Stability of Line Solitons for the KP-II Equation in $\mathbb {R}^2$
Stability of Line Solitons for the KP-II Equation in $\mathbb {R}^2$

Author: Tetsu Mizumachi

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 110

ISBN-13: 1470414244

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The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as . He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward . The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.

Proof of the 1-Factorization and Hamilton Decomposition Conjectures
Proof of the 1-Factorization and Hamilton Decomposition Conjectures

Author: Béla Csaba

Publisher: American Mathematical Soc.

Published: 2016-10-05

Total Pages: 176

ISBN-13: 1470420252

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In this paper the authors prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D≥2⌈n/4⌉−1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that D≥⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ≥n/2. Then G contains at least regeven(n,δ)/2≥(n−2)/8 edge-disjoint Hamilton cycles. Here regeven(n,δ) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case δ=⌈n/2⌉ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

The $abc$-Problem for Gabor Systems
The $abc$-Problem for Gabor Systems

Author: Xin-Rong Dai

Publisher: American Mathematical Soc.

Published: 2016-10-05

Total Pages: 116

ISBN-13: 1470420155

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A longstanding problem in Gabor theory is to identify time-frequency shifting lattices aZ×bZ and ideal window functions χI on intervals I of length c such that {e−2πinbtχI(t−ma): (m,n)∈Z×Z} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above abc-problem for Gabor systems.

Rohlin Flows on von Neumann Algebras
Rohlin Flows on von Neumann Algebras

Author: Toshihiko Masuda

Publisher: American Mathematical Soc.

Published: 2016-10-05

Total Pages: 128

ISBN-13: 1470420163

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The authors will classify Rohlin flows on von Neumann algebras up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III0 factors. Several concrete examples are also studied.

Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations
Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations

Author: Volker Bach

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 134

ISBN-13: 1470417057

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The authors study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. They specify assumptions that ensure the global existence of its solutions and allow them to derive its asymptotics at temporal infinity. They demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

Moduli of Double EPW-Sextics
Moduli of Double EPW-Sextics

Author: Kieran G. O'Grady

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 188

ISBN-13: 1470416964

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The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of ⋀3C6 modulo the natural action of SL6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK 4-folds of Type K3[2] polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. The author will determine the stable points. His work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds.

Classification of $E_0$-Semigroups by Product Systems
Classification of $E_0$-Semigroups by Product Systems

Author: Michael Skeide

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 138

ISBN-13: 1470417383

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In these notes the author presents a complete theory of classification of E0-semigroups by product systems of correspondences. As an application of his theory, he answers the fundamental question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial.

Classes of Polish Spaces Under Effective Borel Isomorphism
Classes of Polish Spaces Under Effective Borel Isomorphism

Author: Vassilios Gregoriades

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 102

ISBN-13: 1470415631

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The author studies the equivalence classes under Δ11 isomorphism, otherwise effective Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and he shows the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of Δ11-reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals.

The Fourier Transform for Certain HyperKahler Fourfolds
The Fourier Transform for Certain HyperKahler Fourfolds

Author: Mingmin Shen

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 178

ISBN-13: 1470417405

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Using a codimension-1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH∗(A). By using a codimension-2 algebraic cycle representing the Beauville-Bogomolov class, the authors give evidence for the existence of a similar decomposition for the Chow ring of Hyperkähler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a K3 surface. They indeed establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.