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# M28-43

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16 Sep 2014, 01:31
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There are two schools in the village. The average age of pupils in the first school is 12.2 years and the average age of pupils in the second school is 13.1 years. What is the average age of all school pupils in the village?

(1) There are 40 more pupils in the second school than there are in the first.

(2) There are three times as many pupils in the second school as there are in the first.

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16 Sep 2014, 01:31
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Official Solution:

This is a weighted average question. Say $$x$$ and $$y$$ are the number of pupils in the first and the second schools in the village, respectively: $$\text{average}=\frac{12.2x+13.1y}{x+y}$$.

(1) There are 40 more pupils in the second school than there are in the first. Given: $$x+40=y$$. Not sufficient to find the average.

(2) There are three times as many pupils in the second school as there are in the first. Given: $$3x=y$$. Hence, $$\text{average}=\frac{12.2x+13.1*3x}{x+3x}=\frac{12.2+13.1*3}{4} \approx 12.9$$. Sufficient.

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08 Feb 2018, 02:17
Hi Bunuel,

Why ist't statement 1 sufficient? Is it because x doesn't cancel out once it's plugged in, in place of x?

Thanks as per usual...I'd also be grateful if you can give me a tip on studying...I've tried jeff sackmann's math bible and also tried his additional sets of questions...I thought I was half decent as quants by the time I finished it...then did the real test and found myself scoring a 29...I then did the economist gmat and it was helpful in terms of a few tricks like reverse plugging in etc...but I've now found that I've forgotten a few things I've learned via Sackmann...would you suggest a targeted approach on mathclub like subject by subject sets of 10 questions on the quizzes? I'd really appreciate your advice. Best wishes,

Tosin
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08 Feb 2018, 02:35
ttaiwo wrote:
Hi Bunuel,

Why ist't statement 1 sufficient? Is it because x doesn't cancel out once it's plugged in, in place of x?

Thanks as per usual...I'd also be grateful if you can give me a tip on studying...I've tried jeff sackmann's math bible and also tried his additional sets of questions...I thought I was half decent as quants by the time I finished it...then did the real test and found myself scoring a 29...I then did the economist gmat and it was helpful in terms of a few tricks like reverse plugging in etc...but I've now found that I've forgotten a few things I've learned via Sackmann...would you suggest a targeted approach on mathclub like subject by subject sets of 10 questions on the quizzes? I'd really appreciate your advice. Best wishes,

Tosin

For (1): if we substitute $$x+40=y$$ into $$\text{average}=\frac{12.2x+13.1y}{x+y}$$ we'd get $$\text{average}=\frac{25.3 x + 524}{2x+40}$$. Without knowing x or y, we cannot get the single numerical value of this expression, so cannot answer the question.

Check Study Plans for Novice and Advanced GMAT Takers.

For all other QUANT subjects:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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08 Feb 2018, 03:33
Bunuel wrote:
ttaiwo wrote:
Hi Bunuel,

Why ist't statement 1 sufficient? Is it because x doesn't cancel out once it's plugged in, in place of x?

Thanks as per usual...I'd also be grateful if you can give me a tip on studying...I've tried jeff sackmann's math bible and also tried his additional sets of questions...I thought I was half decent as quants by the time I finished it...then did the real test and found myself scoring a 29...I then did the economist gmat and it was helpful in terms of a few tricks like reverse plugging in etc...but I've now found that I've forgotten a few things I've learned via Sackmann...would you suggest a targeted approach on mathclub like subject by subject sets of 10 questions on the quizzes? I'd really appreciate your advice. Best wishes,

Tosin

For (1): if we substitute $$x+40=y$$ into $$\text{average}=\frac{12.2x+13.1y}{x+y}$$ we'd get $$\text{average}=\frac{25.3 x + 524}{2x+40}$$. Without knowing x or y, we cannot get the single numerical value of this expression, so cannot answer the question.

Check Study Plans for Novice and Advanced GMAT Takers.

For all other QUANT subjects:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.

Thanks bro
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01 Jun 2018, 12:02
Hi Bunuel, great breakdown of statement 1. I completely understand why statement 1 is insufficient (ie x can be anything), but what I do not understand is why (25.3x+5242)/(2x+40) is not considered one variable, one equation situation. Can you please explain? Is it because average itself is considered an unknown?
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02 Jun 2018, 03:48
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norovers wrote:
Hi Bunuel, great breakdown of statement 1. I completely understand why statement 1 is insufficient (ie x can be anything), but what I do not understand is why (25.3x+5242)/(2x+40) is not considered one variable, one equation situation. Can you please explain? Is it because average itself is considered an unknown?

Because it's NOT an equation. If it were (25.3x+5242)/(2x+40) = (some number), then yes, we'd be able to solve for x. For example, imagine that we want to find the value of x + 10. We have one variable x but we cannot find the value of x + 10 because for that we need the value of x. If it were x + 10 = 11, then we'd be able to find that x = 1.

Hope it helps.
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Re: M28-43 &nbs [#permalink] 02 Jun 2018, 03:48
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# M28-43

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# Wrong numbers when using \footnotemark

When adding a footnote to a frame title in Beamer using the code from another question, I get wrong numbering. Here you can see 1 above and 0 below:

This is an imaginary example. In reality, I need footnotes to cite the source of the material.

Here is the code:

\documentclass{beamer}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}

% http://tex.stackexchange.com/questions/22323/footnote-in-block-statement
\makeatletter
\renewcommand*{\footnoterule}{\kern -3pt \hrule \@width 2in \kern 8.6pt}
\makeatother

\begin{document}
\begin{frame}{SCLP example\footnotemark}
\footnotetext{To be continued.}
An example of a soft constraint logic program.
\end{frame}
\end{document}

There is a related question "Enumeration of footnotes wrong", but if I add

\addtocounter{footnote}{+1}

before the text of my footnote, I get wrong enumeration once again: 2 above and 1 below.

On the other hand, I tried to replace numbers with stars in the whole document as described here:

\def\thefootnote{\fnsymbol{ctr}}

However, pdfLaTeX wouldn't compile this code.

What else can I try?

-

This might form part of how the frame title is set. In fact, looking at your analysis, it is clear that the frame title is set after the frame contents. As such, avoiding insertion of counter-related commands in the title might be a workable quick-fix.

Here I've forced the counter to be incremented (\stepcounter{footnote}) to allow for correct display when using \footnotetext, and only typeset the number in the frame title using \textsuperscript{\thefootnote}:

\documentclass{beamer}% http://ctan.org/pkg/beamer

%\usepackage[english]{babel}
%\usepackage[utf8]{inputenc}
%\usepackage{times}
%\usepackage[T1]{fontenc}

% http://tex.stackexchange.com/questions/22323/footnote-in-block-statement
\makeatletter
\renewcommand*{\footnoterule}{\kern -3pt \hrule \@width 2in \kern 8.6pt}
\makeatother

\begin{document}
\begin{frame}{SCLP example\textsuperscript{\thefootnote}}
\stepcounter{footnote}\footnotetext{To be continued.}
An example of a soft constraint logic program.
\end{frame}
\end{document}

-

At least in your example, \footnote works inside the frame title.

\documentclass{beamer}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}

% http://tex.stackexchange.com/questions/22323/footnote-in-block-statement
\makeatletter
\renewcommand*{\footnoterule}{\kern -3pt \hrule \@width 2in \kern 8.6pt}
\makeatother

\begin{document}
\begin{frame}{SCLP example\footnote{To be continued.}}
An example of a soft constraint logic program.
\end{frame}
\end{document}

EDIT: In response to your comment: Try to insert the following code snippet in your "full" document:

   \begin{frame}{SCLP example\addtocounter{footnote}{-1}\footnotemark}
\stepcounter{footnote}\footnotetext{To be continued.}
An example of a soft constraint logic program.
\end{frame}

-
Indeed. But in my full document \footnote adds the footnote exactly in the same block with the title and not in the footer. I think, it's style-dependent. –  sdk Mar 2 '12 at 20:44

format: plain


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# Improved calculation of warming-equivalent emissions for short-lived climate pollutants

## Abstract

Anthropogenic global warming at a given time is largely determined by the cumulative total emissions (or stock) of long-lived climate pollutants (LLCPs), predominantly carbon dioxide (CO2), and the emission rates (or flow) of short-lived climate pollutants (SLCPs) immediately prior to that time. Under the United Nations Framework Convention on Climate Change (UNFCCC), reporting of greenhouse gas emissions has been standardised in terms of CO2-equivalent (CO2-e) emissions using Global Warming Potentials (GWP) over 100-years, but the conventional usage of GWP does not adequately capture the different behaviours of LLCPs and SLCPs, or their impact on global mean surface temperature. An alternative usage of GWP, denoted GWP*, overcomes this problem by equating an increase in the emission rate of an SLCP with a one-off “pulse” emission of CO2. We show that this approach, while an improvement on the conventional usage, slightly underestimates the impact of recent increases in SLCP emissions on current rates of warming because the climate does not respond instantaneously to radiative forcing. We resolve this with a modification of the GWP* definition, which incorporates a term for each of the short-timescale and long-timescale climate responses to changes in radiative forcing. The amended version allows “CO2-warming-equivalent” (CO2-we) emissions to be calculated directly from reported emissions. Thus SLCPs can be incorporated directly into carbon budgets consistent with long-term temperature goals, because every unit of CO2-we emitted generates approximately the same amount of warming, whether it is emitted as a SLCP or a LLCP. This is not the case for conventionally derived CO2-e.

## Introduction

Comprehensive climate policies must appraise a range of greenhouse gases and aerosols, which can differ significantly in their radiative efficiencies and atmospheric lifespans, and hence the nature of their climate impacts.1 To reflect this, different climate pollutants are often expressed using a common emission metric. Emissions reporting under the United Nations Framework Convention on Climate Change (UNFCCC) now requires the use of 100-year Global Warming Potential (GWP100) to account for all gases as carbon dioxide equivalent (CO2-e) quantities. Despite its prevalence in the UNFCCC and national climate policies, GWP has received criticism,2,3,4 not least that it cannot be used to appraise temperature-related goals,5 and other equivalence metrics have been proposed.6,7,8,9 Indeed, Shine3 notes that strong caveats were in place when GWP was introduced in the Intergovernmental Panel on Climate Change’s First Assessment Report10: “It must be stressed that there is no universally accepted methodology for combining all the relevant factors into a single [metric]… A simple approach [i.e., the GWP] has been adopted here to illustrate the difficulties inherent in the concept.” Working Group 1 of the Fifth Assessment Report, AR5, did not recommend any metric and emphasised that the choice of metric depends on the specific goal of the climate policy. In AR4, however, the GWPs were the recommended metric to compare the effects of long-lived greenhouse gases,11 and AR5 values of GWP100 have now been adopted for emissions reporting (see the textual proposal from 12 December 2018 on the transparency framework for action and support referred to in Article 13 of the Paris Agreement: https://unfccc.int/process/bodies/subsidiary-bodies/ad-hoc-working-group-on-the-paris-agreement-apa/information-on-apa-agenda-item-5).

The temperature response to emissions is ambiguous under GWP1,12,13 and this ambiguity is particularly relevant in the context of the Paris Agreement, given its stated aim of ‘holding the increase in the global average temperature well below 2 °C above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5 °C.’ Beyond the reference to a balance of emissions by sources and removals by sinks well before the end of the century, neither the means by which this is to be achieved nor the metrics used to assess progress are explicitly stated.14 Tanaka and O’Neill15 demonstrate that net-zero aggregate CO2-e emissions based on GWP100 (which is often assumed to be the definition of the balance of sources and sinks described in the Paris Agreement) are not essential to limit warming to 1.5 °C. Wigley16 posits that the balance of sources and sinks in Article 4.1 of the Paris Agreement is scientifically inconsistent with the temperature goals in Article 2.1. These papers show how moving from the temperature goals articulated in the Paris Agreement to emissions targets and profiles is not something that is currently well-handled by conventional carbon accounting; they also show that the area is receiving renewed scrutiny as countries, firms and sectoral bodies try to work out mitigation strategies of their own.

This paper demonstrates a method that unambiguously links aggregated greenhouse gas emissions with their warming outcomes on decade to century timescales, allowing short-lived climate pollutants (SLCPs) to be brought into a carbon budget framework.17 It is designed to be useful for informing policies that specifically aim to limit global warming, as is required under the Paris Agreement. This method builds on the revised usage of GWP, denoted GWP*, proposed in Allen et al.,12,18 building on Shine, et al.6 Specifically, we address a shortcoming in the originally proposed definition of GWP*, in that it did not account for the delayed temperature response to past increases in SLCP emissions, bringing aggregate emissions into closer agreement with both CO2-forcing-equivalent emissions19 and the temperature response.

## Results

### A revised definition of GWP*

A new usage of GWPs, denoted GWP*, allows emissions of short-lived and long-lived climate pollutants (SLCP & LLCPs) to be more consistently expressed within a single metric by equating a change in the emission rate of an SLCP as equivalent to a single emissions pulse of a long-lived pollutant. As originally defined in Allen, et al.,18 a step-change in emission rate of an SLCP (ΔESLCP tonnes per year) is equivalent to a one-off pulse emission of ΔESLCP × GWPH × H tonnes of CO2, where GWPH is the conventional Global Warming Potential relative to CO2, integrated over a time-horizon H years. Emissions of LLCPs, defined here as those having an atmospheric lifetime longer than H, will still behave as a cumulative pollutant within time-horizon H, and therefore equivalent emissions for LLCPs are derived simply by multiplying those emissions by GWPH.

This rate-based equivalence for SLCPs overcomes the problems inherent in GWP (or any pulse-based metric) in not adequately distinguishing their largely non-cumulative behaviour. However, although a sustained SLCP emissions rate will result in a stable atmospheric concentration and hence maintain the same level of forcing, some additional long-term warming will occur while the climate system is still equilibrating to past increases in SLCP emissions. Note this is not a cumulative impact of emissions mirroring that of CO2: it is, rather, the delayed response associated with equilibration to a past increase in forcing. After a sufficiently long period of constant emissions (on the order of centuries), SLCP-induced warming will stabilise, whereas CO2-induced warming continues to increase as long as CO2 emissions remain above zero. After CO2 emissions reach zero, ongoing thermal adjustment in surface temperature is largely balanced by ocean uptake of CO2,20 at least in the absence of strong Earth System feedbacks.21 The multi-century component of the thermal response of the climate system, together with carbon cycle feedbacks,22 act to prolong the warming impact of SLCP emissions.23 As noted in Allen, et al.,12 this can be incorporated ‘by including a small contribution that scales with time-integrated [SLCP] emissions.’

This component was not pursued in Allen, et al.12 for simplicity and because the contribution of this multi-century adjustment to past increases in SLCP emissions is small compared to the impact of current changes in SLCP emissions under the scenarios considered in that paper (see their Fig. 2 and supplementary Fig. 1 to this paper). Nevertheless, it may be significant for individual countries whose SLCP emissions have increased within the past half-century or so and are now approximately stable.

Thus, we propose a re-definition of GWP* to incorporate both timescales, as well as providing a theoretical justification below, we here adjust this empirically to produce the best fit between cumulative CO2-warming-equivalent (CO2-we) emissions and resultant warming (see below and Methods for full details). Calculated using this re-defined GWP*, CO2-we emissions of an SLCP in a given year are defined:

$$E_{{\mathrm{CO}}_2{\mathrm{we}}} = {\mathrm{GWP}}_H \times \left[ {r \times \frac{{\Delta E_{{\mathrm{SLCP}}}}}{{\Delta t}} \times H + s \times E_{{\mathrm{SLCP}}}} \right]$$
(1)

where GWPH is the conventional global warming potential for a given SLCP over time-horizon H, ΔESLCP the change in SLCP emission rate over the preceding Δt years, ESLCP the SLCP emissions for that year, and r and s the weights assigned to the rate and stock contributions, respectively. The only difference between this formulation and that of Allen et al.12,18 is that they used r = 1 and s = 0. Including the time period Δt spreads the CO2-we pulse corresponding to a change in SLCP emission rate over Δt. Allen et al.12 suggest at least 20 years, which has the effect of reducing the volatility in CO2-we emissions and improving the correspondence with temperature response. The first (“rate”) term on the right-hand side, $$r \times \frac{{\Delta E_{{\mathrm{SLCP}}}}}{{\Delta t}} \times H \times {\mathrm{GWP}}_H$$, represents the response to the changing SLCP emission rates. The second (“stock”) term, s × ESLCP × GWPH, is added to represent the long-term equilibration to past increases in forcing, which can be approximated by a small term scaling with cumulative SLCP emissions. In other words, the rate term approximates the short-timescale climate response to a change in radiative forcing; the stock term approximates the long-timescale equilibration which occurs even when there is constant radiative forcing.

The exact values of r and s will depend on the precise timescales of the climate response to radiative forcing, on how long ago the increase in SLCP emissions occurred, and on carbon cycle feedbacks, all of which are uncertain and scenario-dependent. Constraining r + s = 1 ensures that total CO2-we emissions over 100 years corresponding to a steady SLCP emission starting in year 1 are the same as total CO2-e emissions would be, consistent with the original derivation of GWP* presented in Allen, et al.18 The ratio s/(rH) corresponds to the fractional rate of decline of SLCP emissions for these to be considered equivalent to a zero rate of CO2-we emissions and hence not cause further warming. This depends on the SLCP and details of the scenario in question. Our objective is a simple and reliable indicator of the relationship between emissions and recent and near-term future warming associated with the largest non-CO2 climate drivers, so we estimate r and s using a multiple linear regression onto the response to methane emissions in commonly used scenarios (representative concentration pathways, RCPs), focusing on the time period 1900–2100.

The method is applicable to other SLCPs, but the optimal values of r and s could be different for each SLCP dependent on past emissions. As many short-lived industrial gases have only started being released in recent decades, the warming responses to these gases is likely distinct to those for methane, with a greater emphasis on the immediate effects of changing emission rates, and thus not necessarily reflecting the same r and s values as derived for methane here. That said, GWP*, like any metric, depends on strong assumptions of linearity, so the additional precision may not be worth the additional complexity. The application of GWP* to specific other gases is beyond the scope of this paper but warrants further investigation, given their impacts are substantial and potentially growing. For example, the radiative forcing from total halocarbons (long and short-lived) is less than half that from methane1 but may grow with increasing demand for air conditioning.

### Empirical estimation of flow and stock contributions

The global mean surface temperature (GMST) responses (ΔT) to methane radiative forcing from the RCP2.6, 4.5 and 6 scenarios taken from the AR5 database,24 were derived using the default configuration of the FaIR simple climate model.25 Methane was chosen as it has the largest impact on radiative forcing of all the SLCPs, and these scenarios were used as they represent pathways approximately in line with current policies, as well as a more ambitious pathway (RCP 2.6). Multiple linear regression was then used to find values of r and s to best fit the relationship between cumulative emissions of CO2-we and ΔT (using the RCP emissions from the AR5 database), constrained such that r + s = 1. Using this approach, r = 0.75 and s = 0.25 are the mean values based on the three RCPs and are found to provide a good fit for all three (see methods section below).

To demonstrate the improved warming-equivalence of GWP*, cumulative emissions (left axes) are shown alongside the corresponding warming in Fig. 1. The change in GMST calculated from the methane radiative forcing is shown as a time series (black dashed line, right axis) for RCP 2.6 (upper), RCP 4.5 (middle) and RCP 6 (lower). The cumulative CO2-e emissions calculated using GWP100 (cyan) show that there is no agreement between CO2-e emissions and warming when methane emissions are stable or in decline. CO2-we emissions calculated with GWP*, especially when both stock and flow properties are included (purple), show a clear improvement with the cumulative emissions matching the temperature response.

Orange lines show cumulative CO2-we emissions retaining only the flow term i.e., setting r = 1, s = 0 in equation 1, equivalent to the definition in ref.,12 which scales with smoothed annual emissions (dotted line, inner left axis). GWP* defined by only the change in rate of methane emissions (i.e., as originally defined) overestimates the cooling that would occur under decreasing methane emissions because it fails to take into account the century-timescale response to earlier methane emission increases.

Figure 2 shows cumulative CO2-e and CO2-we emissions of methane plotted against the modelled temperature response, relative to 1900 and up to 2100, for the RCPs. The linear relationship between cumulative CO2 emissions and warming is the basis for the carbon budget concept, which describes how much CO2 can be emitted before any given threshold of global mean warming is reached. Previous studies 26,27 have computed CO2 budgets conditioned on specific scenarios for non-CO2 forcing. GWP* allows non-CO2 forcings to be included in the carbon budget itself, as it describes a linear relationship between cumulative CO2-we emissions of SLCPs and warming, as shown in purple in Fig. 2.

Under GWP100 (cyan) this relationship breaks down completely when SLCP emissions start to decline. Unphysically, GWP as traditionally used implies declining methane emission rates still contribute to increasing cumulative CO2-e, when they are in fact causing cooling. Hence, the negative gradient towards the end of the cyan scenarios in Fig. 2. If only the flow properties of methane are considered (orange), reducing methane emission rates are now equivalent to a negative CO2 emission, so the line ‘turns back’ on itself as cumulative emissions decline alongside reducing temperatures. Although much closer to a linear relationship than GWP100, in this exclusively rate-based version there is now more CO2-we removal than would be expected to explain a given amount of cooling, if we consider that the truly equivalent relationship should mimic that of CO2 emissions and temperature response, which is approximately linear.28 This is because the rate-based formulation alone does not account for the ongoing warming from the thermal equilibration to past methane emission increases. Incorporating a term that accounts for this behaviour, as proposed above (purple lines), largely overcomes the problem, providing a close match to a linear relationship for the scenarios tested here. The relationship shown here corresponds to a Transient Climate Response to cumulative carbon Emissions (TCRE, or the amount of warming per unit carbon emitted, shown by the grey lines in Fig. 2) of about 1.8 °C per TtC during the historical period and slightly less in the RCP projections. The slope changes because there is a time lag in the climate’s response, therefore negative emissions do not immediately reverse the effects of the same amount of emissions. This effect was reported by Zickfeld, et al.29 for CO2 removals. The TCRE depends directly on the prescribed TCR of the FaIR model used to generate the temperature: but, crucially, would be similar for all non-CO2 forcings provided their different efficacies are taken into account using effective radiative forcing.1 Adding a small correction to account for the fact that the climate system takes time to equilibrate to higher forcing permits a physically plausible interpretation of “equivalence” in the calculation of carbon budgets. This improves the accuracy of the temperature outcome of the equivalence metric, however it should be noted that this is still an application of GWP100 and therefore does not capture everything that a climate model includes. Collins, et al.9 investigate the impact of methane on the remaining carbon budget using an intermediate complexity climate-carbon model, and note that their results show a close correspondence to the GWP* metric as proposed in Allen, et al.18

### Physical interpretation and justification

We can illustrate the physical interpretation of r and s values by considering some more idealised scenarios. Setting the left-hand side of equation 1 (CO2-we emissions) to zero, we are able to calculate the methane trend required to be equivalent to no further CO2 emissions: ΔESLCPt = −[s/(rH)]ESLCP, which is required to generate a radiative forcing pathway that will approximately stabilise temperatures over the time period Δt. Hence with r = 0.75, s = 0.25 and H = 100 years, [s/(rH)] = 0.3% is the rate at which methane emissions need to decline to give stable methane-induced warming. This makes zero CO2-we emissions under GWP* consistent with stable temperatures, matching the temperature response to zero CO2 emissions.

The definition of CO2-we using GWP* is independent of SLCP lifetime (assumed to be much shorter than H), but it does depend on the SLCP forcing history: if temperatures are close to equilibrium following a very gradual forcing increase over many centuries, a near-zero decline rate (near constant SLCP emissions) would be consistent with no further warming. Faster rates of decline would be required to maintain no further warming following a rapid increase in SLCP forcing, because the climate system would be further from equilibrium. Here we have based the coefficients and therefore the rate of decline on a combination of historical (1900 onwards) and future scenario emissions to encompass climate response in the near future to emissions over the last century.

We have used an empirical method to find a definition of GWP* that preserves the link between an emission and the warming it generates in the medium term up to 2100. The physical interpretation of equation 1 is that the flow term (with coefficient r) represents the fast climate response to a change in radiative forcing, generated by the atmospheric and ocean mixed-layer response.30 The timescale of this response is about 4 years here.31 The stock term (with coefficient s) represents the slower timescale climate response to a change in radiative forcing, due to the deep ocean response. This effect means that the climate responds slowly to past changes in radiative forcing, and is why the climate is currently far from equilibrium. We have approximated this response by treating a quarter of the climate response to a SLCP as “cumulative”. The timescale for this response is uncertain,32 and is of the order a few centuries, as discussed below.

The exponential decline of 0.3% per year corresponds to a time constant of about 300 years, consistent with the equilibration timescale of the climate system. This timescale is largely governed by deep ocean adjustment to relatively recent forcing increases identified by Geoffroy, et al.32 (multi-model mean of 290 years, with a standard deviation of 107 years). If the equilibration timescale of the climate system were shorter, then s would be lower. If the deep ocean response timescale were the same as the atmosphere and mixed ocean timescale (about 4 years), then r would be 1, s would be 0, and the definition of GWP* from Allen, et al.12 would not change.

The rate at which global methane emissions need to decline to reduce the rate of methane-induced warming to zero has not been explored systematically with complex models and would depend on the details of its atmospheric chemistry. An indication of the rate of decline of emissions of a generic SLCP required to stabilise SLCP-induced warming following a linear increase in SLCP emissions over a multi-decade period can be provided by considering the generic response to the following widely studied scenario: a linear increase in forcing to the equivalent of a CO2 doubling over 70 years, followed by constant forcing. If the thermal response of the climate system is characterised by a short (sub-decadal, atmosphere and mixed layer ocean) adjustment time, d1, and a long (multi-century, deep ocean) adjustment time, d2 70 years, temperatures after year 70 adjust exponentially from their value at year 70 (the Transient Climate Response, or TCR) to their long-term equilibrium value (the Equilibrium Climate Sensitivity, or ECS) with an adjustment timescale d2. Hence, warming would rise at a fractional rate of (ECS−TCR)/(d2 × TCR) per year in the decades immediately after forcing is held constant (see supplementary Fig. 2). On multi-decade timescales longer than the SLCP lifetime, SLCP emissions would need to fall at the same fractional rate to yield no further warming, since the rate of SLCP emissions scales with SLCP-induced forcing and the temperature response is linear in forcing.

For representative values, (ECS = 2.75 °C, TCR = 1.6 °C, d2 = 239 years, after Millar, et al.31) this indicates a decline rate, (ECS−TCR)/(d2 × TCR), of 0.3% per year, corresponding to a time-scale of 333 years (the inverse of 0.3%/year). This is consistent with our estimates of r = 0.75 and s = 0.25 for H = 100 years, which give a time-scale (r/s) × H of 300 years. This indicates the approximate rate of decline of methane emissions required for no further warming. However, this timescale depends on the multi-century response of the climate system and carbon cycle feedbacks, all of which are poorly constrained by available observations and modelling: targeted experiments varying methane emission growth and decline rates would give a more precise indication.

The relationship between stable or declining methane emissions is shown in Fig. 3 for a range of model parameters in the simple climate model, FaIR (assuming a constant methane lifetime). Different colours show simulations with a 1-sigma range of d2 values from Geoffroy, et al.,32 and for a range of realised warming fractions (the ratio of TCR:ECS) based on that in the CMIP5 ensemble.33 The dashed lines show the scenario in which methane emissions are kept stable for 130 years after a 70-year ramp up to approximately present day emission rates. For all parameter combinations, constant methane emissions cause a continued warming. The solid lines show scenarios which reduce the emissions by a fractional rate of (ECS−TCR)/(d2 × TCR) per year to compensate for the slow climate response. As predicted, these give stable temperatures over the decades following the emissions peak. Decline rates range from between 0.06% (TCR:1.6 °C, ECS:2.0 °C, d2:397.0 years) and 0.55% (TCR:1.6 °C, ECS:3.2 °C d2:183.0 years). Note that these values have been calculated based on a simple climate model that emulates the response from complex climate models. There remains considerable uncertainty in how the real climate would evolve, for example through feedbacks that are not yet included in climate models, which may not be fully reflected in this range of estimates.

While methane emissions may thus appear to have a cumulative impact on global temperature, this is better interpreted as the delayed response to relatively recent methane emissions increases. Constant anthropogenic methane emissions, if maintained indefinitely, clearly have no further warming impact (being indistinguishable from constant natural emissions). This apparent cumulative impact is important, and captures the potential benefits of early methane mitigation9 not apparent through a solely rate-based equivalence, but is only about 25% (s) of the impact indicated by GWP100 and closer to that indicated by the 100-year Global Temperature-change Potential (GTP) including carbon cycle feedbacks. This should not, however, be interpreted as simple support for a lower metric value than GWP100: most scenarios and policy interventions involve changes in methane emission rates outside the range zero to −0.3%/year, in which case the first term on the RHS of equation 1 (neglected by conventional metrics) dominates.

## Discussion

We have demonstrated how it is possible to represent both the short-lived nature of methane and the long-timescale adjustment of the climate system in a single metric, GWP*. This metric allows SLCP emissions to be converted to CO2-equivalent emissions and preserve an unambiguous link to global warming, which we have therefore termed CO2-warming-equivalent. Just as many different methods have been proposed for the calculation of CO2-equivalent emissions, so there are multiple ways of calculating CO2-warming-equivalent emissions. Some, like GWP*, rely heavily on linearization; others, like CO2-forcing-equivalent emissions19 or explicit modelling of the temperature response,9 have a stronger physical justification and are likely to be more accurate for specific applications, at the cost of simplicity and generality. While complete disambiguation requires all details to be specified, the aim of CO2-we emissions is clear: to calculate the CO2 emission pathway that would yield the same global temperature change on all relevant timescales as that caused by the time-history of some non-CO2 climate forcer. Given this objective, appropriate methodological decisions should always yield broadly similar CO2-we pathways for the same climate forcer, in stark contrast to CO2-e emissions, for which legitimate methodological decisions such as the choice of time-horizon can change results by over an order of magnitude. Conventional pulse-based metrics treat SLCPs like methane as a stock pollutant only, thereby neglecting the rapid climate response to changes in SLCP emission rates, which dominate the temperature response while emission rates are changing.

