instruction stringlengths 22 27.2k |
|---|
Show that over any field $F$, the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors.
Edited:
This is my attempt: Let $f(x)=x^3-3x+1$. Let $a_1,a_2,a_3$ be the roots of $f$. Suppose char $F\neq 2,3$. Suppose also that $f$ is neither irreducible nor splits in $F$. Then $f$ is reducible which im... |
I have this problem about account record pagination. I have to write the function that describes the total pages needed to contain those account records.
This is the problem:
# Account record quantity to store per page
per_page € R
# Account records to insert in pages
tot_acc € R
# The total pages needed to store th... |
How to prove:
if $f: (a,b) \rightarrow R$ is uniformly continuous, then $f$ can be extended to a continuous function $F: [a,b] \rightarrow R$.
It's suffice to show that $f$ can be extended to a continuous function $G: [a,b) \rightarrow R$.
I saw similar theorem (Tietze extension theorem) in topology books, but I'm not... |
Dear all, I am studying symmetry method for ODEs.
The book written by George W.bluman (can be found in google books) discusses the procedure of determining ODE admitting a given group in details. However, as to the converse way, it only gives some theories, but no examples.
I am trying to solve the exercise 3.2 - 9, P1... |
I am reading Boyd's Convex Optimization book and I am stuck on the reasoning behind one of the statements. Specifically, I am looking at page 45, line 3 from the bottom. The statement is:
$A \preceq B \iff \mathcal{E}_A \subseteq \mathcal{E}_B$.
Note that I interpret $A \preceq B$ to mean $B-A \subset \mathbf{S}^n_+$.
... |
Could I please get some help with the following true/sometimes true/false questions?
1) If $A$ is 3-by-4, then for every $\vec{z}$ $\in$ $\mathbb{R^{3}}$ there are infinitely many solutions $\vec{x}^{*}$ to $A^{T}A\vec{x}^{*}$=$A^{T}\vec{z}$
This is related to least-squares, I can see how it could be sometimes true, fo... |
Hi I'm having trouble with this proof. I'm not sure if I did the first part right and how I should use the second part properly. I am following the hint in the book to use the Axiom of Replacement to prove step three below and follow through with Replacement and Union to finalize the proof.
1) Prove that the Cartesia... |
Prove or disprove the following statment: For all integers b, c,and d, if x is a rational number such that $x^2+bx+c=d$, then x is an integer.
This is a homework question from the book Discrete Mathematics for Computer Scientists by Stein, Drysdale and Bogart.
I since x is rational I thought I could start off with:
${(... |
I was reading something which contained the following statement:
It is a well-established mathematical result that theories consisting only of recursive definitions... are inherently consistent.
Does this result have a name? Could anyone point me to literature about this?
|
Problem:
Given $g(x)$, solve the equation $f'(f^{-1}(x)) = \frac{1}{g(x)}$ for an invertible and differentiable function $f(x)$.
So far I have tried setting $y = f^{-1}(x) \Leftrightarrow x = f(y)$, obtaining the differential equation
$$
f'(y) = \frac{dx}{dy} = \frac{1}{g(x)}
$$
which we then solve to obtain $y$ in... |
We all agree that the theory of class fields plays an eminent role in modern number theory.
Nevertheless, what was our main concern is that how to solve various Diophantine equations to which the clues do not appear clear in the theory of class fields. And my question is:
With respect to the beauty and perfection of... |
Suppose M is a 20 by 3 complex matrix, and I'd like to SVD. For example, in Matlab, I can do easily with:
[U, S, V] = svd(M);
where U, S, and V are complex matrices of20x20, 20x3, and 3x3, respectively.
However, I'd like to make U, a 20x20 matrix, to be either real matrix or near-real matrix (i.e., imaginary parts in t... |
How to prove: Continuous function $f : M\to N$ maps compact set to compact set?
The simplest case in real analysis: if $f: [a,b] \rightarrow \mathbb{R}$ is continuous, then I need to show that $f([a,b])$ is closed and bounded, by the Heine-Borel theorem.
I have proved the boundedness, and I need some insight on how... |
I don't know how can I to prove the following (this isn't homework, i'm just flipping through a book):
if $a\in F$, $v\in V$, and $av=0$.
then $a=0$ or $v=0$.
|
I'm asked to find the optimal $k$ such that $\cos{x}-1+x^2/2=O(x^k)\quad (x\to 0)$.
By Taylor's theorem I know that $\cos{x}-1+x^2/2=(1/4!)\xi^4$ for some $\xi\in (0,x)$. So $0<\cos{x}-1+x^2/2<(1/4!)x^4$, that is $\cos{x}-1+x^2/2=O(x^4)$.
My reasoning is as follows: if $0<x<1$ (which can be assumed since we're interes... |
How can I write $\frac{1}{a+x}$, $a$ a non-zero positive constant, in exponential terms in the form of $y = Ce^{-kx}$? I've tried to use to Taylor series but they only seem to work for $x < 1$.
|
I have a set of N elements. I want to select few elements from the set. Now each element i of the set has a probability P(i) of getting selected. Then how can I figure out the probability of selecting atleast K elements from the set?
K elements are required to be different. Some of the probabilities may be zero, but we... |
If I know the formal power series, I know how to find the closed form:
$$\displaystyle F = \sum_{n=0}^{\infty} {X^n} = 1 + X^1 + X^2 + X^3 + ...$$
$$\displaystyle F \cdot X = X \cdot \sum_{n=0}^{\infty} {X^n} = X^1 + X^2 + X^3 + X^4 + ...$$
$$\displaystyle F - F \cdot X = 1 $$
$$\displaystyle F = \frac 1 {1 - X} $$
But... |
I encountered this problem in my probability class, and we weren't able to solve it, so I would like to know the answer.
If you have $m$ balls and $n$ boxes, with $n < m$, and you insert the balls into the boxes randomly, what is the probability that all the boxes have at least one ball in it?
The problem doesn't speci... |
If we are given two sides say $a$, $b$ and an angle $X$. How can we determine whether this angle $X$ is opposite to $a$ or $b$ (i.e. $A$, $B$) or the third included angle $C$ ?
|
For Linear complementarity problems (LCP) like
$\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$
$\mathbf{z} \ge \mathbf{0}$
$\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$
there exists a vast amount of material and algorithms. But what about this kind of problem:
$\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$
$\mathbf{Az} +\m... |
They are similar. Both contain a projection map and one can define sections, moreover the fiber of the fiber bundle is just like the stalk of the sheaf.
But what are the differences between them?
Maybe a sheaf is more abstract and can break down, while a fibre bundle is more geometric and must keep itself continuous. A... |
I have been doing multiply and divide but never really tried to understand it.
E.g. (4 x 12 / 4) could be written as: (12, 4 x 3, 48 / 4) but yet the result is always same, can anyone prove or explain why?
|
I am taking a topology course and we are now learning open and closed set. I am a bit confused to how to prove that a set is closed or opened, how should I approach these kind of problems. For example:
Question 1: Let $(\mathcal{X},d)$ be an arbitrary metric space. Prove that any set which contains a finite number of p... |
Fermat solved the Diophantine equation $(x^2)^2 + (y^2)^2 = z^2$ using descent, the key step was using the Pythagorean triples:
$x^2 = u^2 - v^2$
$y^2 = 2 u v$
$z = u^2 + v^2$
but then it is seen that $x,v,u$ is another Pythagorean triple,
$x = m^2 - n^2$
$v = 2 m n$
$u = m^2 + n^2$
then the observation that $y^2 =... |
If we can use Whitney embedding to smoothly embed every manifold into Euclidean space, then why do we bother studying abstract manifolds, instead of their embeddings in $\mathbb{R}^n$? A vague explanation I have heard is that from this abstract viewpoint, we gain understanding into the intrinsic behavior of the manifol... |
Suppose a source produces an indefinite sequence of positive integers. How can you check whether the numbers are generated truly randomly?
|
Hey, !
In my pre-calculus class the teacher showed the solution of the following example:
\begin{align}
\vert x-3 \vert \lt \vert x - 4 \vert + x
\end{align}
He started by stated the domains needed to be checked:
\begin{aligned}
\lbrack 4, +\infty ) \newline
\lbrack 3, 4 ) \newline
( -\infty, 3)
\end{aligned}
Which... |
Why is the generalized quaternion group $Q_n$ not a semidirect product?
|
Let $r_{2,3}(n)$ and $r_{t,3}(n)$ denote the number of ways to write $n$ as a sum of three positive squares (A063691) and as a sum of three non-negative triangular numbers (A008443), respectively. I have noticed that $r_{2,3}(8k+3) = r_{t,3}(k)$ for $k \geq 1$. For example, $r_{2,3}(11) = 3$ because $11 = 3^2 + 1^2 + 1... |
About a year ago I took a Linear Algebra class that was required for my degree. Unfortunately that class had an unidentified pre-requisite and started at a much higher level then I really needed. Going in I had no prior experience with linear algebra. I can definitely see how understanding linear algebra would be a ... |
Is there a system $\{s_1, \cdots, s_m\}$ of symmetric polynomials of $z_1, \cdots, z_n \in \mathbb{C}$ such that
$$s_1(z_1, \cdots, z_n) = c_1$$
$$s_2(z_1, \cdots, z_n) = c_2$$
$$\cdots$$
$$s_m(z_1, \cdots, z_n) = c_m$$
has at most one solution $(z_1, \cdots, z_n)$ up to permutation, for all choices $c_1, \cdots, c_m \... |
If two sequences $\{a_k\}$ and $\{b_k\}$ are such that
$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(a_k-b_k)=0$$
$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(a_k^2-b_k^2)=0$$
does it mean that the two sequences are asymptotically equally distributed? i.e., you can't distinguish one sequence from the other?
EDIT
As... |
I have got a question on so-called time-dependet Sobolev spaces - in particular as introduced in Evans book on PDE for the treatment of parabolic and hyperbolic PDE.
Let us take a look at a linear hyperbolic PDE in $n$ spatial dimension and $1$ time dimension.
$u_{,tt} - Lu = 0$
where
$Lu = \sum_{i,j} a^{ij} u_{,i} u_{... |
I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out.
By solving I mean fining all solutions and proving there are no more.
Keith Conrad showed how to reduce this equation to a different one which was solved by Fermat in his notes about Fermat descent, oth... |
all. I am reading a paper and I could not understand the dimension of the vector space $Hom_{\Gamma}(\rho_i\otimes \rho, \rho_j)$. The paper is http://pages.uoregon.edu/njp/Lusztig92.pdf, section 1.1, page 113. Here $\rho$ is a 2-dimensional C-vector space. $\Gamma$ is a finite subgroup of the special linear group $SL(... |
I'm a physics grad student working, for my research, in a matrix vector space over the complex numbers. That is, both operators and the vectors they act upon are $n\times n$ complex matrices living in the same space. When it comes to the eigenvalue equation I've been using
$AB-BA=\lambda B$,
where $A$ and $B$ are $n\ti... |
So I am trying to solve this practice problem, and so far I managed to get that the kernel of transformation is zero.
Show that $L$($\vec{x}$)=$A\vec{x}$ from $Im(A^{T})$ to $Im(A)$ is an isomorphism.
Let L(x)=0. Then L(x)=$A$$A^{T}y$ for some y. We know that Ker(A)=$Im(A^{T})^{\perp}$. So if $A$$A^{T}y$=0, $A^{T}y$ i... |
Let $(X_{n})$ be a sequence of nonnegative uniformly integrable random variables. Is it true that $\limsup X_{n} \in L_{1}$? Thanks!
|
Solve $\displaystyle \cos(x-\alpha)\cos(x-\beta) = \cos{\alpha}\cos{\beta}+\sin^2{x}$.
My attempt:
$\displaystyle \cos(x-\alpha)\cos(x-\beta) = \cos{\alpha}\cos{\beta}+\sin^2{x} \Rightarrow \cos(x-\alpha)\cos(x-\beta)-\cos{\alpha}\cos{\beta} = \sin^2{x},$ and
$$\begin{aligned}LHS & = \frac{1}{2}\cos\left(2x-\alpha-\b... |
I was wondering if $\mathbb{Z} \wr S_n$, where $\mathbb{Z}$ is the usual group of integers, $S_n$ the symmetric group on n elements and $\wr$ the wreath product of two groups, contains the braid group $B_n$.
I was also wondering if $n+1$-dimensional matrices of the form :
$$\begin{bmatrix} 1&0&0 \\\\ 1&0&1 \\\\ 1&1&0 \... |
Problem
Solve $f(x) = 6x^3 + 27x^2 + 17x + 20 \equiv 0 \pmod{30}$
My attempt was:
Since $30 = 2.3.5$, we then have:
$$
\begin{cases}
f(x) \equiv 0 \pmod{2}\\
f(x) \equiv 0 \pmod{3}\\
f(x) \equiv 0 \pmod{5}\\
\end{cases}
$$
By inspection, we see that:
$$
\begin{cases}
x \equiv 0 \pmod{2}\\
x \equiv 1 \pmod{2}... |
Consider a particle undergoing geometric brownian motion with drift $\mu$ and volatility $\sigma$ e.g. as in here. Let $W_t$ denote this geometric brownian motion with drift at time $t$. I am looking for a formula to calculate:
$$
\mathbb{P}\big(\max_{0 \leq t \leq n} W_t - \min_{0\leq t \leq n} W_t > z\big)
$$
The... |
Suppose there is a function $f: X \rightarrow \mathbb{R}$, where $X \subseteq \mathbb{R}^n$.
If $x^*$ is a local minimizer of $f$
over $X$, must $x^*$ be either of the
two cases:
if $f$ is differentiable at $x^*$, then $x^*$ must be a stationary point
i.e. $\nabla{f}(x^*)=0$
$f$ is non-differentiable at $x^*$?
In o... |
Let $\pi:E→B$ be a smooth vector bundle. Prove $\pi$ is a submersion.
|
Is there a parameterization by integers of the modular group SL(2,Z)? What I mean is, some expression for a matrix in terms of several variables (a,b,c,...) such that for each n-tuple of integers (a,b,c,...), the expression produces a (preferably distinct) unimodular matrix.
As an example of what I mean, think of Pytha... |
Given the number of combinations w/o repetition for a set of size n from which you choose k is given by:
n! / k! * (n - k)!
How does one calculate the number of these combinations that include a given element. That amount should be the same for all elements of the original set.
For example, given the set {A, B, C, D},... |
So, I'm a little confused about one statement made in class today :
If M is a smooth manifold without boundary such that the tangent
bundle of M is trivial, then M is orientable.
Is this always true ?
|
Given $a, d, n, x$.
Suggest me a suitable algorithm to compute the number of times the digit $x$ appearing in the arithmetic sequence $a, a + d, a + 2 \times d, \cdots, a + n \times d$.
For example : If $ a= 10, d=5,n= 2,x = 0$ the sequence is ($10, 15,20$) so the answer is $2$
Domain specification: $- 1 \le a, n \... |
Fix a complex number $z=x+iy$ where $x,y\in \mathbf{Z}$ Consider the sequence generated by the powers $$z^0, z^1, z^2, z^3,z^4 \ldots$$ The question is whether it is possible to capture any positive integer as either the real part or the imaginary part of this sequence of the powers of a fixed complex number. From some... |
I already know that the function
$$
f(x) =
\begin{cases}
\exp(- \frac{1}{x^2}), \quad x > 0 \\
0 , \quad x \leq 0
\end{cases}
$$
is infinitely differentiable throughout $\mathbb R$. The only real problem, of course, lies in showing that $f^{(k)} (0) = 0$ for any positive integer $k$. What I have not been ab... |
I haven't used my algebra skills much for years and they seem to have atrophied significantly!
I'm having real trouble working out how to re-arrange a formula I've come across to get $x$ by itself on the left hand side. It looks like this:
$\frac{x}{\sqrt{A^{2}-x^{2}}}=\frac{B+\sqrt{C+Dx}}{E+\sqrt{F+G\sqrt{A^{2}-x^{2}}... |
I have the following problem related to a statistics question:
Prove that the function defined for $x\ge 1, y\ge 1$,
$$f(x,y)=\frac{\Gamma\left(\frac{x+y}{2}\right)(x/y)^{x/2}}{\Gamma(x/2)\Gamma(y/2)}\int_1^\infty w^{(x/2)-1}\left(1+\frac{xw}{y}\right)^{-(x+y)/2} dw$$
is increasing in $x$ for each $y\ge 1$ and decreasi... |
The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look at $y=x^{-1}$. How odd, the innocent formula for a power function's antiderivative breaks down but gee, it must have an ... |
This is a concept that seems very intuitive to me, but I feel my proof is messy. Could someone more experienced than I perhaps offer some criticism/tell me if I'm even correct?
Let $A\subset\omega$. I want to show that $A$ is infinite if and only if
$$
\forall_m(m\in\omega\implies\exists_n(n\in A\land (m\lt n))).
$$
... |
I want to determine if a singular point is a local min, max or saddle point.
We are dealing with singular points so we cannot use the hessian matrix.
What I have written, and I think I must of missed something is :
Say we have a function $f(x,y,z)$. To show that $(2,2,2)$ is a saddle point, we want to show :
$\foral... |
This is related to my previous asked question here. The integration can be simplified as (As pointed out by Sivaram, the integral diverges to $-\infty$ as $c\to1+$):
$$\int_{-\pi/2}^{\pi/2} \frac{\sin y}{\sqrt{\sin y+c}}\ \text{d}y,\ \ c>1.$$
I am appreciated if there is a specific answer (it would be great if there ... |
I'm a bit stumped by the exponential family representation of a multi-variate Gaussian distribution. Basically, the exponential form is a generic form for a large class of probability distributions. The standard form is
$$f_X(x) = \exp[\theta' T(x) + F(\theta)]$$
where $\theta$ is a set of parameters (based on $\mu$ an... |
Here is a silly mistake I am making: where exactly is the mistake?
I know that torus cannot hold a metric of constant curvature -1 ( hyperbolic metric ).
But what if I do this:
The upper half-plane and $\mathbb{C}$ are diffeomorphic by a diffeomorphism, say $\phi$; pull the hyperbolic metric from the upper half plane ... |
Is there any way I can calculate derivatives using a Casio Graphing Calculator?
Is there a program available for the calculator which can calculate them?
|
I was working on a scheme in cryptography and came up with the following scenario.
To put it in proper words.
We have an element $\frac{1}{x+m}$.
The 2 elements $x$ and $x_1$ are known.
We want to transform $\frac{1}{x+m}$ to $\frac{1}{x_1+m}$. I.e, we need a $k$ such that $\frac{k}{x+m} = \frac{1}{x1+m}$.
Y... |
This is related to a homework question for a class that I am TAing for. I'm using Sipser terminology here (recognizable for computably enumerable, decidable for computable).
Given that $w^r$ is the reverse of $w$, and that
$$T = \{ \langle M\rangle \mid \mbox{if $M$ accepts $w$ then $M$ accepts $w^r$}\}.$$
Now here ar... |
Let $p$ be prime, and $a$ be integer. When does $(p - 1)! + 1 = p^a$ for some $a$ hold?
For example:
$$p = 5 \implies (5 - 1)! + 1 = 25 = 5^2$$
$$p = 7 \implies (7 - 1)! + 1 = 721 = 7 \cdot 103$$
Any idea?
Thanks,
|
I guess I understand some reasons why we should care for complex structure on manifolds, but what is the reason why product structure is studied? Does it arise naturally somewhere?
The product structure on a smooth manifold $M$ is given by a (1,1)-tensor $E$ such that $E \neq \pm1$ and $E^2 = 1$ and the following integ... |
How can we physically interpret an eigenvalue or an eigenvector in linear algebra?
|
Please help me count.
I have an alphabet with 26 English letters. I can reduce it to 25 letters by representing two of the letters (e.g. O and X) with a new letter (e.g. $\otimes$).
25
To get down to a 25 letter alphabet I have $26 \choose 2$ possibilities. So 325 possible reductions to alphabet of size 25.
2
Getting d... |
$$\lim_{(x,y)\to(0,0)} \frac{xy}{\sqrt{x^2+y^2}}$$
Approaching (0,0) along x or along y both result in the limit approaching 0, so you want to make sure that the limit exists by doing more tests.
My solutions manual uses $x = y^2$ (or $y = x^2$). Why either of those? Why not $y=x$ or $x=y$? Why a parabola?
|
What is the difference between outer measure and Lebesgue measure?
We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\mathbb{R}$.
|
How many symmetric polynomials of n degree with all their coefficients $\ =1 $ are there?Is there a type that computes their number?
|
I am trying to understand the value of $\bar{x_{1} x_{2}}+E(x_{1} x_{2})$. For all $i$, $E(x_{i})$ and $\sigma_{i}$ are given. Wikipedia gives the joint probability density function:
$E(XY) = \int \int x y j(x,y) dx dy$
then I can find out from wikipedia that:
$E(XY) = Cov(X,Y) + E(X)E(Y)$
and by Cauchy-Swartz:
$| Cov(... |
It is a theorem that any function $f$ defined for positive real numbers satisfying
$f(1)=1$
$f(x+1)=x\cdot f(x)$
$f$ is log convex
is identically equal to the gamma function. (Condition 2 means that this function interpolates a shifted factorial function.)
Now, a beginner (such as myself) might ask: What if we weaken... |
I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. Thanks.
|
This problem comes from page 99 in Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows.
I tried to prove condition (ii) for n=1 under condition (i), The author says it follows from Theorems 1.16 and 1.18, but I can not see any relation between condition (ii) and t... |
Hensel's Lemma
Suppose that f(x) is a polynomial with integer coefficients, $k$ is an integer with $k \geq 2$, and $p$ a prime.
Suppose further that $r$ is a solution of the congruence $f(x) \equiv 0 \pmod{p^{k-1}}$. Then,
If $f'(r) \not\equiv 0 \pmod{p}$, then there is a unique integer t, $0 \leq t < p$, such that
$... |
How do you solve this equation for A:
$~~\sin(5A) + \cos(5A)\sin(A) - \cos(3A) = 0$
I've tried expanding it many times, but I can't seem to be able to reduce it to a format I can work with. Is there a simpler method of solution than repeated expansion?
|
I was reading a paper, related to group theory. I came across two doubts (may be simple, but puzzling me):
Let $G$ be a finite group and consider the lattice of subgroups of $G$. On this lattice, we define a Möbius function $\mu$ by $\sum_{H\geq K} {\mu(H)=\delta_{K,G}}$.
I want two verifications:
1) If $G=C_n$ is ... |
Can we say that an isomorphism $V \cong V^*$ is not natural? It seems so intuitively, but formally the notion of a natural transformation between a functor and a cofunctor is not defined (or is it?).
|
Here is an interesting limit of an integral I do not know how to begin. Any help is greatly appreciated.
$\lim_{n\to \infty}n\int_{0}^{\frac{\pi}{2}}(1-\sqrt[n]{\sin(x)})\,\mathrm{d}x$
I know it converges to $\frac{\pi \ln(2)}{2}$, but how?.
Thanks much
|
Follow up to this question. Is $0$ a positive number?
|
$$A:= \{ x^2 | x \in (2,3\sqrt{2}] , x \in \mathbb{R} \textrm{ and } x^2 \in \mathbb{Z}\}$$
$$B:= \{ x^3 | x \in (-2\sqrt[3]{3},3] , x \in \mathbb{R} \textrm{ and } x^3 \in \mathbb{Z}\}$$
s(A) + s(B) = ?
s(A) is number of elements of A set.
For example if $\ A = \{a,b,c\} $
s(A) = 3
|
I want to find out for which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1]), p \in \mathbb{R}_{\geq 1} \cup \{\infty\}$. I first let $p,q \geq 1$ and I looked at the case where $q > p$.
I defined $\tilde{f} := |f|^p$ and $\tilde{p} := \frac{q}{p}$, $f \in C([0,1])$ of course. Also, I let $g... |
Let $f: A \rightarrow B$ be given. Prove:
(a) For each subset $X \subset A, X \subset f^{-1}(f(X)).$
At first I didn't understand why $X$ wouldn't just be equal to $f^{-1}(f(X))$, but I guess that is the difference between an inverse image and an inverse function, right? For instance if I had $A = \{1,2,3\}, B = \{1,... |
What is the fastest way (general method) to calculate the quantity $a^b \!\mod c$? For example $a=2205$, $b=23$, $c=4891$.
|
Let $\mathcal{A}$ denote the collection of all subsets A of an uncountable set $\Omega$ for which either A or $A^c$ are countable. Let $\mu(A)$ denote the cardinality of A. Define $\phi(A)$ equal to $\emptyset$ or $\infty$ according as $A$ is countable or uncountable. Show that $\phi ≪ \mu$. Then show that the Radon-Nik... |
What are the left and right ideals of matrix ring? How about the two sided ideals?
|
I have another short question. Let $U \subseteq \mathbb{R}^2$ be open and $f: U \rightarrow \mathbb{R}$ be continuously differentiable. Also, $\partial_y f(x,y) = 0$ for all $(x,y) \in U$. I want to find a sufficient condition for $U$ such that $f$ only depends on $x$. Of course, the condition shouldn't be too restrict... |
Suppose $(\Omega, \mathcal{A}, P)$ is the sample space and $X: (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}, \mathcal{B})$ is a random variable.
Using language of measure theory,
$P(A \mid X)$, the conditional
probability of an event given a
random variable, is defined from
conditional expectation as $E(I_A
\mi... |
Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution
$$
Z_t=\exp\Big\{W_t^2-\int_0^t{(2W_s^2+1)ds}\Big\} \> .
$$
It is easy to show that $Z_t$ is a local martingale since $P(\int_0^T{(Z_sW_s)^2ds}<\infty)=1.... |
The difference is $a^2 - b^2 = (a - b).(a + b)$
But what about when I have $a^{25} + 1$ ? According to wolfram alpha, the alternate form is:
$(a+1) (a^4 -a^3 + a^2 -a + 1)( a^{20} - a^{15} + a^{10} -a^5 +1)$
However, the square root of 25 is a rational number 5.
But If I had 50, where the square root of 50 is irratio... |
Triangle is equilateral (AB=BC=CA), I need to find AB and R.
Any hints?
I was trying to make another triangle by connecting centers of small circles but didn't found anything
|
I'm facing the problem of solving
$$\sin(x)+x \cdot \cos(x)=0$$
using
$$\tan(x)=\sin(x)/\cos(x)$$
I end at
$$x+\tan(x)=0$$
on the other hand, I also tried
$\cos(x)= \pm \sqrt{1-\sin^2(x)}$
resulting in $x^2-x^2\cdot \sin^2(x)-\sin^2(x)=0$
OR using $\sin(x)= \pm \sqrt{1-\cos^2(x)}$ I get
$1-\cos^2(x)-x^2... |
Is there any example of a Lie Algebra, who has nontrivial radical but contains no abelian ideal?
Here, the radical of a Lie algebra means its maximal solvable ideal.
This question occurs in the proof of the theorem which states that "A Lie algebra is semsimple if and only if its killing form is nondegenerate." In the... |
I want to prove that
$\lim_{h\rightarrow\infty}\left(\int_{0}^{\infty}\left(\cos ht-1\right)\underset{t}{\triangle}\left[\frac{\phi(t)\exp\left(-itx\right)}{it}\right]dt\right)=-\int_{0}^{\infty}\underset{t}{\triangle}\left[\frac{\phi(t)\exp\left(-itx\right)}{it}\right]dt$
where
$\underset{t}{\triangle}\eta(t)=\eta(t)... |
This problem is as titled.
The textbook states that the order of the Weyl group of type $E_8$, $F_4$ are $2^{14}3^55^27$ and 1152 respectively, but I am wondering how are these groups like, namely, how can they be decomposed into simpler groups, or what kind of subgroup or ideals do they have.
Thanks for any attention ... |
What is the sum of all positive integers $n$ for which $2^n + 65$ is a perfect square?
|
I would like to compute $\sum_{k=0}^n S(n,k) k$, where $S(n,k)$ is a Stirling number of the second kind. Any ideas? It is like I am convolving the Stirling numbers of the second kind with the positive integers. Thank you very much!
|
As is known, $B_2$ can be realized as linear Lie algebra of elements of the form $x=
\begin{pmatrix}
0 & b_1 & b_2 \\
c_1 & m & n \\
c_2 & p & q
\end{pmatrix}$, where $c_1=-b_2^t$, $c_2=-b_1^t$, $q=-m^t$, $n^t=-n$, $p^t=-p$.
My problem is, how can I find a Chevalley basis in this algebra?
I think $H$ may be t... |
We have $n$ boxes. And initially there are $x_1, x_2, x_3, \ldots, x_n$ marbles in each box. We randomly (with equal probabilities) select one of the boxes. We take one marble from it and we put it into another (different from the origin) box chosen randomly (with equal probabilities). We continue this process until on... |
I heard an interesting question recently:
What is the minimum number of people required to make it more likely than not that all 365 possible birthdays are covered?
Monte Carlo simulation suggests 2287 ($\pm 1$, I think). More generally, with $p$ people, what is the probability that for each of the 365 days of the y... |
Let $F\subset K$ be an algebraic field extension. Is the set of all elements of $K$ that are purely inseparable over $F$ necessarily a subfield of $K$?
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.