Math O' Man : The Blog of MathematicsMath O' Man : The Blog of Mathematicstag:www.mathoman.com,2017:/en/index.php/2016-03-09T19:52:31+01:00DotCleardaily12016-03-09T19:52:31+01:00Matrix exponential function and square root of a matrix2016-03-09T19:52:31+01:00tag:www.mathoman.com,2016-03-09:/en//1539MathomanA 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 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 functions2014-12-20T13:56:06+01:00tag:www.mathoman.com,2014-12-20:/en//1538MathomanHere 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...Here 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 rotations2014-09-19T14:11:04+02:00tag:www.mathoman.com,2014-09-19:/en//1537MathomanConsider 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?...Consider 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 molecule2014-02-20T16:37:41+01:00tag:www.mathoman.com,2014-02-20:/en//1536MathomanHerr 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...Herr 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 balls2014-01-24T15:56:07+01:00tag:www.mathoman.com,2014-01-24:/en//1535MathomanYesterday 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...Yesterday 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 groups2011-10-30T12:01:23+01:00tag:www.mathoman.com,2011-10-30:/en//1534MathomanThe 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?...The 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 Trivium2011-01-03T12:05:16+01:00tag:www.mathoman.com,2011-01-03:/en//1533MathomanNearly 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,...Nearly 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 series2010-10-29T23:39:45+02:00tag:www.mathoman.com,2010-10-29:/en//1532MathomanTo 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}\;...To 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}\;...Compression of a soccer ball2010-07-11T00:58:40+02:00tag:www.mathoman.com,2010-07-11:/en//1531MathomanToday 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...Today 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 integral2010-06-16T20:05:24+02:00tag:www.mathoman.com,2010-06-16:/en//1530MathomanA 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...A 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 midpoint2010-01-29T13:36:34+01:00tag:www.mathoman.com,2010-01-29:/en//1529MathomanHere 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...Here 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 Riddle2009-12-25T13:57:17+01:00tag:www.mathoman.com,2009-12-25:/en//1528MathomanX-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....X-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 groups2009-12-20T12:45:08+01:00tag:www.mathoman.com,2009-12-20:/en//1527MathomanA 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...A 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 training2009-12-04T14:59:36+01:00tag:www.mathoman.com,2009-12-04:/en//1526MathomanHere 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...Here 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 earth2009-10-23T14:55:30+02:00tag:www.mathoman.com,2009-10-23:/en//1525MathomanHeute 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...Heute 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 beer2009-08-22T11:33:21+02:00tag:www.mathoman.com,2009-08-22:/en//1524MathomanBefore 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:...Before 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 matrix2009-07-14T13:38:38+02:00tag:www.mathoman.com,2009-07-14:/en//1523MathomanThe 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...The 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 Space2009-07-09T14:02:45+02:00tag:www.mathoman.com,2009-07-09:/en//1522MathomanHere 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...Here 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 child2009-07-05T20:08:46+02:00tag:www.mathoman.com,2009-07-05:/en//1521MathomanHere 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...Here 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!2009-07-01T11:30:44+02:00tag:www.mathoman.com,2009-07-01:/en//1520MathomanWe 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.
° ...We 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.
° ...