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Blog – Page 1142

Euler’s Identity:

The most beautiful equation in mathematics that combines five of the most important constants of nature: 0, 1, π, e and i, with the three fundamental operations: addition, multiplication and exponentiation.

It’s mystical.

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as “the most beautiful equation.” It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures “our jewel” and “the most remarkable formula in mathematics.”

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Generative models have had remarkable success in various applications, from image and video generation to composing music and to language modeling. The problem is that we are lacking in theory, when it comes to the capabilities and limitations of generative models; understandably, this gap can seriously affect how we develop and use them down the line.

One of the main challenges has been the ability to effectively pick samples from complicated data patterns, especially given the limitations of traditional methods when dealing with the kind of high-dimensional and commonly encountered in modern AI applications.

Now, a team of scientists led by Florent Krzakala and Lenka Zdeborová at EPFL has investigated the efficiency of modern neural network-based generative models. The study, published in PNAS, compares these contemporary methods against traditional sampling techniques, focusing on a specific class of probability distributions related to spin glasses and statistical inference problems.

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The detailed calculations demonstrate that black holes of 10 may comprise at most 1.2% of dark matter, 100 solar mass black holes—3.0% of dark matter, and 1,000 solar mass black holes—11% of dark matter.

“Our observations indicate that primordial black holes cannot comprise a significant fraction of the dark matter, and simultaneously, explain the observed black hole merger rates measured by LIGO and Virgo,” says Prof. Udalski.

Therefore, other explanations are needed for massive detected by LIGO and Virgo. According to one hypothesis, they formed as a product of the evolution of massive, low-metallicity stars. Another possibility involves mergers of less massive objects in dense stellar environments, such as globular clusters.

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On June 25, China’s Chang’e-6 (CE-6) lunar probe is set to return to Earth, carrying the first surface samples collected from the farside of the moon. In anticipation of this historic event, scientists from the Institute of Geology and Geophysics at the Chinese Academy of Sciences are publishing their predictions for the unique materials that may be found in the CE-6 samples in the journal The Innovation.

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