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# Trying to prove that $\mathcal P(\Bbb R)'\cong\Bbb R^\infty$
Prove that $\mathcal P(\Bbb R)'$ and $\Bbb R^\infty$ are isomorphic vector spaces.
Here $\mathcal P(\Bbb R)'$ is the dual space of the vector space of polynomial functions with real coefficients $\mathcal P(\Bbb R)$. And $\Bbb R^\infty$ is the set of all ... | {
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$$H_m:=\{B_k:k\in\{0,\ldots,m\}\}$$
is a basis of $\mathcal P_m(\Bbb R)'$.
But Im stuck here: my main problem is that I dont know if I can prove (and how, if it would be possible) that $S=\mathcal P(\Bbb R)'$ from the finite case, that is that all the functionals of $\mathcal P_m(\Bbb R)'$ have the form
$$p\mapsto\s... | {
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Note: the notation $\mathbb R^\infty$ is not standard, and in fact some authors use it to refer to the space of sequences for which only finitely many entries are non-zero. (This space is actually isomorphic to $\mathcal P(\mathbb R)$ - can you prove it?) Writing $\mathbb R^\mathbb N$ is more standard, and is a special... | {
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• what means $\ell(x^n)$? The concept of action is not shown in this book (at this moment). May 17 '17 at 17:02
• The phrase "its action on" may be replaced with "what it does to". By $\ell(x^n)$ I mean the real number one gets when applying a linear functional $\ell$ to $x^n$. May 17 '17 at 17:06
• I see but, how you ... | {
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# Math Help - Financial Maths - Future Value Calculation
1. ## Financial Maths - Future Value Calculation
The problem:
Zanele plans to save R50 000 towards buying a new car in three years' time. She makes three equal deposits at the beginning of each year into a savings account, starting immediately. The interest pai... | {
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3. ## Re: Financial Maths - Future Value Calculation
To avoid searching fior the right factor
50000= R(1.0275)^12 +R(1.0275)^8 + R 91.02750^4 which gives R =13362.60
4. ## Re: Financial Maths - Future Value Calculation
Originally Posted by bjhopper
To avoid searching fior the right factor
50000= R(1.0275)^12 +R(1.02... | {
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# An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$
The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$B_n^2 \leq B_{n-1}B_{n+1}$$ for $n \geq 1$, where $B_n$ denotes the $n$th Bell number (i.e. the... | {
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This answer is courtesy of Bouroubi (paraphrased):
Theorem. Define $B(x)=e^{-1}\sum_{k=0}^\infty k^x k!^{-1}$. Dobinski's formula states $B(n)=B_n$ is the $n$th Bell number. Now we let $\frac{1}{p}+\frac{1}{q}=1$. Then $$B(x+y)\le B(px)^{1/p}B(qx)^{1/q}.$$ Proof. Let $Z$ be the discrete random variable with density fu... | {
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Here's a combinatorial argument. Let $S_n$ denote the total number of sets over all partitions of $\{1, 2, \ldots, n\}$, so that $A_n = \frac{S_n}{B_n}$ is the average number of sets in a partition of $\{1, 2, \ldots, n\}$.
First, $A_n$ is increasing. Each partition of $\{1, 2, \ldots, n\}$ consisting of $k$ sets maps... | {
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(Added: The Bender and Canfield paper mentioned below gives this bound as well.)
"The log-convexity of the Bell numbers was first obtained by Engel ["On the average rank of an element in a filter of the partition lattice," Journal of Combinatorial Theory Series A 65 (1994) 67-78] . Then Bender and Canfield ["Log-conca... | {
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# Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$
If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if :
$$(tr(A))^n\geq n^n \det(A)$$
What i have tried is :
As $A\in M_{n\times n} (\mathbb{R})$ a positive defini... | {
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This is really a Calculus problem! Indeed, let us look for the maximum of $h(x_1,\dots,x_n)=x_1^2\cdots x_n^2$ on the sphere $x_1^2+\cdots+x_n^2=1$ (a compact set, hence the maximum exists). First note that if some $x_i=0$, then $h(x)=0$ which is obviously the minimum. Hence we look for a conditioned critical point wit... | {
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For convenience, we use the notation $A\succ 0$ to indicate that a symmetric matrix $A$ is positive definite. We can see the inequality $(tr(A))^n\geq n^n \det(A),\;\forall A\succ 0$ as $$\frac{1}{n}\mathrm{trace}(A)\geq \sqrt[n\,]{\det(A)}, \quad \forall A\succ 0.$$ Note that, if $A=\mathrm{diag}(\lambda_1,\ldots,\lam... | {
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# Prove that for any positive integer n, the following sum is a perfect square
Prove that for any positive integer n, the following sum is a perfect square:
$1+8+16+...+8n$.
I have tried to solve the problem by trying to find the equation that would give me the desired values, but since the values differ by so much wi... | {
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# I Alternating Series, Testing for Convergence
1. Apr 13, 2016
### Staff: Mentor
The criteria for testing for convergence with the alternating series test, according to my book, is:
Σ(-1)n-1bn
With bn>0, bn+1 ≤ bn for all n, and lim n→∞bn = 0.
My question is about the criteria. I'm running into several homework pr... | {
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3. Apr 13, 2016
### pwsnafu
If you can show $\sum_{n=13}^\infty (-1)^n b_n$ converges, then trivially $\sum_{n=1}^\infty (-1)^n b_n$ also converges because all you are doing is adding a finite number of terms. Also $(-1)^{n} = - (-1)^{n-1}$ so it's the same thing.
4. Apr 13, 2016
### Staff: Mentor
That's what I fi... | {
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Page 1 of 1
### Calculating modular inverses of p mod $2^p$
Posted: Thu Mar 11, 2021 8:51 am
Let p an odd prime. We want to calculate $I(p)=p^{-1}\mod 2^p$.
If we use EXGCD, it takes $O(\log_22^p)=O(p)$ time, which is not efficient enough.
Some small values are 3, 13, 55, 931, 3781, 61681, 248347, 4011943, 259179061.... | {
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A variant approach is to use the form of Fermat's little theorem that says that $2^{p-1} \equiv 1 \pmod p$, so that $\frac{2^{p-1} - 1}{p}$ is an integer. We have a $-1$ rather than a $+1$ in the numerator, so square the numerator to get $q = \frac{(2^{p-1} - 1)^2}{p} = \frac{2^{2p-2} - 2^p + 1}{p} = \frac{2^p (2^{p-2}... | {
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ABCD is a quadrilateral with angle ABC a right angle. The point D lies on the perpendicular bisector of AB. The coordinates of A and B are (7, 2) and (2, 5) respectively. The equation of line AD is y = 4x − 26. find the area of quadrilateral ABCD
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5. 🚩
1. Equation of the perpendicular bisector ... | {
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4. 🚩
4. I am not sure how you got 40+4 and 30+35
btw thx for your help
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5. never mind i got it- it's just like matching
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## Similar Questions
1. ### Geometry
ABCD is a quadrilateral inscribed in a circle, as shown below: Circle O is shown with a quad... | {
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"openwebmath_score": 0.6336784362792969,
"tags": nu... |
4. ### Geometry
Angle ABC and ange DBE are vertical angles,the measure of angle ABC =3x+20, and the measure or angle DBE =4x-10. Write and solve an equation to find the measure of angle ABC and the measure of angle DBEin
1. ### Mathematics
Using ruler and a pair of compasses only construt triangle ABC such that AB 8... | {
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# Lining up nonconsecutive multi-line equations
I have the following code:
\documentclass[fleqn, 12pt]{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{graphicx}
\graphicspath{ {./images/} }
\setlength{\parskip}{\baselineskip}%
\setlength{\parindent}{0pt}%
\begin{document}
\raggedright
From the model,
\beg... | {
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Thanks.
• Why not simply put since the first normal equation gives $Y_i = \beta_0 + \beta_1 X_i$ and $\bar{Y} = \beta_0 + \beta_1 \bar{X}$. after the display? There's little added value (if any) from putting the comment in the middle. – egreg Jun 3 '18 at 14:07
• @egreg You mean after the entire multi-line equation? I... | {
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\begin{document}
\raggedright
From the model,
\begin{align*}
\E( Y_i - \bar{Y} )
&= \E[(\beta_0+\beta_1 X_i) - (\beta_0-\beta_1\bar{X})] \\
&\qquad\parbox[t]{0.5\textwidth}{(since the first normal equation gives
$Y_i = \beta_0 + \beta_1 X_i$ and $\bar{Y} = \beta_0 + \beta_1\bar{X}$)}\\
&= \beta_1(X_i-\bar{X}) \,.
\en... | {
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"ta... |
From the model,
\begin{align*}
E( Y_i - \bar{ Y } )
={}& E[ ( \beta_0 + \beta_1 X_i ) - ( \beta_0 - \beta_1 \bar{ X } ) ] \\
&\devioustrick{
(since the first normal equation gives
$Y_i = \beta_0 + \beta_1 X_i$ and
$\bar{Y} = \beta_0 + \beta_1 \bar{X}$
and since the definitive answer on life, the universe
and everything... | {
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# Definition:Convergent Sequence/Metric Space
## Definition
Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
### Definition 1
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}:... | {
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Note the way the definition is constructed.
Given any value of $\epsilon$, however small, we can always find a value of $N$ such that ...
If you pick a smaller value of $\epsilon$, then (in general) you would have to pick a larger value of $N$ - but the implication is that, if the sequence is convergent, you will alw... | {
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# Lecture 10: Survey of Difficulties with Ax = b
Flash and JavaScript are required for this feature.
## Description
The subject of this lecture is the matrix equation $$Ax = b$$. Solving for $$x$$ presents a number of challenges that must be addressed when doing computations with large matrices.
## Summary
Large c... | {
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But now, I want to get real, here, about different situations. So number 1 is the good, normal case, when a person has a square matrix of reasonable size, reasonable condition, a condition number-- oh, the condition number, I should call it sigma 1 over sigma n. It's the ratio of the largest to the smallest singular va... | {
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So we could pick the minimum norm solution, the shortest solution. That would be an L2 answer. Or we could go to L1. And the big question that, I think, might be settled in 2018 is, does deep learning and the iteration from stochastic gradient descent that we'll see pretty soon-- does it go to the minimum L1? Does it p... | {
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Because just the organization of Gram-Schmidt is interesting. So Gram-Schmidt, you could do the normal way. So that's what I teach in 18.06. Just take every column as it comes. Subtract off projections onto their previous stuff. Get it orthogonal to the previous guys. Normalize it to be a unit vector. Then you've got t... | {
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So that's coming. This is section 2.2, now. But there's more. 2.2 has quite a bit in it, including number 0, the pseudo inverse, and including some of these things. Actually, this will be also in 2.2. And maybe this is what I'm saying more about today. So I'll put a little star for today, here. What do you do?
So this... | {
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So you have a matrix that's nearly singular. It's got singular values very close to 0. What do you do then? Well, the world of inverse problems thinks of adding a penalty term, some kind of a penalty term. When I minimize this thing just by itself, in the usual way, A transpose, it has a giant inverse. The matrix A is ... | {
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And then that name I erased is Krylov, and there are other names associated with iterative methods. So that's the section that we passed over just to get rolling, but we'll come back to. So then that one, you never get the exact answer, but you get closer and closer. If the iterative method is successful, like conjugat... | {
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That is the end of this chapter, 2.4. So this is all chapter 2, really. The iterative method's in 2.1. Most of this is in 2.2. Big is 2.3, and then really big is randomized in 2.4. So now, where are we? You were going to let me know or not if this is useful to see. But you sort of see what are real life problems. And o... | {
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The SVD gives you an answer. Boy, where should that have gone? Well, the space over here, the SVD. So that produces-- you have A = U sigma V transposed, and then A inverse is V sigma inverse U transposed. So we're in the case, here. We're talking about number 5. Nearly singular, where sigma has some very small, singula... | {
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And one question, naturally, is, what should delta be? Well, that question's beyond us, today. It's a balance of what you can believe, and how much noise is in the system, and everything. That choice of delta-- what we could ask is a math question. What happens as delta goes to 0? So suppose I solve this problem. Let's... | {
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And let me keep going on this 1 by 1 case. This would be A transpose. A is just a sigma. I think it's just sigma b. So A is 1 by 1, and there are two cases, here-- Sigma bigger than 0, or sigma equals 0. And in either case, I just want to know what's the limit. So the answer x-- let me just take the right hand side. We... | {
