Given an open bounded subset Ω of $mathbbR^n$$ R n, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_infty u = 1$$ — Δ ∞ u = 1 in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C 1(Ω).
We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.
Researchers have developed a new approach for describing the shape of the cerebral cortex, and provide evidence that cortices across mammalian species resemble a universal, fractal pattern.
face_with_colon_three steps towards infinity getting much closer to the solution with reinmans hypothesis: D.
Just as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors—those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.
“At first sight, they look pretty random,” says James Maynard, a mathematician at the University of Oxford. “But actually, there’s believed to be this hidden structure within the prime numbers.”
For 165 years, mathematicians seeking that structure have focused on the Riemann hypothesis. Proving it would offer a Rosetta Stone for decoding the primes—as well as a $1 million award from the Clay Mathematics Institute. Now, in a preprint posted online on 31 May, Maynard and Larry Guth of the Massachusetts Institute of Technology have taken a step in this direction by ruling out certain exceptions to the Riemann hypothesis. The result is unlikely to win the cash prize, but it represents the first progress in decades on a major knot in math’s biggest unsolved problem, and it promises to spark new advances throughout number theory.
A discrepancy between mathematics and physics has plagued astrophysicists’ understanding of how supermassive black holes merge, but dark matter may have the answer.
BHP’s (ASX, NYSE: BHP) Spence copper mine in Chile has celebrated three months of being the company’s first fully autonomous operation, a status reached in April after a two-year journey that included converting its trucks fleet and drilling rigs.
Spence, which produced 249,000 tonnes of copper last year, is BHP’s second largest copper mine behind Escondida, the world’s biggest copper operation. In the three months to July 29, the copper operation has moved 80 million tonnes of material without any safety incidents, surpassing the production plan to date, BHP said.
“While these conditions are necessary for a planet to host life, they do not guarantee it,” said Anthony Atkinson. “Our work highlights the importance of considering a wide range of factors when searching for habitable planets.”
Does a planet just have to be in a star’s habitable zone to be habitable, or are other forces at play? This is what a recent study published in The Astrophysical Journal hopes to address as a team of researchers from Rice University and NASA investigated whether the interaction between a star’s and a planet’s respective magnetic fields could play a role in determining the habitability potential for an exoplanet. This study holds the potential to help scientists better understand the formation and evolution of exoplanets and the necessary conditions for life to emerge on those worlds.
“The fascination with exoplanets stems from our desire to understand our own planet better,” said Dr. David Alexander, who is a professor of physics and astronomy at Rice University, director of the Rice Space Institute and member of the Texas Aerospace Research and Space Economy Consortium, and a co-author on the study. “Questions about the Earth’s formation and habitability are the key drivers behind our study of these distant worlds.”
For the study, the researchers incorporated a star’s stellar activity and magnetic field into longstanding computer models designed to simulate planetary conditions, specifically for habitability. The team then analyzed 1,546 exoplanets to determine the most suitable exoplanets for habitability. In the end, they found that only two exoplanets were potentially habitable: K2-3D and Kepler-186 f. This was based on their size, location within the habitable zone, reside outside the distance where the solar wind separates from the star, and whose magnetic field strengths can shield them from harmful radiation.
For the first time, researchers have demonstrated that not just individual bits, but entire bit sequences can be stored in cylindrical domains: tiny, cylindrical areas measuring just around 100 nanometers. As the team reports in the journal Advanced Electronic Materials, these findings could pave the way for novel types of data storage and sensors, including even magnetic variants of neural networks.
Groundbreaking Magnetic Storage
“A cylindrical domain, which we physicists also call a bubble domain, is a tiny, cylindrical area in a thin magnetic layer. Its spins, the electrons’ intrinsic angular momentum that generates the magnetic moment in the material, point in a specific direction. This creates a magnetization that differs from the rest of the environment. Imagine a small, cylinder-shaped magnetic bubble floating in a sea of opposite magnetization,” says Prof. Olav Hellwig from Helmholtz-Zentrum Dresden-Rossendorf ’s Institute of Ion Beam Physics and Materials Research, describing the subject of his research. He and his team are confident that such magnetic structures possess a great potential for spintronic applications.
An AI model developed by scientists at King’s College London, in close collaboration with University College London, has produced three-dimensional, synthetic images of the human brain that are realistic and accurate enough to use in medical research.
The model and images have helped scientists better understand what the human brain looks like, supporting research to predict, diagnose and treat brain diseases such as dementia, stroke, and multiple sclerosis.
The algorithm was created using the NVIDIA Cambridge-1, the UK’s most powerful supercomputer. One of the fastest supercomputers in the world, the Cambridge-1 allowed researchers to train the AI in weeks rather than months and produce images of far higher quality.
