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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.

0
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0answers
6 views

Understanding iterated covariant derivatives to define Sobolev spaces on manifolds

I'm having big troubles understanding the definition of Sobolev spaces on manifolds. Ok, so we have a Riemannian manifold $(M, g)$, and then we can define a natural riemannian measure (which I will ...
0
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0answers
44 views

Notations in Functional Analysis: $L^p$ and $L_p$ (now also with $\mathscr{L}^p$, $\mathscr{L}_p$, $\mathcal{L}^p$, and $\mathcal{L}_p$)

If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$...
0
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0answers
14 views

Indexing Variables

When picking variables it is common to use prime marks, as in $a, a',a'', ...$, and numbers $a_1, a_2, a_3, ...$. A third option is to use distinct letters $a, b, c, ...$. For example, some people ...
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0answers
31 views

Year to Prepare for the Putnam Exam? [on hold]

I will be taking the Putnam Exam next December and want to prepare well for it. I have never really had competition experience although I have spent time doing brain teasers and math puzzles. ...
0
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0answers
30 views

What is the difference between a “ line ” and a “ straight line ”?

Is there actually a difference between a line and a straight line ? Is figure 1 a line . ? . Should I take help from " Euclid "? I believe according to " Euclid " the above figure is a valid line.
1
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1answer
41 views

Is it trivial that the Levi-Civita connection can be pulled back to a isometric manifold?

Let $M, N$ be Riemannian Manifolds and $\phi: M \to N$ a isometric diffeomorphism. We know that we have unique Levi-Civita Connections $\nabla^M, \nabla^N$ on $M, N$ respectively. One can check that $\...
2
votes
2answers
136 views

Differential equation with “backwards product rule”.

If we have the following differential equation, ($h,f$ known, $y$ unknown): $$f'(x)y(x) + f(x)y'(x) = h(x)$$ it would be easy, since we could spot the derivative for a product: $$(f(x)y(x))' = h(x)$...
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1answer
52 views

How “deep” is the theory of encryption keys? Can a “generalist” approach it or does one need to be a number theorist? [on hold]

How "deep" is the theory of encryption keys? Can a "generalist" approach designing new keys or understanding state of the art "security" or does one need to be a number theorist?
2
votes
2answers
46 views

Integrating $\int \sec xdx$: Why is $\ln|\text{sec}x + \text{tan}x| + C$ preferred over $\tanh^{-1}(\sin x) + C$?

I was trying to integrate $\sec^3x$ and discovered that I would have to integrate $\sec x$ in the process. I had not seen the "standard" approach and came up with my own solution, which is apparently ...
-2
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0answers
18 views

How do I preview my post before posting? [migrated]

How do I preview my post before posting? I do not find any preview button.
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1answer
36 views

(Soft Question) Largest known semiprimes with no known factors

Is there a list, similar to prime numbers and probable primes, of the largest semiprimes with unknown factors? Is there a list of numbers that are either semiprime or prime, with no known factors? Is ...
-1
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0answers
46 views

advice for beginner in representation-theory [on hold]

Thank you for you reading.I am in my second year of my undergraduate and I plan to study representation-theory in the future. I had courses in linear algebra, abstract algebra(Hungerford chapter I-V ...
1
vote
1answer
37 views

Can an algebraic function contain transcendental constants?

Most definitions I've seen [1] [2] are agnostic about what kind of polynomials the function must satisfy. PlanetMath, for example, says A function of one variable is said to be algebraic if it ...
9
votes
0answers
81 views

Peculiar pictures in advanced maths books

I have recently started reading Introduction to Symplectic Topology by McDuff and Salamon and I came across this picture: I find it very funny and really interesting. I read on Wikipedia that Ian ...
3
votes
1answer
52 views

Is there a name for the function $D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$, where $x$ and $y$ are vectors in $\mathbb{R}^n$?

I am studying real analysis and encountered this function $$D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$$ where $x,y$ are $n$-dimensional vectors living in $\mathbb{R}^n$. It is easy to show that is ...
2
votes
1answer
61 views

How is ultrafinitism imprecise?

This is similar to "Why isn't finitism nonsense?" but instead of asking about the practical nature (applications), or instead of asking about the definition, I'm trying to understand how ultrafinitism ...
1
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0answers
120 views
+50

Alpha Test ZoomSpace - A futuristic mathematician's tool.

ZoomSpace v0.1 Alpha Remark. This version only has diagram editing capabilities, and plenty of bugs for you to report. Future versions, maybe in about 2 months, will have basic diagram chasing ...
0
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0answers
24 views

What are some prerequisites for studying Elliptic Curves over $\mathbb Q$?

Suppose you had an undergraduate friend who's looking to take an introductory course on Elliptic Curves over $\mathbb Q$, in the context of Number Theory. The difficulty is around the level of ...
-1
votes
0answers
26 views

I haven’t studied mathematics for a year and forgot everything. How do I re-learn? [closed]

I’m doing GCSE’s and I have been slacking off for the last year. The reason is traveling. I came here to ask in what order do I re-learn high school mathematics without feeling overwhelmed. They are ...
5
votes
1answer
86 views

Having problem with tom Dieck's algebraic topology text

(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.) This question is really soft. I'm having problem reading this text. Let me elaborate. I found this book too ...
0
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0answers
41 views

Learning the fundamentals [closed]

I am a student and I don't feel great in Analysis .... as if I am lacking some early foundation in my stack. so I wanted to go back and brush up all aspects a little bit ... do you guys recommend this ...
4
votes
2answers
74 views

Is there a “solution” to the ordinal game?

Even though I have almost no background in logic, I find the idea of ordinal notation quite interesting. It seems that the idea is to come up with notation to define larger and larger numbers, until ...
3
votes
4answers
149 views

Request for crazy integrals

I'm a sucker for exotic integrals like the one evaluated in this post. I don't really know why, but I just can't get enough of the amazing closed forms that some are able to come up with. So, what ...
1
vote
1answer
46 views

Exercises for commutative algebra

I'm currently studying for my final exam of commutative algebra. In the class we covered Artinian rings, Dedekind domains, Integral closures, Grobner basis, (discrete )Valuation rings... I'm ...
1
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0answers
17 views

Proving inequality relation between two complex numbers and a positive real parameter. [duplicate]

Question: Prove that: $$|z_1+z_2|^2 \le (1+c)|z_1|^2+(1+{1\over c})|z_2|^2$$ where $z_1,z_2$ are complex numbers and $c$ is a positive real parameter. Solution: We can write $$|z|^2=z\...
8
votes
2answers
599 views

Graduate School (Can I be a Mathematician?) [closed]

I am posing this question for myself, but many people surely are wondering the same, so I think this is valuable for others as well as myself. There are similar questions on the site but they do not ...
0
votes
1answer
47 views

Is any cross-fertilization occurring between the disciplines of game semantics and set theory?

I've recently been introduced (at a superficial 'wiki' level) to the Axiom of determinacy and Descriptive set theory. While looking up Witness_(mathematics), I found the link to Game semantics. ...
0
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0answers
70 views

Soundness of Calculus?

In this wikipedia article. It talks about the "soundness of calculus", but it seems to talk about soundness in an informal sense and how analysis/calculus was not very rigorous and not the soundness ...
2
votes
1answer
24 views

Weak Closure, clarification

I'm working on the following question: Show that the closed unit ball $B(0,1)$ in a normed space is also weakly closed. I think I want to show that $$\lim_{n\rightarrow \infty}f(x_n) \in f(B(0,1)) ...
2
votes
1answer
74 views

Can it be useful to think of functors as representing themselves ?

Here's a thought I had and I wonder if it can be of any use, for instance has it ever helped proving any result (however minor the result). Say you're in a situation where you have some objects in ...
6
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3answers
95 views

prove $\sum_{n=0}^{\infty}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$

I am seeking alternate proofs for $$\sum_{n\geq0}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$$ Here's mine: Recall that, for $x\in(0,2)$, $$\frac1x=\sum_{n\geq0}(1-x)^n$$ Hence we have ...
0
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0answers
37 views

Best software to do big number calculations quickly

I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (1000+ digits ). I am currently using python to test these numbers. In the ...
4
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2answers
82 views

What is the significance of Group Automorphism?

