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The weight of the load carried in the wheelbarrow distributed evenly between the wheel and handles.


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A Comprehensive Course in Analysis

Barry Simon California Institute of Technology, Pasadena, CA
Available Formats:
Hardcover ISBN: 978-1-4704-1098-8
Product Code: SIMON-SET
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AMS Member Price: $296.80 Click above image for expanded view A Comprehensive Course in Analysis Barry Simon California Institute of Technology, Pasadena, CA Available Formats:  Hardcover ISBN: 978-1-4704-1098-8 Product Code: SIMON-SET  List Price:$371.00 MAA Member Price: $333.90 AMS Member Price:$296.80
• Book Details

2015; 3259 pp

A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.

Get the Ultimate Companion to A Comprehensive Course in Analysis free with any purchase of the complete set.

Researchers (mathematicians and some applied mathematicians and physicists) using analysis, professors teaching analysis at the graduate level, graduate students who need any kind of analysis in their work.

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• There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopedic nature.

Fritz Gesztesy, Mathematical Reviews
• There is no need to belabor the point that this is a fabulous set of texts and will be a smash hit in graduate programs with good taste and good students willing to work hard and ready for exposure to mathematical culture of a wonderful sort. In addition to being a fine mathematician and teacher of mathematics, Simon is something of a raconteur (in the best sense of the word) with a strong interest in history. The books are peppered with historical asides and human interest material, and this feature adds to their readability. Indeed, they are beautifully and clearly written and certainly make for a major contribution to the literature at the intended level and beyond. When I learned that the AMS (which is to be congratulated with this publication) was launching this series by Barry Simon, I, of course, expected a great deal. I was by no means disappointed; these books are terrific.

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2015; 3259 pp

A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.

Get the Ultimate Companion to A Comprehensive Course in Analysis free with any purchase of the complete set.

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# Glossaries: introduce short name only if used more than once within section/chapter/part

I would like to obtain the following behavior: If an acronym is only used once within a section/chapter/part, at this occurrence only the long name should be used. If however the acronym is used more than one, I want the long-short behavior.

I have the following non working MWE:

\documentclass{book}
\usepackage{xparse}
\ExplSyntaxOn
\NewExpandableDocumentCommand{\fpcompare}{ m m m }
{
% #1 = test to perform
% #2 = text for the true case
% #3 = text for the false case
\fp_compare:nTF { #1 } { #2 } { #3 }
}
\ExplSyntaxOff

\usepackage[acronym,symbols,nomain,toc,sanitizesort,nogroupskip,xindy,numberedsection=autolabel]{glossaries-extra}
\setglossarystyle{super}
\setabbreviationstyle[acronym]{long-short}

\GlsXtrEnableEntryUnitCounting{acronym}{1}{section}
%\renewcommand*{\gls}{\cgls}%
%\renewcommand*{\Gls}{\cGls}%
%\renewcommand*{\glspl}{\cglspl}%
%\renewcommand*{\Glspl}{\cGlspl}%
%\renewcommand*{\GLS}{\cGLS}%
%\renewcommand*{\GLSpl}{\cGLSpl}%

\makenoidxglossaries
\newacronym{wn}{WN}{wireless network}

\glsxtrnewsymbol[description={pi},type=symbols]{symb:pi}{\ensuremath{\pi}}%

\usepackage{xstring}
\renewcommand*{\gls}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%to apply this to acronym only (and not symbols as well)
\fpcompare{\glsentrycurrcount{#1} > 1}{\glsentrydesc{#1}}{\cgls{#1}}% fpcompare to detect how often used
}{%
\cgls{#1}%
}%
}%
\renewcommand*{\Gls}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%
\fpcompare{\glsentrycurrcount{#1} > 1}{\Glsentrydesc{#1}}{\cgls{#1}}%
}{%
\cgls{#1}%
}%
}%
\renewcommand*{\glspl}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%
\fpcompare{\glsentrycurrcount{#1} > 1}{\glsentryplural{#1}}{\cglspl{#1}}%
}{%
\cglspl{#1}%
}%
}%
\renewcommand*{\Glspl}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%
\fpcompare{\glsentrycurrcount{#1} > 1}{\Glsentryplural{#1}}{\cglspl{#1}}%
}{%
\cglspl{#1}%
}%
}%
\renewcommand*{\GLS}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%
\fpcompare{\glsentrycurrcount{#1} > 1}{\GLSentrydesc{#1}}{\cgls{#1}}%
}{%
\cgls{#1}%
}%
}%
\renewcommand*{\GLSpl}[1]{%
\IfEq{\glscategory{#1}}{acronym}{%
\fpcompare{\glsentrycurrcount{#1} > 1}{\GLSentryplural{#1}}{\cglspl{#1}}%
}{%
\cglspl{#1}%
}%
}%

\begin{document}
\section{A}
\gls{wn}\\
\glspl{wn}\\
\glspl{wn}\\
\glspl{wn}\\
count: \glsentrycurrcount{wn}\\
\gls{symb:pi}
\section{B}

\glspl{wn}\\
count: \glsentrycurrcount{wn}

\section{C}
\Glspl{wn}\\
\gls{wn}\\
count: \glsentrycurrcount{wn}
\end{document}

which produces:

I however, would have expected/wanted a count of 4 in section A, and in section C, that at first the full description is printed instead of WNs

How to make this work reliably?

The solution was given by Nicola here in this example:

% arara: pdflatex
% arara: pdflatex
% arara: makeglossaries
% arara: pdflatex
\documentclass{article}

\usepackage{glossaries-extra}

\makeglossaries

\GlsXtrEnableEntryCounting
{abbreviation}% list of categories to use entry counting
{2}% trigger value

\newabbreviation{html}{HTML}{hypertext markup language}
\newabbreviation{xml}{XML}{extensible markup language}

\newglossaryentry{sample}{name={sample},description={sample}}

\begin{document}
This is a sample document that uses entry counting. The entry counting
has been enabled on the \texttt{abbreviation} category.
This means that abbreviations will only be added to the glossary
if they have been used more than $n$ times, where in this
document $n$ has been set to
\glsgetcategoryattribute{abbreviation}{entrycount}.
Entries in other categories behave as normal.

Used once: \gls{html}.

Used twice: \gls{xml} and \gls{xml}.

Used three times: \gls{css} and \gls{css} and \gls{css}.

Used once but this entry is in the general'' category
which doesn't have the entrycount'' attribute set:
\gls{sample}.

\printglossaries

\end{document}

To automatically reset the count e.g. for chapters use this code (copied from manual). Note the \GlsXtrEnableEntryUnitCounting command.

\documentclass{report}
\usepackage{glossaries-extra}
\GlsXtrEnableEntryUnitCounting{abbreviation}{2}{chapter}
\makeglossaries
\newabbreviation{html}{HTML}{hypertext markup language}
\newglossaryentry{sample}{name={sample},description={sample}}
\begin{document}
\chapter{Sample}
Used once: \gls{html}.
Used three times: \gls{css} and \gls{css} and \gls{css}.
Used once: \gls{sample}.
\chapter{Another Sample}
Used once: \gls{css}.
Used twice: \gls{html} and \gls{html}.
\printglossaries
\end{document}

I am just pasting this here for the sake of completeness, please upvote the answer the in the other thread.

• How does this take care of the "more than once within section/chapter/part"-bit of your question? It seems to work when I compile it, but which part of the code defines within which strucutre the counting should be? – sheß Mar 13 '19 at 8:34
• Oh, I see, it's the \GlsXtrEnableEntryCounting{abbreviation}{2} – sheß Mar 13 '19 at 8:35

You could make acronyms section-specific by giving them section-specific names. Of course you could easily automate this:

\documentclass{book}
\usepackage[xindy,style=long,numberline,savewrites=true,acronym,nomain]{glossaries}
\usepackage{pgffor}

\makeglossaries
\glsenableentrycount

%create a separate acrony for each section
\let\oldnewacronym\newacronym   \renewcommand{\newacronym}[3]{\foreach \n in {0,...,10}{\oldnewacronym{c\n-#1}{#2}{#3}}}

%%patch all the commands to use the within-section version of the acronym
\let\oldcgls\cgls \renewcommand{\cgls}[1]{\oldcgls{s\arabic{section}-#1}}
\let\oldcglspl\cglspl \renewcommand{\cglspl}[1]{\oldcglspl{s\arabic{section}-#1}}
\let\oldcGlspl\cGlspl \renewcommand{\cGlspl}[1]{\oldcGlspl{s\arabic{section}-#1}}

\newacronym{wn}{WN}{wireless network}

\begin{document}
\section{A}
\cgls{wn}\\
\cglspl{wn}\\
\cglspl{wn}\\
\cglspl{wn}\\
count: \glsentrycurrcount{s1-wn}\\

\section{B}
\cglspl{wn}\\
count: \glsentrycurrcount{s2-wn}

\section{C}
\cGlspl{wn}\\
\cgls{wn}\\
count: \glsentrycurrcount{s3-wn}
\end{document}

Basically, when the acronym wn is defined, it creates separate acronyms for each section, naming them s1-wn and so on. Now when the acronym is called within section X, it "redirects" and uses sX-wn. The result:

Of course this would need some more tweaking:

1. So far it only works for sections, not chapters, but that should be straighforward
2. It is hard-wired to work for up to 10 sections. You could increase that number, introduce a new counter, or do some .aux-file-magic to automatize this to work for any number of chapters/section/subsections
3. It also needs some further tweaking to deal with symbols and so on.
4. I haven't tried it with glossaries-extra so far, because I don't have that installed.
5. I don't now how you want your list of abbreviations to appear. So for now my proposal ignores this, as did your question.

Credits: Part of my solution is taken from here


text:
# Difference between revisions of "2018 AIME I Problems/Problem 12"

## Problem

For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

## Solution 1

Rewrite the set after mod3 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 All 0s can be omitted Case 1 No 1 No 2 1 Case 2 222 20 Case 3 222222 1 Case 4 12 6*6=36 Case 5 12222 6*15=90 Case 6 1122 15*15=225 Case 7 1122222 15*6=90 Case 8 111 20 Case 9 111222 20*20=400 Case 10 111222222 20 Case 11 11112 15*6=90 Case 12 11112222 15*15=225 Case 13 1111122 6*15=90 Case 14 1111122222 6*6=36 Case 15 111111 1 Case 16 111111222 20 Case 17 111111222222 1 Total 1+20+1+36+90+225+90+20+400+20+90+225+90+36+1+20+1=484+360+450+72=1366 P=1362/2^12=683/2^11 ANS=683

## Solution 2

Consider the numbers {1,4,7,10,13,16}. Each of those are congruent to 1mod3. There is ${6 \choose 0}=1$ way to choose zero numbers ${6 \choose 1}=6$ ways to choose 1 and so on. There ends up being ${6 \choose 0}+{6 \choose 3}+{6 \choose 6}$ possible subsets congruent to 0mod 3. There are $2^6=64$ possible subsets of these numbers.

## Solution 3

Notice that six numbers are $0\pmod3$, six are $1\pmod3$, and six are $2\pmod3$. Having numbers $0\pmod3$ will not change the remainder when $s(T)$ is divided by $3$, so we can choose any number of these in our subset. We ignore these for now. The number of numbers that are $1\pmod3$, minus the number of numbers that are $2\pmod3$, must be a multiple of $3$, possibly zero or negative. We can now split into cases based on how many numbers that are $1\pmod3$ are in the set.

Case 1- $0$, $3$, or $6$ integers: There can be $0$, $3$, or $6$ integers that are $2\pmod3$. We can choose these in $\left(\binom60+\binom63+\binom66\right)\cdot\left(\binom60+\binom63+\binom66\right)=(1+20+1)^2=484$ ways.

Case 2- $1$ or $4$ integers: There can be $2$ or $5$ integers that are $2\pmod3$. We can choose these in $\left(\binom61+\binom64\right)\cdot\left(\binom62+\binom65\right)=(6+15)^2=441$ ways.

Case 3- $2$ or $5$ integers: There can be $1$ or $4$ integers that are $2\pmod3$. We can choose these in $\left(\binom62+\binom65\right)\cdot\left(\binom61+\binom64\right)=(15+6)^2=441$ ways.

Adding these up, we get that there are $1366$ ways to choose the numbers such that their sum is a multiple of three. Putting back in the possibility that there can be multiples of $3$ in our set, we have that there are $1366\cdot\left(\binom60+\binom61+\binom62+\binom63+\binom64+\binom65+\binom66+\right)=1366\cdot2^6$ subsets $T$ with a sum that is a multiple of $3$. Since there are $2^{18}$ total subsets, the probability is $\frac{1366\cdot2^6}{2^{18}}=\frac{683}{2^{11}}$, so the answer is $\boxed{683}$.


text:
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text:
# Simultaneous 2nd Price Item Auctions with No-Underbidding

We study the price of anarchy (PoA) of simultaneous 2nd price auctions (S2PA) under a natural condition of no underbidding. No underbidding means that an agent's bid on every item is at least its marginal value given the outcome. In a 2nd price auction, underbidding on an item is weakly dominated by bidding the item's marginal value. Indeed, the no underbidding assumption is justified both theoretically and empirically. We establish bounds on the PoA of S2PA under no underbidding for different valuation classes, in both full-information and incomplete information settings. To derive our results, we introduce a new parameterized property of auctions, namely (γ,δ)-revenue guaranteed, and show that every auction that is (γ,δ)-revenue guaranteed has PoA at least γ/(1+δ). An auction that is both (λ,μ)-smooth and (γ,δ)-revenue guaranteed has PoA at least (γ+λ)/(1+δ+μ). Via extension theorems, these bounds extend to coarse correlated equilibria in full information settings, and to Bayesian PoA (BPoA) in settings with incomplete information. We show that S2PA with submodular valuations and no underbidding is (1,1)-revenue guaranteed, implying that the PoA is at least 1/2. Together with the known (1,1)-smoothness (under the standard no overbididng assumption), it gives PoA of 2/3 and this is tight. For valuations beyond submodular valuations we employ a stronger condition of set no underbidding, which extends the no underbidding condition to sets of items. We show that S2PA with set no underbidding is (1,1)-revenue guaranteed for arbitrary valuations, implying a PoA of at least 1/2. Together with no overbidding we get a lower bound of 2/3 on the Bayesian PoA for XOS valuations, and on the PoA for subadditive valuations.

## Authors

• 20 publications
• 1 publication
• ### Price of Anarchy of Simple Auctions with Interdependent Values

We expand the literature on the price of anarchy (PoA) of simultaneous i...
11/01/2020 ∙ by Alon Eden, et al. ∙ 0

• ### The Price of Anarchy in Auctions

This survey outlines a general and modular theory for proving approximat...
07/26/2016 ∙ by Tim Roughgarden, et al. ∙ 0

• ### On the Competition Complexity of Dynamic Mechanism Design

The Competition Complexity of an auction measures how much competition i...
09/22/2017 ∙ by Siqi Liu, et al. ∙ 0

• ### The Declining Price Anomaly is not Universal in Multi-Buyer Sequential Auctions (but almost is)

The declining price anomaly states that the price weakly decreases when ...
05/02/2019 ∙ by Vishnu V. Narayan, et al. ∙ 0

• ### Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity

We introduce two new "degree of complementarity" measures, which we refe...
05/11/2018 ∙ by Wei Chen, et al. ∙ 0

• ### Simple combinatorial auctions with budget constraints

We study the efficiency of simple combinatorial auctions for the allocat...
07/28/2020 ∙ by Alexandros A. Voudouris, et al. ∙ 0

• ### Selling to Cournot oligopolists: pricing under uncertainty & generalized mean residual life

We study a classic Cournot market, which we extend to a two-stage game w...
09/27/2017 ∙ by Stefanos Leonardos, et al. ∙ 0

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## 1 Introduction

Simple auctions are often preferred in practice over complex truthful auctions. There has been a vast literature in the last decade studying the performance of simple auctions, using the price of anarchy (PoA) measure. The PoA is the ratio between the worst performance of an auction in equilibrium (for different equilibrium notions, see below) and the optimal outcome, with respect to some objective function.

Starting with the seminal paper of Christodoulou et al. [4, 5], a lot of effort has been given to quantifying the performance of simultaneous item auctions, using the PoA measure with respect to the social welfare objective. In a simultaneous item auctions, bidders submit bids on individual items despite the fact that their valuation is non-linear and exhibit different levels of substitutes and complements. Clearly, these auctions are non-truthful; bidders don’t even have the language to express their true valuations. The main message from [4] and follow-up work [16; 1; 11; 17; 9; 19; 6; 18] is that the performance of simultaneous item auctions is nearly optimal as long as the valuations are subadditive (also known as complement free).

The PoA of simple auctions has been studied with respect to different equilibrium notions in both complete- and incomplete-information settings. The most natural equilibrium notion in complete-information setting is pure Nash equilibrium (PNE), but PoA results have been extended to mixed Nash equilibrium (MNE), correlated equilibrium (CE), coarse-correlated equilibrium (CCE) and learning-based equilibrium notions [2; 16; 1; 17; 9; 19; 6; 18]. For precise definitions of these equilibrium notions, see Section 2.1

. In the incomplete-information setting, the profile of bidder values is drawn from a known probability distribution. Every bidder knows her own value, but only the probability distribution of others’ values. The

Bayesian PoA (BPoA) measures the performance of the worst Bayes Nash equilibrium, where every bidder maximizes her utility in expectation over the other bidders’ values.

In simultaneous item auctions with bidders and items, every bidder has a valuation function , where is the value bidder assigns to set . The valuation function is assumed to be monotone and normalized. Despite the combinatorial structure of the valuation, bidders submit bids on every item separately and simultaneously. Let

be the bid vector of bidder

, where is the bid of bidder for item , and be the bid profile of all bidders. Every item is sold separately via an auction based on bids on item alone.

The two main auction formats studied in the literature are simultaneous first-price auctions (S1PA) and simultaneous second-price auctions (S2PA). As their names suggest, in S1PA, every item is sold in a 1st-price auction; i.e., highest bidder wins and pays her bid, whereas in S2PA every item is sold in a 2nd-price auction; i.e., highest bidder wins and pays the 2nd highest bid.

The PoA and BPoA of simultaneous item auctions depend on the structure of the valuation functions. A hierarchy of complement-free valuations is given in [13]. Four important classes of valuations include unit-demand, submodular, xos, and subadditive valuations, with the following strict containment relation:

A unit-demand valuation is one where there exist values , and . A subadditive valuation is one where , also known as a complement-free valuation, as the value of the union of any two sets never exceeds the sum of their values. The other valuations are formally defined in Section 2.2. Clearly, the PoA can only degrade as one moves to a larger valuation class.

#### PoA and BPoA of simultaneous 2nd price item auctions (S2PA): previous results.

There are some pathological examples showing that the PoA of S2PA can be arbitrarily bad, even in the simplest scenario of single item auction [5]. A common approach toward overcoming such pathological examples is the no overbidding (NOB) assumption, stating that the sum of player bids on the a set of items she wins under bid profile , , never exceeds its value . Consequently, all PoA and BPoA results for S2PA are derived under the NOB assumption.

The PoA of S2PA with NOB for subadditive valuations is [1]. This is tight even with respect to unit-demand valuations. The BPoA of S2PA for xos valuations is [5]. For subadditive valuations the BPoA is at least , and is strictly smaller than [9]. Finally, going beyond subadditive valuation is hopeless: there are instances where the PoA can be as bad as [11; 9]. The above results are summarized in Table 1.

#### No Underbidding (NUB)

Consider the following example (taken from [5]), showing that the PoA for unit-demand valuations is .

###### Example 1.1.

2 bidders, and 2 items: . Bidder 1 is unit-demand with values . Bidder 2 is unit-demand with values . Consider the following bid profile (which is a PNE that adheres to NOB): , and . Under this bid profile, bidders 1 and 2 receive items and , respectively, for a social welfare of 2. The optimal welfare is 4.

In this equilibrium, bidder 1 prefers item , yet bids on item , and gets item instead. The same goes for bidder 2 with respect to item . In what sense is this an equilibrium? To answer this question, we should revisit the foundation of the notion of an equilibrium. A Nash equilibrium is a descriptive, static notion, where no player can make a profitable deviation given the strategies of others. While a NE is a static notion, it is based on the underlying assumption that players engage in some dynamics, where they keep best responding to the current situation until a stable outcome is reached.

Let us revisit the PNE bid profile in Example 1.1 in light of this interpretation. For two sets , the marginal value of given is defined as . Bidder 1’s marginal value for item , given her current allocation (item ) is 1. For bidder , bidding on item is weakly dominated by bidding the marginal value of item given the current outcome, which is . Indeed, if bidder 1 gets item , her marginal value is and she pays at most , so why not?

If a bidder bids on an item less than the item’s marginal value, we say that she underbids. Formally, a bidder is said to underbid on item in a bid profile if , where is the set of items wins under . For example, in Example 1.1, bidder 1 underbids on item (and similarly, bidder 2 underbids on item ). In Section 4 we show that underbidding in a 2nd price auction is weakly dominated in some precise sense.

No-underbidding is not only a mere theoretical exercise. In second price auctions a lot of empirical evidence suggest that bidders tend to overbid, but not underbid (e.g., Kagel and Levin [12], Harstad [10], Cooper and Fang [7] and Roider and Schmitz [15]). It seems that ”laboratory second-price auctions exhibit substantial and persistent overbidding, even with prior experience” [10]. The no-underbidding assumption is also consistent with the assumption made by Nisan et al. [14] that bidders break tie in favor of the highest bid that does not exceed their value. We therefore believe that to get a more accurate measure of the performance of simultaneous 2nd price auctions, we should impose a no underbidding assumption. This begs the following natural question:

Main Question. What is the performance (measured by PoA/BPoA) of simultaneous 2nd price item auctions under a no underbidding assumption?

### 1.1 Our Contribution

We first introduce the notion of item no underbidding (iNUB), where no agent underbids on any item. One might think that by imposing both NOB and iNUB, the optimal welfare will be achieved. This is indeed the case for a single item auction (where the optimal welfare is achieved by imposing any one of these assumptions alone). However, even a simple scenario with 2 items and 2 unit-demand bidders can have a PNE with sub-optimal welfare. This is demonstrated in the following example.

###### Example 1.2.

2 bidders, and 2 items: . Bidder 1 is unit-demand with values . Bidder 2 is unit-demand with values . Consider the following PNE bid profile, which adheres to both NOB and iNUB: . Under this bid profile, bidders 1 and 2 receive items and , respectively, for a social welfare of 4. The optimal welfare is 6. Thus, the PoA is .

Our first result states that is the worst possible ratio for bid profiles satisfying both NOB and iNUB, even for submodular valuations and even in settings with incomplete information (with a product distribution over valuations). In fact, a weaker condition than iNUB suffices for this result, requiring no underbidding only on items , where is the set of items that bidder receives in an optimal allocation under valuation profile , and is the set of items she wins under bid profile .

Theorem [submodular valuations, NOB and iNUB]: For every market with submodular valuations,

• The PoA with respect to CCE and the BPoA (for product or correlated distribution) of S2PA under iNUB are both at least (see Corollary 5.2, which is derived from Theorems 3.5 and 5.1).

• The PoA with respect to CCE, and the BPoA (for product distribution) of S2PA under NOB and iNUB are both at least (see Corollary 5.7, which is derived from Theorems 5.1, 2.16, and 3.6).

The above results are tight, even with respect to PNE and even for unit-demand valuations.

Moreover, the last theorem extends to -submodular valuations, defined as for every . We show that the (B)PoA degrades gracefully with the parameter ; namely the PoA with respect to CCE and the BPoA are at least under iNUB and at least under NOB and iNUB.

Beyond (-)submodular valuations, however, these bounds break. In particular, in Example 6.1 we present an instance with xos bidders and items, where the PoA with iNUB is . We also prove that this is the worst possible PoA, showing that the (B)PoA with xos bidders and items is always at least (see Appendix E). Moreover, in Example E.3 we show an instance with items and xos bidders, where the PoA with NOB and iNUB is , which is no better than the guarantee obtained with NOB alone.

To the best of our knowledge, this is the first PoA separation between submodular and xos valuations in simultaneous item auctions. In fact, the PoA of simultaneous item auctions is often the same for the entire range between unit-demand and xos.

A few remarks are in order. First, the above separation suggests that xos is “far” from submodular. Indeed, in Appendix A we show that xos is not -submodular for any fixed , even in settings with identical items. Second, under iNUB alone (without NOB), the BPoA of -submodular valuations is at least ( for submodular valuations). For xos valuations, however, iNUB alone is not helpful; the PoA may be (see Appendix E).

To deal with valuations beyond submodular, we consider a different no underbidding assumption, which applies to sets of items. A bidder is said to not underbid on a set of items if . The new condition, set no underbidding (sNUB), imposes the set no underbidding condition on every bidder with respect to the set .

With the sNUB definition, the PoA extends to subadditive valuations in full information settings, and to xos valuations even in incomplete information settings (with product distributions).

Theorem [subadditive and xos valuations, NOB and sNUB]: For every market with subadditive valuations, the PoA with respect to CCE of S2PA under NOB and sNUB is at least (see Theorem 7.2). For every market with xos valuations, the BPoA (under product distribution) of S2PA under NOB and sNUB is at least (see Corollary 6.2, which is derived from Theorems 2.15, 4.6 and 3.6). Both results are tight.

For incomplete information we show that the BPoA of subadditive valuations is at least of the optimal social welfare and it can be obtained in a much stronger sense, namely for every bid profile with non-negative sum of utilities (even a non-equilibrium profile) satisfying sNUB. This also holds for markets with arbitrary monotone valuations.

Theorem [Arbitrary valuations, sNUB]:

For every market (arbitrary monotone valuations), the PoA with respect to CCE and the BPoA (for any joint distribution) of S2PA under sNUB is at least

(see Corollary 4.7).

#### Equilibrium existence

PoA results make sense only when the corresponding equilibrium exists. We show that every market with XOS valuations admits a PNE satisfying sNUB and NOB. For subadditive valuations, a PNE might not exist (even without any NOB or NUB conditions). However, under a finite discretized version of the auction, a mixed Bayes Nash equilibrium is guaranteed to exist, and we show that there is at least one bid profile that admits both sNUB and NOB with arbitrary monotone valuation functions.

Interestingly, our results shed new light on the comparison between simultaneous 1st and 2nd price auctions.

#### S1PA vs. S2PA

Table 2 specifies BPoA lower bounds for S1PA and S2PA under NOB, assuming independent valuation distributions. According to these results, one may conclude that S1PA perform better than their S2PA counterparts. Indeed, for S2PA, the PoA is even for unit-demand bidders and even with respect to PNE [5], whereas the BPoA of S1PA is at least , even for the much more general class of xos valuations and with respect to the much more general class of Bayesian NE [19]. Similarly, the BPoA of S2PA may be smaller than for subadditive valuations [9], whereas it is always at least for S1PA[9].

Our new results shed more light on the relative performance of S2PA and S1PA. When considering both no overbidding and no underbidding, the situation flips, and S2PA are superior to S1PA. For xos valuations, the bound for S1PA persists, but for S2PA the bound improves from to . For subadditive valuations and independent valuation distributions, S2PA under sNUB performs as well as S1PA (achieving BPoA of ), however in S2PA the bound holds also for correlated valuation distributions. For valuations beyond subadditive, S2PA performs better ( for S2PA and less than for S1PA). Note that no underbidding is not a reasonable assumption in first price auctions, where bidders pay their bids, therefore no underbidding is only relevant in S2PA.

### 1.2 Our Techniques

The standard technique for establishing performance guarantees for equilibria of simple auctions (i.e., PoA results) is the smoothness framework (see the survey in [18]). Smoothness is a parameterized notion; an auction is said to be -smooth if for any valuation profile and any bid profile there exists a bid for each player , s.t. . It is quite straightforward to show that if an auction is -smooth, then its PoA with respect to PNE is at least .

The power of the smoothness framework is in its extendability. While a lower bound on the PoA with respect to PNE follows easily from the smoothness property, this lower bound extends beyond this equilibrium notion [16; 17; 18; 19]. The first extension theorem shows that smoothness leads to the same PoA bound even with respect to CCE (in full information settings). The second extension theorem shows that smoothness also leads to the same lower bound on the Bayesian PoA in games with incomplete information.

We introduce a new parameterized notion called revenue guaranteed. An auction is said to be -revenue guaranteed if for every valuation profile and bid profile the revenue of the auction is bounded below by .

We show that in every -revenue guaranteed auction, the social welfare in every bid profile with non-negative sum of utilities is at least a fraction of the optimal welfare. Similarly to the smoothness framework, we augment our results with two extension theorems, one for PoA with respect to CCE, and one for BPoA in settings with incomplete information. Moreover, this result holds also in cases where the joint distribution of bidder valuations is correlated (whereas previous BPoA results hold only under a product distribution over valuations).

Combining the two tools of smoothness and revenue guaranteed, we get an improved bound. In particular, we show that in every auction that is both -smooth and -revenue guaranteed, the PoA with respect to CCE is at least . The same holds for the BPoA under product valuation distributions.

With this tool in hand, we analyze simultaneous 2nd price auctions with different valuations functions and different no underbidding conditions, where the goal is to establish revenue-guaranteed parameters that would imply PoA and BPoA bounds.

We first consider submodular and -submodular valuations. We show that every S2PA with -submodular valuations satisfying iNUB is -revenue guaranteed. This directly gives a lower bound of on the BPoA of -submodular valuations (and for submodular valuations). We also show that S2PA with -submodular valuations satisfying NOB are -smooth. Combining -revenue guaranteed with -smoothness gives a bound of on the BPoA for every S2PA with -submodular valuations with NOB and iNUB. For submodular valuations this gives the tight bound.

For valuations beyond -submodular valuations, the iNUB condition is not helpful, so we turn to the stronger sNUB condition. We show that every S2PA with arbitrary monotone valuations satisfying sNUB is -revenue guaranteed for bid profiles with non-negative sum of utilities. This recovers the bound on PoA with respect to CCE and BPoA with correlated distributions for S2PA satisfying sNUB. For XOS valuations, we combine the last result with the known -smoothness to get the bound for S2PA satisfying NOB and sNUB (for PoA with respect to CCE and for BPoA with product distributions). For subadditive valuations, we apply the technique from [1] to yield a tight bound of on the PoA with respect to CCE.

## 2 Preliminaries

### 2.1 Auctions

#### Combinatorial auctions

In a combinatorial auction a set of non-identical items are sold to a group of players. Let be the set of possible allocations to player , the set of possible valuations of player , and the set of actions available to player . Similarly, we let be the allocation space of all players, be the valuation space, and be the action space. An allocation function maps an action profile to an allocation , where is the set of items allocated to player . A payment function maps an action profile to a non negative payment , where is the payment of player . We assume that the valuation function of a player , where , is monotone and normalized, i.e., and also . We let be the valuation profile. An outcome is a pair of allocation and payment and the revenue is the sum of all payments, i.e. . We assume a quasi-linear utility function, i.e. . We are interested in measuring the social welfare, which is the sum of bidder valuations, i.e., . Given a valuation profile , an optimal allocation is an allocation that maximizes the over all possible allocations. We denote by the social welfare value of an optimal allocation.

#### Simultaneous item bidding auction

In a simultaneous item bidding auction (simultaneous item auction, in short) each item is simultaneously sold in a separate auction. An action profile is a bid profile , where is an -vector s.t. is the bid of player to item . The allocation of each item is determined by the bids . We use to denote the items won by player and to denote the price paid by the winner of item . As allocation and payment are uniquely defined by the bid profile, we overload notation and write and .

In a simultaneous second price auction (S2PA), each item is allocated to the highest bidder, who pays the second highest bid, i.e., .

In a simultaneous first price auction (S1PA), each item is allocated to the highest bidder, who pays her bid for that item, i.e., .

#### Full information setting: solution concepts and PoA

In the full information setting, the valuation profile is known to all players. The standard equilibrium concepts in this setting are pure Nash equilibrium (PNE), mixed Nash equilibrium (MNE), correlated Nash equilibrium (CE) and coarse correlated Nash equilibrium (CCE), where . Following are the definitions of the equilibrium concepts. As standard, for a vector , we denote by the vector with the ith component removed. Also, we denote with the space of probability distributions over a finite set .

###### Definition 2.1 (Pure Nash Equilibriun (PNE)).

A bid profile is a PNE if for any and for any , .

###### Definition 2.2 (Mixed Nash Equilibriun (MNE)).

A bid profile of randomized bids is a MNE if for any and for any , .

###### Definition 2.3 (Correlated Nash Equilibriun (CE)).

A bid profile of randomized bids is a CE if for any and for any mapping , .

###### Definition 2.4 (Coarse Correlated Nash Equilibriun (CCE)).

A bid profile of randomized bids is a CCE if for any and for any , .

For a given instance of valuations , the price of anarchy (PoA) with respect to an equilibrium notion is defined as: . For example, the PoA with respect to PNE is . The PoA for the other equilibrium types are defined in a similar manner. For a family of valuations , .

The following lemma will be useful in subsequent sections of this paper.

###### Lemma 2.5.

Consider an S2PA and a valuation . Let be a welfare-maximizing allocation. Then, for every bid profile the following holds:

 n∑i=1∑j∈Si(b)pj(b) ≥ n∑i=1∑j∈S∗i(v)∖Si(b)bij
###### Proof.

Let . Since payments are non-negative, , and each item is sold in a separate second price auction, we get:

 n∑i=1∑j∈Si(b)pj(b) ≥ = n∑i=1n∑l=1,l≠i∑j∈Si(b)∩S∗l(v)maxk≠i bkj  ≥  n∑i=1n∑l=1,l≠i∑j∈Si(b)∩S∗l(v)blj = n∑l=1∑j∈S∗l(v)∖Sl(b)blj

Inequality (2.1) holds since is at most the second highest bid on item . Notice that the term in (2.1) considers for each player all the items she wins in bid profile , which are allocated to some other player in the optimal allocation. Instead, we can change the order of summation and consider for each player all the items which are allocated to her in the optimal allocation, but not in bid profile . This accounts for the last equality. ∎

#### Incomplete information setting: solution concepts and Bayesian PoA

In an incomplete information setting, player valuations are drawn from a commonly known, possibly correlated, joint distribution , and the valuation of each player is a private information which is known only to player . The strategy of player is a function . Let denote the strategy space of player and the strategy space of all players. We denote by the bid vector given a valuation profile .

In some cases, we assume that the joint distribution of the valuations is a product distribution, i.e., . In these cases, each valuation is independently drawn from the commonly known distribution .

The standard equilibrium concepts in the incomplete information setting are the Bayes Nash equilibrium (BNE) and the mixed Bayes Nash equilibrium (MBNE):

###### Definition 2.6 (Bayes Nash Equilibriun (BNE)).

A strategy profile is a BNE if for any , any and any ,

 (2)
###### Definition 2.7 (Mixed Bayes Nash Equilibriun (MBNE)).

A randomized strategy profile is a MBNE if for any , any and any ,

 Ev−i∣viEσ[ui(σi(vi),σ−i(v−i),vi)]≥Ev−i∣viEσ−i[ui(b′i,σ−i(v−i),vi)]

Note that if player valuations are independent, we can omit the conditioning on in Definitions 2.6 and 2.7.

The Bayes Nash price of anarchy is:

 BPoA=infF, σ∈BNEEv[SW(σ(v),v)]Ev[OPT(v)]

The mixed Bayes Nash price of anarchy is defined similarly w.r.t. MBNE.

### 2.2 Valuation Classes

In what follows we present the valuation functions considered in this paper. As standard, for a valuation , item and set , we denote the marginal value of item , given set , as ; i.e., . In a similar manner, the marginal value of a set , given a set , is . Following are the valuation classes we consider:

unit-demand (UD):

A valuation function is UD if there exist values such that for every set , .

submodular (SM):

A valuation function is SM if for every two sets and element , .

xos (also known as fractionally subadditive):

A valuation function is XOS if there exists a set of additive valuations , such that for every set , .

A valuation function is SA if for any subsets , .

monotone (MON):

A valuation function is MON if .

A strict containment hierarchy of the above valuation classes is known: . We now introduce a new class of valuation functions, parameterized by ‘how far’ they are from submodular valuations:

###### Definition 2.8 (α−submodular (α−Sm)).

A valuation function is SM, for , if for every two sets and element , .

###### Lemma 2.9.

For any SM function and any sets :

###### Proof.

Let . As is SM, we have for every . Therefore, . The inequality follows from submodularity, and the last equality is due to telescoping sum. ∎

###### Lemma 2.10.

If a valuation function, , is SM, then there exists a set of additive valuations , such that for every set , and there exists at least one such that .

###### Proof.

The proof is an extension of the proof in Lehmann et al. [13] that any submodular function is XOS. Define additive valuations , one for each permutation of the items in . Let , where is the set of items in permutation preceding item . For any permutation and set with item denoting the the th item of in the permutation ,

 aℓ(S) = ∑j∈Saℓj  =  ∑j∈Sv(j∣Sℓj) ≤ ∑j∈S1α[v({1,2,…,j})−v({1,2,…,j−1})]  =  1αv(S),

where the inequality follows from the definition of SM. For any permutation in which the items of are placed first, we have . ∎

By definition, any SM valuation is SM. On the other hand, there is no , such that XOS is SM. In Appendix A we give an example of identical items, XOS function, sets and such that for every .

### 2.3 Smooth Auctions

We use a smoothness definition based on Roughgarden [16] and Roughgarden et al. [18]:

###### Definition 2.11 (Smooth auction (based on [16], [18])).

An auction is smooth for parameters with respect to a bid space , if for any valuation profile and any bid profile there exists a bid for each player , s.t.:

 ∑i∈[n]ui(b∗i(v),b−i,vi)≥λ⋅OPT(v)−μ⋅SW(b,v) (3)

It is shown in [16; 18] that for every smooth auction, the social welfare of any pure NE is at least . Via extension theorems, this bound extends to CCE in full-information settings and to Bayes NE in settings with incomplete information. These theorems are stated below, and their proofs appear in Appendix B for completeness.

###### Theorem 2.12.

(based on [16], [18]) If an auction is smooth with respect to a bid space , then the expected social welfare of any coarse correlated equilibrium, , of the auction is at least of the optimal social welfare.

###### Theorem 2.13.

(based on [17], [19]) If an auction is smooth with respect to a bid space , then for every product distribution , every mixed Bayes Nash equilibrium, has expected social welfare at least of the expected optimal social welfare.

A standard assumption in essentially all previous work on the PoA of simultaneous second price item auction (e.g., [5], [9], [17], [16], [1]) is no overbidding, meaning that players do not overbid on items they win. Formally,

###### Definition 2.14 (No overbidding (NOB)).

Given a valuation profile , a bid profile is said to satisfy NOB if for every player the following holds,

 ∑j∈Si(b)bij≤vi(Si(b))
###### Theorem 2.15.

(based on [5] and [16]): S2PA with XOS valuations is smooth, with respect to bid profiles satisfying NOB.

###### Proof.

Christodoulou et al. [5] show that for S2PA with XOS player valuations there exists a bid for each player , s.t. Inequality (3) with holds for any bid profile that satisfies NOB. ∎

Theorem 2.15 implies a lower bound of on the Bayesian PoA of S2PA with XOS valuations. This result is tight, even with respect to unit-demand valuations in full information settings [5].

We now extend the last result to -SM valuations.

###### Theorem 2.16.

S2PA with SM valuations is smooth, with respect to bid profiles satisfying NOB.

###### Proof.

Let be an SM valuation profile and let be a PNE satisfying NOB. From Lemma 2.10, for every valuation there exists a set , such that for every set , and there exists such that . Let be a welfare maximizing allocation, and let be an additive valuation such that . Consider the following hypothetical deviation for player : if , and otherwise.

Now let us consider the utility of player when deviating. As for every item , each such item contributes non-negative utility to and we can ignore this contribution while lower bounding ’s utility under . Consider item . If , player wins item . Otherwise, does not win item , and the term is non-positive. Since for every set , and since , we get:

 ∑i∈[n]ui(b∗i(v),b−i,vi) ≥ ∑i∈[n]⎡⎢⎣∑j∈S∗i(v)∩Si(b)(α⋅a∗ij−maxk≠ibkj)+∑j∈S∗i(v)∖Si(b)(a∗ij−maxk≠ibkj)⎤⎥⎦ ≥ ∑i∈[n]∑j∈S∗i(v)(α⋅a∗ij−maxk≠ibkj) ≥ α ∑i∈[n]vi(S∗i(v))−∑i∈[n]∑j∈S∗i(v)maxkbkj ≥ α⋅OPT(v)−∑i∈[n]∑j∈Si(b)maxkbkj = α⋅OPT(v)−∑i∈[n]∑j∈Si(b)bkj ≥ α⋅OPT(v)−∑i∈[n]vi(Si(b)) = α⋅OPT(v)−SW(b,v).

The third inequality follows from the choice of and by the payment structure of 2nd price. The forth inequality follows by the fact that all items are allocated in equilibrium. Finally, the last inequality follows from NOB.

## 3 Revenue Guaranteed Auctions

Following is the definition of revenue guaranteed auctions. We then discuss the implications of this property in both full information and incomplete information settings.

###### Definition 3.1 (Revenue guaranteed auction).

An auction is revenue guaranteed for some with respect to a bid space , if for any valuation profile and for any bid profile the revenue of the auction is at least .

### 3.1 Full Information: Revenue Guaranteed Auctions

The following theorem establishes welfare guarantees on every pure bid profile of a revenue guaranteed auction in which the sum of player utilities is non-negative.

###### Theorem 3.2.

If an auction is revenue guaranteed with respect to a bid space , then for any pure bid profile , in which the sum of player utilities is non-negative, the social welfare is at least of the optimal social welfare.

###### Proof.

Using quasi-linear utilities and non-negative sum of player utilities, we get:

 0≤∑i∈[n]ui(b,vi)=∑i∈[n]vi(Si(b))−∑i∈[n]Pi(b)=SW(b,v)−∑i∈[n]Pi(b)

By the revenue guaranteed property,

 ∑i∈[n]Pi(b)≥γOPT(v)−δSW(b,v).

Punting it all together, we get

 0  ≤  SW(b,v)−∑i∈[n]Pi(b)  ≤  (1+δ)SW(b,v)−γOPT(v) (4)

Rearranging, we get: , as required. ∎

Definition 3.1 considers pure bid profiles, but Theorem 3.2 applies to the more general setting of randomized bid profiles, possibly correlated, as cast in the following extension theorem.

###### Theorem 3.3.

If an auction is revenue guaranteed with respect to a bid space , then for any bid profile , in which the sum of the expected utilities of the players is non-negative, the expected social welfare is at least of the optimal social welfare.

The proof is identical to the proof of Theorem 3.2, except adding expectation over to every term, using the fact the the auction is revenue guaranteed for every in the support of , and using linearity of expectation.

Clearly, in every equilibrium (including CCE) the expected utility of every player is non-negative. It therefore follows that the expected welfare in any CCE is at least of the optimal social welfare.

For an auction that is both smooth and revenue guaranteed, we give a better bound on the price of anarchy:

###### Theorem 3.4.

If an auction is smooth with respect to a bid space and revenue guaranteed with respect to a bid space , then the expected social welfare at any CCE of the auction is at least of the optimal social welfare.

###### Proof.

The proof follows by the proofs of Theorem 2.12 and Theorem 3.3. Let be a CCE of the auction. The proof of Theorem 2.12 shows that:

 ∑i∈[n]Eb[ui(b,vi)]≥λ⋅OPT(v)−μ⋅Eb[SW(b,v)]

From Equation (4) we get,

 Eb[SW(b,v)]−∑i∈[n]Eb[Pi(b)]≤(1+δ)⋅Eb[SW(b,v)]−γ⋅OPT(v)

As utilities are quasi-linear, the left hand side of the above two inequalities are equal. Rearranging, we get: , as required. ∎

### 3.2 Incomplete Information: Extension Theorem for Revenue Guaranteed Auctions

In a similar manner to the smoothness extension theorem, we can prove an extension theorem for the revenue guarantee property, which gives expected welfare guarantees for settings with incomplete information. However, this extension theorem is stronger, in the sense that it holds with respect to correlated distributions and not only product prior distributions.

###### Theorem 3.5.

If an auction is revenue guaranteed with respect to a bid space , then for every joint distribution , possibly correlated, and every strategy profile , in which the expected sum of player utilities is non-negative, the expected social welfare is at least of the expected optimal social welfare.

###### Proof.

We give a proof for pure strategies. The proof for mixed strategies follows by adding in a straightforward way another expectation over the random actions chosen in the strategy profile . As the utility of each player is quasi-linear and the expected sum of player utilities is non-negative, we use linearity of expectation and get,

 0 ≤ ∑i∈[n]Ev[ui(σ(v),vi)] = ∑i∈[n]Ev[vi(S


text:
=>

Thankfully, animal biologists are not supposed to read this blog, as they might be shocked by what follows :

Well, let’s explain !

### * L2A product

(If you are not familiar with the product names, you might find some information here). MAJA is a L2A processor which provides surface reflectances after atmospheric correction with a high quality cloud/shadows mask. MAJA’s products are distributed by THEIA over selected regions of the world, and the processor itself is available. Even if this product has a high quality, it is not fully easy to use, because most of Sentinel-2 images have a significant cloud cover.  The processors which use Sentinel-2 L1C or L2A products must be robust to the presence of spatial or temporal gaps in the coverage.

### * L3A Product

For all the optical remote sensors that provide systematic and frequent acquisitions, It has been a frequent practice to also deliver L3A products, which provide a cloud free image for a longer period of time : one fortnight, one month, one year…Most of the L3A methods select, for each pixel, the best available pixel, using several criteria which are prone to avoid selecting cloudy of shadowed pixels (the most classical one is selecting the date which has the maximal NDVI). Given that these methods work at pixel level, the values of selected contiguous pixel are acquired on different dates, which leads to artefacts related to the use of different dates, different viewing angles…

The good point with L2A products obtained with MAJA is the quality of its cloud/shadows mask. Even if it is not perfect, most of the clouds and shadows are detected, and when they are not detected, it is because the clouds are quite transparent. As a result, it is no longer necessary to use date selection criteria that tend to discard cloud or shadows, as they are already simply discarded using the cloud mask. So we developed WASP, which is the Weighted Average Synthesis Processor. The method was defined in CESBIO for Venµs satellite a long time ago (in 2008). In 2015, it was then included in the Sen2Agri project, funded by ESA, to be applied to Sentinel-2. In that framework, our colleagues from CS-Romania redeveloped the software to make to more operational and provide it as open source. Finally Peter Kettig, at CNES, adapted it to the MUSCATE ground segment and formats, and introduced a few improvements and bug corrections.

### * L3A Method

For each pixel, and each band, WASP almost simply averages the cloud free surface reflectances gathered during a synthesis period of 45 days. For instance, to produce the synthesis of July 15th, it will average all the cloud free L2A observations gathered between June 26th and August 5th. And this is done every month. In details (as already explained in this post) :1- a directional correction is made to homogenise the surface reflectances as if they had been taken from the vertical, avoiding differences where the Sentinel-2 orbits overlap.2- then weights are computed :

• Pixels close to a detected cloud or shadow have a lower weight
• Pixels with a lower aerosol optical thickness have a higher
• A greater weight is given to dates close to the synthesis date (15th of July in the example above)

3- The weighted average is computed The WASP processor is available as an open source software, and Peter Kettig has started producing the Level 3A products over France and a few other tiles. These products are now available on THEIA’s server here. However, the processor is not fully integrated yet within MUSCATE, Peter produces the data himself, so we can’t expect him to deliver them operatinnally in real time. WASP is currently being fully integrated within MUSCATE, so the operational production should happen end of 2018 or beginning of 2019. The detailed processing model for WASP is available here :

### * L3A use case

So, what’s the use of a L3A product :

• as you can see above, the L3A products are almost cloud free over Land, and Theia’s one show a very little amount of artefacts.  They are very convenient to get a good view of a territory, and require less robustness from the processors as the amount of clouds is low (and these clouds are still flagged).
• they can be used to make nice seamless mosaics and posters
• But the knowledge of the exact date of each product is much less accurate than that provided by a L2A product. Although we also provide in the product the weighted average of the dates, it is only an approximation of the surface reflectance of that date, assuming that the evolution with time of surface reflectance along the synthesis period is linear. If you applications requires a good accuracy in terms of date, such as estimating phenology (date of flowering or senescence…), then we advise you  to use Level 2A products.

