text:
### Quick Summary (Probability)

University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.

Random Variables

Suppose $X$ is a random variable which can takes values $x \in \chi$.

$X$ is a discrete r.v. is $\chi$ is countable.
$\Rightarrow p(x)$ is the probability of a value of $x$ and is called the probability mass function.

$X$ is a continuous r.v. is $\chi$ is uncountable.
$\Rightarrow f(x)$ is the probability density function and can be thought of as the probability of a value $x$.

Probability Mass Function

For a discrete r.v. the probability mass function (PMF) is

$p(a) = P(X=a)$, where $a \in \mathbb{R}$.

Probability Density Function

If $B = (a, b)$

$P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx$.

And strictly speaking,

$P(X = a) = \int_a^a f(x) ~dx = 0$.

Intuitively,

$f(a) = P(X = a)$.

Properties of Distributions

For discrete r.v.
$p(x) \ge 0 ~ \forall x \in \chi$.
$\sum_{x \in \chi} p(x) = 1$.

For continuous r.v.
$f(x) \ge 0 ~ \forall x \in \chi$.
$\int_{x \in \chi} f(x) ~dx = 1$.

Cumulative Distribution Function

For discrete r.v., the Cumulative Distribution Function (CDF) is
$F(a) = P(X \le a) = \sum_{x \le a} p(x)$.

For continuous r.v., the CDF is
$F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx$.

Expected Value

For a discrete r.v. X, the expected value is
$\mathbb{E} (X) = \sum_{x \in \chi} x p(x)$.

For a continuous r.v. X, the expected value is
$\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx$.

If $Y = g(X)$, then

For a discrete r.v. X,
$\mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x)$.

For a continuous r.v. X,
$\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx$.

Properties of Expectation

For random variables $X$ and $Y$ and constants $a, b, \in \mathbb{R}$, the expected value has the following properties (applicable to both discrete and continuous r.v.s)

$\mathbb{E}(aX + b) = a \mathbb{E}(X) + b$

$\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)$

Realisations of $X$, denoted by $x$, may be larger or smaller than $\mathbb{E}(X)$,

If you observed many realisations of $X$, $\mathbb{E}(X)$ is roughly an average of the values you would observe.

$\mathbb{E} (aX + b)$
$= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx$
$= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx$
$= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx$
$= a \mathbb{E} (X) + b$

Variance

Generally speaking, variance is defined as

$Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$

If $X$ is discrete:

$Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)$

If $X$ is continuous:

$Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx$

Using the properties of expectations, we can show $Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$.

$Var(X)$
$= \mathbb{E} [(X - \mathbb{E}[X])^2]$
$= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]$
$= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2$
$= \mathbb{E}[X^2] - \mathbb{E}[X]^2$

Standard Deviation

The standard deviation is defined as

$std(X) = \sqrt{Var(X)}$

Covariance

For two random variables $X$ and $Y$, the covariance is generally defined as

$Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$

Note that $Cov(X, X) = Var(X)$

$Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]$

Properties of Variance

Given random variables $X$ and $Y$, and constants $a, b, c \in \mathbb{R}$,

$Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)$

This proof for the above can be done using definitions of expectations and variance.

Properties of Covariance

Given random variables $W, X, Y$ and $Z$ and constants $a, b, \in \mathbb{R}$

$Cov(X, a) = 0$

$Cov(aX, bY) = ab Cov(X, Y)$

$Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)$

Correlation

Correlation is defined as

$Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}$

It is clear the $-1 \le Corr(X, Y) \le 1$.

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.

### Probability Question #4

A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears $k$ times ($k = 1, 2, 3$), he wins $$k$. If his number fails to appear, he loses$1. Calculate the gambler’s expected winnings

### Question of the Day #18

Here is a very very interesting question involving probability that a student saw in her tutorial and asked me. Here it is 🙂

A student is concerned about her car and does not like dents. When she drives to school, she has a choice of parking it on the street in one space, parking it on the street and taking up two spaces, or parking in the lot.
If she parks on the street in one space, her car gets dented with probability 0.1.
If she parks on the street and takes two spaces, the probability of a dent is 0.02 and the probability of a $15 ticket is 0.3. Parking in a lot costs$5, but the car will not get dented.
If her car gets dented, she can have it repaired, in which case it is out of commission for 1 day and costs her $50 in fees and cab fares. She can also drive her car dented, but she feels that the resulting loss of value and pride is equivalent to a cost of$9 per school day.
She wishes to determine the optimal policy for where to park and whether to repair the car when dented in order to minimize her (long-run) expected average cost per school day. What should the student to maximise her utility (minimise her cost)?

This is an interesting question, I guess its good to know some JCs are trying to introduce decision making process in teaching probability.

I’ll post a solution here soon. But to start off, we observe that we have two states here and student has 4 decisions. Have fun! 🙂

### Combinatorics related articles

Here is a compilation of all the Combinatorics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### 2002 A-level H2 Mathematics Paper 2 Question 30 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(a)
(i)
$\frac{7 \times 6 \times 5}{7 \times 7 \times 7} = \frac{30}{49}$
Alternatively, $\frac{^7 P_3}{7^3}$

(ii)
When $n=4$, required probability $= \frac{^7 P_4}{7^3} = \frac{7 \times 6 \times 5 \times 4}{7 \times 7 \times 7 \times 7} = \frac{120}{343}$

(b)
Key GC with either sequence or use table functions.
We want to solve $\frac{^12 C_n \times n!}{12^n} \textless \frac{1}{2}$
Plot $y_1 = \frac{^12 C_x \times x!}{12^x}$ and check for the x that gives a corresponding $y_1$ value that is less than $\frac{1}{2}$

(c)
When $n =21$, probability that all 21 people have different birthdays $= \frac{^365 P_21}{365^21} = 0.55631$

Probability that all 21 people have different birthdays $= \frac{^365 P_23}{365^23} = 0.4927$

Thus, when $n =23$, probability that at least two of the people are the same birthday $= 1 - 0.4927 = 0.5073$ (more than $\frac{1}{2}$)

A very interesting probability question here. This show show counter intuitive probability is! The results shows that the we can expect to find someone with the same birthday in a room of 23, more than half the time.

### 2010 A-level H2 Mathematics (9740) Paper 2 Question 7 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
$\mathrm{P}(A|B') = \frac{\mathrm{P} (A \cap B')}{\mathrm{P}(B')}$
$\Rightarrow \mathrm{P} (A \cap B') = 0.8 \times (1-0.6) = 0.32$

(ii)
$\mathrm{P}(A \cup B) = \mathrm{P}(A \cap B') + \mathrm{P}(B) = 0.32 + 0.6 = 0.92$

(iii)
P(B’|A) $= \frac{\mathrm{P} (B' \cap A)}{\mathrm{P}(A)} = \frac{0.32}{0.7}= 0.457$

(iv)
$\mathrm{P} (A' \cap C) = \mathrm{P} (A')\mathrm{P} (C)$ since A and C are independent
$\Rightarrow \mathrm{P} (A' \cap C) = 0.15$

(v)
$0 \le \mathrm{P} (A' \cap B \cap C) \le 0.15$

### 2015 A-level H2 Mathematics (9740) Paper 2 Question 9 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
$\mathrm{P}(B|A) = \mathrm{P}(B)$ since A and B are independent.

$\therefore, \mathrm{P}(B|A) = \mathrm{P}(B) = 0.4$

(ii)
$\mathrm{P}(A' \cap B' \cap C')$

$= 1 - \mathrm{P}(C \cup B \cup A)$

$= 1 - (0.45 + 0.02 + 0.2 + 0.145)$

$= 0.185$

(iii)

If $\mathrm{P}(A' \cap B' \cap C) = 0$,

$\mathrm{P}(A' \cap B' \cap C')$

$=1 - [0.935 - (0.3 - 0.035 )]$

$= 0.33$ (maximum)

If $\mathrm{P}(A' \cap B \cap C) = 0$,

$\mathrm{P}(A' \cap B' \cap C')$

$= 1 - \mathrm{P}(A \cup B \cup C) - 0.22$

$= 0.165$ (minimum)

$0.165 \le \mathrm{P}(A' \cap B' \cap C') \le 0.33$

### Financial Engineering (I)

Before attempting to read what we have here, students should revise their basic probability and linear algebra first.

Financial Engineering (I) #1 – Overview
Financial Engineering (I) #2 – Introduction to No Arbitrage
Financial Engineering (I) #3 – Interest rates and fixed income instruments
Financial Engineering (I) #4 – Floating Rate Bonds and Term Structure of Interest Rates
Financial Engineering (I) #5 – Forward Contracts
Financial Engineering (I) #6 – Swaps
Financial Engineering (I) #7 – Futures
Financial Engineering (I) #8 – Options
Financial Engineering (I) #9 – Options Pricing
Financial Engineering (I) #10 – The 1-Period Binomial Model
Financial Engineering (I) #11 – Option Pricing in the 1-Period Binomial Model
Financial Engineering (I) #12 – The Multi-Period Binomial Model
Financial Engineering (I) #13 – Pricing American Options
Financial Engineering (I) #14 – Replicating Strategies
Financial Engineering (I) #15 – Dividends, Pricing in the Binomial Model
Financial Engineering (I) #16 – Black-Scholes Model
Financial Engineering (I) #17 – Introduction to Term Structure Lattice Models
Financial Engineering (I) #18 – Cash Account and Pricing Zero-Coupon Bonds
Financial Engineering (I) #19 – Fixed Income Derivatives (1)
Financial Engineering (I) #20 – Fixed Income Derivatives (2)
Financial Engineering (I) #21 – The Forward Equation
Financial Engineering (I) #22 – Model Calibration
Financial Engineering (I) #23 – Pricing in a Black-Derman Toy Model
Financial Engineering (I) #24 – Modelling and Pricing Default-able bonds
Financial Engineering (I) #25 – Credit Default Swaps and Pricing Credit Default Swaps
Financial Engineering (I) #26 – Mortgage Mathematics and Mortgage-Backed Securities
Financial Engineering (I) #27 – Prepayment Risks and Pass-Throughs
Financial Engineering (I) #28 – Principal-Only and Interest Only Mortgaged-Backed Securities
Financial Engineering (I) #29 – Collateralised Mortgage Obligations
Financial Engineering (I) #30 – Pricing Mortgage-Backed Securities

### Geometric Brownian Motion

This is really important for anyone interested in Finance Modelling. As what the movie Wolf on Wall Street says:

They are referring to a geometric brownian motion.

Firstly, we will begin with the definitions.

We say that a random process, $X_t$, is a geometric Brownian motion (GBM) if for all $t \ge 0$
$X_t = e^{(\mu - \frac{\sigma^2}{2})t} + \sigma W_t$
where $W_t$ is a Standard Brownian Motion
Here $\mu$ is the drift and $\sigma$ is the volatility. We write $X_t \sim GBM (\mu, \sigma)$

Also note that
$X_{t+s}$
$= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t+s}}$
$= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t} + (\mu - \frac{\sigma^2}{2})s + \sigma (W_{t+s} - W_t)}$; This is a common technique for solving expectations.
$= X_t e^{(\mu - \frac{\sigma^2}{2})(s) + \sigma (W_{t+s} - W_t)}$. This is very useful for simulating security prices.

Consider $\mathbb{E}_t [X_{t+s}]$

$\mathbb{E}_t [X_{t+s}]$
$= \mathbb{E}_t [X_{t}e^{(\mu - \frac{\sigma^2}{2})s + \sigma(W_{t+s} - W_t)}]$; Notice this expansion is similar to before.
$= X_t e^{(\mu - \frac{\sigma^2}{2})s} \mathbb{E}^t [e^{\sigma (W_{t+s} - W_t)}]$
$= X_t e^{(\mu - \frac{\sigma^2}{2})s} e^{\frac{\sigma^2}{2}s}$
$= e^{\mu s} X_t$
This result tells us that the expected growth rate of $X_t$ is $\mu$.

From the definitions of Brownian Motion introduced earlier, we extend them to Geometric Brownian motion.
1. Fix $t_1, t_2, \ldots , t_n$. Then $\frac{X_{t_2}}{X_{t_1}}, \frac{X_{t_3}}{X_{t_2}}, \ldots \frac{X_{t_n}}{X_{t_{n-1}}}$ are mutually independent.
2. Paths of $X_t$ are continuous as function of $t$, meaning they do not jump.
3. For $s > 0$, $\mathrm{log}(\frac{X_{t+s}}{X_t}) \sim \mathrm{N}((\mu - \frac{\sigma^2}{2})s, \sigma^2 s)$

So now lets try to do some modelling of stock prices as a geometric brownian motion.

Suppose $X_t \sim GBM(\mu, \sigma)$. Clearly
1. $X_t > 0 \Rightarrow X_{t+s} > 0$ for any $s > 0$
This tells us that the limited liability of stock price is not violated.
2. The distribution of $\frac{X_{t+s}}{X_t}$ only depends on s and not on \$latex X_t.
We will look at the Black-Scholes option formula next time and will come back to review the geometric brownian motion for the underlying model.

### Introduction to Brownian Motion

Lets look at brownian motion now. And yes, its the same as what our high school teachers taught about the particles moving in random motion. Here, we attempt to give it a proper structure and definition to work with.

A Brownian Motion is a random process $\{ X_t : t \ge 0 \}$ with parameters $(\mu, \sigma)$ if
For $0 \textless t_1 \textless t_2 \textless \ldots \textless t_{n-1} \textless t_n$, $(X_{t_2} - X_{t_1}), (X_{t_3} - X_{t_2}), \ldots, (X_{t_n} - X_{t_{n-1}})$ are mutually independent. This is often called the independent increments property.
For $s > 0, X_{t+s} - X_t \sim \mathrm{N} ( \mu s, \sigma^2 s)$
$X_t$ is a continuous function of t.
We say that $X_t$ is a $\mathrm{B} (\mu, \sigma)$ Brownian motion with drift $\mu$ and volatility $\sigma$.

For the special case of $\mu = 0$ and $\sigma = 1$, we have a standard Brownian motion. We can denote it with $W_t$ and assume that $W_0 = 0$
If $X_t \sim \mathrm{B}(\mu, \sigma)$ and $X_0 = x$ then $X_t = x + \mu t + \sigma W_t$ where $W_t$ is a standard brownian motion. Thus, $X_t \sim \mathrm{N}(x+\mu t, \sigma^2 t)$

The next concept is important in finance, that is, Information Filtrations.
For any random process, we will use $\mathcal{F}_t$ to denote the information available at time t.
– the set $\{\mathcal{F}_t\}_{t \ge 0}$ is then the information filtration.
$\mathbb{E}[.|\mathcal{F}_t]$ denotes an expectation conditional on time t information available.

Note: The independent increment property of Brownian Motion implies that any function of $W_{t+s} - W_t$ is independent of $\mathcal{F}_t$ and that $(W_{t+s}-W_t) \sim \mathrm{N}(0,s)$.

So let us do a bit of math to obtain $\mathbb{E}_0[W_{t+s}W_s]$ for instance.

Using condition expectation identity, we have
$\mathbb{E}_0 [W_{t+s}W_s]$
$= \mathbb{E}_0 [(W_{t+s} - W_s + W_s)Ws]$
$= \mathbb{E}_0 [(W_{t+s}-W_s)W_s] + \mathbb{E}_0 [{W_s}^2]$
$= \mathbb{E}_0 [\mathbb{E}_s[(W_{t+s} - W_s)W_s]] + s$
$= \mathbb{E}_0 [W_s \mathbb{E}_s[(W_{t+s} - W_s)]] + s$
$= \mathbb{E}_0 [W_s 0] + s$
$= 0 + s$
$= s$


text:
Events

• Sat
12
Dec
2020

• Tue
12
Oct
2021
Thu
14
Oct
2021

Romanian National Podiatry Congress

Go to all podiatry events

multiple choice questions on general topology

Since $A_1$ and $A_2$ are disjoint closed sets, we have a continuous function $g:X\to [0,1]$ such that $g(x)=0$ for $x\in A_1$ and $g(x)=1$ for $x\in A_2$. MCQ quiz on Climate Change multiple choice questions and answers on Climate Change MCQ questions quiz on Climate Change objectives questions with answer test pdf for interview preparations, freshers jobs and competitive exams. Our completely free Topology practice tests are the perfect way to brush up your skills. Each question in the first section is a multiple-choice question with four answer choices. What is the minimum header size of an IP packet? Multiple choice questions with answers. 1. Questions and Answers,Multiple choice Questions And Answers-MCQ sets,Online Test/Quiz,Short Study Notes don’t hesitate to contact us via Facebook,or through our Page 2/15 . Use MathJax to format equations. 2. A directory of Objective Type Questions covering all the Computer Science subjects. Report Error, About Us is topological space. Network topology MCQs, network topology quiz with answers for online learning networking courses.Learn computer networking multiple choice questions (MCQs), network topology quiz questions and answers.Career test on protocols and standards, networking basics, star topology, … When you take the actual test, you will mark your answers on a separate machine-scorable answer sheet. World History Multiple Choice Questions and Answers . PDF | On Jun 1, 2016, Hossein Sahleh and others published Multiple choice... topology I | Find, read and cite all the research you need on ResearchGate Download Networking MCQ Question … Q1. corporate bonds)? iff at least two of the numbers $a_1$, $a_2$, $a_3$ are equal. 0 &\text{if } x\in A_1\\ (2 credits) Answer: The diameter of a subset Aof a metric space (X;d) is supfd(x;y) j(x;y) 2 A Ag. What is the minimum header size of an IP … Asking for help, clarification, or responding to other answers. A nerve in the brain b. MULTIPLE CHOICE QUESTIONS CHAPTERS 1 ... General journal (B) Returns inwards journal (C)Returns outwards journal (D)Petty cash journal . Active 7 years, 3 months ago. Brian M. Scott. is topological space. Any compact space is metrizable. If τ τ is typology on non-empty set X X, then arbitrary ………………. Then what is the difference between discrete and cofinite toplogy on $X$. Any topology on a 3-element set is Hausdor . share | cite | improve this question | follow | edited Sep 16 '12 at 7:01. MCQ Questions and Answers on Data Communication and Computer Networks or Multiple Choice Questions with Answers on Data Communication and Networks from chapter Local Area Network. Wie kann ich meinen computergestützten CAST Permanent-Test umbuchen? When you take the test, you will mark your answers on a separate machine-scorable answer sheet. by admin. Contributors, Except where otherwise noted, content on this wiki is licensed under the following license:CC Attribution-Noncommercial-Share Alike 4.0 International. only if one among the sets $A_1$, $A_2$, $A_3$ is empty. Es gab ein Problem mit einer meiner Multiple-Choice-Fragen in einem Auswahlverfahren für CAST Permanent. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. 300+ General knowledge questions and answers for your virtual quiz (new questions) We may earn commission from links on this page. Generic Information System 3. Good luck! 481 different Multiple Choice Quizzes on JetPunk.com. [2 marks] (ii) Are the following spaces Hausdor ? Your answers are not being recorded. Computer Networking multiple choice questions and answers with explanation for interview, competitive examination and entrance test. 3. Sample Exam, F10PC Solutions, Topology, Autumn 2011 Question 1 (i) Carefully de ne what it means for a topological space Xto be Hausdor . show that A is a locally compact in the product topology, Proving there is one continuous function $F: X \to [0,3]$ such that $F(x) = n$ if $x\in A_n$ for $n=0,1,2,3.$. Multiple Choice Questions Introduction to Geology - Chapter 1 Each chapter will include a few questions designed to test your knowledge of material covered in the chapter and in the Internet-based resources. b. (3 credits) Answer: U X Y is open in the product topology if and only if, given any point (x;y) 2U, there are open sets V in X and W in Y such that x 2V, y 2W and V W U. A. How would I connect multiple ground wires in this case (replacing ceiling pendant lights)? To learn more, see our tips on writing great answers. Viewed 4k times 1. Internet Multiple Choice Questions and Answers Quiz | Internet GK. Who is leader of the Welsh Party Plaid Cymru: (a) Leanne England, (b) Leanne Stone, or (c) Leanne Wood? Multiple Choice Questions of Computer Networking 3-1. This multiple choice questions and answers type Data Communication and Networking Online Test section contains all the suitable and related MCQs of the Network topology ie eight basic topologies: point-to-point, bus, star, ring or circular, mesh, tree, hybrid, or daisy chain Only.All of these Questions have been hand picked from the Question paper of various competitive exams. Computer Networks Multiple Choice Questions and Answers (MCQs): Quizzes & Practice Tests with Answer Key (Computer Networks Quick Study Guide & Course Review Book 1) contains course review tests for competitive exams to solve 1962 MCQs. Sitemap, Follow us on Data manipulation and analysis 4. a. Athens b. Prague c. Rome d. Vienna. Give an example of open set in R with usual topology, which is not an open interval. Here you can find important Mcqs for the preparation Computer Operator, Data entry Operator, Computer Science Jobs exams. Try to answer all questions. 1 &\text{if } x\in A_2\\ When is every closed set (/singleton) the zero set of complex valued functions? We are going to add short questions and MCQs for Topology. Multiple Choice Quiz Questions At the time of Casanova's birth which city was the pleasure capital of Europe: (a) Paris, (b) Rome, or (c) Venice? "Computer Networks MCQ" book helps with fundamental concepts for self-assessment with theoretical, analytical, and distance learning. Question 6: Give the de nition of the diameter of a subset of a metric space. A connection of network topology, represented by replacing all physical elements by lines. (C) τ 1 ∖ τ 2. Professionals, Teachers, Students and Kids Trivia Quizzes to test your knowledge on the subject. Dylan J paid Joel and Company $600 to settle an account of$720. The multiple choice trivia questions and answers are the best way to test your knowledge and other than this you can easily increase your knowledge.. Test-Taking Strategies The questions in the practice test in this book illus-trate the types of multiple-choice questions in the test. 19. General topology grew out of a number of areas, most importantly the following: the detailed study of subsets of the real line (once known as the topology of point sets; this usage is now obsolete); the introduction of the manifold concept; the study of metric spaces, especially normed linear spaces, in the early days of functional analysis. Answer the following questions and then press 'Submit' to get your score. asked Sep 16 '12 at 6:53. poton poton. of member of τ belong to τ. 5. A directory of Objective Type Questions covering all the Computer Science subjects. You will receive incredibly detailed scoring results at the end of your Topology practice test … A. Our editorial is always independent ( learn more ) What is the closure of the set $T=\{1,2,3,4,5 \}\cup (6,7) \cup (7,8]$ in usual topology on $\mathbb{R}$?

format: plain


text:
Overall Objectives
Application Domains
Partnerships and Cooperations
Bibliography
Inria | Raweb 2014 | Exploratory Action
 PDF e-Pub

## Section: New Software and Platforms

### Coqine, Holide, Focalide and Sigmaid

Dedukti comes with four companion tools: Holide , an embedding of HOL proofs through the OpenTheory format [41] , Coqine , an embedding of Coq proofs, Focalide , an embedding of FoCaLiZe certified programs, and Sigmaid , a type-checker for the simply-typed $\varsigma$-calculus with subtyping and a translator to Dedukti. All of the OpenTheory standard library and a part of Coq's and FoCaLiZe's libraries are checked by Dedukti.

A preliminary version of Coqine supports the following features of Coq: the raw Calculus of Constructions, inductive types, and fixpoint deﬁnitions. Coqine is currently being rewritten to support universes. Coqine has been developed by Mathieu Boespﬂug, Guillaume Burel, and Ali Assaf.

Holide supports all the features of HOL, including ML-polymorphism, constant deﬁnitions, and type deﬁnitions. It is able to translate all of the OpenTheory standard theory library. Holide has been developed by Ali Assaf.

Focalide has been improved to support FoCaLiZe proofs found by Zenon using the Dedukti backend for Zenon developped by Frédéric Gilbert. This backend has been improved by a simple typing mechanism in order to work with Focalide. Focalide has also been updated again to work with the latest version of FoCaLiZe.

Sigmaid implements a shallow embedding of the simply-typed $\varsigma$-calculus of Abadi and Cardelli with subtyping in the $\lambda \Pi$-calculus modulo. This translation has been proved[21] to preserve the typing judgments and the semantics of the simply-typed $\varsigma$-calculus and tested on the examples of Abadi and Cardelli.

Focalide and Sigmaid have been developed by Raphaël Cauderlier.

Translators from Version 2.0 of the SMT-LIB standard and from the SMT-solver veriT have been initiated. They are currently developed by Frédéric Gilbert.


text:
Intermediate Algebra

# Practice Test

Intermediate AlgebraPractice Test

### Practice Test

In the following exercises, factor completely.

445.

$80a2+120a380a2+120a3$

446.

$5m(m−1)+3(m−1)5m(m−1)+3(m−1)$

447.

$x2+13x+36x2+13x+36$

448.

$p2+pq−12q2p2+pq−12q2$

449.

$xy−8y+7x−56xy−8y+7x−56$

450.

$40r2+81040r2+810$

451.

$9s2−12s+49s2−12s+4$

452.

$6x2−11x−106x2−11x−10$

453.

$3x2−75y23x2−75y2$

454.

$6u2+3u−186u2+3u−18$

455.

$x3+125x3+125$

456.

$32x5y2−162xy232x5y2−162xy2$

457.

$6x4−19x2+156x4−19x2+15$

458.

$3x3−36x2+108x3x3−36x2+108x$

In the following exercises, solve

459.

$5a2+26a=245a2+26a=24$

460.

The product of two consecutive integers is 156. Find the integers.

461.

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

462.

Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function $h(t)=−16t2+64t+80h(t)=−16t2+64t+80$ models the height, h, of the ball above the ground as a function of time, t. Find: the zeros of this function which tells us when the ball will hit the ground. the time(s) the ball will be 128 feet above the ground. the height the ball will be at $t=4t=4$ seconds.

463.

For the function, $f(x)=x2−7x+5,f(x)=x2−7x+5,$ find when $f(x)=−7f(x)=−7$ Use this information to find two points that lie on the graph of the function.

464.

For the function $f(x)=25x2−81,f(x)=25x2−81,$ find: the zeros of the function the x-intercepts of the graph of the function the y-intercept of the graph of the function.

Order a print copy

As an Amazon Associate we earn from qualifying purchases.


text:
# Stock Splits and Treasury Stock

### Stock Splits

The main objective of a stock split is to reduce the market price of each share of stock.
A stock split is the issuance of a proportional number of additional shares by a corporation that reduces the par value of its common stock. The stock split applies to all shares of common stock, issued shares, unissued shares, and treasury stock. With a reduced market price per share of stock, more investors may be drawn to purchase shares, and the pool of stockholders attracted may become more diverse.

#### Stock Split

For example, Company ABC has 20,000 shares of $50 par common stock outstanding. The current market price is$100 per share. The board of the organization declares a 5-for-1 stock split. As a result:
• 100,000 shares $\left(20\text{,}000\,\text {shares}\times 5\right)$ will be outstanding.
• The par value of each share will be $10 $\left(\50/5\right)$. ### Number and Value of Shares before and after Stock Split Before Stock Split After Stock Split Number of shares issued 20,000 100,000 Par value per share x$50 x $10 Total value$100,000 $100,000 The total amount attributed to common stock outstanding is the same before and after the stock split. Each individual stockholder owns the same total amount of the corporation's stock. For example, if a stockholder owns 5 shares of the$50 par stock before the split (total of $250), then this same stockholder would own 25 shares of the$10 par stock after the split (total par value of $250). Essentially, only the number of shares and the par value of each share have changed, but the total value remains the same. Based on the stock split, the market price of the share should also decrease. With a previous market price of$100 per share, the new market price should now be about $20 per share $\left(\100/5\right)$. Because stock splits only affect the number of shares outstanding and the par value, there is no need for a journal entry. The total amount in the common stock account remains the same. However, the details of the stock split are typically disclosed on the financial statements as a note. ### Treasury Stock Corporations sometimes reacquire their own stock to benefit employees and to help support the market price of its stock. Treasury stock is stock that has been issued and then reacquired by the same corporation. Some of the reasons that a corporation may choose to reacquire its own stock are: • To resell shares to employees • To reissue the shares as bonuses to employees • To assist in supporting the stock's market price The cost method or the par method may be used to record the transactions associated with buying and selling shares of treasury stock. Under the cost method, the treasury stock account is debited for the purchase of the stock and credited when the stock is sold. When selling treasury stock, the credit is always for the same amount as the purchase price, similar to how we treat the par value for common and preferred stock entries. For example, Company ABC has 10,000 shares of$10 par common stock issued and outstanding. On October 1 the corporation purchases 1,000 shares of its own common stock at $25 per share. A journal entry is made by the company to record this change. ### Purchase of Treasury Stock Journal Entry Date Description Debit Credit Oct 1 Treasury Stock (1,000 shares x$25) $25,000 Cash$25,000
To record the purchase of 1,000 shares of treasury stock at $25 On November 1 the corporation sells 700 shares of the treasury stock for$35. A journal entry is made to record the sale.

### Sale of Treasury Stock Journal Entry

Date Description Debit Credit
Nov 1 Cash (700 shares x $35)$24,500
Treasury Stock (700 shares x $25)$17,500
Paid-In Capital from Sale of Treasury Stock   $7,000 To record the sale of 700 shares of treasury stock at$35

In the event that the treasury stock is sold for an amount less than the purchase amount, a decrease in paid-in capital will result. The credit balance in the paid-in capital from sale of treasury stock account would be debited up to the amount in that account. Once that account has been zeroed out, the retained earnings account would be debited. To illustrate, the corporation sells 300 shares of treasury stock for $20 each on December 1. A journal entry is made to record this transaction. ### Sale of Treasury Stock at Reduced Amount Journal Entry Date Description Debit Credit Dec 1 Cash (300 shares x$20) $6,000 Paid-In Capital from Sale of Treasury Stock$1,500
Treasury Stock (300 shares x $25)$7,500
To record the sale of 300 shares of treasury stock at $20 Based on the November 1 journal entry, the credit balance in the paid-in capital from sale of treasury stock account was$7,000. Therefore, the full \$1,500 can be debited to this account on December 1.

As a reminder, to the extent the corporation holds treasury stock, it does not pay itself a dividend.


text:
# Similarity measure

(Redirected from Measure of similarity)

In statistics and related fields, a similarity measure or similarity function is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity measure exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. For example, if two pieces of data have close x, y coordinates, then their “similarity” score, the likelihood that they are similar, will be much higher than two data points with more space between them.[1] In the context of cluster analysis, Frey and Dueck suggest defining a similarity measure

${\displaystyle s(x,y)=-\|x-y\|_{2}^{2}}$

where ${\displaystyle \|x-y\|_{2}^{2}}$ is the squared Euclidean distance.[2]

Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. In machine learning, common kernel functions such as the RBF kernel can be viewed as similarity functions.[3]

## Use in clustering

In spectral clustering, a similarity, or affinity, measure is used to transform data to overcome difficulties related to lack of convexity in the shape of the data distribution.[4] The measure gives rise to an ${\displaystyle (n,n)}$-sized similarity matrix for a set of n points, where the entry ${\displaystyle (i,j)}$ in the matrix can be simply the (negative of the) Euclidean distance between ${\displaystyle i}$ and ${\displaystyle j}$, or it can be a more complex measure of distance such as the Gaussian ${\displaystyle e^{-\|s_{1}-s_{2}\|^{2}/2\sigma ^{2}}}$.[4] Further modifying this result with network analysis techniques is also common.[5]

## Use in sequence alignment

Similarity matrices are used in sequence alignment. Higher scores are given to more-similar characters, and lower or negative scores for dissimilar characters.

Nucleotide similarity matrices are used to align nucleic acid sequences. Because there are only four nucleotides commonly found in DNA (Adenine (A), Cytosine (C), Guanine (G) and Thymine (T)), nucleotide similarity matrices are much simpler than protein similarity matrices. For example, a simple matrix will assign identical bases a score of +1 and non-identical bases a score of −1. A more complicated matrix would give a higher score to transitions (changes from a pyrimidine such as C or T to another pyrimidine, or from a purine such as A or G to another purine) than to transversions (from a pyrimidine to a purine or vice versa). The match/mismatch ratio of the matrix sets the target evolutionary distance.[6][7] The +1/−3 DNA matrix used by BLASTN is best suited for finding matches between sequences that are 99% identical; a +1/−1 (or +4/−4) matrix is much more suited to sequences with about 70% similarity. Matrices for lower similarity sequences require longer sequence alignments.

Amino acid similarity matrices are more complicated, because there are 20 amino acids coded for by the genetic code, and so a larger number of possible substitutions. Therefore, the similarity matrix for amino acids contains 400 entries (although it is usually symmetric). The first approach scored all amino acid changes equally. A later refinement was to determine amino acid similarities based on how many base changes were required to change a codon to code for that amino acid. This model is better, but it doesn't take into account the selective pressure of amino acid changes. Better models took into account the chemical properties of amino acids.

One approach has been to empirically generate the similarity matrices. The Dayhoff method used phylogenetic trees and sequences taken from species on the tree. This approach has given rise to the PAM series of matrices. PAM matrices are labelled based on how many nucleotide changes have occurred, per 100 amino acids. While the PAM matrices benefit from having a well understood evolutionary model, they are most useful at short evolutionary distances (PAM10–PAM120). At long evolutionary distances, for example PAM250 or 20% identity, it has been shown that the BLOSUM matrices are much more effective.

The BLOSUM series were generated by comparing a number of divergent sequences. The BLOSUM series are labeled based on how much entropy remains unmutated between all sequences, so a lower BLOSUM number corresponds to a higher PAM number.


text:
## Illinois Journal of Mathematics

### Regularity of finite $H$-spaces

John R. Harper

#### Abstract

Let $X$ be an $H$-space of the homotopy type of a connected finite CW complex. Suppose the generators of the rational cohomology of $X$ all have dimension $\leq m$. Theorem. If $p$ is a prime satisfying $2p-1\geq m$, then $X$ is mod $p$ equivalent to a product of odd dimensional spheres and generalized Lens spaces $L(p,1,\ldots,1)$ obtained as the orbit space of an action of $Z_{\mathrm{p}}$ on $S^{2\mathrm{p}-1}$.

#### Article information

Source
Illinois J. Math., Volume 23, Issue 2 (1979), 330-333.

Dates
First available in Project Euclid: 20 October 2009

https://projecteuclid.org/euclid.ijm/1256048244

Digital Object Identifier
doi:10.1215/ijm/1256048244

Mathematical Reviews number (MathSciNet)
MR528568

Zentralblatt MATH identifier
0409.55007

Subjects
Primary: 55P45: $H$-spaces and duals

#### Citation

Harper, John R. Regularity of finite $H$-spaces. Illinois J. Math. 23 (1979), no. 2, 330--333. doi:10.1215/ijm/1256048244. https://projecteuclid.org/euclid.ijm/1256048244


text:
# 3. There is a solid cylinder of radius 2 mm inside a hollow cylinder of inner...

###### Question:

3. There is a solid cylinder of radius 2 mm inside a hollow cylinder of inner radius 4mm and outer radius 5 mm as shown in the figure. The inner cylinder has 10 A of current flowing in the direction shown and the outer cylinder has 20 of current in the direction shown. The path of integration is counter clockwise. What is the magnetic field at a point r-3 mm What is the magnetic field at a point r- 7 mm a. b. Space batuwean inner culinoler aned outer elinolar

#### Similar Solved Questions

##### Suppose you intend to perform some statistical inferences at a level of significance, and you draw...
Suppose you intend to perform some statistical inferences at a level of significance, and you draw a random sample of size n = 60 from a population with mean (H) and standard deviation (o). Match each of the numbered items in the list with the numbers in the drop-down menus. 8 Margin of error 2 Samp...
##### QS) (10 Points) sokg object slides down from rest from atop of an inclined rough surface of height 0.Smn: If the velocity of the object is 3m/s at the end of the surface; find the coefficient of kinetic friction betwecn the object and the surfacc:0.5 m300
QS) (10 Points) sokg object slides down from rest from atop of an inclined rough surface of height 0.Smn: If the velocity of the object is 3m/s at the end of the surface; find the coefficient of kinetic friction betwecn the object and the surfacc: 0.5 m 300...
##### True or false?When YOU take a definite Integral, vou need t0 add constant of integration to your final answer:TrueFalseQuastion 45 ptsWhich one of the following is FALSE?f(e)gkr)dz f(r)dz g(r)drfike) g(r)ldz f(c)dzg(r)drf(c)dzf(e)dcf(e)dr
True or false? When YOU take a definite Integral, vou need t0 add constant of integration to your final answer: True False Quastion 4 5 pts Which one of the following is FALSE? f(e)gkr)dz f(r)dz g(r)dr fike) g(r)ldz f(c)dz g(r)dr f(c)dz f(e)dc f(e)dr...
##### Point) Find the area of the surface obtained by rotating the curveX = 3ezyfrom y = 0 to y = 5 about the Y-axis.The area issquare units
point) Find the area of the surface obtained by rotating the curve X = 3ezy from y = 0 to y = 5 about the Y-axis. The area is square units...
##### (1 point) If C is the curve given by r (t) = (1 + 4 sin...
(1 point) If C is the curve given by r (t) = (1 + 4 sin t)i + (1+2 sin’t)j + (1 + 3 sinºt) k, 0 <15 and F is the radial vector field F(x, y, z) = xi + yj + zk, compute the work done by F on a particle moving along C....
##### QUESTION 13 Which of these are true of staphylococcus aepidermidis growths ? produce yellow pus novobioin sensitive, and beta hemolytic none of these are true of staphylococcus epidermidis produce white pus non hemolytic and growth is not prevented by optochin white pus on 'growth non hemolytic and sgrowth prevented by optochin
QUESTION 13 Which of these are true of staphylococcus aepidermidis growths ? produce yellow pus novobioin sensitive, and beta hemolytic none of these are true of staphylococcus epidermidis produce white pus non hemolytic and growth is not prevented by optochin white pus on 'growth non hemolytic...
##### Suppose the price of pork fell from $3.00 to$2.50 per pound. Over the same period,...
Suppose the price of pork fell from $3.00 to$2.50 per pound. Over the same period, the CPI (which reflects the price of all consumer goods) increased by 10%. What is the percentage change in price (%) that we should use to calculate the price elasticity of demand for beef? (Hint: Recall that we use...
##### In an exchange, Julio gave up his business-use real property (FMV $95000, adjusted basis$45000) for...
In an exchange, Julio gave up his business-use real property (FMV $95000, adjusted basis$45000) for smaller piece of business use real property FMV $80000 and$15000 cash. What is gain realized and recognized on exchange? A. $15000;$0 B. $50,000;$0 C. $50,000;$15000 D. $50,000;$50,000...
##### 11 - The following data is given for an ideal vapor-compression refrigeration cycle: At the inlet...
11 - The following data is given for an ideal vapor-compression refrigeration cycle: At the inlet of the compressor h=250 kJ/kg At the exit of the condenser h=400 kJ/kg At the inlet of the evaporator h=75 kJ/kg If the compressor had an isentropic efficiency of 80 %, the heat rejected (kJ/kg) from th...
##### P /500 Problem 2) Combined Loading (30 points The structure shown has a diameter of 40...
P /500 Problem 2) Combined Loading (30 points The structure shown has a diameter of 40 mm. The length 1800 dimensions are a = 500 mm and d = 100 mm. Using load magnitudes of Px = 3000 N, P, = 1000 N, and P, = 1500 N, determine the normal and shear stresses in the structure at point H. (Point H is co...
##### A particular airplane will reach liftoff at a speed of 120 km/h. (a) What minimum constant...
A particular airplane will reach liftoff at a speed of 120 km/h. (a) What minimum constant acceleration does the airplane require for it to liftoff after a takeoff run of 220 m? (Enter the magnitude only.) m/s2 (b) How long does it take the airplane to reach liftoff speed? S...
##### Question 10In a positively skewed (right skewed) data distribution: the median less than the modeNotyeti answeredMarked out of 1.00Select one: TrueRiFlag questionFalse
Question 10 In a positively skewed (right skewed) data distribution: the median less than the mode Notyeti answered Marked out of 1.00 Select one: True RiFlag question False...
##### Number 38 please. Secuon A./ 37. Suppose we have the lottery in Exercise 30. Assume all...
Number 38 please. Secuon A./ 37. Suppose we have the lottery in Exercise 30. Assume all possible tickets are printed and all tickets are distinct. (a) Compute the probability of winning if one ticket is purchased. (b) Compute the probability of winning if seven tickets are purchased. 38. Suppose ...
##### The functions Y1 (t) = sin(2t), Yz(t) = cos(2t) are linearly independent on ( _ o_ 0).Select one: True False
The functions Y1 (t) = sin(2t), Yz(t) = cos(2t) are linearly independent on ( _ o_ 0). Select one: True False...
##### 16. Adjusting entries are needed to correctly measure the (a) ending balance in the cash account;...
16. Adjusting entries are needed to correctly measure the (a) ending balance in the cash account; (b) net income (loss) on the income statement; (c) net income (loss) on the balance sheet; (a) beginning balance in the Cash account. 17. What's the normal balance of Retained Earnings? (a) credit b...
##### Graphing and Finding Zeros (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. $f(x)=\sqrt{3 x-14}-8$
Graphing and Finding Zeros (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. $f(x)=\sqrt{3 x-14}-8$...
##### Probability and statistics for engineers d. 045 Question 2 Not yet answered A continuous random variable...
Probability and statistics for engineers d. 045 Question 2 Not yet answered A continuous random variable X has the following density function (2x+1) f(x)= 1<x<3 10 Find 0. otherwise PC 2 < x < 2.5) Marked out of 2 P Flag question a. 0.275 b.0.1 C 0.05 d. 2.13 3 Let A and B two independen...
##### Factor the greatest common factor from each polynomial.$15 r+35$
Factor the greatest common factor from each polynomial. $15 r+35$...
##### 3. True or False (6 Points) (a) _If a Turing machine M loops in definitely on a string w, M will ...
3. True or False (6 Points) (a) _If a Turing machine M loops in definitely on a string w, M will loop indefinitely on all input strings (b)-E If the language A is recognized by a TM M and the complement of A is recognized (c) E Every Turing machines is a decider. (d) ESuppose a Turing machine contai...
##### EMGOUconducted determine wbether age affccta the lovel of development- of mayors in the Philippineg claasified a8 high _ averege, and low- A orontednobb taken nnd tho rcnulty wre thown below. pample of ono hundred meyoraILayanHigh Avorogo LowAcoYoungOldCan we Jay that ege affects the lovel of oriontedneas of the mayora in tho Philip pinoe? Ube 5% level of uignificance:
EMGOU conducted determine wbether age affccta the lovel of development- of mayors in the Philippineg claasified a8 high _ averege, and low- A orontednobb taken nnd tho rcnulty wre thown below. pample of ono hundred meyora ILayan High Avorogo Low Aco Young Old Can we Jay that ege affects the lovel of...
##### MATH 125, Unit 2, Objective 5, Homework 2.5Section 1 Section 2 Section 3Determine the magnitude, and direction in degrees of v =<9,-6> in the interval 0 < 0 < 3609Round to three decimal places when necessary_ Ivl04
MATH 125, Unit 2, Objective 5, Homework 2.5 Section 1 Section 2 Section 3 Determine the magnitude, and direction in degrees of v =<9,-6> in the interval 0 < 0 < 3609 Round to three decimal places when necessary_ Ivl 04...
##### Add 2.5, 0.025 and 234 and express into correct significant figures. (b) Divide 2.349 by 0.003 and express into correct significant figures_
Add 2.5, 0.025 and 234 and express into correct significant figures. (b) Divide 2.349 by 0.003 and express into correct significant figures_...
##### "98jIns 341 Kq pazaxa 3J10J UOLIOHJ JHEIS WnWIXBU 341 pue aoejuns &41 Kq pauaxa 9210J Jepnolpuadued 341 u3MJaq drysuonepa1 e Ai[eoneuoqiew ssadxg N6 N 0S N 0z N 0z N 0[ NSt NSI NS N â‚¬ N 0[ N 0 WA JJ1OJ (T)J Xpo[q xpoq UOHIJHJ JUEIS UO aJejIns Kq pauaxa Sx-[ a4} 4O pauaxa lunlixew IeInoipuadiad [ 3310J pIBMUMOP BJX] "4po[q a41 uo a3ejins 341Jo jeinwipuadjad 341 SnsuaA 3JJ0 UOQJUJ JEIS WnuIxBU ?41J0 Ydeu3 e MeJp 01 Jq81 941 U! Blep 341 as "B8x [uMop saysnd Suuuds'(Ypo[q p
"98jIns 341 Kq pazaxa 3J10J UOLIOHJ JHEIS WnWIXBU 341 pue aoejuns &41 Kq pauaxa 9210J Jepnolpuadued 341 u3MJaq drysuonepa1 e Ai[eoneuoqiew ssadxg N6 N 0S N 0z N 0z N 0[ NSt NSI NS N â‚¬ N 0[ N 0 WA JJ1OJ (T)J Xpo[q xpoq UOHIJHJ JUEIS UO aJejIns Kq pauaxa Sx-[ a4} 4O pauaxa lunlixew IeIn...
##### OX Consider a particle confined in a one-dimensional box of 'length L shown belowFind the wave function (W) for the system in the range eigenvalue 0 <x <L equation Fwp that satisfies the Ew where H is the Hamiltonian operator for the system, Is this wave function an eigen function of 2 Marks the position operator x? Derive and show, AxApx 1 Mark > h/2 for this problem where;, Ax V(x?) Apx and V(px (px))) Cyp] Th &xpeardion 3 Marks rOlue
OX Consider a particle confined in a one-dimensional box of 'length L shown below Find the wave function (W) for the system in the range eigenvalue 0 <x <L equation Fwp that satisfies the Ew where H is the Hamiltonian operator for the system, Is this wave function an eigen function of 2 M...
##### 0.120 Kg m 0,200 kg; and mnass N( Iv unknown(10 pts ) Draw and labcl frce-body diagrams for m . M and mz below.cuch case (just usc variubles, no numbers yet) (10 pts ) Write the equations of motion for = RemembcT to break your forces into components where applicableequations Io solve for Ti and pls ) Use the given Information and yourmeasured to be 72.9' . Use the given information and your (10 pts ) Now suppose equations solve for M
0.120 Kg m 0,200 kg; and mnass N( Iv unknown (10 pts ) Draw and labcl frce-body diagrams for m . M and mz below. cuch case (just usc variubles, no numbers yet) (10 pts ) Write the equations of motion for = RemembcT to break your forces into components where applicable equations Io solve for Ti and p...
##### (a) Represent f(c) =  ln(2 I) as power series and specify its radlius of convergercesinf&2 Evaluate dr by representing it a8 a power series and determine its radius of convergence.
(a) Represent f(c) =  ln(2 I) as power series and specify its radlius of convergerce sinf&2 Evaluate dr by representing it a8 a power series and determine its radius of convergence....
##### Progressive income taxation is an automatic fiscal stabilizer. Question 7 options: True False
Progressive income taxation is an automatic fiscal stabilizer. Question 7 options: True False...


text:
# Ball Launched from a Horizontal Cliff#

A small ball is launched from a horizontal cliff with initial speed 1.26 $$m/s$$. The side of the cliff is sloped. The ball makes contact with the cliff side after falling a vertical distance of 51.2 cm.

## Part 1#

How long was the ball in the air before hitting the side of the cliff?

Please enter in a numeric value in $$s$$.

## Part 2#

What is the value of the angle $$\theta$$ shown in the figure?

## Part 3#

What is the velocity of the ball just before hitting the side of the cliff? Give your final answer in terms of the $$x$$- and $$y$$-components of the velocity and the unit vectors $$\hat{\imath}$$ and $$\hat{\jmath}$$.

Please enter the coefficient of $$\hat{\imath}$$ (in $$m/s$$).

Please enter in a numeric value in $$m/s$$.

## Part 4#

What is the velocity of the ball just before hitting the side of the cliff? Give your final answer in terms of the $$x$$- and $$y$$-components of the velocity and the unit vectors $$\hat{\imath}$$ and $$\hat{\jmath}$$.

Please enter the coefficient of $$\hat{\jmath}$$ (in $$m/s$$).

Please enter in a numeric value in $$m/s$$.


text:
Self Organizing Maps: How is the location computed and updated?

I have read other similar questions on here, but I am still unsure how SOM deals with the positions/locations of the neurons.

Say that the input space is N-dimensional. I initalise some weights, and then in the competition step, I find the closest neuron to the current input point using for example Euclidean distance. But I don't understand how I update the position of neurons next -- did I initialise random positions for each neuron at the beginning? In what direction do I move the neuron? I know that the position of the neuron will eventually be the 2-D space that I embed the N-dimensional data in, but I can't figure out how it works during training.

Yes, you initialize the weights/positions randomly. The position update during learning consists in moving the position of the best matching unit (BMU) and its neighbors towards the input. From Wikipedia:

The update formula for a neuron v with weight vector Wv(s) is

Wv(s + 1) = Wv(s) + Θ(u, v, s) α(s)(D(t) - Wv(s))

, where s is the step index, t an index into the training sample, u is the index of the BMU for D(t), α(s) is a monotonically decreasing learning coefficient and D(t) is the input vector; Θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s.

Regarding the 2D space, it is just the map coordinates of your neurons. You'll never explicitly convert your N-D space into a 2D space, the SOM neurons' weights will always be N-dimensional.

You may be confused by images like this one:

which represent your data as a 2D lattice, but that's because the input data is also 2D (the points in the figure)! If your input space is N-D (N > 2), you can't use this type of representation. For instance, if you're working with images, you may represent the learned SOM like this: note that the stored data is still N-D (the images), just their organization is 2D.

• Thanks for the answer! However, I still don't quite get it. What is the relation between the weights and the position of the neuron? You said "moving the position of the best matching unit (BMU) and its neighbors towards the input.".. but I thought the position was 2D, and the input is N-dimensional. How do you move it towards it then? I think I understand how the weights are updated, but I don't understand how the positions are updated at each step. – confused00 Apr 11 '16 at 8:04
• Position = weights, same dimensionality as input. Usually we say "position in map space" when referring to the 2D position on the map (this one is not updated explicitly). – rcpinto Apr 11 '16 at 10:42
• but the 2D position is the one used to visualise the final organisation, no? My question relates to how the 2D position is updated every step to culminate in the final 2D positions. – confused00 Apr 11 '16 at 21:04
• It isn't. The neighborhood update rule makes close neurons to be similar implicitly. – rcpinto Apr 11 '16 at 22:22
• in your 2nd image, the 2D position of the images is what gives the final organisation, and is the position that my question relates to. What's the neighbourhood update rule? In the equation posted there is a neighbourhood function which is only used to update the weights, not the 2D position. – confused00 Apr 12 '16 at 13:16


text:
# capacitor after scr dimmer

#### chajla

Joined Jan 26, 2023
6
hi

can somebody please explain, what happens when there is a phase angle regulating circuit, then a bridge rectifier and a resistive load and you connect a capacitor after the rectifier parallel to the load?

thanks

#### Ian0

Joined Aug 7, 2020
6,953
hi

can somebody please explain, what happens when there is a phase angle regulating circuit, then a bridge rectifier and a resistive load and you connect a capacitor after the rectifier parallel to the load?

thanks
How big a capacitor?
Several hundred microfarads = smoothing capacitor for the DC output
several tens of nanofaradds = snubber (in conjunction with a resistor)

#### MisterBill2

Joined Jan 23, 2018
14,185
Is the question about an existing arrangement, wondering what the purpose may be? Or is it wanting to achieve some purpose not yet mentioned? What is the goal to be reached by getting an answer?? Give us more context and you can get better answers. Provide no context and get no answers.

#### chajla

Joined Jan 26, 2023
6
i wanted to keep it simple and not narrow down the focus on a specific situation. i see weird things happening when i do the above and would like to understand the reason. i would have expected the cap to integrate the current, like after a transformer and rectifier.

#### chajla

Joined Jan 26, 2023
6
the cap i used was for smoothing, at least ithought so.

my understanding in this field is clearly limited and my scope is retired in the attic.

(so you have 10 min. on here to edit? even while editing? surprizing.)

#### chajla

Joined Jan 26, 2023
6
dimmer (triac is bta 41-600):

#### Attachments

• 498.9 KB Views: 9
Last edited:

#### Ian0

Joined Aug 7, 2020
6,953
Clearly the 40A amp triac and the 0.5Amp bridge rectifier aren't part of the same circuit.

#### LowQCab

Joined Nov 6, 2012
2,866
My best guess is that the trashy high-frequency Output from the SCR
is making the added-on Capacitor get into a "ringing-Oscillation".

A solution that would alleviate this to some extent, is to treat the SCR's Output
similar to how a Switching-Regulator's Output would be Filtered,
ie, ..........
an Output-Filter consisting of an Inductor,
and a Capacitor-Bank of High-Current-Rated Capacitors.

The biggest problem with this situation is that it's operating at ~60hz.

You can bet on having rude amounts of Noise to contend with
anytime a Phase-Angle-Switched-SCR is involved in any scenario.
There's no free lunch.
If it were that easy, everybody would be doing it that way.

I am assuming that You are dealing with ~204-Peak-Volts DC,
since You did not specify a Voltage.

For ~20-Amps-Capacity use something like a 20-Amp-rated 1mH Inductor,
like this ....... DigiKey HM2131-ND ~$66.oo, For ~10-Amps-Capacity use something like a 10-Amp-rated 1mH Inductor, like this ....... DigiKey RI410PC-ND ~$14.oo,

Following-up this Inductor, You will need a bank of 20- High-Current-rated Electrolytic-Capacitors,
like these ...........
20-X, 1000uF, 250-Volt-rated Electrolytic-Capacitors at ~$20.oo each = ~$400.oo .

And You will still have roughly ~10-Volts of ripple on your DC-Output.

You haven't found a slick-way to create a really cheap, and High-Power, Power-Supply.

In my opinion, Triacs are only good for Incandescent-Light-Bulb-Dimmers

If You want to learn how to create your own high-Power DC Supply, with a well filtered Output,
You will need to provide all the specifications of the item You want to power,
and what your performance expectations will be.

But it's really not worth creating your own Power-Supply,
( unless You just want to learn from the experience ),
because High-Quality Switching-Power-Supplies are very reasonably priced, and all over the Internet.
.
.
.

#### chajla

Joined Jan 26, 2023
6
i have no idea what the db107 is doing in there. can you explain please? i was using a 50a/400v bridge rectifier after the contraption.

thank you, low. im not trying to invent anything. i just want to understand whats happening and why. how does the "ringing-oscillation" work? if i have a 500w/230v lamp dimmed to 0.4adc/20vdc/50w (wattmeter) and connect a 470µf/63v cap after the rectifier the lamp dims even more and strongly flickers at what might be 100hz. even the cables/socket start humming. can you explain what is happening please? cap is not really heating up.

i didnt try at higher power because i only have larger 400v caps i didnt want to risk destroying.

(btw, i have used these and similar \$7 dimmers for many years on lamps, ventilators, hand drills, various heaters and even to run dc motors. they work well, although i never tried testing them at 5kw.)

#### MisterBill2

Joined Jan 23, 2018
14,185
A capacitor will charge up to the highest peak voltage applied to it through the rectifier. That has been a problem with inverter fed high voltage supplies developed by others. That is not new, it just had not been adequately understood.
An SCR or Triac dimmer operates by cutting the power, not the voltage. It chops off a portion of the sine wave, which in some conditions leaves the peak value. That peak will charge a capacitor to the peak voltage, that is how it works. The bridge rectifier assures that peaks in both directions will be able to charge the capacitor.

AND then " i wanted to keep it simple and not narrow down the focus on a specific situation. " So by not providing details we are assured of not having all of the information. Thus it is not likely that we can provide a better answer, nor that you will get that better answer.

#### Sensacell

Joined Jun 19, 2012
3,130
hi

can somebody please explain, what happens when there is a phase angle regulating circuit, then a bridge rectifier and a resistive load and you connect a capacitor after the rectifier parallel to the load?

thanks
The simple answer is that you get huge nasty spikes of current flowing.
This should explain what you are experiencing.

Buzzing, instability and a truly horrible power factor.

#### LowQCab

Joined Nov 6, 2012
2,866
Triacs work kinda-sorta-maybe-acceptable for low-power Electric-Motors, and Incandescent-Light-Bulbs.
But they are an extremely crude and cheap way of reducing AC-Power.
They produce tremendous levels of Electrical-Noise, or "Hash" which can cause all sorts of problems.

This is why You will never see them used in a heavy-duty, Industrial-Environment.
( I should qualify this to say "in a Phase-Angle Power controlling configuration" )
.
.
.

#### Ian0

Joined Aug 7, 2020
6,953
The simple answer is that you get huge nasty spikes of current flowing.
This should explain what you are experiencing.

Buzzing, instability and a truly horrible power factor.
SCR based regulators also need an inductor, between bridge and capacitor.
For 40A at mains voltage, it will need to be in the order of 10mH and will weigh about 25kg.

#### chajla

Joined Jan 26, 2023
6
thank you to everybody who replied. however there is still no exact explanation as to where the "ringing" comes from. what exactly is oscillating and why? i understand an inductance and capacitance will oscillate, but that doesnt seem to be the case here. where do " huge nasty spikes of current flowing" come from that were not there before the cap got connected?

so another example: instead of the lamp that works well there is a series of auto batteries after the rectifier (@60vdc), now no experimental cap connected. same thing, conductors vibrate, 13amp fuse blows with 7 aac showing on the input of the dimmer. for me its surprizing, as the batteries have such a huge capacitance that it should not matter what they are fed.

sorry, but i apparently need an explanation for dumies.

#### LowQCab

Joined Nov 6, 2012
2,866
Anytime there is a very abrupt change in Voltage,
there will be a certain amount of "over-shoot".

Under certain circumstances,
the Voltage "over-shoot" will then continue to over-shoot in the opposite direction,
and keep repeating this over and over again,
until it finally fades-out, and settles-down to the new Voltage.

This is called "Ringing" or "Oscillation", and happens in all Electrical-Circuits.

High levels of Capacitance or Inductance,
can make the Ringing or Oscillation become stronger, and last longer.
It can get so extreme, in some circumstances, that it can destroy Electronic-parts instantly.
.
.
.


text:
# Parsing

## Introduction

Parsing is the process of verifying that a sequence of tokens conforms to the rules of a formal grammar. It also involves constructing a (sometimes implicit) parse tree. A program that performs parsing is known as a parser.

Note: Some compilers combine lexing and parsing into a single phase.

There are three general types of parser covered in these notes:

• Top-down
• Bottom-up
• Universal

Top-down parsers build parse trees from the root to the leaves, whereas bottom-up parsers build parse trees by starting at the leaves and ending at the root [1, P. 192].

## Grammars

In the context of parsing, a grammar is a set of productions that describes how to form valid strings according to a formal language’s syntax.

Note: grammars can’t describe all aspects of a language, e.g., a context-free grammar can’t define that an identifier must be defined before it’s used [1, P. 209].

Terminals are the lexical elements used in specifying the rules for a grammar [1, P. 197].

Nonterminals are syntactic variables that denote sets of strings [1, P. 197].

A production is a rewrite rule specifying a symbol substitution that can be made.

A production head is a number of symbols (with at least one nonterminal). A production body is a sequence of terminals/nonterminals which can replace head.

A CFG (Context-Free Grammar) is a formal grammar where productions can be applied regardless of the context of a nonterminal. In other words, in a CFG a production head can only contain a single nonterminal.

Formally, a CFG $G$ consists of:

1. A finite set of nonterminals $N$.
2. A finite set of terminals $T$.
3. A start symbol $S$ ($S \in N$).
4. A finite set of productions $P$, where each production is in the form $X \rightarrow Y$ and $X \in N$ and $Y_i \in N \cup T \cup \{ \epsilon \}$.

Every construct that can be described by a regular expression can be described by a CFG, but the inverse is not true [1, P. 205].

A language that can be generated by a CFG is a context-free language [1, P. 200].

If two grammars generate the same language then the grammars are said to be equivalent [1, P. 200].

In these notes, productions are written in the following form:

Other productions would then define the nonterminals $expr$ and $stmt$, which are italicized.

A set of productions with a common head can be written using the vertical bar (|):

### Derivation

A derivation of a string for a grammar is a sequence of rule applications (productions) that transform the start symbol into a string.

A derivation can be represented as a tree (a parse tree).

A derivation proves that a string is an instance of an expression [1, P. 200].

The symbol $\overset{*} \Rightarrow$ means “derives in zero or more steps”.

The symbol $\overset{+} \Rightarrow$ means “derives in one or more steps”.

A sentential form is a string that is derivable from the start symbol. Formally, if $S \overset{*} \Rightarrow \alpha$, where $S$ is the start symbol for a grammar $G$, then $\alpha$ is a sentential form of $G$.

A sentence of $G$ is a sentential form with no nonterminals. The language generated by a grammar is its set of sentences [1, P. 200].

In leftmost derivations, the leftmost nonterminal expression in each sentential is chosen (denoted $\alpha \overset{*}{ \underset{\text{lm}} \Rightarrow} \beta$).

In rightmost derivations (canonical derivations), the rightmost nonterminal is chosen (denoted $\alpha \overset{*}{ \underset{\text{rm}} \Rightarrow} \beta$).

At each step in a derivation, there are two decisions to be made:

1. Which nonterminal should be replaced.
2. Which production should replace it.

[1, P. 200]

### Left recursion

A grammar is left-recursive if it has a nonterminal $A$ such that there is a derivation $A \overset{+} \Rightarrow A \alpha$ for some string $\alpha$ [1, P. 212].

Immediate left recursion is where there is a production $A \Rightarrow A \alpha$.

Top-down parsing cannot handle left-recursive grammars. In this case the recursion must be eliminated with the rule:

$A \rightarrow A \alpha \ \vert \ \beta$ becomes

[1, P. 212]

### Left factoring

Left factoring is the process of rewriting productions that share a common prefix, it can be done to make a grammar suitable to predictive parsing [1, P. 214].

For example, $A \rightarrow \alpha \beta_1 \ \vert \ \alpha \beta_2$ becomes:

[1, P. 214]

### Ambiguity

An ambiguous grammar produces more than one parse tree for a given sentence [1, P. 202].

Ambiguous grammars are bad since they leave the meaning of the program undefined.

One way to remove ambiguity is to rewrite grammar unambiguously. Another approach is to use disambiguating declarations. The most popular forms are associativity declarations and precedence declarations, e.g., to declare left associativity in bison: %left +.

### LL grammar

An LL(k) grammar is a grammar that can be parsed by an LL parser with k tokens of lookahead. e.g., LL(1) grammar can be parsed by an LL parser with 1 token lookahead.

An LL parser is a top-down parser that parses from left to right, constructing a left-most derivation.

A grammar is guaranteed not to be LL(1) if it is any of:

• Not left factored
• Left recursive
• Ambiguous

Even if a grammar does not meet one of these criteria, it is still not guaranteed to be LL(1).

### LR grammar

An LR(k) grammar is a grammar that can be parsed by an LR parser with k tokens of lookahead.

An LR parser is a top-down parser that parses from right to left, constructing a right-most derivation.

### BNF

BNF (Backus-Naur Form) is a syntax for context-free grammars.

Example:

<addition> ::= <number>+<number>
<number> ::= <sign><integer>|<integer>
<integer> ::= <digit>|<digit><integer>
<digit>::=0|1|2|3|4|5|6|7|8|9
<sign> ::= +|-

## Parse tree

A parse tree is a graphical representation of a derivation in which each non-leaf node represents the application of a production [1, P. 201].

The leaves of a parse tree are labelled by nonterminals or terminals [1, P. 201].

Parse trees ignore the order in which symbols in sentential form are replaced [1, P. 202].

### AST

An AST (Abstract Syntax Tree) is a tree representation of the syntactic structure of a program.

ASTs are similar to parse trees except that an AST’s interior nodes represent programming constructs, whereas a parse tree’s interior nodes represent nonterminals [1, P. 69].

Parse trees are sometimes called concrete syntax trees to distinguish them from ASTs.

An AST is a key data structure in a compiler.

First and follow sets are used to help construct top-down and bottom-up parsers.

### First sets

To compute $First(X)$ for all grammar symbols $X$:

1. If $X$ is terminal, $First(X) = \{X\}$.
2. If $X$ is nonterminal and $X \rightarrow Y_1, Y_2, ..., Y_k$ is a production for $k >= 1$, add $a$ to $First(X)$ if for some $i$, $a \in First(Y_i)$ and $First(Y_1) ... First(Y_{i-1}) \overset{*} \Rightarrow \epsilon$. If $\epsilon \in First(Y_j)$ for $j = 1, 2,... , k$ then add $\epsilon$ to $First(X)$.
3. If $X \rightarrow \epsilon$ is a production then add $\epsilon$ to $First(X)$.

[1, P. 221]

$Follow(A)$ for a nonterminal $A$ is the set of terminals that can appear immediately to the right of $A$ in some sentential form.

To compute $Follow(A)$ for all nonterminals $A$:

1. Place $in $Follow(S)$ (where $S$ is start symbol and$ is the input right end marker).
2. If there is a production $A \rightarrow \alpha B \beta$, add everything in $First(\beta)$ to $Follow(B)$ (except $\epsilon$ ).
3. If there is a production $A \rightarrow \alpha B$, or a production $A \rightarrow \alpha B \beta$ where $\epsilon \in First(\beta)$, add everything in $Follow(A)$ to $Follow(B)$.

[1, Pp. 221-2]

## Top-down parsing

Top-down parsing is the process of producing a parse tree from an input sequence starting from the root by following the rewriting rules of a grammar [1, P. 217].

At each step of a top-down algorithm, you must determine the production to be applied for a nonterminal. Once a production has been chosen, the rest of the parsing involves matching the production body with the input string [1, P. 217].

### Recursive descent parsing

A recursive descent parser is a top-down LL parser.

A recursive descent parser implements a procedure for each nonterminal. See Recursive descent parser Wiki article for full details.

A left-recursive grammar can cause a recursive descent parser to enter an infinite loop [1, P. 220].

### Predictive parsing

Predictive parsing is a special case of recursive descent parsing that doesn’t require backtracking [1, P. 222].

Predictive parsers work on LL(1) grammars since the proper production to apply for a nonterminal can be determined by looking at only the current input symbol [1, P. 222].

Predictive parsing can be implemented using a predictive parsing table. A predictive parsing table $T[A, \alpha]$ is a 2D array where $A$ is a nonterminal, $\alpha$ is a terminal (or the special end symbol $), and an entry is a production. An LL(1) grammar will produce a parsing table where each entry uniquely identifies a production [1, P. 225]. A table can be constructed for CFG $G$ with the following rules: • For each production $A \rightarrow \alpha$ in $G$: • For each terminal $a \in First(\alpha)$: $T[A, a] = A \rightarrow \alpha$ • If $epsilon \in First(\alpha)$, for each terminal $b \in Follow(A)$: $T[A, b] = A \rightarrow \alpha$. If $\epsilon \in First(\alpha)$ and $\ \in Follow(A)$ do: $T[A, \] = A \rightarrow \alpha$ [1, P. 224] The following pseudocode implements a table-driven predictive parser: let a be the first symbol of w; let X be the top of the stack symbol while X !=$:
if X == a:
pop the stack and let a be the next symbol of w
else if X is a terminal:
error()
else if T[X, a] is an error entry:
error()
else if T[X, a] = X -> Y_1, Y_2, ..., Y_n:
output production X -> Y_1, Y_2, ..., Y_n
pop the stack
push Y_n, ..., Y_2, Y_1 onto the stack with Y_1 on top
let X be the top stack symbol

[1, P. 227]

## Bottom-up parsing

Bottom-up parsing reduces a string to the start symbol by inverting productions (begin with input token sequence, end with start symbol).

Most parser generator tools use bottom-up parsing.

### Shift-reduce parsing

Shift-reduce parsing is a form of bottom-up parsing.

In shift-reduce parsing, a string is split into two substrings. The left substring contains terminals and nonterminals, and the right substring is unexamined. The division can be represented by a bar (|).

There are two primary moves: shift and reduce. Shift reads one token of input. Reduce applies a reduction to the right end of the left string.

A reduction is the inverse of a derivation step. It replaces input sequences with productions. A complete reduction sequence produces a rightmost derivation when reversed [1, P. 235].

A shift-reduce parser is normally implemented with a stack, where shift pushes a terminal onto the stack and reduce pops symbols from the stack and pushes a nonterminal onto the stack.

A handle is a substring that matches the body of a production, and whose reduction allows further reductions back to the start symbol. Handles only appear on the top of the stack, never inside [1, Pp. 235-7].

An item is a production with a dot (.) somewhere in the production body that indicates how much of a production has been seen at given point in the parsing process [1, P. 242].

The items (also known as LR(0) items) for $T \rightarrow (E)$ are:

•  $T \rightarrow .(E)$
•  $T \rightarrow (.E)$
•  $T \rightarrow (E.)$
•  $T \rightarrow (E).$

The production $A \rightarrow \epsilon$ generates an item $A \rightarrow .$ [1, P. 242].

A viable prefix is a prefix of a right-sentential form that does not continue past the right end of the rightmost handle of that sentential form. As long as a viable prefix is on the stack, no parsing error has been detected [1, P. 256].

For any grammar, the set of viable prefixes is a regular language and therefore can be validated by a finite automata. You can build an NFA that takes the stack as its input and either accepts or rejects the stack. The NFA can then be converted to a DFA using the powerset construction.

The rules to construct an NFA for recognizing viable prefixes VPs:

1. Add a dummy production $S' \rightarrow S$ to $G$.
2. NFA states are the items of $G$ (including the extra production).
3. For item $E \rightarrow \alpha . X \beta$, add transition $E \rightarrow alpha . X \beta \overset{X} \rightarrow E \rightarrow \alpha X . \beta$ ($X$ is nonterminal or terminal).
4. For item $E \rightarrow \alpha . X \beta$ and production $X \rightarrow \gamma$ ($X$ is nonterminal) add transition $E \rightarrow \alpha . X \beta \overset{\epsilon} \rightarrow X \rightarrow . \gamma$
5. Every state is an accepting state.
6. Start state is $S' \rightarrow .S$.

When the NFA has been converted into a DFA, each state is a set of items representing the possible current state of the automaton. These states are known as the canonical collections of items.

Item $X \rightarrow \beta . \gamma$ is valid for viable prefix $\alpha \beta$ if $S' \overset{*} \Rightarrow \alpha X \omega \Rightarrow \alpha \beta \gamma \omega$.

A conflict occurs if there are multiple actions leading to a valid parse at a given step. A shift-reduce conflict is when a grammar allows both a shift and a reduce for the same item. A reduce-reduce conflict is when there are two or more possible reductions.

#### LR(0) parser

An LR(0) parser is a shift-reduce parser that uses zero tokens of lookahead to determine what action to take.

Let $s$ be the terminating state when running $DFA(stack)$ and $t$ be the next input token.

At each stage, an LR(0) parser will reduce by $X \rightarrow \beta$ if $s$ contains item $X \rightarrow \beta .$. An LR(0) parser will reduce if $s$ contains item $X \rightarrow \beta. t \omega$.

Since there is zero lookahead, in any configuration of the parser there must be an unambiguous action that it can take.

#### SLR parser

An SLR parser is a type of LR parser.

An SLR parser improves on LR(0) with the modification that a reduction $X \rightarrow \beta$ is only made if $s$ contains $X \rightarrow \beta.$ and $t \in Follow(X)$.

SLR parsing algorithm:

1. Let $M$ be DFA for viable prefixes of $G$.
2. Let $\vert x_1, ..., x_n \$ be the initial configuration.
3. Repeat until configuration is $S \vert \ s$
1. Let $\alpha \vert \omega$ be current configuration.
2. Run $M$ on current stack $\alpha$.
3. If $M$ rejects $\alpha$, report parsing error.
4. If $M$ accepts $\alpha$ with items $I$, let $a$ be next input.
1. Shift if $X \rightarrow \beta . a \gamma \in I$.
2. Reduce if $X \rightarrow \beta . \in I$ and $a \in Follow(X)$.
3. Report parsing error if neither applies.

In the case of conflicts when following this algorithm, the grammar is not an SLR grammar.

## References

1. [1] A. V. Aho, M. S. Lam, R. Sethi, and J. D. Ullman, Compilers: Principles, Techniques, and Tools (2nd Edition). USA: Addison-Wesley Longman Publishing Co., Inc., 2006.


text:
# 6.9 Resolution-of-the-identity MP3

(May 16, 2021)

Joonho Lee working with Martin Head-Gordon developed and added an implementation of RI-MP3 with only cubic-storage requirement. The resulting code can be applied to larger systems than what is possible in CCMAN2 (6.10) due to the improved storage scaling. To use this method, one can set METHOD = RIMP3 along with AUXBASIS. In essence, it applies the RI approximation in Eq. 8.4 to the MP3 correlation energy evaluation. Interested readers are referred to Section 6.3 for further details on the Møller-Plesset perturbation theory.

In addition to RI-MP3 one can incorporate scaled RI-MP2 and RI-MP3 correlation energy contributions and alternative reference molecular orbitals (DFT and OOMP2) as seen in recent works by Bertels et al. and Rettig et al. The use of non-HF orbital references requires a composite job where the first calculation generates the orbital reference and the second first recomputes the SCF energy and Fock with HF on the fixed orbital reference and then proceeds with the correlated calculation.


text:
Middle Years

# 7.07 Similar triangle tests

Lesson

Two triangles are considered to be similar if one of them can be scaled up or down in size, and then rotated and/or reflected to match the other.

### Sides in fixed ratio

We can think of similarity as a weaker version of congruency, where corresponding distances do not need to be equal but instead are in some fixed ratio.

The fixed ratio of distances between two similar figures can be referred to as the scale factor.

### Similarity tests for triangles

There are four tests that we can use to test similarity between two triangles. If any one of these tests is satisfied then the two triangles must be similar. These four tests are:

• AAA: Three pairs of equal angles
• SSS: Three pairs of sides in the same ratio
• SAS: Two pairs of sides in the same ratio and an equal included angle
• RHS: Both have right angles, and the hypotenuses and another pair of sides are in the same ratio

These similarity tests can be proved to work in the same way that the congruency tests work, except with sides in fixed ratio rather than needing to be equal.

It is worth noting that, since all triangles have a fixed angle sum of $180^\circ$180°, having two matching angles is equivalent to having three matching angles.

### Corresponding sides and angles

To determine which sides are corresponding in two potentially similar triangles, we can use their positions with respect to any matching angles. If there is a common sized angle in both triangles, then the sides opposite those angles will be corresponding.

In this case, the sides labelled $x$x and $y$y are corresponding since they are opposite the same sized angle.

Similarly, if side lengths are given and two pairs of sides are in a fixed ratio between the two triangles, the angles between these pairs of sides will be corresponding.

In this case, since the side pairs $5$5 & $8$8 and $10$10 & $16$16 are in the fixed ratio of $1:2$1:2, the angles between these pairs are corresponding and therefore equal.

#### Practice questions

##### Question 1

Consider the two triangles in the diagram below:

1. Are $\triangle ABC$ABC and $\triangle DEF$DEF similar?

Yes, they satisfy AAA.

A

Yes, they satisfy SSS.

B

Yes, they satisfy SAS.

C

Yes, they satisfy RHS.

D

No, they are definitely not similar.

E

Unknown, there is not enough information.

F

Yes, they satisfy AAA.

A

Yes, they satisfy SSS.

B

Yes, they satisfy SAS.

C

Yes, they satisfy RHS.

D

No, they are definitely not similar.

E

Unknown, there is not enough information.

F
##### Question 2

Consider the following:

1. Which two of the following triangles are similar?

A

B

C

D

A

B

C

D
2. What similarity test does this pair satisfy?

AAA

A

SSS

B

SAS

C

RHS

D

AAA

A

SSS

B

SAS

C

RHS

D
##### Question 3

Consider the following diagram:

1. Are the triangles $\triangle ABC$ABC and $\triangle ADE$ADE definitely similar?

Yes

A

No

B

Yes

A

No

B
2. What similarity test does this pair satisfy?

AAA

A

SSS

B

SAS

C

RHS

D

AAA

A

SSS

B

SAS

C

RHS

D
3. Given that $DE=7$DE=7, what will the length of $BC$BC be?


text:
Compute various measures of internal consistencies for tests or item-scales of questionnaires.

Usage

cronbachs_alpha(x)

Arguments

x

A matrix or a data frame.

Value

The Cronbach's Alpha value for x.

Details

The Cronbach's Alpha value for x. A value closer to 1 indicates greater internal consistency, where usually following rule of thumb is applied to interpret the results: α < 0.5 is unacceptable, 0.5 < α < 0.6 is poor, 0.6 < α < 0.7 is questionable, 0.7 < α < 0.8 is acceptable, and everything > 0.8 is good or excellent.

References

Bland, J. M., and Altman, D. G. Statistics notes: Cronbach's alpha. BMJ 1997;314:572. 10.1136/bmj.314.7080.572

Examples

data(mtcars)
x <- mtcars[, c("cyl", "gear", "carb", "hp")]
cronbachs_alpha(x)
#> [1] 0.09463206


text:
## Topics on analysis in metric spaces.(English)Zbl 1080.28001

Oxford Lecture Series in Mathematics and its Applications 25. Oxford: Oxford University Press (ISBN 0-19-852938-4/hbk). viii, 133 p. (2004).
The book is a concise introduction to analysis in metric spaces but most topics make a good foundation also for convex and fractal geometry. The exposition covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems and Sobolev spaces; all these topics are developed in a general metric setting. One chapter is devoted to the minimal connection problem. It includes both the classical problem of the existence of geodesics in finitely compact metric spaces (due to Busemann) and the abstract Steiner problem (the solution of which is based on the Gromov embedding theorem). The last chapter contains a very general description of the theory of integration with respect to a nondecreasing set of functions. The strictly presented material is enlarged by numerous remarks and also by the end-of-chapter exercises.

### MSC:

 28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration 28A78 Hausdorff and packing measures 28A80 Fractals 28B05 Vector-valued set functions, measures and integrals 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 31C15 Potentials and capacities on other spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49J45 Methods involving semicontinuity and convergence; relaxation 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces


text:
Home/Class 11/Chemistry/

## QuestionChemistryClass 11

What volume of $${3}$$ molar $${H}{N}{O}_{{{3}}}$$ is needed to oxidise $${8}{g}$$ of $${F}{e}^{{{3}+}}$$, $${H}{N}{O}_{{{3}}}$$ gets converted to $${N}{O}$$ ?
(A) $${8}{m}{L}$$
(B) $${16}{m}{L}$$
(C) $${32}{m}{L}$$
(D) $${64}{m}{L}$$

Meq.of $${H}{N}{O}_{{{3}}}=\text{Meq.of}{F}{e}^{{{3}+}}$$
$${\left[\text{Eq.of}{H}{N}{O}_{{{3}}}={M}/{3}\right]}$$
or $${3}\times{3}\times{V}=\frac{{{8}}}{{{56}}}\times{1000}$$
$$\therefore{V}={15.87}{m}{L}$$


text:
# zbMATH — the first resource for mathematics

Effective field theory, past and future. (English) Zbl 1336.81007

##### MSC:
 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81V22 Unified quantum theories 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81R40 Symmetry breaking in quantum theory
Full Text:
##### References:
 [1] 1. S. Weinberg, Physica A96, 327 (1979). genRefLink(16, ’S0217751X16300076BIB001’, ’10.1016%252F0378-4371%252879%252990223-1’); genRefLink(128, ’S0217751X16300076BIB001’, ’A1979HB23700034’); genRefLink(64, ’S0217751X16300076BIB001’, ’1979PhyA...96..327W’); [2] 2. Y. Nambu, Phys. Rev. Lett.4, 380 (1960). genRefLink(16, ’S0217751X16300076BIB002’, ’10.1103%252FPhysRevLett.4.380’); genRefLink(128, ’S0217751X16300076BIB002’, ’A1960WQ68000019’); genRefLink(64, ’S0217751X16300076BIB002’, ’1960PhRvL...4..380N’); [3] 3. J. Goldstone, Nuovo Cimento 9, 154 (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962). [4] 4. M. L. Goldberger and S. B. Treiman, Phys. Rev.111, 354 (1956). genRefLink(16, ’S0217751X16300076BIB004’, ’10.1103%252FPhysRev.111.354’); genRefLink(128, ’S0217751X16300076BIB004’, ’A1958WB75000054’); genRefLink(64, ’S0217751X16300076BIB004’, ’1958PhRv..111..354G’); [5] 5. Y. Nambu and D. Lurie, Phys. Rev. 125, 1429 (1962); Y. Nambu and E. Shrauner, Phys. Rev. 128, 862 (1962). [6] 6. The name may be due to Murray Gell-Mann. The current commutation relations were given in M. Gell-Mann, Physics 1, 63 (1964). [7] 7. S. L. Adler, Phys. Rev. Lett. 14, 1051 (1965); Phys. Rev. 140, B736 (1965); W. I. Weisberger, Phys. Rev. Lett. 14, 1047 (1965). [8] 8. I emphasized this point in my rapporteur’s talk on current algebra at the 1968 ”Rochester” conference; see Proceedings of the 14th International Conference on High-Energy Physics, p. 253. [9] 9. S. Weinberg, Phys. Rev. Lett. 17, 616 (1966). The pion-nucleon scattering lengths were calculated independently by Y. Tomozawa, Nuovo Cimento A 46, 707 (1966). [10] 10. M. L. Goldberger, Y. Miyazawa and R. Oehme, Phys. Rev.99, 986 (1955). genRefLink(16, ’S0217751X16300076BIB010’, ’10.1103%252FPhysRev.99.986’); genRefLink(128, ’S0217751X16300076BIB010’, ’A1955WB70400051’); genRefLink(64, ’S0217751X16300076BIB010’, ’1955PhRv...99..986G’); [11] 11. S. Weinberg, Phys. Rev.140, B516 (1965). genRefLink(16, ’S0217751X16300076BIB011’, ’10.1103%252FPhysRev.140.B516’); genRefLink(128, ’S0217751X16300076BIB011’, ’A19656923200036’); genRefLink(64, ’S0217751X16300076BIB011’, ’1965PhRv..140..516W’); [12] 12. S. Weinberg, Phys. Rev. Lett.16, 879 (1966). genRefLink(16, ’S0217751X16300076BIB012’, ’10.1103%252FPhysRevLett.16.879’); genRefLink(128, ’S0217751X16300076BIB012’, ’A19667660100015’); genRefLink(64, ’S0217751X16300076BIB012’, ’1966PhRvL..16..879W’); [13] 13. J. Bernstein, S. Fubini, M. Gell-Mann and W. Thirring, Nuovo Cimento 17, 757 (1960); M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960); K. C. Chou, Soviet Physics JETP 12, 492 (1961). This theory, with the inclusion of a symmetry-breaking term proportional to the $$\sigma$$ field, was intended to provide an illustration of a ”partially conserved axial current,” that is, one whose divergence is proportional to the pion field. · Zbl 0095.43603 [14] 14. For Schwinger’s own development of this idea, see J. Schwinger, Phys. Lett. B 24, 473 (1967). It is interesting that in deriving the effective field theory of goldstinos in supergravity theories, it is much more transparent to start with a theory with linearly realized supersymmetry and impose constraints analogous to setting $$\sigma$$’=F/2, than to work from the beginning with supersymmetry realized non-linearly, in analogy to Eq. (7); see Z. Komargodski and N. Seiberg, to be published. [15] 15. S. Weinberg, Phys. Rev.166, 1568 (1968). genRefLink(16, ’S0217751X16300076BIB015’, ’10.1103%252FPhysRev.166.1568’); genRefLink(128, ’S0217751X16300076BIB015’, ’A1968A745700037’); genRefLink(64, ’S0217751X16300076BIB015’, ’1968PhRv..166.1568W’); [16] 16. S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2239 (1969); C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969). [17] 17. There is a complication. In some cases, such as SU(3)$$\times$$SU(3) spontaneously broken to SU(3), fermion loops produce G-invariant terms in the action that are not the integrals of G-invariant terms in the Lagrangian density; see J. Wess and B. Zumino, Phys. Lett. B 37, 95 (1971); E. Witten, Nucl. Phys. B 223, 422 (1983). The most general such terms in the action, whether or not produced by fermion loops, have been cataloged by E. D’Hoker and S. Weinberg, Phys. Rev. D 50, R6050 (1994). It turns out that for SU(N)$$\times$$SU(N) spontaneously broken to the diagonal SU(N), there is just one such term for N, and none for N=1 or 2. For N=3, this term is the one found by Wess and Zumino. [18] 18. For reviews, see S. Weinberg, in Lectures on Elementary Particles and Quantum Field Theory – 1970 Brandeis University Summer Institute in Theoretical Physics, Vol. 1, eds. S. Deser, M. Grisaru and H. Pendleton (The M.I.T. Press, Cambridge, MA, 1970); B. W. Lee, Chiral Dynamics (Gordon and Breach, New York, 1972). [19] 19. For a while it was not clear why there was not also a chiral U(1) symmetry, that would also be broken in the Lagrangian only by the quark masses, and would either lead to a parity doubling of observed hadrons, or to a new light pseudoscalar neutral meson, both of which possibilities were experimentally ruled out. It was not until 1976 that ’t Hooft pointed out that the effect of triangle anomalies in the presence of instantons produced an intrinsic violation of this unwanted chiral U(1) symmetry; see G. ’t Hooft, Phys. Rev. D 14, 3432 (1976). [20] 20. S. Weinberg, ” Critical Phenomena for Field Theorists,” in Understanding the Fundamental Constituents of Matter, ed. A. Zichichi (Plenum Press, New York, 1977). [21] 21. I thought it is appropriate to publish this in a festschrift for Julian Schwinger; see footnote 1. [22] 22. Unitarity corrections to soft-pion results of current algebra had been considered earlier by L.-F. Li and H. Pagels, Phys. Rev. Lett. 26, 1204 (1971); Phys. Rev. D 5, 1509 (1972); P. Langacker and H. Pagels, Phys. Rev. D 8, 4595 (1973). [23] 23. S. Weinberg, contribution to a festschrift for I. I. Rabi, Trans. N. Y. Acad. Sci. 38, 185 (1977). [24] 24. S. Weinberg, in Chiral Dynamics: Theory and Experiment – Proceedings of the Workshop, held at MIT, July 1994 (Springer-Verlag, Berlin, 1995). The terms in Eq. (13) that are odd in the pion field are given in Section 19.5 ofS. Weinberg, The Quantum Theory of Fields, Vol. II (Cambridge University Press, 1996). [25] 25. S. Weinberg, Phys. Lett. B 251, 288 (1990); Nucl. Phys. B 363, 3 (1991); Phys. Lett. B 295, 114 (1992). [26] 26. C. Ordoñez and U. van Kolck, Phys. Lett. B 291, 459 (1992); C. Ordoñez. L. Ray and U. van Kolck, Phys. Rev. Lett. 72, 1982 (1994); U. van Kolck, Phys. Rev. C 49, 2932 (1994); U. van Kolck, J. Friar and T. Goldman, Phys. Lett. B 371, 169 (1996); C. Ordoñez, L. Ray and U. van Kolck, Phys. Rev. C 53, 2086 (1996); C. J. Friar, Few-Body Systems Suppl. 99, 1 (1996). [27] 27. G. Benfatto and G. Gallavotti, J. Stat. Phys. 59, 541 (1990); Phys. Rev. 42, 9967 (1990); J. Feldman and E. Trubowitz, Helv. Phys. Acta 63, 157 (1990); 64, 213 (1991); 65, 679 (1992); R. Shankar, Physica A 177, 530 (1991); Rev. Mod. Phys. 66, 129 (1993); J. Polchinski, in Recent Developments in Particle Theory, Proceedings of the 1992 TASI, eds. J. Harvey and J. Polchinski (World Scientific, Singapore, 1993); S. Weinberg, Nucl. Phys. B 413, 567 (1994). [28] 28. C. Cheung, P. Creminilli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, J. High Energy Physics 0803, 014 (2008); S. Weinberg, Phys. Rev. D 73, 123541 (2008). [29] 29. F. Wilczek, Phys. Rev. Lett.43, 1570 (1979). genRefLink(16, ’S0217751X16300076BIB029’, ’10.1103%252FPhysRevLett.43.1571’); genRefLink(128, ’S0217751X16300076BIB029’, ’A1979HU10800009’); [30] 30. S. Weinberg, Phys. Rev. Lett.43, 1566 (1979). genRefLink(16, ’S0217751X16300076BIB030’, ’10.1103%252FPhysRevLett.43.1566’); genRefLink(128, ’S0217751X16300076BIB030’, ’A1979HU10800008’); genRefLink(64, ’S0217751X16300076BIB030’, ’1979PhRvL..43.1566W’); [31] 31. This is not true if the effective theory contains fields for the squarks and sleptons of supersymmetry. However, there are no renormalizable baryon or lepton violating terms in ”split supersymmetry” theories, in which the squarks and sleptons are superheavy, and only the gauginos and perhaps higgsinos survive to ordinary energies; see N. Arkani-Hamed and S. Dimopoulos, JHEP 0506, 073 (2005); G. F. Giudice and A. Romanino, Nucl. Phys. B 699, 65 (2004); N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, Nucl. Phys. B 709, 3 (2005); A. Delgado and G. F. Giudice, Phys. Lett. B 627, 155 (2005). [32] 32. The effective field theories derived by integrating out heavy particles had been considered by T. Appelquist and J. Carrazone, Phys. Rev. D 11, 2856 (1975). In 1980, in a paper titled ”Effective Gauge Theories,” I used the techniques of effective field theory to evaluate the effects of integrating out the heavy gauge bosons in grand unified theories on the initial conditions for the running of the gauge couplings down to accessible energies: S. Weinberg, Phys. Lett. B 91, 51 (1980). [33] 33. J. F. Donoghue, Phys. Rev. D 50, 3874 (1884); Phys. Lett. 72, 2996 (1994); lectures presented at the Advanced School on Effective Field Theories (Almunecar, Spain, June 1995), arXiv:gr-qc/9512024; J. F. Donoghue, B. R. Holstein, B. Garbrecth and T. Konstandin, Phys. Lett. B 529, 132 (2002); N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev. D 68, 084005 (2003). [34] 34. This was first proposed in my 1976 Erice lectures; see footnote 20. [35] 35. S. Weinberg, in General Relativity, eds. S. W. Hawking and W. Israel (Cambridge University Press, 1979), p. 700; H. Kawai, Y. Kitazawa and M. Ninomiya, Nucl. Phys. B 404, 684 (1993); Nucl. Phys. B 467, 313 (1996); T. Aida and Y. Kitazawa, Nucl. Phys. B 401, 427 (1997); M. Niedermaier, Nucl. Phys. B 673, 131 (2003). [36] 36. L. Smolin, Nucl. Phys. B 208, 439 (1982); R. Percacci, Phys. Rev. D 73, 041501 (2006). [37] 37. J. Ambjørn, J. Jurkewicz and R. Loll, Phys. Rev. Lett. 93, 131301 (2004); Phys. Rev. Lett. 95, 171301 (2005); Phys. Rev. D 72, 064014 (2005); Phys. Rev. D 78, 063544 (2008); and in Approaches to Quantum Gravity, ed. D. Oríti (Cambridge University Press). [38] 38. M. Reuter, Phys. Rev. D 57, 971 (1998); D. Dou and R. Percacci, Class. Quantum Grav. 15, 3449 (1998); W. Souma, Prog. Theor. Phys. 102, 181 (1999); O. Lauscher and M. Reuter, Phys. Rev. D 65, 025013 (2001); Class. Quantum Grav. 19, 483 (2002); M. Reuter and F. Saueressig, Phys Rev. D 65, 065016 (2002); O. Lauscher and M. Reuter, Int. J. Mod. Phys. A 17, 993 (2002); Phys. Rev. D 66, 025026 (2002); M. Reuter and F. Saueressig, Phys. Rev. D 66, 125001 (2002); R. Percacci and D. Perini, Phys. Rev. D 67, 081503 (2002); Phys. Rev. D 68, 044018 (2003); D. Perini, Nucl. Phys. C (Proc. Suppl.) 127, 185 (2004); D. F. Litim, Phys. Rev. Lett. 92, 201301 (2004); A. Codello and R. Percacci, Phys. Rev. Lett. 97, 221301 (2006); A. Codello, R. Percacci, and C. Rahmede, Int. J. Mod. Phys. A 23, 143 (2008); M. Reuter and F. Saueressig, arXiv:0708.1317; P. F. Machado and F. Saueressig, Phys. Rev. D 77, 124045 (2008); A. Codello, R. Percacci and C. Rahmede, Ann. Phys. 324, 414 (2009); A. Codello and R. Percacci, arXiv:0810.0715; D. F. Litim, arXiv:0810.3675; H. Gies and M. M. Scherer, arXiv:0901.2459; D. Benedetti, P. F. Machado and F. Saueressig, arXiv:0901.2984, arXiv:0902.4630; M. Reuter and H. Weyer, arXiv:0903.2971. [39] 39. F. J. Wegner and A. Houghton, Phys. Rev. A8, 401 (1973). genRefLink(16, ’S0217751X16300076BIB039’, ’10.1103%252FPhysRevA.8.401’); genRefLink(128, ’S0217751X16300076BIB039’, ’A1973Q322600050’); genRefLink(64, ’S0217751X16300076BIB039’, ’1973PhRvA...8..401W’); [40] 40. J. Polchinski, Nucl. Phys. B 231, 269 (1984); C. Wetterich, Phys. Lett. B 301, 90 (1993). [41] 41. A. Codello, R. Percacci and C. Rahmede, Int. J. Mod. Phys. A23, 143 (2008). [Abstract] genRefLink(128, ’S0217751X16300076BIB041’, ’000252898300002’); genRefLink(64, ’S0217751X16300076BIB041’, ’2008IJMPA..23..143C’); [42] 42. A. Codello, R. Percacci and C. Rahmede, Ann. Phys.324, 414 (2009). genRefLink(16, ’S0217751X16300076BIB042’, ’10.1016%252Fj.aop.2008.08.008’); genRefLink(128, ’S0217751X16300076BIB042’, ’000263807700011’); genRefLink(64, ’S0217751X16300076BIB042’, ’2009AnPhy.324..414C’); [43] 43. D. Benedetti, P. F. Machado and F. Saueressig, arXiv:0901.2984, arXiv:0902.4630. [44] 44. A. Bonanno and M. Reuter, Phys. Rev. D 65, 043508 (2002); Phys. Lett. B 527, 9 (2002); M. Reuter and F. Saueressig, J. Cosmol. Astropart. Phys. 09, 012 (2005).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Question using Hubble's Constant

## Homework Statement

Converting Hubble's constant in SI units of s-1 into kms-1 Mpc-1 (kilometers per second per megaparsec)
Number given is 2.31 x 10^-18 s-1.

## Homework Equations

H = v/d (where H is Hubble's constant, v = velocity and d = distance)

## The Attempt at a Solution

I can see how you get the answer but it took me a while to see the reasoning for what to divide/multiply by. I reasoned that if you use H = v/d then 2.31 x 10^-18 s-1 can be re-written as 2.31 x 10^-18 ms-1 m-1 (meters per second per metre). I then treated the numerator and denominator separately. So to get from m/s to km per second you divide by 1000. Then to get from per metre to per megaparsec you multiply by the number of metres in a megaparsec. So overall you get 2.31 x 10^-18 x 3.09 x 10^22 / 1000 which gives 71.4 kms-1 Mpc-1. Using the same logic I don't understand something on the additional notes in the mark scheme for this question. I have uploaded the question and markscheme. They have shown an example of if you forget to convert hubble's constant into s-1 otherwise it will be in years-1. They then work this answer through the same way and give less marks for it however they say that the units are still kms-1 Mpc-1 at the end but that seems wrong to me. If you take hubble's constant in years-1 instead of seconds-1 and then multiply by the same number and divide by the same number surely its impossible to end up with the same final units? or am I missing something?

#### Attachments

• 511.7 KB Views: 120