THE EXTENDED JORDAN'S LEMMA AND THE RELATION BETWEEN LAPLACE TRANSFORM AND FOURIER TRANSFORM

Jordan's lemma can be used for a wider range than the original one. The extended Jordan's lemma can be described as follows. Let f(z) be analytic in the upper half of the z plane (Imz≥0), with the exception of a finite number of isolated singularities, and for P>o, if then where z=Rei and CR is the open semicircle in the upper half of the z plane.With the extended Jordan's lemma one can find that Laplace transform and Fourier transform are a pair of integral transforms which relate to each other.

Implementing Classical Hadamard Transform Algorithm by Continuous Variable Cluster State

Measurement-based one-way quantum computation, which uses cluster states as resources, provides an efficient model to perforrn computation. However, few of the continuous variable （CV） quantum algorithms and classical algorithms based on one-way quantum computation were proposed. In this work, we propose a method to implement the classical Hadamard transform algorithm utilizing the CV cluster state. Compared with classical computation, only half operations are required when it is operated in the one-way CV quantum computer. As an example, we present a concrete scheme of four-mode classical Hadamard transform algorithm with a four-partite CV cluster state. This method connects the quantum computer and the classical algorithms, which shows the feasibility of running classical algorithms in a quantum computer efficiently.

Brushless Doubly-Fed Induction Machine as a Variable Frequency Transformer

VFT （variable frequency transformer） has been recently used as art alternative to HVDC （high voltage direct current） to control power flow between asynchronous networks. VFT consumes less reactive power than a back-to-back HVDC system, provides faster initial transient recovery, and has better natural damping capability. VFT is simply a DFIM （doubly-fed induction machine） where the machine torque controls the power flow from stator to rotor and vice versa. The main disadvantage of this VFT is the slip rings and brushes required for the rotor circuit, especially in bulk power transmission. The BDFM （brushless doubly-fed machine） with nested cage rotor machine is proved to be a comparable alternative to conventional DFIM in many applications with the advantage that all windings being in the stator frame with fixed output terminals. In this paper, the BDFM is used as a BVFT （brushless variable frequency transformer）. A prototype machine is designed and simulated to verify the system validity.

The Numerical Scheme Development of a Simplified Frozen Soil Model

In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change between water and ice, is shown explicitly in both the energy and mass balance equations. The models must be solved by a numerical method with an iterative process, and the rate term of the phase change needs to be pre-estimated at the beginning in each iteration step. Since the rate term of the phase change in the energy equation is closely related to the release or absorption of the great amount of fusion heat, a small error in the rate term estimation will introduce greater error in the energy balance, which will amplify the error in the temperature calculation and in turn, cause problems for the numerical solution convergence. In this work, in order to first reduce the trouble, the methodology of the variable transformation is applied to a simplified frozen soil model used currently, which leads to new frozen soil scheme used in this work. In the new scheme, the enthalpy and the total water equivalent are used as predictive variables in the governing equations to replace temperature, volumetric soil moisture and ice content used in many current models. By doing so, the rate terms of the phase change are not shown explicitly in both the mass and energy equations and its pre-estimation is avoided. Secondly, in order to solve this new scheme more functionally, the development of the numerical scheme to the new scheme is described and a numerical algorithm appropriate to the numerical scheme is developed. In order to evaluate the new scheme of the frozen soil model and its relevant algorithm, a series of model evaluations are conducted by comparing numerical results from the new model scheme with three observational data sets. The comparisons show that the results from the model are in good agreement with these data sets in both the change trend of variables and their magnitude values, and the new scheme, together with the algorithm, is more efficient and saves more computer time.

Weak solutions to one-dimensional quantum drift-diffusion equations for semiconductors

The weak solutions to the stationary quantum drift-diffusion equations （QDD） for semiconductor devices are investigated in one space dimension. The proofs are based on a reformulation of the system as a fourth-order elliptic boundary value problem by using an exponential variable transformation. The techniques of a priori estimates and Leray-Schauder＇s fixed-point theorem are employed to prove the existence. Furthermore, the uniqueness of solutions and the semiclassical limit δ→0 from QDD to the classical drift-diffusion （DD） model are studied.

Exact Wave Function of a Time-Dependent Quantum System

To solve the problem in dispute about a Schrdinger equation with time-depenelent mass and frequency, by means of a simple transformation of variables, the time-dependent Schrdinger equation is transformed into the time-independent one first and then an exact wave function can be found.

