instruction stringlengths 22 27.2k |
|---|
Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by:
$$Tp(x)=(x^2p(x))',$$
for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} (\mathscr{P}_4(\mathbb{R}),\mathscr{P}_3(\mathbb{R}))$ is defined by:
$$Sp(x)=xp''(x),$$
for all $x \in \mathbb{R}$. Let $(1,x,x^2,x^3)$ be a basis for... |
I have met this in diff. geometry class which states:
We are to determine if there exists a regular surface in $ R^3 $, $ S = f(u,v) $ with fundamental forms as follows:
$ I = \begin{bmatrix}
cos^2(u) & 0 \\
0 & 1 \\
\end{bmatrix} $
$ II = \begin{bmatrix}
1 & 0 \\
0 & cos^2(u) \... |
How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$?
So we write it as $R^3=W_1+W_2$ because every $(x_1, x_2, x_3)\in R^3$ can be written as $a((1,1,1)-(1,1,0))+b((1,1,0)-(1,0,0))+c(1,0,0)$
Do I have the correct direction?
|
Let $J_{m \times n}$ be an $m \times n$ matrix of $1$'s (and to abbreviate we write $J_m=J_{m \times m}$) and let $M=\begin{pmatrix} 0_n & J_{n \times m} \\ J_{m \times n} & 0_m \end{pmatrix}$.
I am trying to figure out how to calculate $M^2$. Doing this manually (and always using the fact that multiplying a matrix of... |
Denote by $X := [\mathbb{N}]^\infty$ the set of infinite subsets of $\mathbb{N}$. Recall that the Ellentuck topology is a topology on $X$ generated by sets of the form $\{A\text{ infinite} \mid s\text{ is an initial segment of }A\text{ and }A\setminus s\subset B\}$ for some finite $s\subset \mathbb{N}$ and (infinite) $... |
Say I am given a set of values: \begin{bmatrix}0.9&1.1&1&1.95&2&2.05&2.95&2.95&3.1&3\end{bmatrix}
Is there a way I can get the 'center' or 'average' values 1, 2, and 3 without splitting the data set up? Basically, is there a way to evaluate the entire set of numbers and get an output that says these numbers seem to exi... |
I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n \in \mathbb{Z}_n[x]$ has at least $7$ roots with multiplicity $1$).
How do I approach such a... |
I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting.
(I apologize for not posting inline images and only being able to post 2/4 of my images, but I don't have enough reputation.)
... |
Is it true that if $p: E \to B$ is a $2$-fold covering, the map $p^*$ induced in cohomology is surjective?
|
Question:
Use the Sandwich Theorem to find $$\lim_{n\to ∞} n\sum_{n+1}^{2n} \frac{1}{i^2}$$
Appreciate any guidance.
|
so I've come across this question in preparation for an exam:
Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system $A\underline{x}=\underline{b}$ .
Find the solution space to the homogeneous system of equations $A\underline{x}=\underline{0}$
Find the general... |
I have got some questions regarding matrix norms and inequalities. We only consider square, nonsingular matrices in the following.
If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all $p$ positive integer, where $||A||_p = \max_{||x||_p = 1} ||Ax||_p$?
Given $|C| \leq |A|$, is it true that $\rho(A^{-1}C) \leq \rh... |
https://i.stack.imgur.com/2SxwV.jpg
Why is the answer D? I think the answer is B. How can it be empty - since we are removing the set A?
|
We have a scalene trapezoid. We know AB and CD bases and the diagonal AC. Be P the point of intersection of the two diagonals. Is it possible to find the general expression for AP?
I would like to know the formula for AP, if it exists, depending on AB, CD, AC
|
Given a vector space $K$ of dimension $k$ over a finite field $\mathbb{F}_q$, what is the number of subspaces $T$ of dimension $t<k$ that contain a given subspace $M$ of dimension $m<t$?
This is a homework question, so I'm more looking for a hint than anything else. I conclude that since $K = K' \oplus M$ for some s... |
A quadratic form in the variables $u_i$ is expressed as $u'Du$.
