General Integral Formula Research Articles

Testing an exact diffraction formula with gravitational wave source lensed by a supermassive black hole in binary systems

When it comes to long-wavelength gravitational waves (GWs), diffraction effect becomes significant when these waves are lensed by celestial bodies. Typically, the traditional diffraction integral formula neglects large-angle diffraction, which is often adequate for most of cases. Nonetheless, there are specific scenarios, such as when a GW source is lensed by a supermassive black hole in a binary system, where the lens and source are in close proximity, where large-angle diffraction can play a crucial role. In our prior research, we have introduced an exact, general diffraction integral formula that accounts for large-angle diffraction as well. This paper explores the disparities between this exact diffraction formula and the traditional, approximate one under various special conditions. Our findings indicate that, under specific parameters — such as a lens-source distance of D LS = 0.1 AU and a lens mass of M L = 4 × 106 M ⊙ — the amplification factor for the exact diffraction formula is notably smaller than that of the approximate formula, differing by a factor of approximately rF ≃ 0.806. This difference is substantial enough to be detectable. Furthermore, our study reveals that the proportionality factor rF gradually increases from 0.5 to 1 as D LS increases, and decreases as M L increases. Significant differences between the exact and approximate formulas are observable when D LS ≲ 0.2 AU (assuming M L = 4 × 106 M ⊙) or when M L ≳ 2 × 106 M ⊙ (assuming D LS = 0.1 AU). These findings suggest that there is potential to validate our general diffraction formula through future GW detections.

On some general integral formulae

We repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $\textbf{R}^n$ is represented by a hyperelliptic integral.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

The fundamental solution to \Box_{𝑏} on quadric manifolds – Part 1. General formulas

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N N and the projection onto the nullspace of ◻ b \Box _b . The main application of our formulas is the critical case of codimension two quadrics in C 4 \mathbb {C}^4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

Hermite–Hadamard–Fejér-Type Inequalities and Weighted Three-Point Quadrature Formulae

The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w-harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the first and of the second kind.

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

A simple model of viscous propulsion of a two-dimensional Marangoni boat caused by surface diffusion and sublimation (reaction) of insoluble surfactant is formulated and studied analytically. The speed of the boat is determined explicitly as a function of a Damkohler number by fully resolving the Marangoni-induced bulk flow caused by the release of surfactants onto the surface of a semi-infinite viscous fluid. A complementary general integral formula for the boat speed is also derived using the reciprocal theorem.

On special Riemann xi function formulae of Hardy involving the digamma function

We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.

Isogeometric analysis of the new integral formula for elastic energy change of heterogeneous materials

In recent years, more and more attention has been paid to isogeometric methods, in which shape functions are used to accurately describe CAD models and approximate unknown fields. The isogeometric boundary element method (IGABEM) realizes the integration on the exact boundary of the region, which means there is no geometric discretization error. In this paper, a more general interface integral formula for elastic energy increment of heterogeneous materials is extended from previous work (Dong, 2018), in which the only unknown variable is the displacement located on the interface between matrix and inclusion. This feature makes it more compatible with boundary element method (BEM) because of none of volume parametrization. However, the geometry discontinuity on the boundary called ‘corner point problem’ increases the difficulty and decreases the accuracy when solving complex numerical examples when heterogeneous structures are considered. The discontinuous element method combined with IGABEM is presented and applied to deal with ‘corner point problem’. In the numerical examples, the interface between matrix and inclusion is discretized by quadratic isogeometric elements. Compared with the analytical solution, the numerical results show that the method has higher accuracy and efficiency.

A Simple Analytic Approximation of Luminosity Distance in FLRW Cosmology using Daftardar-Jafari Method

In this paper, the iterative method suggested by Daftardar-Gejji and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for the approximate analytical representation of the luminosity distance in a homogenous Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology. We obtain the analytical expressions of the luminosity distance using the approximate solutions of the differential equation to which the luminosity distance satisfies, subject to the corresponding initial conditions. With the help of this approximate solution, a simple analytic formula for the luminosity distance as a function of redshift is obtained and compared with a numerical solution for the general integral formula by the Maple software. Subsequent comparison of the obtained approximate analytical formula with the corresponding numerical solution for the and quintessential models is provided and showed a high accuracy of the DJM approximations, at least for the certain values of parameters of the models. This comparison demonstrates the efficiency and simplicity of this approach to the problem of calculating the luminosity distance in theoretical cosmology.

