explicit in their proofs. As students self-selected into using Waterproof rather than being randomly assigned, these results are suggestive rather than causal.
Classical methods for solving linear ordinary differential equations, such as superposition, the method of undetermined coefficients, and the annihilator technique, are often presented as heuristic, procedural rules. In this article, we show that these methods admit a coherent algebraic interpretation when constant-coefficient linear differential operators are viewed as elements of the polynomial ring $\mathbb{C}[D]$, acting on spaces of smooth functions.
Without invoking the formalism of $D$-modules or non-commutative operator algebras, we explain how homogeneous solution spaces arise as kernels of linear operators, how particular solutions form affine cosets, and how the search for solutions is an infinite-dimensional eigenvalue problem. Furthermore, we extend this algebraic framework to variable-coefficient equations, resolving the Euler equation through ring isomorphisms and framing d'Alembert's reduction of order as non-commutative operator factorization. We also explore the boundary of this linear theory, demonstrating how diffeomorphic linearization allows certain non-linear equations -- such as those of Bernoulli and Riccati -- to be mapped directly into the $\mathbb{C}[D]$-module framework. Finally, we contrast this framework with the multivariable ring $\mathbb{C}[D_x, D_y]$, using the loss of the principal ideal domain property to explain the intrinsic structural divergence of partial differential equations, and indicate further universal extensions to discrete difference equations and the Weyl algebra.
Mathematical knowledge is organized around statements and their dependencies, but this structure is exposed unevenly: informal papers cite mostly at the document level, while formal libraries record fine-grained dependencies over a much smaller body of mathematics. We introduce TheoremGraph, a unified statement-level dependency graph spanning both informal and formal mathematics. On the informal side, we parse 11.7M theorem-like environments from mathematics arXiv and recover 18.3M candidate directed dependencies, each labeled by the extractor that proposed it so downstream users can trade coverage for precision. On the formal side, we release LeanGraph, a Lean 4 elaborator-level extractor producing 388,105 declaration nodes and 11.3M typed edges across 25 Lean projects. We bridge the two graphs by embedding generated natural-language slogans into a shared semantic space, linking related statements across papers and across the informal/formal divide; an LLM judge affirms 47,952 such matches above a 0.8 cosine floor, with the judge-acceptance rate rising from 48% across the floor to 87% in the >=0.9 tier. On formal concept retrieval, our name-and-signature representation with graph expansion comes within 0.5pp of LeanSearch v2's reranked Recall@10 (0.775 vs. 0.780) without an LM reranker. We release the dataset, extractors, HTTP API, and MCP interface as infrastructure for mathematical search, attribution, and retrieval-augmented reasoning, available at theoremsearch.com and huggingface.co/datasets/uw-math-ai/theorem-matching.
The condition identifies all orders where the only group is the cyclic one.
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We will first discuss the question of which integers $n$ have exactly one group of order $n$, namely the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We will see that these are the integers that are relatively prime to the Euler totient function $\phi(n)$. Then we discuss how many groups there are of order $p^3$ for each prime $p$. We end with a couple of interesting results and conjectures pertaining to groups of squarefree order.
Every group whose order is a cube of a prime falls into one of three abelian or two non-abelian isomorphism types, with explicit maps suppli
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This note discusses our formalisation in Lean 4 of the classification of groups of order $p^3$ for a prime number $p$, using mathlib4. We present the five isomorphism classes and give a detailed account of the formalisation, with particular emphasis on the non-abelian case, which requiring the most substantial formal development. For odd~$p$, the non-abelian groups are the Heisenberg group $\Heis(\Z/p\Z)$ and the semidirect product $\Z/p^2\Z\rtimes\Z/p\Z$; for $p=2$, they are $D_4$ and $Q_8$. We describe the construction of these concrete groups, the structural lemmas about centers, commutators, and exponents, and the explicit isomorphism constructions that classify an arbitrary non-abelian $p^3$-group.
We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation $t_{\text{machine}} = a \cdot t_{\text{human}}^b$ between generated-token cost per successful answer and human time, and find $b < 1$ for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay $p_{\text{success}} = e^{c t_{\text{human}}}$, predicting a linear relation between $\log p_{\text{success}}$ and $t_{\text{human}}$. Using a binning approach for data aggregation we find moderate empirical support (median bin-level $R^2 = 0.92$ across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50\% task-length horizons $h_{50}$; the strongest configurations in our 20 April 2026 snapshot reach roughly $2.5$--$4.3$ hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about $75$~days for the SOTA $h_{50}$.
A three-dimensional solid has the Rupert property if a congruent copy of the solid can pass through a hole cut through it without splitting it. We extend this idea to pairs of convex solids: two solids are called \textit{friends} if each can pass through a suitable hole in the other. A solid is called \textit{lonely} if it has no friends, including itself. We show that a convex solid is lonely if and only if it has constant width. We also show that every convex solid that does not have constant width has a particularly simple friend: an arbitrarily long and arbitrarily thin rectangular cuboid. Finally, we prove that all non-constant-width convex solids lie in a single connected component of the friendship graph. More precisely, any two such solids are connected by a chain of at most ``three handshakes''.
These lecture notes are intended as an accessible introduction to some basic ideas of motive theory for readers with limited background in algebraic geometry. Mathematics often reveals unexpected connections between seemingly distant areas. A simple combinatorial identity may encode geometric structures, arithmetic information, or even sophisticated categorical phenomena.
In these notes, we trace a path from elementary counting arguments to Voevodsky's theory of motives. We show how some classical combinatorial identities emerge naturally from geometry, and how motivic decompositions reveal the deeper geometric and arithmetic structures underlying them. Our guiding example is provided by the Karaji--Pascal identity and its $q$-analogue, which link combinatorics and algebraic geometry.
Thus our primary aim of these lecture notes is to demonstrate that some familiar combinatorial identities can provide non-experts with an entry point to some of the basic ideas of motive theory! Moreover, while introducing the reader to the subject, we also hope to encourage the view that even an elementary mathematical formula may encode a deeper underlying geometric and arithmetic structure. Along the way, the notes offer a gentle introduction to motives through a concrete example rather than through the full technical machinery of modern theory.
Coverage runs from divisibility and modular equations to Möbius inversion for use in discrete math or crypto courses.
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This brief, in the form of an e-book, is a collection of notes that cover elementary and medium level number theory with a target audience of primarily computer science students. It can be used in the number theory portion of a discrete mathematics course, or a course on the mathematical foundations of computer science, or as background material for a cryptography course.
