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## High Probability Generalization Bounds with Fast Rates for Minimax Problems

29 Sept 2021, 00:31 (modified: 14 Mar 2022, 15:38)ICLR 2022 PosterReaders: Everyone
Abstract: Minimax problems are receiving an increasing amount of attention in a wide range of applications in machine learning (ML), for instance, reinforcement learning, robust optimization, adversarial learning, and distributed computing, to mention but a few. Current studies focus on the fundamental understanding of general minimax problems with an emphasis on convergence behavior. As a comparison, there is far less work to study the generalization performance. Additionally, existing generalization bounds are almost all derived in expectation, and the high probability bounds are all presented in the slow order $\mathcal{O}(1/\sqrt{n})$, where $n$ is the sample size. In this paper, we provide improved generalization analyses and obtain sharper high probability generalization bounds for most existing generalization measures of minimax problems. We then use the improved learning bounds to establish high probability generalization bounds with fast rates for classical empirical saddle point (ESP) solution and several popular gradient-based optimization algorithms, including gradient descent ascent (GDA), stochastic gradient descent ascent (SGDA), proximal point method (PPM), extra-gradient (EG), and optimistic gradient descent ascent (OGDA). In summary, we provide a systematical analysis of sharper generalization bounds of minimax problems.
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# HGNChelper Introduction

## Why HGNChelper?

Physicians and biologists like gene symbols and bioinformaticians hate’em. Why? For one thing, they change constantly and are given new names or aliases. For another, some get munged into dates when imported into spreadsheet programs - and not only Excel (Thank you @karawoo for the picture!):

Myself (Levi speaking), I don’t mind them. It’s way easier to remember TP53 than to remember 7157 or ENSG00000141510. They’re a fact of life. So Markus Riester and I wrote HGNChelper to make them a little more pleasant to bioinformaticians.

## HGNChelper functionality

HGNChelper has several functions that seemed useful back in the day when we first wrote it, but really one has withstood the test of time and remained useful:

checkGeneSymbols(x, unmapped.as.na = TRUE, map = NULL, species = "human")

checkGeneSymbols identifies HGNC human or MGI mouse gene symbols which are outdated or may have been mogrified by Excel or other spreadsheet programs. It returns a data.frame of the same number of rows as the input, with a second column indicating whether the symbols are valid and a third column with a corrected gene list.

library(HGNChelper)
human = c("FN1", "tp53", "UNKNOWNGENE","7-Sep", "9/7", "1-Mar", "Oct4", "4-Oct",
"OCT4-PG4", "C19ORF71", "C19orf71")
checkGeneSymbols(human)
#> Maps last updated on: Thu Oct 24 12:31:05 2019
#> Warning in checkGeneSymbols(human): Human gene symbols should be all upper-
#> case except for the 'orf' in open reading frames. The case of some letters
#> was corrected.
#> Warning in checkGeneSymbols(human): x contains non-approved gene symbols
#>              x Approved   Suggested.Symbol
#> 1          FN1     TRUE                FN1
#> 2         tp53    FALSE               TP53
#> 3  UNKNOWNGENE    FALSE               <NA>
#> 4        7-Sep    FALSE            SEPTIN7
#> 5          9/7    FALSE            SEPTIN7
#> 6        1-Mar    FALSE MTARC1 /// MARCHF1
#> 7         Oct4    FALSE             POU5F1
#> 8        4-Oct    FALSE             POU5F1
#> 9     OCT4-PG4    FALSE           POU5F1P4
#> 10    C19ORF71    FALSE           C19orf71
#> 11    C19orf71     TRUE           C19orf71

As you see, it even helps fix capitalization. How does it fix those Excel dates? I imported a column of all human gene symbols into Excel, then exported using a whole bunch of available date formats. Then I kept any that differed from the originals for HGNChelper’s map.

## Mouse gene symbols

Warning the list of valid mouse symbols seems to be incomplete, see below. Mouse genes work the same way, but you need to specify the argument species=mouse:

checkGeneSymbols(c("1-Feb", "Pzp", "A2m"), species="mouse")
#> Maps last updated on: Thu Oct 24 12:31:05 2019
#> Warning in checkGeneSymbols(c("1-Feb", "Pzp", "A2m"), species = "mouse"): x
#> contains non-approved gene symbols
#>       x Approved Suggested.Symbol
#> 1 1-Feb    FALSE             Feb1
#> 2   Pzp    FALSE             <NA>
#> 3   A2m    FALSE         AI893533

I don’t work with mouse data, so use this functionality with care and please let me know if you have any suggestions. For one thing, Pzp in the above example is a valid gene symbol, but it is not in the MGI_EntrezGene.rpt file I used to build a map. Suggestions welcome about how to build a more complete map of valid symbols. To be on the safe side, you could set unmapped.as.na = FALSE to keep unrecognized symbols as-is and only correct ones that have a definitive correction.

## What exactly checkGeneSymbols does

HGNChelper does the following corrections:

1. fix capitalization (for human only). Only orf genes are allowed to have lower-case letters.
2. fix Excel-mogrified symbols
3. fix symbols that are listed as aliases to a more recent symbol in the HGNC or MGI (MGI_EntrezGene_rpt) database.

Numbers 2 and 3 are done by comparing to a complete map of both valid and invalid but mappable symbols, shipped with HGNChelper:

dim(mouse.table)
#> [1] 135053      2
dim(hgnc.table)
#> [1] 98431     2

These are a combination of manually generated Excel mogrifications that remain constant, and aliases that can become out of date with time.

## Updating maps of aliased gene symbols

Gene symbols are aliased much more frequently than I can update this package, like every day. If you want the most current maps of aliases, you can either:

1. use the getCurrentHumanMap() or getCurrentMouseMap() function, and provide the returned result through the map= argument of checkGeneSymbols(), or
2. See the instructions for updating your package locally at https://github.com/waldronlab/HGNChelper (it’s just one command-line command as long as you have R and roxygen2 installed)

## Where do I find HGNChelper?

Please report any issues at https://github.com/waldronlab/HGNChelper/issues

format: plain


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# How to convert km/h to m/s squared

1. Sep 25, 2005

### mike98020205

I guess I'm confused on how to convert km/h to m/s squared.

I understand 1min/60s=360s/1 hr but this seems different. the seconds squared ared throwing me off. I know there's a 1000 meters in a kilometer and 360 seconds in an hour. I guess I don't know how to apply these numbers. Thanks for your help.

2. Sep 25, 2005

### Poncho

1hr = 3600s

The seconds aren't squared

$$\frac{km}{hr}*\frac{hr}{3600s}*\frac{1000m}{1km}=\frac{m}{s}$$

3. Sep 25, 2005

Ty so much.


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Finding a monochromatic cycle in clique

1) what is the minimum $n$ such that in every $2$ - coloring of $K_n$ there exist a monochromatic copy of $C_m$ ?

2) moreover, what is the minimum $n$ such that in every $r$ - coloring of $K_n$ there exist a monochromatic cycle?

I found that the answer to 1 is known, but I don't find it. http://www.mate.unlp.edu.ar/~liliana/lawclique_2016/21.pdf - (page 24)

Let me elaborate on a comment of bof. For positive integers $r$ and $s$, let $c(r,s)$ denote the smallest positive integer $n$ such that if we color the edges of a clique on $n$ vertices with red and blue, either the red subgraph contains a cycle of length $r$ or the blue subgraph contains a cycle of length $s$. The main theorem of Faudree and Schelp [All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974), 313-329] is the following.
(i) If $3\le s\le r$ and $s$ is odd and $(r,s)\ne (3,3)$, then $c(r,s)=2r-1$.
(ii) If $4\le s\le r$ and $s$ and $r$ are even and $(r,s)\ne (4,4)$, then $c(r,s)=r+\frac12s+1$.
(iii) If $4\le s < r$ and $s$ is even and $r$ is odd, then $c(r,s)=\max(r+\frac12s-1,2s-1)$.
So the answer to (1), to guarantee a monochromatic copy of $C_m$, comes by setting $r=s=m$. To complete the answer, we mention that $c(3,3)=c(4,4)=6$.


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# Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular representation theory, Brauer characters, blocks, and defect groups are introduced.

My question is this: How do we find the irreducible Brauer characters and blocks, given a group and a prime?

For instance, let's say we have $p=3$ and the group $G = S_5$, the symmetric group. An example of the precise calculation or method used to determine the characters would be very helpful. Thanks.

-
What have you done so far? Do you know how many irreps there are? Have you determined which ones come from reduction (mod 3)? –  Daniel McLaury Aug 4 '13 at 5:04
There are 5 conjugacy classes with element orders prime to 3, so there are 5 irreducible Brauer characters. But I am unsure how to proceed, given that we do not have the F-representations to begin with. –  BharatRam Aug 4 '13 at 8:25
If you want to learn to compute specific examples, then there is an excellent book that explains how GAP works: math.rwth-aachen.de/~RepresentationsOfGroups –  Jack Schmidt Aug 4 '13 at 15:31

This is a difficult question and you would probably need to learn more theory in order to understand the different methods available.

But one method that is often used in practice is to calculate the representations and then just find the Brauer character directly from the matrices of the representations. Of course, you have to express the traces of the matrices as sums of roots of unity over the finite field, and then lift this sum to the complex numbers to get the Brauer character, but that is not particularly difficult. (That is not completely true - since the lifting is not uniquely defined, you may have to work hard if you want to make it consistent.)

With ordinary (complex) character tables, it is generally much easier to calculate the characters than the matrices that define the representations, but that is not always the case with modular representations. There are fast algorithms for computing representations over finite fields, using the so-called MeatAxe algorithm.

I am more familiar with Magma than with GAP, and I expect there are similar commands in GAP, but in Magma I can just type

> G := Sym(5);
> I := AbsolutelyIrreducibleModules(G, GF(3));

and I get the five absolutely irreducible representations in characteristic three as group homomorphisms, and so I can just look at the images of elements from the different conjugacy classes. There is a Magma command that does this for you, giving the Brauer character table:

> [BrauerCharacter(i): i in I];
[
( 1, 1, 1, 0, 1, 1, 0 ),
( 1, -1, 1, 0, -1, 1, 0 ),
( 4, 2, 0, 0, 0, -1, 0 ),
( 4, -2, 0, 0, 0, -1, 0 ),
( 6, 0, -2, 0, 0, 1, 0 )
]

-


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# Can't put formatted tag names in tag wiki excerpt?

If you enter the text [tag:tag] in the tag wiki textbox, it produces .

If you enter the text [tag:tag] in the tag wiki excerpt textbox, it produces [tag:tag].

Why can't you include formatted tag names in the tag wiki excerpt? Both the tag wiki and the tag wiki excerpt make use of MathJax, but why don't they both have the tag name capability?

The important point above is plain text, which means no Markdown. So not only are not available, but even bold, italic, code, and hyperlinks are not processed.


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## Sales, production, purchases, and cash budgets

Rolen, Inc., is in the process of preparing the fourth quarter budget for 2010, and the following data have been assembled: • The company sells a single product at a price of $25 per unit. The estimated sales volume for the next six months is as follows: September 13,000 units October 12,000 units November 14,000 units December 20,000 units January 9,000 units February 10,000 units • All sales are on account. The company's collection experience has been that 30% of a month's sales are collected in the month of sale, 68% are collected in the month following the sale, and 2% are uncollectible. It is expected that the net realizable value of accounts receivable (i.e., accounts receivable less allowance for uncollectible accounts) will be$211,000 on September 30, 2010. • Management's policy is to maintain ending finished goods inventory each month at a level equal to 30% of the next month's budgeted sales. The finished goods inventory on September 30, 2010, is expected to be 3,600 units. • To make one unit of finished product, 5 pounds of materials are required. Management's policy is to have enough materials on hand at the end of each month to equal 40% of the next month's estimated usage. The raw materials inventory is expected to be 25,200 pounds on September 30, 2010. • The cost per pound of raw material is $2, and 70% of all purchases are paid for in the month of purchase; the remainder is paid in the following month. The accounts payable for raw material purchases is expected to be$37,980 on September 30, 2010. (d) Prepare a materials purchases budget in pounds, by month and in total, for the fourth quarter of 2010. (Amounts to be deducted should be indicated with minus sign.) October November December Total Beginning inventory of raw materials Purchases of raw materials Raw materials available for use Desired ending inventory of raw materials (40% of next month's estimated usage) Quantity of raw materials to be used in production

• b)

Months

Total of fourth QUARTER

Sales of

September

October

November

December

January

February

September

$2,11,000$2,11,000

October

$90,000$2,04,000

$2,94,000 November$1,05,000

$2,38,000$3,43,000

December

$1,50,000$1,50,000

January

February

Total

$3,01,000$3,09,000

$3,88,000$9,98,000

c)

Months

Total of fourth QUARTER

Production

September

October

November

December

January

February

Expected unit sales

12000

14000

20000

9000

10000

46000

Add Desired ending finished goods units

4,200

6,000

2,700

3,000

2700

Total required units

16,200

20,000

22,700

12,000

48,700

Less: Beginning finished goods units

3600

4,200

6,000

2,700

3600

Required production units

12,600

15,800

16,700

9,300

45,100

Jan na dfeb numbers are just to show the ending and beginning inventory.


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# Business Law and the Legal Environment Study Set 1

## Quiz 11 :Cyberlaw

Question Type
On July 24, 2002, the Recording Industry Association of America (RIAA) served its first subpoena to obtain the identity of a Verizon subscriber alleged to have made more than 600 copyrighted songs available for downloading over the Internet through peer-to-peer file transfer software provided by KaZaA. Verizon claimed that because RIAA's subpoena related to material transmitted over Verizon's network-rather than stored on it-it fell outside the scope of the subpoena power. Should the subpoena be quashed as Verizon requests, or should it be honored? [In re Verizon Internet Services, Inc., 257 F Supp 2d 244 (DDC)]
Free
Essay

Refer to the case In re Verizon Internet Services, Inc., (257 FSupp2d 244)
Facts of the case:
Verizon motioned to quash a subpoena of RIAA for a list of its subscribers who allegedly made available music illegally in a peer-to-peer sharing system.
Opinion
The court denied Verizon's motion. The subpoena will be served. It doesn't matter that the information need to be "stored" in Verizon's system, the subpoena order is to provide a "copy of a description" of the information. The court also failed to find any undue burden on Verizon for providing such information.

Tags
Choose question tag
Colleges and universities continue to work to help students understand that what they post on the Web is not private information and can often have unintended consequences. The following examples resulted in student disciplinary proceedings: ? Several students at The Ohio State University boasted on Facebook (a networking/socializing site) that they had stormed the field after Ohio State beat Penn State and had taken part in what erupted into a riot. Law enforcement officials were able to trace the students through the university system, and 50 Ohio State students were referred to the office of judicial affairs. ? Students at the University of Mississippi stated on an open site that they wanted to have sex with a professor. ? A student at Fisher College threatened to take steps to silence a campus police officer. Another problem with the open sites is that the students are posting personal information with which stalkers and others can access them. These nefarious individuals can then easily obtain students' cell phone numbers, addresses, whereabouts, and other information. The most popular college site, Facebook, indicates that students spend an average of 17 minutes per day on the site. A great deal of information can be conveyed during that time period. Students do so without thinking through the possibility that outsiders with bad intentions could be seeking and using information about them that is posted there. What legal and ethical issues do you see in the types of comments that students make on these sites and in the sites themselves? Why and how can the colleges and universities obtain information from these sites without a warrant?
Free
Essay

In this age of growing social media almost nothing seems private anymore. When college students sign up for social media sites and allow public access or loose privacy settings they can't expect privacy. A warrant is mostly necessary for things involving privacy, such as search of a house, locker, and password protected files etc. However, social media profiles effectively become public data when privacy settings are not set, and anybody can legally obtain the information publicly displayed by their profile without warrant.
People need to be aware that if their profiles are public to any groups of people, the things they say may incriminate them, e.g. posting malicious or false information on someone is defamation, and even posting criminal thoughts can be viewed as conspiracy to commit a criminal act.

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Choose question tag
Jane Doe filed a complaint against Richard Lee Russell and America Online (AOL) to recover for alleged emotional injuries suffered by her son, John Doe. Doe claimed that in 1994, Russell lured John Doe, who was then 11 years old, and two other minor males to engage in sexual activity with each other and with Russell. She asserted that Russell photographed and videotaped these acts and used AOL's chat rooms to market the photographs and videotapes and to sell a videotape. In her six-count complaint, Doe claimed that AOL violated criminal statutes and that AOL was negligent per se in distributing an advertisement offering "a visual depiction of sexual conduct involving [John Doe]" and by allowing Russell to sell or arrange to sell child pornography, thus aiding in the sale and distribution of child pornography, including obscene images of John Doe. Does Mrs. Doe have a cause of action? What laws discussed in this chapter apply? [Doe v America Online, Inc., 783 So2d 1010 (Fla)]
Free
Essay

Refer to the case Doe v America Online (AOL) (783 So2d 1010) to answer the following question as below:
Facts of the case:
(1) Doe (plaintiff, anonymous) sued AOL for actions Russell (defendant). Russell solicited sexual act from Doe's son recorded it. Doe claimed AOL provided a medium for Russell to distribute the child pornography.
(2) Doe charged AOL tort for negligence , by failing to monitor its services. AOL motioned to dismiss which was accepted by trial court and affirmed by appeals, both argued that federal internet statute pre-empted civil action by barring suits against internet services. On review by the Florida Supreme Court
The Florida Supreme Court found that federal statute and a previous ruling in Zeran v AOL barred Doe from filing civil law tort against AOL. AOL can't be held liable for the doings of a third party. Doe will probably have a claim against Russell who was also charged for criminal acts.

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Choose question tag
Customers of a chat room are using the chat room, Maphia, for access to each other and to transfer Sega games to each other. They are able to avoid paying the $19 to$60 the games cost for purchase in the stores. The users say they are simply transferring files and that there is no crime. The chat room says it cannot stop customers from interacting. Do you think there are any civil or criminal law violations in their conduct? [Sega Enterprises, Ltd. v Maphia, 857 F Supp 679 (ND Cal)]
Essay
Tags
Choose question tag
Immunomedics, Inc., has discovered sensitive information about its technology posted on various Web sites and chat rooms. The information is so proprietary that it could have come only from company employees, all of whom have signed agreements not to disclose such information. Those who posted the information used screen names, and Immunomedics has asked the court to issue a subpoena to the ISP so that it can determine the identity of those posting the information and recover for breach of contract and trade secret infringement. Should the court issue the subpoena? [Immunomedics, Inc. v Does 1-10, 2001 WL 770389 (NJ Super 2001)]
Essay
Tags
Choose question tag
Glenayre Electronics announced to its employees that it could inspect the laptops it furnished for its employees to use. An employee challenged the inspection of his laptop as a violation of his privacy. Could the company search the laptops? [Muick v Glenayre Electronics, 280 F3d 741 (7th Cir)]
Essay
Tags
Choose question tag
Discuss whether employees would have the right of privacy in the following e-mail situations: a. E-mail sent in a company in which there is no warning given about the lack of privacy in e-mails. [Smyth v Pillsbury, 914 F Supp 97 (ED Pa 1996)] b. An e-mail sent to co-workers from home using the employee's AOL account. c. An e-mail sent from a laptop while the employee is traveling for the company. d. An e-mail sent to a coworker over a company Internet system in a company in which the employer has promised privacy in e-mail. [Commonwealth v Proetto, 771 A2d 823 (Pa Super Ct 2001)] e. Employer monitoring of the e-mails of any employee when those e-mails were stored in a file folder marked "Personal." [Mclaren v Microsoft Corp., 1999 WL 339015 (Tex App- Dallas 1999)] f. Employees using company e-mail for union organization purposes. [Pratt Whitney, National Labor Relations Board General Counsel Advisory Memorandum Cases 12-CA-18446, 12-CA-18722, 12-CS-18863 (February 23, 1998)]
Essay


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# Tag Info

29

The clouds can be seen moving but of course it's subtle at first glance. Frame 16: Frame 20:

27

They are moving, but not fast enough to notice at the distance shown. From the NASA page: These images were taken between 3:50 p.m. and 8:45 p.m. EDT on July 16, showing the moon moving over the Pacific Ocean near North America. The North Pole is in the upper left corner of the image, reflecting the orbital tilt of Earth from the vantage point of the ...

23

In a nutshell: The instrument measures microwave radiances (after calibration) If we know the sea surface temperature, we can use radiances to calculate emissivity. The emissivity at 1.4 GHz is itself a function of near-surface ocean salinity. However, in reality, it's more complicated, because there are other factors that come into the equation, such as ...

13

I'm not familiar with land-based methods, but for global measurements, one method is to use satellite altimetry (I'm more familiar with the geodesy side, but many of the same satellites are used). I think many of the current methods interpolate global or regional currents from a sparse network of buoys. As more radar satellites are launched, however, ...

13

Neither. The Sahara Desert is heterogenous. It’s yellow, red, orange, brown, black, and even green in certain places and at certain times of year. Slightly less cheeky answer: neither. Both examples are both approximations of human vision, displayed on computer screens that can be highly variable. The 2002 version of the Blue Marble is a composite of 4 ...

12

Remember from school when you learned that light is actually photons oscillating at specific frequencies, and we happen to perceive some of these frequencies as specific colors? Technically, satellites measure reflected light in spectral bands, where each band is actually a range of wavelengths or light frequencies. The raw data that is sent back to earth ...

11

The image is real, and the clouds are moving, they just move slow, as clouds do. The unnatural appearance arise because they are taken by DSCOVR at an extraordinarily large distance, from a place called L1, more than one million kilometers away!!!. The rotation of the Earth in the animation makes it difficult to distinguish the movement of the clouds, but ...

10

The structure looks similar to this photograph of a "Japanese land retention system" mentioned in passing towards the bottom of this webpage. From the linked page: Land retention systems in Japan, for example, are often designed as heavy waffle grids which are molded to the topography and cover it to a uniform structural depth. This seems to correspond ...

10

One way to approach this is to treat the Earth as an oblate ellipsoid. This would mean the errors arise from the uncertainties in the Earth's equatorial radius and the flattening. From Groten, "Fundamental Parameters and Current (2004) Best Estimates of the Parameters of Common Relevance to Astronomy, Geodesy, and Geodynamics," Journal of Geodesy 77:10-11, ...

10

Civilian earth observing SAR satellites do not always operate over empty swaths of oceans (specially in the middle of Indian Ocean) in order to save power. They are not designed to track a fast moving target and have repeat cycle of at least a few days. Then there is the issue of resolution. High res acquisitions modes (with satellites like TerraSAR-X) are ...

10

The WMO report Instruments and Methods of Observation has a chapter on Measurement of humidity and a separate chapter on Measurement of upper-air pressure, temperature, humidity. The first deals with instruments used for surface measurements. The second deals with instruments used for measuring vertical profiles, which have to deal with a larger range of ...

9

The Tropical Rainfall Measuring Mission (TRMM) satellite is a single satellite, carrying multiple instruments on board, as described here. It makes a total of 16 orbits/day at a low altitude of 400 km, with different swath widths for each instrument. Thus, the measurement product that you get is not a "continuous" measurement of a single atmospheric ...

8

As casey said, the satellite isn't an instrument in itself, it is a platform hosting many instruments. What you're talking about the discovery of oceans on Titan and Europa, you're referring to the the Cassini-Huygens mission aimed at, among other things : Determine the composition of the satellites surfaces (such as Titan) Prove or disprove the existence ...

8

First OLR map is usually graded in flux units (W/m2) thus it is quantitative data, not just a visual picture. Clouds themselves are not a very reliable indicator of high convection and thunderstorm activity. They could form elsewhere and get blown into a dry region. Also with an OLR map students would not be distracted with the shapes of the clouds. AIRS ...

8

Weather stations and airports around the world almost always include an anemometer. Because of ground friction, the wind speed varies with height, so instrumental deployment is set at a standard height of 10 metres in open rural areas. This may require some adjustment in urban or forested areas. In fact, measuring accurate wind speeds above tree canopies is ...

8

I can engage in some educated speculation about what you think you see, but I'm confident you were fooled by the shadows in the image, which are caused by a low sun angle and distorted by the terrain. These are not inuksuit (which is the correct plural of inuksuk). They are all natural, not man-made. Let's point out a few things we know and can see on the ...

8

This is an interesting and common question, so I'll try to give a good answer here, that I think is a good place, as well as GIS and Open Data SE. First, regarding the format, you will never find data natively stored in text or tabular form, because the large data volume would lead to huge files very difficult to handle. Data is usually stored in binary ...

7

Earthquake prediction have been discussed before in this forum and is rather dispiriting. Distortion in transmitted signals could be an early indication of earthquake, but seismometers (geophones) are still a far better instrument to detect vibrations. In future satellites could be an option in remote areas. However, damage from an earthquake can be ...

7

Talking about the volume of the Earth, we have to ignore that it is not static but in fact highly dynamic. The tides alone make the height differ by up to 50 cm. The information I take is from a 2003 article from esri by Witold Fraczek. As you said, we have different understandings of the Earth. Approximations are made to make calculations easier (and ...

7

You ask, How to remove these abnormal value for more precise data? The short answer is that you might be able to make your data look more representative of the natural 'truth', but you will probably have to pay for this with some precision. The problem is that the measurement itself is imperfect: this looks like an artifact of the data collection method. ...

7

Short answer Different plants reflect light at different wavelengths with specific patterns. If you know the reflection pattern of a broad-leaved forest and that of a needle-leaved forest, you can compare them with the pattern observed with a satellite and conclude to which forest it is more similar. Long answer Optical remote sensing Optical sensors ...

7

$\rm \frac{mol}{m^2}$ shows the amount of $\rm{NO_2}$ in the atmosphere over a square meter of surface area - in mols. The molar mass of the $\rm{NO_2}$ is $14+2\cdot 16=46$. It means, the mass of 1 mol of $\rm{NO_2}$ is $\rm{46g}$. The surface area of the Earth is 510million $\rm{km^2}$. Thus, 1 $\rm \frac{mol}{m^2} \rm{NO_2}$ translates to \$\rm{46 \frac{...

6

Hydrosource Associates (no personal relation) claims to use the following: HSA’s use of geophysical instruments and methods includes seismic, electrical resistivity, ground penetrating radar, electromagnetic, magnetic, gravimetric, self-potential, and magnetotelluric. Since you're looking for a list, you could investigate each of those technologies above....

6

From what @Richard and @JoeKington said in their answers, it seems that the usual way of finding water is through a hydrological map. However, for the sake of completeness, I'll include here a bit about what geophysical methods could be used to find water. I'll not comment on the practical issue of cost-benefit as that was nicely explained by @JoeKington in ...

6

Joe Kington has provided an excellent answer re sensing from space. Torbjørn T. pointed out in a comment that the site that I linked for context actually has a good explanation... so after blushing slightly, I shall summarise it here. People with more specialist knowledge are, of course, welcome to elaborate or correct any misunderstanding. Incidentally, it ...

6

I found a natural-color image of the same area: 68.28 South, 60.47 East, North is up, resolution is 130m/px, which makes the whole image about 100 km x 100 km. Looking at the image i'd say that: sea ice is water with pieces of ice in it, the boundary between the two regions is clear water, probably caused by winds and currents moving the floating ice away ...

6

I can't imagine a moving object more suitable for remote sensing tracking than Iceberg A-68, with such slow displacement and huge size. So I don't think it will be particularly useful to install any instrumentation to track its position on site. However, there would be plenty of other measurements that can be done, about mass balance, stress fields, etc. I'...

6

Water molecules do interact with microwaves and in microwave ovens that operate at 900MHZ- 2.5GHz. This Radar instrument operates using Very High Freq Radio Waves at 193MHz [1]. At lower-than-microwave frequencies, the loss factor is much lower, and as a result the wave can penetrate deeper into the ice sheet with minimal absorption. The difference in ...

6

We use something like an Onset HOBO electronic sensor. They've got logging (and in some cases, networking) built in. They'll typically log both temperature and relative humidity at regular intervals specified by the user. Humidity sensors have a tendency to wander off calibration, so you will need to take that into account when designing your monitoring ...

6

The unit mole / m² has dimensions count per area; the unit µg / m³ has dimensions mass per volume. To go from a quantity per area to a quantity per volume, you need to understand some properties of the instrument. The spectrometer on Sentinel-5P is TROPOMI. TROPOMI is an imaging spectrometer, meaning it measures the spectrum of top-of-atmosphere ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
## Calculus, 10th Edition (Anton)

18$x^{2}$-$\frac{3x}{2}$+12
According to the product rule, if f(x)=h(x)g(x), then f'(x)=h(x)g'(x)+g(x)h'(x). If we set h(x)=3$x^{2}$+6, and g(x)=2x-$\frac{1}{4}$, then using the product rule, f'(x)=(3$x^{2}$+6)$\frac{d}{dx}$[2x-$\frac{1}{4}$]+(2x-$\frac{1}{4}$)$\frac{d}{dx}$[3$x^{2}$+6] $\frac{d}{dx}$[2x-$\frac{1}{4}$]=2, using the power rule $\frac{d}{dx}$[3$x^{2}$+6]=6x, using the power rule Therefore, f'(x)=(3$x^{2}$+6)(2)+(6x)(2x-$\frac{1}{4}$) = 6$x^{2}$+12+12$x^{2}$-$\frac{3x}{2}$ =18$x^{2}$-$\frac{3x}{2}$+12


text:
# Héctor D. Abruña

## Summary

The Abruña Group focuses on the development and characterization of new materials using a wide variety of techniques for fuel cells, batteries, and molecular assemblies for molecular electronics.

## Research Focus

Our research effort takes an interdisciplinary approach to the study of electrochemical phenomena. We employ electrochemical techniques as probes of a variety of chemical systems, and we use other techniques such as x-ray based methods, differential electrochemical mass spectrometry, in-situ FT-IR, scanned probe microscopies, scanning electrochemical microscopy, low temperature conductance and spectroscopic techniques to address problems of electrochemical interest.  Current areas of research include:

1.  Fuel cells:

•  The use of ordered intermetallics, such as BiPt for the electrocatalytic oxidation of formic acid, methanol, ethanol and other small organic molecules of potential utility as fuels in fuel cells.

•  Use of Differential Electrochemical Mass Spectrometry (DEMS), in-situ FT-IR in for mechanistic studies related to fuel cells.

•  Development of in-situ TEM techniques for the study of fuel cell and battery materials

2. Electrical Energy Storage (EES): Batteries and Supercapacitors

•  Computational screening synthesis and characterization of organic molecules for EES

•  In-situ testing of battery systems using in-situ x-ray based technique (XRD, EXAFS, XANES)

•  Lithium/sulfur batteries

3.  Molecular electronics:

•  The synthesis of nanometric building blocks and their use in molecular electronic devices.

•  Transport measurements using mechanical break junctions and break junctions made through electromigration

•  Investigations of graphene as an experimental electrochemical platform

•  Surface diffusion and surface dynamics on single-layer graphene using scanning electrochemical microscopy.


text:
Articles written in Bulletin of Materials Science

• Electrical properties of praseodymium and samarium co-doped ceria electrolyte for low-temperature solid oxide fuel cell application

In this study, we aimed to investigate the structural, morphology and electrical properties of praseodymium and samarium co-doped ceria electrolytes, synthesized using the sol–gel method, for use as an electrolyte in lowtemperature solid oxide fuel cell (LT-SOFC) applications. The X-ray diffraction result shows that all of the samples crystallized into a single-phase cubic fluorite form. The average relative densities of samples sintered at 1400°C is 97.9% of theoretical densities, indicating that they can be used as an electrolyte in LT-SOFC applications. A fascinating and maximum value of ionic conductivity 1.94${\times}$10$^2$ S cm$^{–1}$ and least activation energy ($E_a$ = 0.55 eV) were found for the composition Ce$_{0.8}$Sm$_{0.1}$Pr$_{0.1}$O$_{1.9}$ at a temperature of 500°C. The ionic conductivity and activation energies of compositions Ce$_{0.85}$Sm$_{0.1}$Pr$_{0.05}$O$_{1.925}$ and Ce$_{0.9}$Sm$_{0.05}$Pr$_{0.05}$O$_{1.95}$ found at 500°C were (1.41${\times}$10$^2$ S cm$^{–1}$, $E_a$ = 0.59 eV) and (5.95${\times}$10$^3$ S cm$^{–1}$, $E_a$ = 0.64 eV), respectively. Moreover, all the Praseodymium and samarium co-doped ceria (PrSDC) samples conduct at lower temperature of 300°C. All these results confirmed that praseodymium and samarium co-doped ceria can be useful as a solid electrolyte in LT-SOFC applications.

• # Bulletin of Materials Science

Volume 45, 2022
All articles
Continuous Article Publishing mode

• # Dr Shanti Swarup Bhatnagar for Science and Technology

Posted on October 12, 2020

Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru
Chemical Sciences 2020


text:
# Book Notes: Neural Network Methods for Natural Language Processing -- Part 3 Specialized Architectures, Ch13 CNN

### Overview on CNN and RNN for NLP

• CNN and RNN architectures explored in this part of the book are primarily used as feature extractors

• CNNs and RNNs as Lego bricks: one just needs to make sure that input and output dimensions of the different components match

# Ch13 Ngram Detectors: Convolutional Neural Networks

## CNN Overivew

### CNN overview for NLP

• CBOW assigns the following two sentences the same representations

• “it was not good, it was actually quite bad”
• “it was not bad, it was actually quite good”
• Looking at ngrams is much more informative than looking at a bag-of-words

• This chapter introduces the convolution-and-pooling (also called convolutional neural networks, or CNNs), which is tailored to this modeling problem

### Benefits of CNN for NLP

• CNNs will identify ngrams that are predictive for the task at hand, without the need to pre-specify an embedding vector for each possible ngram

• CNNs also allows to share predictive behavior between ngrams that share similar components, even if the exact ngrams was never seen at test time

## Basic Convolution $$+$$ Pooling

### Convolution

• The main idea behind a convolution and pooling architecture of language tasks is to apply a non-linear (learned) function over each instantiation of a $$k$$-word sliding window over the sentence

• This function (also called “filter”) transforms a window of $$k$$ words into a scalar value

• Intuitively, when the sliding window of size $$k$$ is run over a sequence, the filter function learns to identify informative $$k$$grams
• Several such filters can be applied, resulting in $$\ell$$ dimensional vector (each dimension corresponding to one filter) that captures important properties of the words in the window

### Pooling

• Then a “pooling” operation is used to combine the vectors resulting from the different windows into a single $$\ell$$-dimensional vector, by taking the max or the average value observed in each of the $$\ell$$ dimensions over the different windows

• The intention is to focus on the most important “features” in the sentence, regardless of their location
• The resulting $$\ell$$-dimensional vector is then fed further into a network that is used for prediction

### 1D convolutions over text

• A filter is a dot-product with a weight vector parameter $$\mathbf{u}$$, which is often followed by nonlinear activation function

• Define the operation $$\oplus(\mathbf{w}_{i:i+k-1})$$ to be the concatenation of the vectors $$\mathbf{w}_i, \ldots, \mathbf{w}_{i+k-1}$$. The concatenated vector of the $$i$$th window is then $\mathbf{x}_i = \oplus(\mathbf{w}_{i:i+k-1}) = [\mathbf{w}_i; \mathbf{w}_{i+1}; \ldots; \mathbf{w}_{i+k-1}] \in \mathbb{R}^{k \cdot d_{\text{emb}}}$

• Apply the filter to each window-vector, resulting scalar value $$p_i$$: \begin{align*} &p_i = g(\mathbf{x}_i \cdot \mathbf{u})\\ &p_i \in \mathbb{R},\quad \mathbf{x}_i \in \mathbb{R}^{k \cdot d_{\text{emb}}}, \quad \mathbf{u} \in \mathbb{R}^{k \cdot d_{\text{emb}}} \end{align*}

### Joint formulation of 1D convolutions

• It is customary to use $$\ell$$ different filters $$\mathbf{u}_1, \ldots, \mathbf{u}_{\ell}$$, which can be arranged into a matrix $$\mathbf{U}$$, and a bias vector $$\mathbf{b}$$ is often added

\begin{align*} &\mathbf{p}_i = g(\mathbf{x}_i \cdot \mathbf{U} + \mathbf{b})\\ &\mathbf{p}_i \in \mathbb{R}^{\ell},\quad \mathbf{x}_i \in \mathbb{R}^{k \cdot d_{\text{emb}}}, \quad \mathbf{U} \in \mathbb{R}^{k \cdot d_{\text{emb}} \times \ell}, \quad \mathbf{b} \in \mathbb{R}^{\ell} \end{align*}

• Ideally, each dimension captures a different kind of indicative information

• The main idea behind the convolution layer: to apply the same parameterized function over all $$k$$grams in the sequence. This creates a sequence of $$m$$ vectors, each representing a particular $$k$$gram in the sequence

### Narrow vs wide convolutions

• For a sentence of length $$n$$ with a window of size $$k$$

• Narrow convolutions: there are $$n-k+1$$ positions to start the sequence, and we get $$n-k+1$$ vectors $$\mathbf{p}_{1:n-k+1}$$

• Wide convolutions: an alternative is to pad the sentence with $$k-1$$ padding-words to each side, resulting in $$n+k+1$$ vectors $$\mathbf{p}_{1:n+k+1}$$

• We use $$m$$ to denote the number of resulting vectors

### Vector pooling

• Applying the convolution over the text results in $$m$$ vectors $$\mathbf{p}_{1:m}$$, each $$\mathbf{p}_i \in \mathbb{R}^{\ell}$$

• These vectors are then combined (pooled) into a single vector $$\mathbf{c}\in \mathbb{R}^{\ell}$$ representing the entire sequence

• During training, the vector $$\mathbf{c}$$ is fed into downstream network layers (e.g., an MLP), culminating in an output layer which is used for prediction

### Different pooling methods

• Max pooling: the most common, taking the maximum value across each dimension $$j = 1, \ldots, \ell$$ $\mathbf{c}_{[j]} = \max_{1\leq i \leq m} \mathbf{p}_{i,[j]}$

• The effect of the max-pooling operation is to get the most salient information across window positions
• Average pooling $\mathbf{c} = \frac{1}{m}\sum_{i=1}^m \mathbf{p}_i$

• $$K$$-max pooling: the top $$k$$ values in each dimension are retained instead of only the best one, while preserving the order in which they appeared in the text

### Variations

• Rather than a single convolutional layer, several convolutional layers may be applied in parallel

• For example, we may have four different convolutional layers, each with a different window size in the range 2-5, capturing $$k$$gram sequences of varying lengths

## Hierarchical Convolutions

### Hierarchical convolutions

• The 1D convolution approach described so far can the thought of as a ngram detector: a convolution layer with a window of size $$k$$ is learning to identify indicative $$k$$-gram in the input

$\mathbf{p}_{1:m} = \text{CONV}^k_{\mathbf{U}, \mathbf{b}}(\mathbf{w}_{1:n})$

• We can extend this into a hierarchy of convolutional layers with $$r$$ layers that feed into each other \begin{align*} \mathbf{p}_{1:m_1}^1 & =\text{CONV}^{k_1}_{\mathbf{U}^1,\mathbf{b}^1}(\mathbf{w}_{1:n})\\ \mathbf{p}_{1:m_2}^2 & =\text{CONV}^{k_2}_{\mathbf{U}^2,\mathbf{b}^2}(\mathbf{p}_{1:m_1}^1)\\ &\cdots\\ \mathbf{p}_{1:m_r}^r & =\text{CONV}^{k_r}_{\mathbf{U}^r,\mathbf{b}^r}(\mathbf{p}_{1:m_{r-1}}^{r-1}) \end{align*}

### Hierarchical convolutions, continued

• For $$r$$ layers with a window of size $$k$$, each vector $$\mathbf{p}_i^r$$ will be sensitive to a window of $$r(k-1)+1$$ words

• Moreover, the vector $$\mathbf{p}_i^r$$ can be sensitive to gappy-ngrams of $$k+r-1$$ works, potentially capturing patterns such as “not ___ good” or “obvious ___ predictable ___ plot”, where ___ stands for a short sequence of words

### Strides

• So far, the convolution operation is applied to each $$k$$-word window in the sequence, i.e., windows starting at indices $$1, 2, 3, \ldots$$. This is said to have a stride of size $$1$$

• Larger strides are also possible. For example, with a stride of size $$2$$, the convolution operation will be applied to windows starting at indices $$1, 3, 5, \ldots$$

• Convolution with window size $$k$$ and stride size $$s$$: \begin{align*} \mathbf{p}_{1:m} &= \text{CONV}^{k,s}_{\mathbf{U}, \mathbf{b}}(\mathbf{w}_{1:n})\\ \mathbf{p}_i &= g\left(\oplus \left( \mathbf{w}_{1+(i-1)s : (s+k)i} \right) \cdot \mathbf{U} + \mathbf{b} \right) \end{align*}

### References

• Goldberg, Yoav. (2017). Neural Network Methods for Natural Language Processing, Morgan & Claypool


text:
## Duke Mathematical Journal

### Blow-up results and localization of blow-up points in an $N$-dimensional smooth domain

#### Article information

Source
Duke Math. J. Volume 88, Number 2 (1997), 391-405.

Dates
First available in Project Euclid: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077241584

Digital Object Identifier
doi:10.1215/S0012-7094-97-08816-5

Mathematical Reviews number (MathSciNet)
MR1455526

Zentralblatt MATH identifier
0884.35071

#### Citation

Rial, D. F.; Rossi, J. D. Blow-up results and localization of blow-up points in an N -dimensional smooth domain. Duke Math. J. 88 (1997), no. 2, 391--405. doi:10.1215/S0012-7094-97-08816-5. http://projecteuclid.org/euclid.dmj/1077241584.

#### References

• [1] Bei Hu and Hong Ming Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc. 346 (1994), no. 1, 117–135.
• [2] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319–334.
• [3] J. López-Gómez, V. Márquez, and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Differential Equations 92 (1991), no. 2, 384–401.
• [4] W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal. 6 (1975), 85–90.
• [5] Ming Xin Wang and Yong Hui Wu, Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 24 (1993), no. 6, 1515–1521.
• [6] N. Wolanski, Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), no. 2, 317–326.


text:
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### Re: Several widgets disappear when .mo file installed

z-20
Member

@z-20

Hello all.

I had this same problem when I tried to use the Portuguese translation and this is how I fixed it.

1) Remove all your bp widgets.

2) Open translation.po

3) Find bp-core/bp-core-widgets.php (it is mentioned many times and write down the line numbers)

4) Open mu-plugins/bp-core/bp-core-widgets.php

5) Translate manually the lines where the translation file was supposed to do it.


text:
# matrix unit

A matrix unit is a matrix (over some ring with $1$) whose entries are all $0$ except in one cell, where it is $1$.

For example, among the $3\times 2$ matrices,

 $\begin{pmatrix}1&0\\ 0&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&1\\ 0&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 1&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&1\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&0\\ 1&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&0\\ 0&1\end{pmatrix}$

are the matrix units.

Let $A$ and $B$ be $m\times n$ and $p\times q$ matrices over $R$, and $U_{ij}$ an $n\times p$ matrix unit (over $R$). Then

1. 1.

$AU_{ij}$ is the $m\times p$ matrix whose $j$th column is the $i$th column of $A$, and $0$ everywhere else, and

2. 2.

$U_{ij}B$ is the $n\times q$ matrix whose $i$th row is the $j$th row of $B$ and $0$ everywhere else.

Remarks. Let $M=M_{m\times n}(R)$ be the set of all $m$ by $n$ matrices with entries in a ring $R$ (with $1$). Denote $U_{ij}$ the matrix unit in $M$ whose cell $(i,j)$ is $1$.

• $M$ is a (left or right) $R$-module generated by the $m\times n$ matrix units.

• When $m=n$, $M$ has the structure of an algebra over $R$. The matrix units have the following properties:

1. (a)

$U_{ij}U_{k\ell}=\delta_{jk}U_{i\ell}$, and

2. (b)

$U_{11}+\cdots+U_{nn}=I_{n}$,

where $\delta_{ij}$ is the Kronecker delta and $I_{n}$ is the identity matrix. Note that the $U_{ii}$ form a complete set of pairwise orthogonal idempotents, meaning $U_{ii}U_{ii}=U_{ii}$ and $U_{ii}U_{jj}=0$ if $i\neq j$.

• In general, in a matrix ring $S$ (consisting of, say, all $n\times n$ matrices), any set of $n$ matrices satisfying the two properties above is called a full set of matrix units of $S$.

• For example, if $\{U_{ij}\mid 1\leq i,j\leq 2\}$ is the set of $2\times 2$ matrix units over $\mathbb{R}$, then for any invertible matrix $T$, $\{TU_{ij}T^{-1}\mid 1\leq i,j\leq 2\}$ is a full set of matrix units.

• If we embed $R$ as a subring of $M_{n}(R)$, then $R$ is the centralizer of the matrix units of $M_{n}(R)$, meaning that the only elements in $M_{n}(R)$ that commute with the matrix units are the elements in $R$.

## References

• 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
Title matrix unit MatrixUnit 2013-03-22 18:30:35 2013-03-22 18:30:35 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 15A30 msc 16S50 ElementaryMatrix full set of matrix units


text:
# Using propensity scores from twang in coxph

I've used the twang package to calculate propensity score weights in my observational data set. The twang vignette uses the survey package to apply the weighted PS in analyses, but does not provide an example for Cox proportional hazard estimation.

The survey package provides a function called svycoxph() which which seems to do the trick. However, when i use coxph() from the survival package an set the weight= argument to the extracted weighted PS from twang's ps() object i get an indentical HR estimate, albeit a much more narrow 95%CI.

Can I use the extracted propensity score weights from a ps object from the twang package as a vector in the weight= argument of the survival package's coxph()?

Should i include an example?

Update: When i added a cluster(patient ID) term in the the coxph formula, both the HR and the 95% CI became identical to the syvcoxph output. So I guess that did the trick. But can i conclude that this is ok to use?

I just came across your post because I used twang to calculate inverse probability weights and wanted to use a coxph model afterwards. I had used svyglm before, but wasn't aware of svycoxph. So thanks for this hint.

I tried what you did and got almost identical results with coxph and cluster(patient ID) as with svycoxph.

In the description of svycoxph, they say:

The main difference between ‘svycoxph’ function and the ‘robust=TRUE’ option to ‘coxph’ in the survival package is that this function accounts for the reduction in variance from stratified sampling and the increase in variance from having only a small number of clusters.

In the description of coxph, they give the following explanation:

The term ‘+ cluster(id)’ where each value of ‘id’ is unique is equivalent to specifying the ‘robust=T’ argument, and produces an approximate jackknife estimate of the variance.

From these two explanations I conclude that coxph using cluster(id) is fine for large numbers of clusters (what you probably had). If you have a small number of clusters and want to account for this, svycoxph will be safer. In my case, with 67 clusters, it made a small difference so I will use svycoxph.


text:
# Roman numbering, cleardoublepage and table of content not working

I'm currently doing the thesis report and I cant seem to get the numbering to work as well as the along with some other stuff.

What I want is Abstract(eng) Abstract(swe) and Acknowledgments to all be on the right side, with roman numbering. I can accomplish this with

\newpage
\thispagestyle{empty}
\mbox{}
\newpage

but I wanna use \cleardoublepage, which is not working, between the chapters as well.

Another thing I want is the numbering in the toc to be roman as well, but it isn't. I also want the abstract to show in the toc with the roman page number ref but they all show up with arabic number page 1.

\documentclass[a4paper]{report}

\usepackage[english,swedish]{babel}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{subfig}
\usepackage[table]{xcolor}
\usepackage{lscape}
\usepackage[cp1252]{inputenc}
\usepackage{cite}
\usepackage{appendix}
\parindent 0pt

\begin{document}

\pagenumbering{roman}

\chapter*{Abstract}
\input{abseng}
\pagenumbering{plain}
\cleardoublepage                    % not working

\chapter*{Sammanfattning}
\input{abssve}
\pagenumbering{plain}
\cleardoublepage

\chapter*{Acknowledgment}
\input{ack}
\pagenumbering{plain}
\cleardoublepage

\tableofcontents

\pagenumbering{arabic}
\cleardoublepage                                     %<-- not working
\include{Introduction}

%\include{other chapters}

\bibliographystyle{unsrt}       % (uses file "plain.bst")
\bibliography{myrefs}

\listoffigures[toc]

\begin{appendix}
\include{appendix}

\end{document}

-
cleardoublepage only works in a double sided layout, so adding twoside to your class options should fix that part. –  ArTourter Apr 17 '12 at 10:37
Although I am saying no to the helpful question, I find the information beneficial. However the problem for me is where does this code go? What is the OS and WORD generation for this code? Is it 2003, 2007 or 2010? I would need this information prior to trying to fix my problems which seem the same as this post. If I can manage to get the roman numerals and alpha numeric values to work in my doctoral proposal I would be thrilled. Right now it is a major frustration. So if this code is applicable to OS 2007 and MS Word 2010, great! Now how would it be applied, to become affective? –  user14358 May 8 '12 at 13:27

As I said in my comment, \cleardoublepage only works in a double sided layout, so adding twoside to your class options should fix that part.

However you problems are not all fixed by this. The other issue is that\pagenumbering{plain} is not a valid counter scheme. What you probably meant is \pagenumbering{arabic}.

Once these two changes are done, your document will compile but it will still not behave the way you probably want it to.

The effects of you changing your numbering scheme after every chapter is that the page counter is reset every time you call \pagenumbering{}, but in your document, \cleardoublepage is always used after resetting your page numbers. calling \cleardoublepage before changing your numbering schemes, should make it behave properly.

EDIT:

The easiest way to have empty pages really empty with a non KOMA class is to use the emptypage package. If you use the new KOMA scrreprt class you get access to the \cleardoubleempty command which does the same thing (as well as \cleardoubleplain and \cleardoublestandard depending on the pagestyle you want to have).

Regarding the table of content, you need to reshuffle your document a bit and put the \addccontentsline{}{}{} just below the corresponding \chapter*{} call. And the one for "Contents" just below the \tableofcontents calls. However you will then have to move the

\addtocontents{toc}{~\hfill\textbf{Page}\par}

to the beginning of your document (just after \begin{document} works) so that it does not appear between two contents lines.

-
Maybe he meant \pagelayout{plain} instead of \pagenumbering{plain}? –  Martin Scharrer Apr 17 '12 at 11:14
Ya thats what i meant, copypasted wrong :( sorry, the twoside works fine so it solves the cleardoublepage problem. The reason for \pagelayout{plain} so the number would show as I used begin{abstract} but changed to chapter because of the title being small. Anyways i used \thispagestyle{empty} for the empty page as i dont want number on the empty pages, and i get that now. Another thing is about showing in the toc the abstract page in roman numbers. Abstract iii ... Introduction 9 –  Mamed Apr 17 '12 at 12:12
@Mamed thanks for the clarifications, see the edited answer –  ArTourter Apr 17 '12 at 14:05

It works now, thanks for all the help, I will add the full code below for others in the future.

\documentclass[twoside,a4paper]{report}
...same...
\usepackage{emptypage}

\begin{document}

\pagenumbering{roman}\setcounter{page}{3}
\chapter*{Abstract}
\input{abseng}
\cleardoublepage

\chapter*{Sammanfattning}

\input{abssve}
\cleardoublepage

\chapter*{Acknowledgment}
\selectlanguage{english}             < needed as the "Contents" changed to swedish
\input{ack}
\cleardoublepage

\tableofcontents                     < here before: "add & clear" or else arabic numbers
\cleardoublepage

\pagenumbering{arabic}%\setcounter{page}{9}

\include{Introduction}

It gives

Contents
Page
Abstract                       iii
Sammanfattning                  v
Acknowledgments                vii
Contents                       ix
Introduction                    1
etc

-


text:
# Is it possible for a religion with >1 billion followers to die out?

The religion of Zensunnistianity has over a billion followers in the Alt-World that is otherwise identical to our Earth. Its precepts are similar to the major monotheistic religions from our world, give or take a bit of tree worship here and there. It just happens to use the ensō sign as its holy symbol, placed on ambulances and used by the red ensō humanitarian charities:

Now, given that there are so many followers, is it conceivable that the religion of Zensunnistianity could die out (let's define that as being reduced to fewer than a million worshipers)? Presumably, while humanity continues to exist?

Thank you for all the answers! Having read them all carefully, I am now inclined towards a negative answer to this question.

• What do you count as a follower ? All people declaring themselves followers, or people actually following the dogmas ? If it is the second, it already happened to Catholicism. Seriously, who still obey to the Pope ? Also (at least in Europe), there are way more people describing themselves as Christian, than people actually going to the church on Sunday. – Kolaru Jan 20 '16 at 21:27
• What if they have a belief that forbid say, vaccination, and then there is a new plague? – Per Alexandersson Jan 21 '16 at 3:04
• By die out do you mean that the followers of the religion die out or the concept of that religion dies out ? – Mohammad Ghazanfar Jan 21 '16 at 6:21
• @Would-be-closers, please explain how this is not on-topic for worldbuilding. Whether it is realistic to have a world a few decades removed from ours that is missing a major religion is spot-on-topic, in my view. – Serban Tanasa Jan 21 '16 at 12:44
• @Kolaru I guess by that measure catholicism never got to the 1 billion mark. – Paŭlo Ebermann Jan 22 '16 at 19:43

### No, it’s not possible in our current cultural and geopolitic situation¹.

To reach a billion followers, a religion needs to have certain, related properties:

1. It needs to be a “family religion”, i.e., it must be part of the religion (not necessarily its dogmata, but its culture) that followers procreate and indoctrinate their children. “Automatic” procreation is an essential evolutionary trait for species as well as religions – religions that do not have it are too unfit for survival. Acquiring followers in adulthood would not work, as people are too unlikely to switch or acquire a religion at that age. A religion without this feature would die out or be dominated by another religion before it can reach a billion followers.

2. It must address all of society, i.e., it cannot be exclusive to a certain group of people such as a particular profession, class, age or gender. First of all, there aren’t that many groups of people that contain a billion persons to begin with. There are the genders, but being exclusive to one gender would not make you a “family religion”. There are the poor, but inevitably, some poor people will get rich and adapt the religion to allow it.

3. It must at some time be the dominant religion in some larger area (usually a state). Nothing helps a religion prosper as good as general acceptance and social pressure. Without this, a religion can never reach a billion followers. This is also extremely helpful to wear down the edges (i.e., highly impractical dogmas) of a religion (see also the next point). As a rough estimate, half of the followers of any dominant religion only follow it due to social pressure (or such mechanisms as belief in belief) and not due to actually believing in its dogmata. This half is strong enough to shape the religion to be more feasible.

4. It must be flexible and not entirely homogeneous. A popular religion is inevitably intertwined with culture and cannot be popular without adapting to the culture to some extent. To reach a billion followers, a religion must span several cultures and adapt to all of them. The only alternative would be no cultural homogeneity¹, which is not really possible as culture is shaped by the actual lives of people, which are not homogeneous.

Almost all popular religions nowadays are very flexible and inhomogeneous. To address the more “difficult” examples: Many Islamic countries were much more liberal half a century ago and features flavours such as Ahmadiyya; Catholicism varies a lot over the world.

(Sidenote: If you define the same religion so strict that this sort of flexibility is not allowed, you cannot have a religion reach one billion followers in the first place.)

5. It must promise exclusive correctness at least to some extent; relatedly, there must be some line to draw between this religion and others. This is one of the evolutionary advantages of monotheism over classical polytheism. The promise of exclusive correctness is essential for most of the above features and is another evolutionary key factor as it keeps believers as they are less open to other religions.

Note that I do not understand this in a way that excludes Hinduism, for example. While Hinduism is very diverse, there are still some central concepts that define it. Rather this excludes concepts such as omnism (not the discworld religion) or polytheistic religions with a very open pantheon.

Now, if there is anything like cultural mixing¹, such a religion will form diasporas all over the world (due to poperty 3), simply because adherents migrate. While this is something very prominent nowadays, it also happened historically, for example to Judaism, Zoroastrianism and Christianity – without these religions gathering that many adherents.

While some of these diasporas will die due to adopting the dominant religion or areligiousness, totalitarianism or other hostile environments, chances are high that at least some of them adapt and survive (due to properties 1, 2, 4, and 5). Some of these diasporas may manage to grow in population and sprout diasproas themselves (see, e.g., the History of Russian Jews).

Thus, even if the religion is largely eliminated in its homeland due to some reason, it is very likely that it can still accumulate a million adherents in diasporas. While it may decline in relative population, it is very unlikely to fall below your threshold of one million given the general increase of the world’s population¹.

A very good example of how (not that) this can work is Zoroastrianism, which was ironically considered a counter-example in another answer, but it never had a billion adherents (and some sources even consider it to still have a million adherents today).

• Celibacy – strongly violates point 1, the religion could never even remotely reach a billion followers.

• Rituals that nurture epidemics – some diasporas will resist and the religion can adapt those rituals (point 4).

• Explicit disproof – many religious people are comfortable with believing contradictions and then there is point 4. To give a practical example: Jehovas Witnesses survived many doomsdays.

• Mass suicide – Won’t work due to points 1 and 4.

¹ which I consider to be part of the premise that the world is “otherwise identical to our Earth”.

• This carefully considered answer deserves far more upvotes than it has – Serban Tanasa Feb 5 '16 at 17:16

It can absolutely die out if you give it enough time.

The most common reason a religion dies, and the only reason that consistently works is supplantation.

Take a look at some of the world's pagan religions. If we look at Greek, German, Norse and Egyptian polytheism all were replaced by monotheistic religions, namely Christianity, and later (in the case of Egypt) Islam.

These religions were supplanted through various methods. The growth of scientific thought and experimentation undercut the validity of polytheistic pantheons. Why worship Zeus or Thor to spare you from the storms when you know that storms are predictable and explained by natural phenomenon.

The growth of monotheistic religions was also supported by government. It is much easier to control a single hierarchical church and the people that follow it than it is to manage people of varying belief systems that may come to physical confrontation. Political leaders made certain religions the 'state' religion to promote it and often times to punish those that did not follow.

Its also important to keep absorption as part of this process. Is it coincidence that Jesus' death and birth fall on the same days as major pagan holidays? Hint...it's not Absorbing aspects of local religions that are being replaced helps bring people into the new faith.

A good monotheistic example is Zoroastrianism. If you follow any modern monotheistic religion you can thank Zoroaster for starting the trend. Zoroastrianism is arguably the source and predecessor of Christianity, Islam and Judaism. It was mainly followed during the times of the ancient Persian Empires by millions of people, today estimates put the number of followers at under 200,000.

Notes

• Isolated methods of destroying or controlling a religion are unlikely to be long term effective. As @AndyD273 mentioned persecution is not a great way to get rid of a religion...humans are stubborn like that.

• It takes time. You are not going to be able to plausibly kill off a religion in less than 2 or 3 generations (at a bare minimum).

• They can always come back. Unless it is somehow completely forgotten (which is virtually impossible, especially in the modern era) there will likely be people who keep following the faith.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monica Cellio Jan 20 '16 at 23:40
• Okay, but there is no substantial evidence that Zoroastrianiam provided a basis for other religions. – Xandar The Zenon Jan 21 '16 at 4:32
• It would be more accurate to say that Jesus' birth is celebrated on/near a major pagan holiday. It isn't uncommon to cite the birth as well before Christmas, for a variety of reasons. As for the death, it follows Passover. If you want to make a claim about that being near a pagan holiday, you'll need to go back much further. – jpmc26 Jan 21 '16 at 5:25
• Christianity also wasn't "government sponsored" for the first 300+ years of its existence. Rather the opposite, in most of the Roman world for a large portion of that time. Storms weren't really predictable until the advent of the telegraph, and true forecasting requires computer modeling - long after Zeus and Thor were supplanted. General knowledge of weather patterns has been around for millennia, including some rudimentary understanding, which didn't dent belief, however. – Clockwork-Muse Jan 21 '16 at 8:37
• Folks, please take the discussion to the chat room that was already created. If you right-click on the timestamp for a comment you get its URL, which you can paste into the room. Expect these comments to be deleted in the not-too-distant future. – Monica Cellio Jan 22 '16 at 20:13

There are a few different methods I can think of that would allow a large religion to entirely (or almost entirely) die out.

1. Genocide. Other religions have actively and purposefully decided to exterminate the religion. This doesn't have to simply be the extermination of followers of that religion, but rather a purposeful attempt to eliminate the religion.

There are a number of historical examples of attempts to exterminate religions, but probably the most relevant historical examples would be the willful extermination of the religions of Native Americans. When the Spanish conquistadors arrived, their destruction of the Native civilizations - through extermination, disease, enslavement, and forced religious conversion essentially eliminated those religions entirely.

1. Schism. The religion fragments into multiple sects. Over time, the individual sects diverge to the point that neither truly represents the original religion. Examples of this would be the Orthodox / Roman Catholic schism, the Protestant Reformation, or the fracture of Islam into Sunni and Shiite sects.

Over time, cultural shifts could reduce the number of followers of the original religion to the point that the original religion no longer exists. This will likely involve appropriation of practices of the original religion.

1. Subsumation. Rather than a schism, a new religion may come along that absorbs the followers of the original religion. As a substantial number of followers convert to the new religion, the original religion is eventually reduced to nothing.

Real life examples of this would be subsumption of the Celtic polytheism religions by Christianity.

1. War. While this would technically fall under genocide, a particularly bloody war - like a modern nuclear war - could effectively destroy a religion if the followers of that religion were sufficiently localized.

I suppose complete extinction of humanity would be a trivial solution to the problem of 'how to destroy a religion' as well.

• The Native American religions didn't fully die out, they got integrated through persecution. Even today you can find, in rural areas, people praying in a church in front of a statue of St. George... but addressing their prayers to the horse. – Davidmh Jan 22 '16 at 11:19
• That's a load of nonsense. i doubt there are a dozen statues of St. George in churches in the US in its enitrety and most of those would be in the cities. Protestants don't go in for statues, and Catholics don't have mounted statues of saints. The US never spent a lot of time persecuting the Native American religions. They just persecuted the people. – Oldcat Jan 22 '16 at 23:56

TLDR: celibacy through reformation.

One possibility is that the religion, at its core, is unsustainable. For instance, the Shakers sprung up during a time of religious fervor (can't remember what that was called). Wikipedia claims that At its peak in the mid-19th century, there were 6,000 Shaker believers; last I checked, there were about two of them left. This was due, in part, to people like Mother Ann Lee being hard acts to follow, but also due to the fact that the Shakers were celibate. No new Shaker babies means they had to rely solely on new recruits, who eventually stopped coming as interest waned.

Now, the fact that your religion got to 1 billion followers probably means that its members can breed. But it's possible that over time, that could change; just like Catholicism underwent many changes after the Vatican II, perhaps the leaders of your religion thought it was time to shake things up (pun intended).

If they start calling for wholesale celibacy for all followers, there will be two effects: one, the faithful will stop breeding, and two, the casuals will leave. This will give you a big initial dropoff, followed by a gradual drain over a few generations. Eventually, like the Shakers, there should only be a few left.

As for why anyone would call for celibacy, well, it depends on the tenets of your religion, and the culture that surrounds it. There have been a few cases of celibate Christian sects that thought an end to sex meant an end to man's domination over women (again, going off the Shakers, they were all about sexual equality, also the Acts of Paul and Thecla seem to imply that celibacy empowers women). Other religions tend to discourage sex since it's an indulgence in non-religious pleasure. I'm sure there are more reasons, but honestly you could make up just about anything and in the right circumstances, people would believe it.

• Na, I don't think that would work on a large scale. Look at the christians: They are not allowed to have sex before marriage, yet I know no Christian who really had never sex before marriage and I know lots of Christians, since I'm living in a christian country (even some "real believers"). – Bounce Jan 21 '16 at 10:25
• That's really just an example of difficult standard. That you haven't met any who abide by it doesn't mean that there aren't millions who do. More priests are celebate than pederasts in Catholic hierarchies. They just aren't either newsworthy or interested in talking about it as common discourse. I don't see how your counterargument is compelling. By what definition would you declare someone's faith "real" if not how it effects behavior? – The Nate Jan 21 '16 at 12:40

Ebola

If rituals with dead are important in your religion, and some highly infectious disease like Ebola comes around, the only solution is to abolish the religious practices. Which might be possible if a secular scientific civilization is nearby, and develops and enforces new rules.

If mortality is high enough, whole civilization will collapse before new rules can be developed.

Present the followers overwhelming, undeniable proof their religion is dead, such as:

1. The Great Spaghetti Monster, a beast the size of the moon, appears in orbit carrying the broken bodies of Zen, Sunni, God, Jesus, whoever. "I have crushed your gods. Worship me!"

2. Time travel is invented and it turns out Zensunnistianity is a load of garbage invented by a cabal of greedy tea merchants.

• Welcome to the site, @steg! I edited your answer ever-so-slightly so the top line reads like an answer, so your intended meaning is more obvious. At least, that's what I hope I've done. If my edit is not what you meant, please feel free to revert it! – type_outcast Jan 21 '16 at 0:15
• The body of Zen? There's an interesting mental image. – The Nate Jan 21 '16 at 12:47
• @TheNate The Zen which has a body is not the true Zen? – pydsigner Jan 21 '16 at 18:07
• That's not the moon, it's a noodle pointing at the moon. – The Nate Jan 22 '16 at 3:13

One could try to answer with memetics by comparing it to genetics.

In this model, it proves very difficult to have 1 billion followers of an identical religion, when you look closer you will see sects, sub sects, divisions, internal fighting and suchlike. And a major religious identity like the abhramaic religion has many many divisions such as the three massive divides of Islam, Christianity and Judaism.

Continuing on the evolutionary model, you can see that it's basically impossible for the state of the religion in its present form to remain stable for very long at all. It will grow, shrink, adapt, different sects will speciate and recombine over time such that the beliefs of this population 1000 years from now will be more or less entirely incompatible with the state it's in now. And if you look at the behaviours, biases, practices of today's religions 1000 years ago, you will find them to be abhorrent compared to what we have now.

So to answer the question, yes, the religion of today will die out, but most likely due to it morphing into something different.

Not sure if this is too dark, but how about Mass Suicide? You could have the religion believe a comet is a god, and the only way to join him is to commit suicide at the right time (or make up whatever reasoning you want). You could even have someone from the religion rise up as a prophet and call everyone to do it. Those that do, die, those that don't, probably will mostly switch religions.

This has happened before

Edit: You could also make it so one of the leaders and a smaller group of followers believes the mass suicide is necessary. Then they could poison the other people without them knowing because they're afraid the followers won't have the stomach to do it themselves. (They could do this by poisoning a wine the entire group is required to drink on a holiday. The wine could be produced by the leading body of the religion, so they have easy access in order to poison it).

• It seems like scale is an important part of the original question, and Heaven's Gate didn't claim a billion followers. Any reasoning to supports your theory working at such a large scale? – nitsua60 Jan 21 '16 at 2:59
• @nitsua60 Consider that a religion that teaches peace had 2/3 of the male European population go on a suicide killing spree during the crusades. Consider that 1 billion present-day people will rather let a child starve than eat a cow because cows are holy animals. Another billion will eat cows, but not pork. That same billion believes that the body must be buried intact or the soul finds no peace. But it's OK being torn to pieces by an explosive belt, in return you get 12 virgins. No soul problem there. Do you really think there is anything absurd enough so a billion people won't pick it up? – Damon Jan 22 '16 at 12:58
• @Damon these are good points--I hope Martin_xs6 incorporates some of them! – nitsua60 Jan 22 '16 at 13:42
• There is a difference between "absurd" and "suicidal." There's a reason that of the second billion, only a small, antisocial fraction want to blow themselves up. – Obie 2.0 Jan 22 '16 at 19:30
• @nitsua60 Does the edit address the scale problem? – Martin_xs6 Jan 22 '16 at 20:12

Irrelevance would be my top vote.

Organised religion is a minority interest in Europe, and in the West generally. People (and countries) tick the "Christianity" box in surveys by default, but actual church-goers are a tiny. Wikipedia suggests 10% in the UK, 22% in the US. Go back 50 years and that was high 90-something% for both. People haven't lost their built-in need for "spirituality", but they have recognised that they have no need for organised religion.

I'm not going to go into why it's happened, because that's typically the start of a flame war. Just take the numbers and extrapolate.

• Interesting, i can see how that would lead to a decline, but would it be drastic enough? How about religious renaissances? – Serban Tanasa Jan 21 '16 at 19:15
• You could probably say a similar thing about Paganism in Rome. – Oldcat Jan 22 '16 at 23:57
• @Oldcat Perhaps, but that got a serious kicking from Christianity becoming the official state religion. The interesting thing for Christianity is that it's faded out without anything trying to take its place. For me, that's ample evidence of the truth of humanism - not only can we run a working system of morality without needing to refer to some holy teachings, but the result is a better system of morality that leaves religions in its dust. – Graham Jan 25 '16 at 16:31
• @Serban Tanasa Have there been any in the West? Sure there have been some in the ex-Communist countries, but that's in reaction to an oppressive state religion (communism; yep, that's a religion on the "walks like a duck" rule) suddenly being relaxed, and people being allowed self-expression for once. In the West we've had free choice of religion for a few decades, and the only way church attendances are going is down. – Graham Jan 25 '16 at 16:45

Yes, but it would take a considerable amount of force to do so, and would probably have to be replaced by something else. Henry VIII did something similar when he wiped out Catholicism. To achieve this, he shut down all catholic monasteries, churches and forced the new members into anglicism. He was successful in his effort and managed to establish himself as the new head of the church

• Didn't work, even just in England, so that's a useless example for anything besides the great difficulties involved. He triggered centuries of wars and there are still Catholics in England. – The Nate Jan 21 '16 at 12:43
• Actually religion thrives on persecution. – user1751825 Jan 22 '16 at 6:29

Some of it depends upon the tenets and the tenor of the movement and its adherents current context at the time of the start of the decline. This is question begging as I am assuming your question is possible.

For example, some historical times in a religion's history could be more mystical in nature versus a more empirical time period. It depends a lot on how the person and culture defines truth.

For example, Christianity. It has gone through many periods and some times in that history collapse might have happened, but for differing reasons. Today, many cultures are more relativistic, but in some area empirically based, etc. Yes those are sweeping generalizations and not exhaustive.

But, for the sake of argument, lets say you could produce the body of Jesus. Christianity would collapse. That's its entire base. And, you'd have to have a culture that would believe that is the reason for its existence and the continued requirement for its existence.

This is not to debate anything. I promise. It's just to look at the complexity of what you are asking. So, maybe you could have some prime condition that created the religion and therefore you can create, perhaps, the demise of that religion. It depends upon the complexity of your system.

You could do this with any religion. If the current cultural milieu would allow it. The timing of everything would have to be perfect.

Aliens turn up. This would totally demolish most if not all world religions dependent on an earth centric view. In fact it would probably lead to science becoming the dominant "religion" of choice, or more specifically a belief in the Standard Model (Revised 2018 Edition) being THE explanation for all phenomena in the Universe.

Yes, it is conceivable, thanks to a little thing called Education.

• This does not answer the question, adds nothing to the conversation, and is needlessly inflammatory. Graham's answer by way of contrast is none of that. – Jared Smith Jan 22 '16 at 19:00
• Incorrect, Grahamas post steadfastly refuses to answer the question and thus is not an answer. Perhaps his fear of downvotes is stronger than mine, but I am not going to shy away from giving the correct answer because some people do not want to hear it. I explained a real world example where exactly this is happening and thus answered the question. – Scott Jan 23 '16 at 22:11
• Public education in America has a long history. The decline of Christianity in America started a mere handful of decades ago. The reasons for that decline are varied, whole books have been written on the topic. The quality of American public education is arguably worse now than it was at Christianity's peak, again for reasons that are far too lengthy to explore here. So education cannot be the cause. Nor is any of this relevant to the question, which is how to construct this sort of situation in a narrative without writing an entire book on just the backstory of how it happened. – Jared Smith Jan 25 '16 at 13:23
• I don't mind downvotes, but I'd like them to be because I'm wrong, not just because of dogma. ;) Christianity has been on the slide since WWI demolished a lot of Western certainties, and WWII took that further - not education, but a change in social attitudes. Europe was where that demolition took place, and it's no accident that Europe is further down that road than the US, although the US has its own battlegrounds in civil liberties. The narrative of how to get there is long and conflicted. But I don't have to cover all that - all I have to do is show it happening. – Graham Jan 25 '16 at 16:40

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text:
# integration

• February 24th 2009, 08:21 AM
jarny
integration
Hello,

I am having trouble integrating the following equation, I thought I could do it but it is not the same as the answer in the back of the book. Can someone help me?

The integral of (2x+3)/(x+1)dx thanks
• February 24th 2009, 08:37 AM
Reckoner
Quote:

Originally Posted by jarny
Hello,

I am having trouble integrating the following equation, I thought I could do it but it is not the same as the answer in the back of the book. Can someone help me?

The integral of (2x+3)/(x+1)dx thanks

$\int\frac{2x+3}{x+1}\,dx$

$=\int\frac{2x+2+1}{x+1}\,dx$

$=\int\frac{2(x+1)+1}{x+1}\,dx$

$=\int\left[\frac{2(x+1)}{x+1}+\frac1{x+1}\right]dx$

$=\int\left(2+\frac1{x+1}\right)dx$

You could have also used polynomial long division.
• February 24th 2009, 08:38 AM
janvdl
Quote:

Originally Posted by jarny
Hello,

I am having trouble integrating the following equation, I thought I could do it but it is not the same as the answer in the back of the book. Can someone help me?

The integral of (2x+3)/(x+1)dx thanks

$\int{\frac{2x+3}{x+1}} dx$

$= 2 \int{ \frac{x}{x+1} }dx + 3 \int{ \frac{1}{x+1} }dx$

$= 2 \int{ \frac{x+1-1}{x+1} }dx + 3 \ln{(x+1)} + C$

$
= 2 \left( \int{ \frac{x+1}{x+1} }dx + \int{ \frac{-1}{x+1} }dx \right) + 3 \ln{(x+1)} + C$

$= 2 \left( \int{ 1 }dx + \int{ \frac{-1}{x+1} }dx \right) + 3 \ln{(x+1)} + C$

$= 2 \left( x - \ln{(x+1)} \right) + 3 \ln{(x+1)} + C$

$= 2x + \ln{(x+1)} + C$
• February 24th 2009, 10:19 AM
Krizalid
Of course janvdl meant to say $\ln|x+1|.$ :)
• February 24th 2009, 10:33 AM
jarny
thanks a lot guys
• February 25th 2009, 01:49 AM
janvdl
Quote:

Originally Posted by Krizalid
Of course janvdl meant to say $\ln|x+1|.$ :)

I even wrote it like that on the paper I worked it out on... Sorry about that. (Speechless)