Introducing the Invariance Theorem

This is from a series of lectures by by Hector Zenil and Narsis A. Kianib entitled From Networks to Cells. It is sponsored by the Sante Fe Institute.

Text Transcription below:

So we saw in the last lectures how we can quantify algorithmic randomness by using computer programs running on some reference universal Turing machine. But one could think that it is always possible to find some language in which a particular object had a short encoding, no matter how random. For example, Alice and Bob could agree to compress all the content of the Encyclopaedia Britannica in a few symbols, so when Alice presents Bob with those symbols, Bob would know Alice meant the Encyclopaedia Britannica.

Algorithmic Information Theory (AIT) would mean little if the complexity of something could be determined so arbitrarily by renaming things with any number of symbols. AIT requires the program to reconstruct the original object from scratch so there is no coding cheating. Bob would need a computer program to reconstruct the Encyclopaedia Britannica without any external help from Alice or anyone else. It would also look like one would need to specify the specific programming language and the particular universal Turing machine on which those computer programs would run for algorithmic complexity to work or make sense. Otherwise, one would be able “rename” things and make anything look random or simple by changing the underlying language and Turing machine. So let’s consider the following sequence that has 3 symbols, 0, 1, and 2: We can show that this sequence can be generated by a small computer program like this one: whose length is no more than 56 bytes no matter how long the sequence to produce the same pattern. However that is in the Wolfram Language running on Mathematica. What if instead we had used Lisp, Java or Visual basic? It seems that the result would then depend on the computer language chosen.

And even in a single computer language there may be different computer programs of different size can produce the same object. For example, the following two computer programs can generate any number of digits of the Thue-Morse sequence and both are different in length even though both are small and generate the same sequence. So how to deal with different computer programs in possibly different computer languages with different lengths? Fortunately, the Invariance Theorem as proven by Kolmogorov, Solomonoff and Chaitin, tells us that the difference between any two computer programs producing the same object is at most a constant from each other. More formally, the so-called invariance theorem establishes that the difference of the lengths of the minimal programs on any two computer languages L1 and L2 is always bounded by a constant that depends on L1 and L2 but not of s: The way to see this is by thinking in a translator program between two languages L1 and L2.

Because what the invariance theorem says is that one can always write a 3rd computer program of fixed length capable of translating between any two languages L1 and L2. This is actually very clear with the computer language Java because one of the innovations of Java when it was introduced was that there was a Java Virtual Machine or JVM that made Java multiplatform in the sense that you could run your Java program on any platform without changing any language code. That is because the many JVMs were acting as a translator, or officially called a compiler, between the Java programming language and each machine code for different platforms such as Unix and Windows. So if the translator is shown in blue, one has to only add the translator to every computer program written in some language to convert it into another program in another language, both languages producing the same sequence of interest.

The length of the compiler or the length of the blue part is the same for all input strings and so it is constant. So if L1 is the Wolfram Language and L2 is Java, then the invariance theorem tells that finding the shortest computer programs to measure the algorithmic complexity of a string is about the same length up to a constant. In another more practical example, what the “c” in the invariance theorem characterises in a biological context, for example, is that the DNA and a set of reading and writing chemical machines called ribosomes form a couple of language and compiler able to transcribe and translate DNA into proteins effectively from one language into another. So ribosomes can be thought as some sort of compilers between the DNA, RNA and the protein space with the ribosomes always of same complexity independent of what they are reading or writing. can be reproduced with a code no longer than 56 bytes no matter how long the sequence with this pattern is required.

Well, that is not entirely true. Actually the code has to grow by a little bit if we want to produce longer sequences with the same pattern. It does grow a little bit because one has to determine the end of the algorithm, that is when do we wish the sequence to come to an end. And that number does make the code to grow a bit. In fact, classical information theory tells us that we can encode an integer n in about logarithm of n bits, because remember we can always guess a number in about logarithm of n yes and no answers. You can see how running this code with very large numbers, the byte count does actually change for very large numbers: So despite how trivial this extremely little piece of information may look and how kind of little difference it makes, it is not only mathematically precise but it will turn out to be the most important little piece of information in the field of Algorithmic Information Dynamics, and you will see soon why.

Not to keep you waiting but not getting into details, the main idea is that, unlike a deterministic system that has a well-defined generating mechanism such as this sequence and its generating formula, other systems that may be subject to some source of randomness or are interacting with other systems, this logarithmic change will not be longer be respected and the more removed from the logarithmic term the more it will tell you some unvaluable about the system’s behaviour and its underlying causes. To understand this better, in the next unit we will cover the theory of dynamical systems for you to understand what a dynamical system is before we finally get into the subject of Algorithmic Information Dynamics. .

Additional Resources:
Some great Maths Lecture on British TV – to access them from outside the UK then you’ll need a VPN. This post shows how it’s done – ITV Hub abroad.

To test functions of random generation in tandem with online programs and devices then you’ll probably need to switch IP addresses randomly while conduction the tests by perhaps using rotating proxies – https://residentialip.net/what-are-rotating-proxies-and-how-can-they-make-me-rich/

Solving Flows in Network Formats

Flows in Networks

Flows in Networks are used in problems such as traffic on roads and oil in pipelines. This graph is called a digraph where the arcs have a direction denoted by an arrow. The weights on the arcs are the capacities of the arc and show the limit of the flow that can be passes down that arc.

Consider a network with N nodes. Node 1 will be the source (all flow will come from node 1). Node N will be the sink (all flow will be to node N). For all the other nodes we will assume conservation of flow, i.e. no flow is lost.

The object is to maximise the total flow from node 1 to node N.

Intuitive Approach

Step 1. Find any flow-augmenting path from source to sink such that each of the arcs has a positive flow capacity.

Step 2. The arc with the smallest flow capacity limits the flow along the path. Assign (i.e. ‘send’) this flow (f) and reduce the capacities of the arcs on this path by f.

Step 3. Repeat steps 1 and 2 until no suitable path can be found.

e.g. Initial Flow-augmenting Path 1 ® 2 ® 4 ® 6 f1 = 4

Second Flow-augmenting Path 1 ® 3 ® 5 ® 6 f2 = 4

Third Flow-augmenting Path 1® 3 ® 4 ® 6 f3 = 1

Maximal flow = 4 + 4 + 1 = 9 (this is obviously maximal since all the arcs from the source are saturated).

It’s often difficult to see useful applications for these diagrams but they’re actually very useful in real life.   For instance creating such structure would have definitely helped programmers and designers who created the Netflix algorithm which creates your suggested option screens.  Using the data from previous selections it can use flow format theorems to create a likely pattern of other shows.  Just look carefully how the structure works, if you haven’t got Netflix access then just use this trial to watch the US version!

If however our initial flow-augmenting path was 1® 3 ® 4 ® 6, f1 = 5, then the network would be stuck but not at maximal flow. We must therefore build in a refinement to stop us getting stuck. The idea is to allow a ‘fictitious’ back flow in the wrong direction along an arc in order to cancel out all, or part, of a previously assigned flow. e.g. in the above stuck network we could send up to 5 from node 4 to node 3 and still end up with a real flow from node 3 to node 4. So the second path is 1 ® 2 ® 4 ® 3 ® 5® 6, f2 = 4. Now continuing as before we could again have a flow of 9. (The real flow along (3,4) is 1 from node 3 to node 4).

A different notation can be used called the labelling procedure and appears to be preferred by the examination board.  They are commonly used in network diagrams too if you want to illustrate the flow from something like a e-commerce server to an ATC proxy, or other network devices. In this method when the capacity is reduced after flow has been assigned a back flow in the opposite direction is shown. This can help in two ways; one is by showing the total flow after a few iterations along a certain arc, even if it is the “wrong” direction. The second is it may help with the fictitious flows mentioned above.

Linear Programming – Graphical Methods

Graphical Solution

When the L.P. problem involves only two decision variables its solution may be obtained by graphical means. Consider the earlier example.


The shaded region OABCD contains all the points (x,y) which satisfy the constraints. The points form the set of feasible solutions. Clearly there are an infinite number of such points. We require that one which gives a maximum value of z. From z = 8x + 5y with rearranging we get y = – 8/5x + z/5, for different values of z this equation represents a family of straight lines each with a gradient -8/5. Z will be greatest when the family of parallel lines leave the feasible region i.e. at C (2.5,4.5). Point C’s values can be found using simultaneous equations of the lines which cross at that point. 3x +y = 12 and x + y = 7. By subtraction 2x = 5 \ x = 2.5 y = 4.5.

Hence Zmax = 8(2.5) + 5(4.5) = 42.5.

Integer Solutions

In the above problem we are producing bicycles and trucks. Yet our graphical solution gave us x being 2.5 and y being 4.5. Not very sensible, it is difficult to make half a truck! Some problems obviously must be integer solutions. In these cases we must “test” integer points within the feasible region. The integer point will usually be near the location were the family of profit lines left the feasible region. In our case near point C. Integer points near C yet still within the feasible region are (2,4), (2,5), (3,3). Putting these points in our objective function z = 8x + 5y gives z to be 36, 41 and 39 respectively. Hence our integer solution is a point (2,5) there the profit is £41.

If you find these difficult to follow there are lots more examples online.  Many of the best lessons used to come from the Open University Math’s models which were broadcast on the BBC.  I’m not sure if you can still find them as they are quite old now, but it might be worth checking the BBC’s archive – BBC iPlayer.   If you’re not able to access the site because of location restrictions then this site can help to access via a BBC iPlayer DNS service.

The Simplex Method

The Simplex Method is an algebraic rather than graphical approach to solving Liner Programming problems. The advantage of the Simplex Method is that it can cope with grater than 2 variables which cannot be solved graphically.

The first step is to state the general Linear Programming problem in a standard form.  We require a L.P. problem to be put into a form in which

  1. the object function is to be maximised e.g. Maximise z =
  2. all the variables are to be non-negative i.e. zero or positive because you cannot have negative amounts

III. all the constraints are to be equations (except for the non-negativity constraints) i.e. no >, , <,  but only =.

  1. the right hand side constant of the constraints is positive or zero e.g. not x = 2 but x = 2 i.e. the RHS is +ve.

For further help see the below resources:

Additional: BBC iPlayer Access – Free Trial Offer 

Algorithm Primer

An algorithm is simply a sequence of precise instructions to solve a problem.  Precise means that there should be no ambiguity to any instruction or which instruction is next.

An example of an algorithm is Zeller’s Algorithm which is used to work out which day of the week a date is

Zeller’s Algorithm                                        Example      15/05/1991

Let Day number = D                                                        D = 15

Month number = M                                                         M = 5

and Year = Y                                                                  Y = 1991

If M is 1 or 2 add 12 to M and subtract 1 from Y.

Let C be the first 2 digits of Y                                          C = 19

and Y’ be the last 2 digits of Y                                        Y’ = 91

Add together the integer parts of

(2.6M5.39),(Y’4) and (C4), then                             7+22+4+15

add on D and Y’ and subtract 2C                                    +9138=101

{Note: Integer part of 2.3 = 2 and 6.7 = 6

1.7= 2 and 3.1 = 4}

Find the remainder when this quantity is divided by 7        1017=14r3

If the remainder is 0 – Sunday, 1 – Monday etc.

Communicating an Algorithm

We can communicate algorithms by ordinary language as above, by pseudo-code (stylised English), computer languages or flow charts.

Let’s take the example of finding the real roots of the quadratic equation

Ax2 + Bx + C = 0  (assume A0)

Using the equation

Using pseudo-code we could write

calc ; note its value;

if this is negative    [no real roots – stop]

else                       [calc root of value

calc (B+root) (2A)

calc (Broot) (2A) – stop].

For programmable calculators we could write

Casio Graphic                                                       Texas TI-81

?A                                                                     :Input A

?B                                                                     :Input B

?C                                                                     :Input C

D                                                    : D

(B+D) (2A)                                                     : (B+D) (2A)

:Pause

(BD) (2A)                                                     (BD) (2A)

{Note: These assume real roots}

In BBC Basic we could write

10      Input “A,B,C”;A,B,C

20      D=B*B4*A*C

30      IF D<0 THEN PRINT “No real roots”: STOP

40      PRINT (B+SQR(D))/(2*A)

50      PRINT (BSQR(D))/(2*A)

Using a flow chart we diagrammatically represent the sequence of steps in an algorithm.

