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1
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1answer
31 views

multiplication and division first, then addition and subtraction in equations written in prose

This might be a pretty stupid question, but the following was asked in a german quiz show: "How much is two times two plus two divided by two minus two?" My question now is, is there any literature, ...
0
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0answers
46 views

Equality on classes

On every set $A$ there is an equality $=_A$ defined. This is necessary to speak about injectivity, surjectivity, the group axioms and so ... In category theory, one uses classes to be collections of ...
3
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2answers
39 views

Comparing Coefficients for Partial Fractions

See this simple example : $$\frac{x+1}{(x-1)(x-2)}\equiv \frac{A}{(x-1)}+\frac{B}{(x-2)}$$ Then we can get $x+1 \equiv A(x-2)+B(x-1)$ for $x \neq 1 ,2$ My Question : Is it correct to put $x=1$ ...
-1
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1answer
41 views

Can I make it without times table memorization? [closed]

I am 24 years old and I am planing to get into mathematics sense it is essential In science. I have big plans and Math is the very first thing I want to focus on. I have very important question that ...
1
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0answers
23 views

what can we say if we just know the global section has a given universal algebra structure?

Given an universal algebra A, how to reconstruct a topological space X with some universal (and natural) property (e.g initial,terminal...)in the category of topological spaces whose global sections ...
3
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2answers
65 views

Why do we require that a simple Lie algebra be non-abelian?

We say that a Lie $k$-algebra is simple if it is a simple object in the category of Lie algebras, and also nonabelian. The only simple object which we do not consider to be a simple Lie algebra under ...
4
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4answers
103 views

Should I be concerned if I cannot solve most exercises in my textbook?

I'm self studying introductory real analysis. Out of the proofs in each lesson section and the proofs in each exercise section, I can usually work out$\frac{1}{4}$ of them on my own. After trying to ...
4
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0answers
52 views

Is there a dictionary for math notation?

As a non-mathematician who loves the elegance of mathematics, I'm often confused about certain syntax I see. For example, $2+2$ is "2 plus 2" obviously. But things get more complicated when, for ...
2
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0answers
23 views

Very Basic Numerical Methods Book for Freshman students

To cut a long story short; the nature of this degree (it's not a college degree) is such that numerical methods is treated shortly after Calc I (single-var) and linear algebra, but before multi-var ...
1
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1answer
27 views

Group morphism or group homomorphism?

I apologize for my question that might sound stupid, but i noticed that my lecturer in abstract algebra course uses always "group morphism" instead of "group homomorphism". In the books i see it ...
8
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3answers
141 views

How should a mathematically-inclined person learn descriptive statistics?

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page ...
4
votes
1answer
40 views

Situations in which interchanging the order of integration fail.

Suppose that our underlying space is $\Bbb R^2$ and $f:\Bbb R^2\to \Bbb R$, for concreteness. It is not hard to artificially construct such a function $f$ such that $$ \int_Y\int_X f(x,y)dxdy\ne ...
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0answers
121 views

Soft question: Do I suffer from something? Suicidal because of Math [closed]

I won't tell you the background, but I'm twenty years old. Always been at the top of the class, 99 percentile in our country during the testing before University. Always bright. However, I chose the ...
1
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0answers
41 views

Is knowledge of PDE useful for SDE?

I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have ...
1
vote
1answer
74 views

Guests leaving Hilbert's Hotel?

I am a layman in this field so my understanding of the problem of "Hilbert's Hotel" is limited to the popular version presented to the public. We know that Hilbert's Hotel can accommodate any finite ...
2
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1answer
44 views

Books with SAGE portions

I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through ...
0
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1answer
74 views

Which groups of a given order have the “most complicated” structure?

Let $G$ be a finite group. The size of the group $G/[G,G]$ is a measure of "how abelian" the group is. The larger the size of this group, "the more abelian" it is. Is there some similar measure ...
2
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2answers
78 views

How should I start solving a calculus problem? [closed]

There are so many books teaching how to take derivative and integration of a function. I think I'm good enough (enough for me lol) in those parts, my problem is that I can't start solving a question ...
3
votes
1answer
114 views

Hartshorne or Vakil's notes

I believe Hartshorne and Vakil's notes are two most popular text currently, so my question is about how to choose the text. I have worked through the first 4 chapters of Vakil's notes and now I am ...
4
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0answers
80 views

Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
5
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4answers
132 views

Questions that started new mathematics [closed]

Most mathematical theories (Exceptions are unknown to me) were created by answering one or more open ended non-trivial questions. For example, The Brachiostome problem lead to the calculus of ...
2
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0answers
50 views

What kind of magic would math correspond to? [closed]

Lets pretend for a moment that magic actually exists. Say there is a machine which could turn on your magic "abilities". Say furthermore that proficiency in non-magic skills corresponds to proficiency ...
55
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7answers
4k views

What does strength refer to in mathematics?

