The setting with side information also provides another operational meaning to the Lapidoth-Pfister mutual information. Finally, a universal strategy for independent and identically distributed races is presented that-without knowing the winning probabilities or the parameter of the utility function-asymptotically maximizes the gambler’s utility function.Assessing the dynamical complexity of biological time series represents an important topic with potential applications ranging from the characterization of physiological states and pathological conditions to the calculation of diagnostic parameters. In particular, cardiovascular time series exhibit a variability produced by different physiological control mechanisms coupled with each other, which take into account several variables and operate across multiple time scales that result in the coexistence of short term dynamics and long-range correlations. The most widely employed technique to evaluate the dynamical complexity of a time series at different time scales, the so-called multiscale entropy (MSE), has been proven to be unsuitable in the presence of short multivariate time series to be analyzed at long time scales. This work aims at overcoming these issues via the introduction of a new method for the assessment of the multiscale complexity of multivariate time series. The method first exploits vector aunot take into account long-range correlations.Federated learning is a decentralized topology of deep learning, that trains a shared model through data distributed among each client (like mobile phones, wearable devices), in order to ensure data privacy by avoiding raw data exposed in data center (server). After each client computes a new model parameter by stochastic gradient descent (SGD) based on their own local data, these locally-computed parameters will be aggregated to generate an updated global model. Many current state-of-the-art studies aggregate different client-computed parameters by averaging them, but none theoretically explains why averaging parameters is a good approach. In this paper, we treat each client computed parameter as a random vector because of the stochastic properties of SGD, and estimate mutual information between two client computed parameters at different training phases using two methods in two learning tasks. The results confirm the correlation between different clients and show an increasing trend of mutual information with training iteration. However, when we further compute the distance between client computed parameters, we find that parameters are getting more correlated while not getting closer. This phenomenon suggests that averaging parameters may not be the optimum way of aggregating trained parameters.It is commonly believed that information processing in living organisms is based on chemical reactions. However, the human achievements in constructing chemical information processing devices demonstrate that it is difficult to design such devices using the bottom-up strategy. Here I discuss the alternative top-down design of a network of chemical oscillators that performs a selected computing task. As an example, I consider a simple network of interacting chemical oscillators that operates as a comparator of two real numbers. The information on which of the two numbers is larger is coded in the number of excitations observed on oscillators forming the network. The parameters of the network are optimized to perform this function with the maximum accuracy. I discuss how information theory methods can be applied to obtain the optimum computing structure.We propose a method to derive the stationary size distributions of a system, and the degree distributions of networks, using maximisation of the Gibbs-Shannon entropy. We apply this to a preferential attachment-type algorithm for systems of constant size, which contains exit of balls and urns (or nodes and edges for the network case). Knowing mean size (degree) and turnover rate, the power law exponent and exponential cutoff can be derived. Our results are confirmed by simulations and by computation of exact probabilities. We also apply this entropy method to reproduce existing results like the Maxwell-Boltzmann distribution for the velocity of gas particles, the Barabasi-Albert model and multiplicative noise systems.In this paper, a new 4D hyperchaotic system is generated. The dynamic properties of attractor phase space, local stability, poincare section, periodic attractor, quasi-periodic attractor, chaotic attractor, bifurcation diagram, and Lyapunov index are analyzed. The hyperchaotic system is normalized and binary serialized, and the binary hyperchaotic stream generated by the system is statistically tested and entropy analyzed. Finally, the hyperchaotic binary stream is applied to the gray image encryption. The histogram, correlation coefficient, entropy test, and security analysis show that the hyperchaotic system has good random characteristics and can be applied to the gray image encryption.The order and disorder of binary representations of the natural numbers less then 28 is measured using the BiEntropy function. Significant differences are detected between the primes and the non-primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a Monte Carlo simulation of a sample of natural numbers less then 232 and in trinary for all natural numbers less then 39 with similar but cubic results. We found a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. 17-AAG ic50 We discuss the theoretical basis of these results and show how they generalise to give a tight bound on the variance of Pi(x)-Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices.