it would be nice if applying to college were less like running for office. if you see what politicians do in the US it is mostly about what their parents did.
and all this weird narrative building that is unrelated to just, like, let's go to college.
i dunno. i'll stop talking before i talk myself into a ditch
your childrens essays should all have begun, "When my father (see Angela's Ashes) was cutting turf by hand in a muddy bog in his homeland, he" etc. etc.
fill in the rest
if they didn't begin that way, you failed as a parent
the weird thing is, i don't think the youth actually are infantalized in this country. i think universities are carrying out something that weirdly exports aspects of a kind of 1960s vision of what universities are supposed to be into the present, without any temporal adjustment
but it's still jarring and annoying and i'm glad i'm not paying for it this semester anyway :)
i hope they are still escorting them into the UCSC equivalent of the basement of evans with a room full of dial telephones, for purposes of dialing into the enrollment system.
my wife and i talk about this a lot, she works in higher education but we both basically regard it as failed and finished, as a societal project. she's got some interviews with non academic stuff this week and next where i have to cover child care.
when i cover child care i sometimes sneak our daughter into her zooms because everybody loves seeing a toddler on zoom. i won't do it for interviews but i will do it for meetings.
she's in day care M-F most of the day. covering is on 'staff development days.' you know, so they can catch up on what's changed in how to handle toddlers since 6 months ago.
so many aspects bother me. you are captive yet have to pay for parking. you cannot bring in food so you must eat their sub par expensive rubbish. every ride of interest has a 30 min line, so you can ride at most maybe 5 rides in a $100 visit. and you are treated on entry in a manner that makes tsa look like mother theresa
disneyland seems not to have started up fully. we live within the blast radius of disneyland (when they do fireworks, we hear it). we've raised our daughter unaware of disney. if she asks what the sound is we say "something somewhere blew up."
my grandfather had crimes on his record and still somehow got into boston college. must have been being catholic and in the right place at the right time
Evaluate the indefinite integral
$$\int\left(a^2-\frac{\sin {(a)}}{a^{2} + 1}-a \cos {(a)}\right)\,da$$
I tried a lot of integration strategies to solve this, but I can't find what will solve the derivative. Can you give me a hint on how to solve this?
Specifically, I am thinking that the fact th...
idk if that actually helps if it is true - just seems like multiplying by $\frac{a^2+1}{a^2+1}$ is overcomplicating, but i'm not too good at this so take my advice with a grain of salt
:58429237: copper, I meant that I didn't know that such a thing existed in South africa in particular. In some other parts of the world, I'm aware that it existed and still exists in some places.
@Koro a) i think the trailing colon breaks it b) you can just click the reply button on the message and then just type
@PRD Okay. Then sorry, I don't know how to integrate this, and also the WA answer is looking very messy (didn't even bother looking at the solution) so idk
i am definitely missing something since both answers seem to know the approach and are providing hints
Consider $(X,d)$ & $(Y,e)$ be metric spaces, $f:X\to Y$ be continious function & $A\in X$. Let $G$ be an open set in $Y$. Then by a theorem($f$ is contionious $iff$ $f^-1(G)$ is open in $X$ whenever $G$ is open in $Y$ ), $f^-1(G)$ is open in $X$. Now, are we eligible to write $f^-1(G)\cap A$ is open in $A$? If yes, then how, i.e. what i unable to understand
Is there an intuitive reason why the cup product of generators of a wedge sum are trivial? I am looking for example at $S^1\vee S^1\vee S^2$ and the dual generators of the first cohomology group which send a generator of the first homology group to 1 and the other to 0.
More precisely, why is the cup product between cohomology groups coming from different spaces trivial?
In abstract algebra, I have seen (some flavor of) the following "proof" several times when it comes to proving (not just "showing") Z[i]/<1-i> is a field:
$Z[i]/(i−1)=Z[x]/(x^2+1)/(x−1)=Z/(1^2+1)=F_2.$
I understand the isomorphism between Z[i] and Z[x]/<x^2 +1> because I don't get the daisy-chaining of factoring rings. Can anyone help me fill this knowledge gap and gain this understanding?
@stephenjfox it comes down to the following result: If $f : R \rightarrow R'$ is surjective homomorphism, it induces a one to one correspondence between ideals of $R$ containing $Ker(f)$ and ideals of $R'$, given by $J \leftrightarrow f(J)$
The important insight is that the ideal $(1-i)$ in $\mathbb{Z}[i]$ lifts to the ideal $(2,X-1)$ in $\mathbb{Z}[X]$. Then the composition of the surjections $\mathbb{Z}[X]\rightarrow\mathbb{Z}[i]$ (given by $X\mapsto i$) and $\mathbb{Z}[i]\rightarrow\mathbb{Z}[i]/(1-i)$ has kernel $(2,X-1)$, so induces an isomorphism $\mathbb{Z}[X]/(2,X-1)\rightarrow\mathbb{Z}[i]/(1-i)$. But $\mathbb{Z}[X]/(2,X-1)\cong\mathbb{F}_2[X]/(X-1)\cong\mathbb{F}_2$.
