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This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since the smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.

One problem that arises in computation involving large numbers is precision. In certain situations, the result might be represented by the standard data type, but arithmetic precision might be compromised when dealing with large numbers in the course to the result. Binomial coefficients are an example that suffer from this torment. In the present paper, we review numerical methods to compute the binomial coefficients: Pascal’s recursive method; prime factorization to cancel out terms; gamma function approximation; and a simplified iterative method that avoids loss in precision. Acknowledging that the binomial coefficients might be obtained by a successive convolution of a simple discrete rectangular function, we propose a different approach based on the discrete Fourier transform and another based on its fast implementation. We analyze and compare performance and precision of all of them. The proposed methods overcome the previous ones when computing all coefficients for a given level n, attaining better precision for large values of n.

In order to make the teaching of signal and system course more vivid, philosophical thinking is introduced in the process of classroom teaching by introducing the principle of universality and particularity of contradiction into the interpretation of Fourier series, the principle of unity and opposites of contradiction into the explanation of rectangular function and shock function, the principle of quantitative change and qualitative change into the theory of square wave function expansion, and the unity of finite and infinite opposition into the spectrum analysis of shock function. It can be closely combined. Teaching practice shows that this teaching method can effectively mobilize students’ learning enthusiasm, expand students’ thinking depth, and achieve better teaching effect.

This paper presents a magneto-structural combined dimensional and topology optimization technique for interior permanent magnet synchronous machine (IPMSM) rotors. Dimensional changes to the permanent magnet (PM) location or size are accomplished by interpolating or projecting a smoothed Heaviside rectangular function representing the presence of PM material onto the IPMSM rotor design domain mesh. A density based Solid Isotropic with Material Penalization (SIMP) topology optimization approach is then used to vary the presence of electrical steel in mesh elements to form flux barriers around the PM. The proposed method enforces a defined shape for the PM without requiring the mesh in the design domain to be deformed. Three examples are presented to demonstrate the technique: two flat bar IPMSM and one V-shaped IPMSM.

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of $L_p$-the approximation convergence rate by these methods, when $n \to \infty$, for $1 < p < \infty$, and lower bounds of the quantity of best approximation of this class by arbitrary linear combinations of $n$ translates of arbitrary function, for the particular case $p=2$.

PURPOSE: In previous work, a theoretical model of the point spread function (PSF) for oblique x-ray incidence in amorphous selenium (a-Se) detectors was proposed. The purpose of this paper is to develop a complementary model that includes two additional features. First, the incidence angle and the directionality of ray incidence are calculated at each position, assuming a divergent x-ray beam geometry. This approach allows the non-stationarity of the PSF to be modeled. Second, this paper develops a framework that is applicable to a digital system, unlike previous work which did not model the presence of a thin-film transistor (TFT) array. METHODS: At each point on the detector, the incidence angle and the ray incidence direction are determined using ray tracing. Based on these calculations, an existing model for the PSF of the x-ray converter [Med. Phys. 22(4), 365-374 (1995)] is generalized to a non-stationary model. The PSF is convolved with the product of two rectangle functions, which model the sampling of the TFT array. The rectangle functions match the detector element (del) size in two dimensions. RESULTS: It is shown that the PSF can be calculated in closed form. This solution is used to simulate a digital mammography system at two x-ray energies (20 and 40 keV). Based on the divergence of the x-ray beam, the direction of ray incidence varies with position. Along this direction, the PSF is broader than the reference rect function matching the del size. The broadening is more pronounced with increasing obliquity. At high energy, the PSF deviates more strongly from the reference rect function, indicating that there is more blurring. In addition, the PSF is calculated along the polar angle perpendicular to the ray incidence direction. For this polar angle, the shape of the PSF is dependent upon whether the ray incidence direction is parallel with the sides of the detector. If the ray incidence direction is parallel with either dimension, the PSF is a perfect rectangle function, matching the del size. However, if the ray incidence direction is at an oblique angle relative to the sides of the detector, the PSF is not rectangular. These results illustrate the non-stationarity of the PSF. CONCLUSIONS: This paper demonstrates that an existing model of the PSF of a-Se detectors can be generalized to include the effects of non-stationarity and digitization. The PSF is determined in closed form. This solution offers the advantage of shorter computation time relative to approaches that use numerical methods. This model is a tool for simulating a-Se detectors in future work, such as in virtual clinical trials with computational phantoms.

