text:
# Constraining the $\Lambda$CDM and Galileon models with recent cosmological data

Abstract : The Galileon theory belongs to the class of modified gravity models that can explain the late-time accelerated expansion of the Universe. In previous works, cosmological constraints on the Galileon model were derived, both in the uncoupled case and with a disformal coupling of the Galileon field to matter. There, we showed that these models agree with the most recent cosmological data. In this work, we used updated cosmological data sets to derive new constraints on Galileon models, including the case of a constant conformal Galileon coupling to matter. We also explored the tracker solution of the uncoupled Galileon model. After updating our data sets, especially with the latest \textit{Planck} data and BAO measurements, we fitted the cosmological parameters of the $\Lambda$CDM and Galileon models. The same analysis framework as in our previous papers was used to derive cosmological constraints, using precise measurements of cosmological distances and of the cosmic structure growth rate. We showed that all tested Galileon models are as compatible with cosmological data as the $\Lambda$CDM model. This means that present cosmological data are not accurate enough to distinguish firmly between both theories. Among the different Galileon models, we found that a conformal coupling is not favoured, contrary to the disformal coupling which is preferred at the $2.3\sigma$ level over the uncoupled case. The tracker solution of the uncoupled Galileon model is also highly disfavoured due to large tensions with supernovae and \textit{Planck}+BAO data. However, outside of the tracker solution, the general uncoupled Galileon model, as well as the general disformally coupled Galileon model, remain the most promising Galileon scenarios to confront with future cosmological data. Finally, we also discuss constraints coming from Lunar Laser Ranging experiment and gravitational wave speed of propagation.
Document type :
Preprints, Working Papers, ...
Domain :

http://hal.in2p3.fr/in2p3-01313580
Contributor : Sabine Starita <>
Submitted on : Tuesday, May 10, 2016 - 10:24:52 AM
Last modification on : Thursday, January 11, 2018 - 6:14:22 AM

### Identifiers

• HAL Id : in2p3-01313580, version 1
• ARXIV : 1605.02627

### Citation

J. Neveu, V. Ruhlmann-Kleider, P. Astier, M. Besançon, J. Guy, et al.. Constraining the $\Lambda$CDM and Galileon models with recent cosmological data. 2016. ⟨in2p3-01313580⟩

Record views

format: plain


text:
# Honours Seminar

## Date & time

1–2pm 28 April 2017

## Location

John Dedman Building 27, Room G35

## Speakers

Edmund Heng and Stephen Deng (MSI/ANU)

## Contacts

Bai-Ling Wang

Algebraic objects are often abstract and intuitions have mostly came from experience. The Category of Affine Schemes over $K$, being anti-equivalent to the Category of $K-algebra$ provides us a way of looking at abstract algebraic objects - Algebras, as concrete Geometrical objects - Affine Schemes. Pushing the same idea a little bit further, we can similarly look into a more specific set of Geometrical objects - affine group schemes, which corresponds to the algebraic object - Hopf Algebra.

In this talk I'll first introduce the idea of an Affine scheme through the functorial point of view and show how continuing on this idea allows us to view a more specific type of Algebra - Hopf Algebras, as Geometrical objects too. To save time and since Sam had talked about Categories, I'll assume we all know what it is and skip defining it.

Updated:  18 August 2017/Responsible Officer:  Director/Page Contact:  School Manager


text:
# Small touches that inspire

It's the littlest of things that can really brighten my mood when I notice them. In this case I was watching Fallout: New Vegas' DLC trailer for Honest Hearts. In the trailer you see the player with a pistol and on one side of the pistol at least is written:

καὶ ἡ σκοτία αὐτὸ οὐ κατέλαβεν

This is Greek and refers to the second part of the verse of John 1:5 in the New Testament of the bible, meaning in English: "and the darkness did not comprehend it". In my opinion a great way to bring enlightenment by the bullet.


text:
# volume inside sphere but outside hyperboloid

I am trying to find the volume inside the sphere $x^2 + y^2 + z^2 = 9$, but outside the hyperboloid $x^2 + y^2 - z^2 = 1$. by using a triple integral. for some reason i just cant seem to come up the bounds of integration for this problem. To be more precise, its the region lying to the side of the hyperboloid, that wraps around it, creating a sort of donut shape.

• Hint: Although it's rarely advisable, try integrating in the order $d\theta dr dz$. – Ted Shifrin Mar 17 '14 at 0:09

Just a picture, not an answer.

• pretty picture, doesn't solve the problem though. – Guy Mar 17 '14 at 7:26
• I get $64 \pi /3$ by another method, rotating the region produced by the intersection of hyperbola and circle around the axis . – Alan Mar 17 '14 at 23:45
• I checked Alan's computations:$$\int_{-2}^2 2\pi(4-z^2)dz=64\pi/3$$ -check! – LeoTheKub Mar 18 '14 at 1:27
• Could you comment what I am doing wrong?. You seem to make sense. In my method, Could you let me know how the volume of the sphere above the hyperboloid is found? – Satish Ramanathan Mar 18 '14 at 1:44
• Somewhere in my calculation, I hit your answer. But I do not know where I got lost – Satish Ramanathan Mar 18 '14 at 1:45

Hint:

The volume inside sphere but outside hyperboloid can be seen as the volume of hyperboloid subtracted from volume of the sphere. In the cartesian coordinates, the volume of the hyperboloid could be written as the below iterated integrals.

Equating both conics such as $$z^2 = 9 - x^2 -y^2 = x^2 + Y^2 -1 => 2(x^2 + y^2) = 10$$ $$=> x^2 + y^2 = 5, implies z^2 = 9-5 = 4 => z = +2, -2$$. Let us slice the solid in the xy plane and we get the limits of z is (-2,2). Each slice is the disk enclosed by a circle $x^2 + y^2 = z^2 + 1$, which is the circle of radius $\sqrt{z^2 + 1}$. Now finding the limits of x and y, we slice the hyperboloid in the vertical direction and this amounts to slicing $$[-\sqrt{z^2 + 1},\sqrt{z^2 + 1}]$$ on the x-axis. Along each slice, y goes from bottom of the circle $y = -\sqrt{z^2 + 1 - x^2}$ to the top of the circle $y = \sqrt{z^2 + 1 - x^2}$.

Putting this altogether, the volume of the hyperboloid =

$$V_{hyperboloid} = \int_{-2}^{2} \int_{-\sqrt{z^2 + 1}}^{\sqrt{z^2 + 1}} \int_{-\sqrt{z^2 + 1 - x^2}}^{\sqrt{z^2 + 1 - x^2}} dydxdz$$

Now the volume of the sphere with a radius of 3 $$V_{sphere} = \frac{4}{3}\pi 3^3$$

Thus the volume inside sphere but outside hyperboloid $$= V_{sphere} - V_{hyperboloid} - V=36\pi - \frac{28}{3}\pi - V = \frac{80}{3}\pi - V$$ where V is the volume of shphere around z = 2 to 3. The volume of which is

$$\int_{0}^{2\pi} \int_{2.236}^{3} (\sqrt{9-r^2})rdrd\theta = \frac{16\pi}{3}=16.756$$

Thus the required volume$= \frac{80}{3}\pi - \frac{16\pi}{3} =\frac{64\pi}{3}$

• Are you sure? This is a hyperboloid of one sheet. – Ted Shifrin Mar 17 '14 at 0:07
• its supposed to be the donut shape that wraps around the hyperboloid, inside the sphere. – user2253455 Mar 17 '14 at 0:20
• @user2253455, Atlast, I found the correct setup. See if it is helpful from the picture that is drwan by the other responder. Goodluck – Satish Ramanathan Mar 17 '14 at 22:00


text:
## Mean-Field Variational Inference Made Easy

I had the hardest time trying to understand variational inference. All of the presentations I’ve seen (MacKay, Bishop, Wikipedia, Gelman’s draft for the third edition of Bayesian Data Analysis) are deeply tied up with the details of a particular model being fit. I wanted to see the algorithm and get the big picture before being overwhelmed with multivariate exponential family gymnastics.

### Bayesian Posterior Inference

In the Bayesian setting (see my earlier post, What is Bayesian Inference?), we have a joint probability model $p(y,\theta)$ for data $y$ and parameters $\theta$, usually factored as the product of a likelihood and prior term, $p(y,\theta) = p(y|\theta) p(\theta)$. Given some observed data $y$, Bayesian predictive inference is based on the posterior density $p(\theta|y) \propto p(\theta, y)$ of the the unknown parameter vector $\theta$ given observed data vector $y$. Thus we need to be able to estimate the posterior density $p(\theta|y)$ to carry out Bayesian inference. Note that the posterior is a whole density function—we’re not just after a point estimate as in maximum likelihood estimation.

### Mean-Field Approximation

Variational inference approximates the Bayesian posterior density $p(\theta|y)$ with a (simpler) density $g(\theta|\phi)$ parameterized by some new parameters $\phi$. The mean-field form of variational inference factors the approximating density $g$ by component of $\theta = \theta_1,\ldots,\theta_J$, as

$g(\theta|\phi) = \prod_{j=1}^J g_j(\theta_j|\phi_j)$.

I’m going to put off actually defining the terms $g_j$ until we see how they’re used in the variational inference algorithm.

### What Variational Inference Does

The variational inference algorithm finds the value $\phi^*$ for the parameters $\phi$ of the approximation which minimizes the Kullback-Leibler divergence of $g(\theta|\phi)$ from $p(\theta|y)$,

$\phi^* = \mbox{arg min}_{\phi} \ \mbox{KL}[ g(\theta|\phi) \ || \ p(\theta|y) ]$.

The key idea here is that variational inference reduces posterior estimation to an optimization problem. Optimization is typically much faster than approaches to posterior estimation such as Markov chain Monte Carlo (MCMC).

The main disadvantage of variational inference is that the posterior is only approximated (though as MacKay points out, just about any approximation is better than a delta function at a point estimate!). In particular, variational methods systematically underestimate posterior variance because of the direction of the KL divergence that is minimized. Expectation propagation (EP) also converts posterior fitting to optimization of KL divergence, but EP uses the opposite direction of KL divergence, which leads to overestimation of posterior variance.

