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11
votes
1answer
110 views

How to stay productive while you are studying math? [closed]

Not sure that this question is a good fit for this site, but I will try. When I am working through a chapter of a mathematical book first two hours are normally very productive (easily remember ...
1
vote
1answer
87 views

What is the difference (or relationship) between geometric length and arithmetic numbers?

In Abbott's Understanding Analysis there was a phrase like, "Ancient Greeks did not understand the difference (or relationship) between geometric length and arithmetic numbers." What is this ...
3
votes
1answer
144 views

Advice on mathematical cources and profession

I am a class 9 student studying in India. I deeply love mathematics and I am aspiring to become a mathematician ( I am aiming for research and development). My hobbies include learning and trying to ...
-1
votes
1answer
112 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
5
votes
2answers
81 views

What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
0
votes
0answers
34 views

Connections between probability theory and algebraic topology?

Are there any substantial connections between probability theory and algebraic topology? In particular, are there any current research areas in algebraic topology that involve the use of probabilistic ...
2
votes
0answers
27 views

On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
1
vote
0answers
24 views

Why the generalized root test for series does not use $\liminf$ to get rid of the inconclusive part?

I wonder for what reasons the generalized root test for series (Roughly: $\limsup a_{n}^{1/n} < 1$ implies convergence; $\limsup a_{n}^{1/n} > 1$ implies divergence; $\limsup a_{n}^{1/n} = 1$ ...
55
votes
9answers
3k views

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
1
vote
0answers
39 views

Best books about notation

I am reading A History of Mathematical Notations and I would love to read further about the trends and advantages of different notations in mathematics, pure or applied. Is there any good book or ...
3
votes
0answers
104 views

To prove $\phi(mn)\phi(d)=\phi(m)\phi(n)d$ without explicitly computing the phi function values

If $m,n$ are positive integers with g.c.d.$(m,n)=d$ , then we can show by explicitly computing respective totients that $\phi(mn)\phi(d)=\phi(m)\phi(n)d$, I want to know, is there any more elegant way ...
1
vote
0answers
35 views

Comparison of abstract algebra structures

I made a comparison table of the different abstract algebra structures to help myself refresh with these concepts. Can some kind souls here help cross check the basic correctness? (Any other ...
0
votes
1answer
18 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
2
votes
3answers
76 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
-1
votes
2answers
67 views

Is there a proper term for these?

The 'square root' sign is formally the 'radical symbol'. 'Brackets' are formally 'parentheses' Is there a formal term for the 'fraction sign' or say the 'absolute value bars'? EDIT: I can see some ...
0
votes
0answers
36 views

Question about further studies

I am new here so I am not sure if these type of questions are allowed, or where exactly they should be posted. So if not my apologizes and just delete. I am studying math and some other subjects, in ...
0
votes
3answers
53 views

Working out a reverse formula

My math skills are getting rusty. I am trying to work out what the formula should be for calculating price, $P$, based on a formula I used to calculate margin, $\mu$, with a parameter, cost, $C$. ...
1
vote
1answer
52 views

Finding other problems similar to a math contest problems?

*I don't know if I can ask these type of questions here. Tell me and I will delete it right away if it's doesn't belong here. I'm preparing for a math contest, but I'm done answering all the previous ...
4
votes
1answer
68 views

Maximum number of points you can put on grid $ n\times m$ with no equidistant?

Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length ...
1
vote
1answer
22 views

Concerning ideals of $\mathbb Z[\sqrt m]$ and $\mathbb Z[\sqrt m] [x] $

For a given integer $m<-1$ or non-square integer $m>1$ , how do we calculate the quotient ring $\mathbb Z[\sqrt m]/I$ , for example its order or whether it is a field or has zero divisors or not ...
1
vote
0answers
30 views

References about algebraic/differential geometry in French

Aside from learning mathematics, I am learning French, so I would like to practise both at the same time if possible. Do you know of any good references about complex/algebraic geometry or ...
12
votes
1answer
191 views

Why are “algebras” called algebras?

There's a mathematical object called an "algebra" (e.g. an algebra over a ring), but why does this particular object have such an "important" name (which makes it sound like the most important concept ...
3
votes
1answer
134 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
5
votes
2answers
76 views

Examples of functors that preserves products but not equalizers, and vice versa.

What are simple examples, for student consumption, of A functor which preserves products (or at least finite products) but not equalizers. A functor which preserves equalizers but not products. ...
1
vote
0answers
50 views

Basic questions regarding group theory and symmetry in practice

I have for some months been interested in group theory. I was very fascinated by the level of abstraction I first met when working with groups. Another aspect that has fascinated me lately is ...
6
votes
1answer
31 views

Why are separable and normal field extensions so called?

To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in ...
2
votes
0answers
33 views

What does it mean when a field of mathematics has 'differential' in its name?

Topology and geometry, for example, have their 'differential counterparts' (as well as 'algebraic'). What is required of a mathematical discipline for there to exist such a subfield? For example, why ...
3
votes
1answer
88 views

How to learn about the Rubik's Cube?

So I just started learning Abstract Algebra and I've recently appreciated an introduction to group theory. Among the interesting avenues of group theory is found in tackling a Rubik's Cube. I've ...
7
votes
1answer
54 views

Applications of the following theorem in the real world

We know that every permutation can be expressed as a product of transpositions ( cycles with length 2). As a class project I'm looking for the applications of this fact in the real world; especially ...
2
votes
0answers
31 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
2
votes
3answers
91 views

A useful math project for high school [closed]

I need a project that will be useful for anything or anybody and it must be related to maths. It can also be related to other subjects. The math project should also be able to complete in 2 weeks. ...
4
votes
1answer
73 views

How should we think about equations like $dy = 2x \cdot dx$ from the viewpoint of modern geometry?

We've just started learning about (smooth) manifolds at uni, and I'm kind of hoping this will finally help me get a handle on the dreaded Leibniz notation. Now I've read that expressions $dy$ like can ...
0
votes
1answer
26 views

needing general information (algebra)

I plan to study algebraic number theory (arithmetic and Galois theory) in order to start research in cryptography and coding theory and all the staff... Also, i would like to introduce myself to ...
1
vote
0answers
39 views

What are metric spaces examples of?

A metric space is a very strange kind of object. It is not simply a set with some operations that satisfy some operations, like a group. In the higher reaches of abstract algebra, an algebra is ...
2
votes
3answers
112 views

What is an Isomorphism: Linear algebra [duplicate]

this is a rather soft question to I will tag it as such. Basically what I am asking, is if anyone has a good explanation of what a homomorphism is and what an isomorphism is, and if possible ...
1
vote
2answers
20 views

Methods to quickly compute percentages

Yesterday, talking with a friend of mine, she asked me what is a quick (and – of course – correct) way to compute percentages, say $3.7 \%$ of $149$. Frankly, I was sort of dumbfounded, because I use ...
2
votes
4answers
67 views

motivating diagonalization of a matrix [duplicate]

I have to teach about diagonalization of a matrix to a first year undergrad student and I was wondering what would be a good way to motivate this concept. I would appreciate any suggestions. Thanks!
3
votes
4answers
132 views

Good way to convince a young kid that $0*0 = 0$?

My little brother (6 years old) asked me a question ("What is $0*0$?") and gave an answer to his own question which I found ridiculous so I refuted it but he still thinks he is right. He says that ...
1
vote
0answers
35 views

Why are free objects related to tensors important?

Let $R$ be a commutative ring and $M$ be an $R$-module. Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any ...
4
votes
1answer
113 views

Quality of Videos Lectures and Lectures vs Textbooks

I am a student trying to learn different subjects by watching video lectures and reading on my own time. I was wondering if the lectures from ICTP and nptelhrd are a great use of my time. I tried ICTP ...
5
votes
3answers
203 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
0
votes
2answers
60 views

Are the angles adjacent?

$\angle 1 $ and $\angle 2$ are adjacent . So are $\angle AOD$ and $\angle BOD$ also adjacent. ? I am confused. EDIT: Suppose the point $C$ is not in the diagram , then in this case are ...
3
votes
2answers
91 views

Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra

Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally ...
2
votes
3answers
55 views

Open Source Software for Creating Mathematical Diagrams

I work as a software engineer at a company developing navigation systems. As I have a mathematical background I normally get assigned the more mathematical problems and I find myself regularly having ...
4
votes
2answers
55 views

Looking for at least one surjective ring homomorphism from $M_n(R)$ to $R$

Let $R$ be a ring , I am looking for a surjective ring homomporphism from $M_n(R)$ to $R$ . Please help . Thanks in advance .
0
votes
1answer
55 views

How to interpret a matrix operation?

Suppose we have a 3-term vector $v_1 = [1, 2, 3]$, and another 3-term vector $v_2 = [16, 20, 22]$. Now suppose we find a 3 x 3 matrix $$M = \begin{pmatrix} -2 & 3 & 7 \\ 9 & -2 & 6 \\ ...
0
votes
0answers
60 views

Number of non-isomorphic groups of order $n$

$G$ is a group of order $n=p_1p_2....p_k$ then how to find the number of non-isomorphic groups of order $n$ where $p_i's$ are distinct primes I can find the number of number of non-isomorphic ...
1
vote
0answers
43 views

Visualization of theorem relationships

A few minutes ago a thought for a website/program crossed my mind and now I somehow hope that such a thing already exists. I thought of: A set of theorems linked by arrows telling the viewer/reader ...
2
votes
1answer
44 views

Suggestions to solve for $A$ in the equation $e^{-\frac{(A-B)^{2}}{D}}=C$

This seems almost silly to ask but I am stuck with it. I have the following equation \begin{equation}\tag{*} e^{-\frac{(A-B)^{2}}{D}}=C \end{equation} I know $A,B \in \mathbb{R}, ...
0
votes
3answers
36 views

Cross products and determinants in $\mathbb{R}^3$

I know that the absolute value of determinant of three vectors in $\mathbb{R}^3$ is the volume of the parallelepiped determined by the three vectors. The volume can be computed by basic calculation ...