Thematically it is split into five areas that map to chapters. The first chapter is introductory and covers topics including divisibility, prime numbers, and modular arithmetic including modular linear equations. The second chapter covers additional topics such as Euler's totient function, units and inverses, the Chinese remainder theorem, and Fermat's and Euler's theorems. The following chapter covers primitive roots, quadratic residues, the Jacobi and Legendre symbols, Gauss's lemma and Eisenstein's theorem, and briefly discusses applications of number theory to cryptography. The fourth chapter is focused on traditional primality testing methods covering Miller's algorithms, Rabin's conversion of a Miller algorithm into a probabilistic primality test algorithm, Solovay-Strassen's algorithm and several other peripheral results including Carmichael numbers and the equivalence of Miller's two algorithms. Finally the last brief chapter can be viewed as an introduction to more advanced elements of number theory and its coverage includes multiplicative functions, the M\"obius function, Dirichlet products and Dirichlet and M\"obius inversions.
Different parts of this e-book are for freshman to senior undergraduate students in computing and in particular computer science. Graduate students with limited exposure to number theory can use it to acquire a background suitable for typical cryptography courses at the master's level.
We study Dice Bingo, a game in which players fill a $3\times3$ bingo board whose entries are possible sums of two fair dice. After each roll, a player marks one matching square, and the goal is to complete a row, column, or diagonal. We model optimal play for a fixed board as a finite Markov decision process and derive Bellman equations that compute the exact expected number of rolls required to obtain a bingo. Using this framework, we identify a unique optimal board up to natural symmetries and determine its exact expected completion time.
We then investigate head-to-head competition in which two players observe the same sequence of dice rolls. By analyzing a joint Markov chain that tracks both boards simultaneously, we compute (in exact arithmetic) win, loss, and tie probabilities. Surprisingly, a board with a worse expected completion time can nevertheless be favored in head-to-head competition. Motivated by this phenomenon, we exhibit nontransitive triples of bingo boards: board $A$ is favored against board $B$, board $B$ is favored against board $C$, and board $C$ is favored against board $A$.
Finally, we consider strategic play in which players adapt their choices to their opponent's board rather than merely minimizing their own completion time. In this setting, optimal decisions depend on the opponent's state, leading naturally to game-theoretic analysis. We present a position with no pure Nash equilibrium and compute an explicit mixed Nash equilibrium.
f(p/q) = p + Ω^(q) matches data while equating Gradus to weighted harmonic coincidence counts.
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Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + \Omega^(p) + \Omega^(q), where \Omega^(n) = \sum_i e_i(p_i - 1) sums the weighted prime exponents of n. We propose the simpler asymmetric formula f(p/q) = p + \Omega^(q), which treats numerator and denominator differently and performs comparably on standard consonance data. We also show that, under a model in which harmonics are integer-indexed and counted uniformly up to a fixed truncation level, Gradus is equivalent to a weighted harmonic coincidence count with weights w(n) = \Omega^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + \Omega^(k), whose rightmost diagonal gives the two-stage dissonance of the superparticular (consecutive-harmonic) intervals. The formula f admits a simple two-stage interpretation in terms of harmonic context and partial recognition, which we offer as a speculative perceptual hypothesis.
Specializations of the dyadic Cauchy-kernel identity yield duplication formulas, Hasse-Sondow series, and eta zeros
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This is an expository note. We take the dyadic Cauchy-kernel identity of Castillo-Costin-Costin, a global rational/factorial decomposition built on the polylogarithm, and follow it down several specializations. In each direction it returns to a classical landmark: the polylogarithm duplication formula and Hurwitz's Fourier-series formula; representations of the zeta function at the special argument pi and at rational arguments, in the neighborhood of Hurwitz's multiplication theorem; the Hasse-Sondow globally convergent series; and, through its discrete scale invariance, the extra zeros of the Dirichlet eta function together with the harmonic-sum asymptotics of Flajolet-Gourdon-Dumas, with Dirichlet L-values emerging as the amplitudes of a log-periodic oscillation. The aim is unification: to exhibit one compact identity as an organizing center from which these classical results may be read off. We claim no new theorems; where an identity may not previously have been displayed in exactly this form, we say so and explain why it is nonetheless a recombination of known ingredients.
The method produces flat metrics with twelve cone points of deficit pi/3 and parametrizes their space of shapes.
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Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\pi/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/\Gamma$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $\Gamma$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\pi/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.
The geometric transition introduces the physicist's key idea and its place in modern theory on the centennial of his birth.
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On the centennial of Erdal \.In\"on\"u's birth, this article reflects on his scientific legacy and his role in shaping modern theoretical physics in T\"urkiye. We briefly discuss his life, scientific vision, and contributions to academic institutions, and then turn to his most celebrated scientific achievement: the \.In\"on\"u-Wigner contraction. Through the simple geometric example of a sphere becoming a plane, we present an accessible introduction to this important idea and its significance for modern physics.
New collection of 100 questions with known answers shows top models succeeding after repeated evaluation stages.
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Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.
Extending Cauchy's theorem to annular domains in 1843 replaced avoidance strategies with expansions whose negative powers encode singularity
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For the better part of a century, isolated singularities were treated as pathological obstructions requiring elaborate avoidance strategies. Pierre Alphonse Laurent (1813--1854), a French military engineer at Le Havre, ended this avoidance in 1843 by extending Cauchy's Taylor-type theorem to doubly connected (annular) domains, producing the doubly infinite power series that now bears his name. Negative-power terms in the expansion encode precise geometric information about the singularity rather than signaling a breakdown of the formalism. Laurent's contribution arrived through an unhappy institutional trajectory -- submitted after a prize deadline, subjected to a priority claim by Cauchy, and issued in full only posthumously in 1863 -- yet it became indispensable to every branch of mathematics and mathematical physics that touches on complex function theory. We reconstruct the mathematical problem Laurent solved, place it within Cauchy's analytic program of the 1830s--1840s, examine the institutional failure that prevented publication, document the independent parallel proof by Weierstrass (1841, published 1894), and trace the series' absorption into the standard toolkit via Briot and Bouquet and the residue calculus. Drawing on Laurent's 1843 Comptes rendus notice, Cauchy's Academy report, Bertrand's memorial notice (1890), and the secondary literature (Neuenschwander 1978, 1981; Manning 1975; Bottazzini 1986; Gray 2015), we analyze the philosophical significance of the series, which we term ``exile mathematics'', and survey its reach into perturbation theory, number theory, probability, and quantum field theory. Readers familiar with the theorem but not its institutional history will find here a documented account of why a foundational result was withheld for two decades and how it nevertheless achieved canonical status.
Rich formatting and clear input areas abstracted for integration into varied educational tools for proof assistants and languages.
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Waterproof Editor provides an educational environment specifically targeted to teaching with proof assistants or programming languages. It arose from Waterproof, educational software targeted at helping students acquire the skill of giving mathematical proofs. Its original features such as enabling rich formatting and providing clear input areas are now abstracted away in an npm package and can be used in different educational contexts. We invite interested parties to use this component in their educational software, and offer to assist with this.
