SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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Data dependence of approximate current-vortex sheets near the onset of instability

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891617500175 ◽

2017 ◽

Vol 14 (03) ◽

pp. 517-534

Author(s):

Alessandro Morando ◽

Paolo Secchi ◽

Paola Trebeschi

Keyword(s):

Smooth Solutions ◽

Helmholtz Instability ◽

Small Perturbations ◽

Vortex Sheets ◽

Weakly Nonlinear ◽

Linearized Stability ◽

Stability And Instability ◽

The Cauchy Problem ◽

Time Existence ◽

Strong Norm

The paper is concerned with the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [On the weakly nonlinear Kelvin–Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equations 8(4) (2011) 691–726] have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was shown in [Approximate current-vortex sheets near the onset of instability, J. Math. Pures Appl. 105(4) (2016) 490–536; Existence of approximate current-vortex sheets near the onset of instability, J. Hyperbolic Differ. Equations]. In the present paper, we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

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Real-time FPGA-based implementation of the AKAZE algorithm with nonlinear scale space generation using image partitioning

Journal of Real-Time Image Processing ◽

10.1007/s11554-021-01089-9 ◽

2021 ◽

Author(s):

Parastoo Soleimani ◽

David W. Capson ◽

Kin Fun Li

Keyword(s):

Real Time ◽

Memory Management ◽

Scale Space ◽

Image Resolution ◽

Hardware Architecture ◽

Frame Rate ◽

Time Sharing ◽

Scale Invariant ◽

Nonlinear Scale Space ◽

Nonlinear Scale

AbstractThe first step in a scale invariant image matching system is scale space generation. Nonlinear scale space generation algorithms such as AKAZE, reduce noise and distortion in different scales while retaining the borders and key-points of the image. An FPGA-based hardware architecture for AKAZE nonlinear scale space generation is proposed to speed up this algorithm for real-time applications. The three contributions of this work are (1) mapping the two passes of the AKAZE algorithm onto a hardware architecture that realizes parallel processing of multiple sections, (2) multi-scale line buffers which can be used for different scales, and (3) a time-sharing mechanism in the memory management unit to process multiple sections of the image in parallel. We propose a time-sharing mechanism for memory management to prevent artifacts as a result of separating the process of image partitioning. We also use approximations in the algorithm to make hardware implementation more efficient while maintaining the repeatability of the detection. A frame rate of 304 frames per second for a $$1280 \times 768$$ 1280 × 768 image resolution is achieved which is favorably faster in comparison with other work.

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Rayleigh waves from high-gain long-period stations: Signal extraction, amplitude determination, and separation of overlapping wave trains

Bulletin of the Seismological Society of America ◽

10.1785/bssa0650061761 ◽

1975 ◽

Vol 65 (6) ◽

pp. 1761-1778 ◽

Cited By ~ 1

Author(s):

Eduard Berg

Keyword(s):

Rayleigh Waves ◽

Signal To Noise Ratio ◽

High Gain ◽

Clear Separation ◽

Novaya Zemlya ◽

Oceanic Surface ◽

Single Station ◽

Wave Trains ◽

Short Time

abstract For a signal-to-noise ratio between 0.2 and 0.1 on the original single-component records, amplitudes for Rayleigh waves over oceanic paths of 155° at station MAT and 98° at station KIP have been determined as 12 mμ and 24 mμ peak-to-peak, respectively, with a standard error of less than 11 per cent. In each case the processed correlation signal is the highest in a half-hour record. The method makes use of preliminary high-pass filtering and normalized reference earthquake-matched filtering, and takes full advantage of the well-dispersed oceanic surface wave. The method also provides high resolution of co-located events with short time separation, or of widely spaced events with Rayleigh waves arriving nearly simultaneously at a single station, when the summed vertical and radial matched filtered components are used. Examples include: (1) clear separation and amplitude determination at stations KIP and MAT of two MS = 6.5 earthquakes located 0.7° and 145 sec apart off the coast of central Chile; (2) clear separation at station KIP of a Novaya Zemlya mb = 4.8 event from interfering Rayleigh waves of an mb = 5.0 Kermadec Island earthquake arriving 120 to 140 sec prior to the searched event, with almost complete elimination of interference on the summed vertical and radial processed components; and (3) clear separation at station KIP of two co-located mb = 4.4 and 4.5 earthquakes 6 min apart off the coast of Chile, with determination of their amplitudes in the presence of interfering Rayleigh waves from two central Alaska earthquakes, the first (mb = 4.1) arriving 15 min prior to the first Chile Rayleigh wave and the second between the two Chile arrivals. The single-station threshold reached (10 and 25 digital units, p-p) for stations MAT and KIP at 155° and 98°, respectively, corresponds to an MS = 3.3 and probably can be improved further.

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Short time existence for the elastic flow of clamped curves

Mathematische Nachrichten ◽

10.1002/mana.201700368 ◽

2018 ◽

Vol 291 (13) ◽

pp. 2115-2116

Keyword(s):

Short Time ◽

Time Existence

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THE METHOD OF CONCENTRATION COMPACTNESS AND DISPERSIVE HAMILTONIAN EVOLUTION EQUATIONS

XVIIth International Congress on Mathematical Physics ◽

10.1142/9789814449243_0010 ◽

2013 ◽

pp. 174-196

Author(s):

W. SCHLAG

Keyword(s):

Evolution Equations ◽

Concentration Compactness ◽

Hamiltonian Evolution

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Ricci flow on manifolds with boundary with arbitrary initial metric

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0060 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Tsz-Kiu Aaron Chow

Keyword(s):

Ricci Flow ◽

Positive Curvature ◽

Curvature Operator ◽

Manifolds With Boundary ◽

Convex Boundary ◽

Uniqueness Of The Solution ◽

Positive Time ◽

Natural Boundary Conditions ◽

Short Time ◽

Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

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