Questions tagged [intuition]

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.

To have intuition about a mathematical truth is to have some insight into why it is true, and to understand the motivation for talking about that truth in the first place. This is usually stated in contrast with merely having a superficial knowledge of a mathematical truth as a fact, or only having skills at applying a mathematical truth to solve a problem without having the conceptual understanding of solution.

For a nice explanation of mathematical intuition with examples, and links to other articles on developing mathematical understanding, see Developing Your Intuition For Math on BetterExplained.com.

3739 questions
3
votes
1 answer

Gradient descent scaling

If gradient descent converges with a learning rate of 0.1 for f(x), should the same learning rate work for g(x) = f(10x)? I think that g changes more quickly than f, so want to take smaller steps. The norm of the gradient of g will be 10 times the…
3
votes
1 answer

Intuition of the Surface Integral of a Real-Valued Function

I'm having trouble understanding the idea of a surface integral of a real valued function $f$. I've read some of the other answers here on Stack Exchange, but they seem to be focused on the surface integral of a vector field. If I have a surface…
Snowball
  • 959
  • 3
  • 11
3
votes
1 answer

Cardinality of set of $a_r$?

Question So I conjectured a formula which was proven: Let $b_r = \sum_{d \mid r} a_d\mu(\frac{m}{d})$. We prove that if the $b_r$'s are small enough, the result is true. Claim: If $\lim_{n \to \infty} \frac{\log^2(n)}{n}\sum_{r=1}^n |b_r| = 0$ and…
3
votes
2 answers

Why is $\lambda$ called the *instantaneous* rate of change in the exponential distribution?

In the following paramterisation of the exponential distribution $${\displaystyle f(t;\lambda )={\begin{cases}\lambda e^{-\lambda t}&t\geq 0,\\0&t<0.\end{cases}}},$$ $\lambda$ is called the "rate" parameter. If $T \sim \text{Exp}(\lambda)$, I think…
3
votes
1 answer

Intuition behind the concept of a topology

So I have tried to understand the basics of topology, but I have some trouble getting a good intuition for it. I know that the idea is supposed to be that we have various open sets telling us something about the "nearness" of the points in the…
Felis Super
  • 153
  • 4
3
votes
0 answers

How to think of cohomology classes

I guess some of my problems come from the fact that I can hardly visualise cohomology. For homology in dimensions 1–3, I claim to have at least some intuition what homology classes look like, and when two cocycles are homologous. But for…
Bubaya
  • 1,746
  • 7
  • 16
3
votes
0 answers

Behaviour of sine sums

I was considering the following function $$f_a(x)=\sum_{n=0}^{\lfloor x\rfloor}\sin^2(an)$$ and, as expected, $f_a(x)\approx x/2$ for every $a$ (except $2\pi$ and similar). This is because the function $\sin^2(ax)$ is "on average" equal to $1/2$.…
augustoperez
  • 2,788
  • 8
  • 21
3
votes
1 answer

Basic Question about linearity of expectation

I am going through some introductory notes on probability here http://www.stat.berkeley.edu/~aldous/134/gravner.pdf In Chapter 8, page 89, there is a problem where you get a bag containing 10 Black, 7 red and 5 white balls. What i find surprising is…
user74057
  • 95
  • 1
  • 5
3
votes
1 answer

Eigen Vector Method Vs Correlation-Free Coordinates Transformation Method

So, I general, Normed eigen-vector aims to transforms the original problem to uncorrelated state. But, according to the following, figure the transformation to new coordinates system is done using (sines and cosines): Assuming the equation…
GENIVI-LEARNER
  • 504
  • 1
  • 3
  • 14
3
votes
2 answers

Distributive Property of Dot Products - Geometrically, but a different approach.

The Question: The distributive property for dot-products says that: $$\vec{a} \cdot (\vec{b} + \vec{c})=\vec{a} \cdot \vec{b} + \vec{a} \cdot\vec{c}$$ If we think of the dot product as the projections of $\vec{b}$ and $\vec{c}$ onto $\vec{a}$ scaled…
joshuaronis
  • 1,339
  • 7
  • 20
3
votes
2 answers

Interpretation of the heat equation

Let $u=u(x,t)$ a solution of $$\begin{cases}\partial _tu=\partial _{xx}u\\ u(x,0)=f(x)\end{cases}$$ I can compute the solution, but I can't interpret this sort of equation. For an ODE $v'(t)=f(v(t))$, I see it as : we look at the mouvement of a…
3
votes
3 answers

Orthogonal Projections Are Symmetric - Geometric Intuition

Let us denote the projection matrix onto the column space of $A$ by $\pi_A = A(A^T A)^{-1} A^T$. I am looking for geometric intuition as to why it is symmetric. It is very clear to me due to plenty of algebraic reasons (taking transpose, showing…
3
votes
0 answers

Why are compact objects "small"?

Typically in categories of algebraic objects (I will use $\mathsf{Grp}$ as an example), every object is the direct limit of its finitely generated subobjects. Based on this observation I was wondering how general this phenomenon is, and it turns out…
Alessandro Codenotti
  • 11,165
  • 2
  • 26
  • 51
3
votes
2 answers

Intuition behind why is unit speed parametrization and arc length parametrization the same?

I have found a bunch of simple and not so simple proofs about why a vector function ($f(t)$) parametrized in such a way that it's derivative is always 1 ($|f'(t)|=1$) is the same as parametrizing it by arc length ($f(s) \iff |f'(s)|=1$). Just to…
3
votes
2 answers

Intuitive/combinatorial explanation of Delannoy summand

The $n$th central Delannoy number $D_n$ is the number of lattice walks from $(0,0)$ to $(n,n)$ taking only steps up, right, and northeast, to neighbor lattice points. From Wikipedia: $$D_n=\sum_{k=0}^n{n \choose k}{{n+k} \choose k}.$$ I am wondering…
1 2 3
99
100