So it is worthwhile to meditate thoroughly upon the choice of axioms to ensure that they are wholly true and fully consistent so as to not produce any contradictions.
Doing algebra inherently accepts the underlying axioms underlying the system and is working out further consequences. There's nothing different in kind between solving x+2=4 and determining in a given philosophical system whether one shouls sacrifice themselves for the greater good
This conversation reminds me of a Pol-sci class I was in (feeling like a complete alien) and there were basically 4 or 5 political nerds just arguing/debating over who knows what
@AMDG Have you studied non Euclidean geometries? We have multiple different geometries, that say very different things about the way shapes work. The evidence of our own eyes wasn't enough to detect that Euclidean was wrong until the 19th/20th century. Yet we still teach it to this day
I'm actually just getting over a really, really bad day/week, and relaxing with idle chatter. Anyone who thinks I"m actually arguing/debating in the sense of thinking I am going to convince anyone else...well, I'm no :)
@Alan I'm going to assume you meant "should" and not "souls" or "one's souls". This is axiomatically false: it is very plain that thinking about whether one should sacrifice himself for the greater good, and solving x + 2 = 4, are in fact two different levels of abstraction, the former in fact being more abstract, and the latter, less so.
But yeah, we teach each new generation that the sum of the interior angles of a triangle are 180 degrees, even though our current understanding of reality says this is false.
Ahh, sorry about that, I honestly am too out of focus to have really processed what you meant by fixing a typo. Sitting at a bar at red lobster, idly picking away at breadcrumbs and downing water as I blissfully ignore the stress of a huge academic dishonesty flare up in my doctoral program (of Education)
@Alan It is true within the Euclidean system. It is not true outside of it. The proposition's truth itself has not changed per se. It is true, but the extent to which it is true or can be predicated has been more defined.
Sadly, my "We have to throw out the negative time solution to the quadratic equation for "when does the projectile hit the ground" joke answer of "Unless we have a Delorean with a flux capacitator." gets less and less people recognizing it....today I had 0
In other words, the reality was more clearly defined and modeled, e.g., upon observing that a triangle in non-Euclidean geometry (according to that geometry's species of triangle), if treated as the same triangle in Euclidean geometry, the proposition fails.
@ペガサスSeiya Hmm. Yeah, I'm not familiar with Ohio's status in popular culture. I've been cursed to live in Florida for most of my life, though at least the relatively sane southeast part. So Florida Man stuff all over the place."
Actually, idleness pertains to action that is done due to the absence of purpose, but to act for the sake of some purpose therefore means the act is not idle.
@Alan so the next time you see something extreme, make an Ohio meme that fits it. For example let's say you eat some really awful tasting food. Well, then you'll say, "Tastiest meal in Ohio"
I used to be active in the tabletop board game/rpg/wargaming community, that kept me up to date with memes across generations. Alas no time now between teaching an overload schedule, actually being in a relationship, and taking an accelerated doctoral program in a social science.... BLeh. Speaking of which, I've taken my 90 minutes or so off I really need to go finish that lit review on "Expulsion policies in Preschools in America""
I'm struggling with isomorphisms. Say if I have an operator $\hat{A}$ with eigenfunctions $v_i \in W$ that have distinct eigenvalues $E_i \in E$, how do I construct an isomorphism from $W$ to the space of functions $F$ on $E$. Can I just say, $$T : \sum_i c_i v_i -> \sum_i [c_i]\{\hat{A}\} v_i =\sum_i [c_i]\{\ E_i\} v_i$$ and then I state that everything in curly brackets goes to the argument of the function $F$ i.e. $F(E_i)$ and everything in the square brackets goes in front i.e. $c F(E_i)$?
So we are left with $$T:\sum_i c_i v_i -> \sum_i c_i F(E_i)$$
@DIRAC1930 NOt quite following the setting, is $E$ just your collection of eigenvalues, or something else? What is the space $W$ that this thing is operating on??
it might make more sense to go the other way. given a complex valued function f on E, you might want to associate to it the operator sum_i f(E_i) P_i, where P_i is orthogonal projection onto v_i. this associates a set of complex valued functions on E with a set of operators on W, such that the function f(z) = z is sent to A, f(z) = z^2 is sent to A^2, and so on - the image of a function f roughly corresponds to what you'd want "f(A)" to be.
strictly speaking, eigenfunctions (or any other elements of W) are not in the image of this map, because this map is operator-valued, not vector-valued. but the projections onto the spans of the eigenfunctions are in the image of this map. P_i is the image of the function that takes the value 1 at E_i and zero everywhere else on E
the reason why i say it might make sense to go this way is that it's maybe difficult or not the point, at wherever you are in this, to identify the range of this map, and there are definitely operators on W that aren't in the range. so it's an "isomorphism" onto something, but you'd maybe need to work to find out what.
but its inverse (on whatever domain that it has one) would be a map from some set of operators on W, to complex functions on E. i'm suppressing a lot of details.
