instruction stringlengths 22 15.8k |
|---|
Tom prefers A over B and B over C. Also, Tom is indifferent between a lottery in which he gets C with probability p and A with probability 1-p and a lottery in which he gets B with a probability p and C with probability 1-p. The value of p in both lotteries is the same. For what values of p would Tom's preference be ra... |
Given a graph with an adjacency matrix $\bf A$, powers of this matrix give the number of walks from vertices. That is, $({\bf A}^k)_{ij}$ gives the number of walks from nodes $i$ to $j$ in $k$ steps. Is there any nice physical iterpertation for the powers of the Laplacian matrix?
|
I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?
$$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$
Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks.
|
The question is this:
In how many partitions of $\{1,2,\ldots,n\}$ into three disjoint sets, $A$, $B$, and $C$, are there no two consecutive numbers in the set $A$?
thanks!
|
Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.
$$\int (\cos^3x)(\sin^2x)dx = \int(\cos x)(\cos^2x)(\sin^2x)dx
= \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$
let $u = \sin x$, $du = \cos xdx$
$$\int(1-u^2)u^2du = \int(u^2-u^4)du
= \frac{u^3}{3} - \frac{u^5}{5... |
If ${\bf x}(s)$ is a curve in $\mathbb{R}^3$ on a surface parameterized by its arc length $s$, and ${\bf N}$ is the surface normal at ${\bf x}$, consider the following equality (with "$\cdot$" being the dot product):
\begin{align*}
\frac{d{\bf x}}{ds} \cdot \frac{d{\bf N}}{ds} = \frac{d{\bf x} \cdot d{\bf N}}{d{\bf x} ... |
Let $\{f_j\}$ be a family of self-homeomorphisms of $\overline{\mathbb{R}}^n$ and $x,y\in\overline{\mathbb{R}}^n$, where $\overline{\mathbb{R}}^n$ is the one-point compactification of $\mathbb{R}^n$. What does it mean for $\{f_j\}$ to converge c-uniformly to y in $\overline{\mathbb{R}}^n\setminus \{x\}$? That is, what... |
Hw do i estimate which pattern is the most likely in the distribution of the number of cards in each suit in a hand among 5-3-3-2; 4-4-3-2 and 4-3-3-3. This is a Bridge Game related problem
|
Given $a^{b^c}\mod p = x$, where $a,b,c$ are real positive integers and $p$ is a prime, what is the most efficient algorithm to compute $x$ (ideally in polynomial time)?
|
How do the roots of unity form a group with respect to multiplication (closure, association, identity) ?
|
Let $x$ and $b$ represent cardinals. Assume that $b\geq x > 1$ and $b^2=b$.
Prove that $x^b=2^b$. Thanks!
|
Suppose $m(E)=1$ and $f,g \geq 0$ with $fg \geq 1$. Show that $\displaystyle \int_E f dm \int_E g dm \geq 1$
Can someone please point me to the right direction?
|
In the few linear algebra texts I have read, the determinant is introduced in the following manner;
“Here is a formula for what we call $\det A$. Here are some other formulas. And finally, here are some nice properties of the determinant.”
For example, in very elementary textbooks it is introduced by giving the co-fact... |
I'm currently attending a somewhat disorganized seminar on combinatorics that follows no textbook. So far we have covered the orbit-stabilizer theorem, some recursion, and we're heading into the Möbius inversion formula.
Can anyone suggest a text that approaches combinatorics at this level for a 2nd-3rd year undergrad... |
If $X\le Y$, that is, $Y-X$ is positive semi-definite, does it imply that $Tr\{XZ\}\le Tr\{YZ\}$?
If it's true, how should one go about proving it? Is it a result from some other theorem?
*edit: $Z$ is also positive, all matrices are Hermitian.
|
Problem:
$$\text{ If } 2^{n} - 1 \text{ is prime then n is prime}$$
Proof 1:
$$\text{If } n = kl \text{ with } 2 \leq k, l < n \text{ then } (2^{k} - 1)|(2^{n} - 1). \text{ Hence if } 2^{n} - 1 \text{ is prime, n must be prime }$$
Proof 2: Using the identity:
$$2^{n} - 1 = ( 2 - 1 )( 2^{k-1} + 2^{k-2} + ... + 1 )$$
$... |
This was a question on an assignment presented to my Logic & Mathematics for computer science course, and I am truly baffled as to go on to prove this by double induction:
Consider a string consisting of one or
more decimal digits (0-9). Suppose you
repeatedly insert a 0 to the right of
the leftmost digit and then rep... |
For each given $p$, let $X$ have a binomial distribution with parameters $p$ and $N$. Suppose $p$ is distributed according to a beta distribution with parameters r and s. Find the resulting distribution of $X$. When is this distribution uniform on $x=0,1,2,...N$?
|
It seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of course, for moduli reasons, it is clear why one might want such an invariant, but the actual formula has always seemed quit... |
If $X$ is a compact Riemann surface, $Y$ is a universal cover of $X$, and $f\colon X\rightarrow X$ is a biholomorphic map, then, can $f$ be lifted to a $biholomorphic$ map on $Y$? (I mean, topologically, it can be lifted to a homeomorphism from $Y$ to $Y$, just by using "lifting criteria". But can the lift be "biholomo... |
The rule $(v_1,w_1)⋅(v_2,w_2)=(v_1+v_2,w_1+w_2+(v_1∧v_2))$ defines a group structure on the vector space $V⊕(V∧V)$ whenever $V$ is itself a vector space over some field $F$.
What is a more common name for this group?
The group is nilpotent of class at most $2$, with commutator $[(v_1,w_1),(v_2,w_2)] = (0,2(v_1∧v_2))$.... |
For a non-negative integer $\ell$, define the arithmetic functions $\sigma_{\ell}(k) = \sum_{d \mid k} d^{\ell}$ and $\bar{\sigma}_{\ell}(k) = \sum_{k = dd^{\prime}} (-1)^{(d^{\prime} - 1)/2} d^{\ell}$.
The function $\sigma(k) = \sigma_{1}(k)$ is the divisor-sum function.
Question: Let $k$ be an odd integer. Is the f... |
Number Composition studies the number ways of compositing a number.
I wanna know the number of compositions of $m$ with $n$ parts with the size of the max part equal to or less than $k$.
Is there a closed form for this problem?
|
How to prove that, if a is an ordinal and b is in a, then b is an ordinal?
Here are the definitions I'm using.
A set is an ordinal number if it is transitive and well-ordered by ∈.
A set T is transitive if every element of T is a subset of T.
My difficulty is mainly that I can't prove b is transitive. What's the magic?... |
I was looking at some source code that was described as 'maintaining a moving average over the previous 10 minutes', and noticed it uses the exponential function. Now its been a while since my university study, and i didnt take as many numerical analysis type math topics as I wish I had (mostly vector calc and diff equ... |
I made this observation and it seems reasonable to me to ask :if $n$ is a natural number then the number of the primes less than or equal to $n$ is denoted by $π(n)$ . is that true that in any interval of length $n$ there are at most $π(n)+1$ primes?(the $+1$ is needed for the trivial occasion where $n=p-1$ and the int... |
Suppose we choose a random number n from 0..100.Y and X are discrete random independent variables that have uniform distribution with 0...n.If Y+X equals to 15 what is expectancy of n?
|
I recently came across two properties that imply that a group is abelian:
If the order of each element except the identity element of a group is two then the group is abelian.
If every element of the group is its inverse then the group is abelian.
Obviously this comes from my module with no proof but I also looked in... |
I'm trying to show why it isn't possible to define an order of magnitude on $\mathbb Z_n$ (modular arithmetic) that satisfies the ordering properties of $\mathbb Z$.
By letting addition to be $\oplus$ and multiplication $\otimes$, I know the following about $\mathbb Z_n$:
Closed under $\oplus$ and $\otimes$.
$\oplus$... |
Ok so i am stuck at this:
I need to calculate distance between $2$ points...
For example:
I have $30\times 30$ square and point$1$ is at $X4,Y5$ and point$2$ is at $X30,Y23$ now I need to get the shortest distance from point$1$ to point$2$. By wich way is the shortest North, East, West, South...
I know i have to do tha... |
I have a problem solving an integration. This is my approach:
\begin{align*}
u &= \frac{1}{T}\cdot\intop_{0}^{T}(U_{0}+\hat{u}\cdot \sin(\omega t))dt\qquad\text{with }\omega=\frac{2\pi}{T}\\
u &= \frac{1}{T}\cdot\intop_{0}^{T}(U_{0}+\hat{u}\cdot \sin(\frac{2\pi}{T}\cdot t))dt\\
u &= \frac{1}{T}\cdot\left[U_{0}t-\hat{u}... |
Seems like I still don't get it, I think I am missing something important.
Let $V$ be an $n$ dimensional inner product space ($n \geq 1$), and $T\colon\mathbf{V}\to\mathbf{V}$ be a linear transformation such that:
$T^2 = T$
$||T(a)|| \leq ||a||$ for every vector $a$ in $\mathbf{V}$;
Prove that a subspace $U \subsete... |
Z is a 1D lattice
Gaussian and Eisenstein integers are 2D lattices
But the golden integers (for example) are dense on the real line.
Are there rings of integers which have 3D, 4D, ... lattices?
Here is a plot of $(a + \tfrac{1}{2}(1+\sqrt{5})b,a + \tfrac{1}{2}(1-\sqrt{5})b)$ for $-10\le a,b \le 10$.
which is the la... |
If I have three lines $a,$ $b$ and $c$ in the euclidean 3D space, which are pairwise non-coplanar, is there always a fourth line $x$, that intersects theses three lines?
|
What goes wrong with this argument?
Suppose I have a curve $\gamma: [a, b] \rightarrow R^n$. We express this in polar coordinates as $\gamma(t) = (r(t), \theta(t))$. The derivative of $\gamma$ is just taken componentwise, so $\gamma'(t) = (r'(t), \theta'(t))$. Now the length of $\gamma$ is given by $\int_a^b \| \gamma'... |
Apparently this should be a straightforward / standard homework problem, but I'm having trouble figuring it out.
Let $D$ be a square-free integer not divisible by $3$. Let $\theta = \sqrt[3]{D}$, $K = \mathbb{Q}(\theta)$. Let $\mathcal{O}_K$ be the ring of algebraic integers inside $K$. I need to find explicitly elemen... |
I am working on a texturing algorithm for 3d graphics and I am trying to rotate triangles that are attached at a point so that they are connected (share a single edge) I am using
$\tan^{-1}\left( \frac{m_1 - m_2}{1 + m_1m_2}\right )$
where $m_1 \gt m_2$
This is working well in some test cases but not in all... I have ... |
Grandma has 8 grandchildren, and 4 different types of popsicles:
6 Vanilla popsicles
6 Strawberry popsicles
5 Banana popsicles
3 Chokolate popsicles
This morning, all of her grandchildren came together and asked for one popsicle each (every grandchild asked for a particular flavor). What is the total number of differ... |
Consider the problem of a point p rotating around a given axis.
The axis of rotation is ω , |ω| = 1 , and q is a point on the axis.
Assuming that the point rotates around the axis with unit angular velocity, then the velocity of the point p(t) is
p'(t) = ω x ( p(t) - q )
I'm looking for a more detailed explanation for ... |
While doing some self-study, a friend posed this problem to me:
Let $a$ be a sequence, defined as following:
$$a_0 = 0,\quad a_1= 1, \qquad a_{n+2}=\frac{a_n+a_{n+1}}2$$
Figure out, whether $a$ converges, and if yes to which value. Find a closed form for $a_n$. If you want to, you may also have a look at the common ca... |
I have to prove that $f \colon x \mapsto e^{4x} + x^5 + 2$ ($f\colon \mathbb{R}\to\mathbb{R}$) is bijective. The argument given in the solution is that since the first two summands of the image is a bijective function of $x$, then so is $f$. Nonetheless, this "proof" doesn't seem at all rigorous to me, since there are ... |
In combinatorics, there are several different of the form "How many integer solutions does the equation $\Sigma_{1\le i\le k}X_i=n$ have?"
Some variations include:
$X_i\in\{0, 1\}$
$0\le X_i$
$1\le X_i$
$0\le X_i\le m$
$m_1\le X_i \le m_2$
...
Is there a complete guide to solving these kinds of questions?
|
Let $A$ be a (commutative) noetherian ring, and let $I \subseteq A$ be an ideal. It is not hard to show that if $I$ is generated by a length $n$ regular sequence, then the $A/I$-module $I/I^2$ is a free module of rank $n$. Is the converse true? if we know that $I/I^2$ is free, does it follows that $I$ is generated by a... |
If $a$ is a positive constant,then show that $\displaystyle \lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists and is strictly positive.
|
Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?
Motivation: I took this for granted for a long time, as I thought collapsing the contractible subspace does not change the homotopy type. Now it seems that this is only true for a CW pair...
|
Let $I(f)$ be defined as $\int_{0}^{1}f(x)dx$ from $C$[0, 1] to $\mathbb{R}$, where $C$[0, 1] is a linear space of all continuous functions from closed interval [0, 1] to $\mathbb{R}$. Is there a systematic way to find the kernel of $I$? Clearly, the zero function is there, as integral of $0$ from 1 to 0 will be equal ... |
I am not getting a fact whether if $X\,\times\,Y$ is homeomorphic to $Y\,\times\,Y$ then is it true that $X$ is homeomorphic to $Y$? Actually I want to know the result for say $X=[0,1]$ and $Y=[0,1)$.
|
I got a little stuck on a simple proof of the following probability identity.
Given
$\mathbb{P}(A^c \cap B^c)=1-\mathbb{P}(A)-\mathbb{P}(B)+\mathbb{P}(A\cap B)$
how to prove for any set $X$,
$\mathbb{P}(X \cap A^c \cap B^c)=\mathbb{P}(X)-\mathbb{P}(X\cap A)-\mathbb{P}(X\cap B)+\mathbb{P}(X\cap A\cap B)$
Looks very int... |
My birthday this year (2011) is on a Friday. In most years, one's birthday the following year is on the subsequent day of the week, and in that pattern, my birthday next year (2012) it is on a Saturday.
However, due to 2012 being a leap year, my birthday in 2013 will be on Monday - I miss out on having a Sunday birthda... |
i need to figure out how can i add given number of squares in a rectangle
i want to create this.
https://i.stack.imgur.com/MHoda.jpg
The whole scenario is to create a function that take a number and then create a layout of it, for a quick reference i did this with predefined values for rows, columns, x&y scale etc.
Th... |
"Given $\vec{u}_1,\ldots ,\vec{u}_n$ mutually orthogonal non-zero vectors, explain why for $\vec{v}=c_1\vec{u}_1+\ldots +c_n\vec{u}_n$ $c_k=\frac{\vec{v} \cdot \vec{u}_k}{\vec{u}_k \cdot \vec{u}_k}$"
This I explained by dotting both sides with $\vec{u}_k$ and simplifying everything. However, the question I now have is,... |
Let $u$ and $v$ be two $L^1(\mathbb{R})$ functions such that
$\|u\|_{L^1} \le \|v\|_{L^1}$ and $f$ is non-negative $L^1(\mathbb{R})$
with non-negative inverse Fourier transform.
Is it true that for the convolution
$\|u*f\| \le \|v*f\|$?
If not, maybe someone know additional condition that
will give the last inequality... |
I am starting to see a pattern in my math studies, and I'm interested in your thoughts on the subject.
It seems like there are certain things that are much easier for me to learn than others. For instance, I am a pretty quick learner when I can "visualize" what I'm learning, and logical thinking and rigid formalism see... |
I did some experiments, using C++, investigating the values of $\sqrt{1+24n}$.
n: 1 -> 5
n: 2 -> 7
n: 5 -> 11
n: 7 -> 13
n: 12 -> 17
n: 15 -> 19
n: 22 -> 23
n: 35 -> 29
n: 40 -> 31
n: 57 -> 37
n: 70 -> 41
n: 77 -> 43
n: 92 -> 47
I wonder, if
$$\sqrt{1+24n}$$
is an integer, will it also be a prime?
Is ... |
"Let $P$ be the change-of-basis matrix
from a basis $S$ to a basis $S'$ in a
vector space $V$. Then, for any vector
$v \in V$, we have $$P[v]_{S'}=[v]_{S}
\text{ and hence, } P^{-1}[v]_{S} =
[v]_{S'}$$
Namely, if we multiply the coordinates
of $v$ in the original basis $S$ by
$P^{-1}$, we get the coordi... |
Let $\mathbb{R}(t)$ be the field of rational functions over $\mathbb{R}$ (the fraction field of $\mathbb{R}[x]$).
I am looking for elements in the Brauer group of the field, and the current idea I have to follow on is to find infinitely many cyclic field extensions, and use those to create cyclic division algebras.
My ... |
I am solving these problems in preparation to a midterm. Here's the problem I have solved before tackling the one I am asking help with:
I was given 1 basis vector for $\mathbf{W}$ - $\begin{bmatrix}
1\\
2\\
3\\
4\\
5
\end{bmatrix}$, and I had to find a basis for $\mathbf{W}^{\perp}$. I set up an equation $\be... |
This must be a very basic question but I am a finance student just learning some basics about Fourier Transformation to apply to time series analysis. I did a fourier transform on a function in time domain to get the following function in frequency domain:
$Y_1[\omega] = \frac{1}{1-\phi_1 e^{-jw}}$
$Y_2[\omega] = \frac... |
Given an $n$-sided polygon, how many ways can you color the vertices using $k$ colors so that no two adjacent vertices have the same color?
(Inspired by 2011 AMC 12 A #16 – I'm able to do this for small cases but not in general.)
|
My book introduces $\mathbf{C}^{\infty}$ as subspace of $F(\mathbb{R},\mathbb{R})$ that consists of 'smooth' functions, that is, functions that are differentiable infinitely many times. It then asks me to tell whether or not $\mathbf{C}^{0}$=$(f\in(\mathbb{R},\mathbb{R})$ such that $f$ is continuous$)$ is a subspace. ... |
Is there any efficient way , to find for a particular n, the cardinality of set consisting of all numbers coprimes to n, but bigger than m(assuming i know the prime factorisation of n and m)
I am looking for the implementation which is simple+fast (like the euler totient/phi function, which given the factorisation of n... |
Ok this is probably the most absurd question you'll ever read, but it came to my mind, and I cant shake it off. Eulers Identity states that:
$e^{i\pi}+1=0$. So my ridiculous question is why was it stated this way? Why couldnt it have been $e^{i\pi}=-1$? Are there any reasons for this, or it could have been either of th... |
f(1) = 2
f(2) = 2^2
f(3) = 2^(2^2)
f(4) = 2^(2^(2^2))
etc
how can I find a closed form solution to this? Thanks.
|
Number theory-question again.
Let $K$ be the biquadratic field $K=\mathbb{Q}[\sqrt{m},\sqrt{n}]$ where $m,n$ are distinct squarefree integers. Let $\mathcal{O}_K$ denote the ring of integers of $K$. If $\alpha \in K$, prove that $\alpha \in \mathcal{O}_K$ if and only if the trace and the norm of $\alpha$ over $\mathbb{... |
For an assignment, I was asked to find the vertical asymptote of the function $$g(x)= \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84}.$$
According to my text, a reliable method of finding the asymptote is to factor the numerator and denominator, and what left in the denominator that was not cancelled out is the asymptote.
... |
Let $T(f(t))$=
$\begin{bmatrix}
f(0) & f(1)\\
f(2) & f(3)
\end{bmatrix}$ from $P_2$ to $\mathbb{R}^{2\times 2}$. To show that it is not an isomorphism, I need to show that either kernel of the transformation is not equal to the zero element only, or that the image is not the whole target space. I am struggling in s... |
I'm still confused about order of operations on exponents:
Is $x^{a^b}$ equal to $x^{(a^b)}$ or $(x^a)^b$?
*Hmm... $x^{a^b}$ has the source of x^{a^b}. Whenever I try x^a^b, it requires me to "clarify by using braces".
|
If I have $m$ distinct boxes, and $n$ distinct balls. I put all of these balls into the boxes with one box possibly containing more than one balls. What is the expected maximum number of balls in one box?
Appreciate your thoughts on solving this problem.
|
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$.
