A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

The jamming constant of uniform random graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a “parking process” on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given–possibly unbounded–degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.Fil: Bermolen, Paola. Universidad de la República; UruguayFil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Moyal, Pascal. Northwestern University; Estados Unidos. Universite de Technologie de Compiegne; Franci

Artificial neural networks allow the use of simultaneous measurements of Alzheimer Disease markers for early detection of the disease

BACKGROUND: Previous studies have shown that in platelets of mild Alzheimer Disease (AD) patients there are alterations of specific APP forms, paralleled by alteration in expression level of both ADAM 10 and BACE when compared to control subjects. Due to the poor linear relation among each key-element of beta-amyloid cascade and the target diagnosis, the use of systems able to afford non linear tasks, like artificial neural networks (ANNs), should allow a better discriminating capacity in comparison with classical statistics. OBJECTIVE: To evaluate the accuracy of ANNs in AD diagnosis. METHODS: 37 mild-AD patients and 25 control subjects were enrolled, and APP, ADM10 and BACE measures were performed. Fifteen different models of feed-forward and complex-recurrent ANNs (provided by Semeion Research Centre), based on different learning laws (back propagation, sine-net, bi-modal) were compared with the linear discriminant analysis (LDA). RESULTS: The best ANN model correctly identified mild AD patients in the 94% of cases and the control subjects in the 92%. The corresponding diagnostic performance obtained with LDA was 90% and 73%. CONCLUSION: This preliminary study suggests that the processing of biochemical tests related to beta-amyloid cascade with ANNs allows a very good discrimination of AD in early stages, higher than that obtainable with classical statistics methods