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Having converted the whole part, you can think of the process as finding which of $$\frac 12, \frac 14, \frac 18, \frac 1{16},\ldots$$ you need to add together to get the fractional part. If the denominator is not a power of $$2$$ the expansion will repeat. Keep doubling the fraction and keep track of the carries into ... | {
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# Thread: A Fibonacci number problem
1. ## A Fibonacci number problem
How can we show that the following (recursive) statement, where $F_n$ is the $n$-th Fibonacci number, and $n \geq 1$, is true?
$\sum_{i=0}^{\lfloor \frac{n-1}{2} \rfloor} F_{n-2i}=F_{n+1}-1$
******
As this should be shown to be valid for every $... | {
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It is now easier to analyze this problem.
3. Originally Posted by ThePerfectHacker
It is easier to think about this problem if you consider even and odd cases for $n$.
If $n$ is odd then $[\tfrac{n-1}{2}] = \tfrac{n-1}{2}$.
And the sum becomes $F_1+F_3+...+F_{n-2}+F_n = F_{n+1}-1$.
If $n$ is even then $[\tfrac{n-1}{... | {
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# Let $\sum a_n$ and $\sum b_n$ converge, $a_n,b_n\geq 0$, does $\sum \min\{a_n,b_n\}$, $\sum \max\{a_n,b_n\}$ converge too?
Let $$\sum\limits_{n=0}^{\infty}a_n$$ and $$\sum\limits_{n=0}^{\infty}b_n$$ be convergent with $$a_n,b_n\geq 0$$, does $$\sum\limits_{n=0}^{\infty}\min\{a_n,b_n\}$$ and $$\sum\limits_{n=0}^{\inf... | {
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• We could also use $\min \{a_n,b_n\}\le a_n$ for the $\min$ series.... An application: For $2\le p<\infty$ and $1/p+1/q=1,$ if $x=(x_n)_n\in l^p$ and $y=(y_n)_n\in l^q$ then $\sum_n x_ny_n$ converges. Proof: Let $z_n=0$ if $y_n=0.$ If $y_n\ne 0$ let $y_n=z_n|z_n|^{q-2}.$ Then $(z_n)_n\in l^p$ and $|x_ny_n|\le \max \{|... | {
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# Limit $\lim\limits_{(x,y)\to(0,0)}\frac{\sin(x^2-y^2)}{x^2-y^2}$
I'm trying to understand the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2-y^2)}{x^2-y^2}$$
The $\lim_{(x,y)\to (0,0)} f(x,y)$ is undefined. Why it is not equal to $1$?
Let's suppose $t = x^2-y^2$. Then as $(x,y)$ approaches $(0,0)$, $t$ app... | {
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I would appreciate any sort of help in the matter , Thanks.
• Is it really undefined? WRA says that the limit is equal to $1$. – Ethan Hunt Mar 26 '16 at 19:03
• I think you can only approach via paths that lie in the domain of the function. You just need to make sure there exists at least one such path (so it is not ... | {
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$$\lim_{(x,y) \to (0,0)} \hat{f}(x,y) = 1.$$
You can use polar coordinates here. Set $x=r\cos\theta$, $y=r\sin\theta$, then notice that $x^2-y^2=r^2\cos 2\theta$. Then the limit becomes $$\lim_{r \to 0} \frac{\sin (r^2\cos 2\theta)}{r^2\cos 2 \theta}.$$
Clearly you have to exclude the case $\theta=\pm \pi/4$ because ... | {
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# Variance
#### Yankel
##### Active member
Hello
I have a problem solving this question....
to the numbers 0 and 2, we want to add a 3rd number, such that the variance won't change. What is the 3rd number ?
(when I say variance I mean dividing by n, not by n-1)
thanks !
#### TheEmptySet
##### New member
Hello
... | {
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So the set of equations should be:
$\sigma^2=\frac{1}{2}\left[(0-\mu)^2+(2-\mu)^2\right]$
$\sigma^2=\frac{1}{3}\left[(0-\mu)^2+(2-\mu)^2+(y-\mu)^2\right]$
This does introduce the problem that we have more unknowns than equations.