We document the evolution of Karl Marx's take on infinitesimals. We contrast his initial favorable stance with later criticisms, and examine the differing perspectives of Marx and Engels on the subject. Marx's favorable assessment was based on his study of Sauri's textbook. Later, influenced by Boucharlat's textbook, Marx reversed his position to an unfavorable stance, describing belief in infinitesimals as a `chimera'. Marxist scholar Guglielmo Carchedi claims that ``Marx differentiates with the eyes of the social scientist, of the dialectician'' but fails to note dialectician Engels' endorsement of infinitesimals.
Struik linked Marx to Abraham Robinson, but missed the fact that the link passes via ... Fermat. Namely, there may be an affinity, as per Struik, between Marx's comments on the calculus and Robinson's nonstandard analysis, but the kernel of such an affinity resides in the techniques already found in the context of Fermat's adequality. To adapt Carchedi's metaphor, we could say that Marx may have differentiated with the eyes of adaequo of Pierre de Fermat.
The first editor who worked on some of Marx's mathematical manuscripts in the mid-1920s was Emil J. Gumbel, though he is not mentioned in either the 1933 or the 1968 Soviet edition of Marx's mathematical manuscripts.
The universal group completion of a commutative monoid produces Euler characteristics, K0 of modules, and representation rings as the start
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In this expository paper, we develop the basic ideas underlying Grothendieck groups and to illustrate their appearance across algebra, topology, representation theory, and homological algebra. Motivated by the universal construction associated to a commutative monoid, we define the Grothendieck groups abelian categories and rings. Along the way we study several fundamental examples, including Euler characteristics, projective modules, and representation rings. We conclude with a discussion of $K$-theory and its applications, indicating how the elementary construction of $K_0$ serves as the first layer of a much richer homotopical theory.
Fifteenth-century text claims the procedure as innovation while earlier works applied similar squaring only to specific calculations
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The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a technique for fast exponentiation commonly used in modern cryptography and computational number theory. Despite its prominence, the historical origins of the algorithm are not known with certainty. This paper critically examines the origins and formalization of the algorithm through primary source analysis. We focus on Jamshid al-Kashi's fifteenth-century Miftah al-Hisab where the algorithm is articulated explicitly as a general method and claimed by al-Kashi as his own innovation. To contextualize this, we trace earlier instances of successive squaring in the works of al-Uqlidisi and al-Biruni, who applied these techniques for specific calculations, but did not formalize them into a general procedure. The earliest known work on this method of computation is found in Pingala's prosodic studies in ancient India (c. 200 BCE). Even though it was not fully developed as a general technique, Pingala's work seems to contain the conceptual foundation of the algorithm which is to employ the binary representation of a positive integer. By mapping this intellectual progression, the paper illustrates the historical background of an algorithm that is prominent in modern computation.
Literature trace finds no explicit conjecture in 1995 Carbou paper, showing how names attach through repeated use.
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The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a 1995 paper by Carbou and to subsequent developments in the literature. We argue that the attribution likely arose from informal communication rather than from a formally stated conjecture, illustrating how mathematical terminology may develop through transmission and collective usage.
These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis.
The manuscript develops the basic theory of measurable sets, measurable functions, measures on $\sigma$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory.
The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.
In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some mathematicians. From that time I have been asked several times for references. So I decided to write this. Specially explaining the wonderful formulas \[\frac{\zeta'(\frac12)}{\zeta(\frac12)}=\frac{\pi}{4}+\frac{\gamma}{2}+\frac{\log(8\pi)}{2},\quad \frac{\zeta''(\frac12)}{\zeta(\frac12)}-\Bigl(\frac{\zeta'(\frac12)}{\zeta(\frac12)}\Bigr)^2=8-\frac{\pi^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{\alpha_n^2}\]
The anthyphairetic process leads from a pair (a,b) of coprime positive integers to the pair (1,1) by successive subtractions of the smaller number from the bigger one. This process, which is a slow version of Euclid's algorithm applied to the pair (a,b), corresponds naturally to the process of successive blowups leading to the minimal embedded resolution of the plane curve defined by y^a - x^b = 0. This blowup process may be represented graphically by a special two-dimensional simplicial complex called a lotus. This allows to localize the various numbers appearing either during the anthyphairetic process or during the Euclidean algorithm at precise positions inside the lotus. In this introductory article, I recall first the construction of this lotus starting from the sequence of quotients generated by the Euclidean algorithm. I present then an alternative way of constructing it directly from the sequence of pairs of coprime integers generated by the anthyphairetic process, using what I call anthyphairetic rectangles. I conclude by explaining how to reconstruct from a lotus the corresponding sequence of pairs of coprime integers. This is a simple illustration of the way lotuses may serve as computational architectures.
While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by endowing their vertices with an internal algebraic anatomy -- specifically, treating them not as empty geometric nodes, but as defined subsets of a cyclic base space Z_n. This internal structure allows us to formalize and classify a highly restricted class of graph isomorphisms: those strictly induced by global affine bijections f(x) = ax+b (mod n) operating directly on the underlying base set. By applying this framework to generalized tone networks (Tonnetze) unrolled via the Chinese Remainder Theorem in composite dimensions -- specifically the classic 12-TET (3x4) and the decaphonic 10-TET (2x5) -- we reveal absolute geometric anchors for these spaces, namely the (9,4) and (6,5) systems respectively. We completely classify the topological orbits of these structured graphs, proving a fundamental architectural dichotomy: while the isomorphic landscape of 12-TET splits into an orientation-preserving family and an orientation-reversing chiral mirror (providing a rigorous foundation for musical Negative Harmony), the 10-TET space is unconditionally orientation-preserving. Finally, we demonstrate that these abstract combinatorial properties manifest as rigidly coherent, parallel auditory universes through explicit structural voice-leading maps and acoustic physical modeling synthesis.
This paper discusses digital online mathematics examinations -- a discussion ranging from high school to university level examinations. In particular, we consider the nature of mathematical writing, what is distinctive about mathematical writing, and how mathematics can be typed into a machine. This includes a review of features of notation and layout unique to mathematics and a survey of current technology for typed mathematics, including LaTeX and contemporary proof-checkers such as Lean. Artificial intelligence has already been highly successful for optical character recognition, generating text from hand writing and is even increasingly applied to assess students' work itself. A human-editable text-based format in the middle is important, but neglected. The design of digital online mathematics examinations, which take this text as the source of truth for students' work, will have a profound effect on how mathematics is perceived and how mathematical activity is mediated. Moving examinations on-screen effectively is an important design challenge and responsibility to future generations. The challenge is to design software tools which support mathematics, that is tools which recede into the background and support the generation of mathematical work. We argue for a human-editable text-based format, which includes semantic elements, at the heart of the process.
We trace a conceptual genealogy from Abraham de Moivre's derivation of the normal curve (1733) to the modern distributional approach to statistics. De Moivre's Approximatio ad Summam Terminorum Binomii gave the first systematic derivation of the Gaussian density, its normalising constant (completed by Stirling's identification of $B = \sqrt{2\pi}$), and its tail probabilities computed to six decimal places -- more than seventy years before Gauss. His method -- extracting information from probability laws by evaluating sums against indicator probes -- is recognisably an instance of the operational viewpoint that underlies distributional statistics.
