Algorithms for enumeration of single-transition serial sequences

Sets of n-valued single-transition serial sequences consisting of two serial subsequences (an increasing one and a decreasing one) determined by constraints on the number of the series and on their lengths and heights are considered. Enumeration problems for sets of finite sequences in which the difference in height between the neighboring series is not less than some given value are solved. Algorithms that assign smaller numbers to lexicographically lower-order sequences and smaller numbers to lexicographically higher-order sequences are obtained.     

Solving enumeration problems for serial sequences with a constant difference in adjacent series heights

In this paper, the sets of n-valued serial sequences are considered. The structure of such series is defined by constraints on the number of series, the length of series, and the height of series. The problem of recalculation, numeration, and generation is solved for the sets of ascending, descending, and one-transitive sequences with constant differences in the adjacent series heights.     

Enumeration problems for oriented serial sequences

Sets of n-valued m-sequences of a serial structure are considered. In addition to the conventional concepts of the length of a series and of the number of series in a sequence, the concepts of a series height and of a sequence of series heights are introduced. The structure of sequences, which will be referred to as oriented, is determined from restrictions on the number and length of series and on the order of sequencing of series of various heights. A general approach to solving enumeration problems for sets of such sequences is proposed. The approach is based on formulas for the number of arrangements of elements in cells and the power of a set of height sequences. Exact solutions are derived for some restrictions, which are important for applications.     

The unification of certain enumeration problems for sequences

A unified treatment is presented for certain enumeration problems involving sequences over a finite set. The problems concern permutations, derangements, partitions and compositions, and include the derangement, Smirnov and Simon Newcomb problems and their various generalizations. A single theorem includes all of these problems as special cases.     

Enumeration problems of sets of increasing and decreasing n-valued serial sequences with double-ended constraints on series heights

Enumeration problems for n-valued serial sequences are considered. Sets of increasing and decreasing sequences whose structure is specified by constraints on lengths of series and on the difference in heights of the neighboring series in the case when this difference lies between δ ₁ and δ ₂ are examined. Formulas for powers of these sets and algorithms for the direct and reverse numerations (assigning smaller numbers to lexicographically lower order sequences or smaller numbers to lexicographically higher order sequences) are obtained.     

Truncation method for countable-point boundary-value problems in the space of bounded number sequences

We consider possible methods for the reduction of a countable-point nonlinear boundary-value problem with nonlinear boundary condition on a segment to a finite-dimensional multipoint problem constructed on the basis of the original problem by the truncation method. The results obtained are illustrated by examples.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1203–1230, September, 2004.     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Extremal theorems for Riemann spaces with curvature bounded above. I

Siberian Mathematical Journal -     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002     

The enumeration of sequences with respect to structures on a bipartition

We give a method for enumerating sequences over a finite alphabet with respect to certain maximal configurations. The required generating functions are obtained as solutions of systems of linear equations. The method utilizes a combinatorial decomposition of sequences into maximal sub-configurations.     

TWO FEEDBACK PROBLEMS FOR GRAPHS WITH BOUNDED TREE-WIDTH

Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree decomposition. In this paper,it is discussed how to solve the minimum feedback vertex set problem and the minimum vertex feedback edge set problem efficiently by using dynamic programming on a tree-decomposition.     

Absolute convergence of Fourier series with respect to bounded systems

In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order log N, where N^?1 is the approximation step.     

Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above

The hypothesis of local compactness of the target is removed from an earlier result about interior Hölder continuity of locally energy minimizing maps ? from a Riemannian polyhedron (X, g) to a suitable ball B of radius R <  π/2 (best possible) in a geodesic space with curvature ≤ 1. Furthermore, the variational Dirichlet problem for harmonic maps from an open set \(\Omega \Subset X\) to B is shown to be uniquely solvable, and the solution is continuous up to the boundary ?Ω at any regular point of ?Ω at which the prescribed boundary map is continuous.     

A central limit theorem for trigonometric series with bounded gaps

In this paper it is proved that there exists a sequence {n _k } of integers with 1 ?? n _k+1 ? n _k ?? 5 such that the distribution of ${(\cos 2\pi n_1 x + \dots + \cos 2\pi n_{N}) / \sqrt N}$ on ([?0, 1?], B, dx) converges to a Gaussian distribution. It gives an affirmative answer to the long standing problem on lacunary trigonometric series which ask the existence of series with bounded gaps satisfying a central limit theorem.     

Modified quasilinearization algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation R is negative, the decrease inR is guaranteed if is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value.Here, the state inequality constraint is handled in a direct manner. A predetermined number and sequence of subarcs is assumed and, for the time interval for which the trajectory of the system lies on the state boundary, the control is determined so that the state boundary is satisfied. The state boundary and the entrance conditions are assumed to be linear inx and , and the modified quasilinearization algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration and along all of the subarcs composing the trajectory.At first glance, the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory. Actually, this is not the case. The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variables.In order to start the algorithm, some nominal functionsx(t),u(t), and nominal multipliers (t), (t), , must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to (t), (t), , . Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Dr. R. R. Iyer and Mr. A. K. Aggarwal for helpful discussions as well as analytical and numerical assistance. This paper is a condensation of the investigations described in Refs. 1–2.     

Sequential gradient-restoration algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functionsx(t),u(t), obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable.Here, the state inequality constraint is handled in a direct manner. A predetermined number and sequence of subarcs is assumed and, for the time interval for which the trajectory of the system lies on the state boundary, the control is determined so that the state boundary is satisfied. The state boundary and the entrance conditions are assumed to be linear inx and , and the sequential gradient-restoration algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration of the gradient phase and the restoration phase along all of the subarcs composing the trajectory.At first glance, the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory. Actually, this is not the case. The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variables.To facilitate the numerical solution on digital computers, the actual time is replaced by the normalized timet, defined in such a way that each of the subarcs composing the extremal arc has a normalized time length t=1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times ₁, ₂, at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal boundary is reached are regarded to be components of the parameter being optimized.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.This paper is based in part on a portion of the dissertation which the first author submitted in partial fulfillment of the requirements for the PhD Degree at the Air Force Institute of Technology, Wright-Patterson AFB, Ohio. This research was supported in part by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Professor H. Y. Huang, Dr. R. R. Iyer, Dr. J. N. Damoulakis, Mr. A. Esterle, and Mr. J. R. Cloutier for helpful discussions as well as analytical and numerical assistance. This paper is a condensation of the investigations reported in Refs. 1–2.     

A maximum principle for time-lag control problems with bounded state

A maximum principle is obtained for control problems involving a constant time lag τ in both the control and state variables. The problem considered is that of minimizing $$I(x) = \int_{t^0 }^{t^1 } {L (t,x(t), x(t - \tau ), u(t), u(t - \tau )) dt} $$ subject to the constraints 1 $$\begin{gathered} \dot x(t) = f(t,x(t),x(t - \tau ),u(t),u(t - \tau )), \hfill \\ x(t) = \phi (t), u(t) = \eta (t), t^0 - \tau \leqslant t \leqslant t^0 , \hfill \\ \end{gathered} $$ 1 $$\psi _\alpha (t,x(t),x(t - \tau )) \leqslant 0,\alpha = 1, \ldots ,m,$$ 1 $$x^i (t^1 ) = X^i ,i = 1, \ldots ,n$$ . The results are obtained using the method of Hestenes.     

Necessary conditions for problems with higher derivative bounded state variables

This paper combines the separate works of two authors. Tan proves a set of necessary conditions for a control problem with second-order state inequality constraints (see Ref. 1). Russak proves necessary conditions for an extended version of that problem. Specifically, the extended version augments the original problem by including state equality constraints, differential and isopermetric equality and inequality constraints, and endpoint constraints. In addition, Russak (i) relaxes the solvability assumption on the state constraints, (ii) extends the maximum principle to a larger set, (iii) obtains modified forms of the relationH =H _t and of the transversality relation usually obtained in problems of this type, and (iv) proves a condition concerning (t ¹), the derivative of the multiplier functions at the final time.Russak's work was supported by a NPS Foundation Grant.Tan is indebted to his thesis advisor, Professor M. R. Hestenes, for suggesting the topic and for his help and guidance in the development of his work. Tan's work was supported by the Army Research Office, Contract No. DA-ARO-D-31-124-71-G18.     

Solutions to boundary Cauchy problems with infinite delay

In this paper, we investigate the existence, uniqueness of solutions to boundary Cauchy equations with infinite delay, which are more general than the previous studies. The conclusions are applied to an age dependent population with delay in the birth process.