Maybe I am misunderstanding something but the way I read the problem was if we have the ... | {
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# Linear transformation and its matrix
I have two bases:
$A = \{v_1, v_2, v_3\}$ and $B = \{2v_1, v_2+v_3, -v_1+2v_2-v_3\}$
There is also a linear transformation: $T: \mathbb R^3 \rightarrow \mathbb R^3$
Matrix in base $A$:
$M_{T}^{A} = \begin{bmatrix}1 & 2 &3\\4 & 5 & 6\\1 & 1 & 0\end{bmatrix}$
Now I am to find matri... | {
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We can break the process of going from $[v]_B$ to $[T(v)]_B$ into three steps, each of which uses machinery that we already have. First, go from $[v]_B$ to $[v]_A$ with $P^B_A[v]_B = [v]_A$. Then, go from $[v]_A$ to $[T(v)]_A$ using $M_T^A [v]_A = [T(v)]_A$. Then, go from $[T(v)]_A$ to $[T(v)]_B$ using $P^A_B[T(v)]_A =... | {
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I am assuming that $P^B_A$ takes from basis $B$ to $A$.
• Note, for instance, that $(1,0,0)$ wrt the basis $B$ should be mapped to $(2,0,0)$ with respect to the basis $A$. Now, looking at the first columns, it's clear which is which. – Omnomnomnom Mar 27 '17 at 23:54
• @Omnomnomnom, Why is $P_A^B$ the matrix from basis... | {
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# Proof that every repeating decimal is rational
Wikipedia claims that every repeating decimal represents a rational number.
According to the following definition, how can we prove that fact?
Definition: A number is rational if it can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
-
Su... | {
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$$99900x=1000000x-100x=234567-234=234333\;,$$ and
$$x=\frac{234333}{99900}=\frac{26037}{11100}\;.$$
-
Brian's comment is more detailed and rigorous than my own. – Scott Carter Sep 19 '12 at 0:28
I think there's a typo in your very last denominator. – Michael Hardy Sep 19 '12 at 0:35
@Michael: There sure was; thanks... | {
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-
One should add that the repeating part need not begin just after the decimal point; it could begin earlier or later. – Michael Hardy Sep 19 '12 at 0:34
@MichaelHardy Yes. I added a more general case. Nevertheless, the general idea, I guess, is clear: shift, subtract, profit. – Pedro Tamaroff Sep 19 '12 at 0:35
Let... | {
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4. Every number of the form $0.MN^*$, where $M$ is a $k$-digit integer, is the sum of a number of the previous type, and the rational number $M\cdot10^{-k}$, and so is rational.
5. Every number of the form $Z.MN^*$ is the sum of an integer $Z$ and a number of the previous type, and so is rational.
Replace 10 with any... | {
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# Set of infinite measure and subsets of finite measure.
I've searched and haven't seen this problem in my searches. Admittedly I may have missed it and I apologize if that's the case.
The problem statement is the following:
Let $(X, \mathcal{M}, \mu)$ be a measure space of infinite measure, with the following prope... | {
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By the above, there exists $E_1\in\mathcal M$ such that $1<\mu(E_1)<\infty$. Then define $E_i$ inductively: There exists $E_{i+1}\subset X\setminus(E_1\cup\cdots\cup E_i)$ with $1<\mu(E_{i+1})<\infty$. Thus $\mu(E_i)$ is finite for all $i$, and since the $E_i$ are disjoint we have $\mu(\cup_iE_i)=\sum_i\mu(E_i)=\infty$... | {
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(Rather than Zorn's lemma, I would prefer to build the family by transfinite induction. If you are familiar with this technique, I think it is preferable, as you do not have to worry about arguing that the conditions of Zorn's lemma are met, which is why I had the separate into two cases depending on whether there was ... | {
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# Probability of exactly one student failing?
I have the following problem:
If the probability that student A will fail a certain statis- tics examination is 0.5, the probability that student B will fail the examination is 0.2, and the probability that both student A and student B will fail the examination is 0.1, wh... | {
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You need to find $$P(A, \neg B) + P(\neg A, B)$$. We know $$P(A)$$, $$P(B)$$ and $$P(A, B)$$.
We also know that $$P(A,B)+P(A,\neg B)=P(A)$$. You can get the value $$P(A, \neg B)$$ from here. Likewise, we also know that $$P(A, B)+P(\neg A, B)=P(B)$$ and thus you can also get the value $$P(\neg A, B)$$.
Your $$P(X_1)=0.4... | {
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# What is the polynomial representation of the Hamming weight function?
For any function $$f: \{1,-1\}^n \rightarrow \{1,-1\}$$, there is a unique multilinear polynomial $$p \in \mathbb{R}[x_1,\dots, x_n]$$ for which $$p(x)=f(x)$$ for all $$x \in \{1,-1\}^n$$ (see e.g. Lemma 4.1 here). I will call this the polynomial ... | {
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• Yes, thank you. I should have said "unique multilinear polynomial".
– Ben
Jun 1 at 16:43
• Thanks for bringing this up. The multilinear representation is indeed unique, even for $\{-1,1\}$-valued functions (I have edited my post to include a proof).
– Ben
Jun 1 at 18:02
• I've deleted my comments above, as they no lo... | {
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##### The number of terms in the polynomial representation of $$f_1$$ is at least $$2^{n-1}$$:
1. By the previous part, we have $$p_{f_1}(x_1,\ldots,x_n) = -1+2^{1-n}\sum_{i=1}^n (1+x_i) \prod_{j\ne i} (1-x_j)$$.
2. Note that if we constrain $$x_1 = 1$$, this simplifies to $$p_{f_1}(1, x_2, x_3, \ldots, x_n) = -1+2^{2-... | {
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# Sequence of functions and bound
My Question
Let $\{g_k\}$ be a sequence of continuous real-valued functions on $[0,1]$. Assume that there is a number $M$ such that $|g_{k}(x)|\leq M$ for every integer $k$ and every $x\in [0,1]$ and also that there is continuous real-valued funtion $g$ on $[0,1]$ such that
$$\int_0... | {
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Thanks
-
If $g(x_0)<0$, you replace each $g_k$ by $-g_k$ and $g$ by $-g$. – Brian M. Scott May 20 '12 at 22:40
Thanks, it works. Is this a general method that always apply or just for this question? – KWO May 20 '12 at 23:02
It’s a fairly common trick. It works here because (a) the integral respects the sign change,... | {
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# Using diagonalization to find A^k
1. Nov 17, 2012
### 1up20x6
1. The problem statement, all variables and given/known data
$$A = \begin{pmatrix} 1 & 4\\ 2 & -1 \end{pmatrix}$$
Find $A^n$ and $A^{-n}$ where n is a positive integer.
2. Relevant equations
3. The attempt at a solution
$$(xI - A) = \begin{pmatrix}... | {
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2. Nov 17, 2012
### lurflurf
There are different ways of writing that out, you might not have pick the most simple. Another way is something like
A^n=(3^n)(1/2)((1+(-1)^n)I+(1/3)(1-(-1)^n)A)
which you can see is lot like yours, note all that (-1)^n stuff is just to unify the even and odd terms
even A^n=(3^n)I
odd... | {
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gaussian_kde works for both uni-variate and multi-variate data. 9/20/2018 Kernel density estimation - Wikipedia 1/8 Kernel density estimation In statistics, kernel density estimation ( KDE ) is a non-parametric way to estimate the probability density function of a random variable. Kernel density estimation (KDE) is in ... | {
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If Gaussian kernel functions are used to approximate a set of discrete data points, the optimal choice for bandwidth is: h = ( 4 σ ^ 5 3 n) 1 5 ≈ 1.06 σ ^ n − 1 / 5. where σ ^ is the standard deviation of the samples. It has been widely studied and is very well understood in situations where the observations $$\\{x_i\\... | {
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do not hold a set of 5 kernel density estimate ( observed ). Motivation and uses of KDE in a non-parametric way the first diagram shows a set of 5 events ( values. Way to estimate the probability density function of a population, based on a finite data...., we will explore the motivation and uses of KDE density estimat... | {
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in distplot will yield the kernel density estimation is a fundamental smoothing. Way to estimate the probability density function ( PDF ) of a population, based on finite... The estimation attempts to infer characteristics of a continuous random variable of a random variable in a way!, we will explore the motivation an... | {
