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# Tag Info

1

You are correct in one thing: if an atom in an isotropic medium spontaneously emits a photon, it can do so in any direction at all, and the overall emission will be evenly spread over the unit sphere. However, lasers work using stimulated emission, which is slightly different: if an atom is excited, you can induce it to emit its energy by shining an initial ...

4

Have a look at my answer to Slit screen and wave-particle duality because this covers a lot of topics relevant to your question. You're correct that if we imagine the photon as a little ball then if the arms of the interferometer are different lengths the two "halves" of the little ball cannot arrive at the detector at the same time. But this is not how the ...

2

When one says "photon" one is in the quantum mechanical frame. Quantum mechanics does not follow the rules of classical mechanics if one tries to consider the photon one classical entity, like a bag of energy flowing. The photon is a point like elementary particle in the standard model, it has no extent and when it hits the detector it registers at a ...

5

If we assume you are a sphere in space, at the same distance from the sun as Earth, then we can calculate the heat absorbed - and we can calculate how hot you need to be so heat in = heat out (assuming uniform surface temperature, and radiative heat transfer only). For this, we need the Stefan-Boltzmann expression for total emission at a given temperature: ...

0

If a single red photon hits your telescope from the direction of a planet in the Andromeda Galaxy, then all you know that the planet emitted a red photon. Was it caused by a fire? A scattering of starlight through its atmosphere? An Andromedan with a laser pointer? A single photon of light from an unknown source has about as much information as a random ...

2

Suppose you are using a CCD or a photographic plate to record your image. The interaction with the light occurs when the detector absorbs a photon, and this happens at a point. So the image is built up from a collection of points - one for each photon that is detected. In everyday life, e.g. taking pictures with the CCD in your phone, the intensity of the ...

1

I think a simple view is this: The solar cell must have a PN junction, which is a junction between p-type (many holes, no electrons) and n-type (many electrons, no holes) materials. Right where they meet there is actually a "depletion width" within which there is hardly any of either. Within this region, as photons come in they generate electron-hole pairs, ...

8

It's tempting to think of the light as a little ball (the photon), and since little balls have a definite position the little ball has to be in a superposition of a state where it goes through one slit and a state where it goes through the other. However this is not a good description of what actually happens. The light is not a photon, and it's not a wave ...

0

Is it correct to think that the speed of light does not depend on the speed of light source because photons have no mass In a certain sense, yes. The Lorentz transformations guarantee that the speed $c$ is invariant; an object with speed $c$ in one inertial reference frame (IRF) has speed $c$ in all IRFs. But an object with invariant mass $m$ cannot ...

0

the photons will travel at the speed of light relative to both the moving light source and an object in another frame of reference. time dilation will bridges the gap so that both may co exist.

0

photons have inertia. They move in a straight line when no net force act on them,their momentum and energy also remains constant.(Frequency too). Otherwise they donot move in a straight line which can be seen in gravitational lensing.( http://en.m.wikipedia.org/wiki/Gravitational_lens ). So your reasoning is incorrect. But you are indirectly correct because ...

2

Theoretically yes, the laser principle does not consume any material. There is a light source that excites the electrons in the material to higher levels, they deexcite to some intermediate one, here the avalanche of photons appears producing the laser light and leaving the electrons in the ground state. And you can repeat the process without a loss.

2

Q1: For photons of energies much less gamma rays, the quantum mechanical photon-photon interaction is negligible. This is consistent with the classical electrodynamic description where the principle of superposition holds (electromagnetic waves pass through each other unchanged, as well as through electric/magnetic fields). Q2: in reality, charge is defined ...

1

theoretically if its components never wore out then yes. however in practice things do wear out eventually and so no it could not be done in the same way that a perpetual motion machine can work in theory but not in practice.

1

It is the energy of the incident photon. At low Energy (Long wavelengths as in Radio waves and Microwaves, visible light) Rayleigh scattering would dominant, in which the scattered photon would have the same wavelength as the incident photon. At higher energy (shorter wavelengths as in Ultraviolet spectrum), the electron would absorb the energy of the ...

1

You start with $$E' = \frac{E-up}{\sqrt{1-u^2/c^2}} = \\ \frac{E-up}{\sqrt{(1-u/c)(1+u/c)}}.$$ You know that $p = \frac{E}{c}$, so $$\frac{E-uE/c}{\sqrt{(1-u/c)(1+u/c)}} = \\ \frac{E(1-u/c)}{\sqrt{(1-u/c)(1+u/c)}} = \\ \frac{E\sqrt{1-u/c}}{\sqrt{1+u/c}} = \frac{E\sqrt{c-u}}{\sqrt{c+u}},$$ now using $E = hf$, $$f' = f\sqrt{\frac{c-u}{c+u}}. ... 1 To understand how light is affected by gravity, it helps to think of light as energy. So let's ask a basic question: when it come to light, what is energy? By the Planck-Einstein relation, we know$$E = h \nu$$where \nu is the frequency of the light and h is Planck's constant. So when we talk about the energy of light, keep that in mind. Also note ... 0 It sometimes occurs. It is called sonochromism. Here is my source. 0 That is interesting. At the very least, you should be able to tell the difference between the mirror and the world if you put your hand out to it - the mirror should be cold. Also you could blow on the mirror - it will either come back at your face (the atoms in each version of you's breath will bounce off each other) or show up as condensate, depending on ... 1 What they actually measured was not particle behavior. It was just a quantized energy transfer to the probing electrons. That corresponds to the absorption of individual photons, but it doesn't mean the Surface Plasmon Polariton (SPP) field was acting as a particle. It just interacted locally with the electron, as it must. Typically particle-like behavior ... 1 So, I wonder why is it usually said that photons do not interact, or hardly interact? As far as we know photons do not directly interact with each other. Mathematically, this is manifest in the fact that the equations of motion for electromagnetism are linear: given two sources A and B of electromagnetic radiation, the resulting EM field is precisely ... 2 When you have only one electron then \Delta S=0 makes intuitive sense: you can change the angular momentum l of the atom by changing it's internal structure (by pushing the electron in "another orbit" if you will), while you certainly can't change the internal structure of the electron to change s. Would it be possible to change s then you could ... 11 Absolutely. If Compton scattering occurred in first order in e, the only contributing diagram would be the obvious one. Say we're in a frame with the electron initially at rest and an incoming photon in the z direction. Then the electron 4-momentum is$$p^\mu_{\text{in}} = (m,0,0,0)$$while the photon 4-momentum is$$k^\mu_{\text{in}} = ...

5

The E=mc^2 formula only applies to an object at rest, and light is never at rest. You want to use the more general formula: $E^2={m_0}^2c^4+p^2c^2$ Then you can set the mass to zero. $E=pc$ What this says is that light has momentum, which is related to its energy.

1

This is because instead of $$\dfrac{1}{2}mv^2$$ or $$E = mc^2$$ the energy of light is given by $$E = hf$$ Where h is a number called Planck's constant and f is frequency (sometimes v is used) Here is an example, as requested: Imagine red light with $620. nm$ wavelength. The frequency of this light is $0.483$ x $10^{15}Hz$ This makes the energy of a ...

2

We know that solar cells generate electricity by utilizing the energy of the photon, This is an every day language, electricity. It means things electrical in general every day language. but how does it generate electricity forever? What is generated when the photons hit any material, is heat, and the sun's energy is at maximum 1300Watts per ...

-2

But why is there no such interaction between the electrons spin $\vec{s}$ and photons? Actually, maybe there is. See Compton scattering. The incident photon is partially absorbed and decelerated in the vector sense, whilst the free electron moves. IMHO you can visualize this by drawing repeated circles on a piece of paper without lifting your pencil. Now ...

0

An LED is a collection of discrete atoms. When a current flows, an electron excites one every so often. The excited atom decays after a while and emits a photon. Assuming the current is uniform, variations in intensity come from randomness in arrival times and excited state lifetimes. Some time slices have many photons, some fewer. This distribution is ...

1

There is an equation that helps a lot understanding this issue: Fermi's golden rule $$W_{i\rightarrow f}=\frac{2\pi}{\hbar} \left|\left<f\right|H'\left| i\right> \right|^2 \rho$$ It describes the transition rates from one state to another. $\rho$ is the so called Density of States (DOS) of final states. This system has only two states: The initial ...

0

There is indeed a way wherein "one photon" can be thought of as a Maxwellian wave. We are dealing here with the quantized electromagnetic field. If the EM field is in a one photon state then one can compute two vector fields from the EM field's state that: Fulfill Maxwell's equations for freespace propagation and Uniquely define the one photon state. ...

2

The maxwellian wave is an emergent phenomenon from a great multitude of photons with the frequency of the maxwellian wave. This is explained in this blog entry by Lubos Motl. I will give you my experimentalist's interpretation of this: A photon as a quantum mechanical entity has a wavefunction. This wavefunction is a solution of a form of Maxwell's ...

0

It turns out that light can be thought of as a wave and a particle under different conditions. For example, as Cort Ammon described, the Double Slit experiment showed that light had properties of diffraction. On the other hand, the photoelectric effect considers light as packets of light called photons with certain energy $hf$. Another such experiment is the ...

0

Another way of thinking of this is to imagine that space-and-time is an emergent phenomenon, NOT a pre-existent framework that holds photons, be they waves or particles. If space/time emerges for example at the quantum level of individual photons, effectively there may be no immediate space and time there for a wave to propagate through. But at larger ...

0

Look into Wave-particle duality. It is a major part of Quantum Mechanics which answers your question. A quick summary: light is not just a wave, not just a particle. In some situations it behaves like a wave; in others it behaves like a particle. In some situations, neither wave nor particle sufficiently describe reality. A solid example of this is the ...

0

Good discussion of this question here. In short: because momentum p imparted by light with energy E is given by $E = pc$ then if we compare a photon "shot" to a bullet with the same energy the momentum generated by a photon gun is on the order of $10^{-6}$ the momentum of the traditional gun. But felt recoil is a tricky subject, because it incorporates ...

1

The negative potential works against the $K$ of the ejected electrons. So at very high negative voltages there is no photocurrent. As the voltage moves closer to zero, we hit the point where $f_2$ produces a current. If it's able to produce a current, this means that the electrons are ejected with a high kinetic energy, and thus the work function that binds ...

0

The simplified answer for this is that gravity basically means that you are in elevator which is accelerating: even if you are standing on the ground. So if you take that constant gravity field is pretty much the same as a constant acceleration, then if a beam of light bends downward in an accelerating elevator, it must bend downward if it is in an constant ...

2

In the two-dimensional rubber sheet visualization, it is wrong to think that things fall towards the massive object because they are "rolling down the hill" of the curved spacetime. There is no perpendicular gravity pulling things down into the well. What happens is that you are moving along your world line at a constant velocity, "into the future at one ...

0

Yes, photons are affected by those curves. They also curve space-time due to something called a stress-energy tensor. Remember that the energy of a photon is given by $$E = hf$$, so photons do have energy. This energy lets them be affected by gravity.

0

"Do two photons traveling in opposite directions emit gravitational waves?" Does virtual graviton count as gravitational waves? for sure there are gravitational effects so call it gravitational field. And two photons can interact indirectly through other fields. "If so, does it mean that any volume filled with photon gas will eventually degrade into ...

1

When you say the stopping voltage determines the kinetic energy of the electrons you have that the wrong way round. For a given kinetic energy of photoelectron you must apply a potential to bring it to rest. So it is true that increasing the intensity of the incident light will increase the photocurrent as you have more electrons ejected from the metal, ...

