69 papers
This paper establishes sharp, up to polylogarithmic factors, small cap square function and decoupling estimates for the parabola using axis-parallel rectangles of dimensions for $0 \leq \beta \leq 11 \leq \beta \leq 2$.
This paper demonstrates that partial differential equation propagation acts as a spectral preconditioner that improves Fourier ratio bounds, thereby enabling stable recovery of discretized fields from incomplete random spatial samples with sampling rates that are effectively independent of grid resolution.
This paper establishes sharp uniform bounds on the eigenvalues of time-frequency localization operators and coherent state transform localizations near their spectral phase transition, revealing a qualitative difference in their decay rates where the former exhibits a logarithmic dependence while the latter decays quadratically.
This paper establishes a necessary and sufficient condition for the convergence of infinite products of rational inner functions on the polydisk and extends the concepts of Malmquist-Takenaka bases and unwinding to higher dimensions.
This paper defines and analyzes a class of continuous functions on constructed via an infinite stochastic matrix and specific parameters, establishing criteria for their monotonicity, differentiability, and singularity while investigating the properties of their level sets.
This paper establishes necessary conditions for the existence of tensor invariants in general nonlinear dynamical systems, with a specific focus on semi-quasihomogeneous systems, thereby extending the foundational work of Poincaré and Kozlov on integrability.
This paper establishes an optimal result for the differentiable normal linearization of partially hyperbolic diffeomorphisms by constructing a local conjugacy that is on the center manifold to achieve Takens' normal form without requiring non-resonant conditions, overcoming decoupling difficulties through a novel semi-decoupling method and advanced extension techniques.
This paper resolves an open question by proving weak type bounds for the variation and jump operators associated with rough singular integrals, which subsequently establishes the weak type boundedness of their maximal truncation operators.
This paper proves that every semi-normalized unconditional Schauder frame in an infinite-dimensional Hilbert space contains a subsequence that forms a frame, and applies this result to resolve open questions regarding the non-existence of such frames for specific Gabor systems, exponential systems, and translation-invariant subspaces under critical density conditions.
This paper introduces a new class of generalized Hilbert matrix operators induced by a positive finite Borel measure on (0,1) acting on weighted sequence spaces and establishes a necessary and sufficient condition for their boundedness, thereby extending recent related results.
This paper establishes the first sharp bounds for eigenfunctions of the Laplacian on the square torus for with , proving the discrete restriction conjecture without loss by refining the circle method and extending these sharp results to spectral projectors and additive energy.
This expository article explores the phenomenon where different definitions of fractal dimension yield conflicting results for simple examples and introduces "dimension interpolation" as a framework to unify these isolated numerical answers into a coherent geometric picture by viewing classical dimensions as boundary points of continuous families.
This paper establishes a rigorous framework and a convergent numerical algorithm for solving the continuous inverse problem of reconstructing a geometric -rough path from a discrete observed trajectory by formulating it as a limit of discrete inverse problems driven by piecewise linear paths.
This paper investigates a mathematical model of bacterial biofilms in a continuously stirred tank reactor with wall attachment, establishing global well-posedness and analyzing the stability and existence of both trivial and nontrivial equilibrium states.
This paper investigates the construction of positive definite thresholding functions for correlation matrices that preserve positive semidefiniteness, establishing existence results and bounds while proving that any geometrically unbiased soft-thresholding operator necessarily induces a geometric collapse in the feature space, thereby limiting the recoverable signal.
This paper establishes that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable -Sobolev energy can always be strongly approximated by smooth maps, thereby extending Hang's density result to integer and providing proofs for higher-order and fractional Sobolev spaces as well as cases governed by the Bethuel-Demengel-Colon-Hélein cohomological criterion.
This paper establishes explicit criteria, based on Yavicoli-thickness, -uniformity, and Newhouse thickness conditions, under which compact subsets of (particularly in the plane) are guaranteed to contain similar copies of any linear 3-point configuration or any triangle.
This paper introduces the -transformation, a new integral operator constructed via kernel fusion of the fractional Fourier transform, and establishes its fundamental properties alongside a comprehensive survey of its associated uncertainty principles, including Heisenberg's, Hardy's, Pitt's, and Beurling-Hörmander's theorems.
This paper demonstrates that twisted sectors in the vanishing cohomology of one-parameter deformations of Calabi-Yau type Fermat polynomial singularities, along with the genus zero Gromov-Witten generating series of the corresponding varieties, are components of automorphic forms for certain triangular groups, utilizing mixed Hodge structures, the Riemann-Hilbert correspondence, and genus zero mirror symmetry.
This paper establishes rigorous a priori bounds for the exact period of Rayleigh's stretched string equation to prove that Rayleigh's approximation overestimates the true period, providing explicit inequalities that demonstrate the relative error is directly proportional to the initial displacement and inversely proportional to the initial stretch.