text:
[This article was first published on R on The Jumping Rivers Blog, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
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Regular expressions. How they can be cruel! Well we’re here to make them a tad easier. To do so we’re going to make use of the stringr package

install.packages("stringr")
library("stringr")

We’re going to use the str_detect() and str_subset() functions. In particular the latter. These have the syntax

function_name(STRING, REGEX_PATTERN)

str_detect() is used to detect whether a string contains a certain pattern. At the most basic use of these functions, we can match strings of text. For instance

jr = c("Theo is first", "Esther is second", "Colin - third")
str_detect(jr, "Theo")
## [1]  TRUE FALSE FALSE
str_detect(jr, "is")
## [1]  TRUE  TRUE FALSE

So str_detect() will return TRUE when the element contains the pattern we searched for. If we want to return the actual strings that contain these patterns, we use str_subset()

str_subset(jr, "Theo")
## [1] "Theo is first"
str_subset(jr, "is")
## [1] "Theo is first"    "Esther is second"

To practise our regex, we’ll need some text to practise on. Here we have a vector of filenames called files

files = c(
"tmp-project.csv", "project.csv",
"project2-csv-specs.csv", "project2.csv2.specs.xlsx",
"project_cars.ods", "project-houses.csv",
"Project_Trees.csv","project-cars.R",
"project-houses.r", "project-final.xls",
"Project-final2.xlsx"
)

I’m also going to give us a task. The task is to be able to grab the files that have a format “project-objects” or “project_objects”. Let’s say of those files we want the csv and ods files. i.e. we want to grab the files “project_cars.ods”, “project-houses.csv” and “project_Trees.csv”. As we introduce more regex we’ll gradually tackle our task.

## Regex: The backslash, \

Here we go! Our first regular expression. When typing regular expressions, there are a group of special characters called metacharacters that have other functions. These are:

.{()\^$|?*+ The backslash is SUPER important because if we want to search for any of these characters without using their built in function we must escape the character with a backslash. For example, if we wanted to extract the names of the name of all csv files then perhaps we would think to search for the string “.csv”? Then we would do str_subset(files, "\.csv") # xXx error = TRUE isn't working Hang on a second, what? Ah yes. The backslash is a metacharacter too! So to create a backslash for the function to search with, we need to escape the backslash! str_subset(files, "\\.csv") ## [1] "tmp-project.csv" "project.csv" ## [3] "project2-csv-specs.csv" "project2.csv2.specs.xlsx" ## [5] "project-houses.csv" "Project_Trees.csv" Much better. With regards to our task, this is already useful, as we want csv and ods files. However, you’ll notice when we searched for files contained the string “.csv”, we got files of type “.xlsx” as well, just because they had “.csv” somewhere in their name or extension. Up step the hat and dollar… ## Regex: The hat ,^, and dollar, $

The hat and dollar are used to specify the start and end of a line respectively. For instance, all file names that start with “Proj” (take note of the capital “P”!)

str_subset(files, "^Proj")
## [1] "Project_Trees.csv"   "Project-final2.xlsx"

So what if we wanted specifically just “.csv” or “.ods” files, just like in our task? We could use the dollar to search for files ending in a specific extension

str_subset(files, "\\.csv$") ## [1] "tmp-project.csv" "project.csv" ## [3] "project2-csv-specs.csv" "project-houses.csv" ## [5] "Project_Trees.csv" str_subset(files, "\\.ods$")
## [1] "project_cars.ods"

Now we can search for files that end in certain patterns. That’s all well and good, but we still can’t search for both together. Up step round parentheses and the pipe…

## Regex: Round parentheses,(), and the pipe, |

Round parentheses and the pipe are best used in conjuction with either other. The parentheses specify a group and the pipe means “or”. Now, we could search for files ending in a certain extension or another extension. For our task we need “.csv” and “.ods” files. Using the pipe

str_subset(files, "\\.csv$|\\.ods$")
## [1] "tmp-project.csv"        "project.csv"
## [3] "project2-csv-specs.csv" "project_cars.ods"
## [5] "project-houses.csv"     "Project_Trees.csv"

Alternatively we can use a group and pipe

str_subset(files, "\\.(csv|ods)$") ## [1] "tmp-project.csv" "project.csv" ## [3] "project2-csv-specs.csv" "project_cars.ods" ## [5] "project-houses.csv" "Project_Trees.csv" Now we don’t have to write surrounding expressions more than once. Of course there are other csv and ods files that we don’t want to collect. Now we need a way of specifiying a block of letters. Up step the square parentheses and the asterisk… ## Regex: Square parentheses,[], and the asterisk, * The square parentheses and asterisk. We can match a group of characters or digits using the square parentheses. Here I’m going to use a new function, str_extract(). This does as it says on the tin, it extracts the parts of the text that match our pattern. For instance the last lower case letter in each element of the vector, if such a thing exists str_extract(files, "[a-z]$")
##  [1] "v" "v" "v" "x" "s" "v" "v" NA  "r" "s" "x"

Notice that one of the files ends with an upper case letter, so we get an NA. To include this we add “A-Z” (to add numbers we add 0-9 and to add metacharacters we write them without escaping them)

str_extract(files, "[a-zA-Z]$") ## [1] "v" "v" "v" "x" "s" "v" "v" "R" "r" "s" "x" Now, this is obviously useless at the moment. This is where does the asterisk comes into it. The asterisk is what is called a quantifier. There are three other quantifiers (+, ? and {}), but won’t cover them here. A quantifier quantifies how many of the characters we want to match and the asterisk means we want 0 or more characters of the same form. For instance, we could now extract all of the file extensions if we wished to str_extract(files, "[a-zA-Z]*$")
##  [1] "csv"  "csv"  "csv"  "xlsx" "ods"  "csv"  "csv"  "R"    "r"    "xls"
## [11] "xlsx"

So we go backwards from the end of the line collecting all the characters until we hit a character that isn’t a lower or upper case letter. We can now use this to grab the group letters preceeding the file extensions for our task

str_subset(files, "[a-zA-Z]*\\.(csv|ods)$") ## [1] "tmp-project.csv" "project.csv" ## [3] "project2-csv-specs.csv" "project_cars.ods" ## [5] "project-houses.csv" "Project_Trees.csv" Obviously we still have some pesky files in there that we don’t want. Up step the… only joking! We now actually have all the tools to complete the task. The filenames we want take the form project-objects or project_objects, so we know that preceeding that block of letters for “objects” we want either a dash or an underscore. We can use a group and pipe for this str_subset(files, "(\\_|\\-)[a-zA-Z]*\\.(csv|ods)$")
## [1] "tmp-project.csv"        "project2-csv-specs.csv"
## [3] "project_cars.ods"       "project-houses.csv"
## [5] "Project_Trees.csv"

We still have two pesky files sneaking in there. How do those two files and the three files we want differ? Well the files we want all start with “project-” or “project_” where as the other two don’t. We must also take note that the project could have a capital “P”. We can combat that using a group!

str_subset(files, "(P|p)roject(\\_|\\-)[a-zA-Z]*\\.(csv|ods)$") ## [1] "project_cars.ods" "project-houses.csv" "Project_Trees.csv" If we had a huge file list, we’d want to stop files such as “2Project_Trees.csv” filtering in as well. So we can just use the hat to specify the start of a line str_subset(files, "^(P|p)roject(\\_|\\-)[a-zA-Z]*\\.(csv|ods)$")
## [1] "project_cars.ods"   "project-houses.csv" "Project_Trees.csv"

Regular expressions are definitely a trade worth learning. They play a big role in modern data analytics. For a good table of metacharacters, quantifiers and useful regular expressions, see this microsoft page. Remember, in R you have to double escape metacharacters!

That’s all for now. Cheers for reading!


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# Journal of Operator Theory

Volume 77, Issue 2,  Spring  2017  pp. 391-420.

On polynomial $n$-tuples of commuting isometries

Authors:  Edward J. Timko
Author institution: Department of Mathematics, Indiana University, Bloomington, IN, 47401, U.S.A.

Summary:  We extend some of the results of Agler, Knese, and McCarthy in \textit{J. Operator Theory} \textbf{67}(2012), 215--236, to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let $\mathrm{Ann}(\VV)$ denote the set of all $n$-variable polynomials $p$ such that $p(\mathbb{V})=0$. When $\mathrm{Ann}(\mathbb{V})$ defines an affine algebraic variety of dimension 1 and $\mathbb{V}$ is completely non-unitary, we show that $\mathbb{V}$ decomposes as a direct sum of $n$-tuples $\mathbb{W}=(W_1,\dots,W_n)$ with the property that, for each $i=1,\dots,n$, $W_i$ is either a shift or a scalar multiple of the identity. If $\mathbb{V}$ is a cyclic $n$-tuple of commuting shifts, then we show that $\mathbb{V}$ is determined by $\mathrm{Ann}(\mathbb{V})$ up to near unitary equivalence, as defined in \textit{J. Operator Theory} \textbf{67}(2012), $\mbox{215--236}$.

DOI: http://dx.doi.org/10.7900/jot.2016apr24.2122
Keywords:  polynomial, commuting, isometries

format: plain


text:
# Bibtex export

13»
• Given that the last post was quite some time ago, I don't know how useful this will be, but I wanted to share my solution (due in large part to the collective posts here!). My solution is to use the "short title" field for my own citation keys. If that field is empty, then zotero will use it's own default. Note that I don't know the first thing about Java, so I just tinkered until I got functionality.

MODIFY THE CITATION KEY FOR EXPORTED BIB FILES SO THAT IT USES THE CONTENT OF THE SHORT TITLE FIELD. IF THE SHORT TITLE FIELD IS BLANK, ZOTERO WILL USE IT'S DEFAULT.
1) Open your BibTeX.js translator file (in ....firefox/profiles/####.default/zotero/translators) in a text editor and scroll to (near the end) where you will see, "function doExport() {." A few lines below the function definition, you will see the line, "var citekey = buildCiteKey(item, citekeys);". Replace that line with the following:
if (item.shortTitle==''){
var citekey = buildCiteKey(item, citekeys);
} else {
var citekey = item.shortTitle
}

ADD THE BIBTEXCITEKEY ONLY AND MAKE SAME MODIFICATION
I found the BibTexCiteKeyOnly.js file on another discussion (http://www.zotero.org/message/viewdiscussion/23291). It's a great little hack that allows you to literally drag your reference from zotero into your favorite latex editor! Once you get the BitTexCiteKeyOnly.js file, open it with a text editor and again scroll all the way to the bottom of the file. Replace the line "var citekey = buildCiteKey(item, citekeys);" with the code snippet above and presto.
• I tried Scaffold, but the Bibtex translator does not show up in the list for some reason.

I managed to customise the output key and a few other changes, by opening and editing the BibTeX.js file manually. You can modify the key by changing the citeKeyFormat variable. You are restricted to %a %y and %t format specifiers, but you can extend these by adding new functions to citeKeyConversions. It should not be all that hard if you are familiar with js.

HTH

- Sid
• I'm finding that the standard BibTeX export is doing the wrong thing with HTML entities in notes: it outputs something like this:

{{\textless}p{\textgreater}Nice} pictures showing how well the {JUMP} math programme works.{\textless}/p{\textgreater}
{{\textless}p{\textgreater}See} {\textless}a href="http://jumpmath.org/"{\textgreater}http://jumpmath.org/{\textless}/a{\textgreater} for (maybe?) some details.\

That is really not the desired effect.

I tried the "BibTex generic citation style (dev)" and it works more consistently, but doesn't export the Notes annotation fields at all.
• edited April 26, 2011
It is not suggested that you use the BibTeX CSL style. The right thing to do would be to intelligently transform the html markup into LaTeX markup. This is not trivial. So, perhaps we should strip html tags for now, e.g.:myString.replace(/<[^>]*>?/g, '');It'd be easy enough to put this in the BibTeX translator, but most of the other export formats probably don't expect html markup either. Not sure whether we should strip RIS and other formats or not.
• Stripping HTML tags would be a big improvement. Transforming HTML markup to LaTeX markup could be done with a pandoc webservice, e.g. see http://johnmacfarlane.net/pandoc/try
• pandoc isn't a bad idea in the long run, but there are issues with setting up a host (if zotero was interested in doing that), degrading gracefully when there was no network connection (it is easy to just strip tags, but end-users may be surprised), and handling data privacy appropriately.

If you need help implementing stripping, let me know. I'll probably think a bit about whether this should be part of other translators before submitting a patch & it may be sometime before that such a patch would be pushed out....
• edited April 27, 2011
There a few related problems,

(1) the rich text editor saves invisible HTML formatting from copy and paste.

(2) Exporting as RIS produces valid HTML, which is nice, but some of the links are strangely formatted:

social learning theory

Not sure where best to report issues like this.
• @CPLuser: Thanks a million!!!!

This was a serious problem for me! I was retrofitting Zotero on a big project where I have already 250 citations, and I have spent almost a day searching for the old BibTeX keys manually defined and replacing them with Zotero's auto generated keys (as I found no way to change them). However, this beautiful hack solves the whole issue and I have a one-to-one transparency with the information provided in Zotero ("Short Title") and the BibTeX tag.
• Hi!
I want to modify the BibTex.js to get in the final file.bib this "\\~{a}" instead of this "ã".
How can I do this?

Thanks,
• The question is: do you really need this? The point is that the BibTeX versions I have used all accept Unicode characters in in the text fields (but not in the tags, so that is why it was good to be able to define the tags in the "Short Title" field).

Unicode characters are not automatically accepted by TeX though, so you will need to include the following two packets at the beginning of your TeX file:

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

Does this solve your problem? Including the two packages does not harm if you do not use special characters like ã. However, if they appear in your references this solves the typesetting issue when the *.bbl file is imported to produce the actual bibliography - and saves you the time to figure out where in the translator this change should be made (you would probably need a big translator table to handle all different accents in all languages).
• What Geir said, and then, if you insist, enable the export encoding option in the Export pane of the Zotero preferences and choose something other than Unicode.
• Geir Horn
I included the two packages to my .tex, but an error appears: "! Latex Error: Option clash for package inputenc"

Only in the .bib I cannot have accented characters. In the plain text (.tex) I can put ã, õ, ..., normally.
I'm using this packages:
\usepackage[brazil]{babel}
\usepackage[latin1]{inputenc}
\usepackage{lmodern}
But in the .bib I need put \~{o} instead of õ.

Dan Stillman
Thanks a lot. Worked very well.

Best regards,
• OK, it seems that you have found the solution. Otherwise, I would recommend that you try to comment out the two first packages (% in front) and load the only the two I suggested.

At least the second one since that is the "inputenc" package I also suggested you to include and it will clash with the second include.

Good luck!
• Hello,

I know this thread has been inactive for some time, but it seemed to be the appropriate place to ask my questions anyhow.
Please correct me if I'm wrong.

I have a number of bibtex export issues, a few of which I have fixes for which I have maintained by myself up till now.

for hyphenation, bibtex uses babel and for this to work, the language field in zotero must be defined properly. This means lower case. I enforced this by changing
for(var field in fieldMap) {
if(item[fieldMap[field]]) {
writeField(field, item[fieldMap[field]]);
}
}
at line 1180 into:
for(var field in fieldMap) {
if(item[fieldMap[field]]) {
// Force lower case for language field
if(field == "language"){
writeField(field, item[fieldMap[field]].toLowerCase());
}
else {
writeField(field, item[fieldMap[field]]);
}
}
}

In a page range a single or multiple '-' are replaced by '--'
I changed this to also remove space between the numbers and the dashes
if(item.pages) {
writeField("pages", item.pages.replace(/[-\u2012-\u2015\u2053]+/g,"--"));
}
at line 1305 becomes
if(item.pages) {
writeField("pages", item.pages.replace(/\s*[-\u2012-\u2015\u2053]+\s*/g,"--"));
}

I added an alternate 'µ' in the mapping table
"\u03BC":"{\\textmu}", // MICRO SIGN

I added a character which happened to be in the name of one of my colleagues
"\u0111":"{\\dj}", // LATIN SMALL LETTER d WITH STROKE

I fixed the comment at line 2380
"{\\O}" : "\u00D8", // LATIN SMALL LETTER O WITH STROKE
to become
"{\\O}" : "\u00D8", // LATIN CAPITAL LETTER O WITH STROKE

I added an option for a master's thesis type. Bibtex supports this, zotero doesn't. Use the 'type' field for definition
At line 1163
// determine type
var type = zotero2bibtexTypeMap[item.itemType];
// Add mastersthesis variant. Default is phdthesis
if (item.itemType == "thesis"){
if ((item.type == "M.Sc.") || (item.type == "M.Sc. thesis")) {
type = "mastersthesis";
}
}

A remaining unsolved problem is that \LaTeX, \TeX or \METAFONT cannot be placed in a title. I had fixed this, but since recently, capitalized words are enclosed in {} and now I get \{LaTeX} (with my mod in place) which does not give the desired result....
The current state of the stock translator returns {\textbackslash}{LaTeX}} which is even worse

An other problem is a paper:
"S-D Modulators--Stability Analysis and Optimization" where S and D are actually the capital greek letters Sigma and Delta.
In the bib file the title is abbreviated to '-', where 'modulators' would be more appropriate.
In general, titles like "A-D conversion....." give no meaningful results. It seems to me no word less than 3 characters (no matter which alphabet, so before parsing a capital greek Delta to \Delta) should be allowed as title-key, or alternatively use ALL the words until a minimum of x characters is found and then finish the current word.

I hope my suggestions are useful and maybe someone can help with the two remaining issues

Paul
• I'll look at these, but may take a little; some are obvious, others I'm less sure about -- e.g. I don't think mastersthesis is standard bibtex. We do have a biblatex translator that already does this.
Same applies for the language field and babel: IIRC, biblatex tries hard to get that right.

I'm not sure we'll end up fixing the \LaTeX issue. It's a pain (and personally, I think it's stupid). But I suppose you could just run a .replace after everything else is done for whatever it ends up being then.

Citekeys -- wouldn't be hard to do, would likely have been the right thing to do from the start, but I'm not too keen on changing the citekey algorithm, since people are using those in documents that'd break.
• Hello,

Glad to see such a fast response.
I just found an other issue:
Σ-Δ Modulators--Stability Analysis and Optimization
is translated to
{$\Sigma$-$\Delta$ {Modulators}--{Stability} {Analysis} and {Optimization}}
which is correct, but the citation style converts the uppercase greek characters to lower case.
Enclosing either {$\Sigma$} or ${\Sigma}$ fixes this. It would be nice to have that done automatically.

Best regards,

Paul
• edited April 24, 2015
That last one may be more difficult to do than it may see, but I'll give it a shot.

On a slightly unrelated note, I was looking around at best practices for protecting capitals in BibTeX and came across http://stackoverflow.com/a/2897666 I don't have strong opinion on BibTeX things, but if protecting only capital letters is a better choice (rather than whole words) we can adjust that fairly easily.
• To react on your remark about masterthesis, all the sources I find mention both mastersthesis and phdthesis
As far as I understand Oren Patashnik is the (original) developer and he wrote in 1988
http://ftp.snt.utwente.nl/pub/software/tex/biblio/bibtex/base/btxdoc.pdf (page 9)
so I think we can assume it is a standard entry type

Regarding the capitalization issue, maybe what I'm talking about is more specific: preserve the case of non-latin characters. There is a high probability that these are not parts of words, so capitalization rules for words may not be applicable.
On the other hand, in its current state the bibtex export DOES preserve case in latin-alphabet words, but NOT in greek (single characters? words?) Apparently uppercase/lowercase is only correctly detected for latin characters.

Best regards,

Paul
• Hi,

First of all I reject my suggested addition of \u03BC (micro sign) since I found it was already defined in the list with greek letters as \mu.

I was thinking about the implementation of preserving the upper case of greek letters.
Maybe it's an idea to simply replace the mapping table entry
"\u0394":"$\\Delta$", // GREEK Delta
by
"\u0394":"{$\\Delta$}", // GREEK Delta

Paul


text:
## Silicon nanowire fabric as high-energy anode material for Li-ion batteries

##### 02 December 2011

Researchers at the University of Texas at Austin have developed a non-woven fabric of silicon nanowires; the material has the look and feel of tissue paper, yet is composed entirely of crystalline silicon (Si).

Thermal annealing of the nanowire fabric in a reducing environment results in good performance of the resulting material as an anode in a Li-ion battery without the addition of conductive carbon or binder. In a paper published in the Journal of the American Chemical Society, the team reports achieving anode capacities of more than 800 mAh g–1.

