Math-for-industry Education&Research Hug

Abstract: A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.

File：2012-4

Abstract. This paper introduce simple and general theories of compressed sensing and LASSO. The novelty is that our recovery results do not require the restricted isometry property(RIP). We use the notion of weak RIP that is a natural generalization of RIP. We consider that the proposed results are more useful and flexible for real data analysis in various fields.

File：2012-3

Abstract. This paper investigates the Gaussian quasi-likelihood estimation of an exponentially ergodic multidimensional Markov process, which is expressed as a solution to a Levy driven stochastic differential equations whose coefficients are supposed to be known except for the finite-dimensional parameters to be estimated. We suppose that the process is observed under the condition for the rapidly increasing experimental design. By means of the polynomial type large deviation inequality, the mighty convergence of the corresponding statistical random fields is derived, which especially leads to the asymptotic normality at rate of the square root of the terminal sampling time for all the target parameters, and also to the convergence of their moments. In our results, the diffusion coefficient may be degenerate, or even null. Although the resulting estimator is not asymptotically efficient in the presence of jumps, we do not require any specific form of the driving Levy measure, rendering that the proposed estimation procedure is practical and somewhat robust to underlying model specification.

File：2012-2

Abstract. Special values ζ_Q(k) (k = 2, 3, 4, ...) of the spectral zeta function ζ_Q(s) of the non-commutative harmonic oscillator Q are discussed. Particular emphasis is put on basic modular properties of the generating function w_k(t) of Apery-like numbers which is appeared in analysis on the first anomaly of each special value.
Here the first anomaly is defined to be the "1st order" difference of ζ_Q(k) from ζ(k), ζ(s) being the Riemann zeta function. In order to describe such modular properties for k ≥ 4, we introduce a notion of residual modular forms for congruence subgroups of SL_2(Z) which contains the classical notion of Eichler integrals as a particular case. Further, we define differential Eisenstein series, which are residual modular forms. Using such differential Eisenstein series, for example, one obtains an explicit description of w_4(t). A certain Eichler cohomology group associated to such residual modular forms plays also an important role in the discussion.

File：2012-1