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1 over sigma. So now, let delta go to 0. So that approaches 1 over sigma, because delta disappears. Sigma over sigma squared, 1 over sigma. So it approaches the inverse, but what's the other possibility, here? The other possibility is that sigma is 0. I didn't say whether this matrix, this 1 by 1 matrix, was invertible... | {
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But at sigma equals 0, it's 0. You see, that's a really strange kind of a limit. Now, it would be over there. What have I found here, in this limit? Say it again, because that was exactly right. The pseudo inverse. So this system-- choose delta greater than 0, then delta going to 0. The solution goes to the pseudo inve... | {
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The limit-- as delta goes to 0, this thing is suddenly discontinuous. It's this number that is growing, and then suddenly, at 0, it falls back to 0. Anyway, that would be the statement. Actually, statisticians discovered the pseudo inverse independently of the linear algebra history of it, because statisticians did exa... | {
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And we saw what happened in L2. Well, we saw it for 1 by 1. Would you want to extend to prove this for any A, going beyond 1 by 1? How would you prove such a thing for any A? I guess I'm not going to do it. It's too painful, but how would you do it? You would use the SVD. If you want to prove something about matrices, ... | {
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So that's the logic, and it would be in the notes. Prove it first for 1 by 1, then secondly for diagonal. This, and finally with A's, and they're using the SVD with and U and V transposed to get out of the way and bring us back to here. So that's the theory, but really, I guess I'm thinking that far the most important ... | {
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# Math Help - Hard math problem
1. ## Hard math problem
My friend gave me a good question today so I want to see if you smart geniuses can answer it
Consider a fixed line AB=4. Also consider a line CD, such that
AB=CD=4, and that AB is the perpendicular bisector of CD AND CD is the perpendicular bisector of AB.
Now,... | {
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I said it was a cirlce with area 4pi, but he said it was wrong?
What other familar shape can it possibly be? If i draw it it seems like a circle (i just plotted a lot of points)
1. Please don't double post. It's against the rules and it wastes everybodies time.
2. See attachment
5. It is an ellipse, with foci at A an... | {
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Question
# 11. What are the possible combination outcomes when you toss a fair coin three times? (6.25...
11. What are the possible combination outcomes when you toss a fair coin three times? (6.25 points) H = Head, T = Tail a {HHH, TTT) Ob. (HHH, TTT, HTH, THT) c. {HHH, TTT, HTH, THT, HHT, TTH, THH) d. (HHH, TTT, HT... | {
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We know that,
Probability = Number of favourable outcomes / Total number of outcomes
The probability of getting no heads at all for tossing a fair coin three times is,
$P[B]=n(B)/n(S)$
$=1/8$
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• ### A far coin is tossed thre... | {
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• ### Need help: For the experiment described, write the indicated event in set notation
Need help: For the experiment described, write the indicated event in set notation. A coin is tossed three times. Represent the event "the first two tosses come up the same" as a subset of the sample space. A) {(tails, tails), (he... | {
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• ### 15. How many possible combination outcomes consist of two heads when you toss a fair coin...
15. How many possible combination outcomes consist of two heads when you toss a fair coin four times? (6.25 points) a. 4 b. 5 c. 6 d. 7 e. None of these
• ### My No O-5 points LarPCac 92012 Determine whether the sequenc... | {
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# Expected remaining minutes given fixed number of events
An interview question that was relayed to me by a colleague:
A machine beeps exactly 5 times within a time span of 10 minutes. However, the distribution of these 5 beeps across the entire time span is fully randomized - any distribution of the 5 beeps across t... | {
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These plots summarize a simulation of 10,000 independent iterations of this beeping experiment. The histogram at left confirms the beeps were uniformly distributed (within random variation). The five scatterplots compare each gap with its successor (in terms of the total time, so one unit is ten minutes). (Gap 0 is the... | {
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n <- 1e4
n.sample <- 6 # One larger than the number of beeps
#
# Generate data.
# This is a fast way to generate uniform[0,1] order statistics,
# avoiding an explicit sorting operation.
#
x <- matrix(rexp(n*n.sample), nrow=n.sample) # IID exponential
x <- apply(x, 2, function(y) cumsum(y))
x <- t(x) / x[n.s... | {
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# Partial sum of divergent series
I am trying to find the nth partial sum of this series: $$S(n) = 2(n+1)^2$$
I found the answer on WolframAlpha:
$$\sum_{n=0}^m (1+2n)^2 =\frac{1}{3}(m+1)(2m+1)(2m+3)$$
How can I calculate that sum, without any software?
