Mathematics Survey

The Maths of Graphene

Graphene was made in 2004 by Andre Geim and other scientists. It a carbon allotrope consisting of a single molecule thick layer of carbon atoms held together by strong covalent bonds. Graphene has been described as a 2D material. It is the first material to be so-called and throws up certain theoretical questions.

Previously all matter was considered, no matter how small as belonging in 3 dimensions. This was the flaw of traditional paper based geometry – it was only in 2 dimensions and so could only exist in a theoretical form, much in a similar way that Plato’s ideal forms could not exist in the real world but somehow informed all matter and allowed us to grasp the beauty of things that approximated closer to the ideal form.

In physics graphene is viewed as a 2 dimensional substance that exists in 3D euclidean space. It sometimes referred to as an abstract surface. However, to exist such 2 dimensional surfaces must be curved and strained. This curvature and strain act on the charge carriers to create fields. These fields set up a magnetic influence that form cycles comparable to ’Lamor Cycles’.

The mathematician M. V. Karasev at the Moscow Insistute of Electronics and Mathematics has written up much of the math formulas to map the resulting forces caused by the curvature and strain set up by a two dimensional surface. You can find a copy of his important paper at: springerlink.com.

Discussing the math involved in dealing with 2D material also gives rise the possibility of being able to study and mathematically fix materials that can be detected in 4 or more dimensions. What magentic and and electrical characteristics would such material have?

The future may be a mystery, but the future seems to be always providing us with new ways to test our theoretical understanding of the world.

Playing The Numbers Game

Mathematics in business is usually left to the financial team but numbers are getting ever increasingly important in tracking the development and popularity of a business be it with a new starter company or an established company that is seeking information on its customer base.

One specific way we are seeing this in practice is through social media with most companies having Facebook pages or Twitter accounts. This allows them to stay in contact with their customers as well as gauge responses to certain things. For example following a tesco facebook marketing campaign you would be able to see how liked the specific campaign is as well as read comments to really see what is being said about it.

The number of followers or ‘likes’ on a Facebook page can obviously show how popular a company or brand is but can also be of use by putting information direct to the consumer allowing them to take advantage of deals.

This can work well for companies, although, as was experienced by Tesco in the Scottish town of Greenock it can also mean information about mistakes can spread like wildfire. On discovery that a deal regarding beer was put through wrong offering the customer a ridiculously low price word spread across social media sites leading to crowds of people attempting to take advantage of the mistake before it was rectified.

Here we can see the power of Social media and that is why more and more companies are trying to harness that power and use it to benefit both themselves and their customers. Keeping a good open communication has proved to be very beneficial to the companies that are already taking advantage of this simple way of getting into peoples homes.

I believe in the future we will see a greater dependence on this sort of interaction.

A Basic Introduction to Cryptanalysis

When you look at a cryptogram for the first time, they nearly always look rather daunting after all how are you expected to decipher all that code! However there are certain techniques that can make the task much less daunting and in some senses it is actually quite fun to work them out.

One of the most basic procedures is that of frequency analysis. In fact without this technique you won’t get very far in understanding any of the procedures behind substitution cryptanalysis. So let’s try and briefly explain what’s behind this technique in an effort to expand our mathematical education.

Cryptanalysis relies on the fact that all letters of a specific language have specific characteristics or personalities. To the ordinary observer all the letters might look fairly similar, but to the analyst they will know the specific traits and characters of each individual letter.

GJXV EHRT NUCOT WMOHY UWGK HWIE

Don’t try and decipher this example though – there’s nothing there. The cryptanalyst would initially begin by counting each letters frequency and it’s contacts (the contacts are the letters which are adjacent. They would then construct a frequency table based on the text. This can then be compared with a standard frequency table based on a similar number of words. This table will list the likely frequency of each individual letter – i.e how often it would be expected to appear.

Unfortunately it is rarely as simple to line up your standard frequency table and the one you created to solve the cryptogram. They are very unlikely to be identical even for the reason that they will be based on a different number of characters. However it is surprising how little relative frequencies shift from one piece of text to another. You will almost always find e,t,a,o,n,r, i, s and h in the high frequency areas whereas d,l,u, c and m will normally be found in the medium frequency group.

It’s useful to see how the basics work in encryption, especially if you use such technology to protect yourself online. For example I can use a technology to spoof my IP called a Virtual private network which creates an encrypted tunnel between the client, effectively hiding the IP address of my client.

Using frequency you can narrow down to distinct groups each individual letter, often with very accurate results. However you need more than this to focus more specifically on a possible solution and this is where ’contacts’ are important. Every letter has a cluster of associations that are likely to occur. In fact an experienced cryptanalyst can spot these associations almost without thinking when presented with a frequency distribution and a tally chart.

Is Math too Abstract?

One of the things that puts young people off studying maths is that it is too abstract. Maths is hard to grasp for many because it appears too theoretical. Math appears to inhabit a parallel universe to the everyday world, and this puts many people off.

For others this is the attraction of mathematics – it is a world that has its own immutable laws and its own internally coherent logic. Math seems neater and more precise than the real world.

However, this characterization as math being distinct, separate and autonomous from nature is a very much mistaken one. Math obviously originated as a means of dealing with amounts. This is the common way to present math problems to kids – you have 10 apples and 5 people – how many apples can each person eat? From this example it can be seen that math allows us to symbolize amounts of actual things in numbers and make calculations. The next step is to substitute unknown amounts for letters (algebra) and to use math equations to work out the unknown quantity.

Math is fundamental to the world we live in. Math is the foundation behind scientific equations. These equations allow engineers to turn science into technology that can benefit our lives. The same is true for medicine. We wouldn’t have any of the life saving drugs we do have without maths and chemistry.

Man’s greatest achievement has been to make models of the world and through understanding these models to manipulate reality. These models whether computer simulations, scientific theories or chemical equations are based on maths and cannot exist without math. It is not wrong to describe math as abstract, but it is very wrong to consider math as separate and irrelevant to reality. Indeed for anyone living in a city what they see out of their window in the product of math and applied math.

Further interest:

New Proxies

Math and Wine

I don’t think many of us draw the connection between what amounts to our favorite alcohol and math. It’s there if you take the time to look though.

To start, think about your favorite bottle of wine. Or more simply, think about any of the 90 point wine clubs which exist only to find those exact bottles.

The real question is how winemakers are able to craft wine which is of that high quality. There is, without a doubt more science involved than you might think. Most of us believe that a winemaker walks through a vineyard throughout October and tastes grapes. When the grapes taste just right they are picked and then left alone to ferment and create the wine we drink on a nightly basis.

While that certainly happens in some places-at most wineries these days winemaking is more science than art. Wineries employ full time chemists (they call them enologists, but they’re nothing more than chemists) who test the sugar levels of grapes before the winemaker becomes involved at all.

I think we all know that math is incredibly important, but did you know how much math really went into that bottle of wine you’re enjoying with dinner?

Archive for May, 2012

The Maths of Graphene

Playing The Numbers Game

A Basic Introduction to Cryptanalysis

Is Math too Abstract?

Math and Wine