The work represents another approach to devising a commercially viable solution for practically applying silicon, with its much higher theoretical storage capacity than the graphitic anode materials commonly used, in Li-ion batteries (LIB).

The LIB capacity is limited in part by the intercalation of Li+ by the graphitic anode material; thus, higher capacity batteries require anode materials that can accommodate more Li+. The theoretical capacity of the graphite anode is 372 mAh g-1. Lithium-alloying materials have the potential for significantly higher storage capacities. For instance, Si alloys with Li+ at room temperature to form Li15Si4, which corresponds to a substantially higher theoretical capacity of 3579 mAh g-1. However, Si undergoes an enormous volume expansion of nearly 300% when fully lithiated.

Bulk crystalline Si cannot tolerate the stresses associated with these lithiation/delithiation cycles and crumbles, resulting in battery failure.

On the other hand, Si nanostructures have been found to tolerate extreme changes in volume with cycling. Thin Si films have realized capacities above 2000 mAh g-1. Si nanoparticles have also been explored, but with overall specific capacities that have been limited by the need for a conductive carbon matrix to ensure electrical contact with the electrode. Si nanowires have been promising. For example, Cui and co-workers achieved a capacity of 2725 mAh g-1 with very good stability, retaining more than 1400 mAh g-1 after 700 cycles using an interconnected amorphous Si hollow nanosphere thin-ﬁlm electrode. Unfortunately, these electrode materials are very expensive at present and cannot be produced in the significant quantities needed for commercial LIB applications.

Solvent-based processes for nanowire synthesis, such as supercritical-fluid-liquid-solid (SFLS) and solution-liquid-solid (SLS) growth, can produce large amounts of Si nanowires (SiNWs) at relatively low cost...Here we demonstrate the creation of a Si nanowire fabric and show that it can function as a standalone anode material without the need for additional conductive fillers (activated carbon) or polymeric binders.

—Chockla et al.

The nanowires, made by the supercritical-fluid–liquid–solid process, are crystalline, range in diameter from 10 to 50 nm with an average length of >100 μm, and are coated with a thin chemisorbed polyphenylsilane shell. About 90% of the nanowire fabric volume is void space.

The findings show that the surface composition of Si nanowires plays a critical role in their effectiveness as an anode material in an LIB, the researchers conclude.

Future research needs to focus on understanding the cycling durability and the factors that limit performance, including the formation and stability of the solidelectrolyte interface layer, they suggest.

Another issue is that the since the nanowire fabric has 90% void volume, the volumetric capacity of the nanowire fabric is relatively low. The specific capacity of 800 mAh g-1 corresponds to 186 mAh cm-3, which is significantly lower than the volumetric capacity of 777–867 mAh cm-3 for graphite.

However, they suggest, it should be possible to increase the volumetric capacities of the nanowire fabric by densifying the films with pressure. They estimate that the maximum volumetric capacity of a densified Si nanowire fabric would be 1864 mAh cm-3—more than a factor of 2 higher than that of graphite. However, the influence of volumetric expansion and contraction upon lithiation and delithiation on cycling stability would still need to be understood.

Resources

• Aaron M. Chockla, Justin T. Harris, Vahid A. Akhavan, Timothy D. Bogart, Vincent C. Holmberg, Chet Steinhagen, C. Buddie Mullins, Keith J. Stevenson, and Brian A. Korgel (2011) Silicon Nanowire Fabric as a Lithium Ion Battery Electrode Material. Journal of the American Chemical Society doi: 10.1021/ja208232h

Interesting potential avenue for future improved batteries. A 1000 Wh/kg energy density battery could become a reality for extended range EVs by 2020.

Improved mass production techniques will be developed to lower production cost.

Meanwhile, wireless charging units are also developing at a fast rate.

Much lighter, composites/aluminium bodies, frame and wheels etc will soon contribute to higher efficiency, improved design, lower cost, future vehicles. Fold down, very light weight disappearing doors, could be part of it.

Future weight based progressive registration fees could help to lower operation cost of lower weight vehicles. It could be a bonus-malus budget neutral system, with zero or even negative registration fees for very light vehicles compensated by extra fees for heavy vehicles.

".. high-energy anode material for Li-ion batteries.."

Sites have published such statements for years and years, yet it seems that there's no 'better battery' without a complimentary better cathode.

Perhaps GCC could require electrode pair information and probable power output implications/specs achieved with the latest new high-energy battery articles.

good to read but will it yield superlative batteries in 2020, as wished by some who keep repeating the same thing? I don't think so, that kind of lab finding will takes decades to make to product if ever.

Or. will private industries sit on the fence until public funds come their way? What happened to their good old free enterprise initiative?

Very perceptive Harvey.

Free enterprise has let us down. It’s that simple.

Fortunately Cuba, Iran and North Korea are providing the synergistic advances that we need.

China was coming along well but are now sliding into capitalism - with it's cut throat culture.

I understand China now lets some unsuccessful industries fail and actually let successful ones keep some of their gains instead of confiscating them for the common good..

HarveyD

You should look at the history of the chemistry we use to day in batteries and when they were discovered. Li-ion was discovered in 80s, NaS in the 70s. Pb/PbO in 10s

These somewhat erratic advances may or may not be synergistic.

Our free enterprise system naturally leaves the useless advances to wither (or reemerge as science advances in unexpected directions) and constantly and automatically selects the successful strategy.

Unless, of course YOU know what is best and “the others” must be ignored; but that only works (but not for the best) if you are Emperor or King.

I remain confident that the BEV (or some unexpected alternate) will emerge - but no one knows and if we are disappointed it does not mean someone was delinquent or evil.

Build an affordable battery and the world will pave the road to your factory and make you rich. No one can stop you - in the free world.

That is evolution; that is survival of the fittest and that is free enterprise – but some things will never be.

Because cost is, and MUST remain part of the equation.

The arrogance of picking winners should be done with your OWN money, not other's.

Gosh, Tom, you sounded somewhat reasonable until that last sentence.

Of course the idea that anyone knows what is going to work at the outset means that both free market and "Emperor" will sometimes fail.

What has made this country a world leader is that we have pooled our money and taken chances on long shots, on technologies that took decades to bring to market.

A billion dollars invested in a new battery approach costs each of us $3. Just imagine what our world would be like had we not put a few$3 dollar bills into computers and modern medicine.

There are may good examples of excellent new products without massive public $$. 1. Apple Tablets and iPhones with many copy cats. 2. CFL and many copy cats 3. LEDs and many copy cats and many future applications. 4. Image sensors & digital cameras (from Kodak ...not followed up) and many excellent copy cats. 5. 3D multi-layers printers, a near future boom, 2011 onward. . 6. Lithium batteries (on-going) from many sources. 7. Post-lithium batteries (on-going) from many sources. 8. HEVs from Toyota from 1997 to 2011 (2,400,000 units) 9. PHEVs from GM and on-going from many others. 10.BEVs, 1999 from GM/Toyota (failed) from Nissan 2011 (first successful BEV with 25,000 first year sales) plus on-going from 30+ others. 11. Wind mills, worldwide applications. 12. Solar cells, increased efficiency, worldwide. 12. etc etc For the near future, should government invest more, indirectly or directly (hand outs) or as JVs with private industries to promote and accelerate mass production of new products? The public should be getting back part or all the funds invested. Low interest loans and/or JVs could do it. Those of you who keep quoting stats about how long/slow battery development was for the last hundred years, are missing a VERY big point: The battery companies got rich from doing it that way. It was the way they were incented. Why sell someone a cure, when you can sell them treatments for life? Everyready, Duracell, etc: They make the VAST MAJORITY of their profit from selling you disposable batteries every couple of months. Why would they sell you a rechargeable battery that could last years when the public is only willing to pay an extra 50% in price for that rechargeable over a throwaway battery? It doesn't make economic sense for them to do anything other than improve it enough to claim they are "better than last year's model". Nobody was putting huge percentages of the Revenues into advancing battery tech because it was a negative on their bottom line. Do you think Gillette or Schlick would sell you a razor blade that lasted years? Hell no. Everyone who's ever read a business text book knows the story of razors vs. blade cells. Why do you think batteries were any different? But now, suddenly there is a market ONLY if the battery can be counted on to last for years and not cost more than the rest of your car! Suddenly you see tens of billions of dollars from both public and private sources for research and new advances come out of the woodwork every month. Grow up and realize it takes 3-5 years to bring them to market and not all of them workout. HOWEVER, if you're betting against all of them....you're not too bright. A few numbers to chew on: "13.6% R&D spend as a percentage of sales for the software and Internet industry... 1% R&D spend as a percentage of sales for the chemicals and energy industry" Now, guess which industry Batteries fall under? LOL Still wanna talk about how impossible it is for batteries to advance rapidly? http://www.industryweek.com/articles/rd_spending_by_the_numbers_17988.aspx Uhhh, razor vs. blade "cells"...OK, I never claimed I was taking the time to proof read my posts. Sorry. LOL DaveD...yes, designed obsolescence is how many private enterprises got to be very rich and how the consumption economy kept going strong. We were very easy to convince that a razor blade should last only one day; that a light bulb should be only 10% efficient and last less than 1000 hours; that a steel boat on four wheels should weight 3 to 4-tons, guzzle lots of gas, be about 15% efficient and fall apart after 2 or 3 years, etc etc Our local ICEV got better ONLY when superior imported units from Asia and Europe entered the market and moved the Big-3 units overboard. Will that acquired acceptance of very low quality products and junk foods last forever or even get worse? DaveD yes and no, there were industries and customer who were looking for better batteries well before the current craze on EV : military but also space, medical, mining and other niche applications. HarveyD are you so sure that the list you give was without public investment. People who think that big technology move can be done without public money are just blinded by their ideological view, most of the start up that develop new ideas got this new ideas from government lab or university lab which research was often funded by public funds. Private investor don't put money in basic research or development of new material because it doesn't pay off. Only the government can fund basic material research, and that's what we need to find the perfect anode and cathode that some are drooling about. Developing new material for anode and cathode or electrolyte is incredibly deceptive, only 1 in a 1000 or even less will make it trough, so how can you make it without public money? not a chance in a snowball Treehugger, Yes, but notice the common theme of those industries you name: military, medical and space. They all put a total premium on cutting edge function and a very distant after-thought on cost for those cutting edge components. Harvey, when I read your list I see public funds writ large. Trace each of those items back and you find yourself at government sponsored labs where the initial heavy lifting occurred. You may find some points at which a critical discovery was made in a private lab, but even those would have been built on earlier basic research. Then let's look at wind turbines, solar panels, BEVs and PHEVs. All of those have or are using public funds to build enough market to create economies of scale. Back in the 1980s the government provided large support to get the first generation of turbines installed. They did that, people figured out what worked and what didn't and the next generation required less support. Not many years back the government was about the only purchaser of solar panels. Then they provided substantial assistance for private purchase. Building a market has brought the price down from >50/watt to <1/watt. IMO, the government should increase its funding of basic research and continue to be the funding source for projects which are promising but not close enough to market for private money to support. And if there is a need to pump some money into the economy in order to get things moving, like we've needed as we recover from the most recent recession, putting some of that money into creating American industry I can get behind that as well. We get that money back because those investments create jobs. People with jobs do not collect unemployment checks and pay taxes. They spend their salaries and create more jobs which also pay taxes. Until the 80s we were supporting wind, then Reagan eliminated all of that and Denmark became the world leader. BW...nothing wrong with public$$$supporting basic research and early mass production of essential products required for advancement of the nation and to create long term jobs, etc. However, the public should get pay backs proportional to the funds contributed. That would help to cover numerous other loses without hope of any pay backs and would reduce current and future national budget deficits. Hand outs and subsidies (of public ), to favored few, are not very legal and should be declared illegal, unless a direct pay back mechanism is built in. All investors, including the public-governments, should share the profits equitably. Where does the "favored few" come from? Companies apply for grants, contracts and loan guarantees. It's an open application system, all are free to submit. They get objective reviews and some get funding. Are you saying that we should go to a lottery system for government contracts? We do have a payback system in place for loans. As the case of any loaning agency you lose some through default. There are no 100% guarantees when you make loans. Contracts state a specific payback for money received. We contract with XYZ, Inc. for a container of paper clips and they ship a container of paperclips and then we pay them. Grants, no payback is built into grants. It's assumed that the country will profit from discoveries made from research. The payback comes in terms of taxes which will flow from the successful findings and from the overall benefits to society. We do have a subsidy problem. That's because we have an election funding problem. Until we get the influence of big money out of our election system even the most honorable candidates will be forced to seek campaign money from those who have "political needs". That's how the incredibly profitable oil industry hangs on to its subsidies. Bob Wallace I agree with you this discussion is getting ridiculous, what better use there is of public money than funding basic research to develop new technologies that are too far off for private investors ? and anyway there is no alternative so what it is the point of arguing. Of course advanced research is high risk investment, very few programs really yield something that the public will use at the end, but it is a misconception to only count the benefit of a research by necessarily a final product that will hit the market. Even if a program doesn't end on a product it generates knowledge, competence, training, skilled people and brains which will be used for other programs of research. Is it so hard to understand that? Tree....do not forget that each major political party needs about$500+M every second year for election purposes. He who contributes the most gets the most. It has very little to do with R & D, jobs etc. Very simple accounting. To get \$500 B they have to hand out 10 to 100 times mores. If that could be cut out, the national deficit could be reduced by half (and more) in a very short time. Shameful, isn't it?

Harvey,

Are you arguing against the basic research that is funded by the government or just objecting to the WAY it is spent based on political contributions?

Stop allowing politicians to run for re-election and the problem goes away.

The comments to this entry are closed.


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# Error: Unresolved reference

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## Problem:

Why do I get unresolved reference errors from the linker when I use the Instrument Inline Functions setting in Visual C ?

## Resolution:

If you turn on Instrument Inline Functions in the TrueCoverage Settings dialog in Visual C or you specify /NMttInlines for NMCL,  a subsequent link step of certain applications might result in unresolved reference errors from the linker. These applications are not linking against certain libraries, because they do not expect to need them.  If you specify Instrument Inline Functions,  the compiler keeps all the inline functions, normally in a header file,  whether or not they are used and they become part of the resulting object module.  If these inline functions in turn call other functions in libraries that are not specified in the link step, then the unresolved references will occur.   If you specify the necessary libraries in the link step, then you will not encounter this problem.

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## On Poincaré gauge theory of gravity, its equations of motion, and Gravity Probe B    [PDF]

Friedrich W. Hehl, Yuri N. Obukhov, Dirk Puetzfeld
Ever since E.Cartan in the 1920s enriched the geometric framework of general relativity (GR) by introducing a {\it torsion} of spacetime, the question arose whether one could find a measurement technique for detecting the presence of a torsion field. Mao et al.(2007) claimed that the rotating quartz balls in the gyroscopes of the Gravity Probe B experiment, falling freely on an orbit around the Earth, should "feel" the torsion. Similarly, March et al.(2011) argue with the precession of the Moon and the Mercury and extend later their considerations to the Lageos satellite.--- A consistent theory of gravity with torsion emerged during the early 1960's as gauge theory of the Poincar\'e group. This Poincar\'e gauge theory of gravity incorporates as simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory (EC), the teleparallel equivalent GR|| of GR, and GR itself. So far, PG and, in particular, the existence of torsion have {\it not} been experimentally confirmed. However, PG is to be considered as the standard theory of gravity with torsion because of its very convincing gauge structure.--- Since the early 1970s up to today, different groups have shown more or less independently that torsion couples only to the {\it elementary particle spin} and under no circumstances to the orbital angular momentum of test particles. This is established knowledge and we reconfirm this conclusion by discussing the energy-momentum law of PG, which has same form for all versions of PG. Therefore, we conclude that, unfortunately, the investigations of Mao et al. and March et al. do not yield any information on torsion.
View original: http://arxiv.org/abs/1304.2769


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# 8.9 Buffer Capacity and Buffer Range

## Buffer Capacity

Buffers are characterized by the pH range over which they can maintain a more or less constant pH and by their buffer capacity, the amount of strong acid or base that can be absorbed before the pH changes significantly. Although the useful pH range of a buffer depends strongly on the chemical properties of the weak acid and weak base used to prepare the buffer (i.e., on $$K$$), its buffer capacity depends solely on the concentrations of the species in the buffered solution. The more concentrated the buffer solution, the greater its buffer capacity. As illustrated in Figure $$\PageIndex{1}$$, when $$NaOH$$ is added to solutions that contain different concentrations of an acetic acid/sodium acetate buffer, the observed change in the pH of the buffer is inversely proportional to the concentration of the buffer. If the buffer capacity is 10 times larger, then the buffer solution can absorb 10 times more strong acid or base before undergoing a significant change in pH.

Figure $$\PageIndex{1}$$: Effect of Buffer Concentration on the Capacity of a Buffer

A buffer maintains a relatively constant pH when acid or base is added to a solution. The addition of even tiny volumes of 0.10 M $$NaOH$$ to 100.0 mL of distilled water results in a very large change in pH. As the concentration of a 50:50 mixture of sodium acetate/acetic acid buffer in the solution is increased from 0.010 M to 1.00 M, the change in the pH produced by the addition of the same volume of $$NaOH$$ solution decreases steadily. For buffer concentrations of at least 0.500 M, the addition of even 25 mL of the $$NaOH$$ solution results in only a relatively small change in pH.

## Selecting proper components for desired pH

Buffers function best when the pKa of the conjugate weak acid used is close to the desired working range of the buffer. This turns out to be the case when the concentrations of the conjugate acid and conjugate base are approximately equal (within about a factor of 10). For example, we know the Ka for hydroflouric acid is 6.6 x 10-4 so its pKa= -log(6.6 x 10-4) = 3.18. So, a hydrofluoric acid buffer would work best in a buffer range of around pH = 3.18.

For the weak base ammonia (NH3), the value of Kb is 1.8x10-5, implying that the Ka for the dissociation of its conjugate acid, NH4+, is Kw/Kb=10-14/1.8x10-5 = 5.6x10-10. Thus, the pKa for NH4+ = 9.25, so buffers using NH4+/NH3 will work best around a pH of 9.25. (It's always the pKa of the conjugate acid that determines the approximate pH for a buffer system, though this is dependent on the pKb of the conjugate base, obviously.)

When the desired pH of a buffer solution is near the pKa of the conjugate acid being used (i.e., when the amounts of conjugate acid and conjugate base in solution are within about a factor of 10 of each other), the Henderson-Hasselbalch equation can be applied as a simple approximation of the solution pH, as we will see in the next section.

Example 1: HF Buffer

In this example we will continue to use the hydrofluoric acid buffer. We will discuss the process for preparing a buffer of HF at a pH of 3.0. We can use the Henderson-Hasselbalch approximation to calculate the necessary ratio of F- and HF.

$pH = pKa + \log\dfrac{[Base]}{[Acid]}$

$3.0 = 3.18 + \log\dfrac{[Base]}{[Acid]}$

$\log\dfrac{[Base]}{[Acid]} = -0.18$

$\dfrac{[Base]}{[Acid]} = 10^{-0.18}$

$\dfrac{[Base]}{[Acid]} = 0.66$

This is simply the ratio of the concentrations of conjugate base and conjugate acid we will need in our solution. However, what if we have 100 ml of 1 M HF and we want to prepare a buffer using NaF? How much Sodium Fluoride would we need to add in order to create a buffer at said pH (3.0)?

We know from our Henderson-Hasselbalch calculation that the ratio of our base/acid should be equal to 0.66. From a table of molar masses, such as a periodic table, we can calculate the molar mass of NaF to be equal to 41.99 g/mol. HF is a weak acid with a Ka = 6.6 x 10-4 and the concentration of HF is given above as 1 M. Using this information, we can calculate the amount of F- we need to add.

The dissociation reaction is:

$HF_{(aq)} + H_2O_{(l)} \rightleftharpoons F^-_{(aq)} + H_3O^+_{(aq)}$

We could use ICE tables to calculate the concentration of F- from HF dissociation, but, since Ka is so small, we can approximate that virtually all of the HF will remain undissociated, so the amount of F- in the solution from HF dissociation will be negligible. Thus, the [HF] is about 1 M and the [F-] is close to 0. This will be especially true once we have added more F-, the addition of which will even further suppress the dissociation of HF.

We want the ratio of Base/Acid to be 0.66, so we will need [Base]/1M = 0.66. Thus, [F-] should be about 0.66 M. For 100 mL of solution, then, we will want to add 0.066 moles (0.1 L x 0.66 M) of F-. Since we are adding NaF as our source of F-, and since NaF completely dissociates in water, we need 0.066 moles of NaF. Thus, 0.066 moles x 41.99 g/mol = 2.767 g.

Note that, since the conjugate acid and the conjugate base are both mixed into the same volume of solution in the buffer, the ratio of "Base/Acid" is the same whether we use a ratio of the "concentration of base over concentration of acid," OR a ratio of "moles of base over moles of acid." The pH of the solution does not, it turns out, depend on the volume! (This is only true so long as the solution does not get so dilute that the autoionization of water becomes an important source of H+ or OH-. Such dilute solutions are rarely used as buffers, however.)

## References

1. Brown, et al. Chemistry:The Central Science. 11th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008.
2. Chang, Raymond. General Chemistry:The Essential Concepts. 3rd ed. New York: Mcgraw Hill, 2003
3. Petrucci, et al. General Chemistry: Principles & Modern Applications. 9th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2007.


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Article | Open | Published:

# Optical emission near a high-impedance mirror

## Abstract

Solid state light emitters rely on metallic contacts with a high sheet-conductivity for effective charge injection. Unfortunately, such contacts also support surface plasmon polariton and lossy wave excitations that dissipate optical energy into the metal and limit the external quantum efficiency. Here, inspired by the concept of radio-frequency high-impedance surfaces and their use in conformal antennas we illustrate how electrodes can be nanopatterned to simultaneously provide a high DC electrical conductivity and high-impedance at optical frequencies. Such electrodes do not support SPPs across the visible spectrum and greatly suppress dissipative losses while facilitating a desirable Lambertian emission profile. We verify this concept by studying the emission enhancement and photoluminescence lifetime for a dye emitter layer deposited on the electrodes.

## Introduction

The development of high-efficiency and high-power solid-state light emitters has led to their dramatically increased use in lighting and display applications. Yet there are still significant opportunities to further boost performance. In particular, organic light-emitting diodes exhibit a low external quantum efficiency (EQE), which can largely be attributed to the excitation of dissipative optical modes in the metallic electrode1,2. These can be classified into guided surface plasmon polariton (SPP) modes and high-spatial-frequency modes termed lossy waves (LWs)3,4 Their relative importance depends on the emitter-to-metal spacing (Supplementary Fig. 1, Supplementary Note 1). The electrodes in light emitting devices perform critical functions by facilitating carrier injection and serving as a mirror that redirects photons to the emitting surface. As such, they cannot be omitted from the device stack and these optical loss channels seem unavoidable.

Currently, optical energy dissipation in metallic electrodes is minimized by inserting a dielectric spacer layer between the metal and emitter layers. However, the use of such spacers increases the device’s contact resistance and the probability of trapping light inside the high-index light emitting materials via total internal reflection5. Significant progress has been made in light extraction from light emitting devices (LEDs)6,7,8,9,10. Different nanophotonic structures and metamaterials have been used to modify the emission of quantum emitters11,12,13. Alternatively, researchers have attempted to reduce the SPP mode loss contribution by using wavelength-scale periodic gratings in the electrode that decouple excited SPPs into free-space radiation14,15,16,17. Although these type of structures can out-couple SPPs very efficiently, they give rise to a highly-directional and wavelength-dependent emission17, which is undesirable in many display and lighting applications. In light of these drawbacks, it would be highly desirable to tackle the dissipative loss in the metal at its root and reduce the coupling of emitters to SPPs and LWs in the electrode without the need for spacer layers and complex light-extraction schemes.

Our approach to solve this problem is inspired by works originally performed in radio frequency engineering to suppress the coupling of radiation to bound surface waves supported by a metallic ground plane18. The key is to create a high-impedance surface that does not support bound surface modes. This is possible by patterning a metal film in such a way that the ratio of the electric over the magnetic field at the surface becomes extremely high. Such a surface was found to naturally avoid the undesirable shorting of an electric dipole antenna placed in close proximity to a high-conductivity metal film. It was also found to suppress the coupling of antenna emission to non-radiative bound surface modes. The history of high-impedance surfaces has its roots in the notion of electromagnetic hard and soft surfaces—a concept borrowed from acoustics and brought into the electromagnetic regime at microwave frequencies for RF antenna feed engineering19,20,21 and for the design of artificial magnetic conducting surfaces suitable for conformal antennas18,22. Such surfaces were later proposed as a ground plane for thin absorbers23,24. With the recent appearance of several high-impedance metasurface mirror designs operating in the visible spectral range25,26,27,28,29,30, it is logical to explore their application to enhance the emission from quantum dipole emitters located near a metallic electrode/reflector.

In this work, we provide a new solution for the longstanding fundamental challenge of using highly conductive metallic electrodes in thin film LEDs and break the usual dichotomy between high electrical and high optical performance of such electrodes. We illustrate how the detrimental excitation of SPPs and LWs can be greatly reduced by implementing a high-impedance nanopatterned metal as the electrode. We transplant the concept of high-impedance ground planes into the visible domain with the specific goal of enhancing emission from optical emitters situated on such nano-patterned metasurfaces, effectively constructing optical conformal nanoemitters and illustrate the benefits of this approach for enhancing the light emission from dye molecules placed on different types of nanopatterned metal surfaces.

## Results

### Radiative decay control with a high-impedance metasurface

We start by showing how the radiative decay of a quantum emitter above a metal film can be modified through the introduction of a subwavelength corrugation. Figure 1a shows the electromagnetic fields generated by a dipole emitter placed in close proximity to a smooth metal surface serving as an electric mirror (EM), oscillating at a frequency corresponding to a free space wavelength of λem = 560 nm. Whereas the emitter radiates some energy into the far-field, it also excites undesired LWs and SPPs. The LWs are highly localized in the near-field of the emitter and physically capture the excitation of an image-dipole in the metal. The SPPs propagate away from the emitter along the surface and reduce in strength as they dissipate energy in the metal. After introducing a subwavelength groove-array into the metal to form a high-impedance metasurface (HIM), there is no noticeable SPP excitation on the electrode surface (Fig. 1b). The same holds true for an electric dipole that is oriented normal to the metal surface (Supplementary Fig. 2). A more detailed investigation also reveals that the HIM can effectively reduce the coupling to LWs. The creation of the grooves locally increases the emitter-to-metal spacing by removing some of the metal and thereby diminishes this loss channel (Supplementary Figs. 3, 4).

One easy way to verify whether the patterned surface serves as a HIM is to calculate the reflection phase of a normally incident plane wave. A high-impedance surface features a reflection phase between –π/2 and π/218, and we find a reflection phase of 0.2π for our grooved metasurface. This reflection phase can also be used to calculate the impedance23 and this gives 3.6η, where η is the characteristic impedance of free space. This is 12 times higher than the surface impedance of the flat silver mirror. It can be seen that the far-field radiation is achieved without significantly affecting the broad angular radiation patterns of electric dipole emitters, which is desirable for many lighting and display applications (Supplementary Fig. 5).

The absence of guided SPP modes on a properly designed HIM can be verified by launching a SPP along a smooth metal film in the direction of a finite-sized HIM region. In Fig. 1d, an SPP wave is traveling toward a set of ten-subwavelength grooves is seen to effectively decouple and radiate into the far-field in a mere 1-μm-long corrugated region. Notably, little backreflection of the SPP wave occurs. Simulations show that the SPP decoupling occurs efficiently across a wide frequency range while generating little SPP reflection or transmission for groove depths ranging from 50 to 120 nm (Supplementary Fig. 6).

### Engineering a broad bandgap for surface plasmon polaritons

The observations in Fig. 1 show that the HIM effectively transforms guided SPPs on a smooth metal surface into leaky waves. This is consistent with the calculated bandstructure of the HIM (Fig. 2a) The patterned surface exhibits a region with no allowed SPP modes in the broad frequency range from 370 to 750 THz (400 to 810 nm) due to the high impedance of the structure. The guided modes at the upper and lower edge of this region correspond to plasmonic modes that feature a high magnitude of the magnetic field at the top of the corrugations and inside the grooves. The magnetic field profile |Hz| for these two modes is shown in Fig. 2b, c. At frequencies between these extremes, no guided SPP modes are allowed, and consequently SPPs incident on a HIM are either reflected or radiated into free space, with the radiation channel dominating as a result of the high impedance of the surface as shown in Supplementary Fig. 6. It is worth emphasizing that for the proposed subwavelength groove arrays (groove spacing λem/4) the formation of a bandgap is linked to the effective decoupling of quasi-guided waves from the surface. This behavior contrasts the behavior of typical plasmonic bandgap structures that feature shallow grooves with a spacing of half a SPP wavelength for which the formation of a bandgap is linked to SPP reflection in the plane of the metal surface, rather than the enhanced radiation away from the surface achieved here.

Next, we experimentally measure the transmission of SPPs across an array of ten subwavelength grooves. The targeted groove dimensions are 100 nm deep, 75 nm wide, and the periodicity is 150 nm. To this end, an optically thick, silver film is deposited on a smooth glass substrate. Focused ion beam (FIB) milling is then used to carve a groove-array as shown in Fig. 2d. The groove-array is flanked on one side by a slit through the metal used to launch SPPs towards the groove-array. A decoupling groove is milled on the opposing side of the groove array to monitor the SPP transmission. Upon illumination of the excitation slit with a collimated white light source, both the groove array and the decoupling groove light up. The decoupling groove features a darker section directly behind the groove-array, indicating that as few as ten grooves can effectively prevent the propagation of SPP waves by converting them to far-field radiation.

The SPP transmission spectrum is shown in Fig. 2e. The transmission coefficient is less than 0.2 for wavelengths in the range from 500 to 800 nm and increases abruptly at the edges of this spectral window. The onsets of high transmission occur close to the calculated upper and lower edges of the bandgap of the patterned surface. The low transmission in the bandgap region is due to the decoupling of SPPs by the HIM across the entire visible range, rather than due to Bragg backreflection of SPPs across a narrow bandwidth, as is observed for plasmonic bandgap structures based on shallow metallic gratings31,32,33(Supplementary Note 2).

### Experimental demonstrations of enhanced light emission

We can now exploit these HIMs to increase the EQE of a light-emitting material. We cover a HIM region as shown in Fig. 3a with a 15-nm-thick layer of Rhodamine 6G (R6G) in polymethyl methacrylate (PMMA). The broad emission spectrum of R6G is peaked at 550 nm, inside the predicted band gap of the HIM discussed in Fig. 2a. In order to maintain a fixed density of emitters per unit area for different groove parameters, we first fill the grooves with SiO2 to create a planar overcoat before spin-casting the dye-doped PMMA layer (see “Methods”). SiO2 deposition in the grooves will slightly move band edges calculated in the Fig. 2 to 300 THz and 700 THz. This broad range still contains the spectral emission peak for R6G molecules at 530 THz. The emitter layer is then excited with a 485 nm pulsed laser. In order to minimize the effect of the excitation enhancement on fluorescence intensity, the laser is linearly polarized along the grooves (transverse electric (TE) polarization) to avoid direct excitation of gap SPPs in the grooves that could result in large absorption enhancements in the dye layer. Photoluminescence (PL) maps are obtained by scanning the laser excitation spot across the sample while collecting the PL emission at each point with a confocal microscope. The maps are created while collecting the PL signal over the entire emission spectrum of the dyes.

As seen in emission maps for detection polarizations oriented both parallel and perpendicular to the grooves (Fig. 3b, c), the emission intensity is relatively uniform across the patterned surface region and significantly higher than in the adjacent flat area. From these maps we obtained a PL intensity enhancement factor by averaging the PL intensity across the patterned area and dividing it by the average PL intensity from a flat sample region. The enhancement is larger for the transverse magnetic (TM) polarized emission normal to the grooves (about 15) than for the TE polarized emission (approximately 5). This is expected as the leaky waves excited by the emitting molecules are longitudinal waves with an electric field in the propagation direction (like SPPs on a smooth metal surface). This field direction is maintained upon decoupling by the HIM. The measured IPL enhancement depends on the groove depth for both TM and TE polarization (Fig. 3e). The ratio between TM and TE emission varies from 1 to 2.5 depending on the groove depth. It is clear that an anisotropic HIM can not only enhance emission, but also control the polarization state of the emitted light directly at the source (without the need for external components like polarizers and wave plates).

### Quantitative analysis of the photoluminescence enhancements

To quantitatively understand the origin of this substantial PL enhancement we consider several possible contributions. Under pulsed excitation the measured PL intensity IPL of an ensemble of molecules is proportional to the number of excited molecules per unit area N immediately after the pulse, the collection efficiency of the optics ηcol, and the EQE ηext of the sample:

$$I_{{\mathrm{PL}}}\hskip-1.5pt \sim \hskip-1.5pt N \times \eta _{{\mathrm{col}}} \times \eta _{{\mathrm{ext}}}$$
(1)

We define the EQE as:

$$\eta _{{\mathrm{ext}}} = \frac{{{{\Gamma }}_{\mathrm{r}}}}{{{{\Gamma }}_{\mathrm{r}} + {{\Gamma }}_{{\mathrm{nr}},{\mathrm{m}}} + {{\Gamma }}_{{\mathrm{nr}}}^0}} = \Gamma _{\mathrm{r}} \times \tau$$
(2)

where Γr is the radiative decay rate for emission into the far-field either directly or after interacting with the metamirror, $$\Gamma _{{\mathrm{nr}}}^0$$ is the intrinsic non-radiative decay rate of the molecules, and $$\Gamma _{{\mathrm{nr}},{\mathrm{m}}}$$ is the non-radiative decay rate quantifying the dissipation of energy into the metal film. Here, we treat this rate as a non-radiative rate as none of this emission is observed in the far-field (Fig. 3d). The EQE can also be written in terms of the measured decay lifetime τ= (Γr+Γnr,m + $${{\Gamma }}_{{\mathrm{nr}}}^0$$)−1.

In light of Eqs. (1) and (2), an enhancement of emitted intensity can be attributed to an increase in the number of excited dye molecules per collection area, the collection efficiency, and the EQE due to changes in radiative and total decay rates. In the rest of this report, we look at the impact of each of these terms.

To compare the excitation of molecules on flat and patterned surfaces, we carried out full-field simulations with different groove depths (Fig. 3f). The absorption of laser light in the emissive layer as a function of groove depth reaches a maximum excitation enhancement factor of approximately 2.5. Our measurements are in the low-pump power regime (far from saturation). In this regime, the number of excited molecules is linearly proportional to the absorption enhancement.

Regarding the collection efficiency, we can assume that ηcol is essentially the same for patterned and unpatterned areas for two reasons. First, we use a high numerical aperture objective (NA = 0.95). Second, our simulations indicate that the angular radiation pattern of a dipole above the patterned HIM is virtually unchanged compared to a smooth metal surface (see Fig. 1c and Supplementary Figs. 2, 5).

To understand the changes in EQE, which depends on the radiative and total decay rates, we measure PL decay traces (Fig. 3g and Supplementary Fig. 7). The traces for the HIM and flat silver mirror both show a largely single-exponential decay, indicating that the PL signal predominantly originates from molecules with a single, fast decay rate. Despite the large observed PL enhancement for some groove depths, the decay rate is found to vary by less than 10% for different groove depths. We attribute the small change in lifetime of the emitters to a relatively high intrinsic non-radiative decay rate of R6G molecules when embedded in our chosen polymer matrix. Indeed, the same molecules on a glass substrate have a fast rate of 2.5 ns−1, close to 2.9 ns−1 measured on an unpatterned silver mirror.

It is thus clear that the small changes in the excitation efficiency (about 2.5) and insignificant changes in the collection efficiency and PL decay rate cannot explain the large changes in the PL intensity (approximately 15). These changes can therefore largely be attributed to an enhancement in the radiative rate, for example a redirection of the emission into the far-field instead of into the metal as expected for a high-impedance surface that suppresses SPPs and LWs. It should be noted that unlike photonic bandgap structures in dielectrics where the local density of optical states goes to zero inside the bandgap, HIMs trade bound surface modes with leaky modes. As a result, an emitter can couple its emission to these leaky modes, which eventually contribute to the farfield radiation. The enhancement in the radiative rate can be assessed by dividing the measured PL intensity enhancement by the simulated absorption enhancement (Fig. 3h): a maximum enhancement of the radiative decay rate of approximately five can be reached with a groove depth of 100 nm. As the relative fractions of the power coupled toSPPs versus free space on a smooth metal just depends on thedistance of the emitter to the metal and not on the internal quantumefficiency, this approach is effective for both low and high internal-quantum-efficiency emitters.

Regarding the electrical performance of such electrodes, given the fact that the resistivity of metals like Ag and Al is two or three orders of magnitude lower than a good transparent conductive oxides (TCOs), the use of patterned continuous metal films can outperform continuous TCOs in terms of their charge injection properties. (Supplementary Note 3).

A periodic array of subwavelength grooves can thus increase the PL intensity by increasing the EQE of the emissive layer, while controlling the polarization state of the emission. However, a polarization-independent enhancement in the PL intensity is also possible. To this end, we fabricated high-impedance surfaces based on dimples filled with SiO2 arranged on a square lattice (Fig. 4). Band structure calculations of such a polarization-independent high-impedance surface predict a clear band gap that spans a broad wavelength range 485–600 nm (Supplementary Fig. 8). We also calculated the reflection phase of such a metasurface for normal incident light at visible frequencies (Supplementary Fig. 8b). We find values that are consistent with observations by others at RF frequencies18. We used the same fabrication process as for the linear grooves to fill the dimples with SiO2 and deposit an emissive layer on top of the high-impedance surface. Again, PL maps demonstrate PL enhancement, but in this case the enhancement is independent of the detection polarization. The enhancement increases with dimple depth and reaches a maximum value of approximately 12 for a groove depth of 180 nm (Fig. 4d), similar to the maximum TM polarization value for grooved samples but now obtained for both polarizations. Similar to the linear groove arrays, the lifetime measurements do not show any significant decay rate differences for the molecules on the dimple arrays compared to those on a flat sample region. We conclude that similar physics underlies the observed PL enhancement for both dimple and groove arrays.

## Discussion

Motivated and inspired by the notion of high-impedance ground planes in the radio frequency regime, we have developed an optical version of this concept for enhancing optical emission. We demonstrated the realization of high electrical conductivity metallic electrodes that combine high reflectivity at visible frequencies with significantly reduced optical losses compared to traditional metallic electrodes. These HIMs can enhance the light extraction efficiency of quantum emitters and reduce undesired optical losses in metallic electrodes by introducing a wide SPP band gap and mitigating dissipation in LWs. Within the gap no guided surface waves exist and the emission is rapidly decoupled to free-space. This approach is effective for both low and high internal-quantum-efficiency emitters. The broadband nature of the SPP band gap observed for these metasurfaces makes them promising candidates to improve the performance of solid-state light-emitting devices for illumination and display applications. Moreover, these metasurfaces provide an excellent platform for engineering and manipulating SPPs along two-dimensional structures (flat photonics), allowing additional functionality in light-matter interaction.

## Methods

### Device fabrication

HIMs were fabricated in a multistep process. Silicon wafers with 300-nm-thick thermally grown oxide were used as a planar and smooth substrate. They were cleaned in several steps. First, they were sonicated with Alconox precision cleaner solution, then rinsed with water three times. After this step, they were sonicated for 5 min in Acetone, Methanol, and Isopropanol, consecutively, and finally dried with nitrogen gas. A smooth, optically thick (300 nm) silver film was then deposited onto the substrate using a 2 nm germanium nucleation layer. The metal was nanopatterned by FIB milling with a FEI Helios 600i dual FIB/SEM tool. A thick layer of silicon oxide was deposited on the patterned area with electron beam assisted deposition for the first 50 nm and ion beam assisted deposition for the next 250 nm. Then the deposited SiO2 layer was milled down to the silver surface with FIB until a 5~10 nm overcoat of SiO2 was left on the high-impedance area and the surrounding flat area.

To deposit a thin light-emitting layer, Rhodamine 6G powder was dissolved in a Polymethyl methacrylate (PMMA) solution. The PMMA was diluted with Anisole to reduce the achievable thickness of the emissive layer. The solution was then put in a vortex mixer for 2 min. The concentration of R6G in the PMMA solution is approximately 3 × 10−4 M. The thickness of the R6G film is around 15 nm, as measured by SEM imaging the cross section of our samples.

### Optical measurements

PL maps were acquired with a Witec alpha 300R confocal microscope with a two-axis piezo stage for sample scanning. A PicoQuant LDH-P-C-485 pulsed laser at wavelength of 485 nm is used for excitation of R6G molecules. The PL signal is detected with an avalanche photodiode (MPD APD) for both lifetime measurement and PL intensity measurement. The spatial resolution of the imaging system in our experiments is about 500 nm. The measured lifetime traces were fitted with a bi-exponential function of the form $$I\left( t \right) = a{\mathrm{e}}^{ - r_1t} + \left( {1 - a} \right){\mathrm{e}}^{ - r_2t}$$ where a is the amplitude of the fast contribution and r1 and r2 are the decay rates of the two contributions. The value of a is larger than 0.99 for all five groove depths (Supplementary Fig. 6)

### Simulations

All simulations were carried out using the Finite-Difference Time-Domain method (Lumerical FDTD Solutions).

### Data availability

All data are available from the authors on reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Change history

• ### 04 September 2018

This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.

## References

1. 1.

Brütting, W., Frischeisen, J., Schmidt, T. D., Scholz, B. J. & Mayr, C. Device efficiency of organic light-emitting diodes: progress by improved light outcoupling. Phys. Status Solidi 210, 44–65 (2013).

2. 2.

Saxena, K., Jain, V. K. & Mehta, D. S. A review on the light extraction techniques in organic electroluminescent devices. Opt. Mater. (Amst.) 32, 221–233 (2009).

3. 3.

Chance, R. R., & Prock, A., & Silbey, R. Molecular fluorescence and energy transfer near interfaces. Adv. Chem. Phys. 37, 1–65 (1978).

4. 4.

Ford, G. W. & Weber, W. H. Electromagnetic interactions of molecules with metal surfaces. Phys. Rep. 113, 195–287 (1984).

5. 5.

Reineke, S., Thomschke, M., Lüssem, B. & Leo, K. White organic light-emitting diodes: status and perspective. Rev. Mod. Phys. 85, 1245–1293 (2013).

6. 6.

Möller, S. & Forrest, S. R. Improved light out-coupling in organic light emitting diodes employing ordered microlens arrays. J. Appl. Phys. 91, 3324–3327 (2002).

7. 7.

Madigan, C. F., Lu, M.-H. & Sturm, J. C. Improvement of output coupling efficiency of organic light-emitting diodes by backside substrate modification. Appl. Phys. Lett. 76, 1650 (2000).

8. 8.

Yamasaki, T., Sumioka, K. & Tsutsui, T. Organic light-emitting device with an ordered monolayer of silica microspheres as a scattering medium. Appl. Phys. Lett. 76, 1243 (2000).

9. 9.

Riedel, B., Kaiser, I., Hauss, J., Lemmer, U. & Gerken, M. Improving the outcoupling efficiency of indium-tin-oxide-free organic light-emitting diodes via rough internal interfaces. Opt. Express 18(Suppl 4), A631–A639 (2010).

10. 10.

Takada, N., Tsutsui, T. & Saito, S. Control of emission characteristics in organic thin-film electroluminescent diodes using an optical-microcavity structure. Appl. Phys. Lett. 63, 2032 (1993).

11. 11.

Tsakmakidis, K. L., Boyd, R. W., Yablonovitch, E. & Zhang, X. Large spontaneous-emission enhancements in metallic nanostructures: towards LEDs faster than lasers [Invited]. Opt. Express 24, 17916 (2016).

12. 12.

Hess, O. et al. Active nanoplasmonic metamaterials. Nat. Mater. 11, 573–584 (2012).

13. 13.

Pelton, M. Modified spontaneous emission in nanophotonic structures. Nat. Photonics 9, 427–435 (2015).

14. 14.

Yates, C. J., Samuel, I. D. W., Burn, P. L., Wedge, S. & Barnes, W. L. Surface plasmon-polariton mediated emission from phosphorescent dendrimer light-emitting diodes. Appl. Phys. Lett. 88, 2–4 (2006).

15. 15.

Vuckovic, J., Loncar, M., Scherer, A. & Lonˇ, M. Surface plasmon enhanced light-emitting diode. IEEE J. Quantum Electron. 36, 1131–1144 (2000).

16. 16.

Hobson, Pa, Wedge, S., Wasey, Ja. E., Sage, I. & Barnes, W. L. Surface plasmon mediated emission from organic light-emitting diodes. Adv. Mater. 14, 1393–1396 (2002).

17. 17.

Worthing, P. T. & Barnes, W. L. Efficient coupling of surface plasmon polaritons to radiation using a bi-grating. Appl. Phys. Lett. 79, 3035–3037 (2001).

18. 18.

Sievenpiper, D., Zhang, L. Z. L., Broas, R. F. J., Alexopolous, N. G., & Yablonovitch, E. High-impedance electromagnetic surfaces with a forbidden frequencyband. IEEE. Trans. Microwave Theory. Tech. 47, 2059–2074 (1999).

19. 19.

Kildal, P. S., Lier, E. & Aas, J. A. Artificially soft and hard surfaces in electromagnetics and their application. 1988 IEEE AP-S. Int. Symp. Antennas Propag. 2, 832–835 (1988).

20. 20.

Kildal, P. S. Artificially soft and hard surfaces in electromagnetics. IEEE Trans. Antennas Propag. 38, 1537–1544 (1990).

21. 21.

Maci, S. & Kildal, P. S. Hard and soft surfaces realized by FSS printed on a grounded dielectric slab. IEEE Antennas Propag. Soc. Symp. 2004 1, 285–288 (2004).

22. 22.

Clavijo, S., Díaz, R. E. & McKinzie, W. E. Design methodology for Sievenpiper high-impedance surfaces: an artificial magnetic conductor for positive gain electrically small antennas. IEEE Trans. Antennas Propag. 51, 2678–2690 (2003).

23. 23.

Engheta, N. Thin absorbing screens using metamaterial surfaces. IEEE Antennas Propag. Soc. Int. Symp. (IEEE Cat. No. 02CH37313) 2, 392–395 (2002).

24. 24.

McVay, J., Hoorfar, A. & Engheta, N. Thin absorbers using space-filling-curve high-impedance surfaces. IEEE Antennas Propag. Soc., AP-S Int. Symp . (Dig.) 2A, 22–25 (2005).

25. 25.

Esfandyarpour, M., Garnett, E. C., Cui, Y., McGehee, M. D. & Brongersma, M. L. Metamaterial mirrors in optoelectronic devices. Nat. Nanotechnol. 9, 542–547 (2014).

26. 26.

Schwanecke, aS. et al. Optical magnetic mirrors. J. Opt. A Pure Appl. Opt. 9, L1–L2 (2007).

27. 27.

Rajasekharan, R. & Roberts, A. Optical “magnetic mirror” metasurfaces using interference between Fabry-Pérot cavity resonances in coaxial apertures. Sci. Rep. 5, 10297 (2015).

28. 28.

Lee, S. & Kim, J. Design of optical metamaterial mirror with metallic nanoparticles for broadband light absorption in graphene optoelectronic devices. arXiv 23, 11089–11096 (2015).

29. 29.

Lin, L. et al. Dielectric nanoresonator based lossless optical perfect magnetic mirror with near-zero reflection phase. Appl. Phys. Lett. 108, 171902 (2016).

30. 30.

Pisano, G., Ade, P. A. R. & Tucker, C. Experimental realization of an achromatic magnetic mirror based on metamaterials. Appl. Opt. 55, 4814–4819 (2016).

31. 31.

Kitson, S., Barnes, W. L. & Sambles, J. Full photonic band gap for surface modes in the visible. Phys. Rev. Lett. 77, 2670–2673 (1996).

32. 32.

Bozhevolnyi, S. I., Erland, J., Leosson, K., Skovgaard, P. M. W. & Hvan, J. M. Waveguiding in surface plasmon polariton band gap structures. Phys. Rev. Lett. 86, 3008–3011 (2001).

33. 33.

Barnes, W. L. Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices. J. Light. Technol. 17, 2170–2182 (1999).

## Acknowledgements

This work was supported by a Multi University Research Initiative (MURI FA9550-12-1-0488) from the AFOSR and a gift from Konica Minolta Laboratory USA. A.G.C. also acknowledges the support of a Marie Curie International Outgoing Fellowship.

## Author information

### Affiliations

1. #### Geballe Laboratory for Advanced Materials, Stanford University, 476 Lomita Mall, Stanford, California, 94305, USA

• Majid Esfandyarpour
• , Alberto G. Curto
• , Pieter G. Kik
•  & Mark L. Brongersma
2. #### Department of Applied Physics and Institute for Photonic Integration, Eindhoven University of Technology, 5600, MB Eindhoven, The Netherlands

• Alberto G. Curto
3. #### CREOL, The College of Optics and Photonics, University of Central Florida, Florida, 32816, USA

• Pieter G. Kik

### Contributions

M.E. performed all simulations and modeling and fabricated the samples in this study. M.E. and A.G.C. performed the optical experiments. M.E., A.G.C., P.G.K., N.E., and M.L.B. were involved in the physical analysis of the measurement results. M.L.B. supervised the project.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Mark L. Brongersma.


text:
# Find maxima and minima of the function

Given: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f\left(x,y \right)=-x^4+x^3-3x^2y+3xy^2-y^3$$ Find all points where gradient is equal to zero. Decide whether in those points function has either maxima or minima.

hess's matrix does not help here as gradient is equal to zero at $(0,0)$ thus hess matrix is zero always. Thus how to check whether function has its minimum or maximum at the point of $(0,0)$?

By considering the $y$-derivative, the $y$-coordinate of the gradient is zero iff $y=x$. The $x$-coordinate is then zero iff $y=0$, so the only gradient-zero point is $(0,0)$.
To determine whether this is a minimum or a maximum (or otherwise), we can examine a small perturbation near $(0, 0)$. That is, for small-in-modulus $x, y$, is $f$ positive, negative, or both?
It's clear that if we set $y=x$, we get $-x^4$, so if we leave $(0,0)$ along the line $y=x$, we go negative in both directions.
If we set $y=-x$, we get $f(x, -x) = 8 x^3 - x^4$, which is positive for positive $x$ and negative for negative $x$. Therefore we must have a saddle: if we leave $(0,0)$ along $y=-x$ for positive $x$, we get a positive $f$, while if we leave along $y=-x$ for negative $x$, we get negative $f$.


text:
# Interval Scheduling Problem with more than One Resource

In order to schedule the $n$ job requests over one resource, you sort the requests in order of finish time, choose the request with earliest finish time, choose the next compatible one, and so on.

Now, let us say we have two resources instead of one. How do I schedule my jobs now? As my idea goes, again you start by sorting the requests in order of finish time. My problem is, how do I proceed after that? Do I choose the resources in sequential or in an alternate fashion?

