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# Knowledge Base

 Navigation: Scripting Language > Functions mqwgain

Calculates gain and spontaneous emission for TE and TM modes in multiple quantum well structures using 4x4 k.p electronic band structure method [1-3]. The conduction band is parabolic, while heavy and light hole valence bands are mixed according to 4x4 k.p method and they are nonparabolic.

The solver includes a material database of common III-V semiconductors, ternary alloys, and quaternary alloys. Material properties may be generated automatically for arbitrary alloy compositions or may be input manually. The supported materials are listed in the table below:

 III-V semiconductors Ternary alloys Quaternary Alloys AlAs AlxGa1-xAs InxGa1-xAsyP1-y GaAs AlxGa1-xP InAs AlxIn1-xP AlP GaAsxP1-x GaP InxAl1-xAs InP InAsxP1-x InxGa1-xAs InxGa1-xP

When database materials are used, the properties of ternary alloys P(AxB1−xD) are interpolated from the corresponding properties of the base materials (P(AD) and P(BD)) according to the formula

$$P\left(A_x B_{1-x}D\right)=xP\left(AD\right)+\left(1-x\right)P\left(BD\right)+x\left(1-x\right)C$$,

where x is the composition fraction and C is the bowing parameter (quadratic coefficient). Quaternary alloys are composed from the interpolation of ternary alloy constituents [4]:

$$P\left(A_xB_{1-x}C_yD_{1-y}\right)=\frac{x\left(1-x\right)\left[\left(1-y\right)P\left(A_xB_{1-x}D\right)+yP\left(A_xB_{1-x}C\right)\right]+y\left(1-y\right)\left[xP\left(AC_yD_{1-y}\right)+\left(1-x\right)P\left(BC_yD_{1-y}\right)\right]}{x\left(1-x\right)+y\left(1-y\right)}$$,

for composition fractions x and y. For example, a combination of the properties of InxGa1−xP, InxGa1−xAs, InAsyP1−y, and GaAsyP1−y is used to define the properties of InxGa1−xAsyP1−y.

 Syntax Description result = mqwgain(stack_properties, simulation_parameters, config); stack_properties: struct with fields that define MQW stack geometry and material properties.   simulation_parameters: struct with fields that define simulation parameters for which the output will be calculated.   config: struct with fields that configure the behaviour of the simulation.   result: cell with 4 datasets containing the spatial band diagram, band structure in (E,k) space, spatial wave functions for each (E,k) state, and emission coefficients. result = mqwgain(stack_properties, simulation_parameters); same as above, but using all default values for the fields in config struct.

stack_properties is a struct with the following fields:

Field

Default

Units

Type

Description

gamma

eV

Scalar

Linewidth broadening due to intraband relaxation rate. Represents full with at half maximum of a Lorentzian.

neff

Matrix

Effective index vs. frequency. Two column matrix: first column is frequency (Hz), second column is effective index. Effective index values will be linearly interpolated on to the photon frequency grid for the simulation.

length

m

Matrix

Thickness of each layer, Nx1 array (N layers)

material

Cell

Material definitions, length N cell array. See below for a description of options to specify material properties.

strain

0

(a0-a)/a

Matrix

Strain in each layer as a fraction, negative values for compressive strain. Nx1 array (N layers).

vb

Not included

Struct

Specification for valence band absolute energy. If not defined than material.vb field is used by default.

stack_properties.material is a cell array (one element per layer) where each element is a struct. The struct can be defined in 2 ways. First way (can be generated automatically by calling buildmqwmaterial script command):

 Coefficient Units Description eg eV Band gap ep eV Energy parameter for the optical matrix element me 1/m0 Electron effective mass gamma1 Luttinger parameter gamma2 Luttinger parameter gamma3 Luttinger parameter ac eV Conduction band deformation potential av eV Valence band deformation potential b eV Valence band deformation potential c11 N/m2 Elastic stiffness coefficient c12 N/m2 Elastic stiffness coefficient lc m Lattice constant vb eV Valence band absolute energy (all layers should have common reference)

stack_properties.material second way:

 Coefficient Type Description database_material String Name of the material x 0

stack_properties.vb is a struct with the following fields:

Field

Default

Units

Type

Description

method

palankovski

String

Method for calculating valence band offsets. If “direct” is specified, the offsets must be provided (see offsets).

offsets

eV

Matrix

Directly specified valence band offsets, Nx1 array (N layers).

simulation_parameters is a struct with the following fields:

Field

Default

Units

Type

Description

T

K

Scalar

Simulation temperature.

V

V

Matrix

Electrostatic potential vs. position. A two-column matrix, with position (m) specified in the first column and potential (eV) specified in the second column. Potential values will be linearly interpolated on to the simulation grid. The first layer is assumed to start at z=0.

kt

linspace(0,2*pi/a*0.1,51)

1/m

Matrix

Transverse k values used in the band structure calculation.

cden

1/m3

Matrix

Carrier density array.

phfreq

Hz

Matrix

Photon frequency array.

config is a struct with the following fields:

Field

Default

Units

Type

Description

bcs

See below

See below

Struct

Boundary conditions struct.

dz

1e-10

m

Scalar

Grid spacing ≥ 1Å.

numeigvals

31

Scalar

Maximum number of bands to calculate by the eignesolver at each kt.

materialdb

String or struct

String specifying the path to the material database or empty struct for the default database.

cbvalley

Gamma

String

Choose the conduction band valley for interpolation of material properties: “Gamma”, “X”, “L”, or “All” (default is “Gamma”; option “All” uses the lowest band gap to select).

config.bcs is a struct with the following fields:

Field

Default

Units

Type

Description

pmlactive

false

Boolean

Enable perfectly matched layer at boundaries.

pmlcutoff

[1e-2.1e-2]

(PML probability density)/(MQW probability density)

Matrix

Threshold ratio to reject eigenstates with excess conduction and valence band probability densities located in the PMLs, 2x1 array

pmllength

[10e-9,10e-9]

m

Matrix

PML thickness for left and right boundaries, 2x1 array.

pmlcoefficient

[0.5+1i*0.5,0.5+1i*0.5,-1+1i*1.4,-1+1i*1.4]

Matrix

PML complex coordinate stretching coefficients. First 2 elements for left and right PML for the conduction band and the other two for the valence band.

hwcutoff

[5e-4,5e-4]

$$A^{-3/2}$$

Matrix

Threshold wave function slope to reject eigenstates that do not decay enough at the left and right hard-wall boundaries, 2x1 array.

result is a struct with the following fields:

 Syntax Type Description banddiagram dataset Conduction and valence band edge including strain, but not including quantum confinement effects. bandstructure dataset (E,kt) band diagram for conduction and valence bands. Attributes are: conduction_band, valence_band_lo, valence_band_up. See below for the meaning of lo and up. Parameters are kt and subband. wavefunction dataset Spatial wavefunction for each (E,kt) point. Attributes are conduction_band_1, valence_band_lo_1, valence_band_lo_2, valence_band_up_1, valence_band_up_2, where the 4x4 k.p basis in the valence band is split into two (lo for lower and up for upper) 2x2 bases (vectors in each 2x2 basis designated with 1 and 2). For more information look at references [1] and [2]. Parameters are coordinate, kt, and subband. emission dataset Gain and spontaneous emission. Attributes are: spontaneous_TE, spontaneous_TM, stimulated_TE, stimulated_TM, where TE and TM stand for electromagnetic modes. Parameters are: frequency/energy/wavelength and ndensity (charge density).

Material Definitions

Material parameters are important for the accurate calculation of the MQW band structure and emission characteristics. Many parameters are used to model the optical and electronic material behavior. Some parameters for alloys of compound semiconductors are not available from experiment and must be generated from interpolation of known values. Experimental results may depend on growth conditions and layer thickness, and adjustment of some material parameters may be necessary to obtain agreement with measurements.

Lumerical provides a default material database with the MQW gain solver. These parameters are used automatically when the layer materials are defined by a name and composition fraction. The following code sets the material in layer 2 as Al0.41Ga0.59As

materials = cell(3);

#...

materials{2} = struct;

materials{2}.database_material = "AlGaAs";

materials{2}.x = 0.41;

You can also choose to use your own material database (in the same format) instead of the default material database supplied by Lumerical. By specifying the path of that database in the simulation configuration struct, you can instruct the solver to use those material definitions, e.g.

config.materialdb = "/home/auser/myfolder/my_material_db.json";

Using this approach, the material parameters of the compound semiconductors can be modified, but the default interpolation used by the solver will still be applied to generate parameters for ternary and quaternary semiconductors. The assignment of materials to layers does not change.

Alternately, a material definition can be read directly from a material database (default or custom) and loaded as a struct into the script workspace using the buildmqwmaterial command. For example,

mymat = buildmqwmaterial("/home/auser/myfolder/my_material_db.json", 300, "InAlAs", 0.47);

will read the necessary properties from the material database and build a material definition at T=300K with composition fraction x=0.47 for In0.47Al0.53As. The result is a structure with the coefficients required by the MQW solver. A struct with these fields can be assigned to a material layer and used directly by the solver, e.g.

materials = cell(3);

#...

materials{2} = buildmqwmaterial("/home/auser/myfolder/my_material_db.json", 300, "InAlAs", 0.47);

References

[1] D. Ahn et al., J. Appl. Phys. 64, 4056 (1988)

[2] S. L. Chuang, Physics of Optoelectronic Devices

[3] Chuang, Phys. Rev. B, 43, 9649 (1991)

[4] Vurgaftman et al., J. Appl. Phys., 89, 5815 (2001)

Example

See application note on our knowledge exchange web site.

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# Integral transform

(Redirected from Kernel (integral operator))

In mathematics, an integral transform is any transform T of the following form:

${\displaystyle (Tf)(u)=\int \limits _{t_{1}}^{t_{2}}K(t,u)\,f(t)\,dt}$

The input of this transform is a function f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator.

There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform.

Some kernels have an associated inverse kernel K−1(u, t) which (roughly speaking) yields an inverse transform:

${\displaystyle f(t)=\int \limits _{u_{1}}^{u_{2}}K^{-1}(u,t)\,(Tf)(u)\,du}$

A symmetric kernel is one that is unchanged when the two variables are permuted.

## Motivation

Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform "maps" an equation from its original "domain" into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.

Also there are many applications of probability that rely on integral transforms, such as "pricing kernel" or stochastic discount factor, or the smoothing of data recovered from robust statistics, see kernel (statistics).

## History

The precursor of the transforms were the Fourier series to express functions in finite intervals. Later the Fourier transform was developed to remove the requirement of finite intervals.

Using the Fourier series, just about any practical function of time (the voltage across the terminals of an electronic device for example) can be represented as a sum of sines and cosines, each suitably scaled (multiplied by a constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). The sines and cosines in the Fourier series are an example of an orthonormal basis.

## Usage example

As an example of an application of integral transforms, consider the Laplace transform. This is a technique that maps differential or integro-differential equations in the "time" domain into polynomial equations in what is termed the "complex frequency" domain. (Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary component ω of the complex frequency s = -σ + iω corresponds to the usual concept of frequency, viz., the rate at which a sinusoid cycles, whereas the real component σ of the complex frequency corresponds to the degree of "damping", i.e. an exponential decrease of the amplitude.) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond to eigenvalues in the time domain), leading to a "solution" formulated in the frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond to power series in the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain.

The Laplace transform finds wide application in physics and particularly in electrical engineering, where the characteristic equations that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially damped, scaled, and time-shifted sinusoids in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.

Another usage example is the kernel in path integral:

${\displaystyle \psi (x,t)=\int _{-\infty }^{\infty }\psi (x',t')K(x,t;x',t')dx'.}$

This states that the total amplitude to arrive at ${\displaystyle (x,t)}$ [that is, ${\displaystyle \psi (x,t)}$] is the sum, or the integral, over all possible values of ${\displaystyle x'}$ of the total amplitude to arrive at the point ${\displaystyle (x',t')}$ [that is, ${\displaystyle \psi (x',t')}$] multiplied by the amplitude to go from x' to x [that is, ${\displaystyle K(x,t;x',t')}$.[1] It is often referred to as the propagator of a given system. This (physics) kernel is the kernel of integral transform. However, for each quantum system, there is a different kernel.[2]

## Table of transforms

Table of integral transforms
Transform Symbol K f(t) t1 t2 K−1 u1 u2
Abel transform ${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}}$ u ${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}}$ t
Fourier transform ${\displaystyle {\mathcal {F}}}$ ${\displaystyle e^{-2\pi iut}}$ ${\displaystyle L_{1}}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle e^{2\pi iut}}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$
Fourier sine transform ${\displaystyle {\mathcal {F}}_{s}}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\sin(ut)}$ on ${\displaystyle [0,\infty )}$, real-valued ${\displaystyle 0}$ ${\displaystyle \infty }$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\sin(ut)}$ ${\displaystyle 0}$ ${\displaystyle \infty }$
Fourier cosine transform ${\displaystyle {\mathcal {F}}_{c}}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cos(ut)}$ on ${\displaystyle [0,\infty )}$, real-valued 0 ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cos(ut)}$ 0
Hankel transform ${\displaystyle t\,J_{\nu }(ut)}$ 0 ${\displaystyle u\,J_{\nu }(ut)}$ 0
Hartley transform ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$ −∞ ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$ −∞
Hermite transform ${\displaystyle H}$ ${\displaystyle e^{-x^{2}}H_{n}(x)}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle 0}$ ${\displaystyle \infty }$
Hilbert transform ${\displaystyle {\mathcal {H}}il}$ ${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$
Jacobi transform ${\displaystyle J}$ ${\displaystyle (1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)}$ ${\displaystyle -1}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle \infty }$
Laguerre transform ${\displaystyle L}$ ${\displaystyle e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)}$ ${\displaystyle 0}$ ${\displaystyle \infty }$ ${\displaystyle 0}$ ${\displaystyle \infty }$
Laplace transform ${\displaystyle {\mathcal {L}}}$ e−ut 0 ${\displaystyle {\frac {e^{ut}}{2\pi i}}}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$
Legendre transform ${\displaystyle {\mathcal {J}}}$ ${\displaystyle P_{n}(x)\,}$ ${\displaystyle -1}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle \infty }$
Mellin transform ${\displaystyle {\mathcal {M}}}$ tu−1 0 ${\displaystyle {\frac {t^{-u}}{2\pi i}}\,}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$
Two-sided Laplace
transform
${\displaystyle {\mathcal {B}}}$ e−ut −∞ ${\displaystyle {\frac {e^{ut}}{2\pi i}}}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$
Poisson kernel ${\displaystyle {\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}}$ 0
Weierstrass transform ${\displaystyle {\mathcal {W}}}$ ${\displaystyle {\frac {e^{-{\frac {(u-t)^{2}}{4}}}}{\sqrt {4\pi }}}\,}$ −∞ ${\displaystyle {\frac {e^{\frac {(u-t)^{2}}{4}}}{i{\sqrt {4\pi }}}}}$ ${\displaystyle c\!-\!i\infty }$ ${\displaystyle c\!+\!i\infty }$

In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, c must be greater than the largest real part of the zeroes of the transform function.

Note that there are alternative notations and conventions for the Fourier transform.

## Different domains

Here integral transforms are defined for functions on the real numbers, but they can be defined more generally for functions on a group.

• If instead one uses functions on the circle (periodic functions), integration kernels are then biperiodic functions; convolution by functions on the circle yields circular convolution.
• If one uses functions on the cyclic group of order n (${\displaystyle C_{n}}$ or ${\displaystyle \mathbf {Z} /n\mathbf {Z} }$), one obtains n × n matrices as integration kernels; convolution corresponds to circulant matrices.

## General theory

Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem).

The general theory of such integral equations is known as Fredholm theory. In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator, the nuclear operator or the Fredholm kernel.


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# 1 Model

## 1.1 Overview

The two-parameter logistic model (2PL) is an item response theory model that includes parameters for both the difficulty and discrimination of items. A hierarchical extension, presented here, models these item parameter pairs as correlated draws from a bivariate normal distribution. This model is similar to the hierarchical three-parameter logistic model proposed by Glas and van der Linden (2003).

$\mathrm{logit} [ \Pr(y_{ij} = 1 | \theta_j, \alpha_i, \beta_i) ] = \alpha_i (\theta_j - \beta_i)$ $\log \alpha_i, \beta_i \sim \mathrm{MVN}(\mu_1, \mu_2, \Sigma)$ $\theta_p \sim \mathrm{N}(0, 1)$

Variables:

• $$i = 1 \ldots I$$ indexes items
• $$j = 1 \ldots J$$ indexes persons
• $$y_{ij} \in \{ 0,1 \}$$ is the response of person $$j$$ to item $$i$$

Parameters:

• $$\alpha_i$$ is the discrimination for item $$i$$
• $$\beta_i$$ is the difficulty for item $$i$$
• $$\theta_j$$ is the ability for person $$j$$
• $$\mu_1$$ is the mean for $$\log \alpha_i$$
• $$\mu_2$$ is the mean for $$\beta_i$$
• $$\Sigma$$ is the covariance matrix for $$\log \alpha_i$$ and $$\beta_i$$

Priors:

• $$\mu_1 \sim \mathrm{N}(0,1)$$ is a weakly informative prior for the mean of the log discrimination parameters.
• $$\mu_2 \sim \mathrm{N}(0,25)$$ is a weakly informative prior for the mean of the difficulty parameters.
• Let $$\tau_1^2 = \Sigma_{1,1}$$ be the variance of the log discrimination parameters. Then $$\tau \sim \mathrm{Exp}(.1)$$ is a weakly informative prior for the standard deviation.
• Let $$\tau_2^2 = \Sigma_{2,2}$$ be the variance of the difficulty parameters. Then $$\tau_2 \sim \mathrm{Exp}(.1)$$ is a weakly informative prior for the standard deviation.
• A weakly informative prior is placed on the covariance, $$\Sigma_{1,2} = \Sigma_{2,1}$$, shrinking the posterior towards zero. This is described in more detail in the next section.

## 1.2Stan program

The Stan program for the model is provided below. It differs from the notation above in that it is written in terms of the Cholesky decomposition of the correlation matrix for better efficiency. This matrix is named L_Omega; Omega because it is the correlation rather than covariance matrix, and L because it is a Cholesky decomposition. The vector tau contains the standard deviations that would be the (square root of the) diagonal of the covariance matrix. (The standard deviations are invariant whether or not the the Cholesky decomposition is used.) In the model block, L_Omega is converted to its covariance equivalent L_Sigma, and item parameter pairs xi[i] are sampled from L_Sigma. The first element of vector xi[i] is the log discrimination for item i, and the second is the difficulty for item i.

Weakly informative normal priors are placed on mu, and weakly informative truncated normal priors are placed on tau. The prior placed on L_Omega using lkj_corr_cholesky() is weakly informative and slightly favors a correlation of zero. See the Stan manual for details.

While more efficient, parameters L_Omega and xi are difficult to interpret. To alleviate this inconvenience, item parameters alpha and beta are derived from xi in the transformed parameters block. (There is some redundancy, as beta[i] and xi[i,2] are equal.) Also, L_Omega is converted to a standard correlation matrix, Omega, in the generated quantities block.

data {
int<lower=1> I;               // # items
int<lower=1> J;               // # persons
int<lower=1> N;               // # observations
int<lower=1, upper=I> ii[N];  // item for n
int<lower=1, upper=J> jj[N];  // person for n
int<lower=0, upper=1> y[N];   // correctness for n
}
parameters {
vector[J] theta;              // abilities
vector[2] xi[I];              // alpha/beta pair vectors
vector[2] mu;                 // vector for alpha/beta means
vector<lower=0>[2] tau;       // vector for alpha/beta residual sds
cholesky_factor_corr[2] L_Omega;
}
transformed parameters {
vector[I] alpha;
vector[I] beta;
for (i in 1:I) {
alpha[i] <- exp(xi[i,1]);
beta[i] <- xi[i,2];
}
}
model {
matrix[2,2] L_Sigma;
L_Sigma <- diag_pre_multiply(tau, L_Omega);
for (i in 1:I)
xi[i] ~ multi_normal_cholesky(mu, L_Sigma);
theta ~ normal(0, 1);
L_Omega ~ lkj_corr_cholesky(4);
mu[1] ~ normal(0,1);
tau[1] ~ exponential(.1);
mu[2] ~ normal(0,5);
tau[2] ~ exponential(.1);
y ~ bernoulli_logit(alpha[ii] .* (theta[jj] - beta[ii]));
}
generated quantities {
corr_matrix[2] Omega;
Omega <- multiply_lower_tri_self_transpose(L_Omega);
}

# 2 Simulation

First, the necessary R packages are loaded.

# Load R packages
library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())
library(ggplot2)

The R code that follows simulates a dataset conforming to the model. The Stan model will be evaluated in terms of its ability to recover the generating values of the parameters when fit to this dataset.

# Set paramters for the simulated data
I <- 20
J <- 1000
mu <- c(0, 0)
tau <- c(0.25, 1)
Omega <- matrix(c(1, 0.3, 0.3, 1), ncol = 2)

# Calculate or sample remaining paramters
Sigma <- tau %*% t(tau) * Omega
xi <- MASS::mvrnorm(I, c(0, 0), Sigma)
alpha <- exp(mu[1] + as.vector(xi[, 1]))
beta <- as.vector(mu[2] + xi[, 2])
theta <- rnorm(J, mean = 0, sd = 1)

# Assemble data and simulate response
data_list <- list(I = I, J = J, N = I * J, ii = rep(1:I, times = J), jj = rep(1:J,
each = I))
eta <- alpha[data_list$ii] * (theta[data_list$jj] - beta[data_list$ii]) data_list$y <- as.numeric(boot::inv.logit(eta) > runif(data_list$N)) The simulated data consists of 20 items and 1000 persons. The log discriminations have mean 0 and standard deviation 0.25. The difficulties have mean 0 and standard deviation 1. The correlation between the log discrimination residuals and difficulty residuals is 0.3. The simulated dataset is fit with Stan. # Fit model to simulated data sim_fit <- stan(file = "hierarchical_2pl.stan", data = data_list, chains = 4, iter = 500) Before interpreting the results, it is necessary to check that the chains have converged. Stan provides the $$\hat{R}$$ statistic for the model parameters and log posterior. These are provided in the following figure. All values for $$\hat{R}$$ should be less than 1.1. sim_summary <- as.data.frame(summary(sim_fit)[[1]]) sim_summary$Parameter <- as.factor(gsub("\\[.*]", "", rownames(sim_summary)))
ggplot(sim_summary) + aes(x = Parameter, y = Rhat, color = Parameter) + geom_jitter(height = 0,
width = 0.5, show.legend = FALSE) + ylab(expression(hat(italic(R))))

The Stan model is evaluated in terms of its ability to recover the generating values of the parameters. The R code below prepares a plot in which the points indicate the difference between the posterior means and generating values for the parameters of main interest. This difference is referred to as discrepancy. The lines indicate the 95% posterior intervals for the difference. Ideally, (nearly) all the 95% posterior intervals would include zero.

# Make vector of wanted parameter names
wanted_pars <- c(paste0("alpha[", 1:I, "]"), paste0("beta[", 1:I, "]"), c("mu[1]",
"mu[2]", "tau[1]", "tau[2]", "Omega[1,2]"))

# Get estimated and generating values for wanted parameters
generating_values = c(alpha, beta, mu, tau, Omega[1, 2])
estimated_values <- sim_summary[wanted_pars, c("mean", "2.5%", "97.5%")]

# Assesmble a data frame to pass to ggplot()
sim_df <- data.frame(parameter = factor(wanted_pars, rev(wanted_pars)), row.names = NULL)
sim_df$middle <- estimated_values[, "mean"] - generating_values sim_df$lower <- estimated_values[, "2.5%"] - generating_values
sim_df$upper <- estimated_values[, "97.5%"] - generating_values # Plot the discrepancy ggplot(sim_df) + aes(x = parameter, y = middle, ymin = lower, ymax = upper) + scale_x_discrete() + geom_abline(intercept = 0, slope = 0, color = "white") + geom_linerange() + geom_point(size = 2) + labs(y = "Discrepancy", x = NULL) + theme(panel.grid = element_blank()) + coord_flip() # 3 Example application # Use data and scoring function from the mirt package library(mirt) sat <- key2binary(SAT12, key = c(1, 4, 5, 2, 3, 1, 2, 1, 3, 1, 2, 4, 2, 1, 5, 3, 4, 4, 1, 4, 3, 3, 4, 1, 3, 5, 1, 3, 1, 5, 4, 5)) The example data (Wood et al. 2003) are from a grade 12 science assessment. 600 students responded to 32 dichotomously scored items. Non-responses were scored as incorrect, so the data contain no missing values. The scored response matrix is converted to list form and fit with Stan. # Assemble data list and fit model sat_list <- list(I = ncol(sat), J = nrow(sat), N = length(sat), ii = rep(1:ncol(sat), each = nrow(sat)), jj = rep(1:nrow(sat), times = ncol(sat)), y = as.vector(sat)) sat_fit <- stan(file = "hierarchical_2pl.stan", data = sat_list, chains = 4, iter = 500) As discussed above, convergence of the chains is assessed for every parameter, and also the log posterior, using $$\hat{R}$$. ex_summary <- as.data.frame(summary(sat_fit)[[1]]) ex_summary$Parameter <- as.factor(gsub("\\[.*]", "", rownames(ex_summary)))
ggplot(ex_summary) + aes(x = Parameter, y = Rhat, color = Parameter) + geom_jitter(height = 0,
width = 0.5, show.legend = FALSE) + ylab(expression(hat(italic(R))))

Next we view summaries of the parameter posteriors.

# View table of parameter posteriors
print(sat_fit, pars = c("alpha", "beta", "mu", "tau", "Omega[1,2]"))
## Inference for Stan model: hierarchical_2pl.
## 4 chains, each with iter=500; warmup=250; thin=1;
## post-warmup draws per chain=250, total post-warmup draws=1000.
##
##             mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
## alpha[1]    0.79    0.00 0.12  0.57  0.71  0.78  0.86  1.03   605 1.00
## alpha[2]    1.44    0.01 0.17  1.13  1.33  1.44  1.56  1.77  1000 1.00
## alpha[3]    1.04    0.00 0.13  0.81  0.95  1.04  1.12  1.31   717 1.00
## alpha[4]    0.59    0.00 0.10  0.40  0.52  0.59  0.66  0.80  1000 1.00
## alpha[5]    0.97    0.00 0.13  0.74  0.88  0.96  1.05  1.25  1000 1.01
## alpha[6]    1.11    0.01 0.15  0.83  1.00  1.11  1.20  1.40   640 1.00
## alpha[7]    0.99    0.01 0.14  0.74  0.90  0.99  1.09  1.28   574 1.00
## alpha[8]    0.69    0.00 0.11  0.48  0.62  0.69  0.76  0.91   600 1.00
## alpha[9]    0.73    0.00 0.12  0.51  0.65  0.73  0.81  1.00   663 1.00
## alpha[10]   0.99    0.00 0.12  0.77  0.91  0.98  1.07  1.25  1000 1.00
## alpha[11]   1.70    0.01 0.39  1.03  1.43  1.64  1.91  2.59   799 1.00
## alpha[12]   0.28    0.00 0.07  0.15  0.23  0.28  0.32  0.43   767 1.00
## alpha[13]   1.08    0.01 0.14  0.83  0.99  1.08  1.17  1.35   700 1.00
## alpha[14]   1.02    0.01 0.14  0.77  0.93  1.02  1.11  1.32   652 1.00
## alpha[15]   1.25    0.01 0.18  0.93  1.12  1.24  1.36  1.62   467 1.01
## alpha[16]   0.71    0.00 0.10  0.53  0.64  0.70  0.77  0.92  1000 1.00
## alpha[17]   1.52    0.01 0.30  1.00  1.32  1.49  1.72  2.15   638 1.00
## alpha[18]   1.65    0.01 0.18  1.33  1.52  1.64  1.77  2.04   548 1.00
## alpha[19]   0.83    0.00 0.11  0.61  0.75  0.82  0.90  1.06  1000 1.00
## alpha[20]   1.46    0.01 0.22  1.08  1.32  1.44  1.59  1.94   493 1.00
## alpha[21]   0.83    0.01 0.15  0.57  0.72  0.83  0.94  1.14   732 1.00
## alpha[22]   1.46    0.01 0.27  0.98  1.27  1.44  1.62  2.03   440 1.01
## alpha[23]   0.63    0.00 0.10  0.45  0.56  0.63  0.70  0.85  1000 1.00
## alpha[24]   1.17    0.01 0.15  0.90  1.06  1.17  1.27  1.51   728 1.00
## alpha[25]   0.75    0.00 0.11  0.54  0.67  0.74  0.83  0.97  1000 1.00
## alpha[26]   1.48    0.01 0.16  1.20  1.36  1.47  1.59  1.79   599 1.00
## alpha[27]   1.79    0.01 0.27  1.31  1.60  1.78  1.98  2.35   400 1.01
## alpha[28]   1.03    0.00 0.13  0.80  0.95  1.03  1.12  1.31  1000 1.00
## alpha[29]   0.82    0.00 0.11  0.60  0.74  0.81  0.89  1.05   689 1.00
## alpha[30]   0.43    0.00 0.08  0.26  0.37  0.43  0.49  0.60   717 1.00
## alpha[31]   2.13    0.02 0.30  1.58  1.93  2.11  2.31  2.75   347 1.00
## alpha[32]   0.37    0.00 0.07  0.25  0.32  0.37  0.42  0.53   415 1.00
## beta[1]     1.34    0.01 0.22  0.97  1.20  1.32  1.46  1.84   598 1.00
## beta[2]    -0.31    0.00 0.08 -0.47 -0.36 -0.30 -0.25 -0.15   354 1.01
## beta[3]     1.09    0.01 0.15  0.82  0.99  1.08  1.18  1.42   683 1.00
## beta[4]     0.92    0.01 0.22  0.57  0.76  0.90  1.05  1.38  1000 1.01
## beta[5]    -0.64    0.01 0.12 -0.89 -0.71 -0.63 -0.56 -0.41   399 1.01
## beta[6]     1.84    0.01 0.21  1.49  1.69  1.82  1.98  2.30   594 1.00
## beta[7]    -1.42    0.01 0.19 -1.85 -1.52 -1.41 -1.30 -1.10   517 1.00
## beta[8]     2.21    0.02 0.36  1.66  1.96  2.17  2.41  3.12   470 1.00
## beta[9]    -3.08    0.02 0.47 -4.11 -3.36 -3.02 -2.75 -2.30   499 1.00
## beta[10]    0.36    0.00 0.10  0.18  0.29  0.35  0.42  0.57  1000 1.01
## beta[11]   -3.19    0.02 0.49 -4.36 -3.46 -3.12 -2.85 -2.38   668 1.00
## beta[12]    1.36    0.02 0.49  0.64  1.01  1.27  1.59  2.60   625 1.00
## beta[13]   -0.79    0.01 0.12 -1.03 -0.86 -0.78 -0.71 -0.58   472 1.00
## beta[14]   -1.16    0.01 0.15 -1.50 -1.25 -1.15 -1.05 -0.91   666 1.00
## beta[15]   -1.55    0.01 0.18 -1.92 -1.66 -1.53 -1.42 -1.23   397 1.02
## beta[16]    0.54    0.00 0.15  0.26  0.43  0.53  0.64  0.83  1000 1.00
## beta[17]   -2.80    0.02 0.44 -3.74 -3.04 -2.74 -2.50 -2.12   495 1.00
## beta[18]    0.51    0.00 0.08  0.35  0.45  0.50  0.56  0.68   479 1.01
## beta[19]   -0.29    0.00 0.11 -0.50 -0.36 -0.29 -0.22 -0.08  1000 1.00
## beta[20]   -1.78    0.01 0.19 -2.19 -1.89 -1.76 -1.64 -1.43   434 1.01
## beta[21]   -3.23    0.02 0.53 -4.49 -3.55 -3.14 -2.84 -2.39   619 1.00
## beta[22]   -2.39    0.02 0.33 -3.10 -2.57 -2.37 -2.17 -1.85   425 1.01
## beta[23]    1.36    0.01 0.26  0.95  1.20  1.34  1.51  1.94   725 1.00
## beta[24]   -1.09    0.00 0.13 -1.38 -1.17 -1.08 -0.99 -0.84   716 1.00
## beta[25]    0.77    0.01 0.16  0.49  0.65  0.75  0.87  1.11   721 1.01
## beta[26]    0.11    0.00 0.08 -0.03  0.06  0.11  0.16  0.26   576 1.00
## beta[27]   -1.52    0.01 0.15 -1.86 -1.62 -1.51 -1.41 -1.26   289 1.01
## beta[28]   -0.17    0.00 0.10 -0.38 -0.23 -0.17 -0.11  0.02  1000 1.00
## beta[29]    0.93    0.01 0.16  0.66  0.81  0.91  1.04  1.27   557 1.00
## beta[30]    0.60    0.01 0.25  0.17  0.44  0.57  0.73  1.19   555 1.01
## beta[31]   -1.26    0.01 0.11 -1.50 -1.32 -1.26 -1.18 -1.06   226 1.01
## beta[32]    4.57    0.04 0.92  3.17  3.90  4.43  5.16  6.43   606 1.00
## mu[1]      -0.05    0.00 0.10 -0.25 -0.11 -0.05  0.02  0.13  1000 1.00
## mu[2]      -0.25    0.01 0.34 -0.91 -0.48 -0.26 -0.02  0.39  1000 1.00
## tau[1]      0.50    0.00 0.08  0.38  0.45  0.50  0.55  0.68   708 1.00
## tau[2]      1.83    0.01 0.28  1.37  1.64  1.80  1.99  2.44   523 1.00
## Omega[1,2] -0.44    0.00 0.15 -0.70 -0.56 -0.45 -0.34 -0.12  1000 1.00
##
## Samples were drawn using NUTS(diag_e) at Tue Apr 19 16:52:58 2016.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).

To visualize the correlation between item parameters, the values of alpha[i] and beta[i] may be plotted against one another. This is presented in the left side of the figure below. Because the correlation is actually between xi[i,1] (which is equal to beta[i]) and xi[i,2] (which is alpha[i] on the log scale), a scatter plot pf xi[i,1] versus xi[i,2] is given on the right side.

# Assesmble a data frame of item parameter estimates and pass to ggplot
ab_df <- data.frame(Discrimination = ex_summary[paste0("alpha[", 1:sat_list$I, "]"), "mean"], Difficulty = ex_summary[paste0("beta[", 1:sat_list$I, "]"),
"mean"], parameterization = "alpha & beta")
xi_df <- data.frame(Discrimination = ex_summary[paste0("xi[", 1:sat_list$I, ",1]"), "mean"], Difficulty = ex_summary[paste0("xi[", 1:sat_list$I, ",2]"),
"mean"], parameterization = "xi")
full_df <- rbind(ab_df, xi_df)
ggplot(full_df) + aes(x = Difficulty, y = Discrimination) + geom_point() + facet_wrap(~parameterization,
scales = "free")

# 4 References

Glas, Cees AW, and Wim J van der Linden. 2003. “Computerized Adaptive Testing with Item Cloning.” Applied Psychological Measurement 27 (4). Sage Publications: 247–61.

Wood, R., Wilson D. T., Gibbons R. D., Schilling S. G., Muraki E., and R. D. Bock. 2003. TESTFACT 4 for Windows: Test Scoring, Item Statistics, and Full-Information Item Factor Analysis. Lincolnwood, IL: Scientific Software International.


text:
wx.dataview.DataViewItemAttr¶

This class is used to indicate to a wx.dataview.DataViewCtrl that a certain item (see wx.dataview.DataViewItem) has extra font attributes for its renderer.

For this, it is required to override wx.dataview.DataViewModel.GetAttr .

Attributes are currently only supported by DataViewTextRendererText.

Class Hierarchy¶

Inheritance diagram for class DataViewItemAttr:

Methods Summary¶

 __init__ Constructor. GetBackgroundColour Returns the colour to be used for the background. GetBold Returns value of the bold property. GetColour Returns this attribute’s colour. GetEffectiveFont Return the font based on the given one with this attribute applied to it. GetItalic Returns value of the italics property. HasBackgroundColour Returns True if the background colour property has been set. HasColour Returns True if the colour property has been set. HasFont Returns True if any property affecting the font has been set. IsDefault Returns True if none of the properties have been set. SetBackgroundColour Call this to set the background colour to use. SetBold Call this to indicate that the item shall be displayed in bold text. SetColour Call this to indicate that the item shall be displayed with that colour. SetItalic Call this to indicate that the item shall be displayed in italic text.

Class API¶

class wx.dataview.DataViewItemAttr(object)

Possible constructors:

DataViewItemAttr()

This class is used to indicate to a DataViewCtrl that a certain item (see DataViewItem) has extra font attributes for its renderer.

Methods¶

__init__(self)

Constructor.

GetBackgroundColour(self)

Returns the colour to be used for the background.

Return type

Colour

GetBold(self)

Returns value of the bold property.

Return type

bool

GetColour(self)

Returns this attribute’s colour.

Return type

Colour

GetEffectiveFont(self, font)

Return the font based on the given one with this attribute applied to it.

Parameters

font (wx.Font) –

Return type

Font

GetItalic(self)

Returns value of the italics property.

Return type

bool

HasBackgroundColour(self)

Returns True if the background colour property has been set.

Return type

bool

HasColour(self)

Returns True if the colour property has been set.

Return type

bool

HasFont(self)

Returns True if any property affecting the font has been set.

Return type

bool

IsDefault(self)

Returns True if none of the properties have been set.

Return type

bool

SetBackgroundColour(self, colour)

Call this to set the background colour to use.

Currently this attribute is only supported in the generic version of wx.dataview.DataViewCtrl and ignored by the native GTK+ and OS X implementations.

Parameters

colour (wx.Colour) –

New in version 2.9.4.

SetBold(self, set)

Call this to indicate that the item shall be displayed in bold text.

Parameters

set (bool) –

SetColour(self, colour)

Call this to indicate that the item shall be displayed with that colour.

Parameters

colour (wx.Colour) –

SetItalic(self, set)

Call this to indicate that the item shall be displayed in italic text.

Parameters

set (bool) –

Properties¶

BackgroundColour

Bold

Colour

Italic


text:
52
Active contributors today

## Evaluate the following?

Mia
Featured 2 months ago

$\frac{7}{11}$

#### Explanation:

Computing the rational number with fractional exponent ${\left(\frac{u}{v}\right)}^{\frac{1}{6}} = {u}^{\frac{1}{6}} / {v}^{\frac{1}{6}}$ is determined by prime factorizing the numerator and denominator .

example of: #sqrtx = x^(1/2#:

$\sqrt{\left(\frac{4}{9}\right)} = {\left(\frac{4}{9}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} = {\left({2}^{2}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} / {\left({3}^{2}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} = \frac{2}{3}$

The rational number above ${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {\left(117649\right)}^{\frac{1}{6}} / {\left(1771561\right)}^{\frac{1}{6}}$

Prime Factorization :
$\textcolor{b l u e}{117649 = 7 \times 7 \times 7 \times 7 \times 7 \times 7 = {7}^{6}}$
$\textcolor{b r o w n}{1771561 = 11 \times 11 \times 11 \times 11 \times 11 \times 11 = {11}^{6}}$

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {\left(117649\right)}^{\frac{1}{6}} / {\left(1771561\right)}^{\frac{1}{6}} = {\left(\textcolor{b l u e}{{7}^{6}}\right)}^{\frac{1}{6}} / {\left(\textcolor{b r o w n}{{11}^{6}}\right)}^{\frac{1}{6}}$

Then we apply the power of a power with base $a$ :
${\left({a}^{m}\right)}^{\frac{1}{n}} = {a}^{\frac{m}{n}}$

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = \left({7}^{\frac{6}{6}} / {11}^{\frac{6}{6}}\right) = \frac{7}{11}$

## The length of a rectangle is 7 feet larger than the width. The perimeter of the rectangle is 26 ft. How do you write an equation to represent the perimeter in terms of its width (w). What is the length?

smendyka
Featured 2 months ago

An equation to represent the perimeter in terms of its width is: $p = 4 w + 14$ and the length of the rectangle is $10$ ft.

#### Explanation:

Let the width of the rectangle be $w$.

Let the length of the rectangle be $l$.

If the length ($l$) is 7 feet longer than the width, then the length can be written in terms of the width as:

$l = w + 7$

The formula for perimeter of a rectangle is:

$p = 2 l + 2 w$ where $p$ is the perimeter, $l$ is the length and $w$ is the width.

Substituting $w + 7$ for $l$ gives an equation to represent the perimeter in terms of its width:

$p = 2 \left(w + 7\right) + 2 w$

$p = 2 w + 14 + 2 w$

$p = 4 w + 14$

Substituting $26$ for $p$ allows us to solve for $w$.

$26 = 4 w + 14$

$26 - 14 = 4 w + 14 - 14$

$12 = 4 w$

$\frac{12}{4} = 4 \frac{w}{4}$

$w = 3$

Sustituting $3$ for $w$ in the equation above, $l = w + 7$ allows us to determine the length:

$l = 3 + 7$

$l = 10$

## The hypotenuse of a right triangle is 39 inches, and the length of one leg is 6 inches longer than twice the other leg. How do you find the length of each leg?

George C.
Featured 2 months ago

The legs are of length $15$ and $36$

#### Explanation:

Method 1 - Familiar triangles

The first few right angled triangles with an odd length side are:

$3 , 4 , 5$

$5 , 12 , 13$

$7 , 24 , 25$

Notice that $39 = 3 \cdot 13$, so will a triangle with the following sides work:

$15 , 36 , 39$

i.e. $3$ times larger than a $5 , 12 , 13$ triangle ?

Twice $15$ is $30$, plus $6$ is $36$ - Yes.

$\textcolor{w h i t e}{}$
Method 2 - Pythagoras formula and a little algebra

If the smaller leg is of length $x$, then the larger leg is of length $2 x + 6$ and the hypotenuse is:

$39 = \sqrt{{x}^{2} + {\left(2 x + 6\right)}^{2}}$

$\textcolor{w h i t e}{39} = \sqrt{5 {x}^{2} + 24 x + 36}$

Square both ends to get:

$1521 = 5 {x}^{2} + 24 x + 36$

Subtract $1521$ from both sides to get:

$0 = 5 {x}^{2} + 24 x - 1485$

Multiply both sides by $5$ to get:

$0 = 25 {x}^{2} + 120 x - 7425$

$\textcolor{w h i t e}{0} = {\left(5 x + 12\right)}^{2} - 144 - 7425$

$\textcolor{w h i t e}{0} = {\left(5 x + 12\right)}^{2} - 7569$

$\textcolor{w h i t e}{0} = {\left(5 x + 12\right)}^{2} - {87}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(5 x + 12\right) - 87\right) \left(\left(5 x + 12\right) + 87\right)$

$\textcolor{w h i t e}{0} = \left(5 x - 75\right) \left(5 x + 99\right)$

$\textcolor{w h i t e}{0} = 5 \left(x - 15\right) \left(5 x + 99\right)$

Hence $x = 15$ or $x = - \frac{99}{5}$

Discard the negative solution since we are seeking the length of the side of a triangle.

Hence the smallest leg is of length $15$ and the other is $2 \cdot 15 + 6 = 36$

## How do you solve #6x + 9= 39#?

smendyka
Featured 2 months ago

$x = 5$

#### Explanation:

Take the following steps to isolate and solve for $x$ while keeping the equation balanced.

Subtract $9$ from each side of the equation:

$6 x + 9 - 9 = 39 - 9$

$6 x + 0 = 30$

$6 x = 30$

Divide each side of the equation by $6$

$\frac{6 x}{6} = \frac{30}{6}$

$1 x = 5$

$x = 5$

## How do you solve #\sqrt { 2x + 48} = x#?

HSBC244
Featured 1 month ago

$\left\{8\right\}$.

#### Explanation:

Square both sides:

${\left(\sqrt{2 x + 48}\right)}^{2} = {x}^{2}$

$2 x + 48 = {x}^{2}$

$0 = {x}^{2} - 2 x - 48$

$0 = \left(x - 8\right) \left(x + 6\right)$

$x = 8 \mathmr{and} - 6$

Check, as extraneous solutions may have been introduced in the solving process.

#sqrt(2(8) + 48) =^? 8#

#sqrt(64) = 8" "color(green)(√)#

AND

#sqrt(2(-6) + 48) =^? -6#

$\sqrt{36} \ne - 6 \text{ } \textcolor{red}{\times}$

Hopefully this helps!

## Rectangle A, (dimensions 6 by 10-x) has an area twice that of rectangle B (dimensions x by 2x+1) . What are the lengths and widths of both rectangles?

HSBC244
Featured 3 weeks ago

•Rectangle A: 6 by 7
•Rectangle B: 7 by 3

#### Explanation:

The area of a rectangle is given by $\textcolor{red}{A = l \cdot w}$.

The area of rectangle A is $6 \left(10 - x\right) = 60 - 6 x$

The area of rectangle B is $x \left(2 x + 1\right) = 2 {x}^{2} + x$

We are given that the area of rectangle A is twice the area of rectangle B. Therefore, we can write the following equation.

$60 - 6 x = 2 \left(2 {x}^{2} + x\right)$

$60 - 6 x = 4 {x}^{2} + 2 x$

$0 = 4 {x}^{2} + 8 x - 60$

$0 = 4 \left({x}^{2} + 2 x - 15\right)$

$0 = \left(x + 5\right) \left(x - 3\right)$

$x = - 5 \mathmr{and} 3$

A negative answer for $x$ is impossible, since we're talking about geometric shapes.

Therefore, the rectangles have the following measurements:

•Rectangle A: 6 by 7
•Rectangle B: 7 by 3

As you can see, rectangle A's area is twice the area of rectangle B, just as the problem indicated.

Hopefully this helps!

##### Questions
• 5 minutes ago · in Radical Equations
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text:
## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

#### View:

 Message: [ First | Previous | Next | Last ] By Topic: [ First | Previous | Next | Last ] By Author: [ First | Previous | Next | Last ] Font: Proportional Font

Subject:

Re: xparse loads graphics

From:

Mailing list for the LaTeX3 project <[log in to unmask]>

Date:

Thu, 17 Oct 2013 18:03:51 +0100

Content-Type:

text/plain

Parts/Attachments:

 text/plain (28 lines)
 On 17/10/2013 15:19, Alexander Grahn wrote: > Consider the following example: > > \documentclass{article} > \usepackage{xparse} > > \begin{document} > \ifdefined\includegraphics > \typeout{Package graphics' was loaded.} > \fi > \end{document} > > Why das xparse load graphics? What is it used for? > > Alexander As Lars notes, it's expl3 rather than xparse that loads graphics. At present, we don't use out 'native' graphics drivers by default for scaling/rotation/... as part of l3box/l3coffins. As such, we rely on graphics to do the back-end work. There is an open issue on this, suggesting we switch to 'native' as standard with the LaTeX2e drivers as an option. That might be a good idea as it means they get a proper workout and also as clipping is only available with the native drivers. The main downside is that we have fewer drivers than graphics does, so would still need a fall-back for example for tex4ht. -- Joseph Wright`


text:
## CONF notes of the ATLAS Collaboration - Jet/EtMiss group

Full Title Ref Code Publication Date Groups
Measurement of the ATLAS Detector Jet Mass Response using Forward Folding with 80 $\text{fb}^{-1}$ of $√s=13$ TeV $pp$ data. ATLAS-CONF-2020-022 2020/07/20 JETM
Optimisation of large-radius jet reconstruction for the ATLAS detector in 13 TeV proton-proton collisions ATLAS-CONF-2020-021 2020/07/14 JETM
Object-based missing transverse momentum significance in the ATLAS detector ATLAS-CONF-2018-038 2018/07/14 JETM
ETmiss performance in the ATLAS detector using 2015-2016 LHC pp collisions ATLAS-CONF-2018-023 2018/06/21 JETM
Constituent-level pileup mitigation performance using 2015 data ATLAS-CONF-2017-065 2017/07/20 JETM
Performance of Top Quark and W Boson Tagging in Run 2 with ATLAS ATLAS-CONF-2017-064 2017/07/19 JETM
In-situ measurements of large-radius jet reconstruction performance ATLAS-CONF-2017-063 2017/07/19 JETM
Jet reclustering and close-by effects in ATLAS Run 2 ATLAS-CONF-2017-062 2017/07/19 JETM
Boosted Higgs (→ bb) Boson Identification with the ATLAS Detector at $√s$ = 13 TeV ATLAS-CONF-2016-039 2016/07/30 FTAG / JETM
Jet mass reconstruction with the ATLAS Detector in early Run 2 data ATLAS-CONF-2016-035 2016/07/19 JETM
Discrimination of Light Quark and Gluon Jets in pp collisions at √s = 8 TeV with the ATLAS Detector ATLAS-CONF-2016-034 2016/07/18 JETM
Measurement of large radius jet mass reconstruction performance at √s = 8 TeV using the ATLAS detector ATLAS-CONF-2016-008 2016/03/15 JETM
Studies of b-tagging performance and jet substructure in a high pT $g→ b\bar{b}$ rich sample of large-R jets from p p collisions at $√s = 8$ TeV with the ATLAS detector ATLAS-CONF-2016-002 2016/02/29 FTAG / JETM
Determination of the jet energy scale and resolution at ATLAS using Z/γ--jet events in data at √s = 8 TeV ATLAS-CONF-2015-057 2015/10/06 JETM
Monte Carlo Calibration and Combination of In-situ Measurements of Jet Energy Scale, Jet Energy Resolution and Jet Mass in ATLAS ATLAS-CONF-2015-037 2015/08/12 JETM
Identification of high transverse momentum top quarks in pp collisions at $√s=8$ TeV with the ATLAS detector ATLAS-CONF-2015-036 2015/08/11 JETM
Performance of jet substructure techniques in early √s = 13 TeV pp collisions with the ATLAS detector ATLAS-CONF-2015-035 2015/08/10 JETM
Selection of jets produced in 13TeV proton-proton collisions with the ATLAS detector ATLAS-CONF-2015-029 2015/07/23 JETM
Data-driven determination of the energy scale and resolution of jets reconstructed in the ATLAS calorimeters using dijet and multijet events at √s=8 TeV ATLAS-CONF-2015-017 2015/04/13 JETM
Jet energy scale and its uncertainty for jets reconstructed using the ATLAS heavy ion jet algorithm ATLAS-CONF-2015-016 2015/04/09 JETM / HION
Jet global sequential corrections with the ATLAS detector in proton-proton collisions at √s = 8 TeV ATLAS-CONF-2015-002 2015/03/13 JETM
Reconstruction and Modelling of Jet Pull with the ATLAS Detector ATLAS-CONF-2014-048 2014/07/10 JETM
Pile-up Suppression in Missing Transverse Momentum Reconstruction in the ATLAS Experiment in Proton-Proton Collisions at √s = 8 TeV ATLAS-CONF-2014-019 2014/05/16 JETM
Tagging and suppression of pileup jets with the ATLAS detector ATLAS-CONF-2014-018 2014/05/12 JETM
Performance of shower deconstruction in ATLAS ATLAS-CONF-2014-003 2014/02/17 JETM
The ATLAS transverse-momentum trigger performance at the LHC in 2011 ATLAS-CONF-2014-002 2014/02/11 TRIG / JETM
"Performance and Validation of Q-jets at the ATLAS Detector in pp Collisions at √s = 8 TeV in 2012" ATLAS-CONF-2013-087 2013/08/11 JETM
Jet Charge Studies with the ATLAS Detector Using √s = 8 TeV Proton-Proton Collision Data ATLAS-CONF-2013-086 2013/08/09 JETM
Performance of pile-up subtraction for jet shapes in pp collisions at √s=8 TeV ATLAS-CONF-2013-085 2013/08/11 JETM
Performance of boosted top quark identification in 2012 ATLAS data ATLAS-CONF-2013-084 2013/08/11 JETM
Pile-up subtraction and suppression for jets in ATLAS ATLAS-CONF-2013-083 2013/08/10 JETM
Performance of Missing Transverse Momentum Reconstruction in ATLAS studied in Proton-Proton Collisions recorded in 2012 at √s = 8 TeV ATLAS-CONF-2013-082 2013/08/08 JETM
Jet energy scale and its systematic uncertainty in proton-proton collisions at root-s=7 TeV with ATLAS 2011 data ATLAS-CONF-2013-004 2013/01/26 JETM
TeV-scale jet energy calibration using multijet events including close-by jet effects at the ATLAS experiment ATLAS-CONF-2013-003 2013/01/11 JETM
Jet energy measurement for inclusive jets and b-quark induced jets ATLAS-CONF-2013-002 2013/01/11 JETM
Light-quark and gluon jets: calorimeter response, jet energy scale systematics and jet properties ATLAS-CONF-2012-138 2012/09/24 JETM
In situ jet pseudorapidity intercalibration of the ATLAS detector using dijet events in √s=7 TeV proton-proton 2011 data ATLAS-CONF-2012-124 2012/08/27 JETM
Reconstruction and Calibration of Missing Transverse Energy and Performance in Z and W events in ATLAS Proton-Proton Collisions at √s=7 TeV ATLAS-CONF-2012-101 2012/07/25 JETM
Studies of the impact and mitigation of pile-up on large-R and groomed jets in ATLAS at √s=7 TeV ATLAS-CONF-2012-066 2012/06/26 JETM
Performance of large-R jets and jet substructure reconstruction with the ATLAS detector ATLAS-CONF-2012-065 2012/06/26 JETM
Pile-up corrections for jets from proton-proton collisions at sqrts = 7 TeV in ATLAS in 2011 ATLAS-CONF-2012-064 2012/07/04 JETM
Probing the measurement of jet energies with the ATLAS detector using photon+jet events in proton-proton collisions at √s = 7 TeV ATLAS-CONF-2012-063 2012/07/02 JETM
Probing the measurement of jet energies with the ATLAS detector using Z+jet events from proton-proton collisions at √s = 7 TeV ATLAS-CONF-2012-053 2012/05/30 JETM
Selection of jets produced in proton-proton collisions with the ATLAS detector using 2011 data ATLAS-CONF-2012-020 2012/03/07 JETM
Validating the measurement of jet energies with the ATLAS detector using Z+jet events from proton-proton collisions at $√s = 7\TeV$ ATLAS-CONF-2011-159 2011/12/02 JETM
Reconstruction and Calibration of Missing Transverse Energy and Performance in $Z$ and $W$ events in ATLAS Proton-Proton Collisions at $√s$=7 TeV ATLAS-CONF-2011-080 2011/05/26 JETM
Determination of the Atlas jet energy measurement uncertainties using tracks in proton proton collisions at s√=7~TeV ATLAS-CONF-2011-067 2011/05/04 JETM
Close-by Jet Effects on Jet Energy Scale Calibration in pp Collisions at √s = 7 TeV with the ATLAS Detector ATLAS-CONF-2011-062 2011/04/17 JETM
Light-quark and Gluon Jets in ATLAS: Calorimeter Response, Jet Energy Scale Systematics, and Sample Characterization ATLAS-CONF-2011-053 2011/03/30 JETM
Jet energy scale and its systematic uncertainty in proton-proton collisions at √s=7 TeV in ATLAS 2010 data ATLAS-CONF-2011-032 2011/03/22 JETM
Determination of the ATLAS jet energy measurement uncertainty using photon-jet events in proton-proton collisions at √s = 7 TeV ATLAS-CONF-2011-031 2011/03/18 JETM
In-situ jet energy scale and jet shape corrections for multiple interactions in the first ATLAS data at the LHC ATLAS-CONF-2011-030 2011/03/22 JETM
Probing the jet energy measurement at the TeV-scale with the multi-jet balance technique in proton-proton collisions at √s=7 TeV with the ATLAS detector ATLAS-CONF-2011-029 2011/03/16 JETM
ATLAS Calorimeter Response to Single Isolated Hadrons and Estimation of the Calorimeter Jet Scale Uncertainty ATLAS-CONF-2011-028 2011/03/20 JETM
Calorimeter response to isolated particles identified using resonances in proton-proton collisions at $√s$=7~TeV with the ATLAS detector ATLAS-CONF-2011-019 2010/03/10 JETM
In-situ pseudorapidity intercalibration for evaluation of jet energy scale uncertainty using dijet events in proton-proton collisions at √s=7 TeV ATLAS-CONF-2011-014 2011/03/10 JETM
Update on the jet energy scale systematic uncertainty for jets produced in proton-proton collisions at s=7 TeV measured with the ATLAS detector ATLAS-CONF-2011-007 2011/02/16 JETM
Performance of the Missing Transverse Energy Reconstruction and Calibration in Proton-Proton Collisions at a Center-of-Mass Energy of $√s=7$ TeV with the ATLAS Detector ATLAS-CONF-2010-057 2010/07/13 JETM
Jet energy scale and its systematic uncertainty for jets produced in proton-proton collisions at $√s$=7 TeV and measured with the ATLAS detector ATLAS-CONF-2010-056 2010/07/22 JETM
In-situ pseudorapidity intercalibration to evaluate jet energy scale uncertainty and calorimeter performance in the forward region ATLAS-CONF-2010-055 2010/07/19 JETM
Jet energy resolution and reconstruction efficiencies from in-situ techniques with the ATLAS Detector Using Proton-Proton Collisions at a Center of Mass Energy sqrts = 7 TeV ATLAS-CONF-2010-054 2010/07/20 JETM
Properties of Jets and Inputs to Jet Reconstruction and Calibration with the ATLAS Detector Using Proton-Proton Collisions at sqrts(s) = 7 TeV ATLAS-CONF-2010-053 2010/07/17 JETM
ATLAS Calorimeter Response to Single Isolated Hadrons and Estimation of the Calorimeter Jet Scale Uncertainty ATLAS-CONF-2010-052 2010/07/16 JETM
Observation of W→lnu and Z→ll production in proton-proton collisions at √s = 7 TeV with the ATLAS detector ATLAS-CONF-2010-044 2010/06/03 STDM / EGAM / JETM / MUON
Observation of Energetic Jets in pp Collisions at √s = 7 TeV using the ATLAS Experiment at the LHC ATLAS-CONF-2010-043 2010/05/31 STDM / JETM
Performance of the missing transverse energy reconstruction in proton-proton collisions at center-of-mass energy of √s =7 TeV with the ATLAS detector ATLAS-CONF-2010-039 2010/06/04 JETM
Data-Quality Requirements and Event Cleaning for Jets and Missing Transverse Energy Reconstruction with the ATLAS Detector in Proton-Proton Collisions at a Center-of-Mass Energy of √s=7 TeV ATLAS-CONF-2010-038 2010/06/04 JETM
Measurement of the missing transverse momentum based on tracks in proton-proton collisions at $√s=900$~GeV centre-of-mass energy with the ATLAS detector ATLAS-CONF-2010-020 2010/05/05 JETM
Properties and internal structure of jets produced in proton-proton collisions at $√s=900~$GeV ATLAS-CONF-2010-018 2010/04/16 JETM
Response of the ATLAS calorimeters to single isolated hadrons ATLAS-CONF-2010-017 2010/04/05 JETM
Inputs to Jet Reconstruction and Calibration with the ATLAS Detector Using Proton-Proton Collisions at √s=900 GeV ATLAS-CONF-2010-016 2010/04/06 JETM
Etmiss distribution based on the calorimeter ATLAS-CONF-2010-008 2010/03/12 JETM
Reconstruction of jets from tracks in proton-proton collisions at centre-of-mass energy $√s=900$ GeV with the ATLAS detector ATLAS-CONF-2010-002 2010/03/05 JETM
Jet kinematic distributions in proton-proton collisions at $√s=$900~GeV with the ATLAS detector ATLAS-CONF-2010-001 2010/03/05 JETM
Number of ConfNotes: 74


text:
# Homework Assignment Day 15: Falling from high places

Due Thursday, 3/29, in class.  Please complete the following three problems related to today’s joint class on falling objects and air resistance.  Detailed notes for today’s lecture are available here, and they include three problems we’d like you to complete on paper and hand in on Tuesday.  The problems are:

1. Exercise 1.4 pages 4 and 5 — we did this problem in class, but we’d like you to do it one more time with a parachute diameter of $d=3m$.
2. Exercise 1.11a, page 11.
3. Exercise 1.12, page 12.

If you have any questions, please post them here.

-Mr. Reitz

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