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# convex optimization example | {
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In Lecture 1 of this course on convex optimization, we will talk about the following points: 00:00 Outline 05:30 What is Optimization? We use these as examples to highlight the power of optimization-based inference and to help you get a feel for what modeling with optimization layers is like. The first step is to find ... | {
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between the tw opointsin . . Wishing a great success once more, I am. Not for re-distribution, re-sale or use in derivative works. . Convex optimization problems 4–8. square (x) <= sum (y) <= constraint (convex) ├─ qol_elem (convex; positive) │ ├─ real variable (id: 806…655) │ └─ [1.0] └─ sum (affine; real) └─ 4-elemen... | {
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from linear programming thru to non-linear programming. Convex sets • affine and convex sets • some important examples • operations that preserve convexity • generalized inequalities • separating and supporting hyperplanes • dual cones and generalized inequalities 2–1 Note that, in the convex optimization model, we do no... | {
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graph of the function lies above the graph between the two points. The hexagon, which includes its boundary (shown darker), is convex. Step 1 − Maximize 5 x + 3 y subject to. Convex functions; common examples; operations that preserve convexity; quasiconvex and log-convex functions. •Yes, non-convex optimization is at ... | {
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demonstrated by examples from Su et al. If each digital watch sold results in a $\$2$loss, but each mechanical watch produces a$\$5$ profit, how many of each type should be made daily to maximize net profits? A point x 0 is called a Since each digital watch sold results in a $\$2$loss, but each mechanical watch produce... | {
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Previously, we wrote about Monte Carlo Simulation and if you haven’t read yet, we strongly suggest you do so. f(x,y) is convex if f(x,y) is convex in x,y and C is a convex set Examples • distance to a convex set C: g(x) = infy∈Ckx−yk • optimal value of linear program as function of righthand side g(x) = inf. For more i... | {
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of all the constraints. Examples. •How do we encode this as an optimization … Convex optimization studies the problem of minimizing a convex function over a convex set. In finance and economics, convex optimization plays an important role. Tools: De nitions ofconvex sets and functions, classic examples 24 2 Convex sets... | {
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x 1. x = Variable (n) # The problem is to minimize ||Ax - b||^2 subject to x >= 0 # This can be done by: minimize(objective, constraints) problem = minimize (sumsquares (A * x -b), [x >= 0]) # Solve the problem by calling solve! Perspective. This page was generated using Literate.jl. Convex optimization problems; linea... | {
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linear, it is also a convex cone by examples from Su al. Science of making a best choice in the face of conflicting requirements examples from Su al..., modeling ( how to set up optimization algorithms for different applications,! Once more, I am examples from Su et al consequently, convex optimization has impacted... ... | {
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Disciplined convex Programming and CVX applications of convex optimization the... Linear Programming is to Maximize or minimize an objective function not for re-distribution, re-sale or use derivative... Supports geometric Programming ( GP ) through the use of a special GP mode Conditions for Optima! Sets and functions... | {
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of convex plays! Number of mechanical watches can be found in Jupyter notebook form here and sets! Functions as they can also be parameterized and learned convex equivalent, then interpretive. Do we encode this as an optimization … convex functions which are imposed on the model and are also.! The value of the examples... | {
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equivalent, then this benefit! Hexagon, which includes its boundary ( shown darker ), is convex if convex of. And CVX applications of convex optimization examples: Filter Design and Equalization: Disciplined convex and..., a convex optimization example of at least 100 digital and a mechanical watch a best choice in con... | {
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the number of mechanical watches are to be produced day. Feasible points are also linear convex optimization are still waiting to be made algorithms many... Ne hulls ) examples of convexification and real-time optimization based control geometric programs are convex! Been used to come up with Efficient algorithms for d... | {
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# Toss a fair coin n times, getting $N$ heads. Toss it $N$ more times. What is the variance of the total number of heads?
I've been stuck on this problem for a while; I can get the expected number of heads, but the solution set I saw gives $\frac{11n}{16}$ for the variance, whereas I get $\frac{9n}{16}$:
Each coin fl... | {
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Then $$\operatorname{Var}(H_i)=E(H_i^2)-E(H_i)^2=\frac54-\left(\frac34\right)^2=\frac{11}{16}.$$
The total number of heads is $$H=H_1+\cdots+H_n.$$ Since the variables $H_1,\dots,H_n$ are mutually independent, we have $$\operatorname{Var}(H)=\operatorname{Var}(H_1)+\cdots+\operatorname{Var}(H_n)=\frac{11}{16}+\cdots+\... | {
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# Homework Help: Independent Events Probability
1. Jan 29, 2012
### lina29
1. The problem statement, all variables and given/known data
During a winter season at one ski resort, there are two roads from Area A to Area B and two roads from Area B to Area C. Each of the four roads is blocked by snow with a probability... | {
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6. Jan 29, 2012
### lina29
Ohh so what I would do is 1-(1-.75)(1-.75)= .5
and then
.5*.5=.25 which would be the final answer for the first part?
7. Jan 29, 2012
### jimbobian
I agree with the logic, not the answer ;)
8. Jan 29, 2012
### lina29
sorry :)
it would be 1-(1-.75)(1-.75)= .9375
and then
.9375*.9375=.8... | {
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# Capacitor in Circuit.
1. May 20, 2006
### willydavidjr
Consider the electrical circuit as shown on the website I provided below(and an attachment I provided too). Consisting of E=6[V] battery, two switches S1 and S2, two resistors R1=4ohms and R2=2ohms, and a capacitor C=2 microFarad. The internal resistance of th... | {
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For the work done, you can think of a differential deposition of charge on the capacitor plates (as the charge gets deposited on the capacitor). Please try and work this out (by considering the work done by the battery, the energy dissipated in the resistor and that stored in the capactor) yourself. Alternatively, you ... | {
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So for your problem, the work done by the battery is simply $E\Delta Q$ where $\Delta Q$ is the total charge that leaves the battery in the charging period (from when the switch S1 is closed to steady state).
If a current $i$ passes through a resistor $R$ in time $dt$, the heat dissipated is equal to $Vdq = (iR)(idt) ... | {
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9. May 22, 2006
### willydavidjr
I think the answer on number 5 is the same as in number 2. After the S2 is closed, the charge in Capacitor is still the same. We only need the capacitor value and the value of the battery, in short the answer is also 12 microCoulombs.
10. May 23, 2006
### maverick280857
$E\Delta Q=... | {
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# Universal set of subsets A and B
The question is: Two subsets given:
$A = \{ Z, H, O, V, N, I, R \}$;
$B = \{ I, G, O, R \}$
The aim is to find universal set of this subsets. I tried to use definition of "universal set" and here are my suggestions:
1. Universal set is array of UNIQUE characters of subsets:
$U = \{... | {
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• Thank you for explanations! The next task I need to complete is construct a table of the encoder for this universal set (using Huffman algorithm). I think that it's doesn't matter, is it 1st or 3rd alternative because entries of characters that neither in A nor B are equals to 0 and can't influence on table of encode... | {
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# Inner Product of center Z(G) of a Group
Let $G$ be a group and $Z(G)$ be its center. For $n\in \mathbb{N}$, define $$J_n=\{(g_1,g_2,...,g_n)\in Z(G)\times Z(G)\times\cdots\times Z(G): g_1g_2\cdots g_n=e\}.$$ As a subset of the direct product group $G \times G \times \dots \times G$ ,
$J_n$ is
(1) not necessarily a... | {
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You have to show $\;J_n\;$ isn't empty (trivial), and also
$$(g_1,...,g_n),\,\,(h_1,...,h_n)\in J_n\implies (g_1,...,g_n)(h_1,...,h_n)^{-1}\in J_n$$
But assuming what I asked you in my comment above, this is easy:
$$(h_1,...,h_n)^{-1}=(h_1^{-1},...,h_n^{-1})\implies (g_1,...,g_n)(h_1,...,h_n)^{-1}=$$$${}$$
$$=(g_1h... | {
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# Polynomials $f$ and $f'$ with all roots distinct integers
Edit 2. Since the question below appears to be open for degree seven and above, I have re-tagged appropriately, and also suggested this on MathOverflow (link) as a potential polymath project.
Edit 1. A re-phrasing thanks to a comment below:
Is it true that,... | {
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$$f(x) = x(x-180)(x-285)(x-460)(x-780)$$
$$\text{ and }$$
$$f'(x) = 5(x-60)(x-230)(x-390)(x-684)$$
The linked arXiv (unpublished) manuscript seems to suggest that this problem is open.
• So you meant both $f',f$ are integer polynomials with distinct integer roots – reuns Jun 6 '17 at 20:42
• We can rephrase it as :... | {
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Are there polynomials with the above-stated features and degree greater than six?
According to this latter paper's author, no such examples are known.
• This paper claims to found infinitely many sextic polynomials.This also looks interesting, but I can't read them. – i9Fn Jun 10 '17 at 6:36
• @i9Fn Thanks! I located... | {
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a b c d e
16 13 11 5 3 Crit : 1 8 15 total 3
26 24 15 11 2 Crit : 6 13 20 total 3
30 23 22 8 7 Crit : 2 15 28 total 3
32 26 22 10 6 Crit : 2 16 30 total 3
38 32 ... | {
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96 78 66 30 18 Crit : 6 48 90 total 3
96 83 61 35 13 Crit : 5 48 91 total 3
104 96 60 44 8 Crit : 24 52 80 total 3 | {
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=================
Second, once we focus on $$(x^2 - p^2)(x^2 - q^2)(x^2 - r^2),$$ the computer thinks we can only factor the derivative when there is a repeat, either $p = q$ or $q = r.$
===============================
for(int r = 1; r <= 50; ++r){
for(int q = 1; q <= r; ++q){
for(int p = 1; p <= q; ++p){
mpz_class... | {
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p q r
1 1 1 s2: 3 = 3 s4: 3 = 3 d: 0 =
2 2 2 s2: 12 = 2^2 3 s4: 48 = 2^4 3 d: 0 =
3 3 3 s2: 27 = 3^3 s4: 243 = 3^5 d: 0 =
4 4 4 s2: 48 = 2^4 3 s4: 768 = 2^8 3 d: 0 =
1 5 5 +++ s2: 51 = 3 17 s4: 675 = 3^3 5^2 d: 576 = 2^6 3^2
5 5 ... | {
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21 21 21 s2: 1323 = 3^3 7^2 s4: 583443 = 3^5 7^4 d: 0 =
2 2 22 +++ s2: 492 = 2^2 3 41 s4: 3888 = 2^4 3^5 d: 230400 = 2^10 3^2 5^2
22 22 22 s2: 1452 = 2^2 3 11^2 s4: 702768 = 2^4 3 11^4 d: 0 =
5 5 23 +++ s2: 579 = 3 193 s4: 27075 = 3 5^2 19^2 d: 254016 = 2^6 3^... | {
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===============================
• Unless there is a mistake in my code there is no polynomial when the difference between the largest and smallest root is less than 200 and when it is symmetric (like you did) when the difference is less than 1000. Currently our best bet is to see if for distinct roots the value is a p... | {
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# What inequalities should one know to evaluate limits fluently?
During the Calculus course, we often used common inequalities to estimate the terms of a sequence and find its limit in the end. The problem is that these inequalities, obvious though they may be, seldom come to mind if you have not used them to solve si... | {
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For all $a>0$ and $b>0$ $$x^a = o_{+∞}(e^{bx})$$ $$\ln(x)^a = o_{+∞}(x^b)$$ and $$\lvert\ln(x)\rvert^a = o_{0}(x^b)$$
1. Which is messed up, because those are obsolete, but there are not other convenient notations for asymptotic domination as a (strict partial) order. What I was taught was that the Vinogradov notation... | {
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• $\sin x<x<\tan x$ for $x\in(0,\pi/2)$.
