Related papers: A comparison principle for convolution measures wi…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…
The concept of sharpness has been successfully applied to traditional architectures like MLPs and CNNs to predict their generalization. For transformers, however, recent work reported weak correlation between flatness and generalization. We…
We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Lo\`eve decomposition, consists in a series of…
We consider the empirical validity of the equivalence principle for non-baryonic matter. Working in the context of the TH\epsilon\mu formalism, we evaluate the constraints experiments place on parameters associated with violation of the…
We give an alternative argument to the application of the so-called Maurey- Nikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple method applies to any compact quadratic…
We prove estimates at infinity of convolutions $f^{n\star}$ and densities of the corresponding compound Poisson measures for a class of radial decreasing densities on $\mathbb{R}^d$, $d \geq 1$, which are not convolution equivalent.…
We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We…
A rigorous connection between large deviations theory and Gamma-convergence is established. Applications include representations formulas for rate functions, a contraction principle for measurable maps, a large deviations principle for…
Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse…
We obtain sharp estimate on $p$-spectral gaps, or equivalently optimal constant in $p$-Poincar\'e inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp $p$-spectral gap.
The equivalence of inertial and gravitational masses is a defining feature of general relativity. Here, we clarify the status of the equivalence principle for interactions mediated by a universally coupled scalar, motivated partly by recent…
Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is straightforward to see that sets for which the Fourier and…
We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional…
We introduce and study a family of axioms that closely follows the pattern of parametrized diamonds, studied by Moore, Hru\v{s}\'ak, and D\v{z}amonja in [13]. However, our approach appeals to model theoretic / forcing theoretic notions,…
A parametric Oka principle for liftings, recently proved by Forstneric, provides many examples of holomorphic maps that are fibrations in a model structure introduced in previous work of ours. We use this to show that the basic Oka property…
We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…
This paper presents some sufficient conditions for the validity of the comparison principle for the weak solutions of non - cooperative weakly coupled systems of elliptic second-order PDEs.
We establish a Zador like theorem for $L^r$-optimal vector quantization when the similarity measure is a twice differentiable Bregman divergence of a strictly convex function. On our way we also prove a similar result when the Bregman…
We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a…