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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8
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1answer
81 views

Is there some sort of trick to show naturality?

This is about natural transformations in category theory. Almost always, I somewhat know why some defined maps or homomorphisms behave naturally, but I am almost never entirely sure (if things get ...
0
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1answer
34 views

A question concerning Jacobians of coordinate transformation

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I ...
0
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1answer
28 views

How do you actually write out the terms in a Cauchy sequence?

For example for $\epsilon>0$ there exist $N$ such $n,m>N$ implies $|s_m-s_n|<\epsilon$. I understand that intuitively, we don't need to know a certain limit and thus this definition for ...
0
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0answers
34 views

What is an intuitive way to think about uniform continuity? [duplicate]

I've been having trouble understanding uniform continuity, as it doesn't seem to describe the same kind of thing as regular continuity. I know what the formal difference between the two is and I know ...
1
vote
1answer
76 views

Does a retraction really “retracts” something?

I wonder what is the intuition behind the definitions of a section and retraction in Category Theory. From Awodey's book: Definition 2.7. A split mono (epi) is an arrow with a left (right) ...
0
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2answers
88 views

What is the intuitive way to understand Dot and Cross products of vectors?

Suppose, we are crossing a river with heavy current using a speed-boat. If $x$ is the vector for the current and $y$ is the vector for the speed-boat, then what do $x \cdot y$ and $x \times y$ ...
1
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0answers
43 views

Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
2
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1answer
65 views

Intuitive way of thinking of reduced homology

I know the definition of it, but is there a more intuitive way of thinking of it that even a layman could (kind of) understand? The $n$-th homology group, for example, can be thought of as the number ...
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7answers
183 views

Why is $\ln 1 = 0 $? [closed]

Yes I know, and believe, and have used it for all the time I have done mathematics as fun as well as a subject. But why is it that $$\ln 1 = 0$$
2
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3answers
136 views

What is a dual space?

I've started studying differential geometry by myself and I ran into dual spaces in a section on 1-forms. I'm not very well versed in linear algebra so any help is much appreciated.
0
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0answers
44 views

Non-analytic proof that Beta-Binomial $BB(1,1,n)$ (i.e. $Binom(n,p)$ with $p\sim Unif(0,1)$) is discrete uniform on $[0,n]$

This is equivalent to showing that $$P_n(k)=\int_0^1\binom n kx^k(1-x)^{n-k}\,dx=\frac 1 {1+n}\tag 1$$ for $k\in [n]$. A single integration by parts yields $P_n(k)=P_n{(k+1)}$ for $k\in[n-1]$ which ...
3
votes
1answer
78 views

Geometric justification for the prime spectrum and “generic points”

I realize there have been plently of discussions about this, but most of them are over my head and I never understand the geometric intuition behind them. I'm trying to make a big list of ...
1
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2answers
68 views

Intuition behind Vacuous proofs

My book says we can quickly prove the conditional statement $P \implies Q $ when we know $P $ is false. This much I'm fine with as I can show it with a truth table. But then I'm asked to, using ...
1
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2answers
53 views

How to read formal proofs

I'm in discrete math and I see the following notation $\forall x (P(x) \implies Q(x))$ and I read it as "for all x, P of x therefore Q of x". I'm struggling to follow proofs with this kind of ...
2
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1answer
35 views

Relation between $SO(n)$ and rotations

We often consider $SO(n)$ as the group of rotations in $\mathbb{R}^n$ in the sense that the usual action of $SO(n)$ on $\mathbb{R}^n$ by matrix multiplication can be interpreted as a rotation ...
0
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0answers
26 views

Where does (remainder)/$(\Delta t)^2$ come from?

Can someone please explain what is going on here? Where does (remainder)/$(\Delta t)^2$ come from? How did we go from $(1/2)\|\boldsymbol{\ddot\gamma(t)}\|(\Delta t)^2$ to just ...
2
votes
2answers
29 views

Convex risk measures

What is the intuitive explanation for convex risk measures represented as: $$\rho(X)=\sup_{P\in Q}\{E_{P}(-X)+\alpha(P)\}$$ where $\alpha(P)$ is a penalty function depending on the plausibility of P. ...
1
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1answer
77 views

an intutive explanation for the radon nikodym derivative

Can anyone please assist with an intuitive explanation for the Radon-Nikodym derivative in application to defining a set of probability measures for set(s)?
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3answers
118 views

[Paradox]How can Godel prove that Godel sentence is unprovable but true, if such proof itself proves that Godel sentence is true?