Matrix $T$ consists of the n characteristic vectors (of matrix $D$): $T = [v_1\quad v_2\quad ...\quad v_n]$. The following transformation is applied to the quadratic form:
$u=Ty$, where y is a vector with n rows. This translates into: $(Ty)'D(Ty)\quad=\qu... |
If r distinct balls are distributed at random into N (N ≤ r) boxes, what is the probability that box 1 will receive exactly j balls ( 0 ≤ ≤ r)?
my solution is
[sample space] =$ N^r $
$$P=\frac{ 1}{N^r}\binom{r}{j}$$
I know there is something wrong. can you help me ?
|
I'm writing code.
My arc invariant is:
$$
\theta_0, \theta_1 \in [-2\pi, 2\pi], \\
\theta_1 - \theta_0 \leq 2\pi
$$
where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle.
The output range of atan2 is $[-\pi, \pi]$ at least according to dlang.org documentation and their implementation of it.
I ... |
Suppose that $X$ is a compact Hausdorff space and I represent $C(X)$ isometrically in $B(L^p(X,\mu))$ as multiplication operators for some finite positive regular Borel measure $\mu$. If I remember correctly, if $X$ is a compact metric space, then I may take $\mu$ to be $\sigma$-finite (?). For a general compact Hausdo... |
$$
\left(
\begin{matrix}
\pi_1 & \pi_2 & \pi_3
\end{matrix}
\right)
=
\left(
\begin{matrix}
\pi_1 & \pi_2 & \pi_3
\end{matrix}
\right)
\begin{bmatrix}
0.6 & 0.3 & 0.1 \\
0.2 & 0.7 & 0.1 \\
0.1 & 0.3 & 0.6
\end{bmatrix}
$$
I know also that $\pi_1$, $\pi_2$ and $\pi_3$ have the property $\pi_1 +\... |
Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of the symplectic form? I ask because I am curious to see how they look in Darboux coordinates.
I know that $\Gamma _{ij} ... |
Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix}
2 & 0 & 1 \\
0 & 1 & 1 \\
3 & 0 & 4 \\
\end{bmatrix}$. Prove that $U$ is the subspace of $\mathbb{M_{3\times 3}}$ (space of matrices $3\times 3$). Check if it contains $span\{I,A,A^2,...\}$. Find the dime... |
1) If $T_a : \Bbb C → \Bbb C$ is given by $T_a(z) = z + a$ then $T^{−1} _a
(z) = z − a = T_{−a}(z)$ for some fixed $a ∈ \Bbb C$.
2) If $ρ_θ : \Bbb C → \Bbb C$ is given by $ρ_θ(z) = e^{iθ}z + a$ then $ρ^{−1}_θ(z) = e^{−iθ}(z − a)$ for some fixed $a ∈ \Bbb C$.
3) If $τ : \Bbb C → \Bbb C$ is given by $τ (z) = e^{iθ}z + a$... |
I'm trying to understand how connections are locally represented, and the definition I have to work with is this:
Let $(x^1,\dots,x^n)$ be local coordinates defined in some chart $U \subset M$ such that $\xi\vert_U$ is trivial, and let $\{s_i\}$ be a frame of sections for $\xi\vert_U$. Then we have $\nabla_Xs_i=\omega... |
Find all pairs $(x,y)$ which solve
$$
\left\{
\begin{align}
9x+20y&\equiv0\mod{29}\\
16x+13y&\equiv0\mod{29}
\end{align}
\right.
$$
So I have written this in the form
$$\begin{pmatrix}9&20\\16&13\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\mod{29}$$
But I notice that $\det{A}\equiv0\mo... |
I need to solve $\vec y'=\begin{pmatrix}
1 & -1 \\
0 & 1\\
\end{pmatrix}\vec y$.
The characteristic polynomial is $(r-1)^2$, so the only eingenvalue is $1$.
I found only one eigenvector - $\begin{pmatrix}
c_1 \\
0\\
\end{pmatrix}$.
(I know it doesn't have to be written wi... |
As a prospective undergraduate who is doing pre-study in preparation for my future endeavours, i recently learnt about analytic functions and would like to know whether $f(x) = x^{-1}$ is analytic in the domain of $x$ ?
My attempt: since $\dfrac{f^{(n)}}{n!}x = -(n-1)!x^{-n-1}$ is undefined at $x=0$, we can't have a La... |
Consider $f(0)=0, f(x) = \cos\left(\frac{1}{x}\right)$ otherwise in $\mathbb{R}$
Find an open set whose preimage is not open.
I think I need an open set not including $0$.
|
{p2 ^ p3 -> p1, p3 ^ p4 -> p2, .... p9 ^ p10 -> p8}
Find the minimal model.
My professor told me the answer is the empty set,
but aren't these minimal models too?:
{p1}, {p2}, {p3}, {p4}, {p5}, {p6}, {p7}, {p8}, {p9}, {p10}
|
Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$.
Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$?
What can we say about $E[f(X)]$? Is there something like continuity of $E[f(X)]$. So, assume $X$ is a uniformly distributed ra... |
I am looking for a way to prove the following limit for integer $x$s:
$$\lim_{x\to\infty}{\frac{\log(x+2)-\log(x+1)}{\log(x+2)-\log(x)}}=\frac{1}{2}$$
I could find the result by using a computer program but I cannot formally establish the above equality.
|
Can someone explain to me in detail how to complete these two problems without using truth tables? I'm having a hard time understanding what to do. I know that I'm supposed to use the laws, etc. But I'm confused on where to even start.
Prove that ((p → q) ∧ ¬q) → ¬p is a tautology without using truth tables. Show int... |
p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud:
If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map
$$\Lambda^j F \otimes \Lambda^j G^* \rightarrow R$$
induced by $\Lambda^j \phi: \Lambda^j F \rightarrow \Lambda^j G$. If we choose bases for $F$ and ... |
I think the answer is $10/216$, but I am not sure. I did it by brute force though, and would like to know the background, as well as knowing if $10/216$ is correct.
Edit: now i think via brute force, the answer is 93/216. if you roll a 1 on any dice there are 3x6 = 18 ways of getting a 6 using any number of dice. simil... |
Let $a>0$, show that for $x>0$, $1<f(x)<2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}.$$
Source (added 20221017): It was the last question in math exam of Jiangxi Province, 2008 China's national college entrance exam (a.k.a. Gaokao).
It was said that no examinee proved $f(x) < 2$. ... |
I've asked wolfram alpha to see if there is any solution and in fact there is. Now I just need a way to do this by hand because my school wants me to know such things. So I would really appreciate any help at all, but at best I would like a step by step guide to solve this.
$2.5(e^{c \cdot 100}+e^{-c \cdot 100})=30$
|
Let $U=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_2=0, x_3+ x_4=0 \}$, $W=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_3=0, x_2+ x_4=0 \}$, how to find $U+W$ and $U\cap W$?
I think $U\cap W=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1=x_2=x_3=x_4 \}$
But I don't know how to express $U+W$
|
So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ somehow?
|
Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb A_K^\times\to\Bbb A_F^\times, \qquad N_{K/F}((a_\mu)_\mu)=(\prod_{\mu|\nu}N_{K_\mu/F_\nu}(a_\mu))_\nu$$ via the local norm ma... |
NOTE: There is another question asking "What is a Lattice?" but when reading the question, it has to do with programming, and that is not what my question has to do with. The answer provided to that question didn't answer MY question either, so please don't list this as a duplicate.
I have been researching set theory a... |
I am reading CLRS3, currently Chapter 4 and Section
4.5, "The master method for solving recurrences."
I understood what is the $\epsilon$ , but I can't understand why they choose $ \epsilon \thickapprox 0.2$ here :
$$T(n) = 3T\left(\frac{n}{4}\right) + n\lg n$$
we have $a=3$, $b=4$, $f(n) = n\lg n$, and $n^{\log_b ... |
I have a $N\times N$ grid of side $L$. Each gridpoint can be black or white and a ratio $r$ of the points is black. I want to predict the mean distance between two black points. The most appropriate is the Manhattan distance, but I think that the Euclidean would work, too.
If they were distributed inside the square ign... |
I want to solve $$\iint_{R} (x+z)dR$$ where R is the first octant of the cylinder $x^2+y^2=9$ and between $z=0$ and $z=4$
I think it could be done in either cylindrical or Cartesian.
I am having trouble finding correct bounds and relating z.
In polar we could have $r^2=9$, and first octant would be $0$ to $\pi/2$ but I... |
I have a relatively simple minimisation problem. I have to minimise a linear function with many variables (more than 20), and I would like all the solutions to be different and in set $ x \in \{1,...,n \}$, where $n$ is the number of variables.
How to set up such a constraint for an integer programming solver?
|
Define
$$\mathbf{c}_t = \begin{bmatrix}
x_{1t} \\
x_{2t} \\
\vdots \\
x_{Nt}
\end{bmatrix}\in \mathbb{R}^N$$
where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$.
I am trying to simplify
$$\sum_{j=0}^{p}\beta_j\mathbf{c}^{T}_{j+1}\mathbf{c}_{k+1} $$
where $k = 0, 1, \dots, p$ and $$\boldsymbol{\beta}=\begin{bm... |
A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) A consequence of this is that two disjoint circles $C_1, C_2 \subset D$ cannot link in their image, that is, $f(C_1)$ i... |
Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties?
$a_n>0$ and $b_n>0$
$\{a_n\}$ and $\{b_n\}$ are both decreasing
$\sum a_n$ and $\sum b_n$ both diverge
$\sum\min\{a_n,b_n\}$ converges
|
I've been trying to figure out a counting problem and can't wrap my head around how to calculate the probability.
If we let $X_{1}, . . . , X_{10}$ be independent random variable with a uniform(0,1) distribution. Find the probability that each of the intervals $[0.2i, 0.2i + .2)$, $i = 0, 1, 2, 3, 4$, contains exactly ... |
Suppose a coin is tossed repeatedly until either two successive heads appear or two successive tails appear. Then, assume that the first coin toss results in a head. I would like to find the probability of this tossing ending on two successive tails. One way I know of doing this is by creating a Markov chain and then d... |
I am trying to learn the proof for Euler's theorem which states:
If $\gcd(a,m)=1$ then $a^{\phi(m)} \equiv 1 \mod m$.
The proof goes like this. Take the reduced residue system modulo $m$. $ar_1,...,ar_k$ is also a reduced residue system modulo $m$ (the text proves this lemma) then they state that,
$$r_1 r_2...r_k \... |
Quick (and basic) group theory question:
Say G, H, K some (Lie) groups, does it in general hold that
$$
(G \times H)/H = G
$$
and that
$$
H = K\times G \to K = H / G
$$
And if so, does it then also hold that
$$
(H/G)/K = (H/K)/G
$$
where throughout for simplicity I assume every time I take a quotient that the gro... |
Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let $S$ be any other operator in $B(H)$. Then I want to show that if $\epsilon\rightarrow 0$, then $$\|T_\epsilon S T_\epsilon -T... |
Can $Cov(X,Y)$ be easily expressed in term of $Var(X), Var(Y), E(X), $ and $E(Y)$ ?
|
I need to find $$\int e^{-\sin^2x}{(\cos x -3 x \sin(x)+2 x
\sin^3(x))}dx$$ .
I know that $$\int e^{g(x)}{(f(x)g'(x)+f'(x))}dx = e^{g(x)}f(x)$$.
But I cannot find cannot $f(x)$ in the above expression.It seems really difficult to guess what $f(x)$ could be.Any other faster methods?How do I get $f(x)$ ?
|
Here is the question $$\lim_{x \to 0} {{(\cos x)^{1/2}-(\cos x)^{1/3}}\over(\sin x)^2}$$
My attempt:-> $\lim_{x \to 0} ({(\cos x)^{1/2}-(\cos x)^{1/3}})\cdot x^2/(\sin x)^2\cdot x^2 $
-> $\lim_{x \to 0} ({(\cos x)^{1/2}-(\cos x)^{1/3}})/x^2 $
->$\lim_{x \to 0} [(\cos x)^{1/6}({(\cos x)^{3}-(\cos x)^{2}})]/x^2 $
->$\lim... |
Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Let $\phi: K(X) \rightarrow K(X)$ be the $K$-morphism such that $\phi (X)=1-X$. We have $L:=\{ f\in K(X) : \phi (f)=f\}$. Find some element $Y \in K(X)$ such that $K(Y)=L$.
