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# What will be the events relative to the ground observer , a time-like or space like?

1. Feb 25, 2012

Please refer to the attached diagram

The train moves in the direction B-A

If the events seen in the ground observer simultaneously, it should be seen at A before B in the train

In special case, the difference in times between A and B according to the train observer could be equal to the time a light signal takes to travel from A to B only if v/c=0.618

However, if the light travel from A to B relative to the train observer, it should do so relative to the ground one as well. But because of the experiment setup, still the ground observer sees the 2 events happen at the same time

So what will be the events relative to the ground observer, a time-like or space like?

Thanks,,,

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2. Feb 29, 2012

Re: what will be the events relative to the ground observer , a time-like or space li

3. Feb 29, 2012

### nitsuj

Re: what will be the events relative to the ground observer , a time-like or space li

seperated spatialy and can be observed as happening simultaniously, spacelike I think.

4. Feb 29, 2012

### Michael C

Re: what will be the events relative to the ground observer , a time-like or space li

If two separate events are simultaneous in a particular frame, they have a space-like separation in all frames.

5. Feb 29, 2012

### nitsuj

Re: what will be the events relative to the ground observer , a time-like or space li

Good point the distinction between spacelike & timelike is invariant.

6. Mar 1, 2012

Re: what will be the events relative to the ground observer , a time-like or space li

OK,,, but according to my calculation attached here, the Δt between A and B for the train observer can be equal to the Δttaken by a light signal from A to B in case of v/c=0.618. This yields a time-like separation !

Last edited: Mar 1, 2012
7. Mar 1, 2012

### Fredrik

Staff Emeritus
Re: what will be the events relative to the ground observer , a time-like or space li

In the first sentence (in the attachment), you're talking about A and B as if they are events. Later you're talking about them as if they are locations in space (specifically in the slice of spacetime that the ground considers space). This makes your argument quite confusing.

If A and B are events, it only makes sense to say that a light signal is sent from A to B if their separation is lightlike, but this would of course immediately contradict the earlier claim that there's an inertial frame in which A and B are simultaneous. Then you start talking about calculating the time it takes the signal to go from A to B. By then you are clearly thinking of A and B as locations, not events. The signal would be sent from an event E on A's world line and received at an event F on B's world line. The separation between E and F is obviously lightlike.

Last edited: Mar 1, 2012
8. Mar 1, 2012

### John232

Re: what will be the events relative to the ground observer , a time-like or space li

The ground observer will see the two "simultanious" events happen at two different times, this is because it will take longer for light to reach the ground observer, but this would only be an illusion since the photon would only be traveling a larger distance from one of the points. Similiar to how we look back in time when we look at the stars, the stars are not younger just because we see them that way. It just takes a certain amount of time for the light to reach the observer since it travels at a limited speed. So then, I would say the seperation of the events is just space like.

9. Mar 1, 2012

### Fredrik

Staff Emeritus
Re: what will be the events relative to the ground observer , a time-like or space li

It has nothing to do with that. When we're talking about what someone "sees" or "experiences", we're not talking about what their eyes or video cameras would pick up. We're talking about the coordinate assignments they would make. An observer on the ground "sees" two events as simultaneous if and only if he assigns the same time coordinate to them.

10. Mar 1, 2012

### nitsuj

Re: what will be the events relative to the ground observer , a time-like or space li

I disagree with that, I can use my head and consider the time that the light has travelled an deduce the star is now older, but from an observed "my reality" perspective what's measureable is all of the information I have ( or could possibly have) regarding that "distant star". The physical reality travels at c. all else is "elsewhere" and sometimes termed unphysical.

That being said, the star is actually "younger" here compared to over "there". not an illusion, a reality. Said differently, at the location here the "distant star" is not old yet, because of the spatial separation more time needs to pass until the star is "old" over here. It would be a safe bet the star is even older over there but that proof does not exist over here, it is not a "reality" here, more an "illusion" of the "mind's eye", specifically an assumption. ( of course physics can predict this future accurately, calculating the age of the star over there, and intuitively is reffered to as "it's age", but comparatively thats relative)

(mumbo jumbo here, but I think it could even be physicaly possible for there to be a "closed timelike curve" (or spacelike? idk) that "transports" the light to the observer location faster than light years)

Last edited: Mar 1, 2012
11. Mar 1, 2012

Re: what will be the events relative to the ground observer , a time-like or space li

I meant A and B as events

If A and B are simultaneous for the ground observer, they must have a spacelike separation for the train observer. This should not permit Δt to be long enough to allow B receiving any light from A. But that may happen if v/c=0.618 !

12. Mar 1, 2012

### Fredrik

Staff Emeritus
Re: what will be the events relative to the ground observer , a time-like or space li

Then your calculation doesn't make much sense. You wrote "AB", meaning "the distance in space between A and B in the ground's(?) rest frame". That's OK, since A and B are by assumption simultaneous in that frame. But then you multiplied AB by $1/\gamma$, expecting to get the distance in space between A and B in the other frame. The concept of distance in space in the train's frame between A and B doesn't make sense, since A and B aren't simultaneous in that frame. What you're calculating this way is the distance in the train's frame between the two points on the ground where A and B occurred. This is not a distance between A and B. It's the distance in the train's frame between A and and an event C on the world line of the point on the ground where B occurred. C would occur significantly later than B, but still not late enough to receive a light signal from A.

13. Mar 2, 2012

Re: what will be the events relative to the ground observer , a time-like or space li

The distance AB for the train observer wont change because he sees the event A happens before B. He always considered A and B simultaneous whenever he measures AB as a distance.
The derivation of Δt according to LT is also based on that assumption and therefore multiplying AB by $1/\gamma$ to yield AB. So multiplication by $1/\gamma$ occurs on both sides of the equations i & ii and it will cancel each other out
And even if you dont want to consider $1/\gamma$, no problem, as the quadratic equation still remains valid because AB will cancel each other out again

Last edited: Mar 2, 2012
14. Mar 2, 2012

### Fredrik

Staff Emeritus
Re: what will be the events relative to the ground observer , a time-like or space li

I don't understand what you're saying here.

$(AB)/\gamma$ isn't the distance between A and B in the train's frame. There's no such thing as a distance between A and B in the train's frame, because A and B are not simultaneous in that frame. $(AB)/\gamma$ is the distance in the train's frame between the points on the ground where events A and B occurred. It's a distance in spacetime along a simultaneity line of the train's frame. There's such a simultaneity line through A, and another one through B, but there's no such simultaneity line through both. So $(AB)/\gamma$ is definitely not the distance in the train's frame between A and B.

I think you need to think about what you're actually calculating using the length contraction formula. You can't plug in the distance between two events and expect to get the distance between those same events in another coordinate system. That's not just wrong, it doesn't make sense. Think about it this way instead: In a spacetime diagram where the axes are the t and x axes of the ground's frame, the world lines of the two points on the ground are vertical lines. A and B are two points on those two lines. Since they are simultaneous, there's a horizontal line that goes through both of them. The length of that line segment from B to A is what you called AB. But $(AB)/\gamma$ is the length that an observer on the train would assign to a line with slope 1/v between two events where such a line intersects the two vertical lines.

15. Mar 2, 2012

### Michael C

Re: what will be the events relative to the ground observer , a time-like or space li

"Won't change" relative to what? The distance between the places where A and B happen will certainly be different when measured in different frames.

If you assign coordinates to the events A and B in the ground frame, you can use the Lorentz transformation to work out the coordinates in the train frame:

THE GROUND FRAME:
Let's say that the distance between the two events in the ground frame is d. To make things easy, we'll put event A at the origin:
Event A: t = 0, x = 0
Event B: t = 0, x = d.

THE TRAIN FRAME:
We define the train frame to have the same origin as the ground frame. It is moving relative to the ground frame along the x axis at a velocity of v. The Lorentz transformation gives us the coordinates of the two events in this frame:
Event A: t' = 0, x' = 0
Event B: t' = -$\gamma$vd/c$^{2}$, x' = $\gamma$d.

Increasing the velocity difference between the two frames will not only increase the time separation between A and B, it will also increase the spatial separation: you can see from the coordinates that the distance in space between the two events has been multiplied by $\gamma$. The space-time interval between A and B stays the same.

16. Mar 2, 2012

### John232

Re: what will be the events relative to the ground observer , a time-like or space li

This may just be too hippy for me. So, if I tied a string around Mars and then held the other side at Earth would their be a reality that exist at the speed of sound?

Shouldn't these be the same two things?

If A and B are not on the train then wouldn't the distance from A to B seen from on the train shorten, since A to B could be "seen" as a single object in motion relative to the train?

17. Mar 2, 2012

### Michael C

Re: what will be the events relative to the ground observer , a time-like or space li

A and B are not objects, they are events.

I'm not sure quite what you mean by saying that "A to B could be "seen" as a single object in motion relative to the train". Are you imagining that a single object could be present at both event A and event B? That won't work, since neither event falls within the light cone of the other.

18. Mar 2, 2012

### Fredrik

Staff Emeritus
Re: what will be the events relative to the ground observer , a time-like or space li

Yes, it obviously doesn't matter if you observe something with your eyes or with a video camera.

You're right, but he wasn't talking about the distance in the train's frame between the points on the ground where A and B occur. That distance changes with speed as $1/\gamma$, i.e. it gets shorter. He was talking about how the coordinate difference $x_A-x_B$ in the train's frame depends on the speed of the train. That difference changes with speed as $\gamma$, i.e. it gets longer.

19. Mar 2, 2012

### John232

Re: what will be the events relative to the ground observer , a time-like or space li

In the orignal post it sounded like A and B where two seperate places outside of the train. So then if you put two objects at a location down the track they would appear to come closer together as the train moved down the track. I was trying to say that the track itself would become shorter from the frame of reference of the train while it was moveing, not longer.

20. Mar 2, 2012

In this occasion, I was not talking about the distance Δx between places where A & B occur, but about the distance between the 2 train ends A and B. Your calculation is right that the Δt=$\gamma$vd/c$^{2}$. OK, so can this time difference allows a timelike separation between A and B if a signal would take off from A to reach B?


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# A family of subsets with a “gluing” property

Somewhat in line with this previous MathOverflow question:

I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call elements of $F$ "connected" subsets. (Think of them like connected sets in a topological space, in a loose way.) This family has to satisfy two properties:

• If any two connected sets have non-empty intersection, their union will be connected. (I call this property "glueability".)

• Singletons are always connected. (This property is too vacuous to deserve a name.)

My question is whether any of you have seen anything like this before, or have any idea how to deal with it. This is a sort of awkwardly vague request to make, but I would be very interested to hear if this structure had arisen in any other contexts.

(As an example of what I've come up with: You can easily prove that the maximal connected sets exactly partition the whole set.)

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What exactly do you mean by 'maximally connected'? In my mind, maximally connected would imply a lot of connectivity, which would mean a lot of overlap between the sets, which would exclude that they are a partition. –  supercooldave Jun 14 '10 at 7:08
Are you assuming S is finite? Otherwise, there need not be any maximal connected sets. For example, take S to be any infinite set, and define the connected sets to be precisely the finite subsets of S. –  Kevin Ventullo Jun 14 '10 at 7:30
Did you look at the exponential principle yet? (chapter 5 of Stanley EC II) Possibly also arxiv.org/abs/0911.3760 –  Martin Rubey Jun 14 '10 at 7:35
@supercooldave: "Maximal connected set" is supposed to mean "one for which no superset is connected". So it's "maximal" in the poset sense. @Kevin Ventullo: That's a good point. I could change the glueability property to apply to any family of subsets with nonempty intersection (instead of just pairs), but for now I think I'll restrict the question to finite sets, which is what I really care about anyway. –  Joshua H Jun 14 '10 at 7:45
Does it help if you make the relation "connected" explicit and formulate things as a combination of properties of the relation and of the sets the relation connects. For example, the relation is certainly reflexive and symmetric, but not transitive as mentioned below. And if S1 connected S2, then (S1 cup S2) connected (S1 cup S2)... Just an idea. –  supercooldave Jun 14 '10 at 8:22

It's just an exercise in the transitivity of relations? You have the subsets where a given function F takes a given value: in other words any partition, or any equivalence relation. I wouldn't call this a structure.

Edit: As Joshua has remarked, it is not all the subsets of a given partition-set that need be "connected". But do we not get all examples by taking the P that are "maximal connected", and then taking certain subsets of that including P itself and the singletons in it? The only condition seems to be that there are enough subsets to make P by transitive closure.]

