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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What is meant in the quotation of Terry Tao?

Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/] "It is true that some mathematicians can be vastly more ...
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Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
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Intuition about the first isomorphism theorem

I'm currently studying group theory and recently I've read about the first isomorphism theorem which can be stated as follows: Let $G$ and $H$ be groups and $\varphi :G\to H$ a homomorphism, then $...
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47 views

Homotopy Equivalence intuition

Can somebody tell me intuitively what does it mean geometrically when we say two spaces are homotopy equivalent ? I understand the technincal definition.
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Why should I expect the product of sum of four squares to be a sum of four squares? How did Euler come up with it?

Euler discovered the lovely identity shown here: https://en.wikipedia.org/wiki/Euler%27s_four-square_identity Is there a natural reason to assume a solution can be found? Any intuition? I saw that ...
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Extension theorem from Guillemin-Pollack, motivated sketch of proof?

Let $W$ be a compact, connected, oriented $k + 1$ dimensional manifold with boundary, and let $f: \partial W \to S^k$ be a smooth map. Could anybody sketch with good motivation that $f$ extends to a ...
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Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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Basic probability. Is the textbook wrong?

A and B are independent events such that P(A)=0.7, P(B)=k, P(A U B)=0.8 Find the value of k. Solution given: P(A U B) = P(A) + P(B) - P(A ∩ B).....(i) [Addition Rule] P(A ∩ B) = P(A).P(B).....(...
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32 views

Is it possible to tell whether a set of functions is equicontinuous from the graph of the function?

I never internalized why the set of functions $f_n(x) = \sin(nx)$ is not equicontinuous I know I can show that it is not equicontinuous by definition, by choice of appropriate $x,y$ for $|x-y|<\...
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Intuition for minimum spanning tree result

Let K$_ n$ denote the complete graph on n vertices. To each edge in K$_n$ independently assign a weight drawn from the uniform distribution on [0,1]$\,$. Finally, define MST(n) to be the expected ...
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What is functional analysis in simple words?

To begin with , I am only a secondary school student (17yo) but I am very interested in higher mathematics. However we only learn so little in my school (only single variable calculus and basic linear ...
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Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find $$\color{red}{\iiint_\limits{\large\text{volume}\,\tau}\nabla\cdot\...
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38 views

How small can we make two numbers $a$ and $b$, with prime factorizations such that…?

Given a number $n$, I'd like to find it using either the sum or difference of two other numbers. The other two numbers, which we can call $a$ and $b$, must have a prime factorization with no primes > ...
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The frog puzzle

So here's the puzzle. You're poisoned in the jungle and the only way to save yourself is to lick a special kind of frog. To make matters worse, only the female of that species will do. Licking the ...
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Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...
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Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
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Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
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Grasping ODE by intuition

Consider following differential equation $$\frac{dy}{dx} = y^2(1+x^2)$$ Its solution is: $$y(x) = \frac{-1}{x+\frac{1}{3}x^3+C}$$ As I read in one book, following must be satisfied: $$\frac{dy(x)}...
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Techniques to find a subset of possible values for the rank of a matrix

The rank of a matrix is the number of linearly independent rows or columns of that matrix. In some exercises I need to find the rank of a matrix, but for some (lol unknown) reason I'm always stuck ...
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117 views

Explaining Newton Polygon in elementary way

In this question of mine Proving irreduciblity over $\mathbb{Z}$ I was recommended to read Newton Polygon. Also this appears to be an interesting topic. Also I am currently studying irreducibility of ...
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58 views

Fuzzy logic vs probability

In reading about fuzzy logic it says that fuzzy logic is different from probability. Can some one please explain how these two differ. How can this be explained to a person with no mathematical ...
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What do the properties of the dot product mean? [closed]

I am having trouble understanding the relevance or meaning of the properties of dot product. For example, the distribution property of dot product states: $$\vec a \cdot (\vec b+\vec c) = \vec a \...
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Intuition for power-set structure of finite Boolean rings

