404 research outputs found

    Rates of mixing in models of fluid devices with discontinuities

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    In the simplest sense, mixing acts on an initially heterogeneous system, transforming it to a homogeneous state through the actions of stirring and diffusion. The theory of dynamical systems has been successful in improving understanding of underlying features in fluid mixing, and how smooth stirring fields, coherent structures and boundaries affect mixing rates. The main stirring mechanism in fluids at low Reynolds number is the stretching and folding of fluid elements, although this is not the only mechanism to achieve complicated dynamics. Mixing by cutting and shuffling occurs in many situations, for example in micro–fluidic split and recombine flows, through the closing and re-orientation of values in sink–source flows, and within the bulk flow of granular material. The dynamics of this mixing mechanism are subtle and not well understood. Here, mixing rates arising from fundamental models capturing the essence of discontinuous, chaotic stirring with diffusion are investigated. In purely cutting and shuffling flows it is found that the number of cuts introduced iteratively is the most important mechanism driving the approach to uniformity. A balance between cutting, shuffling and diffusion achieves a long-time exponential mixing rate, but similar mechanisms dominate the finite time mixing observed through the interaction of many slowly decaying eigenfunctions. The time to achieve a mixed condition varies polynomially with diffusivity rate κ, obeying t ∝ κ^{−η} . For the transformations meeting good stirring criteria, η < 1. Considering the time to achieve a mixed condition to be governed by a balance between cutting, shuffling, and diffusion derives η ∼ 1/2, which shows good agreement with numerical results. In stirring fields which are predominantly chaotic and exponentially mixing, it is observed that the addition of discontinuous transformations contaminates mixing when the stretching rates are uniform, or close to uniform. The contamination comes from an increase in scales of the concentration field by the reassembly of striations when cut and shuffled. Mixing stemming from this process is unpredictable, and the discontinuities destroy the possibility to approximate early mixing rates from stretching histories. A speed up in mixing rate can be achieved if the discontinuity aids particle transport into islands of the original transformation, or chops and rearranges large striations generated from highly non-uniform stretching. The long-time mixing rates and time to achieve a mixed condition are shown to behave counter-intuitively when varying the diffusivity rate. A deceleration of mixing with increasing diffusion coefficient is observed, sometimes overshooting analytically derived bounds

    A survey of Heegaard Floer homology

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    This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology. We construct the Heegaard Floer chain complex as a special case of the above, and try to motivate the role of the various seemingly ad hoc features such as admissibility, the choice of basepoint, and Spin^c-structures. We also discuss the proof of invariance of the homology up to isomorphism under all the choices made, and how to define Heegaard Floer homology using this in a functorial way (naturality). Next, we explain why Heegaard Floer homology is computable, and how it lends itself to the various combinatorial descriptions. The last chapter gives an overview of the definition and applications of sutured Floer homology, which includes sketches of some of the key proofs. Throughout, we have tried to collect some of the important open conjectures in the area. For example, a positive answer to two of these would give a new proof of the Poincar\'e conjecture.Comment: 38 pages, 1 figure, a few minor correction

    Critical Data Compression

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    A new approach to data compression is developed and applied to multimedia content. This method separates messages into components suitable for both lossless coding and 'lossy' or statistical coding techniques, compressing complex objects by separately encoding signals and noise. This is demonstrated by compressing the most significant bits of data exactly, since they are typically redundant and compressible, and either fitting a maximally likely noise function to the residual bits or compressing them using lossy methods. Upon decompression, the significant bits are decoded and added to a noise function, whether sampled from a noise model or decompressed from a lossy code. This results in compressed data similar to the original. For many test images, a two-part image code using JPEG2000 for lossy coding and PAQ8l for lossless coding produces less mean-squared error than an equal length of JPEG2000. Computer-generated images typically compress better using this method than through direct lossy coding, as do many black and white photographs and most color photographs at sufficiently high quality levels. Examples applying the method to audio and video coding are also demonstrated. Since two-part codes are efficient for both periodic and chaotic data, concatenations of roughly similar objects may be encoded efficiently, which leads to improved inference. Applications to artificial intelligence are demonstrated, showing that signals using an economical lossless code have a critical level of redundancy which leads to better description-based inference than signals which encode either insufficient data or too much detail.Comment: 99 pages, 31 figure

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Monitoring Pressure and Saturation Changes in a Clastic Reservoir from Time-lapse Seismic Data Using the Extended Elastic Impedance Method

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    The Extended Elastic Impedance (EEI) concept has been used by the oil industry primarily for lithology and fluid prediction in exploration and development projects. We present a method of reservoir monitoring that identifies and maps changes in pressure and saturation in a producing reservoir by applying EEI to time-lapse seismic data. The method uses time-lapse seismic difference data rotated to specific EEI ? angles which are optimised for the changes expected in a given reservoir

    Dynamic Response of Tunable Phononic Crystals and New Homogenization Approaches in Magnetoactive Composites

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    This research investigates dynamic response of tunable periodic structures and homogenization methods in magnetoelastic composites (MECs). The research on tunable periodic structures is focused on the design, modeling and understanding of wave propagation phenomena and the dynamic response of smart phononic crystals. High amplitude wrinkle formation is employed to study a one-dimensional phononic crystal slab consists of a thin film bonded to a thick compliant substrate. Buckling induced surface instability generates a wrinkly structure triggered by a compressive strain. It is demonstrated that surface periodic pattern and the corresponding large deformation can control elastic wave propagation in the low thickness composite slab. Simulation results show that the periodic wrinkly structure can be used as a smart phononic crystal which can switch band diagrams of the structure in a transformative manner. A magnetoactive phononic crystal is proposed which its dynamic properties are controlled by combined effects of large deformations and an applied magnetic field. Finite deformations and magnetic induction influence phononic characteristics of the periodic structure through geometrical pattern transformation and material properties. A magnetoelastic energy function is proposed to develop constitutive laws considering large deformations and magnetic induction in the periodic structure. Analytical and finite element methods are utilized to compute dispersion relation and band structure of the phononic crystal for different cases of deformation and magnetic loadings. It is demonstrated that magnetic induction not only controls the band diagram of the structure but also has a strong effect on preferential directions of wave propagation. Moreover, a thermally controlled phononic crystal is designed using ligaments of bi-materials in the structure.Comment: PhD mechanical engineering, University of Nevada, Reno (2015

    Developments in fractal geometry

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