Well, the value of the stock for people who essentially do not have any meaningful control of the business must essentially be tied to the expectation of some liquidity event down the line -- future cash flows. So this could come in the form of dividends, sale of the stock, bankruptcy proceedings, or a purchase of the business.
If I knew for certain (big if) that a business would never have a liquidity event and I couldn't transfer my ownership then it's dead capital for all intents and purposes and you could consider its value essentially $0, right?
And you can sell your tulip. But if the mania stopped and you suddenly _couldn’t_ find another person to sell it to, would you now be upset you paid $5000 for a tulip? What’s the value at which you wouldn’t be upset? Ok, that’s the intrinsic value of a tulip to you.
The thing about a profitable business that is different from a tulip is that it can at any point decide to issue a one-time or ongoing dividend. It can sell off parts to create cash. It has lots of optionality. Public companies have even more liquidity, which creates more optionality.
Even if you don't have immediate liquidity, it would obviously be worth something to have a slice of e.g. Rolex SA. That's obviously different than owning a tulip.
Dividends are also one way of income in retirement, much more predictably than selling stock. The yields are worse than bonds, but they can be considered to be mostly to rise with inflation, albeit on a year or two delay. Dividends also act as a discipline to keep management focused on the business, since you need to pay real money to shareholders, instead of just doing whatever good idea you have, regardless of whether it is a net benefit to the company.
I disagree with the last statement. The reason why most companies in the US have at least a nominal (one penny per share) dividend is that many pension funds have a requirement to only hold shares that issue dividends. Pension funds are all tax sheltered, so they don't need to worry about paying taxes on dividends. For retail investors, dividends are mostly worse that share buy backs. Why? Dividends are taxed, and the money needs to be reinvested.
> The only reason to do a dividend is because people like the feels of getting a cash payout.
Not really, when capital entities came up, the initial goal was to deliver return on invested capital,i.e. something "you get out of the business/back".
Or do you think back in 14th wenn Dutch East India Company was created, that you could by shares and sell them later to a higher bidder after the mission was accomplished? :-)
I would actually go so far to say as I am not aware of any good "as it really works" references. Handbooks exist, but they're pretty expensive.
Since you have an EE background, I would recommend a few strategies (in any order, except 0 should be first if you have major deficiencies):
0) Brush up on some of your math if you need to. Linear algebra (just up to Eigenvector/Eigenvalue), vector calculus, differential equations. Mostly just understanding the concepts is OK, because the major derivations for RF engineering are relatively simple problems. That said, RF engineering is just one big love letter to linear algebra.
1) Read Pozar, as another commenter have suggested, but you don't need to cover-to-cover it. You absolutely must know some network theory, the basics of transmission lines (characteristic impedance, propagation, loaded driving point impedance), and simple matching techniques (basic RF design is about 75% just making sure power goes where you want it to). Beyond that you can pick and choose depending precisely on what you're doing.
2) Read older papers (1940s-1980s depending on topic) on whatever you're interested in. They're going to assume relatively little starting information. The only caveats are that notation has changed and that a lot of the design techniques, while still valid, were more useful when simulators weren't readily available (i.e. they assume a really strong mathematical background).
3) Stay low frequency as much as possible early on. <6 GHz for sure, ideally lower. This makes things a lot cheaper (metrology, components) and makes mechanical tolerances less critical. Stuff just gets less "fiddly". There's of course a tradeoff where things start to get pretty big at low (10-100's MHz) frequencies.
4) Tear apart anything you can get your hands on -- broken metrology equipment for one. Try and figure out why people are doing what they're doing. Just because a system's cheap doesn't mean they aren't using some cool tricks.
Focusing in on "grabbing references", it's as easy as drag-and-drop if you use Zotero. It can copy/paste references in BibTeX format. You can even customize it through the BetterBibTeX extension.
If you're not a Zotero user, I can't recommend it enough.
I have a terrible memory for details, I'll admit an LLM I can just tell "Find that paper by X's group on Method That Does This And That" and finds me the paper is enticing. I say this because I abandoned Zotero once the list of refs became large enough that I could never find anything quickly.
To be maximally pedantic, sine waves (or complex exponentials through Euler's formula), ARE special because they're the eigenfunctions of linear time-invariant systems. For anybody reading this without a linear algebra background, this just means using sine waves often makes your math a lot less disgusting when representing a broad class of useful mathematical models.
Which to your point: You're absolutely correct that you can use a bunch of different sets of functions for your decomposition. Linear algebra just says that you might as well use the most convenient one!
