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# 16.7 Damped harmonic motion  (Page 2/5)

 Page 2 / 5

## Damping an oscillatory motion: friction on an object connected to a spring

Damping oscillatory motion is important in many systems, and the ability to control the damping is even more so. This is generally attained using non-conservative forces such as the friction between surfaces, and viscosity for objects moving through fluids. The following example considers friction. Suppose a 0.200-kg object is connected to a spring as shown in [link] , but there is simple friction between the object and the surface, and the coefficient of friction ${\mu }_{k}$ is equal to 0.0800. (a) What is the frictional force between the surfaces? (b) What total distance does the object travel if it is released 0.100 m from equilibrium, starting at $v=0$ ? The force constant of the spring is $k=\text{50}\text{.}0 N/m\text{}$ .

Strategy

This problem requires you to integrate your knowledge of various concepts regarding waves, oscillations, and damping. To solve an integrated concept problem, you must first identify the physical principles involved. Part (a) is about the frictional force. This is a topic involving the application of Newton’s Laws. Part (b) requires an understanding of work and conservation of energy, as well as some understanding of horizontal oscillatory systems.

Now that we have identified the principles we must apply in order to solve the problems, we need to identify the knowns and unknowns for each part of the question, as well as the quantity that is constant in Part (a) and Part (b) of the question.

Solution a

1. Choose the proper equation: Friction is $f={\mu }_{k}\text{mg}$ .
2. Identify the known values.
3. Enter the known values into the equation:
$f=\text{(0.0800)}\left(0\text{.200 kg)}\left(9\text{.80 m}/{\text{s}}^{\text{2}}\text{)}.$
4. Calculate and convert units: $f=\text{0.157 N}.$

Discussion a

The force here is small because the system and the coefficients are small.

Solution b

Identify the known:

• The system involves elastic potential energy as the spring compresses and expands, friction that is related to the work done, and the kinetic energy as the body speeds up and slows down.
• Energy is not conserved as the mass oscillates because friction is a non-conservative force.
• The motion is horizontal, so gravitational potential energy does not need to be considered.
• Because the motion starts from rest, the energy in the system is initially ${\text{PE}}_{\mathrm{el,i}}=\left(1/2\right){\text{kX}}^{2}$ . This energy is removed by work done by friction ${W}_{\text{nc}}=–\text{fd}$ , where $d$ is the total distance traveled and $f={\mu }_{\text{k}}\text{mg}$ is the force of friction. When the system stops moving, the friction force will balance the force exerted by the spring, so ${\text{PE}}_{\text{e1,f}}=\left(1/2\right){\text{kx}}^{2}$ where $x$ is the final position and is given by
$\begin{array}{lll}{F}_{\text{el}}& =& f\\ \text{kx}& =& {\mu }_{\text{k}}\text{mg}\\ x& =& \frac{{\mu }_{\text{k}}\text{mg}}{k}\end{array}.$
1. By equating the work done to the energy removed, solve for the distance $d$ .
2. The work done by the non-conservative forces equals the initial, stored elastic potential energy. Identify the correct equation to use:
${\text{W}}_{\text{nc}}=\Delta \left(\text{KE}+\text{PE}\right)={\text{PE}}_{\text{el,f}}-{\text{PE}}_{\text{el,i}}=\frac{1}{2}k\left({\left(\frac{{\mu }_{k}\mathit{\text{mg}}}{k}\right)}^{2}-{X}^{2}\right).$
3. Recall that ${W}_{\text{nc}}=–\text{fd}$ .
4. Enter the friction as $f={\mu }_{\text{k}}\text{mg}$ into ${W}_{\text{nc}}=–\text{fd}$ , thus
${W}_{\text{nc}}={–\mu }_{\text{k}}\text{mgd}.$
5. Combine these two equations to find
$\frac{1}{2}k\left({\left(\frac{{\mu }_{k}\text{mg}}{k}\right)}^{2}-{X}^{2}\right)=-{\mu }_{\text{k}}\text{mgd}.$
6. Solve the equation for $d$ :
$d=\frac{\text{k}}{{\text{2}\mu }_{\text{k}}\text{mg}}\left({X}^{2}–{\left(\frac{{\mu }_{\text{k}}\text{mg}}{k}\right)}^{2}\right).$
7. Enter the known values into the resulting equation:
$d=\frac{\text{50}\text{.}0 N/m\text{}}{2\left(0\text{.}\text{0800}\right)\left(0\text{.}\text{200}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)}\left({\left(0\text{.}\text{100}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}-{\left(\frac{\left(0\text{.}\text{0800}\right)\left(0\text{.}\text{200}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)}{\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{N/m}}\right)}^{2}\right).$
8. Calculate $d$ and convert units:
$d=1\text{.}\text{59}\phantom{\rule{0.25em}{0ex}}\text{m}.$

Discussion b

This is the total distance traveled back and forth across $x=0$ , which is the undamped equilibrium position. The number of oscillations about the equilibrium position will be more than $d/X=\left(1\text{.}\text{59}\phantom{\rule{0.25em}{0ex}}\text{m}\right)/\left(0\text{.}\text{100}\phantom{\rule{0.25em}{0ex}}\text{m}\right)=\text{15}\text{.}9$ because the amplitude of the oscillations is decreasing with time. At the end of the motion, this system will not return to $x=0$ for this type of damping force, because static friction will exceed the restoring force. This system is underdamped. In contrast, an overdamped system with a simple constant damping force would not cross the equilibrium position $x=0$ a single time. For example, if this system had a damping force 20 times greater, it would only move 0.0484 m toward the equilibrium position from its original 0.100-m position.

This worked example illustrates how to apply problem-solving strategies to situations that integrate the different concepts you have learned. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknowns using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life.

Why are completely undamped harmonic oscillators so rare?

Friction often comes into play whenever an object is moving. Friction causes damping in a harmonic oscillator.

Describe the difference between overdamping, underdamping, and critical damping.

An overdamped system moves slowly toward equilibrium. An underdamped system moves quickly to equilibrium, but will oscillate about the equilibrium point as it does so. A critically damped system moves as quickly as possible toward equilibrium without oscillating about the equilibrium.

## Section summary

• Damped harmonic oscillators have non-conservative forces that dissipate their energy.
• Critical damping returns the system to equilibrium as fast as possible without overshooting.
• An underdamped system will oscillate through the equilibrium position.
• An overdamped system moves more slowly toward equilibrium than one that is critically damped.

## Conceptual questions

Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)

How would a car bounce after a bump under each of these conditions?

• overdamping
• underdamping
• critical damping

Most harmonic oscillators are damped and, if undriven, eventually come to a stop. How is this observation related to the second law of thermodynamics?

## Problems&Exercises

The amplitude of a lightly damped oscillator decreases by $3\text{.}0%\text{}$ during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

#### Questions & Answers

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Jennifer Reply
Ex. On an inclined plane, gravitational potential energy, friction energy/work and spring potential energy. (Let's say that the spring is keeping the box from sliding down the slope.) How do we use this in the equation? I'm so confused
Jennifer
Oh! And if there's kinetic energy that is exerting a force opposite to the spring, what do we do?
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Then we subtract the k.e. from force exerted from newton's 2nd law.
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50 N...(50 *1.732)N
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Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
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# About a simple differential equation

Just a simple differential equation.Even we can hand to solve it.But the Mathematica give a error.

DSolve[{2*y[x]*y''[x] == y'[x]^2 + y[x]^2, y[0] == 1, y'[0] == -1},
y[x], x]

DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution. {}

ps:The answer is y=e^(-x).

• Are you sure of your result? it seems to me that $y=e^x$ is incompatible with $y'(0)=-1$. More generally, what is your question here? – MarcoB Dec 8 '15 at 6:17
• @MarcoB Thanks for your promption,and I have updated the result.My question is how to use Mathematica to solve the differential equation in a right way to get the result? – yode Dec 8 '15 at 6:31
• @MarcoB the solution $y=e^x$ seems to be correct. Maple gives that answer: !Mathematica graphics – Nasser Dec 8 '15 at 8:16
• The solution $y=e^x$ does satisfy the ode and the initial conditions. – Nasser Dec 8 '15 at 8:36
• @Nasser No!The e^x cannot satifay the y'(0)=-1 – yode Dec 8 '15 at 8:50

As pointed out in the other answer/comment by @Lotus, Mathematica can solve the equation in general:

DSolve[{2*y[x]*y''[x] == y'[x]^2 + y[x]^2}, y, x]

(* {{y[x] -> E^-x (E^x + E^(2 C[1]))^2 C[2]}} *)

In particular, if we use the more general initial conditions y[0] == y0 and y'[0] == yp0, then Mathematica can apply these initial conditions with no problem:

soln = DSolve[{2*y[x]*y''[x] == y'[x]^2 + y[x]^2, y[0] == y0, y'[0] == yp0}, y[x], x]

(* {{y[x] -> (E^-x (y0 + E^x y0 - yp0 + E^x yp0)^2)/(4 y0)}} *)

And if you then specify y0 -> 1 and yp0 -> -1, then Mathematica gets the correct result:

(y[x] /. soln[[1, 1]]) /. {y0 -> 1, yp0 -> -1}

(* E^-x *)

Why Mathematica is unable to find this solution without our intervention is a bit mysterious to me. Most likely, DSolve is finding the general solution (with the $C_i$ coefficients) and then noting that there isn't a real-valued solution for the $C_i$'s that yields the desired initial conditions. (The solution $e^{-x}$ is a limit of Mathematica's general solution: $C_1 \to \infty$ and $C_2 \to 0$ in such a way that $e^{4 C_1} C_2 = 1$.) However, if we asking for the solution to be expressed in terms of $y(0)$ and $y'(0)$, the result is (presumably) auto-simplified to the extent that the solution can be found for all $y(0)$ and $y'(0)$. (Except for $y(0) = 0$ and $y'(0) \neq 0$, for which there is no solution.)

This is a Comment and not an answer.

The problem here is that there is no "general solution" for the differential equation. Here is what happens without initial conditions:

In[31]:= gensol = DSolve[{2*y[x]*y''[x] == y'[x]^2 + y[x]^2}, y, x]

Out[31]= {{y -> Function[{x}, E^-x (E^x + E^(2 C[1]))^2 C[2]]}}

The above solution is correct, as can be verified easily:

In[41]:= {2*y[x]*y''[x] == y'[x]^2 + y[x]^2} /. gensol // Simplify

Out[41]= {{True}}

DSolve uses Solve internally to solve for the constants and here is what happens there:

In[38]:= Solve[{y[0] == 1, y'[0] == -1} /. gensol[[1, 1]], {C[1],
C[2]}]

Out[38]= {}

With NDSolve though, it just takes a particular branch based on the range of x specified.

I guess we have to start with the general solution and find constants suitably.

By the way Does Maple solve this ?

• No,The differential equation have a solution,the e^(-x) satisfy every condition and the initial equation. – yode Dec 8 '15 at 11:20
• The "general solution" found by Mathematica has $e^{-x}$ as a limit: $C_1 \to \infty$ and $C_2 \to 0$ in such a way that $e^{4 C_1} C_2 = 1$. – Michael Seifert Dec 8 '15 at 19:53

With a bit of assistance,DSolve can obtain a simpler general answer. Begin by changing dependent variables.

eq = 2*y[x]*y''[x] - y'[x]^2 - y[x]^2
eq2 = eq /. y'[x] -> p[y] /. y''[x] -> p[y]*p'[y] /. y[x] -> y
sol = DSolve[eq2 == 0, p[y], y]
(*{{p[y] -> -Sqrt[y] Sqrt[y + C[1]]}, {p[y] -> Sqrt[y] Sqrt[y + C[1]]}}*)

Boundary conditions are: y[0] == 1, y'[0] == -1. Now we find C[1] constant:

sol1 = Solve[-1 == (p[y] /. sol[[1]]) /. y -> 1, C[1]]
(*{{C[1] -> 0}}*)
sol2 = Solve[-1 == (p[y] /. sol[[2]]) /. y -> 1, C[1]]
(*{no solutions}*)

General Solution of this equation is:

 SolE = First@DSolve[y'[x] - ((p[y] /. sol[[1]]) /. y -> y[x]) == 0, y[x], x]
(*{y[x] -> 1/4 E^(-x - C[2]) (E^C[2] - E^x C[1])^2}*)

$$\left\{y(x)\to \frac{1}{4} e^{-c_2-x} \left(e^{c_2}-c_1 e^x\right){}^2\right\}$$

The above solution is correct, as can be verified easily:

Z = (y[x] /. SolE)
FullSimplify[2*Z*D[Z, x, x] - D[Z, x]^2 - Z^2 == 0]
(*True*)

Now we find C[2] constant:

 sol4 = First@Solve[1 == (y[x] /. SolE /. C[1] -> 0) /. x -> 0, C[2], Reals]
(*{C[2] -> 2 Log[2]}*)

Special solution of this equation is:

 Y[x] == (y[x] /. SolE) /. C[1] -> 0 /. C[2] -> C[2] /. sol4
(*Y[x] == E^(-x)*)

$$y(x)=e^{-x}$$

• @Can you know the function is e^(-x) by your InterpolatingFunction If I don't tell you the result? – yode Dec 8 '15 at 11:18
• @yode See this: mathematica.stackexchange.com/questions/19042/… – Mariusz Iwaniuk Dec 8 '15 at 13:01
• @Thanks your link.But you don't know the result is e^x – yode Dec 8 '15 at 14:31
• @yode. I am sure that e^(-x) is correct answer. – Mariusz Iwaniuk Dec 8 '15 at 15:15
• Sorry for my typo,"Thanks your link.But you don't know the result is e^(-x) by your InterpolatingFunction still." – yode Dec 8 '15 at 15:19

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# Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible must have $0$ as the determinant.

Do you have another approach to this question, in terms of probability for example?

• Dec 14 '13 at 8:16
• Dec 14 '13 at 8:29
• I think most matrices are not even square. Do you mean, are most square matrices invertible?
– bof
Dec 14 '13 at 12:42
• Of course, we're dealing with square matrices Dec 14 '13 at 12:54
• It depends on the distribution you're obligated to define along with your use of the word "most". There's no such thing as a uniform distribution over an infinite range of real numbers. Dec 14 '13 at 15:48

There are at least three ways of saying that a matrix over the real numbers is generically invertible:

The topological one: the set of invertible matrices is a dense open set in the set of all matrices.

The probabilistic one: with the Lebesgue measure on the set of matrices, the non-invertible matrices are of measure zero.

The algebraic one: The set of invertible matrices is open (and non-empty) for the Zariski topology; explicitly, this means that there is a polynomial defined on the coefficients of the matrices, such that the set of invertible matrices is exactly the set where this polynomial is not zero. Of course, here the polynomial is the determinant function.

Remark that your question makes sense for matrices with coefficients in an arbitrary infinite field (for finite fields, we are looking at finite sets...) and that we can still say that a matrix is generically invertible in the algebraic sense.

• On the other hand, only 29% of square matrices over $GF(2)$ are invertible.
– bof
Dec 14 '13 at 12:41
• No, I don't consider finite fields in my answer. But what you say is true. It is well-known that $|GL_n(F_q)| = q^{n^2} \prod_{i=1}^n (1-q^{-i})$ Dec 14 '13 at 14:24
• Your answer is better than mine. So why doesn't it have more upvotes? :( Dec 14 '13 at 21:24
• @JeremyDaniel: but, what kind of global measure do you define in $\mathbb R^{n^2}$ in order to define probabilities? Dec 14 '13 at 21:58
• Well, I take the Lebesgue measure, viewing the matrices as the space $\mathbb{R}^{n^2}$. Of course, this is not a probability measure since it is of infinite total mass. Dec 15 '13 at 0:48

Yes. The non-invertible matrices form a set of measure zero in the space of all $n\times n$ matrices, when considered as a subset of $\mathbb R^{n^2}$.

• What does it mean? I don't know measure theory. Dec 14 '13 at 7:01
• @GabrielR. Here's a simple description of measure zero: A set has measure zero if you can cover it by a countable collection of open balls with total volume less than $\epsilon$, for arbitrary small $\epsilon$. Dec 14 '13 at 7:02
• Since the concept is not being used technically, you may as well say that it's the same as the volume being zero for such a nice (i.e. algebraic) set. Dec 14 '13 at 14:06

To expand on Potato's answer, the space of all $n \times n$ matrices is manifold isomorphic to $\mathbb{R}^{n^2}$. The determinant gives an polynomial function from this space to $\mathbb{R}$, and the zero set of a polynomial function has measure $0$.

(Earlier this asserted that the set of singular matrices is a manifold, which is not true.)

• Nice, and I think this also shows the interior is empty in $\mathbb R^{n^2}$, i.e., the matrices with $Det=0$ are nowhere-dense in $\mathbb R^{n^2}$. Dec 14 '13 at 21:50
• I'm not sure this is true, and I definitely don't think it can be by the regular value theorem. It fails for $n=2$, where the gradient of the determinant vanishes at the zero matrix Mar 21 '18 at 4:49
• @qbert You are right of course, I hope that you are the first person who has read this in the 4 years since I wrote this! I will change this to something correct. Mar 21 '18 at 14:25

Here's how I like to think about it. A matrix $A$ is invertible $\iff$ $det(A) \neq 0$. Since the determinant depends continuously on the entiries, if we have $A$ such that $det(A)=0$, we should be able to perturb the entries slightly so that $det(A)$ is nonzero. Likewise, if $det(A) \neq 0$, then any small perturbation of the entries won't make $det(A)$ suddenly $0$. So in this sense, most matrices are invertible.

• Continuity only gets you one direction of this. The zero function depends continuously on the entries, but you can't perturb the matrix to make it nonzero. Dec 14 '13 at 14:53
• You may be able at times, but not always, since $Det^{-1}(0)$ is not open in $\mathbb R^{n^2}$. The opposite is the case, tho, since $Det^{-1}(\mathbb R-{0})$ is an open subset. Dec 14 '13 at 21:56
• Not all perturbations will bring a singular matrix into invertibility. However, one specific perturbation $t\mapsto A+tI$ will always do the job.
– Vim
May 22 '18 at 11:52

I think this needs work, but there is something to it: take a randomly-selected matrix $A$, and associate to it its characteristic polynomial $Det(A -\lambda I)$; this polynomial is also random. This polynomial will have $0$ as its root iff its constant term is 0. But most random polynomials will have non-zero constant term, so will not have zero as their root. Alternatively, consider a characteristic polynomial with a zero root. Any small change in an entry of the matrix will change the constant term from $0$ to non-zero. This last statement can be made more accurate/rigorous by using topology: the set of matrices with determinant non-zero is an open subset of $Gl(n,\mathbb R)$, since it is the complement in $\mathbb R^{n^2}$ of the closed set $Det^{-1}(0)$, so that each non-zero matrix, seen as a point in Euclidean space, has a radius around which its determinant is non-zero.

For low dimensions, you can see this geometrically: for a $2 \times 2$-matrix, its determinant is $0$ iff "the entries are on the same line through the origin" , meaning that the points $(a_{11}, a_{12})$ and $(a_{21},a_{22})$ are on the same line through the origin. There are $c$ possible slopes, and only one way of both points being on the same line through $0$. For a $3 \times 3$-matrix, there is a similar argument, seeing the matrix as the measure of the volume enclosed by $3$ vectors in $\mathbb R^3$. The determinant is $0$ here if the rows, seen as points, are "degenerate". The degenerate cases are finitely-many compared with the non-degenerate ones.

A square matrix is defined by its column vectors, which are the images of the vectors of the canonical basis through the linear map associated with the matrix. The column vectors are linearly dependent if and only if the matrix is not invertible. So we translated our problem to the problem: are most $n$-tuples of vectors in $\mathbb{R}^n$ linearly independent?

Let us pick an $n$-tuple of vectors in $\mathbb{R}^n$ as randomly as we can. For the first vector we have a whole $\mathbb{R}^n$ of possibilities, we should just avoid the null vector, but that is a very special one. Then we should pick the second vector: of all the possible vectors in $\mathbb{R}^n$ only with the ones on the line determined by the first vector will we end up with a pair of linearly dependent vectors, but those are very special vectors, so we will most likely choose another one. If you think of directions in $\mathbb{R}^n$ as determining a pair of antipodal points on the unit sphere, then apart from the two points determined by the direction of the first vector all other points will do. Then choose a third vector: there is now a plane of directions determined by the first two vectors so that if we choose a vector from this plane we end up with three linearly dependent vectors. On our sphere we have a circle of special directions: much less than the other 'generic' directions. Keep choosing vectors this way and you see that a random choice will give you a linearly independent $n$-tuple with probability $1$.

You could actually translate the original problem to the problem of choosing $n$ points at random on a sphere (or a projective space if you want to have a single point in every direction). Then you would be asking yourself how likely it is that all the points you have chosen lie on a proper subspace of $\mathbb{R}^n$.

No they're not. Think about it, the rank of a $n \times n$ matrix can be any integer $k \in \{0, \dots, n\}$. The only case where the matrix is invertible is when $k = n$.

Joke!

Seriously though, to make a precise answer, one has to define what is meant by "most". If we are talking about matrices with real or complex coefficients, the answer is "yes" for any reasonable definition of "most" (the other answers describe some examples). I think your intuitive answer with the determinant is fine.

Let us see. Any matrix can be represented as a finite list with ","s inserted. If we are talking about real numbers, then there would be $2^{\aleph_0}$ matrices. No let us see how can be inverted. Let us take a function $f_n(x)$, which is the determinant of a given $n \times n$ matrix, where the top right value is $x$, and all the other ones held constant. This is continuous, and not the constant function for $0$. Therefore, there are $2^{\aleph_0}$ invertible matrices of $n \times n$ and therefore the number of all invertible matrices. Therefore, there are as many invertible matrices as matrices themselves.

• Are you saying that there exists a 1-1 function from the sets of the invertible matrices into the sets of those who aren't? Dec 14 '13 at 14:43
• @Gabriel: There must be such a function, mustn't there? Both sets have cardinality $2^{\aleph_0}$. But such a function won't be continuous. Dec 14 '13 at 14:53
• Of course, there might be $2^{\aleph_0}$ uninvertible matrices too, in which case the same number of matrices would be invertible and noninvertible. Cardinal numbers break English. Dec 14 '13 at 15:04

A probabilistic argument: choose $n^2$ real numbers randomly. Then the probability that the matrix formed by that numbers is not invertible is zero.