2

A photon that strikes the metal either has enough energy to release a valence electron from its bond with the atoms in the metal, from a certain frequency and up, or its energy is just not enough to eject the electron from the metal. The surplus of energy over the energy needed to eject the electron is used as kinetic energy of the electron and determines ...

2

I'm not sure about the “pick it up with a laser” part, but let's simplify by assuming that we can hit the golf ball from below the ground with photons. Let's further assume that the mass of the ball is $m_{ball} = 46\,\mathrm g$ and we want to hurl it five yards away ($\approx 4.6\,\mathrm m$). If the ground is level and we neglect air drag and wind, the ...

3

Hurl it several yards away Rather than a simple yes/no, let's figure out some numbers - as usual I work with "round" numbers. Mass of golf ball = 50 gram "several yards" = 5 meters Projectile launched at 45° with initial velocity $v$ travels for a time $$t = 2\frac{v}{\sqrt{2}g}$$ during which time it will travel a distance x = v_x t = ...

-1

If the ball was completely reflective, yes. The concept was discussed e.g. here http://en.wikipedia.org/wiki/Solar_sail edit: e.g.- radiation pressure is what keeps Sun from collapse

-1

Well, photons behave both as a wave and as particles, and the photons in coming from the beam of light "enlighten" the particles in front of you.

1

If the source is far away, light acquires a certain degree of coherence. Have a look at the Van Cittert–Zernike theorem, also in wikipedia: "the wavefront from an incoherent source will appear mostly coherent at large distances". The resulting fringes are different for different colors, but any color has max for straightforward direction. So, you see the ...

1

We know that light is massless so why does a black hole's gravity attract light? Because gravity doesn't just attract objects with mass. It alters the path of light too. Because gravity is caused by a concentration of energy which "conditions" the surrounding space, altering its metrical properties, whereupon we talk about spacetime curvature. But note that ...

1

The Coulomb "interaction" appears as the answer to a very specific question, even in QED, that is "what is the correction to the ground state energy of the electromagnetic field if two charges $q_1$ and $q_2$ are pinned at specific locations separated by a distance $r_{12}$?" . The answer to that question is exactly $\Delta E_0 = \frac{q_1q_2}{4 \pi ... 1 A photon has a rest mass of nought (where the rest mass$m$is the Lorentz-invariant quantity in the four-momentum's Minkowski norm squared$E^2/c^2 - p^2 = m^2 c^2$). However, a lightfield of energy$E$gravitates and itself has a gravitational source equivalent to a mass$E/c^2\$. Also, a system of photons has a nonzero rest mass (see reference), as does ...

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# Clock skew constraints for clock tree synthesis

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#### yuhiub90

##### Member level 2
Hi guys,

I'm wondering whether or not to constrain the clock skew (uncertainty) for placement and clock tree synthesis.
Does this affect the clock path during optimization process when building clock tree, e.g. buffers are added to clock path to satisfy the constrained skew. Or it's just optimization on data path to satisfy setup/hold requirements with the constrained clock skew?

Thanks.

#### raj1435

##### Newbie level 5
Hi,

Uncertainty will be help full to satisfy the setup and Hold requirements. Inserting the buffers on the design will be the way to fix the setup or hold violations. Tool will take care the best approach is to insert the buffer or to reduce the data path delay.

yuhiub90

### yuhiub90

Points: 2

#### yuhiub90

##### Member level 2
I know it. But how optimization is done. Does the synthesizer try to build the clock tree with the constrained skew. Or it still perform zero skew optimization, the specified uncertainty margins (setup/hold skew) is used to prevent skew due to variabilities when real chip is fab. Which one is better, constrain or non-constrain it?

Thanks

#### hoanglongroyal

##### Member level 1
hi Yuhiub90,
as I'm using uncertainty in synthesizing, it's just used as a margin which you want to reserve for Back-End later.
in my case, synthesizer do not build clock tree follow the uncertainty and clock-tree timing is idea.
I put a short report timing to you bellow:
Code:
Point                                       Fanout       Cap     Trans      Incr       Path
----------------------------------------------------------------------------------------------
clock clk_core (rise edge)                                                  0.00       0.00
[B]clock network delay (ideal)[/B]                                                 0.00       0.00
.
.
.
clock clk_core (rise edge)                                                1.125      1.125
clock network delay (ideal)                                                0.000      1.125
[B] inter-clock uncertainty                                                   -0.250      0.875[/B]
output external delay                                                     -0.837      0.038
data required time                                                                    0.038
----------------------------------------------------------------------------------------------
data required time                                                                    0.038
data arrival time                                                                    -0.091
----------------------------------------------------------------------------------------------
slack (VIOLATED)                                                                     -0.053
you can see the uncertainty in example make slack more pessimism.
totally, as I know, uncertainty is used for model clk skew, jitter, ... as margin for design.

- - - Updated - - -

Which one is better, constrain or non-constrain it?
I forgot this thing, if you have right uncertainty, then you're good to meet your timing goal.
- you put an enough uncertainty, can help synthesis tool follow and optimize more in your design.
- if you do not specify uncertainty, maybe your design is met as 0 margin, and synthesis tool do not need to optimize --> back-end guy will kill you if their work become more complex.
- if you specify uncertainty too big, synthesis tool will give up because of impossible optimization.

yuhiub90

### yuhiub90

Points: 2

#### yuhiub90

##### Member level 2
thanks hoanglongroyal,

Your report doesn't show clock skew because you are considering ideal clock network. You can see it when invoking set_propagated_clock and back-annotated the delay information from SDF file.
I got the idea of using uncertainty margin for logic synthesis, it's for timing analysis when clock tree has not been implemented. But for back-end stage, is this constraints necessary?

#### hoanglongroyal

##### Member level 1
thanks hoanglongroyal,

Your report doesn't show clock skew because you are considering ideal clock network. You can see it when invoking set_propagated_clock and back-annotated the delay information from SDF file.
I got the idea of using uncertainty margin for logic synthesis, it's for timing analysis when clock tree has not been implemented. But for back-end stage, is this constraints necessary?

hi yuhiub90,
as I mention, uncertainty is margin/budget timing for your design, it models skew,jitter,... timing tool will make report more pessimism by uncertainty. and this is what we need to safe design. you can't make sure 100% that the skew is the same at any time, any where, as your chip come real because of variation (by temperature, fab fault,...). I mean , always have variant value that you need to cover in skew, uncertainty.
of course in some case, back-end guy can reduce skew uncertainty at some paths, such as paths short clock tree, or some paths which they can make sure less variation.

yuhiub90

Points: 2

format: plain


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# What is missing from MATLAB #2 - the next decade edition

168 views (last 30 days)
Rik on 31 Jul 2020
Answered: cui on 10 Jun 2021 at 8:40
Meta threads have a tendency to grow large. This has happened several times before (the wishlist threads #1 #2 #3 #4 #5, and 'What frustrates you about MATLAB?' #1 and #2).
No wonder that a thread from early 2011 has also kept growing. After just under a decade there are (at time of writing) 119 answers, making the page slow to load and navigate (especially on mobile). So after a friendly nudge; here is a new thread for the things that are missing from Matlab.
Same question: are there things you think should be possible in Matlab, but aren't? What things are possible with software packages similar to Matlab that Matlab would benefit from? (note that you can also submit an enhancement request through support, although I suspect they will be monitoring activity on this thread as well)
What should you post where?
Wishlist threads (#1 #2 #3 #4 #5): bugs and feature requests for Matlab Answers
Frustation threads (#1 #2): frustations about usage and capabilities of Matlab itself
Missing feature threads (#1 #2): features that you whish Matlab would have had
##### 2 CommentsShowHide 1 older comment
Walter Roberson on 31 Jul 2020
Thanks, Rik!

dpb on 10 Sep 2020
Editor won't restrict a substituion to selected area...best it knows of is the function. Uncool in the max! I'm now having to fix up a bunch of stuff shouldn't have to have done... :(

Shae Morgan on 31 Jul 2020
facet_wrap or facet_grid (or general ggplot2 functionality) version of subplots, or some altered, simpler customizability for subplotting subsets of data.
gramm is an available toolbox, but it'd be nice to have it built in

Rafael S.T. Vieira on 31 Jul 2020
Edited: Rafael S.T. Vieira on 31 Jul 2020
I would love to have a command TeX2sym and sym2TeX. With it, we could bring formulas from LaTeX and run them in MATLAB and the other way around. It is a tedious work doing it by hand, and I could bet most MATLAB users do it or will do it eventually.
Another useful feature to add would be arbitrary-precision arithmetic...languages such as python and Java (with Bignum), allows unlimited precision when dealing with integers at least. Granted that It is slow, but I believe that the MATLAB team could do something better. And I honestly feel like MATLAB is missing a feature by not having it...even some competing software has it.
And finally I was considering buying the Text Analytics Toolbox, and it would be nice, if it could have a grammar/spell checker. Even if it is not as advanced or doesn't contain all words. With live scripts, we can write an interactive document, so it would be nice if MATLAB could correct our spelling (even if it required a ToolBox to do it).
Rafael S.T. Vieira on 1 Aug 2020
Thank you for your interest, Walter and Rik. I believe that It is best to teach ourselves to code in a particular way, which will allow MATLAB to convert it to sym, than to be doing the task by hand every time.
Operator precedence could take care some of these issues. If \sin x + y was the input, then left-to-right precedence could dictate the output to be sin(x) + y. And to obtain sin(x+y), we would have to code in LaTeX \sin{x+y} or \sin(x+y).
Implied multiplication is indeed very common in mathematics and latex. On the other hand, it is also almost ubiquitous that variables are written just as single letters (especially if we are writing implied multiplication).
Finally, a command like tex2sym does not need to contemplate every math package imho, just some set of commands and macros. MATLAB could even return its best guess, and let us do the remainder of the task. Of course, ideally, we would just copy the contents from environments like  and $...$ and paste them into MATLAB for a tex2sym conversion.

Walter Roberson on 13 Sep 2020
A "soft interrupt" -- that is, an interrupt that can be handled with try/catch .
For example I am running a long calculation at the moment that iterates several times, and each iteration can take more than an hour. I don't always have the patience to wait through as many iterations as I asked for. For my purposes, the output of the last successful iteration would be "good enough" if I were to ask to interrupt the function.
I can control-C, but that interrupts the function completely, losing all of the outputs, and losing the function variables.
If I could somehow "soft interrupt" and have it return the current values of all variables, then that would be good enough for my current task. But the generalization of that would be the ability to catch a soft interrupt in a try/catch so that the code could make whatever final summariziation it needed to in order to create usable output variables. Furthermore, even though my own routine might be happy to return the "last good" values of the output variables, any soft-interrupt I requested might well get received while some lower-level routine had control that did not know about soft-interrupts, so I would want interrupt (with no return values) to apply to those layers until control reached my handler for a clean run-down.
Sindar on 3 Oct 2020
My solution in a similar case was to check whether a certain file ('FAIL_MODE.txt') exists in the directory at the end of each iteration. If it does, break the loop. Then, to stop the program, all I had to do was create the file (which is pretty trivial unless Matlab has frozen the whole computer)

Tim on 5 Nov 2020
The following changes to the internal volume renderer would make Matlab much more useful for volume visualization:
• True RGB volume visualization (vs. just scalar data + colormap)
• Independent, decoupled specification of alpha values and intensity values
• The ability to combine volume images with standard axes and axes objects like points, lines, patches, etc.

