text:
## BWAWAWA!!! I'm 4346th!!!

Stump your fellow simians.
RCC
Posts: 7019
Joined: Mon Jun 07, 2004 5:28 pm
Location: Here for now.

### BWAWAWA!!! I'm 4346th!!!

So I'm looking through the WSOP prelim results to get the details on one person I know winning two bracelets and another finishing second.

(I'd played with both in ARG events, winding up at final tables with both... thus my curiousity)

I clicked on the bracelet winner's name to see the profile, noticed the 2006 HOE ATLARGE tournament was listed. So for I minor ego boost I clicked on it because I'd finished one place ahead of him...

Turns out my name had a link to a profile. News to me. Just wanted to mention how/why I found it.

So I click on it. Turns out I am as I type this ranked 4346th in the world in tournaments with buy-ins under $1,000. (actually they are missing a FARGO 8th place finish from 2004 and they have two of the tournament structures wrong... but who cares...) I have no idea what this means. It does make me giggle tho... I may have to change the profile for the heck of it. It asks what poker players I respect. I'm putting "Shemp" in there for sure... bjornart Posts: 526 Joined: Fri Aug 06, 2004 1:51 pm Now you're 4353rd. shemp Posts: 7247 Joined: Thu Jun 10, 2004 12:16 pm Title: inbred shit-for-brains Location: Planet X You're slipping dude! CHARLEY_BIGTIME #4408 now. Fuggin has-been. http://i36.photobucket.com/albums/e43/C ... gtosee.gif RCC Posts: 7019 Joined: Mon Jun 07, 2004 5:28 pm Location: Here for now. I'm havin' a rough week. shemp Posts: 7247 Joined: Thu Jun 10, 2004 12:16 pm Title: inbred shit-for-brains Location: Planet X You've dropped to 4527th. YOU SUCK!!! RCC Posts: 7019 Joined: Mon Jun 07, 2004 5:28 pm Location: Here for now. This is what happens when you play 6 tournaments per year. So until the weekend of Oct 12-15 I'm going to take a beating... I hope to keep above 10K... shemp Posts: 7247 Joined: Thu Jun 10, 2004 12:16 pm Title: inbred shit-for-brains Location: Planet X You're up one notch to 4526th today! Probably somebody died. Doctor X Posts: 77246 Joined: Fri Jun 04, 2004 8:09 pm Title: Collective Messiah Location: Your Mom 5470 You suck. --J.D. ed Posts: 41975 Joined: Tue Jun 08, 2004 11:52 pm Title: G_D ### Re: BWAWAWA!!! I'm 4346th!!! RCC wrote:So I'm looking through the WSOP prelim results to get the details on one person I know winning two bracelets and another finishing second. (I'd played with both in ARG events, winding up at final tables with both... thus my curiousity) I clicked on the bracelet winner's name to see the profile, noticed the 2006 HOE ATLARGE tournament was listed. So for I minor ego boost I clicked on it because I'd finished one place ahead of him... Turns out my name had a link to a profile. News to me. Just wanted to mention how/why I found it. So I click on it. Turns out I am as I type this ranked 4346th in the world in tournaments with buy-ins under$1,000.

(actually they are missing a FARGO 8th place finish from 2004 and they have two of the tournament structures wrong... but who cares...)

I have no idea what this means. It does make me giggle tho... I may have to change the profile for the heck of it. It asks what poker players I respect. I'm putting "Shemp" in there for sure...
In isolation this really sounds like the ravings of a lunatic.
RCC
Posts: 7019
Joined: Mon Jun 07, 2004 5:28 pm
Location: Here for now.

### Re: BWAWAWA!!! I'm 4346th!!!

ed wrote:
RCC wrote:So I'm looking through the WSOP prelim results to get the details on one person I know winning two bracelets and another finishing second.

(I'd played with both in ARG events, winding up at final tables with both... thus my curiousity)

I clicked on the bracelet winner's name to see the profile, noticed the 2006 HOE ATLARGE tournament was listed. So for I minor ego boost I clicked on it because I'd finished one place ahead of him...

Turns out my name had a link to a profile. News to me. Just wanted to mention how/why I found it.

So I click on it. Turns out I am as I type this ranked 4346th in the world in tournaments with buy-ins under \$1,000.

(actually they are missing a FARGO 8th place finish from 2004 and they have two of the tournament structures wrong... but who cares...)

I have no idea what this means. It does make me giggle tho... I may have to change the profile for the heck of it. It asks what poker players I respect. I'm putting "Shemp" in there for sure...
In isolation this really sounds like the ravings of a lunatic.
It probably is...
En folkefiende
Posts: 17511
Joined: Thu Aug 12, 2004 4:38 am
Location: Waiting for an electrician

### Re: BWAWAWA!!! I'm 4346th!!!

RCC wrote:In isolation this really sounds like the ravings of a lunatic.

What have they done to the old home place?
Why did they have to tear it down?

:p
RCC
Posts: 7019
Joined: Mon Jun 07, 2004 5:28 pm
Location: Here for now.
Heh.

I cashed twice this trip, so my present 6500ish ranking should improve when the new results get posted.

Not yet though.


text:
What is the justification for this inequality?

1. Dec 23, 2006

quasar987

What is the justification for this inequality?

$$\int_0^1(|u(s)|+|v(s)|)(|u(s)|-|v(s)|)ds\leq \left(\int_0^1(|u(s)|^2+|v(s)|^2)ds\right)^{1/2}\left(\int_0^1(|u(s)-v(s)|^2)ds\right)^{1/2}$$

where u and v are complex-valued square-integrable Riemann integrable functions on [0,1].

Thx.

2. Dec 23, 2006

bham10246

This reminds me of Holder's inequality, and the right hand side of the inequality is the L^2 norm. Go to Wikipedia and given p and q, your p and q equal 2.

format: plain


text:
# OpenStax-CNX

You are here: Home » Content » Sampling and Data: Sampling

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# Sampling and Data: Sampling

Summary: This module introduces the concept of statistical sampling. Students are taught the difference between a simple random sample, stratified sample, cluster sample, systematic sample, and convenience sample. Example problems are provided, including an optional classroom activity.

Note: You are viewing an old version of this document. The latest version is available here.

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods.

There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Two simple random samples contain members equally representative of the entire population. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 32 members including Lisa. To choose a simple random sample of size 3 from the other members of her class, Lisa could put all 32 names in a hat, shake the hat, close her eyes, and pick out 3 names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number as shown below.

Table 1: Class Roster
ID Name
00 Anselmo
01 Bautista
02 Bayani
03 Cheng
04 Cuarismo
05 Cuningham
06 Fontecha
07 Hong
08 Hoobler
09 Jiao
10 Khan
11 King
12 Legeny
13 Lundquist
14 Macierz
15 Motogawa
16 Okimoto
17 Patel
18 Price
19 Quizon
20 Reyes
21 Roquero
22 Roth
23 Rowell
24 Salangsang
26 Stracher
27 Tallai
28 Tran
29 Wai
30 Wood

Lisa can either use a table of random numbers (found in many statistics books as well as mathematical handbooks) or a calculator or computer to generate random numbers. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are:

.94360; .99832; .14669; .51470; .40581; .73381; .04399

Lisa reads two-digit groups until she has chosen three class members (that is, she reads .94360 as the groups 94, 43, 36, 60). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.

The random numbers .94360 and .99832 do not contain appropriate two digit numbers. However the third random number, .14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Macierz, 05 corresponds to Cunningham, and 04 corresponds to Cuarismo. Besides herself, Lisa's group will consist of Marcierz, and Cunningham, and Cuarismo.

Sometimes, it is difficult or impossible to obtain a simple random sample because populations are too large. Then we choose other forms of sampling methods that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.

To choose a stratified sample, divide the population into groups called strata and then take a sample from each stratum. For example, you could stratify (group) your college population by department and then choose a simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department and do the same for the remaining departments. Then use simple random sampling to choose numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department and so on represent the members who make up the stratified sample.

To choose a cluster sample, divide the population into strata and then randomly select some of the strata. All the members from these strata are in the cluster sample. For example, if you randomly sample four departments from your stratified college population, the four departments make up the cluster sample. You could do this by numbering the different departments and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.

To choose a systematic sample, randomly select a starting point and take every nth piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1 - 20,000 and then use a simple random sample to pick a number that represents the first name of the sample. Then choose every 50th name thereafter until you have a total of 400 names (you might have to go back to the of your phone list). Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is nonrandom is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favors certain outcomes) in others.

Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (for example, they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.

In reality, simple random sampling should be done with replacement That is, once a member is picked that member goes back into the population and thus may be chosen more than once. This is true random sampling. However for practical reasons, in most populations, simple random sampling is done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same sample more than once using with replacement is very low.

For example, in a college population of 10,000 people, suppose you want to pick a sample of 1000 for a survey. For any particular sample of 1000, if you are sampling with replacement,

• the chance of picking the first person is 1000 out of 10,000 (0.1000);
• the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999);
• the chance of picking the same person again is 1 out of 10,000 (very low).

If you are sampling without replacement,

• the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);
• the chance of picking a different second person is 999 out of 9,999 (0.0999);
• you do not replace the first person before picking the next person.

Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to 4 place decimals. To 4 decimal places, these numbers are equivalent (0.0999).

Sampling without replacement instead of sampling with replacement only becomes a mathematics issue when the population is small which is not that common. For example, if the population is 25 people, the sample is 10 and you are sampling with replacement for any particular sample,

• the chance of picking the first person is 10 out of 25 and a different second person is 9 out of 25 (you replace the first person).

If you sample without replacement,

• the chance of picking the first person is 10 out of 25 and then the second person (which is different) is 9 out of 24 (you do not replace the first person).

Compare the fractions 9/25 and 9/24. To 4 decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To 4 decimal places, these numbers are not equivalent.

When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough or representative of the population. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.

## Example 1

### Problem 1

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

1. A soccer coach selects 6 players from a group of boys aged 8 to 10, 7 players from a group of boys aged 11 to 12, and 3 players from a group of boys aged 13 to 14 to form a recreational soccer team.
2. A pollster interviews all human resource personnel in five different high tech companies.
3. An engineering researcher interviews 50 women engineers and 50 men engineers.
4. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
5. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.
6. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.

#### Solution

1. stratified
2. cluster
3. stratified
4. systematic
5. simple random
6. convenience

If we were to examine two samples representing the same population, they would, more than likely, not be the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will seem natural.

## Example 2

Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task.

Suppose we take two different samples.

First, we use convenience sampling and survey 10 students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class . The amount of money they spend is as follows:

$128;$87; $173;$116; $130;$204; $147;$189; $93;$153

The second sample is taken by using a list from the P.E. department of senior citizens who take P.E. classes and taking every 5th senior citizen on the list, for a total of 10 senior citizens. They spend:

$50;$40; $36;$15; $50;$100; $40;$53; $22;$22

### Problem 1

Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?

#### Solution

No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average part-time student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample.

### Problem 2

Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?

#### Solution

No. Never use a sample that is not representative or does not have the characteristics of the population.

Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he/she has a corresponding number. The students spend:

$180;$50; $150;$85; $260;$75; $180;$200; $200;$150

### Problem 3

Do you think this sample is representative of the population?

#### Solution

Yes. It is chosen from different disciplines across the population.

Students often ask if it is "good enough" to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes.

## Optional Collaborative Classroom Exercise

### Exercise 1

As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons.

1. To find the average GPA of all students in a university, use all honor students at the university as the sample.
2. To find out the most popular cereal among young people under the age of 10, stand outside a large supermarket for three hours and speak to every 20th child under age 10 who enters the supermarket.
3. To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster.
4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
5. To determine the average cost of a two day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.

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text:
339 views

The sum of the digits of a two digit number is $12$. If the new number formed by reversing the digits is greater than the original number by $54$, find the original number.

1. $39$
2. $57$
3. $66$
4. $93$

recategorized | 339 views

Let the two digits be $x$ and $y$.

Original number$=xy=10x+y$

$x + y = 12 \quad \to(1)$

Given that new number formed by reversing the digits is greater than the original number by $54,$

$y * 10 + x = 10 * x + y + 54$
$10 y + x = 10 x + y +54$
$9y-9x = 54$
$x - y = -6 \quad \to(2)$

From $(1)$ and $(2)$

$2 x = 6$
$x = 3$ and $y = 9$

Hence, Original number $=xy=10x+y= 39$

Correct Answer: $A$
by Boss (35.6k points)
edited
Original no is 54 less than Reversed no.
66 Not possible as reversal of  66 is 66 itself.
93 and 57 also not possible because by adding 54  it will become 3 digit no.

Only option left is 39, which is correct answer. :-P
i.e. 39+54=93
by Veteran (60.5k points)
by (43 points)
x+y=12

10y+x=54+10x+y

solving the above equation we will get 9x-9y=-54

solving both equations we will get x=3,y=9
by Active (1.3k points)
Let's assume the unit digit of the number is $y$ & $x$ is the digit of ten's place.

∴ The number is $10x+y$

Now, given $x+y =12$

if the number is $yx$, then $\{(10y+x)-(10x+y)\}=54$

Now, we know, $x+y =12$

∴ $y=12-x$

Now, $\{(10y+x)-(10x+y)\}=54$

$10y+x-10x-y=54$

$10(12-x)+x-10x-(12-x)=54$

Or,$120-10x+x-10x-12+x=54$

Or, $120-20x+2x-12=54$

Or,$108-18x=54$

Or,$108-54=18x$

Or,$18x=54$

Or,$x=3$

∴$y = 12-x = 12-3 = 9$

The original number is $39$ which is option A)
by Boss (17.7k points)

Option A

too simple just put the given options, you will get your answer.

by Junior (813 points)


text:
# Commutative Algebra (math.AC)

• Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty$ and $J$ is a reduction of $I.$ For all large $n>>0,$ the function $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n,$ known as Rees polynomial of the pair $(I,J).$ We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree, i.e. $l(J)-1.$ We also study the function $\lambda({\bf I}^{\bf n}/{\bf J}^{\bf n})$ where $J_i\subsetneq I_i$ be ideals in $R$ such that $J_1\cdots J_s\neq I_1\cdots I_s,$ $J_i$ is a reduction of $I_i$ and $\lambda(I_i/J_i)<\infty$ for $i=1,\ldots,s.$ On the other hand, for ideals $J\subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e(I^n/J^n)$ which is polynomial in $n$ for all $n>>0.$ In this case also we explicitly express the deg $e(I^n/J^n)$ in some special cases. If $J$ is not a reduction of $I,$ $J$ has analytic deviation one and $\sqrt{J:I}=\sqrt{J^n:I^n}$ for all $n\geq 1,$ we give characterization of reductions in terms of deg $e(I^n/J^n).$
• Using the unfolding method given in \citeHL, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the ${\bf g}$-vectors of any cluster form a basis of $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(\Sigma)$ is given, which is realized as $(\mathcal C^{\widetilde Q},\Gamma)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructed from an unfolding $(Q,\Gamma)$ of the acyclic exchange matrix $B$ of $\mathcal A(\Sigma)$.
• If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.


text:
suppose $x=\mathrm{e}^{\mathrm{i}}$, define $$P=\{a_nx^n+\cdots+a_1x+a_0\mid a_k\in\mathbb{Z} \textrm{ and } a_k\ge0\}$$. Separate $P$ into two non-empty subsets $A$ and $B$ so that $A\cap B=\emptyset$ and $P=A\cup B$, with $A$, $B$ and $P$ are congruent to each other.
suppose the least non-zero power of polynomial $f(x)=a_nx^n+\cdots+a_1x+a_0$ is $\mathrm{ged}(f(x))$(donot know if there is a better notation, so i just reverse "deg"). then $$A=\{f(x)\in P\mid \mathrm{ged}(f) \textrm{ is even}\}, B=P\setminus A$$ then $xA=B$, but $P=(1+x^{-1})B$ scaled $B$ is not a congruent transformation. many modification tried but still no solution, so what have i do to continue or there is another method?
As $e^i$ is transzendental, it is save to assume that the relevant congruences $P\to A$, $P\to B$ will be of the form $z\mapsto e^{ni}z+m$ (corresponding to $f\mapsto x^nf+m$ on the level of polynomials). Wlog. $0\in A$, $0\notin B$. Then necessarily, $m_A=0$ and $n_A\ne0$. Then $A\cap\Bbb Z=\{0\}$, hence $1\in B$ and $m_B=1$. As also $2\in B$, $n_N=0$, i.e., $$B=P+1=\{\,f\in P\mid a_0>0\,\}.$$ Then $e^i\in A$, hence $n_A=1$, i.e., $$A=e^iP=\{\,f\in P\mid a_0=0\,\}.$$
• @LarryEppes The trick to find (or in fact to be forced to find) the conruences was to look for $P\to A$ and $P\to B$ instead of $A\to B$ – Hagen von Eitzen Jun 12 '18 at 8:40


text:
Publication
Title
An oxysulfate $Fe_{2}O(SO_{4})_{2}$ electrode for sustainable Li-based batteries
Author
Abstract
 High-performing Fe-based electrodes for Li-based batteries are eagerly pursued because of the abundance and environmental benignity of iron, with especially great interest in polyanionic compounds because of their flexibility in tuning the Fe3+/Fe2+ redox potential. We report herein the synthesis and structure of a new Fe-based oxysulfate phase, Fe2O(SO4)(2), made at low temperature from abundant elements, which electrochemically reacts with nearly 1.6 Li atoms at an average voltage of 3.0 V versus Li+/Li, leading to a sustained reversible capacity of similar to 125 mAh/g. The Li insertiondeinsertion process, the first ever reported in any oxysulfate, entails complex phase transformations associated with the position of iron within the FeO6 octahedra. This finding opens a new path worth exploring in the quest for new positive electrode materials.
Language
English
Source (journal)
Journal of the American Chemical Society. - Washington, D.C.
Publication
Washington, D.C. : 2014
ISSN
0002-7863
Volume/pages
136:36(2014), p. 12658-12666
ISI
000341544600029
Full text (Publisher's DOI)
UAntwerpen
 Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address


text:
# American Institute of Mathematical Sciences

February  2017, 37(2): 963-982. doi: 10.3934/dcds.2017040

## On the mathematical modelling of cellular (discontinuous) precipitation

 1 Department of Mathematics and the Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK 2 John W. Cahn died on 14 March 2016. There is an obituary at https://www.washingtonpost.com/local/obituaries/john-w-cahn-who-fled-nazi-germanyand-became-a-foremost-materials-scientist-dies-at-88/2016/03/15/890fe246-eac6-11e5-a6f3-21ccdbc5f74estory.html

The first author is not supported by any grant

Received  June 2015 Published  November 2016

Cellular precipitation is a dynamic phase transition in solid solutions (such as alloys) where a metastable phase decomposes into two stable phases : an approximately planar (but corrugated) boundary advances into the metastable phase, leaving behind it interleaved plates (lamellas) of the two stable phases.

The forces acting on each interface (thermodynamic, elastic and surface tension) are modelled here using a first-order ODE, and the diffusion of solute along the interface by a second-order ODE, with boundary conditions at the triple junctions where three interfaces meet. Careful attention is paid to the approximations and physical assumptions used in formulating the model.

These equations, previously studied by approximate (mostly numerical) methods, have the peculiarity that $v,$ the velocity of advance of the interface, is not uniquely determined by the given physical data such as $c_0$, the solute concentration in the metastable phase. It is hoped that our analytical treament will help to improve the understanding of this.

We show how to solve the equations exactly in the limiting case where $v=0$. For larger $v$, a successive approximation scheme is formulated. One result of the analysis is that there is just one value for $c_0$ at which $v$ can be vanishingly small.

Citation: Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040
##### References:

show all references

##### References:
Schematic cross-section of a cellular precipitation front, advancing upwards at velocity $v$. The as yet undisturbed metastable phase is labelled 0; the two (relatively) stable phases into which it separates after the front has passed are labelled $\alpha$ and $\beta$
Notation for the coordinate axes and for angles and arc lengths
The thermodynamic construction for phase equilibrium. The common tangent has slope $\mu_{TJ}$ and touches the curve at the points with abscissas $c_\alpha^{eq}$ and $c_\beta^{eq}.$
Dihedral angles $\delta$, inclinations of arcs $\theta$ and surface tensions $\gamma$ at a triple junction
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## machine learning problems and solutions pdf

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# Quartics

$ax^4 + bx^2 + c\mbox{ can always be factored into real quadratics.}\,$

by landen

After it is factored, you can easily solve it by using the quadratic formula on the two real factors. Or you can use the factors for solving partial fraction integrals. We can take a > 0 and also multiply by 4a to avoid fractions.

Suppose b2 − 4ac > 0 $\begin{matrix} 4a^2x^4 + 4abx^2 + 4ac &= &(2ax^2 + b)^2 -b^2 + 4ac\\ &= &(2ax^2+b - \sqrt{b^2-4ac})\ (2ax^2+b + \sqrt{b^2-4ac}) \end{matrix}$

Suppose b2 − 4ac < 0 then 4ac > 0 $\begin{matrix} a^2\,x^4+a\,b^2\,x^2+a\,c&=&\left(a\,x^2+\sqrt{a\,c}\right)^2-2\,a\,\sqrt{a\,c}\,x^2+a\,b^2\,x^2\\ &=&\left(a\,x^2-\sqrt{2\,a\,\sqrt{a\,c}-a\,b^2}\,x+\sqrt{a\,c}\right) \,\left(a\,x^2+\sqrt{2\,a\,\sqrt{a\,c}-a\,b^2}\,x+\sqrt{a\,c}\right) \end{matrix}$


text:
Multiplicity spectra in e^{+}e^{-} and \overline{p}p collisions in terms of Tsallis and Weibull distributions.

# Multiplicity spectra in e+e− and ¯¯¯pp collisions in terms of Tsallis and Weibull distributions.

S. Sharma, M. Kaur, S. Thakur
###### Abstract

Charged hadron production in the annihilations at 91 to 206 GeV in full phase space and in collisions at 200 to 900 GeV collision energies are studied using non-extensive Tsallis and stochastic Weibull probability distributions. The Tsallis distribution shows better description of the data than the Weibull distribution. The 2-jet modification of the statistical distribution is applied to describe data. The main features of these distributions can be described by a two-component model with soft, collective interactions at low transverse energy and hard, constituent interactions dominating at high transverse energy. This modification is found to give much better description than a full-sample fit, and again Tsallis function is found to better describe the data than the Weibull one pointing at the non-extensive character of the multiparticle production process.

Keywords: Charged Multiplicity; LEP; PDFs; Non-extensive entropy

PACS numbers: 5.90.+m, 12.40.Fe, 13.66.Bc

## 1 Introduction

Charged hadron production in high energy particle interactions can be understood in terms of several theoretical and phenomenological models derived from statistics, empirical relationships, phenomenological concepts or pure theoretical cosiderations [?-?]. To understand the particle production mechanism, some models have used ensemble theory from statistical mechanics to extract information from dynamical fluctuations as a key source of inputs to study the multiplicity patterns. Several distributions based on the statistical analyses have been derived for the understanding of particle production and the multiplicity distributions. However, it was realized that data on many single particle distributions deviate from the distributions expected from the statistical models, based on standard statistical mechanics of Boltzman-Gibbs which treats the entropy as an extensive property. This initiated the idea of including modifications to include possible intrinsic, non-statistical fluctuations. These were identified as the source of the deviations. Such fluctuations are important as possible signals of phase transition(s) taking place in an hadronizing system.

The Tsallis formalism was introduced three decades ago [?]. It is a non-extensive generalization of the statistical mechanics and has been very successful in describing very different physical systems in terms of statistical approach, including multiparticle production processes at lower energies. Specially designed to include self-similar systems and systems with long range interactions, it is a reasonable choice to study hadronisation process. The collisions at collider experiments are expected to fall in this category. Thus the Tsallis statistical approach [?-?], was successfully used for multiparticle production processes. The new approach in the Tsallis -statistics which includes entropy as a non-extensive property to describe the particle production has not been tested with the highest energy data from collisions and collisions, though this has been used to explain heavy-ion and some collision data, see references [?-?]. Most of these studies have focussed on the spectra of the hadrons.

The non-extensive part of entropy is quantified in terms of a parameter , the entropic index in the Tsallis function and envisaged that it should have a value greater than unity. The total Tsallis entropy of two sub-systems and is not equal to the sum of the entropies of the subsystems, but is given by;

 Sq(a,b)=Sa+Sb+(1−q)SaSb (1)

In Tsallis -statistics probability is calculated by using the partition function Z, as

 PN=ZNqZ (2)

where Z represents the total partition function and represents partition function at a particular multiplicity, of the grand canonical ensemble of gas consisting of particles. , the average number of particles, is given by

 ¯N=Vn[1+(q−1)λ(Vnλ−1)−2v0n] (3)

-parameter is related to and excluded volume , by

 1K=(q−1)λ2−2v0V (4)

where is the volume containing particles, is related to the temperature through the parameter as;

 λ(β,μ)=−βn∂n∂β (5)

Details of the Tsallis distribution and how to find the probability distribution can be obtained from [?]. In the hadron-hadron interactions, the dynamics of particle production is centered around the fragmentation process, partons are produced at the intermediate stage which quickly undergo fragmentation into hadrons.

A description of the fragmentation process can also be given in terms of Weibull distribution [?] in which it has been shown that result of a single event fragmentation leading to a branching tree of cracks in the materials that show fractal behaviour [?] and can be described by a Weibull-like distribution. This is related to the particle number distribution developed during the fragmentation, the so called multiplicity distribution in case of particle collisions.

Weibull distribution has been studied to describe multiplicity distributions in collisions up to 91 GeV [?] and also for collisions. Weibull parametrization of the multiplicity distribution has also been used to describe the multi-dimensional fluctuations and genuine multi-particle correlations in [?,?].

The Tsallis distribution [?,?] and the Weibull distribution [?] are based on the concepts of statistical mechanics and stochasticity. The simplicity of these statistically inspired models and the ease of application to data, is the beauty of the analyses. Nowadays the statistical approach is a standard procedure used to model high energy multiparticle production processes [?].

In the present work, we focus on the multiplicity distributions, in full phase space in annihilations up to the highest available center-of-mass energies and in collisions at energies ranging from 200 to 900 GeV in restricted rapidity windows. The presence of a shoulder structure in the multiplicity distribution was observed in at collision energy of =91 GeV [?,?] and in data at 900 GeV [?] and at =1800 GeV [?]. The data from LEP2 also confirmed the presence of the shoulder structure at higher then energies [?,?]. It is the higher energy data where the shoulder structure becomes more pronounced.

Recently, we have studied [?] the data in terms of the Tsallis distribution and the Weibull distribution at lower energies in full phase space and in different rapidity windows. To study the high energy data and to take into account the shoulder structure, we implement a 2-jet modification to improve the comparison between the predicted and the experimental values. The two-component approach has been suggested earlier in reference [?]. The study indicates that the Tsallis function is able to reproduce the experimental results far better than the Weibull one. Further analysis is performed here to assert our conclusions.

In Section 2, an outline of probability distribution functions of Tsallis and Weibull distrbutions and their modified forms are given. Details along with the references can be found in [?]. 2-jet fractions denoted by at various energies have been taken from the references [?,?,?-?] obtained by OPAL and L3/ALEPH experiments. The 2-jet data samples considered are only those for collisions, and no 2-jet modification is made to study data as no 2-jet data-samples being available for these collisions. Section 3 presents the analyses of experimental data and the results obtained by the two approaches. Discussion and conclusion are given in Section 4.

## 2 Tsallis and Weibull distributions and their 2-jet modification

### 2.1 Tsallis and its 2-jet modification

Details and the method of calculating the partition function for particles, in the Grand Canonical Ensemble, Tsallis -index and Tsallis -particle probability distribution can be obtained from [?]. The probability is calculated by using the partition function , as

 PN=ZNqZ (6)

where represents the total partition function and represents partition function at a particular multiplicity .

The probability distribution for the 2-jet modified distribution is calculated by adding a weighted superposition of multiplicity in 2-jet and in multi-jet events as follows;

 PN(α:¯n1,V1,v01,q1:¯n2,V2,v02,q2)=αP2−jetN+(1−α)Pmulti−jetN. (7)

where is a weight factor which gives 2-jet fraction and is determined from a jet finding algorithm.

### 2.2 Weibull and its 2-jet modification

The probability density function of a Weibull random variable is;

 PN(N,λ,k)=kλ(Nλ)(k−1)exp−(Nλ)k,whereN≥0. (8)

Where is the scale parameter and is the shape parameter. These two parameters for the distribution are related to the mean of the function, as

 ¯N=λΓ(1+1/k) (9)

The modified Weibull function can be obtained by the weighted superposition of the two Weibull distributions, as above, namely;

 PN(α:N1,λ1,k1;N2,λ2,k2)=αP2−jetN+(1−α)Pmulti−jetN. (10)

## 3 Results

The experimental data on annihilation at different collision energies by two experiments are analysed here. The details of the data from the L3 [?] and OPAL [?,?,?,?,?] experiments at different energies between = 91 to 206 GeV in the full phase space are given in Table 1. For comparison, we also analyse data from collisions at energies ranging from 200 to 900 GeV in restricted rapidity windows from the UA5 collaboration [?,?] as given in Table 2. The experimental distributions are fitted with the predictions from two probability functions, as described above.

### 3.1 one-component Tsallis versus Weibull functions

The probability distribution for Tsallis function is calculated from equation (2) and Weibull function from eqns. (8) and (9) and applied to fit the data in full phase space, as shown in Figures 1 and 2. Parameters of the fits and values are given in Table 2 and the corresponding p values are given Table 3.  We find that overall, Weibull distribution fails to reproduce the data, particularly in the high multiplicity regions, while Tsallis distribution shows good fit in full phase space.

One observes that the values are significantly lower for the Tsallis fits in comparison to the Weibull fits. This is true for all energies. A careful examination of the values from Table 3 shows that for the data from L3 experiment at all energies from 130 to 206 GeV, the Weibull fits are statistically excluded with . While for the Tsallis fits for the data only at 200.2 and 206.2 are statistically excluded with and is good for all other energies with .  For the data from OPAL experiment, the Weibull fit is statistically excluded systematically for all energies between 91 to 189 GeV with except being good for energy 172 GeV. Again Tsallis fit is excluded only for 91 GeV data and remains a good fit for all energies from 133 to 189 GeV with . A comparison of the and values in Table 3 shows that values for the Tsallis distributions are lower by several orders, confirming that Tsallis distribution fits the data far better than Weibull distribution.

Figures 3 and 4 show the Tsallis and Weibull fits to the data on collisions from UA5 collaboration at energies ranging from 200 to 900 GeV in restricted rapidity windows as well as in full phase space. The parameters of the fits are listed in Table 4. Again the Tsallis distribution shows very good fits at all energies and all rapidities with statistically significant values values in comparison to the Weibull distribution, as shown in Table 5. In each of the data samples of , for the Weibull distribution, values increase with energy and in case of collisions these increase with energy as well as with the size of the central rapidity interval. Similarly for Tsallis distribution, the value which measures the entropic index, of the Tsallis statistics is consistently higher than 1 in each of the above mentioned cases. This confirms the property of the non-extensivity in the data as proposed by the Tsallis q-statistics.

### 3.2 Modified Tsallis versus modified Weibull distributions

As it is observed above, in the case of the Weibull fits, the description is good enough only for the data at some energies, while it gets quite high values for the rest. Also, though the Tsallis fits are better than the Weibull ones, there is still a room for improvements. Following the two-component approach suggested by Giovannini [?] we consider the two-components of the modified Tsallis distribution and modified Weibull distribution for data. The probability functions for the two cases can be derived from the equations (7) and (10). The fit parameters, and values for both modified Weibull and modified Tsallis distributions are given in Tables 3, 6 and 7. Figure 5 and 6 show the modified distributions for the L3 and OPAL data.

One can see that values given in Table 3 show that following the two-component fits improve substantially the description. The modified Tsallis distributions describe the data extremely well at all energies. For the modified Weibull distribution, though the values improve substantially, it still fails for energies above the peak, namely 161, 188.6, 189 and 206.2 GeV data samples for which .

Figure 7 shows plots of the , and values estimated from the Tsallis fits and modified Tsallis fits. The bands shown are the confidence bands. The mean values are , and . The figure shows that the values in both the cases of the Tsallis and modified Tsallis fits, exceed unity, emphasising the non-extensive nature of the Tsallis entropy.

Figure 8 shows the dependence of , , , on c.m.s energy of collisions. As the parameter is connected with the average multiplicity, it is expected to increase with the energy of collision and hence charged multiplicity. The dependence of on c.m. energy is studied as a power law , and are the fit parameters. One finds the following values of the parameters and :

for : ,

for : ,

and for : .

The modified analyses for could not be carried out as the 2-jet fractions for these data in different rapidity intervals as well as in the full phase space are not available.

### 3.3 Discussion

The standard Boltzmann-Gibbs statistics, with =1 produces a distribution which is not consistent with the experimental data at high energy and at high multiplicity tail. It is known that it could lead to unphysical results for systems having long-range interactions. This is the non perturbative regime of QCD which plays an important role in hadronization. The large particle density fluctuations [?,?] present in hadron production, which could not be explained, mandated an alternative formalism to be developed. One such case is the Tsallis formalism.

In the Tsallis distribution, -parameter in eq.(4) is related to , the entropic index, , the excluded volume associated to a particle and the volume of the system. The detailed relationship can be found in [?]. The -parameter measures the deviation from Poisson distribution caused by resonance decay and charge conservation and is related to the variance. The definition of is motivated by the parameter of a negative binomial distribution. The Tsallis statistics for with excluded volume causes a substantial broadening of the distribution, taking it much closer to the data. Thus the role of -parameter is crucial to the value of entropic index, . In some analyses the excluded volume is fixed between 0.3-0.4 . The corresponding value of volume then varies from few to few tens of [?].

The smooth increase of with the center of mass energy can be understood as the expected increase of the influence of the partonic interactions among the hadronic particles produced in the event.

Table 2 shows that the shape parameter for the Weibull function determines the shape of the distribution. For the highest LEP energy range, the value of does not change much within limits of errors, as can be seen in Tables 2 and  6. This behaviour is related to the soft gluon emission and subsequent hadronisation. Within a given rapidity interval, the value of decreases slightly with increasing energy. However, it increases considerably from smaller to larger rapidity intervals, as can be seen in Table 4. The scale parameter of the Weibull distribution determines the width of the distribution. Larger value of scale parameter produces a broader distribution. This again is observed in Tables 2 and 4. The width of the probability distribution depends upon c.m. energy. At higher collision energies, more number of high multiplicity events are produced and the multiplicity distribution becomes broader. As a result, increases to compensate for the width. This trend is endorsed by values in the two Tables. Similar results are observed from the modified Weibull fit distribution parameters in Table 6 where values increase systematically from lower to higher rapidity windows.

For various energies, it is shown that by appropriately weighting the multiplicity distribution with the 2-jet fraction in collisions at = 91 to 206 GeV, both the Tsallis distributions and the Weibull distributions substantially improve the agreement with the data giving the statistically significant results. These modified Tsallis distributions reproduce the data well with for all energies, from both the L3 and OPAL experiments. The modified Weibull distributions, also improve the fits by several orders, but fail to describe the data at most of the energy points, as can be observed from values in Table 3

The highest energy data sets have typically low statistics, due to which fit parameters suffer from large errors, especially for modified Tsallis distribution. Nevertheless the present detailed analysis establishes that the performance of the Tsallis function is better than Weibull function. The value known as entropic index for the Tsallis distribution, accounts for the non-extensive thermo-statistical effects in hadron production and is expected to exceed unity in the Tsallis statistics. This is confirmed from the mean values of , as measured for the total sample of events, events with two-jets and events with multijets. The behaviour is found to be more pronounced in the events with higher multiplicity, as evident from the difference in q-values for 2-jet and multi-jet components. The analyses for the Tsallis distribution and the Weibull distribution for collisions from 200 to 900 in full phase space as well as in restricted rapidity windows have also been done. It is found that the Weibull distribution is excluded in full phase space at all energies and in several rapidity windows. Whereas for the Tsallis, it is acceptable at most of the energy and rapidity values with the exception of 900 GeV in full phase space and at 540 GeV for GeV where the fits have . For Tsallis distribution, in each case value exceeds unity, emphasising the non-extensive nature of the Tsallis entropy.

## 4 Conclusions

Analyses of the multiplicity distributions measured in collisions at LEP at energies, =91 to 206 GeV and in collisions at 200 to 900 GeV have been done in the context of Weibull and Tsallis distributions. It is observed that the the Weibull fits for the data from L3 experiment at all energies from 130.1 GeV to 206.2 GeV for the collisions are statistically excluded with , and for the Tsallis fits the data only at 200.2 GeV and 206.2 GeV are statistically excluded and are good for all other energies. For the data from OPAL experiment, Weibull fits are statistically excluded for all energies between 91 GeV to 189 GeV with in annihilation, except the 172 GeV data. In contrast, the Tsallis fits are excluded only for energy at 91 GeV and remain good for all other energies of 133 GeV to 189 GeV. Tsallis non-extensive statistics gives a good description of the particle production in high energy collisions.

Acknowledgement: We thank Professor Edward K.G. Sarkisyan for important suggestions and comments. One of the authors, S. Sharma is grateful to the Department of Science and Technology, Government of India for the fellowship grant.

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