## Archive for the ‘mathematics’ category: Page 7

This video is about the story of two geniuses, Albert Einstein and the famous logician Kurt Godel. It is about their meeting at IAS, Princeton, New Jersey, when they both walked and discussed many things. For Godel, Einstein was his best friend and till his last days, he remain close to Einstein. Their nature was opposite to each other, yet both of them were very good friends. What did they talk about with each other? What did they share? What were their thoughts? For Godel, Einstein was more like his guide and for Einstein, it was a great pleasure to walk with him.

In the first episode, we discover their first meeting with each other and the development of friendship between them.

Liquid-crystal elastomers (LCEs) are shape-shifting materials that stretch or squeeze when stimulated by an external input such as heat, light, or a voltage. Designing these materials to produce desired shapes is a challenging math problem, but Daniel Castro and Hillel Aharoni from the Weizmann Institute of Science, Israel, have now provided an analytical solution for flat materials that shape-shift within a single plane—like font-changing letters on a page [1]. Such “planar” designs could help in producing rods that change their cross section (from, say, round to square) without buckling.

LCEs consist of networks of polymer fibers containing liquid-crystal molecules. When exposed to a stimulus, the molecules align in a way that causes the material to shrink or extend in a predefined direction—called the director. Researchers can design an LCE by choosing the director orientation at each point in the material. However, calculating the “director field” for an arbitrary shape change is difficult, so approximate methods are typically used.

Castro and Aharoni focused on a specific design problem: how to create an LCE that stretches only in two dimensions. These planar LCEs often suffer from residual stress that causes the material to wrinkle or buckle out of the plane. The researchers showed that finding a buckle-free design is similar to a well-known mathematical problem that has been studied in other contexts, such as minimizing the mass of load-carrying structures. Taking inspiration from these previous studies, Castro and Aharoni provided a method for exactly deriving the director field for any desired planar LCE. “Our results could be readily implemented by a wide range of experimentalists, as well as by engineers and designers,” Aharoni says.

For one, classical physics can predict, with simple mathematics, how an object will move and where it will be at any given point in time and space. How objects interact with each other and their environments follow laws we first encounter in high school science textbooks.

What happens in minuscule realms isn’t so easily explained. At the level of atoms and their parts, measuring position and momentum simultaneously yields only probability. Knowing a particle’s exact state is a zero-sum game in which classical notions of determinism don’t apply: the more certain we are about its momentum, the less certain we are about where it will be.

We’re not exactly sure what it will be, either. That particle could be both an electron and a wave of energy, existing in multiple states at once. When we observe it, we force a quantum choice, and the particle collapses from its state of superposition into one of its possible forms.

Pneumonia is a potentially fatal lung infection that progresses rapidly. Patients with pneumonia symptoms – such as a dry, hacking cough, breathing difficulties and high fever – generally receive a stethoscope examination of the lungs, followed by a chest X-ray to confirm diagnosis. Distinguishing between bacterial and viral pneumonia, however, remains a challenge, as both have similar clinical presentation.

Mathematical modelling and artificial intelligence could help improve the accuracy of disease diagnosis from radiographic images. Deep learning has become increasingly popular for medical image classification, and several studies have explored the use of convolutional neural network (CNN) models to automatically identify pneumonia from chest X-ray images. It’s critical, however, to create efficient models that can analyse large numbers of medical images without false negatives.

Now, K M Abubeker and S Baskar at the Karpagam Academy of Higher Education in India have created a novel machine learning framework for pneumonia classification of chest X-ray images on a graphics processing unit (GPU). They describe their strategy in Machine Learning: Science and Technology.

Describes and demonstrates the MR technique of Diffusion Tensor Imaging and reviews some of the basic mathematics of Tensors including matrix multiplication, eigenvalues and eigenvectors.

Providing increased resistance to outside interference, topological qubits create a more stable foundation than conventional qubits. This increased stability allows the quantum computer to perform computations that can uncover solutions to some of the world’s toughest problems.

