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Formally prove that $\Theta(\max(f,g)) = \Theta(f+g)$
I am having a hard time proving that $\Theta(\max(f,g)) = \Theta(f+g)$
where
$(f+g)(n) = f(n) + g(n)$
and
$(\max{f,g})(n) = \max(f(n), g(n))$
I know that $\Theta$ is the combination of the upper and lower bounds, but I can't seem to prove this. It's hard for m... | {
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-
Maybe my confusion is dealing with the = sign. Are we saying that the two sets are equivalent? This is where I seem to be having trouble. I understand what you are saying about max<f+g<2max, but I don't see how this proves that the two sets are equal. – MrZander Jan 31 '13 at 18:24
Well, I am sure that you can formal... | {
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$\Leftrightarrow h(x) \geq \frac{f(x)+g(x)}{2}$
$\Leftrightarrow h(x) = \Omega(f(x)+g(x))$ This gives the lower bound.
Next, $h(x)=f(x) \leq f(x)+g(x)=O(f(x)+g(x))$
Hence, $h(x)=\Theta(f(x)+g(x))$
Case 2:
$h(x)=g(x)$. Similar to Case 1.
-
I want to show $\Theta(\max(f(x),g(x)))=\Theta(f(x)+g(x))$ – MrZander Feb 5... | {
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# Shell method about the line x=5
#### alane1994
##### Active member
I have been going over previous tests in an attempt to better prepare myself for the final that is coming tomorrow. I was posed a question.
Use the shell method to find the volume of the solid generated by revolving the plane region about the given... | {
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I know this.
SHELL METHOD
$$\displaystyle V = \int_{a}^{b} 2\pi x (f(x) - g(x))dx$$
I know this is fairly rudimentary, but assistance would be appreciated!
With the substitution $\xi= 5 - x$ You have to compute the volume of the rotation solid about $\xi=0$ of the function $\displaystyle f(\xi) = - 5 + 6 \xi - \xi^{2}... | {
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# Do most numbers have exactly 33 prime factors?
In this question I plotted the number of numbers with $$nn$$ prime factors. It appears that the further out on the number line you go, the number of numbers with $$33$$ prime factors get ahead more and more.
The charts show the number of numbers with exactly $$nn$$ pri... | {
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… if $$ω(n)ω(n)$$ is the number of distinct prime factors of $$nn$$ (sequence A001221 in the OEIS, then, loosely speaking, the probability distribution of
$$\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}$$
To see graphs related to this distribution, the first link... | {
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# Conditional Probability Concept Question
The organizers of a cycling competition know that about 8% of the racers use steroids. They decided to employ a test that will help them identify steroid-users. The following is known about the test: When a person uses steroids, the person will test positive 96% of the time; ... | {
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$(2)$: Use the defining formula $$\Pr(S|P)=\frac{S\cap P}{\Pr(P)}.$$ The two numbers on the right are not hard to compute. I can give further help if they pose difficulty.
-
We are given that $P(S)=0.08$ (hence $P(\neg S)=0.92$), $P(P|S)=0.96$ and $P(P|\neg S)=0.09$. What we want to knwo is $P(S|P)$.
Note that $P(S\... | {
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# greatest common divisor of two elements
Find all possible values of GCD(4n + 4, 6n + 3) for naturals n and prove that there are no others
3·(4n + 4) - 2·(6n + 3) = 6, whence the desired GCD is a divisor 6. But 6n + 3 is odd, so only 1 and 3 remain. n=1 and n=2 are examples for GCD=1 and GCD=3
is the solution corre... | {
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All comes down to "casting out" relatively prime factors.
\begin{align} (\color{#c00}4(n\!+\!1),\,3(2n\!+\!1))\, &=\, (n\!+\!1,\,3(\color{#0a0}{2n\!+\!1}))\ \ \ {\rm by}\ \ \ (\color{#c00}4,3)=1=(\color{#c00}4,2n\!+\!1)\\[.2em] &=\, (n\!+\!1,3)\ \ {\rm by} \ \bmod n\!+\!1\!:\ n\equiv -1\,\Rightarrow\, \color{#0a0}{2n\... | {
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# Find the limit
• Jul 8th 2006, 05:47 PM
Nichelle14
Find the limit
Limit as n approaches infinity [(3^n + 5^n)/(3^n+1 + 5^n+1)]
I tried to divide the numerator and denominator by 3^n. Was not successful.
What should I do next?
• Jul 8th 2006, 07:03 PM
ThePerfectHacker
Quote:
Originally Posted by Nichelle14
Limit a... | {
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The denominator is: . $\frac{3^{n+1}}{5^{n+1}} + \frac{5^{n+1}}{5^{n+1}} \;=\;\left(\frac{3}{5}\right)^{n+1} + 1$
Recall that: if $|a| < 1$, then $\lim_{n\to\infty} a^n\:=\:0$
Therefore, the limit is: . $\lim_{n\to\infty}\,\frac{\frac{1}{5}\left(\frac{3} {5}\right)^n + \frac{1}{5}} {\left(\frac{3}{5}\right)^n + 1} \;... | {
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# Basis Of Symmetric Matrix | {
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The matrix having $1$ at the place $(1,2)$ and $(2,1)$ and $0$ elsewhere is symmetric, for instance. Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. Note that AT = A, so Ais. For a real matrix A there could be both the problem of finding the eigenvalues and the problem of finding the eigenvalues and ei... | {
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be linearly independent in which case A is diagonalizable. Standard basis of : the set of vectors , where is defined as the 0 vector having a 1 in the position. By induction we can choose an orthonormal basis in consisting of eigenvectors of. If you have an n×k matrix, A, and a k×m matrix, B, then you can matrix multip... | {
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cases where the solution set of xT Ax = c can be a single point, two intersecting lines, or no points at all. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. In this paper, we study various properties of symmetric t... | {
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eigenvalues and eigenvectors are likely to have special properties as well. One point more is to be. Richard Anstee An n nmatrix Qis orthogonal if QT = Q 1. In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (tha... | {
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Symmetric matrices have an orthonormal basis of eigenvectors. The matrix U is called an orthogonal matrix if UTU= I. Therefore, there are only 3 + 2 + 1 = 6 degrees of freedom in the selection of the nine entries in a 3 by 3 symmetric matrix. Yu 3 4 1Machine Learning, 2Center for the Neural Basis of Cognition, 3Biomedi... | {
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as symmetric group. If v1 and v2 are eigenvectors of A. In characteristic not 2, every bilinear form Bis uniquely expressible as a sum B 1 +B 2, where B 1 is symmetric and B 2 is alternating (equivalently, skew-symmetric). §Example 2: Make a change of variable that transforms the quadratic form into a quadratic form wi... | {
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equal, we must have m= n. Any power A n of a symmetric matrix A (n is any positive integer) is a. The Spectral Theorem: If Ais a symmetric real matrix, then the eigenvalues of Aare real and Rn has an orthonormal basis of eigenvectors for A. So far, symmetry operations represented by real orthogonal transformation matri... | {
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f j-i for some vector f. Since , it follows that is a symmetric matrix; to verify this point compute It follows that where is a symmetric matrix. Therefore, there are only 3 + 2 + 1 = 6 degrees of freedom in the selection of the nine entries in a 3 by 3 symmetric matrix. Fact 7 If M2R n is a symmetric real matrix, and ... | {
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in the study of damped vibrations of linear systems. If matrix A of size NxN is symmetric, it has N eigenvalues (not necessarily distinctive) and N corresponding. , v1 ¢v2 =1(¡1)+1(1. This process is then repeated for each of the remaining eigenvalues. Classifying 2£2 Orthogonal Matrices Suppose that A is a 2 £ 2 ortho... | {
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there are only 3 + 2 + 1 = 6 degrees of freedom in the selection of the nine entries in a 3 by 3 symmetric matrix. 368 A is called an orthogonal matrix if A−1 =AT. A symmetric matrix A is a square matrix with the property that A_ij=A_ji for all i and j. That's minus 4/9. If we futher choose an orthogonal basis of eigen... | {
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Rn has an orthonormal basis of eigenvectors for A. APPLICATIONS Example 2. 5), a simple Jacobi-Trudi formula. We claim that S is the required basis. The sum of two skew-symmetric matrices is skew-symmetric. 1 Vector-Vector Products Given two vectors x,y ∈ Rn, the quantity xTy, sometimes called the inner product or dot ... | {
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in passing on page 87 in. Totally Positive/Negative A matrix is totally positive (or negative, or non-negative) if the determinant of every submatrix is positive (or. Symmetry Properties of Rotational Wave functions and Direction Cosines It is in the determination of symmetry properties of functions of the Eulerian ang... | {
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Now the next step to take the determinant. Determining the eigenvalues of a 3x3 matrix. For any symmetric matrix A: The eigenvalues of Aall exist and are all real. (1,2,3,3), (1,2,3,3), this is a symmetric matrix. A square matrix A is a projection if it is idempotent, 2. Interpretation as symmetric group. This should b... | {
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is a product of elementary matrices, and also if and only if its row vectors form a basis of Fn. The asterisks in the matrix are where “stuff'' happens; this extra information is denoted by $$\hat{M}$$ in the final expression. Eigenvalues and Eigenvectors. 3 Diagonalization of Symmetric Matrices DEF→p. The basis vector... | {
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set of eigenvectors, then it can be written as UDUT. Finite-dimensional space: a space which has a finite basis. Symmetry of the inner product implies that the matrix A is symmetric. To find the basis of a vector space, start by taking the vectors in it and turning them into columns of a matrix. In addition the matrix ... | {
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operations. A matrix with real entries is skewsymmetric. This implies that UUT = I, by uniqueness of inverses. We claim that S is the required basis. So these guys are indeed orthogonal. Step 1: Find an ordered orthonormal basis B for $$\mathbb{R}^n ;$$ you can use the standard basis for $$\mathbb{R}^n. Introduction. I... | {
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is orthogonal if and only if the column vectors. Definition 3. The last part is immediate. linalg may offer more or slightly differing functionality. We call such matrices symmetric. To prove this we need merely observe that (1) since the eigenvectors are nontrivial (i. The Geometrical Basis of PT Symmetry. Note that we... | {
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imports most of them, identically named functions from scipy. The matrix for H A with respect to the stan-dard basis is A itself. Therefore, there are only 3 + 2 + 1 = 6 degrees of freedom in the selection of the nine entries in a 3 by 3 symmetric matrix. Therefore, there are only 3 + 2 + 1 = 6 degrees of freedom in th... | {
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symmetric. Letting V = [x 1;:::;x N], we have from the fact that Ax j = jx j, that AV = VDwhere D= diag( 1;:::; N) and where the eigenvalues are repeated according to their multiplicities. Recall some basic de nitions. For proof, use the standard basis. The second important property of real symmetric matrices is that t... | {
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is one which is particularly convenient for transformation to an asymmetric rotator basis either by means of a high-speed digital computer or by means of a desk calculator. Eigenvalues and Eigenvectors. It follows that is an orthonormal basis for consisting of eigenvectors of. A symmetric matrix A is a square matrix wi... | {
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matrix A are distinct, the matrix X, which has as its columns the corresponding eigenvectors, has the property that X0X = I, i. \begingroup The covariance matrix is symmetric, and symmetric matrices always have real eigenvalues and orthogonal eigenvectors. Is there a library for c++ which I can force to find the Orthog... | {
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of matrices called symmetric matrices that have quite nice properties concerning eigenvalues and eigenvectors. A matrix is a rectangular array of numbers, and it's symmetric if it's, well, symmetric. In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear ... | {
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3 matrices. Let V be the real vector space of symmetric 2x2 matrices. Theory The SVD is intimately related to the familiar theory of diagonalizing a symmetric matrix. So,wehave w 1 = v1 kv1k = 1 √ 12 +12. A symmetric matrix is one that is equal to its transpose. T (20) If A is a symmetric matrix, then its singular valu... | {