• It's a sum of squares. You may learn from math.stackexchang... | {
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"openwebmath_score": 0.8016624450683594,
"ta... |
There are a few ways to do this. I think that this one is intuitive.
In the first triangle, the sum of $$i^{th}$$ row equals $$i^2$$
The next two triangles are identical to the first but rotated 120 degrees in each direction.
Adding corresponding entries we get a triangle with $$2n+1$$ in every entry. What is the $$... | {
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# Show that $\lim\limits_{x\rightarrow 0}f(x)=1$
Suppose a function $f:(-a,a)-\{0\}\rightarrow(0,\infty)$ satisfies $\lim\limits_{x\rightarrow 0}\left(f(x)+\frac{1}{f(x)}\right)=2$. Show that $$\lim\limits_{x\rightarrow 0}f(x)=1$$
Let $\epsilon>0$ , then there exists a $\delta>0$ such that $$\left(f(x)+\frac{1}{f(x)}... | {
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which implies that
$$\lim_{x \to 0} \left( f(x) - \frac{1}{f(x)} \right) = 0.$$
Combining both results together we have
$$2 = 2 + 0 = \lim_{x \to 0} \left( f(x) + \frac{1}{f(x)} \right) + \lim_{x \to 0} \left( f(x) - \frac{1}{f(x)} \right) = \lim_{x \to 0} 2f(x)$$
which implies that $\lim_{x \to 0} f(x) = 1$.
• Ye... | {
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# Spanning set definition and theorem
I need a bit of clarification in regards to the spanning set. I am confused between the definition and the theorem.
Definition of Spanning Set of a Vector Space: Let $S = \{v_1, v_2,...v_n\}$ be a subset of a vector space $V$. The set is called a spanning set of $V$ if every vect... | {
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Theorem 4.7 Span(S) is a subspace of V: If $S = \{v_1, v_2,...v_n\}$ is a set of vectors in a vector space $V$. then $span(S)$ is a subspace of $V$. Moreover, $span(S)$ is the smallest subspace of $V$ that contains $S$, in the sense that every other subspace of $V$ that contains $S$ must contain $span(S)$.
Question: T... | {
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• Who says $\text{span}(S)=V$ in theorem $4.7$? – Mathematician 42 Mar 14 '17 at 18:13
• Elementary Linear Algebra, Sixth Edition, By Larson, Edwards, and Falvo, chapter 4, section 4.4, Spanning Sets and Linear Independence, page# 211. It is not the book I am using but because I was confused so I decided to look it up ... | {
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# Find a solution for the equation $a(4n) = {a(n)\over 4}$ [closed]
I was given the equation $a(4n) = {a(n)\over 4}$ where $a(1) = 1$. I know that ${1\over \sqrt n }$ solves this equation, but I don't know how I would find this solution by hand if I didn't know about it.
Any hints on this matter are greatly appreciate... | {
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If you want a more systematic way, define
$$b_{n}\equiv a_{4^{n}}$$
Then
$$b_{0}=a_{1}=1$$
$$b_{n+1}=a_{4^{n+1}}=a_{4\cdot4^{n}}=\frac{a_{4^{n}}}{2}=\frac{b_{n}}{2}$$
with a solution $b_{n}=\frac{1}{2^{n}}$. Now reverse to get
$$a_{n}=b_{\log_{4}n}=\frac{1}{2^{\log_{4}n}}=\frac{1}{4^{\frac{1}{2}\log_{4}n}}=\frac{... | {
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# Writing a function $f$ when $x$ and $f(x)$ are known
I'm trying to write a function. For each possible input, I know what I want for output. The domain of possible inputs is small:
$$\begin{vmatrix} x &f(x)\\ 0 & 2\\ 1 & 0\\ 2 & 0\\ 3 &0\\ 4 &0\\ 5 &0\\ 6 &1\\ \end{vmatrix}$$ My thought is to start with a function ... | {
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First you'd need to explain how you're using the term "function". Under the standard usage of that term, your first table already defines a function and there would be nothing left to do. Perhaps you're looking for an expression in terms of basic arithmetic operations and/or well-known elementary functions? If so, yo... | {
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Between this and Tanner's reference to Curve Fitting, I have a good sense of how to approach this now. While I much prefer mzuba's simple expression, I suspect that it is more of an ad-hoc solution than this general approach. – PunctuallyChallenged Feb 12 '12 at 19:33
