Three-dimensional maps and subgroup growth.

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index n in Δ + = Z 2 ∗ Z 2 ∗ Z 2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of Δ + on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in Δ + . Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on n ≤ 16 darts.

Machine learning of the prime distribution.

In the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy-Ramanujan Theorem. We also provide a theoretical argument explaining the experimental observations of Y.-H. He about the learnability of primes, and posit that the Erdos-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings.

An Approach to Robust ICP Initialization.

In this note, we propose an approach to initialize the Iterative Closest Point (ICP) algorithm to match unlabelled point clouds related by rigid transformations. The method is based on matching the ellipsoids defined by the points' covariance matrices and then testing the various principal half-axes matchings that differ by elements of a finite reflection group. We derive bounds on the robustness of our approach to noise and numerical experiments confirm our theoretical findings.

Diffuse Reflectance Spectroscopy of the Oral Mucosa: In Vivo Experimental Validation of the Precancerous Lesions Early Detection Possibility.

This article is devoted to the experimental validation of the possibility of early detection of precancerous lesions in the oral mucosa in vivo using diffuse reflectance spectroscopy in the wavelength range from 360 to 1000 nm. During the study, a sample of 119 patients with precancerous lesions has been collected and analyzed. As a result of the analysis, the most informative wavelength ranges were determined, in which the maximum differences in the backscattering spectra of lesions and intact tissues were observed, methods for automatic classification of backscattering spectra of the oral mucosa were studied, sensitivity and specificity values, achievable using diffuse reflectance spectroscopy for detecting hyperkeratosis on the tongue ventrolateral mucosa surface and buccal mucosa, were evaluated. As a result of preliminary experimental studies in vivo, the possibility of automatic detection of precancerous lesions of the oral mucosa surface using diffuse reflectance spectroscopy in the wavelength range from 500 to 900 nm with an accuracy of at least 75 percent has been shown.

A New Microstructural Approach to the Strength of an Explosion Weld.

In this paper, the local stress-strain state in an explosion weld was investigated and the local strength of the welded materials near the weld analyzed. It follows from the experimental data that the explosion weld at the microlevel looks like a wavy line. In the first approximation, this wavy line may be assumed to be periodic. We used the two-scale method to analyze the corresponding interface elasticity problem. We carried out numerical computations for three of the most referenced types of weld geometry: the symmetric wave, the asymmetric wave, and the wave with crest. We found that the wave geometry of the weld leads to increase in local stress in the weld zone. The stress concentration varied from 20% to 200% in dependence on the weld geometry and the macroscopic loading. Explosion welding is accompanied by strain hardening of the materials in the welding zone. In some cases, the strain hardening may compensate for the increasing local stress. As a result, the weld may be both stronger and weaker than the welded materials.