According to computational complexity theory, mathematical problems have different levels of difficulty in the context of their solvability. While a classical computer can solve some problems â in polynomial timeâi.e., the time required for solving P is a polynomial function of the input sizeâit often fails to solve NP problems that scale exponentially with the problem size and thus cannot be solved in polynomial time. Classical computers based on semiconductor devices are, therefore, inadequate for solving sufficiently large NP problems.
In this regard, quantum computers are considered promising as they can perform a large number of operations in parallel. This, in turn, speeds up the NP problem-solving process. However, many physical implementations are highly sensitive to thermal fluctuations. As a result, quantum computers often demand stringent experimental conditions such extremely low temperatures for their implementation, making their fabrication complicated and expensive.
Fortunately, there is a lesser-known and as-yet underexplored alternative to quantum computing, known as probabilistic computing. Probabilistic computing utilizes what are called âstochastic nanodevices,â whose operations rely on thermal fluctuations, to solve NP problems efficiently. Unlike in the case of quantum computers, thermal fluctuations facilitate problem solving in probabilistic computing. As a result, probabilistic computing is, in fact, easier to implement in real life.
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.
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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 planeâlike font-changing letters on a page [1]. Such âplanarâ 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 directionâcalled the director. Researchers can design an LCE by choosing the director orientation at each point in the material. However, calculating the âdirector fieldâ 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. âOur results could be readily implemented by a wide range of experimentalists, as well as by engineers and designers,â 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 isnât so easily explained. At the level of atoms and their parts, measuring position and momentum simultaneously yields only probability. Knowing a particleâs exact state is a zero-sum game in which classical notions of determinism donât apply: the more certain we are about its momentum, the less certain we are about where it will be.
Weâre 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 â such as a dry, hacking cough, breathing difficulties and high fever â 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. Itâs critical, however, to create efficient models that can analyse large numbers of medical images without false negatives.
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 worldâs 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 natureâthey 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-Hamedâs 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.
Background. 0:00 Highlights. 6:55 The specific limits of evolution by natural selection. 10:50 Donâs born in a San Antonio Army hospital in 1955 (and his parentsâ background) 14:44 As a teenager big question he wanted answered, âAre we just machines?â 17:23 Donâs early work as a vision researcher; visual systems construct. 20:43 Carlosâs 3-part series on Fitness-Beats-Truth Theorem.
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.
If youâve been following my channel for a really long time, you might remember that some years ago I made a video about whether faster-than-light travel is possible. I was trying to explain why the arguments saying itâs impossible are inconclusive and we shouldnât throw out the possibility too quickly, but Iâm afraid I didnât make my case very well. This video is a second attempt. Hopefully this time itâll come across more clearly!
