Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem
Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem

Author: Gabriella Pinzari

Publisher: American Mathematical Soc.

Published: 2018-10-03

Total Pages: 104

ISBN-13: 1470441020

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The author proves the existence of an almost full measure set of -dimensional quasi-periodic motions in the planetary problem with masses, with eccentricities arbitrarily close to the Levi–Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V. I. Arnold in the 1960s, for the particular case of the planar three-body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group. The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, a common tool of previous literature.

Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem
Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem

Author: Gabriella Pinzari

Publisher:

Published: 2018

Total Pages: 92

ISBN-13: 9781470448134

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We prove the existence of an almost full measure set of (3n − 2)-dimensional quasi-periodic motions in the planetary problem with (1 + n) masses, with eccentricities arbitrarily close to the Levi-Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V. I. Arnold (1963) in the 60s, for the particular case of the planar three-body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group. The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, common tool of previous literature.

Global Regularity for 2D Water Waves with Surface Tension
Global Regularity for 2D Water Waves with Surface Tension

Author: Alexandru D. Ionescu

Publisher: American Mathematical Soc.

Published: 2019-01-08

Total Pages: 136

ISBN-13: 1470431033

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The authors consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of the authors' analysis is to develop a sufficiently robust method (the “quasilinear I-method”) which allows the authors to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called “division problem”). As a result, they are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of the authors' analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.

On Space-Time Quasiconcave Solutions of the Heat Equation
On Space-Time Quasiconcave Solutions of the Heat Equation

Author: Chuanqiang Chen

Publisher: American Mathematical Soc.

Published: 2019-06-10

Total Pages: 94

ISBN-13: 1470435241

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In this paper the authors first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, they obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain their ideas and for completeness, the authors also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

Interpolation for Normal Bundles of General Curves
Interpolation for Normal Bundles of General Curves

Author: Atanas Atanasov

Publisher: American Mathematical Soc.

Published: 2019-02-21

Total Pages: 118

ISBN-13: 147043489X

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Given n general points p1,p2,…,pn∈Pr, it is natural to ask when there exists a curve C⊂Pr, of degree d and genus g, passing through p1,p2,…,pn. In this paper, the authors give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle NC of a general nonspecial curve of degree d and genus g in Pr (with d≥g+r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H0(NC(−D))=0 or H1(NC(−D))=0), with exactly three exceptions.

Multilinear Singular Integral Forms of Christ-Journe Type
Multilinear Singular Integral Forms of Christ-Journe Type

Author: Andreas Seeger

Publisher: American Mathematical Soc.

Published: 2019-02-21

Total Pages: 146

ISBN-13: 1470434377

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We introduce a class of multilinear singular integral forms which generalize the Christ-Journe multilinear forms. The research is partially motivated by an approach to Bressan’s problem on incompressible mixing flows. A key aspect of the theory is that the class of operators is closed under adjoints (i.e. the class of multilinear forms is closed under permutations of the entries). This, together with an interpolation, allows us to reduce the boundedness.

An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg-Witten Invariants
An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg-Witten Invariants

Author: Paul Feehan

Publisher: American Mathematical Soc.

Published: 2019-01-08

Total Pages: 254

ISBN-13: 147041421X

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The authors prove an analogue of the Kotschick–Morgan Conjecture in the context of monopoles, obtaining a formula relating the Donaldson and Seiberg–Witten invariants of smooth four-manifolds using the -monopole cobordism. The main technical difficulty in the -monopole program relating the Seiberg–Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible monopoles, namely the moduli spaces of Seiberg–Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of monopoles. In this monograph, the authors prove—modulo a gluing theorem which is an extension of their earlier work—that these intersection pairings can be expressed in terms of topological data and Seiberg–Witten invariants of the four-manifold. Their proofs that the -monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze and Witten's Conjecture in full generality for all closed, oriented, smooth four-manifolds with and odd appear in earlier works.

Algebras of Singular Integral Operators with Kernels Controlled by Multiple Norms
Algebras of Singular Integral Operators with Kernels Controlled by Multiple Norms

Author: Alexander Nagel

Publisher: American Mathematical Soc.

Published: 2019-01-08

Total Pages: 156

ISBN-13: 1470434385

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The authors study algebras of singular integral operators on R and nilpotent Lie groups that arise when considering the composition of Calderón-Zygmund operators with different homogeneities, such as operators occuring in sub-elliptic problems and those arising in elliptic problems. These algebras are characterized in a number of different but equivalent ways: in terms of kernel estimates and cancellation conditions, in terms of estimates of the symbol, and in terms of decompositions into dyadic sums of dilates of bump functions. The resulting operators are pseudo-local and bounded on for . . While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators studied here fall outside this class, and reflect a multi-parameter structure.

Continuous-Time Random Walks for the Numerical Solution of Stochastic Differential Equations
Continuous-Time Random Walks for the Numerical Solution of Stochastic Differential Equations

Author: Nawaf Bou-Rabee

Publisher: American Mathematical Soc.

Published: 2019-01-08

Total Pages: 136

ISBN-13: 1470431815

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This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs.

Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems
Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems

Author: Laurent Lazzarini

Publisher: American Mathematical Soc.

Published: 2019-02-21

Total Pages: 122

ISBN-13: 147043492X

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A wandering domain for a diffeomorphism of is an open connected set such that for all . The authors endow with its usual exact symplectic structure. An integrable diffeomorphism, i.e., the time-one map of a Hamiltonian which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory; lower estimates are related to examples of Arnold diffusion. This is a contribution to the “quantitative Hamiltonian perturbation theory” initiated in previous works on the optimality of long term stability estimates and diffusion times; the emphasis here is on discrete systems because this is the natural setting to study wandering domains.