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 about Hyperplane

I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
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why does soundness seem to be less important than consistency for the structuralist?

If I am not wrong, many mathematicians (I believe this is not only restricted to structuralists) agree that an inconsistent formal system does not have any model. By model I mean some kind of set ...
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275 views

How to understand the duality between Dilworth's theorem and Mirsky's theorem?

Dilworth's theorem states that for any partial order, the size of the largest antichains is the size of the smallest chain partitions. Mirsky's theorem states that for any partial order, the size of ...
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283 views

Understanding Laplace Transforms

The Laplace transform of a function $f(t)$ is a function that maps $\mathbb{C} \mapsto \mathbb{C}$. $$f(s) = \int_0^\infty f(t)e^{-st}dt, \text{ with } s=x + iy$$ Since $s = x + iy$ is complex, ...
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Multiply two large numbers in under 1000 instructions using reduced ISA with only 7 registers [closed]

Is it possible to multiply two large (15 bit) numbers efficiently (in under 1000 instructions) using the following ISA: ...
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275 views

Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples

In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
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183 views

Hydra game and quantum superposition

Goodstein's theorem is not provable in Peano Arithmetic showed by Kirby and Harrington in 1982 [Wolfram Mathworld]. Any reference of a "quantum" hydra game where a head can remain in a state of ...
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Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
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how do I derive $1 + 4 + 9 + \cdots + n^2 = \frac{n (n + 1) (2n + 1)} 6$ [duplicate]

Possible Duplicate: Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? I am introducing my daughter to calculus/integration by approximating the area under y = f(x*x) by ...
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283 views

Age of Stochasticity?

Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the ...
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Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
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Represent the following decimal values in 8-bit 2's complement and hexadecimal

-27: 11100101 -128: 10000000 I'm having trouble coming up with the hexadecimal representation. Does it change if my numbers are represented in 2's complement as opposed to just regular binary? ...
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A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
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340 views

Homogeneous Differential Equations Inspiration

Homogeneous first order differential equations can be solved by substituting $y/x = v$. I was wondering what is the inspiration for this. I am trying to understand the thinking behind this ...
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582 views

Intuition behind topological spaces

I'm studying topology since a few months ago and I have never caught a good intuition of the topological spaces, but now I think that I did. My intuition is the next; as many people point out the ...
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406 views

KAM theory and the Ergodic hypothesis

I have seen several authors mentioning that KAM theory contradicts the Ergodic hypothesis. Unfortunately, the authors do not elaborate on this. I have some background in KAM theory but very little in ...
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1k views

Constrained variational problems intuition

Problem: minimise $F(x,y,y')$ over $x$, constrained by $G(x,y,y')=0$. $$J_1(x,y,y')=\large \int_{x_0}^{x_1}F(x,y,y')+ \lambda (x) G(x,y,y')dx$$ I understand the Euler-Lagrange equation and Lagrange ...
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Duality between $[G,G]$ and $Z(G)$? [duplicate]

Possible Duplicate: Center-commutator duality Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation} ...
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245 views

Why are only the first four alternating groups are non-simple?

I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
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100 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
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481 views

Get sum to a closed form ideas on how to start

I have this sum $ 1^2+ 2^2 + 3^2 + \ldots + x^2$ I started getting some sums to their closed forms. But I see that sometimes I start off on the bad track. So I'd like if it's possible some tips on ...
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169 views

Why this space is homeomorphic to the plane?

I'm trying to see why this picture below is homeomorphic to the $\mathbb R^2$. It's really hard, please I need an intuitive idea of this. This seems very weird for me, I need help. Thanks a lot
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Questions Based On A Couple Of Weird Limits

When evaluating certain limits,i get an answer with which i'am not fully convinced despite following steps which i claim is correct.Answers given by analytic method and L'Hopital's rule differs! I ...
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253 views

Intuitive interpretation of the adjacency matrix as a linear operator.

Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
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Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
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This quotient space is homeomorphic to the Möbius strip?

Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$ This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is ...
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Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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169 views

The Method Of Separation Of Variables

Can anybody explain me why the method of separation of variables for linear homogeneous PDE works ? thanks
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795 views

Centralizer, Normalizer and Stabilizer - intuition

What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application. Thanks.
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308 views

Intuition behind isomorphism of algebraic varieties

Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for ...
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254 views

Lie group and SO3 visualisation

Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to ...
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What was Klein working on when he “replaces his Riemann surface by a metallic surface”?

I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting: Look at Professor Klein: he is studying one of the most abstract questions of ...
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On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
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Labeled/unlabeled balls in unlabeled boxes

I was hoping I could receive some clarification into the the four cases: Placing labeled balls in unlabeled boxes with repetition. Placing labeled balls in unlabeled boxes without repetition. ...
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What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
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Using induction to prove P(k) with P(k-1)

Use induction to show $$1+\frac14+\frac19+...+\frac1{n^2} < 2-\frac1{n}$$ Assume $P(k-1)$: $$1+\frac14+\frac19+...+\frac1{(k-1)^2} < 2-\frac1{k-1}$$ Show $P(k)$: I tried to show that ...
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Finding a shortest path between a set of points

Suppose: $x_1,x_2,x_3,x_4$ is a shortest path from $x_1$ to $x_4$ $x_2,x_5,x_6,x_7,x_9$ is a shortest path from $x_2$ to $x_9$ $x_{10},x_5,x_8,x_3,x_9$ is a shortest path from $x_{10}$ to $x_9$ ...
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How many ways are there to choose 10 objects from 6 distinct types when…

(a) the objects are ordered and repetition is not allowed? (b) the objects are ordered and repetition is allowed? (c) the objects are unordered and repetition is not allowed? (d) the objects are ...
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Expressing a relationship in a graph using quantified logic

Express the following using quantified formulae for a simple undirected graph $G = (V,E)$. The predicate P({u,v}) is true when $\{u,e\}\in E$ and false otherwise. The diameter of $G$ is at most 2. ...
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Finding recurrence and an algorithm to represent it

You find yourself in a country with integer coin denominations $c_1 < c_2 < ... < c_r$, where $c_1 = 1$. Unfortunately, the greedy algorithm is not guaranteed to find the optimal way to ...
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242 views

If $f(n) = \Theta (g(n))$, why does $g(n) = \Omega (f(n))$?

Why is this the case? I understand that if $f(n) = \Theta (g(n))$ then $c_1g(n)<f(n)<c_2g(n)$, but why does this show that $g(n)$ is bounded below by $f(n)$? I would think that it would be ...
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How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
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If A $\propto$ B and A $\propto$ C while keeping each other constant, then why is A $\propto$ BC? [duplicate]

Possible Duplicate: Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables I guess some people may find this obvious, but I really don't. My question is: If ...
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Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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Intuitive explanation of Residue theorem in Complex Analysis

The residue theorem that states that if a) $U$ is a simply connected and open subset of the complex plane, b) $a_1,\dots,a_n$ are finitely many points of $U$, c) and $f$ is a function which is ...
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263 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
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What's the most elegant way of rotating a 3-dimensional co-ordinate system?

For two dimensional rotation of $x$ and $y$ axes anticlockwise by $\varphi$, the equation that transforms $P(x,y) \rightarrow P(x',y')$, $x'=x \cos(\varphi)+y \sin(\varphi)$ and $y'=y \cos(\varphi)- ...
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Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
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What's the intuition behind non-integer exponents/powers

Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. Is there some intuition to be had for the number $a^x$? For example the intuition of $a^2$ is obvious; it's $a*a$ which ...