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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
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
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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) = ...
2
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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
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
77 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
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0answers
45 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
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2answers
344 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
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2answers
33 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
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0answers
28 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
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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
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2answers
63 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
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1answer
30 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
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1answer
58 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
110 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
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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 ...
3
votes
2answers
43 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
50 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
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3answers
78 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
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2answers
189 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
164 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
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0answers
19 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
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0answers
28 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
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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
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1answer
45 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 ...
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2answers
26 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
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3answers
114 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
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2answers
19 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
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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
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1answer
39 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
84 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 ...
2
votes
3answers
55 views

Does it make sense to define a “metric topological space” $(M, d, \tau)$

When doing things related to compactness, sometimes you have to switch definition from sequential compactness which is defined on a metric space $(M, d)$, to things related to covering compactness ...
0
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2answers
91 views

Is there a difference between $x=0$ and $0=x$

Is there a difference between $x=2$ and $2=x$ One of my students asked this question . What would be a good answer for this ? Here $x$ is unknown. After calculating we get $x=2$
2
votes
1answer
78 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
0
votes
1answer
66 views

What does $(-2)^x$ really mean?

I think we can all agree that $(-2)^{-1}=-1/2,(-2)^0=1,(-2)^1=-2,(-2)^2=4$ But what does the function $f=(-2)^x$ really mean? It is defined on the integers based on how most people understand ...
1
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1answer
53 views

Matrix product demonstration

Sorry for boring you my friends. I have haunted by a problem of relation between matrix product and cross product. I would like to demonstrate the following equation: $$ (\Omega\cdot r)^T(\Omega\cdot ...
38
votes
3answers
4k views

Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
8
votes
2answers
111 views

Why are mathematicians so interested in finding out the gaps between primes and the distribution (randomness) in primes?

I'm a high school student, and I came across an article that mentioned Kanan Soundararajan's and his student's work regarding the patterns in 'random' primes. And I also read about Yitang Zhang's and ...
2
votes
1answer
29 views

Definition algebra over commutative ring (injectivity needed?)

I'm aware that many differently looking definitions exist for an algebra over a commutative ring $A$. For me, the most natural definition of an algebra over a commutative ring $A$ consists of a tuple ...
1
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2answers
44 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
2
votes
1answer
55 views

Derivatives of nested/iterated functions.

I'm learning about derivatives and I have the following function: $$f(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ Of which its derivative is $$\frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}} = \frac{ \frac{ \frac{ 1 ...
0
votes
1answer
42 views

good source for learning continuity,discontinuity

What is a good online source, sites for learning continuity (specifically) . i have learnt limits and differentiation problem lies within the continuity. Any good websites(only) for learning this . ...
0
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0answers
46 views

Big picture of algebraic geometry

I took a look on some books of algebraic geometry and found that there are nearly no graphs on them, so I got lost, where is the 'geometry'? How is those algebraic stuff describing some geometric ...
0
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1answer
24 views

Is a gradient system considered an ODE or PDE?

When you have a system of the type $$\dfrac{dx(t)}{dt} = \nabla V(x)$$ Is this considered an ODE or a PDE? Because you have a single derivative with respect to $t$ on the lefthand side, whereas on ...
0
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0answers
11 views

Does ray tracing have any speed ups in algorithm running time in the frequency domain?

Could ray tracing be Fourier-transformed so that all calculations are done in the frequency domain? I think ray-tracing a set of rays $S$ from the eye into the view frustum might be more efficient ...
2
votes
1answer
56 views

List of common and uncommon categories

I want to learn more about the category of "super commutative" graded $k$-algebra, for instance, its coproduct. However, I couldn't find anything related material. So, am I be able to get access to ...
4
votes
1answer
93 views

What are some PDE applications in recreational mathematics?

I have to do a final project for my PDE subject and last year I did one about Game Theory (specifically, Prisonner's Dilemma and Snowdrift game) for my ODE subject, which the rest of the students ...
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2answers
130 views

Is there any undergraduate mathematics learning website like edx or coursera?

I want to take courses in mathematics of undergraduate level. But in edx or coursera they have only few undergrad math courses( only calculus or a little bit of algebra). There are no hardcore maths ...