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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?
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In a related question 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 ${\sqrt 2 ^ {\... |
I have statistics of $100$ questions which can be answered either "yes" or "no":
1) $63.3 - 36.7$ ($63.3\%$ respondents answered "yes" and $36.7\%$ answered "no")
2) $30.1 - 69.9$
...
100) $88.0 - 12.0$
That $100$ answers would be our sample.
Then I ask the same $100$ questions to $101^{st}$ respondent and get new set ... |
This was asked on sci.math ages ago, and never got a satisfactory answer.
Given a number of sticks of integral length $ \ge n$ whose lengths
add to $n(n+1)/2$. Can these always be broken (by cuts) into sticks of
lengths $1,2,3, \ldots ,n$?
You are not allowed to glue sticks back together. Assume you have an accu... |
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 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 function which can ... |
I was going to ask this on SO but I think its more math than programming:
Given the sidelength, number of vertices and vertex angle in the polygon, how can I calculate the radius of its circumscribed circle.
The polygon may have any number of sides greater than or equal to 3.
The wikipedia entry only discusses circums... |
A long time ago I was curious about the closed-form solutions to the equation:
\begin{equation*}
\frac{d^{n}y}{dx^n} y = 1.
\end{equation*}
For $n$ an odd number, try $y = A x^k$. Then $y^{(n)} = A k(k-1)...(k-n)x^{k-n}$. This gives the formula
\begin{equation*}
A^2 k(k-1)...(k-n) x^{2k - n - 1} = 1
\end{equation*}
whi... |
Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions:
If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$.
$f$ strictly increasing.
$f(2) = 2$.
Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of course all the condit... |
I have a Image processing application in which I have a Matrix equation as below:
$A \cdot R=I$
where,
$A = 3 \times 3$ matrix (Constants)
$R = 3 \times 1$ matrix (Column Vector) Lets call this as actual output.
$I = 3 \times 1$ matrix . Lets call $I$ as Ideal output
I know the values of matrix $I$, and $R$. I have to ... |
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, beca... |
I've taken an undergraduate course in ring and group theory, but haven't studied number theory formally. I've noticed that many important results in number theory have been generalized in group/ring theory (e.g. Lagrange's Theorem generalizing Fermat's Little Theorem).
My question is, are there any results in elementar... |
When showing that two functors $F:A\rightarrow B$ and $G:B\rightarrow A$ are adjoint, one defines a natural bijection $\mathrm{Mor}(X,G(Y)) \rightarrow \mathrm{Mor}(F(X),Y)$. 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 like we... |
If given the cost to play, and the average win. Can I calculate the edge? (probability of winning)
|
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 ... |
Couple of days ago, i actually asked this question : Existence of a normal subgroup with $\lvert\operatorname{Aut}{(H)}\rvert>\vert\operatorname{Aut}{(G)}\rvert$.
I was thinking about the converse of this statement. Suppose $G$ is a finite group, with subgroups $H$ and satisfies $$|\operatorname{Aut}(G)|=|\operatorname... |
Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).
As a converse of the above, let $B$ be a Ba... |
I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship)
A couple of things I was thinking about:
1) Can create a very simple model where wind is 'one' force acting on the a uniform body.
2) Model wind as vectors. This is where I am a little confused on how to start.
3) Adding in ... |
Let $X$ denote the blowup of $\mathbb P^2$, $E$ the exceptional divisor, and $H$ the pullback of the hyperplane class. How can I compute $H^0(X,mE+nH)$, $H^1(X,mE+nH)$, and $H^2(X,mE+nH)$ for $m,n \in \mathbb Z$? If I'm working analytically, how can I think of these geometrically (e.g. "an element of $H^1(X,2E)$ is e... |
By stressing my manipulative powers and a bit of help from Mathematica I was able to show that the triple integral
$$\frac1{8\pi}\int_0^{2\pi}\int_0^\pi\int_0^\pi\,(2\cos\theta\cos\varphi-\sin\theta\sin\varphi\cos\psi)^{2k}\sin\theta\sin\varphi \,\mathrm{d}\theta\mathrm{d}\varphi\mathrm{d}\psi $$
where k is a positive ... |
Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in $A$ which maps $x$ to $y$"?. It seems like the obvious thing to say is "$x$ and $y$ are symmetric in $A$", but I've never come across this phrasin... |
Let $(0,\infty)$ be equipped with the Lebesgue measure, and let $1 < p < \infty$. For each $ f \in L_{p}(\lambda),$ let $$T(f)(x) = x^{-1} \int f \chi_{(0,x)} \ d\lambda \quad \text{for} \ x >0.$$
Prove that $T$ is a bounded linear operator from $L_{p}(0,\infty)$ into itself.
|
A Liouville number is a number which can be approximated very closely be a sequence of rational numbers (here is the rigorous definition I am working off of: http://en.wikipedia.org/wiki/Liouville_number).
I'm looking for an example of a Liouville number which cannot be approximated by a sequence of rational numbers wi... |
Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?
|
i) Does there exists a nonempty subset of integers $S$, such that for all $a \in S$, there exists $b,c \in S$ such that $a = b + c$, where $a, b, c$ are distinct integers.
Edited to add:
ii) Can an $S$ with these properties be finite, and also have the property that $a \in S$ implies $-a \notin S$
This question is ins... |
In the towers of Hanoi game, how do we know that we have the optimal algorithm for solving it? I thought about this and it seemed like any deviation from the standard strategies would be putting you back a step, but I had no idea how to demonstrate this rigorously.
All I know is that the proof involves the Lucas corres... |
$2$ is the first prime number.
$3$ is the second.
If I give a prime number such as $1151024046313875220631$, is there any software/website which can give the position of the prime number?
I know there are resources to find $N$th prime. But I am having a hard time finding the reverse.
|
I was wondering if the following is true:
In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?
In a topo... |
In the article about metric space on wikipedia:
If $X$ is a complete subset of the metric space $M$, then $X$ is closed in $M$. Indeed, a space is complete if it is closed in any containing metric space.
Do the second sentences say that:
In a metric space, a subset is complete if and only if the subset is closed in... |
Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff
The dimension of $\ker(T)$ is finite,
The codimension of the image $\mathrm{im}(T)$ is finite,
The image $\mathrm{im}(T)$ is closed in $Y$.
Question: Is the third condition redundant?
Some lecture notes I'm working through cl... |
I need to create a simple formula for determining the popularity of an item based on votes and age.
Here is my current formula, which needs some work:
30 / (days between post date and now) * (vote count) = weighted vote
Whenever a vost is cast for an item it checks if its weighted vote is > 300. If an item has a weight... |
Let $G$ be an infinite simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ If so, we have to e... |
I have several points on a graph. The graph is of a continuous curve that flows through points $(0, \infty)$, $(11, 5000)$, and $(3, 200000)$. Can I find the equation for that line?
Let's try this:
I'm looking to find a function that given an input $x$ becomes closer to infinity as $x \to 0$, and the closer to $0$ the ... |
I saw a statement on http://en.wikipedia.org/wiki/Metric_space#Continuous_maps:
f is continuous if and only if it is continuous on every compact subset of M1.
It is stated for the case that f is a mapping between two metric spaces.
I was wondering if the statement is true for mappings between general topological spac... |
I have been fascinated by the Collatz problem since I first heard about it in high school.
Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process indefinitely. The conjecture is that no matter what number you sta... |
Suppose, we have $f_1(n) = o\left(f_2(n)\right)$. (small $o$)
Now, we have to compare $f_1\left(f_2(n)\right)$ and $f_2\left(f_1(n)\right)$ asymptotically..
What can we say about it? Which one will be greater & why?
|
I have an examples book with a limit exercise that I can't understand.
The limit in question is:
$$f(x,y)=\frac{x}{x+y}$$
with $x\ne-y$;
$$\lim_{(x,y)\to(0,0)} f(x,y)$$
And then to solve it, it goes:
$$\lim_{(x,y)\to(0,0)} f(x,y)
= \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$
Can you help me under... |
The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.
The shape of this set is well known, why is it that if you zoom into parts of the filaments you will find slightly deformed copies of the original sh... |
This was asked to me by a friend. I tried this problem for a long time, but i couldn't solve it. Seems interesting though.
Prove that it is impossible to construct two different seven digit numbers, one of which is divisible by the other, out of the digits 1,2,3,4,5,6,7 (All seven digits must be in each number)
|
Consider a cubic polynomial, $p(x)=k(x-a)(x-b)(x-c)$ where $k$ is some constant and $a,b,c$ its $3$ roots (not necessarily distinct, not necessarily real). It is very simple to show that if you average two roots of a cubic polynomial and compute the tangent line at their average that it will intersect the cubic polynom... |
I have a non-mathematician friend who is interested in re-learning algebra. I am more than happy to help, but I am in no position to judge what is a good introductory text --- only to identify when a text is a very bad one.