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I'm not sure I follow you, but if you're saying that the only possibility is for the set S to be partitioned into subsets and a given set to be "connected" iff it is/(is a subset of) one of those subsets, that's not right. For instance, we might have S={1,2,3}, F={{1},{2},{3},{1,2},{1,2,3}}... But please let me know if I am misunderstanding your point. –  Joshua H Jun 14 '10 at 7:58
Oh, and also note that if {a,b} and {b,c} are connected, that doesn't imply that {a,c} is connected (as you may have assumed, based on your mention of transitivity?). All that glueability implies is that {a,b,c} is connected. –  Joshua H Jun 14 '10 at 8:01
Re: your edit -- "transitive closure" isn't quite an appropriate term to use. There is no relation here, other than perhaps the relation "a~b if {a,b} is connected". And that relation is not at all fundamental, since we can have interesting families without any pairs present. Take F2={singletons, {1,2,3}, {2,3,4}, {1,2,3,4}}. And if you mean to say that each maximal connected set can be built from smaller sets via gluing, that's not true either... Take the example I called F above. (continued) –  Joshua H Jun 14 '10 at 9:27
(continuation) Any given connected set can either be built up from smaller sets or not, and these smaller sets don't have to be pairs. The obligation runs in the other direction: if smaller connected sets are arranged in the correct way, they entail the existence of larger connected sets. (Thanks a lot for taking the time to think about this stuff, by the way!) –  Joshua H Jun 14 '10 at 9:27
A way of thinking is in terms of equations a = b, or to be clearer F(x) = F(y). If you give me as axioms such equalities for all pairs of elements of your connected sets, the question for me becomes the scope of what can be proved about F. So my description is: use the partition induced by the values of F being provably equal. Your remark is to do with the sets of axioms. So this is a kind of series of remarks about "equational logic", in my view. The gluing on overlaps is accurately modelled by the conjunction of F(x) = F(y) and F(y) = F(z), which gives F(x) = F(z). –  Charles Matthews Jun 14 '10 at 9:37

I can't really follow the discussion in Charles' answer (perhaps, the question got changed?), but there is a clear way to interpret this structure.

The maximal connected subsets $X_1,\ldots,X_n$ of a finite set $X$ form a partition of $X$. Each connected set $Y$ must be contained in one of the $X_i$s, and so partitioning into maximal connected subsets is repeated for each $X_i,$ each of their maximal connected subsets, and so on, until one arrives at one-element subsets, which are connected by the second axiom. Thus the "connected structure" on an $n$-element set $X$ may be recorded by a descending chain in the partition lattice $\Pi_n$ whose $k$th element is the partition of $X$ into "depth $k$ maximal connected sets" (a singleton is depth $m,m+1,m+2,\ldots$ ). A subset $Y\subset X$ is connected $\iff$ $Y$ is a block in one of the partitions in the chain. The chains that arise in this way satisfy a certain non-degeneracy condition.

This indeed looks similar to an "exponential structure" mentioned by Martin Rubey (Stanley, Enumerative combinatorics Def 5.5.1), but there is no postulated relation between the decompositions of different maximal connected subsets (no regularity dictated by "size"), so it is different.

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Since the whole set $S$ can be added to any family $F$ and still form a connected family, your top-level partition might just be $S$ itself. For the induction step in your analysis, therefore, you should look at "maximal proper" connected subsets. –  Joel David Hamkins Jun 17 '10 at 12:45
You are right, Joel, and it occurred to me as well (offline). What is worse, this will necessarily be the case for each $X_i$ at the first level. Unfortunately, the maximal proper connected subsets are not going to form a partition. I am no longer convinced that there is clean answer. –  Victor Protsak Jun 17 '10 at 16:48
Yep. You are thinking about this on much the same lines as I did. And you're entirely right that the recursive decomposition fails to work, at least in any obvious way. Thanks for your thoughts, though! –  Joshua H Jul 6 '10 at 22:13

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## MAT 136h Calculus I

Policy documents:
Department policies
Instructor:
Nándor Sieben
schedule
Syllabus

Please purchase a copy of the textbook MAT 136 Calculus I Lecture Notes. It is available in the bookstore.

Alternate online book:
Whitman
local copy if the sites above are down.

Alternate online book:
Paul Dawkins

Any standard printed Calculus book is useful.

Test dates are going to be announced here at least a week ahead of time:
Calculators are not needed for the tests. Basic (not scientific) calculators are allowed. No other electronic devices or notes are allowed.
Please review the test taking etiquette.

