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Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts

Friday, 27 March 2020

New mathematical model can more effectively track epidemics



As COVID-19 spreads worldwide, leaders are relying on mathematical models to make public health and economic decisions.

A new model developed by Princeton and Carnegie Mellon researchers improves tracking of epidemics by accounting for mutations in diseases. Now, the researchers are working to apply their model to allow leaders to evaluate the effects of countermeasures to epidemics before they deploy them.

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“We want to be able to consider interventions like quarantines, isolating people, etc., and then see how they affect an epidemic’s spread when the pathogen is mutating as it spreads,” said H. Vincent Poor, one of the researchers on this study and Princeton’s interim dean of engineering.

The models currently used to track epidemics use data from doctors and health workers to make predictions about a disease’s progression. Poor, the Michael Henry Strater University Professor of Electrical Engineering, said the model most widely used today is not designed to account for changes in the disease being tracked. This inability to account for changes in the disease can make it more difficult for leaders to counter a disease’s spread. Knowing how a mutation could affect transmission or virulence could help leaders decide when to institute isolation orders or dispatch additional resources to an area.



“In reality, these are physical things, but in this model, they are abstracted into parameters that can help us more easily understand the effects of policies and of mutations,” Poor said.

If the researchers can correctly account for measures to counter the spread of disease, they could give leaders critical insights into the best steps they could take in the face of pandemics. The researchers are building on work published March 17 in the Proceedings of the National Academy of Sciences. In that article, they describe how their model is able to track changes in epidemic spread caused by mutation of a disease organism. The researchers are now working to adapt the model to account for public health measures taken to stem an epidemic as well.

The researchers’ work stems from their examination of the movement of information through social networks, which has remarkable similarities to the spread of biological infections. Notably, the spread of information is affected by slight changes in the information itself. If something becomes slightly more exciting to recipients, for example, they might be more likely to pass it along or to pass it along to a wider group of people. By modeling such variations, one can see how changes in the message change its target audience.

“The spread of a rumor or of information through a network is very similar to the spread of a virus through a population,” Poor said. “Different pieces of information have different transmission rates. Our model allows us to consider changes to information as it spreads through the network and how those changes affect the spread.”

“Our model is agnostic with regard to the physical network of connectivity among individuals,” said Poor, an expert in the field of information theory whose work has helped establish modern cellphone networks. “The information is being abstracted into graphs of connected nodes; the nodes might be information sources or they might be potential sources of infection.”



Obtaining accurate information is extremely difficult during an ongoing pandemic when circumstances shift daily, as we have seen with the COVID-19 virus. “It’s like a wildfire. You can’t always wait until you collect data to make decisions — having a model can help fill this void,” Poor said.

 “Hopefully, this model could give leaders another tool to better understand the reasons why, for example, the COVID-19 virus is spreading so much more rapidly than predicted, and thereby help them deploy more effective and timely countermeasures,” Poor said.


Bibliography:

Rashad Eletreby, Yong Zhuang, Kathleen M. Carley, Osman Yağan, H. Vincent Poor.

The effects of evolutionary adaptations on spreading processes in complex networks. 

Proceedings of the National Academy of Sciences, 2020; 117 (11): 5664

DOI: 10.1073/pnas.1918529117

Thursday, 5 March 2020

Researchers discovered link between complex numbers and superstring theory



Number theory is a branch of mathematics studying the properties of whole numbers. It is an active area of ​​research in fundamental mathematics because it is located at the interconnection of all other disciplines. By precisely analyzing a particular analytical function of the theory, the modular form, a duo of researchers showed that in certain cases, this function could be described by means of quantum observables in the theory of superstrings. A result that could help confirm certain properties of this theory.

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A collaboration of a mathematician and a physicist has shown that the modular forms associated with elliptic curves with complex multiplications are expressed in terms of observables in superstring theory. The study was published in the journal Communications in Mathematical Physics.

The concept of numbers can be extended from integers and rational numbers to include all real numbers and complex numbers, all at once.