We identify a four-stage chain: coefficient extraction (De Moivre) $\to$ generating functions (Euler, Laplace) $\to$ characteristic functions (Fourier, L\'evy) $\to$ distributional pairings $\langle T, \varphi \rangle$ (Schwartz). At each stage the probes become more flexible and the class of laws that can be studied grows wider. The distributional framework, in which a probability law is represented by a distribution--kernel pair $(T, \varphi) \in \mathcal{S}'(\mathbb{R}) \times \mathcal{S}(\mathbb{R})$, is the natural endpoint of this progression.
We formulate and prove a distributional version of the De Moivre--Laplace theorem: the standardised binomial distribution converges to the Gaussian in $\mathcal{S}'(\mathbb{R})$, with De Moivre's original computation corresponding to the special case of indicator test functions. We also discuss the transversality framework, which provides a geometric explanation -- via infinite codimension of degeneracy strata -- for why pathologies such as moment indeterminacy, non-identifiability, and singular Fisher information are rarely encountered in parametric statistical models.
The Koch snowflake is a classical example of a planar curve with infinite perimeter enclosing a finite, positive area. Although such examples are well known individually, classical treatments typically analyze each construction in isolation and classify them by similarity dimension. This paper develops a unified parameter-space representation for a class of self-similar planar constructions, organized by two integers -- the number of self-similar pieces $N$ and the inverse linear scale factor $r$ -- together with two derived growth ratios $\alpha = N/r$ and $\beta = N/r^2$, governing perimeter and area scaling respectively. The $(N,r)$ parameter space is partitioned into three regimes -- $N \le r$, $r < N < r^2$, and $N \ge r^2$ -- corresponding to qualitatively distinct asymptotic behaviors of perimeter and area jointly. Within the intermediate regime $r < N < r^2$, a construction-class refinement distinguishes additive constructions (region bounded by the iterated curve), which yield positive finite asymptotic area under a stated non-overlap assumption, from subtractive constructions (iterated set itself), which yield zero asymptotic area. This records a structural non-equivalence inside the same dimension class that is not visible from $D = \log N / \log r$ alone. Four worked examples illustrate the framework -- the Sierpinski triangle, Sierpinski carpet, Koch snowflake, and a Koch-style construction on a square invented by the author -- and four further constructions are analyzed predictively to demonstrate that diagnostic outputs follow from $(N, r, \text{construction class})$ without re-derivation. The contribution lies in formulation and synthesis: the paper consolidates several classical results into a single diagnostic representation in which, given $(N, r)$ and construction class, the asymptotic behavior of perimeter and area can be inferred directly.
These are my memories of moments with Dick and Liz Askey in Russia, Wisconsin, Arizona, and abroad. Dedicated to the Askey family, these recollections span over 40 years and encompass many dramatic changes in the world. Due to this, it is challenging to entirely separate my personal thoughts and feelings from the factual historical account.
Extending sine, cosine and their exponential relations into the algebraic setting of racks
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Dedicated to Solomon Marcus, the current paper continues a recent series about our meetings. Trying to recreate the spirit of those meetings, we first propose a discussion which started as a high-school problem. The main part of the current paper consists in a section about racks. It presents elements of trigonometry in racks, and Euler formulas associated in this framework
We describe and axiomatize finite solitaire puzzles and zero sum sequential games graph theoretically. Zermelo's theorem telling that there is a win for one of the players or a draw follows from the definitions. The god number is a geometric quantity that quantifies the number of moves necessary to solve the puzzle. In the solitaire case, the god number is the minimal distance from the initial state $v$ to the solution space $A$. If $v$ and $A$ are not specified, the god number is the graph diameter. God number computations are related to combinatorial sorting problems and is a NP-complete problem in general even when restricted to concrete sliding problems. In the two-player case, the god number is a minimax critical value: it minimizes the maximal game event length over the set of all strategies. A strategy is a sub-graph of the game graph that contains the initial vertex. The definition is done so that a ``mate in k" chess problem has god number k. As for examples: in the solitaire case, we look at group games like Rubik type problems, transposition games related to sorting, at sliding puzzles like the 15 puzzle or rainbow ball, or the tower of Hanoi. For two-player games, we illustrate the story using examples of small chess games, a small card game or tic-tac-toe type problems.
A logic with guises as basic objects shows connections between things as structures of how they are conceived.
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This paper develops a formal logic for guises based on the work of H\'ector-Neri Casta\~neda, who understood relations from an internalist viewpoint, following Leibniz. We introduce a syntax, model theory, and proof theory for an intensional logic in which guises (taken as bundles of properties equipped with intention) serve as primary semantic objects. The system integrates (i) a Leibnizian containment semantics for singular truths, (ii) an intentional operator that captures internal relations among guises, and (iii) a modal layer for possibility and necessity modeled as maximally consistent closures. We establish core metatheoretic results (e.i. soundness and canonical-model completeness sketches) and analyze hyperintensional phenomena such as substitution failure in intentional contexts, quasi-indexicality, and de se reference. We compare the framework to classical intensional semantics (Montague), property theory (Bealer), hyperintensional logics (Fine), situation semantics (Barwise and Perry), and to the Leibniz program for a calculus of concepts. The result is a selfcontained formal framework that demonstrates that relations are not external causal links but intentional internal structures encoded in the guises through which agents and objects are conceived: i.e., they are perspectives.
Writings noted infinite regress in definitions and logic's precondition for conventions years before Quine.
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This essay argues that Alice Ambrose precedes key elements later critiques by Quine, first of Truth by convention (Quine 1936) and later of the analytic-synthetic distinction (Quine 1951). I demonstrate how Ambrose identifies in writing as early as in 1931: (1) the paradox of treating logical principles as mere conventions, (2) the infinite regress in stipulative definitions, (3) the preconditional role of logic for any convention, and (4) the instability of the analytic/synthetic divide. Ambrose, therefore, prefigures Margaret Macdonald (1934) unpublished dissertation (Cf. Spinney 2025) and predates some major contributions to the philosophy of logic by Quine a few years before he made them popular.
Drawing primarily on her early work (1931-1934), I argue that Alice Ambrose develops a philosophical project centered on preserving the rigor of extensional logic while rejecting the metaphysical and epistemological endorsements of logicism because of its commitment to the notion of material infinity. Positioning Ambrose as a transitional figure between formalism (Russell) and the constructivist turn represented by intuitionism (Brouwer), I demonstrate how Ambrose offers a practice oriented statement of finitist extensionalism. Employing only extensional methods (considering classes, relations, and propositions by reference to their members and truth values instead of mental processes), Ambrose reformulates an existential claim about pi as an explicit infinite disjunction of concrete instances insisting, against intensional projects, that such claims gain meaning only through a finite stopping rule that produces a witness.
