Perfect matchings, rank of connection tensors and graph homomorphisms

We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

Inverse Problem for Ising Connection Matrix with Long-Range Interaction

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of its eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.

Exact WKB analysis of the vacuum pair production by time-dependent electric fields

We study the vacuum pair production by a time-dependent strong electric field based on the exact WKB analysis. We identify the generic structure of a Stokes graph for systems with the vacuum pair production and show that the number of produced pairs is given by a product of connection matrices for Stokes segments connecting pairs of turning points. We derive an explicit formula for the number of produced pairs, assuming the semi-classical limit. The obtained formula can be understood as a generalization of the divergent asymptotic series method by Berry, and is consistent with other semi-classical methods such as the worldline instanton method and the steepest descent evaluation of the Bogoliubov coefficients done by Brezin and Izykson. We also use the formula to discuss effects of time-dependence of the applied strong electric field including the interplay between the perturbative multi-photon pair production and non-peturbative Schwinger mechanism, and the dynamically assisted Schwinger mechanism.

Can we conjointly record direct interactions between neurons in vivo in anatomically-connected brain areas? Probabilistic analyses and further implications

Large-scale simultaneous in vivo recordings of neurons in multiple brain regions raises the question of the probability of recording direct interactions of neurons within, and between, multiple brain regions. In turn, identifying inter-regional communication rules between neurons during behavioural tasks might be possible, assuming conjoint activity between neurons in connected brain regions can be detected. Using the hypergeometric distribution, and employing anatomically-tractable connection mapping between regions, we derive a method to calculate the probability distribution of ‘recordable’ connections between groups of neurons. This mathematically-derived distribution is validated by Monte Carlo simulations of directed graphs representing the underlying anatomical connectivity structure. We apply this method to simulated graphs with multiple neurons, based on counts in rat brain regions, and to connection matrices from the Blue Brain model of the mouse neocortex connectome. Overall, we find low probabilities of simultaneously-recording directly interacting neurons in vivo in anatomically-connected regions with standard (tetrode-based) approaches. We suggest alternative approaches, including new recording technologies and summing neuronal activity over larger scales, offer promise for testing hypothesised interregional communication and source transformation rules.

Decoding Analysis of Alpha Oscillation Networks on Maintaining Driver Alertness

The countermeasure of driver fatigue is valuable for reducing the risk of accidents caused by vigilance failure during prolonged driving. Listening to the radio (RADIO) has been proven to be a relatively effective “in-car” countermeasure. However, the connectivity analysis, which can be used to investigate its alerting effect, is subject to the issue of signal mixing. In this study, we propose a novel framework based on clustering and entropy to improve the performance of the connectivity analysis to reveal the effect of RADIO to maintain driver alertness. Regardless of reducing signal mixing, we introduce clustering algorithm to classify the functional connections with their nodes into different categories to mine the effective information of the alerting effect. Differential entropy (DE) is employed to measure the information content in different brain regions after clustering. Compared with the Louvain-based community detection method, the proposed method shows more superior ability to present RADIO effectin confused functional connection matrices. Our experimental results reveal that the active connection clusters distinguished by the proposed method gradually move from frontal region to parieto-occipital regionwith the progress of fatigue, consistent with the alpha energy changes in these two brain areas. The active class of the clusters in parieto-occipital region significantly decreases and the most active clusters remain in the frontal region when RADIO is taken. The estimation results of DE confirm the significant change (p < 0.05) of information content due to the cluster movements. Hence, preventing the movement of the active clusters from frontal region to parieto-occipital region may correlate with maintaining driver alertness. The revelation of alerting effect is helpful for the targeted upgrade of fatigue countermeasures.

An Analysis on Novel Corona Virus by a Plithogenic Approach to Fuzzy Cognitive Map

In this paper a Plithogenic approach to Fuzzy cognitive map has been proposed to analyse the impact of novel corona virus.Contradiction degree is an advantageous feature of Plithogenic sets which highly deals with the uncertainty and it substantially increases the accuracy of results. In this research, a new approach is proposed to accumulate the opinions of experts.Using contradiction degree between opinions of experts and plithogenicoperator,the connection matrices obtained from distinct experts are aggregated to improvise a degree of uncertainty. An Analysis on covid-19 (causes, spread and precaution) is done based on the proposed methodology.