Edit: removed a wrong conjecture.

• Can you give some ground on both arguments? Dec 14 '13 at 9:06
• @Michael, have a look at this easy counter-example: $A=\begin{pmatrix}-46&-54\\35&-17\end{pmatrix},B=\begin{pmatrix}48&51\\-60&20\end{pmatrix}$. $\det(A)=2672,\det(B)=4020,\det(A+B)=-69$. Your result is valid when one considers SPD matrices. Who has given a point to your answer ?
– user91684
Dec 15 '13 at 0:19
• @loupblanc Your'e right, I have something mixed up. Dec 15 '13 at 11:27
• Choose the real numbers randomly with what distribution? There's no uniform probability measure on the reals. Dec 16 '13 at 20:26


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# Geometrically integral

Here is a stupid question about the notion of geometric integrality.

Say I have a smooth, projective variety $X$ over a some field $k$, equipped with a morphism $f: X \to C$ to a smooth, projective curve $C$, such that the generic fibre is geometrically integral.

Assume that there exists a finite (dominant) morphism $\varphi: C \to C$ of degree at least $2$.

Is it true that the generic fibre of the composition $\varphi \circ f$ is not geometrically integral?

• Yes, absolutely. The generic fiber $Y$ is an algebraic variety over the function field $K$ of $C$. Consider $K$ as a finite non-trivial extension of a subfield $L$, then $Y\times_L \bar{L}=Y\times_K (K\otimes_L \bar{L})$, and $K\otimes_L \bar{L}$ is never integral. – Cantlog Jun 11 '14 at 19:22
• @Cantlog: I believe your comment should be an answer. – RghtHndSd Jun 12 '14 at 15:21

Yes, absolutely. The generic fiber $$Y$$ is an algebraic variety over the function field $$K$$ of $$C$$. Consider $$K$$ as a finite non-trivial extension of a subfield $$L$$, then $$Y \times_L (K \otimes_L \overline{L})$$, and $$K \otimes_L \overline{L}$$ is never integral.


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Density of potentially crystalline representations of fixed weight
Eugen HELLMANN, Benjamin SCHRAEN - 2013-11-15 (M/13/37)
Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology. The main hypothesis we need is the existence of a potentially diagonalizable lift, so that in the two-dimensional case, the result is unconditional.
Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds
Kei FUNANO - 2013-10-21 (M/13/36)
We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-s. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multi-way isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multi-way isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux result to the k-th eigenvalue and the -way isoperimetric constant. As a consequence the k-th eigenvalue of the weighted Laplacian and the k-way isoperimetric constant are equivalent up to polynomials of k on closed weighted manifolds of nonnegative Bakry-
Allure of Quotations and Enchantment of Ideas
Misha GROMOV - 2013-10-02 (M/13/35)

Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3
Misha GROMOV - 2013-10-01 (M/13/34)

Poisson varieties from Riemann surfaces
Philip BOALCH - 2013-09-30 (M/13/33)

The elliptic dilogarithm for the sunset graph
Spencer BLOCH, Pierre VANHOVE - 2013-09-24 (P/13/24)

The geometry of variations in Batalin-Vilkovisky formalism
Arthemy KISELEV - 2013-09-15 (M/13/32)
We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "delta(0)=0" and "log delta(0)=0" within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Dirac's delta-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving -but not just "formally postulating"- the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation).
On-shell Techniques and Universal Results in Quantum Gravity
N.E.J. BJERRUM-BOHR, John F. DONOGHUE, Pierre VANHOVE - 2013-09-04 (P/13/23)

Quantum supersymmetric cosmology and its hidden Kac--Moody structure
Thibault DAMOUR, Philippe SPINDEL - 2013-09-04 (P/13/29)

Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation
Donato BINI, Thibault DAMOUR - 2013-09-04 (P/13/30)

Merger states and final states of black hole coalescences: a numerical-relativity-assisted effective-one-body approach
Thibault DAMOUR, Alessandro NAGAR, Loic VILLAIN - 2013-09-04 (P/13/31)

The tensor hierarchy algebra
Jakob PALMKVIST - 2013-09-02 (P/13/27)

The tensor hierarchy simplified
Jesper GREITZ, Paul HOWE, Jakob PALMKVIST - 2013-09-02 (P/13/28)

The Nielsen and the Reidemeister Zeta Functions of maps on infra-solvmanifolds of type (R)
Alexander FELSHTYN, Jong Bum LEE - 2013-08-30 (M/13/26)
We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R).
Anyon wave functions and probability distributions
Douglas LUNDHOLM - 2013-08-06 (P/13/25)
The problem of determining the ground state energy for a quantum gas of anyons in two dimensions is considered. A recent approach to this problem by means of lower bounds is here refined to bring out the dependence on the n-particle probability distributions encoded in the wave functions. Furthermore, a class of states which has been proposed in the context of upper bounds for a related many-anyon problem, is here considered from the point of view of these refined lower bounds. A numerical approach to determining their corresponding probability distributions is employed for a limited number of particles.
Ramification and nearby cycles for l-adic sheaves on relative curves
Haoyu HU - 2013-07-09 (M/13/22)
Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito’s ramification theory.
Conformal and Einstein gravity in twistor space
Tim ADAMO, Lionel MASON - 2013-06-18 (P/13/13)

