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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.

4
votes
1answer
26 views

Functional analysis on manifolds

The basic object of functional analysis is the topological vector space, so vector spaces with some topology, we can add additional structure by introducing metrics etc, but the underlying object is a ...
-4
votes
0answers
47 views

The math of monsters [on hold]

Researching for a science fiction novel. My character loves math. I have dyscalculia. Problem. So I need your help. In my story, there are beings (aliens) who live on a plane of existence parallel ...
4
votes
0answers
24 views

Have similar theories like knot theory been developed in higher dimensions?

Well, my question is kind of basic but I hope it would be taken seriously by the community. Also, I'm very new to this topic and I want to study knot theory in future. Knot theory is the study of ...
2
votes
1answer
31 views

Can you define what it means that a set of functions has a common property?

Suppose $\mathcal{A}$ is a set of functions with a common property. Does this mean that the property is shared pairwise or that the property is shared amongst all elements of $\mathcal{A}$? For ...
0
votes
0answers
17 views

Anirban DasGupta Asymptotic Theory Book

Iam having a huge difficult to understand Asymptotic Theory in Statistic/Probability. Ive Found Anirban DasGupta Asymptotic Theory Book, but there is not the solution for the exercise. First I ...
1
vote
1answer
46 views

Why are dual spaces called “dual spaces”, even when I don't see if they're really “complements”?

Why are dual spaces called "dual spaces", even when I don't see if they're really "complements" of the original space. In optimization the primal-dual distinction seems to rely on logical complement. ...
4
votes
2answers
81 views

Methods of making mathematical discovery

There are lots of methods of proof: direct proof, proof by contrapositive, proof by contradiction, proof by induction, proof by cases, computer proof... On the other hand, the process of mathematical ...
1
vote
6answers
89 views

Intuition behind $ ||x|-|y|| \leq |x-y| $

In my math analysis course we proved that $ ||x|-|y|| \leq |x-y| $. However, unlike the triangle equality that I can visualise with the example of a triangle, I cannot visualize this one nor ...
3
votes
1answer
76 views

(Soft question) How to study for a proof-based math class?

I'm an undergrad at a top US university and I'm currently taking an Introduction to Analysis class. I really really enjoy this class and it's actually getting me more interested in math in general. I ...
28
votes
7answers
5k views

How to attack “if true, prove it; if not true, give a counterexample” question?

I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if ...
1
vote
0answers
27 views

What computer algebra system or package for abstract algebra best suits me?

I am a mathematics student who wants a tool on the side, either specifically aimed at, or well suited for abstract algebra (ring/ideal theory, module theory, but perhaps graphs and combinatorics as ...
1
vote
0answers
17 views

Efficient algorithms for minimization of bi-linear matrix expressions?

Say I want to solve $$\min_{\bf x}\|\bf Ax-b\|_2^2$$ This problem is well defined and easy to solve using linear least squares. But what if we suddenly have $$\min_{\bf x,A}\|\bf Ax-b\|_2^2$$ ...
0
votes
0answers
27 views

In 'Probability: A Graduate Course' by Allan Gut, it is written '[…]in a sense life itself is a martingale.' Can someone explain how?

I hope this question doesn't infringe any rules. Original text (p. 480, 2nd edition): If we interpret a martingale as a game, part (ii) states that, on average, nothing happens, and part (i) ...
0
votes
0answers
11 views

Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
1
vote
1answer
56 views

Is “type theory” the only way to get a computer to “do math” on its own? [on hold]

It seems like one of the direct applications of type theory definitions & algorithms is to implement them on a computer and use it for proof assistants, etc. But why couldn't you just implement ...
0
votes
0answers
13 views

Literature on moduli spaces

I am an undergrad student and I want to learn more about moduli spaces, but this is not part of our curriculum. Can anyone advice some literature which is at undergraduate/graduate level?
2
votes
1answer
91 views

What is the need of negative numbers?

