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2answers
44 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
2
votes
1answer
56 views

Derivatives of nested/iterated functions.

I'm learning about derivatives and I have the following function: $$f(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ Of which its derivative is $$\frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}} = \frac{ \frac{ \frac{ 1 ...
0
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1answer
42 views

good source for learning continuity,discontinuity

What is a good online source, sites for learning continuity (specifically) . i have learnt limits and differentiation problem lies within the continuity. Any good websites(only) for learning this . ...
0
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0answers
46 views

Big picture of algebraic geometry

I took a look on some books of algebraic geometry and found that there are nearly no graphs on them, so I got lost, where is the 'geometry'? How is those algebraic stuff describing some geometric ...
0
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1answer
25 views

Is a gradient system considered an ODE or PDE?

When you have a system of the type $$\dfrac{dx(t)}{dt} = \nabla V(x)$$ Is this considered an ODE or a PDE? Because you have a single derivative with respect to $t$ on the lefthand side, whereas on ...
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0answers
12 views

Does ray tracing have any speed ups in algorithm running time in the frequency domain?

Could ray tracing be Fourier-transformed so that all calculations are done in the frequency domain? I think ray-tracing a set of rays $S$ from the eye into the view frustum might be more efficient ...
2
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1answer
57 views

List of common and uncommon categories

I want to learn more about the category of "super commutative" graded $k$-algebra, for instance, its coproduct. However, I couldn't find anything related material. So, am I be able to get access to ...
4
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1answer
94 views

What are some PDE applications in recreational mathematics?

I have to do a final project for my PDE subject and last year I did one about Game Theory (specifically, Prisonner's Dilemma and Snowdrift game) for my ODE subject, which the rest of the students ...
-2
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2answers
130 views

Is there any undergraduate mathematics learning website like edx or coursera?

I want to take courses in mathematics of undergraduate level. But in edx or coursera they have only few undergrad math courses( only calculus or a little bit of algebra). There are no hardcore maths ...
2
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1answer
28 views

Pronunciation of Indexed Collection of Sets

I'm currently learning about sets and I want to discuss the material. However, I'm unsure of the pronunciation of certain symbols. For instance, I know that (1) $A\cup B$ is read, "A union B." ...
44
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25answers
3k views

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus. My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have ...
4
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0answers
95 views

Notebooks used by Paul Erdős

Paul Erdős was known for living out of two half-full (or half-empty) suitcases; one had a few clothes and the other had mathematical papers. Some of these papers were probably referring to his ...
5
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5answers
825 views

Is there such a thing as backwards sigma?

Is there a math function, similar to sigma, that can run down? For example instead of $\sum\limits_{i=1}^{10}i$ , something that adds from 10 to 1 (like a backwards run)...
1
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2answers
78 views

Algebraic Varieties vs Smooth Manifolds

There are many posts I have read on that subject which seem unclear for me. My main question (it might be silly) is: "Every non-singular algebraic variety over $\mathbb{C}$ is a smooth ...
5
votes
5answers
83 views

Where can I find Galois original paper?

As we all know Galois is an ultimate math prodigy. At age 17 or 18 he published a paper which we now know as Galois theory. I want to just see how he thought mathematics by seeing his original ...
1
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1answer
56 views

What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic ...
1
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2answers
28 views

Simple existence proofs without bounds

Which is/are the most simple proof/s of an existential statement like $$ \exists x P(x) $$ or $$ \forall x \exists y P(x,y) $$ where the variables rage over the integers, such that the proof doesn't ...
0
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3answers
76 views

How does logic and elementary set theory work together to prove $A \cup \varnothing = A$?

In How do I prove $A \cup\varnothing = A$ and $A \cap\varnothing = \varnothing$ A proof was given reproduced here: Prove: $A \cup \varnothing = A$ Let $a\in A\cup \varnothing$. Then $a\in A$ or ...
6
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2answers
120 views

Why a sheaf is an object that permits to get global information from local one?

Is there somebody who can explain/show me why a sheaf is something that can permit us to move from the local to the global? An explanation for the layman would be fine. Usually I tend to abhor them, ...
23
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7answers
2k views

Is there an “intrinsic” difference between a plane and a cylinder?

Since the plane and the cylinder have zero Gaussian curvature, I'm wondering, is there an "intrinsic" way of telling one from the other? By "intrinsic" here I loosely mean a property that can be ...
3
votes
0answers
62 views

Are there any other types of non-euclidean geometry?

We have 2 non-euclidean geometries(ie. not satisfying the fifth postulate ) in hand. But can there be some other models of non-euclidean geometries different from the known two? In other words, do ...
2
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0answers
49 views

What would be a good book to study machine learning from? [closed]

I'm looking for a up to date, readable book that assumes strong mathematical background. Thanks!
1
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2answers
56 views

Relation between left and right eigenvectors corresponding to the same eigenvalue

I have a general question on how the left eigenvectors and right eigenvectors of a matrix are related to each other. Background. It is easy to see that the characteristic polynomial of a $A$ and ...
3
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0answers
35 views

A Cartesian coordinate system is a mathematical or physical thing?

I'm convinced that if I ask what of the coordinates systems in the figure is a Cartesian system almost all say that it is the system $O_1$. This answer comes immediately from our habit and ...
0
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3answers
65 views

Where to start Stochastic processes

This semester I have the course stochastic processes in university but as our instructor is awful I can't rely on him and I should study this course on my own.To start, I need a suitable book for ...
13
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5answers
151 views

Is the word “any” a $\forall$ or an $\exists$?

