Archive for the ‘mathematics’ category: Page 5

Sep 17, 2023

Brain Asymmetry Driven by Task Complexity

Posted by in categories: biotech/medical, life extension, mathematics, robotics/AI

A mathematical model shows how increased intricacy of cognitive tasks can break the mirror symmetry of the brain’s neural network.

The neural networks of animal brains are partly mirror symmetric, with asymmetries thought to be more common in more cognitively advanced species. This assumption stems from a long-standing theory that increased complexity of neural tasks can turn mirror-symmetric neural circuits into circuits existing in only one side of the brain. This hypothesis has now received support from a mathematical model developed by Luís Seoane at the National Center for Biotechnology in Spain [1]. The researcher’s findings could help explain how the brain’s architecture is shaped not only by cognitively demanding tasks but also by damage or aging.

A mirror-symmetric neural network is useful when controlling body parts that are themselves mirror symmetric, such as arms and legs. Moreover, the presence of duplicate circuits on each side of the brain can help increase computing accuracy and offer a replacement circuit if one becomes faulty. However, the redundancy created by such duplication can lead to increased energy consumption. This trade-off raises an important question: Does the optimal degree of mirror symmetry depend on the complexity of the cognitive tasks performed by the neural network?

Sep 16, 2023

Life-Changing Cystic Fibrosis Treatment Wins $3-Million Breakthrough Prize

Posted by in categories: biotech/medical, mathematics

A trio of scientists who developed the combination drug Trikafta are among the winners of five major awards in life sciences, physics and mathematics.

Sep 16, 2023

If you’d bought Apple shares instead of iPhones, you’d now have $147,000

Posted by in categories: mathematics, mobile phones

What would happen if, instead of buying the newest iPhone every time Apple launches one, you bought that same amount of Apple stock? There is a tweet floating around saying that if you had bought Apple shares instead of an iPhone every time they came out, you’d have hundreds of millions of dollars. The math is off (if you’d spent $20k on Apple stock when the rumors of the iPhone first started, you’d have $1.5 million today, at best) but in any case – it’d only make sense if you were clairvoyant in 2007, and knew when Apple would be launching phones, and at which price.

I figured a more fair way of calculating it would be to imagine buy a top-of-the-line iPhone every time Apple releases a new iPhone, or spend the same amount on Apple stock. If you had done that, by my calculations, you’d have spent around $16,000 on iPhones over the years (that’s around $20,000 in today’s dollars). If you’d bought Apple shares instead, you’d today have $147,000 or so — or a profit of around $131,000.

Sep 15, 2023

Mathematicians find 12,000 new solutions to ‘unsolvable’ 3-body problem

Posted by in category: mathematics

Calculating the way three things orbit each other is notoriously tricky — but a new study may reveal 12,000 new ways to make it work.

Sep 14, 2023

We’ve Been Misreading a Major Law of Physics For The Past 300 Years

Posted by in categories: mathematics, physics

When Isaac Newton inscribed onto parchment his now-famed laws of motion in 1,687, he could have only hoped we’d be discussing them three centuries later.

Writing in Latin, Newton outlined three universal principles describing how the motion of objects is governed in our Universe, which have been translated, transcribed, discussed and debated at length.

But according to a philosopher of language and mathematics, we might have been interpreting Newton’s precise wording of his first law of motion slightly wrong all along.

Sep 14, 2023

Toward a Complete Theory of Crystal Vibrations

Posted by in categories: computing, information science, mathematics, particle physics

A new set of equations captures the dynamical interplay of electrons and vibrations in crystals and forms a basis for computational studies.

Although a crystal is a highly ordered structure, it is never at rest: its atoms are constantly vibrating about their equilibrium positions—even down to zero temperature. Such vibrations are called phonons, and their interaction with the electrons that hold the crystal together is partly responsible for the crystal’s optical properties, its ability to conduct heat or electricity, and even its vanishing electrical resistance if it is superconducting. Predicting, or at least understanding, such properties requires an accurate description of the interplay of electrons and phonons. This task is formidable given that the electronic problem alone—assuming that the atomic nuclei stand still—is already challenging and lacks an exact solution. Now, based on a long series of earlier milestones, Gianluca Stefanucci of the Tor Vergata University of Rome and colleagues have made an important step toward a complete theory of electrons and phonons [1].

At a low level of theory, the electron–phonon problem is easily formulated. First, one considers an arrangement of massive point charges representing electrons and atomic nuclei. Second, one lets these charges evolve under Coulomb’s law and the Schrödinger equation, possibly introducing some perturbation from time to time. The mathematical representation of the energy of such a system, consisting of kinetic and interaction terms, is the system’s Hamiltonian. However, knowing the exact theory is not enough because the corresponding equations are only formally simple. In practice, they are far too complex—not least owing to the huge number of particles involved—so that approximations are needed. Hence, at a high level, a workable theory should provide the means to make reasonable approximations yielding equations that can be solved on today’s computers.

Sep 13, 2023

Trajectoids: Creating a shape that rolls along a desired path

Posted by in categories: mathematics, media & arts, robotics/AI, transportation

Normally, when we think of a rolling object, we tend to imagine a torus (like a bicycle wheel) or a sphere (like a tennis ball) that will always follow a straight path when rolling. However, the world of mathematics and science is always open to exploring new ideas and concepts. This is why researchers have been studying shapes, like oloids, sphericons and more, which do not roll in straight lines.

All these funky shapes are really interesting to researchers as they can show us new ways to move objects around smoothly and efficiently. For example, imagine reducing the energy required to make a toy robot move, or mixing ingredients more thoroughly with a unique-looking spoon. While these peculiar shapes have been studied before, scientists have now taken it a step further.

Consider a game where you draw a on a tilted table—similar to tilting a pinball table to make the ball go in a particular direction. Now, try to come up with a 3D object that, when placed at the top of the table, will roll down and exactly follow that path, instead of just going straight down. There are a few other rules of this game: the table needs to be inclined slightly (and not too much), there should be no slipping during rolling, and the initial orientation of the object can be chosen at launch. Plus, the path you draw must never go uphill and must be periodic. It must also consist of identical repeating segments—somewhat like in music rhythm patterns.

Sep 9, 2023

GPT Can Solve Mathematical Problems Without a Calculator

Posted by in category: mathematics

Join the discussion on this paper page.

Sep 8, 2023

Math Reveals Secrets of Synchronization in Complex Systems

Posted by in categories: mathematics, neuroscience

Summary: Researchers delved deep into the mysteries of synchronization in complex systems, uncovering how certain elements effortlessly fall into or out of sync. This dance of coordination can be observed from humans clapping in rhythm to the synchronicity of heart cells.

By studying “walks” through networks, the team discovered the role of convergent walks in diminishing the quality of synchronization. These findings could revolutionize our understanding of everything from power grid stability to brain functions and social media dynamics.

Sep 6, 2023

Scientists reveal the hidden math that governs how neurons cluster in the brain

Posted by in categories: computing, mathematics, neuroscience

The density of neurons in the brain is governed by a fundamental mathematical function, new research finds.

The discovery, which holds true across a variety of mammals, could help researchers make better computer models of the brain in the future.

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