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 behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
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2answers
35 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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1answer
56 views

How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $?

So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its ...
3
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1answer
78 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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0answers
17 views

Developing intuition for the modulus of continuity $\omega$?

I am currently in a course which is going through rigorous definitions of continuity, convergence, integrals, etc. I am trying to develop an intuition in understanding the modulus of continuity, ...
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81 views

If f(x)dx is a rectangle with height f(x) and width dx, what is f(z)dz in complex analysis

I am trying to intuitively understand the multiplication $f(z)dz$ in complex analysis. For instance, $f(x)dx$, we are all aware, is a rectangle with height $f(x)$ and width $dx$ so its multiplication ...
3
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1answer
52 views

Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.
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47 views

F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?
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2answers
81 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
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3answers
52 views

Extensions Fields and Zeros of Polynomials

I've just digested the proof for Kronecker's theorem that states every polynomial $f(x)$ over a field $F$ has a zero in an extension field of $F$, mainly, the factor ring of polynomials ...
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1answer
81 views

Building intuition in group theory

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense. Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is ...
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180 views

Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
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2answers
38 views

Intuitive understanding of relationship between unit vectors and position vector

On Wolfram Mathworld they give a unit vector in the $\textbf{x}_n $ direction as: $$ \hat{\textbf{x}_n} \equiv \dfrac{\frac{\partial \textbf{r}}{\partial x_n}}{\lvert \frac{\partial ...
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3answers
339 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
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1answer
60 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
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2answers
83 views

Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...
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0answers
44 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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2answers
26 views

I don't understand the definition of a base of a metric space

Definition: A collection {$v_n$} of subsets is said to be a base for X if for every x $\in$ $X$ and every open set $G$ $\subset$ $X$, such that x $\in$ $G$ we have x $\in$ {$V_n$} $\subset$ $G$ for ...
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1answer
26 views

A Base of a metric space intuition

From what I have read online and from what I have read in Rudin, a collection of open sets $\lbrace$$V_{n}$$\rbrace$ is said to be a base for a metric space $X$ if every open set in $X$ can be ...
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2answers
125 views

Intuitive Understanding of the First Isomorphism Theorem

I've been reading some things about an intuitive understanding of the first isomorphism theorem, and there's just one more part that I do not understand. Specifically, the first isomorphism theorem ...
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0answers
41 views

Understanding what $P - P \log(P)$ means for an event of probability $P$

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space, $X$ be a random variable, and $E \in \Sigma$ be an event with $\mathbb{P}(E) = P$. Then $P - P \log(P) \in [0, 1]$, for all $P \in (0, 1]$, ...
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4answers
143 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
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0answers
41 views

Intuition for problem involving binomial random variable

Question: The below algebraic solution is simple enough. But is there a way to "see" the answer using a clever trick or intuition? Given the algebraic solution, I feel like there should be. I just ...
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2answers
107 views

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
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3k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
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1answer
71 views

Motivation for Definition of Measurable Function

I'm having trouble understanding why a function is defined as "measurable" if the preimage of every measurable set is measurable. I see the parallel to the definition of continuity, and the latter ...
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1answer
65 views

Why does the order not matter? Partial D

When taking partial derivatives, why does the order not matter as long as the function is continuous? Any proof, intuitive or rigorous?
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1answer
68 views

Physical significance of the fact that the cardinality of the real number line is the same as a finite interval of the real number line

It is known that the cardinality of the real number line is the same as a finite interval of the real number line. Is there a physical meaning of this apparently conter-intuitive statement?
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2answers
77 views

Matrix exponentiation intuition.

What does $x^A$ intuitively mean if $x \in \mathbb{C}$ and $A$ is any matrix? Also, what if we had $x$ being a matrix too? Last but not least, what happens if we have a complex $x$ raised to a ...
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1answer
46 views

general mean value theorem

Can anyone give me the intuitive explanation of the general mean value theorem stated in my notes as under: Let $f:U\rightarrow \mathbb R$ and $U\subseteq \mathbb R^n$ and let $f$ is differentiable ...
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1answer
48 views

Intuition for Euler's Partition Theorem

Euler's Partition Theorem states the following: Every number has as many integer partitions into odd parts as into distinct parts. I played around with small examples (I wrote out the partitions ...
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53 views

Dirichlet Distribution - the underlying intuition.