Mario Malic on 13 Nov 2020
When typing code, if user wants to reference a variable or a function, one could click tab and get a matching list of functions and variables. Would it be useful to split this functionality with modifier if user wants to reference one of the two? As an example Shift +Tab for variables and Tab for functions and variables (not to break the current functionality).

Mikhail on 2 Aug 2020
Some (well, most) dynamic programming languages allow for an experienced developer to have an insight look into how their code is actually executed by the runtime environment.
They are able to see the parse tree and the compiled byte code.
They are able to see what parts of code are JIT-compiled, and how exactly the are compiled. When something is not compiled, they are able to see why.
The developer doesn't have to guess whether a particular optimization has kicked in or not. They know for sure how each and every object in their code is handled (whether it is CoW, passed by reference, passed by value).
I'd love to see these capabilities in MATLAB, too.
Rik on 2 Aug 2020
That doesn't make the function horrible, it just means the function is not suited to your needs. I don't complain that my hypothetical electric stove can't cool things down, even though with a TEC it is possible to both heat and cool down things with a single electrically powered device.
Although you could re-use a lot of the internals of readfile to fix double UTF-8 encoded files, the problem is not the function, it's the double encoding. Fixing double encoding is a different task from reading a file correctly. It wouldn't be a matter of 'mending my function', it would be re-using internal functions for a different goal. The word 'mending' implies my function is broken, but returning a double encoded file in a corrected form is not living up to the contract that the name of the function is offering.
Also, there are more reasons to close down the source than just thinking your users are dumb (suggesting otherwise sounds a lot like assuming bad faith). As an example: if you want to provide a function for a license fee, it would be plain stupid to have your license check in an m-file, since it would be almost trivial to circumvent.

Seth Wagenman on 31 Aug 2020
Ability to convert Python items to MATLAB data types (other than numpy arrays) inside of MATLAB, rather than inside of Python using the admin-rights-required API: https://www.mathworks.com/help/matlab/matlab_external/install-the-matlab-engine-for-python.html?s_tid=srchtitle

dpb on 6 Sep 2020
Edited: dpb on 6 Sep 2020
A ready-built insertion method for tables, arrays, etc., so don't have to do all the under-the-hood indexing manually...given an array of the right size and matching index, if had table (say), then
tMyTable=insert(tMyTable,indices,tNewData);
ideally, an option with keyword 'Empty', T|F would be available as well.
Maybe I'm missing some magic pixie dust, but I've been unable to figure out a way to this without physically moving a section at a time to insert the new row which means working from rear to front to avoid changing the indices to the insertion points or by catenating the pieces starting from the front.
dpb on 6 Sep 2020
Yeah, that's undoubtedly a better way on the construction, Walter -- I hadn't really thought about the implementation too much other than going "dead ahead" to "just get 'er done!" for the immediate task at hand.
Taking a break, I thought I'd throw it out as an idea for an enhancement for discussion...with the sidelight maybe somebody did have a "trick" hadn't thought of.

Walter Roberson on 9 Sep 2020
I know I've said it before, but it's still missing and still important:
We need a way to gather all of the outputs of function call into a cell in the middle of an expression .
I know this may be tricky to implement internally. There are internal rules that are hard to work out, that have to do with how many outputs to request from functions. For example,
[A, B] = cellfun(@C, D, 'uniform', 0)
somehow passes knowledge of "two outputs" to C -- for example if you were to use @max then A would be a cell array of the maxima and B would be a cell array of the indices. The situation can get more complicated than this, and figuring out all the cases can make your head hurt. But we do know that any expression C(D(E)) that D(E) will be evaluated asking for exactly one output and that would be passed into C... but the knowledge of multiple outputs would be passed to C and yet not D.
The number of outputs to use is not inherently clear. If for example you call ode45 and ask to gather the outputs, are you asking for the common TOUT, YOUT case, or the full TOUT,YOUT,TE,YE,IE ? There are some cases where extra outputs can be expensive to calculate, so even though an operation that gathered "maximum" outputs might be useful, it is not always desireable, so ability to select the number would be useful.
Then there are issues with, for example, deal(), where you can have any number of outputs with just a single input:
[A,B,C,D] = deal(123)
would initialize A, B, C, and D all to 123. So if you ask to gather "all" of the outputs from deal(123), that number is not well defined.
Working these things out is not trivial -- but it is a really missing bit of the language.
There might be an opportunity for a syntax such as {}name -- e.g.,
arrayfun(@(X0) {}ode45(@f, tspan, X0), x0s)
meaning to gather all of the outputs of the call. At present, {} is not valid before a name.
##### 2 CommentsShowHide 1 older comment
Walter Roberson on 10 Sep 2020
Bruno,
Imagine that you want to implement
[temp{1:3}] = ndgrid(-1:.1:1);
C = cell2mat(cellfun(@(M) M(:), temp, 'uniform', 0);
That is, you want a single array in which each of the columns is one of the outputs of ndgrid(). And you want to do it as a single expression.
ndgrid() does not have a fixed number of outputs -- it is not like sin() with one fixed outputs, or max with two fixed outputs. nargout(@ndgrid) is -1 -- in other words the declaration is like
function varargout = ndgrid(varargin)
If you wanted to capture all of the outputs of max() then you could query ndgrid(@max) to get 2:
function outputs = gather_outputs(f, varargin)
n = nargout(f);
[outputs{1:n}] = f(varargin{:});
end
and you could
gather_outputs(@max, rand(3,5))
and this would be fine for gathering the two outputs of max into a single cell array.
But if we try to
gather_outputs(@deal, [])
then nargout(@deal) is -1 and if we said "Okay, take the absolute value of that -1" then we would be doing
[outputs{1:1}] = deal([])
which would give you just {[]} as the output.
This shows that you cannot just look at nargout() of the function you are invoking, such as @max or @deal .
Can we just look at nargout of the overall expression to determine the number of outputs to use for deal? No,
C = cell2mat(cellfun(@(M) M(:), gather_outputs(@ndgrid, -1:.1:1), 'uniform', 0));
would at best tell you that nargout is 1 (the C variable). You need something else to tell you the number of outputs you want to get -- something like
function outputs = gather_n_outputs(f, n, varargin)
[outputs{1:n}] = f(varargin{:});
end
and then you could
C = cell2mat(cellfun(@(M) M(:), gather_n_outputs(@ndgrid, 3, -1:.1:1), 'uniform', 0));
The request is to build this kind of facility in to MATLAB instead of having to write true functions like gather_n_outputs and have to pass function handles into them. Some syntax like
C = cell2mat(cellfun(@(M) M(:), {3}ndgrid(-1:.1:1), 'uniform',0));
where the hypothethical new syntax {3} indicates that you are requesting 3 outputs and that you want them gathered in a cell array. A common alternate syntax would be {} to request all of the outputs, like
C = cellfun(@(V) V.^2, {}max(rand(3,5)), 'uniform', 0)
which would hypothetically gather both (all) outputs of the max() call into a cell array that would then be available for further processing.
If you just used
C = cellfun(@(V) V.^2, max(rand(3,5)), 'uniform', 0)
then the knowledge of the single output would be carried through into the cellfun, which would tell max() to only emit a single output, so you would not get the second output processed. And the first output would be numeric not cell, so you would need arrayfun instead of cellfun...)

Walter Roberson on 12 Oct 2020
A "select" clause for readtable() and kin.
For example one user only wanted to read the rows in which one particular variable had a particular value
It could at some point be implemented in terms of a rowfun() type function that got passed the variables for a row and could make arbitrary decisions based upon the row contents.
However an earlier stage could potentially have pairs of variable names (or numbers) and a vector or cell array of values, in which the selection code did an ismember(). Such a facility could be further improved if there were a "sorted" option (so the code could figure out when to give up looking -- any one value should no longer be looked for if a larger value were encountered). Or even "grouped", which would not imply sorted as such but would imply that when you find a value that all the instances of that value that exist will be in adjacent rows and so as soon as you detect a change you can know to give up looking for that value.

Sindar on 5 Nov 2020
Ok, this is pretty minor, but:
For common functions that return values as the first output and indices as the second, it would be nice if there was a direct way of getting the indices, so I could do things like this:
x = [1 2 5 4 5];
y = 1:5;
y_xsorted = y(sortInd(x));
% or
y_xunique = y(unique(x,'ia'));
instead of needing to create temporary variables:
x = [1 2 5 4 3];
y = 1:5;
[~,idx] = sort(x);
y_xsorted = y(idx);
[~,idx] = unique(x);
y_xunique = y(idx);
I know I could make wrappers myself, but this seems like a case where a builtin function could potentially be noticeably optimized. (And, is more straightforward than rebuilding the whole output system)

Bruno Luong on 6 Nov 2020
Implement family of (numerical) data structure with O(1) inserttion, removal, that includes chain-list, binary tree, Fibonachi tree, red-black tree etc, etc, ... the performance must be focus point. Don't care if they are encapsulated or not in the OP, just don't make that reduce the performace.

Bruno Luong on 6 Nov 2020
Implement equivalent to C inline-function (or macro) so as calling this function on small data won't be penalized in speed with the over-head.

Bruno Luong on 17 Nov 2020
Edited: Bruno Luong on 17 Nov 2020
No big deal but I wish SIGN(X) function could return 1 for X=0 instead of 0. May be implementing an option to not break the compatibility.
I just rarely use SIGN because of this exception choice.
Bruno Luong on 18 Nov 2020
Why I prefer sign(0) = 1?
To me y = sign(x) should satisfy these two important properties
norm(y) = 1
x = y * abs(x)
This uniquely determines y for any x ~= 0 (real, complex or even quaternion). For x=0, any y=exp(1i*alpha) with alpha real meet both properties, so just select it as y=1 for arbitrary convention.
However the MATLAB SIGN() function does NOT satisfy the first property for x=0.
Adams's trick sign(x+realmin) is interesting, but it just shift the issue to x=-realmin.

cui on 7 Feb 2021
Edited: per isakson on 31 May 2021 at 20:47
Deep Learning toolbox:
How to visualize the changes in dlarray/weight distribution with histograms in deep learning?
Histogram displays how the trend of tensor (weight, bias, gradient, etc.) changes during the training process in the form of histogram. Developers can adjust the model structures accurately by having an in-depth understanding of the effect of each layer.
cui on 9 Feb 2021
@Adam Danz Thank you for your suggestions, I look forward to the official future version will be greatly improved！

cui on 7 Feb 2021
Edited: cui on 8 Feb 2021
MATLAB
I very much hope that the official version will strengthen the readstruct function in the future! Lack of complete uniformity to support more format requirements.

cui on 7 Feb 2021
Edited: cui on 1 Apr 2021
Deep Learning toolbox:
and so on....
The above are influential applications of deep learning in various aspects, but it is difficult to reproduce in matlab. Although Matlab2019b version supports automatic differentiation mechanism, it is still difficult to implement algorithms in matlab. The efficiency of the differentiation mechanism is not high, and many operators do not support it. I tried to implement the more famous yolov3/v4 algorithm with the latest MATLAB2020a version, but it is still not satisfactory
In summary, my personal suggestions are like my personal answer above, and I hope that future versions can improve a lot!

cui on 19 Feb 2021
Deep Learning toolbox:
groupedConvolution2dLayer since R2019a support, but groupedConvolution3dLayer don't support ??? (until R2020b)
Adam Danz on 23 Feb 2021
@cui could you elaborate on your last comment? How did you contact MathWorks and why do you have that impression of them?