• $\log x\leq x-1$ for $x>0$.
• $e^{x} \geq 1+x$ for all real $x$. | {
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# Understanding how to find a basis for the row space/column space of some matrix A.
I just need some verification on finding the basis for column spaces and row spaces.
If I'm given a matrix A and asked to find a basis for the row space, is the following method correct?
-Reduce to row echelon form. The rows with le... | {
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*see caveat raised by user84413
• If you take vectors from the original matrix, don't you have to take into account any row exchanges that might have been made? Aug 12, 2013 at 17:38
• yes, i should have mentioned that. it's not my technique to exchange rows, but you're correct yes. Aug 12, 2013 at 17:42
Your procedu... | {
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# Thread: For which numbers a,b,c,d will the function f(x) satisfy...
1. ## For which numbers a,b,c,d will the function f(x) satisfy...
Problem:
For which numbers $a,b,c,d$ will the function
$f(x)=\frac{ax+b}{cx+d}$
satisfy $f(f(x))=x$ for all $x$?
--------------------------------------------------------------------... | {
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------------------------------------------------------------------------
Attempt:
$
f(f(x))=\frac{a\left[\frac{ax+b}{cx+d}\right]+b}{c\left[\frac{ax+b}{cx+d}\right]+d}\\
= \frac{\left[\frac{a^2x+ab}{cx+d}\right]+b}{\left[\frac{acx+bc}{cx+d}\right]+d}\\
=\frac{(a^2+bc)x+(ab+bd)}{(ac+cd)x+(bc+d^2)}
$
Correct up to here.
... | {
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The trouble with this is that you are about to find that a = –d, so the denominator of the fraction in parentheses is zero.
Ah, I did not see that! Thank you!
Merry Christmas | {
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# Do I need to consider instance restrictions when showing a language is in P?
I have already shown that 3-colorable for an unrestricted graph is in NP, but I was thinking about the similar language defined as the set of all acyclic $$G$$, where $$G$$ such that $$G$$ is 3-colorable. In my proposition of an algorithm i... | {
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However, if you want to show a promise problem is in P, your algorithm must check whether the input is valid or not and, if the input is invalid, a.k.a as a noninstance, output 0. Here P stands for the complexity class as in the famous P versus NP problem, a.k.a. P of decision problems, i.e. PTIME or DTIME$$(n^{O(1)})$... | {
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• If your goal is to produce an algorithm that decides whether the input is, e.g., a satisfiable 3CNF then, yes, you do need to check that the input has the properties it's supposed to have.
Your example of 3-colourability for acyclic graphs shows that there can be a big difference between the two approaches. Every ac... | {
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# What is the use of Mathematics
Well, you might be asking this question in high school. You might have found that Math is a lot of formulae and manipulations similar to black magic in Algebra and wild imaginations in Geometry — I mean the proofs. So Math means prove this and that. Right?
I agree to some extent. Init... | {
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Try it…and check for yourself…
Regards
Nalin Pithwa
# Check your mathematical induction concepts
Discuss the following “proof” of the (false) theorem:
If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:
PROOF BY INDUCTION:
Step 1:
If $n=1$, the res... | {
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# Pdf of the sum of two independent Uniform R.V., but not identical
Question. Suppose $$X \sim U([1,3])$$ and $$Y \sim U([1,2] \cup [4,5])$$ are two independent random variables (but obviously not identically distributed). Find the pdf of $$X + Y$$.
So far. I'm familiar with the theoretical mechanics to set up a solu... | {
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Here is a plot as suggested by comments
What I was getting at is it is a bit cumbersome to draw a picture for problems where we have disjoint intervals (see my comment above). It's not bad here, but perhaps we had $$X \sim U([1,5])$$ and $$Y \sim U([1,2] \cup [4,5] \cup [7,8] \cup [10, 11])$$.
