Network Theorems

Although the application of KirchhofFs laws offers basic tools to analyze a network, knowledge of certain network theorems, use of network equivalence, and use of reduction procedures simplify the process of network analysis. Basically, these theorems are applicable for linear networks. 

One of the most important strategies to simplify or reduce a linear circuit is superposition. The superposition theorem states that the response of a linear network to a number of simultaneously applied sources is equal to the sum of the individual responses due to each source acting alone. 

Note that only at the output terminals n-n are the Thevenin and Norton equivalents the same. In other words, at the output terminals n-n the voltage and current of the Thevenin equivalent circuit and the Norton equivalent circuit are identical. 

A commonly used network analysis method is loop and mesh analysis, which is generally based on KVL. As defined previously, loop analysis refers to the general method of current analysis for both planar and non-planar networks, whereas mesh analysis is reserved for the analysis of planar networks. In loop or mesh analysis, the circulating currents are selected as the unknowns, and a circulating current is assigned to each independent loop or mesh of the network. Then a series of equations can be formed according to KVL. 

The series of equations in the form of [/][/] = V71 can be established by equating the sum of the externally applied voltage sources acting in each loop to the sum of the voltage drops across the branches forming the loop. The number of equations is equal to the number of independent loops in the network. The general equation in loop or mesh analysis is given by 

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The main point of this section is the proof of a very general network theorem which is frequently referred to as Tellegen s theorem (TELLEGEN 1952). It says that... 

The package does not do Hopf bifurcation analysis nor have any direct way to distinguishing between limit cycle and chaotic attractors. The package contains the Zero Deficiency Theorem, the "knot tree network theorem" as well as some older theorems that identify stable networks. The package solves the general reaction balancing problem whose solution is a convex polyhedral cone of extreme reactions. It handles thermodynamic properties of reactions assuming ideality. 

From Euler s theorem, one can derive the following simple relation between the number and type of cycles n, (where the subscript / stands for the number of sides to the ring) necessary to close the hexagonal network of a graphene sheet ... 

Kolmogorov s Theorem (Reformulated by Hecht-Nielson) Any real-valued continuous function f defined on an N-dimensional cube can be implemented by a three layered neural network consisting of 2N -)-1 neurons in the hidden layer with transfer functions from the input to the hidden layer and (f> from all of... 

To round off this section we note a few unusual applications of Polya s Theorem an application to telecommunications network [CatK75], and one to the enumeration of Latin squares [JucA76]. In pure mathematics there is an application in number theory [ChaC82], and one to the study of quadratic forms [CraT80], being the enumeration of isomorphism types of Witt rings of fields. Finally, we note a perhaps unexpected, but quite natural, application in music theory to the enumeration of chords and tone rows for an n-note scale [ReiD85]. In the latter paper it is shown that for the usual chromatic scale of 12 semitones there are 80 essentially different 6-note chords, and 9,985,920 different tone rows. 

Theorem (Max-Flow, Min-Cut) The maximum flow between the source and the sink of a given network is equal to the minimum capacity of the cut-sets that separate the source from the sink.6... 

Kreinovich, V. Y., Arbitrary nonlinearity is sufficient to represent all functions by neural networks A theorem. Neural Networks 4, 381 (1991). 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

Kretsovalis, A., and Mah, R. S. H. (1988a). Observability and redundancy classification in generalised process networks. I Theorems. Comput. Chem. Eng. 12, 671-687. 

The utility and success of Metabolic Control Analysis is mostly due to a number of simple relationships that interconnect the various coefficients and that bridge between local and global properties of the network. First, the summation theorems relate to the structural properties of the network and are independent of kinetic parameters [96]. Using Eq. (90) and (91), it is straightforward to verify that... 

A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited Liqiu Wang, Mingtian Xu, and Xiaohao Wei, Multiscale Theorems... 

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