#### Test 1 Monday 9/25

Chapters 1 and 2:
Review: basic algebra, systems of equations, exponential and logarithmic equations, inequalities, trigonometry, equations of lines, interval notation, basic functions, domain, range, graphs of functions, function algebra, function composition, inverse function, function transformation.
New material: average velocity, average rate of change, difference quotient, slope of secant line, instantaneous rate of change, slope and equation of tangent line, finding limits algebraically and graphically, local and global continuity, approximating derivatives numerically and graphically, finding the derivative using the limit definition, derivative function, sketching the derivative function given the graph of a function, equation of tangent line, local linearization, direction and convexity

#### Test 2 Monday 10/16

Chapter 3:
derivative of basic functions, constant, power, exponential, logarithm, trigonometric, inverse trigonometric, absolute value (including "proofs"); derivative rules, constant multiple, sum, difference, product, reciprocal, quotient, inverse (including "proofs"); definition of the natural base, limits involving $$(1+x)^{1/x}$$ and $$\frac{\sin(x)}{x}$$; log trick; sketching the derivative function given the graph of a graph;

#### Test 3 Wednesday 11/08

Chapter 4:
L'Hospital, function analysis, derivatives of hyperbolic and inverse hyperbolic functions, optimization, implicit differentiation, related rates, Rolle's Theorem, Mean Value Theorem,

#### Test 4 Monday 12/4

Chapter 5:
definite integral, geometric interpretation, Riemann left and right sums, Fundamental Theorem of Calculus (two versions), antiderivatives, indefinite integral, substitution, integration by parts

#### Comprehensive final

scheduled during finals week in our usual room

Homework:
Webwork

Our department computer lab is in AMB 222. Computers and free tutoring are also available in AMB 137 starting the second week of classes.
Free on campus tutoring
Math Achievement Program Academic Success Centers
Extra credit opportunity:
Problem Of The Week


text:
Transfer Function

The engineering terminology for one use of Fourier Transforms. By breaking up a wave pulse into its frequency spectrum

 (1)

the entire signal can be written as a sum of contributions from each frequency,
 (2)

If the signal is modified in some way, it will become
 (3) (4)

where is known as the transfer function.'' Fourier Transforming and ,
 (5)

 (6)

From the Convolution Theorem,
 (7)


text:
# dihedral_style spherical command

## Syntax

dihedral_style spherical

## Examples

dihedral_coeff 1 1  286.1  1 124  1    1 90.0 0    1 90.0 0
dihedral_coeff 1 3  69.3   1 93.9 1    1 90   0    1 90   0  &
49.1   0 0.00 0    1 74.4 1    0 0.00 0  &
25.2   0 0.00 0    0 0.00 0    1 48.1 1

## Description

The spherical dihedral style uses the potential:

$\begin{split}E(\phi,\theta_1,\theta_2) & = \sum_{i=1}^N\nolimits\ C_i\ \Phi_i(\phi)\ \Theta_{1i}(\theta_1)\ \Theta_{2i}(\theta_2) \\ \Phi_{i}(\phi) & = u_i - \mathrm{cos}((\phi - a_i)K_i) \\ \Theta_{1i}(\theta_1) & = v_i - \mathrm{cos}((\theta_1-b_i)L_i) \\ \Theta_{2i}(\theta_2) & = w_i - \mathrm{cos}((\theta_2-c_i)M_i)\end{split}$

For this dihedral style, the energy can be any function that combines the 4-body dihedral-angle ($$\phi$$) and the two 3-body bond-angles ($$\theta_1$$, $$\theta_2$$). For this reason, there is usually no need to define 3-body “angle” forces separately for the atoms participating in these interactions. It is probably more efficient to incorporate 3-body angle forces into the dihedral interaction even if it requires adding additional terms to the expansion (as was done in the second example). A careful choice of parameters can prevent singularities that occur with traditional force-fields whenever theta1 or theta2 approach 0 or 180 degrees.

The last example above corresponds to an interaction with a single energy minima located near $$\phi=93.9$$, $$\theta_1=74.4$$, $$\theta_2=48.1$$ degrees, and it remains numerically stable at all angles ($$\phi$$, $$\theta_1$$, $$\theta_2$$). In this example, the coefficients 49.1, and 25.2 can be physically interpreted as the harmonic spring constants for theta1 and theta2 around their minima. The coefficient 69.3 is the harmonic spring constant for phi after division by sin(74.4)*sin(48.1) (the minima positions for theta1 and theta2).