One thing would be: defining classes in notebooks (Live Scripts in MATLAB case). That's a normal thing for years now in Python, for example.

Robert Guldi on 29 May 2021 at 18:58
As far as I can tell, there does not seem to be very much in the ways of documentation for Excel-based work (i.e creating and manipulating charts). I'd love to see some of this material added into the Documentation search if at all possible.
dpb on 2 Jun 2021 at 15:50
OK, thanks for the hint -- I had thought that
borders=get(theCell, 'Borders');
theBorder=get(borders, 'Item', border);
set(theBorder,'LineStyle', style,'Weight',weight);
would have returned the wanted border Item object, but apparently not...I still don't fully understand that, but the following does work....
theBorder=borders.Item(whichBorder);
set(theBorder,'LineStyle', style,'Weight',weight);
although one apparently can't write
set(borders.Item(whichBorder),'LineStyle', style,'Weight',weight);
in typical MATLAB syntax fashion.
function SetBorder(sheetReference, ranges, whichBorder, weight, style)
if style==XlLineStyle.xlDouble, weight=XlBorderWeight.xlThick; end % only combination that works
if ischar(ranges), ranges=cellstr(ranges); end
if isenum(whichBorder), whichBorder=int32(whichBorder); end
if isenum(weight), weight=int32(weight); end
if isenum(style), style=int32(style); end
for i=1:numel(ranges)
try
range=sheetReference.Range(ranges{i});
borders=get(range, 'Borders');
theBorder=borders.Item(whichBorder);
set(theBorder,'LineStyle', style,'Weight',weight);
catch ME
fprintf('Error in function SetBorder.\nError Message:\n%s\n', ME.message)
%warning('Error in function SetBorder.\n%s', ME.message)
end
end
end
Experimentation also revealed that one can't use any combination of the enumerations (which limitation is also NOT documented). Only with the "Thick" weight is a double-line rendered; all the others are a single line of medium weight.
All kinds of things aren't documented and left for only trial and error to uncover...
The above could be cleaned up and forget about trying to use enumerations, but it's expedient for the moment and there are far more important issues to fix rather than continuing to throw time down the rat hole...
Thanks, you did get me pointed to what can be made to work--muchly appreciated!!

cui on 10 Jun 2021 at 8:40
Single and multi-objective vision tracking algorithms:
KCF, GOTURN, FairMOT, deepSort ,... are some of the best and successful algorithms in recent years, but unfortunately, why is there no such implementation in the "sensor fusion and tracking toolbox", "deeplearning toolbox" and "computer vision toolbox"?


text:
## Monday, December 05, 2016

### Bulk of Czechoslovak MEPs demand apology, explanation for the EU officials' lionization of Castro

Most of the Czech, Slovak, and other post-communist world's political junkies were stunned by the only tweet that the boss of the EU commission Jean-Claude Juncker wrote about the death of the Cuban dictator Fidel Castro:

This "hero to many" description was his picked summary of a somewhat longer statement about Castro's death by Juncker.

Similar outrageous or at least bizarrely neutral remarks were made by other leftists including Trudeau (Canada's PM), Obama (lame duck U.S. president), and Corbyn (the boss of the U.K. Labour Party) but Juncker's lionization of Castro was almost certainly the most radical one of all.

Many Czech politicians, especially the deputies of the European Parliament (e.g. the former Czech EU commissioner Pavel Telička) quickly reacted and expressed their shock. To make things even worse, Czech Public TV journalist Mr Bohumil Vostal (reporting from Brussels) asked Juncker's spokesman, New Democracy's Margaritis Schinas (Greece), for an apology or an explanation. It didn't end up too well:

Vostal meaningfully asked how it was possible for the boss of the EU commission to be signed under such a shocking statement about a man who has executed at least 7,000 people, forced 10% of his nation to flee the country, and who approved the 1968 invasion of Czechoslovakia, among other things. Was it a mistake or a hack? Is there a different explanation?

The staggering Greek aßhole arrogantly replied that all the unflattering statements about Castro were just Mr Vostal's personal opinions that expressed a "very narrow view of reality".

Just imagine. This guy comes from a political party that paints itself as a conservative one, relatively to the Greek calibration. But he ends up being almost the same intolerable fanatical commie as those in Syriza and similar political cesspools. Schinas was literally harassing the Czech journalist because he dared to mention some facts – that Castro was one of the most terrifying criminals among the 20th century politicians.

The shock among the Czech and Slovak politicians didn't fade away in the following week. Today, the EU observer reported that
Czech, Slovak MEPs [were] 'shocked' by EU comments on Castro
and about one-half of them (largely because they wanted to avoid all commnist-party signatories) have filed an official urgent complaint that the EU commission is obliged to answer in 3 weeks. The anger is clear from the Czech and Slovak text:
As "the elected representatives of the Czech and Slovak citizens, who suffered from political persecutions and whose lives will be forever marked by the 1968 events," the 14 MEPs said that "the response by Mr Schinas was arrogant and portrayed shocking ignorance of facts with a total lack of understanding history."
They asked Juncker to take "clear political distance" from his spokesman's words, "be it in written or in a public speech."

You know, we often use the phrase "political correctness" as a set of taboos by which the leftists are constraining the people's freedom of speech. But it's obvious that such taboos may in principle be imposed not only by leftists – even though the leftists are those who are most excited about the restrictions on the freedom of speech these days. In principle, every society or community may define its red lines.

What Schinas and Juncker said and wrote is simply unacceptably political incorrect. Well, this wouldn't mean much. So I want to phrase this problem in a much more tangible way. What Schinas and Juncker have said is technically criminal according to the Czech law. You may read e.g. this story from 2000 about a guy who was promoting communism and had to face a trial. Among other things, the article says
Czech law bans the promotion of ideologies that aim to suppress basic human rights and freedoms, which communism as the Czech Republic experienced it before 1989 undoubtedly does. This, of course, is not the first case of propagation of hard-line communism in the Czech Republic. However, until now, the police only prosecuted far-right extremists, not left-wing radicals.
Our laws simply prohibit the kind of a political behavior that is likely to result in the long-term suppression of basic constitutionally guaranteed human rights or democracy which could be similar to the long totalitarian epoch from 1939 to 1989 (with at most two short breaks). In practice, this law prohibiting the systemically anti-freedom political activities is mainly used to regulate and harass (mostly shy) teenagers who decided to become skinheads or would-be Nazis.

But the authors of the law have explicitly written that the communist totalitarian ideologies should be treated basically on equal footing with the fascist ones and a significant portion of the relevant people – if not a majority – absolutely agrees that these two ideologies should be legally considered to be on equal footing and the left-right asymmetry in the application of the law is unfair. For example, Schinas' denial of the fact that Castro was a criminal against humanity isn't too different from the denial of the Holocaust. Different people have different opinions about the degree to which these two sick attitudes are "the same" but there's no doubt about the fact that they're considered comparable.

You know, I have used the carefully chosen word "comparable" on a November 21st, 1989 meeting of the leaders and vice-chairmen of the Socialist Youth Union of every class (our class picked the chairman and me, a non-member and a dissident) of our high school who were writing a petition on behalf of our high school reacting to the beating of the students in Prague 4 days earlier. This event was ironically led by these kid-communists and the adult in the room was Ms Morávková, our (rather young) history teacher and the chairwoman of the communist party at the high school (who was driving a Mazda just a few years later). Our demands were basically a filtered subset of demands written by the students of the University of Western Bohemia (the names were different in 1989). When we discussed the point whether the Lidové noviny (then a dissident newspapers) should be liberated, Morávková argued that it was unacceptable – after all, with such a bad tolerance, we could allow Nazi publications, too. I got excited and told her that the ideology of her party's Rudé právo (official communist daily) was comparable to the Nazi ideology. She kicked me out of the meeting, some colleagues have slapped me as a hero, but I am absolutely confident that at most two days later, what I have said on that day was considered common sense or mainstream. The totalitarian ideologies and regimes were at least comparable. After all, Stalin has killed more people than Hitler. And be sure that Castro has executed many more people than Pinochet, too. (And we don't even add the difference that while Castro has crippled the Cuban economy, Pinochet has amazingly optimized the Chilean economy.)

You may imagine how shocked we may be when a spokesman of the chairman of the EU commission not only fails to apologize for the quote that might pretty much place Jean-Claude Juncker in jail if this aßhole dared to insert his stinky foot on the Czech territory. Instead, Schinas has more or less attacked the journalist who just pointed out that Castro was a mega-criminal. No, we're really not on the same frequency.

In fact, I think that even the Czech – largely unreformed – communist party hasn't ever used these great words of praise as Juncker. When it comes to his evaluation of this history, Juncker is a more hardcore Stalinist than the unreformed Czech communists. Well, I have an even better example. Even Slavoj Žižek, an unpleasant hardcore Slovenian/global Marxist, has said that Castro must be forgotten as soon as possible, largely because he has only shown his ability to destroy things but when the normal socialist everyday life began in Cuba, he didn't contribute a damn thing that would be both innovative and positive. When the likes of Žižek don't even consider talking about Castro's "heroism", why the hell the boss of the EU commission does?

Also, Raúl Castro banned naming of the streets after Fidel, reflecting Fidel's own wish, to prevent the building of a cult of personality. Why do we have "leaders" like Juncker who want to build that cult instead of relatively moderate and sensible commies such as the Castro brothers? ;-)

Let me recall that in the U.S., I've met lots of people who wore a T-shirt with Che Guevara, a Bolshevik mass killer who has murdered over a thousand of people with his own hand. You can't really find adult people who would wear these T-shirts in Czechia, not even if you search among adult commies. A teenager may be deluded and worship this kind of criminal stuff and he may sometimes get away with it. But it's pretty much universally agreed that such a teenager is immature and politically and morally perverse.

For example, I've been meeting Marcos Mariño, a staunch Spanish Marxist and very smart string theorist, who has picked a Che Guevara T-shirt, too. Of course, I didn't really give a damn. We were basically friends. Long before I knew him, I knew that lots of people in the West are so ignorant that they openly worship some of the šittiest symbols of communism and mechanisms underlying this ideology – and who would pay thousands of dollars for the Lenin-style books and memorabilia that didn't have the value of a penny after 1989 and we were recycling them by tons. But one necessary condition for this amazing tolerance of mine was that I considered Marcos Mariño and similar folks politically irrelevant and retarded older kids.

