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0answers
72 views

Things you've believed for a long time were true, but are false in reality [duplicate]

Do you have any things (mathematical statements, statements about mathematics) you've believed for a long time were true, but now with enough mathematical knowledge you realize were wrong? For ...
2
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5answers
146 views

Is $0$ the midpoint of $(-\infty,+\infty)$?

Is $0$ the midpoint of $(-\infty,+\infty)$? Intuitively, I'd think so, and trying to refine my intuition as to why I'd think so, I would say that this is the case because there is a one-to-one ...
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1answer
44 views

Books and/or online resources on solving problems.

What are some good resources(online, books) that teach you how to tackle difficult and ugly problems in higher math arranged by subjects(analysis, topology, ODEs, groups etc) or topics(polynomials, ...
0
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1answer
192 views

Starting Calculus with a weak foundation in Pre-Calculus

I am struggling in Pre-Calc mathematics, and I want to know is it ok if I start Calculus I with a weak foundation in Pre-calculus mathematics? I understand the general gist of limits, function ...
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1answer
65 views

good lecture series for complex analysis?

I was wondering if i could teach myself complex analysis. Any good lecture series out there on the internet available for free (preferably)? One that would be suitable for an undergraduate student?
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2answers
93 views

Low Level Books on Conjectures/Famous Problems

I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses ...
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2answers
92 views

What do people mean by “(this piece of maths) is hard/difficult”?

Sometimes when I talk to my maths professors, they would say "this piece of maths (e.g. differential geometry) is hard". What do they exactly mean by "hard" (difficult) - when clearly they're the ...
2
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1answer
83 views

How do you use reference books?

Reference books at the research levels often does not include any problem or exercise. While you can't read these books like novels(you normally need to work on other sheet of paper), I'm just ...
2
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3answers
611 views

What languages to learn for maths? [closed]

For people with "hands-on" experience in mathematical research, what languages are the most beneficial to learn? I know that many graduate programs require some degree of non-native lingual ...
0
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1answer
36 views

Evolution of Relations

In Frege, one finds relations treated as predicates in complex terms. However, modern set theory appears to treat them as two-place relation. Is this correct? If so, when did this shift occur and to ...
0
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1answer
53 views

contact point and point of intersection

I am just unable to understand the definitions of contact point and point of intersection.My doubts can be summed up into the following two questions : 1) Suppose $f(x)=x^2$ and $g(x)=0$ are two ...
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1answer
235 views

Path to Differential Geometry

What do I need to learn to start on the rigorous study of differential geometry? I'm about to start my 3rd undergrad year at school, and have taken Cal 1-3, Linear Algebra, Elementary Number Theory, ...
2
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4answers
143 views

How does one explain basic probability theory to a layman?

I have recently been involved in a number of discussions with people with little or no background in mathematics when we considered a problem of the following shape. A random event is going to ...
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1answer
172 views

Less Terse alternative to Advanced Calculus by Folland.

I am currently in an advanced calculus class in university. We use Advanced Calculus by Folland. When I try to follow along the book I find that it is not verbose enough, and has too few examples. I ...
8
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3answers
595 views

What level of rigour is expected in Real Analysis?

I fail to find a duplicate. I am wondering what level of rigour is needed in a typical undergraduate course in Real Analysis. To clarify my question, I provide an exercise from Rudin and my proposed ...
1
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1answer
54 views

How can I know which theorem to use to prove another one?

In class this year a part of what we do is re learning theorems etc from previous years, but a more rigorous way. However, when I suggest a way to prove those theorems/properties/..., I often get an ...
7
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0answers
104 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
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2answers
169 views

Is the exclusion of uncountable additivity a drawback of Lebesgue measure?

A friend and I were having a discussion about Lebesgue measure. I attempted to be profound by making the following points: Analytic geometry has been a fantastic tool, but the concept of ...
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2answers
5k views

Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
1
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2answers
83 views

Why are all non-polynomial functions are basically exponents?

