text:
# Why there is no name for #1@#2&?

Why does the pure function #1@#2& have no name? Is it due to the way Mathematica works, or is it "just because"?

I'm asking because I find inelegant to write expressions like

MapThread[#@#2 &, {{a, b, c}, {1, 2, 3}}]

{a[1], b[2], c[3]}

I would rather have a name form such as

MapThread[Work, {{a, b, c}, {1, 2, 3}}]

Work is not the best choice but I'm open on suggestions :)

• One may say that there is no name because #1@#2& is short enough. But f/@{1,2,3} is shorter than Map[f,{1,2,3}] and Map exists :)
• Then I've thought, maybe it is because Map gives other possibilities with 3rd argument and this is the difference. Well, it is a difference, but so what, why ther is no Work?
-
There is, but it is deprecated: Compose. You can also use Composition[#1][#2] &, although this is hardly better. I still use Compose myself, but I would not take the responsibility to recommend this as a common practice. –  Leonid Shifrin Jul 18 '13 at 12:11
@LeonidShifrin Oh, it is indeed. I haven't thought because I thought that it works like Composition and Composition does not work like I wanted –  Kuba Jul 18 '13 at 12:13
@LeonidShifrin I think You should post this as an answer since documentations says "Compose has been superseded by Composition" only. What could be misleading –  Kuba Jul 18 '13 at 12:15
Just so novices won't be confused here, I'd like to add that nothings preventing anyone from just defining Work=#1@#2& or similar. Also in your example you can use MapAll in place of Work, but that's just a happy coincidence and will break as soon as you do anything slightly different. :P –  jVincent Jul 18 '13 at 12:21
@Kuba Ok, done. –  Leonid Shifrin Jul 18 '13 at 12:23

### Compose and Composition

There is, but it is deprecated (in favor of Composition): Compose:

MapThread[Compose, {{a, b, c}, {1, 2, 3}}]

(* {a[1], b[2], c[3]} *)

I still use Compose myself, but I would not take the responsibility to recommend this as a common practice. You can also use Composition[#1][#2] &, although this is hardly better than your original suggestion in terms of code brevity.

Composition is more general because the result of Composition is a function which can take several arguments:

Composition[f, h][x, y]

(* f[h[x, y]] *)

Composition also has a Flat attribute, which is a mixed blessing (see e.g. discussion in this answer, where this leads to a huge slow-down for iteratively constructed composition of functions).

### Compose and Composition vs. Function - a clarification

Finally, there was a question in the comments why one can not use Function in a similar manner, since one may get an impression that Function is also constructing function calls. Actually, this is not quite so.

Compose is used basically to construct the square brackets, which is syntactically (and also semantically) non-trivial operation. One doesn't have to tie that to functions - one can think of Compose as a tool for programmatic building of normal expressions with non-trivial heads (the same is also true for Composition). So, where we would type something like

f[a]

or

f @ a

we can now do that programmatically as

Compose[f, a]

or similarly with Composition. This is a non-trivial capability, and it has to do with our ability to programmatically construct normal expressions from symbols / other normal expressions. For example, consider the following expression:

expr = Sin[x + Cos[y*z]];

We can get the symbols it is built with:

syms = Cases[expr, _Symbol, Infinity, Heads -> True]

(* {Sin, Plus, x, Cos, Times, y, z} *)

Here is how one can reconstruct it from symbols, using only Composition[..][..]:

Composition[Sin, Plus][x, Composition[Cos, Times][y, z]]

(* Sin[x + Cos[y z]] *)

Of course, built-in Composition itself is not that magical, and one can write their own version of Composition using e.g. replacement rules (and the same is true for Compose). But it is important to recognize their conceptual significance as functions which encapsulate programmatic expression-building. And they could not care less whether expressions they build are executable code (evaluate non-trivially), or just inert symbolic trees.

Now, Function serves a different purpose - it allows to construct function calls programmatically by generating a function call code from a function (basically a macro with placeholders) and a sequence of arguments at run-time. When we define a function like

plus = Function[#1 + #2]

we in fact define a macro which substitutes the parameters of the actual function call like

plus[1, 2]

(* 3  *)

into the body #1+#2 and only then evaluates the body.

So Function has to use lazy evaluation, to allow us to separate the process of defining a function expression, from calling that function with some arguments. And, in terms of execution time, it allows one to postpone the evaluation from "definition-time" to run-time (thus Function is HoldAll).

This is a different purpose from that of Compose and Composition (which, for example, don't carry Hold*-attributes, because they don't have to prevent any evaluation). By itself, Function is not able to syntactically construct an expression from its head and elements. And MapThread[f,{{a,b},{x,y}}] will return {f[a,x],f[b,y]} for a generic f. Therefore, using Function in MapThread will be no different from any other head, which is what one can observe when substituting Function into MapThread in the original example. Put in other way, Function takes care of slots and the ampersand in #1@#2&, but not of @, which is the important thing here.

-
This is the first case when I do not see any reason to wait with accept :) But I will to be fair :) Thank You. –  Kuba Jul 18 '13 at 12:26
@Kuba No problem :). This question was destined to emerge sooner or later. –  Leonid Shifrin Jul 18 '13 at 12:28
@LeonidShifrin chapeau again and again (+1)...i have still one question, if we say that #@#2& is short for Function[#1[#2]] then i don't see why: MapThread[Function, {{a,b,c},{1,2,3}}] shouldn't work...the intention is clear. –  Stefan Jul 18 '13 at 12:37
@Stefan ... Therefore, using Function in MapThread will be no different from any other head, which is what you can observe when substituting Function into MapThread in the original example. Put in other way, Function takes care of slots and the ampersand in #1@#2&, but not of @, which is the important thing here. –  Leonid Shifrin Jul 18 '13 at 12:50
@Stefan Yes, Function is used to delay evaluation, basically as a run-time macro with delayed evaluation, but not to construct expressions from pieces. I guess this is a Mathematica - specific issue and may be unintuitive when viewed from viewpoints of some other languages. In a sense, there are no functions in Mathematica, just rules and evaluation procedure. –  Leonid Shifrin Jul 18 '13 at 13:07

Another important role played by pure functions (those built with slots #) is that in functional programming (and Mathematica follows that paradigm) many times you need to apply a function that doesn't live outside the specific place where you define/evaluate it. In other words, mapping a function to a data set can be done without to save any unneeded variable/function in memory, just using a pure function.

Names["Global*"]
{}

MapThread[#@#2 &, {{a, b, c}, {1, 2, 3}}]
{a[1], b[2], c[3]}

Names["Global*"]
{"a", "b", "c"}

f[letter_, number_] := letter[number];
MapThread[f[#1, #2] &, {{a, b, c}, {1, 2, 3}}]
{a[1], b[2], c[3]}

same result but now you have many more variable in the memory and they are not needed outside the MapThread itseldf, so the memory used is waste.

Names["Global*"]
{"a", "b", "c", "f", "letter", "number"}
`

For that using pure the variables have no names.

-
This is correct, but does not seem very relevant to the question being discussed. The question seems to be about a very particular construction, and my digression into some particular aspects of pure functions in my answer was motivated by the comments rather than the original question itself. If your answer is a kind of a reply to that part of my answer, then I hasten to comment that I did not intend to describe pure functions in any extensive manner there, but only considered one particular aspect of them, relevant to the original question and answering the questions raised in comments. –  Leonid Shifrin Jul 18 '13 at 18:17
Leonid thanks for your comment. I'm new on SE and I'm learning how it works. I thought your answer, while correct, was not fully centered on the question. Indeed, the question I tried to answer is "Why there is no name for #1@#2&". The use of Compose/Composition is something about a possible alternative to use the pure function, not explaining why pure functions don't use variable's name. At least, this is my interpretation of the question. If I misinterpreted the question and you think my answer is not relevant for the whole thread, I can remove my answer. –  bobknight Jul 19 '13 at 6:45
@bobknight You have convinced me that it is possible to interprete my question this way. However, as Leonid Shifrin has said, it was about very particular construction, that's why I haven't precised it better. Maybe I should edit the title to: "Is there a built-in function identical to pure #@#2&?" ? –  Kuba Jul 19 '13 at 8:16


text:
# Calculation of X̄ & R Charts

Posted by   on

X̄ and R for each sub group:
Let Xij, j= 1, 2, ..., n be the measurement on the ith sample (i = 1, 2,...,k). The mean X̄i, the range R and the standard deviation Si, for the ith sample are given by -

Next, we find X̄̄ (X-bar-bar), R̄ (R-bar) and  S̄ (S-bar), the average of sample means, sample ranges and standard deviation respectively as follows-
----(ii)
Setting the control Limits:
We know if σ is the process standard deviation, then the standard error of sample mean is σ/√n, where n is the sample size i.e.
S.E (X̄i) = σ/√n, i = 1,2,..., k
Also from the sampling distribution of range, we know,
E (R) = d2. σ , where
d2 is the constant depending on the sample size.
Thus an estimate of σ can be obtained from R by the relation -
R̄ (R-bar) = d2. σ
=> σ = R̄ /d2 ---(iii)
Also, X̄̄ (X-bar-bar) gives an unbiased estimate of population mean since

format: plain


text:
CERN-PH-TH-2015-242
KA-TP-20-2015
MCNET-15-21
DESY-15-155
WUB-15-06
ZU-TH-31-15
arXiv:1510.08850
TIF-UNIMI-2015-7
JHEP 1601 (2016) 090

by: Bagnaschi, Emanuele (DESY) et al.

Abstract:
We study the prediction for the Higgs transverse momentum distribution in gluon fusion and focus on the problem of matching fixed- and all-order perturbative results. The main sources of matching ambiguities on this distribution are investigated by means of a twofold comparison. On the one hand, we present a detailed qualitative and quantitative comparison of two recently introduced algorithms for determining the matching scale [1, 2]. On the other hand, we apply the results of both methods to three widely used approaches for the resummation of logarithmically enhanced contributions at small transverse momenta: the MC@NLO and POWHEG Monte Carlo approaches, and analytic resummation. While the three sets of results are largely compatible in the low-p$_{⊥}$ region, they exhibit sizable differences at large p$_{⊥}$. We show that these differences can be significantly reduced by suitable modifications of formally subleading terms in the Monte Carlo implementations. We apply our study to the Standard Model Higgs boson and to the neutral Higgs bosons of the Two-Higgs-Doublet Model for representative scenarios of the parameter space, where the top- and bottom-quark diagrams enter the cross section at different strengths.


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# Knewton Tests Knewton CAT Tests

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Knewton Tests Knewton CAT Tests [#permalink]

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Hi!
I am doing the Knewton course and I really like it.
However, I have finished my second practice test and got a very low score in the math section.
Can you please share your experience with the Knewton test to the real one.
I want to know how real are the Knewton tests.
Thanks!!
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Re: Knewton Tests Knewton CAT Tests [#permalink]

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14 Jul 2010, 14:08
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Look guys, you can ask questions like "why did I miss x questions and get score y from Testprep company 1 when I missed p questions and got score q from Testprep company 2" all day long and you will not get the answer you want. What you will see is reps from these respective test prep companies post over and over again that their test prep software is the best representation of the actual test no matter what "evidence" you lay out. And you'll see other posters post more and more of their personal stats and how it compares to yours. Bottom line - if you're mildly curious on how others have scored and their breakdown and check this thread when taking a break from studying to relax your brain, fine. If you're genuinely trying to figure out answers to certain algorithms or expect test prep companies reps to EXPLAIN why their software spewed out a score different from what you thought you'd get, what do you honestly think the responses will be?

My personal view: Knewton provides a good course and good practice questions and completely random scoring. Manhattan GMAT has great quant and terrible verbal (to the point it doesn't even make sense). People swear by GMATPrep and how representative it is, but I didn't see such "almost perfect" resemblance. Someone else will either share my views, or have completely different views. People say Kaplan tests are harder, but you would be fool to assume say 100 points higher than your actual GMAT.

Don't get me wrong - I've posted threads or responded to threads with my score breakdowns too just to see how it compares with everyone else. What I advice against (and I see some people do this) and to ask over and over again the same questions in different forms trying to figure out how they are actually doing. You won't know until the day of the test. And until then, try to spend more energy drilling down the fundamentals and less time analyzing score algorithms.
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25 Apr 2010, 05:36
Thanks for the question. We at Knewton believe that our CATs are the most test-like you can find.

The gents who wrote the original GMAT algorithm work for us, and designed our tests.

They are designed to mimic your experience -- and success -- on test day.

Let me know if you have any other questions.

Cheers,
Josh
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Free GMAT Club tests ($250 value) in addition to any other discounts or coupons when you buy the Knewton Course with KnewtonBest-GMAT-Club discount code. Use this promo code when you sign up for Knewton: KnewtonBest-GMAT-Club Manager Joined: 26 Feb 2010 Posts: 79 Location: Argentina Followers: 4 Kudos [?]: 9 [0], given: 1 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 06:52 Hi JoshKnewton! Thanks for your answer. Do you know if the test have any penalty if you finish earlier? Manager Joined: 21 Mar 2010 Posts: 94 Followers: 2 Kudos [?]: 54 [0], given: 3 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 09:54 In my experience and from what I've read on the forum Knewton CATS are harsh on your quant score and too nice on your verbal. I used to finish my Knewotn Verbal sections with up to 15 minutes left and scored in the 99th percentile on my last CAT (also about 15 min early)... on sunday I scored in the 77th percentile for verbal on a MGMAT test with no time left over - doesn't make much sense... on other MGMATs I have never scored above 93% and have never had time left over. (Josh I messaged you about this, any news?) Quant on the other hand seems a bit harsh, I never scored above 43% on quant on Knewton CATS, but I consistenly scored about 70% on the MGMAT exams.... Having said that - I think the questions are fantastic, I am currently doing the OG and I feel like I have already seen 90% of the questions - Knewton did a good job mirroring Original questions I am quite a bit relieved, I thought the OG was going to be a pandoras box of sorts, I didn't really know what to expect! Manager Joined: 27 Dec 2009 Posts: 167 Followers: 2 Kudos [?]: 105 [0], given: 3 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 09:59 How many CATs does Knewton offer ? What is the cost ? Manager Joined: 26 Feb 2010 Posts: 79 Location: Argentina Followers: 4 Kudos [?]: 9 [0], given: 1 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 13:30 In their courses they offers 5 full test. I don't think that they offer the test alone. You can also take a free-trial test. I strongly recommend this test because is free and you can see how Knewton works. CEO Status: Nothing comes easy: neither do I want. Joined: 12 Oct 2009 Posts: 2786 Location: Malaysia Concentration: Technology, Entrepreneurship Schools: ISB '15 (M) GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Followers: 238 Kudos [?]: 1731 [0], given: 235 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 13:41 I got Q50 on quant with just one wrong answer and that too was 9th question. I want to ask you why q50? as it is scaled on 60 and with just one wrong answer it should have been atleast q51 as q51 is the highest shown. Moreover on verbal i got v25 which was i think rightly calculated as 2 times 4 consecutive answers were wrong with total of 19 incorrect. I m currently working on my verbal part and I have not given any other diagnostic test. Thus I would like to ask Knewton staff, though i know my quant is better than verbal but scoring good on knewton quant is a good indication of having better performance on real gmat or knewton quant was relatively easy? _________________ Fight for your dreams :For all those who fear from Verbal- lets give it a fight Money Saved is the Money Earned Jo Bole So Nihaal , Sat Shri Akaal Support GMAT Club by putting a GMAT Club badge on your blog/Facebook GMAT Club Premium Membership - big benefits and savings Gmat test review : http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html Manager Joined: 26 Feb 2010 Posts: 79 Location: Argentina Followers: 4 Kudos [?]: 9 [0], given: 1 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 14:19 Good score gurpreetsingh I have 48 and 47 in two previous GMAT. Howevver, in Knewton I scored 41 and 37, don't tell me why. with no more than 7 question wrong. And for my some question are no too complicated, however there are some with a lot of time consuming. My verbal was also my weak part and I have improved a lot I think with the Knewton course. Did you finish the course? CEO Status: Nothing comes easy: neither do I want. Joined: 12 Oct 2009 Posts: 2786 Location: Malaysia Concentration: Technology, Entrepreneurship Schools: ISB '15 (M) GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Followers: 238 Kudos [?]: 1731 [0], given: 235 Re: Knewton Tests [#permalink] ### Show Tags 25 Apr 2010, 14:23 No , it was a trial test. I m preparing on my own. I have time constraint so pursuing a course along with my full time work on client site is nearly impossible for me. _________________ Fight for your dreams :For all those who fear from Verbal- lets give it a fight Money Saved is the Money Earned Jo Bole So Nihaal , Sat Shri Akaal Support GMAT Club by putting a GMAT Club badge on your blog/Facebook GMAT Club Premium Membership - big benefits and savings Gmat test review : http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html Knewton GMAT Representative Joined: 23 Oct 2009 Posts: 112 Location: New York, NY Schools: BA Amherst College, MFA Brooklyn College Followers: 19 Kudos [?]: 37 [0], given: 1 Re: Knewton Tests [#permalink] ### Show Tags 26 Apr 2010, 06:47 Thanks for your collective interest in Knewton. We offer 5 CATS, one of which you can take for free in our 7-Day Trial. There is no penalty if you finish the test early (though we suggest that you take your time!). Again, the Knewton tests are designed to mimic your would-be performance on test day. We spend a lot of time and effort perfecting our CATs. Any large discrepancies might be due to the fact that you answered easier questions incorrect. Happy to answer other questions. Cheers, Josh _________________ Josh Anish Senior Editor Knewton, Inc Free GMAT Club tests ($250 value) in addition to any other discounts or coupons when you buy the Knewton Course with KnewtonBest-GMAT-Club discount code.
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26 Apr 2010, 06:57
JoshKnewton wrote:
Thanks for your collective interest in Knewton.

