The only experience that is worse than the annual condo meeting or queuing up at the post office is, probably, the parent-teacher conference.
It's an ordeal for the parents, forced to wait a long time. It's cause of panic for the students, who are afraid that their parents may be mad at them. But, I assure you, it's a terrible experience for the teachers too.
Obviously, since bad marks in maths are widespread in Italian schools (but the same can be said of schools all over the World), queues to speak to the maths teachers are never-ending, exactly like queues at the post office. In such situations, the long waits result in the best and worse behaviours on could possibly imagine.
As an experienced teacher, I tend to split parents into three categories: the strikers, the compliant and the flatterers.
The strikers are the funniest ones. They sit down, ask you how their child is doing and, before you have even finished to speak, they charge. They do it recklessly, without thinking, coming up with completely made up arguments like:
The flatteres are, instead, the most dangerous ones. They always start with something like:
- "My daughter said that, thanks to you, she finally gets maths. She couldn't get it with last year's teacher." You would like to say: "Look, you have probably said the same thing to the other teacher when you were talking about your other child. Also, too bad that last year's teacher was me.", but you just shut up and pretend to be playing the game. The parent, in the meantime, scrutinizes you, trying to understand whether he fooled that idiot of a teacher or not.
- "Your explanations are excellent", they go on, "and my son is working really hard. It's a shame that last exam didn't go so well. But I'm sure that if you could help out with the grades, he would get back on track. It would be too bad if he lost motivation given that you're such a great teacher." And then the may speak ill of the other teachers, expecting that you'd support them against the guy who teaches philosophy or the one who teaches sciences.
With this lot, I usually keep a low profile. I don't expose myself to their trap and let them talk. Not much, but I do let them talk. Luckily the fact that there's a queue of people waiting to speak with me is a good excuse.
The compliant parents, instead, are the ones you try to plead for compassion.
"My son's going through may health problems, you know. He finds it hard to eat and has been so weak lately." And you say: "Are you talking about Alessandro, madam? The 6'4" tall guy who plays football four times a week with the semi-pro team?"
Then, after explaining the most improbable health and family problems, and saying that their child have experienced many different teachers across their school years, the compliant parent plays what they think is the ace up their sleeve:
"Anyway, sir, we can't expect much from my son. Nobody in the family really understands a thing about maths.
Now you would like to tell them:
"Luckily, neither stupidity nor not understanding maths are hereditary. Actually, your son has many chances to get better." But clearly you don't say that and try to encourage them reminding them that it takes time to catch up with maths.
In the end, this is the problem: we live in a society where it's not a problem to say:
"I never understood maths."
Many are even proud of this. How do they not understand that by saying that openly, what they're actually saying to their children is:
"Give up. Maths is useless. I never studied it and yet here I am."
The problem is with the parents (and in some of the humanities teachers) who candidly say that they do not understand maths. They don't realize that they are indirectly giving permission to their childred to not care about maths!
When this happens, a teacher has almost powerless. As long as family and society will take it as granted that one can live with no maths, and that maths is not part of a good citizen's cultural baggage, this is a hard battle to fight.
You will always meet students who say "Sir, I never got maths" and they will feel justified by the society.
But the parent-teacher conference come to an end too, sooner or later. At the end of the day, the maths teacher goes back home a little beaten up, a little dizzy because of all the words that have been spoken. He thinks with some melancholy about those "normal" parents (there are some obviously) with "normal" children (what a blessing!).
Next day, you start again your classes hoping that you'll convince your students to change their minds, they who are so sure they don't get maths. You start again your classes, trying to help them love as much as they can the topic that you yourself love (thanks to some other good teacher).
P.S.
I am sure that in a few years, when I will attend the parent-teacher conference as a parent, I will fall in one of the categories I've been talking about. There's nothing you can do about it, being a parent is really hard... harder even than mathematics!
(translated by Stefano)
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We can consider, for now onwards, a metric space. By my previously post it is clear that a metric space is the only one in which we can define a distance. If we use an Euclidean metric is simple to prove, but also to understand in an intuitive way, that the shortest distance between two points is a line that directly goes from the first point to the second one.
Now we try to repeat the same with a different metric. We can use a taxi driver metric; with this metric we can draw the minimum length curve between two points. In the following picture the minimum length curve is the line crossing the minor number of blocks.
The first important concept we can appreciate from the picture is that the minimum length distance is not univocal in some metrics. In some metric spaces more than a single minimum length curve is allowed.
Everyone can draw a straight line across a paper, probably it depends on how good you are in drawing. Given two points we can draw the line from one point to other one. Our brain is used to think the outstanding space in an Euclidean metric so we can figure out a segment connecting two points as a piece of straight line. It is so easy.
In some metric spaces a straight line is not a straight line in the sense we are usually to think about it. In some other metric spaces a straight line doesn't exist. In math, in this strange and fantastic world, a straight line has a definition. If in a metric spaces exist a minimum length line connecting two points and if this line is unique it represents a straight line for that metric space.
According to the above definition in the taxi driver metric space there isn't a straight line (the minimum length line is not unique) and in the Euclidean space it is the so common straight line we are used to draw.
In the end we can figure out a particular space made by only 4 lines. The man inside the space can see only the four lines and can move only along one of them. No other paths or lines or space or points are accessible by him.
In this case the red line represent the minimum distance path between the man and the final point. For the man the red line represent a straight line between the two points. Surely it is not a straight line from our point of view but it is due to a better space concept and to the possibility of observing the space from the outside.
In the end, we can conclude saying we are in the same condition of the man described above. We are cheated by our limited space perception. To trick us is more easy than we can think.
You are walking on train rails, straight on; in front of you the way and you walk in the same direction. Are you moving along a straight line? I let you think for few seconds... 1... 2... 3...
No, you are not moving on a straight line. We are walking along a minimum distance path between two points on the Earth surface.
This is possible because we perceive the earth as flat. We are like points in a 2D space. Our height is really small if compared with the earth radius so we are unable to perceive the earth curvature. All we perceive is a bidimensional space around us.
When we move along earth surface we don't perceive the fact we are moving along a curve path. Why? Simply becouse the path we are following is the only path we can see. Surely we can see other paths between two points but they are longer than the one we are moving on, so this path is effectively the straight line between two points for us.
The true concept of Euclidean distance isn't applicable to the path we follow on earth surface. When we move along the earth surface we can measure the purple path as distance between two points. The real Euclidean distance is represented by the red line (see figure above).
The difference between the two lines is as greater as the points are far away. If we move from one end to another of a football stadium the distance we travel is small if compared with earth radius. In this case the purple distance and the red one are really close each other and we can consider them as the same distance.
If the difference between the distance on earth surface and the Euclidean distance is neglectable for small distances it is not in the case of a flygth from Rome to New York.
Looking at a planisphere we can see Rome and New York are almost on the same parallel. Intuitively we can think the shortest path between the two cities is following the parallel on which they are located. Another time the perception of the space tricks us so we are wrong again.
On earth surface the shortest distance between two points is called geodesic. The geodesic is obtained intersecting the earth with a plane passing for the two points and the earth center. The consequence of this is the parallel (except the equator) aren't geodesics. The meridians are geodesics instead. An airplane flying from Rome to New York don't follow the parallel but the geodesics that pass over the two cities (see figure above).
The concept of geodesics was introduced by Georg Friedrich Bernhard Riemann.
When in its books of geometry Euclide try to set up the bases of modern geometry, he fixed some definitions. These are indemostrable but are so intuitive that they don't deserve a proof. They are called postulates or axioms. They give us a basic definition basic geometric entities (points, lines, planes, circles, angles).
The most interesting postulate is the fifth. It say, more or less, that given a line and a point out of it, then these exists only one line passing for the point and parallel to the first line, for this reason it is also called postulate of parallel lines. For a lot of time some mathematics tried to deduce the fifth postulate from the other four because it did not seem very intuitive.
For a moment we can think about the straight lines are in the euclidean definition. The shortest line between two points. On the earth surface these lines are the geodesics. We can then try to apply the fifth postulate to the earth surface. We can draw a geodesics, keep a point on earth surface, not on geodesics, and draw a geodesics passing for this point. The two geodesics we have drawn intersect each other in two points. If we repeat the procedure for all points on earth we can find that given a line and a point not on it, there is no line passing for the point and parallel to the first line. The fifth postulate is wrong.
The hypothesis of the falsity of the fifth postulate give us another geometry. The elliptical one. In this geometry the first four postulate of Euclide are ture, the fifth isn't. Every theorems and consequences of the first fourth postulates is true again but aren't the theorems or consequences of the fifth postulate. We have a strange and not intuitive geometry but a valid geometry. The elliptic geometry is really relevant in all problems in which the distances over the earth surface are not neglectable respect to the radius such as navigation, orientation, positioning and long distance travelling.
In the end there exists another way in which the fifth postulate can fail. Given a line and a point there exists infinite lines passing from the point and parallel to first line. Also in this case we obtain a different geometry. It is consistent and valid and is called hyperbolic geometry. It was postulated by Lobachevskij. This type of geometry with the add of the time give the Minkowski geometry that is the base fo relativity theory of Einstein and is used to describe the behavior of the Universe over large distances.
]]>"The illusion of mind" is a very interesting topic but so hard and subtle to treat.
Let us start with a nice result suggested by Camilla and Ludovica Pisani. The following is one of the most famous paradoxes in mathematics: while reading one realizes that intuition and probability make us give different answers to problems. I invite the reader to carefully follow the steps of the proof and find the mistake!
