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1
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2answers
66 views

Notation seen in “awfully sophisticated proof…” I don't understand

I want to understand what the definition of $f_n$ given here means? I tried to seek on the net but I not succeeded. I precise I do chemistry, maths are "just" a curiosity for me. I should be glad, ...
3
votes
1answer
56 views

High School Geometry Text?

This year I will be teaching 8 hard-working home-educated teens a Geometry course. Back in 1994-1999 I worked full time as a High School educator, taking a turn teaching everything from Pre Algebra ...
0
votes
0answers
14 views

Looking for some good introductory level resources for Gibbs Sampling

In context of a course in bayesian modelling Im following, im looking for some good resources (videos, lecture slides, texts) about Gibbs sampling.
6
votes
0answers
44 views

What should I do with a paper I've translated? [migrated]

Aside from reaping the personal benefits, what should one do after translating a paper? It would be nice to offer it to others, but I assume it is a copyright violation to post it online. Sending it ...
1
vote
0answers
12 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . On the other hand interpolation space which is defined in the wikipedia link: ...
2
votes
0answers
46 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
1
vote
1answer
33 views

Transforming a sequence to distinguish a limit

This might be the wrong place to ask this question, but I figured I might get some creative answers: I have a decreasing sequence $\{a_n\}_{n \geq 1}$ with $a_k \in (0,1)$ for all $k$ and $a_n \to ...
1
vote
3answers
78 views

If $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, must we have always $-\frac{a_0}{a_n} \in \mathbb{Z}$?

Let consider the polynomial with integer coefficients: $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ If $f(x)=0$ and $x \in \mathbb{Z}$ with $a_n\neq 0$ If all the roots are integers, must we ...
0
votes
0answers
76 views

Does there exist a continued-fraction for geometric series

I would like to know if there exists a continued fraction representation of a geometric series.Motivated by the fact that,many elementary functions in math can be represented as such,I wondered if ...
2
votes
1answer
55 views

Can we make a subgroup of a group by selecting exactly one element from each distinct left cosets of a subgroup of the given group?

Let $G$ be a group and $H$ be a subgroup of $G$ ; can we select exactly one element from each distinct left coset of $H$ such that the set of all those elements form a subgroup of $G$ ? How do we ...
3
votes
4answers
128 views

On the near-integer $163/\ln(163)$

This question, concerning the approximation $\frac{163}{\ln(163)}\approx 2^5$, was posted on MO 5 years ago: Why Is 163/ln(163) a Near-Integer?. It was concluded that it had nothing to do with 163 ...
1
vote
0answers
47 views

Why do the mathematicians stated $0!$ to be $1$? [duplicate]

My question is very simple, if just as we say $5! = 120, 4! = 24,$ how can we say that $0! = 1$? Why did the ancient mathematicians conventionally consider $0!$ to be $1$? Then there's coming lot of ...
-2
votes
0answers
14 views

Some difference in notation in a project report [on hold]

I am focusing on a specific model in my study. When I am writing my project paper, for describing the estimation procedure, can I describe it more generally with some difference in notation than my ...
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 ...
3
votes
1answer
105 views

Undergraduate mathematics competitions

I am a freshman (math undergraduate) here in Argentina and I am deeply interested in mathematical olympiads but I really need some advice. Right now, my problem solving skills are good but not that ...
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 ...
3
votes
2answers
98 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 ...
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 ...
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
110 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 ...
10
votes
5answers
308 views

Do people whose native languages are read right-to-left experience mathematical statements differently? [closed]

When I see the equation $$A = B$$ the first idea to occur to me is that $A$ can be transformed into $B$. Although of course $$B = A$$ has the same content, to me it connotes rather that $B$ can be ...
0
votes
0answers
12 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 ...
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 ...
2
votes
2answers
68 views

Finding a particular type of sequence of functions

For every bounded function $f:[a,b] \to \mathbb R$ on a closed bounded interval $[a,b]$ , which is dis-continuous at at most countably many points of its domain ; can we find a sequence of ...
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 ...
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 ...
-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 ...
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 ...
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 ...
8
votes
5answers
239 views

Integration of $e^{-x^2}$

I know that this has been asked many times, but I am very interested in the indefinite integral of $e^{-x^2}$. It is stated and proved that there is not elementary way to solve this. So does this ...
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 ...
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 ...
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) ...
0
votes
0answers
101 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
791 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 ...
5
votes
1answer
54 views

What shall I learn in order to understand Auslander-Reiten theory and tilting theory?

I work on cluster algebras and quivers and hence I need to understand Auslander-Reiten theory and tilting theory as soon as possible. I have read some noncommutative algebra and homological algebra ...
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 ...
-2
votes
1answer
107 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 ...
0
votes
0answers
14 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 ...
4
votes
1answer
112 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
2answers
78 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
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 ...
2
votes
1answer
101 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
68 views

Next step to learn mathematics for a high school student? [on hold]

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 ...
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 ...
2
votes
1answer
48 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 ...