Classroom note: Maple® version of the ‘Indian rope trick’

If the point of suspension of a multiple pendulum is suitably oscillated then the pendulum can remain in motion in an upside-down position. Since such pendulums can model flexible materials, this inverted motion is sometimes referred to as an ‘Indian rope trick’. Despite the complexity of the governing differential equations, this rope trick can be successfully demonstrated using a popular computer algebra system to formulate and solve the equations, and display an animation of the motion.     

Classroom note: Using the binomial series to prove the arithmetic mean–geometric mean inequality

In 1968, Leon Gerber compared (1 + x) ^a to its kth partial sum as a binomial series. His result is stated and, as an application of this result, a proof of the arithmetic mean–geometric mean inequality is presented.     

A note on poroacoustic traveling waves under Darcy’s law: Exact solutions

A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy’s law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are pointed out and possible extensions of this work are briefly discussed.     

The core and nucleolus of games: A note on a paper by Göthe-Lundgren et al.

In the paper “On the nucleolus of the basic vehicle routing game”, Mathematical Programming 72, 83–100 (1996), G?the-Lundgren et al. develop a constraint generation method to compute the pre-nucleolus of a game. Their method assumes that constraints that are redundant in the representation of the core can be ignored in the computation of the pre-nucleolus. We provide an example that shows that for a game with an empty core such an assumption is, in general, not valid. Further, we show that a statement made by G?the-Lundgren et al. about an intuitive interpretation of the pre-nucleolus is misleading. Received: January 1996 / Accepted: February 2000?Published online January 17, 2001     

The mystery of ten wooden blocks: Hjelmslev’s geometry of reality

Mathematische Semesterberichte - I shall argue that a collection of wood models at the University of Copenhagen was made by Johannes Hjelmslev as an aid in his elementary...     

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case

Time-dependent statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble averages in turbulence theory and form the backbone for a mathematical foundation of the theory of turbulence. The two main notions of statistical solutions, previously introduced, are revisited and a new formulation of one of them is given. An existence proof for this new formulation is given, along with a number of useful properties.     

Analysis of Iterated Modular Exponentiation: The Orbits of xα mod N

Let N and α be integers larger than 1. Define an orbit to be the collection of residues in generated byiteratively applying mod N to an element which eventually maps back to itself.An orbit's length is the number of distinct residues in the orbit. When N isa large bicomposite integer, such as is commonly used in many cryptographicapplications, and when certain prime factorizations related to N are known,all orbit lengths and the number of orbits of each possible length can beefficiently computed using the results presented. If the required integerfactorizations are only partially known, the risk that a randomly selectedperiodic element might produce an orbit shorter than some (typically large)divisor of can be bounded. The information needed to producesuch a bound is fully available when the prime factors of N are generatedusing the prime generation algorithm defined in Maurer maur. Resultspresented can assist in choosing wisely a modulus N for the Blum, Blum, andShub pseudo-random bit generator. If N is a bicomposite RSA modulus, theanalysis shows how to quantify the risk posed by an iterated encryptionattack.     

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

Stationary statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble average for turbulent flows in statistical equilibrium in time. They are also a generalization of the notion of invariant measure to the case of the three-dimensional Navier–Stokes equations, for which a global uniqueness result is not known to exist and a semigroup may not be well-defined in the classical sense. The two classical definitions of stationary statistical solutions are considered and compared, one of them being a particular case of the other and possessing a number of useful properties. Furthermore, the so-called time-average stationary statistical solutions, obtained as generalized limits of time averages of weak solutions as the averaging time goes to infinity are shown to belong to this more restrictive class. A recurrent type result is also obtained for statistical solutions satisfying an accretion condition. Finally, the weak global attractor of the three-dimensional Navier–Stokes equations is considered, and in particular it is shown that there exists a topologically large subset of the weak global attractor which is of full measure with respect to that particular class of stationary statistical solutions and which has a certain regularity property.     

Nonlinear differential systems in the 3–space: A note on periodic solutions by the analysis of two examples

In this note, we address the problem of the existence and location of periodic solutions in nonlinear differential systems in the 3–space. Our main motivation is the study, via bifurcation theory, of periodic solutions (especially limit cycles). We study this problem in two simple polynomial (chaotic) systems: The first one, due to Muthuswamy and Chua, is the mathematical model to the simplest chaotic circuit, and the second is due to Sprott et al.     

A note on the paper “Resource-constrained project scheduling: Notation,classification, models and methods” by Brucker et al.

The great variety of project scheduling problems studied in the ever growing literature motivated the recent development of classification schemes. In a recent paper (European Journal of Operational Research 112 (1999) 3–41), Brucker et al. make the claim that, so far, no classification scheme exists which is compatible with what is commonly accepted in machine scheduling and introduce a new classification. In this note, we critically review major shortcomings of the suggested scheme which place heavy limitations on its potential use.