Suppose you flip a coin an infinite number of times. The zero-one law pertains to any questions about the outcome for which the answer does not depend on any finite number of flips. The zero-one law tells us that finite changes cannot disturb phenomena that are infinite in nature.
So the probability of finding an infinite cluster in an infinite network cannot change slightly, such as from 0. To put it another way, an infinite network will either have no infinite cluster a probability of zero for finding an infinite cluster or have an infinite cluster a probability of one.
Thus, switching a finite number of open pipes to closed pipes, or vice versa, does not have any effect on whether an infinite open cluster exists. The probability of finding an infinite cluster is either zero or one. Which is it? The answer depends on the bias of your coin. Imagine you have a dial that controls the bias. The probability of finding an infinite cluster is then one.
If you slowly turn the dial clockwise, the likelihood of pipes being open gradually increases, and it might seem like the chance of finding an infinite cluster should also increase gradually from zero to one. In fact, the change happens instantly because of the zero-one law: it states that the likelihood cannot be somewhere between zero and one.
For the square lattice, the probability snaps from zero to one when the dial is exactly in the middle—when the coin has no bias. This critical position of the dial is known as the percolation threshold. No matter what the shape of the network—whether, for example, it is a triangular lattice or a three-dimensional version of the square lattice—the essential question of percolation theory remains the same: Where is the threshold?
How biased does the coin need to be before enough links are open to guarantee an infinite open cluster? The answer depends on the exact shape of the infinite network and is far from easy to find. Even proving that the threshold for a square lattice—the simplest system—is one half was a daunting challenge, finally solved by mathematician Harry Kesten in And despite decades of effort, the exact percolation thresholds are known only for a few exceedingly simple networks.
Ziff, a statistical physicist at the University of Michigan. The bias for the triangular lattice is roughly 0. Lattices are good models for percolation in physical systems such as fractured rock, where the holes are in fixed locations and the cracks between them form randomly. But other real-world networks are far more complicated.
In the FireChat and Bridgefy mesh networks mentioned earlier, for example, the locations of the nodes—the phones carried by the Hong Kong protesters—changed constantly. The edges in such a network, or connections, form when two phones are near enough to each other—within the tens-of-meters range of the Bluetooth-based apps used to share messages. Such networks are described by a different model, called continuum percolation, because the nodes of the mesh network can be anywhere in a continuous space.
Like any mathematical model, the abstract version of this network is based on simplified assumptions. The smartphones are randomly scattered, without any mimicking of the natural clusters and patterns in a map of people's meanderings, and two smartphones are linked based only on their distance from each other, without any consideration of walls or other interference.
The model nonetheless highlights the central role that percolation theory plays in real mesh networks. There are two ways to increase the connectivity of this continuum percolation network: enable direct connection at a longer range or add more smartphones, increasing the density of users.
These modifications can be thought of as dials like those described for the pipe network; turning either one clockwise will increase connectivity. For designers of mesh-networking apps, finding the percolation threshold is a practical engineering problem. Changing the device's power, which controls the range, is one way to turn a dial.
But the existence of a percolation threshold means there is a risk that the network will suddenly lose connectivity as people move around. The optimal power is one that ensures the network is always connected but does not waste energy. The other dial is the density of phones. Mesh networks with a fixed range need a critical density of users and are most likely to provide widespread connectivity at crowded events such as a music festival, a soccer game or a large protest.
Jorge Rios, Bridgefy's CEO and co-founder, says that the company saw large spikes of new users in Kashmir, Nigeria, Hong Kong and Iran during periods of civil unrest, when people turned to mesh networks to maintain communications in case the government shut down the Internet or large crowds jammed cellular connections. Some neighborhoods, such as Red Hook in Brooklyn, N.
Much of the necessary hardware and routing technology is still evolving, but it is easy to imagine bold, futuristic applications—autonomous vehicles could communicate directly, for example, sharing information about traffic patterns or road hazards without relying on any extra infrastructure. The networks used to model the flow of oil through rocks or direct communication between phones mimic the real spatial structure of these systems: two nodes are connected by an edge if the objects they represent are close to each other in physical space.
But for networks that track the spread of disease from person to person, the links are determined by the ways in which that specific germ is transmitted among them. Such networks are particularly tangled: one infected person spending an hour in a nightclub in a big city may pass a virus to a person who carries it across the country or even across continents in the following days.
The simplest epidemiological models lump everyone into three buckets—susceptible, infected and recovered—and neglect this complex structure of connections.
In such models, infected people pass the disease to random others in the susceptible bucket under the assumption that everyone in that group—students in a dorm or residents in a city—is equally likely to get it. The rate at which susceptible people get infected depends on the basic reproductive number, the average number of new infections caused by a single infected person, abbreviated as R 0.
If R 0 is greater than one, then the virus is spreading, and if it is less than one, then the outbreak is dying out. In practice, however, how people interact with one another influences the overall spread of the disease. Another area of the site reviews math in different cultures through the ages, as well as universally important subjects. And the Mathematicians of the Day page features which scholars were born or died on the day's date.
The Glossary of Mathematical Mistakes We are Much as Phil Plait's Bad Astronomy pages point out stellar errors in popular thought, this site from Paul Cox takes on mathematical illiteracy. The actual glossary itself contains many great examples of twisted statisticsfrom Simpson's Paradox to simple exaggeration. And additional pages offer reader submissions of math gone bad; cases of bad math in advertising and the news; confounding puzzles; further reading; and related links.
Mathmania Tour Not just anyone can take on Fermat and challenges like his last theorem, which required very advanced mathematics to crack. But not all open problems demand the attention of specialists. Inspired by the mathematician Paul Erds, who created a tradition of posing unsolved problems to students and offering prizes for solutions, this site provides a great collection of mathematical mysteriesmostly in graph theory, knot theory, sorting networks and finite state machines.
Don't worry if you don't know anything about these topics going in. The site also offers stories, activities and other materials to help you prepare to work on various problems. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.
I do not know if this is indeed what students feel, but at least on one level it makes sense. At the same time Wilson is quite right that true success in science mostly does not come from mathematics. In many fields math is a powerful tool, but only a tool nonetheless; what matters is a physical feel for the systems to which it is applied. In fact biology can claim many scientists like John Maynard Smith, J. Haldane and W. In my own field of chemistry, math is employed as the basis of several physics-based algorithms that are used to calculate the structure and properties of molecules.
But most chemists like me can largely get away by using these algorithms as black boxes; our insights into problems comes from analyzing the results of the calculations within the unique structure and philosophy of chemistry. Knowledge of mathematics may or may not help us in understanding molecular behavior, but knowledge of chemistry always helps.
Einstein's strength was to imagine thought experiments, Fermi's was to do rough back-of-the-envelope calculations. So while mathematics is definitely key to making advances in fields like particle physics, even in those fields what really matters is the ability to imagine physical phenomena and make sense of them.
Most non-mathematicians can collaborate with a mathematician to firm up their analyses, but without a collaborator in the physical or social sciences mathematicians will have no idea what to do with their equations, no matter how rigorous or elegant they are. The other thing to keep in mind is that an over-reliance on math can also seriously hinder progress in certain fields and even lead to great financial and personal losses. Finance is a great example; the highly sophisticated models developed by physicists on Wall Street caused more harm than good.
In the words of the physicist-turned-financial modeler Emanuel Derman , the modelers suffered from "physics envy", expecting markets to be as precise as electrons and neutrinos. In one sense I see Wilson's criticism of mathematics as a criticism of the overly reductionist ethos that some scientists bring to their work.
0コメント