But it is also possible to extend the concept gradually, by adding the roots of polynomials with rational number coefficients (such as the square root of 2 and the square root of 3) little by little. This special class of complex numbers are referred to as "numbers." The precise details of how the concept of numbers can be extended has been considered as one of the important themes in number theory.

Extension of the concept of integral “numbers”. Black dots are the ordinary integers represented in a complex plane. Adding or multiplying any pair of black dots results in another black point. All the red dots and the blue dots in this figure are solutions to certain quadratic equations with integer coefficients. Purple dots are solutions to certain quantum equations with integer coefficients. We can therefore consider these points as part of the "numbers". The operations of addition and multiplication between the black or red points remain in the “numbers” indicated in black or red points, and similarly, these operations of the black-red-blue or purple points remain in the “numbers” in points black-red-blue or purple. In this way, it is possible to gradually expand the set of integral “numbers”. Credits: Kavli IPMU


The invariance of the inverse Mellin transform of the function L

For several decades, researchers have tried to address and understand this problem. One could specify a geometric object by equations using the "numbers" first, and then consider the set of points in the geometric object whose values are the "numbers." As the concept of numbers is gradually extended, and the set of "numbers" expanded, more and more points in the geometric object come to be counted.

A geometric object given by y² = 4x 3 - x is represented by a thin blue curve. In this object, the three black dots have their values ​​in ordinary integers. On the other hand, the three points in the red triangles have their values ​​in a larger set of "numbers" (the coordinates x are of the form (p + q√2) with rational numbers p and q; the coordinates are more complicated). As the concept of "numbers" is expanded, the number of points with their values ​​in "numbers" increases, even for a given geometric object. Credits: Kavli IPMU


The idea is that the way the number of points in the geometric object increases will shed light on how the set of "numbers" expands. Furthermore, this information of the growth rate of the number of points in the geometric object is packed into a function called the inverse Mellin transform of the L-function, which is a function containing the information of how fast the number of points in a geometric object grows as the concept of numbers is extended. This function has been expected to be a modular form, a function that remains invariant under certain operations. This conjecture is known as Langlands conjecture.

Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) Associate Professor and particle theorist Taizan Watari and arithmetic geometry researcher at Middle East Technical University Northern Cyprus Campus and Kavli IPMU Visiting Scientist Satoshi Kondo dared to ask why such functions are invariant under certain operations.

Modular forms that can be described in terms of observables in superstring theory

In string theory, it is known that a class of observables (a) are invariant under the operations (x) that have been referred to already. The invariance under the operations is an indispensable property in the theoretical construction of superstring theory.

Summary of this study. Credit: Kavli IPMU


So, the researchers showed that the inverse Mellin transforms of the L-functions of geometry objects (b) are expressed in terms of the above class of observables (a) in superstring theory with those geometric objects set as the target spaces. As a result, it follows that the functions containing the information of how the concept of numbers is extended, the inverse Mellin transforms, (b) should be invariant under certain operations, which should be modular forms, (x) for reasons from the perspective of superstring theory.



It should be noted that the result above is obtained only for the class of geometric objects called elliptic curves with complex multiplications. The question remains open to whether the functions for more general class of geometric objects (b) are expressed in terms of observables in superstring theory (a).


Bibliography:

String-Theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication

Satoshi Kondo & Taizan Watari

Communications in Mathematical Physics

volume 367, pages89–126(2019)

https://doi.org/10.1007/s00220-019-03302-0

Wednesday, 27 November 2019

Computers make significant miscalculations because they do not know how to deal with mathematical chaos



In most situations of everyday life, the numerical computation systems we use are amply sufficient for accuracy. But when scientists need to perform complex simulations and calculations, numerical precision is of paramount importance. However, in a new study, researchers have found that complex computational calculations can be biased up to 15% due to a pathological inability to grasp the true mathematical complexity of chaotic dynamical systems.

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"Our work shows that the behavior of chaotic dynamic systems is richer than any digital computer can capture, " says computer scientist Peter Coveney of UCL in the UK. " Chaos is more commonplace than many people think and even for very simple chaotic systems, the numbers used by digital computers can cause errors that are not obvious but can have a significant impact."

