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0
votes
1answer
51 views

Book for Functional Analysis

I need a book or lecture notes for the course functional analysis, which I took this semester, the lecturer mentioned some book at the beginning and currently I'm also reading the book of Vitali ...
2
votes
4answers
60 views

Is there in literature a descriptive abbreviation phrase for “for infinitely many $n$”?

Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ ...
2
votes
1answer
42 views

Vector Integration - Intuition

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a ...
2
votes
1answer
69 views

Undergraduate texts in the style of AoPS books.

I've been using the Art of Problem Solving series as supplemental readings throughout my A-levels(which is the equivalent of high school), and found them to be extremely helpful and informative, both ...
41
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5answers
6k views

What software can draw pictures like this?

I used Mathematica to draw a function picture, but it is not beautiful like the picture below. I want know what software can draw beautiful pictures like the one below. This picture is from the ...
2
votes
0answers
64 views

Studying mathematical analysis [closed]

Okay, this will seem like an extremely stupid question but here it goes: Can you learn mathematical analysis if you have a very limited knowledge of mathematics? In my high school we had only one type ...
13
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2answers
371 views

You've got your head round Basic Category Theory: why look at monads next?

Suppose you have acquired a decent understanding of the usual basics of category theory. You now know about functors and natural transformations, representables, limits, adjunctions, and know ...
3
votes
0answers
52 views

Is It Worth It Working Out Every Practice Problem In Math? (Without a calculator)

I'm bouncing back between trig, algebra, and calc books. I've noticed that most of the problems at some point seem to distill into very tedious arithmetic. It is nice to have the prowess of ...
0
votes
1answer
66 views

Differential equations and classic differential geometry are mostly impossible to understand [closed]

I need advice on my studies of mathematics... I'm really depressed because it's impossible for me to understand many important parts of books such as Tenenbaum & Pollard "Ordinary Differential ...
2
votes
1answer
78 views

Looking for intuition and/or insight regarding the “into-internalization principle” in category theory.

Observation 0. Suppose $I$ is a set. Then the category $\mathbf{Set}^I$ can be explicitly described as the category whose objects are as follows: an object is a set $S$ equipped with a function $$I ...
9
votes
3answers
311 views

Developing Mathematic Intuition

I'm an engineering student, currently working my way through the fundamental mathematics courses. I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, ...
1
vote
1answer
43 views

Recommended Book for Open Book Exam (Algebraic Topology)

This is a very soft question --- I have an upcoming open book exam on Algebraic Topology (includes Fundamental Groups/Homology/ Covering Spaces). Any recommendations on what books/notes are suitable ...
2
votes
1answer
47 views

Significance of having closed range for an operator?

One thing I think is of great importance is that it seems to correspond to possibilities similar to rank nullity theorem and further the fredholm alternative much as in finite dimension. Is this a ...
2
votes
1answer
82 views

Are there any intuitive reasons for Goldbach conjecture to be true?

One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true. Typical heuristic reason is based on probability arguments. Such ...
3
votes
0answers
67 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
1
vote
0answers
39 views

Cartesian Coordinates and Geometry

By introducing the notion of "Groups", it became much clear, whether polynomial equations in one variable are solvable by radicals or not. Come in Geometry. It has long history, and much much before ...
2
votes
2answers
127 views

Why do we care whether a functor is representable?

In the algebraic geometry textbook by Görtz and Wedhorn, the authors prove that several common functors are representable. For example, the Grassmannian functors are representable. Beyond being cute ...
1
vote
0answers
19 views

Order Taylor series prediction

It is easy to add the Taylor expansion, less easy to multiply and even less easy to compose. That said, the main problem lies not in the calculation itself in the prediction orders that need to be ...
2
votes
0answers
29 views

Geometry book for the university with solved exercises (affine space, euclidean space, etc…)

I'm looking for a book with solved exercises of affine space, affine transformations, etc... I found a lot of books and pdf's with theory, but none of them contained solved exercises, and I'm having ...
5
votes
2answers
109 views

Applications of Hodge theory to topology and analysis

I am going to give a talk for the PhD students' seminar at my university. The audience is composed mainly by algebraic topologists, algebraic geometers and analysts. I have decided that I'm going to ...
4
votes
3answers
772 views

Does the phrase “If you don't use it, you lose it” apply to mathematics? [closed]

I'm asking this because I ran into the following particular situation: I took some calc courses over 2013, where I learned, amongst other things, to integrate some pretty nasty functions, and this ...
0
votes
1answer
26 views

How many different arrangement are there for $14-15$ puzzle, so that it become impossible to solve?

Sam Loyd's unsolvable $14-15$ puzzle is well known to everyone. I also found it quite fascinating when I was playing with it in my childhood but didn't know about that one particular arrangement that ...
41
votes
7answers
2k views

Much less than, what does that mean?

What exactly does $\ll$ mean? I am familiar that this symbol means much less than. ...but what exactly does "much less than" mean? (Or the corollary, $\gg$) On Wikipedia, the example they use is ...
1
vote
4answers
104 views

Relationship between Mathematics and Genetics [closed]

I was wondering how can we mathematically define the biological nature around us. How can we mathematically define a real plant or a tree, growing from being a 1 cell to a full grown plant with ...
1
vote
0answers
20 views

$C_{c}(X)$ is complete implies $X$ is compact. [duplicate]

If $X$ locally compact Hausdorff space. Then $C_{c}(X)$ is complete implies $X$ is compact. I know that $C_{c}(X)$ dense in $C_{0}(X)$. So in that case $C_{c}(X)=C_{0}(X)$. I know only Tiez ...
2
votes
2answers
47 views

Basics of fields/field extensions

I am taking an introductory number theory course this term, and I have found that while my algebra has been for the most part sufficient, I am severely lacking in even the basics concerning fields and ...
8
votes
2answers
211 views

Applications of the formula expressing roots of a general cubic polynomial

I know that the mathematics related to finding the general formula by expressing the roots of a third (and fourth) degree polynomial by means of radicals has had an impressive impact on mathematics ...
0
votes
1answer
34 views

Examples of Induction over Sets Lacking Natural Well-Orderings?

