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1answer
45 views

Balkan Olympiad in Mathematics 2001 [closed]

Where can I find the solutions of the problems from the Balkan Olympiad in Mathematics 2001, Belgrade?
0
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0answers
50 views

A good primer for Concrete Mathematics?

I've been watching MIT's Mathematics for Computer Science, from Fall 2010 whilst reading Concrete Mathematics. Honestly the topic seems like a hodgepodge of ideas. I can follow about 2/3 of the ...
2
votes
1answer
74 views

What is the easiest proof you know for the Jordan Canonical Form

I read numerous demonstration of the existence of the Jordan Canonical Form, but all of them involve more than 2 pages of demonstration with numerous lemmas in between. I'm writing some notes for ...
1
vote
0answers
34 views

What is the use of the Rofe-Beketov formula?

Let $y_1(x)$ be an elementary solution of the Sturm-Liouville equation, $$ \frac{d}{dx}\left( p(x)\frac{dy}{dx} \right) + q(x)y = 0 $$ It is well known that a second linearly independent solution ...
0
votes
0answers
47 views

Is there a sense in which a formal theory can be more or less “metamathematically aware”?

For example, in classical logic, because of the law of the excluded middle, all propositions must be true or false, so in order to prove that a proposition is actually independent of the theory, we ...
2
votes
0answers
21 views

Website for sharing solutions/proof verification?

Is there a website for sharing solutions to exercises in math books? I'm self-studying math and I find solution manuals like this very helpful. When I do an exercise, I usually scribble down a few ...
4
votes
1answer
52 views

Can we create an “integration by parts” with quotient rule?

Product rule says that $(uv)' = u'v + uv'$, so $\int (uv)' = \int (u'v + uv')$ implies $uv = \int u'v + \int uv'$ and this implies $$\int uv' ~dx = uv - \int u'v ~dx$$ This is integration by parts. ...
3
votes
1answer
61 views

Why do regulated functions receive so little attention in elementary analysis courses?

The only place where regulated functions (= such with one-sided limits everywhere) occasionally seem to come up in elementary analysis courses is in connection with integration, yet there are clearly ...
0
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0answers
35 views

Geometric and graphical interpretation of the fundamental theorem of calculus

We're learning the fundamental theorem of calculus and I'm trying to wrap my head around the theorem intuitively. These are basically my thoughts, the questions are at the end. Suppose you have some ...
1
vote
2answers
64 views

No Galois Theory in Godement's Cours d'Algebre?

I just procured an English translation of Godement's Cours d'Algebre and was interested in reading the treatment of Galois Theory. I started to look for the relevant chapter in the ToC, but to my ...
1
vote
3answers
139 views

Everything in math that we have found and proved to be TRUE so far will remain true forever?

Is there any mathematical statement or theorem or theory, which was used to be TRUE in the past, but then found out FALSE later? In short, my question is: everything in math that we have found and ...
2
votes
0answers
37 views

Euler's Totient Function in Other Rings

I'm looking for rings other than the integers on which we could define an interesting analogue of Euler's Totient function. E.g., on a Euclidean domain with norm $N$ we could let $\phi(x) = ...
0
votes
1answer
45 views

How many dots do I have to write?

This seems very odd and silly. But I do not know where else to ask. This question occurs to me whenever I write an infinite sequence, sum or decimal points etc. Ex: $ 1.2 + 2.3 + 3.4 + ……………$ Ex: ...
1
vote
2answers
63 views

Intuition for the construction of the product topology and its equivalence to the euclidian metric

While I have been provided a proof for the previous statement, I still cannot fully grasp why the euclidian metric [ $d(x,y)=((x_1-y_1)^2+...(x_{n}-y_{n})^2)^{1/2}$] generates the same topology as the ...
0
votes
1answer
62 views

What does “within the same order of magnitude” convey?

This question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more ...
2
votes
0answers
48 views

Embeddings into symmetric structures

In the recent months I've come across a phenomenon which seems to come up in several areas of algebra making me wonder if there's a larger concept behind it, which I just fail to grasp. Namely, ...
1
vote
1answer
27 views

Do (systems of linear equations with scalars and unknowns from different algebraic structures) occur widely?

Generally in linear algebra one studies systems of linear equations where both coefficients and unknowns belong to the same field. I would not be the first person to notice that a system like ...
7
votes
2answers
122 views

are there different kinds of math? [closed]

I do not mean branches such as functional analysis I mean is math we use in elementary school (which I heard uses Peano's axioms) the 'correct' math? Is there math that uses other axioms? Is ...
1
vote
4answers
82 views

How to approach linear algbera after abstract algbera.

I'm a high school student taking classes at a local college, and because of this I've taken classes in an unusual order. In particular, I took abstract algebra I (focused on group theory) last ...
1
vote
2answers
56 views

How to solve this Table? [closed]

This is a solved, filled table. I'm trying to understand how it was put together. Numerically, the first half of the chart is easy to figure out, (The parts in red are resultant in simple addition.) ...
3
votes
2answers
36 views

parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which ...
1
vote
1answer
70 views

If $L$ is a line bundle on a scheme $X$, what is the ring $\oplus_{n \geq 0} \Gamma(X, L^{ \otimes n})$?

If $L$ is a line bundle on a scheme $X$, what is the ring $A = \oplus \Gamma(X, L^{ \otimes n})$? This ring comes up in an exercise that I am struggling with right now, and I would like some insight ...
2
votes
0answers
19 views

Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
0
votes
0answers
29 views

Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: ...
6
votes
3answers
168 views

What are the disadvantages of non-standard analysis?

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more ...
7
votes
1answer
78 views

How are groupoids richer structures than sets of groups?

This has been bugging me for quite some time: My intuition with categories is, that I can simply identify isomorphic objects. It does for example not matter, whether the entries in a sudoku are the ...
3
votes
3answers
81 views

Why integration constant is real?

OK, we are all taught at school that the undefined integral of a function $f(x)$ is $$\int f(x)\;\text{d}x = F(x) + k$$ where $F'(x) = f(x)$ and $k \in \mathbb R$. But, why $k$ must be real? I know ...
3
votes
1answer
47 views

Meaning of $\vee$ notation after an integral

I have an equation with the $\vee$ notation that I've not come across before. $$ \tilde f(\omega, t) = e^{-2\pi i\omega t}\int_{-\infty}^{\infty} e^{2\pi i v t} \, \bar{g}(v - \omega) \, \hat f(v) ...
12
votes
3answers
281 views

What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure ...
35
votes
4answers
3k views

What's between the finite and the infinite?

I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" ...
0
votes
0answers
49 views

Comparing the methods of applied mathematics to computer graphics

As an applied mathematician working towards my PhD, I have some personal interest in aspects of computer graphics and procedural animation. Looking up people and reading papers in the field of ...
0
votes
1answer
30 views

Continous function approximating the precision of a number.

Let us say we have a number $c \in [0,1]$ in some basis: $$c = \sum_{k=1}^{\infty}a_kb^{-k}$$ For instance b=2, $a_k \in\{0,1\}$ would be binary and $b=10, a_k \in \{0, \cdots, 9\}$ would be decimal. ...
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votes
3answers
177 views

What does $3+2i$ apples mean? Can the simple counting analogy with apples be extended to complex numbers? [closed]

Please read, or at least skim the question. Past attempts at answering the question have ignored vital constraints provided below. Natural Numbers Imagine I have $n$, a positive natural number, of ...
2
votes
0answers
56 views

Increased Math Productivity on Mobile Devices [closed]

I have been answering questions on Math.SE for a (relatively) short time now, and I have found that answering questions on my Android phone is somewhat of a nightmare. I end up commenting much of the ...
2
votes
1answer
33 views

Strategies on a symmetric chess play

The idea is that the difficulty of the game of chess is derived primarily from the asymmetry between the king and queen. all other chess pieces are arranged symmetrically and can move symmetrically, ...
1
vote
0answers
23 views

summer program for college freshman

I'm a college freshman and I'm studying mathematics. Do you know any summer programs that would be suitable for a European student? If you have any personal experience, can you tell me about it? ...
5
votes
1answer
84 views

Why Riemann hypothesis and not Riemann's conjecture

I have a stupid question. We say Erdös's conjecture, Goldbach's conjecture, Beal's conjecture... and so on. But we don't say 'Riemann's conjecture.' Instead we use the word 'hypothesis'. Why?
4
votes
1answer
53 views

Limit of a sequence of algebraic structures

(Disclaimer: I'm a novice in algebra. I'm at Chapter 13 of this free online abstract algebra book. This is a soft-question, because it's a loose "does this idea make sense" kind of question, instead ...
33
votes
7answers
2k views

What are some math concepts which were originally inspired by physics?

There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as ...
1
vote
0answers
63 views

Theorems in math that have lead to significant development in other areas of mathematics? [closed]

Several theorems in mathematics are guided by a sheer curiosity, but at times, certain tools are created out of necessity. Are there any theorems in mathematics, that although bear, have no ...
1
vote
2answers
35 views

Pronunciation of Permutation/Combination Notation and Ordered Pairs

Quick question: How am I supposed to pronounce (or read in my head) the following? $P(n,k)$ or $C(n,k)$ Also, how am I supposed to pronounce (or read in my head) this? $K \times M = \{(x,y): ...
2
votes
10answers
865 views

What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
1
vote
2answers
64 views

example of a function that could only be defined recursively

is there a function that can be proved is only defineable recursively? the converse seems to be trivially false, i.e. every iteritive function is trivially defined recursively with 0 for coefficients ...
2
votes
0answers
29 views

General concept of filtrations of algebraic structures

Question What is the essence, the general idea, behind the concept of a filtration of an algebraic structure? Background From linear algebra, I know that a filtration of an $n$-dimensional vector ...
23
votes
3answers
534 views

List of old books that modern masters recommend

This is a fairly unambigious question but it hasn't been asked before so I thought I would ask it myself: Which old books do the modern masters recommend? There are old books where the mathematical ...
0
votes
1answer
24 views

Writing the needed or the known as clarification?

Suppose $\kappa > 2 \ $ and $a,b \ $ two positive real numbers satisfying $a > b$. If somewhere in a proof about $a, b \ $ I would need the inequality $a^{-\kappa} < b^{-\kappa}$, how should ...
0
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0answers
11 views

What are some applications of transformation semigroups?

I have been told that transformation semigroups have applications to statistics, computer science, and combinatorics. What are some basic (and if possible, simple) examples of transformation ...
2
votes
7answers
657 views

Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...
1
vote
1answer
36 views

Can we assign a number to each theorem stating its complexity?

I was wondering if inside an axiomatic theory it could be possible to assign each theorem a number that indicates its complexity. Theorems with small complexity numbers would be "almost axioms"; if ...
-1
votes
2answers
92 views

How can we apply Algebra in our life?? [duplicate]

We all know that Math is the science's mother and it's applied in all areas, I have an idea about where functions and geometry calculus ans stats are applied (in economic, banks, building ...), but I ...