@ILikeMathematics no because what actually matters is, is your map a group homomorphism
and the answer is, yes
the multiplication is irrelevant in this context as an operation on $\mathbb{Z}$
including it, say $(\mathbb{Z}, +, \cdot)$, would indicate you actually want your map to be a (non-unital) ring homomorphism
listing operations means that those are the operations that need to be preserved
If you had a map between $(A, f_1, ..., f_n)$ and $(B, g_1, ..., g_n)$ then that would indicate that you have a map from $A$ to $B$ which maps operation $f_k$ to operation $g_k$
say, if $f_k$ is an $m$-ary operation, $T:A\to B$ is your map, then you want $g_k$ to also be $m$-ary, and to $T(f_k(a_1, ..., a_m)) = g_k(T(a_1), ..., T(a_m))$ where $a_i\in A$
but listing all the operations, I don't think that's really that popular, unless there is not a lot of them, and in some popular subject like rings or groups
so you may usually say, this map $T$ is a group homomorphism, or ring homomorphism, and so on
Suppose that I have to functions whose range is the set of real numbers. Call these functions $f(x,y)$ and $g(x,y)$, I want to find a solution of this system of equations. By a solution, I mean $(x_0,y_0)$ such that $f(x_0,y_0) = 0$ and $g(x_0,y_0) = 0$. Is there a way to determine if a system of...
Hi everyone. Can someone help me with the interpretation of this numerical
Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack.
. Find the probability that the patient followed a course of meditation and yoga?
Here if I want to take out probability of the patient having a heart attack provided he followed the yoga and meditation course what would it be? The question says this course reduces the risk by 30% so does it mean theres still a risk of 70%? Does that make any sense?
I have observed that across all math textbooks, no one really provides a clear, step-by-step process on how to graph the three reciprocal functions. They often avoid discussing vertical and horizontal stretches and compressions. I have noticed this with my own students and wish to change it.
I am about to send my textbook chapter for grammar revision before releasing it to be seen by many other students around the world. Would anyone here be willing to assist me in proofreading my textbook chapter? This is a very important task, and even a small change could improve it significantly. In exchange, you would gain experience, my personal gratitude, and the right to share this document however you wish.
The grammar doesn't have to be checked only the mathematical content
(alsoe note that chatgpt won't help with this task sicne he snuggle himself to go on a route different from the ones of the textbooks)
I have observed that across all math textbooks, no one really provides a clear, step-by-step process on how to graph the three reciprocal functions. They often avoid discussing vertical and horizontal stretches and compressions. I have noticed this with my own students and wish to change it.
I am...
@FedericoRuck Honestly, I'm not a big fan of your exposition. When I teach this material, I like to go back to working with sets in the plane. A function is a set of ordered pairs, e.g. $$f = \{ (x,y) : y = f(x) \} $$ (note the abuse of notation). Elementary transformations can be described by working directly with the ordered pairs.
For example, it is, I think, pretty clear that the set $g = \{(x+1,y) : (x,y)\in f\}$ is the same set as $f$, but translated to the right by $1$ unit. But if $y = f(x)$, a quick change of variables (take $x'=x+1$, then $x=x'-1$) gives that $g(x) = f(x-1)$.
@think_meaning_buildß This is how I present the idea to other mathematicians. I am a lot more careful with students.
That is, for any real number $x$, there is at most one ordered pair in $f$ which has $x$ as a first coordinate.
A relation is just a set of ordered pairs.
After building some familiarity working with functions and relations as sets, I introduce the idea that a function can be viewed as a kind of "arrow" which maps things in the domain to things in the codomain (which is what I think you mean by a "correspondence"), but that is not typically how I introduce the topic.
To talk about a reciprocal function, given a function $f$, define $$1/f = \{ (x,1/y) : (x,y)\in f\}.$$ Observe that $1/1 = 1$, that if $|t| > 1$, then $1/|t| < 1$ (and the bigger $|t|$ is, the closer to zero $|t|$ is), and if $|t|<1$, then $1/|t| > 1$ (and the smaller $|t|$ is, the bigger $1/|t|$ is).
And if $(x,0)\in f$, then $x$ is not in the domain of $1/f$.
@think_meaning_buildß Sure. The definition I gave above is the vertical line test, once you draw the picture.
In any event, the above is really all you need in order to sketch any reciprocal function. The first examples are $x \mapsto 1/x$ and $x \mapsto 1/x^2$. Then, months later, reciprocal trig functions are no biggie.
