Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

A novel formula for representing the equivalent resistance of the m × n cylindrical resistor network

The problem of solving the equivalent resistance between two points for resistor networks has important significance in physics. This paper mainly changes and rewrites the formula for calculating the resistance between two points of an unconventional m × n cylindrical resistor network with a zero resistor axis and any two left and right boundaries. To enhance the efficiency of calculating the equivalent resistance between two points, Chebyshev polynomials and hyperbolic cosine functions are employed to represent the new formula. And in the inference process, the famous discrete cosine transform of the third kind (DCT-III) is used to process the matrix. We give the equivalent resistance formula for several special cases, and display them by a three-dimensional graph. Subsequently, the calculation efficiency of the original formula and the rewritten formula are compared. At the end of the paper, a heuristic algorithm suitable for robot path planning on cylindrical environment is proposed.

Two optimized novel potential formulas and numerical algorithms for [Formula: see text] cobweb and fan resistor networks

The research of resistive network will become the basis of many fields. At present, many exact potential formulas of some complex resistor networks have been obtained. Computer numerical simulation is the trend of computing, but written calculation will limit the time and scale. In this paper, the potential formulas of a [Formula: see text] scale cobweb resistor network and fan resistor network are optimized. Chebyshev polynomial of the second class and the absolute value function are used to express the novel potential formulas of the resistor network, and described in detail the derivation process of the explicit formula. Considering the influence of parameters on the potential formulas, several idiosyncratic potential formulas are proposed, and the corresponding three-dimensional dynamic images are drawn. Two numerical algorithms of the computing potential are presented by using the mathematical model and DST-VI. Finally, the efficiency of calculating potential by different methods are compared. The advantages of new potential formulas and numerical algorithms by the calculation efficiency of the three methods are shown. The optimized potential formulas and the presented numerical algorithms provide a powerful tool for the field of science and engineering.

Analytical potential formulae and fast algorithm for a horn torus resistor network

In this paper, a (u+1)×v horn torus resistor network with a special boundary is researched. According to Kirchhoff's law and the recursion-transform method, a model of the resistor network is established by the voltage V and a perturbed tridiagonal Toeplitz matrix. We obtain the exact potential formula of a horn torus resistor network. First, the orthogonal matrix transformation is constructed to obtain the eigenvalues and eigenvectors of this perturbed tridiagonal Toeplitz matrix; second, the solution of the node voltage is given by using the famous fifth kind of discrete sine transform (DST-V). We introduce Chebyshev polynomials to represent the exact potential formula. In addition, the equivalent resistance formulae in special cases are given and displayed by a three-dimensional dynamic view. Finally, a fast algorithm of computing potential is proposed by using the mathematical model, famous DST-V, and fast matrix-vector multiplication. The exact potential formula and the proposed fast algorithm realize large-scale fast and efficient operation for a (u+1)×v horn torus resistor network, respectively.

Fast algorithm and new potential formula represented by Chebyshev polynomials for an [Formula: see text] globe network

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an [Formula: see text] globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

Circuit network theory of n-horizontal bridge structure

This research investigates a complex n order cascading circuit network with embedded horizontal bridge circuits with the N-RT method. The contents of the study include equivalent resistance analytical formula and complex impedance characteristics of the circuit network. The research idea is as follows. Firstly the equivalent model of n-order resistance network is established, and a fractional difference equation model is derived using Kirchhoff’s law. Secondly, the equivalent transformation method is employed to transform the fractional equation into a simple linear difference equation, and its particular solution is computed. Then the solution to the difference equation is used to derive the effective resistance of the resistance network of the embedded horizontal bridge circuit, and various special cases of equivalent resistance formula are analyzed and the correctness of the analysis model gets verified. Finally, as an expanded application, the equivalent complex impedance of LC network is studied, and Matlab drawing tool is employed to offer the equivalent impedance with various variables of the graph. Our results provide new research ideas and theoretical basis for relevant scientific researches and practical applications.

Potential formula of an m × n globe network and its application

Searching for the explicit solutions of the potential function in an arbitrary resistor network is important but difficult in physics. We investigate the problem of potential formula in an arbitrary m × n globe network of resistors, which has not been resolved before (the previous study only calculated the resistance). In this paper, an exact potential formula of an arbitrary m × n globe network is discovered by means of the Recursion-Transform method with current parameters (RT-I). The key process of RT method is to set up matrix equation and to transform two-dimensional matrix equation into one-dimensional matrix equation. In order to facilitate practical application, we deduced a series of interesting results of potential by means of the general formula, and the effective resistance between two nodes in the m × n globe network is derived naturally by making use of potential formula.

Potential formula of the nonregular m × n fan network and its application

Potential formula of an arbitrary resistor network has been an unsolved problem for hundreds of years, which is an interdisciplinary problem that involves many areas of natural science. A new progress has been made in this paper, which discovered the potential formula of a nonregular m × n fan network with two arbitrary boundaries by the Recursion-Transform method with potential parameters (simply call RT-V). The nonregular m × n fan network is a multipurpose network contains several different types of network model such as the interesting snail network and hart network. In the meantime, we discussed the semi-infinite fan network and a series of novel and special conclusions are produced, the effective resistance is educed naturally. The discovery of potential formulae of resistor network provides new theoretical tools and techniques for related scientific research.

Two-point resistances in an Apollonian network

The computation of resistance between two nodes in a resistor network is a classical problem in electric theory and graph theory. Based on the Apollonian packing, Andrade et al. introduced a deterministic growing type of networks A(k) [Phys. Rev. Lett. 94, 018702 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.018702]. In this paper, first, a recursive algorithm for computing resistance between any two nodes in A(k) is given. Then as explanations, using the algorithm, explicit expressions for some resistances in A(k) are obtained.