Double Integral Research Articles

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(τ2+h3). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

Second-order accurate BGK schemes for the special relativistic hydrodynamics with the Synge equation of state

This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are presented by removing some terms in the approximate nonequilibrium distribution at the cell interface for the BGK scheme without loss of accuracy. They are practical because the moment integrals of the approximate distribution can be reduced to the single integrals by some coordinate transformations. The relations between the left and right states of the shock wave, rarefaction wave, and contact discontinuity are also discussed, so that the exact solution of the 1D Riemann problem could be derived and used for the numerical comparisons. Several numerical experiments are conducted to demonstrate that the proposed gas-kinetic schemes are accurate and stable. A comparison of the sBGK schemes with the BGK scheme in one dimension shows that the former performs almost the same as the latter in terms of the accuracy and resolution, but is much more efficient.

A 64 × 64 SPAD Flash LIDAR Sensor Using a Triple Integration Timing Technique With 1.95 mm Depth Resolution

This paper presents a 64×64 dual mode flash LIDAR sensor that utilizes a triple integration timing technique. The sensor, fabricated in 130 nm HV CMOS, is capable of operating in both a direct time-of-flight (ToF) mode where the timestamp of the first arriving photon is recorded per-pixel, as well as a photon counting mode where the number of photons is recorded over a time interval. The timing technique utilizes both a time-to-digital converter (TDC) and a time-to-amplitude converter (TAC) with a counter measuring global clock cycles and the triple integration interpolator (TII) measuring between clock cycles. The TII uses an analog integration with an additional two reference integrations allowing the time measurement to be resistant to PVT variation and in turn, allowing the circuit to be miniaturized without causing a large timing non-uniformity across the array. Utilizing the TII, the sensor achieves a state-of-the-art timing performance, with a resolution of 13 ps (1.95 mm depth resolution), a maximum range of 220 μs (32 km), a single-shot jitter of 233 ps, and a differential non-linearity (DNL) of 6 ps (0.47 LSB). The sensor captures at a maximum frame rate of 8,300 fps and consumes 733 mW during operation. Experimental scenarios demonstrating the operation of the sensor are also provided.

Multiple Diamond-Alpha Integral in General Form and Their Properties, Applications

In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.

(2008-6104) Calculation of centroid of high dimensional fuzzy number and application

In this paper, the conception of centroid of n-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.

Optimal computation of anisotropic galaxy three point correlation function multipoles using 2DFFTLOG formalism

We study two key issues militating against the use of the anisotropic three-point correlation function (3PCF) for cosmological parameter inference: difficulties with its computational estimation and high-dimensionality. We show how high-dimensionality may be reduced significantly by multipole decompositions of all angular dependence. This allows deriving the full expression for the multipole moments of the anisotropic 3PCF and its covariance matrix in a basis where the dimensionality reduces from nine to two at each multipole in the plane-parallel limit. We use 2D FFTLog formalism to show how the multipole moments with double momentum integrals over the product of bispectrum and two highly oscillating spherical Bessel functions and its covariance with double momentum integrals over the product of three galaxy power spectra and a combination of four highly oscillating spherical Bessel functions may be computed optimally.

APPLICATION OF THE THEORY OF NEW INFORMATION OPERATORS IN CONDUCTING RESEARCH IN THE FIELD OF INFORMATION TECHNOLOGIES

Due to the rapid development and implementation of the latest information technologies in many fields of science and technology significant changes have taken place. In particular, new mathematical theories have emerged; they can be effectively used in building and improving existing mathematical models of various phenomena and objects. Mathematical modeling is currently one of the main methods in scientific research of technological, social, economic processes and natural phenomena. No research thesis is complete without the use of mathematical methods and algorithms. In view of this it is important to acquaint young scientists with modern trends in mathematical modeling during their training for PhD degree, in particular. The development of information technology has contributed to the emergence of new approaches to obtaining, processing and analyzing information in scientific research. In this way, the theory of new information operators, which is widely used in mathematical modeling, was created. The analysis of the existing research reveals the need to consider the theoretical basics of new information operators and their application to the numerical integration of several variables functions. The example of the problem of numerical integration of two variables functions shows the way how to create cubature formulas depending on the type of the input information about function. The paper presents new cubature formulas for the approximate calculation of double integrals, in the case when the information is given by the set of traces on lines and values of the function at points. The proposed formulas have high accuracy but use much less input information compared to classical formulas for numerical integration of double integrals. All the results are supported by numerous examples made in the computer mathematics system Mathcad. The implementation of the developed methodology of using new information operators in scientific research has shown its efficacy during seminars with Master and PhD students.

