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2answers
56 views

Why are these two lemmas worth including and proving?

The following two lemmas are from Stein and Shakarchi (2005). (p4 Lemma 1.1) If a rectangle is the almost disjoint union of finitely many other rectangles, say $R=\cup_{k=1}^n R_k$, then $|R| = ...
2
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1answer
85 views

Does the partial order on $\mathbb{C}$ induced by $[0,\infty)$ have any non-trivial uses or application?

We can define a binary relation on the complex numbers as follows. $z \leq z'$ iff there exists $x \in [0,\infty)$ with $z+x=z'.$ Equivalently, $z \leq z'$ iff $\mathrm{Re}(z) \leq ...
4
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2answers
101 views

Size Issues in Category Theory

Barr and Wells state in their text Toposes, Triples and Theories (pdf link) It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. ...
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0answers
15 views

Finding the full text of an older article which is not online; specifically Dvoretzky's “Some results on convex bodies and Banach spaces.”

I am looking for the article in the title (more specifically, for one proof in the article), for which the bibliographic data is MR139079 (25 #2518) 52.30 Dvoretzky, Aryeh Some results on convex ...
1
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1answer
62 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
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5answers
2k views

Why teach linear algebra before abstract algebra?

Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...
2
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0answers
23 views

Analogy for Baire categories?

I'm looking for an analogy to grasp the intuitive notion of size that Baire categories on $\mathbb{R}$ provides. For instance, the cardinality of a subset of $\mathbb{R}$ provides a notion of size in ...
24
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2answers
2k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
0
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0answers
20 views

Big picture question about the Thurston Metric on Teichmuller Space

I'm having difficulty in appreciating the significance of the Thurston metric on Teichmuller space. It seems like a lot of work to develop the theory of this asymmetric metric space with fewer ...
0
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0answers
30 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
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7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...
13
votes
2answers
674 views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go . . . ] Here's my description of the game: There's a $4\times 4$ grid with some random, numbered cards on. The numbers are either one, two, or multiples of three. ...
6
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3answers
170 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
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0answers
66 views

Becoming a math's Lecturer?

What is the initial process to become a Mathematics's lecturer at a University in London. I want to be able to lecture in a Russell Group university. What academic ladder do i have to climb? Would ...
3
votes
3answers
122 views

Soft question: Lucrative careers for applied mathematics [closed]

I am currently a math/CS undergraduate. I enjoy applied math a lot and want to go to graduate school for it. I've been thinking that I will do a PhD, but I also know I don't want to stay in academia, ...
3
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1answer
75 views

Is there any way to analyze an absurdly large exponent?

On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was: Given a 1080p TV, how long would it take to view every ...
0
votes
1answer
93 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
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4answers
181 views

Applications of inflection points

Recently, I was teaching maxima, minima and inflection points to first year engineering students. I motived extrema by giving practical examples of optimization problems, but when a colleague asked me ...
0
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1answer
48 views

Is there a limit for how “good” a numerical method can be?

Multiplying two matrices $A \cdot B$ of size $n \times n$ in the trivial way requires $n^3$ computations. However, more efficient algorithms such as the Strassen algorithm have a lower complexity of ...
3
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1answer
66 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
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0answers
81 views

One year vs Two year MSc in Mathematics?

I am trying to choose between the MSc in Mathematics at Warwick and the ALgebra, Geometry and Number Theory(ALGANT) european integrated master program. The Warwick program lasts for one year while the ...
16
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11answers
769 views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
0
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0answers
10 views

Transformation Matrix of an Endomorphism

I am going through my script at the moment and there is an example given which I do not understand: Let $F \in \text{End}(\mathbb{R}^2)$ be represented by $ \begin{pmatrix} 0 & 1 \\ 1 & 0 ...
6
votes
1answer
62 views

Is there any advantage to the $a \equiv b\;\;(\mathrm{mod}\;c)$ notation?

Congruences modulo equivalence classes other than those defined by division remainders are ubiquitous in contemporary mathematics. It is not uncommon for a single mathematical argument to refer to ...
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2answers
53 views

Absorbing Mathematics [closed]

Exams are coming up, and I wanted to get the full spread of opinion here on MSE. What is the best way (in your opinion) to study and retain mathematics?
1
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0answers
35 views

Any quick mental mathematics techniques?

I recently came across a very easy way to find the squares of fairly large numbers ($1$-$100$, but could be extended indefinitely), by using a simple formula. It does require you to know the square ...
1
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0answers
38 views

Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
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0answers
29 views

Good, memorable explanations for relationship between functions and set operations

Recently I was in a position to explain basic relationships between functions and set operations (preimages commute with all set operations, images commute with unions, but they fail to commute with ...
0
votes
1answer
29 views

Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
6
votes
1answer
147 views

Very interesting graph!

I found a VERY interesting graph on http://www.xamuel.com/graphs-of-implicit-equations/. It looked very, very cool, but the equation of the graph is so simple! Here is the image of the graph: My ...
0
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0answers
25 views

If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
1
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1answer
66 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
0
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4answers
40 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
0
votes
1answer
64 views

Topics on Algebra-Ring Theory for an essay-project on undergraduate level!!

I'm an undergraduate in the mathematics field. So i want to be a little more productive and wanted to do an essay or project on Algebra Ring-Theory (because I got it as a course this semester -Ring ...
1
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3answers
64 views

Why are the axes in coordinate geometry perperndicular?

In coordinate geometry, the $x$ and $y$ axis are perpendicular to each other. But is there any special reason for this (other than to make it simple)? Will coordinate geometry have contradictions if ...
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2answers
83 views

How to evaluate this particular infinite sum? [closed]

Show that $$ ∑_{n=0}^∞p^n \cos(nθ)=\frac{1−p\cos θ}{1−2p \cos θ+p^2} $$ if $|p|<1$ Read more: http://www.physicsforums.com
1
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1answer
62 views

How to get better at inequality problems?

I really like inequality problems, but I've found that I have severe difficulties with ones that should be simple. I can prove something simple like this For positive $a, b, c$ with $a + b + c = 6$ ...
2
votes
2answers
92 views

Dr Seuss style prose advanced mathematics text

There are a number of Dr Seuss style mathematics books for elementary school mathematics. It occurred to me that there is no particular reason that that style of prose couldn't be adapted to a ...
0
votes
1answer
65 views

Can you use a calculator on the GRE Math Subject Exam? [closed]

Well, my princeton review GRE Mathematics Practice Book just came today..I am so excited to enroll in MIT (hopefully!). Can we use our calculators on the real test? Thanks.
3
votes
2answers
83 views

De Morgan's laws in logic and set theory

In logic De Morgan's law means $\lnot (A \land B) \Leftrightarrow \lnot A \lor \lnot B$ In set theory De Morgan's law means $(A \cap B)^C = A^C \cup B^C$ I'm surprised that the same idea is true in ...
2
votes
1answer
125 views

A beautiful identity of $\sin(x)$ [duplicate]

When I was in high school, I had proved that $$\sin^2(x)-\sin^2(y)=\sin(x-y)\sin(x+y) $$ I think it is beautiful since it resembles the identity $a^2-b^2=(a+b)(a-b)$. But I can not find it in ...
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0answers
24 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
0
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2answers
53 views

Is the collection of all functions between two sets a set?

Can we say "the set of all functions between two sets" as easily as we could say "the set of all real numbers", for example?
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2answers
142 views

(soft) How hard REALLY is Advanced Calculus I and Abstract Algebra I [closed]

WTF everyone here downvote Arthur Fischer he is asking for it!!! This next fall, I will be taking Advanced Calculus I and Abstract Algebra I, along with Physical Chemistry I, Computer Science I, ...
1
vote
1answer
36 views

The Jacobi nome $q$

Does anyone know why $q = e^{-\pi K'/K} = e^{\pi i \tau}$ is called the nome? Is there a historical reason? Does the word nome mean something in Latin or German?
5
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2answers
136 views

Why is probability so unintuitive to us? [closed]

There are so many famous paradoxes which are examples of how humans are unable to intuitively understand probability -- there's a discrepancy between their supposed actual experience and the ...
2
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0answers
75 views

Should I drop Differential Equations?

For many years, my university offered only one Bachelor's degree in math, "Bachelor in Mathematical Subjects." Here, Differential Equations is a mandatory part of the degree. Further specialization in ...
3
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1answer
104 views

Why are mathematical results discovered by multiple people independently?

This is a meta question. No this isn't a meta question about site, this is a meta question about maths itself. It has been observed quite a lot of times, that around some point in history,maybe ...
59
votes
14answers
5k views

Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?

I'm a second year math student. And I've the following problem. When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the ...
2
votes
1answer
59 views

Is there any interesting interpretation of the set of all functions between two sets?

Is there any way to interpret the set of all functions from a set $X$ to a set $Y$? There is an interpretation of it as the cartesian product of $X$-many copies of $Y$, but I am asking for a more ...