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# Compiler design

200 views

In this why questions why the left recursion is not removed as it is necessary for $LL1$

Why they given the option without removing the left recursion

$S->Aa|b$

$A->Ac|Sd|Null$

edited

Because He is not interested in asking Whether the Grammar is $LL(1)$ Or Not. Of course, The Grammar is Not $LL(1)$ because of Left Recursion. But Why do we say that "The Grammar is Not $LL(1)$ because of Left Recursion" ?

That's because When you make the Predictive Parser table for this Grammar, Some Cell will definitely contain Multiple Entries Which makes it Non-$LL(1)$. But in the Question He wants to ask about what Entries the respective cells will contain.

And One More thing, When Some Grammar is given to you and it does contain Left Recursion, We do not remove left recursion from it and afterwards decide it is $LL(1)$ or not. We decide it for the given Grammar.. Without changing it.
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Ok I got it and one more question that if it asked to find first and follow of particular variable in grammar then at that time we have to remove the left recursion to find the first and follow ?

PlZ correct me if m wrong
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No. When you are asked to Remove Left Recursion, then only you need to remove it.

## Related questions

1 vote
1
334 views
S→ A/a A→ a LL1 or not?

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Tag Info

3

We have $\forall t \geq 0$, \begin{align} M_t &= \int_0^t \int_s^t f(u) dW_u ds \\ &=\int_0^t \left(\int_s^t f(u)dW_u+\int_0^s f(u)dW_u-\int_0^s f(u)dW_u\right) ds \\ &=\int_0^t \left(\int_0^t f(u)dW_u\right)ds-\int_0^t\left(\int_0^s f(u)dW_u\right)ds \tag{By linearity} \\ \end{align} Denote here $I_t= \int_0^tf(u)dW_u$, we have then \begin{...

3

Remember that by the definition of the limit supremum, we have $$\limsup_{t\to\infty} f(t) \leq M \iff \left(\forall \epsilon > 0, \exists T : t > T \implies f(t) \leq M + \epsilon\right)$$ In particular, note that we cannot conclude $f(t) \leq M$ for any $t.$ (e.g. note that $\limsup_{n\to\infty} 1 + \frac{1}{n} \leq 1$) In your context, this means ...

2

Assuming you are talking about probability measures on the reals, and convolution with respect to $(\mathbb R, +)$, the answer is No. A convolution semigroup need not be continuous anywhere. Let $f$ be any discontinuous solution of Cauchy's functional equation $f(x+y)=f(x)+f(y)$ (these are discontinuous everywhere, and unbounded in every neighborhood) and ...

2

For $i\in\{1,\dots,6\}$, let $m_i = E(T_B | Z_t = i)$ be the expected number of steps until reaching $\{3,4,5\}$, starting from state $i$. Trivially, $m_3=m_4=m_5=0$. To obtain a system of linear constraints, apply first-step analysis (conditioning on the first step out of state $i$). For $i=1$, we have \begin{align} m_1 &= \sum_{j\in\{2,6\}} E(T_B | ...

2

Taking two steps in the Markov chain can lead to one of two things, with equal probability: $1 \to 2 \to 3$ or $1 \to 6 \to 5$ and we're done. $1 \to 2 \to 1$ or $1 \to 6 \to 1$ and we're back where we started. We took $2$ steps. In the first case, we have $0$ steps left, and in the second case, we have $\mathbb E[T_B]$ more steps left in expectation. ...

2

You can actually compute the integral with the formula $d(tY_t) = Y_t \, dt + t \, dY_t$. Fixing $Y_t = B_t^2$, you have by Itô formula that $dY_t = 2B_t dB_t + dt$ and therefore \begin{align} \int_0^1 B_t^2 d t &= \int_0^1 d(tB_t^2) - \int_0^1 td(B_t^2)\\ &= B_1^2 - \int_0^12tB_tdB_t - \int_0^1tdt\\\ &= B_1^2 - \frac12 - \int_0^12tB_tdB_t. \end{...

2

You have the right mean and variance for $$\int_0^1 B_t^2 \; dt \; ?$$ But, its distribution is not of any known functions. In fact, it is rather complex. You may find some research results in the link below. https://projecteuclid.org/download/pdf_1/euclid.aop/1020107767

2

This question has been asked before in this forum. My answer in Determining distribution of $X_t = \int_0^t W_s^2 \mathrm{d} s$ gives a couple of references to the original solution of Cameron & Martin, and a different approach of M. Kac.

2

Well, in these sorts of things there's not really much to do except try to exploit the independence of the increments. So here, $W_{t+2}=W_{t}+W_{t+2}-W_{t}$ and $W_{t+1}=W_t+W_{t+1}-W_t$. Thus, we get $$\mathbb{E}W_t(W_{t+1}+W_{t+2})=2\mathbb{E} W_t^2+\mathbb{E}W_t(W_{t+2}-W_{t})+\mathbb{E}W_t(W_{t+1}-W_t)$$ And by independence and Gaussianity, we thus ...

2

The step from finite case to infinite case is not so obvious and requires the very famous Kolmogorov extension theorem.

1

As an alternative approach in the spirit of your initial attempt, you can condition on the number $2k$ of steps to reach $\{3,4,5\}$ from $1$: \begin{align} E(T_B) &= \sum_{k=1}^\infty 2k\ P(\text{$2k$ steps}) \\ &= \sum_{k=1}^\infty 2k\ 2^k \left(\frac{1}{2}\right)^{2k} \\ &= \sum_{k=1}^\infty k \left(\frac{1}{2}\right)^{k-1} \\ &= \frac{1}{...

1

Hints: $\ S_n\$ is odd when $\ n\$ is odd, and even when $\ n\$ is even. If $\ S_n = m\$ then $\ \frac{m+n}{2}\$ of the first $\ n\$ steps must have been to the right and $\ \frac{n-m}{2}\$ of them must have been to the left. The number of rightward steps among the first $\ n\$ follows a binomial distribution with parameters $\ n\$ and $\frac{1}{2}\... 1 Your sollution is overly complicated. The easy path to finding the probability generating function of the product of independent random variables is to use the definition. $$\mathsf G_X(t)=\mathsf E(t^X)$$ Let$Y$be the random variable resulting in$1$if the coin toss is head and$2$if it is tails. So$Z=YX$and$Y\sim\mathcal U\{1,2\}$. $$\begin{... 1 I actually found a way to solve my problem. The rows of the limiting transition matrix are linear combinations of the stationary distributions, \pi_k, k=1,...,K of the K strongly connected components that correspond to a leaf of the condensation graph (i.e. the absorbing states). The coefficients associated with each \pi_k for each row are given by ... 1 This is a birth-death process, so we may instead use the detailed balance equations$$ \lambda(1-\mu)\pi_{n-1} = (1-\lambda)\mu \pi_n,\quad n\geqslant 1. $$This yields the recurrence$$ \pi_n = \left(\frac{\lambda(1-\mu)}{\mu(1-\lambda)}\right)^n\pi_0,\quad n\geqslant 0. $$Now, assuming that \frac{\lambda(1-\mu)}{\mu(1-\lambda)}<1, we have convergence ... 1 Let S be the set of all the states of the Markov chain. Consider a matrix A and B such that A=P^{r-1}=(a_{ij})_{i,j\in S} and B=P^r=(b_{ij})_{i,j\in S} respectively, where P=(p_{ij})_{i,j\in S} is the transition matrix of the Markov Chain. Then, P^{r+1} = BP = \Biggl($$\sum_{k} b_{ik}p_{kj} \Biggr)_{i,j\in S}$ As per the question, \$...

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# Glued PCB component removal

#### letsfixitgarage

Joined Dec 30, 2020
6
Got so frustrated with one repair that decided to register in this forum to look for help..

Have a HP docking station with shorted MOSFET.. thought it will be an easy job heat it up with a hot air station and remove the component.. but was I wrong.. first resoldered the legs and started heating up with 360 degrees C and 20% airflow after some time of nothing happening steadily increased it up to 400 C and 100% airflow still nothing then tried to move a capacitor next to the MOSFET and it didn't move either.. only then I got the idea that it could be glued with some kind of epoxy.. anyone with this kind of experience? What would be the possible, non destructive, ways of removing it..

Attached is the picture of the nice guy.. even scratched it trying to move it..

#### Attachments

• 2.5 MB Views: 19

#### shortbus

Joined Sep 30, 2009
9,322
Probably not hot enough yet. It most likely is soldered with lead free solder and the takes more heat around
217°C/422°F from, https://www.digikey.com/en/maker/blogs/rohs-vs-non-rohs-soldering Also not sure about that component but many big things like that also have a coppr pad under them as a heat sink that my also be soldered to the board.

#### jpanhalt

Joined Jan 18, 2008
11,088
Is it adhered with "hot goo" or "hot snot?' If so, isopropyl alcohol is a miracle release agent for that. Just a drop or two, and the snot removes easily.

#### letsfixitgarage

Joined Dec 30, 2020
6
Probably not hot enough yet. It most likely is soldered with lead free solder and the takes more heat around
217°C/422°F from, https://www.digikey.com/en/maker/blogs/rohs-vs-non-rohs-soldering Also not sure about that component but many big things like that also have a coppr pad under them as a heat sink that my also be soldered to the board.
Thanks for your reply. I applied 400 C and yes the MOSFET has quite big surface on the bottom and the trace where it is soldered on is quite big too but as I mentioned at the same time when I was heating the MOSFET I tried to move a much smaller capacitor (in the picture up from the MOSFET) and it didn't move either that's why I thought that there must be some kind of epoxy glue.. that withstands 400C heat..

#### letsfixitgarage

Joined Dec 30, 2020
6
Is it adhered with "hot goo" or "hot snot?' If so, isopropyl alcohol is a miracle release agent for that. Just a drop or two, and the snot removes easily.
Hi, no the board is beautifully clean as if nothing is there.. didn't notice even protective coating on top.. you can check the attached picture in previous post..

#### jpanhalt

Joined Jan 18, 2008
11,088
1) Get some good solder wick. I use Chemtronics. I have tried other brands, and have found nothing better.
2) Get a "hot-air" desolder-er. It is the only way to heat multiple pads to soldering temperature. I have tried some options, but none work as well as the cheapest hot-air station. My current one netted out to about $70. 3) A cheap HF propane or butane torch can work m a pinch. Thread Starter #### letsfixitgarage Joined Dec 30, 2020 6 1) Get some good solder wick. I use Chemtronics. I have tried other brands, and have found nothing better. 2) Get a "hot-air" desolder-er. It is the only way to heat multiple pads to soldering temperature. I have tried some options, but none work as well as the cheapest hot-air station. My current one netted out to about$70.
3) A cheap HF propane or butane torch can work m a pinch.
1 will not help in this case as the SMD has big pad underneath it that is soldered to the board track as heatsink and a small area of glue between it and the board.. I have an advanced hot air station and was applying 400C with a lot of air flow. With this setting I usually lift off even stubborn components but this time it didn't move at all.. i assume that there is some kind of epoxy applied by the manufacturer to not lose any components in the reflow procedure.. read that some use acetone/paint remover to soften the glue as nothing else effects the epoxy.. although that would damage the board too..

#### jpanhalt

Joined Jan 18, 2008
11,088
If the solder doesn't melt at 400°C, it is either a very unusual solder, or it is not at 400°C (pure tin melts at 232°C). The fact that your advanced hot air station could not move a small capacitor would make me lean toward the latter. Can you remove that capacitor with your regular soldering iron? When you are heating the component, does the exposed solder seem to melt?

There are some extremely high temperature epoxies, but the highest I found was rated to only 343°C and also required curing at high temperature. I think it is unlikely such epoxies or even ceramics are used in assembly.

#### letsfixitgarage

Joined Dec 30, 2020
6
If the solder doesn't melt at 400°C, it is either a very unusual solder, or it is not at 400°C (pure tin melts at 232°C). The fact that your advanced hot air station could not move a small capacitor would make me lean toward the latter. Can you remove that capacitor with your regular soldering iron? When you are heating the component, does the exposed solder seem to melt?

There are some extremely high temperature epoxies, but the highest I found was rated to only 343°C and also required curing at high temperature. I think it is unlikely such epoxies or even ceramics are used in assembly.
Thanks for your thoughts. Decided to check my "advanced" air station it's a Chinese brand "bst 863" quite happy with it, but never checked is it accurate.. so multimeter temperature probe at 300 set shoved 230 actual and at 400 set 280-290 actual.. so not the 400 but should be enough for the job.. as said with other boards this setting was lifting anything..

I added leaded solder to the legs and it melts without problem.. haven't checked the neighbor components to be sure but I think it flowed too will check one more time..

I will try to put the board on the preheater to heat it up completely to 100-150C and then apply heat to the MOSFET.. if this will struggle too I think next step will be exacto knife..

#### jpanhalt

Joined Jan 18, 2008
11,088
I have a cheaper model 8582D hot air station from China. Its hot air temp with a thermocouple is right on. I would set the air temp a bit higher. Temperature control is up to you. Before my hot air station, I used a much hotter blower and even a small propane flame. You just have to keep the flame moving to control temperature.

Will you be using the scalpel to pry it or cut the legs off? Before I would use a scalpel to pry anything, I would use solder wick on the pins, then cut them at the IC's body. That way, only the heatsink is holding it. Good center heat and it should slide right off. A pre-heat plate will help, and/or apply the hot air to the back too.

#### Deleted member 115935

Joined Dec 31, 1969
0
Epoxy fitting is used, but not in my experience on commercial stuff,

Looking at the picture, it also looks like the board has been conformably coated,
so you will have to scratch that off first, else it will go brown and be horrible to re solder !

looking at the PCB picture,
Those chips have large planes attached to them,
so I bet they also have a large plane underneath the chip ( see if on the bottom of the board there are vias in the middle of the chip area.

If it is like that , The amount of soaking at temperature you have to do is tremendous,
Think like 30 minutes soak before you put the boost on, which could be another 15 minutes,

measure the board temperature about 1cm away from the chip,
if that's not above 200 C , you are no where hot enough.

#### letsfixitgarage

Joined Dec 30, 2020
6
Thank you for your tips and suggestions!

Got the chip off, a replacement on and as was suggested problem was heat, not somekind off epoxy.. wanted to preheat it to ~200C but problem with this board (docking station) that it has a lot of connectors (don't know much they can take before melting) so I set the preheater to ~100C so the board was at ~80C and then set the hot air to 450 so real was around 350.. and then I chose a test point/ connection point next to the MOSFET as a test point as it was on the same pcb track as the MOSFET and the same lead free solder as under the MOSFET. So I was heating both of them and when the solder point weakened the MOSFET came off without any problem..

So yes these boards take in a lot of heat.. was quite a surprise for me as I worked first time on this kind off board.

Wanted to share it, maybe it will be helpful to someone.
And thank you everyone for your help!


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Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_univar_robust_1var_mestim_wgt (g07dc)

## Purpose

nag_univar_robust_1var_mestim_wgt (g07dc) computes an M$M$-estimate of location with (optional) simultaneous estimation of scale, where you provide the weight functions.

## Syntax

[theta, sigma, rs, nit, wrk, ifail] = g07dc(chi, psi, isigma, x, beta, theta, sigma, tol, 'n', n, 'maxit', maxit)
[theta, sigma, rs, nit, wrk, ifail] = nag_univar_robust_1var_mestim_wgt(chi, psi, isigma, x, beta, theta, sigma, tol, 'n', n, 'maxit', maxit)

## Description

The data consists of a sample of size n$n$, denoted by x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$, drawn from a random variable X$X$.
The xi${x}_{i}$ are assumed to be independent with an unknown distribution function of the form,
 F((xi − θ) / σ) $F((xi-θ)/σ)$
where θ$\theta$ is a location parameter, and σ$\sigma$ is a scale parameter. M$M$-estimators of θ$\theta$ and σ$\sigma$ are given by the solution to the following system of equations;
 n ∑ ψ((xi − θ̂) / σ̂) i = 1
= 0
 n ∑ χ((xi − θ̂) / σ̂) i = 1
= (n1)β
$∑i=1nψ((xi-θ^)/σ^) = 0 ∑i=1nχ((xi-θ^)/σ^) = (n-1)β$
where ψ$\psi$ and χ$\chi$ are user-supplied weight functions, and β$\beta$ is a constant. Optionally the second equation can be omitted and the first equation is solved for θ̂$\stackrel{^}{\theta }$ using an assigned value of σ = σc$\sigma ={\sigma }_{c}$.
The constant β$\beta$ should be chosen so that σ̂$\stackrel{^}{\sigma }$ is an unbiased estimator when xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ has a Normal distribution. To achieve this the value of β$\beta$ is calculated as:
 ∞ β = E(χ) = ∫ χ(z)1/(sqrt(2π))exp{( − z2)/2}dz − ∞
$β=E(χ)=∫-∞∞χ(z)12πexp{-z22}dz$
The values of ψ ((xiθ̂)/(σ̂)) σ̂$\psi \left(\frac{{x}_{i}-\stackrel{^}{\theta }}{\stackrel{^}{\sigma }}\right)\stackrel{^}{\sigma }$ are known as the Winsorized residuals.
The equations are solved by a simple iterative procedure, suggested by Huber:
σ̂k =
 sqrt(1/(β(n − 1)) (n ) ∑ χ((xi − θ̂k − 1)/(σ̂k − 1))i = 1 σ̂k − 12)
$σ^k=1β(n-1) (∑i=1nχ (xi-θ^k-1σ^k-1) ) σ^k-12$
and
 n θ̂k = θ̂k − 1 + 1/n ∑ ψ((xi − θ̂k − 1)/(σ̂k))σ̂k i = 1
$θ^k=θ^k- 1+1n ∑i= 1nψ (xi-θ^k- 1σ^k) σ^k$
or
 σ̂k = σc $σ^k=σc$
if σ$\sigma$ is fixed.
The initial values for θ̂$\stackrel{^}{\theta }$ and σ̂$\stackrel{^}{\sigma }$ may be user-supplied or calculated within nag_univar_robust_1var_mestim (g07db) as the sample median and an estimate of σ$\sigma$ based on the median absolute deviation respectively.
nag_univar_robust_1var_mestim_wgt (g07dc) is based upon function LYHALG within the ROBETH library, see Marazzi (1987).

## References

Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley
Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust estimation of location and scale in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 1 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

## Parameters

### Compulsory Input Parameters

1:     chi – function handle or string containing name of m-file
chi must return the value of the weight function χ$\chi$ for a given value of its argument. The value of χ$\chi$ must be non-negative.
[result] = chi(t)

Input Parameters

1:     t – double scalar
The argument for which chi must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
2:     psi – function handle or string containing name of m-file
psi must return the value of the weight function ψ$\psi$ for a given value of its argument.
[result] = psi(t)

Input Parameters

1:     t – double scalar
The argument for which psi must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
3:     isigma – int64int32nag_int scalar
The value assigned to isigma determines whether σ̂$\stackrel{^}{\sigma }$ is to be simultaneously estimated.
isigma = 0${\mathbf{isigma}}=0$
The estimation of σ̂$\stackrel{^}{\sigma }$ is bypassed and sigma is set equal to σc${\sigma }_{c}$.
isigma = 1${\mathbf{isigma}}=1$
σ̂$\stackrel{^}{\sigma }$ is estimated simultaneously.
4:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n > 1${\mathbf{n}}>1$.
The vector of observations, x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$.
5:     beta – double scalar
The value of the constant β$\beta$ of the chosen chi function.
Constraint: beta > 0.0${\mathbf{beta}}>0.0$.
6:     theta – double scalar
If sigma > 0${\mathbf{sigma}}>0$, then theta must be set to the required starting value of the estimate of the location parameter θ̂$\stackrel{^}{\theta }$. A reasonable initial value for θ̂$\stackrel{^}{\theta }$ will often be the sample mean or median.
7:     sigma – double scalar
The role of sigma depends on the value assigned to isigma as follows.
If isigma = 1${\mathbf{isigma}}=1$, sigma must be assigned a value which determines the values of the starting points for the calculation of θ̂$\stackrel{^}{\theta }$ and σ̂$\stackrel{^}{\sigma }$. If sigma0.0${\mathbf{sigma}}\le 0.0$, then nag_univar_robust_1var_mestim_wgt (g07dc) will determine the starting points of θ̂$\stackrel{^}{\theta }$ and σ̂$\stackrel{^}{\sigma }$. Otherwise, the value assigned to sigma will be taken as the starting point for σ̂$\stackrel{^}{\sigma }$, and theta must be assigned a relevant value before entry, see above.
If isigma = 0${\mathbf{isigma}}=0$, sigma must be assigned a value which determines the values of σc${\sigma }_{c}$, which is held fixed during the iterations, and the starting value for the calculation of θ̂$\stackrel{^}{\theta }$. If sigma0${\mathbf{sigma}}\le 0$, then nag_univar_robust_1var_mestim_wgt (g07dc) will determine the value of σc${\sigma }_{c}$ as the median absolute deviation adjusted to reduce bias (see nag_univar_robust_1var_median (g07da)) and the starting point for θ$\theta$. Otherwise, the value assigned to sigma will be taken as the value of σc${\sigma }_{c}$ and theta must be assigned a relevant value before entry, see above.
8:     tol – double scalar
The relative precision for the final estimates. Convergence is assumed when the increments for theta, and sigma are less than tol × max (1.0,σk1)${\mathbf{tol}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,{\sigma }_{k-1}\right)$.
Constraint: tol > 0.0${\mathbf{tol}}>0.0$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of observations.
Constraint: n > 1${\mathbf{n}}>1$.
2:     maxit – int64int32nag_int scalar
The maximum number of iterations that should be used during the estimation.
Default: 50$50$
Constraint: maxit > 0${\mathbf{maxit}}>0$.

None.

### Output Parameters

1:     theta – double scalar
The M$M$-estimate of the location parameter θ̂$\stackrel{^}{\theta }$.
2:     sigma – double scalar
The M$M$-estimate of the scale parameter σ̂$\stackrel{^}{\sigma }$, if isigma was assigned the value 1$1$ on entry, otherwise sigma will contain the initial fixed value σc${\sigma }_{c}$.
3:     rs(n) – double array
The Winsorized residuals.
4:     nit – int64int32nag_int scalar
The number of iterations that were used during the estimation.
5:     wrk(n) – double array
If sigma0.0${\mathbf{sigma}}\le 0.0$ on entry, wrk will contain the n$n$ observations in ascending order.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n ≤ 1${\mathbf{n}}\le 1$, or maxit ≤ 0${\mathbf{maxit}}\le 0$, or tol ≤ 0.0${\mathbf{tol}}\le 0.0$, or isigma ≠ 0${\mathbf{isigma}}\ne 0$ or 1$1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, beta ≤ 0.0${\mathbf{beta}}\le 0.0$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, all elements of the input array x are equal.
ifail = 4${\mathbf{ifail}}=4$
sigma, the current estimate of σ$\sigma$, is zero or negative. This error exit is very unlikely, although it may be caused by too large an initial value of sigma.
ifail = 5${\mathbf{ifail}}=5$
The number of iterations required exceeds maxit.
ifail = 6${\mathbf{ifail}}=6$
On completion of the iterations, the Winsorized residuals were all zero. This may occur when using the isigma = 0${\mathbf{isigma}}=0$ option with a redescending ψ$\psi$ function, i.e., ψ = 0$\psi =0$ if |t| > τ$|t|>\tau$, for some positive constant τ$\tau$.
If the given value of σ$\sigma$ is too small, then the standardized residuals (xiθ̂k)/(σc) $\frac{{x}_{i}-{\stackrel{^}{\theta }}_{k}}{{\sigma }_{c}}$, will be large and all the residuals may fall into the region for which ψ(t) = 0$\psi \left(t\right)=0$. This may incorrectly terminate the iterations thus making theta and sigma invalid.
Re-enter the function with a larger value of σc${\sigma }_{c}$ or with isigma = 1${\mathbf{isigma}}=1$.
ifail = 7${\mathbf{ifail}}=7$
The value returned by the chi function is negative.

## Accuracy

On successful exit the accuracy of the results is related to the value of tol, see Section [Parameters].

Standard forms of the functions ψ$\psi$ and χ$\chi$ are given in Hampel et al. (1986), Huber (1981) and Marazzi (1987). nag_univar_robust_1var_mestim (g07db) calculates M$M$-estimates using some standard forms for ψ$\psi$ and χ$\chi$.
When you supply the initial values, care has to be taken over the choice of the initial value of σ$\sigma$. If too small a value is chosen then initial values of the standardized residuals (xiθ̂k)/σ $\frac{{x}_{i}-{\stackrel{^}{\theta }}_{k}}{\sigma }$ will be large. If the redescending ψ$\psi$ functions are used, i.e., ψ = 0$\psi =0$ if |t| > τ$|t|>\tau$, for some positive constant τ$\tau$, then these large values are Winsorized as zero. If a sufficient number of the residuals fall into this category then a false solution may be returned, see page 152 of Hampel et al. (1986).

## Example

```function nag_univar_robust_1var_mestim_wgt_example
isigma = int64(1);
x = [13;
11;
16;
5;
3;
18;
9;
8;
6;
27;
7];
beta = 0.3892326;
theta = 0;
sigma = -1;
tol = 0.0001;
[thetaOut, sigmaOut, rs, nit, wrk, ifail] = ...
nag_univar_robust_1var_mestim_wgt(@chi, @psi, isigma, x, beta, theta, sigma, tol)

function [result] = chi(t)
ps = min(1.5, abs(t));
result = ps*ps/2;
function [result] = psi(t)

if abs(t) < 4.5
if abs(t) < 3
result=min(1.5, abs(t));
else
result=1.5*(4.5-abs(t))/1.5;
end
if t < 0
result = -result;
end
else
result=0;
end
```
```

thetaOut =

10.5487

sigmaOut =

6.3247

rs =

2.4513
0.4513
5.4513
-5.5487
-7.5487
7.4513
-1.5487
-2.5487
-4.5487
16.4513
-3.5487

nit =

8

wrk =

3
5
6
7
8
9
11
13
16
18
27

ifail =

0

```
```function g07dc_example
isigma = int64(1);
x = [13;
11;
16;
5;
3;
18;
9;
8;
6;
27;
7];
beta = 0.3892326;
theta = 0;
sigma = -1;
tol = 0.0001;
[thetaOut, sigmaOut, rs, nit, wrk, ifail] = ...
g07dc(@chi, @psi, isigma, x, beta, theta, sigma, tol)

function [result] = chi(t)
ps = min(1.5, abs(t));
result = ps*ps/2;
function [result] = psi(t)

if abs(t) < 4.5
if abs(t) < 3
result=min(1.5, abs(t));
else
result=1.5*(4.5-abs(t))/1.5;
end
if t < 0
result = -result;
end
else
result=0;
end
```
```

thetaOut =

10.5487

sigmaOut =

6.3247

rs =

2.4513
0.4513
5.4513
-5.5487
-7.5487
7.4513
-1.5487
-2.5487
-4.5487
16.4513
-3.5487

nit =

8

wrk =

3
5
6
7
8
9
11
13
16
18
27

ifail =

0

```


text:
What is $\lim_{n\to\infty} \frac{n}{\sum_{i=1}^n \frac1{a_i}}$ if $a_n=(2^n+3^n)^\frac{1}{n}$?

Let $$\displaystyle a_n=(2^n+3^n)^\frac{1}{n}$$. Compute the limit $$\displaystyle \lim_{n\to\infty}\dfrac{n}{\displaystyle \sum_{i=1}^{n}\dfrac{1}{a_i}}$$

By squeeze theorem, limit of $$a_n$$ is $$3$$. Further the sequence $$\displaystyle \sum_{i=1}^{n}\dfrac{1}{a_i}$$ is strictly monotonic and diverges to infinity since the limit of $$n^{th}$$ term of the series $$\displaystyle \sum_i\dfrac{1}{a_i}$$ is not $$0$$ infact $$1/3$$. Hence by Cesaro Stolz theorem for $$\cdot/\infty$$ case, $$\displaystyle \lim_{n\to\infty}\dfrac{n}{\displaystyle \sum_{i=1}^{n}\dfrac{1}{a_i}}=3.$$

Is my reasoning correct? Are there more easier ways to solve this?

• What you have done is correct and it seems to be the best way to answer the question. – Kavi Rama Murthy Dec 13 '19 at 23:28
• @KaboMurphy Thank you Sir. – Yadati Kiran Dec 13 '19 at 23:29
• This works for any mean and any sequence that approaches a limit. – marty cohen Dec 13 '19 at 23:43
• @martycohen Does that mean I can employ Cauchy's first theorem on limits here? – Yadati Kiran Dec 14 '19 at 0:03
• @YadatiKiran:How you used squeeze theorem to show $a_n \rightarrow 0$? – Styles Sep 29 '20 at 16:43

$$a_n = 3\Big(1+(\frac23)^n\Big)^{1/n}=3\exp\Big(\frac1n\ln(1+(2/3)^n)\Big)\xrightarrow{n\rightarrow\infty}3$$. The rest follows from Cesaro sums: $$\lim_{n\rightarrow\infty}\frac1n\sum^n_{k=1}\frac{1}{a_k}=\frac{1}{3}$$


text:
# Double Pendulum

The equations of motions for the double pendulum is given by

$$\dot{\theta_1} = \frac{6}{ml^2}\frac{2p_{\theta1} - 3\cos(\theta_1 - \theta_2)p_{\theta2}}{16 - 9\cos^2(\theta_1 - \theta_2)}$$

and similarly for the other pendulum. In respect to what does the change in angle for the first pendulum refer to? Is it with respect to time? So that $\dot{\theta_1} = \frac{d\theta}{dt}$?

-

Variables with two or three dots, like $\ddot{\theta}$ and $\dddot{\theta}$, represent second and third time derivatives respectively.


text:
# Locks in PostgreSQL: 4. Locks in memory

• Translation
To remind you, we've already talked about relation-level locks, row-level locks, locks on other objects (including predicate locks) and interrelationships of different types of locks.

The following discussion of locks in RAM finishes this series of articles. We will consider spinlocks, lightweight locks and buffer pins, as well as events monitoring tools and sampling.

# Spinlocks

Unlike normal, «heavy-weight», locks, to protect structures in the shared memory, more lightweight and less expensive (in overhead costs) locks are used.

The simplest of them are spinlocks. They are meant to be acquired for very short time intervals (a few processor instructions), and they protect separate memory areas from simultaneous changes.

Spinlocks are implemented based on atomic processor instructions, such as compare-and-swap. They support the only, exclusive, mode. If a lock is acquired, a waiting process performs busy waiting — the command is repeated («spins» in a loop, hence the name) until it is a success. This makes sense since spinlocks are used in the cases where the probability of a conflict is estimated as very low.

Spinlocks do not enable detection of deadlocks (PostgreSQL developers take care of this) and provide no monitoring tools. Essentially, the only thing we can do with spinlocks is to be aware of their existence.

# Lightweight locks

So-called lightweight locks (lwlocks) come next.

They get acquired for a short time that is needed to work with the data structure (such as a hash table or a list of pointers). As a rule, a lightweight lock is held briefly, but sometimes lightweight locks protect input/output operations, so in general, the time might also be considerable.

Two modes are supported: exclusive (for data modifications) and shared (only for reading). There is actually no wait queue: if a few processes wait for release of a lock, one of them will get the access in a more or less random fashion. In high-concurrency and large-load systems, this can be troublesome (for example, see this discussion).

There are no techniques to check for deadlocks, so this is left to the responsibility of developers of the core. However, lightweight locks have monitoring tools, so, unlike spinlocks, we can «see» them (I will show a bit later how to do this).

# Buffer pin

Yet another type of locks, which we already touched upon in the article on the buffer cache, is a buffer pin.

Different operations, including data modifications, can be performed with a pinned buffer, but under the condition that the changes are not visible to other processes due to multiversion concurrency control. That is, we can, for instance, add a new row to the page, but cannot replace a page in the buffer with another one.

If a buffer pin hinders a process, as a rule, the latter just skips this buffer and chooses a different one. But in some cases, where exactly this buffer is needed, the process queues and «falls asleep»; the system will wake it up when the buffer is unpinned.

Monitoring can access waits related to buffer pins.

# Example: buffer cache

Now, in order to get some (although incomplete!) insight into how and where locks are used, let's consider the buffer cache as an example.

To access a hash table that contains links to buffers, a process must acquire a lightweight buffer mapping lock in a shared mode, and if the table needs to be updated, in an exclusive mode. To reduce the granularity, this lock is structured as a tranche that consists of 128 separate locks, each protecting its own part of the hash table.

The process gets access to the buffer header using a spinlock. Certain operations (such as a counter increment) can also be performed without explicit locking, by means of atomic processor instructions.

In order to read the contents of a buffer, a buffer content lock is needed. It is usually acquired only for the time needed to read pointers to tuples, and after that, a buffer pin provides sufficient protection. To change the contents of a buffer, this lock must be acquired in an exclusive mode.

When a buffer is read from disk (or written to disk), an IO in progress lock is also acquired, which indicates to other processes that the page is being read (or written) — they can queue up if they need to do something with this page.

Pointers to free buffers and to the next victim are protected by one buffer strategy lock spinlock.

# Example: WAL buffers

WAL buffers provide another example.

The WAL cache also uses a hash table that contains mapping of pages to buffers. Unlike for the buffer cache, this hash table is protected by the only lightweight WALBufMappingLock lock since the size of the WAL cache is smaller (usually 1/32 of the buffer cache) and access to the buffers is more regular.

Writes of pages to disk are protected by the WALWriteLock lock, so that only one process can perform this operation at a time.

To create a WAL record, the process must first allocate space in a WAL page. To do this, it acquires an insert position lock spinlock. When the space is allocated, the process copies the contents of its record to the space allocated. Copying can be performed by several processes simultaneously, therefore, the record is protected by a tranche of 8 lightweight wal insert lock locks (the process must acquire any of them).

The figure does not show all WAL-related locks, but this and previous examples must give an idea of how locks in RAM are used.

# Wait events monitoring

Starting with PostgreSQL 9.6, the pg_stat_activity view has built-in events monitoring tools. When a process (system or backend) cannot do its job and waits for something, we can see this wait in the view: the wait_event_type column shows the wait type and the wait_event column shows the name of a specific wait.

Note that the view shows only waits that are properly handled in the source code. If the view does not show a wait, in general, this does not mean with 100 percent probability that the process really waits for nothing.

Unfortunately, the only available information on waits is the current information. No accumulated statistics are maintained. The only way to get the picture of waits in time is sampling the state of the view at a certain interval. No built-in tools are provided to this end, but we can use extensions, such as pg_wait_sampling.

We need to take into account the probabilistic nature of sampling. To get a more or less credible picture, the number of measurements must be pretty large. Low-frequency sampling may fail to provide a credible picture, while use of higher frequencies will increase overhead costs. For the same reason, sampling is useless to analyze short-lived sessions.

All the waits can be divided into several types.

Waits for the locks discussed make up a large category:

• Waits for locks on objects (the value of Lock in the wait_event_type column).
• Waits for lightweight locks (LWLock).
• Waits for a buffer pin (BufferPin).

But processes can also await other events:

• Waits for input/output (IO) occur when a process needs to read or write data.
• A process can wait for the data needed from a client (Client) or another process (IPC).
• Extensions can register their specific waits (Extension).

Sometimes situations arise when a process just doesn't do any productive work. This category includes:

• Waits of a background process in its main loop (Activity).
• Waits for a timer (Timeout).

Usually, waits like these are treated as «normal» and do not indicate any problems.

The wait type is followed by the name of a specific wait. For the complete table, see the documentation.

If the name of a wait is not defined, the process is not in the waiting state. We need to treat this point in time as unaccounted for since we are actually unaware of what exactly is happening at that moment.

However, let's watch this for ourselves.

=> SELECT pid, backend_type, wait_event_type, wait_event
FROM pg_stat_activity;

  pid  |         backend_type         | wait_event_type |     wait_event
-------+------------------------------+-----------------+---------------------
28739 | logical replication launcher | Activity        | LogicalLauncherMain
28736 | autovacuum launcher          | Activity        | AutoVacuumMain
28963 | client backend               |                 |
28734 | background writer            | Activity        | BgWriterMain
28733 | checkpointer                 | Activity        | CheckpointerMain
28735 | walwriter                    | Activity        | WalWriterMain
(6 rows)

It's clear that all background backend processes are idle. Empty values of wait_event_type and wait_event tell us that the process is waiting for nothing; in our example, the backend process is busy executing the query.

## Sampling

To get a more or less complete picture of waits by means of sampling, we will use the pg_wait_sampling extension. We need to build it from source codes, but I will omit this part. Then we add the library name to the shared_preload_libraries parameter and restart the server.

=> ALTER SYSTEM SET shared_preload_libraries = 'pg_wait_sampling';

student$sudo pg_ctlcluster 11 main restart  Now we install the extension in the database. => CREATE EXTENSION pg_wait_sampling;  The extension allows us to look through the history of waits, which is stored in a circular buffer. But what's mostly interesting to us is the waits profile, that is, the statistics accumulated since the server start. This is roughly what we will see a few seconds later: => SELECT * FROM pg_wait_sampling_profile;   pid | event_type | event | queryid | count -------+------------+---------------------+---------+------- 29074 | Activity | LogicalLauncherMain | 0 | 220 29070 | Activity | WalWriterMain | 0 | 220 29071 | Activity | AutoVacuumMain | 0 | 219 29069 | Activity | BgWriterMain | 0 | 220 29111 | Client | ClientRead | 0 | 3 29068 | Activity | CheckpointerMain | 0 | 220 (6 rows)  Because nothing happened since the server start, most waits refer to the types Activity (backend processes wait until there is some work for them) and Client (psql waits until a user sends a request). With the default settings (of the pg_wait_sampling.profile_period parameter), the sampling period equals 10 milliseconds, which means that values are saved 100 times a second. Therefore, to evaluate the duration of waits in seconds, we need to divide the value of count by 100. To figure out which process the waits pertain to, let's add the pg_stat_activity view to the query: => SELECT p.pid, a.backend_type, a.application_name AS app, p.event_type, p.event, p.count FROM pg_wait_sampling_profile p LEFT JOIN pg_stat_activity a ON p.pid = a.pid ORDER BY p.pid, p.count DESC;   pid | backend_type | app | event_type | event | count -------+------------------------------+------+------------+----------------------+------- 29068 | checkpointer | | Activity | CheckpointerMain | 222 29069 | background writer | | Activity | BgWriterMain | 222 29070 | walwriter | | Activity | WalWriterMain | 222 29071 | autovacuum launcher | | Activity | AutoVacuumMain | 221 29074 | logical replication launcher | | Activity | LogicalLauncherMain | 222 29111 | client backend | psql | Client | ClientRead | 4 29111 | client backend | psql | IPC | MessageQueueInternal | 1 (7 rows)  Let's produce some workload using pgbench and see how the picture changes. student$ pgbench -i test

We reset the accumulated profile to zero and run the test for 30 seconds in a separate process.

