Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability

Towards Fractional Gradient Elasticity

An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one-dimension. The second involves the Riesz fractional derivative in three-dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case it is shown that stress equilibrium in a Caputo elastic bar requires the existence of a non-zero internal body force to equilibrate it. In the second case, it is shown that in a Riesz type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.Comment: 10 pages, LaTe

Entomopathogenic nematodes for biological control of codling moth

Entomopathogenic nematodes are often found naturally infecting codling moth larvae. The effect of an autumn treatment with S. feltiae on the fruit damage in the following summer was evaluated by treating 4 different apple orchards in October 2004 and 2005 at application rates of 3.75; 2 and 1.5 billion nematodes in 4000 l / ha. In three of the treated orchards, one treated with 3.75x109 nematodes/ha the other two treated with 2e9 nematode/ha, reduction in fruit damage was around 50%. In the most heavily infested orchard, which was treated with 1.5x109 nematode/ha only 33% reduction in fruit damage was achieved. Compared to previous studies, this was the first assessing the effect on the fruit damage in the summer following the treatment rather than assessing the mortality of sentinel larvae fixed to the treated tree trunks

Heavy quark collisional energy loss in the quark-gluon plasma including finite relaxation time

In this paper, we calculate the soft-collisional energy loss of heavy quarks traversing the viscous quark-gluon plasma including the effects of a finite relaxation time $\tau_\pi$ on the energy loss. We find that the collisional energy loss depends appreciably on $\tau_\pi$ . In particular, for typical values of the viscosity-to-entropy ratio, we show that the energy loss obtained using $\tau_\pi$ = 0 can be $\sim$ 10$\%$ larger than the one obtained using $\tau_\pi$ = 0. Moreover, we find that the energy loss obtained using the kinetic theory expression for $\tau_\pi$ is much larger that the one obtained with the $\tau_\pi$ derived from the Anti de Sitter/Conformal Field Theory correspondence. Our results may be relevant in the modeling of heavy quark evolution through the quark-gluon plasma.Comment: v2: 5 pages, 4 figures, added references. Accepted for publication in Phys. Rev.