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# Browse Dept. of Materials Science and Engineering by Title

• (1995)
We study the adsorption of macromolecules from solution onto single solid surfaces, focusing on the evolution of compositional heterogeneity of the layer. The origin and development of adsorption states within the physisorbed ...

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PDF (5Mb)
• (2012-06-27)
Lithium-ion batteries are commonly used as energy storage devices in a variety of applications. The cathode architectures and materials have a large influence on the performance of lithium-ion batteries, namely power and ...

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PDF (4Mb)
• (1996)
This work reports a systematic series of in situ spectroscopic ellipsometry (SE) studies on silicon film growth by reactive magnetron sputtering of a Si target in (Ar + H$\sb2$) at low temperatures ($\le$600$\sp\circ$C). ...

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PDF (6Mb)
• (1990)
Single-pitted specimens of HF-80, a HSLA steel, were tested in laboratory air and in 1 M NaCl solution to study the influence of a corrosive environment on its fatigue life.

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PDF (2Mb)
• (2015-01-21)
Epidermal electronic devices offer enormous potential as sensors and actuators for biomedical research and clinical application. One such area, cutaneous wound healing management, currently utilizes subjective analyses of ...

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PDF (3Mb)
• (2011-05-25)
Aligned arrays of single-walled carbon nanotubes (SWNTs) are an attractive format for macroelectronics and RF analog electronics with exceptional electrical, mechanical and optical properties. Unlike isolated SWNT, the ...

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PDF (3Mb)
• (2010-08-20)
Interfacial science is a fascinating field in materials science. From grain boundaries in bulk materials to interfaces in thin films, the properties of interfaces can lead to novel functionality in materials. The effect ...

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PDF (4Mb)
• (2011-01-21)
Interference lithography can create large-area, defect-free nanostructures with unique optical properties. In this thesis, interference lithography will be utilized to create photonic crystals for functional devices or ...

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PDF (7Mb)
• (1989)
Three types of stainless steel--austenitic, ferritic and duplex--were cathodically precharged with hydrogen at high temperature in a molten salt electrolyte. Constant strain rate tests and sustained load tests were performed ...

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PDF (6Mb)
• (2012-09-18)
There is much interest in the development of technologies which can enhance the performance of dye-sensitized solar cells (DSCs), due to their promising efficiencies relative to the low cost of materials and manufacturing. ...

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PDF (25Mb)
• (1996)
The implantation damage behavior of GaAs/Al$\sb{0.6}$Ga$\sb{0.4}$As multilayer structures has been investigated by implanting samples with 1 MeV Kr$\sp+$, 1.5 MeV Kr$\sp+$, 1 MeV Ar$\sp+$, and 1.5 MeV Kr$\sp{++}$ at 77 K. ...

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PDF (6Mb)
• (2011-06-27)
The cost of energy used for lighting, heating, and cooling in commercial and residential buildings is increasing annually while wasteful losses and gains through inefficient windows can be avoidable with advanced technology. ...

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PDF (7Mb)
• (1994)
Application of compound semiconductors to electronic and optoelectronic devices offers numerous advantages over elementary semiconductors, such as the feasibility of high quality heterostructures prepared by modern crystal ...

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• (1994)
Kinetics and mechanisms of metastable (C49-TiSi$\sb2)$ and stable titanium disilicides (C54-TiSi$\sb2)$ have been investigated by thermally annealing Ti/polycrystalline Si bilayers using a combination of in situ sheet ...

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• (2013-08-22)
Relaxor-ferroelectric single crystals, such as (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT), have the potential to transform technologies of medical imaging, actuation, and sensors, due to their extraordinary high piezoelectric ...

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• (1995)
The length of the jumps executed by atoms during diffusion is important for understanding atomic transport in crystal growth. Molecular dynamics simulations suggest that atomic jumps longer than a nearest-neighbor spacing ...

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• (1992)
This research investigated the effects of the ion beam assisted deposition process on the interdiffusion and reaction of Ti thin films deposited on Si (100). The analytical techniques employed for the characterization of ...

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• (2014-09-16)
Device quality CdS/CdTe heterostructures and completed solar cells (~12% efficient) have been studied using low-temperature photoluminescence (PL). The PL was studied as a function of temperature (82-295 K) and laser ...

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• (2009-09-25)
Materials and thin film processing development has been and remains key to continuing to make ever smaller, or miniaturized, microelectronic devices. In order to continue miniaturization, conformal, low-temperature deposition ...

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• (1996)
The use of energetic (average energy $\simeq$ 18 eV), rather than thermal, Si beams increased Si(001) epitaxial thicknesses t$\rm\sb{e}$ by up to an order of magnitude over the growth temperature range \$\rm T\sb{s}=80{-} ...

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## Stream: general

### Topic: go to definition failing for some people

#### Kevin Buzzard (Feb 11 2021 at 17:20):

I am teaching right now. For two students, clicking on "go to definition" (the little injective hook-arrow) in the infoview simply doesn't work. I have no idea how to debug. They're live streaming their screens. Is this a known issue?

#### Mario Carneiro (Feb 11 2021 at 17:28):

I haven't seen this before fwiw

Is there a #mwe?

#### Alex J. Best (Feb 11 2021 at 17:43):

Doesn't work for me at all, on windows, are they also on windows?

#### Bryan Gin-ge Chen (Feb 11 2021 at 17:56):

@Alex J. Best if you hit ctrl+shift+p and run "toggle developer tools", do you see any errors in the console when you try to go to definition in the widget infoview?

#### Alex J. Best (Feb 11 2021 at 17:57):

Trying it now :smile: as expected it seems to be some funny buisness with backslashes

console.ts:137 [Extension Host] rejected promise not handled within 1 second: Error: cannot open C:%5CUsers%5Calexj%5C.elan%5Ctoolchains%5Cleanprover-community-lean-3.26.0%5Clib%5Clean%5Clibrary%5Cinit%5Ccore.lean. Detail: Unable to resolve resource C:%5CUsers%5Calexj%5C.elan%5Ctoolchains%5Cleanprover-community-lean-3.26.0%5Clib%5Clean%5Clibrary%5Cinit%5Ccore.lean

#### Bryan Gin-ge Chen (Feb 11 2021 at 18:02):

Interesting... I wonder why there's a difference from the usual go to definition function.

#### Alex J. Best (Feb 11 2021 at 18:04):

Do you know if I need to setup the vscode extension for development to set breakpoints in vscode?

#### Eric Wieser (Feb 11 2021 at 18:13):

I think you do, yes - then you launch a new vscode instance from the one you develop the extension in

#### Alex J. Best (Feb 11 2021 at 18:44):

Ok on windows it seems the URI scheme when we get to https://github.com/leanprover/vscode-lean/blob/master/src/infoview.ts#L285 is set to "C" instead of "file" (https://docs.microsoft.com/en-us/dotnet/api/system.uri.scheme?view=net-5.0) it seems something is parsing "C:/..." as a uri scheme instead of a drive

#### Bryan Gin-ge Chen (Feb 11 2021 at 18:50):

And that function, revealEditorPosition seems to be called from here, being passed Uri.parse(message.loc.file_name).

#### Bryan Gin-ge Chen (Feb 11 2021 at 18:51):

So is Lean returning file_names differently for the usual go to definition command and the go to definition button in the widget?

#### Alex J. Best (Feb 11 2021 at 18:57):

I think it should use Uri.file instead of Uri.parse

#### Alex J. Best (Feb 11 2021 at 18:59):

Building with that change it works for me (as does the usual go to definition via F12)

#### Bryan Gin-ge Chen (Feb 11 2021 at 19:03):

Oh yeah, it looks like Uri.file is used almost everywhere else in the extension. There are a few instances of Uri.parse in src/docview.ts: are there issues there on Windows as well?

#### Alex J. Best (Feb 11 2021 at 19:07):

https://github.com/leanprover/vscode-lean/pull/258

I'll look now

#### Alex J. Best (Feb 11 2021 at 19:09):

I also think upstream https://github.com/microsoft/vscode-uri/pull/12 might fix this too, but I have no idea how to tell when / if that will make it to a vscode release (or if it has already)

#### Eric Wieser (Feb 11 2021 at 19:18):

I think I was looking at making a change in the other direction at one point, Uri.file doesn't work on unsaved buffers as far as I remember

#### Bryan Gin-ge Chen (Feb 11 2021 at 19:21):

@Kevin Buzzard I guess it may be too late now for your course, but Gabriel just released vscode-lean 0.16.24 which includes this fix.

#### Alex J. Best (Feb 11 2021 at 19:23):

Bryan Gin-ge Chen said:

Oh yeah, it looks like Uri.file is used almost everywhere else in the extension. There are a few instances of Uri.parse in src/docview.ts: are there issues there on Windows as well?

what is doc view used for ? Is it that nice thing where you can view a tutorial in vscode as a separate editor pane, if so can someone point me to where I can set that up (I've looked to no avail!)

#### Alex J. Best (Feb 11 2021 at 19:27):

Ah, got it, its built in to the extension lol!

#### Bryan Gin-ge Chen (Feb 11 2021 at 19:27):

Yep, I think if you hit ctrl+shift+p and search for "Lean: open documentation view" it should open a panel with some links you can try.

#### Alex J. Best (Feb 11 2021 at 19:30):

Yeah they seem to work fine as-is.

#### Alex J. Best (Feb 11 2021 at 19:30):

Eric Wieser said:

I think I was looking at making a change in the other direction at one point, Uri.file doesn't work on unsaved buffers as far as I remember

Do you mean completely new buffers, or just any buffers with unsaved changes in?

#### Eric Wieser (Feb 11 2021 at 20:55):

Completely new buffers

#### Eric Wieser (Feb 11 2021 at 20:56):

In particular, try this does not work in those buffers

#### Eric Wieser (Feb 11 2021 at 20:57):

And nor do \-inputs until you leave the new buffer and reenter it, although that's unlikely to be related to URIs

Last updated: May 11 2021 at 00:31 UTC

format: plain


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## College Algebra (10th Edition)

The solution set is $\left\{-3, 3\right\}$
Write 9 as $3^2$ to obtain: $x^2-3^2=0$ Factor the difference of two squares using the formula $a^2-b^2=(a-b)(a+b)$ to obtain: $(x-3)(x+3)=0$ Use the Zero-Product Property by equating each factor to zero and then solving the equation to obtain: $\begin{array}{ccc} &x-3=0 &\text{ or } &x+3=0 \\&x=3 &\text{ or } &x=-3 \end{array}$ Thus, the solution set is $\left\{-3, 3\right\}$.


