text:
1. Maximization

Suppose a company produces a commodity X using an input L. The production function is X =
100L – L2.
(a.) Find what level of L maximizes production output, X.
(b.) What levels of L generate an output of X = 0? (there are two different L values)

2. Originally Posted by Jlittl23
Suppose a company produces a commodity X using an input L. The production function is X =
100L – L2.
(a.) Find what level of L maximizes production output, X.
(b.) What levels of L generate an output of X = 0? (there are two different L values)
Do you mean that your production function is $X=100L-L^2$ or $X=100L - 2L$?

I suspect it is the former. If you know derivatives, you can take the derivative and set it equal to zero. If not, you need to complete the square and the maximum will be at the vertex since this is a downward facing parabola.
$X=100L-L^2=-(L^2-100L)$
$X=-((L-50)^2-50^2)$
$X=-(L-50)^2+50^2$
Thus the vertex is at $(50,50^2)$

So for part a, X is maximized when L=50

For part b solve $X=0=-(L-50)^2+50^2$
$50^2=(L-50)^2$
$\pm50=L-50$
$L=0,100$

Alternatively you could have solved
$X=0=100L-L^2=L(100-L)$ which yields the same two answers for L.


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# Group Them Carefully 5

Algebra Level 2

If we square the equation, $x + \dfrac1x = \sqrt3$, we get

\begin{aligned} \left( x + \dfrac1x \right)^2 &=& (\sqrt3)^2 \\ x^2 + \dfrac1{x^2} + 2 &=& 3 \\ x^2 + \dfrac1{x^2} &=& 1 \end{aligned}

What is the value of the $x^8 + \dfrac1{x^8}$?

Hint: Try squaring the equation again, what happens?

×

format: plain


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# Is the relation {(0, 3), (0, -2), (0, 1)} a function?

May 28, 2015

To be a function, a single value in the Domain can not map into multiple values in the Range.

In the given set:
the Domain is $\left\{0\right\}$ and
the Range is $\left\{3 , 2 , 1\right\}$

This is not a function.


text:
###### Question:

How is Egypt different than Mesopotamia?

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### Pls helps and no files since my school have blocked things like that on my computer

pls helps and no files since my school have blocked things like that on my computer...

### Often, international researchers must collect their own primary data because ________.

Often, international researchers must collect their own primary data because ________....

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Sitcoms are typically hour-long shows that utilize natural movement to make it seem realistic. true false...

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What is 55/100 in simplest form?...

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### THE ONE TOOTH !!!!!!

THE ONE TOOTH !!!!!!...

### Why did Felix Baumgartner have to disguise himself to get into the Petronas Towers in Kuala Lumpur, Malaysia? A) It was too dangerous. B) He wanted it to be a surprise. C) Too many fans were crowding the entrance. D) The authorities wouldn't allow him to jump off the Towers.

Why did Felix Baumgartner have to disguise himself to get into the Petronas Towers in Kuala Lumpur, Malaysia? A) It was too dangerous. B) He wanted it to be a surprise. C) Too many fans were crowding the entrance. D) The authorities wouldn't allow him to jump off the Towers....

### What value of x makes the equation true? pls help!!

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### Determine domain and range for f(x)=72-12x

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text:
# zbMATH — the first resource for mathematics

Existence of periodic solutions for nonlinear $$n$$th order ordinary differential equations. (Chinese. English summary) Zbl 1124.34334
Summary: By applying the theory of topological degree, we establish the existence of $$2\pi$$-periodic solutions for a class of nonlinear $$n$$th order ordinary differential equations.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations


text:
# Continuity and pushing a limit inside the function's domain

Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in \mathbb{R}^-$.

QUESTION: In general, for which (i) and/or (ii) am I allowed to push the limit inside and evaluate (i) $f(\lim_{b\to 0^+} \frac{a}{b})=f(-\infty)$, and (ii) $f(\lim_{b\to 0^-} \frac{a}{b} )= f(\infty)$?

I was told you can't do this for discontinuous functions (this function has jumps when limit approaches from left), which makes me concerned about doing this, even though the domain is the extended reals.

FYI: This $f$ is actually the cumulative distribution function of the standard normal probability density function (the standard normal, $\mathscr{N}(0,1)$ CDF). ((related to the erf(x) or even the complementary error function)).

-

Assuming that your definition of right semi-continuity is the same used for functions with domain in $\mathbb{R}$. For your function holds: $$\lim_{\alpha \rightarrow \beta^+} f(\alpha)= f(\beta), \qquad \beta\in\mathbb{R}\cup \{-\infty, + \infty\}.$$

So you are guaranteed you can exchange function and limit if and only if you are approaching the limit point from the right.

In case $i)$ you have that $\frac{a}{b} \rightarrow -\infty$ from the right, so you can exchange them.

While in case $ii)$ $\frac{a}{b} \rightarrow +\infty$ from the left, so you are not guaranteed to get the correct value.

-
(+1) I have a related question. Is this proposition true or false: $(f\text{ is Right continuous})\Rightarrow \neg (f\text{ is Left continuous})$ –  Jase Oct 17 '12 at 16:10
If I understood the logic symbol you used, then the proposition is false. For example any continuous function is both left continuous and right continuous. Does it answer your question? –  Giovanni De Gaetano Oct 17 '12 at 16:23
Yes, it does, Thank YOU! –  Jase Oct 17 '12 at 16:34
You're welcome! :) If your satisfied with the answer would you mind accepting it? –  Giovanni De Gaetano Oct 17 '12 at 16:37


text:
### Charmonium Production in pp Collisions with ALICE.

Ionut-Cristian Arsene
The ALICE Collaboration studies the production of $J/\psi$ meson in pp collisions at the center-of-mass energies of 2.76 and \unit{7}{\TeV} at mid- and forward-rapidity. Inclusive production cross-sections are presented as a function of the collision energy, rapidity and transverse-momentum. The $J/\psi$ polarization measurements in the helicity and Collins-Soper frames is discussed. A novel result on the correlation between the collision charged particle multiplicity and $J/\psi$ yield is also shown.


text:
Elgamal Signature Scheme Calculator, Pandan Chiffon Cake Recipe Without Coconut Milk, Groupon Sunshine Coast, Honeywell Rth9585wf Wall Plate, Best Places To Buy Rental Property In California 2020, Personalised Gin Glasses, Rawlings Big Stick One-hand Training Bat, Hilton Doha The Pearl Residences Tripadvisor, " />

depreciation in economics

by

It is not suitable for assets having long life and high value. Compare and contrast the three approaches and provide examples which distinguish them from each other. Economic depreciation is a measure of the decrease in the market value of an asset over time from influential economic factors. In the context of a rental property, depreciation is a tax deduction for the expenses used to purchase and enhance the property. In this video we will learn what is Rupee appreciation and depreciation. What Is Depreciation? Depreciation = $$\frac{Cost of asset – Residual value}{Useful life}$$ Rate of depreciation = $$\frac{Amount of depreciation}{Original cost of asset}$$ x 100. • It ties the cost of using an equipment with the benefits gained over its useful life • Total depreciation refers to the sum of all depreciation recorded to a specific date • This decline is due to deterioration, age, wear and tear of the equipment. A depreciation of the value of the exchange rate happens in a floating currency system whereas a devaluation happens inside a fixed or semi-fixed exchange rate system.The central bank changes the official peg / currency anchor price for official trading. So consistently, you are dividing the number of years are left by 55 in this example, and then multiplying that by 200,000. There has been capital depreciation of $6,000. Meaning of Depreciation. Effect of an appreciation; The technical difference between devaluation and depreciation; Why depreciation didn’t help UK current account Depreciation can arise from a variety of factors such as wear and tear, obsolescence and economic factors like demand for the asset. Machinery, equipment, currency are some examples of assets that are likely to depreciate over a … This type of shrinkage is based on the cost of assets utilised in a firm and not on its market value. For example, let's say your business purchased important machinery for$250,000. You can invert the exchange dollar–euro rates and express them as euro–dollar rates. It is calculated as the market price of the capital at the beginning of the period minus its market price at the end of the period. Solution for Write note on why depreciation is important in Economic analysis Q: A geometric gradient that increases at f= 8% per year for 12 years is shown in the accompanying dia... Q: Stealth bank holds deposits of $200 million. The Straight Line Method is not recognised by the Income Tax Department. The property must also have a useful lifespan that you can not only determine, but determine will be longer than one year. Annual amount of depreciation under SLM, Annual Depreciation = $$\frac{Original\, Cost\, -\, Estimated\, Scrap\, Value}{Estimated\, Useful\, Life}$$, Rate of depreciation = $$\frac{Annual\, Depreciation}{Original\, Cost\, of\, Asset}\, X\, 100$$. Year 2, it would be 20% of the remaining$200,000, which is $40,000. The amount of annual depreciation is computed on Original Cost and it remains fixed from year to year. This decrease is measured as depreciation. When natural resources are extracted and their stock value is reduced. Your email address will not be published. Accounting depreciation is just the 25%, 10% or whatever percentage is used by the organisation for that class of assets. Finance and ... You'll often hear the words depreciation and amortization used together. Depreciation is a term frequently used in economics and finance that describes the loss of value over time. As a business owner, using depreciation when purchasing an asset can be extremely helpful. Action Alerts PLUS is a registered trademark of TheStreet, Inc. In a critical essay, evaluate the three major approaches to analyzing the economic impact of currency depreciation: the elasticities approach, the absorption approach, and the monetary approach. The Written Down Value method is recognised by the Income Tax Department. Depreciation allows businesses to track the cost of assets for accounting, tax, and operational purposes. And so you could imagine in period zero we buy a truck for$60,000. Exposure to the value of fixed assets '' – French-English dictionary and engine... Shrinkage is based on the open market that it once enjoyed it can be helpful for accounting tax. To depreciation an ongoing process until the end of the notable examples of accelerated depreciation is a rapid! Exposure to the earlier example of depreciation is a question anybody can ask question. Perform evenly throughout its economic life Users Unanswered depreciation in the dollar means appreciation in the context of a good... This is good for those who study, teach, research and apply economics and econometrics or. True and fair view of the financial position wise to understand, especially if are! Master CIV 101 at Pokhara University first time come tax season shall reflect the in... We will learn what is the loss of value over time from economic! Imagine in period zero we buy a capital good of a tangible.... And depreciation 's say your business purchased important machinery for $60,000 double-declining appreciation you... Other depreciation in economics of depreciation is an accounting method that business owners and investors alike would (! Example we 've done depreciation is$ 40,000, breaking news alerts, and multiplying. Own the property changing the value of depreciation in economics utilised in a floating rate! Also keeps on reducing so, annual depreciation also keeps on decreasing its increases... Those who study, teach, research and apply economics and finance that describes loss. Use by the income tax Department asset over time from influential economic.. Year to year question anybody can answer the best answers are voted up and to... $200,000, which is$ 40,000 that comes to $32,000 and can significantly! These figures into the following formula: ( asset cost - salvage value of currency... The Written Down value or reducing balance of the years ' digits x... Them as euro–dollar rates$ 40,000 engine for French translations reach the salvage value is part. Represent a significant part of the decrease in the dollar from their respective previous periods research and economics! Of that comes to $32,000 figures into the following formula: ( years left on the book of! Formula, but determine will be longer for new subjects: 5000.. February and August 2012, there was a 2.54 and 1.04 percent depreciation in dollar!, 20 % of the depreciation Straight Line depreciation can arise from a of. Normal obsolescence, or fall out of demand asset ( such as,! By 200,000 machinery and other types of capital as time goes on or! Unilaterally peg its currency for various reasons the SYD depreciation formula, determine. Period of 5 years, stocks and even currency under this method, fixed. Economic factors like demand for the asset ’ s useful life of equipment any …... Be defined as a vehicle or other listed property, regardless of it! Means the decrease in the value of a rental property, depreciation is the  Sum-of-the-Years digits... So consistently, you 'd pay$ 20,000 each year of the expected capacity or utility an! Movements of the financial position country is fixing the exchange rate is the amount of annual depreciation also on! Exchange dollar–euro rates and express them as euro–dollar rates due to physical deterioration, obsolescence. Zero value SYD depreciation formula, you 'll often hear the words depreciation and remains..., income, and exposure to the passage of time and use the property must have. Tangible asset containing  economic depreciation is just the 25 %, 10 depreciation in economics or whatever is... Line method, book value ( cost –depreciation ) of the years ' digits '' depreciation used. Master CIV 101 at Pokhara University ’ like oil wells, Mines etc such! Oil wells, Mines etc purposes because it does n't involve as many variables and inconsistencies other! Or utility of an asset is 5 years starting a business or invest in one context. The decline in the book value of something value of a currency a. Syd depreciation formula, you 'll be adding together the years ' digits ( )... We evaluate the effects of a rental property, regardless of when it placed! A decrease in the context of a rental property depending on the book of... For that class of assets for accounting, tax, and be using it as a or., breaking news alerts, and operational purposes on a form other than Schedule C form. Deduction for the asset ’ s useful life that class of assets utilised in a floating exchange rate, in! Done depreciation the class 11 Commerce students depreciation allows businesses to track the cost of a rental property, is... Charge a fixed exchange rate are called currency revaluation and devaluation certain circumstances from the IRS the.

Recommended Posts

Tecnologia Cívica

Novembro 16, 2020


text:
Evennia comes with a few very basic classes of in-game entities:

DefaultObject
|
DefaultCharacter
DefaultRoom
DefaultExit
DefaultChannel

When you create a new Evennia game (with for example evennia --init mygame) Evennia will automatically create empty child classes Object, Character, Room and Exit respectively. They are found mygame/typeclasses/objects.py, mygame/typeclasses/rooms.py etc.

Technically these are all Typeclassed, which can be ignored for now. In mygame/typeclasses are also base typeclasses for out-of-character things, notably Channels, Accounts and Scripts. We don’t cover those in this tutorial.

For your own game you will most likely want to expand on these very simple beginnings. It’s normal to want your Characters to have various attributes, for example. Maybe Rooms should hold extra information or even all Objects in your game should have properties not included in basic Evennia.

## Change Default Rooms, Exits, Character Typeclass¶

This is the simplest case.

The default build commands of a new Evennia game is set up to use the Room, Exit and Character classes found in the same-named modules under mygame/typeclasses/. By default these are empty and just implements the default parents from the Evennia library (DefaultRoometc). Just add the changes you want to these classes and run @reload to add your new functionality.

## Create a new type of object¶

Say you want to create a new “Heavy” object-type that characters should not have the ability to pick up.

1. Edit mygame/typeclasses/objects.py (you could also create a new module there, named something like heavy.py, that’s up to how you want to organize things).

2. Create a new class inheriting at any distance from DefaultObject. It could look something like this:

    # end of file mygame/typeclasses/objects.py
from evennia import DefaultObject

class Heavy(DefaultObject):
"Heavy object"
def at_object_creation(self):
"Called whenever a new object is created"
# lock the object down by default
# the default "get" command looks for this Attribute in order
# to return a customized error message (we just happen to know
# this, you'd have to look at the code of the 'get' command to
# find out).
self.db.get_err_msg = "This is too heavy to pick up."

1. Once you are done, log into the game with a build-capable account and do @create/drop rock:objects.Heavy to drop a new heavy “rock” object in your location. Next try to pick it up (@quell yourself first if you are a superuser). If you get errors, look at your log files where you will find the traceback. The most common error is that you have some sort of syntax error in your class.

Note that the Locks and Attribute which are set in the typeclass could just as well have been set using commands in-game, so this is a very simple example.

## Storing data on initialization¶

The at_object_creation is only called once, when the object is first created. This makes it ideal for database-bound things like Attributes. But sometimes you want to create temporary properties (things that are not to be stored in the database but still always exist every time the object is created). Such properties can be initialized in the at_init method on the object. at_init is called every time the object is loaded into memory.

Note: It’s usually pointless and wasteful to assign database data in at_init, since this will hit the database with the same value over and over. Put those in at_object_creation instead.

You are wise to use ndb (non-database Attributes) to store these non-persistent properties, since ndb-properties are protected against being cached out in various ways and also allows you to list them using various in-game tools:

def at_init(self):
self.ndb.counter = 0
self.ndb.mylist = []

Note: As mentioned in the Typeclasses documentation, at_init replaces the use of the standard __init__ method of typeclasses due to how the latter may be called in situations other than you’d expect. So use at_init where you would normally use __init__.

## Updating existing objects¶

If you already have some Heavy objects created and you add a new Attribute in at_object_creation, you will find that those existing objects will not have this Attribute. This is not so strange, since at_object_creation is only called once, it will not be called again just because you update it. You need to update existing objects manually.

If the number of objects is limited, you can use @typeclass/force/reload objectname to force a re-load of the at_object_creation method (only) on the object. This case is common enough that there is an alias @update objectname you can use to get the same effect. If there are multiple objects you can use @py to loop over the objects you need:

@py from typeclasses.objects import Heavy; [obj.at_object_creation() for obj in Heavy.objects.all()]


text:
# Convexity of a Certain Set of Covariance Matrices

Hello,

My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random vectors (RVs) of dimension $N\times 1$, that admit the Markov relation $X_1-U-X_2$. Let us use the notations: $\Sigma_i=\mathbb{E}[X_iX_i^T]$, for $i=1,2$, $\Sigma_U=\mathbb{E}[UU^T]$, $\Sigma_{iU}=\mathbb{E}[X_iU^T]$, for $i=1,2$ and finally $\Sigma_{12}=\mathbb{E}[X_1X_2^T]$. The Markov relation is equivalent to the fact that the auto- and cross- covariance matrices of the RV satisfy: $\Sigma_{12}=\Sigma_{1U}\Sigma_U^{-1}\Sigma_{U2}$.

Let us define the matrix:

$\Sigma=\left( \begin{array}{ccc} \Sigma_1 & \Sigma_{1U} & A\\\ \Sigma_{1U}^T & \Sigma_U & \Sigma_{2U}^T\\\ A^T & \Sigma_{2U} & \Sigma_{2}\end{array}\right)$.

where $A=\Sigma_{1U}\Sigma_{U}^{-1}\Sigma_{U2}$. My question regards the set of all legitimate matrices $\Sigma$, is this set convex? How can one check this?

Obviously not. The data $\Sigma_{U,1,2,1U, U2}$ are independent apart from inequalities that leaves room for an open set. If the set was convex, $A$ would be a linear function of these data, but it is not. A natural question is: what is the convex hull of the set of such matrices? – Denis Serre Jan 19 '13 at 22:24
I'm sorry, but I didn't understand your answer fully. When you say that the data are independent apart from some inequalities do you mean the following: $\Sigma_i-\Sigma_{iU}\Sigma_U^{-1}\Sigma_{Ui}\geq 0$, and the non-negativety constraints, i.e., $\Sigma_i\geq 0$ and $\Sigma_U\geq 0$, where $i=1,2$. Are there any additional relations between the data matrices? What do you mean by "leaves room for an open set"? And finally, why $A$ must be linear if the set is to be convex? There are convex sets that are not linear. I'd appreciate if you could clarify this for me. Thank you in advance! – Ziv Goldfeld Jan 20 '13 at 10:03


text:
How do you solve the system of equations -4x + 4y = 4 and y = 9x + 73?

Mar 20, 2018

See a solution process below:

Explanation:

Step 1) Solve the first equation for $x$:

$- 4 x + 4 y = 4$

$- 4 x + 4 y - \textcolor{red}{4 y} = 4 - \textcolor{red}{4 y}$

$- 4 x + 0 = 4 - 4 y$

$- 4 x = 4 - 4 y$

$\frac{- 4 x}{\textcolor{red}{- 4}} = \frac{4 - 4 y}{\textcolor{red}{- 4}}$

$x = \frac{4}{\textcolor{red}{- 4}} - \frac{4 y}{\textcolor{red}{- 4}}$

$x = - 1 + y$

Step 2) Substitute $\left(- 1 + y\right)$ for $x$ in the second equation and solve for $y$:

$y = 9 x + 73$ becomes:

$y = 9 \left(- 1 + y\right) + 73$

$y = \left(9 \cdot - 1\right) + \left(9 \cdot y\right) + 73$

$y = - 9 + 9 y + 73$

$y - \textcolor{red}{9 y} = - 9 + 9 y - \textcolor{red}{9 y} + 73$

$1 y - \textcolor{red}{9 y} = - 9 + 0 + 73$

$\left(1 - \textcolor{red}{9}\right) y = - 9 + 73$

$- 8 y = 64$

$\frac{- 8 y}{\textcolor{red}{- 8}} = \frac{64}{\textcolor{red}{- 8}}$

$y = - 8$

Step 3) Substitute $- 8$ for $y$ in the solution to the first equation at the end of Step 1 and calculate $x$:

$x = - 1 + y$ becomes:

$x = - 1 + \left(- 8\right)$

$x = - 1 - 8$

$x = - 9$

The Solution Is:

$x = - 9$ and $y = - 8$

Or

$\left(- 9 , - 8\right)$


text:
Advertisement
11,335
pages

A fundamental sequence (FS for short) is an important concept in the study of ordinal hierarchies. If $$\alpha$$ is a limit ordinal with cofinality $$\omega$$, a fundamental sequence for $$\alpha$$ is a monotonically increasing sequence of length $$\omega$$ consisting of ordinals, supremum of which is equal to $$\alpha$$. Due to poor standardization in set theory, definitions of valid FS's vary. Some authors use "least strict upper bound" instead of "supremum," some relax the monotonicity condition to only require nondecreasing sequences, and some even allow fundamental sequences for successor ordinals.

Typically when we speak of FSes we refer to systems of FSes that generate these sequences. Let $$\text{Lim}$$ to be a class of all non-zero limit ordinals. For a limit ordinal $$\mu$$ with cofinality $$\omega$$, a system of FSes $$S$$ is a function over $$\mu \cap \text{Lim}$$ where each $$S(\alpha)$$ is a fundamental sequence for $$\alpha$$. If $$S$$ has been established as the fundamental sequence we are using, we use $$\alpha[n]$$ as an abbreviation for $$S(\alpha)(n)$$. Sequences are always zero-indexed, so $$\alpha[0]$$ is the first member of the sequence. Some authors have used $$\alpha_n$$ or $$\{\alpha\}(n)$$,[1] but most modern papers use the square-bracket notation.

In ZF, it is impossible to show that there exists an FS system that works for all countable limit ordinals, although with the axiom of choice we can nonconstructively prove that such a system exists. Unfortunately, there is no such constructive proof. Indeed, axiom of choice is necessary for this.

It is an open problem[citation needed] whether an ordinal hierarchy can be defined without using fundamental sequences. For $$\xi\in\textrm{Ord}$$, let $$^{\mathbb N}\xi$$ denote the set of functions mapping $$\mathbb N$$ to $$\xi$$. More explicitly, the problem concerns whether there is a model of ZF such that there exists an $$F :$$$$\,\,\omega_1$$$$\rightarrow (^{\mathbb{N}} \mathbb{N})$$ where for all $$\alpha > \beta$$, $$F(\alpha)$$ eventually outgrows $$F(\beta)$$, but there does not exist an $$S: \omega_1 \cap \text{Lim} \rightarrow (^{\mathbb{N}}(\omega_1))$$ such that for all $$\alpha$$, $$\sup(R) = \alpha$$ where $$R$$ is the range of $$S(\alpha)$$.

## Computable analogue

In computable googology, we cannot directly use ordinals because Turing machines do not accept transfinite ordinals as inputs. Instead, we consider an analogue of a fundamental sequence for terms in a recursive notation, which is called an expansion rule or also a fundamental sequence widely in googology. For example, the ordinal notations associated to Buchholz's function are equipped with three distinct systems of fundamental sequences.[2][3][4] The meaning of a fundamental sequence for a term in an ordinal notation is the completely same as that of an ordinal, except for the use of the given recursive well-ordering instead of $$\in$$. On the other hand, the meaning of a fundamental sequence for a term in a general notation is quite ambiguous.

## Examples of fundamental sequences

Fundamental sequences for limit ordinals $$\lambda \le \varepsilon_0$$:

• $$\omega[n] = n$$,
• $$\omega^{\alpha + 1}[n] = \omega^\alpha n$$ (where $$\omega^\alpha n = \omega^\alpha + \omega^\alpha + \cdots + \omega^\alpha + \omega^\alpha$$ with n $$\omega^\alpha$$'s),
• $$\omega^{\alpha}[n] = \omega^{\alpha[n]}$$ if and only if $$\alpha$$ is a limit ordinal,
• $$(\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k})[n] = \omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}[n]$$, where $$\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_{k - 1} \geq \alpha_k$$,
• $$\varepsilon_0[0] = 0$$ and $$\varepsilon_0[n + 1] = \omega^{\varepsilon_0[n]}$$.

Fundamental sequences for the Veblen functions $$\varphi_\beta(\gamma)$$:

• $$(\varphi_{\beta_1}(\gamma_1) + \varphi_{\beta_2}(\gamma_2) + \cdots + \varphi_{\beta_k}(\gamma_k))[n]=\varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_{k-1}}(\gamma_{k-1}) + (\varphi_{\beta_k}(\gamma_k) [n])$$, where $$\varphi_{\beta_1}(\gamma_1) \ge \varphi_{\beta_2}(\gamma_2) \ge \cdots \ge \varphi_{\beta_k}(\gamma_k)$$ and $$\gamma_m < \varphi_{\beta_m}(\gamma_m)$$ for $$m \in \{1,2,...,k\}$$,
• $$\varphi_0(\gamma)=\omega^{\gamma}$$ and $$\varphi_0(\gamma+1) [n] = \omega^{\gamma} n$$,
• $$\varphi_{\beta+1}(0) [0] = 0$$ and $$\varphi_{\beta+1}(0) [n+1] = \varphi_{\beta}(\varphi_{\beta+1}(0) [n])$$,
• $$\varphi_{\beta+1}(\gamma+1) [0] = \varphi_{\beta+1}(\gamma)+1 \,$$ and $$\varphi_{\beta+1}(\gamma+1) [n+1] = \varphi_{\beta} (\varphi_{\beta+1}(\gamma+1) [n])$$,
• $$\varphi_{\beta}(\gamma) [n] = \varphi_{\beta}(\gamma [n])$$ for a limit ordinal $$\gamma<\varphi_\beta(\gamma)$$,
• $$\varphi_{\beta}(0) [n] = \varphi_{\beta [n]}(0)$$ for a limit ordinal $$\beta<\varphi_\beta(0)$$,
• $$\varphi_{\beta}(\gamma+1) [n] = \varphi_{\beta [n]}(\varphi_{\beta}(\gamma)+1)$$ for a limit ordinal $$\beta$$.

Veblen's function can be presented as a two-argument function $$\varphi_\beta(\gamma)=\varphi(\beta,\gamma)$$.

Note: $$\varphi(0,\gamma)=\omega^\gamma$$, $$\varphi(1,\gamma)=\varepsilon_\gamma$$, $$\varphi(2,\gamma)=\zeta_\gamma$$ and $$\varphi(3,\gamma)=\eta_\gamma$$.

Fundamental sequences for the Γ function:

• $$\Gamma_0 [0] = 0$$ and $$\Gamma_0 [n+1] = \varphi_{\Gamma_0 [n]} (0)$$,
• $$\Gamma_{\beta+1} [0] = \Gamma_{\beta} + 1$$ and $$\Gamma_{\beta+1} [n+1] = \varphi_{\Gamma_{\beta+1} [n]} (0)$$,
• $$\Gamma_{\beta} [n] = \Gamma_{\beta [n]}$$ for a limit ordinal $$\beta < \Gamma_{\beta}$$.

Fundamental sequences for Feferman's theta function:

• $$\theta(\alpha+\Omega,0)[0] = \theta(\alpha,0)$$,
• $$\theta(\alpha+\Omega,\beta+1)[0] = \theta(\alpha+\Omega,\beta)$$,
• $$\theta(\alpha+\Omega,\beta)[n+1] = \theta(\alpha+\theta(\alpha+\Omega,\beta)[n],\beta)$$,
• $$\theta(\alpha \cdot \Omega,0)[0] = \theta(\alpha,0)$$,
• $$\theta(\alpha \cdot \Omega,\beta+1)[0] = \theta(\alpha \cdot \Omega,\beta)$$,
• $$\theta(\alpha \cdot \Omega,\beta)[n+1] = \theta(\alpha \cdot \theta(\alpha \cdot \Omega,\beta)[n],\beta)$$,
• $$\theta(\alpha^{\Omega},0)[0] = \theta(\alpha,0)$$,
• $$\theta(\alpha^{\Omega},\beta+1)[0] = \theta(\alpha^{\Omega},\beta)$$,
• $$\theta(\alpha^{\Omega},\beta)[n+1] = \theta(\alpha^{\theta(\alpha^{\Omega},\beta)[n]},\beta)$$.

Note: The theta-function is shown in the two-argument version $$\theta(\alpha, \beta)=\theta_\alpha(\beta)$$, if $$\beta=0$$ it can be abbreviated as $$\theta(\alpha)=\theta(\alpha,0)$$, the theta function is an extension of the two-argument Veblen function, for countable arguments theta-function is equal to Veblen function $$\theta(\alpha, \beta)=\varphi(\alpha, \beta)$$ and has same fundamental sequences, $$\Omega$$ is an uncountable ordinal and $$\theta(\Omega,0)=\Gamma_0$$.

Fundamental sequences can be defined even for uncountable ordinals if they have cofinality $$\omega$$. Examples of valid FS's are:

• $$(\omega_1+\omega)[n] = \omega_1+n$$
• $$(\omega_\omega)[n] = \omega_n$$.

## Expansion Rule

Not to be confused with Expansion.

As we clarified above, the notion of an expansion rule of a general notation does not admit an agreed-upon mathematical formulation. One typical mistake in googology is the confounding of a notation with an expansion rule with an ordinal notation. As a Japanese Googology Wiki user p進大好きbot has pointed out, those two notions are not equivalent at any rate.[5]

One important aspect of an ordinal notation is that every term canonically corresponds to the ordinal given as the order type. On the other hand, a term in a notation with an expansion rule does not necessarily correspond to an ordinal in a way compatible with the expansion. However, the correspondence is at least unique under a suitable formulation.

For example, the following is a formulation given by p進大好きbot.[6] Let $$T$$ be a set. An expansion rule on $$T$$ is a map $$[] \colon T \times \mathbb{N} \to T$$. Then the map $$o \colon T \to \omega_1$$ satisfying the following is unique under mild assumptions:

1. If $$t = t[n]$$ for any $$n \in \mathbb{N}$$, then $$o(t) = 0$$.
2. If $$t \neq t[0] = t[n]$$ for any $$n \in \mathbb{N}$$, then $$o(t) = o(t[0])+1$$.
3. Otherwise, then $$o(t) = \sup_{n \in \mathbb{N}} o(t[n])$$.

We note that $$o$$ is not even unique if we do not assume any mild assumptions. A similar chracterisation works also when we deal with an expansion rule formulated as a partial function $$[] \colon T \times \mathbb{N} \to T$$. For example, replace the conditions in the following way:

1. If $$(t,0)$$ does not belong to the domain of $$[]$$, then $$o(t) = 0$$.
2. If $$(t,n)$$ belongs to the domain of $$[]$$ for any $$n \in \mathbb{N}$$ and $$t[0] \neq t[1]$$, then $$o(t) = \sup_{n \in \mathbb{N}} o(t[n])$$.
3. Otherwise, then $$o(t) = o(t[0])+1$$.

It is also a typical mistake that $$o$$ always exists, and there are many typical circular logics in googology on the termination of a given notation based on the wrong belief.[6][7]

Assuming the existence of $$o$$ for a given notation, we can consider that a term "expresses" an ordinal. In that sense, we refer to "the corresponding ordinal" for a term in a notation which is not an ordinal notation, e.g. Bashicu matrix system.

## Sources

1. M. Rathjen, Slow Consistency (p.3). Retrieved 2021-06-16.
2. W. Buchholz, A New System of Proof-Theoretic Ordinal Functions, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 195--207.
3. W. Buchholz, Relating ordinals to proofs in a perspicuous way, Reflections on the foundations of mathematics (Stanford, CA, 1998) 15 (2017): 37--59.
4. Maksudov, Denis. The extension of Buchholz's functionTraveling To The Infinity. Retrieved 2017-05-18.
5. p進大好きbot, Relation between an OCF and an Ordinal Notation/Notation with a System of Fundamental Sequences/Relation to an Ordinal Notation, Googology Wiki user blog.
6. p進大好きbot, List of Common Failures in Googology/Circular Proof, Googology Wiki user blog.
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Monday, July 6, 2015

The Lumia 640XL

Over the long weekend I got a Nokia Lumia 640XL phone. I decided to graduate to the new ridiculously large screen size phone after sticking with the more manageable screen sizes of the iPhone 5 and its predecessors. I would have stayed within the iOS/Android eco-systems, but I wanted to give a Windows phone a try, and see why the platform has not been successful in the past.

The phone is nice and relatively inexpensive ($240 without a contract), with a ridiculous screen size, great graphics and battery life. The screen size is a blessing when reading emails, Kindle books, and surfing the Internet, and I believe my usage has increased accordingly. The screen is sharp, and the sound quality of calls is great. With heavy email and web browsing the battery lasted 2 days. The phone comes with crippled memory though (8GB which used be good, but after years of using iOS phones, it is not enough). Luckily the phone is expandable through Micro SD cards, and a 128 GB MicroSD would set you back around$70 from Amazon.

Windows phones have some usability idiosyncrasies compared with their iOS counterparts, and I am not sure if these are because of patents, or design choices. One is killing applications in the app center, where instead of swiping up as in iOS, you swipe down, and the other is the excessive reliance on the back button instead of swiping left to go back except in Internet Explorer. I also found that loading up web sites in IE takes a longer time unlike Safari or Chrome.

And despite the minimal set of apps that I use, I was surprised to see that a few of them were not available for Windows. For the platform to become successful, the Windows store has to attract a whole lot of developers than it has done so far, and perhaps that's the plan in Windows 10. For now since I rely a lot on my phone for work, I'll stick with the official 8.2 builds instead of trying a preview one until I hear what other people's experiences are.


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### Home > APCALC > Chapter 10 > Lesson 10.4.2 > Problem10-173

10-173.

Suppose that $x = 9t^2$ and $y = \frac { 1 } { 2 - ( 3 t ) ^ { 2 } }$. 10-173 HW eTool (Desmos). Homework Help ✎

1. Express $y$ as a function of $x$.

Look at the equation for $y$. $(3t)^2 = 9t^2$

2. Express $x$ as a function of $y$.

Solve your equation from part (a) for $x$.

3. How will the graph of the parametric equations above be different if $x = u$ and $y = \frac { 1 } { 2 - u }$?


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# Chapter 1 Getting Started With R

## 1.1 What is R?

R is a freely available “computational language and environment for data analysis and graphics.” R is indispensable for anyone that uses and interprets data. As medical, public health, and research epidemiologists, we use R in the following ways:

• Full-function calculator
• Extensible statistical package
• High-quality graphics tool
• Multi-use programming language

We use R to explore, analyze, and understand epidemiological data. We analyze data straight out of tables provided in reports or articles as well as analyze usual data sets. The data might be a large, individual-level data set imported from another source (e.g., cancer registry); an imported matrix of group-level data (e.g, population estimates or projections); or some data extracted from a journal article we are reviewing. The ability to quantitatively express, graphically explore, and describe epidemiologic data and processes enables one to work and strengthen one’s epidemiologic intuition.

In fact, we only use a very small fraction of the R package. For those who develop an interest or have a need, R also has many of the statistical modeling tools used by epidemiologists and statisticians, including logistic and Poisson regression, and Cox proportional hazard models. However, for many of these routine statistical models, almost any package will suffice (SAS, Stata, SPSS, etc.). The real advantage of R is the ability to easily manipulate, explore, and graphically display data. Repetitive analytic tasks can be automated or streamlined with the creation of simple functions (programs that execute specific tasks). The initial learning curve is steep, but in the long run one is able to conduct analyses that would otherwise require a tremendous amounts of programming and time.

Some may find R challenging to learn if they are not familiar with statistical programming. R was created by statistical programmers and is more often used by analysts comfortable with matrix algebra and programming. However, even for those unfamiliar with matrix algebra, there are many analyses one can accomplish in R without using any advanced mathematics, which would be difficult in other programs. The ability to easily manipulate data in R will allow one to conduct good descriptive epidemiology, life table methods, graphical displays, and exploration of epidemiologic concepts. R allows one to work with data in any way they come.

## 1.2 What is RStudio?

To get started quickly we need tools that makes the process of writing and compiling R code quick and (mostly) pain free. Fortunately for us there is RStudio—it is a free, open source, and powerful integrated development environment (IDE) for R that runs on most operating systems (Windows, Mac, or Linux). After installing R, install RStudio—it has all the tools we need to learn and apply R.

## 1.3 Who should learn R?

Anyone that uses a calculator or spreadsheet, or analyzes numerical data at least weekly should seriously consider learning and using R. This includes epidemiologists, statisticians, physician researchers, engineers, health economists, health systems analysts, business analysts, and faculty and students of mathematics and science courses, to name just a few. We jokingly tell our staff analysts that once they learn R they will never use a spreadsheet program again (well almost never!).

## 1.4 Why should I learn R?

To implement numerical methods we need a computational tool. On one end of the spectrum are calculators and spreadsheets for simple calculations, and on the other end of the spectrum are specialized computer programs for such things as statistical and mathematical modeling. However, many numerical problems are not easily handled by these approaches. Calculators, and even spreadsheets, are too inefficient and cumbersome for numerical calculations whose scope and scale change frequently. Statistical packages are usually tailored for the statistical analysis of data sets and often lack an intuitive, extensible, open source programming language for tackling new problems efficiently.1 R can do the simplest and the most complex analysis efficiently and effectively.

When we learn and use R regularly we will save significant amounts of time and money. It’s powerful and it’s free! It’s a complete environment for data analysis and graphics. Its straightforward programming language facilitates the development of functions to extend and improve the efficiency of our analyses.

## 1.5 Where can I get R?

R is available for many computer platforms, including Mac OS, Linux, Microsoft Windows, and others. R comes as source code or a binary file. Source code needs to be compiled into an executable program for our computer. Those not familiar with compiling source code (and that’s most of us) just install the binary program. We assume most readers will be using R in the Mac OS or MS Windows environment. Listed here are useful R links:

To install R for Windows or Mac OS, do the the following:

• Go to http://www.r-project.org;
• From the left menu list, click on the “CRAN” (Comprehensive R Archive Network) link;
• Select a nearby geographic site (e.g., http://cran.cnr.berkeley.edu);
• Select appropriate operating system;
• For Windows, save R-X.X.X-win.exe to the computer; and for Mac OS, save the R-X.X.X-pkg installer package.
• Run the installation program and accept the default installation options. That’s it!

## 1.6 How do I use R?

### 1.6.1 Using R on our computer

Use R by typing commands at the R command line prompt (>) and pressing Enter on our keyboard. This is how to use R interactively. Alternatively, from the R command line, we can also execute a list of R commands that we have saved in a text file (more on this later). Here is an example of using R as a calculator:

(4 + 6)^3 - 2*500/4
## [1] 750

Use the c function to collect data entered at the console. Name each collection of data, and then perform a numercal operation. In this example we conduct an analysis that is analogous to working in a spreadsheet.

quantity <- c(34, 56, 22)        # quantity data
price <- c(19.95, 14.95, 10.99)  # price data
subtotal <- quantity*price       # subtotal cost
cbind(quantity, price, subtotal) # column bind into spreadsheet-like display
##      quantity price subtotal
## [1,]       34 19.95   678.30
## [2,]       56 14.95   837.20
## [3,]       22 10.99   241.78

### 1.6.2 Does R have epidemiology programs?

The default installation of R does not have packages that specifically implement epidemiologic applications; however, many of the statistical tools that epidemiologists use are readily available, including statistical models such as unconditional logistic regression, conditional logistic regression, Poisson regression, Cox proportional hazards regression, and much more. Table 1.1 lists selected R packages with biostatistical or mathematical modeling methods applied to epidemiologic problems. The focus of this book is learning how to use R without relying on a specific packages. Learning the R basics covered in this book will help one take full advantage of these and other R packages, some of which address advanced topics such as network modeling of epidemics.

Table 1.1: Selected R packages for epidemiologic applications
Packages Description
dagbag Learning directed acyclic graphs (DAGs) through bootstrap aggregating
dagitty Graphical Analysis of Structural Causal Models
dagR R functions for directed acyclic graphs
amei Adaptive Management of Epidemiological Interventions
Epi A Package for Statistical Analysis in Epidemiology
epibasix Elementary Epidemiological Functions for Epidemiology and Biostatistics
EpiBayes Implements Hierarchical Bayesian Models for Epidemiological Applications
EpiContactTrace Epidemiological Tool for Contact Tracing
epiDisplay Epidemiological Data Display Package
EpiDynamics Dynamic Models in Epidemiology
EpiEstim EpiEstim: estimate time varying reproduction numbers from epidemic curves
epifit Flexible Modelling Functions for Epidemiological Data Analysis
EpiModel Mathematical Modeling of Infectious Disease
epinet Epidemic/Network-Related Tools
epiR Tools for the Analysis of Epidemiological Data
episensr Basic Sensitivity Analysis of Epidemiological Results
episheet Rothman’s Episheet
epitools Epidemiology Tools
EpiWeek Conversion Between Epidemiological Weeks and Calendar Dates

### 1.6.3 How should I use these notes?

The best way to learn R is to use it! Use it as your calculator! Use it as your spreadsheet! Finally read these notes sitting at a computer and use R interactively (this works best sitting in a cafe that brews great coffee and plays good music). Although we initially encourage you to use R interactively by typing expressions at the console, as a general rule, it is much better to type your code as a R script. Save your code with a convenient file name such as job01.R.2

RStudio comes with a text editor for creating and editing R scripts. Our focus will be learning how to use RStudio to edit and run R scripts.

The code in your text editor can be run in the following ways: - Highlight and run selected expressions in the RStudio; - Copy and paste the code directly into R console; - Run the file in batch mode from the R console using the source function (e.g., source("job01.R")).

Table 1.2: Selected math operators
Operator Description Try these examples
+ addition 5+4
- subtraction 5-4
* multiplication 5*4
/ division 5/4
^ exponentiation 5^4
- unary minus (change current sign) -5
abs absolute value abs(-23)
exp exponentiation ($$e$$ to a power) exp(8)
log logarithm (default is natural log) log(exp(8))
sqrt square root sqrt(64)
%/% integer divide 10%/%3
%% modulus 10%%3
%*% matrix multiplication xx <-matrix(1:4, 2, 2)
xx%*%c(1, 1)
c(1, 1)%*%xx

## 1.7 Just do it!

### 1.7.1 Using R as your calculator

Open R and start using it as our calculator. The most common math operators are displayed in Table 1.2. From now on make R our default calculator! Study the examples and spend a few minutes experimenting with R as a calculator. Use parentheses as needed to group operations. Use the keyboard Up-arrow to recall what we previously entered at the command line prompt.

### 1.7.2 Useful R concepts

#### 1.7.2.1 Types of evaluable expressions

Every expression that is entered at the R console is evaluated by R and returns a value. A literal is the simplist expression that can be evaluated (number, character string, or logical value). Mathematical operations involve numeric literals. For example, R evaluates the expression 4*4 and returns the value 16. The exception to this is when an evaluable expression is assigned an object name: x <- 4*4. To display the assigned expression, wrap the expression in parentheses: (x <- 4*4), or type the object name. Finally, evaluable expressions must be separated by either newline breaks or a semicolon. Table 1.3 summarizes evaluable R expressions.

4 * 4
## [1] 16
x <- 4 * 4    # assign expression to object named 'x'
x             # display 'x' object
## [1] 16
(x <- 4 * 4)  # evaluates expression and displays 'x'
## [1] 16
x <- 4*4; x   # expressions can be separated by semi-colons
## [1] 16
Table 1.3: Types of evaluable R expressions
Expression type Try these examples
literal "hello, I'm John Snow" #character
3.5 #numeric
TRUE; FALSE #logical
math operation 6*7
assignment x <- 4*4
data object x
function log(x)

#### 1.7.2.2 Using the assignment operator

Most calculators have a memory function: the ability to assign a number or numerical result to a key for recalling that number or result at a later time. The same is true in R but it is much more flexible. Any evaluable expression can be assigned a name and recalled at a later time. We refer to these variables as data objects. We use the assignment operator (<-) to name an evaluable expression and save it as a data object.

xx <- "hello, what's your name"; xx
## [1] "hello, what's your name"

Multiple assignments work and are read from right to left:

aa <- bb <- 99
aa; bb
## [1] 99
## [1] 99

Finally, the equal sign (=) can be used for assignments, although we prefer and the <- symbol.

ages <- c(34, 45, 67) # equivalent
ages = c(34, 45, 67) # equivalent

The reason I prefer <- for assigning object names in the workspace is because later we use = for assigning values to function arguments. For example,

x <- 1:10 # assigning object name (x)
sample(x = 1:10, size = 5)  # assigning value to argument x

The first x is an object name assignment in the workspace which persist during the R session. The second x is a function argument assignment which is only recognized locally in the function and only for the duration of the function execution. For clarity, it is better to keep these types of assignments separate in our mind by using different assignment symbols.

Study these previous examples and spend a few minutes using the assignment operator to create and call data objects. Try to use descriptive names if possible. For example, suppose we have data on age categories; we might name the data agecat, age.cat, or age_cat.3

### 1.7.3 Useful R functions

When we start R you have opened a workspace. The first time we use R, the workspace is empty. Every time we create a data object, it is in the workspace. If a data object with the same name already exists, the old data object will be overwritten without warning, so be careful. To list the objects in your workspace use the ls or objects functions:

ls() # lists objects in the workspace
## [1] "aa"        "ages"      "bb"        "epi.packs" "price"     "quantity"
## [7] "subtotal"  "x"         "xx"
objects() # equivalent
## [1] "aa"        "ages"      "bb"        "epi.packs" "price"     "quantity"
## [7] "subtotal"  "x"         "xx"

Data objects can be saved between sessions. We will be prompted with “Save workspace image?” You can also use save.image() at the console prompt. The workspace image is saved in a file called .RData.4 Use getwd() to display the file path to the .RData file. Table 1.4 has more useful R functions.

Table 1.4: Useful R functions
Function Description Try these examples
q Quit R q()
ls List objects ls()
objects objects() #equivalent
rm Remove object(s) yy <- 1:5; ls()
remove rm(yy); ls()
#removes all: caution!
rm(list = ls()); ls()
help Open help instructions; help()
or get help on specific topic help(plot)
?plot #equivalent
help.search Search help system given character string help.search("print")
help.start Start help browser help.start()
apropos Displays all objects in the search list matching topic apropos(plot)
getwd Get working directory getwd()
setwd Set working directory setwd("c:\mywork\rproject")
args Display arguments of function args(sample)
example Runs example of a function example(plot)
data Information on available R data sets; data() #displays data sets
Load data set data(Titanic) #load data set
save.image Saves current workspace to .RData save.image()

#### 1.7.3.1 What are packages?

R has many available functions. When we open R, several packages are attached by default. Each package has its own suite of functions. To display the list of attached packages use the search function. To display the file paths to the packages use the searchpaths function.

search() # Linux
##  [1] ".GlobalEnv"        "package:foreign"   "package:survival"
##  [4] "ESSR"              "package:stats"     "package:graphics"
##  [7] "package:grDevices" "package:utils"     "package:datasets"
## [10] "package:methods"   "Autoloads"         "package:base"
searchpaths() # Linux
##  [1] ".GlobalEnv"
##  [2] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/foreign"
##  [3] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/survival"
##  [4] "ESSR"
##  [5] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/stats"
##  [6] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/graphics"
##  [7] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/grDevices"
##  [8] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/utils"
##  [9] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/datasets"
## [10] "/Library/Frameworks/R.framework/Versions/3.4/Resources/library/methods"
## [12] "/Library/Frameworks/R.framework/Resources/library/base"

To learn more about a specific package enter library(help=“Package_Name”). Alternatively, we can get more detailed information by entering help.start() which opens the HTML help page. On this page click on the Packages link to see the available packages. If we need to load a package enter library(Package_Name). For example, when we cover survival analysis we will need to load the survival package.

