Free Endpoint Research Articles

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ℂ+ and such that ImQ(z ≥ 0 and ImzQ(z) ≥ 0 ifz e ℂ+. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf +λ fdm = 0; f′(0−) = 0. In this note we show that the set of functionsQ which are holomorphic in ℂ+ and such that the kernel $$\frac{{Q(z) - \overline {Q(\zeta )} }}{{z - \bar \zeta }}$$ hasκ negative squares of ℂ+ and ImzQ(z) ≥ 0 ifz e ℂ+ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf′ +λf dm +λ 2 fdD = 0 on [0,L),f′(0−) = 0. Hereκ is the number of pointsx whereD increases or 0 >m(x + 0) −m(x − 0) ≥ −∞; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.

Polymer Brushes: From Self-Consistent Field Theory to Classical Theory.

We consider planar brushes formed by end-grafted polymers with moderate to strong excluded-volume interactions. We first rederive the mean-field theory and solve the resulting self-consistent equations numerically. In the continuum limit, the results depend sensitively on a single parameter, beta, whose square is the ratio of the scaling prediction for the brush height to the unperturbed polymer radius of gyration, and which measures therefore the degree to which the polymers are stretched. For large values of beta, the density profile is close to parabolic, as predicted by the infinite-stretching theory of Zhulina et al. and of Milner et al. As beta decreases, the profile deviates strongly from a parabolic one. By calculating the most probable paths and comparing their contribution to various properties with those obtained from the full self-consistent theory, we determine the effect of the fluctuations about such paths. At large values of beta, these effects are very small everywhere. As beta decreases, fluctuation effects on the density profile become increasingly important near the grafting surface, but remain small far from it. For all values of beta, we find that polymer paths which begin far from the grafting surface are strongly, and almost uniformly, stretched throughout their length, including their free end points. Paths which begin close to the grafting surface are also stretched, but they initially move away from the grafting surface before reaching a maximum height and then returning to it. The classical theory is then derived from the self-consistent field equations by retaining, for each end point location, only that single trajectory which minimizes the free energy of the system. This free energy contains an entropy, of relative weight beta-1, which arises from the distribution of end points. Even for brushes which are only moderately stretched, the results of the classical theory for the brush profile and polymer end point distribution agree well with those of the full self-consistent theory except near the grafting surface itself. There the density profile as calculated in the self-consistent theory shows a characteristic decrease which is not captured by the classical theory. However it does capture the fact that the individual polymer paths are stretched in general throughout their length, including the end points, and yields nonmonotonic paths for polymers whose end points are close to the grafting surface. In addition, it reproduces extremely well the form of the density distribution far from the grafting surface, which is essentially Gaussian. This results from the fact that the stretching energy dominates the interaction energy of those polymers which extend far from the grafting surface, so that their behavior is nearly ideal. In the limit of infinite stretching, beta --> infinity, the theory reduces to that of Zhulina et al. and of Milner et al.

Assessment of endpoints for clinical trials for localized prostate cancer

Objectives The AUA Practice Guidelines Panel convened to address the issue of appropriate endpoints for assessment of treatment modalities for localized carcinoma of the prostate. Methods A review of the literature and the design of existing clinical trials produced a consensus, which was presented to and critiqued by the members of the general conference. Results The pitfalls associated with identification of local failure endpoints were discussed, and the more accurate endpoints of freedom from metastatic progression and overall survival were recognized. The strict definition that must be fulfilled for intermediate endpoints to become surrogates for metastasis free and/or survival endpoints was stressed. For more efficient and rapid conduct of future clinical trials, the urgent need to validate such surrogate endpoints by evaluation in randomized control trials is obvious. PSA, while an indicator of disease activity and a critical marker for estimating disease progression or regression in response to therapy, is not a surrogate for metastasis free or overall survival. Conclusion Until surrogate endpoints are validated, the committee has evaluated the endpoints in current use, reviewed their limitations, and stressed the importance of quality-of-life assessment together with the traditional endpoint assessment.

Analytical computation of differential equations involved in dynamical nonlinear optimal problems

We present two programs, written in Reduce, for non-constrained free endpoint nonlinear dynamical optimal problems, in fixed time, in closed loop and in open loop, which compute analytically the optimal feedback laws in terms of differential equations. The open loop case leads to ordinary differential equations (ODEs) and the closed loop leads to partial differential equations (PDEs). In the case of closed loop problems the program uses nonstandard Reduce programming for the declarations of dependencies of u and its partial derivatives. Algorithms are presented for the open loop and closed loop cases and the same example is computed in both these cases.

Epi-derivatives of integral functionals with applications

We study first- and second-order epi-differentiability for integral functionals defined on L 2 [ 0 , T ] {L^2}[0,T] , and apply the results to obtain first- and second-order necessary conditions for optimality in free endpoint control problems.