=> SELECT pg_wait_sampling_reset_profile();

student$pgbench -T 30 test  We need to execute the query while the pgbench process is not finished yet: => SELECT p.pid, a.backend_type, a.application_name AS app, p.event_type, p.event, p.count FROM pg_wait_sampling_profile p LEFT JOIN pg_stat_activity a ON p.pid = a.pid WHERE a.application_name = 'pgbench' ORDER BY p.pid, p.count DESC;   pid | backend_type | app | event_type | event | count -------+----------------+---------+------------+------------+------- 29148 | client backend | pgbench | IO | WALWrite | 8 29148 | client backend | pgbench | Client | ClientRead | 1 (2 rows)  The waits of the pgbench process will certainly differ slightly depending on a particular system. In our situation, a wait for WAL writing (IO/WALWrite) is highly likely to be presented, however, most of the time the process was doing something presumably productive rather than being idle. ## Lightweight locks We always need to keep in mind that if a wait is missing when sampling, this does not mean that there was really no wait. If the wait was shorter than the sampling period (a hundredth of a second in our example), it could just fail to get into the sample. That's why lightweight locks did not occur in the profile, but they will if the data is collected for a long time. To be able to see them for sure, we can intentionally slow down the file system, for example, by using the slowfs project, built on top of the FUSE file system. This is what we can see on the same test if any input/output operation takes 1/10 of a second. => SELECT pg_wait_sampling_reset_profile();  student$ pgbench -T 30 test

=> SELECT p.pid, a.backend_type, a.application_name AS app,
p.event_type, p.event, p.count
FROM pg_wait_sampling_profile p
LEFT JOIN pg_stat_activity a ON p.pid = a.pid
WHERE a.application_name = 'pgbench'
ORDER BY p.pid, p.count DESC;

  pid  |  backend_type  |   app   | event_type |     event      | count
-------+----------------+---------+------------+----------------+-------
29240 | client backend | pgbench | IO         | WALWrite       |  1445
29240 | client backend | pgbench | LWLock     | WALWriteLock   |   803
29240 | client backend | pgbench | IO         | DataFileExtend |    20
(3 rows)

Now the major wait of the pgbench process relates to input/output, more exactly, to WAL writes, which synchronously occur for every commit. Because (as shown in one of the above examples) a WAL write is protected by a lightweight WALWriteLock lock, this lock is also present in the profile — and it's just what we wanted to look at.

## Buffer pin

To see a buffer pin, let's make use of the fact that open cursors hold the pin to faster read the next row.

Let's start a transaction, open a cursor and select one row.

=> BEGIN;
=> DECLARE c CURSOR FOR SELECT * FROM pgbench_history;
=> FETCH c;

 tid | bid |  aid  | delta |           mtime            | filler
-----+-----+-------+-------+----------------------------+--------
9 |   1 | 35092 |   477 | 2019-09-04 16:16:18.596564 |
(1 row)

Let's check that the buffer is pinned (pinning_backends):

=> SELECT * FROM pg_buffercache
WHERE relfilenode = pg_relation_filenode('pgbench_history')
AND relforknumber = 0 \gx

-[ RECORD 1 ]----+------
bufferid         | 190
relfilenode      | 47050
reltablespace    | 1663
reldatabase      | 16386
relforknumber    | 0
relblocknumber   | 0
isdirty          | t
usagecount       | 1
pinning_backends | 1     <-- buffer is pinned 1 time

Now let's vacuum the table:

|  => SELECT pg_backend_pid();

|   pg_backend_pid
|  ----------------
|            29367
|  (1 row)

|  => VACUUM VERBOSE pgbench_history;

|  INFO:  vacuuming "public.pgbench_history"
|  INFO:  "pgbench_history": found 0 removable, 0 nonremovable row versions in 1 out of 1 pages
|  DETAIL:  0 dead row versions cannot be removed yet, oldest xmin: 732651
|  There were 0 unused item pointers.

|  Skipped 1 page due to buffer pins, 0 frozen pages.

|  0 pages are entirely empty.
|  CPU: user: 0.00 s, system: 0.00 s, elapsed: 0.00 s.
|  VACUUM

As we can see, the page was skipped (Skipped 1 page due to buffer pins). Indeed, VACUUM cannot process it because physical deletion of tuples from a page in a pinned buffer is forbidden. But vacuuming will not wait either, and the page will be processed next time.

And now let's perform vacuuming with freezing:

|  => VACUUM FREEZE VERBOSE pgbench_history;

If freezing is explicitly requested, none of the pages tracked in the all-frozen bit can be skipped; otherwise, it is impossible to reduce the maximum age of non-frozen transactions in pg_class.relfrozenxid. So, vacuuming hangs until the cursor is closed.

=> SELECT age(relfrozenxid) FROM pg_class WHERE oid = 'pgbench_history'::regclass;

 age
-----
27
(1 row)

=> COMMIT; -- cursor closes automatically

|  INFO:  aggressively vacuuming "public.pgbench_history"
|  INFO:  "pgbench_history": found 0 removable, 26 nonremovable row versions in 1 out of 1 pages
|  DETAIL:  0 dead row versions cannot be removed yet, oldest xmin: 732651
|  There were 0 unused item pointers.

|  Skipped 0 pages due to buffer pins, 0 frozen pages.

|  0 pages are entirely empty.
|  CPU: user: 0.00 s, system: 0.00 s, elapsed: 3.01 s.
|  VACUUM

=> SELECT age(relfrozenxid) FROM pg_class WHERE oid = 'pgbench_history'::regclass;

 age
-----
0
(1 row)

And let's look into the waits profile of the second psql session, where the VACUUM commands were executed:

=> SELECT p.pid, a.backend_type, a.application_name AS app,
p.event_type, p.event, p.count
FROM pg_wait_sampling_profile p
LEFT JOIN pg_stat_activity a ON p.pid = a.pid
WHERE p.pid = 29367
ORDER BY p.pid, p.count DESC;

  pid  |  backend_type  | app  | event_type |   event    | count
-------+----------------+------+------------+------------+-------
29367 | client backend | psql | BufferPin  | BufferPin  |   294
29367 | client backend | psql | Client     | ClientRead |    10
(2 rows)

The BufferPin wait tells us that VACUUM was waiting for the buffer to get free.

And this is where we will consider discussing locks as finished. Thank you all for attentive reading and comments!
Postgres Professional
Разработчик СУБД Postgres Pro


text:
# Tips for golfing in Io

What general tips do you have for golfing in Io? I'm looking for ideas that can be applied to code golf problems in general that are at least somewhat specific to Io (e.g. "remove comments" is not an answer). Please post one tip per answer.

# You can stick methods onto the back of most literals

"text" print    # 12 bytes
"text"print     # 11 bytes

12 print        # 8 bytes
12print         # 7 bytes

(0<1,0,1) print # 15 bytes
(0<1,0,1)print  # 14 bytes

## Not everything

0x12print # prints nothing

## You can leave out the else part of the if function

This isn't in the documentation... I initially thought that you have to include the else part, like the elvis operator in other languages; turns out that I can leave out the else part. (Please add this to the tutorial/documentation!)

if("bug"size>2,"True",nil)

So, if you don't want the else part to return anything, you could just do

if("bug"size>2,"True")

## Higher-level function shorthand

This seems like a pretty interesting golfing point. E.g.

list(1,2,3)map(i,i+1)print

However, Io is pretty permissive on not specifying the counter; the map body can be used as a point-free function, as Io tries to fill in the operand of this expression. This can be golfed into

list(1,2,3)map(+1)print


text:
# 5 ways to Handle Imbalanced Dataset in Machine Learning

In real world, having an imbalanced dataset is a common thing. Knowing how to handle them is an important skills for any data scientist. In this post, you will learn 5 ways to handle imbalanced dataset for Machine Learning.

## Handling Imbalanced Dataset –

#### Method 1 – Get more data –

The first strategy to handle imbalanced dataset is to get more training data especially for the minority class. Do this will reduce the class imbalance in your data.

#### Method 2 – Use a different Metric for evaluation –

A second strategy to handle class imbalance is to use a metric that is better suited for class imbalance. We all know that using Accuracy to evaluate a model which suffers from class imbalance is a very bad idea. So, we can use other metrics to evaluate the model like confusion matrix, precision, recall, F1 Score, ROC curves.

Here you can find details of all these metrics.

3 . F1 Score

4 . ROC Curve

#### Method 3 – Use an algorithm that can handle class imbalanced –

There are many algorithms in scikit-learn which can handle class imbalanced out of the box like Random Forest. Random Forest has a parameter called class_weight which can helps you reduce the effect of class imbalance. Let’s see how to do that.

First let’s read a dataset to work with. We will use the credit card fraud detection dataset from kaggle.

import pandas as pd
import numpy as np
import plotly.express as px
pd.options.display.max_columns = 999

df.head()

Now, let’s train a Random Forest model without any modification so that we can compare the results later.

from sklearn.model_selection import train_test_split
from sklearn.ensemble import RandomForestClassifier
from sklearn.pipeline import make_pipeline
from sklearn.impute import SimpleImputer
from sklearn.metrics import roc_auc_score
from sklearn.model_selection import cross_val_predict

# separate features and target
X = df.drop('Class', axis=1)
y = df['Class']

# split the data into training and test set
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, stratify=y)

# train a random forest model
rf = make_pipeline(SimpleImputer(),RandomForestClassifier(random_state=42))
rf.fit(X_train, y_train)

# make predictions
y_scores = cross_val_predict(rf, X_train, y_train, cv=3, method='predict_proba')

score = roc_auc_score(y_train, y_scores[:, 1])
print(np.round(score, 2))

output -
0.94

Now, let’s use the class_weight=’balanced’ in random forest and measure the roc auc score.

# train a random forest model
rf = make_pipeline(SimpleImputer(),
RandomForestClassifier(class_weight='balanced',random_state=42))
rf.fit(X_train, y_train)

# make predictions
y_scores = cross_val_predict(rf, X_train, y_train, cv=3, method='predict_proba')

score = roc_auc_score(y_train, y_scores[:, 1])
print(np.round(score, 2))

output -
0.94

Both the model has similar roc auc score. Although handling for imbalanced class does not improved the result on this dataset but you get the idea of how to use it.

#### Method 4 – Downsampling

Another way to handle class imbalance is Downsampling. In Downsampling, we randomly sample without replacement from the majority class to create a new subset of observations equal to the size of the minority class. For example if the minority class has 100 observations and majority class has 900 observations then we will randomly sample 100 observations from the majority class and combine both these samples to create a dataset with 200 observations. Then we train a model on this new dataset.

X = df_down.drop("Class", axis=1)
y = df_down['Class']

# split the data into training and test set
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, stratify=y)

# train a random forest model
rf = make_pipeline(SimpleImputer(),RandomForestClassifier(random_state=42))
rf.fit(X_train, y_train)

# make predictions
y_scores = cross_val_predict(rf, X_train, y_train, cv=3, method='predict_proba')

score = roc_auc_score(y_train, y_scores[:, 1])
print(np.round(score, 2))

output -
0.98

We can see that after downsampling the accuracy has increased from 0.94 to 0.98.

#### Method 5 – Upsampling

In upsampling, for every observation in the majority class, we randomly select an observation from the minority class with replacement. The end result is the same number of observations from the minority and majority classes.