Three types of boxes are common

  1. The start stop box
  2. The instruction box
  3. The question box

Russian Peasant’s Algorithm

Example

Write down side by side the two numbers to be multiplied           163    24

Repeatedly [beneath the LHS number, write down the number    81      48

which is half the number above, ignoring remainders, beneath     40      96

the RHS number write down the number doubled] until you        20     192

reach 1 on the LHS.                                                                  10     384

5       768

2     1536

1     3072

Delete those rows where the number in the LHS is a multiple

of 2.  Add up the numbers left on the RHS                                163      24

81      48

40      96

20     192

10     384

5       768

2     1536

1     3072

This number is the answer                                                                3912

There’s a great demonstration of the Russian Peasants algorithm on the BBC learning websites, and if you’re lucky a couple of explanations on the BBC iPlayer archive.  For many maths lessons for children and adult learners it’s a great resource.  If you’re having trouble accessing from outside the UK then this BBC iPlayer DNS option works great and allows people to access all UK media sites without restrictions.

Another alternative is to use a resource which supplies residential IP addresses, although these aren’t specifically designed for streaming video they work pretty well.  There’s an example of them in this post ATC proxies explained be aware that they can be quite expensive though,

Infinity is bigger than you think – Numberphile

Video Transcript

JAMES GRIME: We’re going to break a rule. We’re break one of the rules of Numberphile. We’re talking about something that isn’t a number. We’re going to talk about infinity. So infinity. Now like I said, infinity is not a number. It’s a idea. It’s a concept. It’s the idea of being endless, of going on forever. I think everyone’s familiar with the idea of infinity, even kids. You start counting 1, 2, 3, 4, 5– you might be five years old, but already you’re thinking, what’s the biggest number I can think of. And you go, oooh, it’s 20. You get a bit older, and you go, maybe it’s a million. It never ends, does it? ‘Cause you can keep adding 1. So that’s the idea of infinity. The numbers go on forever.

But I’m going to tell you one of the more surprising facts about infinity. There are different kinds of infinity. Some infinities are bigger than others. Let’s have a look. The first type of infinity is called countable. And I don’t like the name countable. And Brady gave me a little bit of a hmm, just then. Because if you’re talking about infinity, you can’t count infinity, can you? Because it goes on forever. I think it’s a terrible name. I prefer to call it listable. Can we list these numbers? All right.

Let’s do these simple numbers, 1, 2, 3–

BRADY HARAN: You’re not gonna do all of them, are you James?

JAMES GRIME: 4. How long have we got?

BRADY HARAN: (LAUGHING) 10 minutes.

JAMES GRIME: Right. 5, 6– so you can list the whole numbers. So this is called countable. Listable, I prefer. What about the integers? All the integers. That’s all the negative numbers as well. So there’s 0. Let’s have that. But there’s 1 and minus 1, there’s 2 and minus 2, there’s 3, and minus 3. Now, that is an infinity as well. And in some sense, it’s twice as big, because there seems to be twice as many numbers. But it is infinity as well. They’re both infinity, and they’re both the same type of infinity. They both can be listed.

Perhaps more surprisingly, the fractions can be listed as well. But you have to be a bit clever about this. Let’s try and list the fractions. I’m going to write out a rectangle. 1 divided by 1. That’s a fraction. [INAUDIBLE]. Let’s have 1 divided by 2, 1/3, 1/4, 1/7– OK, that goes on. Let’s do the next row and have two at the top. 2/1, 2/2, 2/3, 2/4. Let’s do the next one. 3/1, 3/2. 4/6, 4/7. That goes on and we can keep going. So here, I’ve made some sort of an infinite rectangle array of fractions. Now if I want to make it a list like this, though, If I went row by row, you’re going to have a problem. If you go row by row, I’ll go– there’s 1, 1/2, 1/3, 1/5, 1/6, 1/7– and I’ll keep going forever. And I’m never going to reach the second row. I can’t list them. Not that way. You can’t list them that way. You’ll never reach the second row.