While qubits can be developed in a variety of ways, the topological qubit will be the first of its kind, requiring innovative approaches from design through development. Materials containing the properties needed for this new technology cannot be found in nature—they must be created. Microsoft brought together experts from condensed matter physics, mathematics, and materials science to develop a unique approach producing specialized crystals with the properties needed to make the topological qubit a reality.

Donald Hoffman interview on spacetime, consciousness, and how biological fitness conceals reality. We discuss Nima Arkani-Hamed’s Amplituhedron, decorated permutations, evolution, and the unlimited intelligence.

The Amplituhedron is a static, monolithic, geometric object with many dimensions. Its volume codes for amplitudes of particle interactions & its structure codes for locality and unitarity. Decorated permutations are the deepest core from which the Amplituhedron gets its structure. There are no dynamics, they are monoliths as in 2001: A Space Odyssey.

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Try out my quantum mechanics course (and many others on math and science) on Brilliant using the link https://brilliant.org/sabine. You can get started for free, and the first 200 will get 20% off the annual premium subscription.

face_with_colon_three year 2017.

First observed in liquid helium below the lambda point, superfluidity manifests itself in a number of fascinating ways. In the superfluid phase, helium can creep up along the walls of a container, boil without bubbles, or even flow without friction around obstacles. As early as 1938, Fritz London suggested a link between superfluidity and Bose–Einstein condensation (BEC)3. Indeed, superfluidity is now known to be related to the finite amount of energy needed to create collective excitations in the quantum liquid4,5,6,7, and the link proposed by London was further evidenced by the observation of superfluidity in ultracold atomic BECs1,8. A quantitative description is given by the Gross–Pitaevskii (GP) equation9,10 (see Methods) and the perturbation theory for elementary excitations developed by Bogoliubov11. First derived for atomic condensates, this theory has since been successfully applied to a variety of systems, and the mathematical framework of the GP equation naturally leads to important analogies between BEC and nonlinear optics12,13,14. Recently, it has been extended to include condensates out of thermal equilibrium, like those composed of interacting photons or bosonic quasiparticles such as microcavity exciton-polaritons and magnons14,15. In particular, for exciton-polaritons, the observation of many-body effects related to condensation and superfluidity such as the excitation of quantized vortices, the formation of metastable currents and the suppression of scattering from potential barriers2,16,17,18,19,20 have shown the rich phenomenology that exists within non-equilibrium condensates. Polaritons are confined to two dimensions and the reduced dimensionality introduces an additional element of interest for the topological ordering mechanism leading to condensation, as recently evidenced in ref. 21. However, until now, such phenomena have mainly been observed in microcavities embedding quantum wells of III–V or II–VI semiconductors. As a result, experiments must be performed at low temperatures (below ∼ 20 K), beyond which excitons autoionize. This is a consequence of the low binding energy typical of Wannier–Mott excitons. Frenkel excitons, which are characteristic of organic semiconductors, possess large binding energies that readily allow for strong light–matter coupling and the formation of polaritons at room temperature. Remarkably, in spite of weaker interactions as compared to inorganic polaritons22, condensation and the spontaneous formation of vortices have also been observed in organic microcavities23,24,25. However, the small polariton–polariton interaction constants, structural inhomogeneity and short lifetimes in these structures have until now prevented the observation of behaviour directly related to the quantum fluid dynamics (such as superfluidity). In this work, we show that superfluidity can indeed be achieved at room temperature and this is, in part, a result of the much larger polariton densities attainable in organic microcavities, which compensate for their weaker nonlinearities.

Our sample consists of an optical microcavity composed of two dielectric mirrors surrounding a thin film of 2,7-Bis[9,9-di(4-methylphenyl)-fluoren-2-yl]-9,9-di(4-methylphenyl)fluorene (TDAF) organic molecules. Light–matter interaction in this system is so strong that it leads to the formation of hybrid light–matter modes (polaritons), with a Rabi energy 2 ΩR ∼ 0.6 eV. A similar structure has been used previously to demonstrate polariton condensation under high-energy non-resonant excitation24. Upon resonant excitation, it allows for the injection and flow of polaritons with a well-defined density, polarization and group velocity.

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