Efficient path tracking methods

Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described.     

Sample path methods in the control of queues

Sample path methods are now among the most used techniques in the control of queueing systems. However, due to the lack of mathematical formalism, they may appear to be non-rigorous and even sometimes mysterious. The goal of this paper is threefold: to provide a general mathematical setting, to survey the most popular sample path methods including forward induction, backward induction and interchange arguments, and to illustrate our approach through the study of a number of classical scheduling and routing optimization problems arising in queueing theory.Z. Liu was supported in part by the CEC DG XIII under the ESPRIT BRA grants QMIPS.P. Nain was supported in part by NSF under grant NCR-9116183 and by the CEC DG XIII under the ESPRIT BRA grants QMIPS.D. Towsley was supported in part by NSF under grant NCR-9116183.     

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest.The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.     

Multiple criteria sorting models and methods. Part II: theoretical results and general issues

4OR - Multiple criteria sorting methods assign objects into ordered categories while objects are characterized by a vector of n attributes values. Categories are ordered, and the assignment of the...     

Multiple UAVs path planning algorithms: a comparative study

Unmanned aerial vehicles (UAVs) are used in team for detecting targets and keeping them in its sensor range. There are various algorithms available for searching and monitoring targets. The complexity of the search algorithm increases if the number of nodes is increased. This paper focuses on multi UAVs path planning and Path Finding algorithms. Number of Path Finding and Search algorithms was applied to various environments, and their performance compared. The number of searches and also the computation time increases as the number of nodes increases. The various algorithms studied are Dijkstra’s algorithm, Bellman Ford’s algorithm, Floyd-Warshall’s algorithm and the AStar algorithm. These search algorithms were compared. The results show that the AStar algorithm performed better than the other search algorithms. These path finding algorithms were compared so that a path for communication can be established and monitored.     

Divergence theorems in path space II: degenerate diffusions

Let x denote an elliptic diffusion process defined on a smooth compact manifold M. In a previous work, we introduced a class of vector fields on the path space of x and studied the admissibility of this class of vector fields with respect to the law of x. In the present Note, we extend this study to the case of degenerate diffusions. To cite this article: D. Bell, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Critical resonance in the non-intersecting lattice path model

We study the phase transition in the honeycomb dimer model (equivalently, monotone non-intersecting lattice path model). At the critical point the system has a strong long-range dependence; in particular, periodic boundary conditions give rise to a resonance phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain.The research leading to this article was conducted in part while the first author was visiting Microsoft.Mathematics Subject Classification (2000): 82B20Revised version: 7 July 2003     

Heuristic methods for single link shared backup path protection

Protecting communication networks against failures is becoming increasingly important as they have become an integrated part of our society. Cable failures are fairly common, but it is unacceptable for a single cable failure to disconnect communication for more than a few seconds—hence protection schemes are employed. In contrast to manual intervention, automatic protection schemes such as shared backup path protection (SBPP) can recover from failure quickly and efficiently. SBPP is a simple but efficient protection scheme that can be implemented in backbone networks with technology available today. In SBPP backup paths are planned in advance for every failure scenario in order to recover from failures quickly and efficiently. Planning SBPP is an NP-hard optimization problem, and previous work confirms that it is time-consuming to solve the problem in practice using exact methods. We present heuristic algorithms and lower bound methods for the SBPP planning problem. Experimental results show that the heuristic algorithms are able to find good quality solutions in minutes. A solution gap of less than \(3.5~\%\) was achieved for 5 of 7 benchmark instances (and a gap of less than \(11~\%\) for the remaining instances.)     

Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model

In this paper, we complete the global qualitative analysis of the well-known FitzHugh–Nagumo neuronal model. In particular, studying global limit cycle bifurcations and applying the Wintner–Perko termination principle for multiple limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.     

搜索路径的优化设计模型

研究对指定区域进行全面搜索用时最少的行进路线.问题一中提出了往复式的搜索方式,结合拐弯处所用的时间,得到最优时间为49.806小时,添加三名队员后,最优时间为46.593小时:对问题二提出了分组进行搜索的方式,当分组队员数分别为22、23、5人时,最优时间为20.45小时.     

Elliptic three-folds II: Multiple fibres

Let be an elliptic fibration with a section, where is a projective surface and is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset to obtain a new elliptic fibration which now has multiple fibres along . This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of .

     

Multiple bifurcations and periodic “bubbling” in a delay population model

In this paper, the flip bifurcation and periodic doubling bifurcations of a discrete population model without delay influence is firstly studied and the phenomenon of Feigenbaum’s cascade of periodic doublings is also observed. Secondly, we explored the Neimark–Sacker bifurcation in the delay population model (two-dimension discrete dynamical systems) and the unique stable closed invariant curve which bifurcates from the nontrivial fixed point. Finally, a computer-assisted study for the delay population model is also delved into. Our computer simulation shows that the introduction of delay effect in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node → stable focus → an lower-dimensional closed invariant curve (quasi-periodic solution, limit cycle) or/and stable periodic solutions → chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling) and the sudden change between two different attractors, etc.     

An origin-based model for unique shortest path routing

Link weights are the main parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem in such a way based on which a more efficient solution algorithm than the existing ones may be developed. In this paper, an exact formulation is first introduced and then mathematically proved correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem.     

A continuous path of singular masas in the hyperfinite II1 factor

Using methods of Tauer, we exhibit an uncountable family ofsingular masas in the hyperfinite II₁ factor R all with Pukánszkyinvariant {1}, no pair of which is conjugate by an automorphismof R. This is done by introducing an invariant (A) for a masaA in a II₁ factor N as the maximal size of a projection eA forwhich A e contains non-trivial centralizing sequences for eNe. The masas produced give rise to a continuous map from theinterval [0, 1] into the singular masas in R equipped with thed_{, 2}-metric. A result is also given showing that the Pukánszky invariantis d_{, 2}-upper semi-continuous. As a consequence, the sets ofmasas with Pukánszky invariant {n} are all closed.