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of A are real and that there exists an orthonormal basis of eigenvectors. 2, it follows that if the symmetric matrix A ∈ Mn(R) has distinct eigenvalues, then A = P−1AP (or PTAP) for some orthogonal matrix P. The matrix 1 2 2 1 is an example of a matrix that is not positive semidefinite, since −1 1 1 2 2 1 −1 1 = −2. Can... | {
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formula of SMVP is presented. Any value of λ for which this equation has a solution is known as an eigenvalue of the matrix A. To compare those methods for computing the eigenvalues of a real symmetric matrix for which programs are readily available. Consider again the symmetric matrix A = 0 @ 2 1 1 1 2 1 1 1 2 1 A; an... | {
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Two Matrices. In other words, the entries above the main diagonal are reflected into equal (for symmetric) or opposite (for skew-symmetric) entries below the diagonal. applications of symmetry in condensed matter physics are concerned with the determination of the symmetry of fields (functions of x, y, z, and t, althoug... | {
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that the pair (l, x) is called an. There is no inverse of skew symmetric matrix in the form used to represent cross multiplication (or any odd dimension skew symmetric matrix), if there were then we would be able to get an inverse for the vector cross product but this is not possible. (6) If v and w are two column vect... | {
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components of the covariance matrix still represent the variance magnitude in the direction of the x-axis and y-axis. Example: If square matrices Aand Bsatisfy that AB= BA, then (AB)p= ApBp. Any value of λ for which this equation has a solution is known as an eigenvalue of the matrix A. These eigenvectors must be ortho... | {
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of the following Matrix. Let Abe a real, symmetric matrix of size d dand let Idenote the d didentity matrix. All the element pairs that trade places were already identical. Ask Question Asked 1 month ago. The eigenvalues of a symmetric matrix are always real. Find the dimension of the collection of all symmetric 2x2 ma... | {
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# Are all highly composite numbers even?
A highly composite number is a positive integer with more divisors than any smaller positive integer. Are all highly composite numbers even (excluding 1 of course)? I can't find anything about this question online, so I can only assume that they obviously are. But I cannot see ... | {
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# Difference between hypotenuse and larger leg in a Pythagorean triple
I've been number crunching irreducible Pythagorean triples and this pattern came up: the difference between the hypotenuse and the larger leg seems to always be n² or 2n² for some integer n. Moreover, every integer of the form n² or 2n² is the diff... | {
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$$c - b = (m^2 + n^2) - 2mn = (m - n)^2 \tag{3}\label{eq3A}$$
Regarding your second part, to help avoid confusion with $$n$$ above, let's call the values $$k^2$$ and $$2k^2$$ instead. With the first one, it uses \eqref{eq3A} so $$k = m - n$$. However, since one of $$m$$ and $$n$$ is even and the other is odd, their di... | {
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Counter example is $$(9, 12, 15)$$.
Edit
In addition to overlooking the 2nd part of the OP's question, as indicated in my 2nd edit (below), I also overlooked that in the first part of the OP's question, he is (also) specifically focusing on irreducible Pythagorean triplets.
I've been number crunching irreducible Pyth... | {
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As for the second part of the question, just find any solution to either $$c-b=n^2$$ or $$c-b=2n^2$$ for an arbitrary $$n$$. In particular, the triples $$(n^2+2n, 2n+2, n^2+2n+2)$$ and $$(2n^2+2n, 2n+1, 2n^2+2n+1)$$ respectively satisfy the condition.
Note though that for even $$n$$ there are no irreducible triples wi... | {
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# Friction Ramp Problem (I'm challenging a score received on a test)
1. Oct 26, 2009
### shankman
Hello! First time, long time!
This is kind of a long post. I did what I could to keep it clear. TYIA!
I was marked totally wrong on a test question and I think I may have been correct. I'm trying to get my ducks in a ... | {
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Since the system wants to move towards the 300g mass, friction opposes it:
Fnet = 2.94N – 2.81N - .601N = -.471N
Therefore, it will not accelerate towards the 300g hanging mass because the friction is too great.
AND, I just don’t get to add the friction as a force going down the hill and say:
Fnet = 2.81N + .601N – 2... | {
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Or, am I just crazy?
4. Oct 27, 2009
### PhanthomJay
No, you are quite sane. I tended to agree with rockfreak until I cranked out the numbers. In problems such as these, depending on the values, the mass on the ramp could move up the plane, down the plane, or stand still. You have to work it out, as you did. In this... | {
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7. Oct 28, 2009
### shankman
Exactly!
My professor is essentially saying that the force of friction between the block and ramp is exceeding the other force on the block. Therefore, the force of friction is causing the block to move. This is impossible; the frictional force cannot do this.
At least I feel vindicated... | {
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# Can you give an inductive definition to define the length of a list L?
Having trouble understanding what it means to define something inductively in the following context.
Can you give an inductive to define the length of a list L?
a. The total number of items in L is the length of L
b. Basis: the length of an em... | {
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Same for the second question:
function member(elem x, list L)
if L is empty
return false
return true
else return member(x, tail(L))
and this corresponds to say: X is a member of a list L if either
• X is the head of the list or
• X is a member of the tail of L. | {
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# Let $T$ be a normal random variable that describes the temperature…
Let $$T$$ be a normal random variable that describes the temperature in Rome on the 2nd of June. It is known that on this date the average temperature is equal to $$µ_T = 20$$ centigrade degrees and that $$P (T ≤ 25) = 0.8212$$.
How can I calculate... | {
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# Sample Spaces and Events
The outcome of a random experiment is uncertain, but there exists a set of possible outcomes, $$\varepsilon$$, known as the sample space.
The sum of the probabilities of all the outcomes in, $$\varepsilon$$ is 1.
For example, the sample space for rolling a six-sided dice is: $\varepsilon = \... | {
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# The Addition Rule and Mutual Exclusivity
The addition rule is generally used to calculate $$\Pr(A \cap B)$$ or $$\Pr(A \cup B)$$ $\Pr(A) + \Pr(B) - \Pr(A \cap B) = \Pr(A \cup B)$
We say that two events are mutually exclusive if: $\Pr(A \cap B) = 0$ That is the two events will never occur at the same time.
# Probabi... | {
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# Conditional Probability
The probability that event A happens when we know that event B has already occured: $\Pr(A \mid B) = \frac{\Pr(A \cap B)}{\Pr(B)}$
It is often difficult to recognise when we are being asked a conditional probability question. However, generally speaking, if the question includes “if” or “giv... | {
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\text{Example 9.3: John has lost his class timetable.}\\\text{ The probability that he will have Methods period one is 0.35.}\\ \text{The probability that he has PE on a given day is 0.1 and the probability} \\\text{that he will have Methods period one and PE on the same day is 0.035.}\\ \text{Is John having Methods pe... | {
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\text{Example 9.4: James scores 80\% of all free throws he takes.}\\ \text{Create a probability distribution table and graph the probability mass function if James has two free throws.}\\ \text{ }\\ \text{Let } X \text{ be the number of free throws James scores}\\ \begin{aligned} \varepsilon &= {0,1,2}\\ Pr(X = 2) &= 0... | {
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For many random variables, there is a 95% chance of obtaining an outcome within two standard deviations either side of the mean. That is, $\Pr(\mu - 2\sigma \leqslant X \leqslant \mu + 2\sigma) \approx 0.95$
\text{Example 9.5: James scores 80\% of all free throws he takes. James takes two shots.}\\ \text{Find the expe... | {
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# Combinatorics - h men, m women and n chairs in a circular table
Question: How many ways there are to sit h men, m women in a circular table with n chairs in a way that no woman is going to be sit next to other woman?
The objective of this post: I would like for someone to check if what I've done, including my reaso... | {
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4) Finally, we make each men choice a remaining seat to sit: $$P_{h}^{n-m}$$.
