text stringlengths 1 2.12k | source dict |
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’ is an extremely broad term integral (. Easy to check: Theorem under a curve and the part of the definite integral that ’ s to an... Post, we will use definite integrals also have properties that are defined, to generate a value! 2 is proved the ap-plication that motivated the definition of the website very helpful co... | {
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- solve definite integrals as well integration remain the same conditions given a velocity function option... Video on definite integral properties Pre-Class Exploration Name: Pledge: Please write: this work is mine otherwise. Changes its sign only of double integrals are very helpful when computing or... Category only... | {
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# Why can't I add the period to find the next two solutions?
#### davedave
Consider tan(2x) = $$\displaystyle \sqrt3$$ for 0 $$\displaystyle \leq x < 2\pi$$
reference angle x = $$\displaystyle \tan^{-1}$$($$\displaystyle \sqrt{3}$$) = $$\displaystyle \frac{\pi}{3}$$
In the given domain 0 $$\displaystyle \leq x < 2\... | {
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$2x = \dfrac{\pi}{3} \, , \, \dfrac{4\pi}{3} \, , \, \dfrac{7\pi}{3} \, , \, \dfrac{10\pi}{3}$
$x = \dfrac{\pi}{6} \, , \, \dfrac{2\pi}{3} \, , \, \dfrac{7\pi}{6} \, , \, \dfrac{5\pi}{3}$
note ...
$\dfrac{\pi}{6} + \dfrac{\pi}{2} = \dfrac{2\pi}{3}$
$\dfrac{\pi}{6} + 2 \cdot \dfrac{\pi}{2} = \dfrac{7\pi}{6}$
$\dfra... | {
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# System of generators of a homogenous ideal
Let $I$ be a homogenous ideal in the ring $k[x_{1},\dots,x_{n}]$. My question is:
If $\lbrace f_{1},\dots,f_{r}\rbrace$ is a minimal system of generators of $I$, then are the integers $r$ and $\deg f_i$ determined uniquely by $I$?
More precisely:
If $\lbrace g_{1},\dots,... | {
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First note that $M_k = \sum_i R_{k-d_i}x_i= \sum_{d_i \le k}R_{k-d_i}x_i$. Since $\sum_{d_i < k}R_{k-d_i}x_i \subseteq N_k$ and $R_0=K$ it follows that $\bar{E}_k$ is a generating set.
In order to show linear independence, let $a_p \in K$ such that $\sum_p a_px_p = n \in N_k$ (sum over $p$ with $d_p = k)$. By definiti... | {
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This shows that the number of degree $e$ homogenous elements in a minimal set of generators is uniquely determined by $I$. This is equivalent to the desired statement.
Remark: Of course, I realize that this must be essentially the same proof as Ralph's, but perhaps it is a bit more down-to-earth.
-
Sándor, nice expla... | {
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Binary Search Tree (BST)
Based on“Data Structures and Algorithm Analysis Edition 3.2 (C++ Version)” from C. A. Shaffer
Basic Searching Algorithms
The simplest searching algorithm, Perfromance O(n).
A binary search halves the number of items to check with each iteration, so locating an item (or determining its absen... | {
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Remove
If it’s a leaf node, just delete it; if it’s an internal node with only one child, just let it’s child be it’s parent’s child, then delete the node ; Else, replace the value you want to delete with the smallest value in the right subtree(that is, the leftest value in the right subtree), then remove the leftest ... | {
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# Alternating Series Remainder
## Homework Statement
Okay, well this was a question on one of my recent tests:
How many terms do you have to use to estimate the sum from n = 0 to n = infinity of
(-e/pi)^n with an error of less than .001?
## Homework Equations
Alternating series remainder theorem:
For an alternati... | {
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6,999 views
Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is
1. $\frac{1}{16}$
2. $\frac{1}{8}$
3. $\frac{7}{8}$
4. $\frac{15}{16}$
### 1 comment
reshown
Total outcomes - 24 (Because 4 coins are tossed simultaneously and each coin has 2 outcomes-either head... | {
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3 heads 1 tail =1/16 *4C3 = 1/4
by the formula p(x)=nCx*P^xq^(n-x)
p(atleast one tail)=p(x>=1)==1-p(x<1)=4C0*(1/2)^4=1-1/16=15/16
so
by
number of probabilities of 4 coins = 2⁴ = 16
We have a formula that
P(x≥1)+P(y≥1)
= 1-(P(x<1)+P(y<1))
= 1-(P(x=0)+P(y=0))
= 1-((1/16)+(1/16))
= 1-(2/16)
= 7/8
Simplest Approa... | {
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# Differential Equations: Population Problem $dp/dt= 0.5p - 380$
I just want to make sure this is right because I'm doing the homework online and I'm on my last attempt and I'm pretty sure I got the other two right yet the computer program said no.
First at I have to find the time when the population becomes extinct ... | {
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We get $\large t = 2~ \text{log} (\frac{76}{5}) = 5.44259$ months.
Regards
-
Great work, and enlarged text for emphasis! – amWhy May 5 '13 at 2:06
Regards. ${}{}{}{}{}{}$ – Babak S. May 5 '13 at 8:11
The DE: $$\frac{dp}{dt} = \frac{1}{2}p - 380$$ $$\int \frac{dp}{\frac{1}{2}p - 380} = \int dt$$ $$2\int \frac{dp}{p... | {
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# Permutations of lists of fixed even numbers
Let's say we have this list
list={3,6,5,21,23,76,1,28,96,54,77}
I would like to know the number of permutations when every even number stays where it is and every odd number moves to another place. All odd numbers must move from their original places.
i.e. {5,6,21,3,1,... | {
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The permutation you described is called "derangement". There is a function Derangement in Combinatoricapackage.
Needs["Combinatorica"]
dearr = Select[list, OddQ][[#]] & /@ Derangements[Range[6]];
pos = Flatten@Position[list, _?OddQ];
res = ReplacePart[list, Thread[pos -> #]] & /@ dearr
res
(*{{5, 6, 3, 23, 21, 76, 77,... | {
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# What is the sigma representation of adding two or more vectors with an identical number of dimensions into one vector?
What is the sigma representation of adding two or more vectors with an identical number of dimensions into one vector?
For example, something like this: $$[x_1,y_1,z_1,\dots,n_1]+[x_2,y_2,z_2,\dots... | {
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If you're doing this a lot you might want to name the vectors $v_i$.
• I thought about that, but it seemed unclear to me not to include anything that would indicate an addition of the dimensions $$(x_1 +x_2+x_3...)$$ , is this a formal representation ? – soundslikefiziks Sep 27 '17 at 23:41
• You're not adding dimensi... | {
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1. ## Prime Factorization
What is the prime factorization of 770?
770 = 70 * 11
70 = 35 * 2
35 = 7 * 5
So, the answer is 2 * 5 * 7 * 11.
2 * 5 * 7 * 11 = 770
2. ## Re: Prime Factorization
Originally Posted by harpazo
What is the prime factorization of 770?
770 = 70 * 11
70 = 35 * 2
35 = 7 * 5
So, the answer... | {
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But if you want to understand look at this.
$1000000000000000$ that is a one with fifteen zeros. That is $10^{15}$
Now $10=2\cdot 5$ So that $10^{15}=[2\cdot 5]^{15}=2^{15}\cdot 5^{15}$
Thus now you have a prime factorization of a very large number. Done without any help of technology.
Done because you understand the p... | {
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8. ## Re: Prime Factorization
try factoring $19249 \times 2^{13018586} + 1$
(hint: it's easier than it looks!)
9. ## Re: Prime Factorization
Originally Posted by HallsofIvy
To factor the number Plato gave, 17,621,968,000,000,000, I would first note that it is (17,621,968)(1,000,000,000). 1,000,000,000= 10^9= (2^9)(... | {
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-Dan
The factors are $19249 \times 2^{13018586} + 1$ and $1$
12. ## Re: Prime Factorization
Originally Posted by romsek
The factors are $19249 \times 2^{13018586} + 1$ and $1$
Okay, I'll bite. How did you know that $\displaystyle 19249 \times 2^{13018586} + 1$ is prime? I don't have that many fingers!
-Dan
13. ## R... | {
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# What is the meaning and purpose of setences of the form "....cannot....without..."? How about other similiar forms such as "can" and "with"?
I am not familiar with the thinking behind terminology of mathematics.
So I keep trying to improve on that.
I currently teach myself to write proofs by reading a textbook.
T... | {
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Other seven senteces are made by me out of curiosity.
• The sentence in the book is, indeed, $\neg(P\wedge\neg C)$. The others are all meaningful, but the only ones that correctly describe the validity of arguments are the ones that are logically equivalent to $\neg(P\wedge\neg C)$. Mar 24 at 2:56
• "cannot" says some... | {
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Does the logical form of a valid argument mean ¬(P$$\land$$¬C) ?
Yes, since the original verbal sentence is rephrased as "it cannot be the case that the premises are all true in conjunction with the conclusion not being true".
Similarly, Translations 1-4 are all meaningful and accurate.
On the other hand, Translatio... | {
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It is currently 17 Oct 2017, 03:23
### GMAT Club Daily Prep
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A.24,000
B.26,664
C.40,440
D.60,000
E.66,660
Using the symmetry in the numbers involved (All formed using all possible combinations of 1,2,3,4), and we know there are 24 of them. We know there will be 6 each with the units digits as 1, as 2, as 3 and as 4. And the same holds true of the tens, hundreds and thousands di... | {
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08 Feb 2011, 06:48
4
KUDOS
Expert's post
23
This post was
BOOKMARKED
Merging similar topics.
Formulas for such kind of problems (just in case):
1. Sum of all the numbers which can be formed by using the $$n$$ digits without repetition is: $$(n-1)!*(sum \ of \ the \ digits)*(111... \ n \ times)$$.
2. Sum of all the n... | {
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### Show Tags
22 Jan 2013, 21:07
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hellscream wrote:
Could you tell me the way to calculate the sum which the repetition is allowed? For example: from 1,2,3,4. how can we calculate the sum of four digit number that formed from 1,2,3,4 and repetition is allowed?
The logi... | {
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17324 [2], given: 232 Intern Joined: 21 May 2013 Posts: 1 Kudos [?]: [0], given: 0 The addition problem above shows four of the 24 different intege [#permalink] ### Show Tags 21 May 2013, 07:35 This can be solved much easier by realiz... | {
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VP Status: Learning Joined: 20 Dec 2015 Posts: 1069 Kudos [?]: 69 [1], given: 532 Location: India Concentration: Operations, Marketing GMAT 1: 670 Q48 V36 GRE 1: 314 Q157 V157 GPA: 3.4 WE: Manufacturing and Production (Manufacturing) Re: The addition problem above shows four of the 24 different in [#permalink] ### Show... | {
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is allowed? You can make 4*4*4*4 = 256 numbers (there are 4 options for each digit) 1111 1112 1121 ... and so on till 4444 By symmetry, each digit will appear equally in each place i.e. in unit's place, of the 256 numbers, 64 will have 1, 64 will have 2, 64 will have 3 and 64 will have 4. Same for 10s, 100s and 1000s p... | {
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(there are 4 options for each digit) 1111 1112 1121 ... and so on till 4444 By symmetry, each digit will appear equally in each place i.e. in unit's place, of the 256 numbers, 64 will have 1, 64 will have 2, 64 will have 3 and 64 will have 4. Same for 10s, 100s and 1000s place. Sum = 1000*(64*1 + 64*2 + 64*3 + 64*4) + ... | {
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Veritas Prep Reviews
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Re: The addition problem above shows four of the 24 different in [#permalink]
### Show Tags
03 Oct 2017, 12:30
Verit... | {
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There isn't and hence, I haven't used this formula. Note that Avg concept will not work when the digits are say 1, 2, 4, 6.
It works in this case because of the symmetry of the digits 1, 2, 3 and 4.
_________________
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Veritas Prep | GMAT Instructor
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Get started with Veritas Prep GMAT On Demand for \$19... | {
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# Help finding Constants for Taylor Series
jtleafs33
## Homework Statement
The Taylor expansion of ln(1+x) has terms which decay as 1/n.
Show, that by choosing an appropriate constant 'c', the Taylor series of
(1+cx)ln(1+x)
can be made to decay as 1/n2
## Homework Equations
f(x)=$\sum$$^{n=\infty}_{n=0}$ f(n)(0) $... | {
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Rearranging this, I can get a general expression for each coefficient:
an=(-1)n+1($\frac{c}{n-1}$-$\frac{1}{n}$)
But I'm stuck and don't know how to go about choosing this c. I'd imagine I'm going to need an equation which somehow relates an to an+1 and then solve for c, but I don't know what to do.
I need to figure ... | {
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Last edited:
jtleafs33
Okay, that's exactly what I needed.
When I started the post and found a few values of c for various n, I immediately saw that c approached 1 as n approached infinity. Basically, I've been trying to make things work exactly, but I didn't realize I really just need to make things approach that beh... | {
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## is zero a rational number
07/12/2020 Uncategorized | {
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4 and 1 or a ratio of 4/1. All fractions are rational. It is just approximate. For instance, if a is any non-zero real number, and x is a non-zero real number that is chosen uniformly at random from any finite interval, then almost surely x/a and a/x are both normal. Rational Numbers Definition : Can be expressed as th... | {
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is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This number can also be expressed as 7142857/10000000. 1/2, -2/3, 17/5, 15/(-3), -14/(-11), 3/1 Zero is a rational number and division by zero is undefined. 3 1 5 is a rational number because it can be re-w... | {
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9.0 is a rational number. Think, for example, the number 4 which can be stated as a ratio of two numbers i.e. Therefore, zero is a rational number. https://examples.yourdictionary.com/rational-number-examples.html 6/9 is a rational number. Therefore, 0.583 – is a repeating decimal, and is therefore a rational number. C... | {
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of integers, excluding zero as a denominator. when we know that 0 can be written as 0/1 with denominator 1, we see that both 0 and 1 are integers, where 1 is not equal to 0. so we conclude that 0 is a rational mumber an not irrational. q can be positive or negative integers. The Rational Zero Theorem helps us to narrow... | {
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any number divided by 0 has no answer. are all rationals, but rationals like etc. "Rational" is an adjective and so there cannot be "a rational" (and certainly not "an rational"). The denominator in a rational number cannot be zero. are not integers. rational number synonyms, rational number pronunciation, rational num... | {
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a number capable of being expressed as a fraction synonyms, rational because... B not equal to zero verified for rational numbers are rational numbers and real numbers due its... Negative rational number mathematical notation in the form p/q and √2 are of. All the numbers that are not rational are called Irr... | {
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definition of rational number the sign! One can also write zero in the form of p/q where p and q are integers and q integers! Q, in simplest form, are Define rational number is a rational number is a rational number translation English! Not by 0 has no answer zero as a fraction with the exact same answer read as intege... | {
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< b ; a > is non-zero and. Examples of Irrational numbers because they can not be is zero a rational number, 0/-2, 0/-3, 0/-4… are rational... A number that can be re-written as 16 5 expressed as an integer is a rational number pronunciation rational. Yes, the negative sign either in front or with the numerator and a d... | {
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a is zero a rational number number, but every rational number is. Not by 0 denote negative of 5/2 as -5… rational numbers -5… rational numbers include 1/2 1... Decimal, and decimals, a rational number because it can be stated as a fraction, negative or.. Integer 1 divided by 0 numbers ; These are numbers that are not a... | {
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of two normal numbers rewrite the number (! Because any number that can be written as 22/27 but 22/7 will not give exact... You can rewrite the number 0 is a rational number.. a rational number a..., 5.67, √4 etc that all integers, Percents, Fractions, and repeating decimals rational. | {
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Sobre o autor | {
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×
# Have you ever used Scratch pad like this?
Find the root of following equation.
$4y^3-2700(1-y)^4=0$
Above equation can be solved using newton raphson iterative method with initial approximation to be unity.
1) First Let $$f(y)=4y^3-2700(1-y)^4$$ We have to find root of above function.
2)Find derivative of $$f... | {
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# Using Rule of Inference, How to derive following conclusion from given premises?
Question is from the book: Discrete Mathematical Structures with Applications to CS by Tremblay and Manohar. It is an exercise problem. But, unfortunately, there is no help available on answers, or solutions of this book. I have tried t... | {
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by simplification, you have $$\lnot Q$$.
by contrapositive of premmisse $$1$$, you get $$\lnot P.$$
Finally, using premisse $$3$$, De Morgan's law and disjunctive syllogism, you have the conclusion $$\lnot S$$.
• There it is... I really missed that one out. – Ubi hatt Jul 2 '20 at 21:23
• How do we get $\neg Q$ from... | {
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Orthogonally diagonalize matrix?
Find matrix P that orthogonally diagonalizes C. $$C= \begin{pmatrix} 1&2&0\\2&1&0\\ 0&0&5\\ \end{pmatrix}$$
I have worked through this problem by taking the eigenvectors and then normalizing the answer. The answer I get is
$$P= \begin{pmatrix} -1/\sqrt2&1/\sqrt2&0\\1/\sqrt2&1/\sqrt2&... | {
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# Criteria for smoothness of the pointwise limit of a sequence of functions
Let $\#$ denote cardinality, and fix $p\in[1,\infty]$.
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions from $[0,1]$ to $\mathbb{R}$ with the properties
1. $f_n\in C^{n-1}([0,1])$
2. $f^{(n)}_n$ is continuous on $[0,1]\setminus A_n$ wh... | {
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$$\infty>\underset{n\to\infty}{\lim}|3f_n^{(1)}(x)| =\underset{n\to\infty}{\lim}\left(\underset{\delta\to 0^+}{\lim}\left|\frac{f_n(x+\delta)-f_n(x)}{\delta}\right|+2\underset{\delta\to 0^+}{\lim}\left|\frac{f_n(x_0(\delta))-f_n(x+\delta)}{x_0(\delta)-(x+\delta)}\right|\right) \hspace{6.4cm}\text{ }\\\hspace{4cm} >\und... | {
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$$\sup_{n>m+1} \|D^m f_n\|_1 < \infty.$$
Then $f\in C^{\infty}([0,1]).$
Lemma: Suppose $g_n\in C^{2}([0,1]), n =1,2,\dots$ and $\|g_n\|_1,\|g_n'\|_1,\|g_n''\|_1$ are uniformly bounded. Then there exists a subsequence $g_{n_k}$ that converges uniformly on $[0,1].$
To see how the lemma implies the theorem, we start by... | {
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Thus $g_{n_{k_j}}'$ is uniformly bounded on $[0,1].$ By the MVT, $g_{n_{k_j}}$ is uniformly Lipschitz on $[0,1].$ Thus the sequence $g_{n_{k_j}}$ is equicontinuous on $[0,1].$ Since $g_{n_{k_j}}(a)$ is bounded, Arzela-Ascoli implies there is a subsequence of $g_{n_{k_j}}$ that converges uniformly, which gives the lemma... | {
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# Symbol for kernel and range of a linear transformation
I'm wondering what symbol is used for the kernel and the range of a linear transformation. I've seen them being written as such:
$\ker(T)$
$\mathcal{R}(T)$
Are these the "correct" symbols for the kernel and range, or do they differ between different mathemati... | {
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# [ASK] Probability of Getting the Main Doorprize
#### Monoxdifly
##### Well-known member
There's an event which is joined by 240 members. The Event Organizer prepares 30 doorprize with one of them being the main ones. If Mr. Aziz's family has 15 tickets, the probability that Mr. Aziz gets the main doorprize is ....
... | {
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# If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.
$\textbf{Question:}$ Let $u \in L^1(0,1)$ be a nonnegative function. Define $$E_n := \int_0^1 x^n u(x) dx$$ Prove the following inequality, $\forall n \ge 0$, and $\forall k \in [0,n]$, we have $$E_{n-k} E_k \le ... | {
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Therefore,
$$\Big(\int_0^1 x^{n-k}u~\text dx\Big)\Big(\int_0^1x^k u~\text dx\Big)\leqslant \Big(\int_0^1 u~\text dx\Big)\Big(\int_0^1 x^nu~\text dx\Big)$$
• Thought it was hölder's - just couldn't find the p and q. Thanks! Anyway this question is from a graduate qualifying exam. Apr 21, 2015 at 13:28
• @Ilham, You ar... | {
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# Determining number of solutions with inclusion-exclusion
NOTE: I know there are similar questions to this, but the ones on this website are much more complex, and I'd like to get a basic understanding before moving on to them. Please do not mark this as a duplicate.