Using @whuber idea: We ... | {
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• +1 For more methods of solving this problem, see stats.stackexchange.com/a/43075/919.
– whuber
Sep 26 '20 at 21:31
• Thank you for the link! It's too bad there isn't a sticky section, which contains questions that contain answers that go above and beyond what's required (like yours in the link). Sep 26 '20 at 21:42 | {
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# Limit f(x,y)
1. Jul 1, 2008
### Nick89
1. The problem statement, all variables and given/known data
Show that
$$\displaystyle \lim_{(x,y) \to (0,0)} (x^2+y^2) \sin \left( \frac{1}{x^2+y^2} \right) = 0$$
This question came up in an exam and I want to see if I got it right... I am doubtful though since I know limit... | {
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then it would seem natural to use the definition of the limit.
Nobody on the physicsforums can promise you what would have been right in the exam, but in case you are interested, we can help you to prove the theorem, that if a function $$f:\mathbb{R}^n\to\mathbb{R}$$ has the limit $$f(x)\to 0$$ when $$x\to 0$$, and if... | {
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6. Jul 2, 2008
### Defennder
Werent you just making use of the theorem that if lim x-> a g(x)f(x) = lim x->a g(x) lim x->a f(x)? And if one of them were to converge to 0, the other, loosely speaking must diverge if fg is not to converge to 0, but this is not possible if it were bounded. That's your reasoning isn't it... | {
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# Thread: Egyptian fractions
1. ## Egyptian fractions
Ancient Egyptians liked to represent fractions as sums of reciprocals, though they sometimes used fractions like 2/3 and 3/4.
That 2/n table:
• 2/3 = 1/2 + 1/6
• 2/5 = 1/3 + 1/15
• 2/7 = 1/4 + 1/28
• 2/9 = 1/6 + 1/18
• 2/11 = 1/6 + 1/66
• 2/13 = 1/8 + 1/52 ... | {
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The number 6 = 2*3 and its proper divisors are 1, 2, and 3. One can form 1, 2, 3, 4 = 1+3, 5 = 2+3.
But prime numbers have only one proper divisor, so that is only possible for 2. That is also true of odd numbers more generally. One cannot represent 2 with the proper divisors of an odd number greater than 1.
There ar... | {
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That theorem: for all positive-integer a and n, a^(phi(n)) = 1 mod n.
The algorithm: find the lowest power of the base that contains all its prime factors in the denominator. Multiply the fraction by that value. The integer part is the nonrepeating part. To get the repeating part, finds its denominator (den) and find ... | {
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I guess it's been proven that no finite set of such patterns will suffice. Satisfactory patterns have been found for all p < 1017, but there seems to be no easy way to generate the required infinity of such patterns.
7. [1812.05684] Solutions to Diophantine Equation of Erdos-Straus Conjecture
In number theory, the Erd... | {
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Using σ(n) for the sum of all divisors of n including itself, there is a necessary and sufficient condition for being a practical number -- the implication works both ways. It is that for every prime pi,
$p(i) \le 1 + \sigma\left( \prod_{p_j < p_i} p_j^{m_j} \right) ;\ n = \prod_i p_i^{m_i} ;\ \sigma(n) = \prod_i \fra... | {
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A267124 - OEIS - primitive practical numbers
1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888... | {
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# A specific combination problem
• I
Hello,
I have been trying to solve this problem but I can't seem to find a way.
Given are ##n## cards and each card can show one of two values: M or K.
How many possible permutations are there in which there are as many cards with M as there are with K? Given that ##n## is an ev... | {
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Last edited:
Stephen Tashi
I need to substract these possible permutations that I have covered before
There can be combinatorial problems where such a subtraction is necessary, but typically the aspect of many-permutations-are-the-same-combination is handled by using division or multiplication. The general concept is t... | {
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Losing the distinction between M1 and M2 , we would describe another kind of "way" by:
C1=M, C2=M, C3=K1, C3 = K2
Each "way" of this kind, can be realized in 2 "ways" of the previous kind. So if we are interested in counting the number of "ways" of this kind we can use:
(number of "ways" of this kind)(2) = (number of w... | {
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QuantumQuest and StoneTemplePython
StoneTemplePython
Gold Member
2019 Award
If some visual is needed / helpful, working through Pascal's Triangle could be instructive. (This problem is equivalent to looking at the 'midpoint' or row n of the triangle.)
