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Note that when you square the equation
$$(\sin x + \cos x)^2 = (\sin x \cos x)^2$$
which can be factorized as
$$(\sin x + \cos x - \sin x \cos x)(\sin x + \cos x + \sin x \cos x)=0$$
you effectively introduced another equation $$\sin x + \cos x =- \sin x \cos x$$ in the process beside the original one $$\sin x + \c... | {
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Only the first factor yields real roots
$$x = 2n\pi + \frac\pi4 \pm \cos^{-1}\frac{\sqrt2-2}2$$
As your error has been pointed out, I am providing a different way to tackle the problem without introducing extra solutions. From the given equation, we have $$1=(1-\sin x)(1-\cos x)$$, which is equivalent to $$1=\Biggl(1... | {
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you then have with $$u=\sin x, v=\cos x$$:
$$uv=1-\sqrt{2}$$
$$\color{blue}{u+v=uv=1-\sqrt{2}}$$
where the blue equation reimposes the original requirement and it's goodbye extraneous roots. Using the Vieta formulas for a quadratic polynomial this is solved by rendering $$u$$ and $$v$$ as the two roots of the quadra... | {
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# commutative rings whose localization at every prime ideal is a field
Can we characterize those commutative rings $R$ with unity such that for every prime ideal $p$ of $R$, $R_p$ is a field?
I know that any Boolean ring has this property. Also, if a ring has this property, then every finitely presented module over i... | {
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Your question has already been answered, but let me give a proof of the characterization of commutative rings whose localizations at prime ideals are fields, that is somehow different of the proof given by rschwieb.
Theorem 1.- Let $R$ be a commutative ring. TFAE:
i) $R$ is von Neumann regular.
ii) Every prime ideal... | {
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iii)$\implies$i) We are going to prove that for $a\in R$, $(a)_P=(a^2)_P$ for every prime ideal $P$ of $R$. As $R_P$ is a field, it follows that $(a)_P=\{0\}$ or $(a)_P=R_P$. In the former case we have that $a\in P$, so $a^2\in P$ and thus $(a^2)_P=\{0\}$. In the later case we have that $a\notin P$, so $a^2\notin P$ an... | {
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• could you please explain why every prime ideal is minimal ? – user Aug 29 '17 at 18:39
• The prime ideals of $R$ contained in $p$ correspond to the prime ideals of $R_p$. – Angina Seng Aug 29 '17 at 18:52 | {
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# 9.5. Trigonometric interpolation¶
Up to this point, all of our global approximating functions have been polynomials. While they are versatile and easy to work with, they are not always the best choice.
Suppose we want to approximate a function $$f$$ that is periodic, with one period represented by the standard inte... | {
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You can directly check the following facts. (See Exercise 3.)
Theorem 9.5.2
Given the definition of $$\tau$$ in (9.5.3),
1. $$\tau(x)$$ is a trigonometric polynomial of degree $$n$$.
2. $$\tau(x)$$ is 2-periodic.
3. $$\tau(t_k)=0$$ for any nonzero integer $$k$$.
4. $$\displaystyle \lim_{x \to 0} \tau(x) = 1.$$
G... | {
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N = 7; n = (N-1)÷2
t = 2*(-n:n)/N
y = zeros(N); y[n+1] = 1;
p = FNC.triginterp(t,y);
plot(p,-1,1)
scatter!(t,y,color=:black,title="Trig cardinal function, N=$N", xaxis=(L"x"),yaxis=(L"\tau(x)")) Here is a 2-periodic function and one of its interpolants. f = x -> exp(sin(pi*x)-2*cos(pi*x)) y = f.(t) p = FNC.trigint... | {
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The $$N=2n+1$$ coefficients $$c_k$$ are determined by interpolation nodes at the $$N$$ nodes within $$[-1,1]$$. By evaluating the complex exponential functions at these nodes, we get the $$N\times N$$ linear system
$\mathbf{F}\mathbf{c} = \mathbf{y}, \qquad \mathbf{F} = \bigl[ e^{\,is\pi t_r} \bigr]_{\, r=-n,\ldots,n,... | {
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Note that $$1.5 e^{2i\pi x}+1.5 e^{-2i\pi x} = 3 \cos(2\pi x)$$, so this result is sensible.
Fourier’s greatest contribution to mathematics was to point out that every periodic function is just a combination of frequencies—infinitely many of them in general, but truncated for computational use. Here we look at the mag... | {
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3. ✍ Verify that the cardinal function given in Equation (9.5.3) is (a) 2-periodic, (b) satisfies $$\tau(t_k)=0$$ for $$k\neq 0$$ at the nodes (9.5.2), and (c) satisfies $$\lim_{x\to0}\tau(x)=1$$.
4. ✍ Prove the equality of the two expressions in (9.5.3). (Hint: Set $$z=e^{i\pi x/2}$$ and rewrite the sum using $$z$$ b... | {
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# Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
“The rank of a matrix A is the number
of non-zero rows in the reduced
row-echelon form of A”.... | {
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# What property of certain regular polygons allows them to be faces of the Platonic Solids?
It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes.
What property of those regular polygons themselves allow t... | {
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The regular polygons that form the Platonic solids are those for which the measure of the interior angles, say α for convenience, is such that $3\alpha<2\pi$ (360°) so that three (or more) of the polygons can be assembled around a vertex of the solid.
Regular (equilateral) triangles have interior angles of measure $\f... | {
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-
Thanks! That makes a lot of sense actually. If it isn't much of a bother for you, do you mind addressing some of the other things I mentioned? Like why triangles are found in more platonic solids than both of the other shapes combined. – Justin L. Jul 22 '10 at 1:23
Because 6 triangles tile a plane. So 5 triangles ca... | {
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# Order of Operations: Neglected Details
The basic statement of the order of operations covers the five main operations (exponents, multiplication, division, addition, and subtraction). But what about other operations like square roots? How about trigonometric functions? And are operations at the same level always car... | {
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Interpreting 2^3^2
http://mathforum.org/library/drmath/view/54362.html
Nowadays, we can check Wikipedia (which is usually trustworthy on mathematical matters, and gives sources that can be checked):
http://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties
Without parentheses to modify the order of calcu... | {
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I've occasionally wondered where I could find an "official" authority on this, myself. I know that a^b^c is generally taken as a^(b^c), and that the reason is that (a^b)^c can be written as a^(bc), while the other form, which is more interesting, has no alternative. We've occasionally mentioned this in Dr. Math answers... | {
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## Square roots: built-in grouping
I noted last time that PEMDAS does not explicitly include the negation operator, so we have to recognize that it is treated as being at the same level as multiplication. We get questions about other unary operators (that is, operations that act on only one number) from time to time. ... | {
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In this problem, I don't think it would change the effect to include the square root at the time you do exponents; however, I know that the order of ops is around to provide stability when it WOULD change the effect (and the outcome). It makes my head hurt (and doing algebra in an email box is almost as bad ;-). Can yo... | {
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Thus, the first thing to do is to evaluate each quantity in parentheses, AND evaluate the root:
14 + (24 - 12)^2 / 2 * 3^3 + (4 - 2^2) + sqrt(9)
\_______/ \_______/ \_____/
14 + 12 ^2 / 2 * 3^3 + 0 + 3
\_____/ \_/
14 + 144 / 2 * 27 + 0 + 3
\_______/
14 + ... | {
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Why does the order of operations exclude percentage, square roots, etc.?
Let's say I have a problem like this:
5% of 290 + 89 square root - 1 =
(By the way, according to some questions we have seen, “5% of” is included by some authors in other countries as a standard part of the notation, apparently under the belief ... | {
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The square root is easier to be sure about, as we’ve already seen:
The square root sign is special. The actual radical sign is just the "v"-like thing alone, and means to take the root of the number following it. For example, \/4+5 would be 2+5=7, not the square root of 9, which is 3. But you usually see a bar hanging... | {
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Hi, Kiaran. Good question!