A course I am taking has started to introduce Boolean rings: rings where every element is idempotent. It was proved that every finite Boolean ring $R$ is isomorphic to a power set ring $\wp (S)$ for ...
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Motivation for the mapping cone complexes

I was reading some topics in Homological Algebra when I came across the concepts of cone of a map of complexes and cylinder. My knowledge of Algebraic Topology is pretty basic so I only used these ...
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Coordinate transformations and interpreting what the Jacobian determinant describes

Apologies for a perhaps rather trivial question, but I really want to get the concept cleared up in my head. I understand that when one changes from one coordinate system there is an appropriate ...
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“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
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Simple Question about Interpretation of Poisson Distribution

So if you know for instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. Can the Poisson distribution ...
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Additional term in solution for wave equation

Given the wave equation: $u_{tt} = c^2u_{xx} \quad x \in \mathbb{R}, t> 0$ $ u(x,0) = \phi(x), x \in \mathbb{R}$ $ u_{t}(x,0) = \psi (x), x \in \mathbb{R}$ The solution is: $u(x,t) = \frac{1}{...
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Understanding conditional probability: What is the reasoning for multiplying the individual probabilities when the condition is a value?

Let's say there are two attributes viz temperature and rainfall. We will have 4 rows which are Temp Rain L Y L N H Y H N Now P(Rain | Temp=L) = 0....
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Trivial power of line bundle

I'm trying to understand the following: Thinking of a line bundle as a bunch of locally generating sections together with transition functions (which in this case are just multiplication by local ...
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How to rigorously determine whether two events are independent?

Consider the following question: Tim has lost his pet in either forest A (with probability 0.4) or in forest B (with probability 0.6). If his pet is in forest A and Tim spends a day ...
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How would look a harmonic and p-harmonic series vector in a multidimensional space?

Learning about the harmonic series divergence and p-harmonics series convergence, I tried to manipulate them globally as follows. $$\lim_{n \to \infty}A \cdot I = \lim_{n \to \infty} \begin{pmatrix} \...
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Intuition behind the “infinite velocity” of a falling ladder

In Calculus there is a "classic" related rates problem involving a falling ladder. Say the ladder is $25$ ft tall and is leaning against a wall. The bottom edge of the ladder is pulled away from the ...
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Abstraction and/or concreteness - What should be emphasized

Alexandar Grothendieck was probably a mathematician focusing on theory developement and abstraction much much more than focusing on concrete examples and/or problems. In his biography, he wrote: ...
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Intuition: groups, quotient groups, cosets, homomorphisms.

If we start with the group of rational numbers $\mathbb{Q}$ and the subgroup of $\mathbb{Q}$; $\mathbb{Z}$ the integers, and then form the quotient group $\mathbb{Q}$/$\mathbb{Z}$ we have that this ...
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Concerning Approximations, Multiplicative Order, and Residue Number Systems…

I'm interested in particular representations of reals in residue number systems. Specifically, if we are given a real $0 \le n \le 1$, we wish to represent that number as a fraction in a residue ...
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Approximating $x=\sqrt{2}+1$

Suppose $y>1$ is some approximation to $x=\sqrt{2}+1$. Give a brief reason (not a proof) why one should expect $(1/y)+2$ to be a closer approximation to $x$ than $y$ is. After testing this out ...
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Is there a geometric meaning associated with the condition “dot product equals $1$?”

Consider $x,y \in \mathbb{R}^n$. Then the condition $x \bullet y = 0$ is easy to understand; it just means that $x$ and $y$ are orthogonal. Question. Does the condition $x \bullet y = 1$ have an ...
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What are the examples to understand L-process and R-process?

R-processes and L-processes R-process on $ [0,\infty) $ to signify a process all of whose paths are right-continuous on $[0,\infty)$ with limits from the left on $(0,\infty)$. R-function or R- ...
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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 $\sigma_{\alpha}|int(\Delta^...
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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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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 means....
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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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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 surface....
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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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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 $\...