Multicomponent steady flows in porous media with phase transitions

Steady multiphase flow of a multicomponent mixture in a porous medium with phase transitions is considered. It is shown that, in a wide class of cases, the thermodynamic problem separates from the filtration problem and the latter is integrated in quadratures. The class of exact solutions which has been found is used to interpret indicator curves. Solutions are presented in an analytic form for systems of the “gas-condensate” and “oil-gas” type.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Recognizing polygonal parts from width measurements

Automatic recognition of parts is an important problem in many industrial applications. One model of the problem is: given a finite set of polygonal parts, use a set of “width” measurements taken by a parallel-jaw gripper to determine which part is present. We study the problem of computing efficient strategies (“grasp plans”), with the goal to minimize the number of measurements necessary in the worst case. We show that finding a minimum length grasp plan is -hard, and give a polynomial time approximation algorithm that is simple and produces a solution that is within a log factor from optimal.     

On the Dissection of Rectangles into Right-Angled Isosceles Triangles

We consider the problem of dissecting a rectangle or a square into unequal right-angled isosceles triangles. This is regarded as a generalization of the well-known and much-solved problem of dissecting such figures into unequal squares. There is an analogous “electrical” theory but it is based on digraphs instead of graphs and has an appropriate modification of Kirchhoff's first law. The operation of reversing all edges in the digraph is found to be of great help in the construction of “perfect” dissected squares.     

Exploiting structure in quantified formulas

We study the computational problem “find the value of the quantified formula obtained by quantifying the variables in a sum of terms.” The “sum” can be based on any commutative monoid, the “quantifiers” need only satisfy two simple conditions, and the variables can have any finite domain. This problem is a generalization of the problem “given a sum-of-products of terms, find the value of the sum” studied in [R.E. Stearns and H.B. Hunt III, SIAM J. Comput. 25 (1996) 448–476]. A data structure called a “structure tree” is defined which displays information about “subproblems” that can be solved independently during the process of evaluating the formula. Some formulas have “good” structure trees which enable certain generic algorithms to evaluate the formulas in significantly less time than by brute force evaluation. By “generic algorithm,” we mean an algorithm constructed from uninterpreted function symbols, quantifier symbols, and monoid operations. The algebraic nature of the model facilitates a formal treatment of “local reductions” based on the “local replacement” of terms. Such local reductions “preserve formula structure” in the sense that structure trees with nice properties transform into structure trees with similar properties. These local reductions can also be used to transform hierarchical specified problems with useful structure into hierarchically specified problems having similar structure.     

The difference in a problem of rations from the Rhind mathematical papyrus

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”     

A comparison of procedures for estimating the parent probability distribution from a given set of fractiles

On one hand, eliciting subjective probabilities (fractiles) is a well-established procedure. On the other hand, knowledge of a subjective variable's central moments or distribution function is often assumed. However, the problem of converting elicited fractiles into the required moments or distribution function has been largely ignored. We show that this conversion problem is far from trivial, and that the most commonly used conversion procedures often produce huge errors. Alternative procedures are proposed; the “Tocher's curve” and “linear function of fractiles” methods are shown to be much more accurate than the commonly used procedures.     

Multi-step quasi-Newton methods for optimization

Quasi-Newton methods update, at each iteration, the existing Hessian approximation (or its inverse) by means of data deriving from the step just completed. We show how “multi-step” methods (employing, in addition, data from previous iterations) may be constructed by means of interpolating polynomials, leading to a generalization of the “secant” (or “quasi-Newton”) equation. The issue of positive-definiteness in the Hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original “single-step” methods. The results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach (by comparison with “single-step” methods), particularly as the dimension of the problem increases.     

Algorithmic Aspects of Tree Amalgamation

The amalgamation of leaf-labeled trees into a single (super)tree that “displays” each of the input trees is an important problem in classification. We discuss various approaches to this problem and show that a simple and well-known polynomial-time algorithm can be used to solve this problem whenever the input set of trees contains a minimum size subset that uniquely determines the supertree. Our results exploit a recently established combinatorial property concerning the structure of such collections of trees.     

Interaction of coupled modes accompanying non-linear flexural vibrations of a circular ring

An analytic investigation of the problem of the non-linear free flexural vibrations of a circular ring taking account of the coupled modes is presented. A system of amplitude-frequency modulation equations is obtained by the method of many scales. The integral of this system enables one to construct an “amplitude-phase pattern” which characterizes the possible dynamic modes in the case of arbitrary initial conditions. It is shown that an energy threshold exists and, when this is exceeded, the appearance of a travelling wave and pronounced amplitude-frequency modulation are possible. The threshold value of the energy depends on the “detuning” of the frequencies of the coupled modes due to the presence of initial imperfections. A general solution of the problem is obtained in elliptic Jacobi functions.     

Landmarks in graphs

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.

Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two.     