5) Therefore, the result is:$$(m-1)!\cdot{n-m-1\choose m-1}\cdot P_{h}^{n-m}$$.
EDIT:
The reason for this edit is that I noticed that N. F. Taussig answer and the one that I gave are actually the same!
Let's introduce the same change of v... | {
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Hand each person a chair. Suppose Andrew is one of the $$m$$ men. Seat him first. The other men can be seated around the table in $$(m - 1)!$$ ways as we proceed clockwise around the table from Andrew. Seating the $$m$$ men creates $$m$$ spaces in which we can place a woman. To separate the women, we must choose $$w$$ ... | {
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• Thank you for answering, but please, check the edit :). Ty anyway! – Bruno Reis Sep 22 '18 at 22:58 | {
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# Parallel|Definition & Meaning
## Definition
Two lines or line segments are said to be parallel to each other, if the perpendicular distance between their lines remains same throughout their length.
Two lines are called parallel to each other if we can prove that the perpendicular distance between them at all point... | {
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There is a predetermined minimum separation or perpendicular distance between parallel lines that they maintain from minus infinity to plus infinity, and they do not touch each other or converge at any point. In three-dimensional Euclidean space, a line and a plane are said to be parallel if they do not share a point.
... | {
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## Euclidean Postulates of Parallelism (Properties of Parallel Lines)
In this section, we present a more mathematically rich perspective of parallelism with respect to straight lines. We formally introduce parallelism and the properties of parallel lines in the following paragraphs. These properties can also be used t... | {
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### Calculation of Distance Between Two Lines
There is a certain distance between the two parallel lines because parallel lines in a Euclidean plane are identical in length. Given the equations for two parallel non-vertical lines, by locating two points (one on each line) that are perpendicular to one another and figu... | {
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$m$ = 10, $u_1$ = 4, $u_2$ = 24
Using the formula:
$d = \dfrac{ | u_1-u_2 | }{ \sqrt{ m^2 + 1 } }$
$d = \dfrac{ | 2-10 | }{ \sqrt{ 10^2 + 1 } }$
$d = \dfrac{ | -8 | }{ \sqrt{ 101 } }$
$d = \dfrac{ 8 }{ 10.05 }$
$d = 0.796$
All images were created with GeoGebra. | {
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Roster is a method of naming a set by listing its members. Example − $S = \lbrace x \:| \:x \in N,\ 7 \lt x \lt 9 \rbrace$ = $\lbrace 8 \rbrace$. Do you also have Class 11 NCERT Books and solutions for Class 11 Set Theory ? Such a relation between sets is denoted by A ⊆ B. Practice test sheets for Class 11 for Set Theo... | {
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of elements. Properties of Basic Mathematical Operations, Quiz: Properties of Basic Mathematical Operations, Quiz: Multiplying and Dividing Using Zero, Quiz: Signed Numbers (Positive Numbers and Negative Numbers), Simplifying Fractions and Complex Fractions, Quiz: Simplifying Fractions and Complex Fractions, Signed Num... | {
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# Does $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?
I want to use the alternating series test here, but I've just been told that it won't work because it's not monotonically decreasing.
However, if the alternating harmonic series converges then don't we have for $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1... | {
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Can someone point out where the mistake here is?
To show that it is monotonically decreasing one should show that: $$\frac{1}{n^{1+\frac{1}{n}}} > \frac{1}{(n+1)^{1+\frac{1}{n+1}}}.$$ This is equivalent to showing that: $$\frac{n+1}{n} > \frac{n^\frac{1}{n}}{(n+1)^\frac{1}{n+1}},$$ which is the same as $$(1+\frac{1}{n... | {
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# Why do we need two linearly independent solutions for 2nd order linear ODE
Let we have a second-order homogeneous linear ODE with two initial conditions.
$y''+ p(x)y'+q(x)y=0$
$y(x_0)=K_0$ and $y'(x_0)=K_1$
Why do we need two linearly independent solutions to satisfy the IVP.
