On the existence of Liapunov functions for the problem of Lurie

Control theory and matrix analysis for existence of Liapunov functions for problem of Luri

Liapunov functions for the problem of Lurie

Popov criteria for asymptotic stability of Lurie system of differential equations and Liapunov function

A note on contact transformations

Contact transformation

Energy functions for Morse Smale system

Construction of energy functions for Morse system of vector fields on manifol

Periodic points of diffeomorphisms

Fixed points of diffeomorphism class of Smale and exponential functio

A Class of Functional Equations of Neutral Type

Class of functional integral equations in space of continuous function

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3

Unsupervised learning of overlapping image components using divisive input modulation

This paper demonstrates that nonnegative matrix factorisation is mathematically related to a class of neural networks that employ negative feedback as a mechanism of competition. This observation inspires a novel learning algorithm which we call Divisive Input Modulation (DIM). The proposed algorithm provides a mathematically simple and computationally efficient method for the unsupervised learning of image components, even in conditions where these elementary features overlap considerably. To test the proposed algorithm, a novel artificial task is introduced which is similar to the frequently-used bars problem but employs squares rather than bars to increase the degree of overlap between components. Using this task, we investigate how the proposed method performs on the parsing of artificial images composed of overlapping features, given the correct representation of the individual components; and secondly, we investigate how well it can learn the elementary components from artificial training images. We compare the performance of the proposed algorithm with its predecessors including variations on these algorithms that have produced state-of-the-art performance on the bars problem. The proposed algorithm is more successful than its predecessors in dealing with overlap and occlusion in the artificial task that has been used to assess performance

The effect of internal gravity waves on cloud evolution in sub-stellar atmospheres

Context. Sub-stellar objects exhibit photometric variability which is believed to be caused by a number of processes such as magnetically-driven spots or inhomogeneous cloud coverage. Recent sub-stellar models have shown that turbulent flows and waves, including internal gravity waves, may play an important role in cloud evolution.Aims. The aim of this paper is to investigate the effect of internal gravity waves on dust cloud nucleation and dust growth, and whether observations of the resulting cloud structures could be used to recover atmospheric density information.Methods. For a simplified atmosphere in two dimensions, we numerically solve the governing fluid equations to simulate the effect on dust nucleation and mantle growth as a result of the passage of an internal gravity wave. Furthermore, we derive an expression that relates the properties of the wave-induced cloud structures to observable parameters in order to deduce the atmospheric density.Results. Numerical simulations show that the density, pressure and temperature variations caused by gravity waves lead to an increase of dust nucleation by up to a factor 20, and dust mantle growth rate by up to a factor 1:6, compared to their equilibrium values. Through an exploration of the wider sub-stellar parameter space, we show that in absolute terms, the increase in dust nucleation due to internal gravity waves is stronger in cooler (T dwarfs) and TiO2-rich sub-stellar atmospheres. The relative increase however is greater in warm(L dwarf) and TiO2-poor atmospheres due to conditions less suited for efficient nucleation at equilibrium. These variations lead to banded areas in which dust formation is much more pronounced, and lead to banded cloud structures similar to those observed on Earth. Conclusions. Using the proposed method, potential observations of banded clouds could be used to estimate the atmospheric density of sub-stellar objects