You can generalise the problem: suppose you kno... | {
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\begin{align} f(x) &= 2 \times \dfrac{(x-1)(x-2) \dots (x-6)}{(0-1) (0-2) \dots (0-6)} + 0 \times (\text{stuff}) + 1 \times \dfrac{x(x-1) \dots (x-5)}{6(6-1)(6-2) \dots (6-5)}\\ &= 2 \dfrac{(x-1) (x-2) \dots (x-6)}{720} + \dfrac{x(x-1) \dots (x-5)}{720} \\ &= \dfrac{(x-1)(x-2)(x-3)(x-4)(x-5)}{720} \left[ 2(x-6) + x \ri... | {
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As long as we're giving silly answers, here's another one: $f = \chi_{\lbrace0\rbrace} + \chi_{\lbrace0,6\rbrace} = 2\chi_{\lbrace0\rbrace} + \chi_{\lbrace6\rbrace}$, where $\chi_A$ denotes the indicator function of the set $A$. – kahen Feb 12 '12 at 19:18
There is absolutely nothing wrong with using the table of v... | {
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In fact, for numerical evaluation purposes, one could use this form directly, with some care needed when $x$ is equal to an interpolation point. (If $x$ is nearly, but not equal to, an interpolation point, the method still performs with good accuracy; see the linked article for a deeper discussion.)
For OP's case, we ... | {
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# Knights and Knaves
A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet four inhabitants: Bozo, Marge, Bart and Zed.
• Bozo says," Bart and Zed are both knights".
• Marge tells you that both Bart is a knight and Zed is a knave
• Bart tells you... | {
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A quick way to solve it is to suppose that Bart is a knight. Then he’s telling the truth, so Marge and Zed are also knights. But that’s impossible, because Marge said that Zed is a knave: if she’s a knight, she’s telling the truth, and Zed isn’t knight. Thus, Bart cannot be a knight and must therefore be a knave. Can y... | {
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You have chosen a good formalization here: with the corrections from Brian M. Scott's answer, you are given that \begin{align} (0) \;\;\; & Bo \equiv Ba \land Ze \\ (1) \;\;\; & Ma \equiv Ba \land \lnot Ze \\ (2) \;\;\; & Ba \equiv Ma \land Ze \\ (3) \;\;\; & Ze \equiv Bo \land Ma \\ \end{align} Now, looking at the sha... | {
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# Roll 5 dice simultaneously
1. Dec 7, 2015
### spacetimedude
1. The problem statement, all variables and given/known data
Calculate
a) P(exactly three dice have the same number)
b) Calculate the conditional probability P(three of the dice shows six|two of the dice shows 5)
2. Relevant equations
3. The attempt at ... | {
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Any help will be appreciated!
2. Dec 7, 2015
### Staff: Mentor
This the probability for "the first 3 dice show the same number, the last two do not have this number" (or any other set of 3 specific dice). It does not matter which three dice show the same number, you have to take this into account.
Also, you include... | {
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55XXX
5X5XX
etc.
5. Dec 7, 2015
### Staff: Mentor
"exactly two", yes.
Note that you can also choose the 3 non-5 dice instead of the 2 dice that are 5, as (5 choose 3) = (5 choose 2). It is still just one choice that fixes both groups.
6. Dec 7, 2015
### spacetimedude
That makes it much more clear! Thanks so much... | {
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# for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed?
for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed in the metric space $(\mathbb{Q},d)$ where $d(x,y) = |x-y|$
my attempt:
I suspected it's closed for all real numbers:
let $x,y \in \mathbb{Q}$ then if $[x,y]\cap \mathbb{Q}$ is closed it means th... | {
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• Could you explain why in all cases the complement would be in that form? why for instance could I not have $(\leftarrow, x] \cup [y, \rightarrow)$ – hellen_92 Jan 26 '15 at 19:21
• I say this because, if $x,y$ where irrational, then wouldn't the set $\mathbb{Q} \cap [x,y]$ infact just be $\mathbb{Q} \cap (x,y)$ and t... | {
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The answer is quite obvious when phrased this way. Do you see how to find $W$?