I'm not a math expert, but I need dealing with some math tools for natural language processing research. One of the most common tools is the Dirichlet distribution. I know that with a multinomial ...
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1answer
33 views

Conflict between geometric intuition and computed answer

Evaluate the integral $\int_{C} z ds$ where C is the intersection of $x^{2}+y^{2}=4$ and $z=0$ (oriented clockwise as viewed from above). My interpretation of this problem yields the following ...
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0answers
27 views

Negative integrals spherical coordinates

Need help interpreting a result prompted by the following question. Solve the triple integral of $\sqrt{x^{2} + y^{2} +z^{2}}$ on the region Q where Q is bound by $z=-\sqrt{9-x^2-y^2}$ and the x-y ...
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0answers
47 views

Geometric intuition for mixed partial derivatives

I'm trying to better understand exactly what $f_{xy}(x,y)$ at a point is geometrically, and possibly understand why $f_{xy}$ and $f_{yx}$ should be equivalent, not just because the math happened to ...
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2answers
52 views

Complete this reasoning? Number theory

I have this really weird confusion with $gcds$ and and basic theory dividing numbers and at the moment, I am stuck at this. If $gcd(a,b) = 1$, it means the biggest number that divides them evenly ...
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1answer
45 views

Use Leibniz' formula to show that the $(2n)$th derivative of $(2x^2 + 3x +1)sinx$ is $(-1)^n(2x^2+3x-8n^2+4n+1)sinx+(-1)^{n+1}(8nx+6n)cosx$ wrt $x$

If I let $f=f(x)=sinx$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\displaystyle ...
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9answers
1k views

I can't understand logical implication

I just started studying logic (high school) anyway...for the truth table of logical implication If sentence $A$ is true and $B$ is true then $A\implies B$ is true. does that mean if $A$ and $B$ are ...
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391 views

Novel approaches to elementary number theory and abstract algebra

As a part of a university course, I'll have to study Herstein's Topics in algebra and Hardy&Wright's Introduction to the theory of numbers. Can you suggest some books (to be used as companions) ...
2
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1answer
79 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
2
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1answer
146 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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1answer
40 views

What does a lattice of the direct power of the two-element chain look like?

In universal algebra, it is known that every finite Boolean lattice is isomorphic to a direct power of the two-element chain. I am having hard time figuring out what a lattice of the direct power of ...
2
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1answer
87 views

Intuition behind combinatorics problems?

I have a hard time understanding when a problem is a combination or permutation. Especially when using the multinomial theorem. For example: three boxes numbered 1,2 and 3 for k = 1 2 and 3, box k ...
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0answers
11 views

Conceptual question about independence and stopping times

Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence of i.i.d. random variables with common distribution function $\mu$. Consider a property $A$, such that $\mu(A)>0$. Define $T$ to be stopping time ...
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0answers
24 views

Is it possible to derive the sum of the tangent or cotangent from this?

So we can suppose that we can find the finite sum $$\sum_{x=a}^b{ \tan{(x)} + \cot{(x)} } \tag{1}$$ for essentially all integer values of $x$. I'm wondering, can we derive either: $$\sum_{x=a}^b{ ...
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4answers
561 views

What does it really mean when we say that the probability of something is zero? [duplicate]

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
5
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3answers
245 views

Intuitively, how do you explain the concept of Flux?

Lately in my physics and mathematics classes, I've come across the concept of Flux. And although I've been able to define them mathematically and figure out how to use them. I'm still not entirely ...
2
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1answer
88 views

counting occurence of subgraphs by counting their occurence in larger subgraphs

I have a mental block in fully understanding the following notion. Let $G$ be a graph of order $n$ and $H$ a fixed small graph of order $k \le n$. Suppose that there are $d$ copies of $H$ as an ...
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2answers
34 views

What is the d in the formula of a plane in $ R^3$

In algebra the formula for a line is $y=ax+b$ the $b$ moves the position of the line up and down the y axis. The formula for a plane is given to me as $ax+by+cz+d=0$ the $d$ must move the position of ...
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1answer
101 views

Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as: A matrix is just a compact notation, which allows you to specify several linear equations at once ...