The drift velocity in reptation models for electrophoresis

The drift velocity (diffusion constant) in the Rubinstein–Duke model with periodic boundary condition is calculated analytically to lowest order in the applied electric field and numerically for the whole scaling regime. The model is modified by restricting the polymer-storing capacity of the cells and for this case again the diffusion constant is determined. The periodic boundary condition decouples the different tube configurations. Thus, with the process of tube renewal removed, only the diffusion of length defects through the tube remains. The effect of the periodic boundary condition on the value of the diffusion constant and the behavior of the scaling function is discussed on the basis of numerical results for both models with free endpoint motion. The results strongly suggest that to linear order in the field the drift velocity is unaffected by the process of tube renewal, i.e., is only determined by the transport of reptons along the tube.

On the computation of the Minkowski dimension using morphological operations

SUMMARYThe Minkowski dimension as a complexity measure, and its relation to the other dimensions in the case of self‐similar fractal objects, is discussed. An implementation of the Minkowski dimension, using morphological operations on a square grid, is presented. The effect of resolution, orientation and the symmetry of the structuring element used on the estimation of the Minkowski dimension is studied. The corrections required for free end points are presented.

Free End-Point Linear-Quadratic Control Subject to Implicit Continuous-Time Systems: Necessary and Sufficient Conditions for Solvability

Abstract For an implicit continuous-time system with arbitrary constant coefficients we derive necessary and sufficient conditions for solvability of the associated free end-point linear-quadratic optimal control problem. In particular, this problem turns out to be solvable if and only if the underlying system is output stabilizable, as is the case for a standard system

Wilson-Polyakov loops for critical strings and superstrings at finite temperature

An open string with end-points fixed at spatial separation L is a string theory analogue of the static quark-antiquark system in quenched QCD. Following a review of the quantum mechanics of this system in critical bosonic string theory the partition function at finite β (the inverse temperature) for fixed end-point open strings is discussed. This is related by a conformal transformation (“world-sheet duality”) to the correlation function of two closed strings fixed at distinct spatial points (a string theory analogue of two Wilson-Polyakov loops). Temperature duality ( β → β′ = 4 π 2/ β) relates this correlation function, in turn, to the finite-temperature Green function for a closed strong propagating between initial and final states that are at distinct (euclidean) space-time points. In addition, spatial duality relates the fixed end-point open string to the familiar open string with free end-points. A generalization to fixed end-points superstrings is suggested, in which the superalgebra may be viewed as the spatial dual of the usual open-string superalgebra. At zero temperature world-sheet duality relates the partition function of supersymmetric fixed end-point open strings to the correlation function of point-like closed-string states. These couple to combinations of the scalar and pseudoscalar states of a type-2b superstring superfield. At finite temperature supersymmetry is broken and this correlation function involves the propagation of non-supersymmetric states with non-zero winding numbers (which formally include a tachyon at temperatures above the Hagedorn transition). Temperature duality again relates the partition function to the finite-temperature Green function describing the propagator for point-like closed-string states of the dual theory, in which supersymmetry is broken. The singularity that arises in the critical bosonic theory as L is reduced below L = 2 π√ α′ is absent in the superstring and the static potential is well defined for all L.

Join- and-cut algorithm for self-avoiding walks with variable length and free endpoints

We introduce a new Monte Carlo algorithm for generating self-avoiding walks of variable length and free endpoints. The algorithm works in the unorthodox ensemble consisting of all pairs of SAWs such that the total number of stepsNtot in the two walks is fixed. The elementary moves of the algorithm are fixed-N (e.g., pivot) moves on the individual walks, and a novel “join- and-cut” move that concatenates the two walks and then cuts them at a random location. We analyze the dynamic critical behavior of the new algorithm, using a combination of rigorous, heuristic, and numerical methods. In two dimensions the autocorrelation time in CPU units grows as N≈1.5, and the behavior improves in higher dimensions. This algorithm allows high-precision estimation of the critical exponentγ.

Non-local Monte Carlo algorithm for self-avoiding walks with variable length and free endpoints

We discuss the dynamic critical behaviour of a Monte Carlo algorithm for self-avoiding walks of variable length and free endpoints. The algorithm works in the unorthodox ensemble consisting of all pairs of SAWs such that the total number of steps N tot in the two walks is fixed. In two dimensions the autocorrelation time in CPU units grows as N ≈1.5, and the behaviour improves in higher dimensions.