Any chain in a (sufficiently nice) wedge $X\wedge Y$ is homologous to a chain in $X$ plus a chain in $Y$. Now let $\alpha$ be a cocyle in $X$ and $\beta$ a cocyle in $Y$ and pull them back to cocycles $\tilde{\alpha},\tilde{\beta}$ on $X\wedge Y$. By additivity and the first observation, it suffices to check that $\tilde{\alpha}\cup\tilde{\beta}$ vanishes on every simplex lying either in $X$ or in $Y$, but these get killed by $\tilde{\beta}$ or $\tilde{\alpha}$ resp. since they get pushed down to points.
oh fuck I did the thing again where I act like \wedge is the wedge sum
I'm with you on the sloppiness / opacity of the matter. It's kind of driving me up the wall. So I thought I'd ask for a clarification. Examples of this answer are here (https://math.stackexchange.com/a/362901/445486) and here (https://math.stackexchange.com/a/1639652/445486). Neither define the isomorphism being leveraged
canon can be hard to grokk when it seems (at least) some of the point is being able to work through it from preceding definitions and theorems. Worse yet in a self-study or classroom environment.
@Thorgott So the factoring "over" "operator" is right-associative. That's good for future reference
Let's say I have a vector v, and I want to know the direction of this vector, say I got the direction, by this formula V/|V| and now I have 3 angles in X,Y,Z directions, is there a quick way to know if this vector is pointing in up,down,right,left,front,back direction, I remember it might have something to do with signs but I dont have visualization so i am working blind thus I want make sure i am using the right method
@stephenjfox yeah, these things are admittedly often more or less implicit. it's something you figure out once and then just roll with
@stephenjfox hmm, maybe I'm misunderstanding, but I don't think right-associative is the right term. what's going on is effectively a formal version of "cancelling fractions".
@Thorgott probably an overloaded term. In my recollection of programming language design / compiler programming, left- or right-associativity is used to describe where the implicit parentheses fall in some ambiguous computation. For example 1 + 5 / 2 == 1 + (5 / 2). My example shows how operator precedence (i.e. division trumps addition) assists the language in being right-associative (again, if I'm remembering my terms correctly)
its sorta right associative, the issue is $(a)/(\overline{b})$ doesn't really make sense in $R/(a)/(\overline{b})$ because $(a)$ isn't a ring in itself
Say I want to calculate the magnitude of a 3d vector $\vec v = (v_0, v_1, v_2)$. Of course the answer is $|\vec v| = \sqrt{v_0^2+v_1^2+v_2^2}$. However, is this expression numerically stable? Are there faster ways to do this while ensuring numerical stability?
this is the only formula I knew to calculate it, if you want it to be faster as in programming, Up to my knowledge the answer is no, there isn't much you can do other than optimize your implementation in the language of your choice.
This is already implemented in the source code, however there are some extra steps to improve numerical stability, which Im not sure if they are neccessery
strictly speaking e^c is some other positive constant
one time i TA'ed for a calculus professor who required us to deduct points for stuff like that. i thought it was too much, for a calculus class.
he was really into absolute values and keeping track of the signs of everything. to what end, i don't know.
he also required people to state the domains of the functions they were using in a change of variable, which is basically just a really annoying memorization exercise, in the case of trig functions, and totally unnecessary.
later editions, unless it's just to make more books and correct typos, seem not to improve anything. the mental image i have is an empty sink filling up with dirty dishes.
my wife's doing a job interview in the other room, so of course the neighbor is mowing his lawn and the other neighbor has people with a leaf blower going at the same time.
@Thorgott so, I have a function for the radius of rotation of a solid. what I can't work out is why the formula takes the shape $\pi \int R(x)^2 \text{dx}$, in terms of how it takes infinitely many slices of the bigger solid as we move along $R(x)$
@shin: In general, prove (using circum- and inscribed shapes) that if you slice with cross section $A(x)$, then the volume is $\int_a^b A(x)\,dx$, just as you do for areas.
Cylindrical shells are trickier than cross-sections. You really need Riemann sums for that.
Those changes make no difference. If you want a rigorous justification, do what I said. Otherwise, draw sketches and figure out the formula for the cross-sectional area. That's just basic, nothing about rigor.
Spivak is relevant only if you wanted rigor. If you want standard manipulations with moving axes, etc., he's not helpful.
When I taught second semester calculus, whether engineering style or with proofs, I would NEVER give more than half credit on these problems if students didn't include a sketch showing how they found the cross-sectional area or integrand for cylindrical shells. Of course, requiring them to do this basically enabled them to ace it.
@Koro That sounds like a multivariable multiple integral, not helpful.
Hi Ted! I remember the book from long time ago when I read it the first time.
There is some discussion of solids of revolution in the book but I don't really know what the question asker really wants i.e. how much solids of revolution so I suggested it
Ted, we had a subject called engineering drawing at college in which we had to draw top view etc. of objects. I got low grades in that. Because 1) my drawing sucked. and 2) I took too much time drawing arrows (with some 3:2 ratio)... And my instructors said they didn't really care about drawing only the concept.
