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2
votes
3answers
56 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
1
vote
2answers
127 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
3
votes
2answers
85 views

Is linear algebra developed any further? [duplicate]

I heard an opinion that linear algebra has ceased to develop 100 years ago because there's nothing else to "discover" in this branch of mathematics, and no scientific activity other than teaching ...
1
vote
0answers
23 views

Calculating the gradient of a scalar field. Am I missing something?

Question Find the gradient of $f$ at each point where it exists. $f:\Bbb{R^3} \to \Bbb{R} ~,~f(x,y,z)=xy^3z-\sin(x).$ Attempted solution $\text{grad } f=\nabla ...
0
votes
2answers
27 views

Understanding order and countability

I am confused with how to defining the order of countable set. Let me express my thoughts by some examples: $\Bbb{N}$ has a 'natural' order, namely $1\lt2\lt3...$ For any countable set, we can define ...
1
vote
5answers
62 views

Does excluding or including zero from the definitions of “positive” and “negative” make any consequential difference in mathematics?

I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative. But today I made a few Google searches, and they all say the same ...
0
votes
0answers
19 views

Persistent Homology. Missing points

I'm working on a project with a professor. This project involves Persistent Homology methods over a point cloud. Recently we found some inconsistencies in the point clouds that we were reading, ...
1
vote
1answer
48 views

Need some suggestion for an introductory talk on 'Local Cohomology'?

Next week i am to give a talk on 'Local Cohomology' and i am writing to request suggestions for some basic interesting results for the talk.The relevant information is as follows: (1) The audience ...
21
votes
10answers
2k views

Is basis change ever useful in practical linear algebra?

In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?
3
votes
1answer
58 views

Gauss: The study of Euler's works…

I keep coming across this quote by Gauss but I haven't actually been able to locate the original source: “The Study of Euler’s works will remain the best school for the various fields of mathematics ...
1
vote
1answer
36 views

Good book for self study of Continued Fractions

Does anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. ...
3
votes
2answers
29 views

Is a Quotient the number of times one value fits inside another, or the value of one of the groups produced by the operation

My question is a simple one, but one I haven't been able to figure out through research. When a simple division is performed suppose 10/2 = 5, is that 5 classified as the frequency or the number of ...
11
votes
2answers
157 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
3
votes
1answer
99 views

Importance of guide/advisor in a PhD [closed]

I am really in a fix in my career. I completed my Masters in 2014. I qualified in a PhD Scholarships Test in 2015 and in the same year joined in a University for research work. My interest lies in ...
0
votes
2answers
49 views

hyperbolic spaces and fractals

Is there a relation between hyperbolic spaces and fractals? In group theory, if we take the Cayley graph of a free group on two generators, we get a fractal quaternary tree, which I'd like to think as ...
1
vote
0answers
45 views

Recommendations: Any Good books to study Path-Integration from 0 again?

I was researching and talking with some friends about I want to start from zero studying path integral, this question, and they recommended I start by studying "Quantum Mechanics and Path Integrals". ...
1
vote
1answer
29 views

Soft question on the notion of connections

We have met the notion of connection in different places: 1.For a vector bundle $E\to M$,a connection is defined to be a way of diffentiating a section of $E$ along a vector field of $M$,by which we ...
0
votes
0answers
16 views

what broad topics in game theory are likely discussed when you say 'game-theoretic analysis' of something?

I actually do not know anything about game theory, and in my current research I think I need to start knowing what it is all about. In the meantime, I always read papers that say they did a ...
0
votes
1answer
28 views

Is my understanding of the $u$ substitution process correct?

I'm just getting a hang of doing integrals, so I was wondering if my understanding of the $u$ substitution process is correct. If we have $$\int f(x) \ dx$$ we write $f(x) = g'(x) \cdot h(x) = ...
2
votes
1answer
55 views

Group actions as colimit of torsors?

I'm learning group theory and I know a little bit about category theory(Mac Lane ch1-3,but have not appreciated what a colimit is). I know a group action can be viewed as a functor from a group as ...
4
votes
1answer
136 views

What is the definition of $n\cdot n\cdot n$?

Intuitively, What does it mean when you multiply numbers? I asked my professor about what does it mean when we multiply $5\cdot 5\cdot 5$. He said there is no definition of this thing in mathematics. ...
0
votes
2answers
75 views

How to get rid of “ox near amorphous mountain” feeling? [closed]

I am an (almost independent [1] learner) mathematics student. I believe in the only way to get used to the ideas is to derive them by yourself, but when I try to derive some good and deep result, my ...
1
vote
0answers
43 views

Is it “okay practice” to exclude the epicness from the definition of extremal / strong epis?

Basically I wonder, whether I "should" include the property of being epi in the definition of extremal epis / strong epis (/...) (dually for extremal monos etc.). One hand it is terminology-wise a ...
8
votes
2answers
340 views

How to know if one problem is more difficult than another one?

I have seen in many places (books, lectures, ...) that people say that some unsolved problem is more (or much more) difficult than another one or sometimes they point some problem as the most ...
0
votes
2answers
29 views

Soft Question - book recommendation - Stochastic Processes

My mother language book on stochastic processes is pretty much complete(~500 pages) but would like another one in English, to have in my library. I'm looking for a similar book containing the ...
1
vote
0answers
27 views

Nested logarithms to represent real intervals.

