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I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked".
So the question: What is a good book, for laymen, which ... |
One of the main ways that sine and cosine come up is as two solutions to the differential equation y'' = -y. Why is this an important differential equation? Well, by Newton's second law it's the differential equation that says "the force is proportional and opposite to the position." For example, this is what happen... |
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are mathematically. Can anyone explain what a monad is using as little category theory as possible? |
There are two descriptions that I know of. The first can easily be found by looking at wiki under Monad. The second is more interesting in my opinion.
I will assume that you don't know the definition of a monoidal action, if you do, just skip ahead.
A monoidal action is a functor from a monoid to the category of ... |
Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is:
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_5?ie=UTF8&s=books&qid=1279755366&sr=8-5
An added benefi... |
There are two descriptions that I know of. The first can easily be found by looking at wiki under Monad. The second is more interesting in my opinion.
I will assume that you don't know the definition of a monoidal action, if you do, just skip ahead.
A monoidal action is a functor from a monoid to the category of ... |
Say we have a function $f(z)=\sum_{n=0}^\inf a_n z^n$ with radius of convergence R>0. Why is the radius of convergence only R? Can we conclude that there must be a pole, branch cut or discontinuity for some z_0 with |z_0|=R? What does that mean for functions like
$f(z)=\begin{cases}
0 & \text{for $z=0$} \\
e^{... |
Why do complex functions have a finite radius of convergence? |
There are two descriptions that I know of. The first can easily be found by looking at wiki under Monad or consulting Harry's nice summary. The second is more interesting in my opinion.
I will assume that you don't know the definition of a monoidal action, if you do, just skip ahead.
A monoidal action is a functo... |
We define Lie algebra's abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the commutator bracket [a,b]=ab-ba. However, the notation suggests that this bracket is the one we think about. Additionally, the ... |
It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes.
What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number ... |
What property of regular polygons allows them to be faces of the Platonic Solids? |
Two questions:
1. Find a bijective function from (0,1) to [0,1]. I haven't find the solution to this since I saw it a few days ago. It strike me as odd--mapping a open set into a closed set.
2. S is countable. It's trivial to find a bijective function f:N -> N-S, (when |N| = |N-S|) f(n) = the nth smallest number ... |
I just got my statistics test back and I am totally confused about one of the questions!
> A study was done that took a simple random sample of 40 people and measured whether
> the subjects were right-handed or
> left-handed, as well as their ages.
> The study showed that the proportion
> of left-handed people... |
why can't we conclude anything from correlation? |
What can we conclude from correlation? |
It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes.
What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number ... |
Just an observation, but it appears the "study" was conducted under false pretenses. Handedness is a false dichotomy, people can also be ambidextrous.
Also, you cannot necessarily conclude that people become right handed. What if left handed people have a shorter life expectancy? I do not believe that there is anyth... |
I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at 2*2 = 4 degrees per hour.
But I'm having a bit more trouble understanding the Chain Rule as applied to mul... |
Intuitive reasoning behind the Chain Rule in multiple variables? |
You cannot necessarily conclude that people become right handed. What if left handed people have a shorter life expectancy? I do not believe that there is anything you can conclude from the facts given, other than what is already directly stated.
Also, just an observation, but it appears the "study" was conducted un... |
I just got my statistics test back and I am totally confused about one of the questions!
> A study was done that took a simple random sample of 40 people and measured whether
> the subjects were right-handed or
> left-handed, as well as their ages.
> The study showed that the proportion
> of left-handed people... |
You cannot necessarily conclude that people become right handed. What if left handed people have a shorter life expectancy? Or perhaps there was a spike in the birth rate of right handed people in the past. Thus, I do not believe that there is anything you can conclude from the facts given, other than what is already d... |
The [discriminant][1] <a href="http://mathurl.com/2dqz7gw"><img src="http://mathurl.com/2dqz7gw.png"></a> of the cubic polynomial <a href="http://mathurl.com/24gdf5r"><img src="http://mathurl.com/24gdf5r.png"></a> [indicates not only if there are repeated roots when Ξ vanishes][2], but also that there are three distinc... |
Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots? |
[This answer][1] suggests that there are explicit polynomial equations for which the existence
(or nonexistence) of integer solutions is unprovable. How can this be?
[1]: http://math.stackexchange.com/questions/250/a-challenge-by-r-p-feynman-give-counter-intuitive-theorems-that-can-be-transl/260#260 |
How can there be explicit polynomial equations for which the existence of integer solutions is unprovable? |
-1 is not 1, so where is the mistake? |
An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.
Doesn't symmetry and transitivity implies reflexivity? Consider the following argument.
