On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation

Communicated by L.N. Shevrin     

On detecting a wavefront described by a 2D eikonal equation when the velocity in the medium depends on one spatial variable

In this paper, the 2D eikonal equation f _x ² + f _y ² = (ky + b)^2α is considered. If a solution is found, the relation f (x, y) = C determines the location of the wavefront. However, finding the solutions is still an open question. In this paper, we propose to directly find a curve in parametric form, which determines the wavefront without solving the equation.     

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.     

On a periodic vehicle routing problem

This paper deals with a study on a variant of the Periodic Vehicle Routing Problem (PVRP). As in the traditional Vehicle Routing Problem, customer locations each with a certain daily demand are given, as well as a set of capacitated vehicles. In addition, the PVRP has a horizon, say T days, and there is a frequency for each customer stating how often within this T-day period this customer must be visited. A solution to the PVRP consists of T sets of routes that jointly satisfy the demand constraints and the frequency constraints. The objective is to minimize the sum of the costs of all routes over the planning horizon. We develop different algorithms solving the instances of the case studied. Using these algorithms we are able to realize considerable cost reductions compared to the current situation.     

Bifurcation for a functional yield chemostat when one competitor produces a toxin

A model, with general yield functions: Fi(S), i=1,2, of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper. The conditions in terms of the relevant parameters for the Hopf bifurcation of the three-dimensional system have been proved, which implies the existence of limit cycles in the 3-D system.     

On the genus distributions of wheels and of related graphs

We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl’s result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels.     

The control of a wheeled mechanical system

Non-holonomic systems with rolling or wheeled systems are investigated. The investigation is restricted to kinematic models and the dynamics of the drive mechanism of the system are taken into account. A control law is constructed which stabilizes the motion of a wheeled system along a specified trajectory (a plane smooth curve). For the basic variables of the system, the property of stabilizability is substantiated in the large.     

On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x] and k[x,y] over any field k of zero characteristic. In particular, if D₁ and D₂ are commuting derivations of k[x,y] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial f∈k[x,y] such that D₁(f)=λf and D₂(f)=μf for some λ,μ∈k[x,y], or (ii) they are Jacobian derivations

     

Limit of the spectra of the interior Neumann problems when a solid domain shrinks to a plane one

Let (Δ + λ) u = 0 in $\text{D}$c$\text{R}$^d, $\text{?u}\text{?N}\text{=0}$ on ?$\text{D}$. How do the eigenvalues λ_j behave when $\text{D}$ shrinks to a domain Δ ? R^{d ? 1} ? The answer depends not only on Δ but on the way $\text{D}$ shrinks to Δ. The limit of λ_j is found. Examples are given.     

On the capacitated vehicle routing problem

 We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. Received: October 30, 2000 / Accepted: December 19, 2001 Published online: September 5, 2002 Key words. vehicle routing problem – integer programming – decomposition algorithm – separation algorithm – branch and cut Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

The 3-D Hopf bifurcation in bio-reactor when one competitor produces a toxin

In this paper, a model of competition in the bio-reactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent is investigated. The conditions of the three dimensional Hopf bifurcation are obtained. The Hopf bifurcation implies the existence of limit cycles in the model that corresponds to the nonlinear oscillation in the reactor.     

On the history of lateral vehicle dynamics

The lateral motion of vehicles on tracks and roads is characterized by the rolling contact between wheels and guideway. Railway vehicles feature passive lateral guidance while road vehicles are operated by steering. Lateral dynamics is an extremely complex problem but the fundamental behaviour can already understood by strongly simplified mechanical models. In this paper the historical development of modelling approaches for lateral dynamics is reviewed what is very helpful for benchmarking and checking more complex problems, too. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On one problem of tracking a given trajectory

A controlled system nonlinear in phase variables is considered. Under the assumption that the system is subject to an uncontrollable disturbance, an algorithm of forming the control is specified such that a given trajectory is tracked.     

Optimal boundary control of vibrations at one endpoint of a string when the second endpoint is free

The generalized solution u(x, t) of the wave equation u _tt (x, t) − u _xx (x, t) = 0 admitting the existence of finite energy at every time instant t is used to find among all W ₂ ¹ [0,T]-functions with a long time interval T the optimal boundary control for a string with a free endpoint that takes the vibration process from a given arbitrary state to a given final state. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 23–36, 2004.     

On the waiting time of arriving aircrafts and the capacity of airports with one or two runways

In this paper we examine a model for the landing procedure of aircrafts at an airport. The characteristic feature here is that due to air turbulence the safety distance between two landing aircrafts depends on the types of these two machines. Hence, an efficient routing of the aircraft to two runways may reduce their waiting time.