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# [texhax] pagestyle{empty}

Thu Mar 30 01:24:39 CEST 2006

Reinhard Kotucha :

>>>>>> "Martin" == Martin Schröder <martin at oneiros.de> writes:
>
>  > The behaviour is not a bug but a feature. The standard LaTeX
>  > classes are written so that initial pages (pages containing a
>  > \maketitle, \part, or \chapter) have a different page style from
>  > the rest of the document; to achieve this, the commands internally
>  > issue \thispagestyle{plain}. This is usually not acceptable
>  > behaviour if the surrounding page style is 'empty'.
>
> It is meant as a feature but it is definitely a bug.
>
> If you say \pagestyle{empty} in the preamble, you should get what you
> asked for.  The bug is that the bevaviour depends on the
> \documentclass you are using and not on the \pagestyle you selected.
>

But how do you want it to behave?

If using \pagestyle{empty} (or another style) in the preamble should
affect the entire document, then how would you get one style for chapter
titlepages, and another for regular pages?

/daleif

``You cannot help men permanently by doing for them
what they could and should do for themselves. ''
-- Abraham Lincoln


text:
# Iron ore rocks

## Redirected from Iron ore rock

36,747pages on
this wiki
[talk] • [view]

Iron ore rocks are protruding rocks containing iron. A player with a Mining level of 15 or higher can mine iron ore from rocks found in various mines, granting 35 Mining experience for each ore mined.

## Rock locationsEdit

Main article: Mines

## Respawn rateEdit

Iron ore, like all other mined rocks, respawns at a rate inversely proportional to the number of people in the rock's RuneScape world. If the world is empty, the ore respawns in 10 seconds; if it is full (with 2000 players), it respawns in 5 seconds. Thus the respawn rate of iron ore is

$10-\frac{x}{400}$ seconds,

where x is the number of players in the world.

A table of respawn rates is given below.

Players 0 200 400 600 800 1000 1200 1400 1600 1800 2000
Respawn time (s) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0

## Trivia Edit

• There was a popular place to mine iron in the mining area near Al Kharid, which has a place in the mine where there are three rocks in a triangle formation, making the player able to mine iron faster due to the three respawns. After the dungeoneering resource dungeon update, the triangle formation changed.
• There are some spots where the graphics for the iron ore rocks have not been updated, including the mining spot to the East of East Ardougne.


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# Write fraction in simplest form

Practice writing fractions in simplest form If you're seeing this message, it means we're having trouble loading external resources for Khan Academy. MY WEBSITE: https://sites.google.com/site/mrheath Disclaimer: I do not own this song. I did not write it. I created a music video that goes along with. Note: When you get a fraction as an answer, you're usually asked to write it in its simplest form. This tutorial shows you how to use the GCF of the numerator and. Simplifying fractions is often required when your answer is not in the form required by the assignment. As a matter of fact, most math instructors will demand. that. Rational and irrational numbers. The principal square root. A proof that square root of 2 is irrational. What is a real number. Simplest Form Calculator (or Fractions in Simplest Form Calculator) is an online tool to write a given fraction in the simplest form. It makes calculation easy and fun. Rename to Lowest Terms uses circles to demonstrate how a fraction can be renamed in lowest terms. The following equivalent fractions were made by Rename to.

Fun math practice! Improve your skills with free problems in 'Write fractions in lowest terms' and thousands of other practice lessons. Solution: The fraction $\,\frac{6}{15}\,$ is not in simplest form, because $\,6\,$ and $\,15\,$ have a common factor of $\,\bf{3}\,$. To simplify the fraction, use. 3. Simplest Radical Form. Before we can simplify radicals, we need to know some rules about them. These rules just follow on from what we learned in the first 2. What is 4.75 as an improper fraction in simplest form - 2416096. Hi there! Have questions about your homework? At Brainly, there are 60 million students who want. Solution: The fraction $\,\frac{6}{15}\,$ is not in simplest form, because $\,6\,$ and $\,15\,$ have a common factor of $\,\bf{3}\,$. To simplify the fraction, use. Rename to Lowest Terms uses circles to demonstrate how a fraction can be renamed in lowest terms. The following equivalent fractions were made by Rename to. Simplifying fractions is the name of this fast-paced race game! Learning how to write fractions in their simplest form is essential for math class and real-world.

## Write fraction in simplest form

3. Simplest Radical Form. Before we can simplify radicals, we need to know some rules about them. These rules just follow on from what we learned in the first 2. Home. Simplifying radicals: Section 2. Back to Section 1. Simplifying powers. Factors of the radicand. Fractional radicand. We can identify with the absolute value. Using Rational Numbers. How to add, subtract, multiply and divide rational numbers. A rational number is a number that can be written as a simple fraction (i.e. as a. 4 stars "Easy to use, excellant calculator" December 25, 2008 | By. Simplest Form Calculator (or Fractions in Simplest Form Calculator) is an online tool to write a given fraction in the simplest form. It makes calculation easy and fun. Rational and irrational numbers. The principal square root. A proof that square root of 2 is irrational. What is a real number.

Practice writing fractions in simplest form If you're seeing this message, it means we're having trouble loading external resources for Khan Academy. Free student math practice. Express as a simple fraction in lowest terms. 3 . A fraction is in its simplest form (this is also called being expressed in lowest terms) if the Greatest Common Factor (GCF), also called the Greatest Common Divisor. EdHelper.com Math Worksheets: Fractions Worksheets and Printables Addition, Subtraction, Multiplication, Division! Fraction Puzzles! Algebraic Fractions. Using Rational Numbers. How to add, subtract, multiply and divide rational numbers. A rational number is a number that can be written as a simple fraction (i.e. as a.

Number line models are used to picture renaming fractions to lowest terms. A fraction is said to be in simplest form if its numerator and denominator are relatively prime, that is, they have no common factors other than 1. What is 4.75 as an improper fraction in simplest form - 2416096. Hi there! Have questions about your homework? At Brainly, there are 60 million students who want. EdHelper.com Math Worksheets: Fractions Worksheets and Printables Addition, Subtraction, Multiplication, Division! Fraction Puzzles! Algebraic Fractions. Fraction Division Worksheets. This page offers plenty of printable fraction worksheets in division. It contains dividing all types of fractions, missing place holders. MY WEBSITE: https://sites.google.com/site/mrheath Disclaimer: I do not own this song. I did not write it. I created a music video that goes along with. Free student math practice. Express as a simple fraction in lowest terms. 3 .

Convert Fractions to Percents. Divide the top of the fraction by the bottom, multiply by 100 and add a "%" sign. The simplest method is to use a calculator. Note: In this tutorial, you'll see how to take the information from a given table and use it to find a ratio. You'll also see how to write the answer in simplest form. Number line models are used to picture renaming fractions to lowest terms. A fraction is said to be in simplest form if its numerator and denominator are relatively prime, that is, they have no common factors other than 1. Quadratic Equations from the quadratic equations worksheets Solving Quadratic Equations (b=0, Whole Number Only Answers) Solving Quadratic Equations (b=0. Home. Simplifying radicals: Section 2. Back to Section 1. Simplifying powers. Factors of the radicand. Fractional radicand. We can identify with the absolute value.

Simplifying fractions is often required when your answer is not in the form required by the assignment. As a matter of fact, most math instructors will demand. that. Fun math practice! Improve your skills with free problems in 'Write fractions in lowest terms' and thousands of other practice lessons. Free student math practice. Express as a simple fraction in lowest terms. 1 + 3 . A fraction is in its simplest form (this is also called being expressed in lowest terms) if the Greatest Common Factor (GCF), also called the Greatest Common Divisor. Simplifying fractions is the name of this fast-paced race game! Learning how to write fractions in their simplest form is essential for math class and real-world. Free online kids math games make learning fun. Topics: addition, subtraction, multiplication, division, fractions, money, algebra, and place value.. Convert Fractions to Percents. Divide the top of the fraction by the bottom, multiply by 100 and add a "%" sign. The simplest method is to use a calculator.

### Media:

Write fraction in simplest form
Rated 5/5 based on 150 student reviews


text:
# attenuation

1. ### Resistance and attenuation

The topic was actually meant to be 'resistance and electromagnetic oscillation'. It is hard to translate specific words. I hope you understand what I meant. I have an oscillating circuit. The data I know are: Capacitance C=20*10^(-6) F Inductance L=180*10^(-6) H Attenuation (logaritmic...


text:
Volume 390 - 40th International Conference on High Energy physics (ICHEP2020) - Parallel: Quark and Lepton Flavour Physics
Flavourful Inert Doublet Dark Matter
L. Mukherjee
Full text: pdf
Pre-published on: January 16, 2021
Published on:
Abstract
In this talk I discuss about some anomalous results in flavour physics and try to correlate them with the Inert Higgs doublet dark matter model. This is achieved by considering an extension of the Inert Higgs doublet model with $SU(2)_L$ singlet vector like fermions. Flavour data in $b \to s \ell \ell$ decay observables (like $R(K)$, $R(K^*)$) and muon $(g-2)$ puts stringent bounds on the model parameters. The model can simultaneously satisfy relevant constraints in the dark matter sector and also abide by the bounds given by the ongoing direct detection experiments. I also discuss the discovery possibilities of such exotics in the future high luminosity (HL) runs of the LHC.
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access


text:
# No magic

In times of rain, fog and drizzles, I take the U-Bahn for commuting to and from work. I'm not as regular as your proverbial clockwork, but still punctual enough to see certain co-commuters on an almost daily basis. Some of them are individual enough to stick out of the crowd. Sir David, for example, a long thin figure with short grey hair and a short, accurately trimmed grey beard, invariably dressed in a checkered grey suit with a light grey shirt and a gunmetal grey tie, dark grey suede shoes and charcoal grey socks. Even his beyerdynamic headphones, which he's sure to wear, are grey, and while listening to Vivaldi, he's studying the daily Frankfurter Allgemeine with great interest and concentration.

Such intense is his concentration that he does not even notice Mike sitting next to him with a battered Nokia cell phone and talking very loudly, as every day, in an unidentifiable language with what I assume to be a subordinate of Mike. Mike is ebony black, 6'4'', 150+ kg, wears a gold Rolex and other golden accessories over a black Savile Row suit but still manages to look like a very serious and very worried business man. When U2 goes underground at Potsdamer Platz and his phone loses the connection (as every day), his worries grow troublesome for his health as accentuated by repeated violent bursts of shouting at his unfortunate subordinate who, luckily for him or her, can't hear the verbal assault because of the severed connection.

Despite this acoustic disturbance, Sir David remains concentrated on his newspaper and entirely misses Audrey stepping into the train. Audrey is a seriously cute American girl in her twenties with dazzling blue eyes and jet black hair she's wearing as an asymmetric bob. Unlike many Americans, she has a soft, melodious voice which is as pleasant to listen to as she's a pleasure to look at, despite her constant fumbling with a white Apple Watch. Today, however, she's discussing a technical subject with one of her colleagues from the advertising agency she's working for. The guy is the living prototype of a Berlin-Mitte hipster with the trade-mark combination of an undercut leaving a 2 cm ponytail stub at the back and a beard of Abrahamian dimensions in front, together with the indispensable oversized horn-rimmed black glasses fulfilling no medical function but representing a fashion statement.

He seems to believe, as many non-native speakers, that an excessive use of the F-word documents an intimate familiarity with both the American language and the American culture, and thus demonstrates that you are on the level. Know what I'm sayin'? Fuck, eh. Right now, he keeps whining, in a high wheezy voice not fitting the beard, that his brochure wasn't accepted because the fucking pdf was too fucking big, fuck eh. Know what I fucking mean, eh? Audrey knows, and contrary to what I would have expected, she's reacting in a totally enthusiastic way. Oh so true, she cheers, and adds immediately that she's recently found the solution for big pdfs that seem to be the major, if not the problem troubling her agency. A wonderful, an absolutely fabulous web service! Her original pdf of 500 MB reduced to 10 MB without any loss of quality! Pure magic! But ... who ... ?, her hipster colleague manages to ask, and she's shouting at him, full of delight: ILOVEPDF DOT COM!

Two days later one of my colleagues (with a PhD in physics) tells me that he used ilovepdf.com to compress the pdf of his recent publication, so that it's of a size suitable for uploading to arXiv. He shows me the result, and I'm impressed: there are no immediately obvious compression artifacts, although the file size has been reduced from more than eight to just one MB. Now I'm really getting curious. Is it possible that these ilovepdf guys are doing something ... clever? Perhaps they employ one of the new image formats, such as webp, bpg, or even flif? That would be most interesting, and I thus set out to get to the core of this business.

Several web services promise to compress images (or, more precisely, pixel graphics or, synonymously, raster image files, or short, bitmaps or pixmaps) of various formats or entire pdf documents (where the compression, of course, boils down to exactly the same: compressing the pixmaps embedded in the pdf container). All of them also promise to respect our privacy. For example, ilovepdf.com (and iloveimg.com) states:

Absolutely all uploaded files on ilovepdf.com are deleted from our servers one, two or twenty-four hours (depending on if a user is non registered, registered or premium) after been processed.

Hmmm. Why only the uploaded files? What about the compressed ones? I like the statement of smallpdf.com better:

Please note that uploaded and processed files are never stored longer than an hour on our servers and then are deleted permanently. During this hour your files are not accessed, copied, analyzed or anything else except we have the explicit permission of the user for example for a support case.

“Analyzed” is the key term here. Even if the uploaded and processed files are deleted, it takes only fractions of a second to extract the text of pdf documents, or the raw pixmaps embedded in them:

        less upload.pdf > text.txt

On tinypng.com, we only read:

Submitted content will not be shared with third parties other than Voormedia’s service providers, unless required to comply with the law or requests of governmental entities. Voormedia uses service providers based in the USA.

For the present case of use (getting a file size acceptable for arXiv), all of that doesn't seem to matter. After all, our intention is to publish our content, not to keep it confidential. The same goes for Audrey and her agency. Still ... if you can do it yourself, why should you become dependent on others? And after you have made yourself depending on this service, what will you do when it really matters?

But can we do it ourselves? Are the results of these services within our reach, or are their makers truly magicians with capabilities beyond the John and Susan Does of the interwebs? To narrow it down to the point which matters most for me: can these services, given a file that I deem to be suitable for publication, significantly compress it further? After all, I know how to treat my images, don't I? Well, at least I believe I do.

I've made a comparison using several pdf documents (including the publication of my colleague above) as well with a few raster images. To my simultaneous relief and disappointment, the most frequent statement I got from the web services under consideration included:

We are sorry, your file is very well compressed and we can't compress it without quality loss.

Or, as honest as cute:

We compressed your file from 30.36 kB to 29.57 kB. That's not that much. Sorry.

No magic, no new formats. What a pity! At the same time, I was impressed by the technical quality of these services. All documents and images returned from them were an excellent compromise between image quality and size. Furthermore, I actually never managed to produce an image of equal quality but smaller file size. But I was always close with very little effort.

To give explicit examples: the publication of my colleague was 8.3 MB in size. The sole reason for the large size were a couple of images embedded as uncompressed tiff. I would normally compress these images prior to generating the pdf with pdfLaTeX, of course, but we can equally well compress them afterwards. Our two services return files of 0.95 (ilove) and 0.86 MB (small). I have to magnify them a five, even a tenfold to see the effects of compression. Yes, these services use lossy compression schemes, specifically jpeg, but they do that expertly.

What could John Doe do against specialized services jam-packed with expertise and knowledge on image compression? Well, I just tried to apply the little I know, like this (well-known) ghostscript one-liner:

            gs -sDEVICE=pdfwrite -dPDFSETTINGS=/default -dNOPAUSE -dQUIET
-dBATCH -sOutputFile=out.pdf in.pdf

resulting in

                gs (default):       1.41 MB     83%
gs (ebook)          1.05 MB     87%
ilovepdf:           0.95 MB     89
gs (screen)         0.94 MB     89%
smallpdf            0.86 MB     90%

With the default setting, image quality is basically indistinguishable from the original and thus even better than the files produced by ilove and small. The higher compression stages of ghostscript with compression ratios rivaling those of ilove and small produce visual artifacts in this particular case, but are always worth a glance when trying to compress a pdf.

The conclusion of this exercise is obvious, I believe. Pixmaps for inclusion in documents should be compressed, most preferably losslessly. At the moment of writing, the most suitable format for images in my trade is png. These images can be compressed further (but lossily!) by reducing the range of colors (with pngquant, for example, or simply by converting them to greyscale). For photographs with soft contrasts and 16 M colors, jpeg remains the reigning format, but we have to be very careful not to reduce the image quality in an obvious way. And if you forgot all that and are facing a giant pdf that cannot be uploaded anywhere, use ghostscript.

That's it. That's all there is to know (at least concerning the current topic). I plan to look at new image formats, though, but those will be the subject of future posts. ;)

# Display manager

To simplify my life, I'm continuously trying to standardize the software configuration of all physical and virtual Linux installations I'm administering. In terms of Linux distributions, I've already reduced the previous diversity to ArchLinux and Debian Sid/Testing. As window manager, I'm using either wmii or OpenBox. The former is started via startx and a corresponding entry in ~/.xinitrc, but what about the latter? Well, of course, I'm using a display manager to start it.

But which one? For some (probably historical) reason, I've set up my four systems with three different display managers: lxdm, lightdm, and slim, and of course I've no idea which system is actually employing which display manager. Surely, the Swiss knife of system information should know.

 ➜  ~ inxi -xx -S
System:    Host: deepgreen Kernel: 4.7.4-1-ARCH x86_64 (64 bit gcc: 6.2.1)
Desktop: Openbox 3.6.1 dm: N/A Distro: Arch Linux

Not necessarily, as you see.

Since I'm also using systemd, /etc/systemd/system/display-manager.service contains the required information. Debian-based systems offer in addition /etc/X11/default-display-manager. And for those not using systemd: you should know what you're doing anyway.

# In the old days

Research places like the one I'm associated with are characterized by a constant turnover of personnel. Currently, our 20 senior scientists are supported by about 50 assistant and associate researchers on temporary positions. Our annual turnover rate is thus as high as 30 to 40%, meaning that I meet about 15 to 20 new people every year.

The level of understanding in physics and material science fluctuates, but does not seem to deteriorate over the years. That's the good news. What does decline significantly is the ability to read and write and to use a computer efficiently. At the same time, the fraction of people with an undue sense of entitlement is growing dramatically.

What do I call an “undue sense of entitlement”? Well, imagine. It's your first day at a new place where you hope to perform top-notch research yielding results important enough to publish them in prestigious journals. You are shown into your office, which you share with some more experienced colleagues, and on your desk sits a brand-new 24 inch Full-HD display connected to an equally brand-new desktop computer. And exactly that's the moment when you demand, loud and clear, two monitors. The bigger the better! And an hour later, you call our IT service to demand the real Office. And Photoshop! When the IT freaks ask for which purpose you need this software, you most strongly express your righteous indignation. First of all, that's none of his business, and second, that should be obvious! After all, you have letters to write. And later, there may be images whose contrast needs to be increased.

Now, that's exactly what you would do, right?

No, of course not. No halfway sensible person would behave in this way. Alas, every year we get more and more young people with this attitude. Experience tells us that the people demanding the most are the ones returning the least. They also tend to create constant trouble: they are more concerned with their own self-importance than with their research, and are generally ignorant, obnoxious, and unproductive.

I still vividly remember my own time as a PhD student at the MPI-FKF. I had previously written my diploma thesis on an HP Vectra, an IBM AT compatible, which I had to share with the six or seven members of our research group. At the weekends, I also had access to my own computer, an Schneider PC1512 that I primarily used for running Pascal programs. God, did it feel slow compared to the Vectra at work! At the MPI, however, computing was not yet “personal”. Instead, VT220 text terminals were offered for the interaction with the VAX station in the basement. I'd say we had perhaps 20 terminals for about 300 scientists.

# Messlatte

The German term 'Messlatte' is usually used figuratively to label something which sets a standard, serves as a reference or constitutes a benchmark with which others of the same kind have to be compared to. The headline of c't 11/2016 reads 'Messlatte MacBook?', and the corresponding article has a remarkably surreal introduction: "Apples MacBooks haben sich in vielen Köpfen als die Referenz für Notebooks festgesetzt: teuer, aber schick und gut. Windows-Notebooks haben allerdings aufgeholt – und bieten in mancherlei Aspekt sogar mehr."