Jean-Claude Juncker is not one of those. He basically considers himself a top (unelected) political representative of the European continent. We simply cannot ignore when such a person tramples upon moral restrictions that are widely considered red lines in at least several nations of the European Union. Fidel Castro was a criminal who has ended or crippled very many lives and one national economy, it's immoral to lionize him, and it's just OK for very many people – including Christians – to celebrate on the day when he died. Nature has restored an infinitesimal fraction of justice – the rest is being compensated in hell if you believe that such a place exists at least in "some sense".


text:
# Equation 20.16 Proof

In studying the Fundamentals of power electronics by Robert Erickson I do not understand how they arrived at the equations for current and voltage

Can someone help me understand how solving these differential eqn led to the solution below? I have started with applying Kirchoffs voltage law but not sure where to go from there. R0 is the tank characteristic impedance

Initial conditions and current/voltages eqns I am trying to derive

It would be appreciated if any one can help. If more clarification is needed or the book link does not work please let me know

Here is the work Ive done, I hope its clear! Ive chosen my initial conditions to be constants so I could understand better myself how to solve these equations

• How did R comes in solution ? What is it ? – user215805 Oct 12 '20 at 7:09
• It is reffered to as the characterisitic impedance of the tank circuit. It is equal to sqrt(L/C). Are you able to access the book? – Quintin Oct 12 '20 at 7:18
• Are $V_1$ and $I_2$ constants or sinusoids ? – AJN Oct 12 '20 at 13:30
• The problem looks to be a straight forward solving of differential equations ? Are you familiar with any methods for solving differential equations ? Have you tried out any method on this set of differential equations ? – AJN Oct 12 '20 at 13:31
• Can you show the result that you have got, including the steps so that we can find the mistake (if any). edit it into the question. – AJN Oct 12 '20 at 16:12

Your solution (to the homogeneous as well as particular) is correct. If we solve the differential equations as usual, we need to plug in the initial conditions at $$\t=0\$$ while the initial conditions are given at $$\t=\alpha/\omega_0\$$ in the book. That is why you seemed to have ended up with an unknown $$\(I_0 - I_2)\$$ in your coefficients and missing a $$\(-\alpha)\$$ in the argument to the $$\\sin()\$$ and $$\\cos()\$$.

To try to get it into the book form, either we need to somehow apply the initial condition at $$\t=\alpha/\omega_0\$$ instead of $$\t=0\$$.

Here is one way. Knowing the answer already, let's use that to our advantage. Re-write the homogeneous (and particular) solution in the following way:

$$\frac{d^2 i(t)}{dt} + \frac{i(t)}{LC} = 0$$

$$i(t) = A_1 \cos(\omega_0 t - \alpha) + A_2 \sin(\omega_0 t - \alpha)\\ \frac{d i(t)}{dt} = -A_1 \omega_0 \sin(\omega_0 t - \alpha) + A_2 \omega_0\cos(\omega_0 t - \alpha)\\ \frac{d^2 i(t)}{dt^2} = -A_1 \omega_0^2 \cos(\omega_0 t - \alpha) + A_2 \omega_0^2 \sin(\omega_0 t - \alpha)\\$$

This re-written solution also satisfies the original homogeneous differential equation

This makes it easy to apply the initial condition defined at $$\\omega_0 t = \alpha\$$

$$A_1 \cos(0) + A_2 \sin(0) + I_2 = I_2\\ A_1 = 0$$

This means that

$$v_2(t) = V_1 - L \frac{d i(t)}{dt}\\ v_2(t) = V_1 - L ((-0) \omega_0 \sin(\omega_0 t - \alpha) + A_2 \omega_0\cos(\omega_0 t - \alpha))$$

Applying the other initial condition here, we get

$$0= V_1 - L A_2 \omega_0\cos(0)\\ A_2 = V_1/(L / \sqrt{LC}) = V_1 / (\sqrt{L/C}) = V_1/R$$

## Note

This is not the only method, but, if we want to get the result into what the book has, then this method can be followed. Here, I skipped the re-writing of the particular solution since it was a constant, but it also should be re-written in a general case just like the homogeneous solution was re-written.

AJN nicely answered (+1) - I'm just putting the following as an answer so i can show detail and pictures.

I find that using Laplace to solve circuit problems algebraically is often times easier than wading through differential equations. I'm much better at algebra than d.e. For example, below i will solve this case for the inductor current:

Transform to Laplace

Now solve using any of our circuits methods/tricks (like source transformation, superposition etc.):

First i'll use source transformation on that current source,

Now solving for the current,

$$i_1(s) = \frac{{\frac{V}{s}+LI_2+\frac{I_2}{s^2C}}}{sL+\frac{1}{sC}}$$

I'll simplify the above and then use Heavyside's method to expand into partial fractions (to make inverse Laplace transformation easy):

$$i_1(s) = \frac{sCV+s^2CLI_2+I_2}{s^3CL+s}$$

$$i_1(s) = \frac{\frac{sV}{L}+s^2I_2+\frac{I_2}{LC}}{s(s^2+\frac{1}{LC})}$$

$$\frac{\frac{sV}{L}+s^2I_2+\frac{I_2}{LC}}{s(s^2+\frac{1}{LC})} = \frac{A}{s} + \frac{B}{s+j\frac{1}{\sqrt{LC}}}+\frac{C}{s-j\frac{1}{\sqrt{LC}}}$$

$$A = \left.\frac{\frac{sV}{L}+s^2I_2+\frac{I_2}{LC}}{s^2+\frac{1}{LC}}\right|_{s=0}\$$

$$A = I_2$$

$$B = \left.\frac{\frac{sV}{L}+s^2I_2+\frac{I_2}{LC}}{s(s-j\frac{1}{\sqrt{LC}})}\right|_{s=-\frac{1}{\sqrt{LC}}}\$$

$$B = \frac{jV\sqrt{LC}}{2L}$$

Since the denominator of the B & C fractions are complex conjugates of each other we know that, $$C = B^*$$

So, $$C = \frac{-jV\sqrt{LC}}{2L}$$

Now, we have expanded the equation for the inductor current out into forms we can easily inverse Laplace transform back to time functions,

$$i_1(s) = \frac{I_2}{s} + \frac{\frac{jV\sqrt{LC}}{2L}}{s+j\frac{1}{\sqrt{LC}}} + \frac{-\frac{jV\sqrt{LC}}{2L}}{s-j\frac{1}{\sqrt{LC}}}$$

Taking the inverse Laplace transform (or looking up in a table):

$$i_1(t)=I_2e^{-0t} + \frac{jV\sqrt{LC}}{2L}e^{\frac{-j}{\sqrt{LC}}t} - \frac{jV\sqrt{LC}}{2L}e^{\frac{+j}{\sqrt{LC}}t}$$

These last 2 terms we can combine thanks to Euler with this identity:

$$sin(x) = \frac{e^{jx}-e^{-jx}}{2j}$$

So,

$$i_1(t) = I_2 + \frac{V\sqrt{LC}}{L}sin(\frac{t}{\sqrt{LC}})$$

and since the characteristic impedance is, $$Z_0 = \sqrt{\frac{L}{C}}$$

we can finally write as,

$$i_1(t) = I_2 + \frac{V}{Z_0}sin(\frac{t}{\sqrt{LC}})$$

• Since the Laplace transform assumed that initial conditions are applied at t=0, while the book has them at $t=\alpha/\omega_0$, a time shifting will bring the argument inside the $\sin(\cdot)$ to the form $\omega_0 t - \alpha$. – AJN Oct 17 '20 at 2:12
• Yes, agree AJN. – relayman357 Oct 17 '20 at 3:13


text:
# [texhax] Problem with eqnarray

Sukomal Modak SUKOMAL at csiberkeley.com
Sun Apr 10 21:34:30 CEST 2005

Dear Modhumita,

> > \begin{eqnarray}
> >   f(x,y,z) & = & P \times \\
> >   \exp \bigl[ -\bigl( \frac{ \cos \theta \cos \phi (x - \mu_x) \\
> >                             + \sin \theta (y - \mu_y) \\
> >                    - \cos \theta \sin \phi (z -
> \mu_z)}{A_x} \bigr)^2 \\
> >             -\bigl(\frac{ \sin \theta \cos \phi (x - \mu_x) \\
> >                         + \cos \theta (y - \mu_y) \\
> >                   + \sin \theta \sin \phi (z - \mu_z)}{A_y}
> \bigr)^2 \\
> >   -\bigl( \frac{\sin \phi (x - \mu_x) + \cos \phi (z - \mu_z)}{A_z}
> > \bigr)^2 \bigr] \end{eqnarray}

You can try any of the following two options:

\usepackage[?]{amsmath}

\begin{eqnarray}
f(x,y,z) = P \times \exp \Bigg[
&-&\left(\frac{ \cos \theta \cos \phi (x - \mu_x)
+\sin \theta (y - \mu_y)
-\cos \theta \sin \phi (z - \mu_z)}
{A_x}
\right)^2 \nonumber \\
&-&\left(\frac{ \sin \theta \cos \phi (x - \mu_x)
+\cos \theta (y - \mu_y)
+\sin \theta \sin \phi (z - \mu_z)}
{A_y}
\right)^2 \nonumber \\
&-&\left(\frac{ \sin \phi (x - \mu_x)
+\cos \phi (z - \mu_z)}
{A_z}
\right)^2 \Bigg]
\end{eqnarray}

\begin{align}
f(x,y,z) = P \times \exp \Bigg[
&-\left(\frac{ \cos \theta \cos \phi (x - \mu_x)
+\sin \theta (y - \mu_y)
-\cos \theta \sin \phi (z - \mu_z)}{A_x}
\right)^2 \nonumber\\
&-\left(\frac{ \sin \theta \cos \phi (x - \mu_x)
+\cos \theta (y - \mu_y)
+\sin \theta \sin \phi (z - \mu_z)}{A_y}
\right)^2 \nonumber\\
&-\left(\frac{ \sin \phi (x - \mu_x)
+\cos \phi (z - \mu_z)}{A_z}
\right)^2 \Bigg] \nonumber
\end{align}

For the second option, you need to use the amsmath package with
\usepackage[?]{amsmath}. You can use multiple package as you need. For
example, AASTeX style package can also be used. I hope amsmath and
aastex are compatible (I have no real experience in using both of them
together).

If you include both the equations, you will be able to see the
difference in spacings before the minus signs. For the equation typeset
with align environment, the spacings are much better than that for plain
eqnarray. This is what Morten and Robin mentioned earlier.

You do not need to put \\ after every line in your input tex file in any
of the equation environment. You need to put this only where you want a
line break in the displayed equation in the output. The use of extra \\
after the group for \frac started with { and before closing the group
with } is confusing. Is it a real line break or a line break within the
numerator of frac? This is what Robin pointed out.

You can use \left( and \right) pair, or similar pairs, across lines.
That is why you were getting error messages. If you open an adjustable
size parenthesis with \left( and if you have to break the line before
closing it with \right), you can use a dummy closing using \right. . For
example:

\left(   blah blah \right. \\
\left.   blah blah \right)

However, sometimes this creates a problem in determining the appropriate
size of the parenthesis automatically. In such case you can use \bigl(,
\big(, \bigg(, \Big(, \Bigg(, etc. This is also mentioned by Morten.

You do not need any escape character to get [ bracket. A simple [ or
\left[ is good enough. \[ has a special meaning. However, you need \{ to
get braces in the equation, as { alone has a special meaning.

You should also consider giving extra spacing after \theta and \phi
using \, or \; to beautify the equations.

I suggest that you read the amsmath documentation. This will pay you big
time.

Hope this helps.

Regards,
--
Sukomal Modak

=================================
Sukomal Modak, Ph.D.

Structural Engineer (R&D)
Computers and Structures, Inc.
1995 University Ave., Suite 540
Berkeley, CA 94704

Tel: (510) 845-2177 (x 234)
Fax: (510) 845-4096
=================================


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# Big Data for Volatility vs.Trend

So different aspects of Big Data — in this case dense vs. tall — are of different value for different things.  Dense data promote accurate volatility estimation, and tall data promote accurate trend estimation.