### * L3A defects

• WASP synthesis method assumes that surface reflectance change very slowly with time. This assumption is wrong over some surfaces such as water or snow.
• It is not always possible to get at least one clear observation for each pixel in a 45 days period. As a result, we still can have a low percentage of cloudy pixels, but these pixels are flagged.
• It is still possible to spot orbit limits on the composites (did you see one on the image above ? I did, but I am trained to see defects, even on my products). These are mainly due to the juxtaposition of different observation dates, but also to directional correction approximations.
• A few clouds or shadows can escape the detection, especially small cloud or shadows of thin clouds, but the weighted average usually decreases their influence, except during very cloudy periods.

### Practical information:

#### Data

python theia_download.py -l 'Bretagne' -a config_hagolle.cfg -d 2018-07-01 -f 2018-07-31 --level LEVEL3A

### Acknowledgements

We would like to thank a lot of people who participated to the development of WASP

• At CESBIO, since 2007, I was successively helped by Vincent Debaecker, Mireille Huc, David Morin and Mohamed Kadiri
• The operational version of WASP derives from the development of Sen2Agri system, funded by ESA (with Benjamin Koetz as appreciated technical officer) and a lively consortium lead by Louvain University (UCL, with Sophie Bontemps as project leader), with CESBIO, CS-France and CS-Romania as partners. It was a pleasure working with WASP developers in Romania (Cosmin Udroiu, Alex Grosu)
• Peter Kettig finalized the development in MUSCATE framework, solved a few issues, defined the final format, optimized the code. He also produces the first data, and supervises the integration within MUSCATE by Cap Gemini
• Joelle Donadieu, at CNES, and the CAP GEMINI Muscate team have smoothly integrated the products within MUSCATE

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text:
PL EN

Journal

## Filozofia (Philosophy)

2018 | 73 | 8 | 589 – 605
Article title

### INDEXICKÁ POVAHA EPISTEMICKÝCH MODALÍT

Authors
Content
Title variants
EN
Indexical nature of epistemic modals
Languages of publication
SK
Abstracts
EN
The aim of the present paper is defending the idea that epistemic modals like “may”, “might”, “must”, etc. are indexical expressions and providing an out-line of their semantics in terms of Kaplanian semantics for indexicals. It is argued that, though closely similar to ordinary indexicals in having both deictic and anaphoric use, epistemic modals are special in having more complicated meaning. This is because their (Kaplanian) content (as well as their character and extension) is twofold – apart from expressing an ordinary intension, like typical indexicals do. They also express a relation between a proposition and a collection of propositions. In particular, assume that SE is a sentence that contains an epistemic modal E, S is a sentence obtained from SE by deleting E (and making all grammatical amendments that are required in order for S to be grammatical), A is an agent and PA is an epistemic perspective of A (i.e. a (sub)set of A’s beliefs); it holds that a speaker’s utterance of SE (where the speaker may, but need not, be identical with A) expresses as its content (relative to a context of utterance) the proposition according to which the proposition expressed by S (relative to the context of utterance) is in a certain relation (like the compatibility relation or the entailment relation, etc.) with PA. The semantic role of E consists in introducing both the epistemic perspective and the relation into the content of utterances of SE.
Keywords
EN
Journal
Year
Volume
Issue
Pages
589 – 605
Physical description
Contributors
author
• Katedra logiky a metodológie vied Filozofickej fakulty Univerzity Komenského v Bratislave, Bratislava, Slovak Republic
References
Document Type
Publication order reference
Identifiers


text:
# Black Holes and Information Loss

Discussion in 'Physics & Math' started by RJBeery, Mar 21, 2017.

1. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
Agreed, but when we consider GR in the context of "observers" and "declarers of existence" and "existing objects" then coordinates do matter. Existence has a necessary temporal element implicit in its definition, and we cannot say that black holes exist in any physical way simply because they exist within the mathematics of general relativity.

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3. ### SchmelzerValued Senior Member

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There is nothing one can do about this. "Exists" in common sense means "exists now", but there is no definition what is "now" in the GR spacetime. So, in a strong sense it makes no sense to say something "exists" if it does not exist for a sufficiently long time so that it can send and receive messages from us now. That means, "exists" is also only sloppy speaking.

All one can say about the existence of BHs is that there may be some observer, who was nearby a million years ago, and send us a good bye message that in a few seconds he will be inside the BH. If classical GR is true, which is quite improbable in this region.

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5. ### RJBeeryNatural PhilosopherValued Senior Member

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Did you read the paper in the OP? I deal with this. While we cannot say what "exists now" we certainly can say what does not. GR allows for ambiguous but not arbitrary definitions of "now".

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7. ### SchmelzerValued Senior Member

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3,703
Sometimes we can, sometimes not. We cannot say if something exists now if there are two time-like coordinates so that in one coordinate it does not exist now, and in the other it does.

GR BHs are of this type. You can have coordinates where they never exist, and others where they exist in short time of the external observer after the collapse has started. You would need a criterion which coordinates define true time, GR does not give anything of this type.

8. ### RJBeeryNatural PhilosopherValued Senior Member

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4,136
This is not true. Whichever definition you give to "now", it cannot involve two events which are time-like separated. There are no "coordinates where [black holes] exist in short time of the external observer after the collapse has started." This is literally a complete contradiction to the GR definition claiming that event horizons do not exist in the past light cone of future null infinity.

9. ### SchmelzerValued Senior Member

Messages:
3,703
Wrong. Take the Kruskal coordinates https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates which have the same light cones as a standard Minkowski space. Put as usual done an infalling line to it which describes the radius of the surface of the collapsing star, I think you have seen that picture somewhere. This line comes from the right down but higher than the X=-T line, moves to the left until it intersects the X=T line (creating the horizon) at some $T_h$ and later hits (creates) the singularity at some $T_s$. Starting with the point where it is behind the horizon the Kruskal time $T>T_h$ will be a time where the BH exists, and a short time later, when the surface hits the singularity, $T>T_s$, is even a time where even the singularity already exists. But T is a normal time-like function, thus, the events on a T=const surface are space-like separated.

10. ### karenmanskerHSIRIBanned

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638
IMHO, RJB is correct!

11. ### SchmelzerValued Senior Member

Messages:
3,703
This is about mathematical facts, not about opinions.

12. ### sweetpeaRegistered Senior Member

Messages:
770
RJB, out of curiosity I followed your link and noticed you give your old or present university ''
University of Nebraska''. Wondering about your qualifications for writing this paper, I found this page...
https://www.researchgate.net/profile/Rhoderick_Beery

I take it you store your 'papers' there. Are your present qualifications to do with the ''Theoretical Physics, Quantum Physics'' subheading on that page ? If yes, what are they? If not, why the mention of your university? To add learned flavour?
Can you do the math which comes with the subjects '' Theoretical Physics, Quantum Physics''?

13. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
Why wouldn't I list my Alma Mater? Researchgate allows for alums to follow each other's work and research. And what to do you mean "do the math"? Yes I can do the math of QM. GR and Christoffel symbols...not so much. Theoretical and Quantum Physics are my areas of interest. I feel like you're trying to make some sort of reverse argument-from-authority...confusing -- logic and reasoning are sufficient to stand on their own without regard for who is authoring them. I think you'll find I'm receptive to errors in logic in anything I've written.
Schmelzer, I can't understand what you're saying. It is against the theory of general relativity for the creation event or the continued existence of an event horizon to reside in an external observer's past light cone. This isn't a controversial claim, this is basic GR.

14. ### SchmelzerValued Senior Member

Messages:
3,703
So what? "Exists now" refers to a time coordinate. Not to some past light cone. The events inside the BH may be in the future light cone, and may be space-like separated, thus, neither in the future nor in the past light cone. For space-like separated events, you have coordinates where the BH already "exists now".

15. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
I'm not talking about third parties making remote declarations. I'm talking about observers (i.e. "us") claiming that events within the black hole are occurring now. If events occur at time T from my perspective then at T+1 they existed in my past.

This is why I tried to focus on whether or not information has been lost to an event horizon. "Has been lost" is necessarily an event occurring in the past.

16. ### sweetpeaRegistered Senior Member

Messages:
770
Thanks for answering.
If you could give your qualifications then I would have some idea how proficient you are at the math of ''Theoretical and Quantum Physics'' .
Knowing that will help me take more interest in your papers, i.e. let me know, and perhaps others, you can do the walk and not just talk the talk. As you are well aware, most models are built from math, understanding the math means you can avoid the pitfalls of layman models.
You must also have noticed how many laypeople 'build' their own 'models' from just an understanding of these layman models.

17. ### karenmanskerHSIRIBanned

Messages:
638
RJB . . . . gullibility warning!!

Last edited: Mar 27, 2017
18. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
I'm not interested in you being interested in my papers, and the idea of doing homework to earn your respect is a bit silly. When math is relevant to the model I provide it. Here's an example: https://goo.gl/uJWO4n

Bras, kets and matrices have no bearing on any papers that I've written to this point. I tend to make aesthetic appeals because my topics are largely philosophical in nature. On the topic in this thread, however, it isn't a matter of philosophy -- if the astrophysicist community believes that black holes have existed in the past (and no longer do, vis-a-vis micro black holes) then they are contradicting the mathematics of general relativity.

19. ### SchmelzerValued Senior Member

Messages:
3,703
Yes, but not in your past lightcone.
Yes, but once contemporaneity is not well-defined but coordinate-dependent in GR, past and future are not well-defined in the same sense. With the exception of the past and future light cones, of course. (And even this only if you ignore solutions with causal loops.)

20. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
Fair. How about this: is there any information which is space-like separated from all external points in spacetime outside of our past light cones?

Last edited: Mar 27, 2017
21. ### karenmanskerHSIRIBanned

Messages:
638
An exercise: Suppose (not real!) that you were able to exist within a BH. As an intelligent, scientific observer, what information would you expect to 'observe' when looking outward from the BH interior (i.e., you are observing incoming information from outside the BH). Would 'space and its contents ' appear to be contracting (toward your POV) or expanding (away from your POV)?

22. ### SchmelzerValued Senior Member

Messages:
3,703
No. Our past lightcone covers all we can see now. Everything which happens later is in some future light cone of some point of our own past light cone.

In particular, what we see if we look at a BH is the surface of the collapsing BH before horizon formation. So, everything behind the BH is in its future.

And as I have already said, we can also use coordinates where the BH horizon never forms, because it remains in the future forever.

So, above positions - the BH exists now and the BH will be created only in the future, or even never - are compatible with GR math, because "exists now" is simply not well-defined.

23. ### RJBeeryNatural PhilosopherValued Senior Member

Messages:
4,136
Please reconcile the above statements with the LHC Safety Assessment report discussing evaporating micro black holes.


text:
# Show that the statement " If $x$ is real number such that $x^3+4x=0$, then $x$ is 0." is true using contra positive method.

Toolbox:
• Contrapositive method of proving a statement "if $p$ then $q$ is prove if ~$q$ then ~$p$.
The given statement is
" If $x$ is real number such that $x^3+4x=0$, then $x$ is 0."
Let $p :$ If $x$ is real number such that $x^3+4x=0.$
and
$q:$ $x=0$.
$\Rightarrow\:$ ~$p:$ $x^3+4x\neq 0$ $\forall\:x \in R$. and
~$q$: $x\neq 0$.
To show that the above statement is true using contra positive method,
Let us assume that $q$ is false.
We have to prove that $p$ is also false.
$i.e.,$ Let $x\neq 0$ and $x\in R$
$x^3+4x=0$
$\Rightarrow\:x(x^2+4)=0$
$\Rightarrow\:$ either $x=0$ or $x^2+4=0$
But our assumption is $x\neq 0$
$\Rightarrow\: x^2+4=0$
$x^2+4$ cannot be $0$ since $x^2 \geq 0\:\: \forall x \in R$
$\therefore$ $x^3+4x$ cannot be equal to zero
$\therefore$ The given statement is true.
edited Jul 26, 2014


text:
# Is f(x)=x^3-x+e^(x-x^2)  concave or convex at x=1 ?

Apr 26, 2018

It is convex.

#### Explanation:

The function will be convex if the second derivative is positive, and concave if it is negative.
Weisstein, Eric W. "Second Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SecondDerivativeTest.html

The first derivative is:
$3 {x}^{2} + \left(- 2 x + 1\right) {e}^{- {x}^{2} + x} - 1$

The second derivative is:
$6 x + {\left(2 x - 1\right)}^{2} {e}^{- {x}^{2} + x} - 2 {e}^{- {x}^{2} + x}$

Calculation detail steps here:
http://calculus-calculator.com/derivative/

At $x = 1$:
$6 + 2 {e}^{2} = 20.78$ positive value


text:
# Alkylation of conjugated nitriles - regioselectivity

Why does the enolate of the nitrile attack the alkyl chloride from the bond shown in the picture and not from the far end pi bond?

• This looks like it can be answered well with orbitals (I’m not the one who could do so, though; it might require calculations). – Jan Jun 9 '15 at 19:21
• The hint I have is that it's the product of kinetic control because of bigger HOMO coefficient. – RBW Dec 30 '15 at 18:43
• According to Fleming's Molecular Orbitals, Huckel's theory can't be applied to these systems as it suggests a bigger coefficient of HOMO at the gamma C atom. – RBW Dec 30 '15 at 19:09
• I don't like this mechanism, It could depend on coordination of lithium which isn't shown. – Mithoron Jan 3 '16 at 18:49
• Marko, do you have any source for this reaction and/or the proposed mechanism? – Martin - マーチン Jan 8 '16 at 11:21

I found this article that sheds some light on deprotonation mechanism - it differs significantly from presented in the question and although it's for acetonitrile not crotonitrile it still provides some useful information.

Lithium amide exists as $\ce{(LiNH2)2}$ dimers which tend to coordinate to nitrile groups creating Li-N bonds. In acetonitrile deprotonation occurs intramolecularly with opening of $\ce{Li2N2}$ rings, not pericyclicly, but for crotonic nitrile analogous mechanism isn't possible what suggests intermolecular reaction.

On the basis of our MP2(full)/6-31+G*//MP2(full)/6-31+G* calculations (Scheme 2 and Figure 1), the adduct, which is formed from CH3CN (7) and the dimeric base 8, is stabilized by about 12 kcal/mol relative to the separated reactants. It is likely that in solution this coordination proceeds by replacement of a solvent molecule from the dimeric lithium amide. The existence of a complex comparable to 9 and the dimeric character of solvated lithium amides 6 are confirmed by NMR spectroscopy and structural studies. To allow hydrogen transfer from the nitrile to the amide moiety, the central four-membered ring of 9 must open. The resulting intermediate 10, an “open dimer”, is also supported by various investigations of related reaction pathways. 7 The cleavage of an Li-N bond requires about 11 kcal/mol, while the transition state for proton transfer, 11, lies only just above slightly more than 1 kcal/mol higher in energy than 10. This low barrier is indicated in a lenghtened C-H bond of the open dimer (1.094 Å, compared with 1.082 Å in acetonitrile). After the formation of another “open dimer”, 12, which is stabilized by 14 kcal/mol relative to the transition state, and extrusion of ammonia, the final intermediate, 14, is a mixed (dianionic) aggregate of LiNH2 and LiCH2CN, comparable to Boche’s QUADAC.

Another article provides some info about anions of nitriles:

The exceptional nucleophilicity stems primarily from the powerful inductive stabilization of metallated nitriles rather than resonance stabilization as is the case with enolates.

and influence of nitrile group on carboanion:

Essentially, the powerful electron withdrawal of the CN group stabilizes the adjacent negative charge in a manner akin to an ylide. Consistent with this inductive stabilization is the range of geometries for the metallated carbon which spans a continuum varying from planar in 12 to pyramidal in the cyclopropanecarbonitrile 13.

Mainly inductive stabilisation suggests that in metallated anion of crotonitrile negative charge is concentrated on alpha carbon and thus it's more nucleophilic then gamma carbon and in low temperature, under kinetic control, attack via alpha carbon should be favoured.

• That sounds very promising, please don't wait too long as I have to assign the bounty soon. I am very much looking forward to reading about this mechanism. I would be very happy, if you could include the full citation in your final review. – Martin - マーチン Jan 6 '16 at 15:51
• I am still looking forward to learning more. ;) – Martin - マーチン Jan 14 '16 at 4:55
• It would be nice to see a conclusion. – RBW Feb 3 '16 at 21:19
• @Marko Conclusion doesn't seem exactly satisfying imo, but maybe I'll find sth else later. – Mithoron Feb 3 '16 at 23:55