My question is bit naive, Related to the above question is: Add, Subtract, Multiply and Divide are fundamental "operations", then why we try to provide a sign to numbers as +ve or -ve, to say why are ...
0
votes
2answers
65 views

direct proofs of inequalities

I completed an entire chapter set on direct proofs only to find my teacher said reject my answers due to a false method. what i did was assume the problem was true and then solved it as such and then ...
4
votes
2answers
413 views

Proof by contradiction in Constructive Mathematics

I’m watching this video on Constructive Mathematics Five Stages of Accepting Constructive Mathematics, and Andrej Bauer makes the following claim: Mathematicians call two different things “Proof by ...
3
votes
1answer
59 views

Why do we need to treat proper schemes, not only projective schemes?

I know that many complete varieties are projective: For example, complete curves, smooth complete surfaces, arithmetic surfaces, and abelian varieties are projective. And even for general proper ...
3
votes
1answer
33 views

A convex function's epigraph is convex. What property has that of an increasing function?

In some sense, convex functions are the second-order version of increasing functions: under suitable hypotheses (for instance if the function is $C^2$), an analytic characterization of convexity is ...
3
votes
0answers
63 views

How similar are permutation groups that are isomorphic as abstract groups?

Let's say that two permutation groups $P_1$ and $P_2$ are isomorphic as abstract groups, but not necessarily permutation isomorphic. How similar will $P_1$ and $P_2$ be, and how much structure will ...
1
vote
3answers
71 views

What's the deal with this $\frac1\pi$?

I recently learned about the very interesting Dirac Delta function, defined as $$\delta(x)=\frac1\pi\lim_{\epsilon\to 0}\frac{\epsilon}{x^2+\epsilon^2}$$ Which is a very majestic definition, as the ...
0
votes
0answers
15 views

Examples where estimating the derivative is easier than estimating the actual quantity.

I recently came across a paper in which they author estimated the deriviative of the quanity he was interested in rather than the quantity itself. This was done because apparntly the derivative ...
0
votes
0answers
27 views

What is possible 6 figure alpha numeric count.

What is possible count for 6 alpha numeric figure? How much it will be if we count each 6 digit numeric and 6 figure alphabet. Combine each figure each time if photoshop has 6 alpha numeric in the ...
7
votes
1answer
78 views

What is it called when the definition of “<adjective> <thing>” does not imply that it is a special case of “<thing>”?

There are a couple of terminologies that I find mildly confusing: Using the definition of graph on wikipedia, a graph is defined as having a finite number of vertices and edges. It states that there ...
1
vote
0answers
71 views

Definition 4.25 on p.94 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. Definition 4.25 Let $f$ be defined on $(a, b)$. Consider any point $x$ such that $a \leq x < b$. We write $$f(...
2
votes
2answers
37 views

Question regarding the First Few Terms of a Fibonacci-like Recurrence.

My friend handed me an interesting problem involving Fibonacci numbers and i was interested in trying to generalize it, and see what known results are around after that. But before I do that, i had a ...
0
votes
2answers
27 views

Resources to help me with convex analysis

My mentor assigned me with the task of studing the content of appendix A and part of appendix B in Bertsekas' Nonlinear Programing, which cover the basics of convex analysis and its prerequisities. ...
0
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0answers
13 views

Operation that selects a unique representative from each class of associate elements

Is there some kind of a standard, uniform, or otherwise natural method for selecting representatives of classes of associate elements in certain rings? I would be interested in a simultaneous ...
3
votes
1answer
41 views

What notation should be used for right adjunct of arrow ($f^{\sharp}$ or $f^{\flat})$?

In the style of CWM let $\mathcal A$ and $\mathcal X$ denote categories and let there be an adjunction $\langle F,G,\phi\rangle$ from $\mathcal X$ to $\mathcal A$. So for every pair $(x,a)$ there is ...
0
votes
2answers
44 views

If the union of two sets is uncountable, what may we conclude?

I have $2$ questions. Here are these: 1) Assume that the union of sets $A$ and $B$ is uncountable. What exactly can we conclude from here? I think, at least one of these sets is not countable. Or ...
2
votes
0answers
79 views

Does probability theory suffer from Gödel's incompleteness theorem?

Let us consider the following two theorems by Gödel: 1) Any consistent formal system $F$ within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are ...
0
votes
1answer
27 views

Handwriting: distinguishing between upper and lowercase “x”.

I'm following along with Kreyszig's intro to functional analysis text, where he uses $X$ to indicate an abstract set, and often uses $x$ to indicate an element of $X$. I want to match his notation, ...
3
votes
1answer
80 views

As someone without a maths background, how should one study for a mathematical logic master's degree? [closed]

I have a background in philosophy and I am currently enrolled in a Master's degree in logic. I prepared as best as I could before I enrolled (read How to Prove It by Daniel Velleman and chapters on ...
20
votes
3answers
3k views

Why is Sesame Street's Count von Count's favorite number $34,\!969$? [closed]

In the 2 minute BBC News audio clip Sesame Street: What is Count von Count's favourite number? "The Count" is asked Do you have a favorite number? to which he replied Thirty four thousand, nine ...
0
votes
0answers
46 views

A person who has not joined IMO, can contribute to mathematics in the future? [duplicate]

The following question is a question that confuses my mind long since. I'm sorry if it looks nonsense. If mathematicians who had made important discoveries in mathematics such as Bernhard Riemann, ...
8
votes
0answers
73 views

What is the precise definition of the prefix “co” in mathematics?

Given a notion "A" in mathematics, in many cases "coA" is also defined. Here are some common examples: sine and cosine; tangent and cotangent; secant and cosecant; function and cofunction; morphism ...
6
votes
0answers
98 views

How to effectively read a mathematical textbook?

I am self learning mathematics and here are some of the tips and techniques I follow while I go through texts like Rudin, Munkres, Artin etc.. I would request the community to mention more ...
1
vote
1answer
68 views

Mathematical Statistics book suggestion

I recently took a very demanding course on statistics. One of my main difficulties (I am an economist) was to "visualize" joint and conditional distributions, and difficulties in calculating integrals....
5
votes
2answers
109 views

What mathematical consequences might there be if Euler Mascheroni constant is rational?

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant. Since we cannot prove ...
5
votes
1answer
53 views

How to interpret objects and symbols in written mathematics?

When reading mathematics how do you intuitively interpret the given objects such as sets, functions, sequences, etc? do you automatically visualize them graphically? When dealing with sets $A, B, X, ...
2
votes
0answers
67 views

Solution of the congruence $X^5 \equiv 1 \pmod {25}$ with lift

Can someone explain me the right steps for the solution of this congruence using the method of the lift? $X^5 \equiv 1 \pmod {25}$ I know that I can write this as $f(x) = X^5-1 \equiv 0 \pmod {...
3
votes
1answer
80 views

Which subspaces of $\mathbb C^n$ are spanned by real vectors?

Which complex $k$-dimensional subspaces of $\mathbb C^n$ are spanned by real vectors? Can we characterise them? (here $1<k<n$). By "complex", I mean that I am interested in subspaces $W \le \...
4
votes
2answers
106 views

Are there 'serious' mathematical problems emerging from data science?

Are there any proper research-level mathematics problems that have come out of the activity known as "data science"? I put the quotes because it really is not so clear what "data science" is to me; I ...
0
votes
0answers
23 views

Are there any proof assistants based on logic programming?

Logic programming is a programming language paradigm. In it, a programmer creates a bunch of axioms in the form of horn clauses, representing computations, which the implementation of the language ...
1
vote
1answer
64 views

All the derivatives of distributions are also distributions, but what about the converse?

Say you have some linear functional $f$ well defined on $\mathscr{D}(\mathbb{R})$: then what if for some test function $\phi$ you have $$ -f(\phi') = g(\phi)? $$ If that $g$ defines a distribution,...
8
votes
0answers
101 views

Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
5
votes
2answers
169 views

“Milk” the integral $\int_0^\infty\left(\frac{x^2}{x^4+2ax^2+1}\right)^r\frac{x^2+1}{x^2(x^s+1)}\mathrm dx$

I found the following integral in chapter $13$ of Irresistible Integrals, and I would like to see which conclusions you can reach from it. My goal in asking this question is to see which methods I can ...
0
votes
2answers
58 views

Proving a theorem by proving a related one

I have a question that might be stupid, but it bugs me for some time. It's quite simple: Let's say we have a conjecture, such that if the conjecture is true then another theorem will be true. Does ...