Principal Value Sense Research Articles

Weyl's law for the eigenvalues of the Neumann–Poincaré operators in three dimensions: Willmore energy and surface geometry

We deduce eigenvalue asymptotics of the Neumann–Poincaré operators in three dimensions. The region Ω is C2,α (α>0) bounded in R3 and the Neumann–Poincaré operator K∂Ω:L2(∂Ω)→L2(∂Ω) is defined byK∂Ω[ψ](x):=14π∫∂Ω〈y−x,n(y)〉|x−y|3ψ(y)dSy where dSy is the surface element, n(y) is the outer normal vector on ∂Ω, and the integral is understood in the principal value sense. Then the ordered eigenvalues λj(K∂Ω) of the Neumann–Poincaré operator satisfy|λj(K∂Ω)|∼{3W(∂Ω)−2πχ(∂Ω)128π}1/2j−1/2asj→∞. Here W(∂Ω) and χ(∂Ω) denote the Willmore energy and the Euler characteristic of the boundary surface ∂Ω. This formula is the so-called Weyl's law for eigenvalue problems of Neumann–Poincaré operators.

Hilbert transform-based semi-analytic meta-model for maximum response envelopes in dynamics of railway bridges

This paper presents an innovative meta-model for fast assessment of dynamic design envelopes of railway bridges. The proposed approach allows the computation of instantaneous amplitudes of dynamic response time series through the integral Hilbert transform (HT). Based upon the semi-analytic solution of the moving load problem of linear bridge structures, which is analytical in the time domain, the HT is obtained in the Cauchy principal value sense in closed-form. Therefore, the proposed HT-based semi-analytic (HTSA) solution provides a direct estimation of the instantaneous response envelopes of railway bridges under high-speed trains. Since the HT demodulates the signals by suppressing the fast-varying oscillating components, the sampling rates needed for tracing the instantaneous amplitudes are considerably lower than those required for sampling the original signals. Thus, the reduction in computational time achieved by the HTSA approach stems from the sub-sampled approximation of instantaneous envelopes. In order to appraise the effectiveness of the proposed HTSA meta-model, four validation case studies are presented, including mono-dimensional and realistic three-dimensional bridge structures. The presented results report substantial reduction in computational cost with limited accuracy loss, and demonstrate the usefulness of the HTSA meta-model for fast comparison of bridge design alternatives during early design stages.

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

On the Increasing Tritronquée Solutions of the Painlevé-II Equation

The increasing tritronqu\'ee solutions of the Painlev\'e-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|<\tfrac{2}{3}\pi$ and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of $\alpha$. Here these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex $\alpha$, all such solutions are asymptotically pole-free along the bisecting ray of the complementary sector $|\arg(-x)|<\tfrac{1}{3}\pi$ that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity and the poles are dealt with in the principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlev\'e-II parameter $\alpha$ is of the form $\alpha=\pm\tfrac{1}{2}+{\rm i}p$, $p\in\mathbb{R}\setminus\{0\}$, one of the increasing tritronqu\'ee solutions has no poles or zeros whatsoever along the bisecting axis.

Effect of inner Lindblad resonance on spiral density waves propagation in disc galaxies: reflection over absorption

Interaction of spiral density waves with stars in the vicinity of the inner Lindblad resonance in galactic discs is investigated using the linear perturbation theory and the leading orders in the epicyclic and WKB approximations. In analogy with shear flows in hydrodynamics, we conjecture that a weak nonlinearity in a narrow resonance region modifies the standard (Landau-Lin) bypass rule of the singularity to the integration in the principal value sense. This indeed leads to the reflection of the spiral wave instead of absorption, but the detailed picture looks awkward: the intervals of the wave weakening alternate with the intervals of the wave growth, so that the net absorption is absent. Incidentally, we rectify the result concerning leading spiral waves obtained earlier for the standard bypass rule.

Nonlocal $s$-minimal surfaces and Lawson cones

The nonlocal $s$-fractional minimal surface equation for $\Sigma \partial E$ where $E$ is an open set in $\mathbb{R}^N$ is given by \[ H_\Sigma^ s (p) := \int_{\mathbb{R}^N} \frac{\chi_E(x) - \chi_{E^c}(x)} {{\lvert x-p \rvert}^{N+s}}\, dx = 0 \; \textrm{for all} \; p\in\Sigma \textrm{ .} \] Here $0 \lt s \lt 1 , \chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s \to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s$-minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $\mathbb{R}^3$, the nonlocal catenoid, an analog of the standard catenoid $\lvert x_3 \rvert \log(r+ \sqrt{r^2 - 1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $\lvert x_3 \rvert r \sqrt{1 - s}$. We also find a two-sheet embedded s-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any $0 \lt s \lt 1 , n , m \geq 1$, $s$-minimal Lawson cones $\lvert v \rvert \alpha \lvert u \rvert, (u, v) \in \mathbb{R}^n \times \mathbb{R}^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n + m 7$, which suggests that unlike the classical theory (or the case $s$ close to $1$), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.

Three Revolutions in The Kernel Are Worse Than One

An example is constructed of a purely unrectiable measure for which the singular integral associated to the kernel K(z) = z z2 is bounded in L 2 ( ). The singular integral fails to exist in the sense of principal value -almost everywhere. This is in sharp contrast with the results known for the kernel 1 (the Cauchy transform).

Eigenvalue problem of Hamiltonian operator matrices

This paper deals with the eigenvalue problem of Hamiltonian operator matrices with at least one invertible off-diagonal entry. The ascent and the algebraic multiplicity of their eigenvalues are determined by using the properties of the eigenvalues and associated eigenvectors. The necessary and sufficient condition is further given for the eigenvector (root vector) system to be complete in the Cauchy principal value sense.

Symplectic approach for the plane elasticity problem of quasicrystals with point group 10 mm

The symplectic approach is introduced into the plane elasticity problem of quasicrystals with point group 10mm. The basic equations of the problem are equivalently written as the Hamiltonian dual equations. It is shown that the generalized eigenvector system of the corresponding Hamiltonian operator matrix is complete in the Cauchy Principal Value (CPV) sense. The analytical solution and related numerical results of the problem are then given.

On the Surface Limit in the Boundary Integral Equations of Electromagnetics

Boundary integral equations of electromagnetics are enforced at a current-carrying surface, where the kernels are strongly singular and give rise to discontinuous integrals, which must be evaluated in the principal-value sense, after the contributions from the neighborhood of the singularity are analytically extracted. There exist two basic techniques of taking the surface limit: one is to make the field point approach a small disk on the current-carrying surface and to subsequently let the disk radius vanish; another is to deform the surface by creating a hemispherical indentation about the field point and to shrink the radius of the dent to zero. In this paper, both methodologies are examined and it is demonstrated that the often used hemispherical indentation approach is flawed and can lead to incorrect results.

The First Negative Moment of Skew-tand Generalized Student'st-Distributions in the Principal Value Sense

The (Cauchy) principal value is a method for assigning values to certain improper integrals which would otherwise be undefined. Using the principal value sense, this study derives an explicit expression of the first negative moment of skew-tand generalized Student'st-distributions for practical applications. Some applications obtained from the FNM of skew-tand generalized Student'st-distributions are also discussed.

A symplectic eigenfunction expansion approach for free vibration solutions of rectangular Kirchhoff plates

The eigenfunction system of the Hamiltonian operator appearing in the free vibration of rectangular Kirchhoff plates with two opposite edges simply supported is studied. The governing differential equations for free vibration of rectangular Kirchhoff plates is rewritten as a Hamiltonian system based on the known results, and the associated Hamiltonian operator is obtained. Then, in the sense of Cauchy's principal value, the completeness of the symplectic eigenfunction system is proved. It offers a theoretical guarantee of the feasibility of variable separation methods based on the Hamiltonian system for the problem. The exact general solution for the corresponding Hamiltonian system of the problem is given by the symplectic eigenfunction expansion method. The general solution is more simple and convenient than the existing result. Examples are given to illustrate that, combining the general solution with the corresponding boundary conditions, the frequency equations and the transverse displacement functions for Lévy-type plates can be directly derived. Furthermore, the boundary conditions for the problem, which can be solved by this approach, are indicated.

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.

An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals

In this paper, a robust method is presented for numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals, which exist in the Cauchy principal value sense in two- and three-dimensional problems. In this method, the singularities involved in integration kernels are analytically removed by expressing the non-singular parts of the integration kernels as power series in the local distance ρ of the intrinsic coordinate system. For three-dimensional boundary integrals, the radial integration method [1] is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals. Some examples are provided to verify the correctness and robustness of the presented method.

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy's principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.

Uniqueness theorems for Cauchy integrals

If $\mu$ is a finite complex measure in the complex plane $\mathbb{C}$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called \emph{reflectionless} if it is continuous (has no atoms) and $C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$ then $\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the weak $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.

The first negative moment in the sense of the Cauchy principal value

Alternative sufficient conditions, for the existence and finiteness of the first negative moment of a continuous density function defined on the real line, are established in the sense of the (Cauchy) principal value. Using the principal value sense for practical applications, we derive explicit expressions of the reciprocal moment of a skew-normal distribution. In addition, some interesting properties discovered from the skew-symmetric model are also obtained. Finally, a case example is used to illustrate the concepts in lifetime data analysis.

Numerical evaluation of two-dimensional singular boundary integrals—Theory and Fortran code

In this paper, an approach is presented for the numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals which exist in the Cauchy principal value sense in two-dimensional problems. In this approach, the singularities involved in integration kernels are analytically removed by expressing the nonsingular parts of the integration kernels as polynomials of the distance r . A self-contained Fortran code is listed and described for implementation of the proposed approach. The attached code is also able to evaluate general regular integrals using Gaussian quadrature, which enables the code to evaluate any two-dimensional boundary integral. Some examples are provided to verify the correctness of the presented formulations and the included code.

Some considerations in connection with Kurepa's function

In this paper we consider the functional equation for factorial sum and its particular solutions (Kurepa's function K(z) [3] and function K1(z)). We determine an extension of domain of functions K(z) and K1(z) in the sense of Cauchy's principal value at point [2]. In this paper we give an addendum to the proof of Slavic's representation of Kurepa's function K(z) [6]. Also, we consider some representations of functions K(z) and K1(z) via incomplete gamma function and we consider differential transcendency of previous functions too.