Statistical method for modeling Knudsen diffusion in nanopores.

This paper presents a statistical method for the calculation of gaseous flux and diffusion coefficients through a Knudsen-regime cylindrical nanopore. A general integral formula for the flux is derived in terms of collision frequency, molecular density, and a scattering path length probability distribution. Under appropriate steady-state assumptions, the general formula simplifies to Fick's first law, from which an expression for the diffusion coefficient is derived. The model is shown to be dimensionally consistent with the Einstein relation. The conditions for agreement with Fick's second law are investigated. Using a model probability distribution the model leads to an expression for the diffusion coefficient for a pore of finite length. This result is shown to compare favorably with a classic formula from the literature.

Exploring elliptical vortex beams with the spatial damping effect.

A general integral formula is analytically derived to represent the wave function of elliptical vortex modes. The derived formula can be straightforwardly employed to take account of spatial damping to explore vortex structure. Spatial damping is found to cause the central degenerate singularity of the circular vortex mode to be split into several singularities with different topological charges. For the non-circular vortex beam, the influence of spatial damping on the distribution of singularities is not significant. Theoretical analyses are confirmed to agree very well with the experimental measurements.

Integrals formulas involving confluent hypergeometric Functions of three variables Ф\(_2^{(3)}\) and Ѱ\(_2^{(3)}\)

The aim of this paper is to establish two general integral formulas involving confluent hypergeometric functions of three variables Ф2(3) and Ѱ2(3) with the help of two extension formulas for Lauricella’s functions of three variables FA(3) and FD(3) due to Atash [1] and Atash and Bellehaj [2]. Some applications of our main results are also presented.

On the generalized quotient integrals on homogeneous spaces

Abstract A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the special homogeneous spaces are derived by using the general quotient integral formula. Finally, our results are supported by some examples.

Boundary value problems of statics of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with boundary value problems of statics of the thermoelasticity theory of isotropic microstretch materials with microtemperatures and microdilatations. For the system of differential equations of equilibrium the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

Odd zeta motive and linear forms in odd zeta values

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\unicode[STIX]{x1D701}(n)$. They naturally include the Beukers–Rhin–Viola integrals for $\unicode[STIX]{x1D701}(2)$ and the Ball–Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.

Existence of weak solutions to degenerate p-Laplacian equations and integral formulas

We study the problem of solving some general integral formulas and then apply the conclusions to obtain results about the existence of weak solutions of various degenerate p-Laplacian equations. We adapt Variational Calculus methods and the Mountain Pass Lemma without the Palais–Smale condition, and we use an abstract version of Lions' Concentration Compactness Principle II.

Analytic Approximation of Luminosity Distance in Cosmology via Variational Iteration Method

This work deals with a new approach in the approximate analytical representation of the luminosity distance in a homogenous Friedmann-Lemaitre-Robertson-Walker (FLRW) model of the Universes by means of the variational iteration method (VIM). For the analytical calculation of the luminosity distance, we obtain the approximate solution of the differential equation which the luminosity distance obeys, using the corresponding initial conditions. On the basis of this approximate solution, a simple analytic formula for the luminosity distance as a function of redshift is obtained and compared with a numerical solution from the general integral formula.

Mathematical problems of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with the linear theory of thermoelasticity for elastic isotropic microstretch materials with microtemperatures and microdilatations. For the differential equations of pseudo-oscillations the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

General integral formulas involving Hubert hypergeometric functions of two variables

In this paper, we have established two general integral formulas involving Humbert hypergeometric functions of two variables ψ2 and 2. The results are obtained with the help of a generalization of classical Kummer's summation theorem on the sum of the series 2F1(-1) due to Lavoie et al. [5]. Some interesting applications are also presented.