instruction stringlengths 22 27.2k |
|---|
In what follows $T$ is the torus and $K$ is the Klein bottle.
I am just looking at Hatcher's example calculation using cellular homology of $T \times S^1$ and $K \times S^1$. Hatcher's homology notes are available here (see page 142 for question, and the commutative diagram and corresponding notes on page 141 for the d... |
My textbook states the following proposition:
Let $f:R \rightarrow S$ be a ring homomorphism and let $s$ be in the image of $f$. Then $\{r \in R \mid f(r) = s\}$ is in one-to-one correspondence with $\ker(f)$.
What does it mean to have one to one correspondence with $\ker(f)$? Does it mean the set $\{r
\in R \mid f(r)... |
I would like to show that as X approaches infinity,
$$\sum_{n\leq X} \frac{1}{\phi(n)} \sim \log(X)\cdot\sum_{k=1}^{\infty} \frac{\mu(k)^2}{k\cdot\phi(k)}.$$
I have already proven
$$\sum_{n\leq X} \frac{1}{\phi(n)} = \sum_{k\leq X} \left[\frac{\mu(k)^2}{k\cdot\phi(k)}\cdot\sum_{t\leq X/k} \frac1t\right].$$
I recognize ... |
How do you find the inverse of $[2+i]_{3}$? I changed it into solving for x in $(2+i)x \equiv 1 \pmod{3}$, and tried to solve for x with extended euclid algorithm, but with no luck. Am I supposed to do something different with complex numbers?
Thanks!
|
Show that the set of all polynomials $f(x)$ of degree at most $5$ with integer coefficients is a ring. Is the set of such polynomials a field?
I don't see how the ring of polynomials with degree at most $5$ is closed under multiplication. If I multiply $x^2$ and $x^5$ I do not get another polynomial of degree at most... |
I'm reading "Elementary Number Theory by Kenneth H.Rosen", and in the book, there is one problem that I couldn't understand the solution.
Problem 9 (Chapter 9.2 - 6th edition)
Problem
Show that if $p$ is a prime and $p \equiv 1 \pmod{4}$, then there is an integer $x$ such that $x^2 \equiv -1 \pmod{p}$.
Solution from... |
Problem
Prove that the congruence equation $ax^2 + bx + c \equiv 0 \pmod{m}$, with $(2a, m) = 1$ is equivalent to a congruence of the form $x^2 \equiv r \pmod{m}$.
I really have no idea where to start. Is there a theorem which relates to this problem? What do we have to show? A hint would be sufficient.
Thanks,
|
So that I don't have to retype them every single time.
e.g. I'll only need to type "Fourier transform" to get the template.
Sure, I could look up the Latex code from the page source in Wikipedia, but it would be nicer if I could search the code directly.
|
Let $\mathbf{L}$ be the set of all languages over $\Sigma=\{0,1\}$ and $E(\Sigma)$ be a lexicographic enumeration of $\Sigma^*$. Then there exists a bijection from $f:\mathbf{L} \rightarrow [0,1)$ where $f(L)$ has its $i$-th bit (to the right of the binary point) set to 1 iff the $i$-th string in $E(\Sigma)$ belongs t... |
I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of Cardinal Arithmetic, which states that for $\kappa$ and $\lambda$ cardinal numbers, the larger infinite and the smaller nonz... |
Lately, I learned about the following continued fraction for the exponential function:
$$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$
I thought it was something new, but evaluating the successive convergents of this continued fraction was a disappointment, as they are nothing ... |
Let $M^m$ and $N^n$ be compact, oriented smooth manifolds without boundary. Then what is the degree of the map
$$ f: M\times N \to N \times M$$
given by $f(x,y) = (y,x)$? I have the feeling it should be $(-1)^{mn}$ (would fit in with some proof I have to give), but the result I come up with is $1$.
My reasoning: If $w_... |
Let $h\colon A\to A'$ be a ring homomorphism between $A,A'$ which are commutative rings with $1$.
Let $P,Q$ be $A$-modules. Then, are there any gaps in my following argument?
$A'\otimes_{A}(P\otimes_{A}Q)=(A'\otimes_{A}P)\otimes_{A}Q=((A'\otimes_{A}P)\otimes_{A'}A')\otimes_{A}Q=(A'\otimes_{A}P)\otimes_{A'}(A'\otimes_{A... |
I'm trying to simplify the following booleans:
$$Y=[\overline{ \overline{(A+B)} \quad \overline{(C+D)}}]$$
My solution is:
$$Y=[\overline{ \overline{(A+B)} \quad\overline{(C+D)}}]$$
$$ = [\overline{ \overline{(A+B)}} \quad \overline{ \overline{(C+D)}}]$$
After this step my module is saying that the next step would be:
... |
This is not mathematician question but I think it's related. How I can get access to some of "Software: Practice and Experience" articles without subscription?
Any advice is welcome. Sorry if I'm posting at wrong site. Thanks.
|
My question is, how can I evaluate the following integral?
$$\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$$
Thanks.
|
I read some book where 'card' symbol is used to signify number of elements in the set. But where is it from? What the full word could be for 'card'?
|
I would like to derive an expression for an Epipolar Line that appears in a right camera (there are two cameras - a left a right one). These cameras are placed in a $(x,y,z)$ vector space. This is a past paper for my computer vision course.
Left camera position = $[-10,0,0]$
Right camera position = $[10,10,0]$
Both o... |
Suppose players can pay 1 unit of payoff to pick what kind of person they meet (ie. football fan or opera fan). How do I calculate the proportion of football fans in the long run?
|
I am not very familiar with how generating functions are used but for something I was needing I ran into the following constructions,
Let $c_{nk}$ be the number of solutions in $\{1,2,3,4...,\}$ for the equation $x_1+x_2+x_3+...+x_k = n$. Then one can easily show that $c_{nk} = ^{n-1}C_{k-1}$ But the following is not... |
I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not possibly confused with another function.
Does anyone say it's bad practice to write $\sin^{-1}(x)$ for $\arcsin(x)$?
|
In many texts charecteristic function is defined as a Fourier transform of probability density (if random variable admits a density function). Also we can define a charecteristic function as Fourier transform of probability measuer instead (like in Jacod J., Protter P. Probability essentials, second edition, page 104).... |
We would like to find an upper bound on the following function:
$\left(\frac{\omega_1}{\omega_2}\right)^{(\alpha_1-1)} \frac{\Gamma(\alpha_1+\alpha_2-1)}{\Gamma(\alpha_1)\Gamma(\alpha_2)} {}_2F_1\left(\alpha_1-1,\alpha_1+\alpha_2-1,\alpha_1,-\frac{\omega_1}{\omega_2}\right)$
for all $\omega_1,\omega_2 > 0$ and $\alpha_... |
the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to relax the constraint by introducting a cost of violating it?
Just to be more clear: ideally, I'd like to be able to solve something... |
QN1 by logical argument verify that {a} is not open for any real number "a".
i guess the set is not open since there is no open interval about "a"
instead the set is said to be closed since any of its complement must be open.
am i correct?
QN2 GIVE TWO COUNTER-EXAMPLES WHICH SHOWS THAT THE IMAGE OF INTERSECTION OF TWO ... |
Consider a matrix Lie group equipped with a left &/or right invariant metric.
The adjoint of linear transformation $A$ with respect to the inner product is denoted as $A^*$.
Here what is actually meant by $A^*$?
How to compute an expression like $(ad(X))^*(Y)$ where X,Y are lie algebra elements.
Edited:
Note the defin... |
I have a system of $N$ linear equations, $Ax=b$, in $N$ unknowns (where $N$ is large).
If I am interested in the solution for only one of the unknowns, what are the best approaches?
For example, assume $N=50,000$. We want the solution for $x_1$ through $x_{100}$ only. Is there any trick that does not require $O(n^{3}... |
It is well known that if $X$ is a Banach space and $X^*$ contains separating sequence (i.e. sequence $(f_n) \subset X^{*}$ such that $f_n x = 0$, $n = 1,2, \dots$, implies that $x=0$) and $K \subset X$ is weakly compact then $K$ is weakly metrizable (i.e. metrizable in weak topology).
Using this we can easily prove, fo... |
I would to solve the Diophantine equation $2x^4 - 2y^4 = z^2$ by descent. This is an exercise from Carmichael Diophantine Analysis but I cannot do it.
Since the LHS is even $2|z$, let $z = 2z'$ and divide both sides by $2$ to get $$(x-y)(x+y)(x^2+y^2)=2z'^2.$$ Congruence conditions tell us that either x and y are both ... |
Is there a simple formula an integer polynomial that $2\sin(2\pi/n)$ satisfies?
For $2\cos(2\pi/n)$ the answer is relatively nice. For any given $n$, we have $2\cos(2\pi/n)= z + z^{-1}$ where $z = e^{2 \pi i/n}$ satisfies a cyclotomic polynomial of degree $\varphi(n) = 2k$,
$$
0 = a_{2k}z^{2k} + a_{2k-1}z^{2k-1} + \l... |
I'd like to determine the function corresponding to the following power series:
$$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},
$$
where $|x|<1$.
|
I'm not a wiz at math and this problem has me a little stumped. I need the average score out of 5 but it's a little different how I do it.
I have 5 fields, field one is rating one, field two is rating two, all the way up to field 5.
I need to grab the average of all these fields together, this is currently what I have.... |
Let $A\in\mathbb{R}^{m\times n}, \ c\in \mathbb{R}^n, \ b\in\mathbb{R}^m$ and consider the linear program $$\max \{ c^Tx : Ax\le b\} \ (1)$$
Its dual is $$\min \{ b^Ty : A^Ty=c, \ y\ge 0\} \ (2)$$ The weak duality theorem states that for $x$ feasible for (1) and $y$ feasible for (2), then $$c^tx\le b^ty$$
The following... |
There is an assertion that if f and g are both differentiable at x, then so is f + g at x. It is safe if $f'(x)$ and $g'(x)$ are both finite, I wonder if it still holds for infinite derivative. To be precise, suppose $f,g: \mathbb R\to\mathbb R, x\in \mathbb R, f'(x)=\lim\limits_{t\to x}\frac{f(t)-f(x)}{t-x}=+\infty, g... |
If $F$ is a finite field such that every element is a square, why must $char(F)=2$?
|
What is the easiest way to add a set number of nautical miles to a known latitude and longitude? I am writing a program in C# that takes a point of origin in decimal notation:
33.4483333, -112.0733333 (Phoenix, AZ)
What I want to to is draw a "box" around that point simulating a sort of "square radius". So if I want a ... |
$x_1$, $x_2$, and $x_3$ are i.i.d. normal random variables with distribution $N(0, \sigma_x^{2})$
$\epsilon_1$, $\epsilon_2$, and $\epsilon_3$ are i.i.d. normal random variables with distribution $N(0, \sigma_n^{2})$
lets define :
$y_1 = x_1 + \epsilon_1$
$y_2 = x_2 + \epsilon_2$
$y_3 = x_3 + \epsilon_3$
given the val... |
Thanks to comments, it should be a plane but why does it look a bit like a fractal? Does my code overlook something or some err in plotting tool? I used Python and GNUplot.
Apparently an animated illusion.
And generated the points with Python and generated graph with GnuPlot like:
$ python copyPasteTheCodeToFile.py > ... |
This function has a closed interval of (-4, 0), (4, 0), while passing through the origin.
I'm struggling to find the maxima and minima of the function, since this finction doesn't have a standalone constant. According to my book, f has a relative minimum if f'(x) changes from negative to positive at (c, f(c)), and a r... |
How does one show that the unit circle admits no polynomial parametrization?
What is needed for this, are there general criteria?
Thanks
|
Is there a method to find the units in $\mathbb{Z}_{n}[x]$?
For instance, take $\mathbb{Z}_{4}$. How do we find all invertible polynomials in $\mathbb{Z}_{4}[x]$? Clearly $2x+1$ is one. What about the others? Is there any method?
|
Is the following identity correct
$\arcsin{x} = \frac{\pi}{2} + i \ln{(x+\sqrt{x^2-1})}?$
Here, $x < 1$. How can we show that it is true? One way to see it is by differentiating, since
$\frac{d}{dx} (LHS) = \frac{d}{dx} (RHS).$
Thanks.
|
The Euler-Maclaurin summation formula is
\begin{eqnarray}
\sum_{k = a}^{b} f(k) = \int_{a}^{b} f(t) \, dt + B_1 (f(a) + f(b)) + \sum_{n = 1}^{N} \frac{B_{2n}}{(2n)!} ( f^{(2n-1)}(b) - f^{(2n-1)}(a) ) + R_{N},
\end{eqnarray}
where $B_{n}$ is the $n^{\text{th}}$-Bernoulli number taking $B_{1} = \tfrac{1}{2}$, and the... |
This is at heart a mathematical problem, but is best motivated in physical terms. I'll introduce a very special case and move on to the general case later.
Special case An object, taken for convenience to be a cube of side $a$, is placed at distance $d$ in front of a convex spherical mirror of radius $R$. What is th... |
Are there two continuous functions ( $f:R^n\to R$, $g:R\to R$ ) which are not differentiable at point $0$, but if you compose them, $h=g(f(x))$ is differentiable at $0$? ($f(0)=0$)
|
It is known that if $M$ is an oriented submanifold of $\mathbb{R}^{n}$ and $\omega$ is a compactly supported form on $M$, then
\begin{align}
\int_{M} \omega = - \int_{-M} \omega,
\end{align}
where $-M$ denotes $M$ with opposite orientation. Is there an analogous identity for the Lebesgue integral of Lebesgue integrable... |
What is the right approach to take and find the moments of the following:
$$\mathcal{Z}_t=\int\mathcal{W}_t^k\,d\mathcal{W}_t=?$$
$$\mathcal{W}_t \sim \mathcal{N}(0, t),\ k=2,3...$$
$$\operatorname{E}(\mathcal{Z}_t)=?$$
$$\operatorname{Var}(\mathcal{Z}_t)=?$$
I know that $\mathcal{Y}_t=\int\mathcal{W}_t\,d\mathcal{W}_... |
From a bank of masters exams:
Say the position of a particle moving
in $\mathbb{R}^n$ is given by a smooth
vector-valued function $\vec{x}(t)$.
Suppose that $\vec{x}(t)$ satisfies a
differential equation,
$$ \frac{d\vec{x}}{dt} = A(t)\vec{x},$$
where $A(t)$ is a
real anti-symmetric matrix depending
s... |
My text wants me to prove that $e^A$ converges for any $n \times n$ matrix and put a bound on it in terms of $|A|$ and $n$ where $| |$ is the norm induced by identification of $M(n, n)$ with $\mathbb{R}^{n^2}$. I can prove convergence and it's obviously bounded by $e^{|A|}$ but I don't see how to get it in terms of $n... |
In the same vein as:
$ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $
Starting with:
$ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = \frac{\sin(\pi q)}{\pi q}$
I've noticed that:
$ - \frac{\pi ^2}{3!} = \displaystyle \sum_{j_1=1}^{\infty} -j_1^{-2} $
$ \frac{\pi ^4}{5!... |
If $R$ is PID and $S$ is an integral domain that contains $R$, why is the gcd of $a$ and $b$ in $R$ the same as the gcd of a and b in $S$ ? Is this a standard result? where can I find the proof? Thanks.
|
Consider a one-dimensional random walk on the positive integers with:
(1) - A fully reflecting boundary at "x = 0".
(2) - Initiation of the walker at "x = 1".
(3) - A P[+1 Step] / P[RHS Step] probability for the walker less than or equal to the P[-1 Step] / P[LHS Step] probability.
What's the average distance of the ... |
I am trying to show that the functions $t^3$ and $|t|^3$ are independent on the whole real line. To do this, I try and prove it by contradiction. So assume that they are dependent. So then there must exists constants $a$ and $b$ such that $at^3+b|t|^3=0$ for all $t \in (-\infty,\infty)$. Now pick two points $x$ and $y$... |
We are supposed to write a program to solve the following initial value problem numerically using 4th order Runge-Kutta. That algorithm isn't a problem and I can post my solution when I finish.
