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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Intuitive concept for eulers number [on hold]

I'm trying to generate an intuitive notion for eulers number and why it's important. I understand it's definition as a limit, it's relation to the trigonometric functions and that the derivative of $...
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41 views

Could the classical Ramsey numbers be successors of primes?

Let $R(n)$ denote the classical Ramsey-number, that is, the smallest natural $N$, for which every blue-red edge-colouring of $K_N$ admits a monochromatic $K_n$. Consider the conjecture: $R(n)-1$ is ...
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25 views

Are there any differences between the mathematical definition of vectors and scalars and how they are defined in physics?

From a purely mathematical perspective, the notion of scalars and vectors and their different roles makes sense to me. Vectors are elements of a given vector space $V$, and scalars are elements of the ...
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1answer
51 views

Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
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108 views

Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
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22 views

Least amount of repetitions s.t. probability greater than 1/2

Assume that for a formula $F$ over $n$ variables, there are exactly $k$ allocations that satisfy it. How many random samples from the set $\{0,1\}^n$ are necessary to find an allocation satisfying the ...
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1answer
69 views

Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD

I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as a ...
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102 views

Why doesn't coordinate difference between two points correspond to distance between two points?

I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\...
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3answers
63 views

Find the first $4$ Hermite polynomials using a recursion relation

Given the Probabilists' Hermite differential equation: $$U''-xU'+\lambda U=0\tag{1}$$ A book question asks me to: Find the first $4$ polynomial solutions (for $...
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71 views

Why is analysis over the complex numbers so useful vs say other fields?

First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime ...
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22 views

cycloid of a unit-speed circle

In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, ...
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0
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25 views

Softmax Derivation Help

I've been reading a paper that derives logistic regression from a few assumptions . Here is the link. If you go to page 5 and look at equation 18 the author claims that this essentially says the ...
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98 views

How interpret the dual lattice $\Gamma^*$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \...
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2answers
102 views

In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
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52 views

Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
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1answer
43 views

Intuition for basic fact surrounding Gaussian integers.

What is the intuition behind the following fact? Among the odd primes: Those that have remainder $3$ upon division by $4$ remain prime in $\mathbb{Z}[i]$. Those that leave remainder $1$ ...
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1answer
45 views

Finding the product of a prime function…

If we take the primes $p_k < n$, and raise them to the highest power possible such that $(p_k)^{r_k} \le n$, what is the lower bounds on $\prod{ (p_k)^{r_k} }$? In other words, what are the ...
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105 views

Understanding the tensor-hom adjunction intuitively

I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i....
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160 views

an outline of “intuitive mathematics”?

This question is related to the third answer in this post. There seems to be a difference between the intuitive idea of a thing (such as a function) and "models" of that thing in mathematics (such ...
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2answers
31 views

Partial differentiability of $f(x, y) := {x^3 - y^3 \over x^2 + y^2}$ at $(0, 0)$

I thought this task up myself, so I'd be good to know whether my solution is correct or not. :-) Given $$f(x, y) := {x^3 - y^3 \over x^2 + y^2}$$ for $(x, y) \in \Bbb R \setminus {0},$ ...
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855 views

Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math? [closed]

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. For example, I have no idea what eigenvalues, ...
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Intuition about heat equation with Neuman boundary data

On a bounded domain, consider the heat equation $u_t - \Delta u = 0$ with $\partial_\nu u = c$ (constant) and initial data $u_0$ which is non-negative. As usual $\nu$ is the outward normal vector. ...
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2answers
42 views

Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
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What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
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Is my intuition of dense sets correct?

I am working with the usual definition of a dense set, which is Let $U$ be any non-empty open subset of $X$. A set $A$ is dense in $X$ iff $A \cap U \neq \emptyset$. My highly informal and ...
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68 views

Is this symbolic statement impossible?

Is this statement logically impossible if x is a single real number (i.e. not a set)? $$(x<5) \land(x>7)$$ it seems to me that x cannot both be greater than 7 and less than 5 if ...
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1answer
17 views

Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
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Intuitive reason why the Euler characteristic is an alternating sum?

The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-...
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47 views

Neumann vs Dirichlet eigenvalue problem - Intuition

What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, ...
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1answer
30 views

Intuition behind the number of roots of a polynomial and dimension of the solutions of a linear scalar differential equation.

I'm sorry if my question is ill-posed but it's something I've been wondering about for quite a while. I understand that a polynomial of degree n must have less than n real roots (because each power ...
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34 views

Intuition behind the finite geometric series formula?

Can anyone give some intuition or insight on why $S_n = a(\frac{1-r^n}{1-r})$ works? (I've seen the proof but I like being able to visualize to think about formulas in different ways.)
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Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
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Counter-examples for these T/F probability questions?

Claim 1: If $f$ is a pdf of a random variable $X$, then $0 \leq f(x) \leq 1 \quad \forall x\in\mathbb{R}$ Why is this False? Is it because $f(x)$ does not need to be defined on the whole real line? ...
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Given $C \subset A \subset X$, why is that $C$ is closed in $X$ if $A$ is closed, $C$ is open in $X$ if $A$ is open?

I want to understand a result discussed here : Subspace of a normal space Let $(X, \mathfrak{T})$ be a topological space. Given $C \subset A \subset X$, let $C$ be a closed set in $A$, then claim ...
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3answers
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Graphical explanation of the difference between $C^1$ and $C^2$ function?

We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary ...
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How quickly can we find a value that has large multiplicative order modulo $n$?

If we're trying to find an element modulo $n$ that has multiplicative order at least $\sqrt{n}$, how quickly can we do this? We don't know if $n$ is prime or composite, only that $n$ definitely has a ...
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75 views

What are permutable equivalence relations intuitively?

What are permutable equivalence relations, and what are they used for? What is the idea behind them? Could someone give me an example and a counterexample for finite sets? I have encountered the ...
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Legitimacy of drawing a complex curve like a plane curve

In algebraic geometry, we often consider a complex algebraic curve, and in order to get some intuition, we often draw it on the plane as if it were a plane curve. In most cases it turns out that the ...
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Finding 8 co-primes $\le 2^n$

We can find 8 co-prime integers $\le 2^n$ for sufficiently large $n$. I'm looking for asymptotic bounds for the minimum distance away from $2^n$ we have to go before finding 8 co-primes. In other ...
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29 views

Good explanation on generators of intersection of two cyclic subgroups?

If $\langle a \rangle$ is a cyclic group and $\langle a^n \rangle$ and $\langle a^m \rangle$ are two subgroups then a generator for $\langle a^n \rangle \cap \langle a^m\rangle$ is $a^{\text{lcm}(n,m)}...
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I need an intuitive explanation of eigenvalues and eigenvectors

Been browsing around here for quite a while, and finally took the plunge and signed up. I've started my mathematics major, and am taking a course in Linear Algebra. While I seem to be doing rather ...
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42 views

Dirichlet conditions - Explanation of the proof of theorem $4$

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
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Bayes' rule application proving my intuition wrong

This is the example I am looking at. Question: Suppose a family has $2$ children and one is a boy, and that the probability of having a child of either sex is equal and independent across ...
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Intuitive explanation of $L^2$-norm

I have to play a lot with the $L^2$-norm defined as $\|w\|=\sqrt{\int_a^b <f,f>}$. However, I don't understand the interpretation of that norm. We know that the euclidean norm measure the length ...
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How do curves consist of points?

According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in ...
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Is counting roots with multiplicites at all a geometric concept?

It is well known that a polynomial of degree $n$ admits $n$ roots when the field is algebraically closed. However, this comes with a caveat, in particular that the roots be counted with multiplicity. ...
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65 views

Propositional Logic - Can you Derive $C \to A$ from $A$ alone, given the introduction rule?

Apparently, according to the Conditional Introduction rule, this is valid: Prove $C \to A$ Source: http://kpaprzycka.wdfiles.com/local--files/logic/W12R Page 5 So before this, the way I viewed ...
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27 views

How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
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57 views

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...