Four mathematicians have estimated the chances that there’s a clear path through a random maze.
Category: mathematics – Page 38
Actually, nothing is wrong with it if you are a computer science major. It’s just that it has no place in the philosophy department.
From the point of anyone wanting to work in natural language, symbolic logic has all of the vices of mathematics and none of its virtues. That is, it is obscure to the point of incomprehensibility (given the weak neurons of this English major at any rate), and it leads to no useful outcome in the domain of human affairs. This would not be so bad were it not for all those philosophy major curricula that ask freshmen to take a course in it as their “introduction” to philosophy. For anyone looking to explore the meaning of life, this is a complete turnoff.
What were the philosophy mavens thinking?
As opposed to black holes, white holes are thought to eject matter and light while never absorbing any. Detecting these as yet hypothetical objects could not only provide evidence of quantum gravity but also explain the origin of dark matter.
No one today questions the existence of black holes, objects from which nothing, not even light, can escape. But after they were first predicted in 1915 by Einstein’s general theory of relativity, it took many decades and multiple observations to show that they actually existed. And when it comes to white holes, history may well repeat itself. Such objects, which are also predicted by general relativity, can only eject matter and light, and as such are the exact opposite of black holes, which can only absorb them. So, just as it is impossible to escape from a black hole, it is equally impossible to enter a white one, occasionally and perhaps more aptly dubbed a “white fountain”. For many, these exotic bodies are mere mathematical curiosities.
A single universal equation can closely approximate the frequency of wingbeats and fin strokes made by birds, insects, bats and whales, despite their different body sizes and wing shapes, Jens Højgaard Jensen and colleagues from Roskilde University in Denmark report in a new study published in PLOS ONE on June 5.
The ability to fly has evolved independently in many different animal groups. To minimize the energy required to fly, biologists expect that the frequency that animals flap their wings should be determined by the natural resonance frequency of the wing. However, finding a universal mathematical description of flapping flight has proved difficult.
Researchers used dimensional analysis to calculate an equation that describes the frequency of wingbeats of flying birds, insects and bats, and the fin strokes of diving animals, including penguins and whales.
ICTP lectures “Topology and dynamics of higher-order networks”
- Network topology: 1 https://youtube.com/watch?v=mbmsv9RS3Pc&t=7562s.
- Network topology:2 https://youtube.com/watch?v=F6m5lPfk5Mc&t=3808s.
-Network geometry.
Topological Dirac equation and Discrete Network Geometry-Metric cohomology Speaker: Ginestra Bianconi (Queen Mary University of London) Higher-order networks [1] capture the many-body interactions present in complex systems and are dramatically changing our understanding of the interplay between topology of and dynamics. In this context, the new field of topological signals is emerging with the potential to significantly transform our understanding of the interplay between the structure and the dynamics in complex interacting systems. This field combines higher-order structures with discrete topology, discrete topology and dynamics and shows the emergence of new dynamical states and collective phenomena. Topological signals are dynamical variables, not only sustained on the nodes but also on edges, or even triangles and higher-order cells of higher-order networks. While traditionally network dynamics is studied by focusing only on dynamical variables associated to the nodes of simple and higher-order networks topological signals greatly enrich our understanding of dynamics in discrete topologies. These topological signals are treated by using algebraic topology operators as the Hodge Laplacian and the discrete Dirac operator. Recently, growing attention has been devoted to the study of topological signals showing that topological signals undergo collective phenomena and that they offer new paradigms to understand on one side how topology shape dynamics and on the other side how dynamics learns the underlying network topology. These concepts and idea have wide applications. Here we cover example of their applications in mathematical physics and dynamical systems. The field is topical at the moment with many new results already established and an already rich bibliography, therefore it is very timely to propose a series of lectures on the topic to introduce new scientists to this emergent field. Here we propose a series of lectures for a broad audience of scientists addressed mostly to physicist and mathematicians, but including also computer scientists and neuroscientists. The course is planned to be introductory, and self-contained starting from minimum set of prerequisites and focus mostly on the mathematical physics aspect of this field. The course will cover 4 lectures and 1 seminar. Ref: [1] Bianconi, G.: Higher-order networks: An introduction to simplicial complexes. Cambridge University Press (2021). [2] Bianconi, G., 2021. The topological Dirac equation of networks and simplicial complexes. Journal of Physics: Complexity, 2, p.035022.[3]Bianconi, G., 2023. The mass of simple and higher-order networks. Journal of Physics A: Mathematical and Theoretical, 57, p.015001.[4] Bianconi, G., 2024. Quantum entropy couples matter with geometry. arXiv preprint arXiv:2404.08556.[5] Millán, A.P., Torres, J.J. and Bianconi, G., 2020. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters, 124(21), p.218301.
A simple concept of decay and fission of “magnetic quivers” helps to clarify complex quantum physics and mathematical structures.
An essay by Steve Nadis and Shing-Tung Yau describes how math helps us imagine the cosmos around us in new ways.
By studying the geometry of model space-times, researchers offer alternative views of the universe’s first moments.
Recently, a team of chemists, mathematicians, physicists and nano-engineers at the University of Twente in the Netherlands developed a device to control the emission of photons with unprecedented precision. This technology could lead to more efficient miniature light sources, sensitive sensors, and stable quantum bits for quantum computing.