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Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals…
We derive a fine-grained uncertainty relation for the measurement of two incompatible observables on a single quantum system of continuous variables, and show that continuous variable systems are more uncertain than discrete variable…
Heisenberg's uncertainty principle is often cited as an example of a "purely quantum" relation with no analogue in the classical limit where $\hbar \to 0$. However, this formulation of the classical limit is problematic for many reasons,…
For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $\Theta(k)$ variables while keeping the complexity preserved at $\Theta(k)$? This question,…
The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a…
Effectively measuring and modeling the reliability of a trained model is essential to the real-world deployment of monocular depth estimation (MDE) models. However, the intrinsic ill-posedness and ordinal-sensitive nature of MDE pose major…
This paper focuses on the complex relationship between Heisenberg's Uncertainty Principle and the nodal structure of wave functions in a variety of quantum systems including the quantum harmonic oscillator, the particle in a 1D box , and…
We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. In particular, we focus on characterizing the variance over values induced by a distribution over MDPs. Previous work…
In this paper, we propose the uncertain volatility models with stochastic bounds. Like the regular uncertain volatility models, we know only that the true model lies in a family of progressively measurable and bounded processes, but instead…
Given a finite set $X$ and a function $f:X\to X$, we define the degree of noninvertibility of $f$ to be $\displaystyle\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from…
Non-Hermitian physics has become a fundamental framework for understanding open systems where gain and loss play essential roles, with impact across photonics, quantum science, and condensed matter. While the role of complex eigenvalues is…
Classical results of second order parabolic quasi-linear equations always require that the nonlinear terms are controlled by a power of the unknown functions and their first derivatives. We improve the previous results. More precisely, in…
The aim of this paper is to prove new uncertainty principles for an integral operator $\tt$ with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Faris's local uncertainty principle…
We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a…
Data structure has a dramatic impact on the properties of neural networks, yet its significance in the established theoretical frameworks is poorly understood. Here we compute the Vapnik-Chervonenkis entropy of a kernel machine operating on…
We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically…
We prove a new version of the Uncertainty Principle of the form $\int |f|^2 \lesssim \int_{E^c} |f|^2 + \int_{\Sigma ^c}|\hat f|^2 $ where the sets $E$ and $\Sigma$ are $\epsilon$-thin in the following sense: $|E \cap D(x, \rho_1(x))| \le…
In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields…
Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report improved new versions of various error-disturbance tradeoff relations…
Assessing the predictive uncertainty of deep neural networks is crucial for safety-related applications of deep learning. Although Bayesian deep learning offers a principled framework for estimating model uncertainty, the common approaches…