In this letter, we deduce the approximate analytical formula for the finite Airy beam propagating in the misaligned slab system with a rectangular aperture by using the generalized Huygens-Fresnel integral equation, the light transfer matrix and expanding the hard-edged rectangular function into a finite sum of complex Gaussian function. The particular cases of this beam propagating in the apertured aligned slab system, unapertured misaligned slab system and unapertured aligned one are further illustrated through numerical examples as well. The effects of aperture size and misaligned factors on beam evolution in these slab structures are developed, which can provide a fast and effective method for investigating other kinds of pseudo non-diffracting laser beams through the rectangularly apertured misaligned slab system because most of the optical devices are slightly misaligned more or less in the practical application. Keywords: Finite Airy beam, Transfer matrix, Generalized Huygens-Fresnel integral equation, Misalignment, Rectangular aperture

In the past, many functions were considered for simulating the lightning return-stroke current. Some of these functions were found to have problems related to their discontinuities or the discontinuities of their derivatives at onset time. Such problems appear in the double exponential function and its modifications. However, other functions like the Pulse function and Heidler function do not suffer from such problems. One of the main objectives of this work is to simulate the lightning return-stroke current full wave, including the decay part, using either Heidler function or the Pulse function. This work is not only necessary for the evaluation and development of lightning return-stroke modeling, but also for the calculation of the lightning current waveform parameters. Although the lightning return-stroke current, measured at the CN Tower, is simulated using the Pulse function and Heidler function, the simulation of the CN Tower lightning current derivative signal is considered using the derivative of the Pulse and Heidler functions. First, we build a modeling environment for each function, which can be described as parameter estimation system. This system, which represents an automated approach for estimating the analytical parameters of a given function, is capable to best fit the function with the measured data. Using these analytical parameters transforms the discrete data into a continuous signal, from which the current waveform parameters can be estimated. This analytical parameters estimation system is recognized as a curve fitting system. For curve fitting technique, the initial value of each analytical parameter and its feasible region, where the optimal value of this analytical parameter is located, must be specified. The more accurate the initial point is the easier and faster the optimal value can be estimated. On choosing the best approach of the initial condition, which gives the nearest location to the optimal point, applying the estimation system and achieving the analytical model that fits the CN Tower measured current derivative, the current waveform parameters can be easily studied. In order to be sure that the analytical parameter extraction system gives the best fit of a function, it needs to be evaluated. Instead of going through the measured data, we first use artificial digital data as a productive way to evaluate the system. Also, a comparison between both the Pulse and Heidler functions is performed. The described fitting process is applied on 15 flashes, containing 31 return strokes. The calculated current waveform parameters were used to form statistics to determine the probability distribution of the value of each parameter, including the range and the 50% probability level, which is fundamental in building lightning protection systems.

The adiabatic quantum information processing task requires that the evolution of a system must be kept in its instantaneous eigenstate. However, normally the adiabaticity will be ruined due to the interaction between the system and the noisy environment in its long evolution time. Here, in this paper, we show that zero-area pulse control can be used to realize the adiabatic process in a nonadiabatic regime. A concrete example is provided where one spin chain is cut into two chains. The pulse function is applied in the laboratory frame and suitable pulse conditions are obtained numerically. We find that compared with the pulse conditions obtained in the adiabatic frame, the results are similar for low-energy-level systems but tend to deviate when the system's energy level increases. We then obtain the pulse conditions theoretically by writing the control Hamiltonian in the adiabatic frame. It is found that a sequence of pulses with intensities tuned by a time-dependent energy difference is required to guarantee an effective adiabatic speedup.