We offer 5 CATS, one of which you can take for free in our 7-Day Trial.

There is no penalty if you finish the test early (though we suggest that you take your time!).

Again, the Knewton tests are designed to mimic your would-be performance on test day. We spend a lot of time and effort perfecting our CATs. Any large discrepancies might be due to the fact that you answered easier questions incorrect.

Happy to answer other questions.

Cheers,
Josh

Ok that explain something to me.
thanks again!!
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26 Apr 2010, 11:08
JoshKnewton wrote:
Thanks for your collective interest in Knewton.

We offer 5 CATS, one of which you can take for free in our 7-Day Trial.

There is no penalty if you finish the test early (though we suggest that you take your time!).

Again, the Knewton tests are designed to mimic your would-be performance on test day. We spend a lot of time and effort perfecting our CATs. Any large discrepancies might be due to the fact that you answered easier questions incorrect.

Happy to answer other questions.

Cheers,
Josh

Ok . thanks for the details.
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28 Apr 2010, 00:48
As for me, I think that in Knewton tests math part is really harder than the real one. For example, I never got Quant score over 42 (though I scored 48 on real test). One time I even got 33.
--
As for Verbal part, diagnostic test showed Verbal Score = 35 (which is exactly the same score from the real one). I hope that my improvements in that part (from 35 to 38) will be reflected in actual GMAT result.
--
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28 Apr 2010, 03:01
I would like to know from knewton staff, is the knewton quant harder than real?
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28 Apr 2010, 06:18
DenisSh wrote:
As for me, I think that in Knewton tests math part is really harder than the real one. For example, I never got Quant score over 42 (though I scored 48 on real test). One time I even got 33.
--
As for Verbal part, diagnostic test showed Verbal Score = 35 (which is exactly the same score from the real one). I hope that my improvements in that part (from 35 to 38) will be reflected in actual GMAT result.
--

thank DeniSh really useful information!
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28 Apr 2010, 09:04
Again, the Knewton CAT is designed to mimic the actual GMAT. So no, our Quant Section is not harder than that of the real test.
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28 Apr 2010, 10:24
gurpreetsingh wrote:
I would like to know from knewton staff, is the knewton quant harder than real?

Don't worry, you are going to score very high in math!
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28 Apr 2010, 20:44
Hey everyone,

I'm just wondering what is everyone's opinion on the Knewton tests. On my first official GMAT I scored a 630 (this was 2 months ago and I was getting ~600 in my practice tests)

Now I have taken three Knewton CATs and I've gotten some odd results.

On my first two Knewton CATs I scored 530 and 540 respectively (my quant score was lower than usual, the verbal score was almost spot on with my official test). My quant score was the same on both tests.

On my third test I scored a 490 (which was quite disheartening since it seems I'm getting worse o_O) but the odd part about it is that I got very few verbal questions wrong (less than half of the questions I got wrong on the first two tests) yet my verbal score was much lower than on the first two tests and the quant score was exactly the same as the first two tests.

am I missing something?
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Re: Knewton Tests Knewton CAT Tests [#permalink]

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30 May 2010, 04:42
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Re: Knewton Tests Knewton CAT Tests   [#permalink] 30 May 2010, 04:42

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The \mathrlap stuff looks like you are used to manual spacing anyway. tex) \begin { flalign* } \setminus Set difference, reverse solidus or reverse slash, like \ (binary) The fleqn option in the document class will apply left aligning setting in all equations of the document put the table inside a div Discussion The syntax could be left in reST (for a set period of time?) Edit and compile if you like: Full explanation in Chapter 9, Creating Graphics: Constructing a flowchart. Sometimes you will probably want the last line to be numbered. Introduction. I also didn't see why the huge parentheses for the argument of det, so I replaced the pair bigg( bigg) \begin {flalign*} & 100000 & + 1 & = 100001 &\\ & 1 & + 1 & = 2 &\\ \end {flalign*} I would like the alignment to be just like the text file. Author: Anonymous User 3293 online LaTeX editor with autocompletion, highlighting and 400 math symbols. documentclass{article} usepackage{babel} usepackage{mathtools} usepackage{geometry} usepackage{titlesec} geometry{ a4paper, total={170mm,257mm}, left=20mm, top=20mm, } To include a TeX file into your document, just use the "\include {file}" command in the "LaTeX" menu. We can use equation environment to create the continue fractions. To learn how to code the accents etc consult the document by Apostolos Syropoulos available from Using Latex to type Greek (as a PostScript file 'usage-a4.ps'). This example produces the following output: Other counter commands. Using the flalign environment (numbered equations) or the flalign* (unnumbered equations) environment, left alignment can be obtained with two Texmaker allows you to work onto documents separated in several files.

Below is the code: \documentclass [8pt] {article} \usepackage {mathtools} \usepackage {xfrac} \begin {document} 4 posts This document gives an overview of the LaTeX code that works within ICC. AMSLaTeX align environment. LaTeX L, H \begin{minipage}{0pt} GitHub Gist: instantly share code, notes, and snippets.

LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems.

The first environment from the amsmath package that we are going to explain is the flalign environment. documentclass{article} usepackage{geometry} usepackage{ctex} usepackage{xeCJK} usepackage{enumitem} usepackage{multirow} usepackage{amsmath,amsthm,amsfonts,amssymb} usepackage{cases} usepackage{graphicx} Pas dinstallation, collaboration en temps rel, gestion des versions, des centaines de modles de documents LaTeX, et plus encore. They are listed with their default values.

You can instead use \begin{flalign} . Code, edit and compile here:

Use ampersands (&) for alignment and a double backslash (\\) to insert a linebreak.

However, sometimes the line numbers for whole paragraphs are missing - this is related to the math environment and occurs depending on whether you have a blank line before your equations or not. The documentation is currently split across several documents: Overview of LaTeXs hook management and core hooks lthooks-doc.pdf. By default, LaTeX typesets text as fully-justified, but occasionally left-aligned or "ragged right" text (for right-to-left languages) may be more appropriatesuch as text within narrow columns. I would not use the equation environment; rather, I would use flalign as such: \documentclass{article} \usepackage{amsmath} \begin{document} \noindent A \hfill For example if you are using article documentclass then you may want to use the following command. Operators spacing. The {} around the equals sign are needed to give it something to space itself against. This works of course only if you have at least two such pairs, meaning that you need at least two &: if you have only one & then you just jumped from the GitHub Gist: instantly share code, notes, and snippets. Open this code fragment in Overleaf. Multiple Alignments in flalign. It shows using styles, placing nodes in a matrix, and drawing arrows using loops.

ReferenceSheetforaThesis withLATEX2eandKOMA-Script Allexamplesweretestedwithpdflatex. The indent applied to flalign from both margins is set with \multlinegap.

Thepackagementionedintheheadingshastobeincluded(seeB.2). \usepackage{amsmath} A short introduction to LaTeX (with some focus on the differencesbetween the old LaTeX 2.09 and current standard [code]\documentclass[fleqn]{article} [/code]This will left align all the equation in the article. LaTeX. latex cheat sheet index card. 4.2 About documents separated in several files. You only have to declare in the code at least two alignment columns. To learn how to code the accents etc consult the document by Apostolos Syropoulos available from Using Latex to type Greek (as a PostScript file 'usage-a4.ps'). Version: 5.x - Scientific WorkPlace & Scientific Word. The "fl" in flalign and flalign* stands for "full length", not "flush left". Generic hooks for document-level commands ltcmdhooks-doc.pdf. Simply put a \: (or bigger space like \quad at the end of \llap. \begin {flalign*} & 100000 & + 1 & = 100001 &\\ & 1 & + 1 & = 2 &\\ \end {flalign*} I would like the alignment to be just like the text file. An online LaTeX editor thats easy to use. Like the other AMS alignments it is designed to set equations as pairs of columns with a left column right aligned (typically to &=) and a left column left aligned, and then the next two columns again being right-left aligned. Meta. \begin {flalign*} & RPC & {}= {}& A+B\tilde {f} +C x &\\ & A & {}= {}& a+\eta &\\ & E & {}= {}& cte & \end {flalign*} Notes: 1.

Hooks available when reading files ltfilehook-doc.pdf. LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. Type I to replace it with another command, or to continue without it. When the text is not fully justified, sometimes it may look "too ragged". documentclass{article} usepackage{babel} usepackage{mathtools} usepackage{geometry} usepackage{titlesec} geometry{ a4paper, total={170mm,257mm}, left=20mm, top=20mm, } latex cheat sheet index card. TeX - LaTeX: I need to have an equation aligned on the left of my page and its tag on the right. The options below control the operation of the TeX input processor that is run when you include 'input/tex' , 'input/tex-full', or 'input/tex-base' in the load array of the loader block of your MathJax configuration, or if you load a combined component that includes the TeX input jax. This removes math and most other nontext material from LaTeX file index.tex and puts the rest in a web page called "index@hide.html". The fleqn option in the document class will apply left aligning setting in all equations of the document. Using the flalign environment (numbered equations) or the flalign* (unnumbered equations) environment, left alignment can be obtained with two ampersands (&). The & sign separates two columns, so an & at the beginning of a line means that the line starts with a blank column. An online LaTeX editor thats easy to use. Texmaker allows you to work onto documents separated in several files. You can instead use \begin{flalign} . In LaTeX, you can add line numbers to your document by adding. documentclass{article} usepackage{geometry} usepackage{ctex} usepackage{xeCJK} usepackage{enumitem} usepackage{multirow} usepackage{amsmath,amsthm,amsfonts,amssymb} usepackage{cases} usepackage{graphicx} flalign {flalign} flalign* {flalign*} alignat {alignat= cols } alignat* {alignat*= cols } multline {multline} multline* {multline*} Table 1: The supported amsmath environments Maybe not the most impressive example, but as you can see the contents of the environment are exactly the same as for the regular align*evironment from ams-math.