The following coefficients must be defined for each dihedral type via the dihedral_coeff command as in the example above, or in the Dihedral Coeffs section of a data file read by the read_data command:

• $$n$$ (integer >= 1)

• $$C_1$$ (energy)

• $$K_1$$ (typically an integer)

• $$a_1$$ (degrees)

• $$u_1$$ (typically 0.0 or 1.0)

• $$L_1$$ (typically an integer)

• $$b_1$$ (degrees, typically 0.0 or 90.0)

• $$v_1$$ (typically 0.0 or 1.0)

• $$M_1$$ (typically an integer)

• $$c_1$$ (degrees, typically 0.0 or 90.0)

• $$w_1$$ (typically 0.0 or 1.0)

• […]

• $$C_n$$ (energy)

• $$K_n$$ (typically an integer)

• $$a_n$$ (degrees)

• $$u_n$$ (typically 0.0 or 1.0)

• $$L_n$$ (typically an integer)

• $$b_n$$ (degrees, typically 0.0 or 90.0)

• $$v_n$$ (typically 0.0 or 1.0)

• $$M_n$$ (typically an integer)

• $$c_n$$ (degrees, typically 0.0 or 90.0)

• $$w_n$$ (typically 0.0 or 1.0)

## Restrictions

This dihedral style can only be used if LAMMPS was built with the USER_MISC package. See the Build package doc page for more info.


text:
Question

# The substance whose resistance gets reduced to virtually zero at very low temperature is called

A
electrical conductor
B
hyper-conductor
C
superconductor
D
insulator

Solution

## The correct option is C superconductor$$\text{Substance whose resistance gets reduced to virtually zero at a very low temperature is called a }superconductor.$$A superconductor is a material that can conduct electricity or transport electrons from one atom to another with no resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing through a loop of superconducting wire can persist indefinitely with no power source.Option C is correct.Chemistry

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# Struggling with writing logical proofs

I am struggling with the way to write a clear and mathematical proof of logical theorems. Take for example the theorem $\Gamma \models A, \Gamma \subseteq \Delta$ implies $\Delta \models A$. I can prove it this way:

If every formula in $\Delta$ is validated, then that validates every formula in $\Gamma$, which validates $A$.

But how do I prove such thing with mathematical language?

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What exactly dies "validated" mean here? Spell that out a bit, and you will have your desired proof. – Peter Smith May 15 '13 at 18:19
If the word "validated" was used in the definition of $\models$ (and was, therefore, defined beforehand), then you might not need to spell it out, but otherwise I agree with Peter that you should. – Andreas Blass May 15 '13 at 20:55

A proof of the above fact needs the definitions :

(i) $Γ \vDash A$ iff for every interpretation $\mathcal M$, if all formulae in $\Gamma$ are true in $\mathcal M$, then also $A$ is

and :

(ii) $\Gamma \subseteq \Delta$ iff for every formula $\alpha$, if $\alpha \in \Gamma$, then $\alpha \in \Delta$.

Consider now an interpretation $\mathcal M$: due to the fact that $\Gamma \subseteq \Delta$, if all formulae in $\Delta$ are true in $\mathcal M$, then $\mathcal M$ will be also a model for all formulae in $\Gamma$ (which are among those in $\Delta$).

But $A$ is true in every model of $\Gamma$; thus, for every interpretation $\mathcal M$, if $\mathcal M$ is a model for all formulae in $\Delta$, then also $A$ is true in $\mathcal M$, i.e.

$\Delta \vDash A$.

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text:
Magnification of a system of lenses

I have a lens system consisting of 7 elements. However, I cannot take the system apart to measure the distance between them or their focal lengths. I do, however, know the effective focal length of the whole system. Is there a way to calculate the magnification of the whole system (for a given image distance)?

If your system is equivalent to a single lens with focal lenght $$f$$, then you can proceed like if you had one only lens with that $$f$$, so $$M_L=\frac{s'}{s}$$, with $$s$$ the distance to the object


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# Limiter

Dynamic range limiter

• Library:
• Audio Toolbox / Dynamic Range Control

## Description

The Limiter block performs dynamic range limiting independently across each input channel. Dynamic range limiting suppresses the volume of loud sounds that cross a given threshold. The block uses specified attack and release times to achieve a smooth applied gain curve.

## Ports

### Input

expand all

• Matrix input –– Each column of the input is treated as an independent channel.

• 1-D vector input –– The input is treated as a single channel.

This port is unnamed unless you specify additional input ports.

Data Types: single | double

#### Dependencies

To enable this port, select Specify from input port for the Threshold (dB) parameter.

Data Types: single | double

#### Dependencies

To enable this port, select Specify from input port for the Knee width (dB) parameter.

Data Types: single | double

#### Dependencies

To enable this port, select Specify from input port for the Attack time (s) parameter.

Data Types: single | double

#### Dependencies

To enable this port, select Specify from input port for the Release time (s) parameter.

Data Types: single | double

### Output

expand all

The Limiter block outputs a signal with the same data type as the input signal. The size of the output depends on the size of the input:

• Matrix input –– The block outputs a matrix the same size and data type as the input signal.