Sophie Germain (1776-1831) was the first woman we know who did important original research in mathematics, specifically in elasticity theory and number theory. Celebrating her semiquincentennial year, we outline Germain's recently unearthed number theory results on Fermat's Last Theorem, in the context of her life, work, and interactions with Lagrange, Legendre, and Gauss. For two centuries her accomplishment on Fermat's Last Theorem was thought to consist of a single theorem attributed to her in a publication by Legendre, the first general result towards proving Fermat's Last Theorem. But recent discoveries in her handwritten manuscripts and correspondence with Legendre and Gauss show that she accomplished much more, albeit forgotten. In particular, she had a grand plan for proving Fermat's Last Theorem in its entirety, and carried this plan a long way, using then new tools, e.g., congruence, modular primitive roots, and permutations.
These notes have been written for a series of lectures to be given at the 44th Finnish Summer School on Probability and Statistics in Lammi, Finland, from 25th to 29th May, 2026. They contain an introduction to Wiener chaos decomposition in finite dimension, a construction of Gaussian fields on the torus, including white noise and the Gaussian free field, and applications to the $\Phi^4$ model. They do not cover other important aspects of the topic, such as stochastic integration, stochastic PDEs and Malliavin calculus.
We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals. The continuum is not necessarily identifiable with R; even if one seeks such an identification, infinitesimals are not ruled out. Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory and in this sense is epistemologically 'safe'. We sketch a recent theory of infinitesimal analysis that formalizes Leibnizian definitions and heuristic principles while eschewing both the axiom of choice and ultrafilters, thus challenging received philosophical views on the nature of infinitesimals.
Real-world focused mathematicians integrate computation readily while theory-focused ones keep it separate, per interviews at a long-experi
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Recent research in mathematics education points to an "epistemic clash" when programming and computational thinking (CT) are leveraged alongside more established forms of mathematical thinking (MT). The emergence of generative AI emphasises the need to understand the mechanisms shaping relations between CT and MT. We address this need by analysing interviews with 15 mathematicians on their use of computations across their teaching and research activities. The interviews were conducted at a critical site with a history of integrating computations across its science and mathematics programs for more than 20 years. Drawing on Cultural Historical Activity Theory and Communities of Practice theory, we consider MT and CT as methodologies grounded in practice. We identify three perspectives shaping how mathematicians position CT: mathematical theory considered as a source of control, computations as a source of pragmatic reach, and real-world impact as a source of legitimacy. This three-perspectives model explains why mathematicians who emphasise real-world impact are most likely to carry programming into teaching, whereas those who position theoretical mathematics as authoritative are least likely to do so. Mathematicians working on numerical algorithms occupy an uneasy intermediate position. Our findings suggest that the perceived clash between MT and CT is not purely epistemic, but also ontological, as it depends on how computations are positioned within the goal of doing mathematics. For mathematics education, this implies that perceived meaningful integration with CT is mediated by context, and that more extensive use can be stabilised by leveraging authentic learning goals external to mathematics.
Telescoping sums, partial fractions, and Fourier analysis make several profound entries accessible and reveal their connections.
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We revisit several entries from Ramanujan's notebooks which follow from more elementary arguments than a first glance may suggest. Our goal is to demystify these results through more accessible proofs, while also shining some light on the web of interconnections within the notebooks and demonstrating the continuing relevance of Ramanujan's methods. Classical and modern tools, such as multisection, telescoping sums, partial fraction decomposition and Fourier analysis, are employed to reprove and extend identities originally presented without explanation. These contributions try not only to enrich our understanding of Ramanujan's intuition but also to offer new avenues for exploration in number theory, special functions and mathematical analysis.
The distance formula, angle sum proof, and equations for special curves now follow directly from the 1868 mapping to a Euclidean disc.
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Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More importantly he showed that figures on the disc would satisfy the identities of hyperbolic geometry characteristic of a surface of negative curvature. However Beltrami did not always give a full explanation of the equations which he used. These notes are an attempt to provide a derivation of some of his principal results, including his formula for hyperbolic distance on the disc, his proof that the sum of the (hyperbolic) angles of a triangle on the disc is less than two right angles and his equations for circles, equidistants and horocycles.
Other focus lies at the diagonal crossing, proving existence exactly when the quadrilateral is orthodiagonal.
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We solve the following problem of W.H. Besant using a formula for the coefficients of an ellipse inscribed in a quadrilateral, $Q$: \enquote{If an ellipse be inscribed in a quadrilateral so that one focus is equidistant from the four vertices(call that point $EP$), the other focus must be at the intersection of the diagonals(call that point $IP$).} We also prove somewhat more than just solving Besant's problem itself, though it would be nice to see the details of the geometric approach proposed by Besant. More precisely, we also prove the converse result and additional results when $Q$ is a trapezoid. Finally, we show that such an inscribed ellipse exists if and only if $Q$ is orthodiagonal.
Pathologies such as non-identifiability arise only from special non-transverse alignments of the kernel feature map.
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These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is well-behaved - in the sense of being identifiable, having non-singular Fisher information, and admitting robust estimation - is reformulated as a question about the geometry of a kernel-induced feature map. Statistical pathologies correspond to geometric degeneracies of this map, and transversality theory provides a precise language for understanding when and why such degeneracies are non-generic.
The exposition is organised in three parts. Part I surveys the statistical phenomena that motivate the geometric treatment: representation failure, non-identifiability, moment indeterminacy, singular information, nuisance parameters, and the Behrens-Fisher problem. Part II develops the necessary geometric toolkit - smooth maps, Sard's theorem, transversality, jets, stratifications, and the parametric transversality theorem - at a level accessible to students with a background in analysis and linear algebra but no prior exposure to differential topology. Part~III returns to the statistical problems of Part~I and shows how each one admits a unified geometric interpretation as a transversality condition on the feature map.
These notes are a pedagogical companion to the research paper Labouriau (2026) "Transversality and Geometric Regularisation in Distributional Statistical Models" (arXiv:2605.04536 [math.ST]), expanding its arguments with motivating examples, geometric intuition, and exercises aimed at advanced Master's and PhD students with a background in mathematical statistics and measure theory. They are designed to support seminars or reading groups.
We propose a continuous measure of tonal ambiguity that extends the established concept of uniqueness. While uniqueness is widely regarded as necessary for tonality, it cannot (i) discriminate among sets that possess it, (ii) capture hierarchical organization in modes of limited transposition, or (iii) account for temporal unfolding. To address these limitations, we introduce a companion measure, grounded in information theory, that quantifies tonal ambiguity on a continuous scale. The measure applies across pitch-class sets and tuning systems, expanding analytic coverage of tonal relationships and offering a practical tool for theory and analysis.