Isn't the proof that Godel sentence is unprovable but true a proof itself that Godel sentence is true? Godel in the preface of his proof remarked: “From the remark that [the unprovable statement] ...
1
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0answers
95 views

Show that the centre of the circle $\dots$ approaches the point $\epsilon(s_0) = \dots$ - Elem Diff Geo Pressley

Help on getting started with this exercise Another approach to the curvature of a unit-speed plane curve $\gamma$ at a point $\gamma(s_0)$ is to look for the `best approximating circle' at this ...
3
votes
5answers
130 views

How to make sense out of the $\epsilon-\delta$ definition of a limit?

The informal intuition for the limit of a function is this: What is the value of the function $f$ as $x$ gets infinitely close to $c$? How on earth does this monster $$ \lim_{x \to c} f(x) = L ...
1
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2answers
141 views

Why do we have to prove $1+1=2$?

I have 0 knowledge on number theory, but I currently have to take a course about proving. So I wonder why do we have to prove something that is "trivial", in the sense that we are using it, without ...
7
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0answers
62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
2
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1answer
34 views

Intuitive interpretation of “co-bundles” i.e objects of $X\downarrow \mathsf{Set}$?

Objects of a slice category $\mathsf{Set}\downarrow X$ are just set functions into $X$, and they can be identified with the partition they induce on their domain with their fibers. Arrows in the slice ...
0
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1answer
33 views

Are conditional probabilities only work on uniform distribution?

The conditional probability of $A$ given $B$ is defined by $\mathrm{Pr}(A\mid B)=\mathrm{Pr}(A\cap B)/\mathrm{Pr}(B)$. I'm trying to understand this intuitively, but the intuition only works for the ...
1
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1answer
30 views

What is the intuition for two player games, mixed strategies are computed with respect to pure strategies instead of mixed strategies?

Let $x$ be the mixed strategy of player $1$ Then the mixed strategy for player $1$ is calculated with respect to $[1, 0], [0, 1]$, the pure strategies of player $2$. i.e. $x^*$ = $\max \min ...
5
votes
2answers
172 views

Determine whether a point lies inside the curve or outside a random curve using pencil and scale

Say, I am given a point and a closed curve. I don't know anything about the curve (where it is, what it is, its size etc.;say it is hidden somewhere)."I just can't see the curve but I can see the ...
4
votes
1answer
48 views

Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
4
votes
1answer
78 views

Can we produce a long exact sequence in cohomology from more than just short exact sequences?

It is well known that given a short exact sequence $$0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0,$$ we can form a long exact sequence in cohomology. (Example: the proof of the ...
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0answers
44 views

Reason for the method of factorization of cyclic expressions.

For example, I am given that factorize: $$a^2b+a^2c+ab^2+2abc+ac^2+b^2c+bc^2$$ So by the traditional method, we take the powers of $a$ $$=a^2(b+c)+a(b+c)^2+bc(b+c)$$ $$=(b+c)(a^2+ab+ac+bc)$$ ...
0
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1answer
38 views

Intuition for a proof that the rationals are incomplete. [duplicate]

Let A be a set of positive rationals $p$ such that $p^2<2$. Now this set contains no upper bound. To prove this, for every rational $p$, a number $p- \frac{p^2-2}{p+2}$ is associated. This ...
2
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0answers
79 views

Homology/cohomology for the uninitiated [closed]

I have heard of (co)homology occurring in many mathematical contexts and I vaguely suspect that it non-trivially relates different subjects. Also that it somehow relates to category/topos theory, ...
0
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0answers
34 views

What is the difference between irreducible and geometrically irreducible?

Let $k$ be a field and take $0 \not = F \in k[x_0, ..., x_n]$, where $F$ is homogeneous. And define $V(F)$ to be the projective hypersurface of zeros of $F$. I was wondering if someone could possibly ...
4
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1answer
35 views

Geometric intuition for left/right exactness

Sheaf cohomology measures the obstruction of the global section functor from being exact. Since it's left exact, it is exact iff it preserves epis. In particular, $H^1$ measure the failure to be ...
0
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2answers
49 views

Why can $(x,y)$ coordinates be defined as $(\cos\theta,\sin\theta)$ for the unit circle?

