Math O' Man : The Blog of Mathematics
http://www.mathoman.com/en/index.php/
en2016-03-09T19:52:31+01:00daily12016-03-09T19:52:31+01:00Matrix exponential function and square root of a matrix
http://www.mathoman.com/en/index.php/1539-matrix-exponential-function
2016-03-09T19:52:31+01:00enMathomanRiddles And ExercisesA little exercice about the matrix exponential made me think abour the following interesting property which makes a link with the sqare root of a matrix. I wish you fun proving ;-)
Let A be a real invertible n×n matrix. One defines
[tex]\sqrt{A}=\{ B\in M_n(\mathbb{R})\;\mid\;...A limit with commutator of invertible analytic functions
http://www.mathoman.com/en/index.php/1538-a-limit-of-a-quotient-of-a-difference-of-invertible-analytic-functions
2014-12-20T13:56:06+01:00enMathomanRiddles And ExercisesHere come a nice exercise for a lecture on complex analysis. Let f and g be two different holomorphic non-linear functions defined in a neighborhood of 0 and such that
f(0) et g(0) vanish and that f'(0) et g'(0) do not vanish. What is the limit in 0 of the function
[tex]\frac{f\circ g-g\circ...Axis and angle of the composition of two rotations
http://www.mathoman.com/en/index.php/1537-axis-and-angle-of-the-composition-of-two-rotations
2014-09-19T14:11:04+02:00enMathomanMaths for mathematiciansConsider two rotations r and r' in three dimensional space R3 (around two axis passing through the origin). It is a known (but not trivial) fact that the composition r'or is another rotation. The question arises naturally how to determine its axis and angle. Is there an easy geometric way to do it?...Compute the angles in a methan molecule
http://www.mathoman.com/en/index.php/1536-compute-the-angles-in-a-methan-molecule
2014-02-20T16:37:41+01:00enMathomanRiddles And ExercisesHerr is an exercise from basic geometry in space. It refers to a question on Pierre Lecomte's Blog and the ensuing discussion.
The methan molekule CH4 is a regular tetrahedron with the carbon atom in the center. Can you determine, with a minimum of calculation, the angle between two...A not so simple riddle with balls
http://www.mathoman.com/en/index.php/1535-a-not-so-simple-riddle-with-balls
2014-01-24T15:56:07+01:00enMathomanRiddles And ExercisesYesterday a collegue in office gave me something to think about in order to re-activate my brain after my lecture. Here it is.
Ten balls are ligned up in front of us and we play the following game. I mark mentally two neighboring balls and you must find out which ones.
To do so you have the...Small exercice about finite groups
http://www.mathoman.com/en/index.php/1534-small-exercice-about-finite-groups
2011-10-30T12:01:23+01:00enMathomanMaths for mathematiciansThe theory of finite groups is full of little, weird exercises. Here is one of them. It seems quite simple... Try it out!
Is it true that every automorphism of a finite group, which sends at least half of the elements of the group to their inverses, is an involution?...Some solutions to Arnold's Trivium
http://www.mathoman.com/en/index.php/1533-some-solutions-to-arnold-s-trivium
2011-01-03T12:05:16+01:00enMathomanMaths for mathematiciansNearly twenty years ago Vladimir Arnol'd has published a collection of exercises which he called Trivium but many of which are not trivial at all. In the french version of this blog you can find most solutions. You are invited to contribute solutions if you wish. Exercises no. 27, 41, 51, 58, 68,...A convergent series
http://www.mathoman.com/en/index.php/1532-a-convergent-series
2010-10-29T23:39:45+02:00enMathomanRiddles And ExercisesTo find out for which [tex]\alpha>0[/tex] the series [tex]\sum_{k=1}^\infty\,\frac1{k^\alpha}[/tex] converges, one can use the following inequality which is valid for all [tex]n\in\llbracket2,\infty\llbracket[/tex],
[tex]\int_2^n\frac{{\rm d}x}{x^\alpha}\;...0[/tex] the series [tex]\sum_{k=1}^\infty\,\frac1{k^\alpha}[/tex] converges, one can use the following inequality which is valid for all [tex]n\in\llbracket2,\infty\llbracket[/tex],
[tex]\int_2^n\frac{{\rm d}x}{x^\alpha}\;...]]>Compression of a soccer ball
http://www.mathoman.com/en/index.php/1531-compression-of-a-soccer-ball
2010-07-11T00:58:40+02:00enMathomanNo MathsToday is the finale of the soccer world championship and the last opportunity ta ask my reader the following question:
When a player receives a ball on his head, the ball is compressed for a very short moment before it is bounced back. Have a guess,
what is the maximal deformation of the ball...Compute an approximate value of an integral
http://www.mathoman.com/en/index.php/1530-compute-an-approximate-value-of-an-integral
2010-06-16T20:05:24+02:00enMathomanRiddles And ExercisesA friend sent me a nice collection of exercises about which I will talk soon on this blog. One of the questions is simply:
Calculate the mean value of sin100(x) with a precision of 10%.