Consider a countable set like $\mathbb{Q}$. Let $f : \mathbb{N} \rightarrow \mathbb{Q}$ be any bijection witnessing the equinumerosity of $\mathbb{Q}$ And $\mathbb{N}$. Then for each predicate $\phi$, ...
0
votes
0answers
28 views

Abduction reasonning

I was wondering if sometimes mathematicians use abduction reasonning and if yes, I've you got an example? I precise I have no knowlegde about logic symbolism. Thank you in advance :)
0
votes
0answers
57 views

Thoughts on Mersenne numbers

Mersenne numbers are numbers of the form $2^n-1$. Let us use the standard symbol $M_n$ for them. I am interested here in opinions about methods for obtaining the answers of these two questions: ...
5
votes
1answer
60 views

How can I improve my problem solving abilities so that I stop missing the obvious?

I'm a generally good math problem solver. I get decent scores on contests, top of my class in math courses, and have a pretty wide array of knowledge from which to relate concepts in order to solve ...
2
votes
0answers
38 views

Functors with ability of “reduction in the formulas”

I am interesting about next class of functors. Let $F : \mathbf{C} \to \mathbf{K}$ - functor between categories. And let $P(f_1,...,f_n)$ predicate in abstract category first-order language. And ...
2
votes
1answer
117 views

Why is it okay to omit the limits on some definite integrals?

To illustrate, here is a textbook example for deriving the formula for the expectation of two independent random variables: If a random variable $x_1$ has probability density ...
1
vote
0answers
27 views

What formulas are available to find the nth digit of a number?

Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have, $$(1) \quad \operatorname{d_n}(x)=\lfloor ...
1
vote
0answers
31 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
0
votes
0answers
20 views

An identity involving Gamma function and its derivative

Reading on wikipedia's page about Gamma function I've read this interesting formula: $$\Gamma^{(n)}(1)=(-1)^nn!\sum_{\pi \vdash n}\prod_{i=1}^r\frac {\zeta^*(a_i)}{k_i!a_i}$$ But I don't fully ...
2
votes
0answers
10 views

Higher-dimensional analogues of the equivalence of compact Riemann surfaces and projective curves

I'm going to be studying this result for a dissertation this year, and I wondered what there were in the way of higher-dimensional analogues? Also, what are some standing research questions on this ...
2
votes
1answer
41 views

What do modern day cryptographers work on? [closed]

I am a student of Pure Mathematics.I want to get some information on the following: $1$.What do modern day cryptographers work on? $2$. How does pure mathematics influence modern day cryptography? ...
2
votes
2answers
58 views

McDougall Litell Issues with notation

My school uses textbooks by a company called McDougall Litell for all math classes. My friend pointed out to me that there was no really decent notation used in any of those books. Would it be ...
2
votes
1answer
27 views

Should you use an equivalence or an implication in a definition?

In a definition like the following: An object $x$ is called $P$ if [and only if] it has properties $p$ and $q.$ should one use an implication (if) or an equivalence (if and only if)? It makes ...
2
votes
2answers
50 views

Very Elementary books on Analytic Number Theory

Till today, I was learning "Algebra", more than other subjects (analysis/topology). I thought, learning number theory may not be difficult for me. Many theorems/statements in number theory are easy ...
5
votes
4answers
141 views

What is the relation between $\Gamma(a)$ and circles?

Looking through my calc textbook, it states that $$\int_0^\infty x^{a-1} e^{-x} \text{d}x =\Gamma(a)$$ As I have read ahead, I can understand most of the fairly basic concepts behind this function, ...
2
votes
2answers
132 views

What mathematical areas lie at the interface of analysis, algebra and geometry? [closed]

Would it be some area that draws on many fields such as algebraic geometry? Is there some sort of unification of these three fields?
1
vote
2answers
37 views

simple proof involving representation of exp(x)

Some days ago I presented some exercises to a bunch of engineers. One of the exercises involved was: Use $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}$ to show that $\lim_{n \to ...
6
votes
1answer
172 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
8
votes
0answers
165 views

What is “Field with One Element”?

I was reading the Wikipedia article about The Field with One Element and I came across the following quotes: "...F1 refers to the idea that there should be a way to replace sets and operations, the ...
1
vote
0answers
10 views

Existence theorems for conformal modeling, and uses thereof

In 2010 Ebenfelt, Khavinson, and Shapiro posted to the Arxiv a paper (Two dimensional shapes and lemniscates) which analyzed the fingerprints of smooth shapes, and showed that for any finite Blaschke ...
2
votes
1answer
79 views

GRE General - How to get faster at basic math?

I've been studying for the GRE quantitative section for months, and I feel like I've hit a wall over which I cannot climb. In short, I always run out of time. As such, I get at least a quarter of the ...
4
votes
0answers
121 views

I have a free summer before university. What should I learn? [closed]

Note: This is a soft question. It may be a bit early to be thinking about this, but I figured I'd ask now and see what responses I get. I'm currently a high school senior, and I quite like pure ...
5
votes
2answers
173 views

“Every geometry is a projective geometry!” So Hyperbolic geometry is a projective geometry?

The great mathematician Arthur Cayley (https://en.wikipedia.org/wiki/Arthur_Cayley ) seems to have said "all geometry is projective geometry" (sorry no exact source, probably it is somewhere in Felix ...