More (No Hesitations blog)

# Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A ${\sigma}$-field ${\mathcal{F}}$ is a non-empty collection of subsets of the sample space ${\Omega}$ closed under the formation of complements and countable unions (or equivalently of countable intesections – note ${\bigcap_{i} A_i = (\bigcup_i A_i^c)^c}$). Hence ${\mathcal{F}}$ is a ${\sigma}$-field if

$1. {A^c \in \mathcal{F}}$ whenever ${A \in \mathcal{F}}$
$2. {\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}}$ whenever ${A_i \in \mathcal{F}, n \geq 1}$

Definition 2. Set functions and measures. Let ${S}$ be a set and ${\Sigma_0}$ be an algebra on ${S}$, and let ${\mu_0}$ be a non-negative set function

$\displaystyle \mu_0: \Sigma_0 \rightarrow [0, \infty]$

• ${\mu_0}$ is additive if ${\mu_0 (\varnothing) =0}$ and, for ${F,G \in \Sigma_0}$,

$\displaystyle F \cap G = \varnothing \qquad \Rightarrow \qquad \mu_0(F \cup G ) = \mu_0(F) + \mu_0(G)$

• The map ${\mu_0}$ is called countably additive (or ${\sigma}$-additive) if ${\mu (\varnothing)=0}$ and whenever ${(F_n: n \in \mathbb{N})}$ is a sequence of disjoint sets in ${\Sigma_0}$ with union ${F = \cup F_n}$ in ${\Sigma_0}$, then

$\displaystyle \mu_0 (F) = \sum_{n}\mu_0 (F_n)$

• Let ${(S, \Sigma)}$ be a measurable space, so that ${\Sigma}$ is a ${\sigma}$-algebra on ${S}$.
• A map $\displaystyle \mu: \Sigma \rightarrow [0,\infty].$ is called a measure on ${(S, \Sigma)}$ if ${\mu}$ is countable additive. The triple ${(S, \Sigma, \mu)}$ is called a measure space.
• The measure ${\mu}$ is called finite if

$\displaystyle \mu(S) < \infty,$

and ${\sigma}$finite if

${\exists \{S_n\} \in \Sigma}$, (${n \in \mathbb{N}}$) s.th.$\displaystyle \mu(S_n)< \infty, \forall n \in \mathbb{N} \text{ and } \cup S_n = S.$

• Measure ${\mu}$ is called a probability measure if $\displaystyle \mu(S) = 1,$ and ${(S, \Sigma, \mu)}$ is then called a probability triple.
• An element ${F}$ of ${\Sigma}$ is called ${\mu}$-null if ${\mu(F)=0}$.
• A statement ${\mathcal{S}}$ about points ${s}$ of ${\mathcal{S}}$ is said to hold almost everywhere (a.s.) if

$\displaystyle F \equiv \{ s: \mathcal{S}(s) \text{ is false} \} \in \Sigma \text{ and } \mu(F)=0.$

Continue reading “Very brief notes on measures: From σ-fields to Carathéodory’s Theorem”

# David Aldous’ review of the Black Swan

The phrase “Black Swan” (arising earlier in the different context of Popperian falsification) is here defined as an event characterized [p. xviii] by rarity, extreme impact, and retrospective (though not prospective) predictability, and Taleb’s thesis is that such events have much greater effect, in financial markets and the broader world of human affairs, than we usually suppose. The book is challenging to review because it requires considerable effort to separate the content from the style. The style is rambling and pugnacious—well described by one reviewer as “with few exceptions, the writers and professionals Taleb describes are knaves or fools, mostly fools. His writing is full of irrelevances, asides and colloquialisms, reading like the conversation of a raconteur rather than a tightly argued thesis”. And clearly this is perfectly deliberate. Such a book invites a review that reflects the reviewer’s opinions more than is customary in the Notices. My own overall reaction is that Taleb is sensible (going on prescient) in his discussion of financial markets and in some of his general philosophical thought but tends toward irrelevance or ridiculous exaggeration otherwise.

# Rio’s Inequality

Let ${X}$ and ${Y}$ be two integrable real-valued random variables and let ${ Q_x(u) = inf\{t: P(|X|>t) \leq u \}}$ be the quantile function of ${|X|}$. Then if ${Q_X Q_Y}$ is integrable over ${ (0,1)}$ we have

$\displaystyle \boxed{|Cov(X,Y)| \leq 2 \int\limits_{0}^{2a} Q_x(u) Q_Y(u) du}$

where ${ a= a(\sigma(X), \sigma(Y)) = \sup\limits_{\substack{B \in \mathcal{B} \\ C \in \mathcal{C}}} |Cov(\mathbb{I}_{\sigma(X)},\mathbb{I}_{\sigma(Y)})|}$ is the a-mixing coefficient.

Proof: Set ${X^{+} = sup(0,X)}$ and ${X^{-} = sup(0,-X)}$ then

$\displaystyle Cov(X,Y) = Cov(X^{+},Y^{+}) + Cov(X^{-},Y^{-}) - Cov(X^{+},Y^{-}) - Cov(X^{-},Y^{+})$

since ${ X = (X^{+} - X^{-})}$ and ${ Y = (Y^{+} - Y^{-})}$

note also that

$\displaystyle Cov(X^+,Y^+) = \int \int_{\mathbb{R}^{2}_{+}} [P(X>u, Y> \upsilon) - P(X>u)P(Y> \upsilon)]du d\upsilon$

which implies that

$\displaystyle |Cov(X^+,Y^+)| \leq \int \int_{\mathbb{R}^{2}_{+}} \inf (a, P(X>u),P(Y> \upsilon))du d\upsilon$

# Kolmogorov’s Maximal Inequality

1. Let ${X_1, X_2,...,X_n}$ be independent random variables with ${ \mathbb{E}[X_i]=0, \mathbb{E}[X_i^2]< \infty }$. Set ${S_n = \sum_{i=1}^{n} X_n}$. Then ${\forall \varepsilon > 0}$

$\displaystyle \boxed{ P \left( \max_{1 \leq k \leq n} |S_K| \geq \varepsilon \right) \leq \frac{\mathbb{E}[S_n^2]}{\varepsilon^2} }$

Proof: Let

$\displaystyle \begin{array}{rcl} A &\equiv& \{ \max_{1 \leq k \leq n} |S_k| \geq \varepsilon \} ,\\ A_k &\equiv& \{ |S_i| < \varepsilon, i=1,...,k-1,|S_k| \geq \varepsilon \}, \qquad 1 \leq k \leq n \end{array}$

Notice that ${\cup_{i=1}^{n}A_k = A}$ and


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Article Text

Technique for insertion of percutaneous central venous catheters in the newborn period
1. G Bayley
1. Bristol School of Anaesthesia, Bristol, UK
1. Correspondence to:
Dr Bayley, Department of Anaesthetics, Bristol Royal Infirmary, Marlborough Street, Bristol, UK;
kateandguy{at}hotmail.com

Statistics from Altmetric.com

The use of percutaneous central venous catheters is of proven value for the provision of parenteral nutrition and intravenous drug treatment in neonates. They have become an integral part of the management of very low birthweight infants in most intensive care units.

At the Royal Children’s Hospital in Melbourne we used a silastic catheter, which has an external diameter of 0.6 mm and comes in a variety of different lengths (Epicutaneo-cava catheter manufactured by Vygon; lengths 15, 30, and 50 cm; ref nos 2184.015, 2184.00, and 2184.005; cost AU$59.10). It is packaged with a metal 19 GA butterfly needle for use in insertion of the line. This technique has some drawbacks. 1. The 19 GA needle is difficult to put directly into neonatal veins because of its large size. 2. It can be difficult to appreciate “flash back” of blood into the metal needle. 3. It is not possible to “flush” the needle to ensure correct positioning of the line as well as patency of the vessel. 4. It is not feasible to place femoral venous lines using this method. We therefore use a method whereby the vein, using the Seldinger technique, is ultimately cannulated with a 20 GA catheter through which the silastic line can be inserted. 1. The procedure should be carried out under optimal conditions using an aseptic technique. If the infant is already ventilated, we advocate the use of a muscle relaxant as well as adequate sedation. This is especially advisable for insertion of femoral venous lines. 2. The vein is initially cannulated with a 24 GA (external diameter 0.7 mm) cannula. The sites most often used are the great saphenous vein at the ankle or knee joint, the femoral vein, the basilic or cephalic veins in the antecubital fossa, or, occasionally, the superficial temporal vein. A transilluminator or “cold light” inserted into the finger of a sterile glove can be of use in locating deep veins as well as protecting the sterile field. 3. A guidewire is then inserted through the cannula into the vein. We use a “duoflex spring wire guide”: diameter 0.45 mm, length 25 cm (duoflex spring wire guide manufactured by Arrow; product no AW-04018; cost AU$13.00; Insyte intravenous catheter manufactured by Becton Dickinson catheter; GA 24, 22, and 20; cost AU\$2.00). This has the advantage of having a soft tip at both ends of the wire and being a snug fit to the smallest catheter. Care must be taken not to advance the wire if any resistance is met.

4. A small nick is made in the skin at the site of wire to facilitate the insertion of the larger intravenous cannulae.

5. A 20 GA (external diameter 1.1 mm) cannula is then threaded over the wire into the vein (a 22 GA (external diameter 0.8 mm) can be used to dilate the vein before the larger cannula is inserted). This can be flushed with saline to ensure patency of the vein.

6. The silastic catheter can then be fed up the vein through the 20 GA cannula with a pair of toothless forceps. Occasionally the silastic line coils up in the hub of the cannula. This can be overcome by cutting the cannula flush to the hub and reinserting the silastic line.

7. The silastic catheter is placed to the required length and the other cannula is withdrawn.

8. The silastic catheter should be placed outside the cardiac outline in accordance with new guidelines.1–3 The position is always confirmed radiologically either by plain radiograph or, if necessary, by injection of radio-opaque dye. We have seen neonates with pericardial tamponade associated with malpositioned catheters, which has been well documented in the literature.1–3

We have found this method to be extremely reliable in the insertion of percutaneous venous catheters.

The use of the guidewire incurs additional costs (see above). In our experience these are partially offset by an improved success rate using the above method. We do not open the silastic catheter until the 20 GA is in place within the vein. This means that a line is not wasted if the vein cannot be cannulated.

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# MSc Thesis

During my MSc in Statistics at Imperial College I undertook a summer research project under the supervision of Dr Andrew Duncan. This project involved investigating the use of Maximum Mean Discrepancy (MMD) for parameter inference in generative models based on ODEs. In particular, an adjoint method for gradient descent in high-dimensional parameter spaces was studied.


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2014 — Regular Convex Bodies

• Introduction — Plan of the progam.
• Session 1 — Warm-up exercises.
• Session 2 — (No notes.) Vector notation. Diameter of high dimensional convex bodies. Stereographic projection.
• Session 3 — Metric spaces. Path metrics on graphs. Path metrics on the edge graphs of Platonic solids.
• Session 4 — Tiling problems. Soma Cubes. Simplicial complexes.
• Session 5 — Euler’s Formula. There can be at most five Platonic Solids.
• Session 6 — The Szilassi Polyhderon. Euler characteristic. There are dodecahedra.
• Session 7 — Groups. Symmetries. (Examples missing due to $\TeX$nical problems.)
• Session 7 — Announcement of final project. Rotational symmetries of the cube. How to read a math paper systematically. (No notes.)

Introductory remarks
The plan for the semester is an ambitious one. We’re going to understand the structure of all the regular convex polytopes in all dimensions, and build up a intuition for dimensions greater than three. We’ll spend most of our time learning the tools we need to understand how a geometric object can be pieced together. These tools will include vectors, metric spaces, symmetry groups, and simplicial complexes. I’m taking a very broad view of what constitutes a tool and counts as information about a space. The final result of the project will be a poster presentation about the solids and some 3-dimensional “nets” of the 4-dimemsional solids (the simplex, cube, cross-polytope, and 24-cell).