Distributional Geometry of Squashed Cones
Dmitri FURSAEV, Alexander PATRUSHEV, Sergey SOLODUKHIN - 2013-06-17 (P/13/21)
A regularization procedure developed in \cite{Fursaev:1995ef} for integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational $O(2)$ symmetry in a subspace orthogonal to a singular surface $\Sigma$ so that the surface is allowed to have extrinsic curvatures. A new feature of squashed conical singularities is that the surface terms in the integral invariants, in the limit of small angle deficit, now depend also on the extrinsic curvatures of $\Sigma$. A case of invariants which are quadratic polynomials of the Riemann curvature is elaborated in different dimensions and applied to several problems related to entanglement entropy. The results are in complete agreement with computations of the logarithmic terms in entanglement entropy of 4D conformal theories \cite{Solodukhin:2008dh}. Among other applications of the suggested method are logarithmic terms in entanglement entropy of non-conformal theories and a holographic formula of entanglement entropy for theories with gravity duals.
Nonholonomic deformation of coupled and supersymmetric KdV equation and Euler-Poincaré-Suslov method
Partha GUHA - 2013-06-14 (M/13/15)
Recently Kupershmidt \cite{Kup} presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al \cite{KKK}. In this paper we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler-Poincarnsional construction to construct nonholonomic deformation of a wide class of coupled KdV equations, all these equations follow from the Euler-PoincarS^1) \bo C^{\infty}(S^1)}}$, where$Diff(S^1)$is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to two component Camassa-Holm equation. We also give a derivation of a nonholonomic deformation of the$N=1$supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler-Poincar Application of Jacobi's last multiplier for construction of singular Hamiltonian of the activator-inhibitor model and conformal Hamiltonian dynamics Partha GUHA, Anindya GHOSE CHOUDHURY - 2013-06-14 (M/13/16) The relationship between Jacobi's last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator and inhibitor model and these are connected to conformal Hamiltonian structure. The role of the Jacobi Last Multiplier in Nonholonomic Systems and Almost Symplectic Structure Partha GUHA - 2013-06-14 (M/13/17) The relationship between Jacobi's last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure is elucidated in this paper. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers, can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. Contiguity relations for linearisable systems of Gambier type Alfred RAMANI, Basil GRAMMATICOS, Partha GUHA - 2013-06-14 (P/13/18) We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations of the solutions of the Gambier mapping. In all cases the resulting contiguity relation is a linearisable equation, involving free functions, and which can be related to the generic Gambier mapping. Quantum aspects of the Liénard II equation and Jacobi's Last Multiplier-II Anindya GHOSE CHOUDHURY, Partha GUHA - 2013-06-14 (P/13/19) This is a continuation of the paper [J. Phys. A: Math. Theor. 46 (2013) 165202], in which we mapped the Lis ordering technique. In this paper we present further results on the construction of three sets of exactly solvable potentials giving rise to bound-state solutions of the Schr Chemotherapy in heterogeneous cultures of cancer cells with interconversion Rui DILãO - 2013-06-14 (M/13/20) Recently, it has been observed the interconversion between differentiated and stem-like cancer cells. Here, we model the \textit{in vitro} growth of heterogeneous cell cultures in the presence of interconversion from differentiated cancer cells to cancer stem cells, showing that, targeting only cancer stem cells with cytotoxic agents, it is not always possible to eradicate cancer. We have determined the kinetic conditions under which cytotoxic agents in \textit{in vitro} heterogeneous cultures of cancer cells eradicate cancer. In particular, we have shown that the chemotherapeutic elimination of \textit{in vitro} cultures of heterogeneous cancer cells is effective if it targets all the cancer cell types, and if the induced death rates for the different subpopulations of cancer cell types are large enough. Nouveaux développements sur les valeurs des caractères des groupes symétriques; méthodes combinatoires Pierre CARTIER - 2013-06-05 (M/13/14) L’étude asymptotique des diagrammes de Young de grande taille a été entreprise cu dans le but de contrôler les algèbres d’opérateurs liées aux groupes libres ; elles ont servi ensuite tude des zéros de la fonction zeta de Riemann. Plus récemment , elles ont été appliquées par Biane aux propriétés asymptotiques des permutations. Nous insisterons surtout sur les formules exactes qui sous-tendent ces formules asymptotiques obtenues par les collaborateurs de Biane (Sniady, Féray), et qui développent de nouveaux domaines de la combinatoire (principalement cartes planaires). Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function Hafedh HERICHI, Michel L. LAPIDUS - 2013-05-15 (M/13/12) mRNA diffusion as a mechanism of morphogenesis in Drosophila early development Rui DILãO - 2013-04-29 (M/13/11) In Drosophila early development, bicoid mRNA of maternal origin is deposited in one of the poles of the egg, determining the anterior tip of the embryo and the position of the head of larvae. The deposition of mRNAs is done during oogenesis by the mother ovary cells and is transported into the oocyte along microtubules. Initially, the oocyte has only one nucleus, but after fertilization and deposition of the egg, nuclear duplication by mitosis is initiated without the formation of cellular membranes. During the first 14 nuclear divisions of the developing embryo, bicoid mRNA of maternal origin is translated into protein in the ribosomes and accumulates near the external nuclear walls of the recently formed nuclei. Here, we show that mRNA diffusion is the main morphogenesis mechanism explaining consistently the establishment of Bicoid protein gradients. Moreover, we show that if diffusion for both bicoid mRNA and Bicoid protein were assumed, a steady distribution of Bicoid protein would result, with a constant concentration along the embryo, contradicting observations. Equivariant operational Chow rings of spherical varieties and T-linear varieties. Richard Paul GONZALES VILCARROMERO - 2013-04-12 (M/13/10) We stablish localization theorems for the equivariant operational Chow rings (or equivariant Chow cohomology) of singular spherical varieties and T-linear varieties. Our main results provide a GKM description of these rings in the case of spherical varieties admitting a BB-decomposition into algebraic rational cells. Our description extends certain topological results to intersection theory on singular varieties. Dynamic trajectory control of gliders Rui DILãO, João FONSECA - 2013-04-04 (M/13/09) A new dynamic control algorithm in order to direct the trajectory of a glider to a pre-assigned target point is proposed. The algorithm runs iteratively and the approach to the target point is self-correcting. The algorithm is applicable to any non-powered lift-enabled vehicle (glider) travelling in planetary atmospheres. As a proof of concept, we have applied the new algorithm to the command and control of the trajectory of the Space Shuttle during the Terminal Area Energy Management (TAEM) phase. Chemotaxis with directional sensing during Dictyostelium aggregation Rui DILãO, Marcus HAUSER - 2013-03-24 (M/13/07) With an in silico analysis, we show that the chemotactic movements of colonies of the starving amoeba Dictyostelium discoideum are driven by a force that depends on both the direction of propagation (directional sensing) of reaction-diffusion chemotactic waves and on the gradient of the concentration of the chemoattractant. It is shown that the directional sensing of amoebae is due to the sensitivity of the cells to the time variation of the concentration of the chemoattractant combined with its gradient. It is also shown that chemotaxis exclusively driven by local concentration gradients leads to unstable local motion, preventing cells from aggregation. These facts show that the formation of mounds, which initiate multicellularity in Dictyostelium discoideum, is caused by the sensitivity of the amoebae to three factors, namely, to the direction of propagation of the chemoattractant, to its gradient, and to the spiral spatial topology of the propagating chemoattractant. The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models Rui DILãO - 2013-03-24 (M/13/08) To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deductible from the mass action law and, in the limit$n \to \infty\$, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability.
Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires: Notes de cours
Laurent LAFFORGUE - 2013-02-18 (M/13/06)