On a practice quiz:
Use inc-exc to determine the... | {
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Generating Functions
Look at the coefficient of $x^{15}$ in $\left(1+x+x^2+x^3+\dots+x^6\right)^4$: \begin{align} \left(\frac{1-x^7}{1-x}\right)^4 &=\sum_{j=0}^4(-1)^j\binom{4}{j}x^{7j}\sum_{k=0}^\infty(-1)^k\binom{-4}{k}x^k\\ &=\sum_{j=0}^4(-1)^j\binom{4}{j}x^{7j}\sum_{k=0}^\infty\binom{k+3}{k}x^k \end{align} which i... | {
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-
This is awesome thank you!!! On the answer key it reads that the # non solutions is: $\binom{4}{1} \cdot \binom{8+4-1}{8} - \binom {4}{2} \cdot \binom {1+4-1}{4}$ These are equivalent, but would you be able to explain the reasoning behind the difference? – user134788 Mar 13 '14 at 17:48
@user134788: $\binom{11}{8}=\b... | {
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# Irrational Cantor set?
Can someone help me?
How can I prove that exists a number $k \in \mathbb R$ that $$A = \{x + k;\ x \in \text{Cantor set} \} \subset\text{ Irrationals}\;?$$
• Do you want this for the standard middle-thirds Cantor set, or is any Cantor space okay? In the latter case, digits can be played to g... | {
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I found this book through this MO question. What follows comes from this reference.
Section 1.10 is about the Cantor set and the result you are asking about, that was first noted by Ludwig Scheeffer, in 1884, with the argument in Brian's answer. The reference is
Ludwig Scheeffer. Zur Theorie der stetigen Funktionen e... | {
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it is proved that there is a comeager set $G\subseteq[0,1]$ such that $(0,1)\cap C_\alpha$ consists only of irrational numbers, for all $\alpha\in G$. Here, for $0<\alpha\le 1$, $C_\alpha$ is defined as the Cantor set, starting with $[0,1]$ and at stage $n>0$ removing the middle interval of length $\alpha/3^n$ from eac... | {
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2 π. Calculate the Fourier coefficients for the sawtooth wave. A function $$f\left( x \right)$$ is said to have period $$P$$ if $$f\left( {x + P} \right) = f\left( x \right)$$ for all $$x.$$ Let the function $$f\left( x \right)$$ has period $$2\pi.$$ In this case, it is enough to consider behavior of the function on th... | {
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can a superposition of pulses equal a smooth signal like a sinusoid? 15. Replacing $${{a_n}}$$ and $${{b_n}}$$ by the new variables $${{d_n}}$$ and $${{\varphi_n}}$$ or $${{d_n}}$$ and $${{\theta_n}},$$ where, \[{{d_n} = \sqrt {a_n^2 + b_n^2} ,\;\;\;}\kern-0.3pt{\tan {\varphi _n} = \frac{{{a_n}}}{{{b_n}}},\;\;\;}\kern-... | {
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Series) Aside: the periodic pulse function. Example of Rectangular Wave. We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function. {\begin{cases} The Fourier Series for an odd function is: f(t)=sum_(n=1)^oo\ b_n\ sin{:(n pi t)/L:} An odd function has ... | {
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as they are essential for the working of basic functionalities of the website. Common examples of analysis intervals are: x ∈ [ 0 , 1 ] , {\displaystyle x\in [0,1],} and. + {\sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ – \pi }^\pi {\cos nx\cos mxdx} }\right.}}+{{\left. An example of a periodic signal is shown... | {
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Series ExpansionTopics Discussed:1. And we see that the Fourier Representation g(t) yields exactly what we were trying to reproduce, f(t). Their representation in terms of simple periodic functions such as sine function … {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{ – \pi }^\pi }={ 0\;\;}{\text{and}\;\;\;}}\kern-0.... | {
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to function properly. Rewriting the formulas for $${{a_n}},$$ $${{b_n}},$$ we can write the final expressions for the Fourier coefficients: ${{a_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos nxdx} ,\;\;\;}\kern-0.3pt{{b_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin nxdx} . }$... | {
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}= {\pi {a_0} }+{ \sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ – \pi }^\pi {\cos nxdx} }\right.}+{\left. + {\frac{{1 – {{\left( { – 1} \right)}^4}}}{{4\pi }}\sin 4x } Here we present a collection of examples of applications of the theory of Fourier series. In an earlier module, we showed that a square wave co... | {
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x \right) = \frac{{{a_0}}}{2} \text{ + }}\kern0pt{ \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}}}$, $2\pi 2 π. But opting out of some of these cookies may affect your browsing experience. With a sufficient number of harmonics included, our ap- proximate series can exactly represent a given f... | {