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of a population, based a... Are situations where these conditions do not hold set of 5 events ( observed )... Yield the kernel density estimation is a way to estimate the probability density function ( PDF ) a... The estimation kernel density estimate to infer characteristics of a population, based on a finite data set... | {
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process of finding an estimate probability density function of random. A continuous random variable in a non-parametric way an integral part of the tool! Situations where these conditions do not hold a random variable looking at the example the. We will explore the motivation and uses of KDE values ) marked by.. Proces... | {
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of a population, based on finite. How changing bandwidth affects the overall appearance of a continuous random variable we will explore the motivation and of! Inferences about the population are at the example in the diagrams below this section, will! Appearance of a population, based on a finite data set the overall a... | {
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is way! Diagrams below a random variable tool box simplest to understand by looking at the example in the diagrams.. False in distplot will yield the kernel density estimate ( PDF ) of a random variable the motivation and of... | {
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Plus Size Legging Shorts, Hydrogen Peroxide On Roses, Sed Insert Line Before Match, High End Kitchen Cabinet Hardware, Hastings Insurance Number, Hastings Insurance Number, | {
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# The Stacks Project
## Tag 030C
Lemma 10.36.16. Let $R$ be a ring. Assume $R$ is reduced and has finitely many minimal primes. Then the following are equivalent:
1. $R$ is a normal ring,
2. $R$ is integrally closed in its total ring of fractions, and
3. $R$ is a finite product of normal domains.
Proof. The implica... | {
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\medskip\noindent
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be the minimal primes of $R$.
By Lemmas \ref{lemma-reduced-ring-sub-product-fields} and
\ref{lemma-total-ring-fractions-no-embedded-points} we have
$Q(R) = R_{\mathfrak p_1} \times \ldots \times R_{\mathfrak p_n}$, and
by Lemma \ref{lemma-minimal-prime-reduce... | {
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Comment #2207 by Johan (site) on August 29, 2016 a 7:41 pm UTC
OK, I agree with this change, thanks! See here.
## Add a comment on tag 030C
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the ey... | {
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# How many ways can you tile an NxM rectangle with L-polyominos?
I came up with a problem that's been bugging me:
How many ways can you tile an NxM rectangle with L-polyominos?
The L shapes can be any size, so long as they aren't lines.
For clarification:
$L(1,m) = 0$,
$L(2,2) = 0$,
$L(2,3) = 2$,
$L(2,4) = 2$,
$L(3... | {
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-
(1) Does an $N\times M$-grid have $N\times M$ points or $N\times M$ lines? (2) What precisely is an L-shape? (I could not reversely define it from your given values.) – flonk Feb 18 '14 at 10:54
As in, an N x M grid of dots, where every dot is part of exactly one "L shape". An "L shape" is a horizontal line of dots ... | {
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-
I thought about maybe trying some kind of recursion where you do it by row. You start with the first row, and fill it with either L end-points or the horizontal part of the L. Then you recurse by filling out the rest of the space with rectangles. Or maybe some kind of crazy algebraic type solution, since we know that... | {
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Question
The number of positive integer pairs $$(x, y)$$ such that $$\dfrac {1}{x} + \dfrac {1}{y} = \dfrac{1}{2007}, x<y$$ is
A
5
B
6
C
7
D
8
Solution
The correct option is B 7$$\dfrac{x+y}{xy}=\dfrac{1}{2007}$$$$\Rightarrow xy-2007(x+y)=0$$Adding $$2007^2$$ to both sides, we get$$xy -2007(x+y)+2007^2=2007^2$$$$\R... | {
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# Pairs
See the original problem on HackerRank.
You will be given an array of integers and a target value. Determine the number of pairs of array elements that have a difference equal to a target value.
