Mathematics
Besov Space Well-Posedness for Reduced Micropolar Fluid Models
This paper establishes well-posedness theory for a two-dimensional reduced model of inviscid micropolar fluids, which couples Euler-type velocity dynamics with scalar microrotation transport-diffusion. Local strong solutions are proven to exist and be unique in Besov spaces B^s_{p,1} with regularity…
Multidimensional Symmetric Cooperative Motion and Porous Medium Equation Convergence
This paper extends the study of symmetric cooperative motion to higher dimensions by employing a novel approach that discretizes the porous medium equation. The authors overcome challenges related to finite difference schemes and unbounded initial data, ultimately establishing a new multidimensional…
Massey Products as the Algebraic Backbone of Spherical T-Duality for Iterated Sphere Bundles
Cavalcanti demonstrates that the cohomological data encoded in the Gysin sequences of iterated sphere bundles can be reformulated in terms of a vanishing Massey product. When such a bundle is equipped with an integral cohomology class of transgressive degree one, it admits a T-dual iterated sphere b…
Fair Division of Goods with Independence Structures and Epistemic Envy-Freeness
This paper studies the fair division of indivisible goods under an independence structure, motivated by network connections. It establishes the existence of epistemic EF1 partitions for agents with arbitrary additive valuations, and provides conditions for envy-freeness up to one good when valuation…
Shifted Twisted Quantum Affine Algebras: Structure, Representations, and Classification
This paper introduces shifted twisted quantum affine algebras, a twisted version of shifted quantum affine algebras, derived from Drinfeld's current presentation. These algebras admit a triangular decomposition and their structure is independent of the specific coweight pair, depending only on the t…
Log-Critical Well-posedness for Two-Phase Muskat Problem with Surface Tension via Schauder Estimates
This paper establishes short-time well-posedness for the two-phase Muskat problem with surface tension, accommodating varying viscosities, arbitrary density contrasts, and rigid boundaries without the Rayleigh-Taylor condition. The core contribution lies in deriving novel Schauder-type estimates for…
Superspace Coinvariants and Inverse Systems for $GL_n(\mathbb{F}_q)$
This paper introduces and analyzes the $GL_n(\mathbb{F}_q)$-superspace coinvariant ring, defined as the quotient $\Omega/SI$. The authors have successfully computed the bigraded Hilbert series for this ring and established an operator-theoretic characterization for its associated inverse system, $SI…
Generalized Convex Integration Framework for Weak Solutions in Geophysical Fluid Dynamics
This paper introduces a generalized convex integration framework to construct weak solutions for PDEs, particularly those relevant to geophysical fluid dynamics. The method produces $L^\infty$ solutions that are weakly time-continuous. This approach has successfully been applied to yield new results…
Topology of H-Structures and Intrinsic Torsion Energy on Lie Groups
This paper investigates the topology of isometric classes of tensorial H-structures on closed, connected Lie groups, particularly focusing on their relationship with Riemannian metrics and intrinsic torsion energy. The authors establish a homotopy equivalence between the space of H-structures and Ri…
Equivalence of Chiral Algebras with Abelian Conformal Structure to Differential Algebras
This paper establishes an equivalence between chiral algebras with an abelian conformal structure and differential algebras. Specifically, it demonstrates that the variety of chiral algebras defined by a binary quadratic operad Var with an abelian conformal part is identical to the variety of differ…
Sharp Convergence Rate Analysis of Circumcentered-Reflection Method for Subspace Projections
This paper analyzes the circumcentered-reflection method (CRM) for projecting onto the intersection of two subspaces, $U$ and $V$. The authors demonstrate that when CRM is initialized in $V$, it achieves a sharp and optimal convergence rate of $\rho_V = (\sin^2\theta_p - \sin^2\theta_F) / (\sin^2\th…
Bounding Automorphism Groups of Algebraic Curves by $p$-Rank in Positive Characteristic
This paper addresses the challenge of bounding the automorphism group size of algebraic curves in positive characteristic $p$. It proves that unlike genera, no general bound exists based solely on the $p$-rank. However, a specific bound of the form $|\Gamma|<900 \left(\frac{p}{p-1}\right)^4 \gamm…
Structural Properties of Constrained Graph Homomorphism Orders
This paper introduces and analyzes partial orders derived from constrained graph homomorphism types, extending standard properties like canonical cores and finiteness to these variants. It identifies specific cores for full and surjective homomorphisms, links them to point-determining graphs, and ch…
New Fourier Restriction Theorem Utilizing L^q-Dimensions Beyond Stein-Tomas
This paper introduces a novel Fourier restriction theorem that replaces the Frostman condition with $L^q$-dimensions, thereby offering a continuous range of estimates that encompass the Stein-Tomas theorem at its endpoint. The proof employs Stein's complex interpolation and yields endpoint estimates…
Layer-Ratio Monte Carlo Estimates of Dedekind Numbers Reveals Non-Unimodal Whitney Number Structure
Chen et al. introduce a layer-ratio Monte Carlo method to estimate Dedekind numbers M(n)—the count of monotone Boolean functions—by reformulating the problem as reconstructing Whitney numbers of the ideal lattice. Their approach uses reversible fixed-layer Markov chains to estimate local averages of…
Bounded Solutions and Critical Thresholds in Quasilinear Reaction-Diffusion Equations
This paper investigates a quasilinear reaction-diffusion equation in $N\geq3$ dimensions. For initial data satisfying specific conditions related to its decay at infinity, the authors prove that the solution remains uniformly bounded. This boundedness is shown to be a critical behavior dependent on …
Bernstein Rigidity for Capillary Minimal Graphs: Linearity in Low Dimensions and Connectivity of Minimizers
Edelen, Li, and Zhu establish a Bernstein-type rigidity theorem for capillary minimal graphs in a half-space, proving that entire capillary minimal graphs must be linear (i.e., affine hyperplanes) in low dimensions, or more generally when no tangent cone at infinity splits off a vertical line. They …
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 …