text: Its domain and range are both (-∞, ∞) or all real numbers as well. The range is the set of possible output values, which are shown on the y -axis. This means that they also all share a common parent function:  y=bx. Let us see the domain and the range … What is the domain and range of f(x)? Which of the following functions do not belong to the given family of functions? The domain of a function is the set of all real values of x that will give real values for y . Domain and Range of Relations and Functions; Finding the Domain Algebraically; More Practice; Note: More advanced topics with functions can be found in the Parent Functions and Transformations and Advanced Functions: Compositions, Even and Odd, and Extrema section. Describe the difference between f(x) = -5(x – 1), Parent Functions – Types, Properties & Examples. Learn . Domain and Range of absolute value function. Range of a quadratic (parent) is y>= 0. voila Which of the following functions do not belong to the given family of functions? Things about the ten parent functions. Based from the graph, we can see that the x and y values of g(x) will never be negative. Sample Problem 3: Use the graph of parent function to graph each function. We can do this by remembering the important properties of each function and identifying which of the parent graphs we’ve discussed match the one that’s given. The sine function takes the reals (domain) to the closed interval (range). Range: c F è W2, è W2 g Domain: :∞,∞ ; Range: k F è W2, è W2 o F è W2 Domain: >1,1 ? From the graph, we can see that it forms a parabola, confirming that its parent function is indeed y = x2. The domain, range… Another way to identify the domain and range of functions is by using graphs. That’s because functions sharing the same degree will follow a similar curve and share the same parent functions. Exponential functions are functions that have algebraic expressions in their exponent. All linear functions have a straight line as a graph. Obviously, that value is x = 2 and so the domain is all x values except x = 2. 1. The domain of a graph consists of all the input values shown on the $$x$$-axis. Domain and range. Range: >1,1 ? The range of a function is the set of output values when all x-values in the domain are evaluated into the function, commonly known as the y-values.This means I need to find the domain first in order to describe the range.. To find the range is a bit trickier than finding the domain. Similar to the square root function, its parent function is expressed as y = ∛x. Refer to this article to learn about the characteristics of parent functions. The vertex of the parent function y = x2 lies on the origin. Converting repeating decimals in to fractions. The graphs of five functions are shown below. Domain: All reals except @ J E 5 6 A è, L0,1, G2 Range: :∞,∞ ; Domain: All reals except @ J E 5 6 A è, J L0, G1, G2 . That leaves us with the third option. We can also see that this function is increasing throughout its domain. All quadratic functions have a domain of all real numbers. To find the range of a function, first find the x-value and y-value of the vertex using the formula x = -b/2a. Date Lesson Assignment Day 1 Monday Aug. 26 Introductory Activities. Odd Powered Parent Cubic. The domain is D=[0,+\\infty[ because \\sqrt{x} exists if and only if x\\geq 0. Parent :! This means that the domain and range of y = √x are both [0, ∞). We use parent functions to guide us in graphing functions that are found in the same family. 0. Identify the parent function of the following functions. Unit 1 - Parent Functions and their Graphs . (^ is before an exponent. Which of the following functions do not belong to the given family of functions? Linear parent functions, a set out data with one specific output and input. The graph of h(x) shows that their x and y values will never be equal to 0. It’s now time to refresh our knowledge about functions and also learn about new functions. This function is increasing throughout its domain. Parent Functions And Transformations Parent Functions: When you hear the term parent function, you may be inclined to think of… Recent Posts. Or the domain of the function f x = 1 x − 4 is the set of all real numbers except x = 4 . PLAY. Edit. Hence, its domain is (0,∞). In Functions and Function Notation, we were introduced to the concepts of domain and range. Lastly, state the domain and range of the transformed function (of the graph given to you) in set-builder notation. Polynomials. The graph extends on both sides of x, so it has a, The parabola never goes below the x-axis, so it has a, The graph extends to the right side of x and is never less than 2, so it has a, As long as the x and y are never equal to zero, h(x) is still valid, so it has both a, The graph extends on both sides of x and y, so it has a, The highest degree of f(x) is 3, so it’s a cubic function. Is the function’s graph decreasing or increasing? What if we’re given a function or its graph and we need to identify its parent function? On simplification, when x ≠ 2 it becomes a linear function f x = x + 1 . Let’s try f(x) = 5(x – 1)2. (y ≥ 0) Y-intercept: (0,1) Also a Step by Step Calculator to Find Domain of a Function and a Step by Step Calculator to Find Range of a Function are included in this website. Range: y is greater than or equal to 0. x-intercept: 0. y-intercept: 0. Domain Range Continuous Increasing Decreasing Constant Left End Right End Symmetry x-intercepts y-intercepts VA HA Bounded Extrema. Menu. Domain: ARN. The parent function of linear functions is y = x and it passes through the origin. Choose from 500 different sets of parent functions domain range flashcards on Quizlet. All linear functions have a straight line as a graph. The idea again is to exclude the values of x that can make the denominator zero. This section will present each of the function types from the library of functions list, give it’s parent function, and then any relevant information on that function. The example below shows two different ways that a function can be represented: as a function table, and as a set of coordinates. . Match. The range of a function is all the possible values of the dependent variable y.. ... Domain and range are real numbers; Slope, or rate of change, is … 9 months ago. Watch Queue Queue Which parent function matches the graph? They also show an increasing curve that resembles the graph of a square root function. Then state what happened to each of the parent functions in your own words – did the parent function move left/right/up/down, flip, stretch or compress – to the best of your ability. The range values for these functions get very small (toward negative infinity) or very large (toward positive infinity) whenever the denominator of the respective ratio gets close to 0. Table of Domain and Range of Common Functions. View Domain and Range Practice.pdf from MATH 250 at George Mason University. Linear parent functions, a set out data with one specific output and input. First four parent functions as its highest degree numbers as well or root! Absolute values can never be equal to zero for y = √x = ∛x is increasing its! Even Powered parent quadratic forms a parabola, confirming that its parent function is also increasing throughout its and! 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# What is the vector field associated with this ODE $\ddot{x}+p(t)\dot{x}+q(t)x=0$? What is the divergence of the vector field?

What is the vector field associated with this ODE $\ddot{x}+p(t)\dot{x}+q(t)x=0$? What is the divergence of the vector field?

My attempt:

Let $y=\dot{x}$

$\ddot{x}+p(t)\dot{x}+q(t)x=0\ \Rightarrow \frac{d}{dt}\begin{pmatrix} x\\y \end{pmatrix}=\begin{pmatrix}\ y\\ -p(t)y-q(t)x\end{pmatrix}=F$

Then $\nabla\ \cdot F=\frac{\partial y}{\partial x}+\frac{\partial (-py-qx)}{\partial y}=-p(t)$.

Is this correct?

• Give some context. Are you taking a course on differential equations? Why do you have no idea? Have you read the relevant part of the textbook or lecture notes? What exactly do you find confusing? – symplectomorphic Oct 27 '16 at 2:37
• @symplectomorphic Let $y=\dot{x}$, $\ddot{x}+p(t)\dot{x}+q(t)x=0\ \Rightarrow \frac{d}{dt}\begin{pmatrix} x\\y \end{pmatrix}=\begin{pmatrix}\ y\\ -p(t)\dot{x}-q(t)x\end{pmatrix}$. Is $(y,-p(t)\dot{x}-q(t)x)$ the vector field associated with this ODE? – User90 Oct 27 '16 at 3:34
• Not quite. The idea is to transform the second-order equation into a system of first-order equations by setting $y=\dot{x}$. Then $\dot{x}=y$ and $\dot{y}=-py-qx$. So the second row of your matrix should be $-py-qx$, not $-p\dot{x}$. – symplectomorphic Oct 27 '16 at 3:39
• @symplectomorphic So the divergence is $\nabla\ \cdot F=\frac{\partial y}{\partial x}+\frac{\partial (-py-qx)}{\partial y}=-p(t)$. Is this correct? – User90 Oct 27 '16 at 4:41

format: plain


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The ‘best‘ way to indicate matrix transpose in my opinion is,

$\mathbf{X}^\top$ is the matrix transpose of $\mathbf{X}$.


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# Does hydrazine crystallize when freezing?

When discussing problems of keeping hydrazine from freezing on spacecraft we stumbled upon a question: can pure hydrazine separate through fraction-freezing from the stabilizing substances like UDMH? That would largely depend on what form it takes. If it crystallizes, it's a risk - it's quite unstable when pure. If it turns amorphous, embedding the stabilizer within the structure, it's moot. Publicly available resources are quite skimpy on descriptions of freezing hydrazine (or I just suck at searching). So, -

Does hydrazine crystallize when freezing? Can it trigger fraction-freezing separation if it's the first freezing substance?

• Pretty much everything crystallizes when freezing. The exceptions are relatively rare. – Ivan Neretin Dec 6 '18 at 10:06
• @IvanNeretin: non-polar substances tend to just grow more dense gradually turning into amorphous solids. – SF. Dec 6 '18 at 10:28
• Even if this were mostly true (which it isn't), it doesn't apply to hydrazine, which is polar. – Ivan Neretin Dec 6 '18 at 10:30
• @SF. - There are a hundred-odd stable(ish) elements that all seem to have crystalline solid phases... – Jon Custer Dec 6 '18 at 14:58
• @JonCuster: And there are millions of complex hydrocarbon compounds with lengthy chains that don't.... – SF. Dec 6 '18 at 15:17

I would take a different position on answering the question, and look at the water-hydrazine binary phase diagram. Looking at J. A. McMillan and S.C. Los, J. Chemical Physics 42(1) 160-161 (1965) one finds the experimentally determined phase diagram. With enough water in the mix, one proceeds smoothly from the pure hydrazine melting point down to a slight eutectic near the monohydride, but only 3K lower in temperature.

Further, the same authors have another paper, J. Chem Phys 42(3) 829-834 (1965) that discusses the nonequilibrium phase transformations. The first sentence in the abstract says

Although neither hydrazine nor water supercool to a large extent, their mixtures do. Aqueous solutions of hydrazine may be strongly supercooled and stabilized as rigid glasses.

In the introductory paragraph, one also finds

Since the phase diagram of this system is known, it was tempting to look for a correlation between the irreversible processes of crystallization and the equilibrium phase transformations. The hydrazine-water system is particularly useful for this purpose, since crystallization can indeed be induced in the entire range of concentration. Samples of different hydrazine content were then supercooled until glass stabilization occurred and then warmed up at different rates. The processes of glass transformation and crystallization were studied, and values of the free-energy barriers could be obtained for the latter process.

The rest of this second paper is devoted to measuring and describing thermodynamically the crystallization curves at different water-hydrazine compostions.

So, yes, you can definitely get a glass, if cooled quickly enough, but it just might decide to phase separate on you if it warms up a bit.

For the case of (only) hydrazine-1,1-dimethylhydrazine, the phase diagram is covered in J. A. McMillan, Journal of Chemical and Engineering Data 12(1) 39-40 (1967). This shows a very slow decrease in the liquidus temperature as 1,1-dimethylhydrazine is added, with the eutectic occurring at 94% mole percent 1,1-dimethylhydrazine at 214K, only 61K below the pure hydrazine melting point. There are no compounds. Further, the article states

Unlike the systems of the components with water, the mixture hydrazine-1,1-dimethylhydrazine does not supercool

which indicates that phase separation and crystallization of the two components will readily occur. On the other hand, the samples were sealed in glass under nitrogen, and if kept in the dark were stable over a 6 month period.

• Unfortunately hydrazine for use as rocket fuel is stabilized with UDMH, not water, which would be very bad for performance of the propellant. – SF. Dec 6 '18 at 17:05
• @SF. - noted. additional information added specifically on hydrazine-1,1-dimethylhydrazine. It isn't a particularly good antifreeze... – Jon Custer Dec 6 '18 at 17:57
• It was good enough for the military :) Huge thanks for the great answer! – SF. Dec 6 '18 at 23:24

I found an old research paper which states that hydrazine forms monoclinic crystalline structure in molecule pairs.

Given the ongoing discussions on whether a rocket-fuel mixture separates or not as it cools, I'm adding this paper which says hydrazine monohydrate forms trigonal crystals. I admit to massive ignorance as to the exact compounds used in fuels.

• The usual hydrazine fuel is stabilized with UDMH (Unsymmetric dimetylhydrazine, $NH_2 - N(CH_3)_2$; properties similar to hydrazine but much more stable, −57 °C freezing point and acts both as stabilizer and antifreeze agent on plain hydrazine.) – SF. Dec 6 '18 at 15:23


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# How to get glossary terms to properly typeset subscripts/superscripts?

I have just begun using the glossaries package. I have created a glossary entry for the coproduct symbol as disjoint union:

\newglossaryentry{coprod}
{
name = {\ensuremath{\coprod}},
description = {The coproduct in the category of sets, i.e., the disjoint union},
sort = coprod,
symbol = {\ensuremath{\coprod}}
}

Where I put either \gls{coprod} or \glssymbol{coprod} in the body of my document, it prints the coproduct symbol correctly, but it doesn't format subscripts and superscripts (e.g. \gls{coprod}_1^r) as it would for the \coprod symbol in math mode (i.e. above and below). Instead, they're to the right, and too close together. How can I fix this?

Addendum: An example of the call to \gls{coprod} in my code looks like:

$[n] = \gls{coprod}_1^r \lambda_i$
• Are you in math mode issuing your \gls? If not that's your problem, but since it is not clear what your context is please provide a MWE. Feb 15, 2017 at 5:27
• Yes the \gls is in math mode. I will try Elad Den's displaystyle idea. Feb 15, 2017 at 17:53
• @TeXnician - I would have tried what Elad Den put in his answer as a MWE, but actually it comes out fine for me too. My issue is happening in a large project (a PhD thesis), using a lot of different packages. I assume at this point that the issue has to do with interaction among the parts. If it is not already a known bug, then I doubt I will be able to isolate the issue without more work than it is worth to me. Feb 16, 2017 at 0:18

I had the exact same problem and when researching found this answer immediately, yet it did not solve my problem.

It seems to be necessary to pass the superscripts or subscript as optional argument to the \gls{} call in the form of \gls{<name>}[_{xx}^{yy}]. Below is an example of this in action:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{glossaries}

\newglossaryentry{sym}{
name = {\ensuremath{\mathbb{R}}},
description = {Real numbers}
}

\makeglossaries

\begin{document}

$\gls{sym}[^3_x]$ in a text.
\begin{align}
\gls{sym}[_t^{2\times3}]
\end{align}

\printglossaries
\end{document}

And this is the result:

It's hard to tell what the problem is. Trying to simulate it I was unable to recreate it.

\documentclass {article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{glossaries}

\newglossaryentry{coprod}
{
name = {\ensuremath{\coprod}},
description = {The coproduct in the category of sets, i.e., the disjoint union},
sort = coprod,
symbol = {\ensuremath{\coprod}}
}

\begin{document}
coprod in a paragraph $\coprod_{1}^{2}$
$\coprod_{1}^{2}$

gls in paragraph $\gls{coprod}_{1}^{2}$
$\gls{coprod}_{1}^{2}$

\end{document}

It sounds like you are not in display style, so either you using \gls in a $$environment, in which case the super and subscript SHOULD be on the side, or there's some other problem. In any case, try defining name = {\ensuremath{\displaystyle\coprod}}, • Thanks (+1). I tried adding the \displaystyle but it didn't help. If I replace the \gls{coprod} in the body of my document with \coprod, it prints as I want it to. Could glossaries be interacting badly with some other package? Feb 16, 2017 at 0:18 • maybe... hard to tell, that's why a MWE is needed... If I had to guess I'd bet on the documentclass rather than a package, are you using some sort of thesis documentclass ? Feb 16, 2017 at 7:37 • I'm using \documentclass{report} Feb 16, 2017 at 13:06 • When I try your code with \documentclass{report} instead of {article} it still looks fine. I think that figuring out a MWE is going to take far more work than it's worth to me. I will either just have to live with the bad formatting, figure out a manual workaround, or else take \coprod out of the glossary. I posted the question hoping it was a known bug. Feb 16, 2017 at 13:10 • In the  environment, I can get around the need to fix this using \underset and \overset. In the$$ environment, it's still a problem because although the subscript/superscript are to the right like they should be, they're further from the \coprod symbol than they should be, and too close together. Feb 16, 2017 at 13:29


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# Dirichlet's principle

(Redirected from Dirichlet principle)
Jump to: navigation, search

In mathematics, Dirichlet's principle in potential theory states that, if the function $u ( x )$ is the solution to Poisson's equation

$\Delta u + f = 0\,$

on a domain $\Omega$ of $\mathbb{R}^n$ with boundary condition

$u=g\text{ on }\partial\Omega,\,$

then u can be obtained as the minimizer of the Dirichlet's energy

$E[v(x)] = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\,\mathrm{d}x$

amongst all twice differentiable functions $v$ such that $v=g$ on $\partial\Omega$ (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet.

Since the Dirichlet's integral is bounded from below, the existence of an infimum is guaranteed. That this infimum is attained was taken for granted by Riemann (who coined the term Dirichlet's principle) and others until Weierstrass gave an example of a functional that does not attain its minimum. Hilbert later justified Riemann's use of Dirichlet's principle.

## References

• Courant, R. (1950), Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer, Interscience
• Lawrence C. Evans (1998), Partial Differential Equations, American Mathematical Society, ISBN 978-0-8218-0772-9


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# Climate sensitivity

Climate sensitivity

Climate sensitivity is a measure of how responsive the temperature of the climate system is to a change in the radiative forcing. It is usually expressed as the temperature change associated with a doubling of the concentration of carbon dioxide in Earth's atmosphere.

The equilibrium climate sensitivity refers to the equilibrium change in global mean near-surface air temperature that would result from a sustained doubling of the atmospheric (equivalent) CO2 concentration (ΔTx2). This value is estimated, by the IPCC Fourth Assessment Report (AR4) as likely to be in the range 2 to 4.5 °C with a best estimate of about 3 °C, and is very unlikely to be less than 1.5 °C. Values substantially higher than 4.5 °C cannot be excluded, but agreement of models with observations is not as good for those values.[1] This is a slight change from the IPCC Third Assessment Report (TAR), which said it was "likely to be in the range of 1.5 to 4.5 °C".[2] More recent work continues to support a best-guess value around 3 °C.[3]

A model estimate of equilibrium sensitivity thus requires a very long model integration. A measure requiring shorter integrations is the transient climate sensitivity which is defined as the average temperature response over a twenty year period centered at CO2 doubling in a transient simulation with CO2 increasing at 1% per year.[4] The transient sensitivity is lower than the equilibrium sensitivity, due to the "inertia" of ocean heat uptake. Fully equilibrating ocean temperatures would require integrations of thousands of model years.

An estimate of the equilibrium climate sensitivity may be made from combining the effective climate sensitivity with the known properties of the ocean reservoirs and the surface heat fluxes; this is the effective climate sensitivity. This "may vary with forcing history and climate state".[5] [6]

A less commonly used concept, the long-term sensitivity, can be defined which includes the effects of slower feedbacks.[7]

Although climate sensitivity is usually used in the context of radiative forcing by CO2, it is thought of as a general property of the climate system: the change in surface air temperature (ΔTs) following a unit change in radiative forcing (RF) and expressed in units of °C/(W/m2). For this to be so, the measure must be independent of the nature of the forcing (e.g. from greenhouse gases or solar variation); to first order this is indeed found to be so.

For a coupled atmosphere-ocean global climate model the climate sensitivity is an emergent property: it is not a model parameter, but rather a result of a combination of model physics and parameters. By contrast, simpler energy-balance models may have climate sensitivity as an explicit parameter.

$\Delta T_s = \lambda \cdot RF$

The terms represented in the equation relate radiative forcing of any cause to linear changes in global surface temperature change.

It is also possible to estimate climate sensitivity from observations; however, this is difficult due to uncertainties in the forcing and temperature histories.

Climate sensitivity can be a useful summary of the sensitivity of the real climate, or of a given model climate. But it is not the same as the expected climate change at, say 2100: the TAR forecasts this to be an increase of 1.4 to 5.8 °C over 1990.

## Essentials

CO2 climate sensitivity has a component directly due to radiative forcing by CO2 (or any other change in Earth's radiative balance), and a further contribution arising from feedbacks, positive and negative. "Without any feedbacks, a doubling of CO2 (which amounts to a forcing of 3.7 W/m2) would result in 1 °C global warming, which is easy to calculate and is undisputed. The remaining uncertainty is due entirely to feedbacks in the system, namely, the water vapor feedback, the ice-albedo feedback, the cloud feedback, and the lapse rate feedback";[8] addition of these feedbacks leads to a value of approximately 3 °C ± 1.5 °C.

### Radiative forcing due to doubled CO2

In the 1979 NAS report[9] (p. 7), the radiative forcing due to doubled CO2 is estimated to be 4 W/m2, as calculated (for example) in Ramanathan et al. (1979).[10] In 2001 the IPCC adopted the revised value of 3.7 W/m2, the difference attributed to a "stratospheric temperature adjustment".[11] More recently an intercomparison of radiative transfer codes (Collins et al., 2006)[12] showed substantial discrepancies among climate models and between climate models and more exact radiation codes in the forcing attributed to doubled CO2 even in cloud-free sky; presumably the differences would be even greater if forcing were evaluated in the presence of clouds because of differences in the treatment of clouds in different models. Undoubtedly the difference in forcing attributed to doubled CO2 in different climate models contributes to differences in apparent sensitivities of the models, although this effect is thought to be small relative to the intrinsic differences in sensitivities of the models themselves (Webb et al., 2006).

### Sample calculation using industrial-age data

Rahmstorf (2008)[8] provides an informal example of how climate sensitivity might be estimated empirically, from which the following is modified. Denote the sensitivity, i.e. the equilibrium increase in global mean temperature including the effects of feedbacks due to a sustained forcing by doubled CO2 (taken as 3.7 W/m2), as x °C. If Earth were to experience an equilibrium temperature change of ΔT (°C) due to a sustained forcing of ΔF (W/m2), then one might say that x/(ΔT) = (3.7 W/m2)/(ΔF), i.e. that x = ΔT * (3.7 W/m2)/ΔF. The global temperature increase since the beginning of the industrial period (taken as 1750) is about 0.8 °C, and the radiative forcing due to CO2 and other long-lived greenhouse gases (mainly methane, nitrous oxide, and chlorofluorocarbons) emitted since that time is about 2.6 W/m2. Neglecting other forcings and considering the temperature increase to be an equilibrium increase would lead to a sensitivity of about 1.1 °C. However, ΔF also contains contributions due to solar activity (+0.3 W/m2), aerosols (-1 W/m2), ozone (0.3 W/m2) and other lesser influences, bringing the total forcing over the industrial period to 1.6 W/m2 according to best estimate of the IPCC AR4, albeit with substantial uncertainty. Additionally the fact that the climate system is not at equilibrium must be accounted for; this is done by subtracting the planetary heat uptake rate H from the forcing; i.e., x = ΔT * (3.7 W/m2)/(ΔF-H). Taking planetary heat uptake rate as the rate of ocean heat uptake, estimated by the IPCC AR4 as 0.2 W/m2, yields a value for x of 2.1 °C. (All numbers are approximate and quite uncertain.)

### Sample calculation using ice-age data

"... examine the change in temperature and solar forcing between glaciation (ice age) and interglacial (no ice age) periods. The change in temperature, revealed in ice core samples, is 5 °C, while the change in solar forcing is 7.1 W/m2. The computed climate sensitivity is therefore 5/7.1 = 0.7 K(W/m2)−1. We can use this empirically derived climate sensitivity to predict the temperature rise from a forcing of 4 W/m2, arising from a doubling of the atmospheric CO2 from pre-industrial levels. The result is a predicted temperature increase of 3 °C."[13] Based on analysis of uncertainties in total forcing, in Antarctic cooling, and in the ratio of global to Antarctic cooling of the last glacial maximum relative to the present, Ganopolski and Schneider von Deimling (2008) infer a range of 1.3 to 6.8 °C for climate sensitivity determined by this approach.[14]

## History of the concept

### Three degrees as the consensus estimate

The standard modern estimate of climate sensitivity – 3 °C, plus or minus 1.5 °C – originates with a committee on anthropogenic global warming convened in 1979 by the National Academy of Sciences and chaired by Jule Charney. Only two sets of models were available; one, due to Syukuro Manabe, exhibited a climate sensitivity of 2 °C, the other, due to James E. Hansen, exhibited a climate sensitivity of 4 °C. "According to Manabe, Charney chose 0.5 °C as a not-unreasonable margin of error, subtracted it from Manabe’s number, and added it to Hansen’s. Thus was born the 1.5 °C-to-4.5 °C range of likely climate sensitivity that has appeared in every greenhouse assessment since..."[15]

Chapter 4 of the "Charney report" compares the predictions of the models: "We conclude that the predictions ... are basically consistent and mutually supporting. The differences in model results are relatively small and may be accounted for by differences in model characteristics and simplifying assumptions."[9]

### Subsequent developments

In 2008 climatologist Stefan Rahmstorf wrote, regarding the Charney report's original range of uncertainty: "At that time, this range was on very shaky ground. Since then, many vastly improved models have been developed by a number of climate research centers around the world. Current state-of-the-art climate models span a range of 2.6–4.1 °C, most clustering around 3 °C."[8]

### Other estimates

Idso, (1998)[16] reprinted in[17] calculated based on eight natural experiments a λ of 0.1 °C/(Wm−2) resulting in a climate sensitivity of only 0.4 °C for a doubling of the concentration of CO2 in the atmosphere.

Andronova and Schlesinger (2001) found that the climate sensitivity could lie between 1 and 10 °C, with a 54 percent likelihood that it lies outside the IPCC range.[18] The exact range depends on which factors are most important during the instrumental period: "At present, the most likely scenario is one that includes anthropogenic sulfate aerosol forcing but not solar variation. Although the value of the climate sensitivity in that case is most uncertain, there is a 70 percent chance that it exceeds the maximum IPCC value. This is not good news." said Schlesinger.

Forest, et al. (2002)[19] using patterns of change and the MIT EMIC estimated a 95% confidence interval of 1.4–7.7 °C for the climate sensitivity, and a 30% probability that sensitivity was outside the 1.5 to 4.5 °C range.

Gregory, et al. (2002)[20] estimated a lower bound of 1.6 °C by estimating the change in Earth's radiation budget and comparing it to the global warming observed over the 20th century.

Shaviv (2005)[21] carried out a similar analysis for 6 different time scales, ranging from the 11-yr solar cycle to the climate variations over geological time scales. He found a typical sensitivity of 0.54±0.12°K/(W m−2) or 2.1 °C (ranging between 1.6 °C and 2.5 °C at 99% confidence) if there is no cosmic-ray climate connection, or a typical sensitivity of 0.35±0.09°K/(W m−2) or 1.3 °C (between 0.99 °C and 2.5 °C at 99% confidence), if the cosmic-ray climate link is real. (Note Shaviv quotes a radiative forcing equivalent of 3.8Wm−2. [ΔTx2=3.8 Wm−2 λ].)

Frame, et al. (2005)[22]] and Allen et al. noted that the range of the confidence limits is dependent on the nature of the prior assumptions made.

Annan and Hargreaves (2006)[23] presented an estimate that resulted from combining prior estimates based on analyses of paleoclimate, responses to volcanic eruptions, and the temperature change in response to forcings over the twentieth century. They also introduced a triad notation (L, C, H) to convey the probability distribution function (pdf) of the sensitivity, where the central value C indicates the maximum likelihood estimate in degrees Celsius and the outer values L and H represent the limits of the 95% confidence interval for a pdf, or 95% of the area under the curve for a likelihood function. In this notation their estimate of sensitivity was (1.7, 2.9, 4.9)°C.

Forster and Gregory (2006)[24] presented a new independent estimate based on the slope of a plot of calculated greenhouse gas forcing minus top-of-atmosphere energy imbalance, as measured by satellite borne radiometers, versus global mean surface temperature. In the triad notation of Annan and Hargreaves their estimate of sensitivity was (1.0, 1.6, 4.1)°C.

Royer, et al. (2007)[25] determined climate sensitivity within a major part of the Phanerozoic. The range of values—1.5 °C minimum, 2.8 °C best estimate, and 6.2 °C maximum—is, given various uncertainties, consistent with sensitivities of current climate models and with other determinations.[26]

## References

1. ^ IPCC (2007). "2.3 Climate sensitivity and feedbacks". In Pachauri, R.K and Reisinger, A. (Eds.). Climate Change 2007: Synthesis Report. Contribution of Working Groups I, II and III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Geneva, Switzerland: Intergovernmental Panel on Climate Change. Retrieved 2010-07-03
2. ^ Prentice, I.C. et al (2001). "Technical Summary: F.3 Projections of Future Changes in Temperature". In Houghton J.T. et al (Eds.). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Geneva, Switzerland: Intergovernmental Panel on Climate Change. Retrieved 2010-07-03
3. ^ Knutti, Reto; Hegerl, Gabriele C. (2008-10-26). "The equilibrium sensitivity of the Earth's temperature to radiation changes". Nature Geoscience 1 (11): 735–743. doi:10.1038/ngeo337. Retrieved 2010-07-03.
4. ^ Randall, D.A., et al (2007). "8.6.2 Interpreting the Range of Climate Sensitivity Estimates Among General Circulation Models, In: Climate Models and Their Evaluation.". In Solomon, S., D. et al (Eds.). Climate Change 2007: Synthesis Report. Contribution of Working Groups I, II and III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge. Retrieved 2010-07-03
5. ^ Prentice, I.C. et al (2001). "9.2.1 Climate Forcing and Climate Response, in chapter 9. Projections of Future Climate Change". In Houghton J.T. et al (Eds.). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.. ISBN 9780521807678. Retrieved 2010-07-03, 2011-10-05(url)
6. ^ Solomon, S., D. et al (Eds.) (2007). "Glossary A-D, Climate sensitivity". Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, 2007. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.. Retrieved 2011-10-05
7. ^ "Target CO2". RealClimate. April 2008.
8. ^ a b c Rahmstorf, Stefan (2008). "Anthropogenic Climate Change: Revisiting the Facts". In Zedillo, E. (PDF). Global Warming: Looking Beyond Kyoto. Brookings Institution Press. pp. 34–53.
9. ^ a b Ad Hoc Study Group on Carbon Dioxide and Climate (1979). "Carbon Dioxide and Climate: A Scientific Assessment" (PDF). National Academy of Sciences.
10. ^ V. Ramanathan, M.S. Lian, and R.D. Cess (1979) (PDF). Increased Atmospheric CO2: Zonal and Seasonal Estimates of the Effect on Radiative Energy Balance and Surface Temperature. Journal of Geophysical Research.
11. ^ O. Boucher, et al (2001). "6.3.1 Carbon Dioxide in: Chapter 6 Radiative Forcing of Climate Change". In Houghton J.T. et al (Eds).. Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge. Retrieved 2010-07-03
12. ^ Collins, W. D., et al. (2006). "Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4)". J. Geophys. Res. 111 (D14317): D14317. Bibcode 2006JGRD..11114317C. doi:10.1029/2005JD006713. Retrieved 2010-07-03.
13. ^ John Farley (2008). "The Scientific Case for Modern Anthropogenic Global Warming". Monthly Review.
14. ^ Ganopolski, A., and T. Schneider von Deimling (2008). "Comment on "Aerosol radiative forcing and climate sensitivity deduced from the Last Glacial Maximum to Holocene transition" by Petr Chylek and Ulrike Lohmann". Geophys. Res. Lett. 35 (L23703): L23703. Bibcode 2008GeoRL..3523703G. doi:10.1029/2008GL033888. Retrieved 2010-07-03.
15. ^ Richard A. Kerr (13 August 2004). "Three Degrees of Consensus". Science 305 (5686): 932–4. doi:10.1126/science.305.5686.932. PMID 15310873.
16. ^ Idso, Sherwood (1998). "CO2-induced global warming: A skeptic’s view of potential climate change". Climate Research 10: 69–82. doi:10.3354/cr010069.
17. ^ Gerhard, L.C.; Harrison, W.E.; Hanson, W.E. (2001). "Geological perspectives of global climate change". AAPG Special Publication SG47: 317–336.
18. ^ Andronova, N.; Schlesinger, M. E. (2001). "Objective Estimation of the Probability Distribution for Climate Sensitivity". J. Geophys. Res. 106 (D19): 22605. Bibcode 2001JGR...10622605A. doi:10.1029/2000JD000259.  as PDF data
19. ^ Forest, C.E.; Stone, P.H.; Sokolov, A.P.; Allen, M.R.; Webster, M.D. (2002). "Quantifying uncertainties in climate system properties with the use of recent observations". Science 295 (5552): 113–7. doi:10.1126/science.1064419. PMID 11778044.  as PDF
20. ^ Gregory, J.M.; Stouffer, R.J.; Raper, S.C.B.; Stott, P.A.; Rayner, N.A. (2002). "An observationally based estimate of the climate sensitivity". Journal of Climate 15 (22): 3117. doi:10.1175/1520-0442(2002)015<3117:AOBEOT>2.0.CO;2.  as PDF
21. ^ Shaviv, N.J. (2005). "On climate response to changes in the cosmic ray flux and radiative budget". J. Geophys. Res. 110: A08105. Bibcode 2005JGRA..11008105S. doi:10.1029/2004JA010866.  (preprint)
22. ^ Frame, D.J.; Booth, B.B.B.; Kettleborough, J.A.; Stainforth, D.A.; Gregory, J.M.; Collins, M.; Allen, M.R. (2005). "Constraining climate forecasts: the role of prior assumptions". Geophysical Research Letters 32 (9): L09702. Bibcode 2005GeoRL..3209702F. doi:10.1029/2004GL022241.
23. ^ Annan, J.D.; Hargreaves, J. C. (2006). "Using multiple observationally-based constraints to estimate climate sensitivity". Geophysical Research Letters 33 (6): L06704. Bibcode 2006GeoRL..3306704A. doi:10.1029/2005GL025259.  as PDF
24. ^ Forster, Piers M. de F.; Gregory, Jonathan M. (2006). "The Climate Sensitivity and Its Components Diagnosed from Earth Radiation Budget Data". Journal of Climate 19 (1): 39–52. doi:10.1175/JCLI3611.1.
25. ^ Royer, Dana L.; Berner, Robert A.; Park, Jeffrey (29 March 2007). "Climate sensitivity constrained by CO2 concentrations over the past 420 million years". Nature 446 (7135): 530–2. doi:10.1038/nature05699. PMID 17392784.
26. ^ Sceptics as Jan Veizer have pointed out that while data for the whole Phanerozoic are available Royer et al. left out the time span younger than 420 Ma with an ice age and extremely high carbon dioxide content during the Hirnantian.

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# Rheology of Semi-dilute Solutions of Calf-thymus DNA

Bandyopadhyay, Ranjini and Sood, AK (2002) Rheology of Semi-dilute Solutions of Calf-thymus DNA. In: Pramana-Journal of Physics, 58 (4). pp. 685-394.

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Rheology_of_Semi-dilute_Solutions.pdf

We study the rheology of semi-dilute solutions of the sodium salt of calf-thymus DNA in the linear and nonlinear regimes. The frequency response data can be fitted very well to the hybrid model with two dominant relaxation times $\tau_0$ and $\tau_1$. The ratio $( \tau_0 / \tau_1) \sim 5$ is seen to be fairly constant on changing the temperature from 20 to $30^o\hspace{2mm}C$. The shear rate dependence of viscosity can be fitted to the Carreau model.


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# Are there any antimatter calculations?

1. Mar 12, 2005

### Gary King

Are there any antimatter calculations, that are in any way related to antimatter? (Please post the equation and the use of it Also, I already am aware of the $$E = mc^2$$ one ;)

2. Mar 13, 2005

### Crosson

All of the equations that apply to matter, also apply to antimatter!

3. Mar 13, 2005

Oh, okay .


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