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# Complete the sentences using a preposition + gerund of the following verbs : cause ; pay ; escape ; pass the exam ; interrupt ; tell ; live ; play ; solve ; spend ; walk ; help?

Английский язык | 5 - 9 классы

Complete the sentences using a preposition + gerund of the following verbs : cause ; pay ; escape ; pass the exam ; interrupt ; tell ; live ; play ; solve ; spend ; walk ; help.

It took us a long time but we succeeded __ this problem.

I’ve always dreamed __ in a small house by the sea.

The driver of another car accused me ___ the accident.

There was a fence around the lawn to stop people __ on the grass.

Forgive me __ you but may I ask you something?

Where are you thinking __ your holiday this year?

The guards were not able to prevent the prisoner ___

I wanted to cook the meal by myself but Dave insisted __ me.

I’m sorry we’ve had to cancel our game of tennis tomorrow.

I was really looking forward ___They insisted __ for the meal.

I congratulated Ann __ the exam.

They accused me __ lies.

Ответить на вопрос
Ответы (1)
Molchanadari 2 янв. 2018 г., 11:21:00

It took us a long time but we succeeded in solving this problem.

I’ve always dreamed of living in a small house by the sea.

The driver of another car accused me of causing the accident.

There was a fence around the lawn to stop people from walking on

the grass.

Forgive me for interrupting you but may I ask you something?

Where are you thinking of spending your holiday this year?

The guards were not able to prevent the prisoner from escaping.

I wanted to cook the meal by myself but Dave insisted on helping me.

I’m sorry we’ve had to cancel our game of tennis tomorrow.

I was really

looking forward to playing.

They insisted on paying for the meal.

I congratulated Ann on passing the exam the exam.

They accused me of telling lies.

Dianaeg2109 20 окт. 2018 г., 20:43:16 | 1 - 4 классы

## Помогите перевести, пожалуйста?

Помогите перевести, пожалуйста.

- Ok.

Then it was last year.

You put your feet into the water last year!

The Wolf shouted.

And you made the water dirty.

- But I am only four months old, the sheep said.

- How could I put my feet into the river a year ago?

- If it was not you, then it was your brother or your father.

The Wolf shouted and jumped on the poor Sheep.

But the Sheep ran away.

And the Wolf fell into the water.

Срочно : - [.

Karinad 16 нояб. 2018 г., 21:33:22 | 5 - 9 классы

## Правильно ли составлены вопросы в Past Continuos : (Подправьте если что предложение, где ошибка) Were you climbing a mountain?

Правильно ли составлены вопросы в Past Continuos : (Подправьте если что предложение, где ошибка) Were you climbing a mountain?

Was the teacher pointing to the board?

Was it raining?

Was he crossing the street?

Was Greg kicking the ball?

Were you opening the door?

Were you playing cards?

Were you repairing the car?

Were you enjoying our holiday?

Lilo4ka 20 янв. 2018 г., 13:38:16 | 1 - 4 классы

## Как переводится стих The Country and the citi Some people live in the citi Where the houses are very tal Some people live in the country But in the country where the houses are small The gardens are v?

Как переводится стих The Country and the citi Some people live in the citi Where the houses are very tal Some people live in the country But in the country where the houses are small The gardens are very big And in the cities where the houses are tall There are no gardens at all.

UliaTurenkova1 18 дек. 2018 г., 06:53:58 | 1 - 4 классы

## Complete the following sentences about yourself using the Gerund?

Complete the following sentences about yourself using the Gerund.

Sabrina22022006sabri 3 янв. 2018 г., 18:37:51 | 10 - 11 классы

## Complete the sentences by putting the verbs in brackets () into the Past Continuos?

Complete the sentences by putting the verbs in brackets () into the Past Continuos.

1)I I tried to explain my problem to her, but she.

(not / listen) 2)He.

(talk) on the phone when I arrived.

3) A lot of people .

(wait) for the 7.

30 bus last night.

4)I. (live) in London when I met them.

5)I nearly had an accident this morning.

A car .

(come) towards me, but I moved quickly out the way.

6)At the end of the first half of the game, they .

(win).

7)It was a sunny afternoon and people .

(sit) on the grass in the park.

8)Which hotel .

0070809 12 нояб. 2018 г., 23:30:07 | 5 - 9 классы

## Помогите пожалуйста ?

Помогите пожалуйста !

Вам не трудно : *

Complete the sentences with the words from the list.

Use the correct tense form of the verbs To solve, to fall, to call, to point, to reduce 1.

You ….

And click when you use the mouse.

2. Can you help me to… this problem?

3. Sales … for the first time to 10 000 copies in 1996.

Nikolai11111 6 февр. 2018 г., 15:50:44 | 5 - 9 классы

## Choose the right word to complete the sentences?

Choose the right word to complete the sentences.

1) - (Was / were) there any money in the box?

₋ I dont know, but there (is / are) no money in it now.

2) Where (is / are) the coins?

(Is / Are) they on the shelf?

3) Ask Jane if the has (many / much) money.

4) Money (was / were) important in the past, money (is / are) important now.

5) Take (this / these) money and buy some fruit / 6) The money (is / are) not in the bag, (it / they) (is / are) in the moneybox.

Димон34636 24 июн. 2018 г., 03:51:14 | 5 - 9 классы

## Вставьте слова : Your work is ?

Вставьте слова : Your work is .

Better.

She could play the piano .

She could walk.

The problem was .

Solved.

This is cinema .

We saw the film.

Can you tell me the way?

Im working .

To pass my exam.

He`s the man .

House the pictures were stolen from.

Worohok 7 нояб. 2018 г., 19:02:24 | 5 - 9 классы

## Переведите СОРОЧНО ?

Переведите СОРОЧНО !

Так что бы было с рифмой !

What is Poetry?

Who Knows?

Not a rose, but the scent of the rose ; Not the sky, but the light in the sky ; Not the fly, but the gleam of the fly ; Not the sea, but the sound of the sea ; Not myself, but what makes me See, hear, and feel something that prose Cannot : and what it is who knows?

Мария110899 23 мар. 2018 г., 11:11:30 | 5 - 9 классы

## Complete the sentences with into over on off up and down?

Complete the sentences with into over on off up and down.

The. page, the picture you are looking for is there.

Don't turn the sound.


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# Can the site-design be improved to be more welcoming to new users?

I think that the design of MSE could be more welcoming to new users. I don't know how much effort it would take and of course that should be considered. My view is as someone who is relatively new to the site, just a few months, and who is relatively active. I am not under the delusion that these ideas could be implemented quickly, but unless efforts are begun they will never happen. The following list is not presented in any particular order and does not pretend to be complete. I should also say that despite what may sound like complaints, what I am presenting here are ideas to improve something that I find very valuable. I'm presenting them now before I become too accepting of the status quo.

• MathJax is an example. The information on using it is generally good but it could use a table of contents with internal links to help users, esp. newcomers, find what they need. The supplementary material presented as comments should be integrated into the main flow and kept up-to-date.
• Policies should be spelled out in one place with a table of contents (and links) for everyone to see. At present one discovers policies randomly in some Meta threads. Some policies are shown in a FAQ page which, for all practical purposes, is hidden.
• We could use a Style Sheet with recommended notation conventions, where there is ambiguity. This "what to" guide would go beyond the MathJax notes, which is primarily a "how to" guide.
• The language used when a question is put on hold or closed could be more considerate. For example the statement that a question is "off-topic" is very off-putting and actually a misnomer in most cases which we may be stuck with. We should explain that in the explanation. And more generally make the explanations kinder. The following response is cold and uninformative.

"This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level."
Moreover, the criteria described above are ignored much more often than they are applied.

• Written criteria for accepting, closing, and deleting a question should be spelled out in one place with a table of contents (and links) for everyone to see. (It could be a special part of Policies.)
• Re: The language used when a question is put on hold or closed could be more considerate. What you ask in this bullet point seems to be related to these feature requests: A feature requested for the purpose of avoiding unintentional newbie-biting and Should “missing context” be under “off topic?”. – Martin Sleziak Nov 28 '17 at 7:02
• So one question is whether the voting on those feature requests already shows sufficient support. (The two feature request on our local meta have currently scores 10 and 16. The feature request on meta.SE is at score 86.) If yes, then the question why they weren't implemented yet is mainly question for SE staff. (And perhaps for moderators, since they also communicate with the people from SE.) Probably I should add that this network is very large and there are probably hundreds of feature request. So that some of them do not get implemented or it takes quite a long time is not that surprising. – Martin Sleziak Nov 28 '17 at 10:12
• I think there's one very good and important point here: it would be useful if we could have a much more basic MathJax introduction. I imagine a student who is studying calculus or discrete math for the first time with no experience with TeX finding our existing guide to be a bit impregnable. Such a guide could be styled like: "To typeset $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, use \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} enclosed by dollar signs," etc. Something with very basic, simple examples would be useful, I think. – Aaron Montgomery Nov 28 '17 at 14:53
• I agree with @AaronMontgomery the the MathJax tutorial provided here on meta is a bit complicated for a newbie. It is currently linked from the How to ask page (which is displayed to a user when asking the first question). When I leave a comment about Mathjax, I use this comment template. It contains several links, the first one is help on mathjax notation - incidentally, it contains exactly the example with quadratic formula. [ed.] – Martin Sleziak Nov 28 '17 at 15:01
• It seems that some of the stuff raised in this question should be discussed - although I think that discussing all of this in one question is probably going to be rather difficult. Anyway, +1 from me for raising some interesting questions. (Despite the fact that I disagree with some parts of the question and in some places it is somewhat unclear what the OP actually suggests.) – Martin Sleziak Nov 28 '17 at 15:07
• Well, then I propose a block, after registering an account, from asking, until such a new user goes through the notice of mathjax with links to learn it, and checks "I understand;" and until such a new user goes through a pop-up of site policies, and checks "I understand"; and until such a new user is presented with a "style sheet" about notations conventions, and a review of mathjax, and a review of site policies, and checking "I understand". And only then, be allowed to ask a question. All such information is available, if a new user takes the initiative to learn the site policies, etc.. – Namaste Nov 28 '17 at 19:04
• Stephan.... cont .... Since most new askers (and some not so new askers) don't check this out, it doesn't mean the site is "not nice." But perhaps we simply need more hoops for new users to jump through by requiring a series of three or four pop-ups, policy, expectations, how to ask a good question, style, and require the would be new asker to check off "I understand" prior to posting. If after that, if they still fail to abide by what they claimed they understand, they should be held accountable, just as any other user is. – Namaste Nov 28 '17 at 19:08
• I think that it's already not that hard to learn the standards of this community. The problem is that there are tons of people posting PSQs [Problem Statement Questions] who do not care about this site and are only interested in having the community solve their homework for them. [ed.] – Qudit Nov 28 '17 at 22:46
• After a short exchange the poster edited the question and I made some additional edits, to focus the discussion on the design of the site and the documentation we provide, and away of things that can be taken as discussing the conduct of individual users. Thus, I also tried to remove all discussion on that aspect. – quid Nov 29 '17 at 19:27
• @quid that "rationale" means very little, especially when you are responsible for having decided what this question should be about, give the asker an ultimatum to make it so, and delete valid responses to the post (given your censorship and deletion of 50 % of the post). so, you designed this post to be what you believe is valid by crafting (and requiring the edit of) the question, and by deleting real and valuable content. Without such content, due to your intervention, you've rendered the question meaningless, inhibiting the very users who care about such questions, from future engagement – Namaste Nov 29 '17 at 23:26
• @quid What you claim is the case hardly has merit, because you've destroyed the very conversation you claim you had with the asker, destroyed the conversation and context of the post, to make it what you think is worth keeping. All good and well, except this is not the question asked by a non-mod user, and is not the conversation about the question you chose to change. Hence: your rationale is meaningless, as you've left no evidence in comments to verify your claims, and censored the comments, which only you and other mods can now see. That's not fair. – Namaste Nov 29 '17 at 23:34
• Any way, I think it deserves mentioning that it rests in the hands of people to be nice and welcoming, not scripts. Most users know that written rules, regulations, specifications, exceptions in print (privacy policies, insurance policies, consumer responsibilities, user agreements) aren't designed to be fun, but rather informative. Words cannot coddle. Only people can be welcoming, can set limits, can be nice, or not nice, so if a user acts in a not nice way, in an unwelcoming way, that's where change needs to take place.... – Namaste Nov 29 '17 at 23:42
• ...To blame documentation for users feeling "unwelcome" is outrageous, especially when the blame comes from users who, like us all, are ultimately responsible for when and how or even if, new users are welcomed, recognizing that "welcoming a new user" $\not \equiv$ "coddling a new user." – Namaste Nov 29 '17 at 23:44
• I do wish for a table of contents on the MathJax tutorial meta post, one that's similar to the "How to ask a good question" meta post. – Simply Beautiful Art Nov 30 '17 at 1:36
• "The following response is cold and uninformative." How do you propose to make it informative? I think it contains all of the necessary things while still being short enough that people will care to read it. And cold? Oddly, I do not think it is cold at all. What, specifically, is so cold about it? – Simply Beautiful Art Nov 30 '17 at 1:41

