Explicit Parametric Form Research Articles

Minimum-time interception of a moving target by a material point in a viscous medium

In this paper we investigated a model that describes the motion of a material point in a viscous medium under a force that is arbitrary in direction, but limited in magnitude. This model was named ”the isotropic rocket” in the early work of Rufus Isaacs. We obtained a parametric description of a reachable set for the isotropic rocket and solved a group of reachability problems for a final configuration that varies in a known time-dependent manner (moving target). To describe the reachable set, we obtained an explicit parametric form of its boundary and all its projections onto all subspaces of the state space. Convergent algorithms have been proposed for computing minimum-time interception in position and velocity spaces. Finally, we numerically investigated particular cases of minimum-time interception, validating the development of this study.

Explicit solutions of conjugate, periodic, time-varying Sylvester equations

Solutions of a group of conjugate time-varying matrix equations are discussed in this paper. Through mathematical derivation, the solutions to this group of equations are equivalent to the solutions to a class of conjugate time-invariant matrix equations. Further, the related conditions of solvability are obtained and the general explicit solutions are represented by using quasicontrollability and quasiobservability matrices. A detailed algorithm is presented to make the calculation process clear, and the effectiveness of the algorithm is verified by a concrete example. The proposed algorithm can provide complete solutions to the considered equation in explicit parametric form and its main computation includes solving an ordinary linear algebraic equation and some matrix multiplication operations.

Similarity reductions of peakon equations: integrable cubic equations

We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa–Holm (mCH) equation and Novikov’s equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov’s equation to a negative mKdV flow and a negative Sawada–Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second order and second degree, while for Novikov’s equation the reduction is a particular case of Painlevé V. Furthermore, we show that each of these two different Painlevé-type equations is related to the particular cases of Painlevé III that arise from analogous similarity reductions of the Camassa–Holm and the Degasperis–Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlevé III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.

On a multivariate IFR and positively dependent lifetime model induced by multiple shot-noise processes

In reliability and survival analysis, the shot-noise processes have been intensively employed as a useful tool for modeling the impact of a dynamic environment (in the form of the point process of stresses) on survival characteristics of systems and organisms. However, the studies on the shot-noise processes have been mostly focused on the univariate analysis. In this paper, we develop a class of multivariate stochastic failure models based on a dynamic shock model induced by multiple shot-noise processes and study its properties. Explicit parametric forms for the multivariate survival functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well.

Multivariate reliability modelling based on dependent dynamic shock models

• This paper develops new classes of multivariate reliability models with positive dependence. • The developed novel methodology is also used to generate multivariate exponential distributions. • Multivariate dependence and ageing properties of the developed models are discussed. • This paper uses advanced shock models to generate multivariate reliability models. In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.

Explicit solutions for a long-wave model with constant vorticity

Explicit parametric solutions are found for a nonlinear long-wave model describing steady surface waves propagating on an inviscid fluid of finite depth in the presence of a linear shear current. The exact solutions, along with an explicit parametric form of the pressure and streamfunction give a complete description of the shape of the free surface and the flow in the bulk of the fluid. The explicit solutions are compared to numerical approximations previously given in Ali and Kalisch (2013), and to numerical approximations of solutions of the full Euler equations in the same situation Teles da Silva and Peregrine (1988). These comparisons show that the long-wave model yields a fairly accurate approximation of the surface profile as given by the Euler equations up to moderate waveheights. The fluid pressure and the flow underneath the surface are also investigated, and it is found that the long-wave model admits critical layer recirculating flow and non-monotone pressure profiles similar to the flow features of the solutions of the full Euler equations.

Topological shape optimization design of continuum structures via an effective level set method

AbstractThis paper proposes a new level set method for topological shape optimization of continuum structure using radial basis function (RBF) and discrete wavelet transform (DWT). The boundary of the structure is implicitly represented as the zero level set of a higher-dimensional level set function. The interpolation of the implicit surface using RBF is introduced to decouple the spatial and temporal dependence of the level set function. In doing so, the Hamilton-Jacobi partial differential equation (PDE) that defines the motion of the level set function is transformed into an explicit parametric form, without requiring the direct solution of the complicated PDE using the finite difference method. Therefore, many more efficient gradient-based optimization algorithms can be applied to solve the optimization problem, via updating the expansion coefficients of the interpolant and then evolving the level set function and the boundary. Furthermore, the DWT is employed to handle the full matrix arising from t...

Stochastic Modeling for Velocity of Climate Change

The velocity of climate change is defined as an instantaneous rate of change needed to maintain a constant climate. It is developed as the ratio of the temporal gradient of climate change over the spatial gradient of climate change. Ecologically, understanding these rates of climate change is critical since the range limits of plants and animals are changing in response to climate change. Additionally, species respond differently to changes in climate due to varying tolerances and adaptability. A fully stochastic hierarchical model is proposed that incorporates the inherent relationship between climate, time, and space. Space-time processes are employed to capture the spatial correlation in both the climate variable and the rate of change in climate over time. Directional derivative processes yield spatial and temporal gradients and, thus, the resulting velocities for a climate variable. The gradients and velocities can be obtained at any location in any direction and any time. In fact, maximum gradients and their directions can be obtained, hence minimum velocities. Explicit parametric forms for the directional derivative processes provide full inference on the gradients and velocities including estimates of uncertainty. The model is applied to annual average temperature across the eastern United States for the years 1963– 2012. Maps of the spatial and temporal gradients are produced as well as velocities of temperature change.

Explicit form of parametric polynomial minimal surfaces with arbitrary degree

In this paper, from the viewpoint of geometric modeling in CAD, we propose an explicit parametric form of a class of polynomial minimal surfaces with arbitrary degree, which includes the classical Enneper surface for the cubic case. The proposed new minimal surface possesses some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into four categories with respect to n=4k-1, n=4k,n=4k+1 and n=4k+2, where n is the degree of the coordinate functions in the parametric form of the minimal surface and k is a positive integer. The explicit parametric form of the corresponding conjugate minimal surface is given and the isometric deformation is also implemented.

A class of parametric polynomial minimal surfaces with arbitrary degree

Minimal surfaces is a kind of special surface in differential geometry with important applications in geometric design. In this paper, we propose a new class of parametric polynomial minimal surfaces with arbitrary degree from the viewpoint of geometric modeling. The proposed minimal surface has explicit parametric form, and some interesting geometric properties such as symmetry, containing straight lines and self-intersections. From the geometric properties, the proposed minimal surface can be classified into four categories with respect to n = 4 k + 1, n = 4 k + 2, n = 4 k + 3, n = 4 k + 4, where n is the degree of minimal surface and k is a positive integral number. The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented.

A class of superconformal surfaces

Superconformal surfaces in Euclidean space are the one for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (C R Acad Sci Paris T Ser A 288:993–995, 1979) and Guadalupe-Rodriguez (Pac J Math 106:95–103, 1983) for any codimension, reaches equality at all points. In this paper, we show that any pedal surface to a $$2$$ -isotropic Euclidean surface is superconformal. Opposed to almost all known examples, superconformal surfaces in this class are not conformally equivalent to minimal surfaces. Moreover, they can be given in an explicit parametric form since $$2$$ -isotropic surfaces admit a Weierstrass-type representation.

Describing Chaos of Continuous Time System Using Bounded Space Curve

The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state; moreover the motion of phase point will not approach infinite at last; system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.

Interval kernel regression

Kernel regression is more attractive when it is not possible to determine explicit parametric form of the model and moreover, it does not depend on probabilistic distribution. This paper introduces kernel regression in which the input data set is described by interval-value variables. Two model families are considered. The first family estimates the bounds of the intervals regarding either a smooth function for center variables of the intervals (first model) or two smooth functions for range and center variables, respectively (second model). The second family performs the estimates of the intervals based on regression mixtures. These mixtures assume either a smooth function for center variables and a linear function based on least squares for range variables (third model) or a smooth function for range variables and a linear function for center variables (fourth model). The predictions of the lower and upper bounds of new intervals are computed and two different simulation studies are carried out to validate these predictions. Five real-life interval data sets are also considered. The prediction quality is assessed by a mean magnitude of relative error calculated from a test data set.

Relativistic path integral and relativistic Hamiltonians in QCD and QED

The proper-time 4d path integral is used as a starting point to derive the new explicit parametric form of the quark-antiquark Green's function in gluonic and QED fields, entering as a common Wilson loop. The subsequent vacuum averaging of the latter allows to derive the instantaneous Hamiltonian. The explicit form and solutions are given in the case of the $q\bar q$ mesons in magnetic field.

