Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.
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37 views
Best way to write the characteristic polynomial
In linear algebra I've seen that there are two major different ways
to write the characteristic polynomial of a mapping $f$: As
$$
...
3
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1answer
45 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
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1answer
25 views
Prove that any circuit contains a cycle
This is a practice question (not HW)
Prove that any circuit in a graph must contain a cycle AND that any circuit that is not a cycle contains at least two cycles.
Note : This is for a first course ...
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2answers
63 views
What exactly is the complex plane, and how is it useful?
A lot of functions are defined on the complex plane, like the Gamma function:
the Lambert W function,
etc.
But I have no idea about what the complex plane means and how it's useful, or just ...
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1answer
44 views
Geometric representation of product rule?
At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule:
However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot ...
4
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1answer
82 views
How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?
Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
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0answers
68 views
Why Riemann integration is needed? [closed]
What is the necessity of the notion of "Riemann Integration" ? Why is normal definite integral is not good enough ?
2
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0answers
29 views
Intuition behind criterion for an irreducible Markov chain to be transient
I have been looking over my notes for Markov chains, and I have come across the following:
Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
7
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1answer
122 views
How to improve mathematical creativity?
To introduce myself: I'm an undergraduate mathematics student in Germany. Currently I'm studying in the second semester and until now I'm doing well, but I still got the feeling that my ability to ...
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1answer
45 views
Volume of a hypersphere
We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$.
Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere?
...
11
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7answers
296 views
$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]
If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
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2answers
112 views
It is possible to define our intuitive notion for probability in subsets of $[0,1]$
I've always heard and read the sentence:
If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.
What is the meaning for that? Is this the "real" ...
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0answers
47 views
What to take from representation of $S_d$?
I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
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4answers
96 views
the role of logic in math and education
My question is somewhat related to this discussion:
Is Mathematics one big tautology?
I have a computer science background and I have always approached math from the logic point of view ...
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1answer
95 views
How information works?
I am really confused after reading wikipedia...
What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information.
For ...
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1answer
60 views
Logical explanation of Euler's formula
This question is a about (if not proving) at least guessing the Euler's formula.
I don't want the proof using the infinite sums.
We can guess by logic that for example that the equation ...
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2answers
104 views
Algebraic geometry in representation theory?
I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
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2answers
130 views
The thought process of derivatives explained (intermediate calculus) “derivatives with respect to what”
My intention here is to contribute, if there is a problem with my solution or explanation--if it is wrong--please add a comment and don't just down vote. My answer represents my understanding and I ...
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1answer
32 views
Intuitive bernoulli numbers
Can somebody explain me or give me a link with a intuitive point of view of Bernoulli numbers?
I mean, somebody just saw a typical sequence of numbers that appears in some taylor expansions, and them ...
1
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1answer
49 views
The relationship between inner automorphisms, commutativity, normality, and conjugacy.
An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$
I have three somewhat broad questions about this:
Why is it related to ...
5
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0answers
73 views
If I'm not studying foundational issues, what should I take for granted?
When I'm reading a proof for a theorem and I find myself asking "why is that true?", I sometimes form a rationalization based on my own intuition. For example, I just read this in "baby Rudin" (page ...
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1answer
37 views
Weak homotopy equivalence
I know a continuous map $f:X\to Y$ between topological spaces is a weak homotopic equivalence if it induces isomorphisms on the corresponding homotopy groups, but what kind of information do I get ...
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3answers
116 views
Verifying a proof that if $x,y,z \geq 0$ and $x+y+z = 1$, then $0 \le xy + yz + zx - 2xyz \le \frac{7}{27}$
I was working some recreational problems from a book (The Art and Craft Of Problem Solving, Zeitz) and came across one from the '84 IMO:
Suppose that $x, y, z$ are non-negative reals, with $x + y ...
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1answer
32 views
An intutive explanation of natural density (asymptotic density)
I was wondering if someone can provide an intuitive explanation to natural density. I understand the concept very basically (pretty much the definition) but I can't seem to understand what natural ...
2
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1answer
46 views
Unclear on relationship between different dimensionalities of Fourier transform
This is probably a silly question, but it's one that's directly relevant to a project of mine and I figured this was the place to go.
I have some objects that contain a 1d and a 2d array of double ...
2
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1answer
57 views
Confusion about Banach Matchbox problem
While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand.
