The triangle, for example, is the upper-case greek letter delta, which in calculus represents 'change of'. You might have heard of 'delta-T' with respect to 'change of time'.
In calculus, upper-case delta means 'change over a finite time' vs lower-case delta meaning 'instantaneous change'. The practical upshot, for example, is that the lower-case is the instantaneous rate-of-change at an instant in time, whereas the upper-case is the change over a whole time (e.g. the average rate of change per second for time = 0 seconds to time = 3 seconds).
If you are trying to grok this, I would suggest an introductory calculus or pre-calculus resource. It doesn't have to be a uni textbook - higher-level high school maths usually teaches this. In this particular case, the Khan Academy would be my recommendation because it is about the right level (we're not talking esoteric higher-level university knowledge here) and it is eminently accessable. For example, this link may be a good starter in this instance:
However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle":
t_c = 5 \middot 10^{-5} \sqrt( V / Dt )
... where the "D" I've used is where the triangle appears in the formula.
But that can't be it, because just two lines above it we have:
"For a pulse of width Dt, the critical time ..."
So that's stating that "Dt" is the width of the pulse, and should be thought of as a single term.
So maybe that's the wrong formula, or maybe it was just a bad example. So trying to be more helpful, the "triangle" is a Greek capital delta and means different things in different places. However, it is often used to mean "a small change in".
https://en.wikipedia.org/wiki/%CE%94T
FWIW ... at a glance I can't see where that result is derived, it appears simply to be stated without explanation. I might be wrong, I've not read the rest of the paper.
You and I both know that it reads as one term, but for someone unfamiliar with calculus but exposed to algebra they are drilled to understand separate graphemes as separate items, because the algebraic 'multiply' is so often implied, e.g. 3x = 3 * x as two individual 'things'.
I think there's merit in explaining the concept of delta representing change, because it's not obvious. For example, when I was taught the concept in school, my teacher explicitly started with doing a finite change with numbers, then representing it in terms of 'x' and 'y', then merged them into the delta symbol. That's a substantial intuitive stepping stone and I think it's pretty reasonable that someone may not find this immediately apparent.
But I think the problem isn't the specifics of the "Δ", it's the meta-problem of believing that symbols have a "one true meaning" instead of being defined by the scope.
I agree that explaining the delta notation would be helpful, but that's like giving someone a fish, or making them a fire. They are fed for one day, or warm for one night, it's the underlying misconceptions that need addressing so they can learn to fish and be fed, or set on fire and be warm, for the remainder of their life.
e: I also think, on reflection, that a signfigicant part of your ability to grok a new paper per your comments is your comfort in approaching these concepts due to your familiarity. Think of learning a new language - once you have a feel for it, you're likely more comfortable exploring new concepts within it, however when you're faced with it from the start you probably feel very lost and apprehensive.
I feel that understanding calculus is a fairly fundamental step in the 'language of maths', teaching that symbols don't necessarily represent numbers but can represent concepts (e.g. delta being change). This isn't something you encounter until then, but once you do you begin to understand the characters associated iwth integrals, matricies, etc. in a way that you may not have previously with algebra alone.
That's literally the whole problem: he sees two symbols.
But most will already be familiar with the family of goniometric functions such as sin and cos, there’s log and possibly exp and sqrt. There’s min and max; advanced math has inf and sup.
The important thing here is that "For a pulse of width Dt" is the definition of this variable, but this can be easily missed if you're not used to this naming convention.
This convention is used in a whole bunch of scientific fields, like quantum mechanics, chemistry, biology, mechanics, thermodynamics, etc.
It’s also very useful in how it relates to derivatives, which is a crucial concept in just about any kind of science you could care to mention.
So yes, there is a learning curve, but we write things this way for good reasons, most of the time.
Multiplication should be represented by a (thin) space in good typography, to avoid this sort of things. Not doing it is sloppy and invites misreading. Same with omitting parenthesis around a function’s argument most of the time (e.g. sin 2πθ instead of sin(2 π θ) ).
I have this same problem with programming, when I have to deal with code written by non-mathematicians. They tend to use all these stupid variables with more than one letter and that confuses the heck out of me.
For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.
> Maybe it's just a too advanced paper for my knowledge.
Maybe it is for now ... the point being that if you start at the beginning, chip away at it, search for terms on the 'net, read multiple times, try to work through it, and then ask people when you're really stuck, that's one way of making progress.
You can, instead, enroll in an on-line course, or night-school, and learn all this stuff from the ground up, but it will almost certainly take longer. Your knowledge would be better grounded and more secure, but learning how to read, investigate, search, work, then ask, is a far greater skill that "taking a course".
