The reconstruction of a subclass of domino tilings from two projections

AbstractWe present a new way of studying the classical and still unsolved problem of the reconstruction of a domino tiling from its row and column projections. After giving a simple greedy strategy for solving the problem from one projection, we introduce the concept of degree of a domino tiling. We generalize an algorithm for the reconstruction of domino tilings of degree two from two projections, to domino tilings of degree three and four

On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence π is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for π to be k-graphic, with k≥ 3 , appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form π= (d, … , d, d- 1 , … , d- 1 , d- 2 , … , d- 2 ) , d≥ 2 , which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k≥ 4 and to other families of degree sequences having simple characterization, such as gap-free sequences

Biomechanical analysis of pedalling for rehabilition purposes: experimental results on two pathological subjects and comparison with non-pathological findings

In this paper the experimental results obtained by means of a prototype measuring device dedicated to the evaluation of the rehabilitation level of the lower limb are presented. The analysis of the experimental data collected on non-pathological subjects allows the identification of the characteristic meaning of the most significant parameters typical of healthy subjects. These data have been employed for a systematic comparison with the same parameters measured on two pathological subjects, in order to define quantitative indicators of the rehabilitation degree of the lower limbs and indicators of the “quality” of the movement

Parametric machines: a fresh approach to architecture search

Using tools from category theory, we provide a framework where artificial neural networks, and their architectures, can be formally described. We first define the notion of machine in a general categorical context, and show how simple machines can be combined into more complex ones. We explore finite- and infinite-depth machines, which generalize neural networks and neural ordinary differential equations. Borrowing ideas from functional analysis and kernel methods, we build complete, normed, infinite-dimensional spaces of machines, and discuss how to find optimal architectures and parameters -- within those spaces -- to solve a given computational problem. In our numerical experiments, these kernel-inspired networks can outperform classical neural networks when the training dataset is small.Comment: 31 pages, 4 figure

Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.Comment: Added references. Extended Section 7. Added 3 figures. Corrected typos. 42 pages, 7 figure

On null 3-hypergraphs

International audienceGiven a 3-uniform hypergraph H consisting of a set V of vertices, and T ⊆ V 3 triples, a null labelling is an assignment of ±1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and −1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed

Persistence modules, shape description, and completeness

Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes. In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the problem. Therefore we introduce geometric homomorphisms between persistence modules, and show that in some cases they perform better. A combinatorial structure, the $H_0$-tree, is shown to be an invariant for geometric isomorphism classes in the case of persistence modules obtained through the 0th persistent homology functor

PIXE characterization of CsI(Tl) scintillators used for particle detection in nuclear reactions

Particle-Induced X-ray Emission has been used to measure Thallium concentration in several CsI(Tl) scintillators from different manufacturers, in order to check their nominal declared values and correlate their behaviour with actual Tl concentration. Indeed, both Tl doping level and its uniformity affect light emission of these detectors, which are largely employed in nuclear physics experiments. In some of the examined crystals Tl concentration values from PIXE measurements came out to be quite different from those declared. This allowed us to explain apparent anomalies in the trend of their a/c-induced light yield ratio versus Tl content. In some cases, the presence of unexpected contaminants was also pointed out. 2008 Elsevier B.V. All rights reserved