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In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0 , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15-16] and Oustaloup and co-workers [14]. [ABSTRACT FROM AUTHOR]

We show that on a metric graph of genus g, a divisor of degree n generically has g (n - g + 1) Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini. [ABSTRACT FROM AUTHOR]

With the rapid development of economic levels, the demand for electricity usage in various industrial sectors has continued to rise. In order to adequately meet the growth in demand for electricity usage at each user end within the power system, accurate forecasting of electricity load is required. This paper uses the K-Means algorithm as the basis, combined with the Canopy algorithm, whose simple and efficient advantages can be used as the initial clustering step for demanding clustering algorithms. The PCA algorithm is used to reduce the dimensionality of the residential multidimensional electricity consumption feature set, and after optimization, the feature vector weights are calculated to obtain the appropriate selection to determine the training and testing samples. The improved K-Means algorithm carries out five clustering tests, and its average accuracy is above 97%. The average relative error is then used as an indicator to compare the algorithm with the time series method, curve extrapolation method, and WKNR method for analysis. The average relative error of the smart grid load forecasting method proposed in this paper is 0.87%, while the average relative errors of the other three algorithms are 1.64%, 1.57%, and 0.93%, respectively. The distribution of relative errors falls less than 1% more than the other methods. It can be seen that the improved K-Means algorithm has higher prediction accuracy and also makes the smart grid load testing platform more practical due to its simple implementation principle.

The circuit analysis approach based on geometric algebra and M , the power definition based on the geometric product between the voltage and the current multivectors, are used here to demonstrate the shortcomings of the traditional definition of the non-sinusoidal apparent power S. The shortcomings of S are illustrated in three ways. Firstly, by showing an example of how the norm of M contains S. Secondly, through six experiments that involve compliance with: Kirchhoff's circuit laws, Tellegen's theorem, the principle of conservation of energy, the equivalency of two terminal networks and the concept of reactive power compensation. Lastly, by showing how the use of S leads the current's physical component power theory astray. The experiments show contradictions between the aforementioned circuit theory fundamentals and the results attained with S but a compelling harmony with the results attained with M . The evidence reveals two unprecedented discoveries: (1) that mathematical models aimed at explaining energy flow in non-sinusoidal circuits shouldn't be based on the decomposition of S—as traditionally done— and, (2) the inappropriateness of extrapolating definitions from sinusoidal conditions to non-sinusoidal settings. [ABSTRACT FROM AUTHOR]

In this paper, we state and prove complete versions of two basic theorems of linear circuit theory, viz. Thevenin–Norton and Maximum power transfer theorems. We show how to compute port behavior of multiports which have port equations of the general form B v P - Q i P = s , which cannot even be put into the hybrid form, indeed may have number of equations ranging from 0 to 2n, where n is the number of ports. The method works for linear multiports which are consistent for arbitrary internal source values and further have the property that the port conditions uniquely determine internal conditions. We present the most general version of maximum power transfer theorem possible. This version of the theorem states that 'stationarity' (derivative zero condition) of power transfer occurs when the multiport is terminated by its adjoint, provided the resulting network has a solution. If this network does not have a solution, there is no port condition for which stationarity holds. This theorem does not require that the multiport has a hybrid immittance matrix. [ABSTRACT FROM AUTHOR]

The notion of resistance distance as a convenient metric for graphs was introduced in Klein (J Math Chem 12:81–95, 1993). It is inspired by the concept of equivalent resistance for electrical circuits and has numerous applications, in particular, in organic chemistry, physics and random walks on graphs. Besides, computing resistance distance of various circuits has always been of interest for electrical engineers. In this paper, we provide a brief review of the concept and a physics perspective on resistance distance, highlighting some useful analytical methods for computing it. To some extend, these methods generalize and build on top of the results presented in Bapat (Math Stud 68(1–4):87–98, 1999, Indian J Pure Appl Math 41(1):1–13, 2010) and Kagan (Am J Phys 83:53–63, 2015). We then illustrate these methods using graphs with rotational symmetry as an example. The same analysis can be applied to computations of the complex impedance of AC-circuits of the same circular topology and can be used to investigate resonance phenomena therein. At the end, we discuss the concept of resistance distance in the context of the Weisfeiler–Leman stabilization. [ABSTRACT FROM AUTHOR]

We consider the Signature Method ( Σ -method) for the structural analysis of differential–algebraic equations (DAEs) that arise in the modeling and simulation of electrical circuits. Different formulations of the set of model equations are considered. We show that for some formulations the structural approach may fail for certain circuit topologies, while other formulations are better suited for a structural analysis. In particular, we show that for the branch-oriented model equations the Signature Method always succeeds with a structural index that corresponds to the differentiation index of the system. The results are illustrated by a number of examples. [ABSTRACT FROM AUTHOR]

Using polynomial interpolation, along with structural properties of the family of rational positive real functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m . Moreover we introduce an easy-to-find parametrization in R 2 m + 3 of a large subset of these interpolating functions. [ABSTRACT FROM AUTHOR]

