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1answer
37 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
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2answers
93 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 ...
3
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0answers
41 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
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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: ...
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1answer
16 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
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1answer
66 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 ...
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0answers
23 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
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1answer
34 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
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0answers
45 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
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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
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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 ...
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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
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1answer
97 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 $$\int_{a}^{b} f(x) \ dx = Area$$...
3
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0answers
79 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
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4answers
793 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
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1answer
106 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
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2answers
55 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
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2answers
228 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
89 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
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1answer
47 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
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1answer
40 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
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0answers
27 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
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1answer
61 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
62 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
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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
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1answer
44 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
51 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
67 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
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2answers
134 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
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0answers
24 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 $\text{grad } f=\nabla f=(y^3z-\cos(x),3xy^2z,xy^3)$ ...
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2answers
28 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 ...
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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 ...
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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
52 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 ...
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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
61 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
41 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
200 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
113 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
54 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 ...
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0answers
47 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
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1answer
31 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
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0answers
17 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 'game-...
0
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1answer
29 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) = g'(...
2
votes
1answer
56 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
137 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
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2answers
78 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 ...