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... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is a numerical beast, but has anybody found any real-world uses for them? For example, quaterions have a nice connection t... |
This problem is taken from International Mathematics Competition for University Students 2009 (IMC 2009), Day 2, Problem 5.
Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S \subset \mathbb{M}$, denote $\delta(S)$ the dimesion of the vector space generated by all columns of a... |
Given a finite group $G$ and its subgroups $H,K$ such that $$G \times H \cong G \times K$$ does it imply that $H=K$.
Clearly, one can see that this doesn't work out for all subgroups. Is there any condition by which this can remain true.
|
The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology.
Now, I've read that the Hilbert Cube is homogeneous. That is, given two points $p, q\in H$, there is a homeomorphism $f:H\rightarrow H$ with $f(p)=q$.
What's confusing to me i... |
There are $N$ bugs in a plane. All bugs are moving at the same constant (nonzero) speed, but no two bugs are moving in the same direction (velocity vectors are of the same speed, but no two are parallel).
Prove that at some point in time $N$ bugs will form convex polygon.
Edit: Can you loosen up any of the conditions ... |
There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends.
What is the expected number of loops at the end of the process?
|
I was wondering if anyone had any experience with an undergraduate course that emphasized the building of mathematical theories or if they'd ever heard of this being done? How did the class work (did the professor list axioms and definitions each week and see what the students were able to prove, did you discuss a spec... |
Some points on (integer, rectangular) grid in a plane are colored white, some black and some are not colored.
In each step, one vertical or horizontal line can be selected, and all colored points on that line would toggle their color.
Prove there is a finite sequence of steps after which on every vertical or horizontal... |
Suppose that $c_k$ is an decreasing sequence of non-negative real numbers, such that $c_0=1$
and $c_{k}\leq \frac{1}{2}(c_{k-1}+c_{k+1})$.
Is it true that the generated function of $c_k$ admit an integral representation as below
$$
\sum_{k=0}^{\infty}c_kz^k=\int_{\partial\mathbb D} \frac{1}{1-\zeta z}d\mu(\zeta),
$$
w... |
If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant.
My question is suppose $f$ is continuous and it satisfies $f(x)=f(2x+1)$, then can the domain of $f$ be restricted so that $f$ doesn't remain a constant. If yes, then give an exa... |
Can one think of an example of a continuous $2\pi$ periodic function whose Fourier series fails to converge on $\mathbb{R}$.
I referred this in the wikipedia page but no avail: It might be interesting to note that Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a ... |
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... |
Someone asked me for a formula for the sum of the harmonic progression.
So I did some calculations and gave him an approximate formula:
$$\int_1^n\frac{dx}{x} = \frac{y_1 + y_2}{2} + \frac{y_2 + y_3}{2} + \cdots +\frac{y_{n-1} + y_n}{2}$$
where $y_i$ is $i$th term of the HP
$$\ln(n) = \frac{y_1}{2} + y_2 +y_3 + \cdots ... |
I am trying to find the answer to the following question:
If I am playing poker and am dealt two cards neither of which are spades, what are the chances of two or more of the flop cards (the flop is the first three community cards) being spades.
I answered this question in the following way:
As I have no spades, there ... |
There are $N>1$ points in a plane. Consider the set of all distances between pairs of the points. Let $n$ be the number of elements of this set.
We know that $$n \leq {N \choose 2}.$$
What is the lower bound estimate on $n$?
For the sake of clearity, if points are $A_1, A_2, \dots, A_N$, and $S = \{d(A_i, A_j)| 1\leq i... |
Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told me about a few, but along with some well-known problems he told me the f... |
We know that $(\mathbb{Q},+,\times)$ is a subfield of $(\mathbb{R},+,\times)$. It is easy to see that the automorphism of $\mathbb{Q}$ is only the identity. For a quick proof, lets go through the main steps:
$f(1)=1 \Longrightarrow f(n)=n$ for all $n \in \mathbb{N}$.
$f(-1)=-1$ which says that $f(x)=x$ for all $x \in ... |
Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, Theorem about a circle, three chords and a midpoint on January $29$th, $2010.$
"Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and $M$ be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$, that pa... |
Given the following:
an $(n \times z)$ matrix $A = {(a_1,a_2, \ldots ,a_n)}^{T}$ where $z \geq n$ and every $a_i$ is a $z$-dimensional row vector.
$a_i = [a_{i1}\, a_{i2}\, \ldots\, a_{iz}]$ where $ \forall j\colon a_{ij} \geq 0$.