For centuries theorists have been considering how very small upstream effects can accumulate to become very important downstream. In chaos theory, the phenomenon is known as the butterfly effect: metaphorically, the hypothetical notion that the infinitely small flapping of a butterfly's wings in one place could help to generate a tornado to another. The results of the study were published in the journal Advanced Theory and Simulations .


Mathematical chaos: the extreme importance of initial conditions

It is a poetic concept, but although it may seem fanciful, mathematical modeling suggests that the notion is rooted in very measurable terms. The butterfly effect is mainly attributed to the American mathematician and meteorologist Edward Norton Lorenz, who, in the 1960s, by repeating a weather simulation, used a mathematical shortcut that will shape the story: he used slightly simplified numbers for the second experiment (0.506 instead of 0.506127).

" I went for a cup of coffee in the hallway and came home after about an hour, during which time the computer simulated about two months of weather. The figures being printed did not look like the old ones, "says Lorenz.

Lorenz's results showed how small changes in initial conditions can produce large changes over time in complex and chaotic systems, in which many variables affect and influence each other.

Chaos and complex calculations: the snowball effect of imprecision errors

Weather forecasting is one example, but the same phenomenon of snowball errors has since been highlighted in many areas, from modeling orbital trajectories to turbulence and molecular dynamics. The fact is, even though the butterfly effect has been known for decades, it remains a fundamental problem in the way computers perform calculations.

Weather forecast simulations require a very large number of parameters and initial variables. The slightest approximation to the value of these entries can lead to an accumulation of inaccuracies and calculation errors. Credits: NASA

" Extreme sensitivity to initial conditions is a defining characteristic of chaotic dynamical systems. Since the first use of numerical calculators for computer science, it is known that the loss of precision due to the discrete approximation of real numbers can significantly alter the dynamics of chaotic systems after a short simulation period, "explain the researchers.

This loss of precision does not occur in simple calculations. The calculator app on your smartphone is probably perfectly adequate for everything you need in everyday life. But in complex calculations with many variables and starting conditions, small rounding errors at the beginning can lead to huge calculation errors at the end of a given simulation.

The problem of using floating point

The researchers explain that the heart of the problem is so-called floating-point arithmetic: the standardized way of understanding real numbers by computers using binary code, which uses approximations and conversions to represent numbers.

In large and complex systems, these approximations can generate significant errors, a problem compounded by the way floating-point numbers are distributed among real numbers, even in a newer, more complex 64-bit format called a double floating point. precision.

Graphs showing the relative errors of the floating calculation of a generalized Bernoulli map for values ​​3, 5 and 7. Credits: Bruce M. Boghosian et al. 2019

" We've long thought that rounding errors were not a problem, especially if we were using double-precision floating-point numbers - binary numbers using 64 bits instead of 32. But in our study, we put highlighting a problem due to the uneven distribution of fractions represented by floating-point numbers, and which is not likely to disappear simply by increasing the number of bits, "explains mathematician Bruce Boghosian, Tufts University.

An accumulation of errors, including in the simplest calculations

As part of the research, the team compared a known chaotic system called the Bernoulli map to a numerical computation of the same system and discovered what it calls systematic distortions and pathologies in the simulation of chaotic dynamical systems. Indeed, while Lorenz discovered his butterfly effect by leaving whole numbers in a calculation, the researchers found their own much more subtle equivalent by simply asking a computer to perform a mathematical calculation.

" For Lorenz, it's a very small change in the last few digits of the numbers used to start a simulation that provoked its divergent results. What neither he nor the others have understood, and which is highlighted in our new work, is that a finite (rational) initial condition describes a behavior that can be statistically highly unrepresentative, "says Coveney.



The researchers recognize that the Bernoulli map is a simple chaotic system that does not necessarily represent more complex dynamic models, but they warn that the insidious nature of their floating-point moth means that no scientist should neglect these risks.

Bibliography:

A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers

Bruce M. Boghosian  Peter V. Coveney  Hongyan Wang

First published: 23 September 2019

https://doi.org/10.1002/adts.201900125

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