GLIVENKO-CANTELLI TYPE THEOREMS ANDTHEIR APPLICATIONS

In this paper, we present a sufficient condition for uniform convergence of means to their expectations over the classes of real functions. Our proof is very simple via connecting this convergence to uniform convergence over the class of indicator functions. Furthermore, we obtain a convergent result concerning a non=VC class with an application to variable transformation for fitting regression model.

Line-element based nonlinear adaptive piecewise compensating correction for LVDT sensors

In order to solve the linear variable differential transformer （LVDT） displacement sensor nonlinearity of overall range and extend its working range, a novel line-element based adaptively seg- menting method for piecewise compensating correction was proposed. According to the mechanical structure of LVDT, the output equation was calculated, and then the theoretic nonlinear source of output was analyzed. By the proposed line-element adaptive segmentation method, the nonlinear output of LVDT was divided into linear and nonlinear regions with a given threshold. Then the com- pensating correction function was designed for nonlinear parts employing polynomial regression tech- nique. The simulation of LVDT validates the feasibility of proposed scheme, and the results of cali- bration and testing experiments fully prove that the proposed method has higher accuracy than the state-of-art correction algorithms.

An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation

In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.

Three-dimensional beam propagation method based on the variable transformed Galerkin's method

A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The infinite Cartesian x-y plane is mapped into a unit square by a tangent-type function transformation. Consequently, the infinite region problem is converted into the finite region problem. Thus, the boundary truncation is eliminated and the calculation accuracy is promoted. The three-dimensional BPM basic equation is reduced to a set of first-order ordinary differential equations through sinusoidal basis function, which fits arbitrary cladding optical waveguide, then direct solution of the resulting equations by means of the Runge-Kutta method. In addition, the calculation is efficient due to the small matrix derived from the present technique. Both z-invariant and z-variant examples are considered to test both the accuracy and utility of this approach.

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac＇s symbolic method

By virtue of the new technique of performing integration over Dirac＇s ket-bra operators, we ex- plore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel- Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, de- riving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel opera- tor （GFO） to correspond to the generalized Fresnel transform （GFT） in classical optics. We derive GFO＇s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum op- tics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product （IWOP） of operators are fully used. All these have confirmed Dirac＇s assertion： ＂...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory＂.

Orthogonal transformation operation theorem of a spatial universal uniform rotating magnetic field and its application in capsule endoscopy

According to the anti-phase sine current superposition theorem, the orientation, the magnetic flux density, the angular speed and the rotational direction of the spatial universal rotating magnetic field (SURMF) can be controlled within the tri-axial orthogonal square Helmholtz coils (TOSHC). Nevertheless, three coupling direction angles of the normal vector of the SURMF in the Descartes coordinate system cannot be separately controlled, thus the adjustment of the orientation of the SURMF is difficult and the flexibility of the robotic posture control is restricted. For the dimension reduction and the decoupling of control variables, the orthogonal transformation operation theorem of the SURMF is proposed based on two independent rotation angular variables, which employs azimuth and altitude angles as two variables of the three-phase sine current superposition formula derived by the orthogonal rotation inverse transformation. Then the unique control rules of the orientation and the rotational direction of the SURMF are generalized in each spatial quadrant, thus the scanning of the normal vector of the SURMF along the horizontal or vertical direction can be achieved through changing only one variable, which simplifies the control process of the orientation of the SURMF greatly. To validate its feasibility and maneuverability, experiments were conducted in the animal intestine utilizing the innovative dual hemisphere capsule robot (DHCR) with active and passive modes. It was demonstrated that the posture adjustment and the steering rolling locomotion of the DHCR can be realized through single variable control, thus the orthogonal transformation operation theorem makes the control of the orientation of the SURMF convenient and flexible significantly. This breakthrough will lay a foundation for the human-machine interaction control of the SURMF.

Magnetically assisted gas-solid fluidization in a tapered vessel:Part Ⅰ. Magnetization-LAST mode

This article presents further experimental results of the Magnetization-LAST mode in magnetically assisted gas-fluidized tapered beds, including external transverse magnetic field control of solid phase movement, central channel formation, spout depth and the pressure drop across the bed. Phase diagrams similar to those recently reported for the Magnetization-FIRST mode were also developed. Dimensional analysis based on ＂pressure transform＂ of the initial set of variables and involving the magnetic granular Bond number pertinent to particle aggregate formation was applied to develop the scaling relationships.

