Tagged Questions

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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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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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 ...
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Intuition for Kuratowski-Mrowka characterization of compactness

Fact. A space $X$ is compact iff for every space $Y$, the projection $X\times Y\rightarrow Y$ is a closed map. The finite subcover definition of compactness seems reasonably intuitive: finite covers ...
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What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...
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Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
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Motivating $y'=y \implies y=Ce^x$

Is there some intuitive reason why one should think that a function which is its own derivative should be of the form $Ca^x$ for some number $a$? Of course I can prove that the unique solution set to ...
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How to see that $\text{gcd}(a,b) = \text{gcd}(a-b,b)$?

I'm trying to understand why $\text{gcd}(a,b) = \text{gcd}(a-b,b)$. What is clear to me is that the $\text{gcd}$ divides $a,b$ and also $a-b$ (let's assume $a\ge b$). But then it seems to me we ...
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What does it mean for a set to be countably infinite?

Why distinguish between countable and uncountable? What advantages does this property have? I haven't studied much set theory but I am writing about the set of algebraic vs transcendental numbers and ...
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Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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The reason I ask is because according to this source: the $\fbox{$\color{blue}{\mathrm{PDF}}$}$ for the sum of two Exponential Density Functions is $$\rho(x_1,x_2)\mathrm{d}x_1 \mathrm{d}... 2answers 131 views Which general physical transformation to the number space does exponentiation represent? Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while 2 + 4 is ... 7answers 3k views Why does the fundamental theorem of calculus work? I've known for some time that one of the fundamental theorems of calculus states:$$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$Despite using this formula, I've yet to see a proof or even a ... 2answers 81 views What is the difference between a Poisson and an Exponential distribution? For a Poisson distribution:$$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$where \mu is the mean number of occurrences. For an Exponential distribution:$$f(x;\lambda) = \begin{cases} \...
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Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? What is the true nature of this paradox ? I don't really understand this ?
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How to see symbol manipulation from an intuitive perspective in math?

I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon ...
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Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...