This paper deals with the application of fractional derivatives in the modeling of electrical circuits RC, RL, RLC, power electronic devices and nonlinear loads, the equations are obtained by replacing the time derivative by fractional derivatives of type Riemann–Liouville, Grünwald–Letnikov, Liouville–Caputo and the fractional definition recently introduced by Caputo and Fabrizio. The fractional equations in the time domain considers derivatives in the range of α ∈ (0; 1], analytical and numerical results are presented considering different source terms introduced in the fractional equation. The resulting solutions modified the capacitance, inductance, also, the resistance exhibits fluctuations or fractality of time in different scales. Furthermore, the results showed the existence of heterogeneities in the electrical components causing irreversible dissipative effects. The classical models are recovered when the order of the fractional derivatives are equal to 1. [ABSTRACT FROM AUTHOR]

In the last years, several mathematical models coupling differential algebraic equations and partial differential equations for describing the behavior of electrical circuits have been proposed in the literature. Most of them investigate the properties of coupled systems that include one-dimensional drift-diffusion equations for describing the highly sensitive semi-conducting elements in the circuit. Here, we extend the results to coupled systems with higher dimensional drift-diffusion models that allow a proper modeling of semiconductor devices constituted of regions with different material properties. For stability reasons, we investigate a monolithic simulation approach and consider two common variants of PDE discretizations for semiconductors in the system: besides a finite element method that may be combined with the Scharfetter-Gummel approach, a mixed finite element discretization. The resulting differential algebraic equations share important properties that allow us to show that their index is always less or equal to two and depends only on the circuit's topology. The information about the index enables us to conclude unique solvability of the associated initial value problems as well as to choose appropriate time-stepping methods for them. [ABSTRACT FROM AUTHOR]

The swarm-based algorithms can be modelled, under suitable assumptions, as equivalent dynamic circuits reproducing the cinematic characteristics of the trajectories followed by the swarm members. This can be made in terms of voltages measured at the terminal of capacitors and currents measured at the terminals of inductors. Through the use of swarm circuits, the role played by the parameters becomes clear since it is possible to apply the stability analysis of continuous systems. This allows us to govern the exploration and/or the exploitation properties of the system simply by tuning its parameters into the convergence range or vice versa. The presented circuital model has been tested on famous benchmarks for optimization and inverse problems. The obtained results show that the swarm circuits are capable to manage exploration as well exploitation and can be used for real-time optimizations such as navigation in unknown ambient of mobile robots and so on. [ABSTRACT FROM AUTHOR]

We introduce a new class of linear systems, the L p-well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study. [ABSTRACT FROM AUTHOR]

Time-domain models of dynamical systems are formulated in many applications in terms of differential-algebraic equations (DAEs). In the linear time-varying context, certain limitations of models of the form E(t)x'(t) + B(t)x(t) = q(t) have recently led to the properly stated formulation A(t)(D(t)x(t))' + B(t)x(t) = q(t), which allows for explicit descriptions of problem solutions in regular DAEs with arbitrary index, and provides precise functional input-output characterizations of the system. In this context, the present paper addresses critical points of linear DAEs with properly stated leading term; such critical points describe different types of singularities in the system. Critical points are classified according to a taxonomy which reflects the phenomenon from which the singularity stems; this taxonomy is proved independent of projectors and also invariant under linear time-varying coordinate changes and refactorizations. Under certain working assumptions, the analysis of such critical problems can be carried out through a scalarly implicit decoupling, yielding a singular inherent ODE. Certain harmless problems for which this decoupling can be rewritten in explicit form are characterized. Some electrical circuit applications, including a linear time-varying analogue of Chua's circuit, are discussed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Without assuming the boundedness, strict monotonicity and differentiability of the activation functions, the authors utilize the Lyapunov functional method to analyze the global convergence of some delayed models. For the Hopfield neural network with time delays, a new sufficient condition ensuring the existence, uniqueness and global exponential stability of the equilibrium point is derived. This criterion concerning the signs of entries in the connection matrix imposes constraints on the feedback matrix independently of the delay parameters. From a new viewpoint, the bidirectional associative memory neural network with time delays is investigated and a new global exponential stability result is given. [ABSTRACT FROM AUTHOR]

A general theory for random walks on transfinite networks whose ranks are arbitrary natural numbers is established herein. In such networks, nodes of higher ranks connect together transfinite networks of lower ranks. The probabilities for transitions through such nodes are obtained as extensions of the Nash-Williams rule for random walks on ordinary infinite networks. The analysis is based on the theory of transfinite electrical networks, but it requires that the transfinite network have a structure that generalizes local-finiteness for ordinary infinite networks. The shorting together of nodes of different ranks are allowed; this complicates transitions through such nodes but provides a considerably more general theory. It is shown that, with respect to any finite set of nodes of any ranks, a transfinite random walk can be represented by an irreducible reversible Makov chain, whose state space is that set of nodes. [ABSTRACT FROM AUTHOR]

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