Online Ramsey games for triangles in random graphs

In the online F-avoidance edge-coloring game with r colors, a graph on n vertices is generated by randomly adding a new edge at each stage. The player must color each new edge as it appears; the goal is to avoid a monochromatic copy of F. Let N₀(F,r,n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he/she loses. Even when F=K₃, the order of magnitude of N₀(F,r,n) is unknown for r≥3. In particular, the only known upper bound is the threshold function for the number of edges in the offline version of the problem, in which an entire random graph on n vertices with m edges is presented to the player to be r edge-colored. We improve the upper bound for the online triangle-avoidance game with r colors, providing the first result that separates the online threshold function from the offline bound for r≥3. This supports a conjecture of Marciniszyn, Spöhel, and Steger that the known lower bound is tight for cliques and cycles for all r.     

On maximum length convolutional codes under a trellis complexity constraint

We look at convolutional codes with maximum possible code length for prescribed redundancy, conditioned on constraints on the free distance and on the bit-oriented trellis state complexity. Rate (n−1)/n codes have been studied to some extent in the literature, but more general rates have not been studied much. In this work we consider convolutional codes of rate (n−r)/n, r⩾1. Explicit construction techniques for free distance d_free=3 and 4 codes are described. For codes with r=2, an efficient exhaustive search algorithm is outlined. For the more general case with r⩾2, a heuristic approach is suggested. Several new codes were found for r=1 and in particular for r=2 and 3. Compared to previously known codes of similar free distance and complexity constraints, the new codes have either strictly higher rate, or the same rate but smaller low distance multiplicities.     

Sequential gradient-restoration algorithm for optimal control problems with nondifferential constraints

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π 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), π 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. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, and the stepsize of the restoration phase by a one-dimensional search on the constraint errorP. If α _g is the gradient stepsize and α _r is the restoration stepsize, the gradient corrections are ofO(α _g ) and the restoration corrections are ofO(α _r α _g ²). Therefore, for α _g sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalÎ at the end of any complete gradient-restoration cycle is smaller than the functionalI at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time ? is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 4 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0 ≤t ≤ 1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control, (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.     

Finding K shortest looping paths with waiting time in a time–window network

A time-constrained shortest path problem is a shortest path problem including time constraints that are commonly modeled by the form of time windows. Finding K shortest paths are suitable for the problem associated with constraints that are difficult to define or optimize simultaneously. Depending on the types of constraints, these K paths are generally classified into either simple paths or looping paths. In the presence of time–window constraints, waiting time occurs but is largely ignored. Given a network with such constraints, the contribution of this paper is to develop a polynomial time algorithm that finds the first K shortest looping paths including waiting time. The time complexity of the algorithm is O(rK²|V₁|³), where r is the number of different windows of a node and |V₁| is the number of nodes in the original network.     

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l₁ exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l₁ exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l₁ exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.     

Synthesis of minimum-order robust inverters

In the present paper, we consider the inversion problem for dynamical systems, that is, the problem of reconstruction of the unknown input signal ξ(t) of a given system on the basis of known information (about either the complete phase vector or a measurable output of the system). An auxiliary dynamical system forming the desired estimate of the signal ξ(t) is called an inverter.In earlier papers of the authors, attention was mainly paid to the possibility of inversion of a dynamical system in different cases in principle. In this relation, a model of dynamical systems with some stabilizing control was used as an inverter for the solution of the problem; moreover, this control was often designed with the use of an additional dynamical system, an observer of the phase vector of the original system or the system in deviations. Thus, a dynamical system whose dimension either coincides with the dimension of the original system or exceeds it was considered as an inverter.In the solution of practical problems, it is often required to synthesize inverters of minimal order. (This requirement is related to constraints on the complexity, cost, and operation speed of automated control systems.) In the present paper, we consider the problem on the possible reduction of the order of the inverter in various cases and the problem on the construction of inverters of minimal order.     

Approximation algorithms and hardness results for the clique packing problem

For a fixed family F of graphs, an F-packing in a graph G is a set of pairwise vertex-disjoint subgraphs of G, each isomorphic to an element of F. Finding an F-packing that maximizes the number of covered edges is a natural generalization of the maximum matching problem, which is just F={K₂}. In this paper we provide new approximation algorithms and hardness results for the K_r-packing problem where K_r={K₂,K₃,…,K_r}.We show that already for r=3 the K_r-packing problem is APX-complete, and, in fact, we show that it remains so even for graphs with maximum degree 4. On the positive side, we give an approximation algorithm with approximation ratio at most 2 for every fixed r. For r=3,4,5 we obtain better approximations. For r=3 we obtain a simple3/2-approximation, achieving a known ratio that follows from a more involved algorithm of Halldórsson. For r=4, we obtain a (3/2+?)-approximation, and for r=5 we obtain a (25/14+?)-approximation.     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

Lower Bounds for Scheduling on Identical Parallel Machines with Heads and Tails

In this paper, we investigate new lower bounds for the P|r _j ,q _j |C _max scheduling problem. A new bin packing based lower bound, as well as several new lifting procedures are derived for this strongly NP -hard problem. Extensive numerical experiments show that the proposed lower bounds consistently outperform the best existing ones.     

A numerical method for solving the Dirichlet problem for the wave equation

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem Aq = f with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator A in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem Aq = f on a uniform grid is reduced to a system of linear algebraic equations A _ll q = F. Using the singular value decomposition, we find singular values of the matrix A _ll and develop a numerical algorithm for constructing the r-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated r-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.     

