Looking for a specific product?

Make a search for products & suppliers, articles & news.

Finalist: Ane Espeseth og Torstein Vik

Ane Espeseth (18 år) og Torstein Vik (16 år), Ålesund
Skole: Fagerlia videregående skole

Motivic Symbols and Classical Multiplicative Functions

 

f we take two integers (also known as whole numbers), we can add them, subtract one from the other, or multiply them. For example:

5 + 3 = 8     5 - 3 = 2     5*3 = 15

In addition to these three operations, we have another one, very useful when computing probabilities, known as a binomial coecient. A binomial coecient
will take a positive integer, for instance 5, and another integer, for instance 3,
and compute how many di erent ways you can choose 3 things out of 5 things.
A computation would look like this:

 

 

 

 

 

This answer tells us that if we must choose 3 students out of a group of 5 students, there are 10 possible choices.

These four operations are related by certain formulas, called lambda-ring axioms. Any "number system" which has four operations related to each other by these axioms is called a lambda-ring.

The first main discovery in our report is a method for constructing infinitely many new "number systems" like this, which are almost as easy to compute with as the integers. The method is built on a technical new idea called a motivic symbol.

A completely different area of mathematics is the study of prime numbers. A prime number is a positive integer that cannot be factored into two smaller positive integers. For example, 7 is a prime number, but 8 is not, since it can be factored into 2 times 4. Understanding the structure of the prime numbers is one of most dicult challenges in all of mathematics. To help in this endeavour,  mathematicians have invented something called multiplicative functions. A typical example is the sigma function, which computes the sum of all factors of a given number.

Many of the most important multiplicative functions can be expressed in terms of the famous Riemann zeta function by a speci c kind of formula. Such functions are called classical in our report.

Now we can formulate our second main discovery, which is a new and surprising connection between the theory of lambda-rings and the theory of multiplicative functions. We have proved that the collection of classical  multiplicative functions is a lambda-ring, in which the operations correspond to certain well-known  number-theoretic operations on multiplicative functions. For example, addition in this lambda-ring corresponds to an operation called Dirichlet
convolution. 

 

Related news

Latest news

New Zealand Navy turns to Vestdavit for hands-on experience

Since 2000, Vestdavit has supplied 14 davits for seven NZ Navy vessels.

DNV GL and AEP Ohio win Energy Award for helping new energy efficient technology in ultra-low temperature freezers reach the market

Stirling ultra-low temperature freezers now ENERGY STAR certified

New 4K Displays Enable Cutting-Edge Multi-Data Fishing Vessel Bridge Solution from Furuno Norway

Newbuild Danish trawler 'Gitte Henning' first to benefit from large-format, ultra high definition bridge display innovation.

New DNV GL class notations aim to improve stern tube bearing performance

In the “Shaft align” class notations and the revised main class requirements for single stern tube bearing installations,

DNV GL provides due diligence to institutional lenders for investment in 659 MW Walney offshore wind project

Through DNV GL’s technical assessment to the group of lenders,

Welcome to the new service centre in Rørvik!

OK Marine will sell and offer service of its recognized cleaning fish products.

DNV GL boosts investment in 3D printing with establishment of Global Additive Manufacturing Centre of Excellence in Singapore

Through the newly established centre, DNV GL is running the recently announced collaboration with Sembcorp Marine,

DNV GL advised Partners Group on technical risks in its investment in Borssele III-IV

Due diligence on Dutch offshore wind project focused on technical risks.

Next generation hull calculation software: DNV GL launches Nauticus Hull version V20

With new ship rules and IACS requirements increasing the scope of hull design,