On distorted diffeomorphisms

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We sketch the construction of diffeomorphisms that are distorted in a group of C^1 maps yet undistorted in a group of C^2 maps.

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Ultrasound color Doppler imaging has been routinely used for the diagnosis of cardiovascular diseases, enabling real-time flow visualization through the Doppler effect. Yet, its inability to provide true flow velocity vectors due to its one-dimensional detection limits its efficacy. To overcome this limitation, various VFI schemes, including multi-angle beams, speckle tracking, and transverse oscillation, have been explored, with some already available commercially. However, many of these methods still rely on autocorrelation, which poses inherent issues such as underestimation, aliasing, and the need for large ensemble sizes. Conversely, speckle-tracking-based VFI enables lateral velocity estimation but suffers from significantly lower accuracy compared to axial velocity measurements. To address these challenges, we have presented a speckle-tracking-based VFI approach utilizing multi-angle ultrafast plane wave imaging. Our approach involves estimating axial velocity components projected onto individual steered plane waves, which are then combined to derive the velocity vector. Additionally, we've introduced a VFI visualization technique with high spatial and temporal resolutions capable of tracking flow particle trajectories. Simulation and flow phantom experiments demonstrate that the proposed VFI method outperforms both speckle-tracking-based VFI and autocorrelation VFI counterparts by at least a factor of three. Furthermore, in vivo measurements on carotid arteries using the Prodigy ultrasound scanner demonstrate the effectiveness of our approach compared to existing methods, providing a more robust imaging tool for hemodynamic studies. Learning objectives: - Understand fundamental limitations of color Doppler imaging. - Understand principles behind advanced vector flow imaging techniques. - Familiarize with the ultrasound speckle tracking technique and its implications in flow imaging. - Explore experiments conducted using multi-angle plane wave ultrafast imaging, specifically utilizing the pulse-sequence mode on a 128-channel ultrasound research platform.

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www.seribangash.com The Mariana Trench is one of the most remarkable geological features on Earth. Here are some details about it: Location: The Mariana Trench is located in the western Pacific Ocean, east of the Mariana Islands. It stretches for about 2,550 kilometers (1,580 miles) and is known as the deepest part of the world's oceans. Depth: The trench reaches incredible depths, with its deepest point known as the Challenger Deep, which plunges down to approximately 10,984 meters (36,037 feet) below sea level. To put this into perspective, if Mount Everest, the tallest mountain on Earth, were placed at the bottom of the Challenger Deep, its peak would still be over 2 kilometers (1.25 miles) underwater. Formation: The Mariana Trench was formed by the subduction of the Pacific Plate beneath the Mariana Plate. This process creates a deep trench as the heavier Pacific Plate is forced beneath the lighter Mariana Plate. Geological Features: The trench is characterized by steep, V-shaped valleys, and its walls are composed of highly compressed sedimentary rock. At the bottom of the trench, there are also large amounts of marine sediment. Pressure: The pressure at the bottom of the Mariana Trench is immense, reaching over 1,000 times the pressure at the surface. This extreme pressure creates a challenging environment for exploration and makes it difficult for organisms to survive. Exploration: Despite its extreme conditions, the Mariana Trench has been the subject of numerous scientific expeditions and explorations. One of the most famous explorations was the dive to the Challenger Deep by Swiss scientist Jacques Piccard and U.S. Navy Lieutenant Don Walsh in 1960. More recently, in 2012, filmmaker James Cameron made a solo dive to the bottom of the Challenger Deep in the Deepsea Challenger submersible. Biological Discoveries: Despite the harsh conditions, the Mariana Trench is home to a surprising variety of life forms, including unique species of deep-sea fish, crustaceans, and microbial life. Some organisms have adapted to survive in the extreme pressure and darkness of the trench. Environmental Importance: Studying the Mariana Trench provides valuable insights into the geology, biology, and oceanography of the deep sea. It also helps scientists better understand the processes that shape the Earth's crust and the distribution of life in the oceans. Conservation: Due to its remote location and extreme depths, the Mariana Trench has remained relatively untouched by human activity. However, there is growing concern about the potential impacts of deep-sea mining and pollution on this fragile ecosystem, highlighting the need for conservation efforts to protect this unique environment. https://seribangash.com/barber-shop-business-complete-guide-for-beginners/ https://seribangash.com/legend-virat-kohli-in-cricket-history/

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On distorted diffeomorphisms

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