A bag contains 2 counters, as to which nothing is known except that each is either black or white. Then, one is black, and the other is white.
We know that, if a bag contained 3 counters, two being black and one white, the chance of drawing a black one would be $2/3$; and that any other state of things would not give this chance.
Now the chances, that the given bag contains (i) BB, (ii) BW, (iii) WW, are respectively , , . Add a black counter.
Then, the chances that it contains (i) BBB, (ii) BBW, (iii) BWW, are, as before, , , .
Hence the chances of now drawing a black one,
Hence the bag now contains BBW (since any other state of things would not give this chance).
Hence, before the black counter was added, it contained BW, i.e. one black counter and one white.
It is based on the American television game show Let's Make a Deal and named after its original host, Monty Hall.
Suppose you're on a game show, and you're given the choice of three doors: behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then asks you: ``Do you want to pick door No. 2?"
Is it to your advantage to switch your choice?
One would answer, following intuition and forgetting the past that since one can choose between two doors, then the probability of choosing the right one is .
This is wrong! If you change, the winning probability increases to !
It concerns the probability that, in a set of randomly chosen people, some pair of them will have the same birthday. However, probability is reached with just people, and probability with people! These conclusions are based on the assumption that each day of the year (except February 29) is equally probable for a birthday.
The first time I heard about this, I was completely shocked but then we (my professor and my university colleagues) computed the probabilities and it came out to be true.
However, to find other interesting paradoxes like the ones mentioned above, you just need to google... for instance: ``doubling the ball" (Banach-Tarski paradox)
and ``The barber of Sevile":
]]>Hi, everyone. Today we’re going to speak about pizza!
Yes, I’m not crazy! I perfectly know it’s a blog on maths!
Indeed, we want to talk of geometry of pizza and why there are some people who prepare it round and some who prepare it squared…who is earning money from this?
But the real answer I want you to propone today is: do you prefer a round or a squared pizza?
I imagine that the average reaction is another question: “What’s the difference? It’s pizza!”
The difference is huge and I’ll show what it is. Have fun!
Dido and oxhide
There are plenty of mathematical problems called maximum/minimum problems. In general they can be solved in many ways but I have no intentions of explaining the general theory. Some of you probably knew about it from school or at University. To people who reads it for the first time: “don’t be scared!”
In a few words, we try to understand how to minimize or maximize quantities under certain circumstances. An example from history: once upon a time there was Dido, Tyre’s queen, arrived in North Africa at the court of King Iarbas as a refugee. Iarbas decided to give her as much and as could be encompassed by an oxhide. Apart from the strange offering, Dido made lots of strips from the oxhide and made a very tiny rope. Finally, she used it to cover an huge land, the land in which Cartago was founded. Now, the question is: what’s the best geometrical form, having a rope, to obtain the maximum area?
In this example, the question is equivalent to: having fixed the perimeter, what’s the figure with maximum area? The answer is: the circle!
Another example: in an opposite way, fixing the area, what is the figure with smaller perimeter? The answer is: the circle again!
Do you know how to show this fact? To simplify the problem let us consider regular polygons, that are polygons with equal edges. It is possible to show that, fixed area A, the perimeter of a regular n-gon is: <<formula>>
So, this is a very well known formula, don’t you know it?
We see that the perimeter decreases as n increases, so we could say that the minimum perimeter is when there are an infinite number of edges…so? The circle!
If you compute the limit as n goes to infinity, you get <<formula>> and it’s the perimeter of a circle of area A!
So, we answered the question!
What a Pizza!
Come back to Pizza! We discovered that, fixed area, a round pizza has less perimeter than any other shape. But…perimeter is a thing, border is another. The border is an area, not a line!!
So, now the real question is? Who has the minimal border? Who maximal?
Let us start writing some formulas. A is the fixed area, n is the number of the edges of the regular polygon. Let b be the measure of the border.
The n-gon can be decomposed in n isosceles triangles. The angle in front of the base is and the area of this triangle is , where is the apothem. Knowing that , we have that and so .
If the pizza was a circle of area , then where is the radius. Hence . Comparing and we get . That’s the relation between a and r.
Now, either to the n-gon and to the circle, we cut a border thick . We avoid to make all computations, the final formula for the internal areas of the little triangles are . So we get: .
For the circle, the situation is similar: .We compare now the two areas computing: . Using the relation between r and a, we get that the difference is positive if has a value between 0 and . The value , if , is greater than and, as n goes to infinity, it goes to .
This means that if b is less than the 87% of the radius, then the internal area of the circle is greater that the n-gon’s one, for all . The converse holds for borders!
So, the question “do you prefer a round or a squared pizza? ” has now an answer. Since the square is a 4-gon the answer is: if you like borders, choose a squared pizza, otherwise a round one!
Conclusion (of the average man)
With all those letters, we lost the gist. Let us fix some values.
An average pizza has an area of . It turns out that, if the border is between 0 and 17 centimeters, the internal area of the square is smaller than the circle’s one. Since the border is 1 or 2 cm thick, than our result holds!
If you’re not crazy/deviated/maniac/sociopathic/my-cousin/curious, it’s enough for today. See you next time!
Conclusion (of an applied mathematician, of a economist, of a financial man and similar)
There are other questions: if a pizza company makes round pizzas, how much money does it lose?
Indeed, having a greater area to cover with ingredients, it should spend more.
I made some computations: fixing and , the internal area of a squared pizza is and of a round pizza is . So a squared pizza has 1.5% less ingredients than a round one.
Simplifying, the pizza company would save the 1.5% of the price of ingredients making squared pizzas instead of round pizzas. This means that, if making 100 pizzas costs 400 euros (100 euros for bases and 300 euros for ingredients), supposing that he sells pizzas at a price of 6 euros, he would earn 600-400=200 euros.
Leaving the price to 6 euros, but making squared pizzas, he would save the 1.5% of the costs of ingredients. Hence he would save 4.50 euros. It’s not too much, but it’s only 100 pizzas! Do you want to save more? Make bigger pizzas!
Conclusion (of a pure mathematician)
What? Have I used numbers?
Conclusive conclusion
Well, this is the end! Sayonara!
]]>In a previous post we talked about Euclidean and other distances...
In that post we introduced the definition of Euclidean, taxi driver and infinity distances. Almost anyone of you remembers the minimum distance, at the end of the post we explained that the minimum distance is not acceptable as a measure because it does not fulfill the distance definition requirements.
We can summarize the requirements a distance need to fulfill in order to be acceptable. It should:
Reading again the previous post, a certain curiosity to know more about the distance and the way it influences our life grew in me... In this post I will try to explain the concept of proximity between two geometrical entities.
For a moment we come back to the company and the biker of the previous post. We supposed that only one employee uses a bike to go to work, what happens if the number of employees that use bike to go to work is , big as you want. Everyone of these bikers has a certain distance from the company, and it doesn't matter which definition of distance we use, the important is that it fulfills the distance requirements. We have a lot of couple defined with the notation where is the company position in the space and on of the n bikers. For each couple we can define a number that represents the distance between the biker and the company.
In maths, the union composed by the company and the bikers is called set. If we add a distance definition to the set we obtain a metrical set. Metrical sets was theorized by Felix Hausdorff at the beginning of 20th century. Hausdorff, when introduced the metrical sets was studying a more general class of sets called topological sets. Metrical sets are a special class of topological ones.
In the previous paragraph I introduced in an easy way a very hard concept to understand. What is a topological set? A topological set is a set in which is defined the concept of proximity. Before introduce more complex concepts can be useful to keep in mind two questions, What does it mean proximity? What does it mean that two points are close? From an intuitive point of view (as often happens in maths, intuition drives men before equations...) we agree if I say that two points are close if the distance between them is really small. But... what is the definition of small? A mathematician could answer "as small as I want".
]]>Surely you're wondering how does Mathematics fit in Game of Thrones? Well I confess, I could use any other TV series to tell you what follows in this post, but honestly ... what is best than Game of Thrones (GoT)?
The connection to Mathematics is not so much considered in the series but in the type of data used. In particular one kind of data that has been used is the set of the comments regarding GoT posted on the IMDb site reviews to produce this infographic, on which I worked with my brother.
Firstly, very briefly, let us see what types of information were extracted from various comments (671).
The map on the left shows the major lineages; each one has been placed in its city of reference, indicating:
Instead, in the chart on the right hand side appear:
The part on the infographics which, however, I would like to focus attention is the one related to the word-cloud in the bottom right side of the picture, where the adjectives that have been mainly used in the reviews of a lineage are reported.
It needs to be explained how to determine whether a word in a speech has to be regarded as an adjective rather than as a preposition or article. And it is here that Mathematics comes to help.
To perform this type of analysis a POS (Part of Speech) Tagger has been used, namely a tool able to make grammatical analysis of a text. The POS Tagger taken into account is based on OpenNLP library, which is essentially based on the Maximum Entropy model that we will analyze in detail.
Before examining the MaxEnt algorithm, I would define the concept of entropy used here.
Most commonly we talk about entropy in the following areas:
To these interpretations of entropy, one can be added and it plays a very important role in information theory (especially in the field of signal processing) and will be the way in which we understand it in our case.
Let's consider a source of messages . The amount of information transmitted from the message increases with the increase of the uncertainty of the product message. The greater our knowledge about the message produced by the source, the lower the uncertainty, the entropy and the transmitted information.
We formulate the concept of entropy, as introduced by Claude Shannon in 1948.
Let be a source and a signal emitted by . The information given by is called autoinformation and is defined as
where is the probability that the event happens.
The entropy of a source is the expected value of the autoinformation, i.e. the average information contained in any signal from , in particular
if is a discrete variable
if is a continuous variable
Let be a discrete source, then
In particular the maximum of the entropy is when all the events are equally probable.
Let's see a simple example.
Example 1
Suppose to have a source that emits with probability and with probability .
Then
and the entropy is equal to if (unpredictable sources), while it is equal to if (predictable sources, i.e. it always outputs or always ).
The classifier Maximum Entropy is a discriminative classifier widely used in the areas of Natural Language Processing, Speech Recognition and Information Retrieval. In particular, it is used in order to solve problems of classification of text such as language detection, topic and sentiment analysis.
The Maximum Entropy algorithm is based on the principle of Maximum Entropy and selects the model that has the maximum entropy (as enunciated by Shannon) on the training set of all tested models.
Recalling Bayes' theorem (see here), the Max Entropy classifier is used when you do not have any information about the prior distribution of the data and it is not correct to make assumptions about.
Unlike the Naive Bayes classifier, the Maximum Entropy has not the hypothesis that the variables are independent of one another, which reflects the nature of the natural text where the variables into account are the words, that of course are not independent of one another since the grammatical rules of the language; moreover, it requires more time in training while providing more reliable results.
Example 2
Before going into the theory behind the MaxEnt, consider an example which clarifies from the outset what will be said in a formal way in the following.
Suppose we want to determine the grammatical form of the word "set."
The word "set" can take the following forms:
We collect a large number of examples from which to extract information to determine the decision-making model . The model we're going to build will assign the word "set" a chance to take a particular grammatical meaning.
As we don't have other information from the data, you can impose for our model:
There are several models that hold previous identity, including:
By analyzing the data set further, let's suppose to get other informations, such as every time the word "set" is preceded by a pronoun is a verb. This, added to the normalization condition, changes the possible chances, reducing the possible models.
The goal of the MaxEnt algorithm is to determine the model uniform as possible (maximizing the entropy), according to the information derived from the data, without making any additional assumptions.
We pass now to the explanation of the algorithm.
Consider a text-based document and have words to each of which corresponds to a particular tag (i.e. a grammatical part of the document: noun, adjective, pronoun, article, etc.). We introduce the concept of "history" of the word as the possible informations arising from the context in which is located and we indicate it with .
We make a small example to explain how you can understand the "story" of a word.
Example 3
Consider the sentence "Today is a beautiful day". The set of words is {Today, is, just, a, beautiful, day} and we call "history" of a word the grammatical information of the previous and next word.
For example, for the word "beautiful"
= {feminine singular indefinite article - "a", feminine singular noun - "day"}
What we have to define is a stochastic model that estimates the conditional probability of getting a tag, given a particular "story" , namely is .
Then we follow the usual classification scheme, i.e. we build our model starting from couples of the training set, where is the "story" of the word and is the class assigned to it (the grammatical part of speech) .
Consider the probability distribution based on the sample
where is the size of the training set, while is the number of occurrences of pair in the training set.
We introduce the indicator function
and consider the features as variables for the construction of our model.
The average value of variable compared to the probability derived from the sample is
where clearly whether each pair of the training set has occurrence .
While the average value of variable with respect to probability model is equal to
We impose the condition that the average value of the model is limited to on the training set, i.e.
We now have so many conditions as the previous ones for each , which can be met by several models. To choose the best conditions, we use the principle of Maximum Entropy, by selecting the closest possible model to standard form (maximization of information) .
In particular, it shows that exists and a well established model that maximizes the entropy of a system with constraints.
In our case the problem is the following:
Determine such that
with
We use Lagrange multipliers to solve it :
obtaining the solution
At this point we insert in the Lagrangian the values of and and get , maximizing the function that follows for which it is stated that (without proving it here):
So, as made the POS Tagger training on a given set (training set), we proceed to make a classification on new words to test (test set). In our case the POS Tagger, already trained on a large enough data set, is used for the classification of words present in IMDb's review; discarded all the words that are not classified as adjectives, then we will assign adjectives to families based on the frequency in the review of a specific lineage.
Now, if you're not a fan of Game of Thrones, I repeat that it can also be applied to other TV series (House of Cards, The Walking Dead, etc.) or in completely different areas, like ... what we will see in my next post.
Until now there is a unique conclusion ... 24.04 the sixth season begins.
Winter is coming ...
In these days I and a collegue of mine had a conversation on a recent law proposed by Ségolèn Royal in French Parlament.
Shortly, this law would provide a monetary refund for those who go to work by bike; the refund is proportional to the distance covered every day.
After a moment, everyone of us started thinking at the main point of it; everyone knows that there are lots of choices of paths when going from a point A to a point B in a city and everyone knows that the paths have different lengths. Which of those paths is used for deciding the amount of money to be refunded? We agreed on the answer...the shortest one; and you, do you agree with us? Probably someone among my readers doesn't know there are a lot of way to calculate the shortest distance between two points. In this post we will talk about distances, the way used to calculate them and some basic concepts in topology and limits.
