Rotation numbers for quasi-periodically forced monotone circle maps

Rotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form „ - 1/n ≤(1/n)(y_n - y₀) ≤„ + 1/n, where (1/n)(y_n - y₀) is an estimate of the rotation number obtained from an orbit of length n with initial condition y₀, and „ is the true rotation number. This allows rotation numbers to be computed reliably and efficiently. Although Herman has proved that quasi-periodically forced circle maps also possess a well-defined rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which y_n - y₀ - „n is not bounded. This renders the estimation of rotation numbers for quasi-periodically forced circle maps much more problematical. In this paper, a new characterization of the rotation number is derived for quasiperiodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satisfies analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, the boundedness of y_n - y₀ - „n is considered. It is shown that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits. Conversely, if for any orbit y_n - y₀ - „n is unbounded either above or below, then there is a residual set of orbits for which y_n - y₀ - „n is unbounded both above and below. In proving these results a min-max characterization of the rotation number is also presented. The performance of an algorithm based on this is evaluated, and on the whole it is found to be inferior to the integral based method.     

An efficient method for reliability evaluation of multistate networks given all minimal path vectors

The multistate networks under consideration consist of a source node, a sink node, and some independent failure-prone components in between the nodes. The components can work at different levels of capacity. For such a network, we are interested in evaluating the probability that the flow from the source node to the sink node is equal to or greater than a demanded flow of d units. A general method for reliability evaluation of such multistate networks is using minimal path (cut) vectors. A minimal path vector to system state d is called a d-MP. Approaches for generating all d-MPs have been reported. Given that all d-MPs have been found, the issue becomes how to evaluate the probability of the union of the events that the component state vector is greater than or equal to at least one of the d-MPs. There is a need for a more efficient method of determining the probability of this union of events. In this paper, we report an efficient recursive algorithm for this union probability evaluation based on the Sum of Disjoint Products (SDP) principle, and name it the Recursive Sum of Disjoint Products (RSDP) algorithm. The basic idea is that, based on the SDP principle and a specially defined “maximum” operator, “⊕”, the probability of a union with L vectors can be calculated via calculating the probabilities of several unions with L-1 vectors or less. The correctness of RSDP is illustrated. The efficiency of this algorithm is investigated by comparing it with an existing algorithm that is generally accepted to be efficient. It is found that RSDP is more efficient than the existing algorithm when the number of components of a system is not too small. RSDP provides us with an efficient, systematic and simple approach for evaluating multistate network reliability given all d-MPs.     

N-particle dynamics of the Euler equations for planar diffeomorphisms

The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this article, we consider the dynamics of N point particles or “blobs” moving under the action of the Euler equations associated with the group of diffeomorphisms of the plane in a variety of different metrics. This dynamical system is already in widespread use in the field of image registration, where the point particles correspond to image landmarks, but its dynamical behavior has not previously been studied. The 2-body problem is always integrable, and we analyze its phase portrait under different metrics. In particular, we show that 2-body capturing orbits (in which the distances between the particles tend to 0 as t  → ∞) can occur when the kernel is sufficiently smooth and the relative initial velocity of the particles is sufficiently large. We compute the dynamics of these “dipoles” with respect to other test particles, and supplement the calculations with simulations for larger N that illustrate the different regimes.     

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

While frame-invariant solutions for arbitrarily large rotational deformations have been reported through the orthogonal matrix parametrization, derivation of such solutions purely through a rotation vector parametrization, which uses only three parameters and provides a parsimonious storage of rotations, is novel and constitutes the subject of this paper. In particular, we employ interpolations of relative rotations and a new rotation vector update for a strain-objective finite element formulation in the material framework. We show that the update provides either the desired rotation vector or its complement. This rules out an additive interpolation of total rotation vectors at the nodes. Hence, interpolations of relative rotation vectors are used. Through numerical examples, we show that combining the proposed update with interpolations of relative rotations yields frame-invariant and path-independent numerical solutions. Advantages of the present approach vis-a-vis the updated Lagrangian formulation are also analyzed.     

A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.     

Fiber/cement interface tailoring with plasma treatment

A comprehensive study on the optimal conditions of plasma treatment of polyethylene fibers on their interfacial property with cement matrix is reported in this paper. Single fiber pullout tests were performed for bond property evaluation. With the optimal treatment conditions, a sixfold increase in bond strength or sevenfold increase in interfacial toughness can be achieved separately. Furthermore, a distinct interfacial chemical bond, compared to common frictional bond was observed for the first time in the polyethylene/cement systems. On the matrix side, matrix modifiers such as high alumina cement and PVA powder were also studied to examine their additional influence on the interface properties. Some preliminary results of the stability of plasma treatment are also reported in this paper.     

Spin operator for quantum light in integrated photonic waveguides

In this work a quantum mechanical derivation of the Spin operator in integrated isotropic and anisotropic photonic waveguides is obtained from both first principles of the electromagnetic theory and conservation laws, and moreover, within a phenomenological quantization approach. The Spin operator will be derived by starting from the spin conservation law to obtain the spin flux density and by means of a quantization based on the vector structure and orthonormality property of guided modes; likewise, since anisotropic guides produce polarization changes, and consequently changes of the expected values of spin, we will also present the Momentum operator for anisotropic guides. The expression for the Spin angular momentum operator shows that both the presence of modal longitudinal components and the modal mismatching reduces the spin values, but it comes into the standard expression when it is used in homogeneous media and in particular with uniform and infinite plane modes. Likewise these results have a direct influence on the Stokes's operators which allow one to analyze quantum polarization in integrated photonics.     

Parallel time scales and two-dimensional manufacturer and individual customer warranties

We deal with a system whose failures depend on several parallel effects, such as the time in use L and the mileage H. Manufacturer warranties are typically described by a two-dimensional region in the (L, H)-plane. A proper determination of the warranty limits must be based on a two-dimensional distribution of time to failure on this plane. The aim of this paper is to demonstrate the possibility of designing individual warranties for a “nontypical” customer who has a very low or very high usage rate b = H/L, and to show a simple way to calculate warranty limits by minimizing the lifetime coefficient of variation. The latter is carried out by introducing the “best” combined time scale in the form K = (1-ε)L+εH which provides the minimal lifetime coefficient of variation.     

A modified effective crack-length formulation in elastic-plastic fracture mechanics

In examining the performance of standard effective crack-length formulations, the authors noted quantitative accuracy up to “high” fractions of limit load under loading conditions for which the elastic T-stress was non-negative, while a pronounced deviation from the corresponding continuum elastic-plastic plane-strain finite-element solutions was seen in shallow-cracked geometries having negative T-stress. This trend can be rationalized by noting that, under modified boundary layer (K_I and T) loading, the maximum plastic zone radius strongly increases as the T-stress decreases from zero (J.R. Rice (1974), J. Mech. Phys. Solids 22, 17–26; S.G. Larsson and A.J. Carlsson (1973), J. Mech. Phys. Solids 21, 263–277; N.P. O'Dowd and C.F. Shih (1991), J. Mech. Phys. Solids 39(8), 989–1015.) Accordingly, we formulate a modified effective crack length to account for the effects of the elastic T-stress.

The new formulation consistently extends the load range for which accurate predictions of compliance, J-integral, and crack-tip constraint are obtained in several plane strain specimen geometries. The magnitude of influence of the T-stress varies with specimen type and relative crack depth. The greatest “improvement” to standard effective crack length approximations occurs in specimens of “moderately” negative T-stress.     

Choosing efficient convergence parameters in the iterative method of conjugate solutions for a contact boundary-value problem. Report no. 3. Thermoelastic problem

An expression is obtained for convergence parameters reflecting local compliance in the region of contact of two bodies. The expression excludes the effects of the temperature component of displacements of points on the surface of the bodies, as well as effects due to their possible rigidbody displacement and (or) rotation. An algorithm is presented to obtain the matrix for rigidbody rotation and the vector for rigidbody displacement.Translated from Problemy Prochnosti, No. 10. pp. 53–58, October, 1994.     

On the stability of periodic orbits in lattice dynamical systems

The variation of the norm of a matrix operator in the space l_q² when q varies is investigated. The linear part of a lattice dynamical system is a matrix operator. The consequences of this variation on the stability of periodic orbits in such systems is given.     

A generalized numerical approximation for the two-dimensional Laplacian operator

A numerical approximation for the two-dimensional Laplacian operator at a point is derived for the general case in which the point is surrounded by an arbitrary number of arbitrarily located coplanar nodes. The expression obtained is shown to generalize the usual Cartesian finite difference expression.     