The growth of technical and economic indicators in industry is ensured by the improvement of existing and the development of new principles of organization and forecasting of production. The current trend is the development of digital models of industrial objects and, in particular, their “digital twins,” of which sensors are a mandatory part. This makes it possible to successfully predict the technical condition of objects, which is especially important in the field of metallurgical and heavy mechanical engineering, where forced stops are accompanied, sometimes, by enormous losses. The article considers a model of a piezo-active acoustic sensor, which is intended for a wide range of tasks and, in particular, for evaluation of the equipment condition of the workshop of metallurgical or machine-building industry enterprises on the basis of periodic registration of acoustic signal spectra in the workshop space. In essence, it is a pressure sensor with a specific bandwidth, sensitivity and directivity pattern in radiation and reception. The problem of developing a mathematical, discrete and digital sensor model is solved. On the basis of non-stationary equations of mathematical physics, a distributed model of a sensor built according to the scheme: plate-piezo cylinder-rod-mass has been developed. It can be truncated to the desired configuration and takes into account piezo-active properties, temperature effects, frictional hysteresis, and elastic characteristic nonlinearity. This distributed model is reduced to an integral view by determining the generalized pulse function (Green) of the plate and then the entire sensor, which provides a signal at the electrical output by convolution of the input pressure function with the generalized pulse function of the sensor. The integral view of the continuous model provides high accuracy of approximation of the discrete model of the sensor, which in turn also determines high accuracy of conversion of input signals. Parametric study is performed, a possibility of correction of sensor frequency response is shown. The author is not concerned with the features of implementing the digital model of the sensor on the basis of the discrete model, as they are well studied and related to the properties of the platform on which the digital model is implemented.

The accuracy of tropospheric temperature measurements with pure rotational Raman (PRR) lidars is affected by the collisional broadening of N2 and O2 PRR lines. In this paper, we intercompare nine calibration functions (CFs) in the traditional PRR lidar technique via simulation. Taking into account the PRR line broadening, the simulation is performed for five sets of spectral filters (SFs) with different passbands in a PRR lidar receiving system. For simplicity of calculations, the transmission function of each SF is approximated by a rectangular function on the wavenumber interval, within which a SF passes the bulk of the backscattered signal intensity in the corresponding PRR lidar channel. A narrow-linewidth laser operating at wavelengths of 354.67 and 532 nm is considered as a lidar transmitter. The CF best suited for tropospheric temperature retrievals from raw PRR lidar data for each set of SFs and laser wavelength is determined by comparative analysis of calibration errors ΔT produced using these CFs. The absolute error |ΔT| does not exceed the value of 3 × 10–3 K when using the best three-coefficient CF, while |ΔT

When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential ϕ depends on the electrostatic charge qe, as expressed by the equation L(qe)=ϕ, where L is an integral operator. Method of moments (MoM) is an excellent candidate for solving L(qe)=ϕ numerically. In fact, considering qe as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, qe=&sum, n=1N&alpha, nfn (with weights &alpha, n and N, number of the subsections of the conductor) defined in the L domain so that ϕ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system Lmn&alpha, n=gm with dense matrix, equivalent to L(qe)=ϕ. Once solved, the linear algebraic system gets &alpha, n and therefore qe is obtainable so that the electrostatic capacitance C=qe/V, where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve Lmn&alpha, n=gm. These methods have, as a basic idea, the projection of a problem related to a matrix A&isin, Rn&times, n, having a number of non-null elements of the order of n, in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine Lmn according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function, (2) pulse function as the basis function as well as testing function, (3) triangular function as the basis function and delta function as the testing function, (4) triangular function as the basis function and pulse function as the testing function, (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five Lmn and five pair qe and C were computed. For each case, for the resolution of Lmn&alpha, n=gm obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab&reg, Toolbox to evaluate qe and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of Lmn and considerations on C, to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss&ndash, Seidel approaches.

Nano-probing system is a common tool in failure analysis (FA) field and it is widely used for nanoscale defect localization, transistor characterization and local circuit functional debug for soft failure verification. While device hard failure can be easily localized or characterized by DC bias. However, some soft failures related with timing failures are not easy to be diagnosed by DC measurement especially it is related to speed issue. These cases need to be applied with pulse signal to manifest the failure signatures by observe the rising time and falling time behaviors. In this paper, a novel application of circuit debugging by using nano-probing technique will be demonstrated. A full adder is a digital circuit that performs a set of binary numbers, and the function can easily be verified from the signals and truth table. Forcing a voltage in the input side of full adder circuit, and measuring its output signals, FA engineers can easily diagnose whether the full adder function is workable or not.

In this work, aluminum nitride (A1N) film with good piezoelectric properties was grown on the silicon (Si) substrate as piezoelectric layer, and the properties of surface acoustic wave (SAW) devices with different interdigital transducer (IDT) structures were researched by using Rectangle function, Hanning function and Kaiser function. MATLAB and e-LINE plus software were used to generate layout files quickly and accurately. Devices with 300nm finger width were tested at room temperature and the results indicated that devices with Kaiser function structure show better resonant waveforms, the center frequency was up to 4.94GHz, the inhibition degree of sidelobe increased obviously to 43.53dB, and insertion loss was -5.87dB. This work play an active role in the design and research of high performance surface acoustic wave devices.

Transit times of a bolus through an organ can provide valuable information for researchers, technicians and clinicians. Therefore, an indicator is injected and the temporal propagation is monitored at two distinct locations. The transit time extracted from two indicator dilution curves can be used to calculate for example blood flow and thus provide the surgeon with important diagnostic information. However, the performance of methods to determine the transit time Δt cannot be assessed quantitatively due to the lack of a sufficient and trustworthy ground truth derived from in vivo measurements. Therefore, we propose a method to obtain an in silico generated dataset of differently subsampled indicator dilution curves with a ground truth of the transit time. This method allows variations on shape, sampling rate and noise while being accurate and easily configurable. COMSOL Multiphysics is used to simulate a laminar flow through a pipe containing blood analogue. The indicator is modelled as a rectangular function of concentration in a segment of the pipe. Afterwards, a flow is applied and the rectangular function will be diluted. Shape varying dilution curves are obtained by discrete-time measurement of the average dye concentration over different cross-sectional areas of the pipe. One dataset is obtained by duplicating one curve followed by subsampling, delaying and applying noise. Multiple indicator dilution curves were simulated, which are qualitatively matching in vivo measurements. The curves temporal resolution, delay and noise level can be chosen according to the requirements of the field of research. Various datasets, each containing two corresponding dilution curves with an existing ground truth transit time, are now available. With additional knowledge or assumptions regarding the detection-specific transfer function, realistic signal characteristics can be simulated. The accuracy of methods for the assessment of Δt can now be quantitatively compared and their sensitivity to noise evaluated.

The intervention method is a time series model which could be used to model data with extreme fluctuation whether up or down. Stock price return tend to have extreme fluctuation which is caused by internal or external factors. There are two kinds of intervention function; a step function and a pulse function. A step function is used for a long-term intervention, while a pulse function is used for a short-term intervention. Modelling a time series data needs to satisfy the homoscedasticity assumptions (variance of residual is homogeneous). In reality, stock price return has a high volatility, in other words it has a non-constant variance of residuals (heteroscedasticity). ARCH (Autoregressive Conditional Heteroscedasticity) or GARCH (Generalized Autoregressive Conditional Heteroscedasticity) can be used to model data with heteroscedasticity. The data used is stock price return from August 2008 until September 2018. From the stock price return data plot is found an extreme fluctuation in September 2017 (T=110) that is suspected as a pulse function. The best model uses the intervention pulse function is ARMA([1,4],0) (b=0, s=1, r=1). The intervention model has a non-constant variance or there is an ARCH effect. The best variance model obtained is ARMA([1,4],0)(b=0, s=1, r=1)–GARCH(1,1) with the AIC value is -205,75088. Keywords: Stock Return, Intervention, Heteroscedasticity, ARCH/GARCH