If I go for sequential manner, I schedule all the possible jobs in the first resource and then do the same for second resource with the jobs yet to be scheduled.

If I go for alternate fashion, I choose the first possible job in the first resource, then second possible job in the second resource and so on.

In each case we will have take in to account the chosen jobs being compatible, needless to say.

I can not decide which of these is going to be optimal.

Any input will be appreciated.

• Thanks for the answer. You have put forth a general solution for $k$ resources. When we consider two resources, is not it the same as going in an alternate fashion? – Masroor Mar 18 '13 at 1:22


text:
Price of call/put is convex in $K$ (strike price)

Let $\lambda\in(0,1)$. Then $$C(T, \lambda K_1 + (1 - \lambda)K_2, S, t) \leq \lambda C(T, K_1, S, t) + (1 - \lambda)C(T, K_2, S, t)$$

$T$ - the maturity

$K_1$,$K_2$ - Strike prices

$S$ - stock price

$t$ - current time

In other words, the price of the call/put option in convex in $K$. Show the same claim for the price of put options, American call options, and American put options.

I think you need to just apply the triangle inequality, but I am not sure. Any suggestions is greatly appreciated.

• This has been answered on math stackexchange. Please check: math.stackexchange.com/questions/112063/… – phdstudent Jan 15 '16 at 18:16
• I will still post it as an answer, so that this does not go unanswered. Tried to vote for close, but the duplicate must be on quant.stackexchange. – phdstudent Jan 15 '16 at 18:29

Let the price of an option at strike $K$ be given by $V(K)$. To say that the price is convex in the strike means that

$$V(K-\delta) + V(K+\delta) > 2 V(K)$$

for all $K>0$ and $\delta>0$. Let's assume that the opposite is true, i.e. that there exist tradeable option contracts expiring on the same date such that

$$V(K-\delta) + V(K+\delta) \leq 2 V(K)$$

I therefore buy a contract at $K+\delta$ and one at $K-\delta$, and finance my purchase by selling two of the options at $K$ (which I can do, because the two options struck at $K$ are at least as expensive as the other two combined).

At expiry the price of the stock is $S$, and my total payout is

$$P = (S-(K-\delta))^+ + (S-(K+\delta))^+ - 2(S-K)^+$$

Now there are four regimes:

• $S<K-\delta$, which means $P=0$
• $K-\delta < S < K$, which means $P=S-(K-\delta) > 0$
• $K < S < K+\delta$, which means $P=S-K+\delta - 2(S-K)=K+\delta-S>0$
• $S>K+\delta$, which means $P = S-K+\delta + S-K-\delta - 2(S-K) = 0$

So I have the possibility of making a profit, but no possibility of making a loss - which is an arbitrage. Since no arbitrages exist, the option price must be convex in the strike price.

• This answer starts from the identity $V (K+\delta)+V (K-\delta) > 2V (K)$ which is only a *particular case* of the full convexity definition stated in the OP's question (corresponds to taking $K_1=K-\delta$, $K_2=K+\delta$ and $\lambda=1/2$). Wouldn't it be more general to show that convexity means $\partial_K^2 C (K) = C^{\prime\prime}(K) \geq 0$ (if this derivative exists of course) and then re-derive the Breeden-Litzenberger identity to show that this is indeed true, because $\text {sign}[C^{\prime\prime}(K)] = \text {sign}[q(S_0;T,S_T=K)] \geq 0$ by definition of a pdf? – Quantuple Mar 28 '16 at 19:13
• @Quantuple Is it though? - a particular case? – Henrik Mar 27 '17 at 12:39
• What do you mean @Henrik? Don't you agree that starting from $$C(T, \lambda K_1 + (1 - \lambda)K_2, S, t) \leq \lambda C(T, K_1, S, t) + (1 - \lambda)C(T, K_2, S, t) \tag{A}$$ letting $K_1 = K-\delta$, $K_2=K+\delta$ and $\lambda=1/2$ one gets: $$C(T, \frac{1}{2} (K + \delta + K - \delta), S, t) \leq \frac{1}{2} \left( C(T, K-\delta, S, t) + C(T, K+\delta, S, t)\right)$$ which is equivalent to writing: $$2 C(T,K, S, t) \leq C(T, K-\delta, S, t) + C(T, K+\delta, S, t) \tag{B}$$ hence inequality $(B)$ is a particular instance of inequality $(A)$ – Quantuple Mar 27 '17 at 13:06
• @Quantuple $K\mapsto C(K)$ needs to be really messed up for the definitions to not be equivalent. See e.g. math.stackexchange.com/q/83383/28157 and math.stackexchange.com/q/83383/28157 – Henrik Mar 28 '17 at 9:26
• Thank you @Henrik for the reference. I guess what I was saying then amounts to put up a "caveat" stating that it's sufficient to prove midpoint-convexity assuming the mapping is continuous (midpoint convexity -> rational convexity -> convexity). – Quantuple Mar 28 '17 at 9:50


text:
Activity: Expectation Values for a Particle on a Ring

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• This activity is used in the following sequences
What students learn
• How to calculate expectation values
• Linear combinations of eigenstates with different eigenvalues are not stationary (yield expectation values that depend on time)
• Media

Expectation Values for a Particle on a Ring: Instructor's Guide

Expectation Values for a Particle on a Ring Handout

Introduction

It could be beneficial to see what students remember about expectation values by having asking them a Small White Board Question about the topic. Use their responses to drive an intro discussion about what expectation value is: not a measurement, but a weighted average.

Student Conversations

• Operators vs Measurement: Students commonly attempt to determine the values resulting from a quantum experiment by allowing the operator corresponding to the observable of interest to act on the initial state. Students who do this should be encouraged to consider the nature of this transformation (it's a vector, not a scalar) and to recognize that the transformation does not necessarily yield an eigenvector (the state of the system should be an eigenstate after the measurement).
• Degeneracy: Students may experience some difficulty due to the degeneracy of some states, in particular, that you have to include all the states that share that eigenvalue. $P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$
• Notation: As with earlier activities, students are usually comfortable at this point with doing expectation values in bra-ket notation, but fewer are comfortable with using wavefunction notation. For groups that finish early, ask them to use the other method to compare.
• Time Dependence: Some students might wonder about the time-(in)dependence of their expectation values. This is a good opportunity to remind students of Time Dependence for a Quantum Particle on a Ring where we discussed that quantities whose opperator commutes with the Hamiltonian will have time independent probabilities.

Wrap-up

This activity provides an opportunity to contrast two methods of finding expectation values.

• Carry out the explicit and messy differentiation and integration on the given state.
• Recast the initial state as a linear combination of eigenstates and carry out the much simpler calculations on these eigenstates.
Generally, students in the class will be mixed in the approach they choose. By emphasizing this when you wrapup this activity, students have the opportunity to sort out for themselves the benefits of each method. Remind them of Energy and Angular Momentum for a Quantum Particle on a Ring where they made this sort of comparison explicitly.

Discuss some sense-making techniques for expectation value such as: units of the expectation value, what the expectation values tells you about the distribution of possible measurement values. Drawing a histogram of possible measurements vs probability might be a good way to illustrate these properties to students.

Extensions and Related Materials

This is a part of Quantum Ring Sequence of activities.

Author Information
Corinne Manogue, Kerry Browne, Elizabeth Gire, Mary Bridget Kustusch, David McIntyre
Keywords
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
Learning Outcomes


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Suppose an algaecide for cleaning swimming pools contains 50 grams (g) of anhydrous copper(II) sulfate, CuSO$$_4$$. If the molecular weight of copper(II) sulfate is 160 grams per mol, how many molecules of CuSO$$_4$$ are in the algaecide?

Hunter B.

In order to answer this question, we first need to know what information we are given. From the question, we know we have 50 grams of copper(II) sulfate and we know its molar mass is 160 grams per mol. Using this information, we need to find the number of molecules in 50 grams of copper(II) sulfate. To solve this, we first need to get the grams of copper(II) sulfate into moles of copper(II) sulfate. To do this, we need to divide the known mass of copper(II) sulfate, 50 g, by the known molar mass of copper(II) sulfate, 160 g per mol, which gives us 50 g CuSO$$_4$$ $$\bullet \frac{1}{160 \frac{g}{mol}}=0.313$$ moles of copper(II) sulfate. We then need to multiply the moles of copper(II) sulfate by Avogadro's number, 6.022 X $$10^{23}$$, which gives us: 0.313 moles of copper(II) sulfate $$\bullet \frac{6.022 X 10^{23}moles \hspace{.08cm} of \hspace{.05cm} copper(II) sulfate}{1 \hspace{.08cm} mol \hspace{.05cm} of \hspace{.08cm}copper(II) \hspace{.05cm}sulfate} =$$ 1.88 X 10$$^{23}$$ molecules of copper(II) sulfate. So, there are 1.88 X 10 $$^{23}$$ molecules of copper(II) sulfate in the algaecide.

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Consider the equilibrium reaction $$2A+B\rightleftharpoons 4C$$. After multiplying the reaction by a multiple of 2, what is the new equilibrium equation and what is the new equilibrium-constant expression, $$K_c$$?

Hunter B.

To find the new equilibrium equation, we simply multiply the coefficient of each species by 2, yielding $$4A+2B \rightleftharpoons 8C$$. To find the new equilibrium-constant expression, we need to recall that the parent formula for the equilibrium-constant expression, the reaction quotient $$Q_c$$, is as follows: $$mA+nB\rightleftharpoons xC+yD$$ = $$\frac{[C]^x[D]^y}{[A]^m[B]^n}$$. Which gives us $$K_c$$ = $$\frac{[C]^8}{[A]^4[B]^2}$$.

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Solve the following equation for $$x$$. $$2x+5=-3x-10$$.

Hunter B.

In order to solve this problem for $$x$$, we first need to collect all of the like terms. To collect the $$x$$ terms, we can either subtract 2$$x$$ from both sides or we can add 3$$x$$ to both sides. By adding 3$$x$$ to each side, we are left with $$5x+5=-10$$. We now need to collect the like terms without the variable $$x$$. To do this, we can either subtract 5 from each side or we can add 10 to each side. By adding 10 to each side, we are left with $$5x+15=0$$. Now, we need to separate these two terms so that we can solve for $$x$$. To do this, we subtract 15 from each side leaving us with $$5x=-15$$. Then, the last step is to solve for $$x$$. By dividing each side by 5, we are left with the answer of $$x=-3$$.

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