$\forall r \in \{1,2,\ldots ,n\}\colon \sum_{i=1}^{z}a_{ri} = 1$.
$\forall p,q\col... |
Is it guaranteed that there will be some $p$ such that $p\mid2^n-1$ but $p\nmid 2^m-1$ for any $m<n$?
In other words, does each $2^x-1$ introduce a new prime factor?
|
Let $H,K$ be proper subgroups of a group $G$ having a complete set $S$ of representatives of left cosets in common, that is,
$$
G = \bigsqcup_{s \in S} sH = \bigsqcup_{s \in S} sK
$$
It seems in general one cannot expect any serious relation on $H,K.$ But I am afraid of overlooking some general result here. Any informa... |
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
|
I was just reading a paper of Ramanujan entitled "On question 330 of Professor Sanjana" when I got confused regarding a proposition which I am unable to answer. The proposition is:
$ \displaystyle f(p) = \frac{\pi}{2^{p+1}} \frac{\Gamma(p+1)}{ \Bigl[ \Gamma(\frac{P+2}{2}) \Bigr]^{2}}$, then $$\log{f(p)}= \log(\frac{\p... |
Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$?
Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question known in general?
|
Sometimes when I am reading a mathematical text I will be unable to decipher the steps necessary to translate one mathematical statement into the next. Usually I have to ask for help, but I was wondering if there is a software package that would let me enter two mathematical statements and the it would find the steps n... |
Let us be given a $n\times n$ matrix containing only zeros and ones.Now, the goal is to remove some 'ones' from the matrix (i.e. replace them with zeros) so that in each row and each column there is at most one 'one'. Of course, it is preferable that we leave as much 'ones' as it is possible (maximum is obviously $n$).... |
Prove that a number in the sequence $2,3,4,...,n \ (n>2$, is relatively prime to all other numbers if and only if it is a prime that exceeds $\displaystyle\frac{n}{2}$. Does such a prime always exist?
|
Everyone knows that $\pi$ is an irrational number, and one can refer to this page for the proof that $\pi^{2}$ is also irrational.
What about the highers powers of $\pi$, meaning is $\pi^{n}$ irrational for all $n \in \mathbb{N}$ or does there exists a $m \in \mathbb{N}$ when $\pi^{m}$ is rational.
|
How is the power series expansion of an analytic function at a point constructed, without using Cauchy's theorem (or formula)?
|
This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak.
Give an example of a bounded function $f:[0,1] \to \mathbb{R}$ which is not Riemann Integrable, but is a derivative of some function $g$ on $[0,1]$.
|
What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the better. All the above can be proved w... |
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?
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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.
This prob... |
I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.
Let $\Sigma = \{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\}$ be the set of usable characters. Every string i... |
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
"The rank of a matrix A is the number
of non-zero rows in the reduced
row-echelon form of A".
The lecturer then explained that if the matrix $A$ has size $m
\times n$, ... |
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a convergent $|\alpha q - p| > |\alpha q_{N-1} - p_{N-1}|$. I'm wondering if there are any nice proofs for this result?
In my b... |
What might be some potential methods to encode a 100-point signal (curve) for input to a Artificial Neural Network?
Example: we have a large number of 100-pt 'curves' ranging from flat-line to approximately a half-sine wave with a wide range of max
amplitude. The curves have some noise and occasional anomalous
... |
I have recently been thnking some about algebraic groups and reading parts of Humphreys book on them, and I was wondering if there is a general process to showing they are simply connected. In particular I was wondering over other fields than $\mathbb{C}$ but if the answer only works in $\mathbb{C}$ I will settle for t... |
I need to find the minimum of a function $f(t) = \int_0^1\!g(t,x)\,dx$. What I did in mathematica is as follows:
f[t_] = NIntegrate[g[t,x],{x,-1,1}]
FindMinimum[f[t],{t,t0}]
However mathematica halts at the first try, because NIntegrate does not work with the symbolic t. Although Plot[f[t],{t,0,1}] works perferctly, F... |
Is it true that $\sin(n^k) ≠ (\sin n)^k$ for any positive integers $n$ and integers $k ≠ 1$?
What if $n > 0, k ≠ 1$ are rational?
|
How can you prove that
$$
\sum_{k=1}^{n-1}\tan^{2}\left(\frac{k \pi}{2n}\right)
=
\frac{\left(n-1\right)\left(2n - 1\right)}{3}
$$
for every integer $n\geq 1$ > ?.
