Related papers: Quantitative Reifenberg theorem for measures

We show that probabilities of results of all possible measurements performing on a quantum system depend on the system's state only through its density matrix. Therefore all experimentally available information about the state contains in…

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers…

We introduce generalized $(\alpha,\beta)$-transformations, which include all $(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2 large…

We give a bound to the precision in the estimation of a parameter in terms of the expectation value of an observable. It is an extension of the Cramer-Rao inequality and of the Heisenberg uncertainty relation, where the estimation precision…

In this paper, I attempt a personal account of my understanding of the measurement problem in quantum mechanics, which has been largely in the tradition of the Copenhagen interpretation. I assume that (i) the quantum state is a…

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is…

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the…

Let $\varepsilon>0$. We construct an explicit, full-measure set of $\alpha \in[0,1]$ such that if $\gamma \in \mathbb{R}$ then, for almost all $\beta \in[0,1]$, if $\delta \in \mathbb{R}$ then there are infinitely many integers $n\geq 1$…

We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online---that is, while the…

By implicitly assuming that all measurements occur simultaneously, Bell's Theorem only applied to local theories that violated Heisenberg's Uncertainty Principle. By explicitly introducing time into our derivation of Bell's theorem, an…

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter $\Omega$ is expressed by the integration with respect to a measure $P(\Omega,\cdot)$ which is equivalent to the one-codimensional…

Let $\beta>1$ be a real number and define the $\beta$-transformation on $[0,1]$ by $T_\beta:x\mapsto \beta x\bmod 1$. Further, define $$W_y(T_{\beta},\Psi):=\{x\in [0, 1]:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\}$$ and…

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg…

Heisenberg's uncertainty principle implies fundamental constraints on what properties of a quantum system can we simultaneously learn. However, it typically assumes that we probe these properties via measurements at a single point in time.…

We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk…

For Y a subset of the complex plane,a beta ensemble is a sequence of probability measures on Y^n for n=1,2,3...depending on a real-valued continuous function Q and a real positive parameter beta.We consider the associated sequence of…

In previous articles we presented a derivation of Born's rule and unitary transforms in Quantum Mechanics (QM), from a simple set of axioms built upon a physical phenomenology of quantization. Physically, the structure of QM results of an…

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the…

We summarize some results of geometric measure theory concerning rectifiable sets and measures. Combined with the entropic chain rule for disintegrations (Vigneaux, 2021), they account for some properties of the entropy of rectifiable…

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval $T_{\beta}(x)=\beta x$ (mod $1$), $x\in[0,1]$, $\beta>0$, which are the so-called…