Demystification of Entangled Mass Action Law

Recently, Gorban (2021) analysed some kinetic paradoxes of the transition state theory and proposed its revision that gave the ``entangled mass action law'', in which new reactions were generated as an addition to the reaction mechanism under consideration. These paradoxes arose due to the assumption of quasiequilibrium between reactants and transition states. In this paper, we provided a brief introduction to this theory, demonstrating how the entangled mass action law equations can be derived in the framework of the standard quasi steady state approximation in combination with the quasiequilibrium generalized mass action law for an auxiliary reaction network including reactants and intermediates. We also proved the basic physical property (positivity) for these new equations, which was not obvious in the original approach.Comment: Minor correction

Rosenblatt's first theorem and frugality of deep learning

First Rosenblatt's theorem about omnipotence of shallow networks states that elementary perceptrons can solve any classification problem if there are no discrepancies in the training set. Minsky and Papert considered elementary perceptrons with restrictions on the neural inputs: a bounded number of connections or a relatively small diameter of the receptive field for each neuron at the hidden layer. They proved that under these constraints, an elementary perceptron cannot solve some problems, such as the connectivity of input images or the parity of pixels in them. In this note, we demonstrated first Rosenblatt's theorem at work, showed how an elementary perceptron can solve a version of the travel maze problem, and analysed the complexity of that solution. We constructed also a deep network algorithm for the same problem. It is much more efficient. The shallow network uses an exponentially large number of neurons on the hidden layer (Rosenblatt's $A$-elements), whereas for the deep network the second order polynomial complexity is sufficient. We demonstrated that for the same complex problem deep network can be much smaller and reveal a heuristic behind this effect

Thermal "portrait" of sportsmen with different aerobic capacity

Nowadays thermography technique is widely used in clinical and diagnostic procedures. Thermal “portrait” fixed by infra-red thermotracer in rest conditions is characterized by mosaic temperature distribution on skin surface and large individual differences in this pattern. We obtained significant correlations between maximal and average temperature on sportsmen upper body part and important indexes of aerobic capacity – VO2max and anaerobic threshold. We suggest mechanisms responsible for its formation and also propose hypothesis assuming possible role of brown adipose tissue influence on thermal “portrait” forming