Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space

This paper focuses on studying Noether’s theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by El-Nabulsi. Both holonomic and nonholonomic systems are studied. First, the fractional variational problem from extended exponentially fractional integral, as well as El-Nabulsi–Hamilton’s canonical equations are established; second, the definitions and criteria of fractional Noether symmetric transformations and fractional Noether quasi-symmetric transformations are presented which are based on the invariance of El-Nabulsi–Hamilton action under the infinitesimal group transformations; finally, the fractional Noether’s theorem is established, which reveals the inner relationship between a fractional Noether symmetry and a fractional conserved quantity.     

An integral equation method for non-self-adjoint eigenvalue problems and its applications to non-conservative stability problems

The aim of the work reported in this paper is to present the new formulation of the integral equation method for non-self-adjoint problems and to apply the method to stability problems of elastic continua subjected to non-conservative loadings. A general non-self-adjoint eigenvalue problem stated in terms of differential operators is transformed into a set of coupled integral equations. Our derivation of integral equations is based on an inverse formulation of a canonical form for the original problem and the corresponding fundamental solution pair. Three well-known non-conservative stability problems in elasticity are examined by this integral equation method as illustrative examples. The approximate values of the critical parameters of sample problems demonstrate a sufficient accuracy through a comparison of other values.     

Temperature field in infinite hollow cylinders heated asymmetrically around their perimeter under general nonuniform boundary conditions

A solution has been found, by the method of finite integral transformations, of the heat conduction equation for a hollow cylinder, heated asymmetrically around its perimeter, under general boundary conditions. Formulas are given which reduce the problem with non-uniform boundary conditions to an equivalent problem with uniform boundary conditions.     

Canonical-covariant Wigner function in polar form

The two-dimensional Wigner function is examined in polar canonical coordinates, and covariance properties under the action of affine canonical transformations are derived.     

Geometry and dynamics of squeezing in finite systems

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.     

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Summary. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamiltons equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.     

载荷垂直于非均匀变截面弹性园环的弯扭问题

本文从基本微分方程出发,利用了内力、位移的微分关系,全面研究载荷垂于非均匀变截面弹性园环的弯扭问题,通过适当的数学变换,获得了抗弯刚度和抗扭刚度为任意函数的通用积分式。并对两个特殊情况——阶梯园环和等截面园环获得了通解。为了说明该公式的具体应用,文末给出了示例,并作了求解。     

Methods of checking general safety criteria in UML statechart specifications

This paper describes methods and tools for safety analysis of UML statechart specifications. A comprehensive set of general safety criteria including completeness and consistency is applied in automated analysis. Analysis techniques are based on OCL expressions, graph transformations and reachability analysis. Two canonical intermediate representations of the statechart specification are introduced. They are suitable for straightforward implementation of checker methods and for the support of the proof of the correctness and soundness of the applied analysis. One of them also serves as a basis of the metamodel of a variant of UML statecharts proposed for the specification of safety-critical control systems. The analysis is extended to object-oriented specifications. Examples illustrate the application of the checker methods implemented by an automated tool-set.     

Nonparaxial wave analysis of three-dimensional Airy beams

The three-dimensional Airy beam (AiB) is thoroughly explored from a wave-theory point of view. We utilize the exact spectral integral for the AiB to derive local ray-based solutions that do not suffer from the limitations of the conventional parabolic equation (PE) solution and are valid far beyond the paraxial zone and for longer ranges. The ray topology near the main lobe of the AiB delineates a hyperbolic umbilic catastrophe, consisting of a cusped double-layered caustic. In the far zone this caustic is deformed and the field loses its beam shape. The field in the vicinity of this caustic is described uniformly by a hyperbolic umbilic canonical integral, which is structured explicitly on the local geometry of the caustic. In order to accommodate the finite-energy AiB, we also modify the conventional canonical integral by adding a complex loss parameter. The canonical integral is calculated using a series expansion, and the results are used to identify the validity zone of the conventional PE solution. The analysis is performed within the framework of the nondispersive AiB where the aperture field is scaled with frequency such that the ray skeleton is frequency independent. This scaling enables an extension of the theory to the ultrawideband regime and ensures that the pulsed field propagates along the curved beam trajectory without dispersion, as will be demonstrated in a subsequent publication.     

A new technique to derive the Green’s type integral formula in thermoelasticity

New closed-form influence functions of a unit point heat source on elastic displacements and new Green’s type integral formula for a boundary-value problem (BVP) for a thermoelastic half-space are presented. The main difficulties in obtaining such results are observed in deriving the influence functions of a concentrated unit force onto elastic volume expansion and, also, in Green’s functions in heat conduction. For canonical Cartesian domains, these functions have been derived successfully for hundreds of BVPs and were published in a handbook. So, this paper shows the way to derive not only thermoelastic influence functions and Green’s type integral formulas for the half-space, but also for many new BVPs in thermoelasticity in other Cartesian canonical domains. Moreover, the technique proposed here may be applied in any orthogonal canonical domain provided by the lists of Green’s functions in heat conduction and influence functions for elastic volume expansion that are known.     

