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Category: mathematics – Page 43

Biological materials are made of individual components, including tiny motors that convert fuel into motion. This creates patterns of movement, and the material shapes itself with coherent flows by constant consumption of energy. Such continuously driven materials are called active matter.

The mechanics of cells and tissues can be described by active matter theory, a scientific framework to understand the shape, flow, and form of living materials. The active matter theory consists of many challenging mathematical equations.

Scientists from the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG) in Dresden, the Center for Systems Biology Dresden (CSBD), and the TU Dresden have now developed an algorithm, implemented in an open-source supercomputer code, that can for the first time solve the equations of active matter theory in realistic scenarios.

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While physics tells us that information can neither be created nor destroyed (if information could be created or destroyed, then the entire raison d’etre of physics, that is to predict future events or identify the causes of existing situations, would be impossible), it does not demand that the information be accessible. For decades physicists assumed that the information that fell into a black hole is still there, still existing, just locked away from view.

This was fine, until the 1970s when Stephen Hawking discovered the secret complexities of the event horizon. It turns out that these dark beasts were not as simple as we had been led to believe, and that the event horizons of are one of the few places in the entire cosmos where meets quantum mechanics in a manifest way.

The quest to unify quantum mechanics and gravity stretches back over a century, soon after the development of those two great domains of physics. What prevented their unification was a proliferation of infinities in the mathematics. Anytime gravity became strong at small scales, our equations diverged to infinity and gave useless non-results. But here we are at the boundaries of black holes, which by definition are places of strong gravity. And because the event horizons are mathematical constructs, not actual surfaces with finite extent, to truly understand them we must examine them microscopically, which plants them firmly in the realm of the quantum.

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One of the most startling scientific discoveries of recent decades is that physics appears to be fine-tuned for life. This means that for life to be possible, certain numbers in physics had to fall within a certain, very narrow range.

One of the examples of fine-tuning which has most baffled physicists is the strength of dark energy, the force that powers the accelerating expansion of the universe.

If that force had been just a little stronger, matter couldn’t clump together. No two particles would have ever combined, meaning no stars, planets, or any kind of structural complexity, and therefore no life.

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Undeterred after three decades of looking, and with some assistance from a supercomputer, mathematicians have finally discovered a new example of a special integer called a Dedekind number.

Only the ninth of its kind, or D, it is calculated to equal 286 386 577 668 298 411 128 469 151 667 598 498 812 366, if you’re updating your own records. This 42 digit monster follows the 23-digit D discovered in 1991.

Grasping the concept of a Dedekind number is difficult for non-mathematicians, let alone working it out. In fact, the calculations involved are so complex and involve such huge numbers, it wasn’t certain that D would ever be discovered.

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Cavities at the sides of a water channel can cause waves to be completely absorbed, suggesting new techniques for protecting coastlines.

If waves of water, light, or sound were to impinge upon a hypothetical object called a perfect absorber, they would be neither reflected nor transmitted; they would simply vanish. Researchers have now demonstrated perfect absorption using ordinary water waves traveling down a narrow channel [1]. The waves are canceled out by their own reflections from cavities built into the side of the channel. With further development, the researchers believe that the effect could be used to reduce erosion or protect sensitive structures by using an array of elements deployed near coastlines.

“We were motivated by the need to control or absorb waves in rivers or to protect coastlines,” says mathematical physicist Agnes Maurel of ESPCI Paris. “Completely absorbing wave energy is even better than redirecting it, and you can also imagine perhaps harvesting such energy.”

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An hypothesized term to fix a small mathematical inconsistency predicted electromagnetic waves, and that they had all the properties of light that were observed before and after him in the Nineteenth Century. Unwittingly, he also pointed science inexorably in the direction of the special theory of relativity

My last two articles, two slightly different takes on “recipes” for understanding Electromagnetism, show how Maxwell’s equations can be understood as arising from the highly special relationships between the electric and magnetic components within the Faraday tensor that is “enforced” by the assumption that the Gauss flux laws, equivalent to Coulomb’s inverse square force law, must be Lorentz covariant (consistent with Special Relativity).

From the standpoint of Special Relativity, there is obviously something very special going on behind these laws, which are clearly not from the outset Lorentz covariant. What i mean is that, as vector laws in three dimensional space, there is no way you can find a general vector field that fulfills them and deduce that it is Lorentz covariant — it simply won’t be so in general. There has to be something else further specializing that field’s relationship with the world to ensure such an in-general-decidedly-NOT-Lorentz covariant equation is, indeed covariant.

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Sperm can “defy the laws of physics”, according to new research.

The laws of motion have helped us to comprehend the behaviours of the natural world for centuries, but sperm appears to go against one of the laws set down by Isaac Newton.

Kenta Ishimoto and his fellow mathematical scientists from Kyoto University have revealed new research which suggests that sperm actually display qualities which don’t follow Newton’s third law of motion.

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Some researchers see formal specifications as a way for autonomous systems to “explain themselves” to humans. But a new study finds that we aren’t understanding.

As autonomous systems and artificial intelligence become increasingly common in daily life, new methods are emerging to help humans check that these systems are behaving as expected. One method, called formal specifications, uses mathematical formulas that can be translated into natural-language expressions. Some researchers claim that this method can be used to spell out decisions an AI will make in a way that is interpretable to humans.

Research Findings on Interpretability.

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