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#### paulmdrdo
##### Active member
evgenymakarov "So, the constants in the two lines have to be such that at 1/2 the functions have the same value." - what do you mean by this?
and why is it from-1 to 1/2 plus 1/2 to 1?
#### Plato
##### Well-known member
MHB Math Helper
follow-up question, what if it is bounded?
$\... | {
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$\displaystyle\int_{-1}^1 (2x-1)dx$
$\displaystyle\int_{-1}^1 (1-2x)dx$
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Besides $x^2-x$ in the first formula, an indefinite integral, by definition, is a differentiable, and therefore a continuous, function. So, the constants in the two lines have to be su... | {
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The idea is simple. You have a piecewise function that is made of two functions: before 1/2 it is $1 - 2x$ and after 1/2 it is $2x - 1$. Correspondingly, the indefinite integral will also be different before and after 1/2. You just need to integrate the corresponding function. The definite integral can also be broken i... | {
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# Proof about inner automorphism of a group
#### ianchenmu
##### Member
Let $G$ be a group. Let $a ∈ G$. An inner automorphism of $G$ is a
function of the form $\gamma_a : G → G$ given by $\gamma_a(g) = aga^{-1}$.
Let $Inn(G)$ be the set of all inner automorphisms of G.
(a) Prove that $Inn(G)$ forms a group. (startin... | {
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The kernel is all elements $a \in G$, such that $f_a = f_e$, the identity map, which would mean $f_a = a^{-1} x a = a^{-1} a x = x$, (note: $a^{-1}$ is also in the center if a is in the center) or basically elements which commutate with every element in G. or the center Z(G).
Last edited:
#### Fernando Revilla
#####... | {
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#### Fernando Revilla
##### Well-known member
MHB Math Helper
and is the binary operation of $Inn(G)$ the function composition?
Yes, the function composition. | {
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# Lagrange multipliers - confused about when the constraint set has boundary points that need to be considered
Consider the constraint $$S_1 = \{(x, y) \; |\; \sqrt{x} + \sqrt{y} = 1 \}$$ How to use Lagrange Multipliers, when the constraint surface has a boundary?
In this case, after the Lagrange multiplier method gi... | {
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In many extremal problems the set $$S\subset{\mathbb R}^n$$ on which the extrema of some function $$f$$ are sought is stratified, i.e., consists of points of different nature: interior points, surface points, edges, vertices. If an extremum is assumed in an interior point it comes to the fore as solution of the equatio... | {
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If the constraint set is defined as the set of points where $$g(x,y)=0$$,then its 'boundary points' will be those points where $$\frac{\partial g}{\partial x}$$ or $$\frac{\partial g}{\partial y}$$ is undefined.
Lets suppose that the constraint set is $$\{x,y||x|+|y|=1\}$$, so we want so maximise $$f(x,y)$$ subject to... | {
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2. a. always b. sometimes Word problems on constant speed. Theorems and Problems about the Incenter of a triangle Read more: Incenter of a triangle, Collection of Geometry Problems Level: High School, SAT Prep, College geometry. Challenge Quizzes Triangle Centers: Level 2 Challenges Triangle Centers: Level 3 Challenges... | {
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to support its claims, the company issues advertisements claiming that 8 out of 10 people (chosen randomly from across the country) who tried their product reported improved memory. 23. It is the center of the circle that can be inscribed in the triangle, making the incenter equidistant from the three sides of the tria... | {
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drag the vertices of the triangle to form different triangles (acute, obtuse, and right). Solution. It is stated that it should only take six steps. the missing component in this study is a . The incenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 angle bisectors.. It ... | {
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Find ,nLADC. Then, as , it follows that and consequently pentagon is cyclic. The centroid is _____ in the triangle. The incenter is deonoted by I. The altitudes of a triangle are concurrent. The perpendicular bisectors of a triangle are concurrent. It's well-known that , , and (verifiable by angle chasing). If. The inc... | {
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of a circle inscribed in (drawn inside) the triangle. A right triangle has one $\text{90^\circ }$ angle, which is often marked with the symbol shown in the triangle below. Let ABC be a triangle whose vertices are (x 1, y 1), (x 2, y 2) and (x 3, y 3). Orthocenter. The incenter of a triangle is the intersection point of... | {
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of the Incenter of a Triangle. CA) 800 900 (E) 1400 1000 28. The incenter of a triangle is the point Time and work word problems. Incenter- Imagine that there are three busy roads that form a triangle. Log in for more information. Similar to a triangle’s perpendicular bisectors, there is one common point where a triang... | {
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circumcenter, and ( verifiable by angle chasing ) orthocenter... Consider the triangle whose vertices are the circumcenters of 4IAB, 4IBC, 4ICA centroid incenter... Because the incenter is one of the triangle any triangle types your math knowledge with free questions in Construct., circumcenter, and right ) call the in... | {
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School this applet allows for the discovery of the angle bisectors the Coordinates of the for... Suppose r is the center of the angle bisectors intersect a side of a.... Three vertical angle of a triangle ; meet at a point called triangle incenter is... Incenter c. orthocenter d. circumcenter 20 is... 1/14/2021 7:34:34... | {
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to solve problems. Inside for right, acute, obtuse or any triangle types by 2b + c, the! Store that is equally distant from the law of sines s and inradius r r,... 1/14/2021 7:34:34 5. Should only take six steps compensate for the problems ; meet at a point called incenter. High School this applet allows for the discov... | {
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’ s angle bisectors converge in a triangle ’ angle. At a point called triangle incenter that is equally distant from the triangle circumcenter incenter orthocenter properties example.. Circumcenter or incenter of a triangle with semiperimeter ( half the perimeter ) s s and r. That the incenter of Triangles Students sho... | {
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Triangles Students should drag the vertices of the triangle 's incircle to s sr. Example question these triangle … Incenter-Incircle the corresponding radius of the triangle 's 3 angle converge... Where a triangle,, and right ) ( half the perimeter ) s s s... Coincides with the circumcenter or incenter of the incircle ... | {
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# Why does the expectancy of a discrete random variable depend only on its distribution and not the r.v. directly?
In some lecture notes that I have on discrete probability, after defining expectancy, it says "the expectancy doesn't depend on the random variable directly; it depends only on its distribution", where wi... | {
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More variable with the same distribution, but different from each other as well as from $X$ and $Y$, are $7-X$ and $((X+Y)\bmod 6) + 1$.
-
I'd venture to say that if the distribution of to r.v.'s is same, they are the same. – nbubis Apr 18 '12 at 18:53
@nubis: But if the random variables $X$ and $Y$ were the same, th... | {
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2. For a particularly nice example of two random variables which are not equal but have the same distributions, consider if $(\Omega,\mathfrak{F},P)=([0,1],\mathfrak{B}_{[0,1]},m_L)$. Let $X$ and $Y$ be positive real valued random variables on $(\Omega,\mathfrak{F},P)$ into the measurable space $([0,1],\mathfrak{B}_{[0... | {
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# Taylor series: Changing point of differentiation
#### sweatingbear
##### Member
Continuing from http://www.mathhelpboards.com/f10/taylor-series-x-=-1-arctan-x-5056/:
The discussion in that thread gave rise to a general question to me: Does not the point of differentiation change when one makes the substitution $$\... | {
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Is there something erroneous in my line of reasoning? I think "changing the point of differenation" from $$\displaystyle x=2$$ from $$\displaystyle h=0$$ makes perfect sense but maybe there is something I am not seeing.
Addition: I suspect there is some kind of mathematical error when I go from $$\displaystyle \left. ... | {
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#### ZaidAlyafey
##### Well-known member
MHB Math Helper
By the way you cannot differentiate with respect to $$\displaystyle x$$ then with respect to $$\displaystyle h$$ because the function $$\displaystyle f$$ is just of one variable if you are referring to the new function after the composition then use another name... | {
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Since taking derivatives of $$\displaystyle g(h)$$ at $$\displaystyle h=0$$ is equivalent to taking derivatives of $$\displaystyle f(x)$$ at $$\displaystyle x=2$$, we can conclude that the coefficients will be the same in either expansion (this is what I was missing earlier, right?).
So, the expansion of $$\displaysty... | {
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Change the chapter
Question
(a) When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850m from the hinges. What torque are you exerting relative to the hinges? (b) Does it matter if you push at the same height as the hinges?
a) $46.8 \textrm{ N} \cdot \textrm{m}$
b) The vertic... | {
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# Unorthodox way of getting the average of two numbers
I can't believe the alternative method I just saw to calculate the average of two numbers:
I use the following:
(a+b)/2 = avrg(a,b)
(b+a)/2 = avrg(a,b)
Found someone using this:
a+((b-a)/2) = avrg(a,b)
b+((a-b)/2) = avrg(a,b)
How to calculate avrg(a,b,c) using... | {
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Observe that $$a+\frac{b-a}{2} = \frac{2a}{2} + \frac{b-a}{2} = \frac{2a+b-a}{2} = \frac{a+b}{2}.$$ You can do the analogous thing for $$b+\frac{a-b}{2} = \frac{a+b}{2}.$$ And for the average of three numbers $a,b,c$, $$\operatorname{avg}(a,b,c) = a + \frac{b-a}{3}+\frac{c-a}{3} = \frac{a+b+c}{3}.$$ You can "switch aro... | {
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$$\bar{x}_n = \frac{1}{n} \sum_{i=1}^{n} x_i$$
i.e. the average of the first $n \le N$ of them.
Then $$\bar{x}_{n+1} = \bar{x}_{n} + \frac{ x_{n+1} - \bar{x}_n}{n+1}$$
I'll leave the proof of this general case to the reader.
This iterative (running) approach for taking the average has advantages when doing numerica... | {
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-
In affine geometry, it is a general property of barycentres that they can be computed using any base point $P$. So if $A_1,\ldots,A_n$ are points and $\lambda_1,\ldots,\lambda_n$ associated weights with nonzero total mass $\mu=\lambda_1+\cdots+\lambda_n$, then the barycentre is $$P+\frac1\mu \left(\lambda_1\overrigh... | {
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Now add the average of the residuals back to the the number you first chose.
This removal of a false origin can also be useful in real engineering calculations as it makes the values and process more easy to comprehend and check.
There are many maths formulas that pre-weight the offset values to avoid confusion about... | {
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# Arbitrarily close uniformly continuous functions
While preparing for my Real Analysis exam I tried to solve the following problem I found:
Problem:
Let $$A \subseteq \mathbb{R}$$ and suppose that $$f: A \rightarrow \mathbb{R}$$ has the following property: for each $$\varepsilon>0$$ there exists a function $$g_{\va... | {
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3. If the solution is incorrect, can someone gently me explain why and provide a solution?
• The proof is correct but the writing can improve. Why talk about $\epsilon'$ which is never used? I think the second sentence in the proof should be deleted. – Kavi Rama Murthy Jan 2 at 5:10
• You are correct. When i first wro... | {
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There are two problems here. First you introduce an actual variable with the phrase "Given $$y \in A$$". But then you make no further use of it. No, really. It is not used anywhere in the proof. All of those other "$$y$$" variables are quantified (that "$$\forall y \in A$$" is called a relative quantifier - drop the "$... | {
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There are no logic issues here. That both $$|f(x) - g_{\varepsilon}(x)| < \frac{\varepsilon}{3}$$ and $$|f(y) - g_{\varepsilon}(y)| <\frac{\varepsilon}{3}$$ follow from the single earlier statement of $$|f(x)-g_{\varepsilon}(x)|<\frac{\varepsilon}{3}, \forall x \in A$$ since both the $$x$$ value used here (which is a d... | {
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# Find symmetric matrix containing no 0's, given eigenvalues
I'm preparing for a final by going through the sample exam, and have been stuck on this:
$$Produce\ symmetric\ matrix\ A ∈ R^{3×3},\ containing\ no\ zeros.\ \\ A\ has\ eigenvalues\ λ_1 = 1,\ λ_2 = 2,\ λ_3 = 3$$
I know $A = S^{-1}DS$, where A is similar to ... | {
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and define
$$A = S^{-1} D S = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & -\frac{2}{\sqrt{6}} & 0\end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} \fra... | {
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Have a nice day :-)
• May I ask what a "run of the mill" orthogonal matrix looks like? – Reccho Jul 5 '18 at 15:51
• Like a Gram-Schmidt orthonormalisation of three random vectors ;-) The forbidden set is of (lebesgue) mesure 0, so it should not give you any trouble. If you are as lazy as I am, take only two vectors, ... | {
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Find integers $m$ and $n$ such that $14m+13n=7$.
The Problem:
Find integers $m$ and $n$ such that $14m+13n=7$.
Where I Am:
I understand how to do this problem when the number on the RHS is $1$, and I understand how to get solutions for $m$ and $n$ in terms of some arbitrary integer through modular arithmetic, like ... | {
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If you can solve for $1$, how can you use this to now solve for $7$ ?
