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2
votes
0answers
11 views

Abstraction and Genaralization

This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...
0
votes
1answer
45 views

What is the name of people who do algebra? [on hold]

People who do topology is called topologists, people who do analysis is called analysts, people who do geometry is called geometers, then how about algebra?
3
votes
1answer
49 views

Any math competitions dedicated to calculations by hand (college level math)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
2
votes
1answer
15 views

Intuition of the distribution of the minimum of exponential random variables

Let $X,Y$ be two independent random variables with exponential distribution with parameters $a$ and $b$ respectively. It is known (see e.g. here) that $Z:=\min\{X,Y\}$ is exponential distributed with ...
28
votes
4answers
3k views

Why do we study real numbers?

I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic ...
2
votes
1answer
79 views

Example of Non-separable stochastic process.

This question is related to the link: https://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process ...
0
votes
2answers
68 views

Product of negative numbers [duplicate]

Why is a negative number multiplied by a negative number a positive number? I'm trying to know what does multiplying by a negative number mean. If you think of multiplication as a "groups of" ...
10
votes
5answers
832 views

Are all numbers real numbers?

If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego ...
104
votes
22answers
11k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
1
vote
0answers
19 views

Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
0
votes
2answers
37 views

Is this element-of_{ij} - looking symbol the Levi-Civita symbol?

I'm reading this formula: from a page Is the symbol that looks like an element-of symbol with two indices i and j the Levi-Civita symbol? Mathematics is my weak-side so I'm not sure. Actually I ...
1
vote
2answers
74 views

When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?

In this question I was shown a very elegant solution based on writing a function as the upper envelope of a family of linear functions: $$f(x) = \sup_{y\in C} f(y) + \langle \nabla f(y), x-y ...
27
votes
10answers
11k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
1
vote
1answer
125 views

Generalization of principle of inclusion and exclusion (PIE)

The PIE can be stated as $$|\cup_{i=1}^n Y_i| = \sum_{J\subset[n], J\neq \emptyset} (-1)^{|J|-1} |Y_J|$$ where $[n]=\{1,2,...,n\}$ and $Y_J=\cap_{i \in J} Y_i$. Problems using it are usually reduced ...
2
votes
1answer
26 views

Is there an agreed upon convention for naming ZFC+Large Cardinal Axioms?

Is there an agreed upon convention in general for what to name ZFC+[Large Cardinal Axiom]? Or would one have to state explicitly which axiom was being added? To explain what I mean, note that anyone ...
-2
votes
3answers
39 views

ordinary differential equation project suggestion [on hold]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
5
votes
3answers
141 views

What mathematical objects permit “taking of limits”?

Background I have been reading a lot of abstract algebra recently (at the level of Artin/Dummit & Foote/Herstein Topics in Algebra for those of you familiar with these books). I have noticed ...
0
votes
0answers
33 views

PhD in Combinatorics (instead of Mathematics) [migrated]

In recent years I have become aware of a few PhD programs specifically in combinatorics and optimization. Most notably, Georgia Tech and Carnegie Mellon both have PhD programs in Algorithms, ...
0
votes
0answers
18 views

Limit Terminology

From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the ...
45
votes
6answers
10k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
5
votes
3answers
557 views

The J programming language: is it useful for mathematics?

I just stumbled upon the J programming language, which has the description: J is particularly strong in the mathematical, statistical, and logical analysis of data. It is a powerful tool in ...
26
votes
6answers
834 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
2
votes
2answers
73 views

Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques, e.g. with substitution. However, whenever I am ...
7
votes
1answer
197 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
10
votes
4answers
2k views

What is the name of the vertical bar in $(x^2+1)\vert_{x = 4}$ or $\left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$?

I've always wanted to know what the name of the vertical bar in these examples was: $f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$) $\int_0^4 (x^2+1) \,dx = ...
0
votes
0answers
25 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...
1
vote
0answers
35 views
+100

Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = ...
7
votes
1answer
67 views

Why is complex analysis so nice? And how is it connected/motivating for algebraic topology?

This is very much a soft question, but after seeing Cauchy's integral formula in lecture today I was really struck by how neat complex analysis is. I don't understand how all of these amazing analytic ...
114
votes
10answers
9k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
4
votes
1answer
521 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of a PDE textbook(e.g. Folland's Introduction to Partial Differential ...
1
vote
1answer
58 views

What exactly is a “chaotic” sequence?

In response to this question, I was told that a certain sequence $a_n$ is "chaotic" and "wandering." The particular sequence in question is defined by $a_0 = z, z \in \mathbb{C}$ and $a_{n+1} = ...
-1
votes
0answers
45 views

Active, beautiful fields of mathematics [on hold]

Some fields of mathematics are attributed more beauty than others. There are many historical quotes of mathematicians celebrating the beauty of number theory -of which I know little- in comparison to ...
13
votes
9answers
858 views

Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
4
votes
1answer
91 views

Why does the Möbius function take its values so often in $\{0,+1,-1\}$?

The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by $$\mu([x,x])=1$$ $$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$ ...
0
votes
1answer
38 views

Rename Real and Complex [on hold]

Let's imagine that the definitions of Real numbers and Complex numbers were discovered today. What would be the most suitable names for those new number systems?
-1
votes
0answers
35 views

Analysis and algebra [on hold]

I'd like to know if there exist a field of the theoretical math that really combines analysis and algebra. Some people say that Model theory combines those two subjects but I personally want ...
1
vote
0answers
15 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
4
votes
3answers
208 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
0
votes
1answer
45 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
38
votes
7answers
2k views

Is speed an important quality in a mathematician? [on hold]

Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a ...
17
votes
4answers
5k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
0
votes
1answer
39 views

Good introductory book in geometric probability

I recently came across the proof of the Buffon theorem and I was fascinated by geometric probability. Could someone indicate me a good introductory book? Maybe with many exercises?
1
vote
0answers
49 views

Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
47
votes
6answers
2k views

What to answer when people ask what I do in mathematics

This is not really a math question, but I think that every mathematician and student in math (specially pure) struggles with this at some point. Inevitably at some point when we're talking with ...
2
votes
1answer
27 views

Reference request: examples where probabilistic methods make theoretical contributions to deterministic fields?

This idea came up during a discussion on whether you need randomness for the concept of probability to be valid, and if not, what is needed. I argued that having a stationary, recurrent sequence, is ...
697
votes
53answers
416k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
-1
votes
0answers
29 views

Visualization of Fubini's Theorem

I understand that Fubini's Theorem is vital to evaluating double and triple integrals (via the equivalence of iterated integrals) especially in elementary multivariable calculus, and I know that it ...
-1
votes
1answer
30 views

What to do if a series is neither arithmetic or geometric? [closed]

If I have an infinite series that is neither arithmetic or geometric (common ratio and difference aren't constant), would this mean it's divergent and cannot be solved?
0
votes
2answers
57 views
24
votes
9answers
38k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...