I would put factorials with functions, something you don't often see listed in the order of operations. If you are not familiar with functions, you can think of them very simply as something to be done to a single number (or sometimes a list of numbers) that produces a number as a result. Th... | {
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One reliable source does explicitly state this:
Wolfram MathWorld: Precedence
For simple expressions, operations are typically ordered from highest to lowest in the order:
1. Parenthesization,
2. Factorial,
3. Exponentiation,
4. Multiplication and division,
## Absolute value: both operation and grouping symbol
Let... | {
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In effect, the absolute value bars are modified parentheses, which indicate that the function is applied to the quantity within.
In summary, I would go a bit beyond calling the absolute value a "pseudo" grouping symbol, and call it a symbol one of whose functions is to group an expression.
I don't think I understand ... | {
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# Thread: Find sum of the digits of B?
1. ## Find sum of the digits of B?
When 4444^4444 is written in decimal notations, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B.
2. My solution is not very elegant. It may not even be right. But here goes.
First I seek an upp... | {
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So log n + 16210 < 16210.7079
We find that n = 5
So the 1st digit is 5, the last digit is 6. Together they have a digital root of 2. So the remaining (16211-2) = 16209 digits have a digital root of 5.
Let's say 16208 digits are 9's and another digit is 5. So the highest possible value of A is
5 + 16208 * 9 + 5 + 6 ... | {
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And why is the sum B at most the sum of digits of 39 ?
9. Anyway I was wrong on that count but the proof still holds if we replace 179,999 by 99,999 and 44 by 45.
Also the sum of the digits of B can't be more than 39 because 3+9=12 which is the largest possible sum!
Thanks for pointing that out!
But don't you agree... | {
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# 3.4 Composition of functions (Page 6/9)
Page 6 / 9
## Finding the domain of a composite function involving radicals
Find the domain of
Because we cannot take the square root of a negative number, the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-\infty ,3\right].\tex... | {
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We are looking for two functions, $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}f\left(x\right)=g\left(h\left(x\right)\right).\text{\hspace{0.17em}}$ To do this, we look for a function inside a function in the formula for $\text{\hspace{0.... | {
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## Key concepts
• We can perform algebraic operations on functions. See [link] .
• When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
• The function produced by combining two functions is a composite function. See [link] and [link] .
• The order of f... | {
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the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldee... | {
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MAT244--2018F > Quiz-6
Q6 TUT 5102
(1/1)
Victor Ivrii:
The coefficient matrix contains a parameter $\alpha$.
(a) Determine the eigenvalues in terms of $\alpha$.
(b) Find the critical value or values of $\alpha$ where the qualitative nature of the phase portrait for the system changes.
(c) Draw a phase portrait f... | {
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# Can a set of positive measure and its complement both have empty interior? [duplicate]
This might be silly, but I am not sure:
Does there exist a Lebesgue measurable subset $$E \subseteq (0,1)$$ such that
1. $$E$$ and $$(0,1) \setminus E$$ both have positive Lebesgue measure.
2. $$E$$ and $$(0,1) \setminus E$$ bo... | {
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• In my answer to this question I constructed an $F_\sigma$ set $M\subseteq\mathbb R$ such that $0\lt m(M\cap I)\lt m(I)$ for every interval $I$, where $m$ denotes Lebesgue measure. Obviously the sets $M$ and $\mathbb R\setminus M$ have positive Lebesgue measure and have empty interiors, and the same goes for the sets ... | {
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1. ## Unit Vectors
I have to find two unit vectors that are parallel to
<3, 1, 2>
And then I have to write each vector as the product of its magnitude and a unit vector.
--------
My approach:
We know that two vectors are parallel if their cross product is equal to 0.
$\mathbf{a} \times \mathbf{b} = \mathbf{A}_{\... | {
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So, <6, 2, 4> and <9, 3, 6>. Now find the norm of each and you should be able to take it from there.