Boundary-value technique to solve linear state regulator problems

In this paper, a method is proposed to solve free endpoint optimal control problems with linear state and quadratic cost. After translating the problem into a two-point boundary-value problem (TPBVP) that arises from the optimization of the Hamiltonian, two pointst 1 andt 2,t 1<t 2, belonging to the given interval of integration [t 0,t f], are chosen and conditions are derived at these points appropriately. Then, letting τ=(t−t 0)/∈, σ=(t f−t)/∈, the τ-scaled, the original, and the σ-scaled TPBVP's are solved on the intervals [t 0,t 1], [t 1,t 2], and [t 2,t f] respectively as boundary-value problems.

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one “effectively independent” sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

Positive Solution of a Problem of Emden–Fowler Type with a Free Boundary

A sufficient condition is given for the existence of a solution for a generalized Emden–Fowler problem with a free end point. In the special case of the Emden–Fowler equation, this condition is als...

A new approach to nonlinear optimal feedback law

For a fixed-time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations. They are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations. These equations, involving Lie brackets of vector fields, are derived from the Maximum Principle. An application of this approach to robotics is given.

The Liouville–Bäcklund transformation for the two-dimensional SU(N) Toda lattice

We describe the Liouville–Bäcklund transformation for the two-dimensional SU(N) Toda lattice with free end points. Integration of this transformation gives us the general solution of this equation, which depends on the N arbitrary solutions of the two-dimensional Laplace equation.

Lie Brackets and Optimal Nonlinear Feedback Regulation

We consider fixed-time, free endpoint optimal control problems, and show that the optimal feedback loops satisfy systems of quasi-linear partial differential equations. In a Bolza problem, where the dimensions of the state and of the control vector are equal, this system is of first order. We arrive at the partial differential equations by eliminating the adjoint vector from a set of conditions involving successive derivatives of the pseudo-Hamiltonians and therefore Lie brackets of vector fields. These conditions may be recovered from the Taylor expansion of the first order Volterra kernels. We show that such kernels are intimately related to the usual Hamiltonian formalism. Our partial differential equations should provide new ways for computing optimal feedback loops. Some very encouraging examples may be found in a joint paper with H. Bourdache-Siguerdidjane (Proc. 6th Internat. Conf. Analysis Optimiz. Systems, Nice, June 1984).

Theory of boundary effects on sine-Gordon solitons

We examine the properties of solutions to the sine-Gordon equation in the presence of various boundary conditions. Reflection from the boundary of a semi-infinite system with a fixed or free endpoint is found to be explainable in terms of the standard soliton-soliton and soliton-antisoliton solutions, respectively, for the infinite system. We also consider both nonlinear standing-wave and solitonic solutions to the sine-Gordon equation on a system of finite length L. Through the use of the application by Costabile et al. of the separation of variables ansatz due to Lamb, analytic expressions in terms of Jacobi elliptic functions are found for these two types of solutions, which assume the forms appropriate to the infinite and semi-infinite system, respectively, as L→∞. Concentrating on the solitonic solution, we examine its symmetry properties, relations among its characteristic parameters, and give plots of its waveform in several cases. The effect of the boundaries on this solution is such that the oscillation of the soliton center closely resembles that of a particle in a symmetric potential, whose shape is determined straightforwardly from the time dependence of the soliton position and of the potential energy of the system.

Optimal Open-Loop Maneuver Profiles for Flexible Spacecraft

Using the Calculus of Variations, optimal slewing profiles minimizing a structural e xcitation criterion are e stablished for a dynamically s imple spacecraft maneuvering between two quiescent states. Two problem types are considered. In the free end point problem, the structural deformation and its time derivative are unconstrained at maneuver's end. For the constrained end point problem, these variables a re r equired to vanish, which necessarily degrades the excitation criterion. Several figures are presented that illustrate both the n ature and the limitations inherent in maneuvering the spacecraft from one attitude state to another. For a given maneuver amplitude, en, the key parameter influencing structural e xcitation is the product of the maneuver time, Ta, and the lowest significant structural frequency, w. It is shown that when wTa 10, however, this penalty is fairly minor, and some reasonable control of terminal conditions is then practical. Thus it is generally d esirable that all maneuver times meet this criterion. When this is the case, it is possible to derive a normalized suboptimal slewing profile, F(x), applicable to a1 1 maneuvers. Given 0 and Ta, the commanded maneuver rate becomes e( t) = eO F(t/Ta)/Ta. Only a minor computational and memory burden is therefore necessary to perform almost optimal re-orientations.

A perturbation method for control systems with multiple switch points

We consider a system arising from an application of the Maximum Principle to a free endpoint trajectory optimization problem arising in control. The system involves a small parameter, and has the property that the optimal control associated with the reduced problem ( \epsilon=0 ) moves on and off the boundary of the control region a finite number of times. We show how a technique involving a nonlinear chgnge of independent variables can be used to obtain uniformly valid parameter expansions for the solution of the full problem for e small, and we establish conditions on the Hamiltonian under which this procedure may be carried out.