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We obtain
$$3S' = 2S - S(a^2+b^2+c^2)$$
For any $$\triangle XYZ$$ and point $$P$$ in the plane, we know the expression $$XP^2 + YP^2 + ZP^2$$ is minimized when $$P$$ is the centroid of $$\triangle XYZ$$. For an equilateral triangle of side $$1$$, the centroid is at a distance $$\frac{1}{\sqrt{3}}$$ from the vertices.... | {
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Drawing a card from a deck
A single card is drawn from a standard 52-deck of cards with four suits: hearts, clubs, diamonds, and spades; there are 13 cards per suit. If each suit has three face cards, how many ways could the drawn card be either a club of any kind or anything else besides a face card?
-
Hint: How man... | {
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How many spades are "good"? How many diamonds?
Now just add them all up.
-
I don't think she is worried about probability here, just the "number of good outcomes". – JavaMan Apr 19 '11 at 3:04
I feel like a total idiot. I made this problem much more difficult than what is there. Thanks DJC. I appreciate it. – user9... | {
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Therefore, if you want to find the number of sides of a die that are (say) even or prime, you count the number of sides which are even (there are $3$ such sides - namely $2$ , $4$, $6$) add to it the number of sides which are prime (again, there are $3$ such sides - namely $2$, $3$, $5$), and then subtract the sides wh... | {
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# Finding the n'th term of fibonacci like sequence
1. Sep 30, 2013
### neerajareen
The fibonacci sequence can be defined as $${F_n} = {F_{n - 1}} + {F_{n - 2}}$$ and specifying the initial conditions as \eqalign{ & {F_1} = 1 \cr & {F_2} = 1 \cr}
Also there exists a general formula for the fibonacci which is given ... | {
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Let $A=\pmatrix{ 1 & 1 \cr 1 & 0 \cr}$
The eigenvalues of $A$ are \eqalign{ & {\lambda_1} = \frac{1+\sqrt{5}}{2} \cr & {\lambda_2} = \frac{1-\sqrt{5}}{2} \cr}
We can diagonalize $A$ , so that it has the form
$A= U \Lambda U^{-1}$
where $\Lambda=\pmatrix{ \lambda_{1} & 0 \cr 0 & \lambda_{2} \cr}$ is a diagonal matrix,... | {
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You can do the same procedure for your Gn - you'll get three roots because you will be left with a cubic equation, so you'll have three unknowns which makes sense because the first three values of Gn will have to be specified. Unless you have a closed form expression for your roots though you will have some floating po... | {
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# Recurring decimals as fractions
Earlier I solved this question and it got me thinking.
$\text{Why do recurring decimals have a fractional equivalent?}$
So I decided to work it out, this is what I got.
Let $$b$$ be a number with $$n$$ digits, it can have both leading and trailing zeros.
Every recurring decimal ca... | {
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That's all for now, hope you found this note interesting.
Note by Jack Rawlin
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# What's the cross product in 2 dimensions? [duplicate]
The math book i'm using states that the cross product for two vectors is defined over $$R^3$$:
$$u = (a,b,c)$$
$$v = (d,e,f)$$
is:
$$u \times v = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ d & e & f \\ \end{vmatrix}$$
and the direction of the... | {
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• Point for you to ponder on - the cross product of two vectors gives a third vector, perpendicular to the original two vectors. Could you define a (meaningful) cross product in R^2 that would imitate that property? Moreover, there are similar threads to this one in MSE, I recommend you to try reading some of them :) –... | {
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E. A. Abbott describes a 2D cross product nicely in his mathematical fantasy book "Flatland":
Flatland describes life and customs of people in a 2-D world: in this universe vectors can be summed together and projected, areas are calculated, rotations are clock-wise or counter clock-wise, reflection is possible... but c... | {
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My Math Forum Reciprocal sum of primes diverges... proof?
Number Theory Number Theory Math Forum | {
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December 27th, 2006, 10:01 PM #1 Newbie Joined: Nov 2006 Posts: 20 Thanks: 0 Reciprocal sum of primes diverges... proof? We'll need the following definitions to solve this problem: definition: For any number x, Nj(x) is the number of numbers less than or equal to x whose only prime divisors are in the set of the firs... | {
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December 28th, 2006, 02:17 AM #2 Newbie Joined: Nov 2006 Posts: 20 Thanks: 0 I have already solved part a), if anyone has idea in solving part b, part c, and part d, please do teach me. Thank you very much.
December 28th, 2006, 02:36 PM #3 Newbie Joined: Dec 2006 Posts: 29 Thanks: 0 For part a), I believe you mean... | {
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Tags diverges, primes, proof, reciprocal, sum
,
,
### prove the summation of the reciprocal primes is divergent
Click on a term to search for related topics.
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Similar Threads Thread Thread Starter Forum Replies Last Post magnuspaajarvi Applied Math 1 February 8th, 2014 11:08... | {
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# Compute $\int_{0}^{\infty}\frac{\sin(x)}{xe^x} \ dx$.
I know one can solve this in many ways and the answer is $$\pi/4$$. However I'm interested in one particular solution involving Laplace transform.
I once saw a solution of this integral where one just did something like directly taking the Laplace transform of t... | {
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Let $$I(a)=\int_0^\infty \frac{\sin(x)}{x}e^{-ax}dx$$ where your integral is $$I(1)$$. $$I'(a)$$ is then: $$I'(a)=-\int_0^\infty \sin(x)e^{-ax}dx$$ which is the Laplace Transform of the sine function. Thus $$I'(a)=-\frac{1}{a^2+1}$$
Integrating back we get: $$I(a)=-\arctan(a)+C$$ To find the constant, we notice that $... | {
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My usual preferred way of solving this sees the same phenomenon. It begins with a Schwinger parametrization, writing $$\frac{1}{x}=\int_0^\infty\exp -sx ds$$. Then the integral becomes $$\int_0^\infty ds\int_0^\infty dx\sin x\exp -(1+s)x$$. Up to a constant variable shift, this yields the deliberately Laplace-based app... | {
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1. ## trigonometric limit problem
lim (sin(x) - x) / x^3
x~>0
so i tried splitting it up into sin(x) / x^3 - x / x^3. sin(x) / x^3 = (1/x^2)(sin(x) / x) and the limit as x approaches 0 of sin(x)/x = 1 so the first term just becomes 1/x^2. for the second term i canceled out an x from the top and bottom so i also got 1... | {
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We can apply L'Hopital: .$\displaystyle \lim_{x\to0}\frac{\cos x - 1}{3x^2} \;=\;\frac{1-1}{0} \:=\:\frac{0}{0}$ . . . indeterminate
At this point, we can: .$\displaystyle \begin{array}{cc}(1) & \text{Apply L'Hopital twice} \\ (2) & \text{Do some clever algebra} \end{array}$
Method 1:
. . L'Hopital again: .$\display... | {
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# Probability that two random letters from a language will be the same?
I'm trying to find the probability that two randomly-selected letters from "average" text in a language will be the same.
For example, if my hypothetical language contains four letters which each occur on average with the following frequency:
A ... | {
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• A derivative question: What frequency distribution minimizes this probability that two randomly-selected letters from "average" text in a language will be the same? Is it all letters equally common distribution? – curious_cat Mar 21 '13 at 6:49
• @curious_cat yes a discrete uniform minimizes the probability. (This is... | {
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If $L_1$ is the first letter, $L_2$ is the second letter, ..., and $L_k$ is the $k^\mathrm{th}$ then
the probability is $\sum_x P(L_1=x) \times P(L_2=x|L_1=x)\times P(L_3=x|L_1=x,L2=x)\times ... \times P(L_k=x|L_1=x,L2=x, ...,L_k=x)$.
If you assume independence (which would be the case if you were selecting from the ... | {
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# How can I write this as a double sum?
$$2 \sum_{1\leq j < k \leq N} P(F_j F_k)$$ as a double sum. Would it be $$\sum_{j=1}^k \sum_{k=n}^N P(F_j F_k)$$?
Also I am just confused about what a double sum would actually mean, I understand that you would sum over the values twice but can someone please explain more in de... | {
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There are more ways you can approach this.
1. We want one variable, say $$j$$, to be independent, and let it independently take all the values it can possibly have that satisfy the condition $$1\le j < k \le N$$. We will then adjust $$k$$ so that this condition is actually met.
Therefore, let $$j$$ vary across all val... | {
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The difference between these two approaches is which variable you assign "independence" to.
• thank you! i have a really stupid question but how come certain double sums have a 2 next to it and other double sums do not? Mar 4 '19 at 4:31
• also, why do you initially take j to be $1 \leq j \leq N-1$ Mar 4 '19 at 4:33
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So, any bijection $$\sigma$$ into a disjoint union as mentioned above will give you a representation as a double sum. You could of course proceed inductively, to get triple, quadruple sums et cetera.
The double sum can hence be understood as a specific grouping of summands/a partition of the index set.
Defining $$I:=\... | {
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# Find the value of x and y
#### Albert
##### Well-known member
x=abc is a three digits number
y=$x^2$
the last 3 digits of y is also equal to abc
that is y=???abc
#### Bacterius
##### Well-known member
MHB Math Helper
Re: find the value of x and y
$$x^2 \equiv x \pmod{1000} \tag{1}$$
[JUSTIFY]Clearly for $x$ c... | {
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Here's an Python 3 script implementing the algorithm outlined above, using recursion, for powers of $10$:
Code:
# Work count
work = 0
def Verify(x, m):
global work
work += 1
return pow(x, 2, 10**m) == x
def Solve(m):
# Solutions
found = []
if m == 1:
for x in [2, 4, 5, 6, 8]: # Precomputed
if Verify(x, m):
found.a... | {
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y-x =$x^2-x =x(x-1)$ must be a multiple of 1000
for 1000=$2^3 \times 5^3 =8\times 125$
further more x and (x-1) are coprime
if x is a multiple of 125 then x-1 must be a multiple of 8---------(1)
if x is a multiple of 8 then x-1 must be a multiple of 125---------(2)
so the possible solutions meet restriction of (1)... | {
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# If there are injective homomorphisms between two groups in both directions, are they isomorphic?
If I have two groups, $G$ and $H$ and two injective homomorphisms $\phi:G \to H$ and $\psi: H \to G$, then by the first isomorphism theorem applied to $\phi$, we have that $G \cong \mathrm{Im} (\phi)$, a subgroup of $H$.... | {
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On the other hand, let $S_{\infty}'$ denote the subgroup of $S_{\infty}$ of permutations of the set $\{3, 4, \ldots\}$, and consider the map $\Phi: S_{\infty} \to S_{\infty}'$ that maps a permutation $(a_1 \cdots a_n)$ to $((a_1 + 2) \cdots (a_n + 2))$ (this amounts to relabeling and so is manifestly an isomorphism). T... | {
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For groups this property - sometimes called the Cantor-Schröder-Bernstein property after the corresponding theorem for plain sets - is wrong. Let $G = F_2$ the free group on two generators $\{a,b\}$ and $H = F_3$ the free group on three generators $x,y,z$. Then there are monomorphisms $f \colon G \to H$, given by $f(a)... | {
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• What about ordered fields? – celtschk Apr 30 '15 at 22:09
• You mean uncountable algebraically closed fields. Look at the algebraic closure of $\Bbb Q(\pi)$ and that of $\Bbb Q$, both are countable and non-isomorphic. – Asaf Karagila May 1 '15 at 11:09
• @celtschk: For ordered fields the analogue of algebraically clo... | {
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Density of a Cylinder Calculator
Created by Gabriela Diaz
Last updated: Nov 27, 2022
Welcome to the density of a cylinder calculator! If you're wondering how do I find the density of a cylinder?, then you've come to the correct place! With this tool, you can expect to determine the density of any right cylindrical ob... | {
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$\small \rho =\cfrac{m}{V}$
💡 Take a look at the density calculator to keep exploring this topic!