There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. For example, $\log$ is simply inverse of ...
7
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4answers
740 views

The role of 'arbitrary' in proofs

Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..." My question is, why ...
8
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1answer
158 views

Follow up to Pinter's abstract algebra

I wanted to learn abstract algebra this summer so I bought Pinter's A book of Abstract Algebra. I was planning on reading it over the course of the summer, but just finished the last problem of its ...
2
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0answers
74 views

Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
2
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1answer
148 views

Most dificult concepts in mathematics [closed]

(Sorry, last soft question!) Borwien, in The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, (probably quite rightly) says of the Riemann Hypothesis, that No layman has ever ...
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51answers
5k views

What was the book that opened your mind to the beauty of mathematics? [closed]

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...
2
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1answer
62 views

Varying definitions of symmetric and selfadjoint operators

There seems to be some disagreement (at least when consulting textbooks), what constitutes a symmetric operator and what constitutes a selfadjoint operator (of course, only the unbounded case is of ...
0
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1answer
57 views

Trying to translate the paper “COHOMOLOGIE ET GROUPE DE STEINBERG RELATIFS” by J P Loday J Alg (54) 178, 1978

I am trying to translate the paper "COHOMOLOGIE ET GROUPE DE STEINBERG RELATIFS" by J P Loday J Alg (54) 178, 1978 using google traslator. In section 1, he writes Un morphisme d’extensions relatives ...
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3answers
58 views

Name of quantity that is not invariant, but only changes in one direction

How do you call a quantity that is not an invariant, but only changes in one direction during the process? Example: The degree of the polynomials go down when Euclidean division is applied, so the ...
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1answer
45 views

Anyone worked with this particular orthogonal matrix

In my recent studies of quaternions, the following orthogonal matrix has come up. For example, it is related to the matrix representation of quaternion multiplication. Has anyone seen it come up in ...
58
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3answers
7k views

Mathematical research of Pokémon

In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some ...
6
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1answer
111 views

Are there categorifications of prime or irreducible elements (of a ring, say)?

I'm very sorry if this is a duplicate in any way or is otherwise a stupid question. I've looked around (for quite a while) but . . . no luck. There's a categorification of what it means to be an ...
6
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1answer
349 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
3
votes
0answers
85 views

Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
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1answer
121 views

having great difficulty in understanding long math problems ! any advices !!! please [closed]

I'm not an English origin I'm good at direct math exercises But I'm having great difficulty in understanding long math problems Such as described below Which contains many sentences and many terms ...
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2answers
121 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
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3answers
367 views

About Zeno's paradox and its answers

I read this question How can Zeno's dichotomy paradox be disproved using mathematics? . The first (ie the one on the top) answer uses the fact that $\sum\limits_{n=1}^\infty\frac{1}{2^n}=1$, ...
0
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1answer
82 views

Where should I go for translations of mathematical texts?

I am currently trying to read Applications algébriques de la cohomologie de groupes. II: théorie des algèbres simples by J-P. Serre. It is very hard for me to read this article since I'm not a native ...
4
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0answers
225 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
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3answers
797 views

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
23
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4answers
959 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
2
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1answer
36 views

Can you interpolate a function $f: \mathbb{R} \rightarrow \mathbb{R}^2$ piecewise (by two interpolations)?

I am currently trying to improve on-line handwriting recognition. On-line means in this case that I have the information how the symbols are written as a list of $n$ tuples of coordinates $(x(t_i), ...
0
votes
0answers
81 views

Difference between isomorphisms, deformations, diffeomorphisms and homeomorphis,s

I am learning about complex algebraic varieties and have encountered several equivalence relations: isomorphism, birational, deformation equivalent, diffeomorphic and homeomorphic. I consider all of ...
1
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1answer
69 views

Concerning a specific family of recursive sequences

There is a family of recursive sequences that I came up with: $$a_n=\text{sum of factors less than $a_{n-1}$ of } a_{n-1}, a_0\in\mathbb{N}$$ First question: Has it been studied before? Here are ...
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2answers
143 views

Book: Functional Calculus

Is there a good book that investigates in detail the various kinds of functional calculus? I'm having now some knowledge about unbounded operators and integration but I would like to understand ...
47
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7answers
5k views

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
3
votes
1answer
59 views

Is there a reason that sine substitution is preferred to cosine substitution?

When integrating by trigonometric substitution that has $\sqrt{a^2-x^2}$ in the integrand, the general recommended approach is to make the substitution $x = a \sin\theta$, and this substitution is the ...
6
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1answer
198 views

How can I choose the best algorithm to integrate ODE's numerically?

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
3
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4answers
2k views

I want to learn math from ground-up, basic to advanced, beginner to expert

I want to learn math. I've learned math long time ago, but i hardly remember anything. I really want to relearn but have no idea where to begin. I want to learn math by reading through good books, ...
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0answers
29 views

What [precisely] is special, algebraically, about fundamental Pell solutions?

I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation. As a concrete [numerical] example, ...
2
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1answer
90 views

Best multivariable calculus text for physics [duplicate]

I will soon start studying electrodynamics from Griffith's Electrodynamics. I tried to learn the math required from the first chapter but found that I couldn't understand it very well. So are there ...