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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432 views

Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?

Let $R$ be a commutative ring, $\mathfrak{p}$ a prime ideal of $R$, $M$ a $R$-module, and $N$ a $R_\mathfrak{p}$-module. Why do we have this isomorphism? $$M \mathbin{\otimes_R} N \cong M_\mathfrak{p} ...
4
votes
2answers
726 views

Deepest theorems with simplest proofs [closed]

Which are the deepest theorems with the most elementary proofs? I give two examples: i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function ii) Proof that the halting problem is ...
1
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1answer
89 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...
10
votes
2answers
254 views

What is the difference between probability and statistics?

Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?) I'm pretty ...
5
votes
1answer
165 views

Sorting through “algebra of random variables,” vs. “probability space,” etc

I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for ...
2
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2answers
163 views

Is it possible to determine that a coin is biased or not, by tossing it a number of times?

Is it possible to determine that a coin is biased or not, by tossing it a number of times ? I am sure that this problem has been studied,I am interested to know about the mathematics behind this ...
1
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3answers
2k views

Pattern-matching puzzle (shapes)

some puzzle that can be found online are typically a 3x3 grid where you've got nine spots filled with some images or symbols or geometric figures and you have to predict what the missing one should ...
2
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0answers
77 views

Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$ m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy $$ where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$ h(x,y) = x^p y^q f(x,y) dx dy ...
6
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3answers
310 views

Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive?

Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive? Whenever $f'$ (the first derivative) is positive the function is increasing,but does that ...
5
votes
6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
1
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1answer
390 views

Is there an intuitive interpretation of $ABA^T$?

I see expressions like this all the time in technical literature. The only $A$ and $B$ can be any size matrices as long as the expression is legal. I believe that the transposition is usually ...
2
votes
3answers
386 views

How many different combinations of $X$ sweaters can we buy if we have $Y$ colors to choose from?

How many different combinations of $X$ sweaters can we buy if we have $Y$ colors to choose from? According to my teacher the right way to think about this problem is to think of partitioning ...
2
votes
2answers
272 views

What is the reasoning behind a multinomial coefficient in a practical sense?

If you want to divide a team of 10 people into teams A, B, and C of sizes 3,5, and 2, how many divisions are possible? If you want to divide them into just teams of sizes 3,5, and 2, how many ...
15
votes
3answers
3k views

Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
3
votes
3answers
139 views

Intuition about the faces in the connected planar graphs

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer ...
5
votes
0answers
180 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
2
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2answers
3k views

Find the area of the curved shape

How to find area of this curved shape?
6
votes
1answer
261 views

Generalisation of dualities, what concept do dualities represent?

Duality is a concept that pops up in different areas of mathematics as well as other science, but besides being a "woo isn't that nice?", is there anything more to duality (than loosely stated some ...
1
vote
2answers
213 views

Factorials and Combinations

I understand that $$n! = n (n-1) (n-2)\cdots 2 \cdot 1.$$ My book says this can also be written as $$n (n-1)!$$ Without telling me why My question is How and why is that? Why can't we leave it as it ...
5
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2answers
2k views

Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
3
votes
1answer
355 views

Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements

Let $X$ be the set of natural numbers $k_i$, $k_i \geq 1$, with the property that at least one of the equations $p_i = $6$ k_i \pm 1$ gives the $i$-th prime number (disregarding $2$ and $3$), and ...
8
votes
2answers
197 views

In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...
2
votes
2answers
915 views

Difference between Logarithms of different bases

Every time i see a logarithmic function and if it so happens that i'am required to take the derivative or the integral of that particular function i get stumped and i tend to avoid that problem. What ...
7
votes
2answers
1k views

where did determinant come from? [duplicate]

Possible Duplicate: What's an intuitive way to think about the determinant? I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$ I ...
18
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5answers
2k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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4answers
2k views

Ten soldiers puzzle

This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each ...
26
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2answers
2k views

Intuition behind looking at permutations of the roots in Galois theory

What I find after reading books is that they explain only the conceptual definition and no one mentions the explanation behind it; I have been reading the Galois theory as many people told me to read, ...
27
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3answers
2k views

Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
2
votes
1answer
240 views

An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
6
votes
3answers
1k views

What is a physical “dimension” - in the sense of “dimensional” analysis?

Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ...
10
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3answers
4k views

Definition of e

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form Is $dy/dx$ not a ratio? I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as ...
16
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7answers
6k views

Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
13
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1answer
2k views

What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
33
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1answer
797 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
10
votes
3answers
611 views

What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?

What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ? $\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation ...
4
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0answers
84 views

Complex graphs in 3D

Does anyone have red-green 3D software for plotting 4D graphs in 3D with 3D glasses? I've seen a 4D hypercube done this way and it's very revealing...
1
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3answers
495 views

I need explanation for this solution for the proof. (Perfect square ends with 0,1,4,5,6,9)

Give a proof to the sentence: "The final decimal digit of a perfect square is 0, 1, 4, 5, 6 or 9." Solution: A integer $n$ can be expressed as $10a+b$, where $a$ and $b$ are positive integers and $b$ ...
1
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3answers
107 views

Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
79
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4answers
6k views

What are the Axiom of Choice and Axiom of Determinacy?

Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple ...
3
votes
1answer
137 views

Looking for an article on general principles of discrete mathematics

In his article 2 cultures Timothy Gowers states that the structure in combinatorics is there in the form of somewhat vague general statements that allow proofs to be condensed in the mind, and ...
8
votes
3answers
660 views

Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
11
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4answers
1k views

What is the intuition behind the “par” operator in linear logic?

I'm $\DeclareMathOperator{\par}{\unicode{8523}}$ trying to wrap my mind around the $\par$ ("par") operator of linear logic. The other connectives have simple resource interpretations ($A\otimes B$ ...
42
votes
5answers
6k views

Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)? ...
3
votes
3answers
150 views

“Contradiction-free” in logic vs. “Contradiction-free” in plain mathematics

In our course we have defined a theory $T$ to be contradiction-free, if there are no formulas $\alpha_1,\ldots \alpha_n\in T$ such that $\neg ( \alpha_1 \& \ldots \ \& \alpha_n )$ is provable ...
6
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1answer
456 views

Intuition behind the scaling property of Fourier Transforms

I had a course in PDE last year where we used fourier transforms extensively; I understand the rules of manipulation and can prove the scaling theorem directly from the definition using a ...
5
votes
1answer
651 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
51
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12answers
5k views

I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
11
votes
4answers
2k views

Why is it that Complex Numbers are algebraically closed?

I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
14
votes
5answers
3k views

Four men, hats and probability

I encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...
6
votes
2answers
2k views

Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$

I came across the following problem in my self-study of real analysis: For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = ...