1 Irrationality 3 1.1 Simple proofs of irrationality . . . . . . . . . . . . . . . . . . . . 3 1.2 Variation on a proof by Fourier (1815) . . . . . . . . . . . . . . . 10 1.2.1 Irrationality of e . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 The number e is not quadratic . . . . . . . . . . . . . . . 11 1.2.3 Irrationality of e √ 2 (Following a suggestion of D.M. Masser) . . . . . . . ...

In this paper we extend the well known Kalman-Yacubovic-Popov (KYP) lemma to the case of matrices with general regular inertia. We show that the version of the lemma that was derived for the case of pairs of stable matrices whose rank difference is one, extends to the more general case of matrices with regular inertia and in companion form. We then use this result to derive an easily verifiable...

This paper proposes a multirate Model Predictive Control (MPC) algorithm for constrained systems where the control effort is adjusted several times during one sampling interval. Stability criterion for this new proposed algorithm is established. The test of the stability condition, the estimation of stability regions and the choice of the terminal weighting term are discussed.

Abstract In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect predation between prey predator with nonlinear functional response. First, we develop results concerning boundedness, existence uniqueness solution. Furthermore, Lyapunov principle Routh–Hurwitz criterion are applied study respectively local global stability results....

The stability of one liquid thread immersed in a fluid in a shear field is considered by linear stability analysis. A constant shear stress is imposed far away from the thread. The shear flow tends to deform and elongate the thread. The stability of the thread is characterized by the growth rate of a random perturbation. The equation for the growth rate leads to an eigenvalue problem with the w...

Abstract Since eroded sediments are produced from different sources distributed throughout a basin, sediment delivery processes at basin scale have to be modelled by a spatially distributed approach. In this paper a new theoretically based relationship is proposed for evaluating the sediment delivery ratio, SDR{, of each morphological unit, i, into which a basin is divided. Then, using the sedi...

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals uid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, thus violating the Lax criterion. The lacking information lies in an additional jump condition, which is rather simple in the case of reversible ...

The lateral instabilities of vehicles are well-known phenomena, see for example, the shimmy motions of bikes, motorbikes or the steered wheels of cars. Another interesting phenomenon is the snaking motion of the skateboard-skater system that is analyzed in this study. A mechanical model of the downhill skateboarding is constructed in order to consider the effect of the slope angle on the stabil...

Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaini...

All the conventional criterions to verify if a matrix is positive definite, Hurwitz or an M-matrix, such as Sylvester condition and Hurwitz theorem, require us to compute n determinants. This paper develops a new and simple criterion, based on a new type of Gaussian elimination process. In contrast to the traditional criterions, the new criterion only computes one determinate. Thus, the computi...