\t\t \t\t\tGenerative artificial intelligence (AI) models like OpenAI’s GPT-4o or Stability AI’s Stable Diffusion are surprisingly capable at creating new text, code, images and videos. Training them, however, requires such vast amounts… \t\t.
Physicists have developed new specialty optical fibers with a micro-structured core to support future quantum computing data transfer needs.
Researchers from Tokyo Metropolitan University have created sheets of transition metal chalcogenide “cubes” connected by chlorine atoms. While sheets of atoms have been widely studied e.g. graphene, the team’s work breaks new ground by using clusters instead. The team succeeded in forming nanoribbons inside carbon nanotubes for structural characterization, while also forming microscale sheets of cubes which could be exfoliated and probed. These were shown to be an excellent catalyst for generating hydrogen.
The findings have been published in Advanced Materials (“Superatomic layer of cubic Mo 4 S 4 clusters connected by Cl cross-linking”).
„ and show the arrangement of the nanosheet when viewed from different directions, respectively. (Image: Tokyo Metropolitan University)
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.”
A discrepancy between mathematics and physics has plagued astrophysicists’ understanding of how supermassive black holes merge, but dark matter may have the answer.
