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# zbMATH — the first resource for mathematics

## Dettweiler, Michael

Compute Distance To:
 Author ID: dettweiler.michael Published as: Dettweiler, M.; Dettweiler, Michael External Links: MGP · Wikidata
 Documents Indexed: 23 Publications since 1999 Reviewing Activity: 17 Reviews
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#### Co-Authors

 3 single-authored 9 Reiter, Stefan 2 Amram, Meirav 2 Catanese, Fabrizio M. E. 2 Sabbah, Claude 2 Teicher, Mina 2 Wewers, Stefan 1 Friedman, Michael 1 Heinloth, Jochen 1 Jöllenbeck, Mirjam 1 Kühn, Ulf 1 Schmidt, Johannes 1 Yun, Zhiwei
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#### Serials

 3 Publications of the Research Institute for Mathematical Sciences, Kyoto University 2 Israel Journal of Mathematics 2 Journal of Algebra 1 Beiträge zur Algebra und Geometrie 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Compositio Mathematica 1 Journal of Differential Equations 1 Journal für die Reine und Angewandte Mathematik 1 Transactions of the American Mathematical Society 1 Journal of Symbolic Computation 1 International Journal of Mathematics 1 Mathematical Research Letters 1 Advances in Geometry 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Oberwolfach Reports
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#### Fields

 17 Algebraic geometry (14-XX) 8 Field theory and polynomials (12-XX) 8 Several complex variables and analytic spaces (32-XX) 6 Ordinary differential equations (34-XX) 5 Group theory and generalizations (20-XX) 3 Number theory (11-XX) 2 Special functions (33-XX) 1 General and overarching topics; collections (00-XX)

#### Citations contained in zbMATH

18 Publications have been cited 122 times in 75 Documents Cited by Year
An algorithm of Katz and its application to the inverse Galois problem. Zbl 1049.12005
Dettweiler, Michael; Reiter, Stefan
2000
Middle convolution of Fuchsian systems and the construction of rigid differential systems. Zbl 1183.34137
Dettweiler, Michael; Reiter, Stefan
2007
Rigid local systems and motives of type $$G_{2}$$. Zbl 1194.14036
Dettweiler, Michael; Reiter, Stefan
2010
Vector bundles on curves coming from variation of Hodge structures. Zbl 1368.14019
Catanese, Fabrizio; Dettweiler, Michael
2016
The direct image of the relative dualizing sheaf needs not be semiample. Zbl 1310.14017
Catanese, Fabrizio; Dettweiler, Michael
2014
Hodge theory of the middle convolution. Zbl 1307.14015
Dettweiler, Michael; Sabbah, Claude
2013
Plane curve complements and curves on Hurwitz spaces. Zbl 1074.14026
Dettweiler, Michael
2004
The classification of orthogonally rigid $$G_{2}$$-local systems and related differential operators. Zbl 1312.32015
Dettweiler, Michael; Reiter, Stefan
2014
Painlevé equations and the middle convolution. Zbl 1134.34057
Dettweiler, Michael; Reiter, Stefan
2007
Variation of local systems and parabolic cohomology. Zbl 1131.14016
Dettweiler, Michael; Wewers, Stefan
2006
On rigid tuples in linear groups of odd dimension. Zbl 0945.12001
Dettweiler, Michael; Reiter, Stefan
1999
Middle convolution and Galois realizations. Zbl 1077.12002
Dettweiler, Michael
2004
On Galois representations via Siegel modular forms of genus two. Zbl 1004.11028
Dettweiler, Michael; Kühn, Ulf; Reiter, Stefan
2001
Erratum to: “Hodge theory of the middle convolution”. Zbl 1408.14036
Dettweiler, Michael; Sabbah, Claude
2018
Rigid $$G_{2}$$-representations and motives of type $$G_{2}$$. Zbl 1346.14027
Dettweiler, Michael; Schmidt, Johannes
2016
The fundamental group for the complement of Cayley’s singularities. Zbl 1184.14004
Amram, Meirav; Dettweiler, Michael; Friedman, Michael; Teicher, Mina
2009
Variation of parabolic cohomology and Poincaré duality. Zbl 1157.14003
Dettweiler, Michael; Wewers, Stefan
2006
Construction of relative motives with interesting étale realization using the middle convolution. Zbl 1098.14518
Dettweiler, M.
2004
Erratum to: “Hodge theory of the middle convolution”. Zbl 1408.14036
Dettweiler, Michael; Sabbah, Claude
2018
Vector bundles on curves coming from variation of Hodge structures. Zbl 1368.14019
Catanese, Fabrizio; Dettweiler, Michael
2016
Rigid $$G_{2}$$-representations and motives of type $$G_{2}$$. Zbl 1346.14027
Dettweiler, Michael; Schmidt, Johannes
2016
The direct image of the relative dualizing sheaf needs not be semiample. Zbl 1310.14017
Catanese, Fabrizio; Dettweiler, Michael
2014
The classification of orthogonally rigid $$G_{2}$$-local systems and related differential operators. Zbl 1312.32015
Dettweiler, Michael; Reiter, Stefan
2014
Hodge theory of the middle convolution. Zbl 1307.14015
Dettweiler, Michael; Sabbah, Claude
2013
Rigid local systems and motives of type $$G_{2}$$. Zbl 1194.14036
Dettweiler, Michael; Reiter, Stefan
2010
The fundamental group for the complement of Cayley’s singularities. Zbl 1184.14004
Amram, Meirav; Dettweiler, Michael; Friedman, Michael; Teicher, Mina
2009
Middle convolution of Fuchsian systems and the construction of rigid differential systems. Zbl 1183.34137
Dettweiler, Michael; Reiter, Stefan
2007
Painlevé equations and the middle convolution. Zbl 1134.34057
Dettweiler, Michael; Reiter, Stefan
2007
Variation of local systems and parabolic cohomology. Zbl 1131.14016
Dettweiler, Michael; Wewers, Stefan
2006
Variation of parabolic cohomology and Poincaré duality. Zbl 1157.14003
Dettweiler, Michael; Wewers, Stefan
2006
Plane curve complements and curves on Hurwitz spaces. Zbl 1074.14026
Dettweiler, Michael
2004
Middle convolution and Galois realizations. Zbl 1077.12002
Dettweiler, Michael
2004
Construction of relative motives with interesting étale realization using the middle convolution. Zbl 1098.14518
Dettweiler, M.
2004
On Galois representations via Siegel modular forms of genus two. Zbl 1004.11028
Dettweiler, Michael; Kühn, Ulf; Reiter, Stefan
2001
An algorithm of Katz and its application to the inverse Galois problem. Zbl 1049.12005
Dettweiler, Michael; Reiter, Stefan
2000
On rigid tuples in linear groups of odd dimension. Zbl 0945.12001
Dettweiler, Michael; Reiter, Stefan
1999
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#### Cited by 90 Authors

 11 Dettweiler, Michael 8 Reiter, Stefan 6 Haraoka, Yoshishige 4 Catanese, Fabrizio M. E. 4 Patrikis, Stefan T. 3 Crawley-Boevey, William 2 Amram, Meirav 2 Bauer, Ingrid C. 2 Gleizer, Oleg 2 Hiroe, Kazuki 2 Kawakami, Hiroshi 2 Kerr, Matthew 2 Lu, Xin 2 Pirola, Gian Pietro 2 Teicher, Mina 2 Torelli, Sara 2 Yamakawa, Daisuke 2 Yun, Zhiwei 1 Akhmedov, Anar 1 Arias-de-Reyna, Sara 1 Armana, Cécile 1 Belavin, Vladimir A. 1 Bibilo, Yulia 1 Bleher, Frauke M. 1 Böckle, Gebhard 1 Bogner, Michael 1 Boxer, George 1 Cadoret, Anna 1 Calegari, Frank 1 Cao, Junyan 1 Ciliberto, Ciro 1 Conder, Matthew J. 1 Dedieu, Thomas 1 Dieulefait, Luis Victor 1 Emerton, Matthew 1 Filipuk, Galina V. 1 Flamini, Flaminio 1 Frediani, Paola 1 Galati, Concettina 1 Ghigi, Alessandro 1 Gong, Cheng 1 González-Alonso, Victor 1 Guralnick, Robert Michael 1 Hall, Chris 1 Hallouin, Emmanuel 1 Hamaguchi, Sho 1 Jöllenbeck, Mirjam 1 Karemaker, Valentijn 1 Kawamata, Yujiro 1 Kazakov, Alexander Yakov 1 König, Joachim 1 Konnai, Shotaro 1 Kostov, Vladimir Petrov 1 Levin, Brandon 1 Litterick, Alastair J. 1 Madapusi Pera, Keerthi 1 Malle, Gunter 1 Mano, Toshiyuki 1 Marin, Ivan 1 Moteki, Takahiro 1 Nakamura, Akane 1 Narváez-Macarro, Luis 1 Oshima, Toshio 1 Pardini, Rita 1 Paun, Mihai 1 Pearlstein, Gregory J. 1 Penegini, Matteo 1 Polizzi, Francesco 1 Rebolledo, Marusia 1 Roberts, David Peter 1 Rollenske, Sönke 1 Sabbah, Claude 1 Sakallı, Sümeyra 1 Santachiara, Raoul 1 Schmidt, Johannes 1 Sevenheck, Christian 1 Shaw, Peter R. 1 Simpson, Carlos Tschudi 1 Stoppino, Lidia 1 Takei, Yoshitsugu 1 Takemura, Kouichi 1 Tan, Sheng-Li 1 Tang, Shiang 1 Taylor, Richard Lawrence 1 Thomas, Lara 1 Vila, Núria 1 Wewers, Stefan 1 Xu, Wan-Yuan 1 Yokoyama, Toshiaki 1 Zuo, Kang
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#### Cited in 38 Serials

 7 Advances in Mathematics 5 Journal of Algebra 5 Transactions of the American Mathematical Society 3 Israel Journal of Mathematics 3 Compositio Mathematica 3 Duke Mathematical Journal 3 Inventiones Mathematicae 3 Mathematische Annalen 2 Communications in Mathematical Physics 2 Mathematische Zeitschrift 2 Publications of the Research Institute for Mathematical Sciences, Kyoto University 2 Journal of Symbolic Computation 2 Advances in Geometry 2 Bollettino dell’Unione Matematica Italiana 1 Journal of Mathematical Analysis and Applications 1 Journal of Mathematical Physics 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Annales de l’Institut Fourier 1 Collectanea Mathematica 1 Funkcialaj Ekvacioj. Serio Internacia 1 Publications Mathématiques 1 Journal of Differential Equations 1 Theoretical Computer Science 1 Topology and its Applications 1 International Journal of Algebra and Computation 1 Expositiones Mathematicae 1 Experimental Mathematics 1 Journal of Mathematical Sciences (New York) 1 Selecta Mathematica. New Series 1 The Ramanujan Journal 1 Journal of Dynamical and Control Systems 1 Journal of Nonlinear Mathematical Physics 1 Journal of the Institute of Mathematics of Jussieu 1 Algebra & Number Theory 1 Japanese Journal of Mathematics. 3rd Series 1 International Journal of Group Theory 1 Forum of Mathematics, Sigma 1 European Journal of Mathematics
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#### Cited in 19 Fields

 37 Algebraic geometry (14-XX) 25 Ordinary differential equations (34-XX) 21 Several complex variables and analytic spaces (32-XX) 14 Group theory and generalizations (20-XX) 12 Number theory (11-XX) 10 Field theory and polynomials (12-XX) 9 Special functions (33-XX) 6 Associative rings and algebras (16-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 3 Nonassociative rings and algebras (17-XX) 2 Convex and discrete geometry (52-XX) 2 Differential geometry (53-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Quantum theory (81-XX) 1 Topological groups, Lie groups (22-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Manifolds and cell complexes (57-XX) 1 Computer science (68-XX)

#### Wikidata Timeline

The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.


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# econometrics

You are currently browsing articles tagged econometrics.

## GMM and its application outside finance

The 2013 Nobel prize in economics was won by Fama, Shiller, and some other dude, according to most media accounts. Fama and Shiller were pretty easy to explain: one of them is at Chicago and is associated with a theory called “efficient markets,” so he’s the free market guy. Shiller criticized the Chicago guy, so we know where to put him on the political spectrum. But this third guy, Hansen, well, he’s at Chicago, but he does some sort of theoretical econometrics, so if we’re the Guardian we’ll just assume he’s “ultra-conservative” and then ignore him, or if we’re anyone else we’ll skip to just ignoring him (even the Economist gives up, complaining they can’t explain his work without “writing all sorts of equations in our newspaper.”) This post attempts to provide a relatively gentle, albeit with all sorts of equations, introduction to part of the third guys’s research, focusing on applications in causal modeling in microeconomics rather than the examples from finance or macroeconomics.

Tags: , ,

## Remarks on Chen and Pearl on causality in econometrics textbooks

Bryant Chen and Judea Pearl have published a interesting piece in which they critically examine the discussions (or lack thereof) of causal interpretations of regression models in six econometrics textbooks. In this post, I provide brief assessments of the discussion of causality in nine additional econometrics texts of various levels and vintages, and close with a few remarks about causality in textbooks from the perspective of someone who does, and teaches, applied econometrics. Like Chen and Pearl, I find some of these textbooks provide weak or misleading discussion of causality, but I also find one very good and one excellent discussion in relatively recent texts. I argue that the discussion of causality in econometrics textbooks appears to be improving over time, and that the oral tradition in economics is not well-reflected in econometrics textbooks.

The Chen and Pearl paper has been around for a while in working paper form and recently came out in the Real World Economics Review, also available here from the authors with much clearer typesetting.

The additional textbooks I discuss below are: Amemiya (1985), Kmenta (1986), Davidson and MacKinnon (1993), Gujarati (1999), Hayashi (2000), Wooldridge (2002), Davidson and MacKinnon (2004), Deilman (2005), and Cameron and Trivedi (2005).

## The intuition of robust standard errors

Commonly econometricians conduct inference based on covariance matrix estimates which are consistent in the presence of arbitrary forms of heteroskedasticity; the associated standard errors are referred to as “robust” (also, confusingly, White, or Huber-White, or Eicker-Huber-White) standard errors. These are easily requested in Stata with the “robust” option, as in the ubiquitous

reg y x, robust

Everyone knows that the usual OLS standard errors are generally “wrong,” that robust standard errors are “usually” bigger than OLS standard errors, and it often “doesn’t matter much” whether one uses robust standard errors.  It is whispered that there may be mysterious circumstances in which robust standard errors are smaller than OLS standard errors. Textbook discussions typically present the nasty matrix expressions for the robust covariance matrix estimate, but do not discuss in detail when robust standard errors matter or in what circumstances robust standard errors will be smaller than OLS standard errors. This post attempts a simple explanation of robust standard errors and circumstances in which they will tend to be much bigger or smaller than OLS standard errors.

## What do we know about the effect of income inequality on health?

This post briefly surveys some of the methods and results in the literature on health and income inequality, closing with some remarks on problems with the existing literature and where future research may take us. It is not intended as anything resembling a comprehensive survey; Lynch et al (2004) provides a useful review of the empirical literature up to that time.

## Patient knowledge, antibiotic abuse, and impolite physicians

Antibiotic overuse causes great social harm yet is largely absent from public discussion of drug policy. There is a textbook external effect of an antibiotic prescription: the more antibiotics are used, the higher the risk we all face of resistant infections. As a result, there tends to be too much use of antibiotics. There have been ongoing efforts to reduce use of antibiotics, particularly in the context of treating respiratory infections, in part by educating GPs, the supply side of the relationship, on appropriate use.

In “Patient knowledge and antibiotic abuse: Evidence from an audit study in China” Janet Currie, Wanchuan Lin, and Wei Zhang consider the demand side of the relationship: what is the effect of patient knowledge on antibiotic use? Read the rest of this entry »

## Will Yelp kill Applebee’s?

A new working paper by Michael Luca estimates the effect of Yelp reviews on Seattle restaurant revenues. Disentangling causality here is difficult, as even if reviews have no effect on revenues we would expect to observe reviews and revenues both moving with changes in underlying relative quality. Luca exploits a quirk in the way Yelp presents information: average scores are reported rounded in 0.5 star bins on a 5 star scale. For example, underlying average scores of 2.76 and 3.24 are both reported as “3 stars,” but a good review which bumps the average up to 3.25 bumps the reported score up to 3.5 stars. The estimates show that Yelp reviews do have a substantial effect on revenues.
Read the rest of this entry »

« Older entries

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## anonymous one year ago A 0.145 kg baseball is thrown horizontally at 15 m/s from a platform, so that it starts at 5.2 m above the ground. How much has its kinetic energy increased when it reaches the ground? 16 J , 24 J , 0 J , or 7.4 J ? :/

1. Michele_Laino

the initial kinetic energy is: $KE = \frac{1}{2}m{v^2} = \frac{1}{2} \times 0.145 \times {15^2} = ...Joules$

2. anonymous

so we get 16.3125? so our solution is 16 J?

3. Michele_Laino

that result is right! Nevertheless it is not our final answer!

4. anonymous

ohh okay! what is netx?

5. Michele_Laino

we have to consider that situation: |dw:1433103218745:dw| our ball, in free falling, has gained a vertical velocity

6. anonymous

yes:)

7. Michele_Laino

now, we have to compute that vertical speed

8. anonymous

ok! how do we do taht?

9. Michele_Laino

such vertical speed is given by the so called formula of Torricelli, which is: $v = \sqrt {2gh} = \sqrt {2 \times 9.81 \times 5.2} = ...m/\sec$

10. anonymous

10.10006 ? what happens now? :O

11. Michele_Laino

ok! Now we have this situation: |dw:1433103512624:dw|

12. Michele_Laino

namely the final velocity has the subsequent magnitude: ${v_f} = \sqrt {{{10}^2} + {{15}^2}} = ...m/\sec$

13. anonymous

18.0277?

14. Michele_Laino

ok! So the final kinetic energy is: $KE = \frac{1}{2}m{v^2} = \frac{1}{2} \times 0.145 \times {18^2} = ...Joules$

15. anonymous

okay! we get 23.49 so about 24 J is our solution?

16. Michele_Laino

no, since that the requested energy change is: $\Delta KE = 23.49 - 16 = ...Joules$

17. anonymous

ohh okay so we get 7.49 so about 7.4 J is our answer? :O

18. Michele_Laino

yes! that's right!

19. anonymous

yay!! thanks@!!

20. Michele_Laino

:)


text:
Home

>

English

>

Class 11

>

Physics

>

Chapter

>

Kinetic Theory Of Gases

>

One mole of an ideal gas under...

Updated On: 27-06-2022

Text Solution

Solution : Here, P= (P_0/ 1+((V/V_0)^2) )
  rArr nRT/ V = (P_0 / 1+((V/V_0)^2))
  [PV= nRT according to ideal gas equation]
  rArr RT/V = (P_0/ (1+ ((V/V_0)^2)))  [ Since n = 1 mole]
  rArr RT/V_0 = (P_0/ 1+ ((V/V_0)^2)) [ at V = V_0]
  rArr (P_0)(V_0) = RT(1+1)
  rArr (P_0)(V_0) = RT xx 2
  T = ((P_0)(V_0)/2R)`

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Transcript

hello everyone that solve the question question says one mole of an ideal gas undergoes a process if we are given some function of P ok is given in some functional pump OK where did not and vinodha constant ok so only is variable ok find the temperature of gas when we cross to Vinod ok so as I said only V is variable ok in this function in and feel so variable ok and we are given number of moles equals to 1 ok so what were so we all know that it is an ideal gas so we can write ideal gas as ok ideal gas equation we all know that it is TV equals to an Aati ok because number of mole = 21 so we can write TV equals to

rnt ok and it is also given ok so we are given that he in some functional form what is given the equals to not divided by 1 + V note Race 2 square ok so we can substitute P value in this function ok ok so we are asked to find the temperature ok be asked to find temperature when we equals to Vinod ok so what we will do we will we will try to get this function in some in in terms of temperature ok so we can use this ideal gas equation and we can substitute P ok so what we can substitute we cancel equal to Aati / ok so if you searched for here we will get it delivered by week was to

ok divided by 1 + we did this to square ok so now we need to solve this ok so if V = 2 sorry it is equals to Vinod ok so let's write it we equals to Vinod ok if we cross to Vinod so you Sab should we not use this function and then then temperature will be ok so what we can get temperature value ok Sophie equals to p not be divided by our bracket 1 + we divide by Vinod Race 2 square ok so now we want to substitute week was to be not so we will get

equal to arduino divide by our bracket 1 + Vinod / Vinod Race 2 square ok so after solving Muni Kata final answer ok so A final final answer will be t = 23 node Vinodh / to ok so this will be our final answer thank you so this will be the temperature when you are volume will be cause to Vinod ok so thank you


text:
### Communities

tag:snake search within a tag
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
created:<1w created < 1 week ago
post_type:xxxx type of post
Challenges

+7
−0

# Challenge

Make a program that takes input of 3 non-negative integers: a result and 2 other values that once calculated results to the 3rd value.

• The program must figure out how to get the 2 first numbers to equivalence itself with the 3rd. It could be either addition, subtraction, multiplication, integer division, modulo and exponentiation.
• You can't change the places of the 2 variables when building the equation, meaning you can't get some value by intertwining in the methods excluding addition and multiplication.
• The returning value is the operator that makes the equation true (it's best to use some distinguishable character to separate it from the other methods).
• Use whatever characters you want to use to determine the method that makes the equation truthy.
• If multiple operators make the equation truthy, then output them in any order.
• What makes it true? If an equation exists within the 2 numbers that results to the value of the 3rd.
• What to return when it's false? Something but not nothing (whitespaces and newlines aren't allowed as such output).
• This is code-golf so the shortest program wins!

# Test Cases

# Input (#, #, #)
// Output (+-*/%^ or .)

2 3 5
+

7 3 2
/

9 8 4
.

1 2 2
*

2 2 4
+ * ^

14 6 8
-

5 0 2
.

0 0 1
^

8 4 0
%

0 0 0
+ - *

Why does this post require moderator attention?
Why should this post be closed?

You are accessing this answer with a direct link, so it's being shown above all other answers regardless of its score. You can return to the normal view.

+1
−0

# Ruby, 57 bytes

->a,b,c{%w[+ - * / % **].select{c==a.send(_1,b)rescue p}}

Attempt This Online!

Why does this post require moderator attention?

+4
−0

# C (gcc), 147 151 bytes

#define L(X)X(+,a+b)X(-,a-b)X(*,a*b)X(/,b&&a/b)X(%,b&&a%b)X(^,pow(a,b))
#define M(o,d)printf("%s",d==c?#o" ":z++==5?".":"");
z;s[99];f(a,b,c){L(M)}

Try it online!

Somewhat naive solution with X macros that builds up a string before printing. The b&&a/b checks are there to deal with division by zero.

Why does this post require moderator attention?

+3
−0

# Python 3, 106 91 bytes

Saved 15 bytes thanks to Moshi in the comments

lambda a,b,c:[p for p in"+ - * // % **".split()if(p in'+-**'or b)and eval(f"{a}{p}{b}")==c]

Try it online!

Why does this post require moderator attention?

Further improvements (1 comment)
91 bytes with list comprehension (2 comments)
+2
−0

# Ruby, 58 bytes

Simple eval solution. Using ** for exponentiation.

->a,b,c{%w[+ - * / % **].select{eval("#{a}#{_1}#{b}")==c}}

Attempt This Online!