# perform upsampling
df_class1_up = df_class1.sample(class0_count, replace=True)
# combine both dataframe
df_up = pd.concat([df_class0, df_class1_up], axis=0)

# train a model on this new dataset
X = df_up.drop("Class", axis=1)
y = df_up['Class']

# split the data into training and test set
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, stratify=y)

# train a random forest model
rf = make_pipeline(SimpleImputer(),RandomForestClassifier(random_state=42))
rf.fit(X_train, y_train)

# make predictions
y_scores = cross_val_predict(rf, X_train, y_train, cv=3, method='predict_proba')

score = roc_auc_score(y_train, y_scores[:, 1])
print(np.round(score, 2))

output -
1.0

Doing upsampling worked even much better than downsampling. This model has a roc auc score close to 1, You are seeing 1 because I was round the scores by 2 decimal places. This is not always going to be the case, so please apply all the method that is described here and choose the one that works best for your dataset.

I hope you liked this post. If you do then please share this post with others and subscribe to our blog below for more articles related to Machine Learning.

Rating: 1 out of 5.


text:
Get premium membership and access revision papers with marking schemes, video lessons and live classes.
OR

# Form 4 Mathematics Paper 1 Exam Revision Questions With Answers Set 2

The figure shows a rectangular metal sheet measuring 1 m long and 80 cm wide. From each of the four corners of the rectangle a square of side x cm is removed. The remaining part is then folded along the dotted line to form an open cuboid. The metal is 2 mm thick, and its density is 2.5 g/cm^3.

a) If A cm^2 is the area of the unshaded part, show that A = 8 000 - 4x^2
b) Given that the area of the unshaded part is 0.76 m^2, find the value of x hence state the dimensions of the cuboid.
c) Calculate the mass of the empty cuboid in kg.
d) The cuboid is filled with a liquid whose density is d g/cm^3. Given that the mass of the cuboid when full of liquid is 39.8 kg, determine the value of d.

(11m 3s)
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NTNUJAVA Virtual Physics LaboratoryEnjoy the fun of physics with simulations!
July 25, 2014, 02:14:59 am

Discovery consists of seeing what everybody has seen and thinking what nobody has thought. ..."Albert von Szent-Gyorgyi(1893-1986, 1937 Nobel Prize for Medicine, Lived to 93)"

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 Author Topic: AC source powered RC circuit  (Read 9755 times) 0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).
Fu-Kwun Hwang
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 « Embed this message on: March 14, 2011, 09:15:23 pm » posted from:Taipei,T\'ai-pei,Taiwan

This is a simulation of AC powered RC circuit.
It can be used to study phase between voltage of capacitor$VR$, voltage of resistor $VR$and voltage of AC power supply $Vs$.

$V_{AC}=V_C+V_R$,

$I(t)=\frac{V_R}{R}=\frac{V_{AC}-V_C}{R}=\frac{V_{AC}-Q/C}{R}$

$V_C(t)=\frac{Q}{C}$

$Q(t)=\int I\, dt$
The current $I$ and voltage $V_C$ are 90 degree out of phase.

The voltage $V_C$ in capacitor is proportional to charge $Q$,
However, change $Q$ need to be charged by current $I$. ($Q=\int I\,dt$).
So current $I$ is leading charge $Q$ (90 degree earlier).
And voltage in resistor $V_R$ is proportional to current $I$.
Voltage in R ($V_R$)is leading Voltage in capacitor($V_C$)

The following simulation assume, $V_{AC}=V_o \sin\,(\omega\,t+\phi)$

$V_R=I\,R,\qquad V_C=\frac{Q}{C}=\frac{1}{C}\,\int I dt$

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)
Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list


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journal

# Journal of Computational Biology: a Journal of Computational Molecular Cell Biology

journal
#1
Brenden K Petersen, Jiachen Yang, Will S Grathwohl, Chase Cockrell, Claudio Santiago, Gary An, Daniel M Faissol
Traditionally, precision medicine involves classifying patients to identify subpopulations that respond favorably to specific therapeutics. We pose precision medicine as a dynamic feedback control problem, where treatment administered to a patient is guided by measurements taken during the course of treatment. We consider sepsis, a life-threatening condition in which dysregulation of the immune system causes tissue damage. We leverage an existing simulation of the innate immune response to infection and apply deep reinforcement learning (DRL) to discover an adaptive personalized treatment policy that specifies effective multicytokine therapy to simulated sepsis patients based on systemic measurements...
January 25, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#2
Sagi Snir
Several studies have pointed out that the tight correlation between genes' evolutionary rate is better explained by a model denoted as the Universal PaceMaker (UPM) rather than by a simple rate constancy as manifested by the classical hypothesis of molecular clock (MC). Under UPM, each gene is associated with a single pacemaker (PM) and varies its evolutionary rate according to this PM ticks. Hence, the relative rates of all genes associated with the same PM remain nearly constant, whereas the absolute rates can change arbitrarily according to the PM ticks...
January 24, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#3
Christopher Barrett, Qijun He, Fenix W Huang, Christian M Reidys
Recently, a framework considering RNA sequences and their RNA secondary structures as pairs led to some information-theoretic perspectives on how the semantics encoded in RNA sequences can be inferred. This pairing arises naturally from the energy model of RNA secondary structures. Fixing the sequence in the pairing produces the RNA energy landscape, whose partition function was discovered by McCaskill. Dually, fixing the structure induces the energy landscape of sequences. The latter has been considered originally for designing more efficient inverse folding algorithms and subsequently enhanced by facilitating the sampling of sequences...
January 17, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#4
Kyu Min Lee, Minhyeok Lee, Junhee Seok, Sung Won Han
Given the continuous advancement in genome sequencing technology, large volumes of gene expression data can be easily obtained. However, the corresponding increase in genetic information necessitates adoption of a new approach for network estimation. Data dimensions increase with the progress in genome sequencing technology, thereby making it difficult to estimate gene networks by causing multicollinearity. Furthermore, such a problem also occurs when hub nodes exist, where gene networks are known to have regulator genes that can be interpreted as hub nodes...
January 17, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#5
Baekdoo Kim, Thahmina Ali, Changsu Dong, Carlos Lijeron, Raja Mazumder, Claudia Wultsch, Konstantinos Krampis
The availability of low-cost small-factor sequencers, such as the Illumina MiSeq, MiniSeq, or iSeq, have paved the way for democratizing genomics sequencing, providing researchers in minority universities with access to the technology that was previously only affordable by institutions with large core facilities. However, these instruments are not bundled with software for performing bioinformatics data analysis, and the data analysis can be the main bottleneck for independent laboratories or even small clinical facilities that consider adopting genomic sequencing for medical applications...
January 17, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#6
Xiaodong Wang, Lei Wang
Biological events like inversions are not automatically detected by the usual alignment algorithms. Alignment with inversions does not have a known polynomial time algorithm and Schöniger and Waterman introduced a simplification of the alignment problem with nonoverlapping inversions, where all regions will not be allowed to overlap. They presented an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^6} )$$ \end{document} algorithm to compute nonoverlapping inversion distance between two strings of length n...
January 14, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#7
Heewon Park, Makoto Yamada, Seiya Imoto, Satoru Miyano
Identifying individual characteristics is a crucial issue in personalized genome research. To effectively identify sample-specific characteristics, we propose a novel strategy called robust sample-specific stability selection. Although stability selection shows effective feature selection results and has attractive theoretical property (i.e., per-family error rate control), the method's results are sensitive to the value of the regularization parameter because the method performs feature selection based only on the particular parameter value that maximizes the selection probability...
January 14, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#8
Roberto Solar, Victor Sepulveda, Alonso Inostrosa-Psijas, Oscar Rojas, Veronica Gil-Costa, Mauricio Marin
Approximate Bayesian computation (ABC) is a useful technique developed for solving Bayesian inference without explicitly requiring a likelihood function. In population genetics, it is widely used to extract part of the information about the evolutionary history of genetic data. The ABC compares the summary statistics computed on simulated and observed data sets. Typically, a forward-in-time approach is used to simulate the genetic material of a population starting from an initial ancestral population and following the evolution of the individuals by advancing generation by generation under various demographic and genetic forces...
January 9, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#9
Yuandi Wang, Qingmei Wen, Zhigang Zhou, Nanjun Yan
Mathematicization for cell modeling provides an effective tool to verify the biological theory, and the existing research mainly focuses on the description of cell structures. This article then addresses the pattern question of cell division or morphogenesis by means of bubble model with weighted membranes. In this study, we show that cell shapes including intersection angles at junction points depend on weights on membranes. For convenience, adhesion and contractile force are considered together as a factor in construction of patterning model...
January 9, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#10
Andreas D M Gunawan, Hongwei Yan, Louxin Zhang
Rooted phylogenetic networks are rooted acyclic digraphs. They are used to model complex evolution where hybridization, recombination, and other reticulation events play a role. A rigorous definition of network compression is introduced on the basis of recent studies of relationships between cluster, tree, and rooted phylogenetic networks. The concept reveals new connections between well-studied network classes, including tree-child networks and reticulation-visible networks. It also enables us to define a new class of networks for which the cluster containment problem is solvable in linear time...
January 9, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#11
Husayn Ahmed P, Vidhya V, Ravi Prabhakar More, Biju Viswanath, Sanjeev Jain, Mahendra S Rao, Odity Mukherjee
Deep sequencing-based genetic mapping has greatly enhanced the ability to catalog variants with plausible disease association. Confirming how these identified variants contribute to specific disease conditions, across human populations, poses the next challenge. Differential selection pressure may impact the frequency of genetic variations, and thus detection of association with disease conditions, across populations. To understand genotype to phenotype correlations, it thus becomes important to first understand the spectrum of genetic variation within a population by creating a reference map...
January 7, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#12
Xiaodan Mao, Yichen Xu, Zhenran Jiang
HColonDB (Human Colon cancer Database) is an important database which integrates genes, pathways, networks, drugs, and other information related to colon cancer. The purpose of the database is to provide a platform for the systematic research of colon cancer. The relationships between genes and pathways, genes and networks, and networks and pathways are obtained from the database KEGG. Furthermore, the information of the drugs used to treat colon cancer is available in HColonDB, which is collected and organized from DrugBank and PubChem database...
January 7, 2019: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#13
Hamid Eghbal-Zadeh, Lukas Fischer, Niko Popitsch, Florian Kromp, Sabine Taschner-Mandl, Teresa Gerber, Eva Bozsaky, Peter F Ambros, Inge M Ambros, Gerhard Widmer, Bernhard A Moser
Clinical decision-making in cancer and other diseases relies on timely and cost-effective genome-wide testing. Classical bioinformatic algorithms, such as Rawcopy, can support genomic analysis by calling genomic breakpoints and copy-number variations (CNVs), but often require manual data curation, which is error prone, time-consuming, and thus substantially increasing costs of genomic testing and hampering timely delivery of test results to the treating physician. We aimed to investigate whether deep learning algorithms can be used to learn from genome-wide single-nucleotide polymorphism array (SNPa) data and improve state-of-the-art algorithms...
December 26, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#14
Qingyu Chen, Xiuzhen Zhang, Yu Wan, Justin Zobel, Karin Verspoor
Duplicate sequence records-that is, records having similar or identical sequences-are a challenge in search of biological sequence databases. They significantly increase database search time and can lead to uninformative search results containing similar sequences. Sequence clustering methods have been used to address this issue to group similar sequences into clusters. These clusters form a nonredundant database consisting of representatives (one record per cluster) and members (the remaining records in a cluster)...
December 26, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#15
Norovirus is the causing agent of acute gastroenteritis disease globally. Efforts in developing therapeutics against virus infection mostly fail due to emergence of drug resistance that is a consequence of presence of high mutation rates in virus genome during virus' life cycle. In this study, we computationally analyzed the affinity of a drug target, wild type VP1 envelope protein and its three variants to a therapeutic antibody FAB5I2. We have found that mutations break important hydrogen bonds and cause high fluctuations in residues that form VP1-FAB5I2 complex interface...
December 20, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#16
Monika Piwowar, Tomasz KuŁaga
Analysis of large data sets is currently a major challenge. Strong efforts are being undertaken to tackle this problem by developing new methods or modifying existing ones. The Z association method is a new method for describing directional association in contingency tables. It allows to arbitrarily group categories for each of the two variables, for which the contingency table is analyzed. The Z coefficient was calculated on a sample data set with gene mutations in different cancer types. Results showed some association with both gene mutations and annotation groups...
December 18, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#17
Gopinath Samykannu, Princy Vijayababu, Christian Bharathi Antonyraj, Perumal Perumal, Sundarabaalaji Narayanan, Syed Ibrahim Basheer Ahamed, Jeyakumar Natarajan
Typhoid fever is an acute illness in humans, caused by Salmonella typhi, a gram-negative bacterium. Outer membrane proteins of S. typhi have strong potential for its use in the development of subunit vaccine against typhoid. In the current study, peptide-based subunit vaccine was constructed from outer membrane protease E (PgtE) against S. typhi. B cell and T cell epitopes were identified at fold level with a validated three-dimensional modeled structure. T cell epitopes from PgtE (IHPDTSANY) have 99.5% binding to a maximum number of major histocompatibility complex class I and class II alleles...
December 14, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#18
Antonio Moretti, Andrew Stirn, Gabriel Marks, Itsik Pe'er
Topographic factor models separate overlapping signals into latent spatial functions to identify correlation structure across observations. These methods require the underlying structure to be held fixed and are not robust to deviations commonly found across images. We present autoencoding topographic factors, a novel variational inference scheme, to decompose irregular observations on a lattice into a superposition of low-rank sources. By exploiting recent developments in variational autoencoders, we replace fixed sources with a nonlinear mapping that parameterizes an unnormalized distribution on the lattice...
December 7, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#19
Sihai Yang, Xianhua Han, Yenwei Chen
Quantitative analysis based on three-dimensional differential interference contrast (DIC) images is currently the mainstream in analyzing gene functions involved in early cell fate specifications. Segmentation and registration are the two most important steps in analysis. Many image segmentation methods have poor performance on embryonic DIC images because of the interference of egg shells, blurs, and nonuniform intensity background. A novel segmentation method is presented based on the shape index (SI) of local intensity variation in DIC images...
December 5, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
#20
Alexandre Bazin, Didier Debroas, Engelbert Mephu Nguifo
When analyzing microbial communities, an active and computational challenge concerns the categorization of 16S rRNA gene sequences into operational taxonomic units (OTUs). Established clustering tools use a one pass algorithm to tackle high number of gene sequences and produce OTUs in reasonable time. However, all of the current tools are based on a crisp clustering approach, where a gene sequence is assigned to one cluster. The weak quality of the output compared with more complex clustering algorithms forces the user to postprocess the obtained OTUs...
December 5, 2018: Journal of Computational Biology: a Journal of Computational Molecular Cell Biology
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# Isomorphism from R^n via n-Term Sequence