I suppose that one must understand calculate the mean value on an interval having the length of a period...Theorem about a circle, three chords and a midpoint
http://www.mathoman.com/en/index.php/1529-theorem-about-a-circle-three-chords-and-a-midpoint
2010-01-29T13:36:34+01:00enMathomanRiddles And ExercisesHere is a nice exercise in plane geometry. As often in mathematics the statement is quite simple but the proof is not!
Let [tex]\scr{C}[/tex] be a circle, A,B two distinct points on [tex]\scr{C}[/tex] and M be the midpoint of the chord [AB]. Take two other chords, [PQ] and [SR], that...Christmas Riddle
http://www.mathoman.com/en/index.php/1528-christmas-riddle
2009-12-25T13:57:17+01:00enMathomanRiddles And ExercisesX-mas is the time to crack nuts. At least that's what normal people do. But mathematicians enjoy cracking very peculiar nuts, like the following one. Show that the equation below is true for every positive integer....An exercise about groups
http://www.mathoman.com/en/index.php/1527-an-exercise-about-groups
2009-12-20T12:45:08+01:00enMathomanRiddles And ExercisesA topological group is a set G with a group structure and a topology such that the binary law
[tex]G \times G \rightarrow G ,\;\; (x,y) \rightarrow xy,[/tex]
and the map of the inverse
[tex]G \rightarrow G ,\;\; x \rightarrow x^{-1},[/tex]
are continuous. In other words both structures, the...Cargo-cult science training
http://www.mathoman.com/en/index.php/1526-cargo-cult-science-training
2009-12-04T14:59:36+01:00enMathomanMaths And SocietyHere is very interesting article about "modern ideas" of teaching, written by João Magueijo a physics professor at Imperial College London. Unfortunately, his observations are so true. It seems that the same problem, i.e., the rise of bureaucrats and educationalists, occurs not only in the UK but...An exercise about the temperature distribution on the earth
http://www.mathoman.com/en/index.php/1525-exercise-about-the-temperature-distribution-on-the-earth
2009-10-23T14:55:30+02:00enMathomanRiddles And ExercisesHeute mal ein bisschen climatology! The following question, funny and useless, is dedicated to my friend A. Wirth, who has left pure mathematics in order to use his mind for applied sciences like oceanography and meteorology ;-)
Exercise: Suppose that the earth is a perfect ball and that the...Mixing wine and beer
http://www.mathoman.com/en/index.php/1524-mixing-wine-and-beer
2009-08-22T11:33:21+02:00enMathomanNo MathsBefore summer I wrote a blog entry about mixing wine and beer. french friends told me that such a mixture can't exist, putting beer into wine is a sacrilege... It is okay to drink one after the other, but never, never simultaneously! The order though, wine first or beer first, depends on the contry:...Comatrix and adjugate matrix
http://www.mathoman.com/en/index.php/1523-comatrix-and-adjugate-matrix
2009-07-14T13:38:38+02:00enMathomanRiddles And ExercisesThe comatrix com(M) of a n x n-matrix M is the n x n-matrix whose entry in (l,k) is [tex]\small{(-1)^{l+k}}[/tex] times the determinant of the matrix which you get by deleting in M the line l and the row k.
It is the transpose of the comatrix which is of interest to us; it is called adjugate...Dimension of the Commutator Space
http://www.mathoman.com/en/index.php/1522-dimension-of-the-commutatorspace
2009-07-09T14:02:45+02:00enMathomanRiddles And ExercisesHere is a beautiful exercise from matrix theory.
Let A be a n x n matrix. Show that the dimension of the commutator space of A (i.e., the set of all matrices that commute with A) is at least n.
There is a proof that works on any field. But in the special cases of real or complex numbers there...A sunday child
http://www.mathoman.com/en/index.php/1521-a-sunday-child
2009-07-05T20:08:46+02:00enMathomanRiddles And ExercisesHere is a personalised birthday gift for my father, in form of a mathematical riddle.
Today, sunday 5th of july 2009, is my father's birthday. He was born on a sunday of a leap year. What is his age today?
In order to solve that exercise you can use the fact that I am older than twenty-three, that...Deserve your dessert!
http://www.mathoman.com/en/index.php/1520-deserve-your-dessert
2009-07-01T11:30:44+02:00enMathomanRiddles And ExercisesWe all know those little riddles that entertain mathematicians at the end of lunch at the cafeteria. Here is one of them:
Connect the following nine points by four straight lines, without lifting your pencil.
° ...