Integral Invariants Research Articles

Effects of manifold correction methods on chaos indicators

The manifold approach of Nacozy et al. (Astrophys Space Sci 14:40–51, 1971), the approximate velocity correction method of Wu et al. (Astron J 133:2643–2653, 2007), and the velocity scaling method of Ma et al. (New Astron 13:216–223, 2008a) are some of the available manifold correction methods. They have been highly successful at maintaining invariant integrals in two-body problems and the Sun–Jupiter–Saturn system. This paper discusses their efficiency on chaos indicators. Because the planar circular restricted three-body problem involves the Jacobi constant $$C_\mathrm{j}$$ and chaotic phenomena, it is preferable to check the numerical performances of manifold corrections. First, we find that a low-order algorithm combined with manifold corrections can greatly improve the precision of the Jacobi constant $$C_\mathrm{j}$$ . Then, numerical experiments show that these manifold correction methods have the same performance in Poincare sections, Lyapunov exponents, fast Lyapunov indicators, smaller alignment indices, and relative finite time Lyapunov indicators. Moreover, manifold corrections not only allow for the use of larger step sizes compared to low-order algorithms without correction but also save substantial computation time compared to the high-order algorithm RKF7(8). In particular, the velocity scaling method of Ma et al. (2008a) lends itself to practical application in long-term integration.

The invariant integral of physical mesomechanics as the foundation of mathematical physics: Some applications to cosmology, electrodynamics, mechanics and geophysics

Phe general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. Phe physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb’s law of electricity generalized for moving electric charges, Newton’s law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes’ law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth’ crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a diskshape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. Phe method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.

A new derivation of the shallow water equations in geographical coordinates and their application to the global barotropic ocean model (the DEBOT model)

The purpose of this paper is to present a new global barotropic ocean model—the DEBOT model. The model is based on the shallow water equations which we newly express in geographical coordinates. The derivation includes the boundary conditions and the Reynolds tensor in a form used commonly in oceanography. The numerical model employs finite differences on an Arakawa-C grid in space and a generalized forward–backward scheme in time with a combined third-order Adams–Bashforth and fourth-order Adams–Moulton step. The validity of the model is demonstrated by the tests based on conservation integral invariants. As a practical application, we present ocean circulation simulations generated by the lunisolar tidal force.

Measured quantum groupoids associated to proper dynamical quantum groups

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang–Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II _1 factors. In this article, we associate to suitable dynamical quantum groups, which are purely algebraic objects, Hopf C* -bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C* -algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU _q (2) studied by Koelink and Rosengren.

The mechanics and physics of fracturing: application to thermal aspects of crack propagation and to fracking.

By way of introduction, the general invariant integral (GI) based on the energy conservation law is presented, with mention of cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter application to demonstrate the approach, including Coulomb's Law generalized for moving electric charges, Newton's Law generalized for coupled gravitational/cosmic field, the new Archimedes' Law accounting for gravitational and surface energy, and others. Then using this approach the temperature track behind a moving crack is found, and the coupling of elastic and thermal energies is set up in fracturing. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding GIs. As applied to the horizontal drilling and fracturing of boreholes, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a fracture are derived in the fluid extraction regime. The theory of fracking in shale gas reservoirs is suggested for three basic regimes of the drill mud permeation, with calculating the shape and volume of the local region of the multiply fractured rock in terms of the pressures of rock, drill mud and shale gas.

A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on [formula omitted

The degenerate Bernoulli polynomials were introduced by Carlitz and rediscovered later by Ustiniv under the name of Korobov polynomials of the second kind. In this paper, we derive a Witt-type formula for the degenerate Bernoulli polynomials which can be represented by the p-adic invariant integral on Zp. In addition, we introduce the λ-Daehee polynomials and give some relations between the degenerate Bernoulli polynomials and the λ-Daehee polynomials.

Integral invariants for space motion trajectory matching and recognition

Motion trajectories provide a key and informative clue in motion characterization of humans, robots and moving objects. In this paper, we propose some new integral invariants for space motion trajectories, which benefit effective motion trajectory matching and recognition. Integral invariants are defined as the line integrals of a class of kernel functions along a motion trajectory. A robust estimation of the integral invariants is formulated based on the blurred segment of noisy discrete curve. Then a non-linear distance of the integral invariants is defined to measure the similarity for trajectory matching and recognition. Such integral invariants, in addition to being invariant to transformation groups, have some desirable properties such as noise insensitivity, computational locality, and uniqueness of representation. Experimental results on trajectory matching and sign recognition show the effectiveness and robustness of the proposed integral invariants in motion trajectory matching and recognition.

On Integral Invariants for Effective 3-D Motion Trajectory Matching and Recognition.

Motion trajectories tracked from the motions of human, robots, and moving objects can provide an important clue for motion analysis, classification, and recognition. This paper defines some new integral invariants for a 3-D motion trajectory. Based on two typical kernel functions, we design two integral invariants, the distance and area integral invariants. The area integral invariants are estimated based on the blurred segment of noisy discrete curve to avoid the computation of high-order derivatives. Such integral invariants for a motion trajectory enjoy some desirable properties, such as computational locality, uniqueness of representation, and noise insensitivity. Moreover, our formulation allows the analysis of motion trajectories at a range of scales by varying the scale of kernel function. The features of motion trajectories can thus be perceived at multiscale levels in a coarse-to-fine manner. Finally, we define a distance function to measure the trajectory similarity to find similar trajectories. Through the experiments, we examine the robustness and effectiveness of the proposed integral invariants and find that they can capture the motion cues in trajectory matching and sign recognition satisfactorily.

On the invariants of Mannheim offsets of timelike ruled surfaces with spacelike rulings

In this paper, we give the characterizations for Mannheim offsets of timelike ruled surfaces with spacelike rulings in dual Lorentzian space [Formula: see text]. We obtain the relations between terms of their integral invariants and also we give new characterization for the Mannheim offsets of developable timelike ruled surface. Moreover, we find relations between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.

Extension of covariant derivative (III): From classical gradient to shape gradient

This paper further extends the generalized covariant derivative from the first covariant derivative to the second one on curved surfaces. Through the linear transformation between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the second class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric structure of the tensor analysis on curved surfaces are revealed.

Symmetry identities of q-Bernoulli polynomials of the second kind

In this paper, we give identities of symmetry for the q-Bernoulli polynomials which are derived from the symmetric properties of the p-adic invariant integrals on ℤp.

FOURTH-ORDER NUMERICAL METHODS FOR THE COUPLED KORTEWEG–DE VRIES EQUATIONS

We compare six fixed-stepsize fourth-order numerical methods for a number of test problems described by a system of coupled Korteweg–de Vries equations. Particular attention is paid to the ability of these methods to preserve fixed points (solitary waves) and the invariants of the system, and establishing to what extent the conservation of integral invariants is indicative of the solution error for these methods.

Shape Description and Matching Using Integral Invariants on Eccentricity Transformed Images

Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately.

On the ABJM four-point amplitude at three loops and BDS exponentiation

We study the three-loop four-point amplitude in ABJM theory. We determine the dual conformal invariant integrals with highest number of propagators and fix their coefficients by two-particle cuts. Evaluating such a combination of integrals in dimensional regularization we provide evidence for exponentiation of the amplitude, including the finite terms. In addition we show that the three-loop amplitude can be expressed in terms of classical polylogarithms of uniform degree of transcendentality.

Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body

The equilibrium problem of the elastic body with a delaminated thin rigid inclusion is considered. In this case, there is a crack between the rigid inclusion and the elastic body. We suppose that the nonpenetration conditions are prescribed on the crack faces. We study the dependence of the energy of the body on domain variations. The formula for the shape derivative of the energy functional is obtained. Moreover, it is shown that for the special cases of the domain perturbations such derivative can be represented as invariant integrals.

Singular invariant integrals for elastic body with delaminated thin elastic inclusion

We consider an equilibrium problem for a 2 D 2D elastic body with a thin elastic inclusion. It is assumed that the inclusion is partially delaminated, therefore providing the presence of a crack. Inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration of the faces. Differentiability properties of the energy functional with respect to the crack length are analyzed. We prove an existence of the derivative and find a formula for this derivative. It is shown that the formula for the derivative can be written in the form of a singular invariant integral.

Fractional Lorentz-Dirac Model and Its Dynamical Behaviors

In the paper, we construct a new kind of fractional dynamical model, i.e. the fractional Lorentz-Dirac model, and explore dynamical behaviors of the model. We find that the fractional Lorentz-Dirac model possesses Lie algebraic structure and satisfies generalized Poisson conservation law, and then a series of Poisson conserved quantities of the model are given. Further, the relation between conserved quantity and integral invariant of the model is studied, and it is proved that, using the Poisson conserved quantities, we can construct a series of integral invariants of the model. Finally, the stability for the manifolds of equilibrium state of the fractional Lorentz-Dirac model is studied.

Algebraic invariant curves and first integrals for Riccati polynomial differential systems

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x′ = 1, y′ = a(x)y2 +b(x)y +c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

Multigrid convergent principal curvature estimators in digital geometry

In many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. In this paper, we investigate a new class of estimators on digital shape boundaries based on integral invariants (Pottmann et al., 2007) [39]. More precisely, we provide both proofs of multigrid convergence of principal curvature estimators and a complete experimental evaluation of their performances.

The J and M integrals for a cylindrical cavity in a time-harmonic wave field

The invariant integrals are being widely used in the study of defects and fracture mechanics, mostly in elastostatics. However, the properties and the interpretation of these integrals in elastodynamics, especially in the case of time-harmonic excitation, remain unexplored. This contribution is focused on the derivation of the time average J integral for a cylindrical inhomogeneity and the M integral for a cylindrical cavity placed in a monochromatic plane elastic wave. It is shown in the context of antiplane linear elasticity that the J integral, or the material force acting on the inhomogeneity, resembles the radiation pressure force exerted on a dielectric cylinder by the normally incident electromagnetic wave. Based on the existing solution of this electrodynamic problem and the corresponding acoustic problem, the J integral is expressed as a function of the nondimensional wave number in the form of the partial wave expansion of the scattering theory. Employing the same classical method as with the J integral, the closed-form solution of the time average M integral for a traction-free cavity is also obtained as a function of the nondimensional wave number. The M integral, i.e., the expansion or scalar moment per unit length on an infinitely long circular cavity, is represented in terms of the scattering phase shifts as in the case of the J integral. Alternative expressions that are more convenient for numerical calculations for the cavity are also derived for both integrals. These calculations are carried out for the J and M integrals in a wide spectrum of frequencies. Asymptotic approximations of both integrals for low and high frequencies are presented. The long wavelength approximation, including the monopole and dipole contributions, has been provided for the J integral in the form of a simple analytical expression. The value of M integral in the vanishing frequency limit is also presented. In the opposite short wavelength limit, the corresponding asymptotic values are derived for both integrals. These solutions, which are valid for the empty cavity, are extended to the case of an inviscid fluid-filled cavity. The obtained results have a variety of engineering and geophysical implications. In particular, their applications to the flaw characterization of materials and further development of non-destructive evaluation techniques are briefly discussed in light of the frequency relationships derived for the J and M integrals.