Beyond Euler angles: exploiting the angle-axis parametrization in a multipole expansion of the rotation operator

Euler angles (alpha,beta,gamma) are cumbersome from a computational point of view, and their link to experimental parameters is oblique. The angle-axis {Phi, n} parametrization, especially in the form of quaternions (or Euler-Rodrigues parameters), has served as the most promising alternative, and they have enjoyed considerable success in rf pulse design and optimization. We focus on the benefits of angle-axis parameters by considering a multipole operator expansion of the rotation operator D(Phi, n), and a Clebsch-Gordan expansion of the rotation matrices D(MM')(J)(Phi, n). Each of the coefficients in the Clebsch-Gordan expansion is proportional to the product of a spherical harmonic of the vector n specifying the axis of rotation, Y(lambdamu)(n), with a fixed function of the rotation angle Phi, a Gegenbauer polynomial C(2J-lambda)(lambda+1)(cosPhi/2). Several application examples demonstrate that this Clebsch-Gordan expansion gives easy and direct access to many of the parameters of experimental interest, including coherence order changes (isolated in the Clebsch-Gordan coefficients), and rotation angle (isolated in the Gegenbauer polynomials).     

Chaos in the relativistic Euler problem

The Euler problem with two fixed point masses and one moving mass is reconsidered in the light of general relativity. The scattering of a particle by two fixed black holes is shown to be strongly chaotic. Two neutral black holes have been used for the study. The particle trajectories have been computed numerically using a modified muffin tin approximation. A plot of the scattering angle against impact parameter showing distinct signs of chaos is presented. (c) 1996 American Institute of Physics.     

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U(y) and with stationary streamlines y=y₀ (such that U^′(y₀)=0), a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.     

Spectral Asymptotic for the High-Dimensional Euler Top

In this paper, we compute the leading coefficient in the asymptotic expansion of the joint eigenvalues for the high-dimensional Euler Top. We also prove a central limit theorem for the same eigenvalues.      

On equivariant Euler characteristics

The Euler characteristic of an orbifold M/G as used in string theory is identified with the Euler characteristic of equivariant K -theory K_G(M).     

On the Euler characteristics of random surfaces

In a Monte Carlo computer experiment, we simulate the Gibbs distribution of nonconnected two-dimensional surfaces isometrically embedded in three-dimensional Euclidean space with fixed boundary and the action given by the area. The simulation involves surfaces built out of plaquettes in a cubical lattice. The foam structure is analyzed in terms of correlations of the local fluctuations in the Euler characteristic and the area. The scaling behavior of the area and the Euler characteristic is discussed by varying the boundary. We show evidence of a phase transition point which is independent of the choice of the boundary. An existence proof is given of the thermodynamic limit for the models considered.Supported in part by the Deutsche Forschungsgemeinschaft and NSF grant No. Phy 81 09110A 01.     

Global Solutions to the Ultra-Relativistic Euler Equations

We show that when entropy variations are included and special relativity is imposed, the thermodynamics of a perfect fluid leads to two distinct families of equations of state whose relativistic compressible Euler equations are of Nishida type. (In the non-relativistic case there is only one.) The first corresponds exactly to the Stefan-Boltzmann radiation law, and the other, emerges most naturally in the ultra-relativistic limit of a γ-law gas, the limit in which the temperature is very high or the rest mass very small. We clarify how these two relativistic equations of state emerge physically, and provide a unified analysis of entropy variations to prove global existence in one space dimension for the two distinct 3 × 3 relativistic Nishida-type systems. In particular, as far as we know, this provides the first large data global existence result for a relativistic perfect fluid constrained by the Stefan-Boltzmann radiation law.     

Global solutions of the relativistic Euler equations

We demonstrate the existence of solutions with shocks for the equations describing a perfect fluid in special relativity, namely, divT=0, whereT ^ij =(p+c ²)u ⁱ u ^j +p ^ij is the stress energy tensor for the fluid. Here,p denotes the pressure,u the 4-velocity, the mass-energy density of the fluid, ^ij the flat Minkowski metric, andc the speed of light. We assume that the equation of state is given byp= ² , where ², the sound speed, is constant. For these equations, we construct bounded weak solutions of the initial value problem in two dimensional Minkowski spacetime, for any initial data of finite total variation. The analysis is based on showing that the total variation of the variable ln() is non-increasing on approximate weak solutions generated by Glimm's method, and so this quantity, unique to equations of this type, plays a role similar to an energy function. We also show that the weak solutions ((x ⁰,x ¹),v(x ⁰,x ¹)) themselves satisfy the Lorentz invariant estimates Var{ln((x ⁰,·)}<V ₀ and for allx ⁰0, whereV ₀ andV ₁ are Lorentz invariant constants that depend only on the total variation of the initial data, andv is the classical velocity. The equation of statep=(c ²/3) describes a gas of highly relativistic particles in several important general relativistic models which describe the evolution of stars.Supported in part by NSF Applied Mathematics Grant Number DMS-89-05205Supported in part by NSF Applied Mathematics Grant Number DMS-86-13450     

Direct Proof of Determinant Representation for Scalar Products of the XXZ Gaudin Model with Generic Non-Diagonal Boundary Terms

After constructing the Bethe state of the XXZ Gaudin model with generic non-diagonal boundary terms,we analyze the properties of this state and obtain the determinant representations of the scalar products for this XXZ Gaudin model.     

Beyond the x-K relations

Explicit formulae are given for all off-diagonal 1–1 and 2–2 resonance terms in the vibrational hamiltonian following the first contact transformation. Such formulae are needed to test the quantitative accuracy of the generalized x-K relation recently published by Della Valle. These formulae are used in a test calculation for linear HCN and DCN and are found only modestly to improve the predicted energy levels for the former, but dramatically improve those of the latter.     

Second order front tracking for the Euler equations

A second order front tracking method is developed for solving the hyperbolic system of Euler equations of inviscid fluid dynamics numerically. Meshless front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution. The linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately. In order to construct a physically correct linearization, the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model. This front interaction model is also extended to include front numbers used in the wave decomposition. It is illustrated numerically for Sod’s Riemann problem, the two interacting blast waves problem, and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions.     

q-Generalized Euler numbers and polynomials

Recently, B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials (see [12]). In this paper, we study new q-extensions of Euler numbers and polynomials by using the method of Kupershmidt. We also investigate the properties of symmetries of these q-Euler polynomials by using q-derivatives and q-integrals.     

Derivation of the Euler Equations from Quantum Dynamics

We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian.©2003 B. Nachtergaele and H.-T. Yau. This article may be reproduced in its entirety for non-commercial purposes.Research partially supported by NSF # DMS-0070774.Research partially supported by NSF # DMS-0072098, the Veblen Fund from the Institute for Advanced Study and a Fellowship from the MacArthur Foundation.     

Singularities in cascade models of the Euler equation

The formation of singularities in the three-dimensional Euler equation is investigated. This is done by restricting the number of Fourier modes to a set which allows only for local interactions in wave number space. Starting from an initial large-scale energy distribution, the energy rushes towards smaller scales, forming a universal front independent of initial conditions. The front results in a singularity of the vorticity in finite time, and has scaling form as function of the time difference from the singularity. Using a simplified model, we compute the values of the exponents and the shape of the front analytically. The results are in good agreement with numerical simulations.