On weakly group-theoretical non-degenerate braided fusion categories
Sonia NATALE - 2013-02-01 (M/13/05)
We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braided fusion category whose Frobenius-Perron dimension is a natural number less than 1800, or an odd natural number less than 33075, is weakly group-theoretical.
Special Functions in Minimal Representations
Toshiyuki KOBAYASHI - 2013-01-22 (M/13/04)
Minimal representations of a real reductive group G are the `smallest' irreducible unitary representations of G. We discuss special functions that arise in the analysis of L^2-model of minimal representations.
Rankin-Cohen Operators for Symmetric Pairs
Toshiyuki KOBAYASHI, Michael PEVZNER - 2013-01-10 (M/13/03)

Sur la correspondance de Simpson p-adique. II : aspects globaux
Ahmed ABBES, Michel GROS - 2013-01-09 (M/13/02)
We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This second article is devoted to the global aspects of the theory.
Sur la correspondance de Simpson p-adique. 0 : une vue d'ensemble
Ahmed ABBES, Michel GROS - 2013-01-09 (M/13/01)
We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. The aim of this article is to give an extensive overview of the theory that has been developped in two articles, the first one (arXiv:1102.5466) devoted to the local aspects and the second one (arXiv:1301.0904) to the global aspects.


text:
# zbMATH — the first resource for mathematics

On minimal dynamical systems on Boolean algebras. (English) Zbl 0697.54021
Summary: Let B be an infinite complete Boolean algebra which admits a countable family H of automorphisms such that the dynamical system (B,H) is minimal. Then B is isomorphic to the completion of the free Boolean algebra of size $$\pi$$ (B).

##### MSC:
 54H20 Topological dynamics (MSC2010) 06E05 Structure theory of Boolean algebras
Full Text:


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# Cantor set

The Cantor ternary set is a remarkable subset of the real numbers named after German mathematician Georg Cantor who described the set in 1883. It has the same cardinality as $$\mathbb{R}$$, yet it has zero Lebesgue measure.

The set can be constructed recursively by first considering the unit interval $$C_{0}=[0,1]$$. In the next step, we divide the set into three equal parts and discard the open middle set. This leads to the new set $$C_{1}=[0,\tfrac{1}{3}]\cup[\tfrac{2}{3},1]$$. This procedure is repeated on the two remaining subintervals $$[0,\tfrac{1}{3}]$$ and $$[\tfrac{2}{3},1]$$, and iteratively on all remaining subsets such that $$C_n = \tfrac{1}{3}C_{n-1} \cup (\tfrac{2}{3} + \tfrac{1}{3}C_{n-1})$$. The Cantor set is defined by the limit as $$n\to\infty$$, i.e., $$\mathcal{C} = \cap_{n=0}^{\infty} C_{n}$$.

The recursion can be illustrated in python in the following way

  import numpy as np

def transform_interval(x, scale=1.0, translation=0.0):
return tuple(map(lambda z: z * scale + translation, x))

def Cantor(n):
if n==0: return {(0,1)}
Cleft = set(map(lambda x: transform_interval(x, scale=1/3), Cantor(n-1)))
Cright = set(map(lambda x: transform_interval(x, translation=2/3), Cleft))
return Cleft.union(Cright)

  print(Cantor(1))

{(0.6666666666666666, 1.0), (0.0, 0.3333333333333333)}

The following code displays the first 5 iterations of the algorithm and clearly illustrates the self-similarity property in the sets $$C_{n}$$.

  def intervalprint(x, n=240):
sym = n*[' ']
val = np.linspace(0, 1, num=n)
for interval in x:
idx = np.where((val >= interval[0]) & (val <= interval[1]))[0]
for i in idx:
sym[i] = u'\u2500'
st = ''.join(sym)
print(st)

x = list(map(Cantor, [0,1,2,3,4,5]))
for i in range(0, len(x)): intervalprint(x[i])

────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────────────────                                                                                ────────────────────────────────────────────────────────────────────────────────
───────────────────────────                           ──────────────────────────                                                                                ──────────────────────────                           ───────────────────────────
─────────         ─────────                           ────────         ─────────                                                                                ─────────         ────────                           ─────────         ─────────
───   ───         ───   ───                           ───   ──         ───   ───                                                                                ───   ───         ──   ───                           ───   ───         ───   ───
─ ─   ─ ─         ─ ─   ─ ─                           ─ ─    ─         ─ ─   ─ ─                                                                                ─ ─   ─ ─         ─    ─ ─                           ─ ─   ─ ─         ─ ─   ─ ─

It is clear that $$\mathcal{C}$$ is bounded (being a subset of the unit interval) and closed since it constructed by removing countably many open intervals from a closed interval. It also has zero measure, meaning that it does not contain any intervals.