My friend is interested in starting from "basics" --- he's comfortable with order-of-operations,... |
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
If you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see that $2$ of the $3$ pairs that have B (for... |
Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here.
Say you have Ax = B. You have B, and A is 30000x30000 elements. Solving: You need to find x given B.
Solving a matrix like this in C code actually introduces an interesting ... |
Problem 63 of the 2001 St. Petersburg Mathematical Olympiad, Second Round, 11th grade:
Are there three different numbers $x, y, z$ in $[0,\pi/2]$ such that the numbers $\sin x$, $\sin y$, $\sin z$, $\cos x$, $\cos y$, $\cos z$ can be divided into three pairs with equal sums?
|
Possible Duplicate:
Behaviour of Polynomials in a PID!
Prove: if $p$ is a prime, and if $n^2+n+p$ is prime for $0\leq n \leq \sqrt{p/3}$, then it is also prime for $0 \leq n \leq p-2$.
This appeared on reddit recently, but no proof was posted. With $p=41$, it is Euler's famous prime-generating polynomial.
|
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by prim... |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that you c... |
Standard deck has $52$ cards, $26$ Red and $26$ Black. A run is a maximum contiguous block of cards, which has the same color.
Eg.
$(R,B,R,B,...,R,B)$ has $52$ runs.
$(R,R,R,...,R,B,B,B,...,B)$ has $2$ runs.
What is the expected number of runs in a shuffled deck of cards?
|
What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up.
Thanks.
|
Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a correct?
$\cos(t\theta_{kl})\approx\frac{1}{2}(\cos(\frac{2\pi kt}{n})+\cos(\frac{2\pi lt}{n}))$
Update: I'm actually interest... |
When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double torus, etc.), together with their associated inverses, was not a group with respect to the operations of connection and... |
I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
|
In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used:
$$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$$
(In the original denoted $\text{ord}$ $_{p}(\cdot)$ instead of $\nu_p(\cdot)$).
where $\nu... |
I'm starting a chapter on Functions and they had the steps shown to reach the p-q equation.
$$ x_{1,2} = -\frac{p}{2} \pm\sqrt{\left(\frac{p}{2}\right)^2 - q}$$
So I wanted to do the same with the Quadratic Equation. I'm using the base linear equation
$$ax+by+c = 0.$$
The solution I have so far is as follows:
$$x^2 + ... |
In $1D$ case, $\mathbb{R}^1$ as a typical case of $1D$ Euclidean space is totally ordered. I was wondering if any $1D$ Euclidean space $E^1$ is ordered as well? What is it
For higher dimensional case, are both $\mathbb{R}^n$ and $E^n$ ordered? What are the orders?
Thanks and regards!
Update:
I meant for an order for a... |
This is a final exam question in my algorithms class:
$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.
I don't know if the problem had a typo or not, because $O(n^3)$ seems more reasonable. The best I can... |
In a densely and totally ordered set, induce a order topology from the order.
Are the dense in terms of the order and the dense in terms of the order topology equivalent?
Thanks and regards!
|
What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic solutions techniques as well as some basic theory.
|
Suppose $p$ and $q$ are discrete distributions over $k$ outcomes, $q$ is given, for which $p$ does the following hold?
\begin{equation*}
p_1 \log q_1 + \ldots + p_k \log q_k \ge q_1 \log q_1 + \ldots + q_k \log q_k\end{equation*}
|
Given a line segment, denoted by it's $2$ endpoints $(X_1, Y_1)$ and $(X_2, Y_2)$, and a circle, denoted by it's center point $(X_c, Y_c)$ and a radius $R$, how can I tell if the line segment is a tangent of or runs through this circle? I don't need to be able to discern between tangent or running through a circle, I j... |
I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper understanding of the algorithms I'll be writing. Does anyone have recommendations for papers, web sites, etc. that present goo... |
Suppose I have a matrix $S$ having a one-dimensional nullspace $\{ e \}$ such that $S + ee^\top$ is a positive definite symmetric matrix.
Now let $b \in Range(S)$ and suppose I solve the equation $(S + ee^\top)x = b$ is there anyway I can derive the solution $x'$ of the equation $Sx' = b$? I was trying a Sherman Morr... |
I am crossposting this question from Stack Overflow since it is more math then programming related.
In Reference to my question about projecting a planar Polygon to a plane, i came to another problem.
I want to project a base contour which is aligned to the x-y-plane along a projection vector from from a point P1 to a ... |
Could someone verify this proof?
Given the following definition of divisibility $(\forall{a,b \in \mathbb{N}})[a \mid b \equiv (\exists{c \in \mathbb{N}})(ac=b)]$.
