I’m already aware of Laplace; I should have been more specific–are there cases in signal processing that involve the use of Ito Calculus? Is it a better method then Laplace in certain situations or not?
:smack: It’s so obvious now!
Ah but…although I “program” using functions in a Web setting, I am really just a journalist who ended up doing “computer science” stuff and therefore I did not come to programming via math like most folks. Luckilly I came to it through some nicely-written Visual Basic, which, is sort of “in English” and works for someone like me - someone who studied journalism because the school didn’t require me to take any math! 
(although they made us take Statistics, which flabbergasted us all)
So, I see what you are saying about functions and how your function examples can never spit out a few numbers (although they can get infinitely close). That last page had some good graphs.
You see, I just finished reading, for the second time, a historical fiction novel in which Newton, Leibniz and Principia Mathematica are important characters. And it pains me to know that they are so important but I don’t know why it is important because I don’t know what calculus is. 
OK you stated you are a PhD student in Math.
That alone is impressive to me.
Do you get job offers regularly? Are the government agencies actively recruiting mathematicians? Google? Illuminati? Wal-Mart? I picture a handful of government agencies/companies chomping at the bit to get a highly skilled mathematicians.
What do you intend to do with such a degree?
I am not questioning you so much as highly skilled mathematicians in general. Especially ones such as yourself. A personal answer would be nice but I am not exactly trying to dig into your life.
Maybe this question would be better aimed at a statistician, but here goes:
Supposing you have noisy information about a value at different times, and also noisy information about its first time derivative. How would you regress a prediction of either the value or the derivative as a function of time, using both sources of information?
To take a physical example, suppose you’re driving along and recording mileposts going by at various times and also recording what the speedometer says second by second, and then want the best estimate of what your location was as a function of time.
You’re looking at panel data, which is not something that I know a lot about. The references in the linked article may be of help.
How much of your graduate work involves writing programs on a computer? If so what type of programming language / computer do you use? Do you use something like Mathematica?
Many people I know get jobs in Academia, that is, a professorship at some university or college. This, like being a grad student, often means that one teaches and does research.
There are also industry jobs. A former fellow student works for Google, another for Boeing (I think). There’s also places like Microsoft or Bell Labs, and Wall Street, too.
It is oft said that the biggest employer of pure mathematicians in the US is the National Security Agency
Oh, and the academic job market is plenty fierce. Fellow students have put in 100 :eek: job applications and gotten 1 offer.
There’s no reason to think that I am highly skilled, as mathematicians go.
Absolutely none.
Though to typeset math (for example, to write my thesis), one uses Latex (pronounced lah-tek)
You may be conflating two concepts here:
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The fact that a functions spits out only one output for each input is the definition of a function .
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All that stuff about stuff getting close to ther stuff was my attempt to define the limit of a function (first paragraph and section labelled ‘motivation’ are relevant). Maybe this page describes the idea better.
Even though most people think of derivatives and integrals is what most people think of when they think of calculus, neither could have been developed (or can be properly explained) without the idea of a limit.
Oh, I noticed, it’s equally hard for me to explain chemistry without drawing. But it’s always worth a try to ask. I’ll give your references a try; I usually refrain from trying to read math because most of the stuff out there assumes that you understand what “the mirror image of E” means.
You mean sigma(Σ)? It means sum.
… why is all that stuff about Groups and Rings a part of Logic?
No, Rysto, the “exists” sign. “the mirror image of E.” Math was the only science class I had where teachers never bothered define their symbols, you had to decipher what each symbol meant from hearing the teacher read the thing out loud. I realize this is a problem with my teachers and not with mathematicians; we had a series of magistral lessons on “mathematics and phylosophy” from a college math professor and he was the only one who actually bothered define everything.
No, he means a backwards E, with the vertical line on the right of the horizontal strokes. It means “there exists”. E.g., “[backwards E]x . x^2 +3 = 5” means “There exists an x such that x^2 + 3 = 5”.
’
No, the mirror image of E would be ∃, or “there exists” .