PS: no, it's not a homework... :-)
|
I'm new at learning boolean algebra and understand the basic laws. But now I'm trying to proof:
$x'z+ xyz + xy'z = z$
Can someone help me with this please ?
Thanks in advance.
|
How does one set out to find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ which satisfy $f^{k}(x)=f(x^k)$ , where $k \in \mathbb{N}$?
Motivation: Is $\sin(n^k) ≠ (\sin n)^k$ in general?
|
I'm doing exercise on discrete mathematics and I'm stuck with question:
If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$.
I've no idea how to prove this, please help me by give ... |
This is a strengthening of Chandru's question:
Example of a function whose Fourier Series fails to converge at One point
Is there a nice and concrete example of a Fourier series that fails to converge on some "big" set of measure zero, for instance on the Cantor ternary set?
|
The number of elements of order 2 in a group is fairly restricted: 0, odd, or infinity. All such possibilities occur already in the trivial group and in dihedral groups.
The number of elements of order 3 in a group can be shown to be similarly restricted: 0, 2 mod 6, or infinity. However, something strange happens: n... |
When flipping a coin to make important decisions in life you can flip once to choose between 2 possible outcomes. (Heads I eat cake, Tails I eat chocolate!)
You can also flip twice to choose between 4 outcomes. (Heads-Heads, Tails-Tails, Heads-Tails, Tails-Heads)
Can you use a coin to choose evenly between three possib... |
I'm looking for the equation of a family of functions that roughly resembles the sketch below (with apologies for the crudeness of said sketch):
Properties I'm looking for:
$\lim_{x\to-\infty}f(x)=y_1$ (i.e. approaches asymptote y=y1)
$\lim_{x\to+\infty}f(x)=y_2$ (i.e. approaches asymptote y=y2)
$|f'(x)|$ is at a... |
How does one prove that ring of Hamilton Quaternions with coefficients coming from the field $\mathbb{Z}/p\mathbb{Z}$ is not a divison ring.
|
I heard that one can construct field based on $p^n$ elements where $p$ is prime. I tried with $p = n = 2$. It seemed easy as there are 2 groups which $\{0,1,2,3\}$ can form and 1 group formed by $\{1,2,3\}$. However each time I get into contradiction. Is there anything I missed or it is not possible to form $p^n$ eleme... |
How can I find the number of $k$-permutations of $n$ objects, where there are $x$ types of objects, and $r_1, r_2, r_3, \cdots , r_x$ give the number of each type of object?
I'm still looking for the solution to this more general problem out of interest.
Here is an example with $n = 20, k = 15, x = 4,$ $ r_1 = 4 \quad ... |
I am learning the axioms of Zermelo-Fraenkel (ZF) set theory.
One axiom schema basically says that given any set S and any formula phi(x), there is a set T consisting of all those elements x of S such that phi(x).
I find this axiom schema unsatisfying because it only guarantees that subsets of S definable by a formula... |
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.
|
How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of something like this.
|
I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?
I know of the ArtofProblemSolving.com. I am currently using the books "Cha... |
How do you calculate the following sum?
\begin{equation*}
\sum_{i=k}^n i
\end{equation*}
|
Familiar Exercise: Suppose $G$ is a finite group and $T$ is an automorphism of $G$ which sends more than three quarters of elements of $G$ onto their inverses, then prove that $G$ is abelian.
The group of Quarternions is an example of a group which sends exactly $\displaystyle \frac{3}{4}$ elements onto their inverses.... |
I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my physics sequence and engineering statistics. I learned a little bit of vector math in physics as well, but I'm not stron... |
Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation rules). Certainly, if the distinction between vectors and dual vectors is ignored, a rank 2 tensor $T$ seems to be simpl... |
The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from some people that there is indeed some connection between the two. But I was never able to figure it out on myself. For ins... |
I had seen this problem a long time back and wasn't able to solve it. For some reason I was reminded of it and thought it might be interesting to the visitors here.
Apparently, this problem is from a mathematics magazine of some university in the United States (sorry, no idea about either).
