Double Integral Research Articles

A Fubini type theorem for rough integration

Jointly controlled paths as used in Hairer and Gerasimovičs (2019), are a class of two-parameter paths Y controlled by a p -rough path X for 2 \leq p < 3 in each time variable, and serve as a class of paths twice integrable with respect to X . We extend the notion of jointly controlled paths to two-parameter paths Y controlled by p -rough and \tilde{p} -rough paths X and \tilde{X} (on finite dimensional spaces) for arbitrary p and \tilde{p} , and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and \tilde{X} , they are integrable with respect to X and \tilde{X} , and moreover we prove a rough Fubini type theorem for the double rough integrals of Y via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science applications, are jointly controlled paths.

Finite time stability of linear time varying delay systems using free matrix based integral inequalities

This paper discusses the problem of finite time stability (FTS) for linear delayed systems where the time-varying delay is bounded. First, based on a Lyapunov–Krasovskii Functional (LKF) which contains terms of triple integrals, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, new approximations of the single and multiple quadratic integrals that appear in the LKF derivative are established using integral inequalities, called free-matrix-based integral inequalities. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore more effective. Finally, less conservative design conditions in terms of Linear Matrix Inequalities (LMIs) are proposed and solved by the LMI Tools of MATLAB to show the advantage and effectiveness of the proposed approach.

Tauberian conditions under which convergence follows from Cesáro summability of triple integrals over $${\mathbb {R}}_{+}^{3}$$

Tauberian conditions under which convergence follows from Cesáro summability of triple integrals over $${\mathbb {R}}_{+}^{3}$$

Mathematical Modeling: Cisplatin Binding to Deoxyribonucleic Acid

The discovery of the cisplatin drug attracted considerable research attention as scientists strove to understand the drug’s mechanism in the human body that is responsible for destroying cancer cells, particularly the coordination between the cisplatin drug and deoxyribonucleic acid. Here, the binding energies of a cisplatin molecule relative to double-stranded deoxyribonucleic acid are obtained. The interactions of the system are determined by performing double integrals, and the analytical expressions are derived from the Lennard–Jones function and the continuum approximation; here, it is assumed that a discrete atomic structure might be replaced by surfaces with a constant average atomic density. The results observed that the cisplatin molecule is binding to the double-stranded deoxyribonucleic acid at either the minor or major grooves. By minimizing the interaction energies between the cisplatin molecule and the minor and major grooves, for arbitrary distances λ and arbitrary tilt angles φ from the axis of the helix of the double-stranded deoxyribonucleic acid, the binding energies are determined, and their values are ≈−6 and ≈−12.5 (kcal/mol), respectively. Thus, we may deduce that the major groove in double-stranded deoxyribonucleic acid is the most preferred groove for linking with the cisplatin molecule. The current analysis might help in the equivalent continuum modeling of deoxyribonucleic acids and nanocomposites.

ОРГАНИЗАЦИЯ ОНЛАЙН-ТЕСТИРОВАНИЯ ПО МАТЕМАТИЧЕСКИМ ДИСЦИПЛИНАМ С ИСПОЛЬЗОВАНИЕМ СЕРВИСА ONLINE TEST PAD

The paper describes the possibility of creating control tests in mathematics in the online service Online Test Pad. The advantages of this service for creating mathematical tests are highlighted: the presence of a convenient formula editor, the ability to create groups of questions of the same type. As an example, the structure and variants of questions in the discipline «Mathematical analysis» on the topic «Double integrals» are given. It is noted that test tasks for online testing should be formulated in such a way that the student cannot use applied mathematical packages when answering them.

Tauberian theorems for the Cesàro summability method of regularly generated double integrals

For a continuous function 1 over R2+ := [1,?) ? [1,?), we denote its integral over [1, x] ? [1, y] by h(x, y) = R x 1 R y 1 1(u, v)dudv and its (C, 1, 1) mean, the average of h(x, y) over [1, x] ? [1, y], by t(h(x, y)) = (xy)?1 R x 1 R y 1 h(u, v)dudv. Analogously, the other means (C, 1, 0) and (C, 0, 1) can be defined. In this paper, we introduce the concept of regularly generated double integrals in senses (1, 1), (1, 0) and (0, 1) and obtain Tauberian conditions in terms of the regularly generated double integrals in senses (1,1), (1,0) and (0,1) under which convergence of h(x, y) follows from that of t(h(x,y)).

