On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere

Let and be the unit disk and the unit sphere, and let be a radially symmetric harmonic map heat flow, whose singularities coincide with downward energy jumps. Then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time.     

An example in holomorphic fixed point theory

If is the open unit ball in the Cartesian product furnished with the -norm , where and , then a holomorphic self-mapping of has a fixed point if and only if for some

     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

A Note on the KAM Theorem for Symplectic Mappings

The mapping version of Pöschel's theory on differentiable foliation structures of invariant tori is presented and the relevant estimates explicitly in terms of the diophantine constant and the nondegeneracy parameters of frequency maps are given. As a direct application of the main result, a generalization of Moser's small twist theorem to high dimensions is given.     

THE HEAT FLOWOF HARMONIC MAPS FROM NONCOMPACT MANIFOLDS

1.IntroductionLetMandNbetwoRiemannianmanifoldsofdimensionmandn.Supposetheirmetricsaregivenbydski~giid-c'dxianddsX~h.odu"duo.Letu:M~Nbeasmoothmap.Theenergydensityfunctionofuisgivenbye(u)~g'jCZh.,~Ical'.ThetotalenergyisdefinedbyE(u)=[e(u)dx.iMAmappingu:M~NiscalledaharmonicmapifitisaclassicalsolutionoftheEulerLagrangeequationofE(u)whichcanbewrittenasT"(U(X))~AU"(X) r3.(U(X))ZZg.j~0,whereT(u)iscalledthetensionfieldofu.Thecorrespondingparabolicsystemwithinitialdata"o(x)knownastheheatequ…     

Kodaira dimension of holomorphic singular foliations

We introduce numerical invariants of holomorphic singular foliations under bimeromorphic transformations of surfaces. The basic invariant is a foliated version of the Kodaira dimension of compact complex manifolds.The author was supported by CNPq-Brazil in 1998 and Conseil Régional de Bourgogne in 1999.     

A 250?GHz, 50?W, CW Second Harmonic Gyrotron

A design concept of a 250 GHz, ≈ 50 W, CW, second harmonic gyrotron for spectroscopy will be presented in this paper. Mode competition, in particular with regard to competition with first harmonic modes, and mode selection procedures are carefully investigated by considering various candidate modes and the TE_6,2 and TE_6,3 modes are chosen as probable operating modes. Cold cavity and self-consistent calculations of power and efficiency are carried out to determine optimum parameters for these two modes. The results indicate that an output power of well over 50 W, CW can be achieved.     

The matrix rate of return

In this paper we give definitions of matrix rates of return which do not depend on the choice of basis describing baskets. We give their economic interpretation. The matrix rate of return describes baskets of arbitrary type and extends portfolio analysis to the complex variable domain. This allows us for simultaneous analysis of evolution of baskets parameterized by complex variables in both continuous and discrete time models.     

An investigation into EMI-induced noise in nanometer multi-quantum well InGaN LEDs

This study investigates the low-frequency noise induced by electromagnetic radiation interference (EMI) in a nanometer multi-quantum well InGaN LED (NMQLED). Theoretical models of the noise spectra and the EMI are constructed. In general, a good agreement is identified between the experimental and theoretical results. Both sets of results reveal that the magnitude of the EMI-induced noise is related to the pulse height, the output load, the parasitic capacitance, the interference frequency and the interference amplitude. It is shown that the harmonic noise increases with an increasing interference amplitude and frequency. The techniques presented in this study provide a systematic approach for obtaining the interference noise and signal-to-noise ratio (SNR) in LEDs and similar wavelength-based semiconductor devices.