So the problem is:
Suppose $... |
A friend of mine is subediting a book whose author died in 1999. The author, at some point, uses the word "trillion" which is, unfortunately, an ambiguous word in the UK: when I was at school it used to mean $10^{18}$ but nowadays it means $10^{12}$. My friend is faced with the following paragraph:
According to Asimov... |
The limit is $$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$$
Experimentally, this limit appears to converge to ${1 \over e}$, but I can't figure out how to solve it.
|
Is there any overarching reason why, after excluding the infinite classes of finite simple groups
(cyclic, alternating, Lie-type),
what remains---the sporadic, exceptional finite simple groups, is in fact a finite list (just $26$)? In some sense, the prime numbers can be viewed as "sporadic," but there is an infinite ... |
How should I rotate a point $(x,y)$ to location $(a,b)$ on the co-ordinate by any angle?
|
I have a function that (more or less) is supposed to select a small number $m$ of random numbers from the range $[1,n]$ (for some $n \gg m$) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the full on chi squared test is not... |
Here's a question. Have there been any theoretical results showing that if $P \neq NP$, there must exist some problems in $NP$ which are not $NP$-complete and which are not in $P$ either? Just curious because I've never seen this question addressed.
|
I'd like to prove that $\forall x\left(x\not\in x\right)$ in the context of Morse-Kelley set theory.
Let's call $A=\left\{y:y\not\in y\right\}$. I can easily prove that $A\not\in A$. In fact, if you suppose that $A\in A$, it follows that $A\not\in A$ by definition of class comprehension. Since this is a contradiction, ... |
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ is the cantor set, then what is the measure of $C+C$?
|
The traditional method 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... |
I was quite impressed by a beautiful proof1 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.e. can someone prove the formula for n-th «k-dimensional simplex number» this way?
For ex... |
Is there a generally accepted formal notation for denoting the number of columns a matrix has (e.g to use in pseudocode/algorithm environment in LaTeX)? 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 descript... |
Is it true that for any $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} - y^{2}$, for $x,y \in \mathbb{N} \cup (0)$?
I was just working out a proof and this turns out to be true from $n=1$ to $n=20$. After that I didn't try, but I would like to see if a counterexample exists for a greater value of $n... |
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'Hopital o... |
Let $G$ be a group. A normal subgroup $N$ of $G$ is called ultracharacteristic 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 automorphisms; i... |
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 sho... |
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^2 - x ... |
A measurable rectangle in the cartesian product $X \times Y$ of two 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 of p... |
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 $f(x_{n})$ rationa... |
When writing software, there are often situations where I need a parameter to be a floating point number $x \in [0,1]$. 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, I hereby declar... |
This question concerns an implementation of the topmoumoute natural gradient (tonga) algorithm as described in page 5 in the paper Le Roux et al 2007 http://research.microsoft.com/pubs/64644/tonga.pdf.
I understand that the basic idea is to augment stochastic gradient ascent with the covariance of the stochastic gradi... |
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 Lin... |
Probably a proof (if any exist) that calls upon Knuth's up-arrow notation or Busy Beaver.
|
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?
|
I just started reading some Schapira notes on Algebra and Topology. Statement 1.7 is the following:
Hom $(Z, \ker(f, g)) ≃ \ker(\text{Hom} (Z, X) ⇉ \text{Hom} (Z, Y ))$
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?
|
I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.
|
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 understand... |
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 I.N. Herstein
Problem:
If $G$ is a group and $a \in G$ if of finite order and has only a finite number of conjugates in $G$, prove that these conjugates of $a$ generate a finite normal subgroup of $G$.
I would like to see a solution for this and also I would like to know whether the conjugate... |
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 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 nothing like tha... |
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 type of que... |
In a talk by V.I. Arnold, 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 the Goursa... |
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 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 take a look in the ... |
I have a symmetric matrix A. How do I compute a matrix B such that $B^tB=A$ where $B^t$ is the transpose of $B$. I cannot figure out if this is at all related to the square root of $A$.
I've gone through wikimedia links of square root of a matrix.
|
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-dimensional ... |
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 the co-ordinates of a points in the 2D plane, will colour all points on that plane black, which are at an irrational dista... |
I'm reading 3D math primer for graphics and game development by Fletcher Dunn and Ian Parberry. On page 170, 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 \mathbf n \... |
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 by introduc... |
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: $X_{n}$ (the ring of "algebraic integers" in $\mathbb Q(\sqrt{-n}))$ is a principal ideal domain for positive $n$ if and ... |
I would like to know if the Lebesgue spaces $L_p$ with $ 0 < p < 1 $ are separable or not.
I know that this is true for $1 \leq p < + \infty$, but I do not find any references for the
case $ 0 < p < 1 $.
Thank you
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