Double Integral Research Articles

Neural Free‐Viewpoint Relighting for Glossy Indirect Illumination

AbstractPrecomputed Radiance Transfer (PRT) remains an attractive solution for real‐time rendering of complex light transport effects such as glossy global illumination. After precomputation, we can relight the scene with new environment maps while changing viewpoint in real‐time. However, practical PRT methods are usually limited to low‐frequency spherical harmonic lighting. All‐frequency techniques using wavelets are promising but have so far had little practical impact. The curse of dimensionality and much higher data requirements have typically limited them to relighting with fixed view or only direct lighting with triple product integrals. In this paper, we demonstrate a hybrid neural‐wavelet PRT solution to high‐frequency indirect illumination, including glossy reflection, for relighting with changing view. Specifically, we seek to represent the light transport function in the Haar wavelet basis. For global illumination, we learn the wavelet transport using a small multi‐layer perceptron (MLP) applied to a feature field as a function of spatial location and wavelet index, with reflected direction and material parameters being other MLP inputs. We optimize/learn the feature field (compactly represented by a tensor decomposition) and MLP parameters from multiple images of the scene under different lighting and viewing conditions. We demonstrate real‐time (512 x 512 at 24 FPS, 800 x 600 at 13 FPS) precomputed rendering of challenging scenes involving view‐dependent reflections and even caustics.

Approximate calculation of double integrals from rapidly oscillating functions of the general form

Approximate calculation of double integrals from rapidly oscillating functions of the general form

An Electromechanical Matching Design Method for Faceted Surface of Mesh Reflector

The faceted surface is used in mesh reflectors to reflect electromagnetic (EM) signals. The matching design method of surface geometry is developed by considering EM performance. The geometrical model of faceted surface is established to avoid the facets overlap by using the force density method, and the far-field scattered by mesh reflector is derived to reveal the relationship between the faceted surface and EM performance. To improve the computational efficiency, the double integral of the far-field is derived in a closed-form expression, and a refinement procedure of EM grids for facets is proposed to homogenize the grids. The electromechanical coupling optimization model is established, and the comparative analysis is carried out, respectively. Numerical results show that the proposed method can obtain the higher gain with a larger surface error of the mesh reflector by adjusting the distribution of the surface faceted error.

Concept Images of University Students for Geometric Representation of the Double Integral Concept

This study has focused on how university students understand the geometric representation of double integral. For this purpose, six participants have been asked six questions. Later, semi-structured interviews were conducted with the participants. In this study, the data obtained from questionnaire form and interviews were analyzed with open and axial coding. As a result of this research, it was observed that the concept images of university students were grouped into two categories as “area” and “volume”. It was determined that the participants acted with an intuitive approach without having to establish a relationship between the concept definition and the concept image, the ∬ in the symbol of the double integral caused the participants to think of it as a two-dimensional geometric structure and their image of the concept of the single integral was very active. The findings obtained in this research shows that there are problems in understanding the concept of the double integral, which is the first step of generalizing to multiple integrals, and that educator should produce solutions for this subject.

Some New Properties of Convex Fuzzy-Number-Valued Mappings on Coordinates Using Up and Down Fuzzy Relations and Related Inequalities

The symmetric function class interacts heavily with other types of functions. One of these is the convex function class, which is strongly related to symmetry theory. In this study, we define a novel class of convex mappings on planes using a fuzzy inclusion relation, known as coordinated up and down convex fuzzy-number-valued mapping. Several new definitions are introduced by placing some moderate restrictions on the notion of coordinated up and down convex fuzzy-number-valued mapping. Other uncommon examples are also described using these definitions, which can be viewed as applications of the new outcomes. Moreover, Hermite–Hadamard–Fejér inequalities are acquired via fuzzy double Aumann integrals, and the validation of these outcomes is discussed with the help of nontrivial examples and suitable choices of coordinated up and down convex fuzzy-number-valued mappings.

Shear deformable plate by SGBEM: Indirect “progenitor matrix” approach

The Mindlin-Reissner's formulation of shear deformable plate is addressed through the Symmetric Galerkin Boundary Element Method (SGBEM). The thick plate's elastic problem is resolved and fundamental solutions matrix is obtained. The application of Betti's theorem in the unlimited domain allows obtaining the Somigliana's Identities (S.I.s). The solving system is obtained by an indirect approach, i.e., through the “progenitor matrix”. The “progenitor matrix” allows simulating all possible plate's load conditions and is the starting point for an automatic calculation code. The coefficients are calculated through the theory of distributions, using double integrals without resorting to regularization techniques but by exploiting the expansion in power series of the Bessel functions present in the fundamental solutions.Particular attention is paid to the presence of domain loads, whose domain integral is transformed into a boundary one using the Radial Integral Method technique (RIM).These strategies lead to a robust procedure that allows obtaining good results even in the presence of boundary sparse discretization. This is demonstrated by the results of the examples carried out which, compared with the analytical solutions present in the literature, show a very good convergence.

Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.

Three-Temperature Boundary Element Modeling of Ultrasound Wave Propagation in Anisotropic Viscoelastic Porous Media

The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, the analytical or numerical solutions to the problems under consideration are always challenging, necessitating the development of new computational techniques. As a result, we use a new BEM model to solve such problems. A time-stepping procedure based on the linear multistep method is obtained after solving the discretized boundary integral equation with the quadrature rule. The calculation of a double integral is required to obtain fundamental solutions, but this increases the total BEM computation time. Our proposed BEM technique is used to solve the current problem and improve the formulation efficiency. The numerical results are graphed to demonstrate the effects of viscosity and anisotropy on the nonlinear ultrasonic stress waves in three-temperature porous media. The validity, accuracy, and efficiency of the proposed methodology are demonstrated by comparing the obtained results to a corresponding solution obtained from the finite difference method (FDM).

Some Properties and Applications of a New General Triple Integral Transform “Gamar Transform’’

The goal of this study is to suggest a new general triple integral transform known as Gamar transform. Next, we compare the current transform to a number of existing triple integral transforms such as those by Laplace, Sumudu, Elzaki, Aboodh, and Laplace–Aboodh–Sumudu. We outline its essential properties and prove some important results, including linearity property, existence theorem, triple convolution theorem, and derivatives properties. Moreover, the proposed new transform is applied to solve some partial differential equations (PDEs) such as Laplace, Mboctara, and Wave equations. The capacity of general triple integral transforms to change PDEs into simple algebraic equations is demonstrated.

An improved triple collocation-based integration of multiple gravity anomaly grids from satellite altimetry: Contribution of ICESat-2

Satellite altimetry has been widely used to determine the marine gravity fields. Different altimeter missions lead to different gravity grids with varying spatial resolutions and accuracies. Marine gravity accuracy in offshore waters is often low. Integrating marine gravity anomalies (MGA) derived from various altimeter missions are key to obtaining high-precision and high-resolution global marine gravity fields. Here an improved triple collocation (ITC) method is first proposed to establish a new 1´×1′ marine gravity anomaly in the South China Sea (SCS), named MGAITC, by merging the grids of MGA from the Haiyang-2A (HY-2A), CryoSat-2, and ICESat-2 altimeter measurements. A combined MGA (MGACOM) is also constructed by using the least squares collocation (LSC) method from three sets of altimeter-derived geoid gradients. The qualities of MGAITC, MGACOM and each altimeter-derived MGA grids are assessed using shipborne gravity and different global gravity models over waters of varying depths and distances from the coastline. The accuracy of the MGAITC gravity against the shipborne gravity is 5.52 mGal, which represents improvements of 7%, 9%, 13%, and 3% relative to the gravity grids derived only from ICESat-2, CryoSat-2, HY-2A and from MGACOM. In all assessments involving different sub-regions, seafloors, and distances from the coastline, MGAITC performs the best, and ICESat-2 yields the best single-mission MGA grid, especially in offshore waters. We find that ICESat-2 has a high potential to improve shallow-water gravity accuracy and the global marine gravity field can be improved by integrating gravity grids from different altimetry missions using ITC.

Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams

In this study, the slope deflection method was presented for structures made of small-scaled axially functionally graded beams with a variable cross section within the scope of nonlocal elasticity theory. The small-scale effect between individual atoms cannot be neglected when the structures are small in size. Therefore, the theory of nonlocal elasticity is used throughout. The stiffness coefficients and fixed-end moments are calculated using the method of initial values. With this method, the solution of the differential equation system is reduced to the solution of the linear equation system. The given transfer matrix is unique and the problem can be easily solved for any end condition and loading. In this problem, double integrals occur in terms of the transfer matrix. However, this form is not suitable for numerical calculations. With the help of Cauchy’s repeated integration formula, the transfer matrix is given in terms of single integrals. The analytical or numerical calculation of single integrals is easier than the numerical or analytical calculation of double integrals. It is demonstrated that the nonlocal effect plays an important role in the fixed-end moments of small-scaled beams.

Sound transmission characteristics of a plate backed by an irregular cavity

The study of the coupling cavity provides an effective theoretical reference for the acoustic design. However, most of the previous studies are based on idealized models and there is little research on the sound radiation power about the irregular shapes. The work in this paper achieves the generalization of the model and explores the impact of irregular back cavities on the sound radiation of the plate structures. The influence on the results of the sound insulation performance testing of the plate is also investigated when irregular cavities involved. The displacement of the plate and the sound pressure inside the cavities are expressed as multidimensional Fourier series with additional terms, which will contribute to maintaining the continuity of pressure and velocity at the boundaries. The sound field of the irregular cavity is considered as a unit cubic one in a new coordinate system by coordinate transformation method. The arbitrary boundary of the cavities is realized by adding the energy terms of impedance into the Lagrange equation. The Hamilton’s principle is employed to realize the complete coupling between the plate and the sound fields on both sides. The sound radiation power of the plate can be obtained by using the Rayleigh integral. By using Fast Cosine Transform, the quadruple integrals of the radiation power are simplified to improve accuracy and efficiency of the calculation. Numerical results from the analytical approach are compared with those given by Finite Element Analysis (FEM) to guarantee the rightness and validity of the models. Moreover, experimental results are also given to discuss the reasons of the bias between theory and practice.

