Pure & Applied Mathematics (Scholarly Publications) http://hdl.handle.net/2262/172 2013-05-15T13:30:25Z 2013-05-15T13:30:25Z ZAITSEV, DMITRI http://hdl.handle.net/2262/64093 2012-06-29T09:31:07Z 2010-01-01T00:00:00Z

Title: On Levi-flat hypersurfaces with prescribed boundary Author: ZAITSEV, DMITRI Abstract: We address the problem of existence and uniqueness of a Levi-flat hypersurface M in ℂn with prescribed compact boundary S for n ≥ 3. The situation for n ≥ 3 differs sharply from the well studied case n = 2. We first establish necessary conditions on S at both complex and CR points, needed for the existence of M. All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity for n = 2 and excluding the possibility of S to contain complex (n - 2)-dimensional submanifolds, we obtain a solution M to the above problem as a projection of a possibly singular Levi-flat hypersurface in ℝ × ℂn. It turns out that S has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of M. There are no more global assumptions on S like being contained in the boundary of a strongly pseudoconvex domain, as it was in case n = 2. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary S must coincide with the constructed solution. Description: PUBLISHED

2010-01-01T00:00:00Z ZAITSEV, DMITRI http://hdl.handle.net/2262/64092 2012-06-29T09:18:31Z 2011-01-01T00:00:00Z

Title: Dynamics of one-resonant biholomorphisms Author: ZAITSEV, DMITRI Abstract: Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in Cn whose differentials have one-dimensional family of resonances in the first m eigenvalues, m ≤ n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions Description: IN_PRESS

2011-01-01T00:00:00Z ZAITSEV, DMITRI http://hdl.handle.net/2262/64092 2012-06-29T09:18:31Z 2011-01-01T00:00:00Z

Title: Dynamics of one-resonant biholomorphisms Author: ZAITSEV, DMITRI Abstract: Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in Cn whose differentials have one-dimensional family of resonances in the first m eigenvalues, m ≤ n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions Description: IN_PRESS

2011-01-01T00:00:00Z ZAITSEV, DMITRI http://hdl.handle.net/2262/64090 2012-06-29T09:06:17Z 2011-01-01T00:00:00Z

Title: Formal and finite order equivalences Author: ZAITSEV, DMITRI Abstract: We show that two families of germs of real-analytic subsets in Cn are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent. Description: IN_PRESS

2011-01-01T00:00:00Z ZAITSEV, DMITRI http://hdl.handle.net/2262/64089 2012-06-29T08:53:17Z 2009-01-01T00:00:00Z

Title: A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear partial differential equations and geometric applications. Author: ZAITSEV, DMITRI Abstract: The main motivation for the work presented in this paper is to construct real hypersurfaces in Cn+1 with maximal Levi number (see below for the definition), a problem that has been open since Levi numbers were introduced in [BHR96]. The examples constructed here are tube hypersurfaces. Moreover, we give a local description of all such hypersurfaces. In order for the real hypersurface M to have the desired properties, Σ must be a non-cylindrical hypersurface whose Gauss map (or equivalently second fundamental form) has rank one. To construct locally defined hypersurfaces in Rn+1 with these properties, indeed to parametrize all such, we prove an existence and uniqueness theorem concerning a Cauchy problem for a class of overdetermined systems of nonlinear partial di erential equations in Rn. Description: PUBLISHED

2009-01-01T00:00:00Z KARAGEORGIS, PASCHALIS http://hdl.handle.net/2262/63768 2012-12-11T13:49:26Z 2012-01-01T00:00:00Z

Title: Dispersion relation for water waves with non-constant vorticity Author: KARAGEORGIS, PASCHALIS Abstract: We derive the dispersion relation for linearized small-amplitude gravity waves for various choices of non-constant vorticity. To the best of our knowledge, this relation is only known explicitly in the case of constant vorticity. We provide a wide range of examples including polynomial, exponential, trigonometric and hyperbolic vorticity functions. Description: IN_PRESS

2012-01-01T00:00:00Z FRY, MICHAEL PHILIP http://hdl.handle.net/2262/62175 2012-02-15T03:32:51Z 2011-01-01T00:00:00Z

Title: Stability of QED Author: FRY, MICHAEL PHILIP Abstract: It is shown for a class of random, time-independent, square-integrable, three-dimensional magnetic fields that the one-loop effective fermion action of four-dimensional QED increases faster than a quadratic in B in the strong coupling limit. The limit is universal. The result relies on the paramagnetism of charged spin-1/2 fermions and the diamagnetism of charged scalar bosons. Description: PUBLISHED

2011-01-01T00:00:00Z LIU, LIUMING VILASECA MAINAR, POL PEARDON, MICHAEL JAMES RYAN, SINEAD MARIE http://hdl.handle.net/2262/61489 2012-01-04T11:58:46Z 2011-01-01T00:00:00Z

Title: Charmonium spectroscopy from an anisotropic lattice study Author: LIU, LIUMING; VILASECA MAINAR, POL; PEARDON, MICHAEL JAMES; RYAN, SINEAD MARIE Abstract: We present a progress report on our study of the charmonium spectrum in full QCD on anisotropic lattices generated by the Hadron Spectrum Collaboration. We adopt a large basis of interpolating operators to extract the excited charmonium states using the variational method. A detailed spectrum of excited charmonium mesons in many $J^{PC}$ channels is obtained. Hybrid states with exotic and non-exotic quantum numbers are determined and preliminary results from a study of disconnected contributions to the $\eta_c$ are presented. Description: PUBLISHED

2011-01-01T00:00:00Z BYKOV, DMITRI VLADIMIROVICH http://hdl.handle.net/2262/60703 2011-11-15T15:48:13Z 2012-01-01T00:00:00Z

Title: Haldane limits via Lagrangian embeddings Author: BYKOV, DMITRI VLADIMIROVICH Abstract: In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(3)/U3(1). We discuss possible other future applications of Lagrangian/isotropic embeddings in this context. Description: PUBLISHED

2012-01-01T00:00:00Z SEN, SIDDHARTHA http://hdl.handle.net/2262/60105 2011-10-13T14:54:08Z 2011-01-01T00:00:00Z

Title: An Effective Theory of Superfluid Turbulence from Local Scale Invariance Author: SEN, SIDDHARTHA Abstract: An effective locally scale invariant model is constructed using Weyl’s method starting from a free Schroedinger equation. The model requires additional gauge and gravitational degrees of freedom. It is suggested that this scale invariant model is an effective theory for superfluid turbulence. The additional degrees of freedom introduced can then be identified with filament excitations or zeros of a Gross-Pitaevski equation which are used to describe superfluid turbu- lence. Qualitative estimates of the way filaments separate after collision are made which agree with observations. Description: PUBLISHED

2011-01-01T00:00:00Z