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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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 ...
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55 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 ...
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37 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 ...
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27 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 ...
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2answers
36 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. ...
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1answer
85 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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138 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] ...
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98 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 ...
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139 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 ...
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144 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 ...
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96 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 ...
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37 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 ...
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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 ...
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1answer
46 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 ...
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251 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 ...
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1answer
50 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 ...
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1answer
104 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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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 $a$ $$=a^2(b+c)+a(b+c)^2+bc(b+c)$$ $$=(b+c)(a^2+ab+ac+bc)$$ ...
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46 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 ...
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81 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, ...
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44 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 ...
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44 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 ...
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50 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 ...
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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 ...
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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 ...
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211 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: $|A\cup B|=|A|+|B|-|A\cap B|$ ...
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33 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 ...
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1answer
43 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 ...
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133 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 ...
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53 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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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 ...
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1answer
30 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?
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70 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 ...
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188 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 ...
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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 ...
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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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31 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 ...
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Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)

Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including ...
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2answers
104 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 ...
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46 views

Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...
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66 views

Distinguish Normal Distribution, Gaussian Distribution and Normalised Gaussian Distribution?

As I understood it, the 'normal distribution' is $$\frac{1}{\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2{\sigma}^2}\right)$$ Now according to this the 'normal probability density function' is ...
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31 views

Intuition of transversality equation

I am studying Differential topology from Guillemin / Pollack and unfortunately i cannot understand ıntuition of Transversality equation Let $f$ be a smooth map between smooth manifolds $X$ and $Y$ ...
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96 views

Partition of Unity for defining Riemannian metric.

Why do we need Partition of Unity for defining a Riemannian metric on a manifold ? What role does it play ?
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36 views

Logarithmic growth and integrability

I was helping a graduate student with a problem, and I realized that I didn't have a good intuitive explanation for the following phenomenon: $$ \int_e^\infty \frac{dx}{x\ln^p(x)}<\infty \quad ...
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1answer
88 views

Geometrical interpretations of SVD

I'm a bit confused by the various geometrical/visual interpretations of SVD or better I'm wondering how to reconcile them. Transformations : As explained here, the 3 matrices produced by the SVD can ...
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2 Cards are drawn from a deck of cards.

I use this site for learning mathematics. I have come across this question. and the explanations are very much clear. Suppose the question is 2 Cards are drawn from a deck of cards. What is ...
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1answer
138 views

surface area of cylindrical spiral

consider $$\{(r\cos(\phi),r\sin(\phi),\phi)|r \in (0,1), \phi \in (0,2\pi)\}.$$ is the surface area the same as that of the unit circle, that is $\pi$? Intuitively yes, maybe not :S
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63 views

fourier series of unknown functions

I am confused in understanding use of fourier expansions of functions. This answer, for example says that we can write voice as a sum of sines and cosines of different frequencies and amplitudes, but ...
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176 views

Intuition for the GP formula.

Okay, so I wanted to ask, whether the formula for a geometric progression also has an intuition. There is an intuition for the AP formula: $$\frac n2(2a+(n-1)d)=\frac n2(a+(a+(n-1)d))$$ So what we ...
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62 views

Fractional Sobolev spaces definition

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...