Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

1999 ◽  
Vol 51 (5) ◽  
pp. 915-935 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Christoph Leuenberger

AbstractConsider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.

1973 ◽  
Vol 10 (3) ◽  
pp. 397-423 ◽  
Author(s):  
Lee A. Bertram

Previously derived shock solutions for a perfectly conducting perfect gas are used to compute shock polars for the one-dimensional unsteady and two- dimensional non-aligned shock representations. A new special-case shock solution, having a downstream particle velocity relative to the shock equal to upstream Alfvén velocity, is obtained, in addition to exhaustive analytical classification schemes for the shock polars. Eight classes of one-dimensional polars and twelve classes of two-dimensional polars are identified.


Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 517
Author(s):  
Mostafa ZAHRI

In this paper, we present a new model for simulating an interesting class of Islamic design. Based on periodic sequences on the one-dimensional manifolds, and from emerging numbers, we construct closed graphs with edges on the unit circle. These graphs build very nice shapes and lead to a symmetrical class of geometric patterns of so-called Islamic design. Moreover, we mathematically characterize and analyze some convergence properties of the used up-down sequences. Finally, four planar type of patterns are simulated.


1995 ◽  
Vol 06 (06) ◽  
pp. 805-823 ◽  
Author(s):  
MIRAN ČERNE

Stationary discs of fibrations over the unit circle ∂D are considered. It is shown that if all fibers of a fibration Σ⊆∂D×Cn over the unit circle ∂D are strongly pseudoconvex hypersurfaces in Cn, then for every stationary disc f of the fibration Σ one can define the partial indices of f. In the case all fibers of Σ are strictly convex, it is proved that all partial indices of a stationary disc f are 0. It is also proved that in the case a stationary disc f of the fibration Σ is non-degenerate, the only possible partial indices of f are 0, 1 and –1. In particular, these results give information on the polynomial hull of Σ and new proofs of results related to the smoothness of the Kobayashi metric on some strongly pseudoconvex domains in Cn.


1984 ◽  
Vol 32 (2) ◽  
pp. 197-205 ◽  
Author(s):  
B. Abraham-Shrauner

An exact solution for a temporally damped sinusoidal electric field which obeys the nonlinear, one-dimensional Vlasov-Maxwell equations is given. The electric field is a generalization of the O'Neil model electric field for Landau damping of plasma oscillations. The electric field is a special case of the form found from the invariance of the one-dimensional Vlasov equation under infinitesimal Lie group transformations. The time dependences of the damping decrement, of the wave-number and of the angular frequency are derived. Use of a time-dependent BGK one-particle distribution function is justified for weak damping where, in general, it is necessary to carry out a numerical calculation of the invariant of which the distribution function is a functional.


2012 ◽  
Vol 09 (03) ◽  
pp. 511-543 ◽  
Author(s):  
JUSTIN HOLMER ◽  
QUANHUI LIN

We show that, for the one-dimensional cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces.


Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.


1995 ◽  
Vol 52 (1) ◽  
pp. 97-105 ◽  
Author(s):  
Miran Černe

Constructed are strictly increasing smooth families Σt ⊆ ∂D × C2, t ∈ [0, 1], of fibrations over the unit circle with strongly pseudoconvex fibers all diffeomorphic to the ball such that there is no analytic selection of the polynomial hull of Σ0 and which end at the product fibration . In particular these examples show that the continuity method for describing the polynomial hull of a fibration over ∂D fails even if the complex geometry of the fibers is relatively simple.


Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.


2015 ◽  
Vol 3 ◽  
Author(s):  
VJEKOSLAV KOVAČ ◽  
CHRISTOPH THIELE ◽  
PAVEL ZORIN-KRANICH

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.


2012 ◽  
Vol 11 (04) ◽  
pp. 1250030 ◽  
Author(s):  
S. N. ETHIER ◽  
JIYEON LEE

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N ≥ 3 and p0, p1, p2, p3 ∈ [0, 1], and let game A be the special case p0 = p1 = p2 = p3 = 1/2. In previous work we investigated μB and μ(1/2, 1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A + B). These means were computable for 3 ≤ N ≤ 19, at least, and appeared to converge as N → ∞, suggesting that the Parrondo region (i.e., the region in which μB ≤ 0 and μ(1/2, 1/2) > 0) has nonzero volume in the limit. The convergence was established under certain conditions, and the limits were expressed in terms of a parameterized spin system on the one-dimensional integer lattice. In this paper we replace the random mixture with the nonrandom periodic pattern Ar Bs, where r and s are positive integers. We show that μ[r, s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern Ar Bs, is computable for 3 ≤ N ≤ 18 and r + s ≤ 4, at least, and appears to converge as N → ∞, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (μB ≤ 0 and μ[r, s] > 0) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.


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