Upside down A, or ∀, is “for every”/“For all”
They are the logical quantifiers, as in:
“for every number, there is a number greater than it”
can be written
“∀x, if x is a number, then ∃y such that y is a number and y>x”
A friend of mine was giving a review session for some class he was teaching. He was at the chalkboard with his 3 year old daughter on his hip. He’s all a-writing math, so she says, “daddy, that A is upside-down.”
Awwwwwww.
It’s not. That’s Algebra .
By “he” in my previous post, I may mean “she”. I’m afraid I can’t recall.
As for groups and rings being a part of logic, as Jamaika a jamaikaiaké said before me, I doubt most people look at it that way; most people put them in their own field of abstract algebra. However, a lot of abstract algebra can be seen to have a logical flavor. What is a group? It’s any structure satisfying a certain small set of axioms. What is a ring? Same thing, only the axioms are different, but of a similar type, in terms of stating various identities that should hold, involving the various constants and functions that are particular to that structure. And so on for various other algebraic structures. And to say that some property holds for all groups is simply to say that it can be proven from the group axioms. So this begins to show connections to logic already.
Then, if you want to study all algebraic structures, in full generality, that becomes a subject called universal algebra. And this is clearly a special subfield of the general study of the relationships between logical theories and the class of structures satisfying those theories, which is called model theory. The sort of logical theories studied in model theory can be composed of the simple sorts of identities that characterize groups, rings, etc., but often have more complex sorts of axioms as well. Clearly, model theory belongs in mathematical logic, and as it happens, work in model theory can often be tapped for results relevant down in group theory or field theory or such in particular.
So that’s possibly why you might see a lot of abstract algebra characterized as a part of or related to logic. I wouldn’t say that it’s conventional to look at things that way, though; while there are lots of connections, as I said, and one needs a certain familiarity with logic to do abstract algebra, most algebraists wouldn’t consider themselves to be logicians, and don’t have broad familiarity with work in logic in general.
First it’s a branch of Logic then it isn’t… guess I’ll stick with using math to figure out how much solvent to put in paint and leave the definitions to those of you who actually get them 
(joking, joking)
Thanks, indistinguishable, will you please stay? (I’m a she but no problem)
Is there such a thing as non-abstract algebra? Would that just be applied algebra, i.e., you usual “find the value of a” problems?
Ah, I see now, you may have been confused into thinking the kind of work done in set theory is the kind of work done in group theory and ring theory, possibly by these lines in the Wikipedia article on set theory: “Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.” Those lines don’t mean that set theory is actually about the structures of abstract algebra. Rather, set theory is a subject of its own, about mathematical objects called sets, and not much else. (What are sets and what can we study about them? Well, I’ll touch on that in a second). But, in the current mathematical climate, set theory plays something of a foundational role for the other branches of math. A common viewpoint is that all the objects/structures that “exist” in the mathematical universe can be considered as particular sets; if you want to assume that some kind of mathematical structure exists, you consider this equivalent to the assumption that some particular kind of corresponding set exists, and then you depend on the set theorists to tell you that, yes, it’s ok to use such a structure, because they’ve established that such a set exists (generally from a particular theory of assumptions about the set theoretical universe, called ZFC).
That is to say, people from other branches of math will use the concepts of set theory to make the definitions at the bottom of their field, and depend on various assertions from set theory to justify their assertions that certain types of objects or structures exist. For example, if I’m doing work with the real numbers, I can start by writing up various axioms about the real numbers that I’ll use in my work (there are infinitely many of them, they have a linear order, every bounded increasing sequence of reals has a least upper bound, etc.); however, the nagging question will remain “How do I know that there actually is some structure of numbers which satisfies all these axioms?”. Or I might want to know whether a particular kind of ring exists, and be unable to come up with any examples or proofs of impossibility on my own. To resolve these existence questions, then, I can turn my descriptions of structures into descriptions of sets, and then ask “Does there exist a set like this in the set theoretical universe?”. And that’s the sort of thing that the axioms of set theory are designed to help me decide (“Yes, ZFC says it does”, “No, ZFC says it doesn’t”, “Hm, well, ZFC doesn’t seem to be enough for us to tell you, but if we add these other commonly studied set theoretic assumptions, we can get an answer”). However, once I’ve finished seeing that, yes, the work I’m doing makes sense, because the structures I’m working with can be interpreted as things in set theory, I generally stop caring about the minutiae of the set theoretical universe, and get back to business dealing directly only with the various properties I asked my structure to have (once I see that there is some set which can be interpreted in some way as the real line, I’ll stop worrying about the details of this interpretation, and just think in terms of real numbers again, secure in the knowledge that they can be made sense of in terms of sets if needed. I’ll know that the logical consistency of my work, if I’m asked to justify it, follows from the logical consistency of the set theoretic assumptions I grounded it in. And the gold standard of mathematics is to ground things in ZFC, so if I can pull that off, there’s no question anymore as to the legitimacy of my work.).