Statement: $(\forall{a,b,c \in \mathbb{N}})[(a\mid b) \rightarrow (ac\mid bc)] $ .
Proof:
Let $a,b \in \mathbb{N}$ be arbitrary. Assuming that $a\mid b$, w... |
I'm am trying to define a data structure to represent road networks. The immediately obvious structure is that of a graph - a set of nodes and edges that connect pairs of nodes. The nodes would represent things like intersections, and the edges would represent lanes.
However, the basic concept of a graph is insuffi... |
A hyperbolic geometry is a non-Euclidian geometry with constant negative curvature. It has the property that given a line and a point, many lines can be drawn containing the point that never meet the given line. The picture below (Circle Limit by M.C.Escher) is a conformal map of hyperbolic geometry to Euclidian plane.... |
If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ is the final topology with respect to the canonical homomorphisms $\psi_i:G_i\rightarrow G$). It is immediate from the ... |
The process below yields a distribution of "response times" (RT), and I'd like to know if there is an analytic solution to obtain the density function of this distribution.
An RT is recorded at the earliest time ($t_1$) at which a function, $f(t)$, reaches a threshold.
The threshold varies across iterations of the proc... |
How can I use the chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ to differentiate an expression like $\frac{d}{dx}(y^3)$ ? The solution in the book is
\begin{equation*}
\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}.
\end{equation*}
I'm not sure how they got it though.
|
Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$,
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example,
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$
becaus... |
Possible Duplicate:
Circular permutations with indistinguishable objects
We can compute the total number of k-ary necklaces/bracelets of length n using the Euler totient function:
http://en.wikipedia.org/wiki/Necklace_%28combinatorics%29
Apparently there is a formula for the number of k-ary necklaces/bracelets when ... |
This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.
The problem reads as follows:
If 3p2 = q2, where $p,q \in \mathbb{Z... |
I solved the following problem from by book, but the answer of this problem at the end of book is $x \leq 3$. Please tell me how I can get this answer.
|
Recently I stumbled upon someone who said he wanted to understand why
$\arctan x = \int\dfrac{dx}{1+x^2}$
At first I was confused. This is an easy result in any integral calculus course. But then he explained that although he understood the proof, he wanted to understand it "intuitively". He wanted to see why it was in... |
Let $f(x)$ be a continuous function from $\mathbb{R}\rightarrow\mathbb{R}$.
Let's denote $k$-times repeated application of the function, $f(f(f(...f(x)...)))$ as $f^{(k)}(x)$.
Let's call any $x$ a periodic point with period $n$ if $f^{(n)}(x)=x$.
Is it true that if a point with period 3 exists, then points with all pos... |
I came across this 'paradox' -
$$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i \cdot 2\pi i}=e^{-4\pi^2}$$
I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why doesn't it work with complex numbers even though it is valid in real case? Is it related to the fact that logarithm of c... |
Regarding this thread,
it is not possible to combine complex powers in the usual way:
$$ (x^y)^z = x^{yz} $$
There was mention of multi-valued functions, is there some theory that makes this all work out correctly?
|
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation.
Here $\det$ denotes the determinant.
Why can the antisymmetrizing bar be inserted in the middle because "there is already antisymmetry in the index lines"?... |
There are at least 2 places in Wikipedia saying that $X_n$ converges to $X$ in mean in and only if $X_n$ converges to $X$ in probability and $X_n$ is uniformly integrable. See the following link for example:
http://en.wikipedia.org/wiki/Convergence_of_random_variables#Properties_4
However, I found a reference in Billin... |
I am looking for an element $(a_n)_{n\in\mathbb Z}\in\ell^1(\mathbb Z)$ with the property
$$
\lVert(a_n)\rVert_{\ell^1(\mathbb Z)} > \lVert\sum_{n\in\mathbb Z}a_n z^n\rVert_\infty,
$$
where the norm on the right hand side denotes the sup-norm on $\mathcal C(\mathbb T)$ ($\mathbb T$ is the 1-Torus).
Motivation: I want t... |
I can understand how the Greek alphabet came to be prominent in mathematics as the Greeks had a huge influence in the math of today. Certain letters came to have certain implications about their meaning (i.e. $\theta$ is almost always an angle, never a function).
But why did $x$ and $y$ come to prominence? They seem li... |
In a proposal, assume $p$ is the condition and $q$ the conclusion, so the proposal is $p \to q$.