• Be aware that this definition is not universal. In some treatments relatively open sets are defined as intersections with the subspace of an open set in the ambient space, and then relatively closed sets are defined as relative complement... | {
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# Two questions on topology and continous functions
I have two questions: 1.) I have been thinking a while about the fact, that in general the union of closed sets will not be closed, but I could not find a counterexample, does anybody of you have one available?
2.) The other one is, that I thought that one could pos... | {
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Edit: Here's my argument for why $C$ is closed. Suppose $(a,b) \in \mathbb{R}^2$ lies inside $\overline{C}$. We will show that $(a,b)$ must be in $C$. Since $\mathbb{R}^2$ is first countable, there exists a sequence $y_n$ in $C$ converging to $(a,b)$. By the definition of $C$, we can write $y_n = (x_n,1/x_n)$. Therefor... | {
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as $C$ is closed. It follows that $f[C]$ equals its closure, hence is closed. So $f$ is a closed map.
Sort of dually: $f$ is continuous iff
$$\forall A \subset X : f[\overline{A}] \subset \overline{f[A]}$$
So the other inclusion then holds.
If $f$ is continuous, and $A \subset X$, $$A \subset f^{-1}[f[A]] \subset f... | {
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# Prove a matrix of binomial coefficients over $\mathbb{F}_p$ satisfies $A^3 = I$.
(This problem is problem $1.16$ in Stanley's Enumerative Combinatorics Vol. 1).
Let $p$ be a prime, and let $A$ be the matrix $A = \left[\binom{j+k}{k} \right]_{j,k = 0}^{p-1}$, taken over the field $\mathbb{F}_p$. Show that $A^3 = I$,... | {
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Lemma: $A_{p - i - 1,j} \equiv (-1)^{j}\binom{i}{j} \pmod{p}$. Proof of Lemma: We note that $A_{p - i - 1,j} = \binom{p - i + j - 1}{j}$ is a polynomial in $p$ with integer coefficients. Thus, when viewed $\pmod{p}$, we have that it is equivalent to its constant term. The constant term is $$\frac{(-i + j - 1)(-i + j - ... | {
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Thus $A^4 = A$. Since $A$ is full rank, this implies that $A^3 = I$, as desired.
• A can be factored as $BB^T$ for a triangular matrix B consisting of elements from pascals triangle using LU decomposition. This should not be hard to show. This might help but I have no idea. – Per Alexandersson Jun 18 '15 at 1:11
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The vectors $e(x)$, $x\in \mathbb F_p$, are linearly independent. Therefore, $A^3 = I$.
Concerning the eigenvalue question, there are two cases depending on whether the equation $x = (1-x)^{-1}$, equivalently, $(2x-1)^2 = -3$, is solvable in $\mathbb F_p$. (A pair of) solutions exist iff $p= 1\pmod 6$.
1. There is no... | {
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• "Differentiate with respect to $x$" I'd rather not. – darij grinberg Aug 16 '15 at 16:41
• @darij grinberg, it's a typo: I differentiate w.r.t. $z$, the identity $\sum_{j=0}^{p-1} {k+j\choose j} z^j = \frac{1-z^p}{(1-z)^{k+1}}$ in $\mathbb F_p(z)$, not the identity (1) in $\mathbb F_p$. – zhoraster Aug 16 '15 at 19:0... | {
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# Does the Extended Euclidean Algorithm always return the smallest coefficients of Bézout's identity?
Bezout's identity says that there are integers $x$ and $y$ such that $ax + by = gcd(a, b)$ and the Extended Euclidean Algorithm finds a particular solution. For instance,
$333\cdot-83 + 1728\cdot16 = \gcd(333, 1728) ... | {
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If $n=1$ we have just one line $a=qb+1$, so the Bezout identity is $a-qb=1$: the coefficients are $x=1$, $y=-q$ and we have $$|x|\le b/2\ ,\quad |y|=q\le qb/2<a/2\ .$$
Now suppose that a procedure of $n-1$ steps gives $$bX+r_1Y=1$$ where by induction we may assume $$|X|\le r_1/2\ ,\quad |Y|\le b/2\ .$$ Then the final ... | {
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• The extension to greatest common divisors of more than two numbers is doubtless an interesting Question, but it's not clear if you are actually answering the Question posed here, whether Extended Euclidean algorithm can promise the optimal size coefficients. The Question is a bit vague (about how size should be measu... | {
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# When I solve for eigenvectors using determinant, is it possible to have an entry in the eigenvector 0 and the other anything?