The baffle is a practical and promising passive damping method of dissipating acoustic energy and increasing acoustic losses in a solid rocket motor (SRM). In this study, numerical studies were conducted conjunction with the acoustic pulse response method (PRM) to evaluate the acoustic damping performance of a full-scale SRM with a baffle. The forced pulse function was imposed according to the frequency of the longitudinal acoustic mode. We comparatively evaluated the acoustic damping performance of the SRM with and without a baffle. The performances were characterized by (1) acoustic growth rate, (2) damping rate, and (3) acoustic energy. The PRM was first validated using data available in the literature. Several numerical investigations were conducted to develop geometry design criteria, which were subsequently used to ensure the effective operation of the baffle to suppress combustion-driven acoustic modes in the SRM. The effects of 1) the baffle axial location ( x / L ) and 2) the relative diameter ( d / D ) on acoustic damping performance were examined in detail. The results indicated that the baffle is effective in suppressing acoustic oscillations only when placed at 1 / 4 ⩽ x / L ⩽ 1 / 2 . Furthermore, when the baffle was placed at x / L = 1 / 2 , a relative improvement of approximately 51% and 15.3% in the growth and damping rates, respectively, was achieved compared with those in the SRM without a baffle. In addition, an annular baffle with a smaller inner diameter was observed to have a good design. A baffle with d / D = 1 / 2 was observed to be associated with a favorable damping effect. This research elucidates the effective design of a baffle in stabilizing combustion in a SRM.

Rational approximations for the Gauss function can be used to construct closed-form expressions of the Voigt function K ( x, y ) in terms of rational functions, logarithms and inverse trigonometric functions. The comparison with accurate reference values indicates a relative accuracy in the percent range for y ≳ 1, but serious problems for smaller y . Furthermore, these expressions are not competitive with other algorithms with respect to computational speed. Both accuracy and speed tests indicate that supposedly “good” approximations of the integrand do not necessarily provide good approximations of the integral, i.e. Voigt function.

In this paper, a simple solution is presented for multifrequency electromagnetic scattering from inhomogeneous bodies. Shifted frequency internal equivalence is used to replace the electromagnetic problem at operating frequency $\omega $ by equivalent volume and current sources at internal frequency $\omega _{0}$ , and hence repeated calculation of system matrix as in the solution of volume integral equation at multiple frequencies is avoided. The volume field equation and surface tangent field equations obtained from the internal and external problems, the substitutes of the original scattering problem internally and externally, are solved by method of moments with pulse function expansion and point matching. Numerical results demonstrate that the proposed method is accurate and efficient within a wideband, fractional bandwidth being typically 100%.

This paper presents an alternative, characteristic function based approach for the Bayesian design of estimators for dynamic linear systems and linear detection problems. For a measurement update, the a posteriori characteristic function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history is obtained as a convolution of the a priori characteristic function of the ucpdf with the characteristic function of the measurement noise. It is shown that this convolution holds for a general measurement structure. Time propagation involves the product of the updated characteristic function of the ucpdf and the characteristic function of the process noise. Some estimation problems are found to be naturally tractable using only characteristic functions, such as the multivariable linear system with additive Cauchy measurement and process noise. It is shown that even the derivation of the Kalman filter algorithm has advantages when formulated using the characteristic function approach. Finally, in some instances the estimation problem can only be formulated in terms of characteristic functions. This is illustrated by a one-update scalar example for symmetric- α -stable distributions.

The Holder convexity of the composition of inverse hyperbolic tangent function and Jacobian sine function is investigated in this paper.

We calculate the coefficients of the Hermite expansion of the characteristic function of $$[-a,a]$$ and [0, a], explicitly. As applications, we calculate the coefficients of the Hermite expansion of the sinc function, the Heaviside function (characteristic function of $$[0,\infty )$$ ) and derivatives of the Dirac’s delta functions.

We prove the existence of a multivariate singular skew-normal density function, derive its moment generating function, and demonstrate that the skewness parameter-vector is confined to the column space of the singular dispersion matrix.

Theoretical relationships of statistical properties of surface slope from statistical properties of the image intensity in remotely sensed images, considering a non-Gaussian probability density function of the surface slope, are shown. Considering a variable detector line of sight angle and considering ocean waves moving along a single direction and that the observer and the sun are both in the vertical plane containing this direction, new expressions, using two different glitter functions, between the variance of the intensity of the image and the variance of the surface slopes are derived. In this case, skewness and kurtosis moments are taken into account. However, new expressions between correlation functions of the intensities in the image and surface slopes are numerically analyzed; for this case, the skewness moments were considered only. It is possible to observe more changes in these statistical relationships when the Rect function is used. The skewness and kurtosis values are in direct re...