I am having a hard time understanding that how/why can we define $(x,y)$ as $(\cos\theta,\sin\theta$. I googled and found out that we can do that because for $\theta\geq\frac\pi2$, a triangle can be ...
5
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0answers
64 views

Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...
0
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3answers
41 views

Intuitively, why does the dot product of any point in the plane with the normal vector always give the same answer?

A few days ago I asked: Help with Proposition $2.3.3$ from Elem. Differential Geometry by Pressley But now I have a similar question: If $\alpha$ and $\beta$ are points on the plane, why does ...
10
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2answers
192 views

Intuition for Inclusion-Exclusion Principle

Many of us are familiar with the inclusion-exclusion principle. I think the principle makes total sense when applied to the two or three sets and we have the following: $|A\cup B|=|A|+|B|-|A\cap B|$ ...
1
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1answer
32 views

Interpretation of $ \int_A f \ d\mu $ in case $\mu (A) = \infty$

Let $\mu$ be a measure and $f$ measurable and real-valued. If $\mu (A) < \infty$ then $\frac{1}{\mu (A)} \int_A f \ d\mu $ (assuming the integral exists) is the $\mu$-average value of $f$. Is ...
0
votes
1answer
38 views

Orthogonality in Hilbert Spaces

For the sake of concreteness, let's say that our Hilbert space is the beloved $\mathscr L^2(\Bbb R)$. Suppose that we have $\psi,\phi\in\mathscr L^2(\Bbb R)$, what's the intuitive meaning to a ...
6
votes
4answers
116 views

In layman's terms: What is a stochastic process?

I'm a software engineering student, so I don't have probability theory in my career, but I hear people near my circle talk about stochastic processes very often. Could someone explain in simple words ...
1
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1answer
43 views

What is the logic behind picking eigenvectors when eigenvalue is zero

Suppose I had this matrix $A = \begin{bmatrix} -1 & 0 \\ 0 & 0 \end{bmatrix}$ eigs of $A$ is $-1$, and $0$ What would be the logic in picking the eigenvector for this scenario?
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0answers
42 views

How did Strassen discover his matrix-multiplication algorithm?

Do we know how Volker Strassen discovered his famous matrix-multiplication algorithm? In particular, is there a systematic way to derive the seven products used in Strassen's algorithm? How did Volker ...
0
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1answer
29 views

Help with Proposition $2.3.3$ from Elem. Differential Geometry by Pressley

Why can we have $\mathbf v \cdot \mathbf N =d$? Why is $\mathbf v \cdot \mathbf N =d$ a plane? Where did $\gamma \cdot \mathbf N=d$ come from? Why can we do this?
4
votes
1answer
63 views

entire functions and multi-valued functions, an easy to understand explanation?

From wikipedia: The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. I have plotted the ...
0
votes
2answers
126 views

Absolute Value of Cosine and Sine in $\mathbb{C}$

Is it generally true that $|\cos(z)|\leq1$, $|\sin(z)|\leq1$ $\forall z \in \mathbb{C}$? I think I'm missing something here (I think it does not hold, only if $z \in \mathbb{R}$). If this were not the ...
0
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0answers
18 views

Convex optimization: how to understand epigraphical projection

In Rockafellar's text on convex optimization: Here we can think of the epigraphical projection as if someone shined a light to $f(x,u)$ and the shadow on the $u$ plane is the projection. My ...
0
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0answers
26 views

Intuition for grad(F(x)) = F(x) grad(log(F(x))?

It is straightforward to prove that $\triangledown f(x) = f(x) \; \triangledown\log(f(x))$. But is there some intuitive way to understand this identity?
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0answers
28 views

Are choice function and axiom of choice equivalent?

Axiom - True for all $X:$ if $X \neq \emptyset,$ then there's some $f: X \to \cup X$ such that for any $A \in X, f(A) \in A.$ But the axiom above looks very much like the choice function. Are they ...
2
votes
2answers
94 views

Derivation of the Dirac-Delta function property: $\delta(bt)=\frac{\delta(t)}{\mid b \mid}$

Considering the case such that $b \gt 0$ and $b \in \mathbb{R^+}$ and making the substitution $t'=bt$, it follows that $$\int_{t=-\infty}^{t=\infty}f(t)\delta(t)\mathrm{d}t ...