## Theorem

Let $G$ be a unitary $R$-module.

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $G$.

Let $R^n$ be the $R$-module $R^n$.

Let $\psi: R^n \to G$ be defined as:

$\displaystyle \map \psi {\sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} } = \sum_{k \mathop = 1}^n \lambda_k a_k$

Then $\psi$ is an isomorphism.

## Proof

By Unique Representation by Ordered Basis, $\psi$ is a bijection.

We have:

 $\displaystyle \sum_{k \mathop = 1}^n \lambda_k a_k + \sum_{k \mathop = 1}^n \mu_k a_k$ $=$ $\displaystyle \sum_{k \mathop = 1}^n \paren {\lambda_k a_k + \mu_k a_k}$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 1}^n \paren {\lambda_k + \mu_k} a_k$

and we have:

 $\displaystyle \beta \sum_{k \mathop = 1}^n \lambda_k a_k$ $=$ $\displaystyle \sum_{k \mathop = 1}^n \beta \paren {\lambda_k a_k}$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 1}^n \paren {\beta \lambda_k} a_k$

thus proving that $\psi$ is also a homomorphism.

$\blacksquare$


text:
# Common emitter amplifier gain with load resistor?

I have a problem I am working on where I have to find the voltage gain of the following circuit with a dc collector current of 0.3mA.

My attempt at a solution involved using the formula A=-(RcIc)/Vt = -(10000*3*10^-4)/0.025 = -120. I am not confident in this solution as I feel like it was too simple and does not account for the "load" resistor. Any advice on where to go from here? I have received a hint to use Thevenin's theorem to transform the circuit but am lost on how that should look.

simulate this circuit – Schematic created using CircuitLab

• What section of that circuit would you feel confident converting to a Thevenin equivalent? – TimWescott Nov 6 '18 at 0:58
• I believe that I am supposed to manipulate the circuit in a way that removes the 10k resistor from collector to ground and replaces it with a voltage source at the base but I am unsure of how to do this. – Burritos Nov 6 '18 at 3:34
• Try this again. "Feel confident" is the key phrase. Put your thumb over the transistor. Does the two resistors with a 5V supply look like something you could reduce to a single source and one resistor? – TimWescott Nov 6 '18 at 16:50

At 0.3 milliamp collector current, the derivative of (Vbe / Iout) which I personally label 'reac', is about 80 ohms.

Simply divide that into the collector load resistance, which is the parallel combination of the two 10 KOhm in parallel

5,000 / 80 = 63X

By the way, the derivative of (Vbe / Iout) is 1/gm.

Thus my 'reac' is just 1/gm.

At 1milliAmp Ic, the gm is 1/26 ohms and the reac is 26 ohms.

If you have an emitter resistance that is not bypassed, as in this schematic below, simply add the reac to the Re, then divide into the collector load.

simulate this circuit – Schematic created using CircuitLab

• Thank you for the response, could you clarify what the variables g and m stand for in the react = 1/gm? If I am assuming correctly, at 0.3mA the gm = 80 and at 1mA gm=26? – Burritos Nov 6 '18 at 6:22
• $g_m$ is the transconductance gain, and has units of Siemens ($\mho$). – TimWescott Nov 6 '18 at 16:48
• The name of the variable is 'gm', and gm is the derivative of Icollecter versus Vbase_emitter. The 'reac' is just 1/gm. By knowing the 'reac' at 1milliAm is 26 ohms, its very easy to walk thru pure-bipolar designs and understand them. FETs have various degrees of freedom (Gate Spacing of Source to Drain, for example), and channel-doping, that prevent detailed analysis on a napkin. – analogsystemsrf Nov 7 '18 at 3:12

If I remember well the transconductance of a bipolar junction transistor (BJT) in common emitter configuration (CE) is $$\1/(40 I_c)\$$. The fact that $$\I_c\$$ is given makes me follow the path of writing the small signal current transfer function (actually it's a transconductance: output current over input voltage) as $$\i_c = 1000/12 \approx 83 \, u_{in}\$$.
For small signals the two $$\10 \, k\Omega \$$ resistors are considered to be in parallel as the one from collector to $$\+5 V\$$ is also considered grounded for AC (small) signals.

This gives us the (small signal) output voltage $$\u_{out} = 5 \, k\Omega * (1000/12) \, u_{in} \$$.
The voltage gain therefore is $$\ u_{out}/u_{in} = 415 \$$.

If you look up Thevenin source transformation you'll find that the current source in the given circuit (the transistor in the small signal model) is paralleled with $$\2\$$ resistors of $$\10 \, k\Omega \$$ each and will form a voltage source with output voltage $$\ (5/12)10^6 \, u_{in} \$$ with a series resistance of $$\ 5 \, k\Omega \$$.

The Thevenin source transformation transforms a bunch of sources, in this case only 1 current source--the transistor, into one voltage source with a series impedance.


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 您现在的位置：首页 > 学术活动 > 专题学术报告/Seminar
 Dark matter in (partially) composite Higgs models
 2018-11-15  【大 中 小】【打印】【关闭】
 CAS Key Laboratory of Theoretical Physics Institute of Theoretical Physics Chinese Academy of Sciences Seminar Title 题目 Dark matter in (partially) composite Higgs models Speaker 报告人 Diogo Buarque Franzosi Affiliation 所在单位 Gottingen U. Date 日期 2018年11月15日（周四）下午16:00 Venue 地点 Conference Room 6620, ITP new building Abstract 摘要 (Based on arXiv:1808.07515) We construct composite and partially composite Higgs models with complex pseudo-Nambu--Goldstone (pNGB) dark matter states  from four-dimensional gauge-Yukawa theories with strongly interacting fermions. The fermions  are partially gauged under the electroweak symmetry, and the dynamical electroweak  symmetry breaking sector is minimal.  The pNGB dark matter particle is stable  due to a $\mathrm{U}(1)$ technibaryon-like symmetry, also present in the technicolor limit of the models. However, the relic density is particle anti-particle symmetric and due to thermal freeze-out as opposed to the technicolor limit where it is typically due to an asymmetry.  The pNGB Higgs is composite or partially composite depending on the origin of the Standard Model  fermion masses, which impacts the dark matter phenomenology. We illustrate the  important features with a model example invariant under an $\SU(4)\times \SU(2) \times \mathrm{U}(1)$ global symmetry. Contact Person 所内联系人 Jing Shu
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### Top 7 Arxiv Papers Today in Representation Theory

##### #1. Classical limit of quantum Borcherds-Bozec algebras
###### Zhaobing Fan, Seok-Jin Kang, Young-Rock Kim, Bolun Tong
Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$ is isomorphic to $U(\mathfrak{g})$ as Hopf algebras. Thus $U_{q}(\mathfrak{g})$ can be regarded as a quantum deformation of $U(\mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(\mathfrak{g})$.
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mathRTb: Zhaobing Fan, Seok-Jin Kang, Young-Rock Kim, Bolun Tong : Classical limit of quantum Borcherds-Bozec algebras https://t.co/fz0qQK3rmd https://t.co/IiwW52H4H8
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##### #2. Specht module branching rules for wreath products of symmetric groups
###### Reuben Green
We review a class of modules for the wreath product S(m) wr S(n) of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products S(m-1) wr S(n) and S(m) wr S(n-1). In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.
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mathRTb: Reuben Green : Specht module branching rules for wreath products of symmetric groups https://t.co/p02RGtHHuz https://t.co/0ITzTuOhwL
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##### #3. Irreducible self-adjoint representation of quantum Teichmüller space and phase constants
###### Hyun Kyu Kim
Quantization of Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichm\"uller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon change of triangulations, one must construct a unitary operator between Hilbert spaces intertwining the quantum coordinate operators and satisfying composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a reducible representation, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of Shale-Weil intertwiners for Schr\"odinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct irreducible representations of the quantum shear coordinate functions and the...
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MathPHYPapers: Irreducible self-adjoint representation of quantum Teichm\"uller space and phase constants. https://t.co/hvLxtnZoes
mathRTb: Hyun Kyu Kim : Irreducible self-adjoint representation of quantum Teichmüller space and phase constants https://t.co/xOtAZ3Mynn https://t.co/MkMJwPzcYb
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##### #4. Three perspectives on categorical symmetric Howe duality
###### Ben Webster
In this paper, we consider the categorical symmetric Howe duality introduced by Khovanov, Lauda, Sussan and Yonezawa. While originally defined from a purely diagrammatic perspective, this construction also has geometric and representation-theoretic interpretations, corresponding to certain perverse sheaves on spaces of quiver representations and the category of Gelfand-Tsetlin modules over $\mathfrak{gl}_n$. In particular, we show that the deformed Webster algebras'' discussed in work of Khovanov-Lauda-Sussan-Yonezawa manifest a Koszul duality between blocks of the category of Gelfand-Tsetlin modules over $\mathfrak{gl}_n$, and the constructible sheaves on representations of a linear quiver invariant under a certain parabolic in the group that acts by changing bases. Furthermore, we show that this duality intertwines translation functors with a diagrammatic categorical action. Includes an appendix by the author and Jerry Guan.
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##### #5. Simple Modules Over The Quantum Affine Space
###### Snehashis Mukherjee, Sanu Bera
The coordinate ring $\mathbb{K}_\mathcal{Q}[x_1,\cdots ,x_n]$ of the quantum affine space is the algebra generated over a field $\mathbb{K}$ by the variables $x_1,\cdots ,x_n$ satisfying the relations $x_ix_j=q_{ij}x_jx_i, \ \forall \ 1 \leq i,j\leq n$. We construct simple $\mathbb{K}_\mathcal{Q}[x_1,\cdots ,x_n]$-modules in a more general setting where the entries $q_{ij}$ lie in a torsion subgroup of $\mathbb{K}^*$ and show analogous results hold as in [5].
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##### #6. The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$
###### Drazen Adamovic, Thomas Creutzig, Naoki Genra, Jinwei Yang
The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ (respectively, $L_k(\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{(p)}$ and we decompose $V^{(p)}$ as an $L_k(\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\mathfrak{gl}_2)$-module. The decomposition of $V^{(p)}$ shows that $V^{(p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{(p)}$ is the logarithmic doublet algebra $A^{(p)}$ introduced in [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of...
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mathRTb: Drazen Adamovic, Thomas Creutzig, Naoki Genra, Jinwei Yang : The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$ https://t.co/gd4vJL4IZZ https://t.co/Olg0994IL6
TheoryPapers: The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$. https://t.co/imH7nFPfqX
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##### #7. On Schur problem and Kostka numbers
###### Robert Coquereaux, Jean-Bernard Zuber
We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n x n matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of SU(n) (Kostka). It is well known that the former yields a semi-classical picture of the latter. We present explicit expressions for low values of n that complement those given in the literature, recall some exact (non asymptotic) relation between the two problems, comment on the limiting procedure whereby Kostka numbers are obtained from Littlewood-Richardson coefficients, and finally extend these considerations to the case of the B2 algebra, with a few novel conjectures.
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text:
At constant temperature, the equilibrium constant (K_p) for the decomposition reactionN_2O_4 ⇋ 2NO_2 is expressed

### Question Asked by a Student from EXXAMM.com Team

Q 2616501470.     At constant temperature, the equilibrium constant (K_p) for the decomposition reactionN_2O_4 ⇋ 2NO_2 is expressed by K_p = (4x^2 p)/((1-x^2)) where, p = pressure, x = extent of decomposition. Which one of the following statement is
true ?
JEE 2001
A

K_ P increases with increase of p

B

K_ P increases with increase of x

C

K_ P increases with decrease of x

D

K_p remains constant with change in p and x

#### HINT

(Provided By a Student and Checked/Corrected by EXXAMM.com Team)

#### Access free resources including

• 100% free video lectures with detailed notes and examples
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text:
# Lesson 6What about Other Bases?

Let’s explore exponent patterns with bases other than 10.

### Learning Targets:

• I can use the exponent rules for bases other than 10.

## 6.1True or False: Comparing Expressions with Exponents

Is each statement true or false? Be prepared to explain your reasoning.

## 6.2What Happens with Zero and Negative Exponents?

Complete the table to show what it means to have an exponent of zero or a negative exponent.

1. As you move toward the left, each number is being multiplied by 2. What is the multiplier as you move toward the right?
2. Use the patterns you found in the table to write as a fraction.
3. Write as a power of 2 with a single exponent.
4. What is the value of ?
5. From the work you have done with negative exponents, how would you write as a fraction?
6. How would you write as a fraction?

### Are you ready for more?

1. Find an expression equivalent to  but with positive exponents.

2. Find an expression equivalent to  but with positive exponents.

3. What patterns do you notice when you start with a fraction to a negative power and rewrite it so that it has only positive powers? Show or explain your reasoning.

## 6.3Exponent Rules with Bases Other than 10

Lin, Noah, Diego, and Elena decide to test each other’s knowledge of exponents with bases other than 10. They each chose an expression to start with and then came up with a new list of expressions; some of which are equivalent to the original and some of which are not.

Choose 2 lists to analyze. For each list of expressions you choose to analyze, decide which expressions are not equivalent to the original. Be prepared to explain your reasoning.

1. Lin’s original expression is and her list is:
2. Noah’s original expression is and his list is:
3. Diego’s original expression is and his list is:
4. Elena’s original expression is and her list is:
• 1
• 0

## Lesson 6 Summary

Earlier we focused on powers of 10 because 10 plays a special role in the decimal number system. But the exponent rules that we developed for 10 also work for other bases. For example, if and , then

These rules also work for powers of numbers less than 1. For example, and . We can also check that .

Using a variable helps to see this structure. Since (both sides have 7 factors that are ), if we let , we can see that . Similarly, we could let  or or any other positive value and show that these relationships still hold.

## Lesson 6 Practice Problems

1. Priya says “I can figure out by looking at other powers of 5. is 125, is 25, then is 5.”

1. What pattern do you notice?
2. If this pattern continues, what should be the value of ? Explain how you know.
3. If this pattern continues, what should be the value of ? Explain how you know.
2. Select all the expressions that are equivalent to .

1. -12
2. 12
3. Write each expression using a single exponent.

4. Andre sets up a rain gauge to measure rainfall in his back yard. On Tuesday, it rains off and on all day.

• He starts at 10 a.m. with an empty gauge when it starts to rain.
• Two hours later, he checks, and the gauge has 2 cm of water in it.
• It starts raining even harder, and at 4 p.m., the rain stops, so Andre checks the rain gauge and finds it has 10 cm of water in it.
• While checking it, he accidentally knocks the rain gauge over and spills most of the water, leaving only 3 cm of water in the rain gauge.
• When he checks for the last time at 5 p.m., there is no change.

Graph A

Graph B

1. Which of the two graphs could represent Andre’s story? Explain your reasoning.
2. Label the axes of the correct graph with appropriate units.
3. Use the graph to determine how much total rain fell on Tuesday.


text:
### Almost Galois $\omega$-Stable classes

by Baldwin and Larson and Shelah. [BlLrSh:1003]

Theorem. Suppose that an aleph_0-presentable Abstract Elementary Class (AEC), boldmath {K}, has the joint embedding and amalgamation properties in aleph_0 and <2^{aleph_1} models in aleph_1 . If boldmath {K} has only countably many models in aleph_1, then all are small. If, in addition, boldmath {K} is almost Galois omega-stable then boldmath {K} is Galois omega-stable.

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text:
# Moerdijk Model Structure on Bisimplicial sets

Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-wise" weak equivalences of simplicial sets (in the usual sense) and cofibrations are the injections.

Is this model structure combinatorial?