text:
2014
01-05

# The Embarrassed Cryptographer

The young and very promising cryptographer Odd Even has implemented the security module of a large system with thousands of users, which is now in use in his company. The cryptographic keys are created from the product of two primes, and are believed to be secure because there is no known method for factoring such a product effectively.
What Odd Even did not think of, was that both factors in a key should be large, not just their product. It is now possible that some of the users of the system have weak keys. In a desperate attempt not to be fired, Odd Even secretly goes through all the users keys, to check if they are strong enough. He uses his very poweful Atari, and is especially careful when checking his boss’ key.

The input consists of no more than 20 test cases. Each test case is a line with the integers 4 <= K <= 10100 and 2 <= L <= 106. K is the key itself, a product of two primes. L is the wanted minimum size of the factors in the key. The input set is terminated by a case where K = 0 and L = 0.

The input consists of no more than 20 test cases. Each test case is a line with the integers 4 <= K <= 10100 and 2 <= L <= 106. K is the key itself, a product of two primes. L is the wanted minimum size of the factors in the key. The input set is terminated by a case where K = 0 and L = 0.

143 10
143 20
667 20
667 30
2573 30
2573 40
0 0

GOOD
GOOD
GOOD
BAD 31

#include<iostream>
#include<stdio.h>
#include<math.h>
#include<string.h>
using namespace std;
#define Max 1100000
int isp[Max];
void isprim()
{
int i,j;
isp[0]=isp[1]=0;
isp[2]=1;
for(i=3;i<Max;i++)
isp[i]=i%2;
int ns;
ns=(int)sqrt(Max*1.0);
for(i=3;i<=ns;i++)
{
if(isp[i])
{
for(j=i*2;j<Max;j+=i)
isp[j]=0;
}
}
}
int main()
{
int len,n,i,j,k,ks;
int num,nums;
char str[1000];
isprim();
while(scanf("%s%d%*c",str,&n)&&(!(str[0]=='0'&&n==0)))
{
len=strlen(str);
for(k=2;k<n;k++)
{
if(isp[k]==0)
{
continue;
}
num=0;
for(i=0;i<len;i+=3)
{
nums=0;
ks=1;
for(j=i;j<i+3&&j<len;j++)
{
ks=ks*10;
nums=nums*10+(str[j]-'0');
}
num=num*ks+nums;
num=num%k;
}
//   printf("num=%d,k=%d\n",num,k);
if(num==0)
{
}


text:
# Where parallels cross

Interesting bits of life

# Mastering Emacs is a slow process: an archeological tour of my (archived) beginnings

If you are reading this, I guess you may be on your way to Emacs enlightenment.

Here I want to give you an (archaeological) taste of how I started with it. I hope that this will give you also a sense of how long it took me to get comfortable with this amazing tool 1. Ideally, new users that read this will feel comfortable even if, after some months of struggling, they have not yet tamed Emacs: as all the best things in life, learning Emacs takes time and dedication.

I started using Emacs towards the end of 2013, the beginning of my Ph.D. There must be a strong connection between academia's need to publish elegant scientific work (i.e., LaTex) and the adoption of Emacs 2.

Anyway, I know that I started roughly then because the oldest Org Archive entry I could find is the following:

* DONE Buy Christmas return ticket <2013-11-09 Sat>
:PROPERTIES:
:ARCHIVE_TIME: 2016-03-25 Fri 14:41
:ARCHIVE_OLPATH: Daily Stuff
:ARCHIVE_TODO: DONE
:END:

Something interesting is that my first archived note tells us what I was using before Org Mode for my calendar: when by Benjamin Crowell 3. You can guess that from the directory in which I was storing my main Org file.

This reminded me how powerful is to archive your tasks for later memory.

I guess by that point I already familiarized with the basic Emacs editing commands: cancelling an action C-g, undoing an edit C-/, killing C-w and yanking text C-y (i.e., copy/pasting), visit files C-x C-f, and load modes specific to programming languages via the init file (I was writing Bash, Java, Haskell, C#, OCaml, Python and Scala during my studies).

I guess Erc, Emacs' IRC mode, entered my life a year after, when I took note of a chat important for my research:

* DONE have chat with IRC ocaml <2014-10-22 Wed>
:PROPERTIES:
:ARCHIVE_TIME: 2016-03-25 Fri 14:40
:ARCHIVE_OLPATH: Ph.D.
:ARCHIVE_TODO: DONE
:END:
@@@ You have joined channel #ocaml                                      [18:39]
@@@ Topic for #ocaml: Discussions about the OCaml programming language |
http://www.ocaml.org and http://caml.inria.fr |
http://ocaml.org/releases/4.02.0.html | Public channel logs at
http://irclog.whitequark.org/ocaml

Notice how I must have still been unaware of Org source blocks and quotations at that time because I was just pasting the raw chat under the heading as a way to log things.

The outcome of a Ph.D. is, mainly, the final thesis, so I started writing my research down in a Org file early on. Emacs' gscholar-bibtex helped a lot with managing references for my LaTex bibliography:

* add gscholar extension to emacs to get bibtex info directly in emacs <2015-11-03 Tue>
:PROPERTIES:
:ARCHIVE_TIME: 2016-03-25 Fri 14:39
:ARCHIVE_OLPATH: On the fly
:END:
use M-x gscholar-bibtex

Today, if you need a bibliography, you should surely look also into org-ref.

I guess 2016 was the turning point for my customization of Emacs. I started using org-contacts for keeping track of birthdays and I finally found the need for Org Archive (all the entries I found were archived on that year).

Immediately after that came the installation of magit and wdired:

* DONE [#B] check out magit and wdired mode for emacs <2016-01-24 Sun>
CLOSED: [2016-02-20 Sat 22:52]
:PROPERTIES:
:ARCHIVE_TIME: 2016-03-25 Fri 14:39
:ARCHIVE_OLPATH: On the fly
:ARCHIVE_TODO: DONE
:END:
- [X] magit
https://www.emacswiki.org/emacs/Magit

| Keys | Effect in Magit mode |
|------+----------------------|
| s    | stage new file       |
| c c  | commit               |
| b b  | change branch        |
| P p  | push changes         |
| F f  | pull changes         |

- [X] dired

after C-f a directory, the dired buffer will open.
Here the commands:

| Keys    | Effect in Dired mode                          |
|---------+-----------------------------------------------|
| m       | marks an entry                                |
| t       | marks all entries that                        |
|         | are not marked, and unmarks those that are    |
| u       | unmarks an entry                              |
| U       | unmarks all entry                             |
| x       | executes command                              |
| C       | copies entry                                  |
| d       | set for deletion                              |
| D       | asks for deletion                             |
| !       | calls shell command on entry                  |
| g       | updates buffer                                |
| * - /   | marks all directories                         |
| A       | search for text                               |
| M - ,   | to see next found results for the text search |
| R       | renames entry                                 |
| S       | changes sorting order of directory            |
| C-x C-q | make buffer editable                          |
| C-c C-c | commit changes to dir - that changes the fs   |
| C-h i   | emacs manual                                  |

Evidently I needed a sweeter and efficient way to both manage Git repositories and modify my file system contents swiftly. Note how I was taking notes of keybindings cheatsheets as Org tables: it really takes time for muscle memory to kick in!

My big challenge as a new Emacs user, which I can still remember with uneasiness, was to setup mu4e to read my emails:

* DONE switch to mu4e as email client                        :emacs
CLOSED: [2016-03-02 Wed 22:18]
:PROPERTIES:
:ARCHIVE_TIME: 2016-03-25 Fri 14:39
:ARCHIVE_OLPATH: On the fly
:ARCHIVE_TODO: DONE
:END:
** THE REASON: Mozilla does not maintain Thunderbird anymore
and the plugin to increase the font ceases to work (blind me!)
** DONE STEP 1: gather information
** DONE STEP 2: OMG, let's install this!
*** DONE understand how to fetch mails
CLOSED: [2016-03-01 Tue 09:28]
**** offlineimap [5/5]
*** DONE understand how to use smtpmail [4/4]
CLOSED: [2016-02-29 Mon 23:27]
- Note taken on [2016-02-29 Mon 23:27] \\
< C-x m > to send an email
*** DONE install mu (if we get to this victory near :D)
CLOSED: [2016-03-01 Tue 10:14]
** DONE STEP 3: now read manual LOL
CLOSED: [2016-03-02 Wed 22:18]
#+BEGIN_SRC sh :results output silent
wget -P /tmp/ https://github.com/djcb/mu-releases/raw/master/mu4e-0.9.16.pdf;
evince /tmp/mu4e-0.9.16.pdf;
#+END_SRC

It felt difficult because I needed to integrate multiple pieces of software while still being a bit uncomfortable with how configuring Emacs itself.

Again it is funny to discover that I had an actual reason to start managing emails from within Emacs other than being a geek: at that time Mozilla ended maintenance for Thunderbird, my previous preferred email client.

Few months later I learned how to maintain my Emacs configuration as an Org Mode file by copying Daniel Mai's approach.

* DONE move to org and use-package
SCHEDULED:<2016-06-21 Tue>
:PROPERTIES:
:ARCHIVE_TIME: 2016-06-21 Tue 01:06
:ARCHIVE_OLPATH: Daily Stuff
:ARCHIVE_TODO: DONE
:END:

More importantly I started using use-package for my dependencies. At that time I introduced use-package mainly because of its clean syntax. Only much later I learned how to use it to load packages lazily and instrument my configuration to find out initialization problems. By the way, that time must have also been when I started preferring the SCHEDULED timestamp format to Org raw dates.

From then on I started installing programming environments within Emacs, like Ensime, Tuareg and Cider. I also bugged a significant amount of times Pandoc's maintainers to support all the features I needed to export my thesis written in Org Mode via the amazing ox-pandoc.

All in all, Emacs was a majestic investment: I basically literate-programmed away my Ph.D. activities via Org Mode and completed the numerous todos with all the extensions Emacs support.

If you are a new user, a possible checklist to familiarize with Emacs is:

1. learn the basics keybindings with the Emacs tutorial C-h t
2. start using Org Mode to keep track of your todos
3. explore how to extend your Emacs to fulfill your needs, where good resources for this are

Again I wish my beginnings (about 3 years actually!) with Emacs let you get that learning this tool is a process. So no rush and enjoy each struggle (and get in touch with the lovely community please)!

Happy enlightening!

## Footnotes:

1

It will take longer or shorter for you, it does not matter: is a joy not a challenge, so take all the time you need.

2

Later I found out even my mother used Emacs for composing her dissertation, but she moved on as soon as she typed the last dissertation character.

3

When is a minimal tool to keep track of your calendar, and, until now, I totally forgot about it.

format: plain


text:
Pattern Recognition from One Example by Chopping
François Fleuret and Gilles Blanchard
In: NIPS 2005, 5-8 Dec 2005, Vancouver, Canada.