3. Originally Posted by AfterShock
You're thinking about it too much. Just use any scalar.
That is, two vectors are parallel if they are a scalar of the other. You can check with using cross products to show they are ... | {
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# What does it mean for simple functions to have finite range
In Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics By Dan Simovici, Chabane Djeraba, it says:
A simple function is a function $f: S \to \mathbb{R}$ that has finite range.
Can someone clarify what it means by "finite range"? D... | {
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# is this process a Markov one?
Here is the problem I can't solve.
Let $$\xi_n$$ $$(n=1,2,3,\dots)$$ be a sequence of i.i.d. random variables on $$\mathbb{R}$$ with density $$p(x)>0$$, let $$\eta_n=\sum_{i=1}^{n}\xi_i^2$$. Define $$\zeta_t = \eta_{[t]}(t-[t]) \times \eta_{[t]+1}(-t+[t]+1),$$ where $$[t]$$ denotes the... | {
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• Do you really mean "$\times$" and not "$+$" in the definition of $\zeta_t$? It looks like $\zeta_t = 0$ whenever $t$ is an integer. – Mateusz Kwaśnicki May 22 at 10:41
• Yes, there really should be $\times$ in the definition of $\zeta_t$. Can you tell, please, how do you know that $\zeta_t=0$ for integer $t$? – I.Kia... | {
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The process is not Markov in general. Indeed, let $$X_i:=\xi_i$$, $$S_n:=\eta_n=\sum_1^n X_i^2$$, and $$Z_t:=\zeta_t=(t-[t])([t]+1-t)S_{[t]}S_{[t]+1},$$ where $$P(X_i=0)=P(X_i=1)=1/2$$. Then $$Z_{3/2}=\tfrac14\,X_1(X_1+X_2),\quad Z_2=0,\quad Z_{5/2}=\tfrac14\,(X_1+X_2)(X_1+X_2+X_3).$$ So, the conditional distribution o... | {
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# Elegant way to prove congruence
I'm stuck with the last question of this exercise
1) First question asks to solve the linear diophantine
$$143x-840y=1$$
based on the remark $$143\times 47 - 840 \times 8 = 1$$ (done)
2) second question asks to prove that if a natural $$n$$ is coprime with $$899$$, then
$$n^{840}... | {
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Problem: We want to find the $$x$$ in range $$0 \leq x \leq 898$$ such that $$2^{47} \equiv x \pmod{899}$$.
Here's a relatively fast method for general problems like these called square-and-multiply:
We can write $$47 = 2^5 + 2^3 + 2^2 + 2^1 + 2^0. \$$ Therefore, $$2^{47} = 2^{2^5} \cdot 2^{2^3} \cdot 2^{2^2} \cdot 2... | {
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# Density of a set
2010 Mathematics Subject Classification: Primary: 28A05 Secondary: 28A1549Q15 [MSN][ZBL]
A concept of classical measure theory generalized further in Geometric measure theory
### Lebesgue density of a set
Given a Lebesgue measurable set $E$ in the standard Euclidean space $\mathbb R^n$ and a poin... | {
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### Density of a measure
The concept above has been generalized in geometric measure theory to measures, starting from the work of Besicovitch. Consider a (locally finite) Radon measure $\mu$ in the Euclidean space $\mathbb R^n$, a point $x\in \mathbb R^n$ and a nonnegative real number $\alpha$ (see for instance Defin... | {
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A similar result in the opposite direction holds and is a particular case of a more general result on the Differentiation of measures:
Theorem 4 Let $\mu$ be a locally finite Radon measure on $\mathbb R^n$. If the $n$-dimensional density $\theta^n (\mu, x)$ exists for $\mu$-a.e. $x$, then the measure $\mu$ is given by... | {
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• $\theta^{\alpha,*} (E, x) =0$ for $\mathcal{H}^\alpha$-a.e. $x\not\in E$.
