Matrix models without scaling limit

In the context of hermitean one--matrix models we show that the emergence of the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result of the lattice characterizing the matrix model. Said otherwise, we are not obliged to take a continuum limit to find these hierarchies. We interpret this result as an indication of the topological nature of them. We discuss the topological field theories associated with both and discuss the connection with topological field theories coupled to topological gravity already studied in the literature.Comment: Latex, SISSA-ISAS 161/92/E

The (N,M)-th KdV hierarchy and the associated W algebra

We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding$M$pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi--field representation of KP hierarchy as sub--systems and naturally appears in multi--matrix models. The N+2M-1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual$W_N$ algebra. We show that there exist M distinct reductions of the (N, M)--th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N, M) algebra is reduced to the W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.Comment: 40 pages, LaTeX, SISSA-171/93/EP, BONN-HE-46/93, AS-IPT-49/9

Sensitivity of Coronal Loop Sausage Mode Frequencies and Decay Rates to Radial and Longitudinal Density Inhomogeneities: A Spectral Approach

Fast sausage modes in solar magnetic coronal loops are only fully contained in unrealistically short dense loops. Otherwise they are leaky, losing energy to their surrounds as outgoing waves. This causes any oscillation to decay exponentially in time. Simultaneous observations of both period and decay rate therefore reveal the eigenfrequency of the observed mode, and potentially insight into the tubes' nonuniform internal structure. In this article, a global spectral description of the oscillations is presented that results in an implicit matrix eigenvalue equation where the eigenvalues are associated predominantly with the diagonal terms of the matrix. The off-diagonal terms vanish identically if the tube is uniform. A linearized perturbation approach, applied with respect to a uniform reference model, is developed that makes the eigenvalues explicit. The implicit eigenvalue problem is easily solved numerically though, and it is shown that knowledge of the real and imaginary parts of the eigenfrequency is sufficient to determine the width and density contrast of a boundary layer over which the tubes' enhanced internal densities drop to ambient values. Linearized density kernels are developed that show sensitivity only to the extreme outside of the loops for radial fundamental modes, especially for small density enhancements, with no sensitivity to the core. Higher radial harmonics do show some internal sensitivity, but these will be more difficult to observe. Only kink modes are sensitive to the tube centres. {Variation in internal and external Alfv\'en speed along the loop is shown to have little effect on the fundamental dimensionless eigenfrequency, though the associated eigenfunction becomes more compact at the loop apex as stratification increases, or may even displace from the apex.Comment: Accepted J. Phys. A: Math. Theor. (Oct 31 2017). 20 pages, 12 figure

Turbulent convection model in the overshooting region: II. Theoretical analysis

Turbulent convection models are thought to be good tools to deal with the convective overshooting in the stellar interior. However, they are too complex to be applied in calculations of stellar structure and evolution. In order to understand the physical processes of the convective overshooting and to simplify the application of turbulent convection models, a semi-analytic solution is necessary. We obtain the approximate solution and asymptotic solution of the turbulent convection model in the overshooting region, and find some important properties of the convective overshooting: I. The overshooting region can be partitioned into three parts: a thin region just outside the convective boundary with high efficiency of turbulent heat transfer, a power law dissipation region of turbulent kinetic energy in the middle, and a thermal dissipation area with rapidly decreasing turbulent kinetic energy. The decaying indices of the turbulent correlations $k$, $\bar{u_{r}'T'}$, and $\bar{T'T'}$ are only determined by the parameters of the TCM, and there is an equilibrium value of the anisotropic degree $\omega$. II. The overshooting length of the turbulent heat flux $\bar{u_{r}'T'}$ is about $1H_k$($H_k=|\frac{dr}{dlnk}|$). III. The value of the turbulent kinetic energy at the convective boundary $k_C$ can be estimated by a method called \textsl{the maximum of diffusion}. Turbulent correlations in the overshooting region can be estimated by using $k_C$ and exponentially decreasing functions with the decaying indices.Comment: 32 pages, 9 figures, Accepted by The Astrophysical Journa

Vibration Induced Non-adiabatic Geometric Phase and Energy Uncertainty of Fermions in Graphene

We investigate geometric phase of fermion states under relative vibrations of two sublattices in graphene by solving time-dependent Sch\"{o}dinger equation using Floquet scheme. In a period of vibration the fermions acquire different geometric phases depending on their momenta. There are two regions in the momentum space: the adiabatic region where the geometric phase can be approximated by the Berry phase and the chaotic region where the geometric phase drastically fluctuates in changing parameters. The energy of fermions due to vibrations shows spikes in the chaotic region. The results suggest a possible dephasing mechanism which may cause classical-like transport properties in graphene.Comment: 9 pages, 5 figure

Anomalous particle-number fluctuations in a three-dimensional interacting Bose-Einstein condensate

The particle-number fluctuations originated from collective excitations are investigated for a three-dimensional, repulsively interacting Bose-Einstein condensate (BEC) confined in a harmonic trap. The contribution due to the quantum depletion of the condensate is calculated and the explicit expression of the coefficient in the formulas denoting the particle-number fluctuations is given. The results show that the particle-number fluctuations of the condensate follow the law $\sim N^{22/15}$ and the fluctuations vanish when temperature approaches to the BEC critical temperature.Comment: RevTex, 4 page