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It is known that there exist mathematical problems of practical relevance which cannot be computed on a Turing machine. An important example is the calculation of the first derivative of continuously differentiable functions. This paper precisely classifies the non-computability of the first derivative, and of the maximum-norm of the first derivative in the Zheng-Weihrauch hierarchy. Based on this classification, the paper investigates whether it is possible that a Turing machine detects this non-computability of the first derivative by observing the data of the problem, and whether it is possible to detect upper bounds for the peak value of the first derivative of continuously differentiable functions. So from a practical point of view, the question is whether it is possible to implement an exit-flag functionality for observing non-computability of the first derivative. This paper even studies two different types of exit-flag functionality. A strong one, where the Turing machine always has to stop, and a weak one, where the Turing machine stops if and only if the input lies within the corresponding set of interest. It will be shown that non-computability of the first derivative is

In 2005, we proposed that the nerve pulse is an electromechanical soliton. This concept represents a challenge to the well-known Hodgkin-Huxley model which is of a purely electrical nature. The soliton theory was criticized by Nimtz and Aichmann in a recent article in Zeitung für Naturforschung A. Here, we wish to comment on some statements that we regard as misinterpretations of our views.

In this paper, we investigate the computational complexity of solutions to the Laplace and the diffusion equation. We show that for a certain class of initial-boundary value problems of the Laplace and the diffusion equation, the solution operator is $\# P_1/ \#P$-complete in the sense that it maps polynomial-time computable functions to the set of $\#P_1/ \#P$-complete functions. Consequently, there exists polynomial-time (Turing) computable input data such that the solution is not polynomial-time computable, unless $FP=\#P$ or $FP_1=\#P_1$. In this case, we can, in general, not simulate the solution of the Laplace or the diffusion equation on a digital computer without having a complexity blowup, i.e., the computation time for obtaining an approximation of the solution with up to a finite number of significant digits grows non-polynomially in the number of digits. This indicates that the computational complexity of the solution operator that models a physical phenomena is intrinsically high, independent of the numerical algorithm that is used to approximate a solution.

New delay on Copernicus satellite pics comes as US ramps up war with Iran again

Astronomers have found the first confirmed atmosphere around a rocky planet in another star’s habitable zone。 The planet, LHS 1140 b, revealed its atmosphere through helium slowly leaking into space。 Located 48 light-years away, the world may have preserved its atmosphere for billions of years, making it a promising target in the search for potenti

"Any potential damage will be assessed when it is safe to do so

Physicists from Heinrich Heine University Düsseldorf (HHU) have examined a fundamental property of quantum mechanics in collaboration with the German Aerospace Center (DLR)。 In the scientific journal Physical Review Letters, they show that this theory does not necessarily need to be formulated with imaginary numbers – real numbers can in fact also

A new study suggests the brain begins making decisions much earlier than scientists previously thought。 Researchers found that even primary sensory regions are influenced by higher brain areas through rapid feedback loops, rather than simply passing information forward。 This more dynamic view of brain function could help engineers design future AI

A new particle detector called PLATON could replace millions of tiny detector components with a single block of light-producing material。 Using a light-field camera, highly sensitive photon sensors, and AI, it reconstructs particle paths in fast, detailed 3D。 Simulations suggest it could match or surpass today’s best detectors while being far easie

China’s Chang’e-6 samples have uncovered a surprising difference between the Moon’s two hemispheres。 Solar wind particles penetrated deeper into the far-side soil because Earth’s magnetosphere slows the particles that reach the near side。 Noble gases locked inside the lunar regolith preserved evidence of this uneven bombardment

Dying Sun-like stars may not fade away quietly。 As blobs of gas erupt unevenly from their swollen surfaces, each burst gives the star a tiny push in the opposite direction。 Thousands of these random kicks could eventually break apart distant stellar pairs or, in rare cases, drive two stars into a violent collision

NASA's PACE satellite captured the Black Sea glowing turquoise during its annual phytoplankton bloom。 The vivid color comes from massive numbers of coccolithophores, microscopic organisms whose reflective shells brighten the water enough to be seen from space。 An astronaut aboard the International Space Station also photographed the bloom spreading

There's no human safety or efficacy data, but that no longer seems to matter

A giant planet had been hiding inside one of astronomy’s most closely studied planetary systems, concealed by a bright disk of cosmic dust。 Webb revealed Beta Pictoris d through atmospheric traces of carbon monoxide, water vapor, and methane, showcasing a promising new method for finding elusive exoplanets

The asteroid that wiped out the dinosaurs was likely an exceptionally rare CO chondrite from a distant region of the solar system。 Its unusual chemistry suggests that planet-cooling dust and debris, rather than sulfur inside the asteroid, may have delivered the deadliest blow

"We can quash the subpoenas, or you could withdraw the subpoenas,” judge told US

JWST has captured unusually detailed images of gas feeding the supermassive black hole at the center of NGC 4696。 A vast filament appears to funnel material into an 800-light-year-wide spinning disk, where gas races around at up to 600 kilometers per second。 The findings suggest black holes may recycle their own fuel by heating gas with jets and la

NASA is ramping up its lunar ambitions by awarding nearly $600 million for four commercial Moon landings planned for late 2028。 Each mission will carry the same trio of science instruments to improve lunar navigation, study dangerous dust kicked up during landings, and map the Moon's radiation environment。 The agency also revealed plans for new rov

The case provides a novel report of the emergence of heterovirulence