Efficient Semi-Analytical Method for the Analysis of Large Finite Connected Slot Arrays

We present an efficient method for the analysis of finite connected slot arrays in the presence of stratified media. The formulation is based on a spectral method of moments, where only one basis function is considered for each array element and one for each slot edge. An expression for the mutual impedance is derived in terms of a double spectral integral. Asymptotic extraction techniques are employed to largely reduce the computation time of one of the spectral integrals. For the other integral, when a guided wave contribution dominates the mutual coupling between two array elements, the result can be approximated as the residue of the spectral polar singularity, providing a closed-form solution of the coupling for elements at electrically large distances. The complete method enables simulations of entire finite arrays with hundreds or even thousands of elements in minutes. The same structure would require impractical computation time when analyzed with general-purpose commercial software. The method allows estimating the performance of finite connected arrays. This is particularly relevant because wideband connected arrays are known to exhibit higher edge effects compared to narrowband arrays, due to the high interelement mutual coupling.

EULER–ZAGIER SUMS VIA TRIGONOMETRIC SERIES

In this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series.

Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions

The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.

New 2D numerical integration formula based on the Legendre wavelets

Based on Legendre wavelets, a new efficient numerical integration method is proposed to estimate double integrals. This new method is expressed in terms of the operational integration matrices. Those which were introduced by Parsian, are calculated in an approximate way. In this work, we also propose the exact computation of these matrices. Better precision of the new method is observed during a comparative numerical study with the composite Simpson method.

Estimates for integrals of derivatives of $n$-valent functions and geometric properties of domains

A number of questions concerning the behaviour of double integrals of the moduli of the derivatives of bounded $n$-valent functions and, in particular, of rational functions of fixed degree $n$ are considered. For domains with rectifiable boundaries the sharp order of growth of such integral means is found in its dependence on $n$. Upper bounds for domains with fractal boundaries are obtained, which depend on the Minkowski dimension of the boundary of the domain. In certain cases these bounds are shown to be close to sharp ones. Lower bounds in terms of the integral means spectra of conformal mappings are also found. These inequalities refine Dolzhenko's classical results (1966) and some recent results due to the authors. Bibliography: 32 titles.

On Certain Triple Integral Relations Involving Elementary Functions

The aim of this paper is to establish three triple integral relations involving elementary functions. A number of triple integrals can be deduced by proper specialization of the unknown functions f and g occurring in these relations. For the sake of illustration, one of our integral relations is applied to evaluate a general triple integral involving Asgar, Gautam and Goyal multivariable A- function.Mathematics Subject Classification: 33C45, 33C60, 26D20

Numerical Solution for 3-D lee wave associated with barotropic mean flow across the Assam-Burma hills

An attempt has been made to obtain a 3-D lee wave numerical solution associated with meso-scale dry mean flow across the Assam-Burma hills (ABH), where the flow is an adiabatic, inviscid, laminar, steady, Bousisnesq, non-rotational.ABH has been synthesized by two three dimensional ellipticalbarriers, separated by a valley of some finite distance, where the two components (U, V) of basic flow and Brunt-Vaisala frequency (N) are idealistic invariant with height. For simplicity the zonal component (U) and meridional component (V) are normal and parallel to the major axis of the elliptical barrier respectively and also perturbation technique has been applied to the governing equations. The perturbation vertical velocity (w' ) and stream line displacement (n' ) are expressed as a double integral. These two integrals have been evaluated numerically. The down-stream variation and updraft/downdraft regions associated with lee waves at different heights has been mapped and discussed and finally have compared to the asymptotic results of earlier investigators.

"THERMOELASTIC DISPLACEMENT DUE TO TRANSIENT SURFACE HEATING "

The starting point in the calculation of normal displacement due to transient heating is the Green’s function for the elastic half-space. Superposition principle leads to a triple integral (double integral over surface and simple integral over time) that can be formally re-written as a three-dimensional convolution product. Given the singularities of the Green’s function in the time/space domain, it is more convenient to employ its spectral counterpart, i.e. the frequency response function (FRF), in the convolution calculation. A special technique for the calculation of the 3D convolution product based on the FRF is advanced in this paper. The resulting algorithm is very efficient from a computational point of view, as the transfers to and from the time/space domain to the frequency domain are handled by the fast Fourier transform. A simulation example is presented, involving the transient thermoelastic displacement due to a uniform heat source that vanishes everywhere except for a square surface domain, and which is applied continuously only in a limited time window. The numerical results predict that the displacement increases with time as long as heat is supplied, and is gradually recovered once the heat is removed. The loaded half-space patch undergoes a growth-release process that is accurately captured by the simulation method. The developed framework anticipates the solution of the contact process with transient heating.