The problem is, separating it out cleanly into something I can put into Runge-Kutta.
$$\exp(-x^\prime)=x^\prime-x+\exp(-t^3)$$... |
I'm having some difficulty doing a proof for the following:
$$\neg A \vee \neg(\neg B \wedge (\neg A \vee B))$$
It is said that you could use the law of excluded middles.
Any help or guidance would be appreciated. Thanks in advance!
|
I would like to prove the following statement: if $A$ is an $R$-algebra (for a commutative ring $R$ with $1$), then the $R$-algebras $M_n(A)$ and $M_n(R)\otimes A$ are isomorphic.
By a proposition (Grillet, Abstract Algebra, p. 529) , we have the situation
where $\varphi((r_{i,j})_{i,j=1}^n):=(r_{i,j}\cdot 1_A)_{... |
I'm researching a potential algorithm, and I'm hoping that someone can verify my calculations.
I have sets of vectors in $\mathbb{R}^6$ that I can use. They have a corresponding value associated with them, but this relation is not necessarily a simple one. I'd like to find the value associated with the vector $(0,0,1... |
Let the sequence $s_n$ be defined by $s_1$ = 1, $s_{n+1}$ = $\frac{1}{4} (2s_n + 5)$ for n $\in $ $\mathbb{N}$. The following is my proof and below it, my concerns.
Proof
(1) We will prove $s_n$ is increasing by induction.
That is $s_n \leq s_{n+1}$ for all n $\in \mathbb{N}$.
Since $$s_1 = 1 < s_2 = \frac{1}{4}(2(1)+5... |
I would like to know if someone can explain in a somehow down to earth (almost logic free) way why is it true that in an axiom system where there is some statement $P$ such that $P$ and its negation $\lnot P$ are true, then every statement in the system is true?
I'm not sure if this can be done, but basically since I d... |
I think we can argue that Sg --the genus-g surface -- has only a
trivial embedding in S4 , since Sg is topologically a wedge of g S1's, and
there are no knotted S1's in S4 (meaning that any two embeddings of S1 in S4 are isotopic.)
But I am not clear on why there are no non-trivial embeddings of Hg ----a 3D
... |
in a previous question, I mistakenly attempted to subtract one cardinal number from another. Anyway, this got me to thinking, suppose I have two sets $X$ and $Y$, with $Y\subseteq X$. Suppose also that $|X|=|Y|=|X-Y|=\kappa$ for some cardinal number $\kappa$. Does there exist some bijection $f\colon X\to X$ such that $... |
would anyone be able to help me with this question?
Consider a random variable $X$ with probability density function
$$f(x) = xe^{−x^2/2}, x\geq 0.$$
Determine x such that
$P(X \lt x) = 0.5$, $P(X \leq x) = 0.5$, and $P(X \gt x) = 0.95.$
What I've tried to do is integrate $f(x)$ to get $F(X)$ and then substitue in the ... |
Suppose a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is analytic at 0, i.e., it is given by its Taylor series on a neighborhood $U$ of 0. Then is it necessarily true that $f$ is analytic at every point in $U$ ?
|
I'm aware of the identity
\begin{align}
\sum_{i = 0}^{k} i! \binom{n+1}{i + 1} S(k,i) = H_{n,-k},
\end{align}
where $H_{n,-k}$ is a generalized Harmonic number defined by $H_{n,m} = \sum_{r = 1}^{n} r^{-m}$. I believe the following sum is related to the generalized Harmonic numbers as well,
\begin{align}
\sum_{i = 0}^{... |
How does one find all positive integer solutions to the equation $x^3+48=y^4$?
|
For example $151x - 294\equiv 44\pmod 7 $. How would I go about solving that?
The answer says to simplify it into the $ax\equiv b\pmod c $ form first, but I have no clue on how to get rid of the $294$... help?
|
The following question was given to as in a recent assignment, namely
Let $$y''+p(t)y'+q(t)y=0~.$$ It is given that $(1+t)^2$ is a solution to this differential equation, and that the wronskian of any two solutions is constant. Find the coefficients of $p$ and $q$.
Say for example the wronskian is a constant, say $C$.... |
I know that there is a formal definition isomorphism but for the purpose of this homework questions, I call two groups isomorphic if they have the same structure, that is group table for one can be turned into the table for the other by a suitable renaming.
