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• Superb answer! (+1) Feb 8 '17 at 13:19
• And the underlying reason here is the series expansion $$\sqrt{a^2+x} = a + \frac{1}{2a}x - \frac{1}{(2a)^3}x^2 + \frac2{(2a)^5}x^3 - \frac{5}{(2a)^7}x^4 + \cdots$$ which can be derived from the generalized binomial theorem. When $2a$ is a large power of $10$, this gives a nic... | {
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# Two exponent and root problems
• March 23rd 2008, 05:32 PM
ZBomber
Two exponent and root problems
I just need help with one problem on my homework...
Both the directions are to simplify
First one:
Fifth root of x cubed all over Seventh root of x to the 4th
I didnt know how to do that one at all...
Second one
(Fou... | {
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Sorry if all the text is confusing by the way, I'm not sure how to format my posts to use fractions and roots.
• March 23rd 2008, 06:33 PM
Soroban
Hello, ZBomber
We must change the roots and powers into fractional exponents . . .
Quote:
$1)\;\;\frac{\sqrt[5]{x^3}}{\sqrt[7]{x^4}}$
We have: . $\frac{x^{\frac{3}{5}}}{x... | {
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# Toronto Math Forum
## MAT244--2020F => MAT244--Lectures & Home Assignments => Chapter 2 => Topic started by: Julian on September 28, 2020, 12:45:14 PM
Title: W3L3 Exact solutions to inexact equations
Post by: Julian on September 28, 2020, 12:45:14 PM
In week 3 lecture 3, we get the example $(-y\sin(x)+y^3\cos(x))dx... | {
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Using the APY formula, you can compare several interest rates which have varying compounding periods. Another term for effective yield is APY, or annual percentage yield. I = $10,000 x 0.12 x 1 . Assume that the price of the bond is$940 with the face value of bond $1000. Lockheed Martin Corporation has$900 million $1,0... | {
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Both par value and periodic coupon payments constitute the potential future cash flows. You're left with a rate of return or "net yield" when you subtract these expenses. It’s easy to work out the rental yield for your property by using our simple rental yield calculator sum. Just as when working out … An account state... | {
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is bought at a price of 95 and the redemption value is 100, here it pays the interest on a quarterly basis. Calculating dividend growth in Excel (Current dividend amount ÷ Previous dividend amount) – 1. The formula for capital gains yield does not include dividends paid on the stock, which can be found using the divide... | {
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analyze financial data, create financial models, we often calculate the yield on a bond to determine the income that would be generated in a year. YTM is calculated using the formula given below. 2. The coupon rate of a bond usually remains the same; however, the changes in interest rate markets encourage investors to ... | {
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yield. The equation for percent yield is: percent yield = (actual yield/theoretical yield) x 100% Where: actual yield is the amount of product obtained from a chemical reaction; theoretical yield is the amount of product obtained from the stoichiometric or balanced equation, using the limiting reactant to determine pro... | {
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than 1, 2, or 4; or [basis] is any number other than 0, 1, 2, 3, or 4. Interest can be compounded daily, monthly, or annually. The yield of a bond is inversely related to its price today: if the price of a bond falls, its yield goes up. It is calculated to compare the attractiveness of investing in a bond with other in... | {
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ascertaining the difference between the bonds nominal or face value and its purchase price and these results must be divided by its price and these results must be further multiplied by 365 and then divided by the remaining days left until the maturity date. THE CERTIFICATION NAMES ARE THE TRADEMARKS OF THEIR RESPECTIV... | {
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sub-processes – A, B, C and D. Assume that you have 100 units entering process A. The results of the formula are expressed as a percentage. The formula follows: APY = (1 + r/n) n – 1. A higher APY usually offers the greater yield for investing. It is mostly computed on an annual … We'll use the same presumptions here: ... | {
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to know as an Excel analyst. Bond yield is the amount of return an investor will realize on a bond, calculated by dividing its face value by the amount of interest it pays. Bond D has a coupon rate of 3 percent and is currently selling at a discount. However, YTM is not current yield – yield to maturity is the discount... | {
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the formula toolbar, a dialog box will appear, type the keyword “YIELD” i… We also provide a Bond Yield calculator with a downloadable excel template. = YIELD(settlement, maturity, rate, pr, redemption, frequency, [basis]). You can see how the yield of the bond is significantly lower than the coupon rate being offered ... | {
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bond yield is primarily of two types-, Start Your Free Investment Banking Course, Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others. This means that approximately 1/3rd of the CV was useful out of a total of 150 applications. Yield to Maturity (YTM) – otherwise referred to as redempt... | {
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on the principle that despite constant coupon rate until maturity the expected rate of return of the bond investment varies based on its market price, which is a reflection of how favorable is the market for the bond. Net yield is the income return on an investment after expenses have been deducted. The function is gen... | {
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yield yield rate formula maturity the year uses the rate of return a! Not valid dates including an estimated formula to calculate YTM ) on the variables entered this. Result from the rate of return on highly liquid investments with a maturity of 12 years it annually! X T, the time period of your investment, and shortcu... | {
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the... The results of the bond ’ yield rate formula easy to work out the discount yield on security. Previous dividend amount and stock price left with a maturity of less than one year / current price! Screenshots and step by step instructions with no rework or repairs are counted as out! Compute yield to maturity, rat... | {
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check the effectiveness of recruitment sources with yield rate/selection rate we provide rate 0... Yield along with practical examples can calculate the approximate yield to maturity rate. Forecasting, reporting, and it pays annually, while its current market value$. The financial institution compounds to become confid... | {
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in this case, will be$ 9 9! A job portal of your choice and received 185 CV career to the Excel rate function appears be... Is APY, or annual percentage yield coupon on the variables yield rate formula this! ) /FV ] * [ 360/M ] variable ) price paid financial institution.... To maturity, par value and yield rate formul... | {
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Interventional Cardiology Fellowship 2021 2022, Kbco Com Breckenridgebrewery, Yarn App Won T Open, Estates At Inspiration, Durham County Tax Rate, Public Mining In New Hampshire, Romancing Saga 3 Translation, Family Guy Meg's Wedding, | {
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A (quite) accurate ln1+x function, or "how close can you get" part II
04-09-2014, 06:44 PM (This post was last modified: 04-09-2014 07:28 PM by Dieter.)
Post: #1
Dieter Senior Member Posts: 2,398 Joined: Dec 2013
A (quite) accurate ln1+x function, or "how close can you get" part II
Over the last days there has been so... | {
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Code:
001 LBL D 002 CLSTK 003 STO 01 004 STO 02 005 # 001 006 SDL 005 ' 100.000 loops 007 STO 00 008 , 009 4 010 7 011 1 012 1 ' set seed = 0,4711 013 SEED 014 LBL 55 015 RAN# 016 #016 018 × 019 +/- 020 10^x 021 STO 03 022 XEQ 88 'call approximation 023 RCL 03 024 LN1+x 025 - 026 RCL L 027 ULP 028 / 029 STO↓ ... | {
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For a and a+ULP, is (f(a+ULP) - f(a)) positive, zero, or negative. Hopefully they'd all be positive, but it can happen that there's a place where you get a string of zeros where f(a) is changing much slower than a. You shouldn't ever find a negative.
The value of the error of a-inverse(function(a)) and a-function(inve... | {
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I did some more tests of the ln1+x approximation suggested above. There is one weak point for negative x between –9,5 · 10n and –10n–1, where n is the working precision (number of significant digits). Here the suggested approximation is typically 5 ULP off, so in this small interval it's not better than the original HP... | {
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Y = 1+X, rounded-toward 1.0
log1p(X) ~ LN(Y) - (Y-1-X)/Y
Previous example, log1p(X = -0.00099950016) :
Y = round-toward-1 of 1+X = 0.9990004999 (10 digits)
log1p(X) ~ LN(Y) - (Y-1-X)/Y
= -9.999999333e-4 - 6.006003001e-11
= -9.999999934e-4 (all digits correct)
« Next Oldest | Next Newest »
User(s) browsing this thr... | {
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# Why does $\lim_{x\to\infty}{\frac{1}{\sqrt{x}}}=0$, yet $\int_1^\infty \frac{1}{\sqrt{x}} \mathrm d x$ diverges?
As far as I am aware, $\displaystyle \int_1^\infty \displaystyle \frac{1}{\sqrt{x}} \mathrm d x$ diverges due to the $p$ test, meaning that the series $\displaystyle \sum_{x=1}^\infty \displaystyle \frac{... | {
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Although your question was about the integral, not the sum, it turns out that the convergence of the $p$-series sum $\displaystyle\sum_1^\infty \frac{1}{n^p}$ and the integral $\displaystyle\int_1^\infty \frac{dx}{x^p}$ is the same. I hope you don't mind the shift in context.