For any a and b,
a R b implies b R a by symmetry. Using transitivity, we have a R a. |
Why isn't reflexivity redundant in the definition of equivalence relation? |
1/7 = 0.(142857)...
with the digits in the parentheses repeating.
I understand that the reason it's a repeating fraction is because 7 and 10 are coprime. But this...cyclical nature is something that is not observed by any other reciprocal of any natural number that I know of (besides multiples of 7). (if I am wr... |
Why is the decimal representation of 1/7 "cyclical"? |
1/7 = 0.(142857)...
with the digits in the parentheses repeating.
I understand that the reason it's a repeating fraction is because 7 and 10 are coprime. But this...cyclical nature is something that is not observed by any other reciprocal of any natural number that I know of (besides multiples of 7). (if I am wr... |
Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls? |
Perhaps it's not an entirely practical application, but Fibonacci numbers can be used to [convert from miles to kilometers and vice versa][1]:
> Take two consecutive Fibonacci
> numbers, for example 5 and 8. And
> you're done converting. No kidding β
> there are 8 kilometers in 5 miles. To
> convert back just re... |
I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that affine transformations (linear + translation) could be represented by matrix multiplication as well.
This got me to think... |
Can non-linear transformations be represented as Transformation Matrices? |
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, constru... |
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, constru... |
Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)?
To be a bit more precise, I guess I should ask if there any interesting examples of semigroups (X, *) for which there is not a monoid (X, *, e) where e is in X. I don't consider an exampl... |
Are there any interesting semigroups that aren't monoids? |
I'm trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the nature of the sort of systems they use. Are they simply based on higher-order logics with Henkin semantics, extended to higehr orders using type theory?
Proof verification programs such ... |
How do proof verifiers work? |
I'm trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the nature of the sort of systems they use. Are they simply based on higher-order logics with Henkin semantics. Though I'm mainly looking for a general answer with useful examples, here are a f... |
I'm trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the exact nature and construction of the sort of systems/proof calculi they use. Are they simply based on higher-order logics that use Henkin semantics, or is there something more to it? Though... |
I'm currently trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the exact nature and construction of the sort of systems/proof calculi they use. Are they essentially based on higher-order logics that use Henkin semantics, or is there something mor... |
The question is more profound than is initially seems, and is really about algebraic structures. The first question you have to ask yourself is *where you're working*:
In general, addition and multiplication are defined on a *structure*, which in this case is a *set* (basically a collection of "things") with two *... |
What's the meaning of this symbol in mathematical notation? :
> β§ |
What is the meaning of this symbol? |
When I tried to approximate $\int_0^1 (1-x^7)^{1/5} - (1-x^5)^{1/7} dx$, I kept getting answers that were really close to 0, so I think it might be true. But why? When I [ask Mathematica][1], I get a bunch of symbols I don't understand!
[1]: http://integrals.wolfram.com/index.jsp?expr=%281-x%5E7%29%5E%281%2F5%2... |
Why is $\int_0^1 (1-x^7)^{1/5} - (1-x^5)^{1/7} dx=0$? |
Many years ago, I had read a book entitled "Think of a Number" by Malcolm E. Lines, and it was an eminently readable and thought provoking book. In the book, there were topics like Fibonacci numbers (along with the live examples from the nature) and Golden Section. Now I'm looking for a similar book. Can anyone recomme... |
The set of all β β β continuous functions is [**c**](http://en.wikipedia.org/wiki/Cardinality_of_the_continuum). How to show that? Is there are bijection between β<sup>n</sup> and the set of continuous functions? |
Just to enlarge on Harry's answer:
Your symbol denotes one of two specified notions of implication in formal logic
$\vdash$ -the **turnstile** symbol denotes **syntactic** implication (syntactic here means related to syntax, the structure of a sentence), where the 'algebra' of the logical system in play (for ex... |
did you try the books of Eastaway and Wyndham? *Why Do Buses Come in Threes?* and *How Long Is a Piece of String?* |
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a limited set of basic geometry operations to find relations between the ele... |
I have read a few proofs that is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that is irrational? |
did you try the books of Eastaway and Wyndham? *Why Do Buses Come in Threes?* and *How Long Is a Piece of String?*
Blurb for the first one says "An amusing explanation of how maths is relevant to almost everything in life. Citing many examples of the way mathematics can explain common phenomena"; for the second, "Th... |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
If all sets were finite, how would mahtematics be like? |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
If all sets were finite, could the real numbers be constructed? |
There is a fragment of mathematics that is given by a set of axioms known as the [Peano axioms](http://en.wikipedia.org/wiki/Peano_axioms). Using these rules you can carry out a vast amount of mathematics relating to natural numbers. For example you can prove lots of theorems in number theory using these axioms. The Pe... |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
There is a fragment of mathematics that is given by a set of axioms known as the [Peano axioms](http://en.wikipedia.org/wiki/Peano_axioms). Using these rules you can carry out a vast amount of mathematics relating to natural numbers. For example you can prove lots of theorems in number theory using these axioms. The Pe... |
Can there be two distinct, continuous functions that are equal at all rationals? |
I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational? |
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number -- n/2 if it is even, 3n+1 if it is odd.
