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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Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
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Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
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2answers
84 views

What is the intuition behind the Cauchy-Schwarz inequality in the real numbers?

The Cauchy-Schwarz inequality states that $$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$ The proof, with the discriminant argument, is ...
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20 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
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2answers
22 views

Relation between Poisson Distribution and Process?

I think I'm confused by the fact that they have similar names and both involve i.i.d. exponential RVs. From what I understand, Poisson distribution: the probability of a particular number of events ...
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32 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
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Finding how large $p$ needs to be to have $n$ unique factors…

If we take a prime $p$, how large does $p$ have to be so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$? (Note that the factors can be prime or composite) Note that I'm looking more for ...
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1answer
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Geometric meaning of the determinant of a matrix

What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know ...
5
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59 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
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1answer
30 views

concentration of volume of hypersphere

I am reading about features of volume of hyperballs, where I see two theorems, Most of the volume of the d-dimensional ball of radius r is contained in an annulus of width $O(r/d)$ near the ...
0
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1answer
62 views

The role of visualization and intuition in graduate and postgraduate math and developing it

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic ...
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26 views

Why is the accuracy of $\Pr($hypothesis) in Bayes's Theorem less important than apparent?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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2answers
48 views

How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
33 views

If the $\Pr($hypothesis) is due only to chance, then what is the problem?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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3answers
315 views

Developing Mathematic Intuition

I'm an engineering student, currently working my way through the fundamental mathematics courses. I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, ...
31
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4answers
861 views

How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...
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2answers
95 views

Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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2answers
50 views

Why is $x^2 \equiv 1 \pmod{x+1}$ for $x > 0$?

One day my mind wandered off and came upon the following. $x^2 \equiv 1 \pmod{x+1}~\forall x>0, x \in \mathbb{Z}$. My markdown might be a little bit broken :) I tested this out in Python for the ...
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4answers
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Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
19
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3answers
5k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
6
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4answers
742 views

Intuitively, why does Bayes' theorem work?

Why does Bayes' theorem work? I'm not looking for a cryptic math demonstration. Rather, I'm interested in the intuition behind the theorem that reveals the a posteriori probability given the prior ...
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0answers
23 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
25 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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2answers
818 views

Intuition behind the Definition of Conditional Probability (for 2 Events)

What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a ...
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1answer
32 views

Intuition behind exponential sum convergence

My textbook states without proof that the summation: $$\sum_{x=0}^{\infty} \frac{1}{x!} e^{ax}$$ converges for all real $a$. I am trying to understand this. I assume the reasoning is that the ...
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4answers
417 views

Is there a more intuitive way to understand matrices and linear algebra?

When learning linear algebar I often blindly follow algorithms to perform operations with matrices without fully understanding concepts behind them. It helped a lot when I started dealing with ...
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2answers
73 views

The significance of failure of uniqueness in differential equations

The nonlinear ODE: $y'(t)=y(t)^{1/2}$ with initial condition $y(0)=1$ has two solutions. Non-uniqueness is not surprising because of the failure of Lipschitz continuity in the $y$ term. While this ...
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2answers
37 views

What is the intuition for adding vs multiplying probabilities?

Caution: I modified this original answer to simplify the examples. You add probabilities when the events you are thinking about are alternatives (eg: A soccer team scores 0 goals or 1 goal or 2 ...
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1answer
26 views

Why does the boundary of a region $D$ have enough information to dictate the value of an integral over $D$?

There are many theorems which say something along the lines of the title: The FTC: $\int_a^bf'(x)dx=f(b)-f(a)$. Green's Theorem: Let $F=(P,Q)$, then $\oint_{\partial D}Fds=\iint_D(\frac {\partial ...
4
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0answers
49 views

How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the ...
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5answers
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The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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1answer
42 views

Limit point Compactness does not imply compactness counter-example

I think that I understand why compactness implies limit point compactness: Suppose $A \subseteq X$ has no limit points. Then $A^{\prime} \subseteq A$. Thus, $A$ is closed. Then for all $a \in A$, ...
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0answers
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Intuition of LHS in Green's thorem for a velocity field

I am trying to get an intuition for the LHS part of Greens theorem. For a potential field like gravity the LHS part is work, but if the vector field is a velocity field then what does the LHS ...
2
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1answer
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compactness requirement for the tube lemma of a product space.

The tube lemma: Let $X,Y$ be topological spaces s.t $Y$ is compact. Let $X_0 \in X$ and let $N$ be an open set in $X \times Y$ so that $x_0 \times Y$ is contained in $N$. Then there exits a ...
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2answers
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How to picture a first countable space?

I find myself forgetting what it means for a space to be first countable on a frequent basis. This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating ...
2
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2answers
68 views

integrals with no analytic answer - intuition and proof

the following integral has no analytic solution it appears: $$\int_0^\pi e^{\sin(x)} \, dx$$ intuitively, what is the reason for this integral having no analytic answer? (is there a way to prove it ...
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How was Zeno's paradox solved using the limits of infinite series?

This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough: A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to ...
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Explain for students: Why does 0 mod n equal 0 (zero)?

I told my students that the mod operator basically gives the remainder of division, so upon seeing: 0 mod 10 Some students (apparently) reasoned that, "10 goes ...
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2answers
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Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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1answer
43 views

Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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7answers
183 views

Why is $\ln 1 = 0 $? [closed]

Yes I know, and believe, and have used it for all the time I have done mathematics as fun as well as a subject. But why is it that $$\ln 1 = 0$$
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Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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0answers
13 views

Geometric intuition for conjunctive spaces

A topological space $S$ will be called conjunctive if for each open set $A$ containing a point $p$, there's a point $q\in S$ satisfying $\overline{\left\{q \right\}}\subset A\cap \overline{\left\{p ...
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Limits Outside of Mathematics.

The concept of the limit is one upon which the whole math of calculus is built. can anyone, however, think of an example in which the concept is applied outside of mathematics? Just anything that ...
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What is meant by finite (infinite) bernoullis?

I came across the following quote while searching for intuition behind Poisson distribution - think about a Poisson process. It really is, in a sense, looking at very, very small intervals of ...
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2answers
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Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?

I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, ...
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1answer
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What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
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4answers
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Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
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1answer
50 views

Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...