### Variational Inference Algorithm

Given the Bayesian model $p(y,\theta)$, observed data $y$, and functional terms $g_j$ making up the approximation of the posterior $p(\theta|y)$, the variational inference algorithm is:

• $\phi \leftarrow \mbox{random legal initialization}$
• $\mbox{repeat}$
• $\phi_{\mbox{\footnotesize old}} \leftarrow \phi$
• $\mbox{for } j \mbox{ in } 1:J$
• $\mbox{{} \ \ set } \phi_j \mbox{ such that } g_j(\theta_j|\phi_j) = \mathbb{E}_{g_{-j}}[\log p(\theta|y)]$.
• $\mbox{until } ||\phi - \phi_{\mbox{\footnotesize old}}|| < \epsilon$

The inner expectation is a function of $\theta_j$ returning a single non-negative value, defined by

$\mathbb{E}_{g_{-j}}[\log p(\theta|y)]$

$\begin{array}{l} \mbox{ } = \int_{\theta_1} \ldots \int_{\theta_{j-1}} \int_{\theta_{j+1}} \ldots \int_{\theta_J} \\[8pt] \mbox{ } \hspace*{0.4in} g(\theta_1|\phi_1) \times \cdots \times g(\theta_{j-1}|\phi_{j-1}) \times g(\theta_{j+1}|\phi_{j+1}) \times \cdots \times g(\theta_J|\phi_J) \times \log p(\theta|y) \\[8pt] \mbox{ } \hspace*{0.2in} d\theta_J \cdots d\theta_{j+1} \ d\theta_{j-1} \cdots d\theta_1 \end{array}$

Despite the suggestive factorization of $g$ and the coordinate-wise nature of the algorithm, variational inference does not simply approximate the posterior marginals $p(\theta_j|y)$ independently.

### Defining the Approximating Densities

The trick is to choose the approximating factors so that the we can compute parameter values $\phi_j$ such that $g_j(\theta_j|\phi_j) = \mathbb{E}_{g_{-j}}[\log p(\theta|y)]$. Finding such approximating terms $g_j(\theta_j|\phi_j)$ given a posterior $p(\theta|y)$ is an art form unto itself. It’s much easier for models with conjugate priors. Bishop or MacKay’s books and the Wikipedia present calculations for a wide range of exponential-family models.

### What if My Model is not Conjugate?

Unfortunately, I almost never work with conjugate priors (and even if I did, I’m not much of a hand at exponential-family algebra). Therefore, the following paper just got bumped to the top of my must understand queue:

It’s great having Dave down the hall on sabbatical this year — one couldn’t ask for a better stand in for Matt Hoffman. They are both insanely good at on-the-fly explanations at the blackboard (I love that we still have real chalk and high quality boards).

### 7 Responses to “Mean-Field Variational Inference Made Easy”

1. Eric Says:

Thanks, Bob. Nice overview. Would it make any sense to try to minimize some symmetric divergence, such as the Jensen-Shannon divergence?

• Bob Carpenter Says:

It certainly makes sense, but the issue is whether you can come up with some computable way to do it. The trick to variational inference is that the hairy integral involved in the componentwise hill climbing can be solved for conjugate priors and approximated elsewhere.

2. michael Says:

I am actually not a fun of any talor expansion based approximation methods including the reference above. They simly break the lower bound condition (except first order approximation).
Actually, there is another paper written by some Japanese researcher that I cannot recall the title. They show that mean field (including non conjugate models) can see from the dual problem, is to minimize bregman divergence block coordinate wise.

3. Aki Vehtari Says:

With power-EP method it is possible to use alpha-divergence which includes both KL divergences and symmetric Hellinger distance as special cases.

4. brendan o'connor (@brendan642) Says:

i thought the presentation in the Koller and Friedman textbook was helpful for variational inference — a bit less tied up with a particular model if i remember right. The Murphy textbook also has a general derivation for mean field, then references out to other chapters for particular instantiations (section 21.3)

5. Rob Says:

Is there any chance you could work through an example, such as the beta-binomial model that you covered in an earlier posting?


text:
# Temporally localized signals¶

## Introduction¶

A case frequently encountered in practice is a stationary signal with an event localized in time.
When the noise properties are understood, and the expected shape of the signal is known, a tool of choice is full forward modeling. That is, here too the analysis includes model selection and parameter estimation steps, and is often called a matched filter search.

In this notebook, we will discuss two simple parametric models:

1. a burst signal

2. a chirp signal

In both examples we assume Gaussian known errors. The generalization to nonparametric models and more complex models can be relatively easily implemented by modifying the code developed for these two examples.

## Import functions¶

We use functions from PyMC3. We will also use FT_continuous, IFT_continuous, and wavelet_PSD from astroML.fourier for wavelet PSD analysis.

import numpy as np
from matplotlib import pyplot as plt

import pymc3 as pm

from astroML.plotting.mcmc import plot_mcmc

from astroML.fourier import FT_continuous, IFT_continuous, wavelet_PSD

# Matched Filter Chirp Search¶

A more complex ten-parameter case of chirp modeling is show in this matched filter chirp search example. The chirp signal is temporally localized and it decays exponentially for t > T:

$y_C(t|T, A, \phi, \omega, \beta) = A sin[\phi + \omega(t - T) + \beta(t - T)^2] exp[-\alpha(t - T)]$

## 1. set up a chirp model example¶

Using equation $$y(t) = b_0 + Asin[\omega t + \beta t^2])$$, we set up a chirp model as an example to show the matched filter search.
The chirp data set is shown in the plot below.

# Set up toy dataset
def chirp(t, T, A, phi, omega, beta):
"""chirp signal"""
signal = mask * (A * np.sin(phi + omega * (t - T) + beta * (t - T) ** 2))
return signal

def background(t, b0, b1, omega1, omega2):
"""background signal"""
return b0 + b1 * np.sin(omega1 * t) * np.sin(omega2 * t)

np.random.seed(0)
N = 500
T_true = 30
A_true = 0.8
phi_true = np.pi / 2
omega_true = 0.1
beta_true = 0.02
b0_true = 0.5
b1_true = 1.0
omega1_true = 0.3
omega2_true = 0.4
sigma = 0.1

t = 100 * np.random.random(N)

signal = chirp(t, T_true, A_true, phi_true, omega_true, beta_true)
bg = background(t, b0_true, b1_true, omega1_true, omega2_true)

y_true = signal + bg

y_obs = np.random.normal(y_true, sigma)

t_fit = np.linspace(0, 100, 1000)
y_fit = (chirp(t_fit, T_true, A_true, phi_true, omega_true, beta_true) +
background(t_fit, b0_true, b1_true, omega1_true, omega2_true))

# show the model
fig = plt.figure(figsize=(9, 9))
ax = fig.add_axes([0.52, 0.7, 0.43, 0.25])
ax.scatter(t, y_obs, s=9, lw=0, c='k')
ax.set_xlim(0, 100)
ax.set_xlabel('$t$')
ax.set_ylabel(r'$h_{\rm obs}$')

## 2. Set up and run the MCMC sampling¶

We use MCMC to compute the posterior parameters in the equation, in order to fit this chirp model. The parameters we estimate using MCMC are $$T$$, $$A$$, $$\omega$$, and $$\beta$$.

# Set up and run MCMC sampling

with pm.Model():

T = pm.Uniform('T', 0, 100, testval=T_true)
A = pm.Uniform('A', 0, 100, testval=A_true)
phi = pm.Uniform('phi', -np.pi, np.pi, testval=phi_true)

b0 = pm.Uniform('b0', 0, 100, testval=b0_true)
b1 = pm.Uniform('b1', 0, 100, testval=b1_true)

log_omega1 = pm.Uniform('log_omega1', -3, 0, testval=np.log(omega1_true))
log_omega2 = pm.Uniform('log_omega2', -3, 0, testval=np.log(omega2_true))

omega = pm.Uniform('omega', 0.001, 1, testval=omega_true)
beta = pm.Uniform('beta', 0.001, 1, testval=beta_true)

y = pm.Normal('y', mu=(chirp(t, T, A, phi, omega, beta)
+ background(t, b0, b1, np.exp(log_omega1), np.exp(log_omega2))),
tau=sigma ** -2, observed=y_obs)

step = pm.Metropolis()

traces = pm.sample(draws=5000, tune=2000, step=step, return_inferencedata=False)

labels = ['$T$', '$A$', r'$\omega$', r'$\beta$']
limits = [(29.75, 30.25), (0.75, 0.83), (0.085, 0.115), (0.0197, 0.0202)]
true = [T_true, A_true, omega_true, beta_true]

## 3. Show estimated model¶

A matched filter search for a chirp signal in time series data.

• The top-right panel shows a simulated data set generated from the chirp model.

• The posterior pdf for the four model parameters ($$T$$, $$A$$, $$\omega$$, and $$\beta$$) is determined using MCMC and shown in the other panels.

• Seven of the parameters can be considered nuisance parameters, and we marginalize over them in the likelihood contours shown here.

# Plot results
fig = plt.figure(figsize=(9, 9))

# This function plots multiple panels with the traces
axes_list = plot_mcmc([traces[i] for i in ['T', 'A', 'omega', 'beta']],
labels=labels, limits=limits,
true_values=true, fig=fig,
bins=30, colors='k',
bounds=[0.14, 0.08, 0.95, 0.95])

for ax in axes_list:
for axis in [ax.xaxis, ax.yaxis]:
axis.set_major_locator(plt.MaxNLocator(5))

ax = fig.add_axes([0.52, 0.7, 0.43, 0.25])
ax.scatter(t, y_obs, s=9, lw=0, c='k')
ax.plot(t_fit, y_fit, '-k')
ax.set_xlim(0, 100)
ax.set_xlabel('$t$')
ax.set_ylabel(r'$h_{\rm obs}$')

## Chirp wavelet PSD¶

The signal in the absence of chirp is taken as

$y(t) = b_0 + b_1 sin(\omega_1 t) sin(\omega_2 t)$

Here, we can consider parameters $$A$$, $$\omega$$, $$\beta$$, and $$\alpha$$ as “interesting,” and other parameters can be treated as “nuisance.”
We will take the same chirp signal above and draw a wavelet PSD for demonstration.

### 1. compute the wavelet PSD of chirp model¶

We use the previous complex ten-parameter case of chirp modeling, and compute its wavelet PSD.

np.random.seed(42)
N = 4096
t = np.linspace(-50, 50, N)
h_true = chirp(t, -20, 0.8, 0, 0.2, 0.02)
h = h_true + np.random.normal(0, 1, N)

f0 = np.linspace(0.04, 0.6, 100)

# compute wavelet PSD
wPSD = wavelet_PSD(t, h, f0, Q=1.0)

### 2. plot signal and wavelet¶

Here, the signal with an amplitude of A = 0.8 is sampled in 4096 evenly spaced bins, and with Gaussian noise with sigma = 1. The two-dimensional wavelet PSD easily recovers the increase of characteristic chirp frequency with time.

# Plot the  results
fig = plt.figure(figsize=(5, 3.75))

# Top: plot the data
ax.plot(t + 50, h, '-', c='#AAAAAA')
ax.plot(t + 50, h_true, '-k')

ax.text(0.02, 0.95, "Input Signal: chirp",
ha='left', va='top', transform=ax.transAxes,
bbox=dict(boxstyle='round', fc='w', ec='k'))

ax.set_xlim(0, 100)
ax.set_ylim(-2.9, 2.9)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_ylabel('$h(t)$')

# Bottom: plot the 2D PSD
ax.imshow(wPSD, origin='lower', aspect='auto',
extent=[t[0] + 50, t[-1] + 50, f0[0], f0[-1]],
cmap=plt.cm.binary)

ax.text(0.02, 0.95, ("Wavelet PSD"), color='w',
ha='left', va='top', transform=ax.transAxes)

ax.set_xlim(0, 100)
ax.set_ylim(0.04, 0.6001)
ax.set_xlabel('$t$')
ax.set_ylabel('$f_0$')

plt.show()


text:
## 53.12 Riemann-Hurwitz

Let $k$ be a field. Let $f : X \to Y$ be a morphism of smooth curves over $k$. Then we obtain a canonical exact sequence

$f^*\Omega _{Y/k} \xrightarrow {\text{d}f} \Omega _{X/k} \longrightarrow \Omega _{X/Y} \longrightarrow 0$

by Morphisms, Lemma 29.32.9. Since $X$ and $Y$ are smooth, the sheaves $\Omega _{X/k}$ and $\Omega _{Y/k}$ are invertible modules, see Morphisms, Lemma 29.34.12. Assume the first map is nonzero, i.e., assume $f$ is generically étale, see Lemma 53.12.1. Let $R \subset X$ be the closed subscheme cut out by the different $\mathfrak {D}_ f$ of $f$. By Discriminants, Lemma 49.12.6 this is the same as the vanishing locus of $\text{d}f$, it is an effective Cartier divisor, and we get

$f^*\Omega _{Y/k} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(R) = \Omega _{X/k}$

In particular, if $X$, $Y$ are projective with $k = H^0(Y, \mathcal{O}_ Y) = H^0(X, \mathcal{O}_ X)$ and $X$, $Y$ have genus $g_ X$, $g_ Y$, then we get the Riemann-Hurwitz formula

\begin{align*} 2g_ X - 2 & = \deg (\Omega _{X/k}) \\ & = \deg (f^*\Omega _{Y/k} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(R)) \\ & = \deg (f) \deg (\Omega _{Y/k}) + \deg (R) \\ & = \deg (f) (2g_ Y - 2) + \deg (R) \end{align*}

The first and last equality by Lemma 53.8.4. The second equality by the isomorphism of invertible sheaves given above. The third equality by additivity of degrees (Varieties, Lemma 33.44.7), the formula for the degree of a pullback (Varieties, Lemma 33.44.11), and finally the formula for the degree of $\mathcal{O}_ X(R)$ (Varieties, Lemma 33.44.9).

To use the Riemann-Hurwitz formula we need to compute $\deg (R) = \dim _ k \Gamma (R, \mathcal{O}_ R)$. By the structure of zero dimensional schemes over $k$ (see for example Varieties, Lemma 33.20.2), we see that $R$ is a finite disjoint union of spectra of Artinian local rings $R = \coprod _{x \in R} \mathop{\mathrm{Spec}}(\mathcal{O}_{R, x})$ with each $\mathcal{O}_{R, x}$ of finite dimension over $k$. Thus

$\deg (R) = \sum \nolimits _{x \in R} \dim _ k \mathcal{O}_{R, x} = \sum \nolimits _{x \in R} d_ x [\kappa (x) : k]$

with

$d_ x = \text{length}_{\mathcal{O}_{R, x}} \mathcal{O}_{R, x} = \text{length}_{\mathcal{O}_{X, x}} \mathcal{O}_{R, x}$

the multiplicity of $x$ in $R$ (see Algebra, Lemma 10.52.12). Let $x \in X$ be a closed point with image $y \in Y$. Looking at stalks we obtain an exact sequence

$\Omega _{Y/k, y} \to \Omega _{X/k, x} \to \Omega _{X/Y, x} \to 0$

Choosing local generators $\eta _ x$ and $\eta _ y$ of the (free rank $1$) modules $\Omega _{X/k, x}$ and $\Omega _{Y/k, y}$ we see that $\eta _ y \mapsto h \eta _ x$ for some nonzero $h \in \mathcal{O}_{X, x}$. By the exact sequence we see that $\Omega _{X/Y, x} \cong \mathcal{O}_{X, x}/h\mathcal{O}_{X, x}$ as $\mathcal{O}_{X, x}$-modules. Since the divisor $R$ is cut out by $h$ (see above) we have $\mathcal{O}_{R, x} = \mathcal{O}_{X, x}/h\mathcal{O}_{X, x}$. Thus we find the following equalities

\begin{align*} d_ x & = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{R, x}) \\ & = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{X, x}/h\mathcal{O}_{X, x}) \\ & = \text{length}_{\mathcal{O}_{X, x}}(\Omega _{X/Y, x}) \\ & = \text{ord}_{\mathcal{O}_{X, x}}(h) \\ & = \text{ord}_{\mathcal{O}_{X, x}}(“\eta _ y/\eta _ x") \end{align*}

The first equality by our definition of $d_ x$. The second and third we saw above. The fourth equality is the definition of $\text{ord}$, see Algebra, Definition 10.121.2. Note that since $\mathcal{O}_{X, x}$ is a discrete valuation ring, the integer $\text{ord}_{\mathcal{O}_{X, x}}(h)$ just the valuation of $h$. The fifth equality is a mnemonic.

Here is a case where one can “calculate” the multiplicity $d_ x$ in terms of other invariants. Namely, if $\kappa (x)$ is separable over $k$, then we may choose $\eta _ x = \text{d}s$ and $\eta _ y = \text{d}t$ where $s$ and $t$ are uniformizers in $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ (Lemma 53.12.3). Then $t \mapsto u s^{e_ x}$ for some unit $u \in \mathcal{O}_{X, x}$ where $e_ x$ is the ramification index of the extension $\mathcal{O}_{Y, y} \subset \mathcal{O}_{X, x}$. Hence we get

$\eta _ y = \text{d}t = \text{d}(u s^{e_ x}) = e s^{e_ x - 1} u \text{d}s + s^{e_ x} \text{d}u$

Writing $\text{d}u = w \text{d}s$ for some $w \in \mathcal{O}_{X, x}$ we see that

$\eta _ y/\eta _ x" = e s^{e_ x - 1} u + s^{e_ x} w = (e_ x u + s w)s^{e_ x - 1}$

We conclude that the order of vanishing of this is $e_ x - 1$ unless the characteristic of $\kappa (x)$ is $p > 0$ and $p$ divides $e_ x$ in which case the order of vanishing is $> e_ x - 1$.

Combining all of the above we find that if $k$ has characteristic zero, then

$2g_ X - 2 = (2g_ Y - 2)\deg (f) + \sum \nolimits _{x \in X} (e_ x - 1)[\kappa (x) : k]$

where $e_ x$ is the ramification index of $\mathcal{O}_{X, x}$ over $\mathcal{O}_{Y, f(x)}$. This precise formula will hold if and only if all the ramification is tame, i.e., when the residue field extensions $\kappa (x)/\kappa (y)$ are separable and $e_ x$ is prime to the characteristic of $k$, although the arguments above are insufficient to prove this. We refer the reader to Lemma 53.12.4 and its proof.

Lemma 53.12.1. Let $k$ be a field. Let $f : X \to Y$ be a morphism of smooth curves over $k$. The following are equivalent

1. $\text{d}f : f^*\Omega _{Y/k} \to \Omega _{X/k}$ is nonzero,

2. $\Omega _{X/Y}$ is supported on a proper closed subset of $X$,

3. there exists a nonempty open $U \subset X$ such that $f|_ U : U \to Y$ is unramified,

4. there exists a nonempty open $U \subset X$ such that $f|_ U : U \to Y$ is étale,

5. the extension $k(X)/k(Y)$ of function fields is finite separable.

Proof. Since $X$ and $Y$ are smooth, the sheaves $\Omega _{X/k}$ and $\Omega _{Y/k}$ are invertible modules, see Morphisms, Lemma 29.34.12. Using the exact sequence

$f^*\Omega _{Y/k} \longrightarrow \Omega _{X/k} \longrightarrow \Omega _{X/Y} \longrightarrow 0$

of Morphisms, Lemma 29.32.9 we see that (1) and (2) are equivalent and equivalent to the condition that $f^*\Omega _{Y/k} \to \Omega _{X/k}$ is nonzero in the generic point. The equivalence of (2) and (3) follows from Morphisms, Lemma 29.35.2. The equivalence between (3) and (4) follows from Morphisms, Lemma 29.36.16 and the fact that flatness is automatic (Lemma 53.2.3). To see the equivalence of (5) and (4) use Algebra, Lemma 10.140.9. Some details omitted. $\square$

Lemma 53.12.2. Let $f : X \to Y$ be a morphism of smooth proper curves over a field $k$ which satisfies the equivalent conditions of Lemma 53.12.1. If $k = H^0(Y, \mathcal{O}_ Y) = H^0(X, \mathcal{O}_ X)$ and $X$ and $Y$ have genus $g_ X$ and $g_ Y$, then

$2g_ X - 2 = (2g_ Y - 2) \deg (f) + \deg (R)$

where $R \subset X$ is the effective Cartier divisor cut out by the different of $f$.

Lemma 53.12.3. Let $X \to \mathop{\mathrm{Spec}}(k)$ be smooth of relative dimension $1$ at a closed point $x \in X$. If $\kappa (x)$ is separable over $k$, then for any uniformizer $s$ in the discrete valuation ring $\mathcal{O}_{X, x}$ the element $\text{d}s$ freely generates $\Omega _{X/k, x}$ over $\mathcal{O}_{X, x}$.

Proof. The ring $\mathcal{O}_{X, x}$ is a discrete valuation ring by Algebra, Lemma 10.140.3. Since $x$ is closed $\kappa (x)$ is finite over $k$. Hence if $\kappa (x)/k$ is separable, then any uniformizer $s$ maps to a nonzero element of $\Omega _{X/k, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)$ by Algebra, Lemma 10.140.4. Since $\Omega _{X/k, x}$ is free of rank $1$ over $\mathcal{O}_{X, x}$ the result follows. $\square$

Lemma 53.12.4. Notation and assumptions as in Lemma 53.12.2. For a closed point $x \in X$ let $d_ x$ be the multiplicity of $x$ in $R$. Then

$2g_ X - 2 = (2g_ Y - 2) \deg (f) + \sum \nolimits d_ x [\kappa (x) : k]$

Moreover, we have the following results

1. $d_ x = \text{length}_{\mathcal{O}_{X, x}}(\Omega _{X/Y, x})$,

2. $d_ x \geq e_ x - 1$ where $e_ x$ is the ramification index of $\mathcal{O}_{X, x}$ over $\mathcal{O}_{Y, y}$,

3. $d_ x = e_ x - 1$ if and only if $\mathcal{O}_{X, x}$ is tamely ramified over $\mathcal{O}_{Y, y}$.

Proof. By Lemma 53.12.2 and the discussion above (which used Varieties, Lemma 33.20.2 and Algebra, Lemma 10.52.12) it suffices to prove the results on the multiplicity $d_ x$ of $x$ in $R$. Part (1) was proved in the discussion above. In the discussion above we proved (2) and (3) only in the case where $\kappa (x)$ is separable over $k$. In the rest of the proof we give a uniform treatment of (2) and (3) using material on differents of quasi-finite Gorenstein morphisms.

First, observe that $f$ is a quasi-finite Gorenstein morphism. This is true for example because $f$ is a flat quasi-finite morphism and $X$ is Gorenstein (see Duality for Schemes, Lemma 48.25.7) or because it was shown in the proof of Discriminants, Lemma 49.12.6 (which we used above). Thus $\omega _{X/Y}$ is invertible by Discriminants, Lemma 49.16.1 and the same remains true after replacing $X$ by opens and after performing a base change by some $Y' \to Y$. We will use this below without further mention.

Choose affine opens $U \subset X$ and $V \subset Y$ such that $x \in U$, $y \in V$, $f(U) \subset V$, and $x$ is the only point of $U$ lying over $y$. Write $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(B)$. Then $R \cap U$ is the different of $f|_ U : U \to V$. By Discriminants, Lemma 49.9.4 formation of the different commutes with arbitrary base change in our case. By our choice of $U$ and $V$ we have

$A \otimes _ B \kappa (y) = \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, y}} \kappa (y) = \mathcal{O}_{X, x}/(s^{e_ x})$

where $e_ x$ is the ramification index as in the statement of the lemma. Let $C = \mathcal{O}_{X, x}/(s^{e_ x})$ viewed as a finite algebra over $\kappa (y)$. Let $\mathfrak {D}_{C/\kappa (y)}$ be the different of $C$ over $\kappa (y)$ in the sense of Discriminants, Definition 49.9.1. It suffices to show: $\mathfrak {D}_{C/\kappa (y)}$ is nonzero if and only if the extension $\mathcal{O}_{Y, y} \subset \mathcal{O}_{X, x}$ is tamely ramified and in the tamely ramified case $\mathfrak {D}_{C/\kappa (y)}$ is equal to the ideal generated by $s^{e_ x - 1}$ in $C$. Recall that tame ramification means exactly that $\kappa (x)/\kappa (y)$ is separable and that the characteristic of $\kappa (y)$ does not divide $e_ x$. On the other hand, the different of $C/\kappa (y)$ is nonzero if and only if $\tau _{C/\kappa (y)} \in \omega _{C/\kappa (y)}$ is nonzero. Namely, since $\omega _{C/\kappa (y)}$ is an invertible $C$-module (as the base change of $\omega _{A/B}$) it is free of rank $1$, say with generator $\lambda$. Write $\tau _{C/\kappa (y)} = h\lambda$ for some $h \in C$. Then $\mathfrak {D}_{C/\kappa (y)} = (h) \subset C$ whence the claim. By Discriminants, Lemma 49.4.8 we have $\tau _{C/\kappa (y)} \not= 0$ if and only if $\kappa (x)/\kappa (y)$ is separable and $e_ x$ is prime to the characteristic. Finally, even if $\tau _{C/\kappa (y)}$ is nonzero, then it is still the case that $s \tau _{C/\kappa (y)} = 0$ because $s\tau _{C/\kappa (y)} : C \to \kappa (y)$ sends $c$ to the trace of the nilpotent operator $sc$ which is zero. Hence $sh = 0$, hence $h \in (s^{e_ x - 1})$ which proves that $\mathfrak {D}_{C/\kappa (y)} \subset (s^{e_ x - 1})$ always. Since $(s^{e_ x - 1}) \subset C$ is the smallest nonzero ideal, we have proved the final assertion. $\square$

Comment #5012 by Elyes Boughattas on

Typo in the Riemann-Hurwitz formula preceding lemma 0C1C: $g_x$ should be $g_X$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Power System Blocks

## Sensitivity computation

The sensitivity computation module is dedicated to compute linearized impact of network small variations on some elements state variables.

A sensitivity value is the numerical estimation of the partial derivative between the variable of impact and the function observed. Sensitivity computation can also be seen as partial derivatives computation on the network model.

# Sensitivity computation inputs

## Network

The main input for sensitivity computation module is an IIDM network.

## Sensitivity factors

Sensitivity factors are the expected partial derivatives to be extracted from the input network.

A standard sensitivity computation input is composed of a list of sensitivity factors, each one composed of a sensitivity variable (variable of impact) and a sensitivity function (function observed).

Currently available sensitivity factors are:

• BranchFlowPerInjectionIncrease : calculates the linear impact of a specific injection increase on a specific branch’s active flow (in MW/MW)
• BranchFlowPerLinearGlsk : calculates the linear impact of a linear combination of injections (GLSK) increase on a specific branch’s active flow (in MW/MW)
• BranchFlowPerPSTAngle : calculates the linear impact of a PST angle increase on a specific branch’s active flow (in MW/°)
• BranchIntensityPerPSTAngle : calculates the linear impact of a PST angle increase on a specific branch’s current (in A/°)

JSON format

Sensitivity computation module provides a JSON mapping to sensitivity factors that can be used as a sensitivity factors provider. The file is a list of JSON objects each one representing a sensitivity factor:

[ {
"@c" : ".factors.BranchFlowPerInjectionIncrease",
"function" : {
"@c" : ".BranchFlow",
"id" : "BRANCH_FLOW_ID",
"name" : "My monitored branch",
"branchId" : "BRANCH_ID_IN_NETWORK"
},
"variable" : {
"@c" : ".InjectionIncrease",
"id" : "INJECTION_INCREASE_ID_1",
"name" : "My impacting injection 1",
"injectionId" : "INJECTION_1_ID_IN_NETWORK"
}
}, {
"@c" : ".factors.BranchFlowPerInjectionIncrease",
"function" : {
"@c" : ".BranchFlow",
"id" : "BRANCH_FLOW_ID",
"name" : "My monitored branch",
"branchId" : "BRANCH_ID_IN_NETWORK"
},
"variable" : {
"@c" : ".InjectionIncrease",
"id" : "INJECTION_INCREASE_ID_2",
"name" : "My impacting injection 2",
"injectionId" : "INJECTION_2_ID_IN_NETWORK"
}
} ]

# Sensisitivity computation outputs

## Sensitivity values

The outputs of sensitivity computation are sensitivity values. It is a list of objects associated to each sensitivity factors:

• The actual value of the partial derivative
• The reference value of the variable at linearization point
• The reference value of the function at linearization point

## Example of interpretation

Let’s imagine that one want to get the impact of an increase on active power generation on generator G on branch B active flow. The sensitivity computation input will contain one sensitivity factor.

After computation, the result is a sensitivity value object composed of three elements:

• a value of -0.05 for the partial derivative
• a variable reference value of 150
• a function reference value of 265

That means that:

• initial generation on generator G is 150MW
• initial active flow on branch B is 265MW from side 1 to side 2
• an increase of 100 MW on generator G may be approximated on branch B as a 5MW decrease of the active flow from side 1 to side 2


text:
Properties

 Label 450.2.h.f Level $450$ Weight $2$ Character orbit 450.h Analytic conductor $3.593$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.h (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} - 3250 x^{3} + 3750 x^{2} - 3125 x + 15625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} -\beta_{4} q^{8} +O(q^{10})$$ $$q -\beta_{7} q^{2} + ( -1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{7} -\beta_{4} q^{8} -\beta_{8} q^{10} + ( \beta_{1} - \beta_{2} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{11} + ( \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{14} + \beta_{6} q^{16} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{17} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{19} -\beta_{2} q^{20} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{22} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{23} + ( -2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{25} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{26} + ( -1 - \beta_{6} - \beta_{8} - \beta_{11} ) q^{28} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{29} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{31} + q^{32} + ( -\beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( -1 + \beta_{3} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{35} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{37} + ( \beta_{1} - \beta_{3} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{38} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{40} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} ) q^{41} + ( 3 + \beta_{1} - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{43} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{10} + \beta_{11} ) q^{44} + ( -3 - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{46} + ( -2 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{47} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{49} + ( 1 + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{50} + ( \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{52} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{55} + ( -1 - \beta_{2} + \beta_{7} + \beta_{9} ) q^{56} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{11} ) q^{58} + ( -\beta_{1} + \beta_{2} - \beta_{5} + 6 \beta_{6} + \beta_{8} + \beta_{9} ) q^{59} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{62} -\beta_{7} q^{64} + ( 2 - \beta_{1} - 2 \beta_{3} - 5 \beta_{4} + 4 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{65} + ( 1 + 2 \beta_{1} - 7 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{67} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{11} ) q^{68} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{70} + ( 5 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 5 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{71} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{73} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{74} + ( 1 - \beta_{1} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{76} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{79} + \beta_{3} q^{80} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{82} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{85} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{86} + ( \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} ) q^{88} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 8 \beta_{4} - 8 \beta_{6} + 7 \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{89} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{8} - \beta_{10} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{92} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{94} + ( -4 - 2 \beta_{1} + \beta_{2} + 8 \beta_{4} - \beta_{5} - 6 \beta_{6} + 7 \beta_{7} + \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{95} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{97} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{2} - 3q^{4} + q^{5} - 2q^{7} - 3q^{8} + O(q^{10})$$ $$12q - 3q^{2} - 3q^{4} + q^{5} - 2q^{7} - 3q^{8} + q^{10} + q^{11} + 4q^{13} + 8q^{14} - 3q^{16} - 8q^{17} - 8q^{19} + q^{20} - 4q^{22} - 11q^{25} - 16q^{26} - 7q^{28} - 6q^{29} - 3q^{31} + 12q^{32} + 2q^{34} - 18q^{35} - 8q^{37} + 2q^{38} + q^{40} + 20q^{41} + 32q^{43} - 4q^{44} - 10q^{46} + 34q^{49} + 9q^{50} + 4q^{52} + 2q^{53} + 44q^{55} - 7q^{56} - 6q^{58} - 19q^{59} - 26q^{61} + 2q^{62} - 3q^{64} + 16q^{65} - 16q^{67} + 12q^{68} - 23q^{70} + 48q^{71} - 30q^{73} - 8q^{74} + 12q^{76} - 39q^{77} - 18q^{79} - 4q^{80} - 40q^{82} - 29q^{83} - 4q^{85} + 12q^{86} + q^{88} + 62q^{89} - 26q^{91} - 10q^{92} + 6q^{95} + 23q^{97} - 6q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} - 3250 x^{3} + 3750 x^{2} - 3125 x + 15625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} + 27 \nu^{10} - 17 \nu^{9} - 8 \nu^{8} - 467 \nu^{7} - 162 \nu^{6} - 195 \nu^{5} + 1350 \nu^{4} + 6075 \nu^{3} - 12700 \nu^{2} - 21250 \nu - 99375$$$$)/4750$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{11} + 124 \nu^{10} + 31 \nu^{9} + 199 \nu^{8} - 1339 \nu^{7} + 35 \nu^{6} - 885 \nu^{5} + 11425 \nu^{4} + 25525 \nu^{3} - 1625 \nu^{2} - 68125 \nu - 228125$$$$)/23750$$ $$\beta_{4}$$ $$=$$ $$($$$$31 \nu^{11} + 134 \nu^{10} - 204 \nu^{9} + 2659 \nu^{8} - 2374 \nu^{7} + 21445 \nu^{6} - 37110 \nu^{5} + 103125 \nu^{4} - 164600 \nu^{3} + 440875 \nu^{2} - 171875 \nu + 1503125$$$$)/118750$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} - \nu^{10} + 6 \nu^{9} - 26 \nu^{8} + 61 \nu^{7} - 120 \nu^{6} + 465 \nu^{5} - 600 \nu^{4} + 1525 \nu^{3} - 3250 \nu^{2} + 3750 \nu - 3125$$$$)/3125$$ $$\beta_{6}$$ $$=$$ $$($$$$73 \nu^{11} - 43 \nu^{10} + 1058 \nu^{9} - 1743 \nu^{8} + 5448 \nu^{7} - 15455 \nu^{6} + 34120 \nu^{5} - 48225 \nu^{4} + 168450 \nu^{3} - 109625 \nu^{2} + 265625 \nu - 568750$$$$)/118750$$ $$\beta_{7}$$ $$=$$ $$($$$$-117 \nu^{11} - 43 \nu^{10} - 367 \nu^{9} + 157 \nu^{8} - 1677 \nu^{7} - 6145 \nu^{6} + 1345 \nu^{5} - 51075 \nu^{4} + 56825 \nu^{3} - 185625 \nu^{2} + 158750 \nu - 925000$$$$)/118750$$ $$\beta_{8}$$ $$=$$ $$($$$$-32 \nu^{11} + 67 \nu^{10} - 577 \nu^{9} + 1092 \nu^{8} - 4037 \nu^{7} + 11150 \nu^{6} - 24255 \nu^{5} + 47050 \nu^{4} - 113175 \nu^{3} + 119500 \nu^{2} - 258125 \nu + 365625$$$$)/23750$$ $$\beta_{9}$$ $$=$$ $$($$$$182 \nu^{11} + 183 \nu^{10} + 877 \nu^{9} + 558 \nu^{8} + 2387 \nu^{7} + 5400 \nu^{6} + 7355 \nu^{5} + 61400 \nu^{4} + 36425 \nu^{3} + 250750 \nu^{2} + 134375 \nu + 759375$$$$)/118750$$ $$\beta_{10}$$ $$=$$ $$($$$$-296 \nu^{11} + 881 \nu^{10} - 1561 \nu^{9} + 9531 \nu^{8} - 18841 \nu^{7} + 43905 \nu^{6} - 106915 \nu^{5} + 170875 \nu^{4} - 196025 \nu^{3} + 677875 \nu^{2} - 181875 \nu + 131250$$$$)/118750$$ $$\beta_{11}$$ $$=$$ $$($$$$-481 \nu^{11} + 636 \nu^{10} - 2216 \nu^{9} + 11486 \nu^{8} - 16046 \nu^{7} + 45850 \nu^{6} - 116440 \nu^{5} + 103050 \nu^{4} - 217900 \nu^{3} + 740250 \nu^{2} + 400625 \nu + 643750$$$$)/118750$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 2 \beta_{10} + \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \beta_{2} - 2 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{11} - \beta_{9} - \beta_{8} + 13 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 7 \beta_{1} - 13$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{10} + 17 \beta_{9} + 7 \beta_{8} - 4 \beta_{6} - \beta_{5} - 32 \beta_{4} + 8 \beta_{3} - 12 \beta_{2} - 7 \beta_{1} - 8$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{11} + \beta_{10} - 3 \beta_{9} + 13 \beta_{8} - 18 \beta_{7} + 48 \beta_{6} + 45 \beta_{5} - 5 \beta_{4} - 24 \beta_{3} + 13 \beta_{2} - 36 \beta_{1} + 14$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - 14 \beta_{10} - 71 \beta_{9} - 36 \beta_{8} - 4 \beta_{7} - 52 \beta_{6} + 36 \beta_{5} + 112 \beta_{4} + 30 \beta_{3} + 5 \beta_{2} + 20 \beta_{1} - 240$$ $$\nu^{8}$$ $$=$$ $$-86 \beta_{11} + 82 \beta_{10} + 85 \beta_{9} - 91 \beta_{8} + 156 \beta_{7} - 339 \beta_{6} - 102 \beta_{5} - 4 \beta_{4} + 55 \beta_{3} - 71 \beta_{2} - 163 \beta_{1} - 184$$ $$\nu^{9}$$ $$=$$ $$-222 \beta_{11} + 400 \beta_{10} - 124 \beta_{9} - 74 \beta_{8} - 188 \beta_{7} + 80 \beta_{6} + 123 \beta_{5} - 84 \beta_{4} - 272 \beta_{3} + 24 \beta_{2} - 172 \beta_{1} + 688$$ $$\nu^{10}$$ $$=$$ $$250 \beta_{11} + 48 \beta_{10} - 692 \beta_{9} - 252 \beta_{8} - 2250 \beta_{7} - 346 \beta_{6} - 24 \beta_{5} - 668 \beta_{4} - 28 \beta_{3} + 182 \beta_{2} + 852 \beta_{1} - 317$$ $$\nu^{11}$$ $$=$$ $$292 \beta_{11} - 376 \beta_{10} + 978 \beta_{9} - 548 \beta_{8} - 532 \beta_{7} - 1448 \beta_{6} - 1720 \beta_{5} - 20 \beta_{4} - 1196 \beta_{3} + 1102 \beta_{2} - 1139 \beta_{1} + 36$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4} - \beta_{6} + \beta_{7}$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 −1.38239 + 1.75756i 0.220100 − 2.22521i 1.97131 + 1.05544i −1.24741 − 1.85579i −1.01542 + 1.99221i 1.95382 − 1.08748i −1.24741 + 1.85579i −1.01542 − 1.99221i 1.95382 + 1.08748i −1.38239 − 1.75756i 0.220100 + 2.22521i 1.97131 − 1.05544i
−0.809017 0.587785i 0 0.309017 + 0.951057i −1.38239 + 1.75756i 0 −0.447412 0.309017 0.951057i 0 2.15144 0.609344i
91.2 −0.809017 0.587785i 0 0.309017 + 0.951057i 0.220100 2.22521i 0 1.64173 0.309017 0.951057i 0 −1.48601 + 1.67086i
91.3 −0.809017 0.587785i 0 0.309017 + 0.951057i 1.97131 + 1.05544i 0 −5.04842 0.309017 0.951057i 0 −0.974449 2.01257i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.24741 1.85579i 0 −2.68284 −0.809017 0.587785i 0 1.37949 1.75983i
181.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.01542 + 1.99221i 0 4.77988 −0.809017 0.587785i 0 −2.20849 0.350097i
181.3 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 1.95382 1.08748i 0 0.757055 −0.809017 0.587785i 0 1.63801 + 1.52214i
271.1 0.309017 0.951057i 0 −0.809017 0.587785i −1.24741 + 1.85579i 0 −2.68284 −0.809017 + 0.587785i 0 1.37949 + 1.75983i
271.2 0.309017 0.951057i 0 −0.809017 0.587785i −1.01542 1.99221i 0 4.77988 −0.809017 + 0.587785i 0 −2.20849 + 0.350097i
271.3 0.309017 0.951057i 0 −0.809017 0.587785i 1.95382 + 1.08748i 0 0.757055 −0.809017 + 0.587785i 0 1.63801 1.52214i
361.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −1.38239 1.75756i 0 −0.447412 0.309017 + 0.951057i 0 2.15144 + 0.609344i
361.2 −0.809017 + 0.587785i 0 0.309017 0.951057i 0.220100 + 2.22521i 0 1.64173 0.309017 + 0.951057i 0 −1.48601 1.67086i
361.3 −0.809017 + 0.587785i 0 0.309017 0.951057i 1.97131 1.05544i 0 −5.04842 0.309017 + 0.951057i 0 −0.974449 + 2.01257i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.f 12
3.b odd 2 1 450.2.h.g yes 12
25.d even 5 1 inner 450.2.h.f 12
75.j odd 10 1 450.2.h.g yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.h.f 12 1.a even 1 1 trivial
450.2.h.f 12 25.d even 5 1 inner
450.2.h.g yes 12 3.b odd 2 1
450.2.h.g yes 12 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{6} + T_{7}^{5} - 29 T_{7}^{4} - 18 T_{7}^{3} + 124 T_{7}^{2} - 24 T_{7} - 36$$ $$T_{11}^{12} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$3$ $$T^{12}$$
$5$ $$15625 - 3125 T + 3750 T^{2} - 3250 T^{3} + 1525 T^{4} - 600 T^{5} + 465 T^{6} - 120 T^{7} + 61 T^{8} - 26 T^{9} + 6 T^{10} - T^{11} + T^{12}$$
$7$ $$( -36 - 24 T + 124 T^{2} - 18 T^{3} - 29 T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$16 + 144 T + 832 T^{2} + 2888 T^{3} + 6808 T^{4} + 8528 T^{5} + 8798 T^{6} - 1222 T^{7} + 693 T^{8} + 28 T^{9} - 8 T^{10} - T^{11} + T^{12}$$
$13$ $$126736 + 87576 T - 61988 T^{2} - 117738 T^{3} + 193653 T^{4} - 65918 T^{5} + 24383 T^{6} - 6508 T^{7} + 1768 T^{8} - 248 T^{9} + 62 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$467856 + 952128 T + 2998296 T^{2} + 1661970 T^{3} + 521545 T^{4} + 71858 T^{5} + 24489 T^{6} + 828 T^{7} + 1530 T^{8} + 80 T^{9} + 36 T^{10} + 8 T^{11} + T^{12}$$
$19$ $$92416 + 350208 T + 454656 T^{2} - 331040 T^{3} + 138640 T^{4} + 31888 T^{5} + 74184 T^{6} + 20368 T^{7} + 5440 T^{8} + 760 T^{9} + 96 T^{10} + 8 T^{11} + T^{12}$$
$23$ $$2560000 - 7040000 T + 8224000 T^{2} - 3448000 T^{3} + 848400 T^{4} - 125600 T^{5} + 49400 T^{6} + 1000 T^{7} + 2260 T^{8} + 180 T^{9} + 70 T^{10} + T^{12}$$
$29$ $$1936 - 16984 T + 56732 T^{2} + 32522 T^{3} + 329733 T^{4} + 94922 T^{5} + 43133 T^{6} + 3672 T^{7} + 608 T^{8} - 28 T^{9} + 22 T^{10} + 6 T^{11} + T^{12}$$
$31$ $$26896 + 254528 T + 950856 T^{2} + 392560 T^{3} + 411640 T^{4} - 70072 T^{5} + 6744 T^{6} + 5418 T^{7} + 2465 T^{8} + 260 T^{9} + 76 T^{10} + 3 T^{11} + T^{12}$$
$37$ $$68492176 + 167191752 T + 166698808 T^{2} + 31337456 T^{3} + 15774833 T^{4} + 686394 T^{5} + 535662 T^{6} + 15116 T^{7} + 4288 T^{8} + 4 T^{9} + 63 T^{10} + 8 T^{11} + T^{12}$$
$41$ $$24010000 + 6860000 T + 8379000 T^{2} + 1930250 T^{3} + 1230525 T^{4} + 155800 T^{5} + 47325 T^{6} - 20600 T^{7} + 4910 T^{8} - 690 T^{9} + 180 T^{10} - 20 T^{11} + T^{12}$$
$43$ $$( 2704 - 14976 T - 836 T^{2} + 1108 T^{3} - 34 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$47$ $$6724000000 + 5658000000 T + 1934800000 T^{2} + 21400000 T^{3} + 60850000 T^{4} - 5530000 T^{5} + 1660000 T^{6} - 44000 T^{7} + 13000 T^{8} + 800 T^{9} + 130 T^{10} + T^{12}$$
$53$ $$31798321 - 32723117 T + 81127466 T^{2} - 14420280 T^{3} + 18074625 T^{4} + 4827888 T^{5} + 716519 T^{6} + 65683 T^{7} + 18805 T^{8} + 525 T^{9} - 59 T^{10} - 2 T^{11} + T^{12}$$
$59$ $$70291456 + 110802944 T + 176090112 T^{2} + 110226688 T^{3} + 38413568 T^{4} + 8411008 T^{5} + 1467688 T^{6} + 226988 T^{7} + 30113 T^{8} + 2918 T^{9} + 252 T^{10} + 19 T^{11} + T^{12}$$
$61$ $$32290652416 + 12809449664 T + 11284636144 T^{2} + 386896060 T^{3} - 12890695 T^{4} + 11808664 T^{5} + 4945991 T^{6} + 667276 T^{7} + 77420 T^{8} + 6760 T^{9} + 514 T^{10} + 26 T^{11} + T^{12}$$
$67$ $$3550253056 - 2589282304 T + 1816095232 T^{2} - 449541568 T^{3} + 36495568 T^{4} + 13317232 T^{5} + 2367568 T^{6} + 222272 T^{7} + 47648 T^{8} + 3312 T^{9} + 242 T^{10} + 16 T^{11} + T^{12}$$
$71$ $$4180398336 - 454660992 T + 1260566208 T^{2} - 555182496 T^{3} + 172968688 T^{4} - 50765264 T^{5} + 12658792 T^{6} - 2238576 T^{7} + 271928 T^{8} - 22584 T^{9} + 1288 T^{10} - 48 T^{11} + T^{12}$$
$73$ $$3906250000 - 546875000 T + 1873437500 T^{2} + 689281250 T^{3} + 109500625 T^{4} + 3165000 T^{5} + 680125 T^{6} + 176000 T^{7} + 51150 T^{8} + 6550 T^{9} + 580 T^{10} + 30 T^{11} + T^{12}$$
$79$ $$15083769856 - 11479857152 T + 4013958656 T^{2} - 698673920 T^{3} + 96988480 T^{4} - 14065632 T^{5} + 2079384 T^{6} + 16468 T^{7} + 32045 T^{8} + 2145 T^{9} + 276 T^{10} + 18 T^{11} + T^{12}$$
$83$ $$27081616 - 53476304 T + 47994744 T^{2} - 10649320 T^{3} + 11199140 T^{4} + 2892316 T^{5} + 2072206 T^{6} + 507474 T^{7} + 68765 T^{8} + 5070 T^{9} + 454 T^{10} + 29 T^{11} + T^{12}$$
$89$ $$1387115536 - 519255848 T + 1398308528 T^{2} - 1465315524 T^{3} + 771047873 T^{4} - 209598446 T^{5} + 39954152 T^{6} - 5374514 T^{7} + 531018 T^{8} - 38136 T^{9} + 1933 T^{10} - 62 T^{11} + T^{12}$$
$97$ $$786185521 - 898033092 T + 864566673 T^{2} - 459325441 T^{3} + 188116133 T^{4} - 45797619 T^{5} + 6081717 T^{6} - 82746 T^{7} + 5813 T^{8} - 844 T^{9} + 288 T^{10} - 23 T^{11} + T^{12}$$


text:
7. Safe and Strong Screening for Generalized LASSO

Safe [GVR11] and strong [TBF+12] scoring rules may be used to filter out variables for LASSO regression. These approaches are a type of feature selection and involve inner-products to reduce the feature space. The safe scoring rule is defined as follows.

$$|x_j^T y| < \lambda - ||x_j||_2 ||y||_2 \dfrac{\lambda_{\mathrm{max}} - \lambda}{\lambda_{\mathrm{max}}}$$

The strong scoring rule tends to remove more variables and is defined as follows.

$$|x_j^T y| < 2\lambda - \lambda_{\mathrm{max}}$$

For these inequalities,

• $$x_j$$ is the vector of values for the j-th,

• $$y$$ is the vector of the dependent variable,

• $$\lambda_{\mathrm{max}} = \mathrm{max}_i |x_i^T y|$$, and

• $$\lambda$$ is the regularization parameter.

7.1. Data

Let’s simulate some data.

The variables $$X_1, X_2, X_3, X_4, X_5$$ are sampled as follows.

• $$X_1 \sim \mathcal{B}(1, 0.8)$$

• $$X_2 \sim \mathcal{B}(1, 0.4)$$

• $$X_3 \sim \mathcal{B}(1, 0.6)$$

• $$X_4 \sim \mathcal{B}(1, 0.3)$$

• $$X_5 \sim \mathcal{B}(1, 0.2)$$

The dependent variable $$y$$ is sampled as follows.

• $$z = 2.0 + 0.5 X_1 + 0.8 X_2$$

• $$p = \dfrac{1.0}{1.0 + \exp(-z)}$$

• $$y \sim \mathcal{B}(1, p)$$

There are 10,000 sampled data points.

[1]:

import numpy as np
import random
from scipy.stats import binom

random.seed(37)
np.random.seed(37)

N = 10_000

x1 = np.random.binomial(1, 0.8, N)
x2 = np.random.binomial(1, 0.7, N)
x3 = np.random.binomial(1, 0.5, N)
x4 = np.random.binomial(1, 0.3, N)
x5 = np.random.binomial(1, 0.2, N)

X = np.hstack((
x1.reshape(-1, 1),
x2.reshape(-1, 1),
x3.reshape(-1, 1),
x4.reshape(-1, 1),
x5.reshape(-1, 1)
))

z = 2.0 + 0.5 * x1 + 0.8 * x2
p = 1.0 / (1.0 + np.exp(-z))
y = binom.rvs(1, p)

7.2. Feature selection

Let’s compute what we need for the inequalities.

[2]:

x_ip = np.abs(X.T.dot(y))                    # inner-product |x_i^T y|
lambda_max = np.max(x_ip)                    # maximum value of inner-products
lambda_v = lambda_max - (0.1 * lambda_max)   # lambda is selected to be lambda_max - (0.1 lambda_max)

x_norm = np.array([np.linalg.norm(X[:,i], 2) for i in range(X.shape[1])])  # l2 norms of each variable/column
y_norm = np.linalg.norm(y, 2)                                              # l2 norm of y

[3]:

lambda_v

[3]:

6833.7

[4]:

lambda_max

[4]:

7593

[5]:

x_ip

[5]:

array([7593, 6770, 4740, 2834, 1890])

[6]:

x_norm

[6]:

array([89.35323161, 83.98214096, 70.79548008, 54.91812087, 44.754888  ])

[7]:

y_norm

[7]:

97.22139682189307

7.2.1. Safe scoring rule

$$\lambda - ||x_j||_2 ||y||_2 \dfrac{\lambda_{\mathrm{max}} - \lambda}{\lambda_{\mathrm{max}}}$$

[8]:

lambda_v - x_norm * y_norm * ((lambda_max - lambda_v)/(lambda_max))

[8]:

array([5964.99540119, 6017.21389479, 6145.41645378, 6299.77835781,
6398.58672739])

The safe scoring rule inequality.

$$|x_j^T y| < \lambda - ||x_j||_2 ||y||_2 \dfrac{\lambda_{\mathrm{max}} - \lambda}{\lambda_{\mathrm{max}}}$$

As you can see, $$X_1$$ and $$X_2$$ are kept, but $$X_3$$, $$X_4$$ and $$X_5$$ are removed by the inequality.

[9]:

x_ip < lambda_v - x_norm * y_norm * ((lambda_max - lambda_v)/(lambda_max))

[9]:

array([False, False,  True,  True,  True])

7.2.2. Strong scoring rule

$$2\lambda - \lambda_{\mathrm{max}}$$

[10]:

2 * lambda_v - lambda_max

[10]:

6074.4

The strong scoring rule inequality.

$$|x_j^T y| < 2\lambda - \lambda_{\mathrm{max}}$$

Again, as you can see, $$X_1$$ and $$X_2$$ are kept, but $$X_3$$, $$X_4$$ and $$X_5$$ are removed by the inequality.

[11]:

x_ip < 2 * lambda_v - lambda_max

[11]:

array([False, False,  True,  True,  True])


text:
# \includegraphics compiling only when in local folder

I just a massive update of my MikTeX in the university computer (previous version was from 2014).

I always work in a way that I can access my files from any machine connected to the university network, but after the update I found that I can compile \includegraphics only if the picture is saved on the local folder (which I recall with \graphicspath).

This is not a terrible problem but still annoying (I depend more on one machine and I have to be more careful not to lose the pictures). Do you have any idea how I can fix this issue?

An example of code is:

\documentclass[svgnames,11pt]{article}
\usepackage{graphicx}

\begin{document}
x
\graphicspath{{C:/.../documents/Pictures/}}
\includegraphics[width=10cm]{heisball44.jpg}
\end{document}

\includegraphics[width=10cm]{./heisball44.jpg}


text:
3 years ago

# Universal disorder-induced broadening of phonon bands: from disordered lattices to glasses.

Eran Bouchbinder, Edan Lerner

The translational symmetry of solids gives rise to the existence of low-frequency phonons. In ordered systems, some phonons characterized by different wavevectors are degenerate, i.e. they share the same frequency $\omega$; in finite-size systems, phonons form a discrete set of bands with $n_q(\omega)$-fold degeneracy. Here we focus on understanding how this degeneracy is lifted in the presence of disorder, and its physical implications. Using standard degenerate perturbation theory and simple statistical considerations, we predict the dependence of the disorder-induced frequency width of phonon bands to be $\Delta\omega\!\sim\!\sigma\,\omega\sqrt{n_q}/\sqrt{N}$, where $\sigma$ is the strength of disorder and $N$ is the total number of particles. This theoretical prediction is supported by extensive numerical calculations for disordered lattices characterized by topological, mass, stiffness and positional disorder, and for computer glasses, where disorder is self-generated, thus establishing its universal nature. The predicted scaling leads to the identification of a crossover frequency $\omega_\dagger\!\sim\!L^{-2/(d+2)}$ in systems of linear size $L$ in $d\!>\!2$ dimensions, where the disorder-induced width of phonon bands becomes comparable to the frequency gap between neighboring bands. Consequently, phonons continuously cover the frequency range $\omega\!>\!\omega_\dagger$, where the notion of discrete phonon bands becomes ill-defined. Two basic applications of the theory are presented; first, we show that the phonon scattering lifetime is proportional to $(\Delta\omega)^{-1}$ for $\omega\!<\!\omega_\dagger$. Second, the theory is applied to the basic physics of glasses, allowing to determine the range of frequencies in which the recently established universal density of states of non-phononic excitations can be directly probed for different system sizes.

Publisher URL: http://arxiv.org/abs/1801.05170

DOI: arXiv:1801.05170v1

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.


text:
# [Learning Notes] Monotonicity of Decision-Making + [Problem Solution] Luo Gu3515: [POI2011]Lightning Conductor

Theme portal
Decision-making monotonicity optimization qwq
The monotonicity of decision is the current state I I i, two decision j, K (j & lt; k) j, K (j & lt; k) j, K (j < k). if K is better, then J will never reverse when I increases
This is the time to satisfy the monotony of decision-making
Then it is obtained that each decision corresponds to an interval so that the decision is the optimal solution for the state in the interval.

A bit of a mystery.
fi=max(aj+∣j−i∣)f_i=max(a_j+\sqrt{|j-i|})fi=max(aj+∣j−i∣)
The way to get rid of absolute value is to do it over and over again.
Open a monotone queue with triples {l,r,pl,r,pl,r,p} to represent the corresponding interval [l,r][l,r][l,r]
Some properties and practices can be obtained.

• All the intervals put together are [1,n][1,n][1,n]
• iii enumerates + + q[h].l once per enumeration, and deletes if the corresponding interval of the team leader is empty
• For each state iii, the optimal decision is the head of the team
• The condition for adding i-decision at the end of the team is that decision I is better for state n than for decision-making end-to-state n.
• Then the corresponding interval is updated by dichotomy search critical value

Code:

#include <bits/stdc++.h>
#define maxn 500010
using namespace std;
struct data{
int l, r, p;
}q[maxn];
int n, a[maxn];
double f[maxn], g[maxn];

int s = 0, w = 1;
char c = getchar();
for (; !isdigit(c); c = getchar()) if (c == '-') w = -1;
for (; isdigit(c); c = getchar()) s = (s << 1) + (s << 3) + (c ^ 48);
return s * w;
}

double calc(int j, int i){ return a[j] - a[i] + sqrt(abs(i - j)); }

int find(data t, int x){
int l = t.l, r = t.r;
while (l <= r){
int mid = (l + r) >> 1;
if (calc(t.p, mid) > calc(x, mid)) l = mid + 1; else r = mid - 1;
}
return l;
}

void Dp(double *dp){
int h = 1, t = 0;
for (int i = 1; i <= n; ++i){
if (h <= t && ++q[h].l > q[h].r) ++h;
if (h > t || calc(q[t].p, n) < calc(i, n)){
while (h <= t && calc(q[t].p, q[t].l) < calc(i, q[t].l)) --t;
if (h > t) q[++t] = (data){i, n, i}; else{
int tmp = find(q[t], i);
q[t].r = tmp - 1;
q[++t] = (data){tmp, n, i};
}
}
dp[i] = calc(q[h].p, i);
}
}

int main(){


text:
# Math Help - Analysis, Dirichlet function

1. ## Analysis, Dirichlet function & Riemann function

Denote by $m$ the Lebesgue measure on $X = (0, 1)$.

1. Prove that the Dirichlet function $d(\cdot)$ is not Riemann integrable.
Since the Dirichlet function is discontinuous at every point $x \in \mathbb{R}$ it is not Riemann integrable by definition.
2. Prove that $d(\cdot)$ is a non-negative Lebesgue measurable simple function.
3. What is the Lebesgue integral $\int d(x)dm$?
I think it is zero but I don't know how to show it.
4. Prove that the Riemann function $r(\cdot)$ is Riemann integrable.

2. For 2, ask yourself this: how might you define the Dirichlet function as the characteristic function of a set?

For 3, I’m not quite sure what you mean by $f_N(x)$.

For 4, I am not sure if you know Lebesgue’s integrability criterion (this is a criterion for Riemann integrability due to Lebesgue, not a condition for Lebesgue integrability). Basically, a function is Riemann integrable iff it is continuous $m$-almost everywhere on the domain. So a straightforward way to show this is to note that the number of discontinuities of $r$ is countable.

3. Originally Posted by Tikoloshe
For 3, I’m not quite sure what you mean by $f_N(x)$.
I fixed it. It should be $d(x)$, not $f_N(x)$

4. So what about the rest? Once you know 2, the answer of 3 follows as the measure of the set whose characteristic function you consider. If you are still having difficulty, try this:

Write the definition of a characteristic function $\chi_E(x)$ for some measurable set $E$. Then write the definition of $d(x)$. Compare the two forms, and see if you find a set whose characteristic function is $d$.


text:
# Gapless chiral spin liquid from coupled chains on the kagome lattice

### Submission summary

 As Contributors: Samuel Bieri · Rodrigo Pereira Arxiv Link: https://arxiv.org/abs/1706.02421v2 (pdf) Date accepted: 2017-12-23 Date submitted: 2017-11-17 01:00 Submitted by: Pereira, Rodrigo Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

Using a perturbative renormalization group approach, we show that the extended ($J_1$-$J_2$-$J_d$) Heisenberg model on the kagome lattice with a staggered chiral interaction ($J_\chi$) can exhibit a gapless chiral quantum spin liquid phase. Within a coupled-chains construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the chain directions. We calculate the low-energy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton mean-field theories with symmetry-protected line Fermi surfaces. These results may be relevant to the state observed in the kapellasite material.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 4, 004 (2018)

The referee has raised very good points. First, we would like to clarify the meaning of right-moving bosons for crossed chains. We label R and L modes in the limit of decoupled chains in such a way that R (L) refers to the modes that propagate in the positive-x (negative-x) direction of q-chains as indicated by the arrows in Fig. 1. Thus, the direction of propagation of R/L modes at a given point depends on the q index. This coordinate system with three different positive-x directions may seem confusing, but it was introduced by Gong et al. in Ref. [37] because it is convenient for analyzing the threefold rotational symmetry of the Hamiltonian.

Second, we agree with the referee that our cosine potential differs significantly from a conventional sine-Gordon model. Nonetheless, we have rewritten the discussion around Eq. (50) to emphasize that the interactions are local in the sense that the operators at different positions (in two dimensions) commute when the distances are larger than the short-distance cutoff. (Note this does not happen for more general cosine potentials with non-integer scaling dimensions.) We interpret this as a sign that the right-moving bosons can be gapped out entirely, leaving only gapless left-moving bosons. We cannot prove the latter statement rigorously, but we show that the low-energy fixed point obtained within this assumption is stable and the theory is consistent. Clearly, the phase in which the coupling between chiral currents is the strongest interaction must be different from the magnetically ordered (cuboc) and valence-bond-crystal phases. In our opinion, our proposal of a gapless chiral spin liquid is the simplest and most plausible picture that emerges from the RG analysis.

Concerning the parton construction, we clarify that the point here was to show that there are at least two mean-field ansatze which are able to reproduce the properties of the chiral spin liquid derived from the coupled-chain approach. This is not a meaningless exercise because, once identified, the parton construction can be useful to analyze the properties of the chiral spin liquid in a regime where the starting point of weakly coupled chains is not reliable. The 1/r^2 decay of the correlation function indeed imposes a strong constraint on the theories that we can select, but it is not the only constraint. Our analysis is also guided by symmetry, as the mean-field ansatz must break time reversal and the reflection by lines perpendicular to the chain directions, but respect the symmetry of reflection by lines parallel to the chains. Out of large number of U(1) chiral spin liquids on the kagome lattice that have been classified in Ref. [45], we have narrowed the choice down to one. The other possible mean-field ansatz uses a Majorana fermion representation with a Z2 gauge structure. Selecting between the Z2 and U(1) parton constructions requires numerical methods to calculate the ground state energy (using e.g. variational Monte Carlo techniques) and it is beyond the scope of this paper. Most numerical studies of chiral spin liquids so far have focused on gapped phases (with the notable exception of Ref. [70]). We hope that our work will encourage the search for gapless chiral spin liquid phases.

### List of changes

1. In the legend of Fig. 1, we now comment that the arrows indicate the positive-x directions, which become the directions of right-moving bosons for each q=1,2,3 in the continuum limit.

2. To address the referee’s concerns about the relevant cosine potential, we have rewritten the discussion around Eq. (50).

3. In section 6.4, we have included the energy magnetization correction in the expression for the thermal Hall conductivity, citing new references [79] and [80]. This correction does not change the conclusion that the thermal Hall response of the gapless chiral spin liquid vanishes by symmetry.

### Submission & Refereeing History

Resubmission 1706.02421v2 on 17 November 2017
Submission 1706.02421v1 on 12 June 2017

## Reports on this Submission

### Anonymous Report 1 on 2017-12-14 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1706.02421v2, delivered 2017-12-14, doi: 10.21468/SciPost.Report.294

### Strengths

1- The authors approach the problem from different perspectives (coupled chains and parton mean fields), and find not inconsistent results.
2- The authors provide a detailed account of their calculation.
3- The topic is timely.
4- The approach is in principle a very good one to address questions such as "is it in principle possible to find this or that phase", as is the goal of the paper.

### Weaknesses

1- The bosonization procedure is still unclear

### Report

The authors have responded well to the comments raised in the first round. The changes in the manuscript in my view did increase the understandability of the paper, and I thus recommend the work for publication.

I have one final comment on Eq. (50). The authors state that the argument of the cosine does not commute with itself at different positions. In bosonization, that means that the field cannot order - I hope the authors agree on that.

The authors then show that the cosine as a whole commutes with itself at different positions and state that "the most plausible scenario" is that the fields are gapped. It is mathematically unclear to me why this "most plausible scenario" is realised, given that the fields themselves cannot be pinned. Is there a way to understand this, or do I have to accept this as, really a guess? In other words, I find the physics plausible, but currently feel like there is a leap of faith from writing down Eq. (50) to saying that the right-moving fields are pinned. Sure, the phase where the right-movers are gapped might be a stable fixed point, but it feels to me like the authors are writing down a Hamiltonian they cannot solve, and then state that there is a stable fixed point, of which they do not show under which conditions the model realises it, and how that is related to the chiral interaction. Is that true?

I thus ask the authors to re-clarify this point.

### Requested changes

1- Clear up the commutation of the argument of the cosine in Eq. (50).

• validity: good
• significance: good
• originality: high
• clarity: good
• formatting: excellent
• grammar: excellent

### Author:  Rodrigo Pereira  on 2018-01-12  [id 200]

(in reply to Report 1 on 2017-12-14)
Category:
remark

We thank the referee for his/her helpful comments. We have added some clarification to emphasize that the full gap in one chiral sector is a conjecture since we cannot pin the bosonic fields in the usual way. Although the gap scenario seems plausible and physically compelling, this particular step in our analysis is indeed an assumption which we are not able to rigorously prove. Nonetheless, our derivation of the field theory from the lattice model and the perturbative renormalization group analysis are completely unbiased. Similarly, the calculation of physical properties is well controlled once we make this assumption about the strong coupling regime. It is worth mentioning that the lack of analytical techniques to handle non-commuting bosonized cosine terms is not unique to our problem. See, for instance, D. Bulmash et al., Phys. Rev. B 96, 045134 (2017) for a recent discussion of other examples.

We stress that the theory relying on our conjecture is consistent as it describes a stable fixed point of the renormalization group flow. Moreover, we clearly state that the chiral spin liquid phase can appear for strong chiral interaction, while valence-bond crystal and magnetic orders are expected to arise for small J_\chi. The validity of the conjecture could be tested by numerical simulations of the lattice model, by computing correlation functions that involve excitations of the gapped modes, such as dimer-dimer or the staggered magnetization mentioned in section 6.1.


text:
# All Questions

2 views

### ScheduledTask with scoped variables. Working with timed evaluation

I guess I have to try harder and focus more but timed evaluation always frustrates me. I will show the issue on simple delayed trigger example. ...
42 views

### Gamma function computation efficiency?

I wonder what kind of algorithm is used to compute the values for the gamma function. Specifically, I am interested in how the computational load increases when the complexity of the input grows. So, ...
57 views

### Reproducing a Plot done with WolframAlpha [duplicate]

I have problem to draw in Mathematica a plot like this: Plot[{Sum[a^10*(-1)^(x-a), {a, 1, x}]}, {x, 1, 10}] The problem is that on the plot there is no graph. As ...
43 views

### How to make GraphicsGrid with different widths of rows?

I want to make GraphicsGrid and I want to use first row for indication of parameters, but in that case the first row remains in same size as others. How can I make ...
38 views

### Using high RasterSize or ImageResolution changes contour pattern

I found that the detailed contour pattern will be altered if we Rasterize the original graphics using high RasterSize or ...
67 views

### Combining 2 or more functions

Maybe the question is stupid but I am newbie in using Mathematica, so I really need your help. I have a periodic smooth function s[ϕ] and $4$ different functions ...
78 views

### How to exit from the innermost enclosing pure function?

Here is what I tried to do: ...
84 views

### Solve Differential Equations using DSolve

I want so solve a differential equation $$\frac{d}{dx}(k_0 (\frac{T[x]}{T_0})^n \frac{dT[x]}{dx})=0,$$ with boundary conditions $T[l]=T_0, T'[0]=-\frac{q_0}{k_0 T[0]^n}$ using ...
44 views

### Infix strange behaviour

Maybe this is a bug or maybe Infix acts differently than the other functions. The fact is that both: TreeForm[Infix[f[x, a]]] and ...
33 views

### how to do variation

I have an equation needs to do variation. I can do it by hand, but I get different result by mathematica. maybe I made some mistakes in programming. can anyone point out for me? thanks This is how I ...
47 views

### How to create 2k sized vector in k sized loop

I want to solve a system of simple differential equations. At some point I need to solve non-linear system of algebraic equations with Newton method. I would like to efficiently create a right hand ...
41 views

### how recover lost data of coordinates from contourplot of mention data?

I have made Contour Plot for a lot of data, like this ...
28 views

### Buttons to generate copyable output

I have a grid as per below, made out of a series of buttons. Upon mouse click, I would like the buttons to (a) change the colour of that square and (b) output the text value of that button ...
20 views

triangle ABC with an arbitrary point M inside it, while MB" , MC" , MA" are distances and AA, BB , CC` are heights of the ABC triangle I need to prove the ...
34 views

### Export all open notebooks and change font size

Is there a simple way to export all open notebooks to pdfs? Also, when I export notebooks the font looks too big / the margins are too big. Can I change these settings when exporting?
49 views

### Export data of a 3D histogram

I want to export the data of a three dimensional histogram in the format # x y z with a blank line between different lines, so that one can plot the ...
56 views

### Sum of first n items in a sequence [on hold]

What's a way to sum over the first n items in a sequence, for example the sequence defined by $2^{k-1}\cdot k$ for $k\in\mathbb{Z},k>0$? I do these by hand all the time but am a bit tired of doing ...
70 views

32 views

### Financial number format

How can I change this function: nFormat=NumberForm[#,{\[Infinity],2},NumberPoint->",",NumberSeparator->".",DigitBlock->3,ExponentStep->10]&; So ...
65 views

### How to make the GroupOpeners visible in the SlideShow regime?

I am preparing a presentation on Mathematica and realized that though I made the GroupOpeners visible in the working regime (through the OptionInspector), they are invisible in the regime of the ...
52 views

### Importing and running a notebook [duplicate]

I have a very long .nb file. I want to subdivide the different sections into separate notebooks, then call them in a common nb ...
80 views

...
59 views

### How to solve this FindFit::nrlnum:

the following code show error message FindFit::nrlnum: "The function value {6.71759 -9.36268*10^-8\ I,6.51797 -8.42641*10^-8\ I,6.31789 -7.49014*10^-8\ I,6.1179 -6.55387*10^-8\ I,5.91794 ...
136 views

### Reducing mesh details while exporting to vector format SVG

I am trying to export a simple cuboid into .SVG format. In the output file, the cuboid is finely meshed, which increases the file size. How to remove this detailed mesh? ...
48 views

### Kernel behaviour in FindMinimum Optimization

I am optimizing very big problem as following: ...
73 views

### Finding Minimum of a list after interpolation

I have a set of lists called Er[l], where l is from 1 to 13. I want to find the minimum point in the plot in these lists after interpolation by spline method. I need the values and the points on the ...
257 views

### Can (compiled) matrix permanent evaluation be further sped-up?

Update  Multiple optimizations that were suggested by members "ssch" and Simon Woods have in aggregate yielded a ~5X code-speedup; and these optimizations now are incorporated in the example ...
58 views

### solving nonlinear pde for response and its derivatives in space and time? [on hold]

It is possible to solve mentioned nonlinear pde, for v(x,t) and its space and time derivatives in Mathematica? Boundary conditions are, v(0, t) = 0; v"(0, t) = 0 @ x= 0 v(l, t) = 0; v"(l, ...


text:
# Digit Sum Condition

Find the sum of all 3-digit positive numbers $$N$$ that satisfy the condition that the digit sum of $$N$$ is 3 times the digit sum of $$N+3$$.

Details and assumptions

The digit sum of a number is the sum of all its digits. For example, $$1123$$ has a digit sum of $$1 + 1 + 2 + 3 = 7$$.

The number $$12=012$$ is a 2-digit number, not a 3-digit number.

×