## 1 Answer

I'm surprised wiser minds than mine have not responded to your bullets in an answer, since there are so many wiser minds than mine here.

• MathJax - I partly agree with you here. Only partly because there is a MathJax tutorial here which is organized enough, IMHO, but if you don't know to search the meta for it, you might not find it: MathJax basic tutorial and quick reference. Another way to find it is to sort the Meta questions by "Frequent". Again, that's not going to be obvious or intuitive for a new user. The inability of the community to create static, non-voted content, at least in the meta, does seem to be a weakness of the SE system that crops up from time to time.
• Policies - I'm not with you on this one. Large numbers of users on all SE sites seem to rush headlong into posting a question in their first ten minutes on SE, but I don't think any of them can make a strong case that they had no way of knowing better. When you're not a member, there is a banner at the top of the Stack with a broad overview of how it works, and no matter who you are, when you are on the Ask a Question page, there's always a sidebar with the header "How to ask" and a link to the help center. I might be more inclined than most to read the instructions for everything before I use it, but I still have a hard time having sympathy for those who would claim to not have been given the opportunity to read the instructions. And the help center is well organized, IMHO, broken down into a sort of table of contents with links.
• Style Sheet - The problem with site-wide notation conventions is most people will be unhappy with it one way or the other and some branches of mathematics may have popular conventions that conflict with those of other branches. Easy example: How should people notate the time derivative of a time-based function like displacement, velocity, or acceleration? That's a huge can of worms, IME.
• On Hold Language - Here is where I'm more of an ally than the other three points. At the same time, it's never going to be possible to write on-hold explanations that cover more than 50% of cases well. It might be better to look at the history of questions that have been put on hold, try to work out the four most common problems, revisit and revise the boilerplate on-hold language to reflect the four most common reasons, and then encourage those with VTC privileges to use the comment option more often. There are many cases where I would say those voting to close should have taken more time to comment on a question with some questions to clarify it or advise edits, but so many times when I go through that trouble, the asker either flat out ignores my comments or only responds in such a way that reinforces the inappropriateness of their question. So it's hard to have a lot of faith in the success of that kind of approach. Most aksers who would be willing and able to improve their question based on comments have already visited the help center and don't need much in the way of comments to fix their questions.
• re MJ: OP wrote "The information on using it is generally good but[...]" and I think they implicitly referred to the meta thread you mentioned. – quid Dec 9 '17 at 16:04
• re On Hold Language: there are two separate issues. The one is that the custom reasons are all under "off-topic" the other is the text of the custom reason. Regarding the latter I agree with you that it can make sense to write ones own reason. It is even built in. off-topic, other. The explanations given there are automatically tracked and if only used more broadly could allow to do what you propose handily. – quid Dec 9 '17 at 16:08

format: plain


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location:  Publications → journals → CJM
Abstract view

# Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds

Published:2010-07-06
Printed: Oct 2010
• William Breslin,
Department of Mathematics, University of Michigan, Ann Arbor, MI, U.S.A.
 Format: HTML LaTeX MathJax PDF

## Abstract

A triangulation of a hyperbolic $3$-manifold is \emph{L-thick} if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic $3$-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of $\pi_1$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic $3$-manifolds.
 MSC Classifications: 57M50 - Geometric structures on low-dimensional manifolds


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## Gelatt, C. D. jun.

Compute Distance To:
 Author ID: gelatt.c-d-jun Published as: Gelatt, C. D. jun.
 Documents Indexed: 1 Publication since 1983 Co-Authors: 2 Co-Authors with 1 Joint Publication 10 Co-Co-Authors

### Co-Authors

 0 single-authored 1 Kirkpatrick, Scott 1 Vecchi, M. P.

### Serials

 1 Science

### Fields

 1 Statistical mechanics, structure of matter (82-XX) 1 Operations research, mathematical programming (90-XX)

all top 5

### Cited by 4,348 Authors

 14 Drezner, Zvi 11 Boysen, Nils 10 Aarts, Emile Hubertus Leonardus 9 Brusco, Michael J. 8 Dörner, Karl F. 8 Drezner, Tammy 8 Hartl, Richard F. 8 Hoffmann, Karl Heinz 8 Ingber, Lester 8 Tavakkoli-Moghaddam, Reza 7 Floudas, Christodoulos Achilleus 7 Liang, Faming 7 Lin, Win-Chin 7 Schaerf, Andrea 7 Winker, Peter 7 Wu, Chin-Chia 6 Albrecht, Andreas A. 6 Burke, Edmund Kieran 6 Ceschia, Sara 6 Gendreau, Michel 6 Salhi, Said 6 Wang, Yongjun 5 Berman, Oded 5 Burkard, Rainer E. 5 Chiang, Wen-Chyuan 5 García-Rodríguez, José Antonio 5 Ghaffarinasab, Nader 5 Gupta, Jatinder N. D. 5 Hertz, Alain 5 Hervás-Martínez, César 5 Lin, Shih-Wei 5 Mladenović, Nenad 5 Ozdamar, Linet 5 Pardalos, Panos M. 5 Parragh, Sophie N. 5 Sahinidis, Nikolaos V. 5 Schneider, Johannes J. 5 Siarry, Patrick 5 Steinley, Douglas L. 5 Tsoulos, Ioannis G. 5 van Laarhoven, Peter J. M. 5 Vázquez Cendón, Carlos 5 Yu, Vincent F. 5 Zhang, Jiangshe 4 Abdalla, Soliman 4 Anandalingam, G. 4 Batle, Josep 4 Battiti, Roberto 4 Blum, Christian 4 Briskorn, Dirk 4 Carrizosa, Emilio 4 de Sousa, Fabiano Luis 4 Deng, Yuefan 4 Di Gaspero, Luca 4 Dorigo, Marco 4 Emde, Simon 4 Feng, Enmin 4 Glover, Fred W. 4 Gong, Guanglu 4 Hansen, Pierre 4 Hedar, Abdel-Rahman 4 Jiang, Tianzi 4 Korst, Jan H. M. 4 Kouvelis, Panagiotis 4 Laporte, Gilbert 4 Lee, Wen-Chiung 4 Mucherino, Antonio 4 Nezamabadi-Pour, Hossein 4 Ooi, C. H. Raymond 4 Paterlini, Sandra 4 Polson, Nicholas G. 4 Potvin, Jean-Yves 4 Qian, Minping 4 Ramos, Fernando Manuel 4 Salamon, Peter F. jun. 4 Santos, Haroldo Gambini 4 Shao, Wei 4 Smith, Robert Longworth 4 Sörensen, Kenneth 4 Steinhöfel, Kathleen K. 4 Talbi, El-Ghazali 4 Torres-Jimenez, Jose 4 Wang, Ling 4 Werner, Frank 4 Wesolowsky, George Orest 4 Yao, Xin 4 Yin, Peng-Yeng 3 Abbasi, Babak 3 Adewumi, Aderemi Oluyinka 3 Akhavan Niaki, Seyed Taghi 3 Al-Refa’ei, Mahmoud H. 3 Alkhamis, Talal M. 3 Amaran, Satyajith 3 Andresen, Bjarne 3 Androulakis, Ioannis P. 3 Arnst, Maarten 3 Barzinpour, Farnaz 3 Baykasoglu, Adil 3 Benedetti, Roberto 3 Błażewicz, Jacek ...and 4,248 more Authors
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### Cited in 361 Serials

 229 European Journal of Operational Research 163 Computers & Operations Research 79 Annals of Operations Research 69 Applied Mathematics and Computation 65 Journal of Global Optimization 54 International Journal of Production Research 32 Information Sciences 29 Mathematical Problems in Engineering 25 Computational Statistics and Data Analysis 24 Mathematical and Computer Modelling 23 Applied Mathematical Modelling 22 Discrete Applied Mathematics 22 Journal of Optimization Theory and Applications 21 Computer Methods in Applied Mechanics and Engineering 21 Pattern Recognition 21 Computational Optimization and Applications 20 Computers & Mathematics with Applications 20 Journal of Statistical Physics 20 Theoretical Computer Science 19 Biological Cybernetics 19 Journal of Computational and Applied Mathematics 17 Journal of Computational Physics 17 International Journal of Computer Mathematics 16 Computer Physics Communications 16 OR Spectrum 15 Neural Computation 12 Artificial Intelligence 12 Journal of Scheduling 12 Algorithms 12 Journal of Theoretical Biology 11 Mathematical Biosciences 11 Computational Statistics 11 Journal of Heuristics 11 Optimization Methods & Software 11 Structural and Multidisciplinary Optimization 11 Journal of Statistical Mechanics: Theory and Experiment 11 Inverse Problems in Science and Engineering 11 Statistics and Computing 10 International Journal of Systems Science 10 Mathematics and Computers in Simulation 10 Physica D 10 Journal of Statistical Computation and Simulation 9 Operations Research Letters 9 International Journal of Approximate Reasoning 9 Asia-Pacific Journal of Operational Research 9 Annals of Mathematics and Artificial Intelligence 8 OR Spektrum 8 Journal of Information & Optimization Sciences 8 Algorithmica 8 Advances in Engineering Software 8 CEJOR. Central European Journal of Operations Research 8 Journal of Applied Mathematics 8 Sādhanā 8 JMMA. Journal of Mathematical Modelling and Algorithms 8 Journal of Industrial and Management Optimization 7 Discrete Mathematics 7 Physica A 7 Psychometrika 7 Journal of Mathematical Psychology 7 Journal of Statistical Planning and Inference 7 Neural Networks 7 Cybernetics and Systems Analysis 7 Computational and Applied Mathematics 7 Top 7 Discrete Dynamics in Nature and Society 7 RAIRO. Operations Research 7 Quantum Information Processing 7 Computational Biology and Chemistry 6 Chaos, Solitons and Fractals 6 Computing 6 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 6 Communications in Statistics. Simulation and Computation 6 Stochastic Processes and their Applications 6 Chaos 6 Computational Management Science 6 Optimization Letters 5 Acta Mechanica 5 Information Processing Letters 5 Journal of Computer and System Sciences 5 Journal of Econometrics 5 Kybernetes 5 Naval Research Logistics 5 Optimization 5 Statistical Science 5 Journal of Computer Science and Technology 5 Applied Mathematics Letters 5 The Annals of Applied Probability 5 International Transactions in Operational Research 5 Journal of Combinatorial Optimization 5 Probability in the Engineering and Informational Sciences 5 International Journal of Modern Physics C 5 Physical Review Letters 5 Archives of Computational Methods in Engineering 5 Natural Computing 5 Journal of Optimization 5 International Journal of Systems Science. Principles and Applications of Systems and Integration 4 International Journal of Engineering Science 4 International Journal of General Systems 4 Automatica 4 Fuzzy Sets and Systems ...and 261 more Serials
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### Cited in 47 Fields

 1,329 Operations research, mathematical programming (90-XX) 422 Computer science (68-XX) 276 Numerical analysis (65-XX) 207 Statistics (62-XX) 180 Biology and other natural sciences (92-XX) 111 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 91 Probability theory and stochastic processes (60-XX) 79 Combinatorics (05-XX) 75 Statistical mechanics, structure of matter (82-XX) 67 Mechanics of deformable solids (74-XX) 66 Systems theory; control (93-XX) 50 Information and communication theory, circuits (94-XX) 36 Calculus of variations and optimal control; optimization (49-XX) 35 Quantum theory (81-XX) 27 Dynamical systems and ergodic theory (37-XX) 24 Geophysics (86-XX) 20 Fluid mechanics (76-XX) 17 Partial differential equations (35-XX) 17 Mechanics of particles and systems (70-XX) 14 Ordinary differential equations (34-XX) 14 Optics, electromagnetic theory (78-XX) 10 Convex and discrete geometry (52-XX) 8 Classical thermodynamics, heat transfer (80-XX) 5 Mathematical logic and foundations (03-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 4 General and overarching topics; collections (00-XX) 4 Functional analysis (46-XX) 4 Astronomy and astrophysics (85-XX) 3 History and biography (01-XX) 3 Number theory (11-XX) 3 Group theory and generalizations (20-XX) 3 Approximations and expansions (41-XX) 2 Operator theory (47-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Algebraic geometry (14-XX) 1 Measure and integration (28-XX) 1 Functions of a complex variable (30-XX) 1 Special functions (33-XX) 1 Difference and functional equations (39-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Integral equations (45-XX) 1 Geometry (51-XX) 1 Differential geometry (53-XX) 1 Manifolds and cell complexes (57-XX) 1 Relativity and gravitational theory (83-XX) 1 Mathematics education (97-XX)


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# Physics at the LHC and beyond

from September 29, 2015 to October 2, 2015
DESY Hamburg
Europe/Berlin timezone
Home > Timetable > Session details > Contribution details

# Contribution

DESY Hamburg - Seminar room 4b

# Dark Matter Mono-$Z$ Production at the LHC beyond Leading Order in the Simplified Models

## Speakers

• Dr. Jian WANG

## Abstract content

We present theoretical predictions for the mono-$Z$ production in the search for dark matter at the LHC with next-to-leading order QCD corrections and parton-shower effects. The calculation is performed in the framework of {\tt MadGraph5$_$aMC@NLO}. We find that the high order QCD corrections are sizable, and can reduce the theoretical uncertainties. We also investigate the discovery potential of this signal at the 13 TeV LHC.


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# Math Help - Poisson Distribution

1. ## Poisson Distribution

If 90% of students are right handed, use the Poisson distribution to estimate the probability that a sample of 80 students has 5 or more right-handed students.
Please help with this problem!

2. Originally Posted by Johny_Junior
If 90% of students are right handed, use the Poisson distribution to estimate the probability that a sample of 80 students has 5 or more right-handed students.
Please help with this problem!
Calculate the mean number of right-handed students in the sample. Then use the Poisson distribution with that mean to calculate the estimated probability.

3. Thanks for your help...but it still doesn't make much sense to me...

4. Originally Posted by Johny_Junior
If 90% of students are right handed, use the Poisson distribution to estimate the probability that a sample of 80 students has 5 or more right-handed students.
Please help with this problem!
Originally Posted by JakeD
Calculate the mean number of right-handed students in the sample. Then use the Poisson distribution with that mean to calculate the estimated probability.
Originally Posted by Johny_Junior
Thanks for your help...but it still doesn't make much sense to me...
Where are you stuck?


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# What type of singularity is this?

$z\cdot e^{1/z}\cdot e^{-1/z^2}$ at $z=0$.

My answer is removable singularity. $$\lim_{z\to0}\left|z\cdot e^{1/z}\cdot e^{-1/z^2}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|=0.$$ But someone says it is an essential singularity. I don't know why.

-
You seem to have made two mistakes in your post. First, $\lim_{z \to 0} z e^{-1/z^2}$ is not zero. Second, you seemed to have lost the $e^{1/z}$ term, so you chould check your algebra. –  JavaMan Oct 29 '12 at 17:11

for $z\cdot e^{\frac{-1}{z^2}}$, note if you approach to the origin along the imaginary line, say $z=ih$, we will get $ihe^{\frac{-1}{(i)^2h}}=ihe^{\frac{1}{h}}$, this obviously does not tends to zero as $h \to 0$

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I think Tao's answer is easiest for me to understand. Thank you very much. –  Sam Oct 29 '12 at 21:25

$$ze^{1/z}e^{-1/z^2}=z\left(1+\frac{1}{z}+\frac{1}{2!z^2}+...\right)\left(1-\frac{1}{z^2}+\frac{1}{2!z^4}-...\right)$$

So this looks like an essential singularity, uh?

I really don't understand how you made the following step:

$$\lim_{z\to 0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|$$

What happened to that $\,z\,$ in the exponential's power?

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Thank you, Don, but I have not figured out how to use the series expansion to determine the singularity. –  Sam Oct 29 '12 at 21:30

First, notice that $$\lim_{z \to 0} e^{1/z}$$ does not exist as you get different values when you approach $0$ along the real line $x + 0i$ from the right and from the left.

From there, it is not difficult to show that $\lim_{z \to 0} z e^{1/z} e^{-1/x^2}$ does not exist either. Finally, we need to show that $\lim_{z \to 0}\frac{1}{f(z)}$ does not exist in order for $f(z)$ to have an essential singularity at $z = 0$.

In other words, you need to examine

$$\lim_{z \to 0} \frac{e^{1/z^2}}{ze^{1/z}}.$$

I'll leave this part to you.

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As JavaMan and Don mentioned, I made several mistakes. Let me carefully study your replies. Also, Thank Tao; I am also trying to understand it. –  Sam Oct 29 '12 at 17:28
Thank JavaMan for your helpful explanation. –  Sam Oct 29 '12 at 21:31


text:
anonymous one year ago Water's heat of fusion is 80. cal/g , and its specific heat is 1.0calg⋅∘C .Some velomobile seats have been designed to hold ice packs inside their cushions. If you started a ride with ice packs that held 1400 g of frozen water at 0 ∘C , and the temperature of the water at the end of the ride was 32 ∘C , how many calories of heat energy were absorbed?

the requested amount $$\large Q_T$$ of heat is given by the heat needed to melting the ice, which is: $\Large {Q_1} = 80 \cdot 1400 = ...cal$ plus the heat needde to raise the temperature of the resulatant water at 0°C to 32°C, which is: $\Large {Q_2} = 1 \cdot 1400 \cdot \left( {32 - 0} \right) = ...cal$ namely: $\Large {Q_T} = {Q_1} + {Q_2} = ...cal$


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## Meteor Watch Day

Today (the 30th of June) is the annual “meteor watch day”, and because of this I am on live TV this evening talking about meteors. I haven’t been able to find out why today (the last day of June) is designated as “meteor watch day”, but as the day seems to be American in origin I’d normally suspect that the greeting card industry were behind it! But, in this case, I cannot see too many cards saying “happy meteor watch day” being sold, so maybe on this occasion it is not an invented holiday by the US greeting card industry 😉

## What is a meteor?

I have discussed meteors several times before, for example here and here. But, just so this background information is all in one place, I’ll repeat myself. A meteor is simply a bit of (natural) space debris which enters into the Earth’s atmosphere. Before it enters the Earth’s atmosphere it is called a “meteoroid” (or, a really big one would be called an “asteroid”), but upon entering the atmosphere a meteoroid becomes a meteor.

The reason a meteor appears bright is because it is burning up in the Earth’s atmosphere due to friction. Most meteors are very small, no larger than grains of sand, but about once a day something the size of a basketball enters the atmosphere, and about once a week something about the size of a car. Larger and larger meteoroids are less and less common, so for example something the size of the meteor which exploded over Chelyabinsk in southern Russia last year (which is thought to have had a size of 17-20 metres as it entered the Earth’s atmosphere) probably enters the Earth’s atmosphere once every 50 or so years. The larger the meteor, the longer it will take to burn up, so most meteors (the sand-grain sized ones) burn up in less than a second.

A “meteorite” is a meteor which makes it to the ground, or at least of which fragments make it to the ground. So, for example, the meteor which exploded in the atmosphere over Chelyabinsk in February of last year (2013) led to several fragments landing (as I discussed <a href="“>here), and these fragments are meteorites.

Not surprisingly, it is only the larger meteors which make it to the ground as meteorites, although precisely what size a meteor needs to be to make it to the ground as a meteorite depends on a number of things including the composition of the meteor, the angle at which it enters the Earth’s atmosphere and the speed at which it enters the Earth’s atmosphere.

As I’ve also mentioned before, there is now strong evidence that it was the impact of an asteroid (remember that is just the name we give to large meteoroids) which wiped out the dinosaurs and most of life on Earth some 70 million years ago. Although the statistics are very paltry and therefore unreliable, we believe the Earth gets hit by an asteroid large enough to cause a mass-extinction event about once every 150-250 million years. Although undoubtedly rare, such events are cataclysmic to life on Earth, and so it is not wasted money to spend time and resources looking for such large “near Earth” objects.

## Meteor showers

There are many meteor showers during the course of each year, they occur when the Earth passes through some debris in its annual orbit about the Sun. This debris is often the material blown off of comets. Some of the better known meteor showers are

Some well-known meteor showers
Name of shower Month in which it occurs
Lyrids late April
Perseids mid August
Orionids late October
Leonids mid November
Geminids mid December

The naming convention is quite simple, they are named after the constellation from where the meteors appear to emanate. So, for example, the Perseids meteor shower in mid-August appears to radiate from the Perseus constellation, as this figure shows.

The Perseids meteor shower in mid-August appears to radiate from the Perseus constellation

The best way to see a meteor shower is not to use a telescope or binoculars, but rather to just look up with your eyes. Although when traced back they appear to emanate from a particular part of the sky, they can appear anywhere and using a telescope or binoculars will restrict your field of view, leading to your possibly missing a meteor.

Lying on one’s back on the ground (or a rug or mat preferably; or reclining in e.g. a deck chair are very effective ways to view a meteor shower. But, be warned that if you are observing one in the winter months it will probably get pretty cold, so have some warm clothes and blankets with you, and a warm drink.

Also, meteor showers are best viewed after midnight. This is because after midnight your particular part of the Earth is facing in the direction of the Earth’s motion about the Sun, and so the meteoroids enter the atmosphere at a steeper angle with a higher speed, and are more numerous than earlier in the night. The image below shows a wonderful image of the Geminids meteor shower, taken in 2012.

A wonderful image of the Geminids meteor shower, which occurs each December.

## Fireballs

If you are very lucky you may see an extremely bright meteor, which is called a “fireball”. I have never seen one, but I have a story of how much luck is involved in seeing one. One November, when I was working at Yerkes Observatory, we arranged a public viewing of the Leonids meteor shower. Several of us working at the Observatory were out for a few hours with members of the public, and we saw several dozen small meteors. But nothing spectacular. My boss had been in his office working all evening, not taking part in our public observing. He lived in George Ellery Hale’s old house (the Observatory’s “Director’s House”), which was all of 2 minutes walk from the Observatory, and at about 9pm he popped home for a break from his work. As he walked the 150 metres or so to the house, he saw a great big fireball streaking across the sky, and the rest of us all missed it as we were inside warming up with a hot drink!

A fireball streaking across the sky

The Perseids meteor shower is in mid-August, and is one of the most popular meteor showers as it occurs during the Northern Hemisphere summer when many people are on their summer holidays and away from city lights. If you get a chance to observe it this year, I highly recommend it, but bear in mind that you do need to be in a dark place. Most meteors are quite faint, and you will miss all but the brightest ones if you are in a city.

Which is the most spectacular meteor shower you have seen? Have you ever seen a fireball?

## Is Suarez’s ban sufficient?

I’m not sure how many of you saw it, but in the World Cup group match between Uruguay and Italy, the Uruguay striker Luis Suarez bit the Italian defender Giorgio Chiellini. This is the third time that this undoubtedly talented player has bitten another player! He did it in 2010 whilst playing for Ajax in the Netherlands when he bit PSV Eindhoven’s Otman Bakkal; and in 2013 he bit Branislav Ivanovic of Chelsea whilst playing for his current team, Liverpool. For his bite in 2010 be served a 7-match ban, and in 2013 he served a 10-match ban.

FIFA acted very quickly over this incident, and Suarez has been given a 4-month ban from all involvement in football, including training with any team or even attending a football match. He has also been given a 9-match ban from playing for Uruguay, and his 4-month ban will mean he will miss the first ten-or-so matches of the club season.

Luis Suarez has been given a four month ban for biting Italian defender Giorgio Chiellini.

My question is, is this ban sufficient? In my opinion the answer is no, it is not. If this were Suarez’s first biting offence then I would consider the punishment about right. Even if it were his second offence, I would think the ban was on the lenient side. But, as this is his third biting offence, I really feel FIFA’s ban is not sufficient. I think Suarez should serve a season ban at the very least; as the new season has not yet started it would isolate him for the best part of 13-14 months. I also think that FIFA should have insisted that Suarez undergoes some kind of psychological assessment and therapy, it is not normal for a 27-year old to bite people. It is the kind of behaviour you might expect from a 5-year old, but not a grown adult.

It also seems that Liverpool are going to off-load this troublesome striker, despite his phenomenal form this last season where he was Liverpool’s and the English Premier League’s top striker and won several footballing awards. Let me remind you, Suarez also served an 8-match ban in the 2011-12 for racially abusing Manchester United’s Patrice Evra using the “n-word”. At the moment, Barcelona have said they wish to buy him, and I would imagine the Liverpool manager Brendan Rogers gave Suarez a final warning after his last biting incident, so I fully expect that he won’t be playing in the English Premiership come next season.

Whether Suarez ever will be able to change his ways remains to be seen. I don’t think he’ll be able to do it without some kind of therapy, but surely he must realise that he is within a hair’s breadth of ruining his career if he keeps up this unacceptable behaviour.

What do you think? Is the FIFA ban too lenient, or about right, or an over-reaction?

## “You’ve Got a Friend” (Carole King – song)

Last week, I was going to post this wonderful song “You’ve Got a Friend” by Carole King, but for no particular reason decided on Thursday to switch it with “Nothing Compares 2 U”. Ironic then that I should wake up on Friday to hear of the sad death of her one-time husband and songwriting partner Gerry Goffin. Together they wrote some well-known songs like “The Loco-Motion”, “Up On The Roof”, “Pleasant Valley Sunday” and “(You Make Me Feel) Like A Natural Woman”. For a complete list of songs they wrote together, see this list here.

By 1968, King and Goffin were divorced, and this led to King starting her own solo singer-songwriter career. In 1971, she released her 2nd solo album, “Tapestry”, which has gone on to become one of the best-selling albums of all time. The album is replete with fantastic songs, and if you haven’t heard this album I highly recommend it.

“You’ve Got a Friend” is taken from King’s hugely successful album “Tapestry”, released in 1971.

King never released her version of “You’ve Got a Friend” as a single, but James Taylor recorded it at pretty much the same time as she was recording “Tapestry” (they actually worked on their respective albums together), and Taylor’s version got to number 1 in the US and number 4 in the Disunited Kingdom.

When you’re down in troubles
And you need some love and care
And nothing, nothing is going right
Close your eyes and think of me
And soon i will be there
To brighten up even your darkest night

You just call out my name
And you know wherever I am
I’ll come running to see you again
Winter, spring, summer or fall
All you got to do is call
And I’ll be there
Yes I will
You’ve got a friend

If the sky above you
Grows dark and full of clouds
And that old north wind begins to blow
And call my name out loud
Soon you’ll hear me knocking at your door

You just call out my name
And you know wherever I am
I’ll come running to see you again
Winter, spring, summer or fall
All you have to do is call
And I’ll be there
Ain’t it good to know that you’ve got a friend

When people can be so cold
They’ll hurt you and desert you
And take your soul if you let them
Oh, but don’t you let them

You just call out my name
And you know wherever I am
I’ll come running to see you again
Winter, spring, summer or fall
All you have to do is call
And I’ll be there
You’ve got a friend

Enjoy!

Which is your favourite Carole King or King/Goffin song?

## What does a 1-sigma, a 3-sigma or a 5-sigma detection mean?

A few days ago, I blogged about the controversy over the BICEP2 result, and the possibility that their measured signal may actually be dominated by contamination from foreground Galactic dust. As Peter Coles’ blog mentions, their paper has now been published in Physical Review Letters. In the abstract to their paper, the BICEP2 team say

Cross correlating BICEP2 against 100 GHz maps from the BICEP1 experiment, the excess signal is confirmed with $3 \sigma$ significance and its spectral index is found to be consistent with that of the CMB, disfavoring dust at $1.7 \sigma$.

What does a phrase like “with $3 \sigma$ significance” actually mean? It is the significance with which scientists believe a result to be real as opposed to a random fluctuation in the background signal (the noise). In order to fully understand why scientists quote results to a particular $\sigma$, and what it means in detail, the first step is to understand something called the normal distribution.

You can read more about the BICEP2 result, and how its conclusions were withdrawn, in my book “The Cosmic Microwave Background – How it Changed Our Understanding of the Universe”. Follow this link for more details.

## The Normal Distribution

If you have a large number of independent measurements, then their distribution will tend towards something called the normal distribution. This distribution looks like the following, where on the x-axis we have some variable (such as the the background noise in a signal), and the y-axis represents the frequency with which that variable occurs. Normal distributions are usually normalised so that the total probability (the area under the curve) is unity (1), as the sum of all probabilities is always equal to one. The curve is often referred to as a bell curve for obvious reasons.

The normal distribution looks life a “bell curve”. In this plot, the x-axis represents the variable being measured, the y-axis is the frequency with which that variable occurs.

The mathematical formula for the normal distribution is given by something called the Gaussian function (and so another name for a normal distribution is a “Gaussian distribution”) and has the form
$f(x,\mu,\sigma) = \frac{ 1 }{ \sigma \sqrt{ 2 \pi} } e^{ - \frac{ (x - \mu)^{2} }{ 2 \sigma^{2} } }$

where $x$ is the variable, $\mu$ is the mean of the distribution, and $\sigma$ is the standard deviation of the distribution. Usually in statistics we have a mean, a median and a mode, but for a normal distribution they are all equal. The standard deviation is related to the width of the curve. For example, in the figure below we show four normal distributions. The blue, red and orange curves all have the same mean (zero), but different standard deviations, which is related to the curve’s width (the diagram actually quotes the variance, which is just the square of the standard deviation). The green curve has a mean of -2 not 0, and it has a different standard deviation to the other three.

Four different normal distributions. The blue, red and orange curves have the same mean (zero) but different standard deviations. The green curve has a mean of -2 and a different standard deviation from the other three.

As can be seen from these diagrams, if the total probability under each curve is unity, then the probability of a value being measured depends on what the mean is and what the standard deviation is. The further a measurement is from the mean (i.e. towards either end of the bell curve), the less and less likely it is of being measured at random, or to put it another way the less and less likely the signal is of being due to a fluctuation in the background.

## So what does a 3-sigma result mean?

We can work out the probability of a particular measurement once we know the mean and the standard deviation of a normal distribution. There are tables to do this, they give the area under the normal distribution function (which remember is related to probability) in terms of a parameter usually written as $Z$. Here is an example of such a table.

A table of probabilities for a normal distribution.

How do we use this table? The first thing to notice is that the normal distribution is symmetrical about the mean, so the probability from $-\infty$ up to the value of the mean is 0.5.

Suppose we have a normal distribution with a mean of $\mu = 2$ and a standard deviation of $\sigma = 0.5$. How would we use this table to calculate the probability of a value greater or equal to e.g. $3$ being real? (that is, any value greater and including 3).

The definition of $Z$ is

$Z = \frac{ | x - \mu | }{ \sigma }$

where the modulus in the numerator is so that $Z$ is always positive. With our example, $Z = (3 - 2)/0.5 = 2.0$. So, finding $Z = 2.0$ in the table gives the cumulative probability $P(Z)$ of the value $x$ being between $-\infty$ and $2$ being $P(Z=2) = 0.9772$. So the probability of a value of $x$ from $-\infty \text{ to } 3 \text{ is } 0.9772 \text{ or } 97.72 \%$.

If we are trying to work out the probability of measuring a value of $x > 3$ then we need to remember that the total probability is 1, so the probability of the value of $x > 3 \text{ is } 1-0.9772 = 0.0228$ or $2.28 \%$. Obviously, with our chosen value of $\sigma = 0.5$, a value of $x=3$ is 2-sigma away from the mean ($Z=2$), so a result quoted as a $2 \sigma$ result (or confidence) means that it has a $2.28 \%$ of being false, and a $97.72 \%$ of being real.

What would we get if we had chosen a value of 1-sigma from the mean, or in other words a value of $x = 2.5$? In this case, $Z = (2.5 - 2)/0.5 = 1$, and so using our table we find $P(Z) = 0.8413$. So the probability of $x$ being equal to or greater than 2.5 is $1 - 0.8413 = 0.1587$ or $15.87\%$. As you can see, a $84.13\%$ chance of a result being real (or a $15.87 \%$ chance of a result being false) is not very good, which is why a $1 \sigma$ detection of a signal is not usually considered good enough to be believed.

What would we get if we had chosen a value of 3-sigma from the mean, or in other words a value of $x = 3.5$? In this case, $Z = (3.5 - 2)/0.5 = 3$, and so using our table we find $P(Z=3) = 0.9987$, so the probability of obtaining a value of equal to or greater than 3.5 is $1 - 0.9987 = 0.0013$ or $0.13\%$. So, when we say that a detection is made at the 3-sigma level, what we are saying is that it is $99.87\%$ certain, or that it has just a $0.13\%$ probability of being false.

Usually in science, a 3-sigma detection is taken as being the minimum to be believed, and quite often 5-sigma is chosen, which is essentially $0\%$ probability of the result being false.

## Summary

The figure below summarises this graphically.

The probabilities of a value lying within 1-sigma, 2-sigma and 3-sigma of the mean for a normal distribution.

To translate between this figure and what we have calculated above, just note that the percentages to the left of the mean all add up to $50\%$, so if we wanted to work out the chance of a result being greater than $1\sigma$ above the mean we would work out $100\% - (50\% + 34.1\%) = 15.9\%$, just as we had above. For $3 \sigma$ we have $100\% - (50\% + 34.1\% + 13.6\% + 2.1\%) = 0.2\%$ (we got $0.13\%$ before, the difference is due to rounding).

And, here is a table summarising the significances, to two decimal places.

The significance of various levels of $\sigma$
$\sigma$ Confidence that result is real
$1 \sigma$ 84.13%
$1.5 \sigma$ 93.32%
$2 \sigma$ 97.73%
$2.5 \sigma$ 99.38%
$3 \sigma$ 99.87%
$3.5 \sigma$ 99.98%
$> 4 \sigma$ 100%

So, going back to the BICEP2 result, they state in their paper that their signal is in excess of the background (noise) signal by $3 \sigma$, which would mean that their signal is real with a $99.87\%$ certainty. But, of course, although there seems to be little doubt that their signal is real, what is still undecided and hotly disputed is whether the signal is nearly entirely due to the CMB or could be mainly due to foreground Galactic dust. We shall have to wait to find out the answer to that question!

## ***UPDATE***

In February 2015 the BICEP2 team withdrew their claim for having discovered primordial B-mode polarisation, and accepted that their detection was of Galactic dust. You can read far more about this fascinating story in my book “The Cosmic Microwave Background – How it Changed Our Understanding of the Universe”.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer. Follow this link for more details.

## The 10 best Bob Dylan songs – no. 8 – “Mr. Tambourine Man” (song)

At number 8 in Rolling Stone Magazine’s 10 best Bob Dylan songs is “Mr. Tambourine Man”, which was on his 1965 album “Bringing It All Back Home”. The song itself was actually written in 1964, and when later recorded by The Byrds, it became a massive hit reaching number 1 in both the US and the Disunited Kingdom. It also brought Bob Dylan’s music to a new audience, once people found out that he was the original composer; and led to the birth of so-called “folk rock”, with The Byrds’ version of the song being an electric version of Dylan’s original acoustic version. It may also have helped influence Dylan to “go electric” himself, although it is also fair to say he was moving in that direction anyway, even before The Byrds’ version of his song.

At number 8 in Rolling Stone Magazine’s 10 greatest Bob Dylan songs is “Mr. Tambourine Man”. The comments accompanying the song are written by David Crosby of The Byrds.

I first heard this song when I was about 15 or 16, and one of the things which struck me most forcibly was the complex internal rhyming; something which reminded me of much of the Welsh poetry I had read, particularly poetry written in cynghanedd (which I explain here). The song is obviously about drugs, and the surrealistic imagery in this song became a hallmark of Dylan’s songwriting during this 1964-66 period of his career.

Hey! Mr. Tambourine Man, play a song for me
I’m not sleepy and there is no place I’m going to
Hey! Mr. Tambourine Man, play a song for me
In the jingle jangle morning I’ll come followin’ you

Though I know that evenin’s empire has returned into sand
Vanished from my hand
Left me blindly here to stand but still not sleeping
My weariness amazes me, I’m branded on my feet
I have no one to meet
And the ancient empty street’s too dead for dreaming

Hey! Mr. Tambourine Man, play a song for me
I’m not sleepy and there is no place I’m going to
Hey! Mr. Tambourine Man, play a song for me
In the jingle jangle morning I’ll come followin’ you

Take me on a trip upon your magic swirlin’ ship
My senses have been stripped, my hands can’t feel to grip
My toes too numb to step
Wait only for my boot heels to be wanderin’
I promise to go under it

Hey! Mr. Tambourine Man, play a song for me
I’m not sleepy and there is no place I’m going to
Hey! Mr. Tambourine Man, play a song for me
In the jingle jangle morning I’ll come followin’ you

Though you might hear laughin’, spinnin’, swingin’ madly across the sun
It’s not aimed at anyone, it’s just escapin’ on the run
And but for the sky there are no fences facin’
And if you hear vague traces of skippin’ reels of rhyme
To your tambourine in time, it’s just a ragged clown behind
I wouldn’t pay it any mind
It’s just a shadow you’re seein’ that he’s chasing

Hey! Mr. Tambourine Man, play a song for me
I’m not sleepy and there is no place I’m going to
Hey! Mr. Tambourine Man, play a song for me
In the jingle jangle morning I’ll come followin’ you

Then take me disappearin’ through the smoke rings of my mind
Down the foggy ruins of time, far past the frozen leaves
The haunted, frightened trees, out to the windy beach
Far from the twisted reach of crazy sorrow
Yes, to dance beneath the diamond sky with one hand waving free
Silhouetted by the sea, circled by the circus sands
With all memory and fate driven deep beneath the waves
Let me forget about today until tomorrow

Hey! Mr. Tambourine Man, play a song for me
I’m not sleepy and there is no place I’m going to
Hey! Mr. Tambourine Man, play a song for me
In the jingle jangle morning I’ll come followin’ you

This song is one of my favourite Dylan songs, and is on one of my favourite albums of his. I highly recommend listening to “Brining It All Back Home”, it is one of the great albums of the 1960s, and has been referred to as one of the most influential albums of its era. There is a great variety of songs on “Brining It All Back Home”, from the “early-rap” “Subterranean Homesick Blues” to the apocalyptic “It’s Alright Ma (I’m Only Bleeding)” to the beautifully cutting “It’s All Over Now, Baby Blue”.

For once, I have managed to find a wonderful version of this song, so difficult to do with most of Dylan’s songs which get removed from YouTube. This version is from the Newport Folk Festival in 1964, many months before the song was released in its recorded version. Enjoy!

## Is the BICEP2 result correct?

I blogged back in March about the announcement by a team of cosmologists that they had discovered evidence for gravitational waves in the Cosmic Microwave Background (CMB). The BICEP2 experiment, which is based at the South Pole, claimed to have detected the so-called “B-mode polarisation” in the CMB, and from the strength of the signal they argued that it was the best evidence yet of both gravitational waves in the very early Universe, and of the theory of Cosmic Inflation.

However, since that announcement there has been considerable controversy in the cosmology community as to whether their result is correct or not. I have re-blogged several other people’s blogs on this controversy, for example Peter Coles’ blog here and Matt Strassler here and here. As Peter and Matt’s blogs indicated, this controversy has been swirling around in the astronomical community for the last several months; but last Thursday (the 19th of June) it made it into the New York Times.

There now seems to be considerable doubt in the cosmological community about the BICEP2 result which was announced in March.

The main concern amongst the skeptics is that the BICEP2 team did not correctly subtract the effects of dust in our own Galaxy from their signal. Our Milky Way has a lot of dust in it, it is this dust which causes the dark clouds in the band of the Milky Way which are familiar to anyone who has looked at the Milky Way in any detail, even with the naked eye. Most of the dust is in the plane of the disk, but some is above and below the plane in what we refer to as “high Galactic latitudes”. The BICEP2 team chose their patch of sky to be well below the plane of the Milky Way to try to minimise the effects of dust.

However, it may be that the amount of dust and its degree of polarisation where BICEP2 made their observations is greater than the BICEP2 team thought. If this is the case, then much of the polarised signal that BICEP2 measured may not be due to primordial gravitational waves, but instead may be dominated by this foreground contamination. As the New York Times story states, the BICEP2 team acknowledge that the foreground contamination may be greater than they assumed, but they are sticking to their claim that it is still small compared to the signal they detected.

We should know the answer to this burning issue within the next 6 to 12 months. The Planck satellite has done a detailed all-sky map of the strength of the polarised emission from dust in our Milky Way, far more detailed than any data currently available, and when these maps are released it should allow astronomers to correctly determine how much of BICEP2’s signal is due to foreground contamination. Planck will also do this at several different frequencies, and as Galactic dust is much warmer than the CMB the ratio of its signal at different frequencies will be different to that of the CMB, allowing for much better separation of the two effects.

In the meantime, we have a great insight into how science really works. Any result in science is closely scrutinised by the community, and is not accepted as being real until (a) it has been confirmed by other experiments and (b) that the community is satisfied that the interpretation of the measurement is correct, and that all other possibilities have been considered. As Carl Sagan once said

extraordinary claims require extraordinary evidence

So far, I think it is fair to say, most people in the astronomical and cosmological communities are treating the BICEP2 result with a good deal of caution, and that caution can only be allayed by further analysis and measurements.

If you wish to read more about the BICEP2 results and the surrounding controversy, an excellent place to start is Peter Coles’ blog here.

## Heartbreak for Wales in South Africa

On Saturday (21st of June), Wales came within three minutes of a remarkable first ever win over South Africa on South African soil. With the final whistle looming, Wales were leading South Africa by 30 points to 24; but one thing Wales have learnt about Southern Hemisphere teams is that they refuse to accept that they are beaten until the final whistle. In the dying moments of this game, in which Wales had never been behind, South Africa dramatically were awarded a penalty try when Wales’ full-back Liam Williams was judged to have illegally prevented South Africa wing Cornal Hendricks from scoring. It led to South Africa winning the game by the smallest of margins, clinching the match by 31 to 30 points.

Wales lost to South Africa in the dying moments of the 2nd Test with SA being awarded a penalty try to clinch the game by one point, 31-30.

After the poor performance in the 1st Test, this match was a dramatic turn around in Wales’ fortunes. As my blog on Saturday suggested, I did not hold out much hope for anything but another heavy defeat. How wrong I was!

Again, like for the 1st Test, I was in Lusaka in Zambia, on the last day of my visit there. After following the first 20 or so minutes on Twitter I decided to see if I could listen to the Radio Cymru live commentary using the mobile phone network, and much to my surprise it worked! So, I was able to follow all but the first 20 minutes of this dramatic Test live, thanks to the Radio Cymru commentary. What a lovely surprise this was, and it meant that I really was able to be engaged in this dramatic match at it unfolded.

From everything I’ve read since getting back, the Wales team were as good in every department of play as they had been poor in the 1st Test. Gatland has said that this defeat is the most painful since he took over coaching Wales in January 2008, and I can believe him. All I can hope is that we learn from this latest agonising defeat, and it strengthens Wales’ resolve to start winning these close games.

This is the fifth or sixth time since about 2011 that Wales have been in a position to beat either South Africa or Australia, but each time Wales have not been able to clinch the match. This is where Southern Hemisphere teams excel over Wales; even when we are within a few points in the last few minutes, they always seem to find a way to clinch victory in the dying moments. It is a situation that Wales desperately need to change, we need to start learning to win these close contests otherwise we will forever be also-rans.

Wales now have a 5 month break before we take on the three Southern Hemisphere giants in Cardiff in November. Certainly the spirit of the Welsh camp will be more positive going into these Autumn games than had we been wiped off the field in the 2nd Test in the same way in which we were in the 1st Test, but it is imperative for our chances in the 2015 World Cup that we get some wins during these Autumn tests. In the meantime, the Welsh players will have a well earned break after a long season, with many of the Welsh players starting the next season playing their club rugby outside of Wales (particularly in France), another big problem facing Welsh rugby which I will discuss in a future blog.


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1. Tukey's Test for Post-Hoc Analysis

After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc analysis is Tukey's Test. The test is known by several different names. Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal.

Tagged as : R statistics
2. Kruskal-Wallis One-Way Analysis of Variance of Ranks

The Kruskal-Wallis test extends the Mann-Whitney-Wilcoxon Rank Sum test for more than two groups. The test is nonparametric similar to the Mann-Whitney test and as such does not assume the data are normally distributed and can, therefore, be used when the assumption of normality is violated. This example will employ the Kruskal-Wallis test on the PlantGrowth dataset as used in previous examples. Although the data appear to be approximately normally distributed as seen before, the Kruskal-Wallis test performs just as well as a parametric test.

Tagged as : R statistics
3. Quadratic Discriminant Analysis of Several Groups

Quadratic discriminant analysis for classification is a modification of linear discriminant analysis that does not assume equal covariance matrices amongst the groups ($$\Sigma_1, \Sigma_2, \cdots, \Sigma_k$$). Similar to LDA for several groups, quadratic discriminant analysis for several groups classification seeks to find the group that maximizes the quadratic classification function and assign the observation vector $$y$$ to that group.

4. Quadratic Discriminant Analysis of Two Groups

LDA assumes the groups in question have equal covariance matrices ($$\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k$$). Therefore, when the groups do not have equal covariance matrices, observations are frequently assigned to groups with large variances on the diagonal of its corresponding covariance matrix (Rencher, n.d., pp. 321). Quadratic discriminant analysis is a modification of LDA that does not assume equal covariance matrices amongst the groups. In quadratic discriminant analysis, the respective covariance matrix $$S_i$$ of the $$i^{th}$$ group is employed in predicting the group membership of an observation, rather than the pooled covariance matrix $$S_{p1}$$ in linear discriminant analysis.

5. Linear Discriminant Analysis for the Classification of Several Groups

Similar to the two-group linear discriminant analysis for classification case, LDA for classification into several groups seeks to find the mean vector that the new observation $$y$$ is closest to and assign $$y$$ accordingly using a distance function. The several group case also assumes equal covariance matrices amongst the groups ($$\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k$$).

6. Linear Discriminant Analysis for the Classification of Two Groups

In this post, we will use the discriminant functions found in the first post to classify the observations. We will also employ cross-validation on the predicted groups to get a realistic sense of how the model would perform in practice on new observations. Linear classification analysis assumes the populations have equal covariance matrices ($$\Sigma_1 = \Sigma_2$$) but does not assume the data are normally distributed.

7. Discriminant Analysis for Group Separation

Discriminant analysis assumes the two samples or populations being compared have the same covariance matrix $$\Sigma$$ but distinct mean vectors $$\mu_1$$ and $$\mu_2$$ with $$p$$ variables. The discriminant function that maximizes the separation of the groups is the linear combination of the $$p$$ variables. The linear combination denoted $$z = a′y$$ transforms the observation vectors to a scalar. The discriminant functions thus take the form:

8. Discriminant Analysis of Several Groups

Discriminant analysis is also applicable in the case of more than two groups. In the first post on discriminant analysis, there was only one linear discriminant function as the number of linear discriminant functions is $$s = min(p, k − 1)$$, where $$p$$ is the number of dependent variables and $$k$$ is the number of groups. In the case of more than two groups, there will be more than one linear discriminant function, which allows us to examine the groups' separation in more than one dimension.

9. Introduction to Rpoet

The Rpoet package is a wrapper of the PoetryDB API, which enables developers and other users to extract a vast amount of English-language poetry from nearly 130 authors. The package provides a simple R interface for interacting and accessing the PoetryDB database. This vignette will introduce the basic functionality of Rpoet and some example usages of the package.

Tagged as : R APIs poetry
10. Calculating and Performing One-way Multivariate Analysis of Variance (MANOVA)

MANOVA, or Multiple Analysis of Variance, is an extension of Analysis of Variance (ANOVA) to several dependent variables. The approach to MANOVA is similar to ANOVA in many regards and requires the same assumptions (normally distributed dependent variables with equal covariance matrices).

Tagged as : R statistics
11. Calculating and Performing One-way Analysis of Variance (ANOVA)

ANOVA, or Analysis of Variance, is a commonly used approach to testing a hypothesis when dealing with two or more groups. One-way ANOVA, which is what will be explored in this post, can be considered an extension of the t-test when more than two groups are being tested. The factor, or categorical variable, is often referred to as the 'treatment' in the ANOVA setting. ANOVA involves partitioning the data's total variation into variation between and within groups. This procedure is thus known as Analysis of Variance as sources of variation are examined separately.

Tagged as : R statistics
12. Computing Working-Hotelling and Bonferroni Simultaneous Confidence Intervals

There are two procedures for forming simultaneous confidence intervals, the Working-Hotelling and Bonferroni procedures. Each estimates intervals of the mean response using a family confidence coefficient. The Working-Hotelling coefficient is defined by $$W$$ and Bonferroni $$B$$. In practice, it is recommended to perform both procedures to determine which results in a tighter interval. The Bonferroni method will be explored first.

Tagged as : R statistics
13. Predicting Cat Genders with Logistic Regression

Consider a data set of 144 observations of household cats. The data contains the cats' gender, body weight and height. Can we model and accurately predict the gender of a cat based on previously observed values using logistic regression?

Tagged as : R statistics
14. PetfindeR, R Wrapper for the Petfinder API, Introduction Part Two

The first post introduced and explored the basic usage of the PetfindeR library. In this post, we take a quick look at some of the additional uses of the library and its methods to extract data from the Petfinder database.

Tagged as : R PetfindeR APIs
15. PetfindeR, R Wrapper for the Petfinder API, Introduction Part One

The goal of the PetfindeR package is to provide a simple and straightforward interface for interacting with the Petfinder API through R. The Petfinder database contains approximately 300,000 adoptable pet records and 11,000 animal welfare organization records, which makes it a handy and valuable source of data for those in the animal welfare community. However, the outputs from the Petfinder API are in messy JSON format and thus it makes it more time-consuming and often frustrating to coerce the output data into a form that is workable with R.

Tagged as : R PetfindeR APIs
16. Analyzing Nationwide Utility Rates with R, SQL and Plotly

R and SQL make excellent complements for analyzing data due to their respective strengths. The sqldf package provides an interface for working with SQL in R by querying data from a database into an R data.frame. This post will demonstrate how to query and analyze data using the sqldf package in conjunction with the graphing libraries plotly and ggplot2 as well as some other packages that provide useful statistical tests and other functions.

Tagged as : R Plotly SQL
17. Cartesian Product and Ordered and Unordered Pairs

A pair set is a set with two members, for example, {2, 3}, which can also be thought of as an unordered pair, in that {2, 3}={3, 2}. However, we seek a more a strict and rich object that tells us more about two sets and how their elements are ordered.

Tagged as : R set theory
18. Algebra of Sets with R

The set operations, union and intersection, the relative complement − and the inclusion relation (subsets) are known as the algebra of sets. The algebra of sets can be used to find many identities related to set relations.

Tagged as : R set theory
19. N-Union and Intersection Set Operations

Set unions and intersections can be extended to any number of sets. This post introduces notation to simplify the expression of n-sets and the set union and intersection operations themselves with R.

Tagged as : R set theory
20. Combined Linear Congruential Generator for Pseudo-random Number Generation

Combined linear congruential generators, as the name implies, are a type of PRNG (pseudorandom number generator) that combine two or more LCGs (linear congruential generators). The combination of two or more LCGs into one random number generator can result in a marked increase in the period length of the generator which makes them better suited for simulating more complex systems.

21. Multiplicative Congruential Random Number Generators with R

Multiplicative congruential generators, also known as Lehmer random number generators, is a type of linear congruential generator for generating uniform pseudorandom numbers. The multiplicative congruential generator, often abbreviated as MLCG or MCG, is defined as a recurrence relation similar to the LCG.

22. Linear Congruential Generator for Pseudo-random Number Generation with R

Linear congruential generators (LCGs) are a class of pseudorandom number generator (PRNG) algorithms used for generating sequences of random-like numbers. The generation of random numbers plays a large role in many applications ranging from cryptography to Monte Carlo methods. Linear congruential generators are one of the oldest and most well-known methods for generating random numbers primarily due to their comparative ease of implementation and speed and their need for little memory.

23. Set Union and Intersections with R

The set operations 'union' and 'intersection' should ring a bell for those who've worked with relational databases and Venn Diagrams. The 'union' of two of sets A and B represents a set that comprises all members of A and B (or both).

Tagged as : R set theory
24. Introduction to Sets and Set Theory with R

Sets define a 'collection' of objects, or things typically referred to as 'elements' or 'members.' The concept of sets arises naturally when dealing with any collection of objects, whether it be a group of numbers or anything else.

Tagged as : R set theory
25. Hierarchical Clustering Nearest Neighbors Algorithm in R

Hierarchical clustering is a widely used and popular tool in statistics

Tagged as : R clustering statistics
26. Factor Analysis with the Iterated Factor Method and R

The iterated principal factor method is an extension of the principal

27. Factor Analysis with Principal Factor Method and R

As discussed in a previous post on the principal component method of factor analysis, the $$\hat{\Psi}$$ term in the estimated covariance matrix $$S$$, $$S = \hat{\Lambda} \hat{\Lambda}' + \hat{\Psi}$$, was excluded and we proceeded directly to factoring $$S$$ and $$R$$. The principal factor method of factor analysis (also called the principal axis method) finds an initial estimate of $$\hat{\Psi}$$ and factors $$S - \hat{\Psi}$$, or $$R - \hat{\Psi}$$ for the correlation matrix.

Tagged as : R factor analysis
28. Factor Analysis with the Principal Component Method and R Part Two

In the first post on factor analysis, we examined computing the estimated covariance matrix $$S$$ of the rootstock data and proceeded to find two factors that fit most of the variance of the data. However, the variables in the data are not on the same scale of measurement, which can cause variables with comparatively large variances to dominate the diagonal of the covariance matrix and the resulting factors. The correlation matrix, therefore, makes more intuitive sense to employ in factor analysis.

29. Factor Analysis with the Principal Component Method and R

The goal of factor analysis, similar to principal component analysis, is to reduce the original variables into a smaller number of factors that allows for easier interpretation. PCA and factor analysis still defer in several respects. One difference is principal components are defined as linear combinations of the variables while factors are defined as linear combinations of the underlying latent variables.

30. Principal Component Analysis with R Example

Often, it is not helpful or informative to only look at all the variables in a dataset for correlations or covariances. A preferable approach is to derive new variables from the original variables that preserve most of the information given by their variances. Principal component analysis is a widely used and popular statistical method for reducing data with many dimensions (variables) by projecting the data with fewer dimensions using linear combinations of the variables, known as principal components.

31. Image Compression with Singular Value Decomposition

The method of image compression with singular value decomposition is

32. Singular Value Decomposition and R Example

SVD underpins many statistical and real-world

33. How to Calculate the Inverse Matrix for 2×2 and 3×3 Matrices

The inverse of a number is its reciprocal. For example, the inverse of 8

Tagged as : R linear algebra matrices

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# Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Learning in the Big Data Regime

Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used over an i.i.d dataset for gradient updates. Our results complement those of [RRT17] and improve on those of [GGZ18].

## Authors

• 1 publication
• 5 publications
• ### Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization: Non-Asymptotic Performance Bounds and Momentum-Based Acceleration

Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stoc...
09/12/2018 ∙ by Xuefeng Gao, et al. ∙ 0

• ### Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization

We provide a nonasymptotic analysis of the convergence of the stochastic...
02/13/2020 ∙ by Ömer Deniz Akyıldız, et al. ∙ 0

• ### Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning

12/22/2020 ∙ by Andrew Lamperski, et al. ∙ 0

• ### Analysis of Langevin Monte Carlo via convex optimization

In this paper, we provide new insights on the Unadjusted Langevin Algori...
02/26/2018 ∙ by Alain Durmus, et al. ∙ 0

• ### Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization

Recent studies on diffusion-based sampling methods have shown that Lange...
01/22/2019 ∙ by Thanh Huy Nguyen, et al. ∙ 0

• ### Train faster, generalize better: Stability of stochastic gradient descent

We show that parametric models trained by a stochastic gradient method (...
09/03/2015 ∙ by Moritz Hardt, et al. ∙ 0

• ### Decentralized Bayesian Learning with Metropolis-Adjusted Hamiltonian Monte Carlo

Federated learning performed by a decentralized networks of agents is be...
07/15/2021 ∙ by Vyacheslav Kungurtsev, et al. ∙ 0

##### This week in AI

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## 1 Introduction

Let

be a probability space where all the random objects of this paper will be defined. The expectation of a random variable

with values in a Euclidean space will be denoted by .

We consider the following optimization problem

 F∗:=minx∈RdF(x), where F(x):=E[f(x,Z)]=∫Zf(x,z)μ(dx), x∈Rd (1)

and is a random element in some measurable space with an unknown probability law . The function is assumed continuously differentiable (for each ) but it can possibly be non-convex. Suppose that one has access to i.i.d samples drawn from , where is fixed. Our goal is to compute an approximate minimizer such that the population risk

 E[F(X†)]−F∗

is minimized, where the expectation is taken with respect to the training data and additional randomness generating .

Since the distribution of is unknown, we consider the empirical risk minimization problem

 minx∈RdFz(x), where Fz(x):=1nn∑i=1f(x,zi) (2)

using the dataset

Stochastic gradient algorithms based on Langevin Monte Carlo have gained more attention in recent years. Two popular algorithms are Stochastic Gradient Langevin Dynamics (SGLD) and Stochastic Gradient Hamiltonian Monte Carlo (SGHMC). First, we summarize the use of SGLD in optimization, as presented in [RRT17]. Consider the overdamped Langevin stochastic differential equation

 dXt=−∇Fz(Xt)dt+√2β−1dBt, (3)

where is the standard Brownian motion in and is the inverse temperature parameter. Under suitable assumptions on , the SDE (3) admits the Gibbs measure as its unique invariant distribution. In addition, it is shown that for sufficiently big , the Gibbs distribution concentrates around global minimizers of . Therefore, one can use the value of from (3), (or from its discretized counterpart SGLD), as an approximate solution to the empirical risk problem, provided that is large and temperature is low.

In this paper, we consider the underdamped (second-order) Langevin diffusion

 dVt = −γVtdt−∇Fz(Xt)dt+√2γβ−1dBt, (4) dXt = Vtdt (5)

where, model the position and the momentum of a particle moving in a field of force with random force given by Gaussian noise. It is shown that under some suitable conditions for , the Markov process is ergodic and has a unique stationary distribution

 πz(dx,dv)=1Γzexp(−β(12∥v∥2+Fz(x)))dxdv

where is the normalizing constant

 Γz=(2πβ)d/2∫Rde−βFz(x)dx.

It is easy to observe that the -marginal distribution of is the invariant distribution of (3). We consider the first order Euler discretization of (4), (5), also called Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), given as follows

 ¯¯¯¯Vλk+1 = ¯¯¯¯Vλk−λ[γ¯¯¯¯Vλk+∇Fz(¯¯¯¯¯Xλk)]+√2γβ−1λξk+1,¯¯¯¯Vλ0=v0, (6) ¯¯¯¯¯Xλk+1 = ¯¯¯¯¯Xλk+λ¯¯¯¯Vλk,¯¯¯¯¯Xλ0=x0, (7)

where is a step size parameter and

is a sequence of i.i.d standard Gaussian random vectors in

. The initial condition may be random, but independent of .

In certain contexts, the full knowledge of the gradient is not available, however, using the dataset

, one can construct its unbiased estimates. In what follows, we adopt the general setting given by

[RRT17]. Let be a measurable space, and such that for any ,

 (8)

where is a random element in with probability law . Conditionally on , the SGHMC algorithm is defined by

 Vλk+1 = Vλk−λ[γVλk+g(Xλk,Uz,k)]+√2γβ−1λξk+1,Vλ0=v0, (9) Xλk+1 = Xλk+λVλk,Xλ0=x0, (10)

where is a sequence of i.i.d. random elements in with law . We also assume from now on that are independent.

Our ultimate goal is to find approximate global minimizers to the problem (1). Let be the output of the algorithm (9),(10) after iterations, and be such that . The excess risk is decomposed as follows, see also [RRT17],

 E[F(X†)]−F∗ = (11)

The remaining part of the present paper is about finding bounds for these errors. Section 2 summarizes technical conditions and the main results. Comparison of our contributions to previous studies is discussed in Section 3. Proofs are given in Section 4.

Notation and conventions. For , scalar product in is denoted by . We use to denote the Euclidean norm (where the dimension of the space may vary). denotes the Borel - field of . For any -valued random variable and for any , let us set . We denote by the set of with . The Wasserstein distance of order between two probability measures and on is defined by

 Wp(μ,ν)=(infπ∈Π(μ,ν)∫Rl∥x−y∥pdπ(x,y))1/p, (12)

where is the set of couplings of , see e.g. [Vil08]. For two -valued random variables and , we denote . We do not indicate in the notation and it may vary.

## 2 Asumptions and main results

The following conditions are required throughout the paper.

###### Assumption 2.1.

The function is continuously differentiable, takes non-negative values, and there are constants such that for any ,

 ∥f(0,z)∥≤A0,∥∇f(0,z)∥≤B.
###### Assumption 2.2.

There is such that, for each ,

 ∥∇f(x1,z)−∇f(x2,z)∥≤M∥x1−x2∥,∀x1,x2∈Rd.
###### Assumption 2.3.

There exist constants such that

 ⟨x,f(x,z)⟩≥m∥x∥2−b,∀x∈Rd,z∈Z.
###### Assumption 2.4.

For each , it holds that

 ∥g(x1,u)−g(x2,u)∥≤M∥x1−x2∥,∀x1,x2∈Rd.
###### Assumption 2.5.

There exists a constant such that for every ,

 E∥g(x,Uz)−∇Fz(x)∥2≤2δ(M2∥x∥2+B2).
###### Assumption 2.6.

The law of the initial state satisfies

 ∫R2deV(x,v)dμ0(x,v)<∞,

where is the Lyapunov function defined in (16) below.

###### Remark 2.7.

If the set of global minimizers is bounded, we can always redefine the function to be quadratic outside a compact set containing the origin while maintaining its minimizers. Hence, Assumption 2.3 can be satisfied in practice. Assumption 2.4 means that the estimated gradient is also Lipschitz when using the same training dataset. For example, at each iteration of SGHMC, we may sample uniformly with replacement a random minibatch of size . Then we can choose where are i.i.d random variables having distribution . The gradient estimate is thus

 g(x,Uz)=1ℓℓ∑j=1∇f(x,zIj),

which is clearly unbiased and Assumption 2.4 will be satisfied whenever Assumption 2.2 is in force. Assumption 2.5

controls the variance of the gradient estimate.

An auxiliary continuous time processes is needed in the subsequent analysis. For a step size , denote by the scaled Brownian motion. Let be the solutions of

 dˆV(t,s,v) = −λ(γˆV(t,s,v)+∇Fz(ˆX(t,s,v)))dt+√2γλβ−1dBλt, (13) dˆX(t,s,v) = λˆV(t,s,v)dt, (14)

with initial condition where may be random but independent of .

Our first result tracks the discrepancy between the SGHMC algorithm (9), (10) and the auxiliary processes (13), (14).

###### Theorem 2.8.

There exists a constant such that for all ,

 W2((Vλk,Xλk),(ˆV(k,0,v0),ˆX(k,0,x0)))≤~C(√λ+√δ). (15)
###### Proof.

The proof of this theorem is given in Section 4.2. ∎

The following is the main result of the paper.

###### Theorem 2.9.

Suppose that the SGHMC iterates are defined by (9), (10). The expected population risk can be bounded as

 E[F(Xλk)]−F∗≤B1+B2+B3,

where

 B1 :=(Mσ+B)(~C(√λ+√δ)+C∗√Wρ(μ0,πz)exp(−c∗kλ)), B2 :=4βcLSn(M2m(b+d/β)+B2), B3 :=d2βlog(eMm(bβd+1)),

where are appropriate constants and is the metric defined in (17) below.

###### Proof.

The proof of this theorem is given in Section 4.3. ∎

###### Corollary 2.10.

Let . We have

 W2(L(Xk),πz)≤ε

whenever

 λ+δ=O(ε2),k=O(1ε2log(1ε)).
###### Proof.

From the proof of Theorem 2.9, or more precisely from (43), we need to choose and such that

 ~C(√λ+√δ)+C∗√Wρ(μ0,πz)exp(−c∗kλ)≤ε.

First, we choose and so that and then

 C∗√Wρ(μ0,πz)exp(−c∗kλ)≤ε/2

will hold for large enough. ∎

## 3 Related work and our contributions

Non-asymptotic convergence rate Langevin dynamics based algorithms for approximate sampling log-concave distributions are intensively studied in recent years. For example, overdamped Langevin dynamics are discussed in [WT11], [Dal17b], [DM16], [DK17], [DM17] and others. Recently, [BCM18] treats the case of non-i.i.d. data streams with a certain mixing property. Underdamped Langevin dynamics are examined in [CFG14], [Nea11], [CCBJ17], etc. Further analysis on HMC are discussed on [BBLG17], [Bet17]. Subsampling methods are applied to speed up HMC for large datasets, see [DQK17], [QKVT18].

The use of momentum to accelerate optimization methods are discussed intensively in literature, for example [AP16]. In particular, performance of SGHMC is experimentally proved better than SGLD in many applications, see [CDC15], [CFG14]. An important advantage of the underdamped SDE is that convergence to its stationary distribution is faster than that of the overdamped SDE in the -Wasserstein distance, as shown in [EGZ17].

Finding an approximate minimizer is similar to sampling distributions concentrate around the true minimizer. This well known connection give rise to the study of simulated annealing algorithms, see [Hwa80], [Gid85], [Haj85], [CHS87], [HKS89], [GM91], [GM93]. Recently, there are many studies further investigate this connection by means of non asymptotic convergence of Langevin based algorithms and in stochastic non-convex optimization and large-scale data analysis, [CCG16], [Dal17a].

Relaxing convexity is a more challenging issue. In [CCAY18], the problem of sampling from a target distribution where is L-smooth everywhere and -strongly convex outside a ball of finite radius. They provide upper bounds for the number of steps to be within a given precision level of the 1-Wasserstein distance between the HMC algorithm and the equilibrium distribution.

Our work continues these lines of research, the most similar setting to ours is the most recent paper [GGZ18]. We summarize our contributions below:

• Diffusion approximation. In Lemma 10 of [GGZ18], the upper bound for the 2-Wasserstein distance between the SGHMC algorithm at step and underdamped SDE at time is (up to constants) given by

 (δ1/4+λ1/4)√kλ√log(kλ),

which depends on the number of iteration . Therefore obtaining a precision requires a careful choice of and even . By introducing the auxiliary SDEs (13, 14), we are able to improve this bound by

 ~C(√λ+√δ),

see Theorem 2.8. This upper bound is better not only in convergence rate for both step size ( vs. ) and variance ( vs. ) but also in the number of iterations. This improves Lemma 10 and hence Theorem 2 of [GGZ18]. Our analysis for variance of the algorithm is also different. The iteration does not accumulate mean squared errors, as the number of step goes to infinity.

• Our proof for Theorem 2.8 is relatively simple and we do not need to adopt the techniques of [RRT17] which involve heavy functional analysis, e.g. the weighted Csiszár - Kullback - Pinsker inequalities in [BV05] is not needed.

• Thanks to the big data regime, dependence structure of the dataset in the sampling mechanism, can be arbitrary, see the proof of Theorem 2.8. The i.i.d assumption on dataset is used only for the generalization error. We could also incorporate non-i.i.d data in our analysis, see Remark 4.5, but this is left for further research.

## 4 Proofs

### 4.1 A contraction result

In this section, we recall a contraction result of [EGZ17]. First, it should be noticed that the constant and the function in their paper are and in the present paper, respectively. Here, the subscript stands for “contraction”. Using the upper bound of Lemma 5.1 for below, there exist constants small enough and such that

 ⟨x,∇Fz(x)⟩≥m∥x∥2−b≥2λc(Fz(x)+γ2∥x∥2/4)−2Ac/β.

Therefore, Assumption 2.1 of [EGZ17] is satisfied, noting that and

 ∥∇Fz(x)−∇Fz(y)∥≤β−1Lc∥x−y∥.

We define the Lyapunov function

 V(x,v)=βFz(x)+β4γ2(∥x+γ−1v∥2+∥γ−1v∥2−λc∥x∥2), (16)

For any , we set

 r((x1,v2),(x2,v2) = αc∥x1−x2∥+∥x1−x2+γ−1(v1−v2)∥, ρ((x1,v1),(x2,v2)) =

where are suitable positive constants to be fixed later and is continuous, non-decreasing concave function such that , is on for some constant with right-sided derivative and left-sided derivative and is constant on . For any two probability measures on , we define

 Wρ(μ,ν):=inf(X1,V1)∼μ,(X2,V2)∼νE[ρ((X1,V1),(X2,V2))]. (17)

Note that and are semimetrics but not necessarily metrics. A result from [EGZ17] is recalled below.

For a probability measure on , we denote by the law of when .

###### Theorem 4.1.

There exists a continuous non-decreasing concave function with such that for all probability measures on , we have

 W2(μpt,νpt)≤C∗√Wρ(μ,ν)exp(−c∗t),∀t≥0, (18)

where the following relations hold:

 c∗ = γ768min{λcLcβ−1γ−2,Λ1/2ce−ΛcLcβ−1γ−2,Λ1/2ce−Λc}, C∗ = √2e1+Λc/21+γmin{1,αc}(max{1,4(1+2αc+2α2c)(d+Ac)β−1γ−1c−1∗min{1,R1}/})1/2, Λc = 125(1+2αc+2α2c)(d+Ac)Lcβ−1γ−2λ−1c(1−2λc)−1, αc = (1+Λ−1c)Lcβ−1γ−2>0, εc = 4γ−1c∗/(d+Ac)>0, R1 = 4⋅(6/5)1/2(1+2αc+2α2c)1/2(d+Ac)1/2β−1/2γ−1(λc−2λ2c)−1/2.

The function is constant on , on with

 f(r) = ∫r∧R10φ(s)g(s)ds, φ(s) = exp(−(1+ηc)Lcs2/8−γ2βεcmax{1,(2αc)−1}s2/2), g(s) = 1−94c∗γβ∫r0Φ(s)φ(s)−1ds,Φ(s)=∫s0φ(x)dx

and satisfies .

###### Proof.

See Theorem 2.3 and Corollary 2.6 of [EGZ17]. ∎

It should be emphasized that , and consequently, contracts at the rate .

### 4.2 Proof of Theorem 2.8

###### Proof.

For each , we define

 Hk:=σ(Uz,i,1≤i≤k)∨σ(ξj,j∈N).

Let be -valued random variables satisfying Assumption 2.6. For , we recursively define , and

 ~Vλ(j+1,i,~v) = ~Vλ(j,i,~v)−λ[γ~Vλ(j,i,~v)+∇Fz(~Xλ(j,i,~x))] (19) +√2γβ−1λξk+1, ~Xλ(j+1,i,~x) = ~Xλ(j,i,~x)+λ~Vλ(j,i,~v). (20)

Let . For each , and for each , we set

 ~Vλk:=~Vλ(k,nT,VλnT),~Xλk:=~Xλ(k,nT,XλnT). (21)

We estimate for ,

 ∥Vλk−~Vλk∥≤λ∥∥ ∥∥k−1∑i=nT(g(Xλi,Uz,i)−∇Fz(~Xλi))∥∥ ∥∥

and

 ∥∥Xλk−~Xλk∥∥≤λk−1∑i=nT∥∥Vλi−~Vλi∥∥. (22)

Denote . By Assumption 2.4, the estimation continues as follows

 ∥Vλk−~Vλk∥ ≤ (23) +λ∥∥ ∥∥k−1∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥+λk−1∑i=nT∥∥gi,nT(~Xλi)−∇Fz(~Xλi)∥∥ ≤ λLk−1∑i=nT∥Xλi−~Xλi∥+λmaxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥ +λ(n+1)T−1∑i=nT∥∥gi,nT(~Xλi)−∇Fz(~Xλi)∥∥.

Using (22), one obtains

 k−1∑i=nT∥Xλi−~Xλi∥ ≤ λT∥VλnT−~VλnT∥+...+λT∥Vλk−1−~Vλk−1∥ (24) ≤ k−1∑i=nT∥Vλi−~Vλi∥,

noting that Therefore, the estimation in (23) continues as

 ∥Vλk−~Vλk∥ ≤ λLk−1∑i=nT∥Vλi−~Vλi∥+λmaxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥ +λ(n+1)T−1∑i=nT∥∥gi,nT(~Xλi)−∇Fz(~Xλi)∥∥.

Applying the discrete-time version of Grönwall’s lemma and taking squares, noting also that yield

 ∥Vλk−~Vλk∥2≤2λ2e2LTλ⎡⎣maxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥2+Ξ2n⎤⎦,

where

 Ξn:=(n+1)T−1∑i=nT∥∥gi,nT(~Xλi)−∇Fz(~Xλi)∥∥. (25)

Taking conditional expectation with respect to , the estimation becomes

 E[∥Vλk−~Vλk∥2∣∣HnT] ≤ 2λ2e2LE⎡⎣maxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥2∣∣ ∣∣HnT⎤⎦ + 2λ2e2LE[Ξ2n|HnT].

Since the random variables are independent, the sequence of random variables , are independent conditionally on , noting that is measurable with respect to . In addition, they have zero mean by the tower property of conditional expectation. By Assumption 2.4,

 ∥g(x,u)∥≤M∥x∥+∥g(0,u)∥

and thus

 E[∥g(~Xλi,Uz,i)∥2|HnT]≤2ME[∥~Xλi∥2]+2E[∥g(0,Uz,i)∥2].

by the independence of from . From Assumptions 2.1, 2.5, and from Lemma 5.1, we deduce that

 E[∥g(0,Uz,i)∥2]≤2E[∥g(0,Uz,i)−∇Fz(0)∥2]+2E[∥∇Fz(0)∥2]≤2δB2+2B2:=c1.

Therefore,

 E[∥g(~Xλi,Uz,i)∥2|HnT]≤2ME[∥~Xλi∥2]+2c1. (26)

Doob’s inequality and (26) imply

 E⎡⎣maxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥2∣∣ ∣∣HnT⎤⎦≤8M(n+1)T−1∑i=nTE[∥~Xλi∥2]+8c1T.

Taking one more expectation and using Lemma 5.3 give

 E⎡⎣maxnT≤m<(n+1)T∥∥ ∥∥m∑i=nTg(~Xλi,Uz,i)−gi,nT(~Xλi)∥∥ ∥∥2⎤⎦ ≤ 8M(n+1)T−1∑i=nTE[∥~Xλi∥2]+8