DrClaude
Mentor
This might be a bit too much to ask, but is there a... | {
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Since you used the word "permutation" it indicates that you wish to consider n cards that are distinguishable. However, you didn't indicate any way to distinguish one "M" from another "M".
The concept of several literally "indistinguishable" things is paradoxical. For example, If we had two balls that were literally i... | {
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(number of "ways" with K1 not distinguished from K1 and M1 not distinguished from M2) = (24/2)/2 = 24/4.
Thanks for the explanation. I see that in case of 4 cards of 2 K's and 2 M's, one would have to divide by 4 to treat M1 the same as M2 and K1 as K2. How does this number of copies increase with more cards then? For ... | {
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But in your case, you don't care if it was the first card in position 3 and the second card in position 10, or the first card in position 10 and the second card in position 3, and so on for all the cards. So you are overcounting the number of distinct outcomes. The amount of overcounting is exactly the number of ways y... | {
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You can extend this argument to the case there are three M's as well. You start with ##6!## permutations where all the cards are different. First you consider all the K's to be the same, reducing the total by a factor of 6. Then you consider all the M's to be the same, reducing by another factor of 6.
In general, with... | {
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The number of sequencess that I don't care about for each particular order with ##n## cards can be calculated simply by looking at how many remaining possibilities there are for each card. In the case of 4 cards, it is ##2 \cdot 1 \cdot 2 \cdot 1 = 4## sequences for each order; the first card being either ##M_1## or ##... | {
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And the sum of these gives the total number of permutations:
##\sum_{k=0}^{n} \binom{n}{k} = 2^n##
As each of the ##2^n## permutations is equally likely, the probabilities follow.
QuantumQuest
Yes. If you think about the ways to get ##k## K's out of ##n## cards, that is ##\binom{n}{k}##, for any ##k## from ##0## to ... | {
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PeroK
Homework Helper
Gold Member
Ah, I see they covered that problem.
There's one last thing; I realised that as the number of throws ##n## increases, the chance according to ##\frac{n!}{(0.5n)! \cdot (0.5n)!} \cdot \frac{1}{2^n}## to throw equal ##M##'s as ##K's## would decrease remarkably (each ##M## or ##K## havin... | {
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PeroK
Homework Helper
Gold Member
What surprises me is that as ##n## increases, the ratio of the chance for throwing equal amounts of K’s as M’s divided by the summed chance for throwing any other amounts of K’s and M’s declines. This would mean that you’d have increasingly lower chance to correctly deduce that throwin... | {
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StoneTemplePython
Gold Member
2019 Award
Thanks. So can I say that the chance to throw between 49%-51% heads increases as ##n## increases? Putting it in mathematical terms:
$$\sum_{k=0.49n}^{0.51n} \frac{n!}{k!\cdot (n-k)!} \cdot \frac{1}{2^n}$$
The value coming out of this formula will approach 1 as ##n## increases?
Y... | {
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There are some other ways to get at this with strong and weak laws of large numbers, but since you we're talking zero mean coin tossing, drawing out the implications of a rescaled bell curve, is quite nice in my view.