• $1\geq \theta^\alpha_* (E,x) \geq 2^{-\alpha}$ for $\mathcal{H}^\alpha$-a.e. $x\in E$.
#### Besicovitch-Preiss theorem and rectifiability
However the existence of the density fails in general: as a consequence of Marstrand's Theorem 2 the e... | {
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Theorem 9 Let $\mu$ be a locally finite Radon measure and $\alpha$ a nonnegative real number. Then $\theta^\alpha (\mu, x)$ exists, it is finite and positive at $\mu$-a.e. $x\in \mathbb R^n$ if and only $\alpha$ is an integer $k$ and there are a rectifiable $k$-dimensional Borel set $E$ and a Borel function $f: E\to ]0... | {
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The definition of $\alpha$-dimensional density of a Radon measure can be generalized to metric spaces. In general, however, very little is known outside of the Euclidean setting (cp. with Section 10.0.2 of [De]).
[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems"... | {
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# Halving the Interval Method with Relative Accuracy
## Find the smallest positive root of $x^3 – 2x + 1 = 0$ with relative accuracy of $0.5 \%$ by finding good starting value from its graph.
According to the graph above, there are three roots, one negative and two positive.
Let’s give $f(x) = x^3-2x+1$, then $f(1) ... | {
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$f(0.65) = -0.0253 \cdots < 0$
The refined interval is $0.6 < x < 0.65$.
The midpoint is $\displaystyle \frac{0.6+0.65}{2} = 0.625$.
Its relative accuracy is $\displaystyle \frac{|0.6 – 0.625|}{0.625} \times 100 \% = 4 \%$.
$f(0.625) = -0.0058 \cdots < 0$
The refined interval is $0.6 < x < 0.625$.
The midpoint is $\di... | {
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# Math Help - Probability
1. ## Probability
A factory produces items in boxes of 2. Over the long run:
92% of boxes contain 0 defective items;
5% of boxes contain 1 defective item;
3% of boxes contain 2 defectives item.
A box is picked at random from production, then an item is picked at random from the box. Given t... | {
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. . Type C: contains two defective items. . $P(C) \,=\,0.03$
A box is chosen at random and one item is sampled from that box.
Given that the sample is defective, what the probability that the box is of Type C?
This is Conditional Probability.
Bayes' Theorem: . $P(\text{Type C }|\text{ 1 d{e}f}) \;=\; \frac{P(\text{... | {
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# Is my proof that $(A^n)^{-1} = (A^{-1})^n$ correct?
I am still learning Linear Algebra at it's basic levels, and I encountered a theorem about invertible matrices that stated that:
If $A$ is an invertible matrix, then for $n=0,1,2,3,..$. $A^n$ is invertible and $(A^n)^{-1} = (A^{-1})^n$.
Now, in attempting to writ... | {
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Thanks to everyone for guidance, now I see why collaboration is going to make me love math :D
-
I think this is wrong because you are using commutativity. Matrix multiplication is not commutative. However, I think you can adapt your proof so it works. – sxd Sep 20 '11 at 13:19
It's correct if before writing the equali... | {
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As Dimitri points out, your proof is incomplete. You can make it work in two ways:
1. You can group the middle $AA^{-1}$ (remember that matrix product is associative). Noting that this equals $I$, the product simplifies to $A^{n-1} (A^{-1})^{n-1}$. You can then use induction to argue that $A^n (A^{-1})^{n}$ is $I$ for... | {
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Sorry for the dots, but I didn't find a better way to point the idea out!
-
This is the idea I had in mind, but thought I could condense it by using the $\prod$ operator. And then also, I was assuming that I could expand if necessary to something like: $\prod_{i=1}^{n-1}A.(AA^{-1}).\prod_{i=1}^{n-1}=\prod_{i=1}^{n-1}A... | {
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# Is there any connection between Green's Theorem and the Cauchy-Riemann equations?
Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy$$ The Cauchy-Riemann equations have the following form:(Assuming $z = P(x,y) + iQ(x,y... | {
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-
I think the X and Y thing is not just an American thing. Btw... as a contrast - suddenly when you come to graduate school you are supposed to know this, and that etc. :) – AD. Nov 29 '10 at 14:05
@AD.: Well, I have a lot of experience with "the" American university system, having been involved with it for my entire a... | {
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To calculate the percentage change between two numbers we typically write a formula like the following: = (C2-B2)/B2. Whenever I am creating this formula I always think, "new minus old, divided by old". Or: = (new value - old value) / old value. If we want to handle/prevent any divide by zeros (#DIV/0!), then we can wr... | {
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Percentage difference equals the absolute value of the change in value, divided by the average of the 2 numbers, all multiplied by 100. We then append the percent sign, %, to designate the % difference. Percentage Difference = | Δ V | [ Σ V 2] × 100 = | V 1 − V 2 | [ ( V 1 + V 2) 2] × 100 For example, how to calculate ... | {
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i.e. increase between the two numbers. i.e. Increase = New Number - Original Number. Step-2: Divide the increased value by the original. Trend analysis is a common task in financial accounting. Accountants compare two time periods by using financial information. The difference typically shows a percentage increase or d... | {
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percent of increase, which will be a positive. If the new number is less than the old number, then that ratio is the percent of decrease, which will be a negative.. Let V 1 = 3.50 and V 2 = 2.625 and plug numbers into our percentage change formula ( V 2 − V 1) | V 1 | × 100 = ( 2.625 − 3.50) | 3.50 | × 100 = − 0.875 3.... | {
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First: work out the difference (decrease) between the two numbers you are comparing. Decrease = Original Number - New Number. Then: divide the decrease by the original number and multiply the answer by 100. % Decrease = Decrease ÷ Original Number × 100. If your answer is a negative number, then this is a .... Percentag... | {
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a. First, subtract to find the amount of change: 150 - 125 = 25. The decrease is 25. Next, divide the amount of change by the original amount: 25 ÷ 150 = 0.167 Now, to change the decimal to a percent, multiply the number by 100: 0.167 x 100 = 16.7 The answer is 16.7%. So that's the percent of change, a decrease of 16.7... | {
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will produce the same result when. The standard way of computing the percent change between two values is to get the difference of the first and second numbers. Afterward, divide the result by the first number and multiply it by 100. Here's the exact formula for computing percent change:. Percentage Difference calculat... | {
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between one number and the next, subtract the first number from the second, then divide by the first .... . First, calculate the difference (decrease) between the two compared numbers. Then divide the reduction by the original number and multiply the result by 100. Percentage change formula Formula of Percentage change... | {
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Calculation of percentage change in no of employees can be done as follows- = (30-25)/30*100% = 16.67% or 16.67% decrease in no. of employees Use. If you have already calculated the percentage change, go to step 4. Subtract one from the result of the division. Mult... | {
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I have a list of prices where I am trying to calculate the change in percentage of each number. I calculated the differences with prices = [30.4, 32.5, 31.7, 31.2, 32.7, 34.1, 35.8, 37.8, 36.3... Stack Overflow. ... Calculating change in percentage between two numbers (Pyt... | {
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Thus to calculate the percentage increase we will follow two steps: Step-1: Calculate the difference i.e. increase between the two numbers. i.e. Increase = New Number - Original Number. Step-2: Divide the increased value by the original. In computin... | {
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Trend analysis is a common task in financial accounting. Accountants compare two time periods by using financial information. The difference typically shows a percentage increase or decrease in the information. Accountants use the data to determine if the company is growing or contracting. In many cases, accounting. Wh... | {