## Abstract

We investigate the learning of the appearance of an object from a single image of it. Instead of using a large number of pictures of the object to recognize, we use a labeled reference database of pictures of other objects to learn invariance to noise and variations in pose and illumination. This acquired knowledge is then used to predict if two pictures of new objects, which do not appear on the training pictures, actually display the same object. We propose a generic scheme called {\it chopping} to address this task. It relies on hundreds of random binary splits of the training set chosen to keep together the images of any given object. Those splits are extended to the complete image space with a simple learning algorithm. Given two images, the responses of the split predictors are combined with a Bayesian rule into a posterior probability of similarity. Experiments with the COIL-100 database and with a database of $150$ degraded \LaTeX{} symbols compare our method to a classical learning with several examples of the positive class and to a direct learning of the similarity.


text:
# Find Research outputs

Search in all content

## Filters for Research & Scholarship

 Search concepts Selected Filters Publication Year 2019 2018 2016 2014 Author Mark Neubauer
• 1 - 50 out of 410 results
• Export search results

## Search results

• ### Z Z → + - ′ + ′ - s-section measurements and search for anomalous triple gauge couplings in 13 TeV p p collisions with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Transverse momentum, rapidity, and centrality dependence of inclusive charged-particle production in sNN=5.02 TeV p + Pb collisions measured by the ATLAS experiment

The ATLAS Collaboration , Dec 10 2016, 763, p. 313-336 24 p.

Research output: Contribution to journalArticlepeer-review

• ### The performance of the jet trigger for the ATLAS detector during 2011 data taking

The ATLAS Collaboration, Oct 1 2016, 76, 10, 526.

Research output: Contribution to journalArticlepeer-review

• ### The differential production cross section of the ϕ(1020) meson in √s = 7 tev pp collisions measured with the atlas detector

The ATLAS Collaboration , Jul 1 2014, 74, 7, 2895.

Research output: Contribution to journalArticlepeer-review

• ### The ATLAS Data acquisition and high level trigger system

Research output: Contribution to journalReview articlepeer-review

Open Access
• ### Test of CP invariance in vector-boson fusion production of the Higgs boson using the Optimal Observable method in the ditau decay channel with the ATLAS detector

The ATLAS Collaboration, Dec 1 2016, 76, 12, 658.

Research output: Contribution to journalArticlepeer-review

• ### Supporting High-Performance and High-Throughput Computing for Experimental Science

Huerta, E. A., Haas, R., Jha, S., Neubauer, M. & Katz, D. S., Dec 2019, 3, 1, 5.

Research output: Contribution to journalReview articlepeer-review

• ### Study of the rare decays of B s 0 and B 0 mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector

The ATLAS Collaboration , Apr 1 2019, 2019, 4, 98.

Research output: Contribution to journalArticlepeer-review

Open Access
• ### Study of the rare decays of Bs0 and B0 into muon pairs from data collected during the LHC Run 1 with the ATLAS detector

The ATLAS Collaboration, Sep 1 2016, 76, 9, 513.

Research output: Contribution to journalArticlepeer-review

• ### Study of the hard double-parton scattering contribution to inclusive four-lepton production in pp collisions at √s=8 TeV with the ATLAS detector

The ATLAS Collaboration , Jan 1 2019, 790, p. 595-614 20 p.

Research output: Contribution to journalArticlepeer-review

Open Access
• ### Study of the Bc+ → J/ψDs+ and Bc+ →J/ψD*s+ decays with the ATLAS detector

The ATLAS Collaboration, Jan 1 2016, 76, 1, p. 1-24 24 p., 4.

Research output: Contribution to journalArticlepeer-review

• ### Study of heavy-flavor quarks produced in association with top-quark pairs at s =7TeV using the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Study of hard double-parton scattering in four-jet events in pp collisions at √s=7 TeV with the ATLAS experiment

The ATLAS Collaboration , Nov 1 2016, 2016, 11, 110.

Research output: Contribution to journalArticlepeer-review

• ### Standalone vertex finding in the ATLAS muon spectrometer

The ATLAS Collaboration , 2014, 9, 2, P02001.

Research output: Contribution to journalArticlepeer-review

• ### Search for W′  →  tb decays in the hadronic final state using pp collisions at √s=13TeV with the ATLAS detector

The ATLAS Collaboration , Jun 10 2018, 781, p. 327-348 22 p.

Research output: Contribution to journalArticlepeer-review

• ### Search for WZ resonances in the fully leptonic channel using pp collisions at s=8 TeV with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for W W/W Z resonance production in ℓνqq final states in pp collisions at √s=13 TeV with the ATLAS detector

The ATLAS Collaboration , Mar 1 2018, 2018, 3, 42.

Research output: Contribution to journalArticlepeer-review

• ### Search for vector-boson resonances decaying to a top quark and bottom quark in the lepton plus jets final state in pp collisions at √s=13TeV with the ATLAS detector

The ATLAS Collaboration , Jan 1 2019, 788, p. 347-370 24 p.

Research output: Contribution to journalArticlepeer-review

• ### Search for top squarks in final states with one isolated lepton, jets, and missing transverse momentum in s =13 TeV pp collisions with the ATLAS detector

(ATLAS Collaboration), Sep 19 2016, 94, 5, 052009.

Research output: Contribution to journalArticlepeer-review

• ### Search for top squarks decaying to tau sleptons in pp collisions at s =13 Te v with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for top-squark pair production in final states with one lepton, jets, and missing transverse momentum using 36 fb−1 of √s=13 TeV pp collision data with the ATLAS detector

The ATLAS Collaboration , Jun 1 2018, 2018, 6, 108.

Research output: Contribution to journalArticlepeer-review

• ### Search for top squark pair production in final states with one isolated lepton, jets, and missing transverse momentum in √ s = 8 TeV pp collisions with the ATLAS detector

The ATLAS Collaboration, Nov 1 2014, 2014, 11, 118.

Research output: Contribution to journalArticlepeer-review

• ### Search for top quark decays t → qH with H → γγ using the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for top-quark decays t → Hq with 36 fb−1 of pp collision data at √s = 13 TeV with the ATLAS detector

The ATLAS Collaboration , May 21 2019, 2019, 5, 123.

Research output: Contribution to journalArticlepeer-review

Open Access
• ### Search for the standard model Higgs boson produced in association with top quarks and decaying into a bb¯ pair in pp collisions at √s=13 TeV with the ATLAS detector

(ATLAS Collaboration), Apr 1 2018, 97, 7, 074015.

Research output: Contribution to journalArticlepeer-review

• ### Search for the standard model Higgs boson produced in association with a vector boson and decaying into a tau pair in pp collisions at s =8 TeV with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for the Standard Model Higgs boson produced by vector-boson fusion and decaying to bottom quarks in √s=8 TeV pp collisions with the ATLAS detector

The ATLAS Collaboration , Nov 1 2016, 2016, 11, 112.

Research output: Contribution to journalArticlepeer-review

• ### Search for the Standard Model Higgs boson decaying into bb¯ produced in association with top quarks decaying hadronically in pp collisions at √s = 8 TeV with the ATLAS detector

The ATLAS Collaboration, May 1 2016, 2016, 5, 160.

Research output: Contribution to journalArticlepeer-review

• ### Search for the production of single vector-like and excited quarks in the Wt final state in pp collisions at √s = 8 TeV with the ATLAS detector

The ATLAS Collaboration, Feb 1 2016, 2016, 2, p. 1-46 46 p., 110.

Research output: Contribution to journalArticlepeer-review

• ### Search for the Production of a Long-Lived Neutral Particle Decaying within the ATLAS Hadronic Calorimeter in Association with a Z Boson from pp Collisions at s =13 TeV

(ATLAS Collaboration), Apr 15 2019, 122, 15, 151801.

Research output: Contribution to journalArticlepeer-review

Open Access
• ### Search for the lepton flavor violating decay Z → eμ in pp collisions at √s = 8 TeV with the ATLAS detector

Atlas Collaboration, Oct 23 2014, 90, 7, 072010.

Research output: Contribution to journalArticlepeer-review

• ### Search for the Higgs boson produced in association with a W boson and decaying to four b-quarks via two spin-zero particles in pp collisions at 13 TeV with the ATLAS detector

Atlas Collaboration, Nov 1 2016, 76, 11, 605.

Research output: Contribution to journalArticlepeer-review

• ### Search for the Higgs boson produced in association with a vector boson and decaying into two spin-zero particles in the H → aa → 4b channel in pp collisions at √s=13 TeV with the ATLAS detector

The ATLAS Collaboration , Oct 1 2018, 2018, 10, 31.

Research output: Contribution to journalArticlepeer-review

• ### Search for the electroweak production of supersymmetric particles in √s = 8 TeV pp collisions with the ATLAS detector

Atlas Collaboration, Mar 4 2016, 93, 5, 052002.

Research output: Contribution to journalArticlepeer-review

• ### Search for the direct production of charginos and neutralinos in final states with tau leptons in √s=13TeV pp collisions with the ATLAS detector

Atlas Collaboration, Feb 1 2018, 78, 2, 154.

Research output: Contribution to journalArticlepeer-review

• ### Search for the direct production of charginos, neutralinos and staus in final states with at least two hadronically decaying taus and missing transverse momentum in pp collisions at √s = 8 TeV with the ATLAS detector

The ATLAS Collaboration , Oct 16 2014, 2014, 10, 96.

Research output: Contribution to journalArticlepeer-review

• ### Search for the Decay of the Higgs Boson to Charm Quarks with the ATLAS Experiment

Atlas Collaboration, May 22 2018, 120, 21, 211802.

Research output: Contribution to journalArticlepeer-review

• ### Search for TeV-scale gravity signatures in high-mass final states with leptons and jets with the ATLAS detector at √s=13 TeV

The ATLAS Collaboration , Sep 10 2016, 760, p. 520-537 18 p.

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in final states with missing transverse momentum and multiple b-jets in proton-proton collisions at √s=13 TeV with the ATLAS detector

The ATLAS Collaboration , Jun 1 2018, 2018, 6, 107.

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in final states with charm jets and missing transverse momentum in 13 TeV pp collisions with the ATLAS detector

The ATLAS Collaboration , Sep 1 2018, 2018, 9, 50.

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in events with large missing transverse momentum, jets, and at least one tau lepton in 20 fb-1 of s \sqrt{s} = 8 TeV proton-proton collision data with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in events with four or more leptons in √s =8 TeV pp collisions with the ATLAS detector

Atlas Collaboration, Sep 4 2014, 90, 5, 052001.

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in events with four or more leptons in √s =13 TeV pp collisions with ATLAS

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry in a final state containing two photons and missing transverse momentum in √s = 13 TeV pp collisions at the LHC using the ATLAS detector

Atlas Collaboration, Sep 1 2016, 76, 9, 517.

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry at √s= 8 TeV in final states with jets and two same-sign leptons or three leptons with the ATLAS detector

Research output: Contribution to journalArticlepeer-review

• ### Search for supersymmetry at √s= 13 TeV in final states with jets and two same-sign leptons or three leptons with the ATLAS detector

The ATLAS Collaboration, May 1 2016, 76, 5, 259.

Research output: Contribution to journalArticlepeer-review

• ### Search for strong production of supersymmetric particles in final states with missing transverse momentum and at least three b-jets at √s = 8 TeV proton-proton collisions with the ATLAS detector

The ATLAS Collaboration , 2014, 2014, 10, 24.

Research output: Contribution to journalArticlepeer-review

• ### Search for strong gravity in multijet final states produced in pp collisions at √s = 13 TeV using the ATLAS detector at the LHC

The ATLAS Collaboration, Mar 7 2016, 2016, 3, 26.

Research output: Contribution to journalArticlepeer-review

• ### Search for squarks and gluinos with the ATLAS detector in final states with jets and missing transverse momentum using √s = 8 TeV proton-proton collision data

The ATLAS Collaboration , Sep 30 2014, 2014, 9, 176.

Research output: Contribution to journalArticlepeer-review

• ### Search for squarks and gluinos in final states with jets and missing transverse momentum using 36 fb-1 of √s =13 TeV pp collision data with the ATLAS detector

Atlas Collaboration, Jun 6 2018, 97, 11, 112001.

Research output: Contribution to journalArticlepeer-review


text:
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# publichadley/scales

### Subversion checkout URL

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file 18 lines (17 sloc) 0.387 kb
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 \name{squish_infinite}\alias{squish_infinite}\title{Squish infinite values to range.}\usage{  squish_infinite(x, range = c(0, 1))}\arguments{  \item{x}{numeric vector of values to manipulate.}  \item{range}{numeric vector of length two giving desired  output range.}}\description{  Squish infinite values to range.}\examples{squish_infinite(c(-Inf, -1, 0, 1, 2, Inf))}
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text:
### - Art Gallery -

In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function. The MFS was developed to overcome the major drawbacks in the boundary element method (BEM) which also uses the fundamental solution to satisfy the governing equation. Consequently, both the MFS and the BEM are of a boundary discretization numerical technique and reduce the computational complexity by one dimensionality and have particular edge over the domain-type numerical techniques such as the finite element and finite volume methods on the solution of infinite domain, thin-walled structures, and inverse problems.

In contrast to the BEM, the MFS avoids the numerical integration of singular fundamental solution and is an inherent meshfree method. The method, however, is compromised by requiring a controversial fictitious boundary outside the physical domain to circumvent the singularity of fundamental solution, which has seriously restricted its applicability to real-world problems. But nevertheless the MFS has been found very competitive to some application areas such as infinite domain problems.

The MFS is also known by different names in the literature, including the charge simulation method, the superposition method, the desingularized method, the indirect boundary element method and the virtual boundary element method.

MFS formulation

Consider a partial differential equation governing certain type of problems

$$Lu=f\left(x,y\right),\ \ \left(x,y\right)\in \Omega ,$$
$$u=g\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{D},$$
$${\displaystyle {\frac {\partial u}{\partial n}}=h\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{N},}$$

where L is the differential partial operator, $$\Omega$$ represents the computational domain, $$\partial \Omega _{D}$$ and $$\partial \Omega _{N}$$ denote the Dirichlet and Neumann boundary, respectively, $$\partial \Omega _{D}\cup \partial \Omega _{N}=\partial \Omega$$ and $$\partial \Omega _{D}\cap \partial \Omega _{N}=\varnothing$$ .

The MFS employs the fundamental solution of the operator as its basis function to represent the approximation of unknown function u as follows

$${{u}^{{*}}}\left(x,y\right)=\sum \limits _{{i=1}}^{N}\alpha _{i}\phi \left(r_{i}\right)$$

where $$r_{i}=\left\|\left(x,y\right)-\left(sx_{i},sy_{i}\right)\right\|$$ denotes the Euclidean distance between collocation points $$\left(x,y\right)$$ and source points $$\left(sx_{i},sy_{i}\right)$$ , $$\phi \left(\cdot \right)$$ is the fundamental solution which satisfies

$$L\phi =\delta \,$$

where $$\delta$$ denotes Dirac delta function, and $${{\alpha }_{{i}}}$$ are the unknown coefficients.

With the source points located outside the physical domain, the MFS avoid the fundamental solution singularity. Substituting the approximation into boundary condition yields the following matrix equation

$$\left[{\begin{matrix}\phi \left(\left.r_{j}\right|_{{x_{i},y_{i}}}\right)\\{\frac {\partial \phi \left(\left.r_{j}\right|_{{x_{k},y_{k}}}\right)}{\partial n}}\\\end{matrix}}\right]\ \cdot \ \alpha =\left({\begin{matrix}g\left(x_{i},y_{i}\right)\\h\left(x_{k},y_{k}\right)\\\end{matrix}}\right),$$

where $$\left(x_{i},y_{i}\right)$$ and $$\left(x_{k},y_{k}\right)$$ denote the collocation points, respectively, on Dirichlet and Neumann boundaries. The unknown coefficients $$\alpha _{i}$$ can uniquely be determined by the above algebraic equation. And then we can evaluate numerical solution at any location in physical domain.
History and recent developments

The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s.[1] However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s,[2] followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.[3][4][5][6]

In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.[7][8] Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients.[9] The MFS has proved particularly effective for certain classes of problems such as inverse,[10] unbounded domain, and free-boundary problems.[11]

Some techniques have been developed to cure the fictitious boundary problem in the MFS, such as the boundary knot method, singular boundary method, and regularized meshless method.

Boundary element method
Boundary knot method
Boundary particle method
Singular boundary method
Regularized meshless method

References

K. VD, A. MA, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput Math Math Phys. 4 (1964) 82–126.
R. Mathon, R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis. (1977) 638–650.
Z. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method[permanent dead link], Computational Mechanics. 44 (2009) 757–763.
W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems Archived 2015-06-06 at the Wayback Machine, Engineering Analysis with Boundary Elements. 35 (2011) 253–257.
W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary Archived 2015-06-06 at the Wayback Machine, Engineering Analysis with Boundary Elements. 34 (2010) 530–532.
JIANG Xin-rong, CHEN Wen, Method of fundamental solution and boundary knot method for helmholtz equations: a comparative study, Chinese Journal of Computational Mechanics, 28:3(2011) 338–344 (in Chinese)
M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61.
M. a. Golberg, C.S. Chen, H. Bowman, H. Power, Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method, Computational Mechanics. 21 (1998) 141–148.
C.M. Fan, C.S. Chen, J. Monroe, The method of fundamental solutions for solving convection-diffusion equations with variable coefficients, Advances in Applied Mathematics and Mechanics. 1 (2009) 215–230
Y.C. Hon, T. Wei, The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. Model. Eng. Sci. 7 (2005) 119–132

A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95.

International Center for Numerical Simulation Software in Engineering & Sciences

Numerical methods for partial differential equations
Finite difference
Parabolic

Forward-time central-space (FTCS) Crank–Nicolson

Hyperbolic

Lax–Friedrichs Lax–Wendroff MacCormack Upwind Method of characteristics

Others

Alternating direction-implicit (ADI) Finite-difference time-domain (FDTD)

Finite volume

Godunov High-resolution Monotonic upstream-centered (MUSCL) Advection upstream-splitting (AUSM) Riemann solver essentially non-oscillatory (ENO) weighted essentially non-oscillatory (WENO)

Finite element

hp-FEM Extended (XFEM) Discontinuous Galerkin (DG) Spectral element (SEM) Mortar Gradient discretisation (GDM) Loubignac iteration Smoothed (S-FEM)

Meshless/Meshfree

Smoothed-particle hydrodynamics (SPH) Moving particle semi-implicit method (MPS) Material point method (MPM) Particle-in-cell (PIC)

Domain decomposition

Schur complement Fictitious domain Schwarz alternating
additive abstract additive Neumann–Dirichlet Neumann–Neumann Poincaré–Steklov operator Balancing (BDD) Balancing by constraints (BDDC) Tearing and interconnect (FETI) FETI-DP

Others

Spectral Pseudospectral (DVR) Method of lines Multigrid Collocation Level-set Boundary element Immersed boundary Analytic element Isogeometric analysis Infinite difference method Infinite element method Galerkin method
Petrov–Galerkin method Validated numerics Computer-assisted proof Integrable algorithm Method of fundamental solutions


text:
# Condensed Matter Physics

## Diffuse scattering

### Small-angle scattering

Apart from characteristic lengths on the atomic scale, diffuse scattering can also reveal nanostructures. The relevant techniques are small-angle neutron scattering (SANS), small-angle x-ray scattering (SAXS) and static light scattering (SLS), depending on whether neutrons, x-rays or visible light is used as the probing beam.

Waves (electromagnetic or particle) interact with structures of the same size as their wavelength (diffraction) as well as larger structures (diffuse scattering). The Fig. indicates what kind of structures are accessible (the area above the lines for photons or neutrons, respectively) for a given type of beam. Diffraction and diffuse scattering processes occur simultaneously and we can only distinguish them due to their different angle dependence. Since scattering takes place in reciprocal space, large structures cause scattering at small angles and vice versa given that there is a Fourier relationship between reciprocal and direct space.

Unlike diffraction, which is a result of the constructive interference of waves scattered by periodic structures, diffuse scattering is simply a consequence of the refraction of rays at interfaces of different optical density. If a ray cuts through an interface at right angles, it will not be refracted. In the case of a flat object (e.g. a piece of sheet glass), there will be no refraction either when the ray enters or when it exits the object - the beam goes straight through (provided the object is transparent). If the ray hits the sheet at an angle, it will be refracted, but the deflection at the entrance and exit points just cancel out - a laser ray shone at a window will be offset sideways but the incident and refracted rays are parallel.

Substituting the flat object with a curved one such as a sphere changes this. If the ray hits the sphere at its equator, the incidence is normal and the beam will pass through without any refraction. However, if the ray enters the sphere away from the equator, it will be refracted, and the refracted ray will leave the sphere at a different angle because of its curvature. As a result it will be refracted back by more than it was refracted on entering, leading to a lensing effect. If the sphere is subjected to many parallel rays, this leads to a distribution of angles under which rays leave the sphere. The smaller the sphere, the larger is its curvature and therefore the wider the angular spread.

Therefore, we can measure the size of a scattering object by measuring the angular spread of the scattered rays emerging from the sample. It is important to note that it doesn't matter whether the scattering object has a smaller or larger optical density than its surroundings since small-angle scattering always measures the result of two coupled refraction processes (when the ray enters and when it leaves the scattering object). Therefore, it is impossible to distinguish between particles and pores in the sample.

A SAXS instrument at a synchrotron x-ray source begins with a bending magnet or insertion device in the storage ring, where x-rays are generated by deflection of moving charges in a magnetic field. Since monochromatic radiation is needed in order to be able to relate the momentum transfer, $$q=\frac{4\pi}{\lambda}\sin{\theta}\qquad,$$ to the scattering angle, $\theta$, the beam passes through a monochromator, which selects one particular wavelength (energy) using a very sharp Bragg reflection. The beam is then passed through a collimator, i.e. a system of slits which limits the divergence of the beam. The beam can also be focused on either the sample or the detector using focusing elements such as curved mirrors or Fresnel zone plates. The thus conditioned beam is delivered to the sample position, after which there is a secondary beam tube of variable length with a two-dimensional detector at its end. Given that the size of the detector is fixed (for a given instrument), the sample-detector distance (or camera length) determines the angular range covered as well as the angular resolution.

A typical SAXS pattern recorded using a two-dimensional detector is shown in the Fig. The count rate is highest at the centre, where those rays that are transmitted through the sample without refraction reach the detector. It decreases very rapidly radially outwards from the centre. The detector has to be protected from the intense primary beam using a beam stop, which shows as a round shape with reduced count rate (green circle in the Fig.) surrounding the centre of the pattern.

In most cases, SAXS patterns are radially symmetric - either because the scattering objects themselves are radially symmetric or because they are randomly distributed such that the different orientations are averaged. Therefore, the two-dimensional pattern can be radially averaged, producing a plot of count rate vs. momentum transfer (or scattering angle) such as the blue curve in the Fig. . The centre of the 2D pattern is on the left of the 1D plot and the edges of the detector image correspond to the right side of the 1D pattern. The count rate is colour-coded in the 2D image and shown numerically on the $y$-axis in the 1D plot. Note the logarithmic axes!

Such a reduced SAXS pattern contains essentially two pieces of information about the sample, and they are contained in two different sections of the curve, represented by the two fit lines shown in red. The Guinier regime (A) is the part of the pattern where a transition occurs from a near constant count rate onto a slope which decreases strongly as the scattering angle (momentum transfer) increases. The count rate in this regime can be described by the function $$I(q)\propto\exp{\left(\frac{R_g^2q^2}{3}\right)}\qquad,$$ where $R_g$ is the Guinier radius, which approximates the radius of gyration, i.e. the radius of a sphere in which a scattering object (whatever its shape is) may rotate freely. Further out lies the Porod regime (B), which describes the slope of the curve in terms of a power law (i.e. a straight line in the log-log plot): $$I(q)\propto q^{-n}\qquad.$$ The exponent $n$ characterises the surface morphology of the scattering object: a smooth interface will produce a slope of $n=4$, while less steep slopes indicate an interface that can be described as a fractal structure.

### Sintering, particle growth and agglomeration

Individual exposures in x-ray scattering experiments are quite fast, so it is possible to do time-resolved experiments to investigate processes in situ. The case study is an example of this; it is an investigation in which a ceramic is formed by sintering alumina and zirconia nano-particles into a glass matrix. The three components are fine powders and are pressed into thin pellets, which are mounted in a furnace with windows transparent to x-rays. The furnace is placed on the sample table of the SAXS beamline. The temperature is gradually increased from $\mathrm{350^oC}$ to $\mathrm{725^oC}$ while SAXS patterns are recorded. As the raw data Fig. (top) shows, the Guinier regime shifts from right to left as the temperature rises. This indicates that the particle size is increasing gradually - the Fourier relationship between direct and reciprocal space means that a feature occurring at a smaller angle (momentum transfer) is a larger feature in direct space.

Above $\mathrm{500^oC}$, a second Guinier shoulder appears, which can be interpreted as a second characteristic length scale corresponding to a type of scattering object not observed at lower temperature. This co-incides with the softening point of the glass matrix used in the experiment - the softening point is the temperature at which a glass becomes just runny enough that it can be worked by a glass blower. When plotting the Guinier radii determined from the SAXS patterns, it appears that there is a gradual and very even increase of the primary particle size, with a much faster increase of the size corresponding to this second feature only present at the higher temperatures.

At the highest temperatures, the raw data curves show a maximum near to the small-angle end of the data set (on the left). The Guinier formula doesn't allow for this behaviour as it predicts an asymptotic limit for $q\to 0$. Such as shape is best modelled by a structure factor, i.e. by scattering objects whose positions in the sample aren't random but more regular. The structure factor maximum corresponds to a correlation length, i.e. a typical or average distance between the scattering objects. This is exactly equivalent to the peaks observed in total scattering patterns, except that, owing to the small angle (momentum transfer), much larger length scales than atomic distances are observed.

The aim of any scattering experiment is to derive a structural model of the material (and its evolution during an in-situ experiment). Taking all the evidence together, it is possible to conclude that the sample begins as a loose collection of individual nano-particles mixed up with larger, irregular glass particles. Around the softening point, the glass particles begin to change their shape and become more regular, with less rugged surfaces. As the glass blobs slowly flow and fill up the porosity, the nano-particles get pressed together and form aggregates, which give rise to the second, larger, Guinier feature. At the highest temperatures, the glass blobs join together, and the aggregates mark out the positions of the original triple junctions between adjacent glass particles, giving rise to a fairly regular spacing of these aggregates as reflected in the structure factor.

### Contrast variation: Anomalous x-ray scattering (ASAXS)

T Barnardo, K Høydalsvik, R Winter, CM Martin, GF Clark
In-situ double ASAXS of the sintering and calcination of sol-gel prepared YSZ ceramics
J Phys Chem C 113 (2009) 10021

### Next...

Scattering techniques, in contrast to microscopy, give a statistical picture of the structure and morphology of a material. This has advantages and disadvantages: data quality is usually good because defects and deformations are averaged out and the typical structure is revealed, but since many of the most interesting properties of materials come about because of the presence of defects and deformations, we need ways to characterise these deviations from the ideal structure, too.


text:
# Copyable math formulas

I'm curious whether there is a package that allows to copy math formulas correctly.

For example, this one should highlight like so

but, not like so

And if I copy one, I should get:

∫₃⁵ (x/(1-x²)) dx

If there is no such a package, I'm looking for an any manual way of doing this.

For example, I can make a phantom text to make the selection highlight correct. But I don't know how to get custom text by copying.

• This reference is not what you are asking for: tex.stackexchange.com/questions/233390/… . However, it may be adaptable for what you need, in that it allows you to copy the tex code from a PDF's equation and paste that. May 20, 2020 at 12:43
• Beyond what I have said above, to do what you are asking may not be possible, because there will be some math (symbols, formats) for which an easy inline pasted portrayal is not possible. This unusual answer of mine, tex.stackexchange.com/questions/332012/… , attempts to do almost the opposite of what you are asking, which is to convert a custom math format into equivalent TeX code (rather than converting TeX code into a custom math format) May 20, 2020 at 12:51
• There is the added problem that thecopyable text inside a PDF is does not cover the full unicode spectrum. Example, which one might think would give the desired behavior: \documentclass{article} \usepackage{accsupp} \begin{document} \BeginAccSupp{method=escape,ActualText={∫₃⁵ (x/(1-x²)) dx}} $\int_3^5\frac{x}{1-x^2}\,dx$ \EndAccSupp{} \end{document} When copy paste is applied to the PDF, the following is the result: ^53X03 965 (x/(1-x²)) dx, such that the non-ASCII things do not represent themselves accurately. May 20, 2020 at 14:49
• @StevenB.Segletes I have the latest version of accsupp, but it doesn't work, I still copy the actual text. Do you know why the problem may occur? May 20, 2020 at 18:20
• I don't think the problem is accsupp, but PDF itself. It is limited in what it can copy/paste...while it can draw a Greek letter, for example, on the visual paper, it does not have the means to store it in the underlying information, I think. May 20, 2020 at 19:03

I managed to do that! Thanks for this answer.

\documentclass{article}
\usepackage{fontspec}
\usepackage{accsupp}

\usepackage{tikz}
\usepackage{stringenc}
\usepackage{pdfescape}
\usetikzlibrary{calc}
\makeatletter
\newcommand{\BeginAccSuppUnicode}[1]{%
\EdefSanitize\asu@str{#1}%
\edef\asu@str{%
\expandafter\expandafter\expandafter\asu@ToSpaceOther
\expandafter\asu@str\space\@nil
}%
\expandafter\let\expandafter\asu@str\expandafter\@empty
\expandafter\asu@ToHexUC\asu@str\relax
\EdefUnescapeHex{\asu@str}{\asu@str}%
\StringEncodingConvert{\asu@str}{\asu@str}{utf32be}{utf16be}%
\EdefEscapeHex{\asu@str}{\asu@str}%
\BeginAccSupp{%
unicode,%
method=hex,%
ActualText=\asu@str
}%
}
\begingroup
\lccode\9=\ %
\lowercase{\endgroup
\def\asu@SpaceOther{9}%
}
\def\asu@ToSpaceOther#1 #2\@nil{%
#1%
\ifx\\#2\\%
\expandafter\@gobble
\else
\asu@SpaceOther
\expandafter\@firstofone
\fi
{\asu@ToSpaceOther#2\@nil}%
}
\def\asu@ToHexUC#1{%
\ifx#1\relax
\else
\pgfmathHex{\the\numexpr#1+"10000000\relax}%
\edef\asu@str{%
\asu@str
0\expandafter\@gobble\pgfmathresult
}%
\expandafter\asu@ToHexUC
\fi
}
\makeatother

\begin{document}

\begin{tikzpicture}[remember picture,overlay]
\node[text opacity=0]at ($(current page.north west)+(70.5ex,-37ex)$){
\BeginAccSuppUnicode{∫₃⁵ (x/(1-x²)) d}
\scalebox{14}[7.2]{x\rule{1pt}{0pt}}
\EndAccSupp{}
\hspace{-23pt}\scalebox{0.1}[7.2]{x}
};
\end{tikzpicture}

$\int_3^5\frac{x}{1-x^2}\,dx$

\end{document}

## Edit

Found another solution that works the same. Although it requires you to manualy convert all symbols to unicode, it is way more compact.

\documentclass{article}
\usepackage{accsupp}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[remember picture,overlay]
\node[text opacity=0]at ($(current page.north west)+(70.5ex,-37ex)$){
\BeginAccSupp{method=hex,unicode,ActualText=222b20832075 00280078002f00280031002d007800b200290029 0064}
\scalebox{14}[7.2]{x\rule{1pt}{0pt}}
\EndAccSupp{}
\hspace{-23pt}\scalebox{0.1}[7.2]{x}
};
\end{tikzpicture}
`


text:
# Capillary electrode array/proposal import 2007

The Capillary Electrode Array: A CMOS DNA Separation System

\section{Introduction}

This proposal is motivated by a key problem in molecular biology and biochemistry: starting from a heterogeneous ensemble of biopolymers in an aqueous solution with a homogeneous concentration profile, create an aqueous solution with a highly heterogeneous concentration profile of localized concentrations of certain biopolymer species. %% (Figure \ref{cea_conc_profile}). Although this procedure is important for all of the biological macromolecules in an organism, DNA molecules are the exclusive focus of this report.

%% \begin{figure}[h] %% \begin{center}

%% \includegraphics{cea_homo_conc.eps}

%% \includegraphics{cea_hetero_conc.eps}

%% \end{center} %% \caption{Concentration profiles in separation process: (\textbf{top}) %% Before separation, three species in solution have spatially homogeneous concentration %% profiles. (\textbf{bottom}) After separation, the three species are %% separated and a spatially heterogeneous concentration profile %% exists. } %% \label{cea_conc_profile} %% \end{figure}

Currently, the most widely used technique for DNA analysis is electrophoresis (see, for example,\cite{viovy00}) in which an electric field is applied to a gel or cross-linked polymer solution. Negatively charged DNA molecules migrate through the solution, with the molecules of the separation medium slowing the DNA molecules through mechanical interactions. Longer DNA molecules experience more obstructions than smaller DNA molecules, and move more slowly through the medium.

In a seminal 1991 paper\cite{prost91}, Ajdari and Prost proposed the use of an inhomogeneous electric field, transverse to the direction of flow, for free-flow (i.e. without a gel-based medium) separation of molecules. Their theoretical analysis indicated separation efficiency significantly greater than that of conventional electrophoresis systems.

Here, we propose the use of CMOS microfabrication and maskless postprocessing to create a molecular separation system that operates on the principle described in \cite{prost91}. This device contains a microchannel, an array of electrode pairs surrounding the microchannel, and a microheater element to actuate, by bubble creation, bulk flow through the microchannel. While the bubble actuator creates the hydrodynamic force to drive DNA molecules through the microchannel, the electrode pairs create electric fields perpendicular to the direction of the microchannel flow. In the following, we describe the basic architecture of the proposed system, and outline areas requiring further development for realization of a practical system for preparative DNA separation.

\section{Fabrication}

In the following, our focus will be directed towards the 1.5 micron ABN process offered by AMI Semiconductor. This process provides two levels of polysilicon for MOS transistor gates and capacitors formed between the polysilicon layers, and two layers of aluminum metallization for electrical interconnection between circuit elements. The minimum width of a polysilicon line in this process is 1.5 $\mu$m, thus the description above. This process is significantly behind the cutting edge of CMOS processes, having been introduced in 1987; currently (2004) MOSIS offers a process with a .13$\mu$m polysilicon line width.

A process flow for fabrication of the proposed device is shown in Figure \ref{cea_process_flow}.

\begin{figure}[h] $\xymatrix{ *+[F]{ \txt{CMOS fabrication} } \ar[d] \\ *+[F]{ \txt{XeF}_2 \txt{ etching} } \ar[d] \\ *+[F]{ \txt{Wafer scribing to expose microchannel}} \ar[d] \\ *+[F]{ \txt{Fluidic connection to microchannel inlet,\\ electrical connections to IC pads}}  }$ \caption{Process sequence for capillary electrode array fabrication.} \label{cea_process_flow} \end{figure}

A cross-section view of the microchannel before and after processing is shown in Figure \ref{cea_cross_section}. Prior to etching with XeF$_2$, a polysilicon-2 minimum-width line has polysilicon-1 and metal-1 electrodes below and above, respectively.

\begin{figure}[h] \begin{center}

\includegraphics[scale=.8]{cea_cross_section.eps}

\end{center} \caption{Microchannel cross section: (\textbf{A}) A

 polysilicon-2 line has electrodes below (polysilicon-1) and above
(metal-1).  (\textbf{B}) After etching with XeF$_2$, the
polysilicon-2 line is removed to form a microchannel.}

\label{cea_cross_section} \end{figure}

Figure \ref{cea_layout_annotated} shows 10 electrode pairs. Each poly-1 electrode is 1.6$\mu$m wide, and each metal-1 electrode is 2.4$\mu$m wide. The spacing between the metal-1 electrodes is 1.6$\mu$m, for a center-to-center electrode spacing of 4$\mu$m.

\begin{figure}[h] \begin{center}

\includegraphics[scale=.8]{cea_layout_annotated.epsi}

\end{center} \caption{Plan view of 10 electrode pairs.} \label{cea_layout_annotated} \end{figure}

The fabrication of the microchannel opening is shown in Figure \ref{cea_opening_cross_section}.

\begin{figure}[h] \begin{center}

\includegraphics[scale=.8]{cea_opening_cross_section.eps}

\end{center} \caption{Microchannel opening cross section (not to scale):

 (\textbf{A}) Polysilicon-2 is exposed to the atmosphere, with a
polysilicon-1 electrode beneath. (\textbf{B}) After etching with
XeF$_2$, the polysilicon-2 line is etched to form the opening to the
microchannel.  }

\label{cea_opening_cross_section} \end{figure}

%% CEA as practical realization of \cite{prost91} dielectric traps

%% bulk flow instead of electrophoretic potential difference \cite{dovichi1999,dovichi2002,yamamoto2000}

A crucial difference between the system analyzed in \cite{prost91} and the device proposed here is the use of bubble-actuated bulk flow instead of an electrophoretic potential to drive molecules through the microchannel. This mechanism is conceptually very similar to that of a bubble-driven inkjet print head, and has been demonstrated in a CMOS-fabricated microsystem \cite{westberg97}. DNA microarray formation using a standard bubble-jet print head has been demonstrated \cite{yamamoto2000}, and it is expected that the mechanism in the proposed system will not damage DNA molecules. One concern is the effect of heating on separation efficiency. An increase in temperature will have the effect of broadening the concentration profile, countering the effect of the separation process.

%% It is expected that this bulk flow will enable the proposed device to %% be used in preparative separation of DNA molecules, by ejecting %% droplets containing purified DNA solutions.

%% \cite{kim96}

\bibliography{cea_proposal-2006-10}

\end{document}

target audience is people familiar with MEMS design and fabrication

mathematical analysis of separation performance not necessary for

  initial proposal

what are the performance requirements for circuitry

 process limitations
straight microchannel -> interelectrode impedance not highly variable
fractal microchannel -> highly variable interelectrode impedance

first present straight microchannel with electrode array

 perhaps not even necessary to go into fractal geometry in first
proposal iteration
fractal microchannel geometry sure to confuse people more :)

specific steps toward realization

 how much circuit design is necessary to justify fabrication?
preliminary device to evaluate microchannel etch clearance?
impedance sensor for evaluating etch clearance / molecular presence?
sensitivity of CMOS impedance sensor - single DNA molecule
detection not possible
difficulty of designing!

technologies required

 CMOS
micron-scale - open layer for simple design
submicron - CMU MEMS a possibility for exposing polysilicon for
etching
reducing complexity / risk of first realizations is important -
minimize process steps
internal fab data would be nice for understanding design... not
necessary
XeF2
bubble actuator
simulation

motivation

 separation in free solution
dispensing unmodified DNA molecules
ideal: printing DNA microarray from DNA solution
preparative / analytical separation

methods

 CMOS microfabrication / micromachining
dielectrophoresis
ratchets
bubble-jet printing mechanism

issues

 addressing complexity of electrokinetic phenomena in CEA
experimental investigation of separation phenomena - too complex for
simulation-based estimates?
relationship between DNA length and channel dimensions / separation
efficiency
clearance of microchannel - long etch times, SiN etched

DNAjet printer: DNA molecules printed on standard surfaces

first step: provide essential insights to enable collaboration with others

 necessary to collaborate with people who can help with system integration
delivering DNAjet printer requires significant effort

essential insights:

 CMOS microfabrication / postprocessing
importance of CMOS economics
importance of CMOS circuitry for complex field generation
transverse inhomogeneous field for separation
DNA separation dynamics

microsystems for CE known to MEMS practitioners dielectrophoresis, transverse-field separation not widely known or used

leverage CMOS microfabrication and novel separation mechanisms to

create DNA processing system with unprecedented(?) capabilities

 dream: integrated upstream system isolates DNA from crude biofluids
for CEA preparation - DNA microarray or integrated high speed sequencing downstream

three dimensional CMOS integration for microfluidics as well as

microelectronics

rotating microfabricated system for molecular separation

 drop-in replacement for DVD, 3D patterned on nanoscale for molecular
information system, molecular analysis system

electromagnetic phenomena in rotating microfabricated systems

potentially relevant to new RF systems...

microantenna array from rotating microscale elements?
DVD player as antenna system? :)

'People always overestimate how much will change in the next three years, and they underestimate how much will change over the next 10 years.'

CEA an incremental step

visual representation difficult in preliminary report

 requires solid modeling
solid model of representative electrode pair / microchannel

representation of electric field in microchannel also difficult

 fractal geometry of microchannel means single electrode pair can create e-field at several points on 'microchannel coordinate'

bubble-jet mechanism

 too complex to satisfactorily address in preliminary effort - ANSYS, other simulation?
key for preparative separation
CMOS-compatible

fractal microchannel geometry

 maximizes length
complex e-field within length of microchannel
manufacturing: XeF2 not perfectly selective
stresses in thin-film layers

charge transport in microchannel

 diffusion
vertical electrophoresis
dependent on switching frequency

molecular transport mechanisms

 vertical electrophoresis
(laminar) bulk flow from bubblejet
diffusion
dielectrophoresis
molecular interaction with microchannel inner surface

\section{Introduction}

This report is focused on an operation with great importance in molecular biology and analytical biochemistry: starting from a heterogeneous ensemble of biopolymers in an aqueous solution with a homogeneous concentration profile, create an aqueous solution with a highly heterogeneous concentration profile of localized concentrations of certain biopolymer species (figure \ref{concentration_profile}). Although this procedure is important for all of the biological macromolecules in an organism, DNA molecules are the exclusive focus of this report.

Current systems for DNA separation require many copies of identical DNA molecules for analysis. However, it is not necessary to have multiple copies of a DNA molecule to characterize the base sequence of the DNA molecule. The polymerase chain reaction provides a straightforward means of creating an essentially unlimited number of copies of a single DNA molecule, making it possible to use current sequencing techniques and PCR to analyze a single isolated DNA molecule. The motivation for analyzing individual or small numbers of DNA molecules is twofold. Single copies of locally unique DNA molecules can contain sequence structure that determines the morphology or fate of a cell or organism (as in the case of an antibody producing B- or T-cell in which DNA rearrangement produces antibody diversity, or a cancerous cell proliferating out of control due to a mutated chromosomal DNA molecule). Secondly, in some cases, such as forensic applications, the amount of DNA present in the orginal sample may be minimal, justifying a process for dealing with small numbers of DNA molecules.

One approach to creating a device that can manipulate single DNA molecules is to use microfabrication \cite{madou97}, which generally refers to the creation of structures with dimensions on the order of a micron or less. Microfabrication is an extremely important technology for our society, and is most notably used to create integrated circuits (ICs) for computer and communications systems.

Generally, microscale devices with mechanical (including fluidic) structures and electrical inputs or outputs are called microelectromechanical systems (MEMS). Many MEMS devices are fabricated using custom processes that are tailored to the production of a specific device. Any microfabrication process typically has high initial costs for startup of the manufacturing process \cite{sze96}. In a custom process with low manufacturing volume, the costs are amortized over fewer devices, increasing the per-unit costs. Additionally, microfabrication processes are typically perfected with extensive use, and a low-volume custom process does not benefit from the gradual improvements seen in high-volume microfabrication processes. The use of standard processes that are adapted to the production of a novel device can improve the manufacturability and economic attractiveness of such a device. Additionally, the use for MEMS of processes that enable the creation of transistor-based circuitry makes it possible to integrate control, sensing, and actuation functions on a single chip.