Perhaps I should use a different example. Can I describe a map to be anything? e.g. can I have a map from the tuple $(1,2)$ such that $T:(1,2) \rightarrow x^1 + y^2$ for some polynomial. Is it enough to say that it invertible just by inspection i.e. I look at the power of $x$ and see that it is $1$ so I put this in the first entry $(1,)$ and similarly for the power of $y$ to get $(1,2)$.
she had a good valentines day, although 3 kids are currently out with covid.
dirac, this sometimes depends on context. physicists (pardon me for assuming that you might be interested in physics) love writing down formulas for maps that do not immediately make their properties obvious, and saying "this map has [all of these properties]" without comment. or with a "proof" that might be some symbolic calculation that, were a math person to glance at it, might be found to assume some of the properties that the "proof" is trying to establish.
but generally speaking you are free to define maps however you like, as long as you are sensitive to things like domains and issues of "well definedness" when they come up.
were you ever able to sustain above 8 hours of math throughout the week? (i happened to write this after leslie's answer but I meant this as a chat question)
so when i "defined" a map up above, for example, you might ask, well, if the set of eigenvalues is infinite, that recipe is asking me to form infinite sums of operators, and why or in what sense can i expect those infinite sums to make sense.
and i just talked about 'functions' on E, do i mean arbitrary functions? do i care about continuity, if E has points that accumulate at another point of E? does any of this affect whether the map i wrote down is one-to-one?
experienced physics folks are very good at choosing the appropriate level of worrying about this kind of stuff, or not worrying about it, but at the beginning it can be very difficult to follow when details matter and when they don't.
dc: i was about to say, "8 hours of math per week? sure." :D
well, it depends on what you mean by 'hours of math.' i would count within that reading, and thinking about stuff, talking to other people, attending lectures or seminars, etc. it's very easy to hit 8 hours that way.
lol at the beginning of the semester i think about trying to do more problems from the book :P but then the psets themselves as well as other coursework (as well as math not being my primary interest) happens
programs vary in how much coursework they expect people to do, and courses vary in how much they expect people to do as 'homework' (whether graded/evaluated in any way or not). outside of a 'homework assignment' type environment, i would agree that it is difficult to put in 8 hours of proofwriting or calculating. you run out of stuff to 'do' pretty quickly. which isn't to say that you run out of math to think about and work on.
is there a genuine difference in the terminology "onto" and "surjective" and "one-to-one" and "injective"? or if not why are there so many redundant terms :P
silly: no, and blame the french. although maybe don't blame them, because 'onto' and 'one-to-one' are pretty bad-sounding phrases for what they represent.
side note, "one-to-one correspondence" is used in some books, mostly older ones, for what would now more commonly be called a "bijection."
So if I have two eigenfunctions $f_i$ that span the whole space $W$ that have one distinct eigenvalue each of the operator $A$ labelled by $a_i$, am I free just to define an isomorphism $T: f_i \rightarrow F(a_i)=x^{a_i}$ and then make it linear by enforcing $T: c f_i \rightarrow F(a_i)=c x^{a_i}$?
if f_1 and f_2 (if you have two of them lets just call them that) span W. if they're also linearly independent (that is, a basis for W), you are generally free to define a linear map T from W into any vector space by arbitrarily deciding what T(f_1) and T(f_2) are going to be, and declaring more generally that T(c_1 f_1 + c_2 f_2) will be c_1 T(f_1) + c_2 T(f_2).
it isn't immediately clear to me from your notation what vector space your T is intended to send the f_i into, or what F(a_i) or x^{a_i} are, but if that's what you're doing, yes, you absolutely can just define T on f_1 and f_2 and "make it linear" that way.
this gets a little subtler if your basis for W is infinite, but not in a way that physicists often keep track of, i am sorry to say. (the issue is that not all lists of scalar coefficients will map to well-defined elements of your space, and you need to check that you're defining T(f_i) so that the infinite sum c_i T(f_i) makes sense as an element of your target space whenever sum c_i f_i makes sense in your domain)
i don't get that at all. it's a reasonable clarifying question to ask, in particular, to see if the OP has spent any time thinking about the problem and maybe what vocabulary/terms they are familiar with.
i could see an annoyed OP reacting like that, but to have some random high rep user pop in and begin doing that is baffling to me.
george bergman was still using troff when i was in grad school but seems to have switched to latex in his retirement.