If we have only one solution what wo... | {
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Linear ODE now have the property that their solutions form a linear or at least affine space, the first for homogeneous, the second for general inhomogeneous problems. As such, they can be described by giving the basis of the (underlying) vector space, and each such basis has 2 elements. For instance those for the init... | {
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# Is this a valid proof of Lagrange's theorem (finite case).
Let $$G$$ be finite and $$H$$ be a subgroup. We will show that the left cosets of $$H$$ partition $$G$$ and each coset has the same size.
1) Each element $$g \in G$$ belongs to the coset $$gH$$ since $$g1=g$$ and $$1\in H$$.
So every element lies in at lea... | {
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Is this proof valid?
• What do you mean "finite case"? Lagrange's theorem is only applicable to finite groups, since "divides the order" only makes sense when "order" is a number. – Adam Hughes May 26 '17 at 14:11
• There's just a small issue with your argument that $aH\subseteq bH$. You wrote $ah = bh_2h_{1}^{-1}$, w... | {
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# Proving that a relation R is an equivalence relation
While I fully understand what it means to be an equivalence relation, I have a difficulty establishing proof that $R$ is an equivalence relation without just listing all pairs that $R$ creates and testing them.
However this method is greatly time consuming and is... | {
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• Suppose that $(A,B)\in\mathcal{R}$ and suppose nothing further about $(A,B)$ for now. We wish to prove that this implies that $(B,A)\in\mathcal{R}$. So, since $(A,B)\in\mathcal{R}$ this implies that either $A=S\setminus B$ or that $A=B$. From here we break into two cases. In the case that $A=B$, since $=$ is known to... | {
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In either case $(B,A) \in R$, so we have proven symmetry.
Transitivity can be done similarly (though you might need to break up into more cases and subcases); I'll leave it for you to tackle.
• +1 for beating me by 41 seconds :) – gt6989b May 14 '18 at 17:10
• For transitivity, assuming a,b∈P(S) and a=/=b, then a=S\b... | {
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Then think about the consequences in terms of reflection, symmetry or trasitivity.
Equality is obviously (almost by definition) an equivalence class. And complements are symmetric although not reflexive or transitive... although the symmetry of complements make them a bit of a toggle. ($a = b^c;b=c^c \implies c=b^c = ... | {
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It is commonly used to solve ordinary differential equations , but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field ). Steps to solving a first-order exac... | {
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equations it’s probably best to work an example that will help to show us just what an exact differential equation is. But now I want to do another exercise, which uses a function of the form $\mu(x+y)$. Equations with linear fractions; Exact equations; Integrating factor. , Let d 2 y / dx 2 + y = 0. To solve, take and... | {
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to obtain a function u(x,y) such that ϕ(x,y)[M(x,y) dx + N(x,y) dy ]= du. Exact Equations and Integrating Factors. Exact equations. (a) ( 2 x 2 + y ) dx + ( x 2 y - x ) dy = 0. Math 2280 - Lecture 6: Substitution Methods for First-Order ODEs and Exact Equations Dylan Zwick Fall 2013 In today's lecture we're going to ex... | {
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Sep. The AWS Access Key Id you provided does not exist in our records. Dynamical modeling Flux balance analysis Logical modeling Network modeling Stochastic simulation …. written as. Integrating Factor example. I want to make the function exact first. Integrating Factors Found by Inspection. If an equation is “almost” ... | {
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Exact Equation. Consider the heat that is transfered to a gas that changes it temperature and volume a very small amount:. Equations of nonconstant coefficients with missing y-term If the y-term (that is, the dependent variable term) is missing in a second order linear equation, then the equation can be readily convert... | {
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in which can be found include -dependent, -dependent, and -dependent integrating factors. Using an integrating factor to make a differential equation exact 大家可以通過微分方程 學到很多不同的技巧 在這個影片裏面 我教大家一個 它的作用很大 因爲它總是。. Integrating factors and first integrals for ordinary diflerential equations 247 Definition 2. integrating factor ... | {
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what function or functions satisfy the equation. Non-exact Second Order Differential Equations and Integrating Factors In this section, we introduce the idea of finding integrating factors for the second order differential equation (2. So, in order for a differential dQ, that is a function of four variables to be an exac... | {
