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I am presently a learner of Hyperbolic Geometry and am using J. W. Anderson's book $Hyperbolic$ $Geometry$. Now the author presents a sketch proof of why every circle preserving homeomorphism in $\overline{\mathbb{C}}$ is an element of the general Möbius group, which is what I am struggling to understand.
First, a bri... |
This are notes from a NN. This is one of many steps to solve differential equations with separate variables. In this step NN does following:
$$g_1(y) = \sum_{n=1}^\infty A_n \sin\left(\frac{n\cdot\pi}{b}y\right)$$
And then, to obtain $A_n$ he does:
$$A_n = \frac{2}{b}\int_0^b g_1(y)\cdot \sin\left(\frac{n\pi}{b}y\right... |
Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say that $|f_k|^a \to |f|^a$ in $L^p$ for some $p$ (depending on $a$ and also possibly depending on $n$)?
Showing the statemen... |
Find a linear transformation $T\colon \mathbb{R}^2 \to P_2(\mathbb{R})$ such that $T(1,2) = 3 + 4x^2$ and $T(3,5) = 1 - 2x + 3x^2$.
I was thinking of approaching the problem the following way:
$$
T \left( \begin{matrix} 1 \\ 2 \end{matrix} \right) = \left( \begin{matrix} 3 \\ 0 \\ 4 \end{matrix} \right)
$$
$$
T \l... |
We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$
How do we create a recurrent decimal with the property of repeating:
$0.\overline{0123456789}$
a) Is there a method to construct such a number?
b) Is there a solution?
c) Is the solution in $\mathbb{Q}$?
According with this Wikipedia page: http://en.wikipedia.or... |
My question is regarding step 3. Why is the partial of $z$ with respect to the partial of $x$ equal to $\dfrac{df}{du}$ rather than the partial of f with respect to the partial of $u$. I don't think I'm understand the basics right. What is the difference in this case, and what exactly are we looking for here. I need in... |
I am building a ranking system. To get a rank of an item, I have considered using ratio and method.
Let's say I have an item with 10 wins and 10 losses.
Ratio: 10/20 = 0.5
Point: 10 - 10 = 0
What other options do I have?
|
Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?
Can someone point me to a proof, or explain if it's a simple answer?
What I'm looking for is the point where it becomes understood that trigonometric functions and pi can be expressed as series. A lot of the information I find when looking for that seems to point back to ... |
Let $G=\{a_2, a_2, a_3...a_n\}$ be a finite abelian group of order $n$. Let $x=a_1 a_2 a_3...a_n$. Prove that $x^2=e$.
My solution: Well since the order of the group is finite, we know the order of the elements of the group are also finite. So there exists a certain $k \in \mathbb{Z}$ such that $a^k = e$. Since the ... |
Possible Duplicate:
Does .99999… = 1?
After reading all the kind answers for this previous question question of mine,
I wonder... How do we get a fraction whose decimal expansion is the simple $0.\overline{9}$?
I don't mean to look like kidding or joking (of course, one can teach math with fun so it becomes more int... |
An ideal $s$ of an Euclidean ring $R$ is maximal if and only if $s$ is generated by some prime element of $R$.
|
I have $a_k=\frac1{(k+1)^\alpha}$ and $c_k=\frac1{(k+1)^\lambda}$, where $0<\alpha<1$ and $0<\lambda<1$, and we have a infinite sequence $x_k$ with the following evolution equation.
$$
x_{k+1}=\left(1-a_{k+1}\right)x_{k}+a_{k+1}c_{k+1}^{2}
$$
I have proven that $x_k$ is bounded and obviously positive. How can I know ... |
Let $v_1, \dots, v_m$ be a linearly dependent list of vectors.
If $v_1 \ne 0$, then there is some $v_j$ in the span of $v_1, \dots, v_{j-1}$
If we let j be the smallest integer with this property, and apply the gram-schmidt procedure to produce an orthonormal list $(e_1, \dots, e_{j-1})$ then $v_j$ is in the span of $(... |
How does one prove $\log{x} < x^n$ for all $n > 0$.
Formally, I would like to show that for all $n > 0$, there exists a $Y$ after which for all $x>Y$, $\log{x} < x^n$.
From there, is it then possible to show $\log{(x)} \cdot \log{(\log{(x)})} < x^n$?
(Unrelated, and just interesting:) In general, is the product $\log(x... |
Can we embedd $GL(2n,q)$ into $GL(n,q^2)$ for $n\in \mathbb{N}$ and $q=p^m$, $p$ a prime? If yes, how?
|
I need to sum the series
$$I + A + A^2 + \ldots$$
for the matrix
$$A = \left(\begin{array}{rr}
0 & \epsilon \\
-\epsilon & 0
\end{array}\right)$$
and $\epsilon$ small. The goal is to invert the matrix $I - A$. The text says to use a geometric series but I had a hard time finding it. I'm studying on my own so I can'... |
Imagine that we have a set of points with defined x coordinate and mass for each of them: {($x_i$, $m_i$)}.
Canonical center of mass is a point $x_0$ that divides points to the left ones and the right ones in such way that
$\sum_{left} m_i |x_i-x_0| = \sum_{right} m_i |x_i-x_0|$
Now I need to find such point $x^*$ (shi... |
The power series $f(z):= \sum_{\nu \geq 0}a_{\nu}z^{\nu}$ converges in a disc $B$ centered at $0$. For every $z\in B$ such that $2z$ is also in $B$, $f$ satisfies $f(2z) = (f(z))^{2}$. Show that if $f(0)\neq 0$, then $f(z)=\mathrm{exp} bz$, with $b:=f'(0)=a_{1}.$
I'm completely lost with this problem. Could someone giv... |
I am trying to formulate in a document some work done. However I have a problem defining the following. If I have a problem defined as the tuple $\langle X,Y \rangle$ such that $(x,(y_{11},y_{12},y_{13},\ldots,y_{1N}))$. How can I define this relation in cartesian products?
For example if X = {$x_{1},x_{2},x_{3}$} and ... |
I'm computing cosine similarities between 2 vectors.
These vectors are information retrieval query and document representations respectively.
They have been computed using tf/idf weights.
Since my documents have different length, tf/idf weights are theoretically unbounded.
The question is: is cosine similarity still a ... |
I haven't been able to find any information on this, but I think that if someone knows it, it's someone here. I need it for some theoretical knowledge about lambda calculus and compiler optimizations.
I'm sorry if this is the wrong stackexchange site, I have been told to go here by some people at stackoverflow.com
|
If $\lambda$ is the largest eigenvalue of a real symmetric $n \times n$ matrix $H$, how can I show that: $$\forall v \in \mathbb{R^n}, ||v||=1 \implies v^tHv\leq \lambda$$
Thank you.
|
I am writing a tech document and I would like to formulate "significantly" close/far. For example a certain suboptimal solution $x^{\dagger}$ is "significantly close" to the optimal solution $x^{*}$.
I was tempted to write something like $x^{*} - x^{\dagger} \simeq 0 $ however this is not correct because the "closeness... |
Good morning,
I would like your help with proving that
$$\lim_{n \to \infty} \frac{e^n}{P(n)} = \infty,$$
where $P(n)$ is a polynomial of degree at least $1$.
Thank you very much.
|
For an image denoising problem, the author has a functional $E$ defined
$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$
which he wants to minimize. $F$ is defined as
$$F = \|\nabla u \|^2 = u_x^2 + u_y^2$$
Then, the E-L equations are derived:
$$\frac{\partial E}{\partial u} = \frac{\partial F}{\partial u} -
\frac{\mathrm ... |
For $m \in L^\infty$, we can define the multiplier operator $T_m \in L(L^2,L^2)$ implicitly by
$\mathcal F (T_m f)(\xi) = m(\xi) \cdot (\mathcal F T_m)(\xi)$
where $\mathcal F$ is the Fourier transform. It is obvious from the defintion that $T_m$ commutes with translations.
How can you show the converse, i.e. every tra... |
Apologies for the (maybe obvious) question. It's come up as part of a proof of the triangle inequality for a metric function I'd like to work with.
For real number $r \geq 1$ and positive integers a, b, c:
Is it generally true that $r^a \leq r^b+r^c$? Given that $a \leq b + c$.
Happy to accept a pointer in the right di... |
I recently graduated with a computer engineering degree and am now going to study neuroscience/neuro-engineering at the graduate level. I need to beef up on my linear algebra, differential equations, numerical analysis, and statistical analysis (bayes what?!). As such, I would like to ask what are some excellent books ... |
Given $a>0$ and $ac-b^2>0$ show
$cy^2+a[(x+\frac{by}{a})^2-(\frac{by}{a})^2] > 0$
I'm completely confused about this, I've tried a few approaches. I end up getting stuck saying that I know $cy^2>0$ using the 2nd of the given inequalities, but I can't show the $a[...]$ part is >0 since all I know is x and y are non-zero... |
I know approx what point i want from below diagram by looking at it(red circle), and im pretty sure there is some smart formula that does what im looking for, but how does the formula look like?
What kinds of math concept should i google on?
Each black dot represent a x/y coordinate, how can i find the range of dots ... |
Suppose $A$, $B$ and $C$ are commutative noetherian rings. Suppose we are given maps $f:A\to B$ and $g:B \to C$.
Suppose further that $f$ is flat and that $g \circ f$ is also flat. Does it follows that $g$ is flat?
Thank you!
|
Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion with respect to the norm.
Is $\overline F$ necessarily algebraically closed?
Thanks.
|
Does
have at most 2 solutions for primes $p\geq 5$?
Mathematica confirms this for the first 10K primes:
Max[
Table[
Length[
Reduce[{
Mod[2*x^2 + 2*x + 3, Prime[i]] == 0, 0 <= x <= Prime[i]}, x,Integers]],
{i,3,10000}]
]
Output: 2
but I wasn't sure if it was true in general, and if it could be p... |
I'm reading the introductory book on L-theory, Algebraic and geometric surgery by Andrew Ranicki, available at 1.
Here is the statement in the page 254 Example 11.18.
If $G$ is a group without $2$-torsion, then (there is) a decomposition
$ G=\lbrace 1\rbrace\cup S\cup S^{-1}$ which determines a $\mathbb{Z}[\mathbb{Z}... |
I saw this in Basic Topology by M.A.Armstrong. It gives three descriptions of real n-dimensional projective space $P^n$. Two of them are:
(a) Begin with the unit sphere $S^n$ in $E^{n+1}$ and identify its antipodal points.
(c) Begin with the unit ball $B^n$ and identify antipodal points of its boundary sphere.
I find ... |
In discussion about the question Is there a way to represent the interior of a circle with a curve?, it was mentioned that such a curve cannot be one-to-one (because $[0,1]$ is not homeomorphic to $[0,1]^2$). I'm curious about in what way the Peano curve is not one-to-one.
The construction of the Peano curve is a recur... |
It seems obvious that in an arbitrary normed space $(X, \|\cdot\|)$ a mapping $T$ defined as
$$ T(x) = \begin{cases}
\frac{x}{\|x\|} & t \neq 0 \\\\
0 & t = 0
\end{cases} $$
is continuous everywhere but 0. However, I'm having difficulty formulating a proof. It's quite simple to see that $\|T(x) - T(y)\| ... |
Suppose you have a matrix $A$. Is there a "standard"/mathematical elegant way to denote all members of the matrix as a set?
For example: suppose there is a matrix $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] $ then I would like to know how to define a function $set( \cdot)$ such that we have:
$$set(... |
So, I know we can get a bound on how long it will take to find a large prime. For example, using the fact that between $N$ and $2N$ there must be a prime. And the fact that all numbers between are prime iff they do not have a divisor less than $\sqrt{2N}$. So, this gives us one bound on the amount of time needed. (Assu... |
I have a function $z = B \sin x \ \sin y+\cos x \ \cos y$. Where $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. I need to find the length of the curve that describes a level set for any value of $B$. That is, if I set $z = A$ (where $A$ is some scalar constant), what is the length of the level curve for any value of the... |
Let us consider a needle of length 1, with one (fixed) end at the coordinate $(x,y,z)=(0,0,0)$. If we make the other end (wich is free) of the needle move, it will describe the unit sphere. We assume the distribution of the free end of the needle is uniform on the sphere, which means it is given by:
$$\cos(\phi) d\phi ... |
i've constructed a couple of equalizers in the category of sets/graphs. If anyone could tell me if my equalizers turned out right, that would be very much appreciated.
example of an equalizer in the category of sets
given:
O1 = {a1, a2, a3, a4} --set 1
O2 = {b1, b2, b3, b4} --set 2
f12 = {(a1->b1), (a2->b2), (a... |
Playing 4 on 4 basketball with up to 12 players (three teams) is easy to manage. But when player thirteen shows up, is there a way to work everyone in such that no one sits two games in a row?
|
Can we prove that every entire one-to-one function is linear?
|
I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an epsilon neighborhood for it to see which other line segments are close to it.
All of the research I have done returns h... |
Let $f\geq 0$ be a measurable function which is finite almost everywhere.
For each $k\in\mathbb{Z}$, define, $E_k=\lbrace x|f(x)>2^k\rbrace, F_k=\lbrace x|2^k\leq f(x)<2^{k+1}\rbrace$.
Is it true that $\sum_{k=-\infty}^{\infty}2^km(E_k)<\infty$ if and only if $\sum_{k=-\infty}^{\infty}2^km(F_k)<\infty$?
|
It is well known how to define standard product topology on a product space $\prod_{i \in I} X_i$.
Assume now that $(X,\lVert \, \cdot \, \rVert_{X})$ and $(Y,\lVert \, \cdot \, \rVert_{Y})$ are normed spaces and that the space $X \times Y$ is also equipped with a norm $\lVert \, \cdot \, \rVert_{X \times Y}$.
Is it tr... |
I'm trying to find an example for $n$ events such that each $k$ ($k < n$) are independent, but the $n$ events together are dependent.
Two events are independent if the probability for both of them to happen equals to the product of the probability of each one separately.
|
$\forall I, A:Set. I\subseteq \bigcup A\to \exists f:I\to A. \forall i\in I. i\in f(i)$
Does this theorem require the Axiom of Choice? To prove, I need to find $\forall i\in I$ a witness of that $i\in \bigcup A$. Any witness $B$ is such that $i\in B\land B\in A$. There may be more than one witness, but I must return on... |
I am trying to find the local min/max and saddle points of the function: $f(x,y) = 9 - 2x + 4y - x^{2} - 4y^{2}$
This is what I have so far:
$f_{x}(x,y) = -2 - 2x = -2 (x + 1) \Rightarrow x + 1 = 0 \Rightarrow x = -1$
$f_{y}(x,y) = 4 - 8y = -4 (2y - 1) \Rightarrow 2y - 1 = 0 \Rightarrow y = 0.5$
Any ideas on how to pr... |
Today, we had a math class, where we had to show, that $a_{100} > 14$ for
$$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$
Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is?
|
I am unsure how to even ask this.
I have a circle and I am at point 0. I add -90, which would bring me at 270. What would this look like as an equation? where -90 could be any negative number from -1 to - 359 and the result would be the positive point.
Sorry, I have not thought about this kind of math since middle scho... |
Suppose $f$ is an integrable $2\pi$-periodic function.
Show that if $\lim\limits_{h\to 0} f(x_0+h)+f(x_0-h)=\infty$ then $\sigma_n f(x_0)=\infty$, where $\sigma_n f$ represents the Cesàro mean, the average of the first $n+1$ Fourier series
My thoughts: if $\lim\limits_{h\to 0} f(x_0+h)+f(x_0-h)=\infty$ then does it imp... |
Let $A$ be an $N$ by $N$ Hermitian matrix with elements $a_{ij}$. What will be the bound on the elements $b_{ij}$ where $B=A^{-1}$?
If $A$ is a diagonal matrix, solution is trivial. Also for tri-diagonal matrix, bounds exists.
|
Why are matrix and scalar multiplication denoted the same way and treated as the same operation in standard mathematical notation? This is always a source of confusion for me because they have completely different properties (specifically commutativity). Multiplying a 1x1 matrix by an NxN matrix isn't even generally ... |
It's probably not at all hard—but at least right now it's not obvious to me—how to determine the asymptotic behavior of
$\sum_{k=1}^n \binom{n}{k} \frac{1}{k}$
(link to OEIS).
|
N people are coming to a conference.
Every person comes or does not with an equal probability, independently of the others.
Among the participants, randomly selected first speaker.
Suppose that N = 4, what is the probability that a person 4 will speak first?
I tried to divide the problem into cases,when in each case... |
Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$.
There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients in $\mathbb R$, which is isomorphic to the De Rham cohomology. As a previous question already revealed, the images of this... |
Can someone help me with proof of the next statement?
Suppose that the group $G$ acts transitively on the set $A$, and let $H$ be the stabiliser of $a\in A$.Then $G$ acts primitively on $A$ if and only if $H$ is a maximal subgroup of $G$.
Proof - Assume that $H$ is not maximal, and choose a subgroup $K$ with $H < K < ... |
Let $G$ be a group and $T$ the set of elements of finite order in $G$.
If $T$ is a subgroup of $G$, then $G/T$ is a torsion-free group.
Suppose $G$ is a compact Hausdorff topological group.
Is it true that $G/cl(T)$ is torsion-free?
(where $cl(T)$ is the topological closure of $T$ in $G$).
|
If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent:
(i)
for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) −\int_0^t Af(X_s)ds$$ is a martingale,
(ii)
for all $f\in C^{1,2}([0,\infty)\times\mathbb{R})$ the process
$$f(t,X_t)−f... |
Let $S=\{r\in \mathbb{Q}\mid r\leq\sqrt{3}\}$.
Now prove that $S$ is bounded above in $\mathbb{Q}$ but it does not have a supremum in $\mathbb{Q}$.
The following is the proof I came up with, but I do not feel confident with it. Please let me know what is incorrect or could be written better.
Proof.
Let $m=\sqrt{3}$.
... |
In a paper, it was claimed that $\lim_{x \to \infty} (1-\frac{f(x)}{x})^x \sim e^{-f(x)}$ when $\frac{(f(x))^2}{x}$ is $o(1)$.
I proved the claim in the following way; however, I'm seeking a simpler proof.
Define $g(x) = \frac{(1-f(x)/x)^x}{x}$ and $h(x) = \frac{e^{-x}}{x}$. To prove the theorem, we must show that $\l... |
Determine whether $a_n =
(1+\frac{2}{n})^{n}$ converges or
diverges. If it converges, find the
limit.
So I tried to say that $a_n = (1+\frac{2}{n})^{n} \Rightarrow \ln(a_n) = n\ln(1+\frac{2}{n})$.
Unfortunately I don't know what the next step is since I think that $n \rightarrow \infty$ as $n \rightarrow \infty... |
I'm a TA for an honors calculus course and I've come upon a homework problem (for which I'm supposed to write solutions!) that has me stumped. This is problem 12, in Chapter 9.13 in volume 2 of Apostol's Calculus book.
Find the critical values of $f(x,y) = (5x+7y-25)e^{-(x^2 + xy + y^2)}$ and classify them as max/min... |
A teaching question. What is or might be simplest form for rational exponents? My search turns up little for the concept other than a translation of the simplest form for n-th root radicals. But that leads to ugly things such as...
$\large{5^\frac{7}{3}\rightarrow 5^2\centerdot 5^\frac{1}{3}}$
$\large{x^{-\frac{2}{3... |
The question:
Is the multiplication result of two unit-length quaternions "perpendicular" to both operands? "Perpendicular" means that dot product between either operand and multiplication result will be zero (as with vector3 cross-product). Operands (say q1 and q2) are not equal to each other.
Reason for asking:
I'm t... |
Let $S$ be a subset of $R$ (the set of Real numbers). Prove/disprove: $\text{cl }S = \text{int }S \cup \text{bd }S$.
The following is my proof, please help me critique it and fix any incorrect assumptions.
Proof:
First, prove $\text{cl }S$ is a subset of $\text{int }S \cup \text{bd }S$ by cases. Case 1: Let $x$ belong... |
The quadratic forms with discriminant -23 up to change of variables are:
A(x,y): $x^2 + xy + 6 y^2$
B(x,y): $2 x^2 - xy + 3 y^2$
C(x,y): $2 x^2 + xy + 3 y^2$
Viewed as number fields it's relatively easy then compute:
A(x,y)A(a,b): $A(xa - 6yb,ya + (x - y)b)$.