Allocation of transmission charges for real-power transactions using markov chains

This work proposes a method to divide the vector of optimal usage of facilities in the grid on account of the transmission of a number of transactions flowing simultaneously into multiple vectors of flows on individual transactions basis. This work focusses on levying charges varying linearly with the power transmitted along various lines. The proposed method is based on the `proportionality assumption`, which is equivalent to compliance of probability laws in the process of flow in the grid, when flows are normalised. A Markov matrix is generated with demand nodes treated as absorbing states. Various powers of the transient sub-matrix allow tracing the fraction of flow from different generation nodes to the demand nodes for any grid flow problem. In this work, various demand nodes are billed for the linearly varying wheeling charges of the grid for the lines actually used. Results for test problems are presented.     

Chaos and nonlinear stochastic dynamics

This article presents a discussion of certain aspects of the interrelationship between the chaotic response of some deterministic nonlinear dynamic system and the stochastic response of the corresponding system obtained by introducing a stochastic perturbation in the form of additive Gaussian white noise. The state space vector of the latter system can then often be represented as a Markov diffusion process. The joint probability density function of the state space vector of this Markov process is closely related to the corresponding chaotic attractor of the underlying deterministic system, and it will be demonstrated that it can be effectively used for prediction purposes.     

Assessment of the dynamics of corrosion fatigue crack initiation applying recurrence plots to the analysis of electrochemical noise data

Electrochemical current oscillations generated during the early stages of corrosion fatigue damage (CFD) were analysed applying recurrence plots. “This novel analysis tool allowed us to assess changes in the dynamics of the CFD process, differentiating pure electrochemical process like of pitting corrosion (PC) from the corrosion fatigue crack formation and initial growth”. The dynamics of CFD initiation was characterized by determining changes in the selected recurrence quantification analysis (RQA) parameter: the percentage of determinism (%D). A significant contribution of this work is that it was possible to separate through changes in %D as a function of the number of cycles (N), the electrochemical process of pitting corrosion from the corrosion fatigue crack initiation and growth, which has a random nature and involved low values of %D of around 5%. A subsequent augment of the %D to values from 75% to 95% with the increase of N could be related to the short fatigue crack arrest. The increment of %D indicates that the electrochemical pitting corrosion process was the predominant contribution to the current oscillations.     

A General method for the construction of difference analogs of a biharmonic operator on irregular nets

We propose a general method for the construction of difference analogs of a biharmonic operator on irregular nets. We show that this method guarantees preserving the invariance of the initial differential operator under linear transformations of a Cartesian coordinate system. Hence follows the uniqueness of the invariant difference analog for a given complex of nodes. We obtain the numerical value of the difference analog of the biharmonic operator for a fifth-order near-boundary complex with a curvilinear boundary. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36, No. 2, pp. 60–64, March-April, 2000.     

Energy-density concept in fracture mechanics

A theory of fracture mechanics is proposed in which attention is focused on the intensity of the energy field in the crack tip region. This energy field possesses a 1/r-type of singularity for both elastic and plastic materials. The strength or amplitude of this field will be referred to as the “energy-density factor”, S. Unlike the stress-intensity factor k in classical fracture mechanics which is only a measure of the local stress amplitude, the energy-density factor is also direction sensitive. The difference between k and S is analogous to the difference between a scalar and vector quantity. In this sense, the critical value S_cr specifies the direction of crack initiation as well as the fracture toughness of the material.     

An alternate finite difference operator for plates of variable stiffness

A finite difference approximation for the equilibrium equation of a thin, laterally loaded, elastic plate of variable stiffness is developed; it uses plate stiffnesses at node points and at points midway between the nodes of the rectangular grids. Displacement and moment results using this operator and an alternate one are compared.     

Dynamic calibration of the relative pose and error analysis in a structured light system

We investigate the problem of dynamic calibration for our structured light system. First, a method is presented to estimate the rotation matrix and translation vector between the camera and the projector using plane-based homography. Then an approach is introduced to analyze theoretically the error sensitivity in the estimated pose parameters with respect to noise in the projection points. This algorithm is simple and easy to implement. Finally, some numerical simulations and real data experiments are carried out to validate our method.