Single-number metrics like GWP typically overestimate the cumulative effects of SLCPs; but there is some apparent cumulative impact of SLCP emissions, which arises not because they accumulate in the atmosphere, but because a component of the climate system’s response to past forcing increases is characterised by a slow equilibration timescale. Based on historical emissions and RCPs 2.6, 4.5 and 6, GWP* is found to best represent the temperature impacts of methane emissions by modifying the definition in Allen, et al.12 to weight the flow response (impact of changing methane emission rates) by 0.75 and the stock response (equilibration of the climate system to past methane emission increases) by 0.25.

The benefits of GWP* are most apparent when SLCP emission rates are declining, as this is when CO2-e emissions derived from conventional GWP100 would indicate a temperature response of the wrong sign (further warming instead of cooling). Under the Paris Agreement, nations have agreed to limit global warming to well below 2 °C, and to pursue efforts to limit it to 1.5 °C. Using GWP* to calculate CO2-we emissions can therefore be useful in linking emissions scenarios with temperature goals. It uses GWP100 in a novel way, and is thus consistent with current requirements that countries use this metric in emissions accounting. GWP* allows the contributions of all climate forcing agents to be aggregated to reach a global total cumulative CO2-we emission, which can then be multiplied by the TCRE to give an estimate of resultant warming over any given time period: ΔT = TCRE × [∑CO2−we + ∑CO2e], where TCRE is the Transient Climate Response to cumulative carbon Emissions, ∑CO2we is the cumulative short-lived GHG emissions aggregated using GWP* and ∑CO2e is the cumulative long-lived GHG emissions aggregated using GWP100. This method provides a simple and transparent mechanism by which to estimate whether countries are on track to meet the Paris Agreement goals in the global stocktake. It also allows SLCPs and cumulative gases to continue to be included together in reporting mitigation ambitions, maintaining fungibility while improving environmental integrity.

## Methods

### Method to derive r and s

Data from the RCP database has been used to investigate how methane emissions relate to warming, and how different emission metrics represent that warming.

The RCP database contains emissions rates and radiative forcings for different greenhouse gases from 1765 to 2100 for the four RCP scenarios (RCP2.6, RCP4.5, RCP6 and RCP8.5). Here, the representation of methane based on GWP100 and GWP* is considered.

The methane emissions time series for each of the RCP scenarios are converted to CO2-e emissions timeseries using a GWP100 of 28.1 The temperature response to the methane radiative forcing (from the RCP database) is calculated using the FaIR model,25 with a factor of 1.65 applied to account for the secondary effects of ozone and stratospheric water vapour as recommended in Myhre, et al.1 Our analysis is consistent with assumptions in Myhre, et al.1 and therefore does not include more recent findings, for example updated radiative forcings from Etminan, et al.34 In all scenarios, the warming trend does not correspond closely to the cumulative CO2-e emissions calculated using GWP100.

The coefficients to weight the long and short term effects are found for each scenario using a linear regression model of the equation ΔT = a.CGWP* + b.CGWP100, where CGWP100 is the cumulative CO2-e emission of methane defined conventionally using GWP100, and CGWP* is the cumulative CO2-e emission of methane defined using GWP* from Allen, et al.12 with Δt= 20 years. Coefficients a and b were found by ordinary least squares multiple linear regression using the statsmodels package in python (http://www.statsmodels.org), and then the normalised coefficients were set by defining r=a/(a + b) and s = b/(a + b), such that r + s = 1.

When radiative forcing is put into the FaIR model, a temperature response is calculated using an impulse response function as described in Smith, et al.,25 with 2 response timescales (239 and 4.1 years as default) and a default TCR and ECS of 1.6 and 2.75 K. The radiative forcings from the IPCC AR5 database used to drive this model have been calculated using the model MAGICC, which includes carbon cycle feedbacks. Note that this model does not include less well understood earth system feedbacks, such as permafrost feedbacks, which could act to accelerate warming. Therefore, the method is most suited for use in more ambitious mitigation scenarios where warming is slower, and abrupt positive feedbacks less likely to be triggered.

Table 1 shows the values for r and s that are generated using the above method, for historical data from 1900 and RCP 2.6, 4.5 and 6 data to 2100. The mean and standard deviation of these values are used in this work for r and s. The time period 1900 to 2100 is chosen as it represents the recent historical increase in methane emissions, as well as capturing three possible futures.

## Data availability

The RCP datasets analysed during the current study are available at http://www.iiasa.ac.at/web-apps/tnt/RcpDb.

## Code availability

The code used to calculate r and s, CO2-we emissions and produce Figs 1 and 2 is available at https://gitlab.ouce.ox.ac.uk/OMP_climate_pollutants/co2-warming-equivalence.

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Myhre, G. et al. Climate change 2013: the physical science basis. In Contribution of Working Group 1 to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (eds Stocker, T. F. et al.) (Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2013).

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Pierrehumbert, R. T. Short-lived climate pollution. Annu. Rev. Earth Planet. Sci. 42, 341–379 (2014).

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7. 7.

Lauder, A. R. et al. Offsetting methane emissions—An alternative to emission equivalence metrics. Int. J. Greenh. Gas. Control 12, 419–429 (2013).

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Tol, R. S. J., Berntsen, T. K., O’Neill, B. C., Fuglestvedt, J. & Shine, K. P. A unifying framework for metrics for aggregating the climate effect of different emissions. Environ. Res. Lett. 7, 044006 (2012).

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Collins, W. J. et al. Increased importance of methane reduction for a 1.5 degree target. Environ. Res. Lett. 13, 054003 (2018).

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IPCC. Climate Change: the Intergovernmental Panel on Climate Change Scientific Assessment. (Cambridge University Press, Cambridge, 1990).

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Forster, P. et al. Climate change 2007: the physical science basis. In Contribution of Working Group 1 to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (eds Solomon, S. et al.) (Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2007).

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Allen, M. R. et al. A solution to the misrepresentations of CO2-equivalent emissions of short-lived climate pollutants under ambitious mitigation. npj Clim. Atmos. Sci. 1, 16 (2018).

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Fuglestvedt, J. S., Berntsen, T. K., Godal, O. & Skodvin, T. Climate implications of GWP-based reductions in greenhouse gas emissions. Geophys. Res. Lett. 27, 409–412 (2000).

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Fuglestvedt, J. et al. Implications of possible interpretations of ‘greenhouse gas balance’ in the Paris Agreement. Phil. Trans. R. Soc. A 376, https://doi.org/10.1098/rsta.2016.0445 (2018).

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Tanaka, K. & O’Neill, B. C. The Paris Agreement zero-emissions goal is not always consistent with the 1.5 °C and 2 °C temperature targets. Nat. Clim. Change 8, 319–324 (2018).

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Wigley, T. M. L. The Paris warming targets: emissions requirements and sea level consequences. Clim. Change 147, 31–45 (2018).

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Allen, M. R. et al. Warming caused by cumulative carbon emissions towards the trillionth tonne. Nature 458, 1163 (2009).

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Allen, M. R. et al. New use of global warming potentials to compare cumulative and short-lived climate pollutants. Nat. Clim. Change 6, 773 (2016).

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Jenkins, S., Millar, R. J., Leach, N. & Allen, M. R. Framing climate goals in terms of cumulative CO2-forcing-equivalent emissions. Geophys. Res. Lett. 45, 2795–2804 (2018).

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Solomon, S., Plattner, G.-K., Knutti, R. & Friedlingstein, P. Irreversible climate change due to carbon dioxide emissions. Proc. Natl Acad. Sci. 106, 1704–1709 (2009).

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Lowe, J. A. & Bernie, D. The impact of Earth system feedbacks on carbon budgets and climate response. Philos. Trans. R. Soc. A 376, 20170263 (2018).

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Smith, C. J. et al. FAIRv1.3: a simple emissions-based impulse response and carbon cycle model. Geosci. Model Dev. 11, 2273–2297 (2018).

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Etminan, M., Myhre, G., Highwood, E. J. & Shine, K. P. Radiative forcing of carbon dioxide, methane, and nitrous oxide: a significant revision of the methane radiative forcing. Geophys. Res. Lett. 43, 614–612,623 (2016).

## Acknowledgements

The authors would like to thank A. Reisinger for helpful discussion that contributed to the development of this paper. M.C. acknowledges support from the Oxford Martin Programme on Climate Pollutants. J.L. acknowledges support from The Wellcome Trust, Our Planet Our Health (Livestock, Environment and People—LEAP), award number 205212/Z/16/Z. M.R.A., M.C. and J.L. acknowledge support from Natural Environment Research Council award number NE/T004053/1–A practical tool and robust framework for evaluating greenhouse gas emissions from land-based activities.

## Author information

Authors

### Contributions

M.C. and M.R.A. initiated the work. J.L. and M.C. developed the work and produced the original draft paper. M.C. undertook the modelling with contributions from M.R.A. and J.L. M.C. produced Figs 1, 2 and 3. M.R.A. produced supplementary Figs 1 and 2. All authors contributed to developing the scientific questions, analysis of the results, subsequent drafts of the paper and in editing the final version.

### Corresponding author

Correspondence to Michelle Cain.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Cain, M., Lynch, J., Allen, M.R. et al. Improved calculation of warming-equivalent emissions for short-lived climate pollutants. npj Clim Atmos Sci 2, 29 (2019). https://doi.org/10.1038/s41612-019-0086-4

• Accepted:

• Published:

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text:
Convergence in $L_1$ implies convergence in measure $f_n \xrightarrow[]{L_1}f \Rightarrow f_n \xrightarrow[]{\mu}f$.

Let $(X, \Sigma, \mu)$ be a measure space. $\mu(X) \le \infty$. Let $f$ and $f_n$ are integrable functions. Prove that $f_n \xrightarrow[]{L_1}f \Rightarrow f_n \xrightarrow[]{\mu}f$.

$f_n \xrightarrow[]{L_1}f := \forall m>0 ,\exists N, \forall n (n > N \Rightarrow \int_X|f_n-f|\mathrm{d}\mu < m)$.

Proof: by definition $$\neg(f_n \xrightarrow[]{\mu}f):=\exists\epsilon>0, \exists \delta>0, \forall N, \exists n(n>N \land \mu( \{x : |f_n - f| \geq \epsilon\}) \geq \delta)$$ Let $m=\epsilon\delta$. Then $$\forall N, \exists n (n>N \land\int_X |f_n - f| \mathrm{d}\mu \geq m)$$ Thus $\neg (f_n \xrightarrow[]{\mu}f) \Rightarrow \neg (f_n \xrightarrow[]{L_1}f)$. And therefore $f_n \xrightarrow[]{L_1}f \Rightarrow f_n \xrightarrow[]{\mu}f$.

Is everything correct? Thanks!

A bit shorter: assume $f \xrightarrow{L^1} f$ and let $\varepsilon > 0$.
$$0 \le \mu(\left\{x \in X : |f_n - f| \ge \varepsilon\right\}) \le \frac{1}{\varepsilon}\int\limits_{\left\{x \in X : |f_n - f| \ge \varepsilon\right\}}|f-f_n|\,d\mu \le \frac1\varepsilon\int_X |f-f_n|\,d\mu \xrightarrow{n\to\infty} 0$$
Hence, the squeeze theorem implies $\mu(\{x \in X : |f_n - f| \ge \varepsilon\}) \xrightarrow{n\to\infty} 0$.
We conclude $f \xrightarrow{\mu} f$.


text:
# Math Help - Sequence of the positive elements in B(H)

1. ## Sequence of the positive elements in B(H)

Let $B(H)$ - the algebra of bounded linear operators on a Hilbert space $H$, and $x\in B(H)$ is positive. Then sequence $x (\frac{1}{n}+x)^{-1}$ is monotone increasing to the range projection of $x:\quad [x].$

2. Originally Posted by karkusha

Let $B(H)$ - the algebra of bounded linear operators on a Hilbert space $H$, and $x\in B(H)$ is positive. Then sequence $x (\frac{1}{n}+x)^{-1}$ is monotone increasing to the range projection of $x:\quad [x].$
This makes no sense. Could you rephrase? What sequence? What projection?

3. What projection?
For each operator $x\in B(H)$ we define the range projection of $x$ as the projection on the closure of $xH$.

What sequence?
The sequence of operators is $x_n=x(\frac{1}{n}+ x)^{-1}$.

4. Originally Posted by karkusha
Let $B(H)$ - the algebra of bounded linear operators on a Hilbert space $H$, and $x\in B(H)$ is positive. Then sequence $x (\frac{1}{n}+x)^{-1}$ is monotone increasing to the range projection of $x:\quad [x].$
Have you come across the functional calculus for selfadjoint operators? If so, you can translate this problem into a problem about functions on the spectrum $S$ of the operator $x$. In fact, you just need to show that the sequence of functions $f_n(t) = t\bigl(\frac1n+t\bigr)^{-1}$ is monotone increasing (on the set $S$) to the function $p(t)$, where $p(0)=0$ and $p(t) = 1$ for $t>0.$

To do this without the functional calculus would be harder. You could start by observing that the operators $x$, $\bigl(\frac1m+x\bigr)^{-1}$ and $\bigl(\frac1n+x\bigr)^{-1}$ all commute with each other. Let $x_n = x \bigl(\frac{1}{n}+x\bigr)^{-1}$. If n>m then $x_n-x_m = \bigl(\frac1m-\frac1n\bigr)x\bigl(\frac{1}{m}+x\bigr)^{-1}\bigl(\frac{1}{n}+x\bigr)^{-1}$. The right side of that is a product of three commuting positive operators and is therefore positive. That shows that the sequence $(x_n)$ is monotone increasing.

To complete the proof, you need to know that $\|x_n\|\leqslant1$. This again is easy if you know a bit of spectral theory. In fact, it follows from the fact that $|f_n(t)|\leqslant1$ on $S$.

First, notice that $1-x_n = \bigl((\frac1n+x)-x\bigr)\bigl(\frac{1}{n}+x\bigr)^{-1} = \frac1n\bigl(\frac{1}{n}+x\bigr)^{-1}$. It follows that if $x\xi$ is in the range of $x$ (where $\xi\in H$) then $(1-x_n)x\xi = \frac1nx_n\xi\to0$ as $n\to\infty$. Thus $x_n\to 1$ strongly on the range of $x$ and hence (since $\|x_n\|\leqslant1$ for all n) on the closure of the range of $x$.

The orthogonal complement of the range of $x$ is the null space of $x$. So if $\eta$ is in that orthogonal complement then $x_n\eta = \bigl(\frac{1}{n}+x\bigr)^{-1}x\eta = 0$. Thus $x_n\to0$ strongly on the orthogonal complement of the range of $x$.

Put that all together to see that $x_n$ converges strongly to the range projection of $x$.


text:
# Electric fields

1. Mar 4, 2008

### Lalitha

Hello Sir ,
i am new to the site and so is not familar of that but i have question and i found u online so wish u to help me with that.

Electric forces are used to accelerate small particles (like ink particles in a printers) but consider electrostatic action on macroscopic objects .Suppose that a 4 kg chicken were to be supported by a vertically upwardly directed electric feild .Since the amximum E strength in air is about 3*10^6.The net charge on each chicken would be unreasonably large since it must be least.

a) 1.3*10^-6

b)1.3*10^-5

c)-1.3*10^-5

d)7.7*10^4

none

Last edited by a moderator: Mar 4, 2008
2. Mar 4, 2008

### Staff: Mentor

In a static situation, the upward electric force must be equal and opposite the weight of the mass being suspended.

3. Mar 4, 2008

### Lalitha

thanks a lot sir i got that.I am very Thankful for your help .

4. Mar 4, 2008

### Lalitha

as we are said that the electric force is equal to mass but in oppostie direction then the force that makes the body stantionary in an electric feild is F=-mg=-4*10 (g=9.8m/sec^2=10 app),

And as E=3*10^6 and using the realtion that F=qE

we get q=F/E=-4*10/(3*10^6)=-1.3*10^-5

5. Mar 4, 2008

### Staff: Mentor

Be careful here.

The force of gravity is down so the force of the electric field must be up, so Fe = -Fg, so if Fg = -40N, the Fe = 40 N. What does that indicate about the polarity of the charge?

6. Mar 4, 2008

### Kurdt

Staff Emeritus
EDIT: beaten by Astro

7. Mar 4, 2008

### Lalitha

Sir here as we consider the force of gravity so it need to be +40N NA SIR as we have given that m=4kg,
and f=mg =4*10=40,
and Fe=-Fg=-40.

Is that not right Sir ?

8. Mar 4, 2008

### Lalitha

Hello Sir Plz let me know am i correct or going wrong there mean as fg=mg=4*10=40 and Fe=-Fg=-40N

9. Mar 4, 2008

### Staff: Mentor

Fe=-Fg= 40N

The gravitational force is downward, and by convention, that is negative.

Acceleration is a vector as is force. By convention a downward vector is negative.

The acceleration of gravity is $$\vec{a}\,=\,-9.81 \hat{j}\,m/s^2$$, where $$\hat{j}$$ is the unit vector in the vertical orientation and is taken as positive.

10. Mar 4, 2008

### Lalitha

Ok thanks a lot for that sir .I got it now .

11. Mar 7, 2008

### Lalitha

The plates are connected to a battery and then the connection is removed so the two plates are left charged and isolated. If a dielectric material such as a sheet of glass is inserted between the plates:

a. no work is done and the energy stored stays constant.
b. work is done, increasing the amount of stored energy in the capacitor.
c. work is done, decreasing the amount of energy stored in the capacitor.

12. Mar 7, 2008

### Lalitha

Plz can anybody help me with this one


text:
## Duke Mathematical Journal

### The characteristic polynomial of a random unitary matrix: A probabilistic approach

#### Abstract

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [8] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results

#### Article information

Source
Duke Math. J., Volume 145, Number 1 (2008), 45-69.

Dates
First available in Project Euclid: 17 September 2008

https://projecteuclid.org/euclid.dmj/1221656862

Digital Object Identifier
doi:10.1215/00127094-2008-046

Mathematical Reviews number (MathSciNet)
MR2451289

Zentralblatt MATH identifier
1155.15025

Subjects
Primary: 15A52
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems

#### Citation

Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M. The characteristic polynomial of a random unitary matrix: A probabilistic approach. Duke Math. J. 145 (2008), no. 1, 45--69. doi:10.1215/00127094-2008-046. https://projecteuclid.org/euclid.dmj/1221656862

#### References

• G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999.
• P. Bourgade, A. Nikeghbali, and A. Rouault, Hua-Pickrell measures on general compact groups, preprint,\arxiv0712.0848v1[math.PR]
• P. Carmona, F. Petit, and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes'' in Exponential Functionals and Principal Values Related to Brownian Motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997, 73--130.
• L. Chaumont and M. Yor, Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning, Camb. Ser. Stat. Probab. Math. 13, Cambridge Univ. Press, Cambridge, 2003.
• P. Diaconis and M. Shahshahani, The subgroup algorithm for generating uniform random variables, Probab. Engrg. Inform. Sci. 1 (1987), 15--32.
• C. P. Hughes, J. P. Keating, and N. O'Connell, On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), 429--451.
• J. Jacod and P. Protter, Probability Essentials, 2nd ed., Universitext, Springer, Berlin, 2003.
• J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Comm. Math. Phys. 214 (2000), 57--89.
• —, Random matrix theory and $L$-functions at $s=1/2$, Comm. Math. Phys. 214 (2000), 91--110.
• M. L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991.
• F. Mezzadri, How to generate random matrices from the classical compact groups, Notices Amer. Math. Soc. 54 (2007), 592--604.
• F. Mezzadri and N. C. Snaith, eds., Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser. 322, Cambridge Univ. Press, Cambridge, 2005.
• A. Nikeghbali and M. Yor, The Barnes G-function and its relations with sums and products of generalized gamma convolution variables, preprint,\arxiv0707.3187v1[math.PR]
• B. Odgers, Random matrix theory and L-functions: Transitions between ensembles, Ph.D. dissertation, University of Bristol, Bristol, United Kingdom, 2006.
• V. V. Petrov, On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm, Theor. Probability Appl. 11 (1966), 454--458.
• —, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Oxford Stud. Probab. 4, Oxford Univ. Press, New York, 1995.
• —, On the law of the iterated logarithm for a sequence of independent random variables, Theory Probab. Appl. 46 (2003), 542--544.
• E. Royer, Fonction $\zeta$ et matrices aléatoires'' in Physics and Number Theory, IRMA Lect. Math. Theor. Phys. 10, Eur. Math. Soc., Zürich, 2006, 165--224.
• A. Selberg, Old and new conjectures and results about a class of Dirichlet series'' in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, Italy, 1989), Univ. Salerno, Salerno, Italy, 1992, 367--385.
• D. N. Shanbhag and M. Sreehari, On certain self-decomposable distributions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), 217--222.
• D. W. Stroock, Probability Theory: An Analytic View, Cambridge Univ. Press, Cambridge, 1993.


text:
brainpy.dyn.synplast.STP#

class brainpy.dyn.synplast.STP(U=0.15, tau_f=1500.0, tau_d=200.0, method='exp_auto', name=None)[source]#

Synaptic output with short-term plasticity.

This model filters the synaptic currents according to two variables: $$u$$ and $$x$$.

$I_{syn}^+(t) = I_{syn}^-(t) * x * u$

where $$I_{syn}^-(t)$$ and $$I_{syn}^+(t)$$ are the synaptic currents before and after STP filtering, $$x$$ denotes the fraction of resources that remain available after neurotransmitter depletion, and $$u$$ represents the fraction of available resources ready for use (release probability).

The dynamics of $$u$$ and $$x$$ are governed by

\begin{split} \begin{aligned} \frac{du}{dt} & = & -\frac{u}{\tau_f}+U(1-u^-)\delta(t-t_{sp}), \\ \frac{dx}{dt} & = & \frac{1-x}{\tau_d}-u^+x^-\delta(t-t_{sp}), \\ \tag{1}\end{aligned}\end{split}

where $$t_{sp}$$ denotes the spike time and $$U$$ is the increment of $$u$$ produced by a spike. $$u^-, x^-$$ are the corresponding variables just before the arrival of the spike, and $$u^+$$ refers to the moment just after the spike.

Parameters
• tau_f (float) – The time constant of short-term facilitation.

• tau_d (float) – The time constant of short-term depression.

• U (float) – The fraction of resources used per action potential.

• method (str) – The numerical integral method.

__init__(U=0.15, tau_f=1500.0, tau_d=200.0, method='exp_auto', name=None)[source]#

Methods

 __init__([U, tau_f, tau_d, method, name]) clear_input() clone() The function useful to clone a new object when it has been used. filter(g) get_delay_data(identifier, delay_step, *indices) Get delay data according to the provided delay steps. load_states(filename[, verbose]) Load the model states. nodes([method, level, include_self]) Collect all children nodes. offline_fit(target, fit_record) offline_init() online_fit(target, fit_record) online_init() register_delay(identifier, delay_step, ...) Register delay variable. register_implicit_nodes(*nodes, **named_nodes) register_implicit_vars(*variables, ...) register_master(master) reset([batch_size]) Reset function which reset the whole variables in the model. reset_local_delays([nodes]) Reset local delay variables. reset_state([batch_size]) Reset function which reset the states in the model. save_states(filename[, variables]) Save the model states. train_vars([method, level, include_self]) The shortcut for retrieving all trainable variables. unique_name([name, type_]) Get the unique name for this object. update(tdi, pre_spike) The function to specify the updating rule. update_local_delays([nodes]) Update local delay variables. vars([method, level, include_self]) Collect all variables in this node and the children nodes.

Attributes

 derivative global_delay_data isregistered State of the component, representing whether it has been registered. mode Mode of the model, which is useful to control the multiple behaviors of the model. name Name of the model.


text:
# Rigidity matroid

In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.[1][2][3]

## Definition

A framework is an undirected graph, embedded into d-dimensional Euclidean space by providing a d-tuple of Cartesian coordinates for each vertex of the graph. From a framework with n vertices and m edges, one can define a matrix with m rows and nd columns, an expanded version of the incidence matrix of the graph called the rigidity matrix. In this matrix, the entry in row e and column (v,i) is zero if v is not an endpoint of edge e. If, on the other hand, edge e has vertices u and v as endpoints, then the value of the entry is the difference between the ith coordinates of v and u.[1][3]

The rigidity matroid of the given framework is a linear matroid that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is linearly independent. A framework is called generic if the coordinates of its vertices are algebraically independent real numbers. Any two generic frameworks on the same graph G determine the same rigidity matroid, regardless of their specific coordinates. This is the (d-dimensional) rigidity matroid of G.[1][3]

## Statics

A load on a framework is a system of forces on the vertices (represented as vectors). A stress is a special case of a load, in which equal and opposite forces are applied to the two endpoints of each edge (which may be imagined as a spring) and the forces formed in this way are added at each vertex. Every stress is an equilibrium load, a load that does not impose any translational force on the whole system (the sum of its force vectors is zero) nor any rotational force. A linear dependence among the rows of the rigidity matrix may be represented as a self-stress, an assignment of equal and opposite forces to the endpoints of each edge that is not identically zero but that adds to zero at every vertex. Thus, a set of edges forms an independent set in the rigidity matroid if and only if it has no self-stress.[3]

The vector space of all possible loads, on a system of n vertices, has dimension dn, among which the equilibrium loads form a subspace of dimension ${\displaystyle dn-{\binom {d+1}{2}}}$. An independent set in the rigidity matroid has a system of equilibrium loads whose dimension equals the cardinality of the set, so the maximum rank that any set in the matroid can have is ${\displaystyle dn-{\binom {d+1}{2}}}$. If a set has this rank, it follows that its set of stresses is the same as the space of equilibrium loads. Alternatively and equivalently, in this case every equilibrium load on the framework may be resolved by a stress that generates an equal and opposite set of forces, and the framework is said to be statically rigid.[3]

## Kinematics

If the vertices of a framework are in a motion, then that motion may be described over small scales of distance by its gradient, a vector for each vertex specifying its speed and direction. The gradient describes a linearized approximation to the actual motion of the points, in which each point moves at constant velocity in a straight line. The gradient may be described as a row vector that has one real number coordinate for each pair ${\displaystyle (v,i)}$ where ${\displaystyle v}$ is a vertex of the framework and ${\displaystyle i}$ is the index of one of the Cartesian coordinates of ${\displaystyle d}$-dimensional space; that is, the dimension of the gradient is the same as the width of the rigidity matrix.[1][3]

If the edges of the framework are assumed to be rigid bars that can neither expand nor contract (but can freely rotate) then any motion respecting this rigidity must preserve the lengths of the edges: the derivative of length, as a function of the time over which the motion occurs, must remain zero. This condition may be expressed in linear algebra as a constraint that the gradient vector of the motion of the vertices must have zero inner product with the row of the rigidity matrix that represents the given edge. Thus, the family of gradients of (infinitesimally) rigid motions is given by the nullspace of the rigidity matrix.[1][3] For frameworks that are not in generic position, it is possible that some infinitesimally rigid motions (vectors in the nullspace of the rigidity matrix) are not the gradients of any continuous motion, but this cannot happen for generic frameworks.[2]

A rigid motion of the framework is a motion such that, at each point in time, the framework is congruent to its original configuration. Rigid motions include translations and rotations of Euclidean space; the gradients of rigid motions form a linear space having the translations and rotations as bases, of dimension ${\displaystyle {\binom {d+1}{2}}}$, which must always be a subspace of the nullspace of the rigidity matrix. Because the nullspace always has at least this dimension, the rigidity matroid can have rank at most ${\displaystyle dn-{\binom {d+1}{2}}}$, and when it does have this rank the only motions that preserve the lengths of the edges of the framework are the rigid motions. In this case the framework is said to be first-order (or infinitesimally) rigid.[1][3] More generally, an edge ${\displaystyle e}$ belongs to the matroid closure operation of a set ${\displaystyle S}$ if and only if there does not exist a continuous motion of the framework that changes the length of ${\displaystyle e}$ but leaves the lengths of the edges in ${\displaystyle S}$ unchanged.[1]

Although defined in different terms (column vectors versus row vectors, or forces versus motions) static rigidity and first-order rigidity reduce to the same properties of the underlying matrix and therefore coincide with each other. In two dimensions, the generic rigidity matroid also describes the number of degrees of freedom of a different kind of motion, in which each edge is constrained to stay parallel to its original position rather than being constrained to maintain the same length; however, the equivalence between rigidity and parallel motion breaks down in higher dimensions.[3]

## Unique realization

The diamond graph, generically rigid but not uniquely realizable

A framework has a unique realization in d-dimensional space if every placement of the same graph with the same edge lengths is congruent to it. Such a framework must necessarily be rigid, because otherwise there exists a continuous motion bringing it to a non-congruent placement with the same edge lengths, but unique realizability is stronger than rigidity. For instance, the diamond graph (two triangles sharing an edge) is rigid in two dimensions, but it is not uniquely realizable because it has two different realizations, one in which the triangles are on opposite sides of the shared edge and one in which they are both on the same side. Uniquely realizable graphs are important in applications that involve reconstruction of shapes from distances, such as triangulation in land surveying,[4] the determination of the positions of the nodes in a wireless sensor network,[5] and the reconstruction of conformations of molecules via nuclear magnetic resonance spectroscopy.[4]

Bruce Hendrickson defined a graph to be redundantly rigid if it remains rigid after removing any one of its edges. In matroidal terms, this means that the rigidity matroid has the full rank ${\displaystyle dn-{\binom {d+1}{2}}}$ and that the matroid does not have any coloops. Hendrickson proved that every uniquely realizable framework (with generic edge lengths) is either a complete graph or a ${\displaystyle (d+1)}$-vertex-connected, redundantly rigid graph, and he conjectured that this is an exact characterization of the uniquely realizable frameworks.[6] The conjecture is true for one and two dimensions; in the one-dimensional case, for instance, a graph is uniquely realizable if and only if it is connected and bridgeless.[7] However, Henrickson's conjecture is false for three or more dimensions.[8] For frameworks that are not generic, it is NP-hard to determine whether a given framework is uniquely realizable.[9]

## Relation to sparsity

Streinu & Theran (2009) define a graph as being ${\displaystyle (k,l)}$-sparse if every nonempty subgraph with ${\displaystyle n}$ vertices has at most ${\displaystyle kn-l}$ edges, and ${\displaystyle (k,l)}$-tight if it is ${\displaystyle (k,l)}$-sparse and has exactly ${\displaystyle kn-l}$ edges.[10] From the consideration of loads and stresses it can be seen that a set of edges that is independent in the rigidity matroid forms a ${\displaystyle (d,{\binom {d+1}{2}})}$-sparse graph, for if not there would exist a subgraph whose number of edges would exceed the dimension of its space of equilibrium loads, from which it follows that it would have a self-stress. By similar reasoning, a set of edges that is both independent and rigid forms a ${\displaystyle (d,{\binom {d+1}{2}})}$-tight graph. For instance, in one dimension, the independent sets form the edge sets of forests, (1,1)-sparse graphs, and the independent rigid sets form the edge sets of trees, (1,1)-tight graphs. In this case the rigidity matroid of a framework is the same as the graphic matroid of the corresponding graph.[2]

In two dimensions, Laman (1970) showed that the same characterization is true: the independent sets form the edge sets of (2,3)-sparse graphs and the independent rigid sets form the edge sets of (2,3)-tight graphs.[11] Based on this work the (2,3)-tight graphs (the graphs of minimally rigid generic frameworks in two dimensions) have come to be known as Laman graphs. The family of Laman graphs on a fixed set of ${\displaystyle n}$ vertices forms the set of bases of the rigidity matroid of a complete graph, and more generally for every graph ${\displaystyle G}$ that forms a rigid framework in two dimensions, the spanning Laman subgraphs of ${\displaystyle G}$ are the bases of the rigidity matroid of ${\displaystyle G}$.

However, in higher dimensions not every ${\displaystyle (d,{\binom {d+1}{2}})}$-tight graph is minimally rigid, and characterizing the minimally rigid graphs (the bases of the rigidity matroid of the complete graph) is an important open problem.[12]

## References

1. Graver, Jack E. (1991), "Rigidity matroids", SIAM Journal on Discrete Mathematics, 4 (3): 355–368, MR 1105942, doi:10.1137/0404032.
2. ^ a b c Whiteley, Walter (1992), "Matroids and rigid structures", Matroid Applications, Encyclopedia of Mathematics and its Applications, 40, Cambridge: Cambridge Univ. Press, pp. 1–53, MR 1165538, doi:10.1017/CBO9780511662041.002.
3. Whiteley, Walter (1996), "Some matroids from discrete applied geometry", Matroid theory (Seattle, WA, 1995), Contemporary Mathematics, 197, Providence, RI: American Mathematical Society, pp. 171–311, MR 1411692, doi:10.1090/conm/197/02540.
4. ^ a b Hendrickson, Bruce (1995), "The molecule problem: exploiting structure in global optimization", SIAM Journal on Optimization, 5 (4): 835–857, MR 1358807, doi:10.1137/0805040.
5. ^ Eren, T.; Goldenberg, O.K.; Whiteley, W.; Yang, Y.R.; Morse, A.S.; Anderson, B.D.O.; Belhumeur, P.N. (2004), "Rigidity, computation, and randomization in network localization", Proc. Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2004), 4, pp. 2673–2684, doi:10.1109/INFCOM.2004.1354686.
6. ^ Hendrickson, Bruce (1992), "Conditions for unique graph realizations", SIAM Journal on Computing, 21 (1): 65–84, MR 1148818, doi:10.1137/0221008.
7. ^ Jackson, Bill; Jordán, Tibor (2005), "Connected rigidity matroids and unique realizations of graphs", Journal of Combinatorial Theory, Series B, 94 (1): 1–29, MR 2130278, doi:10.1016/j.jctb.2004.11.002.
8. ^ Connelly, Robert (1991), "On generic global rigidity", Applied Geometry and Discrete Mathematics, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, 4, Providence, RI: American Mathematical Society, pp. 147–155, MR 1116345.
9. ^ Saxe, J. B. (1979), Embeddability of weighted graphs in k-space is strongly NP-hard, Technical Report, Pittsburgh, PA: Computer Science Department, Carnegie-Mellon University. As cited by Jackson & Jordán (2005).
10. ^ Streinu, I.; Theran, L. (2009), "Sparse hypergraphs and pebble game algorithms", European Journal of Combinatorics, 30 (8): 1944–1964, arXiv:, doi:10.1016/j.ejc.2008.12.018.
11. ^ Laman, G. (1970), "On graphs and the rigidity of plane skeletal structures", J. Engineering Mathematics, 4 (4): 331–340, MR 0269535, doi:10.1007/BF01534980.
12. ^ Jackson, Bill; Jordán, Tibor (2006), "On the rank function of the 3-dimensional rigidity matroid" (PDF), International Journal of Computational Geometry & Applications, 16 (5-6): 415–429, MR 2269396, doi:10.1142/S0218195906002117.


text:
# Question #416e8

Anyways, ${K}_{P}$ is used only for gases, however, ${K}_{C}$ it can be used for both.


text:
# How do i prove that a measure space induced by an outer measure is complete?

Royden Real-analysis 4th edition p.349

Let $X$ be a set. Let $\mu$ be an outer measure on $P(X)$. Define $\mathfrak{M}$ = $\lbrace A\subset X \mid A \text{ is } \mu- \text{ measurable} \rbrace$. Let $\mu^*$ be the restriction of $\mu$ to $\mathfrak{M}$. Then, $(X,\mathfrak{M},\mu^*)$ is a complete measure space.

I know that it is a measure space, but how do i prove that it is complete?

Subset $A$ of a measure 0 set $B$ would have outer measure 0. Given any $E$ then $\mu(A\bigcap E)=0$ because $A\bigcap E\subset B$; and $\mu(A^{c}\bigcap E)=\mu(E)$ because $B^{c}\bigcap E\subset A^{c}\bigcap E\subset E$ and $\mu(B^{c}\bigcap E)=\mu(E)$ since $B$ is measurable with measure 0.
Hence $\mu(A\bigcap E)+\mu(A^{c}\bigcap E)=\mu(E)$ so $A$ is measurable.
• I have a question, "because $A\bigcap E\subset B$; and $\mu(A^{c}\bigcap E)=\mu(E)$ because $B^{c}\bigcap E\subset A^{c}\bigcap E\subset E$ and $\mu(B^{c}\bigcap E)=\mu(E)$ since $B$ is measurable with measure 0" In the above quote, why dont you only use the fact that $\mu(A^c\cap E)\leq \mu(E)$ in order to have, $\mu(E)\geq \mu(A\cap E)+\mu(A^c\cap E)$. – Eduardo Aug 7 '18 at 17:22
• I think there is some problem in my above questioning, because it do not use the mensurability of $B$... Anyway I will be gratefull with some clarification... – Eduardo Aug 7 '18 at 17:31


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Proving $f$ is continuous if $f_1$ and $f_2$ are continuous and the reciprocal

For $$i=1,2$$ let $$(X_i,\tau_i)$$ and $$(X'_i,\tau_i)$$ be topological spaces, and $$f_i: X_i\to X'_i$$ functions. Consider the function $$f:X_1\times X_2\to X'_1\times X'_2$$ defined by $$f(x_1,x_2)=(f_1(x_1),f_2(x_2))$$. Show that if $$f_1,f_2$$ are continuous then $$f$$ is continuous for the product topologies. Prove the reciprocal.

I tried to prove the statement in the following way:

$$\Rightarrow$$ Let $$\mathscr{U}$$ be an open set such that by definition of product topology $$\mathscr{U}=A\times B$$. As $$f_1$$ and $$f_2$$ are continuous functions, then $$f_1^{-1}(A)\in\tau'_1$$ and $$f_2^{-1}(B)\in\tau'_2$$.

So $$f^{-1}(\mathscr{U})=f^{-1}(A\times B)=f_1^{-1}(A)\times f_2^{-1}(B)$$so that by definition of open sets in the product topology are open hence $$f_1^{-1}(A)\times f_2^{-1}(B)\in\tau'_i$$.

$$\Leftarrow$$ Let's define the following functions $$I_1: (X'_i,\tau'_i)\to (X_1\times a_2)$$ and $$I_2: (X'_i,\tau'_i)\to (a_2\times X_2)$$ such that $$(a_1,a_2)\in X'_i$$ By known theorem $$I_1$$ and $$I_2$$ are homeomorphisms. Using the $$p_1$$ and $$p_2$$ as the projection functions. $$f_1=p_1\circ f\circ I_1$$ and $$f_2=p_2\circ f\circ I_2$$. As the composition of continuous functions is continuous hence $$f_1$$ and $$f_2$$ are continuous.

Questions:

1) Is my proof right? Can I give the following step $$f^{-1}(A\times B)=f_1^{-1}(A)\times f_2^{-1}(B)$$?

2) I was advised to define $$I_1$$ and $$I_2$$. Why are the projections not enough? I mean $$f_1=p_1\circ f$$ and $$f_2=p_2\circ f$$.

• $X_1 \times a_2$ is sloppy notation. Jan 20 '19 at 13:19
Firstly, an open set in the product topology is not always of the form $$U \times V$$, it is a union of such open sets, as these form a base for the product topology. So if $$O$$ is product open and so $$O= \bigcup_{i \in I} (U_i \times V_i)$$ we indeed have that $$f^{-1}[U_i \times V_i] = \{(x,x'): f(x,x') \in U_i \times V_i\} =\ \{(x,x'): (f_1(x), f_2(x') \in U_i \times V_i\} = \{(x,x'): f_1(x) \in U_i \text{ and } f_2(x') \in V_i\} =\ \{(x,x'): x\in f_1^{-1}[U_i] \text{ and } x' \in f_2^{-1}[V_i]\} =f_1^{-1}[U_i] \times f_2^{-1}[V_i]$$ which is basic product open, and so $$f^{-1}[O] = \bigcup_i f^{-1}[U_i \times V_i]$$ is a union of open sets and hence open.
The reverse needs the embeddings, as $$\pi_1 \circ f$$ is a map defined on $$X_1 \times X_2$$ and not on $$X_1$$, so is not equal to $$f_1$$. In order to use $$f$$ we need two arguments, so fix $$a \in X_1$$ and $$a' \in X_2$$ and define $$j: X_1 \to X_1 \times X_2$$ by $$j(x)=(x,a')$$. It's continuous because the composition of $$j$$ with the two projections are either the identity on $$X_1$$ or a constant map, so continuous always. And then $$(\pi_1 \circ f \circ j)(x)=\pi_1(f(x,a'))=\pi_1(f_1(x), f_2(a'))= f_1(x)$$ for all $$x\in X_1$$ so $$f_1$$ is then a composition of three continuous maps, hence continuous.
$$f_2$$ is similarly continuous using $$i: X_1 \to X_1 \times X_2$$ defined by $$i(x)=(a,x)$$.


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# Where do I find documentation for packages in a MiKTeX installation?

Is there a systematic way to find documentation for latex packages (e.g. beamer, TikZ, amsmath) on a windows machine with a MiKTeX/TeXnicCenter installation? It seems the package user manuals (pdf files) are scattered around in different folders on my machine. Can one ask MiKTeX or TeXnicCenter to find the documentation for a specific package?

• in texlive the command is `texdoc` I understand it's a bit different in miktex but there are some comments about that here: tex.stackexchange.com/questions/4358/… – David Carlisle Apr 2 '14 at 14:19
• `texdoc` works for `miktex` also. – user11232 Apr 2 '14 at 14:23
• from command prompt `texdoc <package name>` (for example, `texdoc amsmath`) should work on windows and miktex. – user11232 Apr 2 '14 at 14:39
• `texdoc` (and `mthelp`, the MikTeX version) both work directly from the search field in the Start menu I believe, in W7 at least. – Torbjørn T. Apr 2 '14 at 14:41
• Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. – Speravir Apr 3 '14 at 3:28

The original command is `mthelp`, as noted by Torbjørn, to be used on command prompt. Later `texdoc` was introduced as alias for `mthelp`, so Harish’s hint is valid, as well. Execute `mthelp -?` (or with `texdoc`) for command line options (Note, that MiKTeX’s `texdoc` does not understand all command line options from TeX Live’s `texdoc` and vice versa.)

To quote Ulrike Fischer from her answer to the related question How to force MikTeX' texdoc to open package manuals with a PDF reader rather than the DVI viewer? (read also the other answers!):

`texdoc --view packagename` tries to open the "main" documentation file. If this file is a `pdf` it is opened in the pdf-viewer, if it is a `dvi` the dvi-viewer YAP is used.

By simply writing `mthelp ⟨supposed-package-name⟩` it can happen that a local HTML page is produced and opened with your default browser showing the found results. This local HTML file is on my machine saved in `C:\Users\Steffen\AppData\Local\MiKTeX\2.9\miktex\mthelp\` (in general the “UserData” root of MiKTeX, cf. the “Additional Note” in Create a local texmf tree in MiKTeX). It does not matter here, whether I invoked the search with `mthelp` or `texdoc`.

A good example would be `mthelp dickimaw`. This will produce a HTML file with 3 results, if the Dickimaw tutorials are installed:

``````<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<title>dickimaw</title>

dickimaw - Books and tutorials from the â€œDickimaw LaTeX Seriesâ€

The package provides are some of the books and tutorials that form part of the â€œDickimaw LaTeX Seriesâ€. Only the A4 PDF is included here. Other formats, such as HTML or a screen optimized PDF, are available from the package home page. Books included are: â€œLaTeX for Complete Novicesâ€: an introductory guide to LaTeX. â€œUsing LaTeX to Write a PhD Thesisâ€: a follow-on from â€œLaTeX for Complete Novicesâ€ geared towards students who want to use LaTeX to write their PhD thesis. â€œCreating a LaTeX minimal exampleâ€: describes how to create a minimal example, which can be used as a debugging aid when you encounter errors in your LaTeX documents.

C:\LaTeX\MiKTeX\doc\info\dickimaw\dickimaw-minexample.pdf

C:\LaTeX\MiKTeX\doc\info\dickimaw\dickimaw-novices.pdf

C:\LaTeX\MiKTeX\doc\info\dickimaw\dickimaw-thesis.pdf

``````

But typing `mthelp --view dickimaw` will open the first result of them, i.e. `dickimaw-minexample.pdf`.

Note, that there is a certain amount of packages, where the name of main documentation is different from the package name. TikZ is a good example where you will be without success:

```C:\Users\Steffen>mthelp tikz
Documentation for tikz could not be found.
```

You have to know that TikZ is part of of the pgf bundle and described in `pgfmanual.pdf`, so `mthelp pgfmanual` will open this file in you default PDF viewer.

I am meanwhile used to look directly in the `doc` subtree of MiKTeX. But when you use the feature of installing packages “on-the-fly”, these are found in your user profile, according to MiKTeX denomination in “UserConfig” (cf. again “Additional Note” in Create a local texmf tree in MiKTeX).


text:
Doubly Magic Matrices and the MNS

[Edit: For the doubly magic version of the CKM matrix, see CKM as a Magic Unitary Matrix. /Edit ] In the previous post, The MNS Matrix as Magic Square, we applied Kea’s observation that the CKM (quark weak mixing) matrix is approximately the sum of a 1-circulant and 2-circulant matrix to the MNS (lepton weak mixing) matrix. We found that the MNS matrix can also be written that way as well. The squared magnitudes (probabilities) of the MNS matrix are:

Messing around with the results of the previous post, we can obtain these values as the squared magnitudes of the sum of a rather simple 1-circulant and a 2-circulant matrix as follows:

The above matrix has the additional feature that its rows and columns all sum to $(1+\sqrt{2}+i-i\sqrt{2})/\sqrt{6}$. This number is a complex phase, that is, its squared magnitude is 1. So we can multiply by the complex conjugate of this and obtain an MNS matrix whose rows and columns sum to 1, which is the form given in the previous post, (other than I have swapped the columns around to match the literature).

So here’s the problem: Define a complex NxN matrix as “doubly magic” if its rows and columns all sum to unity, and its rows and columns give the same sum when their squared magnitudes are taken before summation. That is, a matrix M is doubly magic if:

for all k (and where the “[ ]” means absolute value, which I can’t fix on the computer I’m using tonight because the POS doesn’t have a modern paint).

How can you characterize such matrices? The solution set for 2×2 matrices can be parameterized by any real number 0 <= a <= 1 as follows:

The above matrix is fully characterized by its top left value. As a increases from 0 to 1, the top left value runs from 0 to (1 +- i)/2, and then on to 1. The squared magnitude is just a, so for each value of a other than 0 and 1, there are two solutions that are complex conjugates. Is there a way to generalize the above solution to 3×3 matrices?

For the case of the tribimaximal matrix, in deriving the doubly magic version of the matrix I found that there was only the given solution (plus the replacement i to -i). This is just like the above general solution for the 2×2 matrices. So, for an arbitrary 3×3 matrix of probabilities that sum to unity is there always a unique (up to complex conjugacy) doubly magic version of the amplitude? Is the MNS matrix just lucky or what? Does the fact that the MNS matrix can be written to be orthonormal play a part in this?

The CKM matrix is usually not written with phases; the top right hand corner (i.e. the Cabibo corner) experimental data is roughly:

To write this as a doubly magic matrix, average the off diagonal values and solve for $0.228 = [1-a+i\sqrt{a-a^2}]$ or therefore $0.228^2 = (1-a)^2 + (a-a^2) = 1-a$ to obtain a = 0.951 and we find that the 2×2 magic version of the Cabibo mixing matrix is approximately:

Can the rest of the CKM matrix be put into doubly magic form? Word is that Kea, who started all this stuff, is working on the problem. Hopefully it will be interesting. Meanwhile, I need to write a post giving the weak hypercharge and weak isospin quantum numbers in 1-circulant and 2-circulant idempotent matrix form. The hope is that this will be a nice fit to the weak mixing angles in the same form, but I suspect that we will have to understand the quark mixing angles before this is all through.

Filed under physics

6 responses to “Doubly Magic Matrices and the MNS”

1. Aw, Carl, I don’t mind if you do it! I’m tired and about to watch a movie.

2. carlbrannen

How about you come up with the high-brow ideas and I’ll crank out the algebra?

3. Pingback: Unitarity and the CKM Matrix « Mass

4. Pingback: CKM as a Magic Unitary Matrix « Mass

5. Pingback: CKM in MNS form! Victory! « Mass


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Zbl 0865.90014
Delbaen, Freddy; Schachermayer, Walter
A general version of the fundamental theorem of asset pricing.
(English)
[J] Math. Ann. 300, No.3, 463-520 (1994). ISSN 0025-5831; ISSN 1432-1807/e

The fundamental theorem of asset pricing is that for a stochastic process $S=(S_t)_{t\in\bbfR}$, the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. The present paper focuses on the term essentially in the above statement. The general idea underlying the no arbitrage condition is that there should be no trading strategy $H$ for the process $S$ such that the final payoff described by the stochastic integral $(H.S)_\infty$ is a nonnegative function strictly positive with positive probability which has been termed as the no free lunch with vanishing risk (NFLVR). The major result of this paper is contained in the following: Let $S$ be a bounded real valued semi-martingale. There is an equivalent martingale measure for $S$ if and only if $S$ satisfies NFLVR. The fact that NFLVR guarantees the existence of an equivalent martingale measure for $S$ allows wide applicability of martingale theory. Several versions of the no free lunch condition are also introduced and their relationship studied.
[A.Mukherji (New Delhi)]
MSC 2000:
*91B24 Price theory and market structure
60G44 Martingales with continuous parameter

Keywords: asset pricing; stochastic process; equivalent martingale measure; no arbitrage condition; no free lunch

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