@TedShifrin speaking of... i can be very naive at times. i was having a drink with a gay couple in Guerneville some time ago, a bar called the rainbow. after a while i surreptitiously leaned over to Jim asn said "i think there is a guy over there checking me out..." Jim shook his head knowingly and said "Joe, you know we ar ein a gay bar, right?"
she was otherwise normal. i tried to tell her, listen, if you want to say that rich people rule the world and that they're arseholes, just do marxism. she would answer that sure, marxism was for the theoreticians in their ivory towers, but conspiracy theory was the tangible, intuitive understanding of society. the marxism wasn't necessary if you had the "flesh" which had been abstracted by university theoreticians
@robjohn Yes but we don't know anything about $h^x(z)$. Then how can we estimate the green function. Here the domain $\Omega$ is arbitrary. If the domain was upper half space or open ball then we know $h^x(z)$ so we can do the estimate.
Anyway I think we need to apply maximum principle to argue that $h^x(z)\geq0$. So $v(z)\leq \Phi(z-x)$ and then we can use the bound for $\Phi$.
How is green function defined for unit balls? What is $h^0$?
Long ago, I studied a theorem in limits of functions, I don't remember correctly but it was something like this: If $f(x)=p(x)q(x)$ is a function such that $p(x)$ is infinitesimal as $x\longrightarrow a$, then $\lim_{x\longrightarrow a}p(x)q(x)=0.$ Is this true in general?
(Even if say, $q(x)\longrightarrow \infty$, then will the same thing hold as well?)
Belly Dancing?...there is no belly dancing......I can understand the objectifying part, even though it is relatively tame. It's really more so the name that is provacative
@TedShifrin Thanks 😊 ! I found it given as a statement in the book Problems in Calculus of One Variable by I.A Maronon pg-69 which states:"The product of an infinitesimal and a bounded function, is an infinitesimal". This is given as a property of infinitesimal functions !
the domain is connected (the reason is subtle: remove a point from R^2, still it is connected; remove a line from R^3 should keep the connectedness too. The explanation involves fundamental group, which I am yet to do a revision of).
f should be from $\mathbb R^2-\{(x,y): xy=0\}$ to Y.
so the domain is not connected
In fact, $Y\simeq \mathbb R^2-\{(x,y): xy=0\}$ via $f$.
So Y is not connected.
W is connected because it is the union of three planes xy-plane, yz-plane, zx-plane and each of these is connected, and has the point (0,0,0) in common so the union is connected.
Thinking about X though. This contains diamonds with holes.
@Jakobian Ohh, I think I understand now what you mean.
A (Hamel) basis of a vector space $V$ is a set of vectors $B\subseteq V$ such that for any $x\in V$ there is a finite amount of distinct $x_1, ..., x_m\in B$ such that there is a unique numbers $a_1, ..., a_m\in\mathbb{K}$ with $x = a_1x_1+...+a_mx_m$. Here $\mathbb{K}$ is a field, for example $\mathbb{R}$ or $\mathbb{C}$
@Koro The book I'm reading is Robot Modeling and Control Second Edition by Mark W. Spong. Every popular robotics textbook. I'm not sure if this definition is accepted by mathematicians.
In the linear algebra book that I've used, the fact that the coordinate representation is unique was proven as a proposition and was not in the definition
But I propose the following revision to Jakobian's definition: If $V\ne \{0\}$, then the definition by Jakobian holds, for $V=\{0\}$, B is defined as the emptyset $\emptyset$.
My knowledge of linear algebra is limited to elementary linear algebra textbooks where bases are not defined abstractly. Good to see different formal definition.
in linear algebra we can have a lot of different way to fix a coordinate system
and you can probably see already why this is a very important concept - both to us and engineers, and physicists etc. etc.
for infinite-dimension this is not quite a choice of coordinates... because we have to interpret this a little differently. But that's irrelevant here I believe
usually its even important in what order do we pick the vectors!
Can anyone explain the literal meaning that a limit doesnot exist? I know this sounds insanely stupid...but I always find whenver a limit at a point of a function to be infinity, the authors claim the limit does not exist! Is this the sole meaning of this seemingly ambiguous terminology?
I understand that in no way, we can calculate the limit of sinx as x tends to infinity, but is this notion only possible to define in these sort of situations?
@Semiclassical so when do we say it does not exist ?
The case of sin(x) is an illustrative one. If we limit ourselves to multiples of pi, then sin(x) is always zero. If we pick a different sequence, we can stick to sin(x)=+1