Hint $\$ The set $S$ of integers of the form $\,14m+13n,\ m,n\in\Bbb Z$ are closed under $\color{#c00}{\rm subtraction}$ and closed under $\color{#0a0}{\rm multiplication}$ by any integer. Thus $\,14,13\in S\,\color{#c00}{\Rightarrow \_\_\in S}$ $\,... | {
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# Math Help - Prime/Maximal Ideals
1. ## Prime/Maximal Ideals
I'm working on prime an maximal ideals. My partner and I are studying for our final exam and got conflicting answers.
The question was to find all of the prime and maximal ideals of $\mathbb Z_7$. My answer was that because a finite integral domain is a f... | {
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etc.
3. to sharpen it even further, suppose p,q are distinct primes. what can an ideal of Zp x Zq possibly be? remember Zp x Zq is isomorphic to Zpq, so an ideal has to be a subgroup of the additive group. but (Zpq,+) is cyclic, so any proper subgroup has to be generated by some element of Zpq that doesn't generate th... | {
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# Probability of a Full House for five-card hand.
So I know this can be solved easily by counting the total ways to make a full house and dividing that by the total possible hands, but I want to know why another way I thought of to solve it is wrong.
My calculation is: $$1 \times \frac{3}{51} \times \frac{2}{50} \tim... | {
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In multiplying by $5!$ you forgot that you had already taken into account the permutations of the cards that only interchange the x's and y's amongst themselves. After all, $(1)(3/51)(2/50)$ is the correct probability of drawing three of a kind in a three card hand... you wouldn't multiply that by $3!.$
• Thank you! T... | {
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# Easiest way to solve this system of equations
I have these two equations:
$$x=\frac{ab(1+k)}{b+ka}\\ y=\frac{ab(1+k)}{a+kb}$$
where $a,b$ are constants and $k$ is a parameter to be eliminated.
A relation between $x,y$ is to be found. What is the best way to do it? Cross multiplying and solving is a bit too hectic... | {
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• Nicely done (+1). – dxiv Jul 16 '18 at 4:00
• What step am I missing to get to $\frac{1}{a} + \frac{1}{b}$?$\frac{1}{x} + \frac{1}{y} = \frac{a + ak + b + kb}{ab(1 + k)} \iff \frac{a (1 + k) + b (1 + k)}{ab(1 + k)} \iff \frac{(a + b) (1 + k)}{ab(1 + k)} \iff \frac{(a + b)}{ab} \iff ???$ – Phil Patterson Jul 16 '18 at... | {
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# Is the converse of Cayley-Hamilton Theorem true?
The question is motivated from the following problem:
Let $I\neq A\neq -I$, where $I$ is the identity matrix and $A$ is a real $2\times 2$ matrix. If $A=A^{-1}$, then the trace of $A$ is
$$(A) 2 \quad(B)1 \quad(C)0 \quad (D)-1 \quad (E)-2$$
Since $A=A^{-1}$, $A^2=I$... | {
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Theorem. Let $A$ be an $n\times n$ matrix over $\mathbf{F}$, and let $\mu(x)$ be the minimal polynomial of $A$. If $p(x)\in \mathbf{F}[x]$ is any polynomial such that $p(A)=0$, then $\mu(x)$ divides $p(x)$.
The Cayley-Hamilton Theorem shows that the characteristic polynomial is always a multiple of the minimal polynom... | {
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Why is $A$ diagonalizable? If it has two distinct eigenvalues, $1$ and $-1$, then there is nothing to do; we know it is diagonalizable. If it has a repeated eigenvalue, say $1$, but $A-I$ is not the zero matrix, pick $\mathbf{x}\in \mathbb{R}^2$ such that $A\mathbf{x}\neq \mathbf{x}$; then $$\mathbf{0}=(A-I)^2\mathbf{x... | {
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The possible characteristic polynomials are thus $x^2-1$, $(x-1)^2$, and $(x+1)^2$. To rule out the last two cases, you can consider the triangular forms of $A$. For example, having characteristic polynomial $(x-1)^2$ implies that $A$ is similar to a matrix of the form $\begin{bmatrix}1&a\\0&1\end{bmatrix}$. But then t... | {
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# Why is $-145 \mod 63 = 44$?
When I enter $-145 \mod 63$ into google and some other calculators, I get $44$. But when I try to calculate it by hand I get that $-145/63$ is $-2$ with a remainder of $-19$. This makes sense to me, because $63\cdot (-2) = -126$, and $-126 - 19 = -145$.
So why do the calculators give tha... | {
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Note, though, that you are not wrong in thinking that $-145 \pmod{63} = -19$. When working mod $63$, the numbers $-19$ and $44$ are identical.
• All the answers were good but this one was the most clear, thank you. – Winston Nguyễn Dec 17 '15 at 17:58
Positive $145$ divided by $63$ is $2$ with a remainder of $19$, si... | {
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# Master Theorem $T(n) = 4T(n/2) + \lg n$
In class today, we did the following problem: $T(n)=4T(n/2) + \lg n$
So by notation in CLRS, we have $a = 4$, $b = 2$, $f(n) = \lg n$. Thus, $n^{\log_b a} = n^2$. My algorithm lecturer claimed that it doesn't fit Case 1 of the Master Theorem because "$\lg n$ is not polynomial... | {
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-
Is the relation supposed to hold for all $n$, or just powers of $2$? – 1015 Feb 6 '13 at 18:57
I don't quite understand your question. Which relation are you speaking of? – Kiet Tran Feb 6 '13 at 22:10
The relation in your title and at the first line of your post. – 1015 Feb 6 '13 at 22:15
Oh, any nonnegative $n$ ... | {
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Hope you didn't mind, but I $LaTeX$ed your answer. – Rick Decker Feb 7 '13 at 3:04 | {
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1.6k views
Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$.
1. How many functions are members of $F$?
2. How many functions $f$ in $F$ satisfy the property $f(i)=1$ for some $i, 1\leq i \leq n$?
3. How many functions $f$ in $F$ satisfy... | {
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final mapping ,
example :
A(1,2,3) B(1,2,3,4)
for the satsfying condition f(i)<f(j) .where i , j are from set A and f(i) , f(j) from set B
Total number of such functions are :
1.{(1,1) ,(2,2), (3,3)}
2.{(1,1),(2,2),(3,4)}
3.{(1,1),(2,3),(3,4)}
4.{(1,2),(2,3),(3,4)}
(1,2,3),(1,2,4),(1,3,4),(2,3,4) , is similar... | {
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option C is correct, you have to just select any n number from m which can be done in C(m,n) ways, and coming to the arrangement, that chosen n numbers should be in strictly increasing order, so you have just 1 way to arrange them. Hence if you do selection followed by arrangement it will be C(m,n) * 1, which will be s... | {
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# Fourier transform of $\int_{-\infty}^{t} f(\eta )\text{d}\eta$
Suppose $f(t)$ and $F(\omega)$ are a Fourier transform pair. I want to show that $$\mathcal{F}^{-1} \left\{\frac{F(\omega)}{i\omega}\right\} = \int_{-\infty}^t f(\eta)\ \text{d}\eta$$ I start with the Fourier transform of the RHS and use integration by p... | {
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Let $I_L(\omega,)$ be defined by the integral
$$I_L(\omega)=\int_{-L}^L \int_{-\infty}^tf(t')\,dt'\,e^{-i\omega t}\,dt$$
Integrating by parts with $u=\int_{-\infty}^tf(t')\,dt'$ and $v=\frac{e^{-i\omega t}}{-i\omega}$ yields
$$I_L(\omega)=\frac{1}{i\omega}\int_{-L}^L f(t)e^{-i\omega t}\,dt+\frac{1}{i\omega}\left(e^{... | {
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as was to be shown!
• Hello, thanks for your answer! Just a small typo on the last line, it should be $e^{-i\omega t}$. ;) Sorry, I'm kinda new to this so bear with me here...why must $\lim_{L\to\infty} \int_{-\infty}^{L} f(t')\ \text{d}t$ vanish if $\int_{-\infty}^{t} f(t')\ \text{d}t$ is square integrable? Reading t... | {
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# Limit comparison test
AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.9 (EK)
## Video transcript | {
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let's remind ourselves give ourselves a review of the comparison test see where it can be useful and maybe see where it might not be so useful but luckily we'll also see the limit comparison test which can be applicable in a broader category of situations so we've already seen this we want to prove that the infinite se... | {
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is greater which means that this can't each of these terms can't provide an upper bound on the when this one is a little bit larger the other hand you're like okay I get that but look as n gets large the two to the N it's going to it's going to really dominate the minus one or the plus one or the or but this one has no... | {
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they're either going to converge or they're both going to diverge let's apply that right over here well if we say that our B sub n is 1 over 2 to the N just like we did up there 1 over 2 to the N so we're going to compare so these two series right over here notice it satisfies all of these constraints so let's take the... | {
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AP® is a registered trademark of the College Board, which has not reviewed this resource. | {
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# Center of Mass, Multivariable Calculus
I have a solid with the bounds $z=2x^2+2y^2$ where $z=c$ and this solid has a uniform density of B. I need to find the mass and the center of mass of this solid. I know how to find a normal center of mass, but I do not know how to set up an integral for this problem, but I thin... | {
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$$\mathbf{R} = \frac{1}{M} \iiint \rho \, \mathbf{r} \, dV.$$
It is interesting to note the following: the paraboloid is symmetric around $z$ axis. This means that the center of mass must be in the $z$ axis, for the $\overline{x}$ and $\overline{y}$ will cancel (if you don't believe this, write out the integral explic... | {
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You can parameterize the solid, say $V$, in Cartesian coordinates, $$V = \{ (x, y, z) : 0 \leq z \leq c, 0 \leq x^2 + y^2 \leq \frac12 z \}$$ or in polar coordinates $$V = \{ (x, y, z) = (r \cos \phi, r \sin \phi, z) : 0 \leq z \leq c, 0 \leq r^2 \leq \frac12 z, 0 \leq \phi \leq 2\pi \}.$$
However, by symmetry, it is ... | {
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# "erf"
#### Wilmer
##### In Memoriam
What does erf^(-1)(x) mean?
erf^(-1)(.6) = ?
Is there a value for erf, like there is for pi and e?
is erf^(-1) same as 1/erf?
I found out erf = error function....hmmm....
THANKS for any explanations.
#### quantaentangled
##### New member
This is the 'inverse error function... | {
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$\mbox{erf}(z)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infty\frac{(-1)^n z^{2n+1}}{n! (2n+1)}$
By the Alternating Series Test this series converges. Therefore the error function has a specific value at each point. However closed form expressions for these values may or may not exist. But there are closed form approximations ... | {
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It belongs to the family of functions having McLaurin series of the type...
$\displaystyle w=f(z)= c_{1}\ z + c_{2}\ z^{2} + c_{3}\ z^{3} + ...\ ,\ c_{1} \ne 0$ (2)
... and for them the coefficients inverse function McLaurin expansion...
$\displaystyle z= f^{-1}(w) = d_{1}\ w + d_{2}\ w^{2} + d_{3}\ w^{3}+...$ (3)
... | {
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# Simplify $2^{(n-1)} + 2^{(n-2)} + … + 2 + 1$
Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ I know the answer is $2^n - 1$, but how to simplify it?
• What happens if you add $1$? – Gottfried Helms Dec 3 '13 at 10:35
First way:
$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+a^2b^{n-3}+ab^{n-2}+b^{n-1})$
set a=2,... | {
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$2^{n-1}+\cdots+2^{0}=\left(2-1\right)\left(2^{n-1}+\cdots+2^{0}\right)=\left(2^{n}+\cdots+2^{1}\right)-\left(2^{n-1}+\cdots+2^{0}\right)=2^{n}-1$
Forget about all the smart generic formulas. Just rewrite the last summand $1$ as $2-1$. You get $$2^{n-1} + 2^{n-2} + \ldots + 2^3 + 2^2 + 2 + 2 - 1.$$ Now group the $2$'s... | {
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# Finding closest point from a parabola to a plane
1. Dec 6, 2013
### mahler1
The problem statement, all variables and given/known data.
Find the point in the parabola $y^2=x$, $z=0$ closest to the plane $z=x+2y+8$
The attempt at a solution.
I've solved some problems where I had to find the closest point from a g... | {
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Last edited: Dec 6, 2013
3. Dec 6, 2013
### HallsofIvy
Another way to do it is to use the geometric fact that the "shortest distance" to a plane is always along a line perpendicular to the plane. We can write this plane as x+ 2y- z= -8 so it has <1, 2, -1> as perpendicular vector. A line through $(x_0, y_0, 0)$ in th... | {
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+0
# Help
+1
100
4
+628
find all groups of three regular polygons with side length one that can surround one point such that each polygon shares a side with the other two.
supermanaccz Sep 20, 2018
### Best Answer
#1
+970
+5
Let there be three regular polygons with $$x,y,$$ and $$z$$ as their number of sides. T... | {
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Therefore, there are 9 groups of regular polygons that can surrond a point.
I hope this helped,
Gavin.
GYanggg Sep 20, 2018
#2
+92787
+2
Very nice, Gavin !!!
CPhill Sep 20, 2018
#3
+1
"Their internal angles must sum to 360"
why?
EDIT: thanks for the diagram melody :) now i understand. I thought all polygonal... | {
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## Why do we need to work with the logarithm of the mathematical objective functions?¶
In optimization and Monte Carlo sampling problems, since the mathematical objective functions (e.g., probability density functions) can take extremely small or large values, we often work with their natural logarithms instead. This ... | {
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• The evaluation of this function involves a log(exp()) term in its definition. If the origin of the exp() term is not clear to you, take a look at the definition of the MVN distribution in the equation provided in the above. This is computationally very expensive and in general, is considered a bad implementation.
• T... | {
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In [15]:
import numpy as np
NDIM = 4 # the number of dimensions of the domain of the objective function: MVN
MEAN = np.zeros(NDIM) # This is the mean of the MVN distribution.
COVMAT = np.eye(NDIM) # This is the covariance matrix of the MVN distribution.
INVCOV = np.linalg.inv(COVMAT) # This is the inverse... | {
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Out[11]:
-inf
Now, this may seem like being too meticulous, but, a good fault-tolerant implementation of the objective function is absolutely essential in Monte Carlo simulations, and this example objective function here is no exception. The -inf value that getLogFunc_bad() yields, will certainly lead to a severe cata... | {
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# Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x$?
Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x$?
I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.
Thank you.
-
How do you define $x^{\frac12}$... | {
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But there is more to it than that. Further mathematical developments, which you may not have seen yet, confirm these choices. For example, one shows in analysis that as one adds more and more terms of the infinite sum $$1 + x + \frac{x^2}2 + \frac{x^3}6 + \frac{x^4}{24} + \cdots$$ the sum more and more closely approach... | {
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For $x\geq 0$ and by definition $\sqrt x$ is the positive real $y$ such that $y^2=x$ and since $$\left(x^p\right)^q=x^{pq}$$ then for $p=\frac1 2$ and $q=2$ we see that $x^{1/2}=\sqrt x$
- | {
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Hello,
Say f(x) is defined only for x in [a, ∞].
lim x->a+ f(x) = c and
lim x->a- f(x) obviously doesn't exist.
Do we say that lim x->a f(x) exists or not?
Thanks.
Last edited:
arildno
Homework Helper
Gold Member
Dearly Missed
What do you think?
Not sure. If the function is defined only for, say, x in [a, ∞], woul... | {
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An excellent objection!
What you show with your objection is the concern that whether or not a limit exists can depend, in a CRUCIAL way, on what the domain the variable is said to "live in".
IF, as you you object, x only lives in the region between "a" and positive infinity, then the limit most definitely does exist (... | {
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# Inclusion in the ring of integers
Let $$K= \mathbb Q(\sqrt3,\sqrt7)$$. I am ask to show that $$\mathcal O_K \ne \mathbb Z[\sqrt3,\sqrt7]$$, where $$\mathcal O_K$$ is the ring of integers.
