Arbitrary Initial Values Research Articles

An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT.

Prescriptions for radiation therapy are given in terms of dose-volume constraints (DVCs). Solving the fluence map optimization (FMO) problem while satisfying DVCs often requires a tedious trial-and-error for selecting appropriate dose control parameters on various organs. In this paper, we propose an iterative approach to satisfy DVCs using a multi-objective linear programming (LP) model for solving beamlet intensities. This algorithm, starting from arbitrary initial parameter values, gradually updates the values through an iterative solution process toward optimal solution. This method finds appropriate parameter values through the trade-off between OAR sparing and target coverage to improve the solution. We compared the plan quality and the satisfaction of the DVCs by the proposed algorithm with two nonlinear approaches: a nonlinear FMO model solved by using the L-BFGS algorithm and another approach solved by a commercial treatment planning system (Eclipse 8.9). We retrospectively selected from our institutional database five patients with lung cancer and one patient with prostate cancer for this study. Numerical results show that our approach successfully improved target coverage to meet the DVCs, while trying to keep corresponding OAR DVCs satisfied. The LBFGS algorithm for solving the nonlinear FMO model successfully satisfied the DVCs in three out of five test cases. However, there is no recourse in the nonlinear FMO model for correcting unsatisfied DVCs other than manually changing some parameter values through trial and error to derive a solution that more closely meets the DVC requirements. The LP-based heuristic algorithm outperformed the current treatment planning system in terms of DVC satisfaction. A major strength of the LP-based heuristic approach is that it is not sensitive to the starting condition.

Solution for the motion of a symmetric Euler gyroscope for arbitrary initial values of the Euler angles using epy kinematic differential Poisson equations

Solution for the motion of a symmetric Euler gyroscope for arbitrary initial values of the Euler angles using epy kinematic differential Poisson equations

Number sequence transition method based on MATLAB BVP solvers for high power Yb3+-doped fiber lasers

A number sequence transition method based on MATLAB BVP solvers (NSTM-BVPs) for high power Yb 3+ -doped fiber lasers with boundary conditions is proposed. The results demonstrate that the NSTM-BVPs can automatically update the initial trial values to deal with the difficulty of setting unreasonable initial values in MATLAB BVP solvers. The number sequence transition method based on bvp6c (NSTM-bvp6c) is then discovered to be superior to NSTM-bvp4c and NSTM-bvp5c. Further, when we set the arbitrary initial value without foreknowledge, the efficiency of NSTM-BVPs is also verified with different pump structures, pump powers, pump reflections and laser output mirror reflectivity . Finally, compared with the shooting method, relaxation method and related experiment, the NSTM-BVPs is valid, and the stimulated Brillouin scattering (SBS) threshold is obtained successfully by NSTM-BVPs. • Number sequence transition method can update automatically the initial trial values. • Comparing other methods, NSTM-BVPs overcome the difficulty of setting initial values. • NSTM-BVPs can be set randomly initial value without foreknowledge. • Average iteration times of NSTM-BVPs are less than 2 for desired high accuracy. • NSTM-BVPs are valid for different laser output mirror reflectivity and SBS threshold.

Fluctuating hydrodynamics and concentration fluctuations in ternary mixtures

We use fluctuating hydrodynamics to derive expressions for the spatial and temporal spectra of concentration fluctuations in a ternary liquid mixture in equilibrium. Our results agree with those found by other investigators who have used a procedure in which arbitrary initial values are adopted. Our method of using fluctuating hydrodynamics has the advantage that it can also be extended to deal with fluctuations in systems out of equilibrium.

波浪データに基づく海面抵抗係数の風速依存性に関する研究

For the purpose of clarifying the appropriate values of the sea surface drag coefficient in high wind speed, the ADWAM was modified to estimate the sea surface drag coefficients as its control variables of the model. In this study, we apply the method to the actual wave observation data to investigate the sea surface drag coefficient. As a result, we confirm that the sea surface drag coefficient was automatically corrected to appropriate values (similar to Mitsuyasu & Honda's equation in wind speed less than 30m/s) from arbitrary initial values with the method. It was also confirmed that the possibility of the estimation of the sea surface drag coefficient under stormy conditions on the basis of the wave data remotely observed from storms.

Limit periodic motions in some systems with aftereffect under resonance condition

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m+1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.

A Gradient Based Iterative Solutions for Sylvester Tensor Equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.

ANTICIPATING SYNCHRONIZATION OF INTEGER ORDER AND FRACTIONAL ORDER HYPER-CHAOTIC CHEN SYSTEM

Such a problem, how to resolve the problem of long-term unpredictability of chaotic systems, has puzzled researchers in nonlinear research fields for a long time during the last decades. Recently, Voss et al. had proposed a new scheme to research the anticipating synchronization of integral-order nonlinear systems for arbitrary initial values and anticipation time. Can this anticipating synchronization be achieved with hyper-chaotic systems? In this paper, we discussed the application of anticipating synchronization in hyper-chaotic systems. Setting integer order and commensurate fractional order hyper-chaotic Chen systems as our research objects, we carry out the research on anticipating synchronization of above two systems based on analyzing the stability of the error system with the Krasovskill–Lyapunov stability theory. Simulation experiments show anticipating synchronization can be achieved in both integer order and fractional order hyper-chaotic Chen system for arbitrary initial value and arbitrary anticipation time.

