Related papers: Accuracy analysis of the box-counting algorithm
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum…
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This…
We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization…
We apply the Principle of Maximum Entropy to the study of a general class of deterministic fractal sets. The scaling laws peculiar to these objects are accounted for by means of a constraint concerning the average content of information in…
While the Quasi-Monte Carlo method of numerical integration achieves smaller integration error than standard Monte Carlo, its use in particle physics phenomenology has been hindered by the abscence of a reliable way to estimate that error.…
We consider a system of classical particles confined in a box $\Lambda\subset\mathbb{R}^d$ with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical…
The problem of local damage diagnosis (based on the detection of impulsive and periodic signals) is discussed. Both features should be checked, as fault frequency must be linked to the true value calculated for a given machine and speed.…
We introduce a method based on the finite size scaling assumption which allows to determine numerically the critical point and critical exponents related to observables in an infinite system starting from the knowledge of the observables in…
Numerical simulations based on electronic structure calculations are finding ever growing applications in many areas of physics. A major limiting factor is however the cubic scaling of the algorithms used. Building on previous work [F. R.…
Neural network models have recently demonstrated impressive prediction performance in complex systems where chaos and unpredictability appear. In spite of the research efforts carried out on predicting future trajectories or improving their…
A major milestone of quantum error correction is to achieve the fault-tolerance threshold beyond which quantum computers can be made arbitrarily accurate. This requires extraordinary resources and engineering efforts. We show that even…
The most critical component of any adaptive numerical quadrature routine is the estimation of the integration error. Since the publication of the first algorithms in the 1960s, many error estimation schemes have been presented, evaluated…
Several two-dimensional studies in spiral galaxies show that HII star formation regions have a fractal distribution, with a fractal dimension of approximately 2.3. In this work, the fractal dimension is calculated through the box-counting…
We explore the conditions under which the particle number conservation constraint deforms the predictions of fragmentation observables as calculated in the grand canonical ensemble. We derive an analytical formula allowing to extract…
Calculating or accurately estimating log-determinants of large positive definite matrices is of fundamental importance in many machine learning tasks. While its cubic computational complexity can already be prohibitive, in modern…
To date, a great deal of attention has focused on characterizing the performance of quantum error correcting codes via their thresholds, the maximum correctable physical error rate for a given noise model and decoding strategy. Practical…
Quantum algorithms for simulation of Hamiltonian evolution are often based on product formulae. The fractal methods give a systematic way to find arbitrarily high-order product formulae, but result in a large number of exponentials. On the…
The calculation of the band-gap by density-functional theory (DFT) methods is examined by considering the behavior of the energy as a function of number of electrons. It is found that the incorrect band-gap prediction with most approximate…
This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a…
We consider a fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from…