BB: $B(xa - (3/2)yb,ya + (x+(1/2)y)b)$
CC: $C(xa - (3/2)y... |
Wikipedia (and my teacher) state Riesz's lemma as follows:
Let $X$ be a normed linear space and $Y$ be a subspace in $X$. If there exists $0 < r < 1$ such that for every $x\in X$ with $||x|| =1$ , one has $d(x, Y) < r$, then $Y$ is dense in $X$.
Wikipedia then goes on to say
In other words, for every proper closed ... |
Let $x_n$, with $n=1,2,\ldots$, be uniformly distributed random variables in $(0,1)$
What is the expected value and probability distribution of the sum
$$\sum_{n=1}^\infty x_n^{n^n}$$
|
let $f\in L^2$ and 2pi periodic
let $f_x(y)=f(x-y)$ and $V = span\{f_x: x\in [0, 2\pi)\}$
show V is dense in $L^2([0,2\pi))$ iff all of the fourier coefficients of f are non zero
i would be grateful for any tips
|
I'm trying to design an algorithm to generate something that I don't know how exactly to call! Ok, I'm not a mathematician, I'm studying computer science and thought this would be a great moment to use some recursive algorithms.
But I'm not asking how to do it, what I really want to know is what's the name of this so I... |
Is $\mathbb{Q}(\sqrt 2,\sqrt 3,\sqrt 5)$ a simple extension?
If yes, what is an explicit generator of this extension?
|
I have seen authors use $\star$, $\ast$, $\cdot$ and $\odot$ to represent arbitrary binary operations on sets. I'm wondering, what is the standard way to read or pronounce something like $a \star b$? How would you read $a \odot b$?
|
Consider two sets $A$ and $B$ with positive measure such that the union is $[0,1]$. Is it true that there is a one-to-one correspondence between $A$ and $B$?
|
Let $x_i$ be uniformly distributed random variables in the interval $[0,a]$, with $a>0$.
Let f(x) be equal to 1 if x=0, and 0 otherwise.
Let $$S(a)=\sum_{n=1}^\infty f(x_n)$$
What is S(1)?
What is $\lim_{a->0} S(a)$ from positive side?
|
Is there any name of a function who is continuous and its inverse function is also continuous?
|
Let $\zeta: \mathbb{R^m} \times \mathbb{R^n} \mapsto \mathbb{R}$ be a smooth function and define $\phi(x) = \inf_{y \in C} \zeta(x,y)$ where $C \subset \mathbb{R^n}$ is compact. Suppose that for every $x \in X \subset \mathbb{R^m}$ there is a unique $y(x) \in C$ such that $\phi(x) = \zeta(x,y(x))$.
Now, according to a ... |
I'm looking at Exercise 2.6b (p.58), which is to prove that for smooth manifolds $M$ and $N$ and a continuous map $F:M\rightarrow N$, we have that $F$ is smooth iff $F^*(C^\infty(N))\subseteq C^\infty(M)$, where $F^*(f)=f\circ F$ is the pullback by $F$. The forwards direction is obvious, but I'm stuck on the backwards ... |
I'm trying to show the following:
Let $I,J$ be ideals of a commutative ring (with 1) $A$ such that $I_{P}=J_{P}$ for every prime ideal P of R. Here $I_{P}$ means the localization of $I$ in $P$. Then $I=J$.
Well I was thinking in using the following result: let $M$ be an $A$-module. If $M_{P}=0$ for every prime ideal th... |
After a fair bit of effort, I managed to prove that
$$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$
where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with the usual restriction of $\Re \alpha > -1$. My proof was rather complicated (involving lots of integration by parts); I'm... |
Can spinors be seen as a generalization of tensors,but with complex numbers?
|
Let $K$ be any subfield of $\mathbb{C}$ and let $m(t)$ be a quadratic polynomial over $K$. show that all zeros of $m(t)$ lie in an extension $K(\alpha)$ of $K$ where $\alpha^2=k\in K$. Thus allowing square roots $\sqrt k$ enables us to solve all quadratic equations over $K$.
|
I'm reading An Introduction to Ordinary Differential Equations by Agarwal and O'Regan. On page 28, I have the expression
$$y\left(x\right)=c\exp\left(-\int^x{p\left(t\right)\, dt} \right)$$
which is equation 5.4.
