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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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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13 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
52 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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33 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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74 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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75 views

Big list of books that focus on intuitive explanations [on hold]

Sometimes, it happens that I come across books that at times give some intuitive explanations of ideas and concepts. But now I would like to ask if you can make a list of books that focus on giving ...
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32 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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12 views

How could one go about constructing this relatively simple contagious diffusion-reaction model?

How could one go about constructing a contagious diffusion-reaction model showing the relationship between disease (e.g. Ebola) and number of available healthcare workers in an unevenly distributed ...
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93 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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22 views

Practical Intuition behind converge of random sequence $X_n \to X$

Can someone come up with a practical example as to what it means for a set of random variables to converge i.e. $X_n(\zeta) \to X(\zeta)$? Specifically, what is the meaning of $n$ and $\zeta$?
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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
20 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
43 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
58 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
62 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
26 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
33 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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16 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
29 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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21 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
24 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
50 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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22 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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885 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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299 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) ...
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1answer
38 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 ...
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1answer
71 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
30 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 ...
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17 views

Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
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1answer
39 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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8 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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21 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
458 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 ...
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3answers
170 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 ...
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1answer
78 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
32 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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81 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 ...
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246 views

What do rationals represent?

While learning about the construction of number systems, I realized that I had many misunderstandings of crucial concepts which I was learning intuitively. I recently learned about the construction of ...
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5answers
46 views

Not clear on what we mean with numbers with infinite digits

I am confused on a rather simplistic question. 1/3 = 0.333333333333 to infinity. So it has infinite digits. How is it possible to multiply such a number with another one and get a finite number? 6/3 = ...
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1answer
46 views

Understanding completness of a metric space

I have a question about completeness of a metric space. The lecture notes that I am reading on my own says the following. PROPOSITION  1. Let X be a complete metric space. Let Y be a closed subset ...
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3answers
280 views

How do you go about formalizing a concept?

I am reading Godel Escher Bach. I love it. In the first few chapters, the author shows what a formal system is and gives examples that eventually lead to a typographical formal system of strings that ...
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0answers
39 views

Intuition behind the definition of Measurable Sets

I started studying "Measure Theory and Integration" and went through the first section which talks about Lebesgue Outer Measure of a set. All was well until I started with the second section which ...
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1answer
39 views

On estimating monthly credit card payment amounts (some pragmatic constraints inside)

Right off the bat, I do hope this question doesn't attract a bunch of derisive comments about my personal affairs. I give the lengthy personal anecdote because I don't have the mathematical training ...
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2answers
54 views

What is the line integral in higher dimensions?

Given a function $\ f:\mathbb{R}^2\to\mathbb{R}$ and some curve $\ \gamma:[a,b]\to\mathbb{R}^2$ it is my understanding that the integral of $\ f$ over $\ \gamma$ is the area of the region "between" $\ ...
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1answer
33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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1answer
77 views

Differential Geometry Intuition Question

Apologies if I get the notation wrong. Still learning this stuff. Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow ...
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1answer
68 views

Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
6
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2answers
168 views

Why should the generalization of a 'sequence' be called a 'net'?

The title says it all, really. Reading through Reed & Simon's book on functional analysis, I have now reached the chapter on topological spaces, and the notion of a net is introduced there to ...
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1answer
93 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
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1answer
28 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...