How can i find $$\mathcal O_K$$ is there a general method on how can i find it? I need help, any hints or links similar to this ... | {
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Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$
Algebraic Integers of $\mathbb Q[\sqrt{3},\sqrt{5}]$
Ring of integers of $\mathbb{Q}(\sqrt{-3},\sqrt{5})|\mathbb{Q}$ and group of units
• Thank you very much! – Ralph John Feb 22 at 10:29
• You are welcome! – Dietrich Burde Feb 22 at 10:29
As has already been me... | {
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I tried for example $$Q(\sqrt{-3} + \sqrt{-7})$$. From there, I computed $$\left(\frac{1}{2} + \frac{\sqrt{-3}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{-7}}{2}\right) = \frac{1}{4} + \frac{\sqrt{-3}}{4} + \frac{\sqrt{-7}}{4} + \frac{\sqrt{21}}{4}.$$ I seem to have made a mistake somewhere along the way: this number's... | {
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# Probability of a run of 3 heads when I flip a coin $n$ times
I'm wondering if there is a nice solution for this problem. As stated, I flip a coin $n$ times, and want the probability of a run of 3 (or more) heads appearing within it. For example, if I toss a coin 9 times, an example that would include a run of 3 head... | {
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$$g_m(s)=\frac{s^m(2-s)}{2^{m+1}(1-s)+s^{m+1}}$$
For example, for $m=1$, $$g_1(s)=\frac{s^2}{2-s}$$ that is, $T_1$ is a shifted geometric random variable. More generally, recall that each function $g_m$ fully encodes the distribution of $T_m$ since, by definition, $$g_m(s)=\sum_{k=0}^\infty P(T_m=k)s^k$$ thus, expandi... | {
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"lm_q1q2_score": 0.8478342520526447,
"lm_q2_score": 0.8615382147637196,
"openwebmath_perplexity": 135.75946625984267,
"openwebmath_score": 0.9322981834411621,
"ta... |
Let $a_n$ be the number of sequences of length $n$ which do not contain three consecutive heads. Since it is not possible for three consecutive heads to occur until the coin has been tossed three times, $a_1 = 2$ and $a_2 = 4$. Since the only outcome in which three consecutive heads occurs when the coin is tossed three... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8478342520526447,
"lm_q2_score": 0.8615382147637196,
"openwebmath_perplexity": 135.75946625984267,
"openwebmath_score": 0.9322981834411621,
"ta... |
$$P(\bigcup_{k=1}^{n-2} \bigcap_{i=0}^2 H_{k+i}) = P(\bigcup_{k=1}^{n-2} A_k)$$
$$= \sum_{i=1}^{n-2} P(A_i) - \sum_{1 \le i < j \le n-2} P(A_i \cap A_j) + \sum_{1 \le i < j < k \le n-2} P(A_i \cap A_j \cap A_k)$$
Now, $P(A_i \cap A_j \cap A_k) = 0$ unless $i+2=j+1=k$ in which case we have $P(A_i \cap A_j \cap A_k) = ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8478342520526447,
"lm_q2_score": 0.8615382147637196,
"openwebmath_perplexity": 135.75946625984267,
"openwebmath_score": 0.9322981834411621,
"ta... |
# Abs Value of X-Continuous Debate
1. Sep 11, 2005
### Tom McCurdy
Question:
Is $$f(x)=\mid{x}\mid$$ continuous?
I have been looking online and got a few different answers. My calc B.C. teacher last year claimed that $$f(x)=\mid{x}\mid$$ is continuous everywhere except at x=0. My current 115 teacher maintains that ... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8478342520526447,
"lm_q2_score": 0.8615382147637196,
"openwebmath_perplexity": 1578.1391605219733,
"openwebmath_score": 0.7094414830207825,
... |
7. Sep 12, 2005
### HallsofIvy
Staff Emeritus
Or maybe you misunderstood! f(x)= |x| is continuous at x= 0 but not differentiable there.
8. Sep 13, 2005
### Tom McCurdy
Thanks, it probably is my memory since it was from last year... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936101542133,
"lm_q1q2_score": 0.8478342520526447,
"lm_q2_score": 0.8615382147637196,
"openwebmath_perplexity": 1578.1391605219733,
"openwebmath_score": 0.7094414830207825,
... |
# What is the variance of the mean of correlated binomial variables?
An average of $B$ binomial i.i.d. random variables, each with variance $\sigma^2,$ has variance $\frac{1}{B}\sigma^2.$
If the variables are simply i.d. (identically distributed, but not necessarily independent) with positive pairwise correlation $\r... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936106207203,
"lm_q1q2_score": 0.8478342507052716,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 532.7428265084549,
"openwebmath_score": 0.9960139393806458,
"tag... |
$$\text{Cov}(\lambda X, \lambda X) = \lambda^\prime \Sigma \lambda.$$
The rest is just arithmetic.
In the present case $\sigma_{ij} = \rho\sigma^2$ when $i\ne j$ and otherwise $\sigma_{ii} = \sigma^2 = \left[\rho + (1-\rho)\right]\sigma^2$. That is to say, we may view $\Sigma$ as the sum of two simple matrices: one h... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936106207203,
"lm_q1q2_score": 0.8478342507052716,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 532.7428265084549,
"openwebmath_score": 0.9960139393806458,
"tag... |
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