2572 papers
This paper introduces a diagonalization method for functions from natural numbers to real numbers and applies it to prove that the equation has only a finite number of solutions for any fixed natural numbers and .
This paper establishes that the explorability of a finitely generated group's Cayley graph by finite automata systems depends on the group's periodicity: periodic groups confine automata to finite areas, groups with non-periodic elements are explorable by automata with three pebbles, and aperiodic groups cannot be explored by any finite automata system.
This paper introduces and develops the concept of ramification within a given modulus, exploring its properties and its potential connections to significant mathematical problems, particularly the Goldbach conjecture.
This paper investigates the distribution of boundary points of expansion to establish a lower bound on the mutual distances between runners on a circular track under specific equidistant spacing conditions, thereby providing a partial application toward the Lonely Runner Conjecture.
This paper introduces the theory of the Collatz process and the method of dynamical balls to formulate the Collatz conjecture, explore its connection to Sophie Germain primes, and develop tools for analyzing the convergence of sequences generated by iterating on fixed integers.
Using the compression method, this paper generalizes lower bounds for the Erdős unit distance and distinct distance problems to higher dimensions ( for ), providing alternative proofs and establishing new estimates for the number of distinct distances.
This paper presents and proves the existence of an efficient algorithm for computing -expansions of classical modular forms of arbitrary weight and level for any congruence subgroup, while also providing the necessary theoretical background and practical implementation details.
This paper provides a comprehensive catalog of Lambert series identities, detailing their formal properties, combinatorial generalizations, and connections to partition functions, while offering a reference compendium of known formulas for various special cases without focusing on rigorous analytic convergence.
This paper improves the current upper and lower bounds for the Heilbronn triangle problem by applying concepts from the geometry of compression, establishing that the minimal triangle area for points on a unit disk satisfies and .
This paper proposes a nonparametric two-sample hypothesis test for low-rank random graphs of differing sizes, utilizing maximum mean discrepancy on optimally transported graph embeddings to determine if the networks originate from the same distribution.
The paper demonstrates the existence of non-formal compact (almost) contact manifolds with specific first Betti numbers and, in certain dimensions, simple connectivity, thereby resolving a key problem in the geography of such manifolds.
This paper establishes a cohomology theory for multipoint connections on complex curves by assuming recursion relations for complex functions, explicitly expressing the results in examples as higher-genus generalizations of elliptic functions derived from functional equations.
This paper establishes that the decomposition theorem components for contraction maps of torus actions of complexity one are intersection cohomology complexes of even codimensional subvarieties, leading to the vanishing of odd-dimensional intersection cohomology for rational complete varieties of this type and providing explicit formulas for the Betti numbers of affine trinomial hypersurfaces based on their defining equations.
The paper proves that any odd-dimensional parallelisable closed manifold admits a contact structure, thereby establishing that odd-dimensional solvmanifolds are contact.
This paper proves the nonexistence of Heronian triangles with three integer medians by deriving a universal identity for arbitrary triangles and establishing a lemma demonstrating that such triangles, if they existed, would necessarily occur in non-similar pairs.
This paper establishes the local-in-time well-posedness of smooth classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters by utilizing new higher-order weighted energy functionals and estimates to handle the degeneracy near the moving boundary.
This paper constructs spectral triples for one- and two-qubit states to define Connes spectral distances, which are then used to propose new measures for quantum discord and coherence, revealing geometric structures such as the Pythagorean theorem in specific two-qubit cases.
This paper establishes results on distinguishing Siegel cusp forms of degree two, demonstrating that a level-one Hecke eigenform can be uniquely determined by its second Hecke eigenvalue under a specific assumption and that two such eigenforms can be distinguished using -functions.
This paper establishes a new Morales-Ramis type theorem for the meromorphic Jacobi non-integrability of general analytic dynamical systems by linking Jacobian multipliers to Lie algebra structures, and applies this criterion to analyze the polynomial integrability of Karabut systems modeling stationary gravity waves.
This paper constructs a family of parametric extensions of coboundary operators for double complexes of meromorphic functions on higher genus Riemann surfaces by utilizing Schottky uniformization and the algebraic completion of an infinite-dimensional Lie algebra module.