Out of curiosity from this question regarding writing real numbers as an iterated sum/difference of square roots I started experimenting with another family of functions: $$\log[ \exp[1] \pm ...
1
vote
0answers
15 views

Is the usage of “pair” or “triple” overly pendatic or redundant

In books you will see that consider the pair $(X, \|\cdot\|)$ or a measurable space is triple $(X, \Sigma, \mu)$ I am writing up a homework and I find the usage of "pair" and "triple" a little bit ...
7
votes
2answers
61 views

Three dimensional spherical excess formula

We all know the spherical excess formula: in a unit sphere, the area of a geodesic triangle is equal to the exceeding from $\pi$ of the sum of the three angles of the triangle. Is there a similar ...
3
votes
1answer
27 views

Why are differential equations with sinusoidal source terms easier to solve than others?

I am a software engineer trying to wrap my tiny human brain around Fourier Transforms for a project I'm currently working on. Although I will ultimately use an open source Math library to do all the ...
-1
votes
0answers
13 views

Looking for examples of interesting contours chosen in contour integrals

As the question states, I am looking for questions that have been answered by picking some rather unique looking contours. Some I have found are: ...
1
vote
1answer
52 views

How to read a mathematics textbook? [duplicate]

How do you best read a textbook such as Hatcher's Algebraic Topology ? Do you start with chapter 0 and make your way through the book ? Can math books such as these be read in a linear fashion in a ...
0
votes
1answer
31 views

Can you take the limit as $x \to \infty$ of an expression such as $\sum_{n \in \mathbb{Z}} \ln(|x - n|)$? [closed]

Consider the function $f(x) = \sum_{n \in \mathbb{Z}} \ln(|x - n|)$ I'm not really concerned with its convergence properties, what I am concerned with is if it is possible to take the limit as $x ...
3
votes
1answer
104 views

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] = ...
3
votes
1answer
47 views

Function is also called mapping : why?

I know many terms like function, mapping, transformation, and morphism are used for describing same mathematical object, functions. But I think that there's more explanation about these names than ...
2
votes
2answers
40 views

Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
2
votes
1answer
48 views

Writing preliminary results in research papers

Suppose we want to put $10$ basic results in the preliminary section of a mathematical researh paper, without proof, but only with reference. It could be done in following way: Theorem 1: If $G$ ...
4
votes
3answers
75 views

How useful are non-square matrices in maths or sciences?

I know that a matrix will be either square or rectangular matrix. I know that square matrices are used to solve a system of linear equations. But what's the use of rectangular matrices, why do we ...
-1
votes
2answers
176 views

Andrew Wiles' Abel Prize for FLT - delayed or not? [closed]

Andrew Wiles was recently awarded the Abel Prize for his work proving Fermat's Last Theorem (FLT). The Abel Prize has existed for 14 years. From my layperson's perspective, it would seem that he ...
7
votes
1answer
157 views

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ?

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ? Is there any additional condition imposing which on the mother space will ...
0
votes
0answers
18 views

Correct notation for Cartesian product

I have already seen two notations for the Cartesian product of a family $\{A_{\lambda}\}_{\lambda\in\Lambda}$: ...
1
vote
0answers
27 views

Find a discontinuous function from $\mathbb R^n \to \mathbb R$ which is weakly differentiable. [duplicate]

For $k \in \mathbb N$, I use the notation $$H^k(\mathbb R^n) = \{ u \in L^2(\mathbb R^n) : D^\alpha u \in L^2(\mathbb R^n) \text{ for all multi-indices } \alpha \text{ with } \lvert \alpha \rvert \le ...
1
vote
1answer
69 views

Where to find about the category theoretic study of manifolds?

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start? Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an ...
1
vote
1answer
44 views

Why is it called the *Inverse* Galois Problem?

This is just a very quick question and hopefully not poorly received. Question: Why is it called the inverse galois problem? The very brief statement given on wikipedia says Is every finite ...
1
vote
2answers
25 views

How does the multiplication theorem correspond to the concept of intersection?

Given two events A and B defined on a sample space S. S : Rolling a six-sided dice A : Getting an even number B : Getting a number ≥ 4 In an elementary sense (the experiment being carried out once), ...
1
vote
3answers
111 views

Dumb question: what is $\sum\limits_{n = 1}^k 1$

Let $\{I_n\}$ be a collection of intervals on $\Bbb {R}$, whose length denoted by $I_n = |I_n|$ Then what is $\sum\limits_{n = 1}^k (I_n + \alpha)$, where $\alpha$ is some real number? The notation ...
1
vote
1answer
48 views

A Soft Question on Differential Geometry

Suppose you have your ambient space $\mathbb{R}^n$ and some subset $M$ of $\mathbb{R}^n$. Now, when I learned differential geometry in college, I learned that the notion of differentiability depends ...
1
vote
2answers
18 views

Convention - Dual $\pm$ Signs

Say I have two equations like $$z+w=x+y$$ $$z-w=x-y$$ Is there any way I can combine these into a statement like $$z\pm w=x\pm y$$ I know the above statement is not correct as that entails 4 different ...
0
votes
2answers
40 views

Cool applications of reaction diffusion equations

I was thinking my undergrad thesis could be about reaction diffusion equations and their application to biology. For example, I know about pattern formation on the coat of animals, but I was told the ...
1
vote
1answer
35 views

Learning complex analysis through problem solving

I am looking for a problem book on complex analysis with solutions that covers/teaches basic and advanced topics of complex analysis (the same topics covered in a standard textbook like Ahlfors with ...
2
votes
3answers
81 views

What does probability signify? Can it ever be verified?

In our probability lecture, our teacher told us that probability is just a number assigned to an event to determine the likelihood of that event but it can never be verified in practice. He even said ...