There are many real-world interactions that can be naturally represented as temporal networks, such as communications in social media and transportation with urban transport facilities. As one of the approaches for analyzing temporal networks, they are often represented as multilayer networks composed of many single-layer networks sharing the same set of nodes. Edges in multilayer networks are divided into two category: intra-layer networks within each layer and inter-layer edges between vertices of different layers. One of the advantages of representing temporal networks as multilayer networks is that each layer is regarded as a snapshot of dynamic interactions and is useful for visualization. Another advantage is that there are some tools for analyzing multilayer networks (such as MuxViz or GenLouvain). On the other hand, many previous research on multilayer networks focus on collection of intra-layer networks, so more discussions are required for inter-layer connections. For example, Mucha et al. propose extended modularity for multilayer networks. Their modularity basically focus on two types of inter-layer connections: ordinal coupling and categorical coupling. Only the corresponding nodes of adjacent layers are connected in ordinal coupling, and all corresponding nodes in each layers are connected in categorical coupling. In order to detect communities, optimization of Mucha's modularity is often employed. However, the modularity is based on an assumption that inter-layer coupling are either ordinal or categorical, which are not enough for representing "decay" of impacts of interactions, which is important for representing temporal networks with multilayer networks. This paper proposes generalization of inter-layer couplings of multilayer networks. The following inter-layer connections are attempted for detecting communities in multilayer networks: (i) rectangular function, (ii) constant function, (iii) exponential decay function and (iv) Gaussian function. Based on our framework, community detection of multilayer networks of several inter-layer connections is performed. Experimental results show that the properties of inter-layer couplings are crucial for the stability of detected communities.

We consider consensus in this paper for hybrid multi-agent systems (HMASs) consisting of continuous-time dynamic agents and discrete-time dynamic agents via designing pulse-modulated protocols. By assuming all the agents communicate with its neighbors only at some sampling instants, and the input of each continuous-time agent is determined by a pulse function, the consensus for first-order HMASs and second-order HMASs are addressed, respectively. According to the model transformation, stochastic matrix theory and the mathematical analysis technique, some sufficient criteria for consensus of these two types of HMASs are derived where it shows that the consensus in the considered HMASs can be achieved when the network structure, pulse function and sampling period respectively satisfy some suitable conditions. And, an algorithm is presented to evaluate the maximum allowable sampling period for achieving consensus. Finally, three numerical simulation examples are performed to validate the main theoretical results.

Quantum tunneling occurs in the isomerization of hydroxymethylene into formaldehyde. Experiments and theory have proven the occurrence of it but both of them have a different result of the half-life of 0.7 hours. The differences the result is the motivation in this study. In theoretical, the probability of quantum tunneling calculates by Wentzel, Kreamer, Berlioun (WKB) approximation. This study calculates probability quantum tunneling use WKB and rectangular function approximation. The calculation of quantum tunneling probability with rectangular function approximation started at 1, 3, 5, 7 until 59 rectangular functions potential barrier use transfer matrix method to simplify the calculation. We use Gaussian function as the potential barrier that the height and width of the potential barrier satisfy the character of the potential barrier in isomerization hydroxymethylene barrier to formaldehyde. And the result, we get the minimum number of rectangular potential barrier to obtain a stable quantum tunneling probability value of 35 potential.

The work addresses the Synthetic Aperture Radar (SAR) geometry, signal model and imaging process. Mathematical description of the observed surface is presented. A waveform with linear frequency modulation (LFM) reflected by the surface is used to produce a two-dimensional (2-D) SAR signal model presented as a sum of Hadamard products of two-and four-dimensional matrices (arrays) with a specially defined rectangular function. It is proven that the SAR signal formation and image reconstruction can be interpreted as direct and inverse projection operations. Based on linearization of the inverse projection, two-dimensional inverse Fourier transform is applied to obtain a complex image known as a Single Look Complex (SLC) image, which can be used for interferogram generation. To verify the proposed SAR kinematics and geometry, surface geometry, signal models and image reconstruction algorithms a numerical experiment is carried out.

In this study, the control of Brushless DC Motors (BLDC), which gained popularity in recent years, is discussed. As a result of insufficient classical control methods, modern control methods have been needed and BLDC is controlled with model reference adaptive control (MRAC) – with MIT law method. For this purpose pulse function, sine wave and step function were applied as a reference speed and the output speed of the system was observed. Simulation results show that the speed of the system follows the speed of the reference model. Then it was thought that in the course of time the resistance value will change because of heated motor and according to this situation the resistance value increased and the output of the system is observed again.

The grating imaging spectrometer has the characteristics of good linearity, wide dispersion range and is widely used in the field of remote sensing. Distortions (including smile and keystone) are one of the important parameters of the grating imaging spectrometer, which directly affects the quality of the image and spectral information obtained by the imaging spectrometer. In order to get the requirements of two kinds of distortions in the design process of the grating imaging spectrometer, the effect of the smile and keystone on the target detection is simulated and analyzed respectively. Based on the spectral response function with the Gaussian, the change of the spectral signal acquired by the grating imaging spectrometer with the amount of the different smile is calculated by combining with the spectral data of the atmospheric in the visible and near-infrared (0.4~1μm). The results show that the amount of smile should be no more than 1nm, 0.6nm and 0.2nm respectively when the spectral resolutions of the imaging spectrometer are 20nm, 10nm and 5nm. With the assumption that the spatial response function is the rectangle function, the effect of the different keystone on spectral signal acquisition of the imaging spectrometer is simulated by using the hyperspectral data. The results indicate that the offset of the keystone should be controlled within 0.04d (d is the pixel width).

In this paper, an asymmetric cryptosystem has been proposed to enhance the security of DRPE. The traditional DRPE scheme is thus tweaked by using fractional Fourier transform (FrFT), a class of structured phase masks called as deterministic phase masks (DMKs) and deploying singular value decomposition (SVD). In specific, we propose to organise the encryption procedure by using two DMKs and FrFT, additionally deploying SVD. On the decryption front, the input image is recovered by utilising the inverse singular value decomposition (ISVD) and an angular portion of the deterministic phase masks. The use of FrFT for encryption and decryption would enhance the robustness of DRPE scheme. Deployment of SVD on our asymmetric cryptosystem provides three components for cipher image is yet another added feature that hardens the security of DRPE scheme. DMKs are formed by the deviation from conventional rectangular function and limited range values which delivers key components with reduced size, better performance and lower complexity. The capability study of defined method, includes analysis on SVD, histogram and correlation coefficient. Our system is subject to an occlusion attack and noise attack to evaluate its performance and reliability. Computational analysis outputs and security investigation are offered in aspect to determine the security potential of proposed system. Comparative results are shown for values of mean-square-error and peak-signal-to-noise ratio of DRPE schemes.

In this chapter we cover various properties of the Fourier transform. In addition to getting a deeper understanding of the machinery of the Fourier transform, by understanding the properties of the Fourier transform we are better fit to deal with new problems and/or deal with older ones more efficiently and faster. The studied properties include linearity, time scaling, reciprocity, time shifting, frequency shifting, time differentiation, time integration, frequency differentiation, frequency integration, time convolution, and frequency convolution. For each of the properties we do a thorough derivation accompanied with one or more examples demonstrating an actual application of the property. We also demonstrate the property flow visually and simultaneously on a time-domain and frequency-domain plot. We single out the convolution property due to its importance and in anticipation of further use throughout the text.

In this paper, the approximate calculation formula is used to calculate the horizontal electric field from lightning which has arbitrary current, with the help of time domain integral equation method solving the lightning induced over voltages on overhead lines. In this paper, the time domain integral equations of horizontal electric field is established. Taking the conductor axis current as the variable, the pulse function is used as the basis function, and the discretization equation is used in space and time. Using time stepping algorithm to solve the resulting linear equations. The calculation process of this method is simple, and it is proved that the method is consistent with the conclusion of the relevant literature.

The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position. When the slit experiment is taken as an apparatus of position measurement, no matter what potential is used to model the slit, only the ground state of the slit-dependent wave function matters., Comment: 4 pages, presentation improved, typos corrected, authors updated

The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position. When the slit experiment is taken as an apparatus of position measurement, no matter what potential is used to model the slit, only the ground state of the slit-dependent wave function matters.

A theoretical investigation of the modulational instability (MI) in a composite system with a nonlocal response function is presented. A composite system of silver nanoparticles in acetone is chosen, whose nonlinearity can be delicately varied by controlling the volume fraction of the constituents, thus enabling the possibility of nonlinearity management. A pump-probe counterpropagation configuration has been assumed, and the interplay between the competing nonlinearities and the nonlocalities in the MI dynamics is systematically explored. A different class of nonlocalities have been considered, and the study reveals that the nonlocality critically depends on the kind of nonlocal function. However, the general behavior is that the strength of nonlocality suppresses the MI gain, while for a rectangular function it assists the emergence of new spectral windows. We also show that the cross coupling effects are significant in enhancing MI, especially in the defocusing nonlinearity. We also emphasize the impact of the relative strength of the nonlinearities in the MI dynamics at different settings of competing nonlinearities. Thus, we emphasize the importance of the different class of nonlocal response in the MI dynamics and explore the interplay between the higher order nonlinear effects and nonlocalities in the counterpropagating configurations.

Recent studies have revealed that multi-soliton solutions of the nonlinear Schr\"odinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multi-solitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multi-solitonic signal and provide the exact solution of the associated Zakharov-Shabat scattering problem for the windowed/doubly-truncated multi-soliton potential. This method further allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multi-solitonic signals. The method devised in this work also applies to general type of signals and may prove to be a useful tool in the theoretical analysis of such systems., Comment: The manuscript is revised for submission to PRE. Also, some typos have been corrected

In this work, using the numerical analysis of the diffraction TM-polarized optical waves by metal gratings under resonance of the electromagnetic field was presented. Modified RCWA method has been proposed. Notably, Toeplitz matrices were modified in the system of differential equations. These matrices are formed on base of complex series of the periodic function of grating's dielectric permittivity and its inverse function. In fact, a rectangular function of permittivity was changed by trapezium function. However grating's relief remains rectangular. Proposed change of the dielectric permittivity function more exactly corresponds to actual gratings. Better convergence of calculation results for the modified system of equations was determined. Modified RCWA was tested for several gratings in order to illustrate the provided advantages.

The auto-correlation function and the frequency power spectral density due to a super-position of uncorrelated exponential pulses are considered. These are shown to be independent of the degree of pulse overlap and thereby the intermittency of the stochastic process. For constant pulse duration and a one-sided exponential pulse shape, the power spectral density has a Lorentzian shape which is flat for low frequencies and a power law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in the pulse function. For a strongly asymmetric two-sided exponential pulse shape, the frequency spectrum is a broken power law with two scaling regions. In the case of a symmetric pulse shape, the power spectral density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distribution of pulse durations is shown to result in apparently longer correlation times but has no influence on the asymptotic power law tail of the frequency spectrum. The effect of additional random noise is also discussed, leading to a flat spectrum for high frequencies. The probability density function for the fluctuations is shown to be independent of the distribution of pulse durations. The predictions of this model describe the variety of auto-correlation functions and power spectral densities reported from experimental measurements in the scrape-off layer of magnetically confined plasmas., 8 figures

We investigate the generation of chaos in economic models through exogenous shocks. The perturbation is formulated as a pulse function where either values or instants of discontinuity are chaotically behaved. We provide a rigorous proof of the existence of chaos in the perturbed model. The analytical results are applied to Kaldor–Kalecki-type models of the aggregate economy subject to export and rainfall shocks, respectively. Simulations are used to demonstrate the emergence and the control of chaos. Our results shed light on a novel source of chaos in economic models and have important implications for policy-making.

In this work we calculate the exact Green's function for arbitrary rectangular potentials. Specifically we focus on Green's function for rectangular quantum wells enlarging the knowledge of exact solutions for Green's functions and also generalizing and resuming results in the literature. The exact formula has the form of a sum over paths and always can be cast into a closed analytic expression. From the poles and residues of the Green's function the bound states eigenenergies and eigenfunctions with the correct normalization constant are obtained. In order to show the versatility of the method, an application of the Green's function approach to extract information of quasi-bound states in rectangular barriers, where the standard analysis of quantum amplitudes fail, is presented., Comment: 9 pages, 7 figures, few typos corrected, matches published version