The easiest way to calculate the distance between two points is the Euclidean distance. The Euclidean distance is the first we learn at school and the one we are confident to. Imagine an employee that lives in an imaginary point A on a bi-dimensional plane. The company is situated in a point B on the same plane. The best choice to go to work for him is to move straight on from A to B. To draw it, we can fix two different points in the plane and connect them with a stright line: this is the Euclidean distance.
For a more detailed discussion suppose to introduce a reference system with the origin in a certain point O. In the reference system both point A and B would have two coordinates (we are using a bi-dimensional plane, in a three dimensional space such as the Earth there are three coordinates).
The formula for Euclidean distance between A and B is
really easy to figure out.
It is well known that the civilization improves and some houses are built on the spece between point A and B.
Our friend, now, has to change the way to go from home to work according to the streets available between the houses.
The employee will pass two houses in vertical direction (up - down direction on your screen) and three houses in horizontal direction (left - right direction) and this the shortest way from home to work and is surely grater than Euclidean distance... a lucky fact for the company who refund the worker.
The distance shown above is called taxi distance or Manhattan distance and is expressed by the following equation
This is a valid alternative to Euclidean distance in everyday life; for example when we drive we use the taxi distance to decide the shortest way to go from our position to the final destination.
But now we have a problem: in order to save some money, the company establishes to refund only higher distance between the vertical and the horizontal. In the case of our employee only the horizontal distance will be refunded (three houses versus two houses in vertical direction). This could seem a stretch of taxi distance but there is a refined equation to express this distance:
this is called distance of infinity.
At the end, there is another distance we can investigate: the minimum distance. It is obtained changing max with min in the distance of infinity; simply only the shortest distance is kept in account and is written as
As we have seen, there are lots of different distances between two points. We are more familiar with some definitions of distance than others but, each definition given above, is a valid distance in the physical word.
An entirely mathematical definition rises up some questions; What is the best way to measure a distance? How many different distances are there?
We will answer these questions in reverse order. With a little imagination we can suppose that there are infinite ways to define a distance. From a mathematical point of view every equation involving subtractions of coordinates of two points is a distance. After this answer, some of us are starting to have doubts the existence of a universal definition for distances.
It is almost totally true. We cannot define a unique distance definition but we can define some criteria a distance definition must be respect.
First of all, a distance must be "positive defined"; in mathematical linguage this means that the value of a distance must be greater tha or, at least, equal to zero but never negative. It sounds so intuitive and everyone agrees that a distance between two points must be positive or zero (if the starting point coincides with the arriving one). I want you notice we are not considering vectorial distance that is positive in one direction and negative in the opposite one.
The second criteria is an extension of first one; it underlines that the distance is zero only if the starting point and the arriving one coincide.
In this case a little deviation is necessary. Suppose our biker travels from his house to another and suppose this new house is in the same street of his company.
Now he has to run three blocks in horizontal direction and zero in vertical direction. If we apply the minimum distance the distance from A to B is zero, the biker could get a little angry for this; this is surely not true.So, we notice the minimum distance is not a good definition and cannot be accepted.
Switching back to our criteria; the third is the symmetry of the distance calculation. It can be written as
The distance from A to B is the same from B to A and this is intuitive too.
The last criterium is the most important one and is called triangular inequality.
Suppose our biker, before going to work, needs to leave his son at school; to figure out the scene we suppose the school is in the point C in the above figure. If C is on the path from A to B there is no distance increase and we can write
If the point isn't on the path from A to B (see figure above) the distance from A to C and then from C to B is higher than distance from A to B. In mathematical form it is
Putting all together we obtain the triangular inequality
With the triangular inequality we have completed the criteria and this discussion too. In the next post we will talk about metrical spaces and topology.
]]>Mathematics has always had a tremendous impact on reality, bigger than you might think, lowering itself perfectly in the present but also anticipating and outlining the future. Even with less advanced than today's technological means, past discoveries changed our perception of the world, simply using theories and formulations.
In this post we won't see complicated formulas (or at least nothing that we haven't already seen at high school) or theorems on the limit of human understanding, I would rather tell a story that struck me deeply: how humans discovered Neptune. It's a story explaining what scientific activity is: a constant challenge to extend the boundaries of our understanding and our knowledge of reality.
Without starting from Adam and Eve (although even there at the beginning there was an apple), I would say that our story can start in 1687 with the publication of the formula of universal gravitation by Isaac Newton: in the universe every material point attracts every other material point with a force that is directly proportional to the product of their masses and inversely proportional to the square of their distance, or, in other words,
where:
• is the Intensity of the Force between two masses;
• is the the Gravitational Constant;
• is the first mass;
• is the second mass;
• is the distance between the centers of masses.