The conjecture states that if this operation is repeated, all n... |
Conjectures that have been disproved with extremely large counterexamples? |
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number -- n/2 if it is even, 3n+1 if it is odd.
The conjecture states that if this operation is repeated, all n... |
Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.
This can be proved using localization: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent. Hence every element of $P$ is a zero-divisor.
Can this be proved without... |
Why is it that a minimal prime ideal consists of zerodivisors? |
The group:
G = < x, y | x^2 = y^3 = (xy)^7 = 1 >
is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a finitely presented group, I could do a coset enumeration, but I don't see how this helps if I want to prove that it's infinite.) |
How do you prove that a group specified by a presentation is infinite? |
Why does a minimal prime ideal consist of zerodivisors? |
Are x*0 = 0 and x*1 = x and -(-x) = x axioms? |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
Okay I burned a lot of reputation points (at least for me) on MathOverflow to gain clarity on how to give some intuition into this problem, so hopefully this answer will be at least be somewhat illuminating.
To gain a deeper understanding of what is going on, first we need to answer the question, "What is a number?... |
Okay I burned a lot of reputation points (at least for me) on MathOverflow to gain clarity on how to give some intuition into this problem, so hopefully this answer will be at least be somewhat illuminating.
To gain a deeper understanding of what is going on, first we need to answer the question, "What is a number?... |
Okay I burned a lot of reputation points (at least for me) on MathOverflow to gain clarity on how to give some intuition into this problem, so hopefully this answer will be at least be somewhat illuminating.
To gain a deeper understanding of what is going on, first we need to answer the question, "What is a number?... |
Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling).
Would this answer be any different if similar but non-identical polygons were allowed? And if only if rotated/translated by rational coefficients?
Would this answer be any differen... |
Is the set of all unique polygons countable? If so, by what bijection to the natural numbers? |
I want to find the least squares solution to **Ax**=<b>b</b> where **A** is a highly sparse square matrix.
I found two methods that look like they might lead me to a solution: [QR factorization](http://en.wikipedia.org/wiki/QR_decomposition), and [singular value decomposition](http://en.wikipedia.org/wiki/Singular_v... |
How can I tell which matrix decomposition to use for OLS? |
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).
This is an old result ... |
Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? |
In intro number theory a key lemma is that if a and b are relatively prime integers, then there exist integers x and y such that ax+by=1. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove that "any ideal... |
Can you find a domain where ax+by=1 has a solution for all a and b relatively prime, but which is not a PID? |
I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone make this a little bit more precise? Are there good reasons for taking this point of view? Can you actually get math ... |
What is "ultrafinitism" and why do people believe it? |
The philosophy is explained in Doron Zeilberger's [article][1].
Basically, it's the belief that there is a largest natural number!
I've heard a funny story (on Scott Aaronson's blog) about someone who was an ultrafinitist.
-Do you believe in 1?
-Yes, he responded immediately
-Do you believe in 2?
-Yes... |
I know some references where I can find this, but they seem tedious(Both Hartshorne and Ueno cover this).
I am wondering if there is an elegant way to describe these. If this task is too difficult in general, how about just $\mathbb{P}^n$?
Thanks! |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say , the funct... |
![complex plot of the zeros][1]
(Diagram and setup from UCSMP _Precaluclus and DIscrete Mathematics_, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Ar... |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say , the funct... |
How does one prove that a simple (steps of length 1 in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in 1 or 2 dimensions returns to the origin with probability 1? |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say , the funct... |
![complex plot of the zeros][1]
(Diagram and setup from UCSMP _Precaluclus and Discrete Mathematics_, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x) = 4x^4 \+ 8x^3 - 3x^2 - 9x \+ c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting... |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say $x_1,x_2,\l... |
I'm not sure if I get the question but I'll wager a guess.
I'll do 1D.
1D walks are building binary strings, 010101, etc.
Say take six steps. Then 111111 is just as likely as 101010.
However, how many of the possible sequences have six ones? 1. How many of the possibly sequences have three ones and three ... |
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