The primary focus of the IC industry's effort is the fabrication of digital and analog complementary metal-oxide-semiconductor (CMOS) integrated circuits, and CMOS IC technology has surpassed all others in miniaturization and cost of fabrication. In addition to providing packaged chips based on designs created in-house, many CMOS IC manufacturers provide fabrication services for designs that are created by their customers. This service dramatically reduces the resources necessary to create new devices using CMOS IC technology. The competitive market for IC devices and IC fabrication services makes it necessary for IC producers to rapidly improve their technology or risk business failure. This economic imperative for continuous improvement of CMOS IC technology has created a rapid increase over time in the number of transistors on a chip, and a rapid decrease in the size and cost of individual transistors in an IC.

CMOS fabrication technology is optimized for the creation of transistors and metal interconnect between transistors. There is considerable flexiblity in the patterning of the thin films on the surface of a silicon wafer that compose the integrated circuit itself, although there is generally no flexibility at all regarding the thickness of the layers. Carefully designed thin film patterns on a CMOS chip can be coupled with processing performed on the chip after fabrication of circuitry to provide the chip with new functionality. Selective removal by etching of certain thin film layers (so-called \emph{sacrificial} layers) in the integrated circuit can create suspended structures and holes in the stack of materials deposited in the fabrication of an integrated circuit. With careful design, these structures can be used to create sensors (to link mechanical inputs to electrical outputs) and actuators (to link electrical inputs with mechanical outputs).

For the purpose of DNA manipulation and analysis, many devices within the CMOS process can be used. Electromagnetic fields created by voltages placed across electrodes can selectively effect the movement of negatively-charged DNA molecules. The impedance between electrodes, which is affected by the DNA molecules and other ions present, can be sensed using on-chip circuitry as a means of analyzing the composition of the solution between the electrodes \cite{ling95}. Resistive heating within on-chip resistors can be used to melt double-stranded DNA, and can vaporize fluid to effect bulk flow of a solution through a microchannel (in a process similar to bubble-jet printers). The temperature dependence of some electrical properties in CMOS devices, such as the resistivity of polysilicon and certain transistor properties, can be used to sense the temperature on the chip. Photodiodes integrated on-chip can be used to sense photons emitted by fluorophores that interact with DNA molecules, and thereby sense the presence of DNA molecules.

In the following, some prior work using novel schemes to electrically separate DNA molecules, and novel microfabricated devices to separate DNA molecules is described. We then describe a new microchannel structure with fractal geometry, fabricated in CMOS, possessing paired electrodes about the body of the microchannel. We sketch a possible application of the device for DNA separation and microarray printing, and propose future theoretical and experimental analyses to verify the utility of the device.

In the following, we present a brief description of

All complementary metal-oxide-semiconductor integrated circuit fabrication processes enable the creation of MOS transistors with positive charge carriers (p-channel, or pMOS) and negative charge carriers (n-channel, or nMOS). The primary advantage of CMOS is the fact that logic elements draw significant current only during transitions from one state to another, and very little current between transitions. This has lead to the overwhelming, and increasing, dominance of CMOS for digital integrated circuit design \cite{rabaey96}. The substantial cost savings that are realized by integrating several functions of an electronic system on a single chip, along with the continuing rapid progress of CMOS technology \cite{sze00}, has motivated the increasing popularity of CMOS for analog integrated circuit design \cite{gray93,razavi01} as well.

A basic circuit in CMOS digital systems is the inverter. Figure \ref{cmos_inverter_schematic} shows a schematic view of a n-channel MOS (abbreviated nMOS) transistor and a p-channel MOS (abbreviated pMOS) transistor connected to form a CMOS inverter.

\begin{figure}[h] \begin{center}

\includegraphics{cmos_inverter_schematic.eps}

\end{center} \caption{A schematic view of the CMOS inverter.} \label{cmos_inverter_schematic} \end{figure}

Figure \ref{cmos_inverter_plan_view} shows a plan view of the layers composing the CMOS inverter. This is the representation of the components of the integrated circuit that is used for layout of the IC. The size of the features on the individual layers, and the placement of different layers relative to one another, is dictated by the \emph{design rules} for the process, which are dependent on the dimensions and tolerances achievable in the process. The design rules are a key part of the well-defined interface between design and fabrication of CMOS integrated circuits \cite{mead80,weste92,pucknell92}. The MOSIS Scalable CMOS (SCMOS) \cite{MOSIS:SCMOS} design rules are intended to be technology and process independent (between CMOS processes), with a scaling factor $\lambda$ used to parameterize the design rules. This makes it possible to use different fabrication processes to realize a particular design, simply by changing the scaling factor to reflect the dimensions achievable in the process used. The layers necessary for specification of an integrated circuit design are generally standard between different processes, due to the basic structure of the MOSFET and the interconnections present in an integrated circuit.

While the two-dimensional shape of the layers in the process can be controlled by the designer, the parameters of the process, such as the sequence of the process steps and the thickness of the layers, are set by the manufacturer, and are not adjustable by the designer. This is particularly relevant for micromachined transducers fabricated in a CMOS process, since the layer thicknesses are rarely optimized for the creation of sensors and actuators. Additionally, the technical details of a process, such as the layer thicknesses, are typically treated by CMOS manufacturers as proprietary secrets that are shared with outside parties either under a restrictive non-disclosure agreement or not at all.

In the following, our focus will be directed towards the 1.5 micron ABN process offered by AMI Semiconductor. This process provides two levels of polysilicon for MOS transistor gates and capacitors formed between the polysilicon layers, and two layers of aluminum metallization for electrical interconnection between circuit elements. The minimum width of a polysilicon line in this process is 1.5 $\mu$m, thus the description above. This process is significantly behind the cutting edge of CMOS processes, having been introduced in 1987; currently (2001) MOSIS offers a process with a .18$\mu$m polysilicon line width.

Figure \ref{cmos_inverter_fabrication_cross_section} shows a cross-section view of the thin film layers comprising the CMOS inverter, realized in the ABN process. In this process, as in virtually all modern CMOS processes, the transistor is formed using a polysilicon gate, with the gate serving as a block for the dopants that form the drain and source regions of the transistor.

The irregular topography of the thin-film stack in figure \ref{cmos_inverter_fabrication_cross_section} limits the resolution achievable during photolithography steps, relative to a flat surface. Modern submicron CMOS processes use a dual damascene process to create planarized surfaces for metallization, as shown in figure \ref{dual_damascene}. In the figure, tungsten is used to fill the via that provides interlayer metal connection, as is common in modern CMOS processes. The dual damascene process avoids the use of potentially troublesome metal etch processes, and instead shapes metal interconnections using oxide trenches and removes excess metal in a chemically-assisted mechanical polishing process.

The metallization schemes for submicron CMOS pose difficulties for MEMS fabrication. The selective etching of certain layers in a CMOS process to create microsystems is dependent on the exposure of the layers to the atmosphere and thereby the wet or dry etchant used. The use of planarization for metal layers makes all but the top layer of metal inaccessible in this processing method. This is in contrast to the AMI ABN process (and generally other processes with feature size larger than 1 micron) which has a non-planar surface and layers that can be exposed to the atmosphere, with an appropriate design. Additionally, the use of tungsten, with its unique etch chemistry, complicates the processing necessary to realize MEMS using CMOS, which has focused on well-known Al, silicon, and silcon oxide etchants.

The central feature of the current design is the microchannel to be formed by sacrificial etching. The microchannel is formed by etching a material within the CMOS thin-film stack, with the material in the shape of a line, coated with silicon oxide along its length, and exposed to the etchant at its ends. The etchant used to create the microchannel must be extremely selective, because of the extreme difference between the width and the length of the microchannel.

CMOS-compatible microchannels have been formed by etching aluminum with an aluminum-specific wet etchant \cite{baltes97_1}. This technique has been used to create a resonant flow sensor \cite{baltes99_1} and a CMOS-compatible droplet ejection system \cite{westberg97}. Aluminum etching is not suited for this application, for two reasons. Due to its low resistivity, aluminum should be used, where possible, for the microelectrodes that create electromagnetic fields within the volume of the microchannel, and for signal transmission within the circuitry on the chip. In addition, wet aluminum etchants, despite their selectivity, etch silicon dioxide at a measurable rate. The microchannel in the current design is an order of magnitude longer than other CMOS-compatible microchannels. This requires a longer etch time to clear the microchannel, resulting in more oxide etching and potential damage to the circuitry that is protected by the oxide passivation.

Polysilicon is another material that can be sacrificially etched in a CMOS process to create microchannels, using xenon difluoride \cite{kim96}. XeF$_2$, which is a white odorless solid at standard temperature and pressure, sublimates at XeF$_2$ vapor rapidly etches mono- and poly-crystalline silicon (at up to 10 $\mu$ per minute \cite{williams96}), but does not measurably etch silicon oxide, aluminum, or photoresist. XeF$_2$ is known to etch titanium and tungsten, two components of many modern metallization processes; however, since these metals are not present in the 1.5 micron AMI ABN process, this issue will not be addressed here.

\section{DNA separation}

In order to effect the transport of electrically charged molecules in controlled way, electric fields can be imposed on a solution, in a process called \emph{electrophoresis} \cite{viovy00} (phoresis, Greek for force). Although this effect occurs with any charged molecule, by far the most widespread application of electrophoresis in analytical chemistry is the separation of DNA.

The most-widely used DNA electrophoresis approach uses a static electric field to force DNA molecules to migrate through separation medium consisting of a cross-linked polymer solution or gel. The constituents of the separation medium act as obstacles, impeding the motion of the DNA molecules (one analogy is the gravitational transport of a rope through a close-packed array of posts extending horizontally from a vertical surface). In this process, longer DNA molecules experience more hindrance to their movement from the gel, and have a lower mobility in the gel than smaller DNA molecules. The mobility $\mu$ of a molecule undergoing electrophoresis is $\bf{V} = \mu \bf{E}$, where $\bf{V}$ is the time-averaged velocity of the molecule, and $\bf{E}$ is the electric field in the solution. A medium is necessary for the electrophoretic separation of DNA in a constant field, because the mobility of DNA molecules in solution is independent of size.

The use of a polymer solution for separation of DNA molecules is less than ideal. The complex structure of the medium makes nearly impossible the development of a quantitative model for predicting molecular dynamics. Although computer simulations have been successful in describing the dynamics of gel electrophoresis of DNA molecules, the results provide only a qualitative understanding of the process. The practical impossibility of a precise model of the gel structure prevents even computer simulations from being used in a predictive manner.

On the other hand, the dynamics of polyelectrolytes in free solution, while still quite complicated, can be much more accurately modeled using hydrodynamic concepts, treating the aqueous solution as a continuum governed by the Navier-Stokes equation.

Several proposals have been made for electrophoretic separation of DNA molecules in solution using asymmetric time-varying electric fields. Called Brownian ratchets or stochastic ratchets\cite{slater97a,slater97b}, these schemes use the Brownian motion of DNA molecules over potential barriers to effect size-dependent transport.

It is well-known that an electric dipole experiences a net force in a nonuniform electric field. This force on a molecule or particle (such as a cell) can be used to effect movement of the object. This process is called dielectrophoresis \cite{pohl78,jones95}.

In a seminal 1991 paper \cite{prost91}, Ajdari and Prost proposed the use of an inhomogeneous electric field, transverse to the direction of flow, for free-flow (i.e. without a gel-based medium) separation of molecules.

It has been shown experimentally that DNA molecules in aqueous solution can be manipulated with microfabricated electrodes \cite{washizu90,washizu95,engh98}. The .

Electrophoresis is the most widely used method for separating DNA molecules by size. In this process, an electric field is applied to a gel or cross-linked polymer solution. Negatively charged DNA molecules

 preparative separation of DNA

what is the current state-of-the-art?

 electrophoresis through viscous medium
microfabricated structures for electrophoresis

what is the key conceptual or technological contribution of this work?

 implementation of novel physical process for separation
use of multiple control elements available in CMOS for separation

 proposal contains, at least, description of device and manufacturing process

what is the plan for success?

 fabrication, experimental process to determine functionality

instead of wafer scribing to expose dielectric tube, use opening on top of tube for XeF2 exposure and fluid discharge

 invert chip for more effective droplet ejection
microscale fluids less affected by gravitational forces
back-side fluid input

somehow use afm nanooxidation technique of \cite{boisen2003} in conjunction with cea to efficiently fabricate dna alumina nanoarrays

how to integrate afm tip with CMOS CEA


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# GRE Math

Reza

## Full-Length GRE Math Practice Test-Answers and Explanations

Did you take the GRE Math Practice Test? If so, then it’s time to review your results to see where you went wrong and what areas you need to improve. GRE Math Practice Test Answers and Explanations Section 1 1- Choice A is correctprime factoring of 77 is: 7 × 11, prime factoring of 136 is: 2 × 2 × 2 × 17, Quantity A = 7 and Quantity B = 2, Quantity A is greater. 2- Choice A is correct$$4\%$$ of $$= 6\%$$ of $$y → 0.04 x = 0.06 y→x=\frac{0.06}{0.04}y→x=\frac{6}{4} y=\frac{3}{2} y=1.5y$$, therefore, $$x$$ is bigger than $$y$$. 3- Choice A is correct.Choose different...


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# implicit differentiation of exponentials containing x and y

So I'm trying to differentiate an equation using implicit differentiation.

I start with $e^{x/y} = 7x - y$

Now the left side of the eqn is where I'm having trouble.

I tried to use differentiation rules for exponentials, but this is incorrect.

Here's what I tried though:

$$(e^x)^{1/y} \ln(e^x) y' = 7 - y'$$

simplfied to:

$$x(e^x)^{1/y} \cdot y' = RS$$

-
Try taking $\ln$ of both sides. –  Baby Dragon Jul 8 '13 at 4:06

## migrated from mathoverflow.netJul 8 '13 at 3:55

This question came from our site for professional mathematicians.

I think the key thing to remember is: $y$ is a function of $x$, so $xy^{-1}$ is a function of $x$, call it $u(x)$: $u(x) = xy^{-1}$. Now the derivative of $e^{u(x)}$ is $e^{u(x)}u'(x)$, so we need $u'(x) = (xy^{-1})' = y^{-1} - xy^{-2}y'$. Thus

$e^{xy^{-1}}(y^{-1} - xy^{-2}y') = 7 - y'$,

an equation which is linear in $y'$, for which we can solve using some simple algebra:

$(1 - e^{xy^{-1}}xy^{-2})y' = 7 - e^{xy^{-1}}y^{-1}$,

or

$y' = \frac{(7 - e^{xy^{-1}}y^{-1})}{(1- e^{xy^{-1}}xy^{-2})} = \frac{(7y^2 - ye^{xy^{-1}})}{(y^2- xe^{xy^{-1}})}$,

which is about as far as we can go without knowing $y(x)$. Of course it should be remembered that, in deriving this formula, we have assumed that $y(x) \ne 0$ is a differentiable function of $x$.

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I deleted and thne re-posted this answer since I accidentally made the previous vs. community wiki, and couldn't figure out how to turn CW off. Sorry, thanks, and enjoy the post! –  1950Robert Lewis Jul 8 '13 at 17:36


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# Math Help - Limit

1. ## Limit

Hi!
How do I show that
$\lim_{n\to\infty}\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx =1$ ?

I've proved that
$\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx=\frac{\sqrt{\pi}}{n} \frac{\Gamma(\frac{1}{n})}{\Gamma(\frac{1}{n}+\fra c{1}{2})}$

Thanks!

2. Originally Posted by DavidEriksson
Hi!
How do I show that
$\lim_{n\to\infty}\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx =1$ ?

I've proved that
$\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx=\frac{\sqrt{\pi}}{n} \frac{\Gamma(\frac{1}{n})}{\Gamma(\frac{1}{n}+\fra c{1}{2})}$

Thanks!
It is fact that $2^{2x-1} \Gamma(x + \frac{1}{2} ) = \frac{ \sqrt{\pi}\Gamma(2x) }{\Gamma(x)}$ Try to use this to finish it .


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#### DOCUMENTA MATHEMATICA, Vol. 17 (2012), 783-805

A. D'Andrea and G. Marchei

A Root Space Decomposition for Finite Vertex Algebras

Let $L$ be a Lie pseudoalgebra, $a \in L$. We show that, if $a$ generates a (finite) solvable subalgebra $S = \langle a \rangle \subset L$, then one may find a lifting $\bar a \in S$ of $[a] \in S/S'$ such that $\langle \bar a \rangle$ is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra $V$ admits a decomposition into a semi-direct product $V = U \sd N$, where $U$ is a subalgebra of $V$ whose underlying Lie conformal algebra $U^\lie$ is a nilpotent self-normalizing subalgebra of $V^\lie$, and $N = V^{[\infty]}$ is a canonically determined ideal contained in the nilradical $\Nil V$.

2010 Mathematics Subject Classification: 17B69

Keywords and Phrases: Pseudoalgebra, vertex algebra

Full text: dvi.gz 43 k, dvi 102 k, ps.gz 294 k, pdf 222 k.


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# An Example of a Nested Decreasing Sequence of Bounded Closed Sets with Empty Intersection

Could someone provide me with an example of a metric space having a nested decreasing sequence of bounded closed sets with empty intersection? I first thought of Cantor set but the intersection is not empty!

• Can you think of sets which are bounded and closed, but not compact? Because if you ever include a compact set in your sequence, Cantor's intersection theorem will imply there is a non-empty intersection. (As is noted in an answer, we can't do this in $\mathbb R$ or $\mathbb R^n$ due to the Heine-Borel theorem) – Milo Brandt Aug 15 '15 at 3:49
• @MiloBrandt: Your comment was indeed what I was hinting at in my answer. – user21820 Aug 15 '15 at 11:02

Let $\mathbb N$ be endowed with the discrete metric. In this metric space, every subset is bounded (although not necessarily totally bounded) and closed. Moreover, the subsets \begin{align*} A_1\equiv&\,\{1,2,3,4,\ldots\},\\ A_2\equiv&\,\{\phantom{1,\,}2,3,4,\ldots\},\\ A_3\equiv&\,\{\phantom{1,2,\,}3,4,\ldots\},\\ \vdots&\, \end{align*} are nested, and their intersection is empty.

However, if you stay within the realm of $\mathbb R$ endowed with the usual Euclidean metric, than you can't have a situation like the one above:

Claim: Suppose that $$A_1\supseteq A_2\supseteq A_3\supseteq\ldots$$ is a countable family of non-empty, closed, bounded subsets of $\mathbb R$. Then, $\bigcap_{n=1}^{\infty} A_n\neq\varnothing$.

Proof: By the Heine–Borel theorem, $A_n$ is compact for each $n\in\mathbb N$. For the sake of contradiction, suppose that $\bigcap_{n=1}^{\infty} A_n=\varnothing$. This is equivalent to $\bigcup_{n=1}^{\infty} A_n^{\mathsf c}=\mathbb R$. In particular, $$A_1\subseteq\bigcup_{n=1}^{\infty} A_n^{\mathsf c}.$$ Since $A_1$ is compact and the sets $(A_n^{\mathsf c})_{n=1}^{\infty}$ form an open cover of it, there must exist a finite subcover. That is, there exists some $m\in\mathbb N$ such that $$A_1\subseteq\bigcup_{n=1}^m A_n^{\mathsf c}=A_m^{\mathsf c},$$ where the second equality follows from the fact that $$A_1^{\mathsf c}\subseteq A_2^{\mathsf c}\subseteq A_3^{\mathsf c}\subseteq\ldots.$$ Now, $A_1\subseteq A_m^{\mathsf c}$ means that if a point is in $A_1$, then it must not be in $A_m$, so that $A_1\cap A_m=\varnothing$. But $A_m\subseteq A_1$ (given that the sets are nested), so that $A_1\cap A_m= A_m=\varnothing$, which contradicts the assumption that $A_m$ is not empty. This contradiction reveals that the intersection $\bigcap_{n=1}^{\infty} A_n$ must not be empty. $\quad\blacksquare$

• Are these sets bounded? What I see is they are bounded from below not from above. Could You please explain why they are bounded? – Fabian Aug 15 '15 at 3:57
• @Fabian You're thinking in terms of the Euclidean metric, in which case each set is unbounded, indeed. However, consider the discrete metric on $\mathbb N$. This is defined as $d(m,n)=1$ if $m\neq n$ and $d(m,n)=0$ if $m=n$. It is not difficult to show that this is a legitimate metric. Also, the distance between any two distinct points is 1. This implies that if you take a “ball” of “diameter” 2 around the point $n=1$, then this ball will contain the whole space! That is, $$\{m\in\mathbb N\,|\,d(m,n)<2\}=\mathbb N,$$ and the whole space fits into a ball, which implies boundedness. – triple_sec Aug 15 '15 at 4:05
• Got it! Great explanation! Thank you triple_sec . – Fabian Aug 15 '15 at 4:09
• @Fabian I added a proof that the intersection may never be empty if you stay in $\mathbb R$. – triple_sec Aug 15 '15 at 4:17

Another simple example is to look at the "punctured line": $(-\infty, 0) \cup (0, \infty)$, which is just the real numbers with $0$ removed. The sets $A_n = \{ x \ \in \Bbb R \,|\, |x| \le 1/n \text{ and } x \ne 0 \}$ are closed and bounded in the punctured line, but their intersection is empty.

Do you know a theorem about nested bounded closed sets having non-empty intersection? If you do then you would need to find a metric space that does not satisfy the conditions of that theorem. $\mathbb{R}$ satisfies that theorem and hence you're not going to find a counter-example there. But there is a smaller metric space sitting inside it, namely $\mathbb{Q}$, that will give you a counter-example.

• I do not understand the downvote. I feel that my hint (without explicitly stating $\mathbb{Q}$) was just right, rather than just giving a complete solution like the other answerers that leave nothing for the asker to try. – user21820 Aug 15 '15 at 11:01
• Hello user21820, I upvoted your answer, and sorry to see users downvote for no reason. Thank you for your hint. – MATH Aug 15 '15 at 17:45
• @MATH: Thanks. Anyway the downvoter removed his downvote already, so perhaps he is happier with the explicit example of $\mathbb{Q}$. =) – user21820 Aug 16 '15 at 2:21

I initially missed the fact that bounded sets were desired.

