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A question I recall from Carl Linderholm's "Mathematics Made Difficult", Chapter 3 Exercise 8.
A farmer acquires and algebraically closed field by extending his field finitely.
What can be said about the original field?
What can be said? I immediately see two fields with this property, ${\Bbb R}$, and $\overline{\Bbb... |
I need help with this recursion question:
|
If $E$ is Lebesgue measurable in $\mathbb{R}^n$ and $I=[a,b]$ how do I show that $E\times I$ is measurable in $\mathbb{R}^{n+1}$?
Jonas:
I'm using $\mu^*(E)=\inf \{ \sum \mathrm{Vol}(I_k) \mid E\subseteq \cup I_k\}$ and for every $\epsilon \gt 0$ there exists an open set $G$ containing $E$ such that $\mu^*(G-E)\lt\epsi... |
a sequence on twin primes In the diagram, How do the stripes come from? Can the prime numbers also interfere like light and wave? When zoom in or zoom out the diagram in Mathematica, the stripes are changing.
|
What does general in general topology really refer to? We use the term all the time without thinking about its origin.
|
I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is?
Edit: No, really, this is a mathematical question. Air blows around the room, which constitutes a vector field. Let's say it blows through a square room ... |
I was thinking about Feynman integrals the other day and in particular about discretizing the paths.
Does anyone know the lay of the land about what happens when you do path integrals over, say, a lattice or graph?
|
another sequence on twin primes The maximal prime factor of average of twin prime pair:
n = 100000;
averageList = Select[Prime[Range[n]], PrimeQ[# + 2] &] + 1;
mpfList = FactorInteger[#][[-1, 1]] & /@ averageList;
ListPlot[%]
Why does this diagram look like a series of trajectories?
|
The Problem:
$$\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \frac{16}{7 \cdot 8 \cdot 9 \cdot 10} + \cdots$$
Any smarter way to solve this may be within a minute or two?
I am adding my solution,If you are a student please don't read-on I find this particular probl... |
Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?
Why do I care: For a number of math kids I know, doing algebraic topology is fine until we get to homology, and then it begi... |
How to prove that in every power of $3$, with natural exponent, the tens digit
is an even number?
For example, $3 ^ 5 = 243$ and $4$ is even.
|
$A$ and $B$ are finite groups.
$H \leq A \times B$.
Can we find some $C \leq A$, $D \leq B$ such that $H \cong C \times D$?
In case the statement is not true: is it true under further assumptions about A and B, such as solvability, nilpotency, etc?
Special cases I can prove:
$A$ and $B$ are abelian (following ideas f... |
Here is a problem from Griffith's book Introduction to E&M.
Let $\theta(x)$ be the step function
$$\theta(x) = \begin{cases}
0, & x \le 0, \\
1, & x \gt 0.
\end{cases}
$$
The question is how to prove $\frac{d\theta}{dx} = \delta(x)$.
I think since the function is discontinuous at $x = 0$, there is no definition of... |
Wikipedia says, A hypercube is an n-dimensional analog of a square/cube. What is a hypercylinder then? An n-dimensional analog of a cylinder?
Constant Approximation Algorithm for MST in Resource Constrained Wireless
Sensor Networks gives a definition,
The Isolation Property. Let c> 0 be a constant. Let E be a set of e... |
the Fourier transformation of a scalar function with respect to one variable might be defined as
$\mathcal{F}\left[w\right](\omega )\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}w(t)e^{-\mathrm{i}\omega t}dt$
In physics, this transformation along with its generalization, the Laplace transformation, has a tremendo... |
Is it possible to characterize the distributional derivative as some sort of "best linear approximation" of a distribution (a la the Fréchet/Gâteaux derivatives), if viewed in the appropriate spaces?
For example, let $f \in D'$, $\phi \in D$, and let $\psi_x \in D$ be a mollified ramp function so that $\psi_x(x,y,z,...... |
Does anybody know how to derive
$$\det(\mathbf A)\cdot \det(\mathbf D + \mathbf E \cdot \mathbf A^{-1} \cdot \mathbf B) = \det(\mathbf D)\cdot \det(\mathbf A + \mathbf B \cdot \mathbf D^{-1} \cdot \mathbf E)$$
where $\mathbf A$, $\mathbf B$, $\mathbf C$, $\mathbf D$, $\mathbf E$ are non-singular matrices?
Most likely,... |
These two appear in my module (without any proof):
$$\sum_{r = 1}^{n} C(n-2,r-2) = 2^{n-1}$$
$$\sum_{r = 0}^{n-1} C(n-2,r) = 2^{n-2}$$
For the first one when when $r=1 \Rightarrow C(n-2,1-2) = C(n-2,-1)$ ?!The same thing for $r=n-1$ in the second one,are they valid if yes,please explain how?
|
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own.
The problem is that I feel like I still don't have any idea how to do algebra. I ... |
Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$
For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$.
Why is the sequence $$\langle a_n\rangle$$ not equidistributed in $[0,1]$?
|
How to sum up this series :
$$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$
Any hint that will lead me to the correct solution will be highly appreciated.
EDIT: Here $C_i = ^nC_i $
|
For example:
What conclusions can be drawn about the relations between two objects with the same group of endomorphism?
Can we tell from End(A) if A is Abelian or not?
Does End(A) contain information about the sub-objects of A?
Any information or references to information about this is highly appreciated.
|
Prove that $$\sum_{\substack{x,y \in \mathbb N \\ ax-by \ne 0 }}\frac1{|ax-by|xy}$$ converges, where $a, b \in \mathbb N$
This 2-D problem can be proved by an integral test. I'm looking for some other proofs that can be easily generalized to higher dimensional cases like below,
Prove that $$\sum_{\substack{x_1,x_2, \c... |
I am reading Andrew Baker's notes on p-adic number theory. I have a question regarding a result he is using. In Lemma 2.19 he says in the third line the
limit
$$\lim_{n\to\infty}N(a_n-b)$$
is positive, where $\{a_n\}$ is Cauchy, $N$ is a non-archimedean norm, and $b$ is not the limit of $a_n$ in the $N$-norm. I am no... |
I saw to prove that $a^2+b^2=c^2$ you can produce the algebraic solutions $a = x^2-y^2, b = 2xy, c = x^2+y^2$. To check that works just multiply it out but it produces infinitely many solutions when you set x and y to be integers.
Can you please show me more examples of this, especially if they show how to find the pol... |
Let $L \subseteq M$ be $A$-modules and let $\mathfrak{p}$ be an associated prime of $M/L$ and assume that the annihilator of $L$ is not contained in $\mathfrak{p}$. How would one go about proving that $\mathfrak{p}$ is an associated prime of $M$?
|
If one case is true but the other false, does that mean that the statement is true or false?
There exists a positive integer $n$ such that $n^2-79n+1601$ is composite
Case 1 - $n$ is even - false.
Case 2 - $n$ is odd - true.
|
I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral:
$$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$
converges. I started by splitting up the integral into an integral running from $0$ to $1$ and another one from $1$ to $\infty$. I first tried to figure... |
If a $1$ meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece?
I got this question as a mathematical puzzle from a friend. It looks similar to MathOverflow question If you break a stick at two points chosen uniformly, the probability the th... |
A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation.
What's the name for the following:
$D=(S,D)$ defined as above except that $D$ is closed under the superset operation?
Crossposted from MO.
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The proof that I know of the theorem goes like this:
Any module $M$ is a quotient of a free module $F$ (over any ring).
Any submodule $K$ of a free module $F$ over a PID $R$ is a free module, so in particular the kernel of the above quotient map is free.
For any free submodule $K$ of a free module $F$ over a PID $R$ w... |
Having: $P = N^{CN + 1}$;
How can I simplify this equation to $N = \cdots$?
I tried using logarithms but I'm stucked...
Any ideas?
|
How do I apply the Doleans-Dade exponential formula for the following Lévy stochastic differential equation:
$$dZ_t=Z_t\left(\theta_1(t)dW_t^{(1)} +\theta_2(t)dW_t^{(2)}+\int_\mathbb R \theta(s,x)\mu(ds,st)-v(ds,dt)\right),$$ where $W_t^{(1)}$ and $W_t^{(2)}$ are two independent Brownian motions, $\mu(ds,st)$ is the ju... |
Prove that $\left \{ 0,1 \right \}^{\mathbb{N}}\sim \left \{ 0,1,2,3 \right \}^{\mathbb{N}}$ and find a direct bijection function.
I got the first part by showing that $\left \{ 0,1 \right \}^{\mathbb{N}} \subseteq \left \{ 0,1,2,3 \right \}^{\mathbb{N}} \subseteq {\mathbb{N}}^{\mathbb{N}}$, which implies that $|\left ... |
Currently I am implementing a DTW (Dynamic Time Warping) algorithm for my project. As some will know it has the complexity of O(n^2). Considering a sound file of length 1-h with 44100 sampling. How long will it take to compute it?
(I know we need some other knowledge such as cpu speed, length of the word we are trying... |
Let $\omega$ denote the countably infinite cardinality and $A$ a cardinality that is strictly larger (i.e., an uncountable one). Is it true that $A^\omega$ has strictly larger cardinality than $A$?
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Greetings!
On a test recently I ended up having to solve this for y:
$$ x = \frac{2y}{y + 1} $$
But I kept getting stuck in cirlces...
$$
\begin{aligned}
x(y + 1 ) = 2y \\
xy + x = 2y \\
\frac{xy + x}{2} = y \\
\end{aligned}
$$
That didn't get me anywhere, so then I started over and tried multiplying both si... |
I have:
$$\ln(ax + b)^4$$
After calculating ($u'$/$u$) I get:
$$4/(ax+b)$$ but the solution is $$4a/(ax+b)$$
What am I doing wrong? Thanks!
PS: $a$ and $b$ are $\mathbb R^+ $
|
A lot of "big" names in analysis (and other fields) seem to be doing some form of combinatorics (without any order, some examples are Tim Gowers, Terence Tao and Jean Bourgain).
So, looking a bit around makes me conclude that combinatorics is a huge field. There must be one "kind" which is the most fruitful in analysis... |
I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me.
If we have 1D wave equation:
$$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$
we say that it's always possible (for physics situations...) to decompose the generi... |
which ways to compute this integral:
$$
\int_{-\infty}^{\infty} e^{tx} \frac{1}{\pi(1+x^2)} \mathrm dx
$$
for different cases of t. t can be any value in $\mathbb{R}$. I was stuck at how to get the antiderivative of the integrand. Thank you!
|
Is there any intuition behind surface integrals and their applications? I get that there's some stuff with fluid flow but it doesn't really stick that well, especially when it comes to surface orientation.
Is there any surefire way to figure out the orientation of a surface? And is there any sort of intuition behind th... |
I've been trying to show the convergence or divergence of
$$ \sum_{n=1}^\infty \frac{\sin^n 1}{n} = \frac{\sin 1}{1} + \frac{\sin \sin 1}{2} + \frac{\sin \sin \sin 1}{3} + \ \cdots $$
where the superscript means iteration (not multiplication, so it's not simply less than a geometric series -- I couldn't find the standa... |
I don't have enough memory to simply create diagonal matrix D x D, getting an 'out of memory' error.
Instead of performing M x D x D operations in the first multiplication, I do M x D operations, but still my code takes ages to run.
Can anybody find a more effective way to multiply these matrices in matlab?
D=20000
M=2... |
Let $\mu$ be a finite positive measure on the borel sets in $\mathbb{R}$ and suppose $\mathcal{L}^{1}(\mathbb{R},\mu) \subset \mathcal{L}^{\infty}(\mathbb{R},\mu)$.
Show that there exists $c>0$ such that if A is a Borel set with $\mu(A) > 0$ then $\mu(A) \geq c$.
No idea at all. Can you please help?
|
What interesting properties of convex sets are retained by star-convex sets?
|
If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line.
Now imagine that instead of concentric circles, concentric ellipses emanate from the rocks' entry points. Again the concentric ellipses will osculate a... |
Is there a good online reference that lists known bounds on Ramsey numbers (and is relatively up to date)? The wikipedia page only has numbers for $R_2(n,m)$.
I am specifically interested in known bounds and values for hypergraph Ramsey Numbers, i.e. 2 colorings of k-subsets (or 2 colorings of the edges of a complete k... |
Question: Which computer programs are useful for a pure mathematician to familiarize themselves with?
Less Briefly: I was once told that, now-a-days, any new mathematician worth his beans knows how to TeX up work; when I began my graduate work one of the fourth year students told me he couldn't TeX, I was horrified! S... |
I'm having trouble with this integral
$$\int\arcsin(\sin x)\,\mathrm dx$$
The problem is with the intervals of definition for each function :/ if someone could dumb it down for me.
$\arcsin\colon [-1, 1] \to [-\pi/2, \pi/2]$.
$\sin:\mathbb{R}\to [-1, 1]$.
right?