This is how you list them. Slightly more clever than that. You take the diagonal lines. Now, I can guarantee that every fraction will appear on one of those diagonal lines. And you list them diagonal by diagonal. So that’s the first diagonal. Then you list the second diagonal– there it is. Then you list the third diagonal, then you take the fourth diagonal, and the fifth. So eventually, you are going to do this every fraction. Every faction appears on a diagonal, and you’re going to list them. Now, if you take all the numbers, right? That’s the whole number line.

Let’s try that. Look, I’m going to draw it. It’s a continuous line of numbers. These are all your decimals. You’ve got 0 there in the middle, and you’ll go 1 and 2 and 3. But it has a 1/3. It will contain pi, and e, and all the irrational numbers as well. Can you list them? How do you list them? 0 to start with, and then 1? But hang on. We’ve missed a half. So we put in the half. Hang on, we’ve missed the quarter. We put in the quarter. But we’ve missed 0.237– so how do you list the real numbers? It turns out you can’t. In fact, rather remarkably, I can show you that we can’t list them, even though were talking about something so complicated as infinity.

BRADY HARAN: Do it, man!

JAMES GRIME: We need paper.

BRADY HARAN: We need an infinite amount of paper here, I think.

JAMES GRIME: (LAUGHING) It’s a big topic. Imagine we could list all the decimals, right? We can’t, actually. But pretend we can. What sort of– what would it look like? We’ll start with all the 0-point decimals. Let’s pick some decimals. 0.121– dot dot dot dot dot. Let’s pick the next one. Let’s say the next one is 0.221–. Next one, let’s do 0.31111129–. And let’s take another one, here. 0.00176–. Now I’m going to make a number. This is the number I’m going to make. I’m going to take the diagonals here.

I’m going to take this number and this number and this number and this number and this number. And I am going to write that down. So what’s that number I’ve made? It’s 0.12101– something, something, something. Now this is my rule. I’m going to make a whole new number from that one. This is the number I’m going to make. If it has a 1, I’m going to change it to a 2. And if it has a 2 or anything else, I will change it to a 1. So let’s try that. So I’m going to turn this into– 0-point. So if it has a 1, I’m going to turn it into a 2. If it’s anything else, I’m going to turn it into a 1. So that will be a 1. I’m going to change 1 here into a 2. I’m going to change that one into a 1. I’m going to change that one into a 2– that was my rule.

And I’ll make something new. That does not appear on the list. That number is completely different from anything else on the list, because it’s not the first number, because it’s different in the first place. It’s not the second number, because it’s different in second place. It’s not the third number, because it’s different in the third place. It’s not the fourth number because it’s different in the fourth place. It’s not the fifth number, because it’s different in the fifth place. You’ve made a number that’s not on that list. And so you can’t list all the decimals, in which case it is uncountable. It is unlistable. And that means it’s a whole new type of infinity. A bigger type of infinity.

BRADY HARAN: Surely we could, James, because all we’ve got to do is keep doing your game and making them and adding them to the list. And if we keep doing that, won’t we get there eventually?

JAMES GRIME: But you could then create another number that won’t be on that list. And so the guy who came up with is a German mathematician called Cantor. Cantor lived ’round about the turn of the 20th century. He was ridiculed for this. For this idea that there were different types of infinity, he was called a charlatan. And he was called– it was nonsense, it was called. And poor old Cantor was treated really badly by his contemporaries, and he spent a lot of his later life in and out of mental institutions, where he died, in the end. Near the end of his life, it was recognized. It was true. It was recognized. And he had all the recognition that he deserved.

BRADY HARAN: And now he’s on Numberphile.

JAMES GRIME: And now he’s on Numberphile, the greatest accolade of all. Georg Cantor.