We can part from this general expression to get a particular equation for the density of a cylinder. Let's see how to obtain it:
1. Begin by expressing the volume formula of a cylinder $V_\text{cyl}$ of radius $r$ and ... | {
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3. Now that you have the volume, enter the Weight/mass to get the value of the Density.
💡 What if you want to calculate the mass of a cylinder 🤔 For this, you'll need to know the cylinder's density and volume. With these, the density of a cylinder calculator will calculate the mass of the cylinder!
Gabriela Diaz
De... | {
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How to adjust vector scaling with a specific aspect ratio
I'm trying to create a vector plot with uniform vector scaling. I need the x and y ranges to be very different, because I need to highlight the angle between the vectors and the y=0 axis. This means that I can't use an automatic aspect ratio, however, changing ... | {
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Is there a way to scale the vectors so that this doesn't happen?
• Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles,... | {
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# What is the difference between independent and mutually exclusive events?
Two events are mutually exclusive if they can't both happen.
Independent events are events where knowledge of the probability of one doesn't change the probability of the other.
Are these definitions correct? If possible, please give more th... | {
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\left.\begin{align}P(A\cap B) &= 0 \\ P(A\cup B) &= P(A)+P(B)\\ P(A\mid B)&=0 \\ P(A\mid \neg B) &= \frac{P(A)}{1-P(B)}\end{align}\right\}\text{ mutually exclusive }A,B
Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other(s). For example: when ... | {
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If two events A and B are independent a real-life example is the following. Consider a fair coin and a fair six-sided die. Let event A be obtaining heads, and event B be rolling a 6. Then we can reasonably assume that events A and B are independent, because the outcome of one does not affect the outcome of the other. T... | {
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If I toss a coin twice, the result of the first toss and the second toss are independent.
However the event that you get two heads is mutually exclusive to the event that you get two tails.
Suppose two events have a non-zero chance of occurring.
Then if the two events are mutually exclusive, they can not be independ... | {
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Events are Independent when happening of one does not influence happening of other. Eruption of volcano on Earth and orbit of Mars do not influence each other, so are independent events.
Growth of human population and preservation of many other species are mutually exclusive, as the one can only happen if the other do... | {
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# In an equilateral triangle, prove that $|BQ| + |PQ| + |CP| > 2l$
I am trying to solve the following problem:
Let $ABC$ be an equilateral triangle with side $l$.
If $P$ and $Q$ are points respectively in sides $AB$ and $AC$, different from the triangle vertices, prove that $$|BQ| + |PQ| + |CP| > 2l$$
I can see tha... | {
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Let $AQ=x$, $AP=y$ and $l=1$.
Thus, $$PQ=\sqrt{x^2-xy+y^2},$$ $$PC=\sqrt{y^2-y+1}$$ and $$BQ=\sqrt{x^2-x+1}$$ and we need to prove that $$\sqrt{x^2-xy+y^2}+\sqrt{x^2-x+1}+\sqrt{y^2-y+1}\geq2.$$ Now, by Minkowwski $$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}+\sqrt{\left(y-\frac{1}{2}\... | {
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# Why log-of-sum-of-exponentials $f(x)=\log\left(\sum\limits_{i=1}^n e^ {x_i}\right)$ is a convex function for $x \in R^n$
How to prove $f(x)=\log(\sum_{i=1}^n e^{x_i})$ is a convex function?
EDIT1: for above function $f(x)$, following inequalities hold:
$$\max\{x_1,x_2,...,x_n\} \le f(x) \le \max\{x_1,x_2,...,x_n\}... | {
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So I achieve that $f(\theta x+(1-\theta)y) \le \theta f(x) + (1-\theta)f(y)$.
Another way to prove the convexity of this function is to use the Jensen's Inequality which states that a function is convex if and only if
$$f(tX+(1-t)Y) \le t f(X) + (1-t)f(Y)$$
Now let $X$ be represented by the vector $({X_1, X_2, X_3,.... | {
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For a multivariate function to be convex, it's equivalent to show that its Hessian matrix is positive semi-definite. That is, you can calculate $$\nabla^2 f(\mathbf{x})$$ here and show it is positive semi-definite.
This can be proved using Cauchy Schwarz inequality as shown here.
• If anyone doesn't understand how CS... | {
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# Math probability problems
## Math probability problems
How likely something is to happen.The probability (chance) is a value from the interval 0;1> or in percentage (0% to 100%) expressing the occurrence of some event.Probability Definition in Math.Probability word problems worksheet.Compute probabilities for simpl... | {
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What is the probability that the problem is solved?Featured on Meta Testing three-vote close and reopen on 13 network sites.The best we can say is how likely they are to happen,.Example 3: There are 20 boys and 30 girls in a class.Many events can't be predicted with total certainty.Hence, the probability that the ball ... | {
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probability problem on coin and read examples.Probability is the likliehood that a given event will occur and we can find the probability of an event using the ratio number of favorable outcomes / total number of outcomes.Probability is a measure of the likelihood of an event to occur.Learn and practice basic word and ... | {
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {
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# How can I simplify $(4ab^{-1})^{-2}$
As part of a wider expression I have a component $$(4ab^{-1})^{-2}$$
I know that using the rules of exponents, if there was no radical within the brackets I could rewrite like this:
$$\frac{(4ab)}{2}$$
I also know that if the only component within the brackets were $$b^{-1}$$ ... | {
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There are several ways to simplify this, but I suggest you work your way from the inside out as this seems to be the easiest way in general, and is what using brackets normally implies doing. Note that $$x^{-n} = \cfrac{1}{x^n}$$. As such, first we get that
$$4ab^{-1} = \cfrac{4a}{b} \tag{1}\label{eq1}$$
Next, using \e... | {
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# What is $26^{32}\bmod 12$? [duplicate]
What is the correct answer to this expression:
$$26^{32} \pmod {12}$$
When I tried in Wolfram Alpha the answer is $$4$$, this is also my answer using Fermat's little theorem, but in a calculator the answer is different, $$0.$$
## marked as duplicate by Carl Mummert, Lee Davi... | {
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$$26\equiv-1\pmod3$$
$$\implies26^{2n}\equiv(-1)^{2n}\equiv1\pmod3$$ where $$n$$ is any integer
$$\implies26^{2n+2}\equiv1\cdot26^2\pmod{3\cdot26^2}$$
$$\implies26^{2n+2}\equiv26^2\pmod{3\cdot2^2}\equiv?$$
Note that $$26\equiv 2\pmod{12}$$, so we can as well compute $$r=2^{32}\bmod 12$$.
We have $$2^{32}=12q+r$$, ... | {
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Remark Numbers like $$4$$ above with $$a^2=a$$ are called idempotents. This implies $$\,a^{\large n} = a\,$$ for all $$n\ge 2$$ either as above or by a simple induction $$\,a^{\large n+1}\! = a\,a^{\large n} = a\cdot a = a.\,$$ Said more conceptually: note $$a$$ is a fixed point of $$\,f(x) = ax\,$$ i.e. $$\,f(a) = a,... | {
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• Splitting the $12$ into the coprime factors $4$ and $3$ is a technique which is substantially generalised in the Chinese Remainder Theorem. The observations in this case are elementary and don't need complex machinery, but it is there if needed to steer the analysis of more complex cases. – Mark Bennet Oct 27 '18 at ... | {
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# Does limit of $x_n=(1+\frac{1}{2})*(1+\frac{1}{4})*…*(1+\frac{1}{2^n})$ exist?
I tried to show that the sequence is increasing and limited, but couldn't find the limit. I also tried with squeeze theorem, but $(1+\frac{1}{2^n})^n<=x_n<=(1+\frac{1}{2})^n$ is not helping and I ran out of ideas.
• – John Dvorak Nov 4 '... | {
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# What's the sum of this series? [duplicate]
I would like to know how to find out the sum of this series:
$$1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \frac{1}{6^2} + \cdots$$
The answer is that it converges to a sum between $\frac 34$ and $1$, but how should we go about estimating this sum?... | {
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$$= \frac{\pi^2}{12}$$
Comment if you have questions.
