Spin chain from membrane and the Neumann-Rosochatius integrable system

We find membrane configurations in AdS_4 x S^7, which correspond to the continuous limit of the SU(2) integrable spin chain, considered as a limit of the SU(3) spin chain, arising in N=4 SYM in four dimensions, dual to strings in AdS_5 x S^5. We also discuss the relationship with the Neumann-Rosochatius integrable system at the level of Lagrangians, comparing the string and membrane cases.Comment: LaTeX, 16 pages, no figures; v2: 17 pages, title changed, explanations and references added; v3: more explanations added; v4: typos fixed, to appear in Phys. Rev.

Integrable discretizations of some cases of the rigid body dynamics

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The construction of discretizations is based on the discrete time Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian reduction. The resulting explicit maps on $e^*(n)$ are Poisson with respect to the Lie--Poisson bracket, and are also completely integrable. Lax representations of these maps are also found.Comment: arXiv version is already officia

Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.Comment: 61 pg

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Microscopic Foundation of Nonextensive Statistics

Combination of the Liouville equation with the q-averaged energy $U_q = _q$ leads to a microscopic framework for nonextensive q-thermodynamics. The resulting von Neumann equation is nonlinear: $i\dot\rho=[H,\rho^q]$. In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy $F_q=U_q-TS_q$ is a stability function for the dynamics. This implies that q-equilibrium states are dynamically stable. The (microscopic) evolution of $\rho$ is reversible for any q, but for $q\neq 1$ the corresponding macroscopic dynamics is irreversible.Comment: revte

Neumann and Neumann-Rosochatius integrable systems from membranes on AdS_4xS^7

It is known that large class of classical string solutions in the type IIB AdS_5xS^5 background is related to the Neumann and Neumann-Rosochatius integrable systems, including spiky strings and giant magnons. It is also interesting if these integrable systems can be associated with some membrane configurations in M-theory. We show here that this is indeed the case by presenting explicitly several types of membrane embedding in AdS_4xS^7 with the searched properties.Comment: LaTeX, 17 pages, no figures;v2: comments and citations added;v3: 20 pages, new subsection, explanations, comments and references added; v4: some typos fixed, to appear in JHE

Monolithic InAs QDs based active-passive integration for photonic integrated circuits

We demonstrated 20 nm relative blue shifted III-V passive waveguides monolithically integrated with InAs QDs active laser diode emitting at 1290 nm through selective area proton implantation and post-annealing method. This work is promising for low-loss monolithic Photonic Integrated Circuits

A finite model of two-dimensional ideal hydrodynamics

A finite-dimensional su($N$) Lie algebra equation is discussed that in the infinite $N$ limit (giving the area preserving diffeomorphism group) tends to the two-dimensional, inviscid vorticity equation on the torus. The equation is numerically integrated, for various values of $N$, and the time evolution of an (interpolated) stream function is compared with that obtained from a simple mode truncation of the continuum equation. The time averaged vorticity moments and correlation functions are compared with canonical ensemble averages.Comment: (25 p., 7 figures, not included. MUTP/92/1

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome

Mapping Connectivity Damage in the Case of Phineas Gage

White matter (WM) mapping of the human brain using neuroimaging techniques has gained considerable interest in the neuroscience community. Using diffusion weighted (DWI) and magnetic resonance imaging (MRI), WM fiber pathways between brain regions may be systematically assessed to make inferences concerning their role in normal brain function, influence on behavior, as well as concerning the consequences of network-level brain damage. In this paper, we investigate the detailed connectomics in a noted example of severe traumatic brain injury (TBI) which has proved important to and controversial in the history of neuroscience. We model the WM damage in the notable case of Phineas P. Gage, in whom a “tamping iron” was accidentally shot through his skull and brain, resulting in profound behavioral changes. The specific effects of this injury on Mr. Gage's WM connectivity have not previously been considered in detail. Using computed tomography (CT) image data of the Gage skull in conjunction with modern anatomical MRI and diffusion imaging data obtained in contemporary right handed male subjects (aged 25–36), we computationally simulate the passage of the iron through the skull on the basis of reported and observed skull fiducial landmarks and assess the extent of cortical gray matter (GM) and WM damage. Specifically, we find that while considerable damage was, indeed, localized to the left frontal cortex, the impact on measures of network connectedness between directly affected and other brain areas was profound, widespread, and a probable contributor to both the reported acute as well as long-term behavioral changes. Yet, while significantly affecting several likely network hubs, damage to Mr. Gage's WM network may not have been more severe than expected from that of a similarly sized “average” brain lesion. These results provide new insight into the remarkable brain injury experienced by this noteworthy patient