Additional Resouces:

For Watching Maths and technology Programs from places like the BBC – http://www.cedps.net/bbc-start-blocking-vpn-access-from-abroad/

Video about BBC VPNs not Working – http://www.youtube.com/watch?v=RQksl7wclZE

Mathematical Concepts – Vedic Mathematics

Sometimes it’s easy to forget that there are different methods of achieving the same goal. Most of us from Western educational systems have learnt mathematics in a very similar way right from primary school to undergraduate level. However that doesn’t mean it’s the best way and there are certainly alternatives especially in the field of mathematics. One of the most famous is the method known as Vedic Mathematics, which I first learnt about in the remarkable BBC documentary – The History of Maths. You may be able to still access on the BBC iPlayer however you will need a residential IP based in the UK to be able to watch it. There’s more help on this at the end of the article if you need it.

What does Vedic mean?

In the introduction to the book Vedic Mathematics Sri Tirthaji describes two senses in which Veda can be understood. The first is as ancient scriptural texts and the second is as true knowledge.

The Vedas are perhaps the oldest known ‘texts’ and form the source of spiritual, philosophical, moral, ethical and secular teachings of the Hindus. It is not possible to determine their age because they were handed down by word of mouth, although they are thought to be more than five thousand years old. The Vedas were originally consisted of three texts – Rigveda, Samaveda and Yajurveda – each dealing with different aspects of human development and conduct. A fourth Veda, Atharvaveda, was included at some ancient epoch. At a time when the power of memory became insufficient amongst the protectors of these verses the Veda were written down. This may have been earlier than 1000 BC.

Each Veda has two portions, one dealing with prayers and mantras and the other (Brahmana) describing the meaning and procedure of the prayers and mantras. The sutras of Vedic Mathematics are supposed to be within an appendix portion (Parishishta) of the Brahmana section of the Atharvaveda. Sri Tirthaji did not give a precise reference for the sutras and to date nobody has found all of them. There are a number of possible reasons for this discussed below.

True Knowledge

The second meaning of Veda is true knowledge which is alive today and relevant to our lives. There are certain principles of conduct, for example, which would appear to be common to all races and religions and applicable at all times throughout history. To cite one example, there is a law written in the hearts of men by which it is natural not to seek harm of anyone. Both the Hindu principle of Ahimsa and the Christian ethic, “Love thy neighbour as thyself”, are direct expressions of this law.

This great principle is as relevant today as ever, irrespective of the date when it was first expressed. What really matters is its permanence and its relevance to the society in which we live. Similarly, with the Vedic mathematical sutras, it does not matter when they were first expressed. The important questions are, are they relevant and how can we not let them be forgotten?

What Sri Tirthaji appears to have discovered is how these sutras apply to the mathematics of his day. This is no doubt the reason for a good deal of arithmetic and algebra. He also applies the sutras to differential calculus, a relatively modern mathematical tool. This fact alone suggests that although the laws expressed by the sutras are unchanging the application of those laws changes with time just as the understanding and nature of mathematical expression evolves or devolves.

Recent research has discovered that the Vedic Mathematical sutras are applicable to any area or topic within mathematics, ancient or modern. The reason for this is that the sutras describe common mental processes of the human mind rather than particular mathematical fashions. For example, the sutras can be found at work within classical Greek geometry, the theory of determinants, in Chaos theory or even in Catastrophe theory.

Additional Reading:
Accessing the BBC – BBC Live VPN

The Mathematics of the Drunken Walk

Lectures on statistics are not always the most exciting ones in the world of maths however certain subjects tend to attract students attention.  One of those is of course alcohol, and not in the over indulgence and problematic way where you end up taking a drug like this one called Selincro to combat it’s effects.   It’s actually revolved around a concept called the Drunkard’s Walk a famous mathematical concept.

It can be best explained in a theoretical example about a drunken man who was walking way too close to a cliff for someone in that state.  The idea is that from his starting position a single step forward would send him over the cliff.   He takes completely random steps oblivious to his own safety in any direction.  His probability of taking a step away is 2/3 and of taking a step towards the cliff is 1/3.  The mathematicians problem – what are the drunkards chance of escaping the cliff?

It’s a classic problem but actually touches on some advanced statistical topics. The particular topic is centered around Stochastic Processes which covers these ‘random walk’ issues, the specific name is called a Markov Chain.

Stochastic Process – a random process which explains how a system or process changes over another unit (commonly time).

Random Walk – a path derived from a series of completely random steps in some defined mathematical space.  Our example is the very drunk man tottering on the edge of a cliff.

Markov Chain – a random walk which actually maintains independent events.  That is the next event is not dependent or related to the previous one.  The drunken man has to be so drunk that his position and last action has no bearing on his next step!

The mathematics of this situation is of course all related to probabilities and how likely the man is to survive his reckless behaviour.   The simplest point is the beginning where he is one step away from the edge, the probability of surviving the next step is 2/3 and he has a 1/3 chance of stepping over the edge.

After that of course it get’s more difficult as the man if he survives will be moving away from the cliff edge and buys himself some time.   The easiest way to visualize this situation is to draw a chart of the probabilities with all the possibilities.  This has to include his relative position from the cliff and an assumption about where he ends up and what position is safe!

The problem is actually not that complex but it can seem so purely because there are so many possibilities after the initial even.  The secret is to define the chart with the possibilities and then try and generalize the problem in order to create a formula. This has to include the probability of stepping towards the cliff edge and stepping away

To solve these problems you normally define the expected probability of the event you are trying to measure.  So in this case it would be defining the probability of falling from the cliff – say P1.

Without too much detailed analysis we can get to the formula as follows:

P1 = (1-P) + (p*P2)

Here the variable P2 is the probability of falling from the cliff on a path consisting of 2 steps!

John Welcome

 

Mathematics Key to 4D Printing

Although some of us are just getting our heads around the amazing potential in 3D printing, the next step is already on the horizon.    A leading mathematician has started working on the formulas required to step into an extra dimension!

Three D printing is already revolutionizing all sorts of areas from manufacturing, medicine to science and engineering.  It’s now fairly simple and inexpensive and has the potential to create all sorts of intricate objects quickly and cheaply.   There are printer parts in our machines and indeed people are having printed body replacement parts transplanted into their bodies with great success.

However there is always a next step, and now mathematicians are working on taking us in to the world of 4d printing.  Just to clarify we are talking about the possibility of fabricating objects with a programmable shape over time.  It’s always been theoretically possible however no-one had really starting looking at working through the complexities involved.

This seems to be changing as Professor Pasquale Ciarletta from Milan has just published a paper in ‘Nature Communications’ where he has started working through the numbers about a specific problem with this.  The professor has been focusing on how to control the sudden nucleation of localised furrows in the soft solids produced in 3d printing.

The advantages and possibilities of these developments may not be initially apparent.  However in addition to the advantages to the field of engineering there is huge potential to have the ability to design and print objects which can morph over time.  The paper related the development to the field of development biology as particular interest.  Here we could look at things like tissue morphogenesis and other areas such as issues in the brain or tumour control.

Ciarletta has acknowledged that there are great complexities behind making this work.  There has already been lots of experimental investigation of the issues involved – the physics behind the concept of ‘creasing’ being particularly challenging.  His study proposes a unique mathematical approach to predicting the experimental conditions required to trigger the onset and how creases change over time.  This is the key to being able to control their appearance on a specific scale and ultimately to be able to print them in 4d.

There are parallel advancements being made in the area of 3d printing too.  You can already sit down and watch the football on Match of the Day live like this on a completely 3d printed television set.  It is also now possible to edit specific printed objects after they have been created.  This is achieved by repeatedly changing the colours of 3d printed objects after thy have been printed.

The concept is currently being developed under the name ColorFab and it involves using a specially created 3d printable ink which can actually change colour under certain conditions – primarily after being exposed to UV light.  This of course has a time delay currently estimated at around 20 minutes, however the researchers are hoping to improve on this substantially in further development.