• How do you know that? – inggumnator Aug 28 '14 at 0:08
• @inggumnator I'll post shortly the solution. – Varun Iyer Aug 28 '14 at 0:08
• @inggumnator do you understand now how it is derived? I hope this helps. – Varun Iyer Aug 28 '14 at 0:15
• @VarunIyer: Nice de... | {
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For example, if we use $1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}$ as an approximation, then the absolute value of the error is less than $\frac{1}{6^2}$. Furthermore, the error is "negative," that is, our estimate is greater than the true value.
• Thanks, this is a nifty trick! – inggumnator Aug 28 '1... | {
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# Math Help - Find the limit of a recursion formula
1. ## Find the limit of a recursion formula
Hello,
I am looking at this recursion formula in my Calculus book.
x_n+1 = 1/2(x_n + 2/x_n)
It goes on to say that we can find the limit of this sequence by doing this:
L = 1/2(L + 2/L) fine
but now it says that this can b... | {
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In fact, it should be clear that if $x_n$ is positive, then $x_{n+1}= (1/2)(x_n+ 1/x_n)$ is also positive while if $x_n$ is negative, then $x_{n+1}= (1/2)(x_n+ 1/x_n)$ is also negative.
That is, if $x_0$ is positive, the limit is $\sqrt{2}$ and if $x_0$ is negative, the limit is $-\sqrt{2}$.
By the way, they get that ... | {
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# Finding all solutions for an equation system
I have a vector of numbers $x_i$ that can only have the values $0$ or $1$.
I need to find all the possible combinations $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that:
$$\begin{cases} x_1+x_5+x_6 \ \mathrm{is\ odd}\\ x_2+x_4+x_6 \ \mathrm{is\ odd} \\ x_3+x_4+x_5 \ \mathrm{is\ od... | {
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### Now we can start by systematically making cases:
• $x_1 = 1$
Equation (I) now reads $1 + x_5 + x_6$ is odd, i.e. $x_5 = x_6$
• $x_5 = x_6 = 1$
Now Equations (II) and (III) combine to $x_2 = x_3 = x_4$, so we have $(1, 0, 0, 0, 1, 1)$ and $(1, 1, 1, 1, 1, 1)$ as solutions
• $x_5 = x_6 = 0$
This means $x_2 \ne x_4$ ... | {
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This matrix is already in reduced row-echelon form, so there are three free variables, and 8 solutions, corresponding to the eight possible values of $x_4$, $x_5$, and $x_6$, and completed by the equations $x_1=1+x_5+x_6$, $x_2=1+x_4+x_6$, and $x_3=1+x_4+x_5$, with all addition being performed in $\mathbb{Z}/2\mathbb{Z... | {
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Distribute 10 white and 10 black balls into 20 distinct boxes s.t. no box is empty?
How many ways are there to distribute 10 white and 10 black balls into 20 distinct boxes such that no box is empty?
Solution is that ${20!}/{10!10!}$. How can we write this? Please can you explain this solution clearly?
-
2 Answers
... | {
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In fact the black marbles bear no such labels. We’ve counted each set of $10$ positions for the black marbles once for every permutation of the black marbles in those positions, so we’ve overcounted by another factor of $10!$. Dividing by $10!$ to compensate finally gives us the correct answer for the unlabelled marble... | {
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# Why round to even integers?
According to the Mathematica help:
Round rounds numbers of the form x.5 toward the nearest even integer.
For example:
Round[{0.5, 1.5, 2.5, 3.5, 4.5}]
gives
{0, 2, 2, 4, 4}
What's the rationale behind this? Why not the usual x.5 always rounds up?
-
It is called bankers' rounding... | {
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Searching for some of those terms returns many good results including:
stackoverflow.com
It does not suffer from negative or positive bias as much as the round half away from zero method over most reasonable distributions. ... But the question was why [] use Banker's actual rounding as default - and the answer is tha... | {
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www.russinoff.com
One consequence of this result is that a midpoint is sometimes rounded up and sometimes down, and therefore, over the course of a long series of computations and approximations, rounding error is less likely to accumulate to a significant degree in one particular direction than it would be if the the... | {
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# Introduction Mathematical Statistics, finding the Mean Square Error of estimators
I'm working on a problem from my introduction to mathematical statistics course. So far, I've done the following work:
Let $X_{1},...,X_{m}$ and $Y_{1},...,Y_{n}$ be independent samples form the Bernoulli distribution with unknown par... | {
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So this is my work so far. I still don't quite understand the meaning of the estimators. Let alone, work out which one is more preferable. I've just try to work from the definition and I think my solution for i) works but I'm not sure. But for ii) I've truely no idea where to start. Any suggestions would be much apprec... | {
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where we've used the independence of all $X_i, Y_i$ to interchange variance with summation, the property that $\mathrm{Var}(aX) = a^2 \mathrm{Var}(X)$ when $a$ is a constant, and the fact that all variables are identically distributed as $\mathrm{Bernoulli}(p)$ to pull out the common factor $p(1 - p)$ out of the sum.
... | {
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# Independent odds, am I (+ friend) seeing this wrong or is there a mistake in the practice exam?
I found this exercise in a practice exam:
Any student has a 90% chance of entering a University. Two students are applying. Assuming each student’s results are independent, what is the probability that at least one of th... | {
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\begin{aligned} {\rm A}:\ {\rm Pr}[\text{one succeeds}] &= 0.5 &\implies& \text{success rate} = 1 - \sqrt{0.5}\phantom0 \approx 0.292893 \\ {\rm B}:\ {\rm Pr}[\text{one succeeds}] &= 0.65 &\implies& \text{success rate} = 1 - \sqrt{0.35} \approx 0.408392 \\ {\rm C}:\ {\rm Pr}[\text{one succeeds}] &= 0.88 &\implies& \tex... | {
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• good effort here – J Tg Aug 31 '17 at 10:38
• Good effort. Just wanted to add that you missed investigating one possibility: 90% was correct, but they intended to ask something else - exactly one student or both students getting admitted. But we can rule out that since none of the probabilities match, and thus your a... | {
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Hence, among the possible answers provided, there should be values like
Pr(success) * Pr(success);
Pr(success) * Pr(fail)
Pr(fail) * Pr(fail)
Pr(success) + Pr(success) [despite this possibly being > 1]
Pr(fail) + Pr(fail) [despite this possibly being > 1]
Pr(success)
AND
1 - any of tho... | {
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# Evaluating the limit $\lim_{x \to 1} (x^3 - 1) / (x - 1)$
$$\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$$
As $x$ approaches to $1$, if I use the substitution method, it will become undefined. Then, I tried to multiply it by its conjugate but I still get undefined answer. How can I solve it?
The other answers are cleaner ... | {
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Hint
$$x^3-1=(x-1)(x^2+x+1)$$ $\ \ \ \ \ \ \ \ \ \$
Use the fact that $x^3-1=(x-1)(x^2+x+1)$.
• How to factorise it? I only know x-1 is one of the factors. – user307537 Jan 24 '16 at 5:27
• So, divide $x-1$ into $x^3-1$. You can do this either by long division or synthetic division. – Tim Raczkowski Jan 24 '16 at 5:... | {
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# Cartesian product of sets with different cardinality
What is the correct way of performing a cartesian product over finite sets with different cardinality? For instance:
Let
$$A = \{1, 2, 3\},\quad B = \{4, 5\},\quad C = \{6, 7\}$$
What would be the result of
$$A \times B \times C$$
What I thought it would corr... | {
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# Math Help - Differentiable points
1. ## Differentiable points
Hi, I'm having a hard time understanding how to determine where a function is not differentiable (the question keeps coming up in my homework though sadly, it didn't in the lesson!) I understand that a point where the function is not differentiable means... | {
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4. Originally Posted by starswept
So, essentially, saying the derivative doesn't exist is exactly like saying the limit does not exist? Does that mean that the right hand limit must equal the left hand limit rule still comes into play?
yes, all that comes into play. and also, if the limit diverges, that is, goes to inf... | {
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# Using dominated convergence to prove partial derivative and integral can be interchanged
Hi guys doing a self study here, and came across this problem. I know the same question with slightly different hypotheses has been asked before but I was a bit confused on the answers given and not sure how to completely adjust... | {
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Yes, that's correct and holds for any continuous function $f: \mathbb{R}^2 \to \mathbb{R}$. For a fixed point $(x_0,y_0)$ continuity at $(x_0,y_0)$ means that for all $\epsilon>0$ there exists $\delta>0$ such that $$|f(x,y)-f(x_0,y_0)| \leq \epsilon \qquad \text{for all} \, \, |(x_0,y_0)-(x,y)| \leq \delta. \tag{1}$$ N... | {
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On the other hand, the mean value theorem shows
$$|u_n(y)| \leq \sup_{\lambda \in [0,1]} |\partial_x f(x+\lambda/n,y)| \leq \sup_{y \in [a,b]} \sup_{|u-x| \leq 1} |\partial_x f(u,y)|.$$
Since $\partial_x f$ is continuous, the right-hand side is a (finite) constant and therefore integrable on the finite interval $[a,b... | {
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# Is there a formula for a simple sum of sums?
Given $f(n) = 1 + (1 + 2) + (1 + 2 + 3)+ \cdots + (1 + 2 + 3 +\cdots + n)$, I am wondering if there is a straightforward formula to compute f(n) and how it may be derived.
The only reduction I thought about so far would be:
n*1 + (n - 1)*2 + (n - 3)*3 ... which seems sy... | {
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# Number Diagonals of Polygon
One of the easiest topics in plain Geometry is to count the number of diagonals in a convex polygon. You might familiar already or even memorized the said formula. So let me show you the technique how to derive it. This tutorial is made with two main reasons. First is to educate and secon... | {
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$d=\displaystyle\frac{n(n-3)}{2}$
Derivation using basic Combinatorics with small common sense:
Consider a convex polygon like the figure shown below.
Removing the segments connecting the points we have the figure in 2, a scattered points.
In Combinatorics, we can connect the points in nC2 ways from n number of poi... | {
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8. With havin so much content and articles do you ever run into any problems of plagorism or copyright infringement?
My blog has a lot of exclusive content I’ve either created myself or
outsourced but it appears a lot of it is popping it up all over the web without my agreement.
Do you know any solutions to help preve... | {
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18. equipments says:
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19. He... | {
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# Matrix Multiplication Divide And Conquer | {
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Use a divide-and-conquer approach as in Strassen’s algorithm, except that instead of getting 7 subproblems of size n/2, get 5 subproblems of size n/2 based on part (a). Divide-and-Conquer Multiplication: Warmup To multiply two n-bit integers a and b: Multiply four ½n-bit integers, recursively. 4-46 faster than the orig... | {
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calls from 8 to 7 and hence, the improvement. These subproblems must be solved and then a method must be found to combine subsolutions into a solution of a whole. The primary topics in this part of the specialization are: asymptotic ("Big-oh") notation, sorting and searching, divide and conquer (master method, integer ... | {
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& Conquer: First Approach II. Divide-and-conquer matrix multiplication c. I want to know when to switch to another algoritm, which in this case is the regular matrix multiplication. Raymund Fischer author of Program of matrix multiplication using function is from Frankfurt, Germany. Divide-and-Conquer Reading: CLRS Sec... | {
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improve the speed of matrix multiplication. Get this from a library! An I/O-Complexity Lower Bound for All Recursive Matrix Multiplication Algorithms by Path-Routing. Note that it is the kth smallest element in the sorted order, not the kth distinct element. In this section we discuss a top-down algorithmic paradigm ca... | {
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of the algorithm. (Divide and Conquer) Reviewed by Huzaif Sayyed on June 16, 2017 Rating: 5 Share This:. ×) by min (resp. Conclusion. I've implemented the O(log_2 7) Strassen algorithm once (which should be really simple after implementing normal divide and conquer) and after benchmarking I've determined that for matri... | {
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"t... |
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