>They're eigenfunctions of linear time-invariant systems
For someone reading this with only a calculus background, an example of this is that you get back a sine (times a constant) if you differentiate it twice, i.e. d^2/dt^2 sin(nt) = -n^2 sin(nt). Put technically, sines/cosines are eigenfunctions of the second derivative operator. This turns out to be really convenient for a lot of physical problems (e.g. wave/diffusion equations).
Interesting, I somewhat of an opposite reaction, although I am certainly not a mathematician. Once everything became definitions, my eyes glazed over - in most cases the rationale for the definitions was not clear and the definitions appeared over-complicated.
It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about maximal abstraction or addressing odd pathological corner cases. This allows me to wade through the complexity without getting overwhelmed like I used to.
My dad always told me growing up today math was like a game and a puzzle, and I hated that. I also hated math at the time. It felt more like torture than a game.
I didn't fall in love with math until Statistics, Discrete Math, Set Theory and Logic.
It was the realization that math is a language that can be used to describe all the patterns of real world, and help cut through bullshit and reckon real truths about the world.
Not the original poster, but I want to push back on one thing -- being capable of something and being one of the best in the world at something are hugely different. Forgive me if I'm putting words in your math -- you mentioned "placing the bar for mathematical skill pretty" low but also mentioned running a sub-10s 100m. If, correspondingly, your notion of mathematical success is being Terence Tao, then I envy your ambition.
I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".
On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.
Five or six years ago my family started went through all the old recipes - from old newspapers, cookbooks, etc. that were in homes across my extended family. They then decided on which to keep, and printed a new cookbook from the compilation of these recipes.
Now if we find (or author) a recipe that we really like, we send it, with any additional annotations, to my parents so that they can include it in the next print edition. It's a relatively time-intensive and expensive process, but from this point forward we should be able to maintain our family's recipes in a physical, living document form.
Maybe we don't get the yellowed pages and flour from grandma's hands on the cover, but I think it's a good system.
I've been doing this, but on my personal blog ( https://www.bbkane.com/recipes/ ). I'm really glad I got to get some of these from my grandma before she passed, and it's been a huge hit to just send a link when someone wants a recipe.
The 600-6GHz range is a rough approximation for some of the most used bands in telecommunications, e.g. Wi-Fi and 5G NR FR1. It's worth noting that the article explicitly mentions that this filter will be useful for FR3, which is "7 GHz to 24 GHz". They do not claim full 600 MHz-6 GHz operation, and as the previous poster noted, the filter was demonstrated from 3.4-11.1 GHz.
More critically: you want to be very very careful about trying to extrapolate this filter down to lower frequencies. We're dealing with "weird physics" here. I am not an expert on spin-wave devices by any means, but a guy in my lab during grad school was working with them, so I do know that the resonant frequencies of the spin-waves are a function of the magnetic bias and the material. The researchers here are tuning the filter by tuning the magnetic bias. Someone more knowledgeable can correct me, but I believe YIG would have trouble propagating spin-waves down at 600 MHz, and so this kind of filter would not be practical.
That's true, Laplace corresponds to a basis of complex exponentials that can grow or decay in time instead purely imaginary exponentials. We restrict the Ae^[(a+jb)t] domain just to Ae^(jbt) for Fourier.
From an circuit analysis standpoint (your problem may be different), but exponentials that decay over time ("a" is negative) corresponds to loss in a circuit, whereas exponentials that grow over time ("a" positive) correspond to something blowing up (this is really a nonphysical result but generally means a circuit is going to oscillate on its own, without a source driving that response). I mostly do electromagnetics/passive RF types of problems, in which you generally want everything to be low-loss. In that case Fourier is perfect, especially since I typically care most about steady-state behavior.
I'm surprised you're one of the only commenters to bring this up. I have an electrical engineering background -- for analysis, lots of systems are assumed to be either linear or very weakly nonlinear, and a lot of our signals are roughly periodic. Fourier transforms are a no-brainer.
Convolution turns into multiplication, differentiation wrt time of the complex exponential turns into multiplication by j*omega. I don't know about you, but I'd rather do multiplication than convolution and time derivatives.
As a corollary, once you accept "we use the Fourier representation because it's convenient for a specific set of common scenarios", the use of any other mathematical transform shouldn't be too surprising (for other problems).
If I knew for certain (big if) that a business would never have a liquidity event and I couldn't transfer my ownership then it's dead capital for all intents and purposes and you could consider its value essentially $0, right?
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