Related papers: VC-Dimension vs Degree: An Uncertainty Principle f…
A recently introduced measure of Boolean functions complexity--disjunc\-tive complexity (DC)--is compared with other complexity measures: the space complexity of streaming algorithms and the complexity of nondeterministic branching programs…
Within quantum electrodynamics we show that the Generalized Uncertainty Principle induces higher-derivative corrections that promote the topological invariant $F_{\mu\nu}\,\widetilde F^{\mu\nu}$ to the dynamical, non-topological operator…
We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $\varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical…
Quantum phase transitions are often embodied by the critical behavior of purely quantum quantities such as entanglement or quantum fluctuations. In critical regions, we underline a general scaling relation between the entanglement entropy…
Since its introduction by Vapnik and Chervonenkis in the 1960s, the VC dimension and its variants have played a central role in numerous fields. In this paper, we investigate several variants of the VC dimension and their applications to…
We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function $f$ is the characteristic vector of the minimum sized set of negated variables needed in any…
Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the "Sensitivity versus Block Sensitivity" question,…
Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic…
We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order…
The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts…
Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a…
The increasingly wide use of deep machine learning techniques in computational mechanics has significantly accelerated simulations of problems that were considered unapproachable just a few years ago. However, in critical applications such…
The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_\epsilon(f)$ (written $\mathsf{N}_\epsilon(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le \epsilon$…
The uncertainty principle lies at the heart of quantum mechanics, as it describes the fundamental trade-off between the precision of position and momentum measurements. In this work, we study the quantum particle in the Boltzmann states and…
A generalized uncertainty principle is obtained from a conformally transformed action containing a scalar field and a unique constraint. The constraint's Lagrange multiplier is found to obey a relativistic diffusion equation transforming…
Uncertainty principle is one of the fundamental principles of quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak…
The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the…
In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of…
Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree,…
Recently Kosaki proved an uncertainty principle for matrices, related to Wigner-Yanase-Dyson information, and asked if a similar inequality could be proved in the von Neumann algebra setting. In this paper we prove such an uncertainty…