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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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 ...
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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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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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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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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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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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Dimension of a space VS dimension of a function in this space [closed]

A colleague and I ran into a problem when we realised that we had a complete different understanding of dimensions. If we consider this function: $z(x,y)=x^2 + y^2$ Person A believes this function ...
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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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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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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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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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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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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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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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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 ...
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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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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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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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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 ...
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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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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 ...
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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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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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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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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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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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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 ...
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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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More intuition on the curl formula

I have a question regarding this quesiton. It says that $3$ simple fields that describe rotations around $x,y,z$ axis are: $$H_1(x,y,z)=(0,−z,y)\\ H_2(x,y,z)=(z,0,−x)\\ H_3(x,y,z)=(−y,x,0)$$ but why? ...
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Order of an Equivalence Class and the number of Coset.

Let $ \mathcal{M} $ be the set of all subsets of finite group $G$ which have $p^{\alpha} $elements. Thus $ \mathcal{M} $ has $ {p^{\alpha}m \choose p^{\alpha}} $ elements. Given $M_1 ,M_2 \in ...
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Intuition for the construction of the product topology and its equivalence to the euclidian metric

While I have been provided a proof for the previous statement, I still cannot fully grasp why the euclidian metric [ $d(x,y)=((x_1-y_1)^2+...(x_{n}-y_{n})^2)^{1/2}$] generates the same topology as the ...
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Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
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Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. The example I have in mind is ...
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What is the geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
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Multiplication principle and permutation

Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as: Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | ...
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What does $3+2i$ apples mean? Can the simple counting analogy with apples be extended to complex numbers? [closed]

Please read, or at least skim the question. Past attempts at answering the question have ignored vital constraints provided below. Natural Numbers Imagine I have $n$, a positive natural number, of ...
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Intuition for orthogonality in $\{0, 1\}^n$

In the beginning of [Kanerva 1988] a boolean algebra over $$ \{0, 1\}^n $$ with bitwise OR and AND is introduced. Example for bitwise OR: $$101 + 001 = 101$$ Example for bitwise AND: $$101 * 001 = ...
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Intuitively, why is compounding percentages not expressed as adding percentages?

I pursue only intuition; please do not answer with formal proofs. I already know the theoretical reason: because each percentage expresses a different base. $1.$ But why not intuitively? My ...
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What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
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Is a series of successive derivatives known/useful?

So, while trying to find something else, it looks like I've found, for many $f(x)$: $$f(x) + f'(x) + f''(x) + f^{(3)}(x) + \dots + f^{(n)}(x)$$ Assuming that there is an easy way to find this sum ...
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How to understand intuition behind compactness? [duplicate]

I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite ...
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Intuitive proof that $U(n)$ isn't isomorphic to $SU(n) \times S^1$

One way to prove this is by comparing their centers. However, I do not feel that this proof gives me much insight into the structures of the groups. (It would make me very happy if I were to be ...
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Can anyone help to explain Benfords Law [duplicate]

Okay so recently I have heard of what is known as Benford's law, and this is the first time I absolutely cannot think of my own inutuion about how this is true. Just trying to think about it makes my ...
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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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Difference between “real functions” and “real-valued functions”

According to my textbook: A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a ...
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59 views

Proper solution of the limit of $\sin(x)/\tan(x)$ as $x \to 0$

In one of my math book, I have a problem where I need to compute $\lim_{x\to0}{\frac{\sin(x)}{\tan(x)}}$ I came up with a solution that I am not able to write formally. The reasoning is the following ...
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Basic question on the infinitely many solutions of a linear system Ax=b,

I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions: If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables ...