Sulla propagazione di onde elettromagnetiche in una colonna di plasma cilindrica circolare indefinita,soggetta ad agitazione termica elettronica e a un campo magnetico assiale uniforme

Summary A study is made about the ways of propagation of electromagnetic waves in a cylindrical plasma column subject to termic and electronic agitation and to an axial uniform field; and it is put in evidence the possibility for four ways of propagation of waves according to the cylinder's axis. Then it is examined a particular case in which the ways of waves propagation are reduced to two.

---

---

A Bruno Finzi nel suo 70^mo compleanno.

---

Lavoro svolto nell'ambito del Gruppo di ricerca n. 3 del Comitato per la Matematica del C.N.R.

---

Entrata in Redazione il 26 gennaio 1970.     

Minimal ‐crystals and isomorphism numbers of isosimple ‐crystals

In this paper we generalize minimal p‐divisible groups defined by Oort to minimal F‐crystals over algebraically closed fields of positive characteristic. We prove a structural theorem of minimal F‐crystals and give an explicit formula of the Frobenius endomorphism of the basic minimal F‐crystals that are the building blocks of the general minimal F‐crystals. We then use minimal F‐crystals to generalize minimal heights of p‐divisible groups and give an upper bound of the isomorphism numbers of F‐crystals, whose isogeny type are determined by simple F‐isocrystals, in terms of their ranks, Hodge slopes and Newton slopes.     

On the uniqueness of promotion operators on tensor products of type <Emphasis Type="Italic">A</Emphasis> crystals

The affine Dynkin diagram of type A _n ⁽¹⁾ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A _n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A _n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

Dynamic steady‐state crack propagation in quasi‐crystals

The steady propagation of planar cracks in quasi‐crystalline bodies with velocity lower than the one of bulk elastic macroscopic waves is under scrutiny. Closed‐form solutions to the balance laws are provided. Unusual Mach number limits are determined. Numerical experiments describing peculiar aspects of the crack propagation in quasi‐crystals are performed by varying parametrically the coupling coefficient between macroscopic deformation and substructural events. In this way, classes of quasi‐crystals are then compared. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence and stability of vector solitary waves for nonlinear Schrödinger systems of Hartree-type with Bessel potential

In this paper, we study a class of two-dimensional nonlinear Schrödinger systems of Hartree-type with Bessel potential kernel, which models the propagation and interaction of two-color light beams in nematic liquid crystals. The global well-posedness is proved by using fixed point argument, Gagliardo-Nirenberg inequality and conservation laws. In addition, we also obtain the existence and orbital stability of ground state vector solitary waves applying variational methods and Concentration-compactness Lemma.     

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem.     

不规则单斜地层中磁弹性剪切波的色散方程

在内夹磁弹性单斜地层中,下界面不规则变化时,研究水平偏振剪切波的传播,该地层夹在两个半无限磁弹性单斜介质之间,得到了闭式的色散方程.不计磁场及介质界面的不规则性,该色散方程与三层介质中经典方程相一致.图示了磁场和界面不规则深度对相速度的影响.     

Conservation of Geometric Structures for Non-Homogeneous Inviscid Incompressible Fluids

In this article we get a result on propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension N ≥ 2. In particular, we investigate conservation of striated and conormal regularity, which generalize the 2-D structure of vortex patches. The results we get are only local in time, even for N = 2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N = 2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.     

Über die Realisierbarkeit eines Laser-gepumpten parametrischen Mikrowellen-Verstärkers

Summary The realizability of a parametric microwave amplifier with a laser-pumped electrooptic crystal as a nonlinear element is investigated theoretically for ideal and lossy crystals. It is found that even with the best presently available crystals a 10-db-travelling wave amplifier needs a pump power flux of the order of 10⁷W/cm². Consequentlycw-operation is impossible. Phase matching can be achieved either by a suitable choice of the propagation directions or by increasing the phasevelocity of the signal with a waveguide. Both possibilities are treated. With given parameters of the nonlinear material, the formulas and graphs permit a quick estimation of the required pump power and the proper crystal- and waveguide dimensions. Finally the introduction of a resonator at signal frequency is being considered. However, it is found, that only a small amount of pump power can be saved, as even with low resonatorQ the amplitude fluctuations of the pump deteriorate the gain stability very much. With high resonatorQ a parametric oscillator is obtained.