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

On the theory of fugoid motions

In 1891 Zhukovslii in his paper “On soaring of birds” [1] solved the problem of the motion of a body of high lift — drag ratio in an atmosphere of constant density. In [2] this problem was considered in greater detail, but the basic assumption of a constant density was made here as well. There have recently appeared numerous papers concerning the analytical solution of the problem of entry into the atmosphere with orbital and escape velocities [3 to 5]. But these studies were concerned primarily with the problems of ballistic entry and entry with low lift — drag ratio. In considering oscillatory states, the authors limited their treatment to small angles between the trajectory and local horizon. In the present paper we consider the problem without imposing any limitations on the slope of the trajectory or initial velocity. The case examined will be that of a hypothetical glider spacecraft of sufficiently high lift — drag ratio. It is interesting to note that the solution of this problem reduces to the solution of Zhukovskii's problem, but for an atmosphere of variable density. The associated trajectories are termed “fugoid”. All of our assumptions about the parameters of such a glider are of a particular hypothetical character.     

Linear optimization over permutation groups

For a permutation group given by a set of generators, the problem of finding “special” group members is NP-hard in many cases, e.g., this is true for the problem of finding a permutation with a minimum number of fixed points or a permutation with a minimal Hamming distance from a given permutation. Many of these problems can be modeled as linear optimization problems over permutation groups. We develop a polyhedral approach to this general problem and derive an exact and practically fast algorithm based on the branch & cut-technique.     

Discrete time Galerkin approximations to the nonlinear filtering solution

This paper concerns discrete time Galerkin approximations to the solution of the filtering problem for diffusions. Two families of schemes approximating the unnormalized conditional density, respectively, in an “average” and in a “pathwise” sense, are presented. L² error estimates are derived and it is shown that the rate of convergence is linear in the time increment or linear in the modulus of continuity of the sample path.     

Bounded realization of asymptotically attainable elements

Problems of the structure of asymptotically attainable elements (AAE) generated by the action of “control functions” that satisfy generally non-convex constraints of a functional character to a high degree of accuracy are considered. No resource type conditions are assumed, which leads to an “unbounded” formulation of the problem concerned with the asymptotic behaviour of attainability domains and their abstract analogues. Necessary and sufficient conditions for an exhaustive realization of the AAE in the class of integrally bounded approximate solutions are established in terms of the generalized problem in the class of finitely additive vector-valued measures.     

Prescribing curvature problem of Bakry-Émery Ricci tensor

We consider the problem of deforming a metric in its conformal class on a closed manifold, such that the k-curvature defined by the Bakry-mery Ricci tensor is a constant. We show its solvability on the manifold, provided that the initial Bakry-mery Ricci tensor belongs to a negative cone. Moveover, the Monge-Ampère type equation with respect to the Bakry-mery Ricci tensor is also considered.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.     

Implicit algorithms for solving the Cauchy problem for the equations of the dynamics of mechanical systems

It is shown that in the numerical solution of the Cauchy problem for systems of second-order ordinary differential equations, when solved for the highest-order derivative, it is possible to construct simple and economical implicit computational algorithms for step-by-step integration without using laborious iterative procedures based on processes of the Newton-Raphson iterative type. The initial problem must first be transformed to a new argument — the length of its integral curve. Such a transformation is carried out using an equation relating the initial parameter of the problem to the length of the integral curve. The linear acceleration method is used as an example to demonstrate the procedure of constructing an implicit algorithm using simple iterations for the numerical solution of the transformed Cauchy problem. Propositions concerning the computational properties of the iterative process are formulated and proved. Explicit estimates are given for an integration stepsize that guarantees the convergence of the simple iterations. The efficacy of the proposed procedure is demonstrated by the numerical solution of three problems. A comparative analysis is carried out of the numerical solutions obtained with and without parametrization of the initial problems in these three settings. As a qualitative test the problem of the celestial mechanics of the “Pleiades” is considered. The second example is devoted to modelling the non-linear dynamics of an elastic flexible rod fixed at one end as a cantilever and coiled in its initial (static) state into a ring by a bending moment. The third example demonstrates the numerical solution of the problem of the “unfolding” of a mechanical system consisting of three flexible rods with given control input.     

Geometric quadrisection in linear time, with application to VLSI placement

We consider the following problem: given a set of points in the plane, each with a weight, and capacities of the four quadrants, assign each point to one of the quadrants such that the total weight of points assigned to a quadrant does not exceed its capacity, and the total distance is minimized.

This problem is most important in placement of VLSI circuits and is likely to have other applications. It is NP-hard, but the fractional relaxation always has an optimal solution which is “almost” integral. Hence for large instances, it suffices to solve the fractional relaxation. The main result of this paper is a linear-time algorithm for this relaxation. It is based on a structure theorem describing optimal solutions by so-called “American maps” and makes sophisticated use of binary search techniques and weighted median computations.

This algorithm is a main subroutine of a VLSI placement tool that is used for the design of many of the most complex chips.