text:
# Module 23 Getting Data into R

This module explains how to enter data into R. Subsequent modules will explain how to use R to create the graphics and compute statistics and test results from these data.

## 23.1 Data in R

### 23.1.1 Create CSV File

For real data (i.e., several variables from many individuals) it is most efficient to enter data into a comma-separated values (CSV) file and then import that file into R. Creating a CSV file with Microsoft Excel is described below, though there are other ways to create CSV files (see FAQs on class webpage). This explanation assumes that you have a basic understanding of Excel (or other spreadsheet software).

The spreadsheet should be organized with variables in columns and individuals in rows, with the exception that the first row should contain variable names. The example spreadsheet below shows the length (cm), weight (kg), and capture location data for a small sample of Black Bears.

Variable names should NOT contain spaces. For example, don’t use “total length” or “length (cm).” If you feel the need to have longer variable names, then separate the parts with a period (e.g., “length.cm”) or an underscore (e.g., “length_cm”). Variable names must also NOT start with numbers or contain “special” characters such as “~,” “!” “&,” “@,” etc. Furthermore, numerical measurements should NOT include units (e.g., don’t use “7 cm”). Finally, for categorical data, make sure that all categories are consistent (e.g., do not have a column that contains both “bayfield” and “Bayfield”).

The spreadsheet is saved as a CSV file by selecting the File..Save As menu item, which will produce the dialog box below. In this dialog box, change “Save as type” to “CSV (Comma delimited),”77, provide a file name (do not put any periods in the name), select a location to save the file (this should be the same location as your assignment template file), and press “Save.” Two “warning” dialog boxes may then appear – select “OK” for the first and “YES” for the second. You can now close the spreadsheet file.78

It is important that each row of the data frame correspond to one individual. This is critically important when data are recorded for two different groups (e.g., for a two-sample t-test; see Module 24). For example, the following data are methyl mercury levels recorded in mussels from two locations labeled as “impacted” and “reference.”

  impacted   0.011  0.054  0.056  0.095  0.051  0.077
reference  0.031  0.040  0.029  0.066  0.018  0.042  0.044

To follow the “one individual per row” rule, these data are entered in stacked format where the “reference” data are stacked underneath the “impacted” data and a column is used to indicate to which group the individuals belong. For example, the Excel file for data entry would look like the following.

### 23.1.2 Read CSV File into RStudio

Use read.csv() in the first R code chunk of your assignment notebook to load the data from the CSV file into RStudio. The EXACT name of the data file is including within quotes within read.csv() and the results should be saved to an object. For example, the code highlighted in yellow in the graphic below was used to read the external “Bears.csv” file into an R object called dat. The circled items in the lower-right pane demonstrate that the “Bears.csv” file is in the same directory as the assignment notebooks file.

One should check the data in this object as described in the Viewing a Data Frame section below.

## 23.2 Viewing a Data Frame

R may be disorienting at first because you generally cannot “see” your data in the same way that you see it in a spreadsheet program. There are, however, several options for viewing your data. First, you can type the name of the data frame object to see its entire contents.

bears
#R>     length.cm weight.kg      loc
#R>  1      139.0       110 Bayfield
#R>  2      138.0        60 Bayfield
#R>  3      139.0        90 Bayfield
#R>  4      120.5        60 Bayfield
#R>  5      149.0        85 Bayfield
#R>  6      141.0       100  Ashland
#R>  7      141.0        95  Ashland
#R>  8      150.0        85  Douglas
#R>  9      166.0       155  Douglas
#R>  10     151.5       140  Douglas
#R>  11     129.5       105  Douglas
#R>  12     150.0       110  Douglas

Typing the name is adequate for small data.frames, but not useful for large data.frames. The entire data.frame is opened in a separate window by double-clicking on the name of the data.frame in the “Environment” tab of RStudio. Alternatively, you can view a subset of 20 rows of a data frame with peek(), the first six rows with head(), the last six rows with tail(), or the first and last three rows with headtail().

peek(bears)
#R>     length.cm weight.kg      loc
#R>  1      139.0       110 Bayfield
#R>  2      138.0        60 Bayfield
#R>  3      139.0        90 Bayfield
#R>  4      120.5        60 Bayfield
#R>  5      149.0        85 Bayfield
#R>  6      141.0       100  Ashland
#R>  7      141.0        95  Ashland
#R>  8      150.0        85  Douglas
#R>  9      166.0       155  Douglas
#R>  10     151.5       140  Douglas
#R>  11     129.5       105  Douglas
#R>  12     150.0       110  Douglas
head(bears)
#R>    length.cm weight.kg      loc
#R>  1     139.0       110 Bayfield
#R>  2     138.0        60 Bayfield
#R>  3     139.0        90 Bayfield
#R>  4     120.5        60 Bayfield
#R>  5     149.0        85 Bayfield
#R>  6     141.0       100  Ashland
tail(bears)
#R>     length.cm weight.kg     loc
#R>  7      141.0        95 Ashland
#R>  8      150.0        85 Douglas
#R>  9      166.0       155 Douglas
#R>  10     151.5       140 Douglas
#R>  11     129.5       105 Douglas
#R>  12     150.0       110 Douglas
headtail(bears)
#R>     length.cm weight.kg      loc
#R>  1      139.0       110 Bayfield
#R>  2      138.0        60 Bayfield
#R>  3      139.0        90 Bayfield
#R>  10     151.5       140  Douglas
#R>  11     129.5       105  Douglas
#R>  12     150.0       110  Douglas

### 23.2.1 Structure and Data Types

Data in R will be designated as an integer (whole numbers), numeric (non-integer numeric values), character (strings), factor (group membership), or logical (TRUE/FALSE). Integer and numeric data types correspond to quantitative data. Character and factor data types correspond to categorical data, with the factor data type having special characteristics in R that we will exploit for certain analyses. These types of data will be discussed further as needed in other modules.

In addition to viewing the contents of a data object, it is useful to examine the structure of the data frame as returned from str().

str(bears)
#R>  'data.frame':  12 obs. of  3 variables:
#R>   $length.cm: num 139 138 139 120 149 ... #R>$ weight.kg: int  110 60 90 60 85 100 95 85 155 140 ...
#R>   $loc : chr "Bayfield" "Bayfield" "Bayfield" "Bayfield" ... In this example, it is seen that three variables were recorded on 12 individuals. The first variables – length.cm – is numeric, the second – weight.kg – is integer, and the last – loc – is a character variable. The unique values in a character variable may be seen with unique(). unique(bears$loc)
#R>  [1] "Bayfield" "Ashland"  "Douglas"

A character variable may be converted to a factor variable with factor(). In this course, the main reason for doing this would be to control the levels of an ordinal variable because R treats levels alphabetically by default. For example suppose that we wanted to order the levels of the loc variable from west to east (so, Douglas, Bayfield, and Ashland). For this purpose the variable is converted to a factor below with factor() and the order of the levels is controlled with levels=.

Make sure the levels are spelled EXACTLY as they appear in the data.

bears$loc <- factor(bears$loc,levels=c("Douglas","Bayfield","Ashland"))
levels(bears$loc) #R> [1] "Douglas" "Bayfield" "Ashland" The levels of a factor variable and their order may be seen with levels(). levels(bears$loc)
#R>  [1] "Douglas"  "Bayfield" "Ashland"

In the previous examples, the $ notation was used to identify a particular variable (i.e., loc) within a data.frame (bears). Think of variables as being nested inside data.frames and, thus, to access the variable you must first identify the data.frame in which it exists and then the name of the variable. The $ simply separates the data.frame from the variable.

bears$length.cm #R> [1] 139.0 138.0 139.0 120.5 149.0 141.0 141.0 150.0 166.0 151.5 129.5 150.0 bears$loc
#R>   [1] Bayfield Bayfield Bayfield Bayfield Bayfield Ashland  Ashland  Douglas
#R>   [9] Douglas  Douglas  Douglas  Douglas
#R>  Levels: Douglas Bayfield Ashland

## 23.3 Vectors

Data.frames are the primary structure in which to store real data. However, much simpler situations that don’t require a data.frame may arise. In R, items of the same data type (see [Data Types]) are stored in a one-dimensional vector. Vectors are usually displayed in one row (with many columns), but they may also be thought of as a single column (with many rows). Items are entered into a vector with c(), where the individual arguments are specific numbers, characters, or logical values.79 Items for a vector of characters must be contained within paired quotes.

