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1answer
52 views

Launching a Plaintext Attack against Affine Cipher

Update 2 Being new to the world of Stack Exchange I did not realize that there exists a site solely devoted to cryptography. In light of this, I hope someone could help me migrate this question to ...
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2answers
41 views

Roots of $f(x)+g(x)$

Question : Let $p,q,r,s \in \mathbb R$ such that $pr=2(q+s)$. Show that either $f(x)=x^2+px+q=0$ or $g(x)=x^2+rx+s=0$ has real roots . My method : To the contrary suppose that both $f(x)$ and ...
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2answers
61 views

“The order of a differential equation is the highest derivative in the equation”. What's wrong with this statement?

I am asked the following "The order of a differential equation is the highest derivative in the equation". What's wrong with this statement? I've checked my text and several other sources, and ...
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1answer
25 views

Study of algebraic structures analogous to the ring of smooth functions and module of vector fields

$\newcommand{\Ga}{\Gamma}$ Let $M$ be a smooth manifold. $\Ga(TM)$ is a module over the ring of smooth (real) functions (which is also an algebra, and denoted by $C^{\infty}(M)$). Also, each $X \in ...
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1answer
62 views

The four colour theorem

I have been reading about the four colour theorem and the fact that it is proved using a computer. My question is whether it is likely that we will ever achieve a proof without the use of a computer? ...
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1answer
101 views

Grothendieck's “Relative” Point of View

I have often read that Grothendieck's insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why ...
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1answer
87 views

How important is the choice of books in studying Analysis?

I am in a fix. I have done a graduate course in Pure Mathematics.I love to study abstract algebra.I want to do postgraduate in Mathematics especially in Abstract Algebra . In order to enter a ...
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1answer
39 views

$\{x|x$ is a positive integer definable in one line of type $\}$

$\{x|x$ is a positive integer definable in one line of type $\}$ I found this example in Enderton's book of set theory ! What does it mean ? I have no idea about "definable" and "in one line of ...
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1answer
36 views

What can we say about a function $g$ if we know${f \over f + g} \to 1$?

This question came to me thinking about the prime counting function, $\pi (x)$. It is known that ${\pi(x) \over \text{li}(x)} \to 1$ but this does not imply $\pi(x) - \text{li}(x) \to 0$. I asked a ...
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1answer
17 views

$a_{A,B}=a_{A,E}+a_{E,B}$ - Relative acceleration

When two balls $A,B$ are moving under gravity , Find acceleration of $A$ relative to $B$ (That is $a_{A,B}$) a) Both $A$ and $B$ are moving down ! b) $A$ moves upwards and $B$ moves downwards (above ...
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When writing proofs, is logical notation a crutch?

I'm near the end of Velleman's How to Prove It, self-studying and learning a lot about proofs. This book teaches you how to express ideas rigorously in logic notation, prove the theorem logically, and ...
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39 views

Comprehensive Mathematics References/Textbooks (Like Bourbaki's Elements, or the Stacks Project)

Are there any comprehensive mathematics reference/textbooks that could be considered somewhat like a modern version of Bourbaki's Elements? "Comprehensive" here could refer to a single area of ...
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20 views

Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
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0answers
42 views

What's the name of this problem? Interesting minimisation of a length.

There is a problem which has to do with minimising the length of a (possibly disjoint) barrier in a region of space (often a 2D circle) such that no straight line can pass through the particular ...
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1answer
46 views

Can I take cryptography after doing pure Mathematics $?$ [closed]

This is not a mathematics problem . May be it's not appropriate to ask this here but I don't know anywhere else to go for advice . I have taken Pure Mathematics at the university and quite enjoying ...
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0answers
36 views

What does it allow to see Differential Geometry from an abstract viewpoint?

I am currently learning Differential Geometry from the "categorical" point of view. We use sheaves, ringed spaces, group objects,... to define smooth manifolds, vector bundles,... My previous course ...
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276 views

Finding the first digit of $2^{74207281}-1$ (new biggest prime record)

I heard that the record for finding the largest prime number was broken a few days ago with the following Mersenne prime $$2^{74207281}-1$$ also called $M_{74207281}$. Now my question is: it is ...
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2answers
56 views

Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
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1answer
54 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
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0answers
26 views

Name for “3D quadrilateral” shape?

I am interested to know the name of the following solid construction - kind of a deformed cube... but I don't know the name, or even a general name: Left and right faces parallel with the $yz$ axes ...
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1answer
77 views

Soft question: Does Spivak encourage students to look up identities?

For a few days i was stuck with the problem of proving $\sum_{k=0}^l\dbinom{n}{k}\dbinom{m}{l-k}=\dbinom{n+m}{l}$ I tried to look up the problem on google, but the answers were extremely complex and ...
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1answer
72 views

Multivariable optimization for time to build a ship in a game, and maybe some possible application in “everyday” life

I precise first that english is not my monther tongue and I may will not be as clear as I would like, just ask me question if you need, thank you. I am playing a game (Galaxy Empire) for a while, ...
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1answer
64 views

Looking Away from the Temptations of the Solution Key [closed]

This is quite a soft question and I believe that it is a very important one and one that many self-learners can relate to. So I recently was going through a problem set in topology and I came across ...
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2answers
173 views

Should I be worried that I can't prove these theorems on my own yet?

I'm a mathematics major, and when I'm studying Real Analysis, Numerical Analysis, etc, when I look at theorems like the Intermediate Value theorem, Mean Value theorem, etc, the results seem obvious ...
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1answer
133 views

Could there be an “$n$-th root” of the category $\mathsf{Set}$?

Here is a thought experiment: Suppose we did not know what sets and functions are. The general idea of a topos is, that it somehow serves as a foundation for mathematics. So let there be an alternate ...
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1answer
56 views

What is the significance of “Homomorphism”?

Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets? Homomorphism relates ...
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1answer
44 views

Balkan Olympiad in Mathematics 2001 [closed]

Where can I find the solutions of the problems from the Balkan Olympiad in Mathematics 2001, Belgrade?
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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 ...
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1answer
73 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 ...
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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 ...
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0answers
45 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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
36 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) = ...
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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: ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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.) ...
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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 ...
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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 ...
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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 ...
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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: ...