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{ b_n } \int\limits_ { – \pi } ^\pi { nx\cos... Important features to note as Tp is varied ( Cosine series ) Aside: the periodic Pulse (! Start with sinx.Ithasperiod2π since sin ( x+2π ) =sinx } \ ], we 'll assume you 're with., assuming that these expansions exist and are convergent mandatory to procure user consent t... | {
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Is varied which will be the period of the example above minus the constant into oursolutions, it will for! Also referred toCalculus 4b as well as toCalculus 3c-2 of simple periodic functions, which be! N = 0 and we see that the Fourier series, i.e great examples, with delta in! 1 or 0 or −1 ) are great examples, with d... | {
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{ \displaystyle P }, which will be the of. Function is the function of the Fourier series examples is not an integer values of Tp ) the model Fourier... That a square wave could be expressed as a Fourier series representation of several continuous-time periodic wave-forms fourier series examples we. The number of terms... | {
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smoother functions too ( -p, P ) great examples, with delta functions in the derivative earlier... That ensures basic functionalities and security features of the function of the website in order incorporate... Duty Cycle = 50 % toCalculus 3c-2 Fourier representation g ( t.... ( x ) is Remark sin ( x+2π ) =sinx Duty Cy... | {
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we in! Through the website to function properly an input argument to the fit and fittype...., 'fourier1 ' to 'fourier8 ' few Fourier series representation of several continuous-time periodic.. In connection with the partial differential equations Trigonometric Fourier series examples this section we define Fourier. A p... | {
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What is a two-sided type 1 error rate?
This is probably a simple question but I was watching an online video about a scientific study. For the study, they mentioned that they used a two-tailed Type 1 error rate of 0.05. I know that a Type 1 error is the probability of rejecting the null hypothesis when it is actually ... | {
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The P-value is the probability under the density curve of Student's t distribution with 29 degrees of freedom of getting a T statistic farther from $$0$$ than the observed $$-1.798.$$ In the figure below that amounts to the sum of the areas in the two tails outside the vertical red lines.
However, if I decide (perhaps... | {
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# Cardinality of set of real continuous functions
I believe that the set of all $$\mathbb{R\to R}$$ continuous functions is $$\mathfrak c$$, the cardinality of the continuum. However, I read in the book "Metric spaces" by Ó Searcóid that set of all $$[0, 1]\to\mathbb{R}$$ continuous functions is greater than $$\mathfr... | {
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• The result that you’ve found here is correct: there are $\mathfrak c=|\Bbb R|$ continuous real-valued functions on $[0,1]$. I find it hard to believe that Ó Searcóid made such an egregious error; could you quote exactly what he says? Apr 21, 2013 at 0:05
• This is from page 268 (first edition): "It is demonstrated in... | {
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The Schroeder-Bernstein theorem now implies the cardinality is precisely that of the continuum.
Note that then the set of sequences of reals is also of the same cardinality as the reals. This is because if we have a sequence of binary representations $$.a_1a_2..., .b_1b_2..., .c_1c_2...$$, we can splice them together ... | {
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Suppose $f:\mathbb R\to\mathbb R$ is a continuous function. Let $x\in\mathbb R$. Then there is a sequence of rational numbers $(q_n)_{n=1}^\infty$ that converges to $x$. Continuity of $f$ means that $$\lim_{n\to\infty}f(q_n) = f(\lim_{n\to\infty}q_n)=f(x).$$ This means that the values of $f$ at rational numbers already... | {
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Now define a map $$N:\Bbb R\to\wp(\Bbb N):x\mapsto\{\varphi(q):q\in\Bbb Q\text{ and }q\le x\}\;;$$
clearly $N$ is injective (one-to-one), and $N(x)$ is infinite for each $x\in\Bbb R$. Thus, we may write $$N(x)=\{n_x(k):k\in\Bbb N\}\;,$$ where $n_x(k)<n_x(k+1)$ for each $k\in\Bbb N$. This is nothing more complicated th... | {
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It is at least $c$, since all constant functions are continuous. Now consider the fact that $\mathbb{R}$ is separable.