Now consider $\mathbf{Z}_2=\{0,1\}$ (group under addition mod... |
Can anybody answer this question?
Thank you.
|
I need to figure out if $\mathbb{Q}$ is either open or closed in $\mathbb{R}$ and prove my assertions. I don't think it's either. I would like a review/critique of my proofs to make sure that there's not an easier way to do it.
To see that it's not open, take some $\frac{m}{n}$ with both $m$ and $n$ perfect squares so ... |
If I find the elements generated by $8$, can I say that the set of these elements is a subgroup of $\mathbf{Z}_9$ provided all these elements are in $\mathbf{Z}_9$? or to show that the set of these elements is a subgroup of $\mathbf{Z}_9$, should I test it for being a subgroup? if your answer is that no test is needed,... |
I'm studying probability for preparing myself for machine learning.
I came across this notion of Expectation of (random variable) or (function of random variable) like $E[X]$ or $E[g(X)]$.
Can anybody explain me the intuitive idea of the above notion (taking into consideration it's implications in machine learning, may... |
Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried searching for 'dual group' and 'opposite group' with no results. It seems that this group is all you need to dispose of ... |
This is copy of statement in this link.
If $f$ is an infinitely differentiable function defined on an open set $D \subset \mathbb{R}$, then the following conditions are equivalent.
$f$ is real analytic.
For every compact set $K \subset D$ there exists a constant $C$ such that for every $x \in K$ and every non-negative... |
As every one knows, the quadratic number fields shares a deep connection with the binary quadratic forms; I have been told this relation when I was a senior high, and now I, learning some difficult theories like the theory of fields of classes, know some basic facts about the algebraic number fields of degree two but I... |
Players $1, 2, 3,\dots, n$ are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who s... |
May I refer you to:
http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/soln62.html#s6201
Can you please explain in detail, problem 1, the inequality:
$[K(u,v): K(u)] \leq n$ ?
EDIT:
Let's see. First the fact that $[K(v): K]=n$, tell us that we can find a minimum polynomial $p(x) \in K[x]$ such that $p(v)=0$,... |
I need to calculate the sums
$$x_1^3 + x_2^3 + x_3^3$$
and
$$x_1^4 + x_2^4 + x_3^4$$
where $x_1, x_2, x_3$ are the roots of
$$x^3+2x^2+3x+4=0$$
using Viete's formulas.
I know that $x_1^2+x_2^2+x_3^2 = -2$, as I already calculated that, but I can't seem to get the cube of the roots. I've tried
$$(x_1^2+x_2^2+x_3^2)(... |
I understand this is a weird question, but I think this could be a place in which I could find the answer.
I'm trying to reprodice this image:
with a java applet. I know that this image is a grid-Fresnel zone plate" but unfortunately I don't know the reference.
I've measured the white square and this is the result (in... |
Where in combinatorics does the procedure of taking diagonals of power series (generating functions) arise?
With diagonal is meant the following: if $f = \sum a_{i_1 ... i_n} x_1^{i_1}...x_n^{i_n}$ is a formal power series in $n$ variables then one can form
for example the diagonal $I_{12}(f) = \sum a_{i_1 i_1 i_3 ... ... |
How can I find all $m,n\in\mathbb{Z}$ satisfying $n^2+n+1=m^3$?
|
Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that:
a. The measure $\mu$ is invariant under all $g \in SL_2(\mathbb R)$ i.e. $\mu(gA) = \mu(A)$.
b. Compute the hyperbolic area of the standard fundamental domain for the modular group ... |
I am trying to understand a proof for critical points of certain energy functions being harmonic functions. It goes as follows: For a function $u(x_1,..,x_n)$, a functional E(u) is defined as
$E(u) = \frac{1}{2}\int |\nabla u|^2 $
For the proof, a curve is constructed in the space of functions by using $u+t\phi$. Rest... |
I think I found this problem in papers of Titu Andeerscu, but I do not remember the solution. I have just added this one to my LTE article since the solution I saw used this method to solve it. Here it is:
Let $m$ and $n$ be positive integers. Prove that for each odd positive integer $b$ there are infinitely many prime... |
I need to find a constant $a$ such that for all $x_1 x_2 > 0$:
$$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$$
that is to say the supremum of the term on the right hand side. My question is how to find it. Of course, $x_1 x_2 > 0$ holds when both variables have the same sign. So I tried to simplify my problem a bit and now... |
Suppose that $H, K$ are subgroups of a finite group $G$, with $|H|$ relatively prime to $|G:K|$. Does it necessarily follow that $H \leq K$, or is there a counterexample?