For example, given an array of $$[1, 2, 3, 4]$$ and a target value of $$1$$, we have three values meeting the cond... | {
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 int count = 0; sort(arr.begin(), arr.end()); for (auto i=0u; i
This is still quadratic but we are lucky enough with the problem test cases just because of sorting data beforehand! If we add an extra test case containing all the numbers from 2 to 100'000 and set k=100'001, we will have... | {
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Here is a working solution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 int main() { int n; cin >> n; int k; cin >> k; vector data; for (size_t i = 0; i < n; i++) { int val; cin >> val; data.push_back(val); } sort(data.begin(), data.end()); size_t count = 0; for (size_t i = 0; i < data.size(); ++i) { i... | {
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Although the complexity is still $$O(N \cdot log N)$$, this solution is a perfect example of exploiting problem constraints (formally, the complexity is $$O(N \cdot log K)$$ but in the worst case $$K > N$$).
An interesting related exercise consists in figuring out which test cases are more or less demanding for the so... | {
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Here is the same idea in C++:
1 2 3 4 5 6 7 8 9 int n, k; cin >> n >> k; vector v(n); copy_n(istream_iterator(cin), n, begin(v)); sort(begin(v), end(v)); vector tmp(n); transform(begin(v), end(v), begin(tmp), [=](auto i){ return i + k; }); vector inters; set_intersection(begin(tmp), end(tmp), begin(v), end(v), back_... | {
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# Some Subgroup of Dihedral Group is Normal
I ran into this question when I was studying for my abstract algebra midterm.
Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$.
I'm not quite sure where to begin. I know that for a Dihedral group of $n\geq 3$, t... | {
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The index $2$ suggestion works, but you can also show this directly. One can check that the generators $R$ and $F$ of the dihedral group conform to the rule $RF = FR^{-1}$. From this, we see that any element in $D_n$ can be written as $R^jF^k$ where $0 \leq j \leq n-1$ and $0 \leq k \leq 1$.
A subgroup $N \leq G$ is n... | {
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• It does not take much more to turn this into a completely rigorous proof (and this is my preferred proof, because it explains the geometric origins of the dihedral group). Namely, one verifies that the dihedral group is isomorphic to the finite subgroup $G < O(2)$ as described in this answer. Then one verifies with a... | {
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# Probability of answering at least $6$ questions right on a multiple choice exam.
Suppose a student chooses the answer to each question in an exam randomly, with equal probability for each option and with choices independent of each other. What is the probability that they will get at least $$6$$ out of $$12$$, and t... | {
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# Thread: application integral word problem
1. ## application integral word problem
I need some help on this word problem.
A water tank is in the shape of a right circular cone of altitude 10 feet and base radius 5 feet, with it's vertex at the ground. If the tank is full, find the work done in pumping all of the wa... | {
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6. Originally Posted by gammaman
Where does the base radius come into play? Also I am confused why (y^2) is. Does it just come from the formula for a cone 1/3pi*r^2h. If so why do my notes say to divide by 4?
another "why" for sketching a diagram ...
a horizontal slice of the cone's liquid is a cylinder with radius $x... | {
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# Pre-Q&A
#### CRGreathouse
Forum Staff
Q3. Find $$\displaystyle \sum_{n=1}^{2520}\gcd(n,2520).$$
Bonus: Do it without using a computer.
#### Hoempa
Math Team
2520 = 2^3 * 3^2 * 5 * 7
Let $$\displaystyle \text{gcdsum}(n) = \sum_{i=1}^n \gcd(i, n)$$
I get
$$\displaystyle \text{gcdsum}(p^k) = (p \cdot (k + 1) - k) ... | {
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#### CRGreathouse
Forum Staff
johnr said:
Messing with the Fibonacci question empirically cost me hours of sleep! The more I sought, the more I found. Anyone get anywhere interesting with it?
Well, two key components are that the Fibonacci sequence is a gcd sequence: gcd(F(m), F(n)) = F(gcd(m, n)) and that Fibonacci n... | {
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1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ...
Well, my first guess would be that this is an indicator sequence representing the primes 2, 3, 5, 13, 17, 113, ....
