Minimum number of edges in a hypergraph guaranteeing a perfect fractional matching and the MMS conjecture

Vladimir Blinovsky

In this paper we prove the Ahlswede-Khachatrian conjecture [1] up to a finite number of cases, which can be checked using modern computers. This conjecture implies the conjecture from [2] and the Manickam-Miklós-Singhi conjecture.

The whole paper is available here.

Modeling networks of spiking neurons as interacting processes with memory of variable length

Antonio Galves, Eva Löcherbach

We consider a new class of non Markovian processes with a countable number of interacting components, both in discrete and continuous time. Each component is represented by a point process indicating if it has a spike or not at a given time. The system evolves as follows. For each component, the rate (in continuous time) or the probability (in discrete time) of having a spike depends on the entire time evolution of the system since the last spike time of the component.

Identifying interacting pairs of sites in Ising models on a countable set

Antonio Galves, Enza Orlandi and Daniel Yasumasa Takahashi

This paper addresses the problem of identifying pairs of interacting sites from a finite sample of independent realizations of the Ising model. We consider Ising models in a infinite countable set of sites under Dobrushin uniqueness condition. The observed sample contains only the values assigned by the Ising model to a finite set of sites. Our main result is an upper bound for the probability of misidentification of the pairs of interacting sites in this finite set.

Powers of Hamilton cycles in pseudorandom graphs

Peter Allen, Julia Böttcher, Hiep Hàn, Yury Person, Yoshiharu Kohayakawa

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε,p,k,ℓ)-pseudorandom if for all disjoint X and Y⊂V(G) with |X|≥ε(pˆk)n and |Y|≥ε(pˆℓ)n we have e(X,Y)=(1±ε)p|X||Y|.

Graph analysis of verbal fluency test discriminate between patients with Alzheimer's disease, mild cognitive impairment and normal elderly controls

Laiss Bertola, Natália B. Mota, Mauro Copelli, Thiago Rivero, Breno Satler Diniz, Marco A. Romano-Silva, Sidarta Ribeiro, and Leandro F. Malloy-Diniz

Verbal fluency is the ability to produce a satisfying sequence of spoken words during a given time interval. The core of verbal fluency lies in the capacity to manage the executive aspects of language. The standard scores of the semantic verbal fluency test are broadly used in the neuropsychological assessment of the elderly, and different analytical methods are likely to extract even more information from the data generated in this test. Graph theory, a mathematical approach to analyze relations between items, represents a promising tool to understand a variety of neuropsychological states. This study reports a graph analysis of data generated by the semantic verbal fluency test by cognitively healthy elderly (NC), patients with Mild Cognitive Impairment—subtypes amnestic (aMCI) and amnestic multiple domain (a+mdMCI)—and patients with Alzheimer's disease (AD).

A model for neural activity in the absence of external stimuli

Aline Duarte, Guilherme Ost

We study a stochastic process describing the continuous time evolution of the membrane potentials of finite system of neurons in the absence of external stimuli. The values of the membrane potentials evolve under the effect of chemical synapses, electrical synapses and a leak current.

The onset of data-driven mental archeology

Sidarta Ribeiro

How did human consciousness arise, in comparison with that of apes? This question has motivated many hypotheses, from psychoanalysis (Freud, 1939) and literature (Bloom, 1999) to evolutionary psychology (Dennett, 1997; Tomasello, 2014). One of the most provocative is that of Julian Jaynes (1976), who understood the question as pertaining to the cultural evolution of introspection during the first millennium BC, known as Axial Age (Jaspers, 1953).

On a toy model of interacting neurons

Nicolas Fournier, Eva Löcherbach

We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-Löcherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation.

Predicting upcoming actions by observation: some facts, models and challenges

C. D. Vargas, M. L. Rangel, A. Galves

Predicting another person's upcoming action to build an appropriate response is a regular occurrence in the domain of motor control. In this review we discuss conceptual and experimental approaches aiming at the neural basis of predicting and learning to predict upcoming movements by their observation.

On the Number of Orientations of Random Graphs with No Directed Cycles of a Given Length

P. Allen, Y. Kohayakawa, G. O. Mota, R. F. Parente

Let H⃗ be an orientation of a graph H. Alon and Yuster proposed the problem of determining or estimating D(n,m,H⃗), the maximum number of H⃗-free orientations a graph with n vertices and m edges may have. We consider the maximum number of H⃗-free orientations of typical graphs G(n,m) with n vertices and m edges. Suppose H⃗ =C↻ℓ is the directed cycle of length ℓ≥3. We show that if m≫n^(1+1/(ℓ−1)), then this maximum is 2^o(m), while if m≪n^(1+1/(ℓ−1)), then it is 2^(1−o(1))m.




The Research, Innovation and Dissemination Center for Neuromathematics is hosted by the University of São Paulo and funded by FAPESP (São Paulo Research Foundation).


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