( v <- c(1,2,5) )
#R>  [1] 1 2 5
( y <- c("Iowa","Minnesota","Wisconsin") )
#R>  [1] "Iowa"      "Minnesota" "Wisconsin"

Single variables from a data.frame are vectors.

bears\$length.cm
#R>   [1] 139.0 138.0 139.0 120.5 149.0 141.0 141.0 150.0 166.0 151.5 129.5 150.0

Vectors that are not extracted from a data.frame will only be used in this course for very simple lists of items, usually as arguments in a function.

1. There are several choices for CSV files here; do NOT choose the one with “UTF-8” in the name.↩

2. You may be asked to save changes – you should say “No.”↩

3. c comes from “concatenate” or “combine.”↩


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# Instructions A 79 year old female patient presents to the Emergency Department with confusion, dehydration, and...

###### Question:

Instructions A 79 year old female patient presents to the Emergency Department with confusion, dehydration, and painful urination. The physician gives a diagnosis of urinary tract infection (UTI). The patient has NKDA and a history of COPD, atrial fibrillation, and shingles. She is given IV fluids to treat the dehydration, 1 gram of Rocephin® IVPB and discharged with a prescription for penicillin drug therapy to treat the UTI. Medication history is as follows: 1. amoxicillin 250mg PO qid x 7 days 2. warfarin 2mg PO q am 3. digoxin 0.25mg PO q am 4. prednisone 10mg q am
What is the mechanism of action for amoxicillin? a. Cell wall synthesis inhibitor O Ở OŠ OŠ O Protein synthesis inhibiitor Folate synthesis inhibitor DNA synthesis inhibitor None of the above are correct
Amoxicillin is an antibiotic. When considering the MOA, is amoxicillin bacteriostatic or bactericidal? Bacteriostatic-inhibits growth and prevents replication Bactericidal-inhibits growth and prevents replication Bacteriostatic-kills the bacteria directly Bactericidal-kills the bacteria directly

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### Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity

Röckner M, Zhang X (2009)
Probability Theory and Related Fields 145(1-2): 211-267.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

Es wurden keine Dateien hochgeladen. Nur Publikationsnachweis!
Autor*in
Einrichtung
Abstract / Bemerkung
In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993-1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.
Stichworte
Invariant measure; Navier-Stokes equation; Ergodicity; Asymptotic; strong Feller property
Erscheinungsjahr
2009
Zeitschriftentitel
Probability Theory and Related Fields
Band
145
Ausgabe
1-2
Seite(n)
211-267
ISSN
0178-8051
Page URI
https://pub.uni-bielefeld.de/record/1633336

### Zitieren

Röckner M, Zhang X. Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probability Theory and Related Fields. 2009;145(1-2):211-267.
Röckner, M., & Zhang, X. (2009). Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probability Theory and Related Fields, 145(1-2), 211-267. https://doi.org/10.1007/s00440-008-0167-5
Röckner, M., and Zhang, X. (2009). Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probability Theory and Related Fields 145, 211-267.
Röckner, M., & Zhang, X., 2009. Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probability Theory and Related Fields, 145(1-2), p 211-267.
M. Röckner and X. Zhang, “Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity”, Probability Theory and Related Fields, vol. 145, 2009, pp. 211-267.
Röckner, M., Zhang, X.: Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probability Theory and Related Fields. 145, 211-267 (2009).
Röckner, Michael, and Zhang, Xicheng. “Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity”. Probability Theory and Related Fields 145.1-2 (2009): 211-267.

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text:
# B-Algebra/Examples

## Examples of $B$-Algebras
### $B$-Algebra Induced by $S_3$
This is a $B$-algebra for the finite set $\set {0, 1, 2, 3, 4, 5}$:
$\begin{array}{c|cccccc} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 2 & 1 & 3 & 4 & 5 \\ 1 & 1 & 0 & 2 & 4 & 5 & 3 \\ 2 & 2 & 1 & 0 & 5 & 3 & 4 \\ 3 & 3 & 4 & 5 & 0 & 2 & 1 \\ 4 & 4 & 5 & 3 & 1 & 0 & 2 \\ 5 & 5 & 3 & 4 & 2 & 1 & 0 \\ \end{array}$


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# jointcal constrained photometry model is not better than processCcd photometry on DC2 run1.2

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## Details

• Type: Bug
• Status: To Do
• Resolution: Unresolved
• Fix Version/s: None
• Component/s:
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None
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• Team:

## Description

From a set of 7 visits from DESC DC2 Run1.2p I am taking 1 visit as an arbitrary reference and for each matched sources in the other visits I compute the difference of magnitude using 1- the processCcd calibration and 2- the jointcal calibration from the default constrained model.

The dispersion (19.9 mmag) is identical in both cases which looks very surprising. The size of the dispersion also looks very large.

In order to reproduce the problem the data are available on lsst-dev:/home/boutigny/2018-10-11 and the notebook to reproduce the plot is on: https://github.com/boutigny/LSST_notebooks/blob/master/DC2/multi_visit_Joincal.ipynb

I am not excluding an error on my side or a misusage of jointcal

## People

• Assignee:
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1. ## Exponential model help

I need to solve this problem
2.5649(1.1096^x)=25
And I am not sure how to?
Any help would be greatly appreciated.

2. Hello, JBondman871!

Solve for $\displaystyle x\!:\;\;2.5649(1.1096^x)\:=\:25$

We have: .$\displaystyle 1.1096^x \;=\;\frac{25}{2.5649}$

Take logs: .$\displaystyle \ln\left(1.1096^x\right) \;=\;\ln\left(\frac{25}{2.5649}\right)$

Then: .$\displaystyle x\cdot\ln(1.1096) \;=\;\ln(25) - \ln(2.5649)$

Finally: .$\displaystyle x \;=\;\frac{\ln(25) - \ln(2.5649)}{\ln(1.1096)}$


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# Area theorem (conformal mapping)

In the mathematical theory of conformal mappings, the area theorem gives an inequality satisfied by the power series coefficients of certain conformal mappings. The theorem is called by that name, not because of its implications, but rather because the proof uses the notion of area.

## Statement

Suppose that $f$ is analytic and injective in the punctured open unit disk $\mathbb D\setminus\{0\}$ and has the power series representation

$f(z)= \frac 1z + \sum_{n=0}^\infty a_n z^n,\qquad z\in \mathbb D\setminus\{0\},$ then the coefficients $a_n$ satisfy

$\sum_{n=0}^\infty n|a_n|^2\le 1.$

## Proof

The idea of the proof is to look at the area uncovered by the image of $f$. Define for $r\in(0,1)$

$\gamma_r(\theta):=f(r\,e^{-i\theta}),\qquad \theta\in[0,2\pi].$ Then $\gamma_r$ is a simple closed curve in the plane. Let $D_r$ denote the unique bounded connected component of $\mathbb C\setminus\gamma[0,2\pi]$. The existence and uniqueness of $D_r$ follows from Jordan's curve theorem.

If $D$ is a domain in the plane whose boundary is a smooth simple closed curve $\gamma$, then

$\mathrm{area}(D)=\int_\gamma x\,dy=-\int_\gamma y\,dx\,,$ provided that $\gamma$ is positively oriented around $D$. This follows easily, for example, from Green's theorem. As we will soon see, $\gamma_r$ is positively oriented around $D_r$ (and that is the reason for the minus sign in the definition of $\gamma_r$). After applying the chain rule and the formula for $\gamma_r$, the above expressions for the area give

$\mathrm{area}(D_r)= \int_0^{2\pi} \Re\bigl(f(r e^{-i\theta})\bigr)\,\Im\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr)\,d\theta = -\int_0^{2\pi} \Im\bigl(f(r e^{-i\theta})\bigr)\,\Re\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr)d\theta.$ Therefore, the area of $D_r$ also equals to the average of the two expressions on the right hand side. After simplification, this yields

$\mathrm{area}(D_r) = -\frac 12\, \Re\int_0^{2\pi}f(r\,e^{-i\theta})\,\overline{r\,e^{-i\theta}\,f'(r\,e^{-i\theta})}\,d\theta,$ where $\overline z$ denotes complex conjugation. We set $a_{-1}=1$ and use the power series expansion for $f$, to get

