Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

1
vote
1answer
32 views

Rigidity Theorem, meaning.

What does it mean to say that a theorem is a rigidity theorem? I'm reading a book (Lipman Bers, a Life in Mathematics) that says for example that Torelli's theorem is a rigidity theorem. So what ...
2
votes
1answer
31 views

Why do we care about the general outer product $\mathbf u\otimes\mathbf v=\mathbf u\mathbf v^{\sf T}$?

Why do we care about dot and cross products? Why do we care about covectors? [essentially is the meaning of the question] Why do we care about dual spaces? So, why do we care about the general outer ...
1
vote
0answers
33 views

Numbers as product of prime numbers raised to exponents $e\in A \subset \mathbb R$

If $\mathbb A^+\subseteq \mathbb R^+$, then let $\mathbb A^- =\{- a : a\in\mathbb A^+\}$, $\mathbb A^* = \mathbb A^+\cup\mathbb A^-$, $\mathbb A=\mathbb A^*\cup\{0\}$. Set of positive integers can be ...
0
votes
0answers
23 views

Conditions on binary representation of a number

I have a range of values from $1$ to $64$. I divide it into $4$ parts $1-16$, $17-32$, $33-48$, $49-64$ and i want to implement certain statements based on a number which lies in one of these four ...
0
votes
0answers
29 views

How to measure repulsion between numbers

How is repulsion measured between two eigenvalues or any two numbers for that matter? Assume that repulsion is $1,$ ($100$ percent) when the two numbers have zero space between them, and repulsion is $...
2
votes
4answers
62 views

Why is do we ignore $(dt)^2$ when computing derivatives by expanding the function?

Consider this function: $s = t^2 + 1$ My question is, in general, while differentiating, why do we ignore the term of second order of smallness? I mean shouldn't that affect our observations? Say I ...
2
votes
1answer
30 views

How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
0
votes
3answers
40 views

Basic calculus question (Intuitive)

So say y is a product of functions of x. For convenience, let us suppose $y = (2x²+1) . (x+2)$ If I were to differentiate this with respect to x, I could probably use this: $If, Y = U.V$ $→...
2
votes
1answer
44 views

Tips for a dum-dum : Introduction to Analysis 1 and 2

First off let me say that math and I don’t get along. I want to be it’s friend, but it certainly doesn’t want to be mine. I have been out of high school for some time (decades) and recently returned ...
0
votes
0answers
14 views

I need resources for elementary Number Theory, Theory of Equations (For KVPY exam), (details in post) [on hold]

So I am currently starting 11th grade (in India). There's this KVPY scholarship exam that I am preparing for, and it needs pretty advanced concepts, like number theory (congruence modulo), theory of ...
1
vote
0answers
34 views

Is there a connection between homomorphism and diagonalization?

The definition of homomorphism is https://en.wikipedia.org/wiki/Homomorphism and definition of diagonalization needs to be understood in the context described in Fourier transform as diagonalization ...
0
votes
1answer
21 views

What fraction of children would have no parents?

This question has pop-culture origins but is mathematical for sure. In the movie "Avengers: Infinity War" (SPOILERS AHEAD) the main villain eliminates half of all life on the Earth (technically the ...
0
votes
0answers
27 views

A D20 (dice) has sides where $N-1,N,N+1$ are always neighbours on surface of solid. For which DN is this possible?

A regular gon D20 dice used for example in various forms of gambling and trading card games is shown below As can be seen each number $N$ residing on some face has two of it's neighbouring faces with ...
-1
votes
0answers
30 views

Is there an example of a non-trivial function with a “nice” closed form expression that can be expressed as a Dirichlet series

Most examples of functions that can be expanded in Dirichlet series in my textbook involve the zeta function. I was wondering how the class of functions that can expanded in a Dirichlet series look ...
3
votes
1answer
76 views

Difference between several books on complex geometry

I would like to learn some complex geometry, especially the interaction between algebraic geometry and complex geometry. I found that there are several famous books: Huybrechts, Complex Geometry; ...
0
votes
0answers
21 views

A problem about Argand diagrams

When plotting an Argand diagram, do I simply plot the points or do I need to draw arrows from the origin to the points? Motivation: I self studied for my FP1 exams, which will be held today and ...
2
votes
0answers
48 views

A surprisingly simple smooth function, is it used as a sigmoid?

Consider the function $$f(x) = \frac{x}{1+|x|}$$ For what we can prove about it's derivative, it exists everywhere, is maximal at $x=0$ ($f'(0)=1$) and we can verify $$\forall x\in \mathbb R : f(x)\...
1
vote
0answers
34 views

Non-English papers in MathSciNet [closed]

Is there any way to find all papers in a specific language (for example German) in MathSciNet?
2
votes
0answers
24 views

Reference request on Combinatorics and set theory

I am thinking about teaching an introductory class in Combinatorics and Set Theory. My view on this is that much of the introductory part in combinatorics can be seen as finite set theory. The issue ...
0
votes
0answers
52 views

Algebra Lecture notes

The question is very simple: are there good Algebra online lecture notes for a first year course (one semester, US) satisfying the following constraints. -They cover basics on groups and rings -They ...
0
votes
1answer
23 views

confusing between rational and integer numbers

I am a bit confused to what is a rational and an integer numbers. These following numbers are integers: $1,2,3,4,... etc.$ but these numbers are also can be written as $\frac{1}{1}, \frac{2}{1}, \...
7
votes
1answer
130 views

What is the best way to study graduate level mathematics?

I am studying a 400/500 level measure theory math book on my own. Right now, when I read it I try to read the proposition then the following proof. And then try to do the exercises on my own. I ...
5
votes
3answers
338 views

Why does a C.D.F need to be right-continuous?

As you may know, if $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space and $X\colon\Omega\to\mathbb{R}^k$ is a random variable, then the cumulative distribution function of $X$ is defined as $$...
5
votes
4answers
227 views

Big list: Free textbooks and resources

We have had multiple book-recommendation, but I have not found a question where the many free (yet legal) books are available under one umbrella. Therefore I propose a thread where all the best, free ...
5
votes
2answers
94 views

Why are |vertical lines| used to mark matrix determinants?

This notation is sometimes used to denote the determinant: $$ \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$$ Why? Where did this notation come from? Was there any relationship between ...
-1
votes
2answers
51 views

Good abstract algebra topic/paper for 1.5 hour university presentation. [closed]

Next week I have to do a presentation during the abstract algebra course I attend. Can you recommand some topic, or paper which would be interesting and I can tell about it within 1-1.5h? The course ...
1
vote
0answers
82 views

Math course after calculus

I'm really enjoying calculus and linear algebra, and I think there are my best subjects. Multivariable calculus was especially interesting, and I hope I can continue my studies in calculus. I'm ...
5
votes
3answers
191 views

Are there “interesting” theorems in Peano arithmetic, that only use the addition operation?

More precisely, are there "interesting" theorems of Presburger arithmetic, other than the following four well-known "interesting" ones? The commutativity of addition. The theorem stating there are ...
0
votes
0answers
37 views

Theorem that converts a multiple integral into a single integral?

Forgive me for the vague question. Several years ago, I read about a theorem that (if I remember well) proved what is written in the title. I guess the theorem is from Wiener and it seemed that this ...
2
votes
1answer
45 views

Connections and metrics on Riemannian manifolds

In a lecture on connections it was claimed that a manifold "attains a shape" when it is equipped with a connection while there were no mentions of a metric. Is it correct to intuitively think that ...
0
votes
0answers
13 views

soft question: solution manual for Demidovich Calculus

Context: I'm taking multivariable calculus this semester and my professor takes exercises from Problems in Mathematical Analysis by Demidovich - a book from a few decades ago, written by russian ...
0
votes
0answers
19 views

Textbook recommendation for measure-theoretic stochastic process?

I am seeking for a great book on graduate-level, measure-theory based book on stochastic process, the ones I can find are this book by Cosma Rohilla Shalizi and Aryeh Kontorovich, and Foundations of ...
1
vote
0answers
37 views

Research topic- Evolutionary Graph Theory

I read Evolutionary Dynamics by Martin A. Nowak and became fascinated with evolutionary graph theory. I'm mentoring an undergraduate student in the subject and would like to give him a research ...
0
votes
0answers
25 views

Is it still unknown whetever any $\mathscr{D}$-class $D$ being a semigroup is bisimple?