Let $\neg$ be the negation on either condition or conclusion.
what is the name for the operation from $p\to q$ to $\neg p\to\neg q$?
And what is the name for the operation from $p \to q$ to $p\to\neg q$? And from $p\to q$ t... |
Exercise 3B.4 on page 84 of Isaacs's Finite Group Theory asks one to prove a precursor of the Sylow theorems for $π$-subgroups, and I only know how to do it using the Sylow theorems, so I have missed the point. I actually needed a similar (false) statement in my research, so I thought it would be wise to get the point... |
Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.
A differential $n$-form (or rather its coefficient function which is obtained by using the canonical one-chart atlas on $\mathbb R^n$) then transforms essentially... |
I am reading p.22 of Dan Bump's Automorphihic forms and representations. A cusp of the congruence group acting on the upper half plane is defined to be an orbit of the action of the congruence subgroup on $\mathbb{Q} \mathbb{P}^1$. It also says that intuitively, the cusps are the places where a fundamental domain of th... |
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can separate a ball $x^2+y^2+z^2 \le 1$ into finitely many disjoint... |
Let $\omega:\mathbb{Z}\to (0,\infty)$ and let $1\leq p<\infty$. Consider the space $\ell^p(\mathbb{Z},\omega)$ of complex valued sequences $f=(a_n)_{n \in \mathbb{Z}}$ such that
$$\|f\|=\|f\|_{\ell^p(\mathbb{Z},\omega)}:=\left(\sum_{n\in\mathbb{Z}}|a_n|^p\omega(n)^p\right)^{1/p}<\infty.$$
Next, given two complex sequ... |
$0$ is finite. If $n$ is finite, then $n+1$ is finite. Hence, by induction, all numbers are finite. What is the catch?
|
Each of n persons draws a card from a shuffled deck of $k$ cards numbered from $1$ to $k$. There are at least as many cards as persons. The winner is the person who is holding the largest card. If everyone is honest, how can they mutually determine the identity of the winner, without any person learning additional i... |
I have this wooden puzzle composed of 25 y-shaped pentominoes, where the objective is to assemble them into a 5x5x5 cube. After spending quite a few hours unsuccessfully trying to solve the puzzle, I finally gave up and wrote a program to try all possible combinations using backtracking. Analyzing the results revealed ... |
Let the class number $h(d)$ denote the number of distinct binary quadratic forms with discriminant $d < 0$.
Is there a better algorithm for $h$ than brute force?
To be precise, by brute force I meant to generate enough forms to completely cover the space and then reducing them down to see how many equivalence classes t... |
I'm working on a game which relies on "flinging" an object. That is, click and hold on the object, and then drag and release it, and it continues on the path you were dragging it. Of course, the most well-known example of flinging is with iPhone and Android devices, where you can quickly scroll down a list by quickly s... |
The elementary real-valued functions are not closed under integration. (Elementary function has a precise definition -- see Risch algorithm in Wikipedia). This means there are elementary functions whose integrals are not elementary. So we can construct a larger class of functions by adjoining all the integrals of e... |
Hi this is my first time posting on here... so please bear with me :P
I was just wondering how I can solve something like this:
$$25x ≡ 3 \pmod{109}.$$
If someone can give a break down on how to do it would be appreciated (I'm a slow learner...)!
Here is proof that I've attempted:
Using definition of modulus we can re... |
In my course of linear algebra I studied that if $V$ is a finite dimensional vector space on the complex field, then every endomorphism of $V$ has an eigenvector. The proof is simple: taken a polynomial $f$ that is null on the endomorphism $A$ (it exists because of the finite dimension of $V$), exploiting the algebraic... |
As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself.
thanks.
|
When I was first learning about the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty eas... |
I am interested in the Gelfand transformation
$$
\Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n.
$$
This is an injective homomorphism of Banach algebras. It is neither isometric nor surjective. However, its image---the Wiener algebra $W$ consisting of all continuous functi... |
I keep coming across calculations like this,
Consider a metric on an $n+2$ dimensional manifold given as,
$ds^2 = 2dudr + 2L(u,r)du^2 -r^2d\Omega_n^2$
Then apparently once can write down the Ricci and Einstein and other tensors as a function of n.
Like for the above the Einstein tensor apparently has the following non... |
I'm not very fluent in mathematical proofs. High School has, sadly, not taught me any kind of proof-theory. That's why I would like your help with my proof of
$$m \equiv S_m \pmod 3.$$
where $S_m$ is the digit sum of $m$ and $\wedge$ is a logical-AND.
I'm going to use the very little that I know: Mathematical induct... |
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