Wavenumber identification of 1D complex structures using Algebraic Wavenumber Identification (AWI) technique under complex conditions

The dynamic characterization of complex structures raises the need for a robust experiment-based wavenumber identification method, especially when they are tested under complex conditions. To this end, this paper presents an optimized Algebraic Wavenumber Identification (AWI) method and describes the dynamic behavior of three complex structures under a series of complex conditions through AWI. The first novelty of this paper is to optimize AWI in terms of computation efficiency and accuracy by two solutions: (a) the multiple integral of AWI, which acts as a filter, is explicitly solved by the least squares fitting method, significantly reducing the computation cost, especially when multiple samples are used as input parameters; (b) a sampling strategy is provided to further improve the robustness of the AWI to measurement errors. On this basis, an AWI implementation procedure is proposed for experimental tests under complex conditions. The second novelty of this paper is on the applications of AWI in wave propagation parameters identification of complex structures, including the damping loss factor estimation of a viscoelastic beam and a honeycomb sandwich beam, the band gap identification of a meta-structure, and experimental dispersion curves extraction of these three complex structures. The third novelty of this paper is the experimental validation of AWI under a series of complex conditions, including signal noise, non-periodic sampling, and uncertainty of measuring points’ geometric coordinates. The experimental and numerical results have been compared to two popular inverse methods, namely, Inhomogeneous Wave Correlation (IWC) and INverse COnvolution MEthod (INCOME), demonstrating the validity of the AWI implementation procedure proposed in this paper.

A contribution to the mathematical theory of diffraction: a note on double fourier integrals

SummaryWe consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of $u$. We conclude the article by validating this theory against full numerical computations for two specific examples.

An Efficient Method of Moments for Thick-Wire Antennas

Detailed formulation of a simple Galerkin method of moments (MoM) is presented. It is based on compact closed-form expressions of mutual impedance among filamentary sinusoidal monopoles. Using piecewise sinusoidal basis functions, the mutual impedance of thick-wire sections can be computed through an azimuthal integration of the mutual impedance between filamentary dipoles. Each element of impedance matrix is calculated through a rapidly convergent single integral only. This is unlike other MoM techniques, where double or triple integrals are needed to evaluate impedance matrix. In our formulation, it has been assumed that the wire current is uniform in azimuth, and changes along the axial direction only. No approximation on the segment length or radius of the thick-wire is made, and hence our formulation remains efficient for arbitrarily long/thick antennas.

New analysis and numerical values for the classical dam problem

The classical, two dimensional, dam problem was solved analytically and independently by Davison in 1932 and Hamel in 1934 and involved double integrals. In 1938 Polubarinova-Kochina produced solutions involving single integrals and in 1940 showed that the Davison-Hamel solutions could be reduced to ones obtained by her method. Missing from the conversion was a linkage between scale constants which is given in this paper. Initial calculations by these authors were difficult because of evaluations of elliptic integrals and numerical quadratures. As a free boundary problem it is now a test case for a multitude of numerical methods. Surveying analytical and numerical approaches, it is found that both are limited in accuracies and are overwhelmingly concerned with the location of the free boundary and its seepage point. Analytical expressions are here given for the gradients near the three critical points of free boundary entry and exit and the intersection of seepage face and tailwater. New numerical results to six decimal places are given for tables and contours on and within the dam boundary for potential and stream functions and orthogonal potential gradients.

Precise calculation of electrical capacitance by means of quadruple integrals in method of moments technique

In this paper, the capacitance of a parallel plate air-gap rectangular capacitor, and a unit cube capacitor have been calculated. Because of its generality and simplicity, the method of moments (MOM) Technique is utilized. In order to improve the accuracy of the calculations, the use of quadratic integrals instead of binary integrals has been proposed. A neat form is provided for the analytical solution of the integrals required for the method of moment. The results show that there is a very small error in calculating the capacity even with coarse boundary division. The described formulas and codes can easily be used for similar purposes.

On the evaluation of electrostatic energy of volume charge distributions and the role of the Coulomb singularity

Methods for evaluating electrostatic interaction energy of volume distributions of charge of different shapes and densities are studied. In our numerical method, a cubic integration volume is divided into smaller sub-cubes, whereupon the pertinent double volume integral is turned into a six-fold sum from which all self-interaction terms exhibiting the Coulomb singularity are duly extracted. More precision is gained by optimising the values of the partial integrals corresponding to the nearest interacting sub-cubes obtained by different discretisation of a cube. Further insight is obtained by modelling a sphere, a cylinder and a cube with cubical elements, for which analytical results are available. We also consider spheres with different radially dependent charge density functions, whose interaction energy is known exactly.