But what about $\arcsin(\sin x)$ ?
|
I am able to measure my distance to a set of (about 6 or 7) fixed but unknown points from many positions.
The difference in position between measurements is also unknown.
I believe that I should be able to work out the relative position of the fixed points, and therefore where I measured from and the path I took.
I hav... |
Short version: if two spherical caps of the same sphere intersect, how can I determined the coordinates of the two "singular points" of this intersection
Long version: On a unit sphere, centered at the original, we consider two spherical cap. Each cap is defined by:
the (unit) vector indicating the top of the spherica... |
In a And\Or-graph induced by the transition function, each node of $G$ corresponds to a state $q$ belonging to a set $Q$ of the state of the Automaton, for $q$ with $\delta(q,a)=q_1*q_2$, the node is a $*$-node with two successors $q_1$ and $q_2$. For $q=\{true,false\}$, the node $q$ is a sink-node. Hence, if $*$ is $\... |
I need to find out for what $\alpha, \beta$ the following sum converges:
$$\sum_{n=2}^\infty n^\alpha (\log n)^\beta$$
I thought I'd do that with the help of the integral criterion, that is to say I considered (after I had substituted $x = e^u$):
$$\int_2^\infty e^{u(\alpha+1)} \cdot u^\beta \mathrm du$$
Then, I real... |
Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which satisfy $f(n) = g(n) + h(n), ~ \forall n \in \mathbb Z$ ?
Evidently it does while $\mathbb Z$ is replaced with the set of a... |
Is $\mathbf{Grp}$ a concrete category? I thought it is, but then the group of symmetries of a square and the quaternion group are both of the order 8, and they are not isomorphic as groups. But sets of the same cardinality are isomorphic as sets, so a standard forgetful functor $(G, \cdot) \mapsto G$ is not faithful! A... |
Problem:
Alex flip a fair coin three times. what is the probability that she gets two heads given that the first is a head?
My solution is based on the argument that from the given information the sample space can be limited to four elements {$HHH,HHT,HTH,HTT$} Of these {$HHH,HHT,HTH$} satisfies the required event.Thu... |
I'm going through an old final from 2003 on MIT's Opencourseware, and problem 6b is giving me a little trouble.
It asks for which primes $p$ is $34$ a square modulo $p$. I approached it like this:
$$\left(\frac{34}{p}\right)=\left(\frac{2}{p}\right)\left(\frac{17}{p}\right)=(-1)^{(p^2-1)/8}\left(\frac{p}{17}\right).$$
... |
Many years ago I wrote this fractal generator:
http://uberto.fractovia.org/
It was shareware but then I put it as open source.
It's written in Delphi, a language that I don't use anymore. So I'm playing lately with the idea of rewriting on some other lang (scala maybe?).
On the mathematical part I'm not expert at all, ... |
I want to write
$P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$
in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the fundamental theorem of symmetric polynomials using the Newton identities.
First I pick out the 'biggest' monomial accordin... |
Possible Duplicate:
Is 0 a natural number?
There seems to be no consensus, although perhaps one is gathering over the centuries to say yes to the first question and identify $\mathbb{N}$ with $\omega$ : the set of finite Von Neumann ordinal numbers. That leaves $\textbf{N}$ = {$1$, $2$, ...} for the counting numbers... |
$ \qquad \qquad $ The greatest term in $\left(1+x\right)^{2n}$ has the greatest cofficient if $\frac{n}{n+1} \lt x \lt \frac{n+1}{n}$
Can we derive something like this for $n$ in general? Please give me some hints.
|
How can I figure out the classical construction (direct sum, product, pullbacks, and in general direct and inverse limits) in the category made by chain complexes and chain maps (of abelian groups or any abelian stuff)? Because of this category is abelian it must have (co)limits, isn't it?
In particular take
$$
\begin... |
I have a question about whether the following two sets on the two sides are the same:
$$
\lim_{a \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\left\{ x \in S \, \Bigm| \, \lim_{n \rightarrow \infty} f_n(x) =\infty \right\}\quad ?
$$
where $\{f_n\}$ is a sequence of r... |
Does there exist a good reference on the algorithms used by calculators, especially on the trigonometric and transcendental functions?
I would still like to know how Casio generates its random numbers. I still wonder if they are any good.
|
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.
I solved this by using a matrix called $S$:
$$\left(\begin{array}{cc}
a& b\\
c& d
\end{array}\right)$$
and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution to $A... |
The current price of a stock can be modeled by $P_0 = \frac{D_{1}}{r-g}$ where $D_1$ is the expected dividend, $r$ is the rate of return, and $g$ is the expected growth rate in the perpetuity. If $r<g$, then
$\displaystyle P_0 = \sum_{t=1}^{N} \frac{D_{0}(1+g)^{t}}{(1+r)^{t}} + \frac{P_N}{(1+r)^{N}}$.
Is there anyway t... |
Fermat's Little Theorem, in Fermat-Euler form, states that: if $\gcd(a,m)=1$, then $a^{\varphi(m)} \equiv 1 \pmod{m}$.
Now, I've been asked to prove it via modular arithmetic. In order to do this I understand that I'm to use two lemmas:
$\varphi(p^n) = p^{n} - p^{n-1}$. This I can prove by working out that there are $... |
$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process.
Find $dZ$ ?
|
We know that $H^p(S^2 \vee S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \vee S^4)$ I want to cup it with itself and get 0. My idea is to use the generator from $H^2(S^2)$(which obviously is zero when squared). H... |
Define $(a,b) < (a',b')$ if $\max(a,b) < \max(a',b')$ or $\max(a,b) = \max(a',b')$ and $b < b'$ or $\max(a,b) = \max(a',b')$ and $b = b'$ and $a < a'$.
Now I want to prove that the order-type of $(\{(b,c) : \max(b,c) = a\}, <)$ is equal to $a + a + 1$, does someone have a hint how to do this? I can't find the bijection... |
Here is what I need to prove:
Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$.
I know that:
1. for any graph $\sum_{v \in V}\ deg(v) = 2e$;
2. for any tree $e=v-1$.
From 1 and 2 it follows that for any tree $\sum_{v \in... |
Can you tell me why the following is true?
A ring $R$ is local if and only if every principal left $R$-module is indecomposable.
(Edit by KennyTM: The above is OP's original question. The latest, completely changed question follows:)
what is the relation between regular ring and this property:
$ab=1$ then $ba=1$ wh... |
Let $K$ be a field and $f(x)$ be an irreducible polynomial over $K$. Suppose, $f(x)$ has degree at least $2$. Is it possible that if $a,b$ are two roots of $f(x)$ with $a\neq b$, then $K(a)=K(b)$. Note I need equality, not isomorphism.
|
It is possible to find the shortest route thanks to algorithms like A*, bread first search, depth first search, etc..
Is there any known algorithm to find how many routes are available if there are more than 1 optimum solution (for example routes with same costs) or any solution at all.
I am working on some graph algor... |
Say I have two algorithms that I don't know how they work but I know what they are meant to achieve. I tried algorithm A once and it succeeded. And tried algorithm B 100 times and succeeded 99 times. So A succeeded 100% times and B succeeded 99% times, but my common sense tells me B is still more "reliable" than A. How... |
Let $R$ be a commutative ring with unity and let $B(R)$ be the set of all idempotent elements in $R$.
Show for $b\in B(R)$, the $R$-modules $R$ and $Rb \times R(1-b)$ are isomorphic to one another.
|
Let $R$ be a commutative ring with identity and $I,J,K$ be ideals of $R$. If $I\supseteq J$ or $I\supseteq K$, we have the following modular law
$$ I\cap (J+K)=I\cap J + I\cap K$$
I was wondering if there are situations in which the modular law holds in which the hypothesis that $I$ contains at least one of $J,K$ is re... |
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does anyone know of any?
|
The following formula for $\pi$ was discovered by Ramanujan:
$$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$
Does anyone know how it works, or what the motivation for it is?
|
This sounds like a simple problem, but I can't get it done. Given the general equation
$ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$
what are the requirements on the coefficients so that this equation represents an ellipsoid? All coefficients are real.
Some more context. To make the representation uni... |
Whilst answering my homework, I have noticed that a question has said that I need to consider the subgroups of $A=\mathbb{Z}_{135}$, including $W=\langle 18,81\rangle$. I was under the impression that for a subset to be a subgroup $a+b$ in the subgroup had to be in the subgroup as well. $18+81 = 99\mod{135}$ which isn'... |
I am working through a review problem asking to find the inverse of
$4\bmod 9 $.
Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step:
gcd(4,9)
9 = 2 * 4 + 1
1 = 9 - 2 * 4
This would sugges... |
I want to create a function $f : [0,255]^3 \rightarrow [0,255]^3$ over the integers such that if you pass in three RGB color values, a triple with a darker or lighter color will be returned, and I want the values to increase in some sort of proportion to how dark or light they are already; lightness and darkness being ... |
One of the rules of logarithms that I've been taught is:
$\log_b(x)$ is equivalent to $\dfrac{\log_k(x)}{\log_k(b)}$.
Recently I've also seen another rule that says:
$\log_b(x)$ is equivalent to $\log_b(k)\log_k(x)$.
Are these equivalent (perhaps via some refactoring) ?
|
Can anyone suggest at least verbally how it is that a torus is constructed from two cells? (Total beginner to topology, been away from math as a whole for awhile...) Thanks for any insights!
|
Could you please help me in this problem?
I have 3 independent variables, $(T,H,t)$, as inputs and
one output $P$ ( I have all data for these inputs and the output, done
experimentally measured every hour during one year).
I want to find a formula of this form:
$$P = f ( T , H , t )$$
where $t$ is the time in hours an... |
Euler's rotation theorem (very informally stated) says that an rotation of a sphere in three dimensional space always has two fixed points. I believe this claim generalizes to (n-1)-hyperspheres in n-dimensional space for odd n, but I am unable to patch the final gap in the proof. My strategy so far is as follows; any ... |
It would be great if someone could explain me the following. If I understand the lecture correctly, given a system of, say, following equations:
$x+2y-2z=1$
$2x+4y+z=3$
$4x+8y+4z=10$
We can say that the they do not describe the whole 3D space because the vectors formed by the first and second columns ( $\begin{pmatr... |
I am looking at the first two examples in Paul Garretts notes on p-adic number theory.
The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower convex hull of $(0,1),(1,0),(2,0.5),(3,0),(4,0),(5,1)$. So I thought the Newton polygon should just be a horizontal line... |
I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance.
Is there a mathematical way to plot all of these on a 2D space, turning the abstract distance into a 2D euclidian distance, and preserving all of their distances?
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Can it be done?
For an arbitrary quadratic field $Q[\sqrt{d}]$, it's easy to show the equations are simply $ f_x = -\sqrt{d} f_y $, where $ f : Q[\sqrt{d}] \to Q[\sqrt{d}]$. I'm working on the case of $Q[\theta]$, when $\theta$ is a root of $\theta^3 - a\theta - b$, but I'm not sure if it's even possible. Has there bee... |
I need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way.
Found out that what I need is a spline, however cannot find one that behaves how I need. $n$-degree Bezier curves and B-splines don't pass through the ... |
This image show a histogram (200 bins) of accumulated distances from a radar distance meter (very noisy).
The peak around 7 meters is an object. At thought this looked kind of like a normal distribution, at least if you ignore values <4m (which for this application is reasonable).
What I am trying to do is to filter ou... |
I randomly pick a natural number n. Assuming that I would have picked each number with the same probability, what was the probability for me to pick n before I did it?
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In high school, when we talked about "combinatorics," we solely meant "mathematics of choice." For instance:
There are 10 people who want to sit around a table. In how many ways is this possible?
We have 50 balls and 20 boxes. In how many ways can we distribute balls into the boxes?
There are 10 apples and 5 oranges. ... |
I am reading a text on maths applied to naturally occurring networks, eg. social networks. The section I am on is "Walks" networks. The text says:
Walks: which allow both nodes and
edges to be used more than once. For
example, in Figure 1,
Mary-Bob-Sue-Ramona-Bob-Mary-Joe is a
walk, but not a trail or a path.... |
I know the following:
my own position
my own facing (the angle im turned)
my targets position
What i would like some help with is how i calculate the shortes way to turn and the angle to turn. If i my position was (4,4) and i faced 90 degrees and my target was to face (1,1) where my target is, i know by drawing it on... |
I'm a lazy type of person. I love experimenting with things before learning the theory. I like to practice, and learn from my mistakes.
In 3-4 months I have a basic University exam on Calculus, and I'd like to learn it in a fun way. I have found these videos to be quite entertaining:
http://www.calculus-help.com/tutori... |
Can anybody give me a hint how to approach the following problem, please?
I actually am having a hard time stating the problem. I think an example would help you understand what the problem is.
Suppose I have 5 numbers each can be max 3 digits. I sum them together. What's the probability of the sum having the last 3 di... |
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