For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

learn more… | top users | synonyms

0
votes
0answers
20 views

$Y$ is a difference of a factor of $X$

To say $Y$ differs by a factor of $X$ means $dY = X$, or equivalently $Y_2 - Y_1 = X$ I believe. But this does not make sense to me, as a factor is what multiplies another number, e.g. $2$ is a ...
-2
votes
2answers
34 views

General results on the change of the parity of a number by repeatedly dividing by 2 [closed]

I know this question may seem open, but I'm a bit interested in figuring out, getting some ideas, or at least getting some sources on how the parity of a number is affected by repeatedly diving it by ...
2
votes
4answers
212 views

Why do we use degrees? [closed]

I see a lot of people who ask why we use radians instead of degrees. But why do we use degrees instead of radians. In the cases we use degrees instead of radians, what convenience does it bring? The ...
0
votes
1answer
63 views

Interpretation help: Showing that Riemann Hypothesis holds “almost surely”

I was perusing this textbook on algorithmic number theory, where I came across this page where they appear to prove that the Riemann Hypothesis holds almost surely. This seems like an odd statement ...
-1
votes
1answer
25 views

Functional analysis as a prerequisite [closed]

Can someone give me an example of a mathematical field in which functional analysis is a prerequisite?!
0
votes
0answers
44 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
1
vote
1answer
35 views

Why are alternating divergent series generally easier to evaluate? [closed]

Why is it that alternating divergent series tend to be easier to evaluate or that there are more ways to evaluate them? Is there a particular reason for the difficulty to evaluate series that don't ...
1
vote
2answers
87 views

a healthy perspective on “knowing everything” [closed]

This is a question about attitude, but related to math studies. I have trouble with two things: 1. making "normal" progress in my learning and 2. having the satisfaction that I understand what is ...
2
votes
0answers
34 views

Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
3
votes
0answers
60 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
0
votes
1answer
15 views

Applying a vector of functions on vectors

Is there an adequate mathematical representation (operator $\star$) to apply a vector of functions to another vector of values, element by element? Something like the following: $$ \left[ f_1, ...
4
votes
1answer
61 views

How Do You Check Your Computations?

I am already a grad student, but sadly I still have problems with careless computations. In my most recent mid-term (multivariable analysis), I lost 21 points out of 100 because of calculation errors ...
0
votes
0answers
15 views

Oka's coherence theorem and Cartan's theorem A and B

So reading up on some articles I've found that whilst attempting to characterize Oka's coherence theorem in sheaf-theoretic language, Cartan was ultimately led to formulating his theorem A and B in ...
0
votes
1answer
29 views

Difference in Notation for Vectors in Linear Algebra & Multivariable Calculus

Often in Linear Algebra we see vectors depicted either in Column or Row Form as : Linear Algebra : Vector in Row Form $$ \vec{V}^{\,} = \left[x_1,\ldots,x_n\right]$$ OR Linear Algebra : Vector in ...
2
votes
0answers
44 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
2
votes
1answer
39 views

Reading a Matrix

This is a softer question, but I'm having trouble keeping straight all of the information that a matrix provides you with straight in my head. All I know is that the rows correspond with the codomain ...
0
votes
0answers
11 views

Spanning Spaces by Different Basis

I have a query related to spanning space by two bases $S_1=\{V_1+V_2, V_3, V_1-V_4,V_3-V_2\}$ $S_2=\{V_1, V_2, V_3, V_4\}$ Can we consider spaces generated by $S_1$ and $S_2$ to be equivalent?? Or ...
1
vote
0answers
23 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
3
votes
1answer
92 views

What is the most general way to think about Integrals?

Given a single-variable scalar function, $f : \mathbb{R} \to \mathbb{R}$ The "area under the curve" (of the graph of the function $f$ in $\mathbb{R^2}$) is given by $$\int_{a}^{b} f(x) \ dx = ...
3
votes
0answers
77 views

Most complicated proof of Pythagoras

Usually a mathematician aims for clarity and elegance when conducting a proof. However, the antimathematician buries all hope of assimilating intuition and reasoning. To illustrate this, I seek the ...
5
votes
4answers
758 views

How important are inequalities?

When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important ...
6
votes
1answer
103 views

How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
1
vote
2answers
53 views

Linear Algebra Trivia: Can anyone identify this class of matrix?

Consider a matrix: \begin{pmatrix} 0 & -y & x \\ x & 0 & -y \\ -y & x & 0 \\ \end{pmatrix} where $x,y$ are positive real numbers I wish to identy the most "specific" class ...
6
votes
2answers
223 views

“There is no set containing everything”? [duplicate]

I was reading this question regarding codomains, and I found something interesting in User134824's answer: "On the other hand, owing to the set-theoretic fact that "there is no set containing ...
3
votes
1answer
87 views

How to figure out the “idea behind” proofs in analysis?

I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty ...
1
vote
1answer
45 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
1
vote
1answer
38 views

What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
0
votes
0answers
26 views

Difficulty during self-studying unique set proofs

I have been following Velleman's How to prove it and working through it on my own. I am working full time now so I can only study after work without any other help. It's been going fairly ok until I ...
1
vote
1answer
58 views

Should I remember the proof of mathematical theorems(every step)?

The problem is, that when I am reading the proof of mathematical theorem(in my case - it is calculus), U understand the idea and every step of proof. But i can't prove the theorem individualy even if ...
3
votes
2answers
59 views

Scalar multiplication as a special form of matrix multiplication

Question What do we gain or lose, conceptually, if we consider scalar multiplication as a special form of matrix multiplication? Background The question bothers me since I have been reading about ...
2
votes
0answers
49 views

Geometric derivative, existance, interpretation and usefulness.

What if one was to define the limit $$\lim_{h\to 0} \sqrt[h]{\frac{f(x+h)}{f(x)}}$$ If we play around with h=1 and for the gamma function this would be a linear function for positive x: $$\Gamma(n+1) ...
3
votes
1answer
59 views

How should one characterize mathematical conclusions? [closed]

I have posted this in Philosophy SE as well because I feel that it is appropriate both here and there. As practiced, mathematical proof seems not to be an explicit formal deduction within a formal ...
1
vote
1answer
43 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
0
votes
1answer
49 views

Sum of integers [duplicate]

I cannot accept that $\sum_{n=1}^\infty n = -\frac{1}{12}$. It should be that such a sum is divergent. That it is divergent is useful for the Test for Divergence in many such problems. I feel like ...
2
votes
3answers
62 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
129 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
87 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
63 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
20 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
50 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 ...
22
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
39 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
30 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
3answers
180 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
104 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
52 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". ...