Mean value of real Dirichlet characters using a double Dirichlet series

AbstractWe study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages—it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.

Application of the Residue Theorem to Euler Integral, Gaussian Integral, and Beyond

The paper is divided into three different parts, which use the residue theorem to solve several different integrals, namely, the Euler integral, the Gaussian integral, the Fresnel integral, and so forth. The process of using the resiude theorem to determine these integrals is to first turn the integrals into convenient forms of complex integrals, and then find integral perimeters so that any integral on one of the curves is the required integral, through the drawing observation of the contour to write the original integral into the form of multiple integral. By studying the resiude theorem to solve the problem of complex integrals, it is demonstrated that the resiude theorem is actually a process that makes the calculation easier. These solved integrals have a wide range of applications including the study of the refraction of light, analytics, probability theory, combinatorial mathematics, and unification of the continuous Fourier transform.

A Generalized Approach of Triple Integral Transforms and Applications

In this study, we introduce a novel generalization of triple integral transforms, which is called a general triple transform. We present the definition of the new approach and prove the main properties related to the existence, uniqueness, shifting, scaling, and inverse. Moreover, relations between the new general triple transform and other transforms are presented, and new results related to partial derivatives and the triple convolution theorem are established. We apply the general triple transform to solve some applications of various types of partial differential equations. The strength of the new approach is that it covers almost all integral transforms of order one, two, and three, and hence no need to find new formulas of triple integral transforms or to study the basic properties.

Generalised Henstock - Kurzweil Integral with Multiple Point

This paper deals with a new Henstock-Kurzweil integral in Banach Space with Bilinear triple n-tuple and integrator function Ψ which depends on multiple points in partition. Finally, exhibit standard results of Generalized Henstock - Kurzweil integral in the theory of integration.

A fluorescent sensing platform based on collagen peptides-protected Au/Ag nanoclusters and WS2 for determining collagen triple helix integrity.

The unique triple helix structure of collagen plays an important role in its biological properties, and the triple helix integrity is closely correlated with its molecular behavior and biological functions. Nevertheless, there is still a lack of convenient, accurate and practical methods for quantitatively determining collagen triple helix integrity. Herein, we first prepared bovine skin collagen peptide (BSCP)-protected Au/Ag nanoclusters (Au/AgNCs@BSCP) with excellent optical properties, high stability and good biocompatibility, which could adsorb on WS2 surface leading to fluorescence quenching. Upon the addition of collagen, the interaction of collagen and Au/AgNCs@BSCP led to the detachment of Au/AgNCs@BSCP from the WS2 surface, causing an increase in the fluorescence signal. Using the difference in the fluorescence recovery of the different samples, we achieved the quantitative determination of collagen triple helix integrity. This developed strategy exhibited excellent accuracy, selectivity, and practicality, thus showing promising potentials in biomedical applications.

Numerical Investigation, Error Analysis and Application of Joint Quadrature Scheme in Physical Sciences

In this work, a joint quadrature for numerical solution of the double integral is presented. This method is based on combining two rules of the same precision level to form a higher level of precision. Numerical results of the present method with a lower level of precision are presented and compared with those performed by the existing high-precision Gauss-Legendre five-point rule in two variables, which has the same functional evaluation. The efficiency of the proposed method is justified with numerical examples. From an application point of view, the determination of the center of gravity is a special consideration for the present scheme. Convergence analysis is demonstrated to validate the current method.

Fractional Laplacian – Quadrature Rules for Singular Double Integrals in 3D

Abstract In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.

Numerical evaluation of 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids

The paper presents a fast and accurate numerical technique for evaluation of the 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids. Following Wang & Achenbach the Green's function is presented in the form of the sum of singular (static) and regular parts, which are both reduced to integrals over a unit sphere. Singular part and its derivatives are then reduced to the integrals over a unit circle, which can be efficiently (fast and accurately) evaluated using the trapezoid rule, which is exponentially convergent for integrals over the period of a periodic integrand. The regular time-harmonic parts are presented through the double integrals. The outer integral is also efficiently evaluated using the trapezoid rule. A special quadrature is developed for evaluation of the inner integral, which accounts for the highly-oscillating behavior of the integrand. Numerical examples are presented, which shows the advantage of the proposed technique over others.