The problem statement is presented below (Source:Here)
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2
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1answer
51 views
Basic Question about linearity of expectation
I am going through some introductory notes on probability here http://www.stat.berkeley.edu/~aldous/134/gravner.pdf
In Chapter 8, page 89, there is a problem where you get a bag containing 10 Black, ...
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0answers
31 views
Schonhage–Strassen algorithm
After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
4
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2answers
54 views
Intuition behind the difference between derived sets and closed sets?
I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
1
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1answer
37 views
Intuition behind symmetric and antisymmetric tensors
I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
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3answers
173 views
Help! Doubt About Uniqueness in Mathematics
Many times in mathematics, as for example when we find the solution of an ODE, we can not claim uniqueness just by construction, instead we have to use a theorem.
The reasoning behind this is that ...
4
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1answer
81 views
Discreteness of eigenvalues for certain operators - can this approach be made rigorous?
I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
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1answer
62 views
Wrong reasoning for uniqueness of solution of ODE?
Sometimes I have seen this argument to prove that a differential equation has an unique solution, but I think it's wrong.
Suppose the differential equation:
$$\mathscr{D}[y(t)]=f(x)$$
where ...
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0answers
24 views
expression that constrain the range of x to a positive interval
For any $x \in R$, I used the exponential $f(x)=e^x$ to constrain the value of $f$ to a positive interval. While serving this purpose, it happens that I cannot use the exponential for some other ...
5
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2answers
172 views
Intuition behind the Axiom of Choice
Why is it different to make one choice or many choices than to make infinite choices from a theoretical point of view in which indeed you are not going to do any?
How could that be different from ...
2
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1answer
43 views
Is this intuition behind product manifolds correct?
I've been studying differential geometry on Spivak's books and recently I proved that the cartesian product of manifolds is another manifold. Right, however, what's the intuition behind this? I've ...
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1answer
118 views
What is duality?
I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
2
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1answer
43 views
proof for the general rule of conversion from base 10 to other bases
I just begin reading the book "what is mathematics" by Richard Courant. He states the general rule for passing from the base ten to any other base B is to perform successive divisions of the number z ...
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1answer
44 views
Turning an ellipse into a parabola
Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
2
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1answer
71 views
Maths branch of logics or vice versa?
Is it logics a branch of maths or vice versa?
From a the point of view of the definition of a logical system, logics is a 'calculus' which has axioms and rules as any branch of maths.
However it ...
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0answers
17 views
Understanding the topology of a variety concretely
My ultimate goal is to understand how to compute the cohomology groups of complex algebraic varieties, without having to know what a scheme is.
Therefore I want to be able to handle simple examples, ...
2
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1answer
48 views
What are central automorphisms used for?
A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$.
It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
4
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0answers
68 views
Geometrical Interpretetion of Half Derivative
How would you understand in a intuitive way the meaning of:
$$D^{\frac{1}{2}}x^2=\frac{\Gamma(3)}{\Gamma(\frac{5}{2})}x^{\frac{3}{2}}=\frac{8}{3\sqrt{\pi}}x^{\frac{3}{2}}$$
or
...
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1answer
73 views
Can anyone comprehend fourth-dimensional space and higher visually? [closed]
When I look at the "Clifford torus," for example, it just looks like a three dimensional object that's morphing/changing shape as it moves. Can anyone actually comprehend fourth-dimensional objects ...
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5answers
108 views
Differentiation confusion
I've been reading my textbook, and it tells me how to go about differentiating from first principles, it goes something like this:
$\eqalign{
& \mathop {\lim }\limits_{h \to 0} {{f(x + h) - ...
3
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1answer
71 views
Intuition behind compact subspaces of a metric space
I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following):
$E$ is compact in $X$ if for every open covering of ...
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2answers
64 views
Whats the connection between functions with curl 0 and holomorphic functions
When I first saw the Cauchy-Riemann differential equations they remind me on the conditions for the curl of a function to be zero.
Here some notation I will use:
$$\frac{\partial f}{\partial x} = ...
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0answers
86 views
Why Markov matrices always have 1 as an eigenvalue
Also called stochastic matrix. Let
$A=[a_{ij}]$ - matrix over $\mathbb{R}$
$0\le a_{ij} \le 1 \forall i,j$
$\sum_{j}a_{ij}=1 \forall i$
i.e the sum along each column of $A$ is 1. I ...
3
votes
3answers
160 views
Why does Newton's method work?
I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks.
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0answers
59 views
Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.
There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