It's stated in my book from 1961 that it's unknown that if a subsemigroup $D$ of semigroup $S$, which is a $\mathscr{D}$-class of $S$ is always bisimple, but it's known for $\mathscr{D}$-classes which ...
1
vote
2answers
62 views

Must any shortest line between two surfaces coincide with normals to both surfaces?

In differential geometry I know of a result which says something along the lines of ...
0
votes
1answer
27 views

Mathematical notation for $i^{th}$ canonical basis vector for the n-dimensional space

One of my constraints in an optimization problem involves using canonical basis vector for the n-dimensional space. How do I precisely write $j^{th}$ canonical basis vector for the n-dimensional ...
8
votes
0answers
89 views

Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
0
votes
1answer
250 views

Origins of 'We' Pronoun in Mathematics

In proofs/books/papers on mathematics the pronoun "we" is usually used. For example: In order to derive the quadratic formula we first complete the square. Or: ... we can deduce this ...
0
votes
0answers
31 views

How do you do a literature review when you don't know the name of the concept?

So this is certainly a soft question, but one that I've encountered a lot of times; just check my post history and you'll find a many questions like this. I frequently play around with random ...
0
votes
2answers
33 views

Operations research vs. constraint programming

What are the differences between these two topics? I took courses a long time ago on these subjects, and the theory seemed to be the same. Yet I sometimes see people making a distinction between them....
1
vote
0answers
36 views

Does this behavior of simple sieves have a name?

The sieve of Eratosthanes sequentially identifies primes by using smaller known primes to discard numbers from a list, eventually leaving behind only other primes. At the nth step, we use $p_n$ and ...
0
votes
1answer
13 views

Transpose method for finding a basis for the row space

My linear algebra textbook outlines a method of finding a basis for the row space of a matrix by finding a basis for the column space of its transpose. Is there any point of using this method? It ...
1
vote
1answer
79 views

(Soft question) Why should one study pure math?

I'm an undergraduate math major and I really enjoy pure math. I don't really have a specific area of interest yet, but I'm looking. That being said, I often ask myself the question "what's the point ...
1
vote
0answers
43 views

Help coming up with a high school research project

I'm going to be a junior next year, and our school requires us to make a science fair project. I've always been interested in math and am heavily involved in math contests. However, for the past few ...
0
votes
0answers
21 views

Is there a standard notation for the Cartesian product of arrays (tensors)?

Note: for the most part, the arrays I am considering are of infinite (sometimes, uncountable) dimension in at least one index, and of rank $\geq2$. e.g. $\aleph_0\times n$ matrices, $|\mathbb{R}|\...
3
votes
1answer
48 views

$(\mathbb{Z}_2\times \mathbb{Z}_3)[X]/ (X^2)$

A very naturel question come in my mind but I do not know how to answer, I need a help. For this ring $(\mathbb{Z}_2\times \mathbb{Z}_3)[X]/ (X^2)$, I was wondreing if thiere elements are like $(0,1)...
2
votes
1answer
44 views

Generalizations of combinatorial quantities for cardinal numbers.

Recently I've came up with an idea, that we can interpret $n!$ in terms of functions, as the amount of bijections of a certain set $X$, where $|X| =n$ (i. e. cardinality of the symmetric group on $X$) ...
2
votes
0answers
20 views

Taylor polynomials in Fourier Domain?

Which types of functions admit Taylor polynomials in the Fourier domain? Ok, this may sound cryptic. Therefore I will try to explain it more in depth. We want to approximate function $$x\to f(x) \...
0
votes
1answer
48 views

Whats the difference between the following? $n+k-1 \choose k$ ${{n+k-1}\choose{k-1}}$ ${{n+k-1}\choose{n}}$

$n+k-1 \choose k$ ${{n+k-1}\choose{k-1}}$ ${{n+k-1}\choose{n}}$ So I've been using $n+k-1 \choose k$ to r pick k objects from n objects allowing repetitions. In the following question though ...
2
votes
3answers
43 views

How does the Pythagorean theorem describe a circle?

The Pythagorean theorem states, for a right triangle with legs $a,b$ and hypotenuse $c$, $$a^2+b^2=c^2$$ By replacing $c$ with $r$, radius this equation becomes the equation of circle at centre $(0,...