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# Resting Membrane Potential

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The human organism is composed of multiple cells, all of them with different components and therefore with differents resting membrane potentials. Some of these cells are excitable (e. g.: cells; neurons; muscle fibers), generating an action potential when subjected to an external stimulus, causing its membrane depolarization. The resting membrane potential (RMP) is due to changes in membrane permeability for potassium, sodium, calcium, and chloride, which results from the movement of these ions across it. Once the membrane is polarized, it acquires a voltage, which is the difference of potentials between intra and extracellular spaces.

## Contents

### edit What is a RMP?

Resting membrane potential is:

• the unequal distribution of ions on the both sides of the cell membrane;
• the voltage difference of quiescent cells;
• the membrane potential that would be maintained if there weren’t any stimuli or conducting impulses across it;
• determined by the concentrations of ions on both sides of the membrane;
• a negative value, which means that there is an excess of negative charge inside of the cell, compared to the outside.
• much depended on intracellular potassium level as the membrane permeability to potassium is about 100 times higher than that to sodium.

### edit Producingand maintaining RMP

RMP is produced and maintained by:

Donnan effect
described as large impermeable negatively charged intracellular molecules attracting positively charged ions (e. g.: Na+ and K+) and repelling negative ones (e. g.: Cl)
Membrane selectivity
is the difference of permeabilites between different ions
Active transport (Na+/K+ ATPase pump)
is the mediated process of moving particles across a biological membrane, against the concentration gradient.
• Primary active transport – if it spends energy. This is how the Na+/K+ ATPase pump functions.
• Secondary active transport – if it involves an electrochemical gradient. This is not involved in maintaining RMP.

### edit Ion affection of resting membrane potential

RMP is created by the distribution of ions and its diffusion across the membrane. Potassium ions are important for RMP because of its active transport, which increase more its concentration inside the cell. However, the potassium-selective ion channels are always open, producing an accumulation of negative charge inside the cell. Its outward movement is due to random molecular motion and continues until enough excess negative charge accumulates inside the cell to form a membrane potential.

### edit Na+/K+ ATPase pump affection of the RMP

The Na+/K+ ATPase pump creates a concentration gradient by moving 3Na+ out of the cell and 2K+ into the cell. Na+ is being pumped out and K+ pumped in against their concentration gradients. Because this pump is moving ions against their concentration gradients, it requires energy.

### edit Ion channels affection of resting membrane potential

The cell membrane contains protein channels that allow ions to diffuse passively without direct expenditure of metabolic energy. These channels allow Na + and K+ to move across the cell membrane from a higher concentration toward a lower. As these channels have selectivity for certain ions, there are potassium- and sodium- selective ion channels. All cell membranes are more permeable to K+ than to Na+ because they have more K+ channels than Na+.

### edit The Nernst Equation

Is a mathematical equation applied in physiology, to calculate equilibrium potentials for certain ions.

$Ei = (\frac{R·T}{F·z})\cdot ln\frac{[X]1}{[X]2}$

• R = Gas Constant
• T = Absolute temperature (K)
• E = The potential difference across the membrane
• F = Faradays Constant (96,500 coulombs/mole)
• z = Valency of ion

### edit The Goldman-Hodgkin-Katz Equation

Is a mathematical equation applied in Physiology, to determine the potential across a cell's membrane, taking in account all the ions that are permeable through it.

$Em = 58 log (\frac{PNa\cdot[Na]out + PK\cdot[K] out}{ PNa\cdot[Na]in + PK\cdot[K]in})$

• E = The potential difference across the membrane
• P = Permeability of the membrane to sodium or potassium
• [ ] = Concentration of sodium or potassium inside or outside

### editMeasuring resting potentials

In some cells, the RPM is always changing. For such, there is never any resting potential, which is only a theoretical concept. Other cells with membrane transport functions that change potential with time, have a resting potential. This can be measured by inserting an electrode into the cell. Transmembrane potentials can also be measured optically with dyes that change their optical properties according to the membrane potential.

### edit Resting membrane potential varies according to types of cells

For example:
• Skeletal muscle cells: −95 mV
• Smooth muscle cells: −50 mV
• Astrocytes: −80/−90 mV
• Neurons: −70 mV
• Erythrocytes: −12 mV

### edit Sources

Web pages:

Books:

• Hall, John E., and Arthur Clifton Guyton. Guyton and Hall Textbook of Medical Physiology. 12th ed. Philadelphia, PA: Saunders/Elsevier, 2011. Print.
• Ward, Clarke, Linden. Physiology at a Glance. 3. Blackwell publishing
• W. Gannon, Review of Medical Physiology. Lange Medical Books/McGraw-Hill, 21st edition
• R. Berne, M. Levy, Physiology. Mosby, 4th edition
• A. Despopoulos, S. Silbernagl, Color Atlas of Physiology. 5th edition

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# Integral trig substitution

1. May 12, 2010

### apiwowar

integral of x/(x2+2x+2)dx

first thing i did was complete the square to get

x/((x+1)2+1

i tried then having x+1 = tanx but that didnt work out

because of the x on top i cant just set w = x+1

what would the right substitution be?

any hints or help would be appreciated

2. May 12, 2010

### TheoMcCloskey

apiwowar - try one more substitution, u = (x+1). This will yield two terms in numerator, thus two fractions, for which you should be able to integrate separately with a little more manipulation.

3. May 13, 2010

### Cyosis

Using the substitution $x+1=\tan u$ will work just fine. However the substitution in post #2 will make it easier. Show us where you got stuck.

4. May 13, 2010

### apiwowar

i did the substitution that the first guy suggested and got 1/2ln((x+1)2+1) - arctan(x+1)

is that right?

5. May 13, 2010

### TheoMcCloskey

That is what I obtained (don't forget constant of integration).

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# Subscript In R

Detecting subscript in command argument - Blogger 2. Move the mouse cursor to any data point and press the left mouse button. This could be a little tricky. 5 to be a subscript to PM. R programing. 0 0 p a t P a_fault 1. Privilege The use of the term 'suitably privileged' in this document means either that the command is in a library (%) routine or the user is privileged. A typical question: How many gifferent committees of 8 can be formed from a group of 12 people? This would be represented as 12C8, where 12 and 8 are both subscripts. 28 x R_{sun}. Most of the stock math commands are written for typesetting math or computer science papers for academic journals, so you might need to dig deeper into LaTeX commands to get the vector. EIF_2 is subscripted, but E_0, k_1, and k_{-1} is not. The individual elements of the array can be referenced by using subscript. Explanation: The answer is B. When you render your file, Pandoc transforms the marked up text into formatted text in your final file format, as below. The lowest subscript value for a dimension is always 0, and the highest subscript value is returned by the GetUpperBound method for that dimension. Creates a line break. What is the exact meaning of the subscript notation $\mathbb{E}_X[f(X)]$ in conditional expectations in the framework of measure theory ? These subscripts do not appear in the definition of conditional expectation, but we may see for example in this page of wikipedia. Sign in Register R, Plot axis label text tips; by Trent Biggs; Last updated almost 4 years ago; Hide Comments (-) Share Hide Toolbars. For a di erentiable kernel K: Rd Rd!R, we denote by r 1K: Rd Rd!Rd (resp. vec_as_subscript: Convert to a base subscript type in vctrs: Vector Helpers rdrr. Although preliminary studies suggested that 5-HT 6 R antagonists have procognitive effects, clinical trials have been disappointing. runs the R code, embeds results and text into. Can anyone help?. 3 years ago. SUBSCRIPT EIGHT. Copy link Quote reply martinez-bakker commented Mar 1, 2015. Not from inside Spotfire. For a di erentiable kernel K: Rd Rd!R, we denote by r 1K: Rd Rd!Rd (resp. In a notebook, a subscript can be created using or. forefinger. You can highlight the 2, then right click and choose font. That will convert the code into a superscript or subscript. • To refer to an array as a whole, use the array name without subscripts. Learn more Subscript in plot title in r. u/noguchisquared. In reply to Marie-Pierre Sylvestre: "[R] Error: subscript out of bounds" Contemporary messages sorted : [ By Date ] [ By Thread ] [ By Subject ] [ By Author ] [ By messages with attachments ] This archive was generated by hypermail 2. For vector arguments, it expands the arguments cyclically to the length of the longest provided none are of zero length. Nat -----Original Message----- From: SAS(r) Discussion [mailto:[email protected] Use it to embed R code and results into slideshows, pdfs, html documents, Word files and more. (Mathcad 15) subscripts in variable names I'd like to write an equation containing variables with subscripts, but my intention is that the subscripts simply identify the variable rather than refer to an array, in other words r(sub)s refers not to element s of array r but simply to a variable named r(sub)s. Precisely speaking, it is Pandoc’s Markdown. In arev, the letter i definitely looks very skinny, even anorexic, when it appears in the second-level subscript position. tex bibtex lesson7a1 pdflatex lesson7a1. Define subscripts. The audit has 8 distinct sections examining the different areas of the plant (shipping, receiving, storage, packaging,etc. Upper R equals StartFraction Uppe. These include superscript, subscript, and Greek letters. The command \texorpdfstring comes to the rescue: \documentclass{article} \usepackage{fixltx2e} \usepackage{hyperref} \begin. But with R[subscript 0] being the mean of a random variable, much more information is contained in the entire probability distribution. So if positive. 09 90 428 520 600 700 780 800 927 t 240 200 180 160 120 100 80 60 40 20 v t N2 N2 t 1 excitation 2 Im t N2 t 1 N2 excitation 1 turn 1 R turn 0. Direct the program to fix the Registry errors automatically. How to solve error the folowing: Error in G[i, ] : subscript out of bounds when using Drivernet R ? Im importing a 412 x 31 matrix: dat<-read. com rmarkdown 0. Detecting subscript in command argument - Blogger 2. Alt + X Shortcuts for Superscript and Subscript On Microsoft Word, enter the hexadecimal code as in the below table, then press alt and x keys together. The subscripts in a chemical equation is the number on the lower right-hand side of a chemical element that tells a chemist how many atoms of that element are present in the equation. New replies are no longer allowed. How to use the LaTeX tables generator? Set the desired size of the table using Table / Set size menu option. Include a superscript in a ggplot2 axis label. Subscript text appears half a character below the normal line, and is sometimes rendered in a smaller font. They are useful in math, chemistry, etc. You can add superscript, subscript, other symbols in google docs, google forms etc. 2 years ago by venu ♦ 6. Ask Question Asked 7 years, 4 months ago. This topic was automatically closed 21 days after the last reply. Formula writing with Polyatomic Ions. ; Enter the table data into the table: copy (Ctrl+C) table data from a spreadsheet (e. Graphing xy plot by group in R 6. markdown documentation: Subscript/Superscript. Hydrocomp Incorporated Mountain View, California 940UO Grant No. I'm trying to create a variable in Manipulate[] with a subscript in its name with the following code: {{Subscript[r, b], 175. Syntax x[real vector r, real vector c] x[|real matrix sub|] Subscripts may be used on the left or right of theequal-assignment operator. 0, Appearance -> "Labeled", Image. Notes: I want to keep the HTML commands list free from hacks resorting to the use of the symbol font, but this can mean that some of the entity codes listed here will not display properly in older browsers. Jump to:navigation, search. where <> is the text-part you want to modify - and you need to take care to add a space after these keywords '\sub', '\super', ' osupersub'. 0 2 H gen ω s ω pu t δ 2 p a t at t=0 t δ 0 0 Suppose that p e (t) drops from 1. Use it to embed R code and results into slideshows, pdfs, html documents, Word files and more. subscript (R) -> Array Slice subscript (R) -> Array Slice Accesses the contiguous subrange of the collection’s elements specified by a range expression. Learn more Special characters and superscripts on plot axis titles. On top of the f there is a, while on the end, there is a 0, and in the middle there is an x. About Us The Simplilearn community is a friendly, accessible place for professionals of all ages and backgrounds to engage in healthy, constructive debate and informative discussions. I use "bquote" to do this. Building on Rick's suggestion, if you want to enter a formula and have subscript and superscript automatically applied you could do so with a macro. (Note that it wasn't always the case, the same page few months ago). Ask Question Asked 8 years, 1 month ago. No extra packages are required to use these symbols. Copy link Quote reply martinez-bakker commented Mar 1, 2015. We’ll show also how to center the title position, as well as, how to change the title font size and color. We are going to use "text replacement" feature in iPhone. More R Subscript. The text in an R Markdown document is written with the Markdown syntax. If you write a text where the underscore is often used in a different context, Org's convention to always interpret these as subscripts can get in your way. a bunch of CG's brought up by Mexican's who sold crack on the streets to support the members of subscript's CS:S addiction. Use a productive notebook interface to weave together narrative text and code to produce elegantly formatted output. I'd like to write the corresponding DisplayLabels in TeX-style so that I can use subscripts, as can be done in regular plots without problem using e. You can only insert certain superscripts or subscripts. I have tried various approaches and suggestions but none of them R › R help. The use of super scripts as you have stated I believe is not very common in machine learning literature. The use of parenthesis with a generic range is optional for all numeric ranges, but is required for symbolic (character) ranges. The command \limits changes the way the limits are displayed in the. It is a toggle switch that turns the subscript on and off. This article explains how to insert spaces of different lengths in mathematical mode. Alt + X Shortcuts for Superscript and Subscript On Microsoft Word, enter the hexadecimal code as in the below table, then press alt and x keys together. You can work around this by using the ZeroWidthTimes option: share | improve this answer | follow | | | |. This tutorial shows how to insert superscripts and subscripts in Microsoft Word using three different tools: 1. Then use the text() function to add text to a plot at xy coordinates. 96 x 10^8 m. indexes, in´dices) (L. It's simple. Hi, I tried this and it works! If the letter is upper-case, the symbol goes through it, but lower case p or x has the symbol above. 0, Appearance -> "Labeled", Image. Definition of subscript in the Definitions. For LaTeX users there is no easy way to write subscript and superscript in text mode. Copy link Quote reply martinez-bakker commented Mar 1, 2015. The command \limits changes the way the limits are displayed in the. Subscripts & Superscripts. Identify the symbol of the cation (first part of the name) and the anion. Imhoff Harley H. On the Insert tab, click Symbol. First, it held that gender identity discrimination is a form of sex stereotyping. As second subscript: repetitive. Below is an example of subscript text, which shows "Subscript Text" halfway below the bottom line of the "Normal Text. I removed my example variable name from definition to expr part of the Module. This is a possibility. Insert Subscript or Superscript. How to add superscript or subscript symbols to a WordPress title Make sure you are logged in as an administrator (It will strip out your HTML otherwise) Navigate to the page, post, or product that you’d like to work on from the backend Add the following code for superscript: ® , and the. The most common way is directly, using the square bracket operator: > x 4 In this example, the user has said "give me the fourth element," and R has said, "you get a vector whose first (and only) element is 4. I'm trying to work out how to have subscript letters in an axis label. Then use the text() function to add text to a plot at xy coordinates. How to Insert a Division Symbol in Microsoft Word By Aaron Parson When writing mathematical equations in Word 2013, you aren't limited to typing on a single horizontal line, nor to using the slash to represent division. May be, like this Subscripts in R plots. md file with knitr rmarkdown-cheatsheet-2. All subscripts below the second nesting will not change font size, and will stay as scriptscriptstyle, while the first subscript will be slightly larger than this, at scriptstyle. Hi, I am using the car package in R to complete a scatterplot matrix with linear regression lines. Open up your. Format text values as superscript or subscript This works well when you want to present data in a visually appealing format. These characters appear smaller than standard text, and are traditionally used for. Node 25 of 25. Below is the complete list of Windows ALT Codes for Math Symbols: Superscript & Subscript Numbers, their corresponding HTML entity numeric character references, and when available, their corresponding HTML entity named character references, and Unicode code points. Superscripting is easy in MS Word. the second line calls position 6,6 which is too big for the first index, and so the subscript is out of bounds. 0024 ohm turn Full winding: R secondary R turn 240. Characters(iChar, 1). Subscripts and superscripts are obtained using the special characters _ and ^ respectively. 1-4, License: GPL-3 Community examples. Subscripts and superscripts can be nested and combined in various ways. Ask Question Asked 3 years, 2 months ago. net dictionary. Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. When a reaction component is a gas, we can also express the amount of that chemical at equilibrium in terms of its partial pressure. Hi there I would like to add a y-axis label that contains a lot of sub/superscript expressions. Let’s say ŷ is the predicted average reading level for a child who has read half an hour a day in preschool. html and provide commented, minimal, self-contained. This is to be expected but the function grm() in package ltm generates an error: "Subscript out of bounds". (The number you multiply or divide. Imhoff Harley H. Although subscripts that reference particular elements are positive, negative subscripts are legal. 0, "Base radius"}, 127. In reply to Marie-Pierre Sylvestre: "[R] Error: subscript out of bounds" Contemporary messages sorted : [ By Date ] [ By Thread ] [ By Subject ] [ By Author ] [ By messages with attachments ] This archive was generated by hypermail 2. As a work around you can use Word 2011 to compose your email and then copy and paste it into Outlook 2011. Browse below to see information about this character, and what fonts you can download for free that have this character in them. Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. This tutorial shows how to insert superscripts and subscripts in Microsoft Word using three different tools: 1. The text in an R Markdown document is written with the Markdown syntax. First, it held that gender identity discrimination is a form of sex stereotyping. forrtl: severe (408): fort: (11): Subscript #3 of the array TABP has value 0 which is less than the lower bound of 1. Is it possible to write Postscript and Subsript in R14 multiline?. So we’d get. tags $blockquote ("Tidy data sets are all the same. I’m having a difficult time finding documentation for the correct syntax to display a subscript in our markdown content files using Blackfriday with Hugo. Type an underscore '_'. frame objects in R (via “R in Action”) The followings introductory post is intended for new users of R. Perhaps tackle this another way — make a statement at the beginning of the document that abc(r) is a registered trademark but you will use abc throughout the document, thus avoiding the (r) issue. Superscripts and Subscripts. subscript (R) -> Array Slice subscript (R) -> Array Slice Accesses the contiguous subrange of the collection’s elements specified by a range expression. Sign in your Google Docs. R also provides a special subscripting method (double brackets) to extract the actual data (in this case a vector) from the data frame:. I'm trying to work out how to have subscript letters in an axis label. Subscripts have square brackets [2] while functions have round brackets (2). Thanks for contributing an answer to TeX - LaTeX Stack Exchange! Please be sure to answer the question. Find the common ratio. LaTeX is a very flexible program for typesetting math, but sometimes figuring out how to get the effect you want can be tricky. This tutorial shows how to insert superscripts and subscripts in Microsoft Word using three different tools: 1. HTML - Subscript. indexes, in´dices) (L. All subscripts below the second nesting will not change font size, and will stay as scriptscriptstyle, while the first subscript will be slightly larger than this, at scriptstyle. Please, see the attached image. Subscript Reduction Operators A reduction operator is a statistical operation (for example, a sum or a mean) that returns a matrix of a smaller dimension. ; Enter the table data into the table: copy (Ctrl+C) table data from a spreadsheet (e. subscript < R >(r: R) -> Self. Are you trying to create a subscript in PowerPoint? It's seems simple, but often these simple tasks take a bit of hunting around to get right. There are two main ways to handle this: 1) use simple mathematical notation and glue the different parts together and 2) use a lot of unicode and hope your typeface (and system) supports all the characters you want to use. Examples: k ro and k rg. On the Insert tab, click Symbol. The intended use when these characters were added to Unicode was to allow chemical and algebra formulas and phonetics to be written without markup, but produce true superscripts and subscripts. The expression arr[12] denotes the value of the 13th element of arr (because subscripts start at 0), while the statement arr[12] = 5 stores the number 5 in the 13th element of arr. Suppose our line is: ŷ = 2. ; The distance from the baseline is very generous (probably too much) by default. Basic Latin 0000-007F is identical to ASCII; Superscript one(¹), two (²) and three (³): See UTF-8_Latin_characters; UTF-8 byte Sequences Code points 1st byte 2nd byte 3rd byte Description U+2070. subscript (r: R) -> Range where R : Range Expression, Self. not something with a Head different from Symbol (e. Likewise subscripts in R involve square brackets [] so x i is written x[i]. When you render your file, Pandoc transforms the marked up text into formatted text in your final file format, as below. * Go to Setting -> General -> Keyboard -> Text Replacement * Press "+" at top right to add a new Text Replacement shortcut; you need to provide: * * Phrase: w. For example, suppose that we wish to typeset the following passage: This passage is produced by the following input:. updateAvailable = nil end elseif self. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. negative 5 percent. Type 'q()' to quit R. Thus "H₂O" (using a subscript character) is supposed to be identical to "H 2 O" (with subscript markup). Subscript symbol is a copy and paste text symbol that can be used in any desktop, web, or mobile applications. Use the subscript tags to place subscripted text onto your websites. Array is a collection of elements of same data type. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Star filled nights. Not from inside Spotfire. This is a possibility. Antonyms: superscript. Below is an example of subscript text, which shows "Subscript Text" halfway below the bottom line of the "Normal Text. For subscript (text below the line) enter ^ before the text you wish to place below the line, then highlight the ^ and the text, right click and then Stack. it is often more convenient to use double subscript notation. Suppose our line is: ŷ = 2. The second is the best choice if you frequently use the same symbols, superscript and subscript (for example, degrees, copyright, squared and cubed). Precisely speaking, it is Pandoc's Markdown. Unicode subscripts and superscripts. 28 x R_{sun}. Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. You can work around this by using the ZeroWidthTimes option: share | improve this answer | follow | | | |. You then select subscript. This is a common cause of error, as programmers often forget that this type is signed on some machines. Superscripted text can be placed onto your website using the tag. Superscript = True ElseIf aUpOrDown(iChar) = "v" Then R. Highlight the text with your cursor. tex bibtex lesson7a1 pdflatex lesson7a1. In this quick tutorial, you'll learn how to create a subscript as well as how to do a superscript in PowerPoint. Subscript = False For iChar = 1 To Len(stNew) If aUpOrDown(iChar) = "^" Then R. There are many flavors of Markdown invented by different people, and Pandoc’s flavor is the most comprehensive one to our knowledge. tags$ blockquote ("Tidy data sets are all the same. Error: invalid subscript type 'list'. Direct the program to fix the Registry errors automatically. The most standard way to write p sub 1 (p 1) is p_1. These features are available in Format Cells option in Font tab. Note that the Unicode spec doesn't contain subscript characters for all of the numbers and letters so in this converter I've had to find the nearest characters which look like superscript versions, or. Ordinary vector subscripts in R start with 1, like FORTRAN and unlike C and its descendants. updateAvailable = nil end elseif self. This seems so basic but i have spent so many hours and couldn't figure out how. Reddit now supports superscript officially with ^ Custom superscript and subscript so you can make formulae like: NH*_4_+* 2H*2O ⇌ H3O+* + OH-a2 + b2 = c2 _1_2D + _1_3T → _2_4He + n0 Here's the syntax: *_subscript_* makes subscript *superscript* makes superscript *_sub_sup* makes _sub_sup *sup_sub_* makes sup_sub_. See Variable subscripts. So far everything is working smoothly -except- the subscript syntax [ '_' ] is only working in the title and not my annotation text box. VAL=/MUSE["PDX"] or VAL=/PDX (see above) Computed fields used. How to add superscript or subscript symbols to a WordPress title Make sure you are logged in as an administrator (It will strip out your HTML otherwise) Navigate to the page, post, or product that you’d like to work on from the backend Add the following code for superscript: ® , and the. 0 The ya-tag and ra-tag, or y and r subscript , and the s after vowels and consonants, were still in force. the rst (resp. Problem 4: What is v R 3 v_{R3} v R 3 v, start subscript, R, 3, end subscript? Reminder: Check the first sign of each element voltage as you walk around the loop. Hello ^just kidding ^the space after "^just" is the non-breaking space. The tag defines the subscript text. It represents the former presence of an [i] offglide after the vowel, forming a so‐called "long diphthong". 1 Anthony S. Table 6Example, leading and trailing superscript and subscript. Harris Funeral Homes v. It can occur with the vowel letters eta η , omega ω , and alpha α. (Note that it wasn't always the case, the same page few months ago). Yes there is. If you use an earlier version of MS Office, you have a slightly more complicated route. If you use a different editor, it can be necessary to execute the bibtex command manually. Subscript text can be used for chemical formulas, like H2O to be written as H 2 O. I would like to insert a superscript in a body of text (e. The first example works as it's just a number, as soon as you have a character in the square brackets it fails. superscript − superscript. In any case, you now have P as the dependent variable, and V as the only independent variable, with x, R, and T being or acting as constants. Hello, Thanks for making this great package. Using special characters []. , CO 2, but not H 2 O), the superscript or subscript can be added with the expression() function, using square brackets to enclose the subscript or a carat (^) to indicate a superscript. A missing value is one whose value is unknown. Below is an example of subscript text, which shows "Subscript Text" halfway below the bottom line of the "Normal Text. The subscript n is used in the combination formula [nCr, where n and r are both subscripts] to indicate the total number from which a group is being selected. Use the subscript tags to place subscripted text onto your websites. Johanson John C. On the other hand, the radius of Alpha Centauri is R_{Alpha Centauri} = 1. You have to repeat inserting the same kind of superscripts or subscripts. Subscript symbol is a copy and paste text symbol that can be used in any desktop, web, or mobile applications. These characters appear smaller than standard text, and are traditionally used for. Ask Question Asked 6 years, 5 months ago. It creates a one to one mapping table with characters and their replacements. Without subscripts, the variable name refers to the whole array. That will convert the code into a superscript or subscript. Note: equations in Excel are floating objects and do not return results. Superscripts and Subscripts. Syntax x[real vector r, real vector c] x[|real matrix sub|] Subscripts may be used on the left or right of theequal-assignment operator. Using special characters []. I can't find a way how to write subscripts in the title or the subtitle in R. If you have a query related to it or one of the replies, start a new topic and refer back with a link. • Examples: • Lab book of multiple readings over several days • Periodic table. That should fix the problem. Superscripts and subscripts can only be done in rich text of some kind, and how you do it can vary from app to app. On the power of PPT-preserving and non-signalling codes Debbie Leung, William Matthews (Dated: August 18, 2014) We derive one-shot upper bounds for quantum noisy channel codes. I could use formatting tips. Determine if the sequence is geometric (Do you multiply, or divide, the same amount from one term to the next?) 2. How to write the superscript in the legend? Please, see the attached image that shows my question As the final presentation of the map is the most important issue that the user sees, it should be perfect in all the items that appear on it How to type the superscript in the legend? Like person/km2 Thank you. Once you have Docs opened to look for “ Format, ” it will be on the top main menu bar with the other options such as Take your mouse cursor on “ Format, ” and you will see a sub-menu appear. 3 R allows users great flexibility in creating and formatting plots. Subscript and superscript in text mode I often need to write various thing in subscript and superscript which have nothing to do with maths so I don't want to have them written in math mode. Find the common ratio. Unicode fonts include special characters that display as superscripted and subscripted versions of the normal number characters. An Introduction to R. Method 3: Add superscript and subscript options to your Quick Access Toolbar. To enable Subscript to follow the four steps from the above and on the 5th step, you will need to select Subscript from the menu, and that is it! The only notable difference will be that your text will start to appear at the bottom now instead of top and will be smaller in size. To summarize the process of writing a recursive formula for a geometric sequence: 1. Step # 1 -- Selecting the Cell First, start Excel and load the sheet. If you use the Count method with the VPageBreaks or HPageBreaks property, you may receive a result of zero. From Free Pascal wiki. That is, if the sequence is 1, 2, 3, 5, 8,. This tutorial shows how to insert superscripts and subscripts in Microsoft Word using three different tools: 1. Markdown Quick Reference R Markdown is an easy-to-write plain text format for creating dynamic documents and reports. To learn more about how to use subscripts, check out our documentation. Identify the symbol of the cation (first part of the name) and the anion. If the kth input coordinates (xWorld(k),yWorld(k)) fall outside the image bounds in the world coordinate system, worldToSubscript sets the corresponding subscripts I(k) and J(k) to NaN. Direct the program to fix the Registry errors automatically. Yes there is. This subreddit is all about notion. For subscript (text below the line) enter ^ before the text you wish to place below the line, then highlight the ^ and the text, right click and then Stack. Subscript and Superscript Actually Sven is a friend of mine whom I used to work with at about the time I started this forum. In a command prompt/shell simply run: pdflatex lesson7a1. It is recommended that you use parentheses when you specify a range. then converts the. Answer / kiran. long time user, 1st time poster. Question: Assume That The Risk-free Rate, Upper R Subscript Upper FRF , Is Currently 7 %, The Market Return, R Subscript Mrm , Is 15% , And Asset A Has A Beta, B Subscript Upper AbA , Of 0. You can add subscripts using the 'add subscript' tool'. x <- rnorm(50) y <- rnorm(50) plot(x, y, xlab=expression(H[2]), ylab=expression(R^4)). V subscript o physics I am eight letter word and i am unavoidable in this world hints 1) letter 8,2 & 6 means dirt or waste 2)letter 4,5,8 &7 means clean 3)letter 1 Suppose that the first one fourth of the distance between two points is covered with an average velocity of 18 m s the average velocity for the remain. Ï‰ s 2 Ï€ 60 Hz Ï‰ s rad s Z S1 1 ohm e 1 85 deg Z S1 Z S2 Z S1 Z S0 1 ohm e 1 80 deg Z S0 Z L1 4 ohm e 1 80 deg Z L1 Z L2 Z L1 Z L0 12 ohm e 1 75 deg Z L0 Z C 4 ohm 3 e-1 90 deg Z C 33% compensation Equivalent impedances for SLG fault analysis (time domain. For higher nested subscripts, we simply extend this to a_{b_{c_{d_e}}} etc. Get your technical queries answered by top developers !. LaTeX is a very flexible program for typesetting math, but sometimes figuring out how to get the effect you want can be tricky. Some examples of using $$\LaTeX$$ in R Markdown documents. 0_01/jre\ gtint :tL;tH=f %Jn! [email protected]@ Wrote%dof%d if($compAFM){ -ktkeyboardtype =zL" filesystem-list \renewcommand{\theequation}{\#} L;==_1 =JU* L9cHf lp. This is a very basic question, and as my code suggests, it should work, but it's not: I want have the variables in these legend keys have subscripts, but when MATLAB displays the graph, they appear as written with the underscore symbol. PK Hx5 5 mathcad/worksheet. The circuit court reached this in two ways. It is necessary to execute the pdflatex command, before the bibtex command, to tell bibtex what literature we cited in our paper. Format text values as superscript or subscript This works well when you want to present data in a visually appealing format. Examples: 1. (1) In programming, a symbol or number used to identify an element in an array. The subscripts in a chemical equation is the number on the lower right-hand side of a chemical element that tells a chemist how many atoms of that element are present in the equation. ProgrammingR. When you render your file, Pandoc transforms the marked up text into formatted text in your final file format, as below. Resources to help you simplify data. 3 The built-in comma operator shall not be used within the subscript operator. Kittle, Jr. Description Subscripts come in two styles. R Markdown Cheat Sheet 1. To get an expression exp to appear as a subscript, you just type _{exp}. Is there an. Making statements based on opinion; back them up with references or personal experience. xml ¢ ( Transient R-L Circuit Example Given a 230kV:34kV, Y-Y, 100MVA transformer supplie d supplied by a source with a short circuit MVA of 1000. (Note that it wasn't always the case, the same page few months ago). updateCheckRunning = false if self. When describing particle algorithms, we use a subscript to denote the time index and brackets to denote the particle index, i. All of the data points should now be highlighted. On a default Hash, subscripts coerce keys into strings, as long as those keys produce something Cool. Sign in Register Superscripts in R & Rmarkdown; by Nathan Brouwer; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. 0 and still, contrary to popular belief, the probability of an exponentially growing pandemic may be. Titanic: Machine Learning from Disaster Start here! Predict survival on the Titanic and get familiar with ML basics. How to add superscript or subscript symbols to a WordPress title Make sure you are logged in as an administrator (It will strip out your HTML otherwise) Navigate to the page, post, or product that you’d like to work on from the backend Add the following code for superscript: ® , and the. The default xlab and ylab do not allow subscripts like r2 or superscripts or xi. indeterminate, since expectations are unreliable. I'm trying to create a variable in Manipulate[] with a subscript in its name with the following code: {{Subscript[r, b], 175. It provides a wide variety of statistical and graphical techniques (linear and nonlinear modelling. Below is the complete list of Windows ALT Codes for Math Symbols: Superscript & Subscript Numbers, their corresponding HTML entity numeric character references, and when available, their corresponding HTML entity named character references, and Unicode code points. the ra (subscript) is seen as variable r with a subscript, not a new. Try not to use subscripts in Mathematica, it doesn't work very nice. These include superscript, subscript, and Greek letters. , or H 2 S0 4. Subscripts in textview. Determinant. In the Professional format:. Type the text you want to be superscripted or subscripted. Answer: The answer is B. It is necessary to execute the pdflatex command, before the bibtex command, to tell bibtex what literature we cited in our paper. We’ll show also how to center the title position, as well as, how to change the title font size and color. In MTEXT: For subscript (text below the line) enter ^ before the text you wish to place below the line, then highlight the ^ and the text, right click and then Stack. md file with knitr 2. Standard & Poor's fund credit quality ratings, identified by the 'f' subscript, are current assessments of the overall credit quality of a fixed-income fund's portfolio. On a default Hash, subscripts coerce keys into strings, as long as those keys produce something Cool. Restart your computer when the program is finished fixing or restoring any corrupted files. LaTeX will happily allow you to insert a list environment into an existing one (up to a depth of four, more levels are available using packages). Numerical subscripts in R start at 1, and continue up to the length of the vector. ----- EPA-600/9-80-015 April 1980 USERS MANUAL FOR HYDROLOGICAL SIMULATION PROGRAM - FORTRAN (HSPF) by Robert C. To Superscript: Repeat all above steps with a carat '^' instead of an underscore. The intended use when these characters were added to Unicode was to allow chemical and algebra formulas and phonetics to be written without markup, but produce true superscripts and subscripts. How can I write v 1,2 with 1,2 as subscripts? Thanks for your help! r plot subscript. 50 Updated: 8/14 1. For example The radius of the sun is R_sun = 6. Center subscript / superscript: Command Input Field Formula {2-4*3} csub ( 123 ) 2−4∗3 123 +18 csup ( plus ) - (1+2+3+4) csub {minus} 18 plus. Use "bquote" in ggplot2 for subscript text. names){ if(sd(data[[f]], na. Thus "H₂O" (using a subscript character) is supposed to be identical to "H 2 O" (with subscript markup). Figures 1 and 2 show the results of six cases of subscripting a particular vector that can be defined with:. First, it held that gender identity discrimination is a form of sex stereotyping. So for the vector above, x[1] is 3, x[2] is 1, etc. 0 The ya-tag and ra-tag, or y and r subscript , and the s after vowels and consonants, were still in force. Subscripts are letters, numbers, words, or phrases that appear slightly below the surrounding text. Once you have Docs opened to look for “ Format, ” it will be on the top main menu bar with the other options such as Take your mouse cursor on “ Format, ” and you will see a sub-menu appear. Imhoff Harley H. Below is the complete list of Windows ALT key numeric pad codes for superscript & subscript letters (Latin and Greek), their corresponding HTML entity numeric character references and, when available, their corresponding HTML entity named character references. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. V subscript o physics I am eight letter word and i am unavoidable in this world hints 1) letter 8,2 & 6 means dirt or waste 2)letter 4,5,8 &7 means clean 3)letter 1 Suppose that the first one fourth of the distance between two points is covered with an average velocity of 18 m s the average velocity for the remain. While we typically aim to apply functions to vectors as a whole, there are circumstances where we want to select only some of the elements of a vector. Subscripts & Superscripts. Subscript may refer to any of the following:. U+1D63 is the unicode hex value of the character Latin Subscript Small Letter R. No need to use extra packages. If you have a query related to it or one of the replies, start a new topic and refer back with a link. The Greek letters "alpha" and "beta" are probably the most common, as they are used in subjects ranging from engineering to finance. Meaning of subscript. Another menu will appear on the. This warning is enabled by -Wall. Superscripting is easy in MS Word. Undo superscript or subscript formatting. This requires only typing both commands after 'Mt'. repeat the steps as necessary. He later found another job which involved less or no web work and has since had his first child. This occurs under the conditions listed in the "Cause" section. In almost all cases, you won’t do this by hand, but rather with software like SPSS or R. The use of parenthesis with a generic range is optional for all numeric ranges, but is required for symbolic (character) ranges. Be sure you are on the worksheet which contains the chart you wish to work with. The individual elements of the array can be referenced by using subscript. Numerical subscripts in R start at 1, and continue up to the length of the vector. The most likely source of the error message is that the data you are putting into the algorithm are not in the format that the function expects. I was hoping there was an undocumented Subscript and Superscript with the TEXT function Steve. If you have a query related to it or one of the replies, start a new topic and refer back with a link. Error message that came up while using SMOTE function from DMwR package in R is captured below, along with its reason and fix. The audit has 8 distinct sections examining the different areas of the plant (shipping, receiving, storage, packaging,etc. Nat -----Original Message----- From: SAS(r) Discussion [mailto:[email protected] 50 Updated: 8/14 1. The first official book authored by the core R Markdown developers that provides a comprehensive and accurate reference to the R Markdown ecosystem. The most standard way to write p sub 1 (p 1) is p_1. R has several ways to subscript (that is, extract specific elements from a vector). subscript (noun) In chemical formulas the number of atoms in a molecule is written as a subscript, so we write HO for water which has two atoms of hydrogen for each one of oxygen. I have a question, not really an issue. Viewed 172k times 95. In Python, there is a method called maketrans. I would like to add the title, " Emissions from PM2. (Note that it wasn't always the case, the same page few months ago). Stoichiometry is the process of balancing chemical equations, and subscripts in chemical formulas play in important role.$^R The result of evaluation of the last successful (?{ code }) regular expression assertion (see perlre). For example, to display H 2 0, you would write it like this:. R's subsetting operators are powerful and fast. You can highlight the 2, then right click and choose font. Preface LATEX [1] is a typesetting system that is very suitable for producing scien- tiﬁc and mathematical documents of high typographical quality. (The number you multiply or divide. Subscript and superscript are important when you are dealing with different types of formulas. Info Unicode Character 'LATIN SUBSCRIPT SMALL LETTER R' (U+1D63) Browser Test Page Outline (as SVG file) Fonts that support U+1D63. ; Enter the table data into the table: copy (Ctrl+C) table data from a spreadsheet (e. Subscript = False For iChar = 1 To Len(stNew) If aUpOrDown(iChar) = "^" Then R. Basic Latin 0000-007F is identical to ASCII; Superscript one(¹), two (²) and three (³): See UTF-8_Latin_characters; UTF-8 byte Sequences Code points 1st byte 2nd byte 3rd byte Description U+2070. You can work around this by using the ZeroWidthTimes option: share | improve this answer | follow | | | |. subscript (noun) In chemical formulas the number of atoms in a molecule is written as a subscript, so we write HO for water which has two atoms of hydrogen for each one of oxygen. Use MathJax to format equations. Let’s say ŷ is the predicted average reading level for a child who has read half an hour a day in preschool. In the expression x[2], the subscript is a vector. The Greek letters "alpha" and "beta" are probably the most common, as they are used in subjects ranging from engineering to finance. subscript (r: R) -> Range where R : Range Expression, Self. The Symbol Codes site is migrating to Sites at Penn State and has been significantly updated. For higher nested subscripts, we simply extend this to a_{b_{c_{d_e}}} etc. Step 2 − Add the following code to res/layout/activity_main. the writer of the TUGboat article is wrong in claiming superscript ordinal suffixes are solely a 'Victorian fetish' peculiar to English; his text, confessedly a 'rant', is riddled with loaded words like 'obscenity' and 'ilk'. way away from the \classic" SIR result rcoming from solving the transcendental equation (= )log(1 r)+r= 0. Active 5 years, 3 months ago. Abbreviated as sub, subscript refers to words or character that are half the height of a standard character and printed lower than the rest of the text. Did you know that you can also insert equations in Excel? On the Insert tab, in the Symbols group, click Equation. To refer to multiple elements in a vector, simply use the c function to create a vector of the indexes you're interested in. I'd like to write the corresponding DisplayLabels in TeX-style so that I can use subscripts, as can be done in regular plots without problem using e. Subscripts of 0 are silently ignored. second) argument. Combining superscripts, subscripts and other symbols in axis labels 6. (1) In programming, a symbol or number used to identify an element in an array. The use of super scripts as you have stated I believe is not very common in machine learning literature. Thanks, Humberto. The command \texorpdfstring comes to the rescue: \documentclass{article} \usepackage{fixltx2e} \usepackage{hyperref} \begin. LaTeX handles superscripted superscripts and all of that stuff in the natural way. A high-resolution flat-crystal spectrometer was used on the SuperEBIT electron beam ion trap to measure the energies of the 2s[subscript 1/2]-2p[subscript 3/2] transitions in lithiumlike through. Hello, Thanks for making this great package. Subscript may refer to any of the following:. 0 2 H gen ω s ω pu t δ 2 p a t at t=0 t δ 0 0 Suppose that p e (t) drops from 1. As an example; need to indicate the units of measurement for some data and therefore need to be able to pull data from the table such as the following: kg/m2, m/s2 etc. So we’d get. I have data labels in a chart that are chemical formula. The Court is also set to hear arguments in R. I want the numbers be formatted with subscripts (in the chart). I have the following code: mtext(text="X[1]") How can I change "[1]" to be subscript? (see picture below) subscript. 5 Markdown syntax. There is an option to have both the superscript and subscript options at a mere one click away, and readily accessible at all times of using PowerPoint, and that is to add them to your Quick Access. Subscript and superscript are important when you are dealing with different types of formulas. Use MathJax to format equations. Preface LATEX [1] is a typesetting system that is very suitable for producing scien- tiﬁc and mathematical documents of high typographical quality. There are many flavors of Markdown invented by different people, and Pandoc's flavor is the most comprehensive one to our knowledge. If you use a different editor, it can be necessary to execute the bibtex command manually. U+1D63 is the unicode hex value of the character Latin Subscript Small Letter R. how to apply subscript in the facet_grid function of ggplot Tag: r , ggplot2 , subscript I would like to use ggplot to plot my result of the association of air pollutant with birth weight changes (95%CI). facebook, tumblr, twitter, reddit, instagram bio etc. Axis Labels Containing Subscripts and Superscripts. R programming language resources › Forums › Graphing › Italicized Subscript This topic has 1 reply, 1 voice, and was last updated 7 years ago by kodyonthekeys. FOR Superscript: press ctrl, shift and plus(+) keys together For Subscript : press ctrl and plus(+) keys together Alternative: Press Ctrl + D and the option box will appear where you can select Superscript and subscript. Hi @Northern,. Making statements based on opinion; back them up with references or personal experience. Info Unicode Character 'LATIN SUBSCRIPT SMALL LETTER R' (U+1D63) Browser Test Page Outline (as SVG file) Fonts that support U+1D63. It's an A and M with subscript r's under it. With this view, we show – again by simple arguments – that R[subscript 0] can be greater than 1. Throughout this book the parentheses are included for consistency and clarity of presentation. 3) If you use floating-point arithmetic to compute an index array, then the array values may not be exact integers. subScripts[id]. SubSequence where R : Range Expression, Self. Question: Assume That The Risk-free Rate, Upper R Subscript Upper FRF , Is Currently 7 %, The Market Return, R Subscript Mrm , Is 15% , And Asset A Has A Beta, B Subscript Upper AbA , Of 0. It's simple. There are many flavors of Markdown invented by different people, and Pandoc’s flavor is the most comprehensive one to our knowledge. I am using Office 2010 Excel. It only takes a minute to sign up. If you used hyperref and simply used \textsubscript in a section heading, hyperref would complain about the math shift. 6k Hi, I need subscript NO3 in plot, not in the x or y title, like this;. Active 3 years, 2 months ago. R Errors Explained: subscript out of bounds r. If you are … ALT Codes for Math Symbols: Superscript & Subscript Numbers Read More ». Question: Q: Use subscripts in text- rd, th, st. Post a new example: Submit your example. One click to add right aligned captions for all tables, equations, figures, charts, and SmartArts in Word. Usually, the subscript is placed in brackets following the array name. After creating a chart in Microsoft Excel, a best fit line can be found as follows: 1. Each messy data set is messy in its own way. numerical subscripts (intended primarily to represent times or time periods; available secondarily as location subscripts or for other purposes) 1,2,3, etc. Type the text you want to be superscripted or subscripted. Find the common ratio. I can't find a way how to write subscripts in the title or the subtitle in R. Viewed 3k times 8. Then, the statement. ; The distance from the baseline is very generous (probably too much) by default. Subsetting is hard to learn because you need to master a number of interrelated concepts: The three subsetting operators. Superscripting is easy in MS Word. Tetrafluoroethane can be used as a solvent in organic chemistry. You may then have do the same and deselect subscript to get the next bit of typing back to normal size. Nt,Rd where t,Rd should be subscript. Hi, How do I write a superscript within gsub? I have the following: gsub("_mgkg",expression(paste("mg kg"^{-1})),names[1]) Thanks -- Shane [[alternative. Then feed that into the procedure you were hoping to invoke. For example, AR[5] identifies element number 5 in an array called AR. xlabel("log10") where "10" is a subscript of "log", but without doing: plt. Apple Footer. 1 Anthony S. example [ I , J , K ] = worldToSubscript( R , xWorld , yWorld , zWorld ) maps points from the 3-D world system to subscript arrays I , J , and K , using 3-D. With R Markdown, you can easily create reproducible data analysis reports, presentations, dashboards, interactive applications, books, dissertations, websites, and journal articles, while enjoying the simplicity of Markdown and the great power of. What does the subscript r stand for in. Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. superscript^2^ subscript~2~ ~~strikethrough~~. Occasionally, though, we run into a three- or higher-dimensional array. R has several ways to subscript (that is, extract specific elements from a vector). Re: Problem with subscript in thermodynamic problem I simply tried to keep as close to the orgiginal file as possible. The data are indicated with a single variable name, and two subscripts, like this x ij The ﬁrst subscript refers to the row that the particular value is in, the second subscript refers to the column. Restart your computer when the program is finished fixing or restoring any corrupted files. Select the Text menu item from the global navigation bar at the very top of your screen. A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. Re: Subscript and superscript letters and numbers by Dave R » Fri May 18, 2012 8:53 pm You can use the insert function from Character Map to insert special characters into the text in SketchUp. From Free Pascal wiki. Resources to help you simplify data collection and analysis using R. If you are … ALT Codes for Math Symbols: Superscript & Subscript Numbers Read More ». Superscript and subscript allow you to type characters that appear above or below the normal text line. In Python, there is a method called maketrans. PK aM*" " mathcad/worksheet. The organization of information into paragraphs is not affected by how the paragraphs are presented: paragraphs that are double-justified contain the same thoughts as those that are left-justified. superscript^2^ subscript~2~ ~~strikethrough~~. How to Subscript and Superscript Text in HTML. I often get asked how can you insert the division sign in Word. Oh, there are suggestions in the online help forums and stuff for what to do in that case. Learn more Special characters and superscripts on plot axis titles. To write italics within L A T E X, there is the command \textit. Viewed 3k times 8. Precisely speaking, it is Pandoc's Markdown. indexes, in´dices) (L. The problem is that I don't know how to include superscripts and subscripts in the axis names of this kind of plots. 1-4 , License: GPL-3 Community examples. subscripts synonyms, subscripts pronunciation, subscripts translation, English dictionary definition of subscripts. R subscript i). Define subscripts. For example, if we have a sequence of numbers and want to use a variable to represent them, we often use a subscript to indicate the index in the sequence. I'm trying to create a variable in Manipulate[] with a subscript in its name with the following code: {{Subscript[r, b], 175. That should fix the problem. , the matrix is nonsingular). What Is a Subscript in Chemistry? In a molecular formula, a subscript is written immediately after the letter denoting an element and refers to the amount of atoms that element has. Sign in your Google Docs. Answer: The answer is B. For K-12 kids, teachers and parents. The entry in the index for the subentry 'History' will be the range of pages between the two \index commands. • Often data come naturally in the form of a table, e. Once you have Docs opened to look for “ Format, ” it will be on the top main menu bar with the other options such as Take your mouse cursor on “ Format, ” and you will see a sub-menu appear.
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# Equisatisfiability

In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not.[1] Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models.

Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept are Skolemization and some translations into conjunctive normal form.

## Examples

A translation from propositional logic into propositional logic in which every binary disjunction ${\displaystyle a\vee b}$ is replaced by ${\displaystyle ((a\vee n)\wedge (\neg n\vee b))}$, where ${\displaystyle n}$ is a new variable (one for each replaced disjunction) is a transformation in which satisfiability is preserved: the original and resulting formulae are equisatisfiable. Note that these two formulae are not equivalent: the first formula has the model in which ${\displaystyle b}$ is true while ${\displaystyle a}$ and ${\displaystyle n}$ are false (the model's truth value for ${\displaystyle n}$ being irrelevant to the truth value of the formula), but this is not a model of the second formula, in which ${\displaystyle n}$ has to be true in this case.

## References

1. M. Krötzsch (11 October 2010). Description Logic Rules. IOS Press. ISBN 978-1-61499-342-1.


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# Why noncompetitive states are so important for understanding the outcomes of competitive elections: the Electoral College 1868–2016.

### Jonathan Cervas and Bernard Grofman

• Cervas, Jonathan, and Bernard Grofman. 2017. Why noncompetitive states are so important for understanding the outcomes of competitive elections: the Electoral College 1868–2016. Public Choice 173(3–4): 251–265. PDF
• ## 1 Introduction

The division between Red America and Blue America has become part of ordinary citizens' understanding of US politics. Colored maps (chloropleths) are now an indispensable aspect of election coverage, visually emphasizing how geography matters. CNN and other broadcasters are able, with the push of a button, to display historical comparisons of voting patterns at various levels of electoral geography. However, institutional rules such as the US Electoral College structure campaigning incentives so that candidates need to allocate their limited resources and time with the goal of increasing their likelihood of gaining the needed 270 Electoral College (EC) majority. Thus, the campaigning of the candidates tends to be focused on the so-called "purple states", i.e., the competitive states where campaigning might be assumed to make a difference (Shaw 1999b, 2006). For example, on the Sunday before Election Day 2016, Donald Trump visited five states; Florida, North Carolina, Pennsylvania, New Hampshire and Michigan. Four of the five states ended up as the four closest states as measured by the final two-party vote margin. On CNN, on election night in 2016, Wolf Blitzer quipped to Jake Tapper that "Jake, [this is] another presidential race where all eyes right now are on Florida", to which Tapper responded "It's one of the critical states in this race. Donald Trump himself has said he doesn't see a path to the presidency for himself without the state of Florida, the 29 electoral votes." Tapper went on to say, "the Clinton campaign knows they need Florida. They have been saying for some time they feel better about Florida than they do about states such as North Carolina,... Ohio, or Iowa". The fifth, North Carolina, had gone to the Democratic candidate in the previous two elections but was a southern state where Republicans were quite successful in state and federal elections. Trump won North Carolina.

The focus of attention on the competitive states is enhanced by the horse-race style coverage of presidential elections by the media, who refer to such states as "battleground" states (Lipsitz 2009). Such states are the ones most likely, over the course of a campaign, to ''swing'' from one candidate to the other. Often such states are taken, at least implicitly, to be the ones determinative of the presidential contest's winner, with the largest of the battleground states in terms of EC votes seen as especially critical. In contrast, outcomes in noncompetitive states, because they will come as ''no surprise'', tend to be treated by the media as completely uninteresting and also largely irrelevant. If, indeed, campaigns focus exclusively on a set of battlegrounds, other states might suffer lower citizen engagement (Gimpel et al 2007; Lipsitz and Teigen 2010), depressed voter turnout (Aldrich 1993; Duffy and Tavits 2008; Geys 2006), and worse representation (Downs 1957; Stokes 1999).

However, while results in these noncompetitive states may not come as surprising, they play an important role in shaping both election outcomes and campaign strategies. The view that the noncompetitive states are largely irrelevant has been strongly challenged by Brams and Kilgour (2017). We will refer to Brams and Kilgour's Public Choice paper by their names and with the B-K acronym interchangeably throughout this essay. These authors point out that each candidate's electoral votes can be thought of as coming from two sources: noncompetitive states—with outcomes effectively decided before the election—and the competitive states that support him or her on Election Day. But it is not simply that the EC votes received in noncompetitive states are just as important in determining the presidential winner as the EC votes received in the competitive states, but also that the readily foreseeable outcomes in noncompetitive states can ''load the electoral dice'' by requiring the candidate with fewer expected easy victories to do remarkably well in the more competitive states in order to win. For example, in 2012, Brams and Kilgour point out (p. 101): ''Because Barack Obama had a 233-191 electoral vote lead over Mitt Romney in the 42 noncompetitive states and the District of Columbia, he needed only 37 of the 114 electoral votes in the competitive states to win with a majority of 270 electoral votes, whereas Romney needed 79''. Indeed, at the extreme, we can imagine that the outcomes in states essentially safe for one party might involve enough electoral votes so as to render outcomes in the more competitive states the ones that are irrelevant. In 1984, Ronald Reagan won 49 out of 51 states (including Washington, DC). Norman Ornstein, writing before the election, said ''Incumbent presidents don't often lose, particularly presidents presiding over 6% real growth and low or non-existent inflation'' (quoted in CQ Press, http://library.cqpress.com/cqresearcher/document.php?id=cqresrre1984091400).

Moreover, when there is a partisan imbalance in EC vote share expected from the noncompetitive states there is also a potential for choice of (slightly) different campaign strategies by the advantaged and the disadvantaged candidate (Strömberg 2008; Shaw and Althaus 2017). The trailing candidate may be forced to campaign in states where the probability of success is low. Another impact of the different degrees of competitiveness across states is tied to the different levels of visible campaign activity in competitive and noncompetitive states. Greater exposure to a campaign can lead to a positive impact on voter interest and political engagement and to higher turnout, with some studies finding the differences across levels of campaign exposure particularly high for low-income individuals (Gimpel et al. 2007; Lipsitz and Teigen 2010).

Brams and Kilgour specify an indicator, Winningness, of the extent to which the virtually certain outcomes in noncompetitive states structure the expected election outcome overall in a two-candidate, plurality rule contest. If we, for simplicity, posit that each of the battleground states is equally likely to go for either candidate, and there are $m$ such states, then Winningnessis the proportion of the $2^{m}$ combinations of zeroes and ones in which the candidate who is ahead in the noncompetitive states is the winner (adding the seats won in competitive states found in that particular combination to the already ''known'' votes in the noncompetitive states). The Winningness value for the Democratic candidate is simply one minus the Winningness value for the Republican candidate.

Note that the greater the advantage a given candidate has in the noncompetitive states, the greater will be the expected proportion of the $2^{m}$ outcomes in which that candidate is the winner of an Electoral College majority, since the candidate ahead in EC votes won in noncompetitive states will need fewer votes from the competitive seats to amass a winning majority than will the other candidate. For example, in 2012, with $m = 8$ competitive states, under the equiprobability assumption, Brams and Kilgour (2017, p. 101) point out that 207 (80.9%) of the 256 splits would result in a win for Obama, whereas only 49 (19.1%) would result in a win for Romney, giving Obama 4.22 times more ways of winning than Romney".

Brams and Kilgour (2017, pp. 101-102) offer two other closely linked indicators that can be used to measure the extent to which outcomes are predictable: Vulnerability and Fragility. Vulnerability is defined as ''the proportion of the coalitions in competitive states in which a single competitive state, by switching to the other candidate, either can cause a change in the winner or create a tie...''; while ''fragility is measured by the expected number of competitive states in a winning coalition that can disrupt victory in this way.'' Both of the latter measures are well defined only for those election years in which no candidate has a large enough EC vote share in the noncompetitive seats to constitute a majority of the Electoral College. Each must be calculated separately for each party. Winningness is defined for all elections.

Brams and Kilgour, using a definition of non-competitive state as one wherein the winner's vote share in a two-party race is expected to be above 53%, In races with third parties, a margin of victory no greater than 6%. For the purposes of this note, we concern ourselves only with the two highest vote earners and calculate accordingly. calculate Winningness, Vulnerability and Fragility for four recent elections: 2000, 2004, 2008 and 2012. We extend their analysis to include all 38 presidential elections in the modern two-party era, from 1868 to 2016. In the next section, we focus on the most important findings of our historical analyses for the Brams and Kilgour measures, evaluating how well each of the three measures (and all three together) allow us to predict EC winners and EC seat shares in these 38 elections.

Table A1 in the on-line Appendix reports the full results of our calculations. In the process of replicating Brams and Kilgour's (2017) analyses, we found a few minor errors that we corrected; those corrections explain the differences in the numbers reported in Table A1 for the elections of 2000 and 2004, and those reported in Table 4 of Brams and Kilgour. In the online Appendix, we consider how analyses would change if we altered the definition of noncompetitive state. While the analyses in the on-line Appendix show that our choice of range to define a competitive state can matter somewhat, to maximize our compatibility with Brams and Kilgour (2017), and because we think this definition is a plausible one in the context of predicting EC outcomes (see Sect. 3 below), we will use the Brams and Kilgour (2017) plus or minus three percentage point definition of competitive state in the remainder of the essay.

In the subsequent section, we offer a simple alternative measure based on the Brams and Kilgour intuition about the importance of the imbalance in partisan breakdown of EC seat shares in the noncompetitive states. We show that this measure, which we label Non-Competitive Advantage, is as predictive of the final EC outcomes and somewhat more predictive of final EC vote percentages than any of the measures proposed by Brams and Kilgour (2017). In sum, we find both Winningness and Non-Competitive Advantage to perform very well.

### 1.1 Winningness, vulnerability and fragility, 1868-2016

Over this entire period, as commonsense would predict, when Winningness is high, Vulnerability and Fragility are both low (with correlations ranging from $-0.88$ to $-0.98$), while the correlations between the latter two variables are quite positive (ranging from $0.80$ to $0.91$). See Tables 1, 2. The Pearson correlations reported in Tables 1, 2 involving Vulnerability and Fragility are calculated only for the elections wherein outcomes can be affected by what happens in the competitive states. In Table 1, Vulnerability and Fragility are defined in all elections that are competitive (17/38), and because the sample is split for Republicans and Democrats, for years in which that party's candidate had a Winningness of 1 (Vulnerability and Fragility are always zero in these cases).

While the various measures proposed by Brams and Kilgour (2017) are of theoretical interest, in and of themselves, we are most interested in how these measures allow us to address the bias imposed on likely Electoral College outcomes from having a substantial proportion of voting outcomes already known in advance in a fashion that favors one political party. Brams and Kilgour note (2017, p. 111) that the sign on the Winningness advantage correctly predicts the winners in all four of the presidential contests they study. When we replicate that analysis for all 38 elections, we find that this holds for all but two elections: 1880 and 1960. This is a very good predictive performance by the Winningness variable. Even if we consider just the 17 elections for which the winner was determined by the competitive states, this is a success rate of 88%. While these two elections were very close in two-party vote margin, and thus might be regarded as hard to predict, they were less so electorally. In 1960, John F. Kennedy won the EC vote by 9.1% and, in 1880, James Garfield won by 7.5%. In neither election were third-party candidacies consequential in affecting relative two-party shares.

A more difficult test for the predictive usefulness of Winningness and the two other variables is to ask how well they, singly or collectively, predict final EC vote share outcomes. Figure 1 plots Winningness, Vulnerability and Fragility scores against the final EC final vote share. These three variables are, in fact, highly correlated with EC outcomes, with the correlation for Winningness at 0.90, that for Republican (Democratic) Fragility at $-0.76 (-0.67)$, while that Republican (Democratic) Vulnerability is $-0.66 (-0.81)$. Because of the frequent occurrence of values of 0 or 1, a perfect linear fit is impossible.

Fig. 1 Comparing Winningness, Vulnerability, and Fragility to Electoral College outcomes.

Note: Candidate's Share of EC is from the Republican perspective in plot one. The Candidate's Share of the EC is labeled ''D'' for the Democratic candidate, and ''R'' for the Republican candidate in the Vulnerability and Fragility plots

We also see from the first plot in Fig. 1 that in most years, Winningness is such that the outcome is expected to be determined solely by what happens in the noncompetitive states, i.e., a Winningness value of zero or one. In the four elections analyzed in Brams and Kilgour (2017), only one, 2008, fell into this category. Had Brams and Kilgour extended their data back somewhat further in time to 1980, however, they would have found that in that election and in each of the four following elections, one of the two candidates had locked up enough votes in noncompetitive states to win the election. In 1992, Bill Clinton was just seven EC votes shy of having enough a majority in noncompetitive states, and could have lost the election in only five of the more than 130,000 different combinations of electoral outcomes among the competitive states, i.e., $Winningness >0.99$.

We have conducted regression analyses with all three Brams-Kilgour measures as independent variables and Democratic EC vote share as the dependent variable, but we do not report results for these regressions since, as expected, the very high correlations among the three variables meant that adding Vulnerability, Fragility, or both, to Winningness did not increase the adjusted $R^2$, and only one of the three variables was statistically significant in any of the models. Also, when we include Vulnerability and Fragility, we require separate equations for each party, and we lose cases. For the 38-election period, we find that the best fitting model in terms of adjusted $R^2$ is the simple bivariate regression in which Winningness alone predicts the EC outcome, with an adjusted $R^2$ value of $0.81$ (see Table A2).

### 1.2 Accuracy of ex-post classification of states as noncompetitive

B-K first justify the use of the ex-post criteria by which they classify competitive and noncompetitive by pointing out that, empirically, the fit between ex ante and ex post evaluations of competitive states is very good. Pre-election polls do a good job of predicting final outcomes to within a small margin of error (Soumbatiants et al. 2006) - though, of course, that margin of error may be enough to generate an erroneous prediction. Still, highly uncompetitive states are unlikely to change partisan direction over the course of a single election cycle. B-K point out that the ±3% value they use to define a competitive state corresponds with the usual pre-election polling margin of error. When a state polls outside this three-percentage point margin, it generally is seen as not winnable by the trailing candidate, although more errors in prediction do occur than would be suggested by the 95% confidence limits (Gelman and King 1993; Shirani-Mehr et al. 2018). Another reason for choosing the ±3% value is a pragmatic one that we found only after we had done robustness checks; over both recent elections and the longer historical data: ±3% value has (marginally) greater predictive power than the often used ±5% definition of competitive state (see on-line Appendix). Collectively, moreover, a large number of competitive states may result in an unexpected outcome if those states go disproportionately for one candidate. Thus, close elections nationally bear resemblances to the flip of a coin.

However, campaigning choices are only ''imperfectly correlated'' with the degree to which a state is competitive (Shaw and Althaus 2017). We would not, in general, expect campaign spending or campaign appearances to be only in competitive states, since candidates also spend some money and make some appearances for reasons not directly related to boosting their own campaign chances, e.g., to help down-ticket candidates or to build for the future. Bartels (1985) has pointed out that campaigns have what he calls both ''instrumental'' and ''ornamental'' reasons for staging campaign events. Attending an event in a swing state, where a candidate's presence could increase turnout is instrumental, while visiting a state to satisfy state parties might be ornamental. Hillary Clinton spent over $600,000 in Arizona, perhaps trying to influence lower ticket races by increasing mobilization efforts. Ultimately, Arizona, a state that has had a strong Republican tradition, became competitive in 2016. Also, some major media markets cover more than one state. And the differential cost of campaigning may increase the desirability of campaigning in some small states where advertising costs are relatively inexpensive (Shaw 1999a, b; Stratmann 2009; Shaw and Althaus 2017). Finally, there is uncertainty about time trends, and the need to have alternative routes to victory. While Shaw and Althaus (2017), who have collected the most complete data on campaign appearances and campaign expenditures by both parties for most of the post-WWII era and show that the candidates of the two major parties were in agreement as to which are the states in which to invest campaign resources (we would not expect perfect symmetry and we do not find such perfect symmetry in the candidates' opinions). In addition to reasons not directly connected with the presidential election contest, a leading presidential candidate and a trailing candidate face somewhat different strategic tasks. Sometimes a trailing candidate must opt for campaigning in a state expected to be won by the opponent, since doing so may open the only possible path to victory and/or may tempt an opponent to divert resources to protect a ''base'' state that could be better spent elsewhere. Strömberg (2008) suggests a hockey metaphor; as a game winds down, a trailing team looking to increase the probability of tying the game pulls their goalie to provide more offensive potential, taking the risk of giving up another goal. A leading team would instead probably act to protect its lead, replacing offensive players with defensively skilled players. As Shaw and Althaus (2017) put it: ''campaigns often hone in on less competitive states when their overall position is weak''. Nonetheless, as both Grofman and Feld (2005) and Strömberg (2008) argue, we would expect to see that competitiveness, along with the number of EC votes at stake in a state, would be key determinants of campaigning. This conclusion differs from that of early political science literature on campaign strategies which claimed that the most populous states would receive the bulk of campaign activities. For example, Brams and Davis (1974) offered a model that predicted campaign allocations proportional to the electoral votes of each state raised to the power of 3/2. For an early critique of the view that campaigning would necessarily focus on the most populous states, see Colantoni et al. (1975). See also Wright (2009) and Miller (2012). Similarly, Shaw and Althaus (2017) posit that ''campaign resources will be disproportionately, but not exclusively, concentrated in battleground states''. In on-line Appendix C, we provide an additional robustness check on our use of an ex-post measure of political competitiveness by relying on Shaw and Althaus (2017) classifications of battleground/target states. We find that their ex ante measure and our ex post competitiveness measure are highly correlated when we include battleground targets from either campaign or from only those in which the campaigns agree about the battleground status of the state. In 2012, B-K note that 99.6% of advertising money was spent in the ten states identified as battlegrounds by FairVote.org. Of those ten states, eight are included in the ex post set of competitive states, while the other two were the next closest states in terms of margin of victory. Similarly, in 2012, 87% of campaign events were held in the set of eight states viewed post hoc as competitive. Data aggregated from FairVote.org, with original data from CNN: http://www.fairvote.org/presidential_tracker_2012#2012_campaign_events We can provide confirmation of the congruence between post hoc measures of competitiveness and ex ante expectations of competitiveness for two additional recent elections, those in 2004 and in 2016. Older elections also largely conform to these expectations. Detailed campaign activities for the 1976 election are available because they were submitted into evidence for the hearing before the Subcommittee on the Constitution of the Committee on the Judiciary (S.J. Res. 28, 1979) on a bill that would abolish the Electoral College and establish a direct popular vote. The data were first used by Bartels (1985). That election shows a similar pattern of campaign activities focused on the competitive states, though there were many more (25) competitive states in 1976 than in the two most recent elections of 2012 and 2016. In 1976, 78% of all campaign events were held in the 25 battleground states, and 78% of all campaign television and radio ads were broadcast there. In the 2016 election, the campaigns and campaign-related Political Action Committees (PACs) spent 82% of advertising money in the states retrospectively classified as competitive. Data compiled from AdAge.com, based on state-specific ad buys between October 21, 2016, and Election Day. http://adage.com/article/campaign-trail/states-where-trump-clinton-spending-most-on-advertising/306377/] Moreover, the only competitive state not targeted by either campaign was Minnesota, a state in which Democratic candidates have the longest winning streak. Similarly, if we look at candidate rallies or events at which the presidential or vice-presidential candidate appeared in 2016, the major party candidates held 79% of all events in the 13 states that we label competitive post hoc. Some studies have claimed that the number of battleground states has narrowed (Gimpel et al. 2007), but what is arguably the most comprehensive study to date, looking from 1952 onward, finds little change in the number of battleground states over time (Shaw and Althaus 2017). We can contribute to this debate by examining the change in the number of competitive states over a much longer time horizon. We show in Fig. 2 the percentage of competitive states as we have measured that concept, with a running average also shown by plotting a locally weighted polynomial regression. What we see is that the post-1952 data are compatible with the Shaw and Althaus's (2017) assertion of little change in the number of battleground states in recent presidential elections, though some evidence exists of fewer competitive electors. However, when we look at the longer time series, what we observe is that we now have relatively few competitive states than in the 1868-1900 period, and the percentage of competitive states is more stable (smaller standard deviation) than it was before 1988. Fig. 2 Percentages of competitive states over time: 1868-2016 Note: Smoothed lines are locally-weighted polynomial regressions with smoothness set at f = 0.5. These lines are intended to show over time patterns among noisy data. Shaw and Althaus (2017) also expect the ability of campaigns to more optimally allocate their resources should increase over time with more sophisticated survey and targeting tools. We relatedly expect that sharper polarization allows for more accurate predictions of which states are likely to be competitive and which are not. We can examine this question by comparing the Shaw and Althaus measure of what states were viewed as battleground states as judged by the behavior of each campaign and our post hoc measure of competitiveness. We show the average level of competitiveness in their battleground states in Table 3. What we see from Table 3 is that, since 1988, the states Shaw and Althaus (2017) find to be battleground states as judged by campaigning, also are consistently highly competitive. However, this consistency does not hold in the election cycles from 1952 to 1984, although low ex post competitiveness in battleground states is found in three of these presidential election years. Thus, at least for the recent period, the only period for which we have relevant campaign data, using post hoc measures of competitiveness as a proxy for campaign strategies is reasonable. In 1964, the Goldwater campaign treated 23 states as battlegrounds (Shaw and Althaus 2017). The Goldwater campaign focused on the South, seeking to mirror the Dixiecrat revolt and pry southern states from the hands of the Democratic party which, except for the Dixiecrat revolt of 1948, had been winning them by large margins. Goldwater's campaign went poorly except in the deep South, winning only a handful of states. All but one of the states he won were states his campaign treated as battlegrounds. The one exception was a very narrow win. ## 2 Using partisan imbalance in noncompetitive states to predict Electoral College outcomes We, like Brams and Kilgour (2017), believe that outcomes in noncompetitive states are critical in understanding final Electoral College winners. In this section, we capitalize on that insight by offering a simple measure that we show jointly performs as well or better than the Brams-Kilgour variables in predicting final EC outcomes. To present our measure, some notation is useful. We may again partition the states into the set of competitive states,$C_j$, and the set of noncompetitive states,$NC_i$, where$i$indicates the election year. The EC votes in a competitive state are labeled as$s(C_j)$and the EC votes in a noncompetitive state are labeled as$s(NC_j)$. We have$s(EC) = s(Cj) + s(NC_j)$. Noncompetitive states won by Democrats are labeled$NC_D$, and the noncompetitive states won by Democrats are labeled$NC_R$. The seats in the noncompetitive states won by the Democrats are labeled$s(NC_D)$and the seats in the noncompetitive states won by Republicans are labeled$s(NC_R)$. We will be interested, on the one hand, in the partisan balance of seats in the noncompetitive states and, on the other hand, in the share of the states that fall into the noncompetitive category. We define our variable of interest as the difference between the two-candidate's noncompetitive electoral totals, divided by the total number of EC votes:$Non-Competitive Advantage=[s(NCD)- s(NCR)]/sEC)$This measure is standardized, thus allowing us to compare its effects across elections. When one party has a big advantage in noncompetitive electoral votes, it will be more likely to win the election. Brams and Kilgour reflect this intuition by examining coalitions among competitive states, and determining outcomes under the explicit assumptions that the competitive state outcomes occur independently of one another and with an equal probability of victory for the two parties in each. We regard both of these assumptions as quite reasonable ones to make for purposes of model tractability, but we might expect them to be falsified if electoral tides sweep in a particular direction and thus create interdependencies in vote outcomes in the competitive states. We do not require either of these strong assumptions. But exactly the same intuition drives our model as that in the work of Brams and Kilgour, namely that the candidate who has a larger advantage in electors from the noncompetitive states will have more options in terms of possible wins in competitive states leading to Electoral College victory. Table 4 shows ex post values for the Democratic and Republican EC vote shares in the noncompetitive states in the first two columns, and it also shows the final EC vote outcome both as a number and as a percentage of the electoral vote total. In addition, we provide a column that reports the difference between the Democratic and Republican EC votes in the noncompetitive states, and a further column showing that difference normalized by total EC votes, i.e., a column that shows Non-Competitive Advantage. Minor party candidacies are likely to be a problem for our analyses only in situations when they receive Electoral College votes. This has not been the case in recent elections, as no minor party candidate has won a state since George Wallace in 1968. In their assessment of minor party impact, Pattie and Johnson (2014) do not find substantial effects, and they also note that such effects have often differed in their partisan impacts. To provide a consistent coding across all elections in our dataset we ignore minor party votes and treat contests as between the two major party candidates in terms of two-party vote share. We first test the predictive usefulness of our Non-Competitive Advantage variable by looking to see how often the party with the advantage in the noncompetitive states wins the EC vote. As does the Winningness measure, in all four of the elections from 2000 through 2012, Non-Competitive Advantage correctly predicts the presidential election outcome. Indeed, we find that in all but two of the 38 elections (1880 and 1960), the party with a Non-Competitive Advantage goes on to win the election, the same strong predictive accuracy as the Winningness measures. Interestingly, the two errors are the same two elections that Winningness fails to predict. The failure of the models to correctly classify states is tied directly to two empirical realities of elections: closely competitive elections (and reversals, where one candidate wins the popular vote and the other wins the Electoral College) are, by definition, more difficult to predict, and candidates who outperform their rivals in battlegrounds can overcome noncompetitive disadvantages. The 1880 election appears to be the former, while 1960 appears to be the latter. Next, we regress Republican EC vote share on the Non-Competitive Advantage variable. Here we find (see Table A2) a very strong and significant relationship between the two measures, and the simple regression between them yields an adjusted$R^2$of$0.96$. We can compare this regression with one that models the same dependent variable with Winningness as the predictive variable. As noted earlier, the adjusted$R^2$of the Winningness model is$0.81$, lower than that for Non-Competitive Advantage at$0.96$. While the very simple Non-Competitive Advantage variable does better in predicting final seat shares than any (or all) of the three variables from Brams and Kilgour (2017), Winningness and Non-Competitive Advantage do equally as well at predicting the directionality of EC outcomes. ## 3 Discussion Brams and Kilgour (2017) begin by suggesting that the road to power through noncompetitive states dictates the terms under which a presidential election is contested. We agree. While competitive states receive the bulk of campaign activities like television and radio advertising, campaign field offices, and visits from the candidates and their surrogates, the media ''horse-race'' coverage about 'swing states' and 'battleground states' takes attention away from the extent to which safe seats matter for electoral outcomes. Partisan balance in noncompetitive states matters since the candidate who enjoys a Non-Competitive Advantage has many additional pathways to the presidency, and thus one candidate can begin the presidential contest severely handicapped. Our results complement a broader literature on the Electoral College (EC), which has both empirical, theoretical and normative components. Normatively, a debate is ongoing between those who see popular vote decisions as the only legitimate way to elect a president, and those who view the Electoral College as a result of a political bargain reflecting federalist efforts to balance popular votes and states as the bases of representation (Hirsch 2008; Edwards 2004; Ross 2012). This debate is tied to proposals about alternative ways of electing the US president. Such proposals tend to surface after each presidential election, especially those (like 2000 and 2016) when a divergence occurs between the popular vote and the EC vote. Theoretically, one can argue about the degree to which the weighted voting rule used in the Electoral College disproportionately empowers citizens of small-population states versus those of more populous states. That controversy is couched most commonly in terms of game-theoretic indices of power, such as the Banzhaf Index or the Shapley-Shubik value (see, e.g., Banzhaf 1968; Owen 1975; Duffy and Matros 2015). Empirical debates have arisen about such issues as the degree of partisan bias imposed by EC apportionment (Grofman et al. 1997; Johnston et al. 2004; Pattie and Johnston 2014; cf. Ladewig and Jasinski 2008), and the nature of optimal campaigning under the Electoral College system (see, especially, Shaw 2006; Strömberg 2008). We have extended Brams and Kilgour's (2017) analyses of Winningness, Vulnerability and Fragility beyond the four recent elections they analyze, to include not just 2016, but all elections between 1868 and 2016. Thus, we have added 34 elections to the investigation. We also added a new and simpler variable based on the logic of the B-K argument, namely, Non-Competitive Advantage, defined as the difference in safe EC votes between the parties, normalized by total EC votes. We find that the candidate holding the edge in Winningness and Non-Competitive Advantage have gone on to win in all but two of the 38 US presidential elections since 1868. In the two mispredicted elections, the partisan advantage in noncompetitive electoral votes was very slim. When we move from attempting to predict a dichotomous outcome variable to seeking to predict final EC vote shares, we found that both Winningness and our new Non-Competitive Advantage variable are strongly predictive of EC vote shares, but now the predictive edge is with our simpler variable ($R^2$of$0.96$versus one of$0.81\$).

In toto, we take these results to be very supportive of recent Public Choice and economics scholarship on optimal campaigning. In particular, campaigns have clear incentives to concentrate resources in the most competitive states rather than focus simply on the most populous ones, and recent campaigns (since the 1980s) show a closer correspondence between post-election closeness of EC votes and the expenditure of campaign resources. However, we have shown that we have relatively few competitive states in more recent election cycles than in those before the twentieth century. More specifically, our results support with a much more extensive dataset the key intuition in Brams and Kilgour (2017) that noncompetitive states play a foundational role in shaping the election of the US president. As with Brams and Kilgour's Winningness, our measure shows that the more potential paths to victory a presidential candidate has, the larger is the candidate's expected EC vote share. Moreover, the candidate who has the edge in the noncompetitive EC votes is almost always elected to the White House.


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# Power set

Which of the following cannot be the size of a power set?

This is a part of the set: Set problems

Try this: All of my problems

×


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# zbMATH — the first resource for mathematics

On prime values of some polynomials. (English) Zbl 1082.11018
The author applies the deformation method of quadratic matrix equations to discuss the solubility and the number of solutions of some inhomogeneous quadratic Diophantine equations involving prime numbers. For example, let $$q \equiv 1$$ mod 8 be a prime number such that $$q^s + 2$$ is also a prime number for some odd integer $$s$$. Then the equation $$2x^2 + 2y^2 - 2q^{\frac{s+1}{2}}x - q = 0$$ has exactly 8 integer solutions.
##### MSC:
 11D09 Quadratic and bilinear Diophantine equations 11D45 Counting solutions of Diophantine equations


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Tea is an aromatic beverage commonly prepared by pouring hot or boiling water over cured leaves of the tea plant, Camellia sinensis.[3] After water, tea is the most widely consumed beverage in the world.[4] It has a cooling, slightly bitter, and astringent flavour that many people enjoy.[5]

Tea likely originated in China as a medicinal drink.[6] It was first introduced to Portuguese priests and merchants in China during the 16th century.[7] Drinking tea became popular in Britain during the 17th century. The British introduced it to India, in order to compete with the Chinese monopoly on the product.[8]

Tea has long been promoted for having a variety of positive health benefits. Recent studies suggest that green tea may help reduce the risk of cardiovascular disease and some forms of cancer, promote oral health, reduce blood pressure, help with weight control, improve antibacterial and antivirasic activity, provide protection from solar ultraviolet light,[9] and increase bone mineral density. Green tea is also said to have "anti-fibrotic properties, and neuroprotective power."[10]Additional research is needed to "fully understand its contributions to human health, and advise its regular consumption in Western diets."[10]

Tea catechins have known anti-inflammatory and neuroprotective properties, help regulate food intake, and have an affinity for cannabinoid receptors, which may suppress pain and nausea and provide calming effects.[11]

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