• 1-D vector input –– The block outputs an N-by-1 matrix (column vector), where N is the number of elements in the 1-D vector.

This port is unnamed until you select the Output gain (dB) parameter.

Data Types: single | double

#### Dependencies

To enable this port, select the Output gain (dB) parameter.

Data Types: single | double

## Parameters

expand all

If a parameter is listed as tunable, then you can change its value during simulation.

Main Tab

Operation threshold is the level above which gain is applied to the input signal.

To specify Threshold (dB) from an input port, select Specify from input port for the parameter.

Tunable: Yes

For soft knee characteristics, the transition area is defined by the relation

$y=x-\frac{{\left(x-T+\frac{W}{2}\right)}^{2}}{\left(2×W\right)}$

for the range $\left(2×|x-T|\right)\le W$, where

• y is the output level in dB.

• x is the input level in dB.

• T is the threshold in dB.

• W is the knee width in dB.

To specify Knee width (dB) from an input port, select Specify from input port for the parameter.

Tunable: Yes

The plot is updated automatically when parameters of the Limiter block change.

Tunable: Yes

Attack time is the time the limiter gain takes to rise from 10% to 90% of its final value when the input goes above the threshold. The Attack time (s) parameter smooths the applied gain curve.

To specify Attack time (s) from an input port, select Specify from input port for the parameter.

Tunable: Yes

Release time is the time the limiter gain takes to drop from 90% to 10% of its final value when the input goes below the threshold. The Release time (s) parameter smooths the applied gain curve.

To specify Release time (s) from an input port, select Specify from input port for the parameter.

Tunable: Yes

• Property –– Make-up gain is set to the value specified by the Make-up gain (dB) parameter.

• Auto –– Make-up gain is applied at the output of the Limiter block such that a steady-state 0 dB input has a 0 dB output.

Tunable: No

Make-up gain compensates for gain lost during limiting. It is applied at the output of the Limiter block.

Tunable: Yes

#### Dependencies

To enable this parameter, set the Make-up gain mode parameter to Property.

When you select this parameter, the block inherits its sample rate from the input signal. When you clear this parameter, specify the sample rate in the Input sample rate (Hz) parameter.

Tunable: No

Tunable: Yes

#### Dependencies

To enable this parameter, clear the Inherit sample rate from input parameter.

When you select this parameter, an additional output port, G, is added to the block. The G port outputs the gain applied on each input channel in dB.

Tunable: No

When you select this parameter, an additional input port SC is added to the block. The SC port enables dynamic range limiting of the input signal x using a separate sidechain signal.

The datatype and (frame) length input to the SC port must be the same as the input to the x port.

The number of channels of the sidechain input must be equal to the number of channels of x or be equal to one.

• Sidechain channel count is equal to one –– The computed gain, G, based on this channel is applied to all channels of x.

• Sidechain channel count is equal to channel count of x –– The computed gain, G, for each sidechain channel is applied to the corresponding channel of x.

Tunable: No

• Interpreted execution –– Simulate model using the MATLAB® interpreter. This option shortens startup time and has a simulation speed comparable to Code generation. In this mode, you can debug the source code of the block.

• Code generation –– Simulate model using generated C code. The first time you run a simulation, Simulink® generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time, but the speed of the subsequent simulations is comparable to Interpreted execution.

Tunable: No

## Block Characteristics

 Data Types double | single Direct Feedthrough no Multidimensional Signals no Variable-Size Signals yes Zero-Crossing Detection no

## Algorithms

The Limiter block processes a signal frame by frame and element by element.

1. The N-point signal, x[n], is converted to decibels:

${x}_{\text{dB}}\left[n\right]=20×{\mathrm{log}}_{10}|x\left[n\right]|$

2. xdB[n] passes through the gain computer. The gain computer uses the static characteristic properties of the dynamic range limiter to brickwall gain that is above the threshold.

If you specified a soft knee, the gain computer has the following static characteristic:

${x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}<\left(T-\frac{W}{2}\right)\\ {x}_{\text{dB}}-\frac{{\left({x}_{\text{dB}}-T+\frac{W}{2}\right)}^{2}}{2W}& \begin{array}{c}\\ \\ \end{array}\left(T-\frac{W}{2}\right)\le {x}_{\text{dB}}\le \left(T+\frac{W}{2}\right)\\ T& {x}_{\text{dB}}>\left(T+\frac{W}{2}\right)\end{array}\text{ },$

where T is the threshold and W is the knee width.

If you specified a hard knee, the gain computer has the following static characteristic:

${x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}

3. The computed gain, gc[n], is calculated as

${g}_{\text{c}}\left[n\right]={x}_{\text{sc}}\left[n\right]-{x}_{\text{dB}}\left[n\right].$

4. gc[n] is smoothed using specified attack and release time parameters:

${g}_{\text{s}}\left[n\right]=\left\{\begin{array}{cc}{\alpha }_{\text{A}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{A}}\right){g}_{c}\left[n\right],& {g}_{\text{c}}\left[n\right]\le {g}_{\text{s}}\left[n-1\right]\\ {\alpha }_{\text{R}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{R}}\right){g}_{c}\left[n\right],& {g}_{\text{c}}\left[n\right]>{g}_{\text{s}}\left[n-1\right]\end{array}$

The attack time coefficient, αA , is calculated as

${\alpha }_{\text{A}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{A}}}\right)\text{\hspace{0.17em}}.$

The release time coefficient, αR , is calculated as

${\alpha }_{\text{R}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{R}}}\right)\text{\hspace{0.17em}}.$

TA is the attack time period, specified by the Attack time (s) parameter. TR is the release time period, specified by the Release time (s) parameter. Fs is the input sampling rate, specified by the Inherit sample rate from input or Input sample rate (Hz) parameter.

5. If the Make-up gain (dB) parameter is set to Auto, the make-up gain is calculated as the negative of the computed gain for a 0 dB input:

$M=-{x}_{\text{sc}}\left({x}_{\text{dB}}=0\right)$

Given a steady-state input of 0 dB, this configuration achieves a steady-state output of 0 dB. The make-up gain is determined by the Threshold (dB) and Knee width (dB) parameters. It does not depend on the input signal.

6. The make-up gain, M, is added to the smoothed gain, gs[n]:

${g}_{\text{m}}\left[n\right]={g}_{\text{s}}\left[n\right]+M$

7. The calculated gain in dB, gm[n], is translated to a linear domain:

${g}_{\text{lin}}\left[n\right]={10}^{\left(\frac{{g}_{\text{m}}\left[n\right]}{20}\right)}$

8. The output of the dynamic range limiter is given as

$y\left[n\right]=x\left[n\right]×{g}_{\text{lin}}\left[n\right].$

## References

[1] Giannoulis, Dimitrios, Michael Massberg, and Joshua D. Reiss. "Digital Dynamic Range Compressor Design –– A Tutorial And Analysis." Journal of Audio Engineering Society. Vol. 60, Issue 6, 2012, pp. 399–408.

## Version History

Introduced in R2016a


text:
# Weak$^*$ convergence of sequences in the dual space of a quotient

Let $M$ be a closed subspace of a Banach space $X$. Let $q:X\to X/M$ be the quotient map, let $q^*: (X/M)^*\to X^*$ be the conjugate operator of $q$, and let $(z^*_n)$ be a sequence in $(X/M)^*$.

Is it true that $(z^*_n)$ is $w^*$-convergent in $(X/M)^*$ $\Leftrightarrow(q^* z^*_n)$ is $w^*$-convergent in $X^*$?

Are the following arguments correct?

The direct implication follows from the $w^*$-$w^*$ continuity of $q^*$.

For the converse, if $(q^* z^*_n)$ is $w^*$-null, then $z^*_n(q\, x) =(q^* z^*_n)(x)$ converges to $0$.

Let $(z_n^*)$ be a sequence in $(X/M)^*$. By definition of $q^*$, $$q^*(z_n^*)(x) = z_n^*(q(x)) \quad \forall x \in X,$$ If $(z_n^*)$ is $w^*$-convergent, then the right-hand side converges, then both sides converge, proving $\Rightarrow$.
Now assume that the mapped sequence $q^*(z_n^*)$ is $w^*$-convergent. By surjectivity, $q$ has a right inverse, which we denote by $q_r$. For $y \in (X/M)$, we choose $x = q_r(y)$ in the equation above, which gives $$q^*(z_n^*)(q_r(y)) = z_n^*(y)$$ By assumption the left-hand side converges, so the right-hand side converges as well, and since $y$ was arbitrary this proves $\Leftarrow$.
• Is the right inverse $q_r$ linear and continuous? – M.González Jan 20 '17 at 19:46


text:
Exact semidefinite programming bounds for packing problems

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $E_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (\sqrt{8}-1)/7$, is sharp by rounding to $Q[\sqrt{8}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.