LaTeX does have built-in commands for changing the typeset alignment of text: together with corresponding environments: However, the ragged2e package provides some refinements which improve upon these standard LaTeX commands and environments.

In LaTeX (and TeX) the word box means an unbreakable unit, whose contents are processed in text mode. The amsmath package also provides the \intertext macro, which you could use to insert Where: while retaining alignment of the equal signs in the equations above and below: \documentclass{article} \usepackage{amsmath} \begin{document} \begin{flalign} A &= 3B \cdot 4C + 5D\\ \intertext{Where:} A &= \text{Something cool}\notag\\ B &= \text{Something Basically align and its siblings are in fact arrays of successive rl columns.flalign then stretches the intercolumn space between these rl pairs to fill the page width. GitHub Gist: instantly share code, notes, and snippets.

TeX - LaTeX: How to draw centered vertical dashed lines in article? There can be problems with accessing the necessary Greek fonts, depending on your TeX installation. In my point of view the constant block should use two (or even three constants) per row, not one large column. This approach avoids explicit spacing commands. \begin{flalign*} Box half filled color Align centered, ragged right and ragged left in align environment label a part of commutative diagram Help rendering a complicated sum/product formula "Illegal unit of measure (pt inserted) Problem with foreach Maths symbols and unicode-math input inside siunitx commands How to evaluate an expression and use the result in newcommand? 1. There are several standard LaTeX commands to change the text alignment. An online LaTeX editor thats easy to use. The standard documentation for the babel package gives some useful information. 4 posts Page 1 of 1 You can instead use \begin{flalign} (This tells Word that this will be a right-aligned tab, just like you want Then, go to the Arrange tool and in the dropdown menu, select Align, and then Distribute Horizontally If the file is not listed in the current directory, either type the directory name in front of the document name, or select the Environments provide a efficient way of modifying blocks of text within your document. LaTeXequationalign, , align(, ), amsmath.sty, . \ (\) Example: This equation \ (x^2-3x+4=0\) is inline. Except for split, each environment has both starred and unstarred forms, where the unstarred forms have automatic numbering using LATEXs equation counter. I need \\subsections inside this flalign, so I used \\intertext. A document can be previewed a number of ways, namely the icon that appears in the top left of an open TeX document (see gif) , or by the shortcut ctrl+alt+v (see also the FAQ for an alternative shortcut). The fl in flalign and flalign* is short for "full length", not "flush left".

I have one flalign environment, so columns are aligned same way whole page. Hello I am using flalign as \begin{flalign} A=&3x+5xy\\ B=&6xy+3x^2y \end{flalign} I want to reference equation number of B, but I don't know how to.

The LaTeX shorthand that declares displayed equations (i.e. doc.tex, line 5 LaTeX Error: Environment equation* undefined. &= 1+1 &\\ I took the liberty to simplify your code: as I didn't see why an array with 20 columns I replaced it with the simpler bmatrix environment. Type I to replace it with another command, or to continue without it. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. I have tried all different combinations of & s and haven't found anything that works quite right. There's probably some better way of doing it, b 1 1 Introduction The amsmath package is a LATEX package that provides miscellaneous enhance- ments for improving the information structure and printed output of documents that contain mathematical formulas. The equation environment does not know the content of the box (flalign) above. This article explains how to use colour in your LaTeX document via the color or xcolor packages. Both packages provide a common set of commands for colour manipulation, but xcolor is more flexible and supports a larger number of colour models, so is the This has nothing to do with algorithm2e so I'll leave that out. 1. It shows using styles, placing nodes in a matrix, and drawing arrows using loops. The amsmath package also provides the \intertext macro, which you could use to insert Where: while retaining alignment of the equal signs in the equations above and below: \documentclass{article} \usepackage{amsmath} \begin{document} \begin{flalign} A &= 3B \cdot 4C + 5D\\ \intertext{Where:} A &= \text{Something cool}\notag\\ B &= \text{Something

11 Other documentation of interest21 Acknowledgments and plans for future work Thanks to all who contributed suggestions, assistance and encouragement. By default, LaTeX typesets text as fully-justified, but occasionally left-aligned or "ragged right" text (for right-to-left languages) may be more appropriatesuch as text within narrow columns. Use ampersands (&) for alignment and a double backslash (\\) to insert a linebreak. And I've changed the differentials d to use upright characters. (2) Using the above suggestion, here's the code (a minimal example) that will generate your left-aligned equation using the align. The align environment provided by AMS-LaTeX is used for two or more equations when vertical alignment is desired; usually equal signs are aligned. & A &{}={}& a+\eta &\\ Placement is one of the following h (Here) - at the position in the text where the table environment appears. By default, LaTeX typesets text as fully-justified, but occasionally left-aligned or "ragged right" text (for right-to-left languages) may be more appropriatesuch as text within narrow columns. The apacite package formats citations according to the complex requirements of the American Psychological Association (APA). Search: How To Set Left Align In Latex. The indent applied to flalign from both margins is set with \multlinegap. \begin{flali Alternativaly you can use the commands provided by the package ragged2e . Short Math Guide for LATEX, version 2.0 (2017/12/22) 5 2.2. Introduction. The default LaTeX environment for producing ragged-right text is flushleft. The equivalent in ragged2e is called FlushLeft (note capitalization). The following example compares the output of those environments: And I need \\label these \\subsections. flalign\subsectionflalign\intertext\label\subsection\label\subsection\subsection. \begin{document}

LaTeX. Hello I am using flalign as \begin{flalign} A=&3x+5xy\\ B=&6xy+3x^2y \end{flalign} I want to reference equation number of B, but I don't know how to. This package provides user control over the layout of the three basic list environments: enumerate, itemize and description. Document 575 The LaTeX apacite package. Short Math Guide for LATEX, version 2.0 (2017/12/22) 5 2.2. formulas, graphs). \usepackage{amsmath}

Answer (1 of 2): Use fleqn as an option in the document class. The trick is to give amsmath the parameter fleqn : \usepackage[fleqn]{amsmath} In a flalign environment the fields alternately flush right and then flushleft. Automatic numbering and cross-referencing. If there are several equations that you need to align vertically, the align environment will do it: \begin{ align* } 2x - 5y & = 8 \\ 3x + 9y & = -12 \end{ align* } Open this amsmath fragment in Overleaf. The following example shows The indent applied to flalign from both margins is set with \multlinegap. See the LaTeX manual or LaTeX Companion for explanation. l.5 \begin{equation*} Your command was ignored. I need \\subsections inside this flalign, so I used \\intertext. A LaTeX environment is one of the following: Text inside a float environment is "floated" according to its placement, an optional parameter. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. gather gather* flalign flalign* multline multline* alignat alignat* split (Although the standard eqnarray environment remains available, it is better to use align or equation+split instead.) The standard report and article classes use the default placement [tbp]. RPC &= A+B\tilde{f} +C x &\\ With a click on his name, Texmaker will open it. \omi flalign is a variant of align but uses the full width of the page (length of the line, hence fl). Also, is there a better alignment solution without using a lot of qquad? MathJax. Multiple Alignments in flalign. The indent applied to flalign from both margins is set with \multlinegap. So everything left aligned, then some space after 1 in the second row to align the + right underneath the + Introduction. gather gather* flalign flalign* multline multline* alignat alignat* split (Although the standard eqnarray environment remains available, it is better to use align or equation+split instead.) (Aside: Just because flalign can be abused to force displayed equations to be typeset left-aligned doesn't mean that it's harmless to do so.). No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. LaTeX forum Math & Science Double Label in 'flalign' Environment Information and discussion about LaTeX's math and science related features (e.g. Using the flalign environment (numbered equations) or the flalign* (unnumbered equations) environment, left alignment can be obtained with two ampersands (&). Let's understand with an example. The flalign environment doesn't align on the left automagically. If the only alignment you want is for the equations to be left-aligned, put an ampersand at the The indent applied to flalign from both margins is set with \multlinegap. Explanation of the downloads: *.tar.gz: Source tar ball of this release *win32*: Windows version, with installer or as usb (= settings stored in the same directory as the executable) version *.deb: Debian version compiled on a developer machine *.dmg.zip: Mac version (usually provided you can use a \parbox and emulate a new paragraph by using a line-changing command \\ together with a \hspace* with proper value (this value, of couse, will be given by \parindent as defined in your document class; to find out this value, you can use \the\parindent ). with regards to the use of the eqnarray environment as well as the. The float environments are figure and table. Type H for immediate help. And I need \\label these \\subsections. This works of course only if you have at least two such pairs, meaning that you need at least two &: if you have only one & then you just jumped from the A flowchart showing how we may choose a math environment.


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# How this expression leads to the given sequence

Here given is a sequence from OEIS.

The sequence is triangle of coefficients from fractional iteration of $e^x - 1$. Few terms are:
1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, 875, 16674
as a triangle this can be written as

 1
1,  3
1, 13,  18
1, 50, 205, 180
1,201,1865,4245,2700

The expression of finding the sequence is also given as:
$A(n;x)$ for n-th row satisfies
$$A(n;x) = \sum_{k=0}^{n-1} \mathrm{Stirling}_2(n, k) * A(k;x)*x,\ A(1;x) = 1.$$

The tabular view shows the entries row wise. $\mathrm{Stirling}_2(n,k)$ is most probably stirling numbers of the second type

I am not able to get how above expression is resulting in the given sequence.
In summation, $k$ begins from 0, but nothing is mentioned about $A(0;x)$. I assumed it to be 0, but still can't get the above values.

Please explain how the first few terms are resulting from the expression.

• Isn't this the same as mathoverflow.net/questions/133271/… ? – Gerry Myerson Jun 13 '13 at 6:02
• The Mathematica function at OEIS works as desired. – Fred Kline Jun 13 '13 at 8:36
• @Fred Kline are you sure of that? Can you explain the the computation manually for the first few terms? I am getting wrong results. – Nadia Jun 13 '13 at 9:02
• I have changed the mathematical expressions to correct LaTeX. Hope this helps. – rem Jun 26 '13 at 8:36

[update] Here -for possibly a nicer reference- I give a screenshot, where the identity by the OEIS-formulae (and kindly expanded by @Carlo Beenakker) is interpreted as matrix-multiplication. It indicates also, that the given formula of relations between the Stirling and A-numbers is near to an eigenvector/eigenmatrix-relation (in fact is a Jordan-decomposition) which I described in my first version of this post (which I kept below).

Here is the matrix-multiplication scheme:

Another approach gave the numbers in a completely surprising context; I've no explanation so far. ([update]:It is clear now. Carlo's expansion gave the key hint!).

For the function $\exp(x)-1$ the matrix $S_2$ of Stirling numbers 2'nd kind is involved. I've just recently begun to exercise with the Jordan-decomposition, and for some exercise I took this matrix

$$\large S_2 \qquad = \qquad \small{ \begin{bmatrix} 1 & . & . & . & . \\ 1 & 1 & . & . & . \\ 1 & 3 & 1 & . & . \\ 1 & 7 & 6 & 1 & . \\ 1 & 15 & 25 & 10 & 1 \end{bmatrix}}$$ and let WolframAlpha Jordan-decompose it such that $$S_2 = A \cdot J \cdot A^{-1}$$

This gave the three matrices:

$$\begin{array} {} A &=&\small { \begin{bmatrix} 1 & . & . & . & . \\ 0 & 1 & . & . & . \\ 0 & 1 & 3 & . & . \\ 0 & 1 & 13 & 18 & . \\ 0 & 1 & 50 & 205 & 180 \end{bmatrix} } \\ J&=& \small {\begin{bmatrix} 1 & . & . & . & . \\ 1 & 1 & . & . & . \\ 0 & 1 & 1 & . & . \\ 0 & 0 & 1 & 1 & . \\ 0 & 0 & 0 & 1 & 1 \end{bmatrix} } \\ A^{-1}&=&\small {\begin{bmatrix} 1 & . & . & . & . \\ 0 & 1 & . & . & . \\ 0 & -1/3 & 1/3 & . & . \\ 0 & 5/27 & -13/54 & 1/18 & . \\ 0 & -301/2430 & 353/1944 & -41/648 & 1/180 \end{bmatrix}}\end{array}$$ (Note: I've transposed the input to Woframalpha and then also the output to keep in line with my usual conventions with that type of matrix-discussion)
The surprise is: that we find the coefficients in the matrix $A$

The relation can also be written as $$S_2 \cdot A = A \cdot J$$ and if $J$ is decomposed in the sum of two (diagonal and subdiagonal) components $J=J_0+J_1$ where $J_0=I$ and $J_1$ the transpose of my $J$ in the update of this post above, then we can reformulate $$S_2 \cdot A = A \cdot (I+J_1) \\ (S_2 - I) \cdot A = A \cdot J_1 \\ S_2^* \cdot A = A \cdot J_1 \\$$ from which -in my opinion- we could formulate a slightly more convenient relation than that given in the OEIS.

Additional remark because there is a relation to the Schröder-function for fractional iteration: for the use for the function $f(x) = \exp(x)-1$ the matrix $S_2$ is factorially similarity-scaled and -when let unscaled- becomes the Bell/(transposed) Carleman-matrix for that function. In that view the scaling $F^{-1} \cdot S_2 \cdot F$ is a matrixoperator; and the matrices $A$ and $A^{-1}$ are in a very similar role as the operators for the Schröder-function of $f(x)$ which is indeed at the heart of the Schröder/Abel-type of fractional iteration...

This is the input for W/A's input field:
JordanForm({{1,1,1,1,1},{0,1,3,7,15},{0,0,1,6,25},{0,0,0,1,10},{0,0,0,0,1}})

Stirling numbers of the second kind $S_2(n,k)$ vanish for $k=0$, so the $k=0$ term in the sum does not contribute. You ask for a manual computation of the first few terms. Here we go:

Stirling numbers: $S_2(n,1)=S_2(n,n)=1$, $S_2(3,2)=3$, $S_2(4,2)=7$, $S_2(4,3)=6$

$$a(2,x)=xa(1,x)=x$$

$$a(3,x)=xa(1,x)+3xa(2,x)=x+3x^2$$

$$a(4,x)=xa(1,x)+7xa(2,x)+6xa(3,x)=x+7x^2+6x^2+18x^3=x+13x^2+18x^3$$

This gives the first few coefficients: $1,1,3,1,13,18$.

• if x means the column-index, then all elements in the matrix $A$ in the first column are zero by $a(2,0)=0 \cdot a(1,0) = 0$. Is that really meant? It would contradict the table-view in OEIS where the first column is 1. Are the indexes for S2 and A are thought differently (for S2 beginning at 0 and for A beginning at 1 ?) – Gottfried Helms Dec 9 '13 at 14:54
• @GottfriedHelms --- my understanding is that $x$ is a variable, not an index; the function $a(n,x)$ is a polynomial of degree $n-1$ in the variable $x$; you are seeking a triangle of coefficients, such that the $n$-th row of the triangle has at the $m$-th position the coefficient of the term $x^m$ in $a(n,x)$. As I worked it out above, you see that this indeed gives the series mentioned by the OP. In particular, the fourth row has elements $1,13,18$, which are the coefficients of $a(4,x)=x+13x^2+18x^3$. – Carlo Beenakker Dec 9 '13 at 16:34
• Carlo, see my accidental beautiful observation in my new answer... – Gottfried Helms Dec 10 '13 at 23:55


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# Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?

Let $G=\mathbb{Z}/2\mathbb{Z}$. Let $f\colon L \to N$ be a smooth map of connected smooth closed $n$-dimensional manifolds such that the induced map

$f^\* \colon H^\*(N,G) \to H^\*(L,G)$

is an isomorphism.

Question: Are the pull back of the Stiefel-Whitney classes of the tangent bundle of $N$ the Stiefel-Whitney classes of the tangent bundle of $L$?.

This is in fact true for the first Stiefel-Whitney class by considering coverings and degrees, but what about the higher degree classes?

Motivation: This came up because (relative) spin is important in defining Floer homology with $\mathbb{Z}$ coefficients. So I am in fact mostly interested in the following sub-question.

Question: In particular what about the second Stiefel-Whitney class in the case where both $N$ and $L$ are also assumed to be oriented? and if the answer is negative: what extra conditions do I need to make it positive?

The idea is that I apriori have to use $G$ coefficients, but can prove that it is a $G$-cohomology equivalence, and want to use that to start the argument over again with other coefficients, but for that I need this property of the second Stiefel-Whitney class.

This sub-question and the relation to Floer homology is related to orientations in real $K$-theory and delooping in the following sense: take a map $h\colon X \to U/O$ by delooping we get a map $\Omega h \colon \Omega X \to \Omega U/O \simeq \mathbb{Z}\times BO$ which classifies a virtual bundle over the loop space of $X$. This bundle is oriented iff the original map composed with the canonical map $U/O \to BO$ classified a virtual $0$-dimensional bundle with vanishing second Stiefel-Whitney class. This is the main point of why orientations in Floer homology is initimitely linked with spin! In the case of a Lagrangian sub-manifold $L\subset T^*N$ the difference of the tangent bundles precisely defines such a map $L \to U/O$ ($U(n)/O(n)$ classifies Lagrangians in $\mathbb{R}^{2n}$) such that the composition to $BO$ classifies the virtual bundle $TN-TL$. So in fact you may add this lifting property as an extra condition to the sub-question if you like, and then I would lose no generality. I believe that this condition implies that all the relative Prontryagin classes vanishes, which may be helpfull.