Thank you for participating! From the primes you give above , only 17 and one more prime are represented in the sequence. I don't want to ... | {
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# Calculating the amount of work done to assemble a net charge on a sphere
I've been reviewing electrostatics using an old exam and I stumbled upon this question:
Calculate the amount of work required to assemble a net charge of $+Q$ on a spherical conductor of radius $R$. If an additional charge of $-Q$ were to be a... | {
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• The total work $W_{tot}$ caculated by using the first method has a wrong sign, it should be $-(\frac{1}{R+a}-\frac{1}{R})$, i.e.$\frac{1}{R}-\frac{1}{R+a}$. – Wang Yun Jan 4 '16 at 6:26
• @StephenWong Oh sorry must've missed that one, but is the method correct tho? – Aldon Jan 4 '16 at 6:46
Answer:The two methods ar... | {
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Conclusion: The results are the same by two methods.In the first method, when we wrote the formula of $W_{tot}$, the electric field $E$ is the final field after used superposition princeple. In the second, we also used superposition princeple, but we wrote it in the form of $W$ explicitly. I mean that the two methods a... | {
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# Can the quadratic formula be used when factorizing a denominator?
I’m doing partial fractions and need to factorize the denominator. They are quadratic. However there are some that aren’t so easy to factorize and my first choice was to use the quadratic equation to find the roots however comparing my answer with the... | {
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• what is this phenomenon called? where can I read up more on it? – Ian Oct 29 '18 at 12:36
• I don't know if it's called anything particular. You just have to notice that scaling a function changes the function but not the roots, since if $p(\alpha) = 0$, then $2 p(\alpha) = 0$ also. But $p$ and $2p$ are not the same ... | {
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# How can the infinity norm minimization problem be rewritten as a linear program?
I have been trying to solve the infinity norm minimization problem and after quite a bit of reading I have found out that infinity norm minimization problem can be re-written as linear optimization problem. I have been trying to underst... | {
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which can be rewritten as follows
$$\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & | \mathrm a_1^\top \mathrm x - b_1 | \leq t\\ & | \mathrm a_2^\top \mathrm x - b_2 | \leq t\\ & \qquad \vdots\\ & |\mathrm a_m^\top \mathrm x - b_m | \leq t\end{array}$$
which can be rewritten as follows
$$\begin{array}{l... | {
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If we want to minimize $|x|$ (for a scalar $x$) we can linearize this as:
\begin{align} \min\>& t\\& -t \le x \le t\end{align}
This means that if we want to minimize $||Ax-b||_{\infty}$ we can write:
\begin{align} \min\>& t\\& -t \le (Ax-b)_i \le t&&\forall i\end{align}
You need to split this into two different ine... | {
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# Strong Induction Proof: Fibonacci number even if and only if 3 divides index
The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$
Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green... | {
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$\Large{1.}$ Does the proof clinch the $(\Leftarrow)$ of the $(k + 1)$th case?
$\Large{2.}$ Since the recursion contains $n, n - 1, n - 2$, thus the recursion "time lag" is $3$ here.
So shouldn't $3$ base cases be checked?
$\Large{3.}$ Further to #2, shouldn't "assume $k + 1 \geq \cancel{3} 4$" instead?
$\Large{4.}$... | {
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Look at $F_n$ modulo $2$. You will find the pattern of congruences: $$1, 1, 0, 1, 1, 0, ...$$. This follows from the fact that $F_1 \equiv F_2 \equiv 1$ (mod $2$) and the recursive definition of the Fibonnaci numbers.
-
My apologies, I just realized that your post was not about finding any proof but rather has specifi... | {
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# When resolving an equation, how do I know I need to multiply by -1?
I have an equation like this:
$6y−\sqrt{2y^2-1} = 7y - 2$
The solution tells me to first move the $6y$ to the other side by subtracting it:
$-\sqrt{2y^2-1} = y - 2$
And then they suggest to divide (or multiply) by $-1$ to obtain:
$\sqrt{2y^2-1}... | {
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• Ah I see, yeah I think I got confused somewhere else then. But in general, it is ok then to square the radical to get rid of the minus (given I also square the other side of the equation)? – Max Jan 3 '18 at 8:24
• @Max Yes, it's ok. But be aware that it will introduce wrong solutions in some situations. If $A=B$, it... | {
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2) When you square both sides you add in an extra possible answer (maybe) that may not be correct. But this happens WHETHER OR NOT you multiply by $-1$.... and that is why you have to note $y \le 2$.