In my experience, nobody with an at least rudimentary ability of logical thinking would view MacBooks as 'reference' for notebooks. I'd suggest 'accessories' instead of 'notebooks', but let's see.

The c't found the MacBook 2016 to be the lightest and slimmest of all contenders. They also report that thanks to this absolutely fantastic and totally unbelievable slimness, there's only space for one port (USB-C), the keyboard has poor tactile feedback due to its insufficient stroke depth, and the CPU performance suffers from insufficient cooling.

How much does it suffer? The (fanless) MacBook 2016 has the same CPU as the (equally fanless) HP Elitebook G1, namely, a Core m5-6y54. In CinebenchR15 (Multi, 64 bit), the Elitebook scores 248 points and the MacBook 122.

One hundred and twenty two. You want to know how bad that is?

In a previous post, I've lamented on the fact that AMD's notebook top model A10-4600M launched in 2012 was not able to deliver a better performance than my low-level Pentium P6200 from 2010. Now, this A10-4600M has a CinebenchR15 (Multi, 64 bit) mark of 187 points, i.e., over 50% more than the current MacBook. Note that the P6200, which I estimate to deliver about 150 CinebenchR15 points, is in my case powering a Fujitsu Lifebook bought in 2011 for €299. Also note that Apple charges €1800 for the MacBook as tested by c't.

You get the same performance for a price tag of €300 in contemporary low-end notebooks powered by Baytrail Atoms, such as the Pentium N3540. Have a look here.

In the same volume, c't also tested two entry-level desktop processors, which are currently available for €60. The Pentium G4400 scores 140 points in CinebenchR15. Single thread. Both cores together achieve 269 points. The quad-core Athlon X4 845 gets 309 points. That's about the performance you get from the contenders of the MacBook equipped with a Skylake generation Core i7, such as the Dell Latitude E7270 (311 points).

Surprised? Always thought an i7 leaves everything else in the dust? Well, i7 is not i7. The performance of ultrabooks is limited by their design factor, and is currently not better than that of desktops available for a quarter of the price—or much worse, as in the case of the MacBook 2016. High-end desktops, by the way, deliver about 900 points and are thus about three times more performant. If you're interested in a comparison of the raw numbercrunching performance of desktop and notebook CPUs, look here.

Interestingly, c't employed an absolute scale and declared all notebooks with a price tag above the €1800 of the MacBook as expensive or very expensive. The Dell mentioned above, for example, was labeled as 'expensive' because of its price tag of €2000. The c't emphasized how much you get for that: namely, a business notebook with options for docking stations, plenty of ports, and three years on-site service, in contrast to the MacBook with none of that.

In my view, that's certainly all very nice to have, but what really counts is the 255% higher performance of the Dell compared to 'Messlatte'. A computer is a computer and should still be mostly defined by its ability to compute.

# Indices

### A ȷupyter notebook

Scientists move in mysterious ways, particularly when they try to measure their individual performance as a scientist. As I've explained in a previous post, the most popular and commonly accepted of these measures is the h index $\mathcal{H}$, which has been declared to be superfluous on both empirical and mathematical grounds. Either of these references relates $\mathcal{H}$ to the square root of the total number of citations $\mathcal{N}$, the first one approximately

$\mathcal{H} \approx 0.5 \sqrt{\mathcal{N}}$

and the second one exactly:

$\mathcal{H}=\sqrt{6}\log{2}\sqrt{\mathcal{N}}/\pi \approx 0.54 \sqrt{\mathcal{N}}$.

Since I anyway wanted to test pandas, seaborn and statsmodel, I gathered $\mathcal{H}$, $\mathcal{N}$, and the i10 index $\mathcal{I}$ from all my coauthors on Google Scholar. It turned out that not even a quarter of my coauthors have a Google Scholar account, but I figured that 71 data points would provide an acceptable statistics.

%matplotlib inline

import pandas as pd
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import statsmodels.formula.api as sm
import seaborn as sns

# read data into a Pandas DataFrame
# check the data head

Citations Hindex i10index
0 952 12 17
1 1913 20 35
2 5327 34 151
3 650 13 14
4 3855 25 68

Looks all right.

logdata = np.log10(data)

The correlation of the data is much more clear when displayed logarithmically:

vars = ["Citations", "Hindex", "i10index"]
sns.pairplot(logdata, vars=vars, size=3, kind="reg");

Now look at that! Two lines of code and seaborn visualizes all correlations in my data set. The diagonal elements of this 3x3 matrix plot show the distributions of $\mathcal{N}$, $\mathcal{H}$, and $\mathcal{I}$ (which seem to be close to normal distributions), and the off-diagonal elements visualize their correlations emphasized by a linear regression (kind="reg"). And how correlated they are! There's indeed no need for a definition of 'indices' if the number of citations is all what it boils down to.

Seaborn is great for visualization, as we have seen, but for quantitative statistical information, it's better to use statsmodel:

hc = sm.ols(formula='Hindex ~ Citations', data=logdata)
fithc = hc.fit()

ic = sm.ols(formula='i10index ~ Citations', data=logdata)
fitic = ic.fit()

hi = sm.ols(formula='Hindex ~ i10index', data=logdata)
fithi = hi.fit()

Let's compare the slope of our data with that predicted above:

fithc.params.Citations

0.45708354021378172

np.sqrt(6)*np.log(2)/np.pi

0.54044463946673071

Solid state phycisists have to work harder!

One can get also get more information, if desired:

fithi.summary()

Dep. Variable: R-squared: Hindex 0.974 OLS 0.974 Least Squares 2592. Mon, 16 May 2016 1.84e-56 14:42:47 114.08 71 -224.2 69 -219.6 1 nonrobust
coef std err t P>|t| [95.0% Conf. Int.] 0.4498 0.019 23.723 0.000 0.412 0.488 0.5454 0.011 50.910 0.000 0.524 0.567
 Omnibus: Durbin-Watson: 3.438 2.391 0.179 2.6 -0.408 0.273 3.462 7.44

And of course, we can display these fits independent of seaborn:

xlist_cit = pd.DataFrame({'Citations': [logdata.Citations.min(), logdata.Citations.max()]})
xlist_i10 = pd.DataFrame({'i10index': [logdata.i10index.min(), logdata.i10index.max()]})

preds_hcit = fithc.predict(xlist_cit)
preds_hcit;

preds_i10cit = fitic.predict(xlist_cit)
preds_i10cit;

preds_hi10 = fithi.predict(xlist_i10)
preds_hi10;

logdata.plot(kind='scatter', x='Citations', y='Hindex')
plt.plot(xlist_cit, preds_hcit, c='red', linewidth=2);

logdata.plot(kind='scatter', x='Citations', y='i10index')
plt.plot(xlist_cit, preds_i10cit, c='red', linewidth=2);

logdata.plot(kind='scatter', x='i10index', y='Hindex')
plt.plot(xlist_i10, preds_hi10, c='red', linewidth=2);

# Ubuntu VSTS

I've used Ubuntu from 2008 to 2012 on actual hardware. I stopped using it mainly because it degraded from a usable Linux distribution to a bugriddled bundle of outdated packages that evolved in the wrong direction. Still, I kept a minimal LTS server as virtual machine which I believed to be potentially helpful to diagnose issues we might have with our Ubuntu server installation at work.

LTS? Stands for "long term support", is released every two years, and is advertised as being supported for five years.

I have known this promise to be false since 2010, and this knowledge has accelerated my departure from Ubuntu. The German computer magazine Hail Ubuntu c't has recently published an online article on Canonical's deception, but they otherwise consequently ignore other Linux distributions and treat Ubuntu as being the most (only?) suitable distribution for newbies. As if newbies wouldn't need security support.

To check the support status of the current version, I've upgraded my existing 14.04 installation to 16.04, which was released just two weeks ago. Canonical does not encourage this update, but recommends to wait until the first point release (16.04.1). To still be able to upgrade, the 'do-release-upgrade' tool needs the command line parameter 'd', directing it to also consider development versions. That's very interesting: the 16.04 LTS release, which is by viewed by many as the prototype of stability and reliability, is officially considered to be a development release.

And to give credit where credit is due: Canonical is right. While the upgrade itself was fast (10 min) and free of errors, and 16.04 boots up to the login prompt faster than ever, the X server exits with a segmentation fault. Just like the old times, and so Ubuntuish!

Fortunately, I do not need an X server to run 'ubuntu-support-status':

Percentage      Time
4.3             3y
2.9             9m
6.4             unsupported

Almost 14% of all packages are not covered by the 5y support (and almost half of those are unsupported at the day of installation). Note that I'm talking about a server installation with a total size of 2 GB and not a single package from multiverse. I don't even want to imagine the situation with a full-blown desktop installation.

# A hierarchical toolchain

I'm an experimental physicist by education. But since quite some time, I do not frequently perform experiments myself. Instead, I'm essentially acting as a consultant. I help to design experiments, to interpret and analyze them. And mostly, I advice my colleagues how to publish the data they got from their experiments or computations. Which often means that I have to analyze data, process images, create graphs, and write text myself, but always in close collaboration with others.

This interactive and iterative process takes time. Since my order book is already filled for the rest of this year, it is important for me to do that efficiently for being able to process each request within a reasonable time frame. In particular, I want to have the same workflow everywhere, whether I'm using the desktop at the office or at home or my netbook at a conference in Nice (basically ruling out all proprietary software).

The following list provides an overview of the applications I've chosen to use for these tasks in a loose hierarchical order (my second choice is given in parentheses). The one-liner below each entry briefly summarizes my personal reasons for the first choice. If I feel inclined to do so, I will elaborate on them in separate, subsequent posts.

Operating system: Archlinux (Debian Testing)
Transparent, simple, up to date. And tons of packages.

Shell: fish (bash, zsh)
Feature complete with minimal configuration.

Remote shell: mosh + tmux
An ssh replacement. Stays connected.

Shell extensions: virtualenv + virtualfish (virtualenvwrapper)
Easiest way to separate different python-based projects (like this blog).

CL Calculator: calc + units
Easiest way to calculate almost everything on the CL.

CL Editor: vim
Easiest way to edit almost everything on the CL.

CL File manager: autojump + ranger (mc)
Super-fast file system navigation and management.

Backup: attic (obnam)
Fast, scrictable, deduplicating.

Notes, Journal, Tasks: zim
My notes for my cloud.

Cloud storage: ownCloud
My cloud with all current projects.

Reference management: Mendeley (JabRef)
Full text search in all pdfs, bibTeX export.

Browser: Chromium (Firefox) in a sandbox
Fast, stable, consistent UI, all extensions I need.

Data analysis: ȷupyter with numpy, scipy, sympy (Mathematica)
Computing without licence hassles. Great UI with MathJax support.

Image processing: ImageJ, Gimp
All I will ever need and more.

Vector graphics: Inkscape
Standard compliant svg, eps and pdf ready for the web or publications.

Figure creation: ȷupyter with matplotlib
Publication quality figures with standard compliant svg, eps, and pdf export.

Typesetting: LaTeX w/ BibTeX
The de facto standard.

Editing: Several. Currently giviving atom a chance.
Syntax highlighting, command and citation completion, syncTeX support.

Version control: mercurial (git)
Keeps track of changes. No more 'manuscript_v17c_XY_NK_LM.tex'.

LaTeX diff generation: scm-latexdiff
Lights up the changes. Essential for not having to read the same stuff again.

# Sharing the cache

I have two systems at home running Archlinux, and even with an almost identical configuration. Isn't it a waste to update these systems independently, and thus to download the same packages twice? But of course it is!

I've searched for 'local repositories', but did not find the possibilities discussed in the pacman wiki entirely convincing. Days later, I stumbled across a discussion on the archlinux forum on pacserve,, which proved to be exactly the solution I wanted. The speed of updates across the wifi is something I can't get enough of. ;)

Highly recommended!


text:
# Stokes phenomenon

In complex analysis the Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is that the asymptotic behavior of functions can differ in different regions of the complex plane. These regions are bounded by Stokes line or anti-Stokes lines.

## Stokes lines and anti-Stokes lines

Somewhat confusingly, mathematicians and physicists use the terms "Stokes line" and "anti-Stokes line" in opposite ways. The lines originally studied by Stokes are what some mathematicians call anti-Stokes lines and what physicists call Stokes lines. (These terms are also used in optics for the unrelated Stokes lines and antiStokes lines in Raman scattering). This article uses the physicist's convention, which is historically more accurate and seems to be becoming more common among mathematicians. Olver (1997) recommends the term "principal curve" for (physicist's) anti-Stokes lines.

Informally the antiStokes lines are roughly where some term in the asymptotic expansion changes from increasing to decreasing, and the Stokes lines are lines along which some term approaches infinity or zero fastest. AntiStokes lines bound regions where the function has some asymptotic behavior. The Stokes lines and antiStokes lines are not unique and do not really have a precise definition in general, because the region where a function has a given asymptotic behavior is a somewhat vague concept. However the lines do usually have well determined directions at essential singularities of the function, and there is sometimes a natural choice of these lines as follows. The asymptotic expansion of a function is often give by a linear combination of functions of the form f(z)e±g(z) for functions f and g. The Stokes lines can then be taken as the zeros of the imaginary part of g, and the antiStokes lines as the zeros of the real part of g. (This is not quite canonical, because one can add a constant to g, changing the lines.) If the lines are defined like this then they are orthogonal where they meet, unless g has a multiple zero.

As a trivial example, the function sinh(z) has two regions Re(z)>0 and Re(z)<0 where it is asymptotic to ez/2 and ez/2. So the antiStokes line can be taken to be the imaginary axis, and the Stokes line can be taken to be the real axis. One could equally well take the Stokes line to be any line of given imaginary part; these choices differ only by change of variables, showing that there is no canonical choice for the Stokes line.

## Example: the Airy function

The Airy function Ai(x) is one of two solutions to a simple differential equation

$y'' - xy = 0 , \,$

which it is often useful to approximate for many values of x – including complex values. For large x of given argument the solution can be approximated by a linear combination of the functions

$\frac{e^{\pm\frac23x^{3/2}}}{x^{1/4}}.$

However the linear combination has to change as the argument of x passes certain values because these approximations are mutli-valued functions but the Airy function is single valued. For example, if we regard the limit of x as large and real, and would like to approximate the Airy function for both positive and negative values, we would find that

Plot of Ai(x) in red and Bi(x) in blue
\begin{align} \mathrm{Ai}(x) &{}\sim \frac{e^{-\frac23x^{3/2}}}{2\sqrt\pi\,x^{1/4}} \\ \mathrm{Ai}(-x) &{}\sim \frac{\sin(\frac23x^{3/2}+\frac14\pi)}{\sqrt\pi\,x^{1/4}} \\ \end{align}
Stokes lines and antiStokes lines for the Airy function

which are two very different expressions. What has happened is that as we have increased the argument of x from 0 to pi (rotating it around through the upper half complex plane) we have crossed an antiStokes line, which in this case is at $\operatorname{arg}\, x = \pi / 3$. At this antiStokes line, the coefficient of $\frac{e^{-\frac23x^{3/2}}}{x^{1/4}}$ is forced to jump. The coefficient of $\frac{e^{+\frac23x^{3/2}}}{x^{1/4}}$ can jump at this line but is not forced to; it can change gradually as arg x varies from π/3 to π as it is not determined in this region.

There are three antiStokes lines with arguments π/3, π. –π/3, and three Stokes lines with arguments 2π/3, 0. –2π/3.

## Example: second order linear differential equations

The Airy function example can be generalized to solutions of second order linear differential equations as follows. By standard changes of variables the equation can usually be changed to one of the form

$\frac{d^2w}{dz^2} = f(z)w$

where f is holomorphic in a simply connected region and w is a solution of the differential equation. Then in some cases w has an asymptotic approximation as a linear combination of functions of the form

$\frac{\exp{\pm\int_{a}^z\sqrt{f(z)}dz}}{f(z)^{1/4}}$

for some constant a, and the antiStokes lines and Stokes lines can be taken as the zeros of the real and imaginary parts of

$\int_{a}^z\sqrt{f(z)}dz.$

If a is a simple zero of f there are three Stokes lines and three anti-Stokes lines meeting, as in the case of the Airy function where f(z)=z has a simple zero at z=0.


text:
# Simple Antiderivative

Icebreaker
[SOLVED] Simple Antiderivative

How would I compute the antiderivative of

$$\int \sqrt{1-\frac{x^2}{2}}$$

It looks familiar, but I can't quite remember how...

James R
Homework Helper
Gold Member
Put

$$x=\sqrt{2} \sin u$$

and go from there.

dextercioby
Homework Helper
U can also put the Riemann measure on $\mathbb{R}$ : $dx$.

Daniel.

Icebreaker
Ah yes, of course. Thanks.

Quick follow-up:

$$\int\log|\sqrt{1-x^2}+x|$$

Last edited by a moderator:
James R
Homework Helper
Gold Member
I think that integrand is related to an inverse hyperbolic trig function... but I'd have to play around with it to work out which one. Maybe somebody else...

Icebreaker
Perhaps it simply cannot be expressed algebraically?

Icebreaker said:
Perhaps it simply cannot be expressed algebraically?
It cannot be expressed in terms of elementary functions, you are correct.

I love Mathematica

Alex


text:
# zbMATH — the first resource for mathematics

On some series containing $$\psi{}(x)-\psi{}(y)$$ and $$(\psi{}(x)- \psi{}(y))^ 2$$ for certain values of $$x$$ and $$y$$. (English) Zbl 0782.33001
The article under review discusses series containing the factors $$\psi(k)- \psi(1)$$, $$(\psi(k)-\psi(1))^ 2$$, $$\psi({1\over 2}\pm k)- \psi({1\over 2})$$ and $$\psi({1\over 2}(k+1)-\psi({1\over 2}k)$$ with $$k\in\mathbb{N}$$. Here $$\psi(x)=d/dx \log \Gamma(x)$$ where $$\Gamma(x)$$ is the gamma function [see L. Lewin, Polylogarithms and associated functions (1981; Zbl 0465.33001)].

##### MSC:
 33B15 Gamma, beta and polygamma functions 40A25 Approximation to limiting values (summation of series, etc.)
##### Keywords:
dilogarithms; psi-function
Full Text:
##### References:
 [1] Gradshteyn, I.S.; Ryzhik, I.M., Tables of integrals, series and products, (1965), Academic Press New York · Zbl 0918.65002 [2] Hansen, E.L., A table of series and products, (1975), Prentice-Hall, Englewood Cliffs, NJ [3] Lewin, L., Polylogarithms and associated functions, (1981), North-Holland Amsterdam · Zbl 0465.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
×

# Something crazy & New to learn !

All the wonder Brilliant users and all my followers, this day I am going to share something really very interesting and may be something new for many of you.

Inspired from a question by @Calvin Lin sir, I have got some discussion points in my mind.

Hope, you all are aware of functions (make mathematics awesome and easy). Here I would like to talk about the function itself and its inverse.

Inverse of a function will exist, if and only if it is one-one(injective) function and onto (surjective) function. Combining one-one and onto together, the function is called Bijective function.

There comes a property of inverse functions :

Graphs of $$y=f(x)$$ and $$y=f^{-1}(x)$$ are symmetrical about the line $$y=x$$ and intersect on the line $$y=x$$.

OR

$$f(x)=f^{-1}(x)=x$$ whenever graphs intersect.

Now here are some points (questions) :

1. So if we talk about $$f(x)=(\frac{1}{16})^x$$ and its inverse functions $$y=log_{\frac{1}{16}}x$$. How many points of intersection will be there ? The above property says that there will be only 1 solution of $$log_{\frac{1}{16}}x=(\frac{1}{16})^x$$ But algebraically two solutions $$x=\frac{1}{4} and x=\frac{1}{2}$$ So now the point is exactly how many roots are there for the equation $$log_{\frac{1}{16}}x=(\frac{1}{16})^x$$

2. Is the above stated property wrong ?

3. Can there be functions for which $$f(x)=f^{-1}(x)$$ has solutions except which lie on the line $$y=x$$. If yes, then finite or infinite number ?

4. Why does the equation $$log_{\frac{1}{16}}x=(\frac{1}{16})^x$$ not follow the above property ?

Note: You can add more discussion points here and analyse it as much as you can.

Thank you !

## ------------------------------------------------

###### Check out more problems which can solved easily by sketching their graphs, instead of going algebraically. So, try the set : Can you draw its graph ?

Note by Sandeep Bhardwaj
2 years, 5 months ago

Sort by:

Am I missing something? Where in property saying that there will be only 1 solution?

Sure it states that $$f(x)$$ and $$f^{-1}(x)$$ intersect on the line $$y=x$$, but that it. It doesn't say $$f(x)$$ and $$f^{-1}(x)$$ ONLY intersect on line $$y=x$$.

$$f(x),~f^{-1}(x)$$ and $$y=x$$ intersect at $$x\approx0.3642$$

So $$f(x)=\left(\dfrac{1}{16}\right)^x$$ follows this property.

Though this property is not always true (see Sandeep Bhardwaj's comment for counterexample), it is true for all real bijective functions. · 2 years, 5 months ago

What I think is that the property stated above is not true, but this property needs to be modified. And it should be that if $$f(x)=x \space \implies f^{-1}(x)=x \space \implies f(x)=f^{-1}(x)=x$$. It is not saying that every $$f(x)$$ and $$f^{-1}(x)$$ intersect on the line $$y=x$$ only. The graphs of $$f(x)$$ and $$f^{-1}(x)$$ may intersect at points not lying on $$y=x$$ and not necessary that these will cut at $$y=x$$. · 2 years, 5 months ago

Let's see if we can find some more example of functions having this sort of rebellion nature. @Sandeep Bhardwaj · 2 years, 5 months ago

Any function that satisfies $$f^2(x) = x$$, will be a good example. This is because we have $$x = f(y), y = f(x)$$ always, and so there are many many solutions.

Apart from $$f(x) = x$$, the next easiest ones to find are $$f(x) = - x$$, and then $$f(x) = - x + c$$.
You might find $$f(x) = \frac{1}{x}$$ and $$f(x) = \frac{ c x - c^2 + 1 } { x - c }$$. Staff · 2 years, 5 months ago

I have a lot of examples contradicting the above property. And one of being is take any line perpendicular to the line $$y=x$$ .For instance, inverse of $$y=-x$$ is the same $$y=-x$$. So here $$f(x)=f^{-1}(x)$$ has infinitely many solutions $$i.e. every Real Number$$. I think the property stated in the note is false.

Also there are many examples against the property, even though if we don't take it perpendicular to the line $$y=x$$.

I would like to share one example here :

$$f(x)=$$$$\begin{cases} x+4 ,if \space x\in[1,2] \\-x+7, if \space x\in[5,6]\ \end{cases}$$

$$f^{-1}(x)=$$$$\begin{cases} x-4 ,if \space x\in[5,6]\\7-x, if \space x\in[1,2]\ \end{cases}$$

Here graphs of $$y=f(x)$$ and $$y=f^{-1}(x)$$ intersect at $$(\frac{3}{2},\frac{11}{2})$$ and $$(\frac{11}{2},\frac{3}{2})$$ and Also the intersection points don't lie on the line $$y=x$$ · 2 years, 5 months ago

Checkout the th function- (-x^3 +1) it have 4 points of intersection other than that with y=x · 7 months, 2 weeks ago

I think the only property we can talk about is that if we have an ecuation we can write as $$f(x)=g(x)$$ the solutions are the values of $$x$$ where the graph of $$f(x)$$ and $$g(x)$$ have intersections. When we have the coincidence that $$g(x)=f^{-1}(x)$$ we can be sure that some of the solutions will be the intersection points of $$y=f(x)$$ with $$y=x$$. Like in this problem · 1 year, 7 months ago


text:
# NEET Questions Solved

A vibration magnetometer consists of two identical bar magnets placed one over the other such that they are perpendicular and bisect each other. The time period of oscillation in a horizontal magnetic field is ${2}^{5}{4}}$ seconds. One of the magnets is removed and if the other magnet oscillates in the same field, then the time period in seconds is

(a) ${2}^{1}{4}}$                          (b) ${2}^{1}{2}}$

(c) 2                              (d) ${2}^{3}{4}}$

Initially magnetic moment of system ${M}_{1}=\sqrt{{M}^{2}+{M}^{2}}=\sqrt{2}M$

and the moment of inertia ${I}_{1}=I+I=2I$

Finally when one of the magnet is removed then

Difficulty Level:

• 17%
• 21%
• 21%
• 43%


text:
Are all derivatives either Lebesgue integrable or improper Riemann integrable?

The function $f:[-1,1] \to \mathbb{R}$ given by $f(x) = x^2\sin\left(\frac{1}{x^2}\right)$ is an example of a function whose derivative is not Lebesgue integrable on $[-1,1]$ but is improper Riemann integrable.

Volterra constructed a function $f:[0,1] \to \mathbb{R}$ whose derivative is bounded (hence Lebesgue integrable) but not Riemann integrable (proper or improper).

Is there a differentiable function $f:[a,b] \to \mathbb{R}$ such that $f'(x)$ is neither improper Riemann integrable nor Lebesgue integrable on $[a,b]$?

Such a derivative is Henstock–Kurzweil integrable , since all derivatives happen to be Henstock–Kurzweil integrable.

The example here unfortunately does not work. It has jump discontinuities, hence isn't a derivative.


text:
# Math Insight

### Related rates

In this section, most functions will be functions of a parameter $t$ which we will think of as time. There is a convention coming from physics to write the derivative of any function $y$ of $t$ as $\dot{y}=dy/dt$, that is, with just a dot over the functions, rather than a prime.

The issues here are variants and continuations of the previous section's idea about implicit differentiation. Traditionally, there are other (non-calculus!) issues introduced at this point, involving both story-problem stuff as well as requirement to be able to deal with similar triangles, the Pythagorean Theorem, and to recall formulas for volumes of cones and such.

Continuing with the idea of describing a function by a relation, we could have two unknown functions $x$ and $y$ of $t$, related by some formula such as $$x^2+y^2=25$$ A typical question of this genre is ‘What is $\dot{y}$ when $x=4$ and $\dot{x}=6$?’

The fundamental rule of thumb in this kind of situation is differentiate the relation with respect to $t$: so we differentiate the relation $x^2+y^2=25$ with respect to $t$, even though we don't know any details about those two function $x$ and $y$ of $t$: $$2x\dot{x}+2y\dot{y}=0$$ using the chain rule. We can solve this for $\dot{y}$: $$\dot{y}=-{x\dot{x} \over y}$$ So at any particular moment, if we knew the values of $x,\dot{x},y$ then we could find $\dot{y}$ at that moment.

Here it's easy to solve the original relation to find $y$ when $x=4$: we get $y=\pm 3$. Substituting, we get $$\dot{y}=-{4\cdot 6\over \pm 3}=\pm 8$$ (The $\pm$ notation means that we take $+$ chosen if we take $y=-3$ and $-$ if we take $y=+3$).

#### Exercises

1. Suppose that $x,y$ are both functions of $t$, and that $x^2+y^2=25$. Express ${ dx \over dt }$ in terms of $x,y,$ and ${ dy \over dt }$. When $x=3$ and $y=4$ and ${ dy \over dt }=6$, what is${ dx \over dt }$?
2. A 2-foot tall dog is walking away from a streetlight which is on a 10-foot pole. At a certain moment, the tip of the dog's shadow is moving away from the streetlight at 5 feet per second. How fast is the dog walking at that moment?
3. A ladder $13$ feet long leans against a house, but is sliding down. How fast is the top of the ladder moving at a moment when the base of the ladder is $12$ feet from the house and moving outward at $10$ feet per second?


text:
statsmodels.regression.rolling.RollingRegressionResults¶

class statsmodels.regression.rolling.RollingRegressionResults(model, store: statsmodels.regression.rolling.RollingStore, k_constant, use_t, cov_type)[source]

Results from rolling regressions

Parameters
modelRollingWLS

Model instance

storeRollingStore

Container for raw moving window results

k_constantbool

Flag indicating that the model contains a constant

use_tbool

Flag indicating to use the Student’s t distribution when computing p-values.

cov_typestr

Name of covariance estimator

Attributes
aic

Akaike’s information criteria.

For a model with a constant $$-2llf + 2(df\_model + 1)$$. For a model without a constant $$-2llf + 2(df\_model)$$.

bic

Bayes’ information criteria.

For a model with a constant $$-2llf + \log(n)(df\_model+1)$$. For a model without a constant $$-2llf + \log(n)(df\_model)$$.

bse

The standard errors of the parameter estimates.

centered_tss

The total (weighted) sum of squares centered about the mean.

cov_type

Name of covariance estimator

df_model

The model degree of freedom.

The dof is defined as the rank of the regressor matrix minus 1 if a constant is included.

df_resid

The residual degree of freedom.

The dof is defined as the number of observations minus the rank of the regressor matrix.

ess

The explained sum of squares.

If a constant is present, the centered total sum of squares minus the sum of squared residuals. If there is no constant, the uncentered total sum of squares is used.

f_pvalue

The p-value of the F-statistic.

fvalue

F-statistic of the fully specified model.

Calculated as the mean squared error of the model divided by the mean squared error of the residuals if the nonrobust covariance is used. Otherwise computed using a Wald-like quadratic form that tests whether all coefficients (excluding the constant) are zero.

k_constant

Flag indicating whether the model contains a constant

llf

Log-likelihood of model

mse_model

Mean squared error the model.

The explained sum of squares divided by the model degrees of freedom.

mse_resid

Mean squared error of the residuals.

The sum of squared residuals divided by the residual degrees of freedom.

mse_total

Total mean squared error.

The uncentered total sum of squares divided by the number of observations.

nobs

Number of observations n.

params

Estimated model parameters

pvalues

The two-tailed p values for the t-stats of the params.

rsquared

R-squared of the model.

This is defined here as 1 - ssr/centered_tss if the constant is included in the model and 1 - ssr/uncentered_tss if the constant is omitted.

rsquared_adj

Adjusted R-squared.

This is defined here as 1 - (nobs-1)/df_resid * (1-rsquared) if a constant is included and 1 - nobs/df_resid * (1-rsquared) if no constant is included.

ssr

Sum of squared (whitened) residuals.

tvalues

Return the t-statistic for a given parameter estimate.

uncentered_tss

Uncentered sum of squares.

The sum of the squared values of the (whitened) endogenous response variable.

use_t

Flag indicating to use the Student’s distribution in inference.

Methods

 conf_int([alpha, cols]) Construct confidence interval for the fitted parameters. Estimated parameter covariance load(fname) Load a pickled results instance plot_recursive_coefficient([variables, …]) Plot the recursively estimated coefficients on a given variable Remove data arrays, all nobs arrays from result and model. save(fname[, remove_data]) Save a pickle of this instance.

Methods

 conf_int([alpha, cols]) Construct confidence interval for the fitted parameters. Estimated parameter covariance load(fname) Load a pickled results instance plot_recursive_coefficient([variables, …]) Plot the recursively estimated coefficients on a given variable Remove data arrays, all nobs arrays from result and model. save(fname[, remove_data]) Save a pickle of this instance.

Properties

 aic Akaike’s information criteria. bic Bayes’ information criteria. bse The standard errors of the parameter estimates. centered_tss The total (weighted) sum of squares centered about the mean. cov_type Name of covariance estimator df_model The model degree of freedom. df_resid The residual degree of freedom. ess The explained sum of squares. f_pvalue The p-value of the F-statistic. fvalue F-statistic of the fully specified model. k_constant Flag indicating whether the model contains a constant llf Log-likelihood of model mse_model Mean squared error the model. mse_resid Mean squared error of the residuals. mse_total Total mean squared error. nobs Number of observations n. params Estimated model parameters pvalues The two-tailed p values for the t-stats of the params. rsquared R-squared of the model. rsquared_adj Adjusted R-squared. ssr Sum of squared (whitened) residuals. tvalues Return the t-statistic for a given parameter estimate. uncentered_tss Uncentered sum of squares. use_t Flag indicating to use the Student’s distribution in inference.


text:
# HiSET: Math : Cones

## Example Questions

← Previous 1

### Example Question #1 : Cones

About the -axis, rotate the triangle with its vertices at , , and the origin. What is the volume of the solid of revolution formed?

None of the other choices gives the correct response.

Explanation:

When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . Substitute these values into the formula for the volume of a cone:

### Example Question #2 : Cones

About the x-axis, rotate the triangle with its vertices at , , and the origin. What is the volume of the solid of revolution formed?

None of the other choices gives the correct response.

Explanation:

When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . Substitute these values into the formula for the volume of a cone:

### Example Question #1 : Cones

A right square pyramid has height 10 and a base of perimeter 36.

Inscribe a right cone inside this pyramid. What is its volume?

None of the other choices gives the correct response.

Explanation:

The length of one side of the square is one fourth of its perimeter, or . The cone inscribed inside this pyramid has the same height. Its base is the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 9, and, as its radius, half this, or .

The volume of a cone, given radius and height , can be calculated using the formula

Set and :

### Example Question #111 : Measurement And Geometry

A right square pyramid has height 10 and a base of area 36.

Inscribe a right cone inside this pyramid. What is its volume?

None of the other choices gives the correct response.

Explanation:

The length of one side of the square is the square root of the area, or  . The cone inscribed inside this pyramid will have as its base the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 6, and, as its radius, half this, or 3.

The volume of a cone, given radius and height , can be calculated using the formula

Set  and :

### Example Question #1 : Cones

A cylinder has volume 120. A cone with base the same size as a base of the cylinder has the same height. Give the volume of the cone.

Insufficient information is given to answer the question.

Explanation:

A cone with the same height as a given cylinder and a base the same radius as those of that cylinder has as its volume one-third that of the cylinder. That makes the volume of the cone one-third of 120, or

### Example Question #11 : Use Volume Formulas To Solve Problems

A right cone has height 10 and slant height 20. Which of the following correctly gives its volume? (Round to the nearest whole number).

Explanation:

The volume of a cone, given radius and height , can be calculated using the formula

.

We are given that , but we are not given the value of . We are given that , and since the cone is a right cone, its radius, height, and slant height can be related using the Pythagorean relation

.

Substituting 10 for  and 20 for , we can find :

, which is what we need in the formula.

Now substitute in the volume formula:

This rounds to 3,142

### Example Question #7 : Cones

A cone has as its base a circle whose radius is twice that of a base of a given cylinder; its height is 20% greater than that of the cylinder. Which of the following is true of the volume of the cone?

The volume of the cone is equal than that of the cylinder.

The volume of the cone is 60% less than that of the cylinder.

The volume of the cone is 20% less than that of the cylinder.

The volume of the cone is 20% greater than that of the cylinder.

The volume of the cone is 60% greater than that of the cylinder.

The volume of the cone is 60% greater than that of the cylinder.

Explanation:

The volume of a cylinder with a base of radius and with height is

The cone has radius twice that of the cylinder, which is . Its height is 20% greater than, or 120% of, that of the cylinder, which is equal to .

The volume of a cone with a base of radius and height is

Set and :

Substitute  for :

.

This means that the volume of the cone is  times that of  the cylinder—or 60% greater.

### Example Question #1 : Cones

About the -axis, rotate the triangle whose sides are along the -axis, the -axis, and the line of the equation

.

Give the volume of the solid of revolution formed.

None of the other choices gives the correct response.

Explanation:

The vertices of the triangle are the points of intersection of the three lines. The -axis and the -axis meet at the origin .

The point of intersection of the -axis and the graph of —the -intercept of the latter—can be found by substituting 0 for :

Divide both sides by 2 to isolate :

The point of intersection is at .

Similarly, the point of intersection of the -axis and the graph of —the -intercept of the latter—can be found by substituting 0 for :

The point of intersection is at .

The three vertices of the triangle are at the origin, , and . When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . Substitute these values into the formula for the volume of a cone:

### Example Question #11 : Use Volume Formulas To Solve Problems

A right cone has height 20; its base has radius 10. Which of the following correctly gives its volume? (Round to the nearest whole number).

Explanation:

The volume of a cone, given radius and height , can be calculated using the formula

.

We are given that and , so we can substitute and calculate:

To the nearest whole, this is 2,094.

### Example Question #10 : Cones

A cone has as its base a circle whose radius is 20% greater that of a base of the cylinder; its height is twice that of the cylinder. Which of the following is true of the volume of the cone?

The volume of the cone is 4% greater than that of the cylinder.

The volume of the cone is 52% less than that of the cylinder.

The volume of the cone is equal than that of the cylinder.

The volume of the cone is 4% less than that of the cylinder.

The volume of the cone is 52% greater than that of the cylinder.

The volume of the cone is 4% less than that of the cylinder.

Explanation:

The volume of a cylinder with a base of radius and with height is

The cone has height twice that of the cylinder, which is . Its radius is 20% greater than, or 120% of, that of the cylinder, which is equal to .

The volume of a cone with a base of radius and height is

Set  and :

Substitute  for :

This means that the volume of the cone is  that of the cylinder—or   less.

← Previous 1


text:
# How to reduce 2-coloring to 2-SAT

Let $G=(V,E)$ be an undirected graph in which $V = \{v_1,v_2,\dots,v_n\}$.

I need to describe a reduction from the 2-coloring problem to the 2-SAT problem (in polynomial time).

I thought of splitting each vertex $v_i$ into $v_i^*$ and $v_i^{**}$, one for each color. Then I was thinking about making for each edge $(u,v)\in E$, two correspond literals: $(u^* \lor v^*)$ and $(u^{**} \lor v^{**})$.

I know I'm on the right track but I can't prove it.

any ideas?

If all you require is a polynomial-time reduction, then test in polynomial time if $G$ is $2$-colourable. If it is, map it to the formula $X$; if it isn't, map it to $X\land\lnot X$.

Consider a graph $G=(V,E)$.

Given a node $v_i \in V$ as you did, you can split into 2 variables $v_{i,1}$ and $v_{i,2}$ representing the 2 colors.

Now you just need 3 kind of clauses:

• each node cannot have more than one color
• Each node must have assigned a color
• $\forall$ edge $(u,v) \in E$, $u$ and $v$ cannot have the same color.

Then you just have to write it as a conjunction of clauses, where each clause is a disjunction of 2 literals.

G admits a 2-coloring $\iff$ there exists a truth assignment that satisfy the formula.


text:
# MESB 374 System Modeling and Analysis Inverse Laplace Transform and I/O Model.

## Presentation on theme: "MESB 374 System Modeling and Analysis Inverse Laplace Transform and I/O Model."— Presentation transcript:

MESB 374 System Modeling and Analysis Inverse Laplace Transform and I/O Model

Inverse Laplace Transform Basic steps Partial fraction expansion (PFE) Residue command in Matlab Input-output model by using Laplace transform

Inverse Laplace Transform Given an s-domain function F(s), the inverse Laplace transform is used to obtain the corresponding time domain function f (t). Procedure: –Write F(s) as a rational function of s. –Use long division to write F(s) as the sum of a strictly proper rational function and a quotient part. –Use Partial-Fraction Expansion (PFE) to break up the strictly proper rational function as a series of components, whose inverse Laplace transforms are known. –Apply inverse Laplace transform to individual components.

Partial Fraction Expansion Case I: Distinct Characteristic Roots Residual Formula Proof

Partial Fraction Expansion Case II: Repeated Roots Residual Formula Proof

Partial Fraction Expansion Residual Formula Case III: General Case

Partial Fraction Expansion Case IV: Order of the Numerator C(s) = Order of the denominator D(s) : n = m

Partial Fraction Expansion Case V: Complex Roots

Residue Command in MATLAB [A, P, K] = residue (num, den) num: vector of coefficients of the numerator; Den: vector of coefficients of the denominator; A: vector of the coefficients in PFE, i.e., P: vector of the roots, i.e., K:constant term.

Residue Command in MATLAB (Example) Ex: Given MATLAB command: >> [A, P, K] = residue ( [1, 0, 0, 2], [1, 2, 1, 0] ) will return the following values: A = [ -4, -1, 2] T, P = [ -1, -1, 0], K = 1 which means that Find: inverse Laplace transform

Obtaining I/O Model Using LT (Laplace Transformation Method) –Use LT to transform all time-domain differential equations into s-domain algebraic equations assuming zero ICs –Solve for output in terms of inputs in s-domain –Write down the I/O model based on solution in s- domain

Step 1: LT of differential equations assuming zero ICs Example – Car Suspension System x p Step 2: Solve for output using algebraic elimination method 1.# of unknown variables = # equations ? 2.Eliminate intermediate variables one by one. To eliminate one intermediate variable, solve for the variable from one of the equations and substitute it into ALL the rest of equations; make sure that the variable is completely eliminated from the remaining equations

Example (Cont.) Step 3: write down I/O model from first equation Substitute it into the second equation

Download ppt "MESB 374 System Modeling and Analysis Inverse Laplace Transform and I/O Model."

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text:
# Homework Help: LC circuit

1. Oct 31, 2016

### RingNebula57

1. The problem statement, all variables and given/known data
In an oscillating circuit consisting of of a parallel-plate capacitor and an inductance coil with negligible active resistance the oscillations with energy $W$ are sustained. The capacitor plates were slowly drawn apart to increase the scillation frequency $\eta$-fold. What work was done in the process?

2. Relevant equations
$C = \frac{\epsilon_0 \cdot S}{d}$
$C'=\frac{\epsilon_0 \cdot S}{\eta^2 d}$

3. The attempt at a solution
So ohm's law is written like this: (before the stretching of the capacitor)

$L \cdot \ddot{q} = -\frac{q}{C}$
This implies:
$\omega^2=\frac{1}{LC}$
$q_{(t)}=q_m \cdot sin(\omega t)$

The electrostatic force exerted by one plate on the other is:

$F = E_{(t)} \cdot q_{(t)}$
$E_{(t)} = \frac{\sigma}{2\epsilon_0} = \frac{q_{(t)}}{2\epsilon_0S}$ $\Bigg\}$ $F = \frac{q_{(t)}^2}{2\epsilon_0S}=\frac{q_m^2 \cdot sin(\omega t)^2}{2\epsilon_0S}$

Because the streching of the capacitor happened slowly it means that we can consider the average force over one period:

$\bar{F}=\frac{q_m^2}{4\epsilon_0S}$
So , the work done is then:
$\Delta W= \bar{F}\cdot d\cdot (\eta^2-1) = \frac{q_m^2d}{4\epsilon_0S} \cdot (\eta^2-1)$

Now,
$W=\frac{q_m^2}{2C}=\frac{q_m^2d}{2\epsilon_0S}$

From here results that :
$\Delta W= \frac{W}{2} \cdot (\eta^2-1)$

This formula implies that the initial charge $q_m$ is not conserved. Is it ok? It looked odd to me...

2. Oct 31, 2016

### TSny

I believe this is correct.

This assumes that $\bar{F}$ remains constant while the plates are separated. But as you found, $q_m$ is not conserved. So, the average force is not constant.

Yes

For notational convenience, I suggest letting $x$ stand for the distance between the plates. Consider the work done by the average applied force for an infinitesimal increase in separation, $dx$.

3. Oct 31, 2016

### RingNebula57

Thank you!

I solved it. Obtaining that :

$\Delta W = W \cdot (\eta-1)$

4. Oct 31, 2016

Looks good!


text:
Dudley, R. M. (1973) Sample Functions of the Gaussian Process
     @article{1973,
jstor_articletype = {research-article},
title = {Sample Functions of the Gaussian Process},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {1},
number = {1},
jstor_formatteddate = {Feb., 1973},
pages = {pp. 66-103},
url = {http://www.jstor.org/stable/2959347},
ISSN = {00911798},
abstract = {This is a survey on sample function properties of Gaussian
processes with the main emphasis on boundedness and continuity, including
Holder conditions locally and globally. Many other sample function
properties are briefly treated. The main new results continue the program
of reducing general Gaussian processes to "the" standard isonormal linear
process $L$ on a Hilbert space $H$, then applying metric entropy methods.
In this paper Holder conditions, optimal up to multiplicative constants,
are found for wide classes of Gaussian processes. If $H$ is $L^2$ of
Lebesgue measure on $R^k, L$ is called "white noise." It is proved that we
can write $L = P(D)\lbrack x\rbrack$ in the distribution sense where $x$
has continuous sample functions if $P(D)$ is an elliptic operator of degree
$> k/2$. Also $L$ has continuous sample functions when restricted to
indicator functions of sets whose boundaries are more than $k - 1$ times
differentiable in a suitable sense. Another new result is that for the
Levy(-Baxter) theorem $\int^1_0(dx_t)^2 = 1$ on Brownian motion, almost
sure convergence holds for any sequence of partitions of mesh $o(1/\log n)$. If partitions into measurable sets other than intervals are allowed,
the above is best possible: $\mathscr{O}(1/\log n)$ is insufficient.},
language = {English},
year = {1973},
publisher = {Institute of Mathematical Statistics},
}

Dudley, R. M. (1978) Central Limit Theorems for Empirical Measures
    @article{1978,
jstor_articletype = {research-article},
title = {Central Limit Theorems for Empirical Measures},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {6},
number = {6},
jstor_formatteddate = {Dec., 1978},
pages = {pp. 899-929},
url = {http://www.jstor.org/stable/2243028},
ISSN = {00911798},
abstract = {Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution
$P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the
empirical measure and let $\nu_n := n^\frac{1}{2}(P_n - P)$. Given a class
$\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of
$\nu_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain
Gaussian process indexed by $\mathscr{C}$. If convergence holds with
respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a
suitable (usually nonseparable) function space, we call $\mathscr{C}$ a
Donsker class. For measurability, $X$ may be a complete separable metric
space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection
of closed sets or open sets. Then for the Donsker property it suffices that
for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$
does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another
sufficient condition is based on metric entropy with inclusion. If
$\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then
$\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.},
language = {English},
year = {1978},
publisher = {Institute of Mathematical Statistics},
}

@article{1979,
jstor_articletype = {research-article},
title = {Corrections to "Central Limit Theorems for Empirical Measures"},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {7},
number = {5},
jstor_formatteddate = {Oct., 1979},
pages = {pp. 909-911},
url = {http://www.jstor.org/stable/2243316},
ISSN = {00911798},
abstract = {},
language = {English},
year = {1979},
publisher = {Institute of Mathematical Statistics},
}

Dudley, R. M. (1987) Universal Donsker Classes and Metric Entropy
     @article{1987,
jstor_articletype = {research-article},
title = {Universal Donsker Classes and Metric Entropy},
author = {Dudley, R. M.},
journal = {The Annals of Probability},
jstor_issuetitle = {},
volume = {15},
number = {4},
jstor_formatteddate = {Oct., 1987},
pages = {pp. 1306-1326},
url = {http://www.jstor.org/stable/2244004},
ISSN = {00911798},
abstract = {Let (X, A) be a measurable space and F a class of measurable
functions on X. F is called a universal Donsker class if for every
probability measure P on A, the centered and normalized empirical measures
n1/2(Pn - P) converge in law, with respect to uniform convergence over F,
to the limiting "Brownian bridge" process GP. Then up to additive
constants, F must be uniformly bounded. Several nonequivalent conditions
are shown to imply the universal Donsker property. Some are connected with
the Vapnik-Cervonenkis combinatorial condition on classes of sets, others
with metric entropy. The implications between the various conditions are
considered. Bounds are given for the metric entropy of convex hulls in
Hilbert space.},
language = {English},
year = {1987},
publisher = {Institute of Mathematical Statistics},


text:
# A seeming paradox in a coin-flipping game

This is related to my other question on a similar topic.

Suppose we play the following game: we flip a coin repeatedly and record the outcomes. For example we might get HHTTTHTTHHTTT.... Now Alice and Bob each choose distinct patterns of the same length, called $A$ and $B$, respectively. Which ever player's pattern appears first wins the game.

Now suppose Alice chooses $A=$HHHH and Bob chooses $B=$HHHT. First, let's notice that the expected number of flips to obtain $A$ is 30, but for $B$ it's only 16. This would seem to imply that Bob is very likely to win this game most of the time.

However, thinking about the game in another way, neither Alice nor Bob can win until HHH occurs. And after this, the game ends on the next flip with each player winning equiprobably.

This seems counter-intuitive to me: we have two events, one expected to occur much sooner than the other, but relative to each other the ordering is 50-50. What am I missing?

-


text:
## Abstract

BACKGROUND: Obtaining laboratory studies to screen for a coagulation disorder prior to tonsillectomy and/or adenoidectomy (T&A) is a controversial procedure. The American Society of Pediatric Otolaryngology advises that such studies only be performed when indicated by history or physical exam. Nevertheless, many surgeons in clinical practice proceed with testing on all children scheduled for T&A. Several observational studies of pre-operative testing have been performed over the past four decades with conflicting results. The cost-effectiveness of this screening has not been studied in a rigorous, formal manner. We constructed a decision analysis model to address the costs and health outcome states of coagulation screening strategies for children undergoing T&A.

METHODS: A 14-day Markov model was constructed to test three pre-operative screening strategies:

1. test all children for coagulation disorders;

2. test only those children with a pertinent history or physical exam (PE) finding; and

3. perform no pre-operative testing on any child.

We performed a literature search to estimate the prevalence of abnormal pre-operative coagulation tests, the prevalence of true coagulation disorders among those patients, the rates of bleeding following T&A, and the mortality following T&A. From published sources, we obtained cost data on laboratory studies, hospital admissions (surgical and observational), hematology consultation, medications for prophylaxis and treatment of a bleeding disorder, and lost parental wages. We estimated quality of life (utility) data from published national sources. Parameters were varied widely in sensitivity analyses based upon the range of reported values. Using a societal perspective and a cycle length of one day, we compared the three strategies based on total costs and quality-adjusted life-years (QALYs).

RESULTS: Total costs for the three strategies were $3200 for testing all children,$3083 for testing only children with a history or PE finding, and \$3077 for not testing children. Total utilities followed a reversed pattern with 0.02579 QALYs for testing all, 0.02654 QALYs for testing some, and 0.02659 QALYs for not testing. These results indicate that the strategy of not testing children prior to surgery dominates the others; it is both less expensive and has a higher utility. Cost-effectiveness ratios were sensitive to variation in the overall prevalence of post-operative bleeding, the cost of post-operative care, and the cost of diagnostic coagulation testing after a post-operative hemorrhage. However, the strategy of not testing children pre-operatively was dominant throughout all sensitivity analyses.

DISCUSSION: Our results demonstrate that universally screening all children for coagulation disorders prior to T&A is not cost-effective. Furthermore, not performing any preoperative testing appears to be the most cost-effective strategy. This study does not prove the superiority of one strategy over another, though it does offer advantages over single institution observational studies. Given the short time duration of the model, absolute cost and utility differences between strategies are small. However, these differences are persistent in sensitivity analyses, indicating that the model is robust. The analysis provides evidence that suggests pre-T&A coagulation testing in children should be used judiciously at most. Detailed conversations between surgeon, patient, and family about the risks, benefits, and costs of such testing should determine its selective use.

Disclosures: No relevant conflicts of interest to declare.

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