In practice, “core” mathematicians (i.e., those who don’t work in logic) just have some intuition about what is and what isn’t allowed, and implicitly assume that the work they’re doing can be formalized in terms of set theory and legitimized by the rules of ZFC, without ever doing so explicitly. Every now and then, though, they may be unable to decide if some particular sort of object exists (e.g., is there a function from the reals to the reals with such and such a property?) and have to go running to the set theorists.
Now, for things like calculus, you can mostly set it up once and be happy for life: show that the structure of real numbers exists, that we can make sense of notions like an infinite sequence of reals, a function from reals to reals, etc., and then once you’ve implemented all this set theoretically, you very rarely have to ask about existence in the set theoretical universe again; you can mostly get by with just the informal understanding of what kind of ways to construct functions, etc., are legitimate. However, for abstract algebra, which is more directly concerned with looking at a wide variety of structures and trying to find ones with special properties, the connections to existence questions from set theory are perhaps more explicit. And I think that’s what the Wikipedia article was trying to say. But, still, for the day to day stuff, even the algebraists can get by on just their intuition about what kind of constructions are legitimate and which aren’t.
Ok, that was long and rambling, and fairly dense, and probably shot to hell any reputation I have as a man of clarity and useful explanation.
Oh, what are these “sets” that everybody keeps going on about? Well, basically, a set is a collection of objects. So, something like {3, my house, George Bush} is a set, a collection of three objects. Or the set of all natural numbers. But, like I said, there’s the catch that we take the point of view that nothing exists except the sets. So {3, my house, George Bush} isn’t the sort of thing the set theorists will study; they’ll look at things like … the empty set with nothing in it, the one-element set containing only the empty set, the two-element set containing both of those. And then more exotic, but really useful things, like the infinite set containing all the sets that can be built up finitely from the empty set, or the even bigger infinite set containing all the sets whose members are drawn from that first infinite set. And so forth. There’s a bunch of rules in ZFC asserting that new and more exotic sets can be built up in various ways starting from the empty set (and also asserting that certain kinds of sets don’t exist at all), and what set theorists do is study the consequences of these rules for the nature of the set theoretical universe, and the consequences of various additions to these rules.
I could make a longer, more explanatory post about set theory and ZFC later, but that should be enough for now.
I probably will, though, after that rambling and dense explanation of the foundational role of set theory, you may no longer want me to.
Sure, there are three big things that get called algebra. There’s normal high-school type algebra (elementary algebra), but nobody really does research work on that, that’s all known and settled. Then there’s abstract algebra, which is some sort of generalization of parts of elementary algebra; instead of being told the various identities that are legitimate for symbolic manipulation, and then using them to reduce or solve equations, you rather choose a bunch of function and constant names and what identities among them you want to consider legitimate, and then look at what properties follows from those identities, and what kind of structures can be made to have those identities. And then there’s the third big thing, linear algebra, which is concerned with vector spaces and linear transforms (which causes them to be concerned with work on matrices). This comes up a lot in physics, naturally, and permeates both work on relativity and work on quantum mechanics.
I’m taking Pre-Calc right now. This is the second time I’ve taken it (took it before in high school). I’m still in school and I seem to GET what I’m supposed to do. I get help when I need it, I study before tests and quizzes. But when I get the test or quiz and I start doing the problems, my mind goes blank and I suddenly have no idea what to do.
Same thing happens when I don’t see a type of problem for a couple days. It might as well be in Chinese because I don’t get it at all.