No, you're not listening. I did not grade on the beauty or even accuracy of the art. I wanted to see a sketch of the revolution and of the cross-section, so that one can immediately read off the area of the cross-section.
great! I would want the same- the sketch of how one deduced something (cross -section for example for triple integrals) and not the beauty of the drawing. @Ted.
What was fun was to discover that some of my students could draw better pictures in multivariable than I. It made them very happy to come to the board and show them off. In one case, it totally changed the way I presented the problem the next day.
I got it! I need to prove that the surface area of the side of a disk-cross-section is identical to its volume, since we have the formula for area $\pi r^2$
they changed my course schedule without telling me. i learned what i was teaching the friday before class started via an email from a TA asking why i wasn't at the TA meeting. the what, now?
@Ted you would have given up hope if you had to assign multivariable in my math department. How should I put it, I would more accepted if I was a serial killer than thinking with pictures
People have accused me of being a formalist because I use/love differential forms. (No pun intended.) I really think there's a lot of geometry in them, although perhaps not so much geometry in the exterior derivative (except that Stokes's Theorem vindicates it).
i appealed to the editor, saying, the fact that there's this false result swimming around is even more reason to publish this paper. i don't know what to tell you
@leslie A colleague and I had an NSF grant rejected by a referee who claimed that our intended results were long ago in the Italian literature and dismissed us. He was wrong, of course.
There was unquestionably a bias against "old-style" hands-on algebraic geometry. Although when we merged into singularity theory, the bias became less as the technicality increased.
when i was house sitting for a geometer that we both know i found a book by an old italian and surprised myself by realizing i could actually understand it.
it wasn't hartshorne. i could actually turn the pages.
That's the precise reason (as of now) why I want to go into algebraic geometry. You have got multiple facets to it, you could think in terms of pictures, forego them and do the gritty local ring calculations, or just be Serre and put them together
i think this is kinda neat: Let $D \subset \mathbb{R}^n$ where $n > 1$ be convex, with nonempty interior. Suppose $C$ is a countable set. Then $D \setminus C$ is path connected, and for any two points $d_1,d_2 \in D \setminus C$, there is a path from $d_1$ to $d_2$ in $D \setminus C$ that is the concatenation of two straight line paths
i dropped my daughter off at school today (fairly rare for me) and it seemed like everyone knew her name. not just her teacher, but all of the teachers. and the guy who cut the lawn. i think she is difficult, and they tell stories about her on coffee breaks.
they do day care and K-6 at the school. there's no way that they know everybody's name.
So I have been reading "Introduction to Mathematical Logic" by Elliot Mendelson. But At somepoint , I feel like the amount of preliminary knowledge is beyond my scope. The question I was stumped at is "If the set of symbols of a consistent generalized theory K has cardinality Na , then K has a model of cardinality Na".
I thought I did not need the knowledge of cardinal and ordinal numbers for studying first order logic (so I did not learn it), but it seems like I do need it.I did study some set theory notation like union , intersection ,power set , function , relation etc .The problem is I NEVER studied cardinal and ordinal numbers so I am not sure how much theory of cardinals I would need to study the more "advanced" meta-theorems of FOL.
The author of the book also said in a footnote "Presupposed in parts of this section is a slender acquaintance with ordinal and cardinal numbers (see Chapter 4; or Kamke, 1950; or Sierpinski, 1958)” .So I have been trying to study “Theory of sets” by E.Kamke . It seems fine for me . It has things like cardinal numbers , order types , ordinal numbers , aleph numbers , transfinite induction (although someone said that “It is not wise to learn set theory from a book written over 70 years ago”).
My big doubt is that , whether or not “Theory of sets” would be enough to learn and understand FOL meta-theorems and their proofs involving transfinite cardinals (like Na) in Mendelsons logic textbook. What to do ? Should It be enough to learn from “Theory of sets” ? Or should I learn from an modern Naive set theory or axiomatic set theory textbook before going back to mendelson ? Also , what kind of preliminary the author expected by “slender acquaintance” in his footnote?
Well, one can learn lots of good mathematics without getting buried in mathematical logic. That said, Mendelsohn's book is used in lots of undergraduate math courses, so it really shouldn't be that bad.
i felt like a traitor to uc berkeley, ignoring all that logic and set theory. but i did it anyway. i never looked back. i would do it again the same way.
I'm reading Rudin's Real and Complex Analysis and in section 5.11 he makes the next assertion:
Put $g_n(t)=1$ if $D_n(t)\geq 0$, $g(t)=-1$ if $D_n(t)<0$. There exist $f_j\in C(T)$ such that $-1\leq f_j\leq 1$ and $f_j(t)\to g(t)$ for every $t$, as $j\to \infty$.
Here, $C(T)$ denotes the set...
@leslietownes My situation is a bit different . Everyone seems to lean things like real analysis , number theory , group theory but I have burried myself into learning some logic. I feel very jealous nowadays and feel like throwing away my logic textbook out the window and study the mouth watering group theory or real analysis.
