Mathematics
Hardy-Type Uncertainty Principles Extended to Variable-Geometry Magnetic Schrödinger Equations
Fanelli et al. establish Hardy-type uncertainty principles and unique continuation results for covariant Schrödinger equations with variable coefficients under bounded electric and magnetic potentials. The central result shows that solutions exhibiting super-quadratic exponential decay at two distin…
Bijection Between Rooted Bicubic Planar Maps and Colored Dyck Paths Yields Constructive Decomposition Theory
Gil and Kaminski establish an explicit bijection between rooted bicubic planar maps on 2n vertices and Dyck paths of semilength 3n whose ascents have lengths divisible by 3, with each 3j-ascent colored by one of g_j colors indexing rooted 3-connected bicubic maps on 2j vertices. This bijection provi…
The Wobbly Table Theorem Gets an Interactive 3D Visualization
Tim Vieira built an interactive 3D visualization of the "wobbly table theorem" — a mathematical result proving that any four-legged table on a continuous, uneven surface can be stabilized by rotation alone (without adjusting leg lengths). The project falls under the "vibe-coding" trend of using AI-a…
Scaling Limits of Random Walks and Intrinsic Metrics in Critical Percolation
This paper demonstrates that simple random walks on critical site percolation clusters on a triangular lattice converge to a continuous diffusion within the gasket of a CLE₆. Furthermore, the intrinsic metric also converges to the geodesic CLE₆ metric. These findings establish the existence of key e…
New Fundamental Lemma for Cubic Shimura Lift to PGL(3)
This paper establishes a new Fundamental Lemma for Hecke correspondences in the context of the cubic Shimura lift. It demonstrates an algebra isomorphism between the spherical Hecke algebras of PGL_3(F) and the cubic cover G' of SL_3(F). This work is crucial for developing a relative trace formula t…
Generalized Finsler Warped Product Metrics with Vanishing Curvature
This paper investigates weakly orthogonally invariant Finsler metrics, providing explicit expressions for their Berwald and Landsberg curvatures. It then establishes a system of partial differential equations that characterize generalized Finsler warped product metrics under conditions of vanishing …
Möbius Strip Diagram Algebras: A New Framework for Nonorientable Cobordisms
This paper introduces novel "Möbius strip diagram algebras" which extend existing partition-style diagram calculi by incorporating handles and Möbius strip features. The authors equate this new diagram category with a linear quotient of a 2D nonorientable cobordism category. The work culminates in t…
Geometric Classification of Steklov Eigenvalues on Trees with Diameter Constraints Completed
This research completes the geometric classification of the first nonzero Steklov eigenvalue (λ₂) on finite trees under diameter constraints. It specifically addresses the previously unresolved cases of odd diameters D ≥ 5, identifying that maximum λ₂ is achieved on generalized almost seesaw trees. …
Modular Functions Decode Monster Group Exponents for Primes >3
Duncan and Swisher introduce a modular function-based method to derive the exponents in the prime factorization of the Monster group's order, specifically for primes larger than 3. This approach leverages properties of modular functions to explain these exponents, connecting number theory and group …
Sol LeWitt's Incomplete Open Cubes: A Mathematical and Artistic Exploration
Sol LeWitt's "Variations of Incomplete Open Cubes" is an artwork that visually represents a mathematical enumeration problem. The artist, a conceptualist, meticulously documented his empirical process of identifying 122 unique cube variations, constrained by connectivity, three-dimensionality, and r…
Complex Exponentials as Dynamic Solutions Unlock Intuitive Solving of Linear Differential Equations
Complex exponentials e^(st) visualize differential equations dynamically: real part of s drives growth/decay, imaginary part ω dictates oscillation frequency. Substituting e^(st) into linear homogeneous DEs with constant coefficients reduces them to algebraic polynomials in s, whose complex roots en…
Arctic Circle Theorem: Central Disorder in Aztec Diamond Domino Tilings Converges to Circle
In random domino tilings of large Aztec diamonds, the central region of coexisting tile orientations forms a shape arbitrarily close to a circle of radius n/sqrt(2), while outer regions exhibit uniform alignment. This Arctic Circle Theorem is proven using interacting particle systems, specifically a…
Pi and Tau in Binary
In binary representation, the digits of Pi and Tau are identical, with Tau's digits being a left-shifted version of Pi's. This mathematical curiosity arises from the fundamental relationship between the two constants (Tau = 2 * Pi) and the nature of binary representation.
Profinite Criterion for Primitive Words in One-Relator Groups with Torsion
This paper introduces a method for identifying surface subgroups within specific one-relator groups that possess torsion. This discovery leads to the derivation of a profinite criterion. This criterion ascertains whether a given word in a free group is primitive, offering a novel tool for analyzing …