Arithmetic dynamics and Generalized Fermat's conjecture
This paper proposes a generalized Fermat's conjecture within the framework of arithmetic dynamics, provides supporting evidence for it, and introduces a multi-indexed version of the conjecture.
🔢 Category
math
2549 papers
A note on the omega-chaos
This paper establishes sufficient conditions for the infinite direct product of a continuous self-map on a compact metric space to be -chaotic and applies these findings to construct examples of unusual -chaotic maps.
On strong law of large numbers for weakly stationary -mixing set-valued random variable sequences
This paper extends the concept of -mixing to set-valued random sequences in Banach spaces and establishes several strong laws of large numbers for such sequences, demonstrating the naturalness and sharpness of the underlying hypotheses through illustrative examples.
Transversal and Hamiltonicity in a bipartite graph collection
This paper establishes improved minimum degree conditions that guarantee the existence of Hamiltonian transversals and Hamiltonian connectivity in both balanced and nearly balanced bipartite graph collections, thereby advancing previous results in the field.
Classifying integer tilings and hypertilings
This paper classifies all tame integer tilings and hypertilings by developing a geometric model based on generalised Farey graphs in the hyperbolic plane, which connects these structures to triangulated polygons, friezes, and the Cayley hyperdeterminant.
A Comparison of Gauge Dimension and Effective Dimension
This paper characterizes the gauge profiles of sets of reals defined by effective dimension and demonstrates a separation between these sets and -well approximable numbers using Hausdorff measure.
Erd\H{o}s Matching (Conjecture) Theorem
This paper proves the long-standing Erdős Matching Conjecture, establishing that the maximum size of a family of -sized subsets of containing no pairwise disjoint sets is bounded by the larger of two canonical extremal constructions.
Almost All Vectorial Functions Have Trivial Extended-Affine Stabilizers
This paper proves that asymptotically almost all vectorial functions over finite fields possess trivial extended-affine stabilizers, implying that the number of equivalence classes matches the naive estimate and that random sampling is a highly effective strategy for cryptographic primitive design due to the exponential rarity of functions with nontrivial stabilizers.
Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion
This paper establishes the existence of global weak solutions for the discrete nonlinear fragmentation equation with degenerate diffusion in arbitrary spatial dimensions by utilizing weak estimates and compactness arguments, thereby extending previous results that were limited to one-dimensional domains with uniformly positive diffusion.
Estimating with a Coin
This paper introduces a novel Monte Carlo method for estimating by tossing a coin, which utilizes a new interpretation of derived from Catalan-number series identities.
Cohomological support varieties of certain monomial ideals
This paper presents a procedure for efficiently computing cohomological support varieties of certain monomial ideals, which leads to the discovery of new examples that are not unions of linear subspaces and a computer-assisted classification of such varieties for homogeneous monomial ideals with six generators over .
Invertibility of the Fourier Diffraction Relation in Raster Scan Diffraction Tomography
This paper establishes that in raster scan diffraction tomography using focused beams, the object's scattering potential is generically uniquely determined in dimensions higher than two, whereas in two dimensions, only a specific subset of Fourier coefficients can be uniquely recovered while others remain ambiguous.
Raster Scan Diffraction Tomography
This paper extends diffraction tomography to accommodate focused beam scanning by modeling incident fields as Herglotz waves, deriving a new Fourier diffraction relation that enables quantitative reconstruction and reveals how different scan geometries impact the imaging results.
Long finite time bubble trees for two co-rotational wave maps
This paper demonstrates that the energy-critical co-rotational wave maps equation from into admits finite-time blow-up solutions containing arbitrarily many concentric concentrating bubbles with alternating signs, thereby confirming that all scenarios predicted by the soliton resolution conjecture can occur.
A problem of Heittokangas-Ishizaki-Tohge-Wen concerning a certain differential-difference equation
This paper resolves an open problem posed by Heittokangas, Ishizaki, Tohge, and Wen by completely characterizing all finite-order entire solutions of the differential-difference equation , where the coefficients are polynomials and .
Overcolored Partition Restricted by Parity of the Parts
This paper extends the recently defined function , which counts multicolored partitions with specific color restrictions for even and odd parts, to the context of overpartitions.
Cohen-Macaulayness of squarefree powers of edge ideals of whisker graphs
This paper investigates the Cohen-Macaulayness, purity, and shellability of the -th squarefree powers of edge ideals of whisker graphs by characterizing these properties in terms of the structural features of the underlying graph, such as its girth and bipartiteness, while also computing depths and verifying a related conjecture.
Mathematical Proof
This paper presents a set of course notes designed to introduce undergraduate students to mathematical proofs, covering foundational topics such as logic, proof techniques, induction, set theory, and real analysis, complete with numerous examples and exercises.
On the intrinsic geometry of polyhedra: Convex polygon coordinates
This paper investigates the intrinsic geometry of polyhedra by utilizing barycentric algebras to characterize coordinate systems, specifically presenting a coalgebra-based algorithm for computing chordal coordinates in convex polygons that naturally yields the Catalan number enumeration of their triangulations.
A finite element continuous data assimilation framework for a Navier--Stokes--Cahn--Hilliard system
This paper develops and analyzes a continuous data assimilation framework using a nudging-based approach and a capped finite element splitting scheme to recover trajectories of a coupled Navier-Stokes-Cahn-Hilliard system with an auxiliary field from coarse spatial observations, demonstrating its effectiveness in synchronizing mismatched initial conditions through numerical experiments.