# Tagged Questions

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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### Why gradient descent works?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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### What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
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### Why does the $2$'s and $1$'s complement subtraction works?

The algorithm for $2$'s complement and $1$'s complement subtraction is tad simple: $1.$ Find the $1$'s or $2$'s complement of the subtrahend. $2.$ Add it with minuend. $3.$ If there is ...
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### Intuition for iterated function for log log log n.

Intuitively, $\log n$ (base 2) is the number of times you have to divide $n$ by 2 before reaching a number around 2. (Waving our hands a little to gloss over floor vs ceiling and $\pm$ 1 errors.) ...
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### The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
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### What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" (...
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### Is there any graphical explanation of the derivative of $\sin x$? [duplicate]

I'm trying to understand in a practical/graphical view the derivative of $\sin(x)$ (that results in $\cos(x)$). Is there any animation or illustration explaining that?
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### Ways to think about the binomial coefficient

Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial ...
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### Average run lengths for large numbers of trials: Intuition and proof

This article states that the formula for the average run lengths for large numbers of trials is:$$\frac{1}{1-Pr(event\ in\ one\ trial)}.$$ My questions What is the intuition behind this formula? Do ...
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### Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?

For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$ I learned this definition some time ago, but I never really understood it. Is there a ...
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### Showing that an integral can not be expressed in terms of elementary functions

I recently encountered an integral of the form: $$\int{\frac{\log(a+bx+\sqrt{x^2+c})}{x}}dx$$ The result involves the dilogarithm function, but I was wondering if there is a fast way of showing that ...
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### Is symbolically solving $a(x)=f(x)g'(x)+f'(x)g(x)$, given $a$, ever easier than integrating $a$?

If we are given a function of $x$, $a(x)$, how hard is it to find an $f(x)$ and $g(x)$ such that $$a(x)=f(x)g'(x)+f'(x)g(x)$$ For comparison, I'd like to know when this is easier than symbolically or ...
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### How can I intuit the role of the central limit theorem in breaking the curse of dimensionality for Monte Carlo integration

I would like to more intuitively understand where the power of Monte Carlo integration comes from for large-dimensional domains of integration. Other questions on this site have referenced the proof ...
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### Can we possibly combine $\int_a^b{g(x)dx}$ plus $\int_c^d{h(x)dx}$ into $\int_e^f{j(x)dx}$?

I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other (...
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### Why should coordinate transformations be reversible?

Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping. ...
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### Euclidean Geometry in Classical Thought - Used for Realization or Representation?

I posted this in the Physics.SE Forum but I figured I'd ask this here as well since it's relevant to the forum subject :] Taken from John J. Roche's "The Mathematics of Measurement: A Critical ...
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### The Duality Functor in Linear Algebra

I'm trying to gain an intuitive understanding of the following construction: For any vector space $M$ over a field $R$, one can define the algebraic dual of $M$ as $M^* := \mathsf{Hom}(M, R)$ and ...
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### Question about equivalent metric spaces

I have studied that topologically equivalent metrics produce the same open and closed sets. They also produce same compact and connected subsets. Does it mean that topologically equivalent metrics ...
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### Intuition on characters of topological groups

I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't ...
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### Forming equation of a plane by solving linear equation set

Given three points on the plane: $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$ and $C(x_3, y_3, z_3)$. I'm trying to obtain the equation of the plane in this format: $ax + by + cz + d = 0$ I ...
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### Understanding isometric spaces

I have studied that an isometry is a distance-preserving map between metric spaces and two metric spaces $X$ and $Y$ are called isometric if there is a bijective isometry from X to Y. My questions ...
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### Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...
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### Why is convexity more important than quasi-convexity in optimization?

In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
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### Signs in binomial expansions

Edit the title as seems fit. \begin{align} (a^3+b^3) &= (a+b)(a^2 -ab+b^2) \\ &= (a+b)^3 -3ab(a+b) \end{align} And so on and so forth. Right now, I only need these expansions in ...
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### Explaining this particular version of the Implicit Function Theorem

I understand the general 'word' definitions of the Implicit Function Theorem and the simple examples such as the one on wikipedia but the version of the Implicit Function theorem given in our lecture ...
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### Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
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### Confusion about proof that first order logic without equality is not contradictory

I am having a problem understanding a proof from the field of mathematical logic. Seems like my brain cannot digest concepts from logic very well. I will quickly define some terminology and then ...
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### $SO(3)$ Lie group

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant ...
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### What is the intuition behind the Fermat-Euler's Theorem?

Can someone give me an intuition behind the working of Fermat-Euler's theorem? I am not looking for definition nor for proof (I know both of them). $$a^{\phi(p)} \equiv 1 \pmod p$$ This is what I ...
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### Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$

This is exercise 1.3.10 in Hatcher's book "Algebraic Topology". Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without ...
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### Perspectives on Riemann Surfaces

So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community ...
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### Problems with infinite $\Omega$, when trying to define product spaces of discrete probability spaces

Definitions In our course we defined a discrete probability space as a tuple $\left(\Omega,P\right)$, where $P:\mathcal{P}(\Omega)\rightarrow\left[0,1\right]$ and $\Omega$ is at most countable, such ...
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### Theories and models

I apologize if my question is not well formed. The reason for it is that I don't understand the concepts enough to be able to ask a completely meaningful question. In the classes we said that a ...
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### Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates. But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
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### Why is better to work with the spectrum of prime ideals than with the maximal one, for example in the definition of affine scheme.

When we have an algebraic variety we can identify the points of the variety with maximal ideals of the coordinate ring. I would like to know why it is more natural to define the main structure of ...
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### Can we think of a chain homotopy as a homotopy?

I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies. The definitions I'm using are: ...