Others have answered your specific question about the delta symbol, but there are deeper processes/problems/questions here:
* Not all concepts or values or represented by a single glyph, sometimes there are multi-glyph "symbols", such as "Δt" in your example.
* When you see a symbol you don't recognise, read the surrounding text. The symbol will almost always be referenced or described.
* The notation isn't universal. Often it's an aid to your memory, to write in a succinct form the thing that has been described elsewhere.
* In these senses, it's very much a language more akin to natural languages than computer languages. The formulas are things used to express a meaning, not things to be executed.
* Specific questions about specific notation can be answered more directly, but to really get along with mathematical notation you need to "read like math" and not "read like a novel".
* None of this is correct, all of it is intended to give you a sense of how to make progress.
But as I say, immediately above the formula it says:
"For a pulse of width ∆t, the critical time ..."
So that really is saying exactly what that cluster of symbols means. There will be things like this everywhere as you read stuff. Things are rarely completely undefined, but you are expected to be reading along.
And you need to work. I just typed this into DDG:
"What does ∆t mean?"
The very first hit is this:
https://en.wikipedia.org/wiki/Delta_%28letter%29
That gives you a lot of context for what the symbol means, and this is the sort of thing you'll need to do. You need to stop, look at the thing you don't understand, read around in the nearby text, then type a question (or two, or three) into a search engine.
Please don't take this the wrong way. It is not meant to be demeaning, and it is not meant to be gatekeeping (quite the contrary!). But: If you do not know what a derivative is, then learning that that symbol means derivative (assuming that it does, I have not actually looked at what you link to) will help you next to nothing. OK, you'll have something to google, but if you don't already have some idea what that is, there is no way you will get through the paper that way.
I hope you take this as motivation to take the time to properly learn the fundamentals of mathematics (such as for example calculus for the topic of derivatives).
Let’s say you go on a journey, and the distance you’ve travelled so far is “x” and the time so far is “t”.
Then your average velocity since the beginning is x / t .
But, if you want to know your current velocity, that would be delta x divided by delta t .
The delta is usually used in a “limiting” sense - you can get a more accurate measurement of your velocity by measuring the change in x during a tiny time interval. The tinier the interval, the more accurate the estimate of current velocity.
What I’m talking about here is the first steps in learning differential calculus. You could look for that at kahnacademy.com. You might also benefit by looking at their “precalculus” courses.
Just keep plugging away at it, the concepts take awhile to seep in. Attaining mathematical maturity takes years.
Before I continued my maths learning after highschool (ie before UK A-levels) I learnt the Greek alphabet to make it easier to understand maths notations as I could 'voice' (internally) all the funny glyphs adopted from Greek.
At uni I learnt how to properly write an ampersand (for logic classes) and how to write Aleph and Beth (for pure maths, particularly transcendental numbers).
Some professors have a fondness for the more confusing Greek letters (lowercase xi, lowercase eta) ... is it n or eta, epsilon or xi, ...
That's the sense in which I'm using ∆, which is common, at least in the US.
One place to start is https://en.wikipedia.org/wiki/Impulse_response
Like in 300 BC today, there's no royal road to geometry.
https://en.wikipedia.org/wiki/List_of_common_physics_notatio...
1. You're reading a journal article. They will assume you know the notation not just of the broader discipline (e.g. physics/electrical engineering), but of the subdiscipline and at times the subsubdiscipline. Journal papers are explicitly written not to be easy to comprehend by beginners.[1] Notation will be only one problem you'll face.
2. As has been pointed out, this is not a mathematics paper. Mathematicians have their own notation, as do physicists and engineers. As I mentioned in the above bullet, they can have their own notation even in subdisciplines (e.g. circuit folks use "j" for the imaginary number, and semiconductor folks use "i"). There is a lot of overlap in notation amongst these parties, but you should never assume because you know one notation that you'll easily understand the math written by other fields.
3. Most introductory textbooks will explain the basic notation. Unfortunately, I often do find gaps where you go to higher level textbooks and they use notation that they don't explain (i.e. they assume you've seen it before), but is not covered in the prior textbooks.
4. Finally, sorry to say this, but "delta" (the triangle) for representing change is used in almost all sciences and engineering. It was heavily used in my high school as well. If you're struggling with this you really need to read some introductory textbooks in, say, physics.
[1] I'm not kidding. I've spent time in academia and I've complained how obtuse some articles are, and almost universally the response is "We write for other experts, not for new graduate students". One professor took pride at the fact that in his field, one can comprehend only about one page of a paper per day - and this coming from someone who is an expert. These people have issues.
If you're trying to get into signal processing, it'll involve calculus in complex numbers, and knowledge of that is often gained through plodding through proofs and exercises over and over.
There is a formula with a triangle and I don't get what's that about, for example.