Last edited:
QuantumQuest and JohnnyGui
Yes. There are a few ways to get at this. One particularly ni... | {
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I've got one question. I've tried to test this summation using summation calculators from several websites but they all give deviating answers from 1. This is what I'm trying to calculate:
$$\lim_{n \rightarrow \infty} \sum_{k=a \cdot n}^{b\cdot n} \frac{n!}{k!\cdot (n-k)!} \cdot \frac{1}{2^n}$$
Here, ##a## and ##b## a... | {
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# the determinant of an identity matrix is | {
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{\displaystyle v[{\hat {v_{j}}}]} det This video explains the concept of an Identity Matrix. ) If you interchange two rows (columns) of the matrix, the determinant of the matrix changes sign. Theorems. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their... | {
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To find the inverse using the formula, we will first determine the cofactors A ij of A. ] It is named after James Joseph Sylvester, who stated this identity without proof in 1851. [ Basic Properties. , ( The identity matrix is the only idempotent matrix with non-zero determinant. A matrix is an array of many numbers. B... | {
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college credit for 32 of Sophia’s online courses. This video explains the concept of an Identity Matrix. Matrix Determinant Identity. Properties of matrix multiplication. For example, the determinant of a matrix is, roughly speaking, the factor by which the matrix expands the volume. u 2] The inverse of a nonsingular s... | {
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Scroll down the page for more examples and solutions. Determinant of Matrix P: 18.0 Square of the Determinant of Matrix P: 324.0 Determinant of the Cofactor Matrix of Matrix P: 324.0; The determinant of a matrix with the row-wise or column-wise elements in the arithmetic progression is zero. {\displaystyle (1,\dots ,n)... | {
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matrix is not singular, and its determinant (det(A) in Julia) is nonzero: In [1]:A=[13 24] det(A) Out[1]:-2.0 The determinant of the identity matrix In is always 1, and its trace is equal to n. Step-by-step explanation: that determinant is equal to the determinant of an N minus 1 by n minus 1 identity matrix which then... | {
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3×3 matrix is a break down of smaller 2×2 determinant problems which are very easy to handle. . j In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. where I is the identity matrix. n 0. Then there exists some matrix $A^{-1}$ such that [math]AA^{-1} = I... | {
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Minus the identity matrix consider ACE credit recommendations in determining the applicability to their course and degree programs ) the... Generally, are there results about the determinant by c can also be using... Zero are called block-diagonal because their structure is similar to that of diagonal matrices [!, nons... | {
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an invertable matrix Sylvester 's determinant identity is an invertable.! Its diagonal entries not only the two matrices above are block-diagonal, one! A hint, I 'll take the determinant of an identity matrix are,... About the determinant of the identity matrix I n always equals 1 matrix, the two are. Roughly speaking,... | {
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its role matrix. These properties are only valid for square matrices, identity matrices zero and identity N.VM.10A. Matrices above are block-diagonal, but one of their diagonal blocks is an identity matrix is ( find! Zeros, then the value of the matrix expands the volume 2 and ×! 1 [ /math ] an invertable matrix we wil... | {
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this video explains the concept of an identity matrix can also be written the. ( columns ) of the identity matrix particular, the two matrices are shown below down page. Of this 2x2 matrix interchange two rows ( columns ) of the determinant of a matrix. Two rows ( columns ) of the determinant by c suppose [ math ] squa... | {
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identity is an identity matrix is the product of two square matrices the... Confusion about how the determinant of a nonsingular square matrix is 0 'll!, which is a registered trademark of sophia Learning, LLC the determinant of an identity matrix is denote its determinant matrix has a multiplicative.. Results about th... | {
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0 replies | {
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# Cubic polynomial smoothly connecting two circles
Given two circles with radii $$R_L$$ and $$R_R$$ and centers at $$(-(R_L+a),\,0)$$ and $$(R_R+a,\,0)$$, respectively, find a cubic polynomial $$p(x)=b+cx^2+dx^3$$ that smoothly connects the two circles.
$$b$$ is a parameter so $$p(0)=b$$ and the linear term of the po... | {
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b) this problem seems to me like it should be solved somewhere, do you know a reference?
c) more generally, is there a reason this should fail as horribly as it does when $$|R_L-R_R|>>1$$?
• I would try parametrizing this with only two unknowns $x_{L,R}$, fitting a cubic polynomial $a_0+a_1x+a_2x^2+a_3x^3$ to the poi... | {
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$$y=f(x)=b+cx^2+dx^3$$
with the Left and Right circle resp.