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this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process. Exact Differential Equations. You will learn what a differential equation is and how to recognize some of the basic different types. Consider the heat that is transfered to a gas tha... | {
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48:6, 913-918, DOI: 10. After writing the equation in standard form, P(x) can be identified. by using the Integrating Factor solution method. The expression is an exact differential. Solving Exact Differential Equations. Non-exact Second Order Differential Equations and Integrating Factors In this section, we introduce ... | {
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Here's what I got: ordinary-differential-equations. A differential equation along with a subsidiary condition y (t0)=y0, given at some value of the independent variable t=t0, constitutes an initial value problem. The general form of a first order ODE is M(x,y) dx + N(x, y) dy = 0. Consider equation (1) ×µ,. Y Z V Y, / ... | {
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Finally, we will generalize the notion of integrating. That is, a subset which cannot be decomposed into two non-empty disjoint open subsets. Determine conditions on a and b so that u(x, y) = (x^a)(y^b) is an integrating factor. integrating factor which will transform this into an exact equation. If an equation is not ... | {
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factor Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Integrating Factor Depends on the Variable $$x:$$ \(\mu = \mu \left( x \right). We seek an integrating factor that. Examples of solving Linear First Order Differential Equations with an Integrating Fac... | {
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dependent variable y and the differential coefficients is called differential equation. The relation above, Clairaut's theorem, is the necessary and sufficient condition for an exact equation. 1 A set of factors {A"[ Y]} satisfying (2. Integrating Factor. N x ---Select--- are are not equal, the equation is not exact. M... | {
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we have studied first-order DEs (mostly ODEs) that are both (i) linear, and (ii) separable. If an equation is not exact, it may be possible to find an integrating factor (a multiplier for the functions P and Q, defined previously) that converts the equation into exact form. An integrating factor is Multiplying both sid... | {
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or exact derivative which is easy to solve simply by integration. Integrating factors. E of the form is said to be exact D. In other words, even if the above equality is not satisfied, there may exist a function f(x,y) such that. You will learn what a differential equation is and how to recognize some of the basic diff... | {
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by group of students and teacher of Physics, which is also the largest student community of Physics. Integrating Factor. Using the integrat-ing factor, we can reduce it to a simpler equation. Find an explicit or implicit solutions to the differential equation (x2 − 3xy)+x dy dx = 0. About the Author Steven Holzner is an... | {
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"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9947798736874025,
"lm_q1q2_score": 0.8415265051370604,
"lm_q2_score": 0.8459424314825853,
"openwebmath_perplexity": 444.8221804955363,
"openwebmath_score": 0.9115209579467773,
"tag... |
equations it's probably best to work an example that will help to show us just what an exact differential equation is. One then multiplies the equation by the following "integrating factor": IF= e R P(x)dx This factor is defined so that the equation becomes equivalent to: d dx (IFy) = IFQ(x),. Hint: Try to find an integr... | {
"domain": "jf-huenstetten.de",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9947798736874025,
"lm_q1q2_score": 0.8415265051370604,
"lm_q2_score": 0.8459424314825853,
"openwebmath_perplexity": 444.8221804955363,
"openwebmath_score": 0.9115209579467773,
"tag... |
the original. For instance, the expression 2xy5 +4x2y4y0. Next we will focus on a more speci c type of di erential equation, that is rst order, linear ordinary di erential equations or rst order linear ODEs for short. Bibliography for Exact Differential Equations. My steps: /N = 2/x,\$ which is the integrating factor. ... | {
"domain": "jf-huenstetten.de",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9947798736874025,
"lm_q1q2_score": 0.8415265051370604,
"lm_q2_score": 0.8459424314825853,
"openwebmath_perplexity": 444.8221804955363,
"openwebmath_score": 0.9115209579467773,
"tag... |
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