ADDED: in light of the answer, all this motivation and these extra possible assumptions are not important nor relevant for the actual question.

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"Floer homology is oriented iff you have spin!" Um... Relative pin structures are sufficient to determine coherent orientations, but I'm not sure I'd care to formulate a converse. Two disjoint Lagrangians have a natural (but trivial!) Floer complex over the integers. –  Tim Perutz May 30 '10 at 18:48
By the way, the (excellent!) idea that one should prove that nearby Lagrangians are mod 2 cohomology equivalent and hence relatively spin is I believe one that Fukaya-Seidel-Smith were aware of when they wrote their papers on nearby Lagrangians. They couldn't use it because they invoke a theorem that requires char $\neq 2$. But it comes up at the very end of Abouzaid's preprint arxiv.org/pdf/1005.0358. –  Tim Perutz May 30 '10 at 19:04
@Tim: The Floer homology I am refering to is the one for the action of a given Hamiltonian on a symplectic manifold $M$, not the intersection Floer homology for two Lagrangians, which I admit is a not obvious from the above since I am explicitly considering a Lagrangian in $T^*N$ (this is also what Viterbo did in "Exact Lagrange submanifolds, periodic orbits and the cohomology of the free loop space", and I am working in the same spirit), in this case the Floer homology can be given coherent orientations without employing any tricks iff $M$ is spin, –  Thomas Kragh May 31 '10 at 7:51
and in the case Viterbo considers the generating functions are related by an oriented bundle iff $L\to N$ is relative spin. Thanks for the reference I had not seen that one, I am looking forward to seing if the passage from homology equivalence to homotopy equivalence can work in my frame work as well. –  Thomas Kragh May 31 '10 at 7:56
I realize now that something is not quite right about this statement because Floer homology can be oriented in the case of $T^*N$ even if $N$ is not spin. I am not sure anymore what precisely the general statement is, but at least the orientation is intimitely linked to spin, because of what I write, and the relative spin statement about generating functions is valid in the Viterbo case. (I have changed the statement) –  Thomas Kragh May 31 '10 at 8:05

## 3 Answers

The answer to the question is positive, due to Wu's formula. See e.g. Milnor-Stasheff, Characteristic classes, lemma 11.13 and theorem 11.14. In fact, all one needs to compute the Stiefel-Whitney classes of a smooth compact manifold (orientable or not) is the cohomology mod 2 (as an algebra) and the action of the Steenrod algebra on it. Both structures are preserved under cohomology isomorphisms induced by continuous maps.

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How does this relate to exotic differentiable structures on the sphere? It seems that none of the things you mention as sufficient to calculate S.-W. classes depend on the differentiable structure at all. Are S.-W. classes of the tangent bundle not enough to detect weird smoothness structures? How is the situation different for Pontryagin classes (which is I think what Milnor used to distinguish them)? –  Ilya Grigoriev May 30 '10 at 18:29
Ilya, homotopy spheres are stably parallelizable. Milnor distinguished exotic 7-spheres via Pontryagin numbers of bounding 8-manifolds. –  Tim Perutz May 30 '10 at 18:51
Ilya -- yes, the (integral) Pontrjagin classes depend on the smooth structure, but the Stiefel-Whitney classes don't and neither do the rational Pontrjagin classes. –  algori May 30 '10 at 18:57
But my last comment does not apply at all to the rational Pontrjagin classes. You can give examples of homotopy-equivalent smooth closed manifolds where one is stably parallelizable and the other has nontrivial Pontrjagin classes: the total spaces of the sphere bundles over two vector bundles over a sphere -- one bundle trivial, the other corresponding to a nontrivial element of the kernel of $\pi_{4k}(BO_j)\to pi_{4k}(BG_j)$. –  Tom Goodwillie Jul 20 '10 at 20:32
algori -- If $E_0\to B$ is a sphere bundle, let $E$ be the mapping cylinder $E=E_0\times[0,1]\cup_{E_0\times0} B$. That's your disk bundle. –  Charles Rezk Jul 21 '10 at 16:57

The most conceptual way of understanding the relation between the mod 2 Wu and Stiefel-Whitney classes of a manifold and the action of the Steenrod algebra on the mod 2 cohomology is to use the homotopy theory of Poincare duality spaces and the Spivak normal fibration, and also the chain homotopy theory of chain complexes with symmetric Poincare complexes and the normal chain bundle expounded in my 1980 paper The algebraic theory of surgery (Part I, Part II). A map $f:L\to N$ of $n$-dimensional manifolds which induces isomorphisms in $Z_2$-coefficient cohomology also induces a chain equivalence of $n$-dimensional symmetric Poincare complexes over $Z_2$. Such a chain equivalence automatically preserves the Spivak normal chain bundles. The mod 2 Wu and Stiefel-Whitney classes of the manifolds are preserved by $f$ because they only depend only on the underlying chain homotopy structure. It is also worth reminding ourselves that Atiyah's 1960 paper Thom complexes established the $S$-duality between the Thom space of the normal bundle of a manifold $X$ and $X_+=X \cup \{*\}$, and so proved a conjecture of Milnor and Spanier: the stable fibre homotopy type of the tangent sphere bundle of a differentiable manifold $X$ depends only on the homotopy type of $X$.

-

There was a fundamental error in my answer.

The error was in misunderstanding the naturality of the $w_i$. $f: L \to N$ inducing an isomorphism in cohomology does not imply anything about the induced map of the tangent bundles. This is where i made my fundamental error. Please see the comments for details or look at earlier versions of this answer.

I had hoped that there would be a more "axiomatic" proof in the sense of Milnor and Stasheff. If anyone comes up with one please feel free to put one here.

Thanks to Tom and Dan for the comments!

-
This doesn't answer the question, because a smooth map $f:N\to L$ need not give a bundle isomorphism, even if it induces isomorphisms on cohomology. –  Tom Goodwillie Jul 20 '10 at 22:24
Are you saying that f being an isomorphism in mod 2 cohomology implies that $f^*(\tau_N) = \tau_L$? –  Dan Ramras Jul 20 '10 at 22:27
(My comment is the same as Tom's) –  Dan Ramras Jul 20 '10 at 22:28
I did not mean to imply that an isomorphism in cohomology implied that the bundles were isomorphic. My mistake. Do the equations i have above actually require that the map of bundles be an isomorphism of bundles? I believe i only need the bundle map. If this is wrong, is there a way it can be fixed? –  Sean Tilson Jul 20 '10 at 23:56
Any map $X\to Y$ of spaces is covered by a bundle map, for any vector bundles on $X$ and $Y$. (Zero map.) Maybe some books say bundle map when they mean fiberwise isomorphism? –  Tom Goodwillie Jul 21 '10 at 1:59


text:
# ruffus

30 thoughts
last posted Dec. 18, 2012, 9:22 p.m.

3 earlier thoughts

0

since we'll be interfacing with SGE and ruffus supposedly offers support for SGE, I think we'll be in good shape. i just need to understand how it works instead of expecting magic

26 later thoughts


text:
# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 8.3.1.5. Let $\operatorname{\mathcal{C}}$ be a small $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. In §5.7.6, we showed that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresented by $X$, and that $\mathscr {F}$ is uniquely determined up to isomorphism (Theorem 5.7.6.13). Proposition 8.3.1.3 can be regarded as a more refined version of this uniqueness assertion: the functor $\mathscr {F}$ is characterized, up to isomorphism, by the requirement that it corepresents the evaluation functor

$\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {G} \mapsto \mathscr {G}(X).$


text:
# Original documents

A requirement under a provision of a federal law for a document to be in its original form is satisfied by an electronic document if
(a) the federal law or the provision is listed in Schedule 2 or 3;
(b) the electronic document contains a secure electronic signature that was added when the electronic document was first generated in its final form and that can be used to verify that the electronic document has not been changed since that time; and
(c) the regulations respecting the application of this section to the provision have been complied with.


text:
# Sum of Fibonacci numbers — speed contest

Write a program that print sum of as many Fibonacci numbers as possible. There is a limit in that your program should not run more than 10s on 2GHz single core PC. (equivalents are acceptable, for instance you have twice as many time on 1GHz PC). Memory consumption is not limited, but provide details, if you exceed normal values.

Your program has to work with different input values. Output can be calculated modulo 10^i, but i has to be at least 9 digits. Other types of results are also welcome as long as they are presented with 9 digits and are not trivially calculated.

Post source, language input value and result.

-
Define "trivially calculated". Are you saying that we can apply mathematical identities to optimise it up to some cutoff point? If so, what precisely is the cutoff point? – Peter Taylor Apr 14 '11 at 18:44
@ralu Instead of giving a PC specification, you can also say print the maximum sum of fibonacci numbers before it times out on ideone.com . That way everyone will have a common ground. The time limit there is 5 secs. – fR0DDY Apr 15 '11 at 3:23
did not knew. That looks fair – ralu Apr 15 '11 at 7:27
What kind of input values are you talking about? – user unknown Apr 16 '11 at 1:40
The question is still unclear to me. For arbitrary input, each machine will reach a limit in 10s, so does 'as many as possible' mean, that the program shall interrupt, and print the sum so far? If you have a timelimit, the program could be interrupted from outside, and show, what it reached. – user unknown Apr 18 '11 at 12:09
show 5 more comments

## Python, 7 chars

It turns out that the period of G(i)=F(i) mod 10^9 is 1.5e9 exactly. So when adding them up, we can group them into groups of size 1.5e9, each of which will have the same sum (mod 10^9). Turns out, that sum is 0. So if we take N to be any multiple of 1.5e9, the sum of the first N fibonacci numbers is 0. So:

print 0

will, given an argument i, print the sum of the first i * 1.5e9 Fibonacci numbers. Super fast and works for an arbitrarily large i.

-
A very nice illustration of the point behind my question about what "trivially calculated" means. You could, of course, extend it to take any i and calculate the sum. – Peter Taylor Apr 14 '11 at 22:11
Indeed, the sum of the first i is the same as the sum of the first i % 1.5e9. – Keith Randall Apr 14 '11 at 22:24
Your program has to work whit different input values – ralu Apr 15 '11 at 0:08
I didn't get any of it. Care to explain to those without a degree in CS as of now? – the_drow Apr 15 '11 at 12:00
– Peter Taylor Apr 15 '11 at 21:46
show 6 more comments

Compensating for my cores being 2.5GHz by allowing just under 8 seconds,

public class FiboSum
{
public static void main(String[] args)
{
long now = System.currentTimeMillis();
System.out.println("Taking F(1) = F(2) = 1, F(n) = F(n-1) + F(n-2):");
long n = 0, x = 1, y = 1, N = 1000000000;
// x = F_n, y = F_{n+1}
while (System.currentTimeMillis() - now < 7800)
{
n++;
long t = (x*x + y*y) % N;
x = x * (2*y - x) % N;
y = t;
}
System.out.println("\\sum_{r=1}^{2^"+n+"} F(r) = " + (N+x+y-1)%N + " mod " + N);
}
}

Output:

Taking F(1) = F(2) = 1, F(n) = F(n-1) + F(n-2):
\sum_{r=1}^{2^94900480} F(r) = 755986263 mod 1000000000

And no sign of Binet's formula or any closed form evaluation of the sum, even though there is a trivial one.

As a closely related point of interest, note that we can use the identity F(m+n) = F(m+1) F(n) + F(m) F(n+1) - F(m) F(n) to compute the nth Fibonacci number in O(lg n) arithmetic operations. Replace int and long with a suitable BigInteger implementation to taste. I've made this tail-recursive.

static long Fibo(int n)
{
if (n > 92) throw new ArithmeticException("Overflow");

long Fm = 1, Fmp = 1;
long Fn = 0, Fnp = 1;

while (n > 0)
{
if ((n & 1) == 1)
{
long t = Fmp * Fn + Fm * (Fnp - Fn);
Fnp = Fmp * Fnp + Fm * Fn;
Fn = t;
}

long t = Fm * (2*Fmp - Fm);
Fmp = Fmp*Fmp + Fm*Fm;
Fm = t;

n >>= 1;
}

return Fn;
}

Or if SML is your cup of tea then

fun fibo x =
let fun f 0 _ _ Fn _ = Fn
| f n Fm Fmp Fn Fnp =
let val Fm2 = Fm * (2 * Fmp - Fm)
val Fmp2 = Fmp * Fmp + Fm * Fm
in if (n mod 2 = 0)
then f (n div 2) Fm2 Fmp2 Fn Fnp
else f (n div 2) Fm2 Fmp2 (Fmp * Fn + Fm * (Fnp - Fn)) (Fmp * Fnp + Fm * Fn)
end
in f x 1 1 0 1
end;

Detailed reasoning at http://www.cheddarmonk.org/Fibonacci.html

-

Knowing that the sum of the fibbonacci series diverges, I can print the result rather trivially with 16 characters in Javascript.

alert(Infinity);

Ok, ok, enough glibness. This probably doesn't meet your criterion about trivial calculation. Here is my full version in Javascript:

var old = new Date().getTime();
var fib1 = 1, fib2 = 1, sum = 0;
while(new Date().getTime() - old < 2){
sum = fib1 + fib2;
fib1 = fib2;
fib2 = sum;
}

What it alerts is unpredictable, it ranges from 1e+100 to Infinity for me.

And here it is golfed (91 chars):

o=new Date().getTime();f=1,g=1,s=0;while(new Date().getTime()-o< 2){s=f+g;f=g;g=s}alert(s);

-
I know that the question says just "print a sum of as many Fibonacci numbers as possible" without saying anything about them being unique, consecutive, or starting at 1, but I think it's implicit that it should be a sum of consecutive Fibonacci numbers starting 1, 1, 2. If you put an alert inside your loop you'll see that the values you output start 2, 5, 12, 29, 70. They should be 2, 4, 7, 12, 20. – Peter Taylor Apr 16 '11 at 7:16
@Peter Taylor That can be easily fixed just by changing the initial value of fib2 to 0. – Peter Olson Apr 16 '11 at 16:36
That gives 1, 2, 5, 12, 29, 70. Still broken. – Peter Taylor Apr 16 '11 at 18:28
+1 for the first code example - :) – Nate Koppenhaver Apr 26 '11 at 17:24
@Peter Taylor I fixed it, I think. I changed sum += fib1 + fib2 to sum = fib1 + fib2. – Peter Olson Apr 27 '11 at 20:12
show 1 more comment

Okay. I took this question as find every fib from fib(0) to fib(n) and take the sum: what's the biggest n you can deal with in 8 sec?

#!/usr/bin/env clisp

(defun sum (list)
(setf s 0)
(loop for foo in list do
(setf s (+ s foo))
)
s
)

(defun fib-list (n)
(loop for x from 1 to n do
(setf *fibs* (append *fibs* (list (sum (last *fibs* 2)))))
)
)

(defun test ()
(progn
;; variable initializations
(setf n 12000)
(setf ti (get-universal-time))
(setf tf (get-universal-time))

(loop while (<= (- tf ti) 8) do ; while the end time minus the start time
; is less than 8 sec...
(progn
(print n)
(setf *fibs* '(1 1))
(setf ti (get-universal-time))  ; record the start time

(fib-list n)
(print (sum *fibs*))

(setf tf (get-universal-time))  ; record the end time
(setf n (+ n 25))
)
)
(print "MAXIMUM NUMBER......")
(print n)
)
)

(test)

Now. this only uses one optimization: my fib calculating method, which uses a list to minimize the repetition of calculations.

The test routine reports that this code can deal with an n of 13000 in 8 seconds under win7 on a 2.35Ghz chip. I discount my dual-cores because clisp is a single-threaded process which derives no benefit therefrom.

no fancy math here, just ANSI CLISP and brute force "trivial calculation".

-


text:
## DMOPC '19 Contest 1 P3 - Simple Math

View as PDF

Points: 12
Time limit: 5.0s
Memory limit: 128M

Author:
Problem type

In math class, Bob is currently studying systems of linear equations. Being bored of his teacher's lectures, he decides to make a problem for himself. In his problem, he is trying to solve for the variables, . He then writes equations, the -th of which being the equation where . Believing that simply finding a solution to this problem would be too easy, he instead wants to find how many solutions of exist if he constrains each of the to be a positive integer less than or equal to . Since this number might be very large, he would be satisfied with this number modulo .

#### Constraints

There is at most solution.

#### Input Specification

The first line contains three space-separated integers, , , and .
lines follow, the -th of which contains three space-separated integers, , , and .

#### Output Specification

Output one line containing one integer, the number of solutions to the system modulo

#### Sample Input

4 3 5
1 4 6
1 3 5
2 3 3

#### Sample Output

2

#### Explanation for Sample Input

The two solutions are and .

#### Sample Input 2

4 4 5
1 2 2
1 3 2
1 4 2
2 4 4

#### Sample Output 2

0

#### Explanation for Sample Input 2

There are no solutions to this system of equations


text:
Why get the baker's hopes up?

When explaining the butler and baker's dreams, Yosef opened with very similar language:

בְּע֣וֹד ׀ שְׁלֹ֣שֶׁת יָמִ֗ים יִשָּׂ֤א פַרְעֹה֙ אֶת־רֹאשֶׁ֔ךָ

בְּע֣וֹד ׀ שְׁלֹ֣שֶׁת יָמִ֗ים יִשָּׂ֨א פַרְעֹ֤ה אֶת־רֹֽאשְׁךָ֙ מֵֽעָלֶ֔יךָ

As Yosef was saying it, the baker probably thought Yosef was going to say the same thing has he said to the butler, only to have his hopes shattered when Yosef said מעליך.

