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The tradition way for two children to divide a piece of cake fairly between them is "you cut, I choose", What process accomplishes this is intuitively clear. Is there a similar process which works for $3$ children.
Suppose that three children $(1),(2)$ and $(3)$ have associated measures (ways of evaluating the p... |
It is said that there is a proof of fundamental theorem of algebra using Lie Theory. I have seen this claim at various places. But I could never find such a proof. Can anybody help me out? |
> How there can be different kinds of infinities?
This is very simple to see. This is because of:
Claim: A given set $X$ and its power set $P(X)$ can never be in bijection.
Proof: By contradiction. Let $f$ be any function from $X$ to $P(X)$. It suffices to prove $f$ cannot be surjective. That means that som... |
I was quite impressed by a [beautiful proof](http://mathoverflow.net/questions/8846/proofs-without-words/8847#8847)<sup>1</sup> of the formula for n'th triangular number — it's kind of a bijective proof built on the top of geometric summation idea.
**Question**: does this argument generalize to higher dimensions? i... |
perhaps a somewhat silly question, but is there a generally accepted formal notation for denoting the number of columns a matrix has? Something I could use in the description of an algorithm like:
if horizontaldim(V) > x then
end if
or
if size(V,2) > x then
end if
or should I just use a... |
Suppose M_1 and M_2 are invariant subspaces of the unilateral shift U such that M_1 subset M_2 and M_1 is of codimension strictly larger than 1 in M_2. Show that there exists M invariant under U satisfying M_1 subset M subset M_2 where the inclusions are strict. All subspaces are closed.
Edit: Perhaps I can take M :... |
I discovered while teaching Calc 2 that if you apply L'Hopital's rule to $\frac{x}{\sqrt{x^2+1}}$ you get $\frac{\sqrt{x^2+1}}{x}$, and if you apply L'Hopital _again_ you get $\frac{x}{\sqrt{x^2+1}}$ back. In other words the L'Hopital _operator_ has a cycle of order two. I suppose the L'Hopital operator should be def... |
Has L'Hopital's Rule been studied as an operator? |
Let $G$ be a group. A normal subgroup $N$ of $G$ is called <i>ultracharacteristic</i> if for every normal subgroup $U$ of $G$ the condition
$$
G/U \cong G/N
$$
implies that $U \ge N.$ What are instances of natural conditions on $G/N$ that make $N$ ultracharacteristic? For example, suppose $G/N$ has only inner autom... |
*Two variables and one constraint*
If we have an objective function of two real variables $f(x,y)$ subject to the constraint $g(x,y)=0$ and want to find the minima or maximum, provided that there is any, then the three following conditions must be satisfied:
$f_{x}(x,y)=0$
$f_{y}(x,y)=0$
$g(x,y)=0$
Hence... |
I discovered while teaching Calc 2 that if you apply L'Hopital's rule to $\frac{x}{\sqrt{x^2+1}}$ you get $\frac{\sqrt{x^2+1}}{x}$, and if you apply L'Hopital _again_ you get $\frac{x}{\sqrt{x^2+1}}$ back. In other words the L'Hopital _operator_ has a cycle of order two.
EDIT (Thanks KennyTM): "I suppose the L'Ho... |
After giving to (1) one of unclaimed pieces, less than 2/3 of cake can be left (in measure of (2) or (3)), so one of them will be unsatisfied.
**Edit:** The right solution is the following.
1. Let (1) divide cake into three equal pieces.
2. Ask (2) and (3) "What is the best one?". If their answers are differen... |
In May's "A Concise Course in Algebraic Topology" I am supposed to calculate the fundamental group of the double torus. Can this be done using van Kampen's theorem and the fact that for (based) spaces $X, Y$: $\pi_1(X\times Y) = \pi_1(X)\times\pi_1(Y)$? Or do I need other theorems to prove this?
I believe that this ... |
The tradition way for two children to divide a piece of cake fairly between them is "you cut, I choose", What process accomplishes this is intuitively clear. Is there a similar process which works for $3$ children.
Suppose that three children $(1),(2)$ and $(3)$ have associated measures (ways of evaluating the pie... |
As other people has said, we model the situation by making a function $F(t)$ that returns the set of balls in the box at time $t$. Within this model, we have $F(\text{11:00}) = \emptyset$. That much is unambiguous.
But it's important to keep in mind that this is only a mathematical model. In the real world, there a... |
This is a pet idea of mine which I thought I'd share. Fix a prime $q$ congruent to $1 \bmod 4$ and define a sequence $F_n$ by $F_0 = 0, F_1 = 1$, and
$\displaystyle F_{n+2} = F_{n+1} + \frac{q-1}{4} F_n.$
Then $F_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}$ where $\alpha, \beta$ are the two roots of $f(x) = x^... |
Quadratic reciprocity via generalized Fibonacci numbers? |
After giving to (1) one of unclaimed pieces, less than 2/3 of cake can be left (in measure of (2) or (3)), so one of them will be unsatisfied.
**Edit:** The right solution is the following.
1. Let (1) divide cake into three equal pieces.
2. Ask (2) and (3) "What is the best one?". If their answers are differe... |
A measurable rectangle in the cartesian product $X \times Y$ of 2 measurable spaces in $\mathbb{R}^{2}$ is of the form $A \times B$, where $A$ and $B$ are measurable subsets of $X$ and $Y$ respectively.
**Question** Can a measurable function on $\mathbb{R}^{2}$ be essentially unbounded on every measurable rectangle... |
* As others have pointed out, there is no mathematical contradiction here. We have a sequence of sets $S_i$ with increasing cardinality whose intersection is the empty set; this is perfectly consistent with the axioms of set theory.
* Nevertheless, there is a serious philosophical problem here. I understand that J.... |
* As others have pointed out, there is no mathematical contradiction here. We have a sequence of sets $S_i$ with increasing cardinality whose intersection is the empty set; this is perfectly consistent with the axioms of set theory.
* Nevertheless, there is a serious philosophical problem here. I understand that J.... |
There is a famous literature on $\displaystyle e = \sum\limits_{n=0}^{\infty} \frac{1}{n!}$. We know that $e$ is irrational as well as transcendental.
Question is: For each $x_{n}$, sequence of $\pm{1}'s$, let $$f(x_{n}) = \sum\limits_{n=0}^{\infty} \frac{x_{n}}{n!}$$ for which sequences $(x_n)$ is $f(x_{n})$ ra... |
Have you tried to modify the proof of the irrationality of $e$ to deal with your similar-looking series? |
<b>Suggestion for the first question</b> [I have not checked all the details myself]: Have you tried to modify the proof of the irrationality of $e$ to deal with your similar-looking series?
Specifically, referring to the standard proof given in
http://en.wikipedia.org/wiki/Proof_that_e_is_irrational
which ... |
When writing software, there are often situations where I need a parameter to be a floating point number x in the range 0 <= x <= 1. I don't know of a name for that category, but I think there must be one because it's such a useful categories. Perhaps there's a name in probability theory?
(If they don't have a name,... |
Is there a name for numbers between zero and one, inclusive? |
Making explicit what is implicit in Qiaochu Yuan's comment, and demonstrating that [someone else's work][1] has failed to evade my eyes. (It is a neat article, read it.) I present this adaptation of Duval's algorithm.
Assign an order to your symbols, say $a_1, a_2, a_1^{-1}, a_2^{-1}$ let first_symbol and _last_sym... |
A Banach Space is a vector space which is normed and is complete with respect to its norm (every cauchy sequence converges). Examples of Banach spaces include $\mathbb{R}^n$ and the space of bounded continuous functions with the norm $\Vert f(x) \Vert = \sup \lbrace \vert f(x) \vert \rbrace$.
The set of all Bounded ... |
What is the biggest number ever used in a mathematical proof? |
Gerhard Michler has worked on a research program to show fairly convincingly that the possibility of infinitely many sporadic groups (with a uniform construction, but highly non-uniform properties) was quite real. Roughly speaking the second round of sporadic groups was discovered looking for special configurations of... |
<b>Suggestion for the first question</b> [I have not checked all the details myself]: Have you tried to modify the proof of the irrationality of $e$ to deal with your similar-looking series?
Specifically, referring to the standard proof given in
http://en.wikipedia.org/wiki/Proof_that_e_is_irrational
which ... |
I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother.
What's the simplest proof or whatever that requires the use of large cardinals? Is there some branch of mathematics that makes particularly heavy use of them? |
What are large cardinals for? |
I just started reading some notes on [Algebra and Topology][1]. Statement 1.7 is the following:
Hom (Z, Ker(f, g)) ≃ Ker(Hom (Z, X) ⇉ Hom (Z, Y ))
where:
Ker(f, g) = {x ∈ X; f(x) = g(x)}
I tried verifying this myself by drawing a diagram, but failed. Can anyone explain why this is true?
[1]: http://pe... |
I was quite impressed by a [beautiful proof](http://mathoverflow.net/questions/8846/proofs-without-words/8847#8847)<sup>1</sup> of the formula for n'th triangular number — it's kind of a bijective proof built on the top of geometric summation idea.
**Question**: does this argument generalize to higher dimensions? i... |
Maybe something is missing in the original statement and is: whose kernel is that in the right hand side?
First of all, let's recall that $f$ and $g$ are maps from $X$ to $Y$.
Then that kernel on the right is obviously
$$
\ker (f_{*}, g_{*}) \subset \mathrm{Hom}(Z,X) \ ,
$$
where
$$
f_{*}, g_{*} : \m... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.
The book doesn't elaborate on this, however. I don't underst... |
Why can't the Polynomial Ring be a Field? |
I will leave the explaining of large cardinals to someone more knowleadgeable and explain one place where they are useful in tidying up things: category theory. In category theory, you are constantly facede with proper classes (the category of all sets, of all groups, etc.). To make things worse, you want to form funct... |
Maybe something is missing in the original statement and is: whose kernel is that on the right hand side?
First of all, let's recall that $f$ and $g$ are maps from $X$ to $Y$.
Then that kernel on the right is obviously
$$
\ker (f_{*}, g_{*}) \subset \mathrm{Hom}(Z,X) \ ,
$$
where
$$
f_{*}, g_{*} : \m... |
Maybe something is missing in the original statement and is: whose kernel is that on the right hand side?
First of all, let's recall that $f$ and $g$ are maps from $X$ to $Y$.
Then that kernel on the right is obviously
$$
\ker (f_{*}, g_{*}) \subset \mathrm{Hom}(Z,X) \ ,
$$
where
$$
f_{*}, g_{*} : \m... |
For a while I've been looking for some kind of software that will let me create and cross-reference definitions. The way I imagine this working would be, for instance: I create a file named "Hopf algebra", I type in "A Hopf algebra over a field k is a k-bialgebra such that..." and follow with the definition of a Hopf a... |
http://groupprops.subwiki.org/wiki/Main_Page is something like this for group theory. There is also the <a href="http://ncatlab.org/nlab/show/HomePage">nLab</a>. Both of them run on wiki software; do you have something else in mind? |
Although i am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was **G.H.Hardy** so much keen to have an elementary proof of the **Prime Number Theorem**. He also stated that producing such a proof, will change the complexion of Mathematics, but n... |
Elementary proof of the Prime Number Theorem - Need? |
Although i am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was **G.H.Hardy** so much keen to have an elementary proof of the **Prime Number Theorem**. He also stated that producing such a proof, will change the complexion of Mathematics, but not... |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) 3 exactly divides |G|.
(ii) The number of 3-Sylows of G is q_G = p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): G surjects onto Z/3Z (and so is a semi-dir... |
For a while I've been looking for some kind of software that will let me create and cross-reference definitions. The way I imagine this working would be, for instance: I create a file named "Hopf algebra", I type in "A Hopf algebra over a field k is a k-bialgebra such that..." and follow with the definition of a Hopf a... |
*This problem is taken from Problem 2.4.31 (page 84) from [Problems in Mathematical Analysis: Integration](http://books.google.com/books?id=OBK7D1qcY0oC&pg=PA84&lpg=PA84&dq=%22Give+an+example+of+a+bounded+function%22+%22which+is+not+Riemann+Integrable,+but+is+a+derivative+of+some+function%22&source=bl&ots=JgTTEAPY3o&si... |
I am working a bit on a collection of Linear Algebra examples,
as well as some examples on induction. This is what is taught freshman year at our university.
I intend to release this to the public, either by selling printed copies or releasing it online.
Since I do not have experience using such material mysel... |
Please see the problem **5** of the given link: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=59&year=2005&sid=722231ab4ec5ce280584eb8f24f07656
It asks us to prove that $$\sum\limits_{n=1}^{\infty} \frac{1}{n^n} = \int\limits_{0}^{1} x^{x} \ dx $$
Natural question which one gets by seeing this... |
In a [talk by V.I. Arnold][1], this is said:
> When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type in t... |
By little Fermat: $\; 6,10|120 \Rightarrow 3^{120} \equiv 1 \pmod{7, 11} \Rightarrow 3^{123} \equiv 3^3 \pmod{77}$
See also these **Fermat-Euler-Carmichael** generalizations of little Fermat-Euler from my post [here][1].
**THEOREM 1**. For natural numbers $ a,e,n $ with $ e,n>1 $
$\;\; n|a^e-a$ for all $a \i... |
Friend told me this one, I'm completely stuck but also completely fascinated:
There are 100 boxes with apples, oranges and bananas (mixed). How to pick 51 boxes and to get at least half of all apples, at least half of all oranges and at least half of all bananas? |
This called Dicman's lemma and is 14.5.7 on page 425 (1st ed) or <a href="http://books.google.com/books?id=lqyCjUFY6WAC&lpg=PA442&pg=PA442">page 442 (2nd ed)</a> of Robinson's Course in the Theory of Groups. |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) 3 exactly divides |G|.
(ii) The number of 3-Sylows of G is q_G = p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): G surjects onto Z/3Z (and so is a semi-dir... |
*Here is my attempt (community wiki style), in case people want to try and repair it:*
If H is a group with 92 elements of order 3, then let G be the subgroup generated by those elements. G has finite order by Dicman's lemma, and so G has a Sylow 3-subgroup P.
P must be cyclic, lest Hall's result imply 46≡4 mod ... |
I recently learned that on Riemannian manifolds, one can define the _sectional curvature_ (<http://en.wikipedia.org/wiki/Sectional_curvature>) of a (2-dimensional) plane section. I was wondering if a similar concept exists for higher dimensional "space sections."
Here is what got me thinking about this: For 2-dimen... |
Are there higher-dimensional analogues of sectional curvature? |
Friend told me this one, I'm completely stuck but also completely fascinated:
There are 100 boxes with apples, oranges and bananas (mixed). How to pick 51 boxes and to get at least half of all apples, at least half of all oranges and at least half of all bananas?
Edit: You can take a look in the boxes. |
Meta: Since there seems be a dearth of questions here, I think we should all try and contribute some questions...
Part a) of the following problem appeared in one of the Putnam Exams (sorry, don't know which year exactly).
If you want to solve Part a) don't read Part b).
You have a painting device, which given... |
Meta: Since there seems be a dearth of questions here (I am used to stackoverflow!), I think we should all try and contribute some questions...
Part a) of the following problem appeared in one of the Putnam Exams (sorry, don't know which year exactly).
If you want to solve Part a) don't read Part b).
You have ... |
Maybe something is missing in the original statement and is: whose kernel is that on the right hand side?
First of all, let's recall that $f$ and $g$ are maps from $X$ to $Y$.
Then that kernel on the right is obviously
$$
\ker (f_{*}, g_{*}) \subset \mathrm{Hom}(Z,X) \ ,
$$
where
$$
f_{*}, g_{*} : \m... |
Friend told me this one, I'm completely stuck but also completely fascinated:
There are 100 boxes with apples, oranges and bananas (mixed). <strike>How to</strike> Prove that you can pick 51 boxes and to get at least half of all apples, at least half of all oranges and at least half of all bananas?
Edit: You can ... |
i'm reading <<3D math primer for graphics and game development>> By Fletcher Dunn and Ian Parberry, on page [170][1], the logarithm of quaternion is defined as
log q = log([cosa nsina]) ≡ [0 an]
i don't see how does log([cosa nsina]) equals to [0 an], can anyone help me out?
thanks.
[1]: http://book... |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) The 3-Sylow subgroup of G is cyclic.
(ii) The number of elements of order 3 in G is 2*p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): 3 exactly divides |G|... |
I found a similar question, answered by Ramanujan. The paper is entitled "Some Integrals in Ramaujan's lost notebook" by Bruce C Brendt. Here is the link: http://www.jstor.org/stable/4097261?cookieSet=1 |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) The 3-Sylow subgroup of G is cyclic.
(ii) The number of elements of order 3 in G is 2*p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): 3 exactly divides |G|... |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) The 3-Sylow subgroup of G is cyclic.
(ii) The number of elements of order 3 in G is 2*p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): 3 exactly divides |G|... |
You're probably thinking, "Why?" Please let me explain...
It is (*very*) well-known that
$$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. $$
For some bizarre reason, I decided to try to solve $ bx + c = 0 $ using this formula ... |
Please see the problem **5** of the given link: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=59&year=2005&sid=722231ab4ec5ce280584eb8f24f07656
It asks us to prove that $$\sum\limits_{n=1}^{\infty} \frac{1}{n^n} = \int\limits_{0}^{1} x^{x} \ dx $$
Natural question which one gets by seeing this t... |
Good example of a polynomial which produces a finite number of primes is: $$x^{2}+x+41$$ which produces primes for every integer $ 0 \leq x \leq 39$.
In a paper H.Stark proves the following result: $X_{n}$(is the ring of "algebraic integers" in $Q(\sqrt{-n})$ is a prinicpal ideal domain for positive $n$ if and only ... |
This is apparently a (hard) problem from the Russian Math Olympiad which no one in the exam solved: See here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=32171
And a solution for that is here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1367869#p1367869
A hint that was given (by Fe... |
i'm reading *3D math primer for graphics and game development* By Fletcher Dunn and Ian Parberry, on page [170][1], the logarithm of quaternion is defined as
\begin{align}
\log \mathbf q &= \log \left( \begin{bmatrix} \cos \alpha & \mathbf n \sin \alpha \end{bmatrix} \right) \\\\
&\equiv \begin{bmatrix} 0 & \alpha... |
This is apparently a (hard) problem from the Russian Math Olympiad which no one in the exam solved.
See here for a list of questions in that exam: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=32171
A solution for this problem is here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1367... |
I just started reading some Schapira notes on [Algebra and Topology][1]. Statement 1.7 is the following:
Hom (Z, Ker(f, g)) ≃ Ker(Hom (Z, X) ⇉ Hom (Z, Y ))
where:
Ker(f, g) = {x ∈ X; f(x) = g(x)}
I tried verifying this myself by drawing a diagram, but failed. Can anyone explain why this is true?
[1]: ... |
Please refer this article. It talks about $L_{p}$ spaces for $0 < p \leq 1$. Link:
http://www.jstor.org/stable/2041603?seq=2 |
Square of an irrational number can be a rational number e.g. $\sqrt{2}$ is irrational but its square is 2 which is rational.
But is there a irrational number square root of which is a rational number?
Is it safe to assume, in general, that negative powers of irrational will always give irrational numbers?
|
Square of an irrational number can be a rational number e.g. $\sqrt{2}$ is irrational but its square is 2 which is rational.
But is there a irrational number square root of which is a rational number?