Let the metric space be $(0,1)$, the set of all real numbers between $0$ and $1$, not including the endpoints, with the usual metric $d(x,y)=|x-y|$.

Then the example can be that the $n$th set is $(0,\ 1/n]$. This is closed within this space; it contain all of its limit points in the space.

The first thing I thought of was $\displaystyle \bigcap_{n=0}^\infty [n,+\infty)$.

If the object called $+\infty$ were included in the space, with the appropriate topology so that $n\to\infty$, then these sets would not be closed, but would become closed if one added $+\infty$ to them as a new member, and then the intersection would not be empty because $+\infty$ would be a member of it.

• The sets should be bounded. – Carl Mummert Dec 3 '15 at 21:30

If $(X,d)$ is a metric space then $$d'(x,y)=\min\{d(x,y),1\}$$ is a metric on $d$, as well. Moreover, the metrics $d$ and $d'$ generate the same topology (hence the same subsets of $X$ are closed in $(X,d)$ and in $(X,d')$).

Every subset is bounded in $(X,d')$.

So it suffices to find a metric space which contains some nested closed subsets with empty intersection. Then you can look at the same system of subsets in the modified metric, and you will have an example of closed bounded sets.

Possibly a simpler example is the usual metric on the rationals, and the closed sets $$C_k=\left[{\lfloor \sqrt{2}k\rfloor\over k}, {\lceil \sqrt{2}k\rceil\over k }\right].$$ Intuitively, the intersection of the $C_k$s is $\{\sqrt 2\}$, but that's not rational, so in $\mathbb{Q}$ their intersection is empty.

• Note that what's really being used here is that $\mathbb{Q}$ is incomplete - I'm essentially taking a non-convergent Cauchy sequence (one "converging" to $\sqrt{2}$) and turning it into a decreasing sequence of bounded closed sets with empty intersection. As e.g. Martin Sleziak's answer shows, this is not necessary - there are complete metric spaces in which nonetheless we have decreasing sequences of bounded closed sets with empty intersection. – Noah Schweber Dec 4 '15 at 0:07


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TitleAuthors / EditorsDate
Search for single production of vector-like quarks decaying into Wb in pp collisions at √s = 13 TeV with the ATLAS detector ATLAS Collaboration 2019
Search for pair production of heavy vector-like quarks decaying into high- p T W bosons and top quarks in the lepton-plus-jets final state in pp collisions at √s = 13 TeV with the ATLAS detector ATLAS Collaboration 2018
Search for pair production of vector-like top quarks in events with one lepton, jets, and missing transverse momentum in √s = 13 TeV pp collisions with the ATLAS detector ATLAS Collaboration 2017
Search for pair production of heavy vector-like quarks decaying to high- p T W bosons and b quarks in the lepton-plus-jets final state in pp collisions at √s = 13 TeV with the ATLAS detector ATLAS Collaboration 2017
Search for the production of single vector-like and excited quarks in the Wt final state in pp collisions at √s = 8 TeV with the ATLAS detector ATLAS Collaboration 2016
Search for magnetic monopoles with the MoEDAL prototype trapping detector in 8 TeV proton-proton collisions at the LHC The MoEDAL collaboration 2016
Search for resonances in the dijet mass spectrum with the ATLAS experiment at the LHC Guescini, Francesco 2016
Search for low-scale gravity signatures in multi-jet final states with the ATLAS detector at √s = 8 TeV ATLAS Collaboration 2015
A search for t t ¯ $$t\overline{t}$$ resonances using lepton-plus-jets events in proton-proton collisions at √s = 8 TeV with the ATLAS detector ATLAS Collaboration 2015
Analysis of events with b -jets and a pair of leptons of the same charge in pp collisions at √s = 8 TeV with the ATLAS detector ATLAS Collaboration 2015
Search for high-mass diboson resonances with boson-tagged jets in proton-proton collisions at √s = 8 TeV with the ATLAS detector ATLAS Collaboration 2015
A search for high-mass resonances decaying to τ+τ in pp collisions at √s=7 TeV with the ATLAS detector ATLAS Collaboration 2013
Search for high-mass states with one lepton plus missing transverse momentum in proton–proton collisions at √s=7 TeV with the ATLAS detecto ATLAS Collaboration 2011


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# 2019 seminar talk: Ramsey Theory of the Henson graphs

Talk held by Natasha Dobrinen (University of Denver, Colorado, USA) at the KGRC seminar on 2019-01-10.

### Abstract

A central question in the theory of ultrahomogeneous relational structures asks, How close of an analogue to the Infinite Ramsey Theorem does it carry? An infinite structure $\mathbf{S}$ is ultrahomogeneous if any isomorphism between two finitely generated substructures of $\mathbf{S}$ can be extended to an automorphism of $\mathbf{S}$. We say that $\mathbf{S}$ has finite big Ramsey degrees if for each finite substructure $A$ of $\mathbf{S}$, there is a number $n(A)$ such that any coloring of the copies of $A$ in $\mathbf{S}$ can be reduced to no more than $n(A)$ colors on some substructure $\mathbf{S}'$ of $\mathbf{S}$, which is isomorphic to the original $\mathbf{S}$.

The two main obstacles to a fuller development of this area have been lack of representations and general Milliken-style theorems. We will present new work proving that the Henson graphs, the $k$-clique free analogues of the Rado graph for $k\ge 3$, have finite big Ramsey degrees. We devise representations of Henson graphs via strong coding trees and prove Millike-style theorems for these trees. Central to the proof is the method of forcing, building on Harrington's proof of the Halpern-Läuchli Theorem.

There is a video recording of this talk on YouTube.

Slides are available, too.


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# Special function

Special function is any mathematical function, supplied with the following:

1. Formal definition, that indicates the way of evaluation of the function. In the most of cases, this refers to a holomorphic function of a single complex variable.

2. Alternative representations of the function, relation of the function to other elementary functions or with other special functions, defined earlier; even if the relation is asymptotic or approximate. [1][2]

3. Properties of the function for the real values of the argument, especially if the function is real–holomorphic: zeros, maxima, minima. Some special values; in particular, value at zero, if it is finite.

4. Asymptotic expansions; for example, that at zero and that at infinity.

5. Numerical implementation, that allows to evaluate the function with several decimal digits. For example, the relatively new special functions tetration to natural base (id est, tet) and the superfactorial (id est, SuFac) are implemented with 14 decimal digits in wide range of values of the argument; this range and the precision are limited, mainly, by the rounding errors and the range of representation of numbers in the complex double variables.

6. Explicit plot of the function for real values of the argument.

7. Complex map of the function; especially if it is holomorphic at least in some part of the complex plane.

8. References to publications, where this function is described or used for some applications, for example, for evaluation of other functions or for formulation of some nontrivial scientific concepts.

## Elementary function

Important subset of special functions is set of elementary functions.

Elementary functions appear as

(1) first three ackermann functions, id est, addition, multiplication and exponentialtion, or

(2) their inverse functions, id est, subtraction, division, logarithm and root finding (for example, sort), or

(3) finite combinations of the functions mentioned in (1) or (2) above.

Several elementary functions are qualified as primary elementary functions:

exp, log and sqrt

The main reason for this classification is, that the primary elementary functions are considered at schools; often, before of define correctly the complex numbers, differentiation, and integration.

sin, arcsin, cos, arccos, tan, arctan are qualified as primary trigonometric functions;

sinh, arcsinh, cosh, arccosh, tanh, arctanh are qualified as primary hyperbolic functions.

## Terminology

Some colleagues tend to call "special functions" those that appear at the textbooks. Some textbooks are qualified as authoritet sources, and only functions described there are considered as special.

For the goals of TORI, the definitions of terms authoritet source and scientific fact slightly deviates from those, suggested in schools or wikipedia; accordingly, the term special function may refers to a function, which is not yet included to the widely spread textbook. However, the description of the special function still should have all the 8 properties mentioned in the preamble, and the concept about that function is supposed to satisfy the TORI axioms, even if the rigorous mathematical proof of some statements is not yet available.

## Justification

Amount of functions, that can be described by a single researcher, is limited. The choice of the objects of investigation is important. The following reasons to investigate the function and to upgrade it to the status of special function are considered as serious.

$u+\mathrm i v =\,$ AuSin$(x\!+\!\mathrm i y)$

1. Function appears in the physical model, that describes or pretends to describe some physical effect.

2. Function appear in any scientific research, is used in description and/or application of any scientific concept.

3. Function is declared in public to be impossible to built-up or to calculate with sufficient precision in realtime.

4. Function can be used as illustration for some more general principle, some non–trivial scientific concept.

5. The complex map of this function or its explicit plot is beautiful.

Often, the range of primary application of a function is narrow; id est, namely for this application, the function appears inly in small region of the complex plane; often, only for the real values of the argument or even for only a piece of the real axis (for example, for positive values of the argument). This narrow area of application does not justify the restriction of consideration of function in the whole complex plane. This extension may be limited only by the explicit cut line, that limits the range of holomorphism of the function.

For example, the complex map of function AuSin (that is Abel function for sin) seems to be limited by the region where the function shows fast oscillations. This can be considered as justification, why the function is not defined (nor implemented) outside some range. Complex map of function AuSin is shown in figure at right.

In other cases, the holomorphic extension, preferably, to the whole complex plane, should be supplied and implemented. Even if the application of the function for complex argument is not yet suggested, one may expect this application to appear in future. In particular, the trigonometric functions, mentioned above, first were introduced only for real values of the argument; the extension to the complex plane begins, roughly, in 1748 with Leonhard Euler, with the Euler identity

$\mathrm e^{\mathrm i \pi} +1=0$

and its more general and more complicated modifications. [3][4].

## References

1. For example, is the dollar, ruble and pound are already defined, then formula One pound of rubles is equivalent of one dollar would be suitable in their descriptions as special functions.
2. One friend of mine found:
Let all constants be round;
Let all, $\pi$ and $\mathrm e$
Be equal to three;
And kilogram be 2 lb.
3. http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch8vol1.pdf EULER'S INTRODUCTIO IN ANALYSIN INFINITORUM Translated and annotated by Ian Bruce. Ian Bruce. Jan. 16th , 2013 latest revision.


text:
# Q=mcT heat problem, were did i go wrong?

by Senjai
Tags: heat, qmct, solved
 P: 104 1. The problem statement, all variables and given/known data If a 45g sample of aluminum at 22 degrees C is given 6.0 x 10^3 J of heat, what will its final temperature be? 2. Relevant equations $$Q = mc \Delta T$$ 3. The attempt at a solution i found in my textbook that aluminum has a specific heat capacity of 900 so c = 900. $$Q = mc \Delta T$$ $$Q = mc( T' - T )$$ $$T' - T = \frac{Q}{mc}$$ $$T' = \frac{Q + T}{mc}$$ $$T' = \frac{(6.0 x 10^3 J) + 22\deg}{(0.045kg)(900)}$$
 P: 104 Q=mcT heat problem, were did i go wrong? so is it $$T' = \frac{Q}{mc} + T$$ ??