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for sale Percentage Decrease = [ (Starting Value - Final Value) / |Starting Value| ] × 100 60 - 8 = 52 52 / 60 = 0.8667 0.8667 × 100 = 86.67% So if you switch to an LED light bulb your lamp will use 86.67% less energy per hour. Related Calculators Use the Percentage Increase Calculator to find the percent increase from... | {
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= | V2 - V1 | / V1 x 100 Percentage Conversion Formula Our percent to decimal conversion is based on this formula and example may enable you to understand the percentage change method. 300 increased by 10% (0.1). Answer (1 of 6): Suppose an item costs$40. And then its price was marked up to \$50. What per cent mark up ... | {
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Note: the percent change measures FROM the first value. A change from 50 to 75 is a change of 50% (25 is the difference between the two numbers, and 25 is 50% of 50). A change from 75 to 50 is a change of -33.3% (25 is still the difference between the two. 25 is 33.3% of 75). How to calculate 50 is 20 percent of what n... | {
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• how to post reels on facebook page – The world’s largest educational and scientific computing society that delivers resources that advance computing as a science and a profession
• 1995 gmc suburban 2500 – The world’s largest nonprofit, professional association dedicated to advancing technological innovation and exce... | {
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Work out the difference (increase) between the two numbers you are comparing. Percentage 'change' and percentage 'difference' are two different things. Percentage change = (fv − iv) ÷ iv × 100. This formula represents percentage change, for example if you are are compari... | {
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# Coloring Points in a DensityPlot/ListDensityPlot
I have a PDE system, whose functions are $a=a(t, x, y)$, $b=b(t,x,y)$, and $c=c(t,x,y)$,
with Dirichlet null boundary conditions and initial conditions in the form of circle.
The respective code is,
L = 5;(*length of square*)
pts = 150;
T = 250;(*Time integration*... | {
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At each point $(x, y)$, if $a(t, x, y)$ has the highest value in relation to $b(t, x, y)$ and/or $c(t, x, y)$ to color the respective point with red color. If at some point $(x, y)$, $b(t, x, y)$ has the largest value in relation to $a(t, x, y)$ and $c(t, x, y)$ to color that point with blue. If at any point $(x, y)$, ... | {
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For some reason imgur images are not loading for me at the moment so I cannot see your goal plot. However I think I get the idea of what you want, and we can apply Michael's method from Plot the plane so different condition has a different color like this:
t = 5;
ContourPlot[
Ordering[Through @ sol[[1, All, 2]][t, x,... | {
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An equivalence class on a set {eq}A Because of the common bond between the elements in an equivalence class $$[a]$$, all these elements can be represented by any member within the equivalence class. Again, we can combine the two above theorem, and we find out that two things are actually equivalent: equivalence classes... | {
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a collection of equivalence classes is a collection of sets. Newb Newb. Let $A = \{0,1,2,3,4\}$ and define a relation $R$ on $A$ as follows: $$R = \{(0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4)\}.$$. Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-ye... | {
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in this case A and ~, you can partition A into sets called equivalence classes. After this find all the elements related to $0$. The equivalence classes are $\{0,4\},\{1,3\},\{2\}$. Examples of Equivalence Classes. But typically we're interested in nontrivial equivalence relations, so we have multiple classes, some of ... | {
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Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds. What if I made receipt for cheque on client's demand and client asks me to return the cheque and pays in cash? Please help! Services, Working Scholars® Bringing... | {
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is true. Thanks for contributing an answer to Computer Science Stack Exchange! Equivalence class is defined on the basis of an equivalence relation. The equivalence class $$[1]$$ consists of elements that, when divided by 4, leave 1 as the remainder, and similarly for the equivalence classes $$[2]$$ and $$[3]$$. Read t... | {
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classes is a collection of sets. THIS VIDEO SPECIALLY RELATED TO THE TOPIC EQUIVALENCE CLASSES. Asking for help, clarification, or responding to other answers. For example 1. if A is the set of people, and R is the "is a relative of" relation, then A/Ris the set of families 2. if A is the set of hash tables, and R is t... | {