I can’t resist saying things about hyperbolic geometry. Therefore, if we get to the end of the proposed project, we’ll take a stab at using out high brow high dimensional intuition to understand the Gieseking manifold. There is more than enough stuff to say about the platonic solids, so we’ll see how far we get.

Session 1

The plan for the semester is to:

• Classify the regular convex polyhedra in all dimensions.
• Learn about negatively curved three dimensional spaces.

This is a slightly edited version of a set of exercises written by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston in 1991 as part of their course Geometry and the Imagination available here: http://goo.gl/7PwbZ7.

I’ve added several homework to the end: ${\circ_{\triangle\bigcirc\square}, \circ_{15}, \square_2, \square_3, \square_4, \circ_?}$. The second last question ${\square_4}$ is difficult, please attempt it, and we’ll discuss your attempts next week. Resist the urge to try and look up solutions. The last question ${\circ_?}$ is quite open ended. Please come up will several ideas and we’ll investigate them next week.

• Cut off each corner of a square, as far as the midpoints of the edges. What shape is left over? How can you re-assemble the four corners to make another square?
• Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What do you get?
• Take two squares. Place the second square centered over the first square but at a forty-five degree angle. What is the intersection of the two squares?
• Mark the sides of a square into thirds, and cut off each of its corners back to the marks. What does it look like?
• How many edges does a cube have?
• Take a wire frame which forms the edges of a cube. Trace out a closed path which goes exactly once through each corner.
• Take a ${3 \times 4}$ rectangular array of dots in the plane, and connect the dots vertically and horizontally. How many squares are enclosed?
• Find a closed path along the edges of the diagram above which visits each vertex exactly once? Can you do it for a ${3 \times 3}$ array of dots?
• How many different colors are required to color the faces of a cube so that no two adjacent faces have the same color?
• A tetrahedron is a pyramid with a triangular base. How many faces does it have? How many edges? How many vertices? How many different colors are required to color the faces so that no two adjacent faces have the same color?
• An octahedron is the shape formed by gluing together equilateral triangles four to a vertex. Balance it on a corner, and slice it halfway up. What shape is the slice?
• How many colors are required to color the faces of an octahedron so that faces which share an edge have different colors?
• Rest a tetrahedron on its base, and cut it halfway up. What shape is the smaller piece? What shapes are the faces of the larger pieces?
• Rest a tetrahedron so that it is balanced on one edge, and slice it horizontally halfway between its lowest edge and its highest edge. What shape is the slice?
• Cut off the corners of an equilateral triangle as far as the midpoints of its edges. What is left over?
• Cut off the corners of a tetrahedron as far as the midpoints of the edges. What shape is left over?
• You see the silhouette of a cube, viewed from the corner. What does it look like?
• Imagine a wire is shaped to go up one inch, right one inch, back one inch, up one inch, right one inch, back one inch, …. What does it look like, viewed from different perspectives?
• The game of tetris has pieces whose shapes are all the possible ways that four squares can be glued together along edges. Left-handed and right-handed forms are distinguished. What are the shapes, and how many are there?
• Someone is designing a three-dimensional tetris, and wants to use all possible shapes formed by gluing four cubes together. What are the shapes, and how many are there?
• Rest an octahedron on a face, so that another face is on top. Slice it halfway up. What shape is the slice?
• Take a ${3 \times 3 \times 3}$ array of dots in space, and connect them by edges up-and-down, left-and-right, and forward-and-back. Can you find a closed path which visits every dot but one exactly once? Every dot?
• Do the same for a ${4 \times 4 \times 4}$ array of dots, finding a closed path that visits every dot exactly once.
• ${\circ_{\triangle\bigcirc\square}}$ What three-dimensional solid has circular profile viewed from above, a square profile viewed from the front, and a triangular profile viewed from the side? Do these three profiles determine the three-dimensional shape?
• ${\circ_{15}}$ Find a path through edges of the dodecahedron which visits each vertex exactly once.
• ${\square_2}$ Consider a square with unit side length. Draw circles of radius one half centered at each vertex. Now inscribe the largest circle possible ${C}$ centered at the center of the square and tangent to the previous circles. What is the area of ${C}$?
• ${\square_3}$ Consider a cube with unit side length. Draw spheres of radius one half centered at each vertex. Now inscribe the largest sphere possible ${C}$ centered at the center of the square. What is the volume of ${C}$?
• ${\square_4}$ Consider a cube in four dimensional space with unit side length. Draw spheres of radius one half centered at each vertex. Now inscribe the largest sphere possible ${C}$ centered at the center of the square. What is the radius of ${C}$? Is there anything counter-intuitive about the number you get?
• ${\circ_?}$ Suppose you have been shrunken down to the size of an ant and are trapped on the surface of a paper tetrahedron, octohedron, or icosahedron, near a vertex, with only a device that with measures area (planimeter). Suppose the surface is vastly larger than yourself. How can you tell which surface you’re solid you are on? What measurements can you make?

Session 2
No notes were distributed. Roughly, introduced vectors and curvature around vertices on polyhedra. We looked at the problem ${\square_4}$ from Session 1. Diameter of ${[0,1]^n}$. Stereographic projection.

Session 3 — Graphs as metric spaces Today we’re going to look at some different metric spaces, and see how they behave under various maps. This is tangentialy to our project of classifying solids, but it is important material.

Definition 1 A metric space ${(X,d)}$ is a set ${X}$ and a distance function ${d : X \times X \rightarrow \mathbf{R}}$ such that:

• Positivity ${d(x,y) \geq 0}$.
• Non-degeneracy ${d(x,y) = 0 \Rightarrow x=y}$ .
• Reflexivity ${d(x,x) = 0}$.
• Symmetry ${d(x,y) = d(y,x)}$.
• Triangle Inequality ${d(x,z) \leq d(x,y) + d(y,z)}$.

The notion of a metric space axiomatizes our notion of distances should be measured. For example symmetry says that the distance from ${x}$ to ${y}$ should be the same as a the distance from ${y}$ to ${x}$. Only the last two items of the definition really have any intuitive geometry content. The other three are there for purely technical reasons.It’s important to point out that there are other ways of formalizing distance where some of the properties of a metric fail. These are different kinds of spaces.

Here are several different structures on ${\mathbf{R}^n}$.

• ${d_1((x_i), (y_i)) = \sqrt{\sum (x_i - y_i)^2 }}$ is the standard Euclidean metric on ${\mathbf{R}^n}$.
• ${d_2((x_i), (y_i)) = \max \{ |x_i - y_i| \}}$
• ${d_3((x_i), (y_i)) = \sum |x_i - y_i|}$

Question 2 For any metric space ${(X,d)}$ we have a notion of balls and spheres. A metric sphere is ${S(x,r) = \{y : d(x,y) = r\}}$. A metric ball is ${B(x,r) = \{y : d(x,y) \leq r\}}$. Draw various metric balls in ${(\mathbf{R}^n, d_i)}$ for ${n = 2,3}$ and ${i = 1, 2, 3}$.

Definition 3 Given a graph ${G = (V,E)}$ we get a natural metric on ${G}$ as follows: A path in ${G}$ from a vertex ${v_1}$ to ${v_2}$ is a sequence of vertices ${\tilde{v}_1, \dots, \tilde{v}_n}$ such that ${v_1 = \tilde{v}_1}$, ${v_2 = \tilde{v}_n}$, and there is an edge from ${\tilde{v}_i}$ to ${\tilde{v}_{i+1}}$ for each ${i = 1, 2, \dots, n-1}$. The length of a path ${\tilde{v}_1, \dots, \tilde{v}_n}$ is ${n}$. The path metric on ${G}$ is given by: ${d(v_1,v_2)}$ is the minimal length of a path from ${v_1}$ to ${v_2}$.

Question 4 Draw some metric balls in the edge graphs of the platonic solids. Graph the number of vertices contained in a ball.

Aside: We noted that once can see the central symmetry of a platonic solid in the growth rate of its metric spheres.

Session 4

We constructed paper models of the dodecahedron and the tiling of planes by regular heptagons.

4.1. Soma Cube Structure

Question 8 (2D Warm-up #1) Consider a chess board with two non-adjacent corners removed. Can you cover the remaining spaces with 3-piece ${L}$-shaped dominoes?

Question 9 (2D Warm-up #2) Consider a chess board with two non-adjacent corners removed. Can you cover the remaining spaces with ${1 \times 2}$ dominoes?

Question 10 (2D Warm-up #3) Can you tile a standard chess board with ${1 \times 2}$-dominoes such that no ${2 \times 2}$-square consists of two dominoes?

Problem #2 is due to Martin Gardner, who inspired countless youth to pursue mathematics through his popular column in Scientific American.

We’ll now explore some of the mathematics of the Soma Cube. We’ll use techniques similar to the colouring we used in the second warm-up exercise.

We’ll use the labellings of the pieces of the cube given here. We’ll name the pieces ${\mathbf{1}, \dots, \mathbf{7}}$.

Note that pieces ${\mathbf{2}}$ and ${\mathbf{3}}$ are somewhat special in that they contain a ${1 \times 1 \times 3}$ bar. This means that they can occupy either ${0,1,2}$ corners of the cube each.

Suppose piece ${\mathbf{2}}$ occupies no corners. Each piece not equal to ${\mathbf{3}}$ can occupy at most one corner, so we can occupy at most ${2 + 5 = 7}$ corners. Thus, ${\mathbf{2}}$ must occupy a corner.

Question 11 Perform a similar analysis to show where ${\mathbf{2}}$ must wind up. You will get, taking in to account the symmetries of the cube, two positions.

4.2. Simplicial complexes

A simplicial complex is the “higher dimensional” version of a graph. They’ll help us to rigorously describe the structure of the high dimensional platonic solids.

Definition 12 A simplicial complex ${X}$ is a set of subsets of ${V}$ such that: if ${A \subseteq B}$ and ${B \in X}$ then ${A \in X}$. The set ${V}$ consists of the vertices of ${X}$, the elements of ${X}$ are the faces of ${X}$, and the dimension of ${B \in X}$ is ${|B|-1}$. An ${n}$-face of ${X}$ is an ${n}$-dimensional face.

The key property of a simplicial complex is that the intersection of two faces of a simplicial complex is again a face.

Example 1 (Graphs)

$\displaystyle \triangle = \left\{ \emptyset, \{1\}, \{2\}, \{3\},\{1,2\}, \{2,3\},\{1,3\} \right\}$

$\displaystyle \square = \left\{ \emptyset, \{1\}, \{2\}, \{3\},\{4\}, \{1,2\}, \{2,3\},\{3,4\}, \{4,1\} \right\}$

Definition 13 Recall the metrics,

• ${d_2((x_i), (y_i)) = \max \{ |x_i - y_i| \}}$
• ${d_3((x_i), (y_i)) = \sum |x_i - y_i|}$

From now on, ${B(0,1,d_3) \subset {\mathbb R}^n}$ will be called the cube, and we will use the notation: ${\square^{n}}$. Similarly, ${B(0,1,d_3) \subset {\mathbb R}^n}$ will be called the cross-polytope, with notation ${\Diamond^{n}}$. We will also use the notation ${\square^n = [0,1]^n}$. Warning: This notation is non-standard.