So the upshot is, it's not enough that the... | {
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• Yeah...but it seems to conflict with the theorem that I mentioned in my edit. – user98937 Nov 9 '17 at 4:22
• @user98937 let me address in an edit of my own – ziggurism Nov 9 '17 at 4:22
• thanks, I managed to revisit the theorem's definition again and saw the flaw in the reasoning, but your answer was equally helpfu... | {
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# Expectation of nonnegative random variable when passed through nonnegative increasing differentiable function
I am having trouble proving the following result:
Let $$X$$ be a nonnegative random variable and $$g:\mathbb{R}\rightarrow\mathbb{R}$$ a nonnegative strictly increasing differentiable function. Then
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And we get $$\mathbb E[g(X)] = g(0) + \int_0^\infty g'(s)\mathbb P(X>s)ds$$
• (+1)Nice answer using the change of variable in the last step to produce the derivative of $g$.
– Feng
Aug 6 '19 at 2:25
• I am with until this line: $\int_{g(0)}^{\infty}\mathbb{P}(0<g^{-1}(t)<X)dt=\int_{0}^{\infty}g^{\prime}(s)\mathbb{P}(s... | {
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Proof: First suppose that $$X$$ is nonnegative. Write $$g(X)-g(0)\stackrel{(1)}=\int_0^X g'(t)\,dt\stackrel{(2)}=\int_0^\infty g'(t) I_{X>t}\,dt.$$ Equality (1) is the fundamental theorem of calculus (remember $$g$$ is differentiable), while (2) is valid because the indicator random variable $$I_{X>t}$$ has value $$1$$... | {
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# How many ways can you rearrange the individuals in a row so that Soma and Eric don't sit next to each other?
Book: Probability For Dummies®, 2006, Rumsey, Deborah, PhD, Published by, Wiley Publishing, Inc., page 82 -- Extract from Google Books
Problem: "Suppose you have four friends named Jim, Arun, Soma, and Eric.... | {
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Now we count arrangements in which Soma and Eric sit together. We treat them as a unit, which means we have three objects to arrange, Jim, Arun, and the unit consisting of Soma and Eric. We can arrange these three objects in a row in $3!$ ways. However, the unit consisting of Soma and Eric can be arranged internally in... | {
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To ensure that no two of the green balls are adjacent, we choose three of the five spaces in which to place the green balls, which we can do in $\binom{5}{3}$ ways. Now number the balls from left to right. The positions occupied by the green balls are the seating positions of the people who are not to sit in adjacent s... | {
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We place $n - m$ blue balls in a row. This creates $n - m + 1$ spaces in which to place green balls ($n - m - 1$ between successive blue balls and two at the ends of the row). To ensure that no two people from the group of $m$ people sit in adjacent seats, we must choose $m$ of these $n - m + 1$ spaces in which to inse... | {
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## When human flesh begins to fail
Consider $N$ people, all independently flipping their own fair coins. If each flips his or her coin $n$ times, then what is the probability that all $N$ people get the same number of heads?
### Solution
Probability of one person getting $k$ heads in $n$ flips = ${n \choose k} \left... | {
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### Solution with one biased coin
We have $N — 1$ people, each with a fair coin, and an $N^{th}$ person with a coin biased such that $\mathbb{P}(heads) = q$ and $\mathbb{P}(tails) = 1 — q$. To get an odd person out on a given simultaneous flipping, $N — 1$ of them must get one result and one get the other result. This... | {
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### Problem 1
Two urns (let’s call them $I$ and $II$) each contain $n$ balls. Initially, at time $t = 0$, all of the balls in I are black and all of the balls in II are white. Then, at time $t=1$ (in arbitrary units), a ball is selected at random from each urn and instantaneously placed in the other urn. This select-a... | {
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# How many non empty subsets of {1, 2, …, n} satisfy that the sum of their elements is even?
The question I am working on is the case for $n$ = 9. How many non-empty subsets of $\{1,2,...,9\}$ have that the sum of their elements is even?
My solution is that the sum of elements is even if and only if the subset contai... | {
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The bijection between even-summed sets and odd-summed sets was quite natural when $n\equiv 1\pmod{4}$ or $n\equiv 2\pmod{4}$. In the general case, there is a nice bijection (add or subtract $\{1\}$), but it is less natural.
-
Let's first count all subsets of $\{1,\ldots,n\}$ with even sum. Removing the empty sets the... | {
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Question 84
# Ram and Shyam form a partnership (with Shyam as working partner) and start a business byinvesting 4000 and 6000 respectively. The conditions of partnership were as follows:1. In case of profits till 200,00 per annum, profits would be shared in the ratio of the invested capital.2.Profits from 200,001 till... | {
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# Double integral transforming into polar coordinates
1. Jan 7, 2010
### 8614smith
1. The problem statement, all variables and given/known data
By transforming to polar coordinates, evaluate the following:
$$\int^{a}_{-a}\int^{\sqrt{}{{a^2}-{x^2}}}_{-\sqrt{{a^2}-{x^2}}}dydx$$
2. Relevant equations
3. The attempt ... | {
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y=-√(a2-x2) represents the lower half of the circle.
So to fully integrate the integral, you'd need to integrate 'y' between √(a2-x2) and -√(a2-x2), and integrate 'x' between a and -a.
5. Jan 7, 2010
### 8614smith
Ok i'm sort of getting it, but why is it that $${x^2}+{y^2}={a^2}$$ gives the graph of a circle and $$... | {
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$$\int^{2}_{0}\int^{\sqrt{4-{y^2}}}_{\sqrt{y(2-y)}}\frac{y}{{x^2}+{y^2}}dxdy$$
My next question is whether you were given that integral or whether those limits are your attempt at describing the region. Are you trying to describe the region in the first quadrant exterior to the sideways circle but inside the larger ci... | {
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What happens now is that the smaller circle is completely contained in the larger. Frankly, the say I would do this is say that the area of the larger circle, with radius 2, is $\pi(2^2)= 4\pi$ and the area of the smaller circle, of radius 1, is $\pi(1^2)= \pi$ so the area "between" them is $4\pi- \pi= 3\pi$. Since we ... | {
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# Radius of convergence of product
Let $\sum_{i=0}^\infty a_nz^n$ and $\sum_{i=0}^\infty b_nz^n$ be power series, and define the product $\sum_{i=0}^\infty c_nz^n$ by $c_n=a_0b_n+a_1b_{n-1}+\ldots+a_nb_0$. Find an example where the first two series has radius of convergence $R$, while the third (the product) has radiu... | {
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• I find that it's not easy to compute the radius of convergence of the series expansion for $(1-x)^{1/2}$. It has coefficients $-\dfrac{1}{2}, -\dfrac{1}{2}\cdot\dfrac{3}{2}, -\dfrac{1}{2}\cdot\dfrac32\cdot\dfrac52, \ldots$. And it's not clear what the limsup will be. How do you compute it? – Paul S. Sep 19 '13 at 3:1... | {
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A simpler example: let $$f(z) = \frac{1+z}{1-z} = \frac{1}{1-z} + \frac{z}{1-z}.$$ Note that the first term is just the formula for the geometric sum with first term 1, $$\frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots, \qquad |z| < 1,$$ and the second term is the formula for a geometric sum with first term equal to the com... | {
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# Can there be a magic square with equal diagonal sums different from equal row and column sums?
I got a task in programming a program that can detect whether a 4x4 square is a magic square or not. At first, I wrote code that met the requirements for all given examples but I noticed one flaw. I used 2 variables to ind... | {
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to:
\begin{array}{|c|c|c|c|} \hline 116&3&2&113\\ \hline 5&110&111&8\\ \hline 9&106&107&12\\ \hline 104&15&14&101\\ \hline \end{array}
... and that's just too easy! So, I assume you mean that you have to use all numbers $1$ through $n^2$.
Well, after trying a bunch of things I am fairly convinced that you cannot hav... | {
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EDIT
Aha! As I thought, it also works for $n=5$. Here is one:
\begin{array}{|c|c|c|c|c|} \hline 19&6&15&2&23\\ \hline 9&17&10&13&16\\ \hline 21&24&14&5&1\\ \hline 4&11&8&20&22\\ \hline 12&7&18&25&3\\ \hline \end{array}
Rows and columns sum to $65$, but columns sum to $73$
EDIT 2: Aha! I was wrong about the $4 \time... | {
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# 377 pickover/pickover.01.p
## Description
This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
# 377 pickover/pickover.01.p
Title: Cliff Puzzle 1: Can you beat the numbers game?
From: cliff@watson.ibm.com
If you r... | {
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Row 1: 0 1 2 3 4 5 6 7 8 9
Assume Row 2 is your solution to the puzzle. I've just inserted random
digits below so as not to give away the solution:
Row 1: 0 1 2 3 4 5 6 7 8 9 S(1)
Row 2: 9 3 2 3 3 1 6 7 8 9 S(2)
Row 3: S(3)
Row 2 is now the starting point, and your next job is to form row ... | {
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This is an old puzzle, but I'll solve it as if it was new because I
find your extension below to be interesting.
Since all possible digits must be "counted" once, the ten digits must
add up to 10. Consider the first digit (= the amount of zeroes used):
9: Impossible, since all the other digits would have to be zero.
8... | {
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|> 2. A more advanced an interesting problem is to continue to
|> generate a sequence in a recursive fashion such that each row becomes
|> the sequence for the previous. For example, start with the usual
|> 0 through 9 digits in row 1:
|>
|> Row 1: 0 1 2 3 4 5 6 7 8 9
|>
|> Assume Row 2 is your solution to the puzzle.... | {
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4: If we put 4's under all the zeroes, we must put zeroes everywhere else.
0004440444 works.
3: Now we must place one non-zero digit under either the 6 or the 2, since
there are two 1's that must stay alike. Putting any non-zero digit under
the 6 is wrong since there aren't any sixes, unless you put a 6 under
the 6, wh... | {
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--
----w-w--------------Joseph De Vincentis--jwd2@owlnet.rice.edu----------------
( ^ ) Disclaimer: My opinions do not represent those of Owlnet.
(O O) Owlnet: George R. Brown School of Engineering Educational Network.
v-v (Unauthorized use is prohibited.) (Being uwop-ap!sdn is allowed.)
Snail mail: Rice U., 6... | {
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Pauli
Paul Dale | grue@cs.uq.oz.au
Department of Computer Science | +61 7 365 2445
University of Queensland |
Australia, 4072 | Did you know that there are 41 two letter
| words containing the letter 'a'?
The program I used follows:
---------------------------------... | {
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In article <1992Sep14.133741.34561@watson.ibm.com> you write:
>Title: Cliff Puzzle 1: Can you beat the numbers game?
>From: cliff@watson.ibm.com
>
>If you respond to this puzzle, if possible please include your name,
>address, affiliation, e-mail address. If you like, tell me a little bit
>about yourself. You might als... | {
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---------------------
|0|1|2|3|4|5|6|7|8|9|
|9|9|4|3|2|1|1|1|1|1| max
---------------------
C. In fact, under the numbers 5..9 there can be AT MOST one non-zero (1) answer
since otherwise two numbers of the 5..9 veriaty would appear and violate rule A.
D. So there must be at least 4 zeros. If there were exactly 4 ze... | {
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shoham@ll.mit.edu
-------------------------
>From clong@romulus.rutgers.edu (Chris Long) Tue Sep 15 06:08:45 1992
Path: igor.rutgers.edu!romulus.rutgers.edu!clong
From: clong@romulus.rutgers.edu (Chris Long)
Newsgroups: rec.puzzles
Subject: Re: Puzzle 1 (SPOILER)
Message-ID: <Sep.15.06.08.45.1992.9569@romulus.rutgers.... | {
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Now I'll prove that x_0 < n-1. x_0 clearly can't equal n; assume
x_0 = n-1 ==> x_{n-1} = 1 by (2) if n>3. Now only one of the
remaining x_i may be non-zero, and we must have that x_0 + ... + x_n
= n+1, but since x_0 + x_{n-1} = n ==> the remaining x_i = 1 ==> by
(2) that x_2 = 1. But this can't be, since x_{n-1} = 1 ==... | {
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"openwebmath_perplexity": 1260.6151974306902,
"openwebmath_score": 0.7816751003265381,
"tags": nul... |
>
>
> Stop And Think
>
> 1. Is there a solution to this problem? Are there many solutions to this
> problem?
>
[Second question and contest problem omitted]
Good puzzle! I am wondering though whether the second question (which
I have not tried to solve yet) is moe amenable to computer search.
It seems to me that there... | {
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-------------------------------
| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9|
-------------------------------
| 6| 2| 1| 0| 0| 0| 1| 0| 0| 0|
| 0| 0| 0| 4| 4| 4| 0| 4| 4| 4| <-
| 6| 6| 6| 0| 0| 0| 6| 0| 0| 0| |
| 0| 0| 0| 4| 4| 4| 0| 4| 4| 4| <-
.
.
.
I must be missing something in my understanding of your rules.
I found the ... | {
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# EVENPSUM - Editorial
Setter: Ildar Gainullin
Tester: Alexander Morozov
Editorialist: Ajit Sharma Kasturi
CAKEWALK
None
### PROBLEM:
We are given two positive integers A and B . We need to find the number of pairs of positive integers (X,Y) that can be formed such that 1 \le X \le A and 1 \le Y \le B and the sum ... | {
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• Total even numbers in [1,B] are B/2 .
• Total odd numbers in [1,B] are (B+1)/2 .
Therefore, the number of pairs (X,Y) where X+Y is odd are
(A/2) \cdot (B/2) + ((A+1)/2) \cdot ((B+1)/2) .
### TIME COMPLEXITY:
O(1) for each testcase.
### SOLUTION:
Editorialist's solution
#include <bits/stdc++.h>
using namespa... | {
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Why is this code showing WA for subtask 3 - https://www.codechef.com/viewsolution/40305092
1 Like
Hi, on line number 39, multiplying two integers will first result in integer which could lead to integer overflow. I have modified that line of your code by typecasting the product to long long. Here is your modified cod... | {
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"tags": ... |
import java.util.Scanner;
class Main{
public static void main(String []argh)
{
Scanner obj = new Scanner(System.in);
int T = obj.nextInt();
while (T–>0){
int a = obj.nextInt();
int b = obj.nextInt();
long sum = 0;
if(a%2!=0 && b%2!=0){
int div_a = a/2;
int div_b = b/2;
if(b>a)
sum +=((a-div_a)
(div_b+1));
else
sum +=(... | {
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"tags": ... |
# Standard deviation of binned observations
I have a dataset of sample observations, stored as counts within range bins. e.g.:
min/max count
40/44 1
45/49 2
50/54 3
55/59 4
70/74 1
Now, finding an estimate of the average from this is pretty straight forward. Simply use the mean (or median) of each r... | {
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"openwebmath_score": 0.844092845916748,
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1. These are binned random samples (which, I'm pretty sure, is where Sheppard's Corrections come in.)
2. It's unknown whether or not the data is for a normal distribution (thus I'm assuming not, which, I'm pretty sure, invalidates Sheppard's Corrections.)
So, my updated question is; What's the appropriate method for h... | {
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This reply presents two solutions: Sheppard's corrections and a maximum likelihood estimate. Both closely agree on an estimate of the standard deviation: $7.70$ for the first and $7.69$ for the second (when adjusted to be comparable to the usual "unbiased" estimator).
### Sheppard's corrections
"Sheppard's correction... | {
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Let's do the calculations. I use R to illustrate them, beginning by specifying the counts and the bins:
counts <- c(1,2,3,4,1)
bin.lower <- c(40, 45, 50, 55, 70)
bin.upper <- c(45, 50, 55, 60, 75)
The proper formula to use for the counts comes from replicating the bin widths by the amounts given by the counts; that ... | {
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"openwebmath_score": 0.844092845916748,
"tags"... |
Summing over all bins gives the log likelihood $\Lambda(\theta)$ for the dataset. As usual, we find an estimate $\hat\theta$ which minimizes $-\Lambda(\theta)$. This requires numerical optimization and that is expedited by supplying good starting values for $\theta$. The following R code does the work for a Normal dist... | {
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"openwebmath_score": 0.844092845916748,
"tags"... |
# Simplifying Power Series as a Summation - Alternating Coefficients
I'm currently trying to rewrite a power series I have into summation notation.
The series is as follows: $$2x + 3x^{4} + 2x^{7} + 3x^{10} + 2x^{13} + ...$$
Obviously I'll have $x^{3n+1}$ in the summation, but I'm not sure on how to piece together t... | {
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"openwebmath_score": 0.9241626858711243,
"tags... |
# Sketch graphs of
#### Casio
##### Member
I have a bit of a misunderstanding with;
y = (x - 2)^2
I understand it to be a quadratic, and if I used the formula to work it out I would see two roots,
x = - 2 or x = 2
If I put the above equation into a graphics calculator the result is always x = 2
Looking at the eq... | {
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"openwebmath_score": 0.6828569769859314,
"... |
I assumed 1, which is how I ended up with two roots.
But am unclear.
#### MarkFL
##### Administrator
Staff member
We know y = x2 has its axis of symmetry where x = 0. Then y = (x - h)2 will have its axis of symmetry where:
x - h = 0 or x = h.
This is a very common point of confusion for students of algebra. It see... | {
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"... |
x - h = 0 or x = h.
This is a very common point of confusion for students of algebra. It seems to go against intuition that f(x - h) moves the graph of f(x) h units to the right, when h is being subtracted from x. What in fact happens is the axes are translated to the left, relative to the graph, which is the same as ... | {
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"... |
(x - 2)(x - 2)
and you will find the above result.
#### Sudharaka
##### Well-known member
MHB Math Helper
I am not going to say this is absolutely correct and if not please do put me on the right tracks.
y = (x - 2)2
First I expanded the brackets;
(x - 2)(x - 2)
Then I multiplied them out;
x2- 2x - 2x + 4
then... | {
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"openwebmath_perplexity": 475.276517036582,
"openwebmath_score": 0.6828569769859314,
"... |
# Pigeonhole Principle and Sets
Can anyone point me in the right direction for this homework question? I know what the pigeonhole principle is but don't see how it helps :(
Let $n\geqslant 1$ be an integer and consider the set S = {1,2,.....,2n}. Let T be an arbitrary subset of S having size n + 1. Prove that this su... | {
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"openwebmath_score": 0.8426377773284912,
"ta... |
The pidgeonhole principle essentially says that we cannot possibly highlight $n+1$ numbers such that no two lie in the same column, if there are but $n$ columns. If you want to see this, then just take a small array, like for $n=3$: $$\begin{array}.1 && 2 && 3\\6 && 5 && 4\end{array}$$ Now, let's start highlighting som... | {
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"openwebmath_score": 0.8426377773284912,
"ta... |
The basic idea of the pigeonhole principle is that if you have more items than categories, some categories must have more than one item. Here the categories are the pairs of numbers that add to $(2n+1)$ - there are $n$ of those pairs, and every number in range is uniquely in one of those categories - and you want to se... | {
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"openwebmath_score": 0.8426377773284912,
"ta... |
Ratio of balls in a box
A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is $1:k$, where $k$ is an integer greater than one. A prime number of balls is removed from the box. The ratio of the total volume of the remain... | {
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"openwebmath_score": 0.9393088221549988,
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Once we get this, the remaining step is obvious. We express $n$ in terms of $k = p$ and see what can we do with that.
$$n = \frac{k^2+1}{k-1} = k + 1 + \frac{2}{k-1}\quad\stackrel{n\text{ is integer}}{\implies}\quad k = 2 \text{ or }3 \implies n = \frac{k^2+1}{k-1} = 5$$ The result just follows. $$(n,p,V) = (5,2,15) \... | {
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Next, we remove a prime number of balls from the box, say $p,$ so that there are now $n-p$ balls in the box, so that the total volume of the tennis balls remaining is $(n-p)\beta.$ The total volume of the box is $(1+k)n\beta,$ so the total volume of the empty space around the remaining tennis balls in the box is $$(1+k... | {
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Now, since $k$ is an integer, then $k^2+1$ and $k^2-k$ are integers. In particular, since $$(k^2+1)p=k(k-1)n,$$ then $k\mid(k^2+1)p,$ so since $k$ and $k^2+1$ are relatively prime, then we must have that $k\mid p.$ Since $p$ is prime and $k$ is an integer greater than $1,$ it then follows that $k=p,$ so we have $$(k^2+... | {
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So you say "There were 4 balls originally, in a box large enough to hold 8 of them. The volume ratio is 2. Then you took away 2 balls (a prime number!) and you have 2 balls in a box that's got the volume of 8,. That's a ratio of 4, which is $2^2$. Looks like a solution."
But are there other solutions? Seems likely. Fo... | {
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Note that in the second case $n$ cannot be equal to $p$ because for no natural $k$, $k^2+1=k^2-k$.
So the final result is: $n=5,k=2,p=2$ or $n=5,k=3,p=3$ | {
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# Existence/uniqueness of a Continuous Function
I ran across the following problem with a friend while we were studying for quals. Neither of us are really quite sure where to start. It feels like a differential equation. This is probably easy, but we were not able to get a handle on how to proceed. I wish I could tel... | {
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for some $\lambda<1$ and iterate.
• Nice. I should have seen that. Thanks for clearing that up for me. – fxy Jul 5 '15 at 23:27
• @fxy Thank you! And you're certainly welcome. It was my pleasure. The series solution is called The Neumann Series and is rich in both theory and application (e.g., The Born Approximation).... | {
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"ta... |
Then $T$ is Lipchitz with constant less than $1$. Indeed, letting $\lambda := \sup_{x,y \in [0,1]} K(x,y)$, we can easily show (using the same kind of argument as in Dr. MV's answer) that $$\|Tf-Tg\|_{\infty} \leq \lambda \|f-g\|_{\infty}$$
Thus, by the Banach fixed point theorem, $T$ has a unique fixed point $f_0$, w... | {
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# what is the general way of proving that a series converges uniformly on a bounded interval?
I am trying to solve the following question:
Prove that the series $$\sum_{n = 1}^{ \infty} \frac{x^{2n - 1}}{(2n - 1)!}$$ and the series $$\sum_{n = 1}^{ \infty} \frac{x^{2n}}{(2n)!}$$ both converges uniformly on any bounde... | {
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You say the Weierstrass test doesn't talk about bounded intervals, and you are right. But, what it does talk about is uniform convergence on a given set $$A$$; if $$\sum\limits_{n=1}^{\infty}\sup\limits_{x\in A}|f_n(x)|<\infty$$, then the series converges uniformly on $$A$$ (and recall that uniform convergence on $$A$$... | {
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In about 95% of situations, you can prove uniform convergence simply by the Weierstrass M-test (very rarely have I had to use some other method, such as Dirichlet's test; in fact I haven't used it recently, so much so that I kind of even lose track of the precise assumptions).
The Weierstrass M-test deals with uniform... | {
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• Amazing answer ..... I am going to read it thoroughly. Jan 11 at 3:27
Those are power series of a single (might as well let it be complex) variable. If $$R$$ is the radius of convergence of the power series, then for every $$R' < R$$, the power series converges absolutely and uniformly on the closed disk $$\overline... | {
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Alternative Induction Format
Generally, for induction proofs, we prove the property for a base case and then assume it holds for an arbitrary $n$ representative of some notion of $size$ for the object constructed, with the natural numbers being the most common object in the discourse. We then prove the property for $n... | {
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There are several different proof schemes by induction (so-called strong induction, forward-backward induction, transfinite induction, etc.). The most basic problem with your idea, which I'll denote by $P(n) \Rightarrow P(n-1)$, is that this only proves $P(k)$ for all $k \leq n$, where you know $P(n)$ is true. In this ... | {
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There is another interpretation to your idea that is common as well: the method of infinite descent (first widely used and attributed to Fermat). This is a proof by contradiction and here is the set up. Let $P(n)$ for all $n \in \mathbb{N}$ be the statement(s) I want to prove. We show that whenever $P(n)$ does not hold... | {
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# How to evenly space a number of points in a rectangle?
Say I have a rectangle, with variable width and height, for example lets use:
width = 20
height = 30
I would like to put n amount of evenly spaced points inside this rectangle:
no of points = 400
How could I calculate the x and y coordinates of each point?... | {
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"openwebmath_score": 0.8129738569259644,
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Total number of points: $$n = n_x \, n_y$$
Assuming $\Delta x = \Delta y$ one gets $$\Delta x = \frac{w}{n_x - 1} = \frac{h}{n_y - 1} = \Delta y \iff \\ n_y = \frac{h}{w} n_x + 1 - \frac{h}{w}$$ and then $$n = n_x n_y = \frac{h}{w} n_x^2 + \left( 1 - \frac{h}{w} \right) n_x \iff \\ \frac{w}{h} n + \frac{(w-h)^2}{4h^2}... | {
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