It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot \rangle$, then the parallelogram equality
$$ 2\lVert u\rVert^2 + 2\lVe... |
I'm interested in why $$\int_0^{\pi/2} \sin x\,dx = 1.$$ I know how to do the integral the conventional way but am more interested in what makes radians special for this problem. If we instead compute $$\int_{0}^{90} \sin x^\circ\,dx,$$ we won't get $1$ as the answer.
What about the definition of radians makes this int... |
If you know the eigenvectors of a specific matrix, is that enough information to get the original matrix back? So for example, for some $2\times 2$ matrix, it is known that its two distinct eigenvectors are $[1, u]$ and $[1, v]$ after normalization, can the original matrix be recovered?
|
I am given a subspace of all polynomials $f(t)$ in $\mathbf{P}_2$ such that $f(1)=0$. I know that a basis for this space is $1-t$, $1-t^{2}$, and when I look at it, it makes perfect sense as to why. I was just wondering what is a systematic way of finding it, without eyeballing. Thanks!
|
Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different irreducible fractions on the boundary, how can I find the corresponding Möbius transformation?
|
Given that:
$$AB= \left[ {\matrix{
3 & 1 \cr
2 & 1 \cr
} } \right]$$
and
$$BA= \left[ {\matrix{
5 & 3 \cr
-2 & -1 \cr
} } \right]$$
find $A$ and $B$.
|
I am working on my homework for a class, and I am really stuck
the question I was given was:
prove that S(a ⊗ b) = (Sa) ⊗ b
Does anyone have any tips on how to solve this?
For further explanation this is problem 6a from this book on the page the link goes to
|
First of all, the polynomial $p(x)=x^{3}-x-4$ must have one real root. Let $\theta$ be one of its roots, L=Q($\theta$), $O$ the integral closure of Z in L.
Then Kummer's theorem tells us then and there that for a prime q not to be nonsplit it is necessary and sufficient that $p(x) \bmod q$ is reducible.
Nevertheless, i... |
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
I think it is probable because we can consider
$$ y =
\begin{cases}
\sin \left( \frac{1}{x} \right), & \text{if } x \neq 0, \\
0, & \text{if } x=0.
\end{cases}
$$
This f... |
Is there a geometric way of looking at the relationship between the positive real numbers $a$, $b$ and $c$ if $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?
|
I know that if $F$ is a field, and $x,y$ are roots of an irreducible polynomial over $F$ lying in an algebraic extension of $F$, then $F(x) \cong F(y)$.
But it seems it is not true (in general) that if $x,y,z$ are roots of an irreducible polynomial in an extension of $F$ then $F(x,y) \cong F(x,z)$.
What might a count... |
Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put
$$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$
If $\alpha$ is irrational, can I find some constant $K$ such that $A_{\alpha}(n) < K \sqrt{n}$ for all $n$? Does the order of $A_{\alpha}(n)$ depend on $\alpha$? Suppose $1/\sqrt{k}$ is repla... |
I would like to show that if $K$ is a field with $p^n$ elements then its characteristic has to be $p$, $p$ prime. I'm not sure where to start. It's clear to me that I can construct a field of order $p^n$ if I have a ring of characteristic $p$ by taking the ring of polynomials and quotienting it by an irreducible polyno... |
I know this is rather simple, but despite my best efforts and quite a bit of searching through google I can't seem to find a satisfactory answer.
An example of what I'm trying to do:
Given a value $N= 6.8\cdot10^{-4}$ and upper and lower errors for that value of $N^{uperr} = 9.1\cdot 10 ^{-4}$ and $N_{lowerr} = 3.9\cdo... |
Given a Lie group, how are you meant to find its Lie algebra? The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them? My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition.
|
A group $G$ is called an FC-group if $|x^G|$ is finite for each $x \in G$ (here $x^G$ is the conjugacy class of $x$ in $G$). Equivalently, $|G:C_G(x)|$ is finite for each $x \in G$.
I think it should be completely obvious that a homomorphic image of an FC-group is an FC-group & maybe I'm just tired, but I've been tryi... |
I've never seen this notation before, and I'm having trouble finding a reference through search. Could someone explain what these notations mean for me?