To see this, note that in the first step the open middle set $$(\tfrac{1}{3}, \tfrac{2}{3})$$ is removed which has measure 1/3. From the remaining two closed sets, $$[0, \tfrac{1}{3}]$$ and $$[\tfrac{2}{3}, 1]$$, the open middle sets are removed in the next iteration: $$(\tfrac{1}{3^{2}}, \tfrac{2}{3^{2}})$$ and $$(\tfrac{2}{3}+\tfrac{1}{3^{2}}, \tfrac{2}{3}+\tfrac{2}{3^{2}})$$, each of measure $$\tfrac{1}{3^2}$$. In the $$n$$th iteration $$2^n-1$$ open intervals are removed each of length $$\tfrac{1}{3^{n}}.$$ Hence, the Lebesgue measure of the Cantor set is $m(\mathcal{C}) = m([0,1])-\sum_{k=0}^{\infty} \tfrac{2^{n}}{3^{n+1}} = 1-\frac{1}{3}\sum_{k=1}^{\infty} \left(\tfrac{2}{3}\right)^{n} = 1 - \frac{1}{3}(1-\tfrac{2}{3})^{-1} = 0.$

In this sense, we can think of the Cantor set just containing “dust”.

It can also be shown that the Cantor set can be represented as $C = \left\{ \sum_{i=1}^{\infty} \frac{x_{n}}{3^{n}} \mid x_{n}\in{0,2}\right\},$ from which it again follows that the set is uncountable. This can also be seen by considering an arbitrary countable subset $N = \{n_{1}, n_{2}, \ldots \}\subseteq \mathcal{C}.$ In the construction of $$\mathcal{C}$$, we must have that $$n_{1}$$ belongs to one of the two closed intervals in $$C_{1}$$. Hence, there is one interval $$I_{1}$$ of length $$\tfrac{1}{3}$$ such that $$n_{1}\not\in I_{1}$$. Similarly, $$C_{2}\cap I_{1}$$ is the union of two new closed intervals of length $$\tfrac{1}{3^{2}}$$, and $$n_{2}$$ can at most be in one of these two sets. Let $$I_{2}\subset I_{1}$$ be this interval s.t. $$n_{2}\not\in I_{2}$$. Continuing in this fashion we construct a series of intervals $$\mathcal{C}\supset I_{1}\supset I_{2}\supset \cdots \supset I_{k} \supset\cdots$$, where $$I_{k}$$ is one of the closed intervals of $$C_{n}$$ of length $$\tfrac{1}{3^{n}}$$ such that $$n_k\not\in I_{k}$$. The limit $$I_{\infty} = \cap_{k=1}^{\infty}I_{k}$$ is non-empty (see below), and by construction $$N\cap I_{\infty} = \emptyset$$. Hence given any arbitrary countable subset of $$\mathcal{C}$$ we can still find elements in $$\mathcal{C}$$ that does not belong to this subset, and thus there cannot exist a bijection between $$\mathbb{N}$$ and $$\mathcal{C}$$ and we conclude that $$\mathcal{C}$$ is uncountable infinite.

Here we used Cantor’s Intersection Theorem: Let $$(X, d)$$ be a complete metric space. Let $$(I_{k})_{k\in N}$$ be a sequence of decreasing non-empty, closed subsets of $$X$$ then $$\bigcap_{k\in N}I_{k} \neq \emptyset.$$

Proof: For each $$k\in N$$ choose $$x_{k}\in I_{k}$$. The diameter of $$I_{k}$$ is defined as $$\operatorname{diam}(I_{k})=\sup\{𝑑(𝑥,𝑦):𝑥,𝑦∈ I_{k}\}$$. We only need to consider the case where $$\operatorname{diam}(I_{k})\to 0.$$ In this case, let $$\epsilon>0$$ then there exists $$N>0$$ such that $$d(x, x’)<\epsilon$$ for all $$x,x’\in I_{N}$$. Let $$m,n>N$$ then $$x_{n}, x_{m}\in I_{N}$$ since $$I_N \supset I_{N+1}\supset … \supset I_{m},I_{n}$$, and thus $$d(x_{n},x_{m})<\epsilon$$. We conclude that $$(x_{k})_{k\in\mathbb{N}}$$ is a Cauchy sequence. Since $$X$$ is complete $$x_n\to x$$ as $$n\to\infty$$ for some $$x\in X$$. Note that $$(x_{n})_{n=N}^{\infty} \subset I_{N}$$ and since $$I_{N}$$ is closed, the limit point also lies in the set $$x\in I_{N}$$. As this holds for any $$N$$, we conclude that $$x\in\bigcap_{k\in N}I_{k}$$. This concludes the proof. Note, that in this case the limit point $$x$$ is in fact the only element. Since if we let $$x’$$ be another element then $$x,x’\in I_n$$ for all $$n\in N$$ but $$d(x,x’)\leq\operatorname{diam}(F_{n})\to 0$$ as $$n\to\infty$$. Hence $$d(x,x’)=0$$, and therefore $$\bigcap_{k\in N}I_{k} = \{x\}.$$