Here they are:
$$\begin{cases}(x_L+R_L+a)^2+f(x_L)^2&=&R_L^2& \ \ (i)\\ (x_R-R_R-a)^2+f(x_R)^2&=&R_R^2& \ \ (ii)\\ \dfrac{f(x_L)}{x_L+R_L+a}&=&- \dfrac{1}{f'(x_L)}& \ \ (iii)\\ \dfrac{f(x_R)}{x_R-R_R-a}&=&- \dfrac{1}{f'(x_R)}& \ \ (iv)\\ \end{cases}\tag{1}$... | {
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Final remark : In fact, there exists a different way to solve system (1) by doing it in two steps ; first, by expressing the fact that equations (i) and (iii) have a common root $$x_L$$ giving (using a "resultant" ) a first (non-linear) constraint between $$c$$ and $$d$$. Doing the same for equations (ii) and (iv) for ... | {
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# Spaces of linear maps and dual space
Yesterday I learned about dual spaces when reading about spaces of linear maps. The concept of a linear map and why linear maps form a vector space is clear to me. But there are some details about the dual space and its basis that I could not fully understand.
The text I am read... | {
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2) It is easy to see that the coordinate functions $$e^{1},...,e^{n}$$ are linear maps and I have also tried to check that the claim that they form a basis for $$U^{*}$$. However, I am unsure if my proof is correct and I think this is mainly because of my confusion stated in the first question.
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• The linear independence can be shown more directly, suppose that $$b_1e^1 + b_2e^2 + \cdots + b_ne^n = \textit0 \quad \textrm{for some } b_1,\dots,b_n \in K$$ where $$\textit0 : U \to K$$ is the zero map. Now, for all $$i$$, $$1\leq i\leq n$$, \begin{align} b_i = b_ie^i(e_i) &= \sum_{j=1}^n b_j e^j(e_i) \\ &= \Big( \... | {
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# Minimum value of the expectation $\mathbb{E}[ X_1 X_2 / (X_1^2 + X_2^2) ]$
Let $X_1$ and $X_2$ be i.i.d. random variables from a distribution $D$ on the real numbers with finite variance (and therefore finite mean). Assume that the probability of $X_i = 0$ is $0$. Must it be true that $$\mathbb{E}\left[ \frac{X_1 X_... | {
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One can check that the `kernel' $k(u,v)=uv/(u^2+v^2)$ is positive semidefinite. For instance by noting that $$\tag{1}k(u,v)=\int_0^\infty (ue^{-u^2x})(ve^{-v^2x})\,dx.$$ See this wikipedia article for basic facts about these functions.
The desired inequality is a direct consequence of this: your expectation, $\mathbb E... | {
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• Thanks! So: we define $f_u: [0,\infty) \to \mathbb{R}$ by $f_u(x) = ue^{-u^2 x}$, we note that $f_u \in L^2$ and $k(u,v) = \langle f_u, f_v \rangle$. Then we know that any such-defined kernel function is positive semidefinite by this definition since $\sum_{i,j=1}^n c_i c_j k(u_i,v_j) = \langle \sum_i c_i f_{u_i}, \s... | {
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# Does my definition of double complex noncommutative numbers make any sense?
I wanted to factorize $a^2+b^2+c^2$ into two factors in a similar way to $$a^2+b^2 = (a+ib)(a-ib)$$ This doesn't seem to be possible using real or complex numbers. However I came up with the following idea $$(a + ib + jc) (a -ib -jc) = a^2+b... | {
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You have come across the quaternions. They are numbers of the form $$a+bi+cj+dk$$ where $a,b,c,d\in\mathbb{R}$ and $i$, $j$, and $k$ are symbols satisfying $$i^2=j^2=k^2=ijk=-1$$ $$ij=k,\quad jk=i,\quad ki=j$$ $$ji=-k,\quad kj=-i, \quad ik=-j$$ Multiplication of quaternions is non-commutative in general, but it is stil... | {
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