Further Reading: Available on British TV

Common Childhood Problems with Maths

During an early phase of a child’s growth, mathematical issues or disabilities could contribute to enormous fear in their own minds. There are lots of unique skills accountable for natural or psychological deficits that could result in your kids requiring mathematics homework help. They could have a negative influence on your child’s cognitive advancement. Let’s attempt to gouge these:

Confusion Managing amounts

Universe fundamental computations like division, multiplication, addition, and subtraction result in math on the entire world. Memorizing and remembering straightforward equations, according to those essential operators, is crucial and where many children falter. These basic calculations develop advance mathematical understanding. It may possibly be prevented by employing an internet math tutor from throughout the world which could come for affordable and possess a comprehensive understanding that could prove very helpful for your children.

Difficulties with computers

There are lots of children around who have great understanding of mathematical theories, but implementing exactly the exact same on computers prove to be a massive undertaking for them. Errors associated with writing numerical, estimating amounts, or misreading personalities, are normal. They wind up fighting doing the fundamental jobs. They surely require help in the shape of online mathematics tutoring at the place where they can learn blending computing and mathematics to enhance their mathematical thinking skills.

Correlating math with reality

This really is among the most frequent issues faced around by children. It’s the inability to associate the realistic way of mathematical theories. Understanding symbols, formulas, shapes, etc., and recalling the same in regards to employing in real life, is 1 hell of a project. It’s the obligation of parents or perhaps somebody who can help your children with math homework assistance, so that actual life program becomes possible with those children.

Putting connections between mathematical entities

It’s hard for children to establish relations between multiple mathematical situations, scenarios, interdependencies, etc. that is essential to make matters relevant. Even an internet math tutor ought to be able to understand that the limitations lying at the mindsets of these children. The mathematics abilities ought to be anchored so they can comprehend items readily.

Learning math wrong way

To get a huge majority, talking, writing, and reading, math itself is the largest hurdle. They confront loads of issues in pronouncing mathematical conditions, facets, terminologies, and much more. They can not comprehend verbal or perhaps written explanations. They find it challenging to interpret things too. Online math tutoring can definitely help sort out these problems. But, in addition, it depends upon how well the children take things favorably.

Useful Links:

BBC BiteSize on Mathematics – http://www.bbc.co.uk/education/subjects/z6vg9j6

Accessing BBC iPlayer from France and Abroad – http://bbciplayerabroad.co.uk/how-do-i-get-bbc-iplayer-in-france/

Has Someone Actually Solved the Riemann Hypothesis

Sometimes a paper comes along which can breath new life into a subject or problem long thought unsolvable. This year a trio of mathematicians looks like they’ve done just that in offering a new tactic to solve the ‘greatest unsolved problem in mathematics – the Riemann Hypothesis.

This paper has just been published in a maths journal called Physical Review and suggests that the analysis is proven correct then it can also be used to prove the Riemann Hypothesis.

 Prime Numbers
Predicting Prime Numbers

For those whose lives are not centred around mathematics this might sound a little obscure.However for mathematicians it represents fame, success and of course cash.The solution to the Riemann hypothesis is one of the seven Millennium Prize problems which cover the most difficult problems in maths.   For more information on this prize have a look on BBC iPlayer where there was a recent maths documentary, this link shows how to access it from outside the UK.  Every one of these problems comes with a one million dollar prize for a solution.

This hypothesis is names after the German born mathematician Bernhard Riemann.It’s such an important problem because it offers a method to understand the distribution of prime numbers. If a method was found it would completely revolutionise mathematics.Being able to work out how may prime exist in any given situation would make many branches of the science much, much easier.

So where is this solution hidden, well it is suggested it lies in quantum mechanics.

An amazing statement from this paper proposes that quantum mechanics could solve the Riemann Hypothesis. This difficult area of physics usually used to try and make sense of some of the smaller scales in nature.

So what’s in the paper? Well the authors have suggested that the existence of a quantum system of energy corresponds to the proposed conditions in the Riemann Hypothesis.They have also defined a specific variable called the Hamiltonian Operator as the crucial part of this system.

If this all works out then the method effectively reduces the huge problem of the Riemann Hypothesis down to the level of the Hamiltonian Operator. A mythical problem that was almost deemed impossible to solve suddenly becomes much closer.The paper is only in the first stages though and peer review is next which could take some time.

But it certainly has created some excitement for anyone who has even a passing interest in mathematics.

Further Information: BBC News Streaming