This is an overview of Nasir al-Din al-Tusi's Treatise of the quadrilateral, an invaluable 13th century document on spherical geometry which was translated into French in 1891. The title we are using here is the one given by the translator (Alexandre Carath{\'e}odory). A title which is closer to the original Arabic is ''Disclosing the secrets of the secant figure.'' The term ''secant figure'', to which the title refers, is the so-called ''complete (spherical) quadrilateral'', that is, the figure that underlies what we call today Menelaus' Theorem. This theorem gives a formula that was extensively used by astronomers in their computations and the establishment of their tables since the first century AD, notably by Ptolemy, in the absence of the spherical trigonometric formulae that were discovered later. Nasir's treatise contains much more than Menelaus' theorem, since we find there a complete system of spherical trigonometric formulae, with complete proofs. The treatise includes at the same time invaluable historical information on the discovery of the trigonometric formulae by the Arab mathematicians of the Middle-Ages and the transformation of the field of spherical trigonometry that this discovery led to. The final version of this paper will appear in the book Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert, edited by Renzo Caddeo and Athanase Papadopoulos, Springer, 2026.
Vectorized doomsdays and Doomyears turn year and month offsets into lookups that exploit 28-year cycles and month gaps.
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Conway's Doomsday Algorithm (1973) determines the day of the week for any date in the Gregorian calendar via three additive components: a century anchor, a year offset, and a month-day offset. The century anchor is a fixed four-entry table. The other two components require live arithmetic: the year offset demands computing $y + \lfloor y/4 \rfloor \pmod{7}$, and the month-day offset requires a subtraction that can produce negative intermediate values. We present two new encoding schemes that replace both arithmetic steps with structured table lookups.
The first, vectorized doomsdays, re-encodes each month's doomsday date as a two-digit number whose tens and units digits represent the backward and forward gaps (respectively) from the nearest multiples-of-seven month anchors. A directional crossing rule (the "square knot rule") pairs the target date's gap with the opposite-direction digit, reducing the month-day offset to a single-digit addition. The second, Doomyears, encodes the year-offset function as a navigational lookup exploiting the 28-year periodicity of the Gregorian weekday cycle. Together with Conway's century anchor table, these form a unified system we call the Calamity Tables. We prove correctness, establish self-verification properties, analyse the internal structure of both encodings, and compare the cognitive complexity of the Calamity Table system against the standard arithmetic method.
The resulting manuscript derives a new explicit formula for a quantum group central element and completes the task in under a minute instead
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This article describes the use of Claude CLI and its Opus 4.6 model, as a tool for writing an entirely AI-generated mathematics research paper. The resulting paper is comparable in scope and quality to papers previously produced by advanced undergraduate students in eight-week summer REU programs advised by the author. The main result is a new explicit formula for a central element of $U_q(\mathfrak{so}_{12})$, which can be used for an interacting particle system with Markov duality. Using SageMath and a sparse PBW-basis pairing matrix that admits symbolic inversion, Claude reduced the central-element computation by several orders of magnitude: a calculation that took 60 hours in a 2023 Python implementation completed in under a minute on a laptop. The article reflects on the implications for undergraduate research mentorship: if generative AI can now produce research of REU caliber, advisors must select problems that better demonstrate the qualities valued by graduate admissions committees. Limitations including poor runtime estimates and literal handling of differing mathematical conventions are documented.
The article is dedicated to the memory and enduring legacy of Professor Robert V. Kohn, Courant Institute, NYU. In this memorial article, we record thoughts and reminiscences of his exemplary life.
Historical review of equivalence relations, Cauchy sequences, and metric spaces suggests their earlier versions offer overlooked benefits.
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The topic is the history of the concepts of equivalence relation, Cauchy sequence, and metric space. The thesis is that disused definitions of these notions could profitably be revived.
Replacing Whitehead's 3D axiom with finite dimensionality yields an exact match between simple matroids and maximal geometrical systems on a
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We give a correspondence between simple matroids and a reconstruction of Alfred North Whitehead's theory of dimension, as developed in "On Mathematical Concepts of the Material World" (1906). In brief, if a geometrical system in the generalized sense of Whitehead has finite ground set and is phi-maximal, then it is a simple matroid. Here "generalized" means that Whitehead's three-dimensional axiom is replaced by finite-dimensionality. Conversely, every simple matroid is a phi-maximal geometrical system in the generalized sense of Whitehead.
Newton's Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law! Here we discuss this and some other interesting features of the inverse cube force law.
The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency structure into a multilayer graph of 308,129 declarations, 8.4 million edges, and 7,563 modules. By introducing graph decompositions that isolate explicit edges from those synthesized by the compiler or driven by proofs, we quantify the structural properties of formalized mathematics. Our analysis reveals three findings. First, taxonomies designed by humans diverge from logical structures, exhibiting a 50.9% coupling across namespaces. Second, developers utilize a median of 1.6% of the imported scope. Third, formalization compresses semantic hierarchies, with network centrality capturing language infrastructure rather than mathematical relevance.
We present the results from Meno, a simple autoformalizer that proves theorems in Lean by systematically exploring the space of both formal and informal proofs, and tactic ablation, a new method for exploring mathematical creativity under constraint. We show these tools in action on simple theorems found in Terrence Tao's Analysis I, selectively ablating solution paths associated with non-constructive proofs, and analyze the properties of the resulting population using Goedel Prover embeddings. Among other things, our analysis of this novel population reveals that they lie on low (one or two) dimensional submanifolds of the much higher-dimensional representation space, and far away from their corresponding human constructions.
Drawing on the Data and Predictions strand of the Indicazioni Nazionali per il curricolo 2012, this study proposes a problem based instructional approach to the teaching of probability. More specifically, the study adopts a design based research methodology structured in a single cycle consisting of two teaching interventions in the same class, carried out in two consecutive years. Within this framework, a set of carefully selected problems is employed to foster students engagement. These problems are designed not only to introduce probabilistic concepts, but also to stimulate students' communicative and argumentative skills. The selected tasks provide opportunities to promote key process goals (such as reasoning and proving, communicating, representing, and making connections) which are often overshadowed by a predominant focus on content goals. This approach aims to support teachers in addressing the difficulties they frequently encounter in guiding students conceptualization processes, particularly in bridging the gap between students intuitive reasoning and formal abstraction. At the same time, it seeks to help students develop more robust and flexible forms of thinking, enabling them to better navigate situations involving uncertainty in everyday life.
Defining areas first, verifying standard integrals by direct sum manipulation, and crossing the fundamental theorem produces the full set of
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We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems about Riemann sums, cross the Fundamental Theorem of Calculus, and harvest the differential calculus on the other side.
This note provides an introduction to selected topics in algebraic graph theory, including strongly regular graphs, Steiner systems, and automorphism groups. We describe constructions and properties of notable graphs such as the Petersen graph, Paley graphs, Hamming graphs, and the Hoffman-Singleton graph, with emphasis on their symmetry and combinatorial structure. Connections with permutation groups are also discussed. Computational examples using SageMath are included to illustrate key concepts and to compute automorphism groups and related invariants.