Thermodynamics and properties of relaxation functions of materials with memory

Consideration is given to thermodynamic restrictions imposed on relaxation functions within the framework of the Coleman theory for materials with memory. It is shown that for the Second Law of Thermodynamics to be held, some integral transformations of relaxation functions should neccessarily be sign defined. This, in particular, implies the requirement of the dissipative property of the relaxation functions which results in some properties of the relaxation functions which restrict their behavior. These properties contain, as particular cases, the restrictions already known and strengthen them. Moreover, the obtained results, in contrast to some earlier ones, are valid for the case of general nonlinear constitutive equations, where relaxation functions describe their linear part.     

General BE approach for three-dimensional dynamic fracture analysis

A general mixed boundary element approach for three-dimensional dynamic fracture mechanics problems is presented in this paper. A mixed traction-displacement integral equation formulation in the frequency domain is used. The hypersingular and strongly singular kernels are regularized by analytical transformations yielding an easy to implement BE approach. Nine-node quadrilateral and six-node triangular continuous quadratic elements are used for external boundaries and crack surfaces. The crack front elements have their mid node at one quarter of the element length allowing for a proper representation of the crack surface displacement. The present approach is intended for the frequency domain analysis of fracture mechanics problems of any general 3D geometry; i.e. boundless or bounded regions, single or multiple, surface or internal cracks. Transient dynamic problems are studied using the FFT algorithm. The numerical results presented show the robustness and accuracy of the approach which requires a reasonable number of elements and degrees of freedom.     

Natural convection of micropolar fluid in an enclosure with boundary element method

The contribution deals with numerical simulation of natural convection in micropolar fluids, describing flow of suspensions with rigid and underformable particles with own rotation. The micropolar fluid flow theory is incorporated into the framework of a velocity–vorticity formulation of Navier–Stokes equations. The governing equations are derived in differential and integral form, resulting from the application of a boundary element method (BEM). In integral transformations, the diffusion-convection fundamental solution for flow kinetics, including vorticity transport, heat transport and microrotation transport, is implemented. The natural convection test case is the benchmark case of natural convection in a square cavity, and computations are performed for Rayleigh number values up to 10⁷. The results show, which microrotation of particles in suspension in general decreases overall heat transfer from the heated wall and should not therefore be neglected when computing heat and fluid flow of micropolar fluids.     

On the free energy of liquid-metal plasma

The free energy of simple liquid metal in the approximation of the integral smallness of the electron–ion interaction has been found on the basis of the model of single-component plasma as the initial system. The result obtained holds at the equivalency of correlation functions determined within Gibbs canonical and large canonical distributions.     

On the solution of hydrodynamical equations via Lie groups

Lie groups of homothetic transformations in the Euclidean space R₂ have been employed to determine and investigate certain classes of solutions of hydrodynamical equations of a perfect fluid. In particular, the conditions for a solution to be regular with respect to the one-parameter group of transformations have been determined. Furthermore, it has been shown that if the regularity conditions mentioned above are satisfied then the problem of obtaining regular solutions reduces to that of solving a system of equations not involving λ in the canonical coordinates (λ, μ) of the subgroup. Some special classes of flows have also been investigated.     

Canonical transformations of skew-normal variates

Conditions are given for linear functions of skew-normal random vectors to maximize skewness and kurtosis. As a direct implication, several measures of their multivariate skewness and kurtosis are shown to be equivalent. An estimator of the shape parameter with good statistical properties is also considered. These results are strictly related to canonical forms of skew-normal distributions and linear transformations to normality.     

On first integrals of second-order ordinary differential equations

Here we discuss first integrals of a particular representation associated with second-order ordinary differential equations. The linearization problem is a particular case of the equivalence problem together with a number of related problems such as defining a class of transformations, finding invariants of these transformations, obtaining the equivalence criteria, and constructing the transformation. The relationship between the integral form, the associated equations, equivalence transformations, and some examples are considered as part of the discussion illustrating some important aspects and properties.     

FREQUENCY DISTRIBUTION OF CONTACT ANGLES IN RANDOM PACKINGS OF GRANULAR MATERIALS

For a certain spectrum of stable grain configurations in randomly packed granular aggregates it is possible to determine the frequency distribution of relative contact angles among neighboring grains. This possibility is explored in the present paper for both two-dimensional and three-dimensional random packings of granular materials. Two relationships are first derived for the local void ratios of any stable “Voronoi Cell“ within the spectrum of stable configurations for the two and the three dimensional random packings, respectively. These relationships depend on the local distribution of relative contact angles, i.e., directions of contact normals. These local void ratios are then related to the gross void ratios of the random 2-D and 3-D assemblies through two integral equations of the Fredholm type of the first kind, their arguments being the frequency distribution functions. The first integral equation corresponding to a two-dimensional random disk packing is solved, exactly by a set of exponential functional transformations. These exact distributions are shown to be generally Maxwellian, with tails favorably biased towards the population of denser “Voronoi Cells” that are statistically more stable. A discussion on the uniform solutions of the integral equation for three-dimensional random packing of spheres is also presented. However, its general solution is left for a future work.