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2
votes
2answers
103 views

Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
6
votes
0answers
79 views

Is everyone intellectually capable of doing higher order maths? [closed]

Maths like differential equations, calculus and very much more. At what point does the math one take start to be limited by intellectual capability? How do I know if I am a math person?
3
votes
1answer
25 views

Intuition for projection along a compact space being closed

Can anyone give me an intuition for why projection along a compact space is a closed map (and in fact this characterizes compact spaces)? In other words, $$\pi: K \times X \to X \,\,\text{ is closed ...
2
votes
2answers
468 views

Stochastic Processes Solution manuals.

Does anyone have a link or a pdf stash of solution manuals for stochastic processes ebooks? I am doing a self-study on this course and I can't seem to find any solution manual online to cross-check ...
0
votes
1answer
381 views

Convergence of a sequence in trigonometric functions

Is the sequence $(a_n)$ defined as $$a_n :=\dfrac {\sin\Big( \dfrac {\pi}4+\dfrac n2 \Big)\sin\Big(\dfrac{n+1}2 \Big)}{\sin\Big(2n+2\Big)\sin\Big( \dfrac {\pi}4+\dfrac {n-1}2 \Big)\sin\Big(\dfrac{n}2 ...
3
votes
2answers
97 views

What to do when confronted with hard problems? [closed]

Well, I'm studying topics in algebra by Herstein these days. Trying to solve all the problems in the book, I realized that some are extremely hard to solve... actually some of them can be solved ...
2
votes
0answers
67 views

What do engineers do when they confront special integrals?

Suppose in a real life situation engineers have to calculate $\int{{2^2}^2}^xdx$ or $\int\sqrt{4-\sin^2x} dx$. The first one doesn't have an integral at all and the second one is an elliptic one. A ...
0
votes
1answer
90 views

How do modern algebraists think about diagonal matrices?

Let $\mathbb{K}$ denote a field and $A$ denote a $\mathbb{K}$-algebra. Then given a $\mathbb{K}$-subalgebra $\Delta$ of $A$, I suppose it make sense to declare that $m \in A$ is ...
1
vote
0answers
11 views

Can we have extension of Mercer theorem to interpolation?

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
1
vote
1answer
28 views

Geometric meaning of the arc length function?

Let $[a,b] \subset \mathbb{R}$ and let $\varphi: [a,b] \to \mathbb{R}^n$ be continuously differentiable. Then the indefinite integral $x \mapsto \int_a^x \| \varphi'(t)\| \, dt$ on $[a,b]$ is the arc ...
18
votes
5answers
666 views

Big list of serious but fun “unusual” books

I would like to have some suggestions about serious (that is, with good mathematical content) but fun books that cover topics (or propose problems) in "recreational mathematics"; in any other field ...
8
votes
3answers
296 views

Elementary topology examples

I'm preparing (to teach) my first class of undergraduate topology and I'm looking for some elementary, motivating applications of topology for the first day. We'll be following Munkres, starting with ...
1
vote
1answer
41 views

How to simplify my arc length formula.

When we use Riemann Sums to evaluate definite integrals, and tend the limit of width of the rectangle to zero, then the area become zero. If I use similar logic to evaluate arc length (not the ...
6
votes
1answer
107 views

Jobs in industry for pure mathematicians

While I love research and academia and would prefer to continue there, I've found myself in industry and haven't felt it's a good match for my interests. Moreover, I'm constantly frustrated by the ...
0
votes
0answers
11 views

Unit Impulse response vs Impulse response in ODE

I'm was watching MIT OCW lectures for Differential Equations and in lecture 23, the professor goes over impulse inputs where impulse is $\int_a^b{f(t)dt}$ where $f(t)$ can vary or be constant. He ...
11
votes
4answers
4k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
5
votes
2answers
103 views

When a homeomorphism can be upgraded into an isometry?

Let $X$ be a metrizable topological space. Let $f:X \rightarrow X $ be a homeomorphism. When can we find some metric $d$ which induces the original topology on $X$ making $f$ an isometry? Partial ...
0
votes
2answers
204 views

Why there is no “Nobel Prize” in mathematics however it is one of the most important fields in sciences in the side of research?

Mathematics is really a field of inventions and research where we find interesting problems some of which we can solve and others which remain open. I'm sorry to ask this question because I see it ...
0
votes
0answers
33 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
2
votes
5answers
73 views

What will change if we admit a different definition of $\sqrt a$

We know that $\sqrt a$ is the non negative solution of the equation $x^2=a$ with $a\geq 0$. So if we want to solve the equation $x^2=a$, we say that $x=\pm\sqrt a$. How will mathematics be affected ...
16
votes
1answer
1k views

Strategies and Tips: What to do when stuck on math?

Since math students will be stuck on some math at some point, what strategies or tips can help (to assuage this recurrent reality of maths)? Certainly, this wonderful website helps; here: one can ...
3
votes
5answers
240 views

Algebra and Analysis [closed]

I just finished my first year of university, double majoring compsci and mathematics. My tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble ...
2
votes
1answer
77 views

What is a good book to learn all of prealgebra?

I am an old man trying to learn mathematics, starting off with prealgebra and need a good comprehensive book for it. The book should NOT contain annoying images like in most American textbooks or ...
0
votes
1answer
35 views

Using Lagrange Multipliers Better?

My question is that "Can we use lagrange multipliers to solve any problem where we need to find local/global minima/maxima?" Also, "Is it much easier to use lagrange multipliers, and if so what cases ...
0
votes
1answer
91 views

Any use of advanced Abstract Algebra in Differential Geometry?

I believe that if someone is going to continue their studies and doing research on Differential Geometry's topics, would never need advanced Abstract Algebra (or maybe not even undergraduate level of ...
4
votes
1answer
128 views

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) ...
-2
votes
0answers
39 views

Are there still good questions to ask? [migrated]

There is no doubt, that if you are long enough on this site, you came across those famous questions with hundreds of likes, thousands of views and tons of answers. Mostly they are asking for proofs of ...
1
vote
1answer
37 views

What this type of identities are called ? e.g. “expression containing no value/constant = value/constant”

What are identities that on one side are free of any values, and just contain relationships/compositions between object/fucntion that do not contain any value at all? For example from: What is ...
0
votes
0answers
100 views

Are Lang's books reliable?

Serge Lang wrote a lot of textbooks on mathematics. However, Goro Shimura criticized him for writing so many books containing a lot of mathematical errors(he did not mention the name of the author, ...
7
votes
4answers
788 views

What do we call a “function” which is not defined on part of its domain?

Before the immediate responses come in, I realize that a properly defined function means that it is defined for every value in its domain. My question is this: if $f:A\to B$ has the property ...
2
votes
1answer
90 views

Failed Calc 2. It's the Algebra, stupid

Well, I took "Multidimensional Math" w/ Linear Algebra and Calculus 2 at the same time over a 6 week period for the Summer session. It was a disaster. I don't think my Algebra and Trig are good ...
0
votes
2answers
58 views

Next step to learn mathematics for a high school student?

I am a high school student. I would like to study mathematics at university. Because I want to be successful in my studies and in productive research so I want to learn and practice more mathematics ...
-2
votes
1answer
106 views

About Dummit's algebra book [closed]

I'm studying Dummit's book and it is quite encyclopedic and the amount is overwhelming. so i wonder if i can skip topics too deep for the undergraduate(me!) so that i can study only topics that are ...
39
votes
2answers
6k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
0
votes
1answer
35 views

A suggestion for Spivak's book

I am starting to venture through the book "Calculus on Manifolds " by Michael Spivak . At the end of the first chapter , it mentions about a transformation matrix. I haven't learned Linear Algebra ...
4
votes
1answer
110 views

Bridge between High School Mathematics and University-level Mathematics?

I've graduated from High School and I am going to major in math at a local University. I've finished High School Calculus and I've self-studied very very basic Multivariable Calculus, Linear Algebra, ...
0
votes
0answers
13 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
0
votes
3answers
76 views

What is the purpose of mixed numbers outside of common usage? [closed]

I am wondering if there is indeed any real usage of mixed numbers such as $3 \frac{1}{2}$ meaning $3+\frac{1}{2}$ in standard Mathematics. Personally I dislike the use of such conventions in schools ...
2
votes
4answers
855 views

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano's Theorem, which, honestly, I find a bit ...
2
votes
1answer
47 views

Alternative proof that base angles of an isosceles triangle are equal

The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows: Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid ...
26
votes
5answers
2k views

When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
1
vote
0answers
16 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
2
votes
1answer
52 views

A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
79
votes
9answers
8k views

Math and mental fatigue

Just a soft-question that has been bugging me for a long time: How does one deal with mental fatigue when studying math? I am interested in Mathematics, but when studying say Galois Theory and ...
7
votes
6answers
459 views

High School Mathematics

Could you recommend any high school mathematics books that are rigorous and present high school mathematics in a higher level.I just realized that I only memorized a lot of things that I was taught in ...
54
votes
11answers
7k views

Why there is no sign of logic symbols in mathematical texts?

Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, ...
28
votes
1answer
914 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
10
votes
6answers
680 views

Suggestions for topics in a public talk about art and mathematics [closed]

I've been giving a public talk about Art and Mathematics for a few years now as part of my University's outreach program. Audience members are usually well-educated but may not have much knowledge of ...
7
votes
6answers
111 views

A proper definition of $i$, the imaginary unit [duplicate]

Back when I was in high school, which was a long time ago, I recall my math teacher telling me that the definition of $i$, the imaginary unit, is $\sqrt{-1}$. Knowing little, at the time, I accepted ...
5
votes
8answers
986 views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...