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Category: mathematics

The octonions themselves will never be “the answer” to how reality works, but they do provide a powerful, generalized mathematical structure that has its own unique properties. It includes real, complex, and quaternion mathematics, but also introduces fundamentally unique mathematical properties that can be applied to physics to make novel — but speculative and hitherto unsupported — predictions.

Octonions can give us and idea of which possibilities might be compelling to look at in terms of extensions to known physics and which ones might be less interesting, but there are no concrete observables predicted by the octonions themselves. Pierre Ramond, my former professor who taught me about octonions and Lie groups in physics, was fond of saying, “octonions are to physics what the Sirens were to Ulysses.” They definitely have an allure, but if you dive in, they may drag you to a hypnotic, inescapable doom.

Their mathematical structure holds an incredible richness, but nobody knows whether that richness means anything for our Universe or not.

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Quadratic equations are polynomials, meaning strings of math terms. An expression like “x + 4” is a polynomial. They can have one or many variables in any combination, and the magnitude of them is decided by what power the variables are taken to. So x + 4 is an expression describing a straight line, but (x + 4)² is a curve. Since a line crosses just once through any particular latitude or longitude, its solution is just one value. If you have x², that means two root values, in a shape like a circle or arc that makes two crossings.

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An international group of scientists, including Andrey Savelyev, associate professor of the Institute of Physical and Mathematical Sciences and Information Technologies of the IKBFU, has improved a computer program that helps simulate the behavior of photons when interacting with hydrogen spilled in intergalactic space. Results are published in the scientific journal Monthly Notices of the Royal Astronomical Society.

Andrey Saveliev states, “In the Universe there are extragalactic objects such as blazars, which very intensively generate a powerful gamma-ray flux, part of photons from this stream reaches the Earth, as they say, directly, and part are converted along the way into electrons, then again converted into photons and only then get to us. The problem here is that say that a certain number of photons should reach the Earth, and in fact it is much less.”

Scientists, according to Andrey Savelyev, today have two versions of why this happens. The first is that a , after being converted into an electron (and this, as is known, in contrast to a neutral photon, a charged particle) falls into a , deviates from its path and does not reach the Earth, even after being transformed again into the photon.

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Ira Pastor, ideaXme exponential health ambassador, interviews Dr. Ronald Mallett, Professor Emeritus, Theoretical Physics, Department of Physics at the University of Connecticut.

Ira Pastor Comments:

Time travel is the concept of movement between certain points in time, analogous to movement between different points in space, by an object or a person, typically with the use of a hypothetical device known as a time machine.

Time travel is a widely recognized concept in philosophy and fiction and the idea of a time machine was originally popularized by H. G. Wells’ 1895 novel The Time Machine.

Forward time travel, outside the usual sense of the perception of time, is an extensively observed phenomenon and well-understood within the framework of special relativity and general relativity and making one body advance a few milliseconds compared to another body has been demonstrated in experiments comparing atomic clocks on jets and satelites versus the earth.

As for backward time travel, it is possible to find solutions in general relativity that allow for it, in a theoretical system known as a “closed timelike curve” (sometimes abbreviated CTC), which is where the world line of an object (the path that an object traces in 4-dimensional space-time) follows a curious path where it eventually returns to the exact same coordinates in space and time that it was at previously. In other words, a closed timelike curve is the mathematical result of physics equations that allows for time travel to the past.

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Scientists hope that the future of gravitational wave detection will allow them to directly observe a mysterious kind of black hole.

Gravitational wave detectors have seen direct evidence of black holes with roughly the mass of giant stars, while the Event Horizon Telescope produced an image of a supermassive black hole billions of times the mass of our Sun. But in the middle are intermediate-mass black holes, or IMBHs, which weigh between 100 and 100,000 times the mass of the Sun and have yet to be directly observed. Researchers hope that their new mathematical work will “pave the way” for future research into these black holes using gravitational wave detectors, according to the paper published today in Nature Astronomy.

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A pair of mathematicians from Australia and France have devised an alternative way to multiply numbers together, while solving an algorithmic puzzle that has perplexed some of the greatest math minds for almost half a century.

For most of us, the way we multiply relatively small numbers is by remembering our times tables – an incredibly handy aid first pioneered by the Babylonians some 4,000 years ago.

But what if the numbers get bigger? Well, if the figures get unwieldy – and assuming we don’t have a calculator or computer, of course – most of us would then turn to long multiplication: another useful trick we learn in school, and a trusty technique for multiplying basically any two numbers together.

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RUDN University mathematicians have proven the Hardy-Littlewood-Sobolev (HLS) inequalities for the class of generalized Riesz potentials. These results extend the scope of these potentials in mathematics and physics because the main tools for working with such potentials are based on HLS inequalities. New mathematical tools can greatly simplify calculations in quantum mechanics and other fields of physics. The results of the study are published in the journal Mathematical Notes.

Modern physics describes the world in terms of fields and their potentials—that is, the values of the field at each point. But the physical quantities that we can measure are forces and accelerations, that is, derivatives of the second-order of the potential of the corresponding field. The problem of reconstructing the field configuration with the available values of forces and accelerations observed in experiments is complex and not always analytically solvable. Differentiation operations in multidimensional space—operators are usually used to describe the correlation between the potential of the field and the forces. In particular, electromagnetic and gravitational interactions are described in the language of operators.

Since the potential of the field can be determined up to a constant value, for the convenience of calculations, the initial value of the potential is taken at some point in multidimensional space, or on the border of any spatial area. But in some cases, mathematical models of such fields lead to a singularity, that is, at some points the value of the field becomes infinite, and therefore loses its physical meaning.

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AMOLF researchers and their collaborators from the Advanced Science Research Center (ASRC/CUNY) in New York have created a nanostructured surface capable of performing on-the-fly mathematical operations on an input image. This discovery could boost the speed of existing imaging processing techniques and lower energy usage. The work enables ultrafast object detection and augmented reality applications. The researchers publish their results today in the journal Nano Letters.

Image processing is at the core of several rapidly growing technologies, such as augmented reality, autonomous driving and more general object recognition. But how does a computer find and recognize an object? The initial step is to understand where its boundaries are, hence edge detection in an image becomes the starting point for image recognition. Edge detection is typically performed digitally using integrated implying fundamental speed limitations and high energy consumption, or in an analog fashion which requires bulky optics.

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