The brachistochrone problem for a disc

The motion of a vertical disc along a curve under the influence of gravity is investigated. On the assumption of regular rolling without slip and separation of contact points, the problem of plotting the curve of most rapid motion of the disc centre from the origin of coordinates to an arbitrary fixed point of the lower half-plane is solved. As usual, the velocity at the initial instant of time is zero, and at the final instant of time it is not fixed. In explicit parametric form, the classical brachistochrone for contact points of the disc is plotted and investigated. The response time, trajectory and kinematic and dynamic characteristics of motion are calculated analytically. Previously unknown qualitative properties of regular rolling are established. In particular, it is shown that the disc centre moves along a cycloid connecting specified points. The envelopes of the boundary points of the disc, produced as its centre moves along the cycloid, are brachistochrones. The feasibility of mechanical coupling of the disc and the curve by reaction forces at the contact point (the normal pressure and dry friction) is investigated.

Parametric eigenstructure assignment by state feedback in descriptor systems

A new complete parametric approach for eigenstructure assignment in the descriptor linear system Eẋ=Ax+Bu via state feedback control u=Kx is presented. Based on a proposed complete parametric solution to the generalised Sylvester matrix equation, explicit parametric forms of the closed-loop eigenvectors and the feedback gain matrix are given. The approach involves mainly a series of singular value decompositions, and is thus numerically simple and reliable. The approach assigns arbitrary rank(E) finite relative eigenvalues to the closed-loop system and guarantees the closed-loop regularity. A numerical example is worked out to demonstrate the effect of the proposed approach.

Managerial Preferences, Corporate Governance, and Financial Structure

Conflicts of interest between insiders (e.g, controlling shareholders) and outsiders (e.g., minority shareholders) are central to the analysis of modern corporation. In an integrated continuous-time contingent claims framework with imperfect corporate governance, we examine a controlling shareholder's optimal choice of capital structure, ownership concentration, private benefit diversion, consumption, and financial market investment. We derive solutions in explicit parametric forms up to numerical integrations. In addition to generating implications consistent with existing empirical evidence, we also show that managerial preference characteristics (such as risk aversion and impatience), corporate governance, and financial market are important determinants of equity value, credit spread, agency costs, capital structure, and ownership concentration. Our model produces many novel empirically testable predictions. For example, it implies that a more risk averse or a less impatient entrepreneur issues less debt and more equity, and that stronger corporate governance leads to higher equity value, lower leverage, less ownership concentration, and lower credit spread. In addition, it also suggests that as the Sharpe ratio in the financial market increases, firm leverage ratio, agency costs, and credit spread all increase.

Synthesis of Universal Motion Curves in Generalized Model

The universal motion curve is synthesized in a generalized model. The acceleration curve is divided into seven intervals of which the curve shapes on varying-acceleration intervals can be prescribed by arbitrary functions. The characteristics of the motion curve are controlled completely by four shape functions of acceleration and seven time increments. To simplify the synthesis derivation, normalized function and symmetric function are defined to represent these shape functions. The displacement, velocity, acceleration, and jerk curves are derived in piecewise function forms and the peak values of velocity and acceleration are also expressed in explicit parametric forms. The designers can quickly construct the motion curves according to the synthesis formulas and the step-by-step procedures. The helpfulness and convenience of these formulas are demonstrated by three examples in symbolic or numerical solution.

Q-warping: Direct computation of quadratic reference surfaces

We consider the problem of wrapping around an object, of which two views are available, a reference surface and recovering the resulting parametric flow using direct computations (via spatio-temporal derivatives). The well known examples are affine flow models and eight-parameter flow models-both describing a flow field of a planar reference surface. We extend those classic flow models to deal with a quadric reference surface and work out the explicit parametric form of the flow field. As a result we derive a simple warping algorithm that maps between two views and leaves a residual flow proportional to the 3D deviation of the surface from a virtual quadric surface. The applications include image morphing, model building, image stabilization, and disparate view correspondence.

Multistate models of postpartum infecundity, fecundability and sterility by age and parity: Methodological issues*

How do hidden physiological processes influence estimates of fecundability and sterility? Does unobserved heterogeneity play a role in these estimates? To address these questions mathematical models of the reproductive process are needed. It is not well known how to evaluate characteristics of reproductive models based on observed reproductive history data, and such models may not be identifiable without ancillary information. However, little is known about how to introduce ancillary information into reproductive models. Furthermore, even if such information was involved, the use of standard software packages for maximization of the likelihood function is often not feasible, because the function cannot be represented in an explicit parametric form. In this paper we propose an approach which represents the likelihood function in a form useful for further analysis. This approach is based on multistate models of the basic physiological processes that influence reproductive outcomes, and it is suitable in applications where ancillary information is given in the form of hazard rates. As an alternative, a competing risks model with incomplete information is discussed.