From this result, it was immediately clear that the motion of the planets of the Solar System is largely determined by the attraction of the sun (just think that the mass of the Sun is 1,000 times the mass of Jupiter, the largest planet of the Solar System).
No time to celebrate this historic achievement, which criticism of Newton's law immediately set off; substantially, this formula gave a good approximation for the motion of the planets, but to describe correctly the dynamics of the Solar System it's needed to take into account the Forces of mutual attraction between all the planets, thus extending the gravitational problem to the case of n-bodies. Newton himself could not resist the obvious objections, feeling so necessary that God put sometimes planetary orbits into place.
Eighteenth century Physicists and mathematicians dedicated themselves to the study of "perturbations" exerted by the planets in the n-body problem:
• Laplace introduced mathematical methods able to treat the problem of planetary perturbations and to show that the observed motion of Jupiter and Saturn could be explained by the mutual attraction of the two planets.
• Although he did considerable simplifications to the overall problem, Joseph Louis Lagrange in 1772 demonstrated the existence of a stable system for the case of three bodies.
These and other achievements in the late 700 and early 800 have laid the foundation for the discovery of the planet Neptune.
In the late eighteenth century Uranus was discovered (from a musician!) and what better time to apply the results obtained by Laplace & Co.? From the beginning it was realized that the actual trajectory of Uranus in the sky was different from the one calculated mathematically; indeed, it was obvious that these diverged.
With the newest scientific knowledge, astronomers concluded that the orbit of Uranus was disturbed by the presence of another body, perhaps a planet, in a farer zone of the Solar System.
It was 1846. The astronomer Galle looked through the telescope in the position suggested by Le Verrier's mathematical formulas and with only 3 nights of observations (a record at the time) was identified Neptune ... eureka! A planet discovered on the table with paper and pen!
Now this story ends here as we arrived to the discovery of Neptune, but it should be stressed how this event, the result of two centuries of research (from Newton Le Verrier, via Laplace and Lagrange), has been another starting point for many of the important scientific discoveries that followed in the coming centuries: chaos theory, relativity, quantum mechanics, and so on; for each of these there is much to tell, but the beauty of every single result is its preceding story and its ability to open a new horizon of discoveries.
Making a final overview, you realize centuries later how all progresses are closely related to each other; as for athletics, scientific research (in which mathematics plays a key role) is like a relay race where the baton passes endless more or less quickly between coats, not falling down ever.
And who knows if someone will get to explain how and why it all began. But...it's another story...
It was October when one of my professors entered in the lesson room and said: “Once upon a time, Snow White wished to prepare cookies for the seven dwarfs. Obviously…with some conditions. She wanted to prepare the least number of cookies such that dividing them into 2 dwarfs there would remain only one, dividing them into 3 dwarfs there would remain only one and so on for 4,5 and 6 dwarfs but…dividing them into all the 7 dwarfs, there would not remain any cookie.” This problem generated, of course, hilarity and some puzzled faces appeared in the room because of the nonsense of the story and because of the absurdity of the question. It was one of the first lessons during my first year at University and I honestly remained astonished.
Since for the moment it is not very important, we’re not going to talk of Snow White’s OCD but we will focus on the mathematical side of the story. What the professor wants is the smaller positive integer number (a number of the set 0,1,2,…) such that, if you divide it by 2,3,4,5 and 6, you get 1 as remainder and, if you divide it by 7, the remainder is 0.
It seems to be only an hard calculation and of course it is! But, behind this story, there’s some beautiful maths waiting to be discovered. It is called Modular Arithmetic. The good thing is that everyone can understand it…well…maybe it is better to say that everyone already knows it! The answer is in the clock.
The clock
Everyone knows what a clock is. If now it’s 9 a.m. and someone tells that you have a meeting in 2 hours, you’ll immediately understand that your meeting is at 11 a.m.. That’s right! And the reason si that 9+2=11. But, if someone tells that your meeting is in 4 hours, you’ll immediately understand that your meeting is at 1 p.m.. However, 9+4=13, not 1. This seems funny but it’s not!
What are we doing? We are restricting our number set! We are not deleting useless numbers but only collecting them into groups of numbers. So, some of them will coincide, as 13 and 1 for example. This relation is called congruence.
So, for examples with small numbers it’s easy, but for ones with big numbers? If now it’s 5, in 1000000 hours, what time is it?
Here’s the problem…this is not so intuitive…so? No problem, Maths is coming to save us!
We can easily identify 1 with 13, 2 with 14, 3 with 15 and so on. But what’s the relationship? Simple, if you divide 14 by 12, the remainder is 2…if you divide 15 by 12, the remainder is 3…and so on. So we identify all the numbers with the same remainder in the division by 12. So the question “If now it’s 5, in 100000 hours, what time will it be?” now has an answer!
