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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.

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Solving the BVP using finite differences and intuition regarding the form of the numerical solution?

Suppose we have $u'' + u' =0$ where $x \in (0,1)$ with the boundary conditions $u(0) =0, u(1) =1$. We consider the BVP finite difference approximation $\frac{u_{j+1} -2u_{j} + u_{j-1}}{h^2} + \...
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Tautological one form assigns a numerical value to the momentum $p$ for each velocity?

In the wiki, it is written that : the tautological one-form assigns a numerical value to the momentum $p$ for each velocity $\dot {q}$, and more: it does so such that they point "in the same ...
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Example for Hilbert quote

Hilbert famously said The art of doing mathematics consists in finding that special case which contains all the germs of generality. Can you give an example of a situation in mathematics where ...
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Connection between fundamental group of circle and division by zero

Going through some of the applications of the proof that $\pi_{1}(S^{1}) \cong \mathbb{Z}$ in Hatcher, I have noticed that almost all of these arguments rely on a contradiction whose assumption ...
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21 views

Difference between a binomial experiment and a Bernoulli Trial?

I'm studying in an introductory statistics textbook and it's confusing me when it mentioned this: "Any experiment that has characteristics two and three and where n= 1 is called a Bernoulli Trial(...
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23 views

Intuition behind why a binomial experiment has to have a fixed number trials

I'm studying an introductory statistics textbook and it mentioned this: "There are three characteristics of a binomial experiment: There are a fixed number of trials. Think of trials as repetitions ...
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54 views

Tossing Coin Expected Value Against Intuition

I have the following two coin toss games: Game 1: A and B tosses a coin. At first the coin is unbiased. Through the game, if heads comes A wins and game stops. If tail comes, the coin is swapped with ...
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How to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, (x^1, …, x^n))$ is a local chart of a smooth manifold.

Let $M$ be a smooth manifold. I am getting lost in the notations. Could someone please explain me how to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, \phi = (x^1, ..., x^n))$ is a local ...
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Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
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51 views

Rigidity Theorem, meaning.

What does it mean to say that a theorem is a rigidity theorem? I'm reading a book (Lipman Bers, a Life in Mathematics) that says for example that Torelli's theorem is a rigidity theorem. So what ...
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Intuition behind the probability of an event and another event's complement

I don't get the intuition when my introductory statistics textbooks mentions this, can someone please explain it to me?: P(D and B`) = P(D) - P(D and B)
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Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
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Power Set and Empty Set question

I have a question regarding the set of functions resulting from a set raised to a power. I think I have part of the understanding correct, however I'm having trouble interpreting $Y^{\emptyset}$. I ...
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76 views

Why is the definition of inductive set well defined?

I've been studying from Enderton's Mathematical Introduction to Logic in which he defines an inductive set as follows: To simplify our discussion, we will consider an initial set $B \subseteq U$ ...
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What it means that multiplication is stretching and addition is sliding the number line?

I have read these definitions but I am not sure that I understand them. Could someone give the intuition behind these definitions?
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The relation between the F1 score and the combined work formula?

I am wondering why we should use the harmonic mean(as stated in Wikipedia: "the traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall") as the F1 measure, ...
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Consider a function $f(x,y,z(x,v))$. Does $\frac{\partial f}{\partial z}$ hold $x$ and $v$ constant?

where $x,y,v$ are independent variables. Basically, I am confused with the fact that a partial derivatives holds all variables constant, other than the variable with which we are taking the ...
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What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
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1answer
63 views

Intuition behind proof of Schwarz's lemma

There is the very well known proof of Schwarz's lemma in complex analysis. When I read it I feel like the answers described here. I'm not sure how I would motivate and explain why one should expect ...
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1answer
26 views

Intuition behind independent events in introductory statistics

I'm learning through an introductory statistics textbook, and I just can't get the intuition when the textbook mentioned this for independent events $A$ and $B$: $\bullet$ P(A|B) = P(A) $\bullet$ P(...
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44 views

Taylor Series - why that specific form?

Something that has bugged me since university. Why does the Taylor Series have that specific form? For example there is a division by n! - why not (say) (n^2)! How does one get to the Taylor Series? ...
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Intuition of finding the formula of Laplace's expansions of the Determinant?

How did Laplace find the formula $\left |A \right |=\sum_{i=1}^{n}(-1)^{i+j}(A)_{ij}M_{ij}$? What is the intuition of the evalution of this formula? Note: I'm not asking for proof that the formula ...
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47 views

Connections and metrics on Riemannian manifolds

In a lecture on connections it was claimed that a manifold "attains a shape" when it is equipped with a connection while there were no mentions of a metric. Is it correct to intuitively think that ...
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Describe the Bidual of Real Polynomials

Trying to understand (bi)duals of infinite-dimension vector spaces, I stumbled over the very concrete example of $\mathbb{R}[X]$, the (formal) polynomials over $\mathbb{R}$ or, equivalently, the space ...
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Problem with estimating a sequence with intuition

I've frequently used "intuition" to solve limits at infinity. For example, if someone asked me what is: $$ \lim_{x \to \infty} f(x) = \frac {x^5 + x^3 + x}{x^2} $$ Or a sequence that can be ...
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Intuition behind joint pdf with transformations with partitioned support

I'm trying to understand this part of Statistical Inference(Casella, Berger) regarding expressing the joint pdf of non-bijective transformations. More specifically, what is the intuition behind (4.3....
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28 views

Interpretation of the pushforward of a vector under a flow.

Suppose $\theta_t(p)$ is the flow of some vector field. I don't really understand how to interpret the meaning of the pushforward $(\theta_t)_*X$ of some vector $X$. What is its intuitive meaning?
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Symplectomorphism, intuitive interpretation.

What is an intuitive way to think about symplectomorphisms? A symplectomorphism between two symplectic spaces is a map $(M_1, \omega_1)\xrightarrow{\phi} (M_2,\omega_2)$ such that for the pullback $\...
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27 views

How to use the condition that $\vec a_i \cdot \vec b_j=2\pi\delta_{ij}$ to find reciprocal lattice vectors, $b_j$, for this rectangular lattice?

Consider a rectangular lattice in two dimensions with primitive lattice vectors $(a,0)$ and $(0,2a)$. Which of the following are reciprocal lattice vectors for this lattice? (a) $\quad\dfrac{...
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162 views

Manhattan distance problem with infinite zig zags

If you turn left/right any finite number of times going from point to point, it will be the same as if you traveled $x$ then turned once and traveled $y$ to get there. I hear that even an infinite ...
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38 views

How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?

I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix. If you can add an intuition for the ...
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Why is line of maximum variance same as the dominant eigenvector of Var-Covar matrix?

PCA was recently introduced to us but I cant seem to wrap my head around the fact that the line of maximum Variance will be same as an Eigenvector of VarCovar matrix, let alone the dominant one. (...
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Intuition for why function composition associative

It is easy to prove that function composition is associative. I think of $f\circ (g\circ h)$ as applying first $h$, then $g$ and finally $f$ to an element of the domain of $h$. But I'm having ...
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Geometric Intuition behind trace and matrix non-commutativity

My intuition of trace is that it is essentially asking, for each basis, where does this basis vector go, and what is the new vector's component in it's original basis. Then, trace is a sum of those ...
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Lie derivative isn't $lim_{t\rightarrow 0} \frac{W_{\theta_t(p)}-W_p}{t}$ in $\mathbb{R^n}$? Why?

The Lie derivative o a vector field $W$ along another vector field $V$ at point p (denoted by $(L_V W)_p$) is defined as the limit when $t \rightarrow 0$ of $\frac{(\theta_{-t})_*W_{\theta_t(p)}-W_p}{...
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Heuristic argument for pointwise convergence imply uniform convergence

I didn't want put the statement precisely in the title of the topic to not have a long title, but the items below describe precisely when pointwise convergence imply uniform convergence: Every ...
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How to be good at coming up with counter example in Topology

This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often ...
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Why is $\mathbb{R}$ unbounded, despite being equinumerous to various bounded sets? Is there a name for this “distinction”? [closed]

$[0, 1] \approx (0,1) \approx \mathbb{R}$, for example. Intuitively, it seems that the infinity of $\mathbb{R}$ is of a different nature than that of the intervals; with $\mathbb{R}$ I can “explode” ...
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Geometrically explained, why do Linear Transformations Take a circle to an ellipse

Short Version: How can it be geometrically shown that non-singular 2D linear transformations take circles to ellipses? (Also, its probably important to state I'd prefer an explanation that doesn't ...
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Does every topological space have an open cover?

Is it guaranteed that any topological space would always have an open cover? I think it should, but I wanted to check why. I feel like it's maybe related to the base of a topology? I know the base ...
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Intuition behind determinant of a matrix with $2$ equal rows [closed]

In my linear algebra course, we have just proved that if a matrix $A$ contains $2$ equal rows, then $\det(A)=0$. I understand how the proof works, but could somebody offer a more intuitive ...
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overdetermined system of non-linear equations

I have to solve an overdetermined system of non-linear equations. My system has a lot of equations, but here, for example, my system has $4$ equations. Because all the variables have to be binary, I ...
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Why are the Galois groups that correspond to extensions which adjoin primitive roots of unity given by the group of units mod n

Considering all the following in the context of Galois theory. I believe, given say the primitive $9^{th}$ root of unity, that this will have as its minimum polynomial , the cyclotomic polynomial $\...
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Motivation for the Arzela-Ascoli's theorem

I'm studying by myself Arzela-Ascoli's theorem and I'm reading this chapter of a lecture notes. Firstly, I would like to be clear that I know that the motivation of Arzela-Ascoli's theorem is ...
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Understanding HNN extensions: intuition, examples, exercises.

What is an HNN extension? What would be some elementary, intuitive examples of them and what exercises involving them would you suggest? The Wikipedia definition is easiest to get to, since neither ...
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Motivation of the von Neumann definition of ordinals

The von Neumann ordinals are defined in such a way that each ordinal is exactly the set of all smaller ordinals. I am wondering about the origin/motivation for this definition of ordinals (that is, ...
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Permutation graph and the number of inversion statistics

Let $K_m$ be the number of m-cliques, $m \in N$, in a random permutation graph $G_n$ with $n$ vertices and $\pi_n$ is the corresponding permutation representation in $S_n$. I am looking for a basic ...
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Intuition behind invariant measure for a stochastic matrix Norris Theorem 1.7.6

I'm trying to figure out the meaning of the following theorem from J. Norris - Markov Chains. Let $$\gamma_i^k=E_k\sum_{n=0}^{T_k-1}1_{\{X_n=i\}}$$ Let $P$ be irreducible and let $\lambda$ be an ...
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Geometric meaning of r-cycles, r-boundaries and homology groups for a geometric simplicial complex.

I just started learning about algebraic topology, and some things are already not so clear to me. If I consider a geometric simplex $K$, I kind of understand what $H_0(K)$ is. it is a set of ...
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Derivations of the trapezoid rule

I know the general method to derive the trapezoid rule is with Taylor series, or, you know, to just look at the trapezoids and figure out the rule. However, I feel that for such a simple rule, there ...