System integration for on-machine measurement using a capacitive LVDT-Iike contact sensor

In this study, an air-bearing capacitive linear variable differential transformer （LVDT）-like contact sensor with a rounded diamond tip was mounted to a desktop machine tool to construct an on-machine （OM） measuring system. The measuring system was capable of decoding the digital signals of linear encoders mounted on the machine tool and acquiring the analog signal of the contact sensor. To verify the measuring system, experimental examinations were performed on an oxygen-free copper （OFC） convex aspheric mold with a diameter of 5 mm and a curve height of 0.46 mm. The acquired signals were processed by the implemented Gaussian regression filter （GRF）, removing the tilt of measured profile, and compensating for the radius of probe tip. The profile obtained was compared to that measured using a commercially available device, and a maximum deviation of 14.6μm was found for the rough cutting. The compensation cutting was then performed according to the form error of OM measurement. As a result, the PV form error compared with the designed profile was reduced from 19.2μm over a measured diameter of 4 mm to 9.7 μm over a measured diameter of 3.1 mm, or a percentage improvement of 35.4% in form accuracy. Through the examination for aspheric machining, the effectiveness of the implemented OM measuring system was demonstrated, and the technical details of system implementation were presented. Further improvement was suggested to reduce the diameter of probe tip and measuring force.

Magnetically assisted gas-solid fluidization in a tapered vessel:Part II Dimensionless bed expansion scaling

The article presents an effort to create dimensionless scaling correlations of the overall bed porosity in the case of magnetically assisted fluidization in a tapered vessel with external transverse magnetic field. This is a stand of portion of new branch in the magnetically assisted fluidization recently created concerning employment of tapered vessels. Dimensional analysis based on ＂pressure transform＂ of the initial set of variables and involving the magnetic granular Bond number has been applied to develop scaling relationships of dimensionless groups representing ratios of pressures created by the fluid flow, gravity and the magnetic field over an elementary volume of the fluidized bed. Special attention has been paid on the existing data correlations developed for non-magnetic beds and the links to the new ones especially developed for tapered magnetic counterparts. A special dimensionless variable Xp = （Ar△Dbt）1/3√RgMQ combining Archimedes and Rosensweig numbers has been conceived for porosity correlation. Data correlations have been performed by power-law, exponential decay and asymptotic functions with analysis of their adequacies and accuracies of approximation.

Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world.In particular,Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics.In this study,the(2+1)-dimensional time-fractional nonlinear Schrodinger equation and(1+1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures.This is shown by constructing appropriate analytic wave solutions.A success-ful implementation of the advised rational(1/φ'(ξ))-expansion method generates new outcomes of the considered equations,by comparing them with those already noted in the literature.On the basis of the conformable fractional derivative,a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme.Finally,the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures.The employed technique is competent,productive and concise enough,making it feasible for future studies.

Stochastic energy-demand analyses with random input parameters for the single-family house

In the analyses of the uncertainty propagation of buildings’energy-demand,the Monte Carlo method is commonly used.In this study we present two alternative approaches:the stochastic perturbation method and the transformed random variable method.The energy-demand analysis is performed for the representative single-family house in Poland.The investigation is focused on two independent variables,considered as uncertain,the expanded polystyrene thermal conductivity and external temperature;however the generalization on any countable number of parameters is possible.Afterwards,the propagation of the uncertainty in the calculations of the energy consumption has been investigated using two aforementioned approaches.The stochastic perturbation method is used to determine the expected value and central moments of the energy consumption,while the transformed random variable method allows to obtain the explicit form of energy consumption probability density function and further characteristic parameters like quantiles of energy consumption.The calculated data evinces a high accordance with the results obtained by means of the Monte Carlo method.The most important conclusions are related to the computational cost reduction,simplicity of the application and the appropriateness of the proposed approaches for the buildings’energy-demand calculations.

Temperature-variable high-frequency dynamic modeling of PIN diode

The PIN diode model for high frequency dynamic transient characteristic simulation is important in conducted EMI analysis. The model should take junction temperature into consideration since equipment usually works at a wide range of temperature. In this paper, a temperature-variable high frequency dynamic model for the PIN diode is built, which is based on the Laplace-transform analytical model at constant temperature. The relationship between model parameters and temperature is expressed as temperature functions by analyzing the physical principle of these parameters. A fast recovery power diode MUR1560 is chosen as the test sample and its dynamic performance is tested under inductive load by a temperature chamber experiment, which is used for model parameter extraction and model verification. Results show that the model proposed in this paper is accurate for reverse recovery simulation with relatively small errors at the temperature range from 25 to 120 ℃.

Solitary Wave Solutions of KP equation,Cylindrical KP Equation and Spherical KP Equation

Three(2+1)-dimensional equations–KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same Kd V equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the Kd V equation have been known already, substituting the solutions of the Kd V equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three(2+1)-dimensional equations can be obtained successfully.