In the winter semester of 1890--1891 Adolf Hurwitz delivered a lecture course at the Albertina University in K\"onigsberg entitled -Theorie der algebraischen Gleichungen-. These lectures contain a particularly clear presentation of the ideas of Evariste Galois and, in particular, a proof of the fundamental theorem of Galois theory formulated in the language of substitutions. The present paper analyzes Hurwitz's treatment of this result on the basis of his lecture notes preserved in the ETH Library in Zurich (Hs 582:66), together with material from his Mathematisches Tagebuch 23 (Hs 582:23). After placing the K\"onigsberg lectures in their historical context, we give an overview of their mathematical content and reconstruct in detail Hurwitz's argument leading to the fundamental theorem.
Possible talks include topology conjectures, self-dual geometry, Boolean algebras, and Yang-Baxter maps.
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I was interested in the work of Solomon Marcus in Mathematical Linguistics as a high-school student. Later, I had the opportunity to discuss with him about many topics. He was a polymath. We wrote a paper together, and I refereed an editorial paper about his work in 2021. Samples of (possible) discussions are presented: some topology conjectures, a self-dual theorem in geometry, results about Boolean algebras, a B-ring Euler formula, Yang-Baxter maps and a discussion on sequences and series. A short appendix on poetry is also included.
We give a method for constructing an interactive art piece which illustrates two different definitions of the braid groups, along with their faithful action on the free group. The box also demonstrates how all motions of points in the plane can be realized by motions in a single T-shaped subspace of the plane. This helps students and those who are not specialists in algebraic topology to understand these important topological objects.
A 12-week concrete-to-abstract program on breaking down large numbers yields bigger, longer-lasting gains than standard teaching.
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This study examines the effectiveness of a structured instructional approach to decomposition and recomposition of large numbers in six primary school classes (three Year 4 and three Year 5, N = 120) using a quasi - experimental design with a control group. The 12 - week intervention is grounded in the Concrete Pictorial Abstract (CPA) progression. The experimental groups achieved average gains of 34.0 points (Year 4) and 29.6 points (Year 5) out of 100, significantly higher than the control groups (16.4 and 11.1 points; p < .001). The Time Group interaction in the mixed ANOVA reached {\eta}^2p = .931. The ANCOVA with the pre - test as covariate estimated an adjusted difference of 18.25 points (F(1,117) = 2,978.10, p < .001, \eta^2p = .962), confirming the robustness of the effect after controlling for baseline differences. Four-week retention exceeded 97% in the experimental group. Internal reliability of the instrument was satisfactory (Cronbach's {\alpha} = .735).
Quadratic modules certify positivity, solving the K-moment problem for compact K via Schmüdgen and Putinar theorems.
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We present an expanded expository account of the $K$-moment problem for polynomial algebras over \(\R^d\), with special emphasis on compact basic closed semialgebraic sets. The central question is to characterize those linear functionals on \(\R[x_1,\dots,x_d]\) which admit representation by integration against a positive Radon measure supported on a prescribed set \(K\subseteq\R^d\). We begin with the classical background and with Haviland's formulation of the multidimensional moment problem, then explain how real algebraic geometry enters through quadratic modules, preorderings, and Positivstellens\"atze. The compact case is treated in detail from two complementary perspectives. The geometric route through Schm\"udgen's theorem and the operator-theoretic route through a Gelfand--Naimark--Segal construction and the spectral theorem. We also discuss Putinar's refinement, compare the roles of \(T(f)\) and \(Q(f)\), and explain how Archimedeanity provides the algebraic shadow of compactness. In order to place the subject in a broader context, we survey determinacy and uniqueness questions, the truncated \(K\)-moment problem and flat extension phenomena, the relation with sums of squares and Hilbert's seventeenth problem, and the special case of algebraic varieties, where positivity modulo an ideal becomes especially transparent.
A review traces their introduction for non-Mendelian rules and the subsequent growth in applications across disciplines.
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Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals with the past and present research on these kind of algebras, together with the most relevant topics regarding them.
Existing knowledge management tools either preserve prose but lose structural relationships, or capture relationships but restrict edge semantics to fixed vocabularies. We introduce Astrolabe, a content-addressable hypergraph for semantic knowledge management. Entries are identified by the SHA-256 hash of their content, carry an ordered reference list of arbitrary width, and store an opaque record string interpreted by plugins. The structure admits two orthogonal decompositions: by width and by depth. We demonstrate the framework with a plugin bridging informal and formal mathematics.
We present a minimal mathematical model for conducting patterns that separates geometric trajectory from temporal parametrization. The model is based on a cyclic sequence of preparation and ictus points connected by cubic Hermite segments with constrained horizontal tangents, combined with a quintic timing law controlling acceleration and deceleration. A single parameter governs the balance between uniform motion and expressive emphasis. The model provides a compact yet expressive representation of conducting gestures. It is implemented as the interactive Wolfram Demonstration "Conducting Patterns" and is used in the Crusis web app.
These personal reminiscences of the great Russian mathematician Sergey K. Godunov (1929-2023) arose from a request by his daughter, Ekaterina, to contribute a piece to a book she is writing about her father's life. I was honoured to accept this invitation and to give written form to the rewarding experience of conducting research on themes pioneered by Professor Godunov, interacting with him personally on several memorable occasions, and helping to establish research collaboration with his Novosibirsk group. Our association began at a conference in Lake Tahoe (USA) in 1995 and was followed by a number of subsequent meetings, notably in Novosibirsk, Manchester, Oxford, and Cambridge. Briefer encounters also took place in the Porquerolles Island (France), in Lyon (France), and in St. Petersburg (Russia). These notes bear witness to the global impact of Godunov's mathematical creativity across multiple branches of science, as well as to its lasting influence on the careers of generations of mathematicians in both academia and industry.
In AI-rich higher education, polished written mathematics has become easier to produce than trustworthy evidence of understanding. This article develops a human-scale methodology for service mathematics, with informatics as its main running case. Its central move is not prohibition of tools but relocation of evidential trust. Students prepare openly, often with digital assistance, but grade-relevant evidence shifts toward live explanation, contingent questioning, and cumulative observation against course outcomes. The design is guided by Realistic Mathematics Education, question-first task construction, short human-scale mathematical tasks, and instructor synthesis after student attempt. It contributes a weekly operational cycle, a realism framework distinguishing professional, disciplinary, and experiential realism, a middle-out white-box / black-box stance on tools, a bounded role for retrieval-grounded AI assistants for students and teachers, and a cumulative oral-evidence model for small and medium cohorts. The paper also provides concrete implementation artifacts: process figures, an ecology of problem types, time-budget estimates, an evidence hierarchy, and a five-grade oral rubric. This is a methodology paper rather than an effectiveness study. Its claim is that the proposed design is pedagogically coherent, operationally plausible for human-scale teaching settings, and responsive to current concerns about AI, oral evidencing, and active learning in undergraduate mathematics education.