Why not break the news gently?

Was the parallel language necessary for an accurate interpretation of the dreams? (I suspect that's the answer, but I don't see where it's implied in the dreams.)

Rav Hirsch explains that Yosef interpreted the dreams from within the occurances of the dream. Thus as he explains in the dream of the butler

If one thinks that through dreams or othe symbolic signs that the One who sends the dream to the mind of a person wishes therebye to tell him something, the kind of symbol in the dream must be of such a natuer that the receiver of it can explain it to himself; it must be clear and apparent. He who hears the explanation must, if it is the correct one, be able to say to himself:"I really ought to have thought that out myself".

Indeed, Rav Hirsch on 40:16 also says that the worth of the explanation is seen in the truth of the words. and that

All that Josph added was that the three tendils which the vine had then to mature to blossoms and grapes were three days which still had to pass.

As Rav Hirsch says on Vayeishev 40:17](http://www.chabad.org/library/bible_cdo/aid/8235#v=17&showrashi=true)

And in the topmost basket were all kinds of Pharaoh's food, the work of a baker, and the birds were eating them from the basket atop my head."

All the food was worthy for Pharaoh and yet it was not he but the birds who ate it, and indeed they had the impudence to eat it away from me from the basket on my head. No birds would do that to a living person. It would be too afraid.

There was no way that he could break it gently to him. All he could do was tell him what the meaning of the dream was so that he could spend the next three days preparing for his death. Indeed, once he had explained it, it became obvious to him what would happen and any attempt to soften the explanation would be regarded as an insult.

As you suspected, the language had to be similar in order to show the accuracy of the two dreams. This is in order to say that these to dreams are closely related in that both will be יִשָּׂ֤א פַרְעֹה֙ אֶת־רֹאשֶׁ֔ךָ after three days with the difference in final results as shown by the ending of the dreams. Note the trop (cantillation marks) on the two phrases to show how they are related.

I haven't seen this directly addressed, but here's my own theory based on the explanation of the Chidah, who explains that from the the baker's saying "from on my head" (which was unnecessary, as he had already mentioned where the baskets were), Yosef understood that subconsciously the baker realized (his mazal saw) that he was going to die and was expressing it with this extra phrase. Maybe Yosef, in confirming this subliminal understanding, chose to employ the same wording of מעליך, and it wasn't really getting the baker's hopes up, as on some level the baker already knew the truth.

As is clear in the posuk, the baker only approached Yosef for an interpretation of his dream upon seeing the butler coming out with good news.

Seeing this it is safe to differentiate between the baker and the butler in that the butler was seeking the true message of his dream whereas the baker was going for the good news only, which means he was masquerading, pretending to be an innocent person seeking for the message relayed to him when in truth he was not which made him a liar.

It is important to note that when the Torah tells of someone lying or any other trait be it good or bad, it is a deeply rooted attribute of that person.For the Torah is very relevant and concise and in a posuk or two the Torah brings out the essence of the person spoken about. This is as we see that the Torah highlights Avraham's attribute of Chesed in a mere few pesukim and in Chazal's eyes it is this facet and virtue which stood as his merit forever.

The Midrash Bereishit Rabba (19:2) points out four that began their remark with the word 'Aff' meaning anger and were lost with that word, one of them being our baker. Now see the pirush Maharza'v on the Midrash who explains that these four were up to cruelty which brought this terminology to their tongue's. This illustrates to us how Chazal took every word that any character uttered and used that to explain their entire predicament.

Well, the posuk in Shmuel b' 22' says "with a crook act crooked".Yosef, replying to the baker's dishonest request in searching for the dream's true meaning, answered him accordingly, by appearing as though ready to glorify his quest for the good news.

Another understanding; the butler dreamt that he was serving the king again which meant that this was good news for him whereas the baker would not even dream of baking for the king again, as we do not see him dreaming that he carried the food to the king, showing us all that good news for him was anything but serving the king again


text:
The Peril of Power when Prioritizing a Point Estimate

I recently noticed that the Pfizer immunobridging trials, presumably set up to demonstrate that their COVID-19 vaccines elicit the same antibody response in children as was seen in 16-25 year olds, for whom efficacy has previously been demonstrated, have a strange criteria for “success”.
COVID-19
non-inferiority trials
clinical trials
power
Author

Lucy D’Agostino McGowan

Published

February 21, 2022

I recently noticed that the Pfizer immunobridging trials, presumably set up to demonstrate that their COVID-19 vaccines elicit the same antibody response in children as was seen in 16-25 year olds, for whom efficacy has previously been demonstrated, have a strange criteria for “success”.

GMT: geometric mean titer. This is a measure of the antibody titers. We use the geometric mean because this data is quite skewed (it is also why you typically see it plotted on the log scale). For those of you who ❤ math, the equation for the geometric mean is just $$\exp\{\frac{\sum_{i=1}^n\textrm{log}(x_i)}{n}\}$$

The primary endpoint of these trials is geomteric mean titer ratio, that is, the ratio between the geometric mean antibody concentration in the younger age groups compared to the 16-25 year olds’ geometric mean antibody concentration.

According to the recent write-up from the 5-11 trial in the New England Journal of Medicine, the trials have been set with two measures of success:

1. The lower bound of the GMT ratio must be $$\ge 0.67$$
2. The point estimate of the GMT ratio must be $$\ge 1$$

The second criteria was originally set by Pfizer to require that the point estimate of the GMT ratio must be $$\ge 0.8$$, however after their data lock the FDA requested this to be changed to the higher threshold. While at first glance, this may seem to make sense, after all we often want to make sure that we hold our pediatric trials to a high standard of efficacy, it turns out this change has statistical implications that change the target in ways that are non-standard for non-inferiority trials.

What do I mean? If we believe that the distribution of antibody concentration in children is exactly the same as what we observed in 16-25 year olds, we would expect this second criteria to fail 50% of the time. Why? When doing any trial, we are observing a sample, not the whole population. We expect a certain amount of uncertainty in our estimates. Here is a small example. Below I have generated 10 samples of 250 people from a log normal distribution with a mean of 1142.5 (log mean of 7.04) and a log standard deviation of 0.8. I am comparing this to an observed sample of 253 individuals drawn from the exact same distribution (incidentally, there were 253 16-25 year olds that led to the “benchmark” of a geometric mean of 1142.5).

Want to try it yourself? You can generate samples from a lognormal distribution in R like this rlnorm(253, mean = 7.040974, sd = 0.8). You can then compare the geometric mean ratio across several samples generated from the same distribution.

Notice in the above plot that even though the true GMT ratio between these groups should be 1 (they were drawn from the exact same distribution!), when using the point estimate as the threshold for success, we “failed” 50% of the time. It was a coin toss whether this trial succeeded or failed by this criteria.

The standard error is just the {standard devation} divided by the square-root of the sample size, $$n$$, $$\frac{sd}{\sqrt{n}}$$

This may not be completely intuitive at first glance, but in fact we can show that the probability of “succeeding” under this criteria is driven by how much better the antibody concentration is in children compared to the benchmark – it is linked to the standard error.

This standard error multiplier is just the critical value derived from a standard normal distribution at a given quantile. You can calculate any success probability in R using the qnorm() function. For example, if we wanted to know the standard error multiplier we’d need to have at least a 70% chance of succeeding by this criteria, you would run qnorm(0.7), revealing we’d need to be at least $$0.52 \times \textrm{standard errors}$$ better than the target.

In order to have a probability of success greater than 80%, for example, the childrens’ antibody response would need to be at least $$0.84 \times$$ standard errors better than the 16-25 year olds’. To have a probability of success greater than 90%, the childrens’ antibody response would need to be at least $$1.28 \times$$ standard errors better than the 16-25 year olds’.

As many have pointed out, it is not uncommon for pediatric trials to be held to a higher standard, often requiring efficacy beyond what is required of adults due to an appropriate caution against intervention in an often vulnerable group. I fully believe that the regulators that requested this had every best intention in doing so. In this particular case, however, this type of threshold can potentially lead to the opposite effect. By requiring the younger children to mount a higher antibody response than the older cohort in order to pass regulatory hurdles, we may be inadvertently pushing towards higher dosing, for example.

It is possible that the choice of 1 was made with the full understanding that the truth had to be better than 1 to pass that threshold with any amount of confidence, but it is important that everyone who contributes to decisions about thresholds understands both the math and the rationale for the choice.

Is this why the 2-4 year old vaccine failed previously? It’s not totally clear since the data hasn’t been released, however based on the tid bits we’ve gotten from media reports, I don’t think so. The New York Times reported, for example, that the 2-4 vaccine only elicited 60% of the response compared to the 16-25 year olds, suggesting that it would have failed by the lower bound criteria alone. So why does this matter? Presumably, these thresholds will be used to compare the post-3rd dose response to the 16-25 year olds as well – does it make sense to require the under 5s to have a stronger antibody response than the 16-25 year olds? Especially with no other option for protection via a vaccine for this age group, I would say no.