Why does this post require moderator attention?

+1
−0

# Japt, 16 bytes

Takes the result as the first input and the other 2 values as an array as the second input. Uses z for floor division & p for exponentiation and outputs " if there's no match (costing 3 bytes :\)

"+-*z%p"Æ¶VrXÃªQ

Try it

Why does this post require moderator attention?


text:
# Thread: ssplitted function integrability prove

1. ## ssplitted function integrability prove

3)b)
there is f(x) a continues and positive in [0,1]

g(x)=f(x) for rational x
g(x)=-f(x) for irational x

prove that g(x) is not integrabile in [0,1]
?

how i tried to solve it:
suppose that this not true and it integrabile
so so every dividing p
inf{S(p)-s(p)}=0
$S(p)=\sum_{i=1}^{n}M_{i}(x_{i}-x_{i-1)}$
$s(p)=\sum_{i=1}^{n}m_{i}(x_{i}-x_{i-1)}$
or i need to prove that infE=supT
now i need to get expresstion and prove that the equations above cannot exist.

what to do here in general
?

2. ## Re: ssplitted function integrability prove

One simple way: the set of discontinuity points of $f$ is $[0,1]$ . Taking into account that $[0,1]$ has not measure zero, $f$ is no Riemann integrable (as a consequence of a well known theorem).

3. ## Re: ssplitted function integrability prove

ok i understand the theory that because of endless points of discontinuety its not differentiable.

how in general could i show this using the above formulas i wrote
?


text:
## Algebra: A Combined Approach (4th Edition)

$\frac{3}{7}$
$\frac{3}{7}$ is already in simplest form, because 3 an 7 have no common factors besides 1.


text:
# Properties

 Label 28T38 Order $$224$$ n $$28$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^2\times F_8$

## Group action invariants

 Degree $n$ : $28$ Transitive number $t$ : $38$ Group : $C_2^2\times F_8$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,24,18,11,5,27,21,16,9,3,26,19,13,7)(2,23,17,12,6,28,22,15,10,4,25,20,14,8), (1,15)(2,16)(5,6)(7,21)(8,22)(11,26)(12,25)(13,28)(14,27)(19,20) $|\Aut(F/K)|$: $4$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
7:  $C_7$
14:  $C_{14}$ x 3
28:  28T2
56:  $C_2^3:C_7$
112:  14T9 x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T9 x 3

## Low degree siblings

28T38 x 5, 28T39 x 3, 32T2228

Siblings are shown with degree $\leq 47$

A number field with this Galois group has 5 arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 5,20)( 6,19)( 7,21)( 8,22)( 9,10)(11,25)(12,26)(13,14)(23,24)(27,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 3, 4)( 5, 6)( 7,21)( 8,22)( 9,23)(10,24)(13,27)(14,28)(17,18)(19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 3, 4)( 5,19)( 6,20)( 9,24)(10,23)(11,25)(12,26)(13,28)(14,27)(17,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $7$ $2$ $( 3,17)( 4,18)( 5, 6)( 7,22)( 8,21)( 9,24)(10,23)(11,26)(12,25)(13,14)(19,20) (27,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ $14, 14$ $8$ $14$ $( 1, 3, 5, 7, 9,11,13,16,18,19,21,24,26,27)( 2, 4, 6, 8,10,12,14,15,17,20,22, 23,25,28)$ $7, 7, 7, 7$ $8$ $7$ $( 1, 3, 5,22,10,11,28)( 2, 4, 6,21, 9,12,27)( 7,24,25,13,15,17,20) ( 8,23,26,14,16,18,19)$ $14, 14$ $8$ $14$ $( 1, 3, 6, 7,24,12,14, 2, 4, 5, 8,23,11,13)( 9,25,28,15,17,19,22,10,26,27,16, 18,20,21)$ $14, 14$ $8$ $14$ $( 1, 3, 6,22,23,12,27,15,17,19, 7, 9,26,14)( 2, 4, 5,21,24,11,28,16,18,20, 8, 10,25,13)$ $7, 7, 7, 7$ $8$ $7$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$ $14, 14$ $8$ $14$ $( 1, 5,23,27, 4, 8,25,16,19,10,13,17,22,12)( 2, 6,24,28, 3, 7,26,15,20, 9,14, 18,21,11)$ $14, 14$ $8$ $14$ $( 1, 5,10,27,18,22,12,15,20,24,14, 4, 7,26)( 2, 6, 9,28,17,21,11,16,19,23,13, 3, 8,25)$ $14, 14$ $8$ $14$ $( 1, 5,24,13, 4, 7,11, 2, 6,23,14, 3, 8,12)( 9,27,17,21,26,15,20,10,28,18,22, 25,16,19)$ $14, 14$ $8$ $14$ $( 1, 7,13,19,26, 3, 9,16,21,27, 5,11,18,24)( 2, 8,14,20,25, 4,10,15,22,28, 6, 12,17,23)$ $7, 7, 7, 7$ $8$ $7$ $( 1, 7,27, 6,26,17,23)( 2, 8,28, 5,25,18,24)( 3, 9,15,22,14,19,12) ( 4,10,16,21,13,20,11)$ $14, 14$ $8$ $14$ $( 1, 7,27,19,25, 4, 9, 2, 8,28,20,26, 3,10)( 5,12,17,24,15,22,14, 6,11,18,23, 16,21,13)$ $14, 14$ $8$ $14$ $( 1, 7,13, 6,25,18,23,15,22,28,19,11, 4, 9)( 2, 8,14, 5,26,17,24,16,21,27,20, 12, 3,10)$ $7, 7, 7, 7$ $8$ $7$ $( 1, 9,18,26, 5,13,21)( 2,10,17,25, 6,14,22)( 3,11,19,27, 7,16,24) ( 4,12,20,28, 8,15,23)$ $14, 14$ $8$ $14$ $( 1, 9,17,25,19,14,21,16,24, 4,12, 5,28, 7)( 2,10,18,26,20,13,22,15,23, 3,11, 6,27, 8)$ $14, 14$ $8$ $14$ $( 1, 9, 4,11,19,28,22,15,23,18,25, 6,13, 7)( 2,10, 3,12,20,27,21,16,24,17,26, 5,14, 8)$ $14, 14$ $8$ $14$ $( 1, 9, 3,12, 5,27,22, 2,10, 4,11, 6,28,21)( 7,16,24,18,25,19,13, 8,15,23,17, 26,20,14)$ $7, 7, 7, 7$ $8$ $7$ $( 1,11, 8, 4,14,24, 6)( 2,12, 7, 3,13,23, 5)( 9,20,16,26,22,17,28) (10,19,15,25,21,18,27)$ $14, 14$ $8$ $14$ $( 1,11,21, 3,13,24, 5,16,26, 7,18,27, 9,19)( 2,12,22, 4,14,23, 6,15,25, 8,17, 28,10,20)$ $14, 14$ $8$ $14$ $( 1,11, 8,17,27,23,20,15,25,21, 3,14, 9, 5)( 2,12, 7,18,28,24,19,16,26,22, 4, 13,10, 6)$ $14, 14$ $8$ $14$ $( 1,11,21,18,28,23,19, 2,12,22,17,27,24,20)( 3,14,10, 5,15,25, 8, 4,13, 9, 6, 16,26, 7)$ $14, 14$ $8$ $14$ $( 1,13,11,23, 8, 5, 4, 2,14,12,24, 7, 6, 3)( 9,21,20,18,16,27,26,10,22,19,17, 15,28,25)$ $14, 14$ $8$ $14$ $( 1,13,12,10,21, 6,17,15,28,26,24, 8,19, 3)( 2,14,11, 9,22, 5,18,16,27,25,23, 7,20, 4)$ $14, 14$ $8$ $14$ $( 1,13,26,10, 7,20,17,16,27,11,23,21, 6, 4)( 2,14,25, 9, 8,19,18,15,28,12,24, 22, 5, 3)$ $7, 7, 7, 7$ $8$ $7$ $( 1,13,25,23,22,19, 4)( 2,14,26,24,21,20, 3)( 5,17,16,27,12,10, 8) ( 6,18,15,28,11, 9, 7)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26) (12,25)(13,27)(14,28)$

## Group invariants

 Order: $224=2^{5} \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [224, 195]
 Character table: Data not available.


text:
# zbMATH — the first resource for mathematics

## Duda, Jakub

Compute Distance To:
 Author ID: duda.jakub Published as: Duda, J.; Duda, Jakub External Links: MGP
 Documents Indexed: 24 Publications since 2001

#### Co-Authors

 12 single-authored 7 Zajíček, Luděk 1 Maleva, Olga 1 Tsaban, Boaz 1 Veselý, Libor
all top 5

#### Serials

 2 Journal of Mathematical Analysis and Applications 2 Czechoslovak Mathematical Journal 2 Journal of Convex Analysis 1 Israel Journal of Mathematics 1 Rocky Mountain Journal of Mathematics 1 Canadian Mathematical Bulletin 1 Colloquium Mathematicum 1 Commentationes Mathematicae Universitatis Carolinae 1 Fundamenta Mathematicae 1 Journal of the London Mathematical Society. Second Series 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Proceedings of the American Mathematical Society 1 Real Analysis Exchange 1 Topology and its Applications 1 Zeitschrift für Analysis und ihre Anwendungen 1 Atti del Seminario Matematico e Fisico dell’Università di Modena 1 Journal of Nonlinear and Convex Analysis
all top 5

#### Fields

 15 Functional analysis (46-XX) 11 Real functions (26-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Global analysis, analysis on manifolds (58-XX) 3 Differential geometry (53-XX) 2 Approximations and expansions (41-XX) 2 Operator theory (47-XX) 1 Measure and integration (28-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

#### Citations contained in zbMATH Open

20 Publications have been cited 64 times in 42 Documents Cited by Year
Metric and $$w^*$$-differentiability of pointwise Lipschitz mappings. Zbl 1143.26006
Duda, Jakub
2007
The various facets of random walk entropy. Zbl 1371.82043
Burda, Z.; Duda, J.; Luck, J. M.; Waclaw, B.
2010
Curves with finite turn. Zbl 1167.46321
Duda, Jakub
2008
On d.c. functions and mappings. Zbl 1072.46025
Veselý, Libor; Duda, Jakub; Zajíček, Luděk
2003
On Gâteaux differentiability of pointwise Lipschitz mappings. Zbl 1152.46030
Duda, Jakub
2008
Differentiability via curves in Banach spaces homeomorphisms. Zbl 1148.26008
Duda, Jakub; Zajíček, Luděk
2007
Smallness of singular sets of semiconvex functions in separable Banach spaces. Zbl 1269.49016
Duda, Jakub; Zajícek, Ludek
2013
On the size of the set of points where the metric projection exists. Zbl 1067.46015
Duda, Jakub
2004
Curves in Banach spaces which allow a $$C^{2}$$-parameterization. Zbl 1225.26059
Duda, Jakub; Zajíček, Luděk
2011
Semiconvex functions: representations as suprema of smooth functions and extensions. Zbl 1168.49015
Duda, Jakub; Zajícek, L.
2009
Cone monotone mappings: continuity and differentiability. Zbl 1135.46046
Duda, Jakub
2008
Second order differentiability of paths via a generalized $$\frac 1 2$$-variation. Zbl 1135.46021
Duda, Jakub
2008
Absolutely continuous functions with values in a metric space. Zbl 1131.26006
Duda, Jakub
2007
On positively differentiable curves with values in normed linear spaces. Zbl 1100.46049
Duda, Jakub
2006
The Banach-Zarecki theorem for functions with values in metric spaces. Zbl 1083.26007
Duda, Jakub; Zajícek, Luděk
2005
On vector-valued curves that allow a $$C^{1,\alpha}$$-parametrization. Zbl 1224.26090
Duda, J.; Zajíček, L.
2010
Generalized $$\alpha$$-variation and Lebesgue equivalence to differentiable functions. Zbl 1191.26003
Duda, Jakub
2009
On a decomposition of Banach spaces. Zbl 1118.28001
Duda, Jakub
2007
On the size of the set of points where the metric projection is discontinuous. Zbl 1101.41032
Duda, Jakub
2006
On inverses of $$\delta$$-convex mappings. Zbl 1053.47522
Duda, Jakub
2001
Smallness of singular sets of semiconvex functions in separable Banach spaces. Zbl 1269.49016
Duda, Jakub; Zajícek, Ludek
2013
Curves in Banach spaces which allow a $$C^{2}$$-parameterization. Zbl 1225.26059
Duda, Jakub; Zajíček, Luděk
2011
The various facets of random walk entropy. Zbl 1371.82043
Burda, Z.; Duda, J.; Luck, J. M.; Waclaw, B.
2010
On vector-valued curves that allow a $$C^{1,\alpha}$$-parametrization. Zbl 1224.26090
Duda, J.; Zajíček, L.
2010
Semiconvex functions: representations as suprema of smooth functions and extensions. Zbl 1168.49015
Duda, Jakub; Zajícek, L.
2009
Generalized $$\alpha$$-variation and Lebesgue equivalence to differentiable functions. Zbl 1191.26003
Duda, Jakub
2009
Curves with finite turn. Zbl 1167.46321
Duda, Jakub
2008
On Gâteaux differentiability of pointwise Lipschitz mappings. Zbl 1152.46030
Duda, Jakub
2008
Cone monotone mappings: continuity and differentiability. Zbl 1135.46046
Duda, Jakub
2008
Second order differentiability of paths via a generalized $$\frac 1 2$$-variation. Zbl 1135.46021
Duda, Jakub
2008
Metric and $$w^*$$-differentiability of pointwise Lipschitz mappings. Zbl 1143.26006
Duda, Jakub
2007
Differentiability via curves in Banach spaces homeomorphisms. Zbl 1148.26008
Duda, Jakub; Zajíček, Luděk
2007
Absolutely continuous functions with values in a metric space. Zbl 1131.26006
Duda, Jakub
2007
On a decomposition of Banach spaces. Zbl 1118.28001
Duda, Jakub
2007
On positively differentiable curves with values in normed linear spaces. Zbl 1100.46049
Duda, Jakub
2006
On the size of the set of points where the metric projection is discontinuous. Zbl 1101.41032
Duda, Jakub
2006
The Banach-Zarecki theorem for functions with values in metric spaces. Zbl 1083.26007
Duda, Jakub; Zajícek, Luděk
2005
On the size of the set of points where the metric projection exists. Zbl 1067.46015
Duda, Jakub
2004
On d.c. functions and mappings. Zbl 1072.46025
Veselý, Libor; Duda, Jakub; Zajíček, Luděk
2003
On inverses of $$\delta$$-convex mappings. Zbl 1053.47522
Duda, Jakub
2001
all top 5

#### Cited by 53 Authors

 8 Zajíček, Luděk 6 Duda, Jakub 3 Bongiorno, Donatella 3 Borwein, Jonathan Michael 3 Frank, Lawrence R. 3 Galinsky, Vitaly L. 2 Ginchev, Ivan G. 2 Williams, Marshall 1 Ambrosio, Luigi 1 De Bernardi, Carlo Alberto 1 Di Bari, Cristina Maria 1 Dubrulle, Bérengère 1 Duda, Jarosław 1 Durand-Cartagena, Estibalitz 1 Estrada, Ernesto 1 Faranda, Davide 1 Frasca, Mattia 1 Gambuzza, Lucia Valentina 1 Giladi, Ohad 1 Gintcheva, Denitza 1 Gorokhovik, Valentin Vikent’evich 1 Gorskiĭ, Aleksandr Sergeevich 1 Guirao, Antonio José 1 Hájek, Petr 1 Kaliaj, Sokol Bush 1 Kral, Q. 1 Kryštof, Václav 1 Lambiotte, Renaud 1 Liu, Hua 1 Maly, Johannes 1 Martínez-Legaz, Juan-Enrique 1 Martini, Horst 1 Masuda, Naoki 1 Mihelich, Martin 1 Nechaev, Sergei K. 1 Niu, Yawei 1 Paillard, Didier 1 Porter, Mason Alexander 1 Preiss, David 1 Rabier, Patrick J. 1 Revalski, Julian Petrov 1 Strobin, Filip 1 Tišer, Jaroslav 1 Trafimovich, Marina A. 1 Valov, A. F. 1 Vanderwerff, Jon D. 1 Veselý, Libor 1 Vetro, Calogero 1 Wang, Guanghui 1 Wu, Senlin 1 Yan, Guiying 1 Zajicek, Ladislav 1 Zhivkov, Nikolay V.
all top 5

#### Cited in 29 Serials

 4 Journal of Mathematical Analysis and Applications 3 Czechoslovak Mathematical Journal 3 Proceedings of the American Mathematical Society 2 Israel Journal of Mathematics 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Rendiconti del Circolo Matemàtico di Palermo. Serie II 2 Acta Mathematica Hungarica 2 Neural Computation 2 Journal of Global Optimization 1 Bulletin of the Australian Mathematical Society 1 Journal of Statistical Physics 1 Mathematical Biosciences 1 Physics Reports 1 Rocky Mountain Journal of Mathematics 1 Studia Mathematica 1 Archiv der Mathematik 1 Commentationes Mathematicae Universitatis Carolinae 1 Mathematica Slovaca 1 Michigan Mathematical Journal 1 Monatshefte für Mathematik 1 Computer Aided Geometric Design 1 Journal of Convex Analysis 1 Abstract and Applied Analysis 1 Journal of High Energy Physics 1 Trudy Instituta Matematiki 1 Journal of the Australian Mathematical Society 1 SIAM Journal on Applied Dynamical Systems 1 Optimization Letters 1 Journal of Physics A: Mathematical and Theoretical
all top 5

#### Cited in 20 Fields

 20 Functional analysis (46-XX) 16 Real functions (26-XX) 9 Calculus of variations and optimal control; optimization (49-XX) 4 Global analysis, analysis on manifolds (58-XX) 4 Probability theory and stochastic processes (60-XX) 4 Biology and other natural sciences (92-XX) 3 Measure and integration (28-XX) 3 Approximations and expansions (41-XX) 3 Operator theory (47-XX) 3 Differential geometry (53-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Combinatorics (05-XX) 2 Functions of a complex variable (30-XX) 2 Convex and discrete geometry (52-XX) 2 Operations research, mathematical programming (90-XX) 1 Ordinary differential equations (34-XX) 1 General topology (54-XX) 1 Statistics (62-XX) 1 Fluid mechanics (76-XX) 1 Quantum theory (81-XX)


text:
Now showing items 1-20 of 24

• #### Age- and tumor subtype-specific breast cancer risk estimates for CHEK2*1100delC carriers

(American Society of Clinical Oncology, 2016)
CHEK2*1100delC is a well-established breast cancer risk variant, most prevalent in European populations. However, there are limited data on risk of breast cancer by age and tumor subtype, limiting its usefulness in breast ...
• #### Annexin A1 expression in a pooled breast cancer series: association with tumor subtypes and prognosis

(2015-07-02)
Abstract Background Annexin A1 (ANXA1) is a protein related with the carcinogenesis process and metastasis formation in many tumors. However, little is known about the prognostic ...
• #### Annexin A1 expression in breast cancer: tumor subtypes and prognosis

(BioMed Central, 2015-06-25)
Background Annexin A1 (ANXA1) is a protein related with the carcinogenesis process and metastasis formation in many tumors. However, little is known about the prognostic value of ANXA1 in breast cancer. The purpose of ...
• #### BRCA2 hypomorphic missense variants confer moderate risks of breast cancer

Breast cancer risks conferred by many germline missense variants in the $\textit{BRCA1}$ and $\textit{BRCA2}$ genes, often referred to as variants of uncertain significance (VUS), have not been established. In this study, ...
• #### Breast cancer risk variants at 6q25 display different phenotype associations and regulate ESR1, RMND1 and CCDC170

(Nature Publishing Group, 2016)
We analysed 3872 common genetic variants across the ESR1 locus (encoding estrogen receptor–alpha) in 118,816 subjects from three international consortia. We found evidence for at least five independent causal variants, ...
• #### Combined effects of single nucleotide polymorphisms TP53R72P and MDM2SNP309, and p53 expression on survival of breast cancer patients

(2009-12-18)
Abstract Introduction Somatic inactivation of the TP53 gene in breast tumors is a marker for poor outcome, and breast cancer outcome might also be affected by germ-line variation in the TP53 gene or its regulators. We ...
• #### Common non-synonymous SNPs associated with breast cancer susceptibility: ﬁndings from the Breast Cancer Association Consortium

(Oxford University Press, 2014-07-04)
Candidate variant association studies have been largely unsuccessful in identifying common breast cancer susceptibility variants, although most studies have been underpowered to detect associations of a realistic magnitude ...
• #### Evidence that breast cancer risk at the 2q35 locus is mediated through IGFBP5 regulation

(NPG, 2014-09-23)
GWAS have identified a breast cancer susceptibility locus on 2q35. Here we report the fine-mapping of this locus using data from 101,943 subjects from 50 case-control studies. We genotype 276 SNPs using the “iCOGS” genotyping ...
• #### Evidence that the 5p12 variant rs10941679 confers susceptibility to estrogen receptor positive breast cancer through FGF10 and MRPS30 regulation

(Elsevier (Cell Press), 2016)
Genome-wide association studies (GWAS) have revealed increased breast cancer risk associated with multiple genetic variants at 5p12. Here, we report the fine-mapping of this locus using data from 104,660 subjects from 50 ...
• #### Fine scale mapping of the 5q11.2 breast cancer locus reveals at least three independent risk variants regulating MAP3K1

(Cell/Elsevier, 18/12/2014)
Genome Wide Association Studies (GWAS) revealed SNP rs889312 on 5q11.2 to be associated with breast cancer risk in women of European ancestry. In an attempt to identify the biologically relevant variants, we analysed 909 ...
• #### Fine scale mapping of the 5q11.2 breast cancer locus reveals at least three independent risk variants regulating MAP3K1.

(2015-12-18)
• #### Fine-Mapping of the 1p11.2 Breast Cancer Susceptibility Locus

(Public Library of Science, 2016-08-24)
The Cancer Genetic Markers of Susceptibility genome-wide association study (GWAS) originally identified a single nucleotide polymorphism (SNP) rs11249433 at 1p11.2 associated with breast cancer risk. To fine-map this locus, ...
• #### Fine-scale mapping of 8q24 locus identifies multiple independent risk variants for breast cancer

(Wiley, 2016)
Previous genome-wide association studies among women of European ancestry identified two independent breast cancer susceptibility loci represented by single nucleotide polymorphisms (SNPs) rs13281615 and rs11780156 at 8q24. ...
• #### Fine-scale mapping of the 5q11.2 breast cancer locus reveals at least three independent risk variants regulating MAP3K1

(Elsevier, 2014-12-18)
Genome-wide association studies (GWASs) have revealed SNP rs889312 on 5q11.2 to be associated with breast cancer risk in women of European ancestry. In an attempt to identify the biologically relevant variants, we analyzed ...
• #### Genetic variation in the immunosuppression pathway genes and breast cancer: a pooled analysis of 42,510 cases and 40,577 controls from the Breast

(Springer, 2015-11-30)
Immunosuppression plays a pivotal role in assisting tumors to evade immune destruction and promoting tumor development. We hypothesized that genetic variation in the immunosuppression pathway genes may be implicated in ...
• #### Genome-wide association analysis of more than 120,000 individuals identifies 15 new susceptibility loci for breast cancer.

(NPG, 2015-03-09)
Genome-wide association studies (GWAS) and large-scale replication studies have identified common variants in 79 loci associated with breast cancer, explaining ~14% of the familial risk of the disease. To identify new ...
• #### High-throughput automated scoring of Ki67 in breast cancer tissue microarrays from the Breast Cancer Association Consortium

(Wiley, 2016-04-06)
Automated methods are needed to facilitate high-throughput and reproducible scoring of Ki67 and other markers in breast cancer tissue microarrays (TMAs) in large-scale studies. To address this need, we developed an automated ...
• #### Identification and characterisation of novel associations in the CASP8/ALS2CR12 region on chromosome 2 with breast cancer risk

(Oxford University Press, 2014-08-28)
Previous studies have suggested that polymorphisms in CASP8 on chromosome 2 are associated with breast cancer risk. To clarify the role of CASP8 in breast cancer susceptibility we carried out dense genotyping of this region ...
• #### Identification of independent association signals and putative functional variants for breast cancer risk through fine-scale mapping of the 12p11 locus

(BioMed Central, 2016)
Background Multiple recent genome-wide association studies (GWAS) have identified a single nucleotide polymorphism (SNP), rs10771399, at 12p11 that is associated with breast cancer risk. Method We performed a ...
• #### Identification of Novel Genetic Markers of Breast Cancer Survival

(Oxford Journals, 2015-04-18)
Background: Survival after a diagnosis of breast cancer varies considerably between patients, and some of this variation may be because of germline genetic variation. We aimed to identify genetic markers associated with ...


text:
# Autoregression#

Here we fit NeuralProphet to data with 5-minute resolution (daily temperatures at Yosemite).

[1]:

if "google.colab" in str(get_ipython()):
!pip install git+https://github.com/ourownstory/neural_prophet.git # may take a while
#!pip install neuralprophet # much faster, but may not have the latest upgrades/bugfixes

import pandas as pd
from neuralprophet import NeuralProphet, set_log_level

# set_log_level("ERROR")

[2]:

data_location = "https://raw.githubusercontent.com/ourownstory/neuralprophet-data/main/datasets/"

[2]:

ds y
0 2017-05-01 00:00:00 27.8
1 2017-05-01 00:05:00 27.0
2 2017-05-01 00:10:00 26.8

## Next-step forecast#

Based on our first contact with the data, we set: * First, we disable weekly_seasonality, as nature does not follow the human week’s calendar. * Second, we increase n_changepoints, and increase changepoints_range, as we are doing short-term predictions.

Further, we can make use of the fact that tomorrow’s weather is most likely similar to yesterdays weather. This means that we will regress our time series on it’s most recent past values, also known as autoregression.

We can achieve this by setting n_lags to the desired number of past observations to regress over. This value is also known as the ‘AR order’.

Here, we predict the temperature in the next 5 minutes based on the last hour:

[3]:

m = NeuralProphet(
n_lags=12,
changepoints_range=0.95,
n_changepoints=30,
weekly_seasonality=False,
)
metrics = m.fit(df)

INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Dataframe freq automatically defined as 5T
INFO - (NP.config.init_data_params) - Setting normalization to global as only one dataframe provided for training.
INFO - (NP.utils.set_auto_seasonalities) - Disabling yearly seasonality. Run NeuralProphet with yearly_seasonality=True to override this.
INFO - (NP.config.set_auto_batch_epoch) - Auto-set batch_size to 64
INFO - (NP.config.set_auto_batch_epoch) - Auto-set epochs to 88

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 5.60E-02, min: 7.11E-01

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 1.09E-01, min: 6.22E-01
INFO - (NP.forecaster._init_train_loader) - lr-range-test selected learning rate: 8.49E-02
Epoch[88/88]: 100%|█████████████████████████████████████████████| 88/88 [00:47<00:00,  1.85it/s, SmoothL1Loss=6.92e-5, MAE=0.302, RMSE=0.606, RegLoss=0]

Please note that a model with an autoregressive component can be harder to fit. The automatic selection of hyperparameters may not lead to ideal results. For best results, consider changing these manually (in order of importance):

• learning_rate

• epochs

• batch_size

The automatically set hyperparameters (printed out as ‘INFO’ level logs) can serve as a good starting point.

[4]:

forecast = m.predict(df)
fig = m.plot(forecast)

INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T
INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T

The predictions are now very precise, but this is not a big surprise, as we are just forecasting the very next 5 minutes.

When plotting the model parameters, the panel ‘AR weight’ displays the weight given to the 12 last observed values, these can be interpreted as our ‘AR coefficients’:

[5]:

# fig_comp = m.plot_components(forecast)
m = m.highlight_nth_step_ahead_of_each_forecast(1)  # temporary workaround to plot actual AR weights
fig_param = m.plot_parameters()

The ‘AR weight’ plot shows that the most recent observations are given significantly more weight compared to more distant observations.

## Multi-step forecast#

To predict multiple steps into the future, we could ‘unroll’ our single-step model, by predicting a step ahead, adding the forecasted value to our data, and then forecasting the next step until we reach the horizon we are interested in. However, there is a better way to do this: We can directly forecast multiple steps ahead with NeuralProphet.

We can set n_forecasts to the desired number of steps we would like to forecast (also known as ‘forecast horizon’). NeuralProphet will forecast n_forecasts steps into the future, at every single step. Thus, we have n_forecasts overlapping predictions of vaying age at every historic point.

When increasing the forecast horizon n_forecasts, we should also increase the number of past observations n_lags to at least the same value.

Here, we forecast the next 3 hours based on the last observed 6 hours, in 5-minute steps:

[6]:

m = NeuralProphet(
n_lags=3 * 12,
n_forecasts=2 * 12,
changepoints_range=0.95,
n_changepoints=30,
weekly_seasonality=False,
)
metrics = m.fit(df)

INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Dataframe freq automatically defined as 5T
INFO - (NP.config.init_data_params) - Setting normalization to global as only one dataframe provided for training.
INFO - (NP.utils.set_auto_seasonalities) - Disabling yearly seasonality. Run NeuralProphet with yearly_seasonality=True to override this.
INFO - (NP.config.set_auto_batch_epoch) - Auto-set batch_size to 64
INFO - (NP.config.set_auto_batch_epoch) - Auto-set epochs to 88

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 9.56E-02, min: 1.39E+00

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 8.37E-02, min: 1.21E+00
INFO - (NP.forecaster._init_train_loader) - lr-range-test selected learning rate: 9.28E-02
Epoch[88/88]: 100%|██████████████████████████████████████████████| 88/88 [01:07<00:00,  1.31it/s, SmoothL1Loss=0.000723, MAE=1.31, RMSE=2.05, RegLoss=0]

[7]:

forecast = m.predict(df)
fig = m.plot(forecast)

INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T
INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T
WARNING - (NP.plotting.plot) - Legend is available only for the ten first handles

We see that our forecast sometimes miss-predicts for a day and then returns again to more accurate forecasts. Let’s have a closer look at that over-prediction on day 6 of our data:

[8]:

fig = m.plot(forecast[144 : 6 * 288])

WARNING - (NP.plotting.plot) - Legend is available only for the ten first handles

On day 6, the redorded temperature drops at nighttime. Based on the observed lower nighttime temperatures, our model predicts a lower daytime peak. However, the actual daytime temperatures are anomalously low, barely higher than during the night. Thus, leading to an overprediction.

We can again visualize the relative importance of the lags:

[9]:

# fig_comp = m.plot_components(forecast)
fig_param = m.plot_parameters()

Note that the ‘AR relevance’ is the relative importance of a lag, averaged over all n_forecasts.

## Reviewing a specific forecast step#

We can have a closer look at a particular forecast horizon by highlighting the n-th step ahead of each forecast. Here we focus on the temperature predicted 3 hours ahead (most distant prediction into the future). Let’s have a closer look at the weights for the different 3 hour ahead forecast:

[10]:

m = m.highlight_nth_step_ahead_of_each_forecast(2 * 12)
fig_param = m.plot_parameters()

Next, we replot the first 6 days and compare it to the forecast one-step-ahead. We observe the single-step ahead forecast to be much more accurate compared to the 3 hour ahead forecast. However, neither is able to forecast the anomaly on day 6.

[11]:

fig = m.plot(forecast[144 : 6 * 288])
fig = m.plot(forecast[144 : 6 * 288])

## Plotting the most recent forecast#

When we are more concerned with our actual forecast than the model fit, we can plot the last most recent forecast:

[12]:

future = m.make_future_dataframe(df, n_historic_predictions=12)
forecast = m.predict(future)
m = m.highlight_nth_step_ahead_of_each_forecast(None)  # reset highlight
fig = m.plot_last_forecast(forecast)

INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 99.995% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T
INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 98.611% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T
INFO - (NP.df_utils._infer_frequency) - Major frequency 5T corresponds to 98.611% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - 5T

The last few historic forecasts can be included to assess how the forecast has changed over time. Here, we focus on the 3-hour ahead forecast, given over the last 2 hours.

[13]:

m = m.highlight_nth_step_ahead_of_each_forecast(2 * 12)
fig = m.plot_last_forecast(forecast, include_previous_forecasts=4)

## Larger forecast horizon#

For predictions further into the future, you could reduce the resulution of the data. Using a 5-minute resolution may be useful for a high-resolution short-term forecast, but counter-productive for a long-term forecast. As we only have a limited amount of data (approx 2 months), we want to avoid over-specifying the model.

As an example: If we set the model to forecast 24 hours into the future (nforecasts=24*12) based on the last day’s temperatures (n_lags=24*12), the number of parameters of our AR component grows to 24*12*24*12 = 82,944. However, we only have about 2*30*24*12 = 17,280 samples in our dataset. The model would be overspecified.

If we first downsample our data to hourly data, we reduce our dataset to 2*30*24=1440 and our model parameters to 24*24=576. Thus, we are able to fit the model. However, it would be better to collect more data.

[14]:

df.loc[:, "ds"] = pd.to_datetime(df.loc[:, "ds"])
df_hourly = df.set_index("ds", drop=False).resample("H").mean().reset_index()
len(df_hourly)

[14]:

1561

[15]:

df_hourly.head(3)

[15]:

ds y
0 2017-05-01 00:00:00 24.891667
1 2017-05-01 01:00:00 16.741667
2 2017-05-01 02:00:00 11.733333
[16]:

m = NeuralProphet(
n_lags=24,
n_forecasts=24,
changepoints_range=0.95,
n_changepoints=30,
weekly_seasonality=False,
)
metrics = m.fit(df_hourly)

INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.936% of the data.
INFO - (NP.df_utils._infer_frequency) - Dataframe freq automatically defined as H
INFO - (NP.config.init_data_params) - Setting normalization to global as only one dataframe provided for training.
INFO - (NP.utils.set_auto_seasonalities) - Disabling yearly seasonality. Run NeuralProphet with yearly_seasonality=True to override this.
INFO - (NP.config.set_auto_batch_epoch) - Auto-set batch_size to 32
INFO - (NP.config.set_auto_batch_epoch) - Auto-set epochs to 171

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 8.52E-02, min: 4.99E-01

INFO - (NP.utils_torch.lr_range_test) - lr-range-test results: steep: 7.25E-02, min: 6.87E-01
INFO - (NP.forecaster._init_train_loader) - lr-range-test selected learning rate: 8.52E-02
Epoch[171/171]: 100%|████████████████████████████████████████████| 171/171 [00:20<00:00,  8.52it/s, SmoothL1Loss=0.00202, MAE=2.13, RMSE=3.4, RegLoss=0]

[17]:

forecast = m.predict(df_hourly)
fig = m.plot(forecast)
# fig_param = m.plot_parameters()

INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.936% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - H
INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.937% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - H
WARNING - (NP.plotting.plot) - Legend is available only for the ten first handles

Finally, we plot the most recent and last ten historic 24-hour forecasts, marking the 24-th hour ahead with an ‘x’.

[18]:

future = m.make_future_dataframe(df_hourly)
forecast = m.predict(future)

INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.936% of the data.


text:
# Chapter Notes: Rotational Motion and Rolling Motion Physics Class 11

Notes for Rotational Motion chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.

Â

CENTRE OF MASS

In a system of extended bodies there is one special point that has some interesting and simple properties no matter how complicated the system is. This point is called the center of mass.

For a system of n particles whose position vectors are ${{\rm{\vec r}}_{\rm{1}}},\,{{\rm{\vec r}}_{\rm{2}}},..........,{{\rm{\vec r}}_{\rm{n}}}$ as shown in the figure, the position vector for the center of mass ${{\rm{\vec r}}_{{\rm{cm}}}}$ isÂ  defined as

${{\rm{\vec r}}_{{\rm{cm}}}} = {{{m_1}{{{\rm{\vec r}}}_{\rm{1}}} + {m_2}{{{\rm{\vec r}}}_{\rm{2}}} + ........ + {m_n}{{{\rm{\vec r}}}_{\rm{n}}}} \over {{m_1} + {m_2} + ........... + {m_n}}}$

or ${{\rm{\vec r}}_{{\rm{cm}}}} = {{\sum {{m_i}{{{\rm{\vec r}}}_{\rm{i}}}} } \over M}$

where $M = \sum {{m_i}}$ is the total mass of the system.
The components of the above equation may be written as

${x_{cm}} = {{\sum {{m_i}{x_i}} } \over M}$, ${y_{cm}} = {{\sum {{m_i}{y_i}} } \over M}$, ${z_{cm}} = {{\sum {{m_i}{z_i}} } \over M}$

The location of the centre of mass is independent of the reference frame used to locate it. The centre of mass of the system of particles depends only on the masses of the particles and the positions of the particles relative to one another.
A rigid body, such as a meter stick, can be thought of as a system of closely packed particles. Hence it also has the centre of mass. The number of particles in the body is so large and their spacing so small, however, that we can treat the body as though it has a continuous distribution of mass. For continuous distribution of mass, the centre of mass is defined as

${{\rm{\vec r}}_{{\rm{cm}}}} = {1 \over M}\int {{\rm{\vec r}}dm}$

where ${\rm{\vec r}}$ is the position vector of the centre of mass of a small mass element dm.
The components of this equation are

${x_{cm}} = {1 \over M}\int {xdm}$, ${y_{cm}} = {1 \over M}\int {ydm}$ and ${z_{cm}} = {1 \over M}\int {zdm}$

Often we deal with homogeneous objects having a point, a line, or a plane of symmetry. Then the centre of mass will lie at the point, on the line, or in the plane of symmetry. For example, the centre of mass of a homogeneous rod will lie at the centre of the rod, the centre of mass of a homogenous sphere will be at the centre of the sphere, the centre of mass of a cone will be at the axis of the cone, etc.

Application 1

Find the center of mass of a uniform semicircular rod of radius R.

Solution

From the symmetry of the body we see at once that the centre of mass must lie along the y axis, so xcm = 0.

In this case it is convenient to express the mass element in terms of the angle $\theta$, measured in radians. The element, which subtends an angle ${d_\theta }$Â at the origin, has a length $R{d_\theta }$Â and a mass $dm = \lambda R{d_\theta }$. (l = mass per unit length)
Its y coordinate isÂ $y = R\sin \theta$.

Therefore, ${y_{cm}} = {1 \over M}\int {ydm}$Â takes the form
${y_{cm}} = {1 \over M}\int\limits_0^\pi {\lambda {R^2}\sin \theta d\theta } = {{\lambda {R^2}} \over M}\left[ { - \cos \theta } \right]_0^\pi = {{2\lambda {R^2}} \over M}$
The total mass of the ring is $M = \pi R\lambda$; therefore, ${y_{cm}} = {{2R} \over \pi }$

Motion of the Centre Of Mass

Now we can discuss the physical importance of the centre of mass concept. Consider the motion of a group of particles m1, m2, â€¦â€¦., mn and whose total mass is M which is a constant. From the definition of the centre of mass, we have

M${{\rm{\vec r}}_{{\rm{cm}}}}$ Â = m1${{\rm{\vec r}}_{\rm{1}}}$ + m2${{\rm{\vec r}}_{\rm{2}}}$ +â€¦â€¦â€¦â€¦â€¦â€¦+ mn${{\rm{\vec r}}_{\rm{n}}}$

Differentiating this equation w.r.t. time, we obtain

$M{{\rm{\vec v}}_{{\rm{cm}}}} = {m_1}{{\rm{\vec v}}_{\rm{1}}} + {m_2}{{\rm{\vec v}}_{\rm{2}}} + ......... + {m_n}{{\rm{\vec v}}_{\rm{n}}}$

where ${{\rm{\vec v}}_{\rm{1}}}\left( { = {{d{{{\rm{\vec r}}}_{\rm{1}}}} \over {dt}}} \right)$Â is the velocity of the first particle, etc., and ${{\rm{\vec v}}_{{\rm{cm}}}}\left( { = {{d{{{\rm{\vec r}}}_{{\rm{cm}}}}} \over {dt}}} \right)$Â is the velocity of the centre of mass.

Differentiating the above equation w.r.t. time, we obtain

$M{{\rm{\vec a}}_{{\rm{cm}}}} = {m_1}{{\rm{\vec a}}_{\rm{1}}} + {m_2}{{\rm{\vec a}}_{\rm{2}}} + ......... + {m_n}{{\rm{\vec a}}_{\rm{n}}}$

where ${{\rm{\vec a}}_{\rm{1}}}$ is the acceleration of the first particle, etc., and ${{\rm{\vec a}}_{{\rm{cm}}}}$ is the acceleration of the centre of mass.Â  Now, from Newtonâ€™s second law the force ${{\rm{\vec F}}_{\rm{1}}}$ acting on the first particle is given by ${{\rm{\vec F}}_{\rm{1}}} = {m_1}{{\rm{\vec a}}_{\rm{1}}}$. Likewise, ${{\rm{\vec F}}_2} = {m_2}{{\rm{\vec a}}_2}$, etc. We can then write the above equation as

$M{{\rm{\vec a}}_{{\rm{cm}}}} = {{\rm{\vec F}}_{\rm{1}}} + {{\rm{\vec F}}_{\rm{2}}} + ......... + {{\rm{\vec F}}_{\rm{n}}}$

Among all these forces the internal forces are exerted by the particles on each other. However, from Newtonâ€™s third law, these internal forces will occur in equal and opposite pairs, so that they contribute nothing to the sum. The right hand sum in the above equation then represents the sum of only the external forces acting on all the particles (system). We can then rewrite the above equation as

$M{{\rm{\vec a}}_{{\rm{cm}}}} = \sum {{{\overrightarrow F }_{external}}}$

This states that the centre of mass of a system of particles moves as though all the mass of the system is concentrated at the centre of mass and all the external forces were applied at that point.
One important situation is that in which $\sum {{{{\rm{\vec F}}}_{{\rm{external}}}}} = \overrightarrow 0$ then ${{\rm{\vec v}}_{{\rm{cm}}}}$ = constant.

Application 2

A dog of mass 10 kg stands on a stationary boat of mass 40 kg so that he is 20 m from the shore. He then walks 8 m on the boat towards the shore and halts. How far is he from the shore now? Assume that there is no friction between the boat and water.

Solution

In the horizontal direction, there is no force on the system (dog + boat).Â  Therefore, the centre of mass of the system does not move in the horizontal direction. Since the dog moves towards the shore, the boat moves away from the shore to keepÂ  centre of mass stationary. Let d be the distance by which the boat moves backwards and let x be the initial distance of the boat from the shore.

The initial x-coordinate of the centre of mass = ${{10 \times 20 + 40.x} \over {10 + 40}}$

The finalÂ  x-coordinate of the centre of mass
= ${{10\left( {20 - 8 + d} \right) + 40\left( {x + d} \right)} \over {10 + 40}}$

Equating the two, we get
d=${8 \over 5}$m
TheÂ  dog is 20 â€“ 8 + ${8 \over 5}$ = 13.6 m from the shore.

Application 3

Two balls with masses m1 = 3 kg and m2 = 5 kg have initial velocities v1 = v2 = 5 m/s in the directions as shown in the figure. They collide at the origin.
(a) Find the velocity of the center of mass 3 s before the collision
(b) Find the position of the center of mass 2 s after the collision

Solution

(a) The given time is of no consequence since vcm is constant for all times.
From equation,Â  in component form

${v_{cmx}} = {{{m_1}{v_{1x}} + {m_2}{v_{2x}}} \over M} = {{\left( {3kg} \right)\left( { - 5\cos {{37}^o}m/s} \right) + \left( {5kg} \right)\left( {0m/s} \right)} \over {8kg}} = - 1.5$m/s

${v_{cmy}} = {{{m_1}{v_{1y}} + {m_2}{v_{2y}}} \over M} = {{\left( {3kg} \right)\left( { - 5\sin {{37}^o}m/s} \right) + \left( {5kg} \right)\left( {5m/s} \right)} \over {8kg}} = + 2$m/s

Thus, ${{\rm{\vec v}}_{{\rm{cm}}}}$= -1.5$\hat i$+ 2$\hat j$m/s

(b) Since the collision occurs at the origin, the position of the center of massÂ  2s later is

${{\rm{\vec r}}_{{\rm{cm}}}}$=${{\rm{\vec v}}_{{\rm{cm}}}}$t = -3$\hat i$+ 4$\hat j$m

Application 4

Â A 75 kg man stands at the rear end of a platform of mass 25 kg and length 4m, which moves initially at 4 m/s over a frictionless surface. At t = 0, he walks at 2 m/s relative to the platform and then stops at the front end. During the period of walking, find the displacement of
(a) the platform,
(b) the man,
(c) the center of mass,

Solution

Initially the man, the platform, and the cm have the same velocity, 4m/s. When he begins to walk forward, his increase in momentum must be compensated by a decrease in the platform's momentum. Let us say that the velocity of the platform relative to the ground while he is walking i.e.

${{\rm{\vec v}}_{{\rm{PG}}}} = {v_p}\hat i$

The velocity of man relative to the ground is then

${{\rm{\vec v}}_{{\rm{MG}}}} = {{\rm{\vec v}}_{{\rm{MP}}}} + {{\rm{\vec v}}_{{\rm{PG}}}} = (2 + {v_p})\hat i$

From the conservation of momentum, we have
(75 + 25) $\times$ 4 = 75( 2 + vp) + 25 vP
Thus, the velocity of the platform is vP = 2.5 m/s, and the velocity of the man

is vm = 4.5 m/s.

Since the velocity of man relative to platform is 2m/s, therefore, it takes Â $t = {{4m} \over {2m/s}} = 2s$Â for him to walk from the rear to the front.

Displacement of platform is Â Â Â Â Â Â Â Â Â Â Â Â Â Â xp = vpt = (2.5)(2) = 5m
Displacement of the man is Â Â Â Â Â Â Â Â Â Â Â Â Â  xm = vmt = (4.5)(2) = 9m
Displacement of the center of mass Â Â xcm = vcmt = (4)(2) = 8m


text:
Volume 398 - The European Physical Society Conference on High Energy Physics (EPS-HEP2021) - T06: QCD and Hadronic Physics
Production of light nuclei in small collision systems measured with ALICE
A. Balbino*  on behalf of the ALICE collaboration
Full text: pdf
Pre-published on: March 07, 2022
Published on: May 12, 2022
Abstract
The energy densities reached in high-energy hadronic collisions at the LHC allow significant production of light (anti)nuclei. Their production yields have been measured as a function of $p_\mathrm{T}$ and charged-particle multiplicity in the ALICE detector for different collision systems and at different center-of-mass energies. One of the most interesting results obtained from such a large variety of experimental data is that the dominant production mechanism of light (anti)nuclei seems to depend solely on the event charged-particle multiplicity. Evidence for this comes from the continuous evolution of the coalescence parameter, the deuteron-to-proton and the $^3$He-to-proton ratios with the event multiplicity across different collision systems and energies. The characterization of the light nuclei production mechanism is complemented by measurements of their production yields in jets, where hard QCD processes are dominant, and in the underlying event, which is dominated by soft QCD processes. In this contribution, recent results on the measurements of light nuclei production in proton-proton and proton-lead collisions at different center-of-mass energies are shown and discussed in the context of the statistical hadronization and coalescence models.
DOI: https://doi.org/10.22323/1.398.0392
How to cite

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

Open Access


text:
# 11.1: How do these fluxes look?

Figure: The mean values and turbulent vertical fluxes of virtual potential temperature, specific humidity, and horizontal momentum for daytime conditions. The top horizontal dashed line indicates the top of the boundary layer; the bottom horizontal dashed line indicates the top of the surface layer. The vertical dashed line is the calculated value of the geostrophic wind extending to the surface if there were no friction. The entrainment zone is an area of mixing between the mixed layer and the free atmosphere. Credit: W. Brune, after Deidonks and Tennekes, 1984

#### Check Yourself

• All these height variations scale with the PBL height, whether it be 1 km or 3 km.
• Eddy fluxes move quantities from higher to lower values. If a mean quantity increases with height, then the eddy flux will be downward (negative); if a mean quantity decreases with height, then the eddy flux will be upward (positive).
• The entrainment zone is a layer of mixing between the boundary layer and the free troposphere.

Before you read on, make sure that you believe these three concepts.

## Virtual Potential Temperature, $$\bar{\theta}_{v}$$

The $$\bar{\theta}_{v}$$ profile is superadiabatic near the surface due to contact with the heated surface, neutral in the middle, and stable above. Eddies can rise from the surface to a height where $$\overline{\boldsymbol{\theta}}_{v}$$ equals its surface value (assuming no cloud formation and condensation). The mixed layer will grow if surface heating or increased humidity by evaporation causes the surface $$\bar{\theta}_{v}$$ to increase, which means that air parcels can rise and be neutrally buoyant at greater $$\bar{\theta}_{v}$$ aloft and thus greater heights.

We see that the eddy flux $$\overline{w^{\prime} \theta}_{v}^{\prime}$$ is greatest at the surface, decreases nearly linearly with height, becomes slightly negative above the PBL height h because eddies are bringing warmer air down from above. Remember, eddy fluxes carry a quantity like virtual potential temperature down the mean gradient. The eddy flux for virtual potential temperature (and all quantities) goes to very low values (essentially zero) above the entrainment zone even though a gradient is present because the eddy energy is much lower there.

## Specific Humidity, $$\bar{q}$$

Specific humidity is greatest at the surface, where moisture sources, such as water bodies and vegetation, are present. The specific humidity falls off slowly with height until it reaches the PBL height, and then falls off rapidly into the free atmosphere. Because $$\bar{q}$$ falls off with height, the humidity flux $$\overline{w^{\prime} q^{\prime}}$$ increases with height until $$\overline{w^{\prime} q^{\prime}}$$ becomes small because $$\bar{q}$$ becomes small.

So what does the daytime convective boundary layer look like? There are isolated convective updrafts surrounded by slower descending air, giving rise to the large-scale eddy circulation, as seen in the following video of the convective boundary layer, viewed from the top. Associated with the large eddies are smaller eddies that come about as the upward and downward air parcels move past each other. If you look closely, you will see eddies of all sizes in the video (:24), some quite large and some quite small, but the smaller ones seem to originate in the larger ones.

Quiz 11-2: State of flux.

1. Find Practice Quiz 11-2 in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
2. When you feel you are ready, take Quiz 11-2. You will be allowed to take this quiz only once. Good luck!


text:
blob: 36774054f6be6df250f8daf3058610a89846b18b [file] [log] [blame]
 /* * Copyright (C) 2010 NXP Semiconductors * * Licensed under the Apache License, Version 2.0 (the "License"); * you may not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ /*! * \file phFriNfc.h * \brief NFC FRI Main Header. * * Project: NFC-FRI * * $Date: Mon Dec 13 14:14:13 2010$ * $Author: ing02260$ * $Revision: 1.20$ * $Aliases:$ * */ #ifndef PHFRINFC_H /* */ #define PHFRINFC_H /* */ #include #include #define PH_HAL4_ENABLE #ifdef PH_HAL4_ENABLE #include #define LOCK_BITS_CHECK_ENABLE #endif #define FRINFC_READONLY_NDEF #ifdef DISABLE_MIFARE_MAPPING #define PH_FRINFC_MAP_MIFAREUL_DISABLED #define PH_FRINFC_MAP_MIFARESTD_DISABLED #define PH_FRINFC_MAP_DESFIRE_DISABLED #else #define PH_NDEF_MIFARE_ULC #endif #ifdef DISABLE_FELICA_MAPPING #define PH_FRINFC_MAP_FELICA_DISABLED #endif #ifdef DISABLE_JEWEL_MAPPING #define PH_FRINFC_MAP_TOPAZ_DISABLED #define PH_FRINFC_MAP_TOPAZ_DYNAMIC_DISABLED #endif #ifdef DISABLE_ISO15693_MAPPING #define PH_FRINFC_MAP_ISO15693_DISABLED #endif #ifdef DISABLE_FORMAT #define PH_FRINFC_FMT_DESFIRE_DISABLED #define PH_FRINFC_FMT_MIFAREUL_DISABLED #define PH_FRINFC_FMT_MIFARESTD_DISABLED #define PH_FRINFC_FMT_ISO15693_DISABLED #endif /* #ifdef DISABLE_FORMAT */ #define PH_FRINFC_FMT_TOPAZ_DISABLED /*! * \name NFC FRI Main Header * * File: \ref phFriNfc.h * */ /*@{*/ #define PH_FRINFC_FILEREVISION "$Revision: 1.20$" /**< \ingroup grp_file_attributes */ #define PH_FRINFC_FILEALIASES "$Aliases:$" /**< \ingroup grp_file_attributes */ /*@}*/ /*! * \ingroup grp_fri_nfc_common * * \brief \copydoc page_cb Completion Routine * * NFC-FRI components that work in an overlapped style need to provide a function that is compatible * to this definition.\n\n * It is \b mandatory to define such a routine for components that interact with other components up or * down the stack. Moreover, such components shall provide a function within their API to enable the * setting of the \b Completion \b Routine address and parameters. * * \par First Parameter: Context * Set to the address of the called instance (component instance context structure). For instance, * a component that needs to give control to a component up the stack needs to call the completion * routine of the \b upper component. The value to assign to this parameter is the \b address of * the context structure instance of the called component. Such a structure usually contains all * variables, data or state information a component member needs for operation. The address of the * upper instance must be known by the lower (completing) instance. The mechanism to ensure that this * information is present involves the structure \ref phFriNfc_CplRt_t . See its documentation for * further information. * * \par Second Parameter: Status Value * The lower layer hands over the completion status via this parameter. The completion * routine that has been called needs to process the status in a way that is comparable to what * a regular function return value would require. * * \note The prototype of the component's \b Process(ing) functions has to be compatible to this * function pointer declaration for components interacting with others. In other cases, where * there is no interaction or asynchronous processing the definition of the \b Process(ing) * function can be arbitrary, if present at all. */ typedef void (*pphFriNfc_Cr_t)(void*, NFCSTATUS); /*! * \ingroup grp_fri_nfc_common * * \brief Completion Routine structure * * This structure finds itself within each component that requires to report completion * to an upper (calling) component.\n\n * Depending on the actual implementation (static or dynamic completion information) the stack * initialisation \b or the calling component needs to inform the initialised \b or called component * about the completion path. This information is submitted via this structure. * */ typedef struct phFriNfc_CplRt { pphFriNfc_Cr_t CompletionRoutine; /*!< Address of the upper Layer's \b Process(ing) function to call upon completion. * The stack initialiser (or depending on the implementation: the calling component) * needs to set this member to the address of the function that needs to be within * the completion path: A calling component would give its own processing function * address to the lower layer. */ void *Context; /*!< Instance address (context) parameter. * The stack initialiser (or depending on the implementation: the calling component) * needs to set this member to the address of the component context structure instance * within the completion path: A calling component would give its own instance address * to the lower layer. */ } phFriNfc_CplRt_t; #define NFCSTATUS_INVALID_DEVICE_REQUEST (0x10F5) #endif /* __PHFRINFC_H__ */


text:
# §15.10(i) Fundamental Solutions

 15.10.1 $z(1-z)\frac{{d}^{2}w}{{dz}^{2}}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$

This is the hypergeometric differential equation. It has regular singularities at $z=0,1,\infty$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_{1}(z)$, $f_{2}(z)$ of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.

# Singularity $z=0$

 15.10.2 $\displaystyle f_{1}(z)$ $\displaystyle=\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right),$ $\displaystyle f_{2}(z)$ $\displaystyle=z^{1-c}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop 2-c};z% \right),$
 15.10.3 $\mathop{\mathscr{W}\/}\nolimits\left\{f_{1}(z),f_{2}(z)\right\}=(1-c)z^{-c}(1-% z)^{c-a-b-1}.$

# Singularity $z=1$

 15.10.4 $\displaystyle f_{1}(z)$ $\displaystyle=\mathop{F\/}\nolimits\!\left({a,b\atop a+b+1-c};1-z\right),$ $\displaystyle f_{2}(z)$ $\displaystyle=(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c-a-b+1}% ;1-z\right),$
 15.10.5 $\mathop{\mathscr{W}\/}\nolimits\left\{f_{1}(z),f_{2}(z)\right\}=(a+b-c)z^{-c}(% 1-z)^{c-a-b-1}.$

# Singularity $z=\infty$

 15.10.6 $\displaystyle f_{1}(z)$ $\displaystyle=z^{-a}\mathop{F\/}\nolimits\!\left({a,a-c+1\atop a-b+1};\frac{1}% {z}\right),$ $\displaystyle f_{2}(z)$ $\displaystyle=z^{-b}\mathop{F\/}\nolimits\!\left({b,b-c+1\atop b-a+1};\frac{1}% {z}\right),$
 15.10.7 $\mathop{\mathscr{W}\/}\nolimits\left\{f_{1}(z),f_{2}(z)\right\}=(a-b)z^{-c}(z-% 1)^{c-a-b-1}.$

(a) If $c$ equals $n=1,2,3,\dots$, and $a=1,2,\dots,n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by (15.10.2) with the interpretation (15.2.5) for $f_{2}(z)$.

(b) If $c$ equals $n=1,2,3,\dots$, and $a\neq 1,2,\dots,n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by $\mathop{F\/}\nolimits\!\left(a,b;n;z\right)$ and

 15.10.8 $\mathop{F\/}\nolimits\!\left({a,b\atop n};z\right)\mathop{\ln\/}\nolimits z-% \sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(1-a\right)_{k}\left(1-b% \right)_{k}}(-z)^{-k}\\ +\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{\left(n\right)% _{k}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(a+k\right)+\mathop{\psi\/}% \nolimits\!\left(b+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop% {\psi\/}\nolimits\!\left(n+k\right)\right),$ $a,b\neq n-1,n-2,\dots,0,-1,-2,\dots$, Symbols: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$: hypergeometric function, $\left(a\right)_{n}$: Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$: psi (or digamma) function, $!$: $n!$: factorial, $\mathop{\ln\/}\nolimits z$: principal branch of logarithm function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer and $m$: integer A&S Ref: 15.5.19. (Compared with (15.10.8) this result has a multiple of $\mathop{F\/}\nolimits\!\left(a,b;n;z\right)$ added to the right-hand side. It also contains an error: the conditions on $a$ and $b$ should be $a,b\neq 1,2,\dots,m$.) Referenced by: 15.10.8 Permalink: http://dlmf.nist.gov/15.10.E8 Encodings: TeX, pMML, png

or

 15.10.9 $\mathop{F\/}\nolimits\!\left({-m,b\atop n};z\right)\mathop{\ln\/}\nolimits z-% \sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(m+1\right)_{k}\left(1-b% \right)_{k}}(-z)^{-k}+\sum_{k=0}^{m}\frac{\left(-m\right)_{k}\left(b\right)_{k% }}{\left(n\right)_{k}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(1+m-k\right% )+\mathop{\psi\/}\nolimits\!\left(b+k\right)-\mathop{\psi\/}\nolimits\!\left(1% +k\right)-\mathop{\psi\/}\nolimits\!\left(n+k\right)\right)+(-1)^{m}m!\sum_{k=% m+1}^{\infty}\frac{(k-1-m)!\left(b\right)_{k}}{\left(n\right)_{k}k!}z^{k},$ $a=-m$, $m=0,1,2,\dots$; $b\neq n-1,n-2,\dots,0,-1,-2,\dots$,

or

 15.10.10 $\mathop{F\/}\nolimits\!\left({-m,-\ell\atop n};z\right)\mathop{\ln\/}\nolimits z% -\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(m+1\right)_{k}\left(\ell+1% \right)_{k}}(-z)^{-k}+\sum_{k=0}^{\ell}\frac{\left(-m\right)_{k}\left(-\ell% \right)_{k}}{\left(n\right)_{k}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(1% +m-k\right)+\mathop{\psi\/}\nolimits\!\left(1+\ell-k\right)-\mathop{\psi\/}% \nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k\right)\right)+% (-1)^{\ell}\ell\,!\sum_{k=\ell+1}^{m}\frac{(k-1-\ell)!\left(-m\right)_{k}}{% \left(n\right)_{k}k!}z^{k},$ $a=-m$, $m=0,1,2,\dots$; $b=-\ell$, $\ell=0,1,2,\dots,m$.

Moreover, in (15.10.9) and (15.10.10) the symbols $a$ and $b$ are interchangeable.

(c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\dots$, then fundamental solutions in the neighborhood of $z=0$ are given by $z^{n-1}$ times those in (a) and (b), with $a$ and $b$ replaced throughout by $a+n-1$ and $b+n-1$, respectively.

(d) If $a+b+1-c$ equals $n=1,2,3,\dots$, or $2-n=0,-1,-2,\dots$, then fundamental solutions in the neighborhood of $z=1$ are given by those in (a), (b), and (c) with $z$ replaced by $1-z$.

(e) Finally, if $a-b+1$ equals $n=1,2,3,\dots$, or $2-n=0,-1,-2,\dots$, then fundamental solutions in the neighborhood of $z=\infty$ are given by $z^{-a}$ times those in (a), (b), and (c) with $b$ and $z$ replaced by $a-c+1$ and $\ifrac{1}{z}$, respectively.

# §15.10(ii) Kummer’s 24 Solutions and Connection Formulas

The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.

 15.10.11 $\displaystyle w_{1}(z)$ $\displaystyle=\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)$ $\displaystyle=(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c};z\right)$ $\displaystyle=(1-z)^{-a}\mathop{F\/}\nolimits\!\left({a,c-b\atop c};\frac{z}{z% -1}\right)$ $\displaystyle=(1-z)^{-b}\mathop{F\/}\nolimits\!\left({c-a,b\atop c};\frac{z}{z% -1}\right).$ 15.10.12 $\displaystyle w_{2}(z)$ $\displaystyle={z^{1-c}}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop 2-c};z\right)$ $\displaystyle={z^{1-c}(1-z)^{c-a-b}}\*\mathop{F\/}\nolimits\!\left({1-a,1-b% \atop 2-c};z\right)$ $\displaystyle={z^{1-c}(1-z)^{c-a-1}}\*\mathop{F\/}\nolimits\!\left({a-c+1,1-b% \atop 2-c};\frac{z}{z-1}\right)$ $\displaystyle={z^{1-c}(1-z)^{c-b-1}}\*\mathop{F\/}\nolimits\!\left({1-a,b-c+1% \atop 2-c};\frac{z}{z-1}\right).$ 15.10.13 $\displaystyle w_{3}(z)$ $\displaystyle=\mathop{F\/}\nolimits\!\left({a,b\atop a+b-c+1};1-z\right)$ $\displaystyle=z^{1-c}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop a+b-c+1};1% -z\right)$ $\displaystyle=z^{-a}\mathop{F\/}\nolimits\!\left({a,a-c+1\atop a+b-c+1};1-% \frac{1}{z}\right)$ $\displaystyle=z^{-b}\mathop{F\/}\nolimits\!\left({b,b-c+1\atop a+b-c+1};1-% \frac{1}{z}\right).$ 15.10.14 $\displaystyle w_{4}(z)$ $\displaystyle=(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c-a-b+1}% ;1-z\right)$ $\displaystyle=z^{1-c}(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({1-a,1-b\atop c% -a-b+1};1-z\right)$ $\displaystyle=z^{a-c}(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({1-a,c-a\atop c% -a-b+1};1-\frac{1}{z}\right)$ $\displaystyle=z^{b-c}(1-z)^{c-a-b}\mathop{F\/}\nolimits\!\left({1-b,c-b\atop c% -a-b+1};1-\frac{1}{z}\right).$ 15.10.15 $\displaystyle w_{5}(z)$ $\displaystyle=e^{a\pi i}z^{-a}\*\mathop{F\/}\nolimits\!\left({a,a-c+1\atop a-b% +1};\frac{1}{z}\right)$ $\displaystyle=e^{(c-b)\pi i}z^{b-c}(1-z)^{c-a-b}\*\mathop{F\/}\nolimits\!\left% ({1-b,c-b\atop a-b+1};\frac{1}{z}\right)$ $\displaystyle=(1-z)^{-a}\mathop{F\/}\nolimits\!\left({a,c-b\atop a-b+1};\frac{% 1}{1-z}\right)$ $\displaystyle=e^{(c-1)\pi i}z^{1-c}(1-z)^{c-a-1}\*\mathop{F\/}\nolimits\!\left% ({1-b,a-c+1\atop a-b+1};\frac{1}{1-z}\right).$ 15.10.16 $\displaystyle w_{6}(z)$ $\displaystyle=e^{b\pi i}z^{-b}\mathop{F\/}\nolimits\!\left({b,b-c+1\atop b-a+1% };\frac{1}{z}\right)$ $\displaystyle=e^{(c-a)\pi i}z^{a-c}(1-z)^{c-a-b}\*\mathop{F\/}\nolimits\!\left% ({1-a,c-a\atop b-a+1};\frac{1}{z}\right)$ $\displaystyle=(1-z)^{-b}\mathop{F\/}\nolimits\!\left({b,c-a\atop b-a+1};\frac{% 1}{1-z}\right)$ $\displaystyle=e^{(c-1)\pi i}z^{1-c}(1-z)^{c-b-1}\*\mathop{F\/}\nolimits\!\left% ({1-a,b-c+1\atop b-a+1};\frac{1}{1-z}\right).$

The $\binom{6}{3}=20$ connection formulas for the principal branches of Kummer’s solutions are:

 15.10.17 $\displaystyle w_{3}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma% \/}\nolimits\!\left(a+b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1% \right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{1}(z)+\frac{\mathop{% \Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits% \!\left(b\right)}w_{2}(z),$ 15.10.18 $\displaystyle w_{4}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma% \/}\nolimits\!\left(c-a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a% \right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{1}(z)+\frac{\mathop{% \Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}% \nolimits\!\left(c-b\right)}w_{2}(z),$ 15.10.19 $\displaystyle w_{5}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma% \/}\nolimits\!\left(a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1% \right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{1}(z)+e^{(c-1)\pi i}% \frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!% \left(a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/% }\nolimits\!\left(c-b\right)}w_{2}(z),$ 15.10.20 $\displaystyle w_{6}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma% \/}\nolimits\!\left(b-a+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-c+1% \right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{1}(z)+e^{(c-1)\pi i}% \frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!% \left(b-a+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/% }\nolimits\!\left(c-a\right)}w_{2}(z).$
 15.10.21 $\displaystyle w_{1}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/% }\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)% \mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+\frac{\mathop{\Gamma\/}% \nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{% \mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b% \right)}w_{4}(z),$ 15.10.22 $\displaystyle w_{2}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma% \/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)% \mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{3}(z)+\frac{\mathop{\Gamma\/}% \nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{% \mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!% \left(b-c+1\right)}w_{4}(z),$ 15.10.23 $\displaystyle w_{5}(z)$ $\displaystyle=e^{a\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)% \mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+e^{(c-b)% \pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}% \nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)% \mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{4}(z),$ 15.10.24 $\displaystyle w_{6}(z)$ $\displaystyle=e^{b\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)% \mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{3}(z)+e^{(c-a)% \pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}% \nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)% \mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{4}(z).$
 15.10.25 $\displaystyle w_{1}(z)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/% }\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)% \mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(z)+\frac{\mathop{\Gamma\/}% \nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop% {\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right% )}w_{6}(z),$ 15.10.26 $\displaystyle w_{2}(z)$ $\displaystyle=e^{(1-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)% \mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!% \left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z)+e^{(1-% c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}% \nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)% \mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z),$ 15.10.27 $\displaystyle w_{3}(z)$ $\displaystyle=e^{-a\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right% )\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!% \left(b\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z)+e^{-b\pi i% }\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}% \nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop% {\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z),$ 15.10.28 $\displaystyle w_{4}(z)$ $\displaystyle=e^{(b-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1% \right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}% \nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(% z)+e^{(a-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)\mathop% {\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-b% \right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{6}(z).$
 15.10.29 $\displaystyle w_{1}(z)$ $\displaystyle=e^{b\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)% \mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+e^{(% b-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(a-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)% \mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}w_{5}(z),$ 15.10.30 $\displaystyle w_{1}(z)$ $\displaystyle=e^{a\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)% \mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{3}(z)+e^{(% a-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)% \mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)}w_{6}(z),$ 15.10.31 $\displaystyle w_{2}(z)$ $\displaystyle=e^{(b-c+1)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c% \right)\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits% \!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{3}(z)+e^% {(b-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/% }\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)% \mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z),$ 15.10.32 $\displaystyle w_{2}(z)$ $\displaystyle=e^{(a-c+1)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c% \right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits% \!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{3}(z)+e^% {(a-c)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/% }\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)% \mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z).$
 15.10.33 $\displaystyle w_{1}(z)$ $\displaystyle=e^{(c-a)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)% \mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{-a% \pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)% \mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(z),$ 15.10.34 $\displaystyle w_{1}(z)$ $\displaystyle=e^{(c-b)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)% \mathop{\Gamma\/}\nolimits\!\left(1-a\right)}{\mathop{\Gamma\/}\nolimits\!% \left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{-b% \pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(1-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)% \mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{6}(z),$ 15.10.35 $\displaystyle w_{2}(z)$ $\displaystyle=e^{(1-a)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)% \mathop{\Gamma\/}\nolimits\!\left(c-b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^% {-a\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}% \nolimits\!\left(c-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)% \mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{5}(z),$ 15.10.36 $\displaystyle w_{2}(z)$ $\displaystyle=e^{(1-b)\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)% \mathop{\Gamma\/}\nolimits\!\left(c-a\right)}{\mathop{\Gamma\/}\nolimits\!% \left(b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^% {-b\pi i}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}% \nolimits\!\left(c-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)% \mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{6}(z).$