@pie you will quickly understand where the name comes from once you look into it
I think it's a great name, honestly :)
as to whether it qualifies as a field: what qualifies as a subfield of mathematics depends on how you feel on any given day, but as a topic it's big enough to warrant having a 1000-page book just written on its foundations link.springer.com/book/10.1007/978-3-031-56334-8
but certainly it's not a field on the level of "number theory", or "algebraic topology", or "analysis"
Let $F\in BV$ be real-valued, and denote by $T_F(x)$ the total variation function of $F$. Then we can write $F=\frac12 (T_F+F)-\frac12(T_F-F)$, where $\frac12 (T_F\pm F)$ are increasing. Now, Folland claims $$\frac12 (T_F\pm F)=\sup\left\{\sum_1^n [F(x_j)-F(x_{j-1})]^\pm :x_0<\cdots<x_n=x\right\}\mp\frac12 F(-\infty),$$where $x^+=\max(x,0)=\frac12 (|x|+x)$ and $x^-=\max(-x,0)=\frac12(|x|-x)$. Could somebody help me verify this equality?
Attempt: Let's just assume $\frac12 (T_F+F)$. Then $$[F(x_j)-F(x_{j-1})]^+=\frac12 |F(x_j)-F(x_{j-1})|+\frac12 (F(x_j)-F(x_{j-1})).$$Now, $\sum_1^n (F(x_j)-F(x_{j-1}))=F(x)-F(x_0)$. But this is where I'm stuck. How do I simplify $$\sup\left\{\frac12\sum_1^n |F(x_j)-F(x_{j-1})|+\frac12 (F(x)-F(x_0)):x_0<\cdots<x_n=x\right\}?$$
Man, my small handful of PhD theorems seem trivial. My dissertation feels like it'd get me a Master's at best. I don't know whether it's imposter syndrome or what. My supervisor thinks I'll prove a conjecture soon but I doubt it heavily. I have until this time next year of supervision, maximum.
@Shaun It definitely sounds like an imposter syndrome. Just continue on what you're doing, you'll prove something you could be proud of, or you won't. Not a tragedy either way
> My supervisor thinks I'll prove a conjecture soon but I doubt it heavily.
Thanks, @Jakobian. The last time I proved something new was three or so weeks ago, after months of getting nowhere. At this rate, I just won't have enough.
BTW may I ask how one can get a master's degree? I know I need to have a CV to apply, but aside from my grades, what else should I do? Do I need to solve a conjecture or something to be accepted into higher education?
@Jakobian I feel like this is an issue with your supervisor tbh, not with you. They're putting unnecessary pressure on you by saying that. You'll explore whatever you are working on further, and maybe you will prove the conjecture you want. Maybe not. Either way, it might be pretty useful to do
Contact who you think might supervise your dissertation, express an honest interest, and perhaps they'll vouch for you in the application process, @pie
pie it really depends on the program. literally every program is different
or maybe not, there may be some countries or regions where the process is so systematized that a bunch of programs within that country or region use basically the same process
but there is no generally applicable answer to this question at the level you have asked it
@Shaun That's it? I thought I would need to write a few books, solve a few conjectures, and publish some research to get accepted. Does this apply to people from outside my country or is it a country specific thing?
To write a book you ideally want to have really good knowledge of a field, and I meant it as, very throughout knowledge, not just confidence that you do
I have a conjecture that a particular function of groups takes on a value almost always for $\operatorname{SL}_2(q)$. It's based off of GAP calculations for $2\le q\le 59$ and what we know of other groups the function applies to. I have proven lower bounds that agree with my conjecture exactly, two thirds of the time; and I have a technique for proving an upper bound that hasn't worked yet.
@Jakobian I can't explain how relieved I feel right now. I thought I had to do all of that in the next three and a half years, and it was stressing me out.
Another question would be: Do you guys work while studying for this degree, or are you part of a scholarship program or something? And if you have to work, how can you find a job in the field of math with only a bachelor's degree?
My first attempt at a PhD was funded by a scholarship, @pie. But then I fell ill with schizophrenia and the pandemic happened, so I left with an MPhil. Now my current attempt is funded by a government doctoral loan that has very favourable repayment plans, like not have to repay if I'm earning below a certain amount. I tutor for the department and train AI to do mathematics.
personally, I don't earn a living. generally, regular work. there's also specific arrangements called 'work student jobs', tailored for these purposes.
@nickbros123 Hmm... that would be hard for someone like me since I plan to be an international student (studying outside my country because the education in math here is poor)."
One thing my supervisor said was that, in one's first year, all the preliminary material is learnt; in the second, small theorems appear; and in the final year (of a UK PhD), the "big" theorems show up in time for the writing up.