Is it safe to assume, in general, that $n^{th}$-root of irrational will always give irrational numbers?
|
Please refer this article. It talks about $L_{p}$ spaces for $0 < p \leq 1$. Link:
http://www.jstor.org/stable/2041603?seq=2
Look at the step functions, the ones that take rational values and whose steps have rational endpoints there should be only countably many of those. And then you can perhaps apply the same ... |
In a related [question][1] we discussed raising numbers to powers.
I am interested if anybody knows any results for raising numbers to irrational powers.
For instance, we can easily show that there exists an irrational number raised to an irrational power such that the result is a rational number. Observe ${\... |
How to calculate correlation between poll statistics and one's answers? |
In a related [question][1] we discussed raising numbers to powers.
I am interested if anybody knows any results for raising numbers to irrational powers.
For instance, we can easily show that there exists an irrational number raised to an irrational power such that the result is a rational number. Observe ${\... |
I've got a fixed solution space defined by a minimum and maximum float which has no divisible slice.
You then have 0..N [Normal distribution][1] probability sets which I need to combine.
In the end I neeed
Method to define probability set with a numeric range (not sliceable) and 0..N Gaussian functions
A func... |
Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions:
1. If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$.
2. $f$ strictly increasing.
3. $f(2) = 2$.
Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of co... |
Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions:
1. If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$.
2. $f$ strictly increasing.
3. $f(2) = 2$.
Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of co... |
Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions:
1. If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$.
2. $f$ strictly increasing.
3. $f(2) = 2$.
Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of co... |
Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions:
1. If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$.
2. $f$ strictly increasing.
3. $f(2) = 2$.
Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of co... |
I have a Image processing application in which I have a Matrix equation as below:
A*R=I
where,
A = 3x3 matrix (Constants)
R = 3x1 matrix (Column Vector) Lets call this as actual output.
I = 3x1 matrix . Lets call I as Ideal output
\* denotes matrix multiplication.
I know the values of matrix I, and... |
Need to compute values of all the entries of a 3x3 matrix? |
In fact, it's a theorem of Erdös that every multiplicative function $f:\mathbb{N}\to\mathbb{R}$ is $f(n)=n^c$ for some constant $c$. This implies that those three conditions are already enough to prove that $f(n)=n$.
You can read a proof by Pierre Pbornsztein [here][1].
[1]: http://www.artofproblemsolving.com... |
In fact, it's a theorem of Erdös that every non-decreasing multiplicative function $f:\mathbb{N}\to\mathbb{R}$ is $f(n)=n^c$ for some constant $c$. This implies that those three conditions are already enough to prove that $f(n)=n$.
You can read a proof by Pierre Pbornsztein [here][1].
[1]: http://www.artofpro... |
Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)?
For example, I think the comparison of modular arithmetic to arithmetic with times, on a clock face, is fantastic, b... |
I've taken an undergraduate course in ring and group theroy, but haven't studied number theory formally. I've noticed many important results in number theory have been generalized in group/ring theory (e.g. Lagrange Thm generalizing Fermats Little Thereom).
My question is are there any results in elementary number... |
When showing that two functors $F:A\rightarrow B$ and $G:B\rightarrow A$ are adjoint, one defines a natural bijection $Mor(X,G(Y)) \rightarrow Mor(F(X),Y)$. What if one do not require the bijection to be natural, what issues would arise? |
I have three questions.
1:
Does the groupification of a semigroup always exist? I believe this should be yes because for every $x$ in the semigroup one could just define an element $x'$ that should work as its inverse. But what would then happen to the product $x'y$ for $x,y$ elements of the semigroup? It feels l... |
If given the cost to play, and the average win. Can I calculate the edge? (probability of winning) |
How do I calculate gambling edge using average win? |
I've recently been interested in Monte Carlo simulations and their uses, unfortunately most of the examples I find are difficult to understand for a beginner. What are some simple examples of using Monte Carlo simulations used to solve math problems? Bonus points for showing a simple example and using it to understand ... |
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