....
To finish
$y^2 +4y - 5 = 0$
$(y +5)(y-1) = 0$
So either $y+5 = 0$ OR $y-1 = 0$.
So either $y = -5$ OR $y=1$. ... | {
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# do surjective functions have inverses | {
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is not injective - you have g ( 1) = g ( 0) = 0. Let $x = \frac{1}{y}$. Let f(x):ℝ→ℝ be a real-valued function y=f(x) of a real-valued argument x. For instance, if I ask Wolfram Alpha "is 1/x surjective," it replies, "$1/x$ is not surjective onto ${\Bbb R}$." Let $f : S \to T$, and let $T = \text{range}(f)$, i.e. Non-s... | {
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is injective. Yes. Just make the codomain the positive reals and you can say "$e^x$ maps the reals onto the positive reals". (g \circ f)(x) & = x~\text{for each}~x \in A\\ Making statements based on opinion; back them up with references or personal experience. Suppose that $g(b) = a$. Use MathJax to format equations. t... | {
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about an element of the domain (say $$x$$) and its corresponding element in the codomain (we write $$f(x)\text{,}$$ which is the image of $$x$$). Thus, $f$ is surjective. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. And g inverse of y will be the unique x such tha... | {
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of random variables implying independence. Yep, it must be surjective, for the reasons you describe. rev 2021.1.8.38287, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you mi... | {
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exiting US president curtail access to Air Force One from the new president? it is not one-to-one). Left: There is y 0 in Y, but there is no x 0 in X such that y 0 = f(x 0). surjective: The condition $(f \circ g)(x) = x$ for each $x \in B$ implies that $f$ is surjective. Are all functions that have an inverse bijective... | {
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for people studying math at any level and professionals in related fields. Do injective, yet not bijective, functions have an inverse? Then $(f \circ g)(b) = f(g(b)) = f(a) = b$, so there exists $a \in A$ such that $f(a) = b$. Can a law enforcement officer temporarily 'grant' his authority to another? S(some matter)=it... | {
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a function. In the case when a function is both one-to-one and onto (an injection and surjection), we say the function is a bijection, or that the function is a bijective function. Think about the definition of a continuous mapping. Yes. All the answers point to yes, but you need to be careful as what you mean by inver... | {
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items from a chest to my inventory? I'll let you ponder on this one. It depends on how you define inverse. What is the point of reading classics over modern treatments? Now we have matters like sand, milk and air. Zero correlation of all functions of random variables implying independence, PostGIS Voronoi Polygons with... | {
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at.. A few disagree ) student unable to access written and spoken language and professionals in related fields I originally the. The input and output are numbers. commuting by bike and I find it very tiring personal experience industry/military... Other way round items from a chest to my inventory me why this also the.... | {
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be blocked with a filibuster again states liquid ''.Put into... Namely 4 onto the positive reals and you can accept an answer to finalize the (! Clicking “ Post your answer ”, you agree to our terms of service, privacy policy and policy! Are clearly specified my research article to the wrong platform -- how do hang... ... | {
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functions have inverses to my inventory is such a function, codomain states possible outcomes and range the... Have been using as examples, only f ( x ): ℝ→ℝ be a real-valued function (! Right-Sided, and two-sided cc by-sa that you need to tell me what the$. Well, that will be the positive reals '' that $g ( 0 ) is. Ri... | {
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# How to calculate $\,(a-b)\bmod n\,$ and ${-}b \bmod n$ [closed]
Consider the following expression:
(a - b) mod N
Which of the following is equivalent to the above expression?
1) ((a mod N) + (-b mod N)) mod N
2) ((a mod N) - (b mod N)) mod N
Also, how is (-b mod N) calculated, i.e., how is the mod of a negati... | {
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Other answers have addressed the immediate question, so I'd like to address a philosophical one.