We consider 1000000 and we add 5. We obtain 1000005 that is 9+12*83333. So the remainder of the division by 12 is 9. It will be 9.
Snow White’s clock
Now, let’s imagine that there exist clock with an arbitrary number of hours. Let n be this number. In this strange world with n-hour-clocks making calculations is different but not so much.
If it’s 0 now, after x hours, what time will it be? We should divide by n and look at the remainder. This will tell what time it is. Amazing, isn’t it?
Let us now give names to things. The numbers a and b are congruent modulo n if the remainders of the divisions by n are the same. I write .
So, two numbers a and b are congruent modulo n if, starting from the 0, after a or b hours, the question “what time will it be?” has the same answer.
This kind of Arithmetic is useful in many cases. It has plenty of applications in pure mathematics (primality tests, Chinese Remainder Theorem, divisibility rules,…) and in applied mathematics (cryptography).
I’m not going to talk about these things now, but I’ll do it in the future!
Chinese soldiers
You now have this brand-new formalism and you can reformulate Snow White’s problem. She wants a number m that is congruent to 1 modulo 2,3,4,5 and 6 and that is congruent to 0 modulo 7. Easier, isn’t it?
But…we haven’t solved it yet…to do this, let’s come back to an ancient past.
Long long time ago, many years before Christ, Chinese people knew about Modular Arithmetic! They used it to count people, especially soldiers!
I’m not talking of small numbers of soldiers but of a land full of people. So, how can you count them?
There’s a smart way to do it. You decide a priori a set of numbers called . Then you ask to soldiers to divide in groups of people and you report the remaining number of them (i.e. the remainder). Let us call it . You see that the number x of soldiers is congruent to modulo .
You do the same with the other numbers and, finally, you obtain the system
Essentially it’s the same thing Snow White needs to do!
Let’s do a simple example just to explain how it works. Let suppose that the number of soldiers satisfies the simple system:
From the first equation I see that , where N is an integer. That’s true because if I divide x by 2, the remainder is 1. Moreover, these are all the numbers with this property.
We substitute x in the second one obtaining , that is .
Multiplying by 2 both sides, we get because 4 is congruent to 1 modulo 3. Hence, for M integer. Substituting again we get . So, x can be a number in the set of the number that are congruent to 5 modulo 6.
The solution to this problem is not unique in principle. Anyway, knowing the upper and lower bounds of the searched number, we can obtain the exact number. For example if someone told us that x lies between 23 and 30, we would have understood that x is 25. This is how experts suppose Chinese people did.
Conclusion
Obviously, I made it simpler than how it is in its fullness. I omitted some details that the reader can search by himself if he/she got curious after reading this post. I suggest to give a look to the Chinese Remainder Theorem, it’s very interesting.
Coming back to Snow White, the number of cookies she needs to prepare must be a number that is congruent to 301 modulo 420. Since she wanted the smaller one, the answer is 301. And…if you don’t believe it…divide and see!
]]>Probably there is no unique definition of applied maths. In this article we report some references that analyze the problem and try to answer this question.
One of the most generic definition is the following:
"Applied (or perhaps Applicable) Mathematics consists of mathematical techniques and results, including those from "pure" math areas such as (abstract) algebra or algebraic topology, which are used to assist in the investigation of problems or questions originating outside of mathematics. " (click here to read more)
In the article " Mathematics Is Biology's Next Microscope", Cohen J. uses a very interesting metaphor:
The landscape of applied mathematics is better visualized as a tetrahedron (a pyramid with a triangular base) than as a matrix with temporal and spatial dimensions. (Mathematical imagery, such as a tetrahedron for applied mathematics and a matrix for biology, is useful even in trying to visualize the landscapes of biology and mathematics.) The four main points of the applied mathematical landscape are data structures, algorithms, theories and models (including all pure mathematics), and computers and software. Data structures are ways to organize data, such as the matrix used above to describe the biological landscape. Algorithms are procedures for manipulating symbols. Some algorithms are used to analyze data, others to analyze models. Theories and models, including the theories of pure mathematics, are used to analyze both data and ideas. Mathematics and mathematical theories provide a testing ground for ideas in which the strength of competing theories can be measured. Computers and software are an important, and frequently the most visible, vertex of the applied mathematical landscape. However, cheap, easy computing increases the importance of theoretical understanding of the results of computation. Theoretical understanding is required as a check on the great risk of error in software, and to bridge the enormous gap between computational results and insight or understanding.
A good presentation of Applied Mathematics is made by the Division of Applied Mathematics at Brown University here.
Further reflections can be found here
Anyway, through this blog, we'll try to tell the various characteristics of the applied mathematics and in particular we'll analyze the vertices of the applied-maths-tetrahedron:
Thoughts on Applied Mathematics, https://www.math.ust.hk/~mahsieh/APMATH.htm
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