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pivoxa15
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How is measure theory associated with number theory, if at all.
If they are connected, can anyone give a link?
If they are connected, can anyone give a link?
Kummer said:There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them.
No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.pivoxa15 said:I assume you are referring to ring theory in algebra.
Kummer said:No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.
pivoxa15 said:All fields are rings, but not conversely.
You said, '...algebra and number theory are related as soon as you start learning about them...'
I assume one does not learn Galois theory as the first exposure to algebra. One does rings and group theory before moving on to Galois theory which uses a combination of them.
pivoxa15 said:How is measure theory associated with number theory, if at all.
If they are connected, can anyone give a link?
O.K. :)Chris Hillman said:And of course we want to see more examples!
Xevarion said:I am not sure exactly what kind of application you want to make here.
Xevarion said:1) There is a generalization of Szemeredi's theorem to Z^n. However as far as I know the only proof of this right now is ergodic in nature, and therefore there are no bounds.
Xevarion said:by the way, you might be interested in reading about some of the recent results of Bourgain, Gamburd, and Sarnak about lattices and orbits. E.g. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6X1B-4KCXJJ4-2&_user=1082852&_coverDate=08%2F01%2F2006&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000051401&_version=1&_urlVersion=0&_userid=1082852&md5=3881f0053692a3bbc99e47aec129fc07" (sorry I don't have a better link)
Really? I find that hard to believe after reading his book on functional analysis.Chris Hillman said:Bela Bolobas, Modern Graph Theory, GTM 184, Springer, 1998 (one of the best books ever published in any subject!)
Measure theory deals with the study of measures, which are used to assign a size or volume to sets in mathematics. It is primarily used in the field of analysis. On the other hand, number theory is a branch of mathematics that focuses on the properties of numbers, particularly integers. It has applications in fields such as cryptography and coding theory.
Measure theory has many practical applications, including in physics, economics, and statistics. For example, it is used in probability theory to define the probability of an event occurring, and in economics to measure consumer preferences and utility. It also has applications in signal processing and image recognition.
The Riemann zeta function is a mathematical function that plays a significant role in number theory. It is defined as the sum of the reciprocals of all positive integers raised to a certain power. Its significance lies in its connection to the distribution of prime numbers, and it has been extensively studied by mathematicians for centuries.
Measure theory provides a rigorous foundation for integration, which is a fundamental concept in calculus. It allows for the integration of a wider range of functions, including those that are not continuous or even defined on a finite interval. Measure theory also helps to define more general notions of convergence, which are essential in the study of integrals.
Number theory is a vast and complex field, with many unsolved problems. Some notable examples include the Goldbach conjecture, which states that every even integer greater than 2 can be expressed as the sum of two prime numbers, and the Riemann hypothesis, which concerns the distribution of prime numbers and has implications for many other areas of mathematics.