My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot ...
7
votes
1answer
78 views

The category of locally $P$ spaces

Let $P$ be a class of topological spaces (for example, compact spaces). The class of locally $P$ spaces consists of those spaces in which every point has a neighborhood basis consisting of $P$ ...
2
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1answer
55 views

How does mathematicians reconcil the fact that the gradient not drawn with respect to the origin

For example, consider the gradient vector of a function Obviously, $\nabla f(x_o,y_o)$ on the above figure is not attached to the origin. But attached to the point for which it is evaluated at. ...
1
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0answers
28 views

Book recommendation: Introduction to Traffic Flow and Traffic Models

I would like to look into the subject of traffic flow and traffic modelling. I have no prior experience with this, so it should be an introduction, preferably one with a level suited for a determined ...
4
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4answers
395 views

Are there any ugly math equations that have far reaching implications? [closed]

We often talk about mathematical beauty and how they end up describing incredible patterns or phenomena. But are there any mathematical equations that are extremely non-trivial and ugly which have ...
0
votes
1answer
75 views

Seemingly contradictory results [duplicate]

The following infinite sums produce remarkable results. $1+2+3+4+...=-\frac{1}{12}$ $1-2+3-4 +...=\frac{1}{4}$ So how are these results compatible with the statement; that integers are closed ...
54
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17answers
3k views

Good “history of mathematical ideas” book?

All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living ...
1
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2answers
46 views

What is a cool topic regarding differential equations that you would suggest for self-studying?

I took a basic course in differential equations and I loved it. I'd like to study them more in depth. However, I'm taking another, more advanced course in a while. For this reason, instead of ...
4
votes
1answer
104 views

Visual references for the Riemann-Stieltjes integral.

I've seen a lot of excellent visual material (gifs, pictures) here, in topics like this, and I used many of them to understand/explain concepts (particularly gifs showing Riemann sums or fourier ...
0
votes
2answers
77 views

How applicable is Goldbach's conjecture to real world scenarios? [closed]

In what scenarios is Goldbach's conjecture, that all even numbers greater than 4 are the sum of two prime numbers; a natural conjecture to research?
1
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2answers
39 views

Are operations upon index-classes (rather than index-sets) allowed in mathematics?

Why we always see extended operations, like arbitrary unions, products, etc. in different parts of mathematics in the form of extensions of finite ones upon arbitrary sets (called index set)? Why ...
0
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0answers
18 views

Weiertrass equation of an elliptic curve.

We know that every elliptic curve is a non-singular $\textbf{cubic}$ projective curve (curve of genus 1), but we can transform this in the Weiertrass form $$y^2 + a_1xy + a_3y = x^3 + a_2x^2 +a_4x ...
8
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0answers
123 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
1
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0answers
65 views

Difference between Advanced Calculus and Calculus on Manifolds?

This is an interesting distinction that I don't fully grasp yet. There's quite some books on the topic of the so-called "Advanced Calculus". Some of the most famous of these are the books by Edwards, ...
8
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0answers
126 views

How can I begin reading journals and papers?

I am an undergraduate CS student but I love Math and spend most of my time doing and reading Maths books. I realise that it's important to get into the habit of reading papers and journals so it will ...
1
vote
1answer
44 views

Is there Any Benefits for Casting a Convex Program Problem into Linear Program Problem?

I'm curious a relative broad question: Suppose I have a convex program problem in hand. (hence, I could use many well-developed software packages to solve this problem for sure; e.g., CVX..) But ...
2
votes
2answers
66 views

Understanding quotient topology and product topology (in the infinite case)

I have troubles understanding the concepts of quotient topology and product topology (in the infinite case). I know that we want to give a topology to new spaces built from the old ones, but the ...
1
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0answers
46 views

How can we recognize if something is a number?

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number. Are there properties that are ...
1
vote
1answer
38 views

Basis for the Definition of a diagonal matrix.

I found the definition of a square matrix on Wikipedia as Well my question is a very stupid one that why we defined the diagonal matrix on the basis of a main diagonal rather than the anti diagonal ...
8
votes
2answers
74 views

Any surviving contemporary manuscripts by ancient mathematicians?

As I understand it, most of what we know about ancient mathematics comes from copies, quotations, and summaries by later scribes and scholars. Medieval Arab mathematicians in particular are given ...
3
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0answers
54 views

What are some of good books in probability? [closed]

I am preparing for Ph.D entrances and my interest lies in probability and statistics. I am trying to learn it from William Feller's book but finding it quite difficult to comprehend. So please suggest ...
3
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0answers
35 views

Learning generalized functions

What is the best book to learn generalized functions and what prerequisites are needed? I would like a book that helps build intuition but it rigorous enough and not overly complex. My background is ...
25
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6answers
3k views

How long to do math each day? [closed]

I have seen some posts math.SE (mkko's answer) indicating that it is the norm for (undergrad?) math majors to study 70-80 hours per week. I'm a little bit shocked by that. For some background on me, ...
2
votes
0answers
36 views

Is there a statement inside of mathematics that is proven only with mathematical induction and with none other method?

Well, the point is that although the method of mathematical induction can be useful and is useful for proving certain statements, I somehow always like things to be proved in some other way than with ...
7
votes
1answer
68 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
3
votes
1answer
58 views

On motivation, focus and mental state [closed]

I have a bit of a meta-question, but first let me give you some background. I used to be a very inquisitive kid, interested a many disciplines, in particular mathematics, computer science and physics. ...
1
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3answers
103 views

Are the assertions “$2 + 2$ equals $4$” and “$2 +2$ is $4$” identical

Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical? Or is this a linguistic, psychological or murky philosophical thing rather than a mathematical thing
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0answers
575 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ ...