This research note reports the surprisingly accurate approximate exponential regimes observed in the time evolution of Susceptible-Infected-Recovered (SIR) epidemiological models. Focusing on the classic SIR model, we demonstrate this phenomenon graphically and through asymptotic analysis. This seemingly overlooked phenomenon enables deriving approximate closed-form solutions to otherwise intractable estimation and prediction problems in SIR modelling. Building on this key observation, we speculate on the presence of approximate exponential evolution regimes in other more complex SIR model variants, with the potential to simplify analysis and implementation across disciplines where compartmental models are utilised.

In this note, we shed light on the physical meaning for the backpropagated error used by the backpropagation training algorithm. Essentially, for a given scalar output of the neural network, its backpropagated error is a linear apportionment of the error at it, in proportion to the linear gain between the outputs of neurons and the output according to a linearised-systems-view of the network. For multiple outputs, superposition provides the total/nett backpropagated error at the outputs of neurons. Subsequently, we present some elementary statistical analysis for backpropagated errors in the network.

Recently, a necessary and sufficient condition for multivaluedness to be implicitly exhibited by counter-cascaded systems was presented. Subsequently, several systems that exhibit multivaluedness were reported. This brief interprets a general information transmission system as a counter-cascaded system with Shannon's noisy-channel coding theorem providing the necessary and sufficient conditions for multivaluedness and is therefore a particular instance of the counter-cascaded network framework.

We propose an analytical method for constructing ergodic and piecewise linear hat maps with a piecewise constant invariant density selected to approximate a prescribed invariant density, and a rational power spectral density with an arbitrary number of prescribed nonrepeating poles. These semi-Markov maps permit a matrix representation of the Frobenius-Perron operator, referred to as the Frobenius-Perron matrix. We prove that the poles of a piecewise linear hat map’s power spectral density coincide with the unique nonzero and nonunity (in absolute value) eigenvalues of the map’s Frobenius-Perron matrix. A  hat map is constructed such that it possesses a Frobenius-Perron matrix with a selectable eigenspectrum, thereby allowing for the prescription of its poles. A topological conjugate of this map is subsequently constructed. The conjugate map’s poles are equal to those of the original map, but its piecewise constant invariant density is selected to approximate the prescribed invariant density. Similar to previous methods, the novel method does not afford the opportunity to prescribe the zeros of the map’s power spectral density. However, unlike these methods, all of the poles are prescribable and no numerical search is performed during construction. We characterize several piecewise linear hat maps constructed using the novel method.

We present a direct solution to the problem of constructing a stochastic matrix with prescribed eigenspectrum, widely referred to as the stochastic inverse eigenvalue problem. The solution uses Markov state disaggregation to construct a Markov chain with stochastic transition matrix possessing the required eigenspectrum. Existing solutions that follow the same approach are limited to constructing matrices with real-valued eigenspectra only. The novel solution directly constructs matrices with complex-valued eigenspectra by applying a new disaggregation technique in tandem with a technique from a previous solution. Due to this generalization, the novel solution is able to successfully model physical systems from a larger family. Furthermore, the novel solution constructs the matrix in a finite and predetermined number of iterations, and without numerical approximation. The solution is demonstrated by deriving an expression for a set of 4 x 4 stochastic matrices sharing the same prescribed complex-valued eigenspectrum and indexed by a real parameter.

This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown nonlinear and ergodic map from causal sequences of probability density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all the map’s branches are monotonically increasing. The original solution constructs a matrix estimate of the map’s Frobenius-Perron operator, which governs the evolution of density functions under iteration of the map, from the density sequences. The one-dimensional map is reconstructed from this matrix. In contrast, the proposed solution constructs a higher-order matrix estimate of the Frobenius-Perron operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing map branches and enlarges the class of maps that can be successfully reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum mode characteristics, regardless of the unknown map’s canonicity or branch monotonicity. Numerical examples that illustrate the proposed solution’s characteristics are presented.

This brief presents the first observations of multivaluedness in four systems: a random process, a nonlinear nondynamical system, a nonlinear dynamical system with nonlinearly sensed input and output and an adaptive linear estimator. The preliminary findings reported here suggest the impact of multivaluedness in different types of networks to range from adverse to benign or even essential.

This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufficient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems. The significance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.

This paper presents some new and explicit stability results for Volterra systems from two different approaches. The first approach is based on monomial domination of the Volterra system's memoryless output nonlinearity and the second on its Lipschitz-norm. The former yields more widely applicable results, but introduces nonconvexity in the signal spaces for certain parameter values.