I have understood the definition of group automorphism and have studied various examples for the same. But what is the significance of an automorphism? When we study isomorphisms, we try to ...
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0answers
40 views

Book recommendation for this type of series

I'm looking for a book which deals with the following type of series.That is, I want a book which explains how to find the sum of the series, for example, like the following: $$1+\frac{2}{6}+\...
6
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0answers
88 views

Where to publish expository mathematics/personal notes?

I have a short expository paper that is more or less my personal take/exploration of a topic that interests me. I submitted to Mathematics Magazine and it was rejected for non-mathematical reasons - ...
4
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0answers
25 views

Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. ...
5
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0answers
71 views

How do you stay focused in advanced (pure) mathematics lectures?

Throughout my entire university career, I've found it incredibly difficult to focus during lectures, so much so that I at one point stopped going because I found it easier to just learn it on my own. ...
1
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0answers
40 views

Relationship between Null Homotopy and Homology of Spheres

I am reading about some of the historical motivations leading up to the discovery of the Hopf Fibration in 1931, but I am having some trouble with some intuition behind why the map was such a shock to ...
2
votes
1answer
95 views

A guide to Algebraic Geometry

I have completed one semester course on Commutative Algebra and Riemann Surfaces, and currently I am trying to read Algebraic Geometry. While reading from different books I feel that I must need a ...
1
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1answer
26 views

If $\sup A\not\in A$ then $A$ contains a countably infinite subset

I am required to show that if a bounded non empty set $A\subseteq \mathbf{R}$ is such that $\sup A\not\in A$, then $A$ contains a countably infinites subset. Now my idea is that for each $k\in\mathbf{...
1
vote
2answers
76 views

Does there exist a similar identity to $\binom{n}{k} = \binom{n}{n-k}$?

I know that $$\binom{n}{k} = \binom{n}{n-k}$$ My question is does there exist a similar identity where you change the top of the choose function, o any similar to the identity above? An example: Can ...
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votes
1answer
47 views

Writing a paper about some findings or not [closed]

If you develop a new mathematical expression for a function in an open usolved problem in mathematics, do you write a paper about it, or do you bring it into the mathematics community first? What ...
1
vote
1answer
18 views

Proof that sum of two subspaces is another subspace

$U_1,U_2$$⊂V$ be subspaces of V (a vector space). Define the subspace sum of $U_1,$ and $U_2$ be defined as the set: $U_1 + U_2$ $=$ {$u_1 + u_2 : u_1 ∈ U_1, u_2 ∈ U_2$}. Let $A$ denote the set $U_1+...
1
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0answers
30 views

Suggestion of a topic to give a seminar on [closed]

I've been attending a seminar series titled Geometric Structures on Manifolds these past few weeks as a part of my semester requirements, and therefore I'm expected to give a seminar as well. The ...
1
vote
1answer
30 views

How to make a kid understand geometry and help him solve problems?

I'm tutoring a 13 year-old boy, a middle school student. He has almost no problem with elementary algebra: he just applies the rules and everything falls into place. However he does struggle with ...
4
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0answers
63 views

Is there a proof of an infinite number of prime numbers using the irrationality of $e$?

That the set of prime integers is infinite can be proved using the irrationality of $\pi$; see this wikipedia link. It analyzes the representation $\tag 1 {\displaystyle {\frac {\pi }{4}}={\frac {3}{...
3
votes
0answers
64 views

Start understanding analysis [closed]

Currently, I have read and re-read the Stephen Abbott Understanding Analysis for about $3$ or $4$ times up to and including the chapter $6$. However, I can't say that now I am feeling more confident ...
0
votes
0answers
91 views

Course recommendation if I Iiked Analysis and Topology

If I have studied Real Analysis, Analysis (with metric spaces) and Topology courses and I really them all, What would be other courses that I would also really like ? Based on this 3 courses ...
2
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0answers
50 views

What's the meaning of characteristic in “characteristic polynomial”?

Why the polynomial that has the eigenvalues as roots called characteristic? Is there any special meaning in the term "characteristic"?
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1answer
46 views

What are the base assumptions we make in mathematics?

In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-...