This study reconstructs the origin of a constant, here called $\Xi$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $\Xi$ arises from the practical necessity of precise measurements on the sky or a circle, through the harmonization of length-measure systems. The analysis of the Nippur measure (with its famous cubit) and the Gudea measure shows that $\Xi = 375/360$ represents the ratio of these established Old Babylonian measure systems. As a precision factor for circumference calculations, it remained in use until today. In Ptolemy's work, we find a slightly refined value of $\Xi = 377/360$. A further refinement of this coefficient led to our modern $\pi$, which still incorporates the two Old Babylonian components of a demonstrably two-stage calculation and refinement process. The accuracy increased by only 0.5\% compared to the first ratio. This factor, attested on several Old Babylonian cuneiform tablets including those from Susa, demonstrates the profound understanding of sexagesimal logic. The relative sexagesimal notation (60 = 1 = 1/60) enabled the universal application of $\Xi$ and its reciprocal for highly accurate calculations of arc-length on circular segments. This investigation leads ultimately to a surprising consequence: the modern radian measure is a direct descendant of this Old Babylonian coefficient.
The transform connects summation identities to Dirichlet series behavior, allowing analogues learned from Yakubovich to be derived.
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W. L. Ferrar seems to have been the first mathematician to clearly draw a connection between the functional aspects of a summation formula and the behavior of the Dirichlet series underlying it. Taking a formula due to him as a starting point, I will describe some new generalizations of Ferrar's formulas and how these were actually obtained after learning a great deal from Semyon. I also present a very concise overview of the underlying theory of summation formulas and how the Mellin transform has been the link between mine and Professor Yakubovich's interests.
Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we further consider how AI may open a grand perspective on mathematics by forging a new route, complementary to mathematical\textbf{ logic,} to understanding the global structure of formal \textbf{proof}\textbf{s}. We begin by providing a sketch of the formal structure of mathematics in terms of universal proof and structural hypergraphs and discuss questions this raises about the foundational structure of mathematics. We then outline the main ingredients and provide a set of criteria to be satisfied for AI models capable of automated mathematical discovery. As we send AI agents to traverse Platonic mathematical worlds, we expect they will teach us about the nature of mathematics: both as a whole, and the small ribbons conducive to human understanding. Perhaps they will shed light on the old question: "Is mathematics discovered or invented?" Can we grok the terrain of these \textbf{Platonic worlds}?
The holiday gift exchange game is a familiar social institution with nontrivial strategic structure. We provide a formal treatment of the game's mechanics, defining the state space, action sets, and the recursive structure of stealing chains; we prove termination and derive an algorithm for counting distinct game trajectories, which grow far faster than the space of possible final allocations. Beyond the base mechanics, we introduce a decorated model incorporating partial information, social costs, and adaptive strategies grounded in discrete choice theory and the frustration-aggression literature. A full factorial simulation of 240,000 games yields three findings of note: implicit social costs are the dominant regulator of aggression, reducing stealing by 27--48\% and outweighing both uncertainty and strategic sophistication; partial information, contrary to expectation, slightly increases stealing through asymmetric uncertainty; correlated valuations amplify every behavioral effect, so that consensus about gift quality, rather than the features themselves, is what intensifies competition. The first-player advantage is robust across all conditions.
Public universities in the US must now meet digital accessibility (DA) standards under 2024 updates to Title II of the ADA. For math instructors, course materials must be screen-reader parsable, which standard LaTeX-to-PDF workflows cannot achieve. Despite MathML's availability as a web standard for accessible math, instructor adoption of DA-compliant workflows remains very low, creating a gap between available technology and classroom practice.
This paper makes three contributions. First, we present a taxonomy of existing approaches to DA-compliant math content, organized by print (PDF) versus web (HTML) output targets, analyzing tradeoffs for instructor adoption. Second, we describe a free workflow using Obsidian (Markdown-based content management), Quartz (static site generator), Git (collaboration and version control), and Cloudflare Pages (free hosting, private source files) that enables math instructors to create and publish DA-compliant course websites with MathML from TeX-based syntax. Setup takes approximately 1-2 hours; thereafter, site updates occur in minutes via a single command. A public setup tutorial is made available. Third, we present an empirical study of student outcomes across 31 sections of Calculus II over 6 semesters. Sections using the proposed system outperformed controls, with the treatment group reaching 2.4 standard deviations above the control mean in the final semester. Although all treatment sections were taught by one instructor, evidence such as acclimation trajectories of other new instructors suggests the system itself meaningfully contributes to performance gains. A student experience survey shows no statistically significant difference between groups, indicating no negative effect on experience. A proposed second study phase will assess barriers to adoption at other institutions.
Without action on values, practice, teaching, technology and ethics, external forces may set the direction of the field.
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Artificial intelligence is transforming mathematics at a speed and scale that demand active engagement from the mathematical community. We examine five areas where this transformation is particularly pressing: values, practice, teaching, technology, and ethics. We offer recommendations on safeguarding our intellectual autonomy, rethinking our practice, broadening curricula, building academically oriented infrastructure, and developing shared ethical principles - with the aim of ensuring that the future of mathematics is shaped by the community itself.
It preserves reasoning rules as far as possible but accepts violations when reasons for breaking them are stronger, as Hamilton did with qu
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The view that Peacock's principle of permanence has been invalidated by Hamilton's introduction of non-commutative algebras has always seemed rather odd, in light of Peacock's favorable reception of quaternions and the endorsement of his principle by Hamilton. But the view is not just odd; it is incorrect. In order to show this, I critically analyze Peacock's attempts to reject possible exceptions to his principle, like the factorial function and an infinite series due to Euler. Then I argue that the principle of permanence is best understood as an expression of a conservative strategy, philosophically grounded in Hume's conception of the laws of reasoning, which advocates their preservation to the furthest extent possible, thus allowing exceptions, i.e., violations of these laws. On this reading, non-commutative multiplication does not invalidate Peacock's principle, if the reasons for violating commutativity outweigh the reasons for its preservation. Finally, I show that Hamilton followed a conservative strategy of precisely this sort when he developed his quaternionic calculus.
Mathematicians need to track these tools and adjust their methods to the new capabilities.
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Recent developments show that AI can prove research-level theorems in mathematics, both formally and informally. This essay urges mathematicians to stay up-to-date with the technology, to consider the ways it will disrupt mathematical practice, and to respond appropriately to the challenges and opportunities we now face.
Periodic orbits on lattices drive domain-colored doubly-periodic functions to repeat seamlessly.
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This paper describes an approach to generating looping animations using the modular flow and elliptic functions. The modular flow is a flow on lattices with many periodic orbits, and elliptic functions are meromorphic, doubly-periodic functions which can be visualized using domain coloring.