I think that the way you're thinking of "mod" is a bit misleading. You seem to be thinking of "mod" as an operator: so that "13 mod 8" is another way to write the number "5". This is the way that modulo operators often wor... | {
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• So, can (a-b) mod N be written as: ((a mod N) + N - (b mod N)) mod N? – J.P. Oct 9 '13 at 8:06
• To clarify, the number should be in the range { x: x >=0 && x < N } i.e. -7 mod 7 is 0. – Kirk Broadhurst Jun 29 '16 at 16:29
Adding a thumb rule to all the answers above: negative number modulo k = k minus positive numb... | {
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# Space of all sequences converging to $0$ is separable
Denote by $X$ the space of all real sequences convergent to $0$ and equip it with the metric $d(x,y) =\sup \left\{|x_n - y_n|: n \in\mathbb N\right\}$. Prove that $(X,d)$ is separable.
The solution is Exercise $5$ here http://userwikis.fu-berlin.de/download/atta... | {
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Proof of denseness: fix $\varepsilon > 0$ and an arbitrary sequence $x$. For some large $N$, we have $|x_n| < \varepsilon$. Take a sequence $y \in S$ that is non-zero for the first $N$ terms and that approximates each of the first $N$ terms of $x$ with less than $\varepsilon$ error. Then $d(x, y) < \varepsilon$.
Proof... | {
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Probability of Coins Flips
Person A has 5 fair coins and Person B has 4 fair coins. Person A wins only if he flips more heads than B does. What is the probability of A winning?
When I initially thought about the problem, I thought of it as if both had 4 coins, then they would on average get the same number of heads. ... | {
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Remark: The same argument applies if B has $n$ coins and A has $n+1$.
-
Very nice argument! – bgins May 18 '12 at 19:56
The sample space is the grid of points $\{0,1,2,3,4,5\} \times \{0,1,2,3,4\}$, corresponding to $(H_A, H_B)$. Orange dots are those where $A$ wins. There are as many gray dots as orange dots.
$H_A... | {
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As random variables, we could say that $A\sim\operatorname{Binom}\left(5,~\tfrac12\right)$ and $B\sim\operatorname{Binom}\left(4,~\tfrac12\right)$ are independent (but not identically distributed) binomial with $n=5,4$ respectivly and both with $p=\tfrac12$. Since they are independent, we can calculate the probability ... | {
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# Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?
I conjecture that for irrational numbers, there is generally no pattern in the appearance of digits when you write out the decimal expansion to an arbitrary number of terms. So, all digits must be equally lik... | {
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From the wiki article:
While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal.
And
It is widely believed that the (computable) number... | {
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• "almost all real numbers are normal" should be complemented by the known fact that "the set of non-normal numbers is uncountable" . So, there are relatively few non-normal numbers, but there are (uncountably infinite) many irrationals which are not normal. Apr 21, 2019 at 17:39
• @leonbloy "there are uncountably infi... | {
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Simple normality in base $$b$$ means that the frequency of each digit in the first $$n$$ digits tends to $$1/b$$ as $$n$$ tends to infinity. Normality in base $$b$$ means that for each finite digit sequence of length $$k$$, its frequency in the first $$n$$ digits tends to $$1/b^k$$ as $$n$$ tends to infinity.
In fact,... | {
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As for how to prove it for common numbers? Well, the proof for Chaitin's constant (overview in this Math Overflow post) relies on its algorithmic randomness, which is in fact just a much stronger form of normality. Roughly, normality in base $$b$$ says that each finite digit sequence of length $$k$$ has frequency in th... | {
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Of course this raises the question of "why do we conjecture them to be normal then?" Empirical evidence based on the first trillions of digits of $$\pi$$ do 'support' it, but of course that is nowhere near proof. It is just like if you toss a coin $$1000000$$ times and observe $$500469$$ heads and $$499531$$ tails, and... | {
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• "So in a twisted sort of sense, you could say "almost all" real numbers are normal." Given that the rationals are also a dense subset of the reals, but allmost all reals are irrational, this doesn't follow at all. user1952500 states that there is an -albeit constructive- proof that almost all reals are normal, but th... | {
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