In context, the statement they're in is the following:
a bounded $f$ is Riemann integrable iff
$$\varliminf_{||C||\to 0}\mathcal{L}(f; C)\ge\varlimsup_{||C||\to 0}\ma... |
This is a dice game where the host throws the dice and the players bet on 3 numbers - numbers below 7, 7 and numbers above 7. On numbers below and above 7 the players get double their stake and on 7 the players get 5 times their stake. For example if i bet 100 bucks on "below" and 7 comes, I will lose, but if 6 comes, ... |
I stumbled across the problem of connecting the points on a $n, n$ grid with a minimal amount of straight lines without lifting the pen.
For $n=1, n=2$ it is trivial. For $n=3$ you can find the solution with a bit trial and error (I will leave this to the reader as it is a fun to do, you can do it with 4 lines). I foun... |
In asymptotic analysis and analytic number theory, one often has to deal with complex integrals over infinite contours in the complex plane, and the required techniques to do so often go beyond the standard courses of complex analysis in one variable. In particular, I am interested in the following type of argument whi... |
I'm taking basic linear algebra now. During a lecture on orthogonal complements, a professor said that there's a thing that is more 'general' than orthogonal complements and it's called the annihilator. He didn't provide further details, so I hopped on Google and found some explanations of this concept. Now I think I u... |
Of the six questions regarding finding vertical asymptotes of graphs, I've had problems with two. The second is using the graph of $$g(x)= \frac{3+x}{x^{2}(3-x)}$$
Now, looking at the function, it seems that both 0 and 3 will be the vertical asymptotes, since $$g(0)= \frac{3+0}{0^{2}(3-0)}= \frac{3}{0(3)}= \frac{3}{0}... |
From the relation $M=E-\epsilon\cdot\sin(E)$, I need to find the value of E, knowing the two other parameters. How should I go about this?
This is part of a computation which will be done quite a number of times per second. I hope there's a quick way to get E out of this equation.
Thank you very much,
MJ
|
Jacobi first noticed the connection between the functions that bear his name and counting the representations of sums-of-squares,
\begin{eqnarray}
\theta_{3}^{n}(q) = \left( \sum_{k \in \mathbb{Z}} q^{k^{2}} \right)^{n} = \sum_{k \geq 0} r_{2,n}(k) \ q^{k},
\end{eqnarray}
where $q = e^{\pi i \tau}$ (or $e^{2 \pi i \ta... |
I have $y$ is a function of $t$.
I have reached a situation here where I need to evaluate
$$\displaystyle \int_0^b{\frac{dy}{dt}dt}$$
Now clearly $y$ has dependence on $t$, otherwise $\displaystyle \frac{dy}{dt}$ should be 0.
So now I write
$$\displaystyle \int_{y(0)}^{y(b)}{dy} = y \rvert_{y(0)}^{y(b)} = y(b) - y(0)$$... |
I know you may find definitions out there for standard curve.
But, how can we define it in a way that makes it more understandable and clear of what it does?
|
Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ its zero section.
I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that $f\circ s_0=i$ and $T_xf$ is bijective for any $x\in s_0(N)$.
How could I prove that there exists an open neighborhood $V$ of... |
Can someone tell me if this is right:
I would like to find the minimal polynomial of
(i) $\sqrt[4]{2}i$ over $\mathbb{Q}$
(ii) $\sqrt[4]{2}i$ over $\mathbb{R}$
(i):
$\sqrt[4]{2}i$ is a root of $f(x) = x^4-2$. This is already monic, so to show that this is a minimal polynomial I only need to show that it is irreducible... |
To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
|
Can anyone help me with this problem?
The random variable $X_n$ takes the values $\frac{k}{n}$, $k=1,2,\ldots,n$, each with probability $\frac{1}{n}$. Find its characteristic function and the limit as $n\to\infty$. Identify the random variable of the limit characteristic function.
Thanks!
|
The first irregular prime is 37. Does FLT(37)
$$x^{37} + y^{37} = z^{37}$$
have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of unity?
Maybe it's not true, but how could I go about finding a counter-example? (for any cyclotomic ring, not necessarily 37)
|
Can anyone help me with it:
Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$
Thanks!
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.