text:
One Way Rotational Clutch - MapleSim Help

Home : Support : Online Help : MapleSim : MapleSim Component Library : 1-D Mechanical : Rotational : Clutches and Brakes : componentLibrary/1Dmechanics/rotational/clutchesBrakes/OneWayClutch

One Way Rotational Clutch

Parallel connection of freewheel and clutch

 Description The One Way Rotational Clutch (or One Way Clutch) component models a one-way clutch, which is a clutch connected in parallel to a free wheel.  This special element is provided because such a parallel connection introduces an ambiguity to the model (the constraint torques are not uniquely defined when both elements are stuck) and this element resolves it by introducing one constraint torque instead of two. See the Clutch model for the basic model description.

Connections

 Name Description Modelica ID ${\mathrm{flange}}_{a}$ Left flange of compliant 1-dim. rotational component flange_a ${\mathrm{flange}}_{b}$ Right flange of compliant 1-dim. rotational component flange_b $\mathrm{heatPort}$ heatPort ${f}_{\mathrm{normalized}}$ Real input; normalized force f_normalized

Parameters

General Parameters

 Name Default Units Description Modelica ID ${\mathrm{\mu }}_{\mathrm{pos}}$ $\left[0.,0.5\right]$ $1$ Table of sliding friction coefficients at given relative velocities mue_pos $\mathrm{peak}$ $1$ $1$ $\mathrm{peak}{\mathrm{\mu }}_{\mathrm{pos}}\left[1,2\right]$ is the static friction coefficient peak ${c}_{\mathrm{geo}}$ $1$ $1$ Geometry constant containing friction distribution assumption cgeo ${f}_{{n}_{\mathrm{max}}}$ $1$ $N$ Maximum normal force fn_max Use Heat Port $\mathrm{false}$ True (checked) means the heat port is enabled useHeatPort

 Name Default Units Description Modelica ID ${\mathrm{\phi }}_{\mathrm{nominal}}$ $1·{10}^{-4}$ $\mathrm{rad}$ Nominal value of ${\mathrm{\phi }}_{\mathrm{rel}}$ (used for scaling) phi_nominal $\mathrm{prefer}$ Prioritize ${\mathrm{\phi }}_{\mathrm{rel}}$ and ${w}_{\mathrm{rel}}$ as states stateSelect ${\mathrm{\omega }}_{\mathrm{small}}$ $1·{10}^{10}$ $\frac{\mathrm{rad}}{s}$ The velocity reinitializes when $\left|\mathrm{\omega }\right|\le {\mathrm{\omega }}_{\mathrm{small}}$ w_small


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# Sagnac Effect and Special Relativity

by KingNothing
Tags: effect, relativity, sagnac, special
 P: 949 http://www.mathpages.com/rr/s2-07/2-07.htm Today I met a man who was claiming that the Sagnac effect, particularly in the case of light moving in a circular pattern with a rotating observing "arm" as in the first figure in the link, is contrary to special relativity. He claimed that since the wave will hit the observing arm at a rate of c+v or c-v, that this contradicts the constancy of c. Now, in my head, I am thinking that this speed of c+v is what we observe from our inertial system which holds the axis of the roation constant, but that an actual observation device placed on the arm would still observe the incoming light as having a speed of 3*10^8 m/s. Is this to be explained by a slight time dilation, making the time difference in the frame of the observer located on the moving arm equal to the time difference observed by us in the reference frame holding the axis constant? $$y = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$
 P: 949 Actually, I have been staying up all night working on this problem. I will post the solution when I am finished.
P: 949
I've made a brief MAPLE worksheet demonstrating just one calculation to show that the Sagnac effect does not imply an inconstancy in the speed of light. Please see the attached PDF.
Attached Files
 Sagnac Effect Explained.pdf (101.3 KB, 25 views)

P: 3,542

## Sagnac Effect and Special Relativity

 Quote by KingNothing Now, in my head, I am thinking that this speed of c+v is what we observe from our inertial system
I think you have it backwards. Inertial frames always observe light traveling at c. In the rotating (non inertial) frame the speed of light can vary. That is what SR says and has been observed so far.
 P: 58 The webpage you quoted contains the time correction in there, but good job on going through the math yourself.
P: 4,491
 Quote by A.T. Inertial frames always observe light traveling at c. In the rotating (non inertial) frame the speed of light can vary. That is what SR says and has been observed so far.
Exactly. Although, the LOCAL velocity of light is allways equal to c, even for rotating and accelerating observers. That is, if you measure the velocity of light traveling NEAR you, you will allways get c. But in the Sagnac effect you don't measure the local velocity of light.

format: plain


text:
This is so cool, the absorb feature has finally landed:

gregoryszorc.com/blog/2018/11/

What it does: looks at your changes in your working directory and correctly absorbs them into the ancestor commits that touched the same files as your changes.

Great way to edit a stack of commits without having to manually sprinkle your changes across each individual commit!

This isn't , but here's a hashtag so that git users may stumble upon this feature.

@JordiGH I bet git will nick this sooner rather than later.

@stsp I hope they do, maybe that way my collaborators' commits will start looking better.

A Mastodon instance for maths people. The kind of people who make $\pi z^2 \times a$ jokes.

Use $ and $ for inline LaTeX, and $ and $ for display mode.


text:
## Git Delete Last Commit

Once in a while late at night when I ran out of coffee, I commit stuff that I shouldn't have. Then I spend the next 10 - 15 minutes googling how to remove the last commit I made. So after third time I wanted to make a record of it so I can refer to it later.

If you have committed junk but not pushed,

git reset --soft HEAD~1

HEAD~1 is a shorthand for the commit before head. Alternatively you can refer to the SHA-1 of the hash you want to reset to. --soft option will delete the commit but it will leave all your changed files "Changes to be committed", as git status would put it.

If you want to get rid of any changes to tracked files in the working tree since the commit before head use --hard instead.

Now if you already pushed and someone pulled which is usually my case, you can't use git reset. You can however do a git revert,

git revert HEAD

This will create a new commit that reverses everything introduced by the accidental commit.


text:
Relative log expression normalization without negative values
1
0
Entering edit mode
6 weeks ago

Hello. I would like to analyze a public RNA-seq data set which the author claims to be normalized with relative log expression normalization. However, in my experience, this generates negative expression values, but all values are positive in his matrix. Am I mistaken or is there a upstream processing not mentioned by the author?

rna-seq • 292 views
2
Entering edit mode
6 weeks ago

RLE does not return negative values, it divides the raw counts by the size or normalization factors.

I expect you are thinking of log2 normalization, which will return negative values for counts below 1. This is why it's usually performed as log2(normalized counts + 1), e.g. the normTransform function in DESeq2.

1
Entering edit mode

I think OP assumes that logcounts were standardizes, e.g. Z-scored so that would be relative expression, but I agree that here (based on how the counts look in the file of this GEO entry) that it is probably RLE-normalized values (that is the DESeq2 normalization method @OP).

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NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Examine solutions to systems of equations in 3 unknowns
Lesson

A system of three linear equations in three unknowns can have no solutions, one solution or infinitely many solutions. In this chapter, we explore the conditions for these three possibilities.

### one equation

A single linear equation in three unknowns represents a plane in a $3$3-dimensional coordinate system. For example, in the equation $3x-y+2z=0$3xy+2z=0, we could choose a fixed value for $z$z, say $z=t$z=t. Then, $y=3x-2t$y=3x2t. But this is the equation of a line in the $x$x-$y$y plane at the level of $t$t. There must be such a line at every possible level of $t$t and we might imagine these infinitely many lines, side by side, forming a plane in $3$3-space.

#### Example 1

There are infinitely many sets of numbers $(x,y,z)$(x,y,z) that satisfy the equation $3x-y+2z=0$3xy+2z=0 but there are also infinitely many sets of numbers that do not satisfy the equation. We can specify the solution set by choosing a parameter $t$t for $z$z and another parameter $s$s for $y$y. Then, we see that $x$x is constrained by the equation to be $x=\frac{1}{3}s-\frac{2}{3}t$x=13s23t.

The solution set can be written in vector form as

The two vectors on the right define a plane and the solution set is the set of linear combinations of the two vectors as $s$s and $t$t range over the real numbers.

### two equations

Two equations represent two planes. The planes might be parallel, in which case they have no points in common and there can be no solution that satisfies both equations. If the planes are not parallel, they intersect along a line and there are infinitely many solutions, all belonging to the line.

#### Example 2

The equations $3x-y+2z=0$3xy+2z=0 and $3x-y+2z=6$3xy+2z=6 represent parallel planes. Any attempt at a simultaneous solution leads to an absurdity like $6=0$6=0 and we must conclude that no solution exists.

On the other hand, a pair of equations like $3x-y+2z=0$3xy+2z=0 and $x+y+z=0$x+y+z=0 can be solved to find a line of solutions. We might observe immediately that $(0,0,0)$(0,0,0) satisfies both equations. That is, both planes pass through the origin.

We could multiply the second equation by $-3$3 and add it to the first equation. This gives $0x-4y-z=0$0x4yz=0.  We now have the equations $x+y+z=0$x+y+z=0 and $4y+z=0$4y+z=0 and these must have the same solution set as the original pair.

As before, we could choose to put $z=t$z=t. Then, from the $4y+z=0$4y+z=0 we find $y=-\frac{t}{4}$y=t4. Finally, from $x+y+z=0$x+y+z=0 we have $x=\frac{t}{4}-t=-\frac{3t}{4}$x=t4t=3t4. All three variables have been expressed in terms of the single parameter $t$t.

This time, the solution set in vector form is

The solution set is the set of scalar multiples of the vector on the right. The solution $(0,0,0)$(0,0,0) corresponds to $t=1$t=1.

### three equations

Three linear equations in three unknowns represent three planes in coordinate space. There are four possibilities.

• All three may be parallel, in which case there is no intersection and no solution. The system of equations would be found to be inconsistent.
• The three planes could intersect pairwise in three different parallel lines, in which case there is no simultaneous solution. Again, the system would be inconsistent.
• All three could intersect in a single line so that there are infinitely many solutions. Solutions would be found as in Example 2.
• The three planes could intersect at a single point.

#### Example 3

Consider the system of three equations

If we subtract the first equation from the second and also subtract twice the first equation from the third, we obtain

Next, subtract $\frac{1}{2}$12 the second equation from the third.

From the third of these equations, we find $z=5$z=5. Then, when this is substituted into the second equation, we get $y=3$y=3. And when these values of $z$z and $y$y are substituted into the first equation, we have $x=2$x=2.

So, the solution set is the single point

There is a more efficient way of setting out these calculations, explained at this link.

### Outcomes

#### M8-8

Form and use systems of simultaneous equations, including three linear equations and three variables, and interpret the solutions in context

#### 91587

Apply systems of simultaneous equations in solving problems


text:
# Thread: Last Problem, I swear

1. ## Last Problem, I swear

Allright, here's the last problem I need help with, thanks to all those that have helped! Solve:

[(3/x^2)+(1/x)]/[2-(4/5x)]

Thanks! You guys rock!

nerd

2. Originally Posted by bandnerd1892
Allright, here's the last problem I need help with, thanks to all those that have helped! Solve:

[(3/x^2)+(1/x)]/[2-(4/5x)]

Thanks! You guys rock!

nerd
$\frac{\frac{3}{x^2}+\frac{1}{x}}{2-\frac{4}{5x}}$

so we want to multiply both the denominator and numerator by the LCD $5x^2$

$\frac{5x^2}{5x^2} \left( \frac{\frac{3}{x^2}+\frac{1}{x}}{2-\frac{4}{5x}} \right)$

after we distribute we get

$\frac{15+5x}{10x^2-4x}=\frac{5(x+3)}{2x(5x-2)}$

3. thank you!

nerd


text:
# Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1

## Theorem

Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Let $\rho$ satisfy definition 2 of the rank axioms:

 $(\text R 1')$ $:$ $\displaystyle \forall X \in \powerset S:$ $\displaystyle 0 \le \map \rho X \le \size X$ $(\text R 2')$ $:$ $\displaystyle \forall X, Y \in \powerset S:$ $\displaystyle X \subseteq Y \implies \map \rho X \le \map \rho Y$ $(\text R 3')$ $:$ $\displaystyle \forall X, Y \in \powerset S:$ $\displaystyle \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y$

Then $\rho$ satisfies definition 1 of the rank axioms:

 $(\text R 1)$ $:$ $\displaystyle \map \rho \O = 0$ $(\text R 2)$ $:$ $\displaystyle \forall X \in \powerset S \land y \in S:$ $\displaystyle \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1$ $(\text R 3)$ $:$ $\displaystyle \forall X \in \powerset S \land y, z \in S:$ $\displaystyle \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set y \cup \set z} = \map \rho X$

## Proof

#### $\rho$ satisfies $(\text R 1)$

We have:

 $\ds 0$ $\le$ $\ds \map \rho \O$ Rank axiom $(\text R 1')$ $\ds$ $\le$ $\ds \card \O$ Rank axiom $(\text R 1')$ $\ds$ $=$ $\ds 0$ Cardinality of Empty Set

Hence:

$\map \rho \O = 0$

$\Box$

#### $\rho$ satisfies $(\text R 2)$

Let $X \subseteq S$.

Let $y \in S$.

We have:

 $\ds \map \rho X$ $\le$ $\ds \map \rho {X \cup y}$ Rank axiom $(\text R 2')$ $\ds$ $\le$ $\ds \map \rho X + \map \rho {\set y} - \map \rho {X \cup \set y}$ Rank axiom $(\text R 3')$ $\ds$ $\le$ $\ds \map \rho X + \map \rho {\set y}$ $\ds$ $\le$ $\ds \map \rho X + \card {\set y}$ Rank axiom $(\text R 1')$ $\ds$ $\le$ $\ds \map \rho X + 1$ Cardinality of Singleton

This proves Rank axiom $(\text R 2)$

$\Box$

#### $\rho$ satisfies $(\text R 3)$

Let $X \subseteq S$.

Let $x, y \in S$.

Let $\map \rho {X \cup \set x} = \map \rho {X \cup \set y} = \map \rho X$.

We have:

 $\ds \map \rho X$ $\le$ $\ds \map \rho {X \cup \set x \cup \set y }$ Rank axiom $(\text R 2')$ $\ds$ $=$ $\ds \map \rho {\paren{X \cup \set x} \cup \paren{X \cup \set y} }$ $\ds$ $\le$ $\ds \map \rho {X \cup \set x} + \map \rho {X \cup \set x} - \map \rho {\paren{X \cup \set x} \cap \paren{X \cup \set y} }$ Rank axiom $(\text R 3')$ $\ds$ $\le$ $\ds \map \rho {X \cup \set x} + \map \rho {X \cup \set x} - \map \rho X$ $\ds$ $=$ $\ds \map \rho X + \map \rho X - \map \rho X$ $\ds$ $=$ $\ds \map \rho X$

Hence:

$\map \rho {X \cup \set x \cup \set y} = \map \rho X$

This proves Rank axiom $(\text R 3)$

$\blacksquare$


text:
# Journal of Quantitative Spectroscopy and Radiative Transfer

Online ISSN: 0022-4073
Publications
Article
A calculation of the A2 sigma --> X2 pi (0, 0) band emission rate factors and line center absorption cross sections of OH applicable to its measurement using solar resonant fluorescence in the terrestrial atmosphere is presented in this paper. The most accurate available line parameters have been used. Special consideration has been given to the solar input flux because of its highly structured Fraunhofer spectrum. The calculation for the OH atmospheric emission rate factor in the solar resonant fluorescent case is described in detail with examples and intermediate results. Results of this calculation of OH emission rate factors for individual rotational lines are on average 30% lower than the values obtained in an earlier work.

Article
This note serves as an introduction to two papers by Klose et al. [Optical tomography using the time-independent equation of radiative transfer, Parts 1 (JQSRT 2002;72:691-713) and 2 (JQSRT 2002;72:715-732)] and provides a brief review of the latest developments in optical tomography of scattering tissue. We discuss advancements made in solving the forward model for light propagation based on the radiative transfer equation, in reconstructing scattering and absorption cross sections of tissue, and in molecular imaging of luminescent sources.

Article
The purpose of this study is to extend the capabilities of multispectral optical imaging techniques for morphological characterization of irregular particles, encountered in several applications. We have utilized the Gaussian random ellipsoid model to quantize the shape elongation and deviation in complex-shaped particles. Compared with the Gaussian random sphere model, it is more realistic and is applicable to wider range of irregular particles with minimum possible of free parameters. By processing the reduced light scattering spectra, the proposed inverse technique can simultaneously recover the size, volume fraction, elongation, and shape deformation of particles that are randomly oriented within a colloidal suspension. The discrete dipole approximation is employed in forward light scattering calculations and input synthetic data generation. To investigate the robustness of our algorithm in coping with real scenarios, we have added different levels of noise to the observed scattering spectra. The results demonstrate the potential of our technique to non-invasively recover the morphology parameters of irregular particles.

Conference Paper
Summary form only given. The manner in which L-K transitions reflect radiation impinging upon a plane layer of given thickness is investigated. It is assumed that the energy distribution of the incident photons is a Planck function. Fluorescence due to K-shell vacancies produced by photoionization is the only effect taken into consideration. Solving the transfer equation allows the reemitted flux and, consequently, the albedo of the layer to be calculated

Article
Absorption lineshapes of OH found in hot post-flame gases were measured using a rapid-tuning single-mode laser. Collisional linewidths due to broadening by H2O and CO2 were determined by fitting the observed lineshapes to standardized Voigt profiles. Lines corresponding to transitions with ground-state angular momentum quantum numbers (J″-values) ranging from 1.5 to 17.5 in the R1-branch of the A2Σ+ ← X2Π (0,0) band near 307 nm were probed. The temperature-dependence of collisional broadening was investigated over the range 1470–2370 K.

Article
Absorption spectra of nitric oxide in the γ(0,0) and γ(1,0) bands have been measured for hard temperature conditions up to in order to validate a model for the simulation of these two bands. The good agreement between experiments and calculations (relative errors of 2–5% for the γ(0,0) band and 10–15% for the γ(1,0) band) consolidates the two important assumptions concerning the intermediate Hund's case between (a) and (b) for the X2Π state of the γ(0,0) and γ(1,0) absorption bands and the use of collisional broadening parameters of γ(0,0) to simulate the γ(1,0) band. Using this simulation, a study of the Beer–Lambert law behavior at high temperature has been carried out. With the instrument resolution used for these experiments, it was shown that a correction of the Beer–Lambert law is necessary. To apply this technique for the measurements of NO concentrations inside the combustion chamber of an optical SI engine, a new formulation of the Beer–Lambert law has been introduced, since the modified form proposed in the literature is no longer applicable in the total column range of interest.

Article
The collision broadening cross sections of the OH (0, 0) (A → X) ultraviolet transition have been measured for 14 broadening gases at 293°K and for water vapor at 378°K. They were obtained by applying corrections for Doppler and instrumental broadening to direct width measurements of OH absorption lines in high resolution spectra. The OH was generated by flash photolysis of H2O2 or H2O in an excess of broadening gas. The assumption that the cross section is independent of the OH rotational quantum number J appears valid for all these gases except water, for which it was necessary to assume that the cross section is a decreasing function of J. The collision widths at 293°K and 580 torr range from a low of 0·04 cm-1 for helium to a high of 0·16 cm-1 for hydrogen. For water vapor at 378°K and 546 torr, the collision width varies from about 0·3 cm-1 for J = ; to about 0·06 cm-1 for J = 7;. The collision widths, when extrapolated to flame temperatures, appear to agree with earlier measurements as well as can be expected. The cross sections obtained are generally larger than those derived from transport property measurements, but all are within a factor of two of the latter values.

Article
Even for the well-studied and ubiquitous species, OH, the current state of theoretical development of broadening theory does not allow extrapolation from low-temperature laboratory measurements to the range of practical combustion devices. We performed a series of experiments at typical combustion conditions to determine the collision broadening of the P1(5) line of the (0,0) band of OH A2Σ+←X2Π transition by Ar in shock-heated H2–O2–Ar mixtures and by air in H2–air flames over a wide range of stoichiometry (φ=0.01–10.0), temperature (T=780–2440 K), and pressure (p=0.7–10.0 atm). The values of the collision width, ΔνC, were acquired by fitting Voigt profiles to the measured spectral line shapes in flames and to the peak absorption coefficients (kν0) in shock tube experiments. Collision broadening parameters (2γAr, 2γN2, and 2γH2O) were then calculated assuming the linear dependence of ΔνC with pressure—the 2γN2 and 2γH2O values were inferred from 2γAir and the equilibrium concentration of N2 and H2O of a given flame. The temperature dependences of 2γi in our temperature range are, respectively, 1.0, 0.75, and 0.87 for Ar, N2, and H2O. The collision broadening cross sections (σ) deduced from 2γi values are expressed with an assumed form, σi(T)=σi,0(T0/T)k, T0=1000 K: for Ar, σAr,0=63.3 (Å2), k=0.50; for N2, σN2,0=68.0 (Å2), k=0.25; for H2O, σH2O,0=188.8 (Å2), k=0.37.

Article
A review of the theory used by Davis et al. for determining the broadening of the NO γ(0,0) spectral lines by N2 has revealed several problems. This note discusses four significant errors and presents methods for their correction.

Article
Quantitative high-resolution spectroscopy was applied to the (0,0) and (1,0) U.V. absorption bands of OH and OD. Known concentrations of OH and OD were prepared in a high-temperature furnace containing a 2:1 mixture of water vapor and oxygen. Band oscillator strengths at 1425 K for the OH (0,0), OH(1,0), OD (0,0) and OD (1,0) bands were 9.6 × 10-4, 2.4 × 10-4, 9.5 × 10-4 and 2.6 × 10-4 with a statistical and systematic uncertainty of about 20 per cent. The use of the OH or OD rotational intensity distribution with vibration-rotation correction factors to determine gas temperature was verified. The collision width of OH or OD lines in the furnace at a total pressure of 580 torr is about 0.14 cm-1 at low J″ and appears to decrease at higher J″.

Article
Individual spectral line parameters including line positions, strengths, and intensities, have been generated for the A2∑-X2Π(0,0) band of OH, applicable to atmospheric and high temperatures. Energy levels and transition frequencies are calculated by numerically diagonalizing the Hamiltonian. Line strengths are calculated using the dipole matrix and eigenvectors derived from energy matrix diagonalization. The line strengths are compared to those calculated from previously published algebraic line strength formulas. Tables of line parameters are presented for 240 and 4600°K.

Article
We present a spectroscopic study of the water vapor continuum absorption in the far-IR region from 10 to 90 cm−1 (0.3–2.7 THz). The experimental technique combines a temperature-stabilized multipass absorption cell, a polarizing (Martin–Puplett) interferometric spectrometer, and a liquid-He-cooled bolometer detector. The contributions to the absorbance resulting from the structureless H2O–H2O and H2O–N2 continua have been measured in the temperature range from 293 to 333 K with spectral resolution of 0.04–0.12 cm−1. The resonant water vapor spectrum was modeled using the HITRAN04 database and a Van Vleck–Weisskopf lineshape function with a 100 cm−1 far-wing cut-off. Within experimental uncertainty, both the H2O–H2O and H2O–N2 continua demonstrate nearly quadratic dependencies of absorbance on frequency with, however, some deviation near the 2.5 THz window. The absorption coefficients of 3.83 and 0.185 (dB/km)/(kPa THz)2 were measured for self- and foreign-gas continuum, respectively. The corresponding temperature exponents were found to be 8.8 and 5.7. The theoretically predicted foreign continuum is presented and a reasonable agreement with experiment is obtained.

Article
Absolute intensities, self-broadening coefficients, and foreign-gas broadening by Ar and N2 were measured at temperatures of 197, 233 and 294 K for the 3001II←0000 band of CO2 at 6348 cm-1. Also, the intensity parameters and total band intensity were calculated. We obtained for the vibration-rotation interaction factor the value F(m) = 1 + (0.26 ± 0.06) × 10-2m + (0.92 ±0.32 × 10-4 m2; for the purely vibrational transition moment, we found |R00003001II|k{cy rillic}(0.4351 ± 0.0014)()10b3 debye; and, for the total band intensity at STP conditions, Sband(3001II←0000)STP = 1255 ± 9 cm-1 km-1 atm-1. Self-broadening coefficients at 197 and 294 K were also measured, as well as broadening by Ar and N2. Foreign-gas-broadening efficiencies (Ar and N2) were determined. Finally, a comparison is made with measurements by other authors and with theoretically calculated values.

Article
Measurements made at temperatures of 197, 233, and 294°K of the absolute intensities and self-broadening coefficients for the vibration-rotation lines of the 201III←000 band of the 12C16O2 molecule, are reported. From these measurements, values have been derived for the vibration-rotation interaction factor (FVR), the purely vibrational transition moment (|R(O)|), and the intensity (SBand). The results are: EVR(m) = 1+(2.2±0.7)×10−3m+(5.6±1.6)×10×5m2, |R(0)| = (2.064±0.017)×10−3 debye, SBand = 21,329±69 cm−1km−1atm−1STP. The results for the self-broadening coefficients are presented in the text.

Article
The NO2 absorption cross-section has been measured from 42 000 to 10 000 cm−1 (238–1000 nm) with a Fourier transform spectrometer (at the resolution of 2 cm−1, 0.01 nm at 240 nm to 0.2 nm at 1000 nm) and a 5 m temperature controlled multiple reflection cell. The uncertainty on the cross-section is estimated to be less than 3% below 40 000 cm−1 (λ > 250 nm) at 294 K, 3% below 30 000 cm−1 (λ > 333 nm) at 220 K, but reaches 10% for higher wavenumbers. Temperature and pressure effects have been observed. Comparison with data from the literature generally shows a good agreement for wavenumbers between 37 500 and 20 000 cm−1 (267–500 nm). Outside these limits, the difference can reach several percent.

Article
Using a complex version of Robert–Bonamy theory (CRBF), the role of the intermolecular potential in the pressure broadening of water perturbed by nitrogen and oxygen is studied. Investigation focuses on questions surrounding the convergence of calculated line widths, i.e., (i) why certain spectral lines are more sensitive than others to short-range interactions, and (ii) whether converged calculations containing short-range interactions represent an improvement over other treatments. Comparison with a large number of experimentally determined halfwidths and line shifts in the (301)←(000) and (221)←(000) bands is provided. It is found that the atom–atom component of the intermolecular potential plays an important role in determining the halfwidth and line shift. To obtain good agreement with measurement, the atom–atom potential needs to be expanded to at least eighth order for all water vapor transitions broadened by oxygen and many broadened by nitrogen.

Article
The energy rediated from an isothermal hydrogen plasma has been calculated considering both line and continuum emission processes. The principal line broadening mechanism for the plasma conditions considered herein was electrons interacting with the radiating system. Detailed line profiles were used for the lower members of the Lyman and Balmer series. The higher members were treated as having dispersion profiles. The continuum radiation was calculated using the well known expressions for the continuum absorption coefficients. Results were obtained for temperatures between 104 and 4 x 104°K, pressures between 10-1 and 101 atm, and plasma thicknesses between 10-1 and 101 cm.

Article
This review summarizes present knowledge of the atomic and molecular mechanisms that produce or remove rotationally, vibrationally, and electronically excited species in heated or excited gases. A logical classification scheme for these mechanisms is introduced, and the extent of present information on each class is indicated. The validity and utility of several general rules and regularities are also examined. An appendix includes a table of rate coefficients for vibrational and electronic excitation and deexcitation reactions, taken directly from or derived from data in the literature. This table provides comprehensive coverage of some classes of reactions and representative coverage of the others.

Article
An extensive tabulation of absorption coefficients of heated air including both molecular and atomic contributions has been carried out with the aid of several large digital computer programs. Tables are presented for temperatures between 1000 and 24 000°K and for eight densities between ten times normal atmospheric to 10-6 times normal. The photon energy range is 0·6–10·7 eV. Absorption coefficients have been averaged over an energy interval of 0·1 eV and are listed at each 0·1 eV step between the range limits.

Article
CO2 is the major constituent of the atmosphere of Venus. Absorption lines due to its 12C16O18O isotopologue have been observed for the first time in Venus spectra in the 2930–3015 cm−1 spectral region, where the HITRAN database does not contain any line from this isotopologue. The measurements were performed by the SOIR instrument, which is part of the SPICAV/SOIR instrument on board the Venus Express mission of ESA. SOIR measured the atmospheric transmission of the upper atmosphere of Venus (z>70 km) by performing a solar occultation experiment using the atmosphere as a gigantic absorption cell. The identification of this newly observed band was first made recently from Mars atmosphere observations by US colleagues. We have made independent theoretical calculations of the positions of the lines of this new 01111–00001 absorption band, which coincide perfectly with the positions of the observed lines. Assuming an oxygen isotopic ratio similar to the one measured previously in the lower atmosphere of Venus, the line strengths of each observed line are deduced and listed.

Article
Pressure-induced foreign-broadening lineshape parameters of the carbon dioxide rovibrational transitions belonging to the (30012)←(00001) overtone band near the 1.573 μm wavelength region are measured by using a tunable diode laser photoacoustic spectrometer. The spectroscopic analysis has concerned the first 11 lines of the R branch. For these lines, the air- and Ar-broadening coefficients are measured at room temperature (∼298 K). The measured broadening coefficients of all the transitions of 12C16O2 are compared with those given in the HITRAN04 database and former measurements with a different spectroscopic method. Agreements and discrepancies are underlined and briefly discussed. The recorded lineshapes are fitted with standard Voigt line profiles in order to determine the collisional broadening coefficient of carbon dioxide transitions.

Article
The study of the isotropiv Raman Q-branch of a hot band Π ← Π allows one to establish a direct connection between the vibration-rotation angular momentum coupling and the resulting spectra. Due to the l-doubling, the Q-branch is split into two subbranches characterized by either even or odd rotational quantum number j. The vibrational bending reduces the rotational transfer rates inside each of these subbranches by a factor of about two and induces an inter-subbranch coupling. The expected propensity rule towards conservation of the parity index for high rotational levels is well observed. Calculated spectra are in excellent agreement with CARS experiments for the v1 + v2 ← v2 band of CO2.

Article
Integrated intensities of acetylene bands at 3·04, 7·53 and 13·7 μm have been measured at 300°K using the Wilson-Wells-Penner-Weber technique and a spectral resolution of 0·6 cm−1. Our best estimates of the intensities are 294 ± 6 cm−2atm−1 for the 3·04 μ bands, 87 ± 2 cm−2atm−1forthe 7·53 μband and 729 ± 28 cm−2atm−1 for the 13·7 μ band at 300°K.

Article
The dependence of pressure broadening upon hyperfine component in the P(10) and P(70) lines of the (17,1) band of the I2 X1Σ(0g+)→B3Π(0u+) has been studied using laser saturation spectroscopy. By limiting absorption to the zero velocity group, Doppler broadening is removed, lineshapes with widths (FWHM) <9 MHz are detectable, and collision-induced broadening is measured at pressures of 0.2–1.2 Torr. The rates for broadening by argon are 8.3±0.3 and 10.7±0.4 MHz/Torr for the P(70) and P(10) lines, respectively. No significant variation in broadening rates is observed for the 15 hyperfine components of these even rotational lines. The effects of velocity cross-relaxation introduce a broad baseline into the spectra, which is strongly dependent on rotational state, pressure, and laser modulation frequency. The observed broadening rates correlate well with prior measurements and the polarizability of the collision partner.

Article
We have measured speeds of planar shocks in gold target foils impacted by ablatively driven gold flyer foils. The drive for the flyer foils was produced in a high-Z hohlraum by the Nova laser. Dynamic pressures inferred from the measurements were 0.75 ± 0.2 Gbar, higher than previous dynamic shocks produced in the laboratory. We suggest a technique which may allow the laboratory investigation of equations of state in the gigabar regime.

Article
It is well established that water plays a fundamental role in various atmospheric phenomena and that the accuracy of its collisional broadening parameters has a crucial influence on reduction of remote sensing data. Nevertheless, in this field the experimental data are still scarce and consequently the estimates reported in spectroscopic databases are not always reliable and/or accurate. In the view of filling this gap, the self-, N2- and O2-broadening parameters of the J=11,1←00,0 rotational transition of water (1.113 THz) have been determined at room temperature. The experimental investigation has also been supported by theoretical calculations.

Article
Tunable DFB InGaAsP diode lasers were utilized to determine water line strengths and self-broadening parameters of selected transitions between 7600 and 7640 cm−1 at temperatures up to 1100 K. The measured line strengths are compared with HITRAN96 and HITEMP96 databases. Measured line strengths for six lines or grouping of lines between 300 and 1100 K agree to within 20 to 40% with HITEMP96 databases. Self-broadening coefficients and their temperature dependence were measured on four lines and compared with theoretical predictions.

Article
Spectroscopic data are noticeably enriched in six spectral regions of the main isotopologue 12C2H2 of the acetylene molecule, namely, in the regions around 3, 2.2, 1.9, 1.7, 1.5, and 1.4 μm. Among these regions, only those at 3 and 1.5 μm were already presented partly in the databases. The results of line intensity measurements, performed for the first time in the 1.4 μm region, are given. Data available in the literature, or obtained in the present work, are compiled to set up line lists usable for applications in the quoted spectral regions. On the whole, 5748 new lines pertaining to 65 bands can be added to the databases.

Article
The self-broadened linewidths of pure rotational lines of water vapor have been investigated in the temperature range 263– over a pressure range from 30– using an optically pumped THz photomixer spectrometer having a relative precision of . The lineshapes of several lines between 12 and were well fit by Lorentzian profiles over the spectral interval spanning ±5 FWHM. The Lorentzian FWHM linewidths varied between 6.6×10−6 and (0.67 and ) and were found to depend on the and Kc quantum numbers as well as temperature. The observed pressure-induced shifts, δν(P), ranged from 1.3×10−7– (0.013–. The measured linewidths also exhibited a (T0/T)n power law dependence on temperature. The best fit exponent n varied from 0.56 to 0.81, dependent on rotational line. A decrease in δν(T) with increasing temperature is also observed, varying between 2×10−8 and (2×10−3 and ) over the 300– temperature range.

Article
The self-broadening coefficients of acetylene at room temperature have been measured for 10 lines in the P branch of the bands of 12C2H2 and 13C12CH2 near 1.533 μm, using a high resolution tunable diode laser spectrometer developed for the Martian space mission PHOBOS-Grunt. The collisional widths are obtained by fitting each recorded line with the Voigt profile as well as the Rautian profile accounting for the collisional Dicke narrowing effect. The standard Voigt model provides slightly smaller broadening coefficients than the Rautian model. Our data are thoroughly compared to the main atmospheric molecule database HITRAN and previous values in various bands of acetylene. Moreover, it is worth noting that a large number of new transitions not listed in the latest HITRAN08 were measured and identified for the first time.

Article
This paper presents a study of absorption in N2-broadened P and R manifolds of the 2ν3 band of CH4 near 6000 cm−1 using high resolution laboratory and atmospheric spectra. This region is of prime importance for the retrieval of methane abundances in the Earth's atmosphere using ground-based or space-borne spectrometers. Recent laboratory investigations have been devoted to the methane spectroscopic parameters in this band, motivated by their previous poor knowledge and their increasing use by remote sensing experiments. In the absence of a better model, previous studies have used Voigt line shapes and thus purposely neglected line mixing (LM). In this paper, we first present direct comparisons between measured laboratory spectra and the results of a model which accounts for LM without adjusting any of the spectroscopic parameters. A good agreement is obtained and the results show that LM does have a significant influence on the shapes of P and R manifolds. Hence, most previously observed discrepancies were not due to improper broadening and shifting coefficients but to the neglect of this effect. This also confirms that widths and shifts derived in recent 2ν3 band studies neglecting LM are “effective” and lack physical meaning, as suggested in a previous work [17] (Frankenberg et al., 2008). In a second step, the conclusions from the laboratory data are tested using ground-based atmospheric solar absorption spectra. The fit residuals obtained confirm the quality of the proposed model and evidence the impact of line mixing on CH4 atmospheric spectra. The present results also confirm that laboratory and atmospheric spectra can alternatively be accurately modeled neglecting LM and using ad hoc broadening and shifting parameters. Conclusions of this exercise can be drawn from two perspectives. From the point of view of spectroscopy and understanding of processes, accurate line parameters will not be deduced from fits of laboratory measurements unless line-mixing effects are included in the spectral-shape model. In the meantime, and from the point of view of atmospheric retrievals, neglecting LM with suitable effective line parameters is convenient and accurate (within current retrieval uncertainties). Note that this is only true if this approach is not used for total pressures significantly above 1 atm (e.g. Jupiter).

Article
The important equilibrium emission processes in a hydrogen plasma have been investigated in the temperature range between 300°K and 10,000°K for pressures up to several hundred atmospheres. Representative emissivity calculations have been carried out for a transparent gas. For a nontransparent gas, the particular case of a total pressure of 100 atmos and a mean beam length of 30 cm has been considered.Important emissivity contributions are made by the pressure-induced fundamental vibration-rotation band and rotational lines of H2 at the lower temperatures, i.e., below approximately 4500°K. Above this temperature, the bound-free and free-free transitions of the H- ion and the continuum and line radiation of the H atom are the most important contributors to the emissivity. The following emission processes also have been investigated: the bound-free transitions of the H+2 ion, free-free transitions of colliding H atoms and H+ ions, electronic transitions of the H2 molecule, quadrupole vibration-rotation transitions of the H2 molecule, and vibration-rotation transitions of the HD molecule. In addition, the effects of the lowering of the ionization potentials by the fields of plasma ions and of the very broad wings of the Lyman α-line have been considered.

Article
We have computed the frequency-dependent absorption coefficient in the spectral range 6000–68,000 cm−1 of equilibrium mixtures of air and carbon for temperatures in the range 3000–10,000°K. These spectral absorption coefficients are used in the computation of a number of “opacity” coefficients over the spectral range considered. In the temperature range 3000–7500°K the main contribution to the absorption comes from molecular electronic transitions, and for equilibrium mixtures contining more than about 25 per cent by weight of carbon the Planck mean opacity is at least 1000 times that of air at the same temperature and density. The reason for this is that the molecules C2, CN,and C3, which are present in relatively large concentration, are much more efficient absorbers in the Planck mean sense than any of the air molecules. The computation takes into account 29 molecular band systems treated by the “Just-Overlapping-Rotational-Line Model” of Patch, Shackleford and Penner.The continuous absorption processes of photo-detachment from O− and C− and free-free absorption due to electrons in the fields of neutral and ionized species are also considered, and they become important at temperatures near 10,000°K. In a non-equilibrium situation in which the carbon is present as soot particles of radius 50–1000 Å, the Planck mean absorption coefficient at 3000–5000°K is another factor of 100 larger than for the case when the same amount of carbon exists as chemical compounds (C2, CN, C3) in the gas phase.

Article
The CH4 overtone bands from 4410 to 9990 Å, long known in the spectra of the major planets, were studied at room temperature with a long-path, high-pressure White cell. Band intensities and profiles were measured, and are more complete than other recent laboratory measurements of these bands. Vibrational assignments of these bands are suggested, which are compatible with the assignments of the overtone bands longward of 1μ; these assignments imply that all the bands studied are combinations increasingly dominated at shorter wavelengths by v1, the symmetric stretch fundamental. In this self-consistent analysis, the weak band at 6825 Å is assigned as 2v1+3v3, providing further laboratory evidence that it is probably not the 5v3 pure overtone that planetary astronomers had hoped it to be.

Article
The technique of nonlinear least squares spectral curve fitting has been used to derive the stratospheric vertical temperature profile from balloon-borne measurements of the 10.4 micron band of CO2. The spectral data were obtained at sunset with the approximately 0.02 per cm resolution University of Denver interferometer system from a float altitude of 33.5 km near Alamogordo, New Mexico, on 23 March 1981. The r.m.s. deviation between the retrieved temperature profile and correlative radiosonde measurements is 2.2 K.

Article
The collision-induced absorption (CIA) spectra of H2–H2 and H2–He are known to play an important role for modelling of low-metallicity cool and dense stellar atmospheres. In this paper we present collision-induced absorption spectra of H2–H2 complexes in the rototranslational (Δv=0), the fundamental (Δv=1), the first (Δv=2) and the second (Δv=3) overtone bands in the temperature range from 1000 to 7000 K, and in the frequency region from 0 to 20 000 cm−1. The translational spectral density functions are computed quantum mechanically, based on: (1) the newly developed ab initio collision-induced H2–H2 dipole functions of Zheng (Computational study of collision induced dipole moments and absorption spectra of H2–H2. Ph.D. thesis, Michigan Technological University, 1997), which account for the short-range H2–H2 intermolecular distances (as small as 2.5 a.u.) and for larger H2 internuclear distances (as large as 2.15 a.u.); (2) semiempirical isotropic H2–H2 potential (Ross et al, J Chem Phys 1983;79(3):1487) suitable for high temperatures. We include the collision-induced absorption coefficient of the vibrational transitions as v1,v2,v′1,v′2≤3 which we computed rigorously. We also give our estimate for the collision-induced absorption coefficients of single vibrational transitions such as in the first and second overtone bands. The dependence of CIA spectra on rotational states of H2 molecules is accounted for in our computations. We have previously (Borysow et al, Astronom Astrophys. 1997;324:185–95) studied the effect of CIA for stars of a wide range of fundamental stellar parameters (effective temperature, gravity, and chemical composition), and determined for which combinations of these parameters it is necessary to include CIA in the model and spectrum computation. These calculations showed that CIA from H2–H2 plays an important, and often even a dominating role for stellar atmospheres of a wide range of stars. The approximate character of the estimates of the H2–H2 absorption coefficient we used in our previous work combined with the large effect CIA had on the stellar atmospheres, were the main inspirations to initiate the more accurate computations of the absorption coefficient we present here. The absorption coefficient we compute in the present analysis is in qualitative agreement with our preliminary estimates (Borysow et al, Astronom Astrophys 1997;324:185–95), but in some spectral regions of high importance for the stellar structure, our computed absorption coefficient is up to a factor of 3 larger than our preliminary estimates. We therefore fully confirm our previous suspicion that H2–H2 CIA will have a pronounced effect on the atmosphere for a wide range of stars. In this paper we therefore quantify the effect the new data have on a typical cool dense stellar atmosphere, and compare our new results with our previous estimates.

Article
We present a high-temperature version, CDSD-1000, of the carbon dioxide spectroscopic databank. The databank contains the line parameters (positions, intensities, air- and self-broadened half-widths and coefficients of temperature dependence of air-broadened half-widths) of the four most abundant isotopic species of the carbon dioxide molecule. The reference temperature is and the intensity cutoff is . More than 3 million lines covering the 260–8310, 418–2454, 394–4662, and 429– spectral ranges for , , , and , respectively, are included in CDSD-1000. The databank has been generated within the framework of the method of effective operators and based on the global fittings of spectroscopic parameters (parameters of the effective Hamiltonians and effective dipole moment operators) to observed data collected from the literature. Line-by-line simulations of several low- and medium-resolution high-temperature (T=800–) spectra have been performed in order to validate the databank. Comparisons of CDSD-1000 with other high-temperature databanks HITEMP, HITELOR, and EM2C are also given. CDSD-1000 is able to reproduce observed spectra in a more satisfactory way than the high-resolution databank HITEMP for temperatures higher than . The databank is useful for studying high-temperature radiative properties of CO2. CDSD-1000 is freely accessible via the Internet.

Article
Absorption coefficients and cross-sections have been determined for the dissociation continuum 3Σ-u←3Σ-g for molecular oxygen as well as for some of the diffuse bands between 1100–1350Å. The regions of minimum absorption or oxygen “windows” for which k <1 cm-1 were remeasured. Both a discrete emission-type light source and a continuous source were used in this investigation. Repeated measurements in the region of continuous absorption (1380–1480 Å) show a k value of 402±20 cm-1 for the peak of the Schumann-Runge continuum.

Article
Molecular band models are used to derive absorption and pressure coefficients for the methane absorption spectrum from 4500 to 10500 Å at intervals of 10 Å. These coefficients provide a necessary basis for the interpretation of the large methane absorptions in the atmospheres of the major planets. Although only coefficients for the “random” or “Goody” band model are derived, the equivalence between these and the “regular” of “Elasser” coefficients is demonstrated. The effects of pressure on the absorption are surprisingly small, leading to large values of the pressure coefficient quite unlike any previous application of the band-model theory. They indicate a pseudo-continuum character of the methane spectrum throughout the visible and near infrared. A spectrum synthesis calculation using the derived coefficients shows the close fit to experimental data that can be realized.

Article
Photoabsorption and fluorescence cross sections of benzene, (o-, m-, p-) xylenes, naphthalene, 1-methylnaphthalene, and 2-ethylnaphthalene in the gas phase have been measured at 106–295 nm using synchrotron radiation as a light source. Fluorescences are observed from the photoexcitation of benzene and xylenes at 230–280 nm and from naphthalene and its derivatives at 190–295 nm. The absolute fluorescence cross section is determined by calibration with respect to the emission intensity of the NO(A-X) system, for which the fluorescence quantum yield is equal to 1. To cross-check the current calibration method, the quantum yield of the SO2(C̃-X̃) system at 220–230 nm was measured since it is about equal to 1. The current quantum-yield data are compared with previously published values measured by different methods.

Article
Measurements of the water absorption coefficient at 193 nm from 300 to 1073 K are reported. The measurements were made using broadband VUV radiation and a monochromator-based detection system. The water vapor was generated by a saturator and metered into a flowing, 99 cm absorption cell via a water vapor mass flow meter. The 193 nm absorption coefficient measurements are compared to room temperature and high temperature shock tube measurements with good agreement. The absorption can be parameterized by a nu3 vibrational mode reaction coordinate and the thermal population of the nu3 mode.

Article
The experimental status of Mössbauer linewidths are reviewed and reported for the three narrowest gamma resonances: 73Ge, 67Zn, and 181Ta. The crucial role played by the solid-state host in achieving narrow emission or absorption linewidths, as well as high recoil-free fraction are emphasized. Factors contributing to line-broadening in experiments with 181Ta and 67Zn gamma resonances are highlighted. Consequences of these observations on a choice of a suitable lasing medium for Mössbauer gamma-ray lasers is commented upon.

Article
We calculate the absorption efficiency of the composite grains, made up of host silicate spheroids and inclusions of ices/graphites/or voids, in the spectral region $7.0-14.0\mu$m The absorption efficiencies of the composite spheroidal grains for three axial ratios are computed using the discrete dipole approximation (DDA) as well as using the effective medium approximation & T-Matrix (EMT-Tmatrix) ap proach. We study the absorption as a function of the volume fraction of the inclusions and porosity. In particular, we study the variation in the $10.0\mu$m feature with the volume fraction of the inclusions and porosity. We then calculate the infrared fluxes for these composite grains and compare the model curves with the average observed IRAS-LRS curve, obtained for several circumstellar dust shells around stars. These results on the composite grains show that the wavelength of the peak absorption shifts and the width of the $10.0\mu$m feature varies with the variation in the volume fraction of the inclusions. The model curves for composite grains with axial ratios not very large (AR$\sim$1.3) and volume fractions of inclusions with f=0.20, and dust temperature of about 250-300$^{\circ}$K, fit the observed emission curves reasonably well.

Article
Absorption cross-sections, kv (cm-1 atm-1), have been measured in the 9.2 and 11.8 μm bands of CFC-11 (CCl3F) using a high-resolution Fourier transform spectrometer. Temperature and total (N2-broadening) pressure have been varied to obtain results at conditions representative of the atmosphere. The measured absolute intensities (in units of 10-17 cm · molecule-1) of the 9.2 and 11.8 μm bands are 2.591 ± 0.013 and 6.974 ± 0.038, respectively.

Article
Spectral absorption coefficients (or absorption cross-sections) kv (cm-1 atm-1) of CFC-11 and CFC-12 have been measured in the spectral region (8–12 μm) known as the atmospheric window. Data obtained with a grating spectrometer, which has wide spectral coverage and adequate spectral resolution (0.4 cm-1), are compared with the so-called NCAR cross-sections that have recently been introduced into the HITRAN Tape. We have also performed measurements, at various temperature-pressure combinations which are chosen to represent tangent heights (as in solar-occultation experiments) or layers in the atmosphere, of the kv of CFC-12 in the 921 and 923 cm-1Q-branches and at 924.1, 927.6, 931.6 935.0 and 1106 cm-1 with the Doppler-limited spectral resolution (∼10-4cm-1) of a tunable-diode laser spectrometer. The latter are especially well suited for atmospheric remote sensing, while the grating spectra are useful in global warming studies. The effects of temperature and pressure on the spectral absorption coefficients are discussed.

Article
Radiative forcing values have been calculated for 11 halogenated compounds which are in current use or which have been suggested as possible replacements for the chlorofluorocarbons. Absorption cross-sections measured over a range of atmospheric temperature and pressure conditions as part of a multi-laboratory programme have been used together with a narrow band radiative transfer model. We provide a “best estimate” radiative forcing taking into account the likely vertical profile of the gas in each case. The Global Warming Potential over a variety of time horizons has also been calculated where the lifetime is available. We present the first such information for 1,2-dichloroethane. For chloroform our radiative forcing is 5 times higher than the value used in previous assessments, possibly because these ignored the effect of absorption outside the 800–1200 cm−1 “window”. For several of the other compounds considered here, our forcing is between 10 and 30% lower than previous assessments. The perfluorocarbons have been found to have large global warming potentials, many times that of CFC-11, due to both strong absorption and long lifetimes. The importance of absorption features at wavenumbers below 800 cm−1 and the effect of temperature variations in absorption cross-section on the radiative forcing are also investigated.

Article
Employing the most recent laboratory absorption-coefficient data1 on the thermal infrared (i.r.) bands of CFC-11 (CFCl3) and CFC-12 (CF2Cl2), which were measured at several temperatures relevant to the troposphere and the stratosphere, the greenhouse effect produced by these two CFCs has been reexamined. The effect upon the atmospheric radiative fluxes due to the temperature dependence of the absorption coefficient, especially in the many hot bands present in this spectral region, has been studied. The validity of the often used optically-thin approximation has been scrutinized in view of the observed enhancement in the absorption by the CFCs at low temperatures. The influence of absorption by water vapor on the radiative transfer through columns of CFC-11 and CFC-12 has also been considered. It has been shown that, even though each of these individual refinements may modify the previously estimated effect due to the CFCs by a small amount only, the collective effect may be a non-negligible 35% change in the surface-troposphere heating for every ppbv of CFC-11 and CFC-12 each introduced into a model atmosphere.

Article
Shock-tube experiments to measure the gas phase FeO fundamental vibration-rotation radiative band intensity at 11.5 μ have been performed. In measurements behind incident shocks, equilibrium amounts of FeO formed quickly, and CO2 formation occured more quickly in the presence of Fe and FeO than can be understood on the basis of previous studies of CO + O2 induction times. Measurements with a calibrated optical system at 10.4–16.3 μ wavelength yielded an integrated band intensity for FeO of 450 ama−1 cm−2 ± 32% (2σ limits). A bound for the rate of the reaction Fe + O2→FeO + Ok ⩾ 5 x 10−12 cm3 molecule−1 sec−1 for T = 2400°K is given and a possible mechanism relating FeO to the rapid CO2 formation is discussed.

Article
The absorption spectrum of N2O, at room temperature, was recorded in the 5400–11 000 cm–1 region at resolutions ranging from 0.008 cm–1 near 5400 to 0.023 cm–1 near 11 000 cm–1 using a Bruker IFS120HR Fourier transform spectrometer. Sample pressure/absorption path length products ranging from 200 to 4700 mbar×m were used. More than 6000 absolute line intensities have been measured in 64 different bands of 14N216O. Using wavefunctions previously determined from a global fit of an effective Hamiltonian to more than 18 000 line positions [Tashkun SA, Perevalov VI, and Teffo JL, to be published], the experimental intensities measured in this work and by Toth [J Mol Spectrosc 1999;197:158–87] were fit using 62 parameters of a corresponding effective dipole moment, with residuals very close to the experimental uncertainty.

Article
The temperature dependence of the infrared absorption cross-sections of CFC-113 (1,1,2-trichlorotrifluoroethane) in a pure vapor phase has been recorded in the 600–1250 cm−1 spectral region using Fourier transform spectroscopy. Spectra at 0.05 cm−1 resolution have been used to derive the integrated band strengths of the five main absorption bands over a range of temperatures from 223 to 283 K. Our results show good agreement with previously published data. The new cross-sections will allow more accurate retrieval of atmospheric CFC-113 concentrations using infrared spectroscopic techniques.Research highlights►Mid-infrared absorption cross-sections of CFC-113 are recorded over a range of relevant atmospheric temperatures. ► Integrated band strengths are compared with values in literature. ►Temperature-induced band centroid shifts are evaluated. ► Conformational interconversions do not affect the spectrum shapes.

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# Permutations and Combinations Compared

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# Permutations and Combinations Compared - Answer Key

## Using Permutations and Combinations to Compare Sudoku and Big Two

### Topic

Using Permutations and Combinations to Compare Sudoku and Big Two

### Vocabulary

• Permutations
• Combinations
• Permutations with Repetition

### Student Exploration

#### Have you ever wondered how many different games are possible in a Sudoku puzzle? How many different ways can you be dealt a hand of cards in Big Two? And which game is a Permutation and which is a Combination?

A Sudoku game.

Sudoku is a logic-based, number-placement puzzle. The board is a 9-by-9 square and is partially completed. The goal of the game is to place the numbers 1 to 9 in each row, each column, and each of the 3-by-3 sub-boxes. For more information on the game visit, http://en.wikipedia.org/wiki/Sudoku.

How many different possible ways are there to play in the first row of numbers?
You can use the counting principle to determine the number of possible ways to make a play in the first row: $9!= 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=362880$. You could also use Permutation notation as follows: $_9P_9 = \frac{9!}{(9-9!)}=\frac{9!}{0!}=9!= 362880$.
How many different possible ways are there to play in first and second row of numbers?
There are still 362880 possibilities for the first row but in the second row you can’t repeat any of the same numbers in the same columns as the numbers in the first row. Therefore there are following possibilities for the second row: $8!= 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1=40320$. Then, to find the different possibilities for the first and second row you multiply the possibilities for each row, 362880 by 40320, which equals 14,631,321,600.
How many different possible ways are there to play the entire game of Sudoku (all nine rows)?
To get the total number of possible ways to play in the entire game of Sudoku (all nine rows) multiplies the possibilities in each row. $1^{st}$ row: 362880, $2^{nd}$ row: 40320, $3^{rd}$ row: 5040, $4^{th}$ row: 720, $5^{th}$ row: 120, $6^{th}$ row: 24, $7^{th}$ row: 6, $8^{th}$ row: 2, and $9^{th}$ row: 1. There are a grand total of 2,548,518,711,000,000,000 ways to play the entire game of Sudoku. There are a lot of ways!

Playing Cards.

The game Big Two, also called Deuces, is a game where all the cards in a deck of 52 cards are dealt to four players. As a result each player gets 13 cards. Then the player with the three of diamonds plays first and rotates to the rest of the players in a counter clockwise direction. Players must play a card or group of cards that are ranked higher than the previous card/s played. The ranking of the cards is specific with the twos holding the highest value and the suits of cards having different values, see the links below for more information. The goal is to be the first player to get rid of all your cards, causing the game to end, and the rest of the players want to have as few cards as possible at the end of the game. For more information on the game visit, http://en.wikipedia.org/wiki/Big_Two or http://www.pagat.com/climbing/bigtwo.html.

How many different ways can you be dealt a hand of cards in the game of Big Two?
The number of ways you can be dealt a hand of cards in Big Two is calculated with $_{52}C_{13} = \frac{52!}{(52-13)! 13!}=\frac{52!}{39! 13!}=\frac{52 \cdot 51 \cdot 50 \cdot \ldots \cdot 41 \cdot 40}{13!}=524,323,122,600$

The game of Big Two starts with the three of diamonds being played.

### Extension Investigation

1. Why dos the Sudoku problem represent a Permutation problem, rather than a Combination? Explain. The Sudoku problem is a Permutation problem because the order that you put the numbers in one row makes a difference. If you start the row with 1, 5, 9, ... that is a different play than 9, 1, 5, ... The objective of the game is to find the correct placement of the numbers, if you could put them in any order it would be a really easy game.
2. Why does the number of ways the cards can be dealt in the game of Big Two represent a Combination problem, rather than a Permutation? Explain. The ways of being dealt the cars in Big Two is a Combination problem because the order that you are dealt the cards doesn’t make a difference. Being dealt a hand with a two of hearts and then a Queen of spades is the same hand as a Queen of spades and a two of hearts.
3. Derive the formula for Combinations. Show your work.
4. What would you do differently to derive the formula for Permutations? Explain.

### Connections to other CK-12 Subject Areas

• Probability and Permutations
• Permutation Problems
• Probability and Combinations
• Combinations
• Combination Problems.

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Zbl 1074.35048
Leblond, Jean-Baptiste
Mathematical results for a model of diffusion and precipitation of chemical elements in solid matrices.
(English)
[J] Nonlinear Anal., Real World Appl. 6, No. 2, 297-322 (2005). ISSN 1468-1218

The model studied by the author is a system of strongly coupled quasilinear equations of parabolic type of the form $$u_t+{\cal A}(u)u=f(\cdot,u,\text {grad}\, u)\quad \text{in }\Omega\times (0,\infty)$$ with the boundary condition $${\cal B}(u)u=0\quad \text{on }\partial\Omega\times (0,\infty).$$ Existence and uniqueness of the state of local thermodynamic equilibrium are established. Moreover, the author proposes an improved version of the original model for which the general theory of H. Amann can be applied to derive the local existence and smoothness of the corresponding initial value-boundary problem.
[Sen-Zhong Huang (Hamburg)]
MSC 2000:
*35K50 Systems of parabolic equations, boundary value problems
35K57 Reaction-diffusion equations

Keywords: strongly coupled quasilinear equations; local thermodynamic equilibrium; local existence

Highlights
Master Server


text:
GCSE Maths Number

Powers And Roots

Fractional Powers

# Fractional Powers

Here we will learn about fractional powers, including what they are, how to simplify and evaluate them, and how to combine them with other laws of exponents.

There are also powers and roots worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are fractional powers?

Fractional powers are a type of index that represents the nth root of a number.

In general, the rule for fractional exponents is x^{\frac{m}{n}}=\sqrt[n]{x^{m}} .

Where m and n are constants and n 0 .

For example,

9^{\frac{1}{2}}=\sqrt{9}=3

m=1, n=2, and x=9 , so we are calculating the second root, or the square root of 9 which is equal to 3 .

## How to use fractional powers

In order to use fractional powers:

1. Simplify any powers using laws of indices.
2. Evaluate or solve (if required).

## Fractional powers examples

### Example 1: simplify and evaluate

Simplify and evaluate the expression

9^{\frac{1}{2}} \times\ 9 .

1. Simplify any powers using laws of indices.

As x^{a} \times x^{b}=x^{a+b} , and 9=9^{1} , we have

9^{\frac{1}{2}}=9^{\frac{1}{2}+1}=9^{1.5}=9^{\frac{3}{2}} .

2Evaluate or solve (if required).

Evaluating the expression, we get

9^{\frac{3}{2}}=(\sqrt{9})^{3}=3^{3}=27 .

So 9^{\frac{1}{2}}\times\ {9}=27 .

### Example 2: algebraic

Write the following expression in its simplest form

(xy)^{a}\div\ {(x)^\frac{a}{2}} .

As we have multiple variables in the first expression raised to the power of a , let’s remove the brackets so we know the power of each variable separately,

(xy)^{a}=x^{a}\times y^{a} .

We can then divide x^{a} \times y^{a} by (x)^\frac{a}{2} to get

x^{a}\times\ {y^{a}}\div{(x)^\frac{a}{2}}=x^{a-\frac{a}{2}}\times\ {y^{a}}=x^{\frac{a}{2}}\times\ {y^{a}} .

(Remember we can only simplify powers if the base is the same.)

So we now have x^{\frac{a}{2}}\times\ {y^{a}} .

The question asks us to simplify the expression and so the solution is,

x^{\frac{a}{2}}y^{a} .

### Example 3: inequalities

Solve the inequality

64^{\frac{1}{3}}\geq{\frac{1}{x}} .

Looking at the left hand side of the inequality sign, we have 64 being raised to the power of \frac{1}{3} and so we can write this as

64^{\frac{1}{3}}=\sqrt[3]{64}=4 .

So we now have the inequality 4\geq{\frac{1}{x}} .

The inverse for dividing by x is multiplying, so we need to multiply both sides by x .

4x\geq 1

This can be simplified by dividing both sides of the inequality by 4 ,

x\geq{\frac{1}{4}} .

This is the solution.

### Example 4: negative fractional exponents

Evaluate the following expression,

216^{-\frac{2}{3}} .

As the power is a fraction and a negative number, we are going to simplify the base by looking at the denominator of the power first.

As the denominator of the fraction is the nth root of the base number we can start to simplify the expression,

216^{-\frac{2}{3}}=(216^{\frac{1}{3}})^{-2}=(\sqrt[3]{216})^{-2}=6^{-2} .

(The power of \frac{1}{3} is exactly the same as the cube root.)

6^{-2}=\frac{1}{6^{2}}

The expression \frac{1}{6^{2}} is the simplified form of the expression, but we can now evaluate the expression as we know that 6^{2}=36 so \frac{1}{6^{2}}=\frac{1}{36} .

So the solution is \frac{1}{36} .

### Example 5: fractional base number

Simplify fully,

(2\frac{113}{256})^{\frac{1}{4}} .

Write your answer as a mixed number.

As the base number is a mixed number, let’s convert it to an improper fraction first,

2\frac{113}{256}=\frac{2\times{256}+113}{256}=\frac{625}{256} .

So we now have (\frac{625}{256})^{\frac{1}{4}} .

As the fraction is raised to the power of \frac{1}{4} , both the numerator and the denominator can be raised to the power of \frac{1}{4} .

(\frac{625}{256})^{\frac{1}{4}}=\frac{625^{\frac{1}{4}}}{256^{\frac{1}{4}}}

As the index is equal to \frac{1}{4} , we can calculate the 4^{th} root of each part of the fraction.

The fourth root of 625 is 5 .

The fourth root of 256 is 4 .

So, \frac{\sqrt[4]{625}}{\sqrt[4]{256}}=\frac{5}{4}=1\frac{1}{4} .

So, (2\frac{113}{256})^{\frac{1}{4}}=1\frac{1}{4} .

### Example 6: expressing a power with a different base

Simplify fully,

(8^{-10})^{\frac{1}{5}} .

Write your answer as a fractions in terms of the power of 2 .

The law of indices we are going to use involves parentheses, namely (x^{a})^{b}=x^{a\times b} , we get

(8^{-10})^{\frac{1}{5}}=8^{-10\times{\frac{1}{5}}}=8^{-2} .

8^{-2}=\frac{1}{8^{2}} .

The question does not want us to state the value of \frac{1}{8^{2}} .

Instead, the question asks for the solution to be written as a power of 2 , so we need to change the expression so that the base number is 2 , instead of 8 .

As 8=2^{3} , we have \frac{1}{8^{2}}=\frac{1}{(2^{3})^{2}}=\frac{1}{2^{3\times{2}}}=\frac{1}{2^{6}}

So the solution is \frac{1}{2^{6}} .

### Common misconceptions

• Fractional power and fraction of amounts

A common error is to mistake a fractional power with multiplying by a fraction.

For example,

27^{\frac{1}{3}}=27\times{\frac{1}{3}}=9 is incorrect.

This is because the denominator of the fraction is the nth root of the base.

The correct answer is

27^{\frac{1}{3}}=\sqrt[3]{27}=3 .

• BIDMAS

When raising a base to a power, the power is associated with a variable or an expression.

For example, let’s look at  3x^{-2} .

Using BIDMAS we can see that x is being raised to the power of -2 , and then is multiplied by 3 to get the answer \frac{3}{x^{2}}

However a common error is to calculate 3x raised to the power of -2 , giving the answer \frac{1}{9x^{2}} which is incorrect. If this was the case, the question would be written as (3x)^{-2}.

• Incorrect application of the exponent

A similar circumstance is applied to fractions that are raised to a power.

For example, let’s look at (\frac{5}{2})^{-3} .

The correct application of the laws of indices would give a correct answer of  (\frac{5}{2})^{-3}=(\frac{2}{5})^{3}=\frac{8}{125} .

However a common error would be to apply the -3 only to the numerator, leaving the denominator unchanged. This would give the incorrect answer of (\frac{5}{2})^{-3}=\frac{1}{125}\times\frac{1}{2}=\frac{1}{250} .

### Practice fractional powers questions

1. Evaluate

4^{\frac{1}{2}}\times\ {4^{\frac{3}{2}}} .

4^{\frac{3}{4}}

12

8

16
4^{\frac{1}{2}}\times\ {4^{\frac{3}{2}}}=4^{\frac{1}{2}+\frac{3}{2}}=4^{\frac{4}{2}}=4^{2}=16

2. Simplify fully,

(x^{a}y)^{\frac{1}{a}} .

\frac{x^{a}}{y^{a}}

(xy)^{a}

xy^{\frac{1}{a}}

\frac{1}{ax^{a}y}
(x^{a}y)^{\frac{1}{a}}=x^{a\times{\frac{1}{a}}}\times\ {y^{\frac{1}{a}}}=x\times\ {y^{\frac{1}{a}}}=xy^{\frac{1}{a}}

3. State the range of values for the following inequality,

8^{\frac{2}{3}}<\frac{x}{5} .

x>20

x<20

x>30

x<30
8^{\frac{2}{3}}=(\sqrt[3]{8})^{2}=2^{2}=4

4<\frac{x}{5}

20<x so x>20

4. Evaluate

625^{-\frac{3}{4}} .

-125

125

\frac{1}{125}

-\frac{1}{125}
625^{-\frac{3}{4}}=\frac{1}{625^{\frac{3}{4}}}=\frac{1}{(\sqrt[4]{625})^{3}}=\frac{1}{5^{3}}=\frac{1}{125}

5. Evaluate

(\frac{1024}{32})^{\frac{1}{5}} .

\frac{1}{8}

2

6.4

160
(\frac{1024}{32})^{\frac{1}{5}}=\frac{\sqrt[5]{1024}}{\sqrt[5]{32}}=\frac{4}{2}=2

6. Simplify the expression below. Write your answer in the form 3^{n}.

(9^{-3})^{\frac{1}{2}}

-\frac{3}{2}

3^{-3}

\frac{1}{27}

27
(9^{-3})^{\frac{1}{2}}=(9^{\frac{1}{2}})^{-3}=3^{-3}=\frac{1}{3^{3}}=\frac{1}{27}

### Fractional powers GCSE questions

1. Solve the inequality 3\div \sqrt[3]{x}<x^{\frac{2}{3}} .

(4 marks)

2. (a) Evaluate 16^{-1\frac{1}{2}} .

(b) Simplify a^{\frac{1}{2}}\times{a^{\frac{1}{2}}} .

(5 marks)

(a)

-1\frac{1}{2}=-\frac{3}{2}

(1)

\sqrt{16}=4

(1)

4^{-3}=\frac{1}{4^{3}}=\frac{1}{64}

(1)

(b)

a^{\frac{1}{2}}\times{a^{\frac{1}{2}}}=a^{\frac{1}{2}+\frac{1}{2}}

(1)

a^{1}

(1)

3. Express the following as a single power of 2 ,

4^{4}\times{\sqrt{8}} .

(6 marks)

4^{4}=(2^{2})^{4}

(1)

(2^{2})^{4}=2^{2\times 4}=2^{8}

(1)

\sqrt{8}=8^{\frac{1}{2}}=(2^{3})^{\frac{1}{2}}

(1)

2^{8}\times{2^{\frac{3}{2}}}

(1)

2^{8+\frac{3}{2}}

(1)

=2^{\frac{19}{2}}=2^{9\frac{1}{2}}

(1)

## Learning checklist

You have now learned how to:

• Consolidate your numerical and mathematical capability from key stage 3 and extend your understanding of the number system to fractional indices
• Calculate with roots, and fractional indices

## Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.

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#### FREE GCSE Maths Scheme of Work Guide

An essential guide for all SLT and subject leaders looking to plan and build a new scheme of work, including how to decide what to teach and for how long

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Logging API

The logging API used by OpenSurgSim code. More...

Collaboration diagram for Logging API:

Modules

Internal logging helpers
Not meant for public consumption.

Files

file  Log.h
The convenience header that provides the entirety of the logging API.

Classes

class  SurgSim::Framework::Logger
An object that can be used to control logging parameters, such as verbosity and log output destination. More...

Macros

#define SURGSIM_LOG(logger, level)
Logs a message to the specified logger with the short level name. More...

#define SURGSIM_LOG_DEBUG(logger)   SURGSIM_LOG(logger, DEBUG)
Logs a message to the specified logger at the DEBUG level. More...

#define SURGSIM_LOG_INFO(logger)   SURGSIM_LOG(logger, INFO)
Logs a message to the specified logger at the INFO level. More...

#define SURGSIM_LOG_WARNING(logger)   SURGSIM_LOG(logger, WARNING)
Logs a message to the specified logger at the WARNING level. More...

#define SURGSIM_LOG_SEVERE(logger)   SURGSIM_LOG(logger, SEVERE)
Logs a message to the specified logger at the SEVERE level. More...

#define SURGSIM_LOG_CRITICAL(logger)   SURGSIM_LOG(logger, CRITICAL)
Logs a message to the specified logger at the CRITICAL level. More...

#define SURGSIM_LOG_IF(condition, logger, level)
Logs a message to the specified logger with the short level name if condition is true. More...

#define SURGSIM_LOG_ONCE(logger, level)
Logs a message to the specified logger with the short level name, but only the first time this statement is reached. More...

#define SURGSIM_LOG_ONCE_IF(condition, logger, level)
Logs a message to the specified logger with the short level name if condition is true, but only the first time this particular condition is true. More...

Enumerations

enum  SurgSim::Framework::LogLevel {
SurgSim::Framework::LOG_LEVEL_DEBUG, SurgSim::Framework::LOG_LEVEL_INFO, SurgSim::Framework::LOG_LEVEL_WARNING, SurgSim::Framework::LOG_LEVEL_SEVERE,
SurgSim::Framework::LOG_LEVEL_CRITICAL
}
Logging levels. More...

Detailed Description

The logging API used by OpenSurgSim code.

Assertions

Macro Definition Documentation

 #define SURGSIM_LOG ( logger, level )

Logs a message to the specified logger with the short level name.

Parameters
 logger Logger used to log the message level Level of this log message (DEBUG, INFO, WARNING, SEVERE or CRITICAL).
Returns
Stream to output the log message

Example

SURGSIM_LOG(logger, WARNING) << messageText;
 #define SURGSIM_LOG_CRITICAL ( logger ) SURGSIM_LOG(logger, CRITICAL)

Logs a message to the specified logger at the CRITICAL level.

Parameters
 logger Logger used to log the message
Returns
Stream to output the log message

Example

SURGSIM_LOG_CRITICAL(logger) << messageText;
 #define SURGSIM_LOG_DEBUG ( logger ) SURGSIM_LOG(logger, DEBUG)

Logs a message to the specified logger at the DEBUG level.

Parameters
 logger Logger used to log the message
Returns
Stream to output the log message

Example

SURGSIM_LOG_DEBUG(logger) << messageText;
 #define SURGSIM_LOG_IF ( condition, logger, level )

Logs a message to the specified logger with the short level name if condition is true.

Parameters
 condition Condition to test. logger Logger used to log the message. level Level of this log message (DEBUG, INFO, WARNING, SEVERE or CRITICAL).
Returns
Stream to output the log message.

Example

SURGSIM_LOG_IF(size >= 100, logger, INFO) << "size is " << size;
 #define SURGSIM_LOG_INFO ( logger ) SURGSIM_LOG(logger, INFO)

Logs a message to the specified logger at the INFO level.

Parameters
 logger Logger used to log the message
Returns
Stream to output the log message

Example

SURGSIM_LOG_INFO(logger) << messageText;
 #define SURGSIM_LOG_ONCE ( logger, level )

Logs a message to the specified logger with the short level name, but only the first time this statement is reached.

Parameters
 logger Logger used to log the message. level Level of this log message (DEBUG, INFO, WARNING, SEVERE or CRITICAL).
Returns
Stream to output the log message.

Example

SURGSIM_LOG_ONCE(logger, level) << messageTextShownOnce;
 #define SURGSIM_LOG_ONCE_IF ( condition, logger, level )

Logs a message to the specified logger with the short level name if condition is true, but only the first time this particular condition is true.

Parameters
 condition Condition to test logger Logger used to log the message level Level of this log message (DEBUG, INFO, WARNING, SEVERE or CRITICAL).
Returns
Stream to output the log message

Example

SURGSIM_LOG_ONCE_IF(condition, logger, level) << messageTextShownOnce;
 #define SURGSIM_LOG_SEVERE ( logger ) SURGSIM_LOG(logger, SEVERE)

Logs a message to the specified logger at the SEVERE level.

Parameters
 logger Logger used to log the message
Returns
Stream to output the log message

Example

SURGSIM_LOG_SEVERE(logger) << messageText;
 #define SURGSIM_LOG_WARNING ( logger ) SURGSIM_LOG(logger, WARNING)

Logs a message to the specified logger at the WARNING level.

Parameters
 logger Logger used to log the message
Returns
Stream to output the log message

Example

SURGSIM_LOG_WARNING(logger) << messageText;

Enumeration Type Documentation

Logging levels.

Please note that most logging macros take an abbreviated version of these enumerations, without the leading LOG_LEVEL_, i.e. one of DEBUG, INFO, WARNING, SEVERE or CRITICAL .

Enumerator
LOG_LEVEL_DEBUG

LOG_LEVEL_INFO

Informational, notify of state changes.

LOG_LEVEL_WARNING

Something failed, but the impact of the failure is not know or minimal (e.g. purely visual).

LOG_LEVEL_SEVERE

Something failed and will impact functionality, some parts of the program will not function correctly.

LOG_LEVEL_CRITICAL

Used by assertion, after using this level the program will not be functional at all.


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# American Institute of Mathematical Sciences

July  2013, 18(5): 1389-1414. doi: 10.3934/dcdsb.2013.18.1389

## Sign-changing solutions of a quasilinear heat equation with a source term

 1 School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China 2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  June 2012 Revised  August 2012 Published  March 2013

The Cauchy problem of a heat equation with a source term $$\psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n$$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $| x| \rightarrow \infty$ with an algebraic rate determined uniquely by $n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.
Citation: Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
##### References:
 [1] B. Bettioui and A. Gmira, On the radial solutions of a degenerate quasilinear elliptic equation in $R^N$,, Ann. Fac. Sci. Toulouse, 8 (1999), 411.   Google Scholar [2] M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited,, Advanced Nonlinear Studies, 6 (2006), 69.   Google Scholar [3] T. Cazenave, F. Dickstein and F.B. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball, Math. Ann., 344 (2009), 431.  doi: 10.1007/s00208-008-0312-6.  Google Scholar [4] C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation,, Nonlinear Anal., 33 (1998), 51.  doi: 10.1016/S0362-546X(97)00542-7.  Google Scholar [5] A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem,, Indiana Univ. Math. J., 31 (1982), 167.  doi: 10.1512/iumj.1982.31.31016.  Google Scholar [6] L.A. Peletier, D. Terman and F.B. Weissler, On the equation $\Delta u+\frac {1}{2}x \nabla u +f(u)=0$,, Arch. Rat. Mech. Anal. 94 (1986), 94 (1986), 83.  doi: 10.1007/BF00278244.  Google Scholar [7] Y.W. Qi, The existence of moving boundary solution of a porous media equation with a source term,, Mathematical Sciences and Applications Gakkōtosho, 6 (1996), 197.   Google Scholar [8] Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation,, Nonlinear Anal. 31 (1998), 31 (1998), 117.  doi: 10.1016/S0362-546X(96)00298-2.  Google Scholar [9] R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity,, J. Differential Equations, 190 (2003), 150.  doi: 10.1016/S0022-0396(02)00086-4.  Google Scholar [10] M. Wang and X. Wang, Existence of positive solutions to a nonlinear initial problem,, Nonlinear Anal. 44 (2001), 44 (2001), 1133.  doi: 10.1016/S0362-546X(99)00344-2.  Google Scholar [11] F.B. Weissler, Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation,, Arch. Rat. Mech. Anal. 91 (1986), 91 (1986), 231.  doi: 10.1007/BF00250743.  Google Scholar [12] F.B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations,, Arch. Rat. Mech. Anal. 91 (1986), 91 (1986), 247.  doi: 10.1007/BF00250744.  Google Scholar [13] F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 38 (1981), 29.  doi: 10.1007/BF02761845.  Google Scholar [14] E. Yanagida, Uniqueness of rapidly decreasing solutions to the Haraux-Weissler equation,, J. Differential Equations 127 (1996), 127 (1996), 561.  doi: 10.1006/jdeq.1996.0083.  Google Scholar

show all references

##### References:
 [1] B. Bettioui and A. Gmira, On the radial solutions of a degenerate quasilinear elliptic equation in $R^N$,, Ann. Fac. Sci. Toulouse, 8 (1999), 411.   Google Scholar [2] M.F. Bidaut-Véron, The $p$-Laplace heat equation with a source term: self-similar solutions revisited,, Advanced Nonlinear Studies, 6 (2006), 69.   Google Scholar [3] T. Cazenave, F. Dickstein and F.B. Weissler, Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball, Math. Ann., 344 (2009), 431.  doi: 10.1007/s00208-008-0312-6.  Google Scholar [4] C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation,, Nonlinear Anal., 33 (1998), 51.  doi: 10.1016/S0362-546X(97)00542-7.  Google Scholar [5] A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem,, Indiana Univ. Math. J., 31 (1982), 167.  doi: 10.1512/iumj.1982.31.31016.  Google Scholar [6] L.A. Peletier, D. Terman and F.B. Weissler, On the equation $\Delta u+\frac {1}{2}x \nabla u +f(u)=0$,, Arch. Rat. Mech. Anal. 94 (1986), 94 (1986), 83.  doi: 10.1007/BF00278244.  Google Scholar [7] Y.W. Qi, The existence of moving boundary solution of a porous media equation with a source term,, Mathematical Sciences and Applications Gakkōtosho, 6 (1996), 197.   Google Scholar [8] Y.W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation,, Nonlinear Anal. 31 (1998), 31 (1998), 117.  doi: 10.1016/S0362-546X(96)00298-2.  Google Scholar [9] R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity,, J. Differential Equations, 190 (2003), 150.  doi: 10.1016/S0022-0396(02)00086-4.  Google Scholar [10] M. Wang and X. Wang, Existence of positive solutions to a nonlinear initial problem,, Nonlinear Anal. 44 (2001), 44 (2001), 1133.  doi: 10.1016/S0362-546X(99)00344-2.  Google Scholar [11] F.B. Weissler, Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation,, Arch. Rat. Mech. Anal. 91 (1986), 91 (1986), 231.  doi: 10.1007/BF00250743.  Google Scholar [12] F.B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations,, Arch. Rat. Mech. Anal. 91 (1986), 91 (1986), 247.  doi: 10.1007/BF00250744.  Google Scholar [13] F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 38 (1981), 29.  doi: 10.1007/BF02761845.  Google Scholar [14] E. Yanagida, Uniqueness of rapidly decreasing solutions to the Haraux-Weissler equation,, J. Differential Equations 127 (1996), 127 (1996), 561.  doi: 10.1006/jdeq.1996.0083.  Google Scholar
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# Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$

It is a theorem that, for every finite group $G$, there exists an $n$ such that $G \leq S_n$. Given a particular group, is there any way to determine or bound the smallest $n$ such that this occurs? The representation of the group is up to you, though bounds that use less information are preferred.

If this problem is too hard, specific special cases are also interesting, as are bounds on $n$.

#### Solutions Collecting From Web of "Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$"

This is not easy for general $G$, but we can immediately produce some crude bounds (on the minimal number $\mu(G)$ such that $G$ embeds into $S_{\mu(G)}$):

If we have an embedding $G \hookrightarrow S_n$, we must trivially have $|G| \leq |S_n| = n!$. Of course, this is sharp in the sense that equality holds for $G = S_n$. On the other hand, the regular representation of $G$ determines an embedding $G \hookrightarrow S_{|G|}$, so $\mu(G) \leq |G|$. This latter bound is sharp:

Theorem If $G$ is a group for which there is no embedding $G \hookrightarrow S_n$ for any $n < |G|$, then $G$ is

1. a cyclic group $\Bbb Z_{p^m}$ of prime power order,
2. a generalized quaternion $2$-group, or
3. the Klein 4-group $\Bbb Z_2 \times \Bbb Z_2$.

There are numerous results available for certain classes of $G$. For example, if $\gcd(|G_1|, |G_2|) = 1$, then $\mu(G_1 \times G_2) = \mu(G_1) + \mu(G_2)$.This and (1) above together resolve the problem for abelian groups:

Theorem If $G$ is abelian, so that we can write it as $\Bbb Z_{a_1} \times \cdots \times \Bbb Z_{a_r}$ for (nontrivial) prime powers $a_1, \ldots, a_r$, then $$\mu(G) = a_1 + \cdots + a_r .$$

(Both of the Theorems here are given by D.L. Johnson and recorded in the reference below, which itself contains a bibliography of this topic.)

The realization of $D_8$ as the group of symmetries of the square determines an embedding $D_8 \hookrightarrow S_4$, and it’s not too hard to prove that $\mu(D_{2n}) = \mu(\Bbb Z_n)$ and $\mu(A_n) = n$ (both for $n > 2$). These facts together determine $\mu(G)$ for all $G$ of order $<16$ except $Q_{12} \cong \Bbb Z_3 \rtimes \Bbb Z_4$:

• $S_1$: $\{ e \}$
• $S_2$: $\Bbb Z_2$
• $S_3$: $\Bbb Z_3$, $S_3$
• $S_4$: $\Bbb Z_4$, $\Bbb Z_2 \times \Bbb Z_2$, $D_8$, $A_4$
• $S_5$: $\Bbb Z_5$, $\Bbb Z_6$, $D_{10}$
• $S_6$: $\Bbb Z_4 \times \Bbb Z_2$, $\Bbb Z_2 \times \Bbb Z_2 \times \Bbb Z_2$, $\Bbb Z_3 \times \Bbb Z_3$, $D_{12}$
• $S_7$: $\Bbb Z_7$, $\Bbb Z_{10}$, $\Bbb Z_{12}$, $\Bbb Z_2 \times \Bbb Z_6$, $D_{14}$
• $S_8$: $\Bbb Z_8$, $Q_8$, $\Bbb Z_{15}$
• $S_9$: $\Bbb Z_9$, $\Bbb Z_{14}$
• $S_{11}$: $\Bbb Z_{11}$
• $S_{13}$: $\Bbb Z_{13}$

This question gives an explicit proof that $\mu(Q_8) = 8$.

Lemiuex, Stephan R., Minimal degree of faithful permutation representations of finite groups. (1999) Master’s thesis.

In general, this is not straightforward. If $G$ acts on a set $A$ of $n$ elements, you know this induces a homomorphism $\varphi : G \to S_n$ with kernel $\cap_{a \in A} \textrm{Stab}(a)$. So your problem is reduced to analyzing stabilizers.

Of course if $G$ acts on itself, say, by left multiplication, you get an embedding of $G$ into $S_{|G|}$, but this is usually not particularly helpful. Take $D_4$, the group of symmetries of a square. Since $|D_4| = 8$ we know a copy of $D_4$ lives in $S_8$, but this isn’t too revealing as also $D_4 < S_4$. A more outrageous example is Cayley’s theorem shows $A_5 < S_{60}$, but we know $A_5 < S_5$.

A nice example of such an embedding is as follows: suppose we want to show any simple group of order $60$ is isomorphic to $A_5$. By the Sylow theorems one can show such a group, say $G$, has a subgroup of index $5$, and then we know this induces an injective homomorphism from $G$ into $S_5$, from the claim follows.

Cayley’s theorem won’t always give you information about a particular group, but it can be used to prove general facts about group theory, i.e. one can reduce a general problem to a specific case concerning $S_n$. For instance, Cayley’s theorem can be used to show for any group $G$ there exists a Galois extension $K$ of $\mathbb{Q}$ with a subfield $F$ such that $\textrm{Gal}(K/F) \simeq G$.

Given a finite group $G$, consider $H$ to be subgroup such that it contains no normal subgroup of $G$ and consider such $H$ of maximum possible order.

Then $G$ permutes cosets of $H$ by left multiplication, and hence gives a homomorphism from $G$ to permutation group of cosets $\{gH\colon g\in G\}$. The kernel of this homomorphism is intersection of all the conjugates of $H$; but, by our assumption, it should be trivial. Hence $G$ embeds in permutation group on $|G/H|$ letters.

This also explains why $H$ is chosen in beginning of maximum order.

This works, as it can be seen for examples in other answers.

A simple upper bound is $\# G$ (when finite). This comes from the fact that for a group $G$, for any $g \in G$, the map $x \mapsto gx$ is bijective (you can check it). So let’s assume for simplicity that $G = \{ 1, \ldots, n \}$. We are looking for an injective group homomorphism $\phi : G \to S_{n}$. To do this, let $\phi(g) = f_{g} = (x \mapsto gx)$, i.e. say $\phi(g)$ is the (bijective) function taking $\{ 1, \ldots, n \} = G$ to itself that sends $x$ to $gx$. To see this is a homomorphism, let $g, h \in G$. Then $$f_{gh} (x) = (gh) x = g(hx) = f_{g}(hx) = f_{g}(f_{h}(x)) ,$$ so $\phi(gh) = \phi(g) \circ \phi(h)$. To see $\phi$ is injective, let $g, h$ be such that $f_{g}(x) = f_{h}(x)$. Then
\begin{align*}
gx & = hx \\
\Rightarrow gx (x^{-1}) & = hx (x^{-1}) \\
\Rightarrow g & = h .
\end{align*}
Thus it’s injective. Similarly, if $G$ is infinite, then we can imbed it into the group of permutations on the set $G$. This is, however, not always the optimal way. For example, if we begin with the group $G = S_{n}$, then we’re imbedding it into $S_{n!}$. So the size of a group bounds the necessary size of the permutation group of which it is a subgroup as stated.