$$\left[\begin{array}{ccc} 4 & 0 \\ 2 & 2 \end{array}\right]$$
I am trying to find the two eigenvalues associated with this matrix. So I find the eigenvalues $\lambda$ that make $\det(A-\lam... | {
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You've found the eigenvectors, for $\lambda =2$ it is $\begin{pmatrix}0 \\ v_2 \end{pmatrix}$ for any $v_2 \neq 0$. Eigenvectors are never unique, as any scalar multiple of an eigenvector is still an eigenvector. If all you need is one eigenvector, you can take $v_2$ to be anything so $\begin{pmatrix}0 \\ 1 \end{pmatri... | {
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# Numerical stability of an integral solved by recursion
The integral $$I_n = \int_0^1 x^n e^x\,\mathrm{d}x \quad n=0,1,2,\cdots$$ suggests a recursion relation obtained by integration by parts: $$I_n = \left[ x^n e^x \right]_0^1 - n\int_0^1x^{n-1}e^x\,\mathrm{d}x = e - nI_{n-1}$$ terminating with $I_0 = e-1$. However... | {
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The figure below shows the forwards algorithm becoming extremely unstable for $n>16$ and fluctuating between very large positive and negative values; conversely, the backwards algorithm is well-behaved. | {
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# Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?
When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition).
As per the calculus wiki: Link, when the Hessian is negative semi-definite then, we can only conclude th... | {
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# Matrices with rotational symmetry
+3
−0
I've seen a claim without proof that the characteristic polynomials of matri... | {
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Consider first a $(2n+1)\times(2n+1)$ block matrix $\begin{pmatrix} A & v & B^{\circ} \\ h & c & h^{\leftarrow} \\ B & v^{\uparrow} & A^{\circ} \end{pmatrix}$ where $A, B$ are $n \times n$, $v$ is $n \times 1$, $h$ is $1 \times n$, and $c$ is $1 \times 1$. We have \begin{eqnarray*}\det \begin{pmatrix} A & v & B^{\circ}... | {
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# Finding particular solutions to PDEs (transport equation)
Suppose I have the following PDEs: (for $u(t,x)$)
(1) $u_{t}-u_{x}=-x$
(2) $u_{t}+2u_{x}=1$
(3) $u_{t}+u_{x}+u=e^{x+3t}$
(4) $2u_{t}+u_{x}=sin(x-t)$
For all equations, it is easy to find the homogenous part of the general solution because they are all tr... | {
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• In (1) try making a substitution $u = v + f(x)$ for some suitable function $f(x)$ so that $-f'(x) = -x$. Same trick works in (2): try $v = u + g(x)$ or $v = u + h(t)$ and pick $g$ or $h$ such that you can cancel the right hand side. – Winther Feb 3 '17 at 16:40
• @Winther, thank for your answer. For (1), I get $u(t,x... | {
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$$\frac{\mathrm dt}{1}=\frac{-\mathrm dx}{1}=\frac{-\mathrm du}{x}$$
With the first and second ratio:
$$\frac{\mathrm dt}{1}=\frac{-\mathrm dx}{1}\implies t=-x+c_1\;;t+x=c_1$$
With the second and the third:
$$\frac{-\mathrm dx}{1}=\frac{-\mathrm du}{x};\;x\mathrm dx=\mathrm du;\;\frac{x^2}{2}+c_2=u$$
Now, the inte... | {
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$$\cos(-x-c_1)+c_2=u\;;u=\cos(x-t)+c_2$$
And with $c_2=g(c_1)$ as before:
$$u(t,x)=cos(x-t)+g(t-2x)$$ | {
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# Maximum Likelihood Estimation of two unknown parameters
Here is a question:
We have a machine A that functions with probability $\theta_1 \theta_2$, and a machine B that functions with probability $\theta_1 \theta_2^2$. A random sample of $n$ A machines and an independent random sample of $n$ B machines are selecte... | {
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If someone could check my solution I would appreciate that.
• This looks right but I'd have said more than this. – Michael Hardy Jan 30 '17 at 16:59
• Do you mean it is unclear how to derive that $\theta_1\theta_2^2= \frac{n_2} n$ and $\theta_1\theta_2= \frac{n_1} n$? – Markoff Chainz Jan 30 '17 at 17:03
• Why doesn't... | {
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