Theory and application of equivalent transformation relationships between plane wave and spherical wave

Based on the governing equations and the equivalent models, we propose an equivalent transformation relationships between a plane wave in a one-dimensional medium and a spherical wave in globular geometry with radially inhomogeneous properties. These equivalent relationships can help us to obtain the analytical solutions of the elastodynamic issues in an inhomogeneous medium. The physical essence of the presented equivalent transformations is the equivalent relationships between the geometry and the material properties. It indicates that the spherical wave problem in globular geometry can be transformed into the plane wave problem in the bar with variable property fields, and its inverse transformation is valid as well. Four different examples of wave motion problems in the inhomogeneous media are solved based on the presented equivalent relationships. We obtain two basic analytical solution forms in Examples I and II, investigate the reflection behavior of inhomogeneous half-space in Example III, and exhibit a special inhomogeneity in Example IV, which can keep the traveling spherical wave in constant amplitude. This study implies that our idea makes solving the associated problem easier.

A Program Study of the Union of Semilattices on the Set of Subsets of Grids of Waterloo Language

The aim is to study the set of subsets of grids of the Waterloo language from the point of view of abstract algebra and graph theory. The study was conducted using the library for working with transition graphs of nondeterministic finite automata NFALib implemented by one of the authors in C#, as well as statistical methods for analyzing algorithms. The results are regularities obtained when considering semilattices on a set of subsets of grids of the Waterloo language. It follows from the results obtained that the minimum covering automaton equivalent to the Waterloo automaton can be obtained by adding one additional to the minimum covering set of grids. .

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

Symmetry groups and Gauss kernels of Schrdinger equations

The relationship between symmetries and Gauss kernels for the SchrSdinger equation iut = uxx ＋ f（x）u is established. It is shown that if the Lie point symmetries of the equation are nontrivial, a classical integral transformations of the Gauss kernels can be obtained. Then the Gauss kernels of Schroedinger equations are derived by inverting the integral transformations. Furthermore, the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.

Robust fault detection filter design for a class of time-delay systems via equivalent transformation

This paper is concerned with the robust fault detection filter （RFDF） design for a class of linear time- invariant systems （LTISs） with output state time delays. Although existing results in literatures study the RFDF for time- delay systems, few is concerned with the output state time-delay systems. The basic idea of our study is to eliminate the time delays of system and transform it to a delay-free system （i.e., a linear time-invariant system without time delays） by the bicausal change of coordinates approach. Then, we design the RFDF for the delay-free LTIS, which is equivalent to the original system with time delays. We first introduce a class of systems with output state time delays, whose fault can be detected by using the RFDF design approach for delay-free systems. Then, since the RFDF design problem can be formulated as a standard H-infinity-model matching problem, it is solved by using H-infinity-optimization LMI techniques. In the last, the adaptive threshold of fault detection is chosen and an illustrative design example is used to demonstrate the validity of the design approach.

A new equivalent circuit model for on-chip spiral transformers in CMOS RFICs

A new compact model has been introduced to model on-chip spiral transformers.Unlike conventional models,which are often a compound of two spiral inductor models（i.e.,the combination of two coupledΠor 2-Πsub-circuits）,our new model only uses 12 elements to model the whole structure in the form of T topology.The new model is based on the physical meaning,and the process of model derivation is also presented.In addition,a simple parameter extraction procedure is proposed to get the elements＇ values without any fitting and optimization. In this procedure,a new method has been developed for the parameter extraction of the ladder circuit,which is commonly used to represent the skin effect.In order to verify the model＇s validity and accuracy,we have compared the simulated and measured self-inductance,quality factor,coupling coefficient and insertion loss,and an excellent agreement has been found over a broad frequency range up to the resonant frequency.

Unified construction of two n-order circuit networks with diodes

In this paper,two different n-order topological circuit networks are connected by diodes to establish a unified network model,which is a previously unexplored problem.The network model includes not only five resistive elements but also diode devices,so the network contains many different network types.This problem can be solved through three main steps:First,the network is simplified into two different equivalent circuit models.Second,the nonlinear difference equation model is established by applying Kirchhoff’s law.Finally,the two equations with similar structures are processed uniformly,and the general solutions of the nonlinear difference equations are obtained by using the transformation technique.As an example,several interesting specific results are deduced.Our study on the network model has significant value,as it can be applied to relevant interdisciplinary research.

Measurement of particle size based on digital imaging technique

To improve the analysis methods for the measurement of the sediment particle sizes with a wide distribution and of irregular shapes, a sediment particle image measurement, an analysis system, and an extraction algorithm of the optimal threshold based on the gray histogram peak values are proposed. Recording the pixels of the sediment particles by labeling them, the algorithm can effectively separate the sediment particle images from the background images using the equivalent pixel circles with the same diameters to represent the sediment particles. Compared with the laser analyzer for the case of blue plastic sands, the measurement results of the system are shown to be reasonably similar. The errors are mainly due to the small size of the particles and the limitation of the apparatus. The measurement accuracy can be improved by increasing the Charge-Coupled Devices （CCD） camera resolution. The analysis method of the sediment particle images can provide a technical support for the rapid measurement of the sediment particle size and its distribution.

Rearrangeability of 7-stage 16 × 16 shuffle exchange networks

It has long been an outstanding conjecture that any(2^(n)×2^(n))-stage shuffle exchange network(Omega net-work)is rearrangeable for 2n 62n.Many researchers have failed to prove this conjecture,including a recent one established by Hasan.However,nobody has pointed out its fallacy.Therefore,as one of the objectives,this paper shall clarify this fact.Since the case of n 53 has been proven by many researchers[1,2],this paper uses a con-structive approach to prove that when n 54,the 7-stage 16616 shuffle exchange network is also rearrangeable.The paper also presents the model of a balanced tree to avoid internal conflict,the representation of permutations using a connection graph and loop graph,and the con-cepts of symmetry graph and identical transform.Based on graphic composition and bipartition,the permutations 16×16 are divided into five classes,with five assignment algorithms proposed.These algorithms are simpler,clearer and easier to program.The techniques used for n=4 may provide hints for the general case of n>4.

Analysis and modeling of on-chip transformers under two ground conditions

Two fabricated on-chip transformers under different ground conditions（i.e.,CG and IG types） have been measured to compare their different characteristics.With the aid of the electromagnetic（EM） solver,we have analyzed the differences from the electric and magnetic aspects,and different effects in these aspects can be described with the lumped capacitor and inductor from the perspective of the equivalent circuit model.A physicsbased equivalent circuit model is proposed to model transformers under different ground conditions.In addition, the simple parameter extraction procedure for the corresponding model is also provided.All the model parameters are extracted and agree with the analysis.In order to verify the model＇s validity and accuracy,we have compared the modeled and measured S-parameters,and an excellent agreement has been found over a broad frequency range.