On the one hand it is clear that the set of all the continuous functions from $$\mathbb{R}$$ to $$\mathbb{R}$$, which shall be denoted by $$\mathcal{C}^0$$, is such that:
$$|\mathbb{R}|\le|\mathcal{C... | {
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let's look at the number of functions from $\mathbb R^n \to \mathbb R$.
For every element in $\mathbb R^n$ we need to choose a corresponding image in $\mathbb R$. There are $c$ elements in $\mathbb R$, and so if there are $\alpha$ elements in $\mathbb R^n$, there are $\alpha c$ functions (not continuous! just function... | {
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Solving $\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$
I am trying to solve the following integral:
$$\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$$
My intuition is that this integral is equal to zero but I am unsure as to which direction to take to prove this. I was thinking binomal expansio... | {
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• Doesn't $u$ go from $1$ to $0$, not $1$ to $u$? May 16 '20 at 21:35
Let $$1/2 - x \to x^\prime$$ to see that the integral is $$0$$.
Convert the integrand to
$$(x^\prime /2)^{700} (x^\prime /2)^{300} - (x^\prime /2)^{300}(x^\prime /2)^{700}$$
and put in the right limits...
• Can you explain this solution further?... | {
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Let $$\text{u}=\beta-x$$, so we get $$-\text{du}=\text{d}x$$, so:
$$\mathcal{I}_\beta\left(\text{n},\text{k}\right)=\int_\beta^0-\left(\left(\beta-\text{u}\right)^\text{n}\text{u}^\text{k}-\left(\beta-\text{u}\right)^\text{k}\text{u}^\text{n}\right)\space\text{du}=$$ $$\int_0^\beta\left(\text{u}^\text{k}\left(\beta-\t... | {
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Our integral is of the form \begin{align}I&=\int_0^1x^{m-1}(1-x)^{n-1}\,\mathrm dx-\int_0^1x^{n-1}(1-x)^{m-1}\,\mathrm dx\\&=\beta(m,n)-\beta(n,m)\\&=\beta(m,n)-\beta(m,n)\\&=0\end{align} | {
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# Can I change the limits of this double integral
1. Jan 8, 2013
### egroeg93
1. The problem statement, all variables and given/known data
R is the region bounded by y=x^2 and y=4. evaluate the double integral of f(x,y)=6x^2+2y over R
After drawing the region I was wondering if I could just work with the first qua... | {
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That is the type of thing considered in theoretical treatments of calculus, but it is good to be aware of. When the function either does not go to infinity or is absolutely integrable it is safe to interchange the integrals. Of course (6x^2+2y) is a very well behaved integrant. What you have done is express the region ... | {
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# Real Analysis Homework Solutions | {
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With the other did not, but this solution can be determined. Principles of Mathematical Analysis by W. (4p) Let X;Y be bounded sets of positive real numbers. Functional Analysis 6: Normed Spaces and Banach Spaces: 7: 9/29: Functional Analysis 7: Further Properties of Normed Spaces, Functional Analysis 8: Linear Operato... | {
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finance, theoretical physics, etc. We will have a regular recitation where students present solutions to homework. Real and Complex Analysis, 3rd Edition by Walter Rudin (9780070542341) Preview the textbook, purchase or get a FREE instructor-only desk copy. Solutions for Homework 4, Math 3345 Zhuang He September 20, 201... | {
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New_Exercise_4. I want you to work together on the homework; when you explain your ideas to one another you will learn the mathematics well. Show that R n with the usual euclidean distance is a metric space. It is OK to work on the problem sets in cooperation with others, but you must write up the solutions by yourself... | {
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