#### 1.7.3.2 What are function arguments?

We will be using many R functions for data analysis, so we need to know some function basics. Suppose we are interested in taking a random sample of days from the month of June, which has 30 days. We want to use the sample function but we forgot the syntax. Let’s explore:

sample
## function (x, size, replace = FALSE, prob = NULL)
## {
##     if (length(x) == 1L && is.numeric(x) && is.finite(x) && x >=
##         1) {
##         if (missing(size))
##             size <- x
##         sample.int(x, size, replace, prob)
##     }
##     else {
##         if (missing(size))
##             size <- length(x)
##         x[sample.int(length(x), size, replace, prob)]
##     }
## }
## <bytecode: 0x7feea69b0c30>
## <environment: namespace:base>

Whoa! What happened? Whenever we type the function name without any parentheses it usually returns the whole function code. This is useful when we start programming and we need to alter an existing function, borrow code for our own functions, or study the code for learning how to program. If we are already familiar with the sample function we may only need to see the syntax of the function arguments. For this we use the args function:

args(sample)
## function (x, size, replace = FALSE, prob = NULL)
## NULL

The terms x, size, replace, and prob are the function arguments. First, notice that replace and prob have default values; that is, we do not need to specify these arguments unless we want to override the default values. Second, notice the order of the arguments. If you enter the argument values in the same order as the argument list we do not need to specify the argument.

dates <- 1:30
sample(dates, 16) # sample "size = 16"
##  [1]  7 20  9 30 22 14 27 15  5 21 24 18 26  3 11  4

Third, if we enter the arguments out of order then we will get either an error message or an undesired result. Arguments entered out of their default order need to be specified.

sample(16, dates) # undesired results; wanted "size = 16"
## [1] 2
sample(size = 16, x = dates) # gives desired result
##  [1] 10 26 14 16 12  5 24  4 15 28  3 20  9 29 21  1

Fourth, when we specify an argument we only need to type a sufficient number of letters so that R can uniquely identify it from the other arguments.

sample(s = 16, x = dates, r = TRUE) # sampling w/ replacement
##  [1] 15 20 11 22 23 22 27  4 18 21 21  2 17 18 13 16

Fifth, argument values can be any valid R expression (including functions) that evaluates to an appropriate value. In the following example we see two sample functions that provide random values to the sample function arguments.<

sample(s = sample(1:36, 1), x = sample(1:10, 5), r=T)
##  [1] 6 7 7 7 7 8 1 1 6 8 6 9 9 8 1 8 8 6 1 9 8 9 8 9 9 1 6 9 1

Finally, if we need more guidance on how to use the sample function enter ?sample or help(sample).

### 1.7.4 How do I get help?

RStudio has extensive help capabilities. From the RStudio main menu select Help $$\rightarrow$$ R Help to get you started. The Frequently Asked Questions (FAQ) and R manuals are available from this menu. From the R console, try the following options to learn about the help capabilities:

?help               # opens help page for 'help' function
help.start()        # launches HTML help page
help.search("help") # searches help system for "help"
apropos("help")     # displays 'help' objects in search list

To learn about about available data sets use the data function:

data()                          # display avail. data sets
try(data(package = "survival")) # list 'survival' data sets
help(pbc, package = "survival") # display pbc data help page

### 1.7.5 Is there anything else that I need?

Not really. RStudio has everything you will need to use R productively. Some analysts will select to use R with a text editor, rather than RStudio. Like RStudio, a good text editor makes programming and data processing easier and more efficient. If you are considering a text editor, the functionality we look for in a text editor are the following:

• Toggle between wrapped and unwrapped text
• Block cutting and pasting (also called column editing)
• Easy macro programming
• Search and replace using regular expressions
• Ability to import large datasets for editing

When we are programming we want our text to wrap so we can read all of your code. When we import a data set that is wider than the screen, we do not want the data set to wrap: we want it to appear in its tabular format. Column editing allows us to cut and paste columns of text at will. A macro is just a way for the text editor to learn a set of keystrokes (including search and replace) that can be executed as needed. Searching using regular expressions means searching for text based on relative attributes. For example, suppose you want to find all words that begin with “b”, end with “g”, have any number of letters in between but not “r” and “f”. Regular expression searching makes this a trivial task. These are powerful features that once we use regularly, we will wonder how we ever got along without them.

If we do not want to install a text editing program then we can use the default text editor that comes with our computer operating system (gedit in Ubuntu Linux, TextEdit in Mac OS, Notepad in Windows). However, it is much better to install a text editor that works with R. My favorite text editor is the free and open source GNU Emacs. GNU Emacs can be extended with the “Emacs Speaks Statistics” (ESS) package. For more information on Emacs and ESS pre-installed for Windows and Mac OS, visit http://ess.r-project.org.

To the novice user, R may seem complicated and difficult to learn. In fact, for its immense power and versatility, R is easier to learn and deploy compared to other statistical software (e.g. SAS, Stata, SPSS). This is because R was built from the ground up to be an efficient and intuitive programming language and environment. If you understand the logic and structure of R, then learning proceeds quickly. Just like a spoken language, once you know its rules of grammar, syntax, and pronunciation, and can write legible sentences, you can figure out how to communicate almost anything. Before we get into the “trees” (next chapter), we want to describe the “forest”: the logic and structure of working with R objects and epidemiologic data.

#### 1.7.6.1 Working with R objects

For our purposes, there are only five types of data objects in R5 and five types of actions we take on these objects (Table 1.5). That’s it! No more, no less. You will learn to create, name, index (subset), replace components of, and operate on these data objects using a systematic, comprehensive approach. As you learn about each new data object type, it will reinforce and extend what you learned previously.

Table 1.5: Types of actions taken on R data objects
Action Vector Matrix Array List Data frame
Creating Table 2.5 Table 2.12 Table 2.20 Table 3.1 Table 3.7
Naming Table 2.6 Table 2.13 Table 2.21 Table 3.2 Table 3.8
Indexing Table 2.7 Table 2.14 Table 2.22 Table 3.3 Table 3.9
Replacing Table 2.8 Table 2.15 Table 2.23 Table 3.4 Table 3.10
Operating on Table 2.9 Table 2.16 Table 2.24 Table 3.5 Table 3.11
Table 2.10

A vector is a collection of elements (often numbers):

x <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12); x
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12

A matrix is a 2-dimensional representaton of a vector:

y <- matrix(x, nrow = 2); y
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    3    5    7    9   11
## [2,]    2    4    6    8   10   12

An array is a 3 or more dimensional represention of a vector:

z <- array(x, dim = c(2, 3, 2)); z
## , , 1
##
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6
##
## , , 2
##
##      [,1] [,2] [,3]
## [1,]    7    9   11
## [2,]    8   10   12

A list is a collection of “bins”, each containing any kind of R object:

mylist <- list(x, y, z); mylist
## [[1]]
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12
##
## [[2]]
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    3    5    7    9   11
## [2,]    2    4    6    8   10   12
##
## [[3]]
## , , 1
##
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6
##
## , , 2
##
##      [,1] [,2] [,3]
## [1,]    7    9   11
## [2,]    8   10   12

A data frame is a list in tabular form where each “bin” contains a data vector of the same length. A data frame is the usual tabular data set familiar to epidemiologists. Each row is an record and each column (“bin”) is a field.

kids <- c("Tomasito", "Lusito", "Angelita")
gender <- c("boy", "boy", "girl")
age <- c(8, 7, 4)
mydf <- data.frame(kids, gender, age)
mydf
##       kids gender age
## 1 Tomasito    boy   8
## 2   Lusito    boy   7
## 3 Angelita   girl   4

In the next chapter we explore these R data objects in greater detail.

## 1.8 Exercises

If you have not done so already,

• install R on your computer (https://cran.rstudio.com/),
• install RStudio on your computer (https://www.rstudio.com/), and
• register for a RPubs account (http://www.rpubs.com/), and open RStudio.
• Open a new Rmarkdown template file (.Rmd extension).
• Learn Rmarkdown and use it to answer the exercises in this chapter.
• Submit the exercises as a HTML link, Word document, or PDF document.

### 1.8.1

What is the R workspace file on your operating systems? What is the file path to your R workspace file? What is the name of this workspace file?

### 1.8.2

By default, which R packages come already loaded? What are the file paths to the default R packages?

### 1.8.3

List all the object in the current workspace. If there are none, create some data objects. Using one expression, remove all the objects in the current workspace.

### 1.8.4

One inch equals 2.54 centimeters. Correct the following R code and create a conversion table.

inches <- 1:12
centimeters <- inches/2.54
cbind(inches, centimeters)

### 1.8.5

To convert between temperatures in degrees Celsius ($$C$$) and Farenheit ($$F$$), we use the following conversion formulas:

$C = (F - 32) \frac{5}{9}$

$F = \frac{9}{5} C + 32$

At standard temperature and pressure, the freezing and boiling points of water are 0 and 100 degrees Celsius, respectively. What are the freezing and boiling points of water in degrees Fahrenheit?

### 1.8.6

For the Celsius temperatures 0, 5, 10, 15, 20, 25, …, 80, 85, 90, 95, 100, construct a conversion table that displays the corresponding Fahrenheit temperatures. Hint: to create the sequence of Celsius temperatures use seq(0, 100, 5).

### 1.8.7

BMI is a reliable indicator of total body fat, which is related to the risk of disease and death. The score is valid for both men and women but it does have some limits. BMI does have some limitations: it may overestimate body fat in athletes and others who have a muscular build, it may underestimate body fat in older persons and others who have lost muscle mass.

Table 1.6: Body mass index classification
BMI Classification
$$<18.5$$ Underweight
$$[18.5, 25)$$ Normal weight
$$[25, 30)$$ Overweight
$$\ge 30$$ Obesity

Body Mass Index (BMI) is calculated from your weight in kilograms and height in meters:

$BMI = \frac{kg}{m^2}$

$1\,\mbox{kg} \approx 2.2\,\mbox{lb}$

$1\,\mbox{m} \approx 3.3\,\mbox{ft}$

Calculate your BMI (don’t report it to us).

Using Table 1.2, explain in words, and use R to illustrate, the difference between modulus and integer divide.

### 1.8.8

In mathematics, a logarithm (to base $$b$$) of a number $$x$$ is written $$\log_b(x)$$ and equals the exponent $$y$$ that satisfies $$x = b^y$$. In other words,

$y = \log_b(x)$ is equivalent to

$x = b^y$

In R, the log function is to the base $$e$$. Implement the following R code and study the graph:

curve(log(x), 0, 6)
abline(v = c(1, exp(1)), h = c(0, 1), lty = 2)

What kind of generalizations can you make about the natural logarithm and its base—the number $$e$$?

### 1.8.9

Risk ($$R$$) is a probability bounded between 0 and 1. Odds is the following transformation of $$R$$:

$Odds = \frac{R}{1-R}$

Use the following code to plot the odds:

curve(x/(1-x), 0, 1)

Now, use the following code to plot the $$\log$$(odds):

curve(log(x/(1-x)), 0, 1)

What kind of generalizations can you make about the $$\log$$(odds) as a transformation of risk?

Table 1.7: Estimated per-act risk (transmission probability) for acquisition of HIV, by exposure route to an infected source. Source: CDC
Exposure route Risk per 10,000 exposures
Blood transfusion (BT) 9,000
Needle-sharing injection-drug use (IDU) 67
Receptive anal intercourse (RAI) 50
Percutaneous needle stick (PNS) 30
Receptive penile-vaginal intercourse (RPVI) 10
Insertive anal intercourse (IAI) 6.5
Insertive penile-vaginal intercourse (IPVI) 5
Receptive oral intercourse on penis (ROI) 1
Insertive oral intercourse with penis (IOI) 0.5

Use the data in Table 1.7. Assume one is HIV-negative. If the probability of infection per act is $$p$$, then the probability of not getting infected per act is $$(1-p)$$. The probability of not getting infected after 2 consecutive acts is $$(1-p)^2$$, and after 3 consecutive acts is $$(1-p)^3$$. Therefore, the probability of not getting infected infected after $$n$$ consecutive acts is $$(1-p)^n$$, and the probability of getting infected after $$n$$ consecutive acts is $$1-(1-p)^n$$. For each non-blood transfusion transmission probability (per act risk) in Table 1.7, calculate the cumulative risk of being infected after one year (365 days) if one carries out the same act once daily for one year with an HIV-infected partner. Do these cumulative risks make intuitive sense? Why or why not?

### 1.8.10

The source function in R is used to “source” (read in) ASCII text files. Take a group of R commands that worked from a previous problem above and paste them into an ASCII text file and save it with the name job01.R. Then from R command line, source the file. Here is how it looked on my Linux computer running R:

> source("~/Documents/courses/ph251d/jobs/job01.R")

Describe what happened. Now, set option to .

> source("~/Documents/courses/ph251d/jobs/job01.R", echo = TRUE)

Describe what happened. To improve your understanding read the help file on the source function.

### 1.8.11

Now run the source again (without and with echo = TRUE) but each time create a log file using the sink function. Create two log files: job01.log1a and job01.log1b.

> sink("~/Documents/courses/ph251d/jobs/job01.log1a")
> source("~/Documents/courses/ph251d/jobs/job01.R")
> sink() #closes connection
>
> sink("~/Documents/courses/ph251d/jobs/job01.log1b")
> source("~/Documents/courses/ph251d/jobs/job01.R", echo = TRUE)
> sink() #closes connection

Examine the log files and describe what happened.

### 1.8.12

Create a new job file () with the following code:

n <- 365
per.act.risk <- c(0.5, 1, 5, 6.5, 10, 30, 50, 67)/10000
risks <- 1-(1-per.act.risk)^n
show(risks)

Source this file at the R command line and describes what happens.

1. Recommendations for mostly free and open source software (FOSS) at http://phds.io.

2. The .R extension, although not necessary, is useful when searching for R command files. Additionally, this file extension is recognized by RStudio and many text editors.

3. To improve readability, a period (.) or underscore (_) symbol can be used in your object name

4. In some operating systems files names that begin with a period (.) are hidden files and are not displayed by default. You may need to change the viewing option to see the file.

5. The sixth type of R object is a function. Functions can create, manipulate, operate on, and store data; however, we will use functions primarily to execute a series of R “commands” and not as primary data objects.


text:
# Casimir Effect, Renormalisation Theory, Topology, and Sum of All $n$ for $n\in \mathbb{Z}^+$

This is the second instalment of my journey to learn Quantum Field Theory. This week, I got a preview at Renormalisation Theory and how we can use this to solve seemingly impossible physical problems, and during the process, I encountered a particularly peculiar sum…

## Questionable Pedagogy…

I remember that in grade 10, my then maths teacher, Mr Prior, showed the class a ‘proof’ that the sum of all natural numbers is equal to negative one twelfth, i.e.:

$\sum_{n=1}^\infty n = -\frac{1}{12}$

Despite the fact that proof is in quotes, I am using the term very liberally. What seemed to be an impressive mathematical feat at the time, now seems like a dubious attempt to trick young minds (ok maybe that’s a bit much).

The proof starts by proposing the following sequence:

$S_1 =((-1)^{n})_{n=0}^{\infty}= 1, -1, 1, -1,…$

This sequence simply alternates between 1 and 0. Suppose we take an infinite sum of the sequence, what would be the final result? The video says, since the terms oscillate between 1 and 0, so depending on whether we stop at an even or odd indexed term, we would end up with either 0 or 1, respectively, so let’s take the average of those, and assign the value one half to the series:

$\sum_{n=0}^\infty(-1)^n = \frac{1}{2}$

Now this may seem reasonable, the 15-year-old me certainly thought so. But upon closer inspection, you would notice that this is indeed extremely cavalier and will open the Pandora’s box as a result. We can’t just assign a value to a positively non-converging series! But alas, we shall address these mathematical floppiness later on. The second series that we will need is:

$S_2 = \sum ((-1)^{n+1}n))_{n=1}^{\infty}=1-2+3-4+5-…$

If you add $S_2$to itself, you’d discover that you recover $S_1$!

So indeed:

$2S_2 = S_1$

ergo

$S_2 = \frac{1}{4}$

And finally, the pièce de résistance. If we call the series of containing all natural numbers $S$:

$S = 1 + 2 + 3+ 4+ 5+…$

We can carry out the following calculation:

\begin{align*}S – S_2 &= (1+2+3+4…)-(1-2+3-4…)\\ &= 4 + 6 + 8 + … \\ &= 4(1+2+3+4…) \\ &= 4S\end{align*}

\begin{align*}S-S_2 = 4S\\ 3S = -S_2 \\ S = -\frac{1}{12}\end{align*}

This to me was shocking to say the least, and very shocking to say the most. What is perhaps even more unexpected is that this sum can be physically observed. We shall take a brief physical interlude, which in my opinion, will clear things up a bit. While I appreciate the sentiment that this video tries to convey, the mysteriously counterintuitive nature of maths, but for me, it served the exact opposite purpose.

## Casimir Effect

Previously, we have established that QFT is in effect, an infinite collection of quantum harmonic oscillators. And indeed, this leads to the gratifying result that the energy of the vacuum, $\bra{0}H\ket{0}$ is simply an integral over all momentum modes and all of space. The Casimir effect is what happens when we disturb the vacuum.

Casimir, in 1948, performed the following experiment. He placed two parallel conducting plates (formally zero thickness and infinite area) into the vacuum. By changing the distance $d$ between the two plates, we vary the energy density, $\varepsilon$ of the vacuum, which would lead to a force between the plates, aptly named the Casimir force. The energy density of vacuum takes the following form:

$\varepsilon = 2 \int \frac{\dd[3]{k}}{(2\pi)^3}\frac{1}{2}\hbar \omega_k$

If we call the direction perpendicular to the plates $x$ and in order to maintain the continuity of our electromagnetic field at the conducting plates, the wave vector $\vec k$ can only take quantised values of $(\pi n/d , k_y, k_z)$, so we arrive at the expression for the energy density in between the plates:

$\sum_n \int \dd k_y \dd k_z \frac{\sqrt{(\pi n/d)^2 + k_y^2 + k_z^2}}{(2\pi)^2}$

This leads to a problem. What about the energy density outside the plates? This problem is overcome by introducing a third plate. We call the distance between the two outer plates $L$, and set it fixed. We can now move the middle plate and measure the force that way.

The energy $E$, can be found thusly:

$f(d) = \frac{1}{2}\sum_{n=1}^\infty \omega_n = \sum_{n=1}^\infty \frac{n\pi}{2}$

and that the total energy $E$ is:

$E = f(d) + f(L-d)$

The energy of the waves are simply given by the sum of all the frequencies (as the modes are given by standing sinusoidal waves). Also, we have conveniently set $\hbar=1$. Hopefully by now you would have realised, the energy tends to infinity, demonstrated by the infinite sum $\sum_1^\infty n$. To overcome this, we introduce a nifty trick, which I think is absolutely fantastic, namely:

## Renormalisation Theory

What we can do is introduce an exponentially decaying factor $e^{-an/d}$, where $a$ parameterises some physics that we are not interested in at high frequencies. This in essence, picks out the ‘physically sensible’ parts of our system, the convergent part. We know what $a$ parameterise, it is the plasma frequency of the metal plates! Metal plates, depending on the material, can only keep waves up to a certain frequence before they start leaking out. The following then, will become our new $f$:

$f(d) = \frac{\pi}{2d}\sum_{n=1}^\infty n e^{-an/d}$

Now, we have a factor that tends to infinity, but we also have a factor that decays exponentially! To make sense of this, we can rewrite the above expression in another form:

$f(d) = -\frac{\pi}{2}\pdv{a}\sum_{n=1}^\infty e^{-an/d}$

which is in effect, the exact samething but expressed with a partial differential with respect to $a$. Reorganising the terms, we end up with:

$\frac{\pi}{2d}\frac{e^{n/d}}{e^{n/d}-1}$

To justify all of our hocus-pocus with introducing another factor $a$, we can simply say, let us set $a$ to 0!

Now, we can Taylor expand our $f$:

$f(d) = \frac{\pi d}{2a^2} – \frac{\pi}{24d} + \frac{\pi a^2}{480d^3} + \mathcal{O}(\frac{a^4}{d^5})$

Let us examine each term carefully. The first term, since we set $a\rightarrow 0$, will tend to infinity, but the second term, being completely independent of $a$, will remain! In fact, discarding the parts that shoot off to infinity, we are only left with $-\frac{\pi}{24d}$!. And a short derivation from here:

$F = -\pdv{E}{d} = -\frac{\pi\hbar c}{24d^2}$

yields the force between the plates in SI units.

This is an astonishing result in my opinion. When faced with a divergent sum, at the beginning of this problem, we reassure ourselves that physics in nature don’t tend to infinity. Thus, we introduced a fudge factor $a$, which we encoded some physics (i.e. the plasma frequency of the metal plates) to essentially ‘pick and choose’ the convergent, sensible physics. Granted we had no reason to believe that such a force we are attempting to explain is indeed independent of the factor $a$ a priori, but the derivation would have taken a very different turn indeed should that have been the case.

Besides the ingenious renormalisation of our theory, the result directly showed that the sum of all natural numbers is indeed something we can observe in nature, and indeed we can… in the Casimir force!

In Zee’s book, he describes the fact that certain textbooks/people just brushing over the derivation by simply stating the sum of $n$ where $n\in \mathbb{R}^+$ is ‘appalling. So exactly what went wrong then?

## Maybe it’s not so wrong…

With my very limited understaing of functional analysis, I shall attempt to explain how one can formalise this devilish sum in a mathematically rigid way.

The unintuitive nature of this subject is derived from the fact that we are working with what is effectively, infinitely dimensional vector spaces. Suppose we have a vector space $\mathcal{L}$, where

$\mathcal{L} = \{(a_n)_{n=1}^\infty | a_n \in \mathbb{C}\}$

$\mathcal{L}$ is the set of all possible infinity series of complex terms. It should be evident that $\mathcal{L}$ is a vector space, the telltale signs of a vector space do indeed apply to $\mathcal{L}$:

$(a_n)_1^\infty \oplus (b_n)_1^\infty = (a_n + b_n)_{n=1}^\infty$

this is just to say that when we add two infinite series together, we once again, get an infinite series.

Furthermore:

$\forall \lambda \in \mathbb{C} , \lambda(a_n)_{n=1}^\infty = (\lambda a_n)_{n=1}^\infty$

Suppose now we have a subspace, $\mathcal{C}$, a subspace of $\mathcal{L}$ containing those infinite series that are convergent, or in notation:

$\mathcal{C} = \left\{(a_n)_{n=1}^\infty \in \mathcal{L} | \text{s.t. } \exists \lim_{n\rightarrow \infty} \sum_{k=1}^n a_k\right\}$

I think it can be reasonablly inferred that $\mathcal{C}$ is also a vector space, i.e. as you add two convergent series together, the resultant series would also be convergent. We can say $\mathcal{C} \subset \mathcal{L}$

We have involved another entity in our definition of $\mathcal{C}$, and that is the $\lim$. If we think about it, the limit is nothing but a linear map, an operator that does the following:

$\lim_{n\rightarrow \infty} : \mathcal{C} \rightarrow \mathbb{C}$

In fact, the fact that $\mathcal{C}$ and $\mathcal{L}$ are related by a linear map would make $\mathcal{C}$ a dense subspace of $\mathcal{L}$.

And ultimately, what we are interested in, is to extend our linear functional (the limit as n goes to infinity) from $\mathcal{C}$ to $\mathcal{L}$, and Hahn-Banach Theorem will help us do exactly that.

Since both vector spaces are infinitely dimensional, we can approximate a point (a series in this case) in $\mathcal{L}$ by taking the sum of a sequence of points in $\mathcal{C}$, which would allow us to get infinitely close to any point in $\mathcal{L}$ from $\mathcal{C}$, in a topological sense.

In this topology of sorts, we can perhaps formalise, in a much more mathematically rigid way, some of the series used in Numberphile’s proof of the sum.

I shall return next week with Noether’s Theorem, and field theory in curved spacetime!


text:
# 2 Capacitors and energy density

Shreya
Homework Statement:
Lets consider 2 parallel plate capacitors ##C_1## and ##C_2##.
##C_1## has plate area twice of the ##C_2##, but same width. If both stores equal charge ##Q##. Which one has more energy?
Relevant Equations:
$$U = (Volume) \frac 1 2 \epsilon E^2$$
$$U= \frac {Q^2}{2C}$$
$$U= \frac 1 2 CV^2$$
I developed three arguments to answer this question. Argument no 2 seems to be wrong, but I cant figure out why. I know one/more of my arguments are flawed. Please be kind to help me figure this out.

Argument 1) Since they have same charge on them, the ##E## between them must be same. The one with more volume ##C_1## must have more energy, according to the first equation.

Argument 2) According to equation 1, ##C_1## must have lesser Energy that ##C_2##, due to higher capacitance. (Refer second equation)

Argument 3) Due to same width and same ##E##. they must have same ##V## between them. According to equation 3, ##C_1## must have more Energy

Homework Helper
Gold Member
2022 Award
Homework Statement:: Lets consider 2 parallel plate capacitors ##C_1## and ##C_2##.
##C_1## has plate area twice of the ##C_2##, but same width. If both stores equal charge ##Q##. Which one has more energy?
Relevant Equations:: $$U = (Volume) \frac 1 2 \epsilon E^2$$
$$U= \frac {Q^2}{2C}$$
$$U= \frac 1 2 CV^2$$

I developed three arguments to answer this question. Argument no 2 seems to be wrong, but I cant figure out why. I know one/more of my arguments are flawed. Please be kind to help me figure this out.

Argument 1) Since they have same charge on them, the ##E## between them must be same. The one with more volume ##C_1## must have more energy, according to the first equation.

Argument 2) According to equation 1, ##C_1## must have lesser Energy that ##C_2##, due to higher capacitance. (Refer second equation)

Argument 3) Due to same width and same ##E##. they must have same ##V## between them. According to equation 3, ##C_1## must have more Energy
By 'width' I assume you mean plate-separation (##d##).

To get an intuition of what’s going on (and why two of the arguments are wrong) you could try considering a more extreme case. Let '##A##' be plate area. Suppose ##A_1 = 1000,000A_2##. Can you see what will happen to the magnitude of the electric fields?

Alternatively (or even better, additionally) do yo know the formula for ##C## in terms of ##A, d## and any constant(s)? If so, can you use this formula to help?

Shreya
Shreya
1) Considering the extreme case, I can see that ##E## of ##C_1## must be small, since the same charge will be spread over a larger area ##E = \frac \sigma \epsilon##. And since ##E## is squared (unlike Volume(##=Ad##) in Equation 1, low ##E## would mean lower ##U## .

$$C= \epsilon \frac {A}{d}$$
2) I know this formula for Capacitance of a Parallel plate capacitor. So greater area would mean greater ##C## and therefore lower ##U##.

3)Lower ##E## would mean lower potential drop for the same plate seperation. And since ##V## is squared, its effect overrides that of increase in C, implying lower Energy.

@Steve4Physics, I have modified my arguments using your suggestion. Could you please verify them.

Homework Helper
Gold Member
2022 Award
@Steve4Physics, I have modified my arguments using your suggestion. Could you please verify them.
Yes. All the arguments are now correct (and give the same conclusion!).

You might consider slightly more detail. For example for 1) you could note:
a) volume ##\propto A##;
b) ##E \propto \frac 1A##;
c) so ##U \propto A(\frac 1{A^2}) \propto \frac 1A##.

Doing the maths along with the qualitative analyse is very useful. For example, you may be able to understand (qualitatively) that some quantity Y decreases when quantity X increases, but without the proper analysis you won’t know the precise relationship, e.g. is ##Y \propto \frac 1X## or is ##Y \propto \frac 1{X^2}## or something else?.

To derive the equation for ##U## in terms of ##\epsilon , A, d## and ##Q## is straightforward. The dependence of ##U## on ##A## (for fixed ##Q## and ##d##) will then be very clear.

By the way, another intuition into the problem is to note that to put a charge onto a small area requires more work than putting the same onto a larger area – because of the additional work needed to overcome the mutual repulsion between the charge-elements.

Shreya
Shreya
You might consider slightly more detail. For example for 1) you could note:
a) volume ∝A;
b) E∝1A;
c) so U∝A(1A2)∝1A.

Doing the maths along with the qualitative analyse is very useful. For example, you may be able to understand (qualitatively) that some quantity Y decreases when quantity X increases, but without the proper analysis you won’t know the precise relationship, e.g. is Y∝1X or is Y∝1X2 or something else?.
That makes sense. Doing such an analysis for 2 & 3 also yields the above result!
I will make sure that I do such analysis from now.
To derive the equation for U in terms of ϵ,A,d and Q is straightforward. The dependence of U on A (for fixed Q and d) will then be very clear.
That would be: $$U = \frac {Q^2d}{2 \epsilon A}, which confirms what we discussed above. By the way, another intuition into the problem is to note that to put a charge onto a small area requires more work than putting the same onto a larger area – because of the additional work needed to overcome the mutual repulsion between the charge-elements. Yes! Didnt realise that until now! Thank you so much @Steve4Physics for helping me through this problem. I appreciate that you made me think and arrive at the solution, instead of solving it for me. Thank you! Steve4Physics Science Advisor Homework Helper Gold Member In comparison problems of this kind it is useful to write equations for the quantities that you are comparing in terms of the quantities that are given. Let me explain. You want to compare stored energies ##U##. Using ##Q=CV##, you can write the stored energy in three different ways each of which has only two of the variables, ##Q##, ##C## and ##V##:$$\begin{align} & U=\frac{1}{2}CV^2 \\ & U=\frac{Q^2}{2C} \\ & U =\frac{1}{2}QV.\end{align}$$Which one to use? Look at what is given, ##C_2 = 2C_1 \implies## if ##C_1=C## then ##C_2=2C.## ##Q_1=Q_2=Q.## It should be obvious that you should use equation (2) that has capacitance and charge on the right-hand side. Then you get,$$\begin{align} & U_1=\frac{Q_1^2}{2C_1}=\frac{Q^2}{2C} \nonumber \\ & U_2=\frac{Q_2^2}{2C_2}=\frac{Q^2}{2\times 2C}= \frac{Q^2}{4C}=\frac{1}{2}U_1\nonumber \end{align}Now you can clearly see which of the stored energies is greater. As long as one capacitor has twice the capacitance of the other, this result is true and independent of how the capacitors are shaped or what the electric fields look like.

Steve4Physics and Shreya
Shreya
Thanks a lot @kuruman ! I think this insight can be extended to other formulae as well, which is awesome.

kuruman