---

---

Gewidmet Herrn Professor Dr. K. P. Meyer zu seinem 60. Geburtstag     

Stability of microstructures for some martensitic transformations

We analyze the stability of laminated microstructure for martensitic crystals that undergo cubic to trigonal, orthorhombic to triclinic, and trigonal to monoclinic transformations. We show that the microstructure is unique and stable for all laminates except when the lattice parameters satisfy certain identities.     

A local radial basis function collocation method for band structure computation of 3D phononic crystals

In this paper, we further extend the local radial basis function collocation method (LRBFCM) for efficient computation of band structures of phononic crystals from 2D to 3D. The proposed LRBFCM uses one fictitious node to tackle instability problems caused by calculation of derivatives of the wave pressure. A few examples of sound pressure wave propagation are modelled to validate the developed method. Comparisons with finite element modeling demonstrate the high stability and efficiency of the new method in computation of band structures of 3D phononic crystals.     

Global existence of the finite energy weak solutions to a nematic liquid crystals model

In this paper, we are concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. For a bounded domain in Ropf³, under the assumption that initial density belongs to , we show the global existence of weak solutions to the nematic liquid crystals model with a penalized system. Furthermore, we also obtain the energy inequality for weak solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Modelling and sensitivity analysis for design of phononic crystals

The two-scale homogenization is employed to study the band gaps in strongly heterogeneous periodic elastic media – the phononic crystals. It was observed that wave propagation can be suppressed for certain frequency intervals. Moreover, the gap distribution w.r.t. the incident frequencies can be modified by changing the shape of inclusions embedded in the matrix. The sensitivity analysis formulae were derived and implemented to obtain optimized microstructures of the phononic crystals. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interval propagation and search on directed acyclic graphs for numerical constraint solving

The fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation have recently been proposed in Schichl and Neumaier (J. Global Optim. 33, 541–562, 2005). For representing numerical problems, the authors use DAGs whose nodes are subexpressions and whose directed edges are computational flows. Compared to tree-based representations [Benhamou et al. Proceedings of the International Conference on Logic Programming (ICLP’99), pp. 230–244. Las Cruces, USA (1999)], DAGs offer the essential advantage of more accurately handling the influence of subexpressions shared by several constraints on the overall system during propagation. In this paper we show how interval constraint propagation and search on DAGs can be made practical and efficient by: (1) flexibly choosing the nodes on which propagations must be performed, and (2) working with partial subgraphs of the initial DAG rather than with the entire graph. We propose a new interval constraint propagation technique which exploits the influence of subexpressions on all the constraints together rather than on individual constraints. We then show how the new propagation technique can be integrated into branch-and-prune search to solve numerical constraint satisfaction problems. This algorithm is able to outperform its obvious contenders, as shown by the experiments.     

Kristallachsenlagen in Hagelkörnern

Summary The big single crystals of a shell of a hailstone are mostly arranged in palisades, directed to a certain point. This preferred orientation of shape of single crystals is identical to the direction of growth of the hailstone at the place of the crystallite refered to. If we look at the statistical deviation of the direction of thec-axis — determined by a Replica method — of the crystallites of a whole shell from the local direction of growth, we find that thec-axis looks mainly in the direction of growth (Figures 7, 10, 12) but the relative height of the maximum of the distribution may vary from one shell to another (Figures 7 and 10). Although in one case no rule of orientation could be found (Figures 14 and 15) and in another thec-axis was perpendicular to the direction of growth (Figure 16). Results of some primary investigations, based upon experiments in the hail tunnel, leed us to the conclusion that the arrangement of the crystallographic axis of the different single crystals depends very much on the growing conditions of the ice deposits. Therefore one may expect, that in the near future an interpretation of the icing conditions of natural hailstones is possible on behalf of their crystallographic structure. Icing experiments in the hail tunnels will give us the rules one has to observe for this procedure.     

On guided waves in 3-D photonic crystals

The propagation of guided waves in 3-D photonic crystals is studied. In this paper, we suppose that the spectrum of reference medium (a perfectly 3-D photonic crystal) has band gaps. If a line defect is introduced along one of the lattice vectors, we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong. Furthermore, we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect.     

The propagation of nonstationary filtration with absorption

We point out a mistake in [1] in the proof of infinite propagation of the weak solution of Cauchy problem for the equationu _t =(A(u)) _xx - (u). A more natural condition is given and the infinite propagation property for boundary value problem is studied. Finally, we discuss the supports of the weak solutions.