$\mathrm{area}(D_r) = -\frac 12\, \Re\int_0^{2\pi} \sum_{n=-1}^\infty \sum_{m=-1}^\infty m\,r^{n+m}\,a_n\,\overline{a_m}\,e^{i\,(m-n)\,\theta}\,d\theta\,.$ (Since $\int_0^{2\pi} \sum_{n=-1}^\infty\sum_{m=-1}^\infty m\,r^{n+m}\,|a_n|\,|a_m|\,d\theta<\infty\,,$ the rearrangement of the terms is justified.) Now note that $\int_0^{2\pi} e^{i\,(m-n)\,\theta}\,d\theta$ is $2\pi$ if $n= m$ and is zero otherwise. Therefore, we get

$\mathrm{area}(D_r)= -\pi\sum_{n=-1}^\infty n\,r^{2n}\,|a_n|^2.$ The area of $D_r$ is clearly positive. Therefore, the right hand side is positive. Since $a_{-1}=1$, by letting $r\to1$, the theorem now follows.

It only remains to justify the claim that $\gamma_r$ is positively oriented around $D_r$. Let $r'$ satisfy $r<r'<1$, and set $z_0=f(r')$, say. For very small $s>0$, we may write the expression for the winding number of $\gamma_s$ around $z_0$, and verify that it is equal to $1$. Since, $\gamma_t$ does not pass through $z_0$ when $t\ne r'$ (as $f$ is injective), the invariance of the winding number under homotopy in the complement of $z_0$ implies that the winding number of $\gamma_r$ around $z_0$ is also $1$. This implies that $z_0\in D_r$ and that $\gamma_r$ is positively oriented around $D_r$, as required.

## Uses

The inequalities satisfied by power series coefficients of conformal mappings were of considerable interest to mathematicians prior to the solution of the Bieberbach conjecture. The area theorem is a central tool in this context. Moreover, the area theorem is often used in order to prove the Koebe 1/4 theorem, which is very useful in the study of the geometry of conformal mappings.


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# Searches for New Physics at HERA

Abstract : The high energy programme of the HERA collider ended in March 2007. During the whole HERA programme, a combined total integrated luminosity of 1 fb$^{-1}$ was collected by the H1 and ZEUS experiments. In this context, an overview of the most recent results of both experiments concerning searches for new physics is presented. The topics covered are searches for contact interactions, leptoquarks and excited leptons, as well as studies of the isolated lepton and multi-lepton topologies, and a general signature based search.
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-00158706
Contributor : Danielle Cristofol <>
Submitted on : Friday, June 29, 2007 - 1:42:49 PM
Last modification on : Tuesday, March 30, 2021 - 3:17:37 AM

### Identifiers

• HAL Id : in2p3-00158706, version 1
• ARXIV : 0706.4114

### Citation

E. Sauvan. Searches for New Physics at HERA. Nuclear Physics B - Proceedings Supplements, Elsevier, 2008, 177-178, pp.246-250. ⟨in2p3-00158706⟩

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## Chemistry (7th Edition)

Published by Pearson

# Chapter 14 - Chemical Equilibrium - Section Problems - Page 594: 55

#### Answer

a. $$K_c = \frac{[ CH_3CHO ]^{ 2 }}{[ C_2H_4 ]^{ 2 }[ O_2 ]}$$ b. $$K_c = \frac{[ CO_2 ][ H_2 ]}{[ CO ][ H_2O ]}$$ c.$$K_c = \frac{[ NO ]^{ 4 }[ H_2O ]^{ 6 }}{[ NH_3 ]^{ 4 }[ O_2 ]^{ 5 }}$$

#### Work Step by Step

1. Write the equilibrium constant expression: - The exponent of each concentration is equal to its balance coefficient. a. $$K_c = \frac{[Products]}{[Reactants]} = \frac{[ CH_3CHO ]^{ 2 }}{[ C_2H_4 ]^{ 2 }[ O_2 ]}$$ b. $$K_c = \frac{[Products]}{[Reactants]} = \frac{[ CO_2 ][ H_2 ]}{[ CO ][ H_2O ]}$$ c.$$K_c = \frac{[Products]}{[Reactants]} = \frac{[ NO ]^{ 4 }[ H_2O ]^{ 6 }}{[ NH_3 ]^{ 4 }[ O_2 ]^{ 5 }}$$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Synopsis

# How to Transmit Light Through a Vapor

Physics 12, s143
By combining two frequencies of light into a pulse called a simulton, a weak signal can travel through a dense atomic vapor as though the medium were nearly transparent.

Researchers have long known that very intense light sources can create self-reinforcing pulses of light called solitons, which can propagate through an opaque medium as though it were nearly transparent. A specific type of soliton called a simulton can also be created by combining a weak signal with an intense pulse of differently colored light, but this had never been demonstrated in a gas. Now, a team led by Charles Adams and Robert Potvliege, at Durham University in the UK, has discovered signs of simultons in a relatively dense atomic vapor. The authors say that finding evidence of the phenomenon in this system was a surprise, as atomic vapors are commonly studied and well-characterized. The result suggests that the medium might also sustain other unusual forms of light pulses.

The researchers beamed a weak continuous-wave (CW) laser and a more powerful pulsed laser into a dense, 2- $𝜇\text{m}$-thick vapor of rubidium atoms. Each laser’s frequency was tuned to trigger a transition of the rubidium atoms to a different energy. While the CW laser alone was quickly absorbed by the medium, when both lasers were used together, some of the CW light was transmitted right through. From their calculations, the researchers interpret the vapor’s increased transparency as being due to interactions between the two lasers and the rubidium atoms. These interactions appear to amplify the CW light into a sequence of pulses, which the researchers identify as simultons.

In the future, the researchers want to study how simultons form and propagate in thicker vapor cells. They also plan to investigate simulton interactions with highly excited atoms, which could lead to applications in optical quantum technologies.

This research is published in Physical Review Letters.

–Sophia Chen

Sophia Chen is a freelance science writer based in Tucson, Arizona.

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# [Solved]: Disproving well-quasi-order by providing an infinite anti-chain

, ,
Problem Detail:

I am currently studying the theory behind Well-Quasi-Orders. However I am having some issues in understanding how an infinite anti-chain can be produced to disprove the claim that a partial order $P$ is a w.q.o.

In particular I am wondering whether it is logically sound to present as proof not the infinite anti-chain but rather an algorithm to produce it.

More specifically, I've been trying to solve an exercise from the following lecture which asks whether the class of $P_{3}\text{-free}$ graphs is a w.q.o or not on the induced subgraph operation $\leq_{i}$

I've proved that said class contains unions of cliques and defined an ordering of this set as follows $P=\lbrace G_{0},G_{1},G_{2},.... \rbrace$ with $G_{i}$ representing all $P_{3}\text{-free}$ graphs on $i$ vertices (and graphs in $G_{i}$ being represented in a touple-like alpharithmetic form $(a_{1},....,a_{i})$ such that $a_{1} \leq a_{2} \leq.... \leq a_{i}$ to avoid repetitions and $\sum_{j=1}^{i} a_{j}=i$ to define an order. For example $(0,0,2,2)$ $\in G_{4}$ encodes the graph with two disconnected $P_{2}$ components.

I have defined an algorithm $B$ that produces an anti-chain that grows bigger and bigger at each step

The algorithm uses gadgets defined as $B_{a,c}$ which represent the graph with $c+1$ connected components, the first $c$ of which are $K_{1}$(i.e a sole vertice) and the last component representing $K_{a}$

The steps of my algorithm are the following:

1. Define $L$ as the sequence

2. Add $B_{a,0}$ to $L$ ($a$ any number $>$ 0)

3. repeat $\infty$ times:

4.Replace each item $B_{a,c} \in L$ by $B_{a^{'}=a+1 + 10^{6},c}$ and produce $L^{'}$ this way.

5.Let $X=B_{a^{'},c}$ be the graph with the maximum $c$ amongst those in $L^{'}$

6.Add to $L=L^{'}$ the graph $B_{a^{'}-2,c+1}$

Certainly, this algorithm does create an anti-chain( in each step I create a graph with more connected components yet less vertices). However I am not sure that such a technique is valid for disproving w.q.o on the infinite anti-thesis claim because of the fact that it works on a step by step basis.

Any help is appreciated. Thank you very much!

#### Answered By : Yuval Filmus

Your algorithm produces an infinite sequence of antichains whose size is unbounded. However, it doesn't produce a single antichain of infinite size.

It is perfectly fine to specify an infinite chain by giving an algorithm that enumerates its members, and even more complicated constructions are possible (e.g. the priority method). However, the end result of the construction should be a single infinite chain rather than a sequence of antichains of unbounded size.

I suggest taking a look at Higman's lemma.

###### Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/63857

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