My problem is the missing lower limit of integration. I don't understand what the notation means. The co... |
If $R$ is a ring with 1 which satisfies $R^r=R^s$ for some $r\neq s$.
Are there any explicit calculation of $K_0(R)$ for such $R$?
I want to know such examples because I think that such $R$ may not have $\mathbb{Z}$-summands.
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Suppose a vector space V is defined on a field F. Does this at all imply that V is also defined on all fields, or does it only dictate that V is defined on F (and could also work with other fields if proven)?
I realize it's sort of silly to assume anything in math, but my confusion comes from examples of vector spaces ... |
I am trying to work out a summation for packet delays which is very similar to the summation for estimating RTT, which is an exponentially weighted moving average. I have modified the estimating RTT summation below (I think I did it correctly) but it does not seem to fit correctly with the part I know is right (where ... |
I'm looking for an information-theoretic proof of Gödel's Theorem that goes something like this, without any reference to diagonalization:
Every axiom system in the scope of Gödel's Theorem has a finite number of bits.
It requires an infinite number of bits to specify the all the truths of number theory.
By the Soundn... |
Why does convergence in higher order mean implies convergence in lower order mean?
Reference: http://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_rth-order_mean
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Let $M$ be an $A$-module and let $\mathfrak{a}$ be an ideal of $A$. Suppose that $\mathfrak{m} \cdot M =0$ for every maximal ideal $\mathfrak{m}$ of $A$ such that $\mathfrak{a} \subseteq \mathfrak{m}$. Is $M$ the trivial module?
Here $\mathfrak{m} \cdot M = \{\sum_{finite} a_{i} x_{i}: a_{i} \in \mathfrak{m}, x_{i} \in... |
I'm interested in creating a mathematical model of an underground (tube) system. The overall aim is being able to efficiently calculate travel times between stations.
Such a system has some interesting properties, for example:
if station $B$ belongs to the quickest route between $A$ and $C$, then the sub-routes $AB$ a... |
Since the antiderivative of $\frac{1}{x}$ is $\ln(|x|)$, the surface under the graph of $\frac{1}{x}$ with $x>1$ is $\infty$.
However, the antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$, so the surface under the graph of $\frac{1}{x^2}$ with $x>1$ is $1$.
Although I understand how one can calculate this, I can't r... |
Let $t \in \mathbb{N}$ and consider the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f_t (m)= 2 \uparrow^{m} t$,
where "$\uparrow$" is Knuth's up-arrow notation (which can be recursively defined - as I can't seem to come up quickly with an online reference for this
recursive definition - by $x\uparrow... |
I am trying to calculate the following
$K^{-1}DK^{-1} * p$
Where $K$ is symmetric positive definite and $D$ is positive diagonal and $p$ is a vector. The problem is that p is very, very small so its rounded to zero by Matlab so I want to take the log of of the calculations of $p$ so that I can actually calculate the m... |
Let $f: \mathbb{R} \to \mathbb{R}$. Prove that if $f$ is bounded such that for all $x, y \in \mathbb{R}$ $x\neq y$ implies $|f(x) - f(y)| \lt |x -y|$ and for all $x \in \mathbb{R}$, $f$ is differentiable at $x$ with $|f'(x)| \lt 1$ then $f(z) = z$ for exactly one number $z \in \mathbb{R}$.
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From the theory of numbers we have the
Proposition:
If $\mathfrak{a}$ and $\mathfrak{b}$ are mutually prime, then the density of primes congruent to $\mathfrak{b}$ modulo $\mathfrak{a}$ in the set of all primes is the reciprocal of $\phi (\mathfrak{a})$ where $\phi$ denotes the Euler function.
And it can be sh... |
Is there an algorithm which will allow me to find an isomorphism between two graphs if I have their adjacency lists?
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Can you find two non-constant meromorphic functions $f,g$ such that $f^3 +g^3=1$?
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Let $x,y\in R$ be two arbitrary scalar numbers. $f_1(x,y),...,f_n(x,y)$ are n scalar functions of $x,y$. Note $x,y$ only have two degree of freedom (DOF), so I think the n functions $f_1(x,y),...,f_n(x,y)$ at most have two DOF. That probably means the vector $[ f_1(x,y),...,f_n(x,y) ]^T$ can not be an arbitrary vector ... |
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