This question arose from considering the following. Suppose an abelian subgroup $K$ of $G$ has prime index $p$, and $q \neq p$ is a prime factor of $... |
Let $x, y >1$ be integers satisfying $2x^2-1=y^{15}$. How can I prove that $5 \mid x$?
|
The Diophantine equation is $n^2 + n + 1 = m^3$ my attempt to solve it shows there is no solutions to this equation, but in fact there are four. I could not find my mistake so I hope someone could point it out please.
Factoring in the Eisenstein integers $\mathbb{Z}[\omega]$ with $\omega^2 + \omega + 1 = 0$: $n^2 + n ... |
I saw $$ \sum_{i \in \mathbb{Z}}
\sum_{j \in \mathbb{Z}} f(i,j) =
\sum_{k \in \mathbb{Z}} \sum_{(i,j)
\in \{i+j =k, i, j \in \mathbb{Z}\}}
f(i,j).$$ I was wondering what
conditions for it to hold are? Is
there a name for the method using
such equality?
I was also wondering if $$\int_{x
\in \ma... |
Its sometimes hard to type it if logarithm is not natural and base is not 10, especially if base is variable. So anyone know rules how to type?
|
Is it possible to collude in a perfect information game?
Exercise 10-7 from book "Algorithms And Networking For Computer Games" - Jouni Smed, Harri Hakonen, p 225
|
Possible Duplicate:
What is the (mathematical) point of geometric constructions?
Are there good motivations to study compass and straightedge constructions?
More specifically I want to know:
1) Are there good motivations to study compass and straightedge constructions in high school?
2) Are there good motivations t... |
I was asked what is the smallest positive integer made of ones (11111...1) that is divisible (no reminder) by a number made up of 100 digits of 9 (9999...9).
I noticed that for 9 the smallest integer made out of ones that will divisible by him, is 111 111 111 (9 digits), and that for 99 the integer will be with 111 111... |
Has anyone studied the real function
$$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so
on) with respect to the Collatz conjecture?
It does what Collatz does on integers, and is defined smoothly on all
the reals.
I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text... |
I am trying to understand why does the limit comprasion test doesn't work for alternating series, is it even true? or is there a counter example? I can't find one. can you help me please?
thanks.
benny
|
Would you please help me in how to solve these questions :
A- Let $H$ be a subgroup of a finite group $G$.Let $\alpha$ and $\beta$ be class function of $G$ and $H$ respectively. Prove that
$$Ind^{G}_{H} (\beta . Res^{G}_{H})= \alpha .Ind^{G}_{H}(\beta) $$
B- Show that
1.Each element in $SU(n)$ conjugate to a diagona... |
I am trying to solve approximately 100 equations in the same number of variables. I wrote a little Mathematica function to do it. It works fine, but usually only 2 to 10 variables end up being something other than 0 and it's hard to read the solution since there are 90+ instances of something like "hn[0]->0". Is the... |
The problem is:
In how many ways can numbers from
[1-K] be placed in a row of N elements
that there is not a triple a, b, c
where a > b > c and pos(a) < pos(b) <
pos(c) in a above mentioned row.
I wrote a small program that calculates this using dynamic programming, but I wonder if there is a simple formula.
... |
Consider the unit disc, complex plane, and Riemann sphere. For each ordered pair of these surfaces, is there a surjective holomorphic map? An injective holomorphic map?
|
Suppose $R$ is a finite ring (commutative ring with $1$) of characteristic $3$ and suppose that for every unit $u \in R\:,\ 1+u\ $ is also a unit or $0$. We need to show that $R$ is a field. Is this true if ${\rm char}(R) > 3$?
Here is what I attempted to do. $\:$ First of all, $\:$ I noticed that the statement is not... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.