Arbitrary initial values and random norm for explosive AR(1) processes generated by stationary errors

This article is concerned with a broad class of explosive AR(1) models. Allowing stationary dependence on the error process, we do not restrict ourselves to independent and identically distributed errors. The model accommodates, as special cases, GARCH errors, AR(1) errors and Gaussian ARMA errors. The error distribution is permitted to be non-normal. To circumvent the effect of initial values, the limit distribution of the least squares estimate using a random norm (rather than a constant norm) is derived. It is shown that the limit distribution using a random norm is free from the initial value provided the error is symmetrically distributed about zero.

Marginal-sum equations and related fixed-point problems

A marginal-sum equation of order p≥2 is a system of nonlinear equations which in turn are linear equations for polynomials of degree p in p variables. Marginal-sum equations typically arise in the construction of a multiplicative tariff in actuarial mathematics.In the present paper we study the existence and the radial uniqueness of solutions of marginal-sum equations and the possibility of computing solutions by iteration. To this end, we first show that the marginal-sum problem is equivalent to several fixed-point problems and we then study these fixed-point problems and the corresponding fixed-point iterations.We show, as a general result, that a marginal-sum equation always has a solution and that the solution cannot be unique. Moreover, for the case p=2 we show that the solution is radially unique and can be computed by fixed-point iteration with respect to a related fixed-point problem and arbitrary initial values. By contrast, for the case p≥3 we present a numerical example in which for certain initial values the fixed-point iteration is cyclic and hence divergent.

General relationship between transit compartments and lifespan models

Transit compartment models (TCM) are important tools in pharmacokinetic/pharmacodynamic (PKPD) modeling. In this work we investigate the relationship between TCMs with arbitrary initial values and lifespan models (LSM) with non-constant past and constant lifespan. We show that the total population in all transit compartments converges to a LSM, if the number of compartments n tends to infinity. The key to obtain this result is to establish a relationship between the initial values of the TCM and the non-constant past of the LSM. We apply the result to two already published PKPD models.

Absolute x-ray energy calibration over a wide energy range using a diffraction-based iterative method

In this paper, we report a method of precise and fast absolute x-ray energy calibration over a wide energy range using an iterative x-ray diffraction based method. Although accurate x-ray energy calibration is indispensable for x-ray energy-sensitive scattering and diffraction experiments, there is still a lack of effective methods to precisely calibrate energy over a wide range, especially when normal transmission monitoring is not an option and complicated micro-focusing optics are fixed in place. It is found that by using an iterative algorithm the x-ray energy is only tied to the relative offset of sample-to-detector distance, which can be readily varied with high precision of the order of 10(-5) -10(-6) spatial resolution using gauge blocks. Even starting with arbitrary initial values of 0.1 Å, 0.3 Å, and 0.4 Å, the iteration process converges to a value within 3.5 eV for 31.122 keV x-rays after three iterations. Different common diffraction standards CeO(2), Au, and Si show an energy deviation of 14 eV. As an application, the proposed method has been applied to determine the energy-sensitive first sharp diffraction peak of network forming GeO(2) glass at high pressure, exhibiting a distinct behavior in the pressure range of 2-4 GPa. Another application presented is pair distribution function measurement using calibrated high-energy x-rays at 82.273 keV. Unlike the traditional x-ray absorption-based calibration method, the proposed approach does not rely on any edges of specific elements, and is applicable to the hard x-ray region where no appropriate absorption edge is available.

Jacobi elliptic functions and the complete solution to the bead on the hoop problem

Jacobi elliptic functions are flexible functions that appear in a variety of problems in physics and engineering. We introduce and describe important features of these functions and present a physical example from classical mechanics where they appear: a bead on a spinning hoop. We determine the complete analytical solution for the motion of a bead on the driven hoop for arbitrary initial conditions and parameter values.

The Riemann–Volterra time-domain technique for waveguides: A case study for elliptic geometry

The method of incomplete separation of variables is applied for solving the wave propagation problems in which the source distribution and the emanated wave are constrained by an elliptic cylinder. Solutions are obtained in the form of expansions in terms of the Mathieu modes, whose completeness makes possible to solve the problem for arbitrary source distribution and initial values of the wavefunction and its time derivative defined within the cylinder. Transient modal amplitudes are found using the Riemann (Riemann–Volterra) method. An important feature of this approach is the straightforward definition of the essentially bounded effective integration areas on the plane spanned by the longitudinal and time coordinates, taking into account the spatiotemporal constraints imposed on the source. For source turned on in a fixed instant, the method is capable to model wave propagation inside the semi-infinite and finite elliptic cylinders provided that the Dirichlet or Neumann boundary conditions are specified on the limiting cross-section(s). Recent techniques of transverse–longitudinal wave decomposition open the prospect of adapting the method to more general cylindrical configurations and to other cases, in which the incomplete separation of variables results in partial differential equations of a known Riemann function (such as the Euler–Poisson–Darboux equation).

Maximum periodic solutions of the Volterra integrodifferential equations in the critical case of a pair of pure imaginary roots

Systems with aftereffect are considered, which are described by integrodifferential equations of the Volterra type in the presence of a small perturbation prescribed by the periodic (or maximum periodic) time function. In the critical case of a pair of pure imaginary roots of a characteristic equation, the question is solved as to existence in the system of maximum periodic motions (i.e., motions tending at the unlimited increase of time to periodic modes) provided that the frequency of the periodic part of disturbance coincides with the natural frequency of a linearized homogeneous system. It is shown that in the analytic case the equations of motion of the system possess a set of the maximum periodic solutions representable by power series in a small parameter specifying a value of the disturbance, and in small arbitrary initial values of uncritical variables of the problem. The conditions of the existence of such solutions are indicated, which are defined by the terms up to the third order of equations.

Active Length Control of Shape Memory Alloy Wires Using Reinforcement Learning

Actively controlled shape memory alloy actuators are useful for a variety of applications that require accurate shape control. For shape memory alloy wires, strain is modulated with temperature, usually by an applied voltage difference across the length. Numerical simulation using reinforcement learning has previously been used for determining the temperature–strain relationship of a shape memory alloy wire and for synthesizing a limited control policy that relates applied temperature to desired strain. However, learning the voltage–strain relationship is of more practical interest in synthesizing feedback control laws for shape memory alloy wires since the control input in practical applications will be an applied voltage that modulates temperature. This article implements a Sarsa-based algorithm for determining a feedback control law in voltage–strain space and validates it experimentally. Experimental results presented in this article demonstrate the ability to control a shape memory alloy specimen from arbitrary initial strains ranging from zero to maximum, including intermediate strains, to an arbitrary intermediate strain. The results also demonstrate theability to control the specimen from similar arbitrary initial values of strain to zero strain. The voltage–strain learning algorithm developed in this article is a promising candidate for synthesizing practical shape memory alloy actuator feedback control laws.

An inverse problem for a wave equation with arbitrary initial values and a finite time of observations

We consider a solution u(p, g, a, b) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T) and ∂itu(p, g, a, b)(x, T0), x ∊ Ω, i = 0, 1, with T0 < T. We prove that if T − T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, …, gN, so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj, bj}1 ⩽ j ⩽ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.

О финальном движении скользящих и вращающихся дисков с однородным кулоновым трением

Abstract We review previous investigations concerning the terminal motion of disks sliding and spinning with uniform dry friction across a horizontal plane. Previous analyses show that a thin circular ring or uniform circular disk of radius R always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed v and angular rotation rate ω , the terminal value of the speed ratio ϵ 0 = v / R ω is always 1.0 for the ring and 0.653 for the uniform disk. In the current study we show that an annular disk of radius ratio η = R 1 / R 2 stops sliding and spinning at the same time, but with a terminal speed ratio dependent on η . For a two-tier disk with lower tier of thickness H 1 and radius R 1 and upper tier of thickness H 2 and radius R 2 , the motion depends on both η and the thickness ratio λ = H 1 / H 2 . While translation and rotation stop simultaneously, their terminal ratio ϵ 0 either vanishes when k > 2 / 3 , is a nonzero constant when 1 / 2 k 2 / 3 , or diverges when k 1 / 2 , where k is the normalized radius of gyration. These three regimes are in agreement with those found by Goyal et al. [S. Goyal, A. Ruina, J. Papadopoulos, Wear 143 (1991) 331] for generic axisymmetric bodies with varying radii of gyration using geometric methods. New experiments with PVC disks sliding on a nylon fabric stretched over a plexiglass plate only partially corroborate the three different types of terminal motions, suggesting more complexity in the description of friction.

The quickest transfer of a spatial dynamic object into a circular domain

The spatial problem of the time-optimal transfer of a point mass by a limited force onto a terminal set in the form of a circle without fixing the final velocity is investigated. The optimal modes of motion are constructed and investigated for arbitrary initial values of the three-dimensional position and velocity vectors using the maximum principle. The governing relations are obtained in the form of fourth-order and eighth-order algebraic equations for the minimum time of motion, which enable the dependence on the initial data to be investigated constructively. The qualitative features of the solution due to a jump discontinuity in the minimum time of motion, which lead to jumps in the control vector, are established. The problem is solved approximately by perturbation methods for the cases of motion close to singular ones. A complete investigation of the control problem for the motion of an object in the plane of a circle and close to it is presented using an original numerical-analytical approach.

An alternative presentation of the method of variation of parameters for higher-order equations

An alternative approach to Lagrange’s method of variation of parameters is presented. Explicit formulas for solutions of arbitrary initial value problems for linear equations of the first, second and third order are provided. These formulas are as simple for practical using as the formula for roots of quadratic equations.