I am fairly certain that $Y=(2X-1)^{2}$ and can easily show that $K(Y) ... |
Suppose V is finite-dimensional and the $\mathscr{E}$ is a subspace of $\mathscr{L}(V)$ such the $ST\in \mathscr{E}$ and $TS \in \mathscr{E}$ for all $S \in \mathscr{L}(V)$ and all $T \in \mathscr{E}$. Prove that $\mathscr{E} = \{0\}$ or $\mathscr{E} = \mathscr{L}(V)$.
|
I´m having trouble writing a demonstration for the Conditional Expected Value using $\sigma$-algebra. I know its really simple and actually logic but I just can´t find the way to write it. Hope anyone can help me. Let X, Y $\in$ $L^1$($\Omega$, $\mathscr F$, P) and $\mathscr G$ $\subset$ $\mathscr F$ we need to prove t... |
I want to show:
Let $(X,d)$ be a metric space and $A \subset X$ be a closed subset. Define $f: X \to \mathbb{R}$ by
$$ f(x) = d(x,A) := \inf_{y\in A}d(x,y), \phantom{.} \forall x \in X.$$
Show that (a) $f$ is continuous and (b) $f(x) = 0$ if and only if $x\in A$.
This function is a little odd to me. By definition... |
would appreciate any hints with the proof for $x^TAx$ using index notation:
Suppose $x$ is an $n$ x 1 vector, $A$ is an $n$ x $n$ matrix. $A$ does not depend on $x$, and also $\alpha = x^TAx$.
Let $\alpha = \sum_{j=1}^{n}\sum_{i=1}^{n} x_i a_{ij} x_j$
Differentiating wrt to the $k^{th}$ element of x:
$\frac{\delta\alp... |
I am interested in finding a formula for the inverse of the function
$$f(x,y) = (x^2 + y^2, xy)$$
which works on the set $S = \{(x,y) \in \mathbb{R}^2 : -x < y < x\}$. I determined that the inverse should be
$$f^{-1}(x,y) = \left(\frac{\sqrt{x + 2y} + \sqrt{x - 2y}}{2}, \frac{\sqrt{x + 2y} - \sqrt{x - 2y}}{2}\right),$$... |
I am trying to do the following problem
Suppose that $X_1,...,X_n\stackrel{iid}\sim N(0,1)$. Define $$\bar{X}_k=\frac{1}{k-1}\sum_{i=1}^{k-1}X_i,\,\,\,\,\,\,\text{for }k=2,3,.....,n
$$
(i) What is the joint distribution of $(X_2-\bar{X}_2,X_3-\bar{X}_3,...,X_n-\bar{X}_n)$?
(ii) What is the distribution of $$\sum_{k=2}... |
I got up to the part where it's $$\frac{9}{125}\int_0^{\large \frac{\pi}{4}}\frac{\sin^2\theta}{\cos\theta}\,\,d\theta$$
but I can't figure out how to finish it off.
By the way the original problem was:
$$\int_0^{0.6}\frac{x^2}{\sqrt{9-25x^2}}\,\,dx$$
|
Find all equations of regular hyperbolas passing through the points $A(\alpha,0),B(\beta,0),$ and $C(0,\gamma)$.
Attempt: I am assuming here that regular means right. We can then write the equation of such as $(x-x_0)^2-(y-y_0)^2 =a^2$. Thus by the given we have $(\alpha-x_0)^2 -y_0^2 = a^2, (\beta-x_0)^2 -y_0^2 = a^... |
Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$?
$P_3(R)=W+ W_1$ because Span$\{1, x\}$+Span$\{x^2, x^3\}$=$a+bx+cx^2+dx^3$, which represents any polynomial.
For $W\cap W_1=\{0\}$, I don't know how to show it.
$P_3... |
Consider the probability generating function for a random variable $X$:
$\varphi_X(s)=\dfrac{7-3s}{15-14s+3s^2}$
Find an expression for $P(X=k)$, for $k\in\mathbb{N}$
My attempt was to break this expression down into a sum via partial fraction decomposition but I wasn't sure where to go from there?
Any idea how to p... |
Suppose V and W are finite-dimensional. Let $v \in V$. Let $$E=\{T \in \mathscr{L}(V,W)\ |\ Tv=0\}.$$
a.) Show E is subspace of $\mathscr{L}(V,W)$
b.) Suppose that $v \neq 0$, what is dim E?
Here is what I have so far.
Proof: Let $v,u \in V$ then $$T(u+v)=T(u) +T(v) = 0+0=0.$$
For some $\alpha \in \mathbb{R}$ we have ... |
Here is a professor's solution to the exercise which states, " Prove that if $x$ is an isolated point of a set $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$."
The professor derived a contradiction upon assuming that for an arbitrary neighborhood $N$ centered at an isolated point $x$ of $S$, we have $N ... |
I have the following problem
A steel company has four different types of scrap metal (called Typ-1 to Typ-4) with
the following compositions per unit of volume
They need to determine the volumes to blend from each type of scrap metal so that
the resulting mixture has the following amounts of Al, Si, C, and Fe:
Al - ... |
Let $n\in \mathbb{N}^+$ be a positive integer. Let $L_n\colon \mathbb{R}\to \mathbb{R}$ be the $n$'th order Legendre polynomial. Let $J_n^{(\alpha,\beta)}\colon \mathbb{R} \to \mathbb{R}$ be the $n$'th order Jacobi polynomial with weights $\alpha$ and $\beta$.
How can I show that
$$2J_n^{(0,-1)}(x) = L_n(x) + L_{n-... |
Which derivatives are eventually periodic?
I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$.
If $f(x)$ was a polynomial, and $\operatorname{deg}(f(x))=n$, note that $f^{(n)}(x)=C$ if $C$ is a constant. This implies that $f^{(n+i)}(x)=0$ for every $i... |
I am stuck on showing that $\varphi : U(55) \to U(55) $ given by $x \mapsto x^3$ is an isomorphism.
I already knwo that $\psi: U(n)\to U(n), \psi(x) = x^k$ is an isomorphism if and only if $\gcd(k,n) = 1$. But I wanted to do a proof without using this knowledge just to practice.
Now I have this:
If $x^3 = y^3$ then $0... |
Let $(a_{n})$ be a Cauchy sequence. Is $c_{n} = (-1)^{n}a_{n}$ also a Cauchy sequence?
|
What is an infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?
|
I've four vectors that spans the $\mathbb{R}^4$ sub-space $W_1$:
$\alpha_1 = \{-1,0,1,2 \}$,
$\alpha_2 = \{3,4,-2,5 \}$,
$\alpha_3 = \{0,4,1,11 \}$,
$\alpha_4 = \{1,4,0,9 \}$
And I'm asked to find an homogeneous system of linear equations such that the solution set equals $W_1$.
I found that $\alpha_1$, $\alpha_2... |
I have the following problem in my homework
Suppose a, b, are two constant paramaters such that the system below
is consistent for any values of f and g. What can you say about the
numbers a and b? Justify your answers.
x1+ax2 = f
bx1+ x2 = g
Am I under thinking this? Wouldn't a or b just have to be r... |
I don't know how to think of this question. Could it be a sequence that eventually repeats the same digit over and over again? How would I explain this?
|
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Assume $u\in L^\infty(\Omega)$. We know a sequence $u_n\in L^\infty(\Omega)$ such that
$$
\sup_{n}\|u_n\|_{L^\infty}<+\infty
$$
and $u_n\to u$ in $L^2$ weakly.
Now, assume that for any $\Omega'\subset\subset \Omega$, i.e., compactly contained in, we ha... |
Could anybody recommend an online course for implementing numerical methods to solve PDEs which can supplement reading? This is with a view to writing an implementation to solve the Monge-Ampere equation.
I have taken basic courses in numerical analysis and PDEs before, so it would be nice to refresh my memory and sol... |
I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only have $f(x)$ related to its extension, not equal to it (Thm: Suppose $f(x)$ is a piecewise smooth on the interval $0\le x... |
Let $F\subset X$ be a finite family of sets of $X$. Is the sigma-algebra generated by $F$ ($\sigma(F)$) also finite?
I was trying to use induction: If $F$ has one element say $A$ then $\sigma(F)=\{\varnothing, X, A, A^c\}$
Suppose that it is true for a family of sets with $n$ elements. Know Let $F$ a family of sets wi... |
I was reading about magic squares and suppose that we speak here only of the magic squares that have in itself numbers from $1$ to $n^2$.