Question 14 Give a description of the face adjacency structure of the tetrahedron, octohedron, and icosahedron.

Question 15 Give a description of the face adjacency structure of ${\Diamond^4}$.

Session 5 — Euler’s Formula and the Platonic Solids

Today we’re going to proceed to the classification of regular convex polyhedra in two dimension. We’re going to use this famous theorem of Euler:

Theorem 16 For any connected planar graph ${G = (V,E)}$ we have:

$\displaystyle |V| - |E| + |F| = 2$

where ${|V|, |E|, |F|}$ denote the numbers of vertices, edges, and faces of ${G}$ when ${G}$ is drawn in the plane. Note that we include the exterior’ face of ${G}$.

Below we write ${V, E, F}$ for ${|V|, |E|, |F|}$.

There are many proofs of this fact. See David Eppstein‘s collection of proofs here.

Note: There are well understood philosophical issues with this proof of Euler’s formula. See Lakatos. I feel that this is a very intuitive proof. You might disagree.

First embed ${G}$ in the plane. During the proof, we will perform a number of operations on ${G}$ which will: preserve the quantity ${V - E + F}$ and eventually lead to a familiar graph for which ${V-E+F}$ is easy to compute.

We will abuse notation and write ${V,E,F}$ for the vertices, edges, and faces of any graph that appears during our proof.

We proceed as follows:

Does ${G}$ have any cycles of length greater than three? If so, pick a vertex on the cycle, and add an edge to another vertex on the cycle. By the Jordan curve theorem (!) this new edge seperates a face. Thus, we add a face and an edge. Note that ${V - E + F}$ is preserved. Continue adding edges until there are no cycles of length greater than three.

We now simplify our graph. In the next two steps we preserve the fact that the boundary of our graph is a simple cycle. At each stage, one of two things may happen:

1. There is face which has only one edge of the boundary of ${G}$. If so, remove the boundary edge. This will remove an edge and a face. Note that this preserves ${V - E + F}$.
2. There is a face with two edges on the boundary of ${G}$. Remove the two boundary edges. This will: remove a vertex, two edges, and a face. Note that ${-1 - (-2) - 1 = 0}$, so again ${V - E + F}$ is preserved.

The final case is when there is a face with three edges on the boundary. Since we’ve ensured that the boundary remains simple, this implies that ${G}$ has been reduce to a triangle. For a triangle, we compute ${V - E + F = 3 - 3 + 2 = 2}$.

It is a subtle point that there is a sequence of moves as above which preserves the boundary.

We can now use Euler’s theorem to prove an upper bound on the number of convex regular polyhedra.

Theorem 17 There are at most five convex regular polyhedra.

Take any convex polyhedra, inscribe it in a sphere, radially project to the sphere, then stereographically project on to the plane from some point which is not a vertex. This will give the edge graph of the polyhedra in the plane.

We now show that regularity strongly constrains the kinds of graphs that can appear.

Regularity implies that each face is bounded by ${k}$ edges, and each vertex is incident to ${\ell}$ edges. We get, by double counting: ${kF = 2E}$ and ${\ell V = 2E}$. Euler’s theorem then gives:

$\displaystyle \frac{ 2E }{ \ell } - E + \frac{2E }{ k } = 2 \Rightarrow \frac{ 1 }{ \ell } - \frac{ 1 }{ 2 } + \frac{ 1 }{ k } = \frac{ 1 }{ E }$

Thus, we want to classify all ${E, k, \ell}$ satisfying:

$\displaystyle \frac{ 1 }{ k } + \frac{ 1 }{ \ell } = \frac{ 1 }{ 2 } + \frac{ 1 }{ E }$

Note that we need ${1/k + 1/\ell \geq 1/2}$ for this to be possible.

$\displaystyle \begin{array}{c|cccccc} & k=3 & 4 & 5 & 6 & 7\\ \hline \ell = 3 & 2/3 & 7/12 & 8/15 & 1/2 &\\ 4 & 7/12 & 1/2 & & & \\ 5 & 8/15 & & & & \\ 6 & 1/2 & & & & \\ 7 & & & & & \\ \end{array}$

We have noted all the solutions ${(k, \ell)}$ such that ${1/k + 1/\ell \geq 1/2}$. Moving down a column, or across a row, can only decrease the value of ${1/k + 1/\ell}$ so this table is complete. There are no solutions of ${1/k + 1/\ell = 1/2 + 1/E}$ where ${1/k+1/\ell = 1/2}$, so we are left with five candidate entries in our table.

Question 18 Write down all solutions of ${1/k + 1/\ell = 1/2 + 1/E}$ and write down which Platonic solids they correspond to.

Question 19 Draw the edge graphs of all the Platonic solids.

Question 20 Suppose Euler’s formula said: ${V - E + F = 0}$. Can you get platonic solid like objects by following the proof above?

Session 6

6.1. Szilassi

For a closed orientable genus ${g}$ surface ${\Sigma_G}$ we have, and any edge intersection free graph whose faces bound topological discs drawn on it, we have:

$\displaystyle \chi(\Sigma_g) = V - E + F = 2 - 2g$

If we take a genus one surface, a torus, we get: ${V - E + F = 0}$. This causes us to ask: are there toroidal equivalents of the Platonic solids? (Note: I’m using terminology very loosely here, what I mean to ask is: are there toroidial solids with a heigh degree of regularity in their face adjacency structures?) Following the proof from the previous session: suppose all the faces are bounded by ${k}$ edges, and each vertex meets ${\ell}$ edges. We get:

$\displaystyle 2E = kF \quad \textrm{and} \quad 2E = \ell V$

We then arrange:

$\displaystyle \frac{ 2E }{ \ell } - E + \frac{ 2E }{ k } = 0 \iff \frac{ 1 }{ \ell } + \frac{ 1 }{ k } = \frac{ 1 }{ 2 }$

We have previously classified all solutions ${1/k + 1/\ell = 1/2}$. They are as follows: ${(k,\ell) = (4,4), (3,6), (6,3)}$. This computation doesn’t tell us about ${(V,E,F)}$. It is interesting to note that all three of these solutions correspond directly to tilings of the plane by regular squares, triangles, and hexagons respectively.

Going back to our question about genus one polyhedra. It turns out that there is a miraculous polyhedron with irregular faces corresponding to the ${(k,\ell) = (6,3)}$ solution. It has a dual which is the ${(k,\ell) = (3,6)}$ solution. These are the Szilassi and Császár polyhedra. They have: ${(V,E,F) = (14,21,7)}$ and ${(V,E,F)=(7,21,14)}$.

One surprising property of the tetrahedron is that all of its faces are adjacent. Its edge graph, when embedded in the plane, shows us that we need at least four colours to the regions of any planar graph. The Szilassi polyhedron provides a higher genus analogue of this phenomena. Suppose a polyhedron has all of its faces adjacent. We have that ${F(F-1) = 2E}$. Also, in such a polyhedron, we’d have three edges at each vertex. Thus, ${3V = 2E}$. We’d then have:

$\displaystyle \begin{array}{lcc} V - E + F & = & 2 - 2g\\ \Rightarrow\frac{ F(F-1) }{ 3 } - \frac{ F(F-1) }{ 2 } + F & = & 2 - 2g\\ \end{array}$

Some further algebraic manipulation gives:

$\displaystyle \frac{ (F-3)(F-4) }{ 12 } = g$

For ${g = 1}$, we have the solution ${F = 7}$, which gives the Szilassi polyhedra.

Question 21 (Open) Is there a polyhedra of genus six with ${F=12}$ such that every face is adjacent to every other face?

It is curious that the lower bound for the chromatic number of of graphs on genus zero and one surfaces both come from polyhedra.

6.2. The Dodecahedron Exists

We give a concrete construction of a dodecahedron.

Consider a regular pentagon of side length one and vertices ${ABCDE}$. Consider the trapezoid ${ABCD}$. We have that ${AB=BC=CD=1}$ and ${AD = 1 + 2\sin\left( \frac{ \pi }{ 10 } \right)}$. Note that the length of ${AD}$ is the fabled “golden ratio”. Construct a square ${PQRS}$ with ${PQ=AD}$. Attach ${ABCD}$ to ${PQ}$ along ${AD}$ and another copy of ${ABCD=UVWX}$ by gluing ${UX}$ to ${RS}$. Now fold the trapezoids in until ${BD=VW}$.

Question 22 Calculate the sum of the dihedral angles at ${PQ}$ and ${QR}$.

One may piece together several of the solids constructe above to form a dodecahedron as details on CutOutFoldUp.com.

Session 7 — Groups and symmetries.

Groups are a vital component of modern mathematics. We’ll use them to describe the symmetries of the Platonic solids rigorously. They were initially invented to describe such symmetries, but they are interesting mathematical objects in themselves.

Definition 26 A group ${G}$ is a set with a map ${\cdot : G \times G \rightarrow G}$ and a distinguished element ${e \in G}$ such that:

• (Identity) For all ${g \in G}$ we have ${e \cdot g = g \cdot e = g}$.
• (Inverses) For all ${g \in G}$ there is ${g' \in G}$ such that ${g \cdot g' = g' \cdot g = e}$.
• (Associativity) For all ${g, h, k \in G}$ we have: ${g \cdot (h \cdot k) = (g \cdot h) \cdot k}$.
• Question 27 Show that there is a unique identity element in any group. Show that for any ${g \in G}$ there is a unique ${g' \in G}$ such that ${g \cdot g' = e}$. In every group ${(g \cdot h)' = h' \cdot g'}$. Give an example of a non-group.

Definition 28 We say that a map ${f : X \rightarrow Y}$ is bijective if:

• Injectvity If ${f(x) = f(y)}$ then ${x = y}$.
• Surjectivity For all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$.
• We usually work with ${S_{\{1, 2, \dots, n\}}}$, which we will abbreviate to ${S_n}$.

Cycle notation It is cumbersome to write down permutations, so we write down the following cycle notation’ instead. We write ${(x_1\ x_2\ \dots\ x_k)}$ for the map ${f(x_1) = x_2,\ f(x_2) = x_3,\ \dots ,\ f(x_k) = x_1}$. A cyclic permutation of two elements is a transposition. Try the following computations:

$\displaystyle (1 2)(3 2 4)(1 2) = (3 1 4) , \quad (1 2 3)(2 3) = (1 2), \quad (12)(23)(34) = (1234)$

Question 29 Show that every permutation can be written as a product of cycles. Show that every permuation can be written as a product of transpositions.

Also, try these computations:

$\displaystyle (12)(23) = (123) ,\ (23)(12) = (132)$

Definition 30 We say that a group ${G}$ is abelian (commutative) if: ${g \cdot h = h \cdot g}$ for all ${g, h \in G}$.

Groups arise as the symmetries of some object. A symmetry is broadly interpreted to mean `a transformation of an object which preserves its structure.’

Question 31 Describe the group of rotational symmetries of a tetrahedron. Describe the group of rotational symmetries which fix a cube.

Definition 32 Given a graph ${\Gamma = (V,E)}$ we say that a map ${f : V \rightarrow V}$ is an isomorphism of the graph if ${f}$ is bijective and ${\{u,v\} \in E}$ implies ${\{f(u), f(v)\} \in E}$.

Question 33 Describe the group of isomorphisms of a cyclic graph.


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# Here, Kitty-Kitty-Kitty-Kitty- Can you please tell me what is the logistic behind this article?

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Can you please help?

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##### Answers 2

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