Higher order stroboscopic averaged functions: a general relationship with Melnikov functions

In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous T-periodic differential equations of the kind x′=ε F(t,x,ε ). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi's which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincaré–Pontryagin–Melnikov functions or just Melnikov functions. While it is known that f1=Tg1, a general relationship between gi and fi is not known so far for i≥ 2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.

Nonuniform average sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces

In this paper, we study nonuniform average sampling problem in multiply generated shift-invariant subspaces of mixed Lebesgue spaces. We discuss two types of average sampled values: average sampled values [Formula: see text] generated by single averaging function and average sampled values [Formula: see text] generated by multiple averaging functions. Two fast reconstruction algorithms for these two types of average sampled values are provided.

Derivation of the Strutinsky method from the least squares principle

The main purpose of this paper is to rigorously establish the Strutinsky method from the least squares principle. Thus, it is the mathematical basis of this method (aspect often neglected) which is revisited in an extensive way. Some formulas previously given without demonstration or in a simplified way are set out here with all the details. In this respect, the most important mathematical properties of the averaging functions are also established in this paper. When some conditions are met, it turns out that Strutinsky’s method is nothing more than a polynomial moving average of the semi-classical level density.

Supervised Learning and Knowledge-Based Approaches Applied to Biomedical Word Sense Disambiguation

Word sense disambiguation (WSD) is an important step in biomedical text mining, which is responsible for assigning an unequivocal concept to an ambiguous term, improving the accuracy of biomedical information extraction systems. In this work we followed supervised and knowledge-based disambiguation approaches, with the best results obtained by supervised means. In the supervised method we used bag-of-words as local features, and word embeddings as global features. In the knowledge-based method we combined word embeddings, concept textual definitions extracted from the UMLS database, and concept association values calculated from the MeSH co-occurrence counts from MEDLINE articles. Also, in the knowledge-based method, we tested different word embedding averaging functions to calculate the surrounding context vectors, with the goal to give more importance to closest words of the ambiguous term. The MSH WSD dataset, the most common dataset used for evaluating biomedical concept disambiguation, was used to evaluate our methods. We obtained a top accuracy of 95.6 % by supervised means, while the best knowledge-based accuracy was 87.4 %. Our results show that word embedding models improved the disambiguation accuracy, proving to be a powerful resource in the WSD task.