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I would like to know why $\mathbb{C}[x,y]$ is not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings.
Thank you! 1
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Find the complete integral of partial differential equation
$$\displaystyle z^2 = pqxy $$
I have solved this equation till auxiliary equation:
$$\displaystyle \frac{dp}{-pqy+2pz}=\frac{dq}{-pqx+2qz}=\frac{dz}{2pqxy}=\frac{dx}{qxy}=\frac{dy}{pxy} $$
But I have unable to find value of p and q.
EDIT:
p = ∂z/∂x
q = ∂z/∂... |
If $f(x)$ is a function satisfying $ \displaystyle f(x+y) = f(x) \cdot f(y) \text { for all } x,y \in \mathbb{N} \text{ such that } f(1) = 3 \text { and } $ $ \sum_{x=1}^{n} f(x) = 120 $, then find the value of $n$.
How to approach this one?
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Let $(X,M,\mu)$ be a measure space. Why are the following statements equivalent?
i) There exists a sequence of pairwise disjoint measurable sets $\{A_{n}\}$ in $M$ , each of finite measure, such that $\mu(A) = \sum_{n=1}^{\infty} \mu(A \cap A_{n})$ for every measurable set $A \in M$.
ii) $\mu$ is a countable sum of pai... |
From Statistical Inference Second Edition (George Casella, Roger L. Berger)
"My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get a least one call each day?"
The answer is .2285, but I don't know how they got it. My reasoning was as follows:
Th... |
I have a vector A and a point P. My problem is how to find out if point is on the left or right side of vector looking from the point of origin of a vector in direction of it.
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What does D = max[0; M(x)] mean?
M(x) is a function.
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Can somebody prove that when we add 2 bit integers and if there is an overflow then the result always will be lesser than the 2 operands used??
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Is there a mathematical function such that;
f(3, 5) = 3
f(10, 2) = 2
f(14, 15) = 14
f(9, 9) = 9
It would be even more cool if there's a function that takes three (3) parameters, but that one could be solved by using recursive functionality;
f( f(3, 5), 4) = 3
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We know that the set of continuous functions and the set of smooth functions in $\mathcal{L}^1$ or $\mathcal{L}^2$ are dense in $\mathcal{L}^1$ and $\mathcal{L}^2$. Is the set of $\mathcal{C}^k$ functions dense in $\mathcal{L}^1$ or $\mathcal{L}^2$ for any $k\in\mathbb{N}$?
EDIT
Is the set $\mathcal{C}^k\setminus \math... |
I recently made a model to solve a conflict problem in sports.
The model is based on Bayesian Games: a complete definition can be found here.
The thing is that I am now looking to compute the equilibria of the system, but I don't know any algorithm to do so.
Do you know of any?
Which language do you think I should use?... |
Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$ and $V(G)=V(H)$.
1) What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$?
2) In the unlikely event that I am the first to think about using that operation, which sy... |
I'm having trouble understanding this proof:
Lemma: Let $\lambda$ be the Lebesgue measure restricted to the Borel sets of $[0,1]$. Then there exists a decreasing sequence $(E_n)_{n\ge0}$ of subsets of $[0,1]$ such that each of them is $\lambda$-thick and $\bigcap E_n = \emptyset$.
Proof: Let $E_n = [0,1]\cap\{f\ge n\}... |
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e \cdot i} = -1 $$ When we take the $\frac{1}{i}$'th power, we see that
$... |
I'm currently reading Artificial Intelligence, by Russel & Norvig. They state that:
A) A sentence is valid if it is true in all models
B) The deduction theorem: "For any sentences $\alpha$ and $\beta$, $\alpha \models \beta$ if and only if ($\alpha \implies \beta$) is valid.
My mental blockage consists of the fact that... |
As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half their differen... |
I am stuck in trying to solve the following:
Given two points $(x_{1}; y_{1})$ and $(x_{2}; y_{2})$, to determine the parameters $a$ and $b$ in the equation:
$$y=\frac{e^{a+bx} - e^{a}}{1+e^{a+bx}}.$$
In other words, I have two unknowns and two set of points, so there's for sure a solution, but how can the following sy... |
I hope you can help me with this excersise:
Lets have finite sequence of characters $0$ and $1$. We are doing following algorithm:
If the sequence starts with $0$, we add $00$ to the end.
If the sequence starts with $1$, we add $1101$ to the end.
Then we remove first three characters.
Prove, that if the sequence... |
I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques which are more generally applicable to arbitrary Lie groups as well.)
Here's the current state of my knowledge on this issue... |
I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be?
In the below image, I have d and n. How can I get r?
Thanks.
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Let $A$ and $B$ be two symmetric matrices of the same size, and let $\alpha_k$ and $\beta_k$ be the $k$th largest eigenvalues of $A$ and $B$ respectively. What can we say about $|\beta_k - \alpha_k|$ in terms of $||B - A||_2$? To be clearer, I want the tightest possible upper bound on $|\beta_k - \alpha_k|$ using onl... |
Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.
Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or do... |
How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?
I need to formulas, unless you're using the ardor3d api's functions/methods.
Matrix is set up like this
xx, xy, xz,
yx, yy, yz,
zx, zy, zz
A Quaternion is fine too.
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I recently learned about finding density functions of functions of random variables using the cumulative distribution function. For example, computing the density function for the difference of two uniform random variables on $[0,1]$ (as in Grinstead & Snell's free online book). I gave myself the following problem.
S... |
I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition.
The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky decomposition, must be symmetric and positive-definite. Checking whether a matrix is symmetric is easy, but the positive part pro... |
In logic theory, if we have a finite domain A of k elements, then we can construct only finitely many structures (finite structures) each of which has A as domain. I think the number of structures will be $2^{k}$. How to prove it - if my claim is right?
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I have some kind of "homework" question. I have the following theorem:
Theorem (Transfinite Recursion over the class of of ordinals $\mathbf{ON}$: Let $\mathbf{V}$ be the class of all sets. If $\mathbf{F} : \mathbf{V} \to \mathbf{V}$ then there exists a unique $\mathbf{G} : \mathbf{ON} \to \mathbf{V}$ such that for all... |
How would I go about it?
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I have a function $S(x)$ which is bounded above and below as follows:
$f(x) + C_1 + \mathcal{O}(g(x)) < S(x) < f(x) + C_2 + \mathcal{O}(g(x))$ as $x \rightarrow \infty$
Can I conclude that $$S(x) = f(x) + C + \mathcal{O}(g(x))$$ as $x \rightarrow \infty$? Can anyone give a short proof for this fact?
EDIT:
(To clear up... |
If I have the Keplerian elements for two orbits, how do I compute the shortest distance between these two orbits in 3D space? Is there any analytical expression to compute that?
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It all about maths I don't understand, how can I solve this exercise, someone say that is a $2$-dimensional problem, but I can not figure out for myself, I already understand what to do, but I don't know why is the formula, can someone explain this exercise for a dummy? http://www.codeforces.com/contest/1/problem/A
Why... |
I'm given a number (n), an interval (i), and the number of steps (s). I want to distribute these numbers over that criteria as best as possible. The upper bounds would be n (which is critical) and lower bounds is greater than or equal to 0.75 * n (but divisible by the interval). So, if n=100, i=5, and s=6, you'd hav... |
Okay, so almost 3 months into my abstract algebra, we just started rings. I have a few questions.
A "trivial ring" is a ring with only one element. So $R={0}$ is a trivial ring. Understandable.
Then a definition states: Let $R$ be a ring. If there is an element $x \in R$ s.t. for all $a \in R$ you have $a * x = a = x ... |
This might be an easy question, but I haven't been able to up come up with a solution.
The image of the map $$f : \mathbb{R} \to \mathbb{R}^2, a \mapsto (\frac{2a}{a^2+1}, \frac{a^2-1}{a^2+1})$$
is the unit circle take away the north pole. $f$ extends to a function $$g: \mathbb{C} \backslash \{i, -i \} \to \mathbb{C}^... |
I know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X}$ is the universal covering space of $X$, and $V$ is a $G$-module? How do you recover the locally free sheaf from $\tilde... |
I am reading Joseph Bernstein's notes on D-modules which are available online here
In section 1 (page 1 of the pdf) Bernstein writes "By Serre's theorem this condition is local."
I was wondering to what theorem Bernstein is referring and wondering where I could find a statement and proof of this theorem.
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Let $R$ be a non-zero commutative ring. Prove that the ideal $(x)$ of $R[x]$ is prime if and only if $R$ is an integral domain.
I'm working on the left-to-right direction right now. I know that $R[x]/(x)$ is an integral domain since $(x)$ is prime. So I want to fix $r\in R$ and use the evaluation map $e_r:R[x]\to R$ gi... |
I have a helix in parametric equations that wraps around the Z axis and a point in space. I want to determine the shortest distance between this helix and the point, how would i go about doing that?
I've tried using Pythagorean theorem to get the distance and then taking the derivative of the distance function to find ... |
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm dx$$ is also $\frac{\pi}{2}.$ Many proofs of this latter one are already in this post.
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Let $\displaystyle X$ be a Polish space. Let $\displaystyle d,d'$ be two complete compatible metrics. Is $\displaystyle 1:(X,d)\to(X,d')$ uniformly continuous.
What I really want to know is: can we unambiguously speak of $\displaystyle f:X\to K$ from one Polish space to a compact Polish space as being uniformly continu... |
I have some problems understanding a question in a math class test for 10 year olds.
The question is:
Someone pays in 200€ to a bank. The interest rate is 3.75%. Some time later, he looks at the account balance and he sees that there are 235€ now. How much time passed?
I would calculate this using logarithms, but the... |
What does it mean to say that an algebra is complete with respect to a filtration? Thanks!
|
The question is how to calculate for the general problem.
The specific problem is this:
Given a door code of $n$ numbers on a 10 digit panel, how many combinations do you have to try?
The catch is that the mechanism that senses whether the correct code has been entered has no concept of the start and end of the code.... |
Problem:
In a school competition,the probability of hitting the target bu Dick is $\frac{1}{2}$,by Betty is $\frac{1}{3}$ and by Joe is $\frac{3}{5}$.If all of them fire independently,calculate the probability that the target will be hit.
This general approach for solving is to find the complement of the probability ... |
Repeating for my exam in commutative algebra.
Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma:
Let H be the intersection of all neighborhoods of $0$ in $G$. Then $H$ is a subgroup.
The proof in the books is the followi... |
I've read the formula written on the title on a mathematics' book and it doesn't seem correct for me:
For the first part of the formula [A ∨ (B∧C)] I have the following possible values:
A; B and C; A and B and C (the Or is not exclusive)
For the second part of the formula [(A ∨ B) ∧ (A ∨ C)] I have the following values... |
A friend of mine introduced me to the following question:
Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$ ($f \in C^{\infty}$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?
He has been able to prove that such a polynomial (with degree at least 2) doesn't e... |
(Disclaimer: I'm an engineer)
Hi everybody, I found this “riddle” posted on the internet:
It's meant as a joke, but I do think it deserves an answer :)
A bit of background: the orange and blue ellipses are a citation from the videogame Portal. They are "portals" connected so that everything that goes into one of them ... |
Let $(X,d),(Y,d)$ be polish spaces and $Y$ compact. Consider the spaces $C_{u}(X,Y)$ of uniformly continuous functions, and $C(X,Y)$ of continuous functions. We can endow these with the uniform-metric.
Can $C(X,Y)$ be generated from $C_{u}(X,Y)$ using only point-wise limits of $\omega$-sequences? I.e. Define $\mathcal{... |
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave.
There could be infinitely many decompositions of that sort.
Anyone knows where I can see a proof or this, or knows of a proof of this?
Than... |
What is the best program to factor large arbitrary-form integers on a single computer, or on a few disjointed computers? "Best" is obviously subjective, but what do you recommend?
I'm working on a project to factor general-form numbers that I know are composite, with a scale of 100 - 1000 digits. I have a few computers... |
Lets say I take a computation that involves only addition and multiplication:
(a+b)*(c+d)
which can be done in many other ways, eg.
a*(c+d) + b*(c+d)
a*c + a*d + b*c + b*d
In terms of additions and multiplications the number of operations required for each of the three examples shown are (2,1) (3,2) (3,4) respectivel... |
So I have this problem to solve...
Let X denote the number of paint defects found in a square yard section of a car body painted by a robot.
These data are obtained:
8, 5, 0, 10, 0, 3, 1, 12, 2, 7, 9, 6
Assume that X has a Poisson distribution with parameter λs.
a) Find an unbiased estimate for λs.
b) Find an unbiased... |
Alright, so I got two points in 3d space, so they have a x,y, and z.
Now if the line's y - which I get like so:
Vector3 v = new Vector3();
v = a.subtract(b, v);
v.normalizeLocal();
float cosine = (float) v.dot(v);
float angle = (float) Math.toDegrees(Math.acos( cosine ));
Now I have a third point (the c point), which... |
Please help me, I can't find the theorem anywhere that states something like:
For a bounded set $U\subset\mathbb R$ there exists a non-descreasing sequence $(a_n)_{n\in \mathbb N}$ with $a_n \in U$ with $\lim_{n\to\infty}a_n=\sup U$.
Thank you!
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I have a camera looking at a computer monitor from varying angles. Since the camera is a grid of pixels, I can define the bounds of the monitor in the camera image as:
I hope that makes sense. What I want to do is come up with an algorithm to translate points within this shape to this:
I have points within the same d... |
I tried to show the following estimation and am not sure, if it holds:
Given $U\subseteq\mathbb R$ and $f:\mathbb R\times \mathbb R\to\mathbb R$. Its derivative with respect to the second argument exists and is denoted by $f_b(a,b)$:
$\displaystyle\frac{d}{db}\left(\sup_{a\in U}f(a,b)\right)\ge \inf_{a\in U}f_b(a,b)$
... |
Given a continuous function $f(x)$ and
$\forall a\gt 0, |\int_{0}^{a}f(x)dx| \leq M$, show
$\int_{0}^{\infty}f(x^2)dx$ exists.
I tried substituting $t=x^2$ which gave me $\displaystyle \int_{0}^{\infty}f(x^2)dx=\frac{1}{2}\int_{0}^{\infty}\frac{f(t)}{\sqrt t}dt$ but I can't see how to use the fact that $f(x)$ has... |
Given the following expression:
$$(7y-1)(y-7) \le 0$$
To me this inequality implies $y \le 7$ and $y \le \frac{1}{7}$ but the correct expression (from my module) happens to be $\frac{1}{7} \le y \le 7$
Where exactly I am wrong?
|
I want to find a nice simple expression for the definite integral
$$\int_0^\infty \frac{x^2\,dx}{(x^2-a^2)^2 + x^2}$$
Now, I can numerically compute this integral, and it seems to converge to $\pi/2$ for all real values of $a$. Is this integral actually always equal to $\pi/2$? How can I show this?
Also, why does Wolf... |
Let R be a commutative ring and I and J be two ideals of R, how to show that IJ= the intersection of I and J if R=I+J?
|
By definition, the weak law states that for a specified large $n$, the average is likely to be near $\mu$. Thus, it leaves open the possibility that $|\bar{X_n}-\mu| \gt \eta$ happens an infinite number of times, although at infrequent intervals.
The strong law shows that this almost surely will not occur. In particul... |
How would you expand a moment generating function with a negative fractional power as a power series??
What method would you be able to use?
|
Are there solutions in $\mathbb{Q}-\mathbb{Z}$ to $X^2+2y^2=1$?
I'm pretty sure that there isnt but Im not sure how to show this. I don't have much experience (yet) with $p$-adics.
|
I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas?
Let $X_1, X_2, \cdots$ be an infinite sequence of iid sequence of integrable random variables
and let $N$ be a positive, integer-valued integrable random variable which... |
I need to find the two roots (x-intercepts) from the equation:
$$(x + 7)(x - 3) = (x + 7)(5 - x)$$
How would I go about this? I would greatly appreciate a step-by-step explanation.
|
$$\int_0^2 \dfrac{\mathrm dx}{\sqrt{x}(x-1)}$$
I want to determine whether this integral converges or diverges. Now usually problems like these are easy, but this one is kind of tricky since it is discontinuous at both 0 and 1. Whereas with one situations where the integral is only discontinuous at 1 value, I could ju... |
The book I am studying uses the notation $w_k$ to denote the number of roots of unity contained in $K$, an imaginary quadratic number field.
Supposedly, this is the same as the size of the unit group in $O$, the ring of integers in $K$. I am well familiar with the standard arguments for finding the units in $O$.
Im ... |
Let $K$ be an imaginary quadratic field and $U$ denote the unit group in the ring of integers in $K$. Are there $\alpha \in K-U$ with finite multiplicative order? That is, is there $n \in N$ such that $\alpha^n=1$?
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I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2zdy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
$z=y$
$y=y'=y''$
???
now, I'm pretty sure I did something wrong. could you please correct.
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The product of two univariate generating functions is simply given by the Cauchy product.
$$ A(x) = \sum_{n=0} a_n x^n $$
$$ B(x) = \sum_{n=0} b_n x^n $$
$$ A(x)B(x) = C(x) = \sum_{n=0} x^n c_n $$
with $c_n = \sum_{k=0}^n a_k b_{n-k}$.
What is the resulting generating function for the bivariate case?
$$ A(x,y) = \s... |
$\displaystyle\sum_{n=0}^{\infty}\sqrt{n}(\sqrt{n^4+1}-n^2)$
Ok I was really hesitant to post this because it's such an elementary question, and I really should be able to do this. However, I've been looking at this for over half an hour and I've tried many methods, but I'm not able to get the series into a nice form ... |
Let $\rho$ be a representation of $G$ on $V$. Why are its eigenvalues roots of unity?
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I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.
|
Given two random samples $a$ and $b$ of a normal distribution, how can I calculate the probability that their difference $|a-b|$ is bigger (or lesser) than $c$?
|
I seem to remember reading somewhere an "abstract nonsense" proof of the well-known fact that the fundamental group of a connected topological group is abelian. By "abstract nonsense" I mean that the proof used little more than the fact that topological groups are the group objects in the category of topological space... |
$$\sum_{i=0}^{n-1} (i+1)(n-1)$$
Is that $O(n^2)$ or $O(n^3)$? Can you explain me how you found it? Thanks.
|
A version of the Riesz Representation Theorem says that a continuous linear functional on the space of continuous real-valued mappings on a compact metric space, $C(X)$, can be identified with a signed Borel measure on the set $X$. Are there any similar results when we replace $C(X)$ by the space of continuous function... |
Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one.
My math experience is calculus through differential equations, twenty five years ago. I also have some computer science rel... |
Given a circle, a point $H$ outside the circle, segments $\overline{HE}$ and $\overline{HT}$ tangent to the circle at $E$ and $T$, respectively, and points $I$ and $G$ on the circle such that $I$, $G$, and $H$ are collinear (all as shown above), knowing the measures of $\angle EHG$ and $\angle GHT$ (call them $\alpha$... |
I am having issues solving the following equation:
$$\frac{x}{2x-3} - \frac{1}{2x} = \frac{3}{4x-6}$$
The resolution of this is 1.
It is in the section of parabolas and it should be pretty easy to solve.
My steps are as follows:
$$\frac{x}{2x-3} - \frac{1}{2x} = \frac{3}{4x-6}$$
Multiply by $2x(4x-6)$
$$4x^2 -1(4x-... |
Let $X = (X_t: t \in [0,T])$ be a stochastic process satisfying a CIR model
$$
dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t,
$$
where $B_t$ is a standard Brownian motion, $\beta$ is a negative constant, $\gamma, \sigma$ are positive constants. In order for the SDE to make sense, assume that $X_t > 0$ for a... |
Consider the set $A$ of natural numbers which are of the form $k^3-p$, for $k$ a positive integer and $p$ a positive prime. Does $A$ have a density (of any of the usual kinds for sets of natural numbers) and if so, what is it?
$A$ contains a lot of numbers, and it seems somewhat difficult to me to prove that a particul... |
Here is an exercise, on analysis which i am stuck.
How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$?
|
Here is a problem in Griffiths Introduction to Electrodynamics as follows.
Check the divergence theorem for the function $\mathbf{v} = r^2\mathbf{\hat{r}}$, using as your volume, the sphere of radius R, centered at the origin?
Here is my calculation for the surface intergral part, please help me finding the error in it... |
I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question.
There are many different techniques one might use to show a manifold is simply conencted, I am only interested in a specifi... |
I have a circle of radius $r$, and I have a bounding box with 4 sides, $a$, $b$, $c$ and $d$ as illustrated in the below diagram:
Here are a few constraints
$$r \geq a$$
$$r \geq b$$
$$c >> a$$
$$d >> a$$
$$c >> b$$
$$d >> b$$
$a$ and $b$ must be perpendicular to $d$ segment.
Another condition is that the the ci... |
This is a very simple question but I believe it's nontrivial.
I would like to know if the following is true:
If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.
Thanks!
If there isn't a proof (or disproof) of the general result, I would be interested to know if t... |
I saw an exercise which was meant to find integers all integers $m,n$ satisfying $2m^2 + 5n^2 = 11(mn-11)$. I found them using the factorization $(m-5n)(2m-n)=-11\cdot 11$. However, what kind of methods there are to solve the original problem, find all $(m,n)\in\mathbb{Z}\times\mathbb{Z}$ satisfying $2m^2+5n^3=11(mn-11... |
I have just discussed with student and he told me him task, and I think it is interesting.
The task.
There are file with points like:
Point0: x=1; y=4;
Point1: x=199; y=45;
Point2: x=42; y=333;
Point3: x=444; y=444;
...
PointN: x=nnn; y=mmm;
You should find polygons and draw them. Each polygon present as internal
I me... |
In Goldstern and Judah's The Incompleteness Phenomenon we are asked to prove that any model of the first two Peano Axioms:
$$\forall x [Sx\neq0]$$ $$\forall x\forall y[Sx=Sy\implies x=y]$$
must be infinite. S is a unary function intended as successor. If the model is finite there must be $m$ and $n$ with $S^m0=S^n... |
Can someone suggest an idea for a school project in knot-theory for a 13 year old?
Thanks
|
My basic question is: "Do longer tautologies take longer to prove?" But obviously this is underdetermined. If you are allowed an inference rule "Tautological Implication" then any tautology has a 1 line proof.
But let's say we're working in natural deduction, is it true that longer tautologies (tautologies with more l... |
I just attended a seminar on the chromatic filtration of the Burnside ring. I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration actually is, and I have had some trouble finding one. Assuming I have the necessary categorical background, i.e. knowledge ... |
I have been asked to expand (1-2t)^(-n/2). So far I have found the log of this function and then differentiated twice in order to find my mean and variance as n and 2n respectively.
How do I write my final answer as a power series in t as far as the term in t^2?
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We have a group $G$ of order $2p$, $p$ an odd prime, which has a nontrivial center. We wish to show the group is cyclic.
This is a simple exercise which can be instantly killed using heavy cannons. However, since I was asked about it by students which barely reached quotients, I wonder what simple ways are to prove thi... |
I have a 3D object which is in its simplest form consisting of an origin in 3D space and a set of vertices that are all local to this origin.
I then transform this 3D origin into 2D camera coordinates using a perspective transform, but I need some way to also transform the local vertices of this object which has the ef... |
Suppose that $f(z)$ is complex analytic on $|z| \leq 1$ and satisfies
$|f(z)| < 1$ for $|z|=1$.
(a) Prove that the equation $f(z)=z$ has exactly one root
(counting multiplicities) in $|z|<1$.
(b) Prove that if $|z_0| \leq 1$, then the sequence $z_n$ defined recursively
by $z_n= f(z_{n-1}) , n=1,2,...$, converges to the... |
This question is motivated by my previous post about sinc function.
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere)
function $f(x)$ on $\mathbb{R}$ such that $$\int_0^\infty f(x) dx=\int_0^\infty f(x)^2 dx$$
or $$\int_{-\infty}^\infty f(x) dx=\int_{-\infty}^\infty f(x)^2 ... |
When one writes $\zeta_n$ which of the n roots of unity is meant here? Does it matter?
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