Sometimes I get to the point where I just . . . I just want to know the &$^#ing answer to my research question and not have to bang my head against a wall, trying to find it.
@Thorgott Yeah there's a misunderstanding. All in all, including tuition, food, hostel, I have to pay close to 100k rs. I get 80k rs from the government, so I have to put 20k of my own money, extra. That's ~200 usd
I get benefits from the English government because I have schizophrenia. It's a lot. It can be pretty debilitating to live with sometimes, so it helps with costs; like with my avolition symptom, sometimes cooking feels like climbing a mountain, so I get takeaways or eat out, so it comes in handy. Normal jobs would be difficult unless they have the flexibility of a research degree, again because of the avolition. This time last year, I was getting out of bed at 16:30 most days.
@nickbros123 Thanks. It got scary a few times. But it's been over 9 months since I heard a voice, the longest it's been so far. They used to accuse me of horrible things, some of which taboo and others extreme, like murdering someone. At first, I thought it was the police interrogating me.
It felt real.
I tried to flee the country once, thinking the voices were real. I almost dropped £1000+ on a last minute ticket to New York from London. I must have looked suspicious, because security intervened. The next thing I knew, I was in a mental hospital.
Let $NBV=\{F\in BV:F\text{ is right continuous and }F(-\infty)=0\}$. Here $BV$ is the space of functions of bounded variation. Then it is claimed that $G(x)=F(x+)-F(-\infty)$ is in $NBV$, where $F(x+)=\lim_{t\searrow x}F(t)$. I see why $G\in BV$ and why it is right continuous, but why is $G(-\infty)=0$?
Why isn't there a site that lists the best-known algorithms for approximating functions? I think it would be a great idea to create something like this, but I’m not sure who to approach about it. Maybe this community could build something like it? I posted about this idea long ago here.
there are few websites in mathematics of this kind
as for "could this community build something like it", I think the question you linked answers that to completion; it's as much as you're going to get out of the "community"
of course, if you want a website like this it's primarily your job to build it, and then maybe people will come and participate, or maybe not
and that's why there's few such websites out there :)
pie: even within pure mathematics there is no universal notion of 'best for approximating functions', right? or even 'approximating.' like, a ton of choices go into variations on what you might want or mean by that. and even if you fix the pure math choices, the "best" implementation of a specific algorithm in specific software or hardware usually depend not just on the math but on the software and hardware.
you're basically asking "why don't places exist on the internet for discussing math, and algorithms, and their implementations" well, they do. just not in a simple form of like, here's the problem, and here's the top 10 list of ways to solve that problem
@psie I think I understand why. $F(-\infty)$ is the limit of $F$ as we approach $-\infty$ from the right. $F(x+)$ denotes the limit at $x$ as we approach $x$ from the right. Now, if we "plug in" $x=-\infty$ in $G(x)$, we should get $0$, since $F(-\infty+)=F(-\infty)$. I don't know what's a more appropriate way of showing this.
one reason this stuff isn't kept in list form is because the more precise it is at the level of problem specification, the more quickly it is likely to become irrelevant to anybody as there are changes in software and hardware and the problems people care about
at the high level you have stuff like 'numerical recipes in c' and other books and resources as people were commenting on in that post you linked. harder to go lower level than that without running into stuff with a very short shelf life
even a lot of stuff in that example book is not particularly relevant if, for example, you aren't working in C, or more generally if you have different processing or storage considerations than the ones that formed a kind of background understanding of what scientific computing had or needed at the time that book was written
I'm struggling to understand the statement of Theorem 2.18 in Basic Algebraic Geometry I (Shafarevich), page 125. In particular, it's not particularly clear to me what an "affine open set" $X'$ is, or what $k[X']$ should mean.
It differs from laziness or a simple lack of motivation. Often, I want to do things and there's stuff I can do alright, like my studies or volunteering here, but with stuff I have to do, there's a huge barrier in the way. Does that sound familiar?
It's everyday things, like tidying up, cooking, getting out of bed, socialising, washing, laundry, etc., that I can't do so well.
well, my intuition says $\lim_{x\to -\infty}F(x+) =F(-\infty)=\inf_{x\in\mathbb R}F(x)$, but the definition of the expression $\lim_{x\to -\infty}F(x+)$ is $\inf_{x\in\mathbb R}\inf_{t>x}F(t)$
right, I should have probably said $F$ should be increasing. So my intuition is correct, that's great to hear :) yeah, it's that last equality that I'm investigating