I was wondering how should the word "any" be used in mathematical context. Is it a "for all" or an "it exists"? For example, assume I stated something like A set $X$ is called nice if $P(x)$ ...
4
votes
2answers
103 views

Why study dimensions?

I am quite new to the forum so please feel free to correct/give pointers if I am posting something in the wrong place. I have been perusing "Dimension Theory" by Witold Hurewicz & Henry Wallman ...
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2answers
59 views

Math journals for Calculus students

I have a student in my Calculus 2 class who might be interested in majoring in Mathematics. He asked be if there were any math journals suited for a less experienced readers. There have been ...
0
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1answer
87 views

Gerd Faltings Interview

I wanted to read about Gerd Falting's journey as a mathematician . I was searching for a link having his interview but unable to find any . Can anyone here suggest any link for Gerd Falting's ...
3
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1answer
90 views

Examples of how to apply algebraic number theory

I am reading about algebraic number theory mainly following milne's notes. But currently I really wonder how such theory can help solve problems of number theory. One example I know is we can use ...
1
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0answers
47 views

``angle" between two group elements

For the group $\mathbb{Z}^n$, we may embed them in $\mathbb{R}^n$ and then it is clear that for any two elements in $\mathbb{Z}^n$, we may treat them as vectors and hence the notion of ``angle" ...
0
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1answer
63 views

Soft question: Can one learn Fourier Analysis without a working knowledge of Integration Theory

As the title indicated, I am wondering if one (probably as an undergraduate math major) can learn much of Fourier Analysis, without taking a course in integration theory. I am taking a very light ...
1
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0answers
63 views

Why is the decimal number system so popular?

This probably isn't a great question, but I was just wondering that why is the decimal number system used around everywhere( not talking about machine languages). My first thought was that it appears ...
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2answers
79 views

The role of the Zariski topology in algebraic geometry

I am having trouble understading the relevance of the Zariski topology being a topology. Every time I see the proof that sets of the form $V(I)=\{p\in\mathbb{A}^n\mid f(p)=0 \ \forall f\in I\}$ ...
4
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0answers
35 views

(Soft Question) Formal name for something one is taking the limit of

When you are taking integral you have an integrand. When you are taking a sum you have a summand. Is there an analog of this for limits? How should one refer to $f(x)$ in the expression ...
0
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2answers
108 views

Ideas for a Talk [closed]

I'm trying to think of a suitable topic for a math talk. I'll have 15 minutes to present, and the audience is math grad students of all different specialties. My talk should be accessible to any ...
4
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2answers
95 views

Differentiation and integration

Which came first : Differentiation or Integration? If one of them was developed to solve certain types of problems, was the other developed for backward compatibility, or was it an independent ...
3
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1answer
60 views

Texts on the History of Linear Algebra

I thought I hated math at first, but linear algebra really changed my outlook on mathematics. What I really didn't like was calculus, which is fine, there are plenty of folks who would love to focus ...
5
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2answers
103 views

Is there an advantage to using polish notation in terms of human readability?

Lately I've been reading a lot of questions and answers related to logic and I have found some of them in the style of this one. As I'm not a fan of using Polish notation, I honestly just skip them. ...
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3answers
140 views

How does one interpret statements like: “The traveling salesman problem is NP-complete?”

The world abounds with statements like: The traveling salesman problem is NP-complete. But when I follow try to follow the Internet's links "down the rabbit hole," I don't get a truly sensible ...
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3answers
51 views

If $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$.

If $a,l\neq0$ , $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$. How can we exactly say (how to prove) that $p(x)$ is a quadratic ? What methods can be used to ...
0
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0answers
46 views

inquiry about differential geometry texts.

i am very interested in diff geometry as a math undergraduate student and want to study the diff geometry in the graduate school. so what would be a good book for the advanced diff geometry to get ...
6
votes
4answers
120 views

Could we “invent” a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we “invented” $i=\sqrt{-1}$? [duplicate]

$\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$. $1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new ...
0
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1answer
25 views

the main measure

I'm looking for the following notion in english if exists, called in french "la mesure principale" ( the main measure ) Let $\theta$ be an angle in standard position the main measure for $\theta$ ...
1
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2answers
37 views

Looking for websites to brush up on algebra skills needed for calculus

I've enrolled in an 8 week online Calculus 1 class, we're currently in week 2 and while I understand the calculus concepts (average rate of change, limits) I'm having a hard time on my homework due to ...
1
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1answer
19 views

Term for using a number as an exponent (complementing “raise to the power of …”)

If we have an equation we want to solve such as $\sqrt{x} = 3$, we can say something such as square both sides or "raise both sides to the power of 2", to arrive at $x = 9$. So $3 \rightarrow 3^2$ ...
5
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2answers
75 views

Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...
3
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0answers
44 views

Failing to recognize the importance of compactness, connectedness, and other topological notions in Real Analysis [closed]

I'm currently taking a course in Real Analysis that uses Principles of Mathematical Analysis by Rudin, and having a somewhat difficult time on tests. I always read that notions like compactness, ...
62
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7answers
5k views

Is linear algebra more “fully understood” than other maths disciplines?

In a recent question, it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood, and hence a ...
0
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0answers
35 views

Strange type of matrix equivalence, $\bf P=Q$. What applications or properties can it have?

Stemming from this question when actually searching for matrix similarities, having found this matrix equivalence: $$\bf A = PBP$$ That is neither transpose nor inversion on either of the $\bf P$s. ...