A PERFORMANCE EVALUATION OF TRAPEZOIDAL VARIANTS FOR NUMERICAL CUBATURE

In this work, double integration cubature schemes of Trapezoid type have been focused. Recently, some derivative-based Trapezoid-type schemes have been proposed in literature incorporating derivatives at means of the limits of integration. We carry out the exhaustive performance evaluation of the existing closed Newton-Cotes Trapezoidal (CNCT) double integral scheme with its derivative-based variants in recent literature. The derivative-free and derivative-based rules are discussed in basic forms with local error terms and composite forms with global error terms. The performance of the rules on some double integrals in the form of observed order of accuracy, computational costs and error drops demonstrates the encouraging performance of the derivative-based trapezoidal variants over the derivative-free scheme performing numerical experiments.

The Evaluation Model Efficiency of Ladle Graphitizing Inoculation of Cast Iron

The model for evaluation of quality of ladle graphitizing inoculation of cast iron, according to the most common technological schemes, using the triple integral, is proposed and implemented. The necessary boundary conditions and parameters of the integral model are presented, described and justified. A concrete example confirms the practical conclusion that graphitizing inoculation of cast iron, melt into a stream in the process of filling a mold, is preferable.

The semirecumbent position for high-resolution esophageal manometry. Results of a feasibility study.

Normative values for high-resolution manometry (HRM) have been obtained with the patient lying supine. The aim of the study was to compare supine, semirecumbent and sitting positions during HRM in terms of variation in normative metrics, diagnostic yield, and patient's comfort. A prospective, single-center feasibility study was planned in consecutive patients referred to the esophageal function laboratory. In each of the three positions, 10 consecutive 5 ml water swallows and three 10 ml multiple rapid swallows were administered. Validated reflux questionnaires were administered prior to the test, and a visual analogue scale (VAS) assessing the patient's comfort after the test. Twenty patients presenting with gastroesophageal reflux symptoms completed the study protocol. The intra-abdominal segment of the lower esophageal sphincter was significantly longer in the sitting position (P = 0.013), and the multiple rapid swallow distal contractile integral was lowest in the supine position (P = 0.012). The VAS comfort score did not significantly differ in the three body positions (P = 0.295). The concordance in the final diagnosis was 80% for semirecumbent vs. sitting (kappa = 0.15; P = 0.001), 70% for supine vs. sitting and 65.0% for semirecumbent vs. supine. Compared to the supine position, both the semirecumbent and sitting position seems to provide similar advantages. HRM metrics and the final manometric diagnosis may be affected by body position, but complementary maneuvers, such are the rapid drink challenge, can resolve diagnostic discrepancies and improve the overall accuracy of the test.

Simplified Closed-Form Single-Scatter Path Loss Model of Non-Line-of-Sight Ultraviolet Communications in Noncoplanar Geometry

Considering the wide acceptability and practicability of single-scatter path loss (PL) model of non-line-of-sight (NLOS) ultraviolet communications (UVC) in noncoplanar geometry and the corresponding complexity of the triple integrals in the exact single-scatter PL model, a novel simplified PL model is proposed in this work with better analytical tractability and high accuracy. By approximately regarding the differential volume as a circular and thin plate, the triple integrals in the exact single-scatter PL model can be reduced to a single integral. With different baseline ranges between transmitter and receiver, beam divergence angles, receiver field-of-view (FOV) angles, inclination angles and azimuth angles taken into account, the results of the proposed PL model are compared with those of an exact single-scatter PL model, a Monte Carlo numerical model, and another recently reported approximate single-scatter PL model, which is called spherical crown (SC) model. Compared with the SC model, the proposed single-scatter PL model is obtained in closed form. Besides, taking the exact single-scatter PL model as the benchmark, the root-mean-square error (RMSE) of the proposed model and the SC model are calculated for comparison. It is demonstrated that the proposed model could achieve better accuracy than SC model under most conditions of our interest. Besides, it is shown that the RMSE of the proposed model is less than 0.4 dB for all considered transceiver geometry and the execution time of the proposed PL model is significantly reduced to less than 1% of the exact single-scatter PL model’s, which manifests the effectiveness of the proposed model. Finally, the experimental results are also provided to verify the correctness of the proposed model.

A thermal form factor series for the longitudinal two-point function of the Heisenberg–Ising chain in the antiferromagnetic massive regime

We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.

Estimates for Schur Multipliers and Double Operator Integrals—A Wavelet Approach

We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular, with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $$B^{1/p-1/2}_{p,p}(\mathbb R,L_2(\mathbb R))$$ . This recovers, extends, and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals with bounded symbol in $$B^{1/p-1/2}_{2p/(2-p),p}(\mathbb R,L_\infty(\mathbb R))$$ , which extends Birman and Solomyak’s result to symbols without compact domain.

Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

We consider a stationary sequence \((X_n)\) constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter \(\beta \in (0,1)\) quantifying the conservativity of the system. This parameter \(\beta \) together with the order of the integral determines the decay rate of the covariance of \((X_n)\). The goal of the paper is to establish limit theorems for the partial sum process of \((X_n)\). We obtain a central limit theorem with Brownian motion as limit when the covariance decays fast enough, as well as a non-central limit theorem with fractional Brownian motion or Rosenblatt process as limit when the covariance decays slowly enough.

Minimisers of a general Riesz-type problem

We consider sets in RN which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel g:RN∖{0}→R+. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers in the perimeter-dominated regime, for a wide class of functions g.

AN EFFICIENT TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE WITH HERONIAN MEAN DERIVATIVE

This study focuses on the Heronian mean derivative-based numerical cubature scheme to better evaluate double integrals’ infinite limits. The proposed modifications rely on the Trapezoidal-type quadrature and cubature schemes. The aforementioned proposed scheme is important to numerically evaluate the complex double integrals, where the exact value is not available but the approximate values can only be obtained. With regards to higher precision and order of accuracy, the proposed Heronian derivative-based double integral scheme provides efficient results. The discussed scheme, in basic and composite forms, with local and global error terms is presented with necessary proofs with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The consequently observed error distributions of the aforementioned scheme are found to be lower than the conventional Trapezoidal cubature scheme in composite form

A NEW AND EFFICIENT CENTROIDAL MEAN DERIVATIVE-BASED TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE

This research presents a new and efficient Centroidal mean derivative-based numerical cubature scheme which has been proposed for the accurate evaluation of double integrals under finite range. The proposed modification is based on the Trapezoidal-type quadrature and cubature rules. The approximate values can only be obtained for some important applications to evaluate the complex double integrals. Higher precision and order of accuracy could be achieved by the proposed scheme. The schemes, in basic and composite forms, with local and global error terms are presented with necessary supporting arguments with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The simultaneously observed error distributions of the proposed schemes are found to be lower than the conventional Trapezoidal cubature scheme in composite form

Energetic boundary element method for accurate solution of damped waves hard scattering problems

The paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains.

Calculate a special case of singular integrals using the base of a trapezoid

This research article has presented a numerical method to find out the values of numerical double integrals with singular partial derivative integrands in case of the integral is defined in every point on integral regions. But, one of the partial derivations of the integral function is undefined in one point of the inside dimension of every point of the outside dimension of the integral. This could be achieved by using the Trapezoidal rule on the two dimensions when the number of the subintervals, on which the two dimensions are partitioned, are equal. This way would be referred to (RTrap2) in the next time. The activity of this rule has been tested by applying it to some of the different double integrals. Then its results have been compared with the analytical values. Finally, this rule has introduced high accuracy in a few subintervals and within a short time.

Development of Double Closed-loop Vector Control Using Model Predictive Control for Permanent Magnet Synchronous Motor

Vector control has been widely used to manipulate the permanent magnet synchronous motor (PMSM), in which the traditional double proportional integral (PI) controllers have commonly been utilized. However, it is difficult to set the suitable parameters of PI and maintain the PMSM performance under the different operating regimes. In order to solve these problems, a double closed-loop model predictive control (MPC) strategy based on the vector control has been presented for PMSM in this paper. Firstly, a simplified discrete model of PMSM is established in the d-q coordinate system without considering the interference. Then, the MPC is developed to substitute the PI controller in the speed-loop and the q-axis current-loop, respectively, and a double closed-loop MPC control system is formed. Finally, some case studies are carried out. The simulation and experimental results demonstrate that the presented MPC strategy improves the dynamic and steady-state performance of PMSM.