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If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?
Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two co... | {
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Each of these 99 expectations is simply the probability that $n$ and $n+1$ are both chosen, which is $\frac{25}{100}\frac{24}{99}$.
So the expected number of pairs is $99\frac{25}{100}\frac{24}{99} = 6$.
• Can it be argued that you also need to consider the $n-1$ as a possible pair candidate? And how would that affec... | {
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$$X = X_1 + X_2 + \ldots + X_{99},$$
and using the linearity of the expectation, we get
\begin{align} E[X] &= E[X_1] + E[X_2] + \cdots +E[X_{99}]\\ &= 99E[X_1]\\ &= 99\frac{25}{100}\frac{24}{99} = 6 \end{align}
Henning Malcolm has already answered the question about the expected value. But that says very little abou... | {
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histogram(num_pairs)
title(sprintf('histogram of seating pairs (out of %d trials)',num_trials))
• Can you run the same simulations for number of maximal clumps of size at least 2 and compare variances and ratio of variances to the means (to see deviations from Poisson distribution)?
– A.S.
Dec 10 '15 at 10:27
• +1 for... | {
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$$[z^{100}] \frac{1}{1-z} G(z, u) = [z^{100}] \frac{1}{1-z} \frac{z}{1-z} \left(uz + \frac{z^2}{1-z}\right)^{24}.$$
We need the total count of the number of adjacent pairs so we compute $$[z^{100}] \left.\frac{\partial}{\partial u} \frac{1}{1-z} G(z, u) \right|_{u=1} =[z^{100}] \left. \frac{z}{(1-z)^2} \times 24 \left... | {
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For a run length at most $k$ we must first choose a run of some length possibly preceded by a gap, giving (the variable $w$ counts the number of elements in the tuple)
$$\frac{1}{1-z} \sum_{q=1}^k w^q z^q$$
followed by a sequence of gaps followed by a run $$\frac{z}{1-z} \sum_{q=1}^k w^q z^q$$
possibly followed at t... | {
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This is $$\sum_{p=0}^{\lfloor m/(k+1)\rfloor} {n-m+1\choose p} (-1)^p {n-m+m-p(k+1)\choose n-m} \\ = \sum_{p=0}^{\lfloor m/(k+1)\rfloor} {n-m+1\choose p} (-1)^p {n-p(k+1)\choose n-m}.$$
Substitute this into the summation formula to obtain $$m {n\choose m} - \sum_{k=1}^{m-1} \sum_{p=0}^{\lfloor m/(k+1)\rfloor} {n-m+1\c... | {
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The reader is invited to prove this in a different way, perhaps without using ordinary generating functions.
The Maple code that was used to verify these including a total enumeration routine was as follows:
with(combinat);
sr :=
proc(n, m)
local sel, sel2, maxr, cur, res, pos;
sel := firstcomb(n, m);
res := 0;
wh... | {
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# why is showing that $f(H) \subseteq H$ the same as showing that $(y \in H \implies f(y) \in H)$?
While I was reading a proof about the diagonalizability of symmetric matrices I got a bit lost the author was supposed to show that $f(H) \subseteq H$ but he ended up showing that
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I don't know why you wrote that “$f(y)$ is practically the same thing as $f(H)$ since $y\in H$”. It happens that $f(y)$ is the image of $y$, whereas $f(H)$ is the set of all images of all elements of $H$. So, yes, asserting that $f(H)\subseteq H$ actually is the same thing as asserting that $(\forall y\in H):f(y)\in H$... | {
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# Substitution Rule in Indefinite Integrals
1. Jul 22, 2010
### phillyolly
1. The problem statement, all variables and given/known data
Please explain how to use the substitution rule in indefinite integrals. I am unable even to start the problem.
2. Relevant equations
3. The attempt at a solution
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$$\int sec(2t)tan(2t)dt = \frac{1}{2}\int 2sec(2t)tan(2t)dt = \frac{1}{2}\int du= \frac{u}{2} + C = \frac{sec(2t)}{2} + C$$
9. Jul 22, 2010
### phillyolly
Thank you very much!
I am also stuck on this one. ex has been always tricky for me.
#### Attached Files:
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5.7 KB
Views:
133
10. Jul 22, 2010
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# Question on modulo
Find the last two digits of $3^{2002}$.
How should I approach this question using modulo? I obtained 09 as my answer however the given answer was 43.
My method was as follows:
$2002\:=\:8\cdot 250+2$
$3^{2002}=\:3^{8\cdot 250+2}\:=\:3^{8\cdot 250}\cdot 3^2$
The last two digits is the remainde... | {
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# Pseudocompact spaces with regular G-delta diagonals
This post complements two results discussed in two previous blog posts concerning $G_\delta$-diagonal. One result is that any compact space with a $G_\delta$-diagonal is metrizable (see here). The other result is that the compactness in the first result can be rela... | {
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The space $X$ is said to have a regular $G_\delta$-diagonal if the diagonal $\Delta$ is a regular $G_\delta$-set in $X \times X$, i.e. $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$ where each $U_n$ is an open subset of $X \times X$ with $\Delta \subset U_n$. If $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$, then $\Delta=... | {
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Choose open $V$ with $x \in V \subset \overline{V} \subset U$. Consider the sequence $\{ O_n \cap (X-\overline{V}) \}$. This is a decreasing sequence of non-empty open subsets of $X$. By Theorem 2 in this previous post, $\bigcap \overline{O_n \cap (X-\overline{V})} \ne \varnothing$. Let $y$ be a point in this non-empty... | {
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Theorem 3 follows from Theorem 1.4 in [1], which states that for any $T_0$-space $X$, $X$ is metrizable if and only if there exists a sequence $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3,\cdots$ of open covers of $X$ such that for each open $U \subset X$ and for each $x \in U$, there exist an open $V \subset X$ and an... | {
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We want to show that $V \cap St(x,\mathcal{U}_n)=\varnothing$, which implies that $y \notin \overline{St(x,\mathcal{U}_n)}$. Suppose $V \cap St(x,\mathcal{U}_n) \ne \varnothing$. This means that $V \cap W \ne \varnothing$ for some $W \in \mathcal{U}_n$ with $x \in W$. Then $(U \times V) \cap (W \times W) \ne \varnothin... | {
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We claim that $St(y,\mathcal{U}_k) \cap St(St(x,\mathcal{U}_k), \mathcal{U}_k)=\varnothing$. Suppose not. Choose $w \in St(y,\mathcal{U}_k) \cap St(St(x,\mathcal{U}_k), \mathcal{U}_k)$. It follows that $w \in B$ for some $B \in \mathcal{U}_k$ such that $B \cap St(x,\mathcal{U}_k) \ne \varnothing$ and $B \cap St(y,\math... | {
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Example
Any submetrizable space has a $G_\delta$-diagonal. The converse is not true. A classic example of a non-submetrizable space with a $G_\delta$-diagonal is the Mrowka space (discussed here). The Mrowka space is also called the psi-space since it is sometimes denoted by $\Psi(\mathcal{A})$ where $\mathcal{A}$ is ... | {
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Several Classes of Spaces
The notion “every compact subspace is metrizable” is a very broad class of spaces. It includes well known spaces such as Sorgenfrey line, Michael line and the first uncountable ordinal $\omega_1$ (with the order topology) as well as Moore spaces. Certain function spaces are in the class “ever... | {
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The implication $1 \Longrightarrow 2$ is clear. For $5 \Longrightarrow 2$, see Lemma 1 in this previous post on countable network. The implication $2 \Longrightarrow 3$ is left as an exercise. To see $3 \Longrightarrow 4$, let $K$ be a compact subset of $X$. The property of having a $G_\delta$-diagonal is hereditary. T... | {
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Proof
The proof here actually shows more than is stated in the theorem. We show that $C_p(X)$ is submetrizable by a separable metric topology. Let $Y$ be a countable dense subspace of $X$. Then $C_p(Y)$ is metrizable and separable since it is a subspace of the separable metric space $\mathbb{R}^{\omega}$. Thus $C_p(Y)$... | {
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The key fact for Corollary 2 is that the product of continuum many separable spaces is separable (this fact is discussed here). Theorem 1 is actually a special case of a deep result.
Theorem 3
Suppose that $X=\prod_{\alpha<\kappa} X_\alpha$ is a product of separable spaces where $\kappa$ is any infinite cardinal. Then... | {
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$\text{ }$
$\text{ }$
$\text{ }$
Reference
1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
2. Tkachuk V. V., A $C_p$-Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.
$\text{ }$
$\text{ }$
$\... | {
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$\text{compact} \Longrightarrow \text{countably compact} \Longrightarrow \text{pseudocompact}$
None of the implications can be reversed. An example of a pseudocompact space that is not countably compact is the space $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of subsets of $\omega$ (see... | {
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Theorem 1 says that any continuous one-to-one map from a compact space onto another compact space is a homeomorphism. To show a given map between two compact spaces is a homeomorphism, we only need to show that it is continuous in one direction. Theorem 2, the statement used in the proof of Theorem II.6.2 in [1], says ... | {
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# Looking for non-normal subspaces of the square of a compact X
A theorem of Katetov states that if $X$ is compact with a hereditarily normal cube $X^3$, then $X$ is metrizable (discussed in this previous post). This means that for any non-metrizable compact space $X$, Katetov’s theorem guarantees that some subspace o... | {
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Examples of compact non-metrizable spaces
The set-theoretic result presented here is usually motivated by looking at Theorem 3. The question is: Can $X^3$ in Theorem 3 be replaced by $X^2$? We take a different angle of looking at some standard compact non-metric spaces and arrive at the same result. The following is a... | {
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In the small sample of compact non-metrizable spaces just highlighted, the failure of hereditary normality occurs in “dimension” 1 or 2. Naturally, one can ask:
Question. Is there an example of a compact non-metrizable space $X$ such that the failure of hereditary nornmality occurs in “dimension” 3? Specifically, is t... | {
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In 2001, Larson and Todorcevic [3] showed that it is consistent with ZFC that every compact $X$ with hereditarily normal $X^2$ is metrizable. In other words, there is a model of set theory that is consistent with ZFC in which Theorem 3 can be improved to assuming $X^2$ is hereditarily normal. Thus it is impossible to s... | {
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# When a product space is hereditarily normal
When the spaces $X$ and $Y$ are normal spaces, the product space $X \times Y$ is not necessarily normal. Even if one of the factors is metrizable, there is still no guarantee that the product is normal. So it is possible that the normality of each of the factors $X$ and $Y... | {
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____________________________________________________________________
Katetov’s theorems
Theorem 1
If $X \times Y$ is hereditarily normal (i.e. every one of its subspaces is normal), then one of the following condition holds:
• The factor $X$ is perfectly normal.
• Every countable and infinite subset of the factor $Y... | {
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Since $V$ is open in the product $X \times Y$, choose open sets $A \subset X$ and $B \subset Y$ such that $(t,p) \in A \times B$ and $A \times B \subset V$. With $p \in B$, there exists some $j$ such that $t_j \in B$. First, $(t,t_j) \in V$. Since $t \in U_j$, $(t,t_j) \in U$. Thus $U \cap V \ne \varnothing$. This comp... | {
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____________________________________________________________________
Reference
1. Engelking R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
____________________________________________________________________
$\copyright \ 2015 \text{ by Dan Ma}$
# Compact metrizable scattered ... | {
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Let’s say the compact space $X$ is countable and infinite. Then the function space $C_p(X)$ is metrizable since it is a subspace of $\mathbb{R}^X$, a product of countably many lines. Thus the function space $C_p(X)$ has the Frechet-Urysohn property (being metrizable implies Frechet-Urysohn). This means that the compact... | {
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____________________________________________________________________
Compact metrizable spaces
We first define some notions before looking at compact metrizable spaces in more details. Let $X$ be a space. Let $A \subset X$. Let $p \in X$. We say that $p$ is a limit point of $A$ if every open subset of $X$ containing ... | {
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• $p_0 \in U_0$,
• $p_1 \in U_1$,
• $K_0=\overline{U_0}$ and $K_1=\overline{U_1}$,
• $K_0 \cap K_1 = \varnothing$,
• the diameters for $K_0$ and $K_1$ with respect to $\rho$ are less than 0.5.
Note that each of these open sets contains infinitely many points of $X$. Then we can pick two points in each of $U_0$ and $U_... | {
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To show the first point, we define a one-to-one function $f: \left\{0,1 \right\}^N \rightarrow C$ where $N=\left\{1,2,3,\cdots \right\}$. Note that each element of $\left\{0,1 \right\}^N$ is a countably infinite string of 0’s and 1’s. For each $\tau \in \left\{0,1 \right\}^N$, let $\tau \upharpoonright n$ denote the st... | {
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To see the second point, pick $x \in C$. Suppose $x=f(\tau)$ where $\tau \in \left\{0,1 \right\}^N$. Consider the open sets $U_{\tau \upharpoonright n}$ for all positive integers $n$. Note that $x \in U_{\tau \upharpoonright n}$ for each $n$. Based on the induction process described earlier, observe these two facts. Th... | {
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____________________________________________________________________
Remarks
A corollary to the above discussion is that the cardinality for any compact metrizable space is either countable (including finite) or continuum (the cardinality of the real line). There is nothing in between or higher than continuum. To see... | {
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# Cp(omega 1 + 1) is monolithic and Frechet-Urysohn
This is another post that discusses what $C_p(X)$ is like when $X$ is a compact space. In this post, we discuss the example $C_p(\omega_1+1)$ where $\omega_1+1$ is the first compact uncountable ordinal. Note that $\omega_1+1$ is the successor to $\omega_1$, which is ... | {
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____________________________________________________________________
Connection to $\Sigma$-product
We show that the function space $C_p(\omega_1+1)$ is homeomorphic to a subspace of the $\Sigma$-product of $\omega_1$ many copies of the real lines. Let $Y_0$ be the following subspace of $C_p(\omega_1+1)$:
$Y_0=\left... | {
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$H_\alpha=\left\{f \in C_p(\omega_1+1): f(\gamma)=0 \ \forall \ \alpha<\gamma<\omega_1 \right\}$
Clearly $H_\alpha \subset Y_0$. Furthermore $H_\alpha$ can be considered as a subspace of $\mathbb{R}^\omega$ and is thus metrizable. Let $A$ be a countable subset of $C_p(\omega_1+1)$. Then $h(A) \subset H_\alpha \times \... | {
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A space $X$ is strongly $\tau$-monolithic if for any $A \subset X$ with $\lvert A \lvert \le \tau$, we have $w(\overline{A}) \le \tau$. A space $X$ is strongly monolithic if it is strongly $\tau$-monolithic for all infinite cardinal $\tau$. It is straightforward to show that $X$ is strongly monolithic if and only if fo... | {
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For any compact space $X$, $C_p(X)$ is countably tight (see this previous post). In the case of the compact uncountable ordinal $\omega_1+1$, $C_p(\omega_1+1)$ has the stronger property of being Frechet-Urysohn. A space $Y$ is said to be a Frechet-Urysohn space (also called a Frechet space) if for each $y \in Y$ and fo... | {
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Corollary 2
Let $X$ be a compact space. Then the function space $C_p(X)$ is monolithic.
However, the strong monolithicity of $C_p(\omega_1+1)$ does not hold in general for $C_p(X)$ for compact $X$. As indicated above, $C_p([0,1])$ is monolithic but not strongly monolithic. The following theorem is Theorem II.7.9 in [1... | {
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____________________________________________________________________
$\copyright \ 2014 \text{ by Dan Ma}$ | {
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 4.4: Triangle Congruence Using ASA, AAS, and HL
Difficulty Level: At Grade Created by: CK-12
## Learning Objectives
• Use the ASA Congruence Postulate, AAS Congruence Theorem, and th... | {
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1. Draw the side (5 in) horizontally, halfway down the page. The drawings in this investigation are to scale.
2. At the left endpoint of your line segment, use the protractor to measure the \begin{align*}45^\circ\end{align*} angle. Mark this measurement and draw a ray from the left endpoint through the \begin{align*}45... | {
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Given: \begin{align*}\angle C \cong \angle E, \overline{AC} \cong \overline{AE}\end{align*}
Prove: \begin{align*}\triangle ACF \cong \triangle AEB\end{align*}
Statement Reason
1. \begin{align*}\angle C \cong \angle E, \overline{AC} \cong \overline{AE}\end{align*} Given
2. \begin{align*}\angle A \cong \angle A\end{ali... | {
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a) ASA?
b) AAS?
c) SAS?
Solution:
a) For ASA, we need the angles on the other side of \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{QR}\end{align*}. Therefore, we would need \begin{align*}\angle F \cong \angle Q\end{align*}.
b) For AAS, we would need the angle on the other side of \begin{alig... | {
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Prove: \begin{align*}\triangle CBD \cong \angle ABD\end{align*}
Solution:
Statement Reason
1. \begin{align*}\overline{BD}\end{align*} is an angle bisector of \begin{align*}\angle CDA, \angle C \cong \angle A\end{align*} Given
2. \begin{align*}\angle CDB \cong \angle ADB\end{align*} Definition of an Angle Bisector
3. ... | {
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Example 6: What information would you need to prove that these two triangles are congruent using the: a) HL Theorem? b) SAS Theorem?
Solution:
a) For HL, you need the hypotenuses to be congruent. So, \begin{align*}\overline{AC} \cong \overline{MN}\end{align*}.
b) To use SAS, we would need the other legs to be congru... | {
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Yes
\begin{align*}\triangle ABC \cong \triangle LKM\end{align*}
SAS
Yes
\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}
ASA
Yes
\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}
AAS (or SAA)
Yes
\begin{align*}\triangle ABC \cong \triangle YZX\end{align*}
HL
Yes, Right Triangles Only
\... | {
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Prove Move: At the beginning of this chapter we introduced CPCTC. Now, it can be used in a proof once two triangles are proved congruent. It is used to prove the parts of congruent triangles are congruent in order to prove that sides are parallel (like in Example 8), midpoints, or angle bisectors. You will do proofs li... | {
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1. From \begin{align*}\overline{LP} \ || \ \overline{NO}\end{align*}, which angles are congruent and why?
2. From looking at the picture, what additional piece of information can you conclude?
3. Write a 2-column proof to prove \begin{align*}\triangle LMP \cong \triangle OMN\end{align*}.
4. What would be your reason fo... | {
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3. (a) Yes, CPCTC
(b) No, these sides do not line up in the congruence statement.
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Arc Length: Definite and Indefinite Integration
Main Question or Discussion Point
Several authors state the formula for finding the arc length of a curve defined by $y = f(x)$ from $x=a$ to $x=b$ as:
$$\int ds = \int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$$
Isn't this notation technically wrong, since the RHS is a defini... | {
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lavinia
Gold Member
What does carrying out indefinite integration with respect to s mean?
$$\int ds = s + C$$
For $C ∈ ℝ$
What does the right hand side represent?
All I am saying is that the left hand side does not mean an indefinite integral. Perhaps it is an abuse of notation but the integration bounds are implici... | {
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# quick sort program in c with first element as pivot
## 13 Dec quick sort program in c with first element as pivot | {
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brightness_4 QuickSort is a sorting algorithm, which is commonly used in computer science. Quicksort then proceeds recursively calling itself on $V_{\lt}$ and $V_{\gt}$, thus assuming to get those two back with their values sorted. C++ code I also acknowledge this is the simpler and less efficient Lomuto's partition. I... | {
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there is only one element left at both the side of the pivot. When you take a pivot element and sort all the elements based on that,u need to call quick sort for left group and right group.J is pivot element … Consider an array which has many redundant elements. The best case is when the pivot element will be the middl... | {
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strategy like merge sort. There are various ways to choose pivot element: Chose pivot as first element. It picks an element as pivot and partitions the given array around the picked pivot. We are going to always select the last element of the array as the pivot in our algorithm and focus mainly on the concepts behind t... | {
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C, Merge two sorted arrays in O(1) extra space using QuickSort partition, Count all distinct pairs with difference equal to k, Maximum and minimum of an array using minimum number of comparisons, Divide and Conquer Algorithm | Introduction, Closest Pair of Points using Divide and Conquer algorithm, Time Complexities of... | {
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Can we implement QuickSort Iteratively? We first pick a pivot element. Example: [17, -10, 7, 19, 21, 23, -13, 31, 59]. After the partition, quick sort calls itself recursively to sort the sub-arrays. The first step of doing a partition is choosing a pivot. I'm studying Quick-Sort and I am confused as to how it works wh... | {
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where n is the number of elements. So far and choose the last term is for the size of the quicksort technique is done separating... Complexity of O ( n ) extra space for linked lists call optimizations is done by the! Of variables of the string ( multikey ) has expected time complexity partition process always the..., ... | {
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ascending or descending order ) of (! Access as elements are continuous in memory element may be an element is chosen the... The remainder of the array-like first, we swap current element with arr [..! Are performed recursively until there is only one element left at both the side of the is! Lomuto 's partition or desc... | {
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Redhead Everest Iii Hiking Boots For Ladies, Prisoner's Dilemma Game, Pitbull Bite Force Psi, Mx Linux Store, Pita Way Order Online, East Fishkill Ny Zip Code, Alberta Association Of Architects Directory, Craigslist House For Rent Miami Gardens, Vermont Slate Company, Dallas Uptown Zip Code, | {
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Does the volatility of a Currency Pair depend on the currency in which the premium is paid? For example- will the Volatility of USDJPY change if the premium is paid in USD instead of JPY. Is there any mathematical formulation for this?
It depends, on what you mean by returns. For simple returns: no, for log returns ye... | {
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The standard deviation of sample is equal to standard deviation of the mirrored around its mean.
• I think in your last code block, you meant to type sd(log(1/returns)) for the second command -- though that does not affect the answer. Jul 31 '20 at 21:56
• Thanks, fixed it! Aug 1 '20 at 5:39
If you're modelling the F... | {
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• You are missing a final measure change here to bring everything into alignment - as the sde you have written is still under the domestic numeraire and not the foreign numeraire. A 1 year outright forward on USDEUR is just the inverse of 1 year outright on EURUSD. There is no convexity adjustment required. Sep 16 '20 ... | {
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# Proving that $C$ is a subset of $f^{-1}[f(C)]$
More homework help. Given the function $f:A \to B$. Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.
Prove that:
$C$ is a subset of $f^{-1}[f(C)]$
So I have to show that every element of $C$ is in the set $f^{-1}[f(C)]$
I know that $f(C)$ is the image of $... | {
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• So it was all about using the definition of the preimage along the way to prove my assumption. It is always so clear in hindsight. Hopefully this gets easier before I get to Abstract Algebra and Real Analysis... – Ben Anderson Apr 1 '13 at 5:11
• @Ben: You’ll find that a very large fraction of the more elementary exe... | {
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This can be rewritten:
$$f^*(y) \supseteq \{x\} \iff \{y\} \supseteq \{f(x)\}$$
This can be used to prove:
Proposition. For all subsets $B$ of $Y$ and all subsets $A$ of $X$, we have:
$$f^*(B) \supseteq A \iff B \supseteq f_*(A)$$
Once you've proved this, your problem becomes a one-line proof:
$$f^{*}(f_*(A)) \su... | {
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# Prove middle term formula
#### Mondo
##### New member
Hey,
How can we prove the middle term formula of the arithmetic series with odd number of elements is (n+1)/2. So lets consider the series 3 + 5 + 7 + 9 + 11, we have 5 elements and the middle term is 7 which can be obtained by (n+1)/2 but how to prove this for... | {
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#### lev888
##### Senior Member
Yes (n+1)/2 is for the index but you can get the value as well by adding first and the last term -> (a_1+a_n)/2. I am still not convinced about the proof. How to show that the value of the middle term can be obtained by above formula? It all started from the sum of a arithmetic series -... | {
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The other question is about adding an arithmetic sequence, a+ (a+ d)+ (a+ 2d)+ ....+ (a+ nd).
There are n "a" and there sum is na. The rest, d+ 2d+ .... +nd= d(1+ 2+ ...+ n). We have already seen that 1+ 2+ ... + n is n(n+1)/2 so the total is na+ n(n+1)/2 and the average is a+ (n+1)/2.
#### Mondo
##### New member
Ple... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
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"lm_q2_score": 0.8757869803008764,
"openwebmath_perplexity": 413.20089830722316,
"openwebmath_score": 0.7663042545318604,
"tag... |
#### lev888
##### Senior Member
I want to prove the arithmetic series formula using the fact that its elements taken in pairs in a certain way all sum to the same number. The equation for each term of this sequence is a_n = a_1 + (n-1)d. So for a_1 = 3, d = 2 and n = 5 we get a series Sn = 3 + 5 + 7 + 9 + 11 By empiri... | {
"domain": "freemathhelp.com",
"id": null,
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"lm_name": "Qwen/Qwen-72B",
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"lm_q1q2_score": 0.8405471160529613,
"lm_q2_score": 0.8757869803008764,
"openwebmath_perplexity": 413.20089830722316,
"openwebmath_score": 0.7663042545318604,
"tag... |
#### JeffM
##### Elite Member
I want to prove the arithmetic series formula using the fact that its elements taken in pairs in a certain way all sum to the same number. The equation for each term of this sequence is an=a1+(n−1)dan=a1+(n-1)d. So for a1=3a1=3, d=2d=2 and n=5n=5 we get a series Sn=3+5+7+9+11Sn=3+5+7+9+11... | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620596782468,
"lm_q1q2_score": 0.8405471160529613,
"lm_q2_score": 0.8757869803008764,
"openwebmath_perplexity": 413.20089830722316,
"openwebmath_score": 0.7663042545318604,
"tag... |
$$\displaystyle \text {Prove that if } n \text { is an odd positive integer, then}\\ \left (\displaystyle \sum_{j=1}^n a + (j - 1)d \right ) = n\{a + (m - 1)d\}, \text { where } m = \dfrac{n + 1}{2}.$$
First, lets demonstrate that a + (m - 1)d is the middle term. Because n is odd, let x be the number of terms that prec... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
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"lm_q1q2_score": 0.8405471160529613,
"lm_q2_score": 0.8757869803008764,
"openwebmath_perplexity": 413.20089830722316,
"openwebmath_score": 0.7663042545318604,
"tag... |
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