The bounds of restricted isometry constants for low rank matrices recovery

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p ? 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ _4r ^A < 0.558 and δ _3r ^A < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ _2r ^A < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ _2r ^A < 0.4931 and δ _2r ^A < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ _2r ^A > 1/√2 or δ _r ^A > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ _2r ^A and δ _r ^A . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.     

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

This paper is concerned with the stabilization problem of Timoshenko beam in the presence of linear dissipative boundary feedback controls. Using C₀-semigroups theory we establish the existence and the uniqueness of solution of the proposed closed loop system. In order to consider the asymptotic behavior of the closed loop system, we first discuss the existence of nonzero solution of a closely related boundary value problem. Then we derive various necessary and sufficient conditions for the system to be asymptotically stable. Finally, we prove the equivalence between the exponential stability and the asymptotic stability for the closed loop system.     

On a generalization of Kaplansky's game

In this paper, games of the following general kind are studied: Two players move alternately by selecting unselected integer coordinate points in the plane. On each move, the first player selects exactly r points and the second player selects exactly one point. The first player wins if he can select p points on a line having none of his opponent's points before his opponent selects q points on a line having none of his own. If this latter eventuality occurs first, the second player wins. It is shown that if p ? c(r)q, then the second player can always win.     

Non-linear multiparameter eigenvalue problems for ordinary differential equations

In this paper we consider the non-linear multiparameter eigenvalue problem in ordinary differential equations: y″_r(x_r) + q_r(x_r)y_r(x_r) + M_r(y_r) + ∑_{s = 1}ⁿμ_s(a_rs(x_r + P_rs) y_r(x_r) = 0, r = 1,…, n, where the non-linear functions M_rand P_rs will be required to satisfy certain Fréchet differentiability conditions. We study bifurcation from the simple eigenvalues of the corresponding linear problem and state a completeness theorem for the eigenfunctions of these systems.     

Integer-friendly formulations for the r-separation problem

In this paper, we consider different formulations for the r-separation problem, where the objective is to choose as as many points as possible from a given set of points subject to the constraint that no two selected points can be closer than r units to one another. Our goal is to devise a mathematical programming formulation with an LP-relaxation which yields integer solutions with great frequency. We consider six different formulations of the r-separation problem. We show that the LP-relaxations of the most obvious formulations will yield fractional results in all instances of the problem if an optimal solution contains fewer than half of the given points. To build computationally effective formulations for the r-separation problem, we write dense constraints with unit right-hand-sides. The LP formulation that performs the best in our computational tests almost always finds 0–1 solutions to the problem.     

The M/Ek/r machine interference model steady state equations and numerical solutions

This paper studies the machine interference problem in which the running times follow the negative exponential distribution, the repair times the Erlang distribution and the number of operatives is more than one. The steady state equations are derived and it is shown that unlike the case of the M/E_k/r ordinary queueing model, the solution cannot be taken in closed form. An efficient numerical procedure is developed instead, based on a decomposition principle. Tabulated results of the average number of machines running and the operative utilization for a range of the problem parameters are given, for the cases M/E₃/2 and M/E₃/3. A tentative conclusion for a closeness in performance between the models M/M/r and M/E_k/r is drawn.     

Syzygies of surfaces of general type

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type X embedded by adjoint line bundles L _r = K + r B, where B is a base point free, ample line bundle. Our main results determine the r for which L _r has N _p property. In corollaries, we will relate the bounds on r to the regularity of B. Examples in the last section show that several results are optimal.     

Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities

The consecutive k-out-of-r-from-n: F system was generalized to multi-state case. This system consists of n linearly ordered components which are at state below j if and only if at least k_j components out of any r consecutive are in state below j. In this paper we suggest bounds of increasing multi-state consecutive-k-out-of-r-from-n: F system (k₁ ? k₂ ? ? ? k_M) by applying second order Boole–Bonferroni bounds and applying Hunter–Worsley upper bound. Also numerical results are given. The programs in V.B.6 of the algorithms are available upon request from the authors.     

Finite group actions on products of spheres

In this work we make some contributions to the theory of actions of finite groups on products of spheres. Suppose that the groupZ _q ^r acts freely on the product of k copies of spheres. Question: Isr≤k? We solve this question for several values ofr andk.     

An exact solution approach for the interdiction median problem with fortification

Systematic approaches to security investment decisions are crucial for improved homeland security. We present an optimization modeling approach for allocating protection resources among a system of facilities so that the disruptive effects of possible intentional attacks to the system are minimized. This paper is based upon the p-median service protocol for an operating set of p facilities. The primary objective is to identify the subset of q facilities which, when fortified, provides the best protection against the worst-case loss of r non-fortified facilities. This problem, known as the r-interdiction median problem with fortification (IMF), was first formulated as a mixed-integer program by Church and Scaparra [R.L. Church, M.P. Scaparra, Protecting critical assets: The r-interdiction median problem with fortification, Geographical Analysis 39 (2007) 129–146]. In this paper, we reformulate the IMF as a maximal covering problem with precedence constraints, which is amenable to a new solution approach. This new approach produces good approximations to the best fortification strategies. Furthermore, it provides upper and lower bounds that can be used to reduce the size of the original model. The reduced model can readily be solved to optimality by general-purpose MIP solvers. Computational results on two geographical data sets with different structural characteristics show the effectiveness of the proposed methodology for solving IMF instances of considerable size.