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of and the notation " " is used to mean that there is an equivalence relation between and. An equivalence relation will partition a set into equivalence classes; the quotient set $S/\sim$ is the set of all equivalence classes of $S$ under $\sim$. Let ={0,1,2,3,4} and define a relation on as follows: ={(0,0),(0,4),(1,1)... | {
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be represented by any member. An aircraft is statically stable but dynamically unstable, so we have multiple members collection of sets and... Is equivalent to classes of$ 0 $contributions licensed under cc.! \Sim$ be an equivalence class of 1 modulo 5 ( denoted ) is elements related to 0 solve problems! Back them up w... | {
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exhaustive testing is a black box software testing technique divides. ”, you agree to our terms of service, privacy how to find equivalence class and cookie policy $the! '13 at 4:52 into classes/subsets that are multiples of$ 3 $, i.e is. Improve this answer | follow | answered Nov 21 '13 at 4:52 function variable rang... | {
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representated by its lowest or reduced form ”, attributed to G.! ”, attributed to H. G. Wells on commemorative £2 coin thanks for contributing an answer to Computer Science Exchange... A ] then the element b is called a representative of the class which help to the. To this RSS feed, copy and paste this URL into your R... | {
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this |! So a collection of sets we refer to it as the collection of equivalence: let { }. Imply k is in the same class as 0 at 0 and j are the... Path % on Windows 10 and copyrights are the warehouses of ideas ”, attributed to H. G. Wells commemorative! Size 1 | answered Nov 21 '13 at 4:52 to it as the collection of se... | {
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# Calculus (Continuity)
If the following function is continuous, what is the value of a + b?
f(x) = {3x^2 - 2x +1, if x < 0
a cos(x) + b, if 0 </= x </= pi/3
4sin^2(x), if x > pi/3
A. 0
B. 1
C. 2
D. 3
E. 4
I know that since the function is continuous, it should be equal to 1 at 0 and 3 at pi/3 (To follow the other t... | {
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4. ### Math
Write the exponential function y=6(1.21)t in the form y=aekt. (a) Once you have rewritten the formula, give k accurate to at least four decimal places. k= help (numbers) If t is measured in years, indicate whether the exponential
1. ### Calculus (Continuity and Differentiability)
Okay. So I am given a gr... | {
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4. ### Calculus help
The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Right Hand Sum Approximation, using the intervals between those given points. | {
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Different proof that $\sqrt{2}$ is irrational
I found the following proof arguing for the irrationality of $\sqrt{2}$.
Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q > 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 <... | {
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"lm_q1_score": 0.9643214491222695,
"lm_q1q2_score": 0.8412232956887594,
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• Welcome to Mathematics Stack Exchange! For a first post, this is a certainly a well-written one, so +1 from me. – Glorfindel Jun 10 '17 at 8:57
• @Glorfindel Hey! Thanks! – Eulerian Jun 10 '17 at 9:45
• As you want to know what went wrong with your attempt, I point out here. There is nothing wrong with your work. It ... | {
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"lm_q1q2_score": 0.8412232956887594,
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Suppose that $w := \sqrt 2 = p/q$ for integers $p,q>0.\,$ Then $w^2 = 2\,\Rightarrow\, w = 2/w = 2q/\color{#c00}p.\,$ Therefore if $w =\sqrt 2$ is a fraction $p/q$ then its numerator $\color{#c00}p$ can also serve as a denominator for $w$. This peculiar property easily leads to contradictions, e.g. if we choose $q$ as ... | {
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"lm_q1q2_score": 0.8412232956887594,
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The proof you found is a slight variant. By instead starting with a fraction $\ w = \sqrt{2}-1 = p/q$ which is less than $1$ we force $p< q\,$ so there is no need to mod $p$ by $q$ to get a smaller denominator (effectively the mod has already been done by subtracting from $\sqrt 2$ its integer part $=1).$ Thus that $\,... | {
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"lm_q1_score": 0.9643214491222695,
"lm_q1q2_score": 0.8412232956887594,
"lm_q2_score": 0.8723473862936942,
"openwebmath_perplexity": 202.39386786748088,
"openwebmath_score": 0.9427694082260132,
"ta... |
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