It is easy to see that we cannot have $2$x$2$ magic square because we cannot arrange numbers $1,2,3,4$ in such a square so that the sum of numbers in every row, column and diagonals ... |
The return is a given, but what about the other implication? We couldn't solve it in class. Sorry about the formatting, I hope it isn't an issue.
|
The composition of convex functions is not necessarily convex or concave: For example, composing $f_1(x) = x^2-1$ and $f_2(y) = y^2$ gives $f_2(f_1(x)) = (x^2-1)^2$. Or consider $f_1(x) = x^2$ and $f_2(y) = e^{-y}$, which yield $f_2(f_1(x)) = e^{-x^2}$. (Of course, the composition is convex if the outer function is ... |
Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I seem to have proved that $R_P$ is regular if and only if $R_{(P)}$ is regular. Do you agree? Also, is there any reason to... |
Using the Lorentzian as the delta function
$$\delta(x) ~=~ \lim_{\epsilon\rightarrow 0} \frac{1}{\pi}\frac{\epsilon^2}{\epsilon^2+x^2}$$
Is there a way to rigorously prove the sifting property, namely
$$\int^{\infty}_{-\infty} f(x) \delta(x-t) dx = f(t)$$
|
I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the fact that $f(1) =1$,$f(-1)=-f(1)$, $f(1+1) = f(1) +f(1)$, .... etc. to prove the claim? Thanks for your time as always guys... |
How would you connect each black box once to each colored box without any lines overlapping, this is racking my brain so please help.
Note that you can move the boxes where ever you want.
Maybe there's some math trickery involved as I have spent hours trying to crack it but cannot seem to without having no option but t... |
Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment
$$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, \mathcal{G}|U)$$
makes a sheaf of $\mathcal{O}_X$-modules. (The restriction should be the natural one.)
The question is to s... |
Let $S_1$ and $S_2$ be linearly dependent subsets of vector space $V$, are $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent?
The counterexample for the first one I can think of is $S_1=\{(1,0),(0,1), (1,1)\}$ and $S_2=\{(1,0),(0,1), (0,0)\}$.
The counterexample for the second one I can think of is $S_1=\... |
I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges).
$G$ orderable means there is a total ordering s. t. for all $g,h$, $g\leq h$ implies $gk\leq hk$ and $kg\leq kh$ for all $k$.
My idea is to take the following set of formulae:
$$\{p... |
Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with this one, because my proof begins with an existential neighborhood and ultimately considers a different route in proving th... |
Let $v: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function such that $v(y) \neq 0,\forall y \in \mathbb{R^n}$ that is differentiable at $ x \in \mathbb{R^n}$.
(a) Show that the value-wise magnitude $|v|: \mathbb{R^n} \rightarrow \mathbb{R}$ defined by $|v|(y) := |v(y)|, \forall y \in \mathbb{R^n}$
is differentiable ... |
I have a random walk Markov chain that has states from $0$ to $N$. The conditions are that when the chain is at $0$, the chain will go to state $1$ with probability $1$. When the chain is at state $N$, it will go to state $N-1$ with probability $1$ as well. However, for everything between, for any other states $k$, the... |
$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$.
Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$.
Then, find $$E[X^k|Z=z]$$.
I need help getting started.
$$P[X|Z=z] = \frac{P[X, Z=z]}{P[Z=z]}$$
$$= \frac{({\lambda e^{-\lambda x}})({\lambda e^{... |
My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula.
However, while I have been successful in understanding addition, subtraction and multiplication of 'g' and 'f':
Say
$f(x) = 2x^2 + 15x -8$
$g(x) = x^2 + 10x + 16$
I ... |
Let $V$ be a vector space and $\beta= \{ u_1,\dots ,u_n \}$ be a subset of $V$.
$\Rightarrow$ $\beta$ is a basis for $V$ iff each vector $v\in V$ can be unquiley expressed as a linear combination of vectors in $\beta$ , that is , can be expressed in the form
$$v=a_1*u_1+\dots +a_n u_n $$
Def of basis $\beta$ for a v... |
Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$.
So my teacher told us to use the formula $\Delta u=\frac{1}{r}\frac{\partial}{\partial r}(r \frac{du}{dr})$ and solve the equ... |
Let $T(a,b) = (a+b,2a-b,3a)$.
a)Find the matrix representing $T$.
b)Find the image of $T$ (as a span of vectors).
So I found that $T$ is a linear transformation. Now would the matrix just be
$A$= $\begin{bmatrix}1 & 1\\2 & -1\\ 3 &0\end{bmatrix}$?
I feel like this would be too easy.